Download DETERMINING SPATIAL MODES OF SEMICONDUCTOR LASERS

DETERMINING SPATIAL MODES OF
SEMICONDUCTOR LASERS USING SPATIAL
COHERENCE
DISSERTATION
Presented in Partial Fulfillment of the Requirements for
the Degree Doctor of Philosophy in the
Graduate School of The Ohio State University
By
Carolyn May Warnky, B.S.E., M.S.
*****
The Ohio State University
2002
Dissertation Committee:
Approved by
Dr. Betty Lise Anderson, Adviser
Dr. Bradley D. Clymer
Dr. Charles A. Klein
Adviser
Department of Electrical
Engineering
ABSTRACT
This dissertation explains our method of finding the structure and modal weights of
spatial modes in lasers by analyzing the spatial coherence of the laser beam. Most previous methods for determining modal weights have relied on assumed shapes for the spatial
modes.
Using a twin-fiber interferometer we measured correlation data from each laser. The
two arms of a single-mode fiber coupler sample the laser beam and send the summed output onto a detector. The detected intensity is a correlation measurement, giving the spatial
coherence of the two sample points. With computer control of fiber positioning and automated data acquisition we were able to measure a matrix (or partial matrix) of correlation
data.
The correlation data is processed using an algorithm based on Jacobian control. This
algorithm extracts the basis modes and weights from the matrix. The algorithm is iterative
and uses information from the data for its initial guess instead of a start based on prior
knowledge of the laser.
We tested eight semiconductor lasers which varied from having a single lateral mode to
four lateral spatial modes. The lasers were ridge-waveguide, Fabry–Perot cavity, quantum
well (or multiple quantum well) GaAs lasers that varied in stripe width from 3 to 30 m. We
verified that the wider confinement layers in one MQW laser design allowed two or three
transverse modes compared to one transverse mode for the other design. We noted that the
ii
wider stripe lasers had more dominant higher-order modes but there were multiple lateral
spatial modes even for the smallest width. One important consideration in determining the
spatial modes and weights was whether the laser was operating in a single longitudinal
mode. If not, the total coherence was lowered, the modal weights were not accurately
determined, and there may not have been enough stability to retrieve some of the lower
weighted spatial modes.
iii
Dedicated to the memory of my father,
Robert W. Beukema,
1928–2001
iv
ACKNOWLEDGMENTS
First, and foremost, I thank my adviser, Betty Lise Anderson, for her vision, enthusiasm
and technical help. She has always been an encouragement to me and I have benefited
greatly from being able to discuss my experimental results, a.k.a. “weird things”, with her.
I also appreciate the material support that we received, initially under the direction of
John Loehr, from the Air Force Research Laboratory at Wright Patterson Air Force Base. I
specifically thank Bill Siskaninetz who fabricated the lasers and packaged them for us, and
answered my questions for me.
Professor Klein was an invaluable resource for the numerical analysis, especially for
pointing out in the beginning that we needed to work with the non-normalized mutual
intensity to extract the spatial modes. He also provided many hours of additional help.
Jim Jones and Bill Thalgott were both helpful in their technical assistance, especially
in making custom mounts for our laser packages.
I thank my husband, Chris, who has been loving and encouraging me for more than half
my lifetime.
Last, but in no way least, I thank God for the beauty and elegance of His creation when
He said, “Let there be light.”
v
VITA
May 6, 1958 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Born - Panama City, Panama
1979 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.S.E. Electronic Engineering,
Southern Illinois University–E
Edwardsville, IL
1994 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M.S. Electrical Engineering,
The Ohio State University
1998-present . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ph.D. candidate,
The Ohio State University.
PUBLICATIONS
Research Publications
C. M. Warnky, B. L. Anderson, and C. A. Klein, “Determining spatial modes of lasers with
spatial coherence measurements,” Applied Optics, vol. 39, no. 33, pp. 6109–6117, 2000.
B. L. Anderson, C. M. Warnky, C. A. Klein, “Laser diode modes: a coherence method
to measure the shapes and the weights,” Invited paper, Proceedings of the International
Conference on Microelectronics and Packaging, Sociedade Brasileira de Microelectrônica
and International Microelectronic and Packaging Society, Campinas, Brazil, 1999.
C. M. Warnky, B. L. Anderson, C. A. Klein, “Spatial coherence for experimental measurement of the shapes and weights of spatial modes in multimode lasers,” Proceedings of IEEE
Laser and Electro-Optics Society Annual Meeting, Orlando, Florida, vol. 2, pp. 289–290,
December, 1998.
FIELDS OF STUDY
Studies in Electrical Engineering: Betty Lise Anderson
vi
TABLE OF CONTENTS
Page
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ii
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
Chapters:
1.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1
1.2
1.3
1.4
2.
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature survey . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Non-coherence methods of determining modal weights
1.3.2 Spatial coherence methods to determine modal weights
Organization of dissertation . . . . . . . . . . . . . . . . . . .
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Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1
2.2
2.3
2.4
Introduction . . . . . . . . . . . . . . . . . .
Laser modes . . . . . . . . . . . . . . . . .
Hermite–Gaussian modes . . . . . . . . . .
Spatial coherence . . . . . . . . . . . . . . .
2.4.1 Spatial coherence overview . . . . .
2.4.2 Spatial coherence and spatial modes .
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15
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2.5
2.6
2.7
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31
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Introduction . . . . . . . . . . . . .
Experimental layout and equipment
Procedures . . . . . . . . . . . . .
Chapter summary . . . . . . . . . .
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Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.1
4.2
4.3
4.4
4.5
4.6
4.7
Introduction . . . . . . . . . . . . . . . . . . .
Computer simulations . . . . . . . . . . . . .
Overall coherence measurements . . . . . . .
Optical spectra . . . . . . . . . . . . . . . . .
Frequency mixing experiments . . . . . . . . .
Laser with m stripe width . . . . . . . . . .
Semiconductor lasers of varying widths . . . .
4.7.1 LD03: Laser with 3 m stripe width .
4.7.2 LD04: Laser with 4 m stripe width .
4.7.3 LD10 : Laser with 10 m stripe width
4.7.4 LD10 : Laser with 10 m stripe width
4.7.5 LD15: Laser with 15 m stripe width .
4.7.6 LD20: Laser with 20 m stripe width .
4.7.7 LD30: Laser with 30 m stripe width .
HeNe laser . . . . . . . . . . . . . . . . . . .
4.8
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Experimental Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.1
3.2
3.3
3.4
4.
2.4.3 Spatial coherence and longitudinal modes .
2.4.4 Propagation of spatial modes . . . . . . .
2.4.5 Spatial coherence summary . . . . . . . .
Twin-fiber interferometer . . . . . . . . . . . . . .
Numerical analysis . . . . . . . . . . . . . . . . .
2.6.1 Jacobian solution . . . . . . . . . . . . . .
2.6.2 Orthogonality condition . . . . . . . . . .
2.6.3 Data-matching condition . . . . . . . . . .
2.6.4 Implementing the Jacobian algorithm . . .
Chapter summary . . . . . . . . . . . . . . . . . .
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64
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Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.1
5.2
5.3
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Conclusions from experimental results . . . . . . . . . . . . . . . . . . 126
Recommendations for future work . . . . . . . . . . . . . . . . . . . . . 130
viii
Appendices:
A.
MATLAB M-files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
A.1 M-file Jacob.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
A.2 M-file makeJs.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
A.3 M-file adJacob.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
ix
LIST OF FIGURES
Figure
1.1
Page
Intensity profiles of a Gaussian beam (solid curve) and a beam with 5%
power in the first-order Hermite–Gaussian mode and 95% in the fundamental mode (dashed curve). . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.1
The coordinate system used for a semiconductor laser.
2.2
Interference fringes for coherent light, partially coherent light and incoherent light. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3
Twin-fiber interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.4
How
3.1
Experimental layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.2
Input ends of the fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3
Comparison of fringe data with translation along (top) and piezoelectric
stretching of fiber (bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.4
Ridge-waveguide design of lasers provided by Wright Patterson Air Force
Base, after [70]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.1
(Top) The complete noise-free correlation matrix on the left and the two
Hermite–Gaussian modes on the right. (Bottom) The correlation matrix
with 21 diagonals missing on the left and the extracted modes (small circles) on the right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
and
. . . . . . . . . . . 17
in the beam relate to the correlation matrix. . . . . . . . . . 40
x
4.2
4.3
4.4
Noise simulations with 50 realizations superimposed. Background noise is
% in (a), and % in (b), (d), and (f). Intensity
0% in (c) and (e),
% in (c) and (d), and % in (e)
dependent noise is 0% in (a) and (b),
and (f). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
The mutual intensity at fixed points, measured by , for LD03
plotted vs. current. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
The mutual intensity at two fixed points, measured by , for LD03
plotted vs. current. The plot for increasing current is marked with small
circles and labeled ’Up’ and for decreasing current labeled ’Down’. . . . . 72
4.5
The total beam power (solid line) of LD03 vs. current compared to the
mean of and (small circles). . . . . . . . . . . . . . . . . . . . . 73
4.6
Contour plots of the LD03 beam for three different coherent levels. The
current increases from top to bottom. The middle plot corresponds to a
current level with low overall coherence. The contour lines indicate increasing power differences from top to bottom. . . . . . . . . . . . . . . . 75
4.7
Optical spectra for LD03 for various currents, starting below threshold at
the top plot and increasing in current towards the bottom plot. . . . . . . . 78
4.8
Optical spectra for LD03 for various currents, starting at 40 mA for the top
plot and increasing in current towards the bottom plot. . . . . . . . . . . . 79
4.9
Optical spectra for LD15 at various current levels. . . . . . . . . . . . . . . 81
4.10 Optical spectra for (left) LD
and (right) LD20 at two different current
levels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.11 Frequency mixing pattern for LD
with 6 GHz detector for current less
than 42 mA (top) and for current greater than 43 mA (bottom). . . . . . . . 84
4.12 Beat frequency spectrum for LD
at 47 mA, 11/28/01, 6 GHz detector . . 85
4.13 Beat frequency spectra for LD
on 3/1/02, with 25 GHz detector, at current levels of 47 mA (top) and 60 mA (bottom). . . . . . . . . . . . . . . . 87
4.14 Beat frequency spectrum of LD
at 40 mA, 3/1/02 . . . . . . . . . . . . 88
xi
4.15 Beat frequency spectra of LD03 at 18.7 mA (top), 19.8 mA (center), and
20.4 mA (bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.16 Beat frequency spectra of LD15 at 23 mA (top) and 40 mA (bottom). . . . . 90
4.17 Beat frequency spectrum of LD20 at 27 mA, or 1.08 . . . . . . . . . . . . 91
4.18 Beat frequency spectra of LD30 at 46 mA (top) and 55 mA (bottom). . . . . 92
4.19 Frequency peaks from optical cavity between laser and detector. . . . . . . 93
4.20 (Left) The transverse spatial mode of the 5 m stripe width laser extracted
by singular value decomposition (circles). (Right) The corresponding intensity profile (circles) compared to the measured intensity (pluses). . . . . 96
4.21 (Left) The lateral spatial modes of the 5 m stripe width laser extracted by
the Jacobian algorithm (circles and squares) compared to fitted Hermite–
Gaussian modes (dashed line). (Right) The reconstructed intensity profile
(circles) compared to the measured intensity profile (+’s). . . . . . . . . . . 98
4.22 (Left) The transverse spatial modes of LD03 for 27 mA (top), 31.6 mA
(middle), and 40 mA (bottom). (Right) The reconstructed intensity profiles
(circles) compared to measured intensity profiles (+’s). . . . . . . . . . . . 102
4.23 (Left) The transverse spatial modes of LD03 for 48 mA (top), 50.7 mA
(middle), and 52 mA (bottom). (Right) The reconstructed intensity profiles
(circles) compared to measured intensity profiles. . . . . . . . . . . . . . . 103
4.24 (Left) The lateral spatial modes of LD03 for 27 mA (top), 31.6 mA (middle), and 48 mA (bottom). (Right) The reconstructed intensity profiles (circles) compared to measured intensity profiles (+’s). . . . . . . . . . . . . . 105
4.25 (Left) The lateral spatial modes of LD04 for 25 mA (top), 28 mA (middle),
and 30 mA (bottom). (Right) The reconstructed intensity profiles (circles)
compared to measured intensity profiles (+’s). . . . . . . . . . . . . . . . . 107
4.26 (Left) The lateral spatial modes of LD
for 40 mA (top), and 47 mA
(bottom). (Right) The reconstructed intensity profiles (circles) compared
to measured intensity profiles (+’s). . . . . . . . . . . . . . . . . . . . . . 109
xii
4.27 Intensity profiles for laser LD
for current levels from 25 mA to 40 mA.
Circles indicate the current levels used for spatial coherence tests. . . . . . 110
4.28 (Left) The lateral spatial modes of LD
for 26 mA (top) and 27 mA
(bottom). (Right) The reconstructed intensity profiles (circles) compared
to measured intensity profiles for the LHS fiber (+’s) and the RHS fiber (x’s).111
4.29 (Left) The lateral spatial modes of LD
for 30 mA (top) and 40 mA
(bottom). (Right) The reconstructed intensity profiles (circles) compared
to measured intensity profiles for the LHS fiber (+’s)and the RHS fiber (x’s). 112
4.30 (Left) The lateral spatial modes of LD15 for 23 mA (top) and 28 mA (bottom). (Right) The reconstructed intensity profiles (circles) compared to
measured intensity profiles (+’s). . . . . . . . . . . . . . . . . . . . . . . . 114
4.31 (Left) The lateral spatial modes of LD15 for 38 mA (top) and 48 mA (bottom). (Right) The reconstructed intensity profiles (circles) compared to
measured intensity profiles (+’s). . . . . . . . . . . . . . . . . . . . . . . . 115
4.32 Intensity profiles for laser LD20 for current levels from 26 mA to 60 mA.
The profiles marked with circles indicate the current levels for spatial coherence tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.33 (Left) Lateral spatial modes of LD20 at current levels of 30 mA (top) and
34 mA (bottom). (Right) The reconstructed intensity profiles (circles) compared to measured intensity profiles (+’s). . . . . . . . . . . . . . . . . . . 118
4.34 (Left) Lateral spatial modes of LD20 at current levels of 40 mA (top) and
55 mA (bottom). (Right) The reconstructed intensity profiles (circles) compared to measured intensity profiles (+’s). . . . . . . . . . . . . . . . . . . 119
4.35 (Left) Lateral spatial modes of LD20 at a current level of 44 mA. (Right)
The reconstructed intensity profile (circles) compared to the measured intensity profile (+’s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.36 Intensity profiles for laser LD30 for current levels from 35 mA to 55 mA.
The profiles marked with circles indicate the current levels for spatial coherence tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
xiii
4.37 (Left) The lateral spatial modes of LD30 at currents of 40 mA (top), 46 mA
(middle), and 55 mA (bottom). (Right) The reconstructed intensity profiles
(circles) compared to measured intensity profiles (+’s). . . . . . . . . . . . 123
4.38 (Left) Lateral spatial modes of the HeNe laser extracted by the Jacobian
algorithm (circles and x’s) compared to fitted Hermite–Gaussian modes
(dashed line). (Right) The reconstructed intensity profile (circles) compared to the measured intensity profile (x’s) and the reconstructed Hermite–
Gaussian intensity (dashed line). . . . . . . . . . . . . . . . . . . . . . . . 124
xiv
LIST OF TABLES
Table
Page
3.1
Manufacturers and model numbers for the equipment used . . . . . . . . . 59
4.1
The relative field and power weights of transverse spatial modes of laser
LD03. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.2
The relative field and power weights of lateral spatial modes of laser LD03. 106
4.3
The relative field and power weights of lateral spatial modes of laser LD
4.4
The relative field and power weights of lateral spatial modes of laser LD15. 116
4.5
The relative field and power weights of lateral spatial modes of laser LD20. 120
4.6
The relative field and power weights of lateral spatial modes of laser LD30. 120
xv
. 110
CHAPTER 1
INTRODUCTION
1.1 Overview
Both designers and users of optical devices or systems have an interest in ways to determine the quality of a laser beam. The designer wants to specify the optimum characteristics
of the laser for a particular application, and the consumer needs a standard for comparison
of different lasers. In addition, some assessment of quality is necessary in the manufacturing process to determine the criteria for acceptance or rejection.
It is difficult to define an all-purpose quality factor for a laser beam and, if quality
is measured by a single number, such as the
factor, the results may be ambiguous
[1, 2]. There are several different, often interrelated, characteristics that play a part in
determining the laser quality. These include wavelength, spectral width, power (or energy),
minimum beam width, location of this minimum and the beam divergence [1]. Various
optical systems will have tighter specifications on some of these beam characteristics than
others. The spatial and temporal coherence of the beam may also be important for certain
applications. Low spatial coherence lasers are more compatible with multimode fiber in
short-haul data links due to reduced modal noise [3], but the trade-off for lower spatial
1
coherence is reduced temporal coherence [4], which broadens the spectral width. Usually,
high temporal coherence is a desirable characteristic.
A common standard for an ideal laser is a single Gaussian spatial mode. This fundamental mode can be focused to a smaller spot size than multimode beams. A smaller
spot size equates to higher irradiance, beneficial for nonlinear processes and industrial
drilling. Some ways to measure overall beam quality based on this standard are the
second-moment method mentioned earlier, knife-edge scanning, slit scanning, and energy
measurement in a variable aperture [1, 5]. Older commercial beam analyzers (e.g. [6])
commonly give statistics relating the fit of the beam’s intensity profile to a fundamental
Gaussian. Newer beam analyzers also make use of
analysis [7].
A laser beam can appear to be Gaussian and still be multimode. The most common
example in the literature is the Gaussian–Schell or Collett–Wolf model [8, 9, 10]. In this
model, a large number of Hermite–Gaussian modes with exponentially decreasing weights
are summed to produce a Gaussian intensity profile. Another example is given in Ref. [7]
where there is a case of a beam with a Gaussian fit of 0.97 with no fundamental mode and
a combination of five higher-order modes. A simpler example is depicted in Figure 1.1,
where a Gaussian beam appears similar to a beam with 95% of the total power in the Gaussian mode and 5% in the first-order Hermite–Gaussian mode. In the figure the Gaussian is
represented with a solid curve and the two-mode beam is the dashed curve. A beam with
an intensity profile that looks similar to a Gaussian beam may have divergence properties
that are significantly different than expected for a Gaussian beam [11].
It is also possible for a laser to be single-mode and not be Gaussian. For example,
thin-junction double heterostructure (DH) semiconductor lasers have an approximately
Lorentzian-shaped spatial mode in the plane perpendicular to the junction that may be
2
Normalized intensity
1
0
−2
−1
0
1
2
Distance normalized to spot size
Figure 1.1: Intensity profiles of a Gaussian beam (solid curve) and a beam with 5% power in
the first-order Hermite–Gaussian mode and 95% in the fundamental mode (dashed curve).
truncated by packaging because of the large divergence [12, 13, 14]. Even parallel to the
junction plane there may be asymmetries or step-index changes and gain guiding that would
cause the fundamental mode to differ from the Gaussian model [11].
An alternative way to analyze the quality of a laser beam is by the form of the spatial
modes and the distribution of power over the modes. A laser designer can use this analysis
to examine variations in the refractive-index profile, gain distribution, or structure of the
laser. Similarly, the systems designer or manufacturer can use this analysis to specify the
lasers needed or accepted.
The modes of a laser come from the geometry of the resonant cavity. The resonant
field variations are defined as modes. A laser cavity may have multiple longitudinal modes
or multiple spatial modes or both, depending on the dimensions of the cavity. The different modes generally oscillate at slightly different frequencies, although there may be
frequency-degenerate modes. Multiple longitudinal modes in semiconductor lasers may
3
produce peaks in the spectrum that are separated by tenths of nanometers or less [15, 16].
Spatial modes may have a spectral separation on the order of hundredths of nanometers
[15, 17], thus multiple spatial modes increase the overall spectral width of the laser even
when only one longitudinal mode is present. Higher order spatial modes also increase the
spatial width of the beam. In this research, the focus was on spatial modes, sometimes
called transverse modes to distinguish them from the longitudinal modes.
Multimode lasers can be characterized by a basis set of spatial modes and the weights of
each mode. Usually, techniques of determining modal weights rely on assumed expressions
for the spatial modes, such as Hermite–Gaussian modes or possibly Laguerre–Gaussian
modes for circularly symmetric lasers. Although these modes have the advantage of mathematical tractability, they may not adequately represent the physical characteristics of the
laser. If the Gaussian model is physically inconsistent with the laser, as for thin-junction
edge emitting semiconductor lasers, the results will be erroneous for the number of modes
and their weights.
In the research described in this dissertation we have developed a new technique that
makes it possible to find the weights and forms of the spatial modes without any a priori
assumptions of the shape of the modes. The only limitation is that the modes oscillate at
distinct frequencies. If two or more modes are frequency degenerate, the extracted mode
will be the sum of the corresponding degenerate spatial modes. The spatial modes found in
this experimental method may show greater insight into the physical characteristics of the
laser. This insight would be especially useful in laser analysis and design, particularly of
semiconductor lasers with their possibly non-standard mode structure.
4
1.2 Objective
The objective of this dissertation is to describe our method of determining the spatial
mode structure and weights of lasers and analyze the experimental results. This technique
involves measuring the spatial coherence of a laser beam in an axial plane using a twin-fiber
interferometer. These measurements comprise a mutual intensity matrix that, when decomposed into eigenvectors and eigenvalues, leads to the spatial modes and mode weights. The
experimental technique is particularly suited to semiconductor lasers or other lasers that
have a few low-order modes that would be difficult to distinguish by intensity measurements. To analyze the data, we have developed an algorithm to decompose the correlation
data and extract the modal weights and structures.
1.3 Literature survey
Most previous work to experimentally determine information about the spatial modes of
lasers has relied on assumed models for the shape of the modes. These methods retrieve the
number and weights of the modes with varying degrees of success but are always limited
by how closely the physical spatial modes of the laser in question match the model. To our
knowledge, the only other method analyzing spatial modes without making assumptions
about mode structure involves using near field scanning microscopy (NSOM) [17, 18]. The
following sections give a brief survey of other work that has been done to determine modal
weights.
1.3.1 Non-coherence methods of determining modal weights
Some general methods of mode analysis that do not measure spatial coherence are
intensity curve fitting [19],
analysis [20, 21], matrix inversion [22, 23, 24], fractional
5
Fourier transform [25], frequency mixing [26] and spectral analysis [15, 27, 17]. The first
three methods, which use intensity measurements in a plane, require additional information
to uniquely reconstruct the weights [28]. Specific work in these methods is discussed below.
