I Theory of Optical Parametric Amplification and Oscillation

E. WOLF, PROGRESS IN OPTICS XV @ NORTH-HOLLAND 1977

I

THEORY OF OPTICAL PARAMETRIC AMPLIFICATION AND OSCILLATION BY

W . BRUNNER and H. PAUL Zentralinstirut fur Optik und Spektroskopie der Akademie der Wissenschaften der DDR, Berlin, DDR

CONTENTS PAGE

§ 1. INTRODUCTION

. . . . . . . . . . . . . . . . . . . . . . . .

5

. . . . . .

. . .

16

MECHANICAL DESCRIPTION PARAMETRIC AMPLIFICATION. . . ’. . , .

. . .

20

8 5. CLASSICAL TREATMENT OF PARAMETRIC AMPLIFICATION. . . . . . . . . . . . . . . . . . . .

25

.

31

. .

53

8 8. EXPERIMENTAL WORK ON OPTICAL PARAMETRIC OSCILLATORS . . . . . . . . . . . . . . . . .

66

§ 2 . BASIC EQUATIONS, .

. .

3

,

§ 3. PARAMETRIC FLUORESCENCE

. .

8 4. QUANTUM

OF

$ 6 . THE OPTICAL PARAMETRIC OSCILLATOR (OPO)

9 7 . RADIATION CHARACTERISTICS OF AN OPO

REFERENCES

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,

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73.

0 1. Introduction Nonlinear optical phenomena can generally be ascribed to nonlinear contributions to the polarization of a suitable medium, i.e., contributions which are quadratic, cubic, etc. in the electric field strength of the applied electromagnetic field. Since these nonlinear polarization terms become large enough to produce observable effects only for fields comparable in their amplitude with the interatomic field acting upon the electron (about 3 x lo8 V/cm), it was only after the advent of the laser, as a new and powerful light source, that nonlinear optics became accessible to experimental study, whereas nonlinear effects due to static electric fields - the Pockels and Kerr effect - were already well established for some time. It was just one year after the first laser had been successfully operated (MAIMAN[ 1960]), that two pioneering experiments demonstrating the nonlinear response of matter to an intense optical wave, were performed, name!y, the observation of a two-photon absorption process (KAISERand GARRETT[1961]), and the generation of the second harmonic (FRANKEN, HILL, PETERSand WEINREICH [1961]). Since then rapid developments in this field, both experimental and theoretical, have taken place. Among the nonlinear optical effects, especially those which allow light to generate at new frequencies and thus to extend the spectrum of known laser frequencies to the uv as well as the ir region, became of high practical importance. In this case, the basic mechanism is that of frequency mixing which can be described as follows: Due to a polarization term quadratic in the electric field strength, two electromagnetic waves with different frequencies* w1 and w 2 give rise to polarization waves oscillating at the mixed frequencies w , + w2 and lo1- w 2 ( , respectively. This polarization, acting as a source for electromagnetic radiation, in turn, produces new electromagnetic waves at the sum or difference frequency, provided the so-called phase matching condition is fulfilled. The physical meaning of the latter is that the polarization wave and the electromagnetic wave radiated by it, must coincide in their phase velocities.

* By frequency,

we always mean circular frequency. 3

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The special case o l= w z (only one incident wave) corresponds to the generation of the second harmonic (frequency doubling). In this paper, we are concerned with the creation of the difference frequency. Here, the fundamental process is most easily understood in the photon picture as the splitting of one photon belonging to the incident wave with the higher frequency (the pump wave) into two photons of the type of the incident wave with the lower frequency (the signal wave), and of the wave created by the nonlinear polarization at the difference frequency (the so-called idler wave), respectively. Hence, energy is transferred from the pump wave to both the signal wave and the idler wave, and this mechanism can be utilized to amplify a weak signal wave by transmitting it through a nonlinear crystal together with a strong pump wave*. Since this process is reminiscent of parametric amplification known from microwave technique - the “parameter” is now the dielectric constant for the signal wave and the idler wave, which is modulated by the intense pump wave - it is often called optical parametric amplification, and, more generally, one speaks of parametric interaction among pump, signal and idler waves. Pump photons can also split spontaneously into signal and idler photons, i.e., in the absence of signal and idler waves (parametric fluorescence). Starting from this “noise”, signal and idler waves may build up due to parametric amplification. To make this experimentally feasible, it will be necessary to provide a suitable feedback for the newly created waves (or, at least, one of them). This is achieved by means of optical resonators. For sufficiently high pump power, the system will pass from the amplifying stage to an oscillating one. (In this case, of course, it is a matter of arbitrariness to decide which of the two generated waves should be termed signal or idler, respectively.) Now, the essential point is that the ratio of the signal and idler frequencies, for a given pump frequency, is wholly determined by the above mentioned phase matching condition, and, hence, depends upon the orientation and the temperature of the nonlinear crystal. Therefore, the signal (and, accordingly, the idler) frequency can be tuned continuously by either turning the crystal or varying its temperature. This light source, named optical parametric oscillator, proves to be a valuable tool for spectroscopic studies, especially in the ir region. In the following, the basic theoretical aspects of this device will be discussed. *The energy fed into the idler wave is lost, as far as the amplification of the signal is concerned; this cxplains the term “idler” wave.

1,

B 21

BASIC EQUATIONS

5

The outline of the paper will be as follows. In 4 2 we introduce the nonlinear susceptibilities characteristic for the nonlinear response of the medium and briefly list their formal properties. Starting from Maxwell’s equations, we deduce the equations governing the spatial and temporal behaviour of three waves coupled by parametric interaction. In 9 3 the main features of the spontaneous parametric process, i.e., the parametric fluorescence, will be described. In § 4 and 0 5 the parametric amplification process will be treated in some detail, both quantum mechanically (with special emphasis laid on the noise characteristics of such an amplifier) and classically. The following sections are devoted to the optical parametric oscillator. In 9 6 typical devices are studied, especially with respect to the oscillation thresholds and the conversion efficiency. In 9 7 the fluctuation properties of the generated radiation which originate, on the one hand, from spontaneous decay of pump photons (parametric fluorescence) and, on the other hand, from phase and amplitude fluctuations of the pump wave are investigated. Finally, in § 8 a short review of the experimental progress in the field of optical parametric oscillators is given.

0 2. Basic Equations 2.1. THE ANHARMONIC OSCILLATOR

In order to gain some insight into the physical mechanism from which nonlinear optical phenomena originate, we adopt for the electron bound in the atom (molecule) the simple model of a slightly anharmonic (one[ 19651, PAUL[1973a]). If we dimensional) oscillator (cf. BLOEMBERGEN modify Hooke’s law for the restoring force by adding a small term which is quadratic in the displacement x from the equilibrium position, the motion of the electron under the action of an external electromagnetic field is governed by the equation

e E(t). m

X+O~X+EX~=-

(2.1)

Here, 0 0 denotes the natural frequency, rn the mass and e the electric charge of the electron. The applied field is assumed to be a superposition of different monochromatic waves

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where the sum extends over a set of discrete frequencies, any of them taken with both the positive and the negative sign. Equation (2.1) may be solved by successive approximation with respect to the small perturbation E X ’ . In the zeroth order we obtain the wellknown result

In the next step, we replace the term &x2 by EX")^. Formally, this gives rise to a contribution to the driving force which is quadratic in the electric field strength and, hence, oscillates at the combination frequencies w’ a”.(Since w ‘ and 0‘‘may be positive as well as negative, a’+0’’may be a sum or a difference frequency.) This effective new force, in turn, leads to the following nonlinear contribution, in addition the linear term ( 2 . 3 ) , to the displacement

+

x(’Yr)=

C x(~)(&!,w ” ) exp {i(o’+o”)t),

(2.4)

O’,”’’

where

x (1) ( W ‘ , W ” ) = --E-

e2

E(w’)E(o’’) (2.5) m2 [oi- ( w ’ + w”)~](w?,- w f 2 ) ( w ;- d2) ‘

Noticing that ex is the induced dipole moment and, hence, n,ex the macroscopic polarization (n, number of bound electrons per unit volume), we may rewrite eqs. (2.4), (2.5) in the following form P(”(w)=

C

w’+”’=

w

~ ( ~ ’ ( ww’ ”f ; a’,w”)E(w‘)E(o’’),

(2.6)

where P(2)(w)is the Fourier amplitude for the second order polarization term (with respect to the electric field strength). The factor x ( ~in) eq. (2.6) has the physical meaning of a second order susceptibility. According to eq. (2.5) it is given by

(2.7) The reason for considering x ( ~as ’ a function of three, rather than two, frequencies will become clear from the discussion of symmetry relations (see 0 2.2). (Note that o’,a” and o ’ + w ’ ’ enter the formula (2.7) in precisely the same manner!)

I,

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BASIC EQUATIONS

7

Obviously, we can readily extend the above consideration to also include higher power corrections to the harmonic restoring force, thus obtaining contributions to the polarization of higher order with respect to the electric field strength. Throughout this article we are mainly concerned with the second order polarization, since the latter is responsible for the parametric interaction.

2.2. NONLINEAR SUSCEPTIBILITIES

Taking account of the vector character and the spatial variation of the electric field strength, we have to generalize eq. (2.6) in the form

or, written more compactly, P(2)(r,o)=

where

2

o”) :E(r, o ’ ) E ( r ,o”),

~ ‘ ~ ’ o’, ( 0 ; o‘,”‘

(2.9)

is different from zero only if

~ ( ~ ’o’, ( 0 o”) ;

0 =@’+ON.

(2.10)

Here r designates the space vector, and the indices p, p‘, p“ have been used to characterize the Cartesian components. The summation over o f , w ” , as in 0 2.1, extends over all frequencies, taken both with the positive sign and the negative sign, which are present in the applied electromagnetic field. In practical cases, e.g., for crystals, the nonlinear susceptibility (which is a third rank tensor due to the vector character of P and E ) is a more complicated function of the frequencies than that following from the simple model of an anharmonic oscillator, and its absolute values can be taken, in practice, only from experiments. A theoretical analysis, however, ) have some characteristic formal properties (cf. reveals that x ( ~ must BLOEMBERGEN [19651, AKHMANOV and KHOKHLOV[ 19641, BUTCHER [1965], PAUL[1973a]). These can be described as follows. (a) Since the product E,,(r, w’)E,,.(r,o”)remains unchanged by interchanging p r and p”, and, simultaneously, o’and o”,only that part of the susceptibility tensor x(*)which has the same symmetry property, will contribute in eq. (2.8). Hence, we may assume x ( ~to) obey the symmetry

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relation

xjZ;,$.(o;

o f ,of')= xf;,&;

of',0 ' ) .

(2.11)

(Otherwise we pass from x'" to the corresponding symmetrized tensor.) (b) While eq. (2.11) is a rather trivial one from the physical point of view, a second symmetry relation holds which is of high physical importance, namely

x:;*#&J; (Because of eq. (2.11)

* p", w+

o f ,off) = x:f)pw+-of; -0, off).

(2.12)

x(') is invariant with respect to the substitution

-off,off-+-0, also.) Equation (2.12) can be substantiated from an explicit calculation of x ( ~ ) (for isolated atoms or molecules) by means of quantum mechanical perturbation theory. (Classically, it follows already from a treatment of the three-dimensional anharmonic oscillator.) On the other hand, the BLOEMBERGEN, DUCUING relation (2.12) is easily seen (cf. ARMSTRONG, and PERSHAN[1962], PAUL[1973a]) to be just the condition for the differential

p

dF= -

~~'(o)dE~(o)+c.c.

(2.13)

w,er

to be a total one. Hence, a state function F can be defined which has the physical meaning of a nonlinear part of the time-averaged, free enthalpy density of the medium. From the symmetry relation (2.12) the so-called Manley-Rowe relations (first established by MANLEYand ROWE[1959] in the microwave field) can be derived which in the generation of light waves at new frequencies determine the fractions in which the pump energy is fed from the pump wave(s) to the generated waves. It is very interesting to note that these predictions are in perfect agreement with the quantum mechanical photon picture. Since the perturbative treatment leading to eq. (2.12) and, hence, to the Manley-Rowe relations, deals with classical electromagnetic fields, this means that nonlinear optics affords an unexpected access to the photon concept-in the framework of classical electrodynamics! Another aspect of the symmetry relation (2.12) is that it connects directly observable coefficients characteristic for different nonlinear effects (e.g., optical rectification and the Pockels effect, cf. BUTCHER [1965], PAUL. [1973a]).

1,

§ 21

BASIC EQUATIONS

9

(c) Fortunately, the large number of elements of the nonlinear susceptibility tensor is, in general, greatly reduced due to the spatial symmetry properties of the media (crystals). In fact, since the physical situation remains unchanged when a symmetry operation is applied, the nonlinear susceptibility tensor must be invariant with respect to any of the symmetry operations characteristic for the crystallographic class to which the material belongs. Considering, in particular, the inversion operation (x, +-xF, p = 1, 2, 3), one easily concludes that inversion symmetry of the medium implies the quadratic (not the cubic!) susceptibility term to vanish. Hence, materials suitable for a study of parametric three-wave interaction necessarily must lack an inversion center, i.e., must be anisotropic crystals. As is seen from eq. (2.7),resonance between the field and the medium will take place, if any of the Fourier components of the field (including those waves at combination frequencies which may be generated as a result of the nonlinear interaction) coincides in frequency with the natural frequency of the harmonic oscillator. In a more realistic model of the atom, the latter frequency has to be replaced by the atomic or molecular frequencies corresponding to different level spacings. In the case of resonance, energy will be exchanged between the field and the medium, and processes which cause the induced individual dipole moments to be damped (e.g., relaxation processes) will become important. Formally, the energy transfer from the field to the medium, or vice versa, will be described by a nonvanishing imaginary part of the susceptibility. In the off-resonance case, on the other hand, the medium will not contribute to the energy balance, and damping mechanisms will exert practically no influence on the interaction which then effectively takes place between the light waves only, the material playing a role similar to that of a catalyser in chemical reactions. Accordingly, the nonlinear susceptibility will be real under these conditions which apply, in particular, to the parametric three-wave interaction we have in mind. Combining this result with the relation (2.14) indicating the nonlinear polarization to be a real quantity, we arrive at the useful symmetry relation (2.15)

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2.3. FIELD EQUATIONS

Until now we have been concerned with the nonlinear response of a medium to applied electromagnetic fields. In the following, the reaction of the induced nonlinear polarization on the field, manifesting itself in the creation of waves at new frequencies, will be studied. To this end, we start from Maxwell’s equations for a lossless and nonmagnetic medium. Substituting in these equations the dielectric displacement vector D by

D

=E

+ 4 n-(P(l’”)+ P‘“’)),

(2.16)

where P“”)is the nonlinear and P‘l”’) the linear polarization term which, for a dispersive and anisotropic medium, reads r, t) =

P(’ln)(

I,

,y(ll*’)(

t’) * E( r, t - t’) dt’

(2.17)

(cf. AKHMANOV and KHOKHLOV[1964]), we find the fundamental equation

(2.18) (The dot indicate$ differentiation with respect to time.) Obviously, the nonlinear polarization term (more strictly speaking, its second time derivative) plays the role of a source term for the electromagnetic field. Let us now specialize to parametric three-wave interaction, i.e., we assume three waves of different (central) frequencies w, ( j = 1, 2, 3 ) to be present, where w 3 = w l+ w 2 , and specify to the second order polarization (2.8) Unless the intensities of the waves are extremely high, the nonlinear source term, due to its relative smallness, will give rise only to a slight modification of the free waves (which correspond to p l n l l = 0 in ey. (2.18)). This suggests the following ansatz for the electric field strength of the wave j , El(r,t ) = A,(r, t)e, exp{i(k,r-o,t))+c.c..

