eralised analysis of double-coupler doubly- esonant optical circuit

L.N.Binh

Indexing terms Coupled waveguides, Optical couplers, Ring resonator, Signal flow graph

Abstract: A generalised analysis of doubly- resonant ring resonators using a double-coupler structure with optical feedback paths, in which lightwaves travelling in opposite directions are coupled with reverse-wave feedback optical waves, is described. The double-coupler doubly- resonant optical circuits are shown to exhibit unique resonant characteristics, with the resonance degeneration coincidentally leading to a significantly sharp optical pass or depleted band for narrow bandpass or notch filtering, respectively. Pole and zero patterns of the transmitted and reflected output transfer functions are used to analyse the resonance and interference of the resonator. Thus tunable and switchable optical resonators are proposed.

1 introduction

In recent years several theoretical and experimental studies of optical fibre resonators [l-51 have been reported with practical applications to optical commu- nications as well as optical signal processing. One spe- cial optical resonant circuit is the double-coupler S- shape [l], sometimes defined as the Yin-Yang, Specta- cles [2] or Z-shape configurations, are of great interest because it exhibits both transmitted and reflected opti- cal characteristics. This type of optical resonator con- sists of two couplers of different coupling coefficients and their input and output ports are interconnected in such a way that its geometrical shape would have the form of an S-shape or, Z-shape or the well known Yin- Yang symbol in Chinese philosophy as shown in Figs. 1 and 2, respectively. The Yin-Yang symbol has been considered over the last two thousand years to repre- sent a balanced system in a body or natural system. In both configurations there is one ring formed by two optical paths through two optical couplers 1 and 2. It is the length of this ring as compared with that of the delay feedback path inside the ring that would make the shape of the resonator to be an S-, Z- or Yin-Yang shape. To this end, the optically resonant system inves- tigated in this paper is considered as a balanced optical 0 IEE, 1995 IEE Proceedings online no 19952241 Paper first received 23rd March 1995 and in revised from 17th July 1995 The author is with the Laboratory for Optical Communications and Applied Photonics, Department of Electrical and Computer Systems Engineering, Moiiash University, Clayton, VIC 3 168, Melbourne, Aus- tralia

296

system in a geometrical sense, as that of a Yin-Yang structure.

coupler 1 - k b r e delay 1 L - 7 fibre delay 3 )

I 1

coupler 2 Fig. 1 Schematic diagram of reverse-coupled optical resonatov using S- shape double coupler

coupler 1 I

doubly resonant ring

coupler 2 Fig. 2 Schematic diagram of reverse-coupled optical resonator using Yin- Yang double coupler

In previous works [l, 2, 5, 61 the classification of these types of resonator is not clear. In fact, they all have a common feature that the resonant structure is formed by using a double-coupler structure with opti- cal delay feedback paths interconnecting certain sec- tions of the ring resonator. Lightwaves thus propagate through the two common resonant rings in opposite directions. Another configuration of such a resonator can be schematically represented in Fig. 3 which has a figure-8 shape formed by twisting the interconnecting optical paths between the ports 2, 4, 5 and 7 of two optical couplers C1 and C2. Further explanation of this figure is given in the following Section. This resonator can be reconfigured to transform it to be a Yin-Yang symbol by twisting the loops 2, 4, 5 , 7 such that it is a planar ring, i.e. it has no other path crossing over, by turning the two couplers upside down. The feedback path 3, 6 is thus inside this ring and the figure becomes a Yin-Yang-like symbol. Similarly, the figure-of-8 reso- nator can be transformed to an S- or Z- shape configu- ration.

IEE Proc -0ptoelectron , lfol 142, No 6, December 1995

The S-shape optical resonator has been investigated theoretically [ 11. However, the optical field representa- tion and resonance conditions have been obtained by solving a series of interrelated optical field equations. The disadvantage of this technique is that although the final equations for the resonance condition are obtained, users cannot inspect the behaviour of the res- onance system. They thus would not be able to visual- ise potential applications of the resonators. On the other hand, the recently developed technique employ- ing a graphical representation in the form of a signal- flow graph (SFG) of the optical resonance circuits not only eases the derivation procedure but also enhances understanding and enables identification of the per- formance of the resonators [6, 71. Further, the SFG representation allows the illustration of the paths of travelling lightwaves, in particular through a direc- tional coupler with counterpropagating lightwaves. However, these studies have not identified an impor- tant class of optical resonator in which the transfer functions of the optical system exhibit common double complex conjugate poles (or resonance), that is when the operating optical phases fall at the optical reso- nance or depletion of the optical energy circulating in the optical system. Our theoretical analysis would also give an insight of the resonance conditions observed in PI.

