Effect of fabrication errors on superresolution property of a pupil filter

doi:10.1364/OE.14.007024

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Effect of fabrication errors on superresolution property of a pupil filter

Weiqian Zhao, Lirong Qiu, and Zhengde Feng »View Author Affiliations

Optics Express, Vol. 14, Issue 16, pp. 7024-7036 (2006)
http://dx.doi.org/10.1364/OE.14.007024

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▸ Article Outline
– Abstract
1. Introduction
2. Pupil filter structure
3. Analytical models for and of pupil filters with fabrication error
4. Effect of fabrication errors on superresolution property
5. Conclusion
– References and links

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Abstract

Three analytical models have been established for superresolution parameters G_Ae , G_Te , and S_e related to transmission function A(ρ), phase function of ϕ(ρ), and the structural parameters with fabrication errors of an N-zone circular-symmetrical superresolution pupil filter. These new models established, directly relate the superresolution parameters of an N-zone super-resolution pupil filter to its fabrication errors to make the quantitative analyses of the effect of fabrication errors easier, thereby providing a theoretical basis for the analysis, design, and fabrication of an N-zone super-resolution pupil filter. The models established for G_Ae , G_Te , and S_e have been used to analyze the effect of the fabrication errors of a three-zone phase-only pupil filter on its superresolution property, to verify their validities.

1. Introduction

Superresolution pupil filters have been widely used to improve the resolution property of an optical system [1–5

T. Wilson, Confocal Microscopy (Academic Press. London, 1990).

]. The superresolution properties of a pupil filter is mainly characterized by axial spot size G_A , lateral spot size G_T , and Strehl ratio S of the intensity point spread function (PSF) of a superresolution optical system, and they can be expressed as [6

T. R. M. Sales, Phase-only Superresolution Elements (University of Rochester. Ph.D. Dissertation, 1997).

{\begin{matrix} G_{A} = \frac{u_{1}}{u_{0}} \\ G_{T} = \frac{v_{1}}{v_{0}} \\ S = {∣ h (0, u_{F}) ∣}^{2} \end{matrix}

(1)

where u ₁ and v ₁ are coordinates of the first intensity minimum in the axial and lateral directions corresponding to superresolution pattern, respectively; u ₀ and v ₀ are coordinates of the first intensity minimum in the axial and lateral directions corresponding to Airy pattern, respectively; and |h(0,u_F )|² is the intensity of the PSF main lobe of a superresolution optical system.

The goal of using a superresolution pupil in an optical system is to make S as large as possible but make G as small as possible. For this purpose, the theoretical models for G and S must be first established, and some scholars have done some works.

In Ref [7

C. J. R. Sheppard and Z. S. Hegedus, “Axial behaviour of pupil-plane filters,” J. Opt. Soc. Am. A 5, 643–647 (1988). [CrossRef]

], the G_A , G_T and S of a superresolution optical system with an annular pupil filter are expressed using the coefficients of the intensity distribution expanded in series near the focus. In Ref. [8

T. R. M. Sales and G. M. Morris, “Diffractive superresolution elements,” J. Opt. Soc. Am. A 14, 1637–1646 (1997). [CrossRef]

], when a binary diffractive element is used to achieve lateral superresolution, the intensity function determined by this two-value phase filter is expanded in series near the geometrical focus to the second order, and G_T is defined as a coordinate ratio of the super-resolved pattern to the Airy disk pattern by the first minimum. In Ref. [9

D. M. de Juana, J. E. Oti, V. F. Canales, and M. P. Caqiqal, “Design of superresolving continuous phase filters, ” Opt. Lett. 28, 607–609 (2003). [CrossRef] [PubMed]

], the models are established for G_A , G_T and S of a complex amplitude pupil filter using the second order intensity expansion similar to the method in Ref. [7

C. J. R. Sheppard and Z. S. Hegedus, “Axial behaviour of pupil-plane filters,” J. Opt. Soc. Am. A 5, 643–647 (1988). [CrossRef]

]. In Ref. [10

S. Ledesma, J. Campos, J. C. Escalera, and M. J. Yzuel, “Simple expressions for performance parameters of complex filters with application to super- Gaussian phase filter,” Opt. Lett. 29, 932–934(2004). [CrossRef] [PubMed]

], based on the models in Ref. [9

D. M. de Juana, J. E. Oti, V. F. Canales, and M. P. Caqiqal, “Design of superresolving continuous phase filters, ” Opt. Lett. 28, 607–609 (2003). [CrossRef] [PubMed]

], superresolution parameters G_A , G_T , and Strehl ratio S, are extended to the case in which the best image plane is not near the paraxial focus, and the models are generalized for a super-Gaussian phase filter in the surroundings of the shifted focus; the super-Gaussian phase filters depend on several parameters that modify the shape of the phase filter and are capable of producing a wide range of optical effects by changing these parameters. The several parameters are transverse superresolution with high depth of focus, 3-D superresolution, and transverse apodization with different axial responses. In Ref. [11–12

H. T. Liu, Y. B. Yan, D. Yi, and G. F. Jin. “Design of three-dimensional superresolution filters and limits of axial optical supperresolution,” Appl. Opt. 42, 1463–1476 (2003). [CrossRef] [PubMed]

], a diffractive 3-D superresolution filter is designed using the theory of linear programming to optimize its parameters based on the intensity function, and parameters G and S are preestablished as variable parameters in the equation. In Ref. [13

M. Y. Yun, L. R. Liu, J. F. Sun, and D. A. Liu, “Three-dimension superresolution by three-zone complex pupil filters,” Opt. Soc. Am. A 22, 272–277 (2005). [CrossRef]

], a three-zone complex amplitude pupil filter is used to realize optical 3-D superresolution through the design of the essential parameters of such filters, the transmittance and radius of the first zone.

