Sinusoidal excitations in reduced Maxwell-Duffing model
aa r X i v : . [ n li n . PS ] N ov Sinusoidal excitations in reduced Maxwell-Duffing model
Utpal Roy , T. Soloman Raju and Prasanta K. Panigrahi ∗ Physical Research Laboratory, Navrangpura, Ahmedabad-380 009, India Physics Group, Birla Institute of Technology and Science-Pilani, Goa, 403 726, India
Sinusoidal wave solutions are obtained for reduced Maxwell-Duffing equations describing the wavepropagation in a non-resonant atomic medium. These continuous wave excitations exist when themedium is initially polarized by an electric field. Other obtained solutions include both mono-frequency and cnoidal waves.
PACS numbers: 42.50.Md, 42.65.Sf
An atomic medium in general conditions is modelled byN-level atoms. In the two-level resonant approximation,the system is characterized by the Bloch equations, whichis inaccurate in several physical situations [1, 2], likedense atomic media and systems involving three or morelevel atoms. Thus the resonant model needs to be gen-eralized and extended to the non-resonant scenario. Oneof the well studied approaches is to consider the responseof the medium as weakly nonlinear. Such situation leadsto the Duffing oscillator model, where the nonlinear re-sponse of the medium is assumed to be cubic. This is thesimplest generalization of the Lorentz model.On the other hand, Maxwell wave equation, for a lin-early polarized light, allows propagation in both the di-rections. However, this can be approximated to unidirec-tional wave propagation when anharmonic contributionto the polarization is very small. As a result, the waveequation reduces from second order to a first order equa-tion. For a non-resonant medium, this approximationresults in the reduced Maxwell-Duffing model (RMD).Different excitations in non-resonant atomic media arecurrently attracting considerable attention, because oftheir relevance to ultra-short regime. Detailed reviews ofvarious aspects of non-linear excitations in atomic mediacan be found in [3, 4]. In case of two level atoms, the lo-calized soliton solutions of the Maxwell-Bloch equationsexplained the physical phenomenon of self-induced trans-parency [5]. In the same system, general cnoidal waveshave also been found as exact solution [6, 7, 8]. It wasobserved [9] that, these waves can be naturally excited inthe presence of relaxation. Such shape preserving Jacobielliptic pulse train solutions have been experimentally ob-served [10]. More general periodic solutions in multi-levelsystems have also been reported [11].In case of Maxwell-Duffing model, a class of exact lo-calized soliton solutions have been recently obtained [12].We present here mono frequency, sinusoidal wave exci-tations for RMD system. This excitation exists only inthe presence of a polarizing background. General cnoidalwave solutions are found both with and without back-ground. ∗ e-mail: [email protected] Below a brief summery of the reduced Maxwell-Duffingmodel is presented after which a procedure to find ex-act solutions of this system through a fractional lineartransform is outlined. We then present the novel si-nusoidal wave solutions including single frequency andcnoidal waves.
Reduced Maxwell-Duffing Model:
The propagation of electromagnetic waves in a mediumis described by the wave equation: ∂ E∂z − c ∂ E∂t = 4 πc ∂ P∂t , (1)where P is polarization of the medium. For unidirec-tional wave propagation, the above equation can be re-duced to a first order equation, ∂E∂z + 1 c ∂E∂t = − πc ∂P∂t . (2)In the Duffing oscillators model, the nonlinear responseof the medium is cubic. The corresponding equation forthe motion of electrons in the presence of an externalelectric field is given by ∂ X∂t + ω X + κ X = em E. (3)Here X represents the displacement of an electron fromits equilibrium position, ω is the oscillator frequency, κ is anharmonicity constant, and m is the effective massof the electron of charge e . The medium polarization isdefined as P = neX , where n is the number density ofthe oscillators in the medium.We choose new variables τ = z/l , x = ω ( t − z/c ) andnormalize the independent variables as˜ e = E/A , q = X/X . (4)In terms of the new variables, Eq. (3) then takes the form ∂ q∂x + q + 2 µq = ˜ e, (5)where 2 µ = κ X / ø and A = mω X /e = mω e − (2 µ/ | κ | ) / . X can be expressed as X =(2 µω / | κ | ) / . Similarly Eq. (2) is transformed to ∂ ˜ e∂τ = − ∂q∂x . (6) - - - Η H a L - - - Η- - H b L FIG. 1: Mono-frequency solutions for µ = 0 . α = 1 . α = 3 .
0. The second case implies A = 0. Here l is defined as l − = 2 πne / ( mc ø ) = ø p / c ø , (7)where ω p = (4 πne /m ) / is the plasma frequency.Eq. (5) and (6) together are called reduced Maxwell-Duffing equations. For finding propagating solutions, onedefines η = x − τ /α, (8)where α is related to the velocity of the pulse. Eq. (6)can be integrated with respect to the single variable η toyield ˜ e ( t, x ) = αq ( t, x ) + C, (9)where C is a constant, which signifies the backgroundelectric field, when electron amplitude q goes to zero.Non-linear equation of motion in Eq. (5) then takes theform: d qdη + (1 − α ) q + 2 µq = C. (10)For a wave propagating in the right direction, α > µ >
0, which is considered below. The other propagationdirection can be like wise studied.
