Due to Fabrication Errors in AWGs

Transcription

1 An Estiation of Perforance Degradation Due to Fabrication Errors in AWGs Thoas Kaalais, Thoas Sphicopoulos and Diitris Syridis The authors are with the departent of Inforatics and Telecounications of the Uniersity of Athens Greece, GR-5784 Abstract: Arrayed Waeguide Gratings AWGs are iportant coponents for the realization of waelength diision ultiplexing optical networs. Their filtering perforance is liited by the existence of phase errors in the grating waeguides due to fabrication iperfections. In this paper the statistical properties of the phase errors are related to the waeguide iperfections using a ariation of the effectie index ethod. The filtering quality of the AWG is then inestigated by considering the behaior of its transfer function in the presence of rando phase errors. The probability density function of the transfer function s sidelobes is ealuated nuerically and the results are justified using theoretical considerations. Finally the behaior of the axiu sidelobe leel is also analyzed nuerically and uniersal diagras are presented which allow the estiation of its ean alue, standard deiation and cuulatie distribution function for eery specific AWG. Index Ters: Optical filters, waeguide filters, gratings, integrated optics, crosstal, tolerance analysis, waelength-diision ultiplexing. --

2 I. INTRODUCTION Arrayed Waeguide Gratings AWGs [],[] are iportant coponents for the realization of odern optical counication networs eploying waelength diision ultiplexing [3] on which they can sere as waelength ultiplexers, deultiplexers and routers. These deices hae been ade coercially aailable and there are techniques for achieing polarization insensitie operation [4]. There are also techniques that allow passband flattening []. The transfer function of the conentional AWG without passband flattening is of Gaussian shape at the passband and ideally has ery low sidelobes, around 6dB. Such a low sidelobe leel would ean that networs eploying AWGs would not suffer fro the accuulation of in-band and out-band crosstal noise in the receier. Unfortunately practical AWGs exhibit a uch higher sidelobe leel, posing liitations in the achieed Bit Error Rate of the receier due to non ideal waelength isolation. Seeral studies hae been ade in order to understand the origin of crosstal in Arrayed Waeguide Gratings [5]-[8]. It has been noted that it is ainly due to the phase errors between the different ars of the grating [6]. Although the aboe studies focus on sall phase errors, these can be quite significant in the case of an AWG with narrow channel spacing such as GHz. In addition the statistics of the phase errors ary fro ar to ar since the waeguide lengths are different. This is especially true for AWGs with sall Free Spectral Range FSR because the difference ΔL, between the lengths of two adjacent grating waeguides is inersely proportional to the FSR. In this paper the behaior of the sidelobe leel directly related to the crosstal of a conentional AWG with Gaussian passband and its relation to arious design paraeters and fabrication tolerances is described, een in the presence of larger phase errors with arying statistics fro ar to ar. Waeguide fabrication errors, lie non ertical sidewalls [9], are --

3 taen into account with the Effectie Refractie Index ethod ERI. Next, the behaior of the transfer function of the AWG in the presence of the induced phase errors is analyzed. The statistical behaior of the sidelobes of the transfer function is shown, both by coputer siulations and theoretical considerations, to be approxiately that of an exponential rando ariable. Finally, using coputer siulations, the behaior of the axiu sidelobe leel is analyzed and uniersal diagras are gien which enable the estiation, fro the fabrication tolerances, of the ean alue, the standard deiation and the Cuulatie Distribution Function CDF of the axiu sidelobe leel for ost types of AWGs. II. RELATION BETWEEN FABRICATION IPERFECTIONS AND PHASE ERRORS As stated in [7], waeguide fabrication errors can produce a ariation Δn eff in the effectie index, n eff, of the fundaental TE and T ode of a waeguide, thus causing phase errors in the grating ars of the AWG. Because of the etching process the waeguide, while originally specified to be rectangular is actually a trapeziu Fig a and there can be soe deiation in the alues of its width w and its height h. The refractie index of each region can also be different fro its specified alue due to errors in the gap waelength λ g. Fig. b shows the ariation of n eff, calculated using the ERI for the fundaental TE ode of a waeguide which was specified to hae core size 7μ x 7μ, core refractie index n.5, n n and index contrast Δ.75%, when only one of the seen paraeters pa,b,w,h,n,n,n is odified. The ERI is applied by segenting, as shown in the Fig a, the waeguide in horizontal layers typically to 3. The width of each layer of the trapeziu is assued constant. The effectie indexes of the arious layers are found and a ertical ultilayered structure is obtained, fro which the waeguide n eff, both for the TE and the T odes, is calculated as usual. Fig. b indicates that if the deiations Δp of the -3-

4 waeguide paraeters p are sall ~.μ for w,a,b,h and ~ -4 for n i then Δn eff is linear with respect to Δp, that is neff neff neff neff neff neff neff Δ neff Δw + Δa + Δb + Δh + Δn + Δn + Δn w a b h n n n As a result the square of σ neff of Δn eff can be written as TE.T TE,T σ n / neff p eff p σ Δp where σ Δp is the standard deiation of Δp. The alues of the deriaties of eq. and can be calculated using the aboe ethod. The deiations of the waeguide paraeters Δp can be assued Gaussian rando ariables with zero ean alue <Δp> for p a,b and standard deiations σ Δp expressing the fabrication tolerances that can be roughly estiated directly or indirectly [7]. Because of the linearity entioned aboe and because a linear cobination of independent Gaussian rando ariables is also a Gaussian rando ariable, Δn eff will be Gaussian with standard deiation gien by. The ean alue of Δn eff, which is due only to Δa and Δb, is sall and does not play any significant role in the results. Hence, <Δn eff > will be assued equal to zero both in the theoretical calculations and the nuerical siulations in this paper. In ost AWGs the bending radii of the grating waeguides are large and do not affect the alue of the deriaties in. Howeer if the bending radii becoe sall, ore elaborate D ode solers [] along with conforal apping ust be used to deterine these deriaties. III. EFFECT OF WAVEGUIDE COUPLING The aboe ethod treats each waeguide separately without including the presence of its adjacent waeguides. One ight expect, that since adjacent grating ars are coupled, the -4-

5 phase errors in one waeguide could affect the phase error on its neighboring waeguides, or in other words, that coupling in the grating ars could correlate the phase errors. The effect of waeguide coupling can be estiated using coupled ode theory []. Assuing the effects of only adjacent waeguides, the coplex aplitude A z of the signal in a waeguide obeys: d dz A z z jβ A z jc A z jc A 3 where A z is the coplex aplitude of the considered waeguide, A z and A z are the coplex aplitudes of its two adjacent waeguides, c and c are the corresponding coupling coefficients while β is the propagation constant βπn eff /λ of the waeguide in the uncoupled case. Diiding by A z and integrating with respect to z the following result is obtained: z z A z A z A z A exp jβ z j c z dz j c z dz 4 A z A z Fro 4 it is deduced that the phase of the optical signal in a grating waeguide depends on the coplex signal aplitudes of the adjacent waeguides and the coupling coefficients. A change in the paraeters of the adjacent waeguides will affect the phase of the optical signal through the coupling coefficients c and c. These coefficients are functions of z since in AWGs, the grating waeguides start oing apart in the region after the first star coupler and oe close to each other at the input of the second star coupler. The deriaties of the coupling coefficients with respect to the waeguide paraeters can be used to easure the change in the coupling coefficients and hence the phase error correlation. If the width w of waeguide changes by a sall aount Δw then the change of the coupling coefficient c will be linear and will be gien by Δ w c /. The induced phase change Δφ in the signal of waeguide will be w -5-

6 z c A z Δφ Δwdz 5 w A z Fig. depicts the ariation of the deriatie c / n and c / w of the coupling coefficient c of two waeguides haing core widths, w w 4μ, core heights h h 4μ and core indexes n.5 where n n i is the core index of the i th waeguide, calculated using the effectie index ethod. The relatie difference between the core indexes and the surrounding index is Δ.33% n.495 and the distance between the centers of the waeguides is initially d7μ resulting in a gap of 3μ between the cores. The corresponding alues of the deriaties of the propagation constant β of the waeguides are β / n.9, 4 β / w 8x μ and will be used to copare the change in the phase introduced by the change in β and by the change in the coupling coefficients. Fro the figure it is deduced that although the deriaties of the coupling coefficients are initially of the sae order as the deriaties of the propagation constant, they gradually diinish as the waeguides oe further apart. Also since the waeguides typically reain uncoupled in the cured section of the grating, it is expected that the phase error in waeguide, caused by the fabrication iperfections of the adjacent waeguide, will be sall copared to the phase error caused by the fabrication iperfections of waeguide. It should also be noted that in this exaple, the waeguides are placed close together and the index contrast Δ is low which iplies that the coupling is soewhat strong. In practical AWGs, where a sharper index contrast Δ should be preferred [] in order to reduce the chip size, and where the grating waeguides often hae larger initial spacing, the coupling induced phase error correlations will be een lower. As a result the phase errors will be assued uncorrelated throughout this paper. -6-

7 IV. STATISTICS OF THE PHASE ERRORS Once σ neff is estiated by the fabrication tolerances σ Δp, the standard deiation σ of the phase error δ, of the th waeguide can be calculated by σ πσ L / λ 6a neff where λ is the central waelength and L the length of the th ar -. By replacing L L +ΔL in 6a where L is the length of the shortest ar, the following equation is obtained for the standard deiations of the grating ars: σ σ + Δσ / 6b where ΔσπΜ-σ neff ΔL/λ, ΔLc/n eff,o FSR the difference in the lengths of two consecutie ars, FSR the Free Spectral Region, c the elocity of light in acuu, n eff,o the noinal effectie refractie index of the fundaental waeguide ode and the nuber of grating ars. It is deduced by 6a and 6b that the phase error in each ar can hae different standard deiation if ΔL is not negligible copared to L as in the case of narrow channel spacing e.g. GHz. Equation 6 does not account for the phase error induced by the iperfect photoas resolution. The effect of the photoas resolution can be incorporated by adding to the lengths of the waeguides a sall rando length Δl with ean alue equal to zero and standard deiation equal to the resolution of the photoas σ ph. The equation giing the standard deiation will change and will becoe π σ σ neff L + neff, oσ ph 7 λ This equation exhibits a linear behaior with respect to, for practical alues of its paraeters. Using as an exaple, the alues of the fabrication tolerances easured in [7], which are σ n σ n x -6 for the ariation of the aterial refractie index and σ w σ h x - -7-

8 3 μ for the ariations in the waeguide core diensions, the alue of σ neff, calculated using the ERI is equal to.x -6. The alues of σ obtained by 7 are plotted in Fig 3a, for L 4 μ, which is a typical alue for the shortest grating ar, ΔL5μ, which corresponds to a 6x6 AWG with channel spacing equal to GHz, and photoas resolution σ ph.5μ [8]. The AWG is assued to hae 65 waeguides, while n eff,o is taen approxiately equal to.5. In Fig 3b σ is also plotted for a GHz spacing 6x6 AWG with the sae nuber of waeguides and ΔL5μ. It is seen that in both cases σ is approxiately linear and can be written in the for of 6b using cure fitting. As a result the standard deiations σ of the phase errors in the grating ars will obey 6b and depend on three paraeters: the initial phase error σ the difference sallest alue of σ and the nuber of ars. Δσ between the largest and the V. STATISTICAL BEHAVIOR OF THE TRANSFER FUNCTION The transfer function between the central input and output ports of an AWG in the presence of the phase errors δ is gien by [5], H C exp jπ exp jδ where C is the optical power of the th grating waeguide noralized to the total power and f-f /FSR is the noralized optical frequency, f being the actual optical frequency and f the central frequency of the AWG. In a conentional AWG, the coefficients C obey a Gaussian law, and in practical AWGs the ratio R t inc /axc should be ery sall to ensure that the sidelobes of the AWG in the absence of phase errors δ, would be ery low. Throughout the nuerical siulations in this paper, the ratio R t is set to be % in which case the sidelobes of the AWG, in the absence of phase errors, are about 6dB. Different alues of this ratio do not alter the sidelobe leel behaior noticeably. By arying R t fro % -8-

9 to 5% for 5 only a.5db ariation in the axiu sidelobe leel was obsered considering a ariety of different alues for σ and Δσ. This fact is also deonstrated in [8]. Fro the aboe rears and equations 6a-6b together with the frequency noralization by FSR, it is deduced that the statistical behaior of H of conentional AWGs will practically depend only on Δσ, σ and. In the case of flattened AWGs the distribution of C is no longer Gaussian and as a result the statistical behaior of the sidelobe leel ay be different fro that of conentional AWGs. The ean alue of the transittance T H of the AWG, is gien by < T >, l C C exp jπ l < exp j δ δ > a l l Since the phase errors are independent rando ariables <expjπδ δ n ><expjπδ ><expjπδ n > if n and <expjπδ δ n > if n. The phase errors δ are Gaussian rando ariables with < δ > and using the fact that σ <expjδ >exp-σ κ /, <T> can be written as: T C l C C exp jπ l exp l exp σ / exp jπ + C σ + σ + exp σ l C b As seen by the third part of equation b the ean alue of T can be decoposed into two parts: one that is frequency dependent and one that is constant. If the phase errors hae all the sae standard deiation, σ σ, then is reduced to T + ideal exp σ C exp σ T where T ideal is the transittance of the AWG with no phase errors δ. Since the sidelobes of the ideal transittance T ideal of the AWG are practically negligible, it is -9-

10 deduced that outside the ain lobe of the transfer function the ean alue of T is independent of the frequency, C T exp σ 3a This eans that the sidelobes of T hae constant ean alue. In the case where the phase errors do not hae the sae standard deiation, the ean alue of the sidelobes is written C T exp σ 3b since the first su in can again be ignored. This is because it is equal to the transittance T i of an ideal AWG with no phase errors with power distribution equal to C exp-σ /. Since the sidelobes of the ideal transfer function are caused by the discontinuities of the power distribution and C exp-σ / is a sooth function of, it is deduced that the sidelobes of T i are ery low. At the central frequency of the AWG the ean alue of the transittance, assuing σ σ, is gien by T exp σ 4 since in this case the first ter in doinates and T ideal. Equation 4 iplies that the phase errors can cause soe additional insertion losses. For σπ/ these insertion losses are.4db but for σπ/4 they becoe.67db. Since, theoretically, the fiber-to-fiber losses of an ideal router can be ept below.5db with proper waeguide design [3], the insertion losses due to phase errors could hae an iportant contribution to the total insertion losses of the deice. After considerable atheatical anipulation a ery lengthy expression for the standard deiation, stdt of the transittance can also be deried. In Fig. 4a the ean alue and the standard deiation of T with 8 and are plotted assuing that σ π/ and Δσπ/. It is obious that the standard deiation and the ean alue of T --

11 outside the ain lobe, are equal with a high degree of accuracy. This is a general result and suggests that the sidelobes are rando ariables with exponential distribution. In Fig. 4b the CDF i.e. PT x, of the transittance T of an AWG at. solid line, calculated by using coputer siulations and the CDF of an exponential distribution gien by - exp-x/<t>, with ean alue <T.> gien by dashed line are plotted for the case of 8, σ π/ and Δσπ/. Fro the excellent agreeent between the two cures it is deduced that the CDF of the sidelobes of T at a gien frequency can be calculated using an exponential approxiation. This result sees to be general and it is further illustrated in Fig 5a-b where the ean alue and the standard deiation of T at.3 hae been plotted for arious alues of and σ assuing that Δ σ. These quantities hae been calculated using coputer siulations by generating saples of the transfer function in each case with Gaussian distributed phase errors. By coparing Fig 5a and 5b, it is confired that the ean alue and the standard deiation of T.3 can be considered approxiately equal, iplying that the distribution of T.3 will reseble an exponential distribution. This behaior can also be theoretically justified. For sall phase errors, the ters expjδ can be approxiated by +jδ and the transfer function can be written as H C exp jπ + j C δ exp jπ 5 The first su is the transfer function of the AWG without phase errors, which has negligible sidelobes. Consequently the sidelobes of the transittance T will be caused by the second su, T H C δ exp jπ 6 The real part R and the iaginary part Y of H are approxiately gien by --

12 -- sin C R π δ 7 cos C Y π δ 8 Since Y and R, are linear cobinations of the Gaussian rando ariables δ, their joint distribution will also be Gaussian. The ariances of R and Y are gien by cos4 sin C C C R π σ σ π σ 9a + cos4 cos C C C Y π σ σ π σ 9b while their correlation is gien by sin4 sin cos C C Y R π σ π π σ By inspecting 9 it is deduced that the ariances of R and Y consist of a constant ter which is the first su in the right hand side of the equations and a transfer function H d with a sooth power distribution equal to C σ. Unless is near ½ or this transfer function can be neglected copared to the constant ter and thus > < T C Y R σ Using the sae reasoning it can be deduced that Y R Y R << and consequently that if is not near ½ or, R and Y can be regarded as independent Gaussian rando ariables with equal standard deiations. By assuing the transforation cos W T R 3a sin W T Y 3b

13 with W [-π π] and applying the theore of transforation of rando ariables [4], the cobined Probability Density Function PDF f T,W of T and W will be gien by ft, W t, w f R, Y r, y J 4 In the aboe equation f R, Y is the cobined PDF of R and Y and J is the deterinant of the Jacobian atrix of transforation 3. By carrying out the atheatical calculations it can be shown that, since R and Y are approxiately independent Gaussian rando ariables with equal standard deiations, the PDF f T of T is gien by t f T t exp 5 < T > < T > and as a result, the sidelobes of T exhibit the behaior of an exponential distribution. VI. STATISTICAL BEHAVIOUR OF THE AXIU SIDELOBE LEVEL Another characteristic easure of crosstal degradation is the axiu sidelobe leel T ax of the transittance. To analyze its behaior for arious alues of σ, Δσ and, a large nuber of saple transfer functions with rando Gaussian phase errors hae been generated for each case and T ax was easured for each saple. The ean alue μ, the standard deiation s and CDF PT ax x of T ax hae been ealuated based on the results. An interesting result obsered by the siulations is that μ and s in db, can be approxiately written as the su of two functions, one depending on σ and Δσ and the other depending on, that is μ, σ, Δσ μ + μ σ, Δ in db 6a σ s, σ, Δσ s + s σ, Δ in db 6b σ -3-

14 Equations 6a and 6b are deried by inestigating the results of the siulations. The functions μ and s are calculated by aeraging out the functions μ and s respectiely, with respect to Δσ and σ. In Fig. 6a, μ and s are plotted, while in Fig. 6b and 6c the sae is done for the functions μ and s. Using the diagras in Fig. 6, the alues of μ and s can be calculated by applying 6a and 6b for eery AWG once, σ and Δσ are deterined. In order to illustrate the accuracy of 6a and 6b the exact and approxiate alues of μ and s hae been plotted in Fig. 7, for σ π/8, Δσπ/8 and σ π/4, Δσπ/4. It is deduced that the accuracy is satisfactory in all cases and iproes considerably when the ean crosstal is not too large. By inestigating the result of the siulations perfored it was obsered that, the shape of the CDF of T ax, i.e. PT ax x was priarily dependent on the nuber of ars. In particular by using the transforation yx-μ/s the transfored CDF was obsered ainly to depend on and was not strongly influenced by σ and Δ σ, that is P T ax x P y 7 The functions P y, depending only on and y are plotted in Fig. 8, each displaced artificially on the horizontal axis by t. The alue of t is equal to the one tenth of the aerage alue of the ratio between the ean alue of the axiu sidelobe leel and its standard deiation, <μ/s>/.36 and is chosen so that the graphs of Fig. 8 becoe distinguishable. The exact dashed line and approxiate solid line CDFs hae been plotted in Fig. 9a and 9b for σ π/8, Δσπ/8, 8 and σ π/4, Δσπ/4, 5 respectiely. The ean alues μ and standard deiation s were calculated using expressions 6a and 6b and the diagras of Fig. 6. Then the approxiate CDFs were coputed using the functions P 5 and P 8 of Fig. 8 by setting xsy+μ-ts. As before it is obsered that the approxiation is quite accurate when the ean crosstal is not too large. Een in the presence of higher crosstal the approxiation reains satisfactory. -4-

15 VII. SUARY OF THE AWG CROSSTALK EVALUATION PROCEDURE The diagras of Fig 6a,6b and 8 can be used to estiate the crosstal perforance for any AWG gien the design paraeters and the waeguide fabrication tolerances. In particular, these fabrication tolerances, expressed in ters of σ Δp can be used in order to deterine σ neff, by applying and the ERI. For each AWG, the paraeters σ and Δσσ Μ- -σ can be calculated by applying 6. The CDF of the sidelobes will be that of an exponentially distributed rando ariable with ean alue gien by. The ean alue and standard deiation of T ax can also be found using Fig. 6 and eq. 6a and 6b. Finally, the CDF of T ax can be found fro eq. 7 and Fig. 8. This procedure can be applied for the TE and T polarization separately, for eery type of AWG with gien geoetrical characteristics L, ΔL, and by using the sae diagras obtained in this paper. For exaple, in section IV the standard deiation of the effectie index for the TE ode was found to be σ neff.x 6 μ -. Using σ ph.5μ, which is the photoas resolution, n eff,o.5 which is an approxiate alue for the effectie index of the fundaental TE ode, and equation 7 for, we find σ.7rad. For a 6x6 AWG with 65 waeguides and ΔL5μ, applying equation 7 for 64, we hae σ Μ- σ 64.rad and Δσσ Μ- -σ.5rad. The expected axiu sidelobe leel can now be calculated fro the functions μ and μ. Fro Fig. 6a one finds μ Μμ 65.5dB. Using the cure of Fig 6b corresponding to n, μ σ,δσμ.7,.5 μ.7,π/4-5db. Consequently the expected axiu sidelobe leel is μμ +μ -.5dB. With a siilar procedure, the standard deiation of the axiu sidelobe leel turns out to be approxiately s -7.5dB. Finally, the CDFs of the axiu sidelobe leel can be used to further appreciate the ipact of the fabrication tolerances. For exaple, since P 65 y P 7 y8% for y 3.8 and -5-

16 xsy+μ.4-8.5db we expect that 8% of the fabricated deices will hae axiu sidelobe leel below 8.5dB. VIII CONCLUSION In this paper, the relation of the phase errors in the grating ars of the AWG, induced by the fabrication iperfections, and the sidelobe leel of its transfer function was inestigated. A siple nuerical tool was proposed in order to relate the fabrication tolerances of indiidual waeguides with the induced phase errors based on the Effectie Index ethod. Waeguide coupling was studied and it was deduced that it does not cause noticeable correlation between the phase errors in the different waeguides. The sidelobes of the AWG were shown to be exponential rando ariables both by using nuerical siulations and theoretical considerations. The losses introduced on the pea of the transfer function due to the phase errors were also studied. Finally the behaior of the axiu sidelobe leel was analyzed using nuerical siulations and uniersal diagras were presented that allow the estiation of its ean alue, standard deiation and cuulatie distribution function fro the fabrication tolerances and the paraeters of the AWG design. APPENDIX Calculation of the standard deiation of T The standard deiation of T is gien by <T >-<T>. The expectation <T> has been coputed in. The coputation of <T > is uch ore inoled since -6-

17 < T > C C C C < exp j δ + δ δ δ > A, n,, l n where we hae defined C C exp jπ. The expectation <expjδ +δ n -δ -δ l > can be coputed using the fact that l < exp j δ + δ δ δ >< exp jδ >< exp j δ δ δ > Aa n l in the case where n, and l. If the aforeentioned condition does not hold the right hand side of Aa will be different. If, for instance n, and l < exp j δ + δ δ δ >< exp jδ >< exp j δ δ > Ab n l Continuing this way, one can write the express <T > as sus of sus containing ters of the for < exp j q δ + q δ + q δ > where the q, q n and q are integers. n n The ters < exp j q δ + q δ + q δ > can be expressed in a way siilar to A n n and so on, until <T > is expressed as sus of sus containing ters of the for < exp jq δ > exp q σ /. The final expression is ery coplicated and will not be gien here. It should be noted howeer that since, as shown theoretically in section V, the distribution of T is exponential with a high degree of accuracy, its standard deiation should be approxiately equal to the square of its ean alue, that is <T > <T> <T> n n l l l REFERENCES []. Sit and C. an Da PHASAR-Based WD-Deices: Principles, Design and Applications, IEEE J. Select. Topics in Quant. Elect. ol. no. June 996 pp 36-5 [] C. Dragone C.A. Edwards and R.C Kistler Intergrated optics NxN ultiplexer on silicon, IEEE Photon. Technol. Lett. ol. 3, 99, pp

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20 Figure : a Geoetry of a fabricated dielectric waeguide. b Variation of n eff with respect to Δp Δa, Δb, Δh, Δw, AΔn, AΔn, AΔn with A 3 Figure : Variation of the coupling coefficient s deriaties with respect to the center-tocenter distance d of two waeguides. Figure 3: The ariation of σ with respect to for the cases of a 6x6 AWG with a GHz spacing and b GHz spacing assuing σ neff.x -6, σ ph.5μ, L 4 μ and 65. Figure 4: a The ean alue and the standard deiation of the transittance of two AWGs haing 8 and. In both cases σ π/ and Δσπ/. b The CDF of T. for 8 σ π/ and Δσπ/ solid line and the CDF of an exponential distribution with ean alue equal to <T.> dashed line. Figure 5: a The ean alue of the transfer function T at.3 and b the standard deiation of T at.3 for arious alues of the phase error σ and the nuber of waeguides. Figure 6: a The functions μ μ solid line and s s dashed line in db, b the function μ μ σ,δσ in db and c the function s s σ,δσ in db, for Δσ Δσ n nπ/4 where n designates the cures fro the lower n to the higher n. --

21 Figure 7: The alues of μ and s calculated with nuerical siulations dashed line and their alues calculated using the functions μ,μ and s,s using approxiation 6 solid line for σ π/8, Δσπ/8 and σ π/4 and Δσπ/4. Figure 8: The functions P used in the calculation of the CDF of T ax. The leftost cure is for 3, the second left cure for 4, and so on up until the rightost which is for 3. Figure 9: The nuerically calculated CDFs dashed lines of T ax in the case of a σ π/8, Δσπ/8, 8 and b σ π/4, Δσπ/4, 5 and their approxiations solid line using the functions P using 6. --

22 a n h n ERI Layers a w b n b neff Δa or Δb Δh AΔn.495 Δw AΔn AΔn Δp --

23 c/ n a d b c/ w μ -.E+ -.E-4-4.E-4-6.E-4-8.E d -3-

24 a.. b.8.6 σ..9 σ

25 - - a <T> stdt stdt 8 <T> PT. x b x -5-

26 σ π/4-4 σ π/4 <Τ.3> -8-3 stdτ σ π/ -36 σ π/

27 a n μ or s db μ sδσ,σ db c s b μσ,δσ db Δσπ/ Δσπ/ σ Δσ n n σ Δσ n -7-

28 . μπ/4,π/4,μ μ or s db sπ/4,π/4,μ -. μπ/8,π/8,μ sπ/8,π/8,μ

29 a. P y -9-

30 a. PTax x x b..8 PTax x x -3-