Vol. 3, No. 12/December 1986/J. Opt. Soc. Am. A 2033

Image formation in polychromatic phase conjugation

Goran Manneberg

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

Received March 10, 1986; accepted August 21, 1986

The image-formation process in polychromatic phase conjugation is examined. The image-size reducing (orenlarging) property is discussed, and the spatial-filtering characteristics are developed, showing that a narrow-bandpass filter can be made with this technique. Possible applications involving frequency conversion of imagesfrom the infrared to the visual part of the spectrum are also discussed briefly.

1. INTRODUCTION

Polychromatic phase conjugation (PPC) was recently re-ported as being able to create the conjugate of a Gaussianbeam by means of nondegenerate four-wave mixing.' Asolution of the nonlinear problem of steady-state operationhas also been derived and discussed. 2 The effects of phasemismatching of various origins are presented in this paper,in which the conjugating effects on an image-bearing beamare also treated.

It is shown below that a reduction of image size can beexpected by a factor equal to the ratio of the probe wave-length , to the conjugate wavelength 3 = [(2/X2) - (1/X)]-', where X2 is the wavelength of the pump wave.

2. BACKGROUND

PPC is the result of third-order nonlinear difference-fre-quency generation with a new geometry for noncollinearphase matching, as seen in Fig. 1. The phase-matchingscheme can be seen in Fig. 2.

The input probe can, in the plane-wave approximation, bedescribed by

A1 = E, expi(wlt + klz). (1)

The pump field can be seen as the superposition of twowaves being created by one beam E 2 reflected back and forthbetween dielectrical mirrors surrounding the medium.They are given by

A 2 a = E 2 expfi[w2t + k 2(x cos 0 - z sin 0)]}, (2a)

A 2 b = E 2 expli[w2t - k 2(x cos 0 + z sin 0)11. (2b)

It is implicit in this formulation that the transverse varia-tions of E are assumed to be negligible. In the expression forthe third-order nonlinear polarization in a volume near the zaxis, where all three fields overlap, a term

Pnl = 6OX 3 El*E2aE2b (3)

is found. If phase matched, this polarization will build up afield

A3 = E3 exp[i(w 3t -k3z)],

where w3 = 2W2 - W1.The phase-matching condition in this case is that

(4)

k3 = (2k2 sin 6)u -k1, (5)

where uz is a unit vector in the z direction and 0 is the angleinside the medium between the pump beam and the z axis,given by

n3W3 - n~sin O=

2n 2w 2

(6)

where n1, n2, and n3 are refractive indices of the nonlinearmedium at the corresponding frequencies. It can now beseen from Eq. (5) that A3 is the phase conjugate of Al buthaving a new frequency W3 [k3 is equal to k, with reversedsigns plus a longitudinal component; the amplitude can beseen from Eq. (3) to be proportional to the conjugate of theprobe amplitude]. From Eq. (6) it can be seen that, with apump wavelength around 700 nm, phase matching can beachieved for wavelengths from the middle infrared to justabove half of the pump wavelength, provided that a materialwith high third-order coefficients, which is transparentthroughout this wavelength interval, can be found.

3. IMAGE FORMATION

The principle for image formation is the same as in phaseconjugation with degenerate four-wave mixing (DFWM) de-scribed by numerous authors 3 (see Fig. 3). In the case ofPPC the magnification is no longer one to one, but the size isreduced by a factor X3/X,, as is shown below.

If an object can be understood as being built up by asuperposition of sinusoidal gratings, it is sufficient to consid-er the imaging of such a grating with arbitrary period. Thenature of the imaging process is most easily understood withthe methods of Fourier optics as described, e.g., in Ref. 4.First consider the transformation of the field from one focalplane of a lens with focal length f (at the probe wavelength)to the other. The general expression for the propagationoperator for distance d and transverse coordinate x is

D(xld) = exp(-4irid/X)exp(-i7rx2 Ad). (7)

The operator for a lens of focal length f, and for transversecoordinate x, connects the fields at the two principal planesof the lens and can be expressed as

H(xif) = exp(-47rif/X)exp(-i7rx 2/Xf). (8)

The connecting equation between the object field UO and the

0740-3232/86/122033-05$02.00 © 1986 Optical Society of America

Goran Manneberg

2034 J. Opt. Soc. Am. A/Vol. 3, No. 12/December 1986

d;L4 I

YI I-. Fz ]~I Z ~ii~

*. - a

Fig. 1. The basic configuration for PPC, in which the pump beam E2 enters the material through a slit in the dielectric mirrors surrounding thethird-order nonlinear material. The propagation angle for this beam inside the material is 0. The probe beam El enters at z =L at the right-hand short end, where the conjugate beam E3 leaves the material.

plane coincides with the focal plane of the lens. This obvi-ously supports the validity of the extension of the plane-wave approximation to the case of an image-bearing probebeam that is done below.

