Full Vectorial Analysis of the Tapered Dielectric Waveguides and Their Application in the MMI Couplers

Transcription

1 Proceedings of the 5th WSEAS Int. Conf. on Microelectronics, Nanoelectronics, Optoelectronics, Prague, Czech Republic, March -4, 006 (pp05-0 Full Vectorial Analysis of the Tapered Dielectric Waveguides and Their Application in the MMI Couplers H. KEIVANI R. GHAYOUR H. ABIRI M. H. SHEYKHI -Islamic Azad University, Branch of Kazeroun College of Electrical Engineering Kazeroun, IRAN -Electrical Engineering Department Shiraz University, Shiraz, IRAN Abstract- Different types of tapered waveguides are analyzed by full vectorial method. Mode epansion and mode orthogonality are also used. Full vectorial without approimation in the wave equation makes the analysis more accurate and closer to the real problem. The proposed method is applied to analyze the loss in different types of simple tapered waveguides with application in tapered MMI couplers. Key-Words: Tapered dielectric waveguide, MMI coupler -Introduction In some of the optical components such as couplers and switches, waveguides with different dimensions (cross section have to be connected directly. This increases the loss and the reflection (return wave at the interconnection point. In recent yours several solutions are proposed to reduce the effect of this problem, where using a tapered waveguide at the inter connection is a well known method. In order to analyze the tapered waveguide, beam propagation method (BPM is used conventionally. However, this method with full vectorial analysis is not accurate due to large approimations []. Several other methods are proposed in references []. In this paper, different types of tapered waveguides are introduced, and then finite difference method is applied to the non approimate wave equation. Analysis is done fully vectorial in 3-dimension by resolving the wave equation into orthogonal modes.. In order to show the accuracy and applicability, the method is used to analyze a tapered 8 MMI coupler, then the result is compared with that of a simple coupler. -Tapered waveguides We have investigated four types of tapered waveguides shown in fig. In fig (a and (b, the widths of the waveguides are varying in z direction, where their heights (thickness are constant. In the waveguides shown in Fig (c and (d, both of the widths and thicknesses of the waveguides are varying in z direction. It is clear that the variations of the width and/or thickness have to follow a fied relation. This relation can be linear as in fig (a and (c or nonlinear (eponential as in Fig (b and (d. In Fig, the connection of two waveguides with different widths are shown. In Fig (a, the connection is done directly without any tapered waveguide. In this case the loss is large and the reflected wave is significant. However, in fig (b, a tapered waveguide is used to connect two different waveguides. In this case the loss and reflected wave are lower. In fact, appropriate dimension of the tapered waveguide and a proper variation of dimension reduce the effect of discontinuity in wave propagation greatly. 3-Analysis of tapered waveguides One approach to analyze the tapered waveguide is dividing the waveguide into m section each having the length Δz in z direction as shown in fig.3. The proper amount of Δz depends on the rate of variation of the tapered waveguide, i. e. the amount of (d -d and l.. In order to find the propagation constant β and field distribution of each propagating

