## LCoS nematic SLM characterization and modeling for diffraction efficiency optimization, zero and ghost orders suppression |

Optics Express, Vol. 20, Issue 16, pp. 17843-17855 (2012)

http://dx.doi.org/10.1364/OE.20.017843

Acrobat PDF (2064 KB)

### 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

## 1. Introduction

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]

2. 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. Express **14**(9), 4169–4174 (2006). [CrossRef] [PubMed]

3. K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics **5**(6), 335–342 (2011). [CrossRef]

4. 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]

8. 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]

9. R. Di Leonardo, F. Ianni, and G. Ruocco, “Computer generation of optimal holograms for optical trap arrays,” Opt. Express **15**(4), 1913–1922 (2007). [CrossRef] [PubMed]

10. F. Wyrowski and O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A **5**(7), 1058–1065 (1988). [CrossRef]

*(sin(x)/x)*envelope [4

^{2}4. 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]

8. 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]

11. 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]

12. 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. Express **16**(21), 16711–16722 (2008). [CrossRef] [PubMed]

14. 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. Express **13**(6), 2111–2119 (2005). [CrossRef] [PubMed]

15. B. Apter, U. Efron, and E. Bahat-Treidel, “On the fringing-field effect in liquid-crystal beam-steering devices,” Appl. Opt. **43**(1), 11–19 (2004). [CrossRef] [PubMed]

17. U. Efron, B. Apter, and E. Bahat-Treidel, “Fringing-field effect in liquid-crystal beam-steering devices: an approximate analytical model,” J. Opt. Soc. Am. A **21**(10), 1996–2008 (2004). [CrossRef] [PubMed]

18. M. Polin, K. Ladavac, S. H. Lee, Y. Roichman, and D. G. Grier, “Optimized holographic optical traps,” Opt. Express **13**(15), 5831–5845 (2005). [CrossRef] [PubMed]

19. S. Zwick, T. Haist, M. Warber, and W. Osten, “Dynamic holography using pixelated light modulators,” Appl. Opt. **49**(25), F47–F58 (2010). [CrossRef] [PubMed]

20. 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]

21. A. Jesacher and M. J. Booth, “Parallel direct laser writing in three dimensions with spatially dependent aberration correction,” Opt. Express **18**(20), 21090–21099 (2010). [CrossRef] [PubMed]

22. D. Palima and V. R. Daria, “Holographic projection of arbitrary light patterns with a suppressed zero-order beam,” Appl. Opt. **46**(20), 4197–4201 (2007). [CrossRef] [PubMed]

*a priori*.

*(sin(x)/x)*envelope. Moreover, depending on the displayed phase pattern, fraction of total power in the zero-order varies from ~2% up to ~40%. This effect is all the more critical when dealing with applications consisting in bi-photonic excitation because of the quadratic dependence of the absorption process on intensity. Further degradation of the diffraction efficiency and limitations of the applicability of the SLM are due to the presence of undesired spots symmetrically displaced in respect to the desired pattern, so called ghost spots [18

^{2}18. M. Polin, K. Ladavac, S. H. Lee, Y. Roichman, and D. G. Grier, “Optimized holographic optical traps,” Opt. Express **13**(15), 5831–5845 (2005). [CrossRef] [PubMed]

## 2. Experiments

23. E. Papagiakoumou, V. de Sars, D. Oron, and V. Emiliani, “Patterned two-photon illumination by spatiotemporal shaping of ultrashort pulses,” Opt. Express **16**(26), 22039–22047 (2008). [CrossRef] [PubMed]

*f*= 1000mm;

_{1}*f*= 500mm) in order that the pupil diameter matches the SLM short axis.

