## Vector diffraction analysis of high numerical aperture focused beams modified by two- and three-zone annular multi-phase plates

Optics Express, Vol. 14, Issue 3, pp. 1033-1043 (2006)

http://dx.doi.org/10.1364/OE.14.001033

Acrobat PDF (317 KB)

### Abstract

Vector diffraction theory was applied to study the effect of two-and three-zone annular multi-phase plates (AMPs) on the three-dimensional point-spread-function (PSF) that results when linearly polarized light is focused using a high numerical aperture refractory lens. Conditions are identified for which a three-zone AMP generates a PSF that is axially super-resolved by 19% with minimal change in the transverse profile and sufficiently small side lobes that the intensity pattern could be used for advanced photolithographic techniques, such as multi-photon 3D microfabrication, as well as multi-photon imaging. Conditions are also found in which a three-zone AMP generates a PSF that is axially elongated by 510% with only 1% change along the transverse direction. This intensity distribution could be used for sub-micron-scale laser drilling and machining.

© 2006 Optical Society of America

## 1. Introduction

1. S. F. Pereira and A. S. van de Nes, “Superresolution by means of polarization, phase and amplitude pupil masks,” Opt. Commun. **234**, 119–124 (2004). [CrossRef]

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

7. H. Wang and F. Gan, “High focal depth with pure-phase apodizer,” Appl. Opt. **40**, 5658–5662 (2001). [CrossRef]

8. H. Y. Chen, N. Mayhew, E. G. S. Paige, and G. G. Yang, “Design of the point spread function of a lens, binary phase filter combination and its application to photolithography,” Opt. Commun. **119**, 381–389 (1995). [CrossRef]

9. G. Yang, “An optical pickup using a diffractive optical element for a high-density optical disc,” Opt. Commun. **159**, 19–22 (1999). [CrossRef]

10. C. Ibáñez-López, G. Saavedra, G. Boyer, and M. Martínez-Corral, “Quasi-isotropic 3-D resolution in two-photon scanning microscopy,“ Opt. Express **13**, 6168–6174 (2005). [CrossRef] [PubMed]

11. C. Ibáñez-López, G. Saavedra, K. Plamann, G. Boyer, and M. Martínez-Corral, “Quasi-spherical focal spot in two-photon scanning microscopy by three-ring apodization,” Microsc. Res. Tech. **67**, 22–26 (2005). [CrossRef] [PubMed]

13. H. Ando, “Phase-shifting apodizer of three or more portions,” Jap. J. Appl. Phys. **31**, 557–567 (1992). [CrossRef]

15. V. F. Canales, J. E. Oti, and M. P. Cagigal, “Three-dimensional control of the focal light intensity distribution by analytically designed phase masks,” Opt. Commun. **247**, 11–18 (2005). [CrossRef]

*NA*) conditions, rays refracted near the periphery of the limiting aperture have a non-zero longitudinal field component (component parallel to the direction of propagation). This contribution to the overall intensity distribution is unaccounted for in scalar methods, so they do not accurately model focusing and DOE performance in a high-

*NA*configuration. A detailed analysis of binary- and multi-phase annular filters in the scalar limit has been reported by Sales and Morris [3

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

*NA*focusing can be achieved using vector diffraction theory [1

1. S. F. Pereira and A. S. van de Nes, “Superresolution by means of polarization, phase and amplitude pupil masks,” Opt. Commun. **234**, 119–124 (2004). [CrossRef]

20. S. Ching-Cherng and L. Chin-Ku, “Ultrasmall focusing spot with a long depth of focus based on polarization and phase modulation,” Opt. Lett. **28**, 99–101 (2003). [CrossRef]

24. E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. Royal Soc. A **253**, 349–357 (1959). [CrossRef]

25. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Royal Soc. A **253**, 358–379 (1959). [CrossRef]

*et al*. applied this method to study how an amplitude DOE alters the transverse intensity distribution [23

23. C. J. R. Sheppard and A. Choudhury, “Annular pupils, radial polarization, and superresolution,” Appl. Opt. **43**, 4322–4327 (2004). [CrossRef] [PubMed]

*et al*. used the vector diffraction method to study axial super-resolution achieved using amplitude-only DOEs [22

22. M. Martínez-Corral, R. Martinez-Cuenca, I. Escobar, and G. Saavedra, “Reduction of focus size in tightly focused linearly polarized beams,” Appl. Phys. Lett. **85**, 4319–4321 (2004). [CrossRef]

*NA*refractive focusing. Emphasis is placed on characterizing changes in the axial extent of the central lobe and changes in the relative intensity of side lobes. These characteristics are most relevant to multi-photon imaging techniques, multi-photon 3D microfabrication, and optical data storage and read-out schemes.