Cutolo et al [19] used measurements of the intensity distribution of a laser beam in the
near field and the far field to determine the excitation coefficients of the transverse modes.
Earlier work was based on intensity measurements in a single plane [29, 30]. The single plane measurements resulted in possibly non-unique results or false solutions and was
more sensitive to noise than the two plane measurements. Their method assumed separable modes with Hermite–Gaussian forms. The theoretical intensity was then a sum over
all the Hermite–Gaussian modes multiplied by power content coefficients (the excitation
coefficients squared) and some possible cross terms, depending on the mutual coherence of
the transverse modes. Then a function
was defined that was a summation of the square
of the difference of the theoretical intensity and the measured intensity. This function was
minimized and the resulting coefficients determined the weights of each Hermite–Gaussian
mode. This method was tested on a HeNe laser and a pulsed Q-switched Nd:YAG laser.
Experimentally this method was straightforward. The majority of the effort was in the
numerical minimization.
There are some difficulties with this method, especially as applied to semiconductor
lasers. Since the forms of the modes were assumed for the numerical analysis, the lasers
are limited to be Hermite–Gaussian. This assumption presents problems with semiconductor lasers, as noted earlier. If the form of the modes is not known a priori, the intensity
distribution is not sufficient to uniquely determine the mode structure because the intensities of different modes add incoherently [9, 10]. The analysis seems to be more suited to
situations where there are significant weights in the higher-order modes. As was shown
6
earlier in Figure 1.1, a laser that is almost single-mode has an intensity distribution that is
nearly indistinguishable from a single-mode laser. Conversely, if the modes were weighted
in such a way that the beam approximated a top hat beam (flat intensity across the top
and steep sides) such as the beam used in Ref. [31], this least squares fit would be more
sensitive to noise.
The
analysis method is a way of quantifying the second moment of the beam by
measuring the intensity profile. This assumes Hermite–Gaussian modes, resulting in a
physical beam width that is proportional to the fundamental mode beam width. Parameters
of the beam, such as the fundamental beam waist, are taken from the geometric properties of the stable resonator. Siegman and Townsend [20] used this method to analyze a
simulated flat-top intensity profile. With the Hermite–Gaussian modes as a basis set, they
determined that higher-order modes were preferred in large-bore, multimode, stable cavity
lasers. They also used experimental data to show how this method gave results that matched
the divergence of a 10 kW CO laser. Lu et al [21] also used
analysis but applied it
to Gaussian–Schell model beams instead of flat-top beams. There were no experimental
results given in this paper.
The
method is straightforward but is only as valuable as the assumptions of the
model. It can be useful for determining divergence for lasers that are highly multimode,
such as those having 10 or more transverse modes. This is typical of high powered lasers,
which are often used in applications where the high irradiance is the most important quality.
For semiconductor lasers, which are small and will not fit the Hermite–Gaussian model, the
analysis is not reliable.
Some drawbacks to the
method are the ambiguity, high sensitivity to scattered
light, and the necessity for physical measurements of the cavity. By ambiguity, we mean
7
that multiple intensity distributions may result in the same
distributions measured in the far field may have different
factor and similar intensity
factors due to diffraction
effects in the near field. Diffraction can cause significant intensity modulation in the near
field that will not be seen in a far-field intensity characterization [32]. Scattered light may
be a problem where there are spherical aberrations, edge diffractions or surface roughness
[2].
analysis is more sensitive to wide angle scattered light because of the quadratic
nature of the second moment definition. Physically measuring the cavity is very difficult,
if not impossible, for semiconductor lasers.
The matrix inversion technique is similar to
analysis but is a mathematical method
which explains how two classes of laser beams, the Gaussian–Schell model and flattened
Gaussian model [22], can be defined in number and weights of Hermite–Gaussian modes
based on the intensity profile and the fundamental spot size [24]. As in
analysis, the
spot size cannot be measured directly in the beam but must be determined from the geometric properties of the resonant cavity. The results given were for numerical simulation, not
experimental data. This method does not try to justify that a particular laser beam actually
fits into one of these two models, other than by comparison of the intensity distribution
with a Gaussian or a flat-top shape. As with the
method, these models mainly apply to
physically large, high powered lasers.
A theoretical approach to recovering the mutual intensity, or spatial coherence, through
intensity measurements was given by Tu and Tamura [25]. The fraction Fourier transform
was used to reconstruct the mutual intensity from the ambiguity function. This approach
has not been implemented experimentally so it is difficult to know how practical it would
be. They also did not extend their discussion to recovering modal weights.
8
Liesenhoff and Rühl [26] employed frequency mixing while analyzing a multimode
laser beam using a Mach-Zehnder interferometer. The beam in the reference arm was
frequency shifted by acousto-optic deflection. Then the fundamental transverse mode was
selected by a mode diaphragm after deflection. The mode diaphragm was an aperture with
a diameter of 30% of the beam diameter. This aperture was located to select primarily the
fundamental mode. The beams were recombined and the two-dimensional fringe pattern
was imaged onto a detector. The fringe pattern was analyzed using phase-measurement
interferometry techniques [33].
The interferometric method was somewhat complex in its experimental configuration
because automatic frequency control was required to control the acousto-optic deflector.
It was never specified how accurate the mode diaphragm was in selecting one pure mode,
just that some of the other peaks in the spectrum “became smaller.” The laser tested was
a slow-flow CW CO laser assumed to have Hermite–Gaussian modes. No quantitative
results were given for modal weights or phase. The main benefit of this method seemed to
be the ability to visualize the interference phase relations in real time, as the results were
visible as parameters were varied.
Spectral analysis was used by Paoli, Ripper, and Zachos to detect the modes of GaAs
junction lasers in a qualitative way [15, 27]. The longitudinal modes were easier to resolve
than the spatial modes because their frequency separation was greater. In high injection
level cases different groups of modes could be superimposed in frequency. Multiple modes
at a particular frequency is one of the basic problems of spectral analysis, even for modern lasers. A high resolution spectrometer is required to distinguish spatial modes. The
spectrometer may be prohibitively expensive for some applications. Frequency degenerate
modes will not be distinguishable [34, 35] and weak modes would probably not be resolved.
9
In addition, some assumptions have to be made to interpret the spectrum. Mode structure
was deduced from fitting the frequencies to those of the assumed Hermite–Gaussian modes
[15, 27].
A modern version of spectral analysis was demonstrated by Knopp et al [17]. They
scanned a vertical cavity surface emitting laser (VCSEL) with a fiber tip used for near
field scanning microscopy (NSOM). The output was directed to a spectrometer with 0.07
nm resolution. The resulting intensity profile at each wavelength for a fixed current level
gave results on the different spatial modes. As with any other spectral analysis, there was
the consideration of frequency degeneracy, resulting in any spatial modes of the same frequency being lumped into one mode. The VCSEL measured had at least seven modes at
high current levels and showed lasing filaments. High resolution equipment is necessary
for this method, both for the scanning tip, which is less than a wavelength from the surface,
and for the spectrometer.
1.3.2 Spatial coherence methods to determine modal weights
Another category of techniques to determine modal weights is that of spatial coherence
measurements. Spatial coherence is a measure of the interference possible between two
points in the same transverse plane of a beam of light. In a laser beam without frequency
degeneracies, each spatial mode is completely coherent and the modes are mutually incoherent, comprising a complete and orthogonal basis set [36]. This will be explained in
more detail in Chapter 2. For now, the consideration is that measuring the spatial coherence
allows us to extract information about the spatial modes of a laser.
One advantage of using spatial coherence is decreased sensitivity to certain types of
noise, which may be of significance in the tails of the beam. Interference measurements
10
filter out the incoherent noise, including spontaneous emission. Similarly, the effect of
scattered light is reduced, especially with respect to the second moment method described
earlier.
Another advantage of the spatial coherence function is its sensitivity to higher order
modes. The intensity of a laser beam with small amounts of higher-order modes may
appear very similar to a pure Gaussian beam but the complex coherence factor will differ
significantly for the two cases (e.g. see Figures 3 and 4 in Ref. [37]).
The spatial coherence methods can be subdivided into vector methods [37, 38, 39, 40]
and matrix methods [31, 41, 42, 43]. The vector methods only interfere the beam with
its mirror image. This produces a vector of coherence information as a function of displacement position. The matrix methods measure spatial coherence as a function of two
positions, thus giving a matrix of data. The data vector in the vector case corresponds to
the antidiagonal of the correlation matrix from the matrix case.
The matrix of correlation data collected in the matrix configuration allows us to determine both the modal weights and the form of the spatial modes, as will be explained in
Chapter 2. Theoretically, any matrix method of measuring spatial coherence could have
been used for determining both the modal weights and and the mode forms. In practice,
no previous method has done more than extract the modal weights based on the assumption of Hermite–Gaussian spatial modes. This research has been the first instance of using
coherence data to find spatial modes that can be of any arbitrary structure. The limitations
of other methods may have been due to experimental difficulties, as some implementations
give better results than others.
The only other matrix method that even quantified modal weights was done by Tervonen et al [31, 44]. They assumed Hermite–Gaussian modes and solved numerically for
11
the weights of the different modes. The results were given for a HeNe laser with fifteen
significant modes and showed good agreement with the experimental data. The published
results do not include any tests on lasers operating primarily in the fundamental mode.
The vector methods, by definition, only acquire enough data to extract modal weights
based on assumed modal forms. Also, they will find incorrect results if all of the modes
are either even or odd [37]. Spano did some experiments with semiconductor lasers but
found that the beams were asymmetric and could not be modeled with Hermite–Gaussian
modes. Without knowing the form of the modes he could not quantitatively determine
the weights. He also did some tests with fibers and was able to fit some curves assuming
Hermite–Gaussian modes [38]. Even for the semiconductor laser cases he was able to
observe changes in mode weights more clearly through the spatial coherence function than
by the intensity profile.
Experimentally, a variety of different interferometers have been used for measuring
spatial coherence. Spano [38, 39] measured the spatial coherence of the laser beam with a
reverse-front interferometer, thus taking a vector of measurements. Anderson [40] and Pelz
[37] employed a twin-fiber interferometer. Both of these implementations measured data in
the vector configuration. In general, it is possible to use a twin-fiber interferometer to take
a matrix of data and that is the configuration that we have been using. Tervonen et al [31]
implemented a computer-controlled version of Young’s two pinhole interferometer in their
technique. This resulted in a matrix of data. Both Iaconis et al [41] and Mendlovic et al
[42] designed their techniques around Sagnac interferometers. In the first case, a rotatable
glass block introduced lateral shear into the interferometer and in the other, a dove prism
and a joint transform correlator produced the necessary displacement and correlation.
12
The pinholes for Tervonen’s interferometer were made by overlapping two masks that
had crossed-slit apertures. As long as the masks were offset by at least the slit width in
both the vertical and horizontal directions, light was only transmitted through two small
squares. Each mask was mounted on computer-controlled 2-D translation stages so the
spatial coherence could be measured as a function of two dimensions. One limitation of
this type of interferometer is the diffraction losses. Only a small portion of the light is
transmitted through the apertures and then the transmitted light is diffracted across a large
area, further decreasing the intensity level of the detected light. These low levels may limit
the amount of data points that have intensity levels significantly above the noise floor.
The twin-fiber interferometer overcomes the problem of diffraction losses observed in
Young’s interferometer. About half of the light collected by the fibers is delivered to the
detector by the fiber coupler, not diffracted over the whole observation plane. (The other
half of the light is coupled into the second, unused, output arm of the fiber coupler.) The
fiber used to sample the beam is still physically small forcing some limitations on the
intensity levels required for good data. Computer-controlled positioning of the two arms
allows the flexibility and speed needed to take a matrix of data. Neither Anderson [40]
nor Pelz [37] had computer-controlled positioning of the fiber arms so they were limited to
the vector configuration. Pelz tested this technique with the vector method on a HeNe laser
with most of the power in the fundamental mode [37]. As previously seen in Figure 1.1, the
intensities are practically indistinguishable in this type of modal distribution. The spatial
coherence function in his measurements, however, was clearly different than the singlemode case even with less than 1% of the power in the first higher-order mode. We will
discuss the twin-fiber interferometer more fully in later chapters.
13
The Sagnac interferometer reflects the light in a closed loop, with a beam-splitter sending one beam clockwise and the other counter-clockwise. It has both positive and negative
characteristics for measuring spatial coherence. On the positive side, by definition the two
arms of the interferometer have the same optical path length so temporal coherence is not
an issue. Also, the interferometer is inherently parallel, allowing a vector of data to be
acquired at one time instead of just a single point, as in the other two matrix methods. The
intensity can be measured with a CCD camera or an array of detectors. The parallelism
was attractive to us for faster data acquisition so we spent several months implementing
the setup outlined by Iaconis [41]. Eventually we returned to the twin-fiber interferometer
because of the experimental trouble with the shearing interferometer. Resolution and range
were issues in the 8-bit CCD camera available. There were also reflections and interference
off of the multiple surfaces of beam-splitters and neutral-density filters. In reality, neither
Iaconis et al nor Mendlovic et al were trying to measure modal weights [41, 42], but instead
were using spatial coherence for other purposes.
1.4 Organization of dissertation
There are a total of five chapters in this dissertation. In Chapter 2 we explain the theoretical background, both of spatial coherence in general and of our experimental technique
and numerical analysis in particular. In Chapter 3 we describe our experimental configuration and procedures. Experimental results are given in Chapter 4 for both physical
measurements and computer simulations. In the final chapter we summarize the results and
draw conclusions. There are also some suggestions for further work in the last chapter.
14
CHAPTER 2
THEORETICAL BACKGROUND
2.1 Introduction
In this chapter we lay the foundation for determining modal weights and structures of
semiconductor lasers by spatial coherence. In Section 2.2 we discuss the topic of laser
modes in general. Section 2.3 covers the specific case of Hermite–Gaussian spatial modes,
including their utility and limitations. We explain the subject of spatial coherence in Section 2.4. Several subsections are included to examine different aspects of spatial coherence
and how they relate to our research. In Section 2.5 we explain the type of interferometer we
used and discuss the limitations in data points, requiring specialized numerical analysis. In
Section 2.6 we present the development of the numerical analysis we used, including the
conditions and resulting algorithms. We conclude the chapter with a summary section.
2.2 Laser modes
One of the fundamental characteristics of lasers is the resonant cavity. As the optical field travels one complete round trip through the cavity, certain field variations, called
modes, reproduce themselves in any cross section in relative shape and relative phase
[45, 46]. The term longitudinal describes modes that have traveled an integral number of
15
round trips through the cavity. Longitudinal modes are separated by a frequency inversely
proportional to the length of the cavity. The number of longitudinal modes contributing to
the laser output is affected by the gain characteristics and geometry of the laser. Unless
specified otherwise, we will assume that there is only one longitudinal mode operating in
the laser beam.
The spatial dependence of the optical field seen in a transverse plane in the laser cavity
is referred to as a transverse or spatial mode. These are the modes of interest in our research.
The order of the spatial mode is generally the same as the number of minima in the intensity
profile of that mode [47, Ch. 14]. The number of peaks in the intensity profile of each mode
is then the order number plus one. The higher the order of the mode, the wider the physical
size of the mode. An increased number of spatial modes also results in a broader spectral
width of a laser, because each mode usually oscillates at a slightly different frequency.
The simple case of a rectangular resonator will have modes that satisfy the following
conditions [48]:
where
, , and
(2.1)
are the width, thickness and length of the resonator,
, and ,
is the refractive index. The longitudinal mode number
integral number of half wavelengths along the optic axis. The mode numbers
is the
and
indicate the number of zeroes of the intensity profile in the two transverse directions.
In a general case, with modes that are separable in
written as
and , the wavefunction can be
&%
%
$
#
$
#
$
$
#
+
)
*
/
,
.
$
0
1
$
3
#
2
"! '!
" !(
! 16
(2.2)
y
x
Laser
optic axis
z
Junction
d
W
L
Figure 2.1: The coordinate system used for a semiconductor laser.
where the indices
%
and
%
and directions, !
indicate the mode orders in the
$# and
!( $# are amplitude functions, and 1 defines the phase. It is possible for a physical
laser to have amplitude functions that cannot be described analytically.
The coordinate system is defined so that the
direction is along the optical axis in the
and axes are oriented to
direction of propagation of the output of the laser and the
fit the geometry of the laser. The origin is usually centered on the output mirror of the
laser. For semiconductor edge emitting lasers, we define the coordinate system as shown
in Figure 2.1. The
axis is in the lateral direction, parallel to the junction, and is in the
transverse direction, perpendicular to the junction plane. This is in keeping with the normal
assumption that the
axis is “horizontal”, but is in conflict with much of the literature on
and .
semiconductor lasers, where the opposite axes are chosen for
%
Strictly speaking, the spatial modes, '!
%
$# and "! # , are only considered to be
modes inside the laser cavity where the geometry and gain characteristics of the laser select
the spatial variations of the optical field. In Section 2.4.4, after some background on spatial
17
coherence is given, we discuss the propagation of spatial modes away from the laser. The
general result is that each mode propagates independently.
If the propagated spatial modes are known for any
, then the results of Sec-
tion 2.4.4 can be used to transform these propagated modes to their forms back at the exit
face of the laser. The modes that leave the laser will usually change shape as they propagate. As an example, an exponential mode described by )+*-, !
#
on the exit face of
a semiconductor laser will have a far-field distribution that is Lorentzian [12], the Fourier
transform of the near-field mode. In the specific case of Hermite–Gaussian modes, the
modes are shape invariant during propagation so the same form is found at any point along
the axis, although with different amplitudes and widths.
2.3 Hermite–Gaussian modes
Laser cavities are often described by separable Hermite–Gaussian spatial modes based
on the work by Fox and Li [49, 50]. Rectangularly symmetric lasers are particularly suited
to this model but lasers that appear to be circularly symmetric may also have physical attributes, such as a Brewster window, that destroy the circular symmetry [20]. For Hermite–
Gaussian modes the amplitude functions of Eq. (2.2) are of the form [45]
%
!
$#
#
"!
! $# ! ! $## ) *,
$ ! and the term 1 from the phase in Eq. (2.2) is defined by
&%
1 ! $#
In these equations,
! %
is the beam width,
!
(# '*),+.-/'*021 ! ! . 3 # is the Hermite polynomial of order ,
is the wavevector,
4
%
*4
! $##
(2.3)
(2.4)
$
! #
is either
or ,
! $# is the radius of the phasefront, and
minimum beam width of the fundamental mode.
18
! $#
is the
This model is often used because the modes are shape invariant with propagation and
can be defined functionally. In addition, any arbitrary symmetric distribution in a plane can
be expanded in terms of these functions. If the Hermite–Gaussian expansion does not physically correspond to the spatial modes of the laser, however, the propagation characteristics
of the physical laser beam will not match the model. Sometimes Hermite–Gaussian modes
describe the physical modes of the laser well, especially when the laser is symmetric, has
low diffraction losses from finite aperture mirrors, and the dimensions of the resonator are
large compared to the wavelength [50].
In the case of semiconductor lasers, some of the preceding criteria may not apply. There
may be asymmetries in the choice of dielectrics [45], or asymmetries may arise in the
fabrication process. Index guiding or gain guiding may confine the extent of the field [11]
limiting the validity of the infinite aperture mirror assumption. The transverse dimension
of the active region may be smaller than the wavelength of the light for thin-junction edge
emitting lasers. The ensuing large divergence of the beam may cause the beam to be clipped
by packaging or the first lens in the optical path [13, 51].
In the edge emitting case where the step-index junction is thinner than the wavelength,
the spatial mode perpendicular to the plane of the junction is closely approximated by
the Lorentzian function in the far field [12, 52]. Other papers [13, 51] have validated the
Lorentzian model and also mentioned the diffraction problem. Naqwi and Durst [13] modeled the beam in the perpendicular plane as a truncated Lorentzian, ignoring the intensity
variation in the beam due to diffraction effects.
Zeng and Naqwi [14] also analyzed the assumption that the far-field distribution is
separable into a product of two independent functions. They concluded that separability
19
off-axis can only be assumed in the paraxial case and gave some quantitative graphs to
show relative error for two different divergence angles.
2.4 Spatial coherence
In Section 2.4.1 we explain spatial coherence and define important terms, including
mutual intensity and quasi-monochromatic conditions. We also describe how the mutual
intensity is found from the detected intensity of a general interferometer. In Section 2.4.2
we discuss how spatial coherence and spatial modes are related. The consideration of longitudinal modes is covered in Section 2.4.3. In Section 2.4.4 we describe the propagation of
spatial coherence and spatial modes. We have included a brief section summary to reiterate
the main points of the spatial coherence section.
2.4.1 Spatial coherence overview
Optical interference, measured by the mutual coherence function, is both spatial and
temporal. The mutual coherence function,
the optical fields at two points, ! #
, is defined as the cross-correlation between
and , with a time difference of [53, Sec. 4.3]1 :
!
#
!
#
(2.5)
where the angle brackets refer to time averaging2 and the asterisk refers to complex conjugation. The subscripts on
do not refer to the mode number but to the two points, and
.
1
Different textbooks, such as Ref. [54, Sec. 10.3] may change which term is conjugated and which term
contains but the results are the same.
2
We are assuming that the light is a wide sense stationary random process and is ergodic, allowing us to
use time averaging in the place of ensemble averaging.
20
Another term used in the study of optical coherence is the complex degree of coherence
. This is defined as the mutual coherence function normalized by the square root of the
auto-correlation at each point with [54, Sec. 10.3]:
! #
! #
# ! # 2 !
.
(2.6)
The normalization factors in the denominator are also the square roots of the intensity at
each point.
We assume cross-spectral purity for our interference process. This assumption is valid
if the spectral content of the light does not change after interference and it allows the mutual
coherence function to be factored into spatial and temporal functions [55, Sec. 5.3].
The time dependence of the mutual coherence function can be broken down into two
factors. One factor is slowly varying and depends on the time difference between the two
sample points. The other factor is sinusoidal and the frequency of the sinusoid is the same
as that of the light. If the beam is narrow band, as is the case for lasers, and is much less
than the coherence time, the beam is temporally coherent so the slowly varying factor is
approximately constant. Light that satisfies these conditions is called quasi-monochromatic
[55, p. 180] and is assumed throughout this dissertation.
