## Abstract

We present the analysis and design of a superprism-based demultiplexer that employs both group and phase velocity dispersion of the photonic crystal (PhC). Simultaneous diffraction compensation and spatio-angular wavelength channel separation is realized in a slab region that divides the PhC. This avoids the excessive broadening of the beams inside the PhC and enhances the achievable angular dispersion of the conventional superprism topology. As a result, a compact demultiplexer with a relaxed requirement for low divergence input beams is attained. The dynamics of the beams envelops are considered based on the curvature of the band structure. Analysis shows at least 36-fold reduction of the PhC area and much smaller propagation length in slab compared to the preconditioned superprism, based on the same design model. PhC area scales as Δω^{-2.5} with Δω being the channel spacing.

©2010 Optical Society of America

## 1. Introduction

The dispersive properties of photonic crystals (PhCs) outside the forbidden band-gap [1,2] have been extensively investigated for demultiplexing applications [3]. This was inspired by the original observation of the large beam steering inside the PhC with a slight change of the incident angle or the wavelength (termed the superprism effect [4]). The s-vector superprism uses the group velocity dispersion (GVD) or the wavelength dependence of the propagation direction of the Bloch modes to spatially separate the channels inside the PhC. However, diffraction induced broadening of the beams and the loss of angular dispersion outside the PhC limit the achievable resolution and integration of the conventional superprism in practice [5].

A number of studies have been undertaken to find an optimum compromise between dispersion and diffraction so as to minimize the PhC area required in the conventional s-vector topology, mainly by optimizing the PhC and incident beam parameters [6–8] or the PhC lattice type [9]. One of the most promising designs, to the best of our knowledge, is based on diffraction compensation [10,11] where the incident beam is first expanded by propagation in a slab region preceding the PhC (preconditioned superprism) [12]. The spatial phase shift arising from diffraction in the slab compensates for the dominant (second order) diffractive phase term of the PhC when the dispersion bands have an opposite curvature compared to the uniform medium. While a two order of magnitude reduction in the PhC area compared to the conventional superprism was demonstrated [13], the mm-scale of PhC section when scaled for dense wavelength division multiplexing (DWDM) applications and even the much larger required propagation in slab reduces the benefits of integration [12].

Another approach is to use the phase velocity dispersion of the PhC to induce GVD in the slab and separate the channels outside the PhC (k-vector superprism) [14,15]. This scheme is challenging to implement and has not received much interest in practice. The propagation direction of the beams and hence the orientation of the waveguides at the output are directly determined by the wave-vector of the PhC Bloch modes and consequently highly fabrication sensitive. Beam-shaping and focusing optics, to collimate the incident beam and project the output to the far field respectively, are also often required in a typical k-vector demultiplexer [16].

The aforementioned superprism designs are based on only one of the available dispersive properties of the PhC. The wave-vector dispersion is neglected in the s-vector, whereas the group velocity dispersion degrades the cross-talk at the output of the k-vector demultiplexer and has to be minimized [16].

We present a superprism-based demultiplexer that exploits both available dispersion sources of the PhC and combines the simplicity of the s-vector with the potentially compact PhC structure of the k-vector superprism for DWDM applications. Adding angular separation to the diffraction compensation function of the slab in the preconditioned scheme, realized by placing the slab between two PhC sections, boosts the effective angular dispersion. This leads to considerably shorter propagation lengths in the PhC and slab compared to the preconditioned superprism. The enhanced dispersion also permits narrower input beams. The all-parallel and collimated beams at the output obviate the peripheral optics commonly required in a k-vector superprism. The demultiplexer is designed by modeling the path and evolution of the beam envelope within the structure based on the gradient and curvature of the equi-frequency contour (EFC) of the dispersion bands of the PhC at each wavelength channel. Based on our analysis, the required PhC area scales with the channel spacing (Δω) as Δω^{-2.5}.

## 2. Composite superprism

#### 2.1 Structure

The proposed demultiplexer, showing the path of two adjacent wavelength channels and the variation of the beam envelope of a single channel are schematically illustrated in Fig. 1 . The structure consists of two prism-shaped sections of an identical PhC placed with parallel bases and separated by a slab region. The PhC and the incident angle are appropriately designed for negative effective index of diffraction [17], i.e. exciting the portion of EFC of the band structure with opposite curvature compared to the slab, over the bandwidth of interest.

