Chiral effects on optical solitons

Mathematics and Computers in Simulation 62 (2003) 149–161

Chiral effects on optical solitons H. Torres-Silva∗ , M. Zamorano Departamento de Electronica, Facultad de Ingeniería, Universidad de Tarapacá, Av. 18 de Septiembre 2222, Casilla 6-D, Arica, Chile

Abstract A basic equation, describing the temporal evolution of fields in a chiral medium with Kerr non-linearity is used to study the effects resulting from the combined action of chirality and non-linearity on optical solitons. The spatial chirality effect is characterized through the Born–Fedorov formalism. Simulations are based in the split-step Fourier method and the numerical results show the chiral effects on solitons with circular and mixed polarization. © 2002 IMACS. Published by Elsevier Science B.V. All rights reserved. PACS: 42.21.Ja; 42.81.Gs; 42.81.Dp Keywords: Chiral; Polarization; Non-linearity; Birefringence; Propagation; Solitons

1. Introduction Non-linear effects resulting from polarization have been of great interest because they lead to various applications including pulse shaping, optical switching, intensity discriminators and all optical logic gates. In optical telecommunication devices, non-linear polarization-dependent effects are also of interest. It has been reported that polarization-dependent losses influence systems containing several elements connected by optical fibers. Also, several non-linear phenomena, including optically induced birefringence, polarization instability and solitons, have led to important advances from the fundamental as well as technological points of view. In addition, interest in non-linear polarization optics is expected to develop further, in view of the current emphasis on photonics-based technologies for information management. In fact, a polarization diversity detection system has been implemented for distributed sensing of polarization mode coupling, in high birefringence fibers, using a pump-probe architecture based on the optical Kerr effect [1]. Thus, a good understanding of polarization and its effects are fundamental to the design and characterization of various devices that use single mode optical fibers. ∗

Corresponding author. E-mail address: [email protected] (H. Torres-Silva). 0378-4754/02/$ – see front matter © 2002 IMACS. Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 4 7 5 4 ( 0 2 ) 0 0 1 7 7 - 5

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In connection with this, Hasegawa and Tappert [2] showed that under slowly varying amplitude, the propagation of an electromagnetic (EM)-pulse in a non-linear Kerr fiber mediums is governed by the completely integrable non-linear Schrödinger (NLS) equation iΨz + Ψtt + C|Ψ |2 Ψ = 0, that admits N-soliton solutions (C = constant) [3]. This was derived from the Maxwell equations under the assumption of weak linear dispersion, and Ψ is the slowly varying amplitude of the electric field of the electromagnetic wave. Later, this was experimentally verified by Mollenauer et al. [4]. For higher channel handling capacity, it is necessary to transmit pulses of the order of sub-picosecond and femtosecond frequencies. But, the propagation of such ultra short pulses experience higher order effects like third order dispersion, self steepening (SS) and stimulated Raman scattering (SRS). In this case, the wave propagation must be described by the higher order non-linear Schrödinger (HNLS) equation iΨz + Ψtt + C|Ψ |2 Ψ + i(C1 Ψttt +C2 |Ψ |2 Ψt + C3 |Ψ |2t Ψ ) = 0 [5]. For example, when C = 0 and when the two inertial contributions of the non-linear polarization namely, the stimulated Raman scattering (SRS) and self steepening (SS) are equal (C2 = C3 = 1) in the absence of third order dispersion (C1 = 0), the HNLS reduces to the completely integrable soliton, having derivative non-linear Schrödinger equation iΨz + Ψtt + C|Ψ |2 Ψ + i(|Ψ |2 Ψ )t = 0 that also admits solitons [6]. The propagation of optical pulses in birefringent fibers has become very useful in the context of non-linear directional couplers and a lot of work has been carried out recently in this direction, where the dynamical equations governing the propagation of signals, in the form
of optical solitons, reduce to the two coupled non-linear Schrödinger family of equations iΨjz + Ψjtt + Cj [ 2k=1 |Ψk |2 ]Ψj = 0, where j = 1, 2. Very recently, the propagation in birefringent optical fibers introduced the new concept of shape changing solitons that share energy amongst themselves during propagation. This energy switching behavior of optical solitons has been used for constructing all optical logic gates. The above models (both single and coupled) support propagation of optical pulses in the pico- and femtosecond ranges that emerge from high intensity lasers. In this paper, we describe a simple approach which takes into account the combined action of chirality and non-linearity. This approach may reproduce the above equations, but also gives rise to novel effects of great significance for future chiral applications. In another paper [7], we have reported on a phenomenological theory describing the self-action of electromagnetic pulses in certain chiral media. The theory is based on the Beltrami–Maxwell formalism extended to non-linear chiral media [8]. Our numerical results show the chiral effects on solitons with circular polarization, and an mixed-polarization spatial solitons in anisotropic cubic media. Chirality was first observed as optical activity and it corresponds to the rotation of the polarization plane, in a linear isotropic material. Phenomenological studies establish that the polarization plane rotation may be predicted by Maxwell’s equations, adding to the polarization P an additional term proportional to  ∇ × E. The Born–Fedorov equations, satisfying the edge conditions [9], allow us to characterize the non-linear chiral media through the equations D = n E + T ∇ × E and B = µ0 (H + T ∇ × H ), where n is the permittivity and T is the chiral coefficient. The pseudo-scalar T represents the measure of chirality and it has units of length. [7,8]. The non-local character of these equations should also be considered, since the polarization P (magnetization M) depends not only on E (H ) but also on the rotor of E (rotor of H ). Even though from an electromagnetic point of view, a homogeneous chiral material may be described by different equations [8,10], in this work, we will use the Born–Fedorov equations in optical fibers, since they are the most adequate for the applications of our interest.

