Abstract

We show that light propagation in a group of degenerate modes of a multi-mode optical fiber in the presence of random mode coupling is described by a multi-component Manakov equation, thereby making multi-mode fibers the first reported physical system that admits true multi-component soliton solutions. The nonlinearity coefficient appearing in the equation is expressed rigorously in terms of the multi-mode fiber parameters.

© 2012 Optical Society of America

1. Introduction

Several years after the publication of the famous Zakharov and Shabat paper [1] which showed that the scalar nonlinear Schrödinger equation (NLSE) is integrable, Manakov [2] discovered that a special form of a two-component vector NLSE in which the nonlinearity is isotropic shares the same property. However, no practical physical system was described by this equation until Wai, Menyuk and Chen [3] found that Manakov’s equation accurately describes the propagation of a polarized optical field in single-mode birefringent optical fibers in the presence of random mode coupling. This implied that such fibers support the existence of vector solitons, as predicted by Manakov almost two decades earlier.

Recently, with fiber-communications exhausting the capacity of single-mode fibers, multimode fibers are being considered for transmission, with the purpose of increasing the throughput through spatial multiplexing. This new paradigm calls for the extension of the existing theoretical framework to the multi-mode fiber case. Random coupling of degenerate or quasi degenerate modes is a distinctive feature of these fibers [4, 5]. We have addressed the linear problem of mode coupling and modal dispersion in multi-mode fibers in a recent paper [6]. In this paper we focus on the nonlinear propagation regime, where we show that the evolution of the electric field in a group of degenerate modes of a multi-mode optical fiber is described by a generalized multi-components Manakov equation [7]. The central feature of the generalized Manakov equation is that it is integrable [7] and hence supports the propagation of solitons. Those are particle-like waveforms that remain unchanged in the process of propagation and preserve their identity when colliding with each other. It is the latter property that distinguishes between solitons and generic solitary waves, whose shape is also unaltered by propagation, but becomes corrupted when two, or more, such waveforms collide. Interestingly, solitary waves in certain multi-mode optical systems were shown to be possible using specific combinations of physical parameters [8, 9]. Yet, unlike the case which we present here, these systems are modeled by non-integrable equations and hence they do not support true soliton propagation.

We consider propagation over a group of degenerate modes in a multi-mode optical fiber. The degeneracy of the modes implies that fiber imperfections in the form of mechanical stress or manufacturing distortions produce strong random coupling between them on a length-scale typically much shorter than the length-scale of the nonlinear evolution [4]. On the other hand, the distinctive difference in wave vector between non degenerate groups of modes makes coupling between them significantly smaller and hence we neglect it in our analysis. The four degenerate LP11 modes [4] of a step-index fiber are a relevant example for a group of degenerate modes of the kind that we consider here. Larger groups of degenerate modes can be encountered in situations where multiple multi-mode fibers are combined in a multi-core fiber structure.

Starting from the coupled NLSE and assuming random mode coupling, we generalize the standard Manakov equation [3] to the multi-mode case. We verify the accuracy of the generalized Manakov equation and demonstrate the existence of multidimansional vector solitons, by solving the complete set of coupled NLSE numerically.

2. Analysis

The electric field in a group of N degenerate spatial modes is represented by a 2N-dimensional complex valued vector E⃗(z,t), which is constructed by stacking the Jones vectors of the N individual spatial modes one on top of the other. The term spatial mode is used here and throughout the paper to refer to the set of two polarization modes sharing, in the weakly guiding fiber approximation [10], the same lateral field profile. The components of E⃗(z,t) then satisfy the following set of coupled NLSE [11, 12]

Ez=iB(0)EB(1)EtiB(2)22Et2+iγjhkmCjhkmEh*EkEme^j,
where B(i)(z), i = 0, 1, 2 are 2N × 2N Hermitian matrices of components βjm(i)(z) [6] and where we use êj with j = 1, . . . , 2N to denote the set of complex orthogonal unit vectors used to represent the electric field.

The fourth term on the right-hand side of Eq. (1) represents the coupling induced by the Kerr nonlinearity of the fiber. The coefficient γ = ω0n2/cAeff is identical to the usual nonlinearity coefficient appearing in the scalar NLSE [11] of a single mode fiber, where n2 is the Kerr coefficient of glass, c is the speed of light in vacuum, and Aeff is the effective area of the fundamental mode at central frequency ω0. The dimensionless constants Cjhkm depend on the details of the spatial mode profiles and are obtained as follows [12]. Defining

Djhkm(1)=dxdy(Fj*Fm)(Fh*Fk)NjNhNkNm,Djhkm(2)=dxdy(Fj*Fh*)(FmFk)NjNhNkNm,Nn2=dxdy|Fn|2nf,
where F⃗n(x, y) describes the lateral mode profiles, and nf (x, y) is the refractive index, we obtain Cjhkm=A0[2Djhkm(1)+Djhkm(2)]/3, with A01=[2D1111(1)+D1111(2)]/3=D1111(1), so that C1111 = C2222 = 1, where the indices 1 and 2 are used to denote the two polarizations of the fundamental mode. Note that A0neff2Aeff, where neff is the effective refractive index of the fundamental mode at the central frequency ω0.