c2 Fig. 3 Schematic diagram of reverse-coupled optical resonator using fig- ure-8 double coupler

This paper presents a complete and generalised anal- ysis for an optical doubly-resonant circuit using a dou- ble-coupler structure as shown schematically in Fig. 3, hereafter denoted as DCDROC. The DCDROC is formed by two optical couplers with optical feedback paths forming a twisted figure-of-8 loop 4, 7', 5', 2' and 7, 2, 4, 5 and an optical feedback path connecting lightwaves in the reverse direction to the doubly reso- nant cavity, denoted by nodes 2, 5, 7, 4. The significant identity of these resonators is a common resonance loop in which lightwaves travel in contradirectional directions with each other. The length of the optical feedback paths compared to that of the halfresonance loop length determines the shape of the resonator.

A graphical representation of the resonator is described and an analytical technique employing the z- transform method is given in Section 2. Identification of the resonance conditions such as the optical narrow bandpass filtering and notch filtering are the main fea- ture of this paper. The following Section gives the sig- nal flow graph (SFG) analysis based on the graphical representation technique in [6]. Analytical solutions are given and potential applications of the resonator under different operating conditions are identified in Section 3 . The pole-zero patterns of the z-transform transfer functions have been used throughout the paper to sys-

IEE Proc -0ptoelectuon , Vol 142, No 6, December 1995

tematically identify the main feature of the DCDROC. This technique is analogous to that commonly used in digital signal processing or control theory [8, 91.

(1 -k2)' " 71---- 11 I ^ .

6' * ( t3G,%-d3

Fig. 4 Graphical representation of yeverse-coupled optical resonafoi

2 Signal flow analysis

2. I Structure and graphical representation The graphical representation of the DCDROC of Fig. 3 is shown in Fig. 4. Two couplers C1 and C2 with field coupling constants kl and k2 are used as the coupling elements in either direction. Optical guided waves can thus travel in both directions. Assume that there is no interaction between these counterpropagat- ing lightwaves. It is also assumed that the couplers are lossless and symmetric. An excess loss factor can be included. However, for the sake of simplicity, this fac- tor is assumed to be zero. Ports 2 and 4 of the coupler C1 are connected by optical delay lines to ports 5 and 7 of the coupler C2, respectively. Due to the bidirectional transmission of the coupler, each coupler is represented by two SFG planar diagrams. For example, coupler CI can be represented by two SFG graphs 1-3-2-4 and 1'- 2'-3'-4' and similarly 5-6-7-8 and 5'4-7'43' for coupler C2. The dashed numbers indicate the reverse propaga- tion direction in a coupler. The optical feedback path is shown as the connection between ports 3 and 6 of the two couplers. It is assumed that the coupling length of the optical couplers C1 and C2 are much shorter that that of the interconnection optical paths and thus coherence condition is hold. The optical fields are denoted with a prime field if it is in the reflection mode. Furthermore, the coupler is assumed to be sym- metrical, hence the coupling transmission coefficients j(kl)1/2 and -j(k2)1/2 in Fig. 4 indicate the cross coupling coefficients of the optical fields between ports of the two couplers. The z-transform parameter z is given by z = exp(j0) where 0 = PL = oT with P = np lc is the propagation constant of the guided fundamental mode, and nf is the effective refractive index of the guided mode propagating in the optical guided wave medium, Tis the propagation time of the lightwaves through the guide length L, and o is the optical radial frequency in vacuum. It is further assumed that the optical waveguide used in the DCDROC is single mode and can be circular of channel types. Only one polarised optical guided wave is considered.

The polarisation properties of the two counterpropa- gating lightwaves are important to satisfy the balance conditions of the DCDROC. If the propagation con-

291

stants of the two polarised optical fields are slightly dif- ferent, the resonant loop 2, 4, 7, 5 would experience two different resonance characteristics and thus the doubly-resonant condition would no longer hold. In this case, inline optical polarisation controllers must be incorporated in the three delay paths 1, 2 and 3 with an appropriate polarisation control mechanism to ensure that the circulating, transmitting and reflecting optical waves are in the same polarisation. Alternatively polar- isation preserving fibres should be used to achieve the same polarisation of the lightwaves. If the DCDROC is implemented in an integrated optical structure, e.g. sil- ica-based planar circuit fabricated by molecular organic chemical vapour deposition, or a LiNbO, diffused or proton exchanged planar system, then the polarisation of optical waves is preserved.

