Preconditioned superprism-based photonic crystal

Preconditioned superprism-based photonic crystal demultiplexers: analysis and design Babak Momeni and Ali Adibi

We present the analysis and design of a new type of photonic crystal (PC) demultiplexers (i.e., preconditioned demultiplexer), in which the simultaneous existence of the superprism effect and the negative effective index for diffraction results in a compact structure by canceling the second-order spectral phase to avoid beam broadening inside the PC. This approach considerably relaxes the requirements for the large area of the structure and the small divergence of the input beam. As a result, the size of the preconditioned demultiplexers varies as N 2.5 (N being the number of wavelength channels) compared to the N4 variation in the conventional superprism-based PC demultiplexers. We use a generalized effective index model to analyze, design, and optimize these demultiplexing structures. This approximate model can be used to extract all the basic properties of the PC device simply from the band structure and eliminates the need to go through tedious simulations especially for three-dimensional structures. Our results show that the preconditioned superprism-based PC demultiplexers have 2 orders of magnitude smaller size compared to the conventional ones. © 2006 Optical Society of America OCIS codes: 130.3120, 350.5500, 050.1940.

1. Introduction

Periodic subwavelength features in a photonic crystal (PC) structure can be engineered to synthesize new optical materials with properties not accessible within natural optical materials.1–5 In this view, applications based on unique dispersive properties of PCs for electromagnetic waves propagating inside the periodic region have received considerable attention recently, and in particular the superprism effect in photonic crystals has been considered for realizing compact optical wavelength demultiplexers. Unique dispersive properties of PCs are the result of the electromagnetic modes of the PCs being adapted to the periodicity that is artificially introduced in the medium. Band folding, band crossing, and band deformations in the vicinity of mode gaps and bandgaps are the main mechanisms that result in band structures (and accordingly, dispersive properties) that are much different from their homogeneous counterparts. In a typical PC band structure shown in Fig. 1 some of these features are highlighted. These unique dispersion

The authors are with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332. B. Momeni’s e-mail address is [email protected]. Received 2 February 2006; accepted 2 May 2006; posted 20 July 2006 (Doc. ID 67720). 0003-6935/06/338466-11$15.00/0 © 2006 Optical Society of America 8466

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properties have been proposed to be exploited in different applications. In particular, we will concentrate on PC demultiplexers based on the superprism effect. After the original demonstration of the superprism effect,6 the idea has been considered for wavelength demultiplexing,7,8 and the basic behavior has been experimentally demonstrated in different platforms.9 –12 However, the limited resolution and fast scaling of the size of the structure with increasing the number of channels have prevented the realization of true spatial separation between channels in a conventional superprism demultiplexer to date.13,14 Efforts have been made by modifying the PC lattice,15 or using the distinction between phase velocities of different wavelength channels,16,17 but an experimental demonstration of a compact integrated demultiplexer with a performance comparable to other integrated approaches has not, to our knowledge, been realized. The reason behind the relatively large propagation length requirement in conventional superprism demultiplexers is that beams diverge as they propagate inside the structure,13,14 thus their spatial separation needs a significant propagation length for a given wavelength resolution. In this paper, we use an alternative configuration for the PC superprism-based demultiplexers that enhances the demultiplexing properties of these structures and relaxes their limitations on resolution and input beam divergence. The broadening of the beam in

Fig. 1. (Color online) In-plane constant frequency contours of the first band of a rotated square lattice photonic crystal in a planar SOI wafer. Coordinates are rotated 45° with respect to the principal lattice vectors of the PC as shown in the right-side figure. This PC has normalized radius of holes of r兾a ⫽ 0.30, and normalized thickness of the silicon layer of h兾a ⫽ 0.60 (a being the lattice constant).

the conventional structure is caused by the secondorder spectral phase (similar to the ordinary diffraction effect in free space),18 and since propagation in a photonic crystal structure allows a negative effective index,18,19 this effect can be combined with the demultiplexing effect to obtain a better spatial separation.20 The structure we investigate in detail in this paper, shown schematically in Fig. 2, consists of a preconditioning region in which the light beam propagates (and thus broadens) in an ordinary medium (unpatterned Si, for example), before entering the PC superprism region. The PC band structure is engineered to have a negative effective index for diffraction (i.e., the opposite sign of the second-order spectral phase compared to the preconditioning region) at all wavelength channels. Thus different wavelength channels of the incident beam are angularly separated from each other inside the PC by the superprism effect and are simultaneously focused to their diffraction-limited spot size due to the negative diffraction. In the rest of this paper, the fundamental requirements and demultiplexing properties of the structure in Fig. 2 will be discussed. For this purpose, we use an extended diffractive index model by generalizing the approximate model described in Ref. 18 to model the beam propagation behavior inside PCs. This model is used to perform quantitative analysis on the demultiplexing performance of these devices, and the results will be used to find optimum demultiplexing PC structures based on the preconditioned superprism

Fig. 2. (Color online) Configuration for a PC working in the preconditioned superprism regime.

