project is supported by the Texas Higher Education Coordi- nating Board under Grants No. 32135-71270 and No. 32134- 71290.

REFERENCES 1. R. S. Tasgal and M. J. Potasek, “Exact Soliton Solutions in a

Periodically Twisted Birefringent Fiber,” in Integrated Photonics Research, OSA Technical Digest Series, Optical Society of Amer- ica, Washington, DC, 1992, Vol. 10, pp. 22-23.

2. M. J. Potasek and J. M. Fang, “Optical Switching for a Nonlinear Directional Coupler Including the Effects of Two Photon Absorp- tion,” in Integrated Photonics Research, OSA Technical Digest Series, Optical Society of America, Washington, DC, 1992, Vol. 10, pp. 12-13.

3. P. A. BClanger and C. Part, “Soliton Switching and Energy Cou- pling in Two-Mode Fibers: Analytical Results,” Phys. Re(,. A, Vol. 41, No. 9, May 1990, pp. 5254-5256.

4. A. K. Chan, C. K. Chui, J. Zha, and Jieren Bian, “Local Cardinal Interpolation Spline Method for Solving Coupled Nonlinear Schrodinger Equations: A Comparison with BPM,” in Integrated Photonics Research, OSA Technical Digest Series, Optical Society of America, Washington, DC, 1992, Vol. 10, pp. 224-225.

5. H. Ghafouri-Shiraz, P. Shurn, and M. Nagata, “A Novel Method for Analysis of Soliton Propagation in Optical Fibers,” IEEE J. Quantum Electron., Vol. QE-31, No. 1, 1995, pp. 190-200. G. P. Agrawal, Nonlinear Fiber Optics, Academic Press, New York, 1989. G. A. Cohen and T. M. Cohen, Mathematic Handbook, McGraw- Hill, New York, 1977. M. Du, A. K. Chan, and C. K. Chui. “A Novel Approach to Solving the Nonlinear Schrodinger Equation by the Coupled Am- plitude Phase Formulation,” IEEE J . Quantum Electron.. Vol. QE-31, NO. 1, 1995, pp. 177-182.

Receiived 11-16-95; rei~ised 1-26-96

Microwave and Optical Technology Letters, 12/2, 68-73 0 1996 John Wiley & Sons, lnc. CCC 0895-2477/96

THREE-COUPLER DOUBLE-RING FIBER RESONATOR FOR LASERS Li Wei Department of Physics University of Waterloo Waterloo, Ontario N2L 3G1, Canada

John W. Y. Lit* Department of Physics and Computing Wilfrid Laurier University Waterloo, Ontario N2L 3C5, Canada

KEY TERMS Fiber resonator, resonance characteristics, sensitivities

ABSTRACT The optimum resonating conditions for a three-coupler double-ring (TCDR) fiber resonator are presented. Special attention is paid to the circulating intensity and the reduction factor. The dependence of the resonance characteristics on the ratio of the lengths of the two rings (RoL), the coupling coeficients of the directional couplers, and the loss

Also Department of Physics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada.

factors are investigated. A comparison of the sensithities for RoL >> 1 and RoL = I is cam’ed out. 0 I996 John Wiley & Sons, Inc.

1. INTRODUCTION

The subject of compound ring resonators [l-41 has become an active research topic in recent years because of its poten- tial to extend the free spectral range (FSR). Compound resonators with a large FSR can be readily used as single- mode fiber lasers [5, 61 and as tunable filters in optical frequency division multiplexing transmission systems [7].

There are two ways to increase the FSR of a double-ring compound resonator. One is to make the lengths of the two rings nearly equal; this means the ratio of the lengths of the two rings (RoL) is almost unity. In this case, the resonator has strong vernier effects, but is also very sensitive to external disturbances, which could cause instabilities and mode hop- ping in applications for single-mode laser operation. The recent theoretical works [3] about compound resonators are focused on this case. The other way is to choose two rings with widely different lengths; this means RoL is much greater than 1. However, little works on the properties of a resonator with RoL B 1 have been published so far, although such resonator has been successfully used in making single-mode laser [8]. In this Letter, we present a three-coupler double-ring (TCDR) resonator. We study the optimum resonating condi- tion of TCDR. The dependence of the resonance characteris- tics on RoL, coupling coefficients of the directional couplers and loss factors is investigated. A comparison of the sensitivi- ties for RoL >> 1 is made. It is found that the TCDR resonator with RoL >> 1 is much less susceptible to external disturbance than that with RoL = 1, so great improvements in stability of single-mode doped fiber lasers could be achieved.

