Design Scheme for Mach-Zehnder Interferometric CWDM. Wavelength Splitters/Combiners

Transcription

1 Desgn Scheme for Mach-Zehnder Interferometrc CWDM Wavelength Spltters/Combners Matteo Cherch Prell Labs Optcal Innovaton, vale Sarca, Mlan, Italy bstract: We propose an analytcal approach to desgn flattened wavelength spltters wth cascaded Mach-Zehnder nterferometers when wavelength dependence of the drectonal couplers cannot be neglected. We start from a geometrcal representaton of the acton of a doubly pont symmetrcal flter, assumng no wavelength dependence of the couplers. Next we derve the analytcal formulas behnd ts workng prncple and we extend them to the wavelength dependent case. We also show how the geometrcal representaton allows to broaden the class of workng structures. He s now wth Dpartmento d Ingegnera Elettrca, Unverstà d Palermo, 908 Palermo, Italy

2 Introducton Wavelength spltters/combners are a partcular knd of wavelength mult/demultplexers, whch are meant to manage two bands only. They can be desgned and mplemented n very dfferent ways. In the context of planar lghtwave crcuts, a convenent choce s the cascadng of Mach-Zehnder nterferometers. Standard technques for the synthess []-[3] of ths knd of flters don t take nto account the wavelength dependence of drectonal couplers. Ths means that they can be appled to DWDM crcuts only, wthn bands where ths dependence s neglgble. On the other hand Jnguj et al. [4] proposed the pont symmetrcal confguraton as an alternatve method for flter synthess. They appled these technque to the two lmtng cases of strong wavelength dependence and no wavelength dependence of couplers. In the frst case wavelength selecton s due to the coupler response, whereas n the second case s due to the extra-length of one of the two arms of each sngle nterferometer. We propose n ths paper an extenson of the last confguraton to the ntermedate case n whch the wavelength dependence of couplers s not neglgble, but not so strong to effectvely allow wavelength selecton. Ths s a typcal stuaton when dealng wth CWDM flters. In the lght of a geometrcal nterpretaton of the results n Ref. [4], we wll show how to correct ths wavelength senstvty to get stll flat passbands. Even though the novel geometrcal and analytcal approach s presented n applcaton to a practcal case, t should be clear that what s proposed s not a partcular structure but a new method to desgn and analyze nterferometrc flters. The presented applcaton wll show how the proposed method can gve more physcal nsght and desgn control than any numercal approach.

3 Wavelength ndependent couplers In ths secton we wll brefly present the synthess technque based on doubly pontsymmetrcal cascadng of Mach-Zehnder nterferometers, supposng wavelength ndependent couplers. Then we wll ntroduce the geometrc representaton of couplers and phase shfters that enables to renterpret prevous numercal results n terms of general analytc formulas. The Doubly Pont-Symmetrcal Confguraton Wavelength spltters/combners are two-port devces that splt/combne two bands centered at two dfferent wavelengths λ and λ. The pont-symmetrc confguraton has been ntroduced n Ref. [4] to get flat flterng wth nterferometrc structures. In partcular when repeated twce (see Fg. ), ths method ensures flat response both on cross-port and through port. In Ref. [4] the general workng prncple of ths technque s explaned analytcally, whereas the lengths of the couplers are calculated usng a numercal optmzaton algorthm. Couplers are supposed to be wavelength ndependent, whereas the extra-length of each Mach-Zehnder nterferometer s chosen so that ts phase shft s n π (n nteger) at λ, and ( n +) π at λ. Clearly, when n s odd (n+) s even and vce versa. For smplcty we wll refer to even and odd phase shfts, dependng on the party of n. We wll now ntroduce a geometrc representaton for the acton of an nterferometer that not only makes t possble to analytcally recover the results of the numercal approach, but also allows to broaden the class of workng structures. 3

4 Geometrc Representaton In two recent papers [5], [6] we have shown how a generalzed Poncaré sphere representaton [7]-[9] can be very helpful when studyng wavelength dependent couplers and nterferometers. The actons of both couplers and phase shfters can be represented as rotatons on a sphercal surface, analogous of the Poncaré sphere for polarzaton states. Fgure dsplays all the ntersectons of the S, S, and S 3 axes wth the sphere. They represent respectvely the sngle wavegudes modes E, E and ther lnear combnatons ( E ) ( E ) E R, L ± E E S, ± E and. lso the generc normalzed state P a α E s plotted. Notce that the relatve phase angle θ s E + ae cosα E + exp( θ ) sn exactly the alttude angle of the pont P wth respect to the equatoral plane on the crcle perpendcular to the S axs passng through P. Consderng the cone havng ths very crcle as bass and the sphere center as vertex, the power-splttng angleα s equal to half the half cone openng angle. Snce we deal wth lossless elements only, all the physcal trajectores are lmted to ponts on the sphercal surface. Furthermore, when consderng transformatons whch are compostons of recprocal elements only (whch s the case when dealng wth drectonal couplers and phase shfters), all trajectores on the sphere can be compostons of rotatons about axes belongng to the equatoral plane only [5],.e. the S S plane. In partcular the acton of a phase shfter s represented by a rotaton about the S axs, and the acton of a synchronous coupler s represented by a rotaton about the S axs. Hence, f we deal wth synchronous couplers and n π (n nteger) phase shfts only, we can restrct ourselves to the projecton of the sphere on the S S 3 plane,.e. a -dmensonal representaton on the crcle layng n that plane. Let us defne the angular expresson of the ampltude couplng rato [4], [5] of a coupler 4

5 ( λ ) = κ ( λ ) [ L + δ L( λ )], () where κ (λ) s the couplng per unt length n the straght part of the coupler and δl (λ) accounts for the contrbuton of the nput and output curves. We wll assume these two quanttes to be the same for all couplers, whereas we wll let the coupler length L vary from coupler to coupler. For the moment we wll also assume any dependence of on λ to be neglgble. Fg. 3 shows the projected trajectores on the S S 3 plane of a Mach-Zehnder nterferometer n the two cases of even and odd multples of π for the phase shft. E and E represent respectvely the stuaton of all power n the nput arm and all power n the other arm. t the lght of what above notced regardng the power splttng angle α, the acton of a coupler s smply represented by a rotaton of twce the couplng angle. nπ phase shft lves the system as t s whereas a ( n + ) π phase shft sends any pont of the crcle to ts mrror mage wth respect to the S axs. Hence, n the case of an even phase shft, the nterferometer follows the path E FG, actng as a coupler wth angle +, whereas n the case of an odd phase shft t wll follow the path E FGH, equvalent to a couplng angle. nalytc formulas Keepng ths n mnd, we go back to the pont-symmetrc structure n Fg.. The flter works f the couplers satsfes the followng condtons: π 4 = t + kπ, () π 4 + = ( t) + mπ 5

6 where t {0, }, and must be postve, k s an nteger and m must be a non-negatve nteger. For t = 0 the frst condton reads that for odd phase shfts power must reman n the through port and the second one means that for even phase shfts lght must go n the cross port. For t = the two ports exchange ther role. Wavelength selectvty s smply obtaned wth a proper choce of the extra length L π β ( λ ) β ( ), beng β (λ) the propagaton constant of the sngle wavegude mode. Solvng () for / λ and, we get π π = + ( m + k) 6 8, (3) π π = + ( m k t) 8 4 where m and k obey the selecton rules m k m k + m k for t = 0 for t = ND k 0. (4) for t = ND k < 0 In partcular for t = 0, m =, and k = we fnd = π and = π that are very close to the optmal values = π and = π numercally found n Ref. [4]. It s straghtforward to rewrte (3) n terms of L X X = κ δl (X =, ), clearly wth the further constrant X κ > δl. Notce that the results of the numercal approach n Ref. [4] depend on the ntal guess. On the contrary our approach allows to know all the solutons at once, and ths may be useful, for example, to fnd the shortest one. In our formalsm the choce t = 0, m = 0, k = 0 gves the 6

7 smaller overall couplng angle 4 + = π, that s fve tmes smaller than the couplng angle of the t = 0, m =, k = case proposed n Ref. [4]. We lke also to pont out that, when the full Poncaré sphere representaton s consdered, t becomes clear why pont-symmetry can guarantee band flatness,.e. tolerance to wavelength changes, or, equvalently, nsenstvty to changes of the phase shfter rotatons. It wll be shown n detal later on that the change of a gven phase shft from ts nomnal value (due to a small devaton of the wavelength from λ ) wll be compensated for by a correspondng change n ts pont-symmetrcal counterpart. Wavelength dependent couplers In ths secton we wll extend prevous results to the case of wavelength dependent couplers. Wth the ad of the full Poncaré sphere representaton we wll also show that, for some choce of parameters t, m and k, the sngly pont-symmetrcal confguraton must be preferred n order to get well flattened response of the flter. Generalzed formulas Whenever t s requred to manage two channels only, we can just focus on the spectral response wthn the two bands about λ and λ.we can take nto account the wavelength dependence rewrtng (3) as π = t + kπ π = ( t) + mπ (5) 7

8 X where κ L + δ L ) κ ( λ )[ L + δ L( λ ) ], and we have assumed λ ( λ ) to be assocated ( X X wth the even (odd) phase shft. Notce that wavelength dependence can be easly accounted for because the frst condton must be satsfed at λ only and the second one at λ only. Solvng for L and L we get L L t + m t + k π δl + 3δL = + κ κ 6 4. (6) t + m t + k π δl 3δL = + κ κ 8 In ths case t s not straghtforward to determne the selecton rules for k and m, because they wll depend on the dfference κ κ. Clearly the smaller ths dfference s, the better the rules n (4) apply also to ths case. Once κ (λ) and δ L (λ) are known, ths smple formulas allow to desgn the drectonal couplers for doubly-pont-symmetrcal flattened flters. Furthermore, many dfferent structures can be found wth dfferent choces of t, k, and m. Numercal Example To confrm our analytcal results we have performed some numercal smulaton. We have consdered slca bured wavegudes, wth a 4.5% ndex contrast and a µm x µm square cross secton, whch guarantees monomodalty n the band of nterest. We have calculated the propagaton constants versus wavelength for the sngle wavegude mode and for the coupler supermodes usng a fully vectoral commercal mode solver [0]. The nner wall wavegude separaton of the couplers have been chosen to be µm. We have also checked wth a commercal beam propagator [] that, wth ths hgh ndex contrast, the contrbuton δl (λ) s not only very small, but ts wavelength dependence s the same of an equvalent straght coupler 8

9 extra-length gvng the same couplng. ll these nformaton can be mplemented n a very accurate and smple transfer matrx model. In Fg 4 t s shown the spectral response for a 490 nm/550 nm spltter wth the choce t =, m =, k = 0. Notce that, by constructon, to the wavelengths 490 nm and 550 nm correspond zeros n the port where they must be suppressed. Clearly, a fne tunng of the analytcal soluton, may gve better performance over the whole channel band. Fg. 5 corresponds to the case t = 0, m = 0, k = 0. In ths case our recept works only locally at the nomnal wavelength, but does not guarantee flatness about 550 nm. Ths can be qualtatvely understood on the Poncaré sphere n Fg. 6. When the wavelength s not exactly 550 nm the phase shft t s not an exact multple of π. Ths gve rse to a non-zero rotaton about the S axs correspondng to each phase shfter. Ths means a departure of the trajectory from the S S 3 plane. Clearly, havng chosen the doubly pont-symmetrc confguraton, the sequence of phase shfts wll be, for example, CC, where (C) means ntclockwse (Clockwse). ut a C rotaton can partally compensate for the effect of an rotaton f and only f they are performed startng from ponts on dfferent hemspheres. On the contrary, when the ponts are n the same hemsphere a C rotaton worsen the effect of the prevous rotaton (n Fg.6 both rotaton move the trajectory far away from the S S 3 plane), and vce versa, gvng a strong departure from the deal trajectory, that s a small tolerance to wavelength changes. In ths case the rght choce s to correct the rotaton wth another rotaton as shown n Fg.7. Ths means that we have to choose a sngly pont-symmetrcal confguraton CC. In Fg. 8 t s shown the spectral response of ths confguraton, whch s found to be well flattened on both ports. So when ( ) and belong to the same hemsphere, flatness may be guaranteed by the CC scheme. ut ths s not always true as, for example, when choosng t = 0, m =, k =. 9

10 We have found a general rule to choose the symmetry that flattens both ports: the CC confguraton must be chosen f and only f ) ( ) and belong to the same hemsphere ND ( + ) and do not belong to adjacent quadrants (of the crcle n the S S plane), 3 OR ) ( + ) and belong to opposte hemspheres ND ( ) and do not belong to opposte quadrants (e.g. when t =, m =, k = 0 ). Notce that do not belong to adjacent quadrants s equvalent to belong to opposte quadrants or to the same quadrant, as well as do not belong to opposte quadrants means belong to adjacent quadrants or to the same quadrant. Conclusons We have presented a geometrcal and analytcal approach to desgn nterferometrc band spltters. Wth ths novel method we have generalzed the doubly pont symmetrcal scheme for flter synthess to the case of spltters made of wavelength dependent couplers. Frst a geometrc nterpretaton of the workng prncple of these structures allows to derve smple analytcal formulas for the case of wavelength ndependent couplers. Then the same formulas can be easly extended to the case of wavelength dependent couplers. lso the geometrcal and physcal nsght helps us to broaden the class of flattened spltters to sngly pont symmetrc structures, whch are needed for certan combnatons of couplng angles. The proposed method can be easly extended to any knd of nterferometrc flter featurng more couplers and phase shfters and/or dfferent symmetres. 0

11 References [] C. K. Madsen, Optcal Flter Desgn and nalyss. Wley, New York, 999, Ch. 4. [] M. Kuznetsov, Cascaded coupler Mach-Zehnder channel droppng flter for wavelengthdvson-multplexed optcal systems, J. Lghtw. Technol., vol., pp. 6-30, Feb [3]. J. Offren, R. Germann, F. Horst, H. W. M. Salemnk, R. eyeler, and G. L. ona, Resonant coupler-based tunable add-after-drop flter n slcon-oxyntrde technology for WDM networks, IEEE J. Select. Topcs Quantum Electron., vol. 5, pp , Sept [4] K. Jnguj, N. Takato, Y. Hda, T. Ktoh, and M. Kawach, Two-port optcal wavelength crcuts composed of cascaded Mach-Zehnder nterferometers wth pont-symmetrcal confguratons, J. Lghtw. Technol., vol. 4, pp , Oct [5] M. Cherch, Wavelength-flattened drectonal couplers: a geometrcal approach, ppl. Opt., vol. 4, pp , Dec [6] M. Tormen and M. Cherch, Wavelength-Flattened Drectonal Couplers for Mrro- Symmetrc Interferometers, J. Lghtw. Technol., 3, (005). [7] R. Ulrch, Representaton of codrectonal coupled waves Opt. Lett., 09- (977). [8] N. Frgo, generalzed geometrcal representaton of coupled mode theory, IEEE J. Quantum Electron., 3 40 (986). [9] S. K. Korotky, Three-space representaton of phase-msmatch swtchng n coupled twostate optcal systems, IEEE J. Quantum Electron., (986). [0] Fmmwave by PhotonDesgn, [] eamprop TM by RSoft Inc.,

12 Fgure captons Fg.. The doubly pont symmetrc structure. Frst the buldng block, composed of a type coupler and a half type coupler, s repeated pont-symmetrcally, resultng n a structure. Ths structure s repeated pont-symmetrcally agan to gve the desred result. Fg.. Generalzed Poncaré sphere for the analyss of two coupled wavegudes. The generc pont P s represented together wth ts relatve phase angle θ and power splttng angle α. Physcal transformaton are represented by composton of rotatons about axes on the S S plane. The ponts on the rotaton axs represent the egenstates of the system. In partcular a synchronous coupler wth couplng angle s represented by a rotaton about the S axs, whereas a θ phase shft s represented by a θ rotaton about the S axs. Fg. 3. Geometrc representaton on the S S 3 plane of the acton of a Mach Zehnder nterferometer n the two cases of (a) even and (b) odd phase shfts. coupler wth couplng angle gves rse to a rotaton, a nπ phase shft lves the system as t s whereas a ( n + ) π phase shft turns over the state wth respect to the S axs. Fg. 4. Smulated spectral response for the bar port (contnuous lne) and the cross port (dashed lne) of a spltter wth t =, m =, k = 0, deally desgned for separatng a 490 nm channel n the bar port from a 550 nm channel n the cross port. The doubly pont symmetrc structure ensures flatness for both ports. Fg. 5. Smulated spectral response for the bar port (contnuous lne) and the cross port (dashed lne) of a spltter wth t = 0, m = 0, k = 0, deally desgned for separatng a 490 nm channel n the cross port from a 550 nm channel n the bar port. In ths case the response s not flat for the bar port.

13 Fg. 6. Geometrcal representaton on the full 3D Poncaré sphere of the response n Fg. 5 for a wavelength slghtly dfferent by 550 nm, correspondng to phase shfts slghtly greater than π. The sequence CC moves the trajectory on the sphercal surface away from the S S 3 plane, snce the angular excess ad up at each stage. Ths means that, unlke the nomnal wavelength case (whch trajectory remans confned n the S S 3 plane), the coupler rotatons are performed at ncreasng dstance from the S S 3 plane, that s far from the nomnal endng pont E. Fg.7. Same as n Fg. 6 when the sequence CC s chosen. In ths case the trajectory remans close to the S S 3 plane. The angular excess of the second (fourth) phase shfter compensates the angular excess of the frst (thrd) phase shfter. In ths way all the coupler rotatons are performed close to the S S 3 plane and they add up to almost zero, lke at the nomnal wavelength. Fg. 8. Spectral response of a t = 0, m = 0, k = 0 structure when the CC confguraton s chosen. In ths case, as explaned n Fg 7, the 550 nm port s very well flattened. 3

14 Fg. 4

15 Fg. 5

16 S 3 G S E F E a) nπ S 3 S H E F E b) G (n+)π Fg. 3 6

17 λ (µm) Fg. 4 7

18 λ (µ m) Fg. 5. 8

19 S 3 S S Fg. 6 9

20 S 3 S S Fg. 7 0

21 λ (µ m) Fg. 8