Ray-tracing method for isotropic inhomogeneous refractionindex media from arbitrary discrete input

Yohei Nishidate1,2,3∗, Takashi Nagata3, Shin-ya Morita2,3, and YutakaYamagata2,3

1University of Aizu, Aizu-Wakamatsu 965-8580, Japan

2Ultra High Precision Fabrication Team, RIKEN, Saitama 351-0198, Japan

3Volume-CAD System Research Program, RIKEN, Saitama 351-0198, Japan

∗Corresponding author: yh.nsdt@gmail.com

A ray-tracing simulation procedure has been developed for optically isotropicgradient refraction index media. The procedure can take discrete points of arbitrarydistribution for definition of refraction index distributions and lens surfaces.It is useful to simulate ray trajectories in real lens systems. The procedure isapplied to ray-tracing simulation of the Luneburg lens and a radial gradientoptical fiber. The simulation results are compared with the analytical solutions, andit is shown that they are in precise agreement. c© 2014 Optical Society of America

OCIS codes: (000.4430) Numerical approximation and analysis; (080.1753) Computa-tion methods; (080.2710) Inhomogeneous optical media; (080.5692) Ray trajectories in inhomo-geneous media; (220.2740) Geometric optical design.

Published inApplied Optics 50(26), pp. 5192-5199 (2011)

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1. Introduction

Lenses fabricated by molding procedures play an important role in modern optical device industry.The mold fabrication is a reasonable way for effective production, but non-uniform (inhomoge-neous) refraction index distributions arise as a consequence of thermal deformations. Engineersand researchers devoted extensive efforts to suppress such effects, and successful optical deviceshave been developed. However, effects of small changes in rays direction due to the refraction in-dex variation in lens media will be a serious issue for further optical devices developments. Severalnondestructive methods were proposed for measuring three-dimensional refraction index distribu-tion [1–5]. Although, an iterative procedure was developed for reconstruction of global refractionindex functions from the measured data [6,7], the result of measurements is obtained as the refrac-tion index map in a resolution of the instrument. In order to evaluate performances of precisionoptical systems, a ray-tracing method for gradient refraction index (GRIN) media should be em-ployed where the refraction index distribution and lens surfaces are represented as sets of discretepoints.

Analytical solutions for specific GRIN distributions were derived, such as spherical and radialiso-indicial media [8]. For general GRIN media, several numerical methods were developed. Mostof numerical solution techniques are based on the ray equation of geometric optics [9]. The fourthorder explicit Runge-Kutta integration scheme was applied by rewriting a parameter of the rayequation to obtain a set of two first-order differential equations [10, 11]. Treatment of rays nearthe exit of GRIN media is also discussed for the method [12]. This technique has been applied tovarious problems, e.g. a GRIN compact disk pick-up system design [13], development of an eyesmodel [14], biological imaging systems [15, 16], an analysis of polymer fiber for refractive-indexsensors [17], and a method for rendering realistic three-dimensional computer graphics [18]. Someimproved procedures for the algorithm were also proposed [19, 20]. Based on the same form oftwo first order differential equations, an algorithm using predictor-corrector method was devel-oped with an automatic step size monitoring scheme [21]. Their step-size monitoring techniqueis based on the Richardson’s scheme [22] for evaluation of the truncation error. The method pro-posed in [23] divides inhomogeneous refraction index media into finite elements assuming that therefraction index is constant within each element. A more sophisticated technique was developedby organizing a point set as adaptively subdivided grids (octree) for GRIN media expression [24].The finite element and the octree grid based techniques can trace rays in GRIN media expressedthrough discrete data, but they require prescribed structured elements. For ray-tracing in humaneyes model, shell series descriptions of GRIN media are sometimes employed [25]. The GRINmedia are represented as a set of concentric shells, and hence iso-indicial surfaces should be givenfor this approach.

Previous ray-tracing methods for GRIN media assume that the distribution is given as a globalfunction, or it is locally reconstructed based on structured grids such as finite elements. Also, lens

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surfaces are assumed to be given as a function, or it is expressed through simple piecewise lineartriangles. In some studies, modeling of lens surfaces is not considered. Recently, we proposeda ray-tracing procedure which is efficient for constant refraction index lenses with its surfacescontaining form errors [26]. The method is based on a piecewise quadratic interpolation methodcalled the Nagata patch [27]. Currently, a precise ray-tracing simulation method is not available forgeneral GRIN media based on a flexible refraction index reconstruction method and a simple lenssurface expression. In this paper, we propose a ray-tracing method for general GRIN media wherethe refraction index distribution and the lens surfaces are given as sets of arbitrary discrete points.