The superresolution parameters G and S mentioned above are all expressed by the coefficients of intensity expansion near the paraxial focus of an optical system, and they vary in different superresolution elements. The superresolution parameters G and S can be established mainly by changing transmission function A(ρ), phase function ϕ(ρ), and structural parameters of a superresolution pupil filter as required. It is therefore of both great theoretical and practical significance to the design of a pupil filter to establish separate generalized models with fabrication errors for G and S directly related to A(ρ), ϕ(ρ), and the structural parameters, and especially to the structural fabrication errors. However, such a direct relationship of the G, S, and the structural parameters with fabrication errors has not been established so far. Consequently, it is very difficult to obtain the design and fabrication parameters of a pupil filter for lack of theoretical basis for definition of fabrication errors. Therefore, the intention is to use the basic definitions of G and S in Eq. (1) to establish new analytic models for G_Ae , G_Te , and S_e directly related to transmission function A(ρ), phase function A(ρ) and the structural parameters of an N-zone circular-symmetrical pupil filter with fabrication errors during the analysis, design, and fabrication of a superresolution pupil filter.

2. Pupil filter structure

The N-zone annular-symmetrical pupil filter with radius R, as shown in Fig.1, can be taken as an optical element with surface relief. If a surface relief is thin, the phase of an incident waveform is delayed by an amount proportional to the structural thickness at each point. Let the thickness of the pupil filter in its central zone along its axis be h ₁, and the thickness of the pupil filter in zone k be h_k , then the total phase delay induced in the wave in zone k passing through the structure in air can be expressed as shown below [14

M. L. Melocchi, Phase apodization for resolution enhancement (Ph.D. Dissertation, University of Rochester, 2003).

]

φ_{k} = \frac{{2 πnh}_{k}}{λ} - \frac{2 π (h_{k} - h_{1})}{λ}

(2)

where λ is the wavelength of an incident light, n is the refractive index of pupil-filter material.

Fig. 1. Circular-symmetrical pupil filter with radius R.

The central zone is usually used as datum, i.e. h ₁=0, and then, Eq. (2) can be rewritten as shown below:

φ_{k} = \frac{2 π (n - 1) h_{k}}{λ}

(3)

3. Analytical models for G_e and S_e of pupil filters with fabrication error

The pupil function of a N-zone circular-symmetrical super-resolution pupil filter P(ρ) is:

P_{k} (ρ) = t_{k} ∙ e^{{iφ}_{k}} (r_{k - 1} < ρ < r_{k}, k = 1,2,3, \dots, N)

(4)

where t_k is the amplitude transmission of the kth zone. P(ρ) is a phase-only pupil filter when t_k ≡1, P(ρ) is an amplitude pupil filter when φ_k ≡0 and P(ρ) is a complex pupil filter when t_k ∈(0,1) and φ_k ∈(0, 2π).

Concentricity is required to be checked N times during the fabrication of an N-zone circular-symmetrical pupil filter, and the transmission error, concentricity error, and depth etching error of an N-zone pupil filter all have their adverse effect on its superresolution characteristic parameters.

As shown in Fig. 2, let the centre of the Nth zone of a pupil filter be origin O in the polar coordinates system, the centre of the kth zone, be O_k , and points O and O_N are the same point. It is assumed that the intersection between polar radius OA_N of the Nth zone and the outer ring of the kth zone be A_k , normalized radius O_kA_k of the kth zone, a_k , eccentricity polar radius OO_k , Δρ_k , which is given by the ratio of centrifugal displacement Δx to actual radius R of a pupil filter, polar radius OA_k of the kth zone, ρ_k . Let ∠O ₁ OO_k =ϕ_k , ∠O ₁ OA_k =θ, ϕ_k ∈ [0,π] and ϕ ₁=0, and the angle is positive when OO ₁ rotates in counter-clockwise direction.

The radial error of the kth zone caused by etching line with Δw is σ_k=Δw/R, the error caused by the variation in the transmission of the kth zone is Δt_k , and the error caused by the variation in the etching depth of kth zone is Δh_k μm.

Fig. 2. Schematic eccentricity of an N-zone pupil filter.

