Electromagnetic beam diffraction by a finite lamellar structure: an ...

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Electromagnetic beam diffraction by a finite lamellar structure: an aperiodic coupled-wave method Brahim Guizal Laboratoire d’Optique P. M. Duffieux, 16, route de Gray, FR-25030 Besanc¸on Cedex, France

Dominique Barchiesi Laboratoire de Nanotechnologie et Instrumentation Optique, 12 rue Marie Curie, B.P. 2060, FR-10010 Troyes Cedex, France

Didier Felbacq Groupe d’Etude des Semiconducteurs, Unite´ Mixte de Recherche, Centre de la Recherche Scientifique 5650, Baˆtiment 21, CC074, Place Eugeˆne Bataillon, 34095 Montpellier Cedex 5, France Received February 14, 2003; revised manuscript received June 6, 2003; accepted July 31, 2003 We have developed a new formulation of the coupled-wave method (CWM) to handle aperiodic lamellar structures, and it will be referred to as the aperiodic coupled-wave method (ACWM). The space is still divided into three regions, but the fields are written by use of their Fourier integrals instead of the Fourier series. In the modulated region the relative permittivity is represented by its Fourier transform, and then a set of integrodifferential equations is derived. Discretizing the last system leads to a set of ordinary differential equations that is reduced to an eigenvalue problem, as is usually done in the CWM. To assess the method, we compare our results with three independent formalisms: the Rayleigh perturbation method for small samples, the volume integral method, and the finite-element method. © 2003 Optical Society of America OCIS codes: 050.1960, 260.1960, 260.2110.

1. INTRODUCTION The problem of the diffraction of electromagnetic waves by gratings has been extensively studied in the past, since these devices are of great practical interest in spectroscopy, photolithography, diffractive optics, etc. Several methods exist to model diffraction from such structures. One can distinguish differential formalisms1,2 from integral ones.3,4 Among these approaches, the coupled-wave method (CWM) is widely used because of its implementation simplicity and effectiveness, at least for lamellar structures. The major part of the published research deals with ideal diffraction gratings, i.e., those with strictly periodic infinite boundaries. However, in some practical situations (such as finite gratings, Fresnel-like lenses, and diffraction from a single defect deposited on an interface), the structures are no longer periodic, and one needs to consider aperiodic scatterers. For the differential method (which is close in principle to the CWM), some authors have used their codes that deal with gratings to treat the aperiodic case.5 They have considered that the structure belongs to a grating whose period is sufficiently large to avoid interaction with neighboring elements. More recently that method has been extended to the case of aperiodic structures and applied to the study of diffraction from rough inhomogeneous dielectric films.6 Other approaches have been introduced to deal with such a problem.7 In the aperiodic approaches the integration 1084-7529/2003/122274-07$15.00

involved in the equations has to be taken only over the interval occupied by the defects. This fact is of fundamental importance from a numerical point of view. Indeed, the large-period approach with the CWM, for example, leads to very small filling ratios that imply numerical problems. In other words, the aperiodic scheme allows an arbitrary choice of the integration step, in contrast to the periodic model, which considers a discretization of the wave vectors as harmonics of the large period of the sample. A similar periodic-to-aperiodic development has been introduced for a perturbation scheme of the Rayleigh method.8,9 Let us add that some authors have claimed that the CWM cannot be used to treat diffraction by finite periodic and aperiodic structures.10 In this paper we propose an alternative approach that extends the CWM to the more realistic case of aperiodic lamellar structures illuminated by a finite incident beam. We address the problem of the diffraction of a limited [TE(s-) or TM- ( p-) polarized] beam by a finite number of rectangular dielectric rods that are deposited on the interface between two dielectric media. The fields are expressed in terms of the Rayleigh expansions above and below the rods. In the rods’ region, the wave equations lead to an infinite integrodifferential system that is transformed into an eigenvalue problem after discretization. Then the boundary conditions are used to derive a set of algebraic equations whose solution gives the unknown Ray© 2003 Optical Society of America

Guizal et al.

