## Effect of fabrication errors on superresolution property of a pupil filter

Optics Express, Vol. 14, Issue 16, pp. 7024-7036 (2006)

http://dx.doi.org/10.1364/OE.14.007024

Acrobat PDF (525 KB)

### Abstract

Three analytical models have been established for superresolution parameters *G _{Ae}
*,

*G*, and

_{Te}*S*related to transmission function

_{e}*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*, and

_{Te}*S*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.

_{e}© 2006 Optical Society of America

## 1. Introduction

*G*, lateral spot size

_{A}*G*, and Strehl ratio S of the intensity point spread function (PSF) of a superresolution optical system, and they can be expressed as [6]:

_{T}*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.

*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.

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]

*G*,

_{A}*G*and

_{T}*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

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

*G*is defined as a coordinate ratio of the super-resolved pattern to the Airy disk pattern by the first minimum. In Ref. [9

_{T}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]

*G*,

_{A}*G*and

_{T}*S*of a complex amplitude pupil filter using the second order intensity expansion similar to the method in Ref. [7

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]

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]

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]

*G*,

_{A}*G*, and Strehl ratio

_{T}*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

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]

*G*and

*S*are preestablished as variable parameters in the equation. In Ref. [13

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]

## 2. Pupil filter structure

*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*, then the total phase delay induced in the wave in zone

_{k}*k*passing through the structure in air can be expressed as shown below [14]

*λ*is the wavelength of an incident light,

*n*is the refractive index of pupil-filter material.

*h*

_{1}=0, and then, Eq. (2) can be rewritten as shown below:

## 3. Analytical models for *G*_{e} and *S*_{e} of pupil filters with fabrication error

_{e}

_{e}

*N*-zone circular-symmetrical super-resolution pupil filter

*P*(

*ρ*) is:

*t*is the amplitude transmission of the

_{k}*k*th zone.

*P*(

*ρ*) is a phase-only pupil filter when

*t*≡1,

_{k}*P*(

*ρ*) is an amplitude pupil filter when

*φ*≡0 and

_{k}*P*(

*ρ*) is a complex pupil filter when

*t*∈(0,1) and

_{k}*φ*∈(0, 2π).

_{k}*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.

*N*th zone of a pupil filter be origin

*O*in the polar coordinates system, the centre of the

*k*th zone, be

*O*, and points

_{k}*O*and

*O*are the same point. It is assumed that the intersection between polar radius

_{N}*OA*of the

_{N}*N*th zone and the outer ring of the

*k*th zone be

*A*, normalized radius

_{k}*O*of the

_{k}A_{k}*k*th zone,

*a*, eccentricity polar radius

_{k}*OO*, Δ

_{k}*ρ*, which is given by the ratio of centrifugal displacement Δ

_{k}*x*to actual radius

*R*of a pupil filter, polar radius

*OA*of the

_{k}*k*th zone,

*ρ*. Let ∠

_{k}*O*

_{1}

*OO*=

_{k}*ϕ*, ∠

_{k}*O*

_{1}

*OA*=

_{k}*θ*,

*ϕ*∈ [0,π] and

_{k}*ϕ*

_{1}=0, and the angle is positive when

*OO*

_{1}rotates in counter-clockwise direction.

*k*th zone caused by etching line with Δ

*w*is σ

_{k}=Δ

*w*/

*R*, the error caused by the variation in the transmission of the

*k*th zone is Δ

*t*, and the error caused by the variation in the etching depth of

_{k}*k*th zone is Δ

*h*μm.

_{k}_{k}, Δ

*t*, and Δ

_{k}*h*can be expressed as

_{k}*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

*k*th zone of an

*N*-zone pupil filter.

*θ*,

*u*=0 when a superresolution pupil filter is used in an optical image system, and the maximum of its intensity has an offset

*u*from the focal plane, i.e. the axial intensity is symmetrical on point (0,

_{Fe}*u*).

_{Fe}*v*=0, the axial intensity PSF of a superresolution optical system obtained using Eq. (7) is:

*e*into Eq. (10), Eq. (10) can be simplified through the quadratic approximation as

^{x}*t*and

_{k0}*φ*

_{k0}are the theoretical parameters required for the design of the

*k*th zone of an

*N*-zone pupil filter.

