Tunable photonic filters: a digital signal processing design approach

Tunable photonic filters: a digital signal processingdesign approach

Le Nguyen BinhDepartment of Electrical and Computer Systems Engineering, Monash University, Clayton,

Victoria 3168, Australia ([email protected])

Received 26 January 2009; revised 30 March 2009; accepted 17 April 2009;posted 20 April 2009 (Doc. ID 106797); published 11 May 2009

Digital signal processing techniques are used for synthesizing tunable optical filters with variable band-width and centered reference frequency including the tunability of the low-pass, high-pass, bandpass,and bandstop optical filters. Potential applications of such filters are discussed, and the design techni-ques and properties of recursive digital filters are outlined. The basic filter structures, namely, the first-order all-pole optical filter (FOAPOF) and the first-order all-zero optical filter (FOAZOF), are described,and finally the design process of tunable optical filters and the designs of the second-order Butterworthlow-pass, high-pass, bandpass, and bandstop tunable optical filters are presented. Indeed, we identifythat the all-zero and all-pole networks are equivalent with well known principles of optics of interferenceand resonance, respectively. It is thus very straightforward to implement tunable optical filters, which isa unique feature. © 2009 Optical Society of America

OCIS codes: 070.2025, 230.2285.

1. Introduction

Tenable optical filters with variable bandwidth andcenter frequency characteristics are important in ap-plications where dynamic changes in the bandwidthand center frequency of the filter are required. Onesuch application is a frequency-division-multiplexing(FDM) or wavelength-division-multiplexing (WDM)optical system, which utilizes the large bandwidth ofoptical fibers to increase the transmission capacity,which is mainly limited by fiber dispersion. In WDMsystems such as in multichannel video distributioncable-television optical systems, tunable optical fil-ters are used as optical demultiplexers at the recei-vers to select one or more desired channels at anywavelength. Further, the operating speed of fibertransmission systems is getting higher and higher,reaching 100Gbits=s and above. Thus it is expectedthat photonic signal processors would play a majorpart in the optical domain, such as phase compari-sons for differentially phase modulated signals, opti-cal equalization, variable delay line, and dispersion

compensation. Optical filters can indeed satisfy anumber of these functionalities depending on the op-erating region of the filter. Optical filters with tun-able bandwidth would thus offer the flexibility foroptical signal processing.

There have been reports of several types of tunableoptical filter such as the optical ring resonator [1],optical transversal filter [2], and Fabry–Perot inter-ferometers [3], [4]. These filters may be referred to asbandpass tunable optical filters because their magni-tude responses have Gaussian-type characteristics.Another type of tunable optical filter is the cascaded-coupler Mach–Zehnder channel adding/droppingfilter whose output ports constitute a power-complementary pair; one port is used as a bandpassfilter, while the other port is used as a bandstop filter[5]. It is believed that there has been no previous re-port of a systematic filter design technique, whichcan be used to design tunable optical filters withthe essential features of variable bandwidth andcenter frequency characteristics as well as generalfiltering characteristics such as low-pass, high-pass,bandpass, and bandstop characteristics.

In this paper, a digital filter design technique isemployed to design tunable optical filters with

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variable bandwidth and center frequency character-istics as well as low-pass, high-pass, bandpass, andbandstop characteristics. The filter design methoddescribed here is adopted from the techniques pre-sented in Refs. [6–9].In brief, an optical filter is determined by the poles

and zeros of the optical networks, the passband, andthe roll-off of the cutoff bands. The passband can betunable if these poles and zeroes can be designedsuch that they can be tuned. Thus if we can representthe transfer transmittance by a cascade of opticalsubsystems with all-zero and all-pole networks, thenit is very straightforward to implement the filter. In-deed an optical interferometer can be considered asan all-zero network, as it is based on the depletion ofthe output beam when its interfered output is in de-struction. Alternatively, a fiber optic interference fil-ter can be considered as a phase comparator with twooutput ports, the constructive and destructive ports.Likewise, an all-pole network can be implementedusing a resonant optical subsystem. At resonancethe output reaches its huge output. This paper thuspresents the design of tunable optical filters based onalternating all-zero and all-pole networks of opticalsubsystems. The design is thus based on the populardigital processing (DSP) technique and the principlesof interference and resonance in optics.This paper is organized as follows: Section 2 gives

the analytical techniques and equivalent DSP in thez-transform domain and then the design strategy oftunable optical filters. Section 3 outlines the struc-ture of optical filters that represent the digital designof Section 2. Section 4 illustrates the design of a num-ber of tunable optical filters whose transfer functionsare very much familiar to designers of electrical anddigital filters in the electronic domain. Finally con-cluding remarks are given.

2. Review of Design Techniques of Recursive OpticalFilters

This section outlines various design techniques andgeneral characteristics of recursive digital filters andtheir applications in the design of tunable opticalfilters.

A. Design Techniques for Recursive Digital Filters

The common approach to the design of recursivedigital filters involves the transformation of a recur-sive analog (or continuous-time) input into a recur-sive digital filter for a given set of prescribedspecifications. This is due to the availability of well-developed techniques for analog filters that are oftendescribed by simple closed-form design formulas. Insuch transformations, the essential properties of thefrequency response of the analog filter are preservedin the frequency response of the resulting digital fil-ter. The two well known techniques used for convert-ing Butterworth, Chebyshev I and II, and ellipticanalog filters to their corresponding digital filtersare the impulse invariance and bilinear transforma-tion methods.

In the impulse invariance method, the impulse re-sponse of a digital filter is determined by samplingthe impulse response of an analog filter. This techni-que requires the analog filter to be band limited toavoid the aliasing (or interference) effect, and thus itis effective only for low-pass and bandpass analogfilters. If it is to be used for high-pass and bandstopanalog filters, then additional band limiting isrequired on these filters to avoid severe aliasingdistortion.

