418
IEEE JOURNAL
OF
QUANTUM ELECTRONICS, VOL. QE-2, NO.
9,
SEPTEMBER
1966
5A2-Theory
of
the
Optical
Parametric
Oscillator
A.
YARIV,
MEMBER,
IEEE,
AND
w.
w.
LOUISELL,
SENIOR
MEMBER,
IEEE
Abstract-A
formalism for describing optical parametric oscil-
lation
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.
INTRODUCTION
N
THIS PAPER,
we
present a general theory for
parametric oscillation in the optical region. Since
this phenomenon is the exact analog of parametric
amplification and oscillation in the microwave region, its
possibility was recognized a number of years ago
[I].
The
practical realization of opt,ical parametric oscillation has
been demonstrated by Giordmaine and Miller
121.
The experimental situat’ion is as follows.
A
nonlinear
optical crystal is placed within an optical resonator. A
“pump” field at
w,
is then fed into the resonator.
It
is
observed that at a certain pumping intensity, oscillation
is set up simultaneously at frequencies
w1
and
w2,
where
w1
+
w2
=
w,.
These frequencies correspond to resonances
of the optical structure
or
are near such resonances.
The theoretical approach takes the following form. The
total electromagnetic field is expanded in a set of basis
functions which consists of the proper field solutions of
the optical resonator in the
absence
of any nonlinear
element’s. In this last case any one of the basis functions
is
a proper solution of Maxwell equations and, conse-
quently, the coefficients of the expansion are time inde-
pendent. When the nonlinearity is “turned on,’’ the in-
dividual basis functions are no longer proper solutions
and the coefficients must be taken as time dependent.
In the limit of small coupling, this time dependence, for
any one mode; contains the full information concerning
the mode energy and its oscillation phase. The main con-
cern of this paper is to obtain the solutions for these
coefficients and, through them, follow the flow of power
from
the pump field into the
w1
and
w2
fields.
THE
ELECTROMAGNETIC
BACKGROUND
The starting point is a solution of Maxwell equations
inside a generalized resonator that contains a nonlinear
dielectric material. The properties of the material are
Manuscript received June
20,
1966. Yariv was supported
by
the
U.
S.
Air Force Laser Technology Group through Contract
AF33(615)-2800. This paper was presented at the 1966 International
Quantum Electronics Conference, Phoenix,
Arix.
Calif.
A.
Yariv is with the California Institute
of
Technology, Pasadena,
Murray Hill,
N.
J.
W.
H.
Louise11
is
with the Bell Telephone Laboratories, Inc.,
characterized by a linear susceptibility
xL
and a suscep-
tibility tensor
of
the third rank
drik.
The induced polari-
zation is given by
Pi~oxLEi
+
dii,EiE,
(1)
i
.k
where,
in
general, the coefficients
dii,
are functions
of
the frequencies of
Pi,
Ei,
and
E,c,
respectively.
If,
how-
ever, the nonlinear medium is transparent in the region
of interest, the coefficients
diik
are frequency independent.
Another consequence
is
that all the
diik
elements which
result from a mere rearrangement of the subscripts are
equal
[3].
Maxwell equations are
with the constitutive equations tal- \en as
j
=
lTE
D
=
EOE+ij+P
(4)
where the total polarization
is
broken into the sum of
the polarization
P,
which is induced by the macroscopic
fields and as given by
(I),
and an applied polarization
P,
which represents the source feeding power into the
resonator.
The electric and magnetic fields in the resonator can be
expanded in terms of Slater’s
[4],
[5]
normal modes
a,
and
Ea.
R(r,
t)
=
=-
Ha(?.)
wq(t)
-
-
a
dG
(5)
The vector functions
E,(?)
and
W,(P)
satisfy the relations
as
well
as
?i
x
E,
=
0
and
fieg,
at the resonator walls.
It
follows from
(6)
and the two boundary ccnditions
previously mentioned, that the sets
E,
and
H,
are
or-
thogonal
[4],
[Fi].
We
are free to choose their amplitudes
so
that they are normal. The orthonormality conditions are
1966
YARIV
AND
LOUISELL:
OPTICAL PBRAMETRIC OSCILLATOR
419
/v
Ea’Eb
du
=
6ab
(7)
He*Hb
dV
=
6ab
where the integration extends over the whole resonator
volume. Substituting
(5)
into
(2)
and taking the ith
component leads to
where use has been made
of
(5)
and
(7).
