Modelado, diseño y análisis de estructuras nanofotónicas basadas en el interferómetro modal

UNIVERSIDAD POLITECNICA DE VALENCIA ESCUELA POLITECNICA SUPERIOR DE GANDIA I.T. Telecomunicación (Sist. de Telecomunicación) “Modelado, diseño y análisis de estructuras nanofotónicas basadas en el interferómetro modal” TRABAJO FINAL DE CARRERA Autor/es: Álvaro Rosa Escutia Director/es: Dr. Pablo Sanchis Kilders GANDIA, 2010 ! !"#$%&' ' !" #$%&'()**+,$--------------------"""-------"+. . • /012%+3'4.-------------------------+3. . . 5" 6)784.,9%+*84---------------------------""!. . • :'$(+*+,$.(2.;'$';'('.98&8.4+;)<8*+,$.2$.5=---"""!. . • >*'9<2.(2.9'%2$*+8.2$%&2.?)784.,9%+*84---------"@. . . . A" B<.+$%2&C2&,;2%&'.;)<%+;'('-----------------"""!D. . • >)%'+;8?2$.2$.2<.+$%2&+'&.(2.<8.24%&)*%)&8------"!@. . • E8&8;2%&'4.(2.(+42F'.(2.<8.24%&)*%)&8--------""!G. . • H)$*+,$.(2.I8%<80.98&8.2<.*8<*)<'.(2.98&J;2%&'4--""!K. . . 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K! >/)*(.6!0)!)'(*:0:!?!2!0)!6:%&0:!1)'!%:!D&</*:!EC"!@:?!"!)'(*:0:!?!#!6:%&0:64!?!)'(*)! %.6! >/)*(.6! 0)! )'(*:0:! ?! 6:%&0:5! )6(8! %:! </=:! -/%(&-.0.! 0.'0)! 6)! +.*-:'! %:6! :/(.&-:<)')6C!L:6!9.-7&':9&.')6!-:6!(=>&9:6!6.'!0)!";2!.!0)!K;KC! radius of curvature should be larger than 24 mm when the pure bending loss is required to be lower than 0.1 dB!90°. For a deeply etched waveguide "hr !5"m#, the pure bending loss is still negligibly small (much smaller than 0.1 dB!90°) even if the radius of curvature is reduced to as little as several tens of micrometers [see Fig. 2(a)]. However, one should note that the transition loss is dominant in the case of deeply etched waveguides. In Fig. 2(b) the transition loss is shown as the bending radius varies for the deeply etched waveguides with different widths. One sees that the transition loss decreases as the bending radius increases. When a compact structure is desired, increasing the bending radius is not an effective method of reducing the transition loss. For example, for the requirement of a transition loss smaller than 0.1 dB, the bending radius has to be larger than 1000 "m when the bending section has the same width as the straight section (i.e., wB !wr!4"m). Fortunately, one can also see that the transition loss is reduced greatly when width wBof the bending section (with the same bending radius) decreases. This is highly advantageous for a compact design. For example, when width w!1.0 "m, the transition loss is $0.008 dB (much smaller than the requirement of 0.1 dB) even when bending radius R is 50 "m. Considering that it is disadvantageous for the reactive ion etching process when the waveguide is too narrow, we chose wB!2.0 "m. From Fig. 2(b) one can see that the bending radius should be larger than 125 "m for a transition loss smaller than 0.1 dB. B. Bilevel Tapered Region The deeply and shallowly etched regions are connected with a bilevel taper [as shown in Fig. 1(b)] through which the field propagates smoothly from the shallowly etched region to the deeply etched reFig. 3. Simulation for the bilevel tapered section: (a) power propagating in the input waveguide along the zdirection, (b) intensity profile at the output end of the taper section. Fig. 4. Field distribution in the MMI section: (a) hr!2"m, (b) hr!5"m. Fig. 5. Field distribution in the output waveguides of the MMI coupler: (a) hr!2"m, (b) hr!5"m. 20 August 2005 !Vol. 44, No. 24 !APPLIED OPTICS 5039 ! "#! $%! %&'()*%+,-! ,.! &-%! /0(/),1%1! 1,! 2%.! +&3%.! *&2')*(1(4! /(0! 2%! 5&%2! &-! 5%*/(! ,-'0%-',!,.!0,/0(1&5)1(!/(0!&-%!(!*62')/2,.!)*7+,-,.!%!)-',08%2(.!/,0)91)5(.!1,! 1).'%-5)%!%!2(!2%0+(!1,!2%!1)0,55)9-!1,!/0(/%+%5)9-!1,!2%!+&3%:! ! ! ;)+&0%!<:<!=!>&3%!9/')5%!*&2')*(1(!,-!2%!?&,!.,!/&,1,!(@.,08%0!5(*(!%'0%8A.!1,2!5%*/(!)-5)1,-',! 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I I1 SOLDANO AND PENNINGS: OPTICAL MULTI-MODE INTERFERENCE DEVICES BASED ON SELF-IMAGING 617 the latter being a consequence of the structural symmetry with respect to the plane y = 0. IV. GENERAL INTERFERENCE This section investigates the interference mechanisms which are independent of the modal excitation, that is, we pose no I restriction on the coefficients c, and explore the periodicity A. Single Images Fig. 3. Multimode waveguide showing the input field *(y,o), a mirrored single image at (3L,), a direct single image at 2(3L,). and two-fold images at ;(3L,) and %(3L,). By inspecting (13), it can be seen that 6(y, L) will be an image of 6(y, 0) if guided modes alone m-1 @(Y,O) = CV+U(Y). (10) u=o The field profile at a distance z can then be written as a superposition of all the guided mode field distributions m-1 Q(Y, .) = CU+U(Y) exp[j(wt - PU.)]