Transient fields in the input coupling region of optical single-mode waveguides

Transcription

1 Transient fieds in the input couping region of optica singe-mode waveguides Werner Kaus 1* and Water R. Leeb 2 1 Nationa Institute of Information and Communications Technoogy (NICT), 4-2-1, Nukui-Kitamachi, Koganei, Tokyo , Japan 2 Institute for Communications and Radio-Frequency Engineering, Vienna University of Technoogy, Gusshausstrasse 25/389, A-1040 Wien, Austria * Corresponding author: kaus@nict.go.jp Abstract: We investigate numericay the optica fied in the region immediatey behind the input facets of dieectric step-index singe-mode sab and fiber waveguides. Visuaization of the intensity distributions gives insight into the formation of the fundamenta mode and of radiation modes. For a more quantitative characterization we determine the amount of optica power and mode purity of the fied in core vicinity as a function of propagation distance. The investigation assists in designing and optimizing waveguides being empoyed as moda fiters, e.g. for astronomica interferometers Optica Society of America OCIS codes: ( ) Guided waves; ( ) Waveguides; ( ) Fiber optics components; ( ) Fibers, singe-mode; ( ) Moda fiter References and inks 1. B. Mennesson, M. Oivier, and C. Ruiier, Use of singe-mode waveguides to correct the optica defects of a nuing interferometer, J. Opt. Soc. Am. A 19, (2002). 2. O. Waner, W. R. Leeb, and R. Fatscher, Design of spatia and moda fiters for nuing interferometry, in Interferometry for Optica Astronomy II, Wesey A. Traub, ed., Proc. SPIE 4838, (2003). 3. J. C. Fanagan, D. J. Richardson, M. J. Foster, and I. Bakaski, "Microstructured fibers for broadband wavefront fitering in the mid-ir," Opt. Express 14, (2006). 4. C. V. M. Fridund, DARWIN The Infrared Space Interferometry Mission, ESA Buetin 103, (2000). 5. O. Waner, et a., Minimum ength of a singe-mode fiber spatia fiter, J. Opt. Soc. Am. A 19, (2002). 6. P. Cheben, D. -X. Xu, S. Janz, and A. Densmore, "Subwaveength waveguide grating for mode conversion and ight couping in integrated optics," Opt. Express 14, (2006). 7. L, Li, New formuation of the Fourier moda method for crossed surface-reief gratings, J. Opt. Soc. Am. A 14, (1997). 8. A. W. Snyder and J. D. Love, Optica waveguide theory (Chapman & Ha, 1983), p. 259 ff. 9. G. Grau and W. Freude, Optische Nachrichtentechnik (Springer, 1991), pp E. Siberstein et a., "Use of grating theories in integrated optics," J. Opt. Soc. Am. A 18, (2001). 11. C. Zhou and L. Li, Formuation of the Fourier moda method for symmetric crossed gratings in symmetric mountings, J. Opt. A: Pure App. Opt. 6, (2004). 1. Introduction In a singe-mode optica waveguide, the fied distribution commony caed fundamenta mode constitutes a steady-state soution of Maxwe's equations. Stricty speaking, it exists ony if the radiation at the guide's input facet aready perfecty matches the mode in both ampitude and phase, or after infinitey ong guidance. Yet, reativey short pieces of singe-mode fibers may act as highy effective moda fiters. A moda fiter has to provide a unique ampitude and phase distribution at its output facet, irreevant of the input fied. Moda fiters constitute key devices in interferometric instruments where pronounced destructive interference of two or (C) 2007 OSA 17 September 2007 / Vo. 15, No. 19 / OPTICS EXPRESS 11808

