1. Introduction

Optical filter design based on nearly-uniform Fiber Bragg gratings (FBGs) have recently attract much attention as it provides convenient, cost-effective, and reliable solutions to some of the important problems in the high bit-rate wavelength-division multiplexed systems [1]–[5]. All the schemes proposed so far, however, involve continuous 1-D gratings in optical fibers. It would be interesting to see how we can carry over the solution into waveguides of other geometries, e.g., is it possible to realize the same grating functionalities in a planar waveguide? In particular, given the recent progress in light confinement in 2-D photonic crystal slab waveguides [6]–[7], we would like to take the next logic step and explore the possibility of etching a discrete, nearly-periodic grating in such a structure.

As a historic note, Laso et al. first proposed a simple discretization procedure in the microwave regime, in a similar effort to set up an equivalence relation between an FBG and a discrete air-hole grating in a microstrip waveguide [8]. This scheme, while simple to implement, works only for purely periodic structures and hence is limited in its applications. In the following we propose a discretization procedure that is much broader in scope and applies to both periodic and nearly-periodic gratings.

2. Background

An 1-D FBG is characterized by the evolution of its effective refractive index n(z) along its length:

where n ₀ is the average effective refractive index and K ₀=(=2π/Λ), Δn(z), and θ(z) specify the grating parameters: Λ is the reference period, Δn(z) accounts for the local grating strength (apodization), and θ(z) determines its phase variation and local period.

Various schemes have been proposed to design continuous FBG’s that can accommodate almost any filter responses. But to the best knowledge of the authors a general scheme for designing discrete gratings is still lacking. In this paper we show that the same grating functionalities can also be realized in any planar waveguide by discretizing the continuous fiber grating into a series of air holes, and the connection between the two is established in terms of grating strength and local grating period.

3. Grating design

We draw our intuition from the discretization procedure outlined in a scheme called “layer peeling algorithm” [9], which, among others, provides a generic formalism for designing continuous, nearly-uniform gratings. Suppose we first design a continuous grating in the form of Eq. (1) (using the “layer peeling algorithm”, say) and then try to create its discrete counterpart by transforming it into a series of discrete air holes, distributed nearly uniformly along the center of a planar waveguide:

The refractive index profile should then be rewritten as:

with

where

ñ ₀=the effective refractive index of the unperturbed waveguide,

a_m=(area of the mth air hole)×(1-n_substrate),

b_m=deviation of the mth air hole from z=mΛ,

W(y)=1 if

$W\left(y\right)=1\mathrm{if}\frac{-t}{2}<y<\frac{t}{2}$

(“t” is the thickness of the planar waveguide)

=0 otherwise,

i.e., W(y) is a windowing function that is nonzero only within the thickness of the waveguide. Also embedded in Eqs. (3-a) and (3-b) is the implicit assumption that the areas of the air holes, hence the a_m’s, are small, for otherwise we would not have been able to approximate the index perturbation by δ functions. Additionally the deviation of the air holes from their respective reference locations, i.e., the b_m’s, are also taken to be small since the grating is assumed to be “nearly uniform”.

 

Fig. 1. Top view of a 3-D planar waveguide (the y-dimension is not shown). A series of air holes (with spacing ~Λ) drilled at the center of the waveguide serve as the perturbation. Note even though we have plotted here a conventional, index-confined rectangular waveguide, our analysis can also be extended to more exotic structures such as the multilayer Bragg waveguide or the 2-D photonic crystal slab waveguide.

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Comparing the functional forms of n(z) in (1) and ñ(x,y,z) in Eq. (3) we arrive at the following intuition:

where θ′(z) is the first order derivative of θ(z).

To show that n(z) and ñ(x,y,z) give the same filter response we go back to the original wave equation. Assuming linearly polarized electric field, say E⃗=E_x, the perturbed wave equation can be written [10]:

where

is the electric polarization due to the perturbation. Since P_pert(r⃗,t) is the only term in the wave equation that provides coupling between the forward and backward propagating waves, n(z) and ñ(x,y,z) are equivalent as long as they give equivalent perturbative coupling term P_pert(r,t).

