## Comparison of generalized phase contrast and computer generated holography for laser image projection

Optics Express, Vol. 16, Issue 8, pp. 5338-5349 (2008)

http://dx.doi.org/10.1364/OE.16.005338

Acrobat PDF (518 KB)

### Abstract

Laser projection based on phase modulation promises several advantages over amplitude modulation. We examine and compare the merits of two phase modulation techniques; phase-only computer generated holography and generalized phase contrast, for the application of dynamic laser image projection. We adopt information theory as a guiding framework and analyze information-relevant metrics such as space-bandwidth product, output display resolution, efficiency, speckle noise, computational load and device requirements. The analysis takes into account the perspective of potential end-users.

© 2008 Optical Society of America

## 1. Introduction

1. C. David, J. Wei, T. Lippert, and A. Wokaun, “Diffractive grey-tone phase masks for laser ablation lithography,” Microelectron. Eng. **57–8**, 453–460 (2001). [CrossRef]

2. M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy: Wide-field fluorescence imaging with theoretically unlimited resolution,” P. Natl. Acad. Sci. USA **102**, 13081–13086 (2005). [CrossRef]

3. M.P MacDonald, G.C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature **426**, 421–424, (2003). [CrossRef] [PubMed]

4. M. Klosner and K. Jain, “Massively parallel, large-area maskless lithography,” Appl. Phys. Lett. **84**, 2880–2882 (2004). [CrossRef]

5. A. W. Lohmann and D. P. Paris, “Binary Fraunhofer holograms, generated by computer,” Appl. Opt. **6**, 1739–1748 (1967). [CrossRef] [PubMed]

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

7. J. Glückstad and P. C. Mogensen, “Optimal Phase Contrast in Common-Path Interferometry,” Appl. Opt. **40**, 268–282 (2001). [CrossRef]

8. D. Gabor, “A new microscopic principle,” Nature **161**, 777–778 (1948). [CrossRef] [PubMed]

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

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

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

11. P. J. Rodrigo, V. R. Daria, and J. Glückstad, “Real-time three-dimensional optical micromanipulation of multiple particles and living cells,” Opt. Lett. **29**2270–2272 (2004). [CrossRef] [PubMed]

12. P. Rodrigo, V. Daria, and J. Glückstad, “Dynamically reconfigurable optical lattices,” Opt. Express **13**, 1384–1394 (2005). [CrossRef] [PubMed]

13. C. A. Alonzo, P. J. Rodrigo, and J. Glückstad, “Photon-efficient grey-level image projection by the generalized phase contrast method,” New J. Phys. **9**, 132 (2007). [CrossRef]

14. J. Glückstad, D. Palima, P. J. Rodrigo, and C. A. Alonzo, “Laser projection using generalized phase contrast,” Opt. Lett. **32**, 3281–3283 (2007). [CrossRef] [PubMed]

15. 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]

## 2. Pattern projection and information theory

*N*independently controllable elements available at the output, which we will refer to as the number of degrees of freedom of the system. If each degree of freedom can have

_{F}*M*distinguishable levels then there are altogether

*N*, is commonly expressed in terms of number of the corresponding binary digits (bits)

_{C}*L*≤

_{x}*x*≤0.5

*L*and -0.5

_{x}*L*≤

_{y}*y*≤0.5

*L*} and an effective band-limiting aperture that transmits spatial frequencies within {-0.5

_{y}*Δ*

*f*≤

_{x}*f*≤0.5

_{x}*Δ*

*f*and -0.5Δ

_{x}*f*≤

_{y}*f*≤0.5Δ

_{y}*f*}. To count

_{y}*N*we follow Lukosz’s thought experiment [17

_{F}17. W. Lukosz, “Optical systems with resolving powers exceeding classical limit,” J. Opt. Soc. Am. **56**, 1463–1472 (1966). [CrossRef]

*x*=0, 1/

*Δ*

*f*, …,

_{x}*m*/

*Δ*

*f*, where the edge of the projection corresponds to

_{x,}*m*=0.5

_{max}*L*

_{x}*Δf*and similarly along the y-axis. Thus, the number of degrees of freedom is given by

_{x}*L*,

_{x}Δf_{x}*L*} then the number of degrees of freedom becomes

_{y}Δf_{y}*N*≈

_{F}*L*

_{x}*L*, which is the familiar space-bandwith product. The +1 term in Eq. (2) guarantees available degrees of freedom when information is encoded only along one single dimension. A conventional communication channel does not use spatially-encoded information but exploits degrees of freedom along the time dimension instead. The temporal degrees of freedom can also be utilized in dynamic pattern projection to expand the degrees of freedom to

_{y}Δf_{x}Δf_{y}*Δf*is the temporal bandwidth and

_{t}*L*is the length of time (duration) of the transmission.

