Download Quadrature frequency doubling

University of Ljubljana
Faculty of mathematics and physics
Seminar
Quadrature frequency doubling
Gregor Tavčar
Mentor
Prof. Dr. Irena Drevenšek Olenik
Ljubljana, October 2008
Abstract
Quadrature frequency doubling is an innovative solution for problems of real life frequency doubling
of laser light. This seminar presents the physical equations governing the ideal and the general nonideal process of frequency doubling; addresses the problems posed by the real life non-ideal
behaviour of frequency doubling; explains the physical principles behind the quadrature frequency
doubling solution; presents an application where this solution has been put to use.
Quadrature frequency doubling
Contents
1
Introduction ......................................................................................................................... 3
2
Light – matter interaction...................................................................................................... 3
3
2.1
Case of frequency doubling .....................................................................................................4
2.2
Type II phase matching............................................................................................................6
Behaviour of frequency doubling in non-ideal conditions ....................................................... 8
3.1
Effects of power imbalance .....................................................................................................9
3.2
Effects of phase mismatch ....................................................................................................11
4
Problems of real life frequency doubling ............................................................................. 12
5
Quadrature frequency doubling .......................................................................................... 14
6
7
5.1
Results ...................................................................................................................................15
5.2
Disadvantages and alternative ways .....................................................................................16
Application in medicine....................................................................................................... 16
6.1
Glaucoma...............................................................................................................................16
6.2
SLT .........................................................................................................................................17
Bibliography ....................................................................................................................... 18
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Quadrature frequency doubling
1 Introduction
The principles of non- linear optics are often found at play in many contemporary laser applications.
Since lasing materials available for practical applications give laser light of only few different
wavelengths the laser beam of desired wavelength is commonly achieved with the help of materials
with nonlinear optical properties. One such, and perhaps the most basic, type of wavelength
modification is the so called frequency doubling or second harmonic generation (SHG). In this case a
laser beam of a given wavelength enters a specific piece of nonlinear optical material and exits it as a
beam of half the original wavelength (twice the original frequency, hence the name frequency
doubling).
This seminar will present some characteristics and problems of real life frequency doubling and also
one “slick trick” called the Quadrature frequency doubling that solves some of the problems.
I encountered the aforementioned problems with real life frequency doubling in the process of
development of a high precision ophthalmological laser apparatus and in the search for some
solution to those problems the Quadrature frequency doubling solution was found in scientific
literature [1] [2].
2 Light – matter interaction
The behaviour of electro-magnetic waves entering and travelling through matter is described by the
electric and magnetic properties of that matter. Since we shall be dealing with materials with
negligible magnetic properties the discussion will be limited only to electrical aspects of light –
matter interaction.
In all physical systems we know of linear and nonlinear response to a given input and the same holds
true for light-matter interaction. In this case we treat the electric field strength of the incident EM
wave as the system input, and the electric polarisation of the matter as the corresponding response.
(1)
where χ and d are electric properties of the given matter. We recognise as the linear electric
susceptibility (which for a general anisotropic matter is a second rank tensor) that manifests itself in
the refractive index. The material property described by the parameter , a third rank tensor, is
responsible for the quadratic response. In general couples all components of incident electric field
with one another, with themselves and with the components of resulting polarisation.
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Quadrature frequency doubling
2.1 Case of frequency doubling
$
!"# &' ()*+,- ./ ./ #
%
For a monochromatic incident plane wave
the quadratic response gives rise to frequency doubling. Multiplication of a harmonic oscillation with
itself gives:
$
(- 0 (- ($ (%-%
(3)
a constant term and a harmonic oscillation of twice the frequency (2ω)
The polarisation oscillating with the double frequency in turn radiates EM waves of that same
double frequency. This is easily seen from the Maxwell’s equations written in the following way:
12 4 2 3 4 2 5
2
2
4 2
. 4
term.
where the linear part of polarisation is contained in the 3 (4)
contains double frequency oscillations so must and so the amplitude can be
+ 2+
considered as being of two parts Since
The nonlinear wave equation (4) with some simplifications translates to the coupled nonlinear
differential equations [3]:
5 3 2
4&2+
78
&+ &+ ' :)
2
46
9+
4&+
46
5 3 2
78
&2+ &0+ ' *:)
2
92+
4
(5)
(6)
Quadrature frequency doubling
+ and
where &
&2+ are the amplitudes of electric fields + and 2+ ; 9+ and 92+ are the
refractive indices for the light of the fundamental and the double frequency respectively; ;! +
!2+ %!+ % (92+ 9+ - is a measure of rate of dephasing (or phase mismatch).
