Abstract

Pixilated spatial light modulators are efficient devices to shape the wavefront of a laser beam or to perform Fourier optical filtering. When conjugated with the back focal plane of a microscope objective, they allow an efficient redistribution of laser light energy. These intensity patterns are usually polluted by undesired spots so-called ghosts and zero-orders whose intensities depend on displayed patterns. In this work, we propose a model to account for these discrepancies and demonstrate the possibility to efficiently reduce the intensity of the zero-order up to 95%, the intensity of the ghost up to 96% and increase diffraction efficiency up to 44%. Our model suggests physical cross-talk between pixels and thus, filtering of addressed high spatial frequencies. The method implementation relies on simple preliminary characterization of the SLM and can be computed a priori with any phase profile. The performance of this method is demonstrated employing a Hamamatsu LCoS SLM X10468-02 with two-photon excitation of fluorescent Rhodamine layers.

© 2012 OSA

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2011 (4)

K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics5(6), 335–342 (2011).
[CrossRef]

S. G. Yang, E. Papagiakoumou, M. Guillon, V. de Sars, C. M. Tang, and V. Emiliani, “Three-dimensional holographic photostimulation of the dendritic arbor,” J. Neural Eng.8(4), 046002 (2011).
[CrossRef] [PubMed]

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photon. Rev.5(1), 81–101 (2011).
[CrossRef]

L. Lobato, A. Lizana, A. Marquez, I. Moreno, C. Iemmi, J. Campos, and M. J. Yzuel, “Characterization of the anamorphic and spatial frequency dependent phenomenon in Liquid Crystal on Silicon displays,” J. Eur. Opt. Soc. Rapid Publ.6, 11012S (2011).

2010 (3)

2009 (2)

H. Zhang, J. H. Xie, J. Liu, and Y. T. Wang, “Elimination of a zero-order beam induced by a pixelated spatial light modulator for holographic projection,” Appl. Opt.48(30), 5834–5841 (2009).
[CrossRef] [PubMed]

L. Golan, I. Reutsky, N. Farah, and S. Shoham, “Design and characteristics of holographic neural photo-stimulation systems,” J. Neural Eng.6(6), 066004 (2009).
[CrossRef] [PubMed]

2008 (4)

C. Lutz, T. S. Otis, V. DeSars, S. Charpak, D. A. DiGregorio, and V. Emiliani, “Holographic photolysis of caged neurotransmitters,” Nat. Methods5(9), 821–827 (2008).
[CrossRef] [PubMed]

V. Nikolenko, B. O. Watson, R. Araya, A. Woodruff, D. S. Peterka, and R. Yuste, “SLM microscopy: scanless two-photon imaging and photostimulation using spatial light modulators,” Front. Neural Circuits2, 1–14 (2008).
[CrossRef] [PubMed]

I. Moreno, A. Lizana, A. Márquez, C. Iemmi, E. Fernández, J. Campos, and M. J. Yzuel, “Time fluctuations of the phase modulation in a liquid crystal on silicon display: characterization and effects in diffractive optics,” Opt. Express16(21), 16711–16722 (2008).
[CrossRef] [PubMed]

E. Papagiakoumou, V. de Sars, D. Oron, and V. Emiliani, “Patterned two-photon illumination by spatiotemporal shaping of ultrashort pulses,” Opt. Express16(26), 22039–22047 (2008).
[CrossRef] [PubMed]

2007 (2)

2006 (1)

2005 (3)

2004 (3)

1999 (1)

V. Arrizon, E. Carreon, and M. Testorf, “Implementation of Fourier array illuminators using pixelated SLM: efficiency limitations,” Opt. Commun.160(4-6), 207–213 (1999).
[CrossRef]

1988 (1)

Anselmi, F.

E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods7(10), 848–854 (2010).
[CrossRef] [PubMed]

Apter, B.

Araya, R.