. (11) u=o Taking the phase of the fundamental mode as a common factor out of the sum, dropping it and assuming the time dependence exp(jwt) implicit hereafter, the field profile 6(y, z) becomes m-1 The first condition means that the phase changes of all the modes along L must differ by integer multiples of 27r. In this case, all guided modes interfere with the same relative phases as in z = 0; the image is thus a direct replica of the input field. The second condition means that the phase changes must be alternatively even and odd multiples of 7r. In this case, the even modes will be in phase and the odd modes in antiphase. Because of the odd symmetry stated in (1 6), the interference produces an image mirrored with respect to the plane y = 0. Taking into account (15), it is evident that the first and second condition of (17) will be fulfilled at L = p(3L,) with p = 0,1,2,. . . (18) q(Y, 2) = c~?l~(Y)exP[j(Po - b).]. (12) A useful expression for the field at a distance 2 = L is then for p even and p odd, respectively. The factor p denotes the periodic nature of the imaging along the multimode waveguide. Direct and mirrored single images of the input field 6(y, 0) will therefore be formed by general interference at distances z that are, respectively, even and odd multiples of the length (3L,), as shown in Fig. 3. It should be clear at this point that the direct and mirrored single images can be exploited in barand cross-couplers, respectively. u=o found by substituting (7) into (12) m-1 (13) u=o The Of '(Y, L), and the Of images and Next, we investigate multiple imaging phenomena, which be determined by the excitation provide the basis for a broader range of MMI couplers. the properties of the mode phase factor (14) exp [j v(v + 3LiT It will be seen that, under certain circumstances, the field @(y,L) will be a reproduction (self-imaging) of the input field 6 (y , 0). We call General Interference to the self-imaging mechanisms which are independent of the modal excitation; and Restricted Interference to those which are obtained by exciting certain modes alone. The following properties will prove useful in later derivations: B. Multiple Images In addition to the single images at distances given by (18), multiple images can be found as well. Let us first consider the images obtained half-way between the direct and mirrored image positions, i.e., at distances P 2 (19) L= -(3L,) with p= 1,3,5,... . The total field at these lengths is found by substituting (19) into (13) even for v even odd for U odd v(v + 2) = { and, +,,(y) for v even (16) with p an odd integer. Taking into account the property of (15) and the mode field symmetry conditions of (16), (20) can be = { -?,hu(y) for v odd ! "#! ! $%&'()!*+,-!!./!012)!3%&'()!10!)4(05%)!6'0!781!4)(9:02(81!;0!;%10<8!18/!7)!)/5='()!>?@A!7)! 78/&%2';>B@A!0/!/C:0(8!;0!0/2();)1!>D@!E!07!;0!1)7%;)1!>F@+! ! 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elf-imaging may exist in three-dimensional multimode structures, for which MPA combined with two-dimensional (finite-element or finite-difference methods) cross-section calculations can provide a useful simulation tool [14]. However, the current trend of etch-patterning produces step-index waveguides, which are, in general, single-moded in the transverse direction. As the lateral dimensions are much larger than the transverse dimensions, it is justified to assume that the modes have the same transverse behavior everywhere in the waveguide. The problem can thus be analyzed using a two-dimensional (lateral and longitudinal) structure, such as the one depicted in Fig. 1, without losing generality. The analysis hereafter is based on such a 2-D representation of the multimode waveguide, which can be obtained from the actual 3-D physical multimode waveguide by several techniques, such as the effective index method (EIM) [15] or the spectral index method (SIM) [16]. A. Propagation Constants Fig. 1 shows a step-index multimode waveguide of width Whl. ridge (effective) refractive index nr and cladding (effective) refractive index nc. The waveguide supports m lateral modes (as shown in Fig. 2) with mode numbers v = 0, 1, ... (m - 1) at a free-space wavelength Xo. The lateral wavenumber k,, and the propagation constant 8, are related to the ridge index n, by the dispersion equation (1) kp, + P; = IC&? with 27r k” = - XO (U + 1)7r k,, = ___ WW where the “effective” width We, takes into account the (polarization-dependent) lateral penetration depth of each mode field, associated with the Goos-Hahnchen shifts at the ridge boundaries. For high-contrast waveguides, the penetration depth is very small so that We, N WM. In general, the effective widths W,, can be approximated by the effective width We, corresponding to the fundamental mode [17], (which shall be noted We for simplicity): where o = 0 for TE and o = 1 for TM. By using the binomial expansion with k& << kgn:, the propagation constants 8, can be deduced from (1)-(3) (5) Therefore, the propagation constants in a step-index multimode waveguide show a nearly quadratic respect to the mode number v. By defining L, as the beat length of the modes 7r 4n, W,“ L 2-w- - ,- PO -PI 3x0 dependance with two lowest-order ItC nr I ID i II 2 D Fig. 1. Two-dimensional representation of a step-index multimode waveguide; (effective) index lateral profile (left), and top view of ridge boundaries and coordinate system (right). v=O 1 2 3 4 5 6 7 8... Fig. 2. Example of amplitude-normalized lateral field profiles I.’