_{t}*N*is invariant for an optical system and one can trade off the degrees of freedom between the different dimensions to achieve certain goals, such as superresolution [17

_{F}17. W. Lukosz, “Optical systems with resolving powers exceeding classical limit,” J. Opt. Soc. Am. **56**, 1463–1472 (1966). [CrossRef]

18. I. J. Cox and C. J. R. Sheppard, “Information capacity and resolution in an optical system,” J. Opt. Soc. Am. A **3**, 1152–1158 (1986). [CrossRef]

19. P. B. Fellgett and E. H. Linfoot, “On the assessment of optical images,” Phil. Trans. R. Soc. London Ser. A **247**, 369–407 (1955). [CrossRef]

## 3. Performance benchmarks

### 3.1 Spatial degrees of freedom

*L*

_{x}*Δ*

*f*)(1+

_{y}*L*

_{x}*Δ*

*f*), describes the number of independently addressable output elements and is a measure of the output display resolution. As with traditional display technology, it is reasonable to demand that laser projection should have a good output display resolution in order to sufficiently render fine details in the projections. The imaging geometry of GPC suggests that it is able to address as many output elements as those available in the SLM. This is subject to the band limits imposed by the lens apertures and can exploit high-resolution input devices using high numerical aperture imaging. At first glance, the Fourier relation between the SLM and projection planes in CGH suggests that the CGH output also matches the SLM pixel count. However, fundamental constraints require reducing the output display resolution in order to avoid speckled outputs.

_{y}20. F. Wyrowski, “Upper bound of the diffraction efficiency of diffractive phase elements,” Opt. Lett. **16**, 1915–1917 (1991). [CrossRef] [PubMed]

*d*and without any interpixel dead space. Its output projection in the Fourier plane is given by [21

21. D. Palima and V. R. Daria, “Effect of spurious diffraction orders in arbitrary multifoci patterns produced via phase-only holograms,” Appl. Opt. **45**, 6689–6693 (2006). [CrossRef] [PubMed]

*Q*(

*f*,

_{x}*f*) represents the desired projection pattern that is discretized on a regular grid and infinitely replicated over the reconstruction plane as a consequence of the sampling at the SLM plane; the sinc function represents an intensity roll-off envelope arising from the diffraction through the finite pixel apertures; and the convolution with

_{y}*A*(

*f*,

_{x}*f*) represents the diffraction broadening of the projected points due to the finite extent of the illumination at the SLM plane. The diffraction-broadened reconstruction points can lead to crosstalk between adjacent reconstruction points. With the output phase treated as an optimization free parameter in a phase-only CGH iterative design, neighbouring points inevitably end up with an unpredictable phase relationship. The pixel crosstalk thus results in speckled reconstructions as neighbouring pixels can randomly interfere constructively or destructively. The phase variation between adjacent reconstruction points can be smoothened as part of the optimization procedure to reduce the speckles even when the reconstruction points overlap [25

_{y}25. H. Aagedal, M. Schmid, T. Beth, S. Teiwes, and F. Wyrowski, “Theory of speckles in diffractive optics and its application to beam shaping,” J. Mod. Opt. **43**, 1409–1421 (1996). [CrossRef]

16. V. Arrizon and M. Testorf, “Efficiency limit of spatially quantized Fourier array illuminators,” Opt. Lett. **22**, 197–199 (1997). [CrossRef] [PubMed]

22. A. J. Waddie and M. R. Taghizadeh, “Interference Effects in Far-Field Diffractive Optical Elements,” Appl. Opt. **38**, 5915–5919 (1999). [CrossRef]

*f*is the focal length of the Fourier lens used,

*λ*is the illumination wavelength,

*T*is the period of the repeated cell and

*D*is the diameter of the illuminating aperture. A suitable compromise is obtained when the relative distance is

*σ*=2. At this spacing, the distance between spots equals the diameter of their central lobes and their first minima coincide. This occurs for

_{R}*D*=2.44

*T*, which requires 2.44 repetitions along the aperture diameter. Since the basic cell for a 512×512 single quadrant target is 1024×1024, the encoding device must have at least 2498×2498 pixels. This can still contain some noise as the rings surrounding the central spot will certainly exert a residual disturbance on the neighbouring spots.