2. Calculation of refraction in GRIN media

2.A. Ray-tracing equation

Ray trajectories in GRIN media can be calculated using the ray equation of geometric optics [9]

d

ds

[n(r)

dr

ds

]= ∇n(r) (1)

where n(r) is a refraction index function of the position vector r, and s is the optical path length.Changing the independent variable s to the ray travel length t defined by dt = ds/n yields thefollowing system of two first-order differential equations [10]

dr

dt= T

dT

dt= n(r)∇n(r).

(2)

where T is the tangent vector of the ray. In order to integrate the equation (2), the refraction indexdistribution in the GRIN medium should be given as a function of position.

2.B. Reconstruction of refraction index distribution

The finite element ray-tracing (FER) [23] and the octree grid techniques [24] used a similar methodfor the reconstruction of refraction index functions as described in Fig. 1-(a). In the FER, GRINmedia are expressed through 3D hexahedral finite elements. The octree grid technique adopted oc-tree data structure for expression of steep and flat gradients by dense and sparse grids, respectively.These techniques may acquire unnatural sampling points, especially in case that aspect ratio ofthe grid is poor. Since refraction index distributions are regarded as a constant value within eachelement, there are possible loss of precision. It is difficult to use points with arbitrary distribution,because sampling points should be located and organized as the grid topology. In addition, raysadvancement should meet Courant-Friedrichs-Levy (CFL) stability condition [28] that no ray trav-els farther than one cell in one integration step. Therefore, dense sampling points may not be used.On the other hand, we apply a distance based refraction index field reconstruction as shown inFig. 1-(b).

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The moving least squares (MLS) approximation is used as the shape function for several mesh-free finite element method implementations [29–31]. The MLS locally approximates some fieldsas a continuous function. We employ the MLS to reconstruct the refraction index function frompoint clouds. In the MLS, the approximated refraction index function n at the coordinates x iscalculated as

n(x) = pT (x)a(x) (3)

where p is the term vector consisting of any nonlinear function pi of the coordinates. The coeffi-cient vector a is obtained from solution of the following equation system

[M(x)]a(x) = [N(x)]n (4)

where n is a vector consisting of the refraction index at sampling points, and the matrices M andN are given as follows

[M(x)] = [PT ][W(x)][P], (5)

[N(x)] = [PT ][W(x)]. (6)

The matrix P comprises a term vector at i-th sampling point coordinates si

P(x) =

pT (s1)...pT (sn)

, (7)

and W is a diagonal matrix defined as

W(x) = diag {w1(x), · · · , wi(x), · · · , wn(x)} (8)

where wi(x) is the weight function of the distance between si and x. The spatial gradient of therefraction index distribution is analytically obtained from derivatives of equation (3)

∂n

∂x=∂pT

∂xa. (9)

Therefore, we can solve the ray equation (2). We use the following tri-cubic basis for the vector p

pT ={

1, x, y, z, xy, yz, zx, xyz, x2, y2, z2, x2y, x2z, y2x, y2z, z2x, z2y, x3, y3, z3}, (10)

and a simple quartic spline weight function [32]

wi(x) = 1− 6l2i + 8l3i − 3l4i (11)

where li is the normalized distance between si and x. If noisy refraction index data are anticipatedas the input, lower degree polynomials, such as a quadratic or a linear basis are preferred forrobustness. It is one of the advantages over grid based techniques that it is quite easy to change thebasis function for fitting of a local field depending on the situation. However, dynamic searchingof sampling points is required at each ray position for calculation of the MLS coefficients. Forefficient searching, the spatial bucketing of points is used as depicted in Fig. 2.