## 2. Method and theory

24. E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. Royal Soc. A **253**, 349–357 (1959). [CrossRef]

*NA*conditions. The optical geometry is depicted in Fig. 2. An

*N*-zone AMP and an aberration free lens (or lens system) are positioned such that their optical axes are collinear with the

*z*-axis of a cylindrical coordinate system whose origin is located at the Gaussian focus of the lens. The numerical aperture is

*NA*= 1.4 in all calculations, unless otherwise stated. Monochromatic linearly polarized plane waves, with electric field vector parallel to the

*x*-axis, propagate along the

*z*-axis, passing through the AMP and entering the pupil of the lens. The light focuses into a medium of refractive index

*n*= 1.5. In the absence of the AMP, the situation is consistent with common applications of high-

*NA*oil-immersion objective lenses.

*P*(

*x*,

*y*,

*z*) in the neighborhood of the focus may be expressed in the cylindrical optical coordinate system [

*u*,

*v*,

*φ*] as

*P*is

*I*∝ ∣

*e*+

_{x}*e*+

_{y}*e*∣

_{z}^{2}, and the PSF is a spatial map of intensity for all values of [

*u*,

*v*,

*φ*] about the focus.

*φ*is defined as the angle subtended by the electric field vector of the incident field and the meridional plane in which the field is calculated. The constant

*A*=

*π*

*l*

_{0}

*f*/

*λ*is defined in terms of the focal length,

*f*, the wavelength within the medium,

*λ*, and

*l*

_{o}, which describes the amplitude distribution of the incident field. It is assumed that uniform amplitude plane waves impinge on the lens, so

*l*

_{o}is set to unity.

*z*and

*r*are the radial and axial coordinates, respectively, of the point in the original coordinate system. The maximum aperture angle,

*α*= arcsin(

*NA*/

*n*), is determined by the numerical aperture of the lens. The wave number

*k*= 2

*π*/

*λ*.

*I*

_{0,1,2}are integrals evaluated over the aperture half-angle

*θ*as

*t*(

*θ*). In applying these formulae, the following approximations are implicit. (1) All inhomogeneous waves are ignored. (2) The Kirchoff boundary conditions are imposed, which is appropriate for AMPs having macroscopic features, as considered here. (3) The Debye approximation is also applied, so only rays falling within the numerical aperture of the lens are considered [25

25. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Royal Soc. A **253**, 358–379 (1959). [CrossRef]

26. J. J. Stamnes, *Waves in Focal Regions: Propagation, Diffraction and Focusing of Light, Sound and Water Waves, in The Adam Hilger Series on Optics and Optoelectronics*, E. R. Pike and W. T. Welford, eds., (Adam Hilger, Bristol, 1986). [PubMed]

*E*(

*u*,

*v*= 0), only depends upon

*I*

_{0}(

*u*,

*v*= 0) giving:

*N*concentric annular zones each having constant differential phase transmittance

*Φ*, (Fig. 1). The radial extent of the AMP,

_{i}*R*, is matched to the limiting aperture of the lens. The radius of the

*i*th zone may be expressed as a dimensionless fraction of

*R*using

*r*= sin(

_{i}*θ*)/sin(

_{i}*α*), where

*θ*is the aperture half-angle of the

_{i}*i*th zone. The optical characteristics of an AMP are determined by the radius and relative phase of each zone. As such, the innermost zone may always be set to

*Φ*

_{1}= 0, and the others may be varied independently over the interval [0, 2

*π*]. The number of independent degrees of freedom is then two for a two-zone AMP (

*r*

_{1}and

*Φ*

_{2}, where 0 <

*r*

_{1}< 1), and it is four for a three-zone AMP (

*r*

_{1},

*r*

_{2},

*Φ*

_{2}, and

*Φ*

_{3}, where 0 <

*r*

_{1}<

*r*

_{2}< 1).

