Download Active Modelocking of an Open-Cavity Helium

Active Modelocking of an Open-Cavity Helium-Neon Laser
A Thesis
Presented to
The Division of Mathematics and Natural Sciences
Reed College
In Partial Fulfillment
of the Requirements for the Degree
Bachelor of Arts
Sam W. Spencer
May 2010
Approved for the Division
(Physics)
Advisor L. Illing
Acknowledgements
I want to begin by thanking Lucas Illing for his continual guidance, clarifications,
arbitration and reassurance during the chaotic and humbling thesis process, and for
helping me remember that not everything needs explanation. There are so many
others who deserve acknowledgement: David Griffiths, for showing me that clarity of
description begets clarity of understanding; Nicholas Wheeler, for teaching me that
physics can be communicated eloquently; Joellyn Johnson, for teaching me how to
move on; Robert Knapp, for letting me realize that an essay is not a proof; Mary
James, for introducing me to wonderful truths; Chris Thoen, for helping me feel
confident; and the 2006 MacNaughtonites and the 2010 physics students for endless friendliness. I want to thank my parents and sister for their conversations and
support. Most importantly, I thank Elisabeth Hawks for making me happy and for
unconditional love.
Table of Contents
Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 1: Theory . . . . . . . . . . . . . . . . . . . . . .
1.1 Laser Resonator Stability . . . . . . . . . . . . . . . .
1.2 Helium-Neon Gain Medium . . . . . . . . . . . . . .
1.3 Acoustooptic Modulator . . . . . . . . . . . . . . . .
1.3.1 Standing vs. Traveling Acoustic Wave AOMs
1.3.2 Diffraction Regimes and Traveling Wave AOM
1.4 Time-domain modeling: Pulse Propagation . . . . . .
1.4.1 Gaussian Pulses . . . . . . . . . . . . . . . . .
1.4.2 Pulse Propagation . . . . . . . . . . . . . . .
1.5 Injection Locking . . . . . . . . . . . . . . . . . . . .
1.5.1 Locked Regime: ωD ≤ . . . . . . . . . . . .
1.5.2 Unlocked Regime: ωD > . . . . . . . . . . .
1.5.3 Consequences of Injection Locking . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
3
7
11
13
13
14
19
19
20
24
26
26
26
Chapter 2: Experimental Design . . . . . . . . . . . . . . . . . . . . . . .
29
Chapter 3: Results and Discussion
3.1 Unmodulated Operation . . . .
3.2 Approaching Modelocking . . .
3.3 Modelocking Behavior . . . . .
.
.
.
.
35
35
41
44
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
Theory
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1
.
.
.
.
Abstract
In this thesis, we investigate the modelocking of a laser, which refers to the simultaneous excitation of multiple modes of light within the laser cavity, and the establishment of a constant uniform phase difference between pairs of these modes, resulting
in the formation of laser pulses. We discuss the physical mechanisms that lead to
modelocking, beginning with descriptions of the necessary laser components and presenting a mathematical model for propagating a Gaussian laser pulse through these
components. We also discuss the nonlinear phenomenon of injection locking and the
role it plays in making modelocking an experimental reality. We then experimentally demonstrate modelocking using a Helium-Neon laser tube with external cavity
mirrors and an acoustooptic modulator. The laser is found to produce pulses of approximately 2 nanoseconds in duration, at a repetition rate of about 160 MHz and
an average output intensity on the order of 110 nanowatts. We present results of the
experiment, including light-intensity timeseries, optical spectra, and radio frequency
spectra of the beat frequencies between modes, to characterize modelocking behavior
and the injection locking process leading to modelocking.
Introduction
Generating short pulses of light from lasers has been an area of intense interest since
the early 1960s, soon after the laser was invented. Short, intense laser pulses have
acquired hundreds of applications in both industry and research roles. Examples in
industry include laser ablation and micromachining [Clark-MXR, 2009], refractive
index variation including lens aberration correction in the human eye [Ding et al.,
2009], high speed fiber optic telecommunications [Wada, 2004], and the generation of
terahertz radiation for security screening [Federici et al., 2005]. Short laser pulses are
also essential in research projects like nonlinear optics studies [Boyd, 2008], inertial
confinement fusion [Pfalzner, 2006], high-harmonic generation [Ferray et al., 1988],
and the time-resolved spectroscopy of chemical reactions [Manescu, 2004].
Many different methods have been developed for extracting short pulses from a
laser, including Q-switching, gain switching, and cavity dumping, each of which relies
on electronic modulation to quickly regulate the output power of the laser [Siegman,
1986]. These techniques typically generate pulses on nanosecond (10−9 s) or picosecond (10−12 s) timescales and rely on the speed of variation of an electrical signal
to form pulses. There is another method, called modelocking, that takes advantage
of the fact that a laser cavity can naturally support multiple frequencies of light simultaneously, and these frequencies, if correctly superimposed, can beat together to
form short pulses. First discussed by Lamb [1964] and first implemented by Hargrove
et al. [1964], modelocked lasers have become the premier choice for the generation of
short, intense laser pulses. Modelocked lasers usually generate ‘ultrafast’ pulses on
the timescale of femtoseconds (10−15 s), with the current record held at pulses under
10 femtoseconds.
This thesis documents an experimental attempt to construct a modelocked laser
using a Helium-Neon laser tube and free-space optical components. We will begin by
introducing the theory of modelocking and give theoretical details about the components with which we will assemble our laser. We will then motivate the design of the
experiment itself. Finally, we will present data obtained while operating the laser,
and analyze our results.
Chapter 1
Theory
This chapter gives an outline of the theory of laser modelocking and an introduction
to the experimental apparatus, including the mathematical description of its components. Thus we will begin with a general description of lasers and the physical
principles behind modelocking, and outline the two broad types of modelocking.
Gain Medium
Mirror Cavity
Pump Process
Figure 1.1: The schematic for a generic laser, including the three main components and a
single standing-wave beam of light.
Every laser requires three main components: a gain medium, an energy pumping
process, and an optical cavity. A gain medium is a substance which is quantummechanically excited into higher energy levels. This energy is released as light, either
randomly in the process of spontaneous emission, or by stimulated emission due to
the passing of light through the substance. Stimulated emission generates photons
that have both the same phase and same frequency as the stimulating photons. An
optical cavity, also called a laser resonator, is a set of mirrors that surrounds the gain
medium and reflects light back into it. If this reflected light encounters the excited
portion of the substance, it can stimulate the emission of more light and generate
a coherent beam. However, if the light instead encounters an unexcited portion of
the substance, the photons may simply be reabsorbed. In order to ensure that more
light is spontaneously emitted than absorbed, an energy pumping process is used to
excite as much of the substance in the gain medium as possible.
Most lasers operate in the ‘continuous wave’ (CW) regime, where operation in the
4
Chapter 1. Theory
Cavity longitudinal mode structure
Power
(a)
Frequency
Power
(b)
Laser Gain
Cavity Loss
Frequency
(c)
Power
Output Spectrum
Frequency
Figure 1.2: (a) The laser cavity supports an infinite Dirac delta frequency comb. (b) The
gain of the laser gain medium (solid curve) has a Lorentzian or Gaussian shape, and must
overcome the cavity losses (dashed curve) for lasing to occur. (c) Overlaying the region
where gain is greater than loss onto the Dirac comb, we end up with a series of narrow
emission bands bounded by a wider envelope.
steady state implies generating a beam with constant frequency and intensity. The
light emitted by a CW laser is ideally monochromatic. The frequency range over which
most gain media amplify, called the gain bandwidth, is usually large enough such
that multiple closely-spaced frequencies may oscillate simultaneously. Imagine a laser
cavity of optical path length L, formed by two inward-facing highly reflective mirrors.
This cavity will support standing light waves of certain evenly-spaced wavelengths,
given by L = mλ
, where m is a positive integer. The frequency corresponding to
2
this standing wave is ν = mc
, implying that the difference in frequency between the
2L
adjacent m and m + 1 modes is given by
∆ν =
c
.
2L
(1.1)
This forms an evenly-spaced comb of Dirac delta peaks in frequency space, representing ‘axial modes’ that will be supported by the cavity. We thus call ∆ν the ‘axial
mode spacing.’ If the cavity is long enough (and thus the allowed modes close enough
in frequency), then multiple axial modes will lie within the laser’s gain bandwidth.
5
Some of these modes will experience enough gain to overcome the losses that they
experience when circulating in the cavity. See Fig. 1.2. In some lasers, called ‘inhomogeneously broadened’ lasers1 , these axial modes can lase simultaneously. Conversely,
in ‘homogeneously broadened’ lasers2 , normal circumstances will allow only one axial
mode to lase at a time, even if multiple modes fall within the region where the gain is
greater than cavity losses. In either case, if multiple axial modes can be forced to lase
simultaneously, and if certain phase relationships between the modes are maintained,
then the modes will beat together to form laser pulses.
Suppose a laser has been forced to lase in multiple axial modes simultaneously.
If each neighboring pair of axial modes has the same relative phase shift, then all
modes will beat together to form pulses. If the mode pairs continue to have the same
relative phase shift over time, then the pulses will remain. We can call this condition
‘constant uniform mode pair phase difference,’ or we can simply say that the laser
is ‘modelocked’. Maintaining a constant uniform mode pair phase difference is the
primary challenge of modelocked laser design. This challenge arises because the phases
of a free-running laser’s modes are typically not linked in any meaningful way. The
relative phases of the modes may fluctuate randomly with respect to one another over
time, a phenomenon called phase noise, which is due to quantum noise, mechanical
vibrations, and temperature fluctuations. Experimentalists cannot hope to eliminate
phase noise completely, but if the noise for each mode is correlated, then pulses will
still form, and we can still consider this state modelocking (see Fig. 1.7) [Wollenhaupt
et al., 2007].
In a practical laser, part of the power of the circulating pulses will be coupled out
of the cavity. The frequency at which pulses are emitted from the cavity, which we
call the ‘round-trip frequency,’ is
νRT =
c
.
2L
(1.2)
Note that this is equal to the axial mode spacing, ∆ν. Thus lasers with long cavities,
and therefore close axial modes, will rapidly emit pulses.
Modelocked lasers fall into two primary classifications, based on the technique used
to bring the axial modes into phase. Some modelocked lasers employ an electronicallydriven modulator to periodically change the cavity loss or frequency of light; this
technique is called active modelocking, because the modulator actively shifts optical
power between axial modes. Loss modulators achieve modelocking by varying the
cavity loss at the round-trip frequency, so that the wings of the pulse experience loss
while the pulse peak does not, and thus the pulse is shortened. Frequency modulators,
on the other hand, the kind used in this thesis, shift part of the light passing through
1
This means that different atoms in the gain medium experience different frequency-shifting
effects, so that different atoms have gain spectra that are slightly shifted relative to one another.
In the HeNe and other gas lasers, this is mostly due to Doppler shifts that arise from the velocity
distribution of the atoms [Siegman, 1986, Section 12.2, pp. 462-465; Section 30.3, pp. 1185-1186].
2
Homogeneous broadening means that all atoms in the gain medium experience the same
frequency-shifting effects, so the gain spectra of the atoms lie on top of one another. In homogeneously broadened lasers, like those with solid-state gain media, the lasing of one mode ‘saturates’
the gain of other modes. We will discuss gain saturation in more detail later.
Chapter 1. Theory
PSD
Power, E
6
0
t
Ν
Figure 1.4: All modes in phase. Ideally,
with a constant mode pair phase difference,
the laser generates a Gaussian pulse train.
The thick dashed line represents power, and
the thin solid line represents electric field.
Power, E
Power, E
Figure 1.3: Power spectral density
(PSD) of a multimode laser. Simulated
as a Dirac delta comb with a Gaussian envelope. The following figures show resulting
waveform for different phase relationships.
0
0
t
t
Figure 1.6: Same phase noise. Although
the phase of the electric field is noisy, pulses
are still formed in the power waveform and
we have modelocking.
Power
Power, E
Figure 1.5: Random phases, no noise.
If phases are random but fixed in time,
then the waveform is deformed but periodic. Mode pair phase differences are constant but nonuniform.
0
t
0
Figure 1.7: Correlated phase noise.
t
Pulses will form, but the waveform will not
be clean. This is the best that experimen- Figure 1.8: Independent phase noise.
talists can accomplish under normal circum- Waveform will be noisy. This is the case
stances.
in a multifrequency continuous wave laser.
