Raghuwanshi, Sanjeev Kumar

Talabattula, Srinivas

IACS Institutional Repository

Indian J. Phys. 82(10) 1373-1383(2008) Applications of degenerate/non-degenerate modes coupling in an optical waveguide Sanjeev Kumar Raghuwanshi* and Srinivas Talabattula Applied Photonics Laboratory, Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore-560 012, India E-mail : sanjeevkr@ece.iisc.ernet.in Received 25 April 200?\ accepted 22 April 2008 Abstract : Mutual coupling between optical modes is essential in the design of integrated optic devices. In this paper we have described how optical energy couples between modes within and between optical waveguides. It has been implicitly assumed so far in the literature that the modes, once formed are unchanging except perhaps through attenuation due to absorption. In reality, simple mechanisms can lead to significant energy exchange among the various modes of a structure. Coupled mode theory describes this energy exchange and serves as the primary tool for designing optical couplers, switches and filters. We have explored a coupling technique that describes the scalar electromagnetic field of a perturbed waveguide in terms of a superposition of modes of the ideal waveguide. In this paper, various applications of degenerate/non-degenerate modes coupling in an optical waveguides are discussed with examples. We have also applied our degenerate coupled mode theory to analyse the co-directional waveguide coupler. The coupling period of the power is verified by Beam Propagation method. Keywords : Optical waveguide structure, coupled mode theory approach. PACS Nos. : 42.79.Dj, 42.79.-e 1. Introduction Consider two proximate single mode optical waveguides. In each waveguide there are two waves, one propagating in the forward direction and one in the backward direction. Energy transfer, i.e. coupling, can occur if the evanescent field from one waveguide extends into the core of the neighboring waveguide. The degree to which two modes exchange energy depends on the design of the coupler and the mode structure of the two waveguides. For example, it seems plausible in the structure that the forward wave in one waveguide (A+) will primarily couple to the forward wave in the adjacent waveguide (8+). We have restricted our discussion to planar or rectangular waveguides 'Corresponding Author © 2 « » I ACS m 1374 Sanjeev Kumar Raghuwanshi and Srinlvas Talabattub in the whole paper. We have followed the development of the coupled mode theory as described by Yariv [1]. Consider the planar step index waveguide structure, with film index n, of thickness h, and substrate and cover indices ns and n0 respectively [7]. The waveguide can support a finite number of guided modes and an infinite number of radiation modes. In fact this paper is the proof of our Beam Propagation method results with coupled mode theory approach [2] for coupling period. The agreements of the results are excellent. When two orthogonal modes have identical values of propagation constant /?, we must use degenerate perturbation methods for solving the equation of motion. Non-degenerate coupled modes theory applies when two modes with different /3s are coupled. Non-degenerate mode generally display weak coupling. There are many examples of structures where two modes with identical propagation constants are coupled. This leads to degenerate coupling and a different approach is required to solve the problem. The primary equation for mode coupling is given by [1-19] ^ e x p { y ( / 3 / z + a ; f ) } ~ ^ e x p { - y ( / 3 / z ~ a ; f ) } + c.a Here Af is the amplitude for mode j and $y(x) is the normalized amplitude distribution for mode j and also A", A \ Ppen are the backward wave, forward wave and perturbation terms respectively. Others terms has usual meaning as in optical waveguide. The detailed derivation and implications of this equation can be found elsewhere [3,7]. We explore non-degenerate coupling first. 