Preconditioned superprism-based photonic crystaldemultiplexers: 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., precondi-tioned demultiplexer), in which the simultaneous existence of the superprism effect and the negativeeffective index for diffraction results in a compact structure by canceling the second-order spectral phase toavoid beam broadening inside the PC. This approach considerably relaxes the requirements for the largearea of the structure and the small divergence of the input beam. As a result, the size of the preconditioneddemultiplexers varies as N2.5 (N being the number of wavelength channels) compared to the N4 variationin the conventional superprism-based PC demultiplexers. We use a generalized effective index model toanalyze, design, and optimize these demultiplexing structures. This approximate model can be used toextract all the basic properties of the PC device simply from the band structure and eliminates the needto go through tedious simulations especially for three-dimensional structures. Our results show that thepreconditioned superprism-based PC demultiplexers have 2 orders of magnitude smaller size compared tothe 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 newoptical materials with properties not accessiblewithin natural optical materials.1–5 In this view, ap-plications based on unique dispersive properties ofPCs for electromagnetic waves propagating insidethe periodic region have received considerable atten-tion recently, and in particular the superprism effectin photonic crystals has been considered for realizingcompact optical wavelength demultiplexers.

Unique dispersive properties of PCs are the result ofthe electromagnetic modes of the PCs being adapted tothe periodicity that is artificially introduced in the me-dium. Band folding, band crossing, and band deforma-tions in the vicinity of mode gaps and bandgaps are themain mechanisms that result in band structures (andaccordingly, dispersive properties) that are much dif-ferent from their homogeneous counterparts. In a typ-ical PC band structure shown in Fig. 1 some of thesefeatures are highlighted. These unique dispersion

properties have been proposed to be exploited indifferent applications.

In particular, we will concentrate on PC demulti-plexers based on the superprism effect. After the orig-inal demonstration of the superprism effect,6 the ideahas been considered for wavelength demultiplexing,7,8

and the basic behavior has been experimentally dem-onstrated in different platforms.9–12 However, thelimited resolution and fast scaling of the size of thestructure with increasing the number of channels haveprevented the realization of true spatial separationbetween channels in a conventional superprism de-multiplexer to date.13,14 Efforts have been made bymodifying the PC lattice,15 or using the distinctionbetween phase velocities of different wavelengthchannels,16,17 but an experimental demonstration ofa compact integrated demultiplexer with a perfor-mance comparable to other integrated approacheshas not, to our knowledge, been realized. The reasonbehind the relatively large propagation length re-quirement in conventional superprism demultiplex-ers is that beams diverge as they propagate inside thestructure,13,14 thus their spatial separation needs asignificant propagation length for a given wavelengthresolution.

In this paper, we use an alternative configuration forthe PC superprism-based demultiplexers that en-hances the demultiplexing properties of these struc-tures and relaxes their limitations on resolution andinput beam divergence. The broadening of the beam in

The authors are with the School of Electrical and ComputerEngineering, Georgia Institute of Technology, Atlanta, Georgia30332. B. Momeni’s e-mail address is momeni@ece.gatech.edu.

Received 2 February 2006; accepted 2 May 2006; posted 20 July2006 (Doc. ID 67720).

0003-6935/06/338466-11$15.00/0© 2006 Optical Society of America

8466 APPLIED OPTICS � Vol. 45, No. 33 � 20 November 2006

the conventional structure is caused by the second-order spectral phase (similar to the ordinary diffrac-tion effect in free space),18 and since propagation in aphotonic crystal structure allows a negative effectiveindex,18,19 this effect can be combined with the demul-tiplexing 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 precondi-tioning region in which the light beam propagates (andthus broadens) in an ordinary medium (unpatternedSi, for example), before entering the PC superprismregion. The PC band structure is engineered to have anegative effective index for diffraction (i.e., the oppo-site sign of the second-order spectral phase comparedto the preconditioning region) at all wavelength chan-nels. Thus different wavelength channels of the inci-dent beam are angularly separated from each otherinside the PC by the superprism effect and are simul-taneously focused to their diffraction-limited spot sizedue to the negative diffraction.