The first three terms on the right-hand side of Eq. (1) account for linear propagation and coupling [6]. While in the ideal case the matrices B(i)(z) (i = 0, 1, 2) are proportional to the identity, unavoidable position dependent perturbations result in the presence of off-diagonal terms, producing linear coupling between the various modes. The strongest coupling results from the term proportional to B(0), and in most cases of practical interest, its characteristic length-scale is shorter by orders of magnitude than the length-scale that characterizes the nonlinear evolution [11]. Under these conditions the orientation of the electric field vector, defined by E⃗/|E⃗| must be uniform, implying that the probability density of the complex field E⃗ may depend only on its modulus |E⃗|. Field propagation is then approximated by averaging the nonlinear terms with respect to this distribution. In the case of N = 1, this condition is rigorously equivalent to the fields Stokes vector’s orientation being uniformly distributed on the surface of the Poincaré sphere. As can be anticipated based on symmetry and dimensionality arguments, the averaged nonlinear term of (1) must reduce to the form jhkmCjhkmEh*EkEme^j=κ|E|2E, where κ is a dimensionless parameter that depends only on the nonlinear coupling coefficients Cjhkm, whose expression can be derived by performing scalar multiplication of both sides by E⃗ and averaging with respect to the field’s orientation. This yields κ = ∑jhkmCjhkmQjhkm, where denotes statistical averaging and Qjhkm=[Eh*EkEmEj*]/|E|4. Since E⃗ can be considered as a constant modulus vector whose orientation is uniformly distributed, the statistical average can be performed introducing an auxiliary random vector X⃗ with 2N complex, statistically independent components. The real and imaginary parts of each component Xi of X⃗ are statistically independent standard Gaussian variables having zero-mean and unit variance. It can be argued that Qjhkm=[Xh*XkXmXj*||X|2]/|X|4. Multiplying both sides by |⃗X|4 and performing another average (with respect to the square modulus |X⃗|2), we find that Qjhkm=[Xh*XkXmXj*]/[|X|4]. Since all components of X⃗ are statistically independent standard complex Gaussian variables, the numerator is 4 (δhkδjm + δhmδjk), where δij is Kronecker’s delta function. The denominator is the second moment of a chi-square random variable having 4N degrees of freedom and hence its value is 4N(4N + 2). This yields

κ=jhkmCjhkmδhkδjm+δhmδjk2N(2N+1).
Equation (1) can thus be reduced to
Ez=iB(0)EB(1)EtiB(2)22Et2+iγκ|E|2E.
This equation is a 2N component generalization of the well-known Manakov-PMD equation describing the propagation in a single-mode optical fiber with random polarization coupling [13]. Indeed, when N = 1, we have C1221 = C2112 = 2/3 and C1111 = C2222 = 1 [11], yielding κ = 8/9, as expected [13]. The non-diagonal terms in the matrices B(1) and B(2) are due to the frequency dependence of the mode coupling, which is typically only a small correction to the coupling contained in the frequency independent part B(0). By neglecting these terms and transforming Eq. (4) to a reference frame evolving with B(0), the final form of the generalized Manakov equation is obtained
Ez=βEtiβ22Et2+iγκ|E|2E.
The terms β′ and β″ are the inverse group velocity and the dispersion coefficient, respectively. The form of Eq. (5) is the multi-component version of the familiar Manakov equation [2].

3. Results

The generalized Manakov equation (5) is integrable by the inverse scattering transform [7], and hence it admits the propagation of solitons. Unlike generic solitary waveforms that were shown to exist in multi-mode fibers with certain parameter combinations [9], true solitons can only exist in the strong coupling regime, where propagation is accurately described by Eq. (5). Thus, in order to test the accuracy of Eq. (5) in describing multi-mode propagation, we verify that the fundamental soliton waveform, which is a rigorous analytical solutions of Eq. (5), indeed forms a solitary solution when the coupled NLSE (1) are solved numerically in the strong mode-coupling regime. In addition, we check that when two such waveforms are launched into different fiber modes and at different central optical frequencies, they collide elastically without leaving a trace in the form of a dispersive wave. As noted earlier, this phenomenon is a distinctive feature of solitons [1] distinguishing them from generic solitary waves. We find that Manakov solitons can be observed when the length-scale characterizing the correlation of mode coupling is smaller than the soliton length Ls = τ2/|β″| (related to the soliton period zs by zs = πLs/2).

In the simulations we consider propagation in the two degenerate LP11 modes of a step-index optical fiber, LP11(a) and LP11(b) [14]. We used a core radius of 7.5μm, a core refractive index of 1.4621, a refractive index step of 9.7 × 10−3 and a dispersion coefficient β″ = −25ps2/km. The nonlinear coefficient in our computations was n2 = 2.6 × 10−20m2W−1, corresponding to γ ≃ 0.835 W−1km−1, κ ≃ 0.76, and the effective area was of 126μm2 for the fundamental mode.