The bilinear transformation method involves analgebraic transformation between the variables sand z that maps the entire imaginary axis in the splane to one revolution of the unit circle in the zplane. Thus a stable analog filter (with poles in theleft-half side of the s plane) can be transformed into astable digital filter (with poles inside the unit circlein the z plane). As a result, unlike the impulse invar-iance method, this method does not suffer from theeffect of aliasing distortion. However, it suffers fromthe effect of nonlinear compression of the frequencyaxis, and thus it is only useful if this undesirable ef-fect can be tolerated or compensated for.

An alternative approach, which is valid for boththe impulse invariance and the bilinear transforma-tion methods, is to design a digital prototype low-pass filter and then perform a frequency transforma-tion on it to obtain the desired low-pass, high-pass,bandpass, and bandstop digital filters. However,the use of the frequency transformation techniquein the design of a digital filter is not so straightfor-ward because the analog prototype low-pass filteris generally not known to the designer. Thus it is ne-cessary to “find” an analog prototype low-pass filtersuch that, after transformation, the resulting digitalfilter would meet a given set of specifications. Due torapid advances in the field of digital signal proces-sing, the digital filter design techniques employedin this paper have been incorporated as standardfunctions in several commercial computer softwarepackages.

B. Properties of Recursive Digital Filters

The three types of recursive digital filter commonlyknown in DSP techniques [10] are the Butterworth,Chebyshev I and II, and elliptic filters, and theirproperties are the following: (i) Butterworth digitalfilters are characterized by a magnitude responsethat is maximally flat in the passband andmonotonicoverall. They sacrifice roll-off steepness for gradualroll-off in the passband and stopband. If the Butter-worth filter smoothness is not required, a Chebyshevor an elliptic filter can generally provide steeper roll-off characteristics with a lower filter order. (ii) Che-byshev I digital filters are equiripple in the passbandand monotonic in the stopband, while Chebyshev IIfilters are monotonic in the passband and equiripplein the stopband. Chebyshev I filters roll off fasterthan Chebyshev II filters but at the expense of pass-band ripple. Chebyshev II filters have stopbands thatdo not approach zero like Chebyshev I filters do but

2800 APPLIED OPTICS / Vol. 48, No. 15 / 20 May 2009

are free of passband ripple. Both the Chebyshev Iand II filters have the same filter order for a givenset of filter specifications. (iii) Elliptic digital filtersare equiripple in both passband and stopband. Theyoffer steeper roll-off characteristics than the Butter-worth and Chebyshev filters but suffer from pass-band and stopband ripples. In general, ellipticfilters, although the most expensive to compute, willmeet a given set of filter specifications with the low-est filter order. These types of digital filter have zeroslocated on the unit circle in the z plane (i.e., jzj ¼ 1),which may significantly simplify the design of tun-able optical filters. However, they may face the pro-blems of stability, as the poles are located on the unitcircle of the z plane [10].

C. Transfer Function of Recursive Digital Filters

For analytical clarity, the variables with a cap [e.g.,HðzÞ] are associated with digital filters, while the cor-responding variables without a cap [e.g.,HðzÞ] are as-sociated with optical filters. The z variable is definedas z ¼ expðjωTÞ ¼ expðjβLÞ, where T is the period ofthe light wave, ω is the frequency of the light wave, βis the propagation of the light waves in the wave-guide, and L is the propagation length of the lightwave. Thus the variable z is indeed a delay variable.The frequency range is in the optical frequency.

However, if the light wave is a modulated wave, thenthe frequency can be that of the envelope that mod-ulates the amplitude of the light waves. In this paperwe mean the optical frequency that is normalizedwith respect to the central frequency of the opticalspectrum. So the frequency or wavelength of the car-riers can be used without loss of generality.The transfer function of the Mth-order recursive

digital filter can be expressed in a rational form as

HðzÞ ¼ AYMk¼1

ðz − zkÞðz − pkÞ

¼ Aðz − z1Þðz − p1Þ

ðz − z2Þðz − p2Þ

…ðz − zMÞðz − pMÞ ;

ð1Þ

where A is a constant and z is the z-transform para-meter, representing the delay function. Furthermore,pk and zk are the kth pole and zero in the z plane,respectively, which can be expressed in the phasorforms as

pk ¼ jpkj exp½jargðpkÞ�; ð0 ≤ jpkj < 1Þ; ð2Þ

zk ¼ jzkj exp½j argðzkÞ�; jzkj ¼ 1; ð3Þ

where “arg” denotes the angle of the argument. Notethat the system stability requires the poles to be lo-cated inside the unit circle in the z plane as describedby the condition given in Eq. (2). Now let the transferfunction of the kth-stage first-order all-pole digitalfilter be defined as

Hap;kðzÞ ¼1

ðz − pkÞ; ð4Þ

and the transfer function of the kth-stage first-orderall-zero digital filter be defined as

Haz;kðzÞ ¼ ðz − zkÞ; ð5Þ

where the subscripts ap and az denote all-pole andall-zero, respectively. The transfer function of theMth-order all-pole digital filter, which is the transferfunction of the cascade of M first-order all-pole digi-tal filters, is given by

HapðzÞ ¼YMk¼1

Hap;kðzÞ: ð6Þ

The transfer function of the Mth-order all-zerodigital filter, which is the transfer function of thecascade of M first-order all-zero digital filters, isgiven as

HazðzÞ ¼YMk¼1

Haz;kðzÞ: ð7Þ

The transfer function of the Mth-order recursivedigital filter, as given in Eq. (1), can thus be writtenalternatively as

HðzÞ ¼ A · HapðzÞ · HazðzÞ: ð8Þ

3. Basic Structures of Tunable Optical Filters

This section describes the basic photonic filter struc-tures, namely, the first-order all-pole optical filter(FOAPOF) and the first-order all-zero optical filter(FOAZOF) of tunable optical filters. The opticaltransmittance or the optical transfer functions ofthese basic subsystems can indeed be cascaded forthe design of optical filters. Depending on the speedof the signals and the envelope containing the opticalcarriers, the size of these subsystems can be as shortas a few hundreds of micrometers. In this case, inte-grated optical structures, the planar light wave tech-nology, must be used.