2 more input beams is required [1-3]. An exampe for such an appication is ESA's DARWIN mission which aims at the investigation of extra-soar panets [4]. The concept reies on astronomica interferometry and asks for an excessive nuing capabiity to suppress the star's radiation whie anayzing the faint ight refected from a nearby panet. In a moda fiter, the swift convergence of the fied distribution towards the one of the fundamenta mode is highy desirabe. It wi depend on waveguide geometry and materia properties, and of course on the form of the incident wave. For an efficient design it is essentia to understand the formation of the fundamenta mode shorty behind the input facet and its dependence on the waveguide parameters. A good design wi be characterized by a waveguide ength that is as short as possibe without sacrificing too much moda fiter functionaity. One attempt to anayticay estimate the minimum ength of an idea fiber waveguide was presented in [5]. That approach, however, coud not take into account many of the rea-word aspects occurring from the viewpoint of both the input fied and the waveguide geometry. Concerning the probem of input couping, more recenty a method was proposed for achieving efficient mode conversion between a fiber and a sub-micrometer waveguide by using a subwaveength grating [6]. Here we present resuts for the fied distribution in the input couping region of singemode waveguides as obtained with a rigorous numerica too based on the Fourier Moda Method [7]. To gain basic insight into the fied behavior shorty behind the input facet, we first anayze the computationay easier two-dimensiona case, i.e. a sab waveguide. We examine how the intensity distribution near the core reaches the steady-state of the fundamenta mode and cacuate power fow and couping efficiency. We further define and cacuate the mode purity which serves as a quantitative measure of the conformity of the fied with the idea fundamenta mode. In a second step, we examine to a esser extent the computationay much more extensive three-dimensiona fiber waveguide, and in particuar ook for any fundamenta differences in fied behavior, power distribution and mode purity compared to the sab waveguide. The paper is structured as foows: In Section 2 we describe the system mode and introduce some definitions required for the proper description of the output fied s quaity. In Section 3 we outine the numerica method empoyed. Section 4 is dedicated to the discussions of the numerica resuts of sab and fiber waveguide. Various input couping situations and the effect of an input pupi positioned just in front of the waveguide s input facet are investigated. In Section 5 we compare the resuts obtained for the sab waveguide with those for the fiber waveguide, whie Section 6 demonstrates the exceent agreement between numerica resuts and anaytica soutions for some specia cases. Finay, in Section 7 we present a summary and concusions. 2. Waveguide modeing and definitions of characteristics Figure 1 shows the waveguide mode and an ideaized optica set-up used to estimate the quaity of the mode at the waveguide output. The step-index singe-mode waveguide consists of a core with refractive index n 1 and thickness 2a, and an infinitey thick cadding with index n 2. Both input and output facets are assumed to carry an idea antirefection coating. A pupi of width D A may be positioned just in front of the waveguide s input facet to hinder part of the input fied from being couped into the cadding and thus removing undesired fied fractions inside the waveguide. Another pupi of width D B paced immediatey behind the waveguide s output facet serves as the exit pupi controing the size of the fied sampe to be used for the evauation of the mode purity. (C) 2007 OSA 17 September 2007 / Vo. 15, No. 19 / OPTICS EXPRESS 11809

3 Fig. 1. System mode. P in.. input power, 2w.. beam width (defined by first zeros in the foca pane), 2a.. core width, D A.. input pupi width, D B.. output pupi width, D C.. near-core power width, p(z).. near-core power, E W.. waveguide fied just before output facet, E f.. fied of fundamenta mode. The constructive and destructive superposition of fied E w with the fundamenta mode E f eading to output power P + and P - serves ony to iustrate the definition of mode purity MP. The right part shows the cross section in case of a fiber waveguide. Tabe 1 ists the waveguide parameters chosen for the numerica cacuations presented in Section 4, with λ 0 representing the shortest waveength under consideration. For both the sab and the fiber, operation at waveength λ = λ 0 is a few percent beow the cut-off of the first higher-order mode. The input pupi is modeed to consist of a metaic ayer of thickness λ 0 with index of refraction n = j10. Tabe 1. Waveguide parameters used for the numerica cacuations (Δ.. reative index difference, V.. normaized frequency, V C.. cut-off frequency). Aso given is the input pupi width D A (if not expicity set to infinity) and the output pupi width D B. 2a n 1 n 2 Δ λ 0 λ 0 / V C D A D B sab 4.5 λ % % 11 λ 0 18 λ 0 fiber 4.5 λ % % 8.8 λ 0 18 λ 0 An idea ens focuses a truncated pane wave to a spot in the foca pane. This pane may be offset from the input pane by a distance Δz ("defocus"). (A positive Δz means that the focus is within the waveguide). In the two-dimensiona case, i.e. the sab waveguide, the input fied distribution is proportiona to f ( x' ) x' sin π w = (1) x' i.e., a sinc distribution with the first zeros at x = w. Thus the spot width of the input beam is 2w. A tit represented by the ange ε 0 in the fied distribution is expressed by using a sighty rotated coordinate system with x instead of x. Since differences in the fied distributions for the two possibe poarization states TE (eectric fied poarized orthogona to the pane of incidence) and TM (magnetic fied poarized orthogona to the pane of incidence) are rather sma, we wi, in the foowing, ony consider the case of TE poarization, i.e. the input fied is poarized aong the y-direction. (C) 2007 OSA 17 September 2007 / Vo. 15, No. 19 / OPTICS EXPRESS 11810