By inverting Eq. (9) it can now be seen that if the conju-gate beam is split off by means of a dichroic filter and causedto pass through a lens that has the same focal length for theconjugate wavelength as the first lens had for the probewavelength, a field distribution

Fig. 2. Phase-matching scheme for the noncollinear difference-frequency generation.

IMAGEFig. 3. Geometry for image formation in PPC. The probe beam ispassed through the object OBJ, which is placed in the focal plane ofa lens L, and then focused into the conjugator. The conjugate beamis split off by a dichroic filter DF and led to pass through a lens L',with the same focal length as LI but for the conjugate wavelength.

field in the other focal plane Uf (inside the conjugator) isthen

Uf = [U0 * D(x, f)]H(x, f) * D(x, f), (9)

Where * denotes convolution. It is then assumed that thedistance between the principal planes of the lens can beneglected. If we write the object field as

Al = El(1 + sin 2rsx), (10)

the field inside the conjugator is

Uf = const.[X(x = -s') + d(0) + a(x = s'1)], (11)

from which, among other things, it is clear that the phasefront at the focal plane is flat (although, of course, the lengthof the k vector varies in the transverse directions), which istrue only when the parameters are so chosen that the Fourier

A3 = E 3 (1 + sin2ir Xsx (12)

is obtained in the focal plane of the second lens. It can thenbe argued that, apart from aberrations in the lenses, anobject will yield a phase-conjugate image that is reduced insize by a factor X3/Xl.

If the design of 0 is reversed (i.e., the pump is coupled intothe left-hand part of the device in Fig. 1), it is, of course,possible to produce a conjugate wave that is downconvertedin frequency and thus enlarges the picture.

The so-called healing theorem of phase conjugation, say-ing that imperfections in the lenses are compensated for, is,however, not applicable in this case, because PPC can nevercompensate for the differences in distortion that are due tochromatic dispersion.

The limitations in the imaging quality are due mainly tothe fact that only low-spatial-frequency components areperfectly phase matched. This is dealt with in detail below.Another important consideration is, of course, that the con-jugator be made physically large enough that higher-spatial-frequency components fall inside the conjugator.

Note also that there is a difference between image forma-tion in phase conjugation with DFWM or stimulated Bril-louin scattering on one side and PPC on the other. In thefirst cases the lens in Fig. 3 can preferably be common be-tween the conjugate and the probe beams. Because of thelarge wavelength separation possible in PPC, chromatic dis-persion and differences in absorption in the lens will createproblems that are best dealt with by using separate lenses, ofthe same focal length, for the two wavelengths.

4. EFFECTS OF PHASE MISMATCH

Consider a case in which we have perfect phase matching forzero spatial frequency, i.e., for the case where 0 in Fig. 4 iszero. From Eq. (6) we then have

n 3w3 = nwl + 2n 2w2 sin 0, (13)

from which the square of the phase mismatch Ak can then becalculated:

X

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Goran Manneberg

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Vol. 3, No. 12/December 1986/J. Opt. Soc. Am. A 2035

Fig. 4. A component with high spatial frequency enters the PPC.

k2 = (n3W3) k 2 (14)

The equation now becomes

02 E 3 Q1Q3aq

2 (Ak) 2 (22)

With K = QQ 3/Ak, the general solution, written with theoriginal independent variable, is

E3 = A exp[K exp(-iAkz)] + B exp[-K exp(-iAkz)]. (23)

If we apply the boundary conditions [Eqs. (20)], we find that

E3 = Q3 E10(exp[K exp(-iAkz)] - expl-K[2 - exp(-iAkz)]})

2KAk eXp{K cosh[1 - exp(-iAkz)]}(24)

where

k= 2 (n w, sin X, 0, nlw1 cos 0 + 2n2w2 sin 0)2 (15)

is the square of the wave vector of the nonlinear polarization.With the help of Eq. (13)

Ak2 = 4nln 2 wlw 2 sin 0(1 - cos ¢P) (16)

which immediately yields

Ak = n1n2W1W2 sin 002 (17)

n 3 w3 c

for reasonably small values of p.The analytical solution of the coupled-wave equations in

which not only the amplitude but also k varies in thetransverse directions is intractable, but a good degree ofunderstanding can be achieved by studying the effects ofphase mismatch on each spatial-frequency component sepa-rately.

