# universally balanced photonic interferometers

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September 15, 2006 / Vol. 31, No. 18 / OPTICS LETTERS 2713

Universally balanced photonic interferometers

Milo A. Popovic, Erich P. Ippen, and Franz X. KrtnerResearch Laboratory of Electronics, Massachusetts Institute of Technology, 77 Massachusetts Avenue,

Cambridge, Massachusetts 02139

Received April 17, 2006; revised June 3, 2006; accepted June 3, 2006; published August 25, 2006

A new general class of optical interferometers is proposed, and the physical principle of their operation isexplained. They split the spectrum entering one input port among the interferometer arms in an arbitrarilychosen wavelength- and/or time-dependent manner but guarantee broadband constructive interference intoa single output port by symmetry. The design relies only on time reversibility of Maxwells equations and aphase condition that holds for lossless, reflectionless four-ports. As an application, a new Vernier scheme isproposed to multiply the tuning and free spectral range of microphotonic adddrop filters. It provides effec-tive suppression of both the amplitude and the phase response of unwanted resonant passbands. 2006Optical Society of America

OCIS codes: 130.3120, 260.3160, 350.2460, 230.3810.

A new class of interferometer is proposed in whichthe spectrum entering one input port may be splitamong the interferometer arms in an arbitrarily cho-sen wavelength- and/or time-dependent manner butis fully recombined in a single corresponding outputport by symmetry. On account of the guaranteed con-structive interference, the proposed devices are re-ferred to as universally balanced interferometers(UBIs). They are a generalization of a bypass switchproposed by Haus and described in Ref. 1. The typesof interferometer that have proved useful in optics(MachZehnder, FabryPerot, Michelson) have gen-erally entailed fixed-reflectivity mirrors and adjust-able arm lengths. In UBIs, illustrated in Fig. 1(a),the mirror design is largely arbitrary [and may con-tain couplers, resonators, switches, and nonrecipro-cal elementssee Fig. 1(b)], while the interferometerarms are fixed. Hence, UBIs are primarily suitablefor microphotonic circuit realization.

An important application of UBIs is as a generalbypass scheme for photonic devices. Inserting an op-tical processing device F into one interferometer arm[Fig. 1(a)] permits the device access to a part of theinput spectrum routed to it by input mirror A. Thepart of the spectrum passing unaffected through thedevice F is recombined with the bypass signal in asingle output port (of A). New generalized filter de-signs emerge that address hitless tuning1 anddispersion-free multiplication of the free spectralrange (FSR) of microresonators, enabling their intro-duction into chip-scale tunable wavelength routers.

We first describe the physical principle of operationand show that UBIs guarantee broadband construc-tive interference into one output port for a generalclass of input mirror choices. Then, we propose aspectrum-slicing UBI that suppresses undesiredresonances and provides a new way to multiply theeffective FSR and wavelength tunability of a micro-photonic adddrop filter, without adding excess dis-persion. We sketch a proof that ideal UBIs using ar-bitrary cascaded MachZehnder lattice filters2 asmirrors contribute zero excess dispersion in general.

Consider a generic interferometer with a splittermirror A and a combiner mirror A [Fig. 1(a) with-out filter F], each having two input and two output

ports. In a folded embodiment [Fig. 1(c)], A and A0146-9592/06/182713-3/$15.00

are coincident. Sufficient conditions to construct aUBI are that combiner A operate as a time-reversedand port-complementary replica of splitter A, A andA be substantially lossless and reflectionless (LR),and the interferometer arms be fixed with a differ-ential phase shift (DPS). A general four-port A that isLR can be represented by a 22 unitary transfermatrix U (with b=U a) of the general form:

U = eio1 ei1 iei2iei2 1 ei1 = ei1+2 11 ii 1 ei2 ei1eio, 1

symmetrized so that o11+22 /2, 11122 /2, and 21221 /2, where elements of Uare labeled umnumneimn. One coupling ratio andthree phases remain as free parameters, the fourthphase being fixed by the requirement of unitarity:

11 + 22 12 21 = . 2

Fig. 1. (Color online) (a) General universally balanced in-terferometer, with a filter inserted in one arm. (b) Illustra-tion of arbitrary splitting mirror A (may be nonrecipro-

cal). (c) Folded UBI (requires a nonreciprocal DPS).2006 Optical Society of America

2714 OPTICS LETTERS / Vol. 31, No. 18 / September 15, 2006

This unitary phase condition contributes in an es-sential way to the construction of a UBI. Otherwise,the design of the mirror A is arbitrary, and A maycontain couplers, resonators, switches, and nonrecip-rocal elements (circulators, gyrators). It may bewavelength dependent and controllable in time (tun-able filter, switch), i.e., U U ,p, where p param-etrizes the configurations. The and p dependence isunderstood and omitted henceforth.

