Abstract

We use a Fourier transform approach to design pupil functions that modify the axial depth of focus for an optical system. We extend previous research in several ways. We first extend the depth of focus to 4cm for a 38cm focal length lens. We show that the transverse size of the focused beam is the same as for an open pupil. We then multiply the pupil function by a circular harmonic window function. The entire depth of focus is now characterized by a vortex beam. Finally we multiply our original pupil function by an edge-enhancing window function. Now the pupil function produces two sharp focus spots at the locations corresponding to the edges of the rectangle function.

© 2009 Optical Society of America

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References

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    [CrossRef]
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2007 (1)

2006 (2)

2001 (1)

1999 (3)

1998 (1)

1996 (3)

1995 (1)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

1993 (1)

S. N. Khonina, V. V. Kotlyar, and V. A. Soifer, “Calculation of the focusators into a longitudinal line-segment and study of a focal area,” J. Mod. Opt. 40, 761-769 (1993).
[CrossRef]

1992 (1)

1974 (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165-190 (1974).
[CrossRef]

1964 (1)

1931 (1)

P. A. M. Dirac, “Quantized singularities in the electromagnetic field,” Proc. R. Soc. London Ser. A 133, 60-72 (1931).
[CrossRef]

Amako, J.

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara and J. Amako, “Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects,” Opt. Eng. 38, 1051-1057 (1999).
[CrossRef]

Bandres, M. A.

Bentley, J. B.

Berry, M. V.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165-190 (1974).
[CrossRef]

Campos, J.

Cottrell, D. M.

Davis, J. A.

Dirac, P. A. M.

P. A. M. Dirac, “Quantized singularities in the electromagnetic field,” Proc. R. Soc. London Ser. A 133, 60-72 (1931).
[CrossRef]

Friese, M. E. J.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Gahagan, K. T.

Gutiérrez-Vega, J. C.

Haist, T.

He, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Heckenberg, N. R.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221-223 (1992).
[CrossRef] [PubMed]

Iemmi, C.

Khonina, S. N.

V. V. Kotlyar, S. N. Khonina, A. A. Kovalev, and V. A. Soifer, “Diffraction of a plane, finite-radius wave by a spiral phase plate,” Opt. Lett. 31, 1597-1599 (2006).
[CrossRef] [PubMed]

S. N. Khonina, V. V. Kotlyar, and V. A. Soifer, “Calculation of the focusators into a longitudinal line-segment and study of a focal area,” J. Mod. Opt. 40, 761-769 (1993).
[CrossRef]

Klein, M. V.

M. V. Klein, Optics (Wiley, 1970), Chap. 9.3.

Kotlyar, V. V.

V. V. Kotlyar, S. N. Khonina, A. A. Kovalev, and V. A. Soifer, “Diffraction of a plane, finite-radius wave by a spiral phase plate,” Opt. Lett. 31, 1597-1599 (2006).
[CrossRef] [PubMed]

S. N. Khonina, V. V. Kotlyar, and V. A. Soifer, “Calculation of the focusators into a longitudinal line-segment and study of a focal area,” J. Mod. Opt. 40, 761-769 (1993).
[CrossRef]

Kovalev, A. A.

López-Coronado, O.

McCutchen, C. W.

McDuff, R.

McNamara, D. E.

Moreno, I.

Nye, J. F.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165-190 (1974).
[CrossRef]

Piestun, R.

Reicherter, M.

Rubinsztein-Dunlop, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Shamir, J.

Sheppard, C. J. R.

C. J. R. Sheppard, “Synthesis of filters for specified axial properties,” J. Mod. Opt. 43, 525- 536 (1996)
[CrossRef]

Smith, C. P.

Smith, D. A.

Soifer, V. A.

V. V. Kotlyar, S. N. Khonina, A. A. Kovalev, and V. A. Soifer, “Diffraction of a plane, finite-radius wave by a spiral phase plate,” Opt. Lett. 31, 1597-1599 (2006).
[CrossRef] [PubMed]

S. N. Khonina, V. V. Kotlyar, and V. A. Soifer, “Calculation of the focusators into a longitudinal line-segment and study of a focal area,” J. Mod. Opt. 40, 761-769 (1993).
[CrossRef]

Sonehara, T.

