Vol. 4, No. 3/March 1987/J. Opt. Soc. Am. B 313Goran Manneberg

Synthesis and analysis of short pulses with polychromaticphase conjugation

Goran Manneberg

Department of Physics 11 and Institute of Optical Research, Royal Institute of Technology, S-100 44 Stockholm,

Sweden

Received May 22,1986; accepted November 10, 1986

A new phase-matching geometry for third-order difference-frequency generation called polychromatic pnase

conjugation is demonstrated to have the ability to compress or elongate optical pulses in the picosecond region in

such a way that the geometry is just scaled in time. The coupled wave equations for polychromatic phase

conjugation are examined for the case of pulsed probe waves and pulsed pump waves. A possible method for

experimental verification is also discussed.

1. INTRODUCTION

Phase-matching considerations in phase conjugation by de-generate four-wave mixing1 require that the pump and theprobe waves be at exactly the same frequency. In addition,

the two pump beams must be exactly counterpropagatingand of the same intensity, resulting in a pump field that is a

standing wave. Recently a new phase-matching geometryfor third-order nonlinear difference-frequency generationhas been reported that uses nondegenerate four-wave mix-

ing2 to create the phase conjugate of a spatially Gaussianbeam. In this geometry the wavelengths of the probe beamand pump beam can differ substantially, the requirement ofcounterpropagating pump beams is relaxed, and a phase-conjugate beam at a new (third) wavelength results. This

phenomenon has been called polychromatic phase conjuga-

tion (PPC). A solution of the nonlinear problem of steady-state operation has been reported. 3 The effects of phase

mismatch in image formation have also been discussed. 4

In this paper, the temporal behavior of the conjugate waveis derived by using pulsed pump and probe waves. It is thenshown that PPC can be used for adjustable elongation or

compression of pulses in the picosecond region.

2. BACKGROUND

PPC is the result of third-order nonlinear difference-fre-quency generation. This takes place in a slab of a nonlinearmaterial whose sides are covered with dielectric mirrors, as

illustrated in Fig. 1. The input probe wave, with amplitudeE1 , is introduced at one of the polished short ends of the slab.

In the plane-wave approximation the input probe wave canbe represented as

A1 = E1 expfi(wlt + klz)]. (1)

The pump wave, of amplitude E2 , is introduced through aslit in the dielectric mirrors, along the flat side of the slab.Inside the slab this wave bounces back and forth betweenthe dielectric mirrors. Assuming lossless propagation, afield is created that is composed of two components, one

going upward and one going downward in the slab:

A 2 a = E 2 expli[w 2t + k 2 (x cos 0 - z sin °)]1, (2)

A 2 b = E2 expi[w 2t - k 2 (x cos 0 + z sin 0)11, (3)

where it is assumed that the variations of the amplitude E2

in the x-y directions can be neglected because the slab is

thinner than the length over which the amplitude makessignificant changes. It can be seen from Eqs. (2) and (3)

that the pump field is composed of one standing-wave com-

ponent (in the x-direction) and one traveling-wave compo-nent (in the z direction). The ratio between these is deter-mined by the propagation angle 0 of the pump field. thusdetermines the phase velocity in the z direction (and, ne-glecting chromatic dispersion, also the group velocity). As a

consequence the z and t dependence of E2 is considered to beequal for A2a and A2b-

In a third-order nonlinear medium this field will create anonlinear polarization2

P. = 6eOX3El*E 2aE2b,

which, if phase matched, will build up a field

A 3 = E 3 exp[i(w 3 t - k3z)],

where w3 = 2w 2 - w1.The phase-matching condition can then be seen to be

k3 = (2k2 sin )u -kl,

(4)

(5)

(6)

where u, is a unit vector in the z direction and 0 is the angle

between the pump beam and the symmetry axis (Fig. 2).From Eq. (6) it follows that

n3 w3 - nWsin 0 =2n 2 w2

(7)

where n1 , n2 , and n3 are the linear refractive indices at thecorresponding frequencies, w1, w2, and W3. From Eq. (7) itcan be seen that in an isotropic medium with normal disper-sion, the phase-matching condition can be met as long as the

probe frequency is substantially less than twice the pumpfrequency. For transient applications it is essential to rec-ognize that the pump field A2 can be separated into a stand-ing-wave component in the x direction and a traveling-wave

