Vol. 3, No. 6/June 1986/J. Opt. Soc. Am. B 853

Polychromatic phase conjugation: a study of the nonlinearproblem

Goran Manneberg

Department of Physics II, Royal Institute of Technology, S-10044 Stockholm, Sweden

Received October 18, 1985; accepted February 10, 1986

The nonlinear problem of polychromatic phase conjugation is described by one compact differential equation,which is solved numerically. The result verifies the possibility of tunable oscillation and also predicts multistabilityon phase-conjugate output.

INTRODUCTION

Polychromatic phase conjugation, in which the probe beam,the pump beams, and therefore also the conjugate beam areseparated substantially in wavelength, was proposed andexperimentally demonstrated in a recent paper,1 where alinearized theory for steady-state operation is described. Inthis paper the full nonlinear theory of steady-state operationis discussed, and the corresponding equation is solved nu-merically.

The geometry of the experiment is shown in Fig. 1. Aplane-parallel slab of glass is coated, on both faces, withdielectric mirrors. On one of the faces there is a slit in themirror where the pump beam (E2) enters. On the short sideof the probe beam (E;) is injected, and on the same side theconjugate beam (E3) emerges.

The field built up by the probe can, in the plane-waveapproximation, be expressed as

Al = Elexp[i(wlt + klz)]. (1)

When the pump beam is introduced into the material itbounces back and forth between the dielectric mirrors. Ifthe width of the beam is properly adjusted, there will be anarea in the middle of the slab where the fields can be de-scribed by

A2 a = E2 expil[w2t + k2(cos Ox - sin Oz)]j, (2a)

A2b = E2 expli[w 2 t - k2(cos Ox + sin Oz)]), (2b)

where 0 is the angle between the direction of the pump beam(inside the glass) and the normal of the face. I have madethe assumption that the amplitude variations are such thatthe two pump beams can be described by the same (z-depen-dent) amplitude. In the expression for the third-order non-linear polarization there is in this case a term

Pn= 6EoX3A,*A2 .A2b (3)

that will yield a field

A3 = E3 exp[i(w 3 t - k3 z)], (4)

where W3 = 2w 2 - w1, which is phase matched if

k3 = (2k2 sin O)u, - ki, (5)

or, if expressed in z components,

k3 = 2k 2 sin 0 -(hl) (5')

(see Fig. 2). This is fulfilled if

n3w3 - nlwsing 22n2w2

(6)

Such a 0 can be found for all values of W3 up to 2w 2 if materialdispersion is neglected. If a reasonable normal dispersion isintroduced, the maximum W3 is given by

W3,max = 2W2 N + n) (7)

The transverse components of k3 are, as can be seen from Eq.(5), reversed in sign, while the z component is reversed insign and multiplied by a factor larger than one. This meansthat the field A3 is the phase conjugate of Al, but in a newwavelength, which has somewhat different consequencesthan in ordinary phase conjugation caused by degeneratefour-wave mixing or stimulated Brillouin scattering.

Seen from a photon point of view, the process can bethought of as a two-photon absorption from the mode de-scribed by E2 , inducing a transition of an electron to a virtuallevel, followed by a stimulated two-photon emission into themodes described by E1 and E3. This concept makes it possi-ble to write the necessary Manley-Rowe relations.

FORMULATION OF THE PROBLEM

In this paper I will develop the nonlinear theory for poly-chromatic phase conjugation in steady state without theundepleted-pumps approximation. This will make a calcu-lation of the efficiency of the oscillation described briefly inRef. 1 possible. The theory will also predict a bistable ormultistable behavior in the experiment when high pump andprobe intensities are used.

The natural starting point in developing the theory is ofcourse the coupled-wave equations in the slowly varyingenvelope approximation (SVEA), which can be written im-mediately with the help of Eqs. (1)-(4):

* - = iQ1 E22E3*, Q1 = i X(3)) (8)

0740-3224/86/060853-04$02.00 © 1986 Optical Society of America

Goran Manneberg

854 J. Opt. Soc. Am. B/Vol. 3, No. 6/June 1986

.dL Y . z

Fig. 1. Geometry of the experiment. A glass plate is covered withdielectric mirrors. The pump E2enters through a slit in the mirror,and the probe El is injected at one of the short ends.

considered wavelength. Then mi is directly proportional toE, 2

/Q,.

If one now considers an interval on the z axis and assumesthat the number of photons entering at a is m2(a) and m3(a)and that at b is ml(b), this must equal the number of photonsleaving the interval, which yields

m2(a) + m3(a) - ml(a) = m2(b) + m3(b) - ml(b). (15)

As a and b are arbitrary it is possible to conclude that

E 112 1E 2

2 1E 3 2Q1 Q2 Q3 = const. =B.

k2b

(16)

It is also possible to obtain separate relations between onlytwo of the modes, but that will not prove necessary.

