1136 IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 10, NO. 8, AUGUST 1998

Efficient Architectures for ExactlyRealizing Optical Filters withOptimum Bandpass Designs

C. K. Madsen

Abstract—Butterworth, Chebyshev, and elliptic bandpass filterdesigns are optimal in the sense of band flatness or equiripplecharacteristics. A new architecture using optical all-pass filters ispresented which can realize these designs exactly and efficientlyusing either ring resonators or reflectors such as Bragg gratingsor thin-film interference filters. Design examples are given for aseventh- and eighth-order elliptic filter, and the new architectureis shown to be tolerant to loss. Previously, reflective filters couldonly approximate optimal responses. An order of magnitudeimprovement in transition width is demonstrated for an ellipticfilter compared to an optimized transmission response for anindividual thin-film filter.

Index Terms—All-pass filters, bandpass filters, dielectric res-onator filters, gratings, lattice filters, optical filters, optical wave-guide filters.

A PPLICATIONS for optical bandpass filters includedemultiplexing and add/drop in WDM systems. Basic

filtering techniques include multipath feed-forward inter-ference using an array of waveguides or a lattice ofcoupled Mach–Zehnder interferometers (MZI’s) and feedbackinterference using thin-film filters, Bragg gratings, or ringresonators. Filters having only feedforward interference pathscontain only zeros in their transfer functions; whereas, filterswith feedback paths contain both poles and zeros. Optimumbandpass designs include the Butterworth, Chebyshev, andelliptic filters, where the design is optimized with respect tocharacteristics of the passband and stopband such as maximumflatness or equiripple. Their transfer functions contain bothpoles and zeros which must be located at particular values.The only optical filter reported to date which can realize theseresponses exactly is a lattice structure interleaving MZI’s andring resonators [1]. A new optical architecture is presentedwhich allows the optimum filter designs to be implementedexactly using thin films, Bragg gratings, or ring resonators asthe basic filtering elements. The new architecture is simplerand more efficient than [1], requiring fewer couplers and phaseshifters, and easily accommodates reflective feedback paths.Reflective feedback paths have a major advantage over ringsin that the period of the frequency response, the free-spectralrange (FSR), is not limited by the bend radius so broad-banddesigns can be easily realized.

Manuscript received February 13, 1998; revised March 24, 1998.The author is with the Lucent Technologies, Bell Laboratories Innovations,

Murray Hill, NJ 07974 USA.Publisher Item Identifier S 1041-1135(98)05354-3.

Optimum bandpass filters can be realized as the sum or dif-ference of two all-pass functions [2]. The frequency responseof each all-pass function is written as

and (1)

The sum and difference of and leads to the followingexpressions for the magnitude response of two new functions,

and , respectively,

(2)

(3)

When both all-pass functions have the same phase, their sumis maximum. When their phases differ by, the sum is zerobut the difference is maximum. The resulting functionsand

are power complementary, i.e.,

(4)

For filter synthesis, it is convenient to work with trans-forms. By replacing with the following expressions areobtained for and :

(5)

(6)

Note that and share a common denominatorpolynomial , but have different numerator polynomials

and . The transform of an all-pass function hasa special symmetry between the numerator and denominatorpolynomials as shown in (7):

(7)

where represents a constant phase. The zero locations aremirror images about the unit circle from the pole locationswhich gives unity magnitude response for all frequencies.

1041–1135/98$10.00 1998 IEEE

MADSEN: EFFICIENT ARCHITECTURES FOR EXACTLY REALIZING OPTICAL FILTERS 1137

(a)

(b)

(c)

Fig. 1. Architectures: (a) schematic for a bandpass filter realized with twoall-pass filters; (b) an even order cascade ring implementation; and (c) an oddorder coupled cavity implementation using reflectors.

The optimal filter designs previously mentioned can bedecomposed into a linear combination of all-pass functions.

, , and have real coefficients and the numeratorpolynomial of has mirror image symmetry. In addition,

is power complementary to whose coefficientshave even or odd symmetry depending on whether the filterorder is even or odd, respectively, where even symmetry refersto and odd symmetry to . The detailsof the synthesis method depend on whether has even orodd order. The linear combination of all-pass functions whichdefine and is expressed in (5) and (6). If haseven order, then and are complex conjugates andhave poles and zeros each.

I. OPTICAL ALL-PASS AND BANDPASS FILTERS

The sum and difference are easily implemented in an opticalfilter using directional couplers. By setting the power couplingratio 50%, the sum and difference of the two all-passfunctions are realized in the cross and bar transmission portsas shown in Fig. 1(a). The directional coupler through andcross amplitude transmission are denoted by and

, respectively. Because is imaginary, the opticalfilter transfer functions are given by and

Several optical all-pass filter architecturesare presented in [3]. A lossless ring resonator with a singlecoupler is a single stage all-pass filter. Its transfer function inthe -domain is given by

(8)

where , the ring circumference iswith radius , and the effective index is for the fundamentalmode. The FSR is given by FSR . Bend loss limits thesmallest radius which can be used, and thus limits the largest

(a)

(b)

Fig. 2. (a) An eighth-order elliptic filter with a transition width of 0.0035� FSR. The coupling ratios and phases for a cascade implementation areshown in the inset. (b) Transmission and reflection responses for an optimizedeighth-order all-pole filter which has a transition width of 0.035� FSR.

achievable FSR. An th-order all-pass filter can be realized bycascading rings as shown in Fig. 1(b). The coupling ratios andphases for each ring are calculated from the all-pass functions.For example, to produce a pole at requires a coupling ratioof and a phase of . The rings arenominally the same circumference with an incremental lengthor index change added to each ring to realize the phase, i.e.,

. For example, heaters may be employedon a section of the ring to locally change the index and createthe desired phase. The new design for optimum bandpass filtersrequires couplers and phase shifts. In contrast,the general waveguide filter architecture [1] requirescouplers and phase shifters to realize anth-order filter.

