simple linear space formalism for polarization-dependent interferometers: theory and application to...

3668 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 32, NO. 20, OCTOBER 15, 2014

Simple Linear Space Formalism forPolarization-Dependent Interferometers: Theory

and Application to Phase-Modulated Photonic LinksNicholas J. Frigo, Fellow, IEEE, Vincent J. Urick, Senior Member, IEEE, and Frank Bucholtz, Senior Member, IEEE

Abstract—We outline a formalism for modeling interferome-ters, such as asymmetric Mach–Zehender interferometers used inboth microwave photonic links and modern transmission systems.The formalism permits modeling elements with birefringence andpolarization-dependent loss. By introducing a coordinate trans-formation between the standard “waveguide” view (coupled po-larizations and independent waveguides) and the “coupler” view(coupled waveguides with independent polarizations), we reducemodeling to a concatenation of block diagonal operators and co-ordinate transformations. This connects to, and generalizes, anearlier approach. We illustrate the formalism by calculating thephase shift in a birefringent interferometer suffering differentialnormal mode losses in the couplers. Such phase shifts can be a sig-nificant source of even-order distortion in phase-modulated linksemploying an interferometer-based receiver.

Index Terms—Microwave photonics, optical polarization.

I. INTRODUCTION

THE development of integrated [1] Mach–Zehnder inter-ferometers (MZI) has enabled advances in digital optical

communications, which increasingly use phase modulation andinterferometric detection to mitigate optical impairments [2] inlong haul transmission systems. This technology has also en-abled advances in phase modulated microwave photonic linksin “phase modulation, interferometric detection” (ΦMID) ap-proaches, which have several performance advantages over “in-tensity modulation, direct detection” (IMDD) systems in analogmicrowave photonic applications [3]. As requirements tightenin both analog and digital systems, it becomes more necessaryto model components and subsystems accurately to capture po-tential system impairments.

One approach to modeling is what we refer to as the “wave-guide view,” which tracks the field amplitudes in evanescentlycoupled waveguides. This view is necessary at the device designlevel since it gives details of process parameters [4]. At the sys-tem modeling level, however, a more useful approach is a gener-alization of the Jones matrix [5], as has been proposed recentlyfor analysis of DPSK demodulators [6]. In that analysis, fields

Manuscript received January 9, 2014; revised June 6, 2014; accepted July 2,2014. Date of publication July 7, 2014; date of current version September 1,2014.

N. J. Frigo is with the Department of Physics, U.S. Naval Academy, Annapo-lis, MD 21402 USA (e-mail: [email protected]).

V. J. Urick and F. Bucholtz are with the Optical Sciences Division, U.S. NavalResearch Laboratory, Washington, DC 20375 USA (e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JLT.2014.2336173

in two waveguides throughout an interferometric structure weretreated as a combined 4-D vector, and couplers, for instance,had a specific 4 × 4 matrix representation. In essence, by ex-tending the waveguide view to include couplers, they reapedthe advantages of the Jones matrix approach and addressed therepresentation of couplers in birefringent networks.

In this paper we generalize that earlier approach [6] in sev-eral ways. First, we observe that the waveguide view is only onebasis for the 4-D space of the fields. We introduce a co-ordinatetransformation between (a) the waveguide view, which treats thearms of an interferometer as independent (i.e., coupled polariza-tions for each waveguide), and (b) the “coupler view” in whichthe polarizations in a coupler are independent (i.e., coupledwaveguides for each polarization). This coordinate transforma-tion enables standard linear space techniques to be used in boththe waveguide and coupler sections of an interferometer. Bypermitting operators for both types of sections to be representedin block diagonal form, we extend the model’s usefulness andease of application. Second, we generalize the operators to in-clude more degrees of freedom (DOF) than the earlier modelrepresented: we add DOFs to operators for propagation, rota-tion, loss, and coupling. We conclude with an application thatincludes both birefringence and polarization-dependent loss. Inparticular, we model normal mode loss in the coupler, which isdifficult to do conventionally.

II. DESCRIPTION OF FORMALISM

A. Overview of Linear Space Description

Our basic approach is illustrated in Fig. 1, upper figure (a) ofwhich is a physical view of a coupler with pigtails, while lowerfigure (b) gives a pictorial view of our method with a detailedview of the waveguides, couplers, and their fields. In both fibersand couplers, the x and y polarizations are indicated by dashedand solid lines, respectively.

For light traveling in the waveguide sections (between pointsA and B before encountering the coupler and between pointsC and D after the coupler), the natural description of the fieldamplitudes is the standard Jones vector: namely, the fields ineach waveguide are represented by 2 × 1 Jones vectors,�s (1) and�s (2) . The components for waveguide “i” are complex numbers,describing amplitude and phase information in the x and ypolarization basis states

�s (i) =

⎡⎣ s

(i)x

s(i)y

⎤⎦ .

0733-8724 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications standards/publications/rights/index.html for more information.

armclean

Sticky Note

change to US gov copyright

FRIGO et al.: SIMPLE LINEAR SPACE FORMALISM FOR POLARIZATION-DEPENDENT INTERFEROMETERS: THEORY AND APPLICATION 3669

Fig. 1. State space approach. (a) Physical view of coupler with input andoutput waveguides. (b) Exploded view showing ‘x’ (dashed) and ‘y’ (solid)polarizations for waveguides and polarization-independent couplers. Matrix ‘S’transforms fields from “waveguide” view to “coupler” view.