In the case of collinear processes the energy-conservationprinciple and the Manley-Rowe equations often yield essen-tially the same relations. Here, however, this is not the case,and one obtains

1E 112 - E212 sinG - E312 = A = const. (17)

Fig. 2. Phase-matching scheme for the process.

SE2 = _iQ2ElE2 *E3, 2 = n2 X sin 6X

from the energy-conservation principle. (Note that the fac-tor sin is included in Q2 in the case of the photon equa-tions.)

It is now straightforward to eliminate El between Eqs.(9) (16) and (7) to yield

5E 3e =-iQ3E,*E2

67rQ3= 7 XMn3 3

with the boundary conditions

El(z = a) = Elo,

E2 (Z = 0) = E20

E3 (Z = ) = -

1E212 = 11E 3

2 + G,

(10) where

F = Q2Q3 - QQ2

Q1 Q3 - Q2 Q3 sin (la)

(llb)

(llc)

SOLUTION

It is interesting to note that the Manley-Rowe relations canbe derived directly from this set of equations, i.e., Maxwell'sequations can be seen as a first quantization of the electro-magnetic field, a possibility that was recognized by Weiss.2To get the first of these equations simply divide Eq. (8) byEq. (10), which yields

d (nlE 12 \ = n3E 3I2 \

W1 W3

G = SA -iBQ-Q - Q2 sin 0

(18)

(19)

El can now be eliminated between the coupled-wave equa-tions as follows: Equation (10) is equivalent to

i 1 6E 3*

Q3 E2* az

which can be derived with respect to z:

6E1 _ 2i 1 6E2* E3* i 1 2E3*6Z Q3 E2 *3 Z 6Z Q3 E2*2 6z 2

(20)

(21)

A shift in the initial phase of E2 corresponds merely to a shiftin the coordinate system in the x direction, and because ofthat I assume in what follows that E2 = E2*-

Equation (21) can now be inserted into Eq. (8), whichgives

(12)

Similar relations can be obtained between El and E2 andbetween E2 and E3 by direct elimination of one of the vari-ables. These relations can be combined to yield

JEll JE21 JE321E11 -E 12 + = const.Q1 Q2 Q3

(13)

The physical meaning of this equation can be visualized withthe help of Fig. 3. First one has to note that the number ofphotons m flowing in the z direction per time and area is

vieoj E~I 2viw EJ2 m i = 1, 2, 3, (14)

where v is the z component of the speed of light at the

A-

Ia b

Fig. 3. The derivation of the photon-number relations and theenergy-conserving equation are explained in the text with the helpof this figure.

'i. I I

Goran Manneberg

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Vol. 3, No. 6/June 1986/J. Opt. Soc. Am. B 855

z2 E2 ( 2 Z Z Q1 Q3E24E3 *. (22)bz2 E2

2\ z z

If Eq. (18) is differentiated, E2 can be eliminated betweenEqs. (18) and (22) to yield

62 E3* 2E3*F (6E3 * \2

1z1 E32 F+_ G = Q1Q3E3*(E3

2 F+ G)2.

(23)

The two constants in the conservation theorems Eqs. (16),and (17)] appear only in the combination A - BQ1, and thevalue of that combination can be derived from Eqs. (b),(11c), and (18):

G = E202 . (24)

NUMERICAL METHOD

As the boundary conditions are given at both boundaries,the method to be used must be iterative and based on sys-tematic guesses.

The values of G and E3 (z = 0) are combined with a guess ofthe value of (6E3/5z)z = 0. From this the function E3(z) canbe obtained by stepwise integration, to yield a value of E3 (z= a). With the help of Eq. (18) E2

2(z = a) is obtainable,which can be used together with Eqs. (lla) and (10) to get(bE 3 /bz)z = a.

This is compared with the value of the same derivativeobtained with stepwise integration. Different possible(6E3/6z)z = 0 are then tested systematically, and in this waythe solution of the equation is found. The program also hasto check that the condition

62E3 E3

o the k3 i

of the SVEA is not violated.

(25)

sin 0 = 0.348. For the input values presented here, threesolutions [(i), (ii), and (iii)] can be seen, which all deplete thepump wave almost totally.

Curve (i) starts off in the same manner as the solutiongiven by the undepleted-pumps approximation but bends asdepletion of the pump gets considerable. Curve (ii) depletesthe pump almost totally in about one tenth of the interactionlength.

It is quite clear that these two solutions to the system ofequations represent two stable states, of which solution (i) isthe one that is reached through slowly increasing the ampli-tudes of the pump and probe (i.e., it is a continuation of thesolution obtainable from the undepleted-pumps approxima-tion). Solution (ii) probably must be reached through atransient in any of the fields. To reveal which, the time-dependent coupled-wave equations must be solved.

