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JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 7, NO. 2, FEBRUARY 1989

Scalar Modes and Coupling Characteristics of Eight-Port Waveguide Couplers ARUN KUMAR, R. K. VARSHNEY, AND R. K. SINHA

Abstract—In this paper, we have given the scalar modes and discussed the coupling characteristics of an eight-port waveguide coupler consisting of four parallel single-mode waveguides. The analysis is based on a rectangular waveguide model which has been known to give accurate results in the case of two waveguide directional couplers. A classification of its various scalar modes is also given.

the eight-port coupler are obtained and its coupling characteristics are discussed. A classification of its scalar modes is given and it is shown that a basic assumption regarding the propagation constants of the guided modes, made in [1], is not correct. ANALYSIS

INTRODUCTION

M

ULTIPORT waveguide couplers have interesting applications in dividing optical power from one channel to many channels and in determining the in-phase and quadrature components of a signal in coherent communication systems and fiber-optic sensors. Although the minimum number of power measurement ports to determine the in-phase/quadrature components of a signal is three, the signal processing is easier with a suitable four detector multiport. Travis and Carroll [1] have suggested an eight-port coupler, using four parallel optical fibers, to realize such a four-detector in-phase/quadrature measuring multiport device. In order to understand the coupling characteristics of such multiport junctions, their modal characteristics should be known. Recently, modes of a 3 x 3 fiber coupler, using three parallel fibers, have been reported by Black et al. [2] and Stevenson and Love [3]. In the case of a 4 X 4 fiber coupler, while Travis and Carroll [1] have discussed some basic features of guided modes of such couplers, no formal method is reported to obtain the coupling characteristics of these couplers. Here, we present a simple method to understand the coupling characteristics of an eight-port coupler which can be obtained either by a) fusing four identical parallel fibers, b) polishing the surfaces of two dual-core fibers and cementing them together, or c) using integrated optical channel waveguides with square cross sections. The analysis is based on a rectangular-core waveguide model which has been known to give an accurate description of two waveguide directional couplers [4], [5]. A similar approach has also been used to study the propagation characteristics of various polarization maintaining fibers [6][8]. The propagation constants of various guided modes of Manuscript received January 27, 1988. This work was partially supported by the Department of Electronics, the Government of India. The authors are with the Department of Physics, Indian Institute of Technology, New Delhi 110016, India. IEEE Log Number 8822048.

An ideal eight-port coupler consists of four identical fibers, with their centers making the four corners of a square, each side of which represents the center to center distance between two nearby fibers. In order to study the coupling characteristics of such a coupler, we use a rectangular core waveguide model (i.e., replace the circular cores by square cores as shown in Fig. l(a)). The dielectric constant distribution of the rectangular-core waveguide model (hereafter referred to as structure 1) is given by n\x, y) = n],

d < \x\ < (a + d) d < \y\ < (a + d)

= n\,

otherwise.

(1)

The above structure can be taken as a perturbed form of a pseudowaveguide coupler (hereafter referred to as structure 2), whose dielectric constant distribution is given by (see Fig. l(b)): n20(x, y) = n20(x) + n20(y) - n\

(2)

where d< | £ | < (a = n2,

otherwise

(3)

where £ stands for x or y. Since the dielectric constant distributions of the two structures differ only in the corner regions (shown shaded in Fig. l(b), where the fractional modal power is rather small) by a small amount (n\ — n\), the modes of structure 1 should be nearly the same as that of structure 2. Further, since the dielectric constant distribution of structure 2 is separable in x and y coordinates, the scalar modes of structure 2 can be obtained analytically. As a result, the propagation constants of various guided modes of structure 1 can be obtained by applying a first-order correction to the propagation constants of the corresponding modes of structure 2.