2. Non-degenerate coupling case : coupling due to a dielectric perturbation We begin with the simple case of a perturbed single mode waveguide and see how a dielectric defect can couple a mode's forward wave to its backward wave. Consider the symmetric slab waveguide shown in Figure 1. z*-e z—a Figure 1. A dielectric waveguide with a notch defect in it acta aa a perturbation to the guided modea. 1375 The waveguide consists of a guiding film index, n1f of thickness ht surrounded by a cladding index, n2. A small dielectric notch of depth /i/10 and length 2a in the core region perturbs the waveguide structure. Assume that the waveguide is a single mode structure and that for z > a there is no backward traveling wave, A~\z > a) = o. The mode expects to see an index of n^ when it gets to the notch region but instead finds the index is n2. This is a perturbation. To begin our analysis, we must determine the unperturbed modes of the waveguide. We can describe the eigenmode of the symmetric waveguide as [7], #y = Cexp{ - 7 (x - /? /2 ) } x>h/2 - ~ cosKX , sinKX uif% ^ ^ u,r> $ = C — or -—ZTZ -h/2<x<h/2 y coStf/i/2 sm«/7/2 ^ = C e x p { 7 ( x + h /2 ) } x < ~ / ) / 2 . (2) where the coefficient, C is chosen to normalize the power in the waveguide to a value of 1 W per unit length. Here 7 refer to the attenuation coefficients in the cover and substrate respectively and « is the transverse component of wave propagation constant in the guiding film. The specific values of «, 7 and (3 are determined from the eigenvalue equation [7] for this structure once /?, n^ and n2 are specified. We assume this has been done and all eigenvalues are known. To determine the coupling, we must describe the perturbation term. In this example, the polarization perturbation, Pm{x) is the product of the change in dielectric constant e0 \nl ~ n i 2 ) ar>d the electric field of the forward wave £,(*) Ppe^x)« ^ ( x ) = e0 (n\ -n*)\ A+$(x)exp{-/(02-u>t)} + c.c. 2 0.4/)< x < 0 . 5 f t , - a < z < < = 0 elsewhere. (3) Notice that the perturbation term exists only in the notch region. Everywhere else in the waveguide the mode sees an index distribution that it would normally see without the perturbation. The amplitude eq. (1) of motion becomes d2 + -1376 Sanjeev Kumar Raghuwanshi and Srinivas Talabattula cos/ex f dx [cos Kh/2) Mo ("! " " i l l A+ exp {_ . (/Jr _ uf)j + 1J |c(i * ' 0.4/1 = jr(A+ exp{-7(/?z-u/f)} + c.c.), (4) where r is the coupling constant defined as J - . ^ W - ^ T l c f " " * ]"tfx,fo, - a < * < a 0JJ C0S/C/7/2J = 0, elsewhere. • (5) Notice that the coupling constant is determined by the overlap of the coupled modes in the perturbation region. In this one-dimensional problem, we only integrate over the x coordinate to determine the coupling coefficient. Multiplying both sides of eq. (4) byexp{/(/3z-u;f)} yields an equation of motion for the amplitudes *£• exp (2//?z) - 5 £ + c.c. = }rA\ (6) To solve eq. (6), we must integrate both sides. The first term average to a small value when integrated over a distance large compared to (2/?)~1. Thus to first order we can drop the A~1 term, leaving the simplified equation of motion »£- + c*-irA+. (7) which, when integrated over z and applying the initial condition that A$ = A ( -a) , has solution ^ = y4+(-a)exp(-yTz). (8) Thus we see that the amplitude of the field does not change (to first order) but the perturbation (a small region of lower index material) alters the phase of the wave. The forward wave in the region of the perturbation will be (note K is negative) E(z) = A*(-%exp{-j(f3 + r)z}. - a < z < a (9) To find a first order solution for A", we must assume that the forward wave, A+, does not change, so ^ / f o - 0 - Except for a slight phase change, this is true. The coupling equation becomes ?