Non-linear optical properties of condensed matter

Rep, Prog. Phys., Vol. 43, 1980. Printed in Great Britain

Non-linear optical properties of condensed matter

D S CHEMLA CNET, 196 rue de Paris, 92220 Bagneux, France

Abstract

This article reviews the non-linear optical properties of condensed matter. The non-linear optical susceptibilities are first introduced in a phenomenological manner and the effects they describe are presented on general grounds. Then the symmetry aspects of non-linear optics are discussed. ‘The propagation of electromagnetic fields in non-linear media are considered and the device applications they result in are described. Finally, the origins of the non-linear optical behaviour of matter, either in the transparency domain or near energy level resonances, are examined.

This review was received in June 1980.

0034-4885/80/101191 f 7 2 $06.50 0 1980 The Institute of Physics

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1192 D S Chemla

Contents

1. Introduction . 2. Phenomenological introduction of non-linear susceptibilities 3. Symmetries and structural aspects of non-linear optics

3.1. Intrinsic symmetries of non-linear susceptibilities 3.2. Geometrical symmetries of non-linear susceptibilities and structural

. . .

aspects of non-iinear optics . 4. Field interactions in non-linear media .

4.1. General aspects of wave propagation in a non-linear medium . 4.2. Wave propagation in second-order effects 4.3. Wave propagation in third-order effects .

5 , Measurement techniques and device applications 5.1. Measurement techniques . 5.2. Material requirements and device applications

6. Origins of non-linear susceptibilities 6.1. Formal expressions . 6.2. The A formulation and the anharmonic oscillator model 6.3. Calculations based on the Unsold approximation 6.4. Ferroelectric oxides . 6.5. Organic molecules and crystals 6.6. Free carrier non-linear optical effects .

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7. Dispersion of the non-linear susceptibilities and non-linear spectroscopy 8. Conclusions .

References .

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1197 1203 1203 1205 1210 1214 1214 1219 1225 1225 1228 1229 1234 1235 1242 1243 1256 1256

Non-linear optical properties of Condensed matter 1193

1. Introduction

Non-linear effects in electromagnetism have been known for a long time, especially in the radiofrequency domain. In the optical part of the spectrum, some effects involving a non-linear response were also known for a long time. For instance, the linear and quadratic electro-optic effects were discovered about a century ago. Never- theless, purely optical non-linear processes had to wait to be observed until the invention of the laser, the reason being that electric charges in matter are held in equilibrium by the atomic electric fields, of the order of 3 x 108 V cm-1, so that anharmonicity in their motion can only be observed by using perturbing fields which are not negligible compared to that value. Such a strong field cannot be provided by conventional light sources, but can be obtained from lasers.

It is customary to consider that the first observation of the transformation of visible light into ultraviolet radiation, by optical second harmonic generation, which was performed by Franken et a1 (1961)) marks the birth of non-linear optics (NLO). After this discovery and for about half a dozen years, a large number of non-linear optical effects were discovered at a very rapid rate. During this period more and more lasers became available, mostly operating on discrete lines. They were used to demonstrate the most important processes of NLO. Three- and four-wave mixing were observed, important concepts such as that of phase matching were developed, it was shown that obtaining gain in the optical part of the spectrum was possible, the symmetry of various processes were investigated, and the most important aspects of light propagation in non-linear media were elucidated. Following this period came another one, about ten years long, during which NLO grew into a mature discipline. The laser technology improved greatly and tunable lasers and ultra-short pulse equipment became available, resulting in an intense and systematic study of non-linear optical processes. Precise measurements of non-linear susceptibilities were performed, initiating an intense investigation of the origins of the non-linear response of matter, for which a large number of relevant mechanisms were singled out. Devices using non-linear optical effects, such as parametric oscillators, and down and up converters, were constructed. Transient coherent processes were observed and new techniques such as non-linear spectroscopy were developed. At the end of this period, NLO emerged as a well-developed field of physics whose fundamental laws were well understood and whose techniques were quite reliable. During the last few years, NLO has started to be used in other fields and is tending to diversify in its applications to match specific needs. This is very apparent when one considers the wide range of conferences and journals in which multiphoton processes are reported. With such a development, it is impossible to cover all aspects of NLO in a review; therefore I have limited the scope of the present article to condensed matter. A very large amount of work has been performed on gases, either of atoms or small molecules, which forms a field of research quite distinct from that dealing with liquids and solids. Even within the restricted field of NLO in condensed matter, I have found it almost impossible to cover in detail all the published literature. It appeared obvious to me that a good introduction to the subject should give the reader a clear insight into the physics of non-linear processes, and it should describe the most important achievements and give a feeling of the evolution of the field, even if this is done at the expense of the description of some specialised developments.