effect. The theoretical model of generalized effective index is developed in Section 2 to describe the wave propagation effects in PCs. This model is then used in Section 3 to analyze preconditioned superprism devices and to define important figures of merit for comparing the performance of different structures and for systematically optimizing the structure. In Section 4, an optimization process is presented to find the optimum operation point and device parameters for designing preconditioned superprism-based demultiplexers. Section 5 covers the final optimization results and their pertinent discussions, and concluding remarks are made in Section 6. 2. Theoretical Background

For dispersion-based applications of PCs, it is essential to study beam propagation effects for the beams going through a PC region. It has been shown18 that for beams with spatial features much larger than the lattice constant (which is usually the case for the practical application of dispersive properties of PCs), the macroscopic behavior of the envelope of the beam during propagation can be described by modeling propagation through the PC as a phase-only transfer function (called the envelope transfer function18) that can be calculated from the PC band structure. Figure 3 shows the propagation of a typical optical beam from plane 1 to plane 2 along an arbitrary direction 共␩兲 inside a PC. Coordinates normal and parallel to the direction of propagation are represented by ␰ and ␩, respectively. For this case, if Pˆ1共k␰兲 and Pˆ2共k␰兲 are Fourier transforms of the beam envelopes at plane 1 and plane 2, respectively, with k␰ being the spatial frequency normal to the direction of propagation, then Pˆ2共k␰兲 ⫽ H共k␰兲Pˆ1共k␰兲,

(1)

H共k␰兲 ⫽ exp关jk␩共k␰兲␩21兴

再 冋

⫽ exp j␩21 k␩0 ⫹ 共k␰ ⫺ k␰0兲⭸k␩兾⭸k␰

册冎

1 ⫹ 共k␰ ⫺ k␰0兲2⭸2k␩兾⭸k␰2 ⫹ · · · 2

,

(2)

Fig. 3. (a) Propagation of an arbitrary beam inside a 2D PC. Coordinates ␰ and ␩ represent directions parallel and perpendicular to the direction of propagation of the beam, respectively. (b) Directions of (a) are shown on the band structure of the PC (which is represented in the form of a constant frequency contour in the 2D wave-vector plane). 20 November 2006 兾 Vol. 45, No. 33 兾 APPLIED OPTICS

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in which ␩21 is the distance between the two planes and k␩ is the component of the wave vector in the direction of propagation. Using the PC band structure at the excitation point (defined by the incident wave vector and the frequency of the beam), the dependence of k␩ on k␰ can be directly found. If we expand k␩共k␰兲 using the Taylor expansion [see Eq. (2)], different terms represent different physical effects on the optical beam. This is in direct analogy with dispersion effects on the shape of a time-domain pulse.21 For example, a zeroth-order term [i.e., k␩0 in Eq. (2)] corresponds to a simple phase shift, a first-order term [i.e., 共k␰ ⫺ k␰0兲⭸k␩兾⭸k␰] represents a spatial drift of the beam envelope from the coordinate axis, and the 1 second-order term [i.e., 2共k␰ ⫺ k␰0兲2⭸2k␩兾⭸k␰2] describes ordinary beam broadening caused by diffraction. The second-order term is the only term present in an ordinary bulk medium. This term is responsible for the well-known beam propagation effects in bulk media resulting in the broadening of the beam during propagation. Considering the propagation of a Gaussian beam inside a PC, the main contribution to the beam broadening is caused by the second-order spectral phase (also called chirp) in Eq. (2). Unlike ordinary bulk media, however, PCs can contribute negative chirp to the signal for a positive propagation length. This comes from the possible different sign of the curvature of the PC bands, as shown in Fig. 1. Beams entering without second-order phase into a material with negative chirp still undergo broadening as in ordinary materials, since the (negative) spectral phase adds up during propagation, resulting in a broadening of the beam. However, if a beam with positive initial chirp (caused, for example, by propagating through a normal bulk medium such as Si) enters a PC structure with negative chirp, the second-order spectral phase gradually cancels out, and as a result, the beam can be focused back to its minimum phase width by propagation inside the PC structure (this process is also known as diffraction compensation18,20). Negative effective index as used here should not be mixed with negative refraction (the terminology that is usually applied to the case of effective negative refractive index in Snell’s law). Negative effective index solely depends on the curvature of the PC bands (second derivative) and can happen for both positive and negative refraction (which is defined based on the normal direction to the interface of the material). Also, negative diffraction index is defined for a single frequency and is different from the group index, which represents the relative amplitude of the group velocity of the modes. In the preconditioned superprism effect, ideally, the beams focus back to their minimum spot sizes at the output of the structure. However, since the PC bands are not perfectly quadratic, the beams have some distortions at the output of the structure caused by higher-order terms in the Taylor expansion of Eq. (2). Such distortions are the limiting parameters that determine the resolution and the performance of the 8468

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device. To analyze the effect of these higher-order terms, we generalize the approximate model (in Ref. 18) to higher-order phase terms in Eq. (2). For the analysis of PC demultiplexers, we are more interested in the spatial size of the optical beams than the details of the beam shape. Thus we define the rms beam width21 for an optical beam with spatial intensity profile I共x兲 as

冕 冕

x2I共x兲dx

wrms2 ⫽

x

.