The amplitude PSF of a pupil-filtering optical system with fabrication errors σ_k, Δt_k , and Δh_k can be expressed as

h (v, u) = 2 \int_{0}^{π} \int_{0}^{ρ_{1}} P_{1} (ρ) ∙ e^{\frac{iu ρ^{2}}{2}} ∙ J_{0} (ρv) ∙ ρdρdθ + \dots + 2 \int_{0}^{π} \int_{ρ_{k - 1}}^{ρ_{k}} P_{k} (ρ) ∙ e^{i \frac{2 π (n - 1)}{λ}} ∙ e^{\frac{{iuρ}^{2}}{2}} ∙ J_{0} (ρv) ∙ ρdρdθ + \dots + 2 \int_{0}^{π} \int_{ρ_{N - 1}}^{1} P_{k} (ρ) ∙ e^{i \frac{2 π (n - 1) Δ h_{N}}{λ}} ∙ e^{\frac{{iuρ}^{2}}{2}} ∙ J_{0} (ρv) ∙ ρdρdθ

= \sum_{k = 1}^{N} 2 \int_{0}^{π} \int_{ρ_{k - 1}}^{ρ_{k}} P_{k} (ρ) ∙ e^{i \frac{2 π (n - 1) Δ h_{k}}{λ}} ∙ e^{\frac{{iuρ}^{2}}{2}} ∙ J_{0} (ρv) ∙ ρdρdθ

(5)

{\begin{matrix} ρ_{k} = Δ ρ_{k} \cos (θ - ϕ_{k}) + \sqrt{{Δ ρ}_{k}^{2}} \cos^{2} (θ - ϕ_{k}) - {Δ ρ}_{k}^{2} + {(\frac{a_{k} + σ_{k}}{1 + σ_{N}})}^{2} \\ ρ_{0} = 0, ρ_{N} = 1 (k = 1, \dots, N) \end{matrix} θ \in [0, π]

(6)

where J ₀ is a zero-order Bessel function, ρ is the normalized polar radius at exit pupil, u is the axial normalized optical coordinate, v is the lateral normalized optical coordinate, and P_k (ρ) is the pupil function of the kth zone of an N-zone pupil filter.

Through the numerical integral of Eq. (5) on θ,

h (v, u) = 2 \frac{π}{M} ∙ \sum_{m = 1}^{M} [\int_{0}^{ρ (1, m)} P_{1} (ρ) ∙ e^{\frac{{iuρ}^{2}}{2}} ∙ J_{0} (ρv) ∙ ρd ρ + \dots + \int_{ρ (k - 1, m)}^{ρ (k, m)} P_{k} (ρ) ∙ e^{\frac{2 π (n - 1) Δ h_{k}}{λ}} ∙ e^{\frac{i {uρ}^{2}}{2}} ∙ J_{0} (ρv) ∙ ρd ρ + \dots + \int_{ρ (N - 1, m)}^{1} P_{N} (ρ) ∙ e^{\frac{2 π (n - 1) Δ h_{N}}{λ}} ∙ e^{\frac{{iuρ}^{2}}{2}} · J_{0} (ρv) ∙ ρd ρ]

= 2 \frac{π}{M} ∙ \sum_{m = 1}^{M} \sum_{k = 1}^{N} \int_{ρ (k - 1, m)}^{ρ (k, m)} P_{k} (ρ) ∙ e^{\frac{2 π (n - 1) Δ h_{k}}{λ}} ∙ e^{\frac{{iuρ}^{2}}{2}} ∙ J_{0} (ρv) ∙ ρd ρ

(7)

where

ρ (k, m) = Δ ρ_{k} \cos (\frac{m}{M} π - ϕ_{k}) + \sqrt{{Δ ρ}_{k}^{2} \cos^{2} (\frac{m}{M} π - ϕ_{k}) - Δ ρ_{k}^{2} + {(\frac{a_{k} + σ_{k}}{1 + σ_{N}})}^{2}}

(8)

{\begin{matrix} ρ (0, m) \equiv 0 \\ ρ (N, m) \equiv 1 \end{matrix} m = 1, \dots, M

(9)

The axial intensity is not symmetrical on u=0 when a superresolution pupil filter is used in an optical image system, and the maximum of its intensity has an offset u_Fe from the focal plane, i.e. the axial intensity is symmetrical on point (0, u_Fe ).

Let v=0, the axial intensity PSF of a superresolution optical system obtained using Eq. (7) is:

I (0, u) = {∣ h (0, u) ∣}^{2} = {∣ 2 \frac{π}{M} ∙ \sum_{m = 1}^{M} \sum_{k = 1}^{N} \int_{ρ (k - 1, m)}^{ρ (k, m)} P_{k} (ρ) ∙ e^{i \frac{2 π (n - 1) Δ h_{k}}{λ}} ∙ e^{\frac{{iuρ}^{2}}{2}} ρd ρ ∣}^{2}

(10)

Substituting the series expansion of e^x into Eq. (10), Eq. (10) can be simplified through the quadratic approximation as

I (0, u) \approx A_{e} u^{4} + B_{e} u^{3} + C_{e} u^{2} + D_{e} u + E_{e}

(11)

where

A_{e} = {[\sum_{k = 1}^{N} \sum_{m = 1}^{M} \frac{ρ^{6} (k, m) - ρ^{6} (k - 1, m)}{24} ∙ t_{k} ∙ \cos φ_{k}]}^{2} + {[\sum_{k = 1}^{N} \sum_{m = 1}^{M} \frac{ρ^{6} (k, m) - ρ^{6} (k - 1, m)}{24} ∙ t_{k} ∙ \sin φ_{k}]}^{2}