1194 D S Chemla

The review is organised as follows. In $2, the non-linear susceptibilities are introduced in a phenomenological manner and the effects they describe are presented on very general grounds. In $3, the symmetry aspects of NLO are dealt with. Some important consequences of intrinsic and geometrical symmetries are discussed and the relation between the non-linear response and the crystal or molecular structure is illustrated. The propagation of electromagnetic fields in non-linear media are considered in $4, using a unified description of the effects associated with second- and third-order processes. The measurement techniques, material requirements and device applications are examined in 95.

Up to that point, the non-linearities are considered as parameters which are given or measured. The second part of the review is concerned with the origins of the non- linear behaviour of matter and the microscopic mechanisms involved in non-linearities. The description of non-linearities far from resonances are examined in 96 and their dispersion around discrete energy levels or bands are discussed in $7.

The references which have been indicated are typical examples which illustrate some developments of the review, and they by no means form a complete list. Indeed, there are a large number of relevant and important contributions which have not been quoted. I sincerely apologise for having omitted them.

2. Phenomenological introduction of non-linear susceptibilities

T o solve the Maxwell equations in a material medium, one has to specify how the electric and magnetic inductions are related to the electric and magnetic fields. I t is customary to express these so-called ‘constitutive relations’ in a form which gives the electric dipole, magnetic dipole, electric quadrupole, etc, densities as a function of the applied fields. For a non-magnetic medium, the most relevant quantity to consider is polarisation P ( t , P), which it is convenient to expand as a power series in the fields :

P( t , P )=P( l ) ( t , P) +is ( 2 ) ( t , 7) +P(3)( t , i.) +. . 9 (2.1)

where P(n)( t , P) is nth order in the fields. The general form of the various terms in this series can be written on the basis of very general principles (Butcher 1965, Flytzanis 1975): the principle of time invariance which states that the properties of the medium are independent of time, the causality principle according to which P(t) depends only on the fields for times earlier than t and the reality principle which ensures that the response of the medium, to a real driving field, is real. Therefore, the most general form of the nth-order polarisation is

(2 * 2)

as a function of the field E at times tl , tz, . . ., tn. According to the above remarks the nth-order response function K(n) is real, it depends only on the differences ~ j = t - t j and it vanishes whenever r j is negative. I t is important to notice that although P ( n ) ( t , 1.) is uniquely determined by equation ( 2 . 2 ) , the response function R (nf is not unique.

The susceptibility tensors naturally appear when transforming the picture from

Non-linear optical properties of condensed matter 1195

the space-time variables to wavevector-frequency variables, through Fourier transformation:

P(t, ?) = J” P ( w , k) exp [i(ti?- ut)] dwdk.

The usefulness of the susceptibility tensors arises from the fact that to be observed the non-linear optical processes require very intense fields, such as those delivered by laser sources. The fields are therefore superpositions of a limited number of mono- chromatic waves. The Fourier transform then involves 6 functions and the expres- sion of the polarisation at frequency w is

F ( w ) = € , { , ( 1 ) ( w ) . E ( w ) + x @ ) ( - w ; w1, w2): E(w1)E(wz) + x ( 3 ) ( - U ; w1’, w2’, w3’)i

E ( w l ’ ) E ( w 2 ’ ) E ( w 3 ’ ) + . . .> (2.3) where the field frequencies are such that w = w l + w 2 = ~ 1 ’ +w2’ +us‘=. . ., and the dot indicates the contracted tensor product. The physical meaning of the above equation is the following : the electromagnetic fields drive distortions of the electric charges, which may oscillate at any combination of the wj, and they in turn radiate at the corresponding frequencies. Therefore, the magnitude of x(”) as well as the physical processes it describes depend strongly upon the positions of the wj with respect to the spectrum of the eigenstates of the medium. This latter usually falls into two well-separated regions : the high-frequency domain where the dominant interaction mechanism is due to electronic transitions, and the low-frequency region where the nuclei vibrations govern the interaction with electromagnetic fields.