Under quasi-monochromatic conditions and cross-spectral purity the mutual coherence
function simplifies to [55, p. 180]
! #
) (2.7)
and the complex degree of coherence simplifies to
! #
) 21
(2.8)
where
!
# is the mutual intensity,
! # is the complex coherence factor
and is the center frequency of the light.3 The mutual intensity is also known as the equal-
time mutual coherence function since it is defined for
see that the complex coherence factor
is a normalized mutual intensity:
where
! #
. From Eq. (2.6–2.8) we can
.
2 (2.9)
refers to the intensity at point
and
refers to the intensity at point
, each measured without any interference between the two points.
Both the mutual intensity and the complex coherence factor may be complex as there
may be a phase difference between the two sample points. The mutual intensity can be
thought of as a phasor of the spatial sinusoidal fringe [55, p. 181]. Mutual intensity can be
separated into a phase term, )+*-, ! 0
# , multiplied by a real term or
)+*-, ! 0 #
(2.10)
where the real term is similar to the modulus of
but also includes the sign if the mutual
intensity is negative. It is not the same as the real part of
.
The complex coherence factor has the property
* The two limits of the inequality represent incoherence, when
coherence, when
*
(2.11)
*
, and complete
. Any fractional values for the complex coherence factor equate
to partial coherence.
Mutual intensity is sometimes represented by the symbol , but we have chosen to keep use the letter J for a matrix symbol in a later section.
3
22
because we
The classical measure of optical interference is fringe visibility,
Sec. 7.3]
where
, defined as [54,
(2.12)
and are the maximum and minimum intensities of the interference fringe.
Under quasi-monochromatic conditions [55, p. 182]
*
* if so
(2.13)
.
The necessary information to extract spatial modes and their weights is determined
from the mutual intensity without normalization. The following discussion and equations
explain how the mutual intensity relates to the detected intensity in a general interferometer.
An interferometer, in the most general sense, interferes two superimposed beams. In
analyzing a particular laser beam, wavefront splitting, or amplitude splitting, or both, are
used to split one beam into two parts, which are then superimposed with a possible time
difference . Amplitude splitting interferes the beam with itself displaced in time and
measures temporal coherence. Wavefront splitting interferes the beam with a spatially
shifted version of itself and measures spatial coherence [55, p. 157]. The beam splitting
may cause different transmissions in each arm, which we will represent by and .
The transmitted fields are summed by the superposition and then the total sum is squared
and averaged over time by the detector. The time average occurs automatically because
any physical detector has a response time that is much longer than the optical period
!
.
Mathematically the intensity at the detector is described by
!
#
!
#
23
!
#
!
#
(2.14)
where is the detected intensity and refers to the field sampled at the two points and
. This simplifies to
where
and , and 0 or
includes the effects of
* + ! #
(2.15)
from Eq. (2.10), any phase difference introduced by
depending on the sign of
. The intensities and
are defined
as for Eq. (2.9). From Eq. (2.15) it follows that
*
The value of the constant . 2 (2.16)
* is not important because it will cancel out when relative
weights are determined.
In solving for the mutual intensity in Eq. (2.16), the phase information of the interference fringes in Eq. (2.15) is not used. As we will see in Section 2.4.2, the phase information
is not necessary, other than any sign changes in
sion points in
, and these can be inferred from inver-
. Ignoring phase simplifies the data acquisition process, especially be
cause the phase differences between the two arms of the interferometer may vary randomly
in the twin-fiber interferometer.
Figure 2.2 displays the appearance of the interference fringes for coherent light, partially coherent light, or incoherent light, assuming . The horizontal axis repre
sents time in arbitrary units with an arbitrary origin and the vertical axis is intensity normal # . The peak-to-peak intensity value in each case is equal to ,
ized to ! with
varying according to the amount of coherence. Incoherent light does not produce
fringes and coherent light has the largest fringes, with the minimum intensity equal to zero
when * .
24
1
coherent
partially
Intensity
coherent
I +I
0.5
1
4|K1||K2||Γ12|
0
2
incoherent
0
1
Time
Figure 2.2: Interference fringes for coherent light, partially coherent light and incoherent
light.
2.4.2 Spatial coherence and spatial modes
The spatial coherence of a laser beam is determined by its composition of spatial modes
and their frequencies. For the specific case of Hermite–Gaussian modes, the spatial modes
usually operate at different frequencies. If there are no frequency degeneracies the modes
form a complete, orthonormal basis set for spatial modes. For physical spatial modes that
do not fit the Hermite–Gaussian model the spatial coherence may not be as straightforward.
To understand the orthogonality properties of general spatial modes in a laser we have
to look at how modes propagate in a laser (see Ch. 14 and 21 in [47] for a more complete
development.) For simplicity we will work in one dimension and will consider the simplest
case of a laser cavity with two mirrors,
and . The wavefunction '! # , after it travels
25
to mirror , is
from mirror
'! #
! # '!
#
(2.17)
# is the prop
agator kernel for the laser cavity. For the return trip, the propagator kernel is !
#
where
mirror, is on the is a position on the
mirror, and
! or the transpose of the original kernel. This is obviously true for the paraxial case and
Ref. [47, Sec. 21.7] explains how to handle nonparaxial apertures. The resulting round
trip propagator kernel is symmetric, in the sense that the transpose of the kernel is equal to
itself. It is not Hermitian because the transpose is not the complex conjugate of the original.
In the Hermitian case the laser modes would be “normal modes” and be orthogonal in
the following sense:
!
# !
#
(2.18)
where is the Kronecker delta. Instead, the modes are biorthogonal, or 4
"! # ! #
(2.19)
This is equivalent to saying the forward traveling wave is power orthogonal to the reverse
traveling wave [47, Ch. 21],
!(! #
# ! #
(2.20)
where the superscript refers to a field traveling in the forward direction and the superscript refers to the reverse direction. Ignoring the phase of gives the real amplitude
%
functions
of Eq. (2.2). These are the same in both directions so we see that the spatial
modes of the laser are spatially orthogonal over any cross-section of the laser, including
the exit mirror.
4
For a simple proof that a symmetric kernel results in biorthogonal modes, see [56, Sec. 3.8], although
this reference uses the term orthogonal, specifying that it is in a non-Hermitian sense.
26
An important consequence of the biorthogonality is that the modes are not necessarily
orthogonal in power. We cannot generally sum the power in each of these modes and claim
that it is equal to the total power. It may be true in specific cases if the modes are also
orthogonal in another sense.
A minor aspect of biorthogonality is that one cannot prove completeness for the set
of modes in a rigorous sense. However, this is not critical in our application of finding
physical modes in a laser.
In the space-frequency domain the modes are orthogonal in power if they each oscillate
at a different frequency [34, 53]. Modes of two different frequencies will produce a beat
frequency on the detector at the difference
of the two frequencies [57]. The frequency
difference is related to the wavelength difference
by
(2.21)
where
is the speed of light in a vacuum and
wavelength of
is the central wavelength. For a nominal
m and mode separation of one hundredth of a nanometer, the difference
frequency is 3 GHz. A detector with a frequency response of at least 6 GHz is necessary to measure this beat frequency. If the wavelength separation is greater, the difference
frequency also increases. When the detector has a frequency response significantly less
than the beat frequency the detector averages over multiple cycles and the modes appear
incoherent to each other. In this sense, they are power orthogonal.
Mandel and Wolf rigorously showed that the cross-spectral density, which is the Fourier
transform of the mutual coherence function, can be expanded into an orthogonal superposition of modes that are completely coherent in the space-frequency domain [53, Sec. 4.7].
27
The cross-spectral density is
as long as
!
#
#$)+*-, 0 ! (2.22)
! # is absolutely integrable with respect to . The expansion of the cross
spectral density can be written in one dimension as
where
!
!
#
1 '! "# !
#
!
# !
"#
#
(2.23)
(2.24)
Here the ’s are eigenfunctions of the cross-spectral density and have a possible frequency dependence, and so may the weights 1 ’s.
Wolf originally developed the expansion of the cross-spectral density in the frequency
domain [36] but it can also apply to the time domain under the right conditions. When the
light is quasi-monochromatic the wavefunctions in the frequency domain are approximately
delta functions and are therefore sinusoids with constant amplitude in the time domain as
shown in Eq. (2.7). We can write the resulting mutual intensity as
where is the power weight of the
!
# !
#
(2.25)
th mode and is the wavefunction of the
th
mode, no longer dependent on frequency.
Removing the phase term, the amplitude part of Eq. (2.25) is
%
%
!
%
# !
#
(2.26)
where the ’s refer to the amplitude functions of Eq. (2.2) and
is defined in the same
way as for Eq. (2.10). Using the homogeneous Fredholm integral equation the weights 28
of the modes can be found from
%
!
#
%
!
#
(2.27)
if the form of the modes is known. The other spatial coherence methods that were discussed
in Chapter 1 base their determination of modal weights on Eq. (2.27), with the Hermite–
%
Gaussian basis set used to define .
For this research, we work directly with Eq. (2.26) to find not only the eigenvalues, or
modal power weights, but also the eigenvectors, which are the mode shapes. The actual
measured quantity is the modulus of the mutual intensity,
The modulus is identical to
change of
radians in )+*-, ! 0
, as defined in Eq. (2.16).
unless there is a sign change, corresponding to a phase
# as
and
vary. A sign change can be inferred from an
inversion point in the mutual intensity plot. The one qualification is that the modes cannot
be frequency degenerate.
If two spatial modes have the same frequency (and polarization) they are frequency
degenerate and are not orthogonal in the power sense of Eq. (2.18), although they are
still spatially orthogonal. The corresponding eigen-function of the cross-spectral density,
Eq. 2.23 and 2.24, or the mutual intensity, Eq. 2.25, is a superposition of the frequency
degenerate spatial modes of the cavity [34, 35]. Because this combination is linear, it is
still spatially orthogonal to the other spatial modes.
The effect of frequency degeneracies on extracting modal weights depends on the a
priori assumptions. The result for our spatial coherence method of mode decomposition is
that the degenerate spatial modes will be extracted as one mode with a shape that is a linear
combination of the individual field modes. For the methods that assume Hermite–Gaussian
modes, however, the mode weights will be determined incorrectly since the degenerate
modes are not orthogonal in power.
29
Frequency degeneracies can arise when different combinations of mode numbers
and
give a constant
. This is obvious in the Hermite–Gaussian case from the phase
term 1 in Eq. 2.4. For thin-junction DH edge emitting semiconductor lasers this is not a
factor because
is constrained to be zero by the size of the junction. Quantum well (QW)
or multiple quantum well (MQW) lasers are not necessarily thin-junction, even though
the active region of the well is much smaller than a wavelength. They have confinement
layers above and below the active region and the dimensions of these layers determine
the number of transverse modes that can be supported [48, Ch. 4]. Other semiconductor
lasers, particularly VCSELs (vertical cavity surface emitting lasers), may have frequency
degenerate modes if they have dimensions larger than a wavelength in both axes orthogonal
to the direction of propagation [17].
There are physical effects that can remove the degeneracy between two modes with
constant
. Irregularities in the laser cavity may cause frequency splitting and there
may be mode repulsion effects from spectral hole burning [57]. Even for modes at the same
frequency, if they have orthogonal polarizations they will be spatially incoherent. VCSELs
often have modes with orthogonal polarizations that are close to frequency degenerate, as
shown experimentally in [58, 18]. The polarization discrimination is an advantage that
spatial coherence measurements have over NSOM, which only separates modes by wavelength, and that by tenths of nanometers [17].
For practical purposes, we can consider the spatial modes of the tested lasers to be
orthogonal in power. Physical perturbations on the different modes tend to separate them
in frequency. With the slow speed detector that we used, two modes would have to have
wavelengths within less than 0.0001 nm to be considered degenerate. That is two orders of
magnitude smaller than the typical separation of lateral modes.
30
2.4.3 Spatial coherence and longitudinal modes
As stated in Section 2.2, we are primarily interested in the spatial modes of a laser
and not the longitudinal modes. The previous sections have assumed there was only one
longitudinal mode. If there are multiple longitudinal modes, however, the overall coherence
of the laser beam will decrease.
Multiple longitudinal modes decrease the temporal coherence in a way similar to multiple spatial modes decreasing the spatial coherence. In a non-rigorous sense, Eq. 2.7, which
defines the mutual coherence function as the mutual intensity times a sinusoidal frequency
term, applies to each longitudinal mode separately. It is beyond the scope of this dissertation to account for all the changes of coherence based on longitudinal modes. One would
have to take into account possible phase differences between the longitudinal modes and
the small changes of frequency between each longitudinal mode family. There could also
be variations in the spatial modes and shapes depending on the order of the longitudinal
mode.
We can think of each longitudinal mode number as representing a family of spatial
modes, whether the family is made up of one or several spatial modes. The term “axial
modes” is used for the longitudinal modes with fundamental spatial modes. Spectrally,
the separation between adjacent axial modes is on the order of ten or more times greater
than the separation between lateral modes [15] so the spatial modes are grouped at each
longitudinal mode spacing. A spectrometer may not be able to resolve the individual spatial
modes but might represent them as one widened longitudinal mode.
31
For a Fabry–Perot cavity, the separation of the axial modes can be derived from the
rectangular resonator case of Eq. 2.1. With
, Eq. 2.1 simplifies to
(2.28)
The wavelength difference between axial modes is then
where
!
- #
!
(2.29)
is the group index of refraction. The group index of refraction varies by material
composition and wavelength but a typical value for GaAs is 4.3 [48]. Using this value, a
Fabry–Perot laser cavity with a typical length of 300 m will have a longitudinal mode
separation of about 0.37 nm at 980 nm.
If there are multiple longitudinal mode families, each of the individual spatial modes
will be incoherent with all of the other spatial modes because of the difference in frequency.
This means that the fundamental spatial mode corresponding to longitudinal mode
will
not interfere with the fundamental spatial mode corresponding to longitudinal mode , nor
will any of the higher order modes.
At the risk of oversimplifying, we can think of a total mutual intensity that is an incoherent sum of the mutual intensities for each longitudinal mode. The number of summation
terms in the total mutual intensity then increases by a factor of the number of active longitudinal modes. Rewriting Eq. 2.25 to account for the number and weights of longitudinal
modes gives
where
!
# !
indexes the longitudinal modes that are lasing,
are the power weights for the active longitudinal modes.
32
#
(2.30)
indexes the lateral modes, and
For our purposes, we will assume the amplitude functions are still the same shape for
each family of longitudinal modes. This assumption is based on the fact that the geometry
of the laser stays the same and the difference in wavelength is small, minimizing changes
in the laser parameters, such as refractive index and gain distribution. Also, because the
spatial modes are tightly grouped at each longitudinal mode, we will treat each longitudinal
mode family as having the same longitudinal power weight for each spatial mode term.
Assuming the phase differences are not significant, we can simplify Eq. 2.30 to
%
'!
%
# '
!
#
(2.31)
Under the assumed conditions, the relative weights of the spatial modes will be the same
for each of the longitudinal mode families. Because the overall coherence is reduced by
multiple longitudinal modes, the magnitude of the total mutual intensity is also reduced.
The number and choice of active longitudinal modes in a semiconductor laser are affected by the injection current and heat sink temperature [59, 60, 61]. The term mode hopping is used to describe random switching between two longitudinal modes where there is
an almost total power transfer. Typically, the dwell times in each mode are on the order of
microseconds [60] to hundreds of microseconds [61]. Mode partition is similar but refers
to mainly single-longitudinal-mode operation with occasional power dropouts due to nonlinear coupling with weak side modes. The duration of the power dropouts is on the order
of nanoseconds [60].
Mode hopping is observed in the coherence measurements as a significant overall decrease in coherence. The time frame is such that the measurements will average the two
modes. Mode partition will also be seen as a decrease in coherence but at a level that could
be disguised by other noise sources. For reliable measurements of spatial coherence, it is
33
important to choose a current level where the laser is operating in a single stable longitudinal mode.
2.4.4 Propagation of spatial modes
Earlier in the chapter we saw that the modes of a laser are considered to be modes
inside the cavity because the field variations are selected by the cavity. The description
of the spatial coherence of the modes defined in the previous section allows us to know
something about how the laser modes propagate away from the cavity.
In general, both the mutual coherence function and the cross-spectral density propagate
through an optical system in the same way as optical fields but they obey a pair of wave
equations instead of just one. Specifically, in free space, the mutual intensity propagates
according to the Helmholtz equations [55, p. 200]
%
! #
where
is the Laplacian operator ( ! %
! #
!"
is the Laplacian operator with respect to point (2.32)
(2.33)
!
%
, and
) with respect to point ,
!
.
Linear optical systems can be defined by their impulse-response functions ! # , also
! # is a point on the input plane and ! #
known as Green’s functions, where is on the output plane. The system is then linear in amplitude with
!
#
!
# ! # (2.34)
The mutual intensity in the output plane is [46, p. 368]
!
#
#
#
# ! ! ! 34
(2.35)
The intensity of a point in the output plane is simply the mutual intensity evaluated at
so it is
! #
! # !
#
!
# (2.36)
This tells us that the intensity at the output of an optical system depends on the mutual
intensity at the input.
In the case of incoherent light, the mutual intensity is zero except when ! # !
, or
# . The system is then linear in intensity, not amplitude, with
! #
The terms !
#
!
#
! # (2.37)
define the point-spread function [46, Ch. 10].
For partially coherent light we can use the expansion of Eq. (2.25) to determine the output of an optical system. Because the modes of the mutual intensity are orthogonal we can
analyze the output as an incoherent sum of the outputs for each mode. Each mode travels
through the system with the amplitude impulse response as in Eq. (2.35) and the resulting
total mutual intensity is a superposition of the mutual intensity from each individual mode.
To evaluate the output intensity, the output intensities for each weighted mode are added
together.
One complication in an optical system is an aperture. If a limiting aperture is part of
the system, we must find the optical field on the incident side of the aperture, multiply by
the aperture function, and then use the truncated field as the input to propagate through the
rest of the system [62]. For a general linear system with partially coherent light the output
mutual intensity at the output of a system that has an aperture at the input is
!
#
#
! # !
!
!
#
!
# !
#
# "!
#
! #
35
!
#
!
(2.38)
#
! #
where one dimension is used for simplicity, !
# is the mutual intensity incident on
the aperture, and is a general aperture function.
As an example, let us look at the situation where a partially coherent beam is diffracted
by a slit of width and then propagates a distance in free space. We will assume paraxial
conditions for the propagation and a distance that is in the Fraunhofer region. Under these
conditions, the impulse response for free space is, in one dimension,
! # "
!0 ! #
%
)+*-, ! 0 # ) *-,
0 )+*-,
Inserting Eq. (2.39) in Eq. (2.38) and suppressing phase gives
!
#
where the !
)+*-,
0 ! #
) *,
0 " 0 "
(2.39)
# !
(2.40)
# ’s are the wavefunctions incident on the aperture. The output intensity is
obviously the sum of the output intensities of each mode.
The aperture function is generally not a symmetric transform in the sense that we discussed in Section 2.4.2. In the simple laser cavity discussed earlier, the optical fields travel
both directions through any apertures, forcing symmetry. A propagating mode outside
of the cavity will usually travel through an aperture in one direction. Therefore, after an
aperture the propagated modes will generally lose their spatial orthogonality even while
retaining their power orthogonality.
Using Eq. (2.35) we can determine the output mutual intensity for any linear optical
system if the input mutual intensity and the impulse-response function are known. If the
impulse-response function is symmetric, or equal in both directions, we can find the mutual
intensity at the laser output mirror from the measured mutual intensity in some other plane
36
after the light has propagated away from the laser. Equivalently, we can use the mutual
intensity in any arbitrary plane to find the propagated modes at that plane. The propagated
modes can be used to find the laser modes at the exit face of the laser using Eq. (2.34), as
long as the impulse-response function is symmetric. Any standard method can be used to
determine the impulse response for systems more complicated than free-space propagation.
For general paraxial optical systems one method is to use the ray-transfer matrix, or ABCD
matrix [62]. This technique is often covered in optics textbooks, including Ch. 15 and 20
in Ref. [47].
2.4.5 Spatial coherence summary
The spatial coherence section contains many important concepts so we include a brief
summary here before going on. In the overview we defined many terms, with the mutual
intensity being the most important for our research. We saw that the mutual intensity is
the equal-time cross-correlation of the optical field at two different points. The mutual
intensity can be measured from the interference fringes that are the detected output of
an interferometer. In Eq. (2.16) the amplitude of the mutual intensity was defined to be
proportional to the peak-to-peak amplitude of the interference fringes.
Section 2.4.2 included our justification for claiming that the spatial modes of a nonfrequency degenerate laser are orthogonal both in power and spatially. Eq. (2.26) defines
mathematically that the mutual intensity can be decomposed into weighted spatial modes.
One condition to avoid is multiple longitudinal modes, as we saw in Section 2.4.3. The
shape and relative weights may still be the same under this condition but there may also be
complicating factors.
37
We explained in Section 2.4.4 how the mutual intensity propagates through any linear
optical system. In a paraxial linear system without apertures, which is symmetric in propagation, the mutual intensity will retain its spatial orthogonality and Eq. (2.35) can be used
to transform the mutual intensity from the output plane to the input plane or vice versa.
The result is that we know how to measure the mutual intensity with an interferometer
and we can collect data for a correlation matrix as mentioned in Section 1.3.2. We also
know that we can decompose the mutual intensity into weighted modes from Eq. (2.26).
These modes can be simply related back to the modes at the exit face of the laser if there are
no intervening apertures. The following section explains how the twin-fiber interferometer
acquires the data matrix. In Section 2.6 we expand on the method we used for the modal
decomposition.
2.5 Twin-fiber interferometer
To acquire data for spatial coherence, some type of optical interference is necessary.
The interferometer can be implemented in a variety of ways as mentioned in Section 1.3.2.
We have chosen to use the twin-fiber interferometer, shown in simplified form in Figure 2.3. The laser beam is sampled at points
and
by two fibers that are the in-
puts to a single-mode fiber coupler. The two fibers of the coupler are cleaved to the
same length [63] to keep the optical path lengths the same, maintaining temporal coherence. The combined output at the other end of the coupler emits onto a detector. The
relationship between the detected intensity and the mutual intensity is given by
* ! # , Eq. (2.15). We introduce the time dif
ference
either by translating one fiber arm along the optic axis or by changing the path
length of one fiber arm with a stretcher or some other phase-changing device. The phase
38
Translation or phase change of one fiber
x1
Laser
Beam
Single−mode fiber coupler
Detector
x2
Figure 2.3: Twin-fiber interferometer
in the fiber interferometer is very sensitive to any mechanical or temperature changes so
multiple cycles are measured to ensure accurate readings for the maximum and minimum
intensities.
