## Abstract

The design of a complete demultiplexer based on the k-vector superprism in a 1-D slab photonic crystal is proposed. This design scales to resolve 32 channels spaced by 0.8 nm (100 GHz) in the C band for a dense wavelength division multiplexing system. It is shown that a prism area of 0.017 mm2 is sufficient for the required wavelength resolution using typical silicon-on-insulator technology and that the total chip size would be 4×3 mm2. In order to achieve this, the modest angular dispersion of a 1-D slab photonic crystal is enhanced by considerably expanding the input beam through the superprism region and employing etched mirrors to collimate and focus the light into and out of the superprism. The plane wave expansion method is used to obtain the wave vector diagram and from this we develop design equations based on conventional ray tracing. We then present an optimization approach which minimizes the prism area whilst maintaining the necessary dispersion. Finally the non-uniformity of phase velocity dispersion across the desired spectral window is addressed.

©2005 Optical Society of America

## 1. Introduction

The richness of the dispersion relation of photonic crystals [1] allows them to be used in many wavefront-shaping applications including beam steering (*S*-vector superprism) [2,3], beam collimation [4] and beam focusing [5]. This wide range of capabilities has been made feasible by a careful analysis of the type of photonic crystal and the region of the band diagram. For example, the curvature of the band diagram will determine whether the beam will be collimated or focused. The huge group velocity dispersion that arises at sharp corners of the band diagram enables the S-vector superprism [2]. As has been well demonstrated by Matsumoto and Baba [5], large group velocity dispersion does not necessarily lead to a large number of resolvable wavelengths. This is due to the fact that we can only use angular beam resolution (due to group velocity dispersion) while the beam is traveling through the photonic crystal. As soon as the beam leaves the photonic crystal, it returns almost to its original direction. The modest lateral displacement achieved in this way does not lead to a high wavelength resolution when the natural beam expansion is taken into account. Using this approach, high wavelength selectivity requires a very large beam width and consequently a large photonic crystal region. This limits the practicality of this approach for applications such as dense wavelength division multiplexing (DWDM) where a wavelength resolution of the order of 0.8 nm is required. The *k*-vector superprism effect [5–8] is a new approach which uses the phase velocity dispersion of photonic crystal. The *k*-vector superprism has advantages over the previous *S*-vector approach in that the beam separation can occur outside the photonic crystal region. In this way it is much closer to a conventional bulk dispersive prism and implies the possibility of using beam expanding and focusing optics with a small prism area. Figure 1 shows a schematic of the proposed device. In this example beam collimation and focusing is accomplished with etched mirrors (other approaches such as tapers or waveguide lenses may also be feasible).

The wave refraction behavior at the interface of the free propagation region (slab region) and the photonic crystal can be described by the wave vector diagram which represents the wave propagation constant for a given wavelength and direction of the propagation (it is also known as the equi-frequency contour diagram). We also found that a normalized wave vector diagram is best suited to our analysis. Figure 2 shows a typical 1-D slab photonic crystal normalized wave vector diagram. The band structure is essentially similar to the band structure of 1-D photonic crystals [9]. Wavefront refraction follows the direction of the phase velocity (**k**-vector) whilst the beam direction (power flow direction) follows the group velocity direction (**S**-vector). The direction of phase velocity in the photonic crystal can be determined by keeping the tangential component of the phase velocities constant on both sides of the interface (in the normalized form *n*
_{eff}sin*φ*
_{1}=*n*_{x}
, where *n*
_{eff} is the effective index of the slab region and *φ*
_{1} is the incident angle). The phase velocity can then be found from **v**
_{p}=*c*(*n*_{x}
**â**
_{x}
+*n*_{z}
**â**
_{z}
)/(${n}_{x}^{2}$
+${n}_{z}^{2}$
) where *c* is the velocity of light in vacuum. The group velocity **v**
_{g}=∇
_{k}
*ω*(*k*), however is perpendicular to the wave vector diagram [10] at the intersection point. This is illustrated in Fig. 2.

