Generation of time-reversed wave fronts by nonlinear refraction*
R. W. Hellwarth Electronics Sciences Laboratory, University of Southern California, University Park, Los Angeles, California 90007
(Received 2 September 1976)
We describe a nonlinear method for generating, nearly instantaneously, a time-reversed replica of any monochromatic-beam wave pattern. The method employs the interaction of the incident beam, of arbitrary wave front, with counter-propagating plane "pump" waves in a homogeneous, transparent, nonlinear medium. Media are shown to exist in which time-reversed waves can be generated with high efficiency using available laser pump sources.

I. INTRODUCTION
For any electromagnetic wave that propagates through an inhomogeneous, nonabsorbing, medium (having no permanent magnetism), there can exist in principle a time-reversed replica of this wave. This means, for example, than an appropriately patterned but irregular wave front can travel through a randomly inhomogeneous medium and emerge as a coherent uniform wave front, providing it is a replica, reversed in time, of a coherent beam that is deformed by the same inhomogeneous medium. Here we propose a new method for generating, nearly instantaneously, the timereversed replica of any monochromatic beam. Our method employs the nonlinear refraction present in any medium and is realizable with existing laser sources.
It is well known to be possible to generate a timereversed wave by nonlinear effects. Zeldovich etal.. showed experimentally that a nearly "time-reversed" wave was produced by stimulated Brillouin scattering (SBS) in the backward direction of a ruby laser beam whose phase front had been deformed by an inhomogeneous medium. This wave was not perfectly time reversed as it was slightly downshifted in frequency by the acoustic frequency. Nosach etal. 2 used SBS to restore the coherence of a laser beam that had been amplified by an inhomogeneous amplifying medium. Recently, Yariv has proposed to "undo" the distortion of images transmitted by multimode optical fibers by parametric mixing in an acentric crystal.3 He has shown that the mixed wave would be a time-reversed version of the propagated wave that, upon further propagation in the fiber, would evolve back into the original pattern at the entrance face of the fiber. 3 This mixing process could also be used to produce an unguided, time-reversed beam. In either case, limitations are placed on the beam-acceptance angles in this process by "phase-matching" requirements on waves that can mix efficiently in the crystal. In the case of the technique using nonlinear refraction, which we discuss below, neither a frequency shift nor phase-matching need play a role, thus allowing a more accurate time-reversed replication than is possible with SBS or parametric mixing. Also, on the basis of nonlinear optical coefficients known to date, the effect we'discuss here can be produced with less laser pump power than either of the other effects.
In Sec. II we will show how, in the presence of counter-propagating pump waves, a beam will cause the generation of its time-reversed wave by the nonlinear refraction which exists in any medium. In Sec.

III we show that the pump power levels required for efficient time-reversed generation are well within the capability of available sources. We also suggest a simple experimental arrangement for demonstrating the generation of a time-reversed wave by nonlinear refraction.

II. THEORY

Consider a monochromatic electromagnetic beam that has a complex wave front and is incident from the left on a transparent slab of nonlinear dielectric, as shown in Fig. 1. We assume that this beam has an electric field ReEl (r) eiwt whose complex amplitude E (i=x,y,z) is known at every point r in space. We will derive the amplitude Fi(r) of the field radiated by the nonlinear electric polarization density RePNL(r) e-ivt that is created in the nonlinear medium by the interaction of this beam with strong forward and backward plane waves at frequency w0 that also exist in the medium. That is, we assume the following electric field to be impressed in a homogeneous nonlinear medium:

Re[Ei(r) etw+t +Gieikoziwot +Hie-iz-iwot].

(1)

By virtue of the (third-order) nonlinear susceptibility that exists in any medium, we have, at v =2wo - w,

PZ =Xij Et.

(2)

Here, as throughout, the summation convention is used for repeated space indices, and

Xij =6Ci k (- V, - wwo, w0 ) GkHj,

(3)

where the Cisl are the nonlinear susceptibility coefficients defined by Maker and Terhune4 and which are
known, at least approximately, for many materials. 5
We will call the oppositely traveling plane waves at w0

y

incident

X

(XI'Y11O)

time-

reversed

./

beam

>

S

nonlinear

'.

medium

pump wavevectors

,

0

Z

z

FIG. 1. Schematic and nomenclature for calculation of timereversed wave fields.

