1. Introduction

Studies of self-similar solutions of the relevant nonlinear differential equations have been of great value in understanding widely different nonlinear physical phenomena [1]. Although self-similar solutions have been extensively studied in fields such as hydrodynamics and quantum field theory their application in optics has not been widespread. Some important results have however, been obtained, with previous theoretical studies considering self-similar behaviour in radial pattern formation [2], self-similar regime of collapse for spiral laser beams in nonlinear media [3], stimulated Raman scattering [4], the evolution of self-written waveguides [5], the formation of Cantor set fractals in soliton systems [6], the nonlinear propagation of pulses with parabolic intensity profiles in optical fibers with normal dispersion [7–10], and nonlinear compression of chirped solitary waves [11, 12].

In this paper we present the discovery of a broad class of approximate but highly accurate self-similar solutions to the generalized linear Schrödinger equation (GLSE) with distributed coefficients. This equation describes the propagation of optical pulses under the influence of dispersion and gain or loss, where all parameters are functions of the distance variable. This class also encloses the set of solitary wave solutions which describes, for example, such physically important applications as the amplification and compression of pulses in optical fibre amplifiers [13]. These linearly chirped solitary wave solutions apply in both the anomalous dispersion regime and in the normal dispersion regime where they may be contrasted with the appropriate asymptotic solutions of the generalized nonlinear Schrödinger equation [14–16]. The importance of these solutions lies in their several potential applications in optical communications systems, in the design of amplifiers and pulse compressors and in the development of tunable sources of amplitude modulated light in situations where the effects of nonlinearity can be neglected.

2. Chirped solutions of the generalized linear Schrödinger equation with distributed coefficients

In this section we search for solutions which are self-similar and have the quadratic phase with respect to the variable τ. The GLSE with distributed coefficients in the form used in fibre optics is given by :

where we suppose that parameters β and g are functions of the propagation distance z. This equation describes the amplification or attenuation (when g(z) is negative) of pulses propagating linearly in a single mode optical fibre where ψ(z,τ) is the complex envelope of the electrical field in a co-moving frame, τ is the retarded time, β(z) is the group velocity dispersion (GVD) parameter, and g(z) is the distributed gain function. The complex function ψ(z,τ) can be written as :

where U and Φ are real functions of z and τ. In general case when the coefficients of the GLSE are functions of the distance z, the amplitudeU(z,τ) of the self-similar solutions has the form :

where the scaling variable T and the function G(z) are :

Here Γ(z) and F(T) are some functions which we seek, where without loss of generality we can assume that Γ(0)=1.

Below we consider the special case when the phase has the quadratic form :

where τ_c is an arbitrary real constant. Equations(1–5) yield the functions c(z) and Γ(z) as :

where c ₀=c(0)≠0 to ensure that the phase is a quadratic function of variable (τ-τ_c ) and the function D(z) is :

Taking into account Eqs. (1–7) we find :

In the general case the coefficient in Eq. (8) is a function of the variable z but the function F(T) depends only on the scaling variable T, hence this equation has nontrivial solutions (F(T)≠0) if and only if the coefficient of the second term is the constant :

Eqs. (6, 9) yield :

We introduce the function F̃(u)=F(T), where the new variable u is :

where τ ₀ is an arbitrary parameter, hence Eq. (8) can be written as :

Here the new parameter s is connected with λ as λ=-s ²/${\tau }_0^2$. Combining Eqs. (5), (6) and (11) we can represent the phase in the form :

where Φ₀(z,τ) and Θ(z;s) are given as :

The amplitude given by Eq. (3) is :

where the function F(u;s) is defined by Eq. (13). Using these results we find in section 3 a general solution of the Eq. (1) with quadratic phase and self-similar form.

3. Self-similar linear propagation of chirped pulses in an optical fiber with varying parameters

As far as Eq. (13) has the solutions cos(us) and sin(us) using Eqs. (14–17)s we find two independent solutions of Eq. (1) as :

Due to the linearity of Eq. (1) one can write a more general solution of Eq. (1) as a linear superposition of the solutions ψ ^(c) and ψ ^(s) :

Here s is an arbitrary real parameter and $\widetilde{𝓡}$(s) is an arbitrary complex function of s. Taking into account Eq. (19) and the linearity of our problem we can represent the general solution of Eq. (1) belonging to the space L (∫^+∞-_∞| ψ(z,τ)|dτ<∞) as :

where the function $\widetilde{𝓡}$(s) can be found from initial conditions. Since u|_z=0=t where t=(τ-τ_c )/τ ₀, using Eq. (20) and Eqs. (15, 17) we can write the initial condition for the function ψ(z,τ) as :

where :

Hence Eq. (22) yields the function $\widetilde{𝓡}$(s) as the Fourier transform of the function 𝓡(t) given by initial condition (Eq. (21)) :

