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Journal of Engineering Mathematics

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01-02-2022

Fast two-beam collisions in a linear optical medium with weak cubic loss in spatial dimension higher than 1

Authors: Avner Peleg, Toan T. Huynh, Quan M. Nguyen

Published in: Journal of Engineering Mathematics | Issue 1/2022

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Abstract

We study the dynamics of fast two-beam collisions in linear bulk optical media with weak cubic loss in spatial dimension higher than 1. The cubic loss arises due to two-photon absorption. We first generalize the perturbation theory that was developed for analyzing two-pulse collisions in spatial dimension 1 to spatial dimension 2. We then use the generalized two-dimensional perturbation theory to show that the collision leads to a change in the beam shapes in the direction transverse to the relative velocity vector. Furthermore, we show that in the important case of a separable initial condition for both beams, the longitudinal part in the expression for the amplitude shift is universal, while the transverse part is not universal. The same behavior holds for collisions between pulsed optical beams in spatial dimension 3. We check these analytic predictions along with other predictions concerning the effects of anisotropy in the initial condition by extensive numerical simulations with the weakly perturbed linear propagation model. The agreement between the perturbation theory and the simulations is very good. Thus, our study extends and generalizes the results of previous works, which were limited to spatial dimension 1. The results are very useful for multiwavelength optical communication systems.

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In particular, the collision length is much smaller than all the length scales associated with the linear processes that are described by the evolution model. For example, in the case of a fast collision between two pulses of the linear propagation model, the collision length is much smaller than the diffraction length (or the dispersion length) [8, 10].

In the current paper, we define the spatial dimension of the problem as the number of the coordinates on which the electric field depends for a given distance z. Therefore, in the three works in Refs. [8‐10], for example, the spatial dimension was 1, since for each value of z, the electric field was a function of one coordinate only.

The dimensionless distance z in Eq. (1) is \(z=Z/L_{D}\), where Z is the dimensional distance, \(L_{D}=x_{0}^{2}/{\tilde{d}}_{2}\) is the diffraction length for a reference beam with width \(x_{0}\) in the x direction, and \(\tilde{d}_{2}\) is the dimensional diffraction coefficient. The dimensionless coordinates x and y are \(x=X/x_{0}\) and \(y=Y/x_{0}\), where X and Y are the dimensional coordinates. \(\psi _{j}=E_{j}/\tilde{P}_{0}^{1/2}\), where \(E_{j}\) is the electric field of the jth beam and \({\tilde{P}}_{0}\) is the peak power. The coefficients \(d_{11}\) and \(\epsilon _{3}\) are given by: \(d_{11}={\tilde{d}}_{11} x_{0}/\tilde{d}_{2}\) and \(\epsilon _{3}={\tilde{\rho }}_{3} {\tilde{P}}_{0} x_{0}^{2}/{\tilde{d}}_{2}\), where \({\tilde{d}}_{11}\) is the dimensional beam-steering coefficient, and \({\tilde{\rho }}_{3}\) is the dimensional cubic loss coefficient.

The separable initial condition is of special importance since it describes the electric fields that are produced by many lasers [4, 5].

In particular, the generalized perturbation method can be used to obtain explicit formulas for the collision-induced changes in the shapes and amplitudes of pulsed optical beams in fast collisions in spatial dimension 3.

The values of \(z_\mathrm{{i}}\) are determined by: \(z_\mathrm{{i}}=z_\mathrm{{c}}+r(z_\mathrm{{f}}-z_\mathrm{{c}})\), where \(r=1/5\), as an example. Thus, \(z_\mathrm{{i}}\) is an intermediate distance that is larger than \(z_\mathrm{{c}}\), at which the collision is not yet completed.

Since in Sect. 3.3 the coefficient \({\tilde{a}}_{1}\) depends on \(A_{1}(0)\) and \(A_{2}(0)\) [instead of on \(A_{1}(z_\mathrm{{c}}^{-})\) and \(A_{2}(z_\mathrm{{c}}^{-})\)], we replace the notation \(\tilde{a}_{1}(z_\mathrm{{c}}^{-})\) by \({\tilde{a}}_{1}\) in this subsection.

Since the fractional intensity reduction factor obtained in the simulation \(\varDelta I_{1}^{(r)(\mathrm{{num}})}(x,y,z)\) shows weak dependence on x, we use two different methods to obtain \(\varDelta I_{1}^{(r)}(y,z)\) from the simulation result. In the first method, we calculate \(\varDelta I_{1}^{(r)}(y,z)\) by averaging \(\varDelta I_{1}^{(r)(\mathrm{{num}})}(x,y,z)\) over the x-interval \([-2,2]\), and in the second method, we use the value of \(\varDelta I_{1}^{(r)(\mathrm{{num}})}(0,y,z)\).

