1152 J. Opt. Soc. Am./Vol. 72, No. 9/September 1982
Evolution of the polarization in codirectional andcontradirectional optical couplers
Otto Schwelb
Department of Electrical Engineering, Concordia University, Montreal, Quebec H3G 1M8, Canada
Received January 16, 1982
Jones and Mueller calculus for both forward- and backward-wave optical couplers is presented. By using Kroneck-er product algebra, the relationship between the system matrices of the amplitude, the coherency, and the polariza-tion vectors is derived. The relationship between the corresponding transfer matrices is also established. Explicitexpressions are given for the Mueller matrix of uniform two-mode devices supporting hermitian and skew-hermi-tian coupling. For uniform devices the eigenvalues and eigenvectors of the propagator matrices of the Jones andStokes vectors are evaluated. Linearly tapered and chirped optical couplers are analyzed, and solutions are givenfor the transfer matrix of the modal amplitude vector.
INTRODUCTION
An important facet of the operation of an optical device is itseffect on the evolution of wave polarization. In this paper Ianalyze the state of polarization in forward-wave and back-ward-wave couplers that can be adequately described bycoupled-mode formalism. Treatment is restricted to mono-chromatic waves and conservative systems.
The state of polarization of an electromagnetic wave is givenby the four-component polarization or Stokes vector whoseelements are determined by the components of the complexelectric field. An analytic definition of the Stokes parametershas been presented by Wolf.1 A concise treatment of the in-terrelation between the characteristics of the polarizationellipse, the Stokes parameters, and their graphic represen-tation on the Poincar6 sphere can be found in Ref. 2. Theevolution of the Stokes vector in an optical device or in amedium is described by a 4 X 4 transfer matrix, the so-calledMueller matrix. A systematic application of matrix methodsto the analysis and design of optical components has beengiven by Gerrard and Burch.3 More recently, the Jones andMueller calculi have been applied to reflection problems andmagnetic materials,4 optical-transmission characteristics ofsingle-mode fibers,5' 6 liquid-crystal twist cells,7 and weaklyinhomogeneous anisotropic dielectrics. 8
In integrated optics as well as in other areas of electro-magnetics, it is often necessary to transfer power from oneguided wave (or mode) to another. One of the simplestmethods to achieve coupling is to let the waves pass throughopen waveguides in close proximity. Gradual transfer ofpower from a forward-propagating wave to a backward-propagating wave can be induced by placing small periodicallyor almost periodically spaced obstacles in the path of propa-gation, such as in hologram gratings. However, power transfercan be achieved by numerous other methods, such as alter-nating /AO techniques or traveling-wave interactions. Aninteresting early survey of the mostly microwave applicationsof conservative codirectional and contradirectional couplinghas been compiled by Barnes.9 Of course, the exchange ofpower between modes of propagation is often an undesirable
phenomenon. Coupling that is due to material or geometricimperfections is the source of cross talk and other types ofsignal distortion resulting from the different group velocitiesof the coupled modes.
Ulrich has treated the state of polarization of codirectionallycoupled waves and described the evolution of a set of char-acteristic parameters on the Poincar6 sphere.10 Vernon andHuggins considered coupling between counterpropagatingwaves but restricted their analysis to the Jones calculus.4 Inthis paper we deal with the Jones and Mueller calculi of bothhermitian and skew-hermitian coupling, i.e. those that con-serve either the sum or the difference of the normalized modalpowers. First, general matrix formulas are derived that ex-pose the relationship between the system and transfer ma-trices within the Jones and Mueller calculi. By using theconcept of a coherency vector and Kronecker product algebra,the connection between Jones and Mueller calculi is also es-tablished. Next, uniform devices are analyzed, their systemand transfer matrices are expressed in terms of the constituentparameters and the length of the coupler, and the appropriateMueller matrices and their eigenvectors are evaluated. Thedevelopment of Jones matrices for couplers exhibiting lineartaper or chirp nonuniformity conclude the paper.
