A Generalized Geometrical Representation of Coupled Mode Theory

The evolufion of a general two-level system under arbitrary operators is cast in a three-dimensional form. An earlier geometrical representation is generalized to include contradirectional, parametric, and skew-Hermitian coupling as well as systems with loss or gain. A formal connection to the theory of rigid body dynamics is made and explicit linearity and transformation properties are rigorously established in a coordinate-free form. The connection with mechanics is shown to permit transformations to rotating coordinate systems, a useful technique in analyzing typical guided wave systems.


I. INTRODUCTION HE conventional description of classical, electromagnetic, and quantum mechanical
physical systems in terms of their normal modes led to the general development of coupled mode theory (CMT) [l], [ 2 ] .Although many physical systems have an infinite number of normal modes, there are a great number of systems of practical importance that can be considered as having only two modes which interact appreciably.As a result, most treatments of CMT are directed at such two-mode systems.With respect to the two modes as a basis set, the modal amplitudes ai evolve in time via the coupled equations where the off-diagonal terms describe the coupling between the modes.
Solutions to (1) for Hermitian or skew-Hermitian matrices Hij obey power sum or power difference conservation laws in the usual CMT, and Barnes has classified such "conservative coupling" into eight categories for Hermitiadskew-Hermitian H, codirectionaUcontradirectional coupling, and direct "(i.e., zero frequency)/parametric interactions [ 3 ] .Of the eight categories, perhaps the most commonly used is that of "codirectional, Hermitian, direct coupling," corresponding to quantum mechanical two-level systems or to codirectional coupled waves under static or slowly varying perturbations.Other model systems have closed-form solutions [ 3 ] , and (1) can always be numerically integrated for arbitrary g.
An alternative to the analytic and numericai descriptions of CMT is a geometrical representation.In such a representation, the CMT "state vector" I$) [with components a; in (l)] and the operator H are both mapped into the representation space and (1) is transformed into a suitable equation of motion.Even though such representations do ". . .not obtain results inaccessible to straightforward calculation, the simplicity of the pictorial representation enables one to gain physical insight and to obtain results quickly which display the main features of interest" [4].The geometrical representation of the quantum mechanical two-level system (Hermitian @) has been used extensively [4] to describe RF and optic4 resonance experiments.A generalized Poincare sphere representation of codirectional coupled waves [5] (extending the usual usage of the Poincare sphere [6]) has had similar impact in guided wave technology, but since the systems it describes are formally identical to a two-level quantum mechanical system, this and the earlier representation [4] differ mostly in emphasis and notation.
This paper: 1) extends the geometrical representation to include operators of arbitrary form and time dependence, 2) establishes rigorous and explicit transformation properties between CMT and its geometrical representation, and 3) establishes and exploits the connection between the geometrical representation and classical mechanics.The connection to classical mechanics, in addition to its intrinsic interest, permits powerful solution techniques, allows one to relax "slowly varying" restrictions of the earlier model, and provides direct physical analogs of CMT solutions.
11. OVERVIEW Fig. 1 presents an overview of the structure and features of the geometrical representation presented in this paper.The four spaces shown consist of vectors (upper two spaces) and operators (lower two spaces) in both the two-dimensional complex CMT space (left two spaces) and the three-dimensional real representation space (right two spaces).Also illustrated are three major types of operations: space mapping, vector evolution, and basis changes.Space mapping is represented by heavy dashed lines and is the vehicle for transformations between the CMT spaces and the representation space.Vector evolu-U.S. Government work not protected by U.S. copyright -~ tion (which takes place in both spaces) is represented by solid lines, and the description of, say, ) evolving into under the influence of operator H is the goal of CMT.Since the CMT states have images ;n the representation space, the CMT state vector evolution is mirrored in the 3-D vector evolution.Operations of the third type, basis changes, also take place in either space and are accomplished by unitary unimodular operators (real orthogonal rotations in the 3-D space).In Fig. 1, basis changes are shown by full dotted arrows, and they transform vectors and operators into the corresponding primed quantities.
There are six permutations of the operations of mapping, evolution, and basis change, and this paper establishes that the operations commute.For instance, a typical application might be a sequence of mapping, basis change, and evolution.This sequence would (Fig. 1) map initial state I$,) to rl, change the basis to a more convenient frame in the geometric space by transforming to y; (basis change), and then allow the vector to evolve to r;.Another permutation of the operation (such as basis change to I $; ) , mapping to _r;, evolution to yi) would result in the same even though, for example, the basis changes occurred in different spaces.Practically speaking, it is the knowledge that these operations can be rigorously permuted that permits one to ignore the formal structure and concentrate on the geometrical solutions.
The three operations of mapping, basis change, and evolution under an equation of motion are treated in Sections 111, IV, and V, respectively.Although the subject may be considered as an exercise in spinor representalion theory, the treatment is simplified immeasurably and made physically accessible by casting it in a way which conveniently exploits the development and notation of classical mechanics [ 7 ] .Once the full connection to mechanics is made, some useful properties and characteristics of the geometrical representation are pursued in Section VI.In particular, the use of rotating coordinate systems is developed.Finally, in Section VII, two examples are presented to demonstrate that the geometrical representation can lead to deeper understanding of coupled mode systems, even if a solution is already known.A summary of the basic results is then given in Table I.

