Wavelength-Selective Switches for Mode-Division Multiplexing: Scaling and Performance Analysis

Transcription

1 JLT-64-4: K.-P. Ho et al, Wavelength-Selective Switches for Mode-Division Multiplexing Wavelength-Selective Switches for Mode-Division Multiplexing: Scaling and Perforance Analysis Keang-Po Ho, Senior Meber, IEEE, Joseph M. Kahn, Fellow, IEEE, Jeffrey P. Wilde, Meber, IEEE Abstract Wavelength-selective switches for odedivision-ultiplexing systes are designed by scaling switches fro single-ode systes. All odes at a given wavelength are switched as a unit, which is necessary in systes with substantial ode coupling, and iniizes the nuber of ports required to accoodate a given traffic volue. When a pure ode is present at the input, odal transission and coupling coefficients are odedependent and ay be coputed using a siple odeclipping odel. When ultiple odes are present, interference between odes alters the transission and coupling coefficients, shifting the passband center frequency and changing its bandwidth. Mode-coupling atrices are used to copute ixed odes having the narrowest or widest bandwidths, or having the largest center-frequency offsets. In a specific design for gradedindex fiber, five ode groups and 5-GHz channel spacing, the one-sided bandwidth ay change up to GHz. In a syste with any cascaded switches and strong ode coupling, the end-to-end response per switch ay be characterized by a ode-averaged transission coefficient. Index Ters Wavelength-selective switch, ultiode fiber, ode-division ultiplexing. I I. INTRODUCTION 3.6 N ode-division-ultiplexed (MDM) systes, ultiple data streas are transitted in different odes of ultiode fiber (MMF) []-[7]. Ideally, transission capacity increases in proportion to the nuber of odes [6][7]. In addition to spatial ultiplexing, MDM systes use wavelength-division ultiplexing (WDM) to fully utilize the bandwidth available in the MMF and inline optical aplifiers. Reconfigurable optical add-drop ultiplexers (ROADMs) [9]-[] are indispensable for dynaically reconfigurable optical networks. To ensure the viability of MDM in such systes, Manuscript received January??, 4, revised April??, 4. This research of JMK was supported in part by National Science Foundation Grant Nuber ECCS-95 and by Corning, Inc. K.-P. Ho is with Silicon Iage, Sunnyvale, CA 9485 (Tel: , Fax: , e-ail: kpho@ieee.org). J. M. Kahn and J. P. Wilde are with E. L. Ginzton Laboratory, Departent of Electrical Engineering, Stanford University, Stanford, CA 9435, USA (eail: jk@ee.stanford.edu, jpwilde@stanford.edu). ROADMs for MDM should achieve functionality and perforance siilar to their counterparts in single-ode fiber (SMF) systes. In ROADMs for long-haul MDM systes, it is desirable to switch all the odes at a given wavelength as a unit between the sae input and output ports [3]-[]. In all long-haul MDM systes to date, ode coupling occurring along the link has been copensated by joint ulti-input ulti-output (MIMO) signal processing of all odes at the receiver []-[7], which requires all odes to be switched as a unit. Moreover, switching all odes as a unit siplifies network anageent and iniizes the nuber of ROADM input and output ports required to accoodate a given aggregate traffic volue [5]. Wavelength-selective switches (WSSs) are a principal coponent in ROADMs [9]-[]. For ipleentation of the switching plane in a WSS, liquid-crystal-on-silicon (LCoS)- based spatial light odulators (SLMs) []-[4] offer several advantages over previous technologies, and have becoe increasingly popular in recent years. The coplex odal profiles of the signals in MMF are the ain coplication in the design of a ultiode WSS. The field distribution at an input port is a speckle pattern deterined by the cobination of odes launched into the MMF and by ode coupling during propagation through the MMF [7][5]. In single-ode fiber, by contrast, regardless of the launched field profile, after propagating just a few eters, the output field profile is always the sae [6]. This paper addresses the design and perforance of LCoSbased ultiode WSSs. Starting with a single-ode WSS, certain physical diensions within the WSS are scaled with the goal of accoodating ultiple odes while aintaining perforance objectives, such as isolation, insertion loss, bandwidth, passband ripple, and passband syetry. Methods to analyze the ode-dependent transission response of ultiode WSSs are developed and applied. Pure odes with different ode sizes are subject to variations in passband shape and bandwidth that are consistent with a siple odeclipping odel. Pure odes also becoe coupled to each other, especially at frequencies near the passband edge. A atrix describing this ode coupling can be used to deterine the ixed odes having the narrowest or widest bandwidths. Another atrix can be used to find the ixed odes having axiu center-frequency offset. The reainder of this paper is organized as follows. Section

2 JLT-64-4: K.-P. Ho et al, Wavelength-Selective Switches for Mode-Division Multiplexing II describes how to scale a WSS fro single- to ulti-ode operation based on the bea spot size. Section III copares passband responses of ultiode WSSs obtained by detailed siulation to those coputed using a siple ode-clipping odel, and derives a ode-coupling atrix using the odeclipping odel. Section IV uses the ode-clipping odel to analyze wavelength-selective filtering of ixed odes, discusses ode-averaged filtering in long-haul systes with strong ode coupling, and presents a ethod to deterine the ode ixtures having the narrowest or widest bandwidths, or the worst center-frequency offsets. Section V discusses the ipact of filtering on optical signals. Section VI concludes the paper. II. WSS SCALING FOR MULTIMODE OPERATION Figure shows a siplified scheatic of an LCoS-based WSS, and is applicable to single- or ulti-ode devices. The input/output ports coprise a linear array of fibers with colliating lenses. The labeling of input and output ports in Fig. assues a drop odule. A ruled grating between the colliating lenses and a Fourier lens aps signals at a given wavelength to/fro the appropriate switching segent on the LCoS SLM, independent of the input and output ports. In Fig., the vertical centerlines of the ruled grating and the LCoS SLM are assued to lie in the two focal planes of the Fourier lens, thereby aking the syste telecentric at the SLM plane (i.e., the chief