Abstract

Spatial engineering of polarization as a new method of beam shaping is analyzed by using scalar diffraction theory. For the one-dimensional case, it is shown that the smallest flattop far-field distribution can be obtained by adopting a linear polarization that changes direction as a linear function of location in the pupil plane. The resulting light field is functionally equivalent to a cosinusoidal function modulation of the wavefront but maintains high efficiency. This polarization beam shaping technique proves to be highly useful in applications where diffraction effects need to be taken into account. The extension of this technique to two-dimensional beam shaping is also demonstrated.

© 2008 Optical Society of America

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References

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  1. F. M. Dickey and S. C. Holswade, Laser Beam Shaping--Theory and Techniques (Marcel Dekker, 2000).
    [CrossRef]
  2. L. A. Romero and F. M. Dickey, “Lossless laser beam shaping,” J. Opt. Soc. Am. A 13, 751-760 (1996).
    [CrossRef]
  3. B. Hao and J. Leger, “Smallest 1D flat-top focus by polarization manipulation,” Proc. SPIE 6290, 629001 (2006).
    [CrossRef]
  4. B. Hao and J. Leger, “Spatially inhomogeneous polarization in laser beam shaping,” Proc. SPIE 6682, 66820Y (2007).
    [CrossRef]
  5. B. Hao and J. Leger, “Polarization beam shaping,” Appl. Opt. 46, 8211-8217 (2007).
    [CrossRef] [PubMed]
  6. S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671-680 (1983).
    [CrossRef] [PubMed]
  7. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2004).
  8. W. H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1978), Vol. 16, pp. 119-232.
    [CrossRef]
  9. Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings,” Opt. Lett. 27, 285-287(2002).
    [CrossRef]

2007 (2)

B. Hao and J. Leger, “Spatially inhomogeneous polarization in laser beam shaping,” Proc. SPIE 6682, 66820Y (2007).
[CrossRef]

B. Hao and J. Leger, “Polarization beam shaping,” Appl. Opt. 46, 8211-8217 (2007).
[CrossRef] [PubMed]

2006 (1)

B. Hao and J. Leger, “Smallest 1D flat-top focus by polarization manipulation,” Proc. SPIE 6290, 629001 (2006).
[CrossRef]

2002 (1)

1996 (1)

1983 (1)

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671-680 (1983).
[CrossRef] [PubMed]

Biener, G.

Bomzon, Z.

Dickey, F. M.

L. A. Romero and F. M. Dickey, “Lossless laser beam shaping,” J. Opt. Soc. Am. A 13, 751-760 (1996).
[CrossRef]

F. M. Dickey and S. C. Holswade, Laser Beam Shaping--Theory and Techniques (Marcel Dekker, 2000).
[CrossRef]

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671-680 (1983).
[CrossRef] [PubMed]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2004).

Hao, B.

B. Hao and J. Leger, “Polarization beam shaping,” Appl. Opt. 46, 8211-8217 (2007).
[CrossRef] [PubMed]

B. Hao and J. Leger, “Spatially inhomogeneous polarization in laser beam shaping,” Proc. SPIE 6682, 66820Y (2007).
[CrossRef]

B. Hao and J. Leger, “Smallest 1D flat-top focus by polarization manipulation,” Proc. SPIE 6290, 629001 (2006).
[CrossRef]

Hasman, E.

Holswade, S. C.

F. M. Dickey and S. C. Holswade, Laser Beam Shaping--Theory and Techniques (Marcel Dekker, 2000).
[CrossRef]

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671-680 (1983).
[CrossRef] [PubMed]

Kleiner, V.

Lee, W. H.

W. H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1978), Vol. 16, pp. 119-232.
[CrossRef]

Leger, J.

B. Hao and J. Leger, “Spatially inhomogeneous polarization in laser beam shaping,” Proc. SPIE 6682, 66820Y (2007).
[CrossRef]

B. Hao and J. Leger, “Polarization beam shaping,” Appl. Opt. 46, 8211-8217 (2007).
[CrossRef] [PubMed]

B. Hao and J. Leger, “Smallest 1D flat-top focus by polarization manipulation,” Proc. SPIE 6290, 629001 (2006).
[CrossRef]

Romero, L. A.

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671-680 (1983).
[CrossRef] [PubMed]

Appl. Opt. (1)

J. Opt. Soc. Am. A (1)

Opt. Lett. (1)

Proc. SPIE (2)

B. Hao and J. Leger, “Smallest 1D flat-top focus by polarization manipulation,” Proc. SPIE 6290, 629001 (2006).
[CrossRef]

B. Hao and J. Leger, “Spatially inhomogeneous polarization in laser beam shaping,” Proc. SPIE 6682, 66820Y (2007).
[CrossRef]

Science (1)

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671-680 (1983).
[CrossRef] [PubMed]

Other (3)

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2004).