1.1. Laser Resonator Stability
7
them to a higher or lower frequency. This shifted light can excite other axial modes,
and, by a process we will discuss later, can even establish the fixed mode pair phase
difference necessary for modelocking.
The other class of modelocked lasers, called passively modelocked lasers, involve
a device that modulates light without being driven. Passive modelocking usually
implies the use of a ‘saturable absorber’: a substance that will absorb light up to a
certain intensity, and transmit more intense light. Like an active loss modulator, this
has the effect of absorbing the wings of a pulse yet letting the pulse peak through.
The advantage of the saturable absorber is that it can react very quickly to intensity
variations, and thus can produce pulses of shorter duration than active loss modulators, which are limited by their driving frequency. It is passively modelocked lasers
that hold the records for shortest pulse duration; Kerr-lens passive modelocking of
Ti:sapphire lasers have produced pulses of under 6 femtoseconds, spanning less than
two optical cycles [Sutter et al., 1999]. Despite the advantages of passive modelocking,
actively modelocked lasers are arguably conceptually simpler, and, with most of the
resources available on-hand in the Reed physics department, the actively modelocked
laser was chosen for this experiment.
In the following sections of this chapter, we will describe in turn the major components of the laser assembled in this thesis. We will begin by examining the laser
cavity and the conditions necessary for the support of a laser beam between the
cavity mirrors. We will then turn to our gain medium, the Helium-Neon discharge
tube, and describe how the medium amplifies light by stimulated emission. With an
understanding of how the gain medium generates coherent light, we will move on to
the active modulator employed in the laser, an acoustooptic modulator, and explain
how the device works as a frequency modulator to bring the axial modes into phase.
With the theory of the components laid out, we will then model the operation of
the modelocked laser by propagating a Gaussian electromagnetic pulse through each
laser component. This argument will make some idealizing assumptions, including
the assumptions that our frequency modulator can be tuned perfectly and that phase
noise is negligible. However, we will see that because the laser system is not perfectly
linear, the reality of imperfect tuning and the presence of phase noise still allow for
modelocking to occur.
1.1
Laser Resonator Stability
Laser resonators, or laser cavities, come in a variety of styles. The simplest case of
a linear optical cavity consists of two spherical mirrors: a ‘high reflector’ (HR) that
maximizes the reflected light, and an ‘output coupler’ (OC) that transmits a small
fraction of the light incident on it. Such a cavity can be designed to support a beam
that is Gaussian in geometry (with transverse intensity described by the Gaussian
function). If a Gaussian beam can be reflected in on itself perfectly at both ends of
the cavity, essentially none of the light in the beam will escape out of the sides of the
8
Chapter 1. Theory
L
Beam Waist
R1
R2
Z1
Z=0
Z2
Figure 1.9: A linear laser cavity with two spherical mirrors, stably supporting a Gaussian
beam. The cavity length, mirror radii, beam waist and distances to mirrors are all indicated.
R1
R2
L < R1
L < R2
Stable
L
L < R1
L > R2
Unstable
L > R1
L > R2
Stable
L < R1+R2
Stable
L = R1+R2
Unstable
L > R1+R2
Figure 1.10: Various conditions under which a Gaussian beam may or may not be supported
by a cavity.
1.1. Laser Resonator Stability
9
cavity. Such a cavity is said to stably support the Gaussian beam.3 Whether a cavity
is stable, and thus able to support a Gaussian beam, depends on its geometry: the
length of the cavity and the curvatures of the mirrors. Consider a cavity composed
of two concave mirrors, with radii of curvature R1 and R2 respectively. We place the
z axis along the optical axis, with z = 0 at the beam waist, the narrowest portion
of the beam. The length of the cavity can be expressed as the sum of the distance
from the R1 mirror to the beam waist, called z1 , and from the beam waist to the R2
mirror, called z2 :
L = z2 − z1 .
(1.3)
Note that z1 is defined to be negative, because it runs from the beam waist inside the
cavity to the mirror R1 , opposite to the direction of the optical axis [Siegman, 1986].
This causes L to always be a positive quantity. See Fig. 1.9.
Suppose that we are given two mirrors, of curvature R1 and R2 , and we want to
find the cavity lengths that will stably support a Gaussian beam of wavelength λ.
To make sure that the beam is stable within the cavity, we need the curvature of
the wavefronts to match the curvature of the mirrors, so that the mirrors reflect the
wavefronts perfectly. Following Chang [2005, p.50], we first write down, for the case
at hand, formulae relating the Gaussian beam curvature to z,
zR2
,
z1
z2
R2 = z2 + R ,
z2
−R1 = z1 +
(1.4)
(1.5)
πw2
where zR = λ 0 is the Rayleigh range, and w0 is the radius of the beam waist. The
minus sign in Eq. 1.4 is needed to settle a discrepancy in conventions. Gaussian
wavefront curvature is taken to be negative for a converging beam, traveling in the
direction of the optical axis. Mirror curvatures, on the other hand, are positive for
concave mirrors (looking out from the inside of the cavity). The minus sign allows
us to relate a negative wavefront curvature to a positive mirror curvature [Siegman,
1986].
We can solve Eq. 1.4 and 1.5 for z1 and z2 ,
q
−R1 1
±
R2 − 4zR2 ,
(1.6)
z1 =
2
2q 1
R2 1
z2 =
±
R22 − 4zR2 .
(1.7)
2
2
and plug them into Eq. 1.3, yielding a formula for the cavity length:
q
q
R2 1
2
2
2
2
L=
±
R2 − 4zR − R1 − 4zR .
2
2
3
(1.8)
Note that so-called unstable cavities can still lase. Such laser cavities are usually designed so
that light will leak out around one of the cavity mirrors. See Siegman [1986] for more details.
10
Chapter 1. Theory
g1 g2
1.0
0.8
0.6
0.4
0.2
20
40
60
80
100
L
Figure 1.11: Stability ranges of L. A graph of the product g1 g2 for a cavity with a
60 cm high reflector and a 45 cm output coupler. The upper stability limit is shown as a
dashed line. The cavity will be stable if 0 < L < R1 or R2 < L < R1 + R2 , so in this case,
0 < L < 45 cm or 60 cm < L < 105 cm.
Some algebra and squaring leads to the following expression for zR :
zR2 =
L (L − R1 ) (L − R2 ) (R1 + R2 − L)
.
(−2L + R1 + R2 )2
(1.9)
The Rayleigh range is positive real by definition, so the right hand side of Eq. 1.9
must be positive as well. The denominator is always positive, by the reality of the
variables involved. The numerator must also be positive, which places restrictions on
the values of R1 and R2 . Either two of the parenthesized terms in the numerator are
negative, or none of them are. This implies two possible inequalities, which represent
two separate cases:
1. R1 + R2 ≥ L ≥ max (R1 , R2 ),
2. min (R1 , R2 ) ≥ L > 0.
Physically, this means that a cavity will be stable in two cases: either the centers of
curvature of both mirrors lie within the cavity, or both centers lie beyond the cavity.4
See Fig. 1.10 for an illustration of these conditions.
A single, useful
inequality
thatapplies for both cases can be derived algebraically.
L
Define g1 = 1 − R1 and g2 = 1 − RL2 . In case 1, the quantities g1 and g2 are
both negative, so their product must be positive. Likewise, in case 2, g1 and g2 are
both positive, and the product must again be positive:
g1 g2 > 0.
(1.10)
Also, note that the length L and curvatures R are always positive, so g1 and g2 must
each be less than 1, and thus so must their product:
1 > g1 g2 .
4
(1.11)
In the case that the curvatures are equal, it is possible for R1 = R2 = L, so both cases are
satisfied simultaneously.
1.2. Helium-Neon Gain Medium
11
Together, Eqs. 1.10 and 1.11 form the ‘stability equation’ for laser resonators:
1 > g1 g2 > 0.
(1.12)
With the mirror curvatures R1 and R2 in hand, we can use Eq. 1.12 to place boundaries
on the length of the cavity. Graphing g1 g2 as a function of L shows that Eq. 1.12 is
equivalent to cases 1 and 2 above: the cavity is stable up to the radius of curvature
of the first mirror, and again from the raius of the second mirror to the sum of the
radii. See Fig. 1.11 for a graph with the mirror curvatures used in this thesis.
1.2
Helium-Neon Gain Medium
This thesis utilizes the common helium-neon gas discharge tube as a gain medium. As
explained in Siegman [1986], the tube is filled with a low-pressure mixture of helium
and neon gas, with about ten times as much helium as neon. The potential difference
between anode and cathode at opposite ends of the tube is over 1000 volts. This
voltage accelerates electrons that collide with the helium atoms, exciting them to the
long-lived 21 S state. If an excited helium atom collides inelastically with a ground
state neon atom, the neon atom can be excited to the 3s state, which has a 632.8 nm
laser transition to a 2p state. Photons of this frequency that interact with excited
neon atoms can stimulate the emission of additional photons of identical frequency
and phase. After many round-trips through the cavity, the photons begin to form a
coherent beam.
Consider the light circulating in a laser cavity as it passes by the gain medium.
The amount by which light is amplified depends critically on frequency, so we express
gain mathematically in terms of frequency. Transforming the electric field of the
light, Ẽ(t),to the frequency domain, Ẽ(ω), we can approximate the effect of the gain
medium by multiplying Ẽ(ω) by the transfer function g(ω):5
αp
.
g(ω) = exp
2 + 4i(ω − ω0 )/∆ω0
(1.13)
Here α is the laser gain coefficient, which is less than 1, p is the gain medium
length, ω0 is the peak laser transition frequency, and ∆ω0 is the full-width halfmaximum (FWHM) gain bandwidth. For the 632.8 nm HeNe transition, values are
0.02 cm−1 < α < 0.1 cm−1 (typically), ω0 ≈ 4.737 × 1014 Hz, and ∆ω0 ≈ 1.56 × 109
Hz (as found by Niebauer et al. [1988]). The real part of Eq. 1.13 is
pα∆ω02
Re(g(ω)) = exp
,
2 (∆ω02 + 4 (ω − ω0 ) 2 )
5
(1.14)
This expression reproduced from Siegman [1986, p.1062], though a factor of 1/2 is introduced as
Siegman’s expression describes an entire round trip, and thus propagation through twice the cavity
length L. Note that atomic collisions and Doppler broadening are ignored in this formula for the
sake of simplicity; these phenomena cause broadening of the gain lineshape.
12
Chapter 1. Theory
ReHgHΩLL
1
0
Ω - Ω0
Figure 1.12: Small-signal gain vs. detuning from resonance. Dependence of laser
gain on the detuning of the incident field from a resonance of the gain medium. Ignoring
line broadening, the lineshape is Lorentzian.
representing the small-signal gain, or the maximum gain of light passing through
the gain medium at frequency ω. A representative plot of this function for an arbitrary gain medium can be seen in Fig. 1.12. This shows that only a narrow band of
frequencies are amplified by the medium.
The gain values given by Re(g(ω)) are those experienced only by very small signals,
and in general, the gain that any particular laser mode experiences is not constant;
otherwise, the mode would continue to gain power forever. Instead, as the mode
increases in intensity, the gain ‘saturates,’ decreasing until it equals the cavity losses,
at which point the mode intensity equals its steady-state value. This saturation is
due to the fact that the atoms in the gain medium need a finite amount of time
to become re-excited. Consider a single mode of light at an intensity greater than
the steady-state intensity. The presence of this light stimulates emission from the
gain atoms more quickly than the atoms can become re-excited by the pump process.
Meanwhile, losses continue at a constant rate and the mode intensity decreases, so at
this instant the gain is less than the losses. As the mode loses intensity, the gain will
increase until it exactly equals the cavity losses. This shows that lasing in the steady
state is an equilibrium point in the dynamics of a laser.
The imaginary part of Eq. 1.13 is
pα∆ω0 (ω − ω0 )
,
(1.15)
Im(g(ω)) = exp −
∆ω02 + 4 (ω − ω0 ) 2
representing the phase shift introduced to each frequency. The gain medium is thus
dispersive, and we will see in a later section that a light pulse will be broadened upon
interacting with the gain medium. A representative plot is given in Fig. 1.13.