In Fig. 1(a) we consider the case in which we launch the waveform describing the fundamental soliton of the Manakov equation (5), P01/2sech(t/τ) with P0 = |β″|/(γκτ2), into each of the two modes LP11(a) and LP11(b). The two launched soliton waveforms are separated by T = 30τ in time and by Δω = 0.4/τ in frequency. As random coupling of polarizations within each spatial mode is always very fast [3], the launch polarization states of the two waveforms is immaterial. We assume a soliton half-width τh = 52.9ps corresponding to τ = 30ps. The soliton length is Ls ≃ 36km and the peak power P0 = 44mW. The correlation length of the mode coupling was taken to be Lc = 100m (Lc/Ls ≃ 2.8 × 10−3). It is evident that the launched waveforms are indeed solitary solutions in this regime and the collision between them is elastic, as expected from true soliton behavior. The collision dynamics shown in Fig. 1(a) is indistinguishable from that obtained integrating the Manakov equation with the same initial conditions.

 figure: Fig. 1

Fig. 1 (a) Collision of two solitons initially in two orthogonal modes in the case of full coupling, and initial angular frequency separation Δω = 0.4/τ. The ratio of the correlation length to the soliton length is Lc/Ls ≃ 2.8 × 10−3. (b) Collision of two solitons initially in two orthogonal modes in the case of absence of coupling, and initial angular frequency separation Δω = 0.4/τ. The time axis is normalized to the soliton time τ, the spatial axis to the soliton length Ls and the vertical axis to the soliton peak power P0.

Download Full Size | PPT Slide | PDF

In Fig. 1(b) we consider the regime in which the spatial modes are perfectly isolated so that no coupling between them occurs, although full coupling occurs between polarizations in each mode. In this case we launch the same sech-waveform, but with P0 = |β″|/(8γC3333γτ2/9), where C3333 ≃ 1.07 corresponds to self-phase modulation in the LP11 modes (see definitions around Eq. (2), where 3 is the index of mode LP11(a)). Since the launched waveform is solitary for the individual modes LP11(a) and LP11(b), the pulses propagate unperturbed until their collision. Yet, upon collision they stick to each other, forming a new (and non-solitary) pulse, while some energy is lost in the form of dispersive radiation. This behavior is in clear contrast with the elastic collision observed in Fig. 1(a), which corresponds to the strong coupling regime, in which the generalized Manakov equation holds.

In Figs. 2(a) and 2(b) we consider cases where the correlation length of the random mode coupling is Lc = 10km (Lc/Ls ≃ 2.8 × 10−1) and Lc = 100km (Lc/Ls ≃ 2.8), respectively. In the case of Lc = 10km shown in Fig. 2(a), the pulse evolution is similar to that observed in the strong coupling case of Fig. 1(a), although some perturbations to the propagating waveforms can be observed due to the insufficient averaging of the random mode coupling in this regime. These perturbations are further exacerbated when the correlation length of the mode coupling is extended to 100km, as shown in Fig. 2(b). In both Figs. 2(a) and 2(b), the launched waveforms were Manakov-equation solitons, identical to those used in Fig. 1(a).

 figure: Fig. 2

Fig. 2 (a) Collision of two solitons initially in two orthogonal modes in the same condition of Fig. 1, with ratio of the correlation length to the soliton length of Lc/Ls ≃ 2.8 × 10−1. (b) Collision of two solitons initially in two orthogonal modes in the same condition of Fig. 1, with ratio of the correlation length to the soliton length of Lc/Ls ≃ 2.8.

Download Full Size | PPT Slide | PDF

The fact that light propagating through degenerate modes in a multi-mode fiber satisfies the generalized Manakov equation can be exploited for the prediction of various properties of signal propagation. For example, the isotropy of the Manakov equation implies that collisions between two polarized pulses (i.e. pulses whose orientation E⃗(t)/|E⃗(t)| is time independent) can always be considered as occurring in a two dimensional (complex) subspace of the 2N dimensional space spanned by the entire set of modes [15]. Consequently, the description of two-pulse interactions becomes identical to the description of two-pulse interactions in single-mode fibers, allowing use of the conventional Stokes-space representation [16]. Pulse collisions can be modeled as a precession of the Stokes vector of each of the pulses about the Stokes vector of the other pulse, identically to the single-mode case described in [17, 18]. When the colliding pulses are orthogonal (implying that their Stokes vectors are antiparallel) their orientations in the 2N-dimensional space of the electric field remain unaltered. Conversely, when the input pulses are not orthogonal, their Stokes vectors are not aligned and precession in Stokes space implies that the pulses’ orientations are modified by the collision, and the distribution of the overall optical power among the various fiber modes changes. This effect, which could not be easily predicted using Eq. (1), may be of importance to space and wavelength multiplexed systems when optical nonlinearities are non negligible.