The total optical gain transmittance denoted as tiG, ( for i = I , 2, 3 with t , as the optical transmittance of the medium and G, as the optical gain or loss) of the DCDROC can be greater than unity if inline optical amplifiers are used. Thus optical amplifiers with a bidi- rectional amplification gain must be employed. Such an amplifier can be an Er- (for 1550nm) or Pr3+- (for 1300nm) doped optical amplifiers in glass [lo-121 or in Er-doped lithium niobate [13]. The optical systems are operating in either coherent conditions, thus optical fields are used throughout and the optical gain G, can take either positive or negative values.

2.2 Optical transfer functions The optical transfer functions for the input-output field H,, (E8/E1) and H1(l (defined as the ratio between the reflected output E,,, and the input field E,) can be obtained by using the Mason rules which can be found in the Appendix. Mason rules have been developed for electrical circuits. They are completely valid for appli- cation to optical circuits because they were derived for the general case of interconnecting circuits represented in graphical form in which the transmittances of paths of the graph [14] are employed. Mason rules were orig- inally derived by using the well known matrix transfer technique in electrical circuit theory. The optical vet- sion of Mason rule is given in the Appendix. This rule has been applied for the first time in [l] and recently it has also been used in [15]. For the sake of clarity it is essential that this rule for the derivation of the optical transfer functions is given again for the optical fields rather than the optical intensity version in [l].

Now apply the optical Mason's rule as outlined in the Appendix to the graph of Fig. 2 to find the optical transfer function between input and output ports 1 and 8. Using eqn. 16 for the five possible optical transmis- sion paths connecting the optical input signal from port 1 to the output port 8, the transmittances for these paths are given by: path 1 : 1 - 3 - 6 - 8

with (P18)l = (K1112)1/2(t3G3)1/22-d3 (la)

with (Pls)2 = - K ~ k 2 a 3 x - D 3 ( I b )

with (P18)3 = - k1K2u3~-D3 ( I C )

with (P18)4 = k1k2a32-D3 (14

path 2 : 1 - 3 - 6 - 7 - 4'

path 3 : 1 - 4 - 7' - 5' - 2 - 3 - 6 - 8

path 4 : 1 - 4 - 7' - 6'- 3 ' - 2' - 5 - 8

298

path 5 : 1 - 4 - 7' - 5' - 2 - 3 - 6 - 7- 4' - 2' - 5 - 8

with (P18)S = IC1 IC2 (KLK2) a2a3z-(D2+D3) (14 The numbers indicate the node order and the connec- tivity of the path is indicated by nodes placed in the anticlockwise direction. There are also two optical loops whose gain transmittances are

l o o p 1 : 2 - 4 - 7 ' - 5 ' - 2

with P11 = (K1K2)1/2a2~-D2 (2a)

loop 2 : 5 - 7 - 4' - 2' - 5

with Pz1 = (K1K2)1/2a2z-DZ (2b) where: K, = l-kl; i = 1, 2, ...

'rk

D , = Cdi n = 1 , 2 , 3 (3b ) i=l

The optical loop gains of loops 1 and 2 are equal because of the symmetrical structure of the DCDROC, provided that any perturbation on the common ring would affect equally on the phase of the propagating lightwaves in both directions. The optical loops are completely overlapped, hence this loop can be consid- ered as a quasiform of the well known optical Sagnac loop. Again, this is true only if the bidirectional optical loops are completely balanced. Further, the bidirec- tional transmission of the signals are feedback with the reverse optical waves in couplers C1 and C2. Therefore there is one possible product of two nontouching loop gains (PI 1P21) given by

(4) P12 = Pl1P,, = K1K2a2z 2 - 2 0 2

The graph determinant is thus

which does not depend on the value of t , and G,. The transmittance of the optical path connecting port 1 of coupler Ci to port 2 of coupler C2 does not contribute to the graph determinant. The roots of the graph deter- minant are the poles of the optical system or equivalent to the resonant frequencies of the optical resonator.

The cofactor (A,,), of the transmitted transmission paths are given by: (Al& A since path 1 does not touch any loop; (Al& = l-Pll = 1-(K1K2)1/2a2~-D2 since path 2 touches loop 2; (AI& = 1-P21 = 1-(Kl,K2)1/2a2z-D2 since path 3 touches loop 1; (Al& = 1 since path 4 touches loops 1 and 2; = 1 since path 5 touches both loops and 2. The input-output system transfer function is therefore given by using eqn. 16:

5

2= 1 a HlS =

(Pls)l(nlS)1 + (P18)2(a18)2

- + (p18)3(n18)3 + (p1S)4(n18)4 + (plS)5(a18)5 - n

( 6 ) substituting eqns. 1-4 and cofactor 5 ) corresponding to each path into eqn. 6 gives

(i = 1, 2, ...,

IEE Pvoc -0ptoelectron Vol 142, No 6, Decembev 1995

In deriving eqn. 7 it is assumed, as mentioned previ- ously, that the doubly-resonant loop experiences the same phase perturbation. The effects of different phase perturbations will be reported in another paper to fea- ture a new effect compared with the renowned Sagnac effect.

