Abstract

We show that the generalized phase contrast method (GPC) can be used as a versatile tool for shaping an incident Gaussian illumination into arbitrary lateral beam profiles with uniform intensity. This energy-efficient technique shapes the beam by redistributing light from designated dark regions and homogenizes the beam by redistributing excess energy from the center to the edges. Results from numerical experiments show efficiencies around 84% for various profiles with minimal intensity inhomogeneities.

© 2008 Optical Society of America

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  1. F. M. Dickey and S. C. Holswade, Laser Beam Shaping: Theory and Techniques (Marcel Dekker, New York, 2000).
    [Crossref]
  2. F. M. Dickey, S. C. Holswade, and D. L. Shealy, eds., Laser. Beam Shaping Applications (CRC Press, 2005).
    [Crossref]
  3. M. A. Karim, A. M. Hanafi, F. Hussain, S. Mustafa, Z. Samberid, and N. M. Zain, “Realization of a uniform circular source using a two-dimensional binary filter,” Opt. Lett. 10, 470–471 (1985).
    [Crossref] [PubMed]
  4. S. P. Chang, J. M. Kuo, Y. P. Lee, C. M. Lu, and K. J. Ling, “Transformation of Gaussian to Coherent Uniform Beams by Inverse-Gaussian Transmittive Filters,” Appl. Opt. 37, 747–752 (1998).
    [Crossref]
  5. B. R. Frieden, “Lossless conversion of a plane laser wave to a plane wave of uniform irradiance,” Appl. Opt. 4, 1400–1403 (1965).
    [Crossref]
  6. P. H. Malyak, “Two-mirror unobscured optical system for reshaping the irradiance distribution of a laser beam,” Appl. Opt. 31, 4377–4383 (1992).
    [Crossref] [PubMed]
  7. J. A. Hoffnagle and C. M. Jefferson, “Design and Performance of a Refractive Optical System that Converts a Gaussian to a Flattop Beam,” Appl. Opt. 39, 5488–5499 (2000).
    [Crossref]
  8. F. Wippermann, U. D. Zeitner, P. Dannberg, A. Bräuer, and S. Sinzinger, “Beam homogenizers based on chirped microlens arrays,” Opt. Express 15, 6218–6231 (2007).
    [Crossref] [PubMed]
  9. C. Y. Han, Y. Ishii, and K. Murata, “Reshaping collimated laser beams with Gaussian profile to uniform profiles,” Appl. Opt. 22, 3644–3647 (1983).
    [Crossref] [PubMed]
  10. M. T. Eismann, A. M. Tai, and J. N. Cederquist, “Iterative design of a holographic beamformer,” Appl. Opt. 28, 2641–2650 (1989).
    [Crossref] [PubMed]
  11. T. Dresel, M. Beyerlein, and J. Schwider, “Design and fabrication of computer-generated beam-shaping holograms,” Appl. Opt. 35, 4615–4621 (1996).
    [Crossref] [PubMed]
  12. J. S. Liu and M. R. Taghizadeh, “Iterative algorithm for the design of diffractive phase elements for laser beam shaping,” Opt. Lett. 27, 1463–1465 (2002).
    [Crossref]
  13. J. Glückstad, “Phase contrast image synthesis,” Opt. Commun. 130, 225–230 (1996).
    [Crossref]
  14. J. Glückstad and P. C. Mogensen, “Optimal phase contrast in common-path interferometry,” Appl. Opt. 40, 268–282 (2001).
    [Crossref]
  15. F. Zernike, “How I discovered phase contrast,” Science 121, 345–349 (1955).
    [Crossref] [PubMed]
  16. J. Glückstad, L. Lading, H. Toyoda, and T. Hara, “Lossless light projection,” Opt. Lett. 22, 1373–1375 (1997).
    [Crossref]
  17. P. J. Rodrigo, R. L. Eriksen, V. R. Daria, and J. Glückstad, “Interactive light-driven and parallel manipulation of inhomogeneous particles,” Opt. Express 10, 1550–1556 (2002).
    [PubMed]
  18. P.J. Rodrigo, V.R. Daria, and J. Glückstad, “Real-time three-dimensional optical micromanipulation of multiple particles and living cells,” Opt. Lett. 292270–2272 (2004).
    [Crossref] [PubMed]
  19. P. J. Rodrigo, V. R. Daria, and J. Glückstad, “Four-dimensional optical manipulation of colloidal particles,” Appl. Phys. Lett. 86, 074103 (2005).
    [Crossref]
  20. I. R. Perch-Nielsen, P. J. Rodrigo, C. A. Alonzo, and J. Glückstad, “Autonomous and 3D real-time multi-beam manipulation in a microfluidic environment,” Opt. Express 14, 12199–12205 (2006).
    [Crossref] [PubMed]
  21. P. C. Mogensen and J. Glückstad, “Phase-only optical encryption,” Opt. Lett. 25, 566–568 (2000).
    [Crossref]
  22. V. R. Daria, P. J. Rodrigo, S. Sinzinger, and J. Glückstad, “Phase-only optical decryption in a planar-integrated micro-optics system,” Opt. Eng. 43, 2223–2227 (2004).
    [Crossref]
  23. D. Palima, C. A. Alonzo, P. J. Rodrigo, and J. Glückstad, “Generalized phase contrast matched to Gaussian illumination,” Opt. Express 15, 11971–11977 (2007).
    [Crossref] [PubMed]
  24. Glückstad, D. Palima, P. J. Rodrigo, and C. A. Alonzo, “Laser projection using generalized phase contrast,” Opt. Lett. 32, 3281–3283 (2007).
    [Crossref] [PubMed]
  25. V. Nourrit, J. L. de Bougrenet de la Tocnaye, and P. Chanclou, “Propagation and diffraction of truncated Gaussian beams,” J. Opt. Soc. Am. A 18, 546–556 (2001).
    [Crossref]
  26. E. Carcole, J. Campos, and S. Bosch, “Diffraction theory of Fresnel lenses encoded in low-resolution devices,” Appl. Opt. 33, 162–174 (1994).
    [Crossref] [PubMed]
  27. A. J. Caley, M. Braun, A. J. Waddie, and M. R. Taghizadeh, “Analysis of multimask fabrication errors for diffractive optical elements,” Appl. Opt. 46, 2180–2188 (2007).
    [Crossref] [PubMed]

