ARTICLE IN PRESS

OpticsOptikOptikOptik 121 (2010) 1075–1083

0030-4026/$ - se

doi:10.1016/j.ijl

�CorrespondE-mail addr

www.elsevier.de/ijleo

Synthesis algorithm of a multi-channel lattice-form optical

delay-line circuit

Shafiul Azam�, Takashi Yasui, Kaname Jinguji

Signal Processing Laboratory, Interdisciplinary Faculty of Science and Engineering, Shimane University, Shimane-ken, Japan

Received 25 September 2008; accepted 23 December 2008

Abstract

A novel synthesis algorithm for multi-channel (MX2) lattice form optical delay-line circuit is presented in this paper.This circuit offers multi-port FIR optical filter with delay time of NDt (Dt: unit delay time). Synthesis algorithm isbased on division of total transfer matrix into unit blocks. Developed method confirms that 1�M optical delay-linecircuit offers same transmission characteristics as 1�M FIR digital filter. Band-pass flat group delay type filter isconsidered as an example in this paper. It is also confirmed that proposed delay-line circuit can realize 100% powertransmittance.r 2009 Elsevier GmbH. All rights reserved.

Keywords: Delay-line circuit; FIR and IIR optical filters; Directional coupler; Phase shifter

1. Introduction

It is well known that optical delay-line circuits arecomposed of delay-lines, directional couplers and phaseshifters have similar filter characteristics to those ofdigital filters. Optical delay-line circuits are roughlyclassified into finite-impulse response (FIR) filters andinfinite-impulse response (IIR) filters. FIR filters consistof feed-forward waveguides and their impulse responsesare limited in finite time whether, IIR filters includefeedback loops such as ring waveguides and theirimpulse responses continue for infinite time [1–3]. Thesecircuits play an important role in optical communicationand act as an optical interleaver, tunable optical add/drop multiplexer (OADM), erbium-doped fiber ampli-fier (EDFA), gain equalizer, group-delay dispersionequalizer and polarization mode dispersion (PMD)

e front matter r 2009 Elsevier GmbH. All rights reserved.

eo.2008.12.028

ing author.

ess: ashafiul@yahoo.com (S. Azam).

compensator [4,5], etc. Thus, it has recently attractedmore attention from researcher in the area of opticalcommunication.

The synthesis of a filter implemented by an optical-delay line circuit has been studied. The results show thatthe arbitrary signal processing function can be realizedby one input-output channel of the optical delay-linecircuit. Recently, one input three output optical delay-line circuit that can realize any arbitrary three-port FIRfilter characteristics [6] is reported. However, this delay-line circuit is restricted to three-port only.

In recent times, optical interleave filter attractedsignificant amount of attention in optical communica-tion. This filter can offer several channels of phaseshifted bandpass transmission simultaneously. So far,several approaches such as optical half-band filters, flat-passband 1� 3 interleave filter designed by 3� 3 fibercoupler, 1-input 3-output optical interleave filter withgroup-delay dispersion equalizer and multi-channelinterleave filters have been proposed [7–10]. However,

ARTICLE IN PRESSS. Azam et al. / Optik 121 (2010) 1075–10831076

optical interleave filter is unable to offer any arbitraryfilter characteristics.

In this paper an M channel optical delay-line circuit isproposed which can realize any arbitrary multi-channelfilter characteristics. The circuit configuration proposedhere consists of a single input port and M output port. Itis composed of M waveguides, ðM � 1Þ � ðN þ 1Þdirectional couplers, ðM � 1Þ � ðN þ 1Þ phase shiftersand one external phase shifter with phase value jex. Thesynthesis algorithm is based on the division of totaltransfer matrix into unit blocks and factorization iscompleted by repeated size reduction. A set of recursionequations are derived to obtain all unknown circuitparameters. The synthesis algorithm is verified by a flatpassband multi-port FIR optical filter. It also showsthat, proposed M-channel optical delay-line circuit canrealize 100% power transmission.