V. Nikolenko, B. O. Watson, R. Araya, A. Woodruff, D. S. Peterka, and R. Yuste, “SLM microscopy: scanless two-photon imaging and photostimulation using spatial light modulators,” Front. Neural Circuits2, 1–14 (2008).
[CrossRef] [PubMed]

Arrizon, V.

V. Arrizon, E. Carreon, and M. Testorf, “Implementation of Fourier array illuminators using pixelated SLM: efficiency limitations,” Opt. Commun.160(4-6), 207–213 (1999).
[CrossRef]

Bahat-Treidel, E.

Bègue, A.

E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods7(10), 848–854 (2010).
[CrossRef] [PubMed]

Bernet, S.

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photon. Rev.5(1), 81–101 (2011).
[CrossRef]

Booth, M. J.

Bryngdahl, O.

Campos, J.

Carreon, E.

V. Arrizon, E. Carreon, and M. Testorf, “Implementation of Fourier array illuminators using pixelated SLM: efficiency limitations,” Opt. Commun.160(4-6), 207–213 (1999).
[CrossRef]

Charpak, S.

C. Lutz, T. S. Otis, V. DeSars, S. Charpak, D. A. DiGregorio, and V. Emiliani, “Holographic photolysis of caged neurotransmitters,” Nat. Methods5(9), 821–827 (2008).
[CrossRef] [PubMed]

Cizmar, T.

K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics5(6), 335–342 (2011).
[CrossRef]

Clark, R. L.

Cole, D. G.

Cooper, J.

Daria, V. R.

de Sars, V.

S. G. Yang, E. Papagiakoumou, M. Guillon, V. de Sars, C. M. Tang, and V. Emiliani, “Three-dimensional holographic photostimulation of the dendritic arbor,” J. Neural Eng.8(4), 046002 (2011).
[CrossRef] [PubMed]

E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods7(10), 848–854 (2010).
[CrossRef] [PubMed]

E. Papagiakoumou, V. de Sars, D. Oron, and V. Emiliani, “Patterned two-photon illumination by spatiotemporal shaping of ultrashort pulses,” Opt. Express16(26), 22039–22047 (2008).
[CrossRef] [PubMed]

DeSars, V.

C. Lutz, T. S. Otis, V. DeSars, S. Charpak, D. A. DiGregorio, and V. Emiliani, “Holographic photolysis of caged neurotransmitters,” Nat. Methods5(9), 821–827 (2008).
[CrossRef] [PubMed]

Dholakia, K.

K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics5(6), 335–342 (2011).
[CrossRef]

Di Leonardo, R.

DiGregorio, D. A.

C. Lutz, T. S. Otis, V. DeSars, S. Charpak, D. A. DiGregorio, and V. Emiliani, “Holographic photolysis of caged neurotransmitters,” Nat. Methods5(9), 821–827 (2008).
[CrossRef] [PubMed]

Dileonardo, R.

Efron, U.

Emiliani, V.

S. G. Yang, E. Papagiakoumou, M. Guillon, V. de Sars, C. M. Tang, and V. Emiliani, “Three-dimensional holographic photostimulation of the dendritic arbor,” J. Neural Eng.8(4), 046002 (2011).
[CrossRef] [PubMed]

E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods7(10), 848–854 (2010).
[CrossRef] [PubMed]

C. Lutz, T. S. Otis, V. DeSars, S. Charpak, D. A. DiGregorio, and V. Emiliani, “Holographic photolysis of caged neurotransmitters,” Nat. Methods5(9), 821–827 (2008).
[CrossRef] [PubMed]

E. Papagiakoumou, V. de Sars, D. Oron, and V. Emiliani, “Patterned two-photon illumination by spatiotemporal shaping of ultrashort pulses,” Opt. Express16(26), 22039–22047 (2008).
[CrossRef] [PubMed]

Farah, N.