If the plane-wave approximation is used as described inSection 3, it is possible to write the coupled-wave equationsin the undepleted-pump-wave approximation as

____ = -iQlE3 exp(-iAkz), (18a)

aE3A_ =-iQ 3 E,* exp(-iAkz), (18b)

where Qi are the coupling coefficients defined by

Qi= 6 X(3)(E2)2.Xini

(19)

The standard boundary conditions of a given probe ampli-tude and zero conjugate input are formulated as

El(z = L) = E0 , E3(Z = ) = O (20)

The mathematical treatment of those equations in whatfollows is inspired by that of Ref. 5.

After derivation of Eq. (18b) with respect to z and subse-quent elimination of El*, we change the independent vari-able from z to

?1= exp(-iAkz). (21)

It has to be noted that this substitution degenerates in thecase of full phase matching.

which gives the conjugate amplitude output

E3 (L) = i Q3 E10 tanh{K[1 - exp(-iAkL)]J

and the conjugate intensity

I3 = 2 C ) IE1012

X cosh[2K(cos AkL - 1)] - cos[2K sin(AkL)]cosh[2K(cos AkL - 1)] + cos[2K sin(AkL)]

(25)

(26)

The photon-number reflectivity

wlIE 3 (L)12

P w 3IE,(L) 2

is plotted for different values of phase mismatch and AQQ3L= 7r/4 in Fig. 5 (corresponding to unit reflectivity for Ak = 0)and for AQ 3L = r/2 (corresponding to near oscillation atAk = 0) in Fig. 6. It can be seen that in the former case thephoton reflectance is of the order of unity for AkLI < 2, andin the latter case the reflectivity is also of the order of unityfor IAkLI < 2 but with a very sharply marked peak aroundAk = 0. By a deliberate offset in the phase-matching anglein such a way that perfect phase matching is achieved forslightly higher spatial frequencies, one can obtain a band-pass filter with sharp enhancement of a distinct spatial fre-quency and a reflectivity of unity for surrounding frequen-cies. Note also that the symmetry is such that only the

4

3

2

1

(-1.

.2-

-41

ogRp

I -916 -64 -32 0 3.2 64 6

Fig. 5. Logarithm of photon number reflectance R, as a function ofphase mismatch at an intensity level that corresponds to Rp = 1 atfull phase matching.

Goran Manneberg

i Alsl

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2036 J. Opt. Soc. Am. A/Vol. 3, No. 12/December 1986

log Rp

-i

-3

-4

95 9.4-32 0 32 64 96

Fig. 6. Same as in Fig. 5, but at an intensity level corresponding tooscillation for zero phase mismatch.

magnitude of the k vector is important for the phase match-ing, i.e., the sign and the direction in the transverse plane areinsignificant. It is also important to remember that wespeak about photon reflectivity. If the probe beam is in theinfrared around 5 m and the conjugate beam is in the greenvisible area, a photon reflectivity of 1 corresponds to anintensity reflectance of 10.

5. SPATIAL FILTERING

Two distinct cases have to be considered.

(1) First, we consider the case when the temporal-fre-quency spread of the probe beam is so small that it is unim-portant in comparison with the spread in k12 due to spread inspatial frequency. It is then obvious from Figs. 5 and 6 thatan optimal range of spatial frequencies that can be effective-ly conjugated is achieved when is so chosen that kL = -2for zero spatial frequency. The reflectivity is then of theorder of unity (or in the case of near oscillation much higher)in the interval kL e (-2, 2).