We used the twin-fiber interferometer to take mutual intensity measurements at all possible combinations of
and
in the beam to acquire as many data points in the correlation
matrix as practical. This is shown graphically for one row of the data matrix in Figure 2.4.
The measurements in the laser beam, shown on the left hand side of the figure, result in a
matrix of data, shown on the right hand side of the figure. The different values of
beam correspond to rows in the matrix and different values of
One fiber is fixed at a point
possible range of positions for
is moved to a different
position of
in the
correspond to columns.
in the beam and the other fiber is then stepped through the
. To take measurements for another row, the first fiber
and the second fiber steps through the new range of
increases, the row index increases; and as the position of
. As the
increases, the
column index increases.
As the fibers cannot physically be superimposed or pass through each other, the data
points are in the upper triangle of the correlation matrix, where
39
, and
is
columns: x2
Step x2
Fix x1
x2
Γ1 2
rows:
x1
Fix x1
y
x
Laser Beam
Figure 2.4: How
Correlation Matrix
and
in the beam relate to the correlation matrix.
the diameter of the fiber. This triangle is located in Figure 2.4 where the
symbol is.
The symmetry of the matrix can be used to fill in the lower triangle of the matrix but
the diagonal and possibly one or more superdiagonals remain unknown. Three missing
diagonals are indicated in Figure 2.4 by dotted lines, corresponding to the main diagonal,
one superdiagonal and its transpose. The number of missing superdiagonals depends on the
diameter of the fiber in comparison to the step size between data points. This experimental
issue could be circumvented by slightly displacing the fibers in the axis (if the modes are
separable) but there may be a reduction in the overall mutual intensity amplitude and the
intensity levels may become too low from having the fibers out of the intensity peak in the direction. With enough redundancy in the data, however, the modes can be recovered even
without these diagonal elements. Because the twin-fiber interferometer does not measure a
complete matrix we have adapted a method from Jacobian control that can be used to find
the modes. This algorithm will be explained more completely in Section 2.6.1.
40
2.6 Numerical analysis
The next step after we measure the mutual intensity is to decompose the data to determine the spatial modes and modal weights. We recognize from Eq. (2.26) that the mutual
intensity is the sum of the outer products of the modes. Rewriting Eq. (2.26) in matrix form
results in
where the
refers to the transpose and boldface signifies a matrix. The
(2.41)
correlation
matrix is symbolized by , which is necessarily real and symmetric. The column vectors of
are the eigenvectors of , and the diagonal values of
%
are the eigenvalues. The eigen-
vectors represent the field modes of the beam and the eigenvalues are the modal power
weights . The modal field weights are found from the square roots of the eigenvalues,
assuming there is only one longitudinal mode.
It is key to note that, with enough data, the forms and weights of the modes can be
found without any assumptions about the modes. The data must extend over the full width
of the beam to keep the spatial orthogonality of the measured modes. The step size should
be small enough to give a reasonable representation of the extracted modes. Any missing
data, such as the missing diagonals discussed above, must be a small enough percentage of
the total data that the redundancy of the matrix is enough to recover the modes.
Our numerical analysis has to overcome the problem of incomplete data because of the
physical limitations of the twin-fiber interferometer. With a complete data matrix, there
are a variety of methods, such as singular value decomposition (SVD), to decompose the
matrix into its eigenvectors and eigenvalues. These methods will not work, however, on an
incomplete matrix.
41
%
Noise is another experimental issue that causes difficulties in finding the modes and
weights from the data matrix. Under the category of noise there are laser output fluctuations, mode partition noise, detector noise, background light variations, and position errors
in the fiber translations. Obviously, noisy data may cause faulty analysis. Noise will distort
the representation of the physical modes of the laser and noise may also produce spurious
higher-order modes that do not correspond to physical modes. We report in Section 4.2
on some computer simulations that we performed to assess the effect of noise on mode
extraction.
In Section 2.6.1 we describe a type of numerical analysis that allows us to extract spatial
modes and weights in the presence of noise, even where there are some missing diagonals
in the data matrix. Some of the details of the analysis are explained more fully in Sections 2.6.2 and 2.6.3. In Section 2.6.4 we discuss some practical issues of implementing
the Jacobian algorithm, including convergence and determining the number of modes in
the presence of noise.
2.6.1 Jacobian solution
Our method used to recover the modes, even with an incomplete correlation matrix, is
known as Newton’s method for systems of equations in numerical analysis [64] or Jacobian control in the field of robotic control [65]. It is an iterative algorithm that starts with
an initial guess for the modes and weights and then updates these to fit two independent
conditions: namely, orthogonality of the modes and a least-squares fit to the measured correlation data. The initial guess for the modes is the column eigenvectors found from a SVD
42
of the incomplete matrix, with zeros filling in the missing diagonal elements. This initialization assures that the results are based on experimental data and not theoretical models,
such as the Hermite–Gaussian model.
The Jacobian solution develops from a Taylor-series approximation. The first-order
Taylor-series approximation of any general vector function with a vector argument
changes by
where
is
!
#
! #
that
(2.42)
is the Jacobian matrix whose elements are defined by
(2.43)
Now suppose we want to find a particular which causes a new desired value of
!
# . The change is found when we solve
#
#
(2.44)
! !
where the difference between the desired and current values of f is called the residual vector
. Because the Jacobian matrix is not generally square, we must use the pseudo-inverse
to solve for
[66]. The resulting set of all possible solutions is
where
is the identity matrix and
!
#
is an arbitrary vector in
(2.45)
space. The second term
represents homogeneous solutions to Eq. (2.44) when the equation is underdetermined.
Our unknowns, the modes and weights, can be represented as the single vector . To do
% this, we first define the weighted spatial modes , where
, to be the same as
the eigenvectors of Eq. (2.41) multiplied by the square root of the appropriate eigenvalues.
We then form a single vector
of length
by concatenating the
43
weighted spatial modes.
To find the set of modes, there are two sets of conditions and therefore two Jacobian
matrices and two residual vectors. To represent the two conditions we define as the
data-matching Jacobian matrix and as the orthogonality condition Jacobian matrix. The
corresponding residual vectors are and . From Eq. (2.44) the two sets of conditions
can be written as
(2.46)
(2.47)
The orthogonality condition is primary and the data-matching condition is secondary.
The calculated modes are required to be orthogonal and within that constraint, a leastsquares fit to the measured data is found. Applying the orthogonality condition in Eq. (2.45)
we obtain
Next we substitute
solve for a
!
# (2.48)
into the Jacobian equation for data matching, Eq. (2.46), and we
that finds a least-squares fit to the data while preserving the orthogonality of
the modes. When we substitute this into Eq. (2.45) and follow the simplifying procedure
of Ref. [65], the result is
The initial guess for
## 1 '! 3
(2.49)
results from the SVD of the measured matrix with zeros for the
missing diagonal elements. After computing Eq. (2.49), we update
process is repeated until either the norm of
criterion.
44
by adding
. This
or the norm of is less than a specified
2.6.2 Orthogonality condition
To see how we calculate the orthogonality Jacobian matrix, first note that orthogonality
requires that the inner product of two different spatial modes be zero. Mathematically this
can be described as a function ! # where
where 0
%
!
#! &
%
for 0
(2.50)
matter, there are !
! #
%
and is the number of modes. Because the order of 0 and does not
%
# ! unique combinations where 0 . The size of is therefore
and is a column vector of length !
#! .
The Jacobian matrix and the residual vector are found by forcing the orthogonality
condition on the updated modes:
0
where %
# # ! # ! ! (2.51)
. The last term of the final line is dropped in a first-order expansion. That
leaves the middle two terms equal to
, based on Eq. (2.44). Ideally the residual
vector is zero but there may be some error from neglecting the second order terms in
previous iterations. To counteract this drift, the residual vector is defined as the negative
of the known error amount
for all allowed combinations of 0
%
(2.52)
.
, and , the orthogonality Jacobian
As an example, if there were three modes, ,
matrix and residual vector would look like:
45
(2.53)
The Jacobian matrix elements are found explicitly by stepping through the rows, in-
dexed by , of 0
and
%
!
0
!0
#
! !% #
and defining for
#
! #
! # #
(2.54)
(2.55)
2.6.3 Data-matching condition
The Jacobian matrix for data matching can be generated in a similar way to the orthog-
onality condition Jacobian matrix. The function ! # used for matching in this case is the
correlation matrix , defined in Eq. (2.41). The ideal
correlation matrix is equal to
%
the sum of the outer products of the weighted spatial modes
, where
mode number. Two equivalent ways of writing
with the weighted vectors
are
or
! # ! 0#
(2.56)
(2.57)
where and 0 are the row and column indices of
and are also used for row indices of
The measured correlation data make up a data matrix . If
from , then there are exactly . !
#
! 0 #
is the
!
#32 ! .
diagonals are missing
unique measured elements in .
The factor of two reflects the symmetry of the matrix. These elements can be indexed
0
by , which is found by stepping through all combinations of
!
# .
! # and
The Jacobian data-matching matrix results from equating the residual vector to the
difference
!
# ! # as in Eq. (2.44). Performing a first-order expansion on
46
Eq. (2.57), similar to the orthogonality expansion in Eq. (2.51), results in the !
of being equal to
! #
0# element
! # ! 0# ! 0# ! # (2.58)
From this result, the Jacobian matrix is generated in the following way:
We start by defining as a . !
# !
#32
!
matrix of zeros. Then for each
row, where the rows are indexed by as described above, we define the non-zero elements
to be
for
%
%
! !
%
'! !
#
#
#
0#
! 0#
! #
(2.59)
(2.60)
.
The residual error for data matching is equal to the difference between the measured
data and the correlation matrix of the current guess for the spatial modes
residual vector is arranged as a column vector with a length of . !
#
! # . The
! # 2! ,
corresponding to the number of unique data points in . Again using as an index, the
elements of the residual vector '! # are equal to
! 0#
! 0 # , where determines
the choice of and 0 .
2.6.4 Implementing the Jacobian algorithm
To use the Jacobian algorithm one has to define the number of modes
that are de-
sired. In a noise-free system the Jacobian will extract the largest weighted modes first
and return modes of zero amplitude if too many modes are requested. The more modes
sought, the slower the process is because of the dependence on
for matrix sizes. In the
presence of noise, sometimes the algorithm will not converge, particularly if more modes
47
have been requested than are physically present. Convergence may also be affected by the
initial starting guess for the modes. When the algorithm does not converge, the resulting
eigenfunctions often have positive and negative spikes that get larger with each iteration.
One variation on the Jacobian algorithm that we used was an adaptive algorithm. It
would request one mode first and then increase the number of modes after the result had
settled down to small successive changes. For a time we tried fixing the modes as they
were found, improving speed, but we eventually discovered that this could give erroneous
results if the higher-order modes had significant weights. The adaptive algorithm, without
fixing the modes, ended up taking longer than one pass of the Jacobian algorithm but it
would sometimes converge better with noisy data.
The Jacobian algorithm returns weighted vectors for the spatial modes. To determine
the relative field weights, we took the norm of each vector and then compared them individually to the sum of each norm. The relative power weights were determined by squaring
the norm of each vector and then comparing that to the sum of the squared norms. The
norm of the vector is defined by
0 )
!
#
! ! # #
(2.61)
2.7 Chapter summary
In this chapter we have discussed the topics necessary to understand our research. In
Section 2.2 we defined longitudinal and spatial modes in lasers. A specific model for spatial
modes is the Hermite–Gaussian model, which we explained in Section 2.3. We discussed
the usefulness and the limitations of this model, especially as the limitations relate to thinjunction edge emitting semiconductor lasers. In Section 2.4.1 we defined spatial coherence
concepts such as mutual intensity and explained how to determine mutual intensity from
48
the detected intensity of an interferometer. In Section 2.4.2 we showed how measurements
of the mutual intensity allow us to extract the spatial modes and weights of a laser, assuming there are no frequency degeneracies. We discussed the decrease in coherence due to
longitudinal modes in Section 2.4.3. The mutual intensity in the measurement plane may
have a different form than at the laser so in Section 2.4.4 we defined how spatial coherence propagates. Section 2.5 explained how the twin-fiber interferometer acquires data for
the correlation matrix. In Section 2.6 we explained the numerical analysis technique that
we have used to analyze a correlation matrix with missing elements and extract the spatial
modes and weights. Now we are ready to see how the experiments are conducted.
49
CHAPTER 3
EXPERIMENTAL IMPLEMENTATION
3.1 Introduction
In this chapter we explain the experimental implementation of our research. We describe the physical layout and equipment in Section 3.2 and the procedures in Section 3.3.
Although we used software for a variety of purposes, including controlling equipment
and analyzing data, a detailed explanation of the software is not pertinent to understanding
this research or its results. Specific implementations for control could be accomplished in
a variety of ways and is obviously specific to the equipment used.
In our lab we use LabVIEW by National Instruments [67] to interface with the automated equipment. This program is extremely simple to use due to its graphical programming. Virtual instruments, the name for the software files that control equipment, are
readily available for most computer-controlled instruments.
Using the basic algorithms for analysis that were defined in Chapter 2, the Jacobian
method could be written in many different software languages. We used MATLAB for
our analysis due to familiarity with it, its availability, its ease of use, and the matrix tools
available. The main MATLAB m-files we used are included in Appendix A to show the
specifics of how the Jacobian algorithm is implemented.
50
3.2 Experimental layout and equipment
In this section we describe the equipment and configuration that we have used in my
research. Table 3.1, near the end of the section, lists the manufacturers and model numbers
for the different pieces of equipment. We do not include standard optical lab equipment
such as fixtures, stages, and mounts.
A simple diagram of our equipment layout is shown in Figure 3.1. In this figure the
laser beam travels from left to right. The single-mode fiber coupler, represented by thick
lines, has two input arms which are pictured sampling the beam in the middle of the figure.
One arm of the coupler is wound around a polarization controller, labeled PC in the figure,
and the other fiber arm is wound around a piezoelectric stretcher, labeled PZS in the figure.
The output of the fiber coupler radiates onto a detector that is connected to an optical power
meter, shown in the top right of the figure. The boxes in the center of the figure, labeled
with
or
, indicate the stages on which the fiber ends are mounted. These are
controlled by motion controllers connected to a computer through the General Purpose
Interface Bus (GPIB), also known by its standard, IEEE-488. The same computer also
interfaces with the optical power meter, collecting data.
A laser mount, used for a semiconductor laser, is shown in Figure 3.1 positioned on
the left hand side of the figure. It is connected to a current source and a temperature
controller and includes a collimating lens. Sometimes the laser beam is also monitored
by another power meter, which is not shown in Figure 3.1. This monitor is to verify that
the overall power output from the laser is not changing over time. A pellicle, which is a
very thin membrane, is placed in the beam and deflects a small portion of the power to
another detector and power meter. The pellicle is positioned in such a way that it deflects
the beam up, away from the table, when it is used with the semiconductor lasers. This
51
Temperature
Controller
Current Source
Optical
Power Meter
GPIB
GPIB
Detector
x1 , y1
PC
Laser
Beam
x2 , y2
Single−mode
Fiber Coupler
PZS
Laser Mount
z
GPIB
x
Power
Supply
Function
Generator
Figure 3.1: Experimental layout
unusual position was chosen so that the pellicle intersects the beam at a constant distance
on the optical axis as
varies. A small amount of interference is introduced between the
front and back surfaces of the pellicle if it is slanted in the – plane and measurements are
being taken in the
axis.
All of the equipment shown in Figure 3.1 is mounted on an optical table with vibration
isolation supports. The table also has an enclosure to reduce background light.
In some experiments we used a Helium Neon laser with a 180 cm long cavity and output
power of approximately 25 mW. This laser was located on an adjacent table to reduce heat
buildup in the enclosure. In the HeNe laser case, the three pieces of equipment shown on
the left hand side of the figure were replaced by the HeNe laser and its exciter, containing
52
the power supply and radio frequency exciter. The beam from the HeNe laser was reflected
from one table to the other with some mirrors to change direction and height.
The single-mode fiber coupler is the defining element of the twin-fiber interferometer.
The bare fiber ends of the input arms are placed in exact positions in the beam by –
– stages, where the movements in
and are controlled by computer. The coupler
sums the fields sampled by the two input arms and then emits the sum onto the detector.
The detector is inherently a square-law device with a time response much slower than the
optical frequency so it squares the fields and averages them over time.
The detector that we used for our spatial coherence measurements was several orders
of magnitude slower than the GHz detectors needed to see beat frequencies. Since the
lateral modes of semiconductor lasers have typically been measured to have wavelength
separations of at least a few hundredths of nanometers [15, 17], the slow speed detector
averaged over many beat frequency cycles detecting the different modes as orthogonal in
frequency.
The coupler sums the fields in the two fiber arms by evanescent coupling [68]. Two
fibers, with most of their cladding removed, are placed together so their fields couple from
their cores. The coupling ratio depends on the interaction length as a fraction of wavelength
and is therefore wavelength dependent. The couplers that we have used have the coupling
region encased in a ferrule and have a fixed interaction length. Three different couplers
were used that were optimized for different wavelengths. The couplers can also be used at
other wavelengths but the splitting ratios will not be 50/50.
The input ends of the fiber coupler are mounted as shown in Figure 3.2 for taking measurements along the
axis. The fibers are taped with removable tape into grooves in metal
plates that are attached to mirror mounts. The mirror mounts are attached perpendicularly
53
y
x−y−z
x
Stages
Fiber ends
Metal plates
Figure 3.2: Input ends of the fibers
to short posts that extend horizontally from the – – stages on the inward facing sides.
To ensure that translation in any direction is aligned with the axes, it is very important that
the mounting plates are parallel to each other and perfectly perpendicular to the table top
and that the fibers are parallel to the table top. We used a small level to facilitate these
adjustments.
The polarization controller is a device that bends and twists a fiber in a controlled
way to induce changes in polarization of the optical field in the fiber [69]. Single-mode
fiber couples light between different polarization states in a random way that is affected by
twists or bends in the fiber [68]. By using the polarization controller, we can match the
polarization of the two arms at the coupler, thus increasing the interference. One fiber arm
of the coupler is wound around the three metal disks of the polarization controller and then
the disks are configured to give maximum visibility of the interference fringes. Another
approach would be to use a fiber coupler made of polarization maintaining fiber but these
are significantly more expensive and more difficult to use.
54
The piezoelectric stretcher has two piezoelectric wafers that expand and contract as the
voltage is varied across them. The two wafers are on the long sides of an oval and the
rounded edges of the oval are fixed. The fiber wraps around the oval and is attached to
the wafers either by epoxy or, in our case, by wax. The stretcher is controlled by a circuit
that is connected to a power supply and a function generator. As a sawtooth voltage is
applied across the wafers, they expand linearly and then contract suddenly. This movement
causes the fiber to stretch and contract, changing the optical path length through the fiber
and introducing the time difference defined in Section 2.4.
The polarization controller and the piezoelectric stretcher are two improvements on the
twin-fiber interferometer that we did not have in the early stages of our research. We had
no way of reducing losses due to polarization changes other than to keep the fibers as fixed
as possible. Instead of using the stretcher to change we translated one of the fibers along
the
axis, using a computer-controlled actuator, to sample the field at different points in
time. Because it only takes a translation of one wavelength, less than 1 m, to pass through
a complete interference cycle, the positioning requirements were very stringent. We were
only able to get seven or eight samples in a cycle and so went through more than ten cycles
to try and verify the maximum and minimum intensities. This source of error has been
greatly reduced by using the stretcher as we can vary the frequency of the stretching and
take as many data points as necessary per cycle.
Another significant improvement over our earlier setup is a new power meter, Newport
2832-C, that is much faster than the older Newport 835. The older meter was limited to
seven measurements per second with the net result that it took two to four hours to take
all the measurements needed for the coherence matrix. The new meter can operate as fast
55
Optical power a.u.
1
data
0.8
0.6
0.4
0.2
fit
0
0
2
4
6
8
10
12
14
16
z position in microns
Optical power a.u.
1
0.8
0.6
0.4
0.2
0
0
10
20
30
40
50
60
70
80
90
100
sample number
Figure 3.3: Comparison of fringe data with translation along
stretching of fiber (bottom).
(top) and piezoelectric
as 1000 Hz so other factors, such as positioning and GPIB handshaking, limit the speed of
data acquisition. We can acquire a larger matrix of data in about half an hour.
Figure 3.3 shows a comparison of an interference fringe acquired by translating one
fiber along the optical axis (top) versus stretching the fiber with the piezoelectric stretcher
(bottom). In both plots the data points are indicated by circles. The top plot also has a
sinusoidal fit drawn in with a solid line, emphasizing the change in phase of the measured
fringe over position. The measurements in the two plots were taken for different lasers and
do not represent the same mutual intensity, which is apparent from the different modulation
depths on the interference fringes. The data in the top plot took approximately 14 seconds
to acquire with the old power meter while the data in the bottom plot was taken in less
56
than one second with the new power meter. The improvement in accuracy in measuring the
maximum and minimum intensities is obvious.
In both cases we can see that phase retrieval would add experimental complexity. In
the top plot the phase varies over time due to the nature of the fiber’s sensitivity to small
external variations such as temperature or motion. That effect is not apparent in the bottom
plot because it covers a shorter amount of time. The discontinuity of the bottom plot is
due to the sawtooth voltage. The voltage also flattened out at the end of each cycle causing
the flattening seen between samples 45 and 65 in the bottom plot. We used a frequency of
about 1 Hz for the sawtooth voltage.
The primary lasers that we tested were prototype lasers provided to us by Wright Patterson Air Force Base under a Cooperative Research and Development Agreement. The
first batch of lasers was fabricated of GaAs and AlGaAs, of varying concentrations and
dopings, with an InGaAs quantum well. The second batch is similar but has three quantum wells of InGaAs and the well thicknesses are 8 nm instead of 5 nm for the first lasers.
Both batches of lasers have Fabry–Perot cavities and ridge waveguides. We were given
several lasers with different stripe widths, with the expectation that the narrowest stripe
width lasers would be single-mode and that the number of modes would increase as the
stripe width increased.
Figure 3.4 shows the basic design of a semiconductor laser with a ridge waveguide. The
drawing is not to scale. The width of the p-contact in the center varied from 3 to 30 m.
