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All-Optical Phase-Independent Logic Elements Based on Phase Shift Induced by Coherent Soliton Collisions Oleg V. Kolokoltsev, R. Salas, and V. Vountesmeri

Abstract—We demonstrate, for the first time to our knowledge, that a fast coherent collision between two Kerr spatial solitons can give rise to a significant phase shift for both interacting beams. takes place The maximal collision-induced phase shift when the amplitudes of the solitons are equal ( 1 2 ) and the length of the interaction zone is comparable with a soliton phase period. Depending on the ratio 2 1 and the collision angle between the solitons, the magnitude of the phase shift can be varied within a reasonable range, for example from 180 to 40 . The analysis of the effect performed by the finite-difference beam-propagation method has shown that it is insensitive to the initial phase difference between the incident beams ( ), even in the case when 1 2 . It has been demonstrated that the phenomenon can be used for all-optical three-soliton logic elements, which are capable of providing more than 3-dB signal amplification and possess -independent output characteristics.

rad =

=

Index Terms—Kerr nonlinearity, nonlinear interactions, optical phase shift, optical spatial solitons, soliton collisions.

A

NUMBER of theoretical and experimental studies conducted in recent years on interactions between self-trapped optical wavebeams, or spatial solitons, have clearly shown their significant potential for prospective ultrafast optical communications and optical computing. Although optical beam self-trapping has been demonstrated to exist, owing to a variety of nonlinear mechanisms [1], [2], among them only the optical Kerr effect in nonresonant third-order materials (Kerr solitons) and two-wave parametric mixing in second-order materials (quadratic solitons) exhibit femtosecond response [3]. From the technological standpoint, however, the optical Kerr effect seems to be more attractive because no symmetry requirements exist for the nonlinear wave structures on which it is based. As is well known, the classical bright Kerr spatial soliton, which is stable only for the (1 1)-dimensional [(1 1)-D] wavebeam geometry [4], represents the fundamental optical mode of the so-called self-induced waveguide channel in Kerr-type medium. It can be formed in planar nonlinear waveguides, when the beam power is sufficient to provide an exact balance between the beam self-focusing and diffraction effects [5], [6]. Two coherent Kerr spatial solitons (to be considered here) can interact through either positive or negative Manuscript received December 12, 2001; revised February 26, 2002. This work was supported by the DGAPA UNAM Science Foundation under Grant IN107100. The authors are with the Centro de Instrumentos, Universidad Nacional Autonoma de Mexico, Ciudad Universitaria, CP 04510, Circuito Exterior, Mexico, D.F. (e-mail: [email protected]). Publisher Item Identifier S 0733-8724(02)05389-6.

inter-soliton forces caused by self-phase modulation and four-wave mixing terms in the nonlinear Schrödinger equation (NLSE) [7]–[10]. The soliton interactions, proposed recently for all-optical applications, can be grouped within two basic subclasses in which they exhibit themselves in a different manner. The first case refers to so-called slow interactions, which take place between two solitons with initially parallel trajectories and slightly overlapping profiles. These interactions strongly depend on the initial relative phase difference ( ) between incident beams and are manifested in such well-known ), periodical phenomena as soliton-soliton repulsion ( ) [8], [9], and energy exchange inside spatial collapse ( ) [10]. The second case is associated the soliton pair ( with so-called fast interaction processes when the solitons collide at a converging angle that is comparable to or larger than their angular spectra. In this case, the solitons “pass through” each other, experiencing a slight position shift, but conserving their propagation angles and profile. In contrast to the previous case, the fast interactions are phase independent. However, any optical switch based on the position shift effect requires an additional output structure of optical bent couplers [11]. Both these cases, including mutual soliton trapping [12], have been recently demonstrated to be effective for multigigabit optical signal routing, optical beam switching, and formation of soliton-based waveguides, which are able to guide and steer a weak optical signal [13]–[15]. As one can see from the above examples, the slow interactions possess a very rich and attractive properties for all-optical applications. However, the nonlinear elements based on the slow interaction effects may operate in a stable manner only if is strictly fixed. Nevertheless, it is easy to image a situation when at least one of two incident beams possesses a random phase. Recently, a polarization modulator [16] and an optical dragging device [17] with multiple-cascade integrated optic schemes, developed to eliminate the influence of on the device characteristics, have been proposed. In this paper, we describe a new property of the fast collisions between coherent spatial solitons, which can open additional possibilities when developing -independent all-optical elements. It consists of a -independent shift in a soliton phase period ( ), which can be induced by the collision. In turn, this effect can be used for controlling the slow interaction processes. The fact that both the soliton envelopes and beams acquire a phase shift after collision is well documented in the exact N-soliton solution to NLSE obtained by Zakharov and Shabat [4] and in the explicit two-soliton solution of Gordon [18]. Experimental demonstration of the effect