Within the first PhC, the multiplexed diffraction-limited input beam is expanded by acquiring negative spatial phase while different wavelengths are partially separated by the GVD. Non-parallel input/output interfaces of the PhC results in wavelength dependant propagation direction in the slab. As in a k-vector superprism, phase velocity dispersion of the PhC magnifies the angular dispersion in slab. In the slab region, spatial separation of adjacent channels continues in the uniform medium with ordinary diffraction and hence after an appropriate propagation length all the beams are focused back to their original width. We also consider an additional slab length beyond the focus plane that precompensates the beams for the subsequent propagation in the second PhC section. The relatively spatially separated beams at the input of the second PhC are coupled to the same Bloch modes as in the first PhC section given the parallel PhC-slab interfaces. While propagating in the PhC, beams continue to separate and refocus with negative diffraction that cancels out the excess spectral phase gained in the slab and finally refract into separate, diffraction-limited beams propagating parallel to the incident beam at the output of the demultiplexer.

#### 2.2 Design

For a given PhC, orientation of the interfaces and the incident angle, the governing parameters of the path and envelope of a Gaussian input beam throughout the structure based on the generalized effective index model [12,17], namely the effective diffractive indices and the propagation direction within each medium are readily extracted from the curvature and gradient of the corresponding EFC of the dispersion bands at the operating point irrespective of the length scale. The excited Eigen mode in each medium is determined from the wave-front matching requirement, i.e. the continuity of the tangential component of the wave-vector along the interfaces, for each wavelength as depicted in Fig. 2 . This approximate model, in analogy with the temporal broadening of a pulse when propagating in a dispersive medium, models the spatial broadening of a beam in an arbitrary medium with a phase transfer function acting on the spatial spectrum of the beam envelope function. The end result is the same as propagation in a uniform medium having second and higher order effective indices corresponding to the curvature and higher order directional derivatives of the dispersion bands of the periodic medium. Modeling the envelope profile of the beam using this model has been verified against direct numerical simulation [17] and been used in designing the preconditioned superprism which is also experimentally demonstrated [13].

We define a unit building block of unit slab length (*L _{S}* in Fig. 3
) which is then scaled to resolve all the input channels with the required cross talk. Compensating for the second order diffraction in the PhC (diffraction compensation) sets the total PhC propagation length (i.e.

*L*in Fig. 3) of an arbitrary channel based on the corresponding second order effective diffractive index (

_{PhC}= L_{PhC1}+ L_{PhC2}*n*

_{e2}) and the beam waist (2

*w*) in the two media as in Eq. (1).

*θ*represents the propagation (group velocity) direction with respect to the interface normal in each medium as depicted in Fig. 3;

*S*and

*PhC*subscripts are used for the slab and PhC respectively. Beam waist in the PhC and slab are related to the incident waist (2

*w*

_{0}) by simple geometric relations given in Eq. (2).

In constructing the unit block, knowing the propagation directions and considering the actual trajectory of the beams, the two PhC sections are appropriately scaled to minimize the residual uncompensated diffraction due to the variation of the propagation lengths and the non-uniformity of effective indices over the considered bandwidth.

The beam profile of a single channel at the output of the unit block, shown in Fig. 1, is identical to that of a preconditioned superprism with the same PhC length; an almost diffraction-limited beam except for the effect of the uncompensated higher order diffractive phase terms presented only along the PhC portion of the path. Consequently, the design procedure of the preconditioned superprism is applicable to the composite demultiplexer by using an effective angular dispersion (Δ_{eff}) that captures the additional spatial separation gained in slab region. The effective dispersion is defined as:

*X*in Fig. 3) to the total contribution from propagation in both PhC sections (i.e. $\mathrm{\Delta}{X}_{1}+|\mathrm{\Delta}{X}_{22}-\mathrm{\Delta}{X}_{21}|$). As a result, a virtual PhC having the diffractive indices and Bloch modes (both wave-vector and the direction of the group velocity) of the actual PhC but with the effective angular dispersion reproduces the envelope and spatial separation of the beams at the output of the composite demultiplexer.

## 3. Analysis

In this section, based on the previously defined virtual PhC model, we formulate the wavelength separation and diffraction compensation conditions in terms of the physical design parameters and analyze the scaling of the composite demultiplexer with the channel spacing.