<
Due to dispersion the speed of light travelling through matter is generally not the same for the light
of fundamental frequency and for the light of double frequency. This difference in speed is
responsible for the dephasing that builds up along the trajectory through the matter (In case of type
II phase matching 9
+ is the average of the ordinary and extraordinary refractive indices: see 2.2).
the incident light the phase matching condition (i.e. ;!
=) can be met.
However, with proper orientation of the optical axes of a suitable nonlinear crystal with respect to
DEE F&+ (=-F6
>2+ (6- >+ (=- ?@AB C
G
9+ .
In the latter ideal case the coupled differential equations (5) and (6) have the following solution [4]:
2
>+ (6 - >+ (=- C$
DEE F&+ (=-F6
?@AB C
GG
9+ .
(7)
2
Where >+ and >2+ denote the intensities of the original and of the frequency doubled light and
DEE is the effective element ofH .
(8)
As required by energy conservation two original frequency photons convert into a single double
frequency photon. The rate of the conversion depends strongly on the intensity (photon density) of
the incident light as seen on Fig 1.
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Quadrature frequency doubling
Fig. 1. Plot of frequency conversion for two different intensities of incident light. Red curves represent
the intensity of base light (8), the green ones represent the intensity of frequency doubled light (7).
The lower intensity - darker curves- and higher intensity – brighter curves-differ by a factor of four.
The frequency conversion is considerably slower for the lower intensity, but both asymptotically
approach complete conversion. In theory practically complete conversion can be achieved with a
long enough path through the SHG medium.
2.2 Type II phase matching
In suitable birefringent crystals phase matching can be achieved by matching the refractive index of
the ordinary polarisation of the double frequency light with the average refractive index of the
ordinary and extraordinary polarisation of the fundamental light [4]. In other words:
!2+IJ
!+IJ
!+)IJ = (see Fig. 2) This condition is called type II phase
matching. Frequency doubling realised via type II phase matching couples the ordinary and
extraordinary component of polarisation of incoming light to produce double frequency light of
ordinary polarisation. In quantum terms: two photons of original light one of ordinary and the other
of extraordinary polarisation produce one double-frequency photon of ordinary polarisation [4]. For
optimal conversion we thus need equal amounts of ordinary and extraordinary polarisation of the
incident light.
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Quadrature frequency doubling
nz
nw xord
n2 w
nw ord
j
ny
Fig. 2. Index ellipsoid. The red line corresponds to the fundamental light, the green line corresponds
to the SHG light. At the angleKbetween the optical axis of the crystal and the direction of the
incident light the mean of nω ord and nω xord reaches exactly n2ω ord.
Fig. 3. Schematic drawing of the SHG process in type II phase matching. Light of frequency ω (red)
having both ordinary and extraordinary components of polarisation and correct angle of incidence is,
traversing SHG crystal, converted into light of frequency 2ω (green) of ordinary polarisation. [5]
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Quadrature frequency doubling
3 Behaviour of frequency doubling in non-ideal conditions
In real life neither perfect phase matching nor perfect power balance (i.e. equal amounts of the two
components of polarisation) can be realised. With phase mismatch the reconversion starts
occurring, that is that the already generated double frequency light undergoes the reverse process,
namely conversion back into the original light. Same holds true for power imbalance.
How bad this reconversion is depends on the degree of phase mismatch and power imbalance. How
far along the path through the SHG material this reconversion starts occurring also depends on the
intensity of the incident light.
The solution of the coupled differential equations (5) and (6) in case of nonzero ;! leads to the
following equation for the efficiency of conversion [2] [3]:
L LM) N92 O8
L
LM)
P QR
N9 is an elliptical Jacobi function
(9)
where [2]:
L
STU
SU
is conversion efficiency
L is the drive – a quantity proportional to the intensity of incident light and to the square of the
LM) is maximum efficiency dependant on relative rate of dephasing Δk and power imbalance δ
Modulus Q is some measure of dephasing and power imbalance.
length of the path traversed through the SHG medium
separately, namely, to observe the N9 function first as a function of drive and power imbalance
To visualise what equation (9) represents it is best to treat phase mismatch and power imbalance
with perfect phase matching assumed and then as a function of drive and rate of dephasing with
perfect power balance assumed [2].