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at a respectively large converging angle . In Fig. 1, the axis coincides with the trajectory of the center of mass of the soliton pair and the axis passes through the center ( ) of , where the interaction zone with the length and are determined from the condition that the distance (where between the soliton centers is is the full width at the half-intensity maximum of the beams). The conditions used for the simulations correspond to the fast . The results in Fig. 1 collision regime, i.e., when were obtained by using the finite-difference beam-propagation method (FD-BPM) applied to the (1 1)-D NLSE, which can be presented in the usual dimensionless form (1) For the self-focusing problem considered here, the variables in , (1) are related to the real ones as follows: , , where is are the linear transverse the soliton width parameter, and electric (TE) wave propagation constant and the film refractive is the nonlinear coefficient, and is the index, respectively, soliton complex amplitude. The simulations were performed for m, propagating in a polythe beams with diacetylene paratoluene-sulfonate (PDA-PTS) nonlinear wave) on a SiO –Si substrate, which possesses the guide ( cm W at m largest nonresonant [22], [23]. ) the collision gives rise As can be seen in Fig. 1(b), (at to the 180 phase shift for both soliton beams, compared with the single soliton case shown in Fig. 1(a). Also, Fig. 1(a)–(f) (at demonstrates that the phase shift for the soliton ) remains equal to 180 at any , i.e., it is the same as in is changed Fig. 1(c), although the initial phase of the soliton from 0 to 180 . Analytically, as follows from soliton inverse-scattering theory [4], the phase shift ( ) induced by the soliton collision, in the approximation when solitons are well separated after collision, can be estimated by Fig. 1. (a) Unperturbed phase distribution of the S -soliton field. (b)–(f) Phase-shift induced for the soliton S , after its collision with the equivalent soliton S , shown for different values of the initial phase difference  (see the cross section Z L). Fig. 1c shows the soliton field intensity for  = 45 .

=

for optical fiber solitons ( 70 phase shift) can be found, e.g., in [19]–[21]. However, to our knowledge, no reports on the phenomenon for spatial solitons and its applications have been presented. Here, we demonstrate the spatial effect and present its possible realization in -independent three-soliton logic-gate schemes. I. THEORETICAL MODEL The numerical simulations of the collision-induced phaseshift effect are illustrated in Fig. 1. It shows the phase evolu-mode equivalent spatial tion of complex amplitudes of two ) with the wavelength m, propagating solitons ( in a planar nonlinear waveguide and colliding in the film plane

(2) and Here, are, respectively, the normalized peak amplitudes of the solitons and their propagation angles, which are the characteristic parameters of the fundamental one-soliton solution to NLSE (1). It is easy to see from (2) that the phase shift takes place when and , as well as that the given exand perimental conditions satisfy the above requirements and provide the theoretical value . Also, it is necessary to note that the corresponding m is compainteraction length m rable with the experimental soliton phase period shown in Fig. 1, which, in turn, is in good agreement with the given by m. theoretical

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Fig. 2. Theoretical (lines) and experimental (boxes) phase shift as a function of the ratio  = , where  is the fixed amplitude of the pump soliton and  is the variable amplitude of the signal soliton (a) the collision angle 1 . (b) = 5 .