A. First-Order All-Pole Optical Filter

A first-order all-pole optical filter is represented by atransfer function (the transmittance) whose numera-tor would be a constant and independent of the zvariable. But its denominator is a first order of z.The numerator can be either a constant or a termz alone. That is, the zero is at the origin on the zplane. A first-order all-pole optical network can beconstructed as shown in Fig. 1. It can representthe kth-stage FOAPOF. This structure can easily beimplemented using either fiber-based or planar lightwave circuit (PLC) silica on silicon technology. TheFOAPOF consists of an optical waveguide circular


ring interconnected by two identical tunable couplers(TCs) and a thermo-optic phase shifter (PS). Thephase shifter can be a thermo-optic section of thewaveguide on which an electrode is deposited, anda voltage applied on it would generate a thermal ef-fects that would change the refractive index of theoptical waveguide and hence a phase delay of thelight waves.The TC has the

complex cross − coupled coefficient

¼ ffiffiffiffiffiffiffiffiffiffiγwakp

expðjθkÞ; ð9Þ

complex direct − coupled coefficient

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiγwð1 − akÞ

pexpðjθkÞ; ð10Þ

where γw (typically γw ¼ 0:89 for an insertion loss of0:5dB) is the intensity transmission coefficient of theTC. The cross-coupled intensity coefficient of the TCis given, from Eq. (9), as

ak ¼ ð1þ cosϕkÞ=2 ð0 ≤ ak ≤ 1Þ; ð11Þfrom which the phase shift of the PS on the upperarm of the TC is given by

φk ¼ cos−1ð2ak − 1Þ ð0 ≤ φk ≤ 2πÞ; ð12Þand the resulting phase shift of the TC is given, fromEq. (10), as

θk ¼ tan−1

�sinϕk

cosϕk − 1

�ð−π=2 ≤ θk ≤ π=2Þ: ð13Þ

The optical transmittances of the upper ðΛ1kÞ andlower ðΛ2kÞ halves of the waveguide ring are definedas

Λ1k ¼ expð−αwL1kÞ expð−jωT1kÞ; ð14Þ

Λ2k ¼ expð−αwL2kÞ expð−jωT2kÞ expðjϕkÞ; ð15Þwhere the amplitude waveguide propagation loss istypically αw ¼ 0:01151 cm−1 as determined from αw ¼

αwðdB=cmÞ=8:686 with αwðdB=cmÞ ¼ 0:1dB=cm. L1kand L2k are the waveguide lengths of the upperand lower halves of the ring, T1k and T2k are the cor-responding time delays of the upper and lower halvesof the ring, j ¼ ffiffiffiffiffiffi

−1p

, ω is the angular optical fre-quency, and ϕkð0 ≤ ϕk ≤ 2πÞ is the phase shift intro-duced by the PS. Note that the lengths of thelower and upper halves of the ring do not necessarilyneed to be the same.

Using the signal-flow graph method described inRef. [8] and Eqs. (9) and (10), the transfer functionof the kth-stage FOAPOF is simply given, by inspec-tion of Fig. 1, as

Hap;kðωÞ ¼Eout

ap;k

Einap;k

¼ γwak expðj2θkÞΛ2k

1 − γwð1 − akÞ expðj2θkÞΛ1kΛ2k;

ð16Þwhere Ein

ap;k and Eoutap;k are the amplitudes of electric-

field components at the input and output ports of theFOAPOF, respectively. It is useful to define the wa-veguide loop length L, the waveguide loop delay T,and the z-transform parameter as

L ¼ L1k þ L2k; ð17Þ

T ¼ T1k þ T2k; ð18Þ

z ¼ expðjωTÞ: ð19ÞSubstituting Eqs. (14), (15), and (17)–(19) into Eq.

(16), the z-transform transfer function of the kth-stage FOAPOF becomes

Hap;kðzÞ ¼Aap;k exp½jð2θk þ ϕkÞ� expð−jωT2kÞz

z − pk; ð20Þ

where the amplitude Aap;k and the pole location pk inthe z plane are given by

Aap;k ¼ γwak expð−αwL2kÞ; ð21Þ

pk ¼ γwð1 − akÞ expð−αwLÞ exp½jð2θk þ ϕkÞ�: ð22Þ

It is useful to express Eq. (22) in the phasor form as

pk ¼ jpkj exp½jargðpkÞ� ð0 ≤ jpkj < 1Þ; ð23Þ

where

jpkj ¼ γwð1 − akÞ expð−αwLÞ; ð24Þ

argðpkÞ ¼ 2θk þ ϕk: ð25Þ

Fig. 1. Schematic diagram of the kth-stage first-order all-pole op-tical filter (FOAPOF) using the PLC technology, microring resona-tor. Continuous lines are optical waveguides.


It is noted here that the ring resonator shown inFig. 1 has two couplers in order to exhibit an all-poleoptical filter. It is a rule of thumb that the number ofpoles or number of solutions of the denominator cor-responds to the number of nontouching rings in theoptical network. That means the resonant frequencyis the frequency at which the total phase of the lightwave circulating in the ring is 2π, which makes theenergy of the light waves resonate or circulate in thering without loss.