Fractional Linear Transform and the solutions:
The above Eq. (10) is in the form, which can be ob-tained from the real part of the non-linear Schrodingerequation with a source. For finding out explicit solutionswe consider Eq. (10): q ′′ + gq + ǫq = C, (11)provided g = 2 µ and ǫ = (1 − α ). Prime indicates differ-entiation with respect to η . It is known earlier [13], thatthis equation can be connected to the elliptic equation f ′′ + af + bf = 0 through the following fractional lineartransformation (FT): q ( η ) = A + Bf ( η, m ) δ Df ( η, m ) δ (12)where A , B and D are real constants, δ is an integer, and f ( η, m ) is a Jacobi elliptic function, with the modulusparameter m (0 ≤ m ≤ δ = 1 and q ( η, m ) = cn ( η, m ). Solutions for other elliptic functions also canbe studied in a similar way. I. General solution: i) The consistency conditions for m = 0 are given by A g + 2 D ( AD − B ) + Aǫ − C = 0 , (13)3 A Bg + AD (1 + 2 ǫ ) + B ( ǫ − − CD = 0 , (14)3 AB g + AD ( ǫ −
1) + BD (1 + 2 ǫ ) − CD = 0 , (15) B g + BD ǫ − CD = 0 . (16)It should be pointed out that cn ( η,
0) = cos ( η ). Onecan see from the above FT (Eq. (12)) that AD = B implies only a constant or trivial solution and is not con-sidered here. An ( AD − B ) factor can be taken out of - - Η- - H a L - - Η- - - H b L FIG. 2: Cnoidal wave solution in presence of source. (a) (Dotted line) g = 2 .
0, (solid line) g = 7 . g = 30 for ǫ = − .
0; b) (Dotted line) ǫ = − . ǫ = − . ǫ = − . g = 5. all the conditions by tactically using the first consistencyin Eq. (13). Other conditions were used to solve for theunknowns A , B and D . The source term ( C ) can bedetermined from the first condition itself. Thus, the so-lution is expressed as q ( η ) = A + Bcos ( η )1 + Dcos ( η ) , (17)where, A , B and D respectively are A = ± ( ǫ + 2) p g (1 − ǫ ) , B = − s − (1 + 2 ǫ )6 g ,and D = ± s − (1 + 2 ǫ )2(1 − ǫ ) . (18)After solving the solution parameters, the source term orthe constant electric field part can be determined fromEq. (13): C = − (1 + 2 ǫ )3 s (1 − ǫ )3 g (19)As has been mentioned earlier, for wave propagation inthe right direction, µ > α >
1, implying g > ǫ < g > ǫ < − /
2, where A , B , D are real and C is a positive quantity.It is worth pointing out that all the solutions are non-singular in nature. This is because the magnitude of D is less than unity. The solutions are depicted in Fig. 1 (a)for µ = 0 . α = 1 .
6. The dotted line corresponds tothe solution for positive value of D and the solid line isfor the negative one. The first plot is with a background, i.e. , A = 0.ii) The consistency conditions, (13-16) support solu-tion for A = 0 if the source is non-zero. In this case, ǫ = − q ( η ) = ± √ g (cid:18) cos ( η ) √ ± cos ( η ) (cid:19) , (20) which is plotted in Fig. 1 (b) for µ = 0 . α = 3 . m = 1in the fractional transform. This is expressed as q ( η ) = A + Bsech ( η )1 + Dsech ( η ) , (21)where A = ± q − (1+ ǫ )3 g D = ± q − ǫ )(1 − ǫ ) , and B = ± q ǫ − g (1 − ǫ ) , with C = − (1 + ǫ )(1 − ǫ ) / (27 g ). This solution existsfor g > ǫ < −
1. For ǫ = 2, the solution goes tothe one, with B = 0. All the solutions are non-singularexcept for B = 0 and D = −√
2, which implies a singularone, signifying self focussing effect.iv) We now analyze the nature of the solutions in theabsence of a polarizing background: C = 0. The resul-tant dynamical equation is the real part of the well knownnon-linear Schr¨odinger equation. In this case, the peri-odic solution for the electron amplitude is of quadraticfractional type ( δ = 2): A = p − /g , B = 0, D = − ǫ = 4. The solution is singular one, implying an in-stability in the electron amplitude or the self focussingof the electric field. This happens for any value of µ < α = − m = 1 / A g + D ( AD − B ) + Aǫ − C = 0 , A Bg + (2 AD + B ) ǫ − CD = 0 , AB g + D ( AD + 2 B ) ǫ − CD = 0 ,B g + BD ǫ + ( AD − B ) − CD = 0 . These equations lead to, A = q − E gǫ , B = − ǫg q gF E , D = q − F ǫ and C = √ q − ǫg (cid:16) ǫ − F √ E (cid:17) , where E = 9 + 2 ǫ − √ ǫ and F = √ ǫ − E and F are positive for all ǫ = 0. These conoidal wavesexist in the domain ǫ < g >
0. Solutions arepicturised in Fig. 2, where the variations of the electronamplitude with g and ǫ are displayed. As µ increases theamplitudes diminish for a fixed value of α = 6 (Fig. 2(a)).Fig. 2(b) shows the nature of the solutions with α forcertain value of µ = 2 .
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