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

Optics Express, Vol. 17, Issue 9, pp. 7348-7360 (2009)

http://dx.doi.org/10.1364/OE.17.007348

Acrobat PDF (232 KB)

### Abstract

An improved first Rayleigh-Sommerfeld method (IRSM1) is intensively applied to analyzing the focal properties of metallic cylindrical focusing micro mirrors. A variety of metallic cylindrical focusing mirrors with different f-numbers, different polarization of incidence, or different types of profiles are investigated. The focal properties include the focal spot size, the diffraction efficiency, the real focal length, the total reflected power, and the normalized sidelobe power. Numerical results calculated by the IRSM1, the original first Rayleigh-Sommerfeld method (ORSM1), and the rigorous boundary element method (BEM) are presented for quantitative comparison. It is found that the IRSM1 is much more accurate than the ORSM1 in performance analysis of metallic cylindrical focusing mirrors, especially for cylindrical refractive focusing mirrors with small f-numbers. Moreover, the IRSM1 saves great amounts of computational time and computer memory in calculations, in comparison with the vectorial BEM.

© 2009 Optical Society of America

## 1. Introduction

1.
Feature issue on “Diffractive optics appliations,” Appl. Opt. **34**, 2399–2559 (1995). [PubMed]

2. D. A. Pommet, M. G. Moharam, and E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” J. Opt. Soc. Am. A **11**, 1827–1834 (1994). [CrossRef]

3. J. N. Mait, “Understanding diffractive optic design in the scalar domain,” J. Opt. Soc. Am. A **12**, 2145–2158 (1995). [CrossRef]

4. J. M. Bendickson, E. N. Glytsis, and T. K. Gaylord, “Scalar integral diffraction methods: unification, accuracy, and comparison with a rigorous boundary element method with application to diffractive cylindrical lenses,” J. Opt. Soc. Am. A **15**, 1822–1837 (1998). [CrossRef]

5. K. Hirayama, E. N. Glytsis, T. K. Gaylord, and D. W. Wilson, “Rigorous electromagnetic analysis of diffractive
cylindrical lenses,” J. Opt. Soc. Am. A **13**, 2219–2231 (1996). [CrossRef]

6. J. S. Ye, B. Z. Dong, B. Y. Gu, G. Z. Yang, and S. T. Liu, “Analysis of a closed-boundary axilens with long
focal depth and high transverse resolution based on rigorous electromagnetic theory,” J. Opt. Soc. Am. A **19**,
2030–2035 (2002). [CrossRef]

7. D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A **14**, 34–43 (1997). [CrossRef]

8. K. Hirayama, E. N. Glytsis, and T. K. Gaylord, “Rigorous electromagnetic analysis of diffraction by finite-number-of-periods gratings,” J. Opt. Soc. Am. A **14**, 907–917 (1997). [CrossRef]

9. H. Haidner, S. Schröter, and H. Bartelt, “The optimization of diffractive binary mirrors with low focal length: diameter ratios,” J. Phys. D **30**, 1314–1325 (1997). [CrossRef]

10. J. B. Judkins and R.W. Ziolkowski, “Finite-difference time-domain modeling of nonperfectly conducting metallic thin-film gratings,” J. Opt. Soc. Am. A **12**, 1974–1983 (1995). [CrossRef]

11. Y. Nakata and M. Koshiba, “Boundary-element analysis of plane-wave diffraction from groove-type dielectric and metallic gratings,” J. Opt. Soc. Am. A **7**, 1494–1502 (1990). [CrossRef]

12. J. M. Bendickson, E. N. Glytsis, and T. K. Gaylord, “Metallic surface-relief on-axis and off-axis focusing diffractive cylindrical mirrors,” J. Opt. Soc. Am. A **16**, 113–130 (1999). [CrossRef]

13. E. Noponen, J. Turunen, and A. Vasara, “Electromagnetic theory and design of diffractive-lens arrays,” J. Opt. Soc. Am. A **10**, 434–443 (1993). [CrossRef]

