《光波导理论与技术 Optical Waveguides Principles and Technologies》课程教学资源（参考文献）Optical Multi-Mode Interference Devices Based on Self-Imaging Principles and Applications

JOURNAL OF LIGHTWAVE TECHNOLOGY,VOL.13.NO.4.APRIL 1995 615 Optical Multi-Mode Interference Devices Based on Self-Imaging:Principles and Applications Lucas B.Soldano and Erik C.M.Pennings,Member,/EEE Invited Paper Abstract-This paper presents an overview of integrated optics outline of the modal propagation analysis (MPA),which routing and coupling devices based on multimode interference. will be used later to describe image formation by general The underlying self-imaging principle in multimode waveguides and restricted multimode interference (Sections IV and V. is described using a guided mode propagation analysis.Special issues concerning the design and operation of multimode inter- respectively).Special design and behavior issues concerning ference devices are discussed,followed by a survey of reported MMI devices are discussed in Section VI.Performances and applications.It is shown that multimode interference couplers compatibility with other components are presented through offer superior performance,excellent tolerance to polarization examples of fabricated MMI couplers and their applications and wavelength variations,and relaxed fabrication requirements in more elaborate optical circuits (Section VII).We conclude when compared to alternatives such as directional couplers adiabatic X-or Y-junctions,and diffractive star couplers. by comparing the properties of MMI devices with those of more conventional routing and coupling devices. I.INTRODUCTION II.THE SELF-IMAGING PRINCIPLE ODAY'S evolving telecommunication networks are in- creasingly focusing on flexibility and reconfigurability Self-imaging of periodic objects illuminated by coherent which requires enhanced functionality of photonic integrated light was first described more than 150 years ago [7].Self- circuits (PICs)for optical communications.In addition,mod- focusing(graded index)waveguides can also produce periodic ern wavelength demultiplexing (WDM)systems will require real images of an object [8].However,the possibility of signal routing and coupling devices to have large optical achieving self-imaging in uniform index slab waveguides was bandwidth and to be polarization insensitive.Also small device first suggested by Bryngdahl [9]and explained in more detail dimensions and improved fabrication tolerances are required by Ulrich [10],[111. in order to reduce process costs and contribute to large-scale The principle can be stated as follows:Self-imaging is a PIC production. property of multimode waveguides by which an input field In recent years,there has been a growing interest in the ap- profile is reproduced in single or multiple images at periodic plication of multimode interference (MMI)effects in integrated intervals along the propagation direction of the guide. optics.Optical devices based on MMI effects fulfil all of the above requirements,and their excellent properties and ease of fabrication have led to their rapid incorporation in more III.MULTIMODED WAVEGUIDES complex PICs such as phase diversity networks [1].Mach- The central structure of an MMI device is a waveguide Zehnder switches [2]and modulators [3].balanced coherent designed to support a large number of modes (typically receivers [4],and ring lasers [5],[6]. 3).In order to launch light into and recover light from that This paper reviews the principles and properties of MMI multimode waveguide,a number of access (usually single- devices and their applications.The operation of optical MMI moded)waveguides are placed at its beginning and at its devices is based on the self-imaging principle,presented end.Such devices are generally referred to as N x M MMI in Section II.Basic properties of multimode waveguides couplers,where N and M are the number of input and output are introduced early in Section III,followed by a short waveguides respectively. A full-modal propagation analysis is probably the most Manuscript received August 8,1994;revised December 19.1994.This comprehensive theoretical tool to describe self-imaging phe- work was supported in part by the Netherlands Technology Foundation (STW) nomena in multimode waveguides.It not only supplies the as part of the programme of