synthesis of a multichannel lattice-form optical delay-line circuit with ring waveguides

Synthesis of a multichannel lattice-form optical delay-line circuit with ring waveguides

Shafiul Azam*, Takashi Yasui and Kaname Jinguji Signal Processing laboratory, Interdisciplinary Faculty of Science and Engineering,

Shimane University, Matsue-shi, 690-8504, Japan * Corresponding author: [email protected]

Abstract: This paper presents a one-input M -output )1( M× circuit configuration and a synthesis algorithm for realizing an optical infinite impulse response (IIR) lattice filter having M -output channels

)2( ≥M .The circuit configuration has a multilayer structure consisting of multiple Mach–Zehnder interferometers with delay time difference of τΔ . It is a natural extension of the conventional two-port optical IIR lattice circuit )2( =M . Synthesis algorithm is derived to obtain all unknown circuit parameters. The proposed synthesis algorithm is based on factorizations of the paraunitary total transfer matrix. Simulation result demonstrates the effectiveness of the proposed multichannel IIR design scheme. The synthesis algorithm for channel−M IIR filter is considered lossless in this paper, which implies that the filter must be power complementary.

©2008 Optical Society of America

OCIS codes: (060.1810) Couplers, switches, and multiplexers; (060.2310) Fiber optics; (060.4230) Multiplexing; (350.2460) Filters, interference.

References and links

1. C. K. Madsen and J. H. Zhao, Optical filter design and Analysis (A Wiley- Interscience Publication, John Wiley & Sons, Inc.,1999).

2. M. Kawachi and K. Jinguji, “Planar lightwave circuits for optical signal processing,” in Tech. Dig. OFC’941994, Paper FB7.

3. Z. Wan and Y. Wu, “Tolerance Analysis of lattice-form optical interleaver with different coupler structures,” J. Lightwave Technol.24, 5013-5018 (2006).

4. M. S. Rasras, D. M. Gill, S. S. Patel, A. E. White, K. Y. Tu, Y. K. Chen, Carothers, D. Pomerene, A. Grove, M. J. Sparacin, D. Michel, J. Beals, and M. Kimerling, “Tunable Narrowband Optical Filter in CMOS,” OFC, 2006 and the 2006 National Fiber Optic Engineers Conference, OFC 2006, pp. 1-4

5. S. Azam, T. Yasui, and K. Jinguji, “Synthesis algorithm of a Multi- channel lattice-form optical delay-line circuit,” submitted to optik (an Elsevier publication).

6. S. Azam, T. Yasui, and K. Jinguji, “Synthesis of 1-input 3-output lattice-form optical delay-line circuit,” IEICE transactions on electronics E90-C,149-156 (2007).

7. P. P. Vaidyanathan, “Multirate Systems and Filter Banks”, Englewood Cliffs, NJ, Prentice- Hall, 1993. 8. K. Jinguji and T. Yasui, “Design Algorithm for multi-channel interleave filters,” J.Lightwave Technol. 25,

2268-2278 (2007). 9. C. K. Madsen, “General IIR Optical Filter Design for WDM Applications using All-Pass Filters,” J.

Lightwave Technol.18, 860-868 (2000). 10. S. Azam, T. Yasui, and K. Jinguji, “Synthesis of 1-input 3-output lattice-form optical delay-line circuit with

IIR architecture,” Recent Patents on Elec. Engin. 1, 214-224 (2008).

11. Q. J. Wang, Y. Zhang, and Y. C. Soh, “Flat-passband 33 × interleaving filter designed with optical directional couplers in lattice structure,” J. Lightwave Technol. 23, 4349-4362 (2005).

12. S. Azam, T. Yasui, and K. Jinguji, “1-input 3-output Optical Interleave Filter with Group-Delay Dispersion Equalizer,” Tech. Dist. EOOC/IOOC, Yokohama, Japan. pp. 766-767(2007).

13. K. Jinguji and M. Oguma, “Optical Half-Band Filters,” J. Lightwave Technol.18, 252-259 (2000). 14. Q. J. Wang, Y. Zhang, and Y. C. Soh “Design of 100/300 GHz optical interleaver with IIR architectures,”

Opt. Express 13, 2643-2652 (2005). 15. K. Jinguji, “Synthesis of coherent two-port optical delay line circuit with ring waveguides,” J. Lightwave

Technol.14, 1882-1898 (1996).