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara and J. Amako, “Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects,” Opt. Eng. 38, 1051-1057 (1999).
[CrossRef]

Spektor, B.

Swartzlander, G. A.

Tiziani, H. J.

Tsai, P.

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara and J. Amako, “Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects,” Opt. Eng. 38, 1051-1057 (1999).
[CrossRef]

Tuvey, C. S.

Wagemann, E. U.

White, A. G.

Yzuel, M. J.

Appl. Opt. (3)

J. Mod. Opt. (2)

C. J. R. Sheppard, “Synthesis of filters for specified axial properties,” J. Mod. Opt. 43, 525- 536 (1996)
[CrossRef]

S. N. Khonina, V. V. Kotlyar, and V. A. Soifer, “Calculation of the focusators into a longitudinal line-segment and study of a focal area,” J. Mod. Opt. 40, 761-769 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara and J. Amako, “Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects,” Opt. Eng. 38, 1051-1057 (1999).
[CrossRef]

Opt. Lett. (7)

Phys. Rev. Lett. (1)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Proc. R. Soc. London Ser. A (2)

P. A. M. Dirac, “Quantized singularities in the electromagnetic field,” Proc. R. Soc. London Ser. A 133, 60-72 (1931).
[CrossRef]

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165-190 (1974).
[CrossRef]

Other (1)

M. V. Klein, Optics (Wiley, 1970), Chap. 9.3.

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Figures (8)

Fig. 1
Fig. 1

Amplitude and phase for pupil functions from rectangles having original lengths of (a), (b) 21 pixels and (c), (d) 61 pixels.

Fig. 2
Fig. 2

Axial intensity (arbitrary units) obtained for (a) a 2 cm pupil, (b) a 21-pixel pupil function, (c) a 61-pixel pupil function.

Fig. 3
Fig. 3

Transverse intensity for a 2 cm pupil at distances from the focal plane of (a)  1.5 mm , (b)  0 mm , (c)  + 1.5 mm ; for a 21-pixel pupil function at distances from the focal plane of (d)  7 mm , (e)  0 mm , (f)  + 7 mm ; for a 61-pixel pupil function at distances from the focal plane of (g)  18 mm , (h)  0 mm , (i)  + 20 mm . Image sizes are 124 μ m × 124 μ m .

Fig. 4
Fig. 4

Phase distribution for a 21-pixel pupil function from Fig. 1 with a circular harmonic phase of (a)  = 1 and (b)  = 3 .

Fig. 5
Fig. 5

Transverse intensity for a 61-pixel pupil function with a circular harmonic phase of = 1 , 3 at distances from the focal plane of (a),(d) 18 mm , (b),(e) 0 mm , and (c),(f) + 20 mm . Image sizes are 124 μ m × 124 μm .

Fig. 6
Fig. 6

Amplitude and phase pupil functions for (a), (b) a 21-pixel pupil function; (c), (d) product of the pupil from (a) and (b) with a Hilbert transform window function; (e), (f) product of the pupil from (a) and (b) with a derivative window function.

Fig. 7
Fig. 7

Axial intensity (arbitrary units) obtained for (a) the Hilbert transform pupil from Figs. 6c, 6d; (b) the derivative window pupil from Figs. 6e, 6f, where β = 0.5 .

Fig. 8
Fig. 8

Transverse intensity for a 21-pixel pupil function with a Hilbert transform window function and derivative window function at distances of (a), (d) 7 mm ; (b), (e) 0 mm ; and (c), (f) + 7 mm . Image sizes are 124 μm × 124 μm .

Equations (4)

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E ( x 2 = 0 , y 2 = 0 , z ) = i λ z 0 a p ( r ) exp ( i k r 2 2 f ) exp ( i k r 2 2 z ) 2 π r d r .
E ( u ) q ( s ) exp ( i 2 π u 0 s ) exp ( i 2 π u s ) d s .
Δ r min 2 2 a n .
Δ z = λ f 2 n a 2 .

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