III. THE 3-D REPRESENTATION OF STATE VECTORS
AND OPERATORS The most natural way to begin is to recast results of [4] in operator form using Dirac notation.For every CMT state vector I$) with complex coordinates, we associate a real vector y given by the expectation value where repeated indexes imply summation.Here the first three gi are the x, y, and z Pauli spin matrices (Appendix I), respectively, g4 is the identity matrix, and the 2, are orthonormal unit vectors.With respect to the chosen basis set, the state vector can be represented as where the ai are positive and real.Then, either power (in CMT) or probability (in QM) is given by a;, and the ith component of y is given in matrix form by xi = 2-ato.a.-1 -2 Equation (3) can be normalized by setting p2 = g l 7 g 2 and factoring out the average phase, so that where 8 is chosen to reflect the relative amplitudes (0 5 0 I x) and the relative phase is given by 4 = (42 -4,).where the reality of the xi = ( $ l g i l $ ) is seen explicitly.
The first three components are plotted in Fig. 2, emphasizing the polar coordinates in (4) and ( 5 ) .(Note that ( 5 ) takes the coordinates of I$) into r , while (4) takes the polar coordinates of y into I $) , up io a phase factor.)Al-  In addition to mapping the state vectors onto the representation space, one can also map operators onto the same space even though they come from a different underlying space (having a 2 X 2 instead of a 2 X 1 representation).By suitable choice of ai, any operator can be represented in matrix form as though it has four components, ( 5 ) does not faithfully represent (4) since the overall phase In other words, the three degrees of freedom expressed are the modulus ( p ) , the relative amplitudes (e), and the relative phase (+), while .x4does not contribute extra information.As ill the earlier representations [4], [ 5 ] , the loss of the overall phase is usually unimportant because it is normally the relative phase which has physical content.As a result, by Ccmsidering aZZ states of the form eiYI$> to be equivalent to I II/> where y is an arbitrary phase, the transformation can be considered as faithful.In Fig. 2, and by using the well-known properties of the spin matrices (Appendix I), the inverse of (6) is found as These wi form the components of a complex vector g.
One may ignore the fourth component of w since it can be removed by a sort of "gauge transformation" (Appendix 11).Thus, both states and operators have a 3-D geometrical vector representation.Contact with earlier representations is established by noting that the wi are real if and only if is Hermitian, in which case (6) is equivalent to [4, eq. ( 5) ] and expresses three (or four) degrees of freedom.However, in general, the operator in ( 6) has eight degrees of freedom, and can be decomposed into a Hermitian, a skew-Hermitian, and a diagonal matrix in the following unique way: where all the variables in the last line are real.The summation on i for the traceless Hermitian (H,) and skew-Hermitian ( E l S H ) matrices is restricted t o three dimensions.AssoGated with the operator are two real vectors wR and in R, and a complex number a4, establishing eight degrees of freedom.The vector components are displayed in two ways [e.g., wR is represented both as the set wRi and as (xR, y R , ZR)] and the components wRi or wI i are found by taking the real or imaginary parts of (7).Thus, for instance, w12 = yI = Im zTr(Hg,,) = gRe(H12 An important feature of ( 6) and ( 7) is that they explicitly prove the linearity property %(El, + g2) = g (H,) + w(H2).--That is, the vector representhg a s;m of operators canbe represented by the sum of the vectors representing each operator.Lest this be considered obvious, Fig. 2 clearly demonstrates that the corresponding property is definitely not true for the state vectors. -- IV. TRANSFORMATION PROPERTIES Although the mapping of CMT states and operators to the representation space offers a visualization of CMT systems, true usefulness requires an assurance of faithfulness between all geometrical transformations and the corresponding CMT state transformations.The groundwork for the connection between basis changes in the two spaces is the homomorphism between the special unitary group SU(2) and the rotation group O:, familiar from the classical mechanical description of gyroscopic motion [7] and the quantum mechanical description of angular momentum 181.We closely follow the results and notation presented in Goldstein's text [7] to enhance the physical intuition afforded by mechanical systems.
Consider a change of basis in the representation space such as E' = R(4, 0 , $) y.Operator R is an orthogonal rotation of the coordinate axes (from U:) described by the Euler angles: 1) a rotation of the axes about the z axis by $, followed by 2) a rotation about the new x axis by 8 , and 3 ) a rotation about the even newer z axis by I) [7].
The matrix corresponding to 8 is uniquely determined by the Euler angles.In a similar-way, a basis change in the state vector space is given by a unitary unimodular operator Q [from SU(2)J whose matrix representation can be cast 2s [7] (9) The matrix elements of Q are the Cayley-Klein parameters of classical mechanic% expressed in terms of the same Euler angles that describe R.This reflects the homomorphism between the Q and-$ operators [7], [SI.Referring to Fig. 1, this m e a h that the Euler angles simultaneously specify unique Q and pi, and if any H is related to the vector gk associated with H ' by = R w R [7].That is, when a coordinate change-is applied, Gectors representing H and 11)) both undergo the same rotation: the matrices land Euler angles are the same.But the skew-Hermitian part of U, once the scalar "i" is removed, also has the same form as HH so that the veal vector also transforms in the same" way.Anticipating the result from the next section, this implies that if the real and imaginary parts of the vector g do not mix in the equation of motion, both wR and gI can be represented in the real representation space.This result may be somewhat surprising, since the imaginary gy might seem likely to exchange real and imaginary components.Finally, the a4 component is seen to be an invariant under the similarity transform so that its removal by the "gauge transformation" of Appendix I1 is not strictly necessary.The effect of similarity transforms on the images of operators is thus seen to be identical to the effect of the associated coordinate transformation on the images of state vectors, namely, gk + iw; = ReR + iRgl.
(1 1) The result is that no special rules are required: given state and operator image vectors in arbitrary positions, one may freely rotate the coordinate system to a more de-