Since the basis states are orthogonal, the light intensity at anypoint is given by �s †�s.

It is convenient [4], [6] to stack the Jones vectors for bothwaveguides into a 4-D state vector which shows the joint polar-ization states at given points.

In particular, the fields at point B, the input to the couplers,can be viewed as

⎡⎣

�s (1)

�s (2)

⎤⎦

B

=

⎡⎢⎢⎢⎢⎢⎢⎣

s(1)x

s(1)y

s(2)x

s(2)y

⎤⎥⎥⎥⎥⎥⎥⎦

B

.

Generally speaking, couplers are analyzed independently of po-larization [3], and they are typically viewed as four-port devices.That is, the field amplitudes at the two output ports are related tothe fields at the two input ports by a 2 × 2 coupling matrix (il-lustrated as C(x) or C(y ) in Fig. 1, and discussed below) whoseelements are determined by the evanescent coupling betweenthe coupler’s waveguides [7].

Since our model aims to incorporate polarization, the physi-cal coupler is viewed conceptually as two couplers, one for eachpolarization, C(x) and C(y ) , shown as planes in Fig. 1(b). Ide-ally, in the absence of non-linearities and polarization mixingin the coupler, one can view the fields at the output ports assuperpositions of the fields through the C couplers, in the usualmanner [3], [7]. We will concentrate on this important case later.For completeness, the possibility of cross-coupling between po-larizations is indicated in Fig. 1(b) by arrows which change theirpolarization characteristics (dashed, solid) in traversing betweenthe two polarization planes. In matrix form, such cross-couplingwould be described by 2 × 2 matrices εxy and εyx . For instance,εxy would describe the x polarized contributions to the outputfields in terms of the y polarized input fields. Both the C(i) and εmatrices are 2 × 2, and only for the ideal case of εxy = εyx = 0can the C(i) matrices have the standard unitary properties of theusual coupled mode formalism [7]. We will argue below (Section

II-B2) that this ideal case should closely represent the physicalsituation.

The natural description of the fields in traversing a coupleris different than those for waveguides, and it is more natural touse a “port” representation for the fields in the “coupler” view:for example, the inputs on the ports of the x plane coupler C(x)

are given by components with x polarization at ports 1 and 2:

�p (x) =

⎡⎣p

(1)x

p(2)x

⎤⎦ .

Considering the inputs to the two conceptual couplers inFig. 1(b), we stack the two input “port” field vectors as wedid for the waveguides’ Jones vectors, only now each of the2 × 1 vectors comprise the complex amplitudes at the ports forthat respective polarization plane

[�p (x)

�p (y )

]

B

=

⎡⎢⎢⎢⎢⎢⎢⎣

p(1)x

p(2)x

p(1)y

p(2)y

⎤⎥⎥⎥⎥⎥⎥⎦

B

.

Again, in the absence of nonlinearities, there will be a 4 × 4matrix relating the input and output amplitudes for the ports inthe coupler description.

It is clear that the “output” waveguide fields at B are the sameas the “input” port fields at B. That is, for instance, s (2)

x = p(2)x

since both expressions represent the x component of the fieldon the second waveguide as it enters the coupler (disregardingtechnical issues such as index changes and losses). Another wayof looking at it is that the fields in both the waveguides and thecouplers are in a 4-D space and that the waveguide and couplerviews are simply different basis sets for that space. As such, itis clear that the coordinate transformation between these twobases is a simple rearrangement matrix S:

⎡⎢⎢⎢⎢⎢⎢⎣

p(1)x

p(2)x

p(1)y

p(2)y

⎤⎥⎥⎥⎥⎥⎥⎦

B

=

⎡⎢⎢⎢⎢⎣

1 0 0 0

0 0 1 0

0 1 0 0

0 0 0 1

⎤⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎣

s(1)x

s(1)y

s(2)x

s(2)y

⎤⎥⎥⎥⎥⎥⎥⎦

B

≡ S

⎡⎢⎢⎢⎢⎢⎢⎣

s(1)x

s(1)y

s(2)x

s(2)y

⎤⎥⎥⎥⎥⎥⎥⎦

B

. (1)

Transformation operator “S” between the waveguide and cou-pler bases is shown schematically as the “switch” box inFig. 1(b) at point B. Once the fields have been transformed tothe coupler basis by (1), the field evolutions to the coupler’s out-put ports are described by a 4 × 4 matrix to yield a 4 × 1 vectorin the coupler basis at point C. Consideration of the situation atthe coupler’s output shows that the inverse transformation, S−1 ,is a reflection about the vertical plane of S. However, since thisoperator performs the same interchange of components, we findthat S−1 = S† = S.