A second ring resonator structure, known as a lattice filter,where the rings are coupled to each other, can also realize anall-pass function [3]. An analog to the lattice ring is a coupledcavity structure [4] created with reflectors. To form an all-passfunction, the last reflector must have an amplitude reflectanceof unity, i.e., 1. Fig. 1(c) shows a bandpass architectureusing two all-pass reflective lattice filters. The FSR is givenby FSR where is the nominal cavity length.The cavity length or index is varied for each stage to createthe relative phase . In the transform description, theamplitude reflectances replace the coupler transmittances,.The partial reflectors must be wavelength independent overthe FSR so that the filter coefficients are constant. Thinfilms with alternating layers of high and low index or Bragggratings can be used to realize the partial reflectors. In eithercase, the index difference must be large enough to satisfythe wavelength independence constraint. Partial reflectors inwaveguides should be realizable with fabrication methods usedto demonstrate photonic bandgaps [5].

1138 IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 10, NO. 8, AUGUST 1998

TABLE IPOLYNOMIALS FOR AN EIGHTH- AND A SEVENTH-ORDER ELLIPTIC FILTER

II. DESIGN EXAMPLES

An eighth-order elliptic filter is designed as a first example.The design was chosen to have a 30-dB stopband rejectionfor both and and a cutoff of 0.1 FSR. Thestopband rejection requirement for one response implies apassband ripple of 0.004 dB for the other response. Typically,only one response, or , is given from a filterdesign program. The details for determining the other responseas well as decomposing them into the all-pass filtersand are covered in [2]. The polynomials , ,and are listed in Table I. The denominator polynomialsfor the all-pass functions are both fourth order. The all-passfunctions are defined asand . The phase factors

and are given in radians. Table I also contains thepolynomials for a seventh-order elliptic design. Now thecoefficients of and are real, so their roots occuron the real axis or in complex conjugate pairs. Fig. 2(a) showsthe magnitude response for the transmission cross and barstates for the eighth-order elliptic design. The normalizedfrequency range of 0 to 1 corresponds to one FSR of thefrequency response. Given and , the couplingratios and partial reflectances are determined via (8) for thecascade architectures or by step-down recursion relations [6]for the lattice structures. The coupling ratios are shown forthe cascade implementation in the inset. The coupling ratiosare identical for and while the phases are complexconjugates of each other. A schematic of a cascade ringconfiguration is shown in Fig. 1(b) for an even ordered filter.As seen for the single ring case in (8), there is a phase constantequal to the sum of the individual phases, i.e. ,associated with each all-pass transfer function which is notequal to in general. In Fig. 1(b), the phase terms- and

- are included in the upper and lower arms, respectively.In Fig. 2(b), an all-pole transmission response is shown whichrepresents an optimized thin-film design with the same order,cutoff frequency, and passband ripple. An all-pole responsecannot meet equiripple requirements in both the passband andstopband. As an example, let the FSR40 nm for both filters.

Fig. 3. An eighth-order elliptic filter with 0.2-dB loss/feedback path length.

Then, the passband to stopband transition width is 1.4 nm forthe all-pole design compared to 0.14 nm for the elliptic design.For WDM systems, filters with small transition widths areimportant because they can accommodate narrower channelspacings.

In the presence of loss, the all-pass property is broken.This is easily verified in (8) by substituting for ,where loss in decibels for one feedback path is .With the eighth-order design, a loss of 0.2-dB/feedback pathwas introduced. The resulting magnitude response is shown inFig. 3. The peak transmission of is reduced, but the30 dB stopband rejection of is retained.

In summary, a new architecture was presented that allowsoptimum bandpass filters to be exactly realized with thin films,Bragg gratings, or ring resonators. Because reflective feedbackstructures can be used, the FSR is not limited by bend radius,as is the case with rings. The new architecture requires fewercouplers and relative phase shifts per stage compared to ageneral waveguide architecture, and it is robust to loss. Inaddition, switching between the two power complementaryresponses is easily achieved by introducing a-phase changebetween the interferometer arms. Some applications includedemultiplexing, adding or dropping single channels as well asfiltering bands of channels for WDM systems.

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[3] C. K. Madsen and G. Lenz, “Optical all-pass filters for phase responsedesign with applications for dispersion compensation,”Photon. Technol.Lett., vol. 10, pp. 994–996, July 1998.

[4] E. Dowling and D. MacFarlane, “Lightwave lattice filters for opticallymultiplexed communication systems,”J. Lightwave Technol., vol. 12,pp. 471–486, Mar. 1994.

[5] J. Foresi, P. Villeneuve, F. Ferrara, E. Thoen, G. Steinmeyer, S. Fan, J.Joannopoulos, L. Kimerling, H. Smith, and E. Ippen, “Photonic-bandgapmicrocavities in optical waveguides,”Nature, vol. 390, pp. 143–145,Nov. 1997.

[6] C. Madsen and J. Zhao, “A general planar waveguide autoregressiveoptical filter,” J. Lightwave Technol., vol. 14, pp. 437–447, Mar. 1996.