--
'However, sinceli) is equivalent to li), the phase ambiguity of the state vectors can absorb any sign ambiguity of Q, although this is usually not required in practice.
sirable orientation for the solution.Since the rotation is specified via the Euler angles, the equivalent basis change Q is found immediately from (9).This permits transitions getween spaces in a natural, rigorous, and explicit way.All that remains is to find the equation of motion.
V. THE EQUATION OF MOTION The goal of the geometrical representation is the ability to transform the CMT problem into the easily visualized representation space, solve the problem there, and transform back to the CMT space if necessary.This requires the repre.sentationspace's version of the equation of motion (1).Given the state vector equation of motion its adjoint equation, [ = ( $ 1 ~~2 ~1 I J ) , and B = aigi, the equation of motion for r I is found to be (Append6 I) (13) which is similar to the familiar quantum mechanical result.The second term on the right side is zero since ~~2~ is fixed.Using the form in (8) for Hand the properties of the spin matrices (Appendix I) allows the right side of ( 13) to be separated into expectation values of 0matrices, which in turn can be expressed as functions ofthe vector r.This yields the equation of motion.By setting Q R = 2 Re w -= 2:R GI = 2 Im = 2gI a = 2 Im a4 = 2 0 ) ~~ (14) and using the results sketched in Appendix I, one finds the equation of motion, which is the central result of the paper: Although it involves both w R and w I , (15) is real and does not mix eR and gI.This afiows the separation implied by (8) and (1 1) and delivers the proof of the coordinate-free form.
The individual terms in the equation of rnotipn are easily identified and given physical significance.The first term corresponds to a rotation of r about the instantaneous axis Q R .This term correspoids to Ulrich's paper [5], except that the classical and unitary transformation connections remove the need for "slowly varying" restrictions on Q R .The remaining terms have not been described in earlier work.The second term corresponds to a growth or decay of c along its axis.This corresponds to loss of probability conservation in QM systems or to common mode gain/loss in coupled mode systems, and it could have been transformed away at the outset (Appendix 11).
Once again, only changes in the magnitude of the state vector appear (Im (04 corresponds to common mode gain/ loss), not changes in the overall phase (Re a4).The third term is entirely unexpected.It allows 1 to change along another axis, namely, wI, and its appearance is due to the skew-Hermitian H S H .This type of operator appears in contradirectional "cupling problems [3] as well as systems with differential mode gain/loss.Examination of (15) shows that the dependence as 1 1 1 leads to hyperbolic solutions, and that there is a "threshold" effect according to whether the effects of w R or w I I dominate.That is, if W R is great enough, it retards a never-ending growth along wI by forcing a change in direction, whereas if wI dominates, the state vector grows without limit.The consequences of this term will be explored in more detail elsewhere [9], however.The remainder of the paper utilizes the apparatus that has been established to develop the model and to solve some special cases.These examples make connections to solution techniques developed from rigid body mechanics [7] and used for decades in magnetic resonance problems [4], [lo], [ll].