The bandwidth of the gain medium is relevant to modelocked laser design, since
the bandwidth limits the minimum duration of pulses produced by the laser. The
minimum pulse duration is approximately equal to the inverse gain bandwidth. For
the HeNe gain medium, with a gain bandwidth of ∆ω0 = 1.56 GHz, this estimation
yields a minimum pulse duration of about 640 picoseconds. The current standard
for research-grade ultrafast pulse generation is the titanium-doped sapphire crystal.
1.3. Acoustooptic Modulator
13
ImHgHΩLL
1
0
Ω - Ω0
Figure 1.13: Phase shift vs. detuning from resonance. Phase shift induced by a gain
medium as a function of the detuning of the incident field from a resonance of the gain
medium.
With a gain bandwidth of about 128 GHz, it can generate pulses lasting under 10 femtoseconds. Comparatively, the HeNe is a relatively mediocre choice of gain medium,
but it is much cheaper and easier to set up.
1.3
Acoustooptic Modulator
This section is intended to describe the variety of acoustooptic modulators (AOMs)
used in active modelocking, and to present a theory of how the AOM in this thesis
interacts with laser light.
An acoustooptic modulator is a mechanical device that consists of a piezoelectric
transducer attached to a transparent crystal. The transducer sends radio-frequency
acoustic vibrations into the crystal, and the compression waves induce changes in
index of refraction to form a ‘phase grating’ off of which laser light can diffract.
Depending on various operational and fabrication parameters, this device can act
either like a standard diffraction grating that produces many diffracted beams, or
as a Bragg scatterer that produces a single diffracted beam at a specified angle, or
the behavior can lie somewhere in between. The distinction between these regimes
will be discussed in a following subsection. In either case, light from the principle
beam is diffracted into one or more higher-order beams. A second and arguably more
important distinction between classes of AOMs is whether the acoustic wave is a
standing wave or a traveling wave.
1.3.1
Standing vs. Traveling Acoustic Wave AOMs
The standing wave AOM creates a stationary phase grating that oscillates in intensity.
Sending a laser through this type of AOM causes the beam to undergo diffraction
periodic in time. Such an AOM can be placed inside the laser cavity and used as a
loss modulator. At zero amplitude of the stationary wave, the beam passes through
unimpeded, but, as the amplitude increases, some of the incident optical power is
14
Chapter 1. Theory
redirected into the diffracted beams, which escape out of the sides of the cavity,
resulting in a loss. If this periodic loss modulation occurs at the round-trip frequency,
a short pulse can form that coincides with low-loss periods. This is because every
time the pulse reaches the AOM, the pulse wings are subject to more loss than the
pulse peak, shortening the pulse. This technique is called modelocking by amplitude
modulation.
The traveling wave AOM, the type employed in the experimental section of this
thesis, creates a moving phase grating. A laser beam passing through this AOM will
constantly be diffracted by the grating, but the traveling nature of the grating causes
higher and lower diffraction orders to become frequency-shifted by various amounts.
This can be thought of, as will be explained in a following subsection, as the addition
(or subtraction) of the energy and momentum of the acoustic waves to the optical
wave. The frequency of each successive diffraction order is shifted by another multiple
of the acoustic wave frequency. The upshot is that these frequency-shifted beams are
generated in phase with the incident beam. If the acoustic frequency is set to equal
the round-trip frequency, then diffracted beams that are reflected back into the laser
cavity will resonate within it as axial modes, and proper alignment can ensure that
the modes stay in phase as light circulates within the cavity. Thus, a traveling wave
AOM can be used to bring the axial modes of a laser into phase. This technique is
called modelocking by frequency modulation. With this application in mind, we will
now turn to focus on the theory of operation of a traveling wave AOM and the use
of this device for shifting the frequency of light.
1.3.2
Diffraction Regimes and Traveling Wave AOM Theory
Based on how an AOM is constructed and employed, the device can operate in two
diffraction regimes: the Bragg regime and the Raman-Nath regime. In the Bragg
regime, light passing through a thick AOM crystal interacts with many acoustic
wavefronts to form a single, intense diffracted beam that only appears at certain
angles of incidence. This is analogous to the diffraction of X-rays off of crystal layers;
thus the reference to Bragg. In the Raman-Nath regime, a relatively thin AOM acts
as a standard diffraction grating, generating many higher-order beams regardless of
incident angle. Although the AOM used in this thesis operates somewhere between
the regimes, it operates closer to the Raman-Nath (diffraction grating) regime, so the
theory of that regime will be discussed in detail.6
Under most circumstances, the regime in which an AOM operates is determined
by geometrical factors7 . For Λλd2 1, the AOM is in the Bragg regime, and for Λλd2 < 1,
we are in the Raman-Nath regime. Here λ is the optical wavelength of incident light,
Λ is the wavelength of the acoustic wave, and d is the width of the crystal.
To understand the origin of this condition, consider first the Bragg diffraction
6
This theory follows Boyd [2008, Section 8.4], who gives detailed descriptions of both diffraction
regimes. The Bragg regime is described in Section 8.4.1, while the Raman-Nath regime is discussed
in Section 8.4.2.
7
For a more accurate method for distinguishing between regimes, the degree of modulation of the
index of refraction must be taken into account. See Moharam and Young [1978].
1.3. Acoustooptic Modulator
15
θ
L
L
Λ
Figure 1.14: Bragg diffraction. A plane wave scatters off of the layers in a crystal. If the
incident angle is correct for constructive interference, the ray diffracted by the lower crystal
face will travel an extra distance 2L = 2Λ sin θ = mλ, where m is an integer, so that the
rays scattered by the upper and lower planes are in phase.
regime, shown in Fig. 1.14: interference between reflections from multiple acoustic
waves generates a diffracted beam, the angle of which is specified by the relative
wavelengths of the optical and acoustic waves:
sin θ =
mλ
,
2Λ
(1.16)
where θ is the half-angle of the diffracted beam away from the acoustic plane, and
m is the diffraction order. Note that the efficiency of diffraction should increase if
the laser beam encounters more acoustic wavefronts before exiting the AOM. Thus
the intensity of the diffracted beam will be greatest if the transverse distance covered
by the beam inside the AOM is much larger than the acoustic wavelength. The
transverse distance, z is found geometrically by examining Fig. 1.15:
z = d tan θ.
(1.17)
Encountering multiple wavefronts requires that z Λ.
Here d is the width of the AOM. In most cases, θ is small, so we can say that
sin θ ≈ tan θ ≈ θ. In this approximation we can combine Eqs. 1.16 and 1.17 and
simplify the result to find the condition that must hold for Bragg scattering to occur:
λd
1.
Λ2
(1.18)
If instead the AOM is thin enough that the beam only encounters one acoustic
wavefront within the AOM, we will have Raman-Nath scattering:
z = d tan θ < Λ.
This is shown in Fig. 1.15.
(1.19)
16
Chapter 1. Theory
d
z = d tan θ θ
Λ
Figure 1.15: Requirements for Raman-Nath diffraction in a traveling-wave AOM.
The wavelength of the acoustic waves, Λ, must be greater than the transverse distance
covered by the light beam through the AOM, d tan θ. This ensures that the beam does not
diffract off of multiple wavefronts.
It will be shown that Raman-Nath scattering generates diffracted beams at angles
similar to a diffraction grating. Assuming for now that this is the case, we can combine
the above inequality with the diffraction grating equation,
sin θ =
mλ
,
Λ
(1.20)
to obtain, for the first order diffracted beam (m = 1), again in the small-angle
approximation,
λd
< 1.
Λ2
(1.21)
Thus the relative dimensions of the AOM and the wavelengths involved give an indication as to the type of scattering the acoustooptic modulation will produce. The
AOM used in this thesis consists of a tellurium dioxide crystal, for which the ratio
λd/Λ2 ≈ 5, placing the AOM between regimes. For the sake of brevity, only the
Raman-Nath theory is presented here; this theory has the advantage of being more
closely related to expressions about general frequency modulators than the Bragg
theory.
We will now present the theory of Raman-Nath scattering. We first note that the
piezoelectric transducer induces a sinusoidal density variation
∆ρ̃ = ∆ρ sin(qz − Ωt).
(1.22)
This will generate a refractive index variation
∆ñ = ∆n sin(qz − Ωt).
(1.23)
To find the amplitude ∆n of the refractive index variation in terms of the known
∆ρ, we can employ the relationship between relative permittivity change and density
1.3. Acoustooptic Modulator
17
change, which we assume is small enough as to be linear:
∆˜ =
∂
∆ρ̃ = γe ∆ρ̃/ρ0 ,
∂ρ
(1.24)
where γe is the electrostrictive constant and ρ0 is the bulk density, both of which are
measurable. Now note that the total refractive index, n = n0 + Ɩ, is
p
n = 0 + ∆˜,
(1.25)
as long as we assume that the relative permeability is 1 (a good approximation for
most dielectrics). Since the relative permittivity variation ∆˜ is small, we can make
√
√
√
the approximation n ≈ 0 + ∆˜/(2 0 ), which results in the relation n0 = 0 , and
∆ñ ≈
∆˜
.
2n0
(1.26)
Using Eq. 1.24 we can finally relate the variation of the refractive index to density
variations of the AOM crystal:
∆n =
γe ∆ρ
.
2n0 ρ0
(1.27)
Now we can figure out what happens to an incident light wave as it passes through
the AOM. Sinusoidal voltage signals applied to the AOM-transducer result in density
variations of the crystal, which lead to variation of the refractive index. This, in
turn, will cause a phase shift. The exact shift depends on both position and time.
To calculate the relative phase shift between different parts of the light wave after
traveling across the AOM, we can use a varying wavenumber, call it ∆κ:
ω
ω
∆nd sin(qz − Ωt)
(1.28)
φ = ∆κd = ∆ñd =
c
c
≡ δ sin(qz − Ωt),
(1.29)
which serves to define δ. If we write the incident field in complex form as Ẽ(x, t),
then the transmitted field Ẽ 0 (x, t) is simply the incident field multiplied by a complex
transmission function eiφ :
Ẽ 0 (x, t) = Ẽ(x, t)eiδ sin(qz−Ωt) .
(1.30)
We can put this result into a more revealing form by using a variation of the JacobiAnger expansion:
eiδ sin θ =
∞
X
Jm (δ)eimθ ,
(1.31)
m=−∞
which employs Bessel functions of order m acting on δ. Applying this expansion to
the transmitted field gives
0
Ẽ (x, t) = Ẽ(x, t)
∞
X
m=−∞
Jm (δ)eim(qz−Ωt) .
(1.32)
18
Chapter 1. Theory
Consider the effect that the modulation has on a plane wave of frequency ω and
wave vector ~k = kx̂. We can assume that the beam covers a large enough transverse
(ẑ-direction) distance to interact with multiple wavelengths of the acoustic wave.
Representing the plane wave in complex notation, we get
0
i(kx−ωt)
Ẽ (x, t) = A[e
−i(kx−ωt)
+e
]
∞
X
Jm (δ)eim(qz−Ωt)
(1.33)
m=−∞
= A
= A
∞
X
m=−∞
∞
X
Jm (δ)[ei[(kx+mqz)−(ω+mΩ)t] + e−i[(kx+mqz)−(ω+mΩ)t] ] (1.34)
Jm (δ) sin[(kx + mqz) − (ω + mΩ)t].
(1.35)
m=−∞
This reveals that the transmitted field is in fact a superposition of plane waves
of varying magnitude, direction and frequency. The diffracted waves are essentially
formed by adding or subtracting the momentum and energy of the acoustic wave
to the incident optical wave. Higher-order diffracted beams vary in frequency in
increments of Ω, the acoustic wave frequency, and vary in wavevector in increments
of ~q, the acoustic wavevector. Thus varying the AOM-transducer frequency provides
control over the frequency of diffracted beams. Since the speed of sound in the
crystal is independent of frequency, changing the transducer frequency also changes
the wave number and thus the deflection angle. (Traveling wave AOMs are often used
as beam deflectors that can precisely alter the direction of diffracted orders without
much dependence on incident beam angle). Imagine the wavevector of the m-th
order output from the AOM as the hypotenuse of a right triangle, with the optical
wavevector ~k as one leg, and an integer number of acoustic wavevectors m~q as the
other leg. Then the angle of the diffracted mode away from the optical wavevector ~k
is given by the trigonometry of the triangle,
tan θm =
mλ
mq
=
,
k
Λ
(1.36)
which at small angles is identical to the diffraction grating equation.