2007 (4)

2006 (1)

2005 (1)

P. J. Rodrigo, V. R. Daria, and J. Glückstad, “Four-dimensional optical manipulation of colloidal particles,” Appl. Phys. Lett. 86, 074103 (2005).
[Crossref]

2004 (2)

V. R. Daria, P. J. Rodrigo, S. Sinzinger, and J. Glückstad, “Phase-only optical decryption in a planar-integrated micro-optics system,” Opt. Eng. 43, 2223–2227 (2004).
[Crossref]

P.J. Rodrigo, V.R. Daria, and J. Glückstad, “Real-time three-dimensional optical micromanipulation of multiple particles and living cells,” Opt. Lett. 292270–2272 (2004).
[Crossref] [PubMed]

2002 (2)

2001 (2)

2000 (2)

1998 (1)

1997 (1)

1996 (2)

1994 (1)

1992 (1)

1989 (1)

1985 (1)

1983 (1)

1965 (1)

1955 (1)

F. Zernike, “How I discovered phase contrast,” Science 121, 345–349 (1955).
[Crossref] [PubMed]

Alonzo, C. A.

Beyerlein, M.

Bosch, S.

Bräuer, A.

Braun, M.

Caley, A. J.

Campos, J.

Carcole, E.

Cederquist, J. N.

Chanclou, P.

Chang, S. P.

Dannberg, P.

Daria, V. R.

P. J. Rodrigo, V. R. Daria, and J. Glückstad, “Four-dimensional optical manipulation of colloidal particles,” Appl. Phys. Lett. 86, 074103 (2005).
[Crossref]

V. R. Daria, P. J. Rodrigo, S. Sinzinger, and J. Glückstad, “Phase-only optical decryption in a planar-integrated micro-optics system,” Opt. Eng. 43, 2223–2227 (2004).
[Crossref]

P. J. Rodrigo, R. L. Eriksen, V. R. Daria, and J. Glückstad, “Interactive light-driven and parallel manipulation of inhomogeneous particles,” Opt. Express 10, 1550–1556 (2002).
[PubMed]

Daria, V.R.

de Bougrenet de la Tocnaye, J. L.

Dickey, F. M.

F. M. Dickey and S. C. Holswade, Laser Beam Shaping: Theory and Techniques (Marcel Dekker, New York, 2000).
[Crossref]

Dresel, T.

Eismann, M. T.

Eriksen, R. L.

Frieden, B. R.

Glückstad,

Glückstad, J.