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2. Basic propagation equation Using the above equations, the corresponding Maxwell’s equations are    ∂(n E) ∂  = n ∂ E + σ E + T ∇ × ∂ E ∇ × H = + σ E + T (∇ × E) ∂t ∂t ∂t ∂t

(1)

∂ B ∂ H ∂(∇ × H ) = −µ0 ∇ × E = − − µ0 T ∂t ∂t ∂t

(2)

Considering the rotor in Eq. (2), the following wave equation is obtained ∂ 2 ∇ 2 E ∂ 2 E ∂ E ∂ 2 E ∂ E = µ  + µ σ ∇ 2 E + µ0 T 2 + (µ  T + µ T )∇ × + µ σ T ∇× 0 n 0 0 n 0 0 ∂t 2 ∂t 2 ∂t ∂t 2 ∂t

(3)

Supposing that the chiral media has a Kerr type non-linearity characterized by a refraction index, such that the permittivity is [11] 2 n = 0 + 2 |E|

(4)

where 0 is the linear part and 2 the non-linear part, respectively, of n . Replacing in Eq. (3) the following wave equation is obtained [12] 2  ∂ 2 ∇ 2 E ∂ 2 E ∂ E  2 ∂ E + 2µ0 T ∇ ∇ 2 E + µ0 T 2 = µ  + µ σ  | E| + µ 0 0 0 2 ∂t 2 ∂t 2 ∂t ∂t 2 2  2   ∂ E  2 ∇ × ∂ E + µ0 σ T ∇ × ∂ E × 2 + µ0 2 T |E| 2 ∂t ∂t ∂t

(5)

The optical electric field E is represented by a right- or left-hand polarized component corresponds to (±) propagating in the z-direction  r, E(  t) = (xˆ ± jy)A( ˆ r,  t)e−j(k± z−ω0 t) = Ψ R,L e−j(k± z−ω0 t)