(2.19)

Here, the quantities e, (unit vector indicating the polarization), k, (wave vector) and o,are those of the free wave j , and the complex amplitude is supposed to vary weakly in space and time, compared to the exponential in eq. (2.19), as a consequence of the nonlinear interaction. Hence, we may split eq. (2.18) into three separate equations, each of them describing the behaviour of one of the waves under the influence of the corresponding resonant nonlinear polarization term, whose explicit form follows

1,

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11

BASIC EQUATIONS

from eq. (2.8). Since the nonlinear polariz,ation will be a small perturbation, we may neglect the dispersion of x") in the small frequency intervals characteristic for the effective line broadening due to the time dependence of the amplitudes A,, i.e., we may write the nonlinear polarization term driving the wave j = 3 , e.g., in the form P%r, t ) = 2 A l ( r , W A r , t)exp{il(kl+k2)r-w3fl}

x x " ' ( w 3 ; w ] , w2):e1e2+c.c., (2.20)

where use has been made of eqs. (2.11) and (2.15), in addition to eq. (2.9). Moreover, in performing the differentiation on the right-hand side of eq. (2.18), we may neglect the time dependence of the amplitudes A,. We thus arrive at the following equation of motion for the wave j = 3 (2.2 1) where P\2) is given by eq. (2.20). Inserting now the ansatz (2.19) (for j = 3 ) into the left-hand side of eq. (2.21) and performing the differential operations, it has to be noticed that, due to the anisotropy of the medium, the direction of wave propagation (indicated by the wave vector k) differs from that of the energy flow, i.e., the ray direction. For most practical purposes it will suffice to study a nonlinear medium which is supposed to fill a half space. When plane waves enter the medium, their amplitudes A will depend on the penetration depth only, irrespective of their direction of propagation. Hence, let the boundary of the medium be described by z = 0, the amplitudes A will be functions of z and t only. Because of the smallness of the parametric coupling we expect only small variations of the A, in space and time to occur. Hence, we may neglect their second order derivatives, and, taking full account of the linear dispersion of the medium, we find from eq. (2.21)

-

dA3 1 dA3 .4rrw: el x ( ' ) ( w 3 ;w l r w2):ele2 -~ cos p 3 -+-=1 7 __ az v3 a t C k3cos ( ~ 3 X A l A 2exp {i(ki+ k2- k,)r)

(2.22)

(cf. AKHMANOV and KHOKtiIOv* [1964]). Here a? denotes the angle between the wave vector and the ray vector, p3 is the angle between the latter and the z-axis, and u3 is the group velocity. The appearance of the

* Note that our notation I S dlfferent from that used by these authors Coincidence achieved by the following replacement of their symbols A,, A,+AY, 2-2x

IS

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group velocity, rather than the phase velocity, is due to the linear dispersion of the medium. Similar equations hold for the other waves j = 1 and j = 2. Now, it is an important consequence of the symmetry relation (2.12) that the coupling constants in all three equations are equal; hence, the parametric interaction among the three waves is governed by one coupling constant which, conveniently normalized, may be written as

(2.23) Since the left-hand side of eq. (2.22) is independent from x and y, mathematical consistency requires the same to hold for the right-hand side, too. Hence, the following conditions must be fulfilled (2.24)

Moreover, the effect of the driving term in eq. (2.22) will be strongest for

Ak,

k3, - ( k i ,

+k2,)=0.

(2.25)

In fact, for A k , f 0 the nonlinear source terms have a different sign in two points separated in z-direction by a distance A z = T ( A k z ) - ' . Hence, the effect on the driven wave will change from amplification to damping, or vice versa, when the wave propagates over a distance Az. In consequence, the net effect on the wave will change periodically with the crystal length L (in z-direction), when L exceeds A z . The physical meaning of a nonvanishing value for Ak, is that the electromagnetic wave and the polarization wave driving it propagate with different phase velocities (in z-direction). Hence, their relative phase which determines whether energy is transferred from the light wave to the polarization wave, or vice versa, changes during the passage through the medium. We thus arrive at the conclusion that the nonlinear interaction will be most effective for waves with wave vectors satisfying the so-called phase matching condition, (2.24) and (2.25). Since the wave numbers are proportional to the (linear) refractive index, the dispersion of the medium plays a decisive role in this condition. For given frequency and direction of propagation for the pump wave j = 3 and fixed direction of propagation for one of the remaining waves, say j = 1 (signal wave), the requirement of phase matching, under favourable circumstances, singles out certain discrete frequencies o1 (and, hence, also o2= o1- ol), whose number is three, at the most (KLEINMAN [1968],

L 8 21

DASIC EQUATIONS

13

see also 0 3). The more frequent case, however, is that the phase matching condition cannot be fulfilled at all. However, if phase matching can be attained in a suitable crystal for three waves propagating in the same direction (collinear interaction), it provides a practicable mechanism for continuously tuning the signal frequency by varying the linear index of refraction. This may be achieved by changing either the temperature of the crystal or its orientation with respect to the propagation direction of the pump wave, and both methods have found successful applications in optical parametric oscillators. (For details see 0 6.2.) In accordance with the experimental situation normally envisaged in optical parametric oscillators, we assume all three waves to propagate in the same direction (parallel to the z-axis). Then eq. (2.22) takes the form dA3 + 1aA3 = i __ az

U! a t

“2:

A,A, exp {-i Akz},

k3cos

( Y ~

(2.26)

where the coupling constant K is defined by eq. (2.23) and vy = u3 cos a3 is the group velocity in the direction of wave propagation. The equations of motion for the signal and idler wave read

aA2

1 aA2

-- +--=i

KW;

-

ATA,exp{iAkz}.

(2.28)

The coupled set of eqs. (2.26H2.28) forms the basis for a theoretical analysis of parametric amplification and oscillation. According to it, however, parametric interaction can take place only if at least two of the waves are present at the beginning of the interaction, since otherwise no nonlinear polarization, and, hence, no coupling will appear. This feature, however, is due only to the classical approach to the problem. Actually, only one wave, the pump wave j = 3, is needed to start the generation of both the signal wave and the idler wave, since pump photons can decay also spontaneously (i.e., in the absence of signal and idter waves) into signal and idler photons. Thus fluctuating (noisy), low intensity signal and idler waves at frequencies determined by the phase matching condition are produced from which coherent; high intensity fields may build up if a sufficiently high feedback is provided by suitable resonators. In fact, it is the spontaneous decay of pump photons which affords a convenient opportunity to generate light fields at new frequencies from a pump wave of fixed frequency. For a formal description of this spontaneous process,

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termed parametric fluorescence, a quantization of the fields appears to be necessary. Before doing so, let us add a more geneial remark. From the classical point of view, the generation of signal and idler waves can start only from fields already existing at these frequencies (at least, at one of them), however small their intensity may be. Hence, the spontaneous decay of pump photons into signal and idler photons may be considered to provide a further instructive example (in addition to the well-known spontaneous emission from atomic systems), of the manifestation of the vacuum fluctuations of the electromagnetic field, as predicted by quantum electrodynamics.

2.4. QUANTUM MECHANICAL WAVE INTERACTION

DESCRIPTION

OF PARAMETRIC THREE-

For a satisfactory theoretical treatment of parametric fluorescence, it will suffice to simply “translate” the equations of motion of the type (2.22) into the quantum mechanical language. A t first, we pass to the mode picture, i.e., we consider an infinite set of waves characterized by discrete values of w and k. The latter are determined by suitable boundary conditions which are well defined for fields existing within resonators. The parametric fluorescence, on the other hand, we want to study in the absence of resonators. As is well known, to describe this situation it is convenient to require the field to be periodic in the x, y and z-direction with large (fictitious) periodicity lengths L,, L, and L,. For convenience we choose, in the following, the mode volume V = L,L,L, to be identical to the volume of the nonlinear crystal. In the mode formalism the parametric interaction between the waves does not give rise to a spatial dependence of the slowly varying amplitudes A,, but manifests itself only in a time dependence of the Ai. (A spatial variation of A, as expressed by eq. (2.22), corresponds, in the mode picture, to a superposition of several neighbouring modes.) In other words, the mode concept yields a Lagrangian (or Hamiltonian) formalism for the field, the amplitudes A, ( t ) being generalized coordinates. Since we are dealing with modes whose wave vectors are close to each other as are their frequencies, allowance has to be made, in the formalism, for the possibility that one mode A is coupled not only to one pair of modes A’, A“, but to several modes A’ and A ” which may fulfil the frequency relation, e.g., wA = wA-+ o,,.,only approximately. Moreover, the explicit spatial dependence

1 9 8

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BASIC EQUATIONS

15

of the right-hand side of eq. (2.22), expressed by the exponential, has no legitimate place in our Lagrangian formalism, and we get rid of it by integrating the exponential over the volume of the crystal V. This leads us to the following modification of eq. (2.22)

where the following abbreviations have been introduced and

F(k)=A

J

V "

exp {-ikr} d'r.

(2.31)

The latter factor obviously reduces the strength of interaction in the case of phase mismatch. Note that the exponential on the right-hand side of eq. (2.29) (in perfect analogy to the space dependent exponential in eq. (2.22)) automatically restricts the interaction to those modes which satisfy the condition 0, = + w,,,. Now, the above mentioned "translation" will proceed by replacing the A, with operators A,, deriving the commutation relations obeyed by them and by constructing an effective interaction Hamiltonian such that Heisenberg's equations of motion following from it are formally identical to the classical equations of the type (2.29) (cf. PAUL[1973a]). In order to find the commutation relations, we need an explicit expression for the unperturbed Hamiltonian H,, (corresponding to x(') = 0). The energy density for a travelling plane wave in a lossless, anisotropic and dispersive medium is given by (cf. KLEINMAN [1968]) (2.32) where n denotes the linear index of refraction, and the meaning of vil is the same as in eq. (2.29). The quantity E l is the amplitude of the positive frequency part of the transverse component (with respect to the propagation direction) of the electric field strength. To obtain a closer correspondence to the quantum mechanical formalism to be developed, we include into E l the time dependence describing the free motion, i.e., write

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according to eq. (2.19)

E l = A cos a exp {-iwt}.

(2.33)

Hence, the operator Ha for one mode of the field, representing the energy contained in the mode volume V, takes the form

nc V Ho = -8’8, 27rvll

s

(2.34)

where is the operator corresponding to the classical quantity (2.33). From the requirement that the energy (per mode volume) must be quantized in portions of plw, one easily derives the following commutation relation 27rhwvI’ [&,&+I = ___ 1. (2.35) cn V From eq. (2.29) the quantum mechanical equation of motion for follows

gA

where x ( A ; A‘, A”)

g(A;A‘, A”) = cos aA cos aApcos

(2.37)

Now, it is readily proved that eq. (2.36) can be interpreted as Heisenberg’s equation of motion corresponding to the following interaction Hamiltonian

Hi,,=-2V

1

A.A’,A”

F ( b - kA,- kA.)x(A; A’, h”)&+&,&.+H.c.. (2.38)

In the next section, this operator will be used to describe in more detail the process of parametric fluorescence.

P

3. Parametric Fluorescence

In order to simplify the treatment, we make the following idealized

PARAMETRIC FLUORESCENCE

17

assumptions : (a) The nonlinear crystal has the shape of a slab (oriented parallel to the x,y-plane) whose breadth is large compared to its thickness L. (b) The pump wave falls perpendicular on the surface z = 0 of the slab. It is described by a one-mode state of the field (travelling plane wave type), the mode being labelled A = 3. Since only one pump photon is “absorbed” in the elementary process, the corresponding transition probability will be proportional to the mean photon number in the pump mode N3 and, therefore, will not be affected by the photon statistical or coherence properties of the pump wave. (For an explicit proof, see GIALLORENZI and TANG[1968a].) Hence, we may choose the initial state of the pump mode (within the crystal) to be the most convenient one for calculation, namely, a state of sharp photon number N3 is connected with the total energy flux W3 through the relation (3.1) We treat the spontaneous decay of pump photons into signal and idler photons by the well-known perturbative formalism. Since the operators Ei, EA,apart from their normalization, are photon creation and annihilation operators, respectively, it follows from the explicit form for the interaction Hamiltonian (2.38) that the fundamental process ( N 3 ) k = 3 1 o ) Al o ) A r +

lN3-

1)A=3I1)AI1)A’

(3.2)

occurs already in the first order of perturbation theory. The correspondis easily evaluated from the knowledge of the ing matrix element of Hi,, commutation relations (2.35), and a straight-forward calculation (KLEINMAN [ 19681) yields the following expression for the probability per unit time A w that a signal photon with given polarization and a frequency falling in the interval w 1 * o1+ A w l is emitted into the small solid angle A01

-

where

R=

2“7r4,f21F(k3- kl - k 2 ) 1 2 0 1 0 /I2 ~I1 1 /I~ 2 ~ 3 - VLW38(03 - 0 1 - 0 2 ) c 3nl n2n3v3

(3.4) (c denotes the velocity of light in vacuum).

18

‘THEORY OF OPTICAL PARAMETRIC AMPLIFICATION

[I,

83

The integration over k2 (wave vector of the idler photon) describes the experimental situation where no observation is made of the idler wave. (Strictly speaking, the two different, but possible, polarization directions of the idler wave must be taken into account, i.e., in eq. (3.3) a similar term corresponding to the second polarization direction must be added.) By virtue of assumption (a) the factor IFI2 representing the phase matching condition takes the simple form

where Ak, is given by eq. (2.25). Due to the appearance of three &functions in the integrand of eq. (3.3), the integration reduces to the immediate neighbourhood of the point k2 which is required to simultaneously satisfy the following three conditions kl,+k2x=0,

kl,+k2,=0,

03=01+02.

(3.6)

A closer investigation reveals that the fulfillment of this set of equations is no trivial matter. It may happen that there exists no solution at all. Assuming, on the contrary, the frequency and propagation direction to be properly chosen, such that eqs. (3.6) are satisfied by a suitable wave vector* k2, we find from eq. (3.3) the power radiated into the frequency interval o1* . * o1+Awl and the solid angle Anl to be II II

4

API = 4hizn1v2v301$, n’n31U2,lV3C

sin’ ($LAk,) ($ Akz)-2W3 Awl Alll,

(3.7)

where use has been made of the relation d o 2 = v2= dk2,

(for fixed k2x and k2y).

(3.8)

It is evident from eq. (3.7) that the emission will become effective only for IAk,JL5 27r;

(3.9)

this means, in a given direction of propagation definite lines will be emitted whose center frequencies are determined by the phase matching condition Ak, = 0 (provided the latter can actually be satisfied!) and whose widths are inversely proportional to the thickness of the crystal L. A more

* As pointed out by KLEINMAN[1968], the more frequent case is that two solutions occur for which, however, the phase mismatch in z-direction is different. Since the latter reduces the transition probability, we reject the solution corresponding to the greater value of IAk, 1.

I,

0 31

PARAMETRIC FLUORESCENCE

19

detailed analysis shows (KI-EINMAN [ 19681) that the phase matching condition, for a given angle S between k3 and k l , singles out three different frequencies, at the most. However, when this angle exceeds a certain it becomes impossible to achieve phase matching maximum value ,,,a, and, hence, the emission ceases abruptly at S=S,,,. The total power corresponding to one emission line follows from eq. (3.7)

and the line width is given by (3.11) Now, it is interesting to note that the scalar product sl(ul - u2) vanishes at S =.,,a, Then both expressions (3.10) and (3.11) will formally diverge. A closer examination (KLEINMAN[1968]) leads to the result that in this case AP:"' increases as L;, and So decreases as L-;. The characteristic properties of the radiation emitted in parametric fluorescence can be summarized as follows: (a) Provided the direction of observation is chosen such that the requirement of phase matching can be met, definite sharp lines (at least one line) may be observed. (b) The centre frequency of such a line varies with the direction of observation. The emission, however, ceases abruptly when the angle S between k, and kl exceeds a critical value 6,,,. At 6 = S,,, both the total power and the bandwidth of the radiation (which, in this case, consists of a single line) are greatly enhanced. (c) Since the transition probability for the elementary process is proportional to the intensity of the pump wave, focusing of the latter will not enlarge the total output power. Finally, let us refer to some experiments on parametric fluorescence. OSHMAN and The first observation of this phenomenon is due to HARRIS, BYER[1967]. A cw argon laser was used for pumping, and the nonlinear material was LiNb03. While the direction of otservation was held fixed (collinear interaction), tuning of the signal frequency was achieved by temperature variation. Utilizing the same pump source and nonlinear [ 19681 studied noncollinear interaction. dielectric, KLISHKOand KRINDACH They detected photographically the radiation emitted in different directions. Coloured rings (see Fig. 1) could be seen even with the naked eye.