TRANSFER CHARACTERISTICS
Consider an electromagnetic transmission system supportingtwo spatially orthogonal monochromatic waves (modes)represented by their complex amplitudes a (z), i = 1, 2, wherez is the axial coordinate. Harmonic time dependenceexp(jwt) is assumed and suppressed. The amplitudes do notdepend on the transverse coordinates of the system and areso normalized that lail2 is the power carried by mode i. Thereal and imaginary parts of ai are conjugate functions andhence Hilbert transforms of each other. The evolution of theamplitudes along the z coordinate is characterized either bythe coupled linear differential equation
where R(z) is the system matrix and the prime indicates dif-ferentiation with respect to z, or by the linear transforma-tion
a(z) = T(z)a(0), (2)
where aT(z) = [al(z), a2(z)] is the two-component state vector(the superscript T indicates transposition) and T(z) is thetransfer matrix. Since modes can propagate in either direc-tion along the transmission medium, T(z) must be nonsing-ular. Equations (1) and (2) characterize the system withinthe framework of the Jones calculus.
As an intermediate step toward the introduction of thepolarization or Stokes vector, one defines a so-called coherencyvector f(z):
fT(z) = [alai*, ala2 *, a2al*, a2a2*] (3)
whose elements are the same as those of the coherency matrixof a monochromatic wave, also known as the projection ma-trix.11 A projection matrix is idempotent, i.e., it is equal toits square, a property satisfied by the coherency matrix whenit is normalized so that la 2 + la212 =1. When one is dealingwith backward-wave couplers for which the law of conserva-tion preserves the difference of powers I a11 2 - a d 2 rather thantheir sum, the elements of a proper projection matrix shouldrather be [aial*, ala2 *, -a 2al*, -a 2a2*1. However, we areconcerned here only with the f(z) vector; the idempotentproperty of the coherency matrix shall not be utilized, andtherefore Eq. (3) is retained for both forward and backwardinteractions.
Various sets of parameters can be used to describe the stateof polarization of the a(z) vector. Generally accepted is theset of Stokes parameters defined by the linear transforma-tion
f(z) = Vs(z), (4)
where
It follows from Eq. (5) that
f'(z) = [T' X T* + T X T*']f(0). (6)
Furthermore, substitution of Eq. (2) into Eq. (1) shows thatT' = -jRT. Applying this latter result to Eq. (6) and per-forming some simple algebraic operations produces the de-sired system equation of the coherency vector
f'(z) = -jF(z)[T(z) X T*(z)]f(0) = -jF(z)f(z), (7)
where F(z) = R(z) X I - I X R*(z) and I is the identity matrixof appropriate dimensionality. Equations (7) and (5) char-acterize the evolution of the coherency vector. The corre-sponding expressions for the Stokes vector are now easy toobtain with the help of transformation (4). Thus on the onehand
determines the system matrix Q(z), whereas on the otherhand
s(z) = V-lf(z) = V-'[T(z) X T*(z)]Vs(0) = M(z)s(0) (9)
determines the transfer matrix M(z) or the Mueller matrixfor the Stokes vector in a coupled transmission system de-scribed by Eqs. (1) and (2).
The properties of the Q(z) and M(z) matrices in many re-spects parallel those of R(z) and T(z). For example, the Qand M matrices must satisfy the equation M'(z) = -jQ(z)M(z). If R(z) and T(z) commute, then it can be shown thatQ(z) and M(z) must likewise commute. Hence when R andT have a common set of eigenvectors, so do Q and M. Fur-thermore, when R is hermitian and T is unitary, then Q is alsohermitian and M is also unitary. This is important becauseit concerns the conservation of the quadratic form sts withinthe system. For, if R = Rt, then
(ata)' = jat(Rt - R)a = 0, (10)
rSOWZ)S(Z = sl(Z) =
S2(Z)
S3WZ
is the Stokes vector,
1V =
1
0
0
-1
0 01 -j
1 j
O O.
and V-1 = 2Vt (t indicates hermitian conjugation). If, forexample, a1 and a2 were the complex amplitudes of thetransverse components Ex and Ey of the electric field, sowould represent the total codirectional power flow,tan-1 (s3 /s2) the phase difference between the amplitudes,1/2 tan1'(s2/sl) the angle between the major axis of the polar-ization ellipse and the x coordinate, (so/s3)1 - [1 - (S3/sO)2]1/21the axial ratio, (so - sl)/(so + sl) the local reflection coefficientif the components were counterpropagating, etc.