D. Palima, C. A. Alonzo, P. J. Rodrigo, and J. Glückstad, “Generalized phase contrast matched to Gaussian illumination,” Opt. Express 15, 11971–11977 (2007).
[Crossref] [PubMed]

I. R. Perch-Nielsen, P. J. Rodrigo, C. A. Alonzo, and J. Glückstad, “Autonomous and 3D real-time multi-beam manipulation in a microfluidic environment,” Opt. Express 14, 12199–12205 (2006).
[Crossref] [PubMed]

P. J. Rodrigo, V. R. Daria, and J. Glückstad, “Four-dimensional optical manipulation of colloidal particles,” Appl. Phys. Lett. 86, 074103 (2005).
[Crossref]

V. R. Daria, P. J. Rodrigo, S. Sinzinger, and J. Glückstad, “Phase-only optical decryption in a planar-integrated micro-optics system,” Opt. Eng. 43, 2223–2227 (2004).
[Crossref]

P.J. Rodrigo, V.R. Daria, and J. Glückstad, “Real-time three-dimensional optical micromanipulation of multiple particles and living cells,” Opt. Lett. 292270–2272 (2004).
[Crossref] [PubMed]

P. J. Rodrigo, R. L. Eriksen, V. R. Daria, and J. Glückstad, “Interactive light-driven and parallel manipulation of inhomogeneous particles,” Opt. Express 10, 1550–1556 (2002).
[PubMed]

J. Glückstad and P. C. Mogensen, “Optimal phase contrast in common-path interferometry,” Appl. Opt. 40, 268–282 (2001).
[Crossref]

P. C. Mogensen and J. Glückstad, “Phase-only optical encryption,” Opt. Lett. 25, 566–568 (2000).
[Crossref]

J. Glückstad, L. Lading, H. Toyoda, and T. Hara, “Lossless light projection,” Opt. Lett. 22, 1373–1375 (1997).
[Crossref]

J. Glückstad, “Phase contrast image synthesis,” Opt. Commun. 130, 225–230 (1996).
[Crossref]

Han, C. Y.

Hanafi, A. M.

Hara, T.

Hoffnagle, J. A.

Holswade, S. C.

F. M. Dickey and S. C. Holswade, Laser Beam Shaping: Theory and Techniques (Marcel Dekker, New York, 2000).
[Crossref]

Hussain, F.

Ishii, Y.

Jefferson, C. M.

Karim, M. A.

Kuo, J. M.

Lading, L.

Lee, Y. P.

Ling, K. J.

Liu, J. S.

Lu, C. M.

Malyak, P. H.

Mogensen, P. C.

Murata, K.

Mustafa, S.

Nourrit, V.

Palima, D.

Perch-Nielsen, I. R.

Rodrigo, P. J.

Rodrigo, P.J.

Samberid, Z.

Schwider, J.

Sinzinger, S.

F. Wippermann, U. D. Zeitner, P. Dannberg, A. Bräuer, and S. Sinzinger, “Beam homogenizers based on chirped microlens arrays,” Opt. Express 15, 6218–6231 (2007).
[Crossref] [PubMed]

V. R. Daria, P. J. Rodrigo, S. Sinzinger, and J. Glückstad, “Phase-only optical decryption in a planar-integrated micro-optics system,” Opt. Eng. 43, 2223–2227 (2004).
[Crossref]

Taghizadeh, M. R.

Tai, A. M.

Toyoda, H.

Waddie, A. J.

Wippermann, F.

Zain, N. M.

Zeitner, U. D.

Zernike, F.

F. Zernike, “How I discovered phase contrast,” Science 121, 345–349 (1955).
[Crossref] [PubMed]

Appl. Opt. (10)

S. P. Chang, J. M. Kuo, Y. P. Lee, C. M. Lu, and K. J. Ling, “Transformation of Gaussian to Coherent Uniform Beams by Inverse-Gaussian Transmittive Filters,” Appl. Opt. 37, 747–752 (1998).
[Crossref]

B. R. Frieden, “Lossless conversion of a plane laser wave to a plane wave of uniform irradiance,” Appl. Opt. 4, 1400–1403 (1965).
[Crossref]

P. H. Malyak, “Two-mirror unobscured optical system for reshaping the irradiance distribution of a laser beam,” Appl. Opt. 31, 4377–4383 (1992).
[Crossref] [PubMed]

