1708 OPTICS LETTERS / Vol. 26, No. 21 / November 1, 2001

Propagation of nonparaxial beams with a modifiedArnoldi method

Q. Luo and C. T. Law

Department of Electrical Engineering and Computer Science, University of Wisconsin—Milwaukee, Milwaukee, Wisconsin 53201

Received June 4, 2001

We have developed a modified Arnoldi method that includes a complex square-root approximation, which excelsat modeling the propagation of highly diverging beams in various media. Simulations of one transversedimensional beam with an ultranarrow width and of cylindrical Gaussian beams with various divergenceangles reveal the strength of this nonparaxial-beam propagation method. © 2001 Optical Society of America

OCIS codes: 350.5500, 000.4430, 260.1960.

Beam propagation has intrigued numerous re-searchers and remains a topic of contemporaryresearch.1 For propagation in bulk media or in polar-ization-insensitive devices, an optical beam satisfiesthe Helmholtz equation. The electric f ield of a beamwith carrier frequency v inside a medium with re-fractive index n can assume the form E cx, y, z 3

exp jk0zexp2jvt, where k0 vn0c and n0 is areference refractive index. If the f ield envelope c isslowly varying, we can simplify the Helmholtz equa-tion with the paraxial approximation. However, fora narrow beam that incurs a large divergence angleor for beam propagation inside inhomogeneous ornonlinear media, rapid variation of the beam size ren-ders the paraxial approximation useless. To remedythis shortcoming, researchers introduced the Lanczosmethod.2 – 5 Unfortunately, that method becomesunstable when the grid spacing is too small.3,5 In thisLetter we elucidate the cause of this difficulty, whicharises from failure to separate the forward and thebackward waves, thus permitting them to mix. Tocircumvent this problem we isolate the forward wavewith a Padé approximation for a matrix square rootoperator6 – 8 and use the isolated wave to construct anovel Krylov subspace (KS) by the Arnoldi method.9,10

For a nonparaxial beam with negligible ref lection,the Helmholtz equation can be reduced to the followingnormalized equation for forward propagation:

≠c≠Z j4k0z01 1 2k0z021H12 2 1c , (1)

where Z z4z0 is the normalized propagation dis-tance, z0 pw0

2n0l is the diffraction length, l is thewavelength, and w0 is the initial beam radius. For theoperator H =t

2 1 V , the transverse Laplacian =t2

and V w02k0

2nn02 2 1 account for linear diffrac-tion and the normalized refractive-index profile, re-spectively. To update c after a DZ step, we need toevaluate a square-root operator and its exponential:

cZ 1 DZ expj4k0z01 1 2k0z021H12 2 1

3 DZcZ . (2)

Obviously, these operators are closely related to theoperator H. H can be discretized with a finite num-ber of bases and represented by a matrix H N with

0146-9592/01/211708-03$15.00/0

elements xijHjxm, where xi and xm are basis vec-tors with i, m 1 2 N2, . . . ,N2 for N grid pointsin each transverse dimension. The exponential of thesquare root operator, 1 1 2k0z021H12, can be per-formed with eigenvalue decomposition of the matrixZ I 1 2k0z021H N , where I is the identity matrix:

exp j4k0z0DZq

Z 2 I S exp j4k0z0DZq

D S21.

(3)

In Eq. (3), S is the similarity transform matrix;p

D isthe diagonal matrix with eigenvalues of

pZ 2 I as its

elements, e.g., bim 1 2 2k0z021ki2 1 km

212 2 1in normalized rectangular coordinates X,Y xw0, yw0 for propagation in homogenous media;ki,km 2piLx,mLy are normalized angular fre-quencies; and Lx,Ly is the size of the computationalwindow.

For large N , direct evaluation of Eq. (3) is imprac-tical. The Lanczos reduction method was introducedto simplify the computational task.2 – 5 The procedureapproximates matrix HN in a reduced M- ,,N- di-mensional KS, formed by vectors c,HN c, . . . ,HM21

N c,where c is the vector that contains samples of c at gridpoints. Then all complicated matrix operations thatinvolve H are conveniently evaluated in KS. Whenthe grid spacing is less than a half-wavelength, bim canbecome imaginary. Waves with such bim are evanes-cent3 and hence cannot propagate.

