Improved first Rayleigh-Sommerfeld method applied to metallic cylindrical focusing micro mirrors

Jia-Sheng Ye; Yan Zhang; Kazuhiro Hane

doi:10.1364/OE.17.007348

1. Introduction

With the development of modern micro photolithography technology, diffractive optical elements (DOEs) can be fabricated with a small feature size, little weight, great versatility and low cost. Because of their great advantages over the conventional optical elements, DOEs are widely used in many practical applications, such as laser-beam focusing, spectral filtering, optical disk readout, wavelength division multiplexing, and so on [1]. In recent years, many researchers have analyzed and designed dielectric DOEs based on the scalar diffraction theory in the scalar domain [2, 3]. As the characteristic scale of the DOEs becomes smaller than the incident wavelength, the scalar diffraction methods gradually appear to be quantitatively inaccurate [4]. Therefore, rigorous electromagnetic theory must be adopted to analyze the optical properties of the dielectric DOEs [5, 6, 7] and perfectly conducting structures [8, 9]. Performance analysis of metallic structures with finite conductivity has also been presented in literatures by the vectorial diffraction methods, such as the finite-difference time-domain method [10] and the boundary element method (BEM) [11, 12].

Although the vectorial theoretical methods can provide accurate analysis of DOEs, a large numerical burden is always encountered in the computations [13, 14], especially for metallic structures owing to a new freedom of electric conductance. Therefore, an approximate method with high accuracy and high efficiency is required. Recently, we proposed an improved first Rayleigh-Sommerfeld method (IRSM1), and applied it to analysis of two-dimensional (2-D) cylindrical microlenses [15]. In the IRSM1, a local boundary approximation model is adopted directly on the actual microlens boundary to determine the local transmitted field. In Ref. [15], since we do not need to solve the complex boundary integral equations in the IRSM1, the IRSM1 costs much less computational time and computer memory than the BEM. In addition, the accuracy of the IRSM1 is much higher than the original first Rayleigh-Sommerfeld method (ORSM1), especially for analysis of cylindrical refractive microlenses with small f-numbers [16]. The strategy of adopting the finite-thickness model of the microlenses made these significant improvements, which was also proved in designs of small f-number microlenses [17].

Despite that the IRSM1 has already been demonstrated to be highly efficient in 2-D dielectric systems, its applicability to 2-D lossy systems with metallic elements or three-dimensional (3-D) systems has not yet been investigated. In these systems, the significance of savings in computational time and computer memory becomes more obvious. In this paper, we firstly generalize the IRSM1 to analyzing the focal properties of various 2-D metallic cylindrical focusing micro mirrors, with different f-numbers (from f/1.5 to f/0.33), different polarization of incidence (TE and TM), and different quantization levels of profiles (from 2-level to 16-level). The focal properties include the focal spot size, the diffraction efficiency, the real focal length, the total reflected power, and the normalized sidelobe power. On taking the thickness of the cylindrical mirrors into account, we can expect that the numerical results obtained by the IRSM1 are much more accurate than those by the ORSM1. To show the superiority of the IRSM1 to the ORSM1, numerical results evaluated by the rigorous BEM are also presented as an accurate reference.

This paper is organized as follows. In Section 2, fundamental theories and integral equations of the BEM, the IRSM1, and the ORSM1 for 2-D scattering of metallic structures are described. Angular spectral expressions for calculating the diffraction efficiency, the total reflected power, and the sidelobe power are also given in this section. In Section 3, the focal properties of various metallic cylindrical focusing micro mirrors calculated by using the three methods mentioned above are presented together with detailed analysis. Section 4 follows with a brief summary and some discussions.

2. Fundamental theories and integral equations

2.1. Boundary element method

Fig. 1. (a) Geometry of a two-dimensional (2-D) metallic cylindrical refractive focusing micro mirror. (b) Geometry of a 2-D metallic cylindrical diffractive focusing micro mirror.

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Figure 1(a) depicts the geometry of a 2-D metallic cylindrical refractive focusing micro mirror in the cross-sectional plane. The boundary Γ represents the mirror boundary, which separates the whole space into two semi-infinite homogenous regions S ₁ and S ₂. The upper region S ₁ is free space and the lower region S ₂ is filled with metal; their refractive indices are n ₁ and ñ₂ = n ₂ - jκ ₂, respectively. The xy plane indicates the incident plane. A unit-amplitude plane wave is normally incident upon the boundary Γ of the cylindrical mirror, and it is finally focused in region S ₁ after reflection. Applying Green’s theorem to Maxwell’s equations and incorporating the boundary conditions, we obtain the integral formulations for the total fields in region S ₁ and S ₂ as [4, 5, 8, 12]

- ϕ_{1}^{t} (r_{1}) + \int_{Γ} [ϕ_{Γ} ({r'}_{Γ}) \frac{\partial G_{1} (r_{1}, {r'}_{Γ})}{\partial \hat{n}} - p_{1} G_{1} (r_{1}, {r'}_{Γ}) \frac{\partial ϕ_{Γ} ({r'}_{Γ})}{\partial \hat{n}}] d l' = - ϕ^{inc} (r_{1}), r_{1} \in S_{1},