0740-3224/87/030313-06$02.00 © 1987 Optical Society of America

314 J. Opt. Soc. Am. B/Vol. 4, No. 3/March 1987 G6ran Manneberg

/7. _- a

Fig. 1. A section of the slab in which the probe pulse E1 and the pump pulse E 2 overlap to generate the conjugate pulse E 3.

k2b

probe pulse and in the conjugate pulse. During the timethat this takes place, the generated conjugate photons travelin the same direction as the probe pulse but with a differencein velocity given by (c/n)(1 + sin ). As a result, the pumppulse propagates a shorter geometrical distance than theconjugate wave. With the help of this simple model, it ispossible to conclude that the conjugate pulse will be a tem-poral copy of the probe pulse, apart from the fact that it iscompressed in time. From this model it is alsostraightfoward to calculate the anticipated compression fac-tor to be

t3 n2 - n3 sin 0t1 n2 + n sin 0

(10)

k3

Fig. 2. Phase-matching scheme for PPC [cf. Eq. (6)].

component in the z direction. An interesting feature of thetraveling-wave part is that its phase velocity (and also itsgroup velocity) is multiplied by a factor of sin 0. Thus theactual length of the envelope of the pump wave in the zdirection is reduced by the factor sin 0.

3. CONJUGATE PULSE GENERATION

This section describes conjugate pulse generation when con-jugation is so weak that reconjugation is negligible. Thecondition for negligible reconjugation is that

12rc 3 t2 IE2 1'dt «< 1, (8)VX1X\3

where t2 is a characteristic duration of the pump pulse. Inthe case when 0 is positive, that is, when W3 > wl, the pumppulse and the probe pulse will propagate in different direc-tions. If the pump pulse is short (a few picoseconds) but ofarbitrary shape, it will be further shortened, because of theaforementioned geometrical compression. The situationcan be visualized as a delta function pulse colliding with theprobe pulse. The pulses travel with a difference in velocitygiven by

c cAV=-+ -sinOnl n2

(9)

if calculated in a rest frame.It can then be qualitatively appreciated (and will be math-

ematically shown below) that, as the pump pulse propagatesthrough the probe pulse, it generates new photons in the

In the opposite case, when W3 < wl and therefore is negative(in this case the pump is coupled into the left-hand part ofthe device in Fig. 1), the probe pulse and the pump pulse willpropagate in the same direction but with a difference inspeed equal to (c/n) ( - sin ). The generating (pump)delta pulse will in this case run in the opposite direction ofthe conjugate photons that it creates, and thus the conjugatepulse will be spread out in time compared with that of theprobe. As long as the geometrical extension of the pumppulse in the z direction is small enough that it can be approx-imated as a delta function, the temporal shape of the probepulse (including asymmetrical parts) will be conserved. Byfollowing the same line of reasoning that leads to Eq. (10), itcan be concluded that the factor of elongation is

t3 n2 + n3 sin tl n2 -n, sin (11)

4. MATHEMATICAL METHODThe full system of coupled wave equations describing PPCas given in Ref. 1 is

E1 n 6E1 iwx 3 ) E22E3*

6E 2 n 2 6E 2 iW2X -3 E- + sinO E36z c sin 0 t 2n 2c sin l

6E 3 n 3 E3 _ n cws(3 )1az +c a6t - - 2n 3 C E2

2El*.

(12a)

(12b)

(12c)

In the approximation discussed here, reconjugation is ne-glected, which is equivalent to letting El and E2 be giventhroughout the material by their shape when they enter theboundary; i.e., neither El nor E2 changes substantially as itpropagates through the slab. The mathematical reason fornot including reconjugation is that the standard method5,6

I I i

I ___J -_ I . I-

Vol. 4, No. 3/March 1987/J. Opt. Soc. Am. B 315Goran Manneberg

used for calculating the impulse response includes Fourier orLaplace transformation, which cannot be done in this case

because the right-hand member of the coupled wave equa-tions contains a product between one known function andone unknown function of z and t. This yields a convolutionin the space to which one wants to transform, which makes

elimination between the equations impossible.Also neglected are other nonlinear effects in the medium,

such as self-steepening and self-phase modulation of theprobe wave. This approximation is adequate, since theprobe wave is usually of a much lower peak power than that

of the pump wave. Nonlinear pulse deformations of thepump will not be crucial, as it can be seen from the results

below that the shape of the conjugate wave is largely inde-

pendent of the shape of the pump pulse, provided that it isshort enough. The above approximations and the mathe-matical method used below follow closely those of Ref. 7.