In the rest of this paper, the fundamental require-ments and demultiplexing properties of the structurein Fig. 2 will be discussed. For this purpose, we use anextended diffractive index model by generalizing theapproximate model described in Ref. 18 to model thebeam propagation behavior inside PCs. This model isused to perform quantitative analysis on the demul-tiplexing performance of these devices, and the re-sults will be used to find optimum demultiplexing PCstructures based on the preconditioned superprism

effect. The theoretical model of generalized effectiveindex is developed in Section 2 to describe the wavepropagation effects in PCs. This model is then used inSection 3 to analyze preconditioned superprism de-vices and to define important figures of merit forcomparing the performance of different structuresand for systematically optimizing the structure. InSection 4, an optimization process is presented to findthe optimum operation point and device parametersfor designing preconditioned superprism-based de-multiplexers. Section 5 covers the final optimizationresults and their pertinent discussions, and conclud-ing remarks are made in Section 6.

2. Theoretical Background

For dispersion-based applications of PCs, it is essentialto study beam propagation effects for the beams goingthrough a PC region. It has been shown18 that forbeams with spatial features much larger than the lat-tice constant (which is usually the case for the practicalapplication of dispersive properties of PCs), the mac-roscopic behavior of the envelope of the beam duringpropagation can be described by modeling propagationthrough the PC as a phase-only transfer function(called the envelope transfer function18) that can becalculated from the PC band structure. Figure 3 showsthe propagation of a typical optical beam from plane 1to plane 2 along an arbitrary direction ��� inside a PC.Coordinates normal and parallel to the direction ofpropagation are represented by � and �, respectively.For this case, if P̂1�k�� and P̂2�k�� are Fourier trans-forms of the beam envelopes at plane 1 and plane 2,respectively, with k� being the spatial frequency nor-mal to the direction of propagation, then

P̂2�k�� � H�k��P̂1�k��, (1)

H�k�� � exp�jk��k���21�

� exp�j�21�k�0 � �k� � k�0��k���k�

�12�k� � k�0�2�2k���k�

2 � · · ·�, (2)

Fig. 3. (a) Propagation of an arbitrary beam inside a 2D PC.Coordinates � and � represent directions parallel and perpendic-ular to the direction of propagation of the beam, respectively. (b)Directions of (a) are shown on the band structure of the PC (whichis represented in the form of a constant frequency contour in the 2Dwave-vector plane).

Fig. 1. (Color online) In-plane constant frequency contours of thefirst band of a rotated square lattice photonic crystal in a planarSOI wafer. Coordinates are rotated 45° with respect to the princi-pal lattice vectors of the PC as shown in the right-side figure. ThisPC has normalized radius of holes of r�a � 0.30, and normalizedthickness of the silicon layer of h�a � 0.60 (a being the latticeconstant).

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

20 November 2006 � Vol. 45, No. 33 � APPLIED OPTICS 8467

in which �21 is the distance between the two planesand k� is the component of the wave vector in thedirection of propagation. Using the PC band struc-ture at the excitation point (defined by the incidentwave vector and the frequency of the beam), the de-pendence of k� on k� can be directly found. If weexpand k��k�� using the Taylor expansion [see Eq. (2)],different terms represent different physical effects onthe optical beam. This is in direct analogy with dis-persion 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., �k� � k�0��k���k�] represents a spatial drift of thebeam envelope from the coordinate axis, and thesecond-order term [i.e., 1

2�k� � k�0�2�2k���k�2] describes

ordinary beam broadening caused by diffraction. Thesecond-order term is the only term present in an ordi-nary bulk medium. This term is responsible for thewell-known beam propagation effects in bulk mediaresulting in the broadening of the beam during prop-agation.

Considering the propagation of a Gaussian beaminside a PC, the main contribution to the beam broad-ening is caused by the second-order spectral phase(also called chirp) in Eq. (2). Unlike ordinary bulkmedia, however, PCs can contribute negative chirp tothe signal for a positive propagation length. Thiscomes from the possible different sign of the curva-ture of the PC bands, as shown in Fig. 1. Beamsentering without second-order phase into a materialwith negative chirp still undergo broadening as inordinary materials, since the (negative) spectralphase adds up during propagation, resulting in abroadening of the beam. However, if a beam withpositive initial chirp (caused, for example, by propa-gating through a normal bulk medium such as Si)enters a PC structure with negative chirp, thesecond-order spectral phase gradually cancels out,and as a result, the beam can be focused back to itsminimum phase width by propagation inside the PCstructure (this process is also known as diffractioncompensation18,20). Negative effective index as usedhere should not be mixed with negative refraction(the terminology that is usually applied to the case ofeffective negative refractive index in Snell’s law).Negative effective index solely depends on the curva-ture of the PC bands (second derivative) and canhappen for both positive and negative refraction(which is defined based on the normal direction to theinterface of the material). Also, negative diffractionindex is defined for a single frequency and is differentfrom the group index, which represents the relativeamplitude of the group velocity of the modes.