(3)

I共x兲dx

x

If A共k␰兲exp关 j⌽共k␰兲兴 is the normalized spatial Fourier transform of the beam profile with

冕

A共k␰兲2dk␰ ⫽ 1,

(4)

k␰

the rms beam width of the beam can be found using Ref. 21 as wrms2 ⫽

冕

A⬘共k␰兲2dk␰ ⫹

k␰

冕

A共k␰兲2⌽⬘共k␰兲2dk␰,

(5)

k␰

in which A⬘共k␰兲 and ⌽⬘共k␰兲 are the first derivatives of A共k␰兲 and ⌽共k␰兲, respectively, with respect to k␰. To analyze the beam propagation effects, we assume that the amplitude of each spectral component of the beam remains intact (which is valid if there is no loss, no gain, and no coupling between modes in the system). The spectral phase term ⌽共k␰兲 is caused by the spatial dispersion during propagation of the optical beam. Starting with a minimum phase Gaussian beam with a beam waist of 2w0, given by A共k␰兲 ⫽

冉冑 冊 冉 w0

2␲

冊

1 exp ⫺ w02k␰2 , 4

1兾2

(6)

entering a material with second-order spectral phase ⌽共k␰兲 ⫽ b2k␰2 and length L, the output rms beam width can be directly calculated from Eq. (5) as wrms,22 ⫽

w02 4b22 , ⫹ 4 w02

(7)

with b2 ⫽

1 d2⌽ 1 d2k␩ L ⫽ L⫽ , 2 2 2 dk␰ 2 dk␰ 2k0ne2

(8)

where ne2 ⫽ 共k0d2k␩兾dk␰2兲⫺1 is the effective index as defined in Ref. 18. Equation (7) can be rewritten as

2 rms,2

w

冉 冊

1 2 L2 ⫽ w0 1 ⫹ 2 , 4 z2

(9)

with z2 ⫽ k0ne2w02兾2 being its effective Rayleigh range. Equation (8) represents the variation (or broadening) of the size of a Gaussian beam in an ordinary medium. This equation also governs the size of the output beam in a conventional PC demultiplexer. On the other hand, the second-order phase is completely compensated in a preconditioned PC demultiplexer. Thus to calculate the output beam size in such compensated structures, we need to use the third-order spectral phase term [i.e., ⌽共k␰兲 ⫽ b3k␰3] in Eq. (5) to obtain

2 rms,3

w

w02 w0 ⫽ ⫹ 4 冑2␲ 2

冕 冉

and the beam-width behavior inside the structure follows the same behavior as in the ordinary secondorder case, with one major distinction that the thirdorder diffractive index 共ne3兲 linearly depends on the beam waist 共2w0兲. In an ideal preconditioned demultiplexer, we are interested in compensating the second-order spectral phase by the PC negative diffraction effect and at the same time, optimizing the PC to have a negligible third-order phase term at the operation point. For such ideal cases, the rms beam width is calculated using the fourth-order spectral phase term as 2 rms,4

w

冊

z4 ⫽

1 exp ⫺ w02k␰2 共9b32k␰4兲dk␰ 2

k␰

2

w0 27b3 ⫽ , ⫹ 4 w04

ne4 ⫽

(10)

(11) wrms,v2 ⫽

Combining Eqs. (10) and (11) results in 2 rms

w

w02 3L2共d3k␩兾dk␰3兲2 ⫽ . ⫹ 4 4w04

(12)

zv ⫽

nev ⫽

We can define z3 ⫽

1

冑3 共

d3k␩兾dk␰3兲⫺1w03,

(13)

冉 冊

(14)

to simplify Eq. (12) as

2 rms,3

w

1 2 L2 ⫽ w0 1 ⫹ 2 . 4 z3

This is in an exact analogy with the second-order form given in Eq. (7). Based on this analogy, we can define the third-order Rayleigh range as z3 ⫽

1 k n w 2, 2 0 e3 0

(15)

to define a third-order diffractive index as ne3 ⫽

2w0

冑3k0 共

d3k␩兾dk␰3兲⫺1,

1 k n w 2, 2 0 e4 0 2冑3w02

d4k 兾dk4兲⫺1. 冑5k0 共 y

(17)

(18)

(19)

In general, for vth order spectral phase, we can calculate the main parameters describing the beam propagation effects as

where 1 d3⌽ 1 d3k␩ b3 ⫽ ⫽ L. 6 dk␰3 6 dk␰3

冉 冊

1 2 L2 ⫽ w0 1 ⫹ 2 , 4 z4

(16)

冉 冊

1 2 L2 w0 1 ⫹ 2 , 4 zv

(20)

1 k n w 2, 2 0 ev 0

(21)

v⫺2 共v ⫺ 1兲 ! w0

dvk 兾dk v ⫺1, 冑共2v ⫺ 3兲!! k0 共 ␩ ␰ 兲

(22)

where 共2v ⫺ 3兲!! stands for factorial over odd numbers up to 2v ⫺ 3 (i.e., 1, 3, 5, . . . , 2v ⫺ 3). To verify the applicability of the approximate model discussed above, we consider incident light coming at a 12° angle to a 45°-rotated square lattice PC of air holes in Si as shown in Fig. 4. The incident beam is preconditioned in this case so that the effect of normal diffraction vanishes at the monitoring plane at the output of the PC region. Using a direct modal approach (mode matching at the interface followed by propagation in a PC),18 the envelope of the output beam profile is calculated and shown in Fig. 4. The result obtained using the approximate effective index method presented in this section is also plotted on the same graph and is in good agreement with direct simulations. Note that for a PC structure, all derivatives in the k␰ ⫺ k␩ plane (e.g., d3k␩兾dk␰3) are calculated at the operation point on the PC band structure and at the specific frequency of operation. Thus by dispersion engineering (i.e., designing the PC to have appropriate dispersion properties at the operation point), we can greatly affect the propagation of an optical beam. 20 November 2006 兾 Vol. 45, No. 33 兾 APPLIED OPTICS

