Realisation of Butterworth-type optical filters using 3 x 3 coupler ring resonators

L.N. Bin h N.X.Thien N.Q. Ngo

Indexing terms: Optical filters, Ring resonators, Design procedures

Abstract: Butterworth-type optical filters are realised using a combination of optical resonators which exhibit a single pole and single zero transfer function, quasi-all zero and quasi-all pole optical circuits. Design procedures as well as filter structures are proposed for optical lowpass, highpass and bandpass optical filters of even and odd orders. In particular, the [3 x 31 planar optical directional coupler has been employed with a direct feedback path and a single optical delay feedback to form a single-pole single-zero optical resonator. This is the basic block of cascaded sections of optical filters. [2 x 21 optical directional couplers have also been employed to construct interferometers and resonators having quasi-all pole or all zeros characteristics for cascading with the double feedback resonators to implement odd- and even-order modular optical filters.

1 Introduction

Optical fibre resonators have recently attracted signifi- cant attention due to their potential for applications to optical filtering [ 11, wavelength division multiplexing and demultiplexing [2], for optical communications and signal processing networks. In these designs, optical resonators have been constructed using [2 x 21 optical fibre couplers. To reduce the required number of cou- plers, multiple input and output port optical couplers have been considered [3]. One type of particular inter- est for constructing optical resonators are the 3 x 3 optical couplers which are equivalent to a six-port opti- cal device.

The 3 x 3 optical couplers have, however, never been considered in a composite optical fibre systems for sig- nal processing purposes. This paper introduces a signif- icant development using of this type of couplers for photonic signal processing. In particular, we consider the use of the [3 x 31 couplers with optical feedback of the output fibre port to input ports so that the resona- 0 IEE, 1996 IEE Proceedings onlune no. 19960079 Paper frst received 21st March 1995 and in revised form 28th September 1995 The authors arc with the Applied Photonics & Optical Communications Laboratory, Department of Electrical and Computer Systems Engineer- ing, Monash University, Clayton, Victoria 3168, Melbourne, Australia

126

tors can have a number of optical transfer functions that are important to designing optical Butterworth- type optical filters. A design for Butterworth-like opti- cal filter has also been reported [4] in which a bow-tie shape optical fibre ring resonator has been used. This configuration is not general and thus does not give a systematic approach for synthesising optical filters.

In signal processing, the optical transfer function in the frequency or wavelength domain plays a major part to determine the filter response in the temporal domain. It is thus important that the frequency response or the transfer function in the frequency domain of these photonic signal processing is specified precisely. A typical technique in digital signal process- ing is partitioning the transfer function into a product of first-order polynomials in the numerator as well as in the denominator. Thus it can be implemented by cascading or connecting in parallel a number of optical devices which represent the first-order optical transfer function.

To reduce the number of optical components required for implementation of optical filters, we intend to design optical circuits that can give a null or a reso- nance in its amplitude response. This would be equiva- lent to an optical section with a transfer function exhibiting a pole and a zero. This arrangement will also lead to a very stable optical network because all the poles can be located inside the unit circle of the z- plane.

Having identified the importance of a network func- tion with a single pole and a single zero, it is required to find an appropriate transfer function of the filter from a basic optical fibre resonator. Optical circuits have been represented in graphical form so that it is easier to identify and construct larger optical circuits [5]. The advantage of this technique is that it allows one to inspect an optical configuration and hence iden- tify newer forms of the optical transfer function.

This paper presents a systematic and novel approach for synthesising optical filters. In particular, the use of a [3 x 31 optical coupler to form an optical resonator such that it forms an optical block with an optical transfer function having a pole and a zero. The poles and zeros of the transfer function, usually defined as the transmittance of one optical port with respect to the optical input port, correspond to the resonance and the depletion of the designated port of the optical sys- tem [l]. The [3 x 31 optical coupler of planar form has been employed with two optical feedback paths form- ing two resonant optical loops: a single fibre feedback with an optical unit delay and an optical feedback path

IEE Proc.-Optoelectron., Vol. 143, No. 2, April 1996

with no delay. The no-delay path is used to indicate a direct optical shunt feedback path connecting two nodes of an optical circuit. This 3 x 3 double feedback forms the basic unit for designing optical filters with a Butterworth-like transfer functions.

