SURFACE-MEDIATED NON-LINEAR OPTICAL EFFECTS IN LIQUID CRYSTALS

Surface-Mediated Non-Linear Optical Effects
In Liquid Crystals

V.O. Kubytskyi kubba@univ.kiev.ua    V.Y. Reshetnyak Physics Faculty, Kyiv National Taras Shevchenko University,
Prospekt Glushkova 2, Kyiv 03022, Ukraine
   T.J. Sluckin School of Mathematics, University of Southampton,
Southampton SO17 1BJ, UK
   S.J. Cox School of Engineering Science, University of Southampton,
Southampton SO17 1BJ, UK
July 7, 2019
Abstract

We make a phenomenological model of optical two-beam interaction in a model planar liquid crystal cell. The liquid crystal is subject to homeotropic anchoring at the cell walls, is surrounded by thin photosensitive layers, and is subject to a variable potential across the cell. These systems are often known as liquid crystal photorefractive systems. The interference between the two obliquely incident beams causes a time-independent periodic modulation in electric field intensity in the direction transverse to the cell normal. Our model includes this field phenomenologically by affecting the potential at the walls of the cell. The transverse periodic surface potential causes spatially periodic departures from a pure homeotropic texture. The texture modulation acts as a grating for the incident light. The incident light is both directly transmitted and also subject to diffraction. The first diffracted beams result in energy exchange between the beams. We find that the degree of energy exchange can be strongly sensitive to the mean angle of incidence, the angle between the beams, and the imposed potential across the cell. We use the model to speculate about what factors optimize non-linear optical interaction in liquid crystalline photorefractive systems.

pacs:
42.15.-i, 42.25.Fx, 42.70.Df, 42.70.Nq, 42.79.Dj, 42.79.Kr, 61.30.-v

I Introduction

It has long been known that nematic liquid crystals act as non-linear optical media Khoo and Wu (1993). The electric fields in a strong light beam reorient the liquid crystal director. In so doing they affect the dielectric properties of the medium, and hence its light transmission and reflection. Thus the liquid crystal reacts differently to a high intensity beam than when the intensity is low. A slab of liquid crystal exhibits analogous properties when irradiated by two beams rather than by a single beam. The liquid crystal responds to the interference pattern between the beams. The result is a grating in the liquid crystal cell, which diffracts the incoming beams. The lowest order diffracted beams from each incident beam act to reinforce the other, leading to the phenomenon of beam amplification. This latter phenomenon is the subject of this paper.

In general beam-coupling is a highly non-linear phenomenon and as such requires intense beams to manifest itself. However, it has been found experimentally that there are circumstances when beam-coupling appears to occur at much lower light intensities. The device possibilities of these high beam-coupling conditions have meant that these systems have attracted much interest. Two particular interesting systems obtain when either the liquid crystal is doped by dye molecules, or when the liquid crystal cell is sandwiched between walls consisting of photoconducting material. Now free charges can play an important role in the non-linear optics – in the former case the ions move inside the liquid crystal itself, whereas in the latter case they affect the boundary conditions to which the liquid crystal is subject. In this paper we consider the second of these cases – that in which the liquid crystal is surrounded by photosensitive layers. An associated feature of such systems is that the degree of beam-coupling is strongly dependent on, and amplified by, a low-frequency voltage across the liquid crystal cell.

The existence of ions in these systems has led to this phenomenon being linked with photorefraction Wiederrecht (2001). In photorefractive systems moving charges lead to non-linear optical effects. These so-called photorefractive liquid crystal systems exhibit large optical non-linearities, at least partly because two non-linear optical processes seem to be manifesting themselves simultaneously. This statement, however, while true, is insufficient even to give the most basic description of the physics of beam-coupling in these systems. In this paper we present a phenomenological model, which although by no means a complete description of the system, provides a basic understanding of some of the most striking features of the experiments. The most important of these features is the observation that the largest beam-coupling effects occur when the grating period is comparable to the cell thickness.