The generation of evanescent waves prevents theLanczos method from determining an appropriate KS.Other conditions, such as with gain or loss media, alsolimit the use of the regular Lanczos method, which re-quires Hermitian HN . Although the KS of a non-Her-mitian HN can be obtained by use of a modifiedLanczos method with biorthogonal basis vectors,4 theorthogonal property of the basis vectors is forgone, andnumerical difficulties may develop.11 To avoid theseproblems, we use an Arnoldi method that main-tains the orthogonality of the basis vectors.9,10 Con-sequently, our method can be used in a wide class ofinteresting applications. We further modif ied theregular Arnoldi method to generate a novel KS, cZ,p

Z cZ, . . . , p

Z McZ, wherep

Z contains only aforward wave; hence our method maintains stability.

© 2001 Optical Society of America

November 1, 2001 / Vol. 26, No. 21 / OPTICS LETTERS 1709

Modifying the package EXPOKIT,10 we implementedthe following modif ied Arnoldi method (MAM):

qp cZjcZj, wp q

Zqp p 1 , (4a)

qp wp21 2

p21Xn1

hn,p21qnhp,p21,

wp Zqp21 2

p21Xn1

hn,p21wnhp,p21

p 2, . . . ,M 1 1 , (4b)

hi,p qi jwp, hp11,p jwpj

i 1, . . . ,p,p 1, . . .M 1 1 , (4c)

where hi,p are elements of an M 1 1 3 M 1 1 upperHeisenberg matrix HN , and wp are auxiliary vectors.Basis vectors qp convert c between KS and the originalspace. Notice that this procedure requires the compu-tation of

pZ once and that the resultant HN is a small,

dense matrix.To compute accurately the matrix square root in the

correct quadrant on the complex plane, we apply thePadé approximation:

qZ

qI 1 X C0I 1

NpXn1

I 1 BnX21AnX , (5)

where X 2k0z021HN and Np is the Padé approxi-mation order. If Z are positive definite, Eq. (5) hasreal-valued coefficients C0 1, An an Np 1

1221 sin2np2Np 1 1, and Bn bn cos2np2Np 1 1. Otherwise, we need to rotate the complexplane by an angle a to achieve an accurate approxima-tion,6– 8 i.e.,

pI 1 X exp2ja2 I 1 Xexpja12.

In this case the three coefficients become complex:

C0 exp ja2Ω1 1

NpXn1

anexp2ja 2 1

1 1 bnexp2ja 2 1

æ,

(6a)

An an exp2ja2 1 1 bnexp2ja 2 122, (6b)

Bn bn exp2ja 1 1 bnexp2ja 2 121. (6c)

The accuracy of Eq. (5) depends not only on the po-larity of eigenvalues of X but on the magnitude of theseeigenvalues. For magnitudes much larger than 1, weneed to precondition Eq. (5) with an approximation ofX, X app that has

pI 1 X app readily available:

qI 1 X

qI 1 X app

qI 1 X0 . (7)

Notice that the second square root in Eq. (7) can beexpanded with a Padé approximation in terms of X0 I 1 X app21I 1 X 2 I, which has smaller eigenvaluesthan do terms X.

To test the MAM we apply it to a one-dimen-sional beam propagation problem.3 Equation (1) withH ≠2≠X2 can be solved analytically in terms of theband-limited Fourier series

cLX,Z N2X

m2N /211Ckmexp jkmX

3 exp j4k0z0bmZ , (8)

where bm 1 2 2k0z021km212 2 1, N is the

number of grid points in X, km 2pmL and L isthe size of a computational window. To simulatepropagation with an extremely large divergence angle,we use a delta-function-like such as a beam at Z 0with Ckm 1 for 1 2 N 02 # m # N 02; otherwise,Ckm 0, where N 0 1008 for N 1024 andN 0 2016 for N 2048. L is chosen such that gridspacing Dx is equal to l2 for N 1024 and to l4 forN 2048. For both cases, the half divergence angleu1/2 sin21N 0N is estimated to be 79.9±. Undersuch extreme condition, the Padé approximation inEq. (5) is not expected to perform well. Therefore weapply a preconditioner X app 0.8HN and measurethe accuracy of our algorithms with the overlaperror: e j1 2 jc jcLjjcL jcLjj.

