Abstract

Generalized Phase Contrast (GPC) is an efficient method for generating speckle-free contiguous optical distributions useful in diverse applications such as static beam shaping, optical manipulation and recently, for excitation in two-photon optogenetics. To fully utilize typical Gaussian lasers in such applications, we analytically derive conditions for photon efficient light shaping with GPC. When combined with the conditions for optimal contrast developed in previous works, our analysis further simplifies GPC’s implementation. The results of our analysis are applied to practical illumination shapes, such as a circle and different rectangles commonly used in industrial or commercial applications. We also show simple and efficient beam shaping of arbitrary shapes geared towards biophotonics research and other contemporary applications. Optimized GPC configurations consistently give ~84% efficiency and ~3x intensity gain. Assessment of the energy savings when comparing to conventional amplitude masking show that ~93% of typical energy losses are saved with optimized GPC configurations.

© 2014 Optical Society of America

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  1. D. Palima, A. R. Bañas, G. Vizsnyiczai, L. Kelemen, P. Ormos, J. Glückstad, “Wave-guided optical waveguides,” Opt. Express 20(3), 2004–2014 (2012).
    [CrossRef] [PubMed]
  2. E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods 7(10), 848–854 (2010).
    [CrossRef] [PubMed]
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    [PubMed]
  4. D. Palima, C. A. Alonzo, P. J. Rodrigo, J. Glückstad, “Generalized phase contrast matched to Gaussian illumination,” Opt. Express 15(19), 11971–11977 (2007).
    [CrossRef] [PubMed]
  5. T. R. M. Sales, R. P. C. Photonics, C. Road, R. Ny, “Structured Microlens Arrays for Beam Shaping,” Proc. SPIE 5175, 109–120 (2003).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  19. M. A. Go, C. Stricker, S. Redman, H.-A. Bachor, V. R. Daria, “Simultaneous multi-site two-photon photostimulation in three dimensions,” J Biophotonics 5(10), 745–753 (2012).
    [CrossRef] [PubMed]
  20. L. Ge, M. Duelli, R. Cohn, “Enumeration of illumination and scanning modes from real-time spatial light modulators,” Opt. Express 7(12), 403–416 (2000).
    [CrossRef] [PubMed]
  21. T. Matsuoka, M. Nishi, M. Sakakura, K. Miura, K. Hirao, D. Palima, S. Tauro, A. Bañas, and J. Glückstad,D. L. Andrews, E. J. Galvez, and J. Glückstad, eds., “Functionalized 2PP structures for the BioPhotonics Workstation,” in Proceedings of SPIE, D. L. Andrews, E. J. Galvez, and J. Glückstad, eds. (2011), Vol. 7950, p. 79500Q.
    [CrossRef]
  22. P. J. Rodrigo, L. Gammelgaard, P. Bøggild, I. Perch-Nielsen, J. Glückstad, “Actuation of microfabricated tools using multiple GPC-based counterpropagating-beam traps,” Opt. Express 13(18), 6899–6904 (2005).
    [CrossRef] [PubMed]
  23. Y. Tanaka, S. Tsutsui, M. Ishikawa, H. Kitajima, “Hybrid optical tweezers for dynamic micro-bead arrays,” Opt. Express 19(16), 15445–15451 (2011).
    [CrossRef] [PubMed]
  24. S. Tauro, A. Bañas, D. Palima, J. Glückstad, “Dynamic axial stabilization of counter-propagating beam-traps with feedback control,” Opt. Express 18(17), 18217–18222 (2010).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  30. A. Bañas, D. Palima, J. Glückstad, “Matched-filtering generalized phase contrast using LCoS pico-projectors for beam-forming,” Opt. Express 20(9), 9705–9712 (2012).
    [CrossRef] [PubMed]
  31. D. Palima, J. Glückstad, “Gaussian to uniform intensity shaper based on generalized phase contrast,” Opt. Express 16(3), 1507–1516 (2008).
    [CrossRef] [PubMed]

2013 (1)

E. Papagiakoumou, “Optical developments for optogenetics,” Biol. Cell 105(10), 443–464 (2013).
[PubMed]

2012 (3)

2011 (2)

2010 (2)

E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods 7(10), 848–854 (2010).
[CrossRef] [PubMed]