Similarly, by following the same procedure described, the reflected transfer function (ElrIEl) for optical input signal reflecting from the input port (port 1) can be obtained as

where

(9a)

(9b)

a2a3 a6 = ~

a3 a5 = -

a1 a1

0 5 = 0 3 - 01 Rearranging eqn. 7 gives the transfer function for the transmitted output

0 6 = 0 2 + 0 3 - 01

BS HI* = - El

2.3 Zeros and poles for resonance and destructive interference By inspection of eqns. 7, 8 and 10, it is clear that the system poles and zeros are dependent on the product of k l and k2, tl and t2, and G1 and GZ. Therefore the loca- tion of inline optical amplifier G1 and G,, tl and t2 or the intensities coupling coefficients k l and k2 of the two couplers can be interchanged without affecting the res- onance conditions of the resonator, provided that the resonator is balanced, that is the order of the delays for the two paths of the double resonant ring are equal and the polarisation of the propagating lightwaves are identical.

If there is a perturbation of the phase, the transmit- tance or gains in the common resonant loop with a dif- ference in the two counterpropagating directions, the resonator performance would be different with the ana- lytical results described here. We will present this case in another paper that will describe the sensitivity of the resonator for such applications as an optical gyroscope, reversed coupled lasers, etc.

The poles and zeros of the system are not affected by the transmission path parameters t3, G3, and d3. Hence they can be used for adjusting the amplitude of the res- onance maxima of the magnitude frequency response. The rest of this Section examines the patterns of the poles and zeros of the DCDROC to determine the con- ditions for resonance and depletion of circulating light- waves travelling in the circuit.

IEE Proc.-Optoelectron., Vol. 142, No. 6, December 1995

2.3. I Zeros of the optical system: The zero loca- tions on the z-plane of the transmitted output transfer function HI* (eqn. 10) are

and z - d 3 = 0 (11)

(12) Eqn. 12 is a quadratic equation of the term a2z-(d1+dz). This quadratic relationship would permit flexible oper- ation of the DCDROC by forming different patterns of positions of the zeros of this transfer function relative to that of the poles in the z-plane. The zeros occur as complex conjugate pairs depending on the delay orders dl and d2 and on whether the roots of this equation are real or complex. Zero locations of the reflected transfer function HItl are

and (13) z,(d2+d3) = 0

2.3.2 System poles: The poles of the system are identical for all optical transfer functions and their locations in the z-plane is determined by the roots of the graph determinant A. This can easily be observed in the DCDROC and indeed for general optical systems because the poles correspond to the storage of energy in a physical system. In DCDROC this is translated as the optical energy preserved and stored in the doubly- resonant loop. As seen from the derivation of the opti- cal transfer functions using Mason rules, the graph determinant is common for all functions. The graph determinant given by eqn. 5 is a perfect square thus the system poles are doubly degenerate, i.e they occur as coincident pairs given by

2.3.3 Interrelationship between transmitted and reflected optical transfer functions: By observing the optical transfer functions of eqns. 10 and 8 for H18 and HI,, , respectively, note the following

0 The order of the zeros of the reflected transfer function eqn. 8 is the same as that of the graph deter- minant eqn. 5, except it has an additional number of zeros at the origin of the z-plane. Thus these zeros are distributed around the z-plane [8] along the radial direction in the same manner as the system poles. The radius of the zeros and poles can thus be tuned accord- ingly for specific applications of the DCDROC.

0 The zeros of the transmitted output function eqn. 11 are located at the origin and those given by eqn. 12 are dependant on the solutions of the quad- ratic equation. Thus the phase angle of these zeros can be tuned by selecting appropriate parameters of the res- onators. The placement of these zeros relative to the system poles is significant as seen in the following Sec- tion.

0 The ratio between the radial position of the poles and zeros of the transmitted output function is (l-kl), thus k , can be used to tune the distance between the pole and zeros so that an optical passband (at reso- nance) or notch filter (at depletion or zero) can be real-

299

ised by assigning a unity value to the pole or zeros in the z-plane. The coupling coefficients k l and k2 can be alternately varied to be close to or moving apart from the pole position to form a Butterworth- or Cheby- shev-like optical bandpass or notch filters. e A special case when k l = k2 = 0.5 would lead to k l

= k2 = Kl = K2 and thus the DCDROC becomes com- pletely balanced. To depart from this condition, such as a fluctuation of the phase or magnitude of the opti- cal field, would lead to fast switching from one reso- nant condition to the other. This case can certainly be of interest for employing the DCDROC as a sensing device or as an optical switching circuit.