~ ( y). corresponding to the first 9 guided modes in a step-Index multimode waveguide the propagation constants spacing can be written as Y(Y + 2)7r 3L, (Po - Pu) = (7) B. Guided-Mode Propagation Analysis An input field profile Q(y, 0) imposed at z = 0 and totally contained within We (Fig. 3), will be decomposed into the modal field distributions ?I,,(y) of all modes: where the summation should be understood as including guided as well as radiative modes. The field excitation coefficients c, can be estimated using overlap integrals e, = /m (9) based on the field-orthogonality relations. If the “spatial spectrum” of the input field q(y, 0) is narrow enough not to excite unguided modes, (a condition satisfied for all practical applications), it may be decomposed into the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elf-imaging may exist in three-dimensional multimode structures, for which MPA combined with two-dimensional (finite-element or finite-difference methods) cross-section calculations can provide a useful simulation tool [14]. However, the current trend of etch-patterning produces step-index waveguides, which are, in general, single-moded in the transverse direction. As the lateral dimensions are much larger than the transverse dimensions, it is justified to assume that the modes have the same transverse behavior everywhere in the waveguide. The problem can thus be analyzed using a two-dimensional (lateral and longitudinal) structure, such as the one depicted in Fig. 1, without losing generality. The analysis hereafter is based on such a 2-D representation of the multimode waveguide, which can be obtained from the actual 3-D physical multimode waveguide by several techniques, such as the effective index method (EIM) [15] or the spectral index method (SIM) [16]. A. Propagation Constants Fig. 1 shows a step-index multimode waveguide of width Whl. ridge (effective) refractive index nr and cladding (effective) refractive index nc. The waveguide supports m lateral modes (as shown in Fig. 2) with mode numbers v = 0, 1, ... (m - 1) at a free-space wavelength Xo. The lateral wavenumber k,, and the propagation constant 8, are related to the ridge index n, by the dispersion equation (1) kp, + P; = IC&? with 27r k” = - XO (U + 1)7r k,, = ___ WW where the “effective” width We, takes into account the (polarization-dependent) lateral penetration depth of each mode field, associated with the Goos-Hahnchen shifts at the ridge boundaries. For high-contrast waveguides, the penetration depth is very small so that We, N WM. In general, the effective widths W,, can be approximated by the effective width We, corresponding to the fundamental mode [17], (which shall be noted We for simplicity): where o = 0 for TE and o = 1 for TM. By using the binomial expansion with k& << kgn:, the propagation constants 8, can be deduced from (1)-(3) (5) Therefore, the propagation constants in a step-index multimode waveguide show a nearly quadratic respect to the mode number v. By defining L, as the beat length of the modes 7r 4n, W,“ L 2-w- - ,- PO -PI 3x0 dependance with two lowest-order ItC nr I ID i II 2 D Fig. 1. Two-dimensional representation of a step-index multimode waveguide; (effective) index lateral profile (left), and top view of ridge boundaries and coordinate system (right). v=O 1 2 3 4 5 6 7 8... Fig. 2. Example of amplitude-normalized lateral field profiles I.’~ ( y). corresponding to the first 9 guided modes in a step-Index multimode waveguide the propagation constants spacing can be written as Y(Y + 2)7r 3L, (Po - Pu) = (7) B. Guided-Mode Propagation Analysis An input field profile Q(y, 0) imposed at z = 0 and totally contained within We (Fig. 3), will be decomposed into the modal field distributions ?I,,(y) of all modes: where the summation should be understood as including guided as well as radiative modes. The field excitation coefficients c, can be estimated using overlap integrals e, = /m (9) based on the field-orthogonality relations. 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