Snell's laws at the interface between nonlinear dielectrics

Volume 129, number 4

PHYSICS LETTERS A

23 May 1988

SNELL’S LAWS AT THE INTERFACE BETWEEN NONLINEAR DIELECTRICS A.B. ACEVES, J.V. MOLONEY’ and A.C. NEWELL Department ofMathematics, University ofArizona, Tucson, AZ 85721, USA Received 7 March 1988; accepted for publication 17 March 1988 Communicated by D.D. Hoim

An equivalent particle theory is developed which is effective in capturing the global behavior of surface waves at the interface between two nonlinear dielectrics.

The interaction of a finite width beam (seif-focused channel) with an interface to which it is obliquely incident at a fixed angle translates into the problem of a nonlinear wavepacket of power P with a fixed initial velocity v, proportional to the angle of incidence, approaching the same interface. In the study, the interface separates two dielectric media whose refractive indices have a self-focusing nonlinear dependence on the electromagnetic field intensity. In short, what we are aiming to provide is the nonlinear analogue of Snell’s laws. An extensive numerical analysis has been performed for this problem covering a wide range of parameters, both ofthe dielectric media and of the incident wavepacket. We have observed three different regimes ofbehavior in the P—v plane. For low powers, the incident wavepacket is broad (its width being proportional to P’) and its reflection properties are given by the familiar linear Snell’s laws. In other words, in the regime where the wavepacket splits into partially reflected and partially transmitted components, each part disperses as it propagates away from the interface. For incident wavepackets with higher powers but whose widths are still greater than ,,,/~,where 4 is the linear refractive index mismatch at the interface, the behavior is fundamentally different. Again, the wavepacket may split into two components, one partially reflectedand the other partially transmitted as it approaches the interface but now each component Permanent address: Physics Department, Heriot-Watt University, Riccarton, Edinburgh EH 14 4AS, Scotland, UK.

propagates as a self-focused channel (soliton) without dispersion. The analysis to determine the amount of reflected and transmitted powers as a function of initial angle (velocity) is work in progress and will be presented in a future paper. Finally, there is a regime where the incident wavepacket width is smaller than or equal to ~/i which we shall refer to as the fully nonlinear regime and this regime is the focus of the present Letter. Here the incident wavepacket exhibits a nontrivial dynamics and does not split into partially reflected and transmitted components as it approaches the interface. At higher incident powers, a new phenomenon can appear in the fully nonlinear regime if the nonlinear mismatch between the two media is large enough. In this case the incident beam can break-up the multiple self-focused channels after entering the new nonlinear medium. The theory we present provides a simple and extremely effective description ofthe behaviorof these wavepackets because the fact that they remain localised allows us to develop an equivalent particle description. Further, the dynamics of each wavepacket (the incident, transmitted or reflectedbeam) can be inferred from the motion and phase plane portraits of an equivalent particle in a potential well. Varying the initial angle ofthe beam is equivalent to choosing different initial velocities of the particle and therefore the nonlinear analogue of Snell’s laws in this regime and their dependence on the incident power and material properties can be read off directly by calculating the asymptotes of the equivalent particle trajectories. Our theory establishes the stability and instability prop231

Volume 129, number 4

PHYSICS LETTERS A

erties of known steady state surface waves, explains the many possible ways in which an incident beam is reshaped by an interfaceand can also predict when beam break-up occurs. Although in principle the theory is valid for (a) large values of a, the ratio of the nonlinear refractive indices, as long as the beam does not cross the interfaceor (b) values ofa close to one, if it does, it turns out that the amount of radiation (that part of a beam which disperses) produced is proportional to (1 ~ ) 2 which even for refractive index ratios of 2:1 is less than 10%. Because of this, —

the theory works extremely well for a large range of a even in cases where the beam crosses and recrosses the interface several times. Slight modifications of the theory allows us to explain situations in which (a) beam break-up occurs, (b) different beams interact or (c) the beams encounter multiple interfaces. The reason for this is that the theory is modular in the sense that the fundamental idea of replacing the beam dynamics with an equivalent particle description still obtains and can be used as the building block in the more complicated descriptions. We discuss two applications, a nonlinear spatial scanner and a fast switch in which a beam initially trapped parallel to one interface can be transferred to a neighbor. Further, although the theory is presented in the context of optics, it should have widespread application in areas of physics in which the nonlinear Schrodinger equation with an additional potential containing information about discontinuities in material properties obtains. The slowly varying envelope of the TE field satisfies 2[/32—n2(x FF*)]F_o (1) 2iflkF +Fxx —k —