It is well-known that both high phase and group velocity dispersion occur near the band edge of 1-D photonic crystals [8]. Furthermore, the band diagram can be an open curve in the direction of grating vector. This fact potentially extends the region of operation to the second Brillion zone, which means that a wider range of angles of refraction may be feasible. Working near the band edge with a relatively sharp corner in the dispersion curve means that we need to avoid the large spatial frequency range of a narrow beam. The purpose of an input collimating mirror is to convert the input beam wavefront into a planar one with a small range of spatial frequencies. This avoids the need for the flat dispersive band diagram that would be required for narrow incident beams [8]. Note that the output beam width after the prism needs to be sufficiently large to provide the required wavelength resolution through the Rayleigh criterion. So the resolution of the demultiplexer can be enhanced if the prism expands the incident beam width considerably. Using the fact that rays follow the group velocity direction but the wavefront refraction will be in the direction of the phase velocity, the photonic crystal can be rotated in such a way that there is a large deflection angle between the incident beam and the refracted beam inside the photonic crystal. This results in an expanded beam width which amplifies the resolution of the demultiplexer.

Whilst the input facet must be large enough to cover the incident beam, the output facet of the prism needs to be of sufficient size to capture all beams in all wavelengths over the desired window of operation. Considering the group velocity dispersion of the prism near the band gap edge, the output side of the prism will usually be larger than the local beam width. The output mirror will collect the transmitted rays from the prism and focus them toward the output waveguides. The displacement of the output beam with wavelength due to the group velocity dispersion will only cause extra coma in the output waveguides, which can be mitigated by a proper mirror design.

In the remainder of this paper we present the details of the design of this system. In section two the wave diagram of 1-D superprism is investigated and we show how the optimum period and polarization for maximizing the phase velocity dispersion is obtained. For illustration purposes, a typical Silicon On Insulator (SOI) wafer is used for simulation. In section three, design equations for a demultiplexer based on *k*-vector superprism are developed. In section four, our optimization scheme that maximizes the phase velocity dispersion whilst minimizing group velocity dispersion as much as possible, is described. It will be shown in this section that maximizing angular dispersion is not necessarily desirable if minimizing the prism area is a concern. We will also show how optimum designs will use both high dispersion and the high beam expansion capability of the prism simultaneously. In section five we apply these results to the design of a 32 channel *δλ*=0.8nm (100 GHz) demultiplexer at a central wavelength of *λ*
_{0}=1.55µmand consider issues such as dispersion non-uniformity across the band of operation. Section six concludes the paper.

## 2. Optimum polarization and period

Polarization independence is an obvious advantage for telecommunications devices. However this goal remains a significant challenge for photonic crystal devices, and we have therefore selected a single polarization that maximizes the angular dispersion. It can be easily shown that the slab TM mode (electric field normal to the plane of incidence) produces a higher dispersion than the case of slab TE mode where the magnetic field is perpendicular to the plane of incidence [11].

In order to minimize the aspect ratio required in the superprism area, the slab height (or waveguide height) must be small. However, if slab height is too small, modes will leak into the upper and lower cladding regions, increasing the equivalent effective index of these regions. Therefore, we will not have sufficient contrast for high dispersion. On the other hand we need to keep the input/output waveguides effectively single mode. For our design examples we have chosen silicon on insulator (SOI) technology with a top silicon layer thickness of 0.5 µm and a Buried OXide (BOX) layer thickness of 2 µm. Figure 3, shows the cross section of the photonic crystal. *y*-axis is normal to the substrate located on *x*-*z* plane. As has been reported in [12], waveguide dimensions of 0.5 µm×0.5 µm can be effectively single mode (higher order modes can be stripped by passing through waveguide bends). The photonic crystal is uniform along the *z*-axis where the propagation constant *k*_{z}
is defined; it is periodic along the *x*-axis where the Bloch wave number *k*_{x}
is defined. A duty factor of *τ*=0.5 is typically selected.

