OPTICAL PHASE CONJUGATION

Chapter 3 OPTICAL PHASE CONJUGATION Robert W . B o y d The Institute of Optics University of Rochester Rochester, New York and Gilbert Grynberg Laboratoire

1. 2.

3.

4.

5.

de Spectroscopic Hertzienne de l'Ecole Normale Université Pierre et Marie Curie Paris, France

Supérieure

Introduction P h a s e Conjugation by Degenerate and Nearly Degenerate F o u r - W a v e Mixing . . 2.1. Degenerate F o u r - W a v e Mixing 2.2. N o n d e g e n e r a t e F o u r - W a v e Mixing 2.3. Polarization of the Phase-Conjugate W a v e 2.4. Q u a n t u m Properties of P h a s e Conjugation by F o u r - W a v e Mixing . . . . 2.5. Resonance E n h a n c e m e n t of Optical Phase Conjugation P h a s e Conjugation by Stimulated Scattering 3.1. Stimulated Brillouin Scattering (SBS) 3.2. P h a s e Conjugation by SBS 3.3. Brillouin-Enhanced F o u r - W a v e Mixing Photorefractive Phase-Conjugate M i r r o r s 4.1. Photorefractive Effect 4.2. F o u r - W a v e Mixing in Photorefractive Materials 4.3. Photorefractive Phase-Conjugate M i r r o r s Applications of Optical P h a s e Conjugation 5.1. O n e - W a y Imaging t h r o u g h Aberrating Media 5.2. Phase-Conjugate Interferometry 5.3. Phase-Conjugate Resonators and Oscillators 5.4. Spectroscopic Applications of P h a s e Conjugation Acknowledgments References

CONTEMPORARY N O N L I N E A R OPTICS

85

86 87 87 92 94 94 96 101 102 104 104 106 106 107 109 Ill Ill 113 114 115 116 116

Copyright © 1992 by A c a d e m i c Press, Inc. All rights o f reproduction in any form reserved. ISBN 0-12-045135-2

Robert W. Boyd and Gilbert Grynberg

86

1.

INTRODUCTION

Optical phase conjugation is a nonlinear optical interaction that generates what is known as a phase-conjugate wavefront. A device that produces such a wavefront is called a phase-conjugate mirror. The idea of phase conjugation is illustrated in Fig. 1. Here, the incident wave E i n c( r , i ) = R e ( i f i n c^

t o t

)

(1)

falls onto a phase-conjugate mirror, and the wave E p c( r , i ) = R e ( r c * f n c e - n

(2)

which is known as the phase-conjugate wave, is generated by means of a nonlinear optical process. The constant rc appearing in Eq. (2) is known as the phase-conjugate amplitude reflectivity. In the process of reflection from a phase-conjugate mirror, the most advanced portion of the incident wavefront generates the most retarded portion of the phase-conjugate wavefront; this property is the opposite of that which occurs in reflection from an ordinary mirror. One of the principal applications of phase conjugation is the removal of aberrations from optical systems. The aberration correction process is illustrated schematically in Fig. 2. Here, an incident wavefront, which we show as a plane wave, passes through an aberrating medium and becomes distorted. After reflection from a phase-conjugate mirror, the sense of the distortion is Ε

\ \ pc

\

^~\{ Fig. 1.

^

\

Ε

·

I

1

inc

\Λ

PCM

N a t u r e of reflection at a phase-conjugate mirror.

aberrating

I

3 ρ

medium

^

aberrating medium

Fig. 2.

^

L J >

r

>

PCM

PCM

Illustration of the aberration-correction properties of phase conjugation.

3. Optical Phase Conjugation

87

reversed, and after passing a second time through the aberrating medium, the wavefront distortion is removed. The two principal means of generating a phase-conjugate wavefront are through degenerate four-wave mixing [1], which is described in Section 2 of this chapter, and stimulated scattering [2], which is described in Section 3. Section 4 describes methods of producing a phase-conjugate wavefront through use of the photorefractive effect; many photorefractive phaseconjugate mirrors combine some of the features of phase conjugation by four-wave mixing with those of phase conjugation by stimulated scattering. In Section 5, we describe some recent results in applications of optical phase conjugation. 2.

2.1. 2.1.1

P H A S E C O N J U G A T I O N BY DEGENERATE A N D NEARLY DEGENERATE FOUR-WAVE MIXING

Degenerate Four-Wave M i x i n g Introduction

Consider a nonlinear medium interacting with an electromagnetic field. The polarization Ρ of the medium can be split into a term P L linear in the applied field, and a nonlinear contribution P N L . For a Kerr medium, the lowest nonlinear contribution appears in the third order of perturbation theory and 1 can be written a s

PNL = R e ^ Z » M i n c ( r ) | A i n c ( r ) e i n c e- ^ . ( 3

2

(3)

( 3)

where χ is the third-order nonlinear susceptibility and e i n c is the field polarization. We assume now that such a medium interacts with three incident electromagnetic waves of the same frequency and polarization: E f (r,i) = Re[/4 f e"- i ( c o r - k - r ) j e E„(r,t) = R e [ A b e

E (r,f) = R e [ V p

-i(coi + k T ) - |

-i(cof-fcx)

je

(4a) e

(4b) (4c)

The two waves E f and E b propagate in opposite directions (Fig. 3). They will be called p u m p beams in the following. The wave E p is the probe beam. When one ik r ikr ikx replaces Ainc(r) by A{e ' + Ahe~ + Ape in Eq. (3), several terms appear. Among these contributions to P N L , one finds a contribution P c that is an 1

χ

( 3)

(3)

The numerical factor 3/4 that appears in Eq. (3) is a consequence of our definition of χ : (3) 3 is independent of ω (at least in a certain range of frequency), P N L(r, t) is equal to €0χ Ε (τ,

If t).

88

Robert W. Boyd and Gilbert Grynberg

Ρ

' - Α χ=0

χ

-L

Fig. 3. Optical phase conjugation by four-wave mixing. T h e phase-conjugate mirror consists of a nonlinear medium interacting with two c o u n t e r p r o p a g a t i n g p u m p beams Af and Ah.

harmonic function of (œt + kx): 2

Pc = R e [ 2 € 0 n ^ A * e {3)

2

where η = ck/ω and κ0 = 3kx AfAJ4n . tion induces the generation of the field

i ( ö ,i +

fex)

]ez,

(5)

This component of the polariza-

E c(r,i) = R e [ X c ^

I ( w i +

^]ez,

(6)

which propagates in the direction opposite to that of E p . The complex amplitude Ac of the new field being proportional to A*, the wave E c is called the phase-conjugate of E p . 2.1.2

Thin Optical

Medium

The amplitude of the phase-conjugate field is obtained by solving the Maxwell wave equation 2

n A

E

c

2

1

Ô EC =

~ ^ ^

^ c ~

2

a pc 2

^ -

)

We assume that the fields and the polarization are given by Eqs. (4)-(6) and that the typical variation length of the complex amplitudes is very large compared to the wavelength λ = 2n/k. This situation corresponds to the slowly-varying-envelope approximation (SVEA), and permits us to neglect 2 2 a A J ax compared to kdAJdx in Eq. (7). In this case, Eq. (7) becomes ^ = -

i

K

o

A * .

(8)

Equation (8) has to be solved with the boundary condition Ac(l) = 0, since there is no input field entering the nonlinear medium from the right-hand side (see Fig. 3) and propagating in the — χ direction. If the variation of Ap within the medium is very small (i.e., Ap(l) ~ Λ ρ(0)), Eq. (8) yields AG = i(K0l)A*9

(9)

( 7

89

3. Optical Phase Conjugation

where Ac and Ap are the fields in the plane χ = 0. Using Eq. (2), we find that 2 r c = iK0l. The phase-conjugate reflectivity for the intensity is Rc = | r c | , i.e., 2 Rc = \κ01\ for the case of a thin optical medium. 2.1.3 2.1.3.1

Thick Optical

Medium

Undepleted Pump Beams of Equal

Intensity

We define a thick optical medium to be a medium for which the probe field amplitude can be strongly modified during propagation. If Ap(l) is significantly different from Ap(0\ it is no longer possible to use Eq. (9) as a solution for Eq. (8). We also need to consider how the four-wave mixing process modifies the amplitude of the probe wave. When the four fields E f , E b , E p , and E c interact in the medium, the component P c of P N L induced by the mixing of E f , E b , and E p creates E c , and the component P p of P N L induced by the mixing of E f , E b , and E c modifies the propagation of E p . More precisely P p is given by an equation similar to Eq. (5), ikx ikx but with A* and e~ replaced by A* and e , respectively. The Maxwell equations with P p as a source term leads to the following equation for Ap\ dA

(10)

ψ = ίκ0Α*. dx

If we assume that the p u m p beams are so intense that their depletion (and their absorption) can be neglected, Ap and Ac can be found from Eqs. (8) and (10). If we want to have the same unknown quantities Ac and A* in the two equations, we transform Eq. (10) by taking its complex conjugate. This last equation and Eq. (8) form a system of two linear differential equations. This system is solved with the boundary conditions Ac(l) = 0 and A*(0) = A*. The solution is *

s i n | K 0| ( / - x )

0

P

" | K 0| c

C O S | K 0| /

c o s | / c 0 | ( / - x)

αϊ(χ)=αγ*::°ζ~c o s \ K

0

\1

(Hb)

The phase-conjugate amplitude reflectivity rc is obtained from the value of 2 ^4c(x) in the plane χ = 0, and the intensity reflectivity Rc = | r c | is 2

R c = tan (|K 0 IO.