14. D. M. Mackie, D.W. Prather, and S. Y. Shi, “Preoptimization improvements to subwavelength diffractive lenses,” Appl. Opt. **41**, 6168–6175 (2002). [CrossRef] [PubMed]

15. J. S. Ye, B. Y. Gu, B. Z. Dong, and S. T. Liu, “Improved first Rayleigh-Sommerfeld method for analysis of cylindrical microlenses with small f-numbers,” Opt. Lett. **29**, 2345–2347 (2004). [CrossRef] [PubMed]

15. J. S. Ye, B. Y. Gu, B. Z. Dong, and S. T. Liu, “Improved first Rayleigh-Sommerfeld method for analysis of cylindrical microlenses with small f-numbers,” Opt. Lett. **29**, 2345–2347 (2004). [CrossRef] [PubMed]

16. J. S. Ye, B. Y. Gu, B. Z. Dong, and S. T. Liu, “Applications of improved first Rayleigh-Sommerfeld method to analyze the performance of cylindrical microlenses with different f-numbers,” J. Opt. Soc. Am. A **22**, 862–869 (2005). [CrossRef]

17. C. Rydberg, B. Y. Gu, and G. Z. Yang, “Design method for small-f-number microlenses based on a finite thickness model in combination with the Yang-Gu phase-retrieval algorithm,” J. Opt. Soc. Am. A **24**, 517–521 (2007). [CrossRef]

*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.

## 2. Fundamental theories and integral equations

### 2.1. Boundary element method

*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

*n*̃

_{2}=

*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

4. J. M. Bendickson, E. N. Glytsis, and T. K. Gaylord, “Scalar integral diffraction methods: unification, accuracy, and comparison with a rigorous boundary element method with application to diffractive cylindrical lenses,” J. Opt. Soc. Am. A **15**, 1822–1837 (1998). [CrossRef]

5. K. Hirayama, E. N. Glytsis, T. K. Gaylord, and D. W. Wilson, “Rigorous electromagnetic analysis of diffractive
cylindrical lenses,” J. Opt. Soc. Am. A **13**, 2219–2231 (1996). [CrossRef]

8. K. Hirayama, E. N. Glytsis, and T. K. Gaylord, “Rigorous electromagnetic analysis of diffraction by finite-number-of-periods gratings,” J. Opt. Soc. Am. A **14**, 907–917 (1997). [CrossRef]

12. J. M. Bendickson, E. N. Glytsis, and T. K. Gaylord, “Metallic surface-relief on-axis and off-axis focusing diffractive cylindrical mirrors,” J. Opt. Soc. Am. A **16**, 113–130 (1999). [CrossRef]

*ϕ*=

*E*and

_{z}*p*= 1 (

_{i}*i*= 1,2) for TE polarization;

*ϕ*=

*H*and

_{z}*p*=

_{i}*n*

^{2}

_{1}(or

*n*̃

^{2}

_{2}) for TM polarization.

*ϕ*

^{t}and

*ϕ*

^{inc}represent the total and incident fields, respectively.

*G*is the 2-D exact Green’s function, i.e.,

_{i}*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*is the wave number in region

_{i}*S*(

_{i}*i*= 1,2). Since the refractive index

*n*̃

_{2}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.

*n*̂ 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.