The main objective of this work is to present a novel design of optical lowpass, bandpass and highpass opti- cal filters using [3 x 31 planar optical couplers with delay and phase modulated path for tuning the effi- ciency of the transfer function, hence mapping to appropriate optical functions.

2 Optical resonators and interferometers as single-pole and single-zero basic optical blocks for Butterworth filter design

In this Section we analyse three basic optical devices which form the core optical components for the design of the Butterworth-type optical filters: two optical reso- nators and one interferometer. For the two resonators, one is constructed by using a [3 x 31 optical directional coupler with two optical feedback paths, and the other is formed by using two [2 x 21 optical directional cou- plers with only one feedback path. Section 2.1 analyses the optical resonant circuit using a 3 x 3 optical cou- pler and Section 2.2 describes the characteristics and performance of the interferometric device and a single pole optical resonator.

112, J@’ t2k

(lei @$-I

Fig. 1 Schematic diagram and optical interconnection paths, cross-sectional structure of planar 3 x 3 coupler shown at right-hand corner. Arrows indicate directions of optical fields and corresponding flowing directions

3 x 3 optical coupler and optical feedback paths as SPSZR

112 -d2 . t2 z e l k

Fig.2 Graphical signal-flow representation. Arrows indicate directions of optical fields and corresponding flowing directions

3 x 3 optical coupler and optical feedback paths as SPSZR

2. I directional coupler

2. I. I Optical transfer matrix of the [3 x 31 cou- pler: A schematic diagram of a [3 x 31 optical fibre directional coupler with two feedback paths is shown in Fig. 1 and its graphical representation for signal flow is shown in Fig. 2. A general transfer matrix for [3 x 31 optical couplers which may have different cross-sec-

Optical resonators using a [3 x 31 optical

tions such as triangular, planar etc. can be derived without difficulty. The [3 x 31 considered in this paper has a planar cross-section (i.e. three optical waveguid- ing regions are in the same line across its section). A planar cross-section type can be considered as a com- posite structure consisting of two [2 x 21 optical fibre couplers sharing a common section. Thus by using a 3 x 3 coupler we can minimise the required number of optical couplers.

The relationship between the input and output fields of a [3 x 31 optical fibre coupler can be expressed by a general matrix with coefficients xII, where the subscripts i and j are integers taking values of 1, 2 and 3. These coefficients are different depending on whether the cross-section is symmetrical or asymmetrical and trian- gular, or planar [6]. We consider, in this paper, only the planar type. This cross-section is featured in Fig. 1. The signal flow graphical representation of the optical circuit of the optical resonators using [3 x 31 optical fibre directional coupler is straightforward [l]. It is, however, noted that in this circuit there is no bidirec- tional propagation of optical waves in the same path (i.e. optical waves travelling only in one direction in an optical path),

[ X I 1 2 1 2 2 1 3 1

x = x 2 1 2 2 2 x 2 3 (1) 1x31 2 3 2 X 3 3 1

There are two optical fibre feedback paths in Fig. 1, a feedback from output port 3 to the input port 3 and similarly for the path between input 2 and output 2. The delay period of the optical fibre can be represented as za* or zd2 where dl and d2 represent the order of delay and z represent the z-transform of the input light- waves, that is the z = exp(jwr) = exp(jPL) where w is the optical radial frequency and L is the delay path length corresponding to the unit delay time T. P is the propagation constant of the guided optical fibre mode. The use of the z-transform notation is most appropri- ate here because it will simplify the representation of optical functions and thus allow quick examination of the performance of optical systems. It is not necessary that the optical system must be operated with digital domain. It is quite straightforward to convert the z- domain transfer functions to its equivalence in the con- tinuous domain.

The transmittances of the fibre feedback paths 1 and 2 are denoted by tl and t2, which may take values less or greater than unity depending on whether optical amplifiers are used or not. The optical field from the feedback path can also be negative because the optical system is assumed to operate under coherence (i.e. its field can be in opposite direction with respect to that of the input field). The phases $1 and +2 are the modu- lated phase factors that can be induced, introduced by phase modulators incorporated in the fibre feedback path. Usually one phase is adjusted and the other is kept constant.