(12)

This formula, obtained independently by Yariv and Pepper [3] and by Bloom and Bjorklund [ 4 ] , shows that, for π / 4 < \κ0\1 < π / 2 , the reflectivity is larger than 100%. Reflectivities much higher than 100% have been experimentally 2 4 observed by several groups [ 5 - 9 ] , and values of Rc of the order of 1 0 ( 1 0 % ) are not uncommon. The possibility of reaching an infinite reflectivity for

90

Robert W. Boyd and Gilbert Grynberg

|fc 0| / = π / 2 is connected with our unphysical assumption of undepleted p u m p waves. Since the energy of the conjugate wave is taken from the p u m p waves, Eq. (12) is valid only as long as \AC\ « \A{\. P u m p depletion has been considered in several papers [ 1 0 - 1 3 ] . One should note that some of these models assume that the probe and the p u m p are copropagating (Θ = 0). In this {3) 2 can modify the propagation case, additional terms of P N L (such as χ Α Α*) of the beams. In fact, in the preceding analysis, we implicitly assumed that the angle θ between k f and k p is sufficiently large for these additional terms to be neglected (because they are not phase-matched). The exact condition under 2 which it is permissible to neglect these terms is 6 kl » 1, which is just the condition for Bragg, and not R a m a n - N a t h , scattering to occur. In the case of degenerate four-wave mixing with beams of the same polarization, the beams can be distinguished only by their directions of propagation, and the angle θ separating the beams should be larger than the divergence angles of the beams (Θ » λ/w, where w is the beam waist radius). If the interaction length / is of the 2 2 2 2 2 order of the Rayleigh length νν /Λ, the quantity /c# /is of the order of θ \ν /λ , which as we have just seen is much greater than unity. Hence, the Bragg condition is automatically fulfilled. (On the other hand, if the beams can be distinguished by their polarization, it is possible to explore the range of small 0, where another theoretical model should be used [14].) Finally, we mention that A{ = Ah is another implicit assumption of the model. This condition ( 3 ) 2 allows us to cancel the relative effect of other terms of P N L (such as χ | A f 1 AP), which modifies the index of each beam. Consider now the transmitted wave. Its value A'p = Ap(l) = tcAp(0) at the exit of the medium is deduced from Eq. ( l i b ) . One can thus calculate the 2 transmission coefficient for the field intensity Tc = \tc\ and find the following relation between Tc and Rc: Tc=l+Rc.

(13)

This relation shows that the difference of intensity between the transmitted and the conjugate wave is constant and equal to the intensity of the incident probe beam. This result suggests that, for each photon emitted in the conjugate wave, a photon is also emitted in the probe wave. We show in Section 2.4 that this interpretation can be justified by a quantum-field description of the process. 2.1.3.2

Extensions

of the Model

In the preceding section, it was assumed that there is no absorption in the Kerr medium. The effects of absorption can be included in our model by adding the terms aRAc and — ocRAp to the right-hand sides of Eqs. (8) and (10), respectively. Here, a R denotes the absorption per unit length for either wave.

3. Optical Phase Conjugation

91

We then find that the intensity reflectivity is given by the expression [ 1 5 ]

/cjsinQc/) 2

R

0

(K0

COS

2

K0l + a R sin κ01)

'

where /c0 = Vl^ol R · When |#c0| > aR, the absorption losses d o not prevent a divergence in Rc, which occurs for tan(jc 0/) = —/c 0 /a R . In the opposite case ( \ K 0 \ < a R ) , the functions of Eq. (14) become hyperbolic functions. Divergence of Rc, which now requires that tanh(jc' 0/) = —/c'0/aR, where K A 2 'o = V R — I κ:01 , can occur only for an inverted medium (a R < 0). It is also possible to extend the theory by taking into account the radial distribution of intensity, the possible imbalance (Af φ Ap) of the p u m p beams [ 1 6 , 1 7 ] , or the fact that the p u m p beams are not precisely counterpropagating. F o r these two last situations, Rc is generally reduced because the phase-matching condition is not exactly fulfilled. We will return to this point in Section 2.2.2. 2—

2.1.4

A

"Time Reversal"

Property of a Phase-Conjugate

Mirror

It has been stated [1] that a phase-conjugate mirror can have an action similar to the time-reversal operation. T o understand this statement, let us consider the wave that is obtained by changing t into its negative in Eq. (4c), that is, E p( r , - i ) = R e [ V

I ( w i + M

]ez.

(15)

We compare this expression with the one obtained from Eqs. (6) and (11a) for the conjugate wave: E c (r,i) = R e ( r c X * e -

i ( e ,f + k

)

* ] e z.

(16)

Since Re(^l) = Re(A*), we note that E c(r, t) and E p(r, — i) coincide (apart from the multiplicative factor r*). The conjugate wave is thus identical to the wave obtained by reversing t to — t in the expression for the probe wave. However, this analysis applies only to monochromatic waves, and cannot be generalized to arbitrary fields. F o r example, r c generally depends on ω. In the case of wavepackets, the conjugate wave is different from the incident wave because the various Fourier components do not have the same reflection coefficient. 2.1.5

Real-Time

Holography and Four-Wave

Mixing

It has been shown [18] that there is a deep analogy between holography and four-wave mixing phase-conjugation. According to this analogy, a hologram is formed by the interference of a reference beam E f and a signal beam E p . The lk r 2 transmission of the hologram is a function of T h = \Afe ' + Ape \ . To reconstruct the image, a second reference wave is sent onto the hologram. We assume that this second wave E b propagates in a direction opposite to E f . The

92

Robert W. Boyd and Gilbert Grynberg

field diffracted from the hologram is proportional to ThAhe (\A{\

2

2

ik r

+ \Ap\ )Ahe '

Hkx 2k r)

+ AhA?Ape - '

I k r

, i.e., to

ikx

+ AfAhA^e~ .

(17)

The first term just corresponds to the transmission of the reference wave Ah, and the second term is associated with a n o n - p h a s e - m a t c h e d emission, which yields a negligible contribution for a thick medium. Finally, the last term (which is perfectly phase-matched) has a mathematical expression identical to that of the phase-conjugate wave. There is thus a profound analogy between holography and four-wave mixing phase conjugation. However, the processes are not totally identical. Holography is a sequential process: One first has to record the hologram, then develop it, and finally illuminate it for constructing the image. In four-wave mixing, all these steps occur simultaneously. The analogy with holography is useful in understanding the physics of the four-wave mixing process. The interference of the p u m p and probe waves E f and E p can create a spatial phase grating in the nonlinear medium. Bragg diffraction of the p u m p beam E b from this grating produces the conjugate wave E c . However, this is not the only process that creates E c . There is also a grating induced by the interference of E b and E p from which E f diffracts. The phase-conjugate emission results from both of these diffraction processes. When the angle θ is small, the two gratings are very different. The grating created by E f and E p has a large period of the order of λ/θ, while the grating created by E b and E p has a small period (of the order of λ/2). When there is no internal motion within the medium, the two gratings make similar contributions to the generation of the phase-conjugate wave. O n the other hand, when the medium consists of moving particles, the small period grating tends to be washed out by the motion, and its influence is often negligible. Finally, one should mention that phase conjugation is not always associated with the formation of a grating. For example, in the case where the nonlinearity arises from a two-photon transition, the conjugate wave is due, at least in part, to the creation of a macroscopic two-photon coherence in the sample [19,20]. 2.2.

