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All-optical circulator based on cross phase modulation in a nonlinear Mach-Zehnder interferometer
Optik 122 (2011) 464–466
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
All-optical circulator based on cross phase modulation in a nonlinear Mach-Zehnder interferometer Punya Prasanna Paltani, S. Medhekar ∗ Department of Applied Physics, Birla Institute of Technology, Mesra, Ranchi 835215, India
a r t i c l e
i n f o
Article history: Received 29 October 2009 Accepted 3 April 2010
a b s t r a c t We show that by exploiting unequal magnitude of cross phase modulation (XPM) in co-propagating and counter-propagating beams, an all-optical circulator (AOC) using a nonlinear Mach-Zehnder Interferometer (MZI) could be constructed. The proposed circulator is novel, simple and polarization independent. © 2010 Elsevier GmbH. All rights reserved.
Keywords: All-optical devices Nonlinear waveguide devices Optical isolator/circulator
1. Introduction The interest in nonlinear waveguide devices has been steadily growing in recent years [1–6] due to their potential use in ultrafast all-optical signal processing and optical computing systems. Nonlinear waveguides possess a variety of novel and exciting features such as power dependent propagation constants and field profiles leading to novel feasibilities for all-optical signal processing and optical computing. Recently, there has been great interest in the Mach-Zehnder Interferometer (MZI) waveguide device. The MZI waveguide device has been developed for use of modulating, switching, and logic gates [3–6]. In this paper, we propose a novel alloptical isolator/circulator using a MZI waveguide device. Optical isolators/circulators are indispensable components in high-speed optical networks [7–11]. Presently, commercially available isolators/circulators (based on magneto-optic effect) are available only in bulk form and an integrated form would be highly desirable to reduce size and cost. Proposals of waveguide isolators using MZI are existing in the literature but those are polarization dependent [12–14]. Polarization independent, single mode inline isolator with two nonreciprocal phase shifters, one for transverse electric (TE) modes and another one for transverse magnetic (TM) modes in a MZI has also been proposed [15] by considering magnetization tangential to the propagation direction, however, such an isolator would need special design and fabrication of the MZI waveguides.
∗ Corresponding author. E-mail address: [email protected] (S. Medhekar). 0030-4026/$ – see front matter © 2010 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2010.04.011
Here in this paper, we show that by exploiting unequal magnitudes of cross phase modulation (XPM) in co-propagating and counter-propagating configurations of two beams, a conventional nonlinear MZI could be made to function as an all-optical circulator (AOC). The proposed circulator is polarization independent and does not need special fabrication as conventional nonlinear MZI is used. The proposed AOC is meant for the digital signals where the intensities of 0 and 1 s are fixed. 2. The device (AOC) The proposed AOC consists of two 3 dB directional couplers at input and output ends of a MZI as shown in Fig. 1. One arm (NLA) of the MZI is made up of a nonlinear material, the other arm (LA) of a linear material. P1 , P2 , P3 and P4 are the four ports of the AOC. The all-optical circulation of an input beam of wavelength 1 could be obtained by using a continuous wave (CW) biasing beam of wavelength 2 provided at port P2 . In presence of the biasing beam, P1 → P4 and P4 → P2 become the forward and reverse paths respectively for the 1 beam. The idea is straight forward. DC1 and DC2 are 3 dB at 1 , while, 2 is chosen such that the whole of it propagates through the NLA. When 1 is launched at P1 , it splits into two equal parts. One part propagates through the NLA and its counterpart through the LA. The part propagating through the NLA experiences XPM due to co-propagation of it with 2 . The intensities of 1 , 2 and other parameters of the MZI are so chosen that split parts of 1 propagating through the LA and the NLA enter into DC2 with the same phase and therefore, whole of the 1 emerges at P4 . However when 1 is injected at P4 it splits into two equal parts at DC2. The part propagating through the NLA, counter propagates with 2 . The
P.P. Paltani, S. Medhekar / Optik 122 (2011) 464–466
465
Here, Ej is transverse field envelop [= Ij sec h(x)] of the beams, Ij the axial intensity, x the transverse coordinate, kj = 2/j the free space propagation constant and n0 is the refractive index of substrate on which MZI switch of refractive index n(x, z) is fabricated. n(x, z) is expressed as,
nj (x, z) =
Fig. 1. The proposed all-optical circulator (AOC) consisting of two 3 dB directional couplers (for input beam) at input and output ends of a MZI.