#102794 - $15.00 USD Received 14 Oct 2008; revised 24 Nov 2008; accepted 24 Nov 2008; published 11 Dec 2008

(C) 2008 OSA 22 December 2008 / Vol. 16, No. 26 / OPTICS EXPRESS 21401

16. P. P. Vaidyanathan, “Passive Cascaded-Lattice Structures for Low-Sensitivity FIR Filter Design, with Applications to Filter Banks,” IEEE Trans. Circuits Syst. cas-33, 1045-1064 (1986).

1. Introduction

In recent years, there has been a growing interest in the application of optical delay-line circuits in optical signal processing. These circuits are composed of directional couplers, phase shifters and delay-lines [1-3]. Optical filters are a good test vehicle for photonic technology since filtering is a fundamental aspect of any form of photonic signal processing. Narrow, box-like, bandpass optical filters are of particular importance for channelizing in RF photonic applications as well as for monitoring optical signals in dense wavelength-division-multiplexed (DWDM) systems [4]. Unlike digital filters, optical filters are roughly classified as FIR and IIR type and they demonstrate similar filter characteristics like digital filters [5, 6].

There exist a number of methods for lattice-form IIR digital filters. Synthesis algorithms were proposed for M×1 IIR digital filters [7]. These methods are based on division of total transfer matrix into unit blocks. It is well known that, optical systems and digital systems are differentiated on two major points. Optical paths necessarily cause phase change whereas the signal paths in digital systems can connect two points without phase change. Directional couplers used in optical circuits are expressed by complex transfer matrices whereas the Givens rotation used in digital systems are expressed by real transfer matrices [8]. Due to these differences, the circuit configurations and synthesis theories developed for digital filters are not applicable to optical filters.

A general design algorithm is presented to implement an IIR optical filter using all-pass ring resonators in a Mach-Zehnder configuration [9]. However, this circuit is a single layer two-port presentation and unable to offer M×1 IIR type optical filter. Recently, 31× IIR type optical delay-line circuit that can offer three-port arbitrary IIR filter characteristics is proposed [10]. Synthesis algorithm is also confirmed by design example. Circuit configuration and synthesis mechanism of a 31× IIR type optical filter is a foundation for channel−M IIR optical filter. In recent times, optical interleave filters attracted significant amount of attention in optical communication [11, 12]. Researchers in this field already reported the circuit structure and synthesis algorithm for two-port and three-port optical interleave filters with IIR architecture [13, 14]. But, these methods are dedicated to interleave filter characteristics and unable to arbitrary filter characteristics. In this correspondence, a novel circuit configuration for M×1 optical delay-line circuit with ring waveguides is presented in this paper that can offer multi-port arbitrary filter characteristics. Each unit element is composed of one symmetric Mach-Zehnder interferometer and one ring waveguide. The symmetric Mach-Zehnder interferometer includes

)1( −M directional couplers and )1( −M phase shifters. The lossless ring resonator with a single coupler and a phase shifter is an all-pass filter. Synthesis algorithm is based on the repeated size-reduction. A set of recursion equations are derived to obtain all unknown circuit parameters. It is assumed in this paper that, the synthesis algorithm for channel−M IIR filter is lossless, which implies that the sum of output powers is unity.

In this paper following notations are used. Boldfaced characters are used to denote vectors

and matrices, ∗A , TA and †A denote the conjugate, transpose and transpose conjugate

of A respectively. The notation )(~

zA represents the para-conjugate of polynomial )(zA which

is defined as ⎟⎠

⎞⎜⎝

⎛= ∗∗

zAzA

1)(

~. )(

~zA denotes the para-conjugate of polynomial matrix

)(zA and is defined by †

1

( )N

kk

k

z z=

=∑A a� when ∑=

−=N

k

kk zz

1

)( aA , where ka is a coefficient

matrix.



An outline of this paper is as follows. Section 2 describes the circuit configuration with transfer function. Section 3 demonstrates synthesis algorithm. Design examples are presented in Section 4. Concluding remarks are written in Section 5.