Nondegenerate Four-Wave M i x i n g

2.2.1

Phase Conjugation with Frequency

Conversion

We consider now the situation where the incident waves have different frequencies co{, cob and ω ρ . We denote their wave vectors by k f , k b , and k p . If {3) the nonlinear susceptibility χ is a very slowly varying function of ω, if there are no intermediate Raman resonances, and if ω ρ , ω{, and œh have comparable values, the preceding formalism can be applied with minor modifications. In particular, the component of the nonlinear polarization that generates the

93

3. Optical Phase Conjugation phase-conjugated wave is Pc = Κ β ί ί β ο χ ^ ^ , , ^ - ' Κ ^ +^ - ^ - ^ + ^ - ^ - '

1

(18)

} .

The phase-conjugate wave has an angular frequency œc = ω{ + œh — ω ρ . Generally, the phase-matching condition is not fulfilled in the nondegenerate case (i.e., k c / k f -f k b — k p). There are, however, situations where the phasemismatch can be kept very small. Consider, for example, the situation where the forward p u m p beam and the probe beam have the same frequency ω{ while the backward p u m p beam and the conjugate waves have frequency eo b. If θ is the angle between k f and k p , α the angle between k f and — k b , a n d β the angle between k p a n d — k c , the phase-matching condition is fulfilled provided that α = — β and . (θ \ ω{ . θ sin - - α = — s i n - . 2 \2 J œh

(19)

Using this geometry, it is possible to obtain efficient phase-conjugation with frequency conversion [21]. It should however be noticed that if the divergence of the probe beam is important, the phase-matching condition (Eq. (19)) cannot be fulfilled for the whole probe beam. 2.2.2

Phase-Conjugate

Reflectivity

Instead of discussing the general situation, we consider now the most commonly studied nondegenerate four-wave mixing process where the two p u m p beams have the same frequency a n d propagate in opposite directions (ω{ = œh = ω; k f = — k b). If we neglect absorption a n d assume that there is neither depletion nor absorption of the p u m p beams, the propagation of E p and E c along the χ direction (we take k p = kpex) are deduced from equations l(Al +A n d ) x similar to Eqs. (8) and (11), but where κ0 is replaced by K0e . The term (3) 2 2 Δ, is equal to 3/c# (| Ah\ — \A{| )/4, and corresponds to the p u m p imbalance effect mentioned in Section 2.1.3.2. The term And is equal to 2(ω — eo p)/c, and describes a contribution to the phase-mismatch due to the nondegenerate situation. The solution to the differential equations yields

R

=

L

ν

V

2

Q

K

J

J

(2

o)

In the case where the p u m p beams have the same intensity (Δ, = 0) the phaseconjugate mirror can act as a frequency filter [22]. If κ01 > π/4, the components of the probe beam that have a frequency close to ω are reflected

94

Robert W. Boyd and Gilbert Grynberg

with a much higher efficiency than those distant from ω. We note also that the frequency difference ω — ω ρ can be compensated by a difference of intensity of the p u m p beams. When Δ, φ 0, the maximum of reflectivity is obtained when 2 ( Δ η ά + Δι) = 0. In that case, we recover the value t a n ( K - 0 / ) for Rc. 2.3.

Polarization of the Phase-Conjugate Wave

In the preceding, we have considered the simple situation in which all the beams have the same linear polarization. However, in a real experiment one can use a probe beam having any polarization, and it is then important to determine the polarization of the phase-conjugate wave [23]. In particular, it can be useful to have a phase-conjugate mirror that conjugates the polarization of the incident beam as well as its phase (vector phase-conjugation). Most phase-conjugate mirrors do not have this property. Several schemes have been proposed to achieve vector phase conjugation. For instance, third-order nonlinearities associated with two-photon transitions coupling S levels in atoms [23,24] can be used. However, vector phase conjugation is obtained only at third-order, and higher order nonlinearities cause the quality of the polarization conjugation to deteriorate [25,26]. In another method [27], the probe beam is split into two polarization components by a Glan polarizer. Each component is sent onto a phaseconjugate mirror, a half-wave plate being placed in the path of one of the components in order that both beams have the same linear polarization at the phase-conjugate mirror. The difference in optical length between the paths followed by the two components of the probe are compensated by the phaseconjugate mirror, and full polarization conjugation is thus achieved. In addition, recent work [28] has demonstrated experimentally that, for κ01 sufficiently large and in the limit 0 « 1, the four-wave mixing process can display bistability in the state of the output polarization, in agreement with an earlier theoretical prediction [10]. 2.4.

Quantum Properties of Phase Conjugation by Four-Wave M i x i n g

Let us now consider a situation where two probe beams E1 and E2 enter the phase-conjugate medium from the two opposite sides (Fig. 4). The amplitudes A\ and A'2 of the fields at the exit of the phase-conjugate medium are equal to A\ = tGA, + M * ,

A\ = rcA*

+ tcA2.

(21)

Equation (21) can be generalized to q u a n t u m fields. Using the expansion of the field in creation and annihilation operators [29], we find the following equations that relate the annihilation operators of the output fields with the

3. Optical Phase Conjugation

\ A

2

Fig. 4.

95

> A;

>· *

**

2

A

G e o m e t r y of the fields reflected and transmitted by the phase-conjugate mirror.

annihilation and creation operators of the incoming fields: a\ = tcal

+ rca2,

a2 = rca\ + tca2.

(22)

The coefficients r c and tc are ordinary numbers (i.e., not Hilbert space operators) when it is possible to neglect the quantum fluctuations of the phaseconjugate medium (a condition that is assumed to be fulfilled in the following). The introduction of the field entering the rear side of the phase-conjugate mirror is necessary to describe the q u a n t u m properties of the output fields; even if there is no real beam incident on this side, the vacuum is nevertheless present, and induces fluctuations on the phase-conjugate emission. The number of photons N'2 = a'la'2 and N\ = a'lax coming from the two sides of the phase-conjugate medium can be calculated from Eq. (22). Using Eq. (13), we find that N'2-N\

= N2-NX.

(23)

This relation shows that the difference between the number of photons at the exit of the phase-conjugate medium is strictly equal to the difference at the entrance. The generation process is thus associated with a pair emission of photons. For each photon emitted in the mode of the conjugate wave, there is also one photon emitted in the mode of the probe wave. Let us now consider the situation in which the state of the field entering the rear side of the phase-conjugate medium is the vacuum. In this situation, we have = 0. Using the commutation relations between annihilation and creation operators ([α,-,α]] = <5l7), the mean number of photons coming out from the phase-conjugate medium is then = Ä c + Ä c .

(24)

This value differs from the classical result by the constant term Rc on the right-hand side of Eq. (24). Even when there is no probe beam (<Ν\> = 0), the mean number of photons at the exit of the medium is nonzero. If we now consider the variance, we find that 2

( A J V ' 2 ) = ( 1 + Ä C) < A T 2 > .

(25)

Robert W. Boyd and Gilbert Grynberg

96

This shows that the photon statistics on the phase-conjugated beam is superpoissonian [30]. O n the other hand, Eq. (23) shows that Δ(ΛΓ'2 - N\) = A(N2 - N,). If the fields entering the phase-conjugation medium satisfy the condition A(N2 — Ni) = 0, the same relation holds for the field at the exit (this is true, in particular, when the two incoming fields are described by number states). In this case, the statistics of the photon difference is clearly subpoissonian. In the simplest case, where the state of the two incoming fields is the vacuum, we have thus superpoissonian statistics on the intensity of each output field 2 and a subpoissonian statistics on the intensity difference. For two independent quasi-classical fields, the mean square deviation of the intensity difference Δ(Λ^ — N2) is equal to V C ^ i ) + < N 2 > . The fact that A(Nl — N2) is smaller than this quantity can thus be considered as a manifestation of squeezing on the intensity difference. Other types of squeezed fields can also be generated by four-wave mixing [ 3 5 ] . 2.5. 2.5.1

Resonance Enhancement of Optical Phase Conjugation Four-Wave Mixing Emission by Motionless

Atoms

Although four-wave mixing is a very general phenomenon that can be observed in various media, many investigations have utilized the response of an atomic or an excitonic resonance. The motivation for these studies is to take advantage of the huge nonlinearities that exist close to these resonances. The first models that were introduced to describe the frequency dependence of the four-wave mixing efficiency assumed motionless atoms. The starting point of all these models is the solution of the optical Bloch equations. In the case of degenerate four-wave mixing for two-level atoms, one can use the well-known formula that describes the nonlinear response of an atom to a monochromatic field of frequency ω. If we call d the matrix element of the electric dipole moment between the ground level and the excited level, δ = ω — ω0 the frequency detuning from resonance, and Ω χ = \dA\/h the resonance Rabi frequency for an incident field of amplitude A, the mean value

2

Subpoissonian statistics have indeed been observed in an experiment d o n e with the phaseconjugate medium placed inside a ring cavity [ 3 1 ] . Actually, this type of experiment is very similar to those done with optical parametric oscillators [ 3 2 , 3 3 ] . F o r this system, it can be shown that one observes subpoissonian statistics on the difference of intensity between the two o u t p u t beams. F o r each beam, one has superpoissonian statistics near threshold, but the statistics become subpoissonian well above threshold [ 3 4 ] .