(12)

B_{e} = \frac{1}{48} {\sum_{k = 1}^{N} \sum_{m = 1}^{M} [ρ^{6} (k, m) - ρ^{6} (k - 1, m)] ∙ t_{k} ∙ \sin φ_{k}} ∙ {\sum_{k = 1}^{N} \sum_{m = 1}^{M} [ρ^{4} (k, m) - ρ^{4} (k - 1, m)] ∙ t_{k} ∙ \sin φ_{k}} - \frac{1}{48} {\sum_{k = 1}^{N} \sum_{m = 1}^{M} [ρ^{6} (k, m) - ρ^{6} (k - 1, m)] ∙ t_{k} ∙ \cos φ_{k}} ∙ {\sum_{k = 1}^{N} \sum_{m = 1}^{M} [ρ^{4} (k, m) - ρ^{4} (k - 1, m)] ∙ t_{k} ∙ \sin φ_{k}}

(13)

C_{e} = \frac{1}{16} {\sum_{k = 1}^{N} \sum_{m = 1}^{M} [ρ^{4} (k, m) - ρ^{4} (k - 1, m)] ∙ t_{k} ∙ \sin φ_{k}}^{2} + \frac{1}{16} {\sum_{k = 1}^{N} \sum_{m = 1}^{M} [ρ^{4} (k, m) - ρ^{4} (k - 1, m)] ∙ t_{k} ∙ \cos φ_{k}}^{2} - \frac{1}{12} {\sum_{k = 1}^{N} \sum_{m = 1}^{M} [ρ^{2} (k, m) - ρ^{2} (k - 1, m)] ∙ t_{k} ∙ \sin φ_{k}} ∙ {\sum_{k = 1}^{N} \sum_{m = 1}^{M} [ρ^{6} (k, m) - ρ^{6} (k - 1, m)] ∙ t_{k} ∙ \sin φ_{k}} - \frac{1}{12} {\sum_{k = 1}^{N} \sum_{m = 1}^{M} [ρ^{2} (k, m) - ρ^{2} (k - 1, m)] ∙ t_{k} ∙ \cos φ_{k}} ∙ {\sum_{k = 1}^{N} \sum_{m = 1}^{M} [ρ^{6} (k, m) - ρ^{6} (k - 1, m)] ∙ t_{k} ∙ \cos φ_{k}}

(14)

D_{e} = \frac{1}{2} {\sum_{k = 1}^{N} \sum_{m = 1}^{M} [ρ^{2} (k, m) - ρ^{2} (k - 1, m)] ∙ t_{k} ∙ \cos φ_{k}} ∙ {\sum_{k = 1}^{N} \sum_{m = 1}^{M} [ρ^{4} (k, m) - ρ^{4} (k - 1, m)] ∙ t_{k} ∙ \sin φ_{k}} - \frac{1}{2} {\sum_{k = 1}^{N} \sum_{m = 1}^{M} [ρ^{2} (k, m) - ρ^{2} (k - 1, m)] ∙ t_{k} ∙ \sin φ_{k}} ∙ {\sum_{k = 1}^{N} \sum_{m = 1}^{M} [ρ^{4} (k, m) - ρ^{4} (k - 1, m)] ∙ t_{k} ∙ \cos φ_{k}}

(15)

E_{e} = {\sum_{k = 1}^{N} \sum_{m = 1}^{M} [ρ^{2} (k, m) - ρ^{2} (k - 1, m)] ∙ t_{k} ∙ \cos φ_{k}}^{2} + {\sum_{k = 1}^{N} \sum_{m = 1}^{M} [ρ^{2} (k, m) - ρ^{2} (k - 1, m)] ∙ t_{k} ∙ \sin φ_{k}}^{2}

(16)

u_{Fe} = - \frac{D_{e}}{{2 C}_{e}}

(17)

{\begin{matrix} φ_{k} = φ_{k 0} + \frac{2 π (n - 1)}{λ} Δ h_{k} \\ t_{k} = t_{k 0} + Δ t_{k} \end{matrix} k = 1 \dots, N

(18)

where t_k0 and φ _k0 are the theoretical parameters required for the design of the kth zone of an N-zone pupil filter.