In a special case where the [3 x 31 cou ler is of planar

the propagation constant of the lightwaves in vacuum and d is the coupling length of the coupler) the transfer matrix becomes

type and a coupling factor of kd = 71: 7 214 (where k is

0 -3 -3

(2)

IEE Proc.-Optoelectron., Vol. 143, No. 2, April 1996 127

2.1.2 Optical transfer functions: In designing and analysing optical circuits using optical fibre or inte- grated optic structures for signal processing applica- tions, it is of great interest that the transfer function is represented in the z-domain [7]. The transfer function between one port to the other can be easily obtained by applying the optical equivalent version of the Mason’s rule as outlined in [l]. The general input-output trans- fer relationship as a function of z is given by

where a = l - x , k f i Z - d l -Xn3 f i Z P d 2 +X,kx,, m z - ( d 1 + d 2 )

(4) Eo(d) and E,(O) are the output and input optical fields evaluated at the start and the end of the coupling region of the [3 x 31 coupler, respectively. A is usually defined as the graph determinant and represents the order of resonance of the optical circuit analogous to that in the electrical domain. The number of roots of A determines the poles of the systems or the resonance conditions of the resonator. The roots of the numerator of eqn. 3 determine the zeros of the resonator or effec- tively the complete depletion of the optical waves (i.e. the interferometric null of the system output).

In optical resonators not only the output-input but the circulating input field transfer function also indi- cates the resonance or depletion condition. Using simi- lar procedures as described above for the circulating input transfer function E,/E,, the circulating field at port j , with respect to the input optical field at port i, is given by

.

For a planar [3 x 31 optical directional coupler with a transfer matrix of eqn. 2 and a direct (or the order of the delay path is zero) shunt feedback from output port 3 to input port 3, that is setting the delay order of the optical feedback path d2 = 0, the output-input transfer function for output port 1 to input port 1 is given by

The second subscript k used here for the transmittance and phase coefficients indicates the kth component when it is incorporated in tandem for forming a cas- caded optical filters. Eqn. 6 thus has only one pole and one zero (i.e. its numerator and denominator exhibit only one root). Therefore, the pole and zero can be independently adjusted by adjusting the coefficients tik and qZk which are the transmittance and phase coeffi- cients of the optical attenuators, optical amplifiers or optical phase modulators incorporated in the fibre feedback paths.

In implementing the optical transfer function of opti- cal Butterworth filters as given in the next two Sec-

128

tions, the transfer function (eqn. 6) can provide an arbitrary pair of pole-zeros denoted by (a, b) given by the roots of the numerator and denominator of eqn. 6 as

(7)

&e”Ik = 2a (8)

1 2

a = -&e341k

or

Substituting eqn. 8 into the numerator of eqn. 6, and manipulating the remaining parameters such that the zero is b, we obtain

Substituting this set of chosen parameters into the transfer function in eqn. 6 gives

Eqn. 10 demonstrates that the resonator would exhibit only one pole and one zero which can be adjusted inde- pendently with each other. The gain of the transfer function given by the z nondependent coefficient of eqn. 10 is dependent on the values of the pole and zero, a and b. This amplitude gain or loss can be compen- sated by an in-line optical amplifier (e.g. an erbium doped fibre amplifier operating in the 155Onm wave- length region).

The circuit of Fig. 1 can be observed to exhibit only one pole due to the fact that there is only one loop with a single delay line in the graph. According to the optical version of the Mason’s rule [I], the number of poles is that of roots of the graph determinant. The graph determinant order is the order of delay of the optical loop. Thus, if there is only one delay in the loop, there must be only one pole for a unit delay or multiple identical poles equally spaced on a concentric circle if the delay order is greater than one.