£- /Wl+exp(-2/7?z). -a<z<a (10) uZ 1377 This can be directly integrated, nothing that A~{a) = 0, to yield / r ( - a ) = - i ^ s i n ( 2 / 3 a ) . (11) To first order, the coupling of the forward wave to the backward is maximized when 2/3a = (q + 1 /2)w, where q is an integer. In terms of a, the maximum reflections occurs when, (<7 + 1/2)7T a-—w~ <12> The behavior of this structure is similar to that of a half wave dielectric stack used to make mirrors. Similarly, choosing a quarter wavelength for a will minimize the reflected term. The maximum amplitude of the reflection depends on y$% which is usually small. Generally speaking, maximum coupling will occur between a forward and backward wave when the spatial frequency of the perturbation is approximately 2/3. 2.1. Coupling due to a symmetric notch in a slab waveguide : Consider the symmetric slab waveguide shown in Figure 3(a). A notch is symmetrically located about z = 0. In this waveguide, we will determine the normalized modes of the structure and calculate the coupling between the fundamental TE mode and the other TE modes of the structure. Assume that A = 1.3jjm, n, = 3 . 5 andn2 =3 .498. The first step is to determine the normalized modes of the ideal waveguide. We find the eigenvalues K and 7 using following equations [7] tan * % = J (For even modes) (13) = =-:" - (For odd modes) (14) There are two even modes and two odd modes supported by this structure. Numerically evaluated values for « and (3 for each of mode are listed in Table 1. The electric field distribution for the modes are given by [7] $y = C e x p { - 7 ( x - f ? / 2 ) } x > / ) / 2 cos/ex c sin KX ry~~co&Kh/2 o r sin/cft/2 g„ = C w » * ^ orC Z ^ ^ - / 7 / 2 < x < / ) / 2 £ / = Cexp{7(x + /7/2)} x < - f t / 2 . (15) 1378 Sanjeev KumarRaghuwanshiandSrinivas Talabattula We know every variable in these equations except the normalization coefficient, C. To normalize, we must adjust the amplitude, C, to satisfy [7] f*n(*)K(x)dx=^jL6i (16) The necessary values of C are listed in Table 1. These mode amplitudes will produce a normalized power of 1 W/cm width in the slab waveguide. Figure 2, shows the calculated mode distributions, for the four normalized modes taken from this data. Table 1. Propagation coefficients for the four allowed modes. Mode designation TE0 TEt TE2 TE3 K - • c m * 1 1335.12 2658.1 3949.8 5158.45 0 -> cm"1 169157 169142 169117 169084 Normalization amplitude (v/cm) 99.7 197.2 287.7 353.3 Having found the modes of the ideal waveguide, we can now calculate the coupling constant for each pair of modes. The polarization perturbation in this example is generated by the electric field of the TE0 mode in the dielectric notches. Explicitly, the perturbation is (for 9 p m <| x 1 1 0 p m and -a<z<a) Ppetx(x)~eo("t-"t)Eo(x,z,t) = - 2 . 6 4 7 X 1 0 ~ 1 3 cos (1335.12x) ~ Af exp{-y {0z - u>t)} + c.c. (17) Knowing Ppert, we can next calculate the coupling coefficients between the modes. Plugging the result into eq. (1), the formula for the general coupling coefficient between the TE0i and TEj mode is 0.5/? -0.5/? 1 J -2 .647 x1(T1 3 COS (1335.12x)#,+ J -2 .647x1(T 1 3 COS (1335.12x)^dx\. 0.45/? -0.45/7 J (18) There are two integrals because there are two dielectric regions in the perturbation. By symmetry, we can see that modes TE, and TE3 will not couple to mode TEQ. The product of an even and odd function in a symmetric integral always yields a null result. Therefore two of the coupling coefficients equal zero. The only mode that the TEQ mode can couple to is the even symmetry TE2 mode. The coupling coefficient will be 10 75 5 -6 -7.5 «-' = 0 1000 2000 3000 4000 5000 (a) 400 200 0 -200 -WO -0.002 -0.001 0 0001 Q002 cm (b) 400 200 I o -200 -400 -0.002 -0001 0 Q0O1 O002 cm (c) -0.002 -0.001 0 cm (d) 0001 0002 400 200 I o -200 -400 -0002 -0001 0 0001 0002 cm <•) 400 300 ' 0 -200 -400 mm*m*mmmmmw*m——*m M M M * A * M M I a li II 1 1 1 1 -01002 -0.001 0 0001 0002 cm (0 Figure 2. (a) Waveguide in Figure 3(a), supports both even {TEQ, TEg.) and odd (7H1f TEJ modes, (b) (c) Even modes, (d) (e) Odd modes. (0 The four mode amplitudes are plotted on the same scale. 