In the preconditioned superprism effect, ideally,the beams focus back to their minimum spot sizes atthe output of the structure. However, since the PCbands are not perfectly quadratic, the beams havesome distortions at the output of the structure causedby higher-order terms in the Taylor expansion of Eq.(2). Such distortions are the limiting parameters thatdetermine the resolution and the performance of the

device. To analyze the effect of these higher-orderterms, we generalize the approximate model (in Ref.18) to higher-order phase terms in Eq. (2). For theanalysis of PC demultiplexers, we are more inter-ested in the spatial size of the optical beams than thedetails of the beam shape. Thus we define the rmsbeam width21 for an optical beam with spatial inten-sity profile I�x� as

wrms2 �

x

x2I�x�dx

x

I�x�dx

. (3)

If A�k��exp� j��k��� is the normalized spatial Fouriertransform of the beam profile with

k�

A�k��2dk� � 1, (4)

the rms beam width of the beam can be found usingRef. 21 as

wrms2 �

k�

A��k��2dk� �k�

A�k��2���k��2dk�, (5)

in which A��k�� and ���k�� are the first derivatives ofA�k�� and ��k��, respectively, with respect to k�. Toanalyze the beam propagation effects, we assumethat the amplitude of each spectral component of thebeam remains intact (which is valid if there is no loss,no gain, and no coupling between modes in the sys-tem). The spectral phase term ��k�� is caused by thespatial dispersion during propagation of the opticalbeam. Starting with a minimum phase Gaussianbeam with a beam waist of 2w0, given by

A�k�� � � w0

�2� 1�2

exp��14 w0

2k�2 , (6)

entering a material with second-order spectral phase��k�� � b2k�

2 and length L, the output rms beamwidth can be directly calculated from Eq. (5) as

wrms,22 �

w02

4 �4b2

2

w02 , (7)

with

b2 �12

d2�

dk�2 �

12

d2k�

dk�2 L �

L2k0ne2

, (8)

where ne2 � �k0d2k��dk�

2��1 is the effective index asdefined in Ref. 18. Equation (7) can be rewritten as

8468 APPLIED OPTICS � Vol. 45, No. 33 � 20 November 2006

wrms,22 �

14 w0

2�1 �L2

z22 , (9)

with z2 � k0ne2w02�2 being its effective Rayleigh

range.Equation (8) represents the variation (or broaden-

ing) of the size of a Gaussian beam in an ordinarymedium. This equation also governs the size of theoutput beam in a conventional PC demultiplexer. Onthe other hand, the second-order phase is completelycompensated in a preconditioned PC demultiplexer.Thus to calculate the output beam size in such com-pensated structures, we need to use the third-orderspectral phase term [i.e., ��k�� � b3k�

3] in Eq. (5) toobtain

wrms,32 �

w02

4 �w0

�2�

k�

exp��12 w0

2k�2 �9b3

2k�4�dk�

�w0

2

4 �27b3

2

w04 , (10)

where

b3 �16

d3�

dk�3 �

16

d3k�

dk�3 L. (11)

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

wrms2 �

w02

4 �3L2�d3k��dk�

3�2

4w04 . (12)

We can define

z3 �1

�3�d3k��dk�

3��1w03, (13)

to simplify Eq. (12) as

wrms,32 �

14 w0

2�1 �L2

z32 . (14)

This is in an exact analogy with the second-orderform given in Eq. (7). Based on this analogy, we candefine the third-order Rayleigh range as

z3 �12 k0ne3w0

2, (15)

to define a third-order diffractive index as

ne3 �2w0

�3k0�d3k��dk�

3��1, (16)

and the beam-width behavior inside the structurefollows the same behavior as in the ordinary second-order case, with one major distinction that the third-order diffractive index �ne3� linearly depends on thebeam waist �2w0�.

In an ideal preconditioned demultiplexer, we areinterested in compensating the second-order spectralphase by the PC negative diffraction effect and at thesame time, optimizing the PC to have a negligiblethird-order phase term at the operation point. Forsuch ideal cases, the rms beam width is calculatedusing the fourth-order spectral phase term as

wrms,42 �

14 w0

2�1 �L2

z42 , (17)

z4 �12 k0ne4w0

2, (18)

ne4 �2�3w0

2

�5k0�d4ky�dk4��1. (19)

In general, for vth order spectral phase, we can cal-culate the main parameters describing the beampropagation effects as

wrms,v2 �

14 w0

2�1 �L2

zv2 , (20)

zv �12 k0nevw0

2, (21)

nev ��v � 1�!