3 Results and discussions

The DCDROCs analysed in this paper can be trans- €ormed to various configurations such as the Yin- Yang, the Z- or S-shape or a twisted figure-of-8 ring resonator illustrated in Figs. 1-3.

300 fi

0 2 & 6 8 UT, rad

Fig. 5 Response of reverse-coupled resonator magnitude Eg/E1 against unit frequency delay

200 r I I 1 1

0 2 4 6 8 o T , rad

Fig. 6 Response of reverse-coupled resonator phase response

1 0

.r" 0.5 3

-1 0 - 2 -1 0 1 2

real axis Fig. 7 Response of reveise-coupled resonator pole-zero plot in z-plane

ring generating a balanced flow of lightwaves if there is no perturbation in its phase and amplitude. A unit delay is assigned for all these paths without losing any generality of the analysis. The main philosophical aspect of the Yin-Yang is that of a naturally balanced flow. As observed from the configuration of the Yin- Yang resonator, the optical ring paths are balanced by the interconnecting feedback path 3-6. Geometrically speaking, to f o m a Yin-Yang resonator all propagst- ing lightwaves paths must follow in harmony.

The twisted figure-of-8 DCDROC resonator ana- lysed in this paper is in fact a mirror of the Yin-Yang provided that the couplers are symmetric. The signifi- cant difference in this analysis is that generalised and balanced aspects of the resonators have been systemati- cally analysed leading to a generic grouping of all reso- nators that exhibit a double-resonant ring with counterpropagating lightwaves.

The Yin-Yang DCDROC configuration degenerates to a Z-shape [7] or S-shape [I] when the delay path d3 is shorter or longer than that of the two equal delay paths of the outer loop. When the delays of the two paths of this loop are not equal then one would have a distorted Yin-Yang resonator. In this paper only the symmetrical Yin-Yang configuration is considered since the analysis of the other two configurations is straightforward.

In the balanced DCDROC the orders of delays d l , d2 and d3 are set to unity. The DCDROC can be set to operate under three cases: (i) as a pure narrowband fil- ter in both the transmitted and reflected modes, (ii) as a narrow bandpass filter in the transmitted output and as a notch filter in the reflected output and (iii) as a tunable bandpass filter.

3. I Case (i): bandpass filtering in transmitted and reflected outputs The resonator can operate under resonance when the poles of the system are positioned on unit circle of the z-plane. In this case, regardless of the transmission and gain coefficients of the path delay d3 as observed from eqns. 13 and 14, the amplitude-frequency response (in arbitrary units) of the transmitted and reflected outputs exhibit a sharp transmission peak as shown in Fig. 5. The phase response and the pole-zero positions in the z-plane are shown in Figs. 6 and 7, respectively. The phase response is consistent with the pole and zero location, that is a change of n: for each time the optical phase w T rotates through a pole position. The magni- tude is expressed in arbitrary units.

A Yin-Yang resonator is defined in this paper as the resonant circuit in which the delay length of the three transmission paths have the same length, that is dl = d2 = d3 and the optical gain and transmittance of its two optical paths are equal to form a balanced system. The dynamic aspect of the Yin-Yang is that two counter- propagating lightwaves are circulating in the resonant

300

wT, rad Fig. 8 Optical notch filtering using k l = k2 = 0 5, tlG1 = t2G2 = t3G3 = 1 0 and unit delay in all optical paths magnitude response, E I ~ I / E I

3.2 Case (ii): optical notch filtering Pole placement is such that the overall transfer func- tion would have resonance peaks andlor the depletion

IEE Proc -0ptoelectron , To1 142, No 6, December 1995

dips by the zeros of the transfer function. The optical system is tested with a set of parameters of kl = k2 = 0.5, tlGl = 1.0 and t2G2 = t3Gg = 1.0. The zeros of the transmitted and reflected optical functions are thus placed on the unit circle of the z-plane as shown in Fig. 10. Its corresponding optical field magnitude and phase responses are shown in Figs. 8 and 9. The magni- tudes of all responses are in arbitrary units. The trans- mitted and reflected optical fields are depleted at the frequency corresponding to those of the zeros.