The linear susceptibilities and thus the dielectric constant depend on one frequency only, so that the contributions of the two regions can be analysed independently. For the higher-order ones, the dispersion is more complicated to analyse, though three parts can be distinguished (Flytzanis 1972): a purely electronic one describing the interaction of the high-frequency fields and their positive combinations (sum mixing) with the electronic clouds, a purely vibrational one arising from the interaction of low-frequency fields and the negative combinations of high-frequency ones (difference mixing) with the ionic lattice, and finally a hybrid term which describes how some fields interact with the lattice which then drives changes in the electronic states interacting with other fields. These latter configurations cannot occur in linear optics ; they are unique features of non-linear processes.

In addition to the frequency dispersion, the susceptibilities exhibit a spatial dispersion, i.e. a k-vector dependence. In the optical domain the wavelengths are usually large compared with the dimension of the polarisable entities within the medium; then the spatial dispersion of the x(”) may be neglected. This is the well- known dipole approximation that we assume from now on when not specified. Its significance is that the polarisation at a point is completely determined by the values of the electric fields at the same point. An exhaustive review of spatial dispersion can be found in the paper by Agranovich and Ginzburg (1966).

Let us now consider some of the effects associated with the second- and third- order susceptibilities. ~ ( 2 ) ( - w i ; wj, wk) is responsible for second harmonic generation (SHG, wj = wk = w , wg = 2w) , optical rectification (wj= - wk, wg = 0), sum or difference mixing (wi = wj rf: w k ) and linear electro-optic or Pockels effect (wg 2: wj, W k 2 ( 0). T o ~ ( 3 ) ( - m i ; wj, wk, W E ) corresponds third harmonic generation (THG, wj = wl = wk, wi = 3w) , quadratic electro-optic or static Kerr effect (06 2: wj, wk N wl N 0), two- photon absorption ( w j = - wk, wl= wi , imaginary part of x ( 3 ) ) , optical Kerr effect (01 = - wk, w z = ut, real part of ~ ( 3 ) ) and a number of four-wave mixings among which

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the most popular one is that used in coherent antistokes Raman spectroscopy (CARS; wj= wk= w1, wz= - w2, wt= 2 w l - 02). Most of these effects will be dealt with in detail in the following sections.

Before we proceed to more specific matters, let us examine briefly how the formulation of non-linear optics relates to the language of other light-matter interactions. Light scattering is usually described in terms of cross sections or transition amplitudes. To see how such quantities are related to non-linear susceptibilities, let us consider a microscopic dipole p which oscillates at the frequency w g = nw under the excitation of the driving field E ( w ) , i.e. p(nu)=&)On(,??(w))n. It in turn radiates at a point M ( R ) a field &U,) N (K2/4m0)p exp ( - iwgt)/R so that the differential cross section is

The cross sections are thus proportional to the modulus squared of the microscopic susceptibilities (or hyperpolarisabilities) K@). Let us notice that this analysis assumes that the applied field is a coherent radiation state, or that the number of photons is large, i.e. the incident intensity is important, Under this assumption the two formal- isms are equivalent. How the cross section relates to the macroscopic susceptibility depends on whether the scattering is coherent or not. If the phases of the microscopic dipoles are locked then the field amplitudes they radiate are summed before the scattered intensity is calculated; if they are randomly distributed then the individual intensities are added to give the total scattered intensity. This has important bearings on the symmetry of the effect (Bonneville and Chemla 1978) as well as its efficiency. This latter is directly related to the so-called 'phase-matching' which is of great importance in device applications and will be examined in detail in $4.