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Fig. 4. (Color online) For a 45° rotated square lattice PC (air holes in Si, r兾a ⫽ 0.40) the profile of the beam envelope at the output is calculated using the direct mode-matching method (solid curve) and the approximate diffractive index method (diamond). The incident light in this calculation is a preconditioned (i.e., broadened) Gaussian beam at normalized wavelength a兾␭ ⫽ 0.197 that illuminates the structure at an angle of 12° with respect to the normal to the interface. The preconditioning is performed so that the effect of the second-order diffraction term vanishes at the output of the PC structure. Good agreement of the accurate and approximate results is clear.

Note also that the overall behavior of the beam width of a given beam inside materials with third-fourth, or higher-order spectral phase terms can be calculated using similar formulas as that of an ordinary Gaussian beam propagating in a bulk medium if we use the appropriate effective index. We use this fact in our analysis of beam propagation in preconditioned superprism devices in Section 3. 3. Analysis of Preconditioned Superprism Devices

The basic topology of the preconditioned superprism demultiplexer with different important parameters is shown in Fig. 5. There are two basic conditions that need to be satisfied in the demultiplexer, namely, spatial separation and diffraction compensation. Spatial separation refers to the different channels being separated in space at the output of the device. This is caused by the propagation of beams of different wavelength in different directions inside the structure. This separation can be quantified by defining cross talk between channels as the sum of the powers of all undesired channels at the location of the desired channel. Diffraction compensation condition is the cancellation of second-order spectral phase from the input diffraction-limited incident beam to the output plane as designed. In this section, we formulate these two conditions in terms of actual physical design parameters. In Fig. 5(a), we have shown the propagation of a Gaussian beam through the structure where 2w0 is the initial waist of the incident beam, ␣ is the incident angle, and L and ␪g are propagation length and propagation angle of the beam corresponding to a single 8470

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Fig. 5. (Color online) (a) Parameters for a preconditioned superprism device are depicted for an incident beam coming at an angle ␣, and for a single channel inside the PC region. (b) The darker pattern trace shows the evolution of an optical beam at a single wavelength throughout the structure without the effect of the second-order diffraction. In this case, ␦3 is the divergence angle of the beam due to the third-order diffraction effect. The brighter pattern is the actual beam profile inside the structure. By compensating the second-order phase, the beam size at the output is the same as that in the assumed structure with zero second-order phase everywhere.

demultiplexing channel in the PC region. In addition, 2wPC ⫽ 2w0 cos ␪g兾cos ␣ is the beam waist corresponding to that beam inside the PC. Here, we consider the third-order spectral phase to be the dominant term in the higher-order effects; validity of this assumption can be easily checked for each design by comparing the contributions of different spectral phase orders. In case other spectral terms become dominant (for example, the fourth-order spectral phase in the ideal structure) the same steps as below can be performed with the corresponding phase term. Figure 5(b) shows the evolution of the beam inside the structure considering only the third-order spectral phase term. The output beam obtained in this way is the actual beam profile only at the output plane of the structure, where the effect of the secondorder spectral phase is designed to vanish. Note that the only difference between the preconditioned PC demultiplexers and the conventional ones (investigated in detail in Ref. 22) is the replacement of the second-order spectral phase with the third-order spectral phase. Thus using the formulation of Section 2, we can apply the same formalism as in Ref. 22 for the calculation of the cross talk in a preconditioned PC device, by using the third-order effective index. As a result, we can calculate the re-

The area of the PC (A) taken by each channel can be estimated as

Table 1. Cross-Talk Parameters

Cross Talk, X (dB)

K(X)

H(X)

⫺20 ⫺30 ⫺40 ⫺50

0.9 0.9 0.9 0.9

0.56 0.83 1.04 1.22

A⫽

quired propagation length L, for achieving a crosstalk level of at most X as22 L ⫽ ␨3 z3,

冉 冊

(24)

In these relations, z3 is the Rayleigh range corresponding to the third-order spectral phase term, ␩3 ⫽ ⌬兾␦3 is the ratio of the angular separation between adjacent channels (⌬) to the divergence angle of each channel due to the third-order diffraction effect inside the PC [as represented by ␦3 in Fig. 5(b)], and K and H are constants given by Table 1 according to the required cross talk. The procedure for calculating K共X兲 and H共X兲 is the same as that in Ref. 22 and is not repeated here. Equations (23) and (24) represent the spatial separation condition of output channels for preconditioned superprism devices. The diffraction compensation condition that describes the cancellation of the overall quadratic phase can be simply put as Lpre npre cos ␣ 2

⫽

L ne2 cos2 ␪g

,

冋 冏 冏册

K ␲wPCⱍne3ⱍ⌬兾2␭ ⫺ H

共

1 2

兲

k0wPC2|ne3| ,

(26)

and therefore, 3

L⫽

2KwPC

ⱍ

ⱍ

wPC ⌬ ⫺ 2冑3H ⭸3k␩兾⭸k␰3 2

.