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J. Opt. Soc. Am., Vol. 67, No. 1,January 1977

Copyright © 1977 by the Optical Society of America

the "pump" waves, as their amplitudes determine the magnitude of the nonlinear susceptibility X,,.
Now we wish to calculate the electromagnetic field amplitude Fi generated at a point (x, y, z) far in front of the medium by PyLy and show that, when a = w0, Fj is proportional to the complex conjugate Et of the incoming wave front at the same point. That is, the generated wave is the "time-reversed" wave. We can, without loss of generality, take the transverse frontal plane, in which we will demonstrate this relation, to be at z =0, as shown in Fig. 1. For simplicity we will assume that, even though the nonlinear medium exists only between z 1 <Z <Z2 (see Fig. 1), the linear refractive indices are the same inside and outside the medium. (Corrections for reflection and refraction at dielectric interfaces can be made later.) The incident wave amplitude at a point (4, i7,z) inside the nonlinear medium may then be written simply in terms of its "initial" amplitude at the point (xt,y',0). We assume these two points to be separated by a distance r much larger than a wavelength The Fresenel-Kirchoff diffraction formula gives
ff Ei(4 , e1z,) =- iRe dx'dy'E1£(x',y', 0)Keikr/(2zrr), (4)
where k =n1 w/c, n1 being the refractive index for this wave, and K = (1 + cose)/2, 0 being the angle between the ray along r and the z axis, as shown in Fig. 1. The nonlinear polarization density, which results from (4) substituted in (2), will radiate to give an electric field amplitude Fi at the point (x, y, 0) in its far field:

f f F1 (x, y, 0) =

f 2 dz 21

ddTi XjkE *X(4,7,z) e QS/s.

(5)

Here q =n2 ./c, n2 being the refractive index for the radiated wave at v, and s is the distance between (x, y, 0)

and (4,27,z). The operator Tij takes the transverse

part of the vector source, which is essentially its projection on a plane perpendicular to the incident beam.

After substituting (4) into (5), one finds that the in-
tegral over 4 and 27can be performed nearly exactly
under conditions which are commonly obtained in practice. To see this, we expand the phase function in powers of [the transverse incident-beam coordinates (4, 77)divided by the distance z between the radiating and initial planes]
qs -kr=Akz +l(qp 2 -kp' 2 )/z +(kx' -qx) *4/z

+ 1Akua2 /z + 0[k 4/z 3] .

(6)

Here, Ale =q - k and x, x', and t are two-dimensional vectors, whose coordinates are (x,y), (x',y'), and (4,q7), and whose magnitudes are p, p', and a, respectively. The first two terms in the rhs of (6) are not functions of
4 and e1; the next is the important term. So that we
may neglect the terms in (6) of order ac2 and higher, we will assume that the incident beam is contained inside a circle of radius d in the i, a7plane, and that both of the following conditions are satisfied:

z >> d2Ak

(7)

and

z» >? /2,

(8)

it is seen that, since Ak <le, these conditions require that the nonlinear interaction region be a minimum
distance from the initial plane at z =0, but not so far
as the Fraunhofer diffraction region. In performing
the integral over 4 and 27we will assume that (rs)'1 does not vary with 4 and ejwithin d and replace it by its
average, which we call (Ts)h'. We treat K similarly, replacing it by K. Then, with (7) and (8), the integral
over 4 and y in (5) gives a delta function. When (4) for
E£ is substituted in (5), the (x', y') integral may be performed trivially to yield

Fi(x,y, O) =2vriq2kl1fdz eiAkz4(z)TijXjkE* R x, q Y,0 ),

where 4=Kz2 /s

(9) . We have assumed that the product

GCHI of the amplitudes of the strong, oppositely travel-

ing, pump waves at w0 was independent of 4 and 71at
given z, at least over the area of the incident beam E,.

However, the amplitudes Gi and Hi (and hence, X1 j) may still vary with z and affect the integral in (9).

The important consequence of (9) is that F1 is proportional to E* at the same point (if k =q), or at a nearby point (if kZtq), in space. That is, if kl=q, a timereversed or phase-conjugate wave is generated in this process. When k *q, a magnified and displaced replica of this wave results. There are no phase-matching requirements here, so a quite divergent beam can be
"time reversed. " We proceed to estimate the beam powers that would be necessary to obtain a given efficiency of generation of the time-reversed wave.

III. NUMERICAL EXAMPLE

To estimate the pump-wave powers necessary to produce a desired time-reversed wave, consider the practical case where (1) the incident beam is nearly colinear with the pump beams (0<< 1), (2) k =q, and (3) the pump amplitudes do not vary appreciably in the region of the nonlinear medium where they overlap the incident beam. Then 4-1, Ak =0, Tij - 6ij, and (9) reduces to

Fi(r) =2rikLXi1j E (r).