3. Symmetries and structural aspects of non-linear optics 3.1. Intrinsic symmetries of non-linear susceptibilities

We noticed in $2 that the response function is not unique. This is because there are n! ways to specify the order of the fields E(tj) in equation (2.2). In order to avoid this inconvenient property, it is possible to define a unique response function R@) by use of a symmetrisation principle:

,... ,(a) (tf; tl?l, . . ., tar'%) = - Ktj,,.,(a) (?; TI?^, . . ., -7nPn) (3.1) n ! ' 2 I

where @ means that the sum is over all the permutations of the triplet of indices ( j , -71, rl) . This fundamental symmetry of the response function is often referred to as the 'intrinsic permutation symmetry'; it induces by Fourier transformation the symmetry of the susceptibility tensor under the permutations of the triplet of indices ( j , w l , AI). The reality condition implies in addition that

x ( n ) ( - w - R ; ~181,. . ., w n k n ) = ~ ( " ) * (oft; - w l - R 1 , . . ., - w n - k n ) . (3.2) Other symmetry relations hold in various particular cases of technological importance. For instance, in non-absorbing materials, the energy flows between the fields and is not dissipated in the medium. It is thus possible to define a 'time-average free energy'


Temperature i"C1

Figure 2. Temperature dependence near the a-f3 transition of the angle fixing the orientation of s i04 tetrahedra in quartz, as determined by second harmonic generation (Crane and Bergman 1976).

exploited before proceeding to microscopical models, and in particular that the complementary exploitation of Cartesian and spherical pictures most often gives a better insight into the physics involved in the process.

4. Field interactions in non-linear media

4.1. General aspects of wave propagation in a non-linear medium

Propagation effects are very important in non-linear optics because of the coherent nature of the exciting fields, which in turn can generate coherent fields at other frequencies. Before analysing in detail the interactions of coupled fields, let us recall briefly some general aspects of coherent and incoherent light scattering. As in the introduction, let us consider an assembly of N microscopic units which can acquire a dipole at the frequency St = nw under the excitation of a field E ( w ) ; p(0 ) = ~ ( n ) On(E(w))n. Each microscopic unit in turn radiates at L2. If the dipole phases are unrelated, they radiate independently, so that the total intensity IT, observed at some point, is the sum of the intensity originating from each dipole: IT= C I ( p ) 2 : NI(p) . This type of scattering is that observed in usual spontaneous Raman scattering, and its efficiency is very small. Non-linear incoherent scattering has been observed either in harmonic generation w + w -+ 2w (Maker 1970) or in hyper-Raman scattering w + w + 2w- wvib (Terhune et a1 1965, Cyvin et ai 1965). In non-resonant configuration they require very intense exciting beams and very sophisticated detec- tion apparatus. Now, under certain circumstances, the phases of the individual dipoles may be locked. Then, at the same point as before, the total field ET is the sum of the fields emitted by each dipole ET = L!?(p)ei$, where the phase factor is independent of the dipole to account for coherence. The total intensity is thus ITCC l,!&]22: N2l(p,), i.e. approximately N times the intensity observed in an incoherent scattering. The possibility that the phase factor is the same for all dipoles occurs only under very specific conditions which are customarily referred to as the 'phase matching conditions'. Most often, they are satisfied in a very small solid angle around a precise direction in which the conversion efficiency can become very large. This opportunity is widely used to observe very weak non-linear phenomena or in

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devices to achieve good efficiency. I n this section, we examine how non-linear optical effects manifest themselves on the macroscopic scale through the coupling among electromagnetic fields in material media. How one can use these manifestations in devices or in spectroscopy will be considered in the next sections.

The description of coherent interactions requires the use of Maxwell equations :

v x E= - a3pt V x R= 2Djat +J. (4.1) Because of the appearance of the time derivatives, the equations must be written independently for each frequency w . Let us consider a non-magnetic medium for which 3 = poR and whose losses are accounted for by the current J = U,!?. The electric induction I) = EOE +P includes all the contributions to the polarisation having the right frequency, i.e. the usual linear polarisation &(U) = ~ o x ( l ) . E ( w ) arising from the reaction of the medium to the field itself and the non-linear polarisation PNL(W) arising from the coupling of fields at other frequencies. Collecting the field and the linear part of the polarisation together, i.e. D- EOE(W) + f i ~ ( w ) +PNL(u) = EE +PNL(w) with E = ~ o ( 1 -t-x(1)(w)), enables us to write equation (4.1) in the following form:

a 22 a 2 v x v x E ( w ) +poa(w) - E ( w ) +/LO€ - E ( w ) = - po at2 PNI&). at at 2