the Foundation for Fundamental Research on basis for numerical modelling and design,but it also provides Matter (FOM). L.B.Soldano is with Delft University of Technology.Department of insight into the mechanism of multimode interference.Other Electrical Engineering.Laboratory of Teecommunication and Remote Sensing approaches make use of ray optics [121.hybrid methods [13], Technology,2628 CD Delft,The Netherlands. E.C.M.Pennings is with Philips Research Laboratories.Wideband or BPM type simulations.We follow here the guided-mode Communication Systems.5656AA Eindhoven.The Netherlands. propagation analysis (MPA),proposed first in [11]for the IEEE Log Number 9409613. formulation of the periodic imaging. 0733-8724/95$04.00@1995EEE

I I1 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 13, NO. 4, APRIL 1995 615 Optical Multi-Mode Interference Devices Based on Self-Imaging : Principles and Applications Lucas B. Soldano and Erik C. M. Pennings, Member, IEEE Invited Paper Abstract-This paper presents an overview of integrated optics outline of the modal propagation analysis (MPA), which will be used later to describe image formation by general and restricted multimode interference (Sections IV and V, MMI devices are discussed in Section VI. Performances and compatibility with other components are presented through examDles of fabricated MMI CouDlers and their aDDlications routing and coupling devices based on multimode interference. The underlying self-imaging principle in multimode waveguides is described using a guided mode propagation analysis. Special ference devices are discussed, followed by a survey of reported applications. It is shown that multimode interference couplers Offer Superior performance, excellent tolerance to polarization issues concerning the design and operation of multimode inter- design and behavior issues concerning I. and wavelength variations, and relaxed fabrication requirements when compared to alternatives such as directional couplers, adiabatic X- or Y-junctions, and diffractive star couplers. in elaborate optical circuits ;Section ~11). we conclude by comparing the properties of MMI devices with those of more conventional routing and coupling devices. I. INTRODUCTION ODAY’S evolving telecommunication networks are in- T creasingly focusing on flexibility and reconfigurability, which requires enhanced functionality of photonic integrated circuits (PICs) for optical communications. In addition, mod￾em wavelength demultiplexing (WDM) systems will require signal routing and coupling devices to have large optical bandwidth and to be polarization insensitive. Also small device dimensions and improved fabrication tolerances are required in order to reduce process costs and contribute to large-scale PIC production. In recent years, there has been a growing interest in the ap￾plication of multimode interference (MMI) effects in integrated optics. Optical devices based on MMI effects fulfil all of the above requirements, and their excellent properties and ease of fabrication have led to their rapid incorporation in more complex PICs such as phase diversity networks [l], Mach￾Zehnder switches [2] and modulators [3], balanced coherent receivers [4], and ring lasers [5], [6]. This paper reviews the principles and properties of MMI devices and their applications. The operation of optical MMI devices is based on the self-imaging principle, presented in Section 11. Basic properties of multimode waveguides are introduced early in Section 111, followed by a short Manuscript received August 8, 1994; revised December 19, 1994. This work was supported in part by the Netherlands Technology Foundation (STW) as part of the programme of the Foundation for Fundamental Research on Matter (FOM). L. B. Soldano is with Delft University of Technology, Department of Electrical Engineering, Laboratory of Telecommunication and Remote Sensing Technology, 2628 CD Delft, The Netherlands. E. C. M. Pennings is with Philips Research Laboratories, Wideband Communication Systems, 5656 AA Eindhoven, The Netherlands. IEEE Log Number 9409613. 