A formalism for describing optical parametric oscillation is developed. The theory is applied to the derivation of the oscillation threshold condition, power output, the Manley-Rowe conditions, index matching, and frequency tuning

Louisell, W. H.

Yariv, A.

English

Caltech Authors - Main

5A2--Theory of the optical parametric oscillator

We have considered the problem of parametric oscillation 

in  the optical region. The mathematical  approach consists 

of solving Maxwell equations  in  the normal-mode formulation for the  amplitudes of the  pump (ω_p), signal (ω_s) and idler (ω_i) modes in  the presence of a   nonlinear optical medium characterized by an SIIG-like tensor x_(ijk) and  with ω+p = ω_i + ω_s.  The derivation 

of the  equations of motion for the normal mode amplitudes 

is similar to those of previous  resonator analyses. 

The original contribution of the  paper, however, is 

the extension of the analysis into  the nonlinear above 

threshold region. This requires that the  pump energy 

density inside the  optical resonator  and  the  pump 

power input be considered as   two independent physical 

quantities.  The analysis shows that below threshold  the 

two are  proportional  to  each other, while above threshold 

the  pump energy   density saturates,  in a manner analogous 

to  the single-pass gain of a  laser oscillator, at 

a constant  value.  The analysis is also used to 

obtain explicit  expressions for the power outputs at 

ω_i and ω_s as a function of the pumping power. The conditions of optimum coupling are derived and the validity 