The simplicity of this formalism is due to the field’s conti-nuity across the boundary and the basic view of coupled modetheory as operating in a linear space. Once established, however,it enables an analysis in either the waveguide or coupler basis


set, with only a simple transformation between the views. Fromthis flows the utility of the formalism: it permits one to cast ananalysis in the most favorable system and then transform to theother view. In this way, component characteristics can be mod-elled individually with simple matrices, rather than requiring an“a priori” analysis of the entire system [4], [6].

B. Operators

As shown earlier [6], the state vector approach allows oneto cast birefringent interferometers in a formal matrix descrip-tion for efficient modeling. In this section, we show that thetransformation in Eqn. (1) can lead to analytic simplifications.

1) Propagation: For propagation in the interferometric armsor lead fibers, for example, from zA to zB , Jones matrices foreach waveguide describe the field evolution as

�s(i)B = J(i)�s

(i)A , where J(i) =

⎡⎣J

(i)11 J

(i)12

J(i)21 J

(i)22

⎤⎦ . (2)

That is, waveguides i = 1, 2 each have their own Jones matrix.Usually [5]–[10] these are taken as unitary matrices, represent-ing lossless propagation. Since there is no coupling betweenguides, the joint evolution in the two fibers can be describedwith 2 × 2 block matrices J(i) and 0 as

[�s (1)

�s (2)

]

B

=

[J(1) 0

0 J(2)

][�s (1)

�s (2)

]

A

= J

[�s (1)

�s (2)

]

A

(3)

where the sans serif “J” is a 4 × 4 block diagonal matrix andthe �s (i) are 2 × 1 Jones vectors.

As shown earlier[8], the most general form for block J(i) isthe unitary matrix

J(φ̄, φ, β̂) = e−j (φ̄+(φ/2)β̂ ·�σ )

= e−j φ̄

[cos φ

2 − j sin φ2 (β3) −j sin φ

2 (β1 − jβ2)

−j sin φ2 (β1 + jβ2) cos φ

2 − j sin φ2 (−β3)

](4)

which has a descriptive geometric representation in Stokesspace (see [9], [10], and Appendix A.) A vector �β = β̄ β̂ de-scribes the differential mode evolution of the state. Its unitvector, β̂ = {β1 , β2 , β3}, is the representation of J’s eigenstate(with eigenvalue +1) on the unit sphere in Stokes space, whileφ = 2β̄z is the rotation J causes about β̂ for any Stokes vectoroperand �s. In addition to this “topological” evolution, φ̄ is anoverall common phase, which can be represented by a “fiducialpaddle” as described in Appendix A. These quantities expressthe degrees of freedom open to unitary matrices. That is, for each2 × 2 block J, formulation (4) has four DOF: two DOF specifyeigenstate β̂, since it is a unit vector; one DOF (φ) describes thedifferential phase between components of the two eigenstates ofJ as a rotation about β̂; and one DOF (φ̄) describes the average(common) phase experienced by both components. In contrast,the equivalent operator[6] “D” has three DOF: a rotation angleand two phase delays. This is a limitation of J’s eigenstates to acircle in Stokes space, rather than the unit sphere. That is, thereis only one vector β̂ about which the states can rotate.

As we show later, the block diagonal form of J makes it easyto apply well-known matrix techniques for more general casesand permits handling smaller matrices. Indeed, the form of Eqn.(4) extends Ref. [6]’s treatment by generalizing the couplingand rotation operators as well.

2) Coupler: In the coupler view, the most general form forthe 4 × 4 matrix operator is

[�p (x)

�p (y )

]

C

=

[C(x) εxy

εyx C(y )

][�p (x)

�p (y )

]

B

= C

[�p (x)

�p (y )

]

B

(5)

where, for example, the 2 × 2 matrix εxy couples light from they polarization at input zB to light in the x polarization at outputzC . If power is conserved, then C is unitary.

We argue here that the εij matrices should be small, and thusthat C should very nearly be in block diagonal form. (i) Thefields throughout the coupler are nearly orthogonal. The x − yfields are not perfectly orthogonal as they would be for planewaves, but the overlap integral [11] in the coupling region willbe small compared to the evanescent coupling to a neighboringguide in the same polarization. (ii) The planar structure of thecoupler may also lead to a birefringence with s and p polarizationeigenstates defined by the planar surface. Any birefringencewould further reduce phase matching between the dissimilarpolarizations.

While this simplification is not necessary for the formalism,we take the εij to be zero on the right side of Eqn. (5) as inearlier treatments [4], [6]. With this assumption, matrix C inEqn. (5) has block diagonal form, just as J does in Eqn. (3)

[�p (x)

�p (y )

]

C

=

[C(x) 0

0 C(y )

] [�p (x)

�p (y )

]

B

= C

[�p (x)

�p (y )

]

B

. (6)

This formulation is both practical and useful: it allows one to ex-ploit the conventional coupler formalism to model polarization-dependent effects in a 2 × 2 form under the usual assumptions[4], [6]. At this point, C is described in the coupler basis: to bringit into the waveguide basis, [6], requires a similarity transformeffected by S, to which we now turn.