2 Proceedings of the 5th WSEAS Int. Conf. on Microelectronics, Nanoelectronics, Optoelectronics, Prague, Czech Republic, March -4, 006 (pp05-0 modes in all directions, finite difference method is used to solve the -dimensional wave equation (. [8]. H n k H n ( H = 0 ( n In this equation n is the matri of refraction coefficient at the points in the cross section of the waveguide. Considering a very low variation of n versus Z, equation ( can be written as: δn δh δh y H n k H ( = 0 n δy δy δ n δ δh y δh H n k H ( y y = 0 n δ δ δy ( We introduce u and w to write H and H y in the following forms: H = u. ep( jβz, H y = w. ep( jβz (3 Fig : Four types of tapered waveguides Fig: Connection of two waveguides with different widths Fig3: The step approimation of a tapered waveguide Where u and w are not functions of z. Now application of finite difference method to both equations given in ( and using eq. (3, results the following equations: n(i,j n(i, j u(i,j ( n k n(i, j n(i,j n(i, j u(i, j u(i,j ( n(i, j n(i,j n(i, j u(i,j u(i,j w(i,j. n(i, j n(i,j n(i, j w(i, j =β u(i, j Δ. y n(i, j (4 n(i,j n(i, j ( w(i,j ( n k n(i, j n(i,j n(i, j w(i, j w(i,j w(i,j n(i, j n(i,j n(i, j w(i,j u(i,j. n(i, j n(i,j n(i, j u(i, j =β w(i, j. n(i, j (5 It is clear that, we have to solve the wave equation in the difference form, (eqs 4 and 5 at all the nodes of the whole waveguide. The resulting system of equations has 3 unknowns β, u, and w. Applying some mathematical manipulations we obtain the following system of eq. B u u w = β N N, w y NN y, N N,N N y y (6 where, N and N y are the numbers of nodes in and y directions respectively. Matri B contains the coefficients of vectors u and w as given in eqs (4 and (5. Solving the system of eq. (6 gives the eign values & eigen vectors, i. e. H, H y and β in the matri form. To include the boundary conditions, we consider the boundary of the cladding region in a place beyond that the amplitude of each mode is almost zero. On the other hand, the number of equations in system (6 depends on the number of mesh points in the cross section of the waveguide. The large number of mesh points makes the result more accurate, but the solution to the problem becomes more complicated or sometimes impossible. To prevent this problem, we do inhomogeneous meshing. In this type of meshing, the mesh size is small where the wave has large variations as in the core,

3 Proceedings of the 5th WSEAS Int. Conf. on Microelectronics, Nanoelectronics, Optoelectronics, Prague, Czech Republic, March -4, 006 (pp05-0 whereas in the places with low variations as in the cladding, the mesh size is large. Fig 4 shows this type of meshing in the core and cladding of a waveguide. Fig4: Nonuniform meshing in the core and cladding of a waveguide. Numerical solution to the Mawell equation and using the matrices of H and H y, give the other vectors of E and H as H y H E z 0 H E z = (,E = ( jw 0 μ y jw y j 0 n β Ez Ey E ( jw,h ( E y 0 0 H = μ y z = jβ y jw0 μ0 y (7 By eq. (7, the field distributions of all the propagating modes in the waveguide are determined. Now we can analyze the tapered waveguide in z direction. The eciting field into the tapered waveguide can be written as: E in = E a E y a y E z a z Hin = Ha H ya y Hza z (8 From Fig 3, it is clear that the input to the tapered waveguide is the input to the section of the staircase model (fig.3. Epansions of the these waves into the modes are epressed as [7] M E in = aμe μ μ=, M H in = aμh μ μ= (9 where E lμ =E lμ E lμy E lμz and H lμ =H lμ H lμy H lμz are the propagating modes in the first section of tapered waveguide of fig 3 and M is the number of the propagating modes in that section. At the - point z= z in the first section, we can write: E(, y, H(, y, z = aμeμ ep( jβz μ= = z aμhμ ep( jβz μ= (0 It is possible to use mode orthogonality to find the field distribution at the point z=z. The electric and magnetic fields in the second section of the tapered waveguide can be written as: E (, y, z H (, y, z = b μ E μ μ = = b μ H μ μ = (, y, z (, y, z ( where E μ = E μ E μy E μz and H μ = H μ H μy H μz are the fields of propagating modes in the nd section of the tapered waveguide (Fig. 3. Now, b μ can be determined as follows [7]: [E * μ * in H Eμ H in ].kddy b μ = A ( [Eμ H * μ E * μ Hμ].kddy A Performing the same operation for all the sections in Fig. 3, we can determine the The wave propagation along the tapered waveguide. We have considered only two modes ( and of the waveguide in fig 3 as the input to first section of the tapered waveguide, thus: E in = E E y E z Ein = E Ey Ez 4-Numerical results: H in = H H y H z Hin = H Hy Hz (3 Before analyzing the tapered waveguides, it is informative to analyze the connection of two waveguides without tapering shown in Fig 5. The dimension of the waveguide shown in Fig. 5(a are: d = 8μm, d =3.μm, t =t = 5μm, n =.49 and n =.46. The amount of power losses of the modes and obtained for the structure of fig. 5 (a are given in table. Fig.5: Interconnection of two waveguides without tapereing