*ΔΦ*= 2

*π*/20 and

*Δr*= 0.05. Specific regions of interest were studied in greater detail as needed by decreasing

*ΔΦ*and

*Δr*. The PSF for each set of AMP parameters was characterized relative to the diffraction limited pattern in terms of (1) the axial (transverse) width of the central lobe; (2) the peak intensity; and (3) the intensity of the largest side lobe(s). The peak in the PSF pattern having the highest intensity was regarded as the central lobe. Under this definition, the central lobe is not necessarily centered at the Gaussian focus. The axial (transverse) extent of the central lobe was quantified using a super-resolution factor,

*G*, defined as the full-width at half-maximum (FWHM) of the central lobe divided by the same in the diffraction limited pattern. The axial Strehl ratio,

*S*, is defined as the peak intensity of the central lobe normalized to that of the diffraction limited pattern. The relative intensity of the largest axial side lobe is quantified using the parameter,

*M*, which is defined as the peak-intensity of the side lobe divided by that of the central lobe.

## 3. Results and discussion

*G*,

*M*, and

*S*all exhibit the greatest variation as a function of

*r*

_{1}along the line

*Φ*

_{2}=

*π*, and the plots are symmetric about this line.

*G*varies from a maximum of 2.31 (

*Φ*

_{2}=

*π*and

*r*

_{1}= 0.54) to a minimum of 0.90 (

*Φ*

_{2}=

*π*and

*r*

_{1}= 0.76). Thus, a two-zone AMP could be used to elongate the central lobe by as much as a factor of two. Where

*G*= 0.90, the axial intensity distribution is comprised of two partly overlapping lobes of equal peak intensity, so

*M*= 1. Given that there are two lobes in the intensity distribution, the PSF cannot reasonably be regarded as super-resolved. This finding is consistent with that reported by Sales, who evaluated super-resolution in terms of the separation of minima in the axial PSF in the confocal mode [4

4. T. R. M. Sales and G. M. Morris, “Axial superresolution with phase-only pupil filters,” Opt. Commun. **156**, 227–230 (1998). [CrossRef]

*S*= 0.35.

*G*and the greatest super-resolution occurs when successive zones of the AMP differ in phase by

*π*. The overall appearance of the PSF is determined by the vector sum of the electric field component of rays that converge near the focus. The greatest overall variation can be expected then when rays recombine with the highest degree of destructive interference, or when they successively differ in phase by

*π*.

*Φ*

_{1}= 0,

*Φ*

_{2}=

*π*, and

*Φ*

_{3}= 0, and the axial PSF was simulated with the [

*r*

_{1},

*r*

_{2}] space discretized by

*Δr*= 0.01. The corresponding plots of

*G*,

*M*and

*S*versus

*r*

_{1}and

*r*

_{2}are shown in Fig. 4. The PSF characteristics are only defined for the upper-left half of the [

*r*

_{1},

*r*

_{2}] space due to the constraint 0 <

*r*

_{1}<

*r*

_{2}< 1.

*G*takes a minimum value of 0.73 at

*r*

_{1}= 0.60 and

*r*

_{2}= 0.77; however,

*M*is approximately unity under these conditions because the axial intensity distribution near the focus actually consists of three lobes having nearly the same peak intensity. This is similar to the circumstances under which

*G*is minimized for a two-zone AMP.

*G*, if only the central lobe has appreciable intensity. Note, however, that if the side lobe intensity exceeds 50%, then the photo-processed volume will be comprised by multiple features.

*G*for all points in the [

*r*

^{1},

*r*

_{2}] space for which

*G*< 1 and

*M*< 0.5. This represents the sub-set of three-zone AMPs that yield an axially super-resolved focus and for which the intensity of the side lobes remains below 50% of the peak intensity. Under these criteria, the maximum axial super-resolution occurs for

*r*

_{1}= 0.58,

*r*

_{2}= 0.73,

*Φ*

_{1}= 0,

*Φ*

_{2}=

*π*, and

*Φ*

_{3}= 0, at which point

*G*= 0.81,

*M*= 0.47, and

*S*= 0.38. The effect of this AMP on the 3D-PSF is shown in Fig. 6. The intensity distribution along the optic axis shows clearly that super-resolution is achieved at the expense of higher side lobes. The transverse intensity distribution also shows that some power is re-distributed into weak side lobes, which is consistent with the decrease in the Strehl ratio. This AMP could be used for 3DM, 3D optical data storage/read-out, or any other application that requires a co-minimized axial and transverse intensity distribution and minimized side lobes.