Experimental work in photorefraction in liquid crystals dates back about decade. In the cases of interest in this paper, the effect results because a spatially modulated light field causes a modulation of the electric field either in the aligning layer itself Miniewicz et al. (1998, 2001), Tabiryan and Umeton (1998); Bartkiewicz et al. (1999); Kaczmarek (2004) or in the interface between the LC and the aligning layer Zhang et al. (2000); Pagliusi and Cipparrone (2001, 2004a). In the first case the liquid crystal cell is lined by photoconductive aligning layers, whose electrical resistance is decreased by light irradiation. This increases the electric field in the liquid crystal bulk, which in turn causes a spatially modulated reorientation of the director in the cell. The effect is reversible; the induced gratings disappear when the incident light is switched off. By contrast, in the second case the photorefraction is controlled by the processes in the interface between LC and aligning surfaces. Both of these layers may be nominally insensitive to light. The resultant spatially modulated electric field induces a reorientation of the director in the bulk and a permanent grating.

Theoretical work on these systems has concentrated on extending existing photorefractive concepts, which have been developed for optically isotropic systems. In this case, the key inputs into an experiment are: the cell thickness , the light wave-length , the grating period , and the dielectric constant of the isotropic medium. Kogelnik Kogelnik (1969) developed a coupled-wave theory which can predict the response of volume holograms (i.e. thick gratings). Klein Collier et al. (1971) gave the criterion for a grating to be thick in terms of the parameter . The coupled wave theory begins to give good results when . Montemezzani and Zgonik Montemezzani and Zgonik (1997) have extended the Kogelnik coupled-wave theory to the case of moderately absorbing thick anisotropic materials with grating vector and medium boundaries arbitrary oriented with respect to the main axes of the optical indicatrix. The dielectric tensor modulation takes the form

(1)

Galstyan et al Galstyan et al. (2006) have also presented a variant of this idea applied to anisotropic thin holographic media.

The key extra piece of physics in liquid crystal cells is that the director is anchored by the cell walls. As a result the spatial modulation of the dielectric function is considerably more complicated than the Montemezzani-Zgonik form. In addition, the liquid crystal cell parameters are often in the so-called Raman-Nath regime Goodman (1996); Raman and Nath (1935), which corresponds to thin gratings. For thin isotropic gratings with a one-dimensional refractive index modulation, the theory is well developed (see for example Goodman (1996)). For such a system, for example, Kojima Kojima (1982) used a phase function method to understand the diffraction problem for weakly inhomogeneous anisotropic materials in the Raman-Nath regime, assuming a dielectric function spatial modulation .

In this paper we shall study the diffraction and energy transfer of two light beams intersecting in a nematic liquid crystal cell with strong homeotropic anchoring at the cell walls. The beam-coupling can be amplified by a DC-electric field, which is applied to the cell perpendicular to the cell walls (Oz-direction). This problem corresponds to that addressed experimentally by Korneichuk et al Korneichuk et al. (2004).

The presence of the two beams causes a periodic lattice in the light intensity field in the cell bulk and its boundaries. In addition, the laterally periodic light intensity also causes a modulation in the dc-electric field potential at the cell boundaries. This paper addresses the photorefraction problem phenomenologically.

Specifically, we consider here a restricted problem with two major caveats. Firstly we shall not examine too closely the origin and mechanism of this modulation. We simply remark that it can and does result from different physico-chemical phenomena taking place at the cell walls. Likewise in this simple approach, we shall suppose that the physics of the electric field in the liquid crystal is driven by dielectric processes, and that charge transport does not play a major role in determining director orientation or light scattering. Elsewhere we shall relax both of these constraints.

The key to understanding beam-coupling in these systems lies in the following observation. The surface potential modulation produces a spatially modulated electric field. The resulting torque on the liquid crystal director distorts the initial homogeneous homeotropic alignment. The consequence is an anisotropic medium with a spatially modulated director and hence optical axis. The test beam – or the beams that write the grating – diffract from the liquid crystal cell, which now possesses a spatially modulated refractive index. One may then calculate beam diffraction and inter-beam energy transfer.

The paper is organized as follows. In Section II we determine the electric field profile in a cell subject to a light-induced periodic modulation of the surface potential. In Section III we calculate the director distribution inside the liquid crystal cell subject to this spatially modulated electric field. Then in Section IV we present results of calculations of beam diffraction and energy transfer. Finally in Section V we present some brief conclusions, and focus on possible extensions of the model.

Figure 1: Schematic picture of a two-beam coupling experiment, showing meaning of quantities used in the paper.