(27)

ⱍ兲

2

.

(28)

1兾2

,

(29)

which consequently results in the optimal propagation length 共Lopt兲 as Lopt ⫽

5K w , 2⌬ 共 PC兲opt

(30)

and the optimum (i.e., minimum) PC area 共Aopt兲 as Aopt ⫽

25K 2 2k0ne2⌬2

共wPC兲opt.

(31)

In terms of the physical parameters of the structure, by using

冉 冊 冉 冊 冉 冊

⭸ ⭸2k␩ ⭸ ⫺cos ␪g 1 ⫽ ⫽ ⫽ 2 2 ⭸k␰ ⭸k␰ ⭸k␰ k0ne2 k0 n1ne22 cos ␣

⭸3k␩ ⭸k␰3

(25)

in which npre and Lpre are the refractive index and the length of the preconditioning region, respectively. Also, ne2 and L are the refractive index and the propagation length of the PC region, respectively. To assess the performance of the preconditioned superprism demultiplexers, we calculate the size of these structures for a given angular channel spacing, ⌬. Starting from Eqs. (14) and (15) for a beam propagating in a medium with third-order diffraction 1 effects, we have ␦3 ⫽ 2␭兾共␲wPC|ne3|兲 and z3 ⫽ 2 k0wPC2 |ne3|, with |ne3| being the magnitude of the thirdorder effective index of the PC. By inserting these relations into Eq. (24) we obtain L ⫽ ␨3 z3 ⫽

ⱍ

10冑3H ⭸3k␩ 共wPC兲opt ⫽ ⌬ ⭸k␰3

(23)

K共X兲 . ␩3 ⫺ H共X兲

ⱍ

The area in Eq. (28) depends explicitly on the diffraction-limited beam waist of the channel inside the PC, 2wPC; thus we can minimize the area directly with respect to this parameter by using ⭸A兾⭸wPC ⫽ 0 to obtain

where ␨3 ⫽

ⱍ

8K 2wPC5 ⭸3k␩兾⭸k␰3 wPC L L⫽ z2 共wPC2⌬ ⫺ 2冑3H ⭸3k␩兾⭸k␰3

⫻

冉 冊

⭸ne2 , ⭸␣

(32)

where 共⭸ne2兾⭸␣兲 is the value calculated at the frequency of operation and over the range of excitation angles, we can rewrite Eqs. (29)–(31) as 共wPC兲opt ⫽

冋

冉 冊册

10冑3H cos ␪g ⭸ne2 1 k0ne2 共⭸␪g兾⭸␻兲 n1 cos ␣ ⭸␣

1兾2

共⌬␻兲⫺1兾2, (33)

Lopt ⫽

Aopt ␭

2

⫽

冋

冉 冊册 冉 冊

5K 10冑3H cos ␪g ⭸ne2 2k0ne2 n1 cos ␣ ⭸␣ ⫻共⌬␻兲⫺3兾2, 25冑10冑3HK 2 2

8␲ ⫻共⌬␻兲⫺5兾2,

冋

1兾2

⭸␪g ⭸␻

⫺3兾2

(34)

冉 冊册 冉 冊

cos ␪g ⭸ne2 n1 cos ␣ ⭸␣

1兾2

1

2 e2

n

⭸␪g ⭸␻

⫺5兾2

(35)

in which we have used ⌬ ⫽ 共⭸␪g兾⭸␻兲⌬␻ as the angular separation between adjacent channels. From Eq. (35), we can see that the area of the structure scales as 共⌬␻兲⫺5兾2, which grows considerably slower than the 20 November 2006 兾 Vol. 45, No. 33 兾 APPLIED OPTICS

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共⌬␻兲⫺4 dependence in the conventional superprismbased demultiplexers.22 We can define the compactness factor for the preconditioned structures as

Cpre ⫽

8␲2ne22 25冑10冑3HK 2

冋

冉 冊册 冉 冊

cos ␪g ⭸ne2 n1 cos ␣ ⭸␣

⫺1兾2

⭸␪g ⭸␻

5兾2

, (36)

which simply relates the spectral spacing between channels (i.e., ⌬␻) to their optimum area (i.e., Aopt) through Aopt ␭

2

⫽

共⌬␻兲⫺5兾2 Cpre

.