2. Circuit formulation

The circuit configuration of a M×1 )2( ≥M lattice-form optical delay-line circuit with ring

waveguides is presented in this section. This circuit includes )1( +M optical waveguides,

)1( −+ MMN directional couplers, )1( −+ MMN phase shifters and an external phase

shifter exϕ . Delay-line with delay time difference τΔ is maintained by the ring waveguides. All optical waveguides are considered lossless with negligible bending loss in this paper. A novel M×1 optical delay-line circuit with ring waveguides that can realize power-complementary multi-port outputs is shown in Fig. 1. This proposed circuit can offer 100% power transmittance. The number of free parameters of an optical delay line circuit is already reported [6] [10]. In this paper the complex expansion coefficients are )~0(~ ,,1 NkCC kMk =

and )~1( NkDk = . Therefore, the degree of freedom for N stages is { }NNM ++ )1(2 .

However, this circuit includes )1( −+ MMN directional couplers and )1( −+ MMN phase

shifters. In addition, restriction conditions take away )12( +N degree of freedom. Hence, Eq.

(1) confirms that an external phase shifter exϕ is requisite in this circuit configuration as below:

{ } 1)1()1()12()1(2 =−+−−+−+−++ MMNMMNNNNM . (1)

The presented multi-port optical delay-line circuit has a number of cascaded unit elements. Each unit element is composed of one symmetric Mach-Zehnder interferometer and one ring waveguide. Transfer function of the thr directional coupler and phase shifter of thn stage MZI can be written as [16],

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

10000

01

00ˆ000

00ˆ000

00

00010

00001

,,

,,,

��

��

��

��

��

��

��

nrnr

nrnrc

nr

KK

KKS . (2a)

{ }11111 ,, �� nrp

nr Pdiag=S (2b)

Where nrK , , nrK ,ˆ and nrP , represents nr,cosθ , nrj ,sinθ− and nrje ,ϕ− respectively. The lossless

ring waveguide with a single coupler and a phase shifter is an all-pass filter. Its thn stage transfer function can be written by the following rational function [10]:

1

1,,

1

1,

1

1

cos)(

,

−

−

−

−−

−

−=

−

−=

z

zPK

z

zezF

n

nana

n

jnaR

n

na

α

αθ ϕ

(3)



where nα indicates the thn pole of the transfer function with nanan PK ,,=α . na,θ is the

coupling angle of the directional coupler connecting the thn stage ring waveguide and the symmetric MZI and na,ϕ indicates the phase shifter value on the thn ring waveguide. The

transfer function of an optical delay-line with a delay time difference τΔ is written as τωΔ− je .

In terms of Z transform τωΔ− je can be replaced by the term 1−z . The transfer function of a thn unit element that includes one MZI and one ring waveguide can be written as follows

[10, 15]:

{ }

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

=

=

−−−−−−−

−−−−−−−−−−−

−−

nanMnannMnMnannnMnMnannMnM

nanMnMnanMnnMnMnanMnnMnMnanMnnMnM

nannnannnnannnnnannnn

nannnannnnannn

nannnann

na

Rn

cn

pn

cn

pn

cnr

pnr

cnM

pnM

Rn

Mnn

UKUKKKUKKKKWKKK

UPKUPKKKUPKKKWPKKK

UPKUPKKUPKKKWPKKK

UPKUPKKWPKK

UPKWPK

U

z

zz

,,1,,2,2,1,,1,2,2,1,,1,2,1

,,1,1,,1,2,2,1,,1,1,2,1,,1,1,2,1

,,3,3,,3,2,3,,3,1,2,3,,3,1,2,3

,,2,2,,2,1,2,,2,1,2

,,1,1,,1,1

,

,1,1,2,2,,),1(),1(

ˆˆˆˆˆˆˆˆ

ˆˆˆˆˆ

00ˆˆˆˆ

000ˆˆ

0000ˆ

1

)(

)()(

��

��

��

��

�

�

�

�� SSSSSSSSS

SSS

(4)

Where, naU , and naW , represent )1( 1−− znα and )(cos 1,

, −−− ze najna

ϕθ respectively.

0,1θ

0,2θ

0,1ϕ

0,2ϕ

exϕ

0,3θ

0,4θ0,4ϕ

1,1θ

1,2θ

1,1ϕ

1,4θ

1,3θ

N,1θ

N,2θ

N,1ϕ

N,4θ

N,3θ0,3ϕ

1,2ϕ

1,3ϕ

1,4ϕ

N,2ϕ

N,3ϕ

N,4ϕ

A1 A2

B2

C2

D2

E2

0S )(1 zS )(zNS

τΔ τΔ

1,aϕNa,ϕ

1,aθ Na,θ

Fig. 1. Circuit configuration of a M×1 ( 5=M ) lattice-form optical delay-line circuit with Ring waveguides proposed in this paper.