The number of zeros of the resonator depends on the number of nontouching loops of the optical circuit. In this case there are two loops in this resonator and they are nontouching, but only one unit delay in one loop, thus this ensures that the order of the numerator is one. It is therefore concluded that this identification of the ,double feedback optical resonator eases the imple- mentation of an optical transfer function with a pair of pole and zero. We can thus name this type of optical resonance circuit the single-pole, single-zero resonator (SPSZR).

Since the SPSZR is the core component for designing optical filters, in particular the digital filters, it is neces- sary to examine closely the feasibility of a direct optical feedback from the output to the input of the [3 x 31 optical coupler. This type of delay is termed as the delay-free feedback. A delay-free loop is formed at the upper part of the 3 x 3 coupler as shown in Figs. 1 and 2 [8]. It is stated in [8] that the necessary and suffi- cient condition for the signal flow graph of the struc- ture to be computable for a digital filter is that there is no delay-free loop. We must make very clear here that temporal delay due to the optical path length of the direct fibre path connecting output port 3 to input port 3 is much shorter than that of the other loop, hence considered as a delay-free path. Furthermore, direct connection loop does not operate under a resonance condition. It is thus reasonable to assume that this loop

c -0ptoelectron , Vol 143, No 2, April 1996

does not have the same meaning as the delay-free loop defined in [SI. This ensures that our derivation for the transfer functions between various ports of the filter is valid. In practice, the delay path of the lower loop is much greater than those of the direct loop and of the coupling length of the 3 x 3 directional coupler.

112,j €I1 n tl Fig. 3 Schematic diagram. Transmission coefficients are indicated along signal flow paths

All-zero optical circuit (AZOC)

.WE2 ; 3 9 6 7 - k 2 l 2

til2 P 2 2 - l Fig.4 All-zero optical circuit (AZOC) Graphical representation and signal flow for all optical paths. Transmission coefficients are indicated along signal flow paths

coupler 1 couder 2

Fi .5 Allpole optical circuit (AZOC) using two 2 x2 optical couplers an! one optical feedback path Schematic diagram. Transmission coefficients are indicated along signal flow paths

Fi .6 All pole optical circuit (AZOC) using two 2 x 2 optical couplers ad one optical feedback path Graphical representation and signal flow for all optical paths. Transmission coefficients are indicated along signal flow paths

2.2 Single-pole and single-zero optical circuits using loop resonator and interferometer Besides the zero-pole pair optical circuit described above, there must be optical circuits or components which exhibit a pole (or higher order multiple poles) or a zero (or multiple order zeros) characteristics so that they can form a set of optical components required to complete the filter structure, especially where an odd

IEE Proc.-Optoelectron., Vol. 143, No. 2, April 1996

order filter is desired. The all-zero optical circuit (AZOC) is an optical interferometer which has been studied in detail in [l]. Similarly, an all pole circuit (APOC) is also given in that work [l]. The schematic diagrams of these all zero and all pole and their graph- ical representations are given in Figs. 3 and 4, and 5 and 6, respectively. The transmission coefficients are denoted as tip or ti, (i = 1, 2) where the subscripts p and z denote the all pole or all zero feature. The phases incorporated in optical paths are y and 0 with appro- priate subscripts.

Following a similar procedure to that described above with reference to Fig. 4 and applying the optical Mason rule El], the transfer functions of the AZOC and its zero position are given in eqns. 11 and 12, respec- tively:

= J(1 - k 1 ) ( 1 - / c 2 ) t l e j e l - J i & G ' e j e 2 . z - 1

(11) which has one pole at origin, p =O and one zero at

Although the transfer function (eqn. 11) of the AZOC contains a pole, it is at the origin in the z-plane and does not affect the performance of this circuit. Thus the position of the zero can be tuned to suit the needs for the design of optical systems. It is thus a quasi-all zero optical circuit.

Similarly, the transfer function of the APOC is obtained by using the graphical method [5] is given as

Thus this transfer function has a zero at origin and a pole at

z = - d t l t 2 / c 1 ~ 2 e j ( ~ 1 + ~ 2 ) (14) Thus this APOC is a quasi-all-pole optical circuit. It is interesting to note that, if an APOC and an AZOC of the same order are cascaded, the overall transfer func- tion exhibit a numerator and denominator of the same pole and zero order because the poles and zeros at the origin cancel each other. The phase coefficients of the AZOC and APOC given by eqns. 12 and 14 indicate the difference in the resulting phase of the pole and zero of their transfer functions. It is the difference in phase for an AZOC, while it is an accumulation for the APOC. Thus care must be taken in assigning the phase modulators to be incorporated in these optical circuits.