11. THE SELF-IMAGING PRINCIPLE Self-imaging of periodic objects illuminated by coherent light was first described more than 150 years ago [7]. Self￾focusing (graded index) waveguides can also produce periodic real images of an object [8]. However, the possibility of achieving self-imaging in uniform index slab waveguides was first suggested by Bryngdahl [9] and explained in more detail by Ulrich [lo], [lll. The principle can be stated as follows: Self-imaging is a property of multimode waveguides by which an input field profile is reproduced in single or multiple images at periodic intervals along the propagation direction of the guide. 111. MULTIMODED WAVEGUIDES The central structure of an MMI device is a waveguide designed to support a large number of modes (typically _> 3). In order to launch light into and recover light from that multimode waveguide, a number of access (usually single￾moded) waveguides are placed at its beginning and at its end. Such devices are generally referred to as N x M MMI couplers, where N and M are the number of input and output waveguides respectively. A full-modal propagation analysis is probably the most comprehensive theoretical tool to describe self-imaging phe￾nomena in multimode waveguides. It not only supplies the basis for numerical modelling and design, but it also provides insight into the mechanism of multimode interference. Other approaches make use of ray optics [12], hybrid methods [13], or BPM type simulations. We follow here the guided-mode propagation analysis (MPA), proposed first in [ll] for the formulation of the periodic imaging. 0733-8724/95$04.00 0 1995 IEEE I -.‘

616 JOURNAL OF LIGHTWAVE TECHNOLOGY.VOL.13.NO.4.APRIL 1995 Self-imaging may exist in three-dimensional multimode nr structures,for which MPA combined with two-dimensional (finite-element or finite-difference methods)cross-section cal- culations can provide a useful simulation tool [14].How- ever,the current trend of etch-patterning produces step-index waveguides,which are,in general,single-moded in the trans verse 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 y analysis hereafter is based on such a 2-D representation of the multimode waveguide,which can be obtained from the actual Fig.I.Two-dimensional representation of a step-index multimode wave. 3-D physical multimode waveguide by several techniques, guide:(effective)index lateral profile (left),and top view of ridge boundaries and coordinate system (right). such as the effective index method (EIM)[15]or the spectral index method (SIM)[16]. A.Propagation Constants Fig.I shows a step-index multimode waveguide of width We Wv.ridge (effective)refractive index n and cladding (effec- tive)refractive index ne.The waveguide supports m lateral modes (as shown in Fig.2)with mode numbers=0. We 1...(m -1)at a free-space wavelength Ao.The lateral +6 wavenumber and the propagation constant B are related to the ridge index n by the dispersion equation +院=品m2 (1) u=0 with Fig.2.Example of amplitude-normalized lateral field profiles (y).cor- responding to the first 9 guided modes in a step-index multimode waveguide. k0= (2) 入0 (w+1)m (3) the propagation constants spacing can be written as Wey where the "effective"width Wer takes into account the (%-)≈+2)m (7) (polarization-dependent)lateral penetration depth of each 3Lz mode field,associated with the Goos-Hahnchen shifts at the ridge boundaries.For high-contrast waveguides,the B.Guided-Mode Propagation Analysis penetration depth is very small so that We WM.In An input field profile (y,0)imposed at z 0 and totally general,the effective widths W can be approximated by the contained within We (Fig.3),will be decomposed into the effective width Weo corresponding to the fundamental mode modal field distributions(y)of all modes: [17].(which shall be noted W for simplicity): o 亚(y,0)=∑c4() (8) Wev≈W.=WM+ (n2-n3)-1/2)(4) where o =0 for TE and o 1 for TM.By using the binomial where the summation should be understood as including expansion withnthe propagation constantsBcan guided as well as radiative modes.The field excitation co- be deduced from (1)-(3) efficients c can be estimated using overlap integrals B≈konm,- (v+1)2TA0 AnrW? (5) 业(y,0)p(y)dy Therefore,the propagation constants in a step-index mul- (9) timode waveguide show a nearly quadratic dependance with 2(）dg respect to the mode number v. By defining L as the beat length of the two lowest-order based on the field-orthogonality relations modes 4n,W2 If the"spatial spectrum"of the input field0)is narrow Lm兰 (6)enough not to excite unguided modes.(a condition satisfied 3入0 for all practical applications),it may be decomposed into the