Following notations are used in this paper. Boldfacedcharacters are used to denote vectors and matrices. A*,AT and A

y denote the conjugate, transpose andtranspose conjugate of A, respectively. The notation

AðzÞ represents the para-conjugate of polynomial A(z)

which is defined as AðzÞ ¼ A�ð1=z�Þ. AðzÞ denotes thepara-conjugate of polynomial matrix A(z) and is defined

by AðzÞ ¼PN

k¼1ay

kzk when AðzÞ ¼PN

k¼1akz�k, where ak

is a coefficient matrix.This paper is arranged in the following order. At first,

circuit structure and related transfer function arepresented in Section 2. A general synthesis algorithmis developed in Section 3. Design example for multi-portFIR optical filer is given in Section 4. Concludingremarks are written in Section 5.

2. Circuit formulation

A multi-port optical delay-line circuit with (MX2) ispresented in this section. This circuit is composed of M

waveguides, ðM � 1Þ � ðN þ 1Þ directional couplers,ðM � 1Þ � ðN þ 1Þ phase shifters and one external phase

Fig. 1. Proposed circuit configuration of an 1

shifter with phase value jex. The top waveguide containsdelay line with a delay time difference of NDt, while theother waveguides maintain equal path length. In thispaper, lossless optical waveguides with negligible bendingloss are assumed.

The multi-port lattice-form optical delay-line circuitconsists one delay-line, (M�1) directional couplers and(M�1) phase shifters in each block as shown in Fig. 1.Transfer function of the optical delay line with delaytime difference Dt can be expressed as,Sd ¼ diagfe�joDt 1 1 � � � 1 1g. Transfer function ofthe rth phase shifter in kth stage can be expressed asS

pr;k ¼ diagf1 1 � � � Pr;k 1 � � � 1 1g with Pr;k ¼ e�jjr;k .

Transfer function of the rth directional coupler in kthstage can be expressed as follows:

Scr;k ¼

1 0 0 � � � � � � � � � 0 0

0 1 0 � � � � � � � � � 0 0

..

. ... ..

. . .. . .

.0 0

0 0 0 � � � Kr;k Kr;k � � � 0 0

0 0 0 � � � Kr;k Kr;k � � � 0 0

..

. ... ..

. . .. . .

.1 0

0 0 0 . .. . .

.0 1

266666666666664

377777777777775, (1)

where Kr;k and Kr;k represent cos yr;k and �j sin yr;k,respectively.

For N stages, this circuit includes N+1 propagationpaths with delay time difference 0�NDt. Transferfunction of a lattice-form optical delay-line circuit canbe expressed in terms of z-transform as below [6]:

X ðzÞ ¼XN

k¼0

xkz�k, (2)

where xk is complex amplitude of the optical signalpropagating along the kth path with delay timedifference kDt. Phase of xk is controlled by the phaseshifters.

By referring Eq. (2), the matrix elements R1ðzÞ�RmðzÞ

of a multi-port optical delay-line circuit can be expressed

�M optical delay-line circuit (M ¼ 5).

ARTICLE IN PRESSS. Azam et al. / Optik 121 (2010) 1075–1083 1077

by the complex expansion coefficients C1;k�Cm;k

(k ¼ 0�N) as below:

R1ðzÞ ¼XN

k¼0

C1;kz�k

R2ðzÞ ¼XN

k¼0

C2;kz�k

..

. ... ..

.

RmðzÞ ¼XN

k¼0

Cm;kz�k. (3)

Degree of freedom, i.e. the number of free parametersof an optical delay-line circuit is already reported in [6].In this paper an M-port optical delay-line circuit withcomplex expansion coefficients C1;k�Cm;k are consid-ered. Therefore, the total number of free parameters forN stages is

M � ðN þ 1Þ � 2 ¼ 2MðN þ 1Þ. (4)

Proposed circuit consists of ðM � 1Þ � ðN þ 1Þ direc-tional couplers and ðM � 1Þ � ðN þ 1Þ phase shifters. Inaddition, restriction conditions take away (2N+1)degree of freedom since the proposed delay-line circuitis considered to be lossless. In Section 3 restrictionconditions are described in details. Hence, it isconfirmed from Eq. (5) that an external phase-shifterjex is required in this proposed circuit configuration.

