New method for nonparaxial beam propagation

1082 J. Opt. Soc. Am. A/Vol. 21, No. 6 /June 2004 A. Sharma and A. Agrawal

New method for nonparaxial beam propagation

Anurag Sharma and Arti Agrawal

Department of Physics, Indian Institute of Technology Delhi, New Delhi-110 016, India

Received August 29, 2003; revised manuscript received January 6, 2004; accepted January 30, 2004

A new method for solving the wave equation is presented that is nonparaxial and can be applied to wide-anglebeam propagation. It shows very good stability characteristics in the sense that relatively larger step sizescan be taken. An implementation by use of the collocation method is presented in which only simple matrixmultiplications are involved and no numerical matrix diagonalization or inversion is needed. The method ishence faster and is also highly accurate. © 2004 Optical Society of America

OCIS codes: 130.2790, 230.7370, 350.5500.

1. INTRODUCTIONRecently several schemes have been suggested for wide-angle and bidirectional beam propagation throughguided-wave devices.1–13 In general, this nonparaxialpropagation would involve solving directly the wave equa-tion, which contains a second-order partial derivativewith respect to z (the general direction of propagation) asagainst the first-order partial derivative in the paraxialwave equation. All the methods for nonparaxial beampropagation discussed in the literature approach thisproblem iteratively; a numerical effort equivalent to solv-ing the paraxial equation several times is involved. Theactual number of iterations depends on the desired accu-racy and the obliquity of the beam. Many of these meth-ods neglect the backward-propagating components andsolve the one-way wave equation; but even methods thatdeal with bidirectional propagation employ special tech-niques either to suppress or to model evanescent modes,which are a source of instability in these methods.8–10 Inall these methods, the square root of the propagation op-erator involved in the wave equation is approximated invarious ways. One of the approximations used is basedon the Pade approximants.1–11 We have recently shownthat a direct numerical solution (DNS) of the scalar waveequation gives very good accuracy and is also numericallyefficient.14 The method is nonparaxial and hence is ap-plicable to wide-angle as well as to bidirectional propaga-tion. We used the collocation method15–17 to formulateour equations. In this paper we present a new method ofsolving the nonparaxial wave equation that uses symme-trized splitting of the operators. Examples show thatthis method is more tolerant to larger step sizes thanother methods, including the DNS.14

2. SPLIT-STEP NONPARAXIAL METHODFor simplicity, we shall confine our discussions in this pa-per to two-dimensional wave propagation for which thescalar wave equation is given by

]2c

]x21

]2c

]z21 k0

2n2~x, z !c ~x, z ! 5 0, (1)

1084-7529/2004/061082-06$15.00 ©

where c (x, z) represents one of the Cartesian compo-nents of the electric field (generally referred to as the sca-lar field) and n2(x, z) defines the refractive-index distri-bution of the medium. The time dependence of the fieldis assumed to be exp(ivt), and k0 5 v/c is the free-spacewave number.

Equation (1) can be rewritten as

]F/]z 5 H~z !F~z !, (2)

where

F~z ! 5 F c]c

]zG , H~z ! 5 F 0 1

2¹t2 2 k0

2n2 0G. (3)

The operator H can be written as a sum of two operators,one representing the propagation through a uniform me-dium of index, say nr , the other representing the effect ofthe index variation of the guiding structure; thus

H~z ! 5 H1 1 H2~z !

5 F 0 1

2¹t2 2 k0

2nr2 0

G 1 F 0 0

k02~nr

2 2 n2! 0G.

(4)

A formal solution of Eq. (2), after use of the symmetrizedsplitting of summation of operators as in Eq. (4), can bewritten as

F~z 1 Dz ! 5 PQ~z !PF~z ! 1 O~~Dz !3!, (5)

P 5 exp~12 H1Dz !, Q~z ! 5 exp~H2Dz !. (6)

The operator P represents propagation in the uniformmedium nr over a distance of Dz/2 and hence can beevaluated by using any method such as the collocation,finite-difference, or fast-Fourier-transform methods. Theoperator Q(z) can also be easily evaluated because of thespecific form of the matrix, and it can be easily seen that

Q~z ! 5 F 1 0

k02~nr

2 2 n2!Dz 1G, (7)

2004 Optical Society of America

A. Sharma and A. Agrawal Vol. 21, No. 6 /June 2004/J. Opt. Soc. Am. A 1083

since

~H2!m 5 0, m > 2, (8)

because of the special form of the matrix H2 . It may benoted that for lossless propagation the matrix P would beHermitian, while the matrix Q always has a determinantvalue equal to unity.