616 JOURNAL OF LIGHTWAVE TECHNOLOGY. VOL. 13, NO. 4, APRIL 1995 Self-imaging may exist in three-dimensional multimode structures, for which MPA combined with two-dimensional (finite-element or finite-difference methods) cross-section cal￾culations can provide a useful simulation tool [14]. How￾ever, the current trend of etch-patterning produces step-index waveguides, which are, in general, single-moded in the trans￾verse 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 (effec￾tive) 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 kp, + P; = IC&? (1) 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 mul￾timode 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 ID I i II 2 D Fig. 1. Two-dimensional representation of a step-index multimode wave￾guide; (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). cor￾responding 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 co￾efficients 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 1-- -- I

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 y=0 This section investigates the interference mechanisms which 平y,0)5 are independent of the modal excitation,that is,we pose no restriction on the coefficients c and explore the periodicity z-02(3Lx) (BLg)(L)2(3L) of(14). y7 A.Single Images Fig.3.Multimode waveguide showing the input field (y,0),a mirrored By inspecting (13).it can be seen that L)will be an single image at (3Lx).a direct single image at 2(3L).and two-fold images image of乎(y,0)if at(3)and(3). [.v(v+2)π exp 3Ln =1or(-1). (17) guided modes alone The first condition means that the phase changes of all the (,0)= cuv(y) (10) modes along L must differ by integer multiples of 2.In this =0 case,all guided modes interfere with the same relative phases The field profile at a distance z can then be written as a as in0:the image is thus a direct replica of the input field. superposition of all the guided mode field distributions The second condition means that the phase changes must be m-1 alternatively even and odd multiples of In this case,the (,2)=】 ,cv吨w()exp[j(ut-Bz (11) even modes will be in phase and the odd modes in antiphase. =0 Because of the odd symmetry stated in (16),the interference Taking the phase of the fundamental mode as a common produces an image mirrored with respect to the plane y=0. factor out of the sum,dropping it and assuming the time de- Taking into account (15),it is evident that the first and pendence exp(jwt)implicit hereafter,the field profile) second condition of(17)will be fulfilled at becomes L=p(3Lx)with p=0,1,2,·… (18) m-1 Ψ(，z)=∑c()cxpi(-B) (12) for p even and p odd,respectively.The factor p denotes =0 the periodic nature of the imaging along the multimode A useful expression for the field at a distance z=L is then waveguide.Direct and mirrored single images of the input found by substituting (7)into (12) fieldy,0)will therefore be formed by general interference m-1 at distances z that are,respectively,even and odd multiples (y,L)=>cv(y)exp w(w+2)π (13) of the length(3L),as shown in Fig.3.It should be clear at =0 3Lx this point that the direct and mirrored single images can be The shape of (y,L),and consequently the types of images exploited in bar-and cross-couplers,respectively. formed,will be determined by the modal excitation c,and Next,we investigate multiple imaging phenomena,which provide the basis for a broader range of MMI couplers. the properties of the mode phase factor 「.(w+2)π exp 3Ls (14) B.Multiple Images In addition to the single images at distances given by (18). It will be seen that,under certain circumstances,the field multiple images can be found as well.Let us first consider (y,L)will be a reproduction (self-imaging)of the input the images obtained half-way between the direct and mirrored field (y,0).We call General Interference to the self-imaging image positions,i.e.,at distances mechanisms which are independent of the modal excitation: and Restricted Interference to those which are obtained by L=3L）wi曲p=135, (19) exciting certain modes alone. The following properties will prove useful in later deriva- The total field at these lengths is found by substituting (19) tions: into (13) w(w+2)= ∫even for v even (15) Lodd for v odd (,号3L)=cw()exp [fju(v+2p()】 (20) and, (-）= (y) for v even with p an odd integer.Taking into account the property of (15) -(y)for v odd (16) and the mode field symmetry conditions of (16),(20)can be