J. A. Hoffnagle and C. M. Jefferson, “Design and Performance of a Refractive Optical System that Converts a Gaussian to a Flattop Beam,” Appl. Opt. 39, 5488–5499 (2000).
[Crossref]

C. Y. Han, Y. Ishii, and K. Murata, “Reshaping collimated laser beams with Gaussian profile to uniform profiles,” Appl. Opt. 22, 3644–3647 (1983).
[Crossref] [PubMed]

M. T. Eismann, A. M. Tai, and J. N. Cederquist, “Iterative design of a holographic beamformer,” Appl. Opt. 28, 2641–2650 (1989).
[Crossref] [PubMed]

T. Dresel, M. Beyerlein, and J. Schwider, “Design and fabrication of computer-generated beam-shaping holograms,” Appl. Opt. 35, 4615–4621 (1996).
[Crossref] [PubMed]

J. Glückstad and P. C. Mogensen, “Optimal phase contrast in common-path interferometry,” Appl. Opt. 40, 268–282 (2001).
[Crossref]

E. Carcole, J. Campos, and S. Bosch, “Diffraction theory of Fresnel lenses encoded in low-resolution devices,” Appl. Opt. 33, 162–174 (1994).
[Crossref] [PubMed]

A. J. Caley, M. Braun, A. J. Waddie, and M. R. Taghizadeh, “Analysis of multimask fabrication errors for diffractive optical elements,” Appl. Opt. 46, 2180–2188 (2007).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

P. J. Rodrigo, V. R. Daria, and J. Glückstad, “Four-dimensional optical manipulation of colloidal particles,” Appl. Phys. Lett. 86, 074103 (2005).
[Crossref]

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

Opt. Commun. (1)

J. Glückstad, “Phase contrast image synthesis,” Opt. Commun. 130, 225–230 (1996).
[Crossref]

Opt. Eng. (1)

V. R. Daria, P. J. Rodrigo, S. Sinzinger, and J. Glückstad, “Phase-only optical decryption in a planar-integrated micro-optics system,” Opt. Eng. 43, 2223–2227 (2004).
[Crossref]

Opt. Express (4)

Opt. Lett. (6)

Science (1)

F. Zernike, “How I discovered phase contrast,” Science 121, 345–349 (1955).
[Crossref] [PubMed]

Other (2)

F. M. Dickey and S. C. Holswade, Laser Beam Shaping: Theory and Techniques (Marcel Dekker, New York, 2000).
[Crossref]

F. M. Dickey, S. C. Holswade, and D. L. Shealy, eds., Laser. Beam Shaping Applications (CRC Press, 2005).
[Crossref]

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Figures (5)

Fig. 1.
Fig. 1.

Optical setup for the generalized phase contrast method.

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Fig. 2.
Fig. 2.

(a). Phasor illustration: Encoding corrected phases, ϕ 1 and ϕ 2, compensates for the amplitude mismatch, a 1 and a 2, resulting in superpositions with matching amplitudes. (b). Phasors of the normalized zero-order, α ¯ , corresponding to PCF phase shifts, θ=π/3, π/2, and π. Vertical dashed line: real part of α ¯ =1/2; horizontal dashed line: maximum value of α ¯ =√3/2. Matched α ¯ and θ guarantees the darkness condition, α ¯ [exp(iθ)-1]=-1.

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Fig. 3.
Fig. 3.

(a,b,c) Phase inputs: false-color images and linescans; (d,e,f) respective GPC outputs: grayscale images and linescans. Efficiencies are indicated near each image. (a,d): binary phase input (0 and π) and θ=π; (b,e): initial phase correction using parameters obtained from binary input; (c,f): refined phase input with matching phase shift. The incident Gaussian is shown, between (d) and (e), as referenced for the output grayscale and spatial scale.

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Fig. 4.
Fig. 4.

Output of numerical experiments implementing GPC-based conversion of an incident Gaussian beam into various flattop profiles. The efficiency (η) and maximum fluctuation (Δ) are indicated below each pattern, followed by the PCF phase shift (θ) used. The scale bar on the lower left indicates the 1/e2 width of the Gaussian beam relative to the patterns.

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Fig. 5.
Fig. 5.

Plot of the maximum fluctuation, Δ, for different phase quantization levels. The inset shows the projected pattern and intensity linescan for 16 quantization levels.

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Equations (25)

Equations on this page are rendered with MathJax. Learn more.