The input of this proposed delay-line circuit is considered as [ ]T0,0,0,1 �� . Therefore, the output can be expressed as follows,



∏=

−⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

0

1

3

2

1

0

0

0

0

1

)(

)(

)(

)(

)(

)(

)(

1

Nk

k

M

M

z

zR

zR

zR

zR

zR

zQ ��S . (5)

The vector elements )(~)(1 zRzR M and )(zQ of a multi-port optical delay-line circuit with

ring waveguides can be expressed by complex expansion coefficients kMk CC ,,1 ~

( 0=k ~ N ) and kD ( 1=k ~ N ) as follows [10, 15]:

∑

∑

∑

∑

=

−

=

−

=

−

=

−

+=

=

=

=

N

k

kk

N

k

kkMM

N

k

kk

N

k

kk

zDzQ

zCzR

zCzR

zCzR

1

0,

0,22

0,11

.1)(

)(

)(

)(

�� (6)

3. Synthesis Algorithm

This section presents a novel synthesis approach to design an M×1 optical delay-line circuit with ring waveguides. The aim of this algorithm is to calculate the unknown expansion coefficients kMk CC ,,1 ~ ( 0=k ~ N ) and kD )~1( Nk = , coupling coefficient angles

kMk ,1,1 ~ −θθ )~0( Nk = and ka,θ ( 1=k ~ N ) of )1( −+ MMN directional couplers, phase

shift values kMk ,1,1 ~ −ϕϕ )~0( Nk = and ka,ϕ ( 1=k ~ N ) of )1( −+ MMN phase shifters.

The whole synthesis algorithm is composed of three stages as illustrated in Fig. 2.

Step 1: The purpose of this step is to find the unit delay time difference )( τΔ from a desired

frequency period )( pf . It is calculated by,

pf

1=Δτ . (7)

Step 2: In step 2 optimum expansion coefficients of )(~)(1 zRzR M and )(zQ are determined. Several methods for designing IIR digital filters can be considered in this regard. For example, Eigenfilter design method, least square approximation method and bilinear transformation methods are mostly used. Note that, all poles of the rational function needs to be inside the unit circle )1( =z in order to be filter stable. Since, the optical delay-line

circuits are passive filters, maximum transmission cannot exceed 100%.

{ } 1)(~

)()(~

)()(~

)()(

~)(

12211 =+++ zRzRzRzRzRzR

zQzQMM�� . (8)

Restriction condition mentioned in Eq. (8) expressed in details by the complex expansion coefficients [5, 10].



Step 3: In this step a set of recursion equations are derived to obtain all unknown circuit parameters. The entire transfer matrix )(zS is decomposed into 1+N unit blocks and finally

all coupling angles kMk ,1,1 ~ −θθ )~0( Nk = and ka,θ )~1( Nk = of )1( −+ MMN

directional couplers, phase values kMk ,1,1 ~ −ϕϕ )~0( Nk = and ka,ϕ ( 1=k ~ N ) of

)1( −+ MMN phase shifters are obtained. Applying factorization, total vector )(zT can be decomposed into the following form,

∏=

−

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

≡

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=1

00121

0

0

0

0

1

)(

0

0

0

0

1

)()()()()(Nk

kNN zzzzzz

�

�

�

�� SSSSSSST . (9)

Here it is assumed that the decomposition processes have proceeded successfully until the 1+n stages as follows. Therefore Eq. (9) can be written as,

.)()()()()( ][11 zzzzz n

nNN TSSST +−= � (10)

where )(][ znT indicates the remaining part after the th)1( +n stage decomposition and the

output at the thn stage is defined as,

∏=

−⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=1

0

][

][1

][3

][2

][1

][][

0

0

0

0

1

)(

)(

)(

)(

)(

)(

)(

1)(

nk

k

nM

nM

n

n

n

nn z

zR

zR

zR

zR

zR

zQz

�

�

��

�� SST . (11)

)(][ zQ n of Eq. (11) can be expressed as below,

∏=

−−=1

1][ )1()(nk

kn zzQ α . (12)

From the definition of the thn zero nα of )(][ zQ n , na,θ and na,ϕ of the thn block can be

obtained as below,

)(cos 1, nna αθ −= ; )(arg, nna αϕ −= . (13)

