. INTRODUCTIONavelength splitters and combiners are a particular type
f wavelength multiplexer and demultiplexer, which areeant to manage two bands only. They can be designed
nd implemented in different ways. In the context of pla-ar lightwave circuits, a convenient choice is the cascad-
ng of Mach–Zehnder interferometers.Standard techniques for the synthesis1–3 of this type of
lter do not take into account the wavelength dependencef directional couplers. This means that they can be ap-lied to dense wavelength division multiplexing circuitsnly, within bands where this dependence is negligible.n the other hand, Jinguji et al.4 proposed the point-
ymmetrical configuration as an alternative method forlter synthesis. They applied this technique to the two
imiting cases of strong wavelength dependence and noavelength dependence of couplers. In the first caseavelength selection is due to the coupler response,hereas in the second case it is due to the extra length ofne of the two arms of each single interferometer.
I propose in this paper an extension of the last configu-ation to the intermediate case in which the wavelengthependence of couplers is not negligible, but not tootrong to effectively allow wavelength selection. This is aypical situation when dealing with coarse wavelength di-ision multiplexing filters. In light of a geometrical inter-retation of the results in Ref. 4, I will show how to cor-ect this wavelength sensitivity to obtain flat passbands.ven though the novel geometrical and analytical ap-roach is presented in application to a practical case, ithould be clear that what is proposed is not a particulartructure but a new method to design and analyze inter-erometric filters. The presented application will showow the proposed method can give more physical insightnd design control than any numerical approach.
. WAVELENGTH-INDEPENDENT COUPLERSn this section I will briefly present the synthesis tech-ique based on doubly point-symmetrical cascading of
ach–Zehnder interferometers, supposing wavelength-ndependent couplers. Then I will introduce the geometricepresentation of couplers and phase shifters that enabless to reinterpret previous numerical results in terms ofeneral analytic formulas.
. Doubly Point-Symmetrical Configurationavelength splitters and combiners are two-port devices
hat split and combine two bands centered at two differ-nt wavelengths �1 and �2.
The point-symmetric configuration has been introducedn Ref. 4 to obtain flat filtering with interferometric struc-ures. In particular, when repeated twice (see Fig. 1), thisethod ensures a flat response on both cross port and
hrough port. In Ref. 4 the general working principle ofhis technique is explained analytically, whereas theengths of the couplers are calculated by using a numeri-al optimization algorithm. Couplers are supposed to beavelength independent, whereas the extra length ofach Mach–Zehnder interferometer is chosen so that itshase shift is n� (n integer) at �1 and �n+1�� at �2.learly, when n is odd, �n+1� is even and vice versa. Forimplicity we will refer to even and odd phase shifts, de-ending on the parity of n.We will now introduce a geometric representation for
he action of an interferometer that not only makes it pos-ible to analytically recover the results of the numericalpproach, but also allows us to broaden the class of work-ng structures.
. Geometric Representationn two recent papers,5,6 we have shown how a generalizedoincaré sphere representation7–9 can be helpful whentudying wavelength-dependent couplers and interferom-ters. The actions of both couplers and phase shifters cane represented as rotations on a spherical surface, analo-ous of the Poincaré sphere for polarization states. Figuredisplays all the intersections of the S , S , and S axes
ith the sphere. They represent, respectively, the singleaveguide modes E1 and E2 and their linear combina-
ions ES,A�1/�2�E1±E2� and ER,L�1/�2�E1± iE2�. Also,he generic normalized state P�a1E1+a2E2�cos �E1exp�i��sin �E2 is plotted. Notice that the relative phasengle � is exactly the altitude angle of the point P withespect to the equatorial plane on the circle perpendicularo the S1 axis passing through P. If we consider the coneaving this circle as a basis and the sphere center as aertex, the power-splitting angle � is equal to half of thealf-cone opening angle. Since we deal with lossless ele-ents only, all the physical trajectories are limited to
oints on the spherical surface. Furthermore, when con-idering transformations that are compositions of recipro-al elements only (which is the case when dealing with di-ectional couplers and phase shifters), all trajectories onhe sphere can be compositions of rotations about the axeselonging to the equatorial plane only,5 i.e., the S1S2lane. In particular the action of a phase shifter is repre-ented by a rotation about the S1 axis, and the action of aynchronous coupler is represented by a rotation abouthe S2 axis. Hence if we deal with synchronous couplersnd n� (n is an integer) phase shifts only, we can restricturselves to the projection of the sphere on the S1S3lane, i.e., a one-dimensional representation on the circleying in that plane.