JOURNAL OF LIGHTWAVE TECHNOLOGY. VOL. 7. NO. 2. FEBRUARY 1989

294

/ /

\ \

\

/

structure 2, and ;' and j stand for 5 (or a) representing symmetric (or antisymmetric) mode in x (or y), respectively. Physically, X(x) (or Y(y)) represents the mode of a directional coupler consisting of two parallel slab waveguides with boundaries parallel to v ( o n ) axis, whose refractive indices are given by (3), and ajx (or ajy) represents the propagation constants of their guided modes. Equations (6a) and (6b) can be solved analytically and ati (£ stands for x or y) can be obtained by solving the following simple transcendental equations [9]. For the mode symmetric in £

/'"

1 1

'

1 1

n?

'

1

til/in

tan (a/is2) - (^. S I)/(M. 5 2

(a)

tan

\s

(8)

For the mode antisymmetric in i 1

— coth (dnaX) =

2

1 + (/W/* a 2) tan (ana2)

(9)

where |

' / 4///// ///, 2

3

\

\

S

Fig. 1. Transverse cross section of (a) an eight-port fiber coupler (dashed circles) and the corresponding rectangular core waveguide model (solid squares), and (b) unperturbed pseudowaveguide coupler. MODES OF STRUCTURE 2

We assume that the fibers used for making the coupler are weakly guiding, and hence, discuss only the scalar modes of the coupler which can be obtained by solving the following scalar wave equation:

= 0.

dy

{y)

(5)

where both X(x) and Y(y) satisfy the following one-dimensional equations:

(klnl(x) - a

=0

LPSS LPsa LPas LPaa

2

- a,-£

(6a)

mode mode mode mode

symmetric in x and y, symmetric in x but antisymmetric in y, antisymmetric in x but symmetric in y, antisymmetric in x and y.

After obtaining aix and ajy, the propagation constants for the unperturbed structure can be obtained by using (7). The propagation constants (ft,-), for the rectangular waveguide model (structure 1), can be obtained after incorporating the difference in the dielectric constants of the two structures in the corner regions through the first order perturbation approach and are given by

(4)

For structure 2, since the refractive index profile is separable in x and y coordinates, the wave function \l/(x, y) is of the form

dx'

2

where / stands for 51 and a. Thus, there would be four different modes of the structure 2 which can be classified as

(b)

dx

i 2

li 12 = konl

U

tfj = Pf + kl(n] - n\)Y

(10)

where

r = r,-r,. with

Yk = (1 + A)/(\ + A + B) A = (2/R)[(tk cos (dnk2)) + B=

sin

[tl{2ank2 + sin {2alxkl))

and

- sin d

2l + [klnliy) - a]})Y = 0

(6b)

where aix and ajy are two separation constants such that ^0

=

{ a i +

C2y^

Jtg«?)1/2

(7)

represents the propagation constant of a guided mode of

2

+ (2iiu/nk2) sin

(ank2) sinh

R = 2CkdnkX + sinh {2dfin) ts = cosh (dixsl),

Ss = sinh (rf/x.vl)

Cv = 1.0

ta = sinh (dnal),

Sa = cosh (dfiaX)

Ca = - 1 . 0

295

KUMAR et al.: SCALAR MODES AND COUPLING CHARACTERISTICS

and k stands for 5 (or a). It is obvious that modes LPsa and LPgs are degenerate (i.e., |3sa = /3as). It should be mentioned that /3j, j82, and |34 of [1] correspond to our {lss, (3Jfl (or Pa,), and &aa, respectively. Further, since LPaa is antisymmetric in x as well as in y, it would be a mode of higher order than LPsa or LPas and hence, /34 should be less than /32 contrary to the assumption made in [1].