The unknown circuit parameters nMn ,1,1 ~ −θθ and nMn ,1,1 ~ −ϕϕ of the thn block can be

acquired by separating )(znS (transfer matrix of thn block) from )(][ znT . Property of para-

unitary ensures that nnn zz ISS =)()(~

where nI is a )( nn × unit matrix. Applying the para-

unitary of )(znS , )(]1[ zn−T can be written as,

.)()(~

)( ][]1[ zzz nn

n TST =− (14)



With this decomposition, )(]1[ zn−T can be expressed as follows,

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

−−

−−

−

−

−

−

)(

)(

)(

)(

)(

)(

1)(

~

)(

)(

)(

)(

)(

)(

1

][

][1

][3

][2

][1

][

]1[

]1[1

]1[3

]1[2

]1[1

]1[

zR

zR

zR

zR

zR

zQz

zR

zR

zR

zR

zR

zQ

nM

nM

n

n

n

nn

nM

nM

n

n

n

n��

S . (15)

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−

∗−

∗∗−

∗−

∗∗∗

∗∗∗−

∗−

∗∗∗∗∗

∗∗∗−

∗−

∗∗∗∗∗∗∗∗

∗∗∗∗−

∗−

∗∗∗∗∗∗∗∗∗

∗

−

−−

−

−

−

)(

)(

)(

)(

)(

000

000

00

0

1

)(

)(

)(

)(

)(

][

][1

][3

][2

][1

,,1,,1,1

,,3,3

,,2,3,2,1,,3,2,3,,2,2

,,1,2,2,1,,3,1,2,3,,2,1,2,,1,1

,,1,2,2,1,,3,1,2,3,,2,1,2,,1,1

,,

]1[

]1[1

]1[3

]1[2

]1[1

zR

zR

zR

zR

zR

UKUPK

UPK

UKKKKUPKKUPK

UKKKKUPKKKUPKKUPK

WKKKKWPKKKWPKKWPK

UU

zR

zR

zR

zR

zR

nM

nM

n

n

n

nanMnanMnM

nann

nannnMnMnannnnann

nannnMnMnannnnnannnnann

nannnMnMnannnnnannnnann

nana

nM

nM

n

n

n

��

��

�

��

��

�

��

��

��

��

��

��

(16)

Since )(~)( ]1[1

]1[ zRzR nnM

−− must be th)1( −n order polynomials, it is required that the numerator of each function is capable of division by the denominator. This required condition

concerning the polynomials of )(~)( ]1[1

]1[ zRzR nnM

−− can be expressed in Eq. (17).

0)()( ][,1

][1,1,1 =+ −−

∗−

∗− n

nMnMn

nMnMnM RKRPK αα (17a)

0)()()( ][,1,2

][1,1,1,2

][2,2,2 =++ ∗

−−−∗

−−−−∗

−∗

− nn

MNMNMnn

MNMNMNMnn

MNMNM RKKRPKKRPK ααα (17b)

��

0)()()( ][,1,2,2,1

][2,2,2,1

][1,1,1 =+++ ∗

−∗

−∗∗∗∗

nn

MnMnMnnnn

nnnnn

nn RKKKKRPKKRPK ααα �� (17c)

0111 ][

,1,2,2,1][

2,2,2,1][

1,1,1 =⎟⎟

⎠

⎞

⎜⎜

⎝

⎛++

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛∗

∗−

∗−

∗∗∗

∗∗∗

∗

n

nMnMnMnn

n

nnnn

n

nnn RKKKKRPKKRPK

ααα�� (17d)

From Eq. (17a) nM ,1−ϕ and nM ,1−θ can be derived as follows,

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

=−

−)(

)(arg

][1

][

,1n

nM

nn

MnM

R

Rj

ααϕ ;

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

=−

−−−

)(

)(tan

][1

,1][

1,1

nn

M

nMnn

MnM

R

PRj

αα

θ . (18)

Substitute the value of nMP ,1− ensure the term ⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−

−

)(

)(][

1

,1][

nn

M

nMnn

M

R

PRj

αα

is real and so nM ,1−θ is

real as well. Note that, circuit parameters obtained from Eq. (17c) and Eq. (17d) are equal [10]

[15]. Consequently, all the circuit parameters nnM ,1,1 ~ ϕϕ − and nnM ,1,1 ~ θθ − can be found by

successively in the order )0~(Nn = as shown in Appendix A. The external phase shifter value can be obtained as follows

{ }∗∗−

∗−

∗∗−−−

∗∗∗ ++++−= 0,10,10

0,0,10,10,20,10

0,10,20,10,20

0,20,10,100,1arg KKCPKKKCPKKCPKC MMMMMMex ��ϕ (19)

Where 00,

00,1 ~ MCC are the 0th stage expansion coefficients.