4 Proceedings of the 5th WSEAS Int. Conf. on Microelectronics, Nanoelectronics, Optoelectronics, Prague, Czech Republic, March -4, 006 (pp05-0 In this table, P out is the output power from the core and P in is the input power to the core. The higher loss of the nd mode in table can be eplained as follows: the power of the first mode is more concentrated at the middle of the core, whereas for the nd mode, the power is concentrated at the sides of the core. During the propagation of mode and mode from section to section, more power of the nd mode enters the cladding or reflected beak to section than those of the l st mode. This makes the nd mode more lossy than the l st one. Fig 6 shows the propagation of the l st mode in the structure of fig 5 (a, where an abrupt change in the field distribution at the discontinuity point is observed. Table : Power loss ( 0*log(P out /P in in fig 5(a for the modes and input Loss in Fig 5(a E in,h in.(db E in,h in 3.8(db Table is obtained from simulation of the first mode in the structure of fig (a and (b in which, d =8μn, d =3.μm, t =t =5μm n =.49 and n =.46. In fig (a profile of the tapered waveguide is linear, whereas in fig (b, the profile has a variation of ep ( z. Table includes different lengths of tapered waveguide (L and also different sizes of Δz and Δd. Generally, the losses in structures of fig (tapered are much less than those in the structures of fig 5 (without tapering. input E in, H in Table : Power loss ( 0*log(P out /P in in fig. (a and fig. (b for the mode Waveguide length L=.5µ L=.5µ Δz=0.µ Δd=0.µ fig(a 0.48(db 0.5(db 0.54(db fig(b 0.55(db fig6: Propagation of the l st mode in the structure of fig 5 (a Investigating the results represented in table, shows that the larger length of the tapered waveguide, the less is the loss, note that larger Δz or Δd (less number of sections in fig 3 increases the loss. From the results represented in table we can see that the linear tapered waveguide has a lower loss than that of the eponential one. Fig 7 shows the propagating of the l st mode in the structure of fig (a, where there is no discontinuity in the field distribution at the interconnection point. This justifies reduction in the loss and reflection of fig. Fig 8 shows the propagation of the nd mode in the tapered waveguide of fig (a. Fig 5 (b, has dimensions: d =4.5μn, d =.8μm, t =4.5 t =.8μm n =.49 and n =.46. Comparing the results given in table with those of table 3 shows that the loss in structure with discontinuity in -dimension (fig. 5 (b is more than the loss in structure with discontinuity in dimension (fig 5 (a. Table 3: Power loss ( 0*log(P out /P in in fig 5(b for the modes and Input E in,h in E in,h in Loss Fig5 (b 3.74(db.6(db Fig 7: Propagation of the l st mode in the structure of fig (a Fig 8: Propagation of the nd mode in the tapered waveguide of fig (a. The results of simulation of the structures shown in fig (c and (d are given in table 4. In these structures, d =.46μn, d =.8μm, t =4.5μm, t =.8μm n =.49 and n =.46. the structure of fig. (c is linearly tapered and that of the fig (b has an eponential variation of ep ( z for and y.