B. First-Order All-Zero Optical Filter

A zero in an optical network corresponds to the de-pletion of the optical field or energy at the outputof such an optical circuit. This depletion is normallyachieved by interfering the light waves of two differ-ent pathswhose phase differencewould be π. Figure 2shows the schematic diagram of an optical interfe-rometer. A phase shifter is incorporated in one ofthe path of the interferometer so as to be tuned todeplete the optical field at the output of the couplerDC2. This interferometer can be the kth-stage FOA-ZOF, which is implementable using PLC technologywithout any difficulty. The FOAZOF, which is anasymmetrical Mach–Zehnder interferometer, con-sists of two waveguide directional couplers DC1and DC2, with cross-coupled intensity coefficientsb1k and b2kð0 ≤ b1k; b2k ≤ 1Þ, respectively, which areinterconnected by two non-equal-length waveguidearms, hence a differential time delay of T. The PSon the lower arm has an externally controlled phaseshift of ψkð0 ≤ ψk ≤ 2πÞ. Thus ψk denotes the addi-tional phase shift imposed on the zeros of this opticalnetwork section. This is not to be confused with thenotation of ϕk as the additional phase shifter usedin Fig. 1.The optical transmittances of the upper ðΛ3kÞ and

lower ðΛ4kÞ waveguide arms are defined as

Λ3k ¼ expð−αwL3kÞ expð−jωT3kÞ; ð26Þ

Λ4k ¼ expð−αwL4kÞ expð−jωT4kÞ expðjψkÞ; ð27Þ

where L3k and L4k are the waveguide lengths of theupper and lower arms, and T3k and T4k are the cor-responding time delays of the upper and lower arms.Using the signal-flow graph method [8], the transfer

function of the kth-stage FOAZOF (for the upper out-put port) is simply given, by inspection of Fig. 2, as

Haz;kðωÞ ¼Eout1

az;k

Einaz;k

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiγazð1 − b1kÞð1 − b2kÞ

pΛ3k

−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiγazb1kb2k

pΛ4k; ð28Þ

where Einaz;k, E

out1az;k , and Eout2

az;k describe the electric fieldamplitudes at the input and output ports of the FOA-ZOF, respectively, and γaz [which is assumed to beγaz ¼ γw ¼ 0:89 (a typical value) for analytical simpli-city] is the intensity transmission coefficient of theFOAZOF. It is useful to define the differential lengthL and the differential time delay T as

L ¼ L4k − L3k; ð29Þ

T ¼ T4k − T3k: ð30Þ

By substituting Eqs. (19), (26), (27), (29), and (30)into Eq. (28), the z-transform transfer function ofthe kth-stage FOAZOF (for the upper output port)becomes

Haz;kðzÞ ¼ Aaz;k expð−jωT3kÞz−1ðz − zkÞ; ð31Þ

where the amplitude Aaz;k and the zero location zk inthe z plane are given by

Aaz;k ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiγazð1 − b1kÞð1 − b2kÞ

pexpð−αwL3kÞ; ð32Þ

zk ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

b1kb2kð1 − b1kÞð1 − b2kÞ

sexpð−αwLÞ expðjψkÞ: ð33Þ

It is useful to express in the phasor form as

zk ¼ jzkj exp½jargðzkÞ�; ð34Þ

where

jzkj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

b1kb2kð1 − b1kÞð1 − b2kÞ

sexpð−αwLÞ; ð35Þ

argðzkÞ ¼ ψk: ð36Þ

Similarly, the z-transform transfer function of thekth-stage FOAZOF (for the lower output port) isgiven by

Fig. 2. Schematic diagram of the kth-stage first-order all-zero op-tical filter (FOAZOF) using the PLC technology.


H�az;kðzÞ ¼

Eout2az;k

Einaz;k

¼ A�az;k exp½−jðωT3k þ π=2Þ�z−1ðz − z�kÞ; ð37Þ

where the amplitude A�az;k and the zero location z�k in

the z plane are given by

A�az;k ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiγazb2kð1 − b1kÞ

pexpð−αwL3kÞ; ð38Þ

z�k ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib1kð1 − b2kÞb2kð1 − b1kÞ

sexpð−αwLÞ exp½jðψk þ πÞ�: ð39Þ

In the phasor form, Eq. (39) becomes

z�k ¼ jz�kj exp½jargðz�kÞ�; ð40Þ

where

jz�kj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib1kð1 − b2kÞb2kð1 − b1kÞ

sexpð−αwLÞ; ð41Þ

argðz�kÞ ¼ ψk þ π: ð42Þ

From Eqs. (36) and (42), the zero zzk (for the upperoutput port) is out of phase with the zero z�zk (for thelower output port) by π. This means that the transferfunctions Haz;kðzÞ and H�

az;kðzÞ constitute a power-complementary pair:

jHaz;kðzÞj2 þ jH�az;kðzÞj2 ¼ 1: ð43Þ

In other words, if Haz;kðzÞ has a low-pass magni-tude response, then H�

az;kðzÞ has a high-pass magni-tude response, an important property that is veryuseful in the design of tunable optical filters. Notethat only the transfer function Haz;kðzÞ is used inthe design of tunable optical filters, as described inthe following subsection.

C. Mth-Order Tunable Optical Filter

The transfer function of the Mth-order tunable all-zero optical filter, which is the transfer function ofthe cascade of M FOAZOFs, is defined as

HazðzÞ ¼YMk¼1

Haz;kðzÞ: ð44Þ

This transfer can thus be rearranged as

HðzÞ ¼ G ·HapðzÞ ·HazðzÞ; ð45Þ

where G is the amplitude optical gain of the inte-grated optical amplifier (IOA). Figure 3 shows the

block diagram representation of Eq. (45), which de-scribes the transfer function between the input portand the upper output port (i.e., Output 1). The loweroutput port (Output 2) is not used in filter design, butits usefulness will be investigated.

