all-optical switching using three-wave-interaction solitons

Edem Ibragimov Vol. 15, No. 1 /January 1998 /J. Opt. Soc. Am. B 97

All-optical switching using three-wave-interactionsolitons

Edem Ibragimov

Department of Mathematical Sciences, Michigan Technological University, Houghton, Michigan, 49931-1295

Received July 21, 1997

The property of three-wave-interaction solitons to preserve their shape upon nonlinear interaction with anarbitrary-shaped radiation is proposed for the creation of all-optical switching devices. It is shown that three-wave-interaction solitons can be used for optical switching in a polarization-gate geometry. This new all-optical logic gate combines the advantages of soliton switching devices with the short length of a second-ordernonlinear interaction. Feasibility of an all-optical logic gate based on two 1.5-cm-long b-barium borate crys-tals and with the intensity of the control pulse 190 MW/cm2 is demonstrated. © 1998 Optical Society ofAmerica [S0740-3224(98)03101-4]

OCIS codes: 060.1810, 190.5530, 190.4360, 190.4410, 190.7710, 060.2330.

1. INTRODUCTIONOptical switching represents one of the major technologi-cal challenges in optics and photonics research. Espe-cially attractive is an idea of using solitons for all-opticalswitching devices. Unique properties of solitons such aselastic collisions and soliton dragging and trapping makethem particularly advantageous for time-domain terabit-rate switching with picosecond or femtosecond pulses.1

The major drawback of all-optical switching devices basedon cubic (x (3)) nonlinearity is the long fiber lengths thatare required by extremely low nonlinear coefficients. In-ability to find lossless nonlinear materials with large non-resonant third-order nonlinearities has led to more de-tailed investigation of the processes of the second order.Recent experiments have demonstrated that nonlinearmaterials with the second-order (x (2)) nonlinearities canbe used not only for harmonic and parametric generationbut also in applications to all-optical switching for whichnonlinear phase distortions are required: spatial soliton-like waves on cascaded x (2) 3 x (2) nonlinearity permit theimplementation of second-order processes to the all-optical switching.2,3 However, in the temporal domain,solitonlike waves on x (2) 3 x (2) nonlinearity are difficultto achieve because of low second-order dispersion in mostnonlinear materials.

A new type of soliton in the x (2) medium was recentlyobserved in an experiment with a synchronously pumpedoptical parametric oscillator.4 Applications of these soli-tons to the second-harmonic and sum-frequency genera-tion were recently considered in theoretical works.5,6

This type of soliton, termed three-wave-interaction (TWI)solitons in Ref. 6, appears in the media with quadraticnonlinearity and can be described analytically by the soli-ton solutions to the TWI equations, discovered 20 yearsago by Zakharov and Manakov7 and by Kaup.8 In Ref. 6these solutions were extended to the case with nonzerophase mismatch, and the potential applications of TWIsolitons to the problem of all-optical switching were dis-cussed.

0740-3224/98/010097-06$10.00 ©

The works5,6 analyze the exact analytical solutions ofTWI equations for the simplest case: interaction of twofirst-order TWI soliton waves. During the interactionthese two waves at the frequencies v1 and v2 generate athird wave at the frequency v1 1 v2 that can carry a sub-stantial part of the entire energy. Because the initialwaves were TWI solitons, at the end of the interaction thesum-frequency wave dies out, leaving two fundamentalswith the shapes and intensities equal to their initialshapes and intensities. Although such analytical solu-tions are a valuable tool for the investigation of TWI pro-cesses and important for the understanding of the processof TWI-soliton interaction, some of the most remarkableproperties of TWI solitons are not reflected by thesesimple two-pole TWI-soliton solutions. Unlike soliton-like waves on cascaded x (2) 3 x (2) nonlinearity, TWI soli-tons are not a combination of three waves at different fre-quencies periodically exchanging energies: each of thetwo fundamental waves can be a TWI soliton, indepen-dent of the parameters two other waves. In the generalcase the TWI soliton is the wave at the fundamental fre-quency (v1 or v2) that asymptotically preserves its shapeupon the interaction with another (arbitrary-shaped)wave at the fundamental frequency.