The split-step nonparaxial (SSNP) method given abovecan be implemented with any of the numerical methodsemployed to solve the wave equation, e.g., the fast-Fourier-transform beam-propagation method (BPM),finite-difference (FD) BPM, or the collocation method. Inthis paper we discuss implementation by use of the collo-cation method; implementation by use of the FD-BPMwill be discussed in a future paper.

3. IMPLEMENTATION IN THECOLLOCATION METHODWe have implemented the SSNP formalism in the colloca-tion method, in which the wave equation is converted to amatrix ordinary differential equation by using the repre-sentation of the field c (x, z) as a linear combination of aset of orthogonal basis functions, fn(x):

c ~x, z ! 5 (n51

N

cn~z !fn~x !, (9)

where cn(z) are the expansion coefficients, n is the orderof the basis functions, and N is the number of basis func-tions used in the expansion. The choice of fn(x) dependson the boundary conditions and the symmetry of the guid-ing structure. The expansion coefficients cn(z) are un-known and represent the variation of the field with z. Inthe collocation method,15–17 these coefficients are effec-tively obtained by requiring that the differential equa-tion, Eq. (1) be satisfied exactly by the expansion, Eq. (9),at N collocation points xj , j 5 1, 2,..., N, which are chosensuch that these are the zeros of fN11(x). Thus, by usingthis condition, and with some algebraic manipu-lations,15–17 one converts the wave equation, Eq. (1), intoa matrix ordinary differential equation:

d2C/dz2 1 @S0 1 k02nr

2I 1 R~z !#C~z ! 5 0, (10)

with

C~z ! 5 F c ~x1 , z !

c ~x2 , z !

]

c ~xN , z !

G ,

R~z ! 5 k02FDn2~x1 , z ! 0 • 0

0 Dn2~x2 , z ! • ]

] ] • 0

0 0 Dn2~xN , z !

G,

(11)

where Dn2(xm , z) 5 n2(xm , z) 2 nr2; m 5 1, 2,..., N;

and S0 is a constant known matrix defined by the basisfunctions.15–17 We refer to Eq. (10) as the collocationequation. In deriving this equation from the wave equa-tion, Eq. (1), no approximation has been made except thatN is finite and Eq. (10) is exactly equivalent to Eq. (1) as

N → `. Thus the accuracy of the collocation method im-proves indefinitely as N increases. The collocation equa-tion is a matrix ordinary differential equation and can besolved as an initial value problem by using any standardmethod, such as the Runge–Kutta method, as we havedone in the DNS.14 In this paper we solve this equationby using the SSNP discussed in Section 2.

We have chosen here a set of sinusoidal functions asthe basis functions,16,17 and, following the procedure out-lined in Section 2, we obtain the formal solution of Eq.(10) as in Eq. (5), with the operators P and Q and the fieldfunction F now being block matrices,

F~z ! 5 F C

dC/dz G ,P 5 expH Dz

2F 0 I

2~S0 1 k02nr

2I! 0GJ ,

Q~z ! 5 F I 0

2R~z ! IG, (12)

where I and 0 are the unit and the null matrices, respec-tively. The operator P represents propagation in a uni-form medium of index nr over a distance Dz/2 and can beeasily obtained as a constant square matrix by using thebasis functions and their properties.17 It has to be evalu-ated only once. Each propagation step thus requires 12multiplications of an N 3 N square matrix with a columnmatrix except at the first and the last steps where eightmore such multiplications are required. We would like toemphasize that using the sinusoidal basis functions in thecollocation method here has an advantage, since no fastFourier transform, matrix inversion, or matrix diagonal-ization need be done for propagation through a uniformmedium, and all matrices involved are obtained analyti-cally; the details are presented in Appendix A.