‘

where /3k is the effective wave index, k=w/c is the free space wavenumber and the refractive indices for x~0are 2( FF*\ 2+ F 2 1 n ~X, ,—n~ a~ , j— We arbitrarily choose n~> n~f,and study the case 0< a0 ~ a, for which trapped nonlinear surface waves are possible. Solutions F(x, z) to eq. (1) whoose modulus has the shape —

—~

23 May 1988

(2~)~ and whose angle of incidence is the inverse tangent of the ratio of the transverse (x) to longitudinal (z) wavenumbers. Steady state solutions to eq. (1) are derived by setting F~= 0 and solving the resulting second order ordinary differential equation subject to the boundary conditions of continuity of F and F~at the interface and F~,F—+0 as x—~±CX). Fig. 1 shows the power P_—5~,,IFI2dx which is independent of z plotted against /3 and the peak of the steady state surface wave ~ for the nonlinear mismatch ratio a=a 0/a, =0.25. Significant features of these diagrams are: (i) the existence of a threshold power P~for the observation of trapped nonlinear surface waves, (ii) regions of the branch of equilibrium solutions where the peak of the surface wave lies to the left (DE) and to the right (AD) of the interface, (iii) the unstable peak can exist in either the left (DE) or the right (CD) medium. The stable peak (CBA) is always on the right. As a—~1, the ~ versus P branch becomes symmetric about .~zz0. Notice that as a becomes slightly less than one, the unstable peak can move from the left to the right of the interface for low power near P~(point C in the diagram). As a 0 the whole branch CDE raises above ~ = 0 with the point D at which the graph intersects ~ = 0 moving to infinity. In this case both peaks lie in the right half plane. The reason for this is that as a—~0the left hand medium becomes linear and can no longer support a steady state self-focused surface wave. We now come to the main point of the Letter, the description of the dynamics of the wavepacket near the interface, veloped in several which,stages. for pedagogic First, imagine purposes, thatis the dewavepacket always remains in the left medium, and —~

let r 2 2 2 \/aOkF(x,z)=~’2A(x,r)exp[ik (/3 —n 0)z] where 213kr=z and A(x, r) satisfies

crN~ ~~—a

~0

2~J\/~sechE2~.(x_X)]j=0, 1, far from the interface will be of particular interest. This shape corresponds to a beam which has a width 232

62

C

5.0

0

~5

Fig. 1. Plot of~versus P and /3 versus P for c~ = 0.25.

5.0

Volume 129, number 4

PHYSICS LETTERS A

A~ iA~~2iA 2A* = i VA. The perturbation potential V, which contains information about the discontinuities in refractive indices, is zero for x <0 and 4—2 (a—’ 1)1 A I 2 for x> 0 where 4 = k2 (n ~ n f) is small and positive and a= a 0aj’ is less than one. From the field equation for A(x, r), we can write down the following exact expressions for the rates of change of normalised power p = fAA * dx, average field intensity location x=p ,fxAA’~dx and velocity v= iP ‘f (AA x A*A~)dx, p7O, 2~V, vT=_2PIJVxAA*dx. (2) —

—

—

—

—

—

-

—

The equivalent particle description is obtained by assuming that the field moves collectively in the sense that AA* is a function only of x—5(r). The motivation for this approximation is as follows. An arbitrary wavepacket, initially far from the interface will break up into and radiationincomponents compatible with thesoliton natural properties the left medium, and each component can be treated separately. For simplicity, we will always take ourincident wavepacket to have a perfect single soliton shape A (x,

t) =

2,io sech{2?io [x—5( r) I }

23 May 1988

the corresponding phase plane diagram (2, v) of the particle. As we shall see, UL (5) has no minima although it can have a maximum (see fig. 2a) corresponding to the unstable surface wave discussed earlier. However, it is also clear that if the incident angle of the beam, or initial velocity of the equivalent particle, is high enough the beam and the partide can cross the interface. Before showing how to deal with this circumstance, we give the dual result for a single soliton wavepacket which always remains in the right hand medium. Here we take