Using the differential of Eq. (11), and let

\frac{\partial I (0, u)}{\partial u} = 0

(19)

The axial coordinates for the first minimum of the axial intensity corresponding to superresolution pattern and Airy pattern are obtained. When the axial intensity is symmetrical on point (0, u_Fe ), axial superresolution parameter G_Ae obtained using Eq. (1) can be expressed as

G_{Ae} = \sqrt{\frac{{D_{e}}^{2} - {4 C}_{e} ∙ E_{e}}{192 \times {C_{e}}^{2}}}

(20)

Let u=u_Fe , the lateral intensity PSF of a superresolution optical system obtained using Eq. (8) is

I (v, u_{Fe}) = {∣ h v, u_{Fe}) ∣}^{2} = {∣ 2 \frac{π}{M} ∙ \sum_{m = 1}^{M} \sum_{k = 1}^{N} \int_{ρ (k - 1, m)}^{ρ (k, m)} P_{k} (ρ) ∙ e^{\frac{2 π (n - 1) Δ h_{k}}{λ}} \cdot e^{\frac{i u_{F e} ρ^{2}}{2}} ∙ J_{0} (ρv) ∙ ρd ρ ∣}^{2}

(21)

Substituting the series expansions of e^x and J ₀(x) into Eq. (21), Eq. (21) can be expressed through the quadratic approximation as

I (v, u_{Fe}) \approx H_{e}^{'} v^{4} - 2 F_{e} v^{2} + X_{e}

(22)

where

F_{e}^{'} = \frac{F_{e}}{8} \frac{u_{Fe}}{24} + {\sum_{k = 1}^{N} \sum_{m = 1}^{M} [ρ^{2} (k, m) - ρ^{2} (k - 1, m)] ∙ t_{k} ∙ \cos φ_{k}} ∙ {\sum_{k = 1}^{N} \sum_{m = 1}^{M} [ρ^{6} (k, m) - ρ^{6} (k - 1, m)] ∙ t_{k} ∙ \sin φ_{k}}

- \frac{u_{Fe}}{24} {\sum_{k = 1}^{N} \sum_{m = 1}^{M} [ρ^{2} (k, m) - ρ^{2} (k - 1, m)] ∙ t_{k} ∙ \sin φ_{k}} ∙ {\sum_{k = 1}^{N} \sum_{m = 1}^{M} [ρ^{6} (k, m) - ρ^{6} (k - 1, m)] ∙ t_{k} ∙ \cos φ_{k}}

+ \frac{u_{Fe}^{2}}{192} {[\sum_{k = 1}^{N} \sum_{m = 1}^{M} [ρ^{4} (k, m) - ρ^{4} (k - 1, m)] ∙ t_{k} ∙ \sin φ_{k}} ∙ {\sum_{k = 1}^{N} \sum_{m = 1}^{M} [ρ^{6} (k, m) - ρ^{6} (k - 1, m)] ∙ t_{k} ∙ \sin φ_{k}}

+ \frac{u_{Fe}^{2}}{192} {\sum_{k = 1}^{N} \sum_{m = 1}^{M} [ρ^{4} (k, m) - ρ^{4} (k - 1, m)] ∙ t_{k} ∙ \cos φ_{k}} ∙ {\sum_{k = 1}^{N} \sum_{m = 1}^{M} [ρ^{6} (k, m) - ρ^{6} (k - 1, m)] ∙ t_{k} ∙ \cos φ_{k}}

(23)

H_{e}^{'} = \frac{H_{e}}{64} + \frac{B_{e}}{2} u_{Fe} + A_{e} ∙ u_{Fe}^{2}

(24)

H_{e} = {\sum_{k = 1}^{N} \sum_{m = 1}^{M} [ρ^{4} (k, m) - ρ^{4} (k - 1, m)] ∙ t_{k} ∙ \cos φ_{k}}^{2} + {\sum_{k = 1}^{N} \sum_{m = 1}^{M} [ρ^{4} (k, m) - ρ^{4} (k - 1, m)] ∙ t_{k} ∙ \sin φ_{k}}^{2}

(25)

F_{e} = {\sum_{k = 1}^{N} \sum_{m = 1}^{M} [ρ^{2} (k, m) - ρ^{2} (k - 1, m)] ∙ t_{k} ∙ \cos φ_{k}} ∙ {\sum_{k = 1}^{N} \sum_{m = 1}^{M} [ρ^{4} (k, m) - ρ^{4} (k - 1, m)] ∙ t_{k} ∙ \cos φ_{k}]

+ {\sum_{k = 1}^{N} \sum_{m = 1}^{M} [ρ^{2} (k, m) - ρ^{2} (k - 1, m)] ∙ t_{k} ∙ \sin φ_{k}} ∙ {\sum_{k = 1}^{N} \sum_{m = 1}^{M} [ρ^{4} (k, m) - ρ^{4} (k - 1, m)] ∙ t_{k} ∙ \sin φ_{k}}

(26)

X_{e} = {\sum_{k = 1}^{N} \sum_{m = 1}^{M} [ρ^{2} (k, m) - ρ^{2} (k - 1, m)] ∙ t_{k} ∙ \cos φ_{k} + \frac{u_{Fe}}{4} ∙ \sum_{k = 1}^{N} \sum_{m = 1}^{M} [ρ^{4} (k, m) - ρ^{4} (k - 1, m)] ∙ t_{k} ∙ \sin φ_{k}}^{2}

+ {\sum_{k = 1}^{N} \sum_{m = 1}^{M} [ρ^{2} (k, m) - ρ^{2} (k - 1, m)] ∙ t_{k} ∙ \sin φ_{k} - \frac{u_{Fe}}{4} ∙ \sum_{k = 1}^{N} \sum_{m = 1}^{M} [ρ^{2} (k, m) - ρ^{2} (k - 1, m)] ∙ t_{k} ∙ \cos φ_{k}}^{2}