Vol. 20, No. 12 / December 2003 / J. Opt. Soc. Am. A

leigh coefficients of the fields in both upper and lower media. It is worth noting that, for the method to be stable, we use the S-matrix algorithm.11,12 After introducing the theory in Section 2, in Section 3 we give details on the numerical implementation. In Section 4 we provide some numerical examples and compare our results with those obtained by use of the Rayleigh perturbation method (RPM), the volume integral method (VIM), and the finite-element method (FEM). The RPM has been used successfully to describe near-field optical microscopy, in which the dielectric contrast is small enough to allow consideration of only the first order of perturbation.9,13 The VIM was developed first in Ref. 14. This method has a larger domain of application than the Rayleigh perturbation scheme, as it is not limited by the dielectric contrast. Both methods, as well as the CWM, use a plane-wave spectrum. In contrast, the FEM is a direct-space method and is based on a Galerkin variational principle to solve Maxwell’s equations in a finite domain with boundaries and boundary conditions. The principle of the method is to replace an entire continuous domain by a number of subdomains in which the electromagnetic field is represented by interpolation functions with unknown coefficients.15 The partial differential equation from Matlab Toolbox has been used for these calculations. Finally, applications of the method to the case of a Fresnel-like lens and a beam splitter are proposed in Section 5.

u 1 共 x, y 兲 ⫽

冕

⫹⬁

⫺⬁

⫹

2275

I 共 ␣ 兲 exp共 ⫺i ␣ x 兲 exp关 i ␤ 1 共 y ⫺ h 兲兴 d␣

冕

⫹⬁

⫺⬁

R 共 ␣ 兲 exp共 ⫺i ␣ x 兲 exp关 ⫺i ␤ 1 共 y ⫺ h 兲兴 d␣ , (1)

where I( ␣ ) is the spectral distribution of amplitudes of the incident wave and R( ␣ ) is that of the backward diffracted wave. Similarly, in the lower medium, the total field can be written as u 3 共 x, y 兲 ⫽

冕

⫹⬁

⫺⬁

T 共 ␣ 兲 exp共 ⫺i ␣ x 兲 exp共 i ␤ 3 y 兲 d␣ ,

(2)

where T( ␣ ) is the spectral amplitude of the transmitted wave, k p ⫽ k 0 冑␧ p , and ␤ p ⫽ 冑k p 2 ⫺ ␣ 2 , p ⫽ 1,3. ␤ p is defined assuming Re(␤p) ⫹ Im(␤p) ⬎ 0. In region 2 the field can be expressed in terms of the Fourier integral: E 2 共 x, y 兲 ⫽

冕

⫹⬁

⫺⬁

e 共 ␣ , y 兲 exp共 ⫺i ␣ x 兲 d␣

for TE polarization,

h 共 ␣ , y 兲 exp共 ⫺i ␣ x 兲 d␣

for TM polarization.

H 2 共 x, y 兲 ⫽

冕

⫹⬁

⫺⬁

(3)

2. THEORY The structure under study is depicted in Fig. 1. The space is divided into three regions: regions 1 (upper halfspace: y ⬎ h) and 3 (lower half-space: y ⬍ 0) are assumed to be dielectric and homogeneous with dielectric permittivities ␧ 1 and ␧ 3 , respectively, and region 2, which consists of a dielectric slab (␧ s ) with P defects (␧ d,1 ,...,␧ d,P ) embedded in it. The structure is invariant along the z direction. The device is illuminated by a TE (electric field parallel to the z axis) or TM (magnetic field parallel to the z axis) monochromatic electromagnetic beam under mean incidence ␪ 0 with vacuum wavelength ␭. This beam is generated by a source located at (x 0 , y 0 ). Throughout this paper we assume an exp(i␻t) time dependence. The z component of the electric or the magnetic field will be denoted by u(x, y). In the upper medium we express the total field in terms of plane waves:

The problem is to determine amplitudes R and T from which the total field can be calculated in regions 1 and 3. For that purpose, one must solve Maxwell’s equation in region 2 and then write the boundary conditions at the interfaces y ⫽ 0 and y ⫽ h. Thus coefficients R( ␣ ) and T( ␣ ) can be derived as is usually done in the CWM (Ref. 11, for example). The only thing that needs to be changed is that the incident amplitudes must be replaced by the wanted distribution I( ␣ ). In the following, we will distinguish the two cases of polarization. A. TE Polarization In this case the electric field in region 2 is a solution of the following equation, ⌬E 2 共 x, y 兲 ⫹ k 0 2 ␧ 2 共 x 兲 E 2 共 x, y 兲 ⫽ 0,

(4)

which can also be written as ⌬E 2 共 x, y 兲 ⫹ k 0 2 关 ␧ 2 共 x 兲 ⫺ ␧ s 兴 ⫻ E 2 共 x, y 兲 ⫹ k 0 2 ␧ s E 2 共 x, y 兲 ⫽ 0.