4. Conclusions

To conclude, we showed that propagation in a degenerate group of randomly coupled spatial modes of a multi-mode optical fiber is the first reported physical scenario described by the generalization [7] of Manakov’s equation [2], and hence admits vector soliton solutions. The nonlinear parameter of the equation was expressed in terms of the standard parameters of a generic optical fiber. This equation constitutes a starting point for analytical and numerical studies of nonlinear effects in multi-mode transmission. Its importance is emphasized by the massive recent interest in spatially multiplexed transmission using multi-mode and multi-core optical fibers.

Acknowledgment

This work has been carried out within an agreement funded by Alcatel-Lucent in the framework of Green Touch. MS also acknowledges financial support from Tera Santa Consortium and from the University of L’Aquila under project Re.C.O.Te.S.S.C. – P.O.R. Regione Abruzzo F.S.E. 2007–2013 Piano 2007–2008.”

References and links

1. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

2. S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).

3. P. K. A. Wai, C. R. Menyuk, and H. H. Chen, “Stability of solitons in randomly varying birefringent fibers,” Opt. Lett. 16, 1231–1233 (1991). [CrossRef]   [PubMed]  

4. R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 ×6 MIMO processing,” J. Lightwave Technol . 30, 521–531 (2012). [CrossRef]  

5. M. Salsi, C. Koebele, D. Sperti, P. Tran, H. Mardoyan, P. Brindel, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Astruc, L. Provost, and G. Charlet, “Mode division multiplexing of 2 × 100Gb/s channels using an LCOS based spatial modulator,” J. Lightwave Technol . 30, 618–623 (2012). [CrossRef]  

6. C. Antonelli, A. Mecozzi, M. Shtaif, and P. J. Winzer, “Stokes-space analysis of modal dispersion in fibers with multiple mode transmission,” Opt. Express, (submitted). [PubMed]  

7. V. G. Makhan’kov and O. K. Pashaev, “Nonlinear Schrödinger equation with noncompact isogroup,” Theor. Math. Phys. 53, 979–987 (1982). [CrossRef]  

8. A. Hasegawa , “Self-confinement of multimode optical pulse in a glass fiber,” Opt. Lett. 5, 416–417 (1980). [CrossRef]   [PubMed]  

9. B. Crosignani and P. Di Porto, “Soliton propagation in multi-mode optical fibers,” Opt. Lett. 6, 329–331 (1981). [CrossRef]   [PubMed]  

10. D. Gloge, “Weakly guiding fibers,” Appl. Opt. 10, 2252–2258 (1971). [CrossRef]   [PubMed]  

11. G. P. Agrawal, Nonlinear fiber optics, 4th ed. (Elsevier Science & Technology, 2006).

12. F. Poletti and P. Horak, “Description of ultrashort pulse propagation in multi-mode optical fibers,” J. Opt. Soc. Am. B 25, 1645–1654 (2008). [CrossRef]  

13. D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 15, 1735–1746 (1997). [CrossRef]  

14. P. Sillard, M. Bigot-Astruc, D. Boivin, H. Maerten, and L. Provost“Few-mode fiber for uncoupled mode-division multiplexing transmissions,” in 37th European Conference and Exhibition on Optical Communication (ECOC), ECOC Technical Digest (Optical Society of America, 2011), Paper Tu.5.7.

15. M. Soljačić, K. Steiglitz, S. M. Sears, M. Segev, M. H. Jakubowski, and R. Squier, “Collisions of two solitons in an arbitrary number of coupled nonlinear Schrödinger equations,” Phys. Rev. Lett. 90, 254102 (2003). [CrossRef]  

16. J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000). [CrossRef]   [PubMed]  

17. L. F. Mollenauer, J. P. Gordon, and F. Heismann, “Polarization scattering by soliton–soliton collisions,” Opt. Lett. 20, 2060–2062 (1995). [CrossRef]   [PubMed]  

18. A. Mecozzi and F. Matera, “Polarization scattering by intra-channel collisions,” Opt. Express 20, 1213–1218 (2012). [CrossRef]   [PubMed]  