20

THEORY OF OPTlCAL PARAMETRIC AMPLIFICATlON

k,

+ pump wave

k2 nonlinear crystal

Q

[I,

84

optco/ axis

fluorescence radiation

Fig. 1. Experimental set-up for the observation of parametric fluorescence.

The observed frequencies varied from 5200 A to 7000 A. Parametric fluorescence in ADP pumped by the second harmonic of a pulsed ruby laser, was investigated in collinear (MAGDEand MAHR [ 1967]), as well as noncollinear interaction (MAGDE,SCARLET and MAHR [1967], MAGDE and MAHR[1968]). In particular, the linear increase of the emitted power with both pump power and crystal length could be confirmed experimentally. GIALLORENZI and TANG[1968b] used a cw argon laser to pump an ADP crystal. Generally, it can be stated that the intensity of the emitted radiation is very low; in cw operation total output powers of order lo-'' ' W are reported. In fact, it was not until an optical parametric oscillator had been operated successfully that parametric fluorescence became an object of experimental study. It seems noteworthy to mention that parametric fluorescence opens an interesting possibility to determine the (linear) refractive index for a nonlinear medium in a frequency range where strong absorption occurs. The method consists in measuring the tuning characteristics for the signal wave from which the refractive index for the idler wave can be evaluated on the basis of the phase matching condition, without a need for observation of the idler wave, i.e., also in the case of strong absorption of the latter.

-

0 4. Quantum Mechanical Description of Parametric Amplification The process leading from microscopic fields generated by parametric fluorescence to coherent macroscopic ones existing in optical parametric oscillators is parametric amplification. In this section we study this process in some detail from the very beginning with special emphasis on its noise characteristics. To this aim, the quantum mechanical description must be adopted. The next section is devoted to a classical treatment, i.e., we

1,5 41

QUANTUM MECHANICAL DESCRIPTION

21

assume the initial signal and idler waves to be intense enough to be considered as coherent classical waves, thus disregarding the amplifier noise which may well be neglected in this case. For a description of the microscopic stage of parametric amplification, we use the following simple model. Because of its high intensity and small bandwidth (laser beam), the pump wave will be approximated by a classical wave with fixed phase and (real) amplitude. We confine ourselves to the linear stage of amplification, i.e., we consider only interaction times for which the depletion of the pump wave is negligibly small. Hence we may assume the phase and amplitude of the pump wave to remain constant during the amplification process. For simplicity, we suppose the signal and idler waves to propagate in the same direction within ring resonators, this enables us to represent them by one-mode states of travelling plane wave type. We disregard resonator losses as well as losses in the nonlinear material (due to diffraction by crystal inhomogeneities, e.g.) and assume the phase matching condition to be exactly satisfied. Introducing, at last, the familiar photon annihilation operators

we find from eqs. (2.27), (2.28) the following equations of motion (cf. LOUISELL, YARIVand SIEGMAN [1961], MOLLOW and GLAUBER [1967])

where (4.4) and E

= exp { - i ( h - q3)}

(4.5)

(cp3 denotes the phase of the pump wave).

The coupled system of equations (4.2) and (4.3) is easily solved to yield

GI( t ) = cosh ( y t ) q1(0) + E*

sinh ( y t ) q:(O),

(4.6)

&(f)=cosh(,t) q2(0)+&*sinh(yt)q:(O).

(4.7)

From these explicit solutions, arbitrary correlation functions for the fields can be calculated. In the case of large amplification, yt >> 1, the difference

22

THEORY OF OPTICAL PARAMETRIC AMPLIFICATION

[I, §

4

between sinh (yf)and cosh ( y t ) can be neglected, and eqs. (4.6), (4.7) take the simpler form

GI(?>=

G[qi(O)+ E*qt(O)I,

(4.8)

& ( t ) = G[q2(O)+ ~*q:(o)I,

(4.9)

G = ;exp { y t}

(4.10)

where the factor is characteristic for the gain. In what follows, we shall assume the initial signal and idler waves to be statistically independent. Introducing the abbreviation

( q : ( f ) q ~ ( f ) ) = N ~ ( t )(A = 1,2L

(4.11)

we thus obtain from eq. (4.8) Ni(t)= G'll+ Ni(O)+Nz(O)+ ~(q~(O))(q~(o)) + ~*(q:(O))(qz(o))I.

(4.12)

Here, the first term in the bracket is due to the spontaneous process analysed in 0 3; it describes the amplified noise starting from the spontaneous decay of pump photons. We emphasize the fact that the amplification process sensitively depends on the initial phase relation among the three interacting waves (the phases of the signal and idler waves are those of the cxpectation values

( 4 m (4Zt))).

The gain is maximum for (P3 = (PI

+ (P2 +;7r,

(4.13)

whereas in the case of the opposite sign of T,the signal and idler waves will be damped rather than amplified. It is interesting to note that the phase relation (4.13) is fulfilled in the spontaneous process already. This becomes evident from an examination of the expectation value ( i j l ( t ) Q 2 ( t ) ) which, in the case of self-excitation of both the signal wave and the idler wave, follows from eqs. (4.6) and (4.7) (cjl(thj2(f)) = E* sinh ( y t ) cosh ( y t ) .

(4.14)

Instead of eq. (4.14),we may also write

Hence, the sum of the phases for the signal wave and the idler wave,

1,s

41

QUANTUM MECHANICAL DESCRlPTlON

23

respectively, is determined by the phase of the pump wave already in the spontaneous decay of one pump photon. The usual situation one encounters in amplification is that only the signal wave, together with the pump wave, is incident, i.e., the idler wave emerges from the vacuum state Then eq. (4.12) reduces to the simple relation N,(t) = G2[1+ N,(O)].

(4.16)

In the approximation (4.8), (4.9), the photon number of the idler wave equals that of the signal wave; hence, the amplified signal may equally well be detected at the idler frequency. Thus, parametric amplification appears to be intrinsically coupled with frequency transformation. This will be of practical importance, if the idler frequency falls in a range which is more convenient for measurement than the signal frequency. To study coherence properties of the amplified field we, at first, calculate the quantity (4.17) which takes its maximum value u = 1 for thermal light (incoherent field), and its minimum value u = 0 for a Glauber state (fully coherent field). CT can be determined directly in an experiment where two independent light beams, which differ only in their direction of propagation, are made to interfere. The visibility of the interference pattern u is then related to u by the simple relation u = 1- u (PAUL[1966]). From eqs. (4.8) and (4.12), the quantity (4.17) for the amplified signal wave becomes (see PAUL[1973b])

This formula greatly simplifies for an idler wave starting from the vacuum state. Then, eq. (4.18) can be written as (4.19) This result indicates that amplification is accompanied by an increase of u1(unless we start from an incoherent field for which a, of course, retains its maximum value u = 1). This loss of coherence must be ascribed to the inevitable amplifier noise originating from the spontaneous decay.

24

THEORY OF OPTICAL PARAMETRIC AMPLIFICATION

11,

§4

While o1 depends on the fluctuations of both phase and (real) amplitude, the quantity

(4.20) which can be measured in an experiment of Hanbury Brown and Twiss type (see HANBURY BROWNand TWISS[1956]), is characteristic for intensity fluctuations and insensitive to phase fluctuations. Also here, the limiting values are D = 1 for thermal light and D = 0 for a Glauber state. Assuming again the idler wave to be in the vacuum state at the beginning of the interaction, we easily find from eqs. (4.8) and (4.12)

(4.21) From this result, it is easily concluded that the intensity fluctuations will be enhanced also as a consequence of amplification. The parametric amplifier closely resembles a laser amplifier. In fact, it can be shown (see PAUL [1973b]) that the formal description of a laser amplifier with a completely inverted, active medium is identical to that of an optical parametric amplifier with no idler wave present at the beginning. However, radiating an idler wave of definite phase (e.g., a suitably attenuated laser beam) into the nonlinear material, in addition to a pump wave and a signal wave, enables one to gain more information from the measurement of intensity correlations for the (amplified) signal wave than that obtainable from a laser amplifier. By repeating the experiment many times with different phases of the idler wave, quantities like ( q f q ) , (q+2q2)-(qtq)2 are found in the form of a Jurier series

(4.22) (cpz denotes the phase of (ql(O))), from which the coefficients f n ( = f!,,) can be determined separately. Thus, quantities of the type ( q l ( 0 ) ) , (q:(O)), (q:(O)q:(O)), . . . , being the constituents of the fn, might be measured. Until now, we have studiea the microscopic stage of amplification. For higher intensities of the signal and/or idler wave, the depletion of the pump wave can no longer be neglected. On the other hand, the specific quantum mechanical features of the radiation will become less relevant in this case. Hence, the amplification of signals of not too low intensity can be described in classical terms.

I , § 51

CLASSICAL TREATMENT

25

0 5. Classical Treatment of Parametric Amplification 5.1. SOIXTION OF THE EQUATIONS OF MOTION

Assuming stationary conditions, we have to consider the slowly varying amplitudes A, in 02.3 to be time independent. Thus the basic eqs. (2.26)-(2.28) governing the parametric three-wave interaction reduce to ordinary differential equations describing the variation of the intensities of the waves with the penetration depth in the nonlinear material z. In contrast to the analysis in § 4, the depletion of the pump wave will now be fully taken into account. The coupled set of equations can be solved rigorously (ARMSTRONG, BLOEMBERGEN, DLJCUING and PERSHAN [1962]). To this end, it is useful to separate the complex amplitude A, into a (real) amplitude and a phase factor, respectively, & ( z ) = ~ A ( z ) exp I-~(PA(Z)) (A = 1,293).

(5.1)

Moreover, we substitute

and

‘ K(T 2aW

=

2 2 2 010203

k , cos2alk2cos2a2k3cosza3

(5.3)

where

denotes the total energy flow per unit area in the direction of wave propagation (2-direction) which is easily shown to be independent of 2. Thus, we find from eqs. (2.26)-(2.28), writing the latter ones in their real and imaginary parts, (5.5)

(5.7)

26

THEORY OF OPTICAL PARAMETRIC AMPLIFICATION

dO d5

-= A S

d + cot 0In (u d5

uz u3),

where the following abbreviations have been introduced

@ ( z )= Akz - C P S ( Z ) + C P ~ ( Z+) ~ p 2 ( ~ ) , AS = Ak< k2 ~

0

~

p.

k3 ~ cos ~ xa32

a:

22

a 3

(5.9) (5.10) (5.11)

From the definitions for uA and W follows the relation

(5.12) which represents the conservation law for the total energy flux (per unit area) in z-direction. Moreover, from eqs. (5.5)-(5.7), the following conservation laws (Manley-Rowe relations) are easily derived u:(z)+ u & ) =

u;'+ u;*= c1,

d ( z ) + u:(z) = z4z;

+ u y = c2,

(5.13)

where the notation uO,= ~ ~ ( has 0 ) been used. BLOEMBERGEN, The solution of eqs. (5.5)-(5.8), as given by ARMSTRONG, DUCUING and PERSHAN [1962], proceeds as follows: At first, eq. (5.8)will be integrated to yield (5.14) where

r=u

~ u cos ~ Oo+% u ~ ASui'.

(5.15)

Inserting the result (5.14) into eq. (5.7) and observing the ManleyRowe relations (5.13), we arrive at a Jacobian elliptic integral. The solution for u$(z) can be written in terms of the elliptic sine (sn), a tabulated function which is the inverse of the elliptic integral. Explicitly we find

I,$

51

CLASSICAL TREATMENT

21

Here the quantities uia 5 U 23 b 5 uzc denote the roots of the cubic equation for u: u ~ ( c- u, : ) ( c ~ - ~ f ) - ( r - i d S ~ : ) ~ = O .

(5.17)

The parameter y 2 is given by

(5.18) and zo is determined by the initial values at z = 0. Similar solutions exist for the signal wave and the idler wave, namely

(5.19) The function sn (x, y 2 ) is periodic, the period depending on y 2 . Elementary special cases of sn (x, y 2 ) are sn (x, 0) = sin x,

(5.20)

sn (x, 1)= tanh x,

(5.21)

and sn (x, y 2 ) vanishes at x = 0

sn (0, y 2 )= 0.

(5.22)

For 1- y 2 << 1, the period is approximately given by xP=41n

4

(1 - y2)$,

(5.23)

5.2. PERFECT PHASE MATCHING (Ak = 0 )

Let us assume the phase matching condition to be exactly fulfilled and the phase difference c p ~ - ( c p Y + c p ~ ) to be equal to *;T. Then we have r=0, and the roots of the cubic equation (5.17) read, for u’;’< uz2,

u:,=o,

Uzb=

u;12+u:2,

u i c = u;’+u;*.

(5.24)

On the other hand, since cos @, = 0 and A k = 0, it follows from eq. (5.14) that 0 is independent of z (apart from an eventual jump by T ) @=

*in.

(5.25)

28

THEORY OF OPTICAL PARAMETRIC AMPLIFICATION

[I,

85

Consequently, sin 0 takes the values +1 or -1, which implies maximum coupling in eqs. (5.5)-(5.7). In case of the positive sign, energy is transferred from both the signal wave and tile idler wave to the pump wave, whereas for the mainly interesting case sin 0 = -1, energy is fed from the pump wave to the signal wave and the idler wave which, therefore, experience an amplification. (We arrived at the same conclusion with respect to t t e phase dependence of the amplification process already in $4.)The amplification of both the signal wave and the idler wave, on the other hand, is accompanied by a depletion of the pump wave, whose amplitude u3, according to eqs. (5.16), (5.22) and (5.24), reaches its minimum value u3,i" =

(5.26)

0

at z = z o . Physically, this means the whole pump power has been converted into signal and idler power. Consequently, at z = zo the amplitudes for the signal wave and the idler wave take their maximum values which follow from eq. (5.19) to be 1

=(uXZ+ u3 )? 02

(A = 1,2).

(5.27)

For z > zo, the function sn2 in eqs. (5.16) and (5.19) increases again. In consequence, the signal wave and the idler wave lose energy, thereby newly creating the pump wave. The amplitude of the latter attains a maximum value u3,,,

at z =,,,z,

where,,,z

= (UY2+ Ug2)f

(5.28)

satisfies the relation

sn2[(u3,- u&)f(zmax-z0)L-', 7 4= 1.

On the other hand, at z =,,,z

the amplitude for the signal wave vanishes, Ul,'.

= 0,

(5.29)

and the amplitude for the idler wave (due to the above assumption ug2> u:') falls to its minimum value uz,,, = (u;* - uf)!