2. THEORY

The configuration of a TCDR resonator is shown in Figure 1. I t has three directional couplers and two ring cavities. The primary ring has a total length L, = I, + I, + I , , and the secondary ring has a total length L, = I , + I,; the latter serves as a subresonator for the primary ring. The two loops share a common part I,. The input energy is coupled to the primary ring by coupler C,, then to the second ring by C, and C,. The three couplers C , , C,, and C, have intensity coupling coefficients k , , k , , and k, , respectively. We assume that the directional couplers have zero length and the whole system is independent on polarization.

The normalized output field of the secondary ring is described by 191

E; jq2 ( t2 + t,A,e’*’) E: 1 + t,t,A,e”& . _ - - (1)

By using a combination of transfer and chain matrices [3, 41, we obtain the expressions of the normalized output field and the circulating field of the primary ring:

E , - q l ( r l + rl t2t3AseLFs - r,r3ApeLFP) E: 1 + t2t3AXeL6s - r , r2r3Ape“~ ’ ( 2 ) - _

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 12, No. 2, June 5 1996 73

In Eqs. (1)-(7), r, = (1 - k f ) 1 / 2 , t , = k,'l2 are, respectively, the effective reflection and transmission coefficients of the ith fiber coupler; 4, = (1 - yL)1'2 and y, (i = 1,2,3) are the amplitude transmission coefficient and the fractional inten- sity loss of the ith fiber coupler, respectively; (Y and /3 are the amplitude attenuation coefficient and the wave propaga- tion constant of the fiber, respectively; 6, and 6,, A , and A , are, respectively, the phase changes and the amplitudes in the primary and secondary rings. The imaginary j in Eq. (1) indicates a v/2 phase change in the field due to coupling.

When the two rings are resonating simultaneously, much optical energy circulates in the primary ring, and no energy comes out of C, and C, from their free ports. By using Eqs. (1) and (2) we obtain the resonating conditions:

where M and N are the resonant numbers; rn and n are positive integers; the superscript r denotes resonance.

The output and circulating intensities can be obtained from Eqs. (2) and (3):

r2r3A,)* - 4r:t , t ,A, sin2(

(13)

(9 I, = (1 + t , t ,A, - r,r2r3A,), - 4t2 t iA, sin2

+ 4rlr2r3Ap sin' (2 ) + 4r ,r2r3t2 t3AsAp sin2

The wavelength responses of the output and circulating intensities are similar to those in Reference [3] . A high-qual- ity resonator with high circulating intensity can be achieved by choosing the optimum coupling coefficients according to Eqs. (10) and (11).

According to Eqs. (8) and (9), when resonance is estab- lished in the two rings simultaneously, we have RoL =

L, : L , = M : N , the ratio of the two resonant numbers. It is obvious that the RoL must be a ratio of an even number to an odd number.

When RoL is close to unity, many spectral orders may be suppressed, resulting in a large FSR. For example, when RoL = L, : L, = 22 : 21 = 1.05, 21 spectral orders are sup- pressed by the vernier effect of the secondary ring. We define the reduction factor as the ratio of the height of the highest side peak to that of the main peak in the circulating intensity [3]. Under conditions governed by Eqs. (10) and (111, there is an optimum value of k , around 0.5 that gives the minimum rcduction factor.

As indicated above, a resonator that has two rings with almost equal lengths can achieve extraordinarily large FSR. Unfortunately, it has serious practical difficulties: difficulty in making two fiber rings with precisely an exact RoL, and difficulty in eliminating the high environmental sensitivity. A more realistic resonator is one the secondary rings that is much shorter than the primary ring (RoL B 1). In this case, wc can expect the resultant FSR to be approximately equal to that of the secondary ring, which should thercfore bc made very small so as to give a large FSR. Such a resonator is less sensitive to environmental perturbations, as will be shown in thc next section. For convenience, wc set N = 1; then L , : L , = M : 1 = M ; thus the resultant FSR is completely deter- mined by the secondary ring:

FSR = M . FSR, = FSR,. (15)

In the next section, we focus on the case of RoL >> 1. By using the above equations, we will give numerical results of

Figure 1 Schematic diagram of a TCDR

74 MICROWAVE AND OPTICAL TECHNOLOGY LEnERS / Vol. 12, No. 2, June 5 1996

the circulating intensity I;/IT and the reduction factor, and discuss the influences of various parameters on them. The sensitivity of the TCDR resonator to external disturbances is studied. A comparison is made on the sensitivities for RoL 3.z 1 and RoL = 1 .