References

  • View by:
  • |
  • |
  • |

  1. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).
  2. S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).
  3. P. K. A. Wai, C. R. Menyuk, and H. H. Chen, “Stability of solitons in randomly varying birefringent fibers,” Opt. Lett. 16, 1231–1233 (1991).
    [Crossref] [PubMed]
  4. R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 ×6 MIMO processing,” J. Lightwave Technol.  30, 521–531 (2012).
    [Crossref]
  5. M. Salsi, C. Koebele, D. Sperti, P. Tran, H. Mardoyan, P. Brindel, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Astruc, L. Provost, and G. Charlet, “Mode division multiplexing of 2 × 100Gb/s channels using an LCOS based spatial modulator,” J. Lightwave Technol.  30, 618–623 (2012).
    [Crossref]
  6. C. Antonelli, A. Mecozzi, M. Shtaif, and P. J. Winzer, “Stokes-space analysis of modal dispersion in fibers with multiple mode transmission,” Opt. Express, (submitted).
    [PubMed]
  7. V. G. Makhan’kov and O. K. Pashaev, “Nonlinear Schrödinger equation with noncompact isogroup,” Theor. Math. Phys. 53, 979–987 (1982).
    [Crossref]
  8. A. Hasegawa , “Self-confinement of multimode optical pulse in a glass fiber,” Opt. Lett. 5, 416–417 (1980).
    [Crossref] [PubMed]
  9. B. Crosignani and P. Di Porto, “Soliton propagation in multi-mode optical fibers,” Opt. Lett. 6, 329–331 (1981).
    [Crossref] [PubMed]
  10. D. Gloge, “Weakly guiding fibers,” Appl. Opt. 10, 2252–2258 (1971).
    [Crossref] [PubMed]
  11. G. P. Agrawal, Nonlinear fiber optics, 4th ed. (Elsevier Science & Technology, 2006).
  12. F. Poletti and P. Horak, “Description of ultrashort pulse propagation in multi-mode optical fibers,” J. Opt. Soc. Am. B 25, 1645–1654 (2008).
    [Crossref]
  13. D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 15, 1735–1746 (1997).
    [Crossref]
  14. P. Sillard, M. Bigot-Astruc, D. Boivin, H. Maerten, and L. Provost“Few-mode fiber for uncoupled mode-division multiplexing transmissions,” in 37th European Conference and Exhibition on Optical Communication (ECOC), ECOC Technical Digest (Optical Society of America, 2011), Paper Tu.5.7.
  15. M. Soljačić, K. Steiglitz, S. M. Sears, M. Segev, M. H. Jakubowski, and R. Squier, “Collisions of two solitons in an arbitrary number of coupled nonlinear Schrödinger equations,” Phys. Rev. Lett. 90, 254102 (2003).
    [Crossref]
  16. J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000).
    [Crossref] [PubMed]
  17. L. F. Mollenauer, J. P. Gordon, and F. Heismann, “Polarization scattering by soliton–soliton collisions,” Opt. Lett. 20, 2060–2062 (1995).
    [Crossref] [PubMed]
  18. A. Mecozzi and F. Matera, “Polarization scattering by intra-channel collisions,” Opt. Express 20, 1213–1218 (2012).
    [Crossref] [PubMed]

2012 (3)

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 ×6 MIMO processing,” J. Lightwave Technol.  30, 521–531 (2012).
[Crossref]

M. Salsi, C. Koebele, D. Sperti, P. Tran, H. Mardoyan, P. Brindel, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Astruc, L. Provost, and G. Charlet, “Mode division multiplexing of 2 × 100Gb/s channels using an LCOS based spatial modulator,” J. Lightwave Technol.  30, 618–623 (2012).
[Crossref]

A. Mecozzi and F. Matera, “Polarization scattering by intra-channel collisions,” Opt. Express 20, 1213–1218 (2012).
[Crossref] [PubMed]

2008 (1)

2003 (1)

M. Soljačić, K. Steiglitz, S. M. Sears, M. Segev, M. H. Jakubowski, and R. Squier, “Collisions of two solitons in an arbitrary number of coupled nonlinear Schrödinger equations,” Phys. Rev. Lett. 90, 254102 (2003).
[Crossref]

2000 (1)

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000).
[Crossref] [PubMed]

1997 (1)

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 15, 1735–1746 (1997).
[Crossref]

1995 (1)

1991 (1)

1982 (1)

V. G. Makhan’kov and O. K. Pashaev, “Nonlinear Schrödinger equation with noncompact isogroup,” Theor. Math. Phys. 53, 979–987 (1982).
[Crossref]

1981 (1)

1980 (1)

1974 (1)

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).

1972 (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

1971 (1)

Agrawal, G. P.

G. P. Agrawal, Nonlinear fiber optics, 4th ed. (Elsevier Science & Technology, 2006).

Antonelli, C.

C. Antonelli, A. Mecozzi, M. Shtaif, and P. J. Winzer, “Stokes-space analysis of modal dispersion in fibers with multiple mode transmission,” Opt. Express, (submitted).
[PubMed]

Astruc, M.

M. Salsi, C. Koebele, D. Sperti, P. Tran, H. Mardoyan, P. Brindel, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Astruc, L. Provost, and G. Charlet, “Mode division multiplexing of 2 × 100Gb/s channels using an LCOS based spatial modulator,” J. Lightwave Technol.  30, 618–623 (2012).
[Crossref]

Bigo, S.

M. Salsi, C. Koebele, D. Sperti, P. Tran, H. Mardoyan, P. Brindel, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Astruc, L. Provost, and G. Charlet, “Mode division multiplexing of 2 × 100Gb/s channels using an LCOS based spatial modulator,” J. Lightwave Technol.  30, 618–623 (2012).
[Crossref]

Bigot-Astruc, M.

P. Sillard, M. Bigot-Astruc, D. Boivin, H. Maerten, and L. Provost“Few-mode fiber for uncoupled mode-division multiplexing transmissions,” in 37th European Conference and Exhibition on Optical Communication (ECOC), ECOC Technical Digest (Optical Society of America, 2011), Paper Tu.5.7.

Boivin, D.