0.5/) r w = 2 £ f -2.647X10-13 C08(1335.12X)287.7 ° ° ? i f ^ f * L dx O J v ' cos (3949.8/7/2) 0.45/1 13B0 Sanjeev Kumar Raghuwanshi and Srinivas Talabattuia = - 3 . 9 1 2 cm \ (19) The coupled mode equation relating the TE2 mode to the TEQ mode is then, 0A* tf/(*«+«') _ 2 * L e-J(k*-*t) + c c 9z dz - j r ^ e - ' ^ ^ \ (20) multiplying both sides byexp{y ( /? 2 z-u ; f ) } converts the equation to a£ ji** - ^L + c.c. = /roa*+e-'(*-*)*. dz ~ az ™" '-*•*- (21) Integrating over z from - a to a, the first term on the left hand side averages to zero to first order. The amplitude of mode TE2 at the end of the perturbation, assuming that /£( -a) = 0, is K (a) = 2jr{ 02 = 7 (ft-A) (7.83 4o"sin(/Jo-02)a I 40 A>"sin(/3o-/?2)a- (22) Squaring this amplitude gives the actual power in the mode. The power in mode TE2 as a function of perturbation length is shown in Figure 4(a). The coupling length ranges from 0 to 0.1 cm in the plot. We can see that a small fraction (<4%) of the mode energy transfers into mode TE2 at the optimum interaction length. 3. Degenerate mode coupling There are many examples of structures where two modes with identical propagation constants are coupled. This leads to degenerate coupling and a different approach is required to solve the problem. Various examples are shown in Figures 3(b), (c). Consider the coupled slab waveguide structure is shown in Figure 3(d). The coupler consists of two identical step index waveguides with core thickness of 10 ^m situated approximately 4 p,m from each other. To calculate the coupling between the waveguides, we must first determine the eigenmodes of the uncoupled waveguides. Using techniques [7], 0 was found to be at A = 1 pm, 0 = 94227 cnrT1 (Figure 5(a)) and the normalized mode for the right-hand waveguide is [7] $ y = C e x p { - 2 8 4 0 ( x - 0 . 0 0 1 2 ) } x > 0 . 0 0 1 2 cm . - cos 1942 (x - 0.0007) *> = ° COS1942X0.0005 0 0 0 0 2 * * * 0 0 0 1 2 Cm Applications of degenerate/non-degenerate modes coupling in an optical waveguide 1381 $ y = C exp {+2840 (x - 0.0002)} x < 0.0002 cm (23) Figure 3. (a) A symmetric waveguide is perturbed by a dielectric notch on the top and bottom of the guiding film. (b) A directional coupler can be made by placing two waveguides Jn close proximity to one another for a finite distance, (c) The coupled waveguides are separated in the y-directkm by a distance of 2a and are 2b wide, (d) Two identical slab waveguides are separated by 4 /im. The amplitude for the left-hand mode is identical in form, requiring only appropriate offset of the coordinates. The modes are norm#teed according to eq. (16). Two identical slab waveguides are separated by 4 ixm. t h e eigenmode for one of the slabs is superimposed on the right waveguide, as shown in Figure 5(b). Eq. (16) was numerically evaluated, yielding C = 433.56. In this example, we assume that initially the mode energy is completely contained in the right hand waveguide. The perturbation therefore only exists in the core of waveguide S, equaling Ppen(x) = # ( * )e 0 ( " I - n? ) [~ A+e-^*-^ + ex:. - 0 . 