(6)

where Ψ R,L represents the complex envelope. In order to solve Eq. (5), the Fourier transform property (∂ 2 /∂t 2 ) ↔ (jω + ∂/∂t)2 is applied, and then ∇ 2 ↔ (jk + ∂/∂z)2 and ∇× are determined. After several algebraic manipulations the result is the following   βω02 ∂ΨR,L ωα 1 βω0 ∂ 2 2 ∂ΨR,L j (1 ± k + j |ΨR,L |2 ΨR,L = 0 (7) + T ) Ψ − |ΨR,L |2 ΨR,L + R,L 0 ∗ ∂z v ∂t 2k0 k0 ∂t 2k0 where v 2 = (1/µ0 ); α = µ0 σ ; k0 = (ω0 /v); β = µ0 2 ; z∗ = z/(1 − k02 T 2 ). In order to get the Eq. (7), the conditions of a slow varying envelope, were considered in the weak guidance approximation, given by            ∂ 2 Ψ   ∂ Ψ   ∂|Ψ |2 Ψ   ∂ 2 |Ψ |2 Ψ   ∂ Ψ             2  j2k  jω0  ;   |jω0 Ψ |;  |jω0 |Ψ |2 Ψ | 2  ∂z         ∂z ∂t ∂t ∂t  The phenomenon of dispersion is included in heuristic form. As the envelope Ψ (z, t) is the slow varying envelope en z and t, the dispersion relation k = k(ω) may be transformed to the domain of spatial

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variations by means of ω = ω − ω0 , which represents a small frequency shift of the side bands in respect to ω0 and through k = k − k0 , which represent the wave number. Using the Fourier transform for k and ω, approximating 1/v  k/ ω and by means of the Taylor series we get k=

1 ∂ ∂k ∂ 1 ∂ 2k ∂ 2 1 ∂ 3k ∂ 3 k0 ∂ = −j ; − j = 2 3 2 3 v ∂t ∂ω ∂t 2 ∂ω0 ∂t 6 ∂ω0 ∂t ω0 ∂t

(8)

replacing this operator, Eq. (7) becomes   βω02 2 ∂φ 1 ∂φ 1  ∂ 2 φ 1  ∂ 3 φ jωα j ∗ +j + k −j k + (1 ∓ Tk) φ− |φ| φ ∂z vg ∂t 2 ∂t 2 6 ∂t 3 2k0 2k0 +(1 ∓ Tk)

∂ (|φ|2 φ) ± kTk0 φ = 0 ∂t 

(9) 

where k  = ∂k/∂ω = 1/vg ; k  = ∂ 2 k/∂ω2 ; k  = ∂ 3 k/∂ω3 ; ω0 = k0 c and ΨR,L = φ. Eq. (9) describes the propagation of pulses in a chiral dispersive and non-linear optical fiber. The analysis of each term is as follows [11]: the first term represents the evolution of pulse with distance. The   second, third and fourth terms represent the dispersion of the optical fiber, that is, k  , k  , k  , respectively. The term k  (= 1/vg ) indicates that the pulses are moving with group velocity.  The dispersion of the group velocity (GVD) is represented by k  , which alters the relative phases of the  frequency components of pulses and produce its temporal widening. In silica fiber, k  is null in the region of 1.3 ␮m, positive (normal dispersion region) for values of λ less than 1.3 ␮ m, negative (anomalous dispersion region) for values higher than 1.3 ␮ m. ´´  The term k represents the slope of the group velocity dispersion, also called cubic dispersion and corresponds to a higher order dispersion; this term is important in ultra short pulses and in the second  optical window, where k  is null. It is also important in fiber with shifted dispersion in the region of 1.5 ␮m. The fifth term is associated with the attenuation of the fiber (α); here this attenuation is weighed by the chirality of the fiber. Non-linear effects are considered in |φ|2 φ and ∂/∂t (|φ|2 φ) terms and are due to the Kerr effect. This latter is characterized by having a refraction index which depends on the intensity of the applied field. In optical fibers, this refraction index means that there is a phase shift which depends on the intensity and that temporal changes of phase. Thus, the Kerr type non-linearity may alter (and widen) the frequency spectrum of the pulse. This term also depends on the chirality of the fiber. The last term, kTk0 φ, is included to account for the chirality of the fiber.