_{2}9. R. Di Leonardo, F. Ianni, and G. Ruocco, “Computer generation of optimal holograms for optical trap arrays,” Opt. Express **15**(4), 1913–1922 (2007). [CrossRef] [PubMed]

10. F. Wyrowski and O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A **5**(7), 1058–1065 (1988). [CrossRef]

## 3. SLM characterization and modeling

### SLM experimental characterization

*φ*, in the zero-order and in ± 1 orders is expected to be

#### SLM phenomenological modeling

15. B. Apter, U. Efron, and E. Bahat-Treidel, “On the fringing-field effect in liquid-crystal beam-steering devices,” Appl. Opt. **43**(1), 11–19 (2004). [CrossRef] [PubMed]

17. U. Efron, B. Apter, and E. Bahat-Treidel, “Fringing-field effect in liquid-crystal beam-steering devices: an approximate analytical model,” J. Opt. Soc. Am. A **21**(10), 1996–2008 (2004). [CrossRef] [PubMed]

15. B. Apter, U. Efron, and E. Bahat-Treidel, “On the fringing-field effect in liquid-crystal beam-steering devices,” Appl. Opt. **43**(1), 11–19 (2004). [CrossRef] [PubMed]

17. U. Efron, B. Apter, and E. Bahat-Treidel, “Fringing-field effect in liquid-crystal beam-steering devices: an approximate analytical model,” J. Opt. Soc. Am. A **21**(10), 1996–2008 (2004). [CrossRef] [PubMed]

*p(*

**r**

*)*, the phase pattern effectively displayed by the SLM,

*f(*

**r**

*)*will be given by:where function

*g*accounts for the cross-talk between pixels induced by the influence of nearby electrodes, and

**r**is the coordinate in the SLM plane. In the frequency domain, linear filtering consists simply in a product by the filter function

*G*, the Fourier transform of function

*g*:

*p*is a periodic function,

*f*is also periodic and both can be expanded in Fourier series.

*P*and

*F*are then sums of Dirac’s functions.

*p(*

**r**

*)*is a crenel-like function then

*f(*

**r**

*)*is a sine series truncated at the fifth harmonic. Therefore, assuming that

*g(*

**r**

*)*does not depend on the phase pattern addressed onto the SLM, we can determine

*G(*

**k**

*)*as the ratio between the Fourier transform of a sine series truncated at the fifth harmonic and the Fourier transform of a crenel-like function:where

*α*and

_{m}*α’*indicate respectively the coefficients of the m-harmonic in the frequency domain for the theoretical crenel function and the experimental sine series (Fig. 3(a), red dots). By interpolating these coefficients, it is possible to trace the filtering function

_{m}*G(*

**k**

*)*for the all spatial frequencies of the displayed pattern (Fig. 3(a), black curve). In the following sections, we propose two strategies to numerically compensate for this filtering effect.

### Rigorous Phase profile correction

*G(*

**k**

*)*(Fig. 3, inset (a)) to compensate the distortions introduced by the SLM. That is, given a theoretical phase pattern,

*P(*

**k**

*),*we generate the corrected one,

*P*

_{corr}(**k**

*)*as:

*P(*

**k**

*)*, of the desired phase profile. However, in practice only spatial frequencies lower than 1/(2

*a)*(

*a*being the pixel pitch) can be corrected. Higher frequencies correspond to inaccessible sub-pixel scales.

*G(*

**k**

*)*. The first harmonic will then be multiplied by

*C(*

**k**

_{1}) = 1/G(**k**

*, where*

_{1})*k*stands for the first harmonic. In the spatial domain, this means that the theoretical modulation depth of the binary grating function,

_{1}*φ*, is multiplied by

*C(*

**k**

_{1}) = 1/G(**k**

*:where*

_{1})*φ’*is the corrected modulation depth. If we refer to the filter value

*G(*

**k**

*= 0.62 (inset a, Fig. 3), then*

_{1})*C(*

**k**

*= 1/0.62 = 1.61. Thus in particular for*

_{1})*φ*= 106, the corrected modulation depth is,

*φ’*= 171. Referring to Fig. 2(a), for such a modulation depth, the fraction of power in the zero order experimentally obtained is ~6%, decreased by 85% with respect to the uncorrected case which exhibit 41% of total power in the zero-order. However, we also notice that the minimum reachable power in the zero-order is ~2%, corresponding to 203 gray level modulation. This suggests that further optimizations could be potentially obtained with a total fraction of zero-order reduction of 95%.