*L*and

_{x}*L*while hologram tiling reduces the effective

_{y}*Δ*

*f*and

_{x}*Δ*

*f*. The steps described above effectively reduce the information capacity by ~24 compared to an imaging system with a similar numerical aperture and band limit. In accordance with the invariance of

_{y}*N*, one can also minimize speckle effects by exploiting temporal degrees of freedom. Instead of reducing the spatial degrees of freedom through the speckle reduction schemes described above, one can repetitively project a pattern with different speckles to an integrating detector for a time interval

_{C}*L*to distinguish the information from an averaged noise background. The reduced information capacity is apparent when one considers that a GPC-based projection can already transmit several different information for the same time interval.

_{t}### 3.2 Temporal degrees of freedom

23. F. Wyrowski, “Diffractive optical elements: iterative calculation of quantized, blazed phase structures,” J. Opt. Soc. Am. A **7**, 961–969 (1990). [CrossRef]

24. P. Senthilkumaran, F. Wyrowski, and H. Schimmel. “Vortex Stagnation problem in iterative Fourier transform algorithms,” Opt. Laser Eng. **43**, 43–56 (2005). [CrossRef]

25. H. Aagedal, M. Schmid, T. Beth, S. Teiwes, and F. Wyrowski, “Theory of speckles in diffractive optics and its application to beam shaping,” J. Mod. Opt. **43**, 1409–1421 (1996). [CrossRef]

### 3.3 Signal-to-noise ratio and light efficiency

*η*is the efficiency and s0 is the noise-free output signal for 100% efficiency. Equation (7) expresses the logical result that a system with zero efficiency has no information capacity. To study the projection efficiency, we again consider a square phase-only SLM containing N×N square pixels, each with side length

*d*, without any interpixel dead space. When this SLM is used to encode phase information into an incident wave the pixelation results in replicated projections having a characteristic intensity roll-off due to the diffraction from the pixel apertures. For single beam scanning, the roll-off pegs the highest possible efficiency at the corners of the projection region to just 16% of the on-axis efficiency. The grayscale/contour plot for the spatial variation of efficiency in CGH single beam deflection is illustrated in the Fig. 3.

16. V. Arrizon and M. Testorf, “Efficiency limit of spatially quantized Fourier array illuminators,” Opt. Lett. **22**, 197–199 (1997). [CrossRef] [PubMed]

*η*′

_{ql}describes the desired efficiency (grey level) at the reconstruction point defined by coordinates (

*q*,

*l*) on a discrete grid separated by

*fλ*/

*T*within the signal window Ω

_{s}at the projection plane. This efficiency limit, which was originally obtained for array illuminators, applies to general CGH-based laser projections as well since the projection pixels require sufficient separation as discussed above. Accounting for the proper separation in Eq. (8) predicts projections with maximum efficiency of ~52% when the image occupies the entire addressable region at the projection plane. The same efficiency limit is obtained when the projection is constrained to a single quadrant signal window. Our tests show that this efficiency limit holds whether the projection is a uniform grey level, a random grey image, or a typical image with a reasonable spread of grey levels like the Lena standard image. Achieving efficiency beyond 52% can only be achieved by a further reduction of the signal window to exclude the outer regions that tend to decrease efficiency. This consequently comes at a price of a further decrease in output display resolution.