ray associated with any port is noral to the SLM plane for all wavelengths in the range of device operation). Applying a linear phase rap along the bea-steering direction (y-axis) switches a signal between different output ports. The syste essentially iages a fiber output onto the SLM with a agnification along the y-axis given by the ratio of the Fourier lens focal length to the colliator lens focal length and along the x-axis by this sae ratio ties a factor associated with the anaorphic scaling of the bea by the grating. Polarization-diversity and additional anaorphic bea-transforation optics are not shown in Fig. for siplicity. In this section, the WSS of Fig. is analyzed taking account of the increased spot size of a ultiode bea copared to a single-ode bea. This analysis yields siple scaling relationships fro single- to ulti-ode WSSs. A. Laguerre-Gaussian Modes We consider graded-index MMF, which has far lower group-delay spread than step-index MMF (assuing ore than two ode groups), which is iportant for iniizing receiver MIMO signal processing coplexity [7][7][8]. Although a practical MMF has a finite core radius to support a finite nuber of odes, for analytical convenience, we consider the eigenodes of an infinite parabolic index profile, which are given in polar coordinates (, ) by the Laguerre- Gaussian (LG) functions: y A B C D Spatial Light Modulator x Fourier Lens Ruled Grating Input Outputs Fig.. Scheatic design of a WSS (drop odule) using an LCoS SLM to switch input signals between output ports. Bea-transforation and polarization-diversity optics are not shown for siplicity. ( ) sin E (, ) C L q exp. () w w w cos The indices q and are the radial and aziuthal orders, respectively, C (,) q!/ ( q )! is a paraeter noralizing the ode to unit energy,,n is the Kronecker delta, equal to only if = n, / ( ) L q ( ) is a generalized Laguerre polynoial, and w is the /e radius of the fundaental LG odal field [setting q = = in ()]. The sine and cosine odes are defined here such that coincides with the SLM frequency-spreading direction (x-axis) in Fig.. In a parabolic-index MMF, all odes with a given value of g = q + + for a group having siilar propagation constants. Although the Herite-Gaussian odes [9]-[3] can describe the eigenodes of a parabolic-index MMF in Cartesian coordinates, the LG odes are easier to separate into groups with siilar propagation constants. For a given ( q, ), with nonzero aziuthal order, sine and cosine odes represent two degenerate odes with identical propagation constants. Including two polarizations, for a given ( q, ) pair, there are two odes for = and four odes for. The total nuber of propagating ode groups is denoted by g ax, and the total nuber of propagating odes in two polarizations is denoted by D, where g ax,,3,4,5, corresponds to D,6,,,3,. The Fourier transfor of an LG ode is a scaled version of the sae LG ode, a property that siplifies the analysis for a syste using a Fourier lens, as in Fig.. When an LG ode propagates fro the input to the output of Fig., the radius w is a function of propagation distance z, w (z). In other words, the LG odes change in size and phase profile as they propagate. In later parts of this paper, our notation does not ake explicit this z-dependent ode radius, since we always copare ode sizes in equivalent planes, e.g. at the SLM surface or in the colliating lens plane. B. Mode Size Scaling The spatial extent of LG odes generally increases with the group nuber g = q + +. Figure shows the intensity profiles of several LG-cosine odes. Figure shows that the

3 JLT-64-4: K.-P. Ho et al, Wavelength-Selective Switches for Mode-Division Multiplexing 3 LG LG LG 4 LG Fig.. Intensity profiles of selected LG-cosine odes. LG 4 and LG odes, both in the g = 5 group with, have larger size than the LG ode. The root-ean-square (RMS) radius of a ode is the square-root of E q, dd, and is equal to g / w. In this section, we discuss scaling of certain diensions within a WSS to accoodate the increased spatial extent of a ultiode bea. We take the fundaental ode radius w as given, and study how the bea size scales with the nuber of odes D. Given a value of w, we quantify the increased spatial extent of a ultiode bea by a scale factor We copute the scale factor based on three different criteria, which are the nuerical aperture (NA), or the radius containing 95% or 99% of the bea energy. In experiental characterization, a fiber s NA is defined as the sine of a half-angle spanning the far-field bea fro its peak intensity to 5% of peak intensity [3]-[34]. For SMF, the NA is defined unabiguously by the half-angle at 5% of the peak intensity. In MMF, the NA in general depends on the ixture of odes excited. Here, the NA is defined by the sine of the half-angle at 5% of peak intensity easured with an over-filled launch that excites all propagating odes with equal power. In Fig. 3(a), the left axis shows the NAs of MMF supporting different nubers of odes D, noralized to the NA of the fundaental LG ode. These noralized NAs are denoted by NA. The right axis in Fig. 3(a) shows the corresponding effective bea radii at 5% of the peak intensity that define the fiber NA. To be consistent with notation below, these are denoted by R eff. For the fundaental LG ode, R eff, =.w. When referring to the fundaental ode, we denote the effective bea radius as R eff,. It is iportant to note that the increase of the noralized NA NA with the nuber of odes does not iply that the NA itself increases. The NA for the fundaental ode ay decrease with an increasing nuber of odes, as explained later. Fro geoetric optics, a fiber s NA is given by ncore n clad, where n and core n are the core and cladding refractive indices, a definition that clad is independent of the core diaeter or the nuber of propagating odes. The NA defined in this way deviates significantly fro the NA as defined here when the nuber of propagating odes is sall. Fig. 3. Options for scale factor as a function of the total nuber of odes D: (a) nuerical aperture for overfilled launch, (b) 95% bea radius for pure odes, (c) 99% bea radius for pure odes (all three are noralized to the corresponding quantities for the fundaental ode). The right axes show the effective bea radii defining the scale factors. In (b) and (c), the sybols show the radius for individual LG odes, while the curves show the largest radius. As an alternative to the NA, we consider scaling a WSS based on the effective bea radius enclosing 95% or 99% of the bea energy of the worst-case (largest) pure ode. For a MMF including ode groups up to g ax, nuerical results show that the largest ode is typically the LGg /, ode for odd g ax and the LG g, ode for even g ax. In Figs. ax / 3(b) and (c), the left axes show the effective bea radii R eff enclosing 95% or 99% of the bea energy for pure odes, noralized to the corresponding radii for the fundaental LG ode. These noralized bea radii are denoted by 95 and 99, respectively. The right axes in Figs. 3(b) and (c) show the effective bea radii R eff. For the fundaental LG ode, ax