XPM induced phase change in this (counter-propagating) case is different than the XPM induced phase change in the earlier (copropagating) case [16]. The difference in the XPM induced phase change makes P4 → P2 the reverse path for the 1 accomplishing all-optical circulation. 3. Theory To simulate AOC, we examine co and counter propagation of 1 and 2 beams in a nonlinear MZI. These beams, in the paraxial approximation, could be described by the coupled equations [17]. ∂Ej ∂z
= −i
1 ∂2 Ej − ikj [nj (x, z) − n0 ]Ej ; j = 1, 2 2kj n0 ∂x2
(1)
Fig. 2. Propagation of input beam (1 ) and biasing beam (2 ) through AOC, when they are launched in same direction (co-propagating beams). (a) Propagation of biasing beam (when it is injected at P2 along with input beam at P1 ) is separately shown in the figure for clarity. As can be seen in figure, whole of the biasing beam propagates through NLA. (b) Propagation of input beam (when it is injected at P1 along with biasing beam at P2 ). Input beam splits in two equal parts, one of the split part co-propagate with biasing beam in NLA and recombines with its counterpart at port P4 .
nL (x, z) nNL (x, z) + nj (x, z)
(for LA) (for NLA)
(2)
where nL (x, z) is the refractive index of the linear arm, nNL (x, z) is the constant part of the refractive index of the nonlinear arm and nj (x, z) is its intensity dependent refractive index change which is expressed as [18]. nj (x, z) ≈ n2 (|Ej |2 + |E3−j |2 ); j = 1, 2
(3)
Here, n2 is the nonlinear coefficient of the material of NLA. is the coupling or interaction coefficient of the two beams. It is worth to be mentioned here that = 2 for mutually incoherent copropagating beams [19] and = 1 for mutually incoherent counterpropagating beams [16]. By solving Eq. (1) using Split Step Fourier Method or Beam propagation Method (BPM) [17], it is shown in this paper that it is possible to select the input intensities I1 of 1 and I2 of 2 and other parameters of the MZI of Fig. 1 so that it can function as an AOC for 1 . The chosen parameters to simulate AOC are corewidth 2a = 8 m, 1 = 1.55 m, 2 = 1 m, I1 = 3 × 1010 W/m2 , I2 = 4.5 × 1010 W/m2 , L = [nL (x, z) − n0 ] = 0.39 % (for the LA) and
Fig. 3. Propagation of input beam (1 ) and biasing beam (2 ) through AOC, when they are launched in opposite direction (Counter-propagating beams). (a) Propagation of biasing beam when it is injected at P2 along with input beam at P4 is shown. The whole of the biasing beam propagates through NLA. (b) Propagation of input beam, when it is injected at P4 along with biasing beam at P2 . Input beam splits in two equal parts, one of the splitted part counter propagate with biasing beam in NLA and recombines with its counterpart at port P2 .
466
P.P. Paltani, S. Medhekar / Optik 122 (2011) 464–466
NL = [nNL (x, z) − n0 ] = 0.21 % (for the NLA), separation of two waveguides is 25 m, branching angle at the combining/splitting region is B = 1.2 ◦ and the chosen length of LA and NLA is Z = 1200 m. Nonlinear coefficient n2 of NLA material is considered to be equal to 2 × 10−14 m2 /W [17,20]. The propagation features of 1 and 2 beams through the AOC obtained by solving Eq. (1) and using above mentioned parameters are shown in Figs. 2 and 3. Fig. 2a shows the propagation of 2 and Fig. 2b shows the propagation of 1 when former is injected at P2 and later at P1 . Both beams are separately shown in Fig. 2a and b for the sake of clarity, however, they are simultaneously propagating into the MZI. As can be seen in Fig. 2a, whole of the 2 propagates through NLA, while the 1 beam splits in two equal parts (splitted part of 1 is co-propagating with 2 in NLA of the MZI) and recombines at port P4 . When 1 is injected into port P4 keeping 2 at P2 , whole of the 2 keeps propagating through NLA (see Fig. 3a). The 1 splits in two equal parts (one of the splitted part counter propagates with the 2 beam) and recombines at port P2 (see Fig. 3b). In other words, the MZI with mentioned parameters acts as a circulator for the 1 beam. 4. Conclusion Exploiting unequal magnitude of cross phase modulation (XPM) in co-propagating and counter-propagating configuration of two beams, a novel all-optical circulator using a nonlinear MachZehnder Interferometer (MZI) is proposed in this paper. The proposed isolator/circulator is meant for the digital signals where the intensities of 0 and 1 s are fixed. The proposed circulator is polarization independent and very simple to fabricate as it uses a conventional Mach-Zehnder interferometer. Acknowledgements Authors are thankful to Oren Cohen, Physics Dept. and Solid State Institute, Technion, Haifa, Israel for providing computer code of counter-propagating beams. Authors acknowledge encouragement from H.C. Pande and P.K. Barhai. SM acknowledges funding