ϕ_{2}^{t} (r_{2}) + \int_{Γ} [ϕ_{Γ} ({r'}_{Γ}) \frac{\partial G_{2} (r_{2}, {r'}_{Γ})}{\partial \hat{n}} - p_{2} G_{2} (r_{2}, {r'}_{Γ}) \frac{\partial ϕ_{Γ} ({r'}_{Γ})}{\partial \hat{n}}] d l' = 0, r_{2} \in S_{2},

where ϕ = E_z and p_i = 1 (i = 1,2) for TE polarization; ϕ = H_z and p_i = n ² ₁ (or ñ² ₂) for TM polarization. ϕ ^t and ϕ ^inc represent the total and incident fields, respectively. G_i is the 2-D exact Green’s function, i.e., G_i(r _i,r ^′ _Γ) = (-j/4)H ⁽²⁾ ₀ (k_i∣r _i -r^′ _Γ∣); H ⁽²⁾ ₀ is the zeroth-order Hankel function of the second kind. r ₁, r ₂, and r ^′ _Γ are the position vectors of a point in region S ₁, in region S ₂, and a source point at the boundary Γ, respectively. k_i is the wave number in region S_i (i = 1,2). Since the refractive index ñ₂ of region S ₂ is complex, we should pay much attention that the wave vector k ₂ is complex and H ⁽²⁾ ₀ is a function with a complex argument. n̂ indicates the boundary unit normal vector towards region S ₁, as shown in Fig. 1(a). dl ^′ denotes the sampling element along the boundary Γ.

When r _i (i = 1,2) approaches r _Γ [18], Eqs. (1a) and (1b) become [4, 5, 8, 12]

(\frac{θ_{Γ}}{2 π} - 1) ϕ_{Γ} (r_{Γ}) + ⨏_{Γ} [ϕ_{Γ} ({r'}_{Γ}) \frac{\partial G_{1} (r_{Γ}, {r'}_{Γ})}{\partial \hat{n}} - p_{1} G_{1} (r_{Γ}, {r'}_{Γ}) \frac{\partial ϕ_{Γ} ({r'}_{Γ})}{\partial \hat{n}}] d l' = - ϕ^{inc} (r_{Γ}),

(\frac{θ_{Γ}}{2 π}) ϕ_{Γ} (r_{Γ}) + ⨏_{Γ} [ϕ_{Γ} ({r'}_{Γ}) \frac{\partial G_{2} (r_{Γ}, {r'}_{Γ})}{\partial \hat{n}} - p_{2} G_{2} (r_{Γ}, {r'}_{Γ}) \frac{\partial ϕ_{Γ} ({r'}_{Γ})}{\partial \hat{n}}] d l' = 0,

where θ _Γ is an angle exterior to region S ₁ at the boundary [5, 16], as shown in Fig. 1(a); ⨏_Γ denotes Cauchy’s principal value of integration. Through a quadratic interpolation technique in the sampling element at the boundary, the boundary fields and their normal derivatives can be numerically calculated [19]. In this paper, we calculate the scattered field in region S ₁ rather than the total field, because the scattered field is related to the focal properties including the focal spot size, the diffraction efficiency, the real focal position, the total reflected power, and the normalized sidelobe power. The scattered field at any point r ₁ in region S ₁ is obtained from Eq. (1a) by the BEM as

ϕ_{1}^{sc (BEM)} (r_{1}) = \int_{Γ} [ϕ_{Γ} ({r'}_{Γ}) \frac{\partial G_{1} (r_{1}, {r'}_{Γ})}{\partial \hat{n}} - p_{1} G_{1} (r_{1}, {r'}_{Γ}) \frac{\partial ϕ_{Γ} ({r'}_{Γ})}{\partial \hat{n}}] d l' .

2.2. Original and improved first Rayleigh-Sommerfeld method

The ORSM1 differs from the BEM in these two aspects. Firstly, in the ORSM1 the boundary fields are approximately specified by the incident fields and the Fresnel reflection coefficients, while the global coupling effects among the boundary fields at different positions are neglected. In contrast, in the BEM the boundary fields are accurately derived from the boundary integral equations. Secondly, in the ORSM1 the surface-relief profile of the mirror is roughly replaced by a straight line boundary Γ_r, instead of the actual curved boundary Γ of the cylindrical mirror in the BEM, as shown in Fig. 1(a).We choose the Dirichlet boundary conditions in the ORSM1, i.e., only the boundary fields are specified. The boundary field amplitude is given by the product of the incident amplitude and the Fresnel reflection coefficient at normal incidence. The optical path difference due to the reflection from a metal surface is taken into account in the phase of the boundary field. Thus, the boundary field at Γ_r in the ORSM1 reads [4, 12]

ϕ_{Γ_{r}} (r_{Γ_{r}}) = R_{0} w (x) \exp [- jδ (x)],

where R ₀ is the Fresnel reflection coefficient at the flat boundary Γ_r, i.e., R ₀ = (n ₁ - ñ₂)/(n ₁+ñ₂) for TE polarization and R ₀ = (ñ₂ −n ₁)/(ñ₂ + n ₁) for TM polarization [20]. r _{Γ_r} represents the position vector at the flat boundary Γ_r, and n̂_t is the unit normal vector at Γ_t. The phase term [12] is given by δ(x) = -2n ₁ k ₀ h(x), where h(x) is the surface-relief depth function of the mirror boundary Γ, as shown in Fig. 1(a). A cosine window function w(x) is introduced to weaken the scattering of the incident fields at the edges of the incident aperture [5].