The energy
U,
in
a
single mode
+(p:
+
can also be expressed,
using (13), as
u,
=
4(p:
+
w:qrr”C>
=
wca:ao.
(15)
Since the energy
of
a
photon is
hwc,
a:a,
is proportional
to the number
of
photons in the mode
c.
Using (13), we can transform the equations
of
motion
(10) and (11) to a form containing the variables
a,
and
a:.
while from
(3)
and
(5)
we
obtain
(9)
Next, we multiply each term in
(8)
by
EOi
and sum over
i(=
1,
2,
3).
Using
(7)
and recalling that
e0(l
+
x”)
=
E,
we obtain
Multiplying
(9)
by
Hci
and summing gives
Equations
(10)
and (11) are the equations
of
motion
for
the mode variables
p,
and
g,.
NORMAL
MODES
Rather than carry out the analysis in terms
of
p,
and
qc,
we introduce the normal mode variables
a,
and
a:
which are defined as
i
a,
=
-
dz
(PC
-
iwcnc),
a?
=
complex
conjugate
of
a,.
(12)
The expressions for
qc
and
p,
become
The advantages of using the normal modes
a,
and
at
rather
than
p,
and
qc
have been discussed previously
[ii],
161.
The total energy stored in the electromagnetic field is
E=p/H.8dv+a/l?-gdz.
(14)
=
$
c
(p:
+
w%q3
c
&!E=
@)*.
dt
Next
we
define normal field variables
A
(t)
and
A:
(t)
by
ac(t)
=
e-iwCtA,(t),
a:(t)
=
eiwo’.4t(t).
(17)
It
is clear, from
(16),
that when
diik
=
0
and
p’
=
0,
~,(t)
=
const.
e-(oa’zqc)t
.
We
will assume that the coupling
is
sufficiently small
so
that
-
dAc
<<
w,A,
dt
(18)
for which case we may replace
and
With these adiabatic approximations, the equations
of
motion (16) become
and
420
IEEE JOURNAL
OF
QUANTUM ELECTRONICS
SEPTEMBER
where
U/E
=
wc/2Q,
is the decay rate for the field variables
of mode
e.
These are the general working equations. They
can be employed to obtain the differential equations which
apply to special cases. In the next section we use them to
obtain the equations describing the paranzetric oscillator.
THE
PARAMETRIC
OSCILLATOR
EQUATIONS
Let us assume that the interactions are limited
to
three modes only, by means to be described shortly. Let
these modes be denoted by the subscripts
I,
2,
and
p,
and let their respective frequencies be
wl,
w2,
and
a.
We
shall refer to modes
1,
2,
and
p
as the signal, idler, and
pump modes, respectively. Furthermore,
we
shall assume
that mode
p
(the pump) is driven by the polarization
source
so
that
P(T,
t)
=
P(v)
sin
wt
(22)
represents the
applied
polarization.
If
we
consider the
equation
of
motion for
al,
i.e.,
(20)
with
e
=
1,
we
notice
that for the case
of
no coupling
(difk
=
O),
the solution
is of the form
al(t)
=
a,(O)e
.
It
follows that
for the case
of
weak coupling the only cumulative con-
tribution to
da,/dt
comes from terms on the right side of
(20)
whose oscillatory factor is The contributions
from terms with frequencies considerably different from
w1
oscillate at a rate equal
to
the difference in frequencies,
thus averaging out to zero in time spans that are long
compared
to
the optical beat periods.
If
the
free (i.e., no
coupling) resonance frequencies satisfy the condition
-iwxte-(wx/2Ql)t
w
=
w,
+
02
(23)
then the product
a%a,
=
A'",(t)A,(t)e
i(w*--W)l
=
A*z(t)Ap(t)eiW't
.on the right side of
(20)
satisfies the synchronism con-
dition. Assuming that this happens for no other pair of
modes, the equation for
al(t)
can be written
as
and similarly
I--
-1
E,,EliE,,
dv
2(-w
+
ol)alaz
(24b)
Using
(23),
these equations are finally written as
-
/
E,,EZiE,,
dv
a5aP
(25)
-
/
E2,EliEP,
dv
ala:
(26)
Equat'ion
(25)
can be simplified further by taking ad-
vantage of the symmetry condition
diik
=
djik
=
d.