(5.30)

For z > z,, the signal wave and the idler wave become amplified again, and the whole cycle described above is repeated again and again. The period in space Az will be called phase coherence length L,. Note that the transition from an amplifying to an attenuating stage, e.g., for the signal wave and the idler wave, is connected with a jump of 0 by T , which

I,

8 51

CLASSICAL TREATMENT

29

t

Fig. 2. Dependence of the amplitudes for the signal wave (u,), the idler wave (u2) and the pump wave ( u g ) on the penetration depth z (for u;> uZ2).

requires one of the waves to vanish (see Fig. 2). The length L, becomes larger as y 2 approaches 1. It is evident that the net amplification of the signal (and idler) wave depends sensitively on the crystal length L. Maximum amplification is achieved if the first zero (or, in principle, any of the zeros) of the amplitude for the pump wave falls on the back surface z = L of the crystal, provided, of course, the pump power is high enough to make this experimentally feasible. For the special case uy = u; which implies y 2 = 1,we have, according to eq. (5.21), no periodic solutions. Maximum energy conversion leading to z3m,n = 0, is achieved at z = zo only. For z > zo, the amplitude w3 increases monotonously. The latter behaviour can be explained as follows. A change from attenuation to amplification for the signal wave and the idler wave would necessitate the amplitude of at least one of these waves to vanish (since otherwise 0 cannot jump). However, since for up= u: the signal wave and the idler wave behave exactly in the same manner, as far as the amplitudes ul, u2 are concerned, the latter ones would vanish simultaneously. In this case, however, parametric interaction ceases to take place (in the classical description!), and a recreation of the signal wave and the idler wave becomes impossible. 5.3. PHASE MISMATCH ( A k # O )

Now we shall extend our analysis to include the effect of phase mismatch ( A k f 0) and/or a deviation of Oo from +in. We thus make the

30

THEORY OF OPTICAL PARAMETRIC AMPLIFICATION

[I,

55

assumption cos Oo # 0 (which, of course, has a definite physical meaning only if all three initial amplitudes u: (A = 1, 2, 3) are nonvanishing; cf. also eq. (S.15)). We see from eq. (5.17) that now uzur in geheral, will be different from zero. While the solutions for u: retain their periodic character, as found in the preceding section, the new feature is that, for u:,>0, the pump power cannot be completely converted into signal and idler power, as becomes obvious from eq. (5.16). Hence, the effective gain in amplification is smaller than that corresponding to A k = 0 , cosOo=O. It is interesting to note that there exists a special case for which u:, equals zero and, hence, full conversion of the pump power becomes possible. In fact, it is easily concluded from eq. (5.17) that u f u vanishes for r=0, i.e., for

A S u; cos oo= --__ 2 uyI.4;.

(5.31)

In this case, however, the distance the pump wave has to transverse in the crystal to become perfectly depleted is larger than for Ak = 0, cos Oo = 0. Thus the effective gain per unit length is smaller. For small values of z , A k z < < 1, a similar conclusion can be drawn directly from eqs. (5.5)-(5.7) which indicate that the gain is maximum for cos Oo = 0 (sin Oo = -1). Generally, it can be stated that the parameter y 2 is smaller for A k f 0 and/or cos Oo f 0 than for A k = 0, cos Oo = 0. Hence, the phase coherence length is smaller in the first case. Now, the usual situation encountered in amplification is that only the pump wave and the signal wave are incident on the nonlinear crystal, while the idler wave builds up from the initial amplitude u i =O. Then cos Oo does not enter the formulas (see eq. (5.15)) - as it must be, since cos 0, is not defined in this case. For A k = 0 we have r=0, and eq. (5.14) implies cos O ( z )= 0. Hence, the phase of the idler wave adjusts so as to afford a maximum coupling among the three waves. As pointed out above, the maximum conversion rate will decrease, as well as the phase coherence length, for growing Ak. Experimentally, parametric amplification has been achieved by WANC and RACETTE [1965]. The experimental set-up is shown in Fig. 3. The pump wave was the second harmonic of a Q-switched ruby laser (output power 30 MW, pulse duration 30 ns) generated in an ADP crystal. It was used to amplify, in a second ADP crystal of 8 cm length, the 0.633 pm line emitted by a He-Ne laser. The initial power was 10-’W, and an amplification by 18% has been observed. This value, however, lies far

I,

5 61

THE OPTICAL PARAMETRIC OSCILLATOR

@3

.

31

signol wave crystal

He -Ne laser ~-06328pm wave

ADP cryslol (frequency doubling) ruby laser A-06937pm

B Fig. 3. Experimental set-up for parametric amplification (after WANGand R A C E ~119651). E

below the theoretical expectation. A remarkably high amplification factor 3-4 was achieved by AKHMANOV, DMITRIEV, KHOKHLOV and KOVRYCIN [ 19661. Pumping with the second harmonic of a glass :Nd3' laser, these authors amplified the original line at A = 1.06 p m in a KDP crystal.

0 6. The Optical Parametric Oscillator (OPO) 6.1. INTRODUCTORY REMARKS

Similar to the situation encountered in laser physics, one may pass from parametric amplification to oscillation, for sufficiently high pump power, by adding a suitable feedback to the signal and/or idler wave, as has been suggested already in 1962 by AKHMANOV and KHOKHLOV [1962], KINGSTON [1962] and KROLL[1962]. We then speak of an optical parametric oscillator (OPO). When a strong pump wave is sent into a nonlinear crystal, signal and idler waves will be created. The starting mechanism is the spontaneous decay of pump photons which has been studied in § 3. The microscopic fields at the signal and idler frequencies thus produced, will be amplified during their passage through the nonlinear crystal, at the cost of the intensity of the pump wave. According to the results of § 5, the gain will be maximum for Ak = 0 and cos Oo = 0. If the crystal is placed within a resonator, e.g., for the signal wave, the latter will pass the crystal many times, thereby becoming more and more amplified, as long as the gain per passage exceeds the losses due to the transmission of the resonator mirrors. Finally, the losses will be exactly balanced by the gain, and,

32

THEORY OF OPTICAL PARAMETRIC AMPLIFICATION

[I,

56

hence, a steady-state oscillation will take place which corresponds to A k = O and cosOo=O. This can be accomplished, however, only if the pump power exceeds a certain minimum value, i.e., the oscillation has threshold character. The pump power at threshold depends on the magnitude of the nonlinear susceptibility, on the resonator losses and on the type of the experimental set-up. We speak of a singly resonant OPO (SRO), when only one of the generated waves, say the signal wave, is resonant, while the remaining two waves pass nonresonantly through the crystal. In this case, the threshold pump power is rather high. A drastic lowering of threshold can be achieved in a doubly resonant OPO (DRO), where both the signal wave and the idler wave resonate. It is mainly for this reason that most of the existing OPOs are of this type. In principle, also the pump wave may be resonant (triple resonance), but experimentally such conditions are very difficult to meet. As to our knowledge, the principle of threefold resonance has found application only in the so-called “internal OPO” which will be described later. Irrespective of the special arrangement, we may in the following theoretical treatment assume that the relations Ak = 0 and cos Oo = 0 hold. This is justified by the fact that the gain is maximum in this case. The requirement of phase matching (Ak = 0 ) ,on the other hand, provides the basic mechanism which allows the OPO to be operated as a light source whose frequency can be tuned continuously.

6.2. TUNING CHARACTERISTICS

As already mentioned in 0 2 , the phase matching condition is sensitively dependent on the linear index of refraction for the pump wave, the signal wave and the idler wave. Hence, it can be changed significantly by varying either the temperature of the crystal or its orientation with respect to the propagation direction for the pump wave. In consequence, the signal frequency determined by the phase matching condition changes, too. Mathematically, the two relations

and

kj = ki + k 2

61

THE OPTICAL PARAMETRIC OSCILLATOR

33

have to be fulfilled simultaneously (see eqs. (2.24), (2.25)). Since k = ( w / c ) n , eq. (6.2) reads, in the case of collinear interaction, o3n3=w,nl+w2nz

(ni=n(wi)).

(6.3)

Making use of eq. (6.1), we find from eq. (6.3) 4 n 3-

n2) = o l ( n *- 122).

(6.4)

This relation cannot be fulfilled, in general, due to the (linear) dispersion of the medium. It can, however, be met in suitable anisotropic media, where the effect of dispersion may be compensated for by that of double [1962]. Taking into refraction, as suggested at first by GIORDMAINE account the temperature dependence of the refractive indices which can be described by power series in the temperature difference AT = T- To, and assuming the pump wave to propagate through the crystal as an extraordinary ray (e), while signal and idler waves are ordinary rays ( o ) , we obtain from eq. (6.4) for w 1 near t w 3 = ma (subharmonic generation)

Thus, starting from w 1 = &w3 at T = TO,the frequency for the signal wave increases as (AT)t, and the tuning range becomes larger, the stronger the refractive indices depend on temperature. According to GIORDMAINE and MILLER[ 19651, the relevant experimental parameters for LiNbOs at the pump frequency w 3 = 3.56 x lO”s-’ (corresponding to the second harmonic of a C a W 0 4 :Nd3+ laser) are d(nS- n:)/aT== 5.7 x lo-’ (“C)-’ and dn:/dw = 3.5 x l o p L 7 s. Hence, eq. (6.5) reads w1 = (178 f 5.4(AT)!)x 1013 s-’

(6.6)

(AT in “C). This dependence of the frequency of the created signal wave on temperature, together with experimental data, is represented in Fig. 4, while the tuning characteristic for ADP and KDP, at a pump wavelength A, = 0.257 pm, is shown in Fig. 5. Tuning of the signal frequency may be accomplished also by rotating the (anisotropic) crystal for fixed direction of propagation of the incident pump radiation. Also in this case, the linear indices of refraction are changed. Similar to eq. (6.5), one finds that the square of the frequency variation w l -10, increases as sin2 cp, where the angle cp is the complement of the internal angle between the optic axis and the direction of

34

THEORY OF OPTICAL PARAMETRIC AMPLIFICATION

Fig. 4. Temperature tuning curve of LiNb03 for a pump wavelength of A, =0.529 Frn (after GIORDMAINE and MILLER[1965]).

O7I A;--\

-20 -M 0

w

20 30 Lo 50

T[ocl

b

Fig. 5. Temperature tuning characteristics for ADP and KDP. The curves were obtained by Dowley using the spontaneous parametric emission technique (from HARRIS [ 19691).

propagation for the pump wave. This has been confirmed experimentally, for LiNb03, by MILLERand NORDLAND[1967] (see Fig. 6). Finally, we mention still another possibility of varying the index of refraction and thus tuning the frequency of the generated radiation. The method uses the electro-optic effect and consists in applying a static electric field whose strength can be varied. In LiNb03 a tuning rate of about 6 . 7 A per kV/cm of the applied field has been observed by KREUZER[1967]. In most practical cases, phase matching is achieved in collinear interaction. Noncollinear phase matching is possible too (cf. e.g., FALKand MURRAY[1969]), but it is less preferable because of the reduction of the effective interaction volume due to the remarkable walk-off of the beams*.

* Strictly speaking, a slight walk-off takes place also in collinear interaction, since the beam direction, in general, differs a little from the direction of wave propagation (cf. 5 2.3).

1,

S 61

35

THE OPTICAL PARAMETRIC OSCILLATOR

'

12 11 la

a

002

au

006

008

aia

sm'y

aiL

016

a18

OM a22 sinz LP

+

0.24 +

Fig. 6 . Angular tuning of a LiNb03 oscillator. Av is the frequency change, A v = (2r)-'(u1 - $ y 3 ) ,and the angle cp is the complement of the internal angle between the optic axis and the direction of propagation of the pump wave (after MILLERand NORDLAND [1967]).

6.3. THE SINGLY RESONANT OPO

Our theoretical treatment of the OPO starts from the rigorous solutions (5.16) and (5.19). As mentioned above, we may set Ak = 0 and cos O o = 0. At first, we study the singly resonant OPO (SRO), i.e., we assume the signal wave to be excited in a resonator with a high quality factor, while both the idler wave and the pump wave nonresonantly pass through the crystal. Then, the intensity of the signal wave will, by far, exceed that of the remaining waves, i.e., we may assume (6.7)

u y >> u y .

U Y 2 >> us2,

Moreover, the quantity uf which is characteristic for the number of idler photons produced near z = 0 by spontaneous splitting of pump photons, can, practically, be set equal to zero. We thus obtain from eq. (5.18) U;'

+?<< u3

+u?'

1;

hence, taking account of eq. (5.20), we may approximately replace the elliptic sine by the usual sine, i.e., we may write, according to eq. (5.19),

i ",")

u ; ( z ) = U i 2 cos2 u y-7 -

.

(6.9)

36

[I, §

THEORY OF OPTICAL PARAMETRIC AMPLIFICATION

The initial condition (at z 0 = u$

= 0)

leads to the relation

(

= U Z ( 0 ) = u;2 cos2 U ] -=

from which follows Z(1

6

u::=&(2n+1)7r, 1

,

n=0,1,2,. *..

(6.10)

(6.11)

It is easily verified that eq. (6.11) ensures also u; and u: to take the correct initial values. Making use of eq. (6.11), we arrive at the following explicit representation for the amplitudes of the signal wave, the idler FISCHER and PAUL[1973]*) wave and the pump wave (BRUNNER,

(

3

u : ( z ) = uy2+u;*sin2 uy= ,

(6.12)

ui(z)=uZ'sin' ( u y ; ) ,

(6.13)

i3

r r i ( z ) = up2cosz u'y= .

(6.14)

Let the resonator for the signal wave be of ring geometry (see Fig. 7). The condition for a stcady-state oscillation will be the requirement for the

A u., U),u,

Fig. 7 . Optical parametric oscillator with ring resonator geometry for the signal wave

(~1).

signal wave to be exactly reproduced after a round-trip through the resonator. This is essentially a condition for the amplitude, since the effect of an eventual phase change, Acpl=cpl(L)-cpl(0), due to the parametric interaction, will be a shift of the resonator frequencies. (Note that for Ak = 0 and cos 0,= 0, no such detuning will occur, since then cos 0 retains its initial value, according to eq. (5.25).) Assuming the output mirror to have a reflectivity R and the remaining two mirrors to be perfectly reflecting, we have to write the condition for the amplitude of the signal wave in the form uy2= Ru:(L), (6.15)

* Note that in the cited paper, the signal wave is labelled by the subscript 2, and the idler wave by 1 .

1 , 8 61

THE OPTICAL PARAMETRIC OSCILLATOR

37

where L denotes the length of the crystal. Substituting here the value for u:(L) following from eq. (6.12), we obtain the relation

(6.16) which defines the stationary value u?= uTt as a function of ui2. Now, it should be remembered that the quantities uA,due to the factor W in the denominator in eq. (5.2), are relative rather than absolute amplitudes. To facilitate the physical interpretation, it will be useful to get rid of this relative character of the amplitudes. To this aim, we substitute

(A = 1, 2, 3)

U A = U A W-'

(6.17) (6.18)

Then, according to eq. (5.11), the parameter 1 (whose reciprocal value is proportional to the strength of the nonlinear coupling) is independent of the total energy flux per unit area W, and it follows from eq. (5.12) that the quantities u i have a simple direct physical meaning; they represent the photon flux per unit area, in units of zt-'. Hence

(6.19)

PA = W A u l f

(f being the transverse cross section of the beam) is the power (the energy flow in the direction of wave propagation) connected with the wave A. In the new amplitudes uA, eq. (6.16) reads 1 R -=-( u : ~ l-R

)

sin (uf'L/l)

(6.20)

From eq. (6.20), the threshold value 2)::h for the onset of the oscillation can easily be evaluated: for given R, the minimum value of vz2 satisfying eq. (6.20) corresponds to the case when (sin (US'LII) becomes maximum which occurs for U ; ~ - + O . We thus find

="(y R

L

(6.21)

and, hence, can replace eq. (6.20) by (6.22)

38

THEORY OF OPTICAL PARAMETRIC AMPLIFICATION

I

1 I

10

20

30

La

50

EO

70

I

80

I

90

Fig. 8. Signal amplitude u;‘ for a singly resonant OPO as a function of relative pump power. In the hatched areas the amplitude is unstable (after BRUNNER, F I T H E R and PAUL[1973]).