��2v � 3�!!w0

v�2

k0�dvk��dk�

v��1, (22)

where �2v � 3�!! stands for factorial over odd numbersup to 2v � 3 (i.e., 1, 3, 5, . . . , 2v � 3).

To verify the applicability of the approximatemodel discussed above, we consider incident lightcoming at a 12° angle to a 45°-rotated square latticePC of air holes in Si as shown in Fig. 4. The incidentbeam is preconditioned in this case so that the effectof normal diffraction vanishes at the monitoringplane at the output of the PC region. Using a directmodal approach (mode matching at the interface fol-lowed by propagation in a PC),18 the envelope of theoutput beam profile is calculated and shown in Fig. 4.The result obtained using the approximate effectiveindex method presented in this section is also plottedon the same graph and is in good agreement withdirect simulations.

Note that for a PC structure, all derivatives in thek� � k� plane (e.g., d3k��dk�

3) are calculated at theoperation point on the PC band structure and atthe specific frequency of operation. Thus by disper-sion engineering (i.e., designing the PC to have appro-priate dispersion properties at the operation point), wecan greatly affect the propagation of an optical beam.

20 November 2006 � Vol. 45, No. 33 � APPLIED OPTICS 8469

Note also that the overall behavior of the beam widthof a given beam inside materials with third-fourth, orhigher-order spectral phase terms can be calculatedusing similar formulas as that of an ordinary Gaussianbeam propagating in a bulk medium if we use theappropriate effective index. We use this fact in ouranalysis of beam propagation in preconditioned super-prism devices in Section 3.

3. Analysis of Preconditioned Superprism Devices

The basic topology of the preconditioned superprismdemultiplexer with different important parameters isshown in Fig. 5. There are two basic conditions thatneed to be satisfied in the demultiplexer, namely,spatial separation and diffraction compensation. Spa-tial separation refers to the different channels beingseparated in space at the output of the device. This iscaused by the propagation of beams of different wave-length in different directions inside the structure.This separation can be quantified by defining crosstalk between channels as the sum of the powers of allundesired channels at the location of the desiredchannel. Diffraction compensation condition is thecancellation of second-order spectral phase from theinput diffraction-limited incident beam to the outputplane as designed. In this section, we formulate thesetwo conditions in terms of actual physical design pa-rameters.

In Fig. 5(a), we have shown the propagation of aGaussian beam through the structure where 2w0 isthe initial waist of the incident beam, � is the incidentangle, and L and g are propagation length and prop-agation angle of the beam corresponding to a single

demultiplexing channel in the PC region. In addition,2wPC � 2w0 cos g�cos is the beam waist corre-sponding to that beam inside the PC. Here, weconsider the third-order spectral phase to be the dom-inant term in the higher-order effects; validity of thisassumption can be easily checked for each design bycomparing the contributions of different spectralphase orders. In case other spectral terms becomedominant (for example, the fourth-order spectralphase in the ideal structure) the same steps as belowcan be performed with the corresponding phase term.Figure 5(b) shows the evolution of the beam insidethe structure considering only the third-order spec-tral phase term. The output beam obtained in thisway is the actual beam profile only at the outputplane of the structure, where the effect of the second-order spectral phase is designed to vanish.

Note that the only difference between the precon-ditioned PC demultiplexers and the conventionalones (investigated in detail in Ref. 22) is the replace-ment of the second-order spectral phase with thethird-order spectral phase. Thus using the formula-tion of Section 2, we can apply the same formalism asin Ref. 22 for the calculation of the cross talk in apreconditioned PC device, by using the third-ordereffective index. As a result, we can calculate the re-

Fig. 4. (Color online) For a 45° rotated square lattice PC (air holesin Si, r�a � 0.40) the profile of the beam envelope at the output iscalculated using the direct mode-matching method (solid curve)and the approximate diffractive index method (diamond). The in-cident light in this calculation is a preconditioned (i.e., broadened)Gaussian beam at normalized wavelength a�� � 0.197 that illu-minates the structure at an angle of 12° with respect to the normalto the interface. The preconditioning is performed so that the effectof the second-order diffraction term vanishes at the output of thePC structure. Good agreement of the accurate and approximateresults is clear.