*kt
.
=
dkii
=
diki
=
dkii
which was previously discussed. Con-
sider, for example, the summation in
(25).
If
we inter-
change
i
and
j,
we obtain
which is equal to the corresponding summation in (26)
since
diik
=
diik.
The same argument can be used to show
that the summation in
(27)
is equal to those of the
first
two.
We
can consequently define
a
single parameter
K
as
to
replace t,he triply-summed factors in (25).
If,
in addi-
tion,
we
define the pumping parameter
X,
as
x
=
8
1
E,@).I'l(i;>
du,
2
2€ (29)
we can rewrite
(as),
(26),
and
(27)
as
da,
w
dt
=
-zwpa,
-
-
a,
-
Kala2
+
zhpe-iO'
2Qp
together with their complex conjugates.
we have
In terms
of
the adiabatic variables
Ai(t)
=
ai(t)eiwit,
-
=
--
A,
+
KA~A,
dA
1
Y1
dt
2
dA$
___
dt
-
-"/z
A$
+
KA,A,*
2
-
dAp
dt
-r,Ap'-
2
KA,A,
+
%X,
1966
YARIV
AND
LOUISELL
:
OPTICAL
PARAMETRIC
OSCILLATOR
and their complex conjugates. The decay rate
yi
is
defined
as
yj
=
wi/Qi,
and is equal to the reciprocal decay time
constant for the mode
j.
The
yi
parameters account not
so much for the losses in the medium as for the fact that
the external coupling (i.e., reflector transmittances) is
ldifferent for the three frequencies.
Equations
(30)
are the main result
of
the preceding
section and constitute the starting point for the following
analysis.
Before proceeding with the solutions of
(30),
it may be
worthwhile to comment on the role
of
the pumping param-
eter
X,.
If
X,
=
0,
no steady-state oscillation can exist
and the energy in the three modes decays with time.
As
X,
is increased from zero, the steady-state solutions
of
(30)
are
A,
=
A,
=
0
and
A,
=
2iQ,X,/w.
We shall
refer to this region as “below threshold.” In this region
the pump field is proportional to the pump parameter
X,,
while the signal and idler modes are not excited.
By increasing
X,
(and
A,)
we
reach a point at which
a
steady-state oscillation at
w1
and
w2
becomes possible.
TO
find the necessary value of
A,,
we set
dA,/dt
=
dA2/dt
=
0
in
(30).
This gives
for the threshold (start oscillation) value of
IA,I2.
Exactly
at
threshold
A,
=
A,
=
0, so
that from the last equation
of
(30)
(with
dA,/dt
=
0)
and
(31),
we obtain
for the (squared) pumping parameter at threshold.
As
the pumping is increased above the threshold value
as given by
(32),
A,
can no longer increase with
it.
This
is due to the fact that no steady-state solution for
A,
and
A,
is possible if
IA,I2
exceeds its threshold value as given by
(31).
The last equation of
(30)
shows that this “clamping”
of
A,
in the face
of
an increasing
X,
is only possible if the
product
KA,A,
“picks up the slack.” The increase
in
pumping above threshold is thus seen to give rise to an
oscillation at
w1
and
02,
with no further increase in the
intensity
of
the pump mode. This is the equivalent of
gain saturation in a laser oscillator, where the population
inversion, and hence, the gain, is proportional to the
pumping rate below threshold but “saturates” once thresh-
old is exceeded. Further increases in the pumping power
give rise
to
higher oscillation (and power output) level.
It
is
evident that a description of the saturation (above
threshold) behavior of the parametric optical oscillator
depends on the use of separate parameters
for
describing
the pump field
(A,)
and the pumping intensity
(X,).
Analyses which employ only
A,
(or
some suitable equiva-
lent thereof) can still be used
to
derive the threshold
conditions. This bas been done by a number of authors.
421
THE
POWER
RELATIONS
From
(30)
we can obtain explicit expressions for the
power output at the signal frequency
w1
and the idler
w2.
Since the stored energy in the ith mode is
wiaiaT
=
wiAjA:,
the total power output is
Pi
=
wiAiA$(wi/Qi),
or
using
~i
=
wi/Qir
Pi
=
yiwiATAi.