This equation has already been derived by KREUZER[1969a] in the so-called parametric approximation. (For the latter, see 0 6.7.) The dependence of the amplitude v? of the (resonant) signal wave on pg/p;th, as obtained from a numerical evaluation of eq. (6.22), is shown in Fig. 8. From this, it becomes obvious that a unique solution exists only for P;/p;th < ( + 3 ~ r )while ~ , for higher pumping powers several branches occur whose number, for

[(2n + 1)412<2< [(2n + 3 ) Tr,

(6.23)

is 2n + 1. An investigation of the stability of these solutions (cf. BRUNNER, and PAUL[1973]) leads to the result that there exist regions FISCHER where the solutions are stable, but these regions are separated by areas of instability, marked by the hatched regions in Fig. 8. The steady-state operation which evolves from the spontaneous process is characterized by the curve A in Fig. 8 which indicates that the quantity v”lL/l may, at best, attain the value IT. Higher oscillation levels (as represented by the curves B and C) cannot be reached without additional experimental effort. One possibility would be the use of an “ignition pulse” which causes the system to pass from a stable state SI (see Fig. 8) into the instability area from which a transition into a higher stable state S y ) , corresponding to the same pump power as S1, becomes possible, provided the quantity AG, = Avl LII, where Av, is the amplitude of the “ignition pulse”, exceeds the distance between S1 and the unstable state lying above it.

1,s

61

THE OPTICAL PARAMETRIC OSCILLATOR

39

Let us now discuss the conversion efficiency q which will be defined as the ratio of the output power for both the signal wave and the idler wave to the input pump power (at z = 0). On account of eq. (6.19), we have (6.24) Making use of the formulas (6.12), (6.13), (6.15) and the frequency relation w 3 = w 1 + 0 2 ,we find from eq. (6.24) (6.25) From this relation, it is evident that a maximum conversion efficiency of 100% will be observed for

These values fall in different stability regions of Fig. 8, and they correspond, as follows from eq. (6.14), to a vanishing amplitude of the pump wave at the end of the crystal, z = L. The conversion coefficient, as a function of the relative excitation P;/P&h, is shown in Fig. 9. Under usual experimental conditions, the operation of the SRO is described by the curve A in Fig. 8. Since then v;'L/l tends to T for p:/p:th4 00, the conversion efficiency strongly decreases, when the relative excitation exceeds a critical value. In the latter case, the large fraction of the pump power which cannot be 4 1

10 -

c- 119

as-

07 06 -

-

05 04 113 r 02 01 -

0 -

Fig. 9. Conversion efficiency for the singly resonant OPO as a function of pump power (after BRUNNER, FISCHER and PAUL[ 19731).

40

THEORY OF OPTICAL PARAMETRIC AMPLIFICATION

I], § 6

converted in single-mode operation will give rise to a multimode operation, unless single-mode operation is ensured by means of frequency selective elements. Experimentally, the SRO has been studied, among others, by KREUZER [ 1969bl. The operation was restricted to the curve A in Fig. 8. As to our knowledge, an excitation of higher levels of operation has not been achieved until now. Of course, the latter problem is of little practical interest, since for a given (sufficiently high) pump power, the resonator losses can always be adjusted, such that the ratio E/P&h corresponds to the maximum conversion efficiency q = 1. The main advantage connected with the excitation of a higher level of oscillation would be to make maximum conversion attainable at a higher relative excitation e.g., in the case of a given pump power, for a higher quality factor of the resonator, which would lead to improved spectral properties of the generated radiation (see 07). Finally, let us remark that the output photon flux is exactly the same for both the signal wave and the idler wave. This is due to the fact that the splitting of any pump photon yields a signal photon as well as an idler photon. Hence, the nonresonant idler wave created in a SRO may equally well be utilized as an intense monochromatic and coherent light beam.

c/&,,

6.4 THE DOUBLY RESONANT OPO

Although the radiation of a SRO has favourable spectral properties (in 07.1, we shall discuss this point in some detail), in the usual operation of an OPO both generated waves are made to resonate, since in this way the threshold pump power is considerably reduced. Accordingly, the first and MILLER operation of an OPO was accomplished by GIORDMAINE [ 19651 in a doubly resonant set-up (DRO). Using the second harmonic of a CaW04 : Nd" laser (A = 0.529 pm) as a pump source, these authors succeeded in tuning the signal wavelength, by means of temperature variation, between 0.97 p m and 1.15 pm. The pump power was 3.8 kW in pulsed operation, and an output power of 15 W could be attained. The experimental set-up is shown in Fig. 10. In the theoretical treatment we must take into account that, as a consequence of the resonance of both the signal wave and the idler wave, the following inequalities hold u;"<

uy2,

u;2<< $ 2 .

(6.27)

I,

8 61

41

THE OPTICAL PARAMETRIC OSCILLATOR

0 -switch cam‘ M’’

dnlecfm

-

o w l t e m p tuning)

cwtings

0142(JL

u : LIMO, filter

hlfer

(frequency

LiNbOp

doubling)

(parametrrc osollofor

Fig. 10 Experimental set-up for the first optical parametric oscillator (after GIORDMAINE and MILLER[19651).

Assuming the phase matching condition to be fulfilled and the photon fluxes for the signal wave and the idler wave to be approximately equal, u’;= u;, we find from eq. (5.18) y 2 = 1.

(6.28)

Hence, by virtue of eq. (5.21), we may approximate sn by tanh. Then eq. (5.19), for A = 1, reads

[

u : ( z ) = ( U ; * + U?*) 1-

(6.29)

while the amplitude for the pump wave varies according to u : ( ~ ) = ( u ~ * + u ; 1 ~ ) t a(nUh;~~ From thc latter relation follows

o2

I

+ U )’~

z - 20

1

1

,

(6.30)

(6.31) Because of (6.27) the left-hand side of this equation is small compared to 1. Hence, we may expand tanh into a power series, and obtain in lowest order (6.32) Since u3 vanishes at z = zo (see eq. (6.30)), which corresponds to a perfect conversion of the pump power, it will suffice to consider values of z S 2z0. Hence, also the function tanh in eq. (6.29) can be expanded to yield the result (6.33)

42

THEORY OF OPTICAL PARAMETRIC AMPLIFICATION

[I,

56

The condition for a steady-state oscillation to exist is that both the signal wave and the idler wave are exactly reproduced after a round-trip through the resonator. This means that the following relations must be obeyed

(6.34) (6.35) where R1, R2 is the reflectivity of the output mirrors for the signal wave and the idler wave, respectively. Now, the above assumption uy = u4 corresponds, in the steady state, to equal resonator losses for the signal wave and the idler wave. Hence, we must set R 1= R, = R, and the equations (6.34) and (6.35) become identical. Inserting here the expression (6.33) for u:(L) and introducing the absolute amplitudes vA (see eqs. (6.17), 16.18)) lead us to the following expression for the steady state photon flux (per unit area) in the signal wave,

(6.36) Obviously, the threshold value for us is

(6.37) and we may rewrite eq. (6.36) in the form

(6.38) where u is defined by

(6.39) Comparing eqs. (6.21) and (6.37) one sees that the threshold pump power for the DRO is smaller than that for the SRO by a factor (1- R)/(4R). For R = 0.98, e.g., this reduction factor is 1/200, and this explains why the DRO can be operated with much less effort. Making use of eq. (6.19), eq. (6.38) can be brought into a form which directly connects the output power at the signal frequency

,rut

1-R R

= (1 - R ) o , v?(L)f = -~

0 1

ur’f

(6.40)

I,

8 61

THE OPTICAL PARAMETRIC OSCILLATOR

43

(cf. eq. (6.34)) with the input pump power P;. Explicitly we find

(6.41) where the threshold input pump power is given by

(6.42) The output pump power follows from eq. (6.30) as

(6.43) Allowing for R , # R2, the conversion efficiency is defined by

0 1 4 ( L ) ( 1 - R d + w 2 & 3 ( 1 -R2)

q=-----

w3

42

(6.44)

In the case R 1= R2= R treated above, one obtains a-1

q=4--.

U2

(6.45)

Evidently, q reaches its peak value q = 1 for (+ = 2 (in this case the pump power P&) vanishes, see eq. (6.43)) and decreases monotonically for u > 2 . The physical reason for this somewhat peculiar effect is that for u > 2 the point zo where the pump power is totally converted lies inside the crystal. Hence, the pump wave will be recreated during the subsequent passage through the remaining part of the crystal. On the other hand, there exists an optimum crystal length Lop, for which total conversion will be achieved. Explicitly, we find from the relation u = 2 which must hold in this case,

1-R 1

Lop=--R

$*

(6.46)

As to be expected, Lo, decreases with decreasing resonator losses and increasing incident pump power. So far, we have assumed the resonators for the signal and/or the idler wave to be of ring geometry which implies that the resonating waves are travelling ones. Experimentally, resonators of the Fabry-Perot type are used most frequently, and, hence, for high reflectivities, l - R < < 1, standing waves will appear. This gives rise to a modification of the theoretical treatment. In particular, it follows that the

44

THEORY OF OPTICAL PARAMETRIC AMPLIFICATION

[I,

86

maximum attainable conversion efficiency is only 50% (SIEGMAN [19621, cf. also BRUNNER, PAULand BANDILLA [1971]). Physically, this can be explained as follows. As is well known, a standing wave may be considered to be a superposition of two counter-running waves. Now, the signal and idler partial waves travelling backwards through the crystal, will recreate the pump wave with a direction of propagation opposite to that of the incident pump wave. Hence, the latter is partly reflected by this mechanism, i.e., a significant part of the pump power remains unconverted. Evidently, this effect cannot occur in the SRO, also if a Fabry-Perot type resonator is used. Hence, the treatment in 0 6.3, actually, is not restricted to a ring resonator. It should be noted that, for not too high reflectivities of the output resonator mirror(s), the backward travelling parts of the signal and/or the idler wave will be less intense than the forward travelling parts. Thus, under these circumstances, the “reflected” pump power will be smaller than in the case l - R < < 1 considered above, and, therefore, the conversion efficiency may considerably exceed 50%. (For details, see BRUNNER and PAUL[ 19761.) Hence, the maximum attainable conversion efficiency can be enlarged by diminishing the quality factor of the resonator(s), at the cost, however, of an enhancement of the threshold pump power.

6.5 THE TRIPLY RESONANT OPO

It has been shown above that a drastic lowering of threshold can be achieved by passing from the SRO to the DRO. Similarly, the threshold pump power can be reduced still further by providing resonance also for the pump wave, as well as for the signal wave and the idler wave. For a theoretical description of such a triply resonant set-up, one might proceed along the same lines as in the preceding sections, including now a round-trip condition for the pump wave. However, the treatment can be significantly facilitated by adopting the mode picture. In fact, the spatial dependence of the wave amplitudes, in the case of highly reflecting resonator mirrors, will be relatively small, and, thus, can be neglected in a good approximation. Hence, only the time dependence of the amplitudes need be taken into account, and we may start from eqs. (2.26)-(2.28), setting dA,/dz = 0 (A = 1 , 2 , 3 ) and A k = 0. It appears natural to choose the normalization of the amplitudes such that their squared absolute values represent the photon numbers. This is accomplished by the

I,#

61

THE OPTICAL PARAMETRIC OSCILLATOR

45

substitution

(6.47) (cf. eq. (4.1)), and we find from eqs. (2.26)-(2.28) Q

_- I- ~ Q ~ Q , ,

(6.48)

6,= iPQT Q3,

(6.49)

Q,= iPQl Q 2 ,

(6.50)

where the effective coupling constant is given by

(6.51) For simplicity, let us assume that the entire space inside the resonators is filled with the nonlinear material. Considering realistic conditions, we have to identify the mode volume V with the volume of that part of the crystal which is actually traversed by the waves of finite beam diameters. Now, the equations (6.48)-(6.50)describe only the parametric interaction among the three waves. They must be complemented by damping terms which are characteristic for the resonator losses. (The latter are due mainly to the transparency of the mirrors.) Moreover, the effect of an incident pump wave which excites the mode A = 3, must be included. This and HAKEN[1968]; a more physical is achieved by writing (see GRAHAM [1966]) treatment has been given by YARIVand LOUISELL

d, = -hQ1 +i@QfQ,,

(6.52)

-bc2Q2+iPQTQ3,

(6.53)

Q2=

Q, = -hQ3 +i@QIQ2+ F,

(6.54)

where the parameters xA(A = 1 , 2 , 3 ) designate the resonator losses and F is the (constant) amplitude of a “driving force” oscillating at the pump frequency. Assuming only one of the resonator mirrors to have a reflectivity RA (A = 1 , 2 , 3 ) different from 1 and denoting the path length corresponding to a round-trip through the resonator by L,, we may express the resonator losses in the form XA

= (1-RA)vi/Lr.

(6.55)

To find the connection between the amplitude F and the incident pump

46

THEORY OF OPTICAL PARAMETRIC AMPLIFICATION

[I, §

6

power, we assume, for the moment, the nonlinear interaction to be switched off. We then obtain from eq. (6.54), in the steady state, Q3

=2F/~3.

(6.56)

Hence, the number of pump photons leaving the resonator per sec, is IQ31*~3

= 41FI21~3.

(6.57)

Under stationary conditions, the same number of photons must enter the resonator. Since, in the steady state, no reflection of the incident pump wave occurs, provided the resonator for the pump wave is correctly tuned to the pump frequency (see, e.g., PAUL[1973b]), the quantity (6.57) equals also the incident pump photon flux

4 = 4)FI2/x3.

(6.58)

We wish to solve eqs. (6.52)-(6.54) for steady-state conditions (Q, = 6, = Q3 = 0). From eqs. (6.52), (6.53), the following relation is easily derived XINI =~2N2, (6.59) where NA= I Q , l 2 denotes the photon number in the mode A. The physical meaning of eq. (6.59) is that of a Manley-Rowe relation. Indeed, it indicates that the numbers of signal and idler photons, respectively, which leave the resonator per sec, are equal, which, in the steady state, necessitates the creation rates for signal and idler photons to be the same. Physically, the latter statement follows from the fact that any pump photon decays into both a signal and an idler photon. After some algebra, we find from eqs. (6.52H6.54) (6.60)

and (6.61)

According to eq. (6.60), the phase of the driving force is transferred to the pump wave, and the phase of the latter, on the other hand, determines the sum of the phases for the signal wave and the idler wave, as is seen from eq. (6.54). (The corresponding phase relation is the same as in QP6.3 and 6.4.) From eq. (6.61) we obtain the threshold value for the incident pump

I,

8 61

THE OPTICAL PARAMETRIC OSCILLATOR

41

photon flux, making use of eq. (6.58),

(6.62) It is interesting to note that the photon number in the pump mode N3 is independent of the pump power, provided, of course, that the latter is high enough to exceed the threshold power. In fact, from eq. (6.60) follows

(6.63) The physical interpretation of this result is that, for growing pump power, more and more pump energy is converted into signal and idler energy, while the excitation level of the pump mode (inside the resonator) does not change. This situation is very similar to that existing in the laser, where, in the steady-state operation, the inversion of the medium is independent of the pump power, due to the fact that higher pump powers give rise to an increased photon production rate by means of stimulated emission. We wish to discuss the expression (6.62) for the threshold pump photon flux in more detail. Since &, is proportional to the product of all three resonator losses, it is evident that the threshold can be lowered significantly with respect to the DRO, by choosing a low-loss resonator for the pump wave. Physically, this effect is explained by the enhancement of the amplitude of the pump wave due to resonance, which leads to a more effective parametric interaction. Seemingly, eq. (6.62) defines the threshold value for the total pump photon flux; in fact, however, it is a condition for the photon flux per unit area. This becomes obvious by rewriting eq. (6.62) in the form 4th

=

II II I1 ( l - Rl)(l -R2)(1- R3)211V2213f , 2pL:

(6.64)

where use has been made of eqs. (6.51), (6.55) and the relation V = L,f (f denotes the beam cross section). Thus, the photon flux per unit area is the relevant physical quantity. Hence, focusing of the pump wave (together with the signal wave, and the idler wave) appears to be preferable from the experimental point of view, since it leads, in practice, to a reduction of threshold pump power by orders of magnitude. The same statement, of course, applies to the SRO and DRO (see eqs. (6.21) and (6.42)).