An alternative Green’s function is developed to match the Dirichlet boundary conditions in the ORSM1 [21]. Assuming that r ^′ ₁ and r ^′ ₂ are the mirror images of each other with respect to the flat boundary Γ_r, i.e., (x ^′ ₁,y ^′ ₁) = (x ^′ ₂,−y ^′ ₂), we can write the alternative Green’s function as [21]

G_{i}^{RS1} (r_{i}, {r'}_{i}) = G_{i} (r_{i}, {r'}_{i}) - G_{i} (r_{i}, {r'}_{j}),

where (i, j) = (1,2) for calculating the reflected fields in region S ₁; G_i(r _i,r ^′ _i) is the exact 2-D Green’s function [4]. For special position vectors r ^′ ₁ and r ^′ ₂ at the flat boundary Γ_r, we have

G_{i}^{RS1} = 0, \frac{\partial G_{i}^{RS1}}{\partial {\hat{n}}_{r}} = 2 \frac{\partial G_{i}}{\partial {\hat{n}}_{r}} .

Substituting Eq. (6) into Eq. (3), we may obtain the scattered field at any point in region S ₁ as

ϕ_{1}^{sc (ORSM 1)} (r_{1}) = \int_{Γ_{r}} [ϕ_{Γ_{r}} ({r'}_{Γ_{r}}) \frac{\partial G_{1}^{RS1} (r_{1}, {r'}_{Γ_{r}})}{\partial {\hat{n}}_{r}}] d l ",

where the integral sampling element dl ^″ in the ORSM1 is along the flat boundary Γ_r.

In the IRSM1, the integral boundary retains the actual metallic mirror boundary Γ, where the boundary fields are approximately specified. In the IRSM1 the boundary field at Γ reads by analogy [15]

ϕ_{Γ} (r_{Γ}) = R (x) w (x) \exp [- j Δ (x)],

where R(x) is a position-dependent Fresnel reflection coefficient at the boundary Γ, which is given by R(x) = (n ₁ cos i ₁ - ñ₂ cos i ₂)/(n ₁ cos i ₁ + ñ₂ cos i ₂) for TE polarization [or R(x) = (ñ₂ cos i ₁-n1 cos i ₂)/(ñ₂ cos i ₁+n ₁ cos i ₂) for TM polarization] [20]; i ₁ and i ₂ represent the local incident and refractive angles at the mirror boundary Γ. It is noted that ñ₂ and cos i ₂ are both complex numbers. The phase of the boundary field at Γ in Eq. (8) is Δ(x) = -n ₁ k ₀ h(x).

For a continuously cylindrical refractive focusing mirror, the scattered field in region S ₁ is calculated by the IRSM1 as [15]

ϕ_{1 c}^{sc (IRSM 1)} (r_{1}) = \int_{Γ} [ϕ_{Γ} ({r'}_{Γ}) \frac{\partial G_{1}^{RS1} (r_{1}, {r'}_{Γ})}{\partial \hat{n}}] d l',

where dl ^′ is along the actual curved boundary Γ of the cylindrical mirror, and the subscript c indicates the continuously refractive mirror profile.

For a multilevel cylindrical diffractive focusing mirror, the mirror boundary consists of the horizonal and the vertical parts, i.e., Γ = Γ_∥+Γ_⊥, as shown in Fig. 1(b). Similar to Eq. (9), the scattered field in region S ₁ calculated by the IRSM1 for a multilevel cylindrical focusing mirror is written as

ϕ_{1 q}^{sc (IRSM 1)} (r_{1}) = \int_{Γ_{∥}} [ϕ_{Γ} ({r'}_{Γ}) \frac{\partial G_{1}^{RS1} (r_{1}, {r'}_{Γ})}{\partial \hat{n}}] d l' + \int_{Γ_{⊥}} [ϕ_{Γ} ({r'}_{Γ}) \frac{\partial G_{1} (r_{1}, {r'}_{Γ})}{\partial \hat{n}}] d l',

where the subscript q stands for the quantized mirror profile.

To quantitatively appraise the accuracy of the IRSM1 (or the ORSM1), the calculated results by the BEM are given as an accurate reference. Here we define the relative root-mean-square error of the focal-plane intensity as [15, 16]

Δ I_{rms} = {\frac{\int_{- \infty}^{+ \infty} {[I_{1} (x) - I_{0}^{(BEM)} (x)]}^{2} d x}{\int_{- \infty}^{+ \infty} {[I_{0}^{(BEM)} (x)]}^{2} d x}}^{1 / 2},

where I ^(BEM) ₀ (x) and I ₁(x) represent the scattered intensities calculated by the BEM and by the IRSM1 (or by the ORSM1) at the preset focal plane, respectively.