P. Sillard, M. Bigot-Astruc, D. Boivin, H. Maerten, and L. Provost“Few-mode fiber for uncoupled mode-division multiplexing transmissions,” in 37th European Conference and Exhibition on Optical Communication (ECOC), ECOC Technical Digest (Optical Society of America, 2011), Paper Tu.5.7.

Bolle, C.

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 ×6 MIMO processing,” J. Lightwave Technol.  30, 521–531 (2012).
[Crossref]

Boutin, A.

M. Salsi, C. Koebele, D. Sperti, P. Tran, H. Mardoyan, P. Brindel, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Astruc, L. Provost, and G. Charlet, “Mode division multiplexing of 2 × 100Gb/s channels using an LCOS based spatial modulator,” J. Lightwave Technol.  30, 618–623 (2012).
[Crossref]

Brindel, P.

M. Salsi, C. Koebele, D. Sperti, P. Tran, H. Mardoyan, P. Brindel, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Astruc, L. Provost, and G. Charlet, “Mode division multiplexing of 2 × 100Gb/s channels using an LCOS based spatial modulator,” J. Lightwave Technol.  30, 618–623 (2012).
[Crossref]

Burrows, E. C.

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 ×6 MIMO processing,” J. Lightwave Technol.  30, 521–531 (2012).
[Crossref]

Charlet, G.

M. Salsi, C. Koebele, D. Sperti, P. Tran, H. Mardoyan, P. Brindel, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Astruc, L. Provost, and G. Charlet, “Mode division multiplexing of 2 × 100Gb/s channels using an LCOS based spatial modulator,” J. Lightwave Technol.  30, 618–623 (2012).
[Crossref]

Chen, H. H.

Crosignani, B.

Di Porto, P.

Esmaeelpour, M.

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 ×6 MIMO processing,” J. Lightwave Technol.  30, 521–531 (2012).
[Crossref]

Essiambre, R.

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 ×6 MIMO processing,” J. Lightwave Technol.  30, 521–531 (2012).
[Crossref]

Gloge, D.

Gnauck, A. H.

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 ×6 MIMO processing,” J. Lightwave Technol.  30, 521–531 (2012).
[Crossref]

Gordon, J. P.

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000).
[Crossref] [PubMed]

L. F. Mollenauer, J. P. Gordon, and F. Heismann, “Polarization scattering by soliton–soliton collisions,” Opt. Lett. 20, 2060–2062 (1995).
[Crossref] [PubMed]

Hasegawa, A.

Heismann, F.

Horak, P.

Jakubowski, M. H.

M. Soljačić, K. Steiglitz, S. M. Sears, M. Segev, M. H. Jakubowski, and R. Squier, “Collisions of two solitons in an arbitrary number of coupled nonlinear Schrödinger equations,” Phys. Rev. Lett. 90, 254102 (2003).
[Crossref]

Koebele, C.

M. Salsi, C. Koebele, D. Sperti, P. Tran, H. Mardoyan, P. Brindel, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Astruc, L. Provost, and G. Charlet, “Mode division multiplexing of 2 × 100Gb/s channels using an LCOS based spatial modulator,” J. Lightwave Technol.  30, 618–623 (2012).
[Crossref]

Kogelnik, H.

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000).
[Crossref] [PubMed]

Lingle, R.

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 ×6 MIMO processing,” J. Lightwave Technol.  30, 521–531 (2012).
[Crossref]

Maerten, H.

P. Sillard, M. Bigot-Astruc, D. Boivin, H. Maerten, and L. Provost“Few-mode fiber for uncoupled mode-division multiplexing transmissions,” in 37th European Conference and Exhibition on Optical Communication (ECOC), ECOC Technical Digest (Optical Society of America, 2011), Paper Tu.5.7.

Makhan’kov, V. G.

V. G. Makhan’kov and O. K. Pashaev, “Nonlinear Schrödinger equation with noncompact isogroup,” Theor. Math. Phys. 53, 979–987 (1982).
[Crossref]

Manakov, S. V.

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).

Marcuse, D.

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 15, 1735–1746 (1997).
[Crossref]

Mardoyan, H.

M. Salsi, C. Koebele, D. Sperti, P. Tran, H. Mardoyan, P. Brindel, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Astruc, L. Provost, and G. Charlet, “Mode division multiplexing of 2 × 100Gb/s channels using an LCOS based spatial modulator,” J. Lightwave Technol.  30, 618–623 (2012).
[Crossref]

Matera, F.

McCurdy, A. H.

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 ×6 MIMO processing,” J. Lightwave Technol.  30, 521–531 (2012).
[Crossref]

Mecozzi, A.

A. Mecozzi and F. Matera, “Polarization scattering by intra-channel collisions,” Opt. Express 20, 1213–1218 (2012).
[Crossref] [PubMed]

C. Antonelli, A. Mecozzi, M. Shtaif, and P. J. Winzer, “Stokes-space analysis of modal dispersion in fibers with multiple mode transmission,” Opt. Express, (submitted).
[PubMed]

Menyuk, C. R.

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 15, 1735–1746 (1997).
[Crossref]

P. K. A. Wai, C. R. Menyuk, and H. H. Chen, “Stability of solitons in randomly varying birefringent fibers,” Opt. Lett. 16, 1231–1233 (1991).
[Crossref] [PubMed]

Mollenauer, L. F.