3 Butterworth-type optical lowpass filters

Having analysed the three basic optical components which have the transfer functions of all zero, all pole and equal pole equal zero types, this Section describes the design of the Butterworth-type lowpass filters of even and odd orders. The order of an optical filters is the order of the denominator of all transfer function.

3. I Even-order filters A fourth-order Butterworth lowpass filter is chosen to illustrate the design and proposed structure for the optical filters. The filter requires a set of poles and zeros which can be derived from procedures given in [l,

129

Table 1: Chosen parameters for the fourth-order lowpass filter

SPSZRI t11=0.1582 t11=0.1366 9,142 @,1=1.?92 p1=j0.1989 Zl=-l

SPSZR2 t,,=0.1582 t2,=0.1366 gI,=3d2 $,,=4.333 p2=j0.1989 z,=-l

SPSZR3 tI3=1.785 t2,=0.640 gI3=d2 $,,=0.642 p3=j0.6681 z3=-I

SPSZR4 tI4=1 ,785 t,,=0.640 gl4=3d2 $24=5.64 p4=j0.6681 Z4z-l

1 .o

0.8 4 27

P g 0.6 U?

2 0.4

2 0.2

c m

0

81. This set contains a number of poles and zeros as follows: four zeros at z1 = z2 = z3 = 24 = -1, that is, on the unit circle of the z-plane and four poles located at p1 = j0.1989 andp2 = -j0.1989 andp, = j0.6681 andp4 = -j0.6681. The poles p 1 and p2 are purely imaginary and conjugated with each other, similarly for the poles p 3 and p4. Thus, we would see later in a plot that the four poles are located on the imaginary axis of the z- plane.

The optical system can thus be easily implemented by cascading four SPSZRs with designed parameters given in the Table 1 where the four resonators are of SPSZR type and denoted as SPSZR 1 to 4. The zeros are placed at the origin.

_ .................... .................... _

_ ............"... ... b -.................... I

I

time

time, ns Fig. 8 Impulse response with arbitrary unit on amplitude vertical m s

Responses of fourth-order Butterworth lowpassjWer

The transfer function can thus be obtained as 4

H g ( z - 1 ) = H,k(z-l) (15) k = l

or

f fFJ(z-1) =

plP2p3P4(1 f .-ll4 (l+pi)(l+pz)(l+p3)(1+P4)(1+~~~-')(l+~~~-')(1+~32-')(1+~42-')

(16) Using the values for p1 - p 2 as listed in Table 1, eqn. 16

130

can be rewritten as

(17) (4) -1 - 0.0117(1 + z - ' ) ~

H L p ( z - 1 + O.4859zr2 + 0 0 1 7 7 r 4 where the superscripts of the transfer functions indicate the order of the filter. An optical amplifier with gain equal to 8.025 can be employed to compensate for the maximum gain at the input or output. Therefore, the final transfer function of the fourth-order lowpass filter is given as

The magnitude, impulse, phase responses and the pole- zero plot of the fourth-order Butterworth-type lowpass filter are shown in Figs. 7-10, respectively. As men- tioned above, the four poles are on the imaginary axis of the z-plane and they are inside the unit circle, thus stability is held. The phase plane of Fig. 9 also con- firms the 180 degrees phase change whenever the opti- cal frequency passing through a zero of the system, that is when there is a depletion of the lightwaves at the output. The side lobe of the impulse response shows the effect of the position of the poles on the time response of the system. To minimise this side lobe effect, the poles should be placed close together.