4 JLT-64-4: K.-P. Ho et al, Wavelength-Selective Switches for Mode-Division Multiplexing 4 TABLE I. SCALING FROM SINGLE- TO MULTI-MODE WSS. HEAVY LINES DENOTE PARAMETERS DEFINING DESIGNS I, II AND III. THE PARAMETERS AND ARE DEFINED IN SECTION II. Coponent Paraeter Design I Design II Design III Design IV Input/output fibers Effective bea radius Reff relative to Reff, Fundaental ode radius w or Reff, Segent width wseg relative to Reff, Iage eccentricity LCoS SLM Pixel pitch / / Iage of fundaental ode radius w / Segent width wseg in frequency direction x Height in bea-steering direction y Nuber of pixels in bea-steering direction y Angular dispersion / Ruled grating Fourier lens Colliator lenses Ports Iage of fundaental ode radius w Overall diensions in both directions Focal length ffourier Radius f-nuber / / / / Focal length fcoll Radius f-nuber Port spacing Port angular separation Nuber of ports the radii containing 95% and 99% of the energy are R eff, =.w and R eff, =.5w, respectively. C. Wavelength-Selective Switch Scaling In this subsection, we discuss how to scale the WSS of Fig. fro single- to ulti-ode operation. In a ultiode WSS, the bea radius is generally larger than that in a singleode WSS, and increases with the nuber of odes D, as shown in Fig. 3. To accoodate the larger bea radius, the optical syste ust be odified if siilar passband perforance is to be aintained. Various options for odifying the design exist, but in all cases, specific optical coponents are scaled by factors related to. As discussed in Sec. II.B, various definitions for exist, and for the reainder of the discussion, it is assued that one has been chosen according to soe design criteria and is siply referred to as without a subscript. Like the fundaental ode radius w (z), the effective bea radius R eff(z) changes with propagation distance z. This z-dependence is suppressed below, since ode sizes are always copared in equivalent planes. Table I suarizes the scaling of key WSS coponent paraeters for four different design approaches. Design I assues that both the LCoS SLM pixel pitch and the ruled grating angular dispersion reain unchanged in scaling fro single- to ulti-ode operation. Design II scales the SLM pixel pitch, Design III scales the grating angular dispersion, and Design IV cobines Designs I and II. These four designs are illustrative, and other designs are obviously possible. We first discuss Design I in detail, and then discuss the other designs. The filtering perforance of the WSS is deterined by the ruled grating angular dispersion and the per-channel segent width of the SLM. In the WSS shown in Fig., when two signals separated by the noinal channel spacing v are input to one port, the corresponding rays are separated by an angle /, where / is the angular dispersion of the ruled grating. On the SLM, located in the focal plane of the Fourier lens with focal length f Fourier, the corresponding iage centroids are separated along the frequency direction (x-axis) by [35][36] w seg f Fourier v. () The SLM is noinally subdivided into switching segents of width w seg. In a single-ode WSS, filtering perforance is deterined substantially by the switching segent width w seg relative to the iage size of the fundaental ode radius w (or

5 K.-P. Ho et al, Wavelength-Selective Switches for Mode-Division-Multiplexed Systes 5 equivalently the effective bea radius R eff, ). High isolation between adjacent WDM channels requires that their iages on the SLM lie on disjoint switching segents. Siilarly, in a ultiode WSS, filtering perforance is deterined by the ratio of w seg to R eff. To derive a practical design guideline, we assue that each odulated WDM signal occupies a two-sided bandwidth B. The frequency separation between the edges of adjacent WDM channels is v B. The corresponding separation along the x-axis on the SLM plane is B w. Obtaining high isolation between adjacent seg channels requires that this separation not be saller than the diaeter of the bea iage on the SLM plane, R eff. Thus, the iniu switching segent width is wseg R eff, (3) B where R eff is obtained fro Fig. 3 based on one of the three criteria, and is replaced by R eff, for SMF. Expression (3) describes the fundaental scaling principle that the switching segent width w seg relative to the iage size of R eff should reain constant. Hence, in converting fro a single- to a ulti-ode bea, the switching segent width w seg relative to the single-ode iage size R eff, should be scaled by. This scaling principle is satisfied by all the designs in Table I. According to (3), the switching segent width w seg ust be increased further in inverse proportion to vb. In an LCoS-based WSS, the ruled grating geoetry and any additional anaorphic optics transfor the iage of a bea on the LCoS SLM into an elliptical spot. The iage spot is copressed along the frequency direction (x-axis) in order to accoodate any WDM channels in an SLM of liited diensions. We assue that in all designs, the iage eccentricity reains approxiately unchanged fro a singleto a ulti-ode WSS. The SLM is used to apply a linear phase rap to an optical bea to steer it to different output ports. For a given axiu steering angle, in a single-ode WSS, the beasteering ability is deterined by nuber of pixels within the fundaental ode radius w or the corresponding R eff,. In a ultiode WSS, assuing nuber of pixels within R eff, reains constant, the nuber of pixels of SLM along the bea-steering direction (y-axis) ust be scaled by to aintain the sae bea-steering perforance for the higherorder odes. In Designs I-III, the nuber of pixels within fundaental ode radius w reains the sae. In Designs I and III, which use the sae SLM pixel pitch, the diensions of the SLM along both the x and