The only remaining equation is now Eq. (12c), in which El is

a function of z + ct/nl and E2 is a function of z - ct sin 0/n 2.

The only necessary boundary condition is that there be noconjugate input, that is,

E 3 (z=O)= O (13)

If for example, both the pump and the probe are of Gaussiantemporal shape, Eq. 12(c) can be written as

3+ 3 E3 = iQ3E2 o2 exp[(ti2 0 + U)2t 2 2l

X E10* exp[-/ 2(t + + v)t_ (14)

The constants u and v are chosen in such a way that the

pulses overlap inside the nonlinear material. Elo and E20

are maximum values of the electrical field of the pump andprobe pulses. The quantity of interest is E3 evaluated at z =a, where a is the active length of the sample. Calculations

are simplified by introducing new coordinates

t = t +-z, 1 = t _-Z' (15)c c

with which Eq. (14) can be simplified to

aE3 [ d -i Q3 exp- (i - )

+ 2 + + u 2t 22 exp[-/ 2 ( + v)2t1-

2]. (16)2~ sin 0

The initial conditions are now given along a line in the new

coordinates

=u, =u, E3 = O. (17)

Standard mathematics can be used to integrate Eq. (16),yielding

E3 = Q exp-{[ 2 (1 sin 0)

+ 2 (1 + 1i ) + u] 2t2 -2}exp[-/2 (t' + v) 2 tl-2]dt'

(18)

which gives E3 for all values of t and -, which, with the

inverse of Eq. (15), easily gives E3 for all values of z and t.

Specifically, the case of interest is the value of E3 at z = a,

which is equivalent to

= t + anC

an' = t -- (19)

An analogous calculation can, of course, be performed for

arbitrary shapes of pump and probe pulses.

5. NUMERICAL RESULTS

The conjugate responses to a variety of input probe waveshave been plotted in Figs. 3 and 4. The length of the nonlin-

ear medium has been assumed to be 4 cm, and the linearrefractive index is 2. All conjugate pulses have been nor-

malized to a peak electric field of unity. In all the figures,the solid line represents the conjugate pulse and the dashedone represents the probe pulse.

The first case treated is when 0 is positive, that is, when

compression of the pulse is expected according to the quali-tative line of reasoning presented in Section 4. From themodel described in Section 3, the compression is expected toimprove with larger 0 according to Eq. (10). It is also ex-

pected that the conjugate pulse will more closely resemblethe probe pulse as the pump pulse becomes shorter. This isbecause the delta function approximation of the pump pulsebecomes more accurate with shorter pump pulses.

Before examining the plots of the pulses, it is also worth-while to point out that in a medium with normal dispersionthe proximity of an absorption edge to the highest involvedfrequency will improve the situation in two ways: first, itwill enhance the nonlinearity, and second, it will help to getlarge phase-matching angles (0) for relatively modest differ-

ences in wavelength between the probe pulse and the conju-gate pulse. To illustrate this point, consider a situation withwavelengths X, = 1500 nm, X2 = 694 nm, and X3 = 450 nm

and refractive indices n1 = 2.4, n2 = 2.6, and n3 = 3.5.Comparable with data for rutile (TiO2 ),8 0 will in this case be

0.96 rad.In Fig. 3(a) it can be seen that with a Gaussian probe pulse

of 200 psec (FWHM understood throughout the rest of thesection) and a Gaussian pump pulse of 10 psec, the compres-sion is modest, with a phase-matching angle of 0.18 rad.