(5)

Introducing the Fourier transform of ␧ 2 (x) ⫺ ␧ s ⫹⬁ ˆ ⫽ 兰 ⫺⬁ ␧ ( ␣ )exp(⫺i␣ x)d␣ and taking the Fourier transform of Eq. (5) leads to ᭙ ␣ 苸 R, Fig. 1.

Geometry of the problem.

⳵ 2e共 ␣ , y 兲 ⳵y2

⫺ ␣ 2e共 ␣ , y 兲 ⫹ k 02␧ se共 ␣ , y 兲 ⫹ k 0 2 ␧ˆ 共 ␣ 兲 * e 共 ␣ , y 兲 ⫽ 0,

(6)

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J. Opt. Soc. Am. A / Vol. 20, No. 12 / December 2003

⳵ 2e共 ␣ , y 兲

᭙ ␣ 苸 R,

⳵y2

Guizal et al.

⳵

⫽ 共 ␣ 2 ⫺ k 02␧ s 兲e共 ␣ , y 兲

⫺ k 02

冕

⫹⬁

⫺⬁

⳵x

This is an integrodifferential set of equations that can be reduced to a set of ordinary differential equations by means of a standard point-matching technique. First, the interval of integration is limited to 关 ⫺␩ k 1 , ␩ k 1 兴 , where ␩ is a positive real scaling number, and then the unknown function is evaluated at N points ␣ n (Fig. 2). The resulting equation is

⫽ 共 ␣ n2 ⫺ k 02␧ s 兲e共 ␣ n , y 兲 ⫺ k 02

兺 e共 ␣

m

冋冕

␣ m ⫹␦ ␣ /2

␣ m ⫺␦ ␣ /2

册

␧ˆ 共 ␣ n ⫺ ␥ 兲 d␥ ,

(8)

d E/dy ⫽ AE共 y 兲 , 2

2

(9)

with E( y) ⫽ 关 e( ␣ 1 , y),...,e( ␣ N , y) 兴 t and A as a matrix whose elements are obtained from Eq. (8). We are now brought back to the case of the standard CWM, and one can apply the eigenvalue technic to solve the last system. Then, writing the boundary conditions allows us to deduce the spectral amplitudes R( ␣ n ) and T( ␣ n ). B. TM Polarization In this case the magnetic field in region 2 is the solution of the following equation:

冋

⳵ x ␧ 2共 x 兲

冋

⫹

1 ⳵ 2 H 2 共 x, y 兲 ␧s

⳵x2

1 ⳵ 2 H 2 共 x, y 兲 ␧s

⫹ k 0 2 H 2 共 x, y 兲 ⫽ 0.

⳵y2

(11)

Introducing the Fourier transform of 1 / 关 ␧ 2 (x) 兴 ⫺ 1 /␧ s ⫹⬁ ˆ ⫽ ␨ (x) ⫽ 兰 ⫺⬁ ␨ ( ␣ )exp(⫺i␣x)d␣ and taking the Fourier transform of the last equation leads to

冕 冋 ⫹⬁

⫺⬁

1 ␧s

␦ 共 ␣ ⫺ ␥ 兲 ⫹ ␨ˆ 共 ␣ ⫺ ␥ 兲

冉

␣2 ␧s

册

⳵ 2h共 ␥ , y 兲 ⳵y2

冊

⫺ k 02 h共 ␣ , y 兲 ⫹ ␣

冕

⫹⬁

⫺⬁

d␥

␥ ␨ˆ 共 ␣ ⫺ ␥ 兲 h 共 ␥ , y 兲 d␥

, y兲

where ␦ ␣ is the discretization step in the ␣ space. This can be written in compact form as follows:

1

⳵x

册 册

(12)

m

⫻

⳵

⳵ H 2 共 x, y 兲

⫹ ␨共 x 兲 ⫹

⫽

2

⳵y2

␨共 x 兲

␧ˆ 共 ␣ ⫺ ␥ 兲 e 共 ␥ , y 兲 d␥ . (7)