Substituting Eqs. (20), (31), and (44) into Eq. (45),the transfer function of the Mth-order tunable opti-cal filter is given by

HðzÞ ¼ exp�j

�XMk¼1

ð2θk þ ϕkÞ − ωXMk¼1

ðT2k þ T3kÞ��

·�AYMk¼1

ðz − zkÞðz − pkÞ

�; ð46Þ

where the amplitude A is defined as

A ¼ GYMk¼1

Aap;kAaz;k: ð47Þ

The proposed Mth-order tunable optical filter hasthe advantage that its poles and zeros can be ad-justed independently of each other. Thus a particularpole-zero pattern can easily be obtained to design fil-ters with general characteristics.

We can now proceed to the design of tunable opticalfilters. We present a detailed design of the second-order Butterworth low-pass, high-pass, bandpass,and bandstop tunable optical filters with variablebandwidth and center frequency characteristics. Ingeneral, the design of a tunable optical filter fromthe characteristics of a digital filter involves the fol-lowing stages: (1) Specify the desired magnitude re-sponse of the optical filter in the optical domain,which is usually described by the spectrum of the op-tical signals to be processed by the filter. (2) Design adigital filter whose magnitude response in the digitaldomain approximates the desired magnitude re-sponse of the optical filter in the optical domain.(3) Design the optical filter structure whose transferfunction is similar in form to the transfer function ofthe digital filter. (4) Determine the parameters of theoptical filter structure using the pole-zero character-istics of the digital filter. (5) Realize the optical filterusing optical interferometers and resonators.

For analytical simplicity, the exponential factor inEq. (46), which represents a linear phase term, canbe neglected because it has no effect on the magni-tude response of the filter. The design of a tunableoptical filter from the characteristics of a digital filterrequires the second factor of Eq. (46) to be equal to 1,such that the following equations hold:

A ¼ A; ð48Þ

pk ¼ pk; ð49Þ


zk ¼ zk: ð50Þ

Substituting Eqs. (21), (32), and (47) into Eq. (48)results in

G ¼ ðγwγ1=2az Þ−MAQMk¼1 ak½ð1 − b1kÞð1 − b2kÞ�1=2 exp½−αwðL2k þ L3kÞ�

:

ð51Þ

Substituting Eqs. (2) and (23)–(25) into Eq. (49) re-sults in

ak ¼ 1 −jpkj

γw expð−αwLÞ; jpkj ≤ γw expð−αwLÞ;

ð52Þ

ϕk ¼�argðpkÞ − 2θk; argðpkÞ − 2θk ≥ 0;argðpkÞ − 2θk þ 2π; argðpkÞ − 2θk < 0:

ð53Þ

From Eq. (53), the largest value of the filter pole(i.e., jpkj) is restricted by the loss of the waveguideloop [i.e., γw expð−αwLÞ]. For practical purposes, a fullcycle phase shift of 2π has been added, without affect-ing the filter performance, to the second equation ofEq. (54) so that ϕk takes a positive value (i.e.,0 ≤ ϕk ≤ 2π). Substituting Eqs. (3) and (34)–(36) intoEq. (50) and using the relation jzkj ¼ jz�kj ¼ jzkj ¼ 1(i.e., the zeros are located exactly on the unit circle)results in

b1k ¼ 11þ expð−2αwLÞ

; ð54Þ

b2k ¼ 1=2; ð55Þ

ψk ¼�argðzkÞ; argðzkÞ ≥ 0;argðzkÞ þ 2π; argðzkÞ < 0:

ð56Þ

To change the center frequency of the tunable op-tical filter without affecting its bandwidth, an addi-tional phase shift of δ0ð0 < δ0 < 2πÞmust be added toEqs. (53) and (56), resulting in

ϕk ¼�argðpkÞ− 2θk þ δ0; argðpkÞ− 2θk þ δ0 ≥ 0;argðpkÞ− 2θk þ δ0 þ2π; argðpkÞ− 2θk þ δ0 < 0;

ð57Þ

ψk ¼�argðzkÞ þ δ0; argðzkÞ þ δ0 ≥ 0;argðzkÞ þ δ0 þ 2π; argðzkÞ þ δ0 < 0:

ð58Þ

In summary, the design equations for the design oftunable optical filters are Eqs. (12), (13), (51), (52),(54), (55), (57), and (58).

D. Design of Second-Order Butterworth Tenable OpticalFilters

To demonstrate the effectiveness of the proposedfilter design technique, this subsection describesthe design of the second-order (M ¼ 2) Butterworthlow-pass, high-pass, bandpass, and bandstop tunableoptical filters with variable bandwidth and centerfrequency characteristics. In this example, a1=T ¼ 5GHz filter is considered, resulting inT ¼ 200ps and L ¼ 4 cm. Using γw ¼ γaz ¼ 0:89,αw ¼ 0:01151 cm−1, and hence expð−αwLÞ ¼ 0:955,Eqs. (52) and (54) become

ak ¼ 1 − 1:18jpkj; jpkj ≤ 0:85; ð59Þ

b1k ¼ 0:523: ð60Þ

Substituting Eqs. (55) and (60) into (51) and (52)leads to

G ¼ ð0:392Þ−MAQMk¼1 ak

; ð61Þ

where exp½−αwðL2k þ L3kÞ� ¼ expð−αwLÞ ¼ 0:955 hasbeen assumed for numerical simplicity. In summary,Eqs. (12), (13), (56), and (58)–(61) are used in the de-sign of the second-order Butterworth tunable opticalfilters.

The following definitions are used hereafter. Thenormalized optical frequency on the frequency axisof the squared magnitude response representsωT=π, the squared magnitude responses are plottedover the Nyquist interval (i.e., 0 ≤ ωT=π ≤ 1), ωc is the3dB angular cutoff frequency of the low-pass andhigh-pass filters, and ωc1ðωc2Þ is the 3dB angularlower (upper) corner frequency of the bandpassand bandstop filters, where ωc1 < ωc2. For the low-pass and high-pass filters, the normalized 3dB band-width is defined as ωcT=π. For the bandpass andbandstop filters, the normalized 3dB bandwidth isdefined as ðωc2 − ωc1ÞT=π.