4 In the three-dimensiona case, i.e. the fiber waveguide, the incident radiation is again assumed to be focused by a ens. However, here we wi ony consider the case of norma incidence (ε = 0). The input distribution is thus proportiona to f () r r J1 1.22π w = with r r = x y, (2) i.e., an Airy distribution whose first zero is at r = w. In the fiber case, the quantity 2w denotes the spot diameter of the input beam. J 1 represents the Besse function of the first kind, order one. For the characterization of the usefuness of a singe-mode waveguide as a moda fiter we introduce the foowing three quantities: 2.1. Near-core power The near-core power p(z) represents the fraction P DC of the tota power in the waveguide that is found within a width D C centered around the core, normaized to the input power P in, i.e. () z P P () z DC p = (3) D C shoud be arge enough to contain essentiay a the power of the steady-state fundamenta mode. Our choice of D C = 8a = 18 λ 0 provides more than 99.99% of the mode s power in P DC for z for an operation at λ = λ 0. Even for a waveength λ = 1.5 λ 0, more than 99.3% of the power of the fundamenta mode is incuded. Diagrams presenting the function p(z) thus iustrate how fast the fied in the waveguide approaches the fundamenta mode. In case of the fiber waveguide the cacuation of power deserves specia attention: Even if the pane wave incident on the ens is ineary poarized, say aong the y-direction, as assumed for the numerica cacuations, the fied in the foca pane wi aso contain components though sma in z and x direction, otherwise Maxwe s equations woud not be satisfied. These components (E z, E x ) are not Airy-distributed. The above defined quantities P DC, P in and hence aso p(z) contain a three fied components. Aong this ine, the exact soution for the fundamenta mode of a step index fiber does aways contain a three fied components [8], [9]. For the fiber parameters given in Tabe 1, the maxima E z component is by a factor of 25 smaer than that of the dominating E y component, and the maximum ampitude of E x is smaer by another factor of 25. The numerica method we appy for cacuating the fied in the foca pane and in the fiber (see Section 3) does automaticay yied the fied components E z and E x Couping efficiency We define the couping efficiency η as the ratio of the power carried by the waveguide s fundamenta mode to the input power P in. However, as the mode extends to infinity in transverse direction, the numerica determination requires a sighty modified definition. We maintain sufficient accuracy when taking instead of the tota power of the fundamenta mode the power P DC for D C = 18 λ 0 in its imit for z, designated P DC, z. The couping efficiency is thus given as η P in DC,z = (4) Pin (C) 2007 OSA 17 September 2007 / Vo. 15, No. 19 / OPTICS EXPRESS 11811

5 2.3. Mode purity So far we introduced quantities which characterize ony the amount of ight couped into the fundamenta mode. In contrast, the foowing definition of mode purity MP(L), with L as the waveguide ength, offers a measure of how we the actua ampitude and phase distribution E w (x) = A w (x)exp(jϕ w (x)) found just before the output pupi resembe the idea distribution of the fundamenta mode E f (x) = A f (x)exp(jϕ f ). In case of the sab, the quantities E w, A w, ϕ w and E f, A f, are functions of x, whie for the fiber they depend on the coordinates r and ϕ. We define mode purity by means of output powers P + and P -, as indicated in the right part of Fig. 1. The first one, P +, is the power resuting from an in-phase superposition of the fied E w with the idea fied E f, where for E f we wi take the numerica soution of the steady-state fundamenta mode. The second one is due to an out-of-phase superposition. For determining P + and P - we set at the waveguide axis the fied ampitude A f equa to A w. In the same way, the phase reationship for cacuating constructive and destructive interference, i.e. ϕ w - ϕ f = 0 and 180, is defined at the waveguide axis. It turned out that the function MP(L) is amost independent on whether or not one incudes the components E z and E x in the cacuation of the three-dimensiona case. To avoid unnecessary compexity, we based the cacuation of mode purity MP of the fiber ony on the ampitude and phase of the dominating y-components. The powers P + and P - depend on the pupi width D B and are reated to the corresponding fied intensities I + and I - by DB 2 ± Psab Isab ( x)dx and P I ( r, ϕ) rdrdϕ DB 2 ± DB 2π 2 fiber fiber (5) ± ± for the sab and the fiber, respectivey. For the sab the intensities I + and I - are in turn given by the eectric fieds as 0 A [ ( )] [ ( )] 2 w (x = 0) j x, z = L ± A (x) exp j x = 0, z L I A (x,z= L)exp ϕ =, (6) sab ± w w f ϕw Af (x = 0) and correspondingy for the fiber. The mode purity MP then foows as 0 P MP = + (7) P The quantity MP tends to infinity in the case where the fied distribution E w approaches that of the fundamenta mode, E f. In genera, a smaer vaue of output pupi diameter D B wi resut in a higher vaue of MP, however at the cost of reduced overa optica throughput. 3. Numerica method To anayze the fied distribution in the waveguide we empoy a rigorous eectromagnetic numerica too that is based on the Fourier moda method (FMM) [7]. The FMM was originay deveoped in the fied of grating theory. In recent years, however, a number of extensions and improvements added to the origina idea have made FMM a versatie and popuar too deaing effectivey with a much wider variety of micro-optica eements. A characteristic of FMM is that it requires the structure under investigation to be divided into segments with respect to the propagation direction, e.g., the z-direction. The thickness of each (C) 2007 OSA 17 September 2007 / Vo. 15, No. 19 / OPTICS EXPRESS 11812