From Fig. 3(b) it can be appreciated that when the pulselengths of both pump and probe are doubled, the interactionlength of 4 cm is not long enough to conjugate the whole

probe pulse, and truncation effects (such as asymmetry) will

occur.Figure 3(c) shows a 200-psec probe pulse and a pump

pulse of 10 psec. The phase-matching angle in this case is

0.8 rad, which gives a more pronounced pulse compression.In Fig. 3(d) the pump pulse is still Gaussian, with a length

of 2 psec. The probe pulse is now changed to an asymmetrictrifunction (as can be seen by the dashed line) with a totallength of 200 psec. The phase-matching angle 0 is now 0.8

rad. It can be seen that the conjugate pulse is a shortenedcopy of the probe pulse, with the asymmetry preserved. Itshould be stressed that pulses of this shape are of courseneither very physical nor compatible with the slowly varyingenvelope approximation (SVEA) used to derive Eqs. (12).Their use here can be thought of merely as illustrative exam-

ples of conjugation of asymmetric pulses (see Appendix B of

Ref. 5 for further discussion of this matter).

316 J. Opt. Soc. Am. B/Vol. 4, No. 3/March 1987 Gbran Manneberg

In Fig. 3(e) it can be seen that the temporal shape of thepump pulse (in this case it is a truncated parabola with alength of 2 psec) has little effect on the shape of the conju-gate pulse. The probe pulse in this case is an asymmetrictrifunction with a total length of 200 psec.

In the case of negative 0, that is, when the pump pulse andthe probe pulse run in the same direction, an elongation ofthe probe pulse is expected. An obvious application for thisis the analysis of the asymmetric parts of short pulses.

In Fig. 4(a) we see the elongation of a Gaussian probepulse (20 psec) with a Gaussian pump pulse (4 psec). As thephase-matching angle is only 0.4 rad, the elongation factor is

E

t = 100 psect2 = 5 psec

-320 -240 -160

(a)

only 2.31, in good agreement with the value of 2.27 obtainedfrom Eq. (11).

With an asymmetric probe pulse of the type shown in Fig.4(b), with a total length of 7.5 psec, a Gaussian pump pulse of4 psec, and 0 = 0.18, the elongated pulse obviously still showsthe asymmetry of the probe. The acquired rounding of thecorners occurs because the pump pulse is no longer muchshorter than the probe pulse and thus cannot be approxi-mated with a delta function any more.

Figure 4(c) shows the elongation of a 100-psec-long asym-metric probe, with a Gaussian pump pulse of 4 psec and =0.8 rad. The resulting pulse is an almost true replica of the

t2 = 1 psec

-160 -80 0(d)

E

t = 200 psect2 = 20 psec

(b)-160 -80 0

(e)160

t = 100 psect2 = 5 psec

-320 160 0(c)

Fig. 3. Plots of the compressed conjugate pulses (solid curves) andthe incoming probe pulses (dashed curves). The curves are dis-cussed in the text.

E

(psec)

Vol. 4, No. 3/March 1987/J. Opt. Soc. Am. B 317

E - input, except for the leading edge, which shows a truncation/ \ \ effect because the length of the "true" elongated pulse would

be longer than what is compatible with the finite length of

./ \ \ t = 10 psec the interaction medium and the speed of the conjugatet2 = 2 psec pulse. This phenomenon can be seen in this case [Fig. 4(c)]

/ \ \ and not in Fig. 4(a) because the elongation factor is muchlarger in Fig. 4(c). With the qualitative reasoning of Section

/ / \ \ ~~~~~~~~3, it can be said that the leading part of the probe pulse/ \ \ enters the medium before the delta-pulse-like pump field/ \ \ does.

6. THEORETICAL EXTENSIONS

There are several directions in which the theory can be(a) developed. First, it is necessary to realize that the approxi-

mation that E2a = E2b, made in connection with Eq. (3), isE-- \ invalid if the width of the waveguide is larger than the

/ \ geometrical extent of the pump pulse. This makes it impos-/ \\ sible to treat pulses in the femtosecond region with this

method. To treat such pulses, the transverse variation of all/ / \ \ four fields must be taken into account.

/ \ \ t2 = 2 psec Further, reconjugation has not been considered in this/ \ \ paper. For moderate pump pulse energies and for most

/ \ \ nonresonant nonlinearities, neglecting reconjugation is avalid approximation. For higher energies and in the case

/ \ \ when either of the involved wavelengths is close to a material/ \ \ resonance, reconjugation must be taken into account.