The total height of the layers above the substrate was less than 3 m. Layers with different
doping are shown separated by thin horizontal lines. The contacts are shown cross-hatched
with different patterns for the n-contact and the p-contact. The active region, where the
quantum wells are located, is labeled with an arrow and depicted by a straight line across
57
plated pads
Si N layer
3 4 p cap
p−contact
p−type cladding
confinement layer
confinement layer
n−type cladding
Optical Beam
Active Region n cap
n−contact
n−contact
substrate
Figure 3.4: Ridge-waveguide design of lasers provided by Wright Patterson Air Force Base,
after [70].
the width of the figure. Above and below the active region are confinement layers, with
cladding layers beyond the confinement layers. These layers act as a waveguide in the
transverse direction. In the lateral direction, the waveguide is determined by the width of
the ridge under the p-contact stripe.
The first batch of lasers fit the description of thin-junction, edge emitting semiconductor
lasers that was given in Chapter 1. The confinement layers above and below the quantum
well were each 90 nm thick. The second batch of lasers had confinement layers that were
185 nm thick and this was enough to allow a second mode in the transverse direction,
perpendicular to the junction.
The first set of lasers were all on one chip that was mounted in an open butterfly package. The leads of the package were bonded to the laser stripe contacts. We mounted the
package on a small shelf attached to the mounting plate that was in contact with the thermoelectric cooling modules. The lasers were highly diverging in the transverse direction
and were apertured by either the hole in the mounting plate or by the collimating lens. The
58
Equipment
Actuators (computer-controlled)
Current source
Detector
Fiber coupler (630 nm)
Fiber coupler (850 nm)
Fiber coupler (980 nm)
Function generator
HeNe laser
HeNe exciter
IR lasers
Laser mount
Micrometers (manual)
Motion controller
Optical power meter
Piezoelectric stretcher
Polarization controller
Power supply
Temperature controller
Manufacturer and model #
Newport 850A and 850B
ILX Lightwave LDX-3207B
Newport 818-IR or 818-SL
Gould 22-30863-50-21201
Gould 22-30685-50-21201
Fiberdyne Labs FSC32025102
Tektronix F6501
Spectra-Physics 125
Spectra-Physics 250
Wright Patterson Air Force Base
ILX Lightwave LDM-4412
Newport SM-05 and SM-D13
Newport PMC 200-P
Newport 835 (old), 2832-C (new)
Canadian Inst. & Res. 915
BT&D MPC 1000
Hewlett Packard 6237B
ILX Lightwave LDT-5412
Table 3.1: Manufacturers and model numbers for the equipment used
5 m laser suffered facet damage after many months of testing and this seemed to affect
the other lasers on the chip too. Although we tested one or two other lasers on the chip we
did not get usable data because they were not very stable, probably due to the damage to
the chip.
The second batch of lasers were mounted separately in open-top TO cans. These packages fit into a hole in the mounting plate so any aperture effects were from the collimating
lens.
59
3.3 Procedures
The main goal of our research was to measure the spatial coherence matrix of a laser
and extract the spatial modes and weights from that data. Most of our tests were on semiconductor lasers that had various stripe widths. For one row, we fixed one fiber at a location
in the beam and then stepped the other fiber through a range of positions
in the beam,
as explained in Section 2.5. At each step we measured the interference between the two
sampled fields by acquiring the maximum intensity and the minimum intensity as one fiber
was stretched by the piezoelectric stretcher. Later, we processed the data to find the modes
and weights.
On the first laser we tested, one with a 5 m stripe from the first batch of lasers, we
measured the spatial coherence both in the lateral and transverse planes, parallel and perpendicular to the heterostructure interface plane. As expected, there was only one mode in
the transverse direction. For the 3 m stripe laser from the second batch we also measured
the spatial coherence in both the lateral and transverse planes. There was a small amount of
power in a second mode for this laser. We did not test all the lasers for multiple transverse
modes.
It is worth noting that the lasers were diffracted in the transverse plane due to aperturing
effects in the laser mount. The quantum-well junction produced a large divergence in the
transverse direction, so the diameter of the collimating lens was too small to pass the beam
without truncating it. To minimize the diffraction, we tried to adjust the collimating lens so
the measurement plane was in the far-field region. Then the transverse profile approximated
a sinc function, without the intensity ripples that could be observed for other lens positions.
The distinguishing characteristic between the various semiconductor lasers tested is the
difference in stripe widths. To assess the effect of stripe widths on the number of spatial
60
modes, we measured the spatial coherence matrix in the lateral plane for lasers with stripe
widths of 3, 4, 5, 10, 15, 20, and 30 m. There were two lasers with 10 m stripe widths
so we labeled them LD
and LD
. The other lasers are labeled LDxx, where ’xx’ is
replaced with the width in m. For most of these lasers we measured the spatial coherence
matrix at different current levels to see if higher current increased the number of modes,
varied their relative weights, or changed the shapes of the modes.
As a baseline measurement, we extracted the spatial modes of a HeNe laser, which
should theoretically be a good fit to the Hermite–Gaussian model. We also tried to vary the
modal content of the HeNe beam by inserting a slit in the beam. Unfortunately, the power
stability of the laser was not adequate for tests comparing results that were more than a few
hours apart. The tube had a center cathode and anodes at both ends for plasma excitation.
Sometimes the front half and sometimes the back half would be excited, and very rarely
the whole tube was excited. The laser power was generally stable over a period of a few
hours.
One procedural question is that of verification. Because we claim to be able to extract
spatial mode shapes, which most other modal weight methods do not, it is difficult to find
a valid way of testing our claim. The other techniques described in Chapter 1 either rely
heavily on the Hermite–Gaussian model, require equipment we do not have (NSOM), or
both. The intensity measurement methods are feasible to implement but would not provide an adequate check since they are based completely on the Hermite–Gaussian model.
Another possibility is using a high-resolution spectrometer to see if frequency peaks could
be distinguished for the different spatial modes. The monochromators available in the
department do not have the resolution necessary to differentiate a few low-order modes.
61
An additional drawback to spectral measurements is that they do not give any information
about the form of the spatial modes.
Using the SPEX 340S monochromator from the Photonics Teaching Lab we did measure the spectra for some of the semiconductor lasers. This monochromator is a CzernyTurner spectrometer that has a 220 mm focal-length input mirror and a 340 mm focal-length
output mirror, giving a resolution of 0.15 nm. For some of the lasers we were able to resolve the longitudinal modes but not the spatial modes. It did indicate, however, that there
were multiple longitudinal modes operating under some conditions.
We performed a simple frequency-mixing test [57] to see if there were indications that
the semiconductor lasers had multiple spatial modes. We focused the beam onto a 6 GHz
photodetector, a New Focus 1537, that was connected to a high speed digital oscilloscope, a
HP Infinium 54845A. The scope sampled the output of the detector at a rate of 8 Gigasamples/sec and performed a fast Fourier transform (FFT) on the samples to give a spectrum
of the beam’s beat frequencies. Obviously, the frequency range was too low to see the
spectrum of the laser modes, but some of the difference frequencies between the modes
were observable. There were also aliasing effects indicating that some of the difference
frequencies were above the range of the equipment. In Chapter 4 we include a discussion
of some of the results.
Although the frequency-mixing test indicated that the lasers did have multiple modes,
the results were not useful for determining weights or shapes of the modes. Also, the expected beat frequencies for multiple spatial modes that was given in Section 2.4.2 is right at
the limit of detection for the photodetector and oscilloscope combination. In addition to the
obvious aliasing effects, it is possible that there were spatial modes beating at frequencies
that were averaged out by the detector.
62
3.4 Chapter summary
In this chapter we have described the experimental implementation of our research
method to extract modes of a laser from spatial coherence measurements. The configuration of equipment was described in Section 3.2. In Section 3.3 we explained how we tested
a group of semiconductor lasers with different stripe widths to see how the modal composition varied with stripe width. We also implemented a frequency-mixing configuration and
measured the optical spectra on some of the semiconductor lasers. In the next chapter we
present our experimental results.
63
CHAPTER 4
RESULTS
4.1 Introduction
In this chapter we describe the results of our work. In Section 4.2 we examine the
computer simulations we performed. From the results of the simulations we can be confident that the analysis can extract low-level modes, even in the presence of moderate noise.
In the next two sections, Section 4.3 and Section 4.4, we present data relating to longitudinal modes, which may complicate the extraction of spatial modes. In Section 4.3, we
present some results on total coherence for the 3 m stripe laser for a range of currents
and in Section 4.4 we display the optical spectra for several of the semiconductor lasers.
In Section 4.5, on frequency mixing, we give results demonstrating the detection of spatial
modes through beat frequencies. The heart of the chapter consists of the data presented in
Sections 4.6 and 4.7, in which we discuss the experimental outcomes for various semiconductor lasers with different stripe widths and varying current levels. For completeness, we
also include a section giving some results on the HeNe laser that we tested.
64
4.2 Computer simulations
We performed a number of computer simulations to test the effectiveness of the numerical analysis. The simulations show how the mode extraction process is affected by different
noise levels. We also varied the number of missing diagonals in the simulated data to see
how the extracted modes were affected.
In these simulations, we generated one-dimensional spatial modes as an example of an
almost single-mode Gaussian beam. We used two Hermite–Gaussian modes with relative
power weights of 95% in the fundamental mode and 5% in the first-order mode. These
weights correspond to 81.3% and 18.7% in terms of the electric field. The modes were
normalized so the total energy was 1. The spatial width of the simulated modes was
,
where
is the diameter of the fundamental mode ! field spot size. The width was
=1, and the range of positions was .
correlation matrix, the size correFrom the simulated modes we generated a
normalized to
sponding to the smallest data matrix used in experimental measurements. The noise-free
correlation matrix was made up of the sum of the outer products of the exact weighted
modes. Using Eq. (2.15), with minimum intensities,
=
=1, we calculated the matrices of maximum and
and , at each point.
First, the number of missing diagonals was varied to see how well the modes could be
retrieved as the available data points were reduced. For an error-free correlation matrix ten
superdiagonals, or a total of 21 diagonals out of 57, could be missing, and the two modes
were still extracted without error. Even when three modes were requested, two modes
were found with the correct weights and the extra mode converged to 0. The incomplete
correlation matrix is shown on the left side of the bottom of Figure 4.1. At the top of the
65
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
2
0
0
−2
−2
0
2
−0.2
−2
−1
0
1
2
Field amplitude
Mutual intensity
0.8
0.6
0.4
0.2
0
2
0.6
0.4
0.2
0
0
−2
−2
0
2
−0.2
−2
−1
0
1
Position normalized to ω0
2
Figure 4.1: (Top) The complete noise-free correlation matrix on the left and the two
Hermite–Gaussian modes on the right. (Bottom) The correlation matrix with 21 diagonals missing on the left and the extracted modes (small circles) on the right.
figure the complete correlation matrix is shown on the left and the constituent Hermite–
Gaussian modes on the right. On the right side of the bottom of the figure are the extracted
modes, marked with small circles, and the original Hermite–Gaussian modes, marked with
dashed lines. In this simulated case, the exact amplitudes of the extracted modes was found.
66
We can see from these results that there is a significant amount of redundancy in the
noise-free correlation matrix. In practice, however, we must take into account the effects of
noise. Due to the physical limitations of the fiber diameter and typical step sizes, the number of missing superdiagonals in a physical experiment is usually two. For the simulated
noise tests that follow, the number of missing superdiagonals was kept at two, for a total of
five missing diagonals.
To test the effects of noise in the data we generated two different types of noise to be
added to the correlation matrix. The first was background noise, with a constant variance
over the whole beam, and the second was intensity-dependent noise, proportional to the
detected intensity at each point. The noisy correlation matrix was then ! # ! ,
where the tilde signifies that noise has been added.
To model the background noise we created a matrix of noise with a uniform distribution
from -0.5 to 0.5 and then multiplied it by a percentage of the peak of the
Different realizations of this noise were added to
background noise: 0%,
%, and
%.
and
matrix.
. We used three levels of
For the intensity-dependent noise, random noise matrices uniformly distributed over -
0.5 to 0.5 were multiplied element by element by a percentage of or . The resulting
noise was added to
noise: 0%,
%,
or , as appropriate. We used 21 levels of intensity-dependent
%, %,
%.
The extracted modes for six of the 63 combinations of noise are illustrated in Figure 4.2. The horizontal axes represent the position in the beam, normalized to spot size.
The vertical axes show electric field amplitude, normalized so that the total energy in the
noise-free beam is 1. The subplots each represent a different set of conditions. Each subplot superimposes the results from 50 different noise realizations of the same conditions.
67
A comparison of the first column of subplots with the second column shows an increase in
background noise. The rows of subplots have increasing intensity-dependent noise as can
be seen from top to bottom. In the first row of plots there is no intensity-dependent noise
and the background noise is
row has
% in Figure 4.2(a) and
% in Figure 4.2(b). The second
% intensity dependent noise with no background noise in Figure 4.2(c) and
% background noise in Figure 4.2(d). In the third row, the intensity-dependent noise is
% background noise
increased to
%, with no background noise in Figure 4.2(e) and
in Figure 4.2(f).
We see in Figure 4.2 that the error in the shape of the extracted modes is most sensitive
to background noise. This is reasonable since the areas of low intensity in the beam will
have a low signal-to-noise ratio, as opposed to the intensity-dependent noise where the
signal-to-noise ratio stays constant. An increase in intensity-dependent noise also increases
the error in the mode forms, but not as significantly. Note also that for all cases the weaker
mode is more corrupted by noise than the fundamental mode.
The mode weights were found correctly to within less than a percentage point for the
worst case shown in Figure 4.2. The relative power weight of the first-order mode in
Figure 4.2(f) is within the range 0.049–0.055, where 0.05 is the noise-free weight. (On
this scale, the weight of the fundamental mode is 1 minus the weight of the first-order
mode.) For all of the possible noise levels tested, even with
%, the relative weight of
the weaker mode stayed within the range of 0.048–0.065. The weight of the weaker mode
tended to increase as the noise levels increased since most of the noise power was included
in the extraction of the weaker mode.
In the experimental results in Sections 4.6 and 4.7, the overall noise was less than
%. Background noise was less than
%. Short-term intensity-dependent fluctuations
68
1
1
(a)
(b)
0
−2
0
−1
0
1
2
−2
1
−1
0
1
2
−1
0
1
2
−1
0
1
2
1
(c)
(d)
0
−2
0
−1
0
1
2
−2
1
1
(e)
(f)
0
−2
0
−1
0
1
2
−2
Figure 4.2: Noise simulations with 50 realizations superimposed. Background noise is 0%
% in (a), and % in (b), (d), and (f). Intensity dependent noise is 0%
in (c) and (e),
% in (c) and (d), and % in (e) and (f).
in (a) and (b),
69
during five to ten minutes were not more than
% at any point in the beam. Taking into
account laser power drift and possible positional errors, the noise was still less than
%.
This most closely corresponds to the case of Figure 4.2(e).
4.3 Overall coherence measurements
In the process of taking spatial coherence measurements, it became apparent that the
absolute magnitude of the measured intensity could not be reliably reconstructed from the
spatial coherence information. Theoretically, if we know the transmission coefficients and of the two arms of the fiber coupler we should be able to take the measured value
of and determine the value of using Eq. (2.16). Then, having extracted the
spatial modes based on Eq. (2.26), we should be able to reconstruct the intensity values for
either or , taking the transmission coefficients into account.
One possible cause of the discrepancy was that the assumption of the slowly varying
character of the temporal coherence was at fault. To test this possibility we moved the fibers
along the optic axis to see if that introduced a change in the coherence measurements based
on the temporal difference between the two fiber inputs. The range of possible locations
was limited by the travel of the micrometers, but we took measurements with the fibers at
their greatest separation along the optic axis and then again with the positions reversed. For
these two extremes there was no difference in the overall coherence for a variety of current
levels.
Another possible explanation was that the assumption of separability for the spatial
modes in the two directions was incorrect. To check this possibility we took three sets of
measurements: two with the fibers at two different fixed displacements in
and the other
with a fixed displacement in . For each configuration we varied the current to laser diode
70
120
y=± 0.2 mm
Imax−Imin in nW
100
80
x=± 0.2 mm
60
40
x=± 0.3 mm
20
0
20
Dip
25
30
35
40
Current in mA
45
Figure 4.3: The mutual intensity at fixed points, measured by vs. current.
LD03 and measured the mutual intensity by recording in Figure 4.3. The plot labeled 50
Dip
55
, for LD03 plotted
. These results are shown
mm is from data taken a week later than the data
for the other two plots.
The obvious feature of the three lines plotted in Figure 4.3 is the large drops in the
mutual intensity at certain current levels. Although the exact location of these dips is not
the same for the top plot and the other two, the general characteristics remain the same.
These similarities seemed to indicate that it was not an issue of spatial coherence changing
rapidly or being inseparable.
Two weeks later we took a more complete set of data with the fibers displaced vertically
so that the positions were
mm. Simultaneously we monitored the power in the
whole beam using a pellicle to direct a portion of the beam to a separate power monitor.
71
70
Imax−Imin in nW
60
50
40
30
Dip
20
10
0
0
10
Up
Down
20
30
40
Current in mA
Dip
50
60
Figure 4.4: The mutual intensity at two fixed points, measured by , for LD03
plotted vs. current. The plot for increasing current is marked with small circles and labeled
’Up’ and for decreasing current labeled ’Down’.
We took one set of measurements with the current increasing and another set with the
current decreasing. The mutual intensity data is plotted in Figure 4.4
In Figure 4.4, the difference between
and
is plotted for increasing and de-
creasing current levels. The increasing current plot is marked with small circles and the
decreasing current plot is a solid line. Again we see that the overall coherence drops radically for certain current levels although these dips are not always of the same depth or in
the same locations as those marked in Figure 4.3. We also see a small hysteresis effect,
noticeable at currents of 25 or 30 mA, near the arrows indicating ’up’ and ’down’ for the
different current directions.
The locations of the coherence dips were very sensitive to temperature changes. By
varying the thermoelectric cooling on the laser diode by
72
C, the location of minimum
25
Mean of
I
+I
max
Power in a. u.
20
15
min
Whole beam
power
10
5
0
0
10
20
30
40
Current in mA
50
60
Figure 4.5: The total beam power (solid line) of LD03 vs. current compared to the mean of
and (small circles).
coherence as a function of current varied by 0.1 to 0.2 mA. The temperature sensitivity and
fluctuations in ambient temperature probably explain the inconsistency of the location and
depth of the coherence fluctuations, as seen in Figures 4.3 and 4.4.
In contrast to the dips and fluctuations of the mutual intensity in Figures 4.3 and 4.4, the
whole beam power shown in Figure 4.5 varies linearly with current from threshold until 45
mA. It continues to vary linearly, albeit in sections, from 45 mA to 60 mA except for a kink
between 50 mA and 55 mA. The average of the maxima and minima also varies linearly
from the threshold current until 45 mA. The average, multiplied by a constant to match the
arbitrary units of the sampled beam power, is plotted in Figure 4.5 with small circles.
The two plots are overlaid in comparison as a reality check that the coherence measurements are correct. The interference fringes should have an average amplitude equal to the
73
total intensity, as seen in Figure 2.2. One possible explanation for the change in proportionality constant is that part of the total beam was cut off from the detector as the beam
widened with increasing current.
The fact that the total beam power varies linearly with current even when the coherence
changes indicates that the differential quantum efficiency is still the same but that the total
power is being redistributed among different modes. This measurement alone is not enough
to say whether the difference in the modes is spatial or longitudinal. The similar behavior
with different displacements in the lateral or transverse directions seems to indicate that the
redistribution is not among spatial modes.
A kink in the power vs. current curve for a laser diode has usually been associated
with an increase in gain for higher-order lateral modes and saturation of lower-order modes
[48, 71]. An alternative interpretation is that multiple modes exist before the kink current
but the beat-length pattern that the modes make in the laser shifts at the kink point [72].
In any case we can see that the mutual intensity measurements pick up on some internal
activity of the laser that the overall intensity measurements miss.
Concurrent with the fluctuations of overall coherence was some indication of beam
steering. We measured the intensity in a range of positions across the whole beam for
three different current levels: 27, 31.6, and 48 mA. At that time, as seen in the dashed
line of Figure 4.3, those current levels corresponded with the beginning of a coherence
dip, the bottom of a coherence minimum, and the highest coherence before another major
coherence dip. Contour plots for the three current levels are given in Figure 4.6, with the
current increasing from top to bottom. The contour lines indicate a difference of 1 nW in
the top plot, 1.5 nW in the middle plot, and 3 nW in the bottom plot. We can see in the
middle frame that the peak of the beam has shifted vertically by about 0.1 mm.
74
1.5
1
0.5
0
−0.5
−1
−1.5
−1
0
1
−1
0
1
Vertical position in mm
1.5
1
0.5
0
−0.5
−1
−1.5
1.5
1
0.5
0
−0.5
−1
−1.5
−1
0
1
Horizontal position in mm
Figure 4.6: Contour plots of the LD03 beam for three different coherent levels. The current
increases from top to bottom. The middle plot corresponds to a current level with low
overall coherence. The contour lines indicate increasing power differences from top to
bottom.
75
Since the overall coherence measurements indicated that there were some other coherence effects besides spatial coherence, we decided to investigate the longitudinal modes of
the lasers, as measured with a monochromator.
4.4 Optical spectra
In this section we present the results of measuring the optical spectra on the semiconductor lasers with stripe widths of 3, 10, 15, and 20 m. The 4 and 5 m stripe width lasers
had been damaged by this point.
We were able to get the sharpest resolution with the 3 and 15 m stripe width lasers,
as seen in Figures 4.7, 4.8, and 4.9. In these spectra one can clearly see the separation
nm, with the uncertainty being equal to the smallest
of longitudinal modes at
step size of the monochromator, 0.02 nm. The longitudinal mode spacing is consistent
with the cavity length of 300 m. Using Eq. 2.29 and a wavelength of 990 nm, the axial
mode separation would be 0.38 nm for
=4.3, or 0.454 nm for
=3.6. Since the group
refractive index is dependent on composition and size of the quantum wells, and with the
limitations of the monochromator step size, the experimental distance is consistent with the
theoretically predicted axial mode separation.
With the two 10 m lasers and the 20 m laser the longitudinal modes were not completely resolved. It appears that this was due to experimental difficulties, possibly because
the spatial extent of the beams was wider for these lasers. The resolution was still adequate
to see that multiple longitudinal modes were lasing.
Figures 4.7 and 4.8 show the optical spectra for the 3 m stripe width laser for various
current levels, starting below threshold at the top of Figure 4.7. Both figures cover a range
76
of 8 nm but Figure 4.7 extends from 984 to 992 nm, and Figure 4.8 extends from 989 to
997 nm.