LPss

LPsa

LPas

LPaa

3 ED 0 ED ED Li] ED t i l Fi

&- 2-

COUPLING CHARACTERISTICS

Light energy propagating in any channel would be cou pled to the other channels through evanescent field coupling which can be understood as follows. If at the input end of the coupler, power is injected only into waveguide 1 (say with an amplitude of 4) it will excite all the four modes discussed above, each with an amplitude 1. The amplitudes and the relative phases in various channels for different modes are shown in Fig. 2. Since different modes propagate with different propagation constants, after a distance z, the total amplitude due to all the four modes in different channels would be different. The field in channel 1 would be given by

£, = exp (-i0uz)[l + 2 exp { - i ( 0 M - 0a)z] + exp{-i(j8aB-0jz}] = exp (-«0Mz) [ 1 + 2 exp {i(z/L2)ir} + exp {i{z/L4)ir}]

(11)

where L 2 [= » / ( 0 M - 0 M )] andL 4 [= x / ( 0lf - fr, fl )] are the coupling lengths between modes LPSS, LPsa, and modes LPSS, LPaa, respectively. Similarly, the fields in channels 2, 3, and 4 would be given by E2 = E3 = exp (~il3 ss z)[ 1 - exp {i(z/L 4 )x}]

Fig. 3. Variation of the normalized propagation constant P of different scalar modes as a function of normalized frequency V for a coupler with n, = 1.4603, n2 = 1.4573, a = 5.0 jim, andd = 2.5 /im.

(12)

and £ 4 = exp ( - i 0 » z ) [ l - 2 exp { I ( Z / L 2 ) T }

+ exp{i(z/L 4 )7r}].

(13)

We now discuss some specific cases of practical interest: a)

z/L2 = 2n and z/L 4 = 2m + 1 m and n are integers. 520

Using equations (11)—(13), it can be seen that in this case the amplitude at in the /th waveguide (/ stands for 1, 2, 3, or 4) would be given by a

\ — a2 = a3 = 2 ,

a 4 = —2

i.e., the power will be equally divided in all the channels. b)

z/L2 = In + 1 and z/L4 = 2m + 1 «1 = - 2 ,

a2 = a 3 = a4 = 2

I.e., power Will be equally divided in all the Channels again.

0.2

0.4

0.6

0.8

NORMALISED SEPARATION ( d / a )

Fig. 4. Variationof coupling lengths L2 and L4 as a function of normalized separation d/abetwen the waveguides of the coupler with «, = 1.4603, n2 = 1.4573, a = 5.0 |tm, and V = 2.3.

296

c)

JOURNAL OF LIGHTWAVE TECHNOLOGY. VOL. 7. NO. 2. FEBRUARY 1989

z/L2 = In + 1 and z/L4 = 2m ax = a2 = a3 = 0,

aA = 4

i.e., the whole power is transferred to channel 4. d)

z/L2 = 2n and z/L4 = 2m «! = 4,

a2 = a3 = a4 = 0

i.e., the incident power would come back in channel 1. Thus, depending upon the values of L2 and LA and the interaction length between the different channels, one can achieve signals of desirable amplitudes/phases in various channels. The values of L2 and L4 depend upon the K[= koa/(n] - n\)xl12\ number of individual waveguides and also on the separation d between the waveguides. Thus, in order to satisfy specific conditions by L2 and L4, the values of Kand d can be selected accordingly. In Fig. 3, we have plotted the variation of the normalized propagation constants Ptj = (/3?: - k\n\)/{kl(n2x n\)} of different modes as a function of Vparameter. The numerical values of various parameters used in the calculations are given below: «, = 1.4603,

n2 = 1.4573

a = 5.0 um,

d = 2.5 /xm.

This figure clearly shows that /3sa (or (3as) is greater than j3aa. In Fig. 4, we have shown the variation of L2 and L4 as a function of normalized separation (d/a) between the waveguides. This figure shows that as (d/a) increases, the coupling lengths L2 and L4 also increase, which is due to decreasing interaction between the waveguides. In summary, we have given a simple method to obtain the scalar modes and the coupling characteristics of an eight-port coupler consisting of four parallel optical waveguides in a cross-sectional square arrangement. ACKNOWLEDGMENT

The authors wish to thank Prof. A. K. Ghatak and Dr. K. Thyagrajan for helpful discussions.

