Reconstruction of optical characteristicsof waveguide lenses by the use of ray tracing

Gleb Beliakov

A method that uses the data of ray tracing for optical waveguide lens diagnostics is described. Thismethod permits a direct reconstruction of the optical characteristics of a waveguide without the optical orthe physical thickness being measured. Conditions are determined for the mathematical problem ofdiagnostics by ray tracing to have a unique solution, and a technique to obtain a numerical solution fromnoisy experimental data is described.

Key words: Optical waveguide, waveguide lens, waveguide diagnostics, ray tracing.

Introduction

A thin-film optical waveguide lens (WGL) is one of themost important optoelectronic devices. One pur-pose of the WGL is to collimate a parallel beam oflight distributed along the waveguide. This effectcan be achieved by the production of a waveguidelayer with a special thickness distribution as well asby other techniques.1-3 The focal properties of theWGL depend considerably on the correspondence ofthe produced waveguide layer (the refractive-indexdistribution) to its theoretical model and on thedevelopment of a nondestructive method of precisecontrol for the focusing properties of the WGL (so-called diagnostics) represents an important technologi-cal problem. The well-known methods of diagnos-tics, such as interferometry,4'5 cannot directly measurethe optical characteristics of the waveguide and intro-duce an additional error that is due to the simplifiedwaveguide model.

In this paper I describe a new method of WGLdiagnostics based on the analysis of traces of thinlaser beams that pass through a sample along thewaveguide. The traces of the rays are visible fromabove, because of scattering, and are easily registered.Such a process is called ray tracing.

If the waveguide layer is homogeneous there is astraight distribution of the light along the waveguide.

When this work was performed, the author was with the RussianPeoples' Friendship University, Moscow, Russia; he is now withthe Departamento de Matematicas, Universidad de Los Andes, A.A. 17308, Santafe de Bogota DE, Colombia.

Received 7 June 1993; revised manuscript received 8 November1993.

0003-6935/94/163401-04$06.00/0.© 1994 Optical Society of America.

If the waveguide has a local inhomogeneous part therays bend in the direction of maximum refractive-index gradient. In the process of diagnostics a set ofrays is forced to pass through a sample in such a waythat the ray traces cover the area where the refractiveindex is to be found. The positions of the points thatform a trace are registered, and then the refractive-index distribution can be calculated by use of thedescribed mathematical methods.

One of the advantages of the proposed method isthat it permits a direct computation of the refractive-index distribution, and thus it avoids introducing anadditional model error in transforming the measuredthickness of the waveguide layer (physical or optical)into the optical characteristics of the WGL. Fur-ther, the precision of our method is higher thanformer methods.

Mathematical Model

The mathematical model of a planar waveguide is aplanar optical medium (Fig. 1) with a given distribu-tion of normalized effective refractive index n(x, y).The ray equation can be written in the followingform6 :

d drds n d = Vn.

(Here ds is an element of geometric path length.)When the ray traces y = y(x) can be parameterizedunambiguously by x, this becomes

_ dy w + w _ (d2y/dx 2 )

dx ax ay 1 + (dy/dx) 2 (1)

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that passes through G there is one and only one pointwith a given value n(x, y).

The solution of Problem 1 can be obtained by thesolution of a Cauchy problem for a system of ordinarydifferential equations of the form

dxy = C(x, y),

Fig. 1. Ray traces in an optical WGL.

where w(x, y) = ln[n(x, y)]. The distributions ofnormalized effective refractive index n(x, y) (which isreferred to as the refractive index hereafter) thatprovide the waveguide with focusing properties werediscussed in Refs. 7-11.

Let us examine a plane rectangular area of thewaveguide G that entirely contains the inhomoge-neous part Q that performs the focusing effect (Fig.1). The complement Q\G, where n -- 1, is called theregular part of the waveguide. We describe thetraces of the rays used to perform the diagnostics bymeans of a family of functions,

{y(x) = Y(x, h), y- < h y+},

parameterized by h. It is convenient to select h to bethe ordinate of the entry point (x_, y(x_)) of each rayh = y(x-).

Next we formulate the mathematical problem ofdiagnostics by ray tracing.

Problem 1. Find the unknown distribution of therefractive index n(x, y) in G such that Eq. (1) holds forevery ray y(x) = Y(x, h) in set (2).

To determine w(x, y) = ln[n(x, y)], Eq. (1) must berewritten in the form

aw wCQx' Y) ) + = F(x, y), (3)

where

C(x, y) = -Yx(x, h) I h=h(x,y))

F(x, y) = Y.(x, h) . (4)+ Yx2(X, h) h=h(x,y)

If function Y(x, h) satisfies the following conditions,(1) Y(x, h) has a continuous second derivative Yx.(x, h),(2) Y(x, h) has a continuous first derivative Yh(x, h) andYh(x, h) • 0 in G, and (3) Y(x, h) has a continuoussecond derivative Yxh(X y), then the following theo-rem holds.

Theorem 1. The solution of Problem 1 exists andis unique when the boundary conditions are given insuch a way that for every characteristic line of Eq. (3)

dw= F(x, y),

with initial conditions

xly=y_ =xO,

(5)

Wy=y = 0.

The first equation of system (5) determines thecharacteristic line of Eq. (3) that passes through thepoint (x0, y-), and the second equation determinesthe logarithm of the refractive index along this line.Because the characteristic lines are orthogonal to therays, they coincide with the wave fronts.