Fig. 5. (Color online) (a) Parameters for a preconditioned super-prism device are depicted for an incident beam coming at an angle�, and for a single channel inside the PC region. (b) The darkerpattern trace shows the evolution of an optical beam at a singlewavelength throughout the structure without the effect of thesecond-order diffraction. In this case, �3 is the divergence angle ofthe beam due to the third-order diffraction effect. The brighterpattern is the actual beam profile inside the structure. By com-pensating the second-order phase, the beam size at the output isthe same as that in the assumed structure with zero second-orderphase everywhere.

8470 APPLIED OPTICS � Vol. 45, No. 33 � 20 November 2006

quired propagation length L, for achieving a cross-talk level of at most X as22

L � 3z3, (23)

where

3 �K�X�

�3 � H�X�. (24)

In these relations, z3 is the Rayleigh range corre-sponding to the third-order spectral phase term,�3 � ���3 is the ratio of the angular separation be-tween adjacent channels (�) to the divergence angleof each channel due to the third-order diffraction ef-fect inside the PC [as represented by �3 in Fig. 5(b)],and K and H are constants given by Table 1 accordingto the required cross talk. The procedure for calcu-lating K�X� and H�X� is the same as that in Ref. 22and is not repeated here. Equations (23) and (24)represent the spatial separation condition of outputchannels for preconditioned superprism devices.

The diffraction compensation condition that de-scribes the cancellation of the overall quadratic phasecan be simply put as

Lpre

npre cos2 �

L

ne2 cos2 g, (25)

in which npre and Lpre are the refractive index and thelength of the preconditioning region, respectively.Also, ne2 and L are the refractive index and the prop-agation length of the PC region, respectively. Toassess the performance of the preconditioned super-prism demultiplexers, we calculate the size of thesestructures for a given angular channel spacing, �.Starting from Eqs. (14) and (15) for a beam pro-pagating in a medium with third-order diffractioneffects, we have �3 � 2����wPC|ne3|� and z3 �

12 k0wPC

2

|ne3|, with |ne3| being the magnitude of the third-order effective index of the PC. By inserting theserelations into Eq. (24) we obtain

L � 3z3 �K

�wPC�ne3���2� � H�12 k0wPC

2|ne3|�, (26)

and therefore,

L �2KwPC

3

wPC2� � 2�3H��3k���k�

3�. (27)

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

A � �wPC

z2L L �

8K2wPC5��3k���k�

3��wPC

2� � 2�3H��3k���k�3��2. (28)

The area in Eq. (28) depends explicitly on thediffraction-limited beam waist of the channel insidethe PC, 2wPC; thus we can minimize the area directlywith respect to this parameter by using �A��wPC� 0 to obtain

�wPC�opt � �10�3H� ��3k�

�k�3��1�2

, (29)

which consequently results in the optimal propaga-tion length �Lopt� as

Lopt �5K2��wPC�opt, (30)

and the optimum (i.e., minimum) PC area �Aopt� as

Aopt �25K 2

2k0ne2�2�wPC�opt. (31)

In terms of the physical parameters of the struc-ture, by using

��3k�

�k�3 �

�

�k���2k�

�k�2 �

�

�k�� 1k0ne2

��cos g

k02n1ne2

2 cos

���ne2

� , (32)

where ��ne2��� is the value calculated at the fre-quency of operation and over the range of excitationangles, we can rewrite Eqs. (29)–(31) as

�wPC�opt �1

k0ne2� 10�3H

��g����cos g

n1 cos ��ne2

� �1�2

�����1�2,

(33)

Lopt �5K

2k0ne2�10�3H cos g

n1 cos ��ne2

� �1�2��g

�� �3�2

������3�2, (34)

Aopt

�2 �25�10�3HK 2

8�2 � cos g

n1 cos ��ne2

� �1�2 1

ne22��g

�� �5�2

������5�2, (35)

in which we have used � � ��g������ as the angularseparation between adjacent channels. From Eq.(35), we can see that the area of the structure scalesas �����5�2, which grows considerably slower than the

Table 1. Cross-Talk Parameters

Cross Talk, X(dB) K(X) H(X)

�20 0.9 0.56�30 0.9 0.83�40 0.9 1.04�50 0.9 1.22

20 November 2006 � Vol. 45, No. 33 � APPLIED OPTICS 8471

�����4 dependence in the conventional superprism-based demultiplexers.22

We can define the compactness factor for the pre-conditioned structures as

Cpre �8�2ne2

2

25�10�3HK2� cos g

n1 cos ��ne2

� ��1�2��g

�� 5�2

,

(36)

which simply relates the spectral spacing betweenchannels (i.e., ��) to their optimum area (i.e., Aopt)through