Because we have a transformation matrix between the two co-ordinate bases (i.e., between the waveguide and coupler views),we can perform a similarity transform to bring an operator inone view into the equivalent operator in the other view. In par-ticular, we can transform the coupler operator C(θx, θy ) in theearlier work [6], which is expressed in the waveguide view, intoits equivalent operator in the coupler view by performing thesimilarity transform S C(θx, θy ) S. That is, from right to left, Stakes coupler co-ordinates (the space we desire) to waveguideco-ordinates, C(θx, θy ) operates on the result in the waveguideview, then S transforms the result back to coupler co-ordinates.This procedure reveals that the earlier 4 × 4 coupler matrix [6]in the waveguide view breaks into two 2 × 2 block diagonalcoupling matrices in the coupler view, where the block matricesboth have the same form. The equivalent to C(x) is found to be

C(x) ∼ C(θx) =

[cos(θx) −j sin(θx)

−j sin(θx) cos(θx)

](7)


for instance, where we have substituted i ⇒ −j to change [6] tothe ejωt convention [8]–[10]. The form of Eqn. (7) used earlier[6] is more restrictive than the general unitary matrix in Eqn.(4): it describes an ideal coupler whose sole DOF is its couplinglength (represented by θx , the product of coupling strength andcoupling length). In the language of coupled mode theory, thedetuning between the coupler’s modes must be perfect: Δ = 0[11].

In a manner identical to the generalization for propagation,above, we generalize our coupling operator C by allowing the2 × 2 coupling matrix blocks C(x) , C(y ) to assume the moregeneral form of J in Eqn. (4), rather than the more restrictedform of Eqn. (7), namely

C(φ̄, φ, κ̂) = e−j2φ̄/2 ×[

cos φ2 − j sin φ

2 (κ3) −j sin φ2 (κ1 − jκ2)

−j sin φ2 (κ1 − jκ2) cos φ

2 − j sin φ2 (−κ3)

]. (8)

Here, the coupler’s eigenstate is given by κ̂ = (κ1 , κ2 , κ3) andthe coupler’s length is described by φ = 2κ̄z. The overall phase,φ̄ is usually unnecessary, but Eqn. (8) makes clear that theoriginal form [6] in Eqn. (7) is restricted to the unit vectorκ̂ = (1, 0, 0).

3) Polarization-Dependent Loss: Polarization-dependentloss (PDL) is usually modeled [5] in Jones space as a partialpolarizer P(0), diagonal in the standard basis with x and y fieldtransmission coefficients 0 ≤ px, py ≤ 1. This polarizer isthen allowed to rotate its pass axis with a matrix R(α), whichrotates a 2-D vector through angle α:

P(0) =

[px 0

0 py

]; R(α) =

[cos α − sin α

sin α cos α

]

where we have corrected a typographical error [6] and usedlinear rather than logarithmic losses.

The standard treatment is to perform a similarity transformwith R(α). PDL at arbitrary angles is then given by

P(α) = R(α)P(0)R−1(α).

These P(α) are diagonal blocks for T (σnm ) after multiplicationby the rotation matrices [6]: their block form permits treatmentas 2 × 2 matrices.

The transform is simple: a polarizer at angle α = θ in Fig. 2is equivalent to rotating the orientation of the field by −θ, thengoing through a polarizer which passes the x axis, and finally ro-tating the result back through angle θ. But as we have discussedabove, this formulation [6] is more restrictive than it need be,since it is limited to one DOF: in terms of Eqn. (4) the R(α)can be cast as

R(α) = J(φ̄ = 0, φ = 2α, β̂ = {0, 1, 0})

limiting that operator’s eigenstate “β̂” to one vector. By puttingthe eigenstate into “normal form” [9], [10], we generalize therotation matrix [6] (and hence the PDL description) by replacingR(α) with the more general

J(φ̄ = 0, φ = 2α, β̂).

Fig. 2. Polarizer in waveguide view. Polarizer is oriented at angle θ withrespect to the x − y fiber bases. Its operator is a similarity transformation of theoperator in the polarizer’s basis.

Fig. 3. Interferometer. Waveguide sections α, β, γ described by Jones matri-ces, while couplers C1 , C2 are each described by two polarization-independentcoupling matrices.

This permits the eigenstate β̂ to be on the entire unit sphere,rather than at a single point.

III. APPLICATION: INTERFEROMETRIC DEMODULATOR WITH

NORMAL MODE LOSS

A. Analysis

A general interferometric structure is shown in Fig. 3 : itconsists of three propagation sections, α, β, and γ, and twocouplers, each of which has the form in Fig. 1(b).

The matrix operator which takes the input states to the outputstates is given by the product of 4 × 4 matrices

M = J(γ ) (SC2S) J(β ) (SC1S) J(α) (9)

read as operators, from right to left, unlike Ref. [6]. As above, theSCS operators grouped in parentheses are similarity transformsfor C (Eqns. (5), (6)) from the coupler to the waveguide co-ordinate system, and correspond to the C(θx, θy ) operators[6].Thus, the first few operators read: propagate along the α wave-guide sections, change from waveguide to coupler co-ordinates,apply the first (left) coupler, change back to waveguideco-ordinates, propagate along the β waveguide sections, etc.,to the end.