3. NUMERICAL RESULTS AND DISCUSSION

3.1. Circulating Intensity and Reduction Factor. Figures 2(a) and 2(b) display the circulating intensity and the reduction factor as functions of the coupling coefficient k , for various values of the fiber propagation loss factor a. We have chosen RoL to be 80:l in Figure 2 and assumed that the couplers have the same coupling loss y. The figures show that the circulating intensity and the reduction factor decrease as k , increases. One reason for this is that the transmission of the secondary ring decreases as k , increases, resulting in a decrease of the main peak circulating intensity. Another reason is that the finesse of the secondary ring increases as k , increases, thus the side-peak intensity will decrease effec- tively faster than the main peak. As a result, the reduction factor decreases with increasing k,. This interesting result shows that a low reduction factor can be achieved by a large coupling coefficient k,. For the case with RoL = 1, however, it has been shown that there is a minimum value in the

0 0.0 0.0 0.2 0.4 0.6 0.8 1.0

k3

(4

100 1.0 main peak - - - - - _

0.0 0.2 0.4 0.6 0.8 1.0

k3 cb)

Figure 2 The reduction factor and the circulating intensity as a function of coupler coefficient k , for various values of the fiber propagation loss factor a with L, = 0.1 m, L, = 8 m, and y = 0.3%. (a) a = 0.5 dB/km, (b) (Y = 1.5 dB/km

reduction factor around k , = 0.5 [3]. Obviously, the main peak intensity and the reduction factor degrade as a in- creases. In applications, it is desirable to have a low reduction factor and a high main peak circulating intensity. Unfortu- nately, Figure 2 shows that a low reduction factor is accompa- nied by a low main peak circulating intensity. Therefore, to obtain optimum performance, k , should be determined by choosing a suitable compromise between the two effects.

3.2. Resonant Numbers. Equations (12) and (13) show that the characteristics of a TCDR resonator depend on the phase changes in the two rings. For a chosen length of the sec- ondary ring, the properties depend on the ratio of the reso- nant numbers, which in turn depend on RoL. We have set one of the resonant numbers N = 1 in Section 2; thus the RoL can only be chosen as even number. The reduction factors are plotted against RoL in Figure 3 for various coupling coefficients k, . The reduction factor increases monotonically with RoL. However, the reduction factor can be reduced effectively by increasing coupling coefficient k, . Thus, it can be concluded that the larger the RoL, the bigger k , has to be, in order to obtain a low reduction factor. In addition, a comparison between Figures 3(a) and 3(b) shows that the reduction factor may also be reduced by a decrease in the coupling loss y.

3.3. Sensitiuity. We have seen in Section 2 that when the two

Figure 3 The reduction factor versus the RoL for two values of the coupling loss y with a = 1 dB/krn, L, = 0.1 rn, and L, = 8 rn. (a) y = 0.3%, (bl y = 0.5%

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 12, No. 2, June 5 1996 75

rings resonate simultaneously, the RoL must be the ratio of an even number and an odd number. In practice, it is very difficult to maintain an exact ratio at all times. External disturbances such as temperature and vibration may easily change the ratio RoL. So it is necessary to investigate the effects of RoL on the intensity of the main peak and on that of the side peak, and consequently on the reduction factor. In Figure 4 the heights of the three most prominent peaks are plotted against the RoL. It is clear that the optimum occurs at a critical ratio RoL = 80. The reduction factor degrades outside this critical ratio. The main peak has the same height as the side peak for RoL = 81. To show clearly the depen- dence of the heights of the peaks on the values of RoL, we introduce a maximum ratio deviation A :

3.5

3.0 E 0 Z 2.5 a

h

.-

.- 4 2.0 0 .I Y

a 1.5

g 1.0 E 3 0.5 2

.I

n n “.V

28 30 32 34 36 38 40 42 44 46 48 50 52 M

(a)

(16)

where (RoL), is the optimum ratio and (RoL), is the ratio when the intensity of the main peak is equal to that of one of the side peaks [for example, in Figure 4, (RoL), = 80, (RoL), = 811. The value of A can be used as a measure of the sensitivity of the compound resonator [l]. In Figure 4 the value of A is 1.25%. This means that a change of less than 1.25% in the length of one of the rings will not alter the fact that the main peak is dominant. This result is obtained for RoL = 80. For different resonant numbers, however, the maximum ratio deviation A will be different.