L. Golan, I. Reutsky, N. Farah, and S. Shoham, “Design and characteristics of holographic neural photo-stimulation systems,” J. Neural Eng.6(6), 066004 (2009).
[CrossRef] [PubMed]

Fernández, E.

Gibson, G.

Glückstad, J.

E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods7(10), 848–854 (2010).
[CrossRef] [PubMed]

Golan, L.

L. Golan, I. Reutsky, N. Farah, and S. Shoham, “Design and characteristics of holographic neural photo-stimulation systems,” J. Neural Eng.6(6), 066004 (2009).
[CrossRef] [PubMed]

Grier, D. G.

Guillon, M.

S. G. Yang, E. Papagiakoumou, M. Guillon, V. de Sars, C. M. Tang, and V. Emiliani, “Three-dimensional holographic photostimulation of the dendritic arbor,” J. Neural Eng.8(4), 046002 (2011).
[CrossRef] [PubMed]

Haist, T.

Ianni, F.

Iemmi, C.

Isacoff, E. Y.

E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods7(10), 848–854 (2010).
[CrossRef] [PubMed]

Jesacher, A.

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photon. Rev.5(1), 81–101 (2011).
[CrossRef]

A. Jesacher and M. J. Booth, “Parallel direct laser writing in three dimensions with spatially dependent aberration correction,” Opt. Express18(20), 21090–21099 (2010).
[CrossRef] [PubMed]

Ladavac, K.

Leach, J.

Lee, S. H.

Liu, J.

Lizana, A.

L. Lobato, A. Lizana, A. Marquez, I. Moreno, C. Iemmi, J. Campos, and M. J. Yzuel, “Characterization of the anamorphic and spatial frequency dependent phenomenon in Liquid Crystal on Silicon displays,” J. Eur. Opt. Soc. Rapid Publ.6, 11012S (2011).

I. Moreno, A. Lizana, A. Márquez, C. Iemmi, E. Fernández, J. Campos, and M. J. Yzuel, “Time fluctuations of the phase modulation in a liquid crystal on silicon display: characterization and effects in diffractive optics,” Opt. Express16(21), 16711–16722 (2008).
[CrossRef] [PubMed]

Lobato, L.

L. Lobato, A. Lizana, A. Marquez, I. Moreno, C. Iemmi, J. Campos, and M. J. Yzuel, “Characterization of the anamorphic and spatial frequency dependent phenomenon in Liquid Crystal on Silicon displays,” J. Eur. Opt. Soc. Rapid Publ.6, 11012S (2011).

Lutz, C.

C. Lutz, T. S. Otis, V. DeSars, S. Charpak, D. A. DiGregorio, and V. Emiliani, “Holographic photolysis of caged neurotransmitters,” Nat. Methods5(9), 821–827 (2008).
[CrossRef] [PubMed]

Marquez, A.

L. Lobato, A. Lizana, A. Marquez, I. Moreno, C. Iemmi, J. Campos, and M. J. Yzuel, “Characterization of the anamorphic and spatial frequency dependent phenomenon in Liquid Crystal on Silicon displays,” J. Eur. Opt. Soc. Rapid Publ.6, 11012S (2011).

Márquez, A.

Maurer, C.

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photon. Rev.5(1), 81–101 (2011).
[CrossRef]

Moreno, I.

Nikolenko, V.

V. Nikolenko, B. O. Watson, R. Araya, A. Woodruff, D. S. Peterka, and R. Yuste, “SLM microscopy: scanless two-photon imaging and photostimulation using spatial light modulators,” Front. Neural Circuits2, 1–14 (2008).
[CrossRef] [PubMed]

Oron, D.

Osten, W.

Otis, T. S.

C. Lutz, T. S. Otis, V. DeSars, S. Charpak, D. A. DiGregorio, and V. Emiliani, “Holographic photolysis of caged neurotransmitters,” Nat. Methods5(9), 821–827 (2008).
[CrossRef] [PubMed]

Padgett, M. J.

Palima, D.

Papagiakoumou, E.

