Exact ray paths in bent waveguides

Exact ray paths in bent waveguides Ajoy Ghatak, Enakshi Sharma, and Jacintha Kompella We present here the Lagrangian formalism for studying the ray paths in cylindrically symmetric media. We have used the analysis to obtain the exact ray paths in bent slabs as well as in bent fibers with a separable profile. 1. Introduction II. The study of ray paths in graded-index multimode media is a subject of considerable interest in the area of fiber optics and graded-index imaging systems.'-' 2 When rays in multimode waveguides encounter bends there are radiation losses; these losses are either by refraction or by tunneling. The fractional loss of power when a ray is reflected from an outer caustic along the ray path is usually calculated by using the WKB method.13"14 Hence it is essential to know the exact ray paths in bent waveguides and thereby know the exact positions of the ray caustics to calculate the bend loss. A number of ray tracing algorithms for inhomogeneous media have been proposed. 3 - 8 However, all these methods are numerical in nature. In this paper, we present an analysis to find ray paths in cylindrically symmetric graded-index media and also in bent waveguides. We use the Lagrangian formulation which makes the analysis simpler and neater than the use of the ray equation d /n dr=Vn dsn ds- Starting from Fermat's extremum principle for ray paths we can define the optical Lagrangian, which when substituted in Lagrange's equations of motion gives the ray equation. This analysis has until now been carried out only in a Cartesian system of coordinates (see, e.g., Refs. 15 and 16). In this section, we define the optical Lagrangian in cylindrical system of coordinates and show that for systems with cylindrical symmetry this approach gives the ray invariants and ray paths in a very straightforward manner. In the cylindrical system of coordinates (r,k,z), the arc length ds along the path of the ray can be written in the form ds = [1 + (r) 2 + (z)2] 1/2dr, (3) where do dr . dz dr (4) Thus Eq. (2) can be written in the form 5 f Ldr = 0, As an application of this method, we have illustrated its use on bent graded-index optical waveguides. (2) f nds =O, (1) in the cylindrical system of coordinates. The equations derived for the ray trajectory have been put in a form convenient for use in numerical calculations. Optical Lagrangian in Cylindrical Coordinates where L = n(r,o,z)[I + (r0)2 + ()2]1/2 (5) represents the optical Lagrangian in the cylindrical system of coordinates. In the following sections, we use the Lagrangian given by Eq. (5) to determine the ray paths for specific types of refractive-index profile which are of considerable practical importance. Enakshi Sharma iswith Delhi University, South Campus, Department of Electronic Science, New Delhi 110021, India; the other authors are with Indian Institute of Technology, Delhi, Physics Department, New Delhi 110016, India. Received 11 December 1987. 0003-6935/88/153180-05$02.00/0. i©1988 Optical Society of America. 3180 APPLIEDOPTICS / Vol. 27, No. 15 / 1 August 1988 Ill. Ray Paths in Cylindrically Symmetric (z-lndependent) Refractive-Index Profiles We assume that the refractive index depends only on the r-coordinate, i.e., n = n(r). (6) This is indeed the case for a straight optical fiber. Even for a bent slab waveguide, the profile is essentially of the form given by Eq. (6), provided we choose our coordinate system appropriately (see Sec. V). Thus 2 L = n(r)[1 + (rk) + ()211/2 circle, we will have a profile of the type given by Eq. (11). Furthermore, it has been shown that a separable profile given by Eq. (12) can describe a large class of optical waveguides (including rectangular core integrated optical waveguides and elliptical core fibers) to and the Lagrange's equations d FaLl ] dr where g = 2A(n0/a2 ), and a represents the radius of the core. If the axis of such a fiber is bent along the arc of a a considerable degree of accuracy.' 