1. Introduction

It is well known that light propagation through a uniform grating is governed by a set of coupled mode equations for the forward and backward propagating mode amplitudes. However, propagation through a uniform grating can also approximately be described by the nonlinear Schrödinger equation (NLSE) when the pulse spectrum is narrow enough to neglect cubic grating dispersion, and the pulse intensity is not too high [2]. This simplified description is useful since results established for the NLSE, such as the existence of solitons, can be applied to light propagation through a grating.

Many grating applications call for nonuniform gratings [3, 4, 5], such as apodized gratings. In apodized gratings the grating strength does not jump at the edges, but increases gradually until it reaches its maximum value [3, 4, 6]. The advantage of such gratings is that the reflection outside the photonic band gap is strongly reduced compared to that of uniform gratings. Pulse propagation through tapered gratings is also important in recent nonlinear grating experiments, in which short, intense light pulse are incident on an apodized grating and form a Bragg grating soliton [6].

Light propagation through nonuniform gratings can be described by the coupled mode equations with slowly varying coefficients [7]. However, the NLSE description is not so simple, because the field is expressed in the grating’s eigenfunctions which vary with position if the grating is not uniform [1, 2]. This dependence is ignored if one simply varies the NLSE coefficients. Issue like this also arise in propagation through a tapered waveguide. There the mode varies with position, and may change significantly over the structure’s length. Such problems are treated using local coupled mode theory which accounts for changes of the basis functions [8]. Here, light propagation through nonuniform gratings is treated similarly, though now the grating’s eigenstates are varying. This leads to an NLS-like equation for the field envelope [Eq. (10)], but with extra terms due to variations in the eigenstates. We also show that the intensity inside a grating is enhanced by a factor v ^-1 compared to that outside. Here v is the instantaneous velocity in units of the group velocity in the grating’s absence [Eq. (7)]. For brevity we consider only apodized gratings in which the Bragg frequency constant, as this is the type of grating used in recent experiments by Eggleton et al. [6].

2. Forward propagation

The electric field in a grating satisfies the coupled mode equations [1]

where ϵ_± are the forward and backward mode amplitudes, V is the group velocity in the grating’s absence, Γ is a nonlinear coefficient that is proportional to the nonlinear refractive index, and κ is the grating strength. For tapered gratings, κ = κ(z) by definition [7]. Though Γ also depends on z, this variation is weak and is ignored here.

We derive the approximate nonlinear Schrödinger results using a multiple scales analysis. Since this has been discussed before [1, 2], we leave the treatment here brief. The key is that the ϵ _± are written as the linear eigenfunctions of Eqs. (1) but with a slowly varying amplitude. By substituting such an ansatz into Eqs. (1) one finds that in a uniform grating the amplitude evolves according to the NLSE [1, 2].

The dispersion relation of a grating can be written as

where γ = 1/√1 - v ², and -1 < v < +1 is the group velocity dΩ₊/dQ in units of V. The eigenstates φ _± associated with these eigenvalues are also well known [1, 2]. Note that Ω² = Q ² + κ ², which is the usual way the dispersion relation is written. Since in a tapered grating κ = κ(z), then also v = v(z) and γ = γ(z). Denoting the vector with elements ϵ _± as ϵ⃗, we write

where

is the rapidly varying contribution to the phase, and a, b and c are envelopes yet to be determined. Here z = z₀ + μz ₁ + μ ² z ₂ and similarly for t, where Z ₀ and t ₀ vary on the length and time scales in the problem, respectively, whereas z ₁, t ₁ and z ₂, t ₂ describe phenomena on increasingly longer scales. In the analysis below these are all taken to be independent. Parameter μ ≪ 1 tracks the size of the various terms [1, 2]. The novel element here is that Q and φ _± vary slowly to account for the grating taper.