Figure 4 shows the characteristics of the tuningparameters versus the normalized bandwidth (i.e.,ωcT=π) of the low-pass and high-pass tunable opticalfilters with variable bandwidth (i.e., 0:1 ≤ ωcT=π ≤ 0:9) and fixed center frequency (i.e., δ0 ¼ 0)characteristics.

The gain curves in Fig. 4(a) show that a large IOAgain, which can be achieved by having two IOAs incascade, is required for a very small or large filterbandwidth (e.g., G ¼ 1716 or 65dB is required atωcT=π ¼ 0:1 for the high-pass filter and at ωcT=π ¼0:9 for the low-pass filter). These curves are symme-trical about the mid-band frequency (i.e., the re-quired gain at ωcT=π for the low-pass filter is the


same as that at 1 − ωcT=π for the high-pass filter).Figure 4(b) shows the intensity coupling coefficientsof the FOAPOFs (i.e., a1 ¼ a2) and FOAZOFs (i.e.,b1k ¼ 0:523 and b2k ¼ 0:5) for both the low-passand high-pass filters. The required coupling coeffi-cients a1 ¼ a2 can be obtained by varying the phaseshifts φ1 ¼ φ2 of the TCs [see the dotted–dotted curvein Fig. 4(b)]. Note that both the curves of a1 ¼ a2 andφ1 ¼ φ2 are symmetrical about the mid-band fre-quency (i.e., the required values at ωcT=π are thesame as those at 1 − ωcT=π). The dotted–dashedcurves in Fig. 4(b) show the required phase shiftsϕ1 and ϕ2 of the FOAPOFs for both the low-passand high-pass filters. The low-pass and high-pass fil-ters require the phase shifts ψ1 ¼ ψ2 ¼ π (see the so-lid curve) and ψ1 ¼ ψ2 ¼ 0 (see the dashed–dashedcurve), respectively, of the FOAZOFs.Figure 5 shows the characteristics of the tuning

parameters versus the normalized bandwidth [i.e.,ðωc2 − ωc1ÞT=π] of the bandpass and bandstop tun-able optical filters with variable bandwidth [i.e.,0:2 ≤ ðωc2 − ωc1ÞT=π ≤ 0:8] and fixed center frequency(i.e., δ0 ¼ 0) characteristics. The gain curves inFig. 5(a) show that a large IOA gain is requiredfor a very small or large filter bandwidth [e.g., G ¼197:5 or 46dB is required at ðωc2 − ωc1ÞT=π ¼ 0:2for the bandstop filter and at ðωc2 − ωc1ÞT=π ¼ 0:8for the bandpass filter]. These curves are symmetri-cal about the mid-band frequency [i.e., the requiredgain at ðωc2 − ωc1ÞT=π for the bandpass filter is thesame as that at 1 − ðωc2 − ωc1ÞT=π for the bandstopfilter]. Figure 5(b) shows the intensity coupling coef-ficients of the FOAPOFs (i.e., a1 ¼ a2) and FOAZOFs(i.e., b1k ¼ 0:523 and b2k ¼ 0:5) for both the bandpassand bandstop filters. The required coupling coeffi-cients a1 ¼ a2 can be obtained by varying the phaseshifts φ1 ¼ φ2 of the TCs that are indicated by thedotted–dotted curve shown in Fig. 5(c). Note thatboth the curves of a1 ¼ a2 and φ1 ¼ φ2 are symmetri-cal about the mid-band frequency [i.e., the requiredvalues at ðωc2 − ωc1ÞT=π are the same as those at1 − ðωc2 − ωc1ÞT=π]. The dotted–dashed curves inFig. 5(c) show the required phase shifts ϕ1 and ϕ2of the FOAPOFs for both the bandpass and bandstopfilters. The bandpass and bandstop filters require thephase shifts of ψ1 ¼ 0::and::ψ2 ¼ π as indicated bythe solid curves and ψ1 ¼ 3π=2::and::ψ2 ¼ π=2 as in-dicated by the dashed–dashed curves, respectively, ofthe FOAZOFs.The above discussions of the tunable optical filters

with low-pass, high-pass, bandpass, and bandstopcharacteristics can be summarized: (i) For a particu-lar filter bandwidth, the phase shifts ðψ1;ψ2Þ of the

FOAZOFs determine the complementary character-istics of the filter. That is, a low-pass filter can betransformed into a high-pass filter or vice versa,and a bandpass filter can be transformed into a band-stop filter or vice versa. (ii) For a particular set ofphase shifts ðψ1;ψ2Þ of the FOAZOFs, the tuningparameters (i.e., the phase shifts φ1 ¼ φ2, ϕ1, andϕ2) of the FOAPOFs determine the bandwidth char-acteristics of a particular filter type (i.e., low-pass,high-pass, bandpass, or bandstop).

E. Magnitude Responses of Tenable Optical Filters withVariable Bandwidth and Fixed Center FrequencyCharacteristics

Figures 6 and 7 show the squared magnituderesponses of the low-pass [Fig. 6(a)], high-pass[Fig. 6(b)], bandpass [Fig. 7(a)], and bandstop[Fig. 7(b)] of the tunable optical filters with variablebandwidth and fixed center frequency (i.e., δ0 ¼ 0)characteristics. Figure 6 shows that the bandwidthof each filter type ( low-pass or high-pass) can be var-ied from ωcT=π ¼ 0:4 to ωcT=π ¼ 0:8 by varying thephase shifts of the FOAPOFs (i.e., φ1 ¼ φ2, ϕ1, andϕ2) and by keeping the phase shifts of the FOAZOFsunchanged (i.e., ψ1 ¼ ψ2 ¼ π for low-pass and ψ1 ¼ψ2 ¼ 0 for high-pass); see Fig. 4. Similarly, Fig. 7shows that the bandwidth of each filter type (band-pass or bandstop) can be varied from ðωc2 −

Fig. 3. Block diagram representation of the Mth-order tunableoptical filter.