7 -' 9 L For n2 (r,z) given by Eq. (11), the Lagrange's equation d az d (OL\_ L0 d (OLA aL ivat dro dr ai / z readily give the following invariants: dL dz - 0 ds 0; . ds (8) '.ds) p2 + 22)1/2] Eqs. (7) and (8) we obtain - 2 r dz [G (10) It should be pointed out that 3 and Tareidentical to the ray invariants as obtained in Ref. 9. However, it is obvious that the derivation of [Eq. (8)] and of the ray equation [Eq. (9)] is very much simplified G2(z) G --n(z) n2- -n2(z)] In Sec. V, we show that the profile given by Eq. (11) is applicable to bent waveguides with a separable profile; i.e., the (unbent) waveguide is characterized by a d n(r)r 2Ido] L n(r z)r2 d= ds Iy. = R-= n(r,z) 2 . (14) 0 (15) (16) d Now, using Eq. (14), [2+ (z)] 1= n2(r,z) (dz)2 ds = (dz d) d-Ods 2 . Substituting for dk/ds from Eq. (16) in the above equation we obtain (12) Obviously, a parabolic-index fiber has a separable profile: 22 =nogx -gy2, = _ n2(z) n The equation defining the exact ray paths can now be obtained. We introduce a new invariant / as refractive-index distribution of the form n2(r) = n2[1- 2A(r/a)2 ] 2 n I a ds a and we get the second ray invariant: [12 n'2 (x) + n" 2 (y). I as dinate gives us exact ray paths. = 0. The Lagrange's equation corresponding to the o-coor- (11) In this section, we show that for a separable profile given by the above equation, it is possible to derive equations, simple integrations of which would give the n2 (Xy) (13) = dZ = [[n(r~z)dsj Ray Invariants and Ray Equations for Profiles of the Formn2 (rz) = nd(r)+ nt(z) We next consider a special class of cylindrically symmetric refractive-index profiles which can be put in a separable form, i.e., n2(rz) = n'(r) + n2(z). dn'(z) dz in the present formalism when compared with their derivation from the ray equation [i.e., from Eq. (1)]. IV. ±2)1/2 the above equation gives Thus we obtain a ray invariant 2 k + 2n(r,z) (9) where f(r) = n2 (r) _ 2 2 G = n(r,z) dz, ds Substituting for the terms on the right-hand side from dr = -~ 1 (r)1'12 , dz +r Defining _ ( d )2]/ ] [1 -(r ds,) dsH [ l dz [n(rz) [ dzl dz ds== dn2(Z) 2 n(rz)I dsjdz Now Eq. (3) can be written as ds ds L1+ r 2 which can be written in the form The quantities Band Tare the two invariants of the ray path. n d[ dr dL __ 2 da. = nr d= d4 gives us (7) dz r2) d~i /3R (17) where F(z) = [n(z) + 12] (18) and r(o) is determined from Eq. (20). Now, using Eq. (16) we may write 1 August 1988 / Vol. 27, No. 15 / APPLIEDOPTICS 3181 dr\2 + 2 + dz 2 rn(rz)r22I ds 2 L ,3R J d¢ \dd¢bJ (19) f= a Substituting for dz/d5 from Eq. (17) we get dr do r [n2(r)- t1Ir2 1/2 /31 J R ?= r R (20) ?= -a Equations (17) and (20) represent (for a separable profile) the rigorously correct ray equations which can be used to determine the ray paths in bent waveguides. V. . Fig. 1. Ray Paths In a Bent Slab Waveguide Slab waveguide bent along the arc of a circle of radius R. We consider a slab waveguide characterized by the followingrefractive-index distribution: n2 =n 2 (X) We assume such a waveguide to be bent along the arc of a circle of radius R as shown in Fig. 1. In the cylindrical system of coordinates defined in Fig. 1 we have 1 3 I n2= n2 (r) = n 2 (r). We can use the analysis of Sec. IV with n2(z) = 0 to I . obtain the ray invariants -S T= [n(r) dz] s (21) n(r) R do, (22) dsJ' 1 ds 0.04/ and the ray equations r2 , dz (P.) (23) 0.06 dr r [[n2(r) - 12r _2 db /3L R~ 1/2 2](4 (24) Introducing the variables p = r - R and t = Rk, we