The third solution (iii) violates the SVEA at the beginningof the interaction length, and it can therefore not be statedwith certainty whether the solution corresponds to a realphysical situation. There are still more solution curves ob-tainable that also violate SVEA, but these have been omit-ted. What can be stated is, however, that the coupled-waveequations predict at least bistability.

-10

-20

-301

RESULTSAll numerical calculations have been carried out for a 2-cm-long sample of the glass SF59 with a nonlinear coefficientreported 3 as 87 X 10-23 m2/V2, which is the sample describedin Ref. 1. The wavelengths used in the calculations are X =633 nm, X2 = 532 nm, and X3 = 459 nm. This gives a phase-matching angle 0 of 0.175 rad.

In some cases below I shall discuss results that includevery high intensities. In those cases there are, of course,many different scattering processes that have to be takeninto account if a full description is wanted. Here, though, Ido not claim to do that; I only describe what would happen iffour-wave mixing were the only nonlinear process going onin the medium. All energies discussed are, however, wellbelow the limit of critical self-focusing, which is 120TW/m

2 .4

Field Distributions Along the z AxisThe variation of E3 along the z axis was calculated for highprobe and pump intensities. It should be noted here that I2in Fig. 4 is the intensity of the pump beam as it entersthrough the slit. To get the actual flow of energy in the zdirection one has to multiply the given figure by a factor 2 X

-40'

10 15

1,= 6TWim'1,=25TWIm'

Fig. 4. Three field distributions of the conjugated wave along theglass slab for the same input values of pump and probe intensity,indicating multistability.

to

1= 10'TWI m2

10 20 10 4b 5b TWImn2

Fig. 5. Conversion efficiency, as a function of pump intensity, at avery low probe intensity. The ring at the end of one of the solutionsindicates that SVEA is not valid beyond this point.

Goran Manneberg

(mm)

856 J. Opt. Soc. Am. B/Vol. 3, No. 6/June 1986

1215 TWIM2

Fig. 6. Conversion efficiency, as a function of probe intensity, forhigh probe intensities.

Conversion Efficiency at OscillationThe conversion efficiency

n2 E112 + 2 J2

IE sin3 (26)

was calculated for different pump energies and a low probeenergy, corresponding to a reasonable level of noise. Theresult, shown in Fig. 5, seems to support the prediction oftunable oscillation. It also opens up the possibility of con-structing a sensitive phase-conjugate image amplifier, con-verting an infrared image to visible light, and amplifying it.

At the pump intensity 2 = 13.8 TW/m2 the conversionefficiency rises abruptly for the fundamental solution.There are also other branches of solution, of which only onedoes not contradict the SVEA. The behavior of that solu-tion over 10 TW/m2 is quite clear and shows that even underthe limit of oscillation for the fundamental solution anotherstable state with a conversion efficiency of about 0.75 exists.Below 10 TW/m2 the field as a function of z violates theSVEA, and the behavior of that branch can be calculated

only with the solution of the coupled-wave equations with-out that approximation.

The Conversion Efficiency for High Probe and PumpEnergiesThe conversion efficiencies, defined in the same way as inthe subsection above, have been calculated for a fixed pumpenergy and varied, high probe intensities (Fig. 6). The bi-stable (or possibly multistable) behavior of the solution ismaintained throughout the different probe intensities.

SUMMARY

The coupled-wave equations for polychromatic phase conju-gation have been solved for the steady state, in the SVEA,but without linearization. The solution shows multistabil-ity and confirms the earlier prediction of tunable oscillation.The necessity for solutions not relying on SVEA was alsopointed out.

ACKNOWLEDGMENTS

M. Breidne is sincerely thanked for valuable suggestions andcomments. K. Biedermann and L. Ostlund are thanked forhelp in many different respects. N. Abramson is thankedfor valuable suggestions.

REFRENCES

1. G. Manneberg, "Polychromatic phase conjugation with noncol-linearly phase-matched difference-frequency generation," J.Opt. Soc. Am. B 3, 849-852 (1986).

2. M. T. Weiss, "Quantum derivation of energy relations analogousto those for nonlinear reactances," Proc. Inst. Radio Eng. 45,1012-1013 (1957).

3. M. Thalhammer and A. Penzkoefer, "Measurements of thirdorder nonlinear susceptibilities by non phase matched third har-monic generation," Appl. Phys. B 32, 137-143 (1983).

4. Calculated according to J. F. Reintjes: Nonlinear Optical Para-metric Processes in Liquids and Gases (Academic, London,1984), pp. 331-334.

Goran Manneberg

l