### Phase Profile correction: zero-order optimization

*C(*

**k**

*)*.

*C*

_{0}(**k**

*) given by the ratio between the phase value corresponding to the minimum of the experimental zero-order diffraction and 106, which represents the theoretical minimum:where*

_{1}*φ*indicates the phase yielding the minimum of the experimental zero order diffraction (203 gray-level for a 2-pixel period grating). This correction will also allow increasing the + 1 and −1 diffraction efficiency from 27% to 31% which represents an 15% increase of power.

_{min}*C*

_{0}(**k**

*/2) for a 4-pixel-periodic grating for which the fundamental frequency is 1/4*

_{1}*a*. Higher harmonics (3/4

*a*, 5/4

*a*…) are not addressable. Phase patterns with larger periods contain more than one harmonic within the addressable compact support and render analysis more complicated.

_{0}correction value cannot be precisely derived in this case, we can argue that it should be higher than 2.5.

*C*

_{0}(**k**

*),*the correction function for any spatial frequencies

**k**. Here, we fitted correcting values with a second order polynomial in both horizontal and vertical directions:

## 4. Experimental results

### Demonstration of SLM Correction efficiency

_{0}and C

_{1}depending on the specific purpose) previously derived, to compensate the spatial smooth filtering effect. However, this numerical calculation, corresponding to a high spatial frequency enhancement, yields some phase patterns with required gray-level values larger than 255 or smaller than 0. Therefore, the corrected gray level maps were truncated for addressing to the SLM. Before truncation, mean value of the output gray profile was centered on 128 to make use of the full gray level dynamic of the SLM.

*a*on the SLM with an expected theoretical maximum diffraction efficiency of 40.57%.

*C*

_{0}(**k**

*)*correction map as described in the previous section. Zero order removal appears to be efficient in every direction. Fraction of power in the zero order can be kept bellow 2% for spots displayed in horizontal and vertical directions, and almost always below 5% for spots in the diagonal direction, whilst uncorrected patterns left around 20% or 40% of total power un-diffracted. Relying on the proposed model, numerical simulations (data not shown) predict a better suppression of the zero order for the diagonal direction. The remaining un-diffracted power must then be attributed to saturation of the gray levels or polarization rotation in the LC matrix.

*C*

_{0}(**k**

*)*, the use of the

*C*

_{1}(**k**

*)*correction map exhibited analogous results, of course increasing diffraction efficiency but also slightly degrading zero-order suppression.

### Two-photon excitation fluorescence

## 5. Conclusion

## Acknowledgments

## References and links

1. | C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photon. Rev. |

2. | 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. Express |

3. | K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics |

4. | L. Golan, I. Reutsky, N. Farah, and S. Shoham, “Design and characteristics of holographic neural photo-stimulation systems,” J. Neural Eng. |

5. | C. Lutz, T. S. Otis, V. DeSars, S. Charpak, D. A. DiGregorio, and V. Emiliani, “Holographic photolysis of caged neurotransmitters,” Nat. Methods |

6. | 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 Circuits |

7. | 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. Methods |

8. | 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. |

9. | R. Di Leonardo, F. Ianni, and G. Ruocco, “Computer generation of optimal holograms for optical trap arrays,” Opt. Express |

10. | F. Wyrowski and O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A |

11. | V. Arrizon, E. Carreon, and M. Testorf, “Implementation of Fourier array illuminators using pixelated SLM: efficiency limitations,” Opt. Commun. |

12. | 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. Express |

13. | 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. |

14. | 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. Express |

15. | B. Apter, U. Efron, and E. Bahat-Treidel, “On the fringing-field effect in liquid-crystal beam-steering devices,” Appl. Opt. |