get dz = dp = p1' ) F 1 (1 + P) (P)]1/2 (26) P) (27) where f(p) = [n2 (p) - a](1 + - ,B2 From the initial launch conditions we can determine the values of the ray invariants A and Tand solve Eqs. (25) and (26) explicitly to obtain the ray paths p = p() and z = z). We consider now the case of a parabolic 2 index bent slab waveguide, i.e., n = n2 (p) = - gp . It is easily seen from Eqs. (25) and (26) that when e=0 the ray path is confined to a single plane, and the zcoordinate of the ray remains constant. We first consider small launch angles and refer to Fig. 2. Variation of f(p) with p for a bent parabolic-index slab waveguide corresponding to F= 0. Curves 1, 2, and 3 correspond to initial on-axis launch angles equal to 0.01 ( = 2.204), 0.10 ( = 2.182), and 0.16 ( = 2.148), respectively. The values of various parameters are R = 2500 um, A = 0.02, n 2= 2.204, a = 25jum. positive for all values of p > p1, and we have what are known as refracting rays which leak out very quickly from the core. It is obvious that for a bent structure there are no bound rays, and all rays launched are either tunneling or refracting. The exact ray paths can be obtained from Eqs. (25) and (26). However, the sign in front of the square root is chosen according to the slope of the ray path and is hence difficult to incorporate into a numerical algorithm. This problem can be overcome by squaring and differentiating the two first-order simultaneous differential equations to obtain two second-order simultaneous differential equations given by d2P2 [n2(P) -12 curve 2 in Fig. 2. We see that f(p) is positive in the regions P2 < p < 3 and for p > p4 and negative in the region 3 < p < 4. This is the case of what is known as tunneling rays which undergo partial reflection at p = p3 and part of the energy tunnels through the evanescent region p3 < p < 4 to leak away beyond p = 4. If the launch angle is sufficiently large, we have a situation similar to the one shown in curve 3 where f(p) is 3182 APPLIEDOPTICS / Vol. 27, No. 15 / 1 August 1988 3 + 1 (1+R) dp -(1 + P) 1 ' d2Z (dz) (dp) ( + p)- 1 d~2 \d~ + 1 d~, R/ (/ +)4 d12(Z) ,~4d-- 21+ (28) R (29) Core cladding 20 P (Pm) ,4/ interface 1 2 . I x 10 0 200 / !o 12( 800 100 m =320 m =440,um = 640pm ( = R0 )In m C Fig. 3. Meridional rays in a bent parabolic-index slab waveguide for different radii of curvatures. Curves 1, 2, 3, 4, and 5 correspond to R equal to 1500, 2000, 2500, 5000, and 25,000 Am, respectively. The launch angle is equal to 0.1 rad. Slab parameters aren = 2.204, A = 0.02, a = 25 Am. z I Pm) -HELICAL RAY (Rzc) -Core claddinginterface Fig. 4. Meridional rays in a bent parabolic-index slab waveguide for different launch angles. Curves 1, 2, 3, 4, and 5 correspond to launch angles 0 = 0.16, 0.12, 0.10, 0.06, and 0.01 rad, respectively. The bend radius is R = 2500,um. which can be solved using Runge-Kutta's Fig. 5. Skew rays in a bent parabolic-index fiber for varying bend radius R as seen in the fiber cross section. Curves 1, 2, 3, and 4 correspond to radius of curvature R equal to 2500 Am, 5000 Am, 12500 Mm, and I, respectively. The initial angle made with the z axis is 1.45 rad. method for simultaneous second-order differential equations. 20 The last term in Eq. (29) is actually absent in a bent slab waveguide; it appears for a bent fiber with a separable profile (see Sec. VI). Meridional rays (for which I = 0) are confined to the plane of the bend and are shown in Figs. 3 and 4. The rays are asymmetric about the axis of the waveguide, and as the radius of curvature increases this asymmetry decreases. For R the ray paths converge to a sinusoidal path of a straight waveguide. VI. We now consider a fiber whose refractive-index profile is described by Eq. (12). We bend this fiber along the arc of a circle so that the y axis of the straight fiber is parallel to the z axis of the coordinate system described in Fig. 1. Now in this coordinate system, the + n2(z). 