(37)

In the optimization process, we use Cpre as the first measure to locate the appropriate operation point on the band structure of a given PC. Examples of the calculated values for different lattice types are shown in Fig. 6. The designs considered throughout this paper are all in the first band of the PC. The motivation behind this choice is the potentially lower propagation loss and lower reflection loss. Nevertheless, the same procedure can be followed for other bands as well. Plots shown in Fig. 6 are very useful in comparing different PC structures and in choosing the optimum structure by looking for the structures with maximum Cpre. From the results in Fig. 6, we can see that the compactness factor becomes larger as the constant frequency contours deviate from the bulk-type circular patterns. Such contours in the first band of triangular lattice PCs occur only at the vicinity of the boundaries of the Brillouin zone. Therefore the bandwidth is limited in these cases. In square lattice PCs, however, the compactness factor can be large even at regions away from the boundaries of the Brillouin zone. Also, by comparing Figs. 6(a) and 6(b) we can see that in the square lattice with interfaces along one of its principal lattice directions, the most appropriate operation points (those with the largest values of compactness factor) have their direction of group velocity (which is normal to the constant frequency contour) parallel to the interface that is not suitable for demultiplexing purposes. However, by rotating the lattice by 45° a relatively large range of the PC band with a large compactness factor can be excited using a single incident angle for demultiplexing. The optimum design operation point in this case is along the |kx| ⫹ |ky| ⫽ ␲冑2兾a directions in the kx–ky plane, due to the intrinsic symmetry of the lattice that implies zero odd-order diffraction effects along this line. Once the operation range on the band structure is selected, the design parameters can be obtained using the process described in detail in Section 4. 8472

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4. Design of Planar Preconditioned Superprism Demultiplexers

In this section we focus on developing a design strategy for planar (slab-type) PCs with 2D in-plane periodicity; nevertheless, the process can be applied with minor modifications to 3D PCs as well. The choice of the planar structures matches the practical realizations like PCs fabricated in a silicon-on-insulator (SOI) wafer. In such structures, the dispersion effect in the unpatterned slab region connected to the PC (which serves as the incident region) cannot be neglected. Note that the optical beams in a preconditioned structure propagate in the slab of unpatterned material (for example, Si) prior to entering the PC region. Knowing the effective index of the unpatterned slab is important both in applying the phasematching condition at its interface with the PC and in designing the lengths of the different parts of the structure (i.e., preconditioning region, and PC region) to achieve the complete compensation of the secondorder diffraction at the output of the structure. A typical dispersion diagram for an asymmetric silicon slab sandwiched between air and silicon oxide (a typical SOI wafer) is shown in Fig. 7(a). Depending on the thickness of the wafer 共h⬘兲, the wavelength on this graph can be scaled [using ␭⬘ ⫽ 共h⬘兾h兲␭] to find the dispersion diagram for an SOI wafer with arbitrary thickness of the Si layer. Using the dispersion diagram of the unpatterned slab that serves as the input region [as shown in Fig. 7(a)], for each incident angle we can find the excited modes inside the PC. An example of the loci of the excited modes at different frequencies inside the PC for four different incident angles (5°, 10°, 15°, and 20°) is shown in Fig. 7(b). For any incident angle, we can find different parameters of the structure, i.e., angle of group velocity 共␪g兲, second-order effective index 共ne2兲, sensitivity factors 关共⭸␪g兾⭸␻兲 and 共⭸ne2兾⭸␣兲], as well as higher-order effective indices associated with each demultiplexing channel in our bandwidth of interest. These parameters describe the propagation behavior for each channel (i.e., direction of propagation, sensitivity to frequency, and the divergence caused by the thirdorder spectral phase term). To get the required cross talk for all channels, the parameter wi (i.e., the beam waist of the incident optical beam consisting of several wavelengths) should be found in such a way that the maximum propagation length required over all channels is minimized. This can be directly performed by reformulating Eq. (27) as Lj ⫽

2Kwi3 cos3 ␪g j

, wi2⌬j cos2 ␪g j cos ␣ ⫺ 2冑3H ⭸3k␩兾⭸k␰3 j cos3 ␣ (38)

ⱍ

ⱍ

in which subscript j stands for the parameters calculated for the jth channel. After finding wi from this process, it is straightforward to set L ⫽ max兵Lj共wi兲其, j

(39)

Fig. 6. Calculated compactness factor (in log10 scale) for different PC lattices on SOI wafers (h is the thickness of the top Si layer, r is the radius of the holes, and a is the lattice constant) are shown along with constant frequency contours of the corresponding PC band. Each contour in the kx–ky plane corresponds to a constant frequency. The value of the normalized frequency 共a兾␭兲 for each constant frequency curve is marked on the contours. In all these cases, the first band of the PC structure is considered. (a) A square lattice slab-type PC with r兾a ⫽ 0.30 and h兾a ⫽ 0.60, and (b) the same square lattice as in (a) with the interface along a direction angled 45° with respect to the principal lattice directions. (c) A triangular lattice with r兾a ⫽ 0.30 and h兾a ⫽ 0.60 with the interface along the ⌫M direction, and (d) the same triangular lattice as in (c) with the interface along the ⌫K direction.

and the length of the preconditioning region is found from Eq. (25) as Lpre ⫽

npre cos2 ␣ ne2 cos2 ␪g

L,

(40)

which completes the design. Here, npre is calculated by taking the exact geometrical properties of the slab

(i.e., material, thickness, layers above and below, etc.) as described in the beginning of this section. 5. Results and Discussion

In this section, we use the procedure of Section 4 to design an optimal preconditioned PC demultiplexer for a dense wavelength division multiplexing (DWDM) system with different wavelength channel spacings 20 November 2006 兾 Vol. 45, No. 33 兾 APPLIED OPTICS