The transfer matrix of the coherency vector is simply theKronecker product12 of T(z) and T*(z):
and similarly
(sts)' = jst(Q - Q)s = 0, (11)
signifying that sts is independent of z. Conversely, when TTt= I, then
at(z)a(z) = at(0)Tt(z)T(z)a(0) = at(0)a(0), (12)
and similarly
st(z)s(z) = st(0)Mt(z)M(z)s(0) = st(0)s(0). (13)
Conditions are somewhat more complicated in couplers, inwhich the difference of the modal powers is conserved. In thiscase
at(Z)[o I a(z) = const., (14)
and as a consequence R is no longer hermitian and T no longerunitary. Rather,
lo 1] [[ - 1]
Tto I OT = [° 1]-
(5) Unfortunately, as we shall see in the next section, these con-
Otto Schwelb
f(z) = [T(z) X T*(z)]f(O).
1154 J. Opt. Soc. Am./Vol. 72, No. 9/September 1982
straints do not carry over in any simple fashion to Q andM.
Imposing no restrictions on the 2 X 2 R and T matrices(Rpq) and (Tpq), the explicit expressions for F, Q, and Mare
j2Ri,
F = -R21*R2L
Lo
R12 °
0 R1 2
R22- R,* -Rl 2 *
-R21* j2R22j
-R12*
Ri - R22*
0
Otto Schwelb
where flo = '/2(131 + /2) is the average phase constant, 42 =(1/2A#)2 L IK12 = IKI 2X2, A: =/1 - 02, X = (Y2 + 1)1/2, Y =A13/21 KI is the asynchronism parameter, and the upper (lower)sign convention corresponds to that used in Eq. (18). In aforward coupler the eigenvalues are real, representing twowaves propagating in the same direction (the group velocitieshave the same sign), one (3-) faster and the other (3+) slowerthan the average phase. The fast and slow waves exchangetheir power periodically. For a backward coupler X can bereal or pure imaginary. When, because of small coupling orlarge differential phase shift, X is real, one again encounters
(15)
(RI, + R22 )i
Q= (R, -R22)i
(R12 + R2 1)iR12 -R 21)r
and
(R, -R22)i
(R,, + R22 )i
(R2 1 -R2)i
-(R12 + R2 1)r
(R12 + R21)i(R12-R2-)i
(R,, + R22)i(R,, -R22)
(R12 - R 21)r
(R12 + R2 1)rI, (16)
(R 22 - Ri1)r
(RI, + R22)i
JT,112 + IT1212 + IT2112 + IT2212
M = ITT112 + IT1212 - IT2 112 - IT2212
2 2(TllT21* + T12T22 *)r
-2(TllT 2 1* + T12T22 *)i
JT,112 - T12 12 + IT 2112 - IT 22 12 2(TllTl2 * + T2 lT2 2 *)r
JT,,1 2 - JT1212 - 1T2 112
+ IT2 2 12 2(TiiT12 * - T2lT 2 2 *)r
2(TllT2*- T 1 2T22*)r 2(TllT2 2 * + Tl2T21*)r
-2(TllT2* - T 2T22*)i -2(TllT2 2 * + T12 T2 1*)i
2(TllTl2 * + T2iT22*)1
2(TiT2* - T2 1T2 2 *)i
2(TllT22* - T 2 T2 1 *)t I2(TllT22*- T 2 T 2 l*)r
respectively. For the uniform conservative forward- andbackward-wave couplers examined in the next section theseexpressions simplify considerably.
UNIFORM COUPLERSUniform conservative couplers are characterized by the con-stant system matrix
R = 1 K](18)
where 3, and /2 are the real loaded phase constants of thecomponent waveguides, K = I K| exp(jo) is the complex couplingcoefficient, and the upper (lower) sign refers to so-calledhermitian (skew-hermitian) coupling. The term loaded isused to describe the phase constant of a transmission linewhose properties are slightly modified by the presence of anadjacent line. The term hermitian coupling is used to de-scribe an ordinary forward-wave coupler, or a contraflow
a fast and a slow wave exchanging their power periodicallyalong the coupler, but now the group velocities of the waveshave opposite signs.9 When X is imaginary, 13+ and 1- arecomplex conjugates, the imaginary part being the attenuation(gain) constant signifying a monotonic rather than periodictransfer of power from one mode to the other. Both modestravel with the same phase velocity, determined by vp =W/0o.