S. Tauro, A. Bañas, D. Palima, J. Glückstad, “Dynamic axial stabilization of counter-propagating beam-traps with feedback control,” Opt. Express 18(17), 18217–18222 (2010).
[CrossRef] [PubMed]

2008 (2)

2007 (1)

2005 (1)

2003 (3)

T. R. M. Sales, R. P. C. Photonics, C. Road, R. Ny, “Structured Microlens Arrays for Beam Shaping,” Proc. SPIE 5175, 109–120 (2003).
[CrossRef]

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[CrossRef] [PubMed]

M. R. Wang, “Analysis and optimization on single-zone binary flat-top beam shaper,” Opt. Eng. 42(11), 3106 (2003).
[CrossRef]

2001 (2)

2000 (2)

1999 (1)

C. Kopp, L. Ravel, P. Meyrueis, “Efficient beamshaper homogenizer design combining diffractive optical elements, microlens array and random phase plate,” J. Opt. A, Pure Appl. Opt. 1(3), 398–403 (1999).
[CrossRef]

1998 (1)

I. Gur, D. Mendlovic, “Diffraction limited domain flat-top generator,” Opt. Commun. 145(1-6), 237-248 (1998).

1997 (1)

1982 (1)

1981 (1)

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–246 (1972).

1970 (1)

1967 (1)

1955 (1)

F. Zernike, “How I Discovered Phase Contrast,” Science 121(3141), 345–349 (1955).
[CrossRef] [PubMed]

Alonzo, C. A.

Anselmi, F.

E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods 7(10), 848–854 (2010).
[CrossRef] [PubMed]

Bachor, H.-A.

M. A. Go, C. Stricker, S. Redman, H.-A. Bachor, V. R. Daria, “Simultaneous multi-site two-photon photostimulation in three dimensions,” J Biophotonics 5(10), 745–753 (2012).
[CrossRef] [PubMed]

Bañas, A.

Bañas, A. R.

Bègue, A.

E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods 7(10), 848–854 (2010).
[CrossRef] [PubMed]

Bøggild, P.

Case, S. K.

Chanclou, P.

Cohn, R.

Daria, V. R.

M. A. Go, C. Stricker, S. Redman, H.-A. Bachor, V. R. Daria, “Simultaneous multi-site two-photon photostimulation in three dimensions,” J Biophotonics 5(10), 745–753 (2012).
[CrossRef] [PubMed]

de Bougrenet de la Tocnaye, J.-L.

de Sars, V.

E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods 7(10), 848–854 (2010).
[CrossRef] [PubMed]

Duelli, M.

Emiliani, V.

E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods 7(10), 848–854 (2010).
[CrossRef] [PubMed]

Gammelgaard, L.

Ge, L.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–246 (1972).

Glückstad, J.

D. Palima, A. R. Bañas, G. Vizsnyiczai, L. Kelemen, P. Ormos, J. Glückstad, “Wave-guided optical waveguides,” Opt. Express 20(3), 2004–2014 (2012).
[CrossRef] [PubMed]

A. Bañas, D. Palima, J. Glückstad, “Matched-filtering generalized phase contrast using LCoS pico-projectors for beam-forming,” Opt. Express 20(9), 9705–9712 (2012).
[CrossRef] [PubMed]

S. Tauro, A. Bañas, D. Palima, J. Glückstad, “Experimental demonstration of Generalized Phase Contrast based Gaussian beam-shaper,” Opt. Express 19(8), 7106–7111 (2011).
[CrossRef] [PubMed]

E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods 7(10), 848–854 (2010).
[CrossRef] [PubMed]

S. Tauro, A. Bañas, D. Palima, J. Glückstad, “Dynamic axial stabilization of counter-propagating beam-traps with feedback control,” Opt. Express 18(17), 18217–18222 (2010).
[CrossRef] [PubMed]

D. Palima, J. Glückstad, “Multi-wavelength spatial light shaping using generalized phase contrast,” Opt. Express 16(2), 1331–1342 (2008).
[CrossRef] [PubMed]

D. Palima, J. Glückstad, “Gaussian to uniform intensity shaper based on generalized phase contrast,” Opt. Express 16(3), 1507–1516 (2008).
[CrossRef] [PubMed]