2. THREE-WAVE-INTERACTIONEQUATIONSThe TWI equations

]A1

]z1

1

v1

]A1

]t1 ig1

]2A1

]t2 5 isA3 A2* exp~iDkz !,

]A2

]z1

1

v2

]A2

]t1 ig2

]2A2

]t2 5 iv2

v1sA3 A1* exp~iDkz !,

]A3

]z1

1

v3

]A3

]t1 ig3

]2A3

]t2 5 iv3

v1sA1A2 exp~2iDkz !,

(1)

1998 Optical Society of America

98 J. Opt. Soc. Am. B/Vol. 15, No. 1 /January 1998 Edem Ibragimov

describe the interaction in a dispersive medium of threepulses with the following parameters: associated electricfields Ej(t,z) 5 Aj(t,z)exp@i(vj 2 kj z)# for j 5 1, 2, 3;center frequencies v j ; group velocities vj ; second-orderdispersion coefficients gj ; wave numbers kj 5 njv j /c,where c is the speed of light and nj is the refractive index;nonlinear coupling constants s 5 2pixnlv1

2/k1c2, wherexnl is the nonlinear dielectric susceptibility; and phase-velocity mismatch Dk 5 k3 2 k1 2 k2 . The pulses con-sidered are sufficiently long that second-order dispersioneffects are negligible. Everywhere in this paper, g15 g2 5 g3 5 0: this assumption was examined inRef. 6.

A consideration of Eq. (1) on the basis of the inversescattering transform is given in Refs. 7–9. In these stud-ies, attention was restricted to the perfectly phase-matched case with Dk 5 0. In Ref. 6 the results ob-tained in Refs. 7–9 for the zero phase mismatch wereextended to the case Dk Þ 0 by applying a simple phasetransform. The basic soliton features of the waves do notchange under this transformation. Therefore every-where in this paper we consider such generalized solu-tions of the system (1) with Dk Þ 0.

Mathematical consideration in Refs. 7 and 8 shows thateach of two fundamental waves in the TWI can containTWI solitons if the group velocity v3 of the wave with thehighest frequency is between the group velocities of fun-damentals: v1 . v3 . v2 . Each TWI soliton corre-sponds to the zero of a diagonal element of the 3 3 3 scat-

tering matrix. Such matrix elements do not change intime. Therefore the TWI solitons possess the qualitycommon for all type of solitons: they asymptotically pre-serve their shape upon interaction with another(arbitrary-shaped) wave at the fundamental frequency.The shape of the soliton wave before (and after) the inter-action (z → 7` and t → 7`) is given by the equation6

Aj~t, z ! 5 Ai,0 sechFAv2v3sAj,0

v1An1,2n j,3

S t 2z

vjD G ; (2)

here z is the propagation coordinate, t is time, j 5 1 cor-responds to the first and j 5 2 to the second fundamentalwave, Aj,0 is the initial amplitude of the jth wave, andn i, j 5 1/vi 2 1/vj , where vj is the group velocity of thejth wave.

The explicit analytical solutions of Eqs. (1) for the casein which both of the fundamental waves are TWI solitonsof the first order are analyzed in Ref. 6. The analogoussolutions for the case when only one of the fundamentalshas a soliton shape become very complicated and in gen-eral are impossible to obtain. Nevertheless, an analyti-cal consideration in Refs. 7 and 8 guarantees that, afterinteraction with an arbitrary-shaped radiation, the fun-damental wave with the soliton shape (2) will remain un-changed. An illustration of this fact is given in Fig. 1,where a small TWI soliton at the frequency v1 interactswith the intense pulse at the frequency v2 . During theinteraction these two pulses generate a third wave with

Fig. 1. Elastic collision of the small TWI-soliton pulse at frequency v1 with an intense pulse at frequency v2 . Time (horizontal axis)is given in picoseconds; the amplitudes of the waves (vertical axis) are in arbitrary units. Pictures show temporal shapes of the inter-acting pulses for different values of z. The profiles shown in the frame are moving with the speed of the wave at sum-frequency, whichis not shown in the figure.