When using a traveling wave AOM for a modelocking experiment, one would,
ideally, capture all diffracted beams and redirect them so that they recirculate inside
the laser cavity. Doing so would eliminate the z-dependence of the phase shift, so the
total phase shift equation would read
φ = δ sin(Ωt).
(1.37)
This idealization will be useful shortly when modeling the behavior of a modelocked
laser. In practice, it is possible to capture just one diffracted beam, typically the
first-order beam, and redirect it so that it circulates in the laser cavity, allowing a
one-directional transfer of power between frequencies in the cavity.
1.4. Time-domain modeling: Pulse Propagation
1.4
19
Time-domain modeling: Pulse Propagation
Having described the components of the modelocked laser and described how each
affects passing light, we can introduce a theory of optical pulses and propagate them
through each component. Following the argument of Siegman [1986, Section 27.3],
this section is intended to demonstrate how an optical pulse can maintain a Gaussian
shape throughout a round-trip through the laser. Unfortunately, some of the approximations used are invalid in the experimental setup described later in this thesis,
but the idealized argument presented here is nonetheless useful for understanding the
generation of Gaussian pulses in a general modelocked laser system.
We will begin by developing the mathematics for the Gaussian pulse, move on
to find the effect each optical element has on the pulse, and finally find a stable
round-trip pulse shape.
1.4.1
Gaussian Pulses
We can approximate the pulses that circulate in the laser as having Gaussian intensity
profiles. In the time domain, we can describe a Gaussian pulse in the form
2
2
Ẽ(t) = e−at ei(ωc t+bt ) = exp(−Γt2 + iωc t),
(1.38)
where a is the envelope parameter, ωc is the carrier frequency, and Γ ≡ a − ib is the
‘Gaussian pulse parameter’. If b is nonzero, then the quadratic dependence of optical
phase on time implies that the instantaneous frequency varies linearly along the pulse.
This linear frequency variation is called ‘chirp,’ and is a result of the shifting of parts
of the pulse spectrum. The instantaneous intensity of the pulse is
2
2
I(t) = e−2at = e−(4 log 2)(t/τ ) ,
(1.39)
p
where τ = 2 log 2/a is the full width at half-maximum (FWHM) of the intensity
profile. Thus a large a parameter implies a short pulse, while a small a implies a long
pulse.
Propagation of the pulse through most optical components is more easily accomplished in the frequency domain than in the time domain. Taking the Fourier
transform of the field results in
Z ∞
Ẽ(ω) =
Ẽ(t)e−iωt dt
(1.40)
−∞
Z ∞
1
2
= √
e−Γt eiωc t e−iωt dt
(1.41)
2π −∞
1
(ω − ωc ) 2
= √ exp −
.
(1.42)
4Γ
2Γ
With this result in hand, we can turn to propagating the pulse through the laser.
20
Chapter 1. Theory
1.4.2
Pulse Propagation
We can propagate our pulse through a simple laser setup by multiplying the field
by either a transmission function in the time domain or a transfer function in the
frequency domain. Consider a linear laser consisting of generic laser gain medium
(with a Lorentzian lineshape) and a generalized frequency modulator within a twomirror cavity. Suppose the cavity has length L. We begin with the standard Gaussian
pulse, in frequency representation, and assume that the carrier frequency is equal to
the peak frequency in the gain medium:
(ω − ω0 ) 2
1
exp −
.
(1.43)
Ẽ0 (ω) = √
4Γ0
2Γ0
First let us consider the effect the gain medium has on the pulse during one roundtrip. We will multiply the field by the transfer function g(ω), from the section on
gain media, but first we can make an approximation that will be useful in analysis
later8 :
αp
(1.44)
g(ω) = exp
2 + 4i(ω − ω0 )/∆ω0
ω − ω0
8
2
≈ exp αp 2 − 4i
(ω − ω0 )
.
(1.45)
−
∆ω0
∆ω02
Here we have made a Taylor expansion approximation based on the assumption that
the spectrum of Ẽ(ω) is much narrower than the total gain bandwidth of the laser
transition.9 Multiplying the Gaussian field by g(ω) yields a new field
(ω − ω0 )
1
8αp
1
2
0
exp 2αp − 4iαp
−
+
(ω − ω0 ) . (1.46)
Ẽ (ω) = √
∆ω0
4Γ0 ∆ω02
2Γ0
The three terms in the exponent represent different physical effects that the gain
medium has on the field. The first term represents the amplification afforded to each
frequency, which is independent of frequency (according to the approximation we
made above, that the bandwidth of the pulse is much narrower than the FWHM of
the gain medium). The second term is imaginary, and thus represents a phase shift
in frequency space, corresponding to a delay of the field in time. We will consider
this term later. The last term is real and quadratic in ω, and corresponds to the new
waveform of the field. This term is, like the exponent of Eq. 1.42, proportional to
−(ω − ω0 )2 , so it is again a Gaussian waveform. We can describe this pulse with a
new Gaussian pulse parameter, Γ0 :
1
8αp
1
=
+
0
4Γ
4Γ0 ∆ω02
Γ0 ∆ω02
=⇒ Γ0 =
.
32αpΓ0 + ∆ω02
8
(1.47)
(1.48)
Note that we are passing through the gain medium twice during the round-trip.
This assumption is invalid for the experimental setup of this thesis, because at practical cavity
lengths, the HeNe’s narrow gain bandwidth covers only a few intermode frequency spacings.
9
1.4. Time-domain modeling: Pulse Propagation
21
Decomposing this result into real and imaginary parts, Γ0 = a0 − ib0 , we have
a0 =
b0 =
∆ω02 (32 (a2 + b2 ) pα + a∆ω02 )
1024b2 p2 α2 + (32apα + ∆ω02 )
b∆ω04
2,
(1.49)
2.
(1.50)
1024b2 p2 α2 + (32apα + ∆ω02 )
These are complicated expressions, and the behavior of a and b over many round
trips depends on the initial values of the parameters. However, we can run a simple
simulation to show that a pulse circulating in a cavity containing only a gain medium
will broaden over time, returning to a sine wave. This is done by choosing values
for the constants and initial values a0 and b0 , and then iterating Equations 1.49 and
1.50. A representative plot of the evolution of a and b after each round-trip can
be seen in Fig. 1.16. Both a and b decrease slowly over time, meaning that the
pulse width increases to infinity and the chirp decreases to zero, approaching a sine
wave. However, if the initial pulse has a high chirp (large b), then there will be a
transient effect where the pulse width decreases before the long-term increase begins.
This suggests that a chirped pulse, and thus a pulse with phase-shifted modes, will
be narrowed in time when passing through the gain medium. However, without
modulation, the gain medium will eventually cause the pulse to broaden, and the
chirp to decrease, due to a narrowing of the pulse spectrum.
a, b
Round
trips
Figure 1.16: Evolution of pulse parameters in a cavity with gain. Calculated iteratively over many round trips. The thick line represents a, and the thin line represents b.
Both a and b decrease slowly over time. The dashed line shows the initial value of a.
Let us now consider the phase shift component of the exponential in Eq. 1.46,
4iαp (ω − ω0 ) /∆ω0 . Taking the inverse Fourier transform of this part of the pulse
spectrum reveals that the constant term, −4iαpω0 /∆ω0 , simply shifts the phase of
the pulse in the time-domain. The frequency-dependent term, 4iαpω/∆ω0 , on the
other hand, actually translates the pulse in the time domain, so that the pulse is
delayed. This small delay, due to the gain medium, means that the round-trip time
is not exactly 2L/c, and the axial mode spacing is not exactly c/2L. However, the
product αp and the difference ω − ω0 are usually quite small compared to ∆ω0 .
22
Chapter 1. Theory
While the perturbation is theoretically nontrivial, its magnitude is usually within the
uncertainty of an experimentalist’s tools for measuring the cavity length or roundtrip frequency. Thus experimentalists usually ignore this perturbation and estimate
the axial mode spacing to be c/2L, as they can tune cavity length and modulation
frequency empirically for best modelocking.
Finally, we consider the effect of the frequency modulator on the pulse. We
developed the modulator theory in the time domain, so we use the time-domain
Gaussian pulse. The gain and phase-shift terms induced by the gain medium are
inconsequential in the steady state10 , so we focus on the term involving the pulse
parameter:
Ẽ(t) ∝ exp[−Γ0 t2 ],
(1.51)
The transmission function from Eq. 1.37, this time replacing the sine function with
a cosine function, is given by T (t) = eiδ cos(Ωt) . We can make a simplifying approximation of the cosine function by Taylor expansion, but we first need to figure out
at which point in time the pulse will cross the modulator. Suppose the pulse crosses
the modulator at t = nπ/2 for odd integer values of n; that is, when cosine sweeps
through zero and the phase changes most rapidly. This would yield a transmission
function approximately proportional to exp[iδΩt]. This would change the instantaneous frequency of the pulse Ẽ(t), bringing the central pulse frequency outside of the
bandwidth of the gain medium after many round-trips. Thus this pulse will not be
amplified and will quickly die off due to cavity losses. To form pulses that survive in
the steady-state, then, we want the pulse to avoid periods of rapidly-changing phase.
If the pulse passes through the modulator when the phase is nearly constant (at the
crest or trough of the cosine function), then we can avoid terms in the transfer function that change linearly with time, and thus avoid shifting the central frequency of
the pulse. If we assume the pulse duration to be much shorter than the modulation
period11 , we can make the following quadratic approximation for the transmission
function:
T (t) ≈ exp iδ 1 − Ω2 t2 /2 .
(1.52)
After passing through the modulator, then, the field is
Ẽ 0 (t) ∝ exp −Γ0 t2 + iδ 1 − Ω2 t2 /2
= exp iδ − Γ0 + iδΩ2 /2 t2 .
(1.53)
(1.54)
The iδ term is a small constant phase shift. The coefficient of t2 is the new Gaussian
pulse parameter: Γ00 = Γ0 +iδΩ2 /2. This shows that |b00 | > |b0 |, so the chirp of the pulse
is increased during modulation, but a00 = a0 , so modulation does not directly compress
10
The gain per round trip depends on the degree to which the gain medium is saturated, and in
the steady state, it is exactly cancelled by the cavity losses. The phase shift accumulates over every
round trip, so emitted pulses generally differ in optical phase.
11
This is another approximation that fails in our experimental case, but is true for modelocked
lasers employing broadband gain media.
1.4. Time-domain modeling: Pulse Propagation
23
the pulse. However, increasing the chirp of the pulse does broaden the spectrum, and
looking back at Eq. 1.49, we see that the when the pulse passes through the gain
medium again, the parameter a increases if the magnitude of b had increased during
the previous round-trip. Thus upon exiting the gain medium, the pulse is narrowed
slightly. This indirect pulse narrowing causes the spectrum to broaden and the pulse
duration to decrease over many round trips, until a steady state is reached.
When we include the modulation in our simulation, as in Fig. 1.17, we see that
both a and b stabilize to reach approximately the same value. This stable value is
independent of the initial pulse parameters; the stable value is reached even when
a0 = b0 = 0.
a, b
Round
trips
Figure 1.17: Evolution of pulse parameters with both gain and modulation. The
thick line represents a, and the thin line represents b. Whenever the lines cross, one of
the values reaches a maximum or a minimum. The dashed and dotted lines represent the
theoretical steady-state values of a and b respectively, as obtained from Eq. 1.58.
Finally, we can solve our formulae analytically for the steady-state pulse parameter. In the steady state,
Γ00 = Γ0
p
2ipαδΩ2 + −4p2 α2 δ 2 Ω4 + ipαδΩ2 ∆ω02
=⇒ Γ =
8pα
(1.55)
(1.56)
(we choose the root such that a is positive). Expanded into real and imaginary parts,
we have
√
iδΩ2
δΩ (16p2 α2 δ 2 Ω4 + ∆ω04 ) 1/4
Γ=
+
(Cos[ψ] + iSin[ψ]),
(1.57)
√
4
8 pα
1
with ψ = argument pαδΩ2 −4pαδΩ2 + i∆ω02 .