It is possible to reduce the period of the photonic crystal down to the limits of minimum feature size of the fabrication technology. However while this will increase the transmittance, the dispersion will not be large. It is a well known fact that the maximum phase and group velocity dispersions occur near the band edge, so the period must be sufficiently large to produce at least a band gap at the desired spectral window. However in order to avoid the possibility of exciting multiple Floquet modes at the slab-prism junction (and multiple reflection angles), small periods where there is no second band are desired. It can be shown that these periods range between 264 and 550 nm for the given material system and centre wavelength.

The plane wave expansion method has been applied to obtain the propagation constant at any wave vector. In order to apply the plane wave expansion method the structure is assumed to be periodic in the *y* direction. Due to the high refractive index contrast in the *y* direction, we have observed that an artificial periodicity in the *y* direction of 8 times of the slab width is sufficient to obtain convergence. The mesh points in the *x* and *y* directions are 64 and 256 respectively. The refractive index mesh points in the *x* and *y* directions are 128 and 256 respectively. The tolerance for eigenvalue calculation is 10^{-12}. For each wave vector diagram, the entire first Brillion zone is scanned in 4×10^{4} points.

High phase velocity dispersion implies that the propagation constant *k*_{z}
should change rapidly with vacuum wave number *k*
_{0}. In Fig. 4 we plot *∂k*_{z}
/*∂k*
_{0} as a function of grating period Λ at the band edge. As the period approaches the first band gap (where the band diagram evolves from no band gap with a closed band diagram, to open band diagram with a band gap), it can be seen that the dispersion increases significantly.

Note that for periods below 264 nm there is no band gap (the band diagram is a closed curve, and the dispersion is very small). Considering the operational wavelength window of the multiplexer and wishing to retain a reasonable operating margin, we have therefore elected to work with a period of Λ=265 nm.

It is the characteristic of 1-D photonic crystal that the first band diagram can be an open curve. Using this capability, and by rotating the crystal, it is possible to excite modes in second Brillion zone, without exciting any other in the first Brillion zone. This unique capability enables us to minimize the prism area considerably by determining regions where the phase velocity dispersion is high, group velocity dispersion is low and the beam expansion capability of prism is well utilized. Our optimization scheme aims toward these goals (see section 4).

## 3. Design equations

In Fig. 5, the schematic diagram of a 1-D photonic crystal superprism is shown and the parameters which are used in this section are represented.

It is assumed that the incident light is the slab TM polarized plane wave (the wavefront after the mirror in Fig. 1 is planar). The wave is confined in the *y* direction by the refractive index contrast of the core with the upper and the lower cladding layers. The conservation of the tangential component of the wave vector through the different interfaces is the key factor determining the direction of refraction. The effective index of the TM mode of the slab and the normalized wave vector diagram of the photonic crystal is used to determine the refraction angle of the incident beam into the photonic crystal and from the photonic crystal into the slab. The rays evidently follow the group velocity directions, which usually differ from phase velocity directions (or that of the wavefront). Group velocity dispersion is defined ned as *∂φ*
_{2}/*∂λ*. The phase velocity dispersion is defined as change of deviation angle *η*=*φ*
_{4}-*φ*
_{1}+*ρ* versus wavelength (*∂η*/*∂λ*, see Fig. 5).

Using the Gaussian approximation, the optical power density in the slab region can be written as:

where *h*
_{0} is the Gaussian effective height of the slab, *θ*
_{0} is the effective Gaussian angular width of the waveguide and is given by [13]:

where *w*
_{0} is the Gaussian effective width of the waveguide at the slab edge, and *n*
_{eff} (slab) is the effective index of the slab. In order to avoid excessive crosstalk, the nominal value for the output waveguide pitch Λ
_{i}
=3.5*w*
_{0} is chosen. Knowing the angular dispersion and Λ
_{i}
, the focal length of the output mirror can be found :

where *δλ* is channel spacing. Knowing the focal length, and restricting the mirror aperture to 2*θ*
_{0}, the minimum aperture size will be:

or using Eq. (3),

The selected aperture size will truncate the field amplitude at 1.8% of its peak value, producing negligible theoretical cross talk [17]. While the minimum output aperture size (or the output size of the prism) is restricted by the angular dispersion and the channel spacing, the real aperture size needs to take the group velocity dispersion into account.