1 .o

aJ V

'c 0.5 (31 z

200 I I I I I

I

i j

I ........ j ................................. ! ... ............. ; .................. .

i I I

wT, rad

Fig. 9 Optical notch filtering using k l = k2 = 0.S, tlG1 = t2G2 = t3G3 = 1.0 and unit delay in all optical paths: phase response

1

-1 - 2 -7 0 1 2

real axis Fig. 10 Optical notch filtering using k l = k2 = 0.5, tlG1 = t2G2 = t3G3 = 1 and unit delay in all optical paths: pole-zero plot in z-plane

0 1 I I

0 2 4 6 8 wT, rad

Fig. 11 Notch filtering effects for k l = k2 = 0.15, t lGl = t2G2 = t3G3 = 1: nzagnitude plot, Eg/EI

-1 0 - 0 5 0 0 5 10 real axis

Fig. 12 Notch filtering effects f o r k ] = k2 = 0.15, t lGl = t2G2 = t3G3 = 1. pole-zero plot in z-plane, Eg'E]

This case is special because the coupling constants kI = k2 = Kl = K2 leading to two zeros in the reflected output transfer function positioned on the unit circle cancelling two poles of the doubly-degenerating poles. When the coefficients k, and k2 are set to be equal but smaller or greater than 0.5 the zeros of the reflected transfer function are still placed on the unit circle of the z-plane. The zeros of the transmitted output trans- fer function are also located on the unit circle. How- ever, they are complex and the phase angles are symmetric. Their values are controlled by these cou- pling coefficients if the gain and transmittance factors are kept to unity.

01 I I I I 0 2 4 6 0

wT, rad Fig. 13 Notch filtering effects for kl = k2 = 0.15, t lGl = t2G2 = t3G3 = 1. magnitude plot, E11/E1

-1.0 -0.5 0 0.5 1.0 real axis

Fig. 14 Notch filtering effects for kl = k2 = 0 15, t lGl = t2G2 = t3G3 = 1. pole-zero plot in z-plane, EII/EI

Figs. 11-14 demonstrate the notch filtering effects for the case where k l = k2 = 0.15 and tlGl = t2G2 = t3G3 = 1. The transmitted output is depleted at optical fre- quencies corresponding to its zeros located in the vicin- ity of z = k 1. The reflected output (or sink node 1') is also depleted at z = f 1 and has a passband filtering at the notch filtering frequencies of the transmitted waves. The pole and zero pattern in the z-plane are shown in Figs. 12 and 14.

It can be easily seen that the zeros of the transmitted output transfer function are moving apart from the pole and zero locations z = k 1 as the coupling coeffi- cients kl and k2 with k , equals to k2 take a value greater than 0.5. Thus one can tune frequencies of the optical notch filter by tuning the coupling coefficients of the two identical couplers. This can be implemented by two couplers in an integrated structure and operat- ing it in a push pull or balanced mode.

The DCDROC can be designed for notch filtering operation with unequal coupling coefficients kI and k2 together with corresponding transmittances and optical gains in all optical paths of the DCDROC in the fol- lowings.

When the zeros of the systems are designed to be close to each other, it is required that the roots of the quadratic eqn. 12 are complex with small imaginary

IEE Proc.-Optoelectron., Vol. 142, No. 6, December 1995 301

parts. Furthermore, when the zeros of the reflected out- put transfer function are placed on unit circles shown in Figs. 16 and 18, the system would exhibit a wide- band notch filtering effect and sharp central optical

-1

1 oo

ill V 3 c .-

k E

10-1 0 2 4 6 8

wT, rad Fig. 15 Tuning optical passband ofnotchfiltering, using k l = 0.23, k 2 = 0.05, t lGl = t3G3 = 0.98, t2G2 = 0.94: output magnitude, E8/Ef

-1.0 -0.5 0 0.5 1 .o real axis

Fig. 16 Tuning optical passband of notch filtering, using kl = 0.23, k2 = 0.05, tlG1 = t3G3 = 0.98, t2G2 = 0.94: pole-zero plot in z-plane of trans- mitted response

0 10

aJ U

.- -1 % I O

E ............... ; .................. ; ................. ; ............... .................................. _. ................................ E r ................................... ; ................. ; ...............

10-2 I I , I I 0 2 4 6 8

UT, rad Fig. 17 Tuning opiicalpassband of notchfilfering, using kl = 0.23, k2 = 0.05, tlG1 = t3G3 = 0.98, t2G2 = 0.94: reflected magnitude, El-/El

- 1 0 I 2 real axis

Fig. 18 Tuning optical passband of notch filtering, using k l = 0.23, k2 = 0.05, tlG1 = t3G3 = 0.98, t2G2 = 0.94: pole-zero plot in z-plane of reflected response

notch filtering on the transmitted and reflected responses, as shown in Figs. 15 and 17, respectively. The parameters used in this operational mode of the DCDROC are kl = 0.23, k2 = 0.05, tlGl = 0.98, t2G2 = 0.94 and t3G3 = 0.98. In this case the optical flow inten- sity as determined by the transmittance tl , t3 and opti- cal gains GI, G3 is unbalanced on the two optical paths (2-4-7 and 7-5-2) of the optical double ring resonator.