Let us define the angular expression of the amplitudeoupling ratio4,5 of a coupler:
���� = �����L + �L����, �1�
here ���� is the coupling per unit length in the straightart of the coupler and �L��� accounts for the contributionf the input and output curves. We will assume these twouantities to be the same for all couplers, whereas we willet the coupler length L vary from coupler to coupler. Forhe moment we will also assume any dependence of � on �o be negligible.
Figure 3 shows the projected trajectories on the S1S3lane of a Mach–Zehnder interferometer in the two casesf even and odd multiples of � for the phase shift. E1 and2 represent, respectively, the situation of all power in
he input arm and all power in the other arm. In light ofhat was noted above regarding the power-splitting angle, the action of a coupler is simply represented by a rota-ion of twice the coupling angle �. A 2n� phase shifteaves the system as it is, whereas a �2n+1�� phase shiftends any point of the circle to its mirror image with re-pect to the S axis.
ig. 1. Doubly point-symmetric structure. First, the buildinglock, composed of a type A coupler and a half-type B coupler, isepeated point symmetrically, resulting in an ABA structure.his structure is repeated point symmetrically to give the de-ired result.
1
Hence in the case of an even phase shift, the interfer-meter follows the path E1FG, acting as a coupler withngle �A+�B, whereas in the case of an odd phase shift itill follow the path E1FGH, equivalent to a couplingngle �B−�A.
. Analytic Formulaseeping the above discussion in mind, we now go back to
he point-symmetric structure in Fig. 1. The filter works ifhe couplers satisfy the following conditions:
ig. 2. Generalized Poincaré sphere for the analysis of twooupled waveguides. The generic point P is represented togetherith its relative phase angle � and power-splitting angle �.hysical transformations are represented by composition of rota-ions about the axes on the S1S2 plane. The points on the rotationxis represent the eigenstates of the system. In particular a syn-hronous coupler with coupling angle � is represented by a 2�otation about the S2 axis, whereas a � phase shift is representedy a � rotation about the S1 axis.
ig. 3. Geometric representation on the S1S3 plane of the actionf a Mach–Zehnder interferometer in the two cases of (a) evennd (b) odd phase shifts. A coupler with coupling angle � givesise to a 2� rotation, a 2n� phase shift leaves the system as it is,hereas a �2n+1�� phase shift turns over the state with respect
here t� 0,1, �A and �B must be positive, k is an inte-er, and m must be a nonnegative integer. For t=0 therst condition reads that, for odd phase shifts, the powerust remain in the through port; and the second condi-
ion means that, for even phase shifts, the light must gon the cross port. For t=1 the two ports exchange theiroles. Wavelength selectivity is simply obtained with aroper choice of the extra length �L�� / ���2�−��1��,here ��� is the propagation constant of the single wave-uide mode. Solving Eqs. (2) for �A and �B, we get
��A =�
16+ �m + k�
�
8
�B =�
8+ �m − k − t�
�
4� , �3�
here m and k obey the selection rules
�m �k�, for t = 0
m k + 1, for t = 1 and k 0
m − k, for t = 1 and k � 0� . �4�
In particular, for t=0, m=2, and k=−1 we find �A
0.1875� and �B=0.875� that are very close to the opti-al values �A=0.1882� and �B=0.8626� numerically
ound in Ref. 4.It is straightforward to rewrite Eqs. (3) in terms of LX
�X /�−�L �X=A ,B�, clearly with the further constraintX /���L.Notice that the results of the numerical approach in
ef. 4 depend on the initial guess. On the contrary our ap-roach allows us to know all the solutions at once, andhis may be useful, for example, to find the shortest one.n our formalism the choice t=0, m=0, k=0 gives themaller overall coupling angle 4�i
A+2�iB=� /2, which is
ve times smaller than the coupling angle of the t=0, m2, k=−1 case proposed in Ref. 4.I also point out that, when the full Poincaré sphere rep-
esentation is considered, it becomes clear why point sym-etry can guarantee band flatness, i.e., tolerance toavelength changes or, equivalently, insensitivity to
hanges of the phase-shifter rotations. It will be shown inetail in Section 3 that the change of a given phase shiftrom its nominal value (due to a small deviation of theavelength from �i) will be compensated for by a corre-
ponding change in its point-symmetrical counterpart.
. WAVELENGTH-DEPENDENTOUPLERS
n this section we will extend previous results to the casef wavelength-dependent couplers. With the aid of the fulloincaré sphere representation we will also show that, forome choice of parameters t, m, and k, the singly point-ymmetrical configuration must be preferred to obtain aell-flattened response of the filter.
. Generalized Formulashenever it is required to manage two channels only, we
an just focus on the spectral response within the twoands about �1 and �2. We can take into account theavelength dependence, rewriting Eqs. (3) as
�4�2A − 2�2
B = t�
2+ k�
4�1A + 2�1
B = �1 − t��
2+ m�� , �5�
here �iX��i�LX+�Li�����i��LX+�L��i��, and we have
ssumed �1 ��2� to be associated with the even (odd)hase shift. Notice that wavelength dependence can beasily accounted for because the first condition must beatisfied at �2 only and the second one at �1 only. Solvingor LA and LB we get
�LA = �1 − t� + 2m
�1+
t + 2k
�2� �
16−
�L2 + 3�L1
4
LB = �1 − t� + 2m
�1−
t + 2k
�2��
8+
�L2 − 3�L1
2� . �6�
In this case it is not straightforward to determine theelection rules for k and m because they will depend onhe difference �2−�1. Clearly the smaller this difference,he better the rules in inequalities (4) also apply to thisase. Once ���� and �L��� are known, these simple formu-as allow us to design the directional couplers for doublyoint-symmetrical flattened filters. Furthermore, manyifferent structures can be found with different choices of, k, and m.
. Numerical Exampleo confirm the analytical results I have performed someumerical simulations. Buried silica waveguides haveeen considered, with a 4.5% index contrast and a 2 m2 m square cross section, which guarantees monomo-
ality in the band of interest. The propagation constantsersus wavelength for the single waveguide mode and forhe coupler supermodes have been calculated by using aully vectorial commercial mode solver.10 The inner-wallaveguide separation of the couplers has been chosen toe 2 m. After checking with a commercial beamropagator,11 it was found that, with this high index con-rast, the contribution �L��� is not only very small, but itsavelength dependence is the same as an equivalent
traight coupler of extra length, which gives the sameoupling. All this information can be implemented in anccurate and simple transfer-matrix model. Figure 4hows the spectral response for a 1490 and 1550 nm split-er with the choice t=1, m=1, k=0. Notice that, by con-truction, the wavelengths 1490 and 1550 nm correspondo zeros in the port where they must be suppressed.learly, a fine-tuning of the analytical solution may yieldbetter performance over the whole channel band. Figurecorresponds to the case t=0, m=0, k=0. In this case our
pproach works only locally at the nominal wavelength,ut does not guarantee flatness at 1550 nm. This can beualitatively understood on the Poincaré sphere in Fig. 6.hen the wavelength is not exactly 1550 nm, the phase
hift it is not an exact multiple of �. This gives rise to aonzero rotation about the S1 axis corresponding to eachhase shifter, which means a departure of the trajectoryrom the S1S3 plane. Clearly, having chosen the doublyoint-symmetric configuration, the sequence of phasehifts will be, for example, ACAC, where A �C� means an-iclockwise (clockwise). But a C rotation can partiallyompensate for the effect of an A rotation if and only if theotations are performed starting from points on differentemispheres. On the contrary, when the points are in theame hemisphere, a C rotation worsens the effect of therevious A rotation (in Fig. 6 both rotations move the tra-ectory far away from the S1S3 plane), and vice versa, giv-ng a strong departure from the ideal trajectory, that is, amall tolerance to wavelength changes. In this case theight choice is to correct the A rotation with another A ro-ation as shown in Fig. 7. This means that we have tohoose a singly point-symmetrical configuration AACC.