**r**

_{i}(

*i*= 1,2) approaches

**r**

_{Γ}[18], Eqs. (1a) and (1b) become [4

4. J. M. Bendickson, E. N. Glytsis, and T. K. Gaylord, “Scalar integral diffraction methods: unification, accuracy, and comparison with a rigorous boundary element method with application to diffractive cylindrical lenses,” J. Opt. Soc. Am. A **15**, 1822–1837 (1998). [CrossRef]

5. K. Hirayama, E. N. Glytsis, T. K. Gaylord, and D. W. Wilson, “Rigorous electromagnetic analysis of diffractive
cylindrical lenses,” J. Opt. Soc. Am. A **13**, 2219–2231 (1996). [CrossRef]

8. K. Hirayama, E. N. Glytsis, and T. K. Gaylord, “Rigorous electromagnetic analysis of diffraction by finite-number-of-periods gratings,” J. Opt. Soc. Am. A **14**, 907–917 (1997). [CrossRef]

12. J. M. Bendickson, E. N. Glytsis, and T. K. Gaylord, “Metallic surface-relief on-axis and off-axis focusing diffractive cylindrical mirrors,” J. Opt. Soc. Am. A **16**, 113–130 (1999). [CrossRef]

### 2.2. Original and improved first Rayleigh-Sommerfeld method

_{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

**15**, 1822–1837 (1998). [CrossRef]

**16**, 113–130 (1999). [CrossRef]

*R*

_{0}is the Fresnel reflection coefficient at the flat boundary Γ

_{r}, i.e.,

*R*

_{0}= (

*n*

_{1}-

*n*̃

_{2})/(

*n*

_{1}+

*n*̃

_{2}) for TE polarization and

*R*

_{0}= (

*n*̃

_{2}−

*n*

_{1})/(

*n*̃

_{2}+

*n*

_{1}) for TM polarization [20].

**r**

_{Γr}represents the position vector at the flat boundary Γ

_{r}, and

*n*̂

*is the unit normal vector at Γ*

_{t}_{t}. The phase term [12

**16**, 113–130 (1999). [CrossRef]

*δ*(

*x*) = -2

*n*

_{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

**13**, 2219–2231 (1996). [CrossRef]

**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]

*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

**15**, 1822–1837 (1998). [CrossRef]

**r**

^{′}

_{1}and

**r**

^{′}

_{2}at the flat boundary Γ

_{r}, we have

*S*

_{1}as

*dl*

^{″}in the ORSM1 is along the flat boundary Γ

_{r}.

15. J. S. Ye, B. Y. Gu, B. Z. Dong, and S. T. Liu, “Improved first Rayleigh-Sommerfeld method for analysis of cylindrical microlenses with small f-numbers,” Opt. Lett. **29**, 2345–2347 (2004). [CrossRef] [PubMed]

*R*(

*x*) is a position-dependent Fresnel reflection coefficient at the boundary Γ, which is given by

*R*(

*x*) = (

*n*

_{1}cos

*i*

_{1}-

*n*̃

_{2}cos

*i*

_{2})/(

*n*

_{1}cos

*i*

_{1}+

*n*̃

_{2}cos

*i*

_{2}) for TE polarization [or

*R*(

*x*) = (

*n*̃

_{2}cos

*i*

_{1}-n1 cos

*i*

_{2})/(

*n*̃

_{2}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

*n*̃

_{2}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*).

*S*

_{1}is calculated by the IRSM1 as [15

**29**, 2345–2347 (2004). [CrossRef] [PubMed]

*dl*

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

_{∥}+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

**29**, 2345–2347 (2004). [CrossRef] [PubMed]

16. J. S. Ye, B. Y. Gu, B. Z. Dong, and S. T. Liu, “Applications of improved first Rayleigh-Sommerfeld method to analyze the performance of cylindrical microlenses with different f-numbers,” J. Opt. Soc. Am. A **22**, 862–869 (2005). [CrossRef]

*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

*y*=

*y*in region

_{i}*S*

_{1}can be expressed in an angular spectrum integral form as [8

**14**, 907–917 (1997). [CrossRef]

*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*, we can express

_{i}*E*

^{sc}

_{z}by a summation as

*A*

_{1}(

*ρ*

_{1n},

*y*) =

_{i}*a*

_{1}(

*ρ*

_{1n})exp(-

*j*

*β*

_{1n}

*y*)Δ

_{i}*ρ*

_{1};