We apply the formalism to calculate the effect of coupler nor-mal mode loss on a phase modulated link with birefringence,


as in Fig. 3. In particular, we examine the port-to-port perfor-mance, ignoring J(γ ) and J(α) , and restricting the analysis toinput on the x axis of port 1. Then the fields at the output portsare given by

�E =

⎡⎢⎢⎢⎢⎣

E1x

E1y

E2x

E2y

⎤⎥⎥⎥⎥⎦

= S C2 S J S C1 S

⎡⎢⎢⎢⎢⎣

1

0

0

0

⎤⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎣

c2x11J(1)11 c1x11 + c2x12J

(2)11 c1x21

c2y11J(1)21 c1x11 + c2y12J

(2)21 c1x21

c2x21J(1)11 c1x11 + c2x22J

(2)11 c1x21

c2y21J(1)21 c1x11 + c2y22J

(2)21 c1x21

⎤⎥⎥⎥⎥⎥⎦

(10)

where, for instance, the “1,2” element of the y polarization plane(see Fig. 1) for the second coupler is given by c2y12 = (C(y )

2 )12 .Keeping track of the matrix elements is easily handled withsoftware: Appendix B shows the Mathematica script used tofind Eqn. (10). Alternatively, one may sum the various pathsbetween an input and output mode by using a diagram similarto Fig. 1(b) to track the possible paths.

These outputs, as before [6], are adequate for CW inputs, butneed to be modified slightly for modulated inputs. For modula-tion frequencies much less than the carrier frequency, standardscalar Fourier analysis shows that the modulation envelope trav-els at the group velocity, while the carrier travels at the phasevelocity [7]. We assume that polarization mode dispersion isunimportant, i.e., that the differential group delay (DGD) be-tween the principal states of polarization [8] is insignificantcompared to the modulation period. Then the modulation en-velopes suffer a common group delay τ while the states ofpolarization of all the spectral components experience essen-tially the same evolution. We can then treat the fields as inEqn. (10) with the understanding that the modulation envelopemust be tracked with the corresponding waveguide operator.For instance, if there is a group delay of τ in arm “1” of theinterferometer, then

J(1)11 E(t) −→ J

(1)11 E(t − τ) (11)

where E(t) includes both amplitude and phase modulation.Then matrix element expansions that include J (i) as in Eqn.(10) can track the fields on which they operate.

The utility of transforming the problem to 2 × 2 blocks isapparent when we model normal mode loss, since we can usestandard techniques: to transform between “port” co-ordinatesand “normal mode” co-ordinates one constructs the matrix Sc

of the normal mode eigenvectors. For a symmetric polarization-independent coupler these normal modes are even and odd su-perpositions of the port states [12] so the transformation matricesbetween the two co-ordinate systems are

Sc =1√2

[1 1

1 −1

]= S†

c = S −1c . (12)

To model differential normal mode loss, we assume that thefield transmission in the even mode is unity while that in theodd mode is p ≤ 1. For an otherwise ideal coupler, the couplingmatrix would then be the similarity transform from the normalmode system to the coupler system:

C = Sc

[e−j π

4 0

0 pe+j π4

]Sc =

e−j π4

2

[1 + jp 1 − jp

1 − jp 1 + jp

]. (13)

Assuming, for algebraic convenience, that both the x andy polarization planes have the same normal mode loss, Eqn.(13) gives the cnkrs coefficients in Eqn. (10): (no summationimplied)

cnkrr =e−jπ/4

2(1 + jp), cnkrs =

e−jπ/4

2(1 − jp) (14)

for coefficients with similar or different indices, respectively.With the identification in Eqn. (11), we can then calculate thepowers on the two output ports

P1 =(1 + p)2

16[I(t) + 2Re

[−B e+j4εp E∗(t − τ)E(t)

]]

(15a)

P2 =(1 + p)2

16[I(t) + 2Re [B E∗(t − τ)E(t)]] (15b)

where

I(t) = E∗(t − τ)E(t − τ) + E∗(t)E(t) (16a)

| J(1)11 |2 + | J

(1)21 |2 + | J

(2)11 |2 + | J

(2)21 |2 =2 (16b)

B = J(1) †11 J

(2)11 + J

(1) †21 J

(2)21 (16c)

εp =π

4− tan−1 p = tan−1 1 − p

1 + p2 . (16d)

Here, Eqn. (16 a) reflects the noninterfering power levels dueto changes in the field’s amplitude only, and Eqn. (16 b) assumeslossless fibers in the interferometer arms.

Eqns. (15) show that εp measures a phase shift imparted di-rectly onto the interference terms. For an input phase modu-lated at RF frequency Ω, the envelope is E(t) = E0e

jφm sin Ωt .Assuming path matching (Ωτ = π) and quadrature biasing(B ∼ j), output at port 1 is given by

P1 =(1 + p2)2 | E0 |2

8[1 + sin(4εp + 2φm sin Ωt)] (17)

which would be interpreted as a bias offset of 0.058 rad for a0.25 dB normal mode differential loss, for instance. This analy-sis, using standard linear space techniques to transform first tothe coupler basis and then to the normal mode basis, would bedifficult to do with other approaches.