/ \ \ Finally, for short pulses, the calculations should be donewithout the SVEA. This would, however, double the degree

0 8 ~4 ° 4 8 t (psec) of all the differential equations that must be solved.(b)

E ̂ 4 7. POSSIBILITIES OF EXPERIMENTAL

118 \ VERIFICATION

/ ! l \ The phenomenon can, of course, be experimentally verified/ ! | \ in the manner described in Ref. 2, but the need for long

/ Z \ interaction length and the absence of the need for pumpI = 2 psec pulses of long coherence length (because of the shortness of

the pulses involved) make an integrated form of PPC inter-esting.

A semiplanar waveguide can be made by the methodshown in Fig. 5 on a glass plate with a high nonlinear coeffi-cient (e.g., Schott No. SF 10 or SF 59). Guiding in the y

/ j j \ direction is achieved by ion implantation or ion exchange,0 ; | t t which creates a planar wave guide covering the whole plate.

400 200 0 200 400(c) t (psec) Confinement in the x direction cannot be achieved by any

Fig. 4. Plots of the elongated conjugate pulses (solid curves) and method utilizing total reflection because the incident anglethe incoming probe pulses (dashed curves). The curves are dis- is generally too small. Instead, Bragg reflectors, designedcussed in the text. - for the phase-matching angle to be used, can be made as

f 3Fig. 5. Experimental setup with an integrated semiplanar waveguide. The beam is confined in the x direction with Bragg reflectors and in they direction with an increase in the index of refraction that is due to ion implantation.

Goran Manneberg

318 J. Opt. Soc. Am. B/Vol. 4, No. 3/March 1987

shown in Fig. 5. To avoid problems that may occur duringattempts to introduce different ions into the glass, the re-flectors are preferably made by implanting (or exchanging) ahigher concentration of the ions used to create the planarwave guide.

The probe pulse is then coupled into the waveguide withdirect end-fire coupling with a cylindrical lens system. Be-cause of the difficulties that other-methods present in cou-pling light into the waveguide at the angles required, thepump is coupled by means of a prism coupler.

The conjugate pulse is then separated from the path of theprobe pulse with the help of a dispersive element, such as theprism depicted in Fig. 5.

8. SUMMARY

By means of the approximation of negligible reconjugation,PPC has been shown to create elongation or compression ofoptical pulses while preserving their shape. The factor ofelongation or compression has been shown to be a simplefunction of the phase-matching angle used. Different fac-tors contributing to a reduction in fidelity of the compres-sion or elongation have been discussed.

A possible method for experimental verification with thehelp of an integrated optical waveguide has also been pre-sented.

ACKNOWLEDGMENTS

I thank N.-E. Molin, L. Ostlund, and K. Biedermann sin-cerely for many valuable suggestions. I also want to expressmy gratitude to the reviewers, who in several instances havemade suggestions that improved the paper.

REFERENCES

1. R. A. Fisher, Optical Phase Conjugation (Academic, New York,1983), and references therein.

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

3. G. Manneberg, "Polychromatic phase conjugation, a study of thenonlinear problem," J. Opt. Soc. Am. B 3, 853-856 (1986).

4. G. Manneberg, "Image formation in polychromatic phase conju-gation," J. Opt. Soc. Am. A 3, 2033-2037 (1986).

5. R. A. Fischer, B. R. Suydam, and B. J. Feldman, "Transientanalysis of Kerr-like phase conjugators using frequency domaintechniques," Phys. Rev. A 23, 3071-3083 (1981).

6. R. C. Shockley, "Simplified theory of the impulse response of anoptical degenerate four wave mixing cell," Opt. Commun. 38,221-224 (1981).

7. T. R. O'Meara and A. Yariv, "Time-domain signal processing viafour wave mixing in nonlinear delay lines," Opt. Eng. 21, 237-242(1982).

8. See, for example, K.-H. Hellwege, ed., Landholt-Bornstein Zah-lenwerte und Funktionen aus Physik-Chemie-Astronomie-Geo-physik und Technik, 6th ed. (Springer-Verlag, Berlin, 1962), Vol.2, p. 146.

Goran Manneberg