Note that \({\bar{\varPsi }}_{20}\) depends on \(y_{20}\). However, for brevity of notation, we did not write this dependence explicitly in Eq. (86) and throughout the paper.

Van Kampen NG (2007) Stochastic processes in physics and chemistry. Elsevier, AmsterdamMATH

Whitham GB (1999) Linear and nonlinear waves. Wiley, New YorkMATHCrossRef

Ishimaru A (2017) Electromagnetic wave propagation, radiation, and scattering. Wiley, HobokenCrossRef

Siegman AE (1986) Lasers. University Science Books, Mill Valley

Kogelnik H, Li T (1966) Laser beams and resonators. Appl Opt 5:1550CrossRef

Merzbacher E (1998) Quantum mechanics. Wiley, New YorkMATH

Lin Q, Painter OJ, Agrawal GP (2007) Nonlinear optical phenomena in silicon waveguides: modeling and applications. Opt Express 15:16604CrossRef

Peleg A, Nguyen QM, Huynh TT (2017) Soliton-like behavior in fast two-pulse collisions in weakly perturbed linear physical systems. Eur Phys J D 71:315CrossRef

Nguyen QM (2021) Collision-induced amplitude dynamics of pulses in linear waveguides with the generic nonlinear loss. Int J Nonlinear Sci Numer Simul, in press

10.

Nguyen QM, Huynh TT, Peleg A (2021) Universality of the amplitude shift in fast two-pulse collisions in weakly perturbed linear physical systems. Indian J Phys, in press

11.

Forghieri F, Tkach RW, Chraplyvy AR (1997) Fiber nonlinearities and their impact on transmission systems (Chapter 8). In: Kaminow IP, Koch TL (eds) Optical fiber telecommunications III. Academic Press, San Diego

12.

Agrawal GP (1997) Fiber-optic communication systems. Wiley, New York

13.

Gnauck AH, Tkach RW, Chraplyvy AR, Li T (2008) High-capacity optical transmission systems. J Lightwave Technol 26:1032CrossRef

14.

Mollenauer LF, Grant A, Liu X, Wei X, Xie C, Kang I (2003) Experimental test of dense wavelength-division multiplexing using novel, periodic-group-delay-complemented dispersion compensation and dispersion-managed solitons. Opt Lett 28:2043CrossRef

15.

Nakazawa M (2000) Solitons for breaking barriers to Terabit/second WDM and OTDM transmission in the next millennium. IEEE J Sel Top Quant Electron 6:1332CrossRef

16.

Dekker R, Usechak N, Först M, Driessen A (2007) Ultrafast nonlinear all-optical processes in silicon-on-insulator waveguides. J Phys D 40:R249CrossRef

17.

Borghi M, Castellan C, Signorini S, Trenti A, Pavesi L (2017) Nonlinear silicon photonics. J Opt 19:093002CrossRef

18.

Ehrlich JE, Wu XL, Lee I-YS, Hu Z-Y, Röckel H, Marder SR, Perry JW (1997) Two-photon absorption and broadband optical limiting with bis-donor stilbenes. Opt Lett 22:1843CrossRef

19.

Liang TK, Nunes LR, Sakamoto T, Sasagawa K, Kawanishi T, Tsuchiya M, Priem GRA, Van Thourhout D, Dumon P, Baets R, Tsang HK (2005) Ultrafast all-optical switching by cross-absorption modulation in silicon wire waveguides. Opt Express 13:7298CrossRef

20.

Jones R, Rong H, Liu A, Fang AW, Paniccia MJ, Hak D, Cohen O (2005) Net continuous wave optical gain in a low loss silicon-on-insulator waveguide by stimulated Raman scattering. Opt Express 13:519CrossRef

21.

Liu A, Rong H, Paniccia MJ, Cohen O, Hak D (2004) Net optical gain in a low loss silicon-on-insulator waveguide by stimulated Raman scattering. Opt Express 12:4261CrossRef

22.

Kivshar YS, Malomed BA (1989) Dynamics of solitons in nearly integrable systems. Rev Mod Phys 61:763CrossRef

23.

Mizrahi V, DeLong KW, Stegeman GI, Saifi MA, Andrejco MJ (1989) Two-photon absorption as a limitation to all-optical switching. Opt Lett 14:1140CrossRef

24.

Aceves AB, Moloney JV (1992) Effect of two-photon absorption on bright spatial soliton switches. Opt Lett 17:1488CrossRef

25.

Peleg A, Nguyen QM, Chung Y (2010) Cross-talk dynamics of optical solitons in a broadband Kerr nonlinear system with weak cubic loss. Phys Rev A 82:053830CrossRef

26.

Okawachi Y, Kuzucu O, Foster MA, Salem R, Turner-Foster AC, Biberman A, Ophir N, Bergman K, Lipson M, Gaeta AL (2012) Characterization of nonlinear optical crosstalk in silicon nanowaveguides. IEEE Photon Technol Lett 24:185CrossRef

27.