from the Department of Science and Technology, DST, Govt. of India through research project [No. 100/(IFD)/7582/2008-2009]. References [1] S. Medhekar, R.K. Sarkar, All-optical passive transistor, Opt. Lett. 30 (2005) 887–889. [2] S. Medhekar, P.P. Paltani, Proposal for optical switch using nonlinear refraction, IEEE Phot. Technol. Lett. 18 (2007) 1579–1581. [3] Y.D. Wu, M.L. Huang, M.H. Chen, R.Z. Tasy, All-optical switch based on the local nonlinear Mach-Zehnder interferometer, Opt. Expr. 15 (2007) 9883–9892. [4] Y.D. Wu, T.T. Shih, M.H. Chen, New all-optical logic gates based on the local nonlinear Mach-Zehnder interferometer, Opt. Expr. 16 (2008) 248–257. [5] S. Medhekar, P.P. Paltani, Novel all-optical switch using nonlinear MachZehnder interferometer, Fiber Int. Opt. 28 (2009) 229–236. [6] S. Medhekar, P.P. Paltani, All-optical transistor using counter propagating beams in a nonlinear Mach-Zehnder interferometer, Fiber Int. Opt. 28 (2009) 268–274. [7] R. Wolfe, J.R. Dillon, R.A. Liberman, V.J. Fratello, Broadband magneto-optic waveguide isolator, Appl. Phys. Lett. 57 (1990) 960–962. [8] K. Ando, T. Okoshi, N. Koshizuka, Waveguide magneto-optics isolator fabricated by laser annealing, Appl. Phys. Lett. 53 (1988) 4–6. [9] H. Hemme, H. Dötsch, P. Hertel, Integrated optical isolator based on nonreciprocal-mode cut-off, Appl. Opt. 29 (1990) 2741–2744. [10] S. Yamamoto, Y. Okamura, T. Makimoto, Analysis and design of semileaky-type thin-film optical waveguide isolator, IEEE J. Quant. Electron. 12 (1976) 764–770. [11] T. Shintaku, Integrated optical isolator based on nonreciprocal higher order mode conversion, Appl. Phys. Lett. 66 (1995) 2789–2791. [12] F. Auracher, H. Witte, A new design for an integrated optical isolator, Opt. Commun. 13 (1975) 435–438. [13] Y. Okamura, T. Negami, S. Yamamoto, Integrated optical isolator and circulator using nonreciprocal phase shifters: a proposal, Appl. Opt. 23 (1984) 1886–1889. [14] N. Bahlmann, M. Lohmeyer, M. Wallenhorst, H. Dötsch, P. Hertel, An improved design for an integrated optical isolator based on nonreciprocal Mach-Zehnder interferometry, Opt. Quant. Electron. 30 (1998) 323–334. [15] O. Zhuromskyy, M. Lohmeyer, N. Bahlmann, H. Dötsch, P. Hertel, A.F. Popkov, Analysis of polarization independent Mach-Zehnder-type integrated optical isolator, J. Lightwave Technol. 17 (1999) 1200–1205. [16] O. Cohen, R. Uzdin, T. Carmon, J.W. Fleischer, M. Segev, S. Odoulov, Collisions between optical spatial solitons propagating in opposite directions, Phys. Rev. Lett. 89 (2002), 133901: 1–4. [17] K. Okamoto, Fundamentals of Optical Waveguides, Academic Press, San Diego, 2002, pp. 274–303. [18] G.P. Agrawal, Nonlinear Fiber Optics, 3rd edition, Academic Press, San Diego, 2001, p. 262. [19] R.D.L. Fuente, A. Barthelemy, Spatial soliton pairing by cross phase modulation, Opt. Commun. 88 (1992) 419–423. [20] C.N. Ironside, J.F. Duffy, R. Hutchins, W.C. Bany, C.T. Seaton, G.I. Stegemen, Waveguide fabrication in nonlinear semiconductor doped glasses, in: Proc. 11th European Conf., Opt. Commun., Venice, 1985, pp. 237–240.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
All-optical circulator based on cross phase modulation in a nonlinear Mach-Zehnder interferometer Punya Prasanna Paltani, S. Medhekar ∗ Department of Applied Physics, Birla Institute of Technology, Mesra, Ranchi 835215, India
a r t i c l e
i n f o
Article history: Received 29 October 2009 Accepted 3 April 2010
a b s t r a c t We show that by exploiting unequal magnitude of cross phase modulation (XPM) in co-propagating and counter-propagating beams, an all-optical circulator (AOC) using a nonlinear Mach-Zehnder Interferometer (MZI) could be constructed. The proposed circulator is novel, simple and polarization independent. © 2010 Elsevier GmbH. All rights reserved.
