Self-phase locking in a type-II phase-matched
optical parametric oscillator
by
Elliott J. Mason, III
Submitted to the Department of Electrical Engineering and
Computer Science
in partial fulfillment of the requirements for the degree of
Master of Engineering in Electrical Engineering and Computer
Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 1996
© Elliott J. Mason, III, MCMXCVI. All rights reserved.
The author hereby grants to MIT permission to reproduce and
distribute publicly paper and electronic copies of this thesis
document in whole or in part, and to grant others the right to do so.
JUN 11 1996
Author .....
........
ES..........
Departmew( of Electrical Engineering and Computer Science
May 28, 1996
Certified by........
Research
Accepted by....
V
..
Suntjs t , Researp
INgal Chuen Wong
Laboratory of Electronics
,
Thesis Supervisor
................
Chairman, Departme
'N eFrederic R. Morgrenthaler
Committee on Graduate Theses
.SA"
Contents
1 Introduction
1.1
Previous Work .....................................
1.2
Self-Phase Locking Experiment .........................
13
2 Theory
2.1
2.2
2.3
Resonant Second-Harmonic Generation . . . . . . . .
.. . . .
.
13
.. . . .
.
13
. . . . . . .
15
. . . . . . .
16
The Doubly Resonant Optical Parametric Oscillator.
. . . . . . .
17
2.2.1
The Double Resonance Threshold . . . . . . .
.. . . .
.
18
2.2.2
Frequency Tuning .
. . . .
.
20
2.1.1
Single-Pass Conversion ...............
2.1.2
Enhancement Factor .. . . . . . .
2.1.3
Intracavity Loss and Optimum Input Coupling
.....
. .....
.
..........
Injection Locking ...........
. ...
2.3.1
The Adler Equation
.
.....
2.3.2
Signal-Idler Mutual Injection
....
........
. . . . . . . ..
.......
21
. . . . . . .
22
. . . . . . .
23
3 Pump Laser System
4
3.1
Cavity Design ..........
3.2
System Performance
25
...
.......
........
.
......
..........
.............
27
Self-Phase Locking Experiment
4.1
Experimental Setup ...
4.1.1
30
..............
.
DRO Cavity Design .....
25
.
....
. . . . . . .
.. . .......
. .
. . . . . ....
.
30
30
Self-phase locking in a type-II phase-matched
optical parametric oscillator
by
Elliott J. Mason, III
Submitted to the Department of Electrical Engineering and Computer Science
on May 28, 1996, in partial fulfillment of the
requirements for the degree of
Master of Engineering in Electrical Engineering and Computer Science
Abstract
Applications in optical frequency metrology often require the synthesis of exact ratios
between input and output frequencies. Specifically, in the realization of an opticalto-microwave frequency chain it is useful to construct an exact 2:1 optical frequency
divider. One approach to optical frequency division is based on optical parametric
oscillators (OPO's). To achieve a 2-to-1 frequency ratio, the OPO must be phaselocked at the frequency degenerate point. It has been demonstrated that a type-I
phase matched OPO can be operated in a self-phase locked regime at frequency
degeneracy. A type-II phase matched OPO, which is better suited for precision frequency measurements, does not in general exhibit self-phase locking. In this thesis
we present results which demonstrate self-phase locking in a type-II phase matched
OPO by the use of intracavity polarization mixing.
Thesis Supervisor: Ngai Chuen Wong
Title: Research Scientist, Research Laboratory of Electronics
Acknowledgments
I would like to express my appreciation to Dr. Franco Wong for his guidance and
support these past two years. It has been a privilege to learn from his experience
and insight which have made this thesis possible. He has truly made the challenges
of research an enjoyable experience.
I would also like to thank all the members of my group including Reggie Brothers
for his support and his friendship, and Joe Teja for helping out at a very critical
moment (and struggling with me through the joys of quantum electronics).
I am
greatful for all of the encouraging words, insightful questions, and well timed jokes.
I would also like to thank those who have supported me in so many ways during
my time at MIT (so far). My parents and grandparents have been a constant source
of encouragement to me through their love and their prayers. It is a blessing to have
friends who make my life here so much more than school. I especially thank Tony
for his kindness and his cheer and Brigette for her love and understanding. Above all
I acknowledge that all my success comes from God who has been my strength, my
teacher, my hope, and my peace.
4.1.2
4.2
Frequency Diagnostic System
. . . . . . . . . . . . . . . . . .
31
Experimental Observations ........................
33
4.2.1
Cavity Losses ...........................
33
4.2.2
Frequency Tuning of SDRO ...................
39
4.2.3
Frequency Tuning and Self-Phase Locking of MDRO
. . . . .
41
5 Conclusion: Summary and Future Work
47
A The Nonlinear Wave Equation
49
B Intensity Servo Circuits
52
List of Figures
1-1
Type-I self-phase locking experimental setup . . . . . . . . . . . . . .
1-2 Type-II self-phase locking experimental setup . . . . . . . . . ... .
2-1
..11
Fundamental and harmonic power in a ring enhancement cavity configuration.. . . . . . . . . . . . . . . .
3-1
10
. . . ..
. . . ... .
16
Pump system schematic. (HWP: half-wave plate; PZT: piezoelectric
transducer)
.............
. ..
..
..
.........
.
4-1
Side view of cavity configuration . . . . . . . . . . . . . . . . .....
4-2
Schematic of DRO setup with local oscillator set up in a heterodyne
..
27
32
detection configuration. (PBS: polarizing beamsplitter; AOM: acoustooptic modulator; QWP: quarter-wave plate; PZT: piezoelectric transducer; TE: thermo-electric) .......
4-3
. . . . . . .
. . . . ......
34
Schematic of DRO setup with local oscillator set up in a homodyne
detection configuration.
(PBS: polarizing beamsplitter; HWP: half-
wave plate; QWP: quarter-wave plate; PZT: piezoelectric transducer;
TE: thermo-electric)
4-4
.......
. . . . . .
. . . . . . ......
.
..
35
The frequency shifted reference beam used as an LO showed beats due
to the overlap of the different diffracted orders of the acousto-optic
modulator...................
4-5
. . .
.......
.......
....
36
Cavity loss as a function of wave plate angle based on finesse measurem ents . . . . . . . . . . .
..
. .
.
. . .
...
. . . . . ..
38
4-6
Spectrum analyzer trace of signal-idler beat note near frequency degeneracy. The beat frequency is around 5 MHz . . . . . . . . . . . .
40
4-7 The reference beat signal at 80 MHz has risen by 8 dB due to the
additional power from the degenerate MDRO output . . . . . . . . .
43
4-8 Comparison of 80 MHz beat signal before and after the MDRO was
self-phase locked reveals sidebands at 20 kHz due to the pump..... .
4-9
45
Interference pattern between the self-phase locked MDRO output and
the LO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
B-1 Fixed set point cavity length servo for the DRO cavity. . . . . . . . .
53
B-2 Dither-and-lock intensity servo for the SHG ring cavity. . . . . . . . .
54
Chapter 1
Introduction
Applications in optical frequency metrology often require the synthesis of exact ratios
between input and output frequencies. Specifically, in the realization of an opticalto-microwave frequency chain it is useful to construct an exact 2:1 optical frequency
divider. One approach to optical frequency division is based on optical parametric
oscillators (OPO's) [1]. In the parametric downconversion process power is transferred
from a pump wave at wp to the "signal" and "idler" subharmonic waves at w, and
wi, respectively, and the conservation of energy requires that wp = w, + wi. The case
where w, = wi is called frequency degeneracy. To achieve a 2-to-1 frequency ratio,
the OPO must be phase locked at the frequency degenerate point.
