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On a new approach to optical solitons in dielectric fibers

In this letter we describe a new and simple approach for modulating the electromagnetic wave propagating in a dielectric medium in the form of solitons by considering the torque developed between the induced dipoles in the medium and external field without taking into account the nonlinear Kerr effect. A reductive perturbation method deduces the Maxwell equation coupled with the Larmor equation of the torque to the derivative nonlinear Schr\"odinger equation that admits optical solitons in the medium.

Hasegawa and Tappert [1] showed that under slowly varying amplitude electromagnetic (EM)-pulse propagating in a nonlinear Kerr fiber medium is governed by the completely integrable nonlinear Schrödinger (NLS) equation iA x +A tt +µ|A| 2 A = 0 that admits N-soliton solutions (µ=constant). This was derived from Maxwell equations under the assumption of weak linear dispersion and A is the slowly varying amplitude of the electric field of the electromagnetic wave (EMW). Later this was experimentally verified by Mollenauer et al [2]. For more channel handling capacity, it is necessary to transmit pulses at the order of sub-picosecond and femto-second frequency levels. But the propagation of such ultra short pulses (USP) experience higher order effects like third order dispersion (TOD), self steepening (SS) and stimulated Raman scattering (SRS). In this case the wave propagation is described by the higher order nonlinear Schrödinger (HNLS) equation iA x +A tt +µ|A| 2 A+ i(µ 1 A ttt +µ 2 |A| 2 A t + µ 3 |A| 2 t A) = 0. For example, when µ = 0 and when the two inertial contributions of the nonlinear polarization namely, the stimulated Raman scattering (SRS) and self steepening (SS) are equal (µ 2 = µ 3 = 1) in the absence of third order dispersion (TOD) (µ 1 = 0), the HNLS reduces to the completely integrable soliton possessing derivative nonlinear Schrödinger (DNLS) equation [3] iA x +A tt +i(|A| 2 A) t = 0, and, however, when µ = 0, it reduces to the completely integrable mixed derivative nonlinear Schrödinger (MDNLS) equation [4] iA x + A tt +µ|A| 2 A +i(|A| 2 A) t = 0 that also admits solitons. When both the TOD and SS effects are included one obtains the Hirota equation for specific parametric explains the propagation of ultra short pulses in the form of soliton. Later Sasa and Satsuma [6] proved that for a particular choice of parametric relations for TOD, SS and SRS (µ 1 = 1, µ 2 = 6, µ 3 = 3), the soliton propagation is supported by another completely integrable The propagation of optical pulses in birefringent fibers have become very useful in the context of nonlinear 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 nonlinear Schrödinger (CNLS) family of equations [7,8] This has been further extended to N-signals [9]. Very recently, the study of propagation in birefringent optical fibers introduced the new concept of shape changing solitons that share energy amongst themselves during propagation [7,8]. This energy switching behaviour of optical solitons has been used for constructing all optical logic gates [10]. The above models (both single and coupled) support propagation of optical pulses in the pico-and femto-second ranges that emerge from high intensity lasers. Now it has also been experimentally proved that visible white light emerging from an incandescent source, namely a quartz-tungsten-halogen bulb when propagating through the photo refractive crystal Sr 0.75 Ba 0.25 Nb 2 O 6 , optical solitons are formed by self trapping [11]. In another direction self modulation of quasi-monochromatic EMW into spatially coherent optical solitons in a dielectric medium is described by the NLS equation iA t + A xx + 2|A| 2 A = 0 and their higher order and coupled versions, however with an interchange of the time(t) and space(x) variables in their derivatives [12,13].
In this letter we describe a new and simple approach to modulate the EMW propagating in a dielectric medium in the form of spatially coherent optical solitons without taking into account the nonlinear Kerr effect in the medium as has been done so far but only by considering the torque associated between the induced dipole moment of the medium and the electric field component of the EM field in the form of Larmor equation. When the dielectric medium is exposed to the weak EM field the induced dipoles, due to polarization when continuously subjected to the external field, give rise to torque. To the lowest order of perturbation the torque due to the electric field component of the EM field is due to the interaction of it with the induced electric dipole moment of the medium. When the medium is anisotropically polarizable the induced dipole moment p and the electric field E are not parallel in general and the torque is given by Γ = p ∧ E. The angular momentum vector is the only physical quantity that can define a unique direction for the polarised atom of the dielectric medium. The classical time evolution equation for the angular momentum can be constructed by writing the time derivative of the angular momentum as equal to the above torque. Thus we have the gyroscopic equation or the Larmor equation for the electric field in the form ∂P(r,t) ∂t = γP ∧ E [14], where the polarization P is the sum i p i over all the atoms in a unit volume and γ is the Larmor frequency. Even though existence of anisotropic polarizability in the dielectric medium is necessary for the presence of an electric field induced torque, it is not sufficient and therefore in addition, it is required that the angular momentum vector should be neither parallel nor orthogonal to the electric field.