B. Discussion

The term 4εp in Eqn. (17) can be viewed as a source of biaserror (caused by normal mode loss (16d)) for the architectureshown in Fig. 3. Thus, the analysis above describes a new phys-ical mechanism for bias phase shifts for such links, which may


Fig. 4. Measured (symbols) and calculated second-order intercept point(OIP2) for an analog optical phase modulated link as a function of interfer-ometer bias phase.

have been attributed to other sources of bias error in previouswork. Such understanding is critical for the implementation ofhigh-fidelity analog optical phase modulated links. Deviationsfrom a precise quadrature bias in an asymmetric interferometersuch as shown in Fig. 3 can lead to significant even-order dis-tortion. This was quantified in [13] using scalar fields, whereexpressions similar to Eqn. (17) were employed to calculatethe response of an analog phase modulated link to a sinusoidalstimulus as a function of interferometer bias phase. In that work,the dc bias phase was expressed as a single parameter, whereasthe analysis here identifies a specific physical contribution tothe dc bias phase. The impact of εp can be readily obtainedby expanding Eqn. (17) into odd- and even-order components.The RF powers of the fundamental and odd-order distortionsare proportional to cos2(εp); the even-order distortion powerscales as sin2(εp). Consequently, the impact on fundamentalsignal transmission is negligible for small εp and the odd-orderintercept points are independent of εp . However, the impact oneven-order intercept points can be significant.

To exemplify the sensitivity of even-order distortion to biasphase, consider the plot of second-order output intercept point(OIP2) as a function of normalized bias phase in Fig. 4. The dataand theory shown there are extracted from the results in [13],where, again, the dc bias phase was parameterized as a singleconstant. Thus, the impact of εp in Eqn. (17) is demonstratedexplicitly by Fig. 4: even small εp can lead to dramatic changesin OIP2. The apparatus utilized to produce these data entailed an80-mW semiconductor distributed feedback laser that was mod-ulated via a LiNbO3 phase modulator followed by an asymmet-ric fiber interferometer such as shown in Fig. 3. The differen-tial delay in the interferometer was nominally τ = 375 ps andthe ideal quadrature condition corresponds to a bias phase ofω τ = (2k + 1)π/2 where ω is the angular optical frequencyand k is an integer. The driving frequencies for the experimentwere 4.0 and 4.5 GHz; the OIP2 is relative to second-orderintermodulation distortion at 8.5 GHz. The bias phase was ad-justed using a heater deposited onto the fiber of one arm of the

interferometer. As can be seen in Fig. 4, the OIP2 exhibits astrong sensitivity to bias phase and, equivalently, εp . Hence, theanalysis presented above has identified a previously unknownsource of bias error in phase modulated systems—an error thatcan be significant for high-fidelity, multi-octave analog links.

IV. CONCLUSION

We have presented a formalism which uses standard lin-ear vector space techniques to analyze interferometric struc-tures with birefringence and PDL. By introducing a simple co-ordinate transformation to move between waveguide and cou-pler descriptions, we showed that the operators could be reducedto 2 × 2 diagonal blocks under usual assumptions [4], [6]. Thistransformation permits system components to be treated con-veniently with conventional matrix techniques. Using generalunitary matrices for operators, we showed that couplers, waveg-uides, and polarization-dependent losses can be introduced witharbitrary eigenstates. We illustrated the formalism by analyzingthe output for a demodulator in a phase modulated link, andfound that differential normal mode loss creates a dc bias phase,which could impair high-fidelity optical links.

APPENDIX AGEOMETRICAL REPRESENTATION OF OPERATORS

It is often useful to view the operations in the text with a3-D geometrical representation. This representation casts boththe coupler and the birefringent propagation in the same de-scription, and permits one to visualize the evolution of states.While it is straightforward for a single fiber or a single coupler,it also has utility for problems in this paper if one imagines twoindependent evolutions occurring. That is, the evolutions occuron two independent spheres, but one may mentally superposethem to gain insight into the differential evolutions in a physi-cal problem. In this Appendix, we present a condensed view ofsome results from earlier work [9], [10], to which the reader isreferred for details.

A 3-D representation of Jones vectors is obtained by castingthem in the normalized form

�s ⇔ e−j φ̄0

[cos θ0

2 e−jφ0 /2

sin θ02 e+jφ0 /2

](18)

where φ̄0 is the common phase between components, φ0 is thedifferential phase, and θ0 describes the relative amplitudes ofthe components. With this identification, a 3-D unit vector, �s,is constructed with “latitude” θ0 and “longitude” φ0 on a 3-D vector basis set. The common phase, φ̄0 , is represented by a“fiducial paddle” that lies on the unit sphere’s surface and rotatesabout vector �s by 2 φ̄0 . Phases of two states are compared bytranslation over a great circle and measuring the difference inpaddle orientations [9], [10].

A geometrical representation of the four DOF for unitary op-erators, such as J in Eqn. (4) of the text, is shown in Fig. 5,which illustrates an operation taking state �si to state �sf . In thisrepresentation, the “slow” eigenstate (eigenvalue +1) of J isrepresented by the unit vector β̂, whose components are the


Fig. 5. Geometrical representation of general unitary matrix. Operator J inEqns. (2,4) is represented by vector �β = β̄β̂ , where the eigenstate with eigen-value +1 is unit vector β̂ . The operator rotates state (vector �s) about β̂ throughangle φ = 2β̄z. Simultaneously, the fiducial paddle representing overall phase,is rotated about �s by angle 2 φ̄.