*ρ*

_{1n}= 2

*nπ*/

*L*; Δ

*ρ*

_{1}= 2

*π*/

*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

*E*

^{sc}

_{z}(

*m*Δ

*x*,

*y*) 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

_{i}*A*

_{1}(

*ρ*

_{1n}) is determined, the total reflected power on the plane

*y*=

*y*is then calculated as [4

_{i}**15**, 1822–1837 (1998). [CrossRef]

**13**, 2219–2231 (1996). [CrossRef]

16. J. S. Ye, B. Y. Gu, B. Z. Dong, and S. T. Liu, “Applications of improved first Rayleigh-Sommerfeld method to analyze the performance of cylindrical microlenses with different f-numbers,” J. Opt. Soc. Am. A **22**, 862–869 (2005). [CrossRef]

**16**, 113–130 (1999). [CrossRef]

*y*=

*f*. The focused power

*P*in the main lobe is [5

_{f}**13**, 2219–2231 (1996). [CrossRef]

*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*.

*a*≤

*x*≤

*b*is calculated as [12

**16**, 113–130 (1999). [CrossRef]

*P*

_{inc}for a TE-polarized normally-incident plane wave in the incident aperture is evaluated by [5

**13**, 2219–2231 (1996). [CrossRef]

*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}(

*ρ*1

*n*) in Eqs. (15)—(18) should be replaced by 1/

*ξ*

_{1},

*β*

^{*}

_{1m},

*A*1(

*ρ*

_{1m}), and [

*A*

_{1}(

*ρ*

_{1n})]

^{*}, respectively.

## 3. Numerical results and analysis

### 3.1. Mirror design and window function

*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

*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 Δ

*h*=

*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

*A*,

*B*} =

*A*-Int[A/B]×

*B*, where

*A*and

*B*are both integers.

**13**, 2219–2231 (1996). [CrossRef]

*l*is a smoothing parameter,

*D*is the diameter of the cylindrical mirror, as shown in Fig. 1(a).

*D*= 30.0

*μ*m; the preset focal lengths are presumed to be 45.0, 30.0, 15.0, and 10.0

*μ*m, 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 = 0.5166

*μ*m. 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

*n*̃

_{2}= 0.130 -

*j*3.07 for

*λ*= 0.5166. The smoothing parameter sets

*l*= 0.5

*μ*m, and the calculating range is selected to be

*L*= 100.0

*μ*m for the convergence of the scattered power.

### 3.2. Effect of f-number

*f*/1.0 [12

**16**, 113–130 (1999). [CrossRef]

*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.

*d*

_{0}= 8

*fλ*/(

*k*

_{1}

*D*) [5

**13**, 2219–2231 (1996). [CrossRef]

*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.

*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.

*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 10.0

*μ*m, respectively.

### 3.3. Effect of incident polarization

*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.

### 3.4. Effect of mirror profile

*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.

## 4. Summary and discussions

*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.

## Acknowledgments

## References and links

1. |
Feature issue on “Diffractive optics appliations,” Appl. Opt. |

2. | D. A. Pommet, M. G. Moharam, and E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” J. Opt. Soc. Am. A |

3. | J. N. Mait, “Understanding diffractive optic design in the scalar domain,” J. Opt. Soc. Am. A |

4. | J. M. Bendickson, E. N. Glytsis, and T. K. Gaylord, “Scalar integral diffraction methods: unification, accuracy, and comparison with a rigorous boundary element method with application to diffractive cylindrical lenses,” J. Opt. Soc. Am. A |

5. | K. Hirayama, E. N. Glytsis, T. K. Gaylord, and D. W. Wilson, “Rigorous electromagnetic analysis of diffractive
cylindrical lenses,” J. Opt. Soc. Am. A |

6. | J. S. Ye, B. Z. Dong, B. Y. Gu, G. Z. Yang, and S. T. Liu, “Analysis of a closed-boundary axilens with long
focal depth and high transverse resolution based on rigorous electromagnetic theory,” J. Opt. Soc. Am. A |