4. NUMERICAL RESULTSWe consider a number of examples to show the effective-ness of the method. In the first example, we consider thepropagation of the fundamental mode through a tilted,graded-index waveguide,5 with index profile given byn2(x) 5 ns

2 1 2nsDn sech2(2x/w), ns 5 2.1455, Dn5 0.003, w 5 5 mm, and l 5 1.3 mm. The computationwas done with 530 collocation points, and the width of thenumerical window was ;185 mm. As a measure of accu-racy, we computed an error (ERR), that includes the ef-fects of both the dissipation in power as well as the loss ofshape of the propagating mode:

ERR 5 1 2

U E cexact* ccalcdxU2

E u c inpu2dx • E u cexactu2dx

, (13)

where c inp , ccalc , and cexact are the input, the propagated,and the exact fields, respectively.18

The first result for a straight waveguide, which wehave plotted in Fig. 1, shows the performance of themethod with respect to the stability of the method forrelatively larger values of Dz. The DNS based on the


Runge–Kutta solution of the collocation equation14 be-comes unstable for Dz . 0.1 mm, whereas the SSNPmethod remains stable even for 1 mm. To the best of ourknowledge, a step size equal to or larger than 1 mm fornonparaxial propagation has not been reported before.Even with such a large step, an accuracy better than0.001 in propagation over a distance of 1000 mm is signifi-cantly better than the accuracies reported in the litera-ture. In Fig. 2 we have plotted the error in propagation(ERR) as a function of the tilt angle. The figure showsthat the SSNP method gives accuracy of the order of 1024

even with a step size of 0.25 mm, which is much betterthan that obtained by Shibayama et al.5 To illustrate thepoint, let us consider the error for a tilt angle of 50°. Theerror in the best results reported by Shibayama et al.5 forthe three-step generalized Douglas (GD) scheme is ;0.04with Dz 5 0.05 mm, whereas in our method the error isless than 0.001 with Dz 5 0.25 mm. This would thusmean much faster and more accurate propagation. Ofcourse, one gets better accuracy with the DNS, since thesingle-step error in the Runge–Kutta method (used in theDNS) is O((Dz)5) as against O((Dz)3) in the SSNPmethod, but then the computation effort is significantlyreduced with the latter method.

We next consider propagation of the TE1 mode in step-index waveguides. Figure 3 shows a plot of ERR as a

Fig. 1. Error in propagation (ERR) as a function of the numberof propagation steps with Dz for the graded-index waveguide.5

Fig. 2. ERR with the tilt angle of the graded-index waveguide5

for propagation up to 100 mm.

function of the propagation steps for the step-indexwaveguide6 with nco 5 1.002, nclad 5 1.000, l 5 1.0 mm,and w 5 15.092 mm. Even with a step size as large as0.4 mm, the propagation is extremely stable and highlyaccurate, while DNS becomes unstable for this step size.Figure 4 shows performance with variation in tilt angle ofthe waveguide. We can see that the curves for thepresent method and DNS14 are very close, except for theSSNP method with step size 0.4 mm at 0°, where error ishigher. However, the error value even with 0.4-mm stepsize is better than that at 50° reported by Yamauchi et al.6

The SSNP method gives better accuracy with twice thestep size used by Yamauchi et al.6; in fact, only 500 com-putation points are required as against 1800 by Yamauchiet al.6

Figures 5 and 6 show performance of the method forthe TE1 mode in the benchmark waveguide19 with nco5 3.3, nclad 5 3.17, l 5 1.55 mm, and w 5 8.8 mm. Asthe refractive-index change from core to cladding is verylarge here, only small step sizes can be taken, yet theSSNP method is stable for a step size of 0.2 mm, as shownin Fig. 5. In fact, the performance at large tilt angleswith 0.2-mm step size is quite close to that for the DNS,14

as shown in Fig. 6. We may note that oscillatory behav-ior in the error curves becomes more pronounced for thestep-index waveguide with larger index jump at the core–