Keywords: All-optical devices Nonlinear waveguide devices Optical isolator/circulator
1. Introduction The interest in nonlinear waveguide devices has been steadily growing in recent years [1–6] due to their potential use in ultrafast all-optical signal processing and optical computing systems. Nonlinear waveguides possess a variety of novel and exciting features such as power dependent propagation constants and field profiles leading to novel feasibilities for all-optical signal processing and optical computing. Recently, there has been great interest in the Mach-Zehnder Interferometer (MZI) waveguide device. The MZI waveguide device has been developed for use of modulating, switching, and logic gates [3–6]. In this paper, we propose a novel alloptical isolator/circulator using a MZI waveguide device. Optical isolators/circulators are indispensable components in high-speed optical networks [7–11]. Presently, commercially available isolators/circulators (based on magneto-optic effect) are available only in bulk form and an integrated form would be highly desirable to reduce size and cost. Proposals of waveguide isolators using MZI are existing in the literature but those are polarization dependent [12–14]. Polarization independent, single mode inline isolator with two nonreciprocal phase shifters, one for transverse electric (TE) modes and another one for transverse magnetic (TM) modes in a MZI has also been proposed [15] by considering magnetization tangential to the propagation direction, however, such an isolator would need special design and fabrication of the MZI waveguides.
∗ Corresponding author. E-mail address: [email protected] (S. Medhekar). 0030-4026/$ – see front matter © 2010 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2010.04.011
Here in this paper, we show that by exploiting unequal magnitudes of cross phase modulation (XPM) in co-propagating and counter-propagating configurations of two beams, a conventional nonlinear MZI could be made to function as an all-optical circulator (AOC). The proposed circulator is polarization independent and does not need special fabrication as conventional nonlinear MZI is used. The proposed AOC is meant for the digital signals where the intensities of 0 and 1 s are fixed. 2. The device (AOC) The proposed AOC consists of two 3 dB directional couplers at input and output ends of a MZI as shown in Fig. 1. One arm (NLA) of the MZI is made up of a nonlinear material, the other arm (LA) of a linear material. P1 , P2 , P3 and P4 are the four ports of the AOC. The all-optical circulation of an input beam of wavelength 1 could be obtained by using a continuous wave (CW) biasing beam of wavelength 2 provided at port P2 . In presence of the biasing beam, P1 → P4 and P4 → P2 become the forward and reverse paths respectively for the 1 beam. The idea is straight forward. DC1 and DC2 are 3 dB at 1 , while, 2 is chosen such that the whole of it propagates through the NLA. When 1 is launched at P1 , it splits into two equal parts. One part propagates through the NLA and its counterpart through the LA. The part propagating through the NLA experiences XPM due to co-propagation of it with 2 . The intensities of 1 , 2 and other parameters of the MZI are so chosen that split parts of 1 propagating through the LA and the NLA enter into DC2 with the same phase and therefore, whole of the 1 emerges at P4 . However when 1 is injected at P4 it splits into two equal parts at DC2. The part propagating through the NLA, counter propagates with 2 . The
P.P. Paltani, S. Medhekar / Optik 122 (2011) 464–466
465
Here, Ej is transverse field envelop [= Ij sec h(x)] of the beams, Ij the axial intensity, x the transverse coordinate, kj = 2/j the free space propagation constant and n0 is the refractive index of substrate on which MZI switch of refractive index n(x, z) is fabricated. n(x, z) is expressed as,
nj (x, z) =
Fig. 1. The proposed all-optical circulator (AOC) consisting of two 3 dB directional couplers (for input beam) at input and output ends of a MZI.