2MðN þ 1Þ � ð2N þ 1Þ � fðM � 1Þ � ðN þ 1Þg

�fðM � 1Þ � ðN þ 1Þg ¼ 1. (5)

3. Synthesis algorithm

The objective of this synthesis technique is to presenta novel synthesis method for multi-port lattice-formoptical delay-line circuit that shows almost similarfrequency response to the desired one. The unknownparameters in this configuration are approximateoptimum coefficients C1;k�Cm;k (k ¼ 0�N), couplingcoefficient angles y1;k�ym�1;k (k ¼ 0�N) of ðM � 1Þ �ðN þ 1Þ directional couplers, phase shift valuesj1;k�jm�1;k (k ¼ 0�N) of ðM � 1Þ � ðN þ 1Þ phaseshifters and one external phase shifter with phase valuejex. The whole synthesis process is described briefly inthree steps as shown below:

Step 1: The aim of the initial step is to get the constantdelay time difference Dt from the desired periodic

frequency f0. It can be calculated by

Dt ¼1

f 0

. (6)

Step 2: The principle of the second step is to obtainthe approximate optimum coefficients C1;k�Cm;k

(k ¼ 0�N). Various functional approximation methodsthat were developed for FIR digital filters can be appliedin this process since the transfer functions ofR1ðzÞ�RmðzÞ are same as those used in the FIRdigital filters. However, in digital filters sometimesthe maximum power transmission obtained fromdiscrete numerical values can exceed 100%. But thispaper deals with a loss-less optical system. So, restric-tion condition written in Eq. (7) needs to be applied onobjective function to keep the total transmittancealways 100%.

R1ðzÞR1ðzÞ þ R2ðzÞR2ðzÞ þ � � � þ RmðzÞRmðzÞ ¼ 1. (7)

By considering the loss-less condition of Eq. (7),following N+1 restriction conditions must be appliedon the objective function:

XN

i¼0

ðC1;iC�1;i þ C2;iC

�2;i þ � � � � � � þ Cm;iC

�m;iÞ ¼ 1

XN�1i¼0

ðC1;iC�1;iþ1 þ C2;iC

�2;iþ1 þ � � � � � � þ Cm;iC

�m;iþ1Þ ¼ 0

XN�2i¼0

ðC1;iC�1;iþ2 þ C2;iC

�2;iþ2 þ � � � � � � þ Cm;iC

�m;iþ2Þ ¼ 0

..

. ... ..

. ...

X1i¼0

ðC1;iC�1;iþðN�1Þ þ C2;iC

�2;iþðN�1Þ þ � � � � � � þ Cm;iC

�m;iþðN�1ÞÞ ¼ 0

ðC1;0C�1;N þ C2;0C�2;N þ � � � � � � þ Cm;0C�m;N Þ ¼ 0. (8)

Step 3: The aim of the last step is to calculatethe transfer function of each block and to drive aset of recursion equations to obtain the coupling anglesof the directional couplers and phase shift values of thephase shifters. Recursion equations are derived byfactorizing the total transfer matrix S(z) into thefollowing N+1 basic block transfer matrix Sk(z).Transfer matrix S(z) can be decomposed into thefollowing form:

SðzÞ ¼ SNðzÞSN�1ðzÞ � � �S2ðzÞS1ðzÞS0 �Y0k¼N

SkðzÞ. (9)

It is noted that, the kth block with transfer matrix Sk(z)is composed of an optical delay-line with delay timedifference Dt, (M�1) directional couplers with couplingangles y1;k�ym�1;k and (M�1) phase shifters with phaseshift values j1;k�jm�1;k. Transfer function of the kth

ARTICLE IN PRESSS. Azam et al. / Optik 121 (2010) 1075–10831078

block is presented as

SkðzÞ ¼ Spðm�1Þ;kS

cðm�1Þ;k � � � � � �S

p2;kSc

2;kSp1;kS

c1;kSd , (10)

SkðzÞ ¼

K1;kP1;kz�1 K1;kP1;k 0 0 0 � � � 0

K2;kK1;kP2;kz�1 K2;kK1;kP2;k K2;kP2;k 0 0 � � � 0

K3;kK2;kK1;kP3;kz�1 K3;kK2;kK1;kP3;k K3;kK2;kP3;k K3;kP3;k 0 � � � 0

..