Aopt

�2 ������5�2

Cpre. (37)

In the optimization process, we use Cpre as the firstmeasure to locate the appropriate operation point onthe band structure of a given PC. Examples of thecalculated values for different lattice types are shownin Fig. 6. The designs considered throughout thispaper are all in the first band of the PC. The motiva-tion behind this choice is the potentially lower prop-agation loss and lower reflection loss. Nevertheless,the same procedure can be followed for other bands aswell.

Plots shown in Fig. 6 are very useful in comparingdifferent PC structures and in choosing the optimumstructure by looking for the structures with maxi-mum Cpre. From the results in Fig. 6, we can see thatthe compactness factor becomes larger as the con-stant frequency contours deviate from the bulk-typecircular patterns. Such contours in the first band oftriangular lattice PCs occur only at the vicinity of theboundaries of the Brillouin zone. Therefore the band-width is limited in these cases. In square lattice PCs,however, the compactness factor can be large even atregions away from the boundaries of the Brillouinzone. Also, by comparing Figs. 6(a) and 6(b) we cansee that in the square lattice with interfaces alongone of its principal lattice directions, the most appro-priate operation points (those with the largest valuesof compactness factor) have their direction of groupvelocity (which is normal to the constant frequencycontour) parallel to the interface that is not suitablefor demultiplexing purposes. However, by rotatingthe lattice by 45° a relatively large range of the PCband with a large compactness factor can be excitedusing a single incident angle for demultiplexing. Theoptimum design operation point in this case is alongthe |kx| � |ky| � ��2�a directions in the kx–ky plane,due to the intrinsic symmetry of the lattice that im-plies zero odd-order diffraction effects along this line.Once the operation range on the band structure isselected, the design parameters can be obtained us-ing the process described in detail in Section 4.

4. Design of Planar Preconditioned SuperprismDemultiplexers

In this section we focus on developing a design strat-egy for planar (slab-type) PCs with 2D in-plane peri-odicity; nevertheless, the process can be applied withminor modifications to 3D PCs as well. The choice ofthe planar structures matches the practical realiza-tions like PCs fabricated in a silicon-on-insulator(SOI) wafer. In such structures, the dispersion effectin the unpatterned slab region connected to the PC(which serves as the incident region) cannot be ne-glected. Note that the optical beams in a precondi-tioned structure propagate in the slab of unpatternedmaterial (for example, Si) prior to entering the PCregion. Knowing the effective index of the unpat-terned slab is important both in applying the phase-matching condition at its interface with the PC and indesigning the lengths of the different parts of thestructure (i.e., preconditioning region, and PC region)to achieve the complete compensation of the second-order diffraction at the output of the structure. Atypical dispersion diagram for an asymmetric siliconslab sandwiched between air and silicon oxide (a typ-ical SOI wafer) is shown in Fig. 7(a). Depending onthe thickness of the wafer �h��, the wavelength on thisgraph can be scaled [using �� � �h��h��] to find thedispersion diagram for an SOI wafer with arbitrarythickness of the Si layer. Using the dispersion dia-gram of the unpatterned slab that serves as the inputregion [as shown in Fig. 7(a)], for each incident anglewe can find the excited modes inside the PC. Anexample of the loci of the excited modes at differentfrequencies inside the PC for four different incidentangles (5°, 10°, 15°, and 20°) is shown in Fig. 7(b).

For any incident angle, we can find different pa-rameters of the structure, i.e., angle of group velocity�g�, second-order effective index �ne2�, sensitivity fac-tors ���g���� and ��ne2���], as well as higher-ordereffective indices associated with each demultiplexingchannel in our bandwidth of interest. These param-eters describe the propagation behavior for eachchannel (i.e., direction of propagation, sensitivity tofrequency, and the divergence caused by the third-order spectral phase term). To get the required crosstalk for all channels, the parameter wi (i.e., the beamwaist of the incident optical beam consisting of sev-eral wavelengths) should be found in such a way thatthe maximum propagation length required over allchannels is minimized. This can be directly per-formed by reformulating Eq. (27) as