(10)

where L =z2-az is the thickness of the nonlinear medium, and r is assumed to be far enough in front of the medium so that conditions (7) and (8) are satisfied. From (10) we see that the ratio R of conjugate-wave power to incident-wave power is

R-I= ,, 2L,

(11)

where

= j19672r 2CCiJklffe~jgkhl I -

(12)

Here, fi, eigi,hi are the complex, normalized, polarization vectors of the waves (ei EiE/IE7EI1/2, etc. ), and IG and 1H are the intensities of the forward and backward pump waves. We have assumed that the timereversed wave has an intensity small enough so as not to perturb appreciably the other wave intensities, i. e.,
R <1.

For CS2, one of the most optically nonlinear of

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J. Opt. Soc. Am., Vol. 67, No. 1, January 1977

R. W.Hellwarth

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camera 2 M I

(c) M3

BS l1

camera 1

L

M2

(a)

FIG. 2. Schematic diagram of apparatus described in text for observing generation of a time-reversed wave by nonlinear refraction in medium A.

liquids, cco5 x(-w, - w, ,co) - 2 x 10-'3 esu. 5 At 6943 A this would imply by (12) that 3- 6 X10- 3 cm/MW. Grischkowsky and Armstrong have observed that for w near the 2P1 /2 resonance in rubidium vapor, the c coefficient (and fi) was nearly four orders of magnitude greater, implying J3- 30 cm/MW near 7950 A. 6 With this latter figure as an example, we see from (11) that to obtain 10% conversion to a time- reversed wave (R = 0. 1) in a 10 cm length of medium, the pump inten-
sities required would be 'G - 1H 1 kW/cm2 . The 2 mm diameter dye-laser beam of several kilowatts, which was used in Ref. (6) to observe the nonlinear index, would certainly satisfy this requirement.
In order to demonstrate the generation of a timereversed wave with a single monochromatic laser source, one might employ an arrangement as shown in Fig. 2. A single-mode monochromatic laser source is directed into the nonlinear medium A. The mirror M2 reflects this beam back through the medium creating forward and backward plane-pump waves of nearly the same amplitude. Beam splitter BS1 and mirror Ml direct a portion of the laser source through a phasedistorting plate P and a focusing lens £ into the pumped region of the medium. Cameras 1 and 2 record the incident and backscattered beam patterns at beam splitter BS2. Camera 1 should show the "single-mode" pattern as in inset (a). If the nonlinear medium is removed and a back -reflecting mirror M3 placed at the focal point of lens 2, adistortedpattern, causedby double-passing P,

will be recordedby camera 2 as indicated in inset (b). However, in the presence of the pumped nonlinear medium, camera 2 should record the same single-mode pattern (c) as recorded by camera 1. That is, the backscattered beam is re-formed by plate P to have a smooth phase front and appear to be the time-reversed image of the incident beam.
In an experimental arrangement such as in Fig. 2, the linear refractive index of the medium A may not match the index outside its front surface. In this case, the length L of the medium must be long enough so that condition (8) is met for most of the interaction volume when the comparison plane z = 0 is considered to be just inside the entrance surface of the medium (i. e., z1 =0 in Fig. 1). Then the time-reversed wave front will have become well formed before exiting the medium A and continue back toward lens C as a time-reversed replica (except for the small reflections at the dielectric interface which can be eliminated by coatings).
ACKNOWLEDGMENT
I would like to thank V. Wang for stimulating discussions on phase-conjugation processes.
* Supported by National Science Foundation Grant No. ENG 751.9335.
'B. Y. Zel'dovich, V. I. Popovichev, V. V. Ragul'skii, and F. S. Faisullov, "Connection between the wavefronts of the reflected and exciting light in stimulated Mandel'stam-Brillouinscattering," Sov. Phys. JETP Lett. 15, 109-113 (1972).
20. Y. Nosach, V. I. Popovichev, V. V. Ragul'skii, and F. S. Faisullov, "Cancellation of phase distortions in an amplifying medium with a 'Brillouin mirror,"' Sov. Phys. JETP Lett. 16, 435-438 (1972).
3A. Yariv, "On transmission and recovery of 3-dimensional image information in optical waveguides, " J. Opt. Soc. Am. 66, 301-306 (1976).
4P. W. Maker and R. W. Terhune, "Study of optical effects due to an induced polarization third-order in the electric field strength," Phys. Rev. 137, A801-A818 (1965).
5R. W. Hellwarth, Third-Order Optical Susceptibilities of Liquids and Solids (Pergamon, New York, 1976).
6 D. Grischkowsky and J. A. Armstrong, "Self-defocusing of light by adiabatic following in rubidium vapor," Phys. Rev. Lett. 6, 1566-1570 (1972).

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