y directions ust be scaled by a factor of. In Design I, the ruled grating angular dispersion / reains unchanged fro a single- to ulti-ode WSS. Fro (), in order to increase w seg by a factor, the Fourier lens focal length f Fourier ust increase by a factor. The iage size of all odes on the ruled grating is deterined by the inverse Fourier transfor perfored by the Fourier lens. The iage of the fundaental ode on the ruled grating is a factor larger in a ultiode WSS than in a single-ode WSS due to the increased Fourier lens focal length. In Design I, to aintain the sae iage size for the fundaental ode with f Fourier increased by, the colliator lens focal length f coll ust also scale by, as the agnification is related to the ratio of the two focal lengths, which in turn iplies that the fundaental ode becoes a factor of larger at the colliator lens. However, for a ultiode bea, the effective bea radius R eff is itself a factor larger, so the radius of the colliator lens ust be a factor larger than that in a single-ode WSS. The iniu spacing between two ports is ainly deterined by the colliator lens radius, so the port spacing ust increase by a factor. Siilarly, the ruled grating diensions along both directions ust increase by a factor. Also, the radius of the Fourier lens should scale by a factor, assuing a fixed nuber of ports, so its f-nuber should scale by a factor /, which ay becoe probleatic for large if the f-nuber becoes ipractically low. An additional factor considered here is that in going fro SMF to MMF, the fundaental ode radius w is scaled by a factor at the fiber input and output facets. Unlike the scaling associated with an increase in the nuber of odes, in which both the bea radius and NA are scaled by, when the fundaental ode radius w changes by a factor, the NA of the fundaental ode changes by a factor /. The NA of the MMF is NA /. A change in w can be accoodated by odifying the colliator lens focal length and radius to keep the colliated fundaental-ode bea radius constant. Under Design I, the colliator lens should agnify the bea to a radius larger. Assuing the fundaental ode radius w scales by a factor of fro SMF to MMF, the colliator lens focal length f coll should also scale by an additional factor of in order to aintain the sae spot size at the SLM. However, because the easured NA of the fundaental ode scales changes by a factor of / fro SMF to MMF, the colliator lens radius is independent of. For Design I, the overall scaling factor for the colliator lens focal length is, and its f-nuber should be scaled by a factor. The port spacing, which is deterined by the colliator lens diaeter, increases by a factor. With the increase in the focal length of the Fourier lens, the angular separation between two adjacent ports increases by a factor Keeping the sae SLM pixel pitch, the axiu bea-steering angle reains unchanged, so under Design I, the nuber of ports is reduced by a factor of as copared to a SMF WSS. Design II scales the pixel pitch of the LCoS SLM by a factor / fro the single-ode WSS design, such that the iage of the fundaental ode on the SLM can be scaled by a factor / fro the single-ode WSS. The nuber of pixels across the iage of the fundaental ode on the SLM along the x- and y-directions reains constant in scaling fro SMF

6 K.-P. Ho et al, Wavelength-Selective Switches for Mode-Division-Multiplexed Systes 6 to MMF. All of the scaled coponent paraeters for Design II are listed in Table I. The overall diensions of the LCoS SLM and the Fourier lens focal length f Fourier are scaled by a factor of / copared to Design I, and are the sae as for the single-ode WSS. But the Fourier lens f-nuber ust be scaled by a factor of / fro the single-ode WSS, which is ore probleatic than the scaling in Design I for large. The nuber of ports is also scaled by a factor of / as copared to the single-ode WSS. Design III scales up the ruled grating angular dispersion / by a factor of fro the single-ode WSS design, so the switching segent width w seg ay scale by a factor without changing the Fourier lens focal length f Fourier, as seen fro (). All of the scaled coponent paraeters for Design III are listed in Table I. Design III reduces any key diensions within the WSS by a factor / as copared to Design I, including the ruled grating size, Fourier lens focal length and radius, colliator lens radius, and port spacing. However, increasing the ruled grating angular dispersion can be probleatic, especially if it has been highly optiized in the initial single-ode WSS design. In Designs I-III, the nuber of ports is reduced as copared to a SMF WSS. To aintain the sae nuber of ports, the axiu bea-steering angle of the SLM needs to be increased. The bea-steering angle is proportional to the slope of the linear phase rap applied to the SLM. When a linear rap is approxiated by a stair-step function, the accuracy of the approxiation is deterined by the product of the step width (the pixel pitch) and the step height (which is proportional to the slope of the linear rap or the beasteering angle). Maintaining the sae phase accuracy, a factor of increase in axiu bea-steering angle can be achieved by scaling the pixel pitch by a factor of /. Design IV is the sae as Design I but with the SLM pixel pitch scaled by a factor of / (as in Design II), so the nuber of ports can be the sae as that of the SMF WSS. Other design choices ay cobine various aspects of the designs shown in Table I, for exaple, increasing the ruled grating angular dispersion by a factor and reducing the SLM pixel pitch by a factor / to obtain the sae perforance as Design IV. The analysis given in this section describes a scaling fro single- to ulti-ode operation based solely on the increased effective bea radius R eff. Unfortunately, a bea of radius R eff cannot be precisely related to a rectangular SLM segent of width w seg. For exaple, satisfying (3) with R eff equal to the 95% bea radius in Fig. 3(b) does not iply a 5% power loss at the SLM, even for the pure odes defining R eff in Fig. 3(b) (the actual power loss is.4%). Because of the coplexity of odal profiles (see Fig. ), different odes with the sae R eff ay be subject to different passband shapes. An analysis ore precise than (3) is required, and is the subject of the following sections. III. FILTERING AND MODE COUPLING FOR PURE MODES Because of the coplex profiles of higher-order pure or ixed odes, evaluating