We substitute Eq. (3) into Eqs. (1) and collect terms with equal powers of μ. The equation at order μ is then satisfied. At order μ² two simultaneous equations result

The middle terms in these equations are new and are due to the grating variations through dv/dz ₁. Eliminating b from Eqs. (5) it is found that, to this order, a satisfies

The general solution of (6) is

where f is an arbitrary function. The numerator simply expresses that the wave propagates at its instantaneous velocity. However, the denominator is due to changes in the grating eigenstates and leads to increases in the wave intensity when propagation. This was pointed out earlier but followed from an ad hoc argument [6]; here it is proven rigorously. This enhancement is clearly important in nonlinear grating experiments [6]. Returning to Eqs. (5) we find by eliminating the ∂a/∂t ₁ terms that

We now take the multiple scales analysis to order μ ³. It is straightforward, though tedious, to show that this leads to two coupled equations for the envelopes, that are of the same general form of Eqs. (5). Eliminating c from these equations gives

Since, to this order, ∂/∂z = (∂/∂z ₀)+(∂\∂z ₁) + (∂/∂z ₂), we find by combining Eqs. (6) and (9) that envelopes a and b satisfy an equation of form of Eq. (9), but with ∂/∂z _i replaced by ∂/∂z, and similarly for t. Using then Eq. (8), leads to the final equation for a. Because of Eq. (7) it is preferable to use A = √v a rather than a, leading to

Here v' = dv/dz, and A_z = ∂A/∂z, etc. Since 1/(κγ ³) is the quadratic grating dispersion, Eq. (10) reduces to the NLSE for a uniform grating [2], for which v' = 0 and v ^-1 = 0. Note the strong v ^-1 velocity dependence of the nonlinear term in Eq. (10) due to the velocity-dependent amplitude of the envelope [see Eq. (7)].

We now find the simplest nontrivial solutions to Eq. (10). We take the field weak enough to neglect the nonlinear term. We further consider CW solutions with frequency Δ > 0 with respect to the Bragg frequency. Choosing Ω₊ = Δ, then the A_t term in Eq. (10) vanishes. Recall further that to lowest order A is constant in the limit we are considering. We therefore drop terms in Eq. (10) that enter at level μ ³ and that contain spatial derivatives of A, as these are small. We are therefore left with

Since the second and third terms in Eq. (11) are real, the lowest order effect of the taper on the CW signal is a phase change that can be calculated by quadrature. To determine its importance we consider the particular apodized grating with [6]

Note that an apodized grating usually has a uniform section between the tapers. Since in such a section v' = 0, it does not affect Eq. (11) and we neglect it here. We take κ̃L = 15 [6], and calculate arg(A). Fig. 1(a) shows the results for Δ/κ̃ = 1.512, for which the minimum velocity within the grating is v = 0.8. Fig. 1(b) is for Δ/ṽ = 1.25, for which this velocity is 0.6. The black lines are exact results, the red lines follow from Eq. (11). The agreement is clearly good in Fig. 1(a), though in Fig. 1(b) the oscillations that are missed are substantial. By considering the exact results it can also be ascertained that the variations in the modulus of A are much smaller than those in the phase. Though the results in Fig. 1 are for κ̃L = 15, those at others apodizations can be found from the scaling arg(A) ∝ 1/(ṽL).

 

Figure 1. Argument of A vs position in a tapered grating. Black lines are exact, red lines follow by integrating Eq. (11). In (a) Δ/ṽ = 1.512, in (b) Δ/vį = 1.25.

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Of course the most relevant phase is that at z = L. Fig. 2 shows the phase at z = L as a function of the normalized detuning Δ/ṽ, for ṽL = 15 as in Figs. 1. The black line gives results following by quadrature from Eq. (11). The dots give exact results obtained from Eqs. (1). Finally, the red line is the analytic approximation

obtained from Eq. (11) by expanding in $\tilde{\kappa }$/Δ. The agreement between the exact results and Eq. (11) is good, except at the smallest detunings. This is due to the grating reflection that was neglected thus far.

 

Figure 2. Absolute value of the argument of envelope A at z = L vs the normalized detuning Δ/ṽ for an apodized grating with ṽL = 15. The black line follows from Eq. (11), the dots are exact results, and the red line is analytic approximation (13).