The lengths of the input and output facets of the prism, *l*
_{1} and *l*
_{2} are related to each other via

The input and output beam widths, *L*
_{1} and *L*
_{2}, are be related by the following equation (see Fig. 6):

Note that at any specific incident angle (*φ*
_{1}) the fraction in Eq. (7) is a function of wavelength because of implicit dependence of phase and group velocity dispersion in photonic crystal. The minimum and the maximum of this coefficient play an important role in the design; let us define them as *m* and *M* respectively at the corresponding transmission angle of *φ*
_{4m} and *φ*
_{4M} (see Fig. 6). In order that the prism facets are sufficiently large to cover the beam width and maintaining the required wavelength resolution, the input and output aperture sizes need to be greater than

where *L*
_{min} is obtained from Eq. (5) by averaging *∂η*/*∂λ* over the desired spectral window. The input and output prism facets need to be greater than

Equations (8) and (4) can be used to design the input and output mirrors. The output mirror profile can be optimized to reduce the effect of coma on the side channels (due to lateral displacement of the beam at the output side of the prism because of group velocity dispersion). Equation (9) can be used to minimize the prism area through *S*=sin*ρ*(*l*
_{1}
*l*
_{2}/2) where ρ is the prism apex angle (see Fig. 5).

## 4. Optimization and numerical demonstration

Having chosen the proper period and polarization, the structural parameters remain to be selected. Our goal is to design a demultiplexer that works at 1550 nm with 32 channels of 0.8 nm spacing. Minimizing the prism area (and hence minimizing the area that must be etched) is the figure of merit for this optimization. The value of prism apex angle appears to have little effect on prism area and so we have chosen a value of 60°. As before, the duty factor is set to 0.5.

Our approach is to vary the slant angle and for each slant angle we seek to find the angle of incidence that minimizes the prism area, whilst respecting all other constraints. Note that whilst the reflection coefficient at the input side of the prism is a function of incident angle (thanks to openness of the wave vector diagram of the 1-D photonic crystal) there will not be any other possibility of internal reflection (except when the incident *k*-vector is the direction of grating vector which is the case of a Bragg grating).

Figure 7 shows minimum prism size and the corresponding angular dispersion (defined as *∂η*/*∂λ*) versus various slant angle *θ* (see Fig. 5). It is noticeable that the maximum dispersion and the minimum prism area do not occur at the same slant angle. This is because the phase velocity and group velocity dispersions both affect the size of the prism. This allows us to reduce the input prism size whilst maintaining the output aperture size for the required wavelength selectivity.

As is shown in Fig. 7, the optimum slant angle occurs at *θ*=-9.9°. Other corresponding optimal parameters can be calculated easily and are listed in Table 1.

As can be see from Table 1, whilst the angular dispersion is modest compared to 2-D S-vector superprisms, the prism size is relatively small. Also due to the large beam expansion (*L*
_{2}/*L*
_{1}≈137) this angular dispersion is enough to resolve 0.8 nm wavelength spacing.

In short, our optimization scheme minimizes the *S*-vector dispersion, using a modest *k*-vector dispersion which is enhanced by the large beam expansion capability of the superprism in order to minimize the prism area.

## 5. Discussion

Phase velocity dispersion which is achievable near the band edge may be substantial, but there is no guarantee of uniformity over the window of interest, especially if the window is relatively large. Fig. 8 shows a section of the normalized wave vector diagram for the lowest, the middle and the highest wavelengths of interest that also dictates the operation of the superprism with the parameters specified in Table 1. Note that the tangential component of the incident wave vector is continuous across the boundaries. The normalized tangential component of the incident wave vector is also plotted in Fig. 8 (The operating point at the different wavelengths and the corresponding group velocity directions are also plotted). As can be seen, the operating points located at the second Brillion zone and the corresponding group velocity dispersions are small. The group velocity dispersion as demonstrated in Fig. 6 shifts the output beams, and considering the proposed structure of Fig. 1, it will lead to increased aberration in the output image. Additionally, note the non-uniformity of phase velocity dispersion in the wavelength range of interest.