A general physical argument for the design of aUBI can be given by considering a canonical physicalmodel for splitter A. Only the leftmost two of thethree matrices in the decomposition in Eq. (1) are rel-evant to interference in Fig. 1(a). Thus, the arbitrarysplitter A may at any one ,p be described as anideal directional coupler with coupling ratio and acharacteristic phase shift o1+2 in one outputarm, shown in Fig. 2(a) excited from one port.

With the objective of recombining all signal fromthe interferometer arms into one waveguide, it is in-structive to think in terms of the time reversibility ofMaxwells equations. Reversing time shows thatproperly phased arm excitations re-enter the upperwaveguide to the left. However, an interferometerconstructed from a splitter A and a combiner that is amirror-image replica with respect to a vertical reflec-tion axis is inconvenient. The arm lengths have aphase difference of 2o, dependent on the arbitrarydesign of A, that must be compensated for, and omay be wavelength (or time) dependent. A more prof-itable approach is to consider the same splitter A ex-cited from the complementary input port [Fig. 2(b)].The splitting ratio is the same, but the phase differ-ence between higher- and lower-intensity outputs, asillustrated, is now /2+o instead of /2o. This isenforced by the phase condition (2). Now, we take thetime-reversed operation of Fig. 2(b), mirror imagedwith respect to both a vertical and a horizontal axisof reflection, for the output combiner A. Then, thearbitrary phases o cancel and only a phase shiftremains [Fig. 2(c)]. If this shift is compensated for byinserting a broadband phase shift in one arm, ageneral UBI device of the form of Fig. 1(a) is derived.

The matrix formalism shows this rigorously. In thetime-reversed solution (indicated by subscript tr) cor-responding to a particular excitation of splitter A, theoutputs become the inputs b* atr, the inputs be-come the outputs a* btr, and the time-reversedtransfer matrix is U tr= U *1 (material-responsephasor tensors are conjugated as * and *).For lossless ( =, = ), reciprocal ( =T, = T)media, the time-forward and time-reversed solutionsare supported by the same structure. For nonrecipro-cal lossless media, the time-reversed solution is sup-ported by a structure with a reversed orientation ofthe built-in (and any applied) DC bias magnetic fields[HDC0 in Fig. 1(a)]. Thus, A and A may be nonre-ciprocal, except in folded arrangement [Fig. 1(c)],where they are coincident.

The total transfer matrix of Fig. 1(a) involves split-ter A U , a DPS matrix, and the matrix of the sec-

ond, time-reversed element U tr, sandwiched by ma-trices representing reflection about a horizontal axis:

T = 0 11 0U tr0 11 01 00 eiU . 3From unitarity, U trU *1=U T, and using Eq. (1),

T = ei2o1 00 1 . 4Thus, all signal entering an input port recombines inone output port, regardless of the splitting within theinterferometer. No assumptions about the particulardesign of splitter A were made beyond unitarity of U .It is interesting, from Eqs. (3) and (4), to see that A,preceded and followed (not shown) by a DPS, is akind of inversein the sense of amplitude onlytothe arbitrary unitary operator corresponding to split-ter A; however, a common-mode phase spectrumo ,p is not canceled by A in general.

Clearly, the phase shift plays an essential role.Hauss hitless switch1 relies on this principle andthus belongs to the general class of UBIs. In earlierwork, Henry et al. employed a DPS to flatten thewavelength response of a MachZehnder Bragg grat-ing filter.3 (Point-symmetric structures for flattenedcross-state passbands4 do not rely on a DPS or fallin this class of devices.) Broadband UBI operation de-pends on a broadband DPS realization. A half-wavearm length difference (at a central wavelength) pro-vides sufficient bandwidth 150 nm to cover com-munications bands.1 Phase shift errors of 10% incur0.2 dB recombination loss.1 The recombining re-

Fig. 2. (Color online) Canonical model of arbitrary LRfour-port mirrors and constructed UBI. (a), (b) Excitationat port 1 or 2 gives /2o phase; (c) cascading the splitterfrom (a) and the time-reversed and 180-rotated version of(b) cancels arbitrary phase o. If the remaining broadband phase shift is compensated for, a UBI is obtained.

sponse is first-order insensitive to small asymmetries

September 15, 2006 / Vol. 31, No. 18 / OPTICS LETTERS 2715

between A and A. For lossy (but reciprocal) A and AEq. (2) is violated, yet the UBI principle still holds inthe sense of zero cross-port transmission.

We next propose a novel filter design making use ofthe UBI principle. The limited FSR and wavelengthtunability of resonant microphotonic adddropfilters5 must

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