The general solution Eq. (20) of the GLSE with distributed coefficients Eq. (1) can be simplified for certain conditions and as result of this reduction admits a self-similar form of solution. This reduction is very interesting because of the high accuracy and simplicity of this self-similar solution and the significant number of applications of such solutions in fibre optics. Let us define the dimensionless function ρ (z) connected with the phase Θ(z; s) (Eq. (15)) as :

Using the expansion of the exponent exp(iΘ(z;s))=1+is ²/ρ(z)+… at |ρ(z)|≫1 one may find the expansion of the integral I(z,u) as :

We will consider below the case when 𝓛_D (z)≫1 where we introduced the function ${𝓛}_D\left(z\right)=\frac{1}{2}\mid \rho \left(z\right)\mid :$ :

where τ ₀ is the width of the pulse at z=0. This function 𝓛_D (z) is a generalization of the dispersion length in dimensionless form, appropriate for the propagation of chirped pulses. In the special case c ₀=0, β(z)=β ₀= constant we find 𝓛_D (z)=L_D/z where L_D =${{\tau }_0}^2$/|β ₀| is the dispersion length. The condition 𝓛_D (z)≫1 yields z≪L_D in this case, indicating that for chirp free pulses, self-similar propagation occurs for distances short compared to the dispersion length as expected. One may prove that only the first term in expansion (25) is essential when the condition 𝓛_D (z)≫1 applies. Hence in this case the general solution given by Eq. (20) reduces to a self-similar form :

where the function 𝓡(u) given by initial condition Eq. (21) (𝓡(u)=𝓡(t)|_t=u). We can see that this self-similar solution satisfies the initial condition (21) since u|_z=0=t and S(0)=1. It follows from Eqs. (20, 25) that this solution can be written more precisely as :

when the function 𝓛_D (z) is not very large compared with 1. In next section we consider numerical results relevant to the solution (27) when the condition 𝓛_D (z)≫1 is satisfied.

4. Numerical simulations of the generalized linear Schrödinger equation with distributed coefficients

In this section, we consider the numerical simulations of Eq. (1) for different initial conditions given by Eq. (21) that corresponds to the propagation of a chirped pulse in a fiber in the linear regime. Actually, we compare the analytical solution given by Eq. (27) with numerical simulations of the propagation using a split-step Fourier method for different values of the initial chirp parameter c ₀ of the input pulse. We notice that these solutions exist in both the normal and anomalous regimes for any pulse shape, so long as it is linearly chirped and the condition 𝓛_D (z)≫1 is satisfied.

We consider numerically the propagation of eigenfunctions A_n (t) of quantum harmonic oscillator because they form complete class of functions in space L ². Let us define A_n (t)=H_n (t)exp(-t ²/2), where H_n (t) are Hermite functions, then any function 𝓡(t) which is integrable with square :

may be represented as :

where :

In the following graphs we consider the function 𝓡(t)=A ₃(t) for β (z)=β ₀=-19ps ².km ^-1, g(z)=g ₀=1.8km ^-1, |c ₀|=0.05THz ² and τ ₀=30ps. Equivalent results could be obtained with any other function 𝓡(t).

 

Fig. 1. Case sign(c ₀)≠sign(β ₀). Top: 𝓛_D (z) for increasing values of c ₀. Left: A ₃(t) spreading. Right: Comparison between the input pulse (thin line), the analytical solution (thick line) and the numerical simulation (thickest line) after propagation.

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First, we consider the case where sign(c ₀)≠sign(β ₀). We can see on Fig.1 (top) that 𝓛_D (z) decreases monotonically along the fibre. In this case, the corresponding pulse spreads (Fig.1 (left)) as for a gaussian pulse [17,18]. We can also check on Fig.1 (right) that its chirp slope decreases during the propagation and both numerical simulation and analytical solution agree with high accuracy so long as the propagation length is not too long and the condition 𝓛_D (z)≫1 applies.

 

Fig. 2. Case sign(c ₀)=sign(β ₀). Top: 𝓛_D (z) for decreasing values of c0. Left: A ₃(t) compressing and then spreading. Right: Comparison between the input pulse (thin line), the analytical solution (thick line) and the numerical simulation (thickest line) after propagation. [Media 1]

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If sign(c ₀)=sign(β ₀), the function 𝓛_D (z) decreases to zero and then increases again along the propagation (Fig.2 (top)). As also is known for a gaussian pulse [17, 18], it corresponds to a compression of the pulse in the first part of the propagation and a broadening of the pulse as soon as 𝓛_D (z) is starting to increase again (Fig.2 (left)). We can check on Fig.2 (right) that its chirp parameter changes sign after propagation through the minimum pulse duration point and the numerical simulation and the analytical solution agree well both before and after the region where the compression is a maximum where the condition 𝓛_D (z)≫1 is not valid.