2.3. Diffraction efficiency, total reflected power, and sidelobe power

For TE polarization, the backward-scattered field on the plane y = y_i in region S ₁ can be expressed in an angular spectrum integral form as [8]

E_{z}^{sc} (x, y_{i}) = \int_{- \infty}^{\infty} a_{1} (ρ_{1}) \exp [- j (ρ_{1} x + β_{1} y_{i})] d ρ_{1},

H_{z}^{sc} (x, y_{i}) = \frac{1}{ξ_{1}} \int_{- \infty}^{\infty} \frac{β_{1}}{k_{1}} a_{1} (ρ_{1}) \exp [- j (ρ_{1} x + β_{1} y_{i})] d ρ_{1},

where a ₁(ρ ₁) denotes the angular spectrum components of the scattered field E ^sc _z(x,y_i); β ₁ = (k ² ₁-ρ ² ₁)^½ if ρ ² ₁ ≤ k ² ₁ and β ₁ = j(ρ ² ₁-k ² ₁)^½ if ρ ² ₁ > k ² ₁; k ₁ = n ₁ k ₀, and ξ ₁ = [μ ₀/(ε ₁ ε ₀)]^½ μ ₀ and ε ₀ represent permeability and permittivity in vacuum, respectively; ε ₁ is the relative permittivity of the region S ₁. Through employing a fast Fourier transform algorithm and taking M sampling points (M = 2^N, where N is an integer) on the plane y = y_i, we can express E ^sc _z by a summation as

E_{z}^{sc} (m Δ x, y_{i}) = \sum_{n = - M / 2}^{M / 2 - 1} A_{1} (ρ_{1 n}, y_{i}) \exp (- j ρ_{1 n} m Δ x), m = - M / 2, \dots, M / 2 - 1,

where A ₁(ρ _1n,y_i) = a ₁(ρ _1n)exp(-j β _1n y_i)Δρ ₁; ρ _1n = 2nπ/L; Δρ ₁ = 2π/L; Δ_x = L/M; L is the calculating range of the sampling region, as shown in Fig. 1. By using the inverse Fourier transform, we obtain

A_{1} (ρ_{1 n}, y_{i}) = \frac{1}{M} \sum_{m = - M / 2}^{M / 2 - 1} E_{z}^{sc} (m Δ x, y_{i}) \exp (j ρ_{1 n} m Δ x), n = - M / 2, \dots, M / 2 - 1 .

The discrete scattered field distributions E ^sc _z (mΔx,y_i) are calculated by Eqs. (3), (7), (9) or (10) for the BEM, the ORSM1, and the IRSM1, respectively, as given in the Subsections 2.1 and 2.2. Once A ₁(ρ _1n) is determined, the total reflected power on the plane y = y_i is then calculated as [4, 5, 16, 12]

P_{r} = Re {\frac{L}{2 ξ_{1}} \sum_{n = - M / 2}^{M / 2 - 1} \frac{β_{1 n}^{*}}{k_{1}} {∣ A_{1} (ρ_{1 n}, y_{i}) ∣}^{2}}

where Re{⋯} represents taking the real part of a complex number; the superscript * implies the complex conjugate.

For a cylindrical focusing mirror, the scattered fields are focused at the preset focal plane y = f. The focused power P_f in the main lobe is [5]

P_{f} = Re {\frac{1}{2} \int_{- \frac{d}{2}}^{\frac{d}{2}} E_{z}^{sc} (x, f) {[H_{x}^{sc} (x, f)]}^{*} d x}

where sinc(x) = sin(x)/x; d denotes the focal spot size, which is defined as the minimum-to-minimum full width of the main lobe at the preset focal plane y = f.

The sidelobe neighbors the main lobe to the left (or to the right) at the preset focal plane. The focused sidelobe power within a ≤ x ≤ b is calculated as [12]

P_{s1} = Re {\frac{b - a}{2 ξ_{1}} \sum_{n, m = - M / 2}^{M / 2 - 1} \exp [j (ρ_{1 m} - ρ_{1 n}) (\frac{b + a}{2})]

The total incident power P _inc for a TE-polarized normally-incident plane wave in the incident aperture is evaluated by [5]

P_{inc} = \frac{1}{2} \int_{- \infty}^{+ \infty} E_{z}^{inc} (x) {[H_{x}^{inc} (x)]}^{*} d x = \frac{1}{2 ξ_{1}} \int_{- \infty}^{+ \infty} w^{2} (x) d x,

where w(x) is the truncated window function. The percentage diffraction efficiency η is defined as η = (P _f/P _inc)×100%, and the normalized sidelobe power is given by P _sl/P _inc. For TM polarization, the corresponding quantities ξ ₁, β _1m, [A ₁(ρ _1m)]*, and A ₁(ρ1n) in Eqs. (15)—(18) should be replaced by 1/ξ ₁, β ^* _1m, A1(ρ _1m), and [A ₁(ρ _1n)]^*, respectively.