16. | E. Bahat-Treidel, B. Apter, and U. Efron, “Experimental study of phase-step broadening by fringing fields in a three-electrode liquid-crystal cell,” Appl. Opt. |

17. | U. Efron, B. Apter, and E. Bahat-Treidel, “Fringing-field effect in liquid-crystal beam-steering devices: an approximate analytical model,” J. Opt. Soc. Am. A |

18. | M. Polin, K. Ladavac, S. H. Lee, Y. Roichman, and D. G. Grier, “Optimized holographic optical traps,” Opt. Express |

19. | S. Zwick, T. Haist, M. Warber, and W. Osten, “Dynamic holography using pixelated light modulators,” Appl. Opt. |

20. | 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. |

21. | A. Jesacher and M. J. Booth, “Parallel direct laser writing in three dimensions with spatially dependent aberration correction,” Opt. Express |

22. | D. Palima and V. R. Daria, “Holographic projection of arbitrary light patterns with a suppressed zero-order beam,” Appl. Opt. |

23. | E. Papagiakoumou, V. de Sars, D. Oron, and V. Emiliani, “Patterned two-photon illumination by spatiotemporal shaping of ultrashort pulses,” Opt. Express |

24. | S. A. Benton and V. M. Bove, in |

25. | J. W. Goodman, in |

26. | I. Moreno, C. Iemmi, A. Márquez, J. Campos, and M. J. Yzuel, “Modulation light efficiency of diffractive lenses displayed in a restricted phase-mostly modulation display,” Appl. Opt. |

**OCIS Codes**

(050.1970) Diffraction and gratings : Diffractive optics

(090.1760) Holography : Computer holography

(230.3720) Optical devices : Liquid-crystal devices

(230.6120) Optical devices : Spatial light modulators

**ToC Category:**

Optical Devices

**History**

Original Manuscript: May 24, 2012

Revised Manuscript: June 29, 2012

Manuscript Accepted: June 29, 2012

Published: July 20, 2012

**Citation**

Emiliano Ronzitti, Marc Guillon, Vincent de Sars, and Valentina Emiliani, "LCoS nematic SLM characterization and modeling for diffraction efficiency optimization, zero and ghost orders suppression," Opt. Express **20**, 17843-17855 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-16-17843

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### References

- 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]
- 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]
- K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics5(6), 335–342 (2011). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- 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]
- F. Wyrowski and O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A5(7), 1058–1065 (1988). [CrossRef]
- 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]
- 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]
- 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).
- 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]
- B. Apter, U. Efron, and E. Bahat-Treidel, “On the fringing-field effect in liquid-crystal beam-steering devices,” Appl. Opt.43(1), 11–19 (2004). [CrossRef] [PubMed]
- E. Bahat-Treidel, B. Apter, and U. Efron, “Experimental study of phase-step broadening by fringing fields in a three-electrode liquid-crystal cell,” Appl. Opt.44(15), 2989–2995 (2005). [CrossRef] [PubMed]
- U. Efron, B. Apter, and E. Bahat-Treidel, “Fringing-field effect in liquid-crystal beam-steering devices: an approximate analytical model,” J. Opt. Soc. Am. A21(10), 1996–2008 (2004). [CrossRef] [PubMed]
- 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]
- S. Zwick, T. Haist, M. Warber, and W. Osten, “Dynamic holography using pixelated light modulators,” Appl. Opt.49(25), F47–F58 (2010). [CrossRef] [PubMed]
- 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]
- 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]
- D. Palima and V. R. Daria, “Holographic projection of arbitrary light patterns with a suppressed zero-order beam,” Appl. Opt.46(20), 4197–4201 (2007). [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]
- 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.
- I. Moreno, C. Iemmi, A. Márquez, J. Campos, and M. J. Yzuel, “Modulation light efficiency of diffractive lenses displayed in a restricted phase-mostly modulation display,” Appl. Opt.43(34), 6278–6284 (2004). [CrossRef] [PubMed]

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