(β1 , β2 , β3) in the matrix representation Eqn. (4). Operator Jrotates states about β̂ in the positive (right-hand rule) sense byangle φ = 2β̄z. This motion changes the latitude and longitudeof the state, and thus, by virtue of Eqn. (18), the relative ampli-tudes and phases of the optical components in the original basisset. (Note that in the basis set defined by ±β̂, this evolution doesnot change the latitude: β̂ is the “North pole” which defines thelatitude for this normal mode basis, and this same evolution ismerely a phase change between components of constant ampli-tude by virtue of Eqn. (18)). While the rotation at uniform rate2β̄ may seem like a special case, it can be shown [8]–[10] thatany concatenation of more general rotations is reducible to thisform.

In addition to changing the relative phases of �s ’s compo-nents by virtue of the changing position of �s, the operator alsochanges the overall optical phase, which is represented by thefiducial paddle attached to �s. Thus, in rotating through angle φ,the orientation of the initial fiducial paddle at �si is changed tothe grey shaded paddle at �sf by rigid rotation about β̂. Simulta-neously, the overall phase change in Eqn. (4) is represented bya rotation of the fiducial paddle by an additional 2φ̄ radians inthe positive sense. The four DOF for this unitary transformation(Eqn. (4)) are thus completely identified in this representation:two DOF for the orientation of β̂, one DOF for the magnitude of�β, represented by angle φ, and one DOF for the overall phase,represented by angle φ̄.

Conversely, any unitary matrix, since it must be reducible tothe form of Eqn. (4), can thus be represented by an equivalent setof β̂, φ̄, and φ. This set can be found by projecting the evolutionmatrix onto the Pauli basis set (σ0 , σ1 , σ2 , σ3) and identifyingthe components [8]–[10].

Note Added in Review: While this paper was under review,Karlsson proposed an extension of the standard Jones/Stokesformalism [14] which expresses the fiducial paddle in moreformal terms as a property of a subspace of a “Stokes statematrix.”

APPENDIX BMATHEMATICA NOTEBOOK

A Mathematica notebook for tracking the matrix operationsin Eqn. (10) is given below:

(* Coupler matrix indices:coupler #, polarization, row, column (for 2 × 2) *)C1 = { {c1x11, c1x12, 0, 0}, {c1x21, c1x22, 0, 0},

{0, 0, c1y11, c1y12}, {0, 0, c1y21, c1y22} };C2 = { {c2x11, c2x12, 0, 0}, {c2x21, c2x22, 0, 0},

{0, 0, c2y11, c2y12}, {0, 0, c2y21, c2y22} };(* propagation matrix indices:

waveguide #, row, column (for 2 × 2) *)J = { {j111, j112, 0, 0}, {j121, j122, 0, 0},

{0, 0, j211, j212}, {0, 0, j221, j222} };S = { {1, 0, 0, 0}, {0, 0, 1, 0},

{0, 1, 0, 0}, {0, 0, 0, 1} };input = {1, 0, 0, 0};mat = S.C2.S.J.S.C1.S;Print[“E field out = ”, mat.input // MatrixForm]

REFERENCES

[1] Y. K. Lize, M. Faucher, E. Jarry, P. Oulette, A. Wetter, R. Kashyap, andA. E. Willner, “Low-loss S-, C-, and L-band differential phase shift keyingdemodulator,” presented at the Conf. Lasers Electro-Opt., Baltimore, MD,USA, May 2007, Paper CMJJ4.

[2] A. H. Gnauck and P. J. Winzer, “Optical phase-shift-keyed trans-mission,” J. Lightw. Technol., vol. 23, no. 1, pp. 115–130, Jan.2005.

[3] V. J. Urick, F. Bucholtz, P. S. Devgan, J. D. McKinney, and K. J. Williams,“Phase modulation with interferometric detection as an alternative tointensity modulation with direct detection for analog-photonic links,”IEEE Trans. Microw. Theory Tech., vol. 55, no. 9, pp. 1978–1985, Sep.2007.

[4] C.-L. Chen and W. K. Burns, “Polarization characteristics of single-modefiber couplers,” IEEE J. Quantum Electron., vol. QE-18, no. 10, pp. 1589–1600, Oct. 1982.

[5] D. Goldstein, Polarized Light. New York, NY, USA: Marcel Dekker,2003.

[6] Y. K. Lize, J.-C. Richard, P. Samadi, and L. R. Chen, “Polarization depen-dent formalism of interferometric structures describing DPSK and DQPSKreceivers,” in Proc. Conf. Lasers Electro-Opt., 2010, Paper CThC2.

[7] H. A. Haus, Waves and Fields in Optoelectronics. Englewood Cliffs, NJ,USA: Prentice-Hall, 1984.

[8] J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization modedispersion in optical fibers,” Proc. Nat. Academy Sci. USA, vol. 97, no. 9,pp. 4541–4550 2000.

[9] N. J. Frigo and F. Bucholtz, “Geometrical representation of optical prop-agation phase,” J. Lightw. Technol., vol. 27, no. 15, pp. 3283–3293, Aug.2009.