=

It should be noted that the effect takes place within a wide range of the collision angle , which satisfies the condition . This was verified numerically, for , where is the critical angle when two solitons can form a spatially oscillating soliton pair (experimental ). Although the theory shows that within the above 180 to 158 , this range of the phase shift changes from change was detected to be smaller in the simulations. Another important question is how the phase shift depends on the difference between the soliton amplitudes. Fig. 2 shows (at ) calculated the phase shift as a function of by (2) and also taken from the data obtained by FD-BPM, in (2), which are shown in Fig. 3. Note that, by varying we have to maintain the well-known condition for the peak Const. Both the theoretical curve and the intensity numerical data in Fig. 2 demonstrate a strong decrease of the changes from 1 to phase shift, from 180 to 40 , as has to be 0.5. This means that, for a given , the ratio well stabilized. However, on the other hand, the data in Fig. 2 demonstrate that the effect can be used not only for simple binary phase controlling, but also for more complex phase coding of the signals. In Section II, we show that by using the , which provides a phase shift for a condition single collision, it is possible to realize the situation when a low-intensity signal soliton can induce a phase shift (which is usually required in optical processing devices) for a certain pump soliton by using a double collision scheme. The main advantage of this collision regime or any multiple-collision scheme is that it can provide significant signal amplification. Moreover, such a solution can significantly stabilize the phase because, as can shift with respect to the fluctuations of be seen in Fig. 2 (curve b), it is possible to decrease the slope by increasing the collision angle. of

II. NUMERICAL SIMULATION OF PHASE-INDEPENDENT ALL-OPTICAL LOGIC GATE To demonstrate the possible practical realization of the phenomenon, we considered the thin-film all-optical logic element shown schematically in Fig. 4. This element is based on the interaction between three coherent Kerr spatial solitons with equal and (see amplitudes. Two of them are the pump beams Fig. 4), with initially parallel propagation directions, whose intersoliton interaction force can be switched from repulsive type and the to attractive type through the fast collision between and are obtained by signal soliton . It is supposed that 3 dB splitting the input pump . Hence, the initial relative and is always constant. For the phase difference for is chosen to be equal to zero. Therefore, scheme considered, when the signal beam is off, the pump beams and , with , attract each other and fuse to form the output signal , as shown in Fig. 4. However, when the signal is on, its gives rise to a 180 phase shif for the fast interaction with is 180 , the soliton will soliton . In this case, since so that both solitons can be deflected to the repel the soliton separate output ports , (see Fig. 4). The main advantage of induced by the this scheme is that the phase shift for signal collision is insensitive to the initial phase difference between and . Fig. 5 shows the 1- m-thick PDA-PTS–SiO –Si waveguide structure used for FD-BPM simulation of the phase-independent logic element based on the scheme presented in Fig. 4. m The length and the width of the structure are m, correspondingly. The input zone of the strucand m m rectangular hollow, which can ture contains a reflect spatial solitons and is used in order to increase the iniand . The output zone tial spatial separation between ,

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Fig. 4. Schematic view of the all-optical logic cell based on the collision between three equivalent solitons.

Fig. 5. Nonlinear structure used for 3-D FD-BPM simulations.

Fig. 3. (a) Unperturbed phase distribution of the soliton S . (b)–(e) Collision-induced phase shift for the soliton S (with the amplitude  ) decreases, as the amplitude of the signal soliton ( ) changes as (b)  0:9 , (c)  = 0:8 , (d)  = 0:7 , and (e)  = 0:6 .

=

of the structure has four 100- m-long rib waveguides used as output ports. The incident angles and phases of the beams and are chosen so that in the operation zone they propagate . In the numerical model, the along parallel paths, with mode were excited by three equivalent optical solitons of beams with Gaussian transversal profile and the beam width pam. Fig. 6 presents the numerical results obrameter tained for a lossless planar waveguide by a three-dimensional (3-D) FD-BPM, applied to (2 1)-D NLSE. As can be seen in is equal Fig. 6(a), when the logic level of the signal soliton and and the to zero, there is a fusion of the pump beams resulting wave, with doubled peak amplitude, is launched into . Hence, in this stage, the scheme provides the output port the logic “1” at . Fig. 6(b) shows another stage of the device, induces the phase shift for the when the control soliton

soliton . As expected, it leads to the mutual repulsion of and , accompanied by their angular deflection to the output and . In this case, the output port is characterized ports by the logic level “0.” At the same time, the signals from the and can amplification be combined to be launched, ports e.g., in a further logic cascade. It is important to note that the scheme provides 3-dB amplification of the signal at the logic and were obtained “1” at , since it was supposed that by 3 dB splitting a certain initial pump with (see Fig. 4). The results shown in Fig. 7 present another solution for the same all-optical device. In this case, the device scheme is modis provided ified so that the 180 phase shift for the soliton and the signal . This corby the double collision between responds to the case when the amplitude of the signal wave and one collision induces phase shift for . The advantage of this scheme is that: 1) the signal soliton does not change its initial propagation direction because the double collision compensates its spatial shift and 2) the theoretical signal amplification, taking into account that , is 4 dB. Finally, we would like to note that the output rib waveguides, shown in Fig. 5, were used here in order to verify the stability of

