Diffraction in multilayer filters between fibers [Fig.1(a)] cannot be evaluated with simple models. In fact,as the Gaussian beam that is approximately the fun-damental mode of the input fiber diffracts, its waistw(z) enlarges with the well-known law
w�z� � w0�1 � z2�zR2, zR �
�w02
�,
where the Rayleigh range is zR and the wavelength is� in the considered medium. For all waves incominginto the output fiber of the filter, after several roundtrips across the interfaces in the filter, differentdistances z are covered, there is a wavelength � ineach crossed medium, and the interfaces have to beconsidered by using the ABCD law. Therefore, theglobal calculation of the diffraction for the trans-mitted power by the filter into the fundamental
mode of the output fiber is not possible in this way.Only a qualitative description can be consideredwith the Gaussian beam diffraction description, as,for example, in [1].
In this paper, it is proposed to consider the propa-gation through the layers of a filter and glue as thepropagation of modes through fictitious fibers [Fig.1(b)] simulated by S matrix analysis. Compared withthe one-dimensional method presented in [1], inwhich neither the diffraction nor the absorptionlosses could be simulated, the method presented hereis fully three-dimensional, and accurate numericalresults are presented for the simultaneous calcula-tion of diffraction and absorption losses.
2. Principle of the Fibered Filter Modal Calculation
Once the layers of the filter and the glue have beenreplaced by fictitious fibers, the beam in each fibercan be expanded on the modes of the fibers: This isthe classical eigenmode expansion (EME) method[2–4]. The modes of an infinitely constant profile fiberalong the propagation direction are first calculated ineach fiber: the input fiber, the high- and low-effective-index thin-film layers of the filter and glue each con-sidered as fictitious fibers, and the output fiber. Then,
the scattering matrix S of each fiber with a fixedlength is calculated. The S matrix of the joints be-tween different fibers are also calculated. The globalS matrix of the filter between the input and the out-put fibers is then calculated, and the transmittedpower and reflected power into each mode of both theoutput and input fibers are deduced in the case wherethe input is the normalized fundamental mode of theinput fiber. Starting from fictitious fiber diametersfar smaller than for the off-stripped input and outputfibers �125 �m�, the diameters are increased until theresponse of the filter converges. Of course, this willwork only for filters that have convergence for ficti-tious fiber diameters below 125 �m. In the case inwhich the convergence happens for diameters clearlylower than 125 �m, it will be shown that the calcu-lation is considerably accelerated, also with a reduceddiameter of the input and output fibers.
A. Considering the Layers as Fictitious Fibers
Throughout this paper, the filters considered alternatebetween H and L layers. H and L are quarter-wavelength optical thickness layers at the wavelengthof � � 1495 nm of a high-refractive-index indexmaterial (ZnS) and a low-refractive-index material�Na3AlF6�.
At first sight, it seems inadequate to use a modalcalculation for the thin-film layers, since negligiblepower reaches the layers’ lateral edges, and a layershould then be considered in a first approximationas laterally infinite. For the same reason, a layercould be considered an optically smooth-surfacedcylinder with a radius large enough so that thebeam is negligible at this radius [whereas the usualtechnologies of deposition result in irregular lateraledge surfaces of the layers: see Fig. 1(a)]. A cylinderof dielectric thin film in air is a uniform-refractive-index fiber. Its modes are calculated here as if thiscylinder was an infinitely invariable profile fiberalong the propagation direction. Thus, the layersare considered fictitious cylindrical fibers with afictitious diameter that, as further calculationshows, can be chosen to be quite smaller than theaverage diameter of the deposited filter, which isusually roughly the 125 �m diameter of the off-stripped fibers [Figs. 1(a) and 1(b)].
B. Calculation of the Modes
Mode calculation in optical fibers with only a core anda cladding [5], as well as more generally in radiallystratified media [6], is well known.
Fig. 1. Multilayer interferential filter inserted between an input and an output fiber. The structure of the filter is composed of Hhigh-refractive-index layers and L low-refractive-index layers (the represented structure is simulated in Subsections 5.A and 5.B.1). (a)Real filter: the filter is deposited directly on the cleaved input fiber end. The output fiber is glued at the other side of the filter. (b) Simulatedfilter: the lengths represented in this diagram and in the last one are not to scale. The diameters of the input and output fibers, of thefictitious fibers, and of the embedding air layer are reduced to accelerate the simulation. (c) Simulated filter with light: the illumina-tion intensity is simulated at 1494.4 nm for the parameters corresponding to Subsection 5.B.1. The horizontal scale is illustrated by the1 �m long input and output fiber represented. The vertical scale is illustrated by the 40 �m diameter of the input, output, and fictitiousfibers.
C14 APPLIED OPTICS � Vol. 47, No. 13 � 1 May 2008
The chosen boundary conditions are the following:Outside the fibers or the filter layers, a surroundingair layer is considered. The outer boundary surface ofthis air layer is at the same radius of the fiber axes inboth the fibers and the layers [Fig. 1(b)]. It is consid-ered that this surface has translucent boundary con-ditions [7], meaning essentially that the light goingout through this surface does not come into the sys-tem again. The surrounding air layer thickness isincreased until this has no consequence on the filtertransmission. The same calculation would run forevery embedding material instead of air that wouldhave the lower refraction index in the simulated sys-tem. The case of an embedding material with a higherrefractive index is not presented here: It causes ad-ditive problems because it involves a system with twohigh-refractive-index regions separated by a low-refractive-index region that are therefore opticallydecoupled.