It has been demonstrated that a type-I phase matched OPO can be operated in a
self-phase locked regime at frequency degeneracy [2]. A type-II phase matched OPO,
which is better suited for precision frequency measurements, does not in general
exhibit self-phase locking. The purpose of this thesis is to present results of an
experiment investigating the the possibility of achieving self-phase locking in a type-II
phase matched OPO by adding intracavity polarization mixing to the OPO system.
In this chapter, previous work is described which will provide the context for the
current results. The experiment which was performed is then briefly introduced.
1.1
Previous Work
One method of achieving phase locking which has been demonstrated is to lock the
signal-idler beat note to a microwave reference frequency source [3]. This was done
using a type-II phase matched KTP OPO. The beat note was obtained by projecting
the orthogonally polarized signal and idler onto a common polarization axis. Then
the beat signal was detected and demodulated at a microwave mixer with respect
to a synthesized microwave reference signal. The phase error was fed to an electricfield (E-field) tuning circuit which corrected the signal-idler beat frequency and phase.
One advantage of this method is that the output frequencies can be tuned to different
values.
In another method it was demonstrated that the signal and idler waves of a type-I
phase matched MgO:LiNbO 3 OPO became self-phase locked at frequency degeneracy [2]. The setup for the experiment is shown in Fig. 1-1. The OPO was operated
near degeneracy so that AŽw = w, - wi
0. Also, in type-I phase matching the
signal and idler waves are co-polarized. Self-phase locking occurred because of the
strong mutual coupling of the indistinguishable signal and idler waves at degeneracy.
The degenerate output was combined with the 1064 nm reference from the Nd:YAG
laser to produce an interference pattern. When the OPO was tuned to degeneracy,
self-phase locking was observed. A stable fringe pattern was held for as long as 20
minutes with no additional tuning or external servo control. The simplicity of this
method is a great advantage. Even though no frequency tuning is possible, the exact
2:1 frequency division can be advantageous in many applications.
1.2
Self-Phase Locking Experiment
It is useful to obtain self-phase locking in a type-II phase-matched OPO system that
achieves a new level of phase locking and retains its frequency tuning capability.
However, because the signal and idler waves are orthogonally polarized, the waves
are not identical at frequency degeneracy and the strong mutual coupling that exists
-f
AinA
4~~~~A~,~~v;Ir
I•1I I
Figure 1-1: Type-I self-phase locking experimental setup.
in type-I phase matching does not take place. This mutual coupling can be imposed
if a means of mixing the ordinary and extraordinary polarizations can be found such
as by the use of an intracavity quarter-wave plate. We have performed such an experiment using an OPO that is pumped by the second harmonic of a diode-laser-pumped
Nd:YAG laser, as shown in Fig. 1-2. The Nd:YAG source provides a frequency stable
pump and also allows for comparison of the degenerate output of the OPO with the
fundamental wave at 1064 nm. (An important consideration is to make sure that none
of the IR light from the Nd:YAG is coupled into the OPO, which might cause phase
locking by the injection locking process.) Since the concept of coupling a portion of
one polarization to the orthogonal polarization is analogous to injection locking, it
will be helpful to examine this process.
Injection locking is the process by which a weak signal is coupled into an oscillator
which can provide gain at that frequency. As the weak injected signal is amplified,
the gain saturates and the frequency and phase characteristics of the injected signal
are imposed onto the oscillator mode. The phase of the oscillator output is said to be
"locked" to that of the injected signal. The injected signal's frequency must be within
a certain range of the resonant frequency of the oscillator. An OPO near degeneracy
can provide gain at a frequency near one half of the pump frequency. (This is why
it is important to insure that none of the IR output of the Nd:YAG is leaked into
high speed
photodiode
(1GHz)
Figure 1-2: Type-II self-phase locking experimental setup.
the OPO.) With the proposed scheme, a portion of the "signal mode" at w, will be
"injected" into the "idler mode" which can provide gain at wi. When the signal and
idler frequencies are close enough they will be "captured" by this process. Thus, a
type of injection locking between the signal and idler can take place.
One part of the experiment was to develop the pump system to be used. As
stated above, a diode-laser-pumped Nd:YAG laser was resonantly doubled to provide
a pump at 532 nm. Since there is only 500 mW of power available from the Nd:YAG
laser, the efficiency of the doubler needs to be fairly high to provide enough power
to the pump (above 100 mW). By using a resonator with low loss mirrors and a
12-mm-long KTP crystal, an overall efficiency above 50% was achieved.
The polarization mixing that is needed to couple the two orthogonal subharmonic
waves was provided by an intracavity quarter-wave plate. When the fast and slow axes
of the quarter-wave plate were rotated with respect to the fixed "crystal" axes, some
portion of the original polarization state was projected onto the orthogonal state.
Using this scheme it was observed that the OPO output became self-phase locked
at frequency degeneracy. Part of the experiment included examining the effects of
different amounts of coupling and the phase characteristics of the self-phase locked
output. The procedure used to obtain these results will be presented as well as a
description of future experiments to be performed with the OPO under the self-phase
locked conditions.
Chapter 2
Theory
In this chapter we will review some key aspects of the theory in order to provide a basis for understanding the experiment from the design process to the final results. Our
starting point for reviewing the fundamental aspects of resonant second-harmonic
generation and optical parametric oscillation will be the nonlinear wave equation.
(For a derivation of the nonlinear wave equation from Maxwell's equations see Appendix A.) The two processes are closely related. Degenerate parametric generation
is the inverse process of second-harmonic generation. The basic equations of injection
locking will also be examined.
2.1
Resonant Second-Harmonic Generation
A theoretical description of second-harmonic generation (SHG) and the effects of
resonance is useful in the pump laser design process. The optimal value of input
coupling can be calculated based on a preliminary estimate of the cavity losses.
2.1.1
Single-Pass Conversion
I will make certain assumptions which simplify the analysis but leave the main features
of the results the same as those of a more detailed analysis. The first assumption is
that the fundamental wave is not depleted in the conversion process in a single pass
(which is a very good assumption when the converted power is compared with the
circulating fundamental power.) Also the Poynting vector walk-off and absorption
loss are neglected. The waves are assumed to be circular Gaussian beams, and all
the interaction is assumed to take place in the near field. The field quantities will
be referenced inside the crystal, ignoring the finite radial extent of the actual crystal.
The nonlinear wave equation under the slowly varying amplitude approximation is
zE
(r, z)=
8z
2
(2.1)
-j W2LOCp(2)ej(Akz+AW)
2n2
where,
p( 2 )
-
odeffEl (r)El (r)
(2.2)
and the subscripts 1 and 2 refer to the fundamental and second harmonic fields,
respectively. Therefore we have w2 = 2w, and Ak = k2- 2k,. The field envelopes have
r
2
spatial dependence Ei(r) = Ele-
r2
and, E 2 (r, z) = E 2 (z)e
. Now, substituting
the second order nonlinear polarization from equation 2.2 into equation 2.1 yields
SE2(r, z) = -wdeff
z
E
E(r)ej (Akz±+ A )
n2C
Now, integrating from z = 0 to z = 1 with initial condition E 2 (r, 0) = 0 gives
E 2 (r, 1) =- j
deff E2(r)l[sin(Akl/2)]ej(Akl/2+A)
n2c
Akl/2
In terms of intensities (defined as I = nceolE 2 ) we have
2w(r)
2
d 12
l2
n2
Srin(Akl/2) 2.
3c
2kAkl/2
2C
0
From here we must integrate over r from 0 to oo:
P1 =
Ii(r)dr =Ii1(0) 2
P2
(r,
I 2l)dr
=
fo
=
I2 (0, 1)
2
2
2r
Since I 2 (r, 1) is proportional to the square of Ii(r), w2 =
w.