. ... ... ... ..

. ... ... ..

.

..

. ... ... ... ..

. ... ... ..

. . .. . .

.

Km�1;kKm�2;k � � � K1;kPm�1;kz�1 Km�1;kKm�2;k � � �K1;kPm�1;k Km�1;kKm�2;k � � � K3;kK2;kPm�1;k. .. . .

.Km�1;kPm�1;k

Km�1;kKm�2;k � � � K2;kK1;kz�1 Km�1;kKm�2;k � � � K2;kK1;k Km�1;kKm�2;k � � � K3;kK2;k � � � � � � Km�1;k

2666666666666664

3777777777777775

.

(11)

In response to an input specified by ½1; 0; 0; � � � � � � ; 0; 0�T, the output responses of the proposed optical delay-linecircuit can be written as

R1ðzÞ

R2ðzÞ

R3ðzÞ

..

. ...

..

. ...

Rm�1ðzÞ

RmðzÞ

26666666666666664

37777777777777775

¼

R½N�1 ðzÞ

R½N�2 ðzÞ

R½N�3 ðzÞ

..

. ...

..

. ...

R½N�m�1ðzÞ

R½N�m ðzÞ

2666666666666666664

3777777777777777775

¼Y0k¼N

SkðzÞ

1

0

0

..

.

..

.

0

0

26666666666666664

37777777777777775

. (12)

The key to solve the unknown circuit parameters y1;N�ym�1;N ; j1;N�jm�1;N of the Nth block is to separate SN(z)(transfer matrix of Nth block) from S(z). Property of para-unitary ensure that, SNðzÞSNðzÞ ¼ IN where IN is a N�N

unit matrix. With this decomposition, ½R½N�1�1 ðzÞ;R½N�1�2 ðzÞ; . . . . . . ;R½N�1�m ðzÞ�T is defined as

R½N�1�1 ðzÞ

R½N�1�2 ðzÞ

R½N�1�3 ðzÞ

..

. ...

..

. ...

R½N�1�m�1 ðzÞ

R½N�1�m ðzÞ

2666666666666666664

3777777777777777775

¼Y0

k¼N�1

SkðzÞ

1

0

0

..

.

..

.

0

0

26666666666666664

37777777777777775

: (13)

ARTICLE IN PRESSS. Azam et al. / Optik 121 (2010) 1075–1083 1079

By employing the para-unitary of SN(z), ½R½N�1�1 ðzÞ;R½N�1�2 ðzÞ; . . . . . . ;R½N�1�m ðzÞ�T can be expressed as

R½N�1�1 ðzÞ

R½N�1�2 ðzÞ

R½N�1�3 ðzÞ

..

. ...

..

. ...

R½N�1�m�1 ðzÞ

R½N�1�m ðzÞ

2666666666666666664

3777777777777777775

¼ SN ðzÞY0k¼N

SkðzÞ

1

0

0

..

.

..

.

0

0

26666666666666664

37777777777777775

0BBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCA

¼ SN ðzÞ

R½N�1 ðzÞ

R½N�2 ðzÞ

R½N�3 ðzÞ

..

. ...

..

. ...

R½N�m�1ðzÞ

R½N�m ðzÞ

2666666666666666664

3777777777777777775

. (14)

Putting SNðzÞ in Eq. (14), the following form can be obtained

R½N�1�1 ðzÞ

R½N�1�2 ðzÞ

R½N�1�3 ðzÞ

..

. ...

..

. ...