48

THEORY OF OPTICAL PARAMETRIC AMPLIFICATION

11,

I 6

It should be remembered that our treatment is based upon the concept of plane waves with a constant intensity across the transverse beam cross sections. In the experiments, however, one deals with light beams whose intensity distributions are Gaussian. The incorporation of this fact into the theoretical description necessitates a modification of the above for[1966], BOYD and KLEINMAN mulas. For details see BOYD and ASHKIN [1968], and the review article of FISCHERand KULEVSKY [1976]. While the triply resonant OPO is favourable because of its low threshold, its conversion efficiency q=

+

%IN1 o, X 2 N 2 0 2

4 ~ IFI2/%3 3 ’

(6.65)

however, is rather small, too. Making use of eqs. (6.59), (6.61) and expressing IF]’ through a, we find u- 1

v = 7 .

(6.66)

This value is smaller, by a factor 4, than the conversion efficiency for the DRO (see eq. (6.45)), and, hence, the peak value for q is only 25%. The physical reason for this is that the incident pump wave is partly reflected by the input mirror. (The depletion of the pump wave provides an effective loss mechanism for the latter which prevents the pump wave from entering the resonator without reflection.) Finally, it should be noted that the above analysis must be modified in the case where both signal and idler beams are standing waves, while the pump beam is a travelling wave. Then, the creation of a backward travelling pump wave by the backward travelling components of both the signal wave and the idler wave, already considered in 0 6.4, must be taken into account, which leads to a reduction of q to half of the value (6.66). Operation of a triply resonant OPO has not been accomplished in the form described above but in a similar arrangement, called “internal OPO”, which, at the same time, is free from two main disadvantages, namely, the complicated experimental set-up, and the low conversion efficiency. 6.6. THE INTERNAL OPO

As early as 1962 it was suggested by KROLL[1962] that the nonlinear crystal be placed inside the resonator for the pump laser, in order to take

I,

8 61

THE OPTICAL PARAMETRlC OSCILLATOR

49

advantage of the higher electric field strength existing within the resonator. Providing resonators also for the signal wave and the idler wave, one thus arrives at a triply resonant arrangement. Evidently, the parametric conversion acts as an additional loss mechanism for the laser operation, which leads to a strong coupling between the parametric and the laser processes. In a theoretical treatment, the driving term in eq. (6.54) has to be replaced by (6.67) (go and g, are coefficients determined by the laser parameters such as pump power, transition probability, relaxation times), which ensures eq. (6.54), for B --+ 0 (switching off the parametric interaction), to reduce to the well-known laser equation. Approximating eq. (6.67) by F = go(1 -s*lQ3l2)03,

(6.68)

OSHMAN and HARRIS [1968] analysed the modified system of eqs. (6.52)(6.54). The main result that they found was that there exist three types of steady-state operation. One type of operation is similar to that described in the preceding section, and is characterized by an efficient conversion of pump power (maximum efficiency 100%) into signal power and idler power. In a second possible regime, the pump radiation drives the phase of the oscillation rather than its amplitude, and, hence, the conversion is insignificant. The third type of operation is a repetitively pulsing regime, where signal and idler waves are emitted as short pulses. The calculations are rather lengthy. For details the reader is referred to the original papers (OSHMAN and HARRIS [ 19681, FALK,YARBOROUGH and AMMANN [197 11). Experimentally, internal optical parametric oscillation was first demon[1970]. Using the experimental arrangestrated by SMITHand PARKER ment shown in Fig. 11, AMMANN, YARBOROUGH, OSHMAN and MONTGOMERY [1970] obtained a remarkable output power of 210mW at a signal laser mirrors

oscillator mirrors

oven

’

LiNb03 Y& M’* LiNbO, 0-swtch laser crystal crystal (OPO)

Fig. 11. Experimental set-up for the internal optical parametric oscillator (after YARBOROUGH, OSHMAN and MONTGOMERY [1970]).

AMMANN,

50

THEORY OF OPTICAL PARAMETRIC AMPLIFICATION

[I, 8

6

wavelength A = 1.96 F m (idler wavelength A = 2.33 pm). In accordance and AMMANN [197 11 with the theoretical prediction, FALK,YARBOROUGH observed a spiking regime with oscillation pulse lengths from 5 to 10 ns, the length of the pumping pulses (from a Q-switched YAG : Nd3+ laser) being 290 ns.

6.7. THE PARAMETRIC APPROXIMATION

In many experiments concerning parametric amplification, or oscillation, the depletion of the pump wave is relatively small, which enables us to neglect, in a good approximation, the spatial variation of the amplitude for the pump wave. Thus the equation for dA3/az can be dropped, which leads to a considerable simplification of the mathematical description (cf., e.g., HARRIS[1969]). The opposite case, namely, a negligible spatial dependence of the amplitudes for the signal wave and the idler wave, is also of practical interest. It corresponds to the experimental situation where the generated waves are excited, at high levels, in resonators with low losses. Then only the equation for aA&z need be retained (BJORKHOLM [1969]). In both cases, we speak of parametric approximation which, in the following, we wish to describe in some detail. It is often used, in the literature, for a convenient calculation of threshold powers. In particular, it allows a rather simple analytical discussion of the effect of phase mismatch. Let us begin with the first of the aforementioned cases, A3(z)=AZ (independent of 2). Then eqs. (2.27) and (2.28) (with aAJat = aA,/dt = 0) are easily integrated to yield cosh (sz) - i

2s

(6.69)

K2 +i A;' S

where A: (A

=

I

2s

sinh (sz) ,

1, 2) denotes the initial value at z

(6.70) =0

and the following

I,

I61

THE OPTICAL PARAMETRIC OSCILLATOR

51

abbreviations have been introduced (6.71) s =[K,K;-(+Akf*]’.

(6.72)

Let us consider the amplification of an incident signal wave with no idler wave present at z = O . From eq. (6.69) we then obtain the signal power, after a penetration depth z in the nonlinear crystal, (6.73) For sz<< 1, this relation approximately takes the form

P l ( z )= Py{ 1+ K1K$ z’[ 1+ i ( K ,K :

- (f hk)’)z2]}.

(6.74)

This means, the signal power increases quadratically with the penetration depth and the coupling constant K , and linearly with the pump power P:. Evidently, a momentum mismatch, A k # 0, gives rise to a reduced gain for the signal wave, as to be expected from physical reasons (cf. § 2.3). The diminished gain, on the other hand, leads to a higher oscillation threshold. The latter is easily evaluated for the SRO, starting from the round-trip condition (6.15) and using eq. (6.74), to yield the result

which, for A k = 0, agrees with eq. (6.21). It should be noted, however, that a similar treatment of the DRO needs considerably more effort. This is partly due to the fact that (for A k # 0) it is no trivial matter to determine the most favourable initial value for cos O,, corresponding to minimum threshold pump power. (In the case of the SRO this problem does not occur, since the phase of the idler wave, being undeterminded at z = 0, for z > 0, automatically adjusts so as to ensure maximum gain for the signal wave and the idler wave; see

P 5.3.)

In the second type of parametric approximation mentioned above, it is assumed that the intensities of both the signal wave and the idler wave >> (A3\.Hence, the remarkably exceed that of the pump wave IA1l, JA2( spatial dependence of the amplitudes for the signal wave and the idler wave may be neglected, A,(z)=Ay, A2(z)=A;. The solution of eq.

52

THEORY OF OPTICAL PARAMETRIC AMPLIFICATION

[I,

86

(2.26) (for aA,/dr=O) is then given by

A3(z)= A:+i

Kw: sin (fA kz) exp {-i+ Akz}AyA; k3 cos2 a3 Ak

(6.76)

(see BJORKHOLM [1969]), where the phases of A?, A;, A: are still arbitrary. Choosing a phase relation, such as to maximize the energy conversion, we find from eq. (6.76) the pump power at the end of the crystal z = L, to be P3( L

+

(6.77)

) = Pg K:PyP; - 2K3(P;P;P:):,

where wlw2w3

klk2k3 cos2 a1cos2 a2 cos’ a)

=p-)

sin (fA kL) f AkL

L 1 sin(4Akl) 1 f’ f A k I ‘

(6.78)

-7.

For an evaluation of the signal power, under steady-state conditions, we make use of the relation

(6.79) indicating the production rates for signal and idler photons to be equal. On the other hand, the complete power balance reads

P; - P3(L)= (1 - R,)P,(L) + (1 - R2)P2(L ) .

(6.80)

Making use of eqs. (6.77) and (6.79), we find from eq. (6.80) the following expression for the output signal power

(6.81) where the threshold pump power is given by P;th

=

(l-Rl)(l-R2) 4

1 w3f(

sin(fAkL) 4AkL

-*

)

.

(6.82)

For R l = R , and A k = 0, the formulas (6.81) and (6.82) coincide with those obtained in the previous treatment (see eqs. (6.41), (6.42)). (Note that the parametric approximation is applicable in the case of highly

1,

P 71

RADIATION CHARACTERISTICS OF AN OPO

53

reflecting mirrors only, i.e., for 1- R 1<< 1, 1 - R2<<1.) In addition, eq. (6.82) reveals the effect of momentum mismatch. It is easily seen that the threshold power for AkL << 1, increases as [1-$(1AkL)']-', in agreement with eq. (6.75). For AkLZ2rr, formula (6.82) appears to be less meaningful. In fact, the requirement IA ,/,/A2!>> IA31 upon which the approximation rests, will not be met in this case.

§ 7. Radiation Characteristics of an

OPO

7.1. BUILDUP OF OSCILLATIONS

In the theoretical treatment of the OPO, as given above, we concentrated on the steady state and disregarded transient oscillations. A discussion of the stability of the stationary solutions leads to the conclusion that the latter are stable with respect to small perturbations, i.e., random deviations of the amplitudes for the signal and/or idler wave from their stationary values become damped. The corresponding transition into the steady state may be a purely exponential damping or an exponentially damped oscillation. In practice, however, it is very difficult to accomplish cw operation. The reasons are, among others, mechanical instabilities of the resonators and the need for a high pump power to be irradiated continuously into the nonlinear material. In fact, most of the OPOs existing nowadays are operated with pulsed excitation (pumping pulses from Q-switched lasers). Therefore, the buildup regime of the OPO plays an essential role. An important question is whether a quasi-stationary state can be reached, since otherwise the parametric interaction will be less effective. Clearly, this requires the duration of the pump pulses to exceed the buildup time of the parametric oscillation. An evaluation of the buildup time proceeds as follows. Starting from initial values for the intensities of the signal wave and the idler wave which originate from the spontaneous decay of pump photons, one has to calculate the number of round-trips needed to reach the stationary state which, in the case of a DRO, is described by eq. (6.36). From this number, it is easy to evaluate the buildup time. A numerical analysis for the DRO has been given by BRUNNER, FISCHER and PAUL[1974]. They obtained a dependence of the signal intensity on the number of roundtrips N which is shown in Figs. 12-15. One recognizes that the value of N

54

THEORY OF OPTICAL PARAMETRIC AMPLIFICATION

I'

1

&;

P 0 -1

-2

-3 -L

-5 -6

0

I

I

3

L lg N ---+

Fig. 12. Ratio of signal power to incident pump power, PYlP:, as a function of the number of round-trips N for different values of the parameter fl/Gth: 104(a); 25 (b); 7.84 (c); 4.00 (d); 1.96 (e); 1.44 (f). The initial signal power has been chosen as :'F = lO-"P;. (After BRUNNER, FISCHER and PAUL119741.)

0 0 a-I a* 9 -1

2

-2

-3 -L

-5

-6

0

1

2

3 /gN

--+

Fig. 13. Growth of the signal power with the number of round-trips. The notation and the initial signal power are the same as in Fig. 12, and the values of P:/P& are: 25 (a); 7.84 (b); 4.00 (c); 1.96 (d): 1.44 (e). (After BRUNNER, FISCHER and PAUL[1974].)

1,5 71

RADIATION CHARACTERISTICS OF AN OPO

I

zP-159

55

0 -1

-2

-3

-5 -6

0

-

2

1

IgN

Fig. 14. The dependence of the signal power on the number of round-trips. The notation and the initial signal power are the same as in Fig. 12, and the parameter P;/P&h has been chosen as: 7.84 (a); 4.00 (b); 1.96 (c); 1.44 (e). (After BRUNNER, FISCHERand PAUL[1974].)

Fig. 15. Number of round-trips N,,,, needed to reach the stationary state, as a function of the relative pump power fi/P&hfor different values of Py/P;: (a); lo-'' (b); (c). (After BRUNNER, FISCHERand PAUL[1974].)

56

THEORY OF OPTICAL PARAMETRIC AMPLIFICATION

[I,

87

needed for a transition to the stationary state, varies between lo2 and some lo’, dependent on the parameters of the system. In particular, one sees that the buildup time T,, decreases with increasing relative excitation P;/&,,. Apparently, this is caused by the fact that P , increases only as (P;IP;,,)f - 3 . Similarly, growinglosses - for fixed relative excitation - lead to a decrease of T~ due to the corresponding increase of the gain. However, if both the pump power and, hence, the gain also are held constant, growing losses give rise to enlarged buildup times. Moreover, it is evident that in order to obtain short buildup times, it will be favourable to use resonators of small length. Provided the threshold for optical damage of the nonlinear crystal is high enough, the best way to shorten the buildup time T~ would be to enhance the losses. A reduction of Th by increasing the relative excitation appears to be less preferable, since this would diminish the conversion efficiency. It is interesting to compare the buildup times for the DRO and the SRO. It can be shown that, for equal relative excitation and equal losses in both cases, q,is smaller for the SRO than for the DRO. It should be noticed, however, that the threshold pump power is much higher for the SRO. Thus, it will be difficult, in general, to achieve equal relative excitations in both cases. From the values for N taken from Figs. 12-15, we obtain buildup times between 10 and 500 ns. The latter exceed, in general, the pulse durations of commonly employed solid-state pulse lasers. Hence, in order to achieve a high conversion efficiency, it may be preferable to use pumping pulses of lower peak power but greater duration which will allow a quasi-stationary regime to be reached. Nowadays, the main practical problem in the pulsed operation of an OPO is not to generate high power pumping pulses but to avoid optical damage of the nonlinear material (production of inhomogeneities) due to the high-intensity fields. For this reason, it will be advantageous to lower the threshold by choosing a suitable low-loss arrangement. On the other hand, damage resistant crystals are preferable which, in addition, should possess a high nonlinearity. The threshold for radiation damage is 300MW/cm2 for KDP and 500MW/cm2 for ADP. For LiNb03 it is smaller by one order of magnitude and depends strongly on temperature. The spectral properties of the radiation generated by an OPO are, to a great extent, influenced by the resonators. The most favourable case is that of a SRO. The resonator has to be tuned to a signal frequency o1for which both the phase matching condition ( 2 . 2 5 ) and the frequency

I, § 71

RADIATION CHARACTERISTICS OF AN OPO

57

relation o3= o1+ o2are fulfilled. Continuous tuning, in a strict sense, of the signal frequency, e.g., by changing the crystal orientation, would necessitate an appropriate continuous adjustment of the resonator mirrors. Otherwise, the signal frequency would jump from one resonator frequency to a neighbouring one. This applies, however, to a strict one-mode regime only. In practice, the gain profile as a function of the phase mismatch A k # 0 is rather broad and, therefore, allows for simultaneous oscillation at several neighbouring resonator frequencies (whose distance is about lo9 clsec). Hence, tuning of the resonator becomes unnecessary, and, in practice, no frequency jumps will be observed. The situation is less satisfactory for an arrangement where both the signal wave and the idler wave are resonant. In this case, the gain for the signal wave and the idler wave becomes optimum for those values of the signal and idler frequencies which satisfy the phase matching condition and the frequency relation o3= o1+ 02,and, moreover, coincide with resonator frequencies. However, all these requirements cannot be met simultaneously in general, since, in practice, only one resonator is used for both waves, which permits the resonator to be tuned to only one frequency. Practically, the DRO oscillates at a set of frequency pairs ol, o2which are resonator frequencies and obey the frequency relation. (The phase matching condition, however, is violated; nevertheless, oscillation takes place due to the small decrease of the gain with increasing phase mismatch.) Thus, the emitted spectrum consists of certain discrete lines (GIORDMAINE and MILLER[ 1965]), and this so-called “cluster” effect makes the frequency tuning of a DRO an intrinsically discontinuous process: the spectrum will suddenly shift from one cluster to another. Moreover, any small change in the pump frequency or the mirror distance will result in a change of the spectral composition of the output. The effective bandwidth of the radiation is about lo1’ clsec. Together with the discontinuous frequency tuning, this prevents the DRO from being employed in high resolution (Lamb dip) spectroscopy where a strictly continuous frequency tuning over a range of about 109c/sec is required. This is not so for the SRO, where the bandwidth can be drastically reduced (from 1OI2 to 107c/sec) by means of wavelength selective elements, e.g., Fabry-Perot etalons. In the one-mode operation thus achieved, the line width will still be influenced by the spontaneous decay of pump photons. On the other hand, fluctuation properties of the pump radiation will be transferred, to some extent, to the generated radiation. We study these problems in the following section.