This allows us to
write the final expression for the second harmonic power in terms of the fundamental
power as
P2
22w2d2ff 12
Sn2 10co-2 27 l
n2
where Ak = 0 for perfect phase matching. For type-II phase matching we have
n o (W2 ) =
(ne1(wi) + no(w 1)). It is helpful to define an effective confocal parameter
b = no(W2 ) 1w . The second harmonic power can now be written as
167r2 dl 1(1/b)
2
Ano(w )ne(w 1)Eoc 1
AP2
(23)
This result is valid in the weak focusing limit (or when I < b). When the crystal
length is comparable to the confocal parameter, the Boyd-Kleinmann reduction factor
must be used [4]. Though the complete analysis is much more detailed, it results in
replacing the factor 1/b with hm(B, () where B is the double refraction parameter and
=
1/b is the focusing parameter. The single-pass conversion factor,YSHG
P2/P
2,
is given by
167r2d2 lhm(B,
)
3
e
=SHG
Ajno(wj)n,(wj)Eoc*
2.1.2
(2.4)
Enhancement Factor
We now consider the enhancement due to cavity resonance of the fundamental wave.
In a high finesse cavity the circulating power is much larger than the input power.
Since the overall conversion efficiency is P2= 7YsHGP (Pl )2, the effect of resonance
is to multiply the single-pass conversion efficiency by the square of -.
The input
fundamental power P 1 , the circulating fundamental power Plc, the generated secondharmonic power P 2 , and the output second-harmonic power P2, are shown in Fig. 2-1.
The value of the resonant enhancement is limited by the total loss in the system.
Fi1r
R1,Ti
Figure 2-1: Fundamental and harmonic power in a ring enhancement cavity configuration.
At resonance, the expression is
Plc
P1
T,
(1 - /R
)2
where R 1 and T1 are the power reflection and transmission coefficients of the input
mirror (mirror 1). The reflectance parameter Rm is defined as the fraction of the
fundamental power left after a round trip in the resonator (not including reflection
from mirror 1). If losses are kept low Rm can be made close to 1. The expression
above can be maximized with respect to T1 - 1 - R 1 .The result is that T1 should be
chosen such that R, = Rm. This condition is known as "impedance matching" since
the reflected fundamental power, given by
Rm) 2
Pir = S(1(1- Vri ,) 2
goes to zero and all of the input power is coupled into the resonator.
2.1.3
Intracavity Loss and Optimum Input Coupling
The main sources of loss are: scattering losses of the four mirrors, transmission
through the mirrors, misalignment loss, and conversion loss. The conversion loss
is unique in that it actually depends on the other losses. It is convenient to separate
the power transmission coefficient of the input mirror, T 1 , from the rest of the loss
terms. The total loss can be expressed as: L + Loony + Ti and the optimum value of
p t = L + Lcon,. The power transmission coefficients of the other mirrors are
Ti is TP'
contained in the term L, along with other crystal and mirror losses and losses due
to slight misalignment of the cavity. In this folded ring geometry, the small angle
of incidence at the curved mirrors introduces astigmatism, which causes some misalignment loss. Also, as previously mentioned, the walk-off of the two fundamental
polarizations causes a spatial separation which leads to additional losses.
The conversion loss is the fraction of the circulating fundamental power which
is converted per pass: Lcon
=
However, Plc is dependent on the overall
S7HGP1c.
conversion efficiency. Making this dependence explicit, we have: Lconv =
Pii
-
P2
=
Pic
is the overall conversion efficiency. So, putting
?7•/SHGP1, where 7 =
everything together, the complete expression for the overall conversion efficiency (to
first order) is
4Ti /'YSHGP1
= [2 - V1 "
T i(2
- L - V/TSHG p) ) 2
(2.5)
This equation can be simplified and expressed in terms of a polynomial in 77 and
solved for 0 < 7 < 1. It also allows the optimum input coupling to be solved without
dependence on 7 [5]:
Tjopt = L/2 +
2.2
(L/2)2 + YSHGP1.
(2.6)
The Doubly Resonant Optical Parametric Oscillator
In order for oscillation to occur there must be a source of gain which can overcome the
loss inside an optical cavity. The process of parametric amplification is what provides
this gain inside an OPO. There is no input being amplified, however, and steady-state
oscillation is built up from noise. In our case both the signal and the idler waves are
resonant inside the cavity so we will be looking at the conditions required for this to
occur. We will also be looking at the different ways in which we can tune the signal
and idler frequencies in order to bring them near degeneracy.
2.2.1
The Double Resonance Threshold
In a doubly resonant OPO (DRO) both the signal and idler waves are resonant in the
cavity at the same time. This means that the effective cavity length for each wave
must be an integer multiple of half wavelengths. Also, each field must satisfy the
condition that the round trip gain is equal to the round trip loss. This loss includes
not only the internal crystal loss, but also any other losses in the cavity and at the
mirrors. We denote the effective single-pass cavity losses for the signal and idler waves
by a, and aj, respectively.
From the coupled wave equations for three-wave mixing an expression can be found
for the threshold pump power needed for doubly resonant oscillation to occur. We
will again make assumptions which simplify the analysis, but preserve the important
physical results. As before I will assume that the interaction is taking place in the
near field of a Gaussian beam and will use a plane-wave approximation. This time,
however, I will include the loss term in the nonlinear wave equation. There are three
coupled nonlinear equations for the pump ('p'), signal ('s') and idler ('i'):
OE
-E,
+ acE, = -j
c8z
-Ej + aE =
Oz
az
where Ak = kp-
Ep + a
j ( A k z + A W)
Wsdeff
,c E Ee-(kz+),
ni~c
j wdef EpE;e-j(Akz+Ap),
nic
=
-E -j wpe EsEie j ( Akz+ A W),
npc
(2.7)
(2.8)
(2.9)
ks - ki.
Now, we will look at the case where the pump wave makes a single pass throughout
the cavity. Since we will be looking for solutions for steady state oscillation, we will
assume the circulating amplitudes Es and Ei are approximately constant over the
length of the crystal. (The coupling losses for the signal and idler are assumed to be
small). Equation 2.9 can now be integrated from z = 0 to z = 1 to give
= E() - jwpdeff E,.El[rsin(Akl/2)]e(Akl/2+A)
Ep()
Enc
Akl/2
Making use of the relationship between the steady state signal and idler waves [6]
E2S
wsnia
E2
winsos
i
the threshold pump intensity required for the parametric gain to overcome the cavity
losses can be found:
sin2(Akl/ 22 )1
a -th=
(Akl/2)
with
(2.10)
2
2d2 l2Wsi
K = 2d ffl w~wi
npnsnic (50
As the phase-mismatch Ak increases, so does the minimum pump power needed
for oscillation. It has been found that the oscillation threshold can be lowered by
using a nonresonant double-pass pump [7]. After reflection from the back mirror, if
the phase relationships are maintained then the interaction length of the crystal can
be increased to up to 21 (and the threshold pump intensity lowered by a factor of
four).
In general the threshold will depend on the relative phases of the three waves and
the phase-mismatch. Assume the mirrors are at z = 0 and z = 1 and the back mirror
has reflectivity R = r2 . Then let AV be the relative phase shift at z = 0 and Apm be
the relative phase shift introduced upon reflection at the back mirror. The threshold
pump intensity will then be proportional to [8]
-2
[sin 0, + r sin(O1 + 02)]
where 0, = Ap + Akl/2 and 02 = Am, + Akl. In steady state the phase shift Ap
will be such that the threshold is minimized for a given AVm and Akl. The threshold
pump intensity will then be
lh =
a[ a
(1+r2 + 2rcos0 2)
[K
sin2 (Akl/2)
(Akl/2) 2
(1
(2.11)
For the case where R = 1 and Ak = 0 the minimum threshold occurs for AVm = 0
(and AV = 7/2). Its value will be 1/4 of the threshold for the single-pass pump. In
effect, the crystal interaction length has been doubled to 21. In general, however,
there will be some transmission (R < 1) and some nonzero relative phase shift (-r
<
AVm < 7r) at the mirror. The threshold can be minimized by tuning Ak. It has been
shown that for R > 0.12 the threshold for a double-pass pump DRO is lower than
that for a single-pass pump DRO for any value of Apm assuming Ak is optimized [8].
