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Calculation of Conversion Efficiency of a Doubly Resonant Optical Parametric Oscillator.

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A4nnalender Phvsik. 7. Folge, Band 33, Heft 5, 1976, S. 387-392
J. A. Barth, Leipzig
Calculation of Conversion Efficiency of a Doubly Resonant
Optical Parametric Oscillator
By W. BRUNNER,
R. FISCHER,
and H. PAUL
Zentralinstitut fur Optik und Spektroskopie der Akademie der Wissenschaften der DDR,
Berlin-Adlershof
Devoted to Prof.Dr. G. Richter on Occasion of the 65th Anniversary of his Birthday
A b s t r a c t . The problem of calculating the conversion efficiency for a doubly resonant optical
parametric oscillator with a Fabry-Perot type resonator configuration is reinvestigated, extending
the usual theoretical treatment, which is restricted to highly reflecting resonator mirrors, to the case
of arbitrary reflectivity R of the output mirror. For decreasing R both the signal and the idler wave
acquire growing portions of travelling waves, and it is shown by numerical analysis that this effect
leads t o a remarkable enhance;ment of the conversion efficiency (as a function of the relative excitation). I n particular, the maximum efficiency may considerably exceed the value 50% to be attained by means of high quality resonators (1- R < 1).
Berechnung des Wirkungsgrades eines doppeltresonanten optischen
parametrischen Oszillators
I n h a l t s u b e r s i c h t . Es wird das Problem der Berechnung des Wirkungsgrades eines doppeltresonanten optischen parametrischen Oszillators mit einer Resonatorkonfiguration vom Fabry-PerotTyp erneut untersncht, vobei die iibliche theoretische Behandlung, die auf hochreflektierende Resonatorspiegel beschrankt ist, auf den Fall eines beliebigen Reflexionsvermogens R des Auskoppelspiegels verallgemeinert wird. Fur abnehmendes R.erhalten sowohl die Signal- als auch die Idlerwelle
in wachsendem MaBe Anteile laufender Wellen, und es wird an Hand numerischer Rechnungen gezeigt, dalj dieser Effekt zu einer beachtlichen VergroOerung des Wirkungsgrades (als Funktion der
Xchwellenuberhohung) fuhrt. Im besonderen kann der maximale Wirkungsgrad den Wert von 50%,
der mit Hilfe von Resonatoren hoher Gute (1- R < 1)erreichbar ist, betrachtlich iibersteigen.
1. Introduction
I n the last years, the optical parametric oscillator (OPO) has become the subject
of a Iarge number of theoretical and experimental studies which indicate that the features of both the OPO and the generated radiation strongly depend on the chosen
experimental arrangement [ 1- 51. We speak of a triply resonant configuration if suitable
resonators are provided for all three waves (signal, idler and pump). I n a doubly resonant
configuration both the signal and the idler wave will resonate, and a singly resonant
OPO corresponds to only one resonating wave. However, also an OPO with all three
waves passing nonresonantly through the nonlinear crystal could already he made to
operate [6]. Generally, the threshold pump power needed to achieve steady-state oscillation strongly increases with a decreasing number of resonating waves. The maximum
conversion efficiency, on the contrary, shows an opposite tendency. It reaches 10076
26*
W. BRUNNER,
R. FISCHER,
and H. PAUL
388
in a singly resonant configuration, whereas in a triply resonant OPO, due to the partial
reflection of the pump wave a t the entrance resonator mirror, a t best 12.5;/, efficiency
might be attained. It should be noted that the actual value for the efficiency to be
observed in a given resonator configuration depends not only upon the relative excitation, but in a sensitive manner also upon whether the waves are running or standing ones,
corresponding to a ring resonator or a Fabi-y-Perot type resonator, respectively. As a
consequence of the trammissivity of the resonator mirrors, the waves in an OPO endowed
with a Fabry-Yerot resonator, however, will contain also travelling parts. This effect
which hitherto has not been discussed in the literature, will affect the conversion efficiency of a doubly resonant OPO, and the present paper is devoted t o a theoretical study
of this problem.