Lj �2Kwi

3 cos3 g j

wi2�j cos2 g j cos � 2�3H ��3k���k�

3�j cos3 ,

(38)

in which subscript j stands for the parameters calcu-lated for the jth channel. After finding wi from thisprocess, it is straightforward to set

L � maxj

�Lj�wi��, (39)

8472 APPLIED OPTICS � Vol. 45, No. 33 � 20 November 2006

and the length of the preconditioning region is foundfrom Eq. (25) as

Lpre �npre cos2

ne2 cos2 gL, (40)

which completes the design. Here, npre is calculatedby 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 todesign an optimal preconditioned PC demultiplexer fora dense wavelength division multiplexing (DWDM)system with different wavelength channel spacings

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 isthe radius of the holes, and a is the lattice constant) are shown along with constant frequency contours of the corresponding PC band. Eachcontour in the kx–ky plane corresponds to a constant frequency. The value of the normalized frequency �a��� for each constant frequencycurve 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 withr�a � 0.30 and h�a � 0.60, and (b) the same square lattice as in (a) with the interface along a direction angled 45° with respect to theprincipal lattice directions. (c) A triangular lattice with r�a � 0.30 and h�a � 0.60 with the interface along the �M direction, and (d) thesame triangular lattice as in (c) with the interface along the �K direction.

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operating around 1550 nm. We also assume that thedemultiplexing structure is fabricated on an SOIwafer with the top Si thickness of 180 nm � h �250 nm, and the underlying SiO2 layer of 3 �m thick-ness. Based on the results shown in Fig. 6, we selectsquare lattice PCs due to their wider bandwidth ofoperation. For each case (with a specified number ofchannels and channel spacing), and for an arbitraryPC lattice and angle of incidence (i.e., the angle be-tween the incident beam direction and normal to thePC interface) we use the compactness factor to choosethe center frequency of operation. Then, around thiscenter frequency, the required propagation length (L)and incident beam waist �2wi� is found so that thedesired cross-talk level for all channels is achieved.This process can be repeated for a range of angles anddifferent PC lattices to find the optimum structure forthe given specifications. For comparison, here we con-sider two classes of square lattice PCs, one with itsinterface along one of the principal lattice vectors[Fig. 6(a)] and the second one, a rotated square latticePC whose interface with Si makes a 45° angle withrespect to the principal lattice vectors [as shown inFig. 6(b)]. The properties of the optimal structures

obtained for these two classes of PCs are listed inTables 2 and 3, respectively.

As the channel spacing becomes larger, the angularseparation between adjacent channels increases (dueto the fixed angular dispersion at the operation pointof the structure, imposed by the PC band structure).Consequently, the larger angular separation betweenadjacent wavelength channels relaxes the restrictionon the diffraction-limited spot size (i.e., the devicecan work with a smaller initial beam waist, wi). Bothof these effects (i.e., larger angular separation andsmaller beam waist) result in a shorter required pro-pagation length and thus a smaller structure for agiven number of channels. This trend can be observedin the results of Table 2 for devices of the same band-width.

Note that increasing the operation bandwidtheventually requires the use of suboptimal structuresand thus lower performance. For example, in eachrow of Table 2, as the number of channels increases,portions of the band structure with less optimal prop-erties need to be included to cover the entire operationbandwidth. As a result, performance deteriorates asthe number of channels increases and thus larger

Fig. 7. (Color online) (a) Dispersion diagram for guiding in anunpatterned SOI wafer with h � 220 nm is shown. (b) Band struc-ture (dotted curves) of a slab-type PC in a SOI wafer (squarelattice, r�a � 0.30, h�a � 0.62) and loci of PC modes (solid curves)excited for the incident wave coming from the unpatterned Si slabat different incident angles (in degrees) are shown.

Table 2. Design Parameters for Optimal Demultiplexers in a SquareLattice PCb

ChannelSpacing(GHz)

Number of Channels

4 8 16 32

1002wi (�m) 40 41 42 48L (mm) 4.9 5 5.3 7.1

2002wi (�m) 28 29 33 N�Pa

L (mm) 1.6 1.9 2.5 N�Pa

4002wi (�m) 22 24 N�Pa N�Pa

L (mm) 0.62 0.9 N�Pa N�Pa

aN�P means not possible.bWith the interface along high-symmetry directions; the thick-

ness of the top Si layer, h � 195 nm; the normalized radius of holes,r�a � 0.30; and the incident angle of � � 10°.