6 segment is chosen such that the materia parameters (e.g., the refractive index n) depend ony on the transverse coordinates x and y. Within an individua segment the soution of Maxwe s equations can then be described in the form of jbmz ( x, y,z) S ( x, y) e E = (8) m which defines the m th (eigen)mode for this specific segment. Mathematicay, S m (x,y) and b m represent the eigenfunction and eigenvaue of the soution. The simpe harmonic z- dependence of the mode is the key to find soutions for ong structures as efficienty as for short structures. The tota fied distribution in each segment is described via the compete basic set of modes which, in case of a waveguide structure, consists in genera of a few guided modes and an infinite number of radiation modes. Fortunatey, most of the higher-order radiation modes die off quicky with increasing propagation distance z and can therefore be negected without sacrificing accuracy. For easy numerica handing, each mode is furthermore represented mathematicay by a Fourier series. The x- and y-dependent materia parameters are described by Fourier series as we. Each segment represents thus an autonomous eigenvaue probem whose soutions are the moda fieds that are specific to the segment s geometry and materia parameters. The compete set of modes in segment is described by E where M P Q ( x, z) = ( A exp( jb z) + B exp( jb z ) S exp( jkpx) exp( jkqy) A m and m m m m pqm m = M p = P q = Q (9) B m stand for the mode ampitudes of forward and backward propagating waves, and K = 2π/Λ is the fundamenta spatia frequency with Λ being the structure s period in transverse direction. The 2M+1 eigenvectors summarized in S pqm and the eigenvaues b m represent now the compete set of soutions to the eigenvaue probem discussed above. Apart from the focusing optics, the structure shown in Fig. 1 consists of three segments. A uniform semi-infinite input segment with a constant index n = 1, a thin metaic segment containing the x- and y-dependent index distribution of the input pupi, and a semi-infinite output segment with the x- and y-dependent index distribution of the waveguide. The output pupi is part of the ideaized optica set-up to compute the mode purity and therefore not considered in the rigorous simuations. The input fied distribution in a uniform segment, such as free space, is represented by a set of pane waves since pane waves are the modes of a uniform segment and the matrix S pqm degenerates therefore to a unitary matrix. On the other hand, the fied distribution inside the waveguide is represented by both guided and radiation modes which are specific to the waveguide structure. The vaues P and Q determine the maximum size of the Fourier representation. The magnitudes of M, P and Q affect the accuracy of the cacuation but aso the amount of computationa time and dynamic memory and therefore need to be chosen carefuy. Finay, by adjusting the mode ampitudes A m and B m such that the fied distributions fufi Maxwe s boundary conditions at each segment interface, we obtain a system of inear equations that is soved numericay by matrix computation. Athough the use of Fourier series makes the system periodic in transverse direction, we are sti abe to anayze singe aperiodic eements such as waveguides by introducing absorbing, perfecty matched ayers aong the rim of the cacuation area parae to the z- direction. The absorption and thickness of these ayers must be chosen such that fied contamination from neighboring waveguide structures due to the inherent periodicity is kept to a minimum. The condition of perfect matching with respect to the materia parameters is needed for suppressing refections from these additiona ayers [10]. (C) 2007 OSA 17 September 2007 / Vo. 15, No. 19 / OPTICS EXPRESS 11813

7 4. Fied behavior for various waveguide geometries and input couping situations In this section we present numerica resuts both for the sab and fiber singe-mode waveguide: First, two-dimensiona coor coded intensity distributions for various input couping situations give a physica insight into how the incident free-space beam is either converted into the waveguide s fundamenta mode or distributed across the cadding in form of radiation modes. In accordance with Fig. 1, the input radiation enters the waveguide from the eft, passes through a transient region whose ength mainy depends on the couping situation, and eventuay approaches the fundamenta mode. To visuaize the fied behavior within a arge dynamic range, the intensity distributions are normaized with respect to their maximum vaues and shown on a ogarithmic scae from 0 to -40 db. The waveguide s core region appears therefore as a red horizonta bar, as this is the ocation of highest intensity. In these figures we have indicated the waveguide input facet and the core-cadding boundaries by thin back vertica and horizonta ines. Second, more quantitative information about couping efficiency and power variation within a cross section of width D C = 18 λ 0 in core vicinity can be read off from graphs showing the near-core power p(z) aong the z-direction. Finay, the waveguide s fiter quaity is estimated with the hep of graphs showing the mode purity MP as a function of waveguide ength L. 4.1 The sab waveguide Using the waveguide parameters specified in Section 2 for the sab waveguide, we found that without input pupi and for operation at λ = λ 0 the maximum couping efficiency into the fundamenta mode is obtained with an input beam width of cose to 2w = 11 λ 0 and perfect aignment (see Fig. 2). This scenario wi serve as a reference. Most of the power not guided by the waveguide is radiated off within the first 1000 λ 0 behind the input facet under an ange of approximatey α = ± 4. Figure 2(b) provides an in x-direction magnified version of Fig. 2(a), covering an area of 100 λ 0 x 4000 λ 0 of the waveguide (again with unequa scae in x- and z-direction) pus an area of 100 λ 0 x 100 λ 0 in front of the facet to iustrate the converging input beam. (C) 2007 OSA 17 September 2007 / Vo. 15, No. 19 / OPTICS EXPRESS 11814