*G*

_{axial}> 1). This can also be viewed as an extended depth of focus. Similar findings have been reported for annular phase DOEs [7

7. H. Wang and F. Gan, “High focal depth with pure-phase apodizer,” Appl. Opt. **40**, 5658–5662 (2001). [CrossRef]

20. S. Ching-Cherng and L. Chin-Ku, “Ultrasmall focusing spot with a long depth of focus based on polarization and phase modulation,” Opt. Lett. **28**, 99–101 (2003). [CrossRef]

*G*

_{axial}takes a maximum value of 2.31 at

*ϕ*

_{2}=

*π*and

*r*

_{1}= 0.54. Under these conditions the lateral extent of the central lobe is

*G*

_{trans}= 1.00, as measured in the transverse plane that contains the peak axial intensity. Even greater PSF elongation can be obtained with a three-zone AMP, which produces a maximum value of

*G*

_{axial}= 6.1 for

*r*

_{1}= 0.43,

*r*

_{2}= 0.69,

*ϕ*

_{1}= 0,

*ϕ*

_{2}=

*π*, and

*ϕ*

_{3}= 0. The axial and transverse intensity distribution in the plane of polarization (

*xz*-plane) is shown in Fig. 7. The elongated PSF appears to be the result of close overlap between a focal-plane centered lobe and four adjacent axial side lobes. The outer side lobes of the set attain the same peak intensity as the focal-plane centered lobe, and the intensity between lobes decreases to only ~60% of the peak value. It is noteworthy that the transverse width of the central lobe remains nearly invariant along the full length of the five-lobe set (

*G*

_{trans}= 0.99 in the Gaussian focal plane). This intensity profile could be used for laser drilling and laser machining applications in which sub-diffraction-limited features are created over an axial distance of several microns. It could also be used in microscopy and imaging applications for achieving sub-diffraction-limited resolution over an extended depth of field.

*G*

_{trans}was considered in some earlier studies of axial super-resolution [17

17. Martínez-Corral, P. Andrés, J. Ojeda-Castañeda, and G. Saavedra, “Tunable axial superresolution by annular binary filters. Application to confocal microscopy,” Opt. Commun. **119**, 491–498 (1995). [CrossRef]

19. Martínez-Corral, M. T. Caballero, E. H. K. Stelzer, and J. Swoger, “Tailoring the axial shape of the point spread function using the Toraldo concept,” Opt. Express **10**, 98–103 (2002). [PubMed]

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

*t*through the change of variables

*ζ*= [(cos

*θ*- cos

*α*)/(1 - cos

*α*)] - 0.5. In these works, centrosymmetric DOEs – those for which

*t*(

*ζ*) is an even function – are shown to leave

*G*

_{trans}= 1. However, the subject does not appear to have been explored to a level that one may conclude

*G*

_{trans}= 1

*if and only if*the DOE is centrosymmetric. We note that the DOEs discussed in the present work are not exclusively centrosymmetric (

*e.g.*the AMPs corresponding to Figs. 6 and 7 are non-centrosymmetric). These results suggest that minimal change to the transverse PSF can also be achieved with certain non-centrosymmetric DOE configurations.

*NA*focal field distributions are not accurately described by scalar theory or methods that employ the paraxial approximation, the magnitude of the discrepancy has not been widely examined. This subject was investigated quantitatively by using both vector diffraction and scalar theory [4

4. T. R. M. Sales and G. M. Morris, “Axial superresolution with phase-only pupil filters,” Opt. Commun. **156**, 227–230 (1998). [CrossRef]

*Φ*

_{1}=

*Φ*

_{3}= 0 and

*Φ*

_{2}=

*π*and then plotting the differences in the characteristic parameters

*G*

_{vector}-

*G*

_{scalar}and

*M*

_{vector}-

*M*

_{scalar}versus [

*r*

_{1},

*r*

_{2}] (Fig. 8). It was found that both levels of theory predict

*qualitatively*similar changes in the PSF as a function of AMP configuration and the values of the characteristic PSF parameters

*G*and

*M*are similar. Yet they differ most in those situations for which the PSF undergoes extreme axial change, be that super-resolution or elongation (compare Figs. 8 and 4). To illustrate the point further, Fig. 9 shows the evolution of the axial intensity distributions calculated using the vector diffraction and scalar methods at four values of

*NA*for a three-zone AMP having

*r*

_{1}= 0.43,

*r*

_{2}= 0.69,

*Φ*

_{1}= 0,

*Φ*

_{2}=

*π*, and

*Φ*

_{3}= 0, which produces the axially stretched PSF shown in Fig. 7 (at

*NA*= 1.4). Although both levels of theory predict that the PSF is axially elongated, the patterns differ significantly as the

*NA*increases. Notably, we find that

*G*

_{vector}-

*G*

_{scalar}= 5.04 at

*NA*= 1.4. Thus, scalar theory may be useful for rapid, qualitative assessment of DOEs under high-

*NA*conditions, but vector diffraction theory appears essential for accurate simulation of the PSF.