One recognises on the left-hand side the usual Maxwell equation describing the propagation of a free wave at frequency w . The right-hand side is a driving term accounting for the non-linear effects. The second-order effects arise from a polarisation of the form fS~~(2)(w)=x(2):E(wl)E(wz) with U = w l + w 2 and third-order effects from &L‘3’(W) = ~ ( 3 ) i E(w1’) E ( ~ 2 ’ ) E (w3‘) with w = wl’ +w2’ +w3’, etc. Of course, other equations of the same form have to be written at all the frequencies w l , w2, wl‘, . . ,, so that one has indeed to deal with a set of coupled equations.

Let us choose the z axis along the propagation direction and express the field as E(?, t ) = E(r) exp [i(wt - kr)]. Because the non-linear polarisations are so small one can make the slowly varying envelope approximation, which specifies that the field amplitude variation owing to the non-linear coupling is small over a wavelength k(aE(w) /az)~((azE(w) laxz) . This leads to

where cy= poco/2n is the linear absorption coefficient and PNL is the total non-linear polarisation at w . This equation has to be satisfied by all the fields interacting in the medium and it is quite complicated to discuss in the most general case. However, for a given set of waves applied to the medium the frequencies arising from second- and third-order effects are most often distinct and in addition the selection rules obeyed by ~ ( 2 ) and ~ ( 3 ) are different. These effects can thus be analysed independently, This is done in the two following paragraphs. For simplicity we assume that the interacting fields are plane waves. This is enough to show the main features of non- linear effects. However, in device design the actual mode structure of laser output is very important, especially because they have finite transverse extension and are usually focused onto the non-linear medium to enhance the intensities. The general properties of laser cavity modes have been investigated by Boyd and Gordon (1961), Boyd and Kogelnik (1962) and Kogelnik and Li (1966). Discussing the influence of the actual mode structure on the detail of the non-linear interaction is beyond the scope of this review. A large number of papers have been devoted to this topic


(Bjorkholm 1966, Kleinman et a1 1966) and the reader can find an exhaustive review of Gaussian beam interaction in the seminal paper of Boyd and Kleinman (1968). Higher-order mode coupling has been considered by Asby (1969a, b, c). As for third-order non-linear effects, beside the analysis of THG with Gaussian beams by Ward and New (1969), there is not, to the best of our knowledge, a detailed discussion of the consequences of the beam structure on the non-linear coupling.

4.2. Wave propagation in second-order effects

Let us consider three fields at wl , wz, w3 such that w i +WZ=W3, interacting through ~ ( 2 ) in a non-linear medium. We assume that the pulse lengths are not too short (> 10-11 s) so that group velocity dispersion can be neglected. Then the field amplitudes satisfy the set of equations

d -- E1 +BalEl= itqE3E2+ exp (iAkx) dz d

dz d - E3 ++a3E3 = ~ K ~ E ~ E z exp ( - iAkz) dz

E2 +&azEz= k i~&3El* exp (iAkz)

where K$ = wtx(2) ( - wt ; wj, wk)/2ncc = wed/nic are the non-linear coupling coefficients and Ak = k3 - k1- k~ is the momentum mismatch. The sign of the right-hand term of equation (4.4(b)) is positive if the three waves propagate in the same direction (forward mixing) and negative if the wave at w2 propagates in the direction opposite to that of the two others (backward mixing). This set of equations has been discussed in length and solved exactly in a number of interaction configurations by Armstrong et a1 (1962). In this case they describe SHG (w1= w2) as well as sum (wl, w2 > O ) or difference (wl<O<w2) mixing. In the case where the absorption is negligible (at< l), the energy is redistributed among the three frequencies via the crystal. By multiplying the right-hand side of the equations by Et*nr/wa one obtains the intensity conservation relation first formulated by Manley and Rowe:

It can be interpreted in terms of the photon number, N(wz)=I&q which vary as

" ( z ) = " ( z ) = - "(z). (4 ' W ) It expresses the view that in each microscopical process every time a high-frequency photon is annihilated one photon at the two other frequencies is created. The case of backward interaction has been discussed by Harris (1966), Gorshkov et aZ(1968), Meadors (1969) and Aslaksen (1970a, b). It presents a number of peculiarities which shall be examined later on. The photon conservation equation is

- 6N1(z) = 6Nz(z) = SN3(z) (4 * 6(b)) in this configuration because the intensity of the wave at w2 increases in the direction opposite to the propagation of the two other waves.

Let us consider first SHG, which is the simplest non-linear effect and which can be used to illustrate the most important features of frequency mixings. An exhaustive

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review of harmonic generation has been given by Akhmanov et al (1975). Let us assume that the losses are small, then the system (4.4) reduces to

2,-1~E,2exp (-iAkz). (4.7) d d - E,=iKE2,,,EWQ exp (iAkx) dx dz

- E - *

The phase mismatch is often expressed as a function of the coherence length

Akz = rr x / l C .

Because of the refractive index dispersion it does not vanish usualIy, so that the conversion efficiency is small and the fundamental intensity can be assumed to remain constant. This is the so-called ‘parametric approximation’ under which the harmonic intensity becomes

It is an oscillating function of the thickness z and is proportional to the squared product of the non-linear susceptibility, the fundamental intensity and the coherence length. Iz , (x) grows over the first lc and then decreases to zero over the next one and so on. Equation (4.8) has been written in a form which allows us to follow the limit Ak + 0. Then I z , exhibits the (sin ~ 1 % ) ~ dependence customarily encountered in interference problems. A strong maximum is observed when Ak= 0, the parametric approximation may fail in such a case, and one has to account for the pump depletion. An exact solution can be found as a function of the fundamental incident intensity: IO= I,(x = 0) = &On ,c I Eo I 2

Iz,(z) = 10 tanh2 (&ox) I,(z) =Io sech2 ( ~ E o x ) . (4.9) Theoretically a total conversion is possible, but in practice a number of limitations appear, related to focalisation and double refraction walk-off. However, conversion efficiencies in excess of 50% can be achieved and have been reported under optimised configurations. Let us now examine how phase matching can be realised. In the case of SHG the condition Ak = 0 gives

nzw=nw.

The most convenient technique to fulfil this condition is to use the birefringence to overcome the dispersion (Maker et a1 1962, Giordmaine 1962). Let us illustrate this with an example. Potassium dihydrogen phosphate (KDP) is a negative uniaxial crystal (ne < no) widely used in NLO and whose point group is T2m. The non-linear coefficient d36 can couple an ordinary fundamental wave propagating with k,,, = n,O- ( w / c ) with an extraordinary harmonic one with kzw=nzwe 2w/c, according to P3(2w)= 2d36El(w) &(U). For a propagation direction making an angle 8 with the optic axis the extraordinary refractive index at 2w is n2,-2(8) = (n2,0)-2 cos2 8 +(nz,e)-2 sin2 8. It is equal to the ordinary fundamental refractive index for the angle Om such that

sin2 O m = [(n,0)-2-(nzw0)-2] [(nz,e)-2- (nz,,,0)-2]-1.

This angle is the phase matching angle. It exists if InuO-n2,,,01 e 1n2,e--n2,,,0I, i.e. if the dispersion is smaller than the birefringence. The refractive indices of KDP are shown in figure 3, together with the I z , phase matching pattern observed in the SHG of a Y A G - N ~ ~ + laser operating at h= 1.0624 pm, as the sample was rotated in the beam so as to continuously modify the angle 8. Phase matching in uniaxial crystals


160

2

152

I L L I I I I

I I I I I I I

A 0 80 160

rl

Figure 3. Variations of the refractive indices against wavelength for the uniaxial crystal KHzP04. The broken lines indicate that the ordinary refractive index at the YAG- Nd3+ laser wavelength (1.064 pm) can be set equal to an extraordinary refractive index at the harmonic wavelength (0.532 pm) for the angle 6,~41.5". On the right- hand side is shown the variation of the harmonic power generated when a thin slab of crystal is rotated in the laser beam, showing the large efficiency which occurs when the exciting and the generated waves' wavevectors become equal.

has been thoroughly investigated by Midwinter and Warner (1965) and their work has been extended to biaxial crystals by Hobden (1967). The expressions of the effective non-linear coefficients in the phase matching direction have been listed by Boyd and Kleinman (1968) and Zernike and Midwinter (1973).