Fig. 2. Elastic collision of the TWI soliton with a pulse with a complicated shape.

the frequency v1 1 v2 , which is not shown in the picture.Intuitively, one can expect that the intense pulse will un-dergo little changes after collision with the small pulseand that the shape of the least intensive wave is morelikely to be destroyed by the interaction. As can be seenfrom Fig. 1, in the course of interaction both waves com-pletely change their shapes. However, after the pulsesseparate, the small soliton pulse remains unchanged incontrast to the intense nonsoliton pulse whose initialshape is completely destroyed during the nonlinear inter-action. Another example of the collision of the solitonpulse and an intense pulse with a complicated shape isshown in Fig. 2. In both pictures one can see another in-teresting feature of the soliton interaction: an appear-ance of a small postpulse at the v2 frequency that exactlyrepeats the shape of the soliton pulse at v1 frequency.This is a reflection of the fact that a TWI-soliton wave in-teracts with its twin inside the intense pulse. Owing tothe temporal soliton shift, the twin pulse at v2 is retardedwith respect to the rest of the radiation at this frequencyand emerges in the form of a separated pulse with theshape and intensity of the initial soliton wave. The smallpedestal that the soliton pulses obtain in both figures af-ter the interaction is caused by the pulses overlapping atz 5 0. If, in the beginning of the interaction, pulses werecompletely separated, the pedestals would not appear.

3. THREE-WAVE-INTERACTION SOLITONSWITCHTWI solitons allow one to overcome most of the challengesin implementing devices that rely on all-optical interac-tions. Strong x (2) nonlinearity allows one to greatly re-duce the size of the device, compared with the x (3) case,and to use low-intensity sources. Because the outputTWI-soliton pulse has exactly the same shape, frequency,signal level, and duration as an input pulse, the devices

on TWI solitons satisfy the requirement of cascadabilityin the sense that the logic output of one gate can be usedto drive the next gate. The behavior of the system is in-dependent of the initial phases between the interactingwaves.

In this paper a simple TWI-soliton switching schemebased on the change in polarization state and analogousto that used by Morioka et al.1,10 is considered. A second-order nonsoliton optical switching in a similar schemewas recently demonstrated in an experiment.11 Asshown in the present paper, TWI solitons allow the usecollinear interaction instead of noncollinear. This makesit possible to use it in fibers, to increase the efficiency ofthe process, and to satisfy the cascadability requirement.TWI solitons do not require equal frequencies of the fun-damental waves, and we can make the energy of the con-trol pulse theoretically as small as we want by adjustingthe frequencies v j and using dispersion of the coefficientsn i, j . In addition, a soliton theory allows one to investi-gate the process analytically and to indicate possible waysfor an optimization.

The configuration for the TWI switch is shown in Fig. 3,where a signal at frequency v2 is gated by a control pulseat v1 . A b-barium borate (BBO) crystal with a x (2) non-linearity and type II oe→e interaction is substituted for athird-order optical Kerr medium used in Ref. 10. Thecontrol pulse is polarized along the e-axis and enters thenonlinear crystal behind the signal that is polarized at45° to the same axis. During the propagation the controle-pulse at the frequency v1 overtakes the o-pulse at thefrequency v2 and they produce the e-wave at the sum fre-quency: v3 5 v1 1 v1 . If the interacting pulses areTWI solitons, in the end of the process the generated SFwave will decay to zero and the two fundamental pulseswill regain their initial shapes and durations, as shown inFig. 4. In the presence of the control pulse the signalwave experiences a phase shift Df2 . (In Fig. 4 the phase


shift Df2 5 p). The wave plates are adjusted so that thepolarizer blocks the signal in the absence of the pump,and the pump increases the probe transmission throughthe system. The power transmission trough the polar-izer is proportional to sin2(Df2/2), where Df2 is a phaseshift caused by the soliton interaction. As was shown inRef. 6, the phase shifts of the fundamental waves causedby the soliton interaction are determined by the followingexpression:

Df j 5 c1 6 c2 ,

c j 5 arctanFsAw2w3~A2,0An2,3 6 A1,0An3,1!

Dkv1An1,2G ;

(3)

here ‘‘1’’ is chosen for index 1 and ‘‘2’’ is for index 2, kjare the wave numbers, and Dk 5 k3 2 k1 2 k2 is thephase-velocity mismatch. The pulses considered are suf-ficiently long that second-order dispersion effects are neg-ligible: this assumption is examined in Ref. 6.

When Dk tends to zero, the denominator in Eq. (3)tends to infinity and c1,2 5 6p/2. In this case one of thefundamentals experiences a phase shift equal to p andthe phase of the other wave does not change. The sign ofc2 determines which of the waves will obtain the phaseshift: if c2 is positive, then according to Eq. (3) the con-

Fig. 3. Schematic of a TWI-soliton logical gate. An opticalpulse at v1 changes the polarization state of a signal at v2 in thecourse of a three-wave quadratic interaction.

Fig. 4. Dynamic of the TWI-soliton interaction along the crystal(intensity versus time). The solid curve is the generated SFwave.

trol wave at v1 obtains the phase shift; if c2 is negative,then the phase shift obtains the signal wave at v2 . Inthe considered scheme the second (signal) wave has to ob-tain a large (;p) phase shift; therefore the condition

ib 5 A2,0An2,3 2 A1,0An3,1 , 0 (4)

has to be satisfied. Here i 5 A21, and b in the caseDk 5 0 is a pure imaginary parameter from Ref. 6. Theparameter b has a simple physical meaning.6 It showsthe difference between the photon fluxes of the first andthe second waves into the interaction zone;12 b 5 0 pro-vides equal photon flux of the first and the second wavesinto the interaction zone. Correspondingly, when ib, 0, there are more photons of the first wave in the in-teraction zone than those of the second. Therefore thecondition ib , 0 leads first to the depletion of the signalwave and then to its recreation with the opposite phase.

In the considered scheme both pulses do not have to beTWI solitons. TWI solitons regain their shape after col-lision with arbitrary-shaped waves; therefore it is usuallyenough if only one of the interacting waves is a TWI soli-ton. If the signal pulse is a TWI soliton, the control pulsecan have an arbitrary shape as long as Eq. (4) is satisfied.In this case the intensity (energy) of the control pulse canbe even smaller than the intensity (energy) of the signal:the relationship is determined by the ratio n2,3 /n3,1 .

If the signal wave is not a TWI soliton, then the controlpulse must be a TWI soliton. In this case its energyshould be minimized. The full width t on the half-maximum of intensity (FWHM) of the TWI-soliton pulsewith the amplitude A1,0 [Eq. (2)] is determined by the foll-wing equation:

t1 51.76v1An1,2n3,1

sA1,0Av2v3

. (5)

The duration of a TWI soliton is inversely proportionalto its amplitude, which means that all solitons have equalareas under the amplitude curve and the shorter solitonshave greater energy. The energy of the TWI-soliton con-trol pulse is E 5 It1 , where I is the intensity. As it fol-lows from Eq. (5), there are three ways to decrease the en-ergy of the control pulse: (1) Minimize the ratiov1 /Av2v3, (2) choose the medium with larger nonlinear-ity, and (3) decrease the parameter An1,2n3,1. However,the parameter n1,2 should not be too small because it willlead to an increase of the interaction length. If the signaland the control pulses have approximately the same du-ration, the length L of the nonlinear crystal increaseswhen n1,2 is small: L ' t/n1,2 . Therefore the energy ofthe TWI-soliton control pulse should be minimized by re-duction of the parameter n3,1 so the interaction length willnot increase.