8 Fig. 2. Sab waveguide operated at λ = λ 0 : Intensity distributions without input pupi for a beam width of 2w = 11 λ 0 and perfect aignment (Δz = 0, ε = 0, Δx = 0). (a) extension in x- direction is 500 λ 0, (b) extension in x-direction is 100 λ 0. Next we investigated the effect of an input pupi on the fied behavior in the transient region. We found that a pupi width of D A = 11 λ 0 exacty matching the beam width (2w = 11 λ 0 ) eads to optimum couping efficiency. Figure 3(a) depicts again the intensity distribution for perfect aignment, whie Figs. 3(b) to 3(d) demonstrate the infuence of input beam misaignments. In Fig. 3(b), a shift of the foca spot by Δz = 50 λ 0 beyond the input facet resuts in an increased amount of power radiated off in a wider anguar range. Figures 3(c) and 3(d) show how anguar and atera misaignment ead to enhanced power oss in the transient region. For an input ange of ε = 1 we find an asymmetric radiation of the unguided modes, whie cose-to-symmetric behavior was observed for a atera dispacement of Δx = 1 λ 0. Note that both misaignments cause a simiar osciatory behavior of the fied around the waveguide axis, resuting in a significant extension of the transient region. When comparing Figs. 2(b) and 3(a) it is evident that a pupi with a propery chosen width is capabe of suppressing the radiation modes to some extent. We concude that for waveguides without input pupi the number of radiation modes excited does not ony increase with increasing mismatch between the sinc distribution s centra obe and the waveguide s fundamenta mode but is aso caused by the sinc distribution s sideobes, haf of which are out-of-phase with the main obe (C) 2007 OSA 17 September 2007 / Vo. 15, No. 19 / OPTICS EXPRESS 11815

9 Fig. 3. Sab waveguide operated at λ = λ 0 : Intensity distributions with input pupi (D A = 11 λ 0 ) for a beam width of 2w = 11 λ 0. (a) perfect aignment (Δz = 0, ε = 0, Δx = 0) (b) defocus Δz = 50 λ 0 (c) anguar misaignment ε = 1 (d) atera misaignment Δx = 1 λ 0. (C) 2007 OSA 17 September 2007 / Vo. 15, No. 19 / OPTICS EXPRESS 11816

10 Figure 4 shows how the near-core power p(z) graduay approaches the steady state for the five cases discussed above. The vaues on the right represent the couping efficiencies η. In case of perfect aignment, these near-core power graphs confirm the positive effect of an input pupi in a twofod way: First, couping efficiency is increased by some 2%, and second, the fied approaches the steady state within a shorter distance. Fig. 4. Sab waveguide: Near-core power p(z) as a function of distance z from the input facet for the cases of Fig. 2 and Figs. 3(a) to 3(d). The numbers at the right give the couping efficiencies η. The presence of an input pupi aso affects the mode purity MP in a favorabe way. Figure 5 shows MP as a function of waveguide ength L for two input pupi widths, namey for the optimized vaue of D A = 11 λ 0 and for a much arger one with D A = 100 λ 0. The atter is practicay identica to the case without input pupi D A =. These graphs again demonstrate that the presence of sideobes in the input beam keeps the fied onger in its transient state and therefore requires a onger waveguide ength to reach the same vaue of mode purity. In our specific exampe the waveguide without pupi woud require about twice the ength to achieve a mode purity of MP = Fig. 5. Sab waveguide operated at λ = λ 0 : Mode purity MP as a function of waveguide ength L for a beam width of 2w = 11 λ 0 and for input pupi diameters of D A = 11 λ 0 and D A = 100 λ 0. The atter case is practicay identica to the case without input pupi (D A = ). (C) 2007 OSA 17 September 2007 / Vo. 15, No. 19 / OPTICS EXPRESS 11817