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2.C. Integration scheme

Sharma [10] used the fourth order explicit Runge-Kutta scheme for numerical integration of equa-tions (2). It is well known that the step size should be small enough in explicit numerical integra-tion. However, floating-point number representations have finite significant digits, and hence theprecision does not improve by selecting any smaller step sizes than certain limit. Excessively smallstep sizes result in unnecessary computational effort as well as loss of precision due to accumula-tion of round-off errors. Therefore, it is preferred to find a reasonable step size to minimize both ofthe numerical error and the computational cost. The Richardson scheme [22] is an automatic stepsize monitoring method which estimates the truncation error through comparing the result withfull and half step sizes. This estimation requires increase in the computational effort by 50%. Theembedded Runge-Kutta-Fehlberg formulae [33] were developed to overcome such computationalcost drawback for the error estimation. Suppose that the first order differential equation of interestis expressed in the following normal form

dx

dt= f(t,x), (12)

and then the embedded Runge-Kutta-Fehlberg formulae of fourth and fifth order precisions(RKF45) are expressed as follows

xt+∆t = xt + ∆t6∑

i=1

biki +O(∆t5), (13)

xt+∆t = xt + ∆t6∑

i=1

biki +O(∆t6) (14)

where ki are obtained through evaluating the right-hand side of equation (12) as

ki = f(t+ ci∆t,xt +i−1∑j=1

aijkj), (15)

and coefficients aij , bi, bi, and ci are given in Table 1.Typically, function evaluations have dominant influence to the integration cost. Since the embed-

ded Runge-Kutta-Fehlberg scheme provides fourth and fifth order precision results using functionevaluations at the same intermediate positions, the truncation error is estimated with minimal com-putational efforts. For the step size optimization, a quantity δ is defined as

δ = maxj

∣∣∣x(j)t+∆t − x

(j)t+∆t

∣∣∣ /ε(j) = maxj

∣∣∣∣∣6∑

i=1

(bi − bi)k(j)i

∣∣∣∣∣ /ε(j) (16)

where the superscript (j) indicates the j-th component of a vector, and ε(j) is the admissible errorfor the corresponding state variable. The position vector components should be normalized prior

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Table 1. The coefficients aij , bi, bi, and ci for the RKF45.

i ciaij

bi bij=1 2 3 4 51 25

21616135

2 14

14

0 0

3 38

332

932

14082565

665612825

4 1213

19322197

−72002197

72962197

21974104

2856156430

5 1 439216

−8 3680513

− 8454104

−15− 9

50

6 12− 8

272 −3544

256518594104

−1140

255

to the calculation since it has a unit of length. If the quantity δ is neither too large nor small(0.8 ≤ δ ≤ 1.0), the integration is accepted without step size modification. Otherwise, an optimalsize ∆tn+1 of the n+ 1-th step is estimated as

∆tn+1 = S∆tnδ15 (17)

where S is the safety factor. We select S = 0.9 and the error tolerance ε(j) as 10−12 for all j. Theoptimal step size found in a current integration step is remembered to be used as the initial stepsize ∆t(0) in the next integration step.

3. Calculation of refraction at lens surfaces

3.A. The Nagata patch

The lens surface expression is very important for precise ray-tracing simulation not only for GRINlenses. We have developed a ray-tracing procedure using a piecewise local quadratic interpolationwhich is called the Nagata patch [27]. The Nagata patch can be applied to general triangular mesheswith arbitrary topology (Fig. 3). It is determined only by position and normal vectors at the verticesof each triangle, and can naturally treat the discontinuity of normal vectors. Interpolation errors inthe position and normal vectors diminish rapidly, and C0 continuity is mathematically assuredacross neighboring patches. Furthermore, a closed-form solution exists for ray-patch intersectionsand an efficient contact search algorithm has been established in our previous study [26].

The Nagata patch interpolant is defined in terms of surface parameters η and ζ as

x(η, ζ) = c00 + c10η + c01ζ + c11ηζ + c20η2 + c02ζ

2 (18)

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where 0 ≤ η ≤ ζ ≤ 1. The vector coefficients cij of the patch are given by

c00 = x00,

c10 = x10 − x00 − c1,

c01 = x11 − x10 + c1 − c3,

c11 = c3 − c1 − c2,

c20 = c1,

c02 = c2,

(19)

where x00, x10, and x11 are position vectors at the patch nodes, and ci is the curvature parameteron the i-th patch edge. The curvature parameter c on the i-th edge is determined from the positionand the normal vectors at its ends as

c(d,n0,n1) =

[n0,n1]

1− c2

1 −c−c 1

n0 · d−n1 · d

(c 6= ±1)

0 (c = ±1)

(20)

where d = x1 − x0, c = n0 · n1. When the normal vectors are parallel at the end points, thecurvature parameter vanishes and hence the edge degenerates into a linear segment.