XPM induced phase change in this (counter-propagating) case is different than the XPM induced phase change in the earlier (copropagating) case [16]. The difference in the XPM induced phase change makes P4 → P2 the reverse path for the 1 accomplishing all-optical circulation. 3. Theory To simulate AOC, we examine co and counter propagation of 1 and 2 beams in a nonlinear MZI. These beams, in the paraxial approximation, could be described by the coupled equations [17]. ∂Ej ∂z
= −i
1 ∂2 Ej − ikj [nj (x, z) − n0 ]Ej ; j = 1, 2 2kj n0 ∂x2
(1)
Fig. 2. Propagation of input beam (1 ) and biasing beam (2 ) through AOC, when they are launched in same direction (co-propagating beams). (a) Propagation of biasing beam (when it is injected at P2 along with input beam at P1 ) is separately shown in the figure for clarity. As can be seen in figure, whole of the biasing beam propagates through NLA. (b) Propagation of input beam (when it is injected at P1 along with biasing beam at P2 ). Input beam splits in two equal parts, one of the split part co-propagate with biasing beam in NLA and recombines with its counterpart at port P4 .
nL (x, z) nNL (x, z) + nj (x, z)
(for LA) (for NLA)
(2)
where nL (x, z) is the refractive index of the linear arm, nNL (x, z) is the constant part of the refractive index of the nonlinear arm and nj (x, z) is its intensity dependent refractive index change which is expressed as [18]. nj (x, z) ≈ n2 (|Ej |2 + |E3−j |2 ); j = 1, 2
(3)
Here, n2 is the nonlinear coefficient of the material of NLA. is the coupling or interaction coefficient of the two beams. It is worth to be mentioned here that = 2 for mutually incoherent copropagating beams [19] and = 1 for mutually incoherent counterpropagating beams [16]. By solving Eq. (1) using Split Step Fourier Method or Beam propagation Method (BPM) [17], it is shown in this paper that it is possible to select the input intensities I1 of 1 and I2 of 2 and other parameters of the MZI of Fig. 1 so that it can function as an AOC for 1 . The chosen parameters to simulate AOC are corewidth 2a = 8 m, 1 = 1.55 m, 2 = 1 m, I1 = 3 × 1010 W/m2 , I2 = 4.5 × 1010 W/m2 , L = [nL (x, z) − n0 ] = 0.39 % (for the LA) and
Fig. 3. Propagation of input beam (1 ) and biasing beam (2 ) through AOC, when they are launched in opposite direction (Counter-propagating beams). (a) Propagation of biasing beam when it is injected at P2 along with input beam at P4 is shown. The whole of the biasing beam propagates through NLA. (b) Propagation of input beam, when it is injected at P4 along with biasing beam at P2 . Input beam splits in two equal parts, one of the splitted part counter propagate with biasing beam in NLA and recombines with its counterpart at port P2 .
466
P.P. Paltani, S. Medhekar / Optik 122 (2011) 464–466
NL = [nNL (x, z) − n0 ] = 0.21 % (for the NLA), separation of two waveguides is 25 m, branching angle at the combining/splitting region is B = 1.2 ◦ and the chosen length of LA and NLA is Z = 1200 m. Nonlinear coefficient n2 of NLA material is considered to be equal to 2 × 10−14 m2 /W [17,20]. The propagation features of 1 and 2 beams through the AOC obtained by solving Eq. (1) and using above mentioned parameters are shown in Figs. 2 and 3. Fig. 2a shows the propagation of 2 and Fig. 2b shows the propagation of 1 when former is injected at P2 and later at P1 . Both beams are separately shown in Fig. 2a and b for the sake of clarity, however, they are simultaneously propagating into the MZI. As can be seen in Fig. 2a, whole of the 2 propagates through NLA, while the 1 beam splits in two equal parts (splitted part of 1 is co-propagating with 2 in NLA of the MZI) and recombines at port P4 . When 1 is injected into port P4 keeping 2 at P2 , whole of the 2 keeps propagating through NLA (see Fig. 3a). The 1 splits in two equal parts (one of the splitted part counter propagates with the 2 beam) and recombines at port P2 (see Fig. 3b). In other