Ii Electric field within the
liquid crystal slab

We consider two equal frequency light beams with wave numbers inside the medium. These beams give rise to electric fields , and intensities proportional to the squares of the respective electric fields:

(2)
(3)
(4)
(5)

The bisector of the beams makes an angle with the cell normal and is the angle between the beams. Initially we suppose that the scattering by the cell is weak, and thus the light transmission through the cell is close to unity.

The beams interfere in the liquid crystal slab, forming a complex intensity pattern with wave number . In principle the pattern must be calculates self-consistently. In practice it may be possible to treat surface-induced and bulk-induced effects separately. In this section we concentrate on the particular effect at the surfaces. At the bottom (i.e. incident) interface, the interference pattern of light takes the form:

(6)

Likewise, at the top substrate we have an analogous pattern, but shifted in phase with respect to the lower substrate:

(7)
(8)

In the absence of the light beams, we suppose a voltage across the liquid crystal cell. This is a key input to the theory. In photorefractive systems the optical non-linear effects are large and strongly amplified by a voltage across the cell. In the simple theory presented here the non-linear effect in the absence of an external field is strictly zero.

We now make the hypothesis that the spatial distribution of light intensity induces a modulation in the surface potentials. The effect of the light beams is to modify these potentials slightly. The boundary conditions on the electric potential at the top and bottom substrates can now be written:

(9)
(10)

where . The parameter is a phenomenological quantity, and in principle is different for each surface configuration. Here we are supposing that the surface preparation of the upper and lower surfaces is identical. The parameter can in principle be determined independently by a Frederiks experiment 111We might imagine a system in which the boundary conditions were homogeneous, for example, and only one surface was attached to a photosensitive layer. Only one of the surfaces would then be affected by an incident light beam. A single light beam would then affect the effective voltage across the cell. The effect would be proportional to the light intensity, and the constant of proportionality would be the parameter . The magnitude of the effect could be measured by monitoring the critical Frederiks field..

We now proceed to determine the electric field potential within the liquid crystal slab. The electric field obeys the equation

(11)

with , where is the nematic director.

We can solve eq.(11) using relation . At this stage we note that the equations for and must be solved self-consistently. However the liquid crystal is subject to homeotropic boundary conditions, and hence except at very strong light intensities the director is closely aligned to the direction perpendicular to the slab: , and then

(12)

The problem to be solved is thus eq.(12), subject to the boundary conditions eq.(10). This problem can be solved analytically, yielding

(13)

where . The potential in the liquid crystal slab consists of the externally imposed voltage plus a contribution linear in the surface perturbation induced by the light-beam interference. This perturbation has the same periodicity in the direction in the cell plane as the initial perturbation. However, the behavior is complex as a result of the competing effects of the out-of-phase surface perturbations.

The resulting electric field, in addition to the imposed external field normal to the cell, has components in both the and directions. The contributions in the direction are particularly important, because they lead to director distortion, and thus to refractive index modulation. It will be this refractive index modulation which induces the beam-coupling which we seek to describe.

The electric field inside the cell bulk is given by taking the gradient of eq.(13). We find:

(14)

where

(15)
(16)
(17)
(18)
(19)

It will be useful later to normalize the field with respect to the externally imposed field:

(20)

where the quantities are given by:

Iii Director profile in the liquid crystal cell

We now determine the director profile in the liquid crystal cell in the presence of the electric fields given by eqs.(14). The bulk free energy of a distorted nematic liquid crystal in an applied electric field takes the form:

(21)

with the electric displacement , , with the anisotropic part of the static dielectric constant .

The electric field felt by the liquid crystal molecules has a number of contributions. The first is the externally-imposed voltage. The second is the periodic modulation in the direction discussed in the last section. This is an indirect effect of the light field acting on the surface layer, transmitted into the bulk as a result of the effect of the Laplace equation. A final contribution comes from the direct effect of the light field on the liquid crystal. We are assuming here that this can be neglected. The justification for this is empirical, and derives from the observation that in the absence of the surface layers, the effect essentially disappears Bartkiewicz et al. (1998); Kaczmarek (2004). The director field is now given by , with small.

The variational problem to be solved consists of minimizing eq.(21), subject to strong anchoring homeotropic boundary conditions at each wall, and subject also to the electric fields given in eq.(14). We simplify further by supposing the so-called one constant approximation, i.e. the splay and bend Frank-Oseen elastic coefficients are equal: .