VI. SOME FEATURES OF THE MODEL A . Basic Properties
In this section, some of the basic properties of the representation are developed and compared to earlier representations [5] , [6].

I ) Projection of States: Consider the projection of
, with image rl, onto another state , with image r2.Equation (4) and Fig. 2 show that the projection of an arbitrary state onto the basis state 11 ) is given by cos 8/2 where 8 is the polar angle in the 3-D space.Choosing an Euler angle set that puts 1 GI) into 1 1 ) also puts rl into 2.
If 8 is the angle between _y; ( -2) and 5, the projection of I $; ) on l$f ) (= 11 )) is given by cos 812 from (4).But since orthogonal rotations preserve angles, 8 is also the angle between rl and c2.Thus, the projection of I $2) onto I $, ) is given by the same cos 8/2 factor as in the Poincare sphere representation [ 5 ] , [6].As a consequence, any pair of oppositely directed vectors in 3-space can represent an orthogonal basis set in the state vector space, and any other such pair can be reached by a rotation through a suitable set of Euler angles.
2) Projections Under Time Evolution: For two states represented by rl and r--, the included angle is given by cos 8 = rl r2/ Ii, 1 1 r,\.Differentiating this expression and employing th"e equation of motion, (15) reveals that if g, = 0, then d(cos 8)ldt = 0. Otherwise, 8 can change in time.This differs from the earlier representation [5] since the presence of contracoupling or differential loss is represented by nonzero %[, and either effect allows initially orthogonal states to evolve into states which lose their orthogonality.

3)
Eigenstates of an Operation: If = 0 and 01 is transformed away, the net results of any state's evolution is a finite rotation (Euler's theorem [5], [7]) about some finite rotation vector w.Vectors proportional to w are un- After transformation to basis set of eigenstates (i.e., normal modes), there is no exchange of power (0 constant), but difference in frequencies (linearly increasing 4).I / altered (except for a possible phase change), and hence a r e eigenstates of the operation.Thus, the unit vectors +.@ are the eigenvectors for the operator whose image is g.Referring to Fig. 1, this corresponds to mapping the CMT operator into the representation space, identifying its eigenstates there, and then mapping back to the state yector'space to find the eigenstates there.In addition, one may run the analysis backwards: the image of any state r can be used to find an operator for which y is an eigenvector.This often enables one to solve problems without the necessity of knowing the exact form of the Hamiltoniqn.For cy # 0, the generalized "eigenstates" can be considered as changing in time (see Appendix II), while for 9; # 0, the eigenvalue system is overdetermined in general.