Mumtaz, S.

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 ×6 MIMO processing,” J. Lightwave Technol.  30, 521–531 (2012).
[Crossref]

Pashaev, O. K.

V. G. Makhan’kov and O. K. Pashaev, “Nonlinear Schrödinger equation with noncompact isogroup,” Theor. Math. Phys. 53, 979–987 (1982).
[Crossref]

Peckham, D. W.

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 ×6 MIMO processing,” J. Lightwave Technol.  30, 521–531 (2012).
[Crossref]

Poletti, F.

Provost, L.

M. Salsi, C. Koebele, D. Sperti, P. Tran, H. Mardoyan, P. Brindel, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Astruc, L. Provost, and G. Charlet, “Mode division multiplexing of 2 × 100Gb/s channels using an LCOS based spatial modulator,” J. Lightwave Technol.  30, 618–623 (2012).
[Crossref]

P. Sillard, M. Bigot-Astruc, D. Boivin, H. Maerten, and L. Provost“Few-mode fiber for uncoupled mode-division multiplexing transmissions,” in 37th European Conference and Exhibition on Optical Communication (ECOC), ECOC Technical Digest (Optical Society of America, 2011), Paper Tu.5.7.

Randel, S.

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 ×6 MIMO processing,” J. Lightwave Technol.  30, 521–531 (2012).
[Crossref]

Ryf, R.

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 ×6 MIMO processing,” J. Lightwave Technol.  30, 521–531 (2012).
[Crossref]

Salsi, M.

M. Salsi, C. Koebele, D. Sperti, P. Tran, H. Mardoyan, P. Brindel, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Astruc, L. Provost, and G. Charlet, “Mode division multiplexing of 2 × 100Gb/s channels using an LCOS based spatial modulator,” J. Lightwave Technol.  30, 618–623 (2012).
[Crossref]

Sears, S. M.

M. Soljačić, K. Steiglitz, S. M. Sears, M. Segev, M. H. Jakubowski, and R. Squier, “Collisions of two solitons in an arbitrary number of coupled nonlinear Schrödinger equations,” Phys. Rev. Lett. 90, 254102 (2003).
[Crossref]

Segev, M.

M. Soljačić, K. Steiglitz, S. M. Sears, M. Segev, M. H. Jakubowski, and R. Squier, “Collisions of two solitons in an arbitrary number of coupled nonlinear Schrödinger equations,” Phys. Rev. Lett. 90, 254102 (2003).
[Crossref]

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Shtaif, M.

C. Antonelli, A. Mecozzi, M. Shtaif, and P. J. Winzer, “Stokes-space analysis of modal dispersion in fibers with multiple mode transmission,” Opt. Express, (submitted).
[PubMed]

Sierra, A.

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 ×6 MIMO processing,” J. Lightwave Technol.  30, 521–531 (2012).
[Crossref]

Sillard, P.

M. Salsi, C. Koebele, D. Sperti, P. Tran, H. Mardoyan, P. Brindel, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Astruc, L. Provost, and G. Charlet, “Mode division multiplexing of 2 × 100Gb/s channels using an LCOS based spatial modulator,” J. Lightwave Technol.  30, 618–623 (2012).
[Crossref]

P. Sillard, M. Bigot-Astruc, D. Boivin, H. Maerten, and L. Provost“Few-mode fiber for uncoupled mode-division multiplexing transmissions,” in 37th European Conference and Exhibition on Optical Communication (ECOC), ECOC Technical Digest (Optical Society of America, 2011), Paper Tu.5.7.

Soljacic, M.

M. Soljačić, K. Steiglitz, S. M. Sears, M. Segev, M. H. Jakubowski, and R. Squier, “Collisions of two solitons in an arbitrary number of coupled nonlinear Schrödinger equations,” Phys. Rev. Lett. 90, 254102 (2003).
[Crossref]

Sperti, D.

M. Salsi, C. Koebele, D. Sperti, P. Tran, H. Mardoyan, P. Brindel, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Astruc, L. Provost, and G. Charlet, “Mode division multiplexing of 2 × 100Gb/s channels using an LCOS based spatial modulator,” J. Lightwave Technol.  30, 618–623 (2012).
[Crossref]

Squier, R.

M. Soljačić, K. Steiglitz, S. M. Sears, M. Segev, M. H. Jakubowski, and R. Squier, “Collisions of two solitons in an arbitrary number of coupled nonlinear Schrödinger equations,” Phys. Rev. Lett. 90, 254102 (2003).
[Crossref]

Steiglitz, K.

M. Soljačić, K. Steiglitz, S. M. Sears, M. Segev, M. H. Jakubowski, and R. Squier, “Collisions of two solitons in an arbitrary number of coupled nonlinear Schrödinger equations,” Phys. Rev. Lett. 90, 254102 (2003).
[Crossref]

Tran, P.