Fig. 4. Characteristics of the tuning parameters versus the nor-malized bandwidth of the low-pass and high-pass tunable opticalfilters with variable bandwidth and fixed center frequency (i.e.,δ0 ¼ 0) characteristics: (a) IOA amplitude gains, G; (b) intensitycoupling coefficients of both low-pass and high-pass filters.


ωc1ÞT=π ¼ 0:2 to ðωc2 − ωc1ÞT=π ¼ 0:6 by varying thephase shifts of the FOAPOFs (i.e., φ1 ¼ φ2, ϕ1, andϕ2) and by keeping the phase shifts of the FOAZOFsunchanged [i.e., ðψ1 ¼ 0;ψ2 ¼ πÞ for bandpass andðψ1 ¼ 3π=2;ψ2 ¼ π=2Þ for bandstop]; see Fig. 5. Notethat the normalized center frequencies are designedat ωT=π ¼ 0 for the low-pass and high-passfilters and at ωT=π ¼ 0:5 for the bandpass and band-stop filters.

F. Magnitude Responses of Tenable Optical Filters withFixed Bandwidth and Variable Center Frequency

Figures 8 and 9 show the squared magnitude re-sponses of the low-pass [see Fig. 8(a)], high-pass

[see Fig. 8(b)], bandpass [see Fig. 9(a)], and bandstop[see Fig. 9(b)] tunable optical filters with fixed band-width and variable center frequency (i.e., δ0 ¼ 0:1π)characteristics. The design parameters are exactlythe same as those in Fig. 4 for the low-pass andhigh-pass filters and in Fig. 5 for the bandpassand bandstop filters, except that an additional phaseshift of δ0 ¼ 0:1π has been added to the phase shiftsof the FOAPOFs [i.e., ϕ1 and ϕ2; see Eq. (57)] and tothe phase shifts of the FOAZOFs [i.e., ψ1 and ψ2; seeEq. (58)].

Figures 8(a) and 8(b) show that the squared mag-nitude responses are shifted by ωT=π ¼ 0:1 to theright of the frequency axis when compared with thecorresponding squared magnitude responses shown

Fig. 5. Characteristics of the tuning parameters versus the nor-malized bandwidth of the bandpass and bandstop tunable opticalfilters with variable bandwidth and fixed center frequency (i.e.,δ0 ¼ 0) characteristics, as obtained from Table 4: (a) IOA ampli-tude gains, G; (b) intensity coupling coefficients of both the band-pass and bandstop filters; (c) optical phase shifts, where ϕ1, ϕ2 andφ1 ¼ φ2 are the phase shifts of both the bandpass and the bandstopfilters.

Fig. 6. Squared magnitude responses of the (a) low-pass and (b)high-pass tunable optical filters with variable bandwidth and fixedcenter frequency (i.e., δ0 ¼ 0) characteristics.The numbers insidethe legend box represent the normalized 3dB cutoff frequencies(i.e., ωcT=π), which also correspond to the normalized filter band-widths. Note that the normalized center frequency is designed atωT=π ¼ 0. The filter parameters are shown in Fig. 4.


in Figs. 6(a) and 6(b). The normalized center fre-quency has been shifted from ωT=π ¼ 0 (Fig. 6) toωT=π ¼ 0:1 (Fig. 8), but the corresponding filterbandwidths of Fig. 6 and Fig. 8 remain unchanged.Similarly, Figs. 9(a) and 9(b) show that the squaredmagnitude responses are shifted by ωT=π ¼ 0:1 tothe right of the frequency axis when compared withthe corresponding squared magnitude responsesshown in Figs. 7(a) and 7(b). The normalized centerfrequency has been shifted from ωT=π ¼ 0:5 (Fig. 7)to ωT=π ¼ 0:6 (Fig. 7), but the corresponding filterbandwidths of Fig. 7 and Fig. 9 remain unchanged.Thus the center frequency of a tunable optical fil-

ter can be tuned, without affecting the filter band-width, to within one free spectral range byapplying an additional phase shift of δ0ð0 < δ0 <2πÞ to the phase shifters of the FOAPOFs and FOA-

ZOFs. From the point of view of the pole-zero pat-terns, the effect of δ0 on the FOAPOFs andFOAZOFs is to rotate the poles and zeros in the an-gular counterclockwise direction relative to the zplane. As a result, the pole-zero pattern of the result-ing tunable optical filter rotates by some angularmovement of δ0 relative to the z plane, and thishas the effect of shifting the filter center frequencyby δ0 to the right of the frequency axis.

G. Summary of Filtering Characteristics of TenableOptical Filters

As shown in Fig. 4, the phase shifts of the low-pass(i.e., ψ1 ¼ ψ2 ¼ π) and high-pass (i.e., ψ1 ¼ ψ2 ¼ 0)tunable optical filters are out of phase with each

Fig. 7. Squared magnitude responses of the (a) bandpass and (b)bandstop tunable optical filters with variable bandwidth and fixedcenter frequency (i.e., δ0 ¼ 0) characteristics. The numbers insidethe legend box represent the normalized 3dB lower and upper cor-ner frequencies ðωc1T=π;ωc2T=πÞ. The normalized filter bandwidthis given by ðωc2 − ωc1ÞT=π. Note that the normalized center fre-quency is designed at ωT=π ¼ 0:5.