2.2.2
Frequency Tuning
Three methods of frequency tuning are used to bring the DRO toward frequency
degeneracy. Angle, cavity-length and temperature tuning provide coarse and fine
tuning of the signal and idler frequencies and thus provide control over the frequency
difference Wd = w,- wi. The signal and idler output frequencies are determined by the
phase matching condition and the cavity resonance frequencies. The phase matching
condition defines a bandwidth within which oscillation may occur. The actual output
frequencies are then determined by the resonance condition which depends on the
effective cavity lengths for the extraordinary signal and ordinary idler waves.
Angle tuning refers to changing the angle between the pump beam propagation
axis and the crystal axis. The angle dependence of the ordinary and extraordinary
indices allows tuning of the phase matching condition:
n o (wp)wp = n+(w)
no(wi)w.
There are practical limitations to the range of angle tuning. As the crystal's angle
is tuned, non-normal angle of incidence for the pump occurs, which may increase
the beam walk-off angle, thus increasing the oscillation threshold. The physical pa-
rameters of the crystal such as transparency range and physical dimensions limit the
tuning range. Also, large angle changes will affect the cavity mode-matching, requiring realignment. In our experiment, the major limitation is the physical size of the
crystal.
If the cavity mirror position is scanned using a piezo-electric transducer (PZT)
there are clusters of discrete longitudinal modes at which the double resonance condition is satisfied. The cavity length may be servo-locked to one of the resonance peaks
for cw operation. Each mode has a different pair of signal and idler wave numbers
and output frequencies. The change in the signal-idler frequency difference between
adjacent modes is twice the free spectral range of the cavity [9]. For an effective
cavity length of 1 cm, the frequency change between adjacent modes is 30/1 GHz.
Thus medium range discrete tuning within the oscillation bandwidth can be achieved
by cavity-length tuning.
Temperature tuning provides fine tuning of the signal-idler difference frequency.
The temperature change causes thermal expansion of the crystal and a change in the
refractive index [10]. The ordinary and extraordinary indices have different temperature dependence so there is a differential index change along with a common-mode
index change. The frequency tuning is usually limited to a range of less than about
0.5 GHz due to the limited ability of the PZT to compensate for the common-mode
shift. Tuning over this range can be accomplished by using a thermo-electric cooler
driven by a current source.
2.3
Injection Locking
Injection locking is a means by which the frequency and phase characteristics of one
oscillator (master) are imposed upon another (slave). If the frequencies are within the
locking range the slave oscillator will lock to a small signal injected from the master
oscillator. The first thing that happens is the amplification of the small injected
signal within the slave oscillator. This only occurs if the frequency of the injected
signal is close enough to the free-running oscillation frequency of the slave oscillator.
The signal is then amplified enough such that the gain saturates and the free-running
oscillation dies out leaving only the signal at the frequency of the master oscillator.
Thus gain and gain saturation are a fundamental part of the injection locking process.
2.3.1
The Adler Equation
The primary equation describing the dynamics of injection locking is the Adler Equation and it follows directly from a general equation for the field amplitude of an
oscillator with resonance frequency wo coupled to an injected signal at frequency
w1inj [11]:
= (o - - - -1 -)1 + 2-in(2.12)
dt
To Te, Irg
dE1
where the fields in terms of their amplitudes and time dependent phases are
6(t) = E(t)(w • ot+O(t ))
Einj (t) ewinit.
Sing (M)=
For simplicity the field amplitudes are normalized such that Einj12 represents the
incident power in the injection signal, while JE12 represents the stored energy in the
cavity. Equation 2.12 takes into account the internal loss (with decay rate 1/To), the
external coupling loss (with decay rate 1/Te), and the internal gain (with growth rate
1/-g). Separating equation 2.12 into real and imaginary parts yields
d =
dt
d
(--- I)E+ V-Eij cos ,
T9
=t (wo
0-
Te
in)
To
(2.13)
Te
- V2E
sin ¢.
(2.14)
Equation 2.14, known as the Adler equation, is the primary equation used to determine injection locking behavior. Looking for solutions under steady-state conditions
we see that the requirement for locked oscillation is
Wo -
=
Einj
sin.
This gives the maximum frequency separation between the oscillation frequency and
injected frequency as
Iwo-
winjl •!5
E "
This equation tells us that when ý/-Eini < E, the frequency separation will be small
compared to the external cavity decay rate 1/Te.
2.3.2
Signal-Idler Mutual Injection
Without the wave plate, the DRO can be thought of as two independent cavities
(one for the extraordinary polarized signal and one for the ordinary polarized idler)
with a common source of gain. Under steady-state conditions, every round trip the
power lost in each cavity is replaced by power from the pump through the parametric
generation process in the crystal. The power added is due to amplification of the field
that is already present in the cavity.
The introduction of a quarter-wave plate into the DRO cavity provides a means
of mixing the orthogonally polarized signal and idler waves. Since each wave will
make two passes through the quarter-wave plate, the overall effect will be to rotate
the linear polarization by a small angle when the axes of the wave plate are offset
slightly from the crystal axes. Each wave will lose a component of its field amplitude
and have a component from the other wave added to it. So the cavities are effectively
coupled.
The fact that the signal and idler fields have a small component along the other
polarization is not sufficient for controlling the output of the two cavities. The interaction between the two coupled cavities is dependent on the amplification of that
small component. The situation is described very well by the injection locking pro-
cess because what the parametric interaction does is amplify a signal entering the
crystal by taking energy from the pump. So when the small component of the idler
wave is put into the polarization state of the signal it plays the same role that the
small injection signal does in the injection-locking process described above. The gain
is provided by the parametric interaction and saturation may occur through pump
depletion. Though this analogy to the laser injection locking process is helpful in
providing the physical reasoning for understanding how self-phase locking may occur,
it is still important to develop a full theoretical description of the dynamics of the
DRO with intracavity polariztion mixing.
Chapter 3
Pump Laser System
The pump laser system was designed to provide over 100 mW of cw power at 532 nm
for use as the OPO pump by means of second-harmonic generation (SHG) of a diodelaser-pumped Nd:YAG laser. The Nd:YAG laser is capable of supplying 500 mW of
power at 1064 nm. To achieve the required SHG output, the fundamental wave is
rosonated in a cavity to enhance conversion efficiency.
3.1
Cavity Design
Various resonance schemes were considered.
The type of resonator geometry can
be a standing-wave cavity or a ring cavity. In the standing-wave configuration the
harmonic power is generated in both directions. This can be beneficial if both harmonic waves are added in phase; however, it requires a reflector designed to yield
constructive interference between the two harmonic waves [12].
An efficient scheme is to resonate both the fundamental and the second-harmonic
waves. The main advantage of the doubly resonant scheme is to produce an increased
effective nonlinear coefficient. However, it also increases the effective round trip loss,
and is in general a more complicated scheme [13]. Potassium titanyl phosphate (KTP)
is a nonlinear optical crystal commonly used for doubling 1064 nm radiation. It has a
relatively large nonlinear coefficient so that a folded ring cavity in which the secondharmonic is generated in one direction can yield a high conversion efficiency. The
traveling-wave ring configuration also avoids spatial hole burning and higher singlefrequency output power can be obtained. We chose to implement the singly-resonant
configuration using a ring cavity.
