Three analytical models have been established for superresolution parameters GAe, GTe, and Se 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 GAe, GTe, and Se 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.
© 2006 Optical Society of America
Superresolution pupil filters have been widely used to improve the resolution property of an optical system [1–5]. The superresolution properties of a pupil filter is mainly characterized by axial spot size GA, lateral spot size GT, and Strehl ratio S of the intensity point spread function (PSF) of a superresolution optical system, and they can be expressed as :
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,uF)|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 , the GA, GT 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. , 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 GT is defined as a coordinate ratio of the super-resolved pattern to the Airy disk pattern by the first minimum. In Ref. , the models are established for GA, GT and S of a complex amplitude pupil filter using the second order intensity expansion similar to the method in Ref. . In Ref. , based on the models in Ref. , superresolution parameters GA, GT, 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], 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. , 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 GAe, GTe, and Se 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 hk, then the total phase delay induced in the wave in zone k passing through the structure in air can be expressed as shown below 
where λ is the wavelength of an incident light, n is the refractive index of pupil-filter material.
The central zone is usually used as datum, i.e. h 1=0, and then, Eq. (2) can be rewritten as shown below:
3. Analytical models for Ge and Se of pupil filters with fabrication error
The pupil function of a N-zone circular-symmetrical super-resolution pupil filter P(ρ) is:
where tk is the amplitude transmission of the kth zone. P(ρ) is a phase-only pupil filter when tk≡1, P(ρ) is an amplitude pupil filter when φk≡0 and P(ρ) is a complex pupil filter when tk∈(0,1) and φk∈(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 Ok, and points O and ON are the same point. It is assumed that the intersection between polar radius OAN of the Nth zone and the outer ring of the kth zone be Ak, normalized radius OkAk of the kth zone, ak , eccentricity polar radius OOk, Δρk, which is given by the ratio of centrifugal displacement Δx to actual radius R of a pupil filter, polar radius OAk of the kth zone, ρk. Let ∠O 1 OOk=ϕk, ∠O 1 OAk=θ, ϕk∈ [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 σk=Δw/R, the error caused by the variation in the transmission of the kth zone is Δtk, and the error caused by the variation in the etching depth of kth zone is Δhk μm.
The amplitude PSF of a pupil-filtering optical system with fabrication errors σk, Δtk, and Δhk can be expressed as
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 Pk(ρ) is the pupil function of the kth zone of an N-zone pupil filter.
Through the numerical integral of Eq. (5) on θ,
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 uFe from the focal plane, i.e. the axial intensity is symmetrical on point (0, uFe).
Let v=0, the axial intensity PSF of a superresolution optical system obtained using Eq. (7) is:
where tk0 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
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, uFe), axial superresolution parameter GAe obtained using Eq. (1) can be expressed as
Let u=uFe, the lateral intensity PSF of a superresolution optical system obtained using Eq. (8) is
Using the differential of Eq. (21), and let
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, uFe), lateral superresolution parameter GTe obtained using Eq. (1) can be expressed as
Strehl ratio S of the intensity PSF symmetrical on point (0, uFe) obtained using Eq. (1) can be expressed as
When Δtk=0, Δw=0, σk=Δw/R=0 and Δhk=0, GTe, GAe, and Se 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
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 tk≡1 is analyzed using the models established for ΔGA, ΔGT, and ΔS to verify their validities.
4.1 Concentricity error
To make analyses easy, let σk=0 and Δhk=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).
Figure 3(a) shows the variation of ΔGA with Δφ 1 and Δρ 2, Fig. 3(b) shows the variation of ΔGT 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, ΔGA, ΔGA, 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 ΔGA, ΔGT, and ΔS with Δρ 2 are larger, and when ϕ=π and Δρ 1=Δρ 2 =Δρ, the variations of ΔGA, ΔGT, and ΔS with Δρ are larger. The variations of ΔGT and ΔS are less than 0.01 when Δρ<0.05, the variation of ΔGA, is less than 0.01 when Δρ<0.03.
When Δρ 1 and Δρ 2 are not in a straight line, the variations of ΔGA, ΔGT, 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 ΔGA 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 ΔGT 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 ΔGA, ΔGT, and ΔS are larger, and when Δρ 1; and Δρ 2 are in a straight line, the variations of ΔGA, ΔGT, and ΔS are larger and the variations are the largest when ϕ=π.
Figures 3 and 4 show that the variations of superresolution parameters ΔGA, ΔGT, 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
where r k0 is the theoretical normalized radius of the kth zone required for the design of a pupil filter.
Figure 5 shows that when σ1=σ2=σ, ΔGA, ΔGT, and ΔS decrease as σ increases if σ>0, and ΔGA, ΔGT, and ΔS increase as the absolute value of σ increases if σ<0. When σ1=-σ2=σ, the trend of change in ΔGA, ΔGT, and ΔS is opposite to that when σ1=σ2=σbut their variable range increase obviously; the variable range in ΔGA and ΔGT 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 ΔGA, ΔGT, 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 ΔGT and ΔS are less than 0.007 when |σ|<0.003, and the variation of ΔGA is less than 0.01 when |σ|<0.002; i.e. the variations of ΔGA, ΔGT, 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.
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 ΔGA, ΔGT, and ΔS with the error caused by etching depth (shown in Fig. 6), are obtained using Eqs. (7)–(31).
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 ΔGA is larger than that in ΔGT and ΔS, and they are ΔGA∈[-0.2, 0], ΔGT∈[-0.014, 0], and ΔS∈[-0.01, 0]; when Δh 1=-Δh 2=Δh, the variation in ΔGA is significant and ΔS slightly decreases as ΔGT decreases whereas ΔS slightly increases as ΔGT increases; when |Δh 1| ≡ |Δh 2|, the trend of change in ΔGT and ΔS is the same as that when Δh 1=Δh 2=Δh and the trend of change in ΔGA is opposite to that when Δh 1=Δh 2=Δh.
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 ΔGA, ΔGT, 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.
Three generalized analytical models have been established for superresolution parameters G Ae, GTe, and Se 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.
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]
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]
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).