3. Numerical results and analysis

3.1. Mirror design and window function

In the nonparaxial approximation, the cylindrical focusing mirror profile can be obtained based on Fermat’s principle. We assume that the phase on the plane y = 0 is zero. For a preset focal position (0, f), the surface-relief depth function h(x) of the continuously cylindrical refractive focusing mirror is

h (x) = \frac{1}{2} (\sqrt{f^{2} + x^{2}} - f),

where f is the preset focal length of the cylindrical refractive focusing mirror. For a quantized mirror profile with equal step depth, the step depth is Δh = h _max/N, where h _max is the largest etching thickness given by h _max = λ/2 and N is the quantization level number. The depth function of a multilevel cylindrical diffractive focusing mirror becomes

h_{q} (x) = Mod {Int [\frac{h (x)}{Δ h}], N} \times Δ h,

where the subscript q indicates quantized profiles; Int[⋯] denotes taking the maximum integer; Mod{A,B} = A-Int[A/B]×B, where A and B are both integers.

To relieve the scattering effect of the incident light at the edges of the incident aperture, a cosine window function is usually introduced as [5]

w (x) = {\begin{matrix} 1, 0 \leq ∣ x ∣ \leq D / 2 - l, \\ \cos^{2} (\frac{∣ x ∣ - D / 2 + l}{4 l} π) \\ 0, D / 2 + l < ∣ x ∣ < \infty, \end{matrix}, D / 2 - l < ∣ x ∣ \leq D / 2 + l,

where l is a smoothing parameter, D is the diameter of the cylindrical mirror, as shown in Fig. 1(a).

Parameters of the cylindrical focusing mirrors are selected as follows: The diameters of all the mirrors are D = 30.0μm; the preset focal lengths are presumed to be 45.0, 30.0, 15.0, and 10.0μm, respectively, and thus the corresponding f-numbers (= f /D) are f /1.5, f /1.0, f /0.5, and f / 0.33. The incident wavelength in free space is λ = 0.5166μm. Region S ₁ is free space with refractive index n ₁ = 1.0 and region S ₂ is filled with silver material [22] with refractive index ñ₂ = 0.130 - j3.07 for λ = 0.5166. The smoothing parameter sets l = 0.5μm, and the calculating range is selected to be L = 100.0μm for the convergence of the scattered power.

3.2. Effect of f-number

Of all the parameters, the f-number is the most important one to characterize the cylindrical focusing mirror. Since the scalar methods (including the ORSM1) are not suitable for the analysis of cylindrical focusing mirrors when the f-number is less than f /1.0 [12], we are very curious to know how the IRSM1 behaves for cylindrical focusing mirrors with small f-numbers. In this Subsection, when the f-number is changed from f/1.5 to f/0.33, the focal properties of the cylindrical refractive mirrors are calculated by the IRSM1, the ORSM1, and the rigorous BEM for TE polarization.

Table1. Focal property of metallic cylindrical refractive focusing mirrors with different f-numbers calculated by the three methods for TE polarization

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Table 1 tabulates several focal properties including the focal spot size, the diffraction efficiency, the real focal length, the total reflected power, the normalized sidelobe power, and the relative root-mean-square error of the focal-plane intensity. The real focal position situates where the maximum axial scattered field intensity is reached. The diffraction-limited spot size is given by d ₀ = 8fλ/(k ₁ D) [5]. It is clearly seen from Table 1 that the relative root-mean-square error of the focal-plane intensity for the IRSM1 is smaller than 1.0% for each f-number, whereas it increases rapidly for the ORSM1 as the f-number decreases. For instance, when the f-number is f/0.33, the relative errors for the IRSM1 and the ORSM1 are 0.67% and 233.1%, respectively. In Table 1, the numerical results calculated by the IRSM1 are very close to those by the BEM; in contrast, the results by the ORSM1 greatly deviate from those by the BEM. It can be ascribed to the following two reasons. First, in the IRSM1 the integral boundary is exactly the same as the actual mirror boundary in the BEM, however, a large error is introduced for replacing the mirror boundary with the straight line boundary in the ORSM1. Second, in the IRSM1 the boundary fields are more accurately modelled by using the local Fresnel reflection coefficient; in contrast, in the ORSM1 the boundary field is coarsely given by the scattered field from a flat boundary. On obtaining much more accurate boundary fields, it is naturally expected that the IRSM1 is much superior to the ORSM1 in calculating the scattered fields in the homogeneous region S ₁ because only an angular spectrum propagation is needed to be imposed.

Fig. 2. The focal-plane intensity distributions of cylindrical refractive focusing mirrors with different f-numbers for TE polarization. (a) f/1.5, (b) f/1.0, (c) f/0.5, and (d) f/0.33. The solid, the dotted, and the dotted-dashed curves represent the scattered intensity profiles calculated by the IRSM1, the BEM, and the ORSM1, respectively.