7. | D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A |

8. | K. Hirayama, E. N. Glytsis, and T. K. Gaylord, “Rigorous electromagnetic analysis of diffraction by finite-number-of-periods gratings,” J. Opt. Soc. Am. A |

9. | H. Haidner, S. Schröter, and H. Bartelt, “The optimization of diffractive binary mirrors with low focal length: diameter ratios,” J. Phys. D |

10. | J. B. Judkins and R.W. Ziolkowski, “Finite-difference time-domain modeling of nonperfectly conducting metallic thin-film gratings,” J. Opt. Soc. Am. A |

11. | Y. Nakata and M. Koshiba, “Boundary-element analysis of plane-wave diffraction from groove-type dielectric and metallic gratings,” J. Opt. Soc. Am. A |

12. | J. M. Bendickson, E. N. Glytsis, and T. K. Gaylord, “Metallic surface-relief on-axis and off-axis focusing diffractive cylindrical mirrors,” J. Opt. Soc. Am. A |

13. | E. Noponen, J. Turunen, and A. Vasara, “Electromagnetic theory and design of diffractive-lens arrays,” J. Opt. Soc. Am. A |

14. | D. M. Mackie, D.W. Prather, and S. Y. Shi, “Preoptimization improvements to subwavelength diffractive lenses,” Appl. Opt. |

15. | J. S. Ye, B. Y. Gu, B. Z. Dong, and S. T. Liu, “Improved first Rayleigh-Sommerfeld method for analysis of cylindrical microlenses with small f-numbers,” Opt. Lett. |

16. | J. S. Ye, B. Y. Gu, B. Z. Dong, and S. T. Liu, “Applications of improved first Rayleigh-Sommerfeld method to analyze the performance of cylindrical microlenses with different f-numbers,” J. Opt. Soc. Am. A |

17. | C. Rydberg, B. Y. Gu, and G. Z. Yang, “Design method for small-f-number microlenses based on a finite thickness model in combination with the Yang-Gu phase-retrieval algorithm,” J. Opt. Soc. Am. A |

18. | M. Koshiba, |

19. | B. Z. Dong, J. Liu, B. Y. Gu, G. Z. Yang, and J. Wang, “Rigorous electromagnetic analysis of a microcylindrical axilens with long focal depth and high transverse resolution,” J. Opt. Soc. Am. A |

20. | M. Born and E. Wolf, |

21. | J. W. Goodman, |

22. | E. D. Palik ed., |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(050.1960) Diffraction and gratings : Diffraction theory

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: January 21, 2009

Revised Manuscript: March 15, 2009

Manuscript Accepted: April 7, 2009

Published: April 20, 2009

**Citation**

Jia-Sheng Ye, Yan Zhang, and Kazuhiro Hane, "Improved first Rayleigh-Sommerfeld method applied to metallic cylindrical
focusing micro mirrors," Opt. Express **17**, 7348-7360 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-9-7348