S. G. Yang, E. Papagiakoumou, M. Guillon, V. de Sars, C. M. Tang, and V. Emiliani, “Three-dimensional holographic photostimulation of the dendritic arbor,” J. Neural Eng.8(4), 046002 (2011).
[CrossRef] [PubMed]

E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods7(10), 848–854 (2010).
[CrossRef] [PubMed]

E. Papagiakoumou, V. de Sars, D. Oron, and V. Emiliani, “Patterned two-photon illumination by spatiotemporal shaping of ultrashort pulses,” Opt. Express16(26), 22039–22047 (2008).
[CrossRef] [PubMed]

Peterka, D. S.

V. Nikolenko, B. O. Watson, R. Araya, A. Woodruff, D. S. Peterka, and R. Yuste, “SLM microscopy: scanless two-photon imaging and photostimulation using spatial light modulators,” Front. Neural Circuits2, 1–14 (2008).
[CrossRef] [PubMed]

Polin, M.

Reutsky, I.

L. Golan, I. Reutsky, N. Farah, and S. Shoham, “Design and characteristics of holographic neural photo-stimulation systems,” J. Neural Eng.6(6), 066004 (2009).
[CrossRef] [PubMed]

Ritsch-Marte, M.

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photon. Rev.5(1), 81–101 (2011).
[CrossRef]

Roichman, Y.

Ruocco, G.

Shoham, S.

L. Golan, I. Reutsky, N. Farah, and S. Shoham, “Design and characteristics of holographic neural photo-stimulation systems,” J. Neural Eng.6(6), 066004 (2009).
[CrossRef] [PubMed]

Tang, C. M.

S. G. Yang, E. Papagiakoumou, M. Guillon, V. de Sars, C. M. Tang, and V. Emiliani, “Three-dimensional holographic photostimulation of the dendritic arbor,” J. Neural Eng.8(4), 046002 (2011).
[CrossRef] [PubMed]

Testorf, M.

V. Arrizon, E. Carreon, and M. Testorf, “Implementation of Fourier array illuminators using pixelated SLM: efficiency limitations,” Opt. Commun.160(4-6), 207–213 (1999).
[CrossRef]

Wang, Y. T.

Warber, M.

Watson, B. O.

V. Nikolenko, B. O. Watson, R. Araya, A. Woodruff, D. S. Peterka, and R. Yuste, “SLM microscopy: scanless two-photon imaging and photostimulation using spatial light modulators,” Front. Neural Circuits2, 1–14 (2008).
[CrossRef] [PubMed]

Woodruff, A.

V. Nikolenko, B. O. Watson, R. Araya, A. Woodruff, D. S. Peterka, and R. Yuste, “SLM microscopy: scanless two-photon imaging and photostimulation using spatial light modulators,” Front. Neural Circuits2, 1–14 (2008).
[CrossRef] [PubMed]

Wulff, K. D.

Wyrowski, F.

Xie, J. H.

Yang, S. G.

S. G. Yang, E. Papagiakoumou, M. Guillon, V. de Sars, C. M. Tang, and V. Emiliani, “Three-dimensional holographic photostimulation of the dendritic arbor,” J. Neural Eng.8(4), 046002 (2011).
[CrossRef] [PubMed]

Yuste, R.

V. Nikolenko, B. O. Watson, R. Araya, A. Woodruff, D. S. Peterka, and R. Yuste, “SLM microscopy: scanless two-photon imaging and photostimulation using spatial light modulators,” Front. Neural Circuits2, 1–14 (2008).
[CrossRef] [PubMed]

Yzuel, M. J.

Zhang, H.

Zwick, S.