For the 1-D continuous grating, assuming Δn(z)≪n ₀, P_pert(r⃗,t) is given by (1):

For sufficiently slowly-varying Δn(z) and θ(z), Δ(n ²) consists of two distinct Fourier components centered respectively at ±K ₀ (see Fig. 2). Without loss of generality we filter out the positive frequency component and apply the sampling theorem on its slowly-varying envelope (including the phase factor e^iθ(z). The end result is an alternative but equivalent expression for the perturbative component at +K ₀ :

where Λ is the sampling period and Γ is one half of the bandwidth of $\Delta {\left(n^2\right)}_{K_0}$ (see Fig. 2).

 

Fig. 2. Fourier spectrum of Δ(n ²). Valid when Δn(z) and θ(z) are slowly varying.

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For the case of a discrete grating, we once again assume Δñ(x, y, z)≪ñ ₀ and from Eqs. (3-a) and (3-b) we get:

Ignoring δ(x) and W(y) for now and once again assuming a_m ’s and b_m ’s are sufficiently slowly varying, we plot the Fourier spectrum of Δ(ñ ²) in Fig. 3. Note only the two components located at K ₀ and -K ₀ can provide coupling between the forward and backward propagating waves and hence are relevant in our analysis.

 

Fig. 3. Fourier spectrum of Δ(ñ ²). Valid when a_m ’s and b_m ’s are slowly varying. Note only the two resonant components at K ₀ and -K ₀ can provide coupling between the forward and backward propagating waves.

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Once again we filter out the component at +K ₀ with a pass-band bandwidth of 2Γ :

Equating $\Delta {\left(n^2\right)}_{K_0}$ with $\Delta {\left({\tilde{n}}^2\right)}_{K_0}$ and assuming b_m ’s are small, we obtain the following relations:

Since a_m is proportional to the area of the mth air hole, Eq. (11-a) confirms our first intuition in Eq. (4-a).

What about the second intuition in Eq. (4-b)? First of all we need to define what “local period” means. In Fig. 4 we plot a few air holes in the vicinity of the mth air hole.

 

Fig. 4. Local period at the mth air hole. The local period at a given air hole is defined to be the average of the distances to the two adjacent air holes, i.e., (Λ₁+Λ₂)/2.

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If we define the local period at the mth air hole as the average of Λ₁ and Λ₂, and recall K ₀=2π/Λ, we get:

where we have substituted in (11-b) for b_m+1 and b _m-1 and made the slowly-varying assumption on θ(z). (12) indeed confirms our second intuition in Eq. (4-b).

One last thing that needs to be addressed is the δ(x) and W(y) functions that we have deliberately left out of Δ(ñ ²) in our derivation so far. We show in the following that the inclusion of the two functions requires the introduction of an extra scaling factor γ into Δñ(x, y, z), i.e.,

where γ is presumably highly dependent on the geometry of the waveguide.

We compute γ by matching coupled mode equations. In the Appendix we derive the coupled mode equations for wave propagation in FBG’s (i.e., the continuous grating), and so long as the goal is to generate the same filter response via a discrete grating in a planar waveguide, we need to reproduce the same set of coupled mode equations for the new waveguide as well. Following through the derivation in the Appendix, we realize that we need to make the following substitution for the discrete grating:

where we have applied equality between Eqs. (7), (8) and (10). Then for the planar waveguide, equation (A-11) becomes:

Since we are considering a TE wave (i.e., E⃗=E_x)), the orthonormality relation is given by [10]:

where the integral is taken over the entire transverse plane and k_m is the propagation constant of the mth mode.

As in the Appendix we replace the ∂²/∂t² operator in Eq. (15) by -ω ², multiply both sides by E*_B(x,y)=E*_F(x,y), integrate over the entire transverse plane, and then apply the orthonormality relation (16). Collecting resonant terms in the resultant equation such as the ones that vary as $e^{-i\frac{K_{0}z}{2}}$ and comparing them with Eq. (A-12), which is the target mode equation we are hoping to reproduce, we realize the following equality needs to be satisfied:

where t is the thickness of the planar waveguide and E_m=E_B=E_F. Or in other words,

where we have approximated ω by ω ₀, the center frequency, and k_m by K ₀/2. c is the speed of light in vacuum and ñ is the effective modal index given by Eq. (3-a).

4. Conclusion

We presented in this paper an efficient scheme for designing discrete and nearly-uniform Bragg gratings in planar waveguides, which in principle are capable of performing any functionalities that can be achieved by continuous FBG’s. One limitation of our design, which we did not explore in this paper, is the potentially significant scattering losses that could be associated with the air holes. More theoretical and experimental work is needed on this subject.