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The phase discussed here is due purely to changes in the eigenstates. Though it is included implicitly in Eqs. (1), in the present treatment it appears explicitly.

3. Contradirectional propagation

In Section 2 we considered forward propagation, neglecting reflections. Here we estimate the reflectivity of an apodized grating. This is a key consideration in Bragg grating soliton experiments, as the apodization is applied to reduce reflection [6]. The derivation is similar to that in Section 2, except that backward propagating modes are included explicitly. The calculation also only needs to be taken to order μ ₂. We thus take

where the eigenstates φ' _± are those for backward propagation. This ansatz is substituted into Eqs. (1) and terms at equal powers of μ are collected. This results in two coupled equations for the envelopes a _± and b _±. Eliminating b _- leads to

We now assume that the reflectivity is small, and thus the forward component is taken to be unaffected by the backward propagating component. Thus the relation between a ₊ and b ₊ is still given by Eq. (8). Using this, the definitions of A, and again taking Ω₊ = Δ, so that any time derivative vanishes, Eq. (15) reduces to

where A _± = √v a _±. But (11) is an approximate relation between A _+z and A. We thus easily find an approximate expression for A _-z

Now since the reflection is taken weak, according to Eq. (7) A ₊ is constant to lowest order. We thus find a simple expression for the reflectivity R of an apodized grating

which can be evaluated by quadrature. As a general observation note that the exponential is generally rapidly varying, and so the reflectivity is small unless Δ/κ̃ → 1. Of course discontinuities in v(z) or κ'(z), caused by discontinuities in the grating strength or its first derivative κ(z) and κ'(z), can also lead to substantial reflections.

Fig. (3)(a) shows the reflectivity as a function of the relative detuning Δ/κ̃ for an apodized grating with κ̃L = 15, as in our previous examples. The black line is the exact result obtained from Eqs. (1), whereas the red line corresponds to approximate result (18). Clearly, for detuning such that Δ < 1.3κ̃ the agreement is quite good. At larger detunings, the exact results indicate resonances (for example around Δ = 1.46κ̃), which the present approximate treatment misses. Nonetheless, for modest detunings Eq. (18) is a good approximation. Fig. (3)(b) is similar to Fig. (3)(a) except that κ̃L = 100, corresponding to a smoother taper. Clearly, the agreement between the exact result and approximation (18) is excellent.

 

Figure 3. Reflectivity vs normalized detuning Δ/κ̃ for two apodized gratings. In (a) κ̃L = 15, in (b) κ̃L = 100. Black lines are exact results, red lines follow from approximation (18).

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Expressions for the reflectivity of nonuninform gratings have been found previously using the WKB method [7]. However, there only the reflectivity due to the photonic band gap, and due to jumps in the grating parameters were considered. Here we consider the more subtle effects that occur wholly outside the photonic band gap.

4. Discussion and Conclusions

In this paper the propagation of light through a tapered grating has been approached in the spirit of a local coupled mode expansion. Unusually, here the local modes are the eigenfunctions of a tapered grating. It should be noted that this is an initial investigation, and that the method needs to be studied more completely before its features can be evaluated. Nonetheless, a rigorous derivation of the enhanced electric field strength inside a tapered grating, and simple expressions for the phase evolution of the field inside the grating, and for the grating reflectivity were obtained.

Even though the method described here reduces the calculations to quadrature, the resulting integrals still needs to be performed numerically. It is likely that for gratings of the form (12) they can be written in terms of hyperelliptic functions, though this is not practical here. However, it would seem likely that good approximations can be obtained using a perturbation method, such as the method of steepest descent.

In conclusion, a novel method for the description of light propagation through tapered gratings has been presented. The method leads to a NLSE, with additional terms that are due to the fact that the eigenstates of the grating change with position. It is concluded that the propagation through a taper leads to a nontrivial phase shift. A simple, approximate expression for the grating reflectivity is also found.