The different sections of Figure 4.7 separated by horizontal dashed lines indicate regions of different scales for the intensity output. Using the bottom section of Figure 4.7,
the spectrum for a current of 28 mA, as a scale of one, the magnification increases by 2,
4, 40, and 80 times for each higher section on the figure. One current level, 20 mA, is
shown with two different magnifications: 4 and 40 . In Figure 4.8 the same scale is used
throughout the figure, demagnified from the bottom plot in Figure 4.7.
The longitudinal modes of the cavity are evident in Figure 4.7, even below threshold.
These resonant modes are determined by the length of the cavity, which is 300 m. Right
at threshold, at 18.7 mA, multiple longitudinal cavity modes experience gain. These modes
are still amplified at 20 mA, as seen in the upper, more magnified plot marked 20 mA. The
longitudinal mode at 986 nm has a much greater amplitude at 20 mA, however, and it is
shown in the lower 20 mA plot, viewed with less magnification. This particular mode is
deleted in the more magnified view to allow the details of the smaller amplitude modes to
appear.
As the current level increases, the longitudinal modes move to longer wavelengths, or
red-shift. This is most easily observed in the dominant mode, such as for currents of 24.5,
25, and 28 mA in Figure 4.7. Between 24.5 and 25 mA the dominant mode red-shifted by
about 0.02 nm, and between 25 and 28 mA the shift was about 0.08 nm.
In certain temperature and current level combinations the lasing mode selection may
become unstable and multiple longitudinal modes may oscillate. Typically, the laser hops
back and forth between multiple modes [60, 61] but on a fast time scale, so the output
may be a time average over the different modes. These multiple modes are clearly seen in
77
18.0 mA
18.7 mA
80x
40x
Intensity (a.u.), different scales
20.0 mA
24.1 mA
24.5 mA
4x
25.0 mA
2x
28.0 mA
1x
984
985
986
987
988
989
Wavelength in nm
990
991
992
Figure 4.7: Optical spectra for LD03 for various currents, starting below threshold at the
top plot and increasing in current towards the bottom plot.
78
40 mA
45 mA
Intensity (a.u.), same scale
46 mA
48 mA
50 mA
52 mA
56 mA
58 mA
989
990
991
992
993
994
Wavelength in nm
995
996
997
Figure 4.8: Optical spectra for LD03 for various currents, starting at 40 mA for the top plot
and increasing in current towards the bottom plot.
79
Figure 4.7 in the plot labeled 24.1 mA and, to a lesser extent, in the 24.5 mA plot. Multiple
modes are also evident in Figure 4.8 in the plots for 46–50 mA, especially for 48 mA,
where there are three modes almost equally weighted.
As the current increased past an unstable region, the laser settled into a single longitudinal mode again, although at a different wavelength. The 28 mA plot in Figure 4.7 and
the 40, 52, and 58 mA plots in Figure 4.8 show single longitudinal modes after a region of
multiple modes.
The optical spectra for the 15 m stripe width laser at several different current levels
are plotted in Figure 4.9. The threshold current for this laser was 15 mA. With the drive
current at less than 30 mA, or 2 , the laser operated in a single longitudinal mode. For
currents of 38 and 40 mA the spectra indicate that the laser was mode hopping.
Figure 4.10 shows the spectra for two different lasers: a 10 m stripe width laser on the
left and the 20 m stripe width laser on the right. Each laser has spectra for two different
current levels plotted in the figure, with the smaller current spectrum indicated with small
circles and the larger current spectrum with a solid line. The threshold current for the 10
m stripe width laser was 26 mA. The two spectra were for current levels of 40 and 47 mA,
or 1.5 and 1.8 . The 20 m laser had a threshold current of 25 mA, so the two spectra
were for current levels of 1.2 and 1.76 . The spectra in this figure were not resolved as
well as for the previous figures. Even so, we can clearly see that the lasers were operating
in multiple longitudinal modes.
In looking at the spectra for all of these semiconductor lasers we can see that they often
operate in multiple longitudinal modes, possibly with overlapping linewidths for the 10 and
20 m stripe width lasers. These multiple longitudinal modes complicate the extraction of
80
LD15
Intensity (a.u.), same scale
23 mA
25 mA
28 mA
38 mA
40 mA
989
990
991
992
993
Wavelength in nm
994
995
Figure 4.9: Optical spectra for LD15 at various current levels.
81
LD10
LD20
1
Intensity (a.u.)
40 mA
44 mA
47 mA
30 mA
988
990
992
994
Wavelength in nm
Figure 4.10: Optical spectra for (left) LD
levels.
984
986
988
Wavelength in nm
990
and (right) LD20 at two different current
spatial modes and weights. For the best experimental results, the spatial coherence matrix
should be measured when the laser is operating in a single longitudinal mode.
If the laser has multiple longitudinal modes for certain test conditions, we can generally
still recover the shapes of the modes. This is based on the assumption that the spatial
modes will stay the same shape for each longitudinal family, due to the geometry of the
laser remaining constant and the small differences in wavelength. The extracted weights,
however, will not be correct.
4.5 Frequency mixing experiments
To verify that the semiconductor lasers under test did indeed have multiple spatial
modes we performed the frequency mixing test described in Section 3.3. As explained
in Section 2.4.2, a fast detector and some spectrum analysis can be used to see the beat
82
frequencies caused by the interference of two or more modes of different frequencies. A
6 GHz detector is required to see the beat frequencies of two modes separated by 0.01 nm.
Due to the frequency limitations and the resulting aliasing, these experiments gave somewhat ambiguous results but did indicate some multiple spatial modes in the lasers, depending on the current level.
One limiting factor in the equipment we used was the oscilloscope. It sampled at a
maximum rate of 8 Gb/s by combining two channels that sampled at a rate of 4 Gb/s. With
discrete time samples, the Fourier transform results in a train of repeating spectra, centered
around integer multiples of the sampling frequency. In this case, the center frequencies
were integer multiples of 4 GHz. The practical frequency limit was then 2 GHz because of
aliasing.
We were able to test most of the semiconductor lasers with the frequency mixing tests.
The exceptions were the 4 m and 5 m stripe width lasers, which were damaged before
the equipment was available. The high-speed detector was only responsive to infrared
wavelengths so we could not use it with the HeNe laser.
For each of the lasers we recorded the frequency mixing results at several different
current levels. Some of the results are given in this chapter, especially those that illustrate
different operating regions.
The laser with the most interesting frequency mixing results was one of the 10 m stripe
width lasers, labeled LD
. The general pattern for the beat frequency spectra is shown in
Figure 4.11. There were two different patterns depending on the laser current. For a current
level less than 42 mA the top pattern was seen and the bottom pattern was seen for current
levels greater than 43 mA. The beat frequency peaks are indicated by vertical lines at DC or
0 Hz,
, 2 , , , and 2 . The relative heights of the vertical lines are representative
83
of the peaks for a particular current level. The small arrows on the figure show which
direction the beat frequencies moved with increasing current. The main difference between
the top and bottom patterns of Figure 4.11 is that and have switched places with
respect to the center.
f
1
f
f
2
1
Intensity a.u.
2f1
∆f
∆f
0 (DC)
1 GHZ
∆f
2 GHz
3 GHz
f1
f2
f1
∆f 2∆f
0 (DC)
4 GHz
2f2
1 GHZ
2 GHz
2∆f
3 GHz
4 GHz
Frequency
Figure 4.11: Frequency mixing pattern for LD
with 6 GHz detector for current less than
42 mA (top) and for current greater than 43 mA (bottom).
There were several mathematical relationships between the beat frequencies. The frequency labeled was equal to 4 GHz minus . The separation between a beat frequency at
and was also
, in the upper pattern, or 2 , in the lower pattern. There was also
a harmonic frequency: either in the upper plot or 84
in the lower plot. The harmonic
was separated from 4 GHz by the same frequency difference as the difference between and .
The patterns seen in Figure 4.11 indicate that there was aliasing of the sampled spectra.
The frequency labeled is the negative frequency response centered about 4 GHz. This
was clearly seen when the current was increased and positive frequency, and moved to the right, to a larger
moved to the left, to a more negative frequency with respect to
4 GHz. Additional satellites appeared when was greater than 2 GHz, as in the lower part
of the figure. The only true beat frequencies in this figure are at DC and . The other peaks
are harmonics of or aliased replicas of and its harmonics. The beat frequency peak at
DC indicates the interference of a mode with itself, although there could be multiple modes
whose self-interference adds incoherently.
Figure 4.12: Beat frequency spectrum for LD
85
at 47 mA, 11/28/01, 6 GHz detector
Figure 4.12 displays a saved screen from the FFT on the detector output when the laser
current was 47 mA, or 1.8 times the threshold current . The beat frequency spectrum
demonstrates the pattern seen in the bottom portion of Figure 4.11. These tests were performed with a New Focus 1537, which is a 6 GHz detector. Similar results were also
obtained with a New Focus 1417, a 25 GHz detector.
Three months later we tested the same laser at various currents. The beat frequency
spectrum displayed at the top of Figure 4.13 is also for 47 mA and it was taken with
the faster New Focus detector. In both Figure 4.12 and 4.13 the horizontal dashed line
marks the DC level of the detected intensity.5 The peak frequencies in the top spectrum
of Figure 4.13 correspond to a peak The actual location of greater than 2 GHz and its aliased counterpart .
is different for the two measurements and the satellite peaks do
not appear in the later test. The difference in frequency could be due to slightly different
operating temperatures or current levels or aging effects in the laser.
The bottom spectrum in Figure 4.13 shows the beat frequencies observed when the laser
LD
was driven with a current of 60 mA, or 2.3 . Labeling the frequencies of the four
peaks from left to right as , , , and , we see that ,
, and and are equidistant from 2 GHz.
One sample of the beat frequency results for the other 10 m stripe width laser is given
in Figure 4.14. The driving current is 40 mA, but instead of seeing aliasing or harmonics like there were for LD
, there is only a large DC component, as indicated by the
horizontal dashed line, and a small peak at 1.35 GHz.
We looked carefully at the beat frequency spectra for the 3 m stripe width laser for
a wide range of current levels. Some of these spectra are shown in Figure 4.15 with the
5
In the monochrome versions of the saved screens the yellow line of the spectrum is not distinguishable
from the gray line marking the limits of the display.
86
Figure 4.13: Beat frequency spectra for LD
levels of 47 mA (top) and 60 mA (bottom).
87
on 3/1/02, with 25 GHz detector, at current
Figure 4.14: Beat frequency spectrum of LD
at 40 mA, 3/1/02
top spectrum being right at threshold, the middle spectrum just above threshold, and the
bottom spectrum at a higher current level. Right at threshold, near 19 mA, a sharp peak
appeared at about 80 MHz. As the current level increased to about 20.4 mA there was also
a broad flat peak that increased from DC to about 2 GHz. At this point it became even less
distinct as the aliased image was superimposed. At higher current levels (not shown) there
were no distinct peaks other than the one near 80 MHz. For current levels below 45 mA
the 80 MHz peak stayed at approximately the same amplitude and frequency.
We were able to visually observe mode hopping in the frequency mixing spectrum
at current levels of 27 mA and 45.7 mA. The spectrum jumped back and forth between
having two peaks at DC and 80 MHz and having more low frequency content between DC
and 80 MHz. Evidently the mode hopping was happening on a relatively slow time scale,
to have the effect be visible to the eye.
88
Figure 4.15: Beat frequency spectra of LD03 at 18.7 mA (top), 19.8 mA (center), and
20.4 mA (bottom).
89
Figure 4.16: Beat frequency spectra of LD15 at 23 mA (top) and 40 mA (bottom).
90
Figure 4.17: Beat frequency spectrum of LD20 at 27 mA, or 1.08 .
Figure 4.16 shows the beat frequency spectra for the 15 m stripe width laser at two
different operating currents, 23 and 40 mA, or 1.5 and 2.67 . These two current levels
were chosen to represent single and multiple longitudinal mode behavior, as shown in Figure 4.9. In the top spectrum of Figure 4.16 there is a large DC component but no distinct
peaks. The bottom spectrum of Figure 4.16 has a gentle slope from DC to 2 GHz. The DC
level is still more than 52 dB greater than the low-frequency response.
For the 20 m stripe width laser there were no sharp peaks for the beat frequencies. The
most distinct spectrum is shown in Figure 4.17, with two small peaks at 640 and 880 MHz.
As the current increased, the peaks flattened out into gentle humps and moved to the right.
Two beat frequency spectra for the 30 m stripe width laser are given in Figure 4.18,
for drive currents of 46 and 55 mA, or 1.4 or 1.67 . These do not have a large DC
component and they both have gradual peaks. The top spectrum in Figure 4.18 also has a
few sharper peaks but these may be due to an experimental problem discussed below and
illustrated by Figure 4.19.
91
Figure 4.18: Beat frequency spectra of LD30 at 46 mA (top) and 55 mA (bottom).
92
Figure 4.19: Frequency peaks from optical cavity between laser and detector.
The last frequency mixing figure, Figure 4.19, is actually an example of an experimental
anomaly to be avoided. For multiple lasers we observed a series of equally spaced peaks
with the same frequency separations independent of laser or current. Eventually we realized
that it was caused by reflections between the laser and the window of the detector. We
verified that this “cavity” was the cause by relocating the detector and observing that the
separation frequency changed due to the new cavity length. The peaks could be removed
by slightly rotating the detector.
Usually the peaks were equally spaced from DC but not for the 3 m laser. In that case
there was a peak in the 50 to 100 MHz range and the cavity peaks were offset from the
initial peak.
The results of the frequency mixing tests are not conclusive for all operating currents.
There were indications of beat frequencies under some conditions. Generally there was a
93
broad peak near DC that appeared near threshold and moved to higher frequencies with
increasing current. Once this peak passed 2 GHz it became difficult to see as it was very
broad and flat and there could be aliasing effects. At this point the drive current was usually
about 10% or 20% above threshold. It is possible that the beat frequency was still there but
it could no longer be detected due to the frequency limitations of the oscilloscope or the
detector or both.
The sharper peaks that were seen in the spectra for LD
may have been due to fre-
quency beating between different transverse mode families. Paoli et al documented optical
spectra where one set of laser modes with the same transverse mode number but multiple
longitudinal modes overlapped with another set with a different transverse mode number
[27]. The separation between mode groups was random between laser diodes but was
on the order of 1 nm. If multiple transverse spatial modes and multiple longitudinal modes
were active it could explain the apparent broadening of the spectra of LD
in Figure 4.10.
There could also be combinations of modes that were close enough in frequency to produce the patterns shown in Figure 4.11, with the additional complications of aliasing and
multiple harmonics.
4.6 Laser with
m stripe width
This section and the following section contain the main results of our research, the extracted spatial modes from different semiconductor laser based on spatial coherence measurements. This section gives the results from the first laser we tested, which was made
with a slightly different design than the other semiconductor lasers we tested. These results
were also reported in our proposal of research and in an article in Applied Optics [43].
94
We refined the experimental procedures after that point to speed up the process and reduce
noise.
The first laser we tested was an experimental one from Wright Patterson Air Force Base
with a nominal wavelength of 967 nm. It was made up of GaAs and AlGaAs with different
doping levels and a single InGaAs quantum well with a thickness of 5 nm. The laser had a
ridge-waveguide design with a
m stripe width and a Fabry–Perot cavity.
We measured the spatial coherence in the transverse, or , direction with the fibers in
the same configuration depicted in Figure 3.2. Normally this configuration would be used
to take measurements in the lateral, or
direction, resulting in a partial data matrix, as
shown in Figure 2.4. For this laser the width of the beam was such that the horizontal offset
introduced minimal changes in intensity and in lateral spatial coherence. This allowed us
to take a complete data matrix, not just the upper triangular portion, with the fibers offset
in the
direction and traveling in the direction. With a complete data matrix we were
able to analyze the data with singular value decomposition (SVD). The analysis in this case
gave results that were virtually identical to the results from the Jacobian algorithm, since
the data matrix was complete.
In the transverse direction we expected the laser to be single-mode, due to the dimensions of the confinement layers. It was also apparent from the intensity profile that the
beam was being diffracted, probably due to the lens aperture or limitations of the mounting
fixture. We adjusted the lens so the measurement plane was in the far field, resulting in an
approximate sinc squared function for the intensity profile.
Both the SVD and Jacobian algorithms extracted one mode, with approximately 1%
of the power in noise if two modes were requested. Figure 4.20 displays the extracted
spatial mode on the left side and the corresponding intensity profile on the right side. The
95
0.8
0.8
y
0.6
0.4
0.2
0
−0.5
0.6
0.4
Me
as
ure
d
Normalized intensity
1
sq
int rt
en
sit
Normalized field amplitude
1
0.2
SVD
mode
Reconstructed
sinc
0
−0.5
0
0.5
Distance from beam center in mm
sinc2
0
0.5
Distance from beam center in mm
Figure 4.20: (Left) The transverse spatial mode of the 5 m stripe width laser extracted by
singular value decomposition (circles). (Right) The corresponding intensity profile (circles)
compared to the measured intensity (pluses).
plots drawn with circles are the extracted mode field and the corresponding reconstructed
intensity profile. The measured intensity profile is indicated by pluses in the right plot with
the corresponding square root values marked by pluses in the left plot. For comparison
purposes, a sinc function is drawn on the left plot and a squared sinc function on the right.
From Figure 4.20 we see that the intensity profile is very similar to a sinc function,
indicating that the diverging beam was truncated in the transverse direction. The coherence
measurements still indicated that the beam was single mode, even though it does not match
a Gaussian profile. There was some background noise in the intensity measurements that
could account for the discrepancy in the minima of the field plots. A small offset from zero
is magnified by the extraction process just as it would be for a square root function.
96
The more interesting experiment for this laser was measuring the spatial coherence in
the lateral, or , direction. The beam was narrower in the transverse direction so it was not
feasible to offset the fibers perpendicular to the measurement plane as we had done for the
transverse measurements. As explained in Section 2.5, the measured correlation data filled
in the upper triangular part of the correlation matrix and, by symmetry, also filled in the
lower triangular part.
The resulting lateral modes for the
m stripe laser are plotted in Figure 4.21 on the
left, with the corresponding reconstructed intensity profile on the right. In the left hand
plot, the extracted modes are depicted with circles and squares. The dashed lines are the
fundamental and first-order Hermite–Gaussian modes with a beam waist radius of 0.45 mm
and the same relative weights as the extracted modes. The relative field weights are 83.4%
and 16.6%, corresponding to power weights of 96.2% and 3.8%. The measured intensity
profile is indicated with pluses in the right hand plot, compared to the intensity profile
reconstructed from the extracted modes, indicated by circles, and the profile reconstructed
from the Hermite–Gaussian fit, drawn with a dashed line.
We included the Hermite–Gaussian modes in Figure 4.21 only to show the type of
modes that most other methods of weighting spatial modes use as a basis set. The physical characteristics of the laser are such that the extracted lateral modes are similar to the
Hermite–Gaussian model. The beam is somewhat asymmetric, as we see in the intensity
profile, so the Hermite–Gaussian model cannot fit it exactly. Most of the other lasers we
tested did not match the Hermite–Gaussian model this well, due to more asymmetry or
other defects.
97
1
0.8
0.8
mode 0
Normalized intensity
Normalized field amplitude
1
0.6
0.4
0.2
0
HG modes
Measured
0.6
0.4
Reconstructed
0.2
HG
0
mode 1
−0.2
−0.2
−0.5
0
0.5
Horizontal position in mm
−0.5
0
0.5
Horizontal position in mm
Figure 4.21: (Left) The lateral spatial modes of the 5 m stripe width laser extracted by
the Jacobian algorithm (circles and squares) compared to fitted Hermite–Gaussian modes
(dashed line). (Right) The reconstructed intensity profile (circles) compared to the measured intensity profile (+’s).
4.7 Semiconductor lasers of varying widths
This section contains the experimental results from the spatial coherence measurements
of the second batch of semiconductor lasers with varying stripe widths. We tested all of the
lasers for lateral spatial modes at multiple current levels. For the 3 m stripe width laser we
also tested for transverse spatial modes. Unless specified otherwise, the procedures were
the same for each of the lasers.
Due to astigmatism and diffraction it was not always an easy decision on where to position the collimating lens. Ideally one would want a collimated beam but the lens position
that collimated the beam in the lateral direction might cause diffraction peaks and valleys
in the transverse direction. One criterion that we used was to have a smooth peak in the
98
transverse direction in the measurement plane. Otherwise a small displacement in height
could significantly change the power level and introduce unintentional variations in the
mutual intensity. The width of the peak was important too since a narrow peak imposed
tight constraints on the vertical alignment and a wide peak would reduce the overall power
level. As we saw in Section 2.4.4, the number of spatial modes and their relative weights
will not change by propagating through a paraxial optical system, although the weights can
be affected by apertures.
These lasers are of a similar design to the 5 m stripe width laser described in the
previous section but have three quantum wells of 8 nm thickness instead of one quantum
well of 5 nm. Also, the second batch has confinement layers that are twice as thick as the
layers in the first laser. The nominal wavelengths for the second batch of lasers was quoted
to us as 980 nm, but the spectral measurements that we took gave operating wavelengths
closer to 990 nm.
In the following subsections we give the results for the different semiconductor lasers.
For each laser we plot the extracted spatial modes on the left side of the corresponding
figure and the reconstructed intensity on the right side, along with the measured intensity
profile of that laser. We have labeled the extracted spatial modes with ‘mode 0’, ‘mode
1’, etc. The labels are for convenience and refer to the order of the relative weights, not
necessarily to the number of zero crossings. The lateral modes do not follow the Hermite–
Gaussian model well as they often have extra humps.
The spatial modes are given in field amplitudes with arbitrary units. Essentially these
units are the square root of the mutual intensity sampled with both the right-hand and lefthand fibers. The power levels depend on the focusing of the laser and the splitting ratio
between the two fibers. These amplitudes are not absolute because they have not been
99
calibrated to a standard, but between different current levels for the same laser they give a
measure of the change in amplitude in each mode. In the tables, the spatial mode weights
are given in relative terms, where the norm of each individual mode is compared to the sum
of the norms for that current level. The relative weights in the tables always add to 100%
by definition and do not take into account the noise level.