The relationship of overlap errors to step sizes isplotted in Figs. 1 and 2 for grid spacings Dx l2 andDx l4, respectively, at a propagation distance z 50l, with l 1 mm. These figures show the conver-gence with small step sizes. They also demonstratethe performance of the MAM and the Padé approxima-tion. For comparison, we also plot results from calcu-lations that use the exact square root, i.e., numericalerrors solely from the MAM. Evidently, the Padé ap-proximation is a limiting factor. Precision improvesdramatically and is far better than that of the Lanc-zos method3 when a preconditioner is used with thePadé approximation, e.g., e , 1029 for Dx 0.5l. Invarious ways of computing the matrix square root, as

Fig. 1. Overlap error e versus step size for various waysto calculate square roots (exact, exact operation with eigen-value decomposition; no preconditioner, approximated op-eration with the Padé expansion and no preconditioner; andwith preconditioner, approximated operation with the Padéexpansion and a preconditioner) for various orders of MAMwith grid spacing Dx l2 and N 1024.

1710 OPTICS LETTERS / Vol. 26, No. 21 / November 1, 2001

Fig. 2. Overlap error e versus step size for the same condi-tions as for Fig. 1, except now with grid spacing Dx l4and N 2048.

Fig. 3. Relative error eZ of paraxial propagation andnonparaxial calculations with the MAM (8th-order methodwith 512 grid points for the Padé approximation without apreconditioner) versus half divergence angle u1/2 for Gauss-ian beams of various initial sizes propagating in free spacefor Z 0.1.

the order of the MAM increases from 4 to 16, its con-vergence accelerates. Naturally, higher-order meth-ods can afford to have a longer step size, e.g., of theorder of l for 16th order and of less than 0.2l for 4thorder. Figure 1 further suggests that 8th order is al-most as good as 16th order and, in fact, its e is 3 orderssmaller than that of the Lanczos method. Obviously,as eigenvalues of X increase with smaller grid spacing,the approximation for the square root deteriorates. Inthe most severe situation, methods that use an approxi-mate square root still exhibit convergence with dimin-ishing step size.

Another example that we compare with the Lanczosmethod is the propagation of a cylindrical Gaussianbeam in free space with various divergence anglesand grid spacings. All the parameters and HNused here are the same as those described for theresearch presented in Ref. 5, for which Luo and Lawobtained beams with different half divergence anglesby varying the initial beam size w0 at l 0.9 mmand computed exact beam profiles ceR,Z by in-

Fig. 4. Relative error eZ versus half divergence angleu1/2 under the same conditions as for Fig. 3, except thatZ 5.

tegrating their angular spectra. We estimate theaccuracy with the relative error as follows: eZ RLf

0 jcR,Z 2 ceR,Zj2dRRLf

0 jceR,Zj2dR, whereLf corresponds to the first zero crossing in the phaseof the exact beam. Figures 3 and 4 illustrate thevariation of relative errors for various grid spacings(Dr 0.5l and Dr 0.2l) and half divergence angleu1/2 at Z 0.1 and Z 5, respectively. As a base-line, we also plot paraxial results. Naturally, theperformance of the MAM matches that of the Lanczosmethod.5 The most remarkable improvement appearsat small spacing and u1/2 . 25±. In fact, eZ becomesindependent of u1/2 and maintains a small value 1026when Dr 0.2l at Z 5.

In summary, we have elucidated and remedied dif-ficulties with the Lanczos method. To achieve higheraccuracy, we developed a modif ied Arnoldi methodthat, combined with a complex Padé square-root ap-proximation, is applicable to propagation of wide-anglebeams in media with loss or gain.

The research was supported by the National ScienceFoundation CAREER program and the Wright Labo-ratory. C. T. Law’s e-mail address is lawc@uwm.edu.

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