D. Palima, C. A. Alonzo, P. J. Rodrigo, J. Glückstad, “Generalized phase contrast matched to Gaussian illumination,” Opt. Express 15(19), 11971–11977 (2007).
[CrossRef] [PubMed]

P. J. Rodrigo, L. Gammelgaard, P. Bøggild, I. Perch-Nielsen, J. Glückstad, “Actuation of microfabricated tools using multiple GPC-based counterpropagating-beam traps,” Opt. Express 13(18), 6899–6904 (2005).
[CrossRef] [PubMed]

J. Glückstad, P. C. Mogensen, “Optimal phase contrast in common-path interferometry,” Appl. Opt. 40(2), 268–282 (2001).
[CrossRef] [PubMed]

J. Glückstad, L. Lading, H. Toyoda, T. Hara, “Lossless light projection,” Opt. Lett. 22(18), 1373–1375 (1997).
[CrossRef] [PubMed]

Go, M. A.

M. A. Go, C. Stricker, S. Redman, H.-A. Bachor, V. R. Daria, “Simultaneous multi-site two-photon photostimulation in three dimensions,” J Biophotonics 5(10), 745–753 (2012).
[CrossRef] [PubMed]

Grier, D. G.

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[CrossRef] [PubMed]

Gur, I.

I. Gur, D. Mendlovic, “Diffraction limited domain flat-top generator,” Opt. Commun. 145(1-6), 237-248 (1998).

Hara, T.

Haugen, P. R.

Hoffnagle, J. A.

Isacoff, E. Y.

E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods 7(10), 848–854 (2010).
[CrossRef] [PubMed]

Ishikawa, M.

Jefferson, C. M.

Kelemen, L.

Kitajima, H.

Kopp, C.

C. Kopp, L. Ravel, P. Meyrueis, “Efficient beamshaper homogenizer design combining diffractive optical elements, microlens array and random phase plate,” J. Opt. A, Pure Appl. Opt. 1(3), 398–403 (1999).
[CrossRef]

Lading, L.

Lee, W. H.

Lohmann, A. W.

Løkberg, O. J.

Mendlovic, D.

I. Gur, D. Mendlovic, “Diffraction limited domain flat-top generator,” Opt. Commun. 145(1-6), 237-248 (1998).

Meyrueis, P.

C. Kopp, L. Ravel, P. Meyrueis, “Efficient beamshaper homogenizer design combining diffractive optical elements, microlens array and random phase plate,” J. Opt. A, Pure Appl. Opt. 1(3), 398–403 (1999).
[CrossRef]

Mogensen, P. C.

Nourrit, V.

Ny, R.

T. R. M. Sales, R. P. C. Photonics, C. Road, R. Ny, “Structured Microlens Arrays for Beam Shaping,” Proc. SPIE 5175, 109–120 (2003).
[CrossRef]

Ormos, P.

Palima, D.

Papagiakoumou, E.

E. Papagiakoumou, “Optical developments for optogenetics,” Biol. Cell 105(10), 443–464 (2013).
[PubMed]

E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods 7(10), 848–854 (2010).
[CrossRef] [PubMed]

Paris, D. P.

Perch-Nielsen, I.

Photonics, R. P. C.

T. R. M. Sales, R. P. C. Photonics, C. Road, R. Ny, “Structured Microlens Arrays for Beam Shaping,” Proc. SPIE 5175, 109–120 (2003).
[CrossRef]

Ravel, L.

C. Kopp, L. Ravel, P. Meyrueis, “Efficient beamshaper homogenizer design combining diffractive optical elements, microlens array and random phase plate,” J. Opt. A, Pure Appl. Opt. 1(3), 398–403 (1999).
[CrossRef]

Redman, S.

M. A. Go, C. Stricker, S. Redman, H.-A. Bachor, V. R. Daria, “Simultaneous multi-site two-photon photostimulation in three dimensions,” J Biophotonics 5(10), 745–753 (2012).
[CrossRef] [PubMed]

Road, C.

T. R. M. Sales, R. P. C. Photonics, C. Road, R. Ny, “Structured Microlens Arrays for Beam Shaping,” Proc. SPIE 5175, 109–120 (2003).
[CrossRef]

Rodrigo, P. J.