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 de￾pendence 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 bar- and 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 deriva￾tions: 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 L= -(3L,) with p= 1,3,5,... . (19) 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

618 JOURNAL OF LIGHTWAVE TECHNOLOGY.VOL.13,NO.4.APRIL 1995 304m boundary),and with periodicity 2W 业n()三∑[(y-2w,0)-(-y+v2W,0明 =-00 (22) 122μm 2.4μm and to approximate the mode field amplitudes by sine-like functions 8.0m p(）≈sin(kv) (23) Based on these considerations,(10)can be interpreted as R±300um a(spatial)Fourier expansion,and it is shown [22]that,at distances L=3u) (24) Fig.4.Schematic layout of a 2 x 2 MMI coupler based on the where p 0 and N 1 are integers with no common divisor, general interference mechanism [19].The multimode waveguide length is the field will be of the form LMMI250 um.Offsets are used to minimize losses at the transitions between waveguides of different curvature.Note the widely spaced access N-1 branches.which decrease coupling between access waveguides and obviates (y,L)= (25) blunting duc to poor photolithography resolution. 90 with written as a=p(2q-N)N (26) (u,3L.)=∑cw()+∑(-Pc() even 9g=p(W-9g) W (27) =1+Yu,0+1--Y(-,0 2 2 where C is a complex normalization constant with C= (21) VN,p indicates the imaging periodicity along z,and g refers to each of the N images along y. The last equation represents a pair of images of (y,0), The above equations show that,at distances z=L,N in quadrature and with amplitudes 1/v2,at distances= images are formed of the extended field in(y),located at the (3Lx),L),..as shown in Fig.3.This two-fold imaging positions each with amplitude 1/N and phaseThis can be used to realize 2 x 2 3-dB couplers. leads to N images (generally not equally spaced)of the inpur Optical 2 x 2 MMI couplers based on the single and two- field (y,0)being formed inside the physical guide (within the fold imaging by general interference have been realized in guide's lateral boundaries),as shown in Fig.5.The multiple III-V semiconductor waveguides [18],[19],in silica-based self-imaging mechanism allows for the realization of N x N dielectric waveguides [201,and in non-lattice matched III-V or N x M optical couplers.Shortest devices are obtained for quantum wells [21],[3]. p=1.In this case,the optical phases of the signals in a NxN Fig.4 shows the schematic layout of the InGaAsP 2 x 2 MMI coupler are,(apart from a constant phase),given by multimode interference coupler reported in [181.[19).The 8- um wide multimode section supports 4 guided modes.Excess 4N(s-1)(2N +r-s)+7 for r+s even (28) losses of 0.4-0.7 dB,with extinction ratios of-28 dB at the and cross state (3=500 um)and imbalances well below 0.1 dB for the 3-dB state (3=250 um)were measured for Prs= 4N(r+8-1)(2N-r-s+1)forr+sodd TE and TM polarizations at Xo=1.52 um.The imbalance of (29) an N x M coupler is defined as the maximum to minimum output power ratio for all M outputs,expressed in dB.This where r 1,2,...N is the (bottom-up)numbering of definition will be used throughout the paper. the input waveguides and s=1.2...N is the (top-down) In general,multi-fold images are formed at intermediate z- numbering of the output waveguides. positions [12].Analytical expressions for the positions and It is important to note that the phase relationships given phases of the N-fold images have been obtained [22]by by (28)and (29)are inherent to the imaging properties of using Fourier analysis and properties of generalized Gaussian multimode waveguides.It appears that the output phases of the sums.A very brief summary of the bases and results of that 4 x 4 coupler satisfy the phase quadrature relationship,and derivation is given here.The starting point is to introduce a that this MMI device can be used as a 90-hybrid which is a field in()as the periodic extension of the input field,0):key component in phase-diversity or image rejection receivers antisymmetric with respect to the plane y=0(which,for and which can be used to avoid the quadrature problem in this analysis,is chosen to coincide with one guide's lateral interferometric sensors