## 4. Conclusion

*NA*focusing of linearly polarized incident light. A systematic approach was adopted in which PSFs were calculated and compared for all possible combinations of phase and zone radius within the discretized two- and four-dimensional space associated with two- and three-zone AMPs, respectively. Two-zone AMP configurations were identified that marginally decrease the axial width of the central lobe, but this is accompanied by a large increase in the intensity of adjacent side lobes that make the achievable intensity distributions unsatisfactory for most applications. Conditions were found for which a three-zone AMP yields an axial intensity distribution that is super-resolved by 19% with minimal change in the transverse profile and sufficiently small side lobes that the intensity pattern could be used for micro-lithographic and micro-imaging applications. Interestingly, conditions were also identified for which the axial PSF is elongated by 510% with only 1% change along the transverse direction. This intensity distribution could be used for sub-micron-scale laser drilling and machining. A comparison of intensity distributions calculated under high-

*NA*conditions using vector and scalar theories shows that the latter is suitable for identifying qualitative changes in the PSF, but the detailed intensity distribution can differ markedly from that computed using the more accurate vector diffraction method.

## Acknowledgements

## References and links

1. | S. F. Pereira and A. S. van de Nes, “Superresolution by means of polarization, phase and amplitude pupil masks,” Opt. Commun. |

2. | A. Diaspro, |

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

4. | T. R. M. Sales and G. M. Morris, “Axial superresolution with phase-only pupil filters,” Opt. Commun. |

5. | X.-F. Zhao, C.-F. Li, and H. Ruan, “Improvement of three-dimensional resolution in optical data storage by combination of two annular binary phase filters,” Chin. Phys. Lett. |

6. | S. Zhou and C. Zhou, “Discrete continuous-phase superresolving filters,” Opt. Lett. |

7. | H. Wang and F. Gan, “High focal depth with pure-phase apodizer,” Appl. Opt. |

8. | H. Y. Chen, N. Mayhew, E. G. S. Paige, and G. G. Yang, “Design of the point spread function of a lens, binary phase filter combination and its application to photolithography,” Opt. Commun. |

9. | G. Yang, “An optical pickup using a diffractive optical element for a high-density optical disc,” Opt. Commun. |

10. | C. Ibáñez-López, G. Saavedra, G. Boyer, and M. Martínez-Corral, “Quasi-isotropic 3-D resolution in two-photon scanning microscopy,“ Opt. Express |

11. | C. Ibáñez-López, G. Saavedra, K. Plamann, G. Boyer, and M. Martínez-Corral, “Quasi-spherical focal spot in two-photon scanning microscopy by three-ring apodization,” Microsc. Res. Tech. |

12. | B. Kress and P. Meyrueis, |

13. | H. Ando, “Phase-shifting apodizer of three or more portions,” Jap. J. Appl. Phys. |

14. | M. P. Cagigal, J. E. Oti, V. F. Canales, and P. J. Valle, “Analytical design of superresolving phase filters,” Opt. Commun. |

15. | V. F. Canales, J. E. Oti, and M. P. Cagigal, “Three-dimensional control of the focal light intensity distribution by analytically designed phase masks,” Opt. Commun. |

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

17. | Martínez-Corral, P. Andrés, J. Ojeda-Castañeda, and G. Saavedra, “Tunable axial superresolution by annular binary filters. Application to confocal microscopy,” Opt. Commun. |

18. | M. Martínez-Corral, P. Andrés, C. J. Zapata-Rodríguez, and M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun. |

19. | Martínez-Corral, M. T. Caballero, E. H. K. Stelzer, and J. Swoger, “Tailoring the axial shape of the point spread function using the Toraldo concept,” Opt. Express |

20. | S. Ching-Cherng and L. Chin-Ku, “Ultrasmall focusing spot with a long depth of focus based on polarization and phase modulation,” Opt. Lett. |

21. | M. Martínez-Corral, C. Ibáñez-López, G. Saavedra, and M. T. Caballero, “Axial gain resolution in optical sectioning fluorescence microscopy by shaded-ring filters,” Opt. Express |