Sales, T. R. M.

T. R. M. Sales, R. P. C. Photonics, C. Road, R. Ny, “Structured Microlens Arrays for Beam Shaping,” Proc. SPIE 5175, 109–120 (2003).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–246 (1972).

Stricker, C.

M. A. Go, C. Stricker, S. Redman, H.-A. Bachor, V. R. Daria, “Simultaneous multi-site two-photon photostimulation in three dimensions,” J Biophotonics 5(10), 745–753 (2012).
[CrossRef] [PubMed]

Tanaka, Y.

Tauro, S.

Toyoda, H.

Tsutsui, S.

Veldkamp, W. B.

Vizsnyiczai, G.

Wang, M. R.

M. R. Wang, “Analysis and optimization on single-zone binary flat-top beam shaper,” Opt. Eng. 42(11), 3106 (2003).
[CrossRef]

Zernike, F.

F. Zernike, “How I Discovered Phase Contrast,” Science 121(3141), 345–349 (1955).
[CrossRef] [PubMed]

Appl. Opt. (6)

Biol. Cell (1)

E. Papagiakoumou, “Optical developments for optogenetics,” Biol. Cell 105(10), 443–464 (2013).
[PubMed]

J Biophotonics (1)

M. A. Go, C. Stricker, S. Redman, H.-A. Bachor, V. R. Daria, “Simultaneous multi-site two-photon photostimulation in three dimensions,” J Biophotonics 5(10), 745–753 (2012).
[CrossRef] [PubMed]

J. Opt. A, Pure Appl. Opt. (1)

C. Kopp, L. Ravel, P. Meyrueis, “Efficient beamshaper homogenizer design combining diffractive optical elements, microlens array and random phase plate,” J. Opt. A, Pure Appl. Opt. 1(3), 398–403 (1999).
[CrossRef]

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

Nat. Methods (1)

E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods 7(10), 848–854 (2010).
[CrossRef] [PubMed]

Nature (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[CrossRef] [PubMed]

Opt. Commun. (1)

I. Gur, D. Mendlovic, “Diffraction limited domain flat-top generator,” Opt. Commun. 145(1-6), 237-248 (1998).

Opt. Eng. (1)

M. R. Wang, “Analysis and optimization on single-zone binary flat-top beam shaper,” Opt. Eng. 42(11), 3106 (2003).
[CrossRef]

Opt. Express (10)

L. Ge, M. Duelli, R. Cohn, “Enumeration of illumination and scanning modes from real-time spatial light modulators,” Opt. Express 7(12), 403–416 (2000).
[CrossRef] [PubMed]

P. J. Rodrigo, L. Gammelgaard, P. Bøggild, I. Perch-Nielsen, J. Glückstad, “Actuation of microfabricated tools using multiple GPC-based counterpropagating-beam traps,” Opt. Express 13(18), 6899–6904 (2005).
[CrossRef] [PubMed]

D. Palima, C. A. Alonzo, P. J. Rodrigo, J. Glückstad, “Generalized phase contrast matched to Gaussian illumination,” Opt. Express 15(19), 11971–11977 (2007).
[CrossRef] [PubMed]

D. Palima, J. Glückstad, “Multi-wavelength spatial light shaping using generalized phase contrast,” Opt. Express 16(2), 1331–1342 (2008).
[CrossRef] [PubMed]

D. Palima, J. Glückstad, “Gaussian to uniform intensity shaper based on generalized phase contrast,” Opt. Express 16(3), 1507–1516 (2008).
[CrossRef] [PubMed]

S. Tauro, A. Bañas, D. Palima, J. Glückstad, “Dynamic axial stabilization of counter-propagating beam-traps with feedback control,” Opt. Express 18(17), 18217–18222 (2010).
[CrossRef] [PubMed]

S. Tauro, A. Bañas, D. Palima, J. Glückstad, “Experimental demonstration of Generalized Phase Contrast based Gaussian beam-shaper,” Opt. Express 19(8), 7106–7111 (2011).
[CrossRef] [PubMed]

Y. Tanaka, S. Tsutsui, M. Ishikawa, H. Kitajima, “Hybrid optical tweezers for dynamic micro-bead arrays,” Opt. Express 19(16), 15445–15451 (2011).
[CrossRef] [PubMed]