of the Manley-Rowe  relations in the presence of losses 

is established

xxx 
1015  sec-1, = 1023  c1n-3) gives 6 = 1.2 X 
10-6 cm stat volt-1, in good agreement  with 
Miller’s experimental value. 
5A-2 Theory of the Optical Parametric 
Oscillator, A. Yariv, California Inslilula 
of Technology,  Pasadena,  Calif., and 
W .  11. Louisell, Bell  ’elephone  Labora- 
tories, Inc., Murray Hill,  N.  J .  
We have considered the problem of para- 
metric oscillation in  the optical region. The 
mathematical approach consists of solving 
Maxwell equations  in the normal-mode 
formulation1 for the  amplitudes of the  pump 
(w,) signal (a,) and idler (w,) modes in  the 
presence of a nonlinear optical medium 
characterized by an SIIG-like tensor xi,e 
and with wP = w i  + wS.  The derivation of 
the  equations of motion for the normal 
mode amplitudes is similar to those of 
previous resonator analyses.1z2 The original 
contribution of the paper, however, is the 
extension of the analysis into  the nonlinear 
above threshold region. This requires that 
the  pump energy den& inside the  optical 
resonator and the pump power input be 
considered as two independent physical 
quantities. The analysis shows that below 
threshold  the two are  proportional  to  each 
other, while above threshold the  pump 
energy densit,y saturates,  in a manner 
analogous to  the single-pass gain of a laser 
oscillator, a t  a constant  value.  The analysis 
is also used to obtain explicit expressions 
for the power outputs a t  w i  and w S  as a 
function of the pumping power. The condi- 
tions of optimum coupling are derived and 
the validity of the Manley-Rowe relations 
in the presence of losses is established. 
Phus. Iiev., vol. 124, p. 1646, 1961. 
I E E E ,  vol. 53 ,  pp. 4-12, January 1965. 
1 W. H. Louisell, A .  Yariv, and .2. E. Siegman, 
P K. H. Kingston and A. L. i\fcWlmrtcr, ,?roc. 
5A-3 Measurement of Optical Second- 
Harmonic  Generation in Centrosym- 
metric  and Noncentrosymmetric Crystals 
Using Focused Gaussian Laser Beams,’ 
J. E. Bjorkholm, G. E. Francois, and 
A. E. Siegman, Microwave  Laboratory, 
Stanford University, Stanford, Calif.  
The optical second-harmonic generation 
(SHG) caused by a focused, lowest order 
Gaussian laser beam in index-matched 
crystals of ADP  and calcite has been ana- 
lyzed in detail, and ommeasured experimen- 
tally  with CW 6325 A laser beams, leading 
to the following results. 
1) For a collimated laser beam, the 
dependence of SIlG upon crystal orienta- 
tion near index matching differs from the 
simple (sin fI)l/e2 dependence in small but 
significant details. These small differences 
are fully explained by the Gaussian plane- 
wave spectrum associated with the finite 
Gaussian transverse variation of t,he laser 
beam. 
2 )  Using a collimated laser beam, the 
nonlinsarity  in  ADP is found to be &s(ADP, 
6328 A) = 1.36 X esu rt 12 percent. 
1 This work has been supported by Air Force 
Contract AFOSR-1625, with previous support from 
other contracts. 
WEDNESDAY AFTERXOOiT 
The  estimated error is significantly smaller 
then previous measurements. The enhance- 
ment  factor (an - l) /n due to n randomly 
phased longitudinal modes is experimentally 
checked. 
3)  A general analytical expression is 
developed for the case of a lowest order 
Gaussian beam focused to a focal spot of 
size zuo in a nonlinear crystal of length I ,  
a t  or  near index-matching. The  amount of 
SHG depends  upon  the  ratios of 1 to a 
“Rayleigh length” z8 = T W ~ ~ / X  character- 
istic of the focal region and a double-re- 
fraction length I ,  = WO/LY,  where LY is the 
angle of double refraction in  the index- 
matched crystal. Going in the direction of 
decreasing focal spot size wo, there are in 
general four asymptotic regions, in which 
the  amount of SI-IG varies as wo-2, wo-t wo, 
and wo2. Between the middle two reglons, 
there  is an optimum degree of focusing, 
for maximum SHG, occurring when zE = l / ~ ;  
the amount of SI-IG at optimum focusing 
varics as I : .  (For  very  short  crystals  or for 
crystals in which LY is  very small not  all of 
the four regions  exist, and the optimum 
eondit,ion is  modified.) 
These analytical results are verified in 
detail using a wide range of degrees of 
focusing in ADP crystals of three different 
lengths. The absolute value of d36 in ADP 
when measured with a focused beam a,lso 
agrees very closely with  the value measured 
with a collimated beam, thlx furbhcr verify- 
ing the analysis. 
4) Quadrupolnr STIG, dc electric-field- 
induced SIlG (ESEIG), and  the classical 
Kerr effcct have been measured in  the 
centrosymmetric croystal calcite  using a 
focused C\V 6328 A laser beam. The ob- 
served quadrupolar effect in calcite, com- 
pared to  ds6 in ADP,  has a magnitude 
~ ~ f ( / d 3 ~  = 0.85 X 10-8 f 1.4 percent, or 
somewhat smaller than  the value of a 
similar (but not identical) quantity meas- 
ured with pulsed ruby lasers by Terhunc 
ct  al. The observed coefficient for ESIIG, 
relative to  ADP, was x51/d36 = 2.1 X lo+& 
24 percent. The measured effective Kerr 
coefiicient (not corrected for t,he combined 
cffects of photoelasticity and stresses pro- 
portional to  the square of the electric field) 
was = 2.0 X esu rt 16 percent. 
The center of the quadratic variation ol 
KSIIG with dc electric field was observed 
to be displaced irom zero dc field, as pre- 
viously observed by  Terhune et  al. One 
previously proposed explanation of this 
displacement as due to inhomogeneous dc 
electric fields is ruled out by these experi- 
ments. Alternative explanations remaining 
are surface harmonic generat,ion (previously 
proposed) and  the effects of a quadrupolar 
correction to  the Poyning theorem. 
5A-4 Theoretical  and  Experimental 
Studies of Optical Second-Harmonic 
Generation  in Birefringent Crystals, 
D. D. Bhawalkar,  R. C. Smith,  and L. S. 
Watkins, Department of Electronics, 
Xouthampton  U iversity, Sou~hampton, 
England. 
A theory has been developed t,o describe 
second-harmonic generation (SIIG) from a 
laser beam for the varions modes of a 
resonator. The  scalar diffraction integral 
is solved for the propagation of an extra- 
ordinary  harmonic wave close to index 
matching.  The  approximations involved are 
identical  to those used in deriving the modes 
of a resonator. The  theory developed is 
completely general and can be applied for 
crystals of any  length, 1. Kleinman et  al.1.2 
have treated two special cases  where 1 >> zo 
and 1 << 20 (220 is the confocal parameter 
of the beam). Our theory can be used for 
these regions but  he experimental and 
t,heoretical results to be presented apply to 
the region  where 1 ‘v zo which is most 
important for practical devices. 
A detailed solution of the diffraction 
integral  in a birefringent medium shows 
that the effect of anisotropy can be des- 
cribed  completely by moving the source 
by z tan LY ( z  is the distance  traveled in  the 
medium and LY is the angle between the 
Poynting  vector  and  the wave normal). 
This  result is applied to the  propagation of 
waves radiating  from  the second harmonic 
polarization generated by a Gaussian beam. 
The expression for the SH polarization is 
obtained  by  squaring  the expression for 
t,he field in the Gaussian beam. The dif- 
fraction  integral  for  the harmonic wave has 
to be integrated over the z and y directions 
which are normal to  the  propagation direc- 
tion z ,  and also along the z direction to 
include SIlG along the entire  length of the 
crystal. The first two integrations can be 
performed exactly but  he final integral 
has  to  be carried out numerically. The 
cffccts of absorption and scatter a t  both 
the fundamental and harmonic frequencies 
are included. 
Two types of calculation are made, one 
of the form of the harmonic beam a t  the 
exit face of the crystal which we define as 
the near-field pattern,  and  the  other of the 
form of the beam at large distances defined 
as the far-field pattern.  Both  the  intensity 
and phase distributions are computed for 
these two cases. 9 number of computations 
have been made of both  the near-field and 
far-field patterns of harmonic from a TEMnop 
beam with 220 = 31.4 mm for ADP crystals 
of thicknesses between 3 and 40 mm. An 
approximation for the near field has been 
found which avoids numerical integration 
but gives results in good agreement with 
the exact theory. Far-field patterns have 
been computed for different  positions of 
the crystal in the Gaussian beam and the 
SHG efficiency obtained. Calculations have 
also been performed to  obtain  the elficiency 
for various crystal thiclmesses. 
The field patterns of second harmonic 
generated by TEMn ‘I have also been in- 
vest,igated. No exact calculations have been 
carried out for this mode since an approxi- 
mate solution has been found adequate  to 
describe the different patterns. I n  this 
description the harmonic wave can  be 
expressed as two orthogonal modes, TEMoo, 
and TJ!” modes, with a phase difference 
between them which is determined by the 
position of the crystal in the beam. As a 
result of this phase difference the far-field 
pattern consists of three maximas when the 
February 1965. 
1 Boyd et  al., Phys. Ih., vol. 137, p. Al305, 
2 Kleinman ei; al., private communication. 


5A2--Theory of the optical parametric oscillator

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