*r*(

_{S}*x*’,

*y*’), with the image of the phase-modulated input,

*p*(

*x*’,

*y*’). For a circular input aperture with radius

*Δr*and a PCF having a circular phase-shifting region with radius

*Δ*

*f*the optimized SRW is given by the Hankel transform [7

_{r}7. J. Glückstad and P. C. Mogensen, “Optimal Phase Contrast in Common-Path Interferometry,” Appl. Opt. **40**, 268–282 (2001). [CrossRef]

*Δ*

*f*, as shown in Fig. 4(a). The SRW can extend beyond the signal region occupied by the image of the modulated input, which results in a residual intensity halo around the projected image. The energy lost to the halo region sets a fundamental limit on the achievable GPC efficiency. The GPC efficiency limit,

_{r}*η*, is obtained by accounting for the energy lost in the SRW tail via numerical integration of the expression:

_{max}### 3.4 Practical SLM devices: performance constraints

### 3.5. Interpixel dead space

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

28. V. Arrizón, E. Carreón, and M. Testorf, “Implementation of Fourier array illuminators using pixelated SLM: efficiency limitations,” Opt. Commun. **160**, 207–213 (1999). [CrossRef]

*r*, is the ratio of area of the active region to the total pixel area. The efficiency declines with decreasing

*r*as plotted in Fig. 5. The efficiency plot, which shows 52% for 100% fill factor, was determined assuming that the CGH projection is constrained to one quadrant of the addressable region in the reconstruction plane. As mentioned in section 3.3, it is possible to project CGH-based images with higher efficiencies by projecting them to a smaller signal window and sacrificing resolution. The figure also includes the efficiency of phase-only CGH-based image projection to one quadrant relative to GPC-based projection to the entire addressable region of the reconstruction plane. GPC-based projection is expected to linearly decrease with

*r*due to the decreasing energy throughput and where the initial efficiency at

*r*=1 is taken to be 0.82 (the efficiency along the dotted line in Fig. 4(b). A horizontal plot of the relative performance would indicate that the fill factor variation has a similar effect on both GPC and CGH efficiency. The actual plot of the relative efficiency in Fig. 5 indicates that GPC is increasingly advantageous over CGH when using SLM devices with decreasing fill factors.

### 3.6 Device modulation transfer function

29. M. L. Hsieh, K. Y. Hsu, E. G. Paek, and C. L. Wilson, “Modulation transfer function of a liquid crystal spatial light modulator,” Opt. Commun. **170**, 221–227 (1999). [CrossRef]

30. E. Hällstig, T. Martin, L. Sjöqvist, and M. Lindgren, “Polarization properties of a nematic liquid-crystal spatial light modulator for phase modulation,” J. Opt. Soc. Am. A **22**, 177–184 (2005). [CrossRef]

31. G. Moddel and L. Wang, “Resolution limits from charge transport in optically addressed spatial light modulators,” J. Appl. Phys. **78**, 6923–6935 (1995). [CrossRef]

33. A. Márquez, C. Iemmi, I. Moreno, J. Campos, and M. Yzuel, “Anamorphic and spatial frequency dependent phase modulation on liquid crystal displays. Optimization of the modulation diffraction efficiency,” Opt. Express **13**, 2111–2119 (2005). [CrossRef] [PubMed]

32. M. Duelli, L. Ge, and R. W. Cohn, “Nonlinear effects of phase blurring on Fourier transform holograms,” J. Opt. Soc. Am. A **17**, 1594–1605 (2000). [CrossRef]

*f*imaging configuration, aberrations in the Fourier plane introduced by the first lens are minimized by using a matching second lens. The ability of a GPC-based projection system to work with the limited modulation transfer function of existing devices is highlighted by our recent work, where we experimentally demonstrated laser projection of the Lena image at 74% efficiency [14

14. J. Glückstad, D. Palima, P. J. Rodrigo, and C. A. Alonzo, “Laser projection using generalized phase contrast,” Opt. Lett. **32**, 3281–3283 (2007). [CrossRef] [PubMed]

### 3.7 Required phase stroke

## 4. Concluding remarks

## Acknowledgement

## References and links

1. | C. David, J. Wei, T. Lippert, and A. Wokaun, “Diffractive grey-tone phase masks for laser ablation lithography,” Microelectron. Eng. |

2. | M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy: Wide-field fluorescence imaging with theoretically unlimited resolution,” P. Natl. Acad. Sci. USA |

3. | M.P MacDonald, G.C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature |

4. | M. Klosner and K. Jain, “Massively parallel, large-area maskless lithography,” Appl. Phys. Lett. |