3. Non-linear Schrödinger equation In order to ease up the solution of the propagation equation, the following change of variables is introduced: t  = t − z∗ vg and z = z∗ . Taking into consideration that SRS and SS effects are not present, then k02 T 2 1 and βω0 /k0 ∂/ ∂t (|φ|2 φ) = 0.

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Thus, Eq. (9) takes the form j

βω02 ∂φ 1  ∂ 2 φ 1  ∂ 3 φ αω0 + k k − j + j (1 ∓ Tk)φ − (1 ∓ Tk)|φ|2 φ ± kTk0 φ = 0 ∂z 2 ∂t 2 6 ∂t 3 2k0 (2k0 )

(10)

Defining the new variables 2/3

ω0 q= β 1/3 φ, 1/3 (2k0 ) 

γ =

2/3

β 1/3 ω0  ξ= z, (2k0 )1/3

ω β 1/6 k   0 ,   6k (2k0 )1/3 k 

1/3

ω0

τ = √  β 1/6 t  , 1/6 k (2k0 )

1/3

1/3

C = 1 ∓ Tk,

Γ =

ω0 α , (2k0 )2/3 β 1/3

D=

kTk0 (2k0 )1/3 2/3

β 1/3 ω0

after some algebraic manipulations, the non-linear Schrödinger equation for a chiral optical fiber is obtained ∂q 1 ∂ 2q ∂ 3q j + − jγ + jΓ Cq ± Dq − C|q|2 q = 0 (11) ∂ξ 2 ∂τ 2 ∂τ 3 With T = 0, C = 1, and D = 0, we obtain the non-linear Schrödinger equation. The q variable may be a right (+) or left (−) hand polarized chiral wave. The wave number k can take any value between k0 /(1 − k0 T ) and k0 /(1 + k0 T ), which are the eingenvalues of the homogeneous Eq. (9) with φ as a constant. With γ = 0√and Γ = 0, the stationary profiles for R or L√fundamental solitons √ have the fol√ lowing forms: qR = 2(kR − D)/ cosh ( 2(kR − D)τ ) and qL = 2(kL + D)/ cosh ( 2(kL + D)τ ), respectively. Also, if we include all components of the non-linear polarization vector P NL in the cubic (3) Kerr medium, which depends of the third-order susceptibility tensor we can obtain coupled
L χ , then, 1 non-linear Schrödinger equations given by: iqrξ + 2 qrτ τ + CR [ s=R |cs qs |2 ]qr = 0 [13], (where r, s = R, L and cs is a numerical factor). These equations can be used to describe polarization changes while the beam propagates along a strongly birefringent optical fiber. Here, we can use the pertubation analysis

Fig. 1. RCP temporal evolution of first-order soliton, kT = 0.

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Fig. 2. RCP spectral evolution of first-order soliton, kT = 0.

to find an approximate analytical expression for the coupled Schrödinger equation near the bifurcation points where, besides the R and L chiral solutions, there are branches of elliptically polarized solutions. 4. Analysis of results Our numerical results are based on the split-step Fourier method, which has a linear and a non-linear operator. Thus, in Eq. (11), the non-linear operator is composed by the last term. First, we consider in our simulation that there is no coupling between R and L solutions. Eq. (11) represents the basic modeling of

Fig. 3. RCP temporal evolution of first-order soliton, kT = 0.5.

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Fig. 4. RCP spectral evolution of first-order soliton, kT = 0.5.

the wave propagation in a chiral optical fiber. It is applicable in both the second and third optical windows.  For the numerical calculations, we use k  = −17.4 ps2 /km, γ = 0 and Γ = 0, which correspond to the anomalous dispersion region for a fiber length equal to 2.9 km. Figs. 1 (temporal evolution) and 2 (spectral evolution) correspond to the first-order soliton with an input peak power P0 = 0.87 W and C = 1. Here, qR = qL . When the chirality factor increases, it is found that the R-pulse remains stable until k0 T = 0.5 (Figs. 3 and 4). This stability gives the possibility of increasing the input power and the dynamic range which preserves the nature of first order soliton.