*u*), axial superresolution parameter

_{Fe}*G*obtained using Eq. (1) can be expressed as

_{Ae}*u*=

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

_{Fe}*e*and

^{x}*J*

_{0}(

*x*) into Eq. (21), Eq. (21) can be expressed through the quadratic approximation as

*u*), lateral superresolution parameter

_{Fe}*G*obtained using Eq. (1) can be expressed as

_{Te}*S*of the intensity PSF symmetrical on point (0,

*u*) obtained using Eq. (1) can be expressed as

_{Fe}*t*=0, Δ

_{k}*w*=0, σ

_{k}=Δ

*w*/

*R*=0 and Δ

*h*=0,

_{k}*G*,

_{Te}*G*, and

_{Ae}*S*obtained using Eqs. (20), (29) and (30) are the theoretical values required for the design of a pupil filter and can be written as

_{e}*G*

_{T0},

*G*

_{A0}and

*S*

_{0}.

## 4. Effect of fabrication errors on superresolution property

*t*≡1 is analyzed using the models established for Δ

_{k}*G*, Δ

_{A}*G*, and Δ

_{T}*S*to verify their validities.

### 4.1 Concentricity error

_{k}=0 and Δ

*h*=0; only the concentricity error of a three-zone pupil filter is taken into consideration.

_{k}*ρ*

_{2}=Δ

*ρ*, when

*ϕ*=

*ϕ*

_{1}-

*ϕ*

_{2}=0 or π i.e. Δ

*φ*

_{1}and Δ

*φ*

_{2}are in a straight line, the variations of superresolution parameters with concentricity error, shown in Fig. 3, are established using Eqs. (7)–(31).

*G*with Δ

_{A}*φ*

_{1}and Δ

*ρ*

_{2}, Fig. 3(b) shows the variation of Δ

*G*with Δ

_{T}*ρ*

_{1}and Δ

*ρ*

_{2}, and Fig. 3(c) shows the variation of Δ

*S*with Δ

*ρ*

_{1}and Δ

*ρ*

_{2}. Figure 3 shows that as Δ

*ρ*increases, Δ

*G*, Δ

_{A}*G*, and Δ

_{A}*S*increase and the variation when

*ϕ*=π is larger than

*ϕ*=0; when

*ϕ*=0 and Δ

*ρ*

_{1}is invariable, the variations of Δ

*G*, Δ

_{A}*G*, and Δ

_{T}*S*with Δ

*ρ*

_{2}are larger, and when

*ϕ*=π and Δ

*ρ*

_{1}=Δ

*ρ*

_{2}=Δ

*ρ*, the variations of Δ

*G*, Δ

_{A}*G*, and Δ

_{T}*S*with Δ

*ρ*are larger. The variations of Δ

*G*and Δ

_{T}*S*are less than 0.01 when Δ

*ρ*<0.05, the variation of Δ

*G*, is less than 0.01 when Δ

_{A}*ρ*<0.03.

*ρ*

_{1}and Δ

*ρ*

_{2}are not in a straight line, the variations of Δ

*G*, Δ

_{A}*G*, and Δ

_{T}*S*with the angle between concentricity errors shown in Fig. 4 are derived using Eqs. (7) and (31). Figure 4 (a) shows the variation of Δ

*G*with

_{A}*ϕ*when Δ

*ρ*

_{1}=Δ

*ρ*

_{2}=0.01, Δ

*ρ*

_{1}=0.01, and Δ

*ρ*

_{2}=0.02, Fig. 4(b) shows the variation of Δ

*G*with

_{T}*ϕ*when Δ

*ρ*

_{1}=Δ

*ρ*

_{2}=0.01, Δ

*ρ*

_{1}=0.01, and Δ

*ρ*

_{2}=0.02, and Fig. 4 (b) shows the variation of Δ

*S*with

*ϕ*when Δ

*ρ*

_{1}=Δ

*ρ*

_{2}=0.01, Δ

*ρ*

_{1}=0.01, and Δ

*ρ*

_{2}=0.02. Figure 4 shows that when Δ

*ρ*≠Δ

*ρ*

_{2}, the variations of Δ

*G*, Δ

_{A}*G*, and Δ

_{T}*S*are larger, and when Δ

*ρ*

_{1}; and Δ

*ρ*

_{2}are in a straight line, the variations of Δ

*G*, Δ

_{A}*G*, and Δ

_{T}*S*are larger and the variations are the largest when

*ϕ*=π.

### 4.2 Etching line width

*w*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*

_{k0}is the theoretical normalized radius of the

*k*th zone required for the design of a pupil filter.