Fig. 3. ERR as a function of the number of propagation stepswith Dz for the step-index waveguide.6

Fig. 4. ERR with the tilt angle of the step-index waveguide6 forpropagation up to 100 mm.


cladding interface (compare Figs. 3 and 5). This may beattributed to the fact that any discretization would ap-proximate the index step by an interpolating curve be-tween two successive sample points around the step. Asexpected this oscillation becomes larger as Dz increasesfrom 0.01 to 0.2 mm [although the log scale deceptivelyshows nearly equal oscillations].

The final example is that of propagation of the TE10mode in the benchmark waveguide19 described above, andwe have obtained the power remaining in the guide afterpropagation of 100 mm at a tilt angle of 20°. Table 1 com-pares the SSNP method with other methods. It is quiteobvious from the table that with fewer points, the SSNPmethod shows higher accuracy. The method is also fasterthan the DNS,14 taking only about half the time. It isalso much easier to implement.

An important parameter to choose is the reference re-fractive index nr . Although, in principle, its value canbe arbitrarily chosen, the value in general affects the ac-curacy. However, as Fig. 7 shows, the accuracy with thenew method is largely insensitive to the choice of nr .

We would like to add that neither the SSNP methodnor the DNS method is very sensitive to perturbations inthe value of the initial field or its derivative. We havecarried out preliminary investigations by adding and sub-tracting a small error (1023) alternately in the initial fieldand its derivative at successive sample points. The errorin the overlap integral was 1.9 3 1022 as against 2.6

Fig. 5. ERR as a function of the number of propagation stepswith Dz for the benchmark step-index waveguide.19

Fig. 6. ERR with the tilt angle of the benchmark step-indexwaveguide19 for propagation up to 100 mm.

3 1025 for propagation of the TE1 mode at 0° for 100 mmwith a propagation step size of 0.1 mm, in the benchmarkwaveguide19 where nco 5 3.3, nclad 5 3.17, l 5 1.55 mm,and w 5 8.8 mm. Thus the propagation remains stable.

5. CONCLUSIONSWe have presented a new method for solving the non-paraxial wave equation based on a symmetrized splittingof the operator. We have implemented this method withthe collocation method. We have also included compari-son with reported results of other methods. The methodshows better stability with relatively larger step sizes be-ing possible. The method involves only simple multipli-cation of matrices, and no numerical diagonalization orinversion of any matrix is needed. It is therefore muchfaster and easier to implement and is more efficient thanother methods.

APPENDIX A: EVALUATION OF exp(H1Dz)This amounts to a solution of the collocation Eq. (10)without the R(z) term, i.e., propagation in a medium ofuniform refractive index nr over a distance Dz, that is, so-lution of the equation

d2C/dz2 1 SC~z ! 5 0, (A1)

Fig. 7. ERR with the reference refractive index for the bench-mark step-index waveguide19 for propagation up to 100 mm withstep size 0.1 mm at 40°.

Table 1. Comparison of Error–Power Loss inPropagation to 100 mm in the Benchmarka

Step-Index Waveguide for TE10 Modeswith Different Methods

Method Nz Nx Power in Waveguide at 20°

SSNP 1000 800 ;0.96DNSa 1000 800 ;0.90

AMIGOb 1429 1311 ;0.95FD2BPMb 1000 2048 ;0.95FTBPMb 1000 256 ;0.55

LETI-FDb 200 1024 ;0.15

a Ref. 14b Ref. 19.


where S 5 S0 1 k02nr

2I is a constant matrix. Equation(A1) can also be written as

]F/]z 5 H1F~z !, (A2)

where F(z) is defined in Eq. (12), and

H1 5 F 0 I

2S 0G

is a constant matrix and has to be evaluated just once. Aformal solution of Eq. (A2) can be written as

F~z 1 Dz ! 5 exp~H1Dz !F~z !. (A3)

The evaluation of exp(H1Dz) can be done by diagonaliza-tion of H1 ; however, H1 is a 2N 3 2N nonsymmetric ma-trix and its diagonalization may involve complex matrixalgebra and hence, present some difficulties. We presenthere a much simpler and analytical method for evaluatingexp(H1Dz).