Fig. 5. RCP temporal evolution of first-order soliton, kT = 0.3.

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Fig. 6. RCP spectral evolution of first-order soliton, kT = 0.3.

For R-pulse, we find that the soliton is stable with P0 = 1.74 W and 0 ≤ k0 T ≤ 0.3. For higher values of k0 T , the soliton shows a positive chirp at z/z0 = 1. (Figs. 5 and 6). Here we have that the chiral factor disturbs the exact balance between the GVD and SPM-induced effects. For the L-pulse the situation is as follows: when |k0 T | increases until 0.3, the intensity |qL |2 increases from one to 1.6 (Figs. 7 and 8). For higher values of k0 T , the soliton breaks and it is found that a mode conversion occurs from N = 1 to 2. Here, we can compare Fig. 9 for a N = 1 soliton with Fig. 10 which corresponds to a typical second order soliton with P0 = 3.49 W (this peak power is required to support the second order soliton without chiral effect). This effect can be explained if one considers that

Fig. 7. LCP temporal evolution of first-order soliton, kT = 0.5.

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Fig. 8. LCP temporal evolution of first-order soliton, kT = 0.3.

in this work the ratio N, between the dispersion length LD and the non-linear length LNL , is given by N = (LD /LNL )(1 ± k0 T ). Another important fact occurs when Tk0 is negative and the losses (Γ ) are included; in this case, the chirality factor can compensate the typical decrease of the pulse power in a normal optical fiber. Fig. 10 (k0 T = 0, P0 = 7.86), and Figs. 11–13 show the behavior of the third-order solitons for R and L polarization, respectively. To understand physically the evolution, it is helpful to look at the spectral evolution in Figs. 12 and 13 for the N = 3 soliton. The spectral changes result from an interplay

Fig. 9. LCP temporal evolution of first-order soliton, kT = 0.7.

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Fig. 10. RCP temporal evolution of second-order soliton, kT = 0.

between SPM, GVD and chirality. SPM generates a positive frequency chirp such, that the leading edge is red-shifted and the trailing-edge is blue-shifted from the central frequency. The SPM-induced spectral broadening is clearly seen in Figs. 11 and 12 as z/z0 = 0.2 with its typical oscillatory structure enhanced by the chiral factor. At z/z0 ∼ 0.35, anomalous GVD and positive chirality shrink the pulse as the pulse is positively chirped. With k0 T = 0.5, the R-soliton is more regular than the L-soliton.

Fig. 11. RCP temporal evolution of third-order soliton, kT = 0.3.

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Fig. 12. RCP spectral evolution of third-order soliton, kT = 0.7.

Now, we consider the system of two coupled non-linear equations. Here we can assume that the total energy of the beam is invariant, (qR2 + qL2 )dτ = UR + UL , and obtain the energy dispersion at the point of bifurcation, so that using the perturbation analysis, it is possible to find approximate expressions for elliptically polarized solutions near the bifurcation points. Let us assume that the solution is nearly circular with qL /qR 1 so the R-pulse has a sech-form solution. In this form, our numerical results for R solitons are useful to obtain the self-consistent equation for [13]: qL : j

1 ∂ 2 qL ∂qL ∂ 3 qL + − jγ + jΓ CL qL − CL ((|qL2 | + A|qR |2 )qL + B|qR2 |qL∗ ) − DqL = 0. ∂ξ 2 ∂τ 2 ∂τ 3

Fig. 13. LCP spectral evolution of third-order soliton, kT = 0.7.

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Fig. 14. Energy dispersion Urc and Url .