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Table 2. Design Parameters for Optimal Demultiplexers in a Square Lattice PCb

Channel Spacing (GHz) 100 2wi (␮m) L (mm) 200 2wi (␮m) L (mm) 400 2wi (␮m) L (mm)

Number of Channels 4

8

16

32

40 4.9

41 5

42 5.3

48 7.1

28 1.6

29 1.9

33 2.5

N兾Pa N兾Pa

22 0.62

24 0.9

N兾Pa N兾Pa

N兾Pa N兾Pa

a

N兾P means not possible. With the interface along high-symmetry directions; the thickness of the top Si layer, h ⫽ 195 nm; the normalized radius of holes, r兾a ⫽ 0.30; and the incident angle of ␣ ⫽ 10°. b

Fig. 7. (Color online) (a) Dispersion diagram for guiding in an unpatterned SOI wafer with h ⫽ 220 nm is shown. (b) Band structure (dotted curves) of a slab-type PC in a SOI wafer (square lattice, r兾a ⫽ 0.30, h兾a ⫽ 0.62) and loci of PC modes (solid curves) excited for the incident wave coming from the unpatterned Si slab at different incident angles (in degrees) are shown.

operating around 1550 nm. We also assume that the demultiplexing structure is fabricated on an SOI wafer with the top Si thickness of 180 nm ⬍ h ⬍ 250 nm, and the underlying SiO2 layer of 3 ␮m thickness. Based on the results shown in Fig. 6, we select square lattice PCs due to their wider bandwidth of operation. For each case (with a specified number of channels and channel spacing), and for an arbitrary PC lattice and angle of incidence (i.e., the angle between the incident beam direction and normal to the PC interface) we use the compactness factor to choose the center frequency of operation. Then, around this center frequency, the required propagation length (L) and incident beam waist 共2wi兲 is found so that the desired cross-talk level for all channels is achieved. This process can be repeated for a range of angles and different PC lattices to find the optimum structure for the given specifications. For comparison, here we consider two classes of square lattice PCs, one with its interface along one of the principal lattice vectors [Fig. 6(a)] and the second one, a rotated square lattice PC whose interface with Si makes a 45° angle with respect to the principal lattice vectors [as shown in Fig. 6(b)]. The properties of the optimal structures 8474

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obtained for these two classes of PCs are listed in Tables 2 and 3, respectively. As the channel spacing becomes larger, the angular separation between adjacent channels increases (due to the fixed angular dispersion at the operation point of the structure, imposed by the PC band structure). Consequently, the larger angular separation between adjacent wavelength channels relaxes the restriction on the diffraction-limited spot size (i.e., the device can work with a smaller initial beam waist, wi). Both of these effects (i.e., larger angular separation and smaller beam waist) result in a shorter required propagation length and thus a smaller structure for a given number of channels. This trend can be observed in the results of Table 2 for devices of the same bandwidth. Note that increasing the operation bandwidth eventually requires the use of suboptimal structures and thus lower performance. For example, in each row of Table 2, as the number of channels increases, portions of the band structure with less optimal properties need to be included to cover the entire operation bandwidth. As a result, performance deteriorates as the number of channels increases and thus larger Table 3. Design Parameters for Demultiplexers in a Square Lattice PCa

Channel Spacing (GHz) 100 2wi (␮m) L (mm) 200 2wi (␮m) L (mm) 400 2wi (␮m) L (mm)

Number of Channels 4

8

16

32

17 1.4

30 2.0

45 2.9

63 4.7

17 0.61

30 0.95

43 1.6

52 3.1

17 0.29

28 0.53

35 1.1

51 2.5

a With the interface at 45° with respect to high-symmetry directions; the thickness of the top Si layer, h ⫽ 242 nm; the normalized radius of holes, r兾a ⫽ 0.30; and the incident angle of ␣ ⫽ 10°.

structures are needed. For larger channel spacings, the structure simply does not support the relatively large bandwidth needed for a large number of channels (listed as not possible, or N兾P in Table 2). By comparing the results in Tables 2 and 3, it is evident that the rotated square lattice offers more compact demultiplexers for the same specifications and supports larger bandwidths. Therefore the rotated square lattice PC structure is preferred for preconditioned superprism demultiplexers. Another important issue is the variation of the effective index over the operation bandwidth. Note that discrepancies encountered in Eq. (40) due to the variations of the effective index 共ne2兲 from channel to channel increase the output beam size for some channels owing to the reminiscent second-order spectral phase term and result in higher cross talk. For 1.6 THz bandwidth around the center frequency, this variation is less than 2% (i.e., the effective index is ⫺0.23 ⫾ 2% over the entire bandwidth) for the rotated square lattice design, while for the same bandwidth the effective index varies between ⫺1.4 and ⫺0.4 in the square lattice with the interface along principal lattice directions (which needs a separate mechanism to compensate it). This is another clear advantage of the rotated square lattice structure. Compared to the previously reported results for conventional superprism structures,13,14,22 the designs in Table 3 require less collimated input beams (the requirement for the input beam waist is relaxed by a factor of more than 2), and there are also almost 2 orders of magnitude improvement in the compactness of the structure (more than 1 order of magnitude improvement in the required propagation length). These improvements bring the superprism-based demultiplexers in the range where their fabrication through conventional techniques is possible. The performance of these devices can be further improved by topology optimization of the PC structure to obtain better demultiplexing properties (our optimization space in this paper is limited to specific lattice types and directions). Also, one can envision that by combining different effects in a heterostructure PC and including contributions from other demultiplexing effects (such as the distinction in wave vectors as in k vector superprisms23), more compact and higherresolution demultiplexers can be realized. Another issue that was not discussed in this paper is the loss due to reflection at the interface of the PC. The effective index used as our main tool in designing preconditioned superprism devices does not provide any information about the reflection loss at the interfaces of the structure. However, several reflection reduction schemes have been proposed to minimize the reflection loss.20,24 –26 In particular, it has been shown that by using adiabatic matching stages it is possible to achieve large angle and wideband coupling of light into and out of PC structures.26 Therefore the design for the optimum superprism demultiplexer can be performed independent of the possible reflection losses as shown in this paper, and efficient matching can be achieved in a subsequent independent stage.