0 0 1 2 < x < -0 .0002 , * 0 Elsewhere (24) 1382 Sanjeev Kumar Raghuwanshi and Srinwas Taiabattula 0 002 004 006 008 01 Couplng Length (cm) 02 04 06 06 1 Propagation Distance (cm) Figure 4. (a) The power transfer from one mode TEQ to mode TEz as a function of notch length in Figure 3(a) The coupling varies periodically with notch length, (b) The power in the original waveguide couples back and forth between the two waveguides. The coupling period for the power is approximately 0.4 cm. The coupling coefficient is found from we* -0.0002 p = ~o0 4 f (1.52 -1.4992) $a(x)^(x)c/x -0 0012 = 3.6214 cnT1 (25) More details data regarding this calculation of coupling coefficient is available with the authors. So how does this result compare to the BPM (Beam propagation method) simulation [19]. Since this is a degenerate coupled system, there will be strong coupling. The amplitude will couple periodically back and forth between the waveguides [8]. The power is proportional to the square of the amplitude. Therefore we would predict the power would couple back and forth as [1-8], I A\2=A* cos(rz)2. (25) Figure 4(b) shows how the power couples back and forth between the waveguides. By inspection, we see that the spatial coupling period is approximately 0.4 cm, which is exactly what we observed in the BPM work [19]. 4. Conclusions In this paper we have explored some applications of amplitude coupling between two waveguides based on a scalar theory. We found that coupling is maximized when the propagation coefficient, 0 for the two modes is equal. We can enhance the coupling Applications of degenerate/non-degenerate modes coupling in an optica/ waveguide 1383 Figure 5. (a) For the waveguide Figure 3(d), there is only one allowed mode, which occurs near K •• 1942.12. (b) The eigenmode for one of the slabs is superimposed on the right waveguide. between dissimilar waveguides by using phase matching technique. The coupling of radiation from a waveguide into free space is an important problem in optoelectronics. Please note that the more details data are available only with authors of this paper, regarding the detail derivation of coupled mode equations for various cases. Reader should have basic knowledge about integrated optical waveguide to understand this paper. Details of derivation are omitted everywhere to avoid any difficulty to understand the mode coupling phenomena. The calculation is difficult because the guided modes couple to free space modes, which are not normalizable. The design of such couplers is as much an art as it is an engineering task. References [1) [2] [3] [4] [5] (6) [7] [81 [9] [10] mi [12J [13] [14] [15] [16] [17] [18] [19] A Yariv IEEE J. Quantum Electron. 9 919 (1973) S K Raghuwanshi, T Srinivas and A Selvarajan J. Indian Inst Sci. 86 667 (2006) S K Raghuwanshi and S Talabattula, XXXII OSI Vdodara, India (2007) W H Louisell Coupled Mode and Parametric Electronics (New York : John Wiley and Sons) (1960) A W Snyder J. Opt. Soc. Amer. 62 1267 (1972) A W Snyder and J D Love Optical Waveguide Theory (London : Chapman and Hall) (1983) D Marcuse Theory of Dielectric Optical Waveguides (New York : Academic Press) (1974) A Hardy and W Streifer IEEE J. Lightwave Technol. 5 1135 (1985) S L Chuang IEEE J. Lightwave Technol. 6 294 (1988) W Streifer, M Osinski and A Hardy IEEE J. Lightwave Technol. 5 1 (1987) H A Haus, W P Huang, S Kawakami and N A Whitaker IEEE J. Ligtwave Technol. 5 16 (1987) H A Haus and W Huang Proc. IEEE7B 1505 (1991) A A Hardy IEEE J. Quantum Electron. 34 1109 (1998) A W Snyder, Y Chen and A Ankiewicz IEEE J. Lightwave Technol. 7 1400 (1989) K Yasumoto Opt. Lett. 7 503 (1993) W P Huang, S T Chu and S K Chaudhuri IEEE J. Quantum Electron. 28 184 (1992) W P Huang J. Opt. Soc. Amer. 3 963 (1994) H A Haus, W P Huang and A W Snyder Opt. Lett. 14 1222 (1989) A N Kireev and T Graf IEEE J. Quantum Electron 39 866 (2003) 

Applications of Degenerate/Non-Degenerate Modes Coupling in an Optical Waveguide

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Applications of Degenerate/Non-Degenerate Modes Coupling in an Optical Waveguide

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