3.B. Intersection of a line with the Nagata patch

A contact search procedure of a line with the Nagata patch has been proposed in our previousstudy [26]. Since the procedure is essential for the present approach, it is briefly reviewed here.The line segment is expressed using a parameter λ which varies from 0 to 1 as

x(λ) = xa + λd (21)

where d = xb − xa is the vector connecting the start xa and the end xb of the line segment. Aspecial local coordinate system with axes e1, e2, and e3 is constructed from the direction vector dof line segment and the quadratic coefficient vector c02 of the Nagata patch as

e3 = d/|d|,e1 =

e3 × c02

|e3 × c02|,

e2 = e3 × e1.

(22)

Equating the position vectors (21) and (18), and subsequent transformation into the special coordi-nate system (22), a system of equations is obtained for the line parameter λ and surface parametersη and ζ

0 = α00 + α10η + α01ζ + α11ηζ + α20η2, (23)

0 = β00 + β10η + β01ζ + β11ηζ + β20η2 + β02ζ

2, (24)

λd = γ00 + γ10η + γ01ζ + γ11ηζ + γ20η2 + γ02ζ

2, (25)

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where

αij =

e1 · (cij − xa) (i = j = 0)

e1 · cij (otherwise), (26)

βij =

e2 · (cij − xa) (i = j = 0)

e2 · cij (otherwise), (27)

γij =

e3 · (cij − xa) (i = j = 0)

e3 · cij (otherwise). (28)

The surface parameters η and ζ for an intersection are found from equations (23) and (24). Then,the line parameter λ is calculated from equation (25).

4. Ray treatment near the exit of GRIN media

In GRIN media, ray trajectories are calculated in the precision assured by the RKF45 schemewithin a specified error tolerance. However, they may be inappropriately calculated when the rayleaves GRIN media (Fig. 4).

It is possible to construct a curve for the ray trajectory approximation using some previouspositions and/or tangent vectors at each position. However, closed-form solution does not existfor intersecting problem of the Nagata patch with curves, and hence rays should be considered aslines. In order to find a better contact position, the ray advancement is adjusted near the exit ofGRIN media using the line parameter λ of equation (21). We use an iterative procedure depictedin (Fig. 5).

If a ray intersects with the lens surface, the step is canceled and enter the iterative procedure.The current step size, the line parameter, and the ray tangent vector are used as initial parameters

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∆t0, λ0, T0, respectively. The iterative procedure is given by the following pseudo-code

//step size, line parameter, and tangent vector

∆t0, λ0,T0

while (∆t0|T0| > ε && !IsZero(λ0 − 1.0)) {if (!IsZero(λ0)) T = T0, break

A = 1.0

do { //loop to get closer to lens surface

T = T0

if (A ≥ 0.1) A = A− 0.1

else A = 0.1A

∆t = Aλ0∆t0

DoFixedStepIntegration(∆t)

} while (CheckLensRayIntersection())

∆t = (1− A)λ0∆t0

do { //loop to find a new contact position

T0 = T

DoFixedStepIntegration(∆t)

} while (!CheckLensRayIntersection())

λ0 = FindLineParameterLambda()

∆t0 = ∆t

}

(29)

The function IsZero returns true if the argument is numerically close enough to zero,DoFixedStepIntegration calculates the fifth order Runge-Kutta integration using the fixed stepsize given as its argument, CheckLensRayIntersection returns true if the ray intersects with atleast one of the Nagata patches, and FindLineParameterLambda returns the line parameter of thelast ray-patch intersection. The iterative procedure consists of an outer loop and two inner loops.In the first inner loop, the ray position is moved forward within the GRIN medium. The fixed stepintegration is performed with its step size Aλ0∆t0. The parameter A is a kind of safety factor toadjust the ray advancement, and it decreases from 0.9 to 0.1 by 0.1. If A becomes smaller than0.1, A is repeatedly multiplied by 0.1. The first inner loop continues until intersection of the raywith the exit surface fails. On the contrary, the second loop continues until the ray intersects withthe exit surface to find an improved contact position with the corresponding line parameter. Aftertwo inner loops are finished, the ray advancement length is estimated by ∆t0|T|0. The outer loopfinishes if it is smaller than a specified tolerance ε, or the line parameter λ is close enough to 0 or

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1. Otherwise, the whole process is repeated using the new line parameter. We use the tolerance εas the square root of the machine precision (1.49 × 10−8 in our environment). Typically, the in-ner loops finish in 1 or 2 iterations. If the interval of the refraction index sampling is much largerthan the ray advancement, the neighbor nodes update and the MLS coefficients calculation are notnecessary during iterations Thus, the iterative procedure is not expensive in general.