R½N�1�m�1 ðzÞ

R½N�1�m ðzÞ

2666666666666666664

3777777777777777775

¼

K1;NP�1;Nz K2;NK�1;NP�2;Nz K3;NK�2;NK�1;NP�3;Nz � � � � � � K�m�1;NK�m�2;N � � �K�2;NK�1;Nz

K�1;NP�1;N K2;NK1;NP�2;N K3;NK�2;NK1;NP�3;N � � � � � � K�m�1;NK�m�2;N � � �K�2;NK1;N

0 K�2;NP�2;N K3;NK2;NP�3;N � � � � � � K�m�1;NK�m�2;N � � �K�3;NK2;N

0 0 K�3;NP�3;N... ...

0 0 0 . ..

..

. ... ..

. ... ...

..

. ... ..

. . ..

0 0 0 � � � � � � K�m�1;NP�m�1;N Km�1;N

26666666666666666664

37777777777777777775

R½N�1 ðzÞ

R½N�2 ðzÞ

R½N�3 ðzÞ

..

. ...

..

. ...

R½N�m�1ðzÞ

R½N�m ðzÞ

2666666666666666664

3777777777777777775

.

(15)

By express R½N�1 ðzÞ�R½N�m ðzÞ given in Eq. (3), R½N�1�m ðzÞ can be written as

R½N�1�m ¼XN�1k¼0

ðK�m�1;NP�m�1;NCm�1;k þ Km�1;NCm;kÞz�k

( )þ ðK�m�1;NP�m�1;NCm�1;N þ Km�1;NCm;N Þz

�N . (16)

The second term of Eq. (16) has to be erased to write R½N�1�m ðzÞ into the following standard form:

R½N�1�m ðzÞ ¼XN�1k¼0

ðC½N�1�m;k Þz

�k. (17)

C½N�1�m;k represents the complex expansion coefficients of R½N�1�m ðzÞ with ðk ¼ 0�N � 1Þ. From this condition, jm�1;N and

ym�1;N can be derived as

jm�1;N ¼ argjCm;N

Cm�1;N

� �, (18a)

ym�1;N ¼ tan�1jCm;NPm�1;N

Cm�1;N

� �. (18b)

Putting the value of jm�1;N in Eq. (18b) confirms that ðjCm;NPm�1;N=Cm�1;N Þ is real. Similarly, R½N�1�m�1 ðzÞ can be

formulated as follows:

R½N�1�m�1 ðzÞ ¼

XN�1k¼0

ðK�m�2;NP�m�2;NCm�2;k

(þ Km�2;NKm�1;NP�m�1;NCm�1;kþKm�2;NK�m�1;NCm;kÞz

�ko

þ ðK�m�2;NP�m�2;NCm�2;N þ Km�2;NKm�1;N � P�m�1;NCm�1;N þ Km�2;NK�m�1;NCm;N Þz�N . (19)

ARTICLE IN PRESSS. Azam et al. / Optik 121 (2010) 1075–10831080

Following the same rule like Eq. (16), phase shift value jm�2;N and coupling coefficient angle ym�2;N can be derivedfrom Eq. (19) as

jm�2;N ¼ argjðKm�1;NP�m�1;NCm�1;N þ K�m�1;NCm;NÞ

Cm�2;N

� �(20a)

ym�2;N ¼ tan�1jðKm�1;NP�m�1;NCm�1;N þ K�m�1;NCm;N ÞPm�2;N

Cm�2;N

� �(20b)

Phase shift values jm�3;N�j1;N and coupling coefficient angle ym�3;N�y1;N can be derived in a similar processdescribed above.

In the next separation step, circuit parameters of (N�1)th block can be obtained. Consequently, all the circuitparameters jm�1;n�j1;n and ym�1;n�y1;n can be found successively in the order n ¼ N; N � 1; N � 2; . . . ; 2; 1; 0:

jm�1;n ¼ argjC½n�m;n

C½n�m�1;n

( )

jm�2;n ¼ argjðKm�1;nP�m�1;nC

½n�m�1;n þ K�m�1;nC½n�m;nÞ

C½n�m�2;n

( )

..

. ... ..

.

j1;n ¼ argjðK2;nP�2;nC

½n�2;n þ K�2;nK3;nP�3;nC

½n�3;n þ � � � � � � þ K�2;nK�3;nK�4;n � � �K

�m�1;nC½n�m;nÞ

C½n�1;n

( )(21a)

Fig. 2. Flowchart diagram of the present synthesis algorithm.