Let us now consider the described scheme for the casewhen the wavelength of the control pulse is l15 1.06 mm and the wavelength of the signal pulse is l25 1.55 mm. During the interaction they generate thewave at the sum frequency l3 5 0.63 mm. The BBOcrystal is chosen here because recently, TWI solitons wereexperimentally observed in BBO under the similarconditions.4 The dispersion formulas13 give the followingvalues of the group speeds: v1 5 1.828 3 1010 cm/s, v35 1.786 3 1010 cm/s, and v2 5 1.756 3 1010 cm/s,


where v1 and v2 are the group speeds of the fundamentalwaves and v3 is the group speed at the sum frequency.These speeds satisfy the relationship v1 . v3 . v2 ,which means that each of the waves can have solitons.4–6

The strong pump pulse in Fig. 4 is a TWI soliton withthe duration t1 5 1 ps. The corresponding intensity cal-culated from Eq. (5) is 190 MW/cm2. Because of the re-markable property of solitons, the control pulse has thesame shape and duration before and after the interaction.Because of the energy conservation, the signal also re-gains its initial shape, in spite of it not being a soliton.(However, in general the energy conservation guaranteesonly an absence of the sum-frequency wave and does notpreserve the shape of the pulse.) The duration of the sig-nal pulse in Fig. 4 is t2 5 510 fs, which is 3.4 times lessthan that required by Eq. (5). The intensity of the signalpulse is 24 MW/cm2. (Because the second pulse does nothave to be a soliton, this parameter can be substantiallychanged without affecting the process.)

Theoretically, soliton pulses regain their shapes onlyasymptotically, and in the beginning of the interactionthey have to be well separated. This would require verylong nonlinear crystals. In practice the crystal’s lengthhas to be optimized. It should be taken as small as pos-sible, but long enough to allow the interacting waves onthe output of the crystal to regain as much as possibletheir initial shapes. Such an optimal length is chosen inFig. 4. In spite of the pulses in Fig. 4 initially overlap-ping, on the output they have almost the same ampli-tudes as the input waves: the amount of the energy ofthe sum-frequency wave after the interaction is only 4%of the energy of the signal pulse. For the considered pa-rameters of the pulses an optimal length of the nonlinearinteraction is equal to 3 cm. In practice, two 1.5-cm crys-tals could be used instead of one 3-cm crystal analogous toRef. 4. Such an arrangement allows one to reduce thespatial walk-off between the beams.

As can be seen from Eq. (3), the total phase shift Df5 c1 2 c2 tends to zero as Dk increases and is close to pfor Dk near zero. It is important for practical purposes toestimate an influence of this parameter on the magnitudeof the phase shift. As calculations show, Dk changesfrom 20.4 cm21 to 0.4 cm21 when the angle u between thewaves and the optical axis changes from 33.35° to 33.42°.Considering that the phase-matching angle in BBO forthis interaction is 33.38°, it gives uDku , 0.4 cm21 whenDu , 1.2 3 1023.

For the Dk 5 0.4 cm21, Eq. (3) gives the following val-ues of the phases: c1 5 4.8 and c2 5 22.5. Thereforethe power transmission though the system when the gateis open is sin2(2.6/2) ' 0.92. Considering that an addi-tional 4% of the energy is lost because of the shortening(optimization) of the crystal’s length, overall transmissionfor the considered example is 88%.