The relevant part of the thermodynamic functional eq.(21) is now given by:

(22)

The Euler-Lagrange equation for this functional is:

(23)

We solve eq.(23) in the limit . This corresponds, roughly speaking, to high voltage or low beam intensities. Expanding to linear order in and , we obtain:

(24)

where the length scale is the relaxation length set by the bulk electric field, with . In order to do this, we linearize eq.(24) for components and of the director reorientation of wave number in the cell plane.

It is convenient at this stage to reformulate the problem in terms of non-dimensional variables. We define a rescaled length along the cell , , a rescaled transverse wave vector , and a rescaled voltage . For some purposes it is convenient to measure length not from the plane of incidence, but rather from the mid-plane of the cell. We can define a length variable , and then inside the cell .

Eq.(24) now reduces to

(25)

with .

This equation can be solved using standard methods and has solution:

(26)

For some purposes it is more convenient to rewrite eq.(III) as:

(27)

The advantage of the expression (27) is that the variation of is expressed in terms of components that are respectively out-of-phase and in-phase with the total optical field intensities on the mid-plane. This expression explicitly exhibits the symmetry of the system around the mid-plane.

We discuss in the next section in detail how to use to calculate non-linear optical effects. However, we note that in general the larger the values of measured in some sense the larger will be the optical effect. It is therefore of some interest to monitor the behavior of as a function of system parameters.

We plot the z-dependence of the out-of-phase component of in Fig.2. One might expect a roughly sinusoidal dependence, with a maximum at the cell mid-plane. And indeed, for closely matched incident beams, with a low wave-number interference pattern, this is what occurs. But when the non-dimensional grating wave-vector is larger than unity, the sinusoidal dependence no longer holds. By , the response is flattened, and by the profile has developed a double hump structure. Not only is the shape unexpected, but the magnitude is reduced in this regime, and as discussed in the last paragraph, this should (and, as we shall see below, does) lead to a reduced non-linear optical effects for larger .

Figure 2: Functional dependence of angular distortion from homeotropic texture, in the limit of low field, for different values of the non-dimensionalized grating wave number . This figure shows the component of the angular distortion out-of-phase with the intensity modulations.

From eq.(27) we observe that can be regarded a figure of merit for the degree of distortion of the liquid crystal. This is a measure of the amplitude of the response at the mid-plane of the cell. We note that technically only measures the out-of-phase distortion, and furthermore even then the magnitude of the distortion is not maximal at the mid-plane of the cell. Nevertheless it serves in a rough and ready way as a surrogate for the magnitude of the grating response to optical probes.

Figure 3: Dependence of the amplitude of the component of the angular distortion out-of-phase with the optical intensity, as a function of non-dimensionalized voltage

In Fig. 3 we plot as a function of voltage.

Iv Diffraction of light beams

iv.1 Formulation of problem

We now consider light beam propagation of each of the two waves through the (now) weakly non-uniform anisotropic liquid crystal cell. We suppose the wave incident from the vacuum to have wave number , with , where the angular frequency and the speed of light take their usual meanings. In the presence of a uniform liquid crystal, the light will be refracted into an ordinary (o-) and an extraordinary (e-) wave. The ordinary wave is polarized perpendicularly to the plane of incidence (in the direction) and the refractive index which corresponds to this wave is . For the extraordinary wave the effective refractive index is given by:

(28)

where is the angle between the director and the direction of propagation inside the medium.

The effect of the non-uniformity will be to modulate the amplitude of the wave in the plane of the outgoing surface, and hence in addition to the refraction, diffraction will also occur. The purpose of this section is to calculate the magnitude of this diffraction. There are in fact two waves, but first we discuss the effect of the modified director on each individual wave.

iv.2 The dielectric function

The dielectric function is given by

(29)

with and director components . In the limit of interest in this paper, the director deviations from the initial homeotropic alignment are small. Then

(30)

The dielectric function now simplifies to

(31)

or alternatively:

(32)

iv.3 Geometrical Optics

The theoretical strategy involves determining perturbations around the transmission through the pure homeotropic (i.e. ) system. The characteristic length for director inhomogeneity in the -direction is the cell thickness . In the -direction the corresponding characteristic length is the grating period . We shall use the Geometrical Optics Approximation (GOA) Panasyuk et al. (2003); Kraan et al. (2007); Kravtsov and Orlov (1990), valid in the limits and .

We seek solutions to the Maxwell equations

(33)

in the following forms:

(34)

The term is the optical path length or eikonal, and the local direction of the wave vector is given by .