The plotting of the fit factor which corresponds to the misfit between the numerical result and the analytical solution also confirms in Fig.3 (left) the importance of the condition 𝓛_D (z)≫1 previously predicted theoretically. We can also notice that the difference between analytical solution and numerical result is less than 0.1% for all distances z where 𝓛_D (z)>30, as occurs in the majority of practical situations as soon as |c ₀| is not too low.

Moreover, we can deduce from Eq. (26) that lim _z→∞ 𝓛_D (z)=2${{\tau }_0}^2$|c ₀| when β (z)=constant. We can then easily verify on Fig.1 (top) and Fig.2 (top) that this limit increases with |c ₀|. Figure 3 (right) shows this limit does not change with the sign of c ₀.We can actually see that the higher is the chirp parameter of the initial pulse, the higher 𝓛_D (z) and the better the above theory is satisfied.

 

Fig. 3. Left: Misfit factor and 𝓛_D (z) function of the propagation when sign(c ₀)=sign(β ₀), c ₀=-0.05THz ². Right: 𝓛_D (z) when sign(c ₀)=sign(β ₀), sign(c ₀)≠sign(β ₀) and c ₀=0.

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The results above have been found for β (z)=β ₀= constant and g(z)=g ₀= constant but they are also valid in the particular cases where g=0 or also for β (z) and g(z) varying along z to simulate respectively a decreasing dispersion fibre and a Raman gain for example. We can then design the whole set-up to compress the pulse more efficiently and self-similarly in the case sign(c ₀)=sign(β (z)) as we can see on Fig.4 (left). The analytical solution now corresponds to the numerical result along the whole propagation in this case (Fig.4 (right)).

 

Fig. 4. Case sign(c ₀)=sign(β (z)). Left: A3(t) compressing. Right: Comparison between the input pulse (thin line), the analytical solution (thick line) and the numerical simulation (thickest line) after propagation.

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Furthermore, results for other shapes of input pulse could lead to other interesting application, such as for the generation of THz amplitude modulated light beams with tunable frequency from a GHz sinusoidal signal (Fig.5 (left)) provided that this signal can be suitably chirped.

Another application could be the compression of hyperbolic secant pulses in telecommunication fibres while remaining in the linear regime (Fig.5 (right)), since these can now be seen to behave in the same way as gaussian pulses.

We can also compare our new approximate solutions to the exact solution already found for a chirped gaussian pulse [17,18] in a fibre with constant β ₀ and no gain. Figure 6 shows that both analytical solutions fit perfectly with the numerical result when sign(c ₀)=sign(β ₀) so long as 𝓛_D (z)≫1. We would obviously get similar results in the case sign(c ₀)≠sign(β ₀).

 

Fig. 5. Compression of a linearly chirped sin function (left) and of a linearly chirped sech pulse (right) when sign(c ₀)=sign(β (z)).

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This confirms that our new analytical solution of the GLSE with distributed coefficients is a very good approximation for a propagating pulse in the linear regime of a fibre. In contrast to the exact analytical solutions already found for gaussian pulses, these new solutions apply for any shape of linearly chirped pulse in any dispersion regime and are indistinguishable from numerically generated solutions in the majority of practical situations.

 

Fig. 6. Comparison between the exact solution from [17,18] (square), the new analytical solution (circle), the numerical simulation (line) and the input (dashed line) for a propagating linearly chirped gaussian pulse when sign(c ₀)=sign(β ₀), g ₀=0 and L=1km.

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5. Discussion

In this paper we present approximate but nevertheless highly accurate self-similar solutions appropriate to the description of pulse propagation in an optical fibre under the influence of dispersion and gain or attenuation. Self-similar techniques have been applied with considerable success recently to the development of solution of the generalized nonlinear Schrödinger equation [13–16] but have not been applied to the the simple problem of pulse propagation under the influence of dispersion only.

It is well known that chirped gaussian pulses propagate self-similarly [17, 18] but we show here that essentially any linearly chirped solitary pulse, including for example a sinusoidally modulated burst of cw light, will propagate self-similarly under certain reasonable conditions. Whilst this is an approximate result it is a highly accurate one and accounts, for example, for the fact that a chirped hyperbolic secant pulse behaves just like a chirped gaussian pulse in most practical applications. Qualitatively the results can be understood on the basis of the changing separation of the different frequency components of a linearly chirped pulse under the influence of dispersion, which yields a linear variation of the group velocity of these components across the pulse. The quantitative treatment presented here, however, enables a clear determination of the regions of propagation where self-similar behavior occurs to high accuracy.

We note also that the analog in the spatial domain of chirped pulse propagation in the temporal domain is the propagation of a spherical wavefront (corresponding to a quadratic phase variation in space). Self-similar propagation in this case occurs in the far field diffraction regime providing a spatial propagation analog of Fig.2 (left).

The results presented here can be expected to assist in the design of pulse compressors and amplifiers, while these similariton solutions may also have applications in other areas, such as the development of new sources of THz modulated beams.