(1.58)
2
This result is complicated. Numerically, we find that b is slightly higher than a for
the values used in the iterative simulation, as seen in the Fig. 1.17.
In this section, we have shown that a steady-state Gaussian pulse solution exists,
where an initial Gaussian pulse maps back onto itself after propagating through the
24
Chapter 1. Theory
gain medium and intracavtiy frequency modulator. We have also provided evidence
that this solution is stable in the sense that any initial Gaussian evolves toward this
solution.
1.5
Injection Locking
There are two practical problems with modelocked laser experiments that we implicitly ignored in the preceding theory. The first is the fact that we cannot hope to
perfectly tune the modulation frequency Ω to line up with the axial mode spacing
∆ν. The frequency-shifted light will, in every practical case, be detuned slightly from
the axial mode frequencies that the laser cavity supports. We expect that the spectra
of both the allowed cavity modes and the frequency-shifted light will be broadened
by noise, but the overlap of these broadened peaks will not be complete, resulting in
low gain of the frequency-shifted light. Thus there would be little power in any of
the modes except the principle mode, forming broad output pulses at best.
The second, and perhaps more important, problem we ignored was that of phase
noise. When we began our discussion of modelocking, we emphasized that a goal of
the technique is to minimize phase noise differences between mode pairs. The pulsepropagation modeling we presented in the previous section showed that the spectrum
of a pulse will be broadened inside our laser (spectral broadening being the other main
goal of modelocking), but it assumed that the pulse was noiseless, and that the optical
components did not contribute any noise. In reality, every component contributes
noise to the system, whether it be from spurious vibrations of the mirrors or AOM,
thermal deformations of those components, or variations of index of refraction in the
air between components.
Thankfully, it turns out that modelocked lasers can operate effectively in the
presence of both detuning of the modulation frequency and standard experimental
noise. This is due to the nonlinearity of the system. Suppose two axial modes, at ω1
and ω2 = ω1 + ∆ν, are oscillating simultaneously in a laser that includes a frequency
modulator that is currently turned off. The two modes have different phase noise.
Now suppose that the frequency modulator is turned on. All we want this modulator
to do is take part of the incident light and increase its frequency by Ω, where Ω is
close but not equal to the axial mode spacing ∆ν. Considering only the ω1 mode for
now, the modulator will therefore output two frequencies of light, ω1 and ω∗ = ω1 +Ω,
where the ω∗ light has some phase relationship with the ω1 signal. We say that the
light at ω∗, which does not lie precisely at any axial mode, has been ‘injected’ into the
cavity. If the laser were a completely linear system, we would have three frequencies
of light superimposed in the cavity: ω1 , ω2 and ω∗. However, because of nonlinear
effects, such as cross-gain saturation, the presence of light at ω∗ affects the generation
of light at ω2 . In fact, if the frequency ω∗ is close enough to ω2 , then the frequency of
the ω2 mode will be ‘pulled’ to the frequency ω∗, and the phase of this pulled mode
will follow the phase of the ω∗ signal. Thus by turning on the modulator, we have
effectively replaced the mode at ω2 with a mode at ω∗, and because the ω∗ mode was
generated from ω1 , the two modes in the cavity have a constant phase relationship.
1.5. Injection Locking
25
This synchronization phenomenon, called ‘injection locking,’ occurs in almost any
nonlinear oscillatory system in which a periodic signal is injected near resonance. It
was described mathematically in a paper by Adler [1973], and this description was
applied to laser oscillators by Kurokawa [1973].
To show how injection locking works, we will consider the behavior of a general selfsustained oscillator when injected with a periodic signal (a more in-depth treatment
of the injection locking of lasers is given in Siegman [1986, Chapter 29]). The system
is described by the so-called Adler’s equation12 ,
dψ
= −ωD + sin ψ,
dt
(1.59)
where ψ is the phase difference between the oscillator and the injected signal, ωD
is the detuning of the injected signal’s frequency from the natural frequency of the
oscillator, and will be shown to be the injection locking range, which has the units
of frequency. We can solve this equation for ψ(t), which tells us how the phase of the
oscillator relates to the phase of the injected signal over time:
"
−
ψ(t) = 2 arctan
ωD
"p
##
p
2
2
ωD
− 2
ωD
− 2
tan
t
ωD
2
(1.60)
There are, then, two general regimes of solutions, which depend critically on the
relative magnitudes of ωD and .
ΨHtL
Π
Ψf
Ψ0
0
T
2T
3T
t
Figure 1.18: Evolution of phase difference, ωD < . Solution to Adler’s equation for
small detuning. The phase of the oscillator approaches a fixed phase difference, ψf , with
respect to the injected signal, so the frequencies are identical and injection locking occurs.
The dashed line shows the final phase difference.
12
The equation in this form was adapted from Pikovsky et al. [2001], which derives Adler’s equation
and provides a generalization that can be applied to many situations.
26
1.5.1
Chapter 1. Theory
Locked Regime: ωD ≤ If the the detuning ωD is less than the parameter , then the radical factor becomes
imaginary:
q
2
ωD
− 2 = ih,
(1.61)
which serves to define h. The solution (1.60) can now be written
ih
ih
ψ(t) = 2 arctan
−
tan
t .
ωD ωD
2
(1.62)
We can solve the problem of the tangent function taking an imaginary argument by
making the replacement tan(iθ) = i tanh(θ):
h
h
ψ(t) = 2 arctan
t .
(1.63)
+
tanh
ωD ωD
2
We plot this function in Fig. 1.18. We can see that the phase
difference
moves from
√
2
+ 2 −ωD
the initial value ψ0 to a final constant value ψf = 2 arctan
, indicating
ωD
that the oscillator is oscillating at the same frequency as the injected signal but with
a shifted phase. Here T is the natural period of the oscillator when undisturbed. This
is the injection locked regime.
Note that if we had set ωD = , then the solution would be ψ(t) = 2 arctan(1) =
π/2, so the phase difference would still be constant and we can still consider this
situation injection locking.
1.5.2
Unlocked Regime: ωD > If the detuning ωD is greater than the parameter , then the radical factor is real, and
we can call Eq. 1.60 our solution. This is plotted in Fig. 1.19 for ωD slightly larger than
(1.2:1). What is happening here is that, as long as ψ evolves slowly, the oscillation
is basically at the injected frequency, but the phase periodically ‘slips,’ changing the
instantaneous frequency of oscillation. If we increase ωD to be much greater than (2:1), shown in Fig. 1.20, then the phase slips become more frequent, and the period
of the phase slips approaches the period of the undisturbed oscillator. Thus, for large
detunings, the oscillator essentially oscillates at the undisturbed frequency.
1.5.3
Consequences of Injection Locking
In the context of our laser, injection locking helps overcome the two difficulties presented in the beginning of this section: modulator detuning and phase noise. As long
as the detuning of the modulator is small enough, injected frequency-shifted modes
will replace the naturally occurring axial modes. Also, the phase noise of the injected
mode replaces the noise of the natural axial mode, essentially coupling the noise of
1.5. Injection Locking
27
ΨHtL
Π
Ψ0
T
2T
3T
t
-Π
Figure 1.19: Evolution of phase difference, ωD & . Solution to Adler’s equation for
detuning slightly larger than . The frequency of the oscillator is usually close to that of the
injected signal, but the phase ‘slips’ periodically, though less frequenty than the undisturbed
period T . The dashed line shows the initial phase, to give a sense of the phase slip period.
ΨHtL
Π
Ψ0
T
2T
3T
t
-Π
Figure 1.20: Evolution of phase difference, ωD . Solution to Adler’s equation for
detuning far larger than . The period of phase slips approach the period of the undisturbed
oscillator, so the oscillator usually oscillates near the undisturbed frequency.
each mode in the cavity. But how small does the detuning have to be? That is, what
can we expect to be, and how can we increase it? Siegman [1986, Eq. 29.30] gives
r
Iinj
=γ
,
(1.64)
I0
where Iinj is the intensity of the injected signal, I0 is the intensity of the undisturbed
mode, both measured inside the cavity, and γ is the ‘cavity decay rate,’ which is
δ
, where δ is the percentage of light that is emitted from
approximately equal to TRT
the output coupler mirror, and TRT is the round-trip time through the cavity, which is
just 1/νRT . What all this means is that we can improve our chances of modelocking
by increasing the intensity of the light we inject into the laser, by increasing the
axial mode spacing (making our cavity shorter), and increasing the amount of light
transmitted through our output coupler mirror. Thus the laser will be more difficult
28
Chapter 1. Theory
to injection-lock if we use a long cavity to reduce the duration of our pulses, so
injecting as much intensity into the cavity is a priority.
In this section, we have seen that the two problems we feared would ruin our
experimental attempts at modelocking, being modulator detuning and phase noise,
are mitigated by injection locking, which results from the inherent nonlinearities of
the laser. On this positive note, we make the assumption that the theory presented
in this chapter will be sufficient for guiding us through the construction and analysis
of a modelocked laser, and turn now to the experimental design of our laser.
Chapter 2
Experimental Design
In this chapter, we will describe and motivate the experimental setup used to achieve
modelocking. First we will consider a simple design that employs a traveling-wave
AOM for modelocking. We will then describe the practical problems with this design
and introduce the actual experimental setup.
HR
L0
er
AOM
HR
t
1s
d
or
0th order
HeNe tube
OC
L0
Figure 2.1: Theoretically simple split cavity for modelocking. Each leg of the cavity is the
same length, so a circulating pulse that splits at the AOM will be recombined at the AOM
with no relative delay between the legs. There are two high-reflecting (HR) mirrors and one
output-coupling mirror (OC). In this configuration, each mirror can ideally have the same
radius, equal to L.
The design of our modelocked laser is centered around the modulator available,
a traveling-wave acoustooptic modulator (AOM). As explained in Chapter 1, the
traveling-wave AOM diffracts incident light into multiple beams, each of which is
frequency-shifted by a certain amount. A modelocked laser can be constructed by
capturing both the 0th order beam, which is not frequency shifted, along with one of
the higher-order frequency shifted beams, such as the 1st order beam, and allowing
both to beat together inside of an optical cavity. A theoretically simple setup that
will contain both 0th and 1st order beams within an optical cavity involves three
30
Chapter 2. Experimental Design
AOM
0,1
AOM
1,1
Reflected
1st order
0,0
Reflected
0th order
0,-1
Figure 2.2: After reflecting off of the output coupler, the 0th order is reflected exactly back in on itself and diffracted once
again in the AOM. The 0th order of this
second diffraction, labelled (0,0), is contained by the cavity, while the other orders are lost.
1,0
1,-1
Figure 2.3: Like the 0th order, the 1st
order is reflected exactly back in on itself
and diffracted once again in the AOM.
This time the 1st order of this second
diffraction, labelled (1,1), continues to
the HeNe tube and is contained by the
cavity, while the other orders are lost.
mirrors and is shown in Fig. 2.1. In this setup, an AOM and a third mirror have
essentially been placed inside of a linear two-mirror laser cavity. The third mirror
reflects the 1st order back into the AOM, so both 0th and 1st orders are reflected
back into the AOM. These reflected beams also diffract. Examining the diffraction
of the reflected 0th order beam in Fig. 2.2, we see that only the unshifted part of
this diffraction pattern (beam 0,0) remains inside the cavity. The diffraction of the
reflected 1st order beam is seen in Fig. 2.3; in this case the first shifted beam of
this diffraction pattern (beam 1,1) remains inside the cavity. Thus the 1,1 beam is
frequency-shifted by the AOM twice: once propagating left and once propagating
right, so we set the AOM modulation frequency to half of the round-trip frequency.
This twice-shifted beam injection locks the modes circulating between the HeNe tube
and the OC, modelocking the laser.
It is important in this configuration that the distance from the AOM to each
mirror on the right-hand side of the the setup is the same, so that the 1st and 0th
order beams traverse the same distance before recombining. If we think of a pulse
propagating rightward in the time domain through this laser, we can see that it will
split at the AOM, the split pulses will traverse each leg separately, and then recombine
again at the AOM. To make sure that the recombined pulses constructively interfere,
each leg of the cavity must be the same length. This length is labeled as L0 on
Fig. 2.1.