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Quadrature frequency doubling
3.1 Effects of power imbalance
Here we assume perfect phase matching (Δk=0) and observe how power imbalance affects the
efficiency of conversion. The quantities from (9) are in this case [2]:
L
2
2
>+ (=-DEE 6 V
%
>I[
ZL
Z
Q
W
M)
$W
>\]\
$ 2
W X$ Y
W
where >I[ is the lower of the two intensities associated with the ordinary and extraordinary
(10)
polarisation. >\]\ is thus the higher of the two intensities. (>\]\ >I[ >+ (=--
Fig. 4. 3D plot of conversion efficiency as a functionof the power imbalance δ and the length of path
traversed through SHG crystal with perfect phase matching assumed (Δk=0). (arbitrary units of x)
At the back end of the plot in Fig. 4 we can quickly recognise tanh2 shape of the ideal power
balanced efficiency (7). Increasing power imbalance causes oscilations in efficiency but they die
away at high power imbalance (δ close to zero). This oscilations in eficiency are not
sinusoidal. Also apparent from the plot is the decreasing of the maximal efficiency with
increasing power imbalance.
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Quadrature frequency doubling
Fig. 5. Plot of frequency conversion for two different intensities of incident light at power imbalance δ
= 0.8 (Δk=0). The lower intensity (darker curve) and higher intensity (brighter curve) differ by a factor
of four.
Similar to the curves in Fig. 1., we see in Fig. 5 that the processes of conversion and reconversion are
slower for lower intensity of incident light. A consequence of this is that the depth at which the
maximum efficiency is reached depends on the intensity of the incident light. The maximum
efficiency itself, however, does not depend on the intensity. A SHG crystal of length that optimally
converts light of particular intensity at conditions as in Fig.5 would perform worse with light of
different intensity.
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Quadrature frequency doubling
3.2 Effects of phase mismatch
Here we assume perfect power balance and observe how phase mismatch affects the efficiency of
conversion. The quantities from eq. (9) are in this case [3]:
LM) ^Q L
2
2
>+ (=-DEE 6
$_
%;! 2 ^%;!
$_ O
2 R
&+ (=-DEE 8
`&+ (=-a2 DEE
2
(11)
Fig. 6. 3D plot of conversion efficiency as a function of the rate of dephasing Δk and length of path
traversed through SHG crystal with perfect power balance assumed (δ=0). (arbitrary units of x and
Δk)
At the back end of the plot in Fig. 6 we again recognise tanh2 shape of the ideal phase-matched
efficiency (7). Increasing Δk causes oscilations in efficiency wihch is what we would expect
from the eiΔkx term in the equations (5) and (6). This oscilations in eficiency are not
sinusoidal. Also apparent from the plot is the decreasing of the maximal efficiency with
increasing dephasing.
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Quadrature frequency doubling
Fig. 7. Plot of frequency conversion for two different intensities of incident light at rate of dephasing
Δk* = 0.05 (δ=0). The lower intensity (darker curve) and higher intensity (brighter curve) differ by a
factor of four. (Δk*= Δk/(Aω(0)deff√2) )
In Fig. 7 we again see a slower rate of conversion/reconversion for the lower intensity of light. Thus,
similarly to the case of power imbalance, the optimal length of the SHG crystal, given the conditions
as in Fig. 7, depends on the intensity of incident light. But in contrast to the case of power imbalance
here the maximal efficiency does depend on the intensity of incident light. This is due to the
following reason: The rate of reconversion depends on the phase mismatch i.e. the phase difference
Δφ = Δk·x that accumulates with depth. Since the higher intensity reaches its efficiency maximum at
a shorter depth (smaller x) the built up phase difference is not as bad as it is for lower intensity that
reaches its efficiency maximum at greater depth (larger x).
4 Problems of real life frequency doubling
From considerations in the previous sections we see that in non-ideal conditions the length of the
SHG crystal is highly important for the efficiency of frequency doubling. This fact brings its own set of
problems in real life frequency doubling.
The cross-sectional intensity distribution in laser beams is rarely uniform. It most commonly has
Gaussian bell-shape. Since different portions of the beam would, due to their different intensity,
require different lengths of frequency doubling element very high efficiencies are in reality not
achievable.