Phase matching using birefringence suffers from an intrinsic limitation owing to double refraction. For an extraordinary wave the wavevector and the energy flow in distinct directions making an angle p, which in the phase matching direction is

For beams with a limited waist the 'walk-off' limits the interaction length and therefore reduces the coupling. For 0, = 0 or 7712 there is no walk-off; then phase matching is said to be non-critical, there is no limitation to the crystal thickness and very efficient conversion can be achieved.

Other phase matching techniques, although not as convenient, have been investigated including those using optical activity (Rabin and Bey 1967) or Faraday rotation (Pate1 and Van Tran 1969). Anomalous dispersion owing to an absorption at a frequency situated between those of the interacting fields has been proposed to allow for phase matching by Franken and Ward (1963) and Bloembergen (1965) and demon-

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strated by Bey et a1 (1967) in a liquid and by Zernike (1969), Matsumoto and Yajima (1973) and Boyd et a1 (1972) in semiconductors. Finally a spatial modulation of the non-linear susceptibility, for example by inverting the crystal axes every coherence length, can also provide a means to achieve a quasi-phase-matching. This was proposed first by Armstrong et al (1962) and demonstrated by Miller (1964a, b) and Armstrong and Rowe (1972).

We can now extend the results of SHG to three-frequency mixing. The coupled wave equations can be solved by use of the photon conservation relations (4.6). According to the boundary conditions different processes can be observed. For the sake of simplicity let us first assume the conversion efficiency is small enough so that the parametric approximation holds.

The first configuration we shall discuss is the forward difference frequency generation. The boundary conditions correspond to a strong field, often called the pump, at the higher frequency applied to a non-linear crystal, together with a weaker one at w l . A field at the difference frequency w2 = w3 - w1 is generated in the medium. If the interaction is phase-matched the low-frequency beams experience an exponential gain according to

I Z ( x ) =? 11(0) sinh (I?,) Il(z)=Il(O) cosh (r,) (4.11) w1

where the gain I? is given as a function of the pump intensity by

(4.12)

giving the parameter dZ/(nln2n3) often referred to as the ‘non-linear figure of merit’ of the crystal.

This process has been used in a number of devices to generate infrared radiation. The occurrence of an exponential gain also suggests the possibility of achieving oscillation. However, this gain cannot be experienced too often before the pump depletion has to be accounted for. The exact solution of the set of coupled equations can be found in terms of the Jacobian elliptic functions. Let us define u=Jb dB (1 - m sin2 6)-1’2 and as usual sn(u, m) =sin rp, cn(u, m) = cos rp. Then the number of photons is

N 3 ( X ) = N3(0) sn2 [r(x +xo), N3(0)/(N3(0) +Ni(O))]

N i ( Z ) =Ni(o) +N3(0) m2 [r(x +Zo), N3(0)/(N3(0) +N1(0))]

N2(x) = N3(0) cn2 [r(x +xo), N3(0)/(N3(0) +N1(0))1

(4.13)

where the two higher-frequency beams contribute to the gain according to

and xo is a characteristic length defined by sn[rxo, N3(0)/(N3(0) +N1(0))] = 1. The variations of the photon number are shown in figure 4. They are periodic functions of the thickness x with a period ZO. The pump number of photons decreases to zero while the number of photons at w1 and w2 increases to their maximum Nl(0) +N3(O) and N3(O) when x varies from 0 to xo. Then the process is reversed and the pump is regenerated. Of course, the approximate solution of equation (4.11) is obtained from the general one in the limit N3(0) BN1(0), because sn(u, 1) = th(u), cn(u, 1) =

Non-linear optical properties of condensed matter

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