[10] N. J. Frigo, F. Bucholtz, and C. V. McLaughlin, “Polarization in phasemodulated optical links: Jones- and generalized Stokes-space analysis,”J. Lightw. Technol., vol. 31, no. 9, pp. 1503–1511, May 2013.

[11] A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE. J. Quan-tum Electron., vol. QE-9, no. 9, pp. 919–933, Sep. 1973.

[12] R. C. Youngquist, L. F. Stokes, and H. J. Shaw, “Effects of normal modeloss in dielectric waveguide directional couplers and interferometers,”IEEE J. Quantum Electron., vol. QE-19, pp. 1888–1896, Dec. 1983.

[13] V. J. Urick, J. D. McKinney, J. F. Diehl, and K. J. Williams,“Equationsfor two-tone analog optical phase modulation with an asymmetric inter-ferometer,” IEEE Photon. Technol. Lett., vol. 25, no. 15, pp. 1527–1530,Aug. 2013.

[14] M. Karlsson, “Four-dimensional rotations in coherent optical commu-nications,” J. Lightw. Technol., vol. 32, no. 6, pp. 1246–1257, Mar.2014.


Nicholas J. Frigo (S’82-M’82-SM’02-F’06) received the B.A. degree in physicsfrom Claremont McKenna College, Claremont, CA, USA, and the M.S. andPh. D. degrees in solid-state physics from Cornell University, Ithaca, NY, USA.In 1983, he joined the Optical Sciences Division of the U.S. Naval ResearchLaboratory, where he was engaged in research on fiber optic sensor technologyfor 6 years. He was the Research Department Head at Litton Industries for 2years. In 1990, he joined ATT Bell Labs as a Member of Technical Staff andwas promoted in 1994 to Distinguished Member Technical Staff, while engagedin research on fundamental impairments for TV transmission over fiber opticlinks, long reach undersea fiber optic systems, and passive optical networkarchitectures for fiber-to-the-home. After the ATT trivestiture, he went to ATTLabs Research, where he was engaged in research on business optical access,polarization mode dispersion, and network diagnostics, holding positions asa Technology Leader, and as the Division Manager of the Optical SystemsResearch Department. In August 2005, he joined the Physics Department, U.S.Naval Academy, Annapolis, MD, USA, as the first Michelson Professor ofPhysics. He is the author or coauthor of more than 60 publications, 90 conferencepresentations, and 40 patents. He has served on various Technical ProgramCommittees, and has been an Associate Editor of Photonic Technology Letters,and is on the Editorial Advisory Committee of OPN.

Vincent J. Urick (M’05-SM’12) received the B.S. degree (magna cum laude)in physics with minors in electronics and mathematics from Bloomsburg Uni-versity, Bloomsburg, PA, USA, in 2001, and the M.S. and Ph.D. degrees fromGeorge Mason University, Fairfax, VA, USA, in 2005 and 2007, respectively.In 2001, he joined the U.S. Naval Research Laboratory, Washington, DC, USA,as a Research Physicist in the Photonics Technology Branch. He is currently theHead of the Applied RF Photonics Section at NRL directing and conducting re-search in microwave photonics, a field where he has more than 90 publications.His present research interests include novel applications of photonics technol-ogy. He received the Department of the Navy Top Scientists and Engineers ofthe Year Award in 2007. He serves or has served on the Technical Program Com-mittees for the IEEE AVIONICS FIBER OPTICS AND PHOTONICS CONFERENCE,the IEEE INTERNATIONAL MEETING ON MICROWAVE PHOTONICS, and the IEEEPHOTONICS CONFERENCE. He served on the 2013 Organizing Committee for theIEEE INTERNATIONAL MEETING ON MICROWAVE PHOTONICS, is a Member ofthe IEEE PHOTONICS SOCIETY TECHNICAL ADVISORY COUNCIL, and Chairs theMicrowave Photonics Committee for the IEEE PHOTONICS CONFERENCE.

Frank Bucholtz (M’83-SM’10) received the B.S. degree in physics and math-ematics from Wayne State University, Detroit, MI, USA, in 1975, and the M.S.and Ph.D. degrees in physics from Brown University, Providence, R1, USA,in 1977 and 1981, respectively. From 1981 to 1983, he was an NRC Postdoc-toral Research Associate at the Naval Research Laboratory where he conductedresearch in ferrimagnetic devices for microwave signal processing. He joinedthe Optical Sciences Division, Naval Research Laboratory in 1983 where heworked on fiber optic magnetic and chemical sensors, and airborn hyperspectralsensing. From 1999–2003, he was the cofounder and the President of AdvancedSensor Technology, a start-up devoted to fiber-optic medical and surgical de-vices. He returned to NRL in 2003 and is currently the Head of the AdvancedPhotonics Section, where he is responsible for the research and developmentin the areas of microwave photonics, micro- and nanooptics, ultrafast lasers,and the application of information theory to DoD problems. He has authored orcoauthored more than 150 publications and conference presentations and holdssix patents.

simple linear space formalism for polarization-dependent interferometers: theory and application to...

Documents