The first decision to make in designing the resonator was to pick the value of the
confocal parameter inside the crystal. A value of 39 mm was chosen. This value is
larger than the optimum value (which maximizes hm(B, ()) so that the effect of the
Poynting vector walk-off between the two fundamental polarizations is reduced. The
key constraints in designing the ring resonator were: achieving a confocal parameter
of 39 mm inside the crystal, the availability of low loss mirrors with specified radii
of curvature, and keeping the physical dimensions practical. For the geometry of the
ring resonator shown in Fig. 2-1, it is possible to use ABCD matrix multiplication to
solve for a stable q-parameter inside the resonator [11]:
q
A-D
2
+
+
A+D
( 2-0
1
where A, B, C, and D are matrix elements representing a round trip through the
resonator. Once the q-parameter at the input mirror was known, the optical beam
from the Nd:YAG laser could be mode-matched to the enhancement cavity.
Using the ABCD matrix, the confocal parameter in the crystal can be plotted as
a function of other parameters such as cavity length. In this way the desired value of
the confocal parameter can be chosen while ensuring that the resonator is not near
instability. The parameters (and their values) used to construct the ABCD matrix
were: the radii of the curved mirrors (20 cm), the distance between the curved mirrors
(21 cm), the distance between the flat mirrors (31 cm), and the angle of incidence on
the mirrors (30).
All of the above calculations are straightforward, but there was an additional
constraint to consider. There must be enough room for the mirror mounts so that the
beam is not blocked. This requires the resonator folding angles to be large enough
to provide clearance; however, the angles should also be kept as small as possible
to reduce astigmatism. So, the critical distances near the mirrors were taken into
account.
from
thermistor
to
TE cooler
32nm
1064nm
Intensity servo
Figure 3-1: Pump system schematic.
transducer)
3.2
(HWP: half-wave plate; PZT: piezoelectric
System Performance
The laser used was a Lightwave Electronics diode-pumped solid-state non-planar ring
laser model 122. The gain medium is Nd:YAG and the output wavelength is 1064 nm.
The maximum power output is around 500 mW. The doubling crystal was a 3 x 3 x 12mm cesium-doped KTP crystal cut for 1064 nm SHG. The KTP crystal was polished
flat-flat and dual-AR coated at 1064 nm and 532 nm. In order to have a source which
could pump the DRO well above threshold (- 65 mW) a conversion efficiency of 50%
was desirable. Using equation 2.6, an optimum input coupler with transmission of
2% was chosen. As seen in Chapter 2 the conversion efficiency depends on the cavity
loss which proved to be highly dependent on the cavity alignment. Thus, it was
difficult to optimize both the crystal alignment and the cavity alignment for optimum
conversion efficiency. Most of the measured conversion efficiencies were lower than
expected when compared to theoretical values assuming perfect alignment. The best
conversion efficiency measured was 68% where 346 mW at 532 nm was measured from
an input power of 507 mW.
In order to stabilize the cavity length such that a stable cw output could be
obtained the cavity should be as mechanically stable as possible. A portion of the
circulating fundamental power which was transmitted through one of the mirrors was
monitored in a cavity length servo that used the PZT-mirror for correcting the cavity
length. The use of a ring cavity geometry presented some difficulty in achieving a
mechanically stable system. Each of the four mirrors was independently positioned on
its own mount. While this allowed the necessary adjustability for cavity alignment,
it also presented problems due to excessive vibrations. Steps were taken to minimize
the mechanical noise introduced into the system. The four mirrors and the crystal
were all mounted on a single aluminum plate which was placed on several damping
pads. Also, the cavity was covered with a box with holes for the beam to enter and
leave to reduce the vibrations due to acoustic noise in the room.
The intensity servo used to stabilize the cavity length was a dither-and-lock
scheme. The PZT-mirror was dithered at 20 kHz and the fundamental power that
escaped one of the high reflectors was monitored and fed to a lock-in amplifier. The
error signal from the lock-in was then integrated and used to drive the PZT. (For a
circuit schematic of the servo see Appendix B). Though the circulating fundamental power was stable (less than 1% peak-to-peak noise), the second-harmonic power
showed residual noise at the dither frequency of 20 kHz.
The temperature of the crystal also had to be stabilized so that the simultaneous
resonance of the ordinary and extraordinary components of the fundamental wave
could be maintained. As described in Chapter 2, changing the temperature produces
a differential change in the extraordinary and ordinary indices which is equivalent to
changing the effective cavity lengths for the two components. The crystal was placed
on a thermo-electric (TE) cooler. The sides of the crystal were covered with a thin
layer of aluminum in order to maintain an even temperature over the surface of the
crystal. A thermistor embedded in aluminum was placed in contact with the crystal.
An LDT-5910 ILX Lightwave temperature controller used the thermistor and TE
cooler to monitor and stabilize the temperature to within 0.2°C.
The typical power available to pump the DRO ranged from around 120 mW to
around 200 mW. This was the power measured after propagation through a Faraday
isolator and lenses, just before entering the DRO. Occasionally at a pump power
of above 150 mW or so the ring cavity intensity servo would become unstable and
oscillate, perhaps due to instabilities in the temperature controller, producing unacceptable noise on the pump. This seemed to be the limiting factor in achieving a
large pump power.
Chapter 4
Self-Phase Locking Experiment
In this chapter we examine the results of the polarization mixing scheme described
in Chapter 2. Observations were made on the DRO both with and without the
intracavity wave plate. The DRO with the intracavity wave plate will be referred to
as the "modified DRO," MDRO. The DRO without the intracavity wave plate will
be referred to as the "standard DRO," SDRO. In one of the experiments the SDRO
was fine tuned smoothly past the frequency degenerate point. The tuning behavior
of the MDRO was very different. However, with the wave plate axes aligned with
the crystal axes such that there was no polarization mixing, the MDRO could also
be tuned past the frequency degenerate point. With the wave plate rotated (-
50)
to provide polarization mixing, once tuned within about 400 MHz of degeneracy, it
was found to jump to the frequency degenerate point without further temperature
tuning. A description of the system and the observations of this self-phase locking
behavior is presented here.
4.1
4.1.1
Experimental Setup
DRO Cavity Design
One of the main goals of the DRO cavity design was to have a mechanically stable
cavity with enough space to insert a wave plate that could be rotated. A 2-element
DRO comprised of a crystal and a mirror was used. One end of the crystal had a
curved mirror surface of radius 40 mm, and the other was polished flat. The curved
surface was highly reflecting for 1064 nm and highly transmitting for 532 nm and it
served as the input coupler for the pump. The flat surface was anti-reflection coated
for 1064 nm and 532 nm. The output coupler had a 25-mm radius of curvature
and was mounted on a PZT stack. It was highly reflecting for both 1064 nm and
532 nm. The crystal and mirror were attached rigidly to an aluminum box. The
crystal platform could be rotated to provide for angle tuning in 0 (0 = 900). The
wave plate was inside a rotation stage which was attached to a spring-loaded mirror
mount and inserted from the top of the box. The mount was attached to the box
with epoxy. The wave plate could then be aligned using the adjustable mirror mount.
A schematic of the cavity configuration is shown in Fig. 4-1.
In order to stabilize the DRO intensity for cw operation the output intensity was
detected and sent to an integrating circuit (see Appendix B for the circuit schematic)
which controlled the PZT voltage. The detector contained a EG&G YAG-100 photodiode and a LM6361 operational amplifier used as a transimpedance amplifier with
a 1.5 kQ feedback resistor. The DRO could be stabilized to oscillate at one of the
double resonance modes; however, the 20 kHz noise from the pump was present on
the signal and idler output intensities.
The KTP crystal length was 12 mm and the physical cavity length was 39 mm.
Using an index of 1.8 for the KTP crystal the effective optical length of the cavity was
about 34 mm. This gives a free spectral range for the cavity of about 4.4 GHz. As
described in Chapter 2, this makes the change in the signal-idler frequency difference
between adjacent cavity modes to be 8.8 GHz.