618 122 prn offset Fig. 4. Schematic layout of a 2 x 2 MMI coupler based on the general interference mechanism [19]. The multimode waveguide length is LMMI cz 250 pm. Offsets are used to minimize losses at the transitions between waveguides of different curvature. Note the widely spaced access branches, which decrease coupling between access waveguides and obviates blunting due to poor photolithography resolution. written as U even uodd The last equation represents a pair of images of Q(y, 0), in quadrature and with amplitudes 1/ a, at distances z = (3L,), ;(3L,), . . . as shown in Fig. 3. This two-fold imaging can be used to realize 2 x 2 3-dB couplers. Optical 2 x 2 MMI couplers based on the single and two￾fold imaging by general interference have been realized in 111-V semiconductor waveguides [ 181, [ 191, in silica-based dielectric waveguides [20], and in non-lattice matched 111-V quantum wells [21], 131. Fig. 4 shows the schematic layout of the InGaAsP 2 x 2 multimode interference coupler reported in [18], [19]. The 8- pm wide multimode section supports 4 guided modes. Excess losses of 0.4-0.7 dB, with extinction ratios of -28 dB at the cross state (3L, = 500 pm) and imbalances well below 0.1 dB for the 3-dB state ($3L, = 250 pm) were measured for TE and TM polarizations at A0 = 1.52 pm. The imbalance of an N x M coupler is defined as the maximum to minimum output power ratio for all M outputs, expressed in dB. This definition will be used throughout the paper. In general, multi-fold images are formed at intermediate z￾positions [ 121. Analytical expressions for the positions and phases of the N-fold images have been obtained 1221 by using Fourier analysis and properties of generalized Gaussian sums. A very brief summary of the bases and results of that derivation is given here. The starting point is to introduce a field Qin(y) as the periodic extension of the input field q(y, 0); antisymmetric with respect to the plane y = 0 (which, for this analysis, is chosen to coincide with one guide’s lateral JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 13, NO. 4, APRIL 1995 boundary), and with periodicity 2W, 03 Si,(Y) [*(y- W2We,0) - *(-y+v2We;0)] 1)=--03 (22) and to approximate the mode field amplitudes by sine-like functions $Ju(Y) 2 sin(k,vy). (23) Based on these considerations, (10) can be interpreted as a (spatial) Fourier expansion, and it is shown 1221 that, at distances P N L = -(3L,) where p 2 0 and N 2 1 are integers with no common divisor, the field will be of the form with where C is a complex normalization constant with IC1 = fi,p indicates the imaging periodicity along z, and q refers to each of the N images along y. The above equations show that, at distances z = L, N images are formed of the extended field qin(y), located at the positions yn, each with amplitude 1/m and phase pn. This leads to N images (generally not equally spaced) of the input field Q(y, 0) being formed inside the physical guide (within the guide’s lateral boundaries), as shown in Fig. 5. The multiple self-imaging mechanism allows for the realization of N x N or N x M optical couplers. Shortest devices are obtained for p = 1. In this case, the optical phases of the signals in a N x N MMI coupler are, (apart from a constant phase), given by r prs = -(s - l)(2N + T - s) + 7r for T + s even (28) 4N and 7l cprS = -(r + s - l)(2N - T - s + 1) for T + s odd 4N (29) where T = 1, 2, ... N is the (bottom-up) numbering of the input waveguides and s = 1, 2,...N is the (top-down) numbering of the output waveguides. It is important to note that the phase relationships given by (28) and (29) are inherent to the imaging properties of multimode waveguides. It appears that the output phases of the 4 x 4 coupler satisfy the phase quadrature relationship, and that this MMI device can be used as a 90O-hybrid which is a key component in phase-diversity or image rejection receivers and which can be used to avoid the quadrature problem in interferometric sensors. 7- r