WSS perforance requires analysis ore detailed than that in Section II. In this section, we discuss filtering of pure odes, while in the following section, we discuss filtering of ixed odes. A. Transission Coefficients and Mode-Clipping Analysis Nuerical siulations of physical optics propagation have been perfored in Zeax for a WSS like that shown in Fig.. The channel spacing is = 5 GHz. The starting point is a single-ode WSS that achieves a one-sided.5-db (94.4% transission agnitude) bandwidth of about. GHz [35]. Using the scaling of Design II in Table I, a ultiode WSS is designed for five ode groups (g ax = 5), a total of D = 3 odes in two polarizations. A scaling factor = is used. Figs. 3(a) and (b) suggest that.8 and.9 based on NA and 95% bea radius criteria, respectively, would suffice. The transission characteristics of a ultiode WSS are soewhat ore coplicated to characterize than those of a single-ode WSS. One ay extend the conventional singleode power transission coefficient by coputing the power transission fro a specific input ode to all propagating odes in the output fiber. Here, in order to be able to study ode-coupling effects, we copute the aplitude transission coefficient fro a specific input ode to a specific output ode. Figure 4 shows the agnitudes of the frequency-dependent aplitude transission coefficients of the ultiode WSS for selected odes, where a specific ode at the input is coupled to an identical ode at the output. Figure 4(a) is for the cosine odes shown in Fig. and Fig. 4(b) is for the corresponding sine odes. Figure 4 shows LG and LG, which are the odes in the two lowest groups, as well as LG and LG 4, which are the odes in the highest group with g = 5 with both sine and cosine odes. The LG ode has a size close to that of the largest LG ode in Fig. 3, but has a ore coplex ode structure. Due to the syetry of the pure odes, the transission coefficients are syetric with respect to the center of the WDM channel, so only the positive-frequency side is shown in Fig. 4. For all these pure odes, the one-sided 6-dB (5%) bandwidth is very close to v / = 5 GHz. The worst-case (iniu) one-sided.5-db bandwidth is about.4 GHz. The worst-case one-sided 3-dB (7.7%) bandwidth is.7 GHz. LG and LG -cosine odes have the sae transission coefficients. LG and LG -sine odes have the sae spatial variation along the x-axis, leading to the sae transission coefficients. Siilarly, the LG -sine and LG 4 - sine odes have the sae transission coefficients.

7 K.-P. Ho et al, Wavelength-Selective Switches for Mode-Division-Multiplexed Systes 7 Transission Coefficient Mag. (linear units) Transission Coefficient Mag. (linear units) Frequency (GHz) (a) LG LG LG LG 4 (b) LG LG LG LG Frequency (GHz) Fig. 4. Transission coefficients of a ultiode WSS for selected (a) cosine odes and (b) sine odes. Results fro siulation and the clipping odel are shown as sybols and curves, respectively In Fig. 4, these siulation results are copared with a ode-clipping odel in which all light outside the noinal SLM switching segent is assued lost. In the clipping odel, the frequency-dependent aplitude transission coefficient for ode ( ) is given by t x l( f ), dydx wseg / ( f ) E y, (4) wseg / where l(f) is the center of the bea at frequency f and is a linear function of f. When a pure ode is input to the WSS, the coefficient (4) yields the coupling back to the sae pure ode, siilar to the odel in [36] for SMF. Energy ay also couple to other odes, as discussed below. Integrating the power over the whole switching segent, expression (4) is also the axiu possible power that can be coupled back to the MMF, with equality only for a MMF with an infinite nuber of odes. In (4), the x-axis corresponds to the angle =. The origin in (4) is the center of the switching segent. Although the bea on the SLM surface is elliptical, only the spatial variation along the frequency direction (x-axis) affects the filtering response. Figure 4 shows that transission coefficients coputed by the ode-clipping odel (4) are consistent with siulation results. On the linear scale of Fig. 4, the difference appears very sall. The difference between the siulation and theoretical results is less than.3 db for noralized losses saller than db. Although the difference easured in db increases at frequencies far fro the center of the passband, the absolute transission and the absolute error are both very sall at those frequencies. Figure 4 deonstrates that the ode-clipping odel can be used to accurately copute the transission characteristics of ultiode WSSs for LG odes. The LG odes given by () are an approxiation to the exact odes of a weakly guiding finite-core graded-index MMF, with discrepancies increasing for higher-order odes. It is possible that the ode-dependent transission coefficients for the higher-order exact odes at the passband edge ay be slightly different fro those shown in Fig. 4. Figure 4 shows that the.5-db bandwidth decreases with increasing ode grou due to an increase in bea radius. For all the odes of Fig. 4, and for all the 5 spatial odes within the first five groups, the narrowest.5-db bandwidth is about.4 GHz, as shown by both the ode-clipping odel and siulation. The coefficient (4) assues that only the SLM segent clips the ode, since other coponents should have a saller effect. Using the 95% or 99% radius R eff defined in Figs. 3(b) or (c) to copute a scaling paraeter 95 or 99 does not iply that all the coponents used in the syste are chosen to have that radius. Instead, it iplies that all the coponents are scaled according to changes in that radius. Nevertheless, if the lens radius were chosen to equal the 95% bea radius R eff in Fig. 3(b), the transission coefficient (4) would be reduced at ost by 5% with respect to its peak value, with the reduction becoing saller near the passband edge. B. Mode-Coupling Coefficients At each frequency, the ode-clipping odel (4) can be generalized to copute a frequency-dependent coupling coefficient fro ode ( ) to ode ( n): c with and ; n ( f ) E E wseg / wseg / * f, E f, dydx E n (5) x l( ) ( f, ) E y, q, f x l( ) ( f, ) E y, p, n n f. The coupling coefficients are syetric, i.e., c ; n c n;. Note that the coupling coefficient (5) reduces to the transission coefficient (4) for q, n. The coupling coefficients (5) are the eleents of a real, syetric ode-coupling atrix, which is used to analyze ixed-ode effects below.