Figure 5 gives A as a function of the resonant number for RoL >> 1 and RoL = 1. We can see by comparing Figure S(a) with Figure S(b) that the values of A for RoL >> 1 are much larger than those for RoL = 1. Therefore, it can be inferred that a resonator constituted by two rings with widely different lengths will be far less sensitive than one consti- tuted by two rings with similar lengths. Moreover, for a particular resonator, Figure 5 shows that it becomes more sensitive as its resonant numbers increase.

4. CONCLUSION We have presented a theoretical analysis for a three-coupler double-ring (TCDR) resonator. The optical phase responses of the circulating intensity and the optimum resonating con- ditions have been investigated. With the use of rings with

h . 5 79.0 79.5 80.0 80.5 81.0 815 RoL

Figure 4 The heights of the three most prominent peaks vary as the RoL with a = 1 dB/km, y = 0.5%, L, = 0.1 m and L, = 8 m. Solid line, main peak; dashed line, left-side peak; dotted line, right- side peak

N=M-1

28 30 32 34 36 38 40 42 44 46 48 50 52 M

(b)

Figure 5 The maximum ratio deviation A against resonant number M . Other parameters of the resonator are the same as in Figure 4. (a) RoL >> 1, (b) RoL = I

unequal lengths, the compound resonator can give a large FSR by vernier effects. The lengths of the two rings may be chosen to be almost equal (RoL = 1) or widely different (RoL B 1) to give a large resultant FSR. For the case of RoL >> 1, it has been shown that both the circulating inten- sity and the reduction factor are dependent on the coupling coefficient k , , the ratio RoL of the lengths of the two rings, and the propagation loss and the coupling loss. A low reduc- tion factor can be obtained by choosing a large k , , or a small RoL, or both. For the case with RoL = 1, the reduction factor has a minimum value, which is determined by the losses. It is shown, moreover, that the resonator with RoL >> 1 is less sensitive to external disturbances than one with RoL = 1. Thus, the TCDR resonator with RoL >> 1 is par- ticularly advantageous for the application of doped fiber lasers. It can employ a large primary ring, which will allow the laser to have a large gain, and a small secondary ring, which will increase the FSR. By appropriately choosing the parameters, a stable single-mode laser operation could be realized.

ACKNOWLEDGMENT This work is supported in part by the National Sciences and Engineering Research Council of Canada.

REFERENCES 1. P. Urquhart, “Compound Optical-Fiber-Based Resonators,” J.

2. J. Zhang and J. W. Y. Lit, “Compound Fiber Ring Resonator: Opf. SOC. Am. Ser. A, Vol. 5, No. 6, 1988, pp. 803-812.

76 MICROWAVE AND OPTICAL TECHNOLOGY LEnERS / Vol. 12, No. 2, June 5 1996

Theory,” J. Opt. SOC. Am. Ser. A, Vol. 11, No. 6, 1994, pp.

3. J. Zhang and J. W. Y. Lit, “All-Fiber Compound Ring Resonator,” J. Lightwave Technol., Vol. 12, No. 7, 1994, pp. 1256-262.

4. G. Barbarossa, M. N. Armenise, and A. M. Matteo, “Theoretical Analysis of Triple-Coupler Ring-Based Optical Guided-Wave Res- onator,” J. Lightwave Technol., Vol. 13, No. 2, 1995, pp. 148-157.

5. J. Zhang and J. W. Y. Lit, “Erbium-Doped Fiber Compound-Ring Laser with a Fiber Ring,” IEEE Photon. Techno/. Lett., Vol.

6. J. L. Zyskind, J. W. Sulhoff, Y. Sun, J. Stone, L. W. Stulz, G. T. Harvey, D. J. Digiovanni, H. M. Presby, A. Piccirilli, U. Koren, and R. M. Jopson, “Single Mode Diode-Pumped Tunable Erbium-Doped Fiber Laser with Linewidth less than 5.5 kHz,” Electron. Lett., Vol. 27, No. 23, 1991, pp. 2148-2149.