The evolution of the Stokes vector is described by either thesystem matrix Q or the Mueller matrix M. In case of a for-ward-wave coupler, these are given by
O 0
Q = j2IKI 0 00 -sin q
LO -cos 0
s ksin 0 cos o
0 -Y
Y O
(20)
1 0
0 1 2s2
0 2s2 Y 2scX2cos-XsinX
2s2 Y . 2sc° - X2sino- Xcoso
0
2.q2
y 92sc-cos 0 + X sin 0X2
I -S2 (2Y2 + 1 -cos 20)
-X2 sin 2k + X-
0
2s 2Y . 2sc- X2sin+-XcosX
- 2sin 20-
1 2 +1 - X2 (2y2 + I + cos 20
coupler of modes that carry negative energy.' 3 Skew-her-mitian coupling takes place in an ordinary contraflow direc-tional coupler or in a co-flow coupler of modes that carrynegative energy. Since passive waveguide modes carry pos-itive energy, we shall generally refer to the upper (lower) signto describe a forward- (backward-) wave coupler.
The eigenvalues of R are
where s = sin (z and c = cos (z. In the case of a backward-wave coupler, the corresponding matrices turn out to be
respectively. As long as I 11 > 1, s and c in Eq. (23) still ab-breviate the same trigonometric functions as in Eq. (21).When IYl < 1, however, we set t = IKIX = -jT = -jIKI(1 -y2)l/2 and replace sin (z and cos (z with - jshTz and chTz,respectively. The Mueller matrix, of course, remains real.For a particular device, Eqs. (21) and (23) are likely to be muchsimpler than the expressions given above. In a synchronouscoupler, for instance, i1 = 12 and Y = 0; when K is real(imaginary), 0 = 0 (0 = '/27'); when the coupler is 1/27r long, i.e.,when (L = 1/2-7r, then s = 1 and c = 0, etc.
The four eigenvalues of Q are q, = q2 = 0 and q3 = -q4 =
2t, whereas those of M are ml = M2 = 1 and m3 = 1/M4 =exp(-j2tz) for both forward- and backward-wave interaction.If we note that R and T commute, the modal matrix consistingof the common eigenvectors of Q and M is
can be represented graphically on the Poincar6 sphere. 8"10
When new variables v, p, and 0 are introduced such that tanv = -cot 0, tan p = (Y/X)tan (z, and sin 0 = X-1 sin (z forexample, Eq. (21) reduces to Eq. (14) in Ref. 8.
A small loss in the transmission medium can be treatednumerically as a perturbation of the system or the transfermatrix, provided that they are well conditioned. In practicethis is usually the case. An alternative method for modelingloss is to connect matched attenuators in cascade with alossless coupler. A matched attenuator is a nonreactive re-ciprocal device whose characteristic impedance matches thatof the lossless device and whose amplitude-transmission loss
1 1aY -aY
Uf = d cos 0 -a cos 0
L- sin 0 a sin P
for the forward case and
aYUb = 1
- cos 0Ld sin k
01
-Y cos o - jX sin 0
Y sin - jX cos o
-a y1
10
acos4 -Ycos -jXsin k-a sin 0 Y sink -jX cos 0
011
-Y cos 0 + jX sin 0Y sin 0 + jX cos 0
o
-Y cos 0 + jX sin 0Y sin 0 + jX cos j
for the backward case, where the real parameter (6Y < 1) inthe first two eigenvectors (columns) indicates that the asyn-chronism between a, and a 2 is Y = (6- 2
- 1)1/2. Thus the firsteigenvector in Uf corresponds to the orthogonal ampli-tudes a1 =[1/2(1 + c3Y)]1/2 exp(j0l) and a 2 = [1/2(l - Y)]1/2 exp(02), with 0, - 02 = ¢. In the case of the second eigenvector,the magnitudes of a, and a2 given above are interchanged, and01 - 02 = 0 + w. In terms of conventional Stokes parameters,the last two eigenvectors in Eq. (24) have no physical inter-pretation. First, these waves have zero total power flow,suggesting the presence of mode pairs, one of which must carrynegative energy.13 When it is modulated, such a mode haslesser power than when it is unmodulated. Second, s2 and s 3are complex, in contradiction to the original definition of s.Because of the hermitian character of Eq. (20), the four ei-genvectors comprising the modal matrix Uf are mutually or-thogonal. Similar considerations apply to the eigenvectorsin Ub; however, in this case all 2 - I a212 represents the totalpower flow. Also, since Eq. (22) is not Hermitian, the foureigenvectors in Eq. (25) are merely pairwise orthogonal. Theevolution of the Stokes parameters in a forward-wave coupler
is exp(-aL), where aL is the total attenuation of the couplerin nepers. For a systematic treatment of the lossy uniformcoupler, see Appendix B. Conservation laws pertinent tolossless couplers are discussed in Appendix A.