The non-uniformity of the phase velocity dispersion becomes more pronounced as the spectral window becomes wider. Figure 9 shows how the angular dispersion varies with wavelength in the desired window of spectrum. Given that we need to demultiplex onto the standard DWDM grid, we need to compensate for this non-uniformity. The easiest way to do this is to make the output channel spacing non-uniform. Figure 9, also shows the output channel spacing for a uniform channel spacing of 0.8 nm. As can be seen from Fig. 9, the channel spacing increases as the channel number grows (or wavelength increases). To maintain the same output power level, the output waveguides at the slab junction must be tapered non-uniformly.

Considering the necessary fan-out for 32 channels with the channel pitch at the chip border of 20µm, the chip total size (including input and output waveguides) for the device of Table 1 would be around 4×3 mm^{2}.

There are many sources of loss in this system, some of which could be mitigated in future designs. Side wall roughness and pattern uniformity of the 1-D photonic crystal structure need to be kept as small as possible. It has been shown that for the minimum feature size of 300 nm, the scattering loss of a ridge waveguide can be small (3.38 dB/mm) [14]. It has also been shown that the loss in 2-D photonic crystal waveguides (with a feature size of 120 nm) using SOI technology is low (3.5 dB/mm) [15] and is mostly due to fabrication imperfections, which also introduce similar loss in ridge waveguides [15,16]. Some of the fabrication imperfection losses can be minimized by aligning the 1-D photonic crystal patterns along the electron beam mesh lines. Considering the absence of lateral mode confinement in our proposed structure, we therefore expect to obtain less scattering loss through the superprism region than has been previously reported for ridge waveguides. Substrate leakage loss can also be minimized by choosing thick enough substrates. Nevertheless considering the small size of the superprism, the main source of loss (also being a matter of concern to others [5]) is the beam coupling into and out of the photonic crystal to the free space propagation regions (the slab regions). Maximizing dispersion usually involves working near the band edge, where reflection is usually high. Approaches such as smoothing the transition by small airholes or projected airholes [17] have been explored for 2-D slab photonic crystals in order to maximize transmission into photonic crystals. Similar techniques together with adding a buffer layer which can act as an antireflection coating should be further explored for 1-D slab photonic crystals. Also we have not attempted to optimize the duty factor here, which may result in both higher dispersion and increased transmission. We have not yet attempted to calculate the coupling loss for this structure due to the absence of a suitable 3-D modeling technique that would be tractable for this relatively large structure and high index contrast.

## 6. Conclusion

In this paper a complete optical design of a demultiplexer based on a 1-D slab photonic crystal *k*-vector superprism has been proposed. A significant advantage of this design is that beam expansion in the prism allows a modest angular dispersion to deliver a high spectral resolution. Exploiting the beam expansion capability of the prism, an optimal design has been obtained that maximizes the phase velocity dispersion while minimizing the group velocity dispersion. The optimum design adjusts the prism area to just fit the path of the beam through the prism within a margin. It results in a prism area of 17,000 µm^{2}, which provides sufficient resolution to demultiplex 32 channels in the C band with a 0.8 nm (100Ghz) channel spacing. To the best of our knowledge, this is the first superprism design that scales to this channel spacing. The corresponding chip size is only 4×3 mm^{2}. The proposed material system is SOI, with a 0.5 µm top silicon layer. Finally we have addressed dispersion non-uniformity and have shown that it can be compensated by using a non-periodic output waveguide spacing.

## Acknowledgments

This work was supported by the Canadian National Science and Engineering Research Council (NSERC) under the strategic research networks program (Agile All-Photonic Networks) and by the Government of Quebec under the Valorization Recherche Quebec program (VRQ NanoQuebec).

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