SOLDANO AND PENNINGS:OPTICAL MULTI-MODE INTERFERENCE DEVICES BASED ON SELF-IMAGING 619 的W X中 504m R=300um 200m 50μm ☑2书 400um Lmm=34 400μm Fig.6.Schematic layout of the 4x490-hybrid and modal propagation analysis within the multimode waveguide [27].The length and the width of the multimode waveguide are Lmmi945 um and Wmmi 21.6 um. respectively. N-fold images will be formed at distances (cf.(24)) (b) (33) Fig.5.Theoretical light intensity patterns corresponding to general or paired interference mechanisms in two multimode waveguides,leading to a mirrored single image (a).and a 4-fold image (b).Note also the multi-fold images at where p≥0andN≥1 are integers having no common intermediate distances.non-equally spaced along the lateral axis.Reproduced by kind permission of J.M.Heaton,British Crown Copyright DRA 1992. divisor. One possible way of attaining the selective excitation of (31)is by launching an even symmetric input field (y,0)(for Several 4 x 4 MMI optical hybrids have been demonstrated example,a Gaussian beam)at y=+We/6.At these positions, in different technologies and sizes,such as 10-25 mm long the modes y=2,5,8,...present a zero with odd symmetry. semi-bulk constructions of sandwiched glass sheets [23],and as shown in Fig.2.The overlap integrals of(9)between the ion-exchanged waveguides on glass substrates [24],[25]. (symmetric)input field and the (antisymmetric)mode fields Recently,ultra-compact (sub-millimeter length)4 x 4 will vanish and therefore c=0 for v=2,5,8,...Obviously deeply etched waveguide couplers were fabricated by reactive- the number of input waveguides is in this case limited to two. ion etching in III-V semiconductor material [26],[27].These When the selective excitation of (31)is fulfilled,the modes devices (shown in Fig.6)attained excess losses below I dB. contributing to the imaging are paired,i.e.the mode pairs 0-1. imbalances from 0.3-0.9 dB and phase deviations of the order 3-4,6-7,...have similar relative properties.(For example of 5, each even mode leads its odd partner by a phase difference of 7/2 at z=L/2-the 3-dB length-,by a phase difference V.RESTRICTED INTERFERENCE of at=L-the cross-coupler length-,etc).This Thus far,no restrictions have been placed on the modal mechanism will be therefore called paired interference.Two- excitation.This section investigates the possibilities and real- mode interference (TMI)can be regarded in this context as a izations of MMI couplers in which only some of the guided particular case of paired interference. modes in the multimode waveguide are excited by the input 2 x 2 MMI couplers based on the paired interference field(s).This selective excitation reveals interesting multiplic- mechanism have been demonstrated in silica-based dielectric ities of v(+2),which allow new interference mechanisms rib-type waveguides with multimode section lengths of 240 through shorter periodicities of the mode phase factor of(14). um (cross state)and 150 um (3-dB state)[30],[31].Insertion loss lower than 0.4 dB,imbalance below 0.2 dB,extinction ratio of-18 dB,and polarization-sensitivity loss penalty of A.Paired Interference 0.2 dB were reported for structures supporting 7-9 modes. By noting that Calculations predict that power excitation coefficients as low as-40 dB for the modes v=2.5,8 can be achieved through a mod3[w(w+2)=0forw≠2,5,8，· (30) correct positioning of the access waveguides,remaining below -30 dB for a 0.1-um misalignment [29]. it is clear that the length periodicity of the mode phase factor of (14)will be reduced three times if Recently,extremely small paired-interference MMI devices were reported [32].The 3-dB (cross)couplers,realized in a cw=0forv=2,5,8., (31) raised-strip InP-based waveguide,are 107-um(216-um)long, and show 0.9-dB(2-dB)excess loss and-28 dB crosstalk. Therefore,as shown in [28],[29].single (direct and inverted) images of the input field (y,0)are now obtained at (cf.(18)) B.Symmetric Interference L=p(L=)with p=0,1,2,... (32) Optical N-way splitters can in principle be realized on the basis of the general N-fold imaging at lengths given by(24). provided that the modes =2,5.8,...are not excited in the However,by exciting only the even symmetric modes,I-to- multimode waveguide.By the same token,two-fold images are N beam splitters can be realized with multimode waveguides found at(p/2)L with p odd.Based on numerical simulations,four times shorter [33]