D. Palima, A. R. Bañas, G. Vizsnyiczai, L. Kelemen, P. Ormos, J. Glückstad, “Wave-guided optical waveguides,” Opt. Express 20(3), 2004–2014 (2012).
[CrossRef] [PubMed]

A. Bañas, D. Palima, J. Glückstad, “Matched-filtering generalized phase contrast using LCoS pico-projectors for beam-forming,” Opt. Express 20(9), 9705–9712 (2012).
[CrossRef] [PubMed]

Opt. Lett. (1)

Optik (Stuttg.) (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–246 (1972).

Proc. SPIE (1)

T. R. M. Sales, R. P. C. Photonics, C. Road, R. Ny, “Structured Microlens Arrays for Beam Shaping,” Proc. SPIE 5175, 109–120 (2003).
[CrossRef]

Science (1)

F. Zernike, “How I Discovered Phase Contrast,” Science 121(3141), 345–349 (1955).
[CrossRef] [PubMed]

Other (3)

T. Matsuoka, M. Nishi, M. Sakakura, K. Miura, K. Hirao, D. Palima, S. Tauro, A. Bañas, and J. Glückstad,D. L. Andrews, E. J. Galvez, and J. Glückstad, eds., “Functionalized 2PP structures for the BioPhotonics Workstation,” in Proceedings of SPIE, D. L. Andrews, E. J. Galvez, and J. Glückstad, eds. (2011), Vol. 7950, p. 79500Q.
[CrossRef]

R. Voelkel and K. J. Weible, “Laser beam homogenizing: limitations and constraints,” in Proc. of SPIE, A. Duparré and R. Geyl, eds. (2008), Vol. 7102, p. 71020J–71020J–12.

J. Glückstad and D. Z. Palima, Generalized Phase Contrast: Applications in Optics and Photonics (Springer Series in Optical Sciences, 2009).

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Figures (7)

Fig. 1
Fig. 1

A GPC system efficiently transforming an incident Gaussian beam into a bright rectangle. Besides a standard imaging or telescopic 4f setup formed by the two Fourier lenses, GPC uses a simple binary phase mask at the input and phase contrast filter at the Fourier plane.

Fig. 2
Fig. 2

(a) Color contour plot of numerically obtained efficiency as a function of (ζ, η). The black crosses mark the highest efficiency for a given ζ. The blue plot corresponds to our condition for the PCF radius to coincide with the first zero crossing of the Fourier transform. (b) The Fourier transform of the phase-only aperture (red plot) can be approximated with a displaced Gaussian (blue plot) near the PCF phase shifting region. The inverse Fourier transform of the ideal output (green plot) is also shown for reference. (The x and y axes are normalized to wf and πw02 respectively.)

Fig. 3
Fig. 3

Optimal ζRect for rectangular phase masks with different aspect ratios are found where the solid (Eq. (28)) and dashed lines (Eq. (29)) with the same color intersect, or where either of these lines intersect with η = 1.1081.

Fig. 4
Fig. 4

Comparison of GPC light shaping to a hard truncated Gaussian delivering 84W on identical rectangular areas. Being only 28% efficient (a), the truncated Gaussian requires 300W and loses 216W (b). The GPC light shaper requires only 100W, saving 200W (c).

Fig. 5
Fig. 5

GPC intensity outputs for circle (a), square (b), rectangle (c). The scale bar in (b) is twice the 1/e2 Gaussian waist, and tick marks in (a)-(c) are separated by half the Gaussian waist. Efficiencies, gain and energy savings are also shown, and are consistently ~84%, ~3x and ~93% respectively. The corresponding intensity line scans are shown in (d)-(f). The x-axis is normalized to w0.

Fig. 6
Fig. 6

Intensity profiles of various arbitrary shapes scaled to satisfy Eq. (21). Corresponding efficiencies, gain and energy savings are also shown, and are consistently ~84%, ~3x and ~93% respectively.

Fig. 7
Fig. 7

(a)-(c) Intensity profiles of various neuron-inspired shapes, directly drawn without scaling, but α-compensated by an outer phase ring. (d)-(f) Intensity profile for a pattern that is branching out. Whereas the efficiencies and intensity gains are highly pattern dependent, the energy savings stays at around ~90%.