6. Conclusions

We presented a systematic analysis and design of a new type of photonic crystal demultiplexers (i.e., preconditioned demultiplexer), in which the simultaneous existence of the superprism effect and the negative effective index of diffraction results in a very compact structure by canceling the second-order spectral phase to avoid beam broadening inside the PC. As a result, the size of the preconditioned demultiplexers varies as N2.5 (N being the number of wavelength channels) is compared to the N4 variation in the conventional superprism-based PC demultiplexers. We analyzed the basic properties of these structures using a generalized effective index model. By using the generalized effective index model in the analysis and design of these structures one can deduce all the basic properties of the structure simply from the band structure without the need to go through tedious simulations, especially for 3D structures. Furthermore, using the effective index technique, we developed a simple and systematic method for designing optimal preconditioned demultiplexers. Our results show that the preconditioned superprism-based PC demultiplexers have 2 orders of magnitude smaller size compared to the conventional superprism-based demultiplexers. This improvement makes it feasible to use these structures for high resolution applications like DWDM systems by fabricating them through normal fabrication techniques. This work was supported by the U.S. Air Force Office of Scientific Research under contract F4962003-1-0362 (G. Pomrenke) and by the National Science Foundation under contract ECS-0239355 (L. Goldberg). References 1. E. Yablonovitch, “Inhibited spontaneous emission in solid state physics and electronics,” Phys. Rev. Lett. 58, 2059 –2062 (1987). 2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486 –2489 (1987). 3. E. Yablonovitch, “Photonic band-gap structures,” J. Opt. Soc. Am. B 10, 283–295 (1993). 4. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U., 1995). 5. S. G. Johnson and J. D. Joannopoulos, “Designing synthetic optical media: Photonic crystals,” Acta Mater. 51, 5823–5835 (2003). 6. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096 –R10099 (1998). 7. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals: toward microscale lightwave circuits,” J. Lightwave Technol. 17, 2032–2038 (1999). 8. K. B. Chung and S. W. Hong, “Wavelength demultiplexers based on the superprism phenomena in photonic crystals,” Appl. Phys. Lett. 81, 1549 –1551 (2002). 9. B. E. Nelson, M. Gerken, D. A. B. Miller, R. Piestun, C. C. Lin, and J. S. Harris, Jr., “Use of a dielectric stack as a onedimensional photonic crystal for wavelength demultiplexing by beam shifting,” Opt. Lett. 25, 1502–1504 (2000). 10. M. Gerken and D. A. B. Miller, “Multilayer thin-film structures with high spatial dispersion,” Appl. Opt. 42, 1330 –1345 (2003). 20 November 2006 兾 Vol. 45, No. 33 兾 APPLIED OPTICS

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19. M. Qiu, L. Thylén, M. Swillo, and B. Jaskorzynska, “Wave propagation through a photonic crystal in a negative phase refractive-index region,” IEEE J. Sel. Top. Quantum Electron. 9, 106 –110 (2003). 20. J. Witzens, T. Baehr-Jones, and A. Scherer, “Hybrid superprism with low insertion losses and suppressed cross-talk,” Phys. Rev. E 71, 026604 (2005). 21. R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Kluwer, 2000). 22. B. Momeni and A. Adibi, “Systematic design of superprismbased photonic crystal demultiplexers,” IEEE J. Sel. Areas Commun. 23, 1355–1364 (2005). 23. T. Matsumoto and T. Baba, “Photonic crystal k-vector superprism,” J. Lightwave Technol. 22, 917–922 (2004). 24. T. Baba and D. Ohsaki, “Interfaces of photonic crystals for high efficiency light transmission,” Jpn. J. Appl. Phys., Part 1 40, 5920 –5924 (2001). 25. J. Witzens, M. Hochberg, T. Baehr-Jones, and A. Scherer, “Mode matching interface for efficient coupling of light into planar photonic crystals,” Phys. Rev. E 69, 046609 (2004). 26. B. Momeni and A. Adibi, “Adiabatic matching stage for coupling of light to extended Bloch modes of photonic crystals,” Appl. Phys. Lett. 87, 171104 (2005).