A (x, r) satisfies the same equation as before except that now V=0, 5>0 and is equal to —4— 2 (a 1) A 12 for 2<0. Taking A(x, r)=2~,sech{2~,[x—s(r)J} —

x exp (~ivx+ 2ic) 2+ l6~ we find ‘i,~=0 and with v~= x~=v,(d/dS) 8o~=UR(5) v where Uk(S), defined only 5= for 2>0, is 4S—’(S—l)tanh(2~,5)+14S—’tanh3(2i~,5) with 4~(1

x exp( ~ivx+ 2ia)

—

—

—

a )S=4. The total power in the wave-

packet is (a,k2) ~ The potential Uk(S) is shown in fig. 2b, and we observe that, for sufficient power, —

with5~=vand8cr~=—v2+ l6~.The effect of the interface is captured by allowing the soliton parameters to vary slowly with t If the wavepacket always remains in the left medium, the perturbation is small because the amount of IA 12 in the right hand me dium is small. Thus, very little radiation is produced and a single wavepacket description is valid for all r. We can therefore use the shape ofAA* to calculate the force in (2) and write the result as the negative gradient d UL(5) /dx of an effective potential (defined only for 2<0) —

(a)

-________________

(b)

~

_________________

0

~.

55

~ ~04 (c) U 0.005

UL(5)=a’AS’(Sa—l) tanh(2~

UR 004 0,04

~

U

_~,,., ___________

0X)

3(2,io5)

+~a’4S’ tanh with 4~(1—a)S=4. The conservation of power p means that ,~remains constant. The total power P in the wave packet is (a 2) ‘87o. The motion of the particle and thereby 0k the dynamics of the beam interaction with the interface can be read off directly from the shape of the effective potential UL(5) and —

-0.02

—008

Fig. 2. Representative effective potentials UL(X), UR(~)and U(s) indicating the variety of possible types of equivalent partide trajectories in the (5, a) parameter plane. Trajectories for (2c) and (2e) are shown in the next figure. Individual curves are labelled by the value ofS (a=O.75, A=0. 1).

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Volume 129, number 4

PHYSICS LETTERS A

Hence if a < ~, a wavepacket crossing from right to left will disperse as radiation in the left hand medium. With these results in hand, we can describe what happens to a wavepacket which cross an interface. If a> ~, a single soliton wavepacket crossing in either direction gives rise to a single soliton wavepacket on the other side and the fractional amount of radiation produced is (1 /~)2 if it crosses from left to right or a—’ (1 ) 2 if it crosses from right to left. A global description of the dynamics for this case is given in the following paragraph. If ~~ a> ~, a left to right crossing wavepacket will decompose into two soliton wavepackets on the right and one can carry out an equivalent two particle analysis. This calculation is given elsewhere. However, it turns out that in most cases the wavepackets separate (the beam splits into two) and can be treated individually. Because the smaller one often has a sufficiently large S value (see fig. 2b) to cause it to be reflected, it can reenter the left medium, and since a> ~,it will do so as a soliton wavepacket, albeit a small, wide one; in other words the reflected beam will be bound with low intensity. On the other hand, a right to left

S< 1 and Uk(S) can have a minimum corresponding to the stable surface wave we have discussed earlier. We now show how to handle the situation in which a wavepacket has sufficient initial velocity to cross the interface from left to right or right to left and develop a global description for the limit of small (1 ...\/~)2 which means that the theory is effective to within 10% for nonlinear mismatches a as low as 0.5. If the wavepacket crosses the interface, it is reshaped in the new medium according to the following prescription. A packet with amplitude 2i~o~.J~/aok and velocity v crossing from left to right will decompose into N soliton wavepackets with 1 ~
~

23 May 1988

—

—

...