We consider a doubly resonant OPO of the following, frequently realized t.ype. The
pumping wave passes nonresonantly through the crystal, while signal and idler wave
resonate in a common Fabry-Perot resonator. I n the limiting case of vanishing reflection
losses at the mirrors (reflectivity R -+ 1)these waves becoine purely standing ones. (In
fact, the calculations carried out for R < 1 until now, deal with standing waves.) As is
well known, a standing wave can be regarded as a superposition of two counterrunning
waves of equal intensities. Assuming the pump ware to traverse the crystal from the
left to the right, those parts of signal and idler wave which travel in the same direction
will be amplified due to their nonlinear interaction with the pump wave, whereas the
counterrunning parts of signal and idler wave produce anew, by means of sum frequency
generation, a counterrunning pump wave, thus diminishing the conversion efficiency
t o about one half of the value to be attained in a ring resonator configuration. I n particular, the niaxiniuni efficiency reduces from 1000/, to 50°&, when replacing the ring
resonator by a Fabry-Perot resonator, as has been shown (for 1 - R
1) in an early
paper of SIEGMAN
[ 71. For decreasing mirror reflectivity, however, the signal and idler
wave in a (doubly resonant) OPO of Fabry-Perot type resonator geometry will contain
also growing portions of travelling waves. I n the following, it will be pointed out that
this effect leads to a considerably increased conversion efficiency.
<
2. Calculation of the Conversion Efficiency
Our treatment of the doubly resonant OPO starts from a consideration of the parametric interaction between three waves propagating in z-direction in a quadratically
nonlinear medium. Using the following ansatz for the electric field strengths of the
waves
+
QA(r,t ) = eAAA(r,t ) exp (ikA;.- ant) c. c. (A = 1, 2, 3),
(1)
where en denotes a unic vector indicating the direction of polarization and An is a slowly
varying amplitude, we may write the equations of motion, in the stationary case, in the
form [8, 91
8A,
Kwii
- -A,A, exp ( - i d k z ) .
a2
k3 c0s2n3
Here, K is an effective coupling constant, ix1 denotes the conipleinent of the angle between el and the z-axis, and AX: stands for k3 - ( k , $ kz). The subscripts 1, 2, 3, in this
order, refer to signal, idler and pump wave, respectively. This set of equations can be
Conversion Efficiency of a Doubly Resonant Optical Parametric OsciIlator
38!J
solved rigorously [8]. Confining ourselves to the phase-matched case (AX: = 0) and snhstituting
UA
k,k2k3 cos2 a1 cos2 a20082 a3
2x.ww:0;4
=
where
denotes the total energy flow per unit area in z-direction which is readily shown to be
independent from z , we find the conveniently normalized intensities for the signal and
idler wave, in their dependence on the penetration depth z , to be
where
Here, UP (A = 1, 2, 3) are the intensities o f the three waves a t the boundary z = 0 of
the nonlinear crystal, and the integration constant zo is determined by these values.
It should be nientioned that the sums uz up' occurring in eqs. (7) and (8)can freely
be replaced by the corresponding values a t any point z , u",z)
u&), since the latter
expressions are subject to conservation laws (the Manley Rowe relations). Moreover,
in the derivation of eqs. (7) it has been assumed that u!' 5 ug and that the phases
of the three waves, at z = 0, are such as to ensure maximum amplification of both signal
and idler wave.
I n order to simplify the mathematical treatment, we assume the losses for signal and
idler wave to be equal, which implies ul(z)= u2(z).Hence y2 equals 1, and the elliptic
sine becomes a hyperbolic tangent. Eq. (7) thus takes the form
+
+
Let us first consider the signal (or idler) wave travelling from the left to the right, i.e.,
in the same direction as the incident pump wave. We then obtain froin eq. (9), putting
there 2 = 0,
Up'
Hy means of the familiar addition theorem
tanh x - tanh y
tanh (z - y) =
1 - tanh x tanh y
it follows from eqs. (9) and (10)
For the signal wave travelling from the left to the right, thereby starting froin the value
u? at z = L (back boundary of the crystal), we readily find, noticing that no pump
W. BRUNNER,
R. FISCHER,
and H. PAUL
390
wave propagating in the same direction exists a t z = L (uk = 0),
Putting here x = L gives us
(14)
which implies
z0= L .
(15)
Hence the intensity for the backward travelling signal (or idler) wave reads
(16)
Supposing now the nonlinear crystal to be placed inside a Fabry-Perot resonator whose
left mirror is totally reflecting and whose right mirror has a reflectivity R < 1(equal for
both the signal and the idler wave), we may describe a full round-trip through the resonator, starting a t z = 0, of the signal (or idler) wave, as follows
up'-+ Z:(L) -+ RZ:(L) = u':(L)-+ u':(O) = Z:(O).
(17)
Steady-state operation of the OPO demands precise reproduction of the signal intensity
after this round-trip, i.e.
up' = 2?(0).