5 Proceedings of the 5th WSEAS Int. Conf. on Microelectronics, Nanoelectronics, Optoelectronics, Prague, Czech Republic, March -4, 006 (pp05-0 Comparing the results given in table with those in table 4 shows that the taper with discontinuity in -dimension has much higher loss than those of the taper with - dimensional discontinuity. Table 4 : Power loss ( 0*log(P out /P in in fig. (c and fig. (d for the modes and Input E in,h in E in,h in fig(c 3.(db.8(db fig(d 3.97(db.55(db 5- MMI couplers The structure of a N simple MMI coupler is shown is fig 9, where n core =.4895,n cladding =.46,w M =60μm,w out =4μ m, w in =3μm d 0 =550μm, N=8. Fig.0 shows the propagation of wave in z direction in this MMI. We can see in fig 0 that at the lengths of 464 μm, 530μm and 630μm of the MMI coupler, we have 8, 7 and 6 outputs respectively. Simulation shows that for the input at the center of the MMI, the y position of 8 output are at.5 μm, 0.5μm, 8.5μm, 6μm, 34μm, 45μm 49.5μm and 57.5μm from the bottom of the MMI waveguide. Table 5: The ratio of output power to the input power ( P r in db for fig. 9 output P r output P r Fig shows the MMI coupler with the tapered waveguides at the output. In the device: nc =.4895, nr =.46, WM = 60μm, Wout = 3μm 0 Win = 3μ m, Wt = 4μm, θ = 60, λ0 = 550nm, N = 8 Table 6 shows the normalized output power in the MMI coupler of fig. Comparing the results of a simple MMI coupler with those of the tapered MMI coupler shows less loss in the tapered one. In fact, in MMI coupler with tapered waveguides reflection at the output gates is reduced and the coupling is improved. For the coupler shown in fig. the output ports power distribution shown in fig., is relatively the same, implying uniform power distributed to the output. fig : N, MMI coupler without tapering Fig.9: N, MMI coupler without tapering In table 5 the relative output power entering to the output waveguides (normalized to the input power are given in db. Ideally, the normalized output power should be: 0*log (/8 =-9db Table6 : The ratio of output power to the input power ( P r in db for fig. output P r output P r Fig 0: Wave propagation along the MMI coupler (z direction fig : represents the simulation of MMI coupler of Fig

6 Proceedings of the 5th WSEAS Int. Conf. on Microelectronics, Nanoelectronics, Optoelectronics, Prague, Czech Republic, March -4, 006 (pp05-0 It is possible to improve the loss in the MMI coupler by tapering the input waveguide as well. Fig 3 shows such a structure, where both input and output waveguides are tapered. The result of Simulation of this structure is represented in table 7, where the amount of loss is lower than that of the structure of fig. The distribution of power in the out put of the 8 coupler is shown in fig 4. Another advantage of tapering the waveguides in couplers is less sensitivity of the response to the manufacturing tolerances. fig 3: : N, MMI coupler without tapered Table 5: The ratio of output power to the input power ( P r in db for fig. 3 output P r output P r fig 4: distribution of power in the out put of the 8 coupler In the simple MMI coupler (Fig 9, the out put power and consequently the loss in the structure strongly depends on the position of the input and output waveguides. In this case any error in manufacturing of the structure makes the loss higher and the output power distribution ununiform. However, in the tapered MMI coupler (fig 3, the response is not strongly dependent to the input and output positions of the coupler. of different cross sections. Tapering of the cross-sections reduce the reflection and loss. In this paper a method is presented to analyze the tapered optical waveguide in 3-dimension by full vectorial approach. In this method, the complete (non approimate equation of the wave propagation is used. In fact, replacing. E = 0 by. D = 0 makes the accuracy of the simulation higher than that of the conventional approimate case. However, many authors stated that. E = 0 is reasonable when the weak guidance approimation is applicable. Meanwhile, application of. E = 0 in devices with a large refractive inde contrast (n core -n cladding, reduces the accuracy greatly, where this is not the case with. D = 0. In addition by doing a proper onohomogenuous meshing in the cross-section of the waveguide more accuracy in results is observed. The results show that using the tapered waveguide in couplers makes the structure more capable in reducing the loss and reflection at the interconnections. We have shown the improvement of performance for MMI couplers using taperd waveguides. 7-Reference [] D. Marcuse, Radiation losses of steptapered channel waveguides, Appl. Opt., vol. 9, pp , Nov [] R. N. Thurston, E. Kapon, and A. Shahar, Two-dimensional control of mode size in optical channel waveguides by lateral channel tapering, Opt. Lett., vol. 6, pp , Mar. 99. [7]R. Syms, J. Cozens, Optical Guied Waves and Devices, International Edition, McGraw- Hill, 99 [8] W.P.Huang, A Vector Beam Propagation Method Based On H Fields IEEE Photon. Technol. Lett., vol. 3, no., December. 99, pp. 7. [9] L. O. Lierstuen and A. Sudb, 8-channel wavelength division multipleer based on multimode interference couplers, IEEE Photon. Technol. Lett., vol. 7, no. 9, Sept. 995, pp Conclusion In optical devices like MMI couplers, we are faced with junctions of dielectric waveguides