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

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References

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Appl. Opt. (2)

Appl. Phys. Lett. (2)

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]

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. 83, 1104-1106 (2003).
[CrossRef]

Applied Optics (1)

H. Liu, Y. Yan, D. Yi and G. Jin, "Design of three-dimensional superresolution filters and limits of axial optical superresolution," Applied Optics 42, 1463-1476 (2003).
[CrossRef] [PubMed]

Chin. Phys. Lett. (1)

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]

Encyclopedia of Modern Optics (1)

S. M. Kuebler and M. Rumi, "Nonlinear optics -- applications: three-dimensional microfabrication," in Encyclopedia of Modern Optics, R. D. Guenther, D. G. Steel and L. Bayvel, eds. (Elsevier, Oxford, 2004).

J. Opt. Soc. Am. A (2)

J. Photopolym. Sci. Technol. (1)

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. 14, 657-668 (2001).
[CrossRef]

Jap. J. Appl. Phys. (1)

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

Microsc. Res. Tech. (1)

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]

Opt. Commun. (8)

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]

M. P. Cagigal, J. E. Oti, V. F. Canales and P. J. Valle, "Analytical design of superresolving phase filters," Opt. Commun. 241, 249-253 (2004).
[CrossRef]

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]

M. 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]

M. Martínez-Corral, P. Andrés, C. J. Zapata-Rodríguez and M. Kowalczyk, "Three-dimensional superresolution by annular binary filters," Opt. Commun. 165, 267-278 (1999).
[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]

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

Opt. Express (2)

Opt. Express (1)

Opt. Lett. (3)

Proc. Royal Soc. A (2)

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

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]

Other (3)

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]

A. Diaspro, Confocal and Two-Photon Microscopy: Foundations, Applications, and Advances (Wiley, New York, 2002).

B. Kress and P. Meyrueis, Digital Diffractive Optics (Wiley, New York, 2000).

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Figures (9)

Fig. 1.
Fig. 1.

Front (left) and profile (right) views of an annular multi-phase plate (AMP). The parameters Φi and ri represent the differential phase transmittance and fractional radius of the ith annular zone, respectively.

Fig. 2.
Fig. 2.

Optical geometry in which an AMP is used to modify the phase front and resulting PSF of a focused optical beam.

Fig. 3.
Fig. 3.

Characteristic changes to the axial PSF affected by a two-zone AMP. Shown are (A) the super-resolution factor G, (B) the sub-space of G < 1 on an expanded scale, (C) the side lobe intensity M, and (D) the Strehl ratio S versus [r 1, Φ 2]. The two-dimensional solutions space was discretized by intervals of ΔΦ = 2π/100 and Δr = 0.01. The inset to B shows the normalized double-peaked axial distribution that results for r 1 = 0.7 and Φ 2 = π.

Fig. 4.
Fig. 4.

Characteristic changes to the axial PSF affected by a three-zone AMP having Φ 1 = 0, Φ 2 = π, and Φ 3 = 0 as a function of radial zone boundaries r 1 and r 2. Shown are (A) the super-resolution factor G, (B) the sub-space G < 1 on an expanded scale, (C) the side lobe intensity M, and (D) the Strehl ratio, S. The two-dimensional solutions space [r 1, r 2] was discretized by intervals of Δr = 0.01.

Fig. 5.
Fig. 5.

Sub-space of G versus [r 1, r 2] for a three-zone AMP having Φ 1 = 0, Φ 2 = π, and Φ 3 = 0 for which axial super-resolution is achieved (G < 1) and side lobe intensity remains below 50% of the peak value (M < 0.5).

Fig. 6.
Fig. 6.

Comparison of the focused PSF generated when a three-zone AMP having Φ 1 = 0, Φ 2 = π, Φ 3 = 0, r 1 = 0.58, and r 2 = 0.73 is placed before the lens. (Left-top) Normalized axial and transverse intensity distribution within the plane of polarization (xz-plane) in the diffraction-limit (no AMP) and (Left-bottom) when the three-zone AMP is present. (Right) Axial and transverse intensity distribution of the AMP-modified beam alone. The AMP-generated PSF is axially super-resolved by G axial = 0.81, with M = 0.47 and S = 0.38, whereas the transverse intensity distribution of the central lobe is minimally broadened by G trans = 1.01.

Fig. 7.
Fig. 7.

(Left) Normalized axial and transverse intensity distribution in the plane of polarization (xz-plane) resulting when a three-zone AMP having r 1 = 0.43, r 2 = 0.69, Φ 1 = 0, Φ 2 = π, and Φ 3 = 0 is placed before the lens. The PSF is axially elongated by a factor of G axial = 6.1 yet remains diffraction limited in the transverse direction (G trans = 0.99). (Right) AMP-modified axial intensity distribution (red trace) versus that computed for diffraction limited focusing (no AMP, blue trace).

Fig. 8.
Fig. 8.

Comparison of the axial PSF parameters G and M as calculated using vector diffraction and scalar theory for three-zone AMPs having Φ 1 = 0, Φ 2 = π, and Φ 3 = 0. (A) G vector - G scalar and (B) M vector - M scalar versus [r 1, r 2].

Fig. 9.
Fig. 9.

Comparison of the normalized axial intensity distribution in the plane of polarization (xz-plane) calculated using vector diffraction (EM) and scalar theory at four values of NA for the case in which a three-zone AMP having r 1 = 0.43, r 2 = 0.69, Φ 1 = 0, Φ 2 = π, and Φ 3 = 0 is placed before the lens.

Equations (9)

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e x u v φ = iA ( I 0 + I 2 cos 2 φ )
e y u v φ = iA I 2 sin 2 φ
e z u v φ = 2 A I 1 cos φ
u = kz sin 2 α
v = kr sin α
I 0 ( u , v ) = 0 α t ( θ ) cos θ sin θ ( 1 + cos θ ) J 0 ( v sin θ sin α ) exp ( iu cos θ sin 2 α )
I 1 ( u , v ) = 0 α t ( θ ) cos θ sin 2 θ J 1 ( v sin θ sin α ) exp ( iu cos θ sin 2 α )
I 2 ( u , v ) = 0 α t ( θ ) cos θ sin θ ( 1 cos θ ) J 2 ( v sin θ sin α ) exp ( iu cos θ sin 2 α )
E ( u , v = 0 ) = iA 0 α t ( θ ) cos θ sin θ ( 1 + cos θ ) exp ( iu cos θ sin 2 α )

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