5. Comparison with analytical solutions

5.A. Luneburg lens

The ray-tracing procedure using the Nagata patch has been evaluated in the previous study [26],and hence the developed algorithm for GRIN media is verified here. The algorithm is applied tothe ray-tracing simulation of the Luneburg lens shown in Fig.6.

The refraction index distribution of the Luneburg lens is defined by the following equation

n(r) =

√2−

(r

R

)2

(30)

where r = (x2 +y2 +z2)1/2, and R = 1.0mm is the radius of the Luneburg lens. The input discretepoints are generated from sampling of the function (30) at intervals of 0.0625mm in x, y, and zdirections to imitate data acquired from a real measurement or molding process simulation. Thespot radius is calculated using 32 incident rays parallel to the optical axis.

Fig. 7 shows the spot radius and the ray positions on the xy plane at the focal point. The spotradius is 9.14×10−6mm (9.14nm). In the Luneburg lens problem, the error monotonically increaseswith the ray travel length. This example gives an estimate of the numerical error anticipated in ourray-tracing simulation procedure using real lens models, because the radius and the thickness areoften in the dimension of millimeters and ray trajectories are similar. It implies that ray-tracingsimulation of the numerical error within the order of nanometers is possible in typical opticalsystems.

5.B. Radial gradient optical fiber

Another example is a radial iso-indicial optical fiber with refraction at the exit surface of the GRINmedium (Fig. 8).

The refraction index distribution is given by the following function of the radial distance on thexy-plane

n(rxy) =

√2.5−

(rxyR

)2

(31)

where rxy is the radial distance, and R = 1.0mm is the radius of the optical fiber. The refractionindex is 1.0 outside the optical fiber, and it is constant. As the Luneburg lens problem, the inputdiscrete points are generated by sampling at intervals of 0.0625mm in x, y, and z directions. A

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ray with initial position {0, y0, 0} and incident direction parallel to the optical axis is traced. Ananalytical solution exists for the radial GRIN media [8], and it can be written for the coordinate yas a function of the coordinate z as

yin(z) = y0cosz

n0

. (32)

where n0 is the refraction index at the ray initial position n(y0). Since analytical solution is notavailable outside the optical fiber, a solution is derived. The end surface of the optical fiber is flatand its normal vector is {0, 0,−1}. Suppose that the length of the optical fiber is zc. Then, the raytangent direction at the exit surface before refraction is given as

v =1

nc

{0, qc, n0} (33)

where qc = −y0sin zcn0

and nc = n(yc) is the refraction index at the contact coordinate yc of the rayon the exit surface. From the Snell’s law, the tangent direction after refraction is obtained as

v′ =1

nc

{0, ncqc,

√n2

0 + (1− n2c)q

2c

}. (34)

The coordinate yc is obtained by substituting zc in equation (32). Finally, the coordinate y outsidethe optical fiber (z > zc) is calculated from the following equation

yout(z) =ncqc√

n20 + (1− n2

c)q2c

(z − zc) + yin(zc) (35)

Fig. 9 shows difference of the ray y coordinate calculated by the ray-tracing simulation andthe analytical solution. In order to demonstrate importance of the iterative procedure (29), theresults are shown with and without it. The ray is well approximated within the error 5 × 10−5mm(50nm) inside the optical fiber. The error linearly grows outside the medium. Without the iterativeprocedure, the error rapidly increases.

The above examples of media with steep refraction index gradient are studied mainly for bench-marking purposes. In existing lenses, however, the variation is typically less than a few percents.Therefore, the simulation of real problems may result in much simpler trajectories with higher pre-cision. The quality of the solution by the proposed scheme rather depends on the accuracy of inputdata.

6. Conclusion

A ray-tracing procedure has been developed for general GRIN media where the refraction indexdistribution and the surface position are given as point sets. Ray trajectories in the GRIN media arecalculated using the ray equation of geometric optics. In order to solve the ray equation using givenarbitrary point sets, the MLS approximation is adopted for the local reconstruction of refraction

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index fields. The RKF45 scheme is employed for numerical integration with automatic step sizecontrol. The GRIN ray-tracing procedure can be used together with the Nagata patch for lenssurfaces expression. Also, an iterative procedure is presented to avoid significant loss of precisionnear the exit surface of GRIN media.