Fig. 8. Squared magnitude responses of the (a) low-pass and (b)high-pass tunable optical filters with fixed bandwidth and vari-able center frequency (i.e., δ0 ¼ 0:1π) characteristics. The numbersinside the legend box represent the new normalized 3dB cutoff fre-quencies (i.e., ω0

cT=π ¼ ωcT=π þ δ0=π). The normalized filter band-widths are still the same as those in Fig. 6 (i.e., ωcT=π ¼ω0cT=π − δ0=π). Note that the new normalized center frequency is

at ωT=π ¼ 0:1.


other by π. As described in Subsection 3.B, the trans-fer function Haz;kðzÞ (for the upper output port) andthe transfer function H�

az;kðzÞ (for the lower outputport) are out of phase with each other by π. Thus,if the upper output port of the tunable optical filterhas a low-pass magnitude response, then its loweroutput port has a high-pass magnitude response.As a result, the tunable optical filter can be usedas a channel adding/dropping filter that passes cer-tain wavelength channels in one output port whileleaving the other channels undisturbed in the otheroutput port. As shown in Fig. 5, the phase shifts ψ1and ψ2 of a particular filter type (bandpass or band-stop) are out of phase with each other by π. As a re-sult, both the output ports of the tunable optical filterhave the same filtering characteristics (bandpass or

bandstop). In summary, the filtering characteristicsof the tunable optical filter are shown in Table 1.

The largest value of the filter pole, which is limitedby the loss of the waveguide loop [see Eq. (53)], is re-stricted to be jpkj ≤ 0:85. As a result, a tunable opticalfilter cannot be designed to have a very narrow orbroad bandwidth, which requires jpkj > 0:85. Thusthe allowable normalized bandwidths of the tunableoptical filter are in the range of 0:1 ≤ ωcT=π ≤ 0:9 forthe low-pass and high-pass filters and 0:2 ≤ ðωc2−

ωc1ÞT=π ≤ 0:8 for the bandpass and bandstop filters.These ranges of filter bandwidths are adequate formany filtering applications.

The normalized filter bandwidth can be extendedto its full range (i.e., between 0 and 1) by incorporat-ing an erbium-doped waveguide amplifier (EDWA)into the waveguide loop of the FOAPOF to compen-sate for the loop loss. However, this has the drawbackof increasing the cost as well as the complexity of thefilter structure, and the latter may degrade the filterperformance unless undesirable effects associatedwith the EDWA are minimized.

Note that the presented design method is applic-able to higher-order filters. The roll-off steepnessof the magnitude responses of the tunable optical fil-ter can be increased by increasing the filter order andhence the number of the FOAPOFs and FOAZOFs.However, the lowest filter order should be used tomeet a prescribed set of filter specifications to keepthe cost and complexity of the filter structure to aminimum.

The filter design technique is presented in a gen-eral manner and is thus applicable in the design ofother types of tunable optical filters such as the Che-byshev I and II and elliptic filters, whose propertieshave been summarized in Section 2. Obviously, thechoice of a particular filter type would depend onthe specific application.

4. Concluding Remarks

A digital filter design technique has been employedto systematically design tunable optical filters withvariable bandwidth and center frequency character-istics as well as low-pass, high-pass, bandpass, andbandstop characteristics. AnMth-order tunable opti-cal filter, which has been designed using integrated-optic structures, consists of a cascade ofM FOAPOFswith a cascade of M FOAZOFs. The effective-ness of the optical filter design method has beendemonstrated with the design of the second-orderButterworth low-pass, high-pass, bandpass, andbandstop tunable optical filters with variable

Fig. 9. Squared magnitude responses of the (a) bandpass and (b)bandstop tunable optical filters with fixed bandwidth and variablecenter frequency (i.e., δ0 ¼ 0:1π) characteristics. The numbers in-side the legend box represent the new normalized 3dB lower andupper corner frequencies ðω0

c1T=π;ω0c2T=πÞ, where ω0

c1T=π ¼ωc1T=π þ δ0=π and ω0

c2T=π ¼ ωc2T=π þ δ0=π. The normalized filterbandwidths are still the same as those in Fig. 7 [i.e.,ðωc2 − ωc1ÞT=π ¼ ðω0

c2 − ω0c1ÞT=π]. Note that the new normalized

center frequency is at ωT=π ¼ 0:6.

Table 1. Filtering Characteristics at the Output Ports of the Second-Order Butterworth Tunable Optical Filter

OutputPorts

FilteringCharacteristics

Upper outputport (output 1)

Low-pass High-pass Bandpass Bandstop

Lower outputport (output 2)

High-pass Low-pass Bandpass Bandstop


bandwidth and center frequency characteristics. Inthis design, for a fixed center frequency, the filterbandwidth can be varied by varying the parametersof the FOAPOFs and by keeping the parameters ofthe FOAZOFs unchanged, and for a fixed bandwidth,the filter center frequency can be varied, to withinone free spectral range, by adding an additionalphase shift to the phase shifters of the FOAPOFsand FOAZOFs.As a verification of the technique, an experimental

development of the first-order Butterworth low-passand high-pass tunable fiber-optic filters has been de-monstrated. In addition to the Butterworth filters,the proposed filter design technique is applicableto the design of other types of tunable optical filtersuch as the Chebyshev I and II and elliptic filters.MATLAB and Simulink can offer designs of moststandard optical filters that can be used directly toobtain the poles and zeroes of the networks, andthese software platforms are commonly availablein university computing networks.

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7. N. Q. Ngo, X. Dai, and L. N. Binh, “Realization of first-ordermonotonic Butterworth-type lowpass and highpass optical fil-ters: experimental verification,” Microwave Opt. Technol.Lett. 8, 306–309 (1995).

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Tunable photonic filters: a digital signal processing design approach

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