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Figure 2 displays the scattered field intensity distributions at the preset focal plane y = f for the same four cylindrical mirrors as in Table 1. The solid, the dotted, and the dotted-dashed curves represent the intensity profiles calculated by the IRSM1, the BEM, and the ORSM1, respectively. It is clearly seen from Fig. 2 that the scattered intensity profiles calculated by the IRSM1 almost overlap those calculated by the BEM for all the f-numbers, whereas the intensity profiles calculated by the ORSM1 significantly deviate from those by the BEM when the f-number is decreased. The ORSM1 completely fails to predict the focal spot size and peak intensity at the preset focal plane when the f-number is less than f /1.0. As the f-number of the cylindrical mirror is increased, the difference between the IRSM1 (or the ORSM1) and the BEM is gradually decreased. When the f-number is larger than f /2.0, numerical error of the IRSM1 (or the ORSM1) can be neglected.

Fig. 3. Same as Fig. 2 except for axial-plane intensity distributions. (a) f /1.5; (b) f /1.0; (c) f /0.5; (d) f /0.33.

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The scattered field intensity distributions on the axial plane x = 0 for the same mirrors as in Fig. 2 are plotted in Fig. 3. It is distinctly seen from Fig. 3 that the axial intensity contours by the IRSM1 almost cover with those by the BEM; in contrast, the intensity contours by the ORSM1 is significantly shifted from those by the BEM. The ORSM1 predicts a much nearer real focal position compared with the BEM, yet this focal shift is rectified by the IRSM1 owing to the consideration of the mirror thickness. For instance, for the f /0.33 cylindrical mirror, the real focal positions calculated by the IRSM1, the BEM, and the ORSM1 are 11.65, 11.64, and 10.0μm, respectively.

3.3. Effect of incident polarization

The applicability of the IRSM1 for TM polarization of incidence is also investigated. It differs from the TE polarization case in the local Fresnel reflection coefficient R(x) in Eq. (8). The focal properties of the cylindrical refractive focusing mirrors calculated by the three methods for TM polarization are presented in Table 2. It is clearly seen from Table 2 that the relative root-mean-square errors of the focal-plane intensity for the IRSM1 are all smaller than 1.0%, whereas they are larger than 100% for the ORSM1 when the f-number is less than f /0.5.

For TM polarization, several focal properties, such as the focal spot size, the diffraction efficiency, and the normalized total reflected power are depicted in Fig. 4. The solid curve with squares, the dotted curve with circles, and the dotted-dashed curve with stars correspond to the numerical results obtained by the IRSM1, the BEM, and the ORSM1, respectively. It is apparently shown in Fig. 4 that the focal spot sizes calculated by the IRSM1 accord well with those calculated by the BEM, whereas the focal spot sizes evaluated by the ORSM1 deviate from them greatly. The focal spot size by the ORSM1 increases monotonically with respect to the f-number, however, the focal spot size by the IRSM1 (or the BEM) varies more complicatedly. It is obvious in Fig. 4 that the diffraction efficiencies and normalized reflected powers by the IRSM1 are very close to those by the BEM, whereas the results by the ORSM1 significantly differ from them as the f-number is decreased. From Fig. 4, as the f-number increases, it is also seen that the difference between the results calculated by the approximate methods (IRSM1 or ORSM1) and those by the rigorous BEM decreases.

Table2. Focal property of metallic cylindrical refractive focusing mirrors with different f -numbers calculated by the three methods for TM polarization

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3.4. Effect of mirror profile

In addition, the properties of 2-level, 4-level, 8-level, and 16-level metallic cylindrical diffractive focusing mirrors with f-number of f /1.0 are studied by the IRSM1, the BEM, and the ORSM1 for TE polarization in detail. The focal properties calculated by the three methods for TE polarization are listed in Table 3. It is apparently seen from Table 3 that the relative rootmean-square errors of the focal-plane intensity for the IRSM1 are all a little smaller than those for the ORSM1, i.e., the IRSM1 is superior to the ORSM1 in analysis of multilevel cylindrical diffractive focusing mirrors. For diffractive focusing mirrors with large step depth including the 2-level and the 4-level mirrors, it is noted that the IRSM1 brings about large errors, which is due to the strong coupling effect at the step edges. For cylindrical diffractive focusing mirrors with eight or more quantization levels whose step depth is less than λ / 16, the IRSM1 is a good approximation to the BEM. For instance, for the 8-level and the 16-level cylindrical focusing mirrors, the relative root-mean-square errors of the focal-plane intensity are 1.51% and 2.15%, respectively; in comparison, the corresponding errors for the ORSM1 are 2.33% and 4.63%. From Table 3, the IRSM1 is accurate in predicting the focal spot size, but the diffraction efficiency calculated by the IRSM1 is slightly lower than that by the BEM.

Fig. 4. Several focal properties of the metallic cylindrical refractive focusing mirrors calculated by the three methods for TM polarization, which include the focal spot size, the diffraction efficiency, and the normalized total reflected power.