Numerical Solution

Given C(x, y) and F(x, y), we can find the numericalsolution of system (5) by applying one of the standardintegration schemes (such as the Runge-Kuttamethod). The difficulty arises when functions C(x, y)and F(x, y) are determined based on the experimentaldata: the digitized noisy representation of set (2).It follows from Eqs. (4) that C(x, y) and F(x, y) have tobe found through a solution of an ill-posed problem ofnumerical differentiation. The well-known methodsof numerical differentiation, such as a method ofsmoothing spline functions, give satisfactory resultsin the approximation of C(x, y) but fail to determineF(x, y), which is important in the calculation ofw(x, y).

To determine the possible cause of this trouble andto find a way around it, we examined the fine struc-ture of the functions y = y(x) from set (2). It turnsout that a specific structure of the refractive-indexdistribution (shown in Fig. 2) leads to a specific formof the second derivative y"(x) (Fig. 3). The standardmethods of numerical differentiation cannot capture

n

Fig. 2. Refractive-index distribution of a Luneburg lens (focaldistance 2, aperturo 0.9).

3402 APPLIED OPTICS / Vol. 33, No. 16 / 1 June 1994

it

y

1.16

X

a b c d e f Fig. 3. Second derivative of one of the functionsy = y(x) from theset Y(x, h). The dashed curve represents the second derivative ofan approximating function (without constraints).

such a structure, and they give approximately theresult shown as the dashed curve.

To introduce the expected form of y"(x) into theprocess of numerical differentiation, we propose touse a recently developed method12 of constrainedsmoothing splines with two side restrictions on thesecond derivative. Such a spline is a unique solutions(x) of the problem of minimization:

min F(4,) = p f q (x)2dxX1

N

+ z qij[4(xi) - yi][q.(xj) - yj]ij=1

over

C = e W 22[X1, XN], 81(X) < iY(x) < 82 (X),

X C X1, XN]}1

where yi are noisy observations of an unknownfunction y(x) at points xi, 81(x) and 2(x) are givenfunctions,p > 0 is a regularization parameter, and qijare the elements of an N x N positive definiteweighting matrix.

To preserve the structure of the second derivative,we introduced the constraints

61(4) =

62W = E > 0

82 = > 0

at (a, f),

at (a, b) U (e, f),

at (c, d),

which ensure that the ray traces are straight in theregular part of the waveguide and close to beingstraight at the center of the lens.

By using this method, we were able to computeF(x, y) with high precision, even from data thatcontained high random noise (y 0.05, normalizedto the lens radius). The method has been exten-sively tested by numerical simulation of ray tracingfor a variety of parameter values (focal length 2-8,

-1 0 1Fig. 4. Comparison between the theoretical refractive-index pro-file of a generalized Luneburg lens (solid curve) and a profilereconstructed from simulated noisy data of ray tracing (dashedcurve). Focal distance 2, aperture 0.9, noise level in the input data8y = 0.05; all values are normalized to the lens radius. Averageerror of reconstruction 8n/n = 0.01.

aperture 0.99-0.8, number of rays 20-100, noise level0-0.05), and it gives stable results. In Fig. 4 thecomparison between the theoretical refractive-indexprofile and the profile reconstructed from numeri-cally simulated noisy data is shown.

Conclusion

The developed method of WGL diagnostics is usefulequally for circular lenses, elliptical lenses, and wave-guides without symmetry. It can be applied to non-destructive diagnostics of gradient-, geodesic-, andLuneburg-type lenses and not only gives the answer,whether the produced sample is good or not, but alsoprecisely determines the extent to which it does notcorrespond to the theoretical model. After the refrac-tive index has been reconstructed, the aberrationsand other important optical characteristics of theWGL can be easily calculated through numericalsimulation of ray tracing.

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mum focal length control," Appl. Opt. 20, 3589-3593 (1981).2. W. H. Southwell, "Inhomogeneous optical waveguide lens

analysis," J. Opt. Soc. Am. 67, 1004-1009 (1977).3. W. H. Southwell, "Geodesic optical waveguide lens analysis,"

J. Opt. Soc. Am. 67, 1293-1299 (1977).4. L. Goldberg, "Interferometric method for measuring diffused

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5. P. Hariharan, B. F. Oreb, and Z. Wanzhi, "Measurement ofaspheric surfaces using a microcomputer-controlled digitalradial-shear interferometer," Opt. Acta 31, 989-999 (1984).

6. W. H. Southwell, "Ray tracing in gradient-index media," J.Opt. Soc. Am. 72, 908-911 (1982).

7. S. Doric and E. Munro, "General solution of the non-full-aperture Luneburg lens problem," J. Opt. Soc. Am. 72,1083-1086 (1983).

8. A. L. Mikaelian, "Self-focusing media with variable index ofrefraction," Prog. Opt. 17, 283-345 (1980).

9. J. Sochacki and C. Gomez-Reino, "Nonfull-aperture Luneburglenses: a novel solution," Appl. Opt. 24, 1371-1373 (1985).

10. J. Sochacki, D. Rogus, and C. Gomez-Reino, "Paraxial design

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ing of planar waveguide. Variable-index focusing elements:Part 1-Lenses of circular symmetry," Fiber Integrated Opt.8, 121-127 (1989).

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