For each channel, the unit block is pre-scaled by the ratio of the propagation length in the virtual PhC to the corresponding total PhC path length in the unit block to ensure the diffraction compensation condition. The largest scaling factor over all the channels gives the final structure. Hence in this model, demultiplexer design reduces to that of setting the required propagation length in the virtual PhC to achieve spatial separation of the adjacent channels with the required crosstalk considering the effective angular dispersion and divergence of the beams. The equivalent virtual PhC representation of the unit block (corresponding to the scaling factor of one) is depicted in Fig. 3 for a single channel.

In the following we assume the third-order spatial phase term to be the dominant uncompensated diffractive effect in the PhC. This conditioned is easily verified in each case by comparing the beam broadening from the fourth-order diffraction. The analysis follows the same steps irrespective of the dominant spectral phase term by substituting the corresponding effective index. In analogy with a conventional s-vector superprism [6,12], the required PhC propagation length for an arbitrary channel is consequently determined from Eq. (4).

Where${z}_{3}=\frac{1}{2}{k}_{0}\left|{n}_{e3,PhC}\right|{w}_{PhC}^{2}$is the Rayleigh range for the third-order diffraction and ${\eta}_{3}={\mathrm{\Delta}}_{eff(\mathrm{min})}/{\delta}_{3}$the ratio of the minimum effective angular separation of each channel with its adjacent wavelengths (${\mathrm{\Delta}}_{eff(\mathrm{min})}$) and the divergence angle of the beam corresponding to the third-order diffraction in the PhC (represented by${\delta}_{3}$in Fig. 1). The latter is related to the Rayleigh range as${\delta}_{3}=2\frac{{w}_{PhC}}{{z}_{3}}$.*K*and

*H*are constants determined based on the crosstalk level and are tabulated in [6]. The third-order effective index (${n}_{e3,PhC}$) is extracted from the directional derivative of the EFC of band structure at the operating point (see Fig. 2) as given in Eq. (5).

The PhC has to be large enough to accommodate the actual beam sizes within the entire path, dominated by the second-order diffraction. An estimate of the PhC area required per wavelength channel in the composite design, noting the expansion of the beams in the first PhC section and the subsequent refocus within the second PhC region and assuming approximately equal propagation length per PhC section (i.e. half the *L _{PhC}* in Eq. (6)), is obtained from Eq. (7).

From this relation, the optimum input beam waist or the corresponding spot radius inside the PhC (see Eq. (2)) that minimizes the PhC area is readily determined from Eq. (8).

^{α}with an α factor of −0.5, −1.5 and −2.5 respectively, where$\mathrm{\Delta}=\frac{{\mathrm{\Delta}}_{eff}}{{\mathrm{\Delta}}_{PhC}}$is the enhancement of the angular separation gained in the composite demultiplexer. Given the proportionality of the effective angular channel separation to the angular dispersion of the PhC (see Eq. (3)) the PhC area in Eq. (10) scales with the channel spacing (Δ

*ω*) as $\mathrm{\Delta}{\omega}^{-2.5}$. This is the same as the preconditioned superprism compared to the Δω

^{−4}scaling of the conventional s-vector superprism.

## 4. Results

Design procedure of the composite superprism developed in the previous sections is applied for demultiplexers with 100GHz channel spacing on the standard ITU C-band wavelength grid. A typical silicon-on-insulator (SOI) platform with 250nm top silicon layer and the square lattice PhC in the lowest photonic band with TM-like polarization (E-field perpendicular to the slab) is considered.

We first determine a set of PhC candidates with potentially high dispersion in the first photonic band around 1550nm wavelength. The design space is limited to the square lattice with potentially higher bandwidth [12] with the lattice period in the typical range of 285-360nm. This is achieved by setting the hole radius for a given period so as to have a small normalized band edge (i.e. ${\kappa}_{y}\text{}=0.\text{2}{k}_{0}$at the *X* symmetry point where $a.{\kappa}_{x}\text{}=\pi $ for κ being the Bloch vector, *k _{0}* the vacuum wave-number and

*a*the lattice period) at the longest wavelength.

Band structure of the PhC is calculated using 3D plane-wave expansion method with the artificial periodicity in the out-of-plane direction [18]. Slab dispersion is also considered by using the effective index of the fundamental slab TM mode at each channel wavelength for the application of the phase matching and diffraction compensation criteria in order to define the unit block. Limiting the PhC-slab interfaces to the symmetry axes of the lattice, sets the apex angle of the PhC sections (ρ in Fig. 3) to 45° with the input interface in either Γ*X* or Γ*Μ* orientation. Finally the diffraction compensation constraint (negative second order effective diffractive index for all the channels) that sets the range of incident angles for each candidate PhC completely defines the design platform.