8 K.-P. Ho et al, Wavelength-Selective Switches for Mode-Division-Multiplexed Systes 8 Coupling Coefficient Magnitude (linear units) LG LG,LG LG sin LG LG,LG LG sin LG LG 3 cos, LG LG sin LG LG 4 cos, LG 3 LG LG 4 LG cos LG 4 LG sin Frequency (GHz) Fig. 5. Coupling coefficients between selected odes for a ultiode WSS supporting five ode groups. For coparison, the blue curve shows the transission coefficients for two odes. Due to syetry of the LG odes in (), sine and cosine odes do not couple to each other in (5). Sine and cosine odes are syetric and anti-syetric along the y-axis, respectively, and the integration of (5) between cosine and sine odes yields zero. The notation for the ode-coupling coefficient (5) ignores the distinction between cosine and sine odes, with the understanding that sine and cosine odes can be analyzed separately. All odes with zero aziuthal order ( = ) couple only to cosine odes and do not couple to sine odes. Here, all odes with = are classified as cosine odes for convenience. Figure 5 shows the agnitudes of the coupling coefficients between selected odes for a MMF WSS designed for five ode groups, as in Fig. 4. For coparison, the blue curve shows the transission coefficients for two odes. In Fig. 5, ode coupling becoes significant only near the passband edge in the frequency range of -3 GHz, siilar to results in [6][7]. Apart fro the syetry c ; n c n;, soe coupling coefficients between different pairs of odes are equal, siilar to the equal transission coefficients for different odes seen in Fig. 4(b). Figure 5 shows that near the passband edge, coupling between odes ay be significant, and ay becoe stronger than the coupling between a ode and itself. The ode-dependent coupling in Fig. 5 causes variations in the transission coefficient at the passband edge, depending on the ixture of odes. C. Scaling of Transission or Coupling Coefficients The transission or ode-coupling coefficients coputed for one value of can be approxiately scaled to obtain the coefficients for other values of. In both (4) and (5), the only frequency dependence is in l(f), the center of the bea at frequency f. The position of l(f) shifts along the x-axis linearly with a change of frequency f. In the ode-clipping odel, assuing a large segent width w R, the coefficients (4) and (5) for different values of are of the sae functional seg eff for, but with a scaling dependent on Consider Design I and assue w R for all values of considered. Given ( f ), one of the coupling coefficients c (5) for a scaling factor, the coupling coefficient for a scaling factor is given by v v c ( f ) c f. (6) The transission coefficients (4) scale in an identical way. We observe that (3) ay be rewritten as B R w. Given a WSS with a segent eff / seg seg eff width w seg and noinal channel spacing v, if the effective bea radius R eff is increased, the sae WSS can be used, provided the signal bandwidth B is reduced in order to increase the ratio B. The scaling (6) is consistent with the approxiation (3) in which the WSS perforance is deterined by. The scaling (6) is valid for the transission or coupling coefficients (4) or (5), and for functions of those coefficients. Equivalently, (6) shows that given values of v and, should be scaled B w seg R eff B inversely proportional to to aintain the sae the transission or coupling coefficients. IV. FILTERING EFFECTS FOR MIXED MODES In long-haul MDM systes, because of ode coupling, signals propagate in ixtures of odes. In this section, we analyze filtering of ixed odes using the ode-clipping odel of (4) and (5). First, a statistical analysis of odeaveraged filtering effects in the strong-coupling regie is given. Then, worst-case odes with extree bandwidth or center-frequency shifts are studied. A. Mode-Averaged Filtering A ixed-ode signal including g ax ode groups ay be described as E E, ix q g ax where the are the coplex odal aplitudes and E are the eigenodes given by (). The notation of (7) does not explicitly separate cosine and sine odes, but both are considered in the analysis. A noralization q g ax is assued. In syste siulation with input signals that are generally tie-dependent ixed odes of the for (7), the frequencydependent ode-coupling coefficients (5) ay be used to find the corresponding tie-dependent output ixed odes. Such tie-dependent siulation ay generally be used to find the tie-dependent distortion induced by a WSS. In a syste with ultiple cascaded WSSs, however, this ethod ay becoe coputationally intensive, and the following statistical analysis ay be useful. (7)

9 K.-P. Ho et al, Wavelength-Selective Switches for Mode-Division-Multiplexed Systes 9 Avg/ Transission Coefficient Mag. (linear units) = g ax = 5 = LG g ax = g ax = 3 g ax = 4 g ax = 5 = 4 g ax = Frequency (GHz) Fig. 6. Mode-averaged transission coefficients of a ultiode WSS for different nubers of ode groups with scaling paraeter = and for five ode groups with = and = 4. In a long-haul syste, a ixed-ode signal (7) ay pass through a cascade of any WSSs. The analysis of filtering by the cascade is siplified by considering a large nuber of WSSs in the strong-coupling regie, which assues full rando coupling of all propagating odes between each WSS. Strong ode coupling is desirable in practice because it iniizes the ipact of ode-dependent gain and loss [7][8], and also iniizes MIMO signal processing coplexity [7][8]. We wish to copute the transission coefficient agnitude of the cascade averaged over the enseble of coplex odal aplitudes appearing in (7). In the strong-coupling regie, the coplex aplitudes of the eigenodes in (7) at the input of each WSS are independent and identically distributed, i.e., they have statistically equal powers and rando phases,. The enseble-average uniforly distributed on correlation between aplitudes is * q, n p, n / D, where denotes enseble average. Because *, n q if p q or n, interference between different eigenodes does not contribute to ode-averaged filtering effects (but does affect filtering of individual rando realizations). Using siple algebra, we find that the average transission coefficient agnitude per WSS in the cascade is equal to the ode-averaged transission coefficient agnitude of one WSS for g ax ode groups, given by t g ( f ) c ax ; n ( f ), (8) D q g ax pn g ax which includes diagonal and off-diagonal eleents of the ode-coupling atrix described by (5). Figure 6 shows the equal-weight ode-averaged transission coefficient agnitude (8) as a