4.1.2
Frequency Diagnostic System
An important part of the experiment was to have the ability not only to tune the
signal and idler frequencies, but also to monitor the signal-idler frequency difference as
it was tuned and eventually locked at frequency degeneracy. The portion of the pump
which was transmitted through the output coupler was filtered out. Then a part of the
quarter-wave plate
KTP c
t mirror
PZT
TE
on stage
Figure 4-1: Side view of cavity configuration.
signal and idler was split off and mode-matched into an 8-THz Newport model SRD SuperCavity optical spectrum analyzer (OSA) with non-TEMoo
00 transverse mode
spacing of ,- 25 GHz. This allowed the monitoring of the signal and idler output
frequencies so that they could be tuned to within the bandwidth of a high-speed
photodiode detector which was around 1 GHz.
The rest of the beam was split with a 50/50 beamsplitter. One portion was focused
onto a photodetector used for intensity stabilization. A polarizing beamsplitter was
placed in front of the detector blocking the idler so that only the signal was detected.
This was done to prevent the signal-idler beat intensity modulation from causing a
problem in the servo since the beat frequency would be within the bandwidth of the
detector when close to frequency degeneracy.
The other portion of the beam was focused onto the high-speed photodiode detector. This time a polarizing beam splitter was used to project the signal and idler
onto the same axis so that the beat note could be detected. The AC portion of the
detector output was sent to an rf spectrum analyzer for analysis. Also, another 50/50
beamsplitter was used to combine the signal and idler with a local oscillator (LO)
from the Nd:YAG laser. This 1064 nm local oscillator was set up first in a heterodyne
(Fig. 4-2), then in a homodyne (Fig. 4-3) detection configuration.
In the heterodyne configuration it was first frequency shifted by 80 MHz with an
acousto-optic modulator (AOM). Due to a lack of complete spatial separation there
was some overlap of different orders of diffracted waves so their beat signals were
visible on the spectrum analyzer as shown in Fig. 4-4. So strictly speaking there was
more than one frequency in our "LO". However, the beat signal between the "0th
order" and "1st order" beams was used as a reference to monitor the behavior of the
MDRO when tuned to degeneracy. It was not used until the last stage of the tuning
process as described below.
In the homodyne configuration the LO was mode-matched (without the AOM) to
the DRO output beam at the beamsplitter. The DC coupled output of the high-speed
detector was monitored directly for observation of the resulting interference. In order
to see the interference pattern, the path length of the LO arm was scanned with a
mirror mounted to a PZT.
4.2
4.2.1
Experimental Observations
Cavity Losses
The sources of the cavity losses were characterized including those caused by the
wave plate. Since the quarter-wave plate removes a component of the ordinary and
extraordinary waves in the cavity, it acts as a source of loss which depends upon the
angle between the crystal axes and the wave plate axes. (This "loss" can also be
sig
Figure 4-2: Schematic of DRO setup with local oscillator set up in a heterodyne detection configuration. (PBS: polarizing beamsplitter; AOM: acousto-optic modulator;
QWP: quarter-wave plate; PZT: piezoelectric transducer; TE: thermo-electric)
PZT
sigr
aetector uu coupieau)
Figure 4-3: Schematic of DRO setup with local oscillator set up in a homodyne
detection configuration. (PBS: polarizing beamsplitter; HWP: half-wave plate; QWP:
quarter-wave plate; PZT: piezoelectric transducer; TE: thermo-electric)
Figure 4-4: The frequency shifted reference beam used as an LO showed beats due
to the overlap of the different diffracted orders of the acousto-optic modulator.
36
thought of as the source of the injected power in self-phase locking process.) An effort
was made to characterize this loss by taking finesse measurements with the wave plate
turned to different angles.
Cold cavity finesse measurements were taken by mode-matching the 1064 nm beam
from the Nd:YAG laser into the "output coupling" mirror of the SDRO. A PZT driven
with about 600 Volts was required to scan one free spectral range (change in cavity
length of 532 nm). First, the finesse of the cavity without the wave plate was found
to be about 630 which implies a total round trip loss of about 1%. Most of this loss
comes from the transmission of the output coupling mirror (- 0.75%).
Once the wave plate was inserted, its fast and slow axes had to be aligned accurately relative to the ordinary and extraordinary axes of the crystal. This was done by
taking advantage of the second-harmonic generation process in the crystal. Initially
the SDRO cavity finesse was measured. With both polarizations entering the cavity,
the input power was turned up (-- 100 mW) such that the converted second-harmonic
output at 532 nm could be seen. An input polarizer was then aligned such that only
the extraordinary wave was input to the cavity and the second-harmonic light was
not generated. In this way, the polarizer's axis was aligned accurately with respect
to the crystal axes. The wave plate was then inserted and the second-harmonic was
again visible. This was due to the rotation of the polarization by the wave plate. The
wave plate could then be rotated until the second-harmonic was no longer visible.
This provided a way to accurately set the offset angle between the crystal and wave
plate axes to zero.
Finesse measurements were then taken with different angles between the crystal
and wave plate axes. About 10 mW of input power was used. The input was still
polarized along the extraordinary axis and a polarizer was placed at the output so
that the loss of the "extraordinary cavity" could be measured as the polarization was
rotated by different amounts by the wave plate. The results of the finesse measurements are summarized in Fig. 4-5. We find that the loss increases linearly with th
erotation angle as expected. The loss is found to saturate at a value of - 1.5% (from
a base value of 1.13%).
1.b
1.45
)KK
1.4
3K
' 1.35
X
a)
0
1.3
. 1.25
I
II
1.2
1.15
11
0
5
10
15
20
25
30
wave plate angle (degrees)
35
40
45
Figure 4-5: Cavity loss as a function of wave plate angle based on finesse measurements.
The oscillation threshold depends on the cavity losses as shown in Chapter 2.
Without the wave plate the oscillation threshold was 65 mW. After the wave plate
was inserted, the MDRO would only start to oscillate again when the wave plate was
aligned normal to the beam. Once this was done, the oscillation threshold was again
65 mW. The wave plate reflectivity was measured to be 0.135% (15%) per surface for
the 1064 nm (532 nm) wavelength. The fact that there is such a large pump reflection
(15%) at the wave plate suggests that the reason the MDRO would only oscillate
when the wave plate was normal to the beam is that the multiple pump reflections
contributed to the parametric conversion process and was probably responsible for
the PZT tuning behavior of the MDRO.
4.2.2
Frequency Tuning of SDRO
First the tuning behavior of the SDRO was examined. When the SDRO cavity length
was scanned, a cluster of longitudinal resonance modes was obtained. A cw output
was achieved by locking the cavity length to one of these modes. There were usually
around 3 to 5 different modes to which the cavity could be locked. However, they
were not adjacent modes so the signal-idler frequency difference could not be well
controlled by cavity-length tuning. The cw output was then mode-matched into the
OSA as described above.
Due to imperfect mode-matching into the OSA, a cluster of transverse modes were
visible when the length of the OSA cavity was scanned. There was one cluster for each
of the signal and idler waves. Since the transverse mode spacing was known (25 GHz)
this provided a reference for measuring how far the signal and idler frequencies were
from degeneracy. At degeneracy the two clusters overlap completely so that only
one cluster is visible. Angle tuning was used to make the transverse mode clusters
for the signal and idler come together. After the angle was changed the SDRO was
locked onto another resonance mode, and the two mode clusters from the OSA could
move closer together (or farther apart depending on the direction of the angle change).
Using angle tuning the two clusters could be moved close enough (- 1 GHz) such that
the signal-idler beat note could be detected by the high-speed photodiode detector.
At this point the two OSA transverse mode clusters appeared to overlap. (When the
beat note did not appear on the rf spectrum analyzer it implied that the transverse
modes for the signal were actually overlapping with different order modes for the
idler, or the MDRO was nearly at frequency degeneracy.)
Once the beat not was detected, temperature tuning was needed to bring the
SDRO closer to degeneracy.