If the zeros of the transmitted output function are then designed to move away from each other, as shown in Fig. 20 with the zeros of the reflected function are still close and just outside to the unit circle as in Fig. 22, the optical passband of the notch filtering response of the transmitted waves displays a ripple of less than l.OdB and a sharp filtering in the reflected waves as shown in Figs. 19 and 22. The optical system is still stable as the poles are inside the unit circle of the z-plane. This approach is similar to the design of a But- terworth optical notched filter as described in [16]. However, it gives a much clearer understanding of the behaviour of the poles and zeros of the system.

I t I 1 0 2 4 6 8

wT, rad Fig. 19 Tuning optical passband of notch filteuing, using k l = 0.45, kz = 0.13, t lGl = t3G3 = 0.98, t2G2 = 0.94: output magnitude,

-1.0 -0.5 0 0.5 1.0 real axis

Fig. 20 Tuning optical passband of notchfiltering, using k1 = 0.45, k2 = 0.13, tlG1 = t3G3 = 0.98, t2G2 = 0.94: pole-zero plot in z-plane of trans- mitted response

l o o ............... ; .................. F .................................. ...................................................... ................ ............................. .......................................

302 IEE Proc.-Optoeleciron.. Vol. 142, No. 6, December 1995

3.3 Case (iii): tunable optical passband filtering The DCDROC can be designed to operate as a ripple filter if the zeros of the transmission output transfer function are tuned close to the pole position of the sys- tem provided that these poles are also close to the unit circle. They are also to be positioned symmetrically rel- ative to the poles. This is shown in Figs. 23-26 where the identical double-coupler configuration is used with a coupling coefficient of 0.12 and a total transmit- tance-gain product of unity for all three optical paths. The passband is now closed to form a wideband filter. The ripples in the optical passband can be designed by tuning the coupling coefficients of the couplers and cas- cading a number of the doubly-resonant circuits.

1 .o

ln 0 . 5 B ._

-1 .o - 2 -1 0 1 2

real axis Fig. 22 Tuning optical passband of notch filtering, using kl = 0.45, k2 =, 0.13, tlG1 = t3G3 = 0.98, t2G2 = 0.94: pole-zero plot in z-plane of reflected response

I I 0 2 4 6 8

wT, rad Fig. 23 Tuning optical passband filtering, with coupling coeficient 0.12, t lGl = t2G2 = t3G3 = I : magnitude, E8iEl

ring are coupled by the feedback beams split from the cavity. The interference of lightwaves at the transmitted and reflected outputs interpreted as the zeros of the transfer functions is affected by the parameters of the couplers and the lightwave paths of the doubly-reso- nant ring only. The order of the delay feedback path and its optical property generate only zeros at the ori- gin of the z-plane.

, 01 I I 1

0 2 4 6 8 wT, rad

Fig. 25 Tuning optical passbana’filtering, with coupling coefficient 0.12, tlG1 = t2G2 = t3G3 = I : magnitude, EII/El

1

.: 0 0 21 n C 07 0 .-

.& - 0

0

5

0

5

0 -1 0 -0 5 0 0 5 1 0

real axis Fig. 26 Tuning optical passband filtering, with coupling coefficient 0. tlG1 = t2G2 = t3G3 = 1: pole-zero plot in z-plane, E11/E1

The DCDROC has been studied with the cases of complete balanced with no loss in the optical transmis- sion paths in the resonator when k l = k, = 0.5, and other cases when there is an offset value from this bal- anced configuration. Furthermore, the coupling coeffi- cients are designed so that the resonators can operate as a narrow bandpass filter, notch filtering devices, and tunable passband filters. The special features of the DCDROC, in particular the reverse-coupled double resonance, allow several possible applications in phot- onic signal processing and optical communications sys- tems. If there is a nonlinear optical element in the DCDROC the optical systems would fall into bistable states and chaotic states, etc. These interesting optical circuits are under study as well as other unbalanced cases of the DCDROC.