Thus, all the circuit parameters )0~(~,~ ,1,1,1,1 NnnnMnnM =−− ϕϕθθ and

)1~(, ,, Nnnana =ϕθ can be obtained by performing the third synthesis steps described

above.

4. Synthesis Example

In this section, two examples of IIR optical frequency filters are demonstrated which are synthesized by design data of IIR optical filters.

A. Optical Elliptic Filter

A fifth-order optical elliptic filter was synthesized using the design data obtained in [15]. Table 1(a) shows the normalized expansion coefficients. Circuit parameters calculated by the present synthesis algorithm are presented in Table 1(b). Fig. 3 shows the cross and through port power transmittance. The transmittance at the stop band was less than -13 dB. It can be confirmed that a maximum transmittance of 100% is realized as expected.

B. Optical IIR Interleave Filter

A three-port optical IIR interleave filter is demonstrated by referring the design data [14]. In this example, the number of expansion coefficients is 12=N . Table 2(a) shows the

Table 1. (a) Normalized expansion coefficients of a two-port fifth-order optical elliptic filter

normalized numerator amplitude expansion coefficients

stage number

through port cross port

denominator amplitude expansion

coefficients

0 0.050892 0.472107 1.000000 1 -0.074644 1.412971 1.843590 2 0.121667 2.298860 2.477256 3 -0.121667 2.262030 1.821912 4 0.074644 1.350477 0.883695 5 -0.050892 0.430801 0.200795

Table 1. (b) Calculated circuit parameters of a two-port fifth-order optical elliptic filter.

stage number

coupling angle πθ ×k,1

phase value πϕ ×k,1

coupling angle πθ ×ka,

phase value πϕ ×ka,

0 2.6363e-001 -2.5191e-001 ------------ ----------- 1 4.8056e-001 3.1617e-001 3.4706e-001 -1.0000e+000 2 2.9921e-002 5.3109e-001 2.4939e-001 -6.8191e-001 3 4.8189e-001 -4.1366e-001 2.4939e-001 6.8191e-001 4 3.8516e-002 5.0593e-001 1.1951e-001 6.0722e-001 5 2.5866e-001 -5.3846e-001 1.1951e-001 -6.0722e-001

External phase shifter value )( exϕ : π×−− 0010768.1 e



Yes No

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

Compute the unit time delay from desired periodic frequency pf : τΔ = .1

pf

Obtain the unknown complex coefficients kMk CC ,,1 ~ )~0( Nk = and

)~1( NkDk = using an approximation method developed for IIR digital filter.

Set initial value: kn

kkMn

kMknk DDCCCC === ][

,][,,1

][,1 and,� with Nn = ,

)~0( Nk = and 10 =D

Compute all poles nα , coupling angles na,θ and phase values na,ϕ )~1( Nn =

From Eq. (12) ~ (13)

?0>n

Calculate the circuit parameters by Eq. (A.1 ~ A.2)

Calculate the circuit parameters by Eq. (A.3 ~ A.4)

1−= nn

Calculate the expansion coefficients using Eq. (B.1)

)~1(,and)~0(~,~ ,,,1,1,1,1 NnNn nananMnnMn ==−− ϕθϕϕθθ are obtained.



Fig. 3. Power frequency response of a two-port fifth order optical elliptic filter. middle band expansion coefficients. Calculated circuit parameters values are listed in Table (2b). It is seen that the last two stages of this design example approaches FIR type that is synthesized by the synthesis algorithm [5] [6]. However, the rest part is synthesized by the synthesis algorithm presented in this paper. Fig. 4 shows the synthesized power frequency response of a three-port IIR optical interleave filter. The 1 dB-down bandwidth of the passband is approximately 0.31, the transmittance at the stopband is less than -30 dB. Presented design example also satisfies the law of energy conservation.