M. Salsi, C. Koebele, D. Sperti, P. Tran, H. Mardoyan, P. Brindel, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Astruc, L. Provost, and G. Charlet, “Mode division multiplexing of 2 × 100Gb/s channels using an LCOS based spatial modulator,” J. Lightwave Technol.  30, 618–623 (2012).
[Crossref]

Verluise, F.

M. Salsi, C. Koebele, D. Sperti, P. Tran, H. Mardoyan, P. Brindel, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Astruc, L. Provost, and G. Charlet, “Mode division multiplexing of 2 × 100Gb/s channels using an LCOS based spatial modulator,” J. Lightwave Technol.  30, 618–623 (2012).
[Crossref]

Wai, P. K. A.

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 15, 1735–1746 (1997).
[Crossref]

P. K. A. Wai, C. R. Menyuk, and H. H. Chen, “Stability of solitons in randomly varying birefringent fibers,” Opt. Lett. 16, 1231–1233 (1991).
[Crossref] [PubMed]

Winzer, P. J.

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 ×6 MIMO processing,” J. Lightwave Technol.  30, 521–531 (2012).
[Crossref]

C. Antonelli, A. Mecozzi, M. Shtaif, and P. J. Winzer, “Stokes-space analysis of modal dispersion in fibers with multiple mode transmission,” Opt. Express, (submitted).
[PubMed]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Appl. Opt. (1)

J. Lightwave Technol (2)

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 ×6 MIMO processing,” J. Lightwave Technol.  30, 521–531 (2012).
[Crossref]

M. Salsi, C. Koebele, D. Sperti, P. Tran, H. Mardoyan, P. Brindel, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Astruc, L. Provost, and G. Charlet, “Mode division multiplexing of 2 × 100Gb/s channels using an LCOS based spatial modulator,” J. Lightwave Technol.  30, 618–623 (2012).
[Crossref]

J. Lightwave Technol. (1)

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 15, 1735–1746 (1997).
[Crossref]

J. Opt. Soc. Am. B (1)

Opt. Express (1)

Opt. Lett. (4)

Phys. Rev. Lett. (1)

M. Soljačić, K. Steiglitz, S. M. Sears, M. Segev, M. H. Jakubowski, and R. Squier, “Collisions of two solitons in an arbitrary number of coupled nonlinear Schrödinger equations,” Phys. Rev. Lett. 90, 254102 (2003).
[Crossref]

Proc. Natl. Acad. Sci. USA (1)

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000).
[Crossref] [PubMed]

Sov. Phys. JETP (2)

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).

Theor. Math. Phys. (1)

V. G. Makhan’kov and O. K. Pashaev, “Nonlinear Schrödinger equation with noncompact isogroup,” Theor. Math. Phys. 53, 979–987 (1982).
[Crossref]

Other (3)

C. Antonelli, A. Mecozzi, M. Shtaif, and P. J. Winzer, “Stokes-space analysis of modal dispersion in fibers with multiple mode transmission,” Opt. Express, (submitted).
[PubMed]

P. Sillard, M. Bigot-Astruc, D. Boivin, H. Maerten, and L. Provost“Few-mode fiber for uncoupled mode-division multiplexing transmissions,” in 37th European Conference and Exhibition on Optical Communication (ECOC), ECOC Technical Digest (Optical Society of America, 2011), Paper Tu.5.7.

G. P. Agrawal, Nonlinear fiber optics, 4th ed. (Elsevier Science & Technology, 2006).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1 (a) Collision of two solitons initially in two orthogonal modes in the case of full coupling, and initial angular frequency separation Δω = 0.4/τ. The ratio of the correlation length to the soliton length is Lc/Ls ≃ 2.8 × 10−3. (b) Collision of two solitons initially in two orthogonal modes in the case of absence of coupling, and initial angular frequency separation Δω = 0.4/τ. The time axis is normalized to the soliton time τ, the spatial axis to the soliton length Ls and the vertical axis to the soliton peak power P0.
Fig. 2
Fig. 2 (a) Collision of two solitons initially in two orthogonal modes in the same condition of Fig. 1, with ratio of the correlation length to the soliton length of Lc/Ls ≃ 2.8 × 10−1. (b) Collision of two solitons initially in two orthogonal modes in the same condition of Fig. 1, with ratio of the correlation length to the soliton length of Lc/Ls ≃ 2.8.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

E z = i B ( 0 ) E B ( 1 ) E t i B ( 2 ) 2 2 E t 2 + i γ j h k m C j h k m E h * E k E m e ^ j ,
D j h k m ( 1 ) = d x d y ( F j * F m ) ( F h * F k ) N j N h N k N m , D j h k m ( 2 ) = d x d y ( F j * F h * ) ( F m F k ) N j N h N k N m , N n 2 = d x d y | F n | 2 n f ,
κ = j h k m C j h k m δ h k δ j m + δ h m δ j k 2 N ( 2 N + 1 ) .
E z = i B ( 0 ) E B ( 1 ) E t i B ( 2 ) 2 2 E t 2 + i γ κ | E | 2 E .
E z = β E t i β 2 2 E t 2 + i γ κ | E | 2 E .

Metrics