Sort: Year | Journal | Reset

### References

- Feature issue on "Diffractive optics appliations,"Appl. Opt. 34, 2399-2559 (1995). [PubMed]
- D. A. Pommet, M. G. Moharam, and E. B. Grann, "Limits of scalar diffraction theory for diffractive phase elements," J. Opt. Soc. Am. A 11, 1827-1834 (1994). [CrossRef]
- J. N. Mait, "Understanding diffractive optic design in the scalar domain," J. Opt. Soc. Am. A 12, 2145-2158 (1995). [CrossRef]
- J. M. Bendickson, E. N. Glytsis, and T. K. Gaylord, "Scalar integral diffraction methods: unification, accuracy, and comparison with a rigorous boundary element method with application to diffractive cylindrical lenses," J. Opt. Soc. Am. A 15, 1822-1837 (1998). [CrossRef]
- K. Hirayama, E. N. Glytsis, T. K. Gaylord, and D. W. Wilson, "Rigorous electromagnetic analysis of diffractive cylindrical lenses," J. Opt. Soc. Am. A 13, 2219-2231 (1996). [CrossRef]
- J. S. Ye, B. Z. Dong, B. Y. Gu, G. Z. Yang, and S. T. Liu, "Analysis of a closed-boundary axilens with long focal depth and high transverse resolution based on rigorous electromagnetic theory," J. Opt. Soc. Am. A 19, 2030-2035 (2002). [CrossRef]
- D. W. Prather, M. S. Mirotznik, and J. N. Mait, "Boundary integral methods applied to the analysis of diffractive optical elements," J. Opt. Soc. Am. A 14, 34-43 (1997). [CrossRef]
- K. Hirayama, E. N. Glytsis, and T. K. Gaylord, "Rigorous electromagnetic analysis of diffraction by finite number-of-periods gratings," J. Opt. Soc. Am. A 14, 907-917 (1997). [CrossRef]
- H. Haidner, S. Schröter, and H. Bartelt, "The optimization of diffractive binary mirrors with low focal length: diameter ratios," J. Phys. D 30, 1314-1325 (1997). [CrossRef]
- J. B. Judkins and R.W. Ziolkowski, "Finite-difference time-domain modeling of nonperfectly conducting metallic thin-film gratings," J. Opt. Soc. Am. A 12, 1974-1983 (1995). [CrossRef]
- Y. Nakata and M. Koshiba, "Boundary-element analysis of plane-wave diffraction from groove-type dielectric and metallic gratings," J. Opt. Soc. Am. A 7, 1494-1502 (1990). [CrossRef]
- J. M. Bendickson, E. N. Glytsis, and T. K. Gaylord, "Metallic surface-relief on-axis and off-axis focusing diffractive cylindrical mirrors," J. Opt. Soc. Am. A 16, 113-130 (1999). [CrossRef]
- E. Noponen, J. Turunen, and A. Vasara, "Electromagnetic theory and design of diffractive-lens arrays," J. Opt. Soc. Am. A 10, 434-443 (1993). [CrossRef]
- D. M. Mackie, D.W. Prather, and S. Y. Shi, "Preoptimization improvements to subwavelength diffractive lenses," Appl. Opt. 41, 6168-6175 (2002). [CrossRef] [PubMed]
- J. S. Ye, B. Y. Gu, B. Z. Dong, and S. T. Liu, "Improved first Rayleigh-Sommerfeld method for analysis of cylindrical microlenses with small f-numbers," Opt. Lett. 29, 2345-2347 (2004). [CrossRef] [PubMed]
- J. S. Ye, B. Y. Gu, B. Z. Dong, and S. T. Liu, "Applications of improved first Rayleigh-Sommerfeld method to analyze the performance of cylindrical microlenses with different f-numbers," J. Opt. Soc. Am. A 22, 862-869 (2005). [CrossRef]
- C. Rydberg, B. Y. Gu, and G. Z. Yang, "Design method for small-f-number microlenses based on a finite thickness model in combination with the Yang-Gu phase-retrieval algorithm," J. Opt. Soc. Am. A 24, 517-521 (2007). [CrossRef]
- M. Koshiba, OpticalWaveguide Theory by the Finite Element Method (KTK Scientific, Tokyo, 1992), pp. 43-47.
- B. Z. Dong, J. Liu, B. Y. Gu, G. Z. Yang, and J. Wang, "Rigorous electromagnetic analysis of a microcylindrical axilens with long focal depth and high transverse resolution," J. Opt. Soc. Am. A 18, 1465-1470 (2001). [CrossRef]
- M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), Chap. 1.
- J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), Chaps. 3 and 4.
- E. D. Palik ed., Handbook of Optical Constants of Solids (Academic Press, INC., Orlando, Florida, 1985), pp. 356.

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.