I SOLDANO AND PENNINGS: OPTICAL MULTI-MODE INTERFERENCE DEVICES BASED ON SELF-IMAGING (b) Fig 5 Theoretical light intensity patterns corresponding to general or pared interference mechanisms in two multimode waveguides, leading to a mrrored single image (a), and a 4-fold image (b) Note also the multi-fold images at intermediate distances, non-equally spaced along the lateral axis Reproduced by kind pernussion of J M Heaton, @British Crown Copynght DRA 1992 Several 4 x 4 MMI optical hybrids have been demonstrated in different technologies and sizes, such as 10-25 mm long semi-bulk constructions of sandwiched glass sheets [23], and ion-exchanged waveguides on glass substrates [24], [25]. Recently, ultra-compact (sub-millimeter length) 4 x 4 deeply etched waveguide couplers were fabricated by reactive￾ion etching in 111-V semiconductor material [26], [27]. These devices (shown in Fig. 6) attained excess losses below 1 dB, imbalances from 0.3-0.9 dB and phase deviations of the order of 5". V. RESTRICTED INTERFERENCE Thus far, no restrictions have been placed on the modal excitation. This section investigates the possibilities and real￾izations of MMI couplers in which only some of the guided modes in the multimode waveguide are excited by the input field(s). This selective excitation reveals interesting multiplic￾ities of v(v+ 2), which allow new interference mechanisms through shorter periodicities of the mode phase factor of (14). A. Paired Integerence By noting that mods[v(v + 2)] = 0 for v # 2,5,8,. . . (30) it is clear that the length periodicity of the mode phase factor of (14) will be reduced three times if Therefore, as shown in [28], [29], single (direct and inverted) images of the input field q(y, 0) are now obtained at (cf. (18)) L = p(L,) with p = 0,1,2,. . . (32) provided that the modes v = 2, 5, 8, . . . are not excited in the multimode waveguide. By the same token, two-fold images are found at (p/2)L, with p odd. Based on numerical simulations, 619 400pm , Lm,,=3Lrd4 , 400pm Fig. 6. Schematic layout of the 4 x 4 90O-hybrid and modal propagation analysis within the multimode waveguide [27]. The length and the width of the multimode waveguide are L,,, N 945 pm and W,,, N 21.6 pm, respectively. N-fold images will be formed at distances (cf. (24)) P N L = -(Lr) (33) where p 2 0 and N 2 1 are integers having no common divisor. One possible way of attaining the selective excitation of (31) is by launching an even symmetric input field @(y, 0) (for example, a Gaussian beam) at y = kWe/6. At these positions, the modes v = 2, 5, 8, . . . present a zero with odd symmetry, as shown in Fig. 2. The overlap integrals of (9) between the (symmetric) input field and the (antisymmetric) mode fields will vanish and therefore c, = 0 for v = 2,5,8, . . . Obviously, the number of input waveguides is in this case limited to two. When the selective excitation of (31) is fulfilled, the modes contributing to the imaging are paired, i.e. the mode pairs 0- 1, 3-4, 6-7, . . . have similar relative properties. (For example, each even mode leads its odd partner by a phase difference of 7r/2 at z = L,/2-the 3-dB length-, by a phase difference of 7r at z = L,-the cross-coupler length-, etc). This mechanism will be therefore called paired interference. Two￾mode interference (TMI) can be regarded in this context as a particular case of paired interference. 2 x 2 MMI couplers based on the paired interference mechanism have been demonstrated in silica-based dielectric rib-type waveguides with multimode section lengths of 240 pm (cross state) and 150 pm (3-dB state) [30], [31]. Insertion loss lower than 0.4 dB, imbalance below 0.2 dB, extinction ratio of - 18 dB, and polarization-sensitivity loss penalty of 0.2 dB were reported for structures supporting 7-9 modes. Calculations predict that power excitation coefficients as low as -40 dB for the modes v = 2,5,8 can be achieved through a correct positioning of the access waveguides, remaining below -30 dB for a 0.1-pm misalignment [29]. Recently, extremely small paired-interference MMI devices were reported [32]. The 3-dB (cross) couplers, realized in a raised-strip InP-based waveguide, are 107-pm (216-pm) long, and show 0.9-dB (2-dB) excess loss and -28 dB crosstalk. B. Symmetric Inte$erence Optical N-way splitters can in principle be realized on the basis of the general N-fold imaging at lengths given by (24). However, by exciting only the even symmetric modes, l-to￾N beam splitters can be realized with multimode waveguides four times shorter [33]