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(a)

(a)

(b) Fig. 6. (a) Two soliton fusion at the conditions when the solitons are equivalent and are launched in the interaction region with  0. (b) Mutual repelling between the solitons, which takes place due to the  (rad) phase shift induced for S by the signal soliton S .

=

the schemes with respect to any possible changes, which can be associated with . It should be stressed that, for both schemes, FD-BPM analysis shows a small variation of the optical power in the ports , , and (it lies within 3%), as the initial phase changes over 2 . Note, at the same time, the difference schemes do not require the control of the initial phase difference and , since the beams are formed from the between the shown in Fig. 4. However, as same optical source, the pump can be seen in Figs. 6 and 7, in the considered structure, there is a small radiation of the lightwave at the input zone of the rib waveguides, the so-called leakage effect. It takes place since the excitation of the given rib waveguides by the soliton beams is not optimal because of the difference between the soliton profile and the profile of the optical guided-wave mode in the channel. It is obvious that these losses can be decreased significantly by

(b) Fig. 7. (a) Soliton–soliton fusion and (b) mutual-soliton repelling caused by the  phase-shift effect induced for the soliton S after its double collision with the signal S (the case when one collision between S and S gives rise to the phase shift of =2 rad).

the optimization of the waveguide structure and the amplification index can approach the theoretical one. III. CONCLUSION It has been demonstrated that the fast collision between two coherent Kerr-type spatial solitons can induce a significant shift in the soliton phase. The maximal phase shift, 180 for both interacting solitons, takes place when the solitons have equal amplitudes and collide at the conditions when the length of the

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interaction zone is comparable with the soliton phase period ( ). The magnitude of the effect can be varied in a wide range of the phase angles, depending on the amplitude difference and the collision angle between the beams. For example, the phase shift changes from 180 to 40 , as the ratio between the solitons amplitude decreases from 1 to 0.5. Also, the phase shift range narrows up to 60 , as the length of the interaction zone . It should be stressed that the phenomenon decreases to is insensitive to the initial relative phase difference between the incident beams. The considered all-optical logic elements based on the phenomenon can provide more than 3-dB signal amplification and possess the characteristics acceptable for all-optical cascaded logic systems. The FD-BPM analysis of the logic elements shows that optical power switched between the device output ports is always the same at any initial phase difference between the control and signal solitons. A deeper analysis of the effect, its physical interpretation, and the stability parameters of the corresponding all-optical devices is required and will be presented soon.