Peleg A, Chakraborty D (2018) Transmission stabilization in soliton-based optical-waveguide systems by frequency-dependent linear gain-loss and the Raman self-frequency shift. Phys Rev A 98:013853CrossRef

28.

Peleg A, Chakraborty D (2020) Radiation dynamics in fast soliton collisions in the presence of cubic loss. Physica D 406:132397MathSciNetCrossRef

29.

Foster MA, Turner AC, Lipson M, Gaeta AL (2008) Nonlinear optics in photonic nanowires. Opt Express 16:1300CrossRef

30.

Ablowitz MJ, Clarkson PA (1991) Solitons. Nonlinear evolution equations and inverse scattering. Cambridge University Press, CambridgeMATHCrossRef

31.

Osborne AR (2010) Nonlinear ocean waves and the inverse scattering transform. Elsevier, AmsterdamMATH

32.

Chakravarty S, Kodama Y (2008) Classification of the line-soliton solutions of KPII. J Phys A: Math Theor 41:275209MathSciNetMATHCrossRef

33.

Gao XY, Guo YJ, Shan WR (2021) Cosmic dusty plasmas via a (3+1)-dimensional generalized variable-coefficient Kadomtsev–Petviashvili–Burgers-type equation: auto-Bäcklund transformations, solitons and similarity reductions plus observational/experimental supports. Wave Random Complex, in press

34.

Gao XY, Guo YJ, Shan WR (2020) Hetero-Bäcklund transformation and similarity reduction of an extended (2+1)-dimensional coupled Burgers system in fluid mechanics. Phys Lett A 384:126788MathSciNetMATHCrossRef

35.

Nguyen QM, Huynh TT (2021) Collision-induced amplitude dynamics of fast 2D solitons in saturable nonlinear media with weak nonlinear loss. Nonlinear Dyn 104:4339CrossRef

36.

Huynh TT, Nguyen QM (2021) Fast soliton interactions in cubic-quintic nonlinear media with weak dissipation. Appl Math Model 97:650MathSciNetMATHCrossRef

37.

McManamon PF, Bos PJ, Escuti MJ, Heikenfeld J, Serati S, Xie H, Watson EA (2009) A review of phased array steering for narrow-band electrooptical systems. Proc IEEE 97:1078CrossRef

38.

Brandl P, Schidl S, Polzer A, Gaberl W, Zimmermann H (2013) Optical wireless communication with adaptive focus and MEMS-based beam steering. IEEE Photon Technol Lett 25:1428CrossRef

39.

Oh CW, Cao Z, Tangdiongga E, Koonen T (2016) Free-space transmission with passive 2D beam steering for multi-gigabit-per-second per-beam indoor optical wireless networks. Opt Express 24:19211CrossRef

40.

Garanovich IL, Longhi S, Sukhorukov AA, Kivshar YS (2012) Light propagation and localization in modulated photonic lattices and waveguides. Phys Rep 518:1CrossRef

41.

Kartashov YV, Vysloukh VA, Torner L (2009) Soliton shape and mobility control in optical lattices. Prog Opt 52:63CrossRef

42.

Pertsch T, Zentgraf T, Peschel U, Bräuer A, Lederer F (2002) Beam steering in waveguide arrays. Appl Phys Lett 80:3247CrossRef

43.

Agrawal GP (2019) Nonlinear fiber optics. Academic Press, San DiegoMATH

44.

Yang J (2010) Nonlinear waves in integrable and nonintegrable systems. SIAM, PhiladelphiaMATHCrossRef

45.

Peleg A, Chung Y (2012) Cross-talk dynamics of optical solitons in multichannel waveguide systems with a Ginzburg–Landau gain–loss profile. Phys Rev A 85:063828CrossRef

46.

Negres RA, Hales JM, Kobyakov A, Hagan DJ, Van Stryland EW (2002) Experiment and analysis of two-photon absorption spectroscopy using a white-light continuum probe. IEEE J Quantum Electron 38:1205CrossRef

47.

Cirloganu CM, Padilha LA, Fishman DA, Webster S, Hagan DJ, Van Stryland EW (2011) Extremely nondegenerate two-photon absorption in direct-gap semiconductors. Opt Express 19:22951CrossRef

48.

Rauscher C, Laenen R (1997) Analysis of picosecond mid-infrared pulses by two-photon absorption in germanium. J Appl Phys 81:2818CrossRef

Title: Fast two-beam collisions in a linear optical medium with weak cubic loss in spatial dimension higher than 1
Authors: Avner Peleg
Toan T. Huynh
Quan M. Nguyen
Publication date: 01-02-2022
Publisher: Springer Netherlands
Published in: Journal of Engineering Mathematics / Issue 1/2022
Print ISSN: 0022-0833
Electronic ISSN: 1573-2703
DOI: https://doi.org/10.1007/s10665-021-10206-3

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