Substituting eqs.(34) into the Maxwell equations (33), we obtain the following pair of equations:

(35)

The magnetic field can now be eliminated, yielding a homogeneous equation for :

(36)

Eq.(36) is a homogeneous system of linear equations for the electric field components, analogous to a vector Helmholtz equation. In general solutions to this equation will be trivial and uninteresting. However, there are non-trivial solutions, corresponding to optical traveling waves, if the determinant ofling waves, if the determinant of this set of equations is null.

In fact the determinant factorizes. An eigenvector in the direction corresponds to the ordinary wave. The perturbations in the dielectric tensor do not affect transmission of the ordinary wave through the sample, and we shall not be interested in this mode of transmission. The eigenvector in the plane (i.e. the plane of incidence) corresponds to the extraordinary (e-) wave. The pair of homogeneous equations are:

(37)

The null determinant condition appropriate to the e- wave now reduces to222This equation is a derivative of what in the literature is usually known as the eikonal equation.:

(38)

iv.4 Perturbation Theory

In the absence of the director modulation, the e-wave is directly transmitted. We consider eq.(38) as a perturbation of this process. We therefore recast eq.(38) to lowest order in :

(39)

In the spirit of the WKB approximation, the solution of eq.(39) can be expressed as the sum of an unperturbed e-wave, plus a small phase change which can be ascribed solely to the modulation. Thus:

(40)

where obeys the equation

(41)

We note also that the effective refractive index can be defined as follows:

(42)

Combining eqs.(41) and (42) yields the well-known expression for the refractive index (28). The wave vector inside the medium is now given by

(43)

with .

Combining eqs.(39),(40) and (41), we can obtain the leading order equation for the phase :

(44)

We now substitute eq.(43) into eq.(44), yielding:

(45)

Eq. (45) can be solved using the method of characteristics. The left hand side of this equation can be transformed into a total derivative:

(46)

Combining eqs.(45) and (IV.4) yields:

(47)

where now the quantities and in this equation are explicitly related by

(48)

defined in eq.(IV.4). Eq.(48) allows a family of solutions

(49)

We note that each member of this family of solutions represents a wave entering the liquid crystal sample at position . The direction of wave propagation is given by the angle . However, because this medium is anisotropic, the angle of the energy propagation, given by the Poynting Vector, is determined by the angle . The family of solutions corresponds to paths in the undistorted anisotropic medium with different travelling in the direction of the Poynting Vector.

Now we can solve eq. (47) directly, by integrating the right hand side, yielding:

(50)

where the integration path is such that , and hence

(51)

Thus

(52)

The key quantity of interest is the phase retardation of the beam as it leaves the cell, i.e. at . We now rewrite eq.(50), so as to express this quantity directly:

(53)

iv.5 Diffraction Pattern

The formula (53) applies to all incident light beams. We confine our interest to the cases in which there are two incident beams, with wave numbers and , with . Eq.(53) permits the calculation the light fields of the beams as they exit the liquid crystal cell. By suitably decomposing these light field into Fourier components, it is possible to identify the amplitudes of particular diffracted beams.

The light field along the plane is modulated by the factor

(54)

where , and ;

(55)

and

(56)

The quantity is the varying component of the additional phase of the incident beam, following from the director modulation in the liquid crystal medium. We now recall that from eq.(27) the director profile can be written in a sinusoidal form: . Combining this result with eq.(56) yields the result:

(57)

where the phase modulation parameters and are respectively the amplitude and phase of the additional phase :

(58)

After complicated but straightforward algebra using eqs. (27), (56), expressions for the quantities and can be derived:

(59)
(60)

Now eq.(IV.5) can be rewritten, using eq.(57), so as explicitly identify different components of the diffracted wave. The key relation is the Jacobi-Anger expansion Abramowitz and Stegun (1972):

(61)

Combining eq.(61) with eqs.(IV.5), (55), (56) and (57) yields the expansion for the electric field at the output surface:

(62)

Now we can identify Born and Wolf (1980) terms in this expansion with the amplitudes and phases of outgoing waves in the diffraction pattern:

(63)

and

(64)

The electric field in the diffracted wave of order then takes the form:

(65)

where is the wave number of the diffracted wave of order , with , and .

iv.6 Beam Coupling

We now return to the original problem (5) in which there are two incident waves with wave numbers , and with . Beam coupling corresponds the diffraction of waves from incident wave to outgoing wave , and from incident wave to outgoing wave . Thus the diffracted wave of order from adds coherently with the directly transmitted wave , and the diffracted wave of order from adds coherently with the directly transmitted wave . Equivalently, using the notation of the last section, and .