ig. 4. (Color online) Simulated spectral response for the barort (solid curve) and the cross port (dashed curve) of a splitterith t=1, m=1, k=0, ideally designed for separating a 1490 nm
hannel in the bar port from a 1550 nm channel in the cross port.he doubly point-symmetric structure ensures flatness for bothorts.
ig. 5. (Color online) Simulated spectral response for the barort (solid curve) and the cross port (dashed curve) of a splitterith t=0, m=0, k=0, ideally designed for separating a 1490 nm
hannel in the cross port from a 1550 nm channel in the bar port.n this case the response is not flat for the bar port.
n
igure 8 shows the spectral response of this configura-ion, which is found to be well flattened on both ports. Sohen 2��i
B−�iA� and 2�i
A belong to the same hemisphere,atness may be guaranteed by the AACC scheme. Buthis is not always true as, for example, when choosing=0, m=1, k=−1.
ig. 6. (Color online) Geometrical representation on the fullhree-dimensional Poincaré sphere of the response in Fig. 5 for aavelength slightly different by 1550 nm, corresponding tohase shifts slightly greater than �. The sequence ACAC moveshe trajectory on the spherical surface away from the S1S3 plane,ince the angular excess adds up at each stage. This means that,nlike the nominal wavelength case (which trajectory remainsonfined in the S1S3 plane), the coupler rotations are performedt an increasing distance from the S1S3 plane, that is, far fromhe nominal ending point E1.
ig. 7. (Color online) Same as in Fig. 6 when the sequenceACC is chosen. In this case the trajectory remains close to the1S3 plane. The angular excess of the second (fourth) phasehifter compensates the angular excess of the first (third) phasehifter. In this way all the coupler rotations are performed closeo the S S plane and they add up to almost zero, like at the
We have found a general rule to choose the symmetryhat flattens both ports: the AACC configuration must behosen if and only if
(1) 2��iB−�i
A� and 2�iA belong to the same hemisphere
nd 2��iB+�i
A� and 2�iA do not belong to adjacent quad-
ants (of the circle in the S1S3 plane), or(2) 2��i
B+�iA� and 2�i
A belong to opposite hemispheresnd 2��i
B−�iA� and 2�i
A do not belong to opposite quad-ants (e.g., when t=1, m=2, k=0).
Notice that “do not belong to adjacent quadrants” isquivalent to “belong to opposite quadrants or to the sameuadrant,” as well as “do not belong to opposite quad-ants” means “belong to adjacent quadrants or to theame quadrant.”
. CONCLUSIONShave presented a geometrical and analytical approach toesign interferometric band splitters. With this novelethod we have generalized the doubly point-
ymmetrical scheme for filter synthesis to the case ofplitters made of wavelength-dependent couplers. First a
ig. 8. (Color online) Spectral response of a t=0, m=0, k=0tructure when the AACC configuration is chosen. In this case,s explained in Fig. 7, the 1550 nm port is well flattened.
eometric interpretation of the working principle of thesetructures allows us to derive simple analytical formulasor the case of wavelength-independent couplers. Thenhe same formulas can be easily extended to the case ofavelength-dependent couplers. Also, the geometricalnd physical insight helps us to broaden the class of flat-ened splitters to singly point-symmetric structures,hich are needed for certain combinations of couplingngles. The proposed method can be easily extended tony kind of interferometric filter featuring more couplersnd phase shifters and/or different symmetries.
The author is now with the Dipartimento di Ingegnerialettrica, Università di Palermo, 90128 Palermo, Italy. M.herchi’s e-mail address is [email protected].
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