(18)
Making use of the relations ( l a ) , (16) and (17), we can write the requirement (18) as
(cf. also [4])
[
(
f2 ",'I
1 = RX 1-ttanh2 R u T T S
,
(19)
where the following abbreviation has been introduced
The conversion efficiency for the parametric process is given by
'71
(1- R)f f ( L )- (1 - R ) uo2s:
Uo'
U r
From the steady-state condition (19) we obtain, in the limit uy -+ 0, the threshold value
for the amplitude of the pump wave up in the form
1
"0,
=
1-R
artnnh -
1 + R'
(22)
Utilizing this result, we may introduce into eqs. (19) and (21) the relative excitation
4
03u&h
as the physically relevant parameter.
(23)
39 1
Conversion Efficiency of a Doubly Resonant Optical Parametric Oscillator
By numerical analysis one finds from eq. (19) the steady-state intensity of the signal
wave, from which the conversion efficiency can be calculated according to eq. (21).
Since we are less interested in the dependence of the signal intensity on the relative
excitation, but rather in the conversion efficiency as a function of g ) we can simplify
the evaluation by eliminating, by means of eq. (2l), uy from eq. (19). Substituting the
expression for S (eq. (20)) into eq. (19) and making use of eq. (23) we find
where
(
31=o - artanh
(e)
.
(25)
Substitution of the expression for uy,as given by eq. (24)) finally yields
where
and
Eq. (26) connects the efficiency q with only the reflectivity R and the relative excitat'ion
and serves as the basis for a numerical analysis.
CJ,
R * 0,7
-
R 0.9
R = 999
R-0.5
0
,
'
1
2
3
L
5
6
7
,
8
9
R-0,3
10a-
Fig. 1 Conversion efficiency 7 versus reIative excitation u for different values of the
reflectivity R of the output mirror (nonresonant pump wave, signal and idler wave
resonating in a common Fabry-Perot resonator)
396
W. BRUXXER,
R. FISCHER,
and H. PAL~L
3. Discussion
The calculated dependence of the conversion efficiency on the relative excitation,
for different values of the reflectivity ZZ, is shown in Fig. 1. One recognizes that also in
an OPO with nonresonant pump wave and signal and idler waves oscillating in a FabryPerot resonator, an efficiency remarkably higher than 5O0&can be attained. As already
mentioned above, this result is due to the tramniissivity of the output mirror, causing the
resonant waves to acquire also travelling parts. It should be noticed, however, that the
efficiency is enhanced by only abont lo(%,when passing from a high reflectivity B & 1
to R = 0.7. I t is only for still smaller reflectivities that the efficiency drastically increases.
One should not forget, however, that a decreased niirror reflectivity leads to an increased threshold puinp power. Hence, an enhanced conversion efficiency is achieved at
the cost of an enlarged pump power (the latter giving rise, however, also to a greater ontp i t power).
Finally, we should like to inention an interesting by-product of our calculation. I n
many cases the doubly resonant OYO is treated in a n approximation which consists in
expanding the hyperbolic tangent iney. (9) up to the second order term. This procedure
can certainly be applied to the case 1 - R << 1. Onr analysis, however, reveals that it
does not lead to a correct description of the effect of partially travelling waves on the
c,onversion efficiency q. I n fact, it predicts 7 to decrease, starting from 7 = 50°, for
1 - R < 1, for decreasing €2.
The authors express their gratitude t o Miss H . HEIMBERG
for performing the niiiiierical calciilations.
Keferences
[I] J. E. BJORKHOL~I,
Appl. Phys. Lett. 18, 53 (1268).
[2] 1,. B. KREUZER.
Proc. Joint Conf. Lasers and Opto-Electronics, 1,ondon 1969. p. 53.
[3] IV. BRUNNEK,
H. PAUL
and A. BANDILLA,
Ann. Physik (Leipzig) B i , 69 and €2(1971).
[4] R. FISCHER.
Exp. Tech. Phys. 21, 21 (1973).
[5] E. 0. h I b t A S h - , J. 81.YARROROUGII, &I. K. OSHXAXand P. C . NONTGOMERY,
Appl. Phys. Lett.
16, 309 (1970).
[GI J. M. Y.4RBOROUGH and G . 9.MASSEY, Appl. Phys. Lett. 18, 438 (1971).
[7] A. E. SIEGIIAN,Appl. Opt. 1, 739 (1962).
[8] J. A. ARMSTRONG,
N. BLOEMBERGEN,
J.DUCUING
and P. S. PERSHAN,
Phys. Rev. l27,lY 18 (1962).
191 W. BRDNNER,
Fortschr. Phys. ?o, G29 (1972).
Bei dcr Redaktion eingegangen am 3. NovrJmber1976.
Anschr. d. Verf.: Dr. IT.BRUNNER,
Dr. R. FISCHER
und Dr. H. PAUL
Zentralinst. f. Optik u. Spektroskopie d. AdW der DDR
DDR-1199 Berlin-Adlershof, Rudower Chaussee 5
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