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4. Summary and discussions

In this paper, the focal properties of various 2-D metallic cylindrical focusing micro mirrors with different f-numbers, different incident polarizations, and different quantization levels of profiles are intensively investigated by the IRSM1. In the IRSM1, the boundary field is approximately specified by using the local Fresnel reflection coefficient, therefore, the complex global coupling effect of the boundary fields is neglected. Several focal properties of the metallic cylindrical focusing micro mirrors including the focal spot size, the diffraction efficiency, the real focal length, the normalized total reflected power, and the normalized sidelobe power are studied in detail. Numerical results calculated by the IRSM1, the BEM, and the ORSM1 are put together for quantitative comparison. The IRSM1 is a very good approximation to the BEM for performance analysis of continuously metallic cylindrical refractive focusing mirrors even when the f-number is decreased to f /0.33, regardless of the incident polarization; in contrast, the ORSM1 completely fails when the f-number is less than f /1.0. Taking the mirror thickness into account is the main reason of the substantial improvements. For 2-D multilevel metallic cylindrical diffractive focusing mirrors, the IRSM1 is also superior to the ORSM1. For a quantized profile with eight or more levels whose step depth is less than λ / 16, the IRSM1 approaches the BEM in high precision, however, the ORSM1 presents much worse results. As the quantization-level number is decreased, the global coupling effect among the boundary fields becomes much stronger and the relative errors for the IRSM1 is increased greatly.

Table3. Focal property of the f /1.0 multilevel metallic cylindrical diffractive focusing mirrors calculated by the three methods for TE polarization

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In addition, the IRSM1 obviously has the following two advantages over the rigorous BEM: Firstly, since it is not necessary to solve the complex boundary integral equations in the IRSM1, large amounts of computing time and computer memory can be saved. The computer resource savings appear to be much more important for analysis of metallic structures owing to a new freedom of finite conductance. For example, in our calculations, on a personal computer with a Pentium IV 1.4GHz CPU and Linux operation system, the BEM costs 327Mbytes of memory and 956min of computing time, whereas the IRSM1 only costs 94Mbytes of memory and 57min. Secondly, in the IRSM1 the scattered fields on the observation plane are in a explicit expression of the boundary fields, so it is expected that the design of metallic DOEs may be implemented in vectorial domain by incorporating the IRSM1 with the simulated-annealing method (or the conjugate-gradient method).

Although the IRSM1 is only applied to a 2-D system in this paper, it is also expected to be applicable to performance analysis of 3-D DOEs. For a 3-D system, a local interface approximation on the surface of the 3-D DOEs needs to be made. From the boundary joining conditions, each component of the electric or magnetic field on the local interface can be determined. Then the field distributions at each point in the whole space will be uniquely calculated from a surface integral of the 3-D DOEs, instead of a curved-boundary integral in the 2-D cylindrical systems.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grants 10604042 and 10674038) and National Basic Research Program of China (Grant 2006CB302901).

References and links

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f/#	Calculated method	Relative error ΔI _rms (%)	Focal spot sizea d (μm)	Diffraction efficiency η (%)	Real focal lengthb y = f_r (μm)	Normalized reflected power P _r/P _inc	Normalized sidelobe power P _sl/P _inc
f / 1.5	IRSM1	0.05	1.61	84.26	45.78	0.9513	0.0283
f / 1.5	BEM	0.00	1.61	84.28	45.77	0.9515	0.0282
f / 1.5	ORSM1	2.13	1.61	84.31	44.75	0.9347	0.0238
f / 1.0	IRSM1	0.15	2.05	82.22	31.43	0.9516	0.0319
f / 1.0	BEM	0.00	2.05	82.27	31.42	0.9517	0.0319
f / 1.0	ORSM1	34.57	1.12	81.78	29.93	0.9160	0.0251
f / 0.5	IRSM1	0.39	1.17	42.76	16.39	0.9523	0.0360
f / 0.5	BEM	0.00	1.17	42.86	16.39	0.9525	0.0360
f / 0.5	ORSM1	175.08	0.68	71.93	14.99	0.8397	0.0289
f / 0.33	IRSM1	0.67	1.42	34.73	11.65	0.9530	0.0312
f / 0.33	BEM	0.00	1.46	35.22	11.64	0.9532	0.0292
f / 0.33	ORSM1	233.11	0.54	62.19	10.00	0.7594	0.0306