0 5 10

Responses of fourth-order Butterworth lowpass filter w T, rad

Fig.9 Phase of fdter against COT

2

1

- E O E -

- 1

-2

pole-zero plot

Re(z) Fig. 10 Distribution of poles and zero on z-plane

Responses of fourth-order Butterworth 1owpassJilter

IEE Proc -0ptoekctron , Vol 143, No 2, Apvd I996

Table 2: Chosen parameters for a fifth-order lowpass filter

SPSZRI t1,=0.4225 t2,=0.297 $,,=nl2 $21=0.99 pl=j0.325 zl=-I SPSZR2 tl2=0.4225 t2,=o.297 $,2=3n12 $,,=5.29 p2=jo.325 z2=-1 SPSZR3 f13=2.11 tzp0.678 $13=~/2 $23~0.6 P,=JU.7265 z3z-l SPSZR4 tI4=2.1 1 t24=0.678 $14=3n12 $24=5.68 pp~D.7265 z4=-I AZOC k1=k2=0.5 tl=t2=1 0,=n e2=0 P5=0 z5=-1

1 .o

dj 0.8 a- c

U) 0.6

P 4 0.4 -E

(71

g 02

0

3.2 Odd-order Butterworth optical filters Odd-order Butterworth type filters can also be imple- mented using SPSZR. However, the optical transfer function would have an additional zero, thus it is nec- essary to cascade with an extra AZOC. This Section gives an example for the design of a fifth-order Butter- worth filter.

The fifth-order Butterworth lowpass filter requires a system of poles and zeros [I] as follows: Five zeros at z1 = z2 = z3 = z4 = z5 = -1 Five poles at p1 = j 0.325, p 2 = -j 0.325,

p3 = j 0.7265, p4 = -j 0.7265

The fifth-order lowpass filter can thus be implemented by cascading four SPSZRs and an AZOC with the cho- sen parameters shown in the Table 2.

With these chosen parameters, the transfer function of the cascaded network is given by

P5 = 0

c I

i - .................... i. .................... 1.

_ ....... .. ........... i. - .................... ;.

I

(19) (5 ) -1 - 0.0165(1 + z - ' ) ~

H L p ( z - 1 + 0.6334r2 + 0.0557r4 These cascaded optical components SPSZR and AZOC should be placed in tandem with an optical amplifier of gain 3.2 to achieve the maximum amplitude of the opti- cal waves at the output to unity. Thus the optical transfer function of the fifth order Butterworth-type lowpass filter is

(20) 0.0528(1+ z - ' ) ~

1 + 0.6334r2 + 0.0557r4 HgL(z-1) =

whenever the optical frequency passes through a zero of the optical circuit. This ringing effect would contrib- ute to the intersymbol interference and hence the sys- tem noise between the transmitted pulses passing through the filter. Hence, an even order optical filter is preferred to reduce the intersymbol interference effect in the digital optical transmission system. The symmet- rical positions of the poles on the imaginary axis ensures the rolling off and the tailing off of the fre- quency response in Fig. 11.

time,ns Fig. 12 Impulse response with arbitrary unit on amplitude vertical axis

Responses offifth-order Butterworth lowpass filter

0 5 10

Responses ofj?j?h-order Butterworth lowpass j?lter WT. rads

Fig. 13 Phase of filter against wT

pole-zero plot

Re(z) Fig. 14 Distribution of poles and zero on z-plane, one additional pole is located at ori- gin as compared to fourth-order filter

Responses offifth-order Butterworth lowpass filter

131

It is interesting to note that the position of the qua- siAZOC in the structure of the fifth-order filter ean be at the end of the third or fifth SPSZR. Thus the spare output port of the AZOC can be used as the output of a third-order Butterworth optical filters if desired. This is the advantage of using an odd-order optical filter as compared with even-order filters.