NONUNIFORM COUPLERS
The transfer matrix of an arbitrary nonuniform coupler can-not be evaluated in closed form. Numerical integrationproduces transfer coefficients for a given coupler length at agiven frequency. Each time the coupler length is adjusted,a new solution to the differential equations is required.Similar computational limitations apply to the Mueller ma-trix. There are certain nonuniformities, however, that areamenable to analytic solution. Some of these have beendiscussed by Milton and Burns.14 Here their treatment isextended to include reflective gratings and gradual variationof the effective grating period, so-called chirp nonuniformity,which plays an important role in nonuniform backward-wavefilter structures.' 5
The modal matrix associated with a z -dependent R of thesame construction as that in Eq. (18) is
1 (23)
(24)
(25)
Otto Schwelb
1156 J. Opt. Soc. Am./Vol. 72, No. 9/September 1982
where 0 is the argument of K and the same sign conventionapplies as in the previous section. For a uniform coupler theU matrix diagonalizes R and T. Thus, for a uniform coupler,a change of basis from a(z) to g(z), according to the lineartransformation
a(z) = Ug(z), (27)
and, as a result,
2 [2 ±exp2-ju)-exp(ju)
0 1(33)
where u =f 2IKXdz. On the other hand, when Y' vanishes,one chooses
decouples the system equation [Eq. (1)]. For a nonuniformcoupler this is no longer the case; instead one finds that thesystem equation on the new g(z) basis is
g'(z) = -j(AR - jU 'U')g(z), (28)
where AR = U-1RU is a diagonal matrix whose diagonal ele-ments /3+(z) and O3(z) are the eigenvalues of R. A secondchange of basis given by
Hi, = exp[I-ij (Z3+ + i) dzI
H22 = exP- JO (- - - ]dzi
to find that
2X i+exp(-ju)
exp(ju)]' '1
H() = [H11(z)H22(Z29)
with the yet undetermined elements Hi, and H22 transformsEq. (28) into
where, in this case, u = Sf (21KIX + YO'/X)dz. Recognizingthat differentiation with respect to the new variable u isproportional to differentiation with respect to z, Eq. (30) canbe considered an equation in u and solution sought in theform
h(u) = D(u)N(u)h(0), (36)
where D is a diagonal matrix. For Eq. (36) to be a solution ofEq. (30), the condition D(u) N(u) + D(u)N(u) = S(u)-D(u)N(u) must be satisfied, where the dot represents dif-ferentiation with respect to u. This condition is indeed ful-filled for tapered couplers (O' = 0) when
D() = exp(j'/2u)
(31)
(37)+exp(-ij/2u
N{u=
where, as before, the prime denotes differentiation with re-spect to z.