f/#	Calculated method	Relative error ΔI _rms (%)	Focal spot sizea d (μm)	Diffraction efficiency η (%)	Real focal lengthb y = f_r (μm)	Normalized reflected power P _r/P _inc	Normalized sidelobe power P _sl/P _inc
f / 1.5	IRSM1	0.05	1.61	84.17	45.80	0.9509	0.0285
f / 1.5	BEM	0.00	1.61	84.14	45.80	0.9506	0.0285
f / 1.5	ORSM1	2.31	1.61	84.31	44.75	0.9347	0.0238
f / 1.0	IRSM1	0.18	2.05	82.04	31.45	0.9507	0.0323
f /1.0	BEM	0.00	2.05	81.97	31.45	0.9502	0.0323
f /1.0	ORSM1	36.32	1.12	81.78	29.93	0.9160	0.0251
f / 0.5	IRSM1	0.42	1.22	43.04	16.41	0.9498	0.0311
f / 0.5	BEM	0.00	1.22	42.93	16.42	0.9491	0.0310
f / 0.5	ORSM1	177.45	0.68	71.92	14.99	0.8397	0.0289
f / 0.33	IRSM1	0.74	1.42	34.53	11.67	0.9490	0.0314
f / 0.33	BEM	0.00	1.37	34.02	11.68	0.9482	0.0334
f / 0.33	ORSM1	236.49	0.54	62.19	10.00	0.7594	0.0306

Quantization level	Calculated method	Relative error ΔI _rms (%)	Focal spot sizea d (μm)	Diffraction efficiency η (%)
2-Level	IRSM1	9.71	1.03	31.71
2-Level	BEM	0.00	1.03	33.74
2-Level	ORSM1	11.38	1.07	31.34
4-Level	IRSM1	4.57	1.12	65.67
4-Level	BEM	0.00	1.12	68.27
4-Level	ORSM1	5.13	1.12	65.46
8-Level	IRSM1	1.51	1.12	76.14
8-Level	BEM	0.00	1.12	76.36
8-Level	ORSM1	2.33	1.12	77.42
16-Level	IRSM1	2.14	1.12	76.18
16-Level	BEM	0.00	1.12	77.62
16-Level	ORSM1	4.63	1.12	80.70

f/#	Calculated method	Relative error ΔI _rms (%)	Focal spot sizea d (μm)	Diffraction efficiency η (%)	Real focal lengthb y = f_r (μm)	Normalized reflected power P _r/P _inc	Normalized sidelobe power P _sl/P _inc
f / 1.5	IRSM1	0.05	1.61	84.26	45.78	0.9513	0.0283
f / 1.5	BEM	0.00	1.61	84.28	45.77	0.9515	0.0282
f / 1.5	ORSM1	2.13	1.61	84.31	44.75	0.9347	0.0238
f / 1.0	IRSM1	0.15	2.05	82.22	31.43	0.9516	0.0319
f / 1.0	BEM	0.00	2.05	82.27	31.42	0.9517	0.0319
f / 1.0	ORSM1	34.57	1.12	81.78	29.93	0.9160	0.0251
f / 0.5	IRSM1	0.39	1.17	42.76	16.39	0.9523	0.0360
f / 0.5	BEM	0.00	1.17	42.86	16.39	0.9525	0.0360
f / 0.5	ORSM1	175.08	0.68	71.93	14.99	0.8397	0.0289
f / 0.33	IRSM1	0.67	1.42	34.73	11.65	0.9530	0.0312
f / 0.33	BEM	0.00	1.46	35.22	11.64	0.9532	0.0292
f / 0.33	ORSM1	233.11	0.54	62.19	10.00	0.7594	0.0306

f/#	Calculated method	Relative error ΔI _rms (%)	Focal spot sizea d (μm)	Diffraction efficiency η (%)	Real focal lengthb y = f_r (μm)	Normalized reflected power P _r/P _inc	Normalized sidelobe power P _sl/P _inc
f / 1.5	IRSM1	0.05	1.61	84.17	45.80	0.9509	0.0285
f / 1.5	BEM	0.00	1.61	84.14	45.80	0.9506	0.0285
f / 1.5	ORSM1	2.31	1.61	84.31	44.75	0.9347	0.0238
f / 1.0	IRSM1	0.18	2.05	82.04	31.45	0.9507	0.0323
f /1.0	BEM	0.00	2.05	81.97	31.45	0.9502	0.0323
f /1.0	ORSM1	36.32	1.12	81.78	29.93	0.9160	0.0251
f / 0.5	IRSM1	0.42	1.22	43.04	16.41	0.9498	0.0311
f / 0.5	BEM	0.00	1.22	42.93	16.42	0.9491	0.0310
f / 0.5	ORSM1	177.45	0.68	71.92	14.99	0.8397	0.0289
f / 0.33	IRSM1	0.74	1.42	34.53	11.67	0.9490	0.0314
f / 0.33	BEM	0.00	1.37	34.02	11.68	0.9482	0.0334
f / 0.33	ORSM1	236.49	0.54	62.19	10.00	0.7594	0.0306

Improved first Rayleigh-Sommerfeld method applied to metallic cylindrical focusing micro mirrors

Abstract

1. Introduction

2. Fundamental theories and integral equations

2.1. Boundary element method

2.2. Original and improved first Rayleigh-Sommerfeld method

2.3. Diffraction efficiency, total reflected power, and sidelobe power

3. Numerical results and analysis

3.1. Mirror design and window function

3.2. Effect of f-number

3.3. Effect of incident polarization

3.4. Effect of mirror profile

4. Summary and discussions

Acknowledgments

References and links

Cited By

Figures (4)

Tables (3)

Equations (26)

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