When the common-mode index change became too
large for the PZT to compensate for the drift, the SDRO angle tuning had to be
adjusted slightly. When the beat note was close enough to zero frequency it could be
temperature tuned smoothly across zero. A spectrum analyzer trace of the beat note
at 5 MHz is shown in Fig. 4-6. Since the cavity finesse was around 630 and the free
spectral range was 4.4 GHz, the signal and idler frequencies were within the cavity
linewidth of about 7 MHz.
Figure 4-6: Spectrum analyzer trace of signal-idler beat note near frequency degeneracy. The beat frequency is around 5 MHz.
An effort was made to prevent 1064 nm radiation that exited the SHG pump
cavity from "leaking" into the DRO cavity. The beamsteering mirror M1 in Fig. 4-2
is highly transmitting at 1064 nm and highly reflective at 532 nm. However, a small
amount of the 1064 nm radiation (17 pW) was detected at the entrance to the DRO.
Assuming the worst case that the beam is perfectly mode-matched into the cavity
and is resonant, there would be about 0.56 mW circulating inside the DRO. The total
circulating power inside the DRO was estimated to be around 120 mW. We tried to
observe self-phase locking in the SDRO when tuned across frequency degeneracy but
we did not succeed even in the presence of the leakage field. We do not believe that
the observed self-phase locking in the MDRO was due to the injection locking by this
very small amount of 1064 nm light.
4.2.3
Frequency Tuning and Self-Phase Locking of MDRO
When the wave plate was inserted, the same procedure was used to align the axes
that was used in the finesse measurements. The relative angle of the wave plate
axes was set at 0' such that no polarization mixing would occur. The cavity was
simultaneously aligned for both the MDRO pump (on the input side) and the 1064 nm
SHG fundamental beam (on the IR output side). Once the wave plate was aligned
using SHG, the fundamental beam could be removed. The MDRO was then tuned
as before, this time with the wave plate inside but with no coupling between the two
polarizations yet.
The angle tuning behavior was somewhat different. The spacing between adjacent
modes in the mode clusters was less uniform. The MDRO could still be locked onto
one of these double resonance modes. In the case of angle tuning the SDRO, once the
OSA transverse mode clusters for the signal and idler had been tuned past each other
(signifying they had been tuned past frequency degeneracy), it had to be angle tuned
in the other direction (back towards the frequency degenerate point). This was not
the case with the MDRO. After the OSA modes had been tuned past degeneracy, if
the crystal angle was changed further in the same direction, there would be another
set of OSA modes for the signal and idler which could be made to cross again. This
tuning behavior, though consistent, was not fully understood. It did, however, help
in the tuning process.
When the OSA modes overlapped, the beat note appeared on the spectrum analyzer as before. It could also be temperature tuned as before. The beat note could
again be tuned across degeneracy with no observable self-phase locking. This again
confirms that the leakage field could not be responsible for the self-phase locking action. The angle between the wave plate and crystal axes was then turned by approximately 50. The signal-idler beat note could again be temperature tuned; however,
the beat note could not be temperature tuned to yield a beat below ~ 425 MHz.
When the beat note was near 425 MHz, further temperature tuning caused the beat
note and sometimes the DRO outputs to disappear. The angle was then readjusted
to compensate for the common-mode temperature drift. The OSA modes could be
made to overlap again, but the beat note would still not appear on the spectrum
analyzer.
One possible explanation for this absence of the beat note was that the selfphase locking process had captured the signal and idler frequencies and they were
degenerate thus producing no beat frequency. To test this assumption, the heterodyne
LO described above was combined with the MDRO output. If the signal and idler
frequencies were degenerate then they would be at the same frequency as the 1064 nm
reference beam from the YAG. The only additional beat signal that could show up
would be at the same frequency (80 MHz) as that of the "reference beats" shown in
Fig. 4-4. If the signal and idler frequencies were not the same then not only would
we see the signal-idler beat note but we would also see additional beats between the
signal (idler) and the LO "0th order" and "1st order" frequencies.
In the heterodyne measurement, it was found that the beat note power at 80 MHz
was increased by 8 dB as shown in Fig. 4-7. No additional beats were detected over
the entire 1 GHz bandwidth of the detector. The MDRO could actually be angle
tuned exactly to degeneracy with no further temperature tuning or means of frequency
control of any kind (besides the intracavity wave plate). As further evidence that what
actually made the reference beat rise by 8 dB was the MDRO output, sidebands are
revealed on the spectrum analyzer trace when the 80 MHz beat signal is expanded.
The beat signal before and after the MDRO was at frequency degeneracy is shown
in Fig. 4-8. As noted above the pump, but not the LO, had residual noise at 20 kHz
due to the dither frequency used in the pump cavity intensity locking process. There
was also significant frequency content around 35 kHz which could also be attributed
to the pump according to an FFT spectral analysis of the noise in the pump system.
The self-phase locking appeared to be stable and could be maintained for around 5
minutes before falling out of lock. The locking was also limited by the stability of the
both the SHG and the MDRO cavity intensity locking servos.
While the MDRO was self-phase locked the temperature was tuned away from
degeneracy. The MDRO output eventually disappeared and the angle was adjusted
Figure 4-7: The reference beat signal at 80 MHz has risen by 8 dB due to the additional power from the degenerate MDRO output.
as before. When the beat note reappeared on the rf spectrum analyzer it was at
375 MHz and rose in frequency as the temperature continued to change. When the
MDRO was tuned toward degeneracy again the beat note was again tuned to 425 MHz
before it became self-phase locked again. Thus it appears that the capture range at
a wave plate angle of - 50 was - 400 MHz.
The relative angle of the wave plate axes was then increased to about 100. This
time no temperature tuning was necessary to achieve self-phase locking. Once the
two OSA mode clusters were tuned together the MDRO became self-phase locked.
This was repeated several times and each time no beat was detected before self-phase
locking occurred.
It was not possible to determine the exact capture range from
the observations that were made, but this suggests an increase in the capture range
caused by an increase of the amount of coupling (via polarization rotation).
When the homodyne measurement was performed on the self-phase locked output,
interference was observed between the MDRO output and the LO. Due to difficulties
in mode-matching and intensity-matching the two beams, the fringe visibility was
reduced. A peak-to-peak modulation of - 40% was seen from the DC coupled out-
put of the 1-GHz photodetector. In Fig. 4-9 the zero level is at the bottom of the
vertical scale and in the horizontal scale the PZT-mirror of the LO was scanned over
approximately one wavelength of the 1064 nm beam.
Figure 4-8: Comparison of 80 MHz beat signal before and after the MNIDRO was
self-phase locked reveals sidebands at 20 kHz due to the pump.
45
Figure 4-9: Interference pattern between the self-phase locked MDRO output and the
LO
46
Chapter 5
Conclusion: Summary and Future
Work
In this thesis, we have achieved preliminary results demonstrating that a type-II
phase-matched DRO can be operated in a self-phase locked regime at frequency degeneracy. A stable pump system was developed via resonant second-harmonic generation of a diode-pumped Nd:YAG laser producing a DRO pump at 532 nm. We
made use of a theoretical calculation of the maximum conversion efficiency to choose
an optimum input coupler. A pump power of up to 150 mW was available to pump
the DRO which had an oscillation threshold of 63 mW.
The DRO cavity was designed with an intracavity quarter-wave plate to provide
polarization mixing of the signal and idler waves. The process of injection locking
was presented as a description of the coupling process that takes place inside the
modified DRO. The tuning characteristics of the empty cavity (SDRO) and the modified cavity (MDRO) were observed. It was shown that with no means of coupling
the extraordinary and ordinary polarizations inside the DRO cavity the signal and
idler frequencies may be tuned smoothly past the frequency degenerate point. With
the addition of an intracavity quarter-wave plate, with its axes turned to an angle of
5' with respect to the crystal axes, a frequency degenerate self-phase locked output
could be maintained with no external frequency control. A 1064 nm reference beam
was used in both a heterodyne and homodyne detection configuration to examine the
self-phase locked output.