4) Choice of Basis:
The above discussion of eigenstates relates closely to the choice of basts sets.As in the analytical representations [2], [7], that choice is a matter of taste and convenience [ 5 ] .For instance, consider the CMT system described by Fig. 3.In Fig. 3(a), the unperturbed basis set is represented by +2, and the perturbation results in the "nondiagonal" vector g.The evolution of the state's initial condition (E) is around circle C about w .The periodic variation of 0 and the limited excursion of + reflects the description of Fig. 3(a) as a "coupled" (periodic exchange of power) and "synchronous" (limited phase variation) system [5].b)].Again, <traverses C, but now 8 is constant and c$ increases linearly.In the new basis, the system is described as "uncoupled" (no power exchange) and "asynchronous" (phase increqsing due to different frequencies) [5].Thus, the same physical system (i.e., r traversing C ) can be described in what seems to be radically different ways without the need to explicitly find Q 5) zChange of Form of Coupled Mode Equations: The matrix in '(1) can always be diagonalized (i.e., vanishing off-diagonal terms) for Hermitian H, leading to uncoupled -- For diagonal terms to vanish reqkres zR = 0 in (8), and the desired form is found by choosing a coordinate system that puts g in the xy plane.Knowledge of w enables this to be done by rotating the coordinate system through the first Euler angle of (4a/2) (setting the x axis perpendicular to the e -' 2 plane) a$ then through the second Euler angle (n/2 -. 0) to make 8 orthogonal to w .The third Eul'er angle is arbitrary, establishing a phase.The rotation shown in Fig. 4 corresponds to the above-mentioned Euler angles Performing these rotations places w in the y ' direction, and the Euler angles give the matrix needed to transform the coordinates and Hamiltonian.

B. The Use of Rotating Coordinate Systems
Operators which are time varying give rise to images which move in the representation space.As in the study of magnetic resonance, the use of rotating coordinate systems is a useful and powerful technique which simplifies visualization pf the solution [4] , [lo], [l 11.The use of such rotating systems can be applied to nonharmonic perturbations [9], but here we restrict ourselves to earlier examples of sinusoidally varying perturbations.The basis of the 'technique is that't$p time derivative of a vector, referenced to a rotating Fame, is given in terms of the stationary frame derivative as where g is the instantaneous angular velocity of the rotating coordinate system [7].Now consider a CMT system which is subject to two perturbations as in Fig. 5 , a static ( L ? ~ and an g F which oscillates as cos Qt.The vector g F can be viewed as the sum of two vectors g> and W F of magnitude q / 2 which counterrotate at angular frequency Q in the yz plane.Associated with each of these vectors is a rotating coordinate system which fixes g> along the Z3 axis.Transforming to the positive rotating axes requires subtraction of Q, and the resulting vector g = g L + g F / 2 -~2 is the ''egective Hamiltonian" [lo], [ll] for the system.As in NMR, the counterrotating gF is ignored.The magnitudes of Q and g L are either equal or not.First, suppose that Q = g L .Then the effective Hamiltonian is represented by g = O F / 2 in the rotating frame, and the trajectories of the states will be circles about 2 : . In particular, an initial state that is represented by in the first frame rotates with angular velocity 9 / 2 around the "equator" in Fig. 5(b).The steady rotation about 2 : in the rotating frame and the initial condition show that there will be a complete and periodic exchange of power between the state represented by and the state orthogonal to it (i.e.7 -Zx in the representation space).Second, suppose that Q < oL [Fig.

5(c)]
. The resultant weff is slightly more complicated, but still easily visualized".An initial state along as above, now traverses a cone with opening angle 0,.In this case, there is a partial and periodic exchange of power between the state represented by and its orthogonal state.In the next section, this technique is applied to two examples that have been described in the literature.