⳵ e共 ␣n , y 兲

冋

⳵ H 2 共 x, y 兲 ⳵x

册 冋 ⫹

⳵

⳵ H 2 共 x, y 兲

1

⳵ y ␧ 2共 x 兲

⳵y

册

⫹ k 0 2 H 2 共 x, y 兲 ⫽ 0

or

兺 m

⫻

⳵ 2h共 ␣ m , y 兲 ⳵y2

再冕 冋 ␣ m ⫹␦ ␣ /2

␣ m ⫺␦ ␣ /2

1 ␧s

⫽

册 冎

␦ 共 ␣ n ⫺ ␥ 兲 ⫹ ␩ˆ 共 ␣ n ⫺ ␥ 兲 d␥

冉

␣ n2 ␧s

⫻

冊

兺 h共 ␣

⫺ k 02 h共 ␣ n , y 兲 ⫹ ␣ n

冋冕

␣ m ⫹␦ ␣ /2

␣ m ⫺␦ ␣ /2

册

␥ ␩ˆ 共 ␣ n ⫺ ␥ 兲 d␥ .

or

, y兲

(13)

Here again we have to transform the set of integrodifferential equations into an ordinary set of differential equations and retrieve the possibility of using the standard CWM. Using the same scheme of discretization as for the TE case, we obtain the system d2 H /dy 2 ⫽ AH共 y 兲 ,

(10)

m

m

(14)

with H( y) ⫽ 关 h( ␣ 1 , y),...,h( ␣ N , y) 兴 t and A as a matrix whose elements are obtained from Eq. (13). As in the TE case, we are now brought back to the case of the standard CWM, and then we can calculate R( ␣ n ) and T( ␣ n ) by using the same procedure. At this stage, it is important to say that, as in Ref. 11, Li’s Fourier factorization rules are used. This leads to the replacement of [兩1/␧兩] by 关 兩 ␧ 兩 兴 ⫺1 . To illustrate the difference between the aperiodic CWM (ACWM) and the classical CWM, let us consider a D-periodical function ␨ (x), which one can write with n integer:

␨共 x 兲 ⫽

兺␨

n

冉

exp ⫺in

n

2␲ D

冊

x .

(15)

Therefore Fig. 2. Discretization of the spectral space. The parameter ␩ is used to control the number of evanescent waves accounted for.

␨ˆ 共 ␣ 兲 ⫽

1 2␲

冕

⫹⬁

⫺⬁

␨ 共 x 兲 exp共 i ␣ x 兲 dx

(16)

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Vol. 20, No. 12 / December 2003 / J. Opt. Soc. Am. A

⫽

1

兺␨ 2␲ n

n

冕

冋冉 冉

process, the number of propagating waves taken into account is 2P ⫹ 1, where P is the integer part of N/2␩ .

⫹⬁

⫺⬁

⫻ exp i ␣ ⫺ n

⫽

兺␨ n

n␦

␣⫺n

2␲ D

2␲ D

冊

冊册

x dx

.

(17)

4. NUMERICAL EXAMPLES

(18)

In all the numerical results presented below, the incident monochromatic beam is assumed to be of the Gaussian type. It is produced by a source located at (x 0 , y 0 ). The distribution of amplitudes in the incident wave is assumed to be of the form

Consequently, the considered values of ␣ are related to the period D, in the periodic case, in contrast to the aperiodic case, in which the discretization of the wave vectors can be independent of the lateral size of the diffractive sample. This fact is of fundamental importance from a numerical point of view. Indeed, the large-period approach with the CWM leads to very small filling ratios that imply numerical problems especially in resonant cases, in which the confinement of light requires high values of N to be accurately described. Moreover, let us note that in the classical CWM, the artificial periodicity of the sample can induce aliasing effects if N is too high, in contrast to the aperiodic model. Therefore, to avoid the aliasing effect, one should increase the artificial period of the sample, and then the filling ratio decreases. If we substitute Eq. (18) into Eq. (13), for example, in TM polarization, we obtain

兺

冋

⳵ 2h共 ␣ m , y 兲 1 ⳵y2

m

⫽

冉

␣ n2 ␧s

␧s

␦ 共 m ⫺ n 兲 ⫹ ␨ m⫺n

冊

⫺ k 02 h共 ␣ n , y 兲 ⫹ ␣ n

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I共 ␣ 兲 ⫽

冋

册

w2 exp共 i ␣ x 0 ⫹ i ␤ y 0 兲 exp ⫺ 共 ␣ ⫺ ␣ 0 兲 2 , 4 2 冑␲ (21) w

where w is the waist of the beam at (x 0 , y 0 ) and ␣ 0 ⫽ k 1 sin(␪0), with ␪ 0 denoting the mean incidence of the beam. As a first example, we consider a defect on an interface (Fig. 3), and we calculate the normalized transmitted intensity I as the ratio of the square modulus of the total electric field to the square modulus of the field that would exist in the absence of the defect (that is due only to the interface):