2(r) fiber profile can be expressed as n2 (rz) = n Using the analysis in Sec. IV, we have two invariants given by Eqs. (14) and (16). Consequently, the ray path can be described rigorously by Eqs. (17) and (20). Once again we have = S + P) Fig.6. 2 m 2 Plots of the ray position .p + z inside the core as a function of distance along the parabolic-index bent fiber axis. These correspond to the plots shown in Fig. 5. tions of the form given by Eqs. (28) and (29) where dz/ dt is now given by Eq. (30). Ray Paths in a Bent Fiber dz -m.(=R4)in (Z)]", (30) where f(z) = n2(z)+ 2 (31) and dp/dt is given by Eq. (26). The exact ray paths can be obtained by solving the two simultaneous first- order differential equations given above. As before, it was found convenient to convert the above equations to two simultaneous second-order differential equa- Figure 5 is a plot of the projection of the ray path on a cross section for skew rays obtained by launch conditions which correspond to a helical ray in a straight fiber.' 6 Figure 6 shows the corresponding variation in total distance from the waveguide axis. As the curvature increases, the rays become more and more skew. This can be explained as being due to the asymmetry induced by the bend. This work got started when two of the authors (A.G. and E.S.) were visiting Gerhard Grau's Institut fur Hochfrequenztechnik and Quantenelektronik at Universitat Karlsruhe (F.R. Germany) during the summer of 1987. The hospitality of Gerhard Grau and Wolfgang Freude is gratefully acknowledged. The authors would also like to acknowledge stimulating discussions with Anurag Sharma. The work was partially supported by a research project sponsored by the National Bureau of Standards (U.S.A.) and the Department of Science and Technology (India). One of the authors (J.K.) would like to acknowledge a research fellowship from the Council of Scientific and Industrial Research. 1 August 1988 / Vol. 27, No. 15 / APPLIEDOPTICS 3183 References 1. A. Gupta, K. Thyagarajan, I. C. Goyal, and A. K. Ghatak, "Theory of Fifth-Order Aberrations of Graded Index Media," J. Opt. Soc. Am. 66, 1320 (1976). 2. A. Rohra, K. Thyagarajan, and A. K. Ghatak, "Aberrations in Curved Graded Index Media," J. Opt. Soc. Am. 69, 300 (1979). 3. A. Sharma, D. V. Kumar, and A. K. Ghatak, "Ray Tracing through Graded-Index Media: a New Method," Appl. Opt. 21, 984 (1984). 11. E. W. Marchand, Gradient Index Optics (Academic, New York, 1978). 4. A.Sharma, "Computing Optical Path Length in Gradient-Index Theory of Optics (U. California, Media: (1985). a Fast and Accurate Method," Appl. Opt. 24, 4367 12. M. J. Nadeau, "Image Analysis of Curved Gradient-Index Rods," M. S. Thesis, Institute of Optics, U. Rochester (1984). 13. A. W.Snyder and J. D.Love, Optical Waveguide Theory (Chapman & Hall, London, 1983). 14. C. Winkler, J. D. Love, and A. K. Ghatak, "Loss Calculations in Bent Multimode Optical Waveguides," Opt. Quantum Electron. 11, 173 (1979). 15. R. K. Luneburg, Mathematical Berkeley, 1964). 16. A. K. Ghatak and K. Thyagarajan, Contemporary Optics (Ple- 5. A. Sharma and A. K. Ghatak, "Ray Tracing in Gradient-Index Lenses: Computation of Ray-Surface Intersection," Appl. Opt. num, New York, 1978). 17. K. Thyagarajan, M. R. Shenoy, and A. K. Ghatak, "Accurate 25, 3409 (1986). 6. L. Montagnino, "Ray Tracing in Inhomogeneous Media," J. Opt. Soc. Am. 58, 1667 (1968). 7. W. H. Southwell, "Ray Tracing in Gradient-Index Media," J. Opt. Soc. Am. 72, 908 (1982). 