3. Optical Phase Conjugation

97

of the atomic polarization Ρ is equal to [36]

P = Rei-δί(5 /^" V,^He. (26) Ω—J lT

2

2

2

+ Γ' +

where Ν is the number of atoms per unit volume, Γ ' is the relaxation rate of the optical coherence, Γ is the radiative width of the excited level ( Γ ' = Γ / 2 for radiative broadening), and ez is the polarization of the incident field. In 2 2 2 the case of a weak incident field ( Ω « (Γ/Γ')(<5 + Γ' )), the linear absorption coefficient a R as deduced from Eq. (26) is _

-2e0h(ô

aR

2

k

Nd r

+ rY

2

2

) ( 2 7

To calculate the four-wave mixing coupling constant f c 0 , we replace A by lk r lkr lkx Afe ' + Ahe~ + Ape , which implies that Ω : is now space-dependent. By expanding Eq. (26) in the weak field limit, we obtain, using Eq. (5), _ k Nd* Γ K

° - e0

Ρ

(δ 2

ΪΓ') 2 2

Γ (δ + Γ )

A f A b

-

)

Comparison of Eqs. (27) and (28) shows that \ κ0\ is always smaller than a R at resonance (δ = 0). There is consequently no divergence of the phase-conjugate reflectivity unless the medium is inverted (see Section 2.1.3.2). O n the other hand, in the dispersive case (\δ\ » Γ'), κ0 can be larger than a R . The study of what occurs at larger p u m p intensity will be briefly described in the next paragraphs; however, the same conclusion holds: \κ0\ can be larger than a R only in the wings of the line. When performing studies of phase conjugation by degenerate four-wave mixing in resonant media, p u m p absorption effects often need to be taken into account [37]. P u m p absorption effects can lead to substantial modification of the optimum values of the laser intensity, detuning, and interaction path length / [38]. The strategy for determining the behavior of the coupling constant at higher values of the p u m p beam intensity consists of generalizing the method developed in the perturbative limit. One first calculates the component of the atomic polarization to first order in Ap and A* (in the usual limit where n e t n en l^pl « l^fl» MbD- O determines the spatial Fourier component of the polarization in the direction of emission of the conjugate wave (to obtain the coupling constant KS) and in the direction of emission of the probe wave (to obtain the absorption coefficient). For example, the value of the coupling constant KS for arbitrary values of the p u m p beam intensity is

( 2 8

98

Robert W. Boyd and Gilbert Grynberg

equal to [ 3 9 - 4 1 ] _ A^El

ê~

((5 -

f~

2

ΐΓ)(Δ

2

IT

Γ' d (

|[ . 4

+

r> L ^ f +

2

+ R )AfAh

M

+

>.ä)]

2

Y'

2

d*

V' 2

- 4 ^ ^ M î }

(29) This formula was first established in the case A{ = Ah by Abrams and Lind [15]. Actually, the case A{ = Ah is very special because the light intensity at the nodes of the standing wave is zero. Thus, even for large p u m p intensities, there are regions in the sample that have a nonsaturated response. In the case where the forward and backward pumps have very different intensities, the medium is almost uniformly saturated. In this situation,/c s is maximum for a detuning from resonance of the order of the resonance Rabi frequency associated with the more intense pump. One can also consider the problem of nondegenerate four-wave mixing [42,43]. In this case, the starting point for the theoretical analysis can no longer be Eq. (26) since the atom does not interact with a monochromatic field. A particularly fruitful method for describing this situation is the dressed-atom approach [44]. Let us consider the situation where the atoms interact with an intense p u m p beam of frequency œh and two weaker beams of frequencies ω{ and ωρ. In the dressed-atom approach, the interaction of the intense p u m p field with a two-level atom leads to an energy spectrum consisting of doublets separated by hcoh. The distance between two levels of the same doublet is 2 2 the Rabi frequency ^Jô + Ω ,, where δ = (œh — ω 0 ) is the detuning from resonance and Q b = \dAh\/h is the resonance Rabi frequency. The resonances of the dressed atom can be probed by the weaker fields. The phase-conjugate emission is particularly intense when all the steps of the four-wave mixing process are resonant in the dressed-atom energy diagram [ 4 5 ] . Examples of fully resonant four-wave mixing emissions are shown in Fig. 5. F o r instance, Fig. 5a describes a resonant emission that occurs when cof = ω ρ = coh — 2 2 Λ/Ο + Ω ,, while Fig. 5e describes another resonance, which appears for 2 2 OF = coh and ω ρ = coh — YJδ + Ω ,. The intensity and the width of these resonances can be calculated using the dressed-atom wavefunctions and the steady-state population of each dressed-atom energy level. These motionless-atom models are very useful for a qualitative analysis of numerous phenomena; however, they often are not quantitatively correct. First of all, when the frequency detuning from resonance and the resonance Rabi frequency are both smaller than the Doppler width, these models generally lead to incorrect results. This point will be clarified in the next section. Secondly, even for frequency detunings that are very large compared with the Doppler width, one should use these models with some caution. In

3. Optical Phase Conjugation (a) _

_

A\ ω

(c)

1

_

_

_

ω

_

99 _

(b)

~*νΨ~

/ I _

^

_

·

^

\ ω

œ T b

_ _ _ _ _ _ _ _ _

"ΊΜ

ω

/

1·

(d)

"

Α\

ρ\ /

| ω

œ Ρ

\

b >

i

"

I J

Μ «τντϊ-

(e)

^

ρΐί /

ω

(f)

^

Κ\

/ I

Fig. 5. Resonant nondegenerate four-wave mixing emission for a two-level a t o m interacting with a saturating p u m p field. The different possible schemes are shown in the dressed-atom 2 energy diagram. T h e energy separation between two levels of the same manifold is HYJΔ + ill.

optical phase conjugation, the various incident beams create a grating in the medium. As an atom moves through this spatial intensity distribution, it feels a time-varying field. If the atom follows adiabatically this variation of intensity, the dipole depends on the local field and the stationary atom model can be used (provided that one neglects the Doppler shift compared to the detuning from resonance). However, collisions create nonadiabatic terms in the nonlinear polarization, and these contributions persist for a time as long as - 1 approximately Γ , which is generally much longer than the time (~λ/ν) over which the intensity seen by the atom varies. In this case, the relation between

Robert W. Boyd and Gilbert Grynberg

100

the dipole and the field depends on the velocity, and one has to average the polarization over the velocity distribution [46]. 2.5.2

Four-Wave Mixing in a Doppler-Broadened

Medium

We consider atoms that interact with three beams whose frequencies are ω(, cob, and ω ρ in the laboratory frame. To calculate the mean dipole moment of an atom of velocity v, one should use the atomic rest frame. In this frame, the incident frequencies are Doppler-shifted and became ω{ — kvx, co b + kvx, and ω ρ — kvx (where oX and ox are the directions of propagation of the pumps and of the probe, respectively, which cross with an angle Θ). Even if the frequencies are degenerate in the laboratory frame (ω{ = œb = ω ρ ) , they appear to be different in the atom's frame. For a Doppler-broadened medium, one should thus perform a calculation of nondegenerate emission for each velocity group and then average over the velocity distribution. Contrary to the case of motionless atoms, there is now a huge difference between the contributions of the two gratings described in Section 2.1.5. Actually, for small values of Ö, the contribution of the grating having a short spatial period is vanishingly small. For a similar reason, the phase-conjugate emission is a very rapidly decreasing function of θ (the motional wash-out becomes more and more efficient when the grating period decreases). M o r e precisely, the intensity -4 of the phase-conjugate emission decreases as ( s i n ö ) when k9u » Γ', where u = yj2kT/m is a measure of the root-mean-squared atomic velocity; thus, ku is the width of the Doppler distribution [47,48]. We now give a few results on the four-wave mixing lineshape when the crossing angle θ is small (kOu « Γ'). First, consider the case of degenerate four-wave mixing with weak fields (\Af I, |>l b|, \AP\ « hT'/d). A calculation made along the lines just described give the following result for the coupling constant [48]:

_J^Nd^ K

°~Se0

h*

_Jn_ (δ - ιΤ') 2

Y(ku) (<5 + r V

.... f

b

*

'

1

By comparison with the result obtained with motionless atoms (Eq. (28)), one sees that, close to resonance, κ0 is here smaller by a factor of the order of Y'/ku. This result is a consequence of the fact that there is only one velocity group 2 that resonantly interacts with the three fields. One can also note that | / c 0 | is a lorentzian of width equal to Γ'. In a thin optical medium (see Eq. (9)), the lineshape of phase-conjugate emission is thus a lorentzian having the homogenous line width. O n e can thus use phase conjugation to obtain Dopplerfree spectra in a sample. Now, consider the case of one saturating p u m p field in degenerate fourwave mixing. Since the emission comes only from the large-period grating, the phase-conjugate efficiency depends upon whether the saturating p u m p field propagates in the same direction as the probe beam (forward saturation) or in

3. Optical Phase Conjugation

101

the opposite direction (backward saturation). In fact, a more intense emission is obtained when the contrast of the grating is larger, i.e., in the case of backward saturation [49]. In this situation (and more precisely when ku » Qh » Γ » Ω Γ , Ω ρ ) the coupling constant can be written €0

h

a

r(ku)Qb

\nbJ

where f(x) is an odd function of χ that tends to 0 as χ oo. (The analytical expression for / can be found in [44,45].) This equation shows that the phase-conjugate emission is zero at resonance, and is maximum for a detuning of the order of the Rabi frequency Qb [50,51]. It can also be noted that κ* depends on δ and Ω,,—only on their ratio ö/Qh. This observation shows that when the backward intensity increases, the only effect is to change the scale on the frequency axis. In particular, the maximum value of the phaseconjugate reflectivity remains constant. Finally, let us compare the coupling constant obtained in the case of backward saturation with the result obtained (for the same intensities) for motionless atoms. Using Eqs. (29) and (31), it is easy to show that the ratio of the maximum values of KS and κ* is of the order of Tku/Ωΐ. When Qb increases, this ratio can become smaller than unity. In contrast to the perturbative case, the degenerate four-wave mixing emission obtained in backward saturation can be larger for moving atoms than for motionless atoms. The origin of this behavior is that the Doppler shift tunes two velocity groups to a fully resonant nondegenerate four-wave mixing process in the atomic frame. Actually, the phase-conjugate emission originates from those two velocity groups that follow the absorption-emission process shown in Figs. 5a and 5b [45]. The relatively large emission observed in this case is thus associated with the energy diagram of the dressed atom. However, this diagram is not really observed on the degenerate four-wave mixing lineshape since, for each detuning δ, there are always two velocity groups that resonantly interact with the incident beams. One can, however, imagine that this spectrum could be attained by using incident beams of different frequency (nondegenerate four-wave mixing in the laboratory frame). This is true, and the dressed-atom energy spectrum can be observed with a natural-width limited precision in the Doppler medium by using nondegenerate four-wave mixing [51]. 3.

P H A S E C O N J U G A T I O N BY S T I M U L A T E D SCATTERING

Another method for generating a phase-conjugate wavefront is through stimulated scattering of light. This method is usually very easy to implement because it simply entails focusing an aberrated beam of light into a material

Robert W. Boyd and Gilbert Grynberg

102

medium in which light scattering can occur. If the power of the beam exceeds some threshold value, stimulated light scattering can occur with good efficiency. Under quite general conditions, and for reasons that are described in what follows, the generated beam is the phase-conjugate of the signal beam. Stimulated Brillouin scattering (SBS) is the stimulated scattering process most often used to generate a phase-conjugate wavefront, and the discussion of this section will assume the case of SBS. However, other processes such as stimulated Rayleigh-wing scattering [52,53] have been used to generate phase-conjugate wavefronts. Stimulated Rayleigh-wing scattering offers the advantages over SBS that (1) it is a very fast process that can form the phaseconjugate of picosecond optical pulses, and (2) it can lead directly to conjugation of the state of polarization of the incident radiation. 3.1.

Stimulated Brillouin Scattering ( S B S )

The process of SBS is depicted in Fig. 6. Here, the laser field £ L( r , f ) = R e [ 4 L e

i ( f c L Z W L i )

"

]

(32)

scatters from a retreating sound wave described by the density disturbance p(r,i) = R e [ p e

i ( i z

i )

i (

-

-" ]

(33)

to generate the scattered, or Stokes, field £ s (r,r) = R e [ / l s é ? "

f c s z W s i )

].

(34)

The frequencies and wavevector amplitudes of these disburbances are related by cos = coL - Ω,

ks = q-kL

(35)

In most phase conjugation experiments, only the laser field is applied externally to the optical medium, and the Stokes and acoustic waves grow from noise [54]. The growth of the Stokes and acoustic waves is a mutually enhancing process, since the Stokes wave is produced by the scattering of the

103

3. Optical Phase Conjugation

laser wave from the acoustic wave, whereas the acoustic wave is produced by the beating together of the laser and Stokes waves through the process of électrostriction. T h e mutual interaction of these waves is described by the following set of equations [ 5 4 , 5 5 ] : dAL

n dAL

dz

c dt

ÔAS

n dAs

~5—·"—^~

Ί ζ - - - ο Ί Γ

dp υ

dp

=

~A

iyeœ 4p0nc

Τ ζ + Έ

2

(36a)

-iyeœ ρ

= ^

Λ

)

^

2

I +

P4s>

i€0yeq Τ

ρ

^

=

Γ

Λ

^

·

)

In these equations, η denotes the refractive index, c the speed of light in vacuum, ye = p(de/dp) the electrostrictive coupling constant, p0 the mean density of the medium, ν the velocity of sound, a n d Γ the p h o n o n intensity decay rate. F o r simplicity, we have ignored the distinction between co L and ω 8 . Hypersonic waves of the sort described by Eq. (36c) tend to be strongly damped, and in most materials the inequality Γ » vq is satisfied. Under these conditions, the first term o n the left-hand side of Eq. (36c), which describes the effects of p h o n o n propagation, can be ignored. If we also assume steady-state conditions, where the time derivatives in Eqs. (36a-c) can be set equal to zero, Eq. (36c) gives an algebraic expression for ρ that can be substituted into Eqs. (36a) and (36b). These equations then straightforwardly give predictions 2 for the spatial variations of the intensities It = ^ - / t ^ 0 c | of the fields, which have the form

^dzΓ = - 0 Λ Λ , dI

* = -gILIs, dz

(37a) (37b)

where 2c 2

9 =

y°

nvc

3—F ρ0Γ

<>iv\

(38)

is the intensity gain factor for SBS. N o t e that Eq. (37b) predicts that the Stokes wave experiences exponential growth in the limit where the laser intensity is spatially invariant. Typically, one finds that the threshold for the occurrence of SBS corresponds to the condition gILL « 25. F o r the case of SBS in methanol at a wavelength of 1.06 μηι, Ω/2π = 2766 M H z , Γ / 2 π = 106 M H z , and g = 0.013 c m / M W . Values for other organic liquids are similar.

104

Robert W. Boyd and Gilbert Grynberg

Fig. 7.

3.2.

Phase conjugation by stimulated Brillouin scattering.

Phase Conjugation by S B S

Zel'dovich et al. [2] discovered that the beam of light generated by SBS can be an accurate phase-conjugate reflection of the incident beam. At first sight, it is not clear why the process of SBS should lead to phase conjugation, since, according to Eqs. (37a, b), SBS can be described solely in terms of the intensities of the two interacting optical waves. The reason why SBS leads to phase conjugation can be understood in terms of the processes that are illustrated in Fig. 7. If the incident beam is badly aberrated, the intensity distribution within the SBS medium will be highly nonuniform. Since, according to Eq. (37a), the amplification experienced for the Stokes wave is proportional to the local value of the laser intensity, a highly nonuniform distribution of Stokes gain is established within the SBS medium. The Stokes wave will be amplified most efficiently if its intensity maxima overlap those of the nonuniform gain distribution. Since the Stokes wave grows from noise, some part of the noise field will have just such an optimum intensity distribution, and this part of the noise field will experience maximum growth. Due to the large amplification of the SBS process (~exp(25) in intensity), the Stokes light leaving the interaction region will be essentially the phase-conjugate of the input wave. This argument can be made quantitative by performing a mode decomposition of the noise field; see, for example, Refs. 2, 56, and 57. 3.3.