册

兺␨

n⫺m ␣ m h 共 ␣ m

, y 兲.

m

(19) By using the above definition of ␨ m , one can deduce Eq. (11) of Ref. 11, which leads to the eigenvalue problem,

兺 m

冋

⳵ 2h共 ␣ m , y 兲 1 ⳵y2

␧s

␦ 共 m ⫺ n 兲 ⫹ ␨ m⫺n

⫽ ⫺k 0 2 h 共 ␣ n , y 兲 ⫹ ␣ n

册

兺 m

⫻

冋

1 ␧s

册

␦ 共 m ⫺ n 兲 ⫹ ␨ m⫺n ␣ m h 共 ␣ m , y 兲 ,

(20)

where (1/␧ s ) ␦ (m ⫺ n) ⫹ ␨ m⫺n corresponds to ˜␧ 2m⫺n of the periodic structure in Ref. 11.

3. NUMERICAL IMPLEMENTATION We checked the convergence of the results by increasing integer N and by enlarging the range of variable ␣ through the parameter ␩. We used the usual criteria of energy balance, reciprocity, and convergence with respect to N. The discretization of the fields in the spectral space implies that they are numerically periodized in the x space (this is a well-known property of the Fourier transform). This means that the choice of the number of samples must be made carefully in order to avoid the aliasing phenomenon. Note that with this discretization

Fig. 3. Intensity below the defect in the plane y ⫽ ⫺␭/20. TE or s polarization, (b) TM or p polarization.

(a)

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the boundaries of the calculation but around the center. This fact could be a drawback of the CWM even if one did the windowing of the calculated signal in order to avoid side effects, like in s polarization. In the second example, we consider a ␭-square dielectric cylinder lying in vacuum. This case cannot be treated by the RPM because it is out of its domain of validity. The comparison will be done with the VIM, the FEM, and the classical CWM. Since the first two methods are direct-space methods, i.e., they use a discretization in xy space, and the size of cells needed to compute the fields accurately is approximately ␭/10. This gives an idea to those who are familiar with the VIM and the FEM about the number of cells to be used with such a structure. In Fig. 4(a) we give the TE-transmitted intensity, at y ⫽ ⫺␭/10, calculated by the four methods. For the ACWM, we used N ⫽ 601 and ␩ ⫽ 10. A good agreement is observed among the four methods. To give an idea about the convergence of the ACWM in this last case, we plot in Fig. 4(b) the error on the intensity I N at (x ⫽ ␭/2, y ⫽ ⫺␭/10) versus the truncation order N:

冏

⌬ ⫽ log10

Fig. 4. (a) Intensity below the ␭-square cylinder in the plane y ⫽ ⫺␭/10. (b) Convergence of the intensity at (x ⫽ ␭/2, y ⫽ ⫺␭/10) for the ␭-square cylinder. The polarization of the incoming light is TE.

I

⫽

冦

兩 E z 共 x, y 兲 兩 2 兩 E z 0 共 x, y 兲 兩 2 兩 E x 共 x, y 兲 兩 2 ⫹ 兩 E y 共 x, y 兲 兩 2 兩 E x 0 共 x, y 兲 兩 2 ⫹ 兩 E y 0 共 x, y 兲 兩 2

for TE polarization

冏

I N ⫺ I 601 I 601

,

where we assume that I 601 is the ‘‘exact’’ value. On the other hand, the classical CWM shows a higher intensity level than other methods, especially at the edges of the domain of calculation. This is due to the grating effect induced by the periodicity. To compare ACWM and CWM calculations, we assume that the period used for the CWM is equal to 10␭ (the number of terms in the Fourier series is N ⫽ 601). Consequently, we observe that a larger period should be considered to get convergence in the CWM. As expected, the ACWM converges more rapidly than the periodic CWM.

. for TM polarization (22)

In this case we set ␪ 0 ⫽ 0 and w ⫽ 40␭. The width of the defect on the glass–air interface is ␭ /2, and its height is ␭ /10. The permittivity of the upper (lower) medium is ␧ 1 ⫽ 2.25 (␧ 3 ⫽ 1). The permittivity of the defect is ␧ 2 ⫽ 2.25, and the source is located at (x 0 ⫽ ␭/4, y 0 ⫽ 10␭). Figures 3(a) and 3(b) show the transmitted intensity, for the two cases of polarization, at y ⫽ ⫺␭/20, calculated by the ACWM (N ⫽ 601, ␩ ⫽ 10), the RPM, and the VIM. It can be seen that the agreement is better among the ACWM, the VIM, and the classical CWM. This can be understood by remembering that the RPM method used is limited to the first order of perturbation9 and is valid if the dielectric contrast of the sample satisfies the relation 2 ␲ (␧ 2 ⫺ ␧ s ) Ⰶ ␭. One may observe a very good agreement between the VIM and the ACWM. In s polarization the level of intensity at the size of the calculated CWM curve is too high with respect to the results of the other models. This shows that the period of the CWM is too small to separate the periodic objects. In p polarization, in contrast, the two arrows locate two minima of the ACWM and the CWM and show that the convergence of the CWM is not yet achieved. In contrast to s polarization, the slow convergence is not visible on