8. D. T. Moore, "Ray Tracing in Gradient-Index Media," J. Opt. Soc. Am. 65, 451 (1975). cal Waveguides Using a Matrix Approach," Opt. Lett. 12, 296 (1987). 18. A. Kumar, K. Thyagarajan, and A. K. Ghatak, "Analysis of Rectangular Core Dielectric Waveguides: An Accurate Perturbation Approach," Opt. Lett. 8, 63 (1983). 19. A. Kumar, R. K. Varshney, and K. Thyagarajan, "Birefringence 9. A.Ankiewiczand C. Pask, "Geometric Optics Approach to Light Acceptance and Propagation in Graded Index Fibers," Opt. Quantum Electron. 9, 87 (1977). Numerical Method for the Calculation of Bending Loss in Opti- Calculations in Elliptical Core Optical Fibers," Electron. Lett. 20, 112 (1984). 20. J. B. Scarborough, Numerical Mathematical Analysis (Oxford and IBH Publishing, Co.,London, 1966). 10. D. Bertilone and C. Pask, "Exact Ray Paths in a Graded-Index Taper," Appl. Opt. 26, 1189 (1987). NASAcontinuedfrompage3169 ,?u =,: sNcu~s1NcsQu, where NCUand Nc, are the counts in the correlation peaks of the unknown and standard detectors, respectively,and Qsand Quare the solid angles subtended by the standard and unknown detectors, respectively. In the second method (see Fig. 4), filters are placed in front of the detectors to pass only the 584-A photons. Since the calibration of the standard detector is good for all 584-A photons, it suffices to count all the photon pulses Nu and N, put out by the unknown and standard detectors, respectively, without having to measure electrons or correlations. Then the efficiency of the unknown detector is given by flu= lsNuJs/Nsfu. This work was done by Santosh K. Srivastava of Caltech for NASA'sJet Propulsion Laboratory. Inquiries concerningrights for the commercial use of this invention should be addressed to the Patent Counsel, NASA Resident Office-JPL, P.F. McCaul, Mail Code 180-801, 4800 Oak Grove Dr., Pasadena, CA 91109. Refer to NPO-15644. Fig. 4. Standard detector is used to calibrate an unknown detector. In this case,the photons are bandpass-filtered, and the coincidencecounting circuitry is not used. General-purpose image-data program The image database computer program IBASE is a general-purpose imagery-information system. The system includes functions The probability p that one of the 584-Aphotons generated by one of the detected 78.8-eV electrons will scatter into the detector in plished either by commands or through a hierarchy of menus. The analytical capabilities of IBASE include contingency tables, image filtering (low-pass,high-pass, and bandpass), proximity maps, clus- - - question is calculated based in part on the known solid angle sub- tended by the detector from the point of view of the beam intersection. Then the efficiency X of the detector is given by for cataloging and managing files of digital images and for conducting analyses of these images. Use of the system may be accom- tering, histograms, regression, slope calculations, scaling, and Bool- ean manipulations. IBASE also has an interface to the Cheshire If the efficiency Is of a detector (the standard detector) has already been measured, it can be used to measure the efficiency fu of Image Classification expert system. All information stored by IBASE is considered part of a GeoCube. A GeoCube is a rectangular region defined by the coordinates of two opposite corners and can store many images called layers. Each an unknown detector in two ways. In the first, the measurement is made on the calibrating apparatus without recording the electron An 7 = NC/pNe. count. 3184 The efficiency of the unknown detector is given by APPLIEDOPTICS / Vol. 27, No. 15 / 1 August 1988 layer contains information such as the name, date, and layer type. IBASE layer can contain a report or data: as used here, report continuedonpage3202