22. | M. Martínez-Corral, R. Martinez-Cuenca, I. Escobar, and G. Saavedra, “Reduction of focus size in tightly focused linearly polarized beams,” Appl. Phys. Lett. |

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

24. | E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. Royal Soc. A |

25. | B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Royal Soc. A |

26. | J. J. Stamnes, |

27. | H. Liu, Y. Yan, D. Yi, and G. Jin, “Design of three-dimensional superresolution filters and limits of axial optical superresolution,” Applied Optics |

28. | S. M. Kuebler and M. Rumi, “Nonlinear optics ╍ applications: three-dimensional microfabrication,” in |

29. | S. M. Kuebler, M. Rumi, T. Watanabe, K. Braun, B. H. Cumpston, A. A. Heikal, L. L. Erskine, S. Thayumanavan, S. Barlow, S. R. Marder, and J. W. Perry, “Optimizing two-photon initiators and exposure conditions for three-dimensional lithographic microfabrication,” J. Photopolym. Sci. Technol. |

30. | H.-B. Sun, K. Takada, M.-S. Kim, K.-S. Lee, and S. Kawata, “Scaling laws of voxels in two-photon photopolymerization nanofabrication,” Appl. Phys. Lett. |

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

**OCIS Codes**

(050.1220) Diffraction and gratings : Apertures

(100.6640) Image processing : Superresolution

(170.5810) Medical optics and biotechnology : Scanning microscopy

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: December 19, 2005

Revised Manuscript: January 21, 2006

Manuscript Accepted: January 23, 2006

Published: February 6, 2006

**Virtual Issues**

Vol. 1, Iss. 3 *Virtual Journal for Biomedical Optics*

**Citation**

Toufic Jabbour and Stephen Kuebler, "Vector diffraction analysis of high numerical aperture focused beams modified by two- and three-zone annular multi-phase plates," Opt. Express **14**, 1033-1043 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-3-1033

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### References

- S. F. Pereira and A. S. van de Nes, "Superresolution by means of polarization, phase and amplitude pupil masks," Opt. Commun. 234, 119-124 (2004). [CrossRef]
- A. Diaspro, Confocal and Two-Photon Microscopy: Foundations, Applications, and Advances (Wiley, New York, 2002).
- T. R. M. Sales and G. M. Morris, "Diffractive superresolution elements," J. Opt. Soc. Am. A 14, 1637-1646 (1997). [CrossRef]
- T. R. M. Sales and G. M. Morris, "Axial superresolution with phase-only pupil filters," Opt. Commun. 156, 227-230 (1998). [CrossRef]
- X.-F. Zhao, C.-F. Li and H. Ruan, "Improvement of three-dimensional resolution in optical data storage by combination of two annular binary phase filters," Chin. Phys. Lett. 21, 1515-1517 (2004). [CrossRef]
- S. Zhou and C. Zhou, "Discrete continuous-phase superresolving filters," Opt. Lett. 29, 2746-2748 (2004). [CrossRef] [PubMed]
- H. Wang and F. Gan, "High focal depth with pure-phase apodizer," Appl. Opt. 40, 5658-5662 (2001). [CrossRef]
- H. Y. Chen, N. Mayhew, E. G. S. Paige and G. G. Yang, "Design of the point spread function of a lens, binary phase filter combination and its application to photolithography," Opt. Commun. 119, 381-389 (1995). [CrossRef]
- G. Yang, "An optical pickup using a diffractive optical element for a high-density optical disc," Opt. Commun. 159, 19-22 (1999). [CrossRef]
- C. Ibáñez-López, G. Saavedra, G. Boyer and M. Martínez-Corral, "Quasi-isotropic 3-D resolution in two-photon scanning microscopy," Opt. Express 13, 6168-6174 (2005). [CrossRef] [PubMed]
- C. Ibáñez-López, G. Saavedra, K. Plamann, G. Boyer and M. Martínez-Corral, "Quasi-spherical focal spot in two-photon scanning microscopy by three-ring apodization," Microsc. Res. Tech. 67, 22-26 (2005). [CrossRef] [PubMed]
- B. Kress and P. Meyrueis, Digital Diffractive Optics (Wiley, New York, 2000).
- H. Ando, "Phase-shifting apodizer of three or more portions," Jap. J. Appl. Phys. 31, 557-567 (1992). [CrossRef]
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