The maximum value of 0 for the wave vector k3 is thengiven by Eq. (17) with (kL)max = 4, which gives

4X1X2n 3 1/2

nl n2X3L2ir sin 6

from which the maximum spatial frequency is obtained:

S. = Xf -ma.

X\If X 1

(27)

vant spatial frequencies. One way to get a rough approxi-mation of the spectral bandwidth necessary to achieve this isto calculate Ak due to a shift in probe frequency Awl (ne-glecting possible shifts in n, because of this). Derivation ofk with respect to wl, remembering that w3 = 2W2 - 3, gives

(n1 + n3 )AwAk = (29)

We then equalize this with k from Eq. (17) and find that

Awl AX, n1 ndX3X1 in (30Ad| | X 1 | n3 (n1 + n 3 )X2 Smax sin0. (30)

With the same wavelengths as above and demanding thatspatial frequencies up to 100 mm-' should be resolved, arelative wavelength spread of 0.03 would be necessary.Note that this is independent of interaction length. Wheth-er this means that a coherent probe source with longitudinalmodes covering that part of the spectrum is necessary orwhether an incoherent source suffices remains to be investi-gated.

A cause of image aberrations in this case is that the size-reduction factor will differ slightly for different spatial-fre-quency components. In what way, and to what extent, thisaffects the imaging also remain to be investigated.

6. APPLICATIONS

There are several applications that concern the detection ofinfrared light, of which I shall briefly mention only a few.

(1) PPC can be used as an infrared image detector, sensi-tive only for a narrow band of wavelengths. This would beuseful if one looks, e.g., for a specific radiation-emittingsubstance. A vibration transition frequency for that sub-stance is then the natural choice of passband for the PPC.

(2) PPC can be used as an infrared image detector forwork with a specific infrared-laser wavelength.

(3) PPC can-possibly-be used as an infrared vidiconfor determining sources of heat. This demands, however,that it be operated close to oscillation, which makes it arather unstable device.

In all these applications the setup should be according toFig. 7. A primary lens L2 images the object into the focalplane of lens L, from which the beam is focused into the

(28)

where xf is a transverse coordinate in the focal plane. For apump wave at 694.3 nm and a probe wave at 5000 nm (yield-ing a conjugate wave at 373 nm) and an interaction length of2 cm, this gives Sm = 2.5 mm-' in the object. For an object 4cm X 4 cm this means 100 X 100 pixels. With future materi-als with higher third-order nonlinear coefficients, this can ofcourse be improved by shortening the interaction length.An interaction length of 0.1 mm would give 1400 X 1400pixels.

(2) Second, we consider the case when the probe wavehas a broad enough spectral bandwidth to have some wave-length that gives maximum conversion efficiency for all rele-

DF L1

E\I H I

OBJ11 L 2

I I

I I

I Fl

Fig. 7. Principal geometry for an infrared-image detector usingPPC. A distant object OBJ forms an intermediate image II througha lens L 2. II is placed in the focal plane of a lens L. Imaging is thendone with PPC through a dichroic filter DF and another lens L' ofthe same focal length at the new wavelength as L for the infrared.

1-

SAlI

-

Goran Marmeberg

I

IL,

Vol. 3, No. 12/December 1986/J. Opt. Soc. Am. A 2037

conjugator, from where it forms an image according to Sec-tion 3 of this paper.

There is also a possible class of applications depending onthe image-size reducing property of PPC. Today this is,however, not an attractive alternative to ordinary opticsbecause of the limitations in spatial frequency, but if theinteraction length could be reduced to micrometers it wouldbe. In all applications, the usefulness of the phenomenon isrestricted by the need for high intensities owing to the lowthird-order nonlinear coefficients of the materials available.The development of nonresonant third-order media wouldfacilitate the methods outlined above.

7. SUMMARY

A theory for image formation in PPC has been developed. Itwas found that a wavelength-converted and size-reduced or-enlarged image was formed. It has also been shown thatthis image could be amplified in intensity. The influence ofphase mismatch resulting from various causes was exam-ined, and a PPC was found to work as a spatial bandpass

filter with resonant amplification for a narrow-frequencyband.

ACKNOWLEDGMENTS

K. Biedermann, N.-E. Molin, and L. Ostlund are sincerelythanked for many valuable comments and suggestions.

REFERENCES

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

2. G. Manneberg, "Polychromatic phase conjugation: a study ofthe nonlinear problem," J. Opt. Soc. Am. B 3, 853-856 (1986).

3. For an extensive list of references, see R. A. Fischer, OpticalPhase Conjugation (Academic, New York, 1983).

4. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill,New York, 1968).

5. D. M. Pepper and R. M. Abrams, "Narrow optical bandpass filtervia nearly degenerate four-wave mixing," Opt. Lett. 3, 212-214(1979).

Goran Manneberg