Table 2. (a) Middle band numerator and denominator amplitude expansion coefficients of a three-port optical IIR interleave filter.

stage number

Numerator amplitude expansion coefficient

Denominator amplitude expansion coefficient

0 1 2 3 4 5 6 7 8 9 10 11

5.6700e-002 1.6740e-001 2.7010e-001 4.0770e-001 4.9730e-001 5.1730e-001 4.6030e-001 3.4970e-001 2.5020e-001 1.3560e-001 4.5900e-002 2.8900e-002

1.0000e+000

0 0

1.4830e+000 0 0

6.3010e-001 0 0

6.9200e-002 0 0



Table 2. (b) Calculated circuit parameters of a three-port optical IIR interleave filter.

St no.

coupling angle

πθ ×k,1

phase value

πϕ ×k,1

coupling angle πθ ×k,2

phase value πϕ ×k,2

coupling angle

πθ ×ka,

phase value

πϕ ×ka,

0

1

2

3

4

5

6

7

8

9

10

11

3.0347e-001

3.2867e-003

1.8678e-003

4.9836e-001

3.1891e-001

2.8650e-001

2.2640e-001

2.5942e-001

2.7590e-001

3.2821e-001

3.2113e-001

3.0409e-001

7.4554e-001

-3.5606e-001

-3.3728e-001

2.6981e-001

2.1084e-001

-2.2671e-001

1.5801e-001

-9.4886e-001

7.4405e-001

6.6667e-001

1.6667e-001

-3.3333e-001

4.7281e-001

7.3363e-002

2.6135e-001

4.9914e-001

9.1548e-005

7.4444e-003

1.2943e-002

1.9099e-014

8.8097e-003

4.5279e-003

2.5000e-001

2.5000e-001

-9.0221e-001

3.6813e-001

-9.5840e-001

-8.2009e-001

5.4852e-001

-8.2466e-001

7.2178e-001

5.9524e-001

1.0000e+000

-1.0000e+000

-5.0000e-001

-1.6667e-001

----------

3.1306e-001

3.1306e-001

3.1306e-001

2.0788e-001

2.0788e-001

2.0788e-001

1.1727e-001

1.1727e-001

1.1727e-001

5.0000e-001

5.0000e-001

---------

3.3333e-001

-1.0000e+000

-3.3333e-001

3.3333e-001

-1.0000e+000

-3.3333e-001

3.3333e-001

-1.0000e+000

-3.3333e-001

0

0

External phase shifter value )( exϕ : π×− 0018121.7 e

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-40

-35

-30

-25

-20

-15

-10

-5

0

5

relative frequency

atte

nuat

ion

[dB

]

Fig. 4. Power frequency response of a three-port optical IIR interleave filter. In practically fabricated optical filters optical waveguides inevitably exhibit propagation loss which is less than 1.0 dB/cm [9,10]. Therefore, the impact of waveguide loss is examined for the proposed optical IIR interleave filter as shown in Fig. 5. The passband transmission decreases proportional to the feedback path loss, but the stopband response is maintained and the filter spectra are in fairly good agreement with the ideal one. With current PLC technology, the minimum allowable diameter for ring waveguides is about 3 mm. It can be estimated that the realizable maximum frequency period is about 20 GHz. The proposed IIR

)(~

)(

)(~

)(''

''

11ωω

ωω

jj

jj

eQeQ

eReR

)(~

)(

)(~

)(''

''

22ωω

ωω

jj

jj

eQeQ

eReR

)(~

)(

)(~

)(''

''

33ωω

ωω

jj

jj

eQeQ

eReR



optical filter is capable of higher-functional optical processing in a small number of stages because of the feedback effect.

Fig. 5. Impact of waveguide loss on the attenuation spectrum of a three-port optical IIR interleave filter.

5. Conclusion

This paper presented a one-input M -output )1( M× circuit configuration for realizing optical

IIR lattice circuits with M -output channels )2( ≥M . The circuit configuration has a multilayer structure composed of MZIs with delay time difference τΔ for designing output responses. Furthermore, synthesis algorithm is also briefly discussed. It is expected that they will be employed as various optical filters and optical adaptive filters in FDM and WDM optical communication.