This split cavity is theoretically sound, but difficult to construct for a number
of practical reasons. Most importantly, the losses that the beam incurs through the
AOM are much greater than the gain from the HeNe tube during each round trip, so
there is no hope that enough photons will circulate in the cavity simultaneously for
lasing to occur. In the standard two-mirror continuous wave configuration, the HeNe
tube used in this thesis has such a low gain that even a glass coverslip, if introduced
into the cavity as a beamsplitter, presents too much loss. If a thin coverslip cannot
be placed into the cavity, then a thicker crystal will definitely not be supported.
The solution to the problem of too much loss is to place the AOM outside of
31
the laser cavity, allowing unimpeded amplification inside the cavity, and then couple
the frequency-shifted light back into this cavity. Figure 2.4 shows the experimental
setup used in this thesis. The setup consists of a ‘primary cavity’ configured like a
standard continuous-wave laser cavity, and a ‘secondary cavity’ formed by using a
third spherical mirror to couple light back into the primary cavity. Modelocking with
an external modulator was first accomplished by Foster et al. [1965], using a tank of
water instead of an acoustooptic device. The setup used in this lab follows the layout
in the graduate lab of Jones [2009] at the College of Optical Sciences of the University
of Arizona.
L
L
R = 60 cm HR
R = 45 cm OC
AOM
R=1m
HR
y
r
vit
e
a
d
r
Edge
yC
to
ar
1s
d
Mirror
n
co
e
S
0th order
Primary Cavity
Scope
ESA
FPI
Fast
Photodiode
Figure 2.4: Experimental setup. The 0th order diffracted from the AOM is reflected by an
edge mirror into either a Fabry-Perot interferometer or a fast photodiode.
The entirety of the optical setup was built on a Newport RS 300 floating optical
table. The primary cavity consists of a HeNe laser tube and an output coupler (OC)
mirror through which the laser beam is emitted. The HeNe tube is a Melles Griot 05LHB-570 powered by the power supply of a Spectra Physics 132 laser. The right end of
the HeNe tube is terminated in a glass window that lies perpendicular to the Brewster
angle of 632.8 nm as it transitions from air to glass. This setup, called a Brewster
window, minimizes the reflectivity of the glass-air interface for light polarized parallel
to the window. The Brewster window thus helps polarize the laser beam. The other
end of the HeNe tube is terminated in a high-reflecting (HR) mirror with a radius
of 60 cm. The spherical output coupler mirror, on the right beyond the Brewster
window in Fig. 2.4, has a radius of curvature of 45 cm, but the manufacturer of the
32
Chapter 2. Experimental Design
mirror is unknown. The output coupler is mounted on a micrometer stage for fine
adjustment of the cavity length. According to Section 1.1 of the Theory chapter, this
pair of mirrors can theoretically support Gaussian beams at cavity lengths from 0 to
45 cm, and from 60 to 105 cm. Maximizing the cavity length is desirable, because
this decreases the axial mode spacing, which increases the number of modes that can
lase simultaneously and thus minimize the duration of our pulses. At 105 cm, the
round-trip frequency is νRT = 157.5 MHz, and the round-trip time is approximately
7 ns. The longest cavity achieved during experimentation was approximately 94 cm
long (νRT ≈ 141 MHz; t ≈ 6.3 ns), at which point the intensity began to fluctuate
and further alignment became difficult.
Light emitted from the output coupler travels a short distance before encountering
a NEOS 46110-1-LTD acoustooptic beam deflector, driven in the range of 80 MHz to
130 MHz by a NEOS 21110-1ASVCO AOM driver. The drive frequency is controlled
by an external voltage, supplied by a Tektronix PS280 power supply. The traveling
acoustic wave in the AOM causes the incident light to diffract into a number of beams;
only the 0th and 1st order beams are of interest to us. The 0th order beam is not
frequency-shifted, so it can be measured for an accurate description of the field inside
the primary cavity. The 0th order beam is deflected by an edge mirror towards either
a photodiode or Fabry-Perot interferometer. A Thorlabs DET10A fast photodiode
is used for measurement of the intensity timeseries and the radio-frequency beats
between laser modes. This detector was chosen for its risetime of 1ns and reasonable
responsivity. The detector is connected to either a Tektronix TDS 620B oscilloscope
or an Agilent E4411B electronic spectrum analyzer.
Alternatively, the 0th order beam can be deflected into a Burleigh SAPlus Laser
Spectrum Analyzer1 . This is a Fabry-Perot interferometer (FPI), which consists of an
optical cavity of two partially-transmitting confocal mirrors in front of a photodiode.
Like the laser cavity, this Fabry-Perot cavity only supports distinct, evenly-spaced
frequencies. Supported frequencies are transmitted through the cavity to the photodiode2 , allowing the measurement of the intensity of a very narrow portion of the
optical spectrum. The distance between mirrors is varied by a piezoelectric transducer over time, allowing the measurement of a range of optical frequencies. By
connecting the photodiode in the interferometer to our scope, we can measure the
relative intensities of different optical frequencies in our laser signal. The mirror set
used in the cavity reflects light in the range 450-700 nm.
An important property of the FPI cavity is the axial mode spacing3 , also called
the ‘free spectral range’ in the case of interferometers. This is important because
it limits the frequency range over which the interferometer can measure. Suppose
light at a frequency of ν, and thus wavelength λ = c/ν, enters the FPI as the cavity
1
The interferometer is equipped with a SA-98-B1 mirror set and driven by a Burleigh RG-91
ramp generator.
2
The photodiode signal runs through a Burleigh DA-100 amplifier.
3
Because of the confocal geometry of the Fabry-Perot interferometer used, it is possible for light
to be injected into the cavity off of the optical axis. In the paraxial approximation, light then travels
f our cavity lengths per round-trip, so the formula for the axial mode spacing is νRT = c/4L, except
in the case that the input beam is aligned exactly on the optical axis, in which case νRT = c/2L.
33
length sweeps from L1 = nλ/2 to L2 = (n + 1)λ/2. Then the light will resonate
in the cavity at both L1 and L2 , so light will be transmitted to the photodiode at
both lengths, and the monochromatic light will cause the FPI to measure a signal at
what appear to be two different frequencies, when in actuality the spectrum is just
repeated. The apparent frequency difference between these transmission peaks is the
free spectral range. The Fabry-Perot cavity we use in this experiment is very short,
with a free spectral range of 8 GHz4 , so the spectrum from a narrow-bandwidth laser
like the HeNe is sure to lie within this range, preventing any part of the spectrum
from overlapping with other parts.
While the 0th order beam emanating from the AOM is used for measurement,
the 1st order beam is used to modelock the laser. This is done by retroreflecting
the 1st order beam back into the AOM with a third spherical mirror, a Newport
10DC1000ER enhanced aluminum mirror with a radius of 1 m. As in the split-cavity
configuration, the 1,1 beam travels along the same path as the light incident on the
AOM, but in the opposite direction. This 1,1 beam is coupled into the primary
cavity. As in the split cavity case, the AOM drive frequency is half of the roundtrip frequency of the primary cavity, so that the twice-shifted beam that is coupled
back into the primary cavity has the same frequency as one of the primary cavity’s
axial modes. The shifted beam then injection-locks the modes inside of the primary
cavity, modelocking the laser. As labeled in Fig. 2.4, the distance between the output
coupler and the third mirror should be as close as possible to the length of the
primary cavity. This is because a pulse circulating in the laser is split at the output
coupler; one part traveling out to the secondary cavity, and the other returning to
the primary cavity. After one round-trip time, these pulses will once again meet at
the output coupler, and if there is any delay between them, the two split pulses will
not maximally constructively interfere. This is similar to the requirement that each
of the legs of the split cavity be the same length, but in this case the splitting at the
output coupler is important, while the splitting at the AOM is not (the 0th order
beam is measured, not reflected back into the cavity). Thus there are three variables
that must be in relatively good agreement for modelocking to occur: the round-trip
frequency of the primary cavity, the round-trip frequency of the secondary cavity,
and the doubled modulation frequency. Thankfully, due to the miracle of injection
locking, these three frequencies do not have to be precisely the same for modelocking
to occur, as we will see in the Results chapter.
Note that, although the AOM operates closer to Raman-Nath regime than the
Bragg regime, generating multiple diffracted beams, the angle of the AOM still affects
the relative intensities of the beams. Specifically, the intensity of a given diffraction
order is maximized when the angle between the acoustic waves and the incident light
is at one of the Bragg angles. For this experiment, the AOM was rotated so that
the intensity of the first diffracted order was maximized, to ensure that sufficient
power was fed back into the laser for modelocking to occur (recall that increasing
the intensity of the injected light can increase the range of modulation frequency
4
The range depends on the cavity length, which changes over time, but we assume that these
changes are very small compared to the total length.
34
Chapter 2. Experimental Design
detunings over which injection locking can occur). While this necessarily decreased
the intensity of the zeroth order beam, and thus decreased the percentage of optical
power redirected to the photodiode, a signal was still resolvable above noise when
analyzed by our measuring equipment.
Chapter 3
Results and Discussion
Having provided a description of the experimental setup in the previous chapter, we
can turn to the results of the experiment and attempt to analyze them. First we will
describe the modelocking procedure and examine the behavior of the laser during this
process.
The first, and most time-consuming, step in the modelocking procedure was to
align the primary cavity for lasing. This was increasingly difficult at longer cavity
lengths. The laser was aligned at a cavity length of about 91 cm, yielding a theoretical axial mode spacing of about ∆ν ≈ 165 MHz. (A small axial mode spacing
was necessary to maximize the number of modes within the modelocked frequency
spectrum.) Once the primary cavity was lasing, the AOM was aligned such that the
1st order mode was most intense. The third mirror, reflecting the 1st order mode
back into the AOM, was then set up at a distance L − 3.5 cm away from the output
coupler, where the correction is necessary because of the high index of refraction of
the AOM crystal.
3.1
Unmodulated Operation
With the components laid out, data collection began. The unmodulated output of
the laser, with the AOM turned off, was measured first. The laser was found to
operate in only one axial mode for most of the time, whereas two-mode operation
was observed only once in a while. We will call these modes ‘primary modes,’ to
distinguish them from the ‘shifted modes’ that will arise later due to modulation.
Fig. 3.1 shows timeseries data of the intensity waveform for one-mode operation, taken
with the fast photodiode. This is essentially continuous-wave output. The mean
voltage was approximately 0.425 mV, while the RMS voltage was 0.450 mV. An
optical power meter, set up just outside of the output coupler mirror, measured the
average power to be about 110 nW, far weaker than the average laser pointer (1-5
mW). We believe that the noise in the output was a combination of electronic noise
(measured to have an RMS voltage of 0.220 mV) as well as fluctuations in laser power
due to changing cavity losses.
Evidence that this output was due to single-mode operation comes from Figs. 3.2
36
Chapter 3. Results and Discussion
0.8
Intensity HmVL
0.6
0.4
0.2
0.0
-20
-10
0
Time HnsL
10
20
Figure 3.1: Timeseries waveform: AOM off, one mode. Single-mode output at 91
cm. Mean voltage is 0.425 mV, RMS 0.450 mV.
-1
0
1
Relative frequency HMHzL
2
3
4
5
6
7
8
9
Optical spectrum HmVL
20
15
10
5
0
0
5
10
Scan time HmsL
15
Figure 3.2: Free spectral range: AOM off, one mode. Power spectral density data
of single-mode laser operation from the FPI, covering the free spectral range (8 GHz) of
the device. The upper horizontal axis gives relative frequency spacing, while the lower axis
gives sweep time. The sweep rate is calculated to be 523.56 MHz per ms.
3.1. Unmodulated Operation
-1000
37
Relative frequency HMHzL
-500
0
500
1000
Optical spectrum HmVL
40
30
20
10
0
-2
-1
0
Scan time HmsL
1
2
Figure 3.3: FPI spectrum: AOM off, one mode. Detail of single-mode laser operation.
The sweep rate is the same as for Fig. 3.2. The peak appears to have a FWHM of about
100 MHz.
1.2
Intensity HmVL
1.0
0.8
0.6
0.4
0.2
0.0
-20
-10
0
Time HnsL
10
20
Figure 3.4: Timeseries waveform: AOM off, two modes. The period of the beats is
about 6 ns, close to the expected period.