The developed procedure is applied to ray-tracing simulation of the Luneburg lens and a radialgradient optical fiber. The spot radius of the Luneburg lens is less than 10 nanometers for 1 mil-limeters radius of the lens. The ratio between the error and the problem dimension is the order of10−6. A ray trajectory inside a radial GRIN optical fiber is calculated within an error less than 50

nm. It is also confirmed that ray trajectories can be significantly improved by finding better contactposition using the iterative procedure.

Using the present algorithm for ray-tracing in GRIN media and the Nagata patch lens surfaceexpression, ray trajectories in real lens systems can be simulated in high precision taking intoaccount inhomogeneity in lens media and form errors on lens surfaces due to fabrication processes.Future study and discussion should focus on applications of our method to ray-tracing in real lenssystems taking lens geometries and refraction index distributions from real measurements.

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List of Figure Captions

Fig. 1. Sampling point acquisitions for reconstructions of the refraction index distribution at posi-tion x. In the grid based acquisition (a), points of label 1 are used but 2 are not. On the otherhand, in the distance based acquisition (b), sampling points are more reasonably selected.

Fig. 2. For local sampling point acquisition, the space is partitioned into cells and all points areregistered into one of the cells according to its position. In this case, calculation of distancesbetween the center (red point) with points belonging to regions 0–8 are enough for MLSsampling points acquisition.

Fig. 3. The Nagata patch interpolant of an edge and a surface. The curvature parameter vector c isuniquely determined for each edge. Accordingly, the surface interpolant is orthogonal to thenormal vectors at the vertices.

Fig. 4. A ray trajectory near the exit of GRIN media. When a ray intersect with the exit surface,an intermediate point of the line connecting start and end points is considered as the contactposition. It can cause significant loss of precision because the curvature of the ray trajectorymay not be negligible.

Fig. 5. Schematic of the iterative procedure to find a better contact position.

Fig. 6. The Luneburg lens is a spherical iso-indicial lens which converges every parallel ray to apoint on the lens surface. The optical axis coincides with the z coordinate axis.

Fig. 7. The spot radius and the incident ray positions on the xy plane at the focal point.

Fig. 8. A radial iso-indicial optical fiber with refraction at the lens surface.

Fig. 9. The error of the ray-tracing simulation in the optical fiber, and comparison of the error withand without treatment near the GRIN media exit.

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(a) Grid based (b) Distance based

Fig. 1. Sampling point acquisitions for reconstructions of the refraction index dis-tribution at position x. In the grid based acquisition (a), points of label 1 are usedbut 2 are not. On the other hand, in the distance based acquisition (b), samplingpoints are more reasonably selected. fig1-a.eps, fig1-b.eps

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Fig. 2. For local sampling point acquisition, the space is partitioned into cells andall points are registered into one of the cells according to its position. In this case,calculation of distances between the center (red point) with points belonging toregions 0–8 are enough for MLS sampling points acquisition. fig2.eps

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Fig. 3. The Nagata patch interpolant of an edge and a surface. The curvature pa-rameter vector c is uniquely determined for each edge. Accordingly, the surfaceinterpolant is orthogonal to the normal vectors at the vertices. fig3.eps

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Fig. 4. A ray trajectory near the exit of GRIN media. When a ray intersect withthe exit surface, an intermediate point of the line connecting start and end points isconsidered as the contact position. It can cause significant loss of precision becausethe curvature of the ray trajectory may not be negligible. fig4.eps

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Fig. 5. Schematic of the iterative procedure to find a better contact position. fig5.eps

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Fig. 6. The Luneburg lens is a spherical iso-indicial lens which converges everyparallel ray to a point on the lens surface. The optical axis coincides with the zcoordinate axis. fig6.eps

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−10

0

10

−10 0 10

y (

nm

)x (nm)

Fig. 7. The spot radius and the incident ray positions on the xy plane at the focalpoint. fig7.eps

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Fig. 8. A radial iso-indicial optical fiber with refraction at the lens surface. Theincident ray grazes the top and the bottom surfaces of the fiber, and it is refracted atthe exit surface. fig8.eps

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−400

−200

0

200

400

0 5 10 15

Err

or

(nm

)

z (mm)

IterativeNo care

Fig. 9. The error of the ray-tracing simulation in the optical fiber, and comparisonof the error with and without treatment near the GRIN media exit. fig9.eps

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