(34)
From the last equation of
(30),
with
d/dt
=
0,
we obtain
A,
-
KA,A,
+
iX,
=
0
2
(35)
and multiplying the first equation of
(30)
with
d/dt
=
0
by
A27
Combining the last two equations leads to
A,
=
2ix,
(37)
Repeating this procedure with the second equation
of
(30)
gives
A:
=
-
2iXp
4K2
YP
+
”yz
IAl12
Equations
(38)
and
(37)
are compatible only if
y1
/All2
=
y2
or,
using
(34),
when
(39)
so
that the number per second
of
photons generated at
w1
is equal
to
that generated at
w2.
This is true regardless
of the respective losses. The losses do, however, determine
the magnitude of the power at
w1
(and
a,).
Equations
(39)
were first derived by Manley and Rowe
[7].
Viewed
from the quantum mechanical point
of
view,
(39)
reflects
the fact that the parametric process can be described as
a scattering event in which a pump photon of energy
hw
is “annihilated” while, simultaneously, two photons-one
at
wl,
the other at w,-are created. Conservation of energy
dictates that
w
=
w1
+
w2,
while the basic nature
of
the
scattering guarantees that the number of photons created
or
annihilated at each of the three frequencies are equal
[8].
To
obtain an expression
for
P,,
we equate the value of
Ad:
as given by
(38)
to
the threshold value
(31).
The
resulting equality
is
then solved for
A,AT
with the result
which, of course, is only valid when
6
Xp/~
~Y~Y~Y,/~K~,
i.e., above threshold.
422
IEEE
JOURNAL
OF
QUANTUM ELECTRONICS
SEPTEMBER
Using
(32)
for
Ap
at threshold we can rewrite
(40)
as
The factor y,y2/4K2 is, according to
(31),
the value
of
IA,I2
at threshold,
so
that (41) can be written
as
%-5-U[Ll]
-
-
(42)
w1
w2
UP
(bh
Equation (42) is the basic power equation.
(P,)th/wn
is
the rate
of
photon dissipation at threshold at
wD.
X,/(XD)th,
the pumping factor, is the factor by which threshold is
exceeded. According to
(42),
each time the pump intensity
is exceeded by an amount equal
to
the threshold intensity,
the photon output at
w1
and
w2
is increased by
(Pp)th/wp.
This linear power relationship is shown in Fig.
1,
in which
the
w1
(and
w3)
photon output, relative
to
the pump
photon
at
threshold,
is
plotted
vs.
the pumping intensity
relative
to
its threshold value.
The
conservation
of
power takes
the
form
of
OPTIMUM
COUPLING
Having obtained in the last section explicit expressions
for the power output as a function of the
loss,
coupling,
and pumping parameters, we are in a position to solve
for
the opt,imum coupling condition which maximizes the
power output.
The starting point is
(41),
The
loss
parameters
y1
,2
are related to resonator Q's by
where
R,,,
are the effective' reflectances and L,
,2
is
the
resonator length at
wl
,2.
y1
,,
can be expressed as the sum
of two terms. The first represents the unaviodable residual
losses and is taken
as
(yl
,2)i.
The second term
is
due to
the useful output coupling
of
power and is talcen as (yl
=
(~1.2)i
+
("/l,da.
(44)
In analogy with the laser coupling analysis
[9],
we define
the ratio
of
the power output
to
the residual losses as
r
PUMP OUTPUT
I
0
I
2
3
4
5
Pr
-
?X,
r"r,
w1
ti
4K2
or,
using
(44)
and
(45),
The part
of
P,
which
is
available
as
output
is
(Pl).
and
is given by
(47)
which, using
(461,
becomes
/('PITH
Maximization of the power output
(P,),
with respect
to
X
yields
Fig.
1.
The power output (in photons per second) at
w1
and
wz
relative to the threshold pump input. The linear relation shows
how,
above threshold, the excess pumping power at
w
is
converted
into oscillation power at
WI
and
w?,
where
w
=
WX
+
WZ.
The
saturation of the pump field intensity inside the resonator
above
horizontal portion
of
the pump output curve corresponds
to the
threshold.
1
i.e., adjusted
so
as
to account
for
the total loss per pass.