38
Chapter 3. Results and Discussion
and 3.3, which show a single mode in the optical spectrum taken from the FabryPerot Interferometer (FPI). Figure 3.2 shows the output of the FPI when the range
of frequencies swept out by the interferometer’s cavity is greater than the free spectral
range of 8 GHz. As a result, the spectrum repeats after 8 GHz. The mirrors in the
FPI oscillate to sweep across a frequency range, so the output of the FPI is actually
a timeseries. We used the free spectral range sweep presented in Fig. 3.2 to find the
ratio between sweep time and frequency spacing. All of the FPI data presented in
this thesis have the same sweep rate as in Fig. 3.2, which is ∼523.56 MHz per ms.
Knowing the sweep rate allows us then to zoom in on small frequency ranges for a
better spectral resolution. Looking in detail at the single mode output by the FPI, as
in Fig. 3.3, we see that the optical mode had a FWHM of about 100 MHz, although
this may be artificially broadened by the combination of the limited risetime of the
FPI’s photodiode and the FPI’s fast sweep rate. The FPI cannot directly measure
absolute frequency, but we can assume that the single lasing mode is close to the
standard HeNe 632.8 nm transition (at a frequency of 474.1 THz).
It is also possible for the laser to switch briefly to two-mode operation. This
was generally an unstable state, and would survive for only a few seconds at most.
A sample timeseries waveform can be seen in Fig. 3.4. The corresponding radiofrequency beat signal on the electronic spectrum analyzer (ESA) is shown in Fig. 3.5,
and a typical FPI spectrum is shown in Fig. 3.6.
In Fig. 3.4 we can see the rather clear beating of the modes, with a periodicity of
about 6 ns, which is close to the round-trip frequency associated with a cavity of 91
cm, T = 6.25 ns. The beat frequency is also visible on the ESA, as in Fig. 3.5, and
gives a far more accurate measure of cavity length than can be achieved with a ruler.
In this case, the cavity appears to have a length of 91.64 cm. The broadened nature
of this peak is due to noise. The apparent width of the peak is deceptive because
Fig. 3.5 is a semi-logarithmic plot; plotting the same data linearly reveals that the
FWHM of the peak is only about 2.42 kHz. The clearest evidence that two modes
can coexist in the cavity is seen in the FPI optical spectrum, as in Fig. 3.6. The
frequency labeled “∆ν” in the upper horizontal axis marks the axial mode spacing
corresponding to the beat frequency that we see on the ESA in Fig. 3.5, at about
163.684 MHz. The distribution of power between the modes was seen to fluctuate
over time, but the mode spacing remained constant, until the spectrum collapsed into
a single mode. Note that the vertical scale is this case is the same as in the plot of
one-mode operation (Fig. 3.3), so the total power in the spectrum is roughly the same
in both cases.
When we discussed laser spectra in the Theory chapter, we predicted that, if the
axial mode spacing was small enough, multiple axial modes would fall within the
laser’s gain bandwidth, and these modes could lase simultaneously, even without the
modulation necessary for modelocking. Why, then, is the presence of multiple modes
so rare? We expect that there is simply too much loss in the cavity for more than
one mode to stably lase at once. That is, only the highest portion of the gain curve
is greater than the cavity losses, and only the modes that lie within this region have
a chance to lase. See Fig. 3.7.
A possible explanation for the relatively instability of multi-mode operation is a
3.1. Unmodulated Operation
39
-6
RF spectrum HdBL
-7
-8
-9
-10
-11
-12
163.675
163.680
163.685
Frequency HMHzL
163.690
Figure 3.5: ESA spectrum: AOM off, two modes. Radio-frequency beat signal visible
on ESA, indicating the presence of two modes in the cavity.
-1000
Relative frequency HMHzL
-500
0 DΝ
500
1000
Optical spectrum HmVL
40
30
20
10
0
-2
-1
0
Scan time HmsL
1
2
Figure 3.6: PSI spectrum: AOM off, two modes. Detail of two-mode laser operation.
The sweep rate is the same as for Fig. 3.2. The mode spacing ∆ν was found from the ESA
plot in Fig. 3.5.
Chapter 3. Results and Discussion
Power
40
Cavity Loss
Only 1-2
modes lase
Laser Gain
Frequency
Figure 3.7: Possible relative position of gain curve to axial modes, with large cavity losses
preventing lasing in more than two modes.
Relative frequency HMHzL
-1000
-500
0
500
1000
1500
Optical spectrum HmVL
40
30
20
10
0
-2
-1
0
1
Scan time HmsL
2
3
Figure 3.8: FPI spectrum: AOM on, significant detuning. Typical spectrum arising
when AOM is turned on and aligned correctly. Each shifted mode is shifted by the same
frequency. The shifted modes are weak because they are not supported by the primary
cavity.
3.2. Approaching Modelocking
41
nonlinear phenomenon called ‘cross-saturation.’ In the Theory chapter we discussed
the general phenomenon of gain saturation, where the gain experienced by any mode
decreases to the level of cavity losses as the mode intensity increases to its steadystate value. This prevents mode intensities from increasing indefinitely. In some gain
media, stimulated emission by one mode not only saturates the gain of that mode, but
also contributes to the saturation of other modes nearby. This effect, called crosssaturation, can cause one particularly intense mode to reduce the gain of nearby
modes below the cavity loss line, so that the other modes die out. Cross-saturation
is usually not stated to be a significant cause for concern in gas lasers like the HeNe
[Siegman, 1986, Section 12.2, pp. 462-465]. However, because our laser has such a
small axial mode spacing of 160 MHz, and because such a small portion of the gain
curve seems to be available above the cavity loss line, we suspect that cross-saturation
could be responsible for the domination of one mode in the spectrum, as well as the
instability observed during two-mode operation.
3.2
Approaching Modelocking
With the laser operational and the third mirror correctly aligned, the final step in
the modelocking procedure was the tuning of the AOM modulation frequency Ω to
coincide with half of the axial mode spacing ∆ν. In this section, we will describe
the effect of injecting frequency-shifted light into the primary cavity for a range of
detunings. When the AOM was turned on, part of the light emitted by the primary
cavity as primary modes was frequency-shifted by the modulation, becoming shifted
modes. These shifted modes were visible not only in the optical spectrum as seen
by the FPI, but also created beat frequencies by interfering with the primary modes.
These beat frequencies were visible on the ESA.
When the modulation frequency was significantly detuned, shifted modes became
visible on the FPI spectrum (Fig. 3.8). The intense mode at 0 MHz is the primary
mode, and the weaker modes to the right are shifted modes. The shifted modes
were weak because they did not line up with axial mode frequencies supported by
the primary cavity. The beating of these shifted modes and the single cavity mode,
shown in Fig. 3.9, was visible on the ESA. The central beat frequency was at twice the
modulation frequency, 2Ω. The sidebands at approximately ±60 kHz are of unknown
origin; the spacing and magnitude of these sidebands appeared to be independent of Ω
and cavity length, so it is conceivable that the AOM itself has an unintended resonance
at this frequency. Fortunately, the sidebands had little effect on laser operation, since
they were about 15 decibels less intense than the central beat frequency. The entirety
of the beat structure could also be seen at higher multiples of 2Ω, corresponding to
the beating of the higher-order shifted modes with the primary mode and with one
another. These higher-frequency beat signals we much weaker than the structure at
2Ω, because the higher-order shifted modes themselves were much weaker than the
primary mode.
Important to modulation was the structure of the central peak of the beat signal.
Unfortunately, the magnitude of the peak was quite noisy. We expect that this noise
42
Chapter 3. Results and Discussion
RF spectrum HdBL
-90
-100
-110
-120
-0.05
0.00
0.05
Relative frequency HMHzL
Figure 3.9: ESA spectrum: AOM-shifted signal. Beat frequencies arising when AOM
is turned on and aligned correctly. The central peak lies near twice the modulation frequency
(2Ω ≈ 180 MHz in this case). Relative frequencies are given since the structure is virtually
independent of Ω.
RF spectrum HmVL
12
10
8
6
4
2
0
175.175
175.180
175.185 175.190
Frequency HMHzL
175.195
175.200
Figure 3.10: ESA spectrum: Detail of AOM-shifted signal. Plot averaged over eight
acquisitions. The linear scale helps emphasize the importance of the central peak compared
to the rest of the beat spectrum generated by the AOM.
3.2. Approaching Modelocking
43
RF spectrum !dB"
!60
a)
Primary
!70
!80
Second Primary
& Shifted
!90
!100
!110
163.0
163.5
164.0
164.5
165.0
Frequency !MHz"
Primary
!70
!80
!90
!100
!110
165.5
RF spectrum !dB"
RF spectrum !dB"
!60
Primary
& Shifted
(2Ω)
b)
166.0
163.4 163.5 163.6 163.7 163.8 163.9 164.0
!60
!70
!80
c)
Primary
& Shifted
(2Ω)
!90
!100
!110
!120
RF spectrum !dB"
!60
!70
163.4 163.5 163.6 163.7 163.8 163.9 164.0
d)
!80
!90
!100
!110
163.4 163.5 163.6 163.7 163.8 163.9 164.0
RF spectrum !dB"
!60
e)
!70
!80
!90
!100
!110
163.4 163.5 163.6 163.7 163.8 163.9 164.0
Frequency !MHz"
Figure 3.11: Beat frequencies visible on the ESA as the detuning of the modulation frequency from the axial mode spacing is reduced. (a) Beat signals when two primary modes
lase simultaneously. Each primary mode beats with the first shifted mode. (b) Detail of
beat between primary modes alone. Dashed line used to indicate ∆ν in following figures.
(c) Beat signal as shifted modes approach axial mode spacings. Dotted line indicates 2Ω.
(d) Noise due to rapid switching in and out of modelocking. (e) Shifted modes within
modelocking region (2Ω ≈ ∆ν).
44
Chapter 3. Results and Discussion
was from the vibrations of the AOM itself. Fig. 3.10 shows the average of eight plots
of the structure of the central peak, taken over about 30 seconds. Note that this
figure is vertically linear, whereas the other ESA plots have been logarithmic, so we
are focusing on only the most intense beat frequencies. The upshot of averaging the
noisy signal over time is that it allows us to estimate an upper bound on the FHWM
of the central AOM beat frequency. In this case, the FWHM is about 3.2 kHz, which
was on the same order of magnitude as the FWHM of the primary-primary beat.
This suggests that the noise in the modulation would not a have a huge impact on
the modelocking of the laser.
With the AOM turned on and detuned, it was still possible (although uncommon)
for two primary modes to lase simultaneously, just as we saw in the previous section
on unmodulated operation. In this case, there were three significant beat frequencies:
one between the two primary modes, one between the first shifted mode and the first
primary mode, and one between the second primary mode and first shifted mode. The
occurrence of three beat signals can be seen in Fig. 3.11 (a). This gave us an indication
about what the axial mode spacing (the primary-primary beat frequency, Fig. 3.11
(b)) was at that point, and how far we would have to tune the modulation frequency
so that the shifted modes lined up with axial modes. Fig. 3.11 (c) through (e) show
the structure of the beat signals as the modulation frequency Ω was decreased. The
dashed line represents the axial mode spacing frequency ∆ν and the dotted line
represents 2Ω. Part (c) shows the beat structure that arose at small detuning. The
entirety of the shifted beat signal was amplified, including the ±60 kHz sidebands
flanking the peak. Part (d) shows the behavior on the edge of the modelocking
region, with detuning around 40 kHz: significant noise appears near 2Ω as the laser
rapidly switches in and out of modelocking. In part (e), the detuning is small enough
(∼20 kHz) that the laser is modelocked. The noise is reduced and the peak of the
beat signal is amplified significantly.
3.3
Modelocking Behavior
When the beat signal was tuned to within about 20 kHz of the axial mode spacing, modelocking occurred. Fig. 3.12 shows the modelocked intensity waveform, with
pulses separated by about 6 ns, with a FWHM of about 2 ns. The laser maintained
this output as long as the modulation frequency did not drift outside of the locking
range. The average output signal measured by the photodiode during this acquisition
was 0.689 mV, which was higher than the unmodulated case of 0.425 mV, probably because modelocking fills more of the gain medium’s bandwidth. For reference,
Fig. 3.13 shows the beat signal for this modelocked data, showing a structure very
similar to Fig. 3.9.