The propagation of EMW in the dielectric, when there are no free stationary and moving electric charges, is described by the Maxwell equations : where the fields E, B, H and D, which are represented by 3-component vectors, bear the usual text book meaning and ǫ is the dielectric constant of the medium. As we consider a dielectric medium for EMW propagation here the constitutive relations between B and H can be written as H = B µ , where µ is the magnetic permeability of the medium. Now taking curl on both sides of the first of Eq. (1b) and using the second of Eq.(1b), and after using the relations for H and D = ǫE + P, we finally obtain ∂ 2 √ µǫ and P = (P x , P y , P z ). To understand the nature of propagation of EMW in the dielectric medium we now have to solve Maxwell equations coupled with Larmor equation.
As fiber is the convenient form of dielectric medium for propagation, it is more meaningful to study the one dimensional version of these equations (say along x-direction). We further consider x as the direction of propagation of EMW, which is taken parallel to the length of the dielectric fiber. Thus we have the following set of coupled equations to be solved. and where e = (1, 0, 0) and time t has been rescaled and velocity c redefined while writing Eq.(2). Now, we solve the set of coupled equations (2) using a reductive perturbation method originally developed by Taniuti and Yajima [15] to understand the modulation of the slowly varying EMW due to nonlinear interactions in the medium. When the amplitude varies slowly over the period of the oscillation, a stretching transformation allows us to separate the system into a rapidly varying part associated with the oscillation and a slowly varying one such as the amplitude. Nonlinear terms modulate the slowly varying amplitude of the wave. Then a formal solution is given in an asymptotic expansion about a uniform value.
A similar study has been performed in the context of EMW propagation in ferromagnetic medium and different types of EM solitons were obtained depending on the nature of the medium [16,17]. We assume that for the set of coupled Eqs. (2) Here F stands for the electric field E(x, t) as well as for the polarization vector P(x, t). Due to the anisotropic character of the molecules in the medium as originally assumed here we have expanded the electric field and the polarization vectors in a nonuniform way about the constant values E 0 and P 0 respectively. The expanded electric field and polarization are functions of the slow variables τ and ξ introduced through the stretching of time (τ = ε 2 t) and the space variable (ξ = ε(x − vt)), to take care of the slow variation in amplitude [18]. Here ε is the same perturbation parameter introduced earlier and v is the group velocity of the propagating EMW. and The above definitions suggest that E x 1 = E y  [3]. It is moreover easy and straightforward to construct the N-solitons using Hirota's bilinearisation procedure [19]. Following ref. [20], we consider the nonlinear transformation ψ = gf * f 2 , so that Eq.(5) can be written in terms of the following bilinear equations.
bilinear operators [19]. In order to obtain soliton solutions, we assume the series expansions for g, f and f * as g = χg 1 + χ 3 g 3 + ..., f = 1 + χ 2 f 2 + χ 4 f 4 + ... and f * = 1 + χ 2 f * 2 + χ 4 f * 4 + ... where χ is an arbitrary parameter. Now, for example to construct a one-soliton solution we set g = χg 1 , f = 1 + χ 2 f 2 and f * = 1 + χ 2 f * 2 and then collect terms with similar powers of χ and solve the resultant equations which admit the following solutions for g, f and f * . where and Ω 1 = Ω 1R + iΩ 1I , and the constant η 1I . Thus the one soliton solution for ψ = gf f * is found to be where ). Similarly we can find two, three ... and N-soliton solutions. As the details of constructing multi-soliton solutions are very lengthy and the form of N-soliton solution is very cumbersome we are not presenting the details here and interested readers can refer to ref. [20]. Using the one soliton solution given in Eq. (8) in the definition of ψ (Eqs.(4)) the components of the electric field vector can be constructed. Thus the optical one soliton in terms of the electric field vector component of the EM field can be written as where Re(Q) and Im(Q) represent the real and imaginary parts of Q respectively and δ xα and δ yα are Kronecker delta functions.
Eqs. (9) show that the electric field component of the propagating plane EMW is mod- However, in our problem where the Kerr nonlinearity is not taken into account, but, by getting support for nonlinearity from the rotation of induced dipoles in the medium due to the torque with the external field the soliton is governed by the DNLS equation. The spatial soliton is formed when the EMW induces a waveguide in the dielectric medium via the nonlinearity of the torque induced, self trapped and in turn is guided in its own wave guide.
In nonlinear optics, multihump solitons corresponding to multimodes propagating without interference and loosing their stability are known [21,22]. Even though we get multihump (two humps) solitons, in our case it corresponds to a single mode which is again a stable soliton corresponding to the completely integrable DNLS equation. Now to understand the effect of the nonlinear term (|ψ| 2 ψ) ξ ) in the optical soliton formation during propagation of EMW we consider Eq.(5) by dropping the dispersion term [23].