Brillouin-Enhanced Four-Wave M i x i n g

The process of Brillouin-enhanced four-wave mixing ( B E F W M ) , which is illustrated in Fig. 8, combines many of the features of degenerate fourwave mixing ( D F W M ) with many of the features of SBS. In this process, the forward-going signal wave at frequency ω 3 = ω2 + Ω beats with the backward-going p u m p wave at frequency ω 2 to drive an acoustic wave that propagates in the forward direction. The forward-going p u m p wave of frequency ωχ scatters from this sound wave to generate the output wave at

3. Optical Phase Conjugation

ω

ι

4

>

Brillouin medium

^

co2

I

ω3=ω2+Ω ωΛ =ω

105

-Ω

1

Fig. 8. G e o m e t r y of Brillouin-enhanced quency of the material medium.

four-wave mixing; Ω denotes the Brillouin fre-

frequency ω 4 = ωχ — Ω. Since this wave is created by a four-wave mixing process, it is the phase-conjugate of the signal wave. However, since the conjugate wave is downshifted from the forward-going p u m p wave by the Brillouin frequency of the medium, it experiences amplification by means of the usual Brillouin gain process. F o r this reason, large phase-conjugate reflectivities and good energy transfer from the forward p u m p wave to the conjugate wave are possible from the B E F W M process. B E F W M can also occur if the signs are reversed in the expressions for ω 3 and ω 4 , that is, if ω 3 = ω2 — Ω and ω 4 = ω χ + Ω. However, in this case the acoustic wave propagates in the backward direction. High phase-conjugate reflectivities can occur in this case also, although the reason is that the signal wave (which drives the F W M process) is amplified as it propagates through the medium. However, good energy transfer to the conjugate wave cannot occur in this case [58]. The mutual coupling among the four interacting waves can be described by the following set of equations [ 5 8 ] : Α Α*

ή Μ Ζ

= ->Wßl4 + 023*' ),

Λ

= -^3(023 +

δ^-' η ^ = Δ

^

= -Λ!(023 +

-^(Öi4 +

Q23e^),

where ß 1 4 = inc€ 0afi4fi4 4,

β 2 3 = inc€0gA2A%.

(39)

Here, g is the usual line-center SBS intensity gain factor given by Eq. (38), and Ak « 2nQ/c is the magnitude of the wavevector-mismatch of the four-wave mixing process. In these equations, Ql4 represents the moving grating formed by the interference of waves 1 and 4; g 2 3 has an analogous interpretation. We see that each of the waves is influenced by two driving terms. The first term in each equation is automatically phase-matched (i.e., is not associated with a phase factor of the sort exp(i Ak z)), and represents the effects of SBS gain or

Robert W. Boyd and Gilbert Grynberg

106

loss. The second term in each equation, which is associated with a phasemismatch factor, describes the process of four-wave mixing. These equations can be solved analytically in the constant p u m p approximation in a manner analogous to that described in Section 2.1. Under more general conditions, they can be solved numerically. Brillouin-enhanced four-wave mixing was first studied by Andreev et al [59] for the case in which the two p u m p waves are at the same frequency (i.e., = ω2) and, consequently, in which the signal and conjugate waves differ ω γ by twice the Brillouin frequency of the medium (e.g., ω 3 = ω1 + Ω, ω 4 = ωλ — Ω). Andreev et al demonstrated phase-conjugate reflectivities as large 6 as 10 . B E F W M was subsequently studied by Skeldon et al. [58] for the case in which the p u m p waves differ by twice the Brillouin frequency of the medium and the probe wave is at the mean of their frequencies. This configuration has the desirable property that there is no frequency shift between the probe and conjugate waves (i.e., ω 3 = ω 4 = ω1 — Ω = ω2 + Ω). Quite recently the noise properties of phase conjugation by B E F W M have been studied by Andreev et al. [60]. These authors have shown that, in a properly optimized system, one can form the phase-conjugate of a signal containing only several photons per mode. The process of B E F W M has recently been reviewed by Scott and Ridley [61]. 4. 4.1.

PHOTOREFRACTIVE PHASE-CONJUGATE

MIRRORS

Photorefractive Effect

The photorefractive effect is a nonlinear optical process that can lead to a large nonlinear response, and requires only milliwatts (or less) of laser power for its use. Response times for the photorefractive effect tend to be rather slow, of the order of 0.1 sec. The photorefractive effect can be explained mathematically by means of a set of equations introduced by Kukhtarev et al. [62]. Here, we shall simply present a qualitative model that explains the nature of the photorefractive effect. The photorefractive effect occurs in crystals that display the linear electrooptic effect and that contain charge carriers (electrons or holes) whose spatial distribution can be modified by the presence of light. The nature of this redistribution is illustrated in Fig. 9 for the case in which the carrier is the electron. Here, an electron, which is initially bound to a donor impurity located at site a, is photoexcited into the conduction band of the material. While in the conduction band, the electron can move through the crystal, due either to thermally-induced diffusion or to drift if a static electric field is present. The electron subsequently recombines with an ionized donor located at site b.

3. Optical Phase Conjugation

107

site a

Fig. 9.

site b

C h a r g e redistribution that leads to the photorefractive effect.

If the intensity I(r) of the light within the crystal is nonuniform, the charge density p(r) within the material will become nonuniform, because electrons will tend to migrate from regions of high intensity to regions of low intensity. This nonuniform charge distribution will produce a nonuniform static electric field E(r) in accordance with Poisson's equation € 0 V . ( E - E ( r ) ) = p(r).

(40)

This static electric field will change the optical properties of the material by means of the linear electro-optic effect. In the scalar approximation, the change in refractive index is given by 3

Δη(Γ) = - | n r e f f£ ( r )

(41)

where η is the mean refractive index and r e ff is the appropriate combination of components of the electro-optic tensor. Let us assume, for example, that two beams of light interfere within the crystals to produce an intensity distribution of the form I(r) = I0 + 7X cos qx. Under many circumstances, the spatial distribution of charge density within the crystal will then be of the form p(r) = px cos qx, where the amplitude ργ of the charge density variation depends on the material properties of the optical medium. According to Eq. (40), the spatial variation of the static electric field will be of the form E(r) = E0 + E1 sin qx and, according to Eq. (41), the change in refractive index will be of the form An(r) = nl sinqx. N o t e that An(r) is shifted in phase with respect to the intensity distribution I(r). This phase shift can lead to the transfer of energy between two beams interacting in a photorefractive crystal. This transfer of energy is often known as two-beam coupling. It plays an important role in the operation of phase-conjugate mirrors that operate by means of the photorefractive effect. 4.2.

Four-Wave M i x i n g in Photorefractive Materials

Fischer et al. [63] (see also C r o n i n - G o l o m b et al [64]), have considered the mutual interaction of four beams of light in a photorefractive crystal, in

108

Robert W. Boyd and Gilbert Grynberg

Λ

^

^3

\ Fig. 10.

\ _

Four-wave mixing in a photorefractive material.

3

the geometry illustrated in Fig. 10. They have shown that, in the approximation where only the grating formed by the interference of beams 1 and 4 and beams 2 and 3 is important, the coupled amplitude equations describing the spatial dependence of the four waves are given by »y

Α Λ

— i = -^{Α,ΑΧ dz /0 dA.^ dz

dA^ ν - ± = -^{Α,ΑΙ dz In

+ A*2A3)At,

y

i

0

dA.^

y

az

10

+

A*2A3)A%, (42)

2

where I0 = Σ * = ι \Aj\ and y is a complex coupling constant that depends on the material properties of the nonlinear medium. F o r those circumstances where the material response can be described by the equations of Kukhtarev et al [62], the coupling constant is given by

2ccosfl 1 +

EO/ESC

where Ε0 = ^

(44) e

is the characteristic field strength of the diffusion process (e being the charge of the mobile carriers), and where E

s

c

Ν e =€ ^ ^

(45)

0*DC
is known as the maximum space charge field. In these equations, q = \k1 — k 4 | is the grating wavevector magnitude, AT t r ap is the density of electron trap centers within the crystal, and 6 d c is the low-frequency dielectric permeability. 3

N o t e that the definition of A3 and AA of Fig. 10 is the opposite of that of Fig. 8. In b o t h cases, we have used the definition used in the original literature ([63] and [ 5 8 ] , respectively).