Figure 3.14 shows the optical spectrum corresponding to Fig. 3.12. We can see
that at least four modes were populated. This figure has the same vertical scale as
in Figs. 3.3 and 3.6, so we can see that the most intense mode has about half the
intensity of the single mode that arises in the unmodulated case, but considering all
four modes, there is a bit more total power in the spectrum now than there was in
3.3. Modelocking Behavior
45
Intensity HmVL
1.5
1.0
0.5
0.0
-20
-10
0
Time HnsL
10
20
Figure 3.12: Timeseries waveform: Modelocked. Modelocked output at 91 cm. FWHM
of pulses is about 2 ns.
-60
RF spectrum HdBL
-70
-80
-90
-100
-110
-120
163.5
163.6
163.7
Frequency HMHzL
163.8
Figure 3.13: ESA spectrum: Modelocked. Beat signal corresponding to Fig. 3.12. This
data was used to find 2Ω ≈ 163.698 MHz. Dashed line shows ∆ν ≈ 163.684 MHz.
46
Chapter 3. Results and Discussion
-500
Relative frequency HHzL
0 2W 4W 500
1000
-1
0
1500
Optical spectrum HmVL
40
30
20
10
0
1
2
Scan time HmsL
3
Figure 3.14: FPI spectrum: Modelocked. Here 2Ω ≈ 163.698 MHz. Note that vertical
scale is same as in Figs. 3.3 and 3.6.
the unmodulated case.
The selection of the zero-point on the horizontal axis of Fig. 3.14 is not arbitrary.
In fact, there is a very weak mode centered at approximately this point, and this mode
lines up with the single primary mode that is visible when the laser is unmodulated.
This is shown in Fig. 3.15, where a modelocked spectrum is shown as a solid line,
and the corresponding unmodulated spectrum is shown as a dashed line, which was
acquired by turning off the AOM. In order to show the detailed structure of the
modelocked spectrum, the FPI’s amplification of the spectrum was increased tenfold
above the other FPI plots in this thesis. In order to show the (far more intense)
unmodulated signal on the same plot, both sets of data were individually rescaled,
so the scales of the two lines should not be considered accurate. What is important,
though, is that the unmodulated mode overlaps with the the first, very weak mode in
the modelocked spectrum. This may seem counterintuitive: modelocking has caused
higher-frequency modes to become more intense than the central primary mode. We
expect that this behavior can be attributed to two facts. The first is the fact that,
because our experimental set up only utilized light that was up-shifted in frequency,
the flow of optical power due to modulation is unidirectional. This creates a Gaussianshaped envelope of mode intensities which becomes wider and is pushed rightwards
in frequency-space as time progresses. The second fact we must consider is the effect
of gain, gain saturation, and cavity losses. While power is being transferred to higher
frequencies, lower frequencies (closer to the gain peak) experiences more gain than
higher frequencies, and this increases the intensity of lower-frequency modes over
time. Uniform cavity losses prevent modes far from the gain peak from becoming
very intense. By considering all of these effects together, it is plausible that the
modelocked laser can achieve the steady states indicated in Figs. 3.14 and 3.15.
3.3. Modelocking Behavior
Optical spectrum Ha.u.L
-250
1.0
0
47
Relative frequency HMHzL
2W
4W
500
750
1000
0.8
0.6
0.4
0.2
0.0
-0.5
0.0
0.5
1.0
Scan time HmsL
1.5
2.0
Figure 3.15: FPI spectrum: Modelocked vs. unmodulated. Amplified modelocked
spectrum (solid line) and unmodulated spectrum (dashed line), showing relative frequencies
of modulated and unmodulated modes. Shows distribution of modelocked mode power
relative to primary mode. Note that the two plots are scaled for simpler comparison.
It is important to reiterate that the existence of a ‘modelocking range,’ a range
of detunings over which the laser stays modelocked, is a direct result of injection
locking. If the laser system were completely linear, no injection locking would occur,
and modelocking would require a perfectly tuned modulation frequency as well as a
stabilization equipment to reduce phase noise. It is difficult to show the frequency
pulling associated with injection locking directly, because the FPI signal is noisy and
the horizontal scale is too coarse to notice small frequency shifts. Frequency pulling
can, however, be seen as a shift of the beat frequency between primary modes on
the ESA, as in Fig. 3.16. The thin line in this figure shows the primary-primary beat
frequency when the AOM is turned off, while the thicker line shows the modelocked
beat signal when the AOM is turned on and correctly tuned. Readers will have to
trust the author that the unmodulated primary-primary beat frequency was never
seen as long as the laser was modelocked, because the injection of frequency-shifted
light pulled the frequencies of the primary cavity modes to coincide with the injected
modes. The best evidence that the thicker line represents modelocking is the fact
that the modulated beat frequency is about 9 decibels greater in magnitude than
the unmodulated beat, indicating that there are more modes, and that the phase
differences between mode pairs is more uniform. This coupling of mode phases was
another result of injection locking.
Assuming that the modelocking range for our laser was about 20 kHz above and
below ∆ν, we can use the injection locking range equation 1.64,
r
Iinj
≈ 20 kHz,
(3.1)
=γ
I0
48
Chapter 3. Results and Discussion
RF spectrum HdBL
-60
-70
-80
-90
-100
-110
-120
163.60
163.65
163.70
Frequency HMHzL
163.75
163.80
Figure 3.16: ESA spectrum: Modelocked vs. unmodulated. Unmodulated beat
signal (thin line) compared to modelocked beat signal (thick line). The displacement of the
peaks indicates that the primary modes were frequency-shifted by about 17.4 kHz in the
presence of the injected modes. This is evidence of the frequency-pulling associated with
injection locking.
to learn more about how our laser operated. Here where Iinj is the intensity of the
injected signal, I0 is the intensity of the undisturbed mode, both measured inside
δ
the cavity. The cavity decay rate γ ≈ TRT
, where δ is the percentage of light that
is emitted from the output coupler mirror, and TRT ≈ 6.25 ns is the round-trip
time through the cavity. Estimating δ ≈ 1%, the ratio of injected mode intensity
to undisturbed mode intensity was about 0.02%. This means that very little light,
after being split by the AOM, reflecting off the third mirror, being split by the AOM
a second time, and transmitting through the ouput coupler, made it back into the
primary cavity. However, this was enough light to broaden the injection locking range
to a point where modelocking could be achieved without too much difficulty.
Conclusion
In this thesis, we have introduced the theoretical components necessary for discussing
modelocking, utilized the theory to motivate the design of a particular modelocked
laser, and presented experimental evidence that such a laser operates as intended.
Although the depth of theoretical material presented may be exhausting, it is hardly
exhaustive. A more general and rigorous form of pulse-propagation modeling can be
accomplished by using Haus’ ‘Master equation of modelocking’ [Haus et al., 1991],
which is a partial differential equation that can accommodate active and passive
modelocking, and take into account complicated effects like intracavity dispersion
and nonlinearities of the gain medium that arise for high peak intensities. This
equation also allows for modeling of pulse shapes other than Gaussian pulses. Another
type of modeling in the frequency domain, using ‘coupled-mode’ equations, presents
analytical solutions that track how power is transferred between modes over time in
an actively-modelocked laser. This frequency-domain modeling can easily account for
detuning of the modulation frequency. If we wanted a more accurate model of how we
should expect our laser to operate, we could have applied both the Master equation
and the coupled-mode equations to our particular setup.
Our laser succeeded in producing pulses that were about 2 nanoseconds long,
although the low intensity of the laser suggests that we could not expect to use
its output for industrial or research applications. There are many other designs of
modelocked lasers that could have been studied and assembled, the limiting factors
being time available to the student and funds available from the institution. Other
laser designs could have been implemented that probably would have yielded shorter,
more intense pulses, such as an erbium-doped fiber ring laser, or a diode laser with AC
pumping at the round-trip frequency. If appreciable output powers could be obtained
with one of these laser designs, a second-harmonic crystal could be used in optical
autocorrelation measurements to discover the shape of pulses that are too fast for
photodiode detectors to measure.
References
Acoustooptics Database: Deflectors, 2001. URL http://acoustooptics.phys.msu.
su/deflectors.asp. MSU, Faculty of Physics, Oscillations Department.
Robert Adler. A study of locking phenomena in oscillators. Proceedings of the IEEE,
61(10):1380–1385, October 1973.
R. W. Boyd. Nonlinear Optics. Academic Press, 3 edition, 2008.
William S. C. Chang. PRINCIPLES OF LASERS AND OPTICS. Cambridge University Press, 2005.
Clark-MXR. Micromachining Handbook. Clark-MXR, Inc., 2009. URL http://www.
cmxr.com/mmhandbook.html. Electronically available publication.
Li Ding, Lana J. Nagy, Lisen Xu, Jack C-H. Chang, Jennifer Swanton, Wayne H.
Knox, and Krystel R. Huxlin. Enhancement of intra-tissue refractive index shaping (iris) of the cornea by two-photon absorption. In Conference on Lasers and
Electro-Optics/International Quantum Electronics Conference, page CWE4. Optical Society of America, 2009.
John F Federici, Brian Schulkin, Feng Huang, Dale Gary, Robert Barat, Filipe
Oliveira, and David Zimdars. Thz imaging and sensing for security applicationsexplosives, weapons and drugs. Semiconductor Science and Technology, 20(7):
S266, 2005.
M Ferray, A L’Huillier, X F Li, L A Lompre, G Mainfray, and C Manus. Multipleharmonic conversion of 1064 nm radiation in rare gases. Journal of Physics B, 21:
L31, 1988.
L. Curtis Foster, M. D. Ewy, and C. Burton Crumly. Laser Mode Locking By An
External Doppler Cell. Applied Physics Letters, 6:6–8, 1965.
L. E. Hargrove, R. L. Fork, and M. A. Pollack. Locking of he-ne laser modes induced
by synchronous intracavity modulation. Appl. Phys. Lett., 5:4–5, 1964.
H. A. Haus, J. G. Fujimoto, and E. P. Ippen. Structures for additive pulse mode
locking. J. Opt. Soc. Am. B, 8(10):2068–2076, 1991.
52
References
H.A. Haus. Mode-locking of lasers. IEEE Journal of Selected Topics in Quantum
Electronics, 6:1173–1185, Nov/Dec 2000.
R. J. Jones. Active Modelocking of a Helium-Neon Laser, 2009. URL http://www.
optics.arizona.edu/opti511l/2009_files/modelocking_Fall2009.pdf.
K Kurokawa. Injection locking of microwave solid-state oscillators. Proceedings of the
IEEE, 61(10):1386–1410, October 1973.
Willis E. Lamb. Theory of an optical maser. Phys. Rev., 134(6A):A1429–A1450, Jun
1964.
Corneliu Manescu. Controlling and probing atoms and molecules with ultrafast laser
pulses. PhD thesis, University of Florida, 2004.
M. G. Moharam and L. Young. Criterion for Bragg and Raman-Nath diffraction
regimes. Appl. Opt., 17(11):1757–1759, 1978.
T. M. Niebauer, James E. Faller, H. M. Godwin, John L. Hall, and R. L. Barger.
Frequency stability measurements on polarization-stabilized He-Ne lasers. Appl.
Opt., 27(7):1285–1289, 1988.
Susanne Pfalzner. An introduction to inertial confinement fusion. Taylor & Francis
Group, Boca Rotan, FL, 2006.
A. Pikovsky, M. Rosenblum, and J. Kurths. Synchronization: A universal concept in
nonlinear sciences. Number 12 in Cambridge Nonlinear Science Series. Cambridge
University Press, 2001.
Anthony E. Siegman. Lasers. University Science Books, 1986.
D. H. Sutter, G. Steinmeyer, L. Gallmann, N. Matuschek, F. Morier-Genoud,
U. Keller, V. Scheuer, G. Angelow, and T. Tschudi. Semiconductor saturableabsorber mirror assisted Kerr-lens mode-locked Ti:sapphire laser producing pulses
in the two-cycle regime. Opt. Lett., 24(9):631–633, 1999.
Osamu Wada. Femtosecond all-optical devices for ultrafast communication and signal
processing. New Journal of Physics, 6(1):183, 2004.
M. Wollenhaupt, A. Assion, and T. Baumert. Section 12.2: Generation of Femtosecond Laser Pulses via Mode Locking. In Frank Trager, editor, Springer Handbook
of Lasers and Optics, chapter C.12. Springer, 2007.