ARTIC

LEIN

PRES

STable 1. Middle band expansion coefficients and calculated circuit parameters of a 5-port optical filter.

Stage number Middle band

expansion

coefficients

Coupling

coefficient angle

y1;k � p

Coupling

coefficient angle

y2;k � p

Coupling

coefficient angle

y3;k � p

Coupling

coefficient angle

y4;k � p

Phase shift value

j1;k � pPhase shift value

j2;k � pPhase shift value

j3;k � pPhase shift value

j4;k � p

k ¼ 0�30

0 0.0295259 5.3165e�002 4.9934e�001 4.9814e�001 4.9992e�001 �9.9503e�015 �1.0000e+000 1.0000e+000 �1.0000e+000

1 0.0628957 1.8273e�001 4.9948e�001 4.9879e�001 3.0839e�001 �1.8906e�014 �1.0000e+000 �6.6626e�014 4.6850e�016

2 0.1062999 2.6386e�001 4.9939e�001 2.7813e�001 2.6101e�001 4.6773e�014 7.3638e�014 �6.7968e�016 �1.6268e�016

3 0.1504811 3.6776e�001 4.1348e�001 4.3127e�001 4.2803e�001 1.0000e+000 1.2258e�015 9.2470e�016 6.7775e�016

4 0.1851230 1.4115e�003 2.9162e�001 3.2259e�001 1.3142e�001 �1.0000e+000 �1.0000e+000 4.6413e�013 6.1610e�014

5 0.1985227 3.2816e�001 4.9988e�001 4.9818e�001 4.9985e�001 �9.1402e�014 1.0000e+000 �1.3039e�013 1.0000e+000

6 0.1853373 2.8777e�001 4.9652e�001 4.9871e�001 4.5527e�001 6.1716e�014 �1.0000e+000 9.1872e�014 3.5450e�015

7 0.1453476 6.4198e�002 4.9916e�001 4.9067e�001 4.4716e�001 �5.6905e�014 1.0000e+000 �7.0211e�015 �8.5217e�015

8 0.0862491 1.7283e�001 4.4151e�001 2.0461e�001 6.1561e�002 �1.0000e+000 5.6635e�015 6.1516e�015 5.8125e�015

9 0.0212753 7.5061e�004 7.3524e�002 3.0095e�001 4.6981e�001 �1.0000e+000 1.6297e�013 �1.0000e+000 �1.0000e+000

10 �0.0343778 1.8630e�001 4.9815e�001 4.9882e�001 4.9907e�001 �1.4616e�014 �1.0000e+000 �1.7024e�013 8.0999e�013

11 �0.0688508 1.0937e�001 4.9451e�001 4.9966e�001 1.6553e�001 2.2419e�014 1.0000e+000 1.0000e+000 �2.3524e�015

12 �0.0762065 3.7085e�002 4.9605e�001 2.6026e�001 3.7842e�001 �1.0000e+000 1.5851e�013 6.5284e�015 �2.1875e�014

13 �0.0587843 1.2751e�002 4.7346e�001 4.8944e�001 4.7268e�001 1.0000e+000 7.1373e�014 9.5891e�014 7.7673e�014

14 �0.0255729 7.5074e�004 3.7981e�001 2.4877e�001 5.5550e�002 1.3888e�013 1.4168e�013 �1.0000e+000 8.8946e�014

15 0.0097832 2.9659e�002 4.9110e�001 4.9587e�001 4.9614e�001 �1.4098e�014 �1.0000e+000 �1.0000e+000 5.1592e�014

16 0.0358630 1.7371e�002 4.8718e�001 4.9876e�001 3.3402e�001 1.5858e�015 �1.0000e+000 �1.0377e�013 �1.9711e�015

17 0.0454558 2.2534e�002 4.9049e�001 3.5008e�001 2.8023e�001 1.0000e+000 �1.0000e+000 �1.1528e�014 �1.1798e�014