Appl. Opt. (6)

Front. Neural Circuits (1)

V. Nikolenko, B. O. Watson, R. Araya, A. Woodruff, D. S. Peterka, and R. Yuste, “SLM microscopy: scanless two-photon imaging and photostimulation using spatial light modulators,” Front. Neural Circuits2, 1–14 (2008).
[CrossRef] [PubMed]

J. Eur. Opt. Soc. Rapid Publ. (1)

L. Lobato, A. Lizana, A. Marquez, I. Moreno, C. Iemmi, J. Campos, and M. J. Yzuel, “Characterization of the anamorphic and spatial frequency dependent phenomenon in Liquid Crystal on Silicon displays,” J. Eur. Opt. Soc. Rapid Publ.6, 11012S (2011).

J. Neural Eng. (2)

S. G. Yang, E. Papagiakoumou, M. Guillon, V. de Sars, C. M. Tang, and V. Emiliani, “Three-dimensional holographic photostimulation of the dendritic arbor,” J. Neural Eng.8(4), 046002 (2011).
[CrossRef] [PubMed]

L. Golan, I. Reutsky, N. Farah, and S. Shoham, “Design and characteristics of holographic neural photo-stimulation systems,” J. Neural Eng.6(6), 066004 (2009).
[CrossRef] [PubMed]

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

Laser Photon. Rev. (1)

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photon. Rev.5(1), 81–101 (2011).
[CrossRef]

Nat. Methods (2)

C. Lutz, T. S. Otis, V. DeSars, S. Charpak, D. A. DiGregorio, and V. Emiliani, “Holographic photolysis of caged neurotransmitters,” Nat. Methods5(9), 821–827 (2008).
[CrossRef] [PubMed]

E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods7(10), 848–854 (2010).
[CrossRef] [PubMed]

Nat. Photonics (1)

K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics5(6), 335–342 (2011).
[CrossRef]

Opt. Commun. (1)

V. Arrizon, E. Carreon, and M. Testorf, “Implementation of Fourier array illuminators using pixelated SLM: efficiency limitations,” Opt. Commun.160(4-6), 207–213 (1999).
[CrossRef]

Opt. Express (7)

M. Polin, K. Ladavac, S. H. Lee, Y. Roichman, and D. G. Grier, “Optimized holographic optical traps,” Opt. Express13(15), 5831–5845 (2005).
[CrossRef] [PubMed]

K. D. Wulff, D. G. Cole, R. L. Clark, R. Dileonardo, J. Leach, J. Cooper, G. Gibson, and M. J. Padgett, “Aberration correction in holographic optical tweezers,” Opt. Express14(9), 4169–4174 (2006).
[CrossRef] [PubMed]

R. Di Leonardo, F. Ianni, and G. Ruocco, “Computer generation of optimal holograms for optical trap arrays,” Opt. Express15(4), 1913–1922 (2007).
[CrossRef] [PubMed]

A. Jesacher and M. J. Booth, “Parallel direct laser writing in three dimensions with spatially dependent aberration correction,” Opt. Express18(20), 21090–21099 (2010).
[CrossRef] [PubMed]

I. Moreno, A. Lizana, A. Márquez, C. Iemmi, E. Fernández, J. Campos, and M. J. Yzuel, “Time fluctuations of the phase modulation in a liquid crystal on silicon display: characterization and effects in diffractive optics,” Opt. Express16(21), 16711–16722 (2008).
[CrossRef] [PubMed]

E. Papagiakoumou, V. de Sars, D. Oron, and V. Emiliani, “Patterned two-photon illumination by spatiotemporal shaping of ultrashort pulses,” Opt. Express16(26), 22039–22047 (2008).
[CrossRef] [PubMed]

A. Márquez, C. Iemmi, I. Moreno, J. Campos, and M. J. Yzuel, “Anamorphic and spatial frequency dependent phase modulation on liquid crystal displays. Optimization of the modulation diffraction efficiency,” Opt. Express13(6), 2111–2119 (2005).
[CrossRef] [PubMed]

Other (2)

S. A. Benton and V. M. Bove, in Holographic Imaging (Wiley Interscience, 2008), p. 59.

J. W. Goodman, in Introduction to Fourier Optics, R. A. Cie, ed. (Englewood, Colorado, 2005), p. 84.

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Figures (8)

Fig. 1
Fig. 1

Experimental setup. L1, L2, L3 achromatic lenses (f1 = 1000mm; f2 = 500mm; f3 = 150mm). O1 Objective Lens Olympus, 60x 0.9NA W, O2 imaging Objective Lens Olympus, 60x 1.2 W; BE Beam Expander; F1 emission filter.