Appendix A: Equation of circuit parameters

When 0>n , the coupling angles and phase values can be obtained as follows,

{ }

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎭⎬⎫

⎩⎨⎧ ∗

−∗∗∗++∗∗+∗

=

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ +

=

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

=

−

∗−−

∗−−

−

−−

)(

)(][,1,4,3,2)(][

3,3,3,2)(][2,2,2

arg

)(

)()(arg

)(

)(arg

][1

,1

][2

][,1

][1,1,1

,2

][1

][

,1

nnn

nn

M

nn

MnMnn

MnMnMnM

nn

M

nn

MnM

R

nn

MRnMKnKnKnKnnRnPnKnKn

nRnPnKj

R

RKRPKj

R

Rj

α

αααϕ

ααα

ϕ

ααϕ

��

��

(A.1)



{ }

{ }

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎭⎬⎫

⎩⎨⎧ ∗

−∗∗∗++∗∗+∗

=

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ +

=

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

=

−

−

−∗

−−∗

−−−−

−

−−−

)(

,1)(][,1,4,3,2)(][

3,3,3,2)(][2,2,2

tan

)(

)()(tan

)(

)(tan

][1

1,1

][2

,2][

,1][1,1,11

,2

][1

,1][

1,1

nnn

nn

M

nMnn

MnMnn

MnMnMnM

nn

M

nMnn

MnM

R

nPnnMRnMKnKnKnKn

nRnPnKnKnnRnPnKj

R

PRKRPKj

R

PRj

α

αααθ

ααα

θ

αα

θ

��

��

(A.2)

When 0=n , the coupling angles and corresponding phase values can be obtained as follows,

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧ ∗−

∗∗∗++∗∗+∗=

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ +

=

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

=

−

∗−−

∗−−

−

−−

]0[0,1

)]0[0,0,10,40,30,2

]0[0,30,30,3,2

]0[0,2,20,2(

arg

)(arg

arg

0,1

]0[0,2

]0[0,0,1

]0[0,10,10,1

0,2

]0[0,1

]0[0,

0,1

C

MCMKKKKCPKnKCnPKj

C

CKCPKj

C

jC

M

MMMMMM

M

MM

��

��

ϕ

ϕ

ϕ

(A.3 )

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧ ∗−

∗∗∗++∗∗+∗−=

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ +

=

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

=

−

−∗

−−∗

−−−−

−

−−−

]0[0,1

0,1)]0[0,0,10,40,30,2

]0[0,30,30,30,2

]0[0,20,20,2(

1tan0,1

)(tan

tan

]0[0,2

0,2]0[0,0,1

]0[0,10,10,11

0,2

]0[0,1

0,1]0[0,1

0,1

C

PMCMKKKKCPKKCPKj

C

PCKCPKj

C

PjC

M

MMMMMMM

M

MMM

��

��

θ

θ

θ

(A.4)

Appendix B: Calculation of expansion coefficients

Comparing the corresponding terms with the same order of z , the )1( −n th stage expansion

coefficients ]1] −nkD )1~1( −= nk of )(]1[ zQ n− and )1~0(~ ]1[

,]1[

,1 −=−− nkCC nkM

nk of

)(~)( ]1[]1[1 zRzR n

Mn −− can be attained as [10],

{ }][1

]1[1

]1[ 1 nk

nk

n

nk DDD +

−+

− −=α



{ }

{ }

{ }

( ){ }][1,,1,3,2,1

][1,2,2,2,1

][1,1,1,1,

]1[1,1

]1[,1

][1,,1,3,2,1

][1,2,2,2,1

][1,1,1,1

]1[1,

]1[,2

][1,,1,2

][1,1,1,1,2

][1,2,2,2

]1[1,1

]1[,1

][1,,1

][1,1,1,1

]1[1,

]1[,

1

1

1

nkMnMnnn

nknnn

nknnna

nkn

nk

nkMnMnnn

nknnn

nknn

nkM

n

nk

nkMnMnM

nkMnMnMnM

nkMnMnM

nkM

n

nkM

nkMnM

nkMnMnM

nkM

n

nkM

CKKKKCPKKCPKPCC

CKKKKCPKKCPKCC

CKKCPKKCPKCC

CKCPKCC

+∗

−∗∗∗

+∗∗

+∗∗−

+∗−

+∗

−∗∗

+∗

+∗∗−

+−

+∗

−−+−∗

−−−+−∗

−∗

−−

+−−−

+−+−∗

−∗

−−

+−

+++−=

−−−−=

−−−=

−−=

��

��

��

α

α

α

α

(B.1)

However the terms ]1[1−

+n

kD and ]1[1,

]1[1,1 ~ −

+−+

nkM

nk CC in Eq. (B.1) will be ignored for 1−= nk .



synthesis of a multichannel lattice-form optical delay-line circuit with ring waveguides

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