H ( f x , f y ) = 1 + [ exp ( i θ ) − 1 ] S ( f x , f y ) .
I ( x ′ , y ′ ) ≈ ∣ a ( x ′ , y ′ ) exp [ i ϕ ( x ′ , y ′ ) ] + α ¯ [ exp ( i θ ) − 1 ] g ( x ′ , y ′ ) ∣ 2 ,
α ¯ = ∣ α ¯ ∣ exp ( i ϕ α ¯ ) = ∫ a ( x , y ) exp [ i ϕ ( x , y ) ] d x dy ∫ a ( x , y ) d x dy ,
g ( x ′ , y ′ ) = ℑ − 1 { S ( f x , f y ) ℑ { a ( x , y ) } } .
a ( x , y ) = a ( r ) = exp ( − r 2 w 0 2 ) .
g ( x ′ , y ′ ) = g ( r ′ ) = 4 π 2 ∫ 0 Δ f r ∫ 0 ∞ exp ( − r 2 w 0 2 ) J 0 ( 2 π f r r ) r J 0 ( 2 π f r r ′ ) f r dr df r .
I ( x ′ , y ′ ) ≈ exp ( − 2 r ′ 2 w 0 2 ) ∣ exp [ i ϕ ( x ′ , y ′ ) ] + α ¯ [ exp ( i θ ) − 1 ] ∣ 2 .
α ¯ [ exp ( i θ ) − 1 ] = − 1 .
∣ exp [ i ϕ ( x ′ , y ′ ) ] + α ¯ [ exp ( i θ ) − 1 ] ∣ 2 = I 0 exp ( 2 r ′ 2 w 0 2 ) A ( x ′ , y ′ )
α ¯ = α ¯ R + i α ¯ I = 1 2 + i 2 cot ( θ 2 ) .
θ = 2 cot − 1 ( 2 α ¯ imaginary ) = 2 cot − 1 [ 2 ∫ a ( x , y ) sin [ ϕ ( x , y ) ] d x d y ∫ a ( x , y ) d x d y ] .
∣ exp [ i ϕ ( x ′ , y ′ ) ] − 1 ∣ 2 = I 0 exp ( 2 r ′ 2 w 0 2 ) A ( x ′ , y ′ ) .
cos [ ϕ ( r ′ ) ] = 1 − I 0 2 exp ( 2 r ′ 2 w 0 2 ) A ( x ′ , y ′ ) .
I ( x ′ , y ′ ) ≈ ∣ exp ( − r ′ 2 w 0 2 ) exp [ i ϕ ( x ′ , y ′ ) ] + g ( x ′ , y ′ ) α ¯ [ exp ( i θ ) − 1 ] ∣ 2
α ¯ [ exp ( i θ ) − 1 ] = − k ,
α ¯ = α ¯ R + i α ¯ I = k 2 + i k 2 cot ( θ 2 ) .
I ( x ′ , y ′ ) ≈ a 0 2 ( x ′ , y ′ ) + k 2 g 2 ( x ′ , y ′ ) − 2 ka 0 ( x ′ , y ′ ) g ( x ′ , y ′ ) cos ( ϕ ) ,
ϕ ( x ′ , y ′ ) = arccos [ − I ( x ′ , y ′ ) + a 0 2 ( x ′ , y ′ ) + k 2 g 2 ( x ′ , y ′ ) 2 k a 0 ( x ′ , y ′ ) g ( x ′ , y ′ ) ]
I 0 = ∣ exp ( − r 0 2 w 0 2 ) + kg ( r 0 ) ∣ 2 .
p ( x , y ) = exp ( − r 2 w 0 2 ) − 2 circ ( r r 0 ) exp ( − r 2 w 0 2 )
α = 2 π ∫ 0 ∞ p ( r ) rdr 2 π ∫ 0 ∞ a ( r ) rdr = 2 exp ( r 0 2 w 0 2 ) − 1 = α R = k 2 .
H ( f r ) = 1 − 2 circ ( f r f r 0 ) .
g ( r ′ ) = ℑ − 1 { circ ( f r f r 0 ) ℑ { a 0 ( r ) } } .
θ = 2 arccot ( ∫ 0 ∞ a 0 ( r ) sin [ ϕ ( r ) ] rdr ∫ 0 ∞ a 0 ( r ) cos [ ϕ ( r ) ] rdr )
A ( f r ) = 2 π ∫ 0 r 0 exp ( − r 2 w 0 2 ) J 0 ( 2 π f r r ) rdr .

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