Fig. 2
Fig. 2

(a) Experimental normalized powers of the zero, ± 1 and ± 2 orders for a horizontal grid with a 2-pixel period displayed on the SLM. (b) Theoretical normalized power of the zero and first order for a crenel and sine function. Gray levels are linearly related to phase shifts (gray level 212 corresponds to φ = 2π for λ = 800nm).

Fig. 3
Fig. 3

Numerical fitting of experimental values of power in zero, first and second orders with a single phase-profile with amplitude proportional to gray-level for a 2-pixel period grating. The phase-profile in question is a Fourier series truncated at the 5th harmonic. Inset (a): Filtering of the spatial frequencies displayed on the SLM. The red dots represent the ratio between the Fourier coefficients of the effective experimental grating phase function and the Fourier coefficients of a crenel-like function. The black line indicates a Lorentzian fitting of the red-dot values. Spatial frequency is in 1/2a unit, a being the pixel size. Inset (b): comparison of a perfect crenel-like phase profile and the phase profile numerically induced from fitting of experimental data.

Fig. 4
Fig. 4

(a) Diagram of correction coefficients measured by displaying grating vertically and horizontally oriented and checkerboard with 2 and 4pixels periods. Values in blue are correction coefficients required to efficiently remove the zero-order. (b) C0(k) interpolated correction function. Spatial frequencies are normalized to 1/2a.

Fig. 5
Fig. 5

Power in the zero-order (a) and in the disk (b) for disks of 10μm diameter displayed at increasing distances from the centre in the vertical, horizontal and diagonal directions, with (hollow symbols) and without (plain symbols) the pattern correction by C0(k). The power is normalized with respect to the total incident power. The inset shows the disposition of the spots in the focus plane of L1.

Fig. 6
Fig. 6

Normalized power in the ghost spots induced by displaying spots of 10μm diameter at various distances from the centre in the vertical and horizontal directions with and without the application of the C0(k) correction map. The power is normalized respect to the total reflected power by the SLM.

Fig. 7
Fig. 7

Fluorescence image displaying a 10μm diameter spot at 60μm from the centre in various directions without (a,c,e) and with (b,d,f) the application of the correction algorithm. Each uncorrected image has been scaled to the maximum intensity of the respective corrected version (Scale bars 10μm). Intensity profiles along the zero-order (g,h,i).

Fig. 8
Fig. 8

Fluorescence image of a neuron (a).2PE fluorescence image of a pattern shaped on a brunch of a neuron (red contours in (a)) induced on a Rhodamine6G layer without (b) and with (c) correction algorithm application. The uncorrected image has been scaled to the maximum intensity of the respective corrected version. Intensity profile along the zero-order(d). (Scale bars 10μm)

Equations (10)

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f(r)=p(r)g(r)
F(k)=G(k)P(k)
G( k m )= α ' m α m
P corr (k)= 1 G(k) P(k)
φ'C( k 1 )φ
C 0 ( k 1 )= φ min /106
C 0 ( k x , k y )=(1 α x k x β x k x 2 )(1 α y k y β y k y 2 )
C 0 ( k x , k y )=(1+0.36 k x +0.56 k x 2 )(1+0.36 k y +0.56 k y 2 )
p 0,corr ( r )= 1 { C 0 ( k ){ p( r ) } }
C 1 ( k x , k y )=(1+0.43 k x 2 +0.10 k x 3 )(1+0.43 k y 2 +0.10 k y 3 )

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