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Quadrature frequency doubling
Fig. 8. A typical crossectional profile of a laser beam
Overcoming the limit of highest possible efficiency may be a goal in applications seeking high
optimization of conversion but that was not the main challenge I was faced with.
The main problem arose when converting pulsed laser beams. A pulsed laser operating at preset
pulse energy emits pulses with their actual energies fluctuating around a mean value within some
tolerance limits. Furthermore the cross-sectional distribution of intensity of those pulsed beams also
varies to some degree from pulse to pulse.
Fig. 9. Crossectional profiles of two consecutive pulses of the same energy but with notably different
intensity distributions. [5]
Since the efficiency of conversion is strongly dependant to the power density of incident light both
these two effects cause the pulses after undergoing the frequency conversion process to fluctuate in
energy much more than originally.
It can be easily imagined that those enhanced fluctuations can fall beyond the limits of operational
deviations allowed by the standards.
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Quadrature frequency doubling
Correlation 532nm/1064nm
single KTP SHG
1.2
1.2
1.15
1.15
1.1
1.1
1.05
1.05
532nm
532nm
Correlation 532nm/1064nm
single KTP SHG
1
1
0.95
0.95
0.9
0.9
0.85
0.85
0.8
0.8
0.8
0.9
1
1.1
1.2
0.96
1064nm
0.98
1
1.02
1.04
1064nm
Fig. 10., Fig. 11. (Fig.11. is a stretched version of Fig. 10.) Energy fluctuations (normalised)of SHG
pulses at λ= 532nm in relation to the fundamental NdYAG pulses at λ=1064nm. Each point measures
the power of one fundamental pulse and the power of the one resulting SHG pulse. Standard
deviations: σ1064=1.5%; σ532=7.1% .Measurements of SLT R &D. [6]
Fig. 10 shows that SHG worsens the fluctuations by a factor of almost five. Since Fig. 11 shows
practically no correlation between deviations of the fundamental and the SHG pulses the SHG pulses
seem to fluctuate mainly due to fluctuations in intensity distribution of fundamental pulses.
5 Quadrature frequency doubling
The “Quadrature frequency doubling” proposes a solution with two differently oriented frequency
doubling elements. The SHG material I was dealing with were KTP crystals (KTiOPO4-Potassium
titanyl phosphate) for frequency doubling the light of Nd-YAG lasers: λ = 1064nm.
The first element is positioned as usual. The second one is placed right behind the first one and is
positioned in similar manner but rotated through 90° around the trajectory of the incident light (see
Fig. 12) [2]. Since the polarisation of fundamental incident light has equal amounts of horizontal and
vertical components it produces vertically polarised SHG light in the first element and horizontally
polarised SHG light in the second one.
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Quadrature frequency doubling
Fig. 12. Schematic drawing of the Quadrature SHG. Fundamental incident light (red) is polarised at
an angle of 45° with respect to the plane in which the optical axis lies (gray) See fig. 3. [5]
This arrangement differs from a single but longer element in that that the double-frequency light
exiting the first element is wrongly polarised to be reconverted in the second element. Thus the
remainder of the original light that was not converted by the first element gets another chance for
conversion in the second element.
To achieve high conversion efficiency of that remainder of original light selecting appropriate length
of the second element is of course important. Furthermore, the two elements being different in
length can be viewed as being responsible for different intensity regions of the beam (the shorter
one for the higher power densities, the longer for lower power densities).
5.1 Results
In my case the implementation of Quadrature frequency doubling method gave very good results in
the laboratory. It reduced the energy fluctuations of the SHG pulses by a factor of two [6]. See Fig.
10 and 13
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Quadrature frequency doubling
Correlation 532nm/1064nm
Quadrature SHG
1.2
1.2
1.15
1.15
1.1
1.1
1.05
1.05
532nm
532nm
Correlation 532nm/1064nm
single KTP SHG
1
1
0.95
0.95
0.9
0.9
0.85
0.85
0.8
0.8
0.8
0.9
1
1.1
1.2
0.8
1064nm
0.9
1
1.1
1.2
1064nm
Fig. 10., Fig. 13. Comparison of energy fluctuations of the Quadrature SHG and the single crystal
SHG. Standard deviations: σsingle = 7.1%; σquad = 3.7%. [6]
5.2 Disadvantages and alternative ways
With quadrature frequency conversion the portion of each polarisation in the final beam varies from
pulse to pulse in a similar way the net energy varies in a single element frequency conversion. Even
more varies the phase difference between the two polarisations. All this makes the net polarisation
very indefinable. For applications requiring accurately polarised light this solution is inappropriate.