The intensity profiles are plotted normalized to the maximum intensity. Because of the
changes in overall coherence, there was not one standard ratio between the reconstructed
intensity profile and the measured intensity profile. Also, the RHS and LHS profiles were
not at the same level since the splitting ratio was not 50/50. Usually the LHS and RHS
profiles were identical in shape and only one is used for the profile plot. Occasionally
there were some differences between the LHS and RHS profiles and these have both been
included on the plots. The differences may be due to a slight mismatch in vertical position
and variations in the beam along the vertical direction. The important comparison between
the reconstructed intensity profile and the measured intensity profiles is the shape of the
profile.
4.7.1 LD03: Laser with 3 m stripe width
In this section we present the spatial modes extracted from LD03, the 3 m stripe width
laser. We start with the extracted transverse modes. For this testing configuration we did
not set up the fiber ends in exactly the same way as Figure 3.2. Instead, we rotated the
metal plates that held the fibers about the optic axis and attached the plates end-on to the
posts mounted on the - - stages. This resulted in the right-hand-side (RHS) fiber being
mounted above the LHS fiber. This limited the range of motion of the two fibers in a
similar way to that discussed in Section 2.5, although in the vertical, instead of horizontal,
100
direction. The beam had too much variation in the lateral direction so we wanted to ensure
the fibers were not offset in that direction.
Figures 4.22 and 4.23 show the extracted transverse modes for LD03 and the corresponding reconstructed intensities for six different current levels. At the top of Figure 4.22
the current level is 27 mA, or 1.44 time the threshold current . The plots in the middle
are for a current level of 31.6 mA, or 1.69 and the plots on the bottom correspond to
40 mA, or 2.14 . In Figure 4.23 the current levels are: (top) 48 mA or 2.6 , (middle)
50.7 mA or 2.7 , and (bottom) 52 mA or 2.8 .
Current (mA)
27.0
31.6
40.0
48.0
50.7
52.0
Relative field weights (%)
mode 0 mode 1 mode 2
100
0
0
89.2
10.8
0
86.3
7.9
5.8
91.3
8.7
0
89.3
10.7
0
87.8
7.3
4.9
Relative power weights (%)
mode 0 mode 1 mode 2
100
0
0
98.6
1.4
0
98.7
0.8
0.5
99.1
0.9
0
98.6
1.4
0
99
0.7
0.3
Table 4.1: The relative field and power weights of transverse spatial modes of laser LD03.
The relative field and power weights are given in Table 4.1. One warning about these
weights is the fact that most of the current levels corresponded to operating regions where
there were multiple longitudinal modes. At the time of these measurements, the only current level listed that was definitely operating in a single longitudinal mode was the 40 mA
level.
The reconstructed intensity profiles still are a close match to the measured intensity
profiles even if the weights are not correct. This is due to the fact that a large percentage of
101
10
1
27 mA
8
0.8
measured
6
0.6
mode 0
reconst.
4
0.4
2
0.2
0
−1
−1
0
0
1
−1
0
1
31.6 mA
8
6
mode 0
4
2
0
−1
mode 1
−1
0
1
Normalized intensity
Field amplitude (a.u.)
10
0.6
reconst.
0.4
0.2
0
1
measured
0.8
−1
0
1
10
1
40 mA
8
reconst.
0.8
mode 0
6
0.6
4
0.4
mode 2
measured
2
mode 1
0
−1
0.2
0
−1
0
1
Vertical position in mm
−1
0
1
Vertical position in mm
Figure 4.22: (Left) The transverse spatial modes of LD03 for 27 mA (top), 31.6 mA (middle), and 40 mA (bottom). (Right) The reconstructed intensity profiles (circles) compared
to measured intensity profiles (+’s).
102
12
48 mA
10
1
reconst.
0.8
8
6
0.6
mode 0
4
measured
0.4
2
mode 1
0
−2
−1
0.2
0
0
1
−1
0
1
8
6
mode 0
4
2
mode 1
0
−2
1
50.7 mA
10
Normalized intensity
Field amplitude (a.u.)
12
−1
0
0.6
0.4
reconst.
0.2
0
1
measured
0.8
−1
0
1
12
1
52 mA
10
0.8
8
6
mode 0
measured
0.6
4
0.4
reconst.
2
0
−2
mode 2
0.2
mode 1
−1
0
1
Vertical position in mm
0
−1
0
1
Vertical position in mm
Figure 4.23: (Left) The transverse spatial modes of LD03 for 48 mA (top), 50.7 mA (middle), and 52 mA (bottom). (Right) The reconstructed intensity profiles (circles) compared
to measured intensity profiles.
103
the power is in the fundamental mode. As we saw in Figure 1.1, if 95% of the power is in
the fundamental mode the other mode does not change the intensity profile by very much.
The field amplitudes on the left sides of the figures are all plotted to the same scale
and we can see that the amplitudes do not increase monotonically with current. Compare
in Figure 4.23, for example, the height of mode 0 for 48 mA with that for 50.7 mA. The
middle plot mode 0 is about 30% lower than the top plot mode 0, even though the total
beam power does increase with current. This discrepancy is due to the decrease in total
coherence, as discussed in Section 4.3.
The lateral spatial modes extracted from LD03 are presented in Figure 4.24. The current
levels are a subset of those used for the transverse mode tests. For the lateral tests, the fibers
were configured in the standard way depicted in Figure 3.2. On the left side of Figure 4.24
the extracted modes are plotted with the reconstructed intensity profiles plotted on the right.
For comparison with the reconstructed intensity profiles we have also plotted the measured
profiles taken with the RHS fiber and the LHS fiber. There was some discrepancy in these
profiles between the fibers, most noticeable near the dip on the right side of the plot. This
discrepancy is probably due to the fibers being slightly misaligned vertically.
As with the transverse modes, we see in Figure 4.24 that the absolute field amplitudes
do not uniformly increase with current. The middle plot was at a point where the overall
coherence was low due to multiple longitudinal modes.
Neither the fundamental modes nor the first-order modes look like Hermite–Gaussian
modes. There could be some defects in the lasers that cause the fluctuations in field amplitude on the right hand side of the plots.
104
12
1
27 mA
10
0.8
8
6
4
0.6
mode 0
reconst.
RHS
0.4
2
0.2
0
−2
mode 1
−1
0
0
1
31.6 mA
10
8
6
4
2
mode 0
0
−1
0
6
1
0.6
reconst.
1
RHS
0.4
0.2
0
12
LHS
−1
0
1
1
48 mA
10
0
0.8
mode 1
−2
8
−1
1
Normalized intensity
Field amplitude (a. u.)
12
LHS
0.8
reconst.
mode 0
0.6
4
RHS
0.4
2
0.2
0
−2
0
−1
0
1
Horizontal position in mm
LHS
−1
0
1
Horizontal position in mm
Figure 4.24: (Left) The lateral spatial modes of LD03 for 27 mA (top), 31.6 mA (middle),
and 48 mA (bottom). (Right) The reconstructed intensity profiles (circles) compared to
measured intensity profiles (+’s).
105
It was somewhat unexpected that this laser, with only a 3 m stripe width, would support multiple lateral modes. We might also expect that the number of modes would increase
with increasing current levels but that is not the case here.
The relative field and power weights are given in Table 4.2, although they are probably
incorrect due to multiple longitudinal mode operation.
Current (mA)
27.0
31.6
48.0
Relative field weights (%)
mode 0
mode 1
84.2
15.8
84
16
100
0
Relative power weights (%)
mode 0
mode 1
96.6
3.4
96.5
3.5
100
0
Table 4.2: The relative field and power weights of lateral spatial modes of laser LD03.
4.7.2 LD04: Laser with 4 m stripe width
For the 4 m stripe width laser, LD04, we measured the spatial coherence in the lateral
direction at 25 mA, or 1.67 , 28 mA, or 1.87 , and 30 mA, or 2 . The Jacobian
algorithm extracted one lateral mode at each of these current levels, shown in Figure 4.25,
and also at 20 mA, which is not shown. The wire bond on this laser was knocked loose so
we were not able to perform any other tests on it, such as the frequency mixing experiment
or measuring the optical spectra.
This laser only had one lateral spatial mode but it was not a Gaussian mode, as seen by
the asymmmetry. The lateral spatial mode did not have the extra bump that laser LD03 did,
even though it had a wider stripe. The amplitude of the mode did increase with increasing
106
25 mA
6
1
0.8
mode 0
4
0.6
reconst.
measured
0.4
2
0.2
−1
0
mode 0
4
2
6
−1
mode 0
0
−1
0
1
1
28 mA
6
0
0
1
Normalized intensity
Field amplitude (a.u.)
0
0.8
0.6
0.4
30 mA
measured
0.2
0
1
reconst.
1
−1
0
1
measured
0.8
4
0.6
0.4
2
0.2
0
0
−1
0
1
Horizontal position in mm
reconst.
−1
0
1
Horizontal position in mm
Figure 4.25: (Left) The lateral spatial modes of LD04 for 25 mA (top), 28 mA (middle),
and 30 mA (bottom). (Right) The reconstructed intensity profiles (circles) compared to
measured intensity profiles (+’s).
107
current, possibly indicating that at these operating points the laser had only one longitudinal
mode.
4.7.3 LD10 : Laser with 10 m stripe width
Figure 4.26 shows the lateral spatial modes for one of the 10 m stripe width lasers,
LD
. The two current levels are 40 mA, or 1.5 , and 47 mA, or 1.8 . Again, as we
saw for LD03, the field amplitudes do not always increase with increasing current levels.
This laser had several longitudinal modes active for both current conditions.
The relative field weights for the lower operating current are 86.4% and 13.6%, and
the relative power weights are 97.6% and 2.4%. For the higher current level, the relative
field weights are 75% and 25%, and the relative power weights are 90% and 10%. The
same relative weights were determined during a later test when the focus of the laser was
different.
The intensity profiles for LD
are similar to LD03 in that they have an unexpected
hump on one side of the profile. In the bottom of Figure 4.26, mode 1 has a positive peak
at the hump and a negative minimum to the left of the inversion point in mode 0. This is
the same type of behavior seen in mode 1 in the top and middle plots of Figure 4.24.
4.7.4 LD10 : Laser with 10 m stripe width
Included here are the results from spatial coherence measurements on the second 10 m
stripe width laser, LD
. The intensity profile on this laser was very asymmetric and
changed shape with different current levels. Figure 4.27 shows a series of plots of the
intensity profile for increasing current levels. The current levels start at 25 mA and increase
by 1 mA to 40 mA. The plots corresponding to the spatial coherence tests are indicated by
small circles.
108
4
40 mA
1
mode 0
3
0.8
2
0.6
1
measured
0.4
reconst.
0
0.2
mode 1
−1
−1
0
0
1
−1
0
1
4
1
3
2
Normalized intensity
Field amplitude (a.u.)
47 mA
mode 0
1
0
mode 1
−1
reconst.
0.8
0.6
measured
0.4
0.2
0
−1
0
1
Horizontal position in mm
−1
0
1
Horizontal position in mm
Figure 4.26: (Left) The lateral spatial modes of LD
for 40 mA (top), and 47 mA (bottom). (Right) The reconstructed intensity profiles (circles) compared to measured intensity
profiles (+’s).
109
60
Power in nW
50
40
30
20
10
0
−1
−0.5
0
0.5
1
Horizontal position in mm
Figure 4.27: Intensity profiles for laser LD
for current levels from 25 mA to 40 mA.
Circles indicate the current levels used for spatial coherence tests.
Figures 4.28 and 4.29 show the lateral spatial modes for the second 10 m stripe width
laser, LD
. The current levels are 26 mA, or 1.13 , and 27 mA, or 1.17 , in Fig-
ure 4.28 and 30 mA, or 1.3 , and 40 mA, or 1.7 in Figure 4.29.
Current (mA)
26
27
30
40
Relative field weights (%)
mode 0
mode 1
66.5
33.5
64.2
35.8
71.4
28.6
69.3
30.7
Relative power weights (%)
mode 0
mode 1
79.7
20.3
76.3
23.7
86.1
13.9
83.6
16.4
Table 4.3: The relative field and power weights of lateral spatial modes of laser LD
110
.
3
1
26 mA
2
RHS
0.8
mode 0
LHS
1
0.6
0
0.4
mode 1
−1
0.2
reconst.
−2
−1
−0.5
0
0.5
0
1
−1
−0.5
0
0.5
1
0.5
1
3
mode 0
2
1
27 mA
LHS
0.8
1
0
−1
RHS
0.6
0.4
mode 1
0.2
reconst.
−2
−1
−0.5
0
0.5
0
1
−1
−0.5
0
Figure 4.28: (Left) The lateral spatial modes of LD
for 26 mA (top) and 27 mA (bottom). (Right) The reconstructed intensity profiles (circles) compared to measured intensity
profiles for the LHS fiber (+’s) and the RHS fiber (x’s).
111
14
1
30 mA
10
reconst.
0.8
5
mode 0
0.6
LHS
RHS
0.4
0
mode 1
0.2
−5
−7
−1
−0.5
0
0.5
14
10
0
1
40 mA
−1
−0.5
0
0.5
1
1
mode 0
0.8
5
0.6
0
0.4
LHS
RHS
mode 1
0.2
−5
−7
reconst.
−1
−0.5
0
0.5
0
1
−1
−0.5
0
0.5
1
Figure 4.29: (Left) The lateral spatial modes of LD
for 30 mA (top) and 40 mA (bottom). (Right) The reconstructed intensity profiles (circles) compared to measured intensity
profiles for the LHS fiber (+’s)and the RHS fiber (x’s).
112
The extracted spatial modes obviously do not fit the Hermite–Gaussian model, or any
other analytic description. The labels of ‘mode 0’ and ‘mode 1’ are more for convenience
than indicative of minima. The top plot of spatial modes in Figure 4.28 is similar to the
bottom plot of spatial modes in Figure 4.26. As the current increases, however, the mode
labeled ‘mode 0’ dips in the middle while the other mode shifts its minimum to the left.
The reconstructions of the intensity profiles are interesting in that they provide a reasonable fit to the right peak but deviate significantly from the rest of the profile. The spectrum
for this laser indicated that there were many longitudinal modes lasing, and it appears that
this complication has corrupted the extraction process of the spatial modes. The relative
field and power weights are given in Table 4.3.
4.7.5 LD15: Laser with 15 m stripe width
This section contains the results of our spatial coherence tests on LD15, a 15 m stripe
width laser. In Figure 4.30 the upper plots correspond to a current level of 23 mA, or 1.5 ,
and the lower plots to 28 mA, or 1.9 . In Figure 4.31 the current levels are 38 and 40 mA,
or 2.5 and 2.7 . The extracted lateral spatial modes are plotted on the left side and the
corresponding reconstructed intensity profiles are plotted on the right, with the measured
intensity included for comparison. Table 4.4 contains the relative field and power weights
for LD15.
The operating conditions pertaining to Figure 4.30 were in the single longitudinal mode
region. With high total coherence we were able to extract three spatial modes, and the
amplitude of the main mode increased with current as expected.
In contrast, the measurements shown in Figure 4.31 were taken in the multiple longitudinal mode region. Also, during the spatial coherence measurements at a current level of
113
18
reconst.
1
23 mA
15
0.8
10
mode 0
measured
0.6
0.4
5
mode 2
0
−2 mode 1
−1
−0.5
0
0.2
0.5
0
−1
1
18
15
1
28 mA
Normalized intensity
Field amplitude (a.u.)
mode 0
10
5
mode 2
0
−2 mode 1
−1
−0.5
0
0.5
Horizontal position in mm
0
0.5
1
measured
0.8
0.6
reconst.
0.4
0.2
0
−1
1
−0.5
−0.5
0
0.5
Horizontal position in mm
1
Figure 4.30: (Left) The lateral spatial modes of LD15 for 23 mA (top) and 28 mA (bottom). (Right) The reconstructed intensity profiles (circles) compared to measured intensity
profiles (+’s).
114
38 mA
20
15
1
measured
mode 0
0.8
0.6
10
reconst.
0.4
5
Field amplitude (a.u.)
20
−0.5
0
mode 0
0.5
40 mA
15
10
5
0
−2
−1
0
−1
1
−0.5
0
0.5
Horizontal position in mm
−0.5
0
0.5
1
1
Normalized intensity
0
−2
−1
0.2
mode 1
0.8
reconst.
0.6
measured
0.4
0.2
0
−1
1
−0.5
0
0.5
Horizontal position in mm
1
Figure 4.31: (Left) The lateral spatial modes of LD15 for 38 mA (top) and 48 mA (bottom). (Right) The reconstructed intensity profiles (circles) compared to measured intensity
profiles (+’s).
115
38 mA, the top plots in Figure 4.31, the monitored beam power dropped suddenly by about
16%, even though the intensity profiles measured before and after the spatial coherence
test were essentially the same. The numerical analysis algorithms were not able to extract
more modes from the correlation matrix but that may have been due to the instability of
the modes. The second mode shown in the upper plot of Figure 4.31 has the appearance
of a bogus mode since it follows the main mode in one area and drops to virtually nothing
for the rest of the positions. The intensity profile of LD15 did not have a distinct hump as
Current (mA)
23
28
38
48
Relative field weights (%)
mode 0 mode 1 mode 2
87.5
7
5.5
88
7.2
4.8
92.3
7.7
0
100
0
0
Relative power weights (%)
mode 0 mode 1 mode 2
99
0.6
0.4
99
0.7
0.3
99.3
0.7
0
100
0
0
Table 4.4: The relative field and power weights of lateral spatial modes of laser LD15.
some of the others did, although there is bulge in the right side of the profile. The central
peak of mode 0, and the intensity profile, had a small dip for the top plot in Figure 4.30 and
some asymmetry for the other plots in Figures 4.30 and 4.31.
4.7.6 LD20: Laser with 20 m stripe width
Laser LD20, with a stripe width of 20 m, also displayed changing intensity profiles
with increasing current levels. Figure 4.32 shows the profiles taken with one fiber for
current levels of 26 mA to 60 mA, in 1 mA steps until 40 mA and then in 5 mA steps.
The profiles marked with small circles indicate the current levels where we also performed
spatial coherence tests.
116
70
60 mA
Optical power in nW
60
55 mA
50
50 mA
40
45 mA
30
40 mA
20
10
0
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Horizontal position in mm
Figure 4.32: Intensity profiles for laser LD20 for current levels from 26 mA to 60 mA. The
profiles marked with circles indicate the current levels for spatial coherence tests.
Figures 4.33–4.35 display the results of the spatial coherence tests on LD20 for various
current levels. In Figure 4.33 the upper plots correspond to a current level of 30 mA, or
1.2 , and the lower plots to 34 mA, or 1.36 . Figure 4.34 has the results from levels of
40 mA, or 1.6 , and 55 mA, or 2.2 . The modes in Figure 4.35 were extracted over a
year earlier, at a current level of 44 mA, or 1.76 . The relative field and power weights for
LD20 are given in Table 4.5. The usual warning about multiple longitudinal modes applies.
The lower current level modes shown in Figure 4.33 are similar to the fundamental and
first-order Hermite–Gaussian modes in the center but have extra humps in the wider areas
of the beam. In contrast to the previous lasers, these humps are on both sides of the profile,
not just one. The corresponding intensity profiles have three peaks. The modes for higher
117
9
1
30 mA
0.8
mode 0
6
measured
0.6
3
0.4
reconst.
0
0.2
mode 1
−3
−2
0
1
0
2
mode 0
9
−2
6
3
0
−2
0
0.8
1
2
measured
0.6
0.4
0.2
mode 1
−3
−1
1
34 mA
Normalized intensity
Field amplitude (a.u.)
−1
reconst.
0
−1
0
1
2
Horizontal position in mm
−2
−1
0
1
2
Horizontal position in mm
Figure 4.33: (Left) Lateral spatial modes of LD20 at current levels of 30 mA (top) and
34 mA (bottom). (Right) The reconstructed intensity profiles (circles) compared to measured intensity profiles (+’s).
118
9
40 mA
1
mode 0
6
measured
0.8
0.6
3
0.4
0
reconst.
0.2
−3
−2
−1
0
1
0
2
−2
−1
0
1
2
9
1
55 mA
Normalized intensity
Field amplitude (a.u.)
mode 0
6
3
0
measured
0.8
0.6
0.4
0.2
mode 1
−3
−2
reconst.
0
−1
0
1
2
Horizontal position in mm
−2
−1
0
1
2
Horizontal position in mm
Figure 4.34: (Left) Lateral spatial modes of LD20 at current levels of 40 mA (top) and
55 mA (bottom). (Right) The reconstructed intensity profiles (circles) compared to measured intensity profiles (+’s).
119
Current (mA)
30
34
40
44
55
Relative field weights (%)
mode 0
mode 1
92.6
7.4
77.8
22.2
100
0
70
30
64
36
Relative power weights (%)
mode 0
mode 1
99.4
0.6
92.5
7.5
100
0
84.4
15.6
76
24
Table 4.5: The relative field and power weights of lateral spatial modes of laser LD20.
current levels, shown in Figures 4.34 and 4.35, have an additional dip in the center of the
modes, resulting in an intensity profile with four peaks.
4.7.7 LD30: Laser with 30 m stripe width
The widest stripe width laser that we tested was 30 m and was labeled LD30. The
intensity profiles taken with one fiber as a function of increasing current levels are plotted in Figure 4.36, with the small circles indicating the drive currents used for the spatial
coherence tests.
Current (mA)
40
46
55
Relative field weights (%)
mode 0
1
2
3
94
6
0
0
67.2
20.3 12.5
0
43.8
24.2 18.1 13.9
Relative power weights (%)
mode 0
1
2
3
99.6
0.4
0
0
88.8
8.1 3.1
0
63.4
19.4 10.9 6.3
Table 4.6: The relative field and power weights of lateral spatial modes of laser LD30.
The extracted lateral spatial modes and the reconstructed intensity profiles for LD30 are
shown in Figure 4.37. The three current levels, from top to bottom, are 40 mA, or 1.2 ,
120
mode 0
44 mA
reconst.
1
6
Normalized intensity
Field amplitude (a.u.)
9
3
0
0.8
RHS
0.6
0.4
LHS
0.2
−3
−2
mode 1
−1
0
1
2
Horizontal position in mm
0
−2
−1
0
1
2
Horizontal position in mm
Figure 4.35: (Left) Lateral spatial modes of LD20 at a current level of 44 mA. (Right) The
reconstructed intensity profile (circles) compared to the measured intensity profile (+’s).
46 mA, or 1.4 , and 55 mA, or 1.67 . The relative field and power weights are given in
Table 4.6. For the lowest current level the reconstructed intensity profile matches well with
the measured intensity profile. This is not true for the higher currents, probably due to the
complication of multiple longitudinal modes. There are multiple humps in the modes for
this laser, especially for the current level of 55 mA.