(27)

Using the differential of Eq. (21), and let

\frac{\partial I (v, u_{F})}{\partial v} = 0

(28)

The lateral coordinates for the first minimum of the lateral intensity corresponding to superresolution pattern and Airy pattern are obtained. When the lateral intensity is symmetrical on point (0, u_Fe ), lateral superresolution parameter G_Te obtained using Eq. (1) can be expressed as

G_{Te} = \sqrt{\frac{F_{e}^{'}}{{8 H}_{e}^{'}}}

(29)

Strehl ratio S of the intensity PSF symmetrical on point (0, u_Fe ) obtained using Eq. (1) can be expressed as

S_{e} = E_{e} + \frac{D_{e}}{2} ∙ u_{Fe}

(30)

When Δt_k =0, Δw=0, σ_k=Δw/R=0 and Δh_k =0, G_Te , G_Ae , and S_e obtained using Eqs. (20), (29) and (30) are the theoretical values required for the design of a pupil filter and can be written as G _T0, G _A0 and S ₀.

The effect of fabrication errors of a pupil filter on the superresolution parameters is

{\begin{matrix} Δ G_{A} = G_{Ae} - G_{A 0} \\ {Δ G}_{T} = G_{Te} - G_{T 0} \\ Δ S = S_{e} - S_{0} \end{matrix}

(31)

4. Effect of fabrication errors on superresolution property

The effect of main fabrication errors caused by eccentricity, etching line width, and etching depth on the superresolution property of a three-zone phase-only pupil filter with t_k ≡1 is analyzed using the models established for ΔG_A , ΔG_T , and ΔS to verify their validities.

4.1 Concentricity error

To make analyses easy, let σ_k=0 and Δh_k =0; only the concentricity error of a three-zone pupil filter is taken into consideration.

Let Δρ ₂=Δρ, when ϕ=ϕ ₁-ϕ ₂=0 or π i.e. Δφ ₁ and Δφ ₂ are in a straight line, the variations of superresolution parameters with concentricity error, shown in Fig. 3, are established using Eqs. (7)–(31).

Fig. 3. Variation of (a) ΔG_A , (b) ΔG_T , and (c) ΔS with concentricity error.

Figure 3(a) shows the variation of ΔG_A with Δφ ₁ and Δρ ₂, Fig. 3(b) shows the variation of ΔG_T with Δρ ₁ and Δρ ₂, and Fig. 3(c) shows the variation of ΔS with Δρ ₁ and Δρ ₂. Figure 3 shows that as Δρ increases, ΔG_A , ΔG_A , and ΔS increase and the variation when ϕ=π is larger than ϕ=0; when ϕ=0 and Δρ ₁ is invariable, the variations of ΔG_A , ΔG_T , and ΔS with Δρ ₂ are larger, and when ϕ=π and Δρ ₁=Δρ ₂ =Δρ, the variations of ΔG_A , ΔG_T , and ΔS with Δρ are larger. The variations of ΔG_T and ΔS are less than 0.01 when Δρ<0.05, the variation of ΔG_A , is less than 0.01 when Δρ<0.03.

When Δρ ₁ and Δρ ₂ are not in a straight line, the variations of ΔG_A , ΔG_T , and ΔS with the angle between concentricity errors shown in Fig. 4 are derived using Eqs. (7) and (31). Figure 4 (a) shows the variation of ΔG_A with ϕ when Δρ ₁=Δρ ₂=0.01, Δρ ₁=0.01, and Δρ ₂=0.02, Fig. 4(b) shows the variation of ΔG_T with ϕ when Δρ ₁=Δρ ₂=0.01, Δρ ₁=0.01, and Δρ ₂=0.02, and Fig. 4 (b) shows the variation of ΔS with ϕ when Δρ ₁=Δρ ₂=0.01, Δρ ₁=0.01, and Δρ ₂=0.02. Figure 4 shows that when Δρ≠Δρ ₂, the variations of ΔG_A , ΔG_T , and ΔS are larger, and when Δρ ₁; and Δρ ₂ are in a straight line, the variations of ΔG_A , ΔG_T , and ΔS are larger and the variations are the largest when ϕ=π.

Fig. 4. Variations of (a) ΔG_A , (b) ΔG_A and (c) ΔS with angle between concentricity errors.

Figures 3 and 4 show that the variations of superresolution parameters ΔG_A , ΔG_T , and ΔS are less than 0.2% when ϕ=π and Δρ=0.005. Δρ=0.005 is therefore taken as an extreme error, and centrifugal displacement Δx<12.5μm when R<2.5 mm.

4.2 Etching line width

Δw can easily cause a change in normalized radii σ of a three-zone pupil filter in the process of fabrication and therefore, it has an adverse effect on the superresolution property. The normalized radii of a pupil filter with extreme σ are

r_{k} = \frac{r_{k 0} + σ_{k}}{1 + σ_{3}} (k = 1,2,3)

(32)

where r _k0 is the theoretical normalized radius of the kth zone required for the design of a pupil filter.