Equation (30) can be solved for linear tapers (Y'/X2 =constant and q5' = 0) or for linear chirp nonuniformity (U/X= constant and Y' = 0). In either case, the H matrix is sochosen as to eliminate the diagonal elements of S. Thus,when qY vanishes, the elements of H are
cos(1/2ru) - r sin('/2ru) F sin(r/2 Fu)r F
,(38)
s sin('/2ru) cos('/2ru) + J sin(1/2ru)
where y = +y'/X2 = [IKI(A3)' - AOI3Ki]/202 and r2 = 1 +
l 2. For chirped couplers (Y' = 0), a solution is obtainedwhen
D(u) = exp(j1l/2u) (39)Fexp(-j1/2u)j
and
- r sin(Q/ 2Fu)IF
T'r sin(Q/2ru)
cOs('/2Fu) + J sin(Q/2ru)
(40)
where now -y = jq'/X and 12 = 1 F y2.The transfer matrix on the original a basis is retrieved by
successive back transformations from h to g to a. Considering2) that H(z)D(z) = exp(-jfJ f3odz)or (for the definition
g(z) = H(z)h(z),
(34)
(35)
N(u) =
Hi, = exp(-i .f O+dz H22 = exp(-j fo I-dz)|
Otto Schwelb
-A1
Otto Schwelb
of D- 1 (0) = o, see Appendix A), and N-1 (0) = H-1(0) = I,the transfer matrix is found to be
zT(z) = exp( J' fodz U(z)oN(z)U-1(0). (41)
Substitution of Eq. (41) into Eq. (9) provides the requisiteelements of the Mueller matrix. It is c~umbersome to performthe matrix multiplications involved by hand, but there is noobstacle to a numerical evaluation or to a graphical repre-sentation of the matrix elements as a function of length or ofany other variable parameter.
CONCLUSION
The Jones and Mueller calculi of two-mode conservativeforward- and backward-wave optical couplers have beenpresented. Closed-form expressions of transfer matrices havebeen given for uniform and for linearly tapered and chirpeddevices that can be adequately treated by coupled-modeanalysis. For uniform devices, the eigenvalues and eigen-vectors of the propagator matrices of the Jones and Stokesvectors have also been evaluated.
APPENDIX A: CONSERVATION LAWS
Lossless couplers conserve energy. Analytically, this is ex-pressed by one of the equivalent conditions
K1K2 = K2K1 = K2, and K1K3 = K3K1 = -K 3 , i.e., K1 is in-volutory and K2 and K3 are idempotent projection matriceswhose eigenvalues are either 0 or 1.
Proof that any one metric given above satisfies Eqs. (A3)can be obtained by first expressing R and T in terms of themetrics:
and then substituting these formulas into Eqs. (A3) andmaking use of the properties listed above.
Much as R and T have been expressed in terms of Kj, so canF, Q, and M. For example, by using the first expansion of Rand T in Eqs. (A5) and (A6), it can be shown that, for real
F = t[K1 X I - I X K1 *] (A7)
and
M(z) = cos 2 tzI + sin2 (z[V-1 (Kl X Kl*)V]- j4-' sin tz cos (zQ, (A8)
where Q = V- 1 FV.
APPENDIX B: LOSSY UNIFORM COUPLERS
Consider a lossy coupler characterized by the system ma-trix
a#(z)Kja(z) = a#(O)Kja(O), (Alb)
where the Kj, j = 0, 1, 2, 3 matrix is a metric of the system andthe operation denoted by # is defined by16
A# = oAto, a= [° 'j1] (A2)
where the upper (lower) sign is applicable to forward- (back-ward-) wave interaction. When the # operation is appliedto a two-component vector, it is interpreted as a# = (a,*,±a2 *). By substituting Eqs. (1) and (2) into the appropriateconservation law [Eqs. (Al)], it can be shown that a propermetric must satisfy the conditions
R#Kj = KjR, T#(z)KjT(z) = Kj-
Proper metrics for the forward- and backward-wave couplerscharacterized by Eq. (18) include the identity matrix I = K0 ,and
K1=X L+exp(-jo~)
K2 =I2X +iexp(-jo)
K3 = I X- Y
exp(jok)]
exp(j~p)X - Yj'-exP(i1)
X+ Y J.