[6] R. K. Varshney and A. Kumar, "Effect of depressed inner cladding on the propagation characteristics of elliptical core fibers," Opt. Lett., vol. 9, pp. 522-524, 1984. [7] R. K. Varshney and A. Kumar. "Birefringence calculation in sidetunnel optical fibers: A rectangular core waveguide model," Opt. Lett., vol. 11, pp. 45-47, 1986. [8] A. Kumar, M. R. Shenoy, and K. Thyagrajan, "Polarization characteristics of elliptical-core fibers with stress induced anisotropy," J. Lightwave Technol., vol. LT-5, pp. 193-198, 1987. [9] A. Kumar, A. N. Kaul, and A. K. Ghatak, "Prediction of coupling length in a rectangular-core directional coupler: An accurate analys i s , " Opt. Lett., vol. 10, pp. 86-88, 1985.

Arun Kumar was born in Saharanpur, India, on August 1, 1952. He received the M.Sc. and Ph.D. degrees in physics from the Indian Institute of Technology (I.I.T.), New Delhi, India, in 1972 and 1976, respectively. He was a Research Associate in the School of Bioengineering and Biosciences, I.I.T., New Delhi, during 1976-1977. Since 1977, he has been on the faculty of the Physics Department, I.I.T., New Delhi, where he is currently an Assistant Professor. During 1980-1981, he worked at the Technical University, Hamburg-Harburg, West Germany, as an Alexander von Humboldt Research Fellow in the field of fiber optics. At present he is at Strathclyde University, Glasgow, U.K., under a British Council Research Fellowship. He has 45 research papers to his credit. His current research interests are in the field of fiber and integrated optics and fiberoptic sensors.

R. K. Varshney was born in Atrauli (Uttar Pradesh), India, on July 1, 1961. He received the B.Sc. and M.Sc. degrees from Agra University, Agra, India, in 1979 and 1981, respectively, and the Ph.D. degree from the Indian Institute of Technology, Delhi, India, in 1987. During 1985-1986, he worked at the Department of Electrical Engineering, University of Florida, Gainesville, FL, as a Fulbright Research Fellow in the field of fiber optics. In 1987, he joined the Department of Physics, Indian Institute of Technology, Delhi, India, as a Senior Scientific Officer. His current research interests are in the fields of fiber and integrated optics and fiberoptic sensors.

REFERENCES [1] A. R. L. Travis and J. E. Carroll, "Possible fused fiberin-phase/quadrature measuring multiport," Electron. Lett., vol. 21, pp. 954-955, 1985. [2] R. J. Black, L. Gagnon, R. C. Youngquist, and R. H. Wentworth, "Modes of evanescent 3 x 3 couplers and three-core fibers," Electron. Lett., vol. 22, pp. 1311-1312, 1986. [3] A. J. Stevenson and J. D. Love, "Vector modes of six-port couplers," Electron. Lett., vol. 23, pp. 1011-1013, 1987. [4] I. Yokohama, K. Okamoto, and J. Noda, "Analysis of fiber-optic polarizing beam splitters consisting of fused-taper couplers," J. Lightwave Technol., vol. LT-4, pp. 1352-1359, 1986. [5] F. P. Payne, C. D. Hussey, and M. S. Yataki, "Modelling fused single-mode fiber couplers," Electron. Lett., vol. 2 1 . pp. 461-462, 1985.

R. K. Sinha received the B.Sc. Honors degree from Jamshedpur Cooperative College, Jamshedpur (Ranchi University), India, in 1982, and the M.Sc. degree in physics from the Indian Institute of Technology (I.I.T.), Kharagpur, India, in 1984. In August 1984, he joined I.I.T., Delhi, as a Research Scholar. Presently he is working as Senior Research Fellow of the Council of Scientific and Industrial Research (CSIR), India, towards the Ph.D. degree. His research interests include characterization of optical waveguides and devices.