-z8. This results in H&)(z-l) having zero at z = +1 and the system of poles are the same as in Hfj2)(f1) because the system poles consist of pairs of conjugate pales and a pole at the origin. This gives a distribution of zeros in opposite position compared to those of the lowpass types. Figs. 15-18 and Figs. 19-22 s respectively, the amplitude, impulse, phase responses and the location of the poles and zeros in the z-plane for the fourth- and fifth-order high pass filters, respec- tively. They are similar to those of the lowpass filters described in Section 3, except that the zeros are on the opposite side of the origin in the z-plane to those of the lowpass type. The magnitude response indicates the hghpass characteristics and consistent with the prop- erty of a highpass filter.

pole-zero plot

0 1 2 3 UT, rads

Fig. 15 Magnitude in arbitrary unit against optical radial frequency o and unit time delay T, Note location of zero is in opposite posltion of that of corresponding low pass filter type

Responses of Butterworth highpassJilter of order 4

Re, ( z ) Fig.18 Disinbution of poles and zero on z-plane, Note location of zero is in opposite position of that of correspondmg low pass filtm type

Responses of Butterworth hLghpassfilter of ordeer 4

time,ns Fig. 16 Responses ofBaterworth hghpTs fdter ofordw 4 Impulse response with arbitrary umt OD ainphtude vertical ax~q NoQ location of zero is in oppos~te position of that of corresponding low pass fdkr type

0 5 10

Responses of Butteyworth higkpasjlier of order 4 wT, rads

Fig. 17 Phase of the filter versus UT; Note location of zero is m apposite pasition of that of correspondmg low pass filter type

4 Optical Butterworth-type highpass filters

In designing highpass digital filters, it requires a fre- quency transformation process to convert a known

transfer function to a Let the function of a n the corresponding highpass tr

is given by H&)(z-'), which can be obtained by substi- tuting the parameter 2-l of the lowpass function by

132

..... ...........,

Fig. 19 Responses ofgth-order Magnttude m arbxtrary umt agains tme delay T, Note locatim of zero i of corresponding lowpass type

and the unit ared wlth that

time,n Fig. 20 Responses ofj@h-order Butter Impulse response with arbitrary unit on am of zero in opposite position as compared type

e location g lowpass

IEE Proc -0ptoeJectron , Vol. 143, No. 2, April 1996

0- 5 10 wT, rads

Fig. 21 Phase of the filter versus cor Note location of zero in opposite position as compared with that of corresponding lowpass type

Responses offifth-order Butterworth higkpms filter

pole-zero plot

Re(z) Fig. 22 Distribution of poles and zero on z-plane; Note location of zero in opposite position as compared with that of corresponding lowpass type

Responses ofjiifth-order Butterworth highpuss Jilter

4. I Hardware design The structure of the optical circuit of the highpass filter is very similar to that of the lowpass filter by cascading a number of basic SPSZR, AZOCs and APOCs, with the necessary changes of the phases incorporated in the transmission paths as follows:

4. I . I For SPSZR: To obtain the zero at z = +1 instead of -1, from eqn. 9 we obtain

(21) -a

0.5 - a &&2" = ~

instead of d[t2kd$2k = a/[0.5 + a] for the case of lowpass filter design. Since the two successive subsystem SPSZRs have their poles which are purely imaginary and are conjugate to each other, all the SPSZRs in the design of lowpass filter are kept the same, except that their phase coefficients are interchanged:

These phases which are obtained in the case of lowpass filters are effectively swapped between the two succes- sive subsystems. Thus an in-line fibre optical phase modulator can be used to tune or switch the low pass into a high pass optical filters and vice versa without changing the structure.

4.7.2 f o r AZOC: The parameters for highpass type are kept the same, except that the phase parameter el = 0 instead of el = n as in the case of lowpass filter. Again, an optical phase modulator can be used here for switching the lowpass type into a highpass optical filter type.

421 * 422 and 423 * (624 (22)

4.2 Transfer function and responses of Butterworth highpass filters As discussed above, the transfer functions of the fourth- and fifth-order highpass filters can be obtained easily by replacing z-I by -2-l from the corresponding lowpass filter transfer functions given in eqns. 17 and 19:

(23) 0.0939(1- z - ' ) ~

1 + 0 . 4 8 5 9 ~ ~ ~ + O.0177zr4 H g ( X - 1 ) =

(24) 0.0528(1 - z-1)5

1 + 0.6334r2 + 0.0557r4 H g ( z - 1 ) =

The responses for eqns. 23 and 24 for the fourth- and fifth-order Butterworth highpass filters are plotted in Figs. 15-18 and 19-22, respectively. They show sym- metrical properties as compared to those of the lowpass type as described in Section 3.