18 0.0375615 4.0145e�002 2.6792e�001 2.8662e�001 3.3837e�001 1.3808e�015 �1.5809e�015 �2.3810e�015 �2.1424e�015

19 0.0174654 6.4232e�004 3.0434e�001 3.0692e�001 3.1572e�001 9.3644e�015 9.1140e�015 �1.0000e+000 �1.0000e+000

20 �0.0050006 2.0808e�002 3.8631e�001 4.9992e�001 4.7254e�001 1.0000e+000 1.0000e+000 1.0000e+000 1.0000e+000

21 �0.0222571 1.7352e�003 4.8666e�001 4.3454e�001 4.7889e�001 �7.9243e�014 �1.8009e�013 �9.4911e�014 1.0000e+000

22 �0.0286092 4.2220e�002 4.9665e�001 4.0868e�001 2.8080e�001 5.6272e�014 7.7769e�014 �1.0641e�015 �4.7959e�015

23 �0.0231120 7.7654e�002 9.2877e�002 1.4066e�001 2.4606e�001 4.5322e�015 4.2916e�015 3.8949e�015 2.7888e�016

24 �0.0099935 8.2548e�004 4.1911e�001 4.6188e�001 4.7374e�001 2.7578e�014 3.0554e�014 �1.0000e+000 9.9295e�014

25 0.0043307 2.7033e�002 4.9586e�001 4.9645e�001 4.9817e�001 1.0000e+000 9.9757e�015 1.0000e+000 1.7918e�014

26 0.0147639 2.0535e�002 4.9800e�001 4.8923e�001 2.2143e�002 �1.2232e�014 2.5864e�014 1.0000e+000 3.0462e�015

27 0.0177248 8.5864e�002 4.9939e�001 1.0833e�001 4.8388e�001 5.7191e�016 1.2074e�014 8.0907e�016 �2.4240e�014

28 0.0133850 1.2408e�001 2.3809e�001 4.9282e�001 4.1785e�001 �1.0000e+000 �4.5319e�017 8.1821e�015 �1.4065e�016

29 0.0045268 7.6947e�004 4.8641e�001 4.2634e�001 2.7188e�001 9.8628e�015 �1.0000e+000 4.9238e�015 1.9060e�014

30 �0.0043484 1.1918e�002 4.9433e�001 4.9424e�001 4.9352e�001 �2.1144e�015 1.0000e+000 6.1992e�014 �1.0000e+000

k ¼ 31�39

31 �0.0098836 8.2457e�002 4.9968e�001 4.9729e�001 3.3054e�001 1.2120e�015 1.0000e+000 1.0000e+000 �1.4354e�016

32 �0.0104696 1.5538e�001 4.9949e�001 2.2428e�001 2.0746e�001 6.3452e�015 �1.0000e+000 1.0789e�016 �1.7786e�017

33 �0.0067480 1.4800e�001 3.3818e�001 4.0018e�001 4.2421e�001 �1.1917e�015 �1.7053e�016 �1.3943e�016 �1.4348e�016

34 �0.0007215 5.7877e�004 3.3833e�001 3.9504e�001 1.9995e�001 �9.8282e�015 �5.6139e�015 1.7045e�015 �1.9643e�015

35 0.0041121 9.9087e�002 4.9900e�001 4.9912e�001 4.9970e�001 �2.0240e�001 �2.0240e�001 �2.0240e�001 �2.0240e�001

36 0.0069805 1.8390e�001 4.9935e�001 4.9849e�001 2.5072e�001 �4.7433e�001 �4.7433e�001 5.2567e�001 �2.7594e�001

37 0.0062372 1.9754e�001 4.9949e�001 3.0463e�001 1.9207e�001 �7.7763e�001 2.2237e�001 �5.7746e�001 �2.8513e�001

38 0.0021830 8.2042e�002 3.3371e�001 2.7082e�001 1.9374e�001 �1.0093e�001 �9.0174e�001 �5.7007e�001 �2.7733e�001

39 �0.0084076 3.5272e�001 3.3310e�001 3.0403e�001 2.4942e�001 3.9751e�001 2.9878e�001 1.9876e�001 9.8733e�002

S.Azam

etal./Optik

121(2010)1075–1083

1081

ARTICLE IN PRESSS. Azam et al. / Optik 121 (2010) 1075–10831082

ym�1;n ¼ tan�1jC½n�m;nPm�1;n

C½n�m�1;n

( )

ym�2;n ¼ tan�1jðKm�1;nP�m�1;nC

½n�m�1;n þ K�m�1;nC½n�m;nÞPm�2;n

C½n�m�2;n

( )

..