function of frequency. In Fig. 6, ode-averaged transission coefficients t ( ) for five ode groups (g ax = 5) are shown for scaling 5 f paraeter values =,, and 4. The choice = is equivalent Transission Coefficient Mag. (linear units) LG g ax = g ax = 3 g ax = 4 g ax = Frequency (GHz) Fig. 7. Transission coefficients for ixed beas with the narrowest bandwidths (solid curves) or widest bandwidths (dashed curves) for different nubers of ode groups, copared to the LG ode. Curves with the sae color represent the sae nuber of ode groups. to that in Figs. 4 and 5. The choice = (using a single-ode WSS without scaling) causes the passband shape to be degraded significantly, and would lead to substantial bandwidth narrowing, as well as substantial interference fro adjacent channels. Conversely, the choice = 4 akes the passband shape ore nearly ideal than for =. The scaling for different is consistent with both (3) and (6). Also shown in Fig. 6 are ode-averaged transission coefficients (8) for the fundaental LG ode and for two to five ode groups ( g ax 5), for =. The passband shape is continuously degraded as the nuber of ode groups increases, also consistent with (3) with R eff taken fro Figs. 3. B. Miniu- or Maxiu-Bandwidth Modes The following three sections discuss worst-case ixed odes, which ay be useful in conservative syste design. The ixed odes with the narrowest or widest bandwidths can be coputed using a atrix whose eleents are given by the ode-coupling coefficients (5). As illustrated in Fig. 5, the coefficients (5) are frequency-dependent. The extreebandwidth ixed odes are of the for (7) with frequencydependent odal aplitudes (f). The narrowest-bandwidth ode has (f) given at each frequency f by the eigenvector of the ode-coupling atrix (5) that has the sallest eigenvalue, corresponding to the iniu transission coefficient. Siilarly, the widest-bandwidth ode has (f) given at each f by the eigenvector of (5) that has the largest eigenvalue, corresponding to the axiu transission coefficient. Figure 7 shows the agnitudes of the transission coefficients of the iniu- and axiu-bandwidth ixed odes for a WSS designed for five ode groups with scaling paraeter =, as in Figs. 4 and 5. The fundaental ode radius w is held constant, and the nuber of ode groups is varied fro two to five ( g ax 5). The fundaental LG ode is also shown for coparison. For a given nuber of ode groups, any transission coefficient between the iniu and axiu is possible for soe set of ixed

10 K.-P. Ho et al, Wavelength-Selective Switches for Mode-Division-Multiplexed Systes LG gax = gax = 3 gax = 4 gax = 5 Fig. 8. Intensity patterns at SLM plane for axiu-offset odes for different nubers of ode groups, copared to the LG ode. The offset increases with an increasing nuber of groups. odes. The transission coefficients in Fig. 7 are syetric with respect to zero frequency, so only the positive-frequency side is shown. In Fig. 7, including five ode groups, the narrowest and widest 6-dB (5%) bandwidths are.3 and 8.6 GHz, respectively. Any 6-dB bandwidth in the range GHz is possible. The narrowest and widest.5-db bandwidths are. and 7. GHz, respectively, varying also in the range of GHz. The 3-dB bandwidth lies in the range GHz. The extree-bandwidth odes in Fig. 7 are all given by cobinations of only cosine odes (including LG ode and other odes with = ). The six sine odes have zero correlation with those cosine odes, and thus do not contribute to the extree odes in Fig. 7. The extree-bandwidth ixed odes correspond to specific frequency-dependent odal aplitudes (f). In MDM systes with high-order odal dispersion [7][37][38], data signals have frequency-dependent odal aplitudes, and it is possible, though unlikely, for a data signal to align with an extree-bandwidth ode at all different frequencies. A signal aligned with the iniu-bandwidth ode would be subject to strong distortion, while a signal aligned with the axiu-bandwidth ode would be subject to strong interference fro adjacent channels. Any transission coefficient between the iniu- and axiu-bandwidth odes is possible, and the ratio between the defines the worst-case or peak-to-peak ode-dependent loss. C. Maxiu-Offset Modes In the frequency-dependent ode-coupling coefficient (5), the coupling coefficient agnitude decreases as the bea center l(f) shifts further fro the center of the switching segent along the frequency direction (x-axis). If we construct ixed odes whose centroid is axially shifted along the xaxis, those odes will have a saller bandwidth to one side and a larger bandwidth to the other side, and should have approxiately the largest shift in passband center frequency. Unlike the extree-bandwidth odes in Sec. IV.B, these axiu-offset odes are independent of frequency. To find the ixed odes with axiu frequency offset, the objective is to find the odal aplitudes in (7) to axiize the ean offset x x Eix dxdy, (9) where the integrations are fro negative to positive infinity. The ean offset (9) is a bilinear function of the aplitude coefficients, and is axiized by defining an xcorrelation atrix and choosing to be the eigenvector having the largest eigenvalue. The x-correlation atrix has eleents given by () c~ xe E * dxdy, q, ; p, n n where q g ax, p n g ax, and the integrations are fro negative to positive infinity. The xcorrelation atrix () has only a sall nuber of non-zero eleents, which are listed in Table II for cosine odes (including odes with = ) up to five ode groups (g ax = 5). For sine odes with, only those ode pairs arked in Table II have non-zero eleents, which are of opposite sign fro the values in Table II. Figure 8 shows the intensity profiles on the SLM for the axiu-offset odes for a WSS designed for five ode groups with scaling paraeter =. As in Figure 7, the fundaental ode radius w is held constant, and the nuber of ode groups is varied fro two to five ( g ax 5). The fundaental LG ode is shown for coparison. If only two groups (LG and LG) are used to for an offset ode, both LG and LG -sine odes have the sae aplitudes, and the axiu ean offset is x w /. If three groups (LG, LG, LG, and LG) are used to for an offset ode, the aplitude ratios for LG, LG, LG, and LG odes are, 3,, and, and the axiu ean offset is x 3 / w. The offset increases with the nuber of ode groups, as shown in Fig. 8. In Fig. 8, the offset odes with four and five ode groups have ean offsets of x 3 / 3 / w.65 w. w, respectively. and x 5 / 5 / w