4~c~

~

a 0

—

a,

~,

‘hlr,

which is equal to (4,io/ao) (1 _~~/~N)2.For N= 1, i.e., ~
(l_\/~)2.

crossing wavepacket will produce only one wavepacket in the left medium. If its S value is less than one and its velocity sufficiently small (see fig. 2a), it can be reflected and reenter the right medium where the scenario we have just discussed occurs. Thus a beam crossing from right to left can either reflect as two wavepackets with a very small amount (proportional to a~’(1 \/~) 2) of transmitted radation or it can reflect one and transmit the other. If ~ > a> ~ and the beam crosses from left to right, again two soliton wavepackets are created although in this case, if one is reflected, it will not form a soliton wavepacket (a self-focused channel) in the left medium but will disperse as radiation. The important thing to stress is that in all cases, the intensities and the initial directions of the transmitted and reflected beams can be calculated and their asymptotic behavior found by, in most cases, a single equivalent particle analysis, but at most a finite particle analysis. All the events we have described in this paragraph have been observed and will be reported in detail elsewhere. We have mentioned that when a> ~, no new particles are created and a small smount of radiation is created on each crossing. Ignoring the radiation, we can therefore develop a ,

—

Volume 129, number 4

PHYSICS LETTERS A

global description of the motion for all z using the composite effective potential with

UL(S),

UR(S)givenasbeforeand81~

0and87, approximated by a0F and a,P respectively. In figs. 2c and 2d we draw two typical composite potentials and in fig. 3 we show a comparison between trajectories computed from theory and by a direct integration of the full p.d.e. (1). The agreement is excellent even when the wavepacket crosses the interface more than once. Using these ideas we can give an exact expression for the nonlinear analogue of the Goos—Haanchen shift. Even when the left medium is linear, that is a0 = 0, a slight modification of the theory (simply decompose the gaussian wavepacket into its soliton components in the right medium) allows us to understand many of the numerical observations of Tomlinson et al. [2] and explain the numerical observations of Akhmediev et al. [3] regarding the sensitive behavior of trajectories starting near unstable stationary states. Moreover, the theory can be used as the basic building block for the important problems of beam switching and/or reflection and transmission at multiple interfaces. Further, our knowledge ofthe displacement produced by the nonlinear interaction of two wavepackets can be exploited in envisaging a simple and controllable optical switch which could function as a spatial scanning element, or in the multiple interface context as a nonlinear directional coupler. A switching wavepacket of amplitude 12 (beam intensity) and veloc7.6

ity v2 (angle of incidence) will displace a stationary wavepacket with parameters ~,, v1 =0 (trapped surface wave) corresponding to fig.2+’v2 2d by an amount 1 /( + ) M=—sgn(v2)~--—ln~( )2~I~,2 ~1i ‘ii ~12 4 2 The effect of a collision in which the switching beam enters from the left is to move to the left the stationary wavepacket from the minimum in the potential well in fig. 2d, and thus give it a higher potential energy. The three possibilities for its subsequent motion depend on the amount of displacement (and therefore on the angle of incidence and power of the switching beam) and can be read off from fig. 3. Depending on the power and angle ofthe incidence the beam can switch to the left, remain Oscillating near the interface or switch to the right. Therefore the arrangement acts as an optical spatial scanner. In particular, if it switches to the right, it can be trapped at a neighboring interface (by using a second collision, radiation losses and/or a gradual ramping of the linear refractive index beyond the second interface). Again, the events described here have been observed and the theoretical predictions verified by numerical experiment. Full details of the theory, the multiple interface calculations, the nonlinear Goos—Haanchen formula, comparison with numerical experiments and the calculation of the critical value of a, for total internal reflection discussed in ref. [21 will be published in detail elsewhere. Here we have sketched the principle ideas and verified their validity with extensive numerical cornputation, a cross-section of which, representing each of the separate cases, is given in fig. 3. This

06

2

23 May 1988

work

was

supported

by

F4962086-

CO13OAFOSR, DAAG-29-85K09 1 ARO, N00 DMS-84O3187NSF, 1 4-84-K-04200NR Physics. The au-

) W~SUp~Or~the Moloney-Newdl

V°

References Fig. 3. Phase planetrajectories (solid lines) correspondingto the potentials (2c) and (2d) in fig. 2. These illustrate beam transmission, reflection and oscillation of a trapped nonlinear surface wave near the interface. Dashed lines are corresponding numerically generated peak intensity trajectories from solving eq. (1) directly.

[I] D.J. Kaupand AC. Newell, Proc. R. Soc.A 361 (1978) 413. [2] Wi. Tomlinson, J.P. Gordon, P.W. Smith and A.E. Kaplan, AppI. Opt. 21(1982) 2041. [3] N.N. Akhmediev, V.1. Korneev and Y.V. Kuzmenko, Soy. Phys. JETP 61(1985) 62.

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