W. H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1978), Vol. 16, pp. 119-232.
[CrossRef]

F. M. Dickey and S. C. Holswade, Laser Beam Shaping--Theory and Techniques (Marcel Dekker, 2000).
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Typical far-field laser beam shaping setup. The optical element can spatially modulate both the amplitude and the phase of the incident light.

Fig. 2
Fig. 2

Arbitrary valley-shaped function.

Fig. 3
Fig. 3

Normalized flattop intensity distribution produced by convolution of double δ functions and a Gaussian profile. As the phase difference between the double δ function increases from 0 to π, the size of the flattop becomes smaller.

Fig. 4
Fig. 4

Normalized flattop intensity distribution produced by convolution of double δ functions and a truncated plane wave. As the phase difference between the double δ function increases from 0 to π, the size of the flattop becomes smaller.

Fig. 5
Fig. 5

Cosine function to produce the smallest flattop focal pattern for plane wave illumination.

Fig. 6
Fig. 6

Simulated annealing result for a one- dimensional polarization plate. The inset shows a visualization of the one-dimensional polarization plate. The polarization rotation angle is a linear function of the wavefront position.

Fig. 7
Fig. 7

Effect of two different starting and ending points on the focal pattern: (a) starting at 198 ° , ending at 46.8 ° ; (b) starting at 252 ° , ending at 100.8 ° . (c) Flattop intensity focal pattern from the polarization plate of either (a) or (b).

Fig. 8
Fig. 8

Polarization plate starting at 210.6 ° , ending at 59.4 ° . The solid curve shows the x-channel plate and the dashed curve shows the y-channel plate. The focal pattern from either channel is a flattop profile.

Fig. 9
Fig. 9

One-dimensional polarization plate for a flattop focal spot size 31% larger (FWHM) than the smallest value.

Fig. 10
Fig. 10

Two-dimensional polarization plate to produce a diamond-shaped flattop focus.

Fig. 11
Fig. 11

Two-dimensional beam shaping: (a) two-dimensional diamond flattop intensity in the focal plane; (b) flattop intensity profiles along 0 ° and 90 ° cuts.

Fig. 12
Fig. 12

Experimental results showing the shape of the spot in the focal plane of a lens produced by a three-section half-wave plate. (a) Two-dimensional pseudocolor plot, showing a flattop response in the horizontal direction and a Gaussian response in the vertical direction. (b) Graph of experimentally measured intensity across the horizontal direction of the spot compared with theory.

Fig. 13
Fig. 13

Intensity distribution of diamond-shaped light at the focal point of a lens. (a) Two-dimensional diamond shape. (b) One-dimensional scan through the x axis.

Equations (12)

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β = 2 2 π r 0 y 0 λ f ,
1 2 exp ( i ϕ / 2 ) δ ( x d λ f ) + 1 2 exp ( i ϕ / 2 ) δ ( x + d λ f ) .
d = s 2 1 + cos ϕ .
g ( x ) = cos ( 2 π d x / λ f ϕ / 2 ) ,
cos ϕ = cos ( 2 π d / s ) ( π d / s ) 2 2 sin ( 2 π d / s ) ( π d / s ) + 3 sin 2 ( π d / s ) ( π d / s ) 2 sin 2 ( π d / s ) ,
g ( x ) = ± cos ( 2 π x / 2.41 + π / 4 ) rect ( x ) .
θ = 2 π a x rect ( x x m d ) ,
cos ( 2 π a x ) rect ( x x m d ) ,
sin ( 2 π a x ) rect ( x x m d ) .
| F { cos ( 2 π a x ) } F { rect ( x x m d ) } | 2 = | 1 2 [ δ ( f x a ) + δ ( f x + a ) ] exp ( j 2 π f x x m ) d sinc ( d f x ) | 2 = | 1 2 { exp [ j 2 π ( f x a ) x m ] d sinc [ d ( f x a ) ] + exp [ j 2 π ( f x + a ) x m ] d sinc [ d ( f x + a ) ] } | 2 ,
| 1 2 j { exp [ j 2 π ( f x a ) x m ] d sinc [ d ( f x a ) ] exp [ j 2 π ( f x + a ) x m ] d sinc [ d ( f x + a ) ] } | 2 .
I ( f x ) = | d sinc [ d ( f x - a ) ] | 2 + | d sinc [ d ( f x + a ) ] | 2 ,

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