Tables (1)

Tables Icon

Table 1 Efficiency, intensity gain and energy savings of GPC shaped light compared with a hard truncated or amplitude masked Gaussian for a circle and different rectangles.

Equations (32)

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

p ( x , y ) = a ( x , y ) exp [ i ϕ ( x , y ) ] .
H ( f x , f y ) = 1 + [ exp ( i θ ) 1 ] circ ( f r / Δ f r ) .
I ( x ' , y ' ) | a ( x ' , y ' ) exp [ i ϕ ( x ' , y ' ) ] + [ exp ( i θ ) 1 ] α ¯ g ( x ' , y ' ) | 2 .
g ( x ' , y ' ) = 1 { circ ( f r / Δ f r ) { a ( x . y ) } }
α ¯ = | α ¯ | exp ( i ϕ α ¯ ) = a ( x , y ) exp [ i ϕ ( x , y ) ] d x d y / a ( x , y ) d x d y
2 g ( 0 , 0 ) | α ¯ | | sin ( θ / 2 ) | = 1.
a ( r ) = exp ( r 2 / w 0 2 ) .
g ( r ' = 0 ) = 1 { circ ( f r / Δ f r ) π w 0 2 exp ( f r 2 / w f 2 ) } = 1 exp ( η 2 ) .
η = Δ f r / w f = λ f Δ f r / ( π w 0 )
a ( r ) exp [ i ϕ ( r ) ] = exp ( r 2 / w 0 2 ) [ 1 2 c i r c ( r / Δ r ) ] .
α ¯ = 2 exp ( ζ 2 ) 1
ζ = Δ r / w 0 .
[ 2 exp ( ζ 2 ) 1 ] [ 1 exp ( η 2 ) ] = 1 / 2.
GPC eff = S I ( x , y ) d x d y / a 2 ( x , y ) d x d y .
{ exp ( r 2 / w 0 2 ) [ 1 2 c i r c ( r / Δ r ) ] } π w 0 2 { exp ( f r 2 / w f 2 ) 2 [ 1 exp ( ζ 2 ) ] }
exp ( η 2 ) 2 [ 1 exp ( ζ 2 ) ] = 0.
ζ = ln [ 2 1 + 1 / 2 ] = 0.3979 ,
η = ln { 2 [ 1 exp ( ζ 2 ) ] } = 1.1081.
[ 1 exp ( η 2 ) ] α ¯ = 1 / 2 ,
exp ( η 2 ) + α ¯ 1 = 0.
α ¯ = 1 / 2 = 0.7071 ,
η = ln ( 1 1 / 2 ) = 1.1081.
a ( x , y ) exp [ i ϕ ( x , y ) ] = exp [ ( x 2 + y 2 ) / w 0 2 ] [ 1 2 rect ( x / W ) rect ( y / H ) ] .
α ¯ = π w 0 2 2 π w 0 2 erf ( W / 2 w 0 ) erf ( H / 2 w 0 ) π w 0 2 = 1 2 erf ( ζ Rect ) erf ( ζ Rect / A R ) .
ζ Rect = erf 1 ( 1 1 / 2 2 ) = 0.3533.
[ 1 exp ( η 2 ) ] [ 1 2 erf ( ζ Rect ) erf ( ζ Rect / A R ) ] = 1 / 2 ,
exp ( η 2 ) 2 erf ( ζ Rect ) erf ( ζ Rect / A R ) = 0.
η = ln ( 1 1 2 4 erf ( ζ Rect ) erf ( ζ Rect / A R ) ) ,
η = ln ( 2 erf ( ζ Rect ) erf ( ζ Rect / A R ) ) .
E . savings = ( 1 ( 100 % GPC eff % ) ( GPC gain ) × ( 100 % Amp masking eff % ) ) × 100 % .
α ¯ = 1 2 { b ( x , y ) exp [ ( x 2 + y 2 ) / w 0 2 ] d x d y } / π w 0 2 = 1 / 2 .
R comp = w 0 × ln [ ( 1 / 2 1 / 8 ) { b ( x , y ) exp [ ( x 2 + y 2 ) / w 0 2 ] d x d y } / π w 0 2 ] .

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