. ... ..

.

y1;n ¼ tan�1jðK2;nP�2;nC

½n�2;n þ K�2;nK3;nP�3;nC

½n�3;n þ � � � � � � þ K�2;nK�3;nK�4;n � � �K

�m�1;nC½n�m;nÞP1;n

C½n�1;n

( )(21b)

The expansion coefficients can be calculated as

C½n�1�m;k ¼ K�m�1;nP�m�1;nC

½n�m�1;k þ Km�1;nC

½n�m;k

C½n�1�m�1;k ¼ K�m�2;nP�m�2;nC

½n�m�2;k þ Km�2;nKm�1;nP�m�1;nC

½n�m�1;k þ Km�2;nK�m�1;nC

½n�m;k ðk ¼ 0; 1; 2 . . . ; n� 1Þ

..

. ... ..

.

C½n�1�1;k ¼ K1;nP�1;nC

½n�1;kþ1 þ K�1;nK2;nP�2;nC

½n�2;kþ1 þ � � � þ K�1;nK�2;nK�3;n � � �K

�m�1;nC

½n�m;kþ1. (22)

C½n�1;k�C

½n�m;k indicate the kth expansion coefficients of R

½n�1 ðzÞ�R½n�m ðzÞ, respectively. Note that, Eq. (22) should be omitted

when n ¼ 0. Therefore, all coupling coefficient angles ym�1;n�y1;n (n ¼ 0�N) of ðM � 1Þ � ðN þ 1Þ directionalcouplers, phase shift values jm�1;n�j1;n(n ¼ 0�N) of ðM � 1Þ � ðN þ 1Þ phase shifters can be obtained from the abovesynthesis process that is summarized in Fig. 2.

4. Design example

In this section a five-port optical FIR filter issynthesized using the design data obtained in [10]. Thehighest degree N of z polynomials is 39. The relativefrequency is normalized with respect to the FSR.

Table 1 shows the middle band expansion coefficients ofa 5-port optical delay-line circuit. It also shows thecalculated circuit parameters y1;k�y4;k and j1;k�j4;k with(k ¼ 0�39). The coupling coefficient angles of the direc-tional couplers and the phase shift values of the phaseshifters are written in radian unit. Calculated phase valuesof the phase shifters are non-zero due to complexexpansion coefficients. Fig. 3 shows the synthesized power

5

0

-5

-10

-15

-20

-25

-30

-35

-40

-45

-50-0.5 -0.4 -0.3 -0.2 -0.1 0

relative frequency

0.1 0.2 0.3 0.4 0.5

atte

nuat

ion

[dB

]

Fig. 3. Power frequency response of a 5-port optical filter.

frequency response of a 5-port optical filter. This powercharacteristic shows 0dB at the center frequency of eachband while the stop band transmittance is less than�26dB. It also be confirmed that 100% power transmit-tance is realized.

5. Conclusion

A multi-channel optical delay line circuit is proposed in this paper. This circuit approaches similarfilter characteristic as those of a 1�M digitalFIR filter. Besides this, an algorithm for synthesizingthe M-channel optical delay line circuit is also derived.Synthesis method was based on division of total transfermatrix into the basic unit transfer matrices. A set ofrecursion equations were derived to obtain the unknowncircuit parameters. Proposed synthesis algorithm hasbeen tested by example and work properly. It isobserved from the synthesis example, that the proposedoptical filter is capable of higher optical signalprocessing in a good number of stages and is expectedto be employed in FDM and WDM optical commu-nication.

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ARTICLE IN PRESSS. Azam et al. / Optik 121 (2010) 1075–1083 1083

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