58

THEORY OF OPTICAL PARAMETRIC AMPLIFICATION

[I, $

7

7.2. QUANTUM MECHANICAL DESCRIPTION OF FLUCTUATIONS

A theoretical analysis of the fluctuation characteristics of the generated radiation is most conveniently performed in the mode formalism, as it has been applied to the triply resonant OPO (see § 6.5). Similar to the procedure adopted by several authors in laser theory (see for instance LAX[1966], PAUL[1969]), we shall supplement the equations of motion, after quantization, by fluctuating forces of Markoffian type which represent both the effect of the spontaneous decay of pump photons and the fluctuations connected with the damping of the fields due to the resonator losses. (In a consistent quantum mechanical treatment, any energy dissipation is necessarily accompanied by fluctuations [dissipation fluctuation theorem].) In addition, phase and intensity fluctuations of the pump wave will be incorporated into the theory. Starting from eqs. (6.52)-(6.54), we thus arrive at the following set of modified equations

4

I

--1

-

2x141

+ip4:43+a,,

(7.1)

G2 = -;x2q2 + ipij:g3 + R2, (7.2) G 3 = - $ ~ ~ i j ~ + i p i jr ~Z ,4+~~+, e x p ( - i l ~ ~ ~ } + ~ x ~ e ~ , e x p { - i c (7.3) p,), where &, 4: are the common photon annihilation and creation operators, after separation of the free time dependence (cf. eq. (4.1)). The fluctuating forces (often named Langevin forces) R, originate from both the loss mechanisms and, generally speaking, the spontaneous production of signal and idler photons in many different pairs of modes, in accordance with the finite bandwidth and angular spread of the spontaneous radiation, as described in § 3 . To be more precise, the fluctuating force fi, driving the signal mode, will represent those spontaneous processes where a signal photon is emitted into the mode A = 1, and the corresponding idler photon into any of the modes, except A = 2, which can be excited due to the broad gain profile (as a function of the phase mismatch). Note that the spontaneous decay of pump photons into both a signal photon A = 1 and an idler photon A = 2 follows, formally, from the commutation relation for qP,4: ( p = 1,2). The meaning of R, is similar to that of R , . In f i 3 , on the other hand, the effect of all spontaneous processes will be involved. Let us assume the fluctuating forces to be Markoffian. This means

(R*(f)>= 0,

(7.4)

I,

5 71

59

RADIATION CHARACTERISTICS OF AN OPO

(fil(tI)&(tZ))=

UA

(7.5)

-t2),

(fi,(t,)R:(r,))= bAs(t, - r2)

(A

= 1,2,3).

(7.6)

Adopting a heuristic approach known from laser theory, we determine the coefficients a,, b, ( p = 1 , 2 ) in the following way. After formally switching off the parametric interaction described by the terms being proportional to p, we deduce from the equation of motion for 4, a rate equation which allows a physical interpretation of a,. From the requirement of quantum mechanical consistency, on the other hand, we find the difference a, - b,. The rate equation in question is easily determined from eq. (7.1) or (7.2) - after setting formally p = 0 - t o be d dt

- +’”) = --x ,(4,4,)+((l;RJ+(ff:L?,)

- (9’-

(F = 1,3.

(7.7)

To evaluate the latter two expectation values (which are complex conjugates of one another), we formally integrate eq. (7.1) or (7.2) (for p = O), to find ij,(t) =

I:

exp{-$x,(t-

r’)}k,(t’) dt’.

(7.8)

Making use of eq. (7.5), we can rewrite eq. (7.7) in the form

(7.9) where N , =(ij:ij,) denotes the mean photon number. From eq. (7.9) it becomes evident that a, has the physical meaning of a spontaneous production rate for photons in the mode p. Hence the ratio u,l x,, = wz

(7.10)

is the mean number of spontaneously produced photons in the mode p, except those photons, however, which originate from a simultaneous spontaneous excitation of the modes A = 1 and A = 2. Quantum mechanical consistency, on the other hand, requires, strictly speaking, the commutation relation [ij,, 4:] = 1 to be satisfied. Since our knowledge of the fluctuating forces is restricted to correlation functions only, this condition must be relaxed to the form ([ij,, ijt])= 1. Inserting here for ij,, ij; the expression (7.8) and its Hermitian conjugate, respectively, we obtain

b, - a, = x,.

(7.11)

60

THEORY OF OPTICAL PARAMETRIC AMPLIFICATION

[I, §

7

The influence of spontaneous processes on the pump wave A = 3 may be described more simply. Since the transition probability for a spontaneous decay is proportional to the mean number of pump photons (cf. § 3), the effect of spontaneous decay is merely to slightly increase the loss constant xj. In the following, we assume this small correction to be included in x 3 . Hence, we may rewrite eqs. (7.9, (7.6) into the final form

(fi;(f,)a,(f,)> = x,N",p s(f1-

f2)

( P = 1,

a,

(7.13)

(R:(tI>R3(f2)>= 07 (R,(t,)R=(f2))=xCL(N"~+ 1)s(ti-t2) (R3(tl)Ri(t2)>=x3

(7.12)

( P = 1,2),

-f2)*

(7.14)

(7.15)

According to the physical interpretation of the fluctuating forces given above, R , and R2 refer to spontaneous processes which are physically independent.* Hence, no cross correlations can exist. (A more sophisticated analysis has been given by BANDILLA [1973].) In particular, the vanishes. correlation function (k,(t)fi,(t)> Let us now consider the fluctuation properties of the pump radiation which is assumed to be generated by a one-mode laser. Because of the well-known amplitude stability and the relative smallness of the phase fluctuations of laser radiation, we may decouple the fluctuations of (real) amplitude and phase by writing the photon annihilation operator for the laser field in the form (cf. HAKEN[1966])

& = ( r L + e d exp{-i[cp,+ PL(f)l}.

(7.16) Here, rL is a constant positive number, cpL is the classical phase, & denotes the phase operator, and the Hermitian operator eLdescribes the deviation of the real amplitude from its mean value r,. Assuming, for simplicity, the resonator losses for the laser and the OPO (at the pump frequency 03) to be equal and dropping terms of the form eL$L,eL& etc., we find from eqs. (7.16) and (6.56) the following expression for the force F driving the pump mode in the triply resonant OPO, due to the incident laser beam, F=FLexp{-i+L}+$xj~Lexp {-ipL} (FL=4~jrLexp(-i(~~)). (7.17)

* It should be noted that for short cavities (resonators of some cm length), as they are usually employed in OPOs, spontaneous excitation, due t o parametric fluorescence, of the signal mode A = 1 is accompanied always by spontaneous excitation of precisely the idler mode A = 2. This means, in practical cases, the operators R, (A = 1,2) describe only the fluctuations due the loss mechanisms, and we have NT = 0 ( p = 1 , 2 ) .

RADIATION CHARACTERISTICS OF AN OPO

61

This explains the appearance of the last two terms in eq. (7.3). Because of the high stability of the amplitude of the laser radiation, the line width of the latter, AoL(<< x 3 ) , is determined by the phase fluctuations only (exp {i+dfl))exp {-i+~(b)I)= exp {-+ A W L ~-~ tzll I

(7.18)

or

([+L(f+ 7)- +L(f)I2)

= AOLId.

(7.19)

The fluctuation characteristics for the amplitude, on the other hand, are represented by the relations

(eJ = 0,

(7.20) (7.21)

Here, NL denotes the mean photon number in the laser cavity, and is the mean square deviation of the photon number which can be written as (ANLI2 = S N L ,

(7.22)

where the parameter 6 is characteristic for the number of photons emitted spontaneously. Realistic values are N L 2 los, 5 2 lo3, and the damping constant aLis of order x 3 . Similar to the procedure in laser theory, we solve the basic equations (7.1)-(7.3) by substituting the ansatz Q(t)=[rA + ei(t)lexp{-i[~A+Jlh(t)l1.

(7.23)

Here, the quantities QA = r, exp {-iq,} (both r, and qA = constant cnumbers) are the classical complex amplitudes, i.e., the steady-state solutions of the equations of motion without fluctuation terms Note that (p3 equals QL (see eq. (6.60)). The physical meaning of the operators eA(t), +,(t) is the same as that of er.and I / J ~ . Writing eqs. (7.1)-(7.3) in their real and imaginary parts, we arrive at the following set of equations (cf. GRAHAM and HAKEN[1968])

1 1m rl

a:,

(7.24)

(I2 -- 21~ 2 ( + ~ - + ~ - + 2 ) + -11 m R:,

(7.25)

4

-2~1(+3-1

$1

- 4z)+-

r2

62

THEORY OF OPTICAL PARAMETRIC AMPLIF’ICATION

[I,

57

(7.27) (7.28)

In the derivation of these equations terms of higher than first order in $A, PA, eL have been neglected. This is justified by the smallness of the fluctuations. Moreover, the fluctuation operators R: have been combined with the factor exp {-in}. The resulting operators have the same formal properties (7.4)-(7.6) like 2: and, hence, have been labelled R: again.

7.3. EVALUATION OF THE LINE WIDTH

Because of the small laser line width, AWL<< x3, the phase operator JIL, as a function of time, is a slowly varying quantity compared to exp { - x 3 t } . Making use of this fact, we find from eqs. (7.24)-(7.26) the simple relation $l(t)+

$2(f)

= $L(f).

(7.30)

Strictly speaking, in the derivation of eq. (7.30) contributions to q2of the type const It, Im k:(t’)exp {-a(?-t‘)} dt’ (a being of the order of the x ’ s ) have been disregarded, because they give rise to exponentially damped terms (as functions of IT[), with a damping constant of the order of the x ’ s , in the calculation of ( [ $ p ( t + T ) - $ F ( t ) ] 2 ) ( p = 1 , 2 ) and, hence, have only a negligible influence on the line width. The phase operator takes the explicit form

(7.31) Applying the formulas (7.18), (7.19) to the radiation of an OPO and making explicit use of eq. (7.19), we obtain the following expression for

I,

8 71

63

RADIATION CHARACTERISTICS OF AN OPO

the line width of the signal wave (GRAHAM and HAKEN[1968])

Evidently, the first term on the right-hand side of eq. (7.32) describes the effect of a finite line width of the pump radiation, while the second term represents the contribution to the line width originating from spontaneous processes and energy dissipation. One recognizes that for x l = x 2 the fourth part of the line width of the pump radiation is transferred to the signal wave. For xI<< x 2 however, the effect of the laser line width is strongly reduced. A physical interpretation of this result will be based upon eq. (7.30) indicating that the phase fluctuations of the pump wave are transferred to the sum of the fluctuating phases for the signal wave and the idler wave. Because of the tendency of a resonator to preserve the phase of the field, the wave with lower feedback - in the extreme case, a wave propagating nonresonantly through the crystal - will follow the phase fluctuations of the pump wave more readily than the strongly resonating wave. Hence, in a SRO the line width of the pump radiation will, practically, not be impressed on the signal wave, in contrast to the DRO for x l = x 2 . The second term on the right-hand side of eq. (7.32) is negligibly small compared to the first one, due to the small values for W?' and WZpto be observed under realistic conditions, and the large factor N , in the denominator. In fact, an upper limit for Wlp, WZpis readily found from experimental data. A typical observation is a spectral noise power (parametric fluorescence) of about 4 x lo-'' W/A, where the radiation of 1 8 , bandwidth is emitted into a solid angle of 10-4sr. For xl- x* = lo's-' we find N;P,Nip to be much smaller than 1 (see also the footnote on page 60). For N1= lo8, the second term on the right-hand side of eq. (7.32), representing the contribution of the spontaneous processes to the line width, amounts to about 0.1 s-', while A q under typical conditions (one-mode operation), will exceed 106s-'. Hence, only the first term on the right-hand side of eq. (7.32) is of importance. From the experimental point of view, it should be emphasized that temperature fluctuations and mechanical vibrations give rise to great frequency shifts and, hence, to a large effective line broadening. For LiNb03, e.g., a temperature change of O.Ol"C, according to eq. ( 6 . 3 , causes a frequency shift of about lo'* s-l.

-

64

THEORY OF OPTICAL PARAMETRIC AMPLIFICATION

[I,

57

7.4. AMPLITUDE FLUCTUATIONS

Eqs. (7.27)-(7.29) can be solved in a straightforward manner (see BRUNNER [1972]). The procedure, however, is a little cumbersome, and we will present the results only. For x1 = x2 = x the fluctuation operator el reads Ql(t)=$

[-Re [ff:(t’)+R,’(t’)]exp{-al(t-t’)}dt’ [k;(t’)+fil(t’)]exp{-az(t-t’)}dt’

+?Re 2 1

+---

a2-a3

-kZ

2

I‘

I)];

{[xx3(rr-

--m

R e [ffT(t’)+ffl(?’)] exp {-a3(t- t’)} dt’,

(7.33)

where the operators appearing in the integrands are always to be taken at time t‘ and the following abbreviations have been used a1 = 4 = X I , x2)7

(7.34) x

(7.35)

Here, u 2 denotes the relative excitation (see eq. (6.61)). From eq. (7.33) we find the correlation function in the form

(el ( t + 7)el(t)>= ___ 4NL

(

1 aL,

a29

a3; 171)+f2(%

a 2 3 a3;

bl), (7.36)

where the symbols f l and f2 stand for sums of terms proportional to } exp {--aj(~\; ( j = 1 , 2 , 3 ) , respectively. Near threshold, exp ( - ( Y ~ ~ T ( and (0 - 1<< 1, fi reduces to the simple expression (7.37)

We thus obtain the mean square deviation of the photon number in the

I,

0 71

65

RADIATION CHARACTERISTICS OF A N OPO

signal mode, (AN,)*= 4Nl(e:), to be (BRUNNER and PAUL[1971])

[

(AN,)’= N1

fl(%

(7.38)

a’,a3; O)+ Np+xp+l}. 4(a-

The function fi(aL, a2,a3;0) is represented in Fig. 16. One sees that fi, in most cases, lies between 0.1 and 1. Hence, the amplitude fluctuations of the pump radiation are transferred without significant reduction to the generated radiation. Since (ANL)2/NL> lo3, the term due to the spontaneous processes can be neglected. Together with the similar statement in § 7.3, this means that the spontaneous decay of pump photons, practically, does not affect the fluctuations of either the phase or the amplitude of the generated radiation. It is interesting to note that the aforementioned transfer of fluctuations from the pump to the generated radiation does not take place in a laser pumped by thermal light. The reason for this is that in a laser the intensity fluctuations of the pump radiation are not reflected by the occupation number of the excited atomic level, N,,,, since the pumping processes for different atoms may be considered, in a good approximation, as statistically independent. Thus, we note that the relawill decrease as N& where tive mean square fluctuation (ANexG)’/NZXc Nto, is the total number of atoms (molecules). In an OPO, on the contrary, the nonlinear polarization responsible for the parametric interaction, instantaneously follows all fluctuations of both the phase and the amplitude of the pump wave.

t

0

1

2

3

1

5

6

7

8

9

1

1% .__ .