Note that these are not the only, merely the most convermetrics to satisfy Eqs. (A3). Kj, j = 1, 2, 3 satisfy th4lowing properties: I K1l = -1, I K21 = IK31 = 0, K1
2 = I, IK2, K3
2 = K3, K2 + K 3 = I, K 2 - K3 = K1 , K2K3 = K3K,
R = R- JR=[1 = k 21 (B1)
where kj =j - jaj, j = 1, 2 is the complex loaded wavenumber, K = KI exp(j0) is the coupling coefficient, RB is givenby Eq. (18), and R, = diag(al, a 2). For brevity, the followingadditional notation is introduced: Aa = a 1 - a2, ao = 112(al+ a 2), Ak = k,-k 2, ko = 1/2(k1 + k 2), A = ao4KI, Z = Aa/21K1,Yc = Y - jZ, and X = Xr - jXi = (Y,2 L 1)1/2.
The eigenvalues of the lossy R are k+ = ko + t and k- = ko- , where = -ji = IKIX is now complex. When Yc issubstituted for Y, Eq. (26) represents the modal matrix of thelossy R. Similarly, when Y is replaced by Y, in the expres-sions of Kj, j = 1, 2, 3 in Eq. (A4), all the listed properties ofthe metrics remain intact and in terms of those modifiedmetrics R and T(z) assume forms analogous to those in Eqs.(A5) and (A6), namely,
R = koI + (K1 = k+K 2 + k-K 3 (B2)
and
T(z) = exp(-jkoz)[cos (zI -j sin (zKf]
= exp(-jk+z)K2 + exp(-jk-z)K 3. (B3)
In the lossy case, T' = -jRT and TR = RT are still valid;however, the conservation laws of Eqs. (A3) break down. The
(A4) system matrices of the coherency and Stokes vectors, F andQ, respectively, each acquire an additional term; thus, for lossy
iient, couplers,e fol-K22 =
?2 = 0,
F = FB-jF, Q = QQ - jQa, (B4)
where F, and Qg are those of the lossless case, whereas
1158 J. Opt. Soc. Am./Vol. 72, No. 9/September 1982
a1 0 0 0
0 ao 0 0
FOO 0 aojLo 0 0 a 2A Z 0 0
Q 2 = 2 K[Z A A . (B5)
The new eigenvalues of the Q matrix, both for the forward andfor the backward coupler, are q = -j2(ao + (i), q2 = -j 2 (ao- ti), q3 = 2(r - jao), and q4 = 2(-r - jao), respectively.The corresponding eigenvalues of the Mueller matrix are ml= exp[-2(ao + WZ)Z], m2 = exp[-2(ao - W)z], m3 = exp[-2(cao+ itr)Z], and m 4 = exp[-2(ao - jr)Z1, respectively. Theseeigenvalues indicate that the first pair of eigenvectors issubject to a power-loss coefficient +2ti, in addition to theaverage power-loss coefficient 2ao, whereas the second pairof eigenvectors represents a forward and a backward harmonicwave attenuated at the average rate.
The common eigenvectors of Q and M are given by theirmodal matrix, which in the codirectional case is
Z (Xi 2 + y2)
Xi2 + Y2
Uf = rXi+ FY
.CXi - 4Y
Z(Xi2+ y2)
Xi-(Xi2 + y2)
4Xi-CY
£Xi + 4Y
whereas in the contradirectional case it is
Xi 2 + y2
Z(Xi2+ Y2)
Ub = x
-4Xi -,CY-CXi + 4Y
-(Xi2 + y2)
Z (Xi2 + y2)Xi
-Xi + CY-cXi -. Y
fi (Xr 2 - y2)XrXr 2 -y2
(j4Xr + CY)
(JzCXr -Y)
-(Xr 2 - y2)
i(Xr2- y2)
Xr
i4Xr-,Y
-ijcXr + 4Y
XJ(Xr2 y 2 )Xr
Xr 2 - Y2
j4Xr -,Y
-(Xr 2 - y2)
JX (Xr 2 - Y)
Xr +
I4Xr -CYI~CXr + 4YJ
where , = sin40 and c = coso. Equations (B6) and (B7) reduceto Eqs. (24) and (25), respectively, when a,, a 2 - 0. Theexplicit expression of the Mueller matrix for lossy couplers istoo cumbersome to be given here; however, a simple computerprogram to evaluate M, written in FORTRAN, is available fromthe author.
This research was supported by the National Sciences andEngineering Research Council, Canada.
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