[10] M. Shalaby, F. Reynaud, and A. Barthelemy, “Experimental observation of spatial soliton interactions with a =2 relative phase difference,” Opt. Lett., vol. 17, no. 11, pp. 778–780, 1992. [11] T.-T. Shi and S. Chi, “Nonlinear photonic switching by using the spatial soliton collision,” Opt. Lett., vol. 15, no. 20, pp. 1123–1125, 1990. [12] M. Shalaby and A. Barthelemy, “Experimental spatial soliton trapping and switching,” Opt. Lett., vol. 16, no. 19, pp. 1472–1474, 1991. [13] F. Garzia, C. Sibilia, and M. Bertolotti, “All-optical soliton based router,” Opt. Commun., vol. 168, pp. 277–285, Sept. 1, 1999. [14] L. Friedrich, G. Stegeman, P. Millar, and J. S. Aitchison, “1 4 optical interconnect using electronically controlled angle steering of spatial solitons,” IEEE Photon. Technol. Lett., vol. 11, pp. 988–990, Aug. 1999. [15] L. Lefort and A. Barthelemy, “All-optical demultiplexing of a signal using collisions and waveguiding of spatial solitons,” IEEE Photon. Technol. Lett., vol. 9, pp. 1364–1366, Oct. 1997. [16] G. Cancellieri, F. Chiaraluce, E. Gambi, and P. Pierleoni, “All-Optical polarization modulator based on spatial soliton coupling,” J. Lightwave Technol., vol. 14, no. 3, pp. 513–523, 1996. [17] F. Chiaraluce, E. Gambi, and P. Pierleoni, “A nonlinear device for optical dragging with compensation of the initial phase difference,” in Proc. 8th Mediter. Electrotechnical Conf., vol. 3, 1996, pp. 1497–1500. [18] J. P. Gordon, “Interaction forces among solitons in optical fibers,” Opt. Lett., vol. 8, no. 11, pp. 596–598, 1983. [19] S. R. Friberg, S. Machida, and Y. Yamamoto, “Quantum-nondemolition measurement of the photon number of an optical soliton,” Phys. Rev. Lett., vol. 69, no. 22, pp. 3165–3168, 1992. [20] K. Watanabe, H. Nakano, A. Honold, and Y. Yamamoto, “Optical nonlinearities of excitonic self-induced-transparency solitons: Toward ultimate realization of squeezed states and quantum nondemolition measurement,” Phys. Rev. Lett., vol. 62, no. 19, pp. 2257–2260, 1989. [21] S. R. Friberg, T. Mukai, and S. Machida, “Dual quantum nondemolition measurements via successive soliton collisions,” Phys. Rev. Lett., vol. 84, no. 1, pp. 59–62, 2000. [22] B. L. Lawrence, M. Cha, and J. U. Kang et al., “Large purely refractive nonlinear index of single crystal P-toluene sulphonate (PTS) at 1600 nm,” Electron. Lett., vol. 30, no. 5, pp. 447–448, 1994. [23] M. Thakur, R. C. Frye, and B. I. Greene, “Nonresonant absorption coefficient of single-crystal films of polydiacetylene measured by photothermal deflection spectroscopy,” Appl. Phys. Lett., vol. 56, no. 12, pp. 1187–1188, 1990.

ACKNOWLEDGMENT The authors would like to thank A. Grishin from the Royal Institute of Technology of Stockholm and N. Bruce from the National University of Mexico for helpful comments. REFERENCES [1] G. I. A. Stegeman, D. N. Christodoulides, and M. Segev, “Optical spatial solitons: Historical perspectives,” IEEE J. Select. Topics Quantum Electron., vol. 6, pp. 1419–1427, Nov.–Dec. 2000. [2] A. W. Snyder, “Guiding light into the millennium,” IEEE J. Select. Topics Quantum Electron., vol. 6, pp. 1408–1412, Nov.–Dec. 2000. [3] D. F. Eaton, “Nonlinear optical materials,” Science, vol. 253, pp. 281–287, July 19, 1991. [4] V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional selffocusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JEPT, vol. 34, no. 1, pp. 62–69, 1972. [5] A. Barthelemy, S. Maneuf, and C. Froehly, “Propagation soliton et autocontinement de faisceaux laser par nonlinearité optique de Kerr,” Opt. Commun., vol. 55, no. 3, pp. 201–206, 1985. [6] J. S. Aitchison et al., “Observation of spatial optical solitons in a nonlinear glass waveguide,” Opt. Lett., vol. 15, no. 9, pp. 471–473, 1990. [7] G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: Universality and diversity,” Science, vol. 286, no. 5444, pp. 1518–1523, 1999. [8] F. Reynaud and A. Barthelemy, “Optically controlled interaction between two fundamental soliton beams,” Europhys. Lett., vol. 12, no. 5, pp. 401–405, 1990. [9] J. S. Aitchison, A. M. Weiner, Y. Silberberg, D. E. Leaird, M. K. Oliver, J. L. Jackel, and P. W. E. Smith, “Experimental observation of spatial soliton interactions,” Opt. Lett., vol. 16, no. 1, pp. 15–17, 1991.

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Oleg V. Kolokoltsev, photograph and biography not available at the time of publication.

R. Salas, photograph and biography not available at the time of publication.

V. Vountesmeri, photograph and biography not available at the time of publication.