5. EXAMPLES OF APPLICATION To illustrate the effectiveness of the method, we present two examples of application. The first one is dedicated to focusing with a micro-Fresnel lens and the second to a beam splitter. A. Micro-Fresnel Lens Fresnel lenses have been used for a long time to concentrate light to a focus. Figure 5(a) shows a transmission Fresnel-like lens that is invariant along the z axis. The profile is formed by a set of rods whose widths and locations are determined by the values x n ⫽ 冑nL, where L is related to the focal length f, the wavelength in vacuum ␭, and the angle of incidence ␪ i by L ⫽ 冑f ␭/cos(␪i). In our example we choose a lens etched in a dielectric material (␧ ⫽ 2.25) with f ⫽ 10 ␮ m. The height of the rods is h ⫽ 1 ␮ m. This device is illuminated by a large TE-polarized beam under normal incidence. The wavelength is ␭ ⫽ 1 ␮ m, which implies that L ⫽ 冑10 ␮ m. By retaining five rods in the simulation, we obtain the nearfield plot (square modulus of the electric field), shown in Figs. 5(b) and 5(c), where the electric field is normalized by its maximum value. It can be seen that the incident beam is focused at the expected location. Figure 5(b) shows a comparison between ACWM and FEM calculated

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intensities in the focal plane. As expected, the central peaks are of the same size. The small differences between the two curves are due to the boundary conditions on the side of the domain of calculation we considered for the FEM. Actually, we consider nonreflecting boundary conditions on the boundaries of the domain, and the domain of calculation is of the same size in both models; therefore the FEM furnishes rough calculations, and we can observe edge effects. In this computation we used a truncation order N ⫽ 401, with ␩ ⫽ 10, and a beam waist w ⫽ 40␭ for the ACWM. B. Beam Splitter As a second application, we consider the beam splitter shown in Fig. 6(a). It is made up of two Fresnel lenses spaced by ␭. This device is illuminated under normal incidence by a TE-polarized beam whose beam waist is w ⫽ 10␭. Figure 6(b) shows the normalized near-field intensity (square modulus of the electric field) at y ⫽ ⫺10␭. The beam is split into two beams propagating in opposite directions. In this case we used the same parameters, N and ␩, as for the Fresnel lens. As exFig. 6. (a) Geometry of the Fresnel beam splitter. (b) Intensity in the focal plane of the two Fresnel lenses ( y ⫽ ⫺10␭). The polarization of the incoming light is TE.

pected, the ACWM predicts the beam splitting, obtained with two ␭-separated Fresnel lenses. We also plot, in Fig. 6(b), the FEM results so that comparison can be made with the ACWM. Let us note that the calculation time of the FEM is almost 10 times higher than that of the ACWM. As a consequence, in the FEM simulation, we choose a calculation window that is only 34␭ wide. This explains the differences observed between the ACWM and the FEM results. Yet, let us note that the main peaks are of the same size and are at the same position.

6. CONCLUSION In this paper we have presented an aperiodic version of the coupled-wave method (CWM) and compared its results with other existing methods. The formulation takes advantage of the versatility of the CWM and its effectiveness. We showed that it can be applied to, for instance, the study of optical diffractive elements, microlenses, and beam splitters. The numerical examples that have been given to illustrate the method are not restrictive; the shape of the beam as well as the lengths of the defects and the spacings between them can be changed. This method can also be applied to nonlamellar aperiodic structures by use of the so-called staircase approximation. In this last case, in order to have a stable method, it is obligatory to use the S-matrix algorithm (used in our implementation) or an equivalent formulation.

REFERENCES Fig. 5. (a) Geometry of the Fresnel-like lens. (b) Intensity in the focal plane of the lens ( y ⫽ ⫺10␭). (c) Map of the intensity around the lens. The polarization of the incoming light is TE.

1.

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