7. K. Oda, N. Takato, and H. Toba, “A Wide-FSR Waveguide Double-Ring Resonator for Optical FDM Transmission System,” J. Lightwave Technol., Vol. 9, No. 6, 1991, pp. 128-736.

8. J. Zhang, C. Yue, G. W. Schinn, W. R. L. Clements, and J. W. Y. Lit, “Stable Single-Mode Compound-Ring Erbium-Doped Fiber Laser,” J. Lightwave Technol., to be published.

9. Y. H. Ja, “Generalized Theory of Optical Fiber Loop and Ring Resonators with Multiple Couplers. 1: Circulating and Output Fields,” Appl. Opt., Vol. 29, No. 24, 1990, pp. 3517-3523.

1867-1873.

PTL-6, NO. 5, 1994, pp. 588-590.

Receiued 1-1 7-96

Microwave and Optical Technology Letters, 12/2, 73-77 0 1996 John Wiley & Sons, Inc. CCC 0895-2477/96

A COMPACT CIRCULARLY POLARIZED MICROSTRIP ANTENNA WITH INTEGRATED FEED A. C. Tarot and C. Terret

Universit6 de Rennes I Campus de Beaulieu 35 042 Rennes Cedex. France

LSRI LAM-URA CNRS 834

KEY TERMS Microstrip antenna, circular polarization, integrated feed

ABSTRACT A circularly polarized antenna, realized in a low-cost technology with an integrated feed, is presented for the land-mobile satellite seruice. This antenna presents large bandwidth, high gain, and good axial ratio in all frequency ranges. 0 1996 John Wiley & Sons, Inc.

1. INTRODUCTION

A broadband, circularly polarized, low-profile antenna is needed for surface mounting on the roofs of cars. We have chosen a multilayered printed antenna with two stacked radiating elements which provides high gain (close to 9 dB) and large bandwidth (around 15%). Besides, as a circular polarization is required, the antenna is usually fed in quadra- ture phase by hybrid couplers, located under the ground plane.

This Letter presents the integration of this feed inside the antenna, to realize a compact antenna with a single access.

II. DESCRIPTION OF ANTENNA

The antenna has been optimized with the spectral-domain approach. The antenna is realized with the printed antenna concept and a low-cost technology [l]. The structure of the antenna is described in Figure 1. The first and third layers are made of polypropylene and the second one of foam; these dielectric substrates have been chosen because they have low loss and low permittivity. Both printed radiating elements are squares of dimensions L , = 64 mm and L , = 71 mm. They are obtained by hot pressing of a copper foil directly on polypropylene. The presence of a metallic rivet introduced in the first layer during hot pressing allows the excitation of the lower patch. Next, the three layers are assembled by hot pressing, through the use of a copolymer film [2]. The an- tenna is thus perfectly compact.

The lower radiating element is fed on the diagonal by two orthogonal accesses in quadrature phase, in order to obtain the required circular polarization (cf. Figure 2).

Ill. FEED INTEGRATION

A. Principle. The principle is to introduce the feeding system between layers 1 and 2, using the lower patch as the ground plane of the feeding circuit (cf. Figure 3). The size of the feeding circuit is obviously the essential parameter, because it must be confined in a square of size L , X L , . In order not to modify radio-electric characteristics of the antenna, the exci- tation of the feeding system through the first layer must be realized in a null field, that is, at the center. The two feed outputs are soldered to the antenna ground plane in order to excite the cavity under the lower patch.

B. Description of the Feeding System. The feed is a 90” branch line, widely used in microwave circuits, playing the role of power divider. Conventional hybrid couplers consist of four quarter-wavelength transmission lines of characteristic impedance 2, and Z,fi (cf. Figure 4). But, at 1.5-GHz frequency, the sizes of the quarter-wavelength lines are too large compared to L , . Therefore we used a size-reduction method [3].

C. Reduced-Size Coupler. A quarter-wavelength transmission line consists of distributed series inductance and shunt capac- itances. The coupler has then the structure shown in Figure 5.

Next, lumped elements are redistributed:

A high-impedance transmission line is employed as an inductive element. Its length 1 is given by the equation

C

layer 3 : polypropylene

layer 2 : foam

layer 1 : polypropylene i t 4-

\ / external feed

Figure 1 Antenna with external feed

MICROWAVE AND OPTICAL TECHNOLOGY LElTERS / Vol. 12, No. 2, June 5 1996 77