109

3. Optical Phase Conjugation

If we consider waves 1 and 2 to be the p u m p waves and wave 4 to be the applied signal wave, we see from the equation for dA3/dz in (42) that the generated wave 3 is driven by two contributions. The first contribution, which is proportional to AXA2A%, is a normal four-wave mixing term, and tends to produce an output wave that is the phase-conjugate of the input wave. The 2 second contribution, which is proportional to A3\A2\ , leads to the amplification (or attenuation) of the conjugate wave due to its interaction with p u m p wave 2. This contribution hence is responsible for the process of twobeam coupling. 4.3.

Photorefractive Phase-Conjugate Mirrors

Phase-conjugate mirrors of several different configurations have been constructed that exploit the properties of the photorefractive effect. Several of these designs are illustrated in Fig. 11. The designs are described briefly in what follows. 4.3.1

Externally

Pumped

PCM

Part (a) of Fig. 11 shows the externally pumped P C M . Here, the p u m p waves A1 and A2 are applied externally. The output wave A3 is generated by four-wave mixing processes, and hence is the phase-conjugate of the incident signal wave A4. Externally pumped phase conjugation in photorefractive media has been studied by Huignard et al. [65] and Feinberg and Hellwarth [66]. 4.3.2

(Linear)

Externally

Self-Pumped

In the externally self-pumped P C M , illustrated in part (b) of Fig. 11 for the case of a linear cavity, only the signal wave A4 is applied externally. This wave transfers energy to wave Ax by means of the process of two-beam coupling. The wave Ax is confined by an optical resonator formed by two external mirrors, and grows to an appreciable amplitude. Wave Al and its reflection A2 form counterpropagating p u m p waves that drive a four-wave mixing process, which creates the output conjugate wave A3. The (linear) externally selfpumped P C M was first demonstrated by White et al. [ 6 7 ] . 4.3.3

Internally Self-Pumped

PCM

In the internally self-pumped P C M , illustrated in part (c) of Fig. 11, only the signal wave A4 is applied externally. By means of a sort of self-focusing process, the details of which are not entirely understood at present, two (or more) auxiliary waves Αγ and A2 are created. Reflection of these waves at the corners of the crystal feeds these waves back into the path of the applied wave A4. Four-wave mixing processes driven by the waves Ax and A2, which act as

110

Robert W. Boyd and Gilbert Grynberg

(a)

(b)

(c)

(d)

(e)

Fig. 11. Geometries of several types of photorefractive phase-conjugate mirrors: (a) externally p u m p e d ; (b) (linear) externally self-pumped; (c) internally self-pumped; (d) mutually p u m p e d or double phase-conjugate mirror; (e) stimulated photorefractive scattering.

p u m p waves, and the incident wave A4 create the output, phase-conjugate wave A3. The internally self-pumped P C M was first demonstrated by Feinberg [68] and has been analyzed theoretically by M a c D o n a l d and Feinberg [69]. The stability characteristics of this device have been investigated by Gauthier et al. [70].

3. Optical Phase Conjugation 4.3.4

111

Mutually Pumped (Double) Phase-Conjugate

Mirror

The mutually pumped (or double) phase-conjugate mirror, illustrated in part (d) of Fig. 11, is pumped by two input waves Αγ and A3. These input waves need not be, and preferably are not, mutually phase-coherent. The waves A2 and A4 are both generated by the nonlinear interaction. One can see from the second equation of (42) that, because A2 is created by the scattering of A3 off of the grating A4A%, A2 will be the phase-conjugate of Ax but will be phase-coherent with A3. Likewise, A4 is the phase-conjugate of A3 but is phase-coherent with Αγ. The mutually pumped (double) phase-conjugate mirror is potentially very useful because it can form an amplified phaseconjugate reflection of a signal beam (say beam 1) even though the beam supplying the energy (say beam 3) is incoherent with respect to beam 1. The mutually pumped (double) phase-conjugate mirror was predicted theoretically by Cronin-Golomb et al [64], and was demonstrated experimentally by Weiss et al. [71]. 4.3.5

Stimulated

Photorefractive

Scattering

Part (e) of Fig. 11 shows a P C M that operates by means of stimulated photorefractive scattering. Here, the back-scattered wave grows from noise and is amplified by the process of two-beam coupling. This wave forms the phase-conjugate of the incident wave for the same reason that in SBS the Stokes wave forms the phase-conjugate of the p u m p wave (see Section 3.2). Phase conjugation by stimulated photorefractive scattering has been studied experimentally by Chang and Hellwarth [72].

5. 5.1.

APPLICATIONS OF OPTICAL PHASE CONJUGATION O n e - W a y Imaging through Aberrating M e d i a

Optical phase conjugation is typically used in the standard double-pass configuration shown in Fig. 2 to remove the effects of aberrations from optical systems. In many circumstances, it would be desirable to remove the effects of aberrations from optical systems in which the image-bearing beam passes only once through the optical system. Several of the techniques that have been developed for single-pass aberration correction are illustrated in Fig. 12. In each case, images are transmitted from the left-hand side of the figure to the right-hand side through an intervening aberrating medium. Part (a) of Fig. 12 shows the method of one-way imaging proposed by Yariv and Koch [73], demonstrated by Fischer et al. [74], and subsequently re-analyzed by Feinberg [75]. In this method, plane wave A4 passes to the left through the aberrator where it "samples" the wavefront aberration by

112

Robert W. Boyd and Gilbert Grynberg (a)

, phase aberrator

image-bearing wave 3

plane w a v e plane w a v e observation plane

, phase aberrator

(b) plane w a v e

image-bearing wave

plane w a v e

A

4 observation plane

thick aberrator \
plane w a v e

observation plane Fig. 12.

Three techniques for one-way imaging t h r o u g h an aberrating medium.

acquiring the additional phase φ. This beam then interacts with the plane wave Αγ and the image-bearing beam A2 in the nonlinear medium to generate the {3) ίφ output wave A3, which is proportional to χ ΑίΑ2(Α4β )*. In propagating to the right through the aberrating medium, this wave acquires the additional phase φ, thus rendering a beam at the observation plane that is unaffected by ίφ the wavefront aberration β . This method requires that the aberrator be a thin aberrator so that it can be imaged into the nonlinear mixing crystal. This technique possesses two additional limitations: Light must pass through the aberrating medium in both directions, and the nonlinear processing apparatus and the observation plane must be on opposite sides of the aberrator.

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Part (b) of Fig. 12 shows the method of single-pass aberration correction proposed and demonstrated by M a c D o n a l d et al. [ 7 6 ] . In this technique, light passes through the aberrating medium in one direction only. In addition, this method is passive in the sense that the nonlinear processing and the plane of observation are on the same side of the aberrator. As indicated in the figure, the image bearing beam A4 acquires the additional phase φ in passing through a thin phase abberator. However, a plane wave A1 also passes through the ίφ aberrator and becomes Αχβ . These waves interact with the plane wave A2 in a nonlinear medium that is placed at an image of the aberrator. The nonlinear interaction generates an output wave A3 that is proportional to (3) ίφ ίφ 0) χ (Α^ )Α2(Α^ )* = χ ΑιΑ2Α%; note that this output wave contains the i spatial information carried by beam 4 but is unaffected by the aberration e . Part (c) of Fig. 12 shows the method of single-pass aberration correction proposed and demonstrated by Alley et al. [ 7 7 ] . This technique requires special preparation of the transmitted laser beam, but utilizes only unidirectional transmission and removes the influence of even thick aberrators (thick in the sense that significant reshaping of the transverse beam profile can occur within the aberrating medium). Here, a laser beam is split into two orthogonal polarization components, one of which serves as a p u m p wave (Αγ) and the other of which has spatial information impressed upon it and serves as an image-bearing beam (A4). These beams are rendered collinear and propagate through the aberrator, where each acquires the additional phase φ. At the region of observation, these beams are again split into orthogonal components, and interact with a plane p u m p wave A2 in a nonlinear medium ) ΐφ ίφ ¥ to generate the wave A3 that is proportional to χ° (Α1β )Α2(ΑΑ,β ) 9 which is independent of the wavefront aberration φ. 5.2.

Phase-Conjugate Interferometry

The remarkable properties of phase-conjugate mirrors can be exploited to construct interferometers that possess unique and desirable operating characteristics. In a conventional (i.e., non-phase-conjugate) interferometer used for wavefront sensing, an u n k n o w n wavefront is allowed to interfere with a reference wavefront, and the resulting intensity distribution is used to determine the structure of the unknown wavefront. In contrast, phase-conjugate interferometers possess the desirable property of being selfreferencing: The unknown wavefront is allowed to interfere with a phaseconjugate replica of itself. Consequently, no externally imposed reference wavefront is needed. Phase-conjugate interferometry was proposed by Hopf [78] and by Basov et al. [79], and has subsequently been studied by many workers [ 8 0 - 8 3 ] . A recent development in phase-conjugate interferometry is the phase-conjugate

Robert W. Boyd and Gilbert Grynberg

114 wave front

Fig. 13.