4.8 HeNe laser
In this section we present some results for the HeNe laser that we tested. We took a
large amount of data with this laser, hoping to have a good baseline to validate our experimental process. We also tried various other configurations such as launching into a fiber
or inserting a slit, and a necessary lens, into the beam. Reluctantly we were forced to the
conclusion that the laser was simply not stable enough for reliable tests. Launching into
121
35
55 mA
30
Optical power in nW
50 mA
25
46 mA
20
45 mA
15
40 mA
10
5
0
−3
−2
−1
0
1
2
3
Horizontal position in mm
Figure 4.36: Intensity profiles for laser LD30 for current levels from 35 mA to 55 mA. The
profiles marked with circles indicate the current levels for spatial coherence tests.
a fiber also added severe alignment constraints and the problem of launching conditions
changing over time.
In the following data, the laser was tested after it had been on for a week to stabilize.
The weights extracted during this test are not identical to some of the other results that we
had, although the shapes of the modes were generally the same, apart from noise fluctuations. The relative field weights from this test were 81% in the fundamental mode, 13%
in the first order mode and 6% in the second order mode. The relative power weights are
97%, 2.3%, and 0.7%. The extracted second order mode is very noisy and has only a vague
resemblance to a second order Hermite–Gaussian mode. The other two modes are a better
match to the Hermite–Gaussian model.
122
9
mode 0
1
40 mA
6
0.8
3
0.6
0
0.4
mode 1
−3
−2
0
measured
reconst.
0.2
0
2
−2
0
2
mode 0
46 mA
6
3
mode 1
0
−3
mode 2
−2
0
1
Normalized intensity
Field amplitude (a.u.)
9
0.8
0.6
measured
0.4
0.2
0
2
reconst.
−2
0
2
9
mode 0
1
55 mA
measured
6
3
0.8
mode 2
0.6
0.4
0
−3
0.2
mode 3
mode 1
−2
0
2
Horizontal position in mm
0
reconst.
−2
0
2
Horizontal position in mm
Figure 4.37: (Left) The lateral spatial modes of LD30 at currents of 40 mA (top), 46 mA
(middle), and 55 mA (bottom). (Right) The reconstructed intensity profiles (circles) compared to measured intensity profiles (+’s).
123
1
HG 0
0.8
0.8
Normalized intensity
Normalized field amplitude
1
0.6
mode 0
0.4
0.2
0.6
HG fit
0.4
Reconstructed
0.2
Measured
HG 2
0
−0.2
0
HG 1
mode 2
mode 1
−1
0
1
Position normalized to spot size
−0.2
−1
0
1
Position normalized to spot size
Figure 4.38: (Left) Lateral spatial modes of the HeNe laser extracted by the Jacobian algorithm (circles and x’s) compared to fitted Hermite–Gaussian modes (dashed line). (Right)
The reconstructed intensity profile (circles) compared to the measured intensity profile (x’s)
and the reconstructed Hermite–Gaussian intensity (dashed line).
124
CHAPTER 5
SUMMARY AND CONCLUSIONS
In this chapter we summarize the main points of the dissertation, draw some conclusions
from the experimental results, and give some ideas for future work.
5.1 Summary
In the preceding chapters we have explained our technique of taking spatial coherence
measurements along one axis of a laser beam and extracting the shape and relative weights
of the spatial modes in that axis. We used a twin-fiber interferometer to measure the mutual
intensity for an array of positions in the laser beam. These measurements resulted in a
correlation matrix that could be decomposed into eigenvalues and eigenvectors, giving the
weights and forms of the spatial modes.
Under most measurement conditions, the twin-fiber interferometer can only measure a
partial matrix of correlation data. We explained in Section 2.6 our method of numerical
analysis to extract the weighted eigenvectors from this partial matrix. As we saw from
Figure 4.1, there is a lot of redundancy in a correlation matrix, allowing us to retrieve the
eigenvectors even when some of the matrix is missing. In Section 4.2 we provided the
results of computer simulations we performed to see how well the algorithm worked in the
presence of different levels of noise.
125
In Chapter 4 we presented the extracted spatial modes and reconstructed intensity profiles for eight semiconductor lasers and one HeNe laser. For seven of the lasers we gave
the results at multiple current levels. In the following section we discuss these results and
draw some conclusions.
5.2 Conclusions from experimental results
We tested a total of eight semiconductor lasers, all made from GaAs, with Fabry–Perot
cavities and ridge-waveguide designs. One of the lasers, with a stripe width of 5 m, was
fabricated about two years before the other lasers. The first laser had a single quantum well
of InGaAs with a thickness of 5 nm. The second group of lasers had three quantum wells
that were each 8 nm thick. The confinement layers were also thicker than the confinement
layers of the first laser. The stripe widths for the second group of lasers were 3, 4, 10, 15,
20, and 30 m, with two samples of the 10 m width.
The normal intensity profile expected for a single spatial mode laser is a profile similar
to a Gaussian profile, which has one peak. The intensity profiles of lasers with just a
few modes and a dominant fundamental mode would also be expected to look similar to a
Gaussian profile but wider. In the transverse direction we expected the lasers to be singlemode due to the thin junction. The stripe widths along the lateral direction of the tested
lasers were narrow enough that most were expected to operate in only a few lateral spatial
modes with an almost Gaussian intensity profile.
In the transverse direction, perpendicular to the diode junction, all the semiconductor
lasers had diffraction effects due to the diverging beam being truncated by the mounting
plate or collimating lens. The first laser still operated in a single mode, even though that
mode was similar to a sinc function due to diffraction. A representative sample of the
126
second batch of lasers, LD03, had two transverse modes with a possible third mode at some
current levels. The modes in this laser matched the general pattern of the Hermite–Gaussian
modes. The additional modes for this batch of lasers were due to the wider confinement
layers.
In the lateral direction, parallel to the diode junction, we found two lateral spatial modes
for the first laser, with power weights of 96% and 4%. These modes were very similar to the
fundamental and first-order Hermite–Gaussian modes. Out of the second batch of lasers,
however, all except LD04 and LD15 had intensity profiles and extracted modes that had
secondary peaks on one or both sides of the beam. For lack of a better term we have
referred to these secondary peaks as humps.
One possibility for these unexpected humps is that they were caused by radiation filaments. Filaments are produced by a self-focusing effect, when the intensity of the radiation
creates a secondary waveguide by changing the minority carrier concentration [48]. Typical diameters are given as 2 m to 20 m, which would be too big to explain the effects
in these lasers. Another possible explanation is that there was some type of defect that was
inhibiting radiation next to the humps. If we look at the contour plots of Figure 4.6, the
humps on the right side of each plot look more like islands that were left when some of
the surrounding height had been removed. The view that we get from this figure is that the
intensity levels are generally reduced on the right hand side of the figure. This same type
of pattern was seen in some of the other lasers also, indicating that the fault was not caused
by external damage to a specific laser.
The spatial coherence measurements showed that the dominant spatial modes themselves exhibited these additional humps and they were not the result of additional spatial
modes. If it were not for the humps one would naturally call these modes low-order modes.
127
The extracted modes do not match what one would expect from the Hermite–Gaussian
model. If one were using the Hermite–Gaussian model to analyze these profiles with the
humps, one would add in higher-order modes to account for the peaks.
One surprising result of our research was the two lateral spatial modes for laser LD03,
especially when LD04 only had one. Since the stripe width was only 3 m it seemed narrow
enough to be single mode. It is possible that whatever mechanism caused the intensity dip
or hump also perturbed the resonant cavity resulting in a second mode. Looking back at
Figure 4.24 we see that the second mode is essentially zero for positions left of center,
where there is no unusual dip in the intensity profile. The negative valley and positive peak
in mode 1 only occur on the right side of the beam. The other lasers have wider stripe
widths and could be expected to have multiple lateral modes anyway.
The next wider stripe laser, LD04, had only one lateral mode even for different current
levels. It was asymmetric and so not strictly Gaussian but similar in shape.
Laser LD10 was like LD03 in the extra hump on one side of the intensity profile. It
also had two lateral modes that were somewhat similar to the modes for LD03. Mode 1 did
extend into the left side of the beam in this case.
The intensity profile of LD10 seemed to be a combination of three peaks that changed
relative amplitudes depending on the current level. There were two extracted modes that
each had contributions to the three peaks, contrary to what one would get from the Hermite–
Gaussian model.
In comparing the reconstructed intensity profiles from this laser to the measured profiles, as in Figures 4.28 and 4.29, we see some large discrepancies. This was evidently due
to erroneous weighting and other complications caused by having multiple longitudinal
modes.
128
One important result from this research is that we have to be alert to the presence of
multiple longitudinal modes and how they will reduce overall coherence. When the spatial
modes are extracted during multiple longitudinal mode operation the spatial weights will
be incorrect, even though the shape of the spatial modes should still be the same for the
different axial mode families.
For some of the wider lasers, such as LD10 , LD20, and LD30, which were operating
in multiple longitudinal modes, the reconstructed intensity profiles did not match the measured intensity profiles if the relative weights of the higher-order modes were more than a
few percentage points. When the reconstruction was a bad fit it was generally too sharp in
its minima. This can be clearly seen in the top plots of Figure 4.28 and Figure 4.34.
The measured intensity profiles could be an incoherent superposition of different intensity profiles if the laser was operating in multiple longitudinal modes while the profile was
measured. There could be some steering of the beam peaks as we saw in Figure 4.6. If the
output was hopping back and forth between different peak locations the time-integrating
intensity measurement would incoherently add the multiple profiles. The coherence measurements would only measure the coherent interaction, by definition. This could explain
the discrepancy in the minima of the reconstructed intensity profiles with the measured
profiles.
The other trend we see in the low coherence regions is that fewer spatial modes are
extracted. This is true for LD15 in Figures 4.30 and 4.31. In the first figure the laser
was operating in a single longitudinal mode and three spatial modes were extracted. In the
second figure there were multiple longitudinal modes according to the spectra of Figure 4.9
and only one realistic mode was extracted for each current level. It is possible that this is
due to the inconsistency of the data as the mode hops from one wavelength to another,
129
resulting in lower coherence for some matrix elements and higher coherence for others.
The larger amplitude of the dominant mode allows it to be retrieved but the lower amplitude
modes may be perceived as noise by the numerical analysis.
So we see that our spatial coherence method did successfully extract spatial modes
but was hampered by the condition of multiple longitudinal modes. The results that we
have during single longitudinal mode operation show that the spatial modes extracted give
a more physical picture of what is occurring in the laser than if we relied on fitting the
intensity profile to the Hermite–Gaussian model.
5.3 Recommendations for future work
In this section we list some of the ideas that could be implemented for future work on
using spatial coherence measurements to extract spatial modes of lasers.
It would be instructive to use this technique of extracting spatial modes on a wide
selection of semiconductor lasers. This method could be used in its current form as an
analysis tool to study many different aspects of semiconductor lasers. For instance, one
could test which designs actually produced single spatial modes, as opposed to beams that
looked Gaussian. One could also test how the manufacturing process or quality control
affect the characteristics of the spatial modes. For example, were the intensity humps of
the lasers we tested from defects in the material, flaws in the manufacturing process, or
from contamination after the lasers were made?
Another avenue for future work would be to improve on the current method for extracting spatial modes. Improvements could include: a better way of dealing with multiple
130
longitudinal modes; expanding the testing into two dimensions; and implementing a different type of interferometer that would be inherently faster by taking an array of data at one
time instead of a single point.
The current testing method had no way of dealing with the problem of multiple longitudinal modes other than to set the drive current at a different level. Since some of the
lasers had multiple longitudinal modes at all current levels there was no way to avoid that
operating region. One way to avoid the problem is to use lasers that are known to be single
longitudinal mode. Distributed feedback (DFB) lasers or distributed Bragg reflector (DBR)
lasers suppress most of the longitudinal modes and the lasers operate in a single longitudinal mode [48]. Vertical-cavity surface-emitting lasers (VCSELs) also tend to operate in
single longitudinal mode due to the large free spectral range of the cavity [45]. Another way
of addressing the problem would be to continue to use lasers that may oscillate in multiple
longitudinal modes but to use a grating to separate the longitudinal modes spatially. Then
one could take spatial measurements for one longitudinal mode at a time. The assumption
that the spatial modes are the same for each axial mode family could be tested in this way.
The spatial coherence testing could also be expanded into a two-dimensional process
instead of relying on the assumption of separability of modes. The two hurdles to be overcome would be the length of time 2-D testing would take with the twin-fiber interferometer
and the expansion of the numerical analysis to cover the 2-D case.
A different direction to expand the testing in 2-D would be to implement a usable form
of the lateral shear interferometer. It would have the advantage of taking parallel measurements and so could be expanded more easily to two dimensions compared to the twin-fiber
interferometer. The first step would be to get a lateral shear interferometer working for one
dimension. By using antireflection coated optics that were optimized to the wavelength
131
under test the problem of multiple reflections could be resolved. It would also be necessary
to have a CCD camera or array with a large, linear dynamic range.
132
APPENDIX A
MATLAB M-FILES
In this appendix we include some basic MATLAB files that we have used for the Jacobian algorithm. The data matrix is the difference between maximum and minimum intensities measured by the twin-fiber interferometer as defined in Section 2.5. The orientation
and indexing of the matrix are also defined in that section. The data is typically measured in
nW of power, so we multiply the raw data by
before processing it. This is an important
step before using these algorithms, as there is at least one criterion that is simply a number,
not a ratio, and out-of-range data could give erroneous results.
Sections A.1 and A.2 give the m-files for the Jacobian algorithm. A similar algorithm
we implemented was the adaptable Jacobian algorithm, and the m-file for that algorithm is
given in Section A.3. The two residual vectors are defined as row vectors in the m-files, as
opposed to column vectors in Chapter 2, but they are transposed before being multiplied
by the pseudoinverse matrices.
A.1
M-file Jacob.m
function [modes,moded,nro,nrm,vdv,ndv,ndvo,ndvm,maxit]=...
Jacob(dif,q,p,n,stepfract)
% usage [modes,moded,nro,nrm,vdv,ndv]=...
Jacob(dif,q,p,n,stepfract)
133
%
%
%
%
%
%
%
%
%
%
%
%
%
function file for extracting n Jacobian (modes)
and SVD (moded) modes from a difference matrix, dif,
that is q by q and missing 1+2p diagonals
stepfract determines how much to change v (0 to 1)
nro is the norm of the residual orthogonality vector
nro is the norm of the residual data matching vector
vdv is the max. of dv for each iteration
ndv is the norm of dv for each iteration
ndvo is the norm of dvo (change in v due to orthog.)
for each iteration
ndvm is the norm of dvm (change in v due to data match)
for each iteration
maxit is the largest number of completed iterations
[u,s,v]=svd(dif);
moded=u(:,1:n)*sqrt(s(1:n,1:n));
v=zeros(1,q*n);
for i=1:n
v((i-1)*q+1:i*q)=sqrt(s(i,i))*u(:,i);
end
modes=reshape(v,q,n);
ndv(1)=1;
ndvm(1)=1;
plot(modes,’--’),hold on
for i=1:50;
i
[Jo,Jm,ro,rm]=makeJs(q,v,p,n,dif);
nro(i)=norm(ro);
nrm(i)=norm(rm);
if nro(i) > 45 & i>3,
plot(modes,’ro’),hold off,
i,nro,ndvo,ndvm,ndv,vdv,
error(’orthogonality conditions not being met’),
end
nJm(i)=norm(Jm);
Jop=pinv(Jo);
dvo=Jop*ro’;
dvm=pinv(Jm*(eye(n*q)-Jop*Jo))*(rm’-Jm*Jop*ro’);
ndvo(i)=norm(dvo);
ndvm(i+1)=norm(dvm);
if abs(ndvm(i+1)-ndvm(i)) < .1,
134
stepsize=stepfract;
elseif abs(ndvm(i+1)-ndvm(i)) < 15,
stepsize=abs(ndvm(i+1)-ndvm(i))*stepfract/2;
elseif i>3
plot(modes,’ro’),hold off,
i,nro,nrm,ndvo,ndvm,ndv,vdv,
error(’not converging’)
else
stepsize=stepfract/10;
end
dv=dvo+stepsize*dvm;
ndv(i+1)=norm(dv);
vdv(i)=max(abs(dv));
if vdv(i) < vdv(1)/50000,
i,
plot(modes,’ro’),
maxit=i;’Eureka!’,
break,
elseif n>1 & vdv(i)<0.00002 & i>2
modes=reshape(v,q,n);
plot(modes,’ko’),
maxit=i, ’Stay with current mode’
break,
elseif i >10 & nrm(i)<1.005*min(nrm(i-10:i-1)) ...
& nrm(i)>0.995*min(nrm(i-10:i-1))
modes=reshape(v,q,n);
plot(modes,’ko’),
maxit=i,
’Mode may be oscillating ...
or else this is good enough’
break,
end
v=v+dv’;
maxit=i;
modes=reshape(v,q,n);
plot(modes), hold on
end;
nro,nrm,ndv,vdv
plot(modes,’ro’)
hold off
135
A.2
M-file makeJs.m
function [Jo,Jm,ro,rm]=makeJs(q,v,p,n,D)
% function [Jo,Jm,ro,rm]=makeJs(q,v,p,n,D)
% D is q x q cross-corr matrix with 1+2p missing diags
% n is number of modes wanted
% Jo is (n-1)n/2 x q orthogonality Jacobian matrix
% Jm is ((q-p)ˆ2-(p-q))/2 x nq data match Jacobian
matrix
% ro is (n-1)n/2 orthogonality residual vector
% rm is ((q-p)ˆ2-(p-q))/2 data match error residual
vector
% v is nq vector for storage of v’s
d1=(n-1)*n/2;
Jo=zeros(d1,n*q);
ro=zeros(1,d1);
m=1;
for i=1:n-1;
for j=i+1:n;
for k=1:q;
Jo(m,(i-1)*q+k)=v((j-1)*q +k);
Jo(m,(j-1)*q+k)=v((i-1)*q +k);
end
ro(m)=-v((i-1)*q+1:(i-1)*q+q)...
*v((j-1)*q+1:(j-1)*q+q)’;
m=m+1;
end
end
d2=((q-p)ˆ2-(q-p))/2;
d3=n*q;
Jm=zeros(d2,d3);
rm=zeros(1,d2);
m=1;
for i=1:q-p-1;
for j=i+p+1:q;
sums=0;
for k=1:n;
Jm(m,(k-1)*q+i)=v((k-1)*q +j);
Jm(m,(k-1)*q+j)=v((k-1)*q +i);
136
sums=sums+v((k-1)*q+i)*v((k-1)*q+j);
end
rm(m)=D(i,j)-sums;
m=m+1;
end
end
A.3
M-file adJacob.m
function [modes,nro,nrm,vdv,ndv,ndvo,ndvm,maxit]=...
adJacob(dif,moded,q,p,n,stepfract)
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
usage [modes,nro,nrm,vdv,ndv,ndvo,ndvm,maxit]=
adJacob(dif,moded,q,p,n,stepfract)
function file for extracting n Jacobian (modes)
using 10 SVD (moded) modes with a difference matrix,
dif, that is q by q and missing 1+2p diagonals
stepfract determines how much to change v
nro is the norm of the residual orthogonality vector
nro is the norm of the residual data matching vector
vdv is the max. of dv for each iteration
ndv is the norm of dv for each iteration
ndvo is the norm of dvo (change in v due to orthog.)
for each iteration
ndvm is the norm of dvm (change in v due to data match)
for each iteration
maxit is the largest number of completed iterations
v=zeros(1,q*n);
for i=1:n
v((i-1)*q+1:i*q)=moded(:,i);
end
for n1=1:n;
clear maxit
modes=zeros(q,n1);
maxloop=50;
ndv=[1 zeros(1,maxloop)];
vdv=[1 zeros(1,maxloop)];
ndvo=[1 zeros(1,maxloop)];
ndvm=[1 zeros(1,maxloop)];
nrm=zeros(1,maxloop);
137
nro=zeros(1,maxloop);
for i=1:maxloop;
i,n1
modes=reshape(v(1:q*n1),q,n1);
plot(modes,’--’), hold on
[Jo,Jm,ro,rm]=makeJs(q,v(1:q*n1),p,n1,dif);
nro(i)=norm(ro);
if nro(i) > 55 & i > 3,
plot(modes,’ko--’),hold off,
i,nro(1:i),nrm(1:i),ndvm(1:i-1),ndv(1:i-1)
error(’orthogonality conditions not being met’),
end
nrm(i)=norm(rm);
nJm(i)=norm(Jm);
Jop=pinv(Jo);
dvo=Jop*ro’;
dvm=pinv(Jm*(eye(q)-Jop*Jo))*(rm’-Jm*Jop*ro’);
ndvo(i+1)=norm(dvo);
ndvm(i+1)=norm(dvm);
nv(i)=norm(v((n1-1)*q + 1:(n1-1)*q + q));
if abs(ndvm(i+1)-ndvm(i)) < .1,
stepsize=stepfract
elseif i > 4
plot(modes,’ro’),hold off,
i,nro(1:i),nrm(1:i),
ndvo(1:i+1),ndvm(1:i+1),
ndv(1:i+1),vdv(1:i),
error(’not converging’)
else
stepsize=stepfract/10;
end
dv=dvo+stepsize*dvm;
ndv(i+1)=norm(dv);
vdv(i)=max(abs(dv));
if vdv(i) < vdv(1)/50000
modes=reshape(v(1:q*n1),q,n1);
plot(modes,’ko’),
maxit=i,’Eureka!’
break,
elseif n>1 & ndv(i)<0.00002 & i>2
138
modes=reshape(v(1:q*n1),q,n1);
plot(modes,’ko’),
maxit=i, ’Stay with current mode’,pause,
break,
elseif i >10 & nrm(i)<1.00005*min(nrm(i-10:i-1))...
& nrm(i)>0.99995*min(nrm(i-10:i-1))
modes=reshape(v(1:q*n1),q,n1);
plot(modes,’ko’),
maxit=i,
’Mode may be oscillating or else ...
this is good enough’,
break,
end
v(1:q*(n1-1)+ q)=v(1:q*(n1-1)+q)+dv’;
maxit=i;
modes=reshape(v(1:q*n1),q,n1);
plot(modes), hold on
end;
nro(maxit-2:maxit),nrm(1:maxit),
vdv(maxit-2:maxit),ndv(1:maxit)
plot(modes,’ko’)
hold off
end
139
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