The inverse situation can be obtained. Here, the parameter CL is the chiral factor, the terms with coefficients A and B are referred to as cross-phase modulation and four-wave mixing (energy exchange), respectively. The coefficient B in cubic media is non-zero and can take values independent of A (∼2). This fact leads to the appearance of new soliton state with non-rotating polarization. To illustrate this fact, the curves corresponding to the fundamental solutions are shown on the energy–dispersion diagram (UR,L versus k) (Fig. 14) with D = 1. Our preliminary results show that the branches of elliptically polarized solutions (Uer , Uel ) start close to 0.5 ≤ k ≤ 1.5 for B = −1. Here, the balance between the non-linearity and chiral gyrotropy results in the existence of elliptically polarized solitons. Also, this study can be extended to linearly polarized solitons. If we consider a plane-polarized vector soliton as a sum of two R–L handed components of the same amplitude, we can isolate at least two characteristics of solitons in non-linear chiral media: (i) the polarization state of the vector soliton changes continuously by an angle-dependent on the soliton’s amplitude, and (ii) this angle depends on the spatial variable—i.e. at a fixed point in space, the polarization state must rotate with time. However, exhaustive numerical results are needed to establish the differences between linear and circular polarizations, caused by the combined action of chirality and non-linearity. 5. Conclusions In this paper we describe a simple approach, which takes into account the joint action of chirality and non-linearity of a Kerr media. The non-linear Schrödinger equation describing propagation in an

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optical fiber is obtained using the Born–Fedorov formalism to apply Beltrami–Maxwell’s postulates to non-linear chiral media. This approach reproduces early results, but also gives rise to novel effects of great significance for future chiral applications. Our simulations, based in the split-step Fourier method and the numerical results thus obtained, show the chiral effect k0 T on solitons with circular polarization and on mixed polarization spatial solitons. For high kT values, the chiral factor disturbs the balance between the GVD and SPM-induced effects in right- and left-hand polarized waves and modifies the normal ratio N, between the dispersion length LD and non-linear length LNL . The stability versus the chiral factor is better for R-solitons. In the L-pulse case, a mode conversion may occur. Also, the numerical study of propagation in birefringent optical fibers shows the existence of shape changing solitons, that share energy amongst themselves during propagation, giving rise to branches of elliptically polarized solitons. Acknowledgements This work have been supported by projects No. 8723-01, 8724-01 and “Proyecto de Internacionalización” of the Universidad de Tarapacá, and FONDECYT, Grant 1010300, Chile. References [1] S. Baker, J. Elgin, Quantum Semiclass. Opt. 10 (1998) 251. [2] A. Hasegawa, F.D. Tappert, Appl. Phys. Lett. 23 (1973) 142; A. Hasegawa, F.D. Tappert, Appl. Phys. Lett. 23 (1973) 171. [3] F.T. Hioe, Phys. Rev. Letts. 82 (1999) 1152. [4] L.F. Mollenauer, R.H. Stolen, J.P. Gordon, Phys. Rev. Lett. 45 (1980) 1095. [5] R. Radhakrishman, M. Lakshmanan, Phys. Rev. E60 (1999) 2317. [6] S.L. Liu, W.Z. Wang, Phys. Rev. E48 (1993) 3054. [7] H. Torres Silva, P.H. Sakanaka, N. Reggiani, C. Villarroel, Pramana J. Phys. 48 (1997) 1. [8] H. Torres Silva, Chiro-plasma surface wave, Advances in Complex Electromagnetic Materials, vol. 38, Kluwer Academic Publishers, The Netherlands, 1997, pp. 249. [9] A. Lakhtakia, V.K. Varadan, V.V. Varadan, Time-harmonic electromagnetic fields in chiral media, Lecture Notes in Physics, vol. 335, Springer, Berlin, 1985. [10] K. Hayata, M. Koshiba, IEEE Transactions on Microwave Theory and Techniques 43 (1995) 1814. [11] G. Agrawal, Non-linear Fiber Optics, Academic Press, New York, 1995. [12] M. Zamorano, H. Torres, C. Villarroel, R. Mexicana de Física 46 (1) (2000) 62. [13] N.N. Akhmediev, E.A. Ostrovskaya, Optics Commun. 132 (1996) 190.