0

*

K3

Fig. 16. The function fl(aL,a*,a 3 ;0) in eq. (7.37) versus 4 a L / x 3 for different values of the parameter a = 8 ( a - 1)x/x3 (after BRUNNERand PAUL[1971]).

66

THEORY OF OPTICAL PARAMETRIC AMPLIFICATION

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P8

Summarizing our results, we can state that the fluctuation characteristics of the OPO output, in one-mode operation and under ideal experimental conditions, such as precisely constant temperature, high mechanical stability of the resonator configuration, etc., are wholly determined by the noise properties of the pump wave.

§ 8.

Experimental Work on Optical Parametric Oscillators

8.1. NONLINEAR CRYSTALS

Since the first experimental investigations of nonlinear optical effects by FRANKEN, HILL,PETERSand WEINREICH [1961] a large number of materials has proved to be convenient to allow the observation of parametric interaction. The requirements to be met in the OPO, of low threshold pump power, high conversion efficiency and tunability over a wide range, however, reduce the number of suitable candidates. The latter should have the following properties: (a) a high nonlinearity, which gives rise to an effective parametric interaction, (b) a birefringence and dispersion large enough to allow for phase matching over a wide frequency range, (c) high resistance to optical damage (this requirement is of special importance in case of high threshold pump powers), (d) small absorption over a wide frequency range, since the absorption suppresses oscillation and, hence, reduces the tunability range. Moreover, the growing of crystals of suitable quality and size should not be too expensive. In Table 1, nonlinear materials are listed which meet these needs in a more or less satisfactory manner and have been employed in parametric oscillators. For a more comprehensive list of data on optical nonlinearities, the reader is referred to the excellent book of ZERNIKEand MIDWINTER [1973]. One recognizes that the tunability range of parametric oscillators extends into the near ir region. For a further extension towards longer. wavelengths, much effort in growing new crystals would be needed. The crystals quoted in the table are either uni- or biaxial ones. In many cases, they can be grown by the Czochralski method which is technically well

1,s

81

67

EXPERIMENTAL WORK O N OPO

TABLE1 Second order optical nonlinearities x!:k), as defined by eq. (2.8), and transparency range for several materials used in parametric oscillators. (The data for x!z are 119711.1 taken from YARIVand PEARSON ~~

~

~

Crystal

Crystallographic class

NH4H2P04(ADP)

S2m

KH2PO4 (KDP)

42m

LiNbO, Ba,NaNb5OlS

3m mm2

X q( 2k )

[ lO-'CGSe]

Transparency range [ ~ m l

xY;~= 1.35*0.06

0.25-1.2

* 0.09

0.25-1.2

x:%= 1.35*0.06 = 1.35

,y\23 = 1.35 * 0.09 #j), = 14.3* 1.5 X$22=6.9*33.0 =46.5*3.3 x\22=45.6rt4.5 xi2,),= 39.0rt7.8

0.4-4.5 0.4-4

controlled nowadays. Utilizing this technique, it becomes possible to grow crystals of about 5 cm length, 1 cm thickness and high optical quality. Theoretically, it is rather difficult to estimate the nonlinear susceptibilities for a given material. Miller's empirical rule (MILLER[ 19641) which connects the nonlinear with the linear susceptibilities in the simple form (8.1) X$k)(W ; 0 2 , 0 3 ) = XI:'(0i)Xc'(O2)X'k"(o3) Avk, has proved to be a useful guide in the search for new suitable crystals. According to Miller's empirical findings Allk is almost constant for a wide range of materials. In the meantime, a considerable advance in the calculation of nonlinear susceptibilities came from a theory of LEVINE [1969, 19701. A very useful technique of experimentally determining the optical nonlinearities of a given material and, moreover, answering the question whether phase matching can be achieved, has been developed by KURTZand PERRY[1968]. The main advantage of this method is that the investigations are made with crystal powders, thus avoiding the efforts of crystal growing. d.2. SHORT SURVEY OF EXPERIMENTAL RESULTS

In most cases, the optical parametric oscillators are operated in doubly resonant configurations. In the last few years, however, the SRO has become more attractive because of its favourable spectral characteristics, especially with regard to applications in high resolution spectroscopy.

68

THEORY OF OPTICAL PARAMETRIC AMPLIFICATION

[I,

58

Generally, the pump is represented by a laser beam or its second harmonic. In pulsed excitation, pump power densities between lo4 and lo8 W/cmZare needed at pulse durations of order 10 ns. Repetition rates up to 10 pulses per sec have been attained. In cw operation, the required pump power densities lie between 0.5 and 50 W/cm2, and pumping is accomplished by means of an argon laser or utilizing the second harmonic from a YAG : Nd” laser. The frequency is tuned by either varying the crystal temperature or rotating the crystal. The first to operate an OPO were GIORDMAINE and MILLER[1965]. In their apparatus (see Fig. 10) the dielectric coatings representing the resonator mirrors, have been evaporated directly to the nonlinear crystal (LiNb03). The tunability range between 0.97 p m and 1.15 p m observed in their first experiment, was extended in a subsequent experiment (GIORDMAINE and MILLER[1966]) to the region between 0.73 F m and 1.193 pm. To this end, a temperature variation between 50°C and 260°C was needed. At a pumping power of 1O’W the output power of 1O’W was attained. MILLERand NORDLAND [1967] used external resonator mirrors ( R = 99’/0) and provided the crystal faces with antireflection coatings. The pump wave was the second harmonic of a glass : Nd” laser (A = 0.53 pm, power P = 5 X lo4 W). The length of the LiNbO, crystal was 0.9 cm, and that of the resonator 2.8 cm. The output power was as high as 50 W, and tunability (for the signal wave and the idler wave, respectively) over the range from 0.684 p m to 2.355 p m was demonstrated. A similar device, with KDP as a nonlinear material, had already been operated by AKHMANOV, DMITRIEV, KHOKHLOV and KOVRYCIN [19661. Since the threshold powers for the doubly resonant devices considered so far are relatively high, optical parametric oscillation could be achieved only in pulsed excitation with pulse duration up to 100 ns. This situation changed favourably, when the new crystal Ba2NaNbS0 became available, whose nonlinearity considerably exceeds that of KDP and LiNb03 and, hence, leads to lower threshold powers. A further experimental LEVINSTEIN, RUBIN,SINCHand VAN improvement is due to SMITH,GEUSIC, UITERT[1968], who replaced the resonator with plane mirrors by a confocal resonator, thus focusing the light beams into the nonlinear crystal (see Fig. 17). In this way, the threshold power could be made as low as 45 mW (later on, a reduction even beyond 3 mW became possible), and from a pump power of 300mW, the authors obtained an output power of 3 mW in a spiking regime. By varying the temperature between

EXPERIMENTAL WORK ON OPO

laser mirrors

YAG Nd3' Bo2r$Nb50,5 loser crystol (frequency doubling)

69

oscillotor mirrors

.i

Su2;hhNb50,5' Lantireflection crystal cwting (porometric osollot o r )

Fig. 17. Doubly resonant quasi-cw OPO with focused beams (after SMITH,GEUSIC,LEVINSTEIN, RUBIN, SINGHand VAN UITERT[1968]).

97°C and 103"C, they tuned the generated frequency in the range between 0.98 p m and 1.16 p m . The conversion efficiency can be enhanced by replacing the FabryPerot type resonators by ring resonators, since in the latter case no backward travelling components of the signal wave and the idler wave will occur (cf. § 6.4). An experiment of this type has been performed by BYER,KOVRIGIN and YOUNG [1969], who achieved cw operation using a LiNb03 crystal of 3.4 cm length, the threshold power being 150 mW. The pump radiation from an argon laser (A =0.515 pm, P = 3 0 0 m W ) has been converted into signal and idler radiation with an efficiency of 60%. Tuning between 0.66 p m and 0.7 p m has been accomplished by temperature variation. Similar to the aforementioned case, the output showed a spiking behaviour. No cw operation, in a strict sense, of a DRO has hitherto been reported. The reason will be found in the high sensitivity of doubly resonant configurations to external perturbations, mechanical vibrations, temperature changes, etc. It is especially the cluster effect (see 5 7.1) and the mode hopping originating from it, which present a serious obstacle to cw operation. Although the singly resonant OPO does not suffer from this drawback, its remarkably higher threshold, nevertheless, does not allow steady-state operation, at least at the present status of cw laser technology. Actually, it can be operated only in pulsed excitation. The first experimental study of [1968a] who attained an a SRO has been performed by BJORKHOLM output power in the nonresonant wave of 250 kW at a peak pump power of 900 kW, which corresponds to a conversion efficiency of 22%. The pump source was a ruby laser emitting pulses of 1011s duration which produced OPO output pulses of about 2 ns duration. LiNb03 was used as a nonlinear material. The observations confirmed that the spectral properties of a SRO are essentially more favourable than those of a DRO

70

THEORY OF OPTICAL PARAMETRIC AMPLIFICATION

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88

(BJORKHOLM [1968b]). Apart from the absence of the cluster effect, the tuning accuracy could be enhanced by a factor 5. At a line width of 10 A, a reproducibility corresponding to an error of *l A has been attained. This uncertainty is mainly due to temperature fluctuations (10 A / T ) occurring in the crystal in the absence of temperature stabilization. At a higher excitation level, theoretically at a relative pump power P:/P&,=4.6, the SRO will start to oscillate in several modes ( KREUZER [ 1969al). By placing an additional wavelength selective element (e.g., of the Fabry-Perot type) into the resonator, it becomes possible, however, to ensure one-mode operation, as demonstrated experimentally by KREUZER[1969b]. ISRAILENKO, KOVRIGIN and NICKLES [19701 utilized the materials LiJO? and a-HJO? to operate a SRO. For pumping they used the second harmonic of a glass:Nd3* laser ( P = 2 MW). By rotating the crystal, the wavelength of the generated radiation could be tuned between 0.68 p m and 1.06 pm (signal wave) and between 0.96 p m and 1.2 p m (idler wave). As a remarkable development in the OPO field, we might mention a device fabricated for sale. A LiNb03 crystal is pumped with the second harmonics of different lines from a Y A G : Nd'+ laser. For different frequency ranges of the generated radiation different mirror combinations are employed. Utilizing 3 (frequency-doubled) lines of the YAG : Nd3+ laser, it becomes possible, by means of temperature tuning, to cover the wavelength range between 0.65 pm and 3 p m . The maximum repetition rate for the pulsed operation is 75 pulses per sec. The maximum peak power amounts to 700 W. Due to its favourable spectral properties, the SRO gains more and more importance as a tunable light source, especially well suited for spectroscopic applications, in the ir and far ir frequency range. Utilizing a CdSe crystal pumped with the 1.83 p m radiation of a YAG : Nd3+ laser, HERBSTand BYER [1972] achieved oscillation at a signal wavelength tunable between 9 . 8 p m and 1 0 . 4 p m by rotation of the crystal. In the same manner DAVYDOV, KULEVSKY,PROKHOROV, SAVEL'EV, SMIRNOV and SHIRKOV [1973] tuned a SRO with CdSe as a nonlinear material and a CaF2 : Dy2+ laser (A = 2.36 pm) as a pump source. At a pulse duration of 40ns and a repetition rate of 1 pulse per second the observed tuning range for the signal wave extended from 2.8 p m to 3.36 p m at a bandwidth of 1.5 cm-' (correspondingly, the idler wavelength varied between 7.88 p m and 13.7 pm).

1,

$81

EXPERIMENTAL WORK ON OPO

71

With special respect to spectroscopic applications of the SRO it is desirable to obtain a high output power at a narrow line width (the latter can be achieved by inserting wavelength selective elements into the resonator), in order to have the capability of transforming the emitted frequency by means of frequency doubling or mixing (production of the difference frequency between, e.g., the signal and the idler wave) at a remarkable conversion efficiency. This becomes possible through the availability of crystals of high optical quality which recently have been developed by different groups. In this context, we refer t o the high gain SRO built by HERBST, FLEMING and BYER[1974]. Its basic element is a 5 cm long by 1.5 cm diameter LiNbO, crystal pumped with the 1.06 p m radiation of a YAG : Nd3+ laser at an energy of 20 mJ in a 20 ns pulse at a repetition rate of 10 pulses per second. The frequency could by tuned between 1.4 p m and 4.4 p m by crystal rotation over an angular range of about 4". The output energy attained 1mJ per pulse at a conversion efficiency of 15%. BYER,HERBST and FLEMING [1975] are about to utilize this SRO as the central element in a widely tunable coherent spectrometer system. By further improvement of the pumping technique (a series of amplifiers follows the YAG : Nd3+ Q-switched laser) they obtained an output energy of 16 mJ corresponding to a conversion efficiency of 40%. Using a 1 mm thick tilted etalon, they could reduce the line width to less than 0.1 cm-' at the resonated signal wavelength. In a similar manner, HORDVIK and SACKEIT[1974] had already achieved a line width of only cm-' (true one mode operation) with a long term frequency stability of cm-'. Since the threshold power increases remarkably if one passes from a multiply resonant configuration to a singly resonant one, one might doubt whether parametric oscillation could be achieved in the perfectly nonresonant case. It should be emphasized, however, that the oscillation threshold is determined not only by the nonlinearity and the resonator configuration but, to a great extent, also by the size and optical quality of the crystal. Therefore, if suitable crystals of great length are available (which is indeed the case, as mentioned above), the situation may not be hopeless. In fact, YARBOROUGH and MASSEY El9711 attained parametric oscillation in an ADP crystal of 5 cm length and excellent optical quality, without any feedback being provided for the waves (see Fig. 18). By varying the temperature between 50°C and 105°C the wavelength could be tuned between 0.42 p m and 0.73 p m . The pump beam was the fourth harmonic of a YAG : Nd3+ laser (pump wavelength A = 0.266 p m , peak

72

THEORY OF OPTICAL PARAMETRIC AMPLIFICATION

0266pm ~

YAG Nd ’* !aser

88

oven

oven

106Lpm 0532pm

[I,

tumble output

KDPcrystal ADPcrysta! ARPcrysfa/ (frequency (frequency (parametrr doublmng) doublicg) generator)

Fig. 18. Optical parametric oscillator with no feedback (after YARBOROUGH and MASSEY [ 1971 1).

power 200 kW corresponding to a power density of 50 MW/cm2). The bandwidth of the output consisting of pulses of 2 ns duration, was found to be 5A. Most of the nonlinear materials available nowadays allow parametric oscillation in the visible and infrared spectral region (see Table 1). Yet it appears desirable, especially with respect to spectroscopic applications of the OPO, to extend the frequency range into the middle and far infrared. To meet this requirement, new crystals must find application, which necessitates crystal growing efforts. Promising candidates are listed in Table 2, together with their transparency range. TABLE2 Nonlinear materials with a transparency range extending far into the ir region Crystal

Crystallographic class

Transparency range [ ~ m l

3m

0.6-13

6mm 42m 42m

0.75-20 2.4-17 0.7-12

&,As%

(Proustite) CdSe CdGeAs, ZnGeP,

8.3. CONCLUDING REMARKS

In the present article, it was our intention to give a rather comprehensive review of the theoretical aspects concerning optical parametric amplification and oscillation, together with some instructive information on the experimental situation in this field. In our opinion, it is impressive to recognize how tiny corrections to the dielectric constant of a medium, as represented by optical nonlinearities, can be made to show up on a

I1

REFERENCES

73

macroscopic scale, once technology has attained an advanced stage. In the present case, it was the advent and further development of the laser which made nonlinear optics experimentally feasible and, in particular, led to the creation of new, tunable light sources, the optical parametric oscillators, which will prove to be valuable tools especially in ir spectroscopy.

Acknowledgement The authors express their gratitude to Dr. R. Fischer, Berlin, for valuable discussions.

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