Design of an Arrayed Waveguide Grating with flat spectral response Thoas Kaalakis, Thoas Sphicopoulos and Diitris Syvridis (Departent of Inforatics and Telecounications, University of Athens, Panepistiiopolis, Athens GR-157 84, Greece) Abstract: Arrayed Waveguide Gratings (AWGs) are key coponents in current and future optical network realizations. In order to prevent the need of accurate wavelength control the transfer function of the AWG should ideally have a rectangular shape. Several techniques have been proposed in order to flatten the Gaussian-like transfer function of the conventional AWG. In this paper we propose a new technique is based on the odification of the arrayed waveguide lengths and their positions on the Free Propagation Regions. The above technique is siilar to the deterinistic tapering technique used in the design of antenna arrays, since the spatial transfer function of the latter has the sae characteristics as the spectral transfer function of an AWG. Therefore, proble is reduced to that of atching the integral of a sinc function with a discrete step function and the optial waveguide lengths are obtained by solving a set of equations nuerically. The perforance of this technique (in ters of transfer function flatness, sidelobe level and insertion losses copared to a conventional AWG) depends on the values given to several initial design paraeters related to the AWG geoetry. The results obtained show that it is feasible to fabricate AWGs with rectangular transfer function with proper adjustent of certain structural paraeters. 1. INTRODUCTION The perforance of odern Wavelength Division Multiplexing [1] optical networks depends greatly on the quality of their individual coponents. A key coponent of these networks in the wavelength ultiplexer and deultiplexer and various solutions have been proposed in order to achieve this functionality []. A proising solution is the Arrayed Waveguide Grating (AWG)[3], which can serve as a ultiplexer, deultiplexer and wavelength router. The fabrication of the conventional AWG does not ipose any severe probles and these devices have been ade coercially available. One drawback of using conventional AWGs in an optical network is that their transfer function is inherently gaussian shape. This results in the need of accurate wavelength control of the LASERs especially when any AWGs are concatenated one after the other in a larger network because the concatenation of two or ore identical gaussian functions results in a narrower gaussian transfer function [4]. This behavior has led any researchers towards the goal of divising a ethod for the flattening of the transfer function of an AWG. The proposed solutions include, the use of an Multi Mode Interference coupler in the input ports of the AWG [5], the use of spatial filtering techniques [6], asyetric Mach Zenhder filters at the output of the AWG [7], a design based on ultiple Rowland Circles [8], and chirping the grating waveguide lengths [9]. In a conventional AWG the lengths of the adjacent waveguide ars is set constant. If we abandon this restriction and allow for non constant adjacent length difference we can gain an extra degree of freedo in the design. In fact, the transfer function then assues a for siilar to the spatial transfer function of an non unifor antenna and techniques used in the design of non unifor antennas [1] can prove of soe use. One of the, the deterinistic tapering technique, can be used in order to ake the transfer function of the AWG ore rectangular, by choosing the waveguide lengths so that the difference between two functions, one that describes the ideal rectangular functions and one its step-wise approxiation with a finite nuber of points becoes as sall as possible [11]. In this paper we will deonstrate how the resultant transfer function can becoe periodic, a property that is iportant when the AWG is used as a wavelength router in a NxN interconnection [1]. We begin by a brief theoretical description of the ethod followed by the obtained results.
. DESCRIPTION OF THE METHOD Following the steps outlined in [11] the optiization procedure described above is reduced to the solution of the following set of equations: A ( t ) = C i 1 t = A ( t ) = C ( 1) i 1 k= ( b t ) a1 exp 1 /. ( b t ) a ( b t ) i a ( 1) exp k i ( 1) exp k + (1) A ( t P ) = C P 1 k= ( 1) ak exp. ap ( 1) exp ( ) ( b t P) b t + k The above syste of equations is to be solved with respect to t which is related to the waveguide lengths L (easured with respect to the length of the central waveguide =) by the relation t = nl / c () where n is the effective index of the fundaental ode of the grating waveguides and c the velocity of light in vacuu. The coefficient C is the power that is intercepted by the central waveguide divided by the total power and the paraeter b is given by b πσ f ch = (3) y out where σ is the standard deviation of the gaussian distribution exp(-y //σ ) that approxiates the fundaental ode of the waveguides, f ch is the channel spacing and y out the spacing between adjacent output and input waveguides (see Figure 1). The ideal function A (t) is defined by A ( t ) t = h( τ )dτ = t f sin c ( f τ) dτ p p (4) where f p is the width of the rectangular transfer function we wish to approxiate. In order to ensure that the transfer function of the AWG will have the sae transfer function for every input / output port cobination the double chriping technique is used which requires that the length of a grating waveguide L is proportional to the position y of the waveguide easured in the output of the first star coupler and the input of the second star coupler fro that of the central waveguide, that is where the paraeter u is given by y = ut (5)
y L y qo FPR waveguide grating FPR input ports Figure 1: An NxN AWG ultiplexer output ports y out u ch = (6) f where R is the radius of the star couplers, f is equal to the central frequency of the AWG. It is also required that the solutions t should be ade proportional to the half of the inverse of the central frequency t cr f n y out = i /( f ) (7) where i is an integer. This ensures that the AWG transfer function H(f), given by can be written as H( F ) P ( j πt f) H( f ) = C exp (8) = C = P P a + ( 1) C cos( = 1 πt F ) (9) with F=f-f and ( i,) a = od (1) In order to derive (9) fro (8) and (7) we have supposed that the power distribution of C, noralized to the total power, of the grating waveguide is syetrical, C =C- and t =-t - which iplies that the distribution of the waveguide lengths easured with respect to the length of the central waveguide is antisyetrical. The integers i are also assued to have a syetrical distribution. Under these assuptions the solutions of (1) iniize the error function
1 E= 4π + H ( F) H F ( F) df = + A( t) A ( t) dt= + A( t) A ( t) dt (11) where H (F) denotes the ideal rectangular transfer function and A(t) the approxiation of A (t) deterined solely by t, i and C 1 F f p / H ( F ) = (1) otherwise A( t ) = C ( 1 ) (13) t t It should be noted that if the fundaental ode of the waveguide is assued gaussian, then the coefficient C can be proved to obey a gaussian law with C a = C exp( a y ) (14) πnσ a= f (15) cr Fro (11) we deduce that the a easure of the error between the function H(F) and H (F) is the square error between A(t) and A (t). An exaple of the functions A(t) and A (t) is plotted in Figure..8.6.4... 1.. 3. 4. t f p Figure : The functions A (t f p ) (drawn with dotted lines) and A(t f p ) (drawn with solid lines). The initial step size is chosen C =.5, the attenuation constant b is equal to.5thz and f p =.16THz.
3. RESULTS As seen by equation (1) there are four factors that deterine the outcoe of the ethod: The fraction of the power intercepted by the central waveguide C, the paraeter b which is related to the channel spacing by (3), the nuber of grating waveguides M=P+1 and finally the choice of the a. It is evident that the choice of a should be such that the A(t f p ) follows A (t f p ) as closely as possible, that is one should select a =, which corresponds to phase shift of a signal at the central frequency f should be an integer ultiple of π or that the length L is proportional to λ =c/f ) when A (t f p ) is increasing and a =1 if A (t f p ) is decreasing (the length L is an odd ultiple of λ/). In figure we have a = in [,1) and a =1 in for t f p >1. In figures 4(a),4(b) and 4(c) the obtained transfer function has been plotted for b=.9 and for three different values of the initial paraeters C. Also shown are the corresponding function A(t) in figure 4(d).. H(f) in db -1. -. -3. H(f) in db -1 - -3-4. 193. 193.4 193.6 193.8 194. -4 193. 193.4 193.6 193.8 194. H(f) in db -1 - -3.6.4. C =.16 C =. C =. -4 193. 193.4 193.6 193.8 194.. 1 3 4 t f p Figure 4: The noralized transfer function of the tapered AWG for b=.9thz and for different values of the initial step C : (a) C =.16 (b) C =. (c) C =.. (d) A(t f p ) (solid lines) and A (t f p ) (dashed line) for the three cases. Figure 4 indicates that the shape of the transfer function is intiately related to the choice of the initial paraeters. In figure 4(a) the paraeters are chosen so that the function A(t) follows A (t) only in [,1) in which A (t) is increasing. This results in a Gaussian shaped transfer function very siilar to that of a conventional AWG (plotted with bolder dashed lines in the figure). The transfer function becoes ore rectangular as C and reaches the value of C =. because the ideal function is better approxiated near this value. As C further increases the transfer function looses its flatness and a ripple appears in its passband. The lighter dashed lines in figures 4(a)-4(c) correspond to a
AWG with C =.14 that has a = for every and is uch siilar to that of figure 4(a). The nuber of waveguides is equal to M=P+1=81 for all designs. The sidelobe level of the transfer function outside the ain lobe is also of uch interest. If the sidelobe level is high then the channel isolation is poor resulting in large out-of-band and in-band crosstalk. To exaine the sidelobe level the transfer functions of the three designs has been plotted in figure 5. It is deduced that the transfer functions have sidelobes below db. Further iproveents in the sidelobe level can be ade possible by oitting soe of the grating waveguides as discussed in [11]... -1. -1. H(f) (db) -. H(f) (db) -. -3. -3. -4. 191 19 193 194 195 196 (a). -4. 191 19 193 194 195 196-1. H(f) (db) -. -3. -4. 191 19 193 194 195 196 Figure 5: The noralized transfer function of the tapered AWG for b=.9thz and for (a) C =.16 (b) C =. (c) C =. plotted in a wider frequency range. In figure 6 the solutions t f p have been plotted for C =. and b=.9thz. Since by equation (5) the positions of the waveguides on the input of the first star coupler it is deduced that the waveguides are spaced closely together near the central waveguide and that there is a gap after waveguides on the right and the left of the central waveguide. (c) 4. PERIODICITY OF THE TRANSFER FUNCTION
One very interesting application of the AWG is as a passive wavelength router. In order to reduce the nuber of wavelengths required in an NxN interconnection, the periodicity of the transfer function of the conventional AWG is used and the nuber of wavelengths required becoes equal to N [ 1]. However, the transfer function obtained with this technique is non periodic due to the fact that the solutions t are ultiples of the half of the central wavelength while in a conventional AWG the t are ultiples of the inverse of the Free Spectral Range (FSR) that the router is specified to have, that is t = (16) FSR This fact suggests that the flattened transfer function can becoe periodic if we odify solutions t obtained by solving the syste of equations (1) so that they becoe integer ultiples of the inverse of the desired FSR, that is [ t FSR] t = (17) FSR where [.] denotes rounding to the closest integer. The resultant transfer function appears in figure 7 for C =., b=.15thz, M=11 and FSR=4THz. Because of the odification of the optiu solutions the sidelobes have risen and their axiu level has reached 16dB. An iproveent of the sidelobe level could be obtained by reoving soe of the waveguides whose ters in (9) as in the case of the non periodic transfer function in [11]. 1.6 1. t fp.8.4. 1 3 4 5 Figure 6: The variation of t with respect to 5. CONCLUSION We have analyzed a new technique for the design of an AWG with flat spectral response which is based on the odification of the lengths and the positions of the waveguides of a conventional AWG ultiplexer. The optiu lengths are deterined by an optiization procedure siilar to that of the deterinistic tapering technique used in the design of non unifor antenna arrays. The resultant transfer function is flat with sidelobes below db and can be ade periodic like in the case of the conventional AWG router.
. H(f) in (db) -1. -. -3. 188 19 19 194 196 f Figure 7: A periodic flattened AWG router 6. REFERENCES [1] C.A. Brackett, Dense wavelength division ultiplexing networks: Principles and applications, IEEE J. Select. Areas Coun., vol. 8, pp 948-964 June 199. [] Rajiv Raaswai and Kuar N. Sivarajan Optical Networks A Practical Perspective Morgan Kaufann Publishers 1998 [3] C. Dragone An NxN Optical Multiplexer Using a Planar Arrangeent of Two Star Couplers, IEEE Photon. Technol. Lett. vol. 3 pp 81-815 no. 9 Septeber 1991 [4] M.K. Sit and C van Da PHASAR Based WDM Systes: Principles Design and Applications IEEE Jour. Sel. Topics. Quantu Elec. Vol No June 1996 pp 36-5. [5] M.R. Aersfoort, J.B.D. Soole, H.P. LeBlance, N.C. Andreadakis, A. Rajhel and C. Caneau Passband Broadening of integrated arrayed waveguide filters using ultiode interference couplers. Electron. Lett., vol. 3 pp 449-451, 1996 [6] C. Dragone, T. Strasser, G.A. Bogert, L.W. Stulz and P. Chou Waveguide grating router with axially flat channel passband produced by spatial filtering Electron. Letters, vol. 33, no. 15 pp. 131-1314 July 1997. [7] K. Okaoto K. Tagikuchi adn Y. Ohori, Eight-Channel Flat Spectral Response Arrayed Waveguide Multiplexer with Asyetrical Mach-Zehnder Filters IEEE Photonics Tech. Lett. Vol. 8 No. 3 March 1996 pp 373. [8] Y.P. Ho, H. Li and Y. Chen, Flat channel passband wavelength ultiplexing and deultiplexing devices by ultiple Rowland circle design, IEEE Photon. Technol. Lett. vol. 9 pp 34-344, Mar. 1997 [9] M.C. Parker and S.D. Walker Design of Arrayed Waveguide Gratings Using Hybrid Fourier-Fresnel Transfo Techniques, IEEE J. Select. Areas Quantu. Electron., vol. 5, no. 5, pp 1379-1384, Septeber/October 1999. [1] R.E. Collin and F.J. Zucker, Antenna Theory Part I, pp 1-19, McGraw Hill 1969 [11] T. Kaalakis and T. Sphicopoulos An Efficient Technique for the Design of an Arrayed Waveguide Grating with Flat Spectral Response Vol 19 No 11 Noveber 1 pp 1716-175