The self-phase locking behavior which has been demonstrated is very promising for
applications such as 2:1 optical frequency division. The use of intracavity elements
in the DRO cavity has not been widely explored.
Better characterization of the
device both experimentally and theoretically is desirable. Also, some improvements
can be made upon the experiments which have been performed already. With better
separation of the different order beams from the AOM a true heterodyne detection
can be performed. Also, the alignment of the homodyne detection scheme can be
improved to obtain a much better fringe visibility in the interference pattern.
Other experiments can be performed to characterize the dynamics of the selfphase locking process.
More observations can be made using temperature tuning
to determine the capture range as a function of wave plate angle (or amount of
polarization rotation). Since the LO used in the homodyne detection determines the
phase of the DRO pump (due to the SHG process) the interference between the selfphase locked DRO output and the LO can be used to look for a quantum tunneling
effect which has been predicted for the phase of the degenerate output of an OPO [14].
The properties of the self-phase locked output can also be examined by performing
noise measurements and correlation experiments with different components of the
output.
Along with more experimental observations, a detailed theoretical analysis of the
OPO in this configuration is desired. A better understanding from both a classical
and quantum perspective of the mechanisms which lead to self-phase locking will help
in taking advantage of these results. The simplicity of this self-phase locking scheme
is a great advantage over other phase locking methods and will hopefully make it a
useful tool in the operation of the type-II phase-matched OPO for precision frequency
measurements.
Appendix A
The Nonlinear Wave Equation
In this Appendix we will derive the classical nonlinear wave equation in terms of
a general nonlinear polarization,PNL, for (nonmagnetic) media. The fields will be
assumed to be quasi-monochromatic plane waves. We start with Maxwell's equations,
where the only source is conduction current (J = crE) and there is no polarization
charge density (-V - P = 0),
BB
it(A.1)
Vx E =
0D
VxH
= aD+J
09t
V-D = 0
V-B = 0
(A.2)
(A.3)
(A.4)
and the constitutive relations
B =- oH
(A.5)
D =
(A.6)
E + P = EE + PNL
Using equations A.1 and A.5 we have
Vx Vx E = V(V -E) - V 2 E = -po
(VxH)
Then using equations A.2, A.3 and A.6 we can write
VE
V2E
2EE2
-•oott
- o-O- t2
oU OE
-
= to 1922at
PNL
2
(A.7)
This is the nonlinear wave equation in terms of the real time-dependent quantities E
and PNL.
We will now make the slowly varying envelope approximation and write the nonlinear wave equation in terms of the complex field envelopes. To facilitate the analysis
of Gaussian beams in the near field, we will write the time-dependent quantities as
the product of a monochromatic plane wave and a slowly varying envelope:
- j ( z - wt +
E(r, z, t) = eRe{Ee k ±)}
PNL(r, z, t)
=
pz - u p t + p)}
w
PRe{Pe-j(k
where E and PNL have Gaussian dependence in the radial dimension, r, and a slow
z and t dependence.
Plugging into equation A.7 with k 2
=
pOw 2 ,
two terms cancel immediately. Then
under the slowly varying envelope approximation we will keep only the most significant terms according to the following relationships:
a2E
k OE
2
-az
a 2E
2
9t2
z
w-
<
aPNL
< WPPNL
WP 9
a2pNL
at2
This yields
aE
< w2 E
a
<
.
OE
2jk 2 E +-jjtoawE + joe2w
where Ak = k - kp and A
= ýp-
=
- .)pow
2
pNLe j
(A
kz+Ap)
pop. (We will only be interested in cases where
the field and the polarization have the same frequency, w = wp, such that energy is
conserved). Finally we have the complex amplitude nonlinear wave equation
(z
Oz
where n =
o = + aE
+ -c-N)E
c~t
Poc (. -p)PNgLej(Akz+Ap)
2n
(A.8)
o/, c = 1/V/ioF0, and a = agoc/2n which represents the loss in the
media. The time-derivative term can be neglected in the steady state cw analysis. For
a particular nonlinear process it is necessary to work with a set of equations like A.8
coupled by the nonlinear polarization PNL.
Appendix B
Intensity Servo Circuits
The following figures are simplified block diagrams of the intensity servo circuits used
for the SHG ring cavity and for the DRO cavity. For each system the input comes
from a EG&G YAG-100 photodiode detector set up as a transimpedance amplifier.
The high voltage PZT drivers contain APEX PA85 and PA84 high voltage operational
amplifiers.
The intensity servo for the DRO cavity is simpler than the one for the SHG ring
cavity. The DRO cavity length was more stable to begin with since there are only two
elements (the crystal and the mirror) attached to an aluminum box. The ring cavity
has five elements (four mirrors and the crystal) attached to an aluminum plate, with
each mirror in a spring loaded mount.
The DRO cavity length was stabilized by taking the voltage level of the detected
IR output and subtracting it from a fixed (but adjustable) voltage set point. This
was used as an error signal which was integrated and fed to the high voltage PZT
driver.
The ring cavity was stabilized with a dither-and-lock scheme where the PZTmirror was dithered at 20 kHz and the power that escaped one of the high reflectors
was detected and used as the input to the circuit as shown below. An EG&G lock-in
amplifier model 5209 was used to demodulate the input and create an error signal
which was integrated and fed back to the PZT driver.
The stages labelled with "s1 / 2 1 are "half-integrator" circuits with a system re-
DC bias
variable gain stage
half-integrator
from
detect(
>r
high voltage driver
(-150 to +150 V)
Figure B-1: Fixed set point cavity length servo for the DRO cavity.
sponse magnitude which increases at 10 dB/decade.
lock-in amplifier
variable gain stage
from
dete
high voltage driver
(0 - 400 V)
Figure B-2: Dither-and-lock intensity servo for the SHG ring cavity.
half-integrator
Bibliography
[1] N. C. Wong, B. Lai, P. Nee, and E. Mason.
Experimental progress toward
realizing a 3:1 optical frequency divider. Proc. Fifth Symposium on Frequency
Standards and Metrology, 1995.
[2] C. D. Nabors, S. T. Yang, and R. L. Byer. J. Opt. Soc. Am. B, 7, 815, 1990.
[3] D. Lee and N. C. Wong. Opt. Lett., 17, 12, 1991.
[4] G. D. Boyd and D. A. Kleinmann. J. Appl. Phys., 39, 3597, 1968.
[5] E. S. Polzik and H. J. Kimble. Opt. Lett., 16, 1400, 1991.
[6] J. E. Bjorkholm. IEEE J. Quantum Electron., QE-5, 293, 1969.
[7] T. Debuisschert, A. Sizmann, E. Giacobino, and C. Fabre. J. Opt. Soc. Am. B,
10, 1668, 1993.
[8] J. E. Bjorkholm, A. Ashkin, and Smith R. G.
IEEE J. Quantum Electron.,
QE-6, 797, 1970.
[9] D. Lee and N.C. Wong. J. Opt. Soc. Am. B, 10, 1659, 1993.
[10] Dicky Lee. Optical parametric oscillators and precision optical frequency measurements. Master's thesis, Massachusetts Institute of Technology, 1996.
[11] Hermann A. Haus. Waves and Fields in Optical Electronics. Prentice-Hall, New
Jersey, 1984.
[12] R. Paschotta, P. Kiirz, R. Henking, S. Schiller, and J. Mlynek. Otp. Lett., 19,
1325, 1994.
[13] Z. Y. Ou and H. J. Kimble. Opt. Lett., 18, 1053, 1993.
[14] P. D. Drummond and P. Kinsler. Phys. Rev. A, 40, 4813, 1989.