Since Eq. (A1) represents propagation in a uniform me-dium, the propagation can be obtained by an eigenvaluedecomposition method. Thus the solution of Eq. (A1)over a single step can be written as

C~z 1 Dz ! 5 cos~ASDz !C~z ! 11

ASsin~ASDz !C8~z !,

(A4)

C8~z 1 Dz ! 5 2AS sin~ASDz !C~z ! 1 cos~ASDz !C8~z !.(A5)

Using this solution in Eq. (A3) gives

exp~H1Dz ! 5 F cos~ASDz !1

ASsin~ASDz !

2AS sin~ASDz ! cos~ASDz !

G .

(A6)

To evaluate the functions of the matrices involved in Eq.(A6), we use the diagonalization procedure. Thus, let S5 VLV21 where V and L are the eigenvectors and eigen-values of S, respectively. Then, we have

ASDz 5 V~ALDz !V21, AL 5 diag.~AL i! (A7)

and

cos~ASDz ! 5 V cos~ALDz !V21, (A8)

sin~ASDz ! 5 V sin~ALDz !V21. (A9)

Thus

exp~H1Dz ! 5 FV 0

0 VG3 F cos~ALDz !

1

ALsin~ALDz !

2AL sin~ALDz ! cos~ALDz !

G3 FV2 0

0 V21G . (A10)

The operator P in Eq. (12) is thus given by Eq. (A10) withDz replaced by Dz/2.

In the case of sinusoidal basis functions in the colloca-tion method,16 the form of S is such that the eigenvalue

decomposition required as per Eq. (A7) is simply doneanalytically. In this case, we choose the basis functionsas

fn~x ! 5 cos~vnx ! for n 5 1, 3, 5,...N 2 1,

5 sin~vnx ! for n 5 2, 4, 6,...N,(A11)

where vn 5 np/2L, with the computation window run-ning from 2L to L. The collocation points are at

xj 5 S 2j

N 1 12 1 DL, j 5 1, 2, 3...,N. (A12)

The matrix S in this case is then given by16,17

S 5 AGA21 1 k02nr

2I 5 A~G 1 k02nr

2I!A21,(A13)

where A is a constant square matrix with elements asAij 5 f j(xi) and the matrix G is given by

G 5 diag.~2v12 2v2

2 2v32...2vN

2!. (A14)

Thus we have

V 5 A, L i 5 k02nr

2 2 vi2. (A15)

Further, it can be shown that

V21 5 A21 5 S 2

N 1 1 DAT. (A16)

Thus no matrix eigenvalue equation need be solved.With these values of V and L, one obtains the followingfrom Eq. (A10):

exp~H1Dz ! 5 S 2

N 1 1 D FA 0

0 AG

3 3c1 0 ¯ 0 s1 0 ¯ 0

0 c2 ¯ 0 0 s2 ¯ 0

] ] � ] ] ] � ]

0 0 ¯ cN 0 0 ¯ sN

s1 0 ¯ 0 c1 0 ¯ 0

0 s2 ¯ 0 0 c2 ¯ 0

] ] � ] ] ] � ]

0 0 ¯ sN 0 0 ¯ cN

43 FAT 0

0 ATG, (A17)

where

ci 5 cos~AL iDz !, si 5 1/AL i sin~AL iDz !,

s i 5 2AL i sin~AL iDz !.

In cases where L i is imaginary [see Eq. (A15)], the quan-tities ci , si , and s i remain real, and sine and cosine func-tions are evaluated through the corresponding hyperbolicfunctions.


ACKNOWLEDGMENTSThis work was partially supported by grant 03(0976)/02/EMR-II from the Council of Scientific and Industrial Re-search (CSIR), India. A. Agrawal is a CSIR research fel-low.

Corresponding author A. Sharma’s e-mail address [email protected].

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New method for nonparaxial beam propagation