Table 3. Design Parameters for Demultiplexers in a Square Lattice PCa

ChannelSpacing(GHz)

Number of Channels

4 8 16 32

1002wi (�m) 17 30 45 63L (mm) 1.4 2.0 2.9 4.7

2002wi (�m) 17 30 43 52L (mm) 0.61 0.95 1.6 3.1

4002wi (�m) 17 28 35 51L (mm) 0.29 0.53 1.1 2.5

aWith the interface at 45° with respect to high-symmetry direc-tions; the thickness of the top Si layer, h � 242 nm; the normalizedradius of holes, r�a � 0.30; and the incident angle of � � 10°.

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structures are needed. For larger channel spacings,the structure simply does not support the relativelylarge bandwidth needed for a large number of chan-nels (listed as not possible, or N�P in Table 2).

By comparing the results in Tables 2 and 3, it isevident that the rotated square lattice offers morecompact demultiplexers for the same specificationsand supports larger bandwidths. Therefore the ro-tated square lattice PC structure is preferred forpreconditioned superprism demultiplexers. Anotherimportant issue is the variation of the effective indexover the operation bandwidth. Note that discrepan-cies encountered in Eq. (40) due to the variations ofthe effective index �ne2� from channel to channel in-crease the output beam size for some channels owingto the reminiscent second-order spectral phase termand result in higher cross talk. For 1.6 THz band-width around the center frequency, this variation isless than 2% (i.e., the effective index is �0.23 � 2%over the entire bandwidth) for the rotated squarelattice design, while for the same bandwidth the ef-fective index varies between �1.4 and �0.4 in thesquare lattice with the interface along principal lat-tice directions (which needs a separate mechanism tocompensate it). This is another clear advantage of therotated square lattice structure.

Compared to the previously reported results forconventional superprism structures,13,14,22 the de-signs in Table 3 require less collimated input beams(the requirement for the input beam waist is relaxedby a factor of more than 2), and there are also almost2 orders of magnitude improvement in the compact-ness of the structure (more than 1 order of magnitudeimprovement in the required propagation length).These improvements bring the superprism-based de-multiplexers in the range where their fabricationthrough conventional techniques is possible. The per-formance of these devices can be further improved bytopology optimization of the PC structure to obtainbetter demultiplexing properties (our optimizationspace in this paper is limited to specific lattice typesand directions). Also, one can envision that by com-bining different effects in a heterostructure PC andincluding contributions from other demultiplexing ef-fects (such as the distinction in wave vectors as in kvector superprisms23), more compact and higher-resolution demultiplexers can be realized.

Another issue that was not discussed in this paperis the loss due to reflection at the interface of the PC.The effective index used as our main tool in designingpreconditioned superprism devices does not provideany information about the reflection loss at the inter-faces of the structure. However, several reflection re-duction schemes have been proposed to minimize thereflection loss.20,24–26 In particular, it has been shownthat by using adiabatic matching stages it is possibleto achieve large angle and wideband coupling of lightinto and out of PC structures.26 Therefore the de-sign for the optimum superprism demultiplexer canbe performed independent of the possible reflectionlosses as shown in this paper, and efficient matchingcan be achieved in a subsequent independent stage.

6. Conclusions

We presented a systematic analysis and design of anew type of photonic crystal demultiplexers (i.e., pre-conditioned demultiplexer), in which the simulta-neous existence of the superprism effect and thenegative effective index of diffraction results in a verycompact structure by canceling the second-orderspectral phase to avoid beam broadening inside thePC. As a result, the size of the preconditioned demul-tiplexers varies as N2.5 (N being the number of wave-length channels) is compared to the N4 variation inthe conventional superprism-based PC demultiplex-ers. We analyzed the basic properties of these struc-tures using a generalized effective index model. Byusing the generalized effective index model in the anal-ysis and design of these structures one can deduce allthe basic properties of the structure simply from theband structure without the need to go through tedioussimulations, especially for 3D structures. Further-more, using the effective index technique, we devel-oped a simple and systematic method for designingoptimal preconditioned demultiplexers. Our resultsshow that the preconditioned superprism-based PC de-multiplexers have 2 orders of magnitude smaller sizecompared to the conventional superprism-based de-multiplexers. This improvement makes it feasible touse these structures for high resolution applicationslike DWDM systems by fabricating them through nor-mal fabrication techniques.

This work was supported by the U.S. Air ForceOffice of Scientific Research under contract F49620-03-1-0362 (G. Pomrenke) and by the National Sci-ence Foundation under contract ECS-0239355 (L.Goldberg).

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