words, the MZI with mentioned parameters acts as a circulator for the 1 beam. 4. Conclusion Exploiting unequal magnitude of cross phase modulation (XPM) in co-propagating and counter-propagating configuration of two beams, a novel all-optical circulator using a nonlinear MachZehnder Interferometer (MZI) is proposed in this paper. The proposed isolator/circulator is meant for the digital signals where the intensities of 0 and 1 s are fixed. The proposed circulator is polarization independent and very simple to fabricate as it uses a conventional Mach-Zehnder interferometer. Acknowledgements Authors are thankful to Oren Cohen, Physics Dept. and Solid State Institute, Technion, Haifa, Israel for providing computer code of counter-propagating beams. Authors acknowledge encouragement from H.C. Pande and P.K. Barhai. SM acknowledges funding
from the Department of Science and Technology, DST, Govt. of India through research project [No. 100/(IFD)/7582/2008-2009]. References [1] S. Medhekar, R.K. Sarkar, All-optical passive transistor, Opt. Lett. 30 (2005) 887–889. [2] S. Medhekar, P.P. Paltani, Proposal for optical switch using nonlinear refraction, IEEE Phot. Technol. Lett. 18 (2007) 1579–1581. [3] Y.D. Wu, M.L. Huang, M.H. Chen, R.Z. Tasy, All-optical switch based on the local nonlinear Mach-Zehnder interferometer, Opt. Expr. 15 (2007) 9883–9892. [4] Y.D. Wu, T.T. Shih, M.H. Chen, New all-optical logic gates based on the local nonlinear Mach-Zehnder interferometer, Opt. Expr. 16 (2008) 248–257. [5] S. Medhekar, P.P. Paltani, Novel all-optical switch using nonlinear MachZehnder interferometer, Fiber Int. Opt. 28 (2009) 229–236. [6] S. Medhekar, P.P. Paltani, All-optical transistor using counter propagating beams in a nonlinear Mach-Zehnder interferometer, Fiber Int. Opt. 28 (2009) 268–274. [7] R. Wolfe, J.R. Dillon, R.A. Liberman, V.J. Fratello, Broadband magneto-optic waveguide isolator, Appl. Phys. Lett. 57 (1990) 960–962. [8] K. Ando, T. Okoshi, N. Koshizuka, Waveguide magneto-optics isolator fabricated by laser annealing, Appl. Phys. Lett. 53 (1988) 4–6. [9] H. Hemme, H. Dötsch, P. Hertel, Integrated optical isolator based on nonreciprocal-mode cut-off, Appl. Opt. 29 (1990) 2741–2744. [10] S. Yamamoto, Y. Okamura, T. Makimoto, Analysis and design of semileaky-type thin-film optical waveguide isolator, IEEE J. Quant. Electron. 12 (1976) 764–770. [11] T. Shintaku, Integrated optical isolator based on nonreciprocal higher order mode conversion, Appl. Phys. Lett. 66 (1995) 2789–2791. [12] F. Auracher, H. Witte, A new design for an integrated optical isolator, Opt. Commun. 13 (1975) 435–438. [13] Y. Okamura, T. Negami, S. Yamamoto, Integrated optical isolator and circulator using nonreciprocal phase shifters: a proposal, Appl. Opt. 23 (1984) 1886–1889. [14] N. Bahlmann, M. Lohmeyer, M. Wallenhorst, H. Dötsch, P. Hertel, An improved design for an integrated optical isolator based on nonreciprocal Mach-Zehnder interferometry, Opt. Quant. Electron. 30 (1998) 323–334. [15] O. Zhuromskyy, M. Lohmeyer, N. Bahlmann, H. Dötsch, P. Hertel, A.F. Popkov, Analysis of polarization independent Mach-Zehnder-type integrated optical isolator, J. Lightwave Technol. 17 (1999) 1200–1205. [16] O. Cohen, R. Uzdin, T. Carmon, J.W. Fleischer, M. Segev, S. Odoulov, Collisions between optical spatial solitons propagating in opposite directions, Phys. Rev. Lett. 89 (2002), 133901: 1–4. [17] K. Okamoto, Fundamentals of Optical Waveguides, Academic Press, San Diego, 2002, pp. 274–303. [18] G.P. Agrawal, Nonlinear Fiber Optics, 3rd edition, Academic Press, San Diego, 2001, p. 262. [19] R.D.L. Fuente, A. Barthelemy, Spatial soliton pairing by cross phase modulation, Opt. Commun. 88 (1992) 419–423. [20] C.N. Ironside, J.F. Duffy, R. Hutchins, W.C. Bany, C.T. Seaton, G.I. Stegemen, Waveguide fabrication in nonlinear semiconductor doped glasses, in: Proc. 11th European Conf., Opt. Commun., Venice, 1985, pp. 237–240.