5. | A. W. Lohmann and D. P. Paris, “Binary Fraunhofer holograms, generated by computer,” Appl. Opt. |

6. | J. Glückstad, “Phase contrast image synthesis,” Opt. Commun. |

7. | J. Glückstad and P. C. Mogensen, “Optimal Phase Contrast in Common-Path Interferometry,” Appl. Opt. |

8. | D. Gabor, “A new microscopic principle,” Nature |

9. | F. Zernike, “How I discovered phase contrast,” Science |

10. | J. Glückstad, L. Lading, H. Toyoda, and T. Hara, “Lossless light projection,” Opt. Lett. |

11. | P. J. Rodrigo, V. R. Daria, and J. Glückstad, “Real-time three-dimensional optical micromanipulation of multiple particles and living cells,” Opt. Lett. |

12. | P. Rodrigo, V. Daria, and J. Glückstad, “Dynamically reconfigurable optical lattices,” Opt. Express |

13. | C. A. Alonzo, P. J. Rodrigo, and J. Glückstad, “Photon-efficient grey-level image projection by the generalized phase contrast method,” New J. Phys. |

14. | J. Glückstad, D. Palima, P. J. Rodrigo, and C. A. Alonzo, “Laser projection using generalized phase contrast,” Opt. Lett. |

15. | D. Palima, C. A. Alonzo, P. J. Rodrigo, and J. Glückstad, “Generalized phase contrast matched to Gaussian illumination,” Opt. Express |

16. | V. Arrizon and M. Testorf, “Efficiency limit of spatially quantized Fourier array illuminators,” Opt. Lett. |

17. | W. Lukosz, “Optical systems with resolving powers exceeding classical limit,” J. Opt. Soc. Am. |

18. | I. J. Cox and C. J. R. Sheppard, “Information capacity and resolution in an optical system,” J. Opt. Soc. Am. A |

19. | P. B. Fellgett and E. H. Linfoot, “On the assessment of optical images,” Phil. Trans. R. Soc. London Ser. A |

20. | F. Wyrowski, “Upper bound of the diffraction efficiency of diffractive phase elements,” Opt. Lett. |

21. | D. Palima and V. R. Daria, “Effect of spurious diffraction orders in arbitrary multifoci patterns produced via phase-only holograms,” Appl. Opt. |

22. | A. J. Waddie and M. R. Taghizadeh, “Interference Effects in Far-Field Diffractive Optical Elements,” Appl. Opt. |

23. | F. Wyrowski, “Diffractive optical elements: iterative calculation of quantized, blazed phase structures,” J. Opt. Soc. Am. A |

24. | P. Senthilkumaran, F. Wyrowski, and H. Schimmel. “Vortex Stagnation problem in iterative Fourier transform algorithms,” Opt. Laser Eng. |

25. | H. Aagedal, M. Schmid, T. Beth, S. Teiwes, and F. Wyrowski, “Theory of speckles in diffractive optics and its application to beam shaping,” J. Mod. Opt. |

26. | R. W. Cohn, “Fundamental properties of spatial light modulators for the approximate optical computation of Fourier transforms: a review,” Opt. Eng. |

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

28. | V. Arrizón, E. Carreón, and M. Testorf, “Implementation of Fourier array illuminators using pixelated SLM: efficiency limitations,” Opt. Commun. |

29. | M. L. Hsieh, K. Y. Hsu, E. G. Paek, and C. L. Wilson, “Modulation transfer function of a liquid crystal spatial light modulator,” Opt. Commun. |

30. | E. Hällstig, T. Martin, L. Sjöqvist, and M. Lindgren, “Polarization properties of a nematic liquid-crystal spatial light modulator for phase modulation,” J. Opt. Soc. Am. A |

31. | G. Moddel and L. Wang, “Resolution limits from charge transport in optically addressed spatial light modulators,” J. Appl. Phys. |

32. | M. Duelli, L. Ge, and R. W. Cohn, “Nonlinear effects of phase blurring on Fourier transform holograms,” J. Opt. Soc. Am. A |

33. | A. Márquez, C. Iemmi, I. Moreno, J. Campos, and M. Yzuel, “Anamorphic and spatial frequency dependent phase modulation on liquid crystal displays. Optimization of the modulation diffraction efficiency,” Opt. Express |

**OCIS Codