IE14;E TRANSACTIONS ON MICROWAVE THEORY AND T1;CHNIQUES, JANUARY 1975 i76 Letters ~~ ~Method for Measuring the Refractive Tliin-Fibn Waveguid&s R. MITTRA AND T. ITOH Index Profile of Absfracf~A method is described for determining the refractive index profile of an optical waveguide. AII optimization algorithm has been employed to obtain the index profile from the measured reflec- tion ~oMi&ent data. Several numerical experiments have been performed to prove the accuracy of the present method. Some of the rbsults are included. An impor~a,nt problem in integrated optical design is the deter- mination of the refractive index profile of an optical waveguide, since the propagation and dispersion characteristics of the guide ~re strongly dependent on the profile function. A method for per- forming this measurement has recently been proposed by Tien et at. [1] and is based upon the use of a prism coupler. This lett6r presents an alternative approach, based on computer processing of the angular scattering coefficient data. obtained by reflecting a collimated Iight beam from a section of the waveguide. The method is applicable to both continuous and discretely stratified profiles. The geometry of the setup k shown in Fig. 1. ~he computer processing of the measured reflection coefficient data proceeds as follows. As a first step, the reflection coefficient as a function of the incident angle 6, of the collimated beam is com- puted for a profile function which is described in terms of a suitable set of unknown parameters. For the multilayer case, shown in Fig. 1 (a), these parameters are taken to be the thickness dl ,2..., loss factor U1,2,..., and the dielectric co fistant cl ,2,... of the various layers. For a normalized incident wave of unit magnitude, the complex reflection coefficient may be readily expressed in terms of these structural parameters and the incident angle 8, as P = (No — yl)/(NO + ~’1)’, No = COS0~/1207r Y~+l + Ar~, tanh -r~d~ Y. = lvm m = 1,2,. ... M N~ + Y,,+l tanh v~,d~’ f% = (27r/~) {em ,– sin’ o, + [(em – sin’ @,)’ + unL’]’/’ ]’j2/@ %7 = (2T/~) ‘ant/ (Z3m) . An estimate of the unknown paramete~s d.,, u~, and .mjm = 1,2,... is now obtained’ by making use of the parameter optimization tdchnique on a digital computer. This requires the minimization of the so-called performance index function F which is defined by ,00(o, ) * measured reflection coefficient, It is evident that F represents the mean-square norm of the devia- tion between the measured angular variation of tbe reflection co- efficient and the couputed values for the trial medium. The above procedure for discrete media can also be applied to the case of continuous profiles with only slight modifications. Although the computation of the reflection coefficient for the trial AIR x ~=, % ,’% ‘3 d, m=2 d2 ● z *C2 (cl) m= !.! . ‘MY ‘M Z!. AIR v z + EY AIR 88 x ————— ————— —-—_—- T (b) ● (z), cnz) ————— d Fig. 1. Structure of optical waveguides and a method of scattering coefficient measurement. TABLE I 1 Layer Case (A = 1.0 Um) Actual Value Computed Value .0 d(um) L c ‘3 d(vm) ,C.se 1 2.5 0 5 2.50031 0 4.9996 case 2 2.5 10-4 5/ 2.5002 0.87 x 10-4 4.9997 TABI,J; H 2 L.y,r C,se (k = 0.6328 W) A,uml WI., ! Computed Value Trial 1 Trial 2 : Lay., c 5( X10-2) d(!lm) c .( X10-2) d(um) c o(k10-2) d(m) 1 2.5 0.1 1.2 2.4996 0.1197 1.2006 2.5004 0.1036 1.200’ 2 2.2 0.1 1000 ~ 2.19.53 0.1126 1000.9 2.1998 0.1199 997.0 medium is slightly more involved for a continuous medium, standard algorithms can still be used td evaluate this quantity for assumed profiles of e(z) and u (z). An optimization algorithm is again used to minimize the performance index F defined in a like manner. The accompanying tables show the. results of computer simula- tion of the experimental scheme outlined above. The measured values of the ,complex reflection PO were simulated from computer evaluation of the reflection coefficient for a given set of e~, c,,,, and d-. The optimization technique was then employed to recover the values of parameters. The results of the numerical experiments are quite favorable as may be seen by reference to Tables I and II. The practical application of the method requires the measure- ment of the magnitude and phase of PO. Although the magnitude of the reflection coefficient is obtainable in a fairly straightforward Manuscript received March 8, 1974; revised May S, 1974. This work was sup orted in part by LT. S Army Research Grant DA-AR O.~-3 1- rnanner, extracting the phase information is usually not easy at c1’ 124-71- 77. The authors are with the Electromagnetic Laboratory, Ele6tricaI optical frequencies. One of the proposed systems of obtaining the Engineering Department, University of Illinois, Urbana, Ill. 61S01. phase information is based on the use of holographic technique as