VII. EXAMPLES A . Fiber Coil Isolator
The system shown in Fig. 6(a) is analyzed here in some detail to exercise the model.A single-mode optical fiber with linear birefringence is coiled and placed in a uniform magnetic field B .The magnetic field induces the Faraday effect [ 13]-[16j7 a circular birefringence which leads to a rotation of the plane of polarization of linearly polarized light.Because the eigenstates are the circularly polarized states, the w F that represents the Faraday effect must be along the z axis in a representation with circular polarization states as bases [5].The natural birefringence. is linear, and since the linear states of polarization lie on the equator [ 5 ] , the eigenstate and operator images must be along two points lying on the equator.We choose the g L representing the birefringence to be along the x axis.By the linearity of Section 111, these vectors may be added to obtain a resultant vector g.The Poincare sphere representation of spatially uniform linear and circular birefringence has been developed [17], and it is unnecessary to repeat it here: its representation looks like Fig. 5(c) with g L and 9 / 2 representing the linear and circular birefringence, respectively.The geometry of the coil in Fig. 6, however, leads to a spatially varying Faraday effect.(In this discussion, the independent variable t becomes the spatial variable z , corresponding to length along the fiber coil.)As light propagates along the fiber, the tangential component of the magnetic field varies sinusoidally due to the coiling of the fiber.Since it is the tangential field which is responsible for the Faraday effect, the Faraday rotation per unit length "freezes" the positive rotating vector.The coil is "tuned" when the linear birefringence of the fiber matches the spatial oscillation in the Faraday effect wL = Q , and the situation is as in Fig. 5(b).Light originally launched with polarization along the x axis has a polarization which rotates uniformly about 2 ; in analogy to simple uniform circular birefringence [ 171.This motion, when transformed back to the fiber frame, leads to a spiral (see [16,Fig. 11).Ultimately there is a complete power transfer to the y polarization in real space as _r rotates to --Zx in the representation space.As in NMR [ 111, the counterrotating vector (rotating at 2Q in the primed frame) has only a minor effect.At this point, the geometrical representation permits reinterpretation of some of the results in [15] and [16].First, the equation of motion [16, eq. (l)] cannot be used in its form there, but must be evaluated in a differential form [18] possibly by using the " N matrices" [19].Second, the reason for the magnitude of the Faraday rotation being half of that for the uniform case [ 141- [16] is that only half the magnitude can be in the corotating coordinate system, while the other half is in the counterrotating system.Third, the presence of the unexplained ripples in the numerical solution (e.g., [ 15, Fig.21 and [ 16, Fig. 31) is seen to be the effect of the counterrotating vector.As one would expect from the NMR analogy, the variations average to nearly zero (i.e., the rotating wave approximation) and vary at twice the frequency of the physical turns in the coil, as is apparent in the graph of the numerical solution.Additionally, the amplitude of the ripples may be estimated from the analogy to the NMR Bloch-Siegert shift [20].
Turning to the situation for an untuned coil, the representation is similar to Fig. 5(c); the linear birefringence is not completely cancelled, and the effective in the rotating frame rotates the initial 1: along the circle shown.One sees geometrically that 1: can never be farther away than 28, from the x axis as it traverses the circle C .When this motion is transformed back to the fiber frame, the pitch of the spiral decreases and turns back as it reaches the top of the circle in Fig. 5(c).The numerical solution [16,Fig. 21 shows this process.The utility of the geometrical representation is illustrated in this example since we can easily estimate the maximum power transfer without the necessity of a full analysis.From Section VI-AI), one can write directly that the maximum power transfer to the orthogonal mode is i.e., a Lorentzian in the detuning.Even such an elementary analytic result is difficult to obtain from the numerical simulation.This illustrates the point that often one does not need all the information contained in the full analytic result.