We are thus able to use terms from the diffraction expression eq.(65) to evaluate the amplitudes of the outgoing waves directions and . We find:

(66)

We note that in principle the quantities depend on the value of the incident wave number. However, in the two-beam coupling case discussed here, the incident waves have wave numbers very nearly equal to each other, and so we may consider the quantities to be the same for each incident wave.

From eq.(IV.6), we can evaluate the outgoing wave intensities, using the relation . Using eq.(63), We find:

(67)

or

(68)

where is as defined in eq.(58).

It is usual to consider one of the beams as the pump beam and the other as the probe beam. Without loss of generality, we shall suppose that corresponds to the pump beam and to the probe beam. In line with the literature, we define

(69)

We can now rewrite the formulas for the outgoing beam intensities in terms of the quantity as follows:

(70)

The degree of beam coupling can now be characterized by the Gain . This is the ratio of the intensity of the outgoing beam in the direction of the probe beam in the presence of the pump beam to the intensity of the same beam in the absence of the pump beam. In the context of this paper, in which we do not consider reflection and refraction at the cell walls, the quantity is defined as:

(71)

A related quantity is the Diffraction Efficiency , which measures the strength with which the grating diffracts the probe beam. The formal definition is the ratio of the intensity of the diffracted probe beam (i.e. in the direction of the pump beam) to that of the incoming probe beam. From eqs.(63) and (65), this is:

(72)

For ease of presentation of our results, it is also convenient to define quantities and . These quantities are respectively analogous to and , but with the roles of the pump and probe beams exchanged.

V Results

v.1 Analytical study of the behavior of intensities.

First we discuss diffraction from a single beam. The transmitted energy is divided between beams of different orders . The amplitude of diffracted beams is given by (eq.(63)), where we recall, from eq.(58) that is the amplitude of the additional phase variations induced by the spatial modulations in the liquid crystal layer. We introduce the energy conservation parameter

(73)

The quantity describes the proportion of transmitted energy that is distributed between diffracted beams of orders from to . We note that in the limit necessarily .

In Fig.4 we show the dependence of on the phase modulation parameter for low . For small (), . In this regime almost all transmitted energy is either in the directly transmitted beam, or in the first-order diffracted beam. This is the regime in which energy exchange between two beams is most effective.

As increases, an increasing proportion of the transmitted energy is transfered to outlying diffracted beams. The quantity remains essentially unity until , while only noticeably departs from unity at . We shall return to the problem of the asymptotic behavior of for large elsewhere. For this study, however, we shall be interested in the small regime.

Figure 4: The dependence of energy conservation parameter on phase modulation parameter . Solid line: , dashed line and dotted line . See text for further discussion.

We now analyze the effect of energy exchange in the presence of both incident beams. We shall take equal intensities in eq.(70) for incident beams . In principle the parameter [eq.(69)], measuring the ratio of the intensities of the beams, can take any value. In our calculations we shall suppose ; this corresponds to equal intensity beams. This will enable us to make contact with previous studies Wiederrecht (2001), which have also used this value. In addition it is easy to monitor energy transfer between beams. We note that in devices we may well expect that , so that a large reservoir of pump beam energy is available to amplify a given probe beam.

The degree of energy transfer is critically dependent on the quantity , defined in eq.(58). When , the modulation of the phase retardation is in-phase with the intensity modulation due to the beam interference. If , eq.(70) implies that there is no net energy exchange between the beams. This is consistent with general intuition Yariv and Yeh (1984) that a phase difference between the intensity and dielectric modulations is required for beam coupling. Interestingly, although we do not pursue this here, this rule no longer holds for the case.

The maximum energy transfer between beams occurs when . In this case the two outgoing beams obey the following rule:

(74)

The behavior of these functions is shown in Fig.5. The function (solid curve on the graph) reaches a maximum value of at . We also plot , and note that this reaches a minimum(at zero) for . The quantity

(75)

denotes that proportion of the energy of the incident beams which remains in the two initial beams directions. The quantity is unity for (at which there is, however, no energy exchange) and reduces steadily with B. Close to the maximum