Because lower neff modes have higher spatial fre-quencies, the modes are always simulated in order ofdecreasing neff and considered in the filter simulationuntil its transmittance and reflectance converge to-ward stable values for a fixed wavelength.
Whereas the input and output fibers are chosen tobe single mode at the considered wavelength, theuniform fictitious fibers replacing the filter layer andglue are strongly multimode. In fact, it will be seenthat their diameter is far higher than the input andoutput fiber core diameter. Further in the paper, thefictitious fibers replacing the layers will sometimes besimply named layers. In the input and output fibers,the bases are constituted of the fundamental guidedmode in the core complemented by cladding modes,which propagate both in the cladding and in the coreand are attenuated after a short distance of un-stripped fiber (usually, the protection polymer clad-ding has a refractive index higher than in the opticalcladding to eliminate them).
C. Calculating the Response of the Filter
The S matrix has been successfully used previously toanalyze guided-wave optical systems. It is particu-larly well described in [7,8], where the advantages ofthe S matrix used with EME are compared with othermethods. In particular, it points out the advantagesof this method in the case in which high reflectionoccurs in the system, which fits particularly well thecase of multilayer interferential filters. A summary ofthe theory adapted to the analysis of a filter betweenfibers is presented here in order that the reader caneasily understand the presented simulations and tointroduce the notations.
The filter is an optical system inserted between theinput fiber where NI modes are considered and theoutput fiber where NO modes are considered. The Smatrix describes the response of the system to inputand output source electromagnetic fields sI and sO
incident on the filter through its input and outputfibers. It gives response electric fields rs
I and rsO out-
ward of the system according to the following scheme
completed with arrows:
)sI�x, y�rs
I�x, y�(
� System �(sO�x, y�rs
O�x, y�)
.
The sources and response fields are expanded onthe basis set of modes of the input and output fibers:
sI ��s1
I
s2I
É
sNI
I�, sO ��
s1O
s2O
É
sNO
O�, rs
I ��rs1
I
rs2I
É
rsNI
I�, rs
O ��rs1
O
rs2O
É
rsNO
O�.
By using the transmittance and reflectance intoeach mode for the source fields sI and sO, the S matrixcan be decomposed into four submatrices:
Y TI represents the transmittance of the compo-nents of sI by the filter,
Y RI represents the reflectance of the componentsof sI by the filter,
Y TO represents the transmittance of the compo-nents of sO by the filter,
Y RO represents the reflectance of the compo-nents of sO by the filter.
Therefore, RI and RO are invertible NI � NI andNO � NO square matrices, whereas TI has NO linesand NI columns, and TO has NI lines and NO columns.The global scheme of the S matrix represented by itssubmatrices is then
�rsI
rsO� �RI
TITO
RO�sI
sO.
The filter is composed of elementary systems: fic-titious fiber sections invariable along the propagationdirection and joints. When considering two sub-systems denoted with indices prime and double primeand having S� and S� matrices, the S matrix of thetwo subsystems in series is not the product of S� andS� but is calculable from the submatrices of the sub-systems and by considering the sources and re-sponses of the system and subsystems [7]. Thematrices of the fiber sections are simply propagationmatrices [7,8]. Depending on its configuration, theelectromagnetic field is not necessarily conserved at ajoint (in that case, a joint matrix would simply havebeen a converting basis matrix). As predicted by theMaxwell equations, an interface between two fibers Aand B introduces continuity only for the tangentialelectric field Et and also in absence of any source forthe tangential magnetic field Ht. It is shown in [7]that the coefficients of the submatrices of a joint Smatrix are all functions of the overlap integrals thatappear in the scalar product
1 May 2008 � Vol. 47, No. 13 � APPLIED OPTICS C15
�mode m�mode n� � S
Em,t � Hn,tdS,
where S is a infinite plane surface parallel to thejoint.
The filters studied here are between identical inputand output classical telecommunication fibers, andthe only considered source is the normalized funda-mental mode indexed 1 of the input fiber. The prin-ciple would be the same for other modes and fibers.The scheme of the system is then simplified, andsome elements of the matrices TI and RI that wehereafter simply call T and R appear as the compo-nents of rs
O and rsI:
sI ��1000�, sO � 0 )
)
�10É
0�
�r1,1
É
É
rNI,1
�(
System
(
�00É
0�
�t1,1
É
É
tNI,1
�)
.
This yields the following arrowed power scheme:
)1
R1 � �i�1
NI
Ri,1
(� System �
(0
T1 � �i�1
NI
Ti,1
)
,
where Ti,j � |ti,j|2 and Ri,j � |ri,j|
2.The simulations reported in this paper are essen-
tially those of T1,1, R1,1, R1, and T1. These four param-eters are of particular interest because they are moreuseful and easily accessible by optical power mea-surement: T1 and R1 include the cladding modes ofthe fiber and are measured near the filter, whereasT1,1 and R1,1 are measured far away from the filter,where only the single guided mode of the fiber re-mains (the measurement of R1 and R1,1 would need acoupler). All the simulations in this paper have beencomputed on an ordinary personal computer (Pen-tium 4, 1.8 GHz, 750 Mbytes of RAM) with a codedeveloped by the authors of [8].
D. Verification of the Modal Method
Whereas the modal method would be perfect withinfinite bases of modes in each fiber, and a spatialsampling of the modes also infinitely accurate, this ispractically not the case. Thus, limitations are takento allow achievable calculation in reasonable time.Verification of the method will mainly consist of ver-ifying
Y that no power is lost for a filter without absorp-tion loss,
Y that the response of the filter effectively con-verges by assimilating its layers to cylinders withlarge enough and increasing diameters,
Y that the simulations conform to the expectedresults presented in Section 3.
3. Expected Results
Before showing some numerical results, it is useful tothink about what they should be, in order to analyzethem, and also to contribute to the validation of thecalculation that would be wrong if it were not inaccord with the previsions. The fundamental inputfiber mode diffraction for each round trip consideredsequentially in the filter can be considered as manydiffractions of Gaussian beams: This yields no nu-merical results but provides interesting explanationsof the expected behavior of the filter:
Y The fundamental Gaussian beam of the inputfiber diffracts in the filter where it is not guided.Therefore, for a given transmitted round trip in thefilter ending at the output fiber, it becomes a Gauss-ian beam with a larger Gaussian radius, which isdecomposed into several modes of the output fiber.For a round trip ending reflected into the input fiber,the beam is decomposed into several modes of theinput fiber. Thus, the transmission and reflection ofthe filter have to be considered into sufficiently nu-merous modes of the output and input fibers.
Y The diffraction of a Gaussian beam is strongerat higher wavelengths. Thus, the transmission spec-trum of the filter into the fundamental mode of theoutput fiber should be attenuated at the higher wave-lengths, and therefore, its vertex should be shiftedtoward lower wavelengths.
Y On the contrary, as for plane waves, the inter-ferences of all round trips in the filter cannot produceexactly the Airy transmittance (transmittance of aFP filter between broad substrates). In fact, becausethe waves all have the same spatial distribution atevery plane parallel to the layers of the filter for planewaves in a FP filter between broad substrates, it isnot the case for a Gaussian beam diffracting withdifferent broadness for each round trip in a FP filterbetween fibers. Therefore, the interference is partial,and the transmission of the filter into the fundamen-tal mode of the output fiber looks like a broadenedAiry peak whose vertex does not transmit 100%. Forhighly resonant filters, part of the power diffractsmore, and this effect is increased.
Y Filters with little diffraction must have similarspectral responses considered between broad sub-strates as well as between fibers. Filters with littlediffraction are filters with little resonance or betweenfibers with broad cores.
Y When considering a linearly polarized planewave polarized 0% or 100% TE in normal incidenceon a plane filter, its polarization remains unchangedby the filter. If we now consider the propagation of alinearly fundamental Gaussian mode emerging from
C16 APPLIED OPTICS � Vol. 47, No. 13 � 1 May 2008
the input fiber into a plane filter, we must considerthat after a propagation distance z, the radius ofcurvature of the beam does not remain infinite but is
R�z� � z�1 �zR
2
z2 .
Despite that, the radius of curvature of the beamremains high because, for the main part of the beampower, the length of the round trip of the beam is farlower than the Rayleigh range for most of the filters.It can then be expected that the polarization is nota-bly changed by the filter only for a negligible part ofthe beam and only for highly resonant filters.
4. Bringing the Modal Method into Operation
A. Parameters to Be Set
Y Outer diameter of the embedding air layer �DB�
Soon defined with the boundary conditions for themode calculation, it must be the same in the fiber andlayer regions. As will be seen, it must be at leastapproximately 2 times the diameter of the input andoutput fibers.
Y Lower limit of neff
In the input and output fibers and in each layer of thefictitious fiber of the filter, the correct decompositionof every beam supposes an infinite orthogonal modebasis. The modes are calculated in order of decreasingneff, starting from the core refractive index of eachfiber. The more neff decreases, the higher the spatialfrequencies the modes have. Therefore, for each fiber,a desired accuracy of the decomposition correspondsto a lower limit of neff that is directly related to themode number of the basis.
Y Spatial accuracy of the modes (�), number ofsamples (N)
The modes propagate in the z direction and are sam-pled in the x and y directions. The modes of the basismust have enough spatial accuracy so that they areorthogonal. There are N2 samples inside a square ofside DB, so N � DB��.
Y Polarization of the mode basis
Limiting the mode basis only to useful modes is ex-tremely effective by restricting the polarization of themodes in the selected basis. Therefore, the injectedfundamental mode of the input fiber is chosen, forexample, to be linearly polarized 0% TE, and themodes of the basis in the input and output fibers andin the filter layers are selected under a given percent-age of TE polarization called TEMAX.
Y Higher number of modes of the mode basis
After having suppressed the modes polarized TEabove TEMAX, some other useless modes are sup-pressed too. The remaining modes are indexed by
increasing neff in each fiber or layer. There are layerswith high and low effective indices in the filter. Thenumber of modes remaining in the basis of the inputand output fibers is called NF, and it is called NH andNL in the layers.
Y Diameter of the input and output fibers’ opticalcladding �DF�
It would be natural to choose the usual value DF
� 125 �m. In fact, it leads to high DB and N, increas-ing the time calculation in terms that are not ade-quate with the ordinary computer used. Thus, thevalues DF � 80 �m and 40 �m have been simulated.DF � 80 �m is currently used in miniaturized deviceswith strongly curved fibers. DF � 40 �m is not cur-rently manufactured but is interesting for simulationin the case where it gives the same results as forDF � 80 �m or 125 �m with strongly reduced timecalculation.
Y Diameters of the fictitious fibers (DH and DL)simulating the H and L layers
They are of course smaller than the diameter of theinput fiber on which the filter is deposited [Fig. 1(b)].Starting from diameters far smaller than 125 �m, aconvergence of the filter response is researched forincreasing DH and DL. This convergence is the re-sponse of the filter deposited on all the surfaces of a125 �m diameter fiber cleaved end. Simulation withsmaller diameters than this of the convergence areuseful also for themselves because they introduce in-teresting new devices. Nevertheless, it must be takencare that the simulation supposes perfect lateral edgesurfaces of the layers, which does not correspond tothe simplest usual technology. Moreover, far smallerdiameters lead to lower neff of the mode basis, corre-sponding to leaky modes (not simulated in this paper)and thus to devices with losses, even without absorp-tion loss.
B. Criteria for Setting the Parameters
The criterion chosen for the parameter optimizationis evaluated for a filter without absorption loss and isthe convergence of the total transmission and reflec-tion sum T1 � R1 toward 100%. It is always observedthat, if this criterion is verified well enough, a con-vergence also occurs for the transmission and reflec-tion Ti,1 and Ri,1 into each considered mode of thefiber.
Of course, the length of the mode basis and thespatial resolution must be optimized to get simulta-neously an acceptable accuracy of the simulation anda short time calculation. This is obtained throughseveral steps:
First simulation:
DF, DH, DL, and DB are fixed as explained before,and the wavelength is also fixed. A low resolutionis chosen for preliminary explorations: for example,� � �. The simulation of Ti,1 is then effectuated with
1 May 2008 � Vol. 47, No. 13 � APPLIED OPTICS C17
a reasonable number of modes in each basis (100–300classed in order of decreasing neff). The more numer-ous modes are 50% TE. Selecting a value of TEMAX 50% is attempted, so that a considerable number ofmodes is eliminated. This is accepted if the effect onTi,1 is negligible.
Second simulation and intermediary tests:
The first simulation could not be done with neff aslow as an acceptable limit for each basis because themodes in all polarizations are far too numerous. Be-cause many of the first considered modes are noweliminated, the number of modes that has consider-ably dropped can be increased again but with theselected good polarization limit. NF is researched bycalculating the coefficients Ti,1 and Ri,1 of the filterand keeping only the modes i of the fiber that corre-spond to appreciable coefficients. The modes of thelayers and NH and NL are not checked directly bythe calculation of the filter: An appropriate test is thecalculation of Ti,j and Ri,j for devices consisting of onlythe input fiber and a single layer.
Third simulation:
� is now decreased until the orthogonality of thebasis is checked with an overlap of two modes below1% for every mode of the basis. � must be decreasedin particular if the number of modes in the basis isincreased, since modes with lower neff correspond tohigher spatial frequencies.
Verifications:
The result can finally be certified by verifying thatincreasing DB, the basis lengths, or decreasing � hasnegligible effect on Ti,1. It is not possible to verifysimultaneously that all the modes in high-TE polar-ization are useless, but it is credible since the radiusof curvature of the Gaussian beam diffracted out ofthe input fiber and considered in each round trip inthe filter remains high.
5. Numerical Calculation
The input and output fibers have a core diameter of2a � 9 �m, a core refractive index of nco � 1.49, anda numerical aperture of NA � 0.11 (typical values forsilica telecommunication fibers), which yields a clad-ding refractive index of ncl � 1.4859396 [5]. The re-fractive indices nH � 2.3 and nL � 1.3 of the H and Llayers are considered sensibly invariable for thenarrow-bandpass FP filters studied. Their physicalthicknesses that are exactly eH � 0.1625 �m and eL
� 0.2875 �m are chosen so that they correspond toquarter-wavelength optical thickness at � � 1.495�m (the choice of � is not exactly 1550 nm becauseexact layer thickness values are critical and take pri-ority over the � value to avoid any ambiguity by ex-amining unknown three-dimensional properties ofrezoning filters). The simulated filters are representedby the formula FIN�HL�n�LH�nFOUT, where FIN and
FOUT denote the input and output fibers beside thefilter and where �HL�n means HL repeated n times.These are FP filters with 2n 1 layer mirrors besidea 2L rezoning layer. Taking the periodic aspect of themirrors into account reduces the time calculation byaccelerating exponentially the S matrix calculation.
The parameters of the simulations at a fixed wave-length are first optimized for a strongly rezoning fil-ter �n � 5�. The chosen wavelength is 1495 nm andcorresponds to the transmission spectrum vertex ofthe FP filters that would have been deposited oninfinitely broad plane substrates.
Then, the spectral response of this filter is calcu-lated and compared with the spectral response ofless-rezoning FP filters (single and also triple FP fil-ters with n � 3). For those less-rezoning filters, thesimulation parameters are kept unchanged becausethey fit better more for those filters in terms of accu-racy but still give a reasonable time calculation (thosetime calculations are also much lower because themodes calculated for the first filter can be used di-rectly).
In Subsections 5.A and 5.B, only the diffraction isconsidered. In Subsection 5.C, the absorption lossesand glue are considered simultaneously with the dif-fraction.
A. Optimizing the Parameters of the Simulation at aFixed Wavelength
The parameters are optimized for the filter FIN
�HL�5�LH�5FOUT, which is a strongly resonant filter.Less-rezoning filters are simulated afterwards withthe same parameters. This filter can be writtenFIN��HL�4H�2B�H�LH�4� FOUT and is strongly rezoningbecause of the high reflectance of its mirrors [R11
� 0.9049 for the mirror FIN�HL�4HFOUT]. The wave-length is � � 1495 nm where the layers have quarter-wavelength optical thickness. The parameters of thesimulation are moved to get as quickly as possibleT1 � R1 � 1 and a convergence of T11, R11, T1, and R1for H and L layers with diameters of DH � DL �� �.The parameters required are the following:
Y Transverse spatial resolution of � � 0.5 �m.Y Outer air diameter around the fiber and sam-
ple number:
fiber diameter of DF � 40 �m ➜ DB � 74 �mand N � 148,
fiber diameter of DF � 80 �m ➜ DB � 200 �mand N � 400.
Y TEMAX � 48%. This reduces the number ofmodes useful for the convergence of the filter’s re-sponse from approximately 2000 to only NF � 28modes for the input and output fibers (neff �1.487729 for the guide mode and neff � 1.366875 forthe last cladding mode), and NH � NL � 45 modes forthe layers H and L (by keeping all the modes with neffhigher than its lower value in each basis). NF hasbeen determined so that no transmission coefficientTi,1 of the filter above 106 is ignored.
C18 APPLIED OPTICS � Vol. 47, No. 13 � 1 May 2008
Several verification tests have been effectuated byimproving separately one or several parameters andchecking that it induces variations of T1,1, R1,1, T1, R1,and T1 � R1 below 1%:
Y Increasing the considered number of modes toNF � 38 and NH � NL � 65.
Y Compacting the basis by suppressing some iso-lated modes with neff above its lower value. This fi-nally yields NH � 34 and NL � 29.
Y Changing the spatial resolution to � � 0.2 �m.This corresponds to � � ��7.5 at 1495 nm.
Y Increasing DB by 10%.
With the selected simulation parameters, the fil-ter is simulated until convergence for increasingDH � DL deposited on two different fibers with DF
� 40 �m and DF � 80 �m (Fig. 2). A complete de-scription of Fig. 2 points out the following:
Y The condition T1 � R1 � 1 is best obtainedfor the highest values of DL and DH for the fiberwith DF � 40 �m. For the fiber with DH � 80 �m,T1 � R1 so that T1,1, R1,1, T1, and R1 are in perfectcontinuity for DH � DL � 40 �m. For increasing val-ues of DH � DL, whereas the convergence is sensiblyobtained, a small linear decay of all the parameters isobservable (lower than 2% for each of T11, R11, T1, R1,and T1 � R1).
Y For reasonably quick calculation of the spec-tra, the convergense of the response of the filter forhigh DH � DL is assimilated to its response for DF
� DH � DL � 40 �m. This yields 1 h of time calcu-lation for one wavelength instead of one day for
DF � 80 �m. Considering the decay and all otherapproximations in the determination of the param-eters, an overestimation of the inaccuracy is cer-tainly below 5%.
Y Whereas it would need considerably higher-power calculation for verification, the most plausibleinterpretation of the decay for high DH � DL is thefollowing. Because the mode number in the layersstays fixed in the calculation while the layer diametersare increased, the spatial frequency of the basis modesextend in a shorter range, and thus, the high-spatial-frequency part of the beam is progressively less takeninto account in its decomposition on the basis.
Y As expected in Section 3, T11 is not 100% as thetransmission of a FP filter between broad substratesand without absorption losses would be.
Y A small accident in the converging curves forhigh DL � 60 �m could not be corrected: trials with alonger basis show that the requested length of thebasis is beyond the possibilities of the computer.Therefore, accidents are simply avoided when re-searching the limit of the filter response for broadlayers.
Y For the lower DL values, the filter behaves as areflecting object because the beam cannot enter thefilter. The fictitious L layers are also only weaklymultimode so that their basis contains only a reducednumber of guided modes lower than 45 (see Table 1).Because there is no cladding, the following modes arenot cladding modes but directly leaky modes withcomplex neff. Because the code used does not actuallycompute leaky modes, the basis is far too short for theL layers. Moreover, the 0.5 �m spatial resolution
Fig. 2. T1,1, R1,1, T1, and R1 for the filter �HL�5�LH�5 at � � 1495 nm, for two fiber diameters (DF � 40 �m and DF � 80 �m), and forlayers deposited on the fibers with various diameters DH � DL. F40�HL12 means DH � DL � 12 �m diameter layers centered on aDF � 40 �m diameter fiber. F80�H80�L76 means DH � 80 �m diameter H layers and DL � 76 �m diameter L layers centered on aDF � 80 �m diameter fiber. The index n corresponds to calculations with power normalization at the joints.
1 May 2008 � Vol. 47, No. 13 � APPLIED OPTICS C19
starts to be inaccurate in regard to the layer diame-ter. Thus, for lower DL and DH, only a good idea of thetendency is shown.
Y For intermediary DL, an optimal value �28 �m�maximizes T1,1 a little above its limit for high DL. Thisis because the beam can enter the filter and is keptfavorably forward propagating inside the layersthanks to the reflections of its edges. Although layersare usually deposited without any care about theiredge surface, they must have an optical surface qual-ity if this vertex of T1,1 is required.
Y It would be considerable work to verify whetherthe limit of T1,1, R1,1, T1, and R1 is affected by the badquality of the filter edge surface because, for most ofthe defects, it would involve nonuniform fibers in thepropagation direction (this verification is not criticalbecause the field vanishes at the edge surface of aDF � 125 �m fiber). Nevertheless, a simple calcula-tion introducing a strong defect in the filter edgeskeeping uniform fiber sections in the propagation di-rection shows that it is not sensible for the consideredexample: By alternating H and L layers with respec-tive DL � 80 �m and DL � 76 �m, almost no differ-ence is found (comparison of F80�HL80 to F80�H80�L76 on Fig. 2).
B. Spectra of Several Filters without Absorption Loss
1. Filter FIN(HL)5(LH)5FOUTThe simulation is performed with the parametersdetermined in Subsection 5.A and optimized to findthe limit of T1,1, R1,1, T1, and R1: DF � DH � DL
� 40 �m and N � 148. The spectral properties of thisfilter are shown in Fig. 3, and its illumination at1494.4 nm is shown in Fig. 1(c). For each wavelength,T1 � R1 � 1. As expected after Section 3, whereas thevertex of the transmission T of the filter deposited ona broad substrate is T � 1 and is at the wavelengthwhere the layers are optically quarter wavelength�1495 nm�, the FP interferometer between fibers hasaltered visibility and the transmission peak of T1,1 isattenuated for a long wavelength:
Y The T1,1 vertex is only 0.617 instead of 1.Y It is shifted to a lower wavelength (1494.4 nm
instead of 1495 nm).Y The T1,1��� curve has no symmetry near
1494.4 nm: Going away from that wavelength, it isattenuated more sharply on the right side.
The diffraction effects are strong for this filter: At1494.4 nm, the power T1 T1,1 � 0.155 is transmit-ted by the filter into cladding modes, and the filterreflects a power R1 � 0.208 instead of transmittingall the power in the fundamental mode in the ab-sence of diffraction. Thus, for the reliability of an
industrial technology, not only must the fiber coreand optical cladding optical properties be repeti-tively well defined but also those of the polymercoating that controls the propagation of the clad-ding modes!
To control the deposition of the filter, two usualthickness monitoring methods are available: opticaland quartz monitoring [9]. The quartz monitoringtechnique measures the physical thickness of the lay-ers. Thus, it is possible to compensate the shift of theresonance by shifting the desired physical thicknessof the layers. The optical monitoring technique mea-sures the transmission or the reflection of the filterduring the deposition. Since it determines the end ofan optically quarter-wavelength layer as a vertex ofthe transmission, this monitoring technique appliedat the 1495 nm control wavelength compensates au-tomatically for the shift of the filter response vertex.Moreover, the optical monitoring technique is theo-retically more adapted to control the deposition of FPfilters [9] and practically too as the measurement isdone directly on the filter at the fiber end and not ona control substrate apart [10] (other references on theexperimental deposition of filters at fiber ends are[11] and [12]). Thus, the simulation of the expectedshift contributes to the validation of the modal sim-ulation of filters between fibers but would easily bequenched by the control during the deposition of thefilter.
Figure 1(c) shows that the intensity of the electro-magnetic field is the highest at the frontiers betweenthe rezoning 2L layer and the adjoining H layers.This is also shown in Fig. 4, where the integratedforward and backward power is represented as afunction of the z propagation coordinate, which hasits origin at the beginning of the 1 �m long input fiber
Fig. 3. Spectra of the filter FIN�HL�5�LH�5FOUT. The vertex of T1,1
coincides exactly with a step of the vertical grid of 0.4 nm. Thepoints for � � 1495 nm are the same as in Figs. 3 and 2 forF40�HL40.
Table 1. Number of Guided Modes (Ng) in the Basis for the L Layers ofWeak Diameters
DL (�m) 12 16 20 24 28
Ng 12 16 20 24 39
C20 APPLIED OPTICS � Vol. 47, No. 13 � 1 May 2008
in the simulation. In Fig. 4, it appears that the totalpower inside the rezoning layer (forward and back-ward) is approximately 200 times the power injectedinto the input fiber (forward power at z � 0). Forhigh-power application, the highest damage riskstays there, where the intensity is the highest, andalso in the glue, because glues have low fusion points.The beam is also broadly distributed in the rezoninglayer [Fig. 1(c)]. The choice of 40 �m diameter ficti-tious fibers appears then to be well adapted becausethe intensity is sensibly confined just inside the re-zoning layer.
2. Filters FIN(HL)2(LH)2FOUT andFINL[(HL)2(LH)2L]3FOUTThese filters have been simulated with DF � 40 �mand DH � DL � 36 �m. These parameters have beenselected to avoid the configuration of an accident andparticularly the next one corresponding to DF � DH
� DL � 40 �m for those filters. This is verified inFigs. 5 and 6, where T1 � R1 � 1 at all wavelengths.
Figure 5 shows that the low-rezoning filter FIN
�HL�2�LH�2FOUT exhibits much fewer diffraction ef-fects than the highly rezoning filter FIN�HL�5�LH�5
FOUT and more precisely quasi-nondiffraction effectseven near of the rezoning wavelength. In fact, forFIN�HL�2�LH�2FOUT, T1,1 � T1 and R1,1 � R1 at everywavelength: the beam stays almost in the fundamen-tal mode of the input and output fibers. A shift of thevertex of the spectra is not observable for this filter.At the rezoning wavelength, T1 � 1 and R1 � 0 as fora FP filter between broad substrates and withoutabsorption losses. More generally, non-rezoning fil-ters between fibers (antireflecting, edge filters, etc.)exhibit negligible diffraction effects.
The filter L��HL�2�LH�2L�3 deposited on a broadsubstrate is a stack of three �HL�2�LH�2 FP filtersseparated by L antirezoning layers and with L layersat both ends (those two layers at both ends allow thefilter to be calculated more rapidly as a periodic struc-ture, without having a significant effect on the re-sponse of the filter). This triple FP filter is morerezoning than the single one. Between broad sub-strates, it maintains 100% transmission while sharp-ening the edges of the transmission peak [9]. Figure6 shows that the spectral response of filter FIN
L��HL�2�LH�2L�3 has the well-known and typicalshape of the same filter between broad substrates [9].The response of this filter between fibers exhibits a
Fig. 4. Power in the filter FIN�HL�5�LH�5FOUT at 1494.4 nm as afunction of z. The forward power is the solid line, and the backwardpower is the dashed line. They are sampled only one time in eachfiber (input, output, and fictitious fiber). Thus, the power in the Hand L layers can be easily distinguished in the figure. Fig. 5. Spectra of the filter FIN�HL�2�LH�2FOUT.
Fig. 6. Spectra of the filter FINL��HL�2�LH�2L�3FOUT.
1 May 2008 � Vol. 47, No. 13 � APPLIED OPTICS C21
small amount of diffraction effects, especially aroundthe rezoning wavelength. In fact, T1,1 is slightlysmaller than T1 near of 1495 nm, and also R1,1 issmaller compared with R1. Whereas some diffractioneffects occur for that filter, they are too small to shiftthe vertex of the transmission peak, and the symme-try of the spectra near 1495 nm, is barely altered (athigher wavelengths, it can be observed that thetransmission is barely lowered by comparing the cal-culated points of the spectra at 1465 and 1525 nmsymmetrically near the vertex of the spectrum at1495 nm in Fig. 6).
C. Diffraction with Absorption Losses and Glue
Till now, the filters have been considered withoutabsorption losses: This allows one to verify that nopower is lost by any inaccuracy in the calculation.Nevertheless, optical layers such as, for example, ZnSand Na3AlF6 have absorption losses depending on thedeposition conditions and typically correspond to acomplex refractive index n ik with an extinctioncoefficient of k � 104 for both of those materials [9].It is expected that absorption losses are stronger inhighly resonant filters where long round trips arecovered by the beam before going out. This corre-sponds well to the simulation of Fig. 7 for the filterFIN�HL�5�LH�5FOUT with absorption losses. Except forthe absorption losses, the parameters of the simula-tion are the same as in Fig. 3, where T1��� � R1���� 1. Thus, the total absorption losses are evaluatedby examining the variation of T1��� � R1��� betweenFigs. 3 and 7: It corresponds to 10.3% absorption atthe resonance {a rougher visual impression of theabsorption losses is quickly obtained by consideringmore simply 1 �T1��� � R1���� in Fig. 7 and in thefollowing figures}. As the wavelength goes away from
resonance, the absorption losses vanish progres-sively.
Figures 8 and 9 resume the effect simultaneously ofthe diffraction and absorption losses at the resonancerespectively for single and triple FP filters and sev-eral n values. The abscissa is the finesse F related ton for the single FP filter for both figures. For multipleFP filters, F is the finesse of the single FP filterembedded. F is a more general parameter than nbecause it is not bound to the materials of the layers.In the figures, the effects of losses appear as T1� R1 drops below 1, and the diffraction effects appearas T1,1 drops below T1 and R1,1 drops below R1. For theclassical telecommunication fiber simulated here, thefundamental-mode Gaussian diameter is 10.51 �m.In that case, the simulations show that both the dif-fraction effects and losses each induce drops of T1,1
Fig. 7. Spectra of the filter FIN�HL�5�LH�5FOUT with absorptionlosses in the layers.
Fig. 8. Response of the filters FIN�HB�n�BH�nFOUT for n � 2, 3, 4,5 at � � 1494.4 nm and with absorption losses. The simulatedpoints have finesses increasing with n.
Fig. 9. Response of the filters FIN��HB�n�BH�n�3LFOUT for n �
2, 3, 4 at � � 1494.4 nm and with absorption losses.
C22 APPLIED OPTICS � Vol. 47, No. 13 � 1 May 2008
above 5% as soon as F � 40 for a single FP filter andas soon as F � 20 for a triple FP filter. For fibers withbroader cores (e.g., strongly multimode) only thelosses have to be considered. Here are some detailsabout the simulations of Figs. 8 and 9:
Y The wavelength is fixed at 1494.4 nm, which isaccurately the resonance for n � 5 that correspondsto the sharpest T1,1 peak. For lower n, the peak isbroader and 1494.4 nm is a good approximation forthe resonance (thus, the modes are calculated onlyone time at � � 1494.4 nm).
Y DH � DL � 36 �m avoids any accident.Y The reflection coefficients in the finesses have
been simulated as R11 for the mirror between fibersFIN�HL�n1HFOUT. For n � 2, 3, 4, and 5, it yieldsrespectively R1,1 � 0.5335, 0.7739, 0.8713, and0.9049.
Y Figure 9 does not show the case of n � 5 be-cause the filter is much more resonant and wouldneed a computer more powerful than that available toprepare this paper.
Y Whereas there is little diffraction in Fig. 9 forn � 2 that corresponds to almost all the power trans-mitted in the fundamental mode (T1,1 � 94% and allother coefficients, each lower than 0.6%), the strongdiffraction for n � 3 corresponds to T1,1 � 47% onlyand to appreciable transmission and reflectance insome cladding modes of the fibers: T2,1 � 15%, T3,1� 11%, R1,1 � R2,1 � 1.9%, and R3,1 � 4%.
Till now, the filters had also been simulatedwithout the glue bounding the output of the filterto the output fiber. The response of the filterFIN�HL�5�LH�5FOUT with absorption loss and succes-sively with a null, quarter, and a half optical thick-ness of glue having a refractive index of 1.52 and anextinction coefficient of 104 is represented in Table 2.It shows that the glue has no effect for a half-wavelayer and very small effects with a quarter-wave gluelayer. Thus, the glue effects are quasi-negligible forevery thickness. This is what was expected becausethe glue is outside the rezoning filter.
6. Conclusion
The modal calculation of filters between single-modefibers established in this paper by assimilating thelayers of a filter to narrower fictitious uniform fibersallows diffraction to be simulated together with ab-sorption losses. The response of the filter is obtainedby increasing progressively the diameter of the ficti-
tious fibers until the spectral response of the filterconverges.
For FP filters between fibers, the accuracy dependson the finesse and multiplicity of the FP filter, on thecomputer power and memory, and on the calculationduration. Between classical single-mode telecommu-nication fibers, the spectral response of a single FPfilter with a finesse of F 65 for a triple FP filter withF 45 is calculable in less than 1 day with a Pentium4 and 750 Mbytes of RAM with an accuracy betterthan 5%. The convergence is soon reached for 40 �mdiameter fictitious fibers. Thus, the lateral edge ir-regularities of 125 �m diameter layers have no de-tectable influence on the response of the filter. For thefilters simulated, the absorption losses should not betroublesome for most applications (remaining below15%).
The main effects of diffraction in FP filters in re-sponse to the guided mode of the input fiber as asource are
Y transmission by the filter into cladding modesof the input fiber (rather than into the guided modefor high finesse of the filter!),
Y reflectance by the filter into cladding modes ofthe output fiber and also into its guided mode even atthe rezoning wavelength of the filter,
Y in consequence, attenuation of the guided-mode transmittance into the output fiber,
Y additively, stronger attenuation at higher wave-lengths and therefore with a transmission vertex ofthe transmittance spectrum shifted toward the lowerwavelength.
The shift of the transmittance vertex is considerableonly for highly rezoning and narrow-bandpass filtersbut is compensated by the optical control during thedeposition. The excitation of optical cladding modesresults in attenuation after a short distance for clas-sical fibers conceived to eliminate them. Thus, thechoice of the optical properties of the polymer clad-ding embedding the fibers is crucial for some appli-cations. Filters with finesse that is too high betweensingle-mode fibers are prohibited since the guidedmode is poorly transmitted. (For such filters, it mustalso be taken care that the diameter of the fictitiousfibers giving the convergence increases: It cannot ex-ceed the diameter of the real layers.) In contrast,filters involving no resonance or FP between broad-core fibers would show negligible diffraction and ab-sorption losses for materials with approximately 104
extinction coefficients. The power calculation with anordinary computer has been troublesome for thehighest finesse achievable (the limit of finesse is gov-erned by absorption losses); yet, it should not be any-more in the future.
The simulations for narrow fictitious fiber diame-ters below this giving the convergence are also usefulto the study of new devices where the technologyensures a controlled lateral surface during the depo-sition of the filters. In particular, an optimal filterdiameter exists for the transmission of the funda-
Table 2. Response of the Filter FIN(HL)5(LH)5FOUT at � � 1494.4 nmwith Absorption Losses as a Function of the Glue Optical Thickness
mental mode of the fiber through a FP filter embed-ded in air.
References1. J. Bittebierre and B. Lazaridès, “Narrow-bandpass hybrid fil-
ters with broad rejection band for single-mode waveguides,”Appl. Opt. 40, 11–19 (2001).
2. S. Peng and A. Oliner, “Guidance and leakage properties ofopen dielectric waveguides. I. Mathematical formulations,”IEEE Trans. Microwave Theory Tech. 29, 843–855 (1981).
3. A. Sudbo, “Numerically stable formulation of the transverseresonance method for vector mode-field calculations in dielectricwaveguides,” IEEE Photon. Technol. Lett. 5, 342–344 (1993).
4. G. Sztefka and H. Nolting, “Bidirectional eigenmode propaga-tion for large refractive index steps,” IEEE Photon. Technol.Lett. 5, 554–557 (1993).
5. A. W. Snyder and J. D. Lowe, Optical Waveguide Theory(Chapman & Hall, 1983).
6. C. Yeh and G. Lindgren, “Computing the propagation charac-
teristics of radially stratified fibers: an efficient method,” Appl.Opt. 16, 483–493 (1977).
7. P. Biensman, “Rigourous and efficient modelling of wavelengthscale photonic components,” Ph.D. thesis (University of Gent,Belgium, 2001).
8. D. F. G. Gallagher and T. P. Felici, “Eigenmode expansionmethods for simulation of optical propagation in photonics:pros and cons,” Proc. SPIE 4987, 69–82 (2003).
9. H. A. MacLeod, Thin Film Optical Filters (Institute of PhysicsPublishing, 2001).
10. J. Lumeau, M. Cathelinaud, J. Bittebierre, and M. Lequime,“Ultranarrow bandpass hybrid filter with wide rejection band,”Appl. Opt. 45, 1328–1332 (2006).
11. M. B. Alsous, J. Bittebierre, R. Richier, and H. Ahmad,“Construction of all-fiber Nd3� fibre lasers with multidielectricmirrors,” Pure Appl. Opt. 5, 777–790 (1996).
12. H. Yanagawa, T. Ochiai, H. Hayakawa, and H. Miyazawa,“Filter-embedded design and its applications to passive com-ponents,” J. Lightwave Technol. 7, 1646–1653 (1989).