Phase-conjugate interferometer using the Fizeau configuration.

Fizeau interferometer, which was proposed and demonstrated by Gauthier et al. [83], and which is illustrated in Fig. 13. In this design, interference occurs between the reflection from a semitransparent conventional mirror and the reflection from a phase-conjugate mirror placed immediately behind the conventional mirror. This design shares the generic properties of all phaseconjugate interferometers, but is more compact and consequently is less susceptible to environmental disturbances due to the close proximity of the two mirrors. In one implementation of this design, the surface reflection from a single crystal of barium titanate served as the conventional reflection and the wave generated internally within the crystal served as the phase-conjugate reflection. 5.3.

Phase-Conjugate Resonators and Oscillators

A phase-conjugate resonator is a resonator in which one mirror has been replaced by a phase-conjugate mirror (Fig. 14). We consider here the case of a linear resonator. Phase-conjugate resonators can correct the phase perturbations due to imperfect optical elements inside the cavity [ 8 4 - 8 6 ] . More precisely, if the phase-conjugate resonator is transversely unbounded, any wavefront with a phase surface matching the surface of the conventional mirror is self-reproduced after one round trip. If there is gain inside the resonator, the generated wave coming out from the conventional mirror then matches the state of the surface of this mirror. A beam of excellent quality can thus be obtained from this side of the resonator. O n the other hand, the beam coming out from the phase-conjugate mirror has a wavefront distorted by the phase perturbations of the optical elements inside the cavity. Oscillation can be achieved either by inserting a gain medium inside the cavity [84,8,87,88]

115

3. Optical Phase Conjugation

PCM

optical elements Fig. 14. Scheme of a phase-conjugate resonator with a conventional m i r r o r ( C M . ) at one end and a phase-conjugate mirror (P.C.M.) at the other end. Various optical elements m a y be placed between the mirrors.

or by using a phase-conjugate mirror with reflectivity larger than 100% [5-9,89-91]. If phase conjugation is due to four-wave mixing, degenerate oscillation is obtained for any length of the cavity, because the phase accumulated during the propagation in one direction is cancelled by the effect of the phaseconjugate mirror and propagation in the opposite direction. In the case of nondegenerate emission, an incident beam of frequency ω ' is reflected into a beam of frequency 2ω — ω' (where ω is the frequency of the p u m p beams). Thus, two reflections from the phase-conjugate mirror and two round trips inside the cavity are required to recover the initial beam. This explains why the free spectral range of the resonator is c/4L (where L is the length of the resonators) instead of c/2L as for a conventional resonator. Such behavior has been observed experimentally [8]. It is also possible to place the phase-conjugate medium inside a conventional resonator. Because of the four-wave mixing gain, an oscillation can be obtained that exhibits some interesting properties [ 9 2 - 9 4 ] . Applications of phase conjugation to the design of solid-state lasers have recently been reviewed by Rockwell [ 9 5 ] . 5.4.

Spectroscopic Applications of P h a s e Conjugation

As shown in Section 2.5.2, it is possible to obtain Doppler-free spectra by four-wave mixing phase conjugation. By comparison with other techniques used for single-photon transitions, such as saturation spectroscopy [ 3 6 ] , phase conjugation has the advantage that the signal is not superimposed on a large background. O n the other hand, the fact that the angle between the probe and the pumps is nonzero may decrease the interaction length and increase the width of the transition (if the residual Doppler effect is larger than the natural width of the transition). To avoid this problem of angle, one can distinguish the probe from the p u m p by the frequency rather than by the direction. This is the principle of heterodyne spectroscopy [96,97] which is done in a collinear geometry.

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In the case of two-photon transitions, it is also possible to detect the Doppler-free two-photon excitation [98] by optical phase-conjugation [99]. This technique may be particularly useful whenever the detection by fluorescence is difficult (in a discharge for example) or even impossible (if the excited state is metastable). Acknowledgments R. W. Boyd gratefully acknowledges discussions with M. Ewbank and the support of the National Science Foundation and the U.S. Army Research Office. The Laboratoire de Spectroscopic Hertzienne de l'Ecole Normale Supérieure is "associé au Centre National de la Recherche Scientifique." References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

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Robert W. Boyd and Gilbert Grynberg J. P. Huignard, J. P. Herriau, P. Aubourg, and E. Spitz, Opt. Lett. 4, 21 (1979). J. Feinberg a n d R. W. Hellwarth, Opt. Lett. 5, 519 (1980). J. O . White, M. C r o n i n - G o l o m b , B. Fischer, and A. Yariv, Appl. Phys. Lett. 40, 450 (1982). J. Feinberg, Opt. Lett. 7, 486 (1982). K. R. M a c D o n a l d and J. Feinberg, J. Opt. Soc. Am. 73, 548 (1983). D. J. Gauthier, P. N a r u m , and R. W. Boyd, Phys. Rev. Lett. 58, 1640 (1987). S. Weiss, S. Sternklar, and B. Fischer, Opt. Lett. 12, 144 (1987). T. Y. C h a n g and R. W. Hellwarth, Opt. Lett. 10, 408 (1985). A. Yariv and T. L. Koch, Opt. Lett. 7, 113 (1982). B. Fischer, M. C r o n i n - G o l o m b , J. O. White, and A. Yariv, Appl. Phys. Lett. 41, 141 (1982). J. Feinberg, Appl. Phys. Lett. 42, 30 (1983). K. R. M a c D o n a l d , W. R. T o m p k i n , and R. W. Boyd, Opt. Lett. 13, 485 (1988). T.G. Alley, M. A. K r a m e r , D . R. Martinez, and L. P. Schelonka, Opt. Lett. 15, 81 (1990). F. A. Hopf, J. Opt. Soc. Am. 70, 1320 (1980). N . G. Basov, I. G. Zubarev, A. B. Mironov, S. I. Mikhailov, and A. Yu. Odulov, Sov. Phys. JETP 52, 847(1980). I. Bar-Joseph, A. Hardy, Y. Katzir, and Y. Silberberg, Opt. Lett. 6, 414 (1981). J. Feinberg, Opt. Lett. 8, 569 (1983). W. L. Howes, Appl. Opt. 25, 473; ibid. 25, 3167 (1986). D . J. Gauthier, R. W. Boyd, R. K. Jungquist, J. B. Lisson, and L. L. Voci, Opt. Lett. 14, 323 (1989). J. Auyeung, D. Fekete, D . M . Pepper, and A. Yariv, IEEE J. Quantum Electron Q E - 1 5 , 1180 (1979). J. F. Lam, and W. P. Brown, Opt. Lett. 5, 61 (1980). A. E. Siegman, P. A. Belanger, and A. Hardy, in "Optical Phase Conjugation" (R. A. Fisher, ed.), Academic Press, New York, 1983. J. Feinberg and G. D . Bacher, Opt. Lett. 9, 420 (1984). M. C r o n i n - G o l o m b and A. Yariv, Opt. Lett. 11, 455 (1986). J. Feinberg and R. W. Hellwarth, Opt. Lett. 5, 519 (1980). S. O d u l o v and M. Soskin, JETP Lett. 37, 289 (1983). H. Rajbenbach and J. P. Huignard, Opt. Lett. 10, 137 (1985). D . Grandclement, G. Grynberg, and M. Pinard, Phys. Rev. Lett. 59, 44 (1986). D . Grandclement, M . Pinard, and G. Grynberg, IEEE J. Quantum Electron. QE-25, 580 (1989). B. Fischer, S. Sternklar, and S. Weiss, IEEE J. Quantum Electron. QE-25, 550 (1989). D . A. Rockwell, IEEE J. Quantum Electron Q E - 2 4 , 1124 (1988). R. K. Raj, D . Bloch, J. J. Snyder, G. Camy, and M. Ducloy, Phys. Rev. Lett. 44, 1251 (1980). D. Bloch, R. K. Raj, and M. Ducloy, Opt. Commun. 37, 183 (1981). G. G r y n b e r g and B. Cagnac, Rep. Prog. Phys. 40, 791 (1977). D . Bloch, M. Ducloy, and E. Giacobino, J. Phys. Β 14, L819 (1981).