5 Optical Butterworth-type bandpass filters

Realisation of bandpass filters requires the cascade of the lowpass and highpass optical filters as designed in the above Sections. The transfer functions and responses of bandpass filters of an order 1 can thus be a combination of low and highpass filters of orders rn and n, respectively:

where 1 = IPZ + a. It is therefore very straightforward to implement optical bandpass filters by cascading appro- priate optical low and highpass filters. Another require- ment for bandpass filters design is the stopband region which can also be dependent on the roll-off of the low and highpass characteristics. Our design procedure for the lowpass and highpass filters thus leads to a very modular structure for implementing many types of optical filters. One can construct optical filtering banks so that several optical filter structures can be config- ured. As discussed in Sections 3 and 4, the positions of the zeros of the low and highpass filters are in opposite locations on the unit circle of the z-plane and the poles are in the vertical axis with a phase variation of n/2 and 4 2 . A bandpass filter would thus require cascad- ing of low and highpass filters preferably with the sin- gle pole optical blocks placed in tandem, such that the effect due to any delay difference between them is min- imal. Detailed analyses of these filters will be reported in the future.

6 Conclusions

The [3 x 31 planar optical coupler has been used as the main component in forming an optical resonant circuit, which can exhibit a transfer function of a single-pole and a single-zero property. This can be achieved only through the use of a direct feedback from one of the output port to an input port and a first-order delay in a second optical delay feedback delay loop. The SPSZR allows us to implement a number of odd and even optical Butterworth-type filters in modular form,

In addition, a quasi-all pole and quasi-all zero optical circuits (AZOC and APOC) are described for incorpo- ration with the SPSZR to form any order optical fil- ters.

Optical lowpass and highpass filters have been described, and a simple transformation from a lowpass design to a highpass one is given. In particular, the cas- cading feature of APOC and AZOC in a cascaded opti-

IEE Proc.-Optoelectron., Vol. 143, No. 2, April 1996 133

cal filter structure leads to a possibility of forming several optical filters of different order in a single struc- ture. A brief algorithm for bandpass filter design is also given for realisation of these filters.

The synthesising techniques for Butterworth optical filters described here would find important applications in signal processing, optical fibre communications such as optical equalisers, and optical grating filters as well as in thin film multilayer filtering systems etc.

7 References

1 NGO, N.Q., and BINH, L.N.: ‘Novel realisation of monotonic Butterworth-tvoe low oass. hinhoass and bandoass filters usinn phase-moduked fibre-optic inkiferometers and“ring resonators: J. Lightwave Technol., 1994, LT-12, pp. 827-841

2

3

4

5

6

7

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DOWLING, E.M., and MACFARLANE, D.L.: ‘Lightwave lat- tice filters for optically multiplexed communications systems’, J. Lightwave Technol., LT-12, pp. 471486 ABD-HAMID, G., and DAVIES, P.A.: ‘Ring resonator using a 3 x 3 flat optical fibre coupler’, Proc. SPIE, 1992, 1792, (Compo- nents for fibre optic applications VII), pp. 2-12 JA, Y.H.: ‘Design and characteristics of periodic Butterworth-like filters using a bow-tie-shaped optical fibre ring resonator’, Opt. Eng., 1994, 33, (9), pp. 2912-2918 B I ” , L.N., NGO, N.Q., and LUK, S.F.: ‘Graphical representa- tion and analysis of the Z-shape double coupler optical resona- tors’, J. Lightwave Technol., 1993, 11, pp. 1782-1792 SHEM, S.K.: ‘Optical fibre interferometers with [3 x 31 direc- tional coupler: analysis’, J. Appl. Phys., 1981, 52, pp. 3865-3872 MOSHELI, B., GOODMAN, J.W., TUR, M., and SHAW, H.J.: ‘Fibre optic lattice signal processing’, Proc. IEEE, 1981, 72, pp. 909-930 MITRA, S.K., and KAISER, J.F. (Ed.): ‘Handbook of digital signal processing’ (John Wiley, New York, 1993), p.321

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