4. FUNDAMENTAL–SUM-FREQUENCY–FUNDAMENTAL AND SUM-FREQUENCY–FUNDAMENTAL–FUNDAMENTALREGIMES IN THREE-WAVE INTERACTIONAs mentioned above, both fundamental waves can containsolitons only if the sum-frequency wave has an interme-

diate speed. Mathematical analysis9 shows two substan-tially different regimes for TWI depending on whether ornot the group velocity of the sum-frequency-wave vSumfalls between the group velocities of the two low-frequency fundamental waves vFund1 and vFund2 : we la-bel the fundamentals to give vFund1 , vFund2 . The firstregime, termed FSF since the velocity of the sum-frequency pulse is between the fundamental velocities,satisfies vFund1 , vSum , vFund2 . The second regime,termed SFF, occurs if vSum is either the largest or thesmallest of the velocities. The main difference betweenthe FSF and SFF regimes is that in FSF both fundamen-tal waves can contain TWI solitons, while in SFF only thefundamental pulse with the extreme velocity can containTWI solitons. As shown above, a TWI-soliton switch gen-erally requires only one of the pulses to be a TWI soliton.Therefore the same scheme can be used for both FSF andSFF regimes. There are no explicit soliton solutions forthe SFF interaction, and expression (3) cannot be used forthe estimations of the phase shift. However, numericsshow that expression (3) gives a good approximation evenfor the SFF case. Furthermore, in the SFF case the fun-damental wave with the intermediate speed develops anunusual stability with respect to nonlinear collisions. Al-though this intermediate wave can never contain solitonsand consists entirely of radiation, it behaves as a quasi-soliton and approximately preserves its shape in nonlin-ear interactions. Unlike a real soliton, the pulse of thewave with an intermediate speed preserves its shape forthe very wide range of intensities. (However, in contrastto the FSF case the initial and final shapes are slightlydifferent.) This makes the SFF regime especially attrac-tive for switching. Another attractive feature of the SFFregime is that it allows the use of new nonlinear materi-als that have recently appeared on the market. For ex-ample, periodically poled lithium niobate uses a thirdtype of nonlinear interaction: e 2 e → e. For such aninteraction in the media with normal dispersion, thegroup velocity of the SF wave will always be less than thevelocities of the fundamental waves. Therefore only SFFinteraction is possible in such media. The nonlinear co-efficient of the periodically poled lithium niobate is ;10time higher than that of BBO. Correspondingly, the in-tensities (energies) of the control pulse required forswitching will be a 100 times smaller than those for BBO.

5. CONCLUSIONSA consideration of TWI interaction shows that TWI soli-tons can be used in the all-optical switching devices. Inparticular, the feasibility of an all-optical switch based onTWI solitons is demonstrated on the example of apolarization-gate beam geometry. In the considered ar-rangement, only one pulse (signal or control) has to be aTWI soliton. This makes it possible to use TWI-solitonswitching in both FSF and SFF regimes. Calculationsshow that for the BBO crystal, when l1 5 1.06 mm andl2 5 1.55 mm, the intensity of the control pulse (l1) canbe as low as 190 MW/cm2, and the nonlinear length asshort as 3 (1.5 1 1.5) cm. The BBO crystal was chosenfor the theoretical demonstration because of the recentexperimental demonstration of TWI-soliton interaction in


this crystal.4 For a wide variety of available media withquadratic nonlinearity, the intensity (energy) of the con-trol pulse can be greatly reduced. In particular, the useof periodically poled lithium niobate would allow one todecrease an intensity (energy) of the control pulse by twoorders of magnitude. Numerical investigation revealsunusual quasi-soliton behavior of the wave with the inter-mediate velocity under SFF interaction. Such behaviormakes the SFF regime especially attractive for switching.In this paper the spatial structure of the beams is nottaken into account. This means that the control beammust have a homogeneous or super-Gaussian distributionin space. The spatial structure of the signal (nonsoliton)wave is not important. For future research it would beinteresting to consider the proposed scheme in a wave-guide geometry. This would remove difficulties with thespatial distribution and at the same time increase thelength of interaction with the simultaneous reduction ofthe energy of the control pulse.

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all-optical switching using three-wave-interaction solitons

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