B. Fibers with Twist and Linear Birefringence
As a final example, we consider the case of twisted optical fiber with linear birefringence which has been treated by Ulrich and Simon [17].We use the same coordinate system as before, referring the linear axes to the initial space axes.The linear birefringence, represented by op, is fixed in the local coordinates of the fiber, but since the fiber is twisted with angular frequency 7, those local coordinates are rotating with angular velocity 7 in real space, and thus wp moves around the equator with angular velocity 27 in the 3-D representation space [5], [6], [ 171.Strains from the twisted fiber give rise to a circular birefringence = g7 [ 171.For analytic convenience, one transforms to the system rotating at 27 to "freeze" gp, as shown in Fig. 7(b).Consideration of (15) and ( 16) then shows that the effective Hamiltonian is @ = ~~( 0 ) + 9 -27.Light initially launched along the x axis, for instance, will follow circular trajectories about this @ in the rotating frame, and cycloidal trajectories in the lab frame.This result agrees with [17] and gives a clear picture of how the 27 term arises by appealing to (16) rather than solving a pair of simultaneous vector differential equations for w and _r.
Furthermore, by showing the source of this term in @, one can appreciate how small perturbations or nonuniform 7 will affect the solution.For instance, variations in 7 can be accounted for in a frame that rotates at the nonuniform speed 27.The effective @ will tilt as a result, and the effects can be estimated using geometric arguments.Without (1 6), one would have to solve a much more difficult set of equations than those solved in [ 171.
VIII.SUMMARY In summary, the geometrical representation of the dynamics of two-level systems [4], [5] was generalized to include arbitrary Hamiltonians, allowing analysis of contracoupling, time-varying, and nonconserving (both common mode and differential loss) coupled mode systems in CMT.By making direct connections to the well-established mechanics of rigid bodies, the representation was put in a coordinate-free form with direct physical analogs, allowing proofs of the transformation and linearity properties which make the representation easy to use.By establishing the correspondences summarized in Table I, the rigorous connections between CMT and the geometrical representation were made explicit.In addition, since the results of rigid body dynamics apply directly, earlier "slowly varying" restrictions were lifted and powerful solution techniques were demonstrated.Finally, several examples were shown in which the geometrical representation led to deeper insights into problems that had previously been solved in other ways.

dt
Manuscript received January 14, 1986.The author is with the Naval Research Laboratory, Washington, DC IEEE Log Number 8610275.20375.

Fig. 2 .
Fig. 2 .Three-dimensional representation of CMT state vectors I$).Polar angle 8 represents relative amplitudes of the two components of the state vector, while azimuthal angle 6 represents their relative phase (4).The equivalent state vectors for the six cardinal points are also shown.

-
then its image y is transformed under &(C#I, 0, 4) to the corresponding _r' .In terms of the geometrical representation, then, mapping and basis changes commute.An apparent complication is that the Q's are double valued: a change of 27r in 4, for -instance,"leaves R unchanged and changes the sign of I -In considering the transformation of operators, (6) and (7), the discussion begins with the usual similarity transform associated with coordinate change Q.If &! is a traceless Hermitian operator, then the vectof g R associated with

Fig. 3 .
Fig. 3. Change of basis set.State represented by follows curve C under operator represented by w.(a) In original basis set, there is a periodic exchange of power (oscillating 0) and limited phase excursion (4).(b)

Fig. 4 .
Fig. 4. Change of CMT form.Operator represented by vector with polar and azimuthal angles 0 and 4 , respectively.Unprimed coordinate system is first rotated by

Fig. 6 .
Fig. 6.(a) Fiber coil isolator subjected to static magnetic field B .Polarized light is initially launched in the x axis of the fiber and analyzed at A .(b) Representation of the isolator.Linear birefringence of the fiber is represented by wL; spatially varying circular birefringence from Faraday effect is represented by oscillating wF.Light initially launched with x polarization is represented in Fig. 5(b) for tuned isolator and Fig. 5(c) for untuned isolator.
[ 151, [ 161 and the Hamiltonian which describes it both have a cos z/R dependence.In the 3-D representation, then, the g representing the Faraday rotation oscillates up and down along the z axis [Fig.6(b)].The reason for the notation in Section VI-B is now clear.Transformation to the frame rotating at the spatial frequency determined by the coil geometry Cl = 1/R

Fig. 7 .
Fiber with twist and linear birefringence.(a) In lab coordinates, twist 7 gives rise to circular birefringence E, and causes linear birefringence axes (yD) to rotate at 27 in representation space.(b) In frame rotating with axes, the effective perturbation is given by 6 = ~~( 0 ) + e -27.