Let σ₃=0, the variations of ΔG_A , ΔG_T , and ΔS with the radial error caused by etching line width (shown in Fig. 5), are obtained using Eqs. (7)–(31).

Figure 5 shows that when σ₁=σ₂=σ, ΔG_A , ΔG_T , and ΔS decrease as σ increases if σ>0, and ΔG_A , ΔG_T , and ΔS increase as the absolute value of σ increases if σ<0. When σ₁=-σ₂=σ, the trend of change in ΔG_A , ΔG_T , and ΔS is opposite to that when σ₁=σ₂=σbut their variable range increase obviously; the variable range in ΔG_A and ΔG_T are almost double of that when σ₁=σ₂=σ, and the variable range in ΔS is nearly half as many again as that when σ₁=σ₂=σ When σ₂=σ and σ₁=0.005, the trend of change in ΔG_A , ΔG_T , and ΔS is the same as when σ₁=σ₂=σ but their variable range increases.

Figure 5 indicates that the variations of superresolution parameters are most significant when σ₁=-σ₂=σ, the variations of ΔG_T and ΔS are less than 0.007 when |σ|<0.003, and the variation of ΔG_A is less than 0.01 when |σ|<0.002; i.e. the variations of ΔG_A , ΔG_T , and ΔS all are less than 1% when the etching-line width is less than 5 μm during fabrication process of a pupil filter of R<2.5 mm.

The analyses mentioned above indicate that the effect of the radial error caused by etching-line width on superresolution properties is larger than that caused by concentricity error during the fabrication process of a pupil filter.

Fig. 5. Variations of (a) ΔG_A , (b) ΔG_T and (c) ΔS with radial error caused by etching line width.

4.3 Error caused by variation of etching depth

The material used to fabricate a three-zone phase-only pupil filter is K9 glass with n=1.51466. When only the error caused by etching depth is considered, the variations of ΔG_A , ΔG_T , and ΔS with the error caused by etching depth (shown in Fig. 6), are obtained using Eqs. (7)–(31).

Fig. 6. Variations of (a) ΔG_A , (b) ΔG_T and (c) ΔS with phase error caused by etching depth.

Figure 6 shows that only the range of [0, 1] is considered, because the error curve is close to symmetrical on Δh=0 when Δh∈[-1, 1]. Also, Fig. 6 indicates that when Δh ₁=Δh ₂=Δh, the variation in ΔG_A is larger than that in ΔG_T and ΔS, and they are ΔG_A ∈[-0.2, 0], ΔG_T ∈[-0.014, 0], and ΔS∈[-0.01, 0]; when Δh ₁=-Δh ₂=Δh, the variation in ΔG_A is significant and ΔS slightly decreases as ΔG_T decreases whereas ΔS slightly increases as ΔG_T increases; when |Δh ₁| ≡ |Δh ₂|, the trend of change in ΔG_T and ΔS is the same as that when Δh ₁=Δh ₂=Δ_h and the trend of change in ΔG_A is opposite to that when Δh ₁=Δh ₂=Δ_h.

The comparisons of analyses mentioned above show that the effect of Δh on superresolution properties is obvious when Δh ₁=-Δh ₂=Δh, and the variations of ΔG_A , ΔG_T , and ΔS are less than 0.1 when the error of etching depth is less than 50 nm. |Δh|=50 nm is therefore an extreme depth etching error.

The analyses mentioned above indicate that the effect of etching depth error on superresolution properties and main-lobe intensity are both obvious. Therefore, a higher accuracy is required for the etching depth in the pupil fabrication process.

5. Conclusion

Three generalized analytical models have been established for superresolution parameters G _Ae, G_Te , and S_e related to transmission function A(ρ), phase function of ϕ(ρ), and the structural parameters with fabrication errors of an N-zone circular-symmetrical superresolution pupil filter. These models established at first relate the superresolution parameters of an N-zone superresolution pupil filter to its structural parameters to make its analyses, design, and fabrication easier. The analyses of the superresolution properties and fabrication errors for a three-zone phase-only pupil filter using the models established indicate that these models can provide an effective theoretical basis for the fabrication of a pupil filter, thereby providing a novel model for the design and fabrication of a superresolution pupil filter.

Acknowledgments

This work was supported by National Natural Science Foundation of China (No.50475035), the Doctoral Program of Higher Education of China (No.20050213035) and the Program for New Century Excellent Talents in University of China (Grant No.NCET-05-0348).