The optimization process aims at finding the demultiplexer with the smallest chip footprint or approximately minimum total device length within the specified platform capable of resolving the multiplexed input beams with the required crosstalk. This is achieved by finding the optimum PhC candidate, incident angle and input beam width combination that minimizes the largest aggregate propagation lengths in PhC and slab among all the channels.

In a systematic optimization, for each PhC candidate, orientation of the input interface and incident angle we construct the unit block based on the effective indices and the trajectories of all the channels numerically extracted from band structure. The two PhC sections are scaled to minimize the residual uncompensated diffraction due to the variation of the path length and effective indices over the input bandwidth.

The effective angular dispersion of the unit blocks, as defined in section 2.2, at the center wavelength is shown in Fig. 4 and compared with that of the conventional superprism topologies of the same PhC. In case of the preconditioned (s-vector) and the k-vector superprism designs, the angular dispersion is simply the group velocity dispersion of the PhC Bloch modes and the induced group velocity dispersion in the slab as a result of the wave-vector dispersion inside the PhC, respectively. For a fixed wavelength, a well-defined operating point (incident angle) of a certain PhC exhibits the maximum dispersion. The effective dispersion of the composite structure (see Eq. (3)) however, combines the angular dispersion of the PhC with the spatial channel separation within the slab that separates the PhC sections. The required propagation length in the slab is directly determined by the diffraction compensation condition. As a result of that interplay, the effective dispersion of the composite superprism exhibits a noticeably different behavior. An order of magnitude enhancement of the moderate angular dispersion of the preconditioned (s-vector) or k-vector designs within the first band is achievable using the proposed superprism. The dispersion peaks correspond to the points with strong curvature on the band structure (strong diffraction inside the PhC). As a result, the excessively long slab propagation offsets the reduction in the PhC area. Indeed the composite designs with minimum chip footprint (see Table 1 ) have moderate effective dispersion.

Next, we find the optimum incident beam width for a given unit block by considering the corresponding virtual PhC. To this end, we calculate the required propagation length in the virtual PhC and the corresponding scale factor of the unit block that resolves each pair of adjacent channels with the specified crosstalk using Eq. (5). We iterate over a range of beam widths above the minimum required waist size set by the singular point of Eq. (5). The optimum beam size minimizes the corresponding maximum scale factor found over all the channels.

The results of the optimization process for 4 and 8 channel demultiplexers with adjacent channel crosstalk of less than 20 dB are summarized in Table 1. The input PhC/slab interface is assumed along the Γ*X* axis which results in minimum non-uniformity of the effective indices. Compared with the preconditioned counterparts, also given in the table, up to 6-fold reduction in the PhC length (4-channel) with at least 36 times reduced PhC area (8-channel), smaller incident beam waist and much shorter required slab propagation are achievable using the composite superprism.

The effective index model used in this study does not provide any measure of the insertion losses, particularly due to the reflection at the PhC/slab interfaces. The composite scheme, while resulting in a smaller device footprint for a given wavelength resolution, incorporates an additional pair of PhC/slab interfaces. Hence a higher insertion loss compared to a conventional superprism with comparable dimensions is expected. Recently an s-vector demultiplexer with PhC insertion loss of less than 2 to 7 dB over 100nm bandwidth for about 100μm propagation inside the PhC and working in the first photonic band was demonstrated [9]. The experimental insertion loss of the preconditioned superprism with comparable dimensions was estimated at about 7dB [13]. Both insertion losses were obtained with no interface treatment measures to enhance the PhC slab coupling efficiency. We estimate an insertion loss of about 10dB for a 4 channel composite demultiplexer. Considering the first band operation, already demonstrated efficient reflection reduction schemes [19–21] and improvement in fabrication process, further reduction of the insertion losses of the superprism demultiplexers is foreseeable.

## 5. Conclusion

We have presented the design procedure and scaling properties of the proposed superprism-based demultiplexer topology where a slab region divides the PhC. Simultaneous spatio-angular separation and diffraction compensation in slab enhances the effective angular dispersion, reducing the on-chip footprint and relaxing the input beam size requirements without peripheral optics. The PhC area is at least 36 times smaller than that required for an equivalent preconditioned superprism. This may lead to practical superprism devices with DWDM channel spacing.

## Acknowledgements

This work was in part supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).

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