5 References -1.0 -0.5 0 0.5 1.0

real axis Fig. 24 Tuning optical passband filtering, with coupling coeflicient 0.12, t lGl = t2G2 = t3G3 = I : pole-zero plot in z-plane, Es/El

4 Conclusions

The pole and zeros patterns of the optical transfer functions for the transmitted and reflected outputs have been used to demonstrate the resonance of the DCDROC. The double degeneracy of the poles of the DCDROC resulting from the resonance of lightwaves travelling in opposite directions creates the sharp reso- nant characteristics. Lightwaves in the doubly-resonant

1 JA, Y.H., and DAI, X.: ‘Experimental studies of an S-shape two- coupler optical fibre ring resonator’, Opt. Eng., 1994, 33, pp. 1056-1060

2 JA, Y ,H.: ‘Densely spaced two-channel wavelength-division demultiplexer using an S-shape two coupler optical fibre ring res- onator’, Appl. Opt., 1993, 32, pp. 6679-6683

3 ODA, K., TAKATO, N., and TOBA, H.: ‘A wide ,FSR waveguide double ring resonator for optical FDM transmission systems’, IEEE J. Lightw. Technol., 1991, 9, pp. 728-736

4 URQUART, P.: ‘Compound optical fibre based resonators’, J. Opt. Soc. Am. A , 1988, 5, pp. 803-182

5 CAPMANY, J., and CASCON, J.: ‘Discrete time fibre optic sig- nal processors using optical amplifiers’, I E E E J. Lightw. Technol., 1994, 12, pp. 106-117 BINH, L.N., NGO, N.Q., and LUK, S.F.: ‘Graphical representa- tion of a Z-shape double coupler optical resonator’, IEEE J . Lightw. Technol., 1993, 11, pp. 1782-1791

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NGO, N Q , and BI”, L N ‘Novel reahation of monotonic Butterworth-type optical filters using phase modulated resona- tors’, IEEE J Lightw Technol, 1994, 12, pp 827-841 MITRA, S K , and KAISER, J F ‘Handbook for digtal signal processing’ (Wiley, N Y , 1993) KUO, B C ‘Digital control systems’ (Saunders College, Ft m r Worth, 1992)

HRAD, M ‘Bidirectional fibre amplifiers’, I E E E Photonics Tech- m m nol L e t t , 1992, 4, pp 911-913 BIERLEY, M ‘Amplification in the 1300 nm telecommunication window in a Nd-doped fluoride fibre’, Electron L e t t , 1990, 26, vu 329-330

mission paths from j to node k, and A is the graph detemlnant defined as

(17a)

(17b)

a = 1 - (-1)?+1 yj-y-. BARNARD, C W , CHROSTOWSKI, J , and KAVE- = l - c P m I + p m 2 - .

= 1 - (sum of all optical loop gains) -t (sum of all gain product of two

nontouching optical loop gains) - (sum of all gain product of three

nontouching loop gains)

OHISHI, Y ‘Laser diode pumped Pr-doped and Pr-Yb-codoped fluoride amplifiers operating at 1 3pm’, Electron L e t t , 1991, 27, pp 1995-1996 SOHLER, W , and SUCHER, H ‘Optical parametric amplifica- tion in Ti-diffused LiNb03 waveguides’, AppZ Phys Lett., 1980, 37, pp 255-280 MASON, S J ‘Feedback theory - further properties of signal graphs’, Proc IRE, 1956, 44, pp 920-926 MINKOV, D ‘Flow graph approach for optical analysis of pla- nar structures’, Appl O p t , 1994, 33, pp 7698-1103 JA, U H ‘Design and characteristics of penodic Butterworth-like filters using a bow-tie-shaped optical fibre ring resonator’, Opt Eng , 1994, 33, pp 2912-2978

- where (P~k)i is the transmittance (optical gain) of the ith transmitted path from node J to node k, Pmr is the rnth possible product of nontouching optical loop gains The cofactor of the ith transmitted optical transmit- tance path (Alk), is defined as (AJk), = A evaluated with the all optical loops touching the transmitted path (P$, excluded. An optical loop is a closed-loop transmit-

Appendix: optical Mason‘s rule

This method is to find all possible transmission paths of the complex field transmitting from input port to output port and all optical loops in the system. Mason’s rule states that a linear transfer function between the independent node j and the independent node k in the optical signal flow graph can be deter- mined as

N

where N is the total number of possible optical trans-

tance path in which the optical nodes can -only touch once per traversal. Two optical loops are nontouching iE they do not have any common optical node which is defined as the branching point of optical lightwaves. The optical signal in the loop must be going in only one direction. Thus if lightwaves are travelling in both directions of an optical path, the signal flow must be represented with two paths corresponding to their direction provided that the optical system is operating in a linear regime, i.e. there is no interaction between the lightwaves and the optical medium. The advantage of this Mason’s rule technique is that a large system can be analysed with a large number of state variables.

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