References and links

1.	T. Wilson, Confocal Microscopy (Academic Press. London, 1990).
2.	L. R. Qiu, W. Q. Zhao, Z. D. Feng, and X. M. Ding, “An approach to higher spatial resolution in a laser probe measurement system using a phase-only pupil filter,” Opt. Eng. 45 (to be published).
3.	M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, and M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun. 165, 267–278 (1999). [CrossRef]
4.	C. J. R. Sheppard and A. Choudhury, “Annular pupil, radial polarization, and superresolution,” Appl. Opt. 43, 4322–4327 (2004). [CrossRef] [PubMed]
5.	W. Q. Zhao, J. B. Tan, and L. R. Qiu, “Bipolar absolute differential confocal approach to higher spatial resolution,” Opt. Express 12, 5013–5021 (2004). [CrossRef] [PubMed]
6.	T. R. M. Sales, Phase-only Superresolution Elements (University of Rochester. Ph.D. Dissertation, 1997).
7.	C. J. R. Sheppard and Z. S. Hegedus, “Axial behaviour of pupil-plane filters,” J. Opt. Soc. Am. A 5, 643–647 (1988). [CrossRef]
8.	T. R. M. Sales and G. M. Morris, “Diffractive superresolution elements,” J. Opt. Soc. Am. A 14, 1637–1646 (1997). [CrossRef]
9.	D. M. de Juana, J. E. Oti, V. F. Canales, and M. P. Caqiqal, “Design of superresolving continuous phase filters, ” Opt. Lett. 28, 607–609 (2003). [CrossRef] [PubMed]
10.	S. Ledesma, J. Campos, J. C. Escalera, and M. J. Yzuel, “Simple expressions for performance parameters of complex filters with application to super- Gaussian phase filter,” Opt. Lett. 29, 932–934(2004). [CrossRef] [PubMed]
11.	H. T. Liu, Y. B. Yan, D. Yi, and G. F. Jin. “Design of three-dimensional superresolution filters and limits of axial optical supperresolution,” Appl. Opt. 42, 1463–1476 (2003). [CrossRef] [PubMed]
12.	H. T. Liu, Y. B. Yan, Q. F. Tan, and G. F. Jin, “Theories for the design of diffractive superresolution elements and limits of optical superresolution,” Opt. Soc. Am. A 19, 2185–2193 (2002). [CrossRef]
13.	M. Y. Yun, L. R. Liu, J. F. Sun, and D. A. Liu, “Three-dimension superresolution by three-zone complex pupil filters,” Opt. Soc. Am. A 22, 272–277 (2005). [CrossRef]
14.	M. L. Melocchi, Phase apodization for resolution enhancement (Ph.D. Dissertation, University of Rochester, 2003).

OCIS Codes
(100.6640) Image processing : Superresolution
(120.2440) Instrumentation, measurement, and metrology : Filters

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: May 11, 2006
Revised Manuscript: June 28, 2006
Manuscript Accepted: July 14, 2006
Published: August 7, 2006

Citation
Weiqian Zhao, Lirong Qiu, and Zhengde Feng, "Effect of fabrication errors on superresolution property of a pupil filter," Opt. Express 14, 7024-7036 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-16-7024

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References

T. Wilson, Confocal Microscopy (Academic Press. London, 1990).
<jrn>L. R. Qiu, W. Q. Zhao, Z. D. Feng, and X. M. Ding, "An approach to higher spatial resolution in a laser probe measurement system using a phase-only pupil filter," Opt. Eng. 45 (to be published).</jrn>
M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, and M. Kowalczyk, "Three-dimensional superresolution by annular binary filters," Opt. Commun. 165, 267-278 (1999). [CrossRef]
C. J. R. Sheppard and A. Choudhury, "Annular pupil, radial polarization, and superresolution," Appl. Opt. 43, 4322-4327 (2004). [CrossRef] [PubMed]
W. Q. Zhao, J. B. Tan, and L. R. Qiu, "Bipolar absolute differential confocal approach to higher spatial resolution," Opt. Express 12, 5013-5021 (2004). [CrossRef] [PubMed]
T. R. M. Sales, Phase-only Superresolution Elements (University of Rochester. Ph.D. Dissertation, 1997).
C. J. R. Sheppard and Z. S. Hegedus, "Axial behaviour of pupil-plane filters," J. Opt. Soc. Am. A 5, 643-647 (1988). [CrossRef]
T. R. M. Sales and G. M. Morris, "Diffractive superresolution elements," J. Opt. Soc. Am. A 14, 1637-1646 (1997). [CrossRef]
D. M. de Juana, J. E. Oti, V. F. Canales, and M. P. Caqiqal, "Design of superresolving continuous phase filters, " Opt. Lett. 28, 607-609 (2003). [CrossRef] [PubMed]
S. Ledesma, J. Campos, J. C. Escalera, and M. J. Yzuel, "Simple expressions for performance parameters of complex filters with application to super- Gaussian phase filter," Opt. Lett. 29, 932-934(2004). [CrossRef] [PubMed]
H. T. Liu, Y. B. Yan, D. Yi, and G. F. Jin. "Design of three-dimensional superresolution filters and limits of axial optical supperresolution," Appl. Opt. 42, 1463-1476 (2003). [CrossRef] [PubMed]
H. T. Liu, Y. B. Yan, Q. F. Tan, and G. F. Jin, "Theories for the design of diffractive superresolution elements and limits of optical superresolution," Opt. Soc. Am. A 19, 2185-2193 (2002). [CrossRef]
M. Y. Yun, L. R. Liu, J. F. Sun, and D. A. Liu, "Three-dimension superresolution by three-zone complex pupil filters," Opt. Soc. Am. A 22, 272-277 (2005). [CrossRef]
M. L. Melocchi, Phase apodization for resolution enhancement (Ph.D. Dissertation, University of Rochester, 2003).

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