11 JLT-64-4: K.-P. Ho et al, Wavelength-Selective Switches for Mode-Division-Multiplexed Systes TABLE II. NON-ZERO ELEMENTS OF THE X-CORRELATION MATRIX () BETWEEN COSINE MODES. THE MODE PAIRS MARKED BY ASTERISKS ALSO HAVE NONZERO CORRELATION BETWEEN SINE MODES. E q, E p, n c ; n c n; LG LG w / *LG *LG w / LG LG w / *LG *LG3 3w / *LG *LG w / LG LG w *LG3 *LG4 w *LG3 *LG / *LG LG *LG LG w 3w / w Figure 9 shows the agnitudes of the transission coefficients of the axiu-offset odes of Fig. 8 calculated using the ode-clipping odel for two through five ode groups ( g ax 5). The transission for the LG ode is shown for coparison. The axiu-offset odes have clearly asyetric passbands, with bandwidths slightly larger than that of the LG ode. On the negative-frequency side, the 6-dB bandwidths are larger than 5 GHz, increasing to 8.5 GHz for g ax = 5, potentially increasing interference fro an adjacent channel. On the positive-frequency side, the 6-dB bandwidths are saller than 5 GHz, decreasing to.5 GHz for g ax = 5, potentially increasing signal distortion. The curves in Fig. 7 with narrowest bandwidth are siilar to the positive-frequency side of the curves in Fig. 9. Likewise, the curves in Fig. 7 with widest bandwidth are siilar to a folding of the negative-frequency portion of Fig. 9 to positive frequency, corresponding to changing the odes of Fig. 8 fro axiu positive to axiu negative offset. The transission coefficients in Fig. 9 exhibit soe differences fro those in Fig. 7, particularly ripples on the negative- and positive-frequency sides at large and sall transission coefficient values, respectively. When the 6-dB bandwidths in Fig. 9 are copared to those in Fig. 7, the differences are less than. GHz (in the worst cases,.5 versus.3 GHz and 8.5 versus 8.6 GHz). The narrowest.5-db bandwidths in Figs. 9 and 7 are alost identical. D. Other Mixed-Mode Characteristics Unlike the iniu- or axiu-bandwidth odes found in Sec. IV.B, which are frequency-dependent ixed odes, the axiu-offset odes found in Sec. IV.C using () are frequency-independent ixed odes. Siilar ethods ay be used to find frequency-independent ixed odes having other characteristics. For exaple, the ixed ode having axiu RMS radius is found as an eigenvector of a - correlation atrix, where is the radial coordinate. Using at least four ode groups, we are able to find ixed odes with Transission Coefficient Mag. (linear units) LG g ax = g ax = 3 g ax = 4 g ax = Frequency (GHz) Fig. 9. Asyetric transission coefficients of the axiu-offset odes fro Fig. 8 for different nubers of ode groups, copared to the LG ode. The axiu-offset odes have the largest shifts in center frequency. RMS radius slightly larger than the axiu RMS radius of the pure odes in those ode groups. These ixed odes only include LG odes that have the sae aziuthal order [the sae in ()], and thus have non-zero -correlation aong the. The passband bandwidth for these ixed odes is typically wider than the narrowest bandwidth aong pure odes in Fig. 4, but the transition band, e.g., between the - and -db bandwidths, is typically wider. The ixed odes with the iniu or axiu.5-db two-sided bandwidth ay be found approxiately as eigenvectors of an x -correlation atrix (or an x -correlation atrix) with axiu or iniu eigenvalues, respectively. V. DISCUSSION For odeling a cascade of any WSSs with strong ode coupling, the ode-averaged transfer function of Fig. 6 represents a typical response obtained by the law of large nubers. Nevertheless, it is possible to encounter certain ixed odes that are subject to ore signal distortion or adjacent-channel interference than the typical case. If the tolerable syste outage probability is low, the worst-case ixed odes shown in in Fig. 7 ay be used for conservative syste design. By adjustent of the scaling paraeter, the worst-case iniu bandwidth ay be designed to be larger than the signal bandwidth to ensure reliable syste perforance. For siulation of tie- and frequency-dependent signals in a link, the frequency-dependent WSS ode-coupling coefficients (5) ay be used in conjunction with rando realizations of fiber propagation atrices to obtain realizations of tie- and frequency-dependent output signals. Even if signals occupy a bandwidth ore than the iniu bandwidth shown in Fig. 7, siulation results should typically correspond to the ode-averaged transfer function of Fig. 6, since the worst-case ixed odes are unlikely to be encountered, with probabilities of 6 or even lower. If one frequency-independent ixed ode is required for characterizing WSS perforance, especially in experiental easureent, the frequency-independent axiu-offset

12 JLT-64-4: K.-P. Ho et al, Wavelength-Selective Switches for Mode-Division-Multiplexed Systes ixed odes of Fig. 8 ay be used to approxiate the frequency-dependent worst-case ixed odes of Fig. 7. The difference in 6-dB bandwidth is less than. GHz. The coupling coefficients (5), which are iportant for syste perforance analysis, ay be deterined in siulation or easureent by launching into the WSS the inphase and quadrature su and difference between two pure odes, E E n and E je n, a total of four cobinations. The coupling coefficients can be obtained fro the difference between the power transission coefficients for the su and the difference between the two odes, e.g., the real part of 4 * E n E E E E. n E is obtained fro n VI. CONCLUSION WSSs for MDM systes are designed starting with a single-ode WSS and scaling up certain physical diensions to accoodate the larger size of a ultiode bea. All odes at a given wavelength are assued to be switched as a unit, which is necessary in systes with ode coupling, and iniizes the nuber of switch ports required to accoodate a given traffic volue. When a pure ode is present at the switch input, odal transission coefficients or coupling coefficients are odedependent, and ay be coputed with reasonable accuracy using a siple ode-clipping odel. For a given switch design, the bandwidth generally becoes narrower with an increasing nuber of ode groups. When ultiple odes are present at the switch input, coupling between odes alters the odal transission and coupling coefficients. Analysis of a syste with any cascaded switches with strong ode coupling in between shows that the response of the cascade ay be characterized by the ode-averaged transission coefficient of a single switch. This ode-averaged response incorporates the effect of ode coupling within the switch. 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