_{3}=0, the variations of Δ

*G*, Δ

_{A}*G*, and Δ

_{T}*S*with the radial error caused by etching line width (shown in Fig. 5), are obtained using Eqs. (7)–(31).

_{1}=σ

_{2}=σ, Δ

*G*, Δ

_{A}*G*, and Δ

_{T}*S*decrease as σ increases if σ>0, and Δ

*G*, Δ

_{A}*G*, and Δ

_{T}*S*increase as the absolute value of σ increases if σ<0. When σ

_{1}=-σ

_{2}=σ, the trend of change in Δ

*G*, Δ

_{A}*G*, and Δ

_{T}*S*is opposite to that when σ

_{1}=σ

_{2}=σbut their variable range increase obviously; the variable range in Δ

*G*and Δ

_{A}*G*are almost double of that when σ

_{T}_{1}=σ

_{2}=σ, and the variable range in Δ

*S*is nearly half as many again as that when σ

_{1}=σ

_{2}=σ When σ

_{2}=σ and σ

_{1}=0.005, the trend of change in Δ

*G*, Δ

_{A}*G*, and Δ

_{T}*S*is the same as when σ

_{1}=σ

_{2}=σ but their variable range increases.

_{1}=-σ

_{2}=σ, the variations of Δ

*G*and Δ

_{T}*S*are less than 0.007 when |σ|<0.003, and the variation of Δ

*G*is less than 0.01 when |σ|<0.002; i.e. the variations of Δ

_{A}*G*, Δ

_{A}*G*, and Δ

_{T}*S*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.

### 4.3 Error caused by variation of etching depth

*n*=1.51466. When only the error caused by etching depth is considered, the variations of Δ

*G*, Δ

_{A}*G*, and Δ

_{T}*S*with the error caused by etching depth (shown in Fig. 6), are obtained using Eqs. (7)–(31).

*h*=0 when Δ

*h*∈[-1, 1]. Also, Fig. 6 indicates that when Δ

*h*

_{1}=Δ

*h*

_{2}=Δ

*h*, the variation in Δ

*G*is larger than that in Δ

_{A}*G*and Δ

_{T}*S*, and they are Δ

*G*∈[-0.2, 0], Δ

_{A}*G*∈[-0.014, 0], and Δ

_{T}*S*∈[-0.01, 0]; when Δ

*h*

_{1}=-Δ

*h*

_{2}=Δ

*h*, the variation in Δ

*G*is significant and Δ

_{A}*S*slightly decreases as Δ

*G*decreases whereas ΔS slightly increases as Δ

_{T}*G*increases; when |Δ

_{T}*h*

_{1}| ≡ |Δ

*h*

_{2}|, the trend of change in Δ

*G*and Δ

_{T}*S*is the same as that when Δ

*h*

_{1}=Δ

*h*

_{2}=Δ

_{h}and the trend of change in Δ

*G*is opposite to that when Δ

_{A}*h*

_{1}=Δ

*h*

_{2}=Δ

_{h}.

*h*on superresolution properties is obvious when Δ

*h*

_{1}=-Δ

*h*

_{2}=Δ

*h*, and the variations of Δ

*G*, Δ

_{A}*G*, and Δ

_{T}*S*are less than 0.1 when the error of etching depth is less than 50 nm. |Δ

*h*|=50 nm is therefore an extreme depth etching error.

## 5. Conclusion

## Acknowledgments

## References and links

1. | T. Wilson, |

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. |

3. | M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, and M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun. |

4. | C. J. R. Sheppard and A. Choudhury, “Annular pupil, radial polarization, and superresolution,” Appl. Opt. |

5. | W. Q. Zhao, J. B. Tan, and L. R. Qiu, “Bipolar absolute differential confocal approach to higher spatial resolution,” Opt. Express |

6. | T. R. M. Sales, |

7. | C. J. R. Sheppard and Z. S. Hegedus, “Axial behaviour of pupil-plane filters,” J. Opt. Soc. Am. A |

8. | T. R. M. Sales and G. M. Morris, “Diffractive superresolution elements,” J. Opt. Soc. Am. A |

9. | D. M. de Juana, J. E. Oti, V. F. Canales, and M. P. Caqiqal, “Design of superresolving continuous phase filters, ” Opt. Lett. |

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. |

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. |

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 |

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 |

14. | M. L. Melocchi, |

**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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