An obvious alternative way of designing a frequency doubled laser apparatus is to place the SHG
element within the laser’s resonator itself. The mirrors of the resonator in this case are such that
one totally reflects lights of both frequencies and the other one reflects only the light of original
frequency. Such arrangement utilises the laser’s own self-adjusting feedback that takes care of
stable operation.
6 Application in medicine
6.1 Glaucoma
Raised intraocular pressure causes damage to the optical nerve which is a disease called glaucoma
[7] [8]. One of the causes of elevated intraocular pressure is blockage of the trabecular meshwork.
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Quadrature frequency doubling
The trabecular meshwork, located around the cornea of the eye, is a sort of a micro sieve through
which intraocular fluid drains back into veins. If this drainage is obstructed intraocular fluid
accumulates in the eye and consequently raises the intraocular pressure [8].
Fig. 14. Schematic depiction of a normal drainage of intraocular fluid through the trabecular
meshwork. [8]
6.2 SLT
Selective laser trabeculoplasty (SLT) is an ophthalmologic procedure in which a certain type of
pigmented cells in the trabecular meshwork are targeted with pulses of laser light of appropriate
wavelength. The consequent death of those cells triggers an immune response that results in
unblocking of the trabecular meshwork. The light of 532nm wavelength turns out to be very
appropriate for absorption in those pigmented cells. The 532nm light is obtained by frequency
doubling the 1064nm light produced by NdYAG laser which is a wide spread commercial laser
source.
The intra-cavity SHG laser seems to be the preferred design choice in this case since it does not
introduce enhanced fluctuations of energy of output pulses. This solution, however, requires a
development and design of an entirely new laser head, meaning, it cannot be just a simple upgrade
of the basic laser.
The company I worked for was already producing 1064nm NdYAG lasers and laser apparatuses
incorporating such lasers so the goal was to develop a frequency doubled SLT laser apparatus by
using the already existing Nd YAGs. This of course called for the solution of extra-cavity placed SHG
element(s).
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Quadrature frequency doubling
7 Conclusion
Frequency doubling is a technique commonly used in the design of various laser applications. It, in its
simplest version (extra-cavity arrangement), suffers from rather low efficiency of frequency
conversion and poor energy stability of output light. The discussed quadrature frequency doubling is
a smart solution addressing these problems. It has proven so useful that we have recently used it in
the design of a new generation of a sophisticated medical laser apparatus where the main goal was
to minimise the pulse-to-pulse energy fluctuations. The use of the quadrature frequency doubling
reduced those energy fluctuations by a factor of almost two compared to the previous generation of
that laser apparatus.
8 Bibliography
1. Koechner, Walter. Solid-state laser engeneering. Berlin : Springer, 1996. ISBN 3-540-60237-2.
2. Eimerl, David. Quadrature frequency conversion. IEEE journal of quantum electronics. 1987, Vols.
QE-23 No. 8.
3. Perlov, Dan and Roth, Michael. Efficiency of non-phas-matched second-harmonic generation by
Gaussian pulses. Applied optics. 1997, Vols. 36, No. 21.
4. Wikipedia. Second harmonic generation. Wikipedia, the free encyclopedia. [Online] [Cited: 14
October 2008.] http://en.wikipedia.org/wiki/Second_harmonic_generation.
5. Tavčar, Gregor. SLT. Ljubljana : Optotek d.o.o. (internal report), 2006.
6. —. SLT izboljšave. Ljubljana : Optotek d.o.o. (internal report), 2007.
7. Wikipedia. Glaucoma. Wikipedia, the free encyclopedia. [Online] [Cited: 18 October 2008.]
http://en.wikipedia.org/wiki/Glaucoma.
8. Adatia, Feisal A. and Damji, Karim F. Chronic open-angle glaucoma. Canadian Family Physician.
[Online] September 2005. [Cited: 18 October 2008.] http://www.cfpc.ca/cfp/2005/Sep/vol51-sepcme-3.asp.
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