11 In Fig. 6 we present mode purity graphs for the four cases of input couping covered in Fig. 3 for two different waveengths of the input radiation, i.e. for the design waveength λ = λ 0 and for a waveength increased by 50% (λ = 1.5 λ 0 ). We assume the focusing optics to be one and the same and thus the beam width to increase by a factor of 1.5 when operating at λ = 1.5 λ 0, i.e. 2w = 16.5 λ 0. Whie the mode purity is strongy affected by misaignments when operating the waveguide at the design waveength (i.e. just beow the waveguide s singe mode cut-off frequency), these infuences are ess pronounced for operation at onger waveengths. However, this comes at the cost of a reduced couping efficiency η: For a fixed input width of D A = 11 λ 0 we found that η is reduced by 0.58 db for operation at λ = 1.5 λ 0. Fig. 6. Sab waveguide: Mode purity MP as a function of waveguide ength L for the four cases of aignment specified in Fig. 3. (a) for operation at λ = λ 0, (b) for operation at λ = 1.5 λ Fied behavior of the sab waveguide with core inhomogeneities Next we investigated the effect of sma variations of the core diameter and core refractive index on mode purity and near-core power p(z). To this end, we divided the waveguide into three arge sections. The first and third section consist of a homogeneous waveguide region equa to the one discussed in Section 4.1. The midde section is subdivided into 400 sma waveguide segments whose engths vary randomy between 3 λ 0 and 7 λ 0. Furthermore, each segment contains a core whose refractive index varies randomy between and (C) 2007 OSA 17 September 2007 / Vo. 15, No. 19 / OPTICS EXPRESS 11818

12 and whose width varies randomy between 4 λ 0 and 4.5 λ 0 (see Fig. 7). The tota ength of the inhomogeneous section is approximatey 2000 λ 0. Fig. 7. Schematic of inhomogeneous waveguide. The mode purity as shown in Fig. 8 was evauated with respect to the fundamenta mode in the homogeneous (first or third) region as a function of waveguide ength L. Shorty after the onset of the inhomogeneous region the mode purity coapses to vaues beow 10 4 and stays there as ong as inhomogeneities are present. Recovery to higher vaues is ony possibe after the fied enters again the homogeneous region. Fig. 8. Sab waveguide operated at λ = λ 0 : Mode purity as a function of L for a waveguide with (bue) and without (red) an inhomogeneous core with a ength of about 2000 λ 0, starting at z = 2000 λ 0. The input beam parameters are 2w = 11 λ 0 and perfect aignment. Whie the effect of inhomogeneities on the mode purity is quite dramatic, the oss in nearcore power remains sma (of the order of 1%), as shown in Fig. 9. Ceary, refections from the inhomogeneous section (2000 λ 0 < z < 4000 λ 0 ) affect the fied in the first homogeneous region (0 < z < 2000 λ 0 ). The seemingy ong period of some 200 λ 0 of the excited standing wave in this region is a numerica artifact caused by undersamping (one sampe per 2 λ 0 ) of the function p(z) in direction of increasing distance z and cannot be directy reated to the waveguide geometry or materia parameters. (C) 2007 OSA 17 September 2007 / Vo. 15, No. 19 / OPTICS EXPRESS 11819

13 Fig. 9. Sab waveguide operated at λ = λ 0 : Near-core power p(z) as a function of distance z for a waveguide with (bue) and without (red) inhomogeneities. The input beam parameters are 2w = 11 λ 0 and perfect aignment. The vaues at the right of the diagram represent the couping efficiencies η to the fundamenta mode in the third section Fiber waveguide In a simiar vein, as with the sab waveguide, we characterize the fied in a fiber waveguide s transient region by cacuating the two-dimensiona intensity distributions in the pane containing the fiber axis z, the couping efficiencies, the near-core power, and the mode purity. The definitions for these characteristics are the same as the ones given by Eqs. (4), (5) and (7) except that the power is now obtained by integrating over an axis-centered circuar area of diameter D C in case of near-core power and couping efficiency, and diameter D B in case of the mode purity. The huge requirements on dynamic memory and computation time for such three-dimensiona simuations were party reaxed by expoiting mirror symmetries [11] of the input fied and the waveguide structure with respect to the transverse coordinates x and y. On the other hand, the use of such symmetries restricted our present study to on-axis input fied iumination at norma incidence ony, i.e. to ε = 0 and Δx = 0. For the fiber parameters specified in Section 2, the maximum couping efficiency into the fundamenta mode of a waveguide without input pupi and operated at λ = λ 0 is obtained for an input beam diameter cose to 2w = 8.8 λ 0. The corresponding intensity distribution is shown in Fig. 10. Simiar to the case of the sab waveguide, the mismatch between the incident Airy distribution and the distribution of the fundamenta mode gives rise to severa radiation modes which spread into the cadding, with cone anges of up to some 5. Obviousy, for wave propagation in three dimensions, these radiation modes die off much faster than their two-dimensiona counterparts, thus reducing the ength of the transient region compared to the sab guide. As expected, if the input matching is reduced by defocusing [see Fig. 10(b)], more power is radiated off into the cadding. (C) 2007 OSA 17 September 2007 / Vo. 15, No. 19 / OPTICS EXPRESS 11820

14 Fig. 10. Fiber operated at λ = λ 0 : Intensity distributions without input pupi, input beam diameter 2w = 8.8 λ 0. (a) perfect aignment (b) defocus Δz = 50 λ 0. As with the sab waveguide, we expect aso for the fiber to find an improvement of the transient fied by pacing a (circuar) pupi in front of the waveguide s input facet. Figure 11 demonstrates that an input pupi which is matched to the input beam diameter significanty reduces both the amount of radiated power into the cadding and the ength of the transient region. Here, too, defocusing wi resut in both suboptima ampitude and phase distribution at the input and thus increases the amount of power going into the radiation modes, as evidenced by Fig. 11(b). (C) 2007 OSA 17 September 2007 / Vo. 15, No. 19 / OPTICS EXPRESS 11821

15 Fig. 11. Fiber operated at λ = λ 0 : Intensity distributions with input pupi, input beam diameter 2w = 8.8 λ 0. (a) perfect aignment (b) defocus Δz = 50 λ 0. The near-core power p(z) for both cases is shown in Fig. 12. Whether with or without misaignment, the power converges quicky, i.e. within the first 200 λ 0, towards the vaue suggested by the couping efficiency. The curves aso confirm the positive effect of the input pupi on the couping efficiency, whose improvement amounts to neary 3% in this specific exampe. Fig. 12. Fiber operated at λ = λ 0 : Near-core power p(z) as a function of distance z for 2w = 8.8 λ 0. The numbers at the right give the couping efficiencies η. Dotted red ine: perfect aignment without input pupi (D A = ) Soid red ine: perfect aignment with input pupi (D A = 8.8 λ 0 ) Bue ine:defocus Δz = 50 λ 0, with input pupi (D A = 8.8 λ 0 ). Finay, the dependence of mode purity MP on fiber ength L is shown in Fig. 13 and Fig. 14. Figure 13 compares the mode purity for input pupis that strongy differ in their (C) 2007 OSA 17 September 2007 / Vo. 15, No. 19 / OPTICS EXPRESS 11822

16 diameter. Simiar to the sab waveguide, the presence of an input pupi increases the mode purity by an order of magnitude for a given fiber ength. Figure 14 shows two pairs of MP curves for the cases of two different input couping conditions (with and without a foca shift of Δz = 50 λ 0 ) and two different waveengths. The curves for the onger waveengths are ess affected by misaignment, however the couping efficiency is reduced by some 1.7 db due to the arger input beam size and the fixed input pupi diameter. As with the sab waveguide, the increase of waveength by a factor of 1.5 entais a arger spot size, i.e. 2w = 13.2 λ 0. Fig. 13. Fiber operated at λ = λ 0 with input beam diameter 2w = 8.8 λ 0 : Mode purity MP as a function of waveguide ength L for input pupi diameters of D A = 8.8 λ 0 and D A = 100 λ 0. The atter case is practicay identica to the case without input pupi (D A = ). Fig. 14. Fiber: Mode purity MP as a function of waveguide ength L for perfect aignment (red ine) and for a defocus of Δz = 50 λ 0 (bue ine) when operated at λ = λ 0. The dotted ines give MP for operation at λ = 1.5 λ 0 (green ine: perfect aignment, magenta ine: defocus of Δz = 50 λ 0 ). 5. Comparison of fieds in sab and fiber waveguide Figure 15 compares the near-core power p(z) for the sab waveguide (dotted ines) and the fiber waveguide (soid ines) for norma incidence and an input pupi width equa to the beam width (D A = 2w = 11 λ 0 for the sab, D A = 2w = 8.8 λ 0 for the fiber). The red ines represent perfect aignment, the bue ines a defocus of Δz = 50 λ 0. The near-core power for the fiber waveguide shows a sighty faster convergence towards the couping efficiency vaue. This may be attributed to the fact that here unguided radiation is distributed over the tota waveguide voume, i.e. in three-dimensiona space, whie for the sab geometry it is (C) 2007 OSA 17 September 2007 / Vo. 15, No. 19 / OPTICS EXPRESS 11823

17 distributed ony over the x-z pane. As expected, defocusing causes smaer couping efficiency η. Fig. 15. Comparison of near-core power p(z) for sab (dotted ines) and fiber (soid ines) with input pupis, operated at λ = λ 0. The red ines are for perfect aignment, the bue ines for a defocus of Δz = 50 λ 0. In Fig. 16 we compare mode purity MP as a function of waveguide ength L for the sab waveguide and the fiber waveguide. Again, both cases are evauated for a system with input pupi widths equa to the optimum beam widths. The output aperture width was taken as D B = 18 λ 0. As expected, for given ength L perfecty aigned input fieds (red ines) ead to higher mode purity than sighty defocused fieds (bue ines). To achieve the same vaue of mode purity, the fiber waveguide (soid ines) requires a onger ength. This behavior may be attributed to the different integration areas in the two-dimensiona and three-dimensiona case when determining the power vaues P + and P -. Fig. 16. Comparison of mode purity MP for sab (dotted ines) and fiber (soid ines) with input pupis, operated at λ = λ 0. The red ines are for perfect aignment, the bue ines for a defocus of Δz = 50 λ 0. (C) 2007 OSA 17 September 2007 / Vo. 15, No. 19 / OPTICS EXPRESS 11824

18 6. Comparison of numerica resuts and anaytica soutions In case of perfect aignment of the input fied we can compare some of the resuts obtained by the numerica fied cacuation with anaytica soutions. This comparison aso serves as vaidity check for our numerica resuts where anaytica soutions are not avaiabe. As demonstrated in Tabe 2, except for two out of the 16 cases, exceent agreement is obtained, giving a cear indication of the correctness of our numerica approach. For the two cases, highighted in grey, we expain the difference as foows: Due to the arger foca spot size, the incident fied shows at the metaic pupi a non-zero tangentia fied component in case of the sab waveguide and non-zero tangentia and norma fied components in case of the fiber waveguide. This causes some absorption and fied distortion by the pupi which changes the fied shape at z = 0 (the ocation of the waveguide s facet) and therefore aso affects the couping efficiency η. Of course this effect coud not be taken care of in the anaytica cacuation. Note that in the fiber case the numerica resuts took into account a three fied components E x, E y and E z, whie the anaytica resuts are based on the assumption of the existence of the singe fied component E y ony. Tabe 2: Comparison of resuts (perfect aignment). waveguide sab fiber waveength λ λ λ 0 λ λ 0 item numerica resut anaytica resut p(z = 0) for D A = (no input pupi) 2) p(z = 0) for D A = 11 λ couping efficiency η for D A = ) couping efficiency η for D A = 11 λ ) p(z = 0) for D A = 2) p(z = 0) for D A = 11 λ couping efficiency η for D A = ) couping efficiency η for D A = 11 λ ) p(z = 0) for D A = 2) p(z = 0) for D A = 8.8 λ ,8378 couping efficiency η for D A = ) couping efficiency η for D A = 8.8 λ ) p(z = 0) for D A = 2) p(z = 0) for D A = 8.8 λ couping efficiency η for D A = ) couping efficiency η for D A = 8.8 λ ) 1) incuding truncation by output aperture of width D B = 18 λ 0 2) here truncation is due to the definition of "near-core power", i.e. by the width D C 7. Summary and concusion When the fied at the input facet of a singe-mode waveguide does not perfecty match the one of the waveguide's fundamenta mode, the actua formation of the fundamenta mode is a process which may take pace over a ength of few hundreds to a few thousands of waveengths. Using the Fourier Moda Method we have cacuated the transient fieds after the input facets of both sab and fiber step-index waveguides. Whie coor-coded diagrams (C) 2007 OSA 17 September 2007 / Vo. 15, No. 19 / OPTICS EXPRESS 11825

19 showing the intensity distribution in the meridiona pane give a good quaitative picture of how the unguided power is radiated off, the near-core power is a quantitative measure of the deveopment of the fundamenta mode aong the waveguide axis. Lasty, our definition of mode purity aows us to determine to a very high degree the identity of the actua waveguide fied with the idea mode. In this context, the definition of an exit pupi turned out to be essentia. For the numericay evauated exampes we have assumed parameters eading to a fundamenta mode with a normaized frequency a few percent beow the next mode s cut-off. We showed that the estabishment of the fundamenta mode strongy depends on both the distribution of the input fied and any misaignment of the input wave. Mode purity is very sensitive to even sma misaignments of input fied and to any variations in waveguide geometry. For the typica input fied distribution containing side obes, the pacement of a pupi just in front of the waveguide s facet is advantageous when striving for singe-mode purity in short waveguide engths.such a pupi coud be reaized by evaporating a meta fim onto the input facet of the sab waveguide or fiber. The pupi heps to remove unwanted fied components from the input fied that may interfere with the fundamenta mode. It aso somewhat reaxes the dependence of mode purity on misaignments. However, it owers the couping efficiency for arger waveengths in a broadband appication. The treatment of the three-dimensiona case (i.e. the fiber) is computationay much more extensive than that of the two-dimensiona case (i.e. the sab). However, we found that fiber waveguides do not exhibit significanty different behavior in the characteristics of near-core power and mode purity when compared to their two-dimensiona counterparts. Thus an insight into the performance of the fiber can indeed be obtained by first studying the equivaent sab probem. Acknowedgments The authors thank Lifeng Li of Tsinghua University, China, for fruitfu discussions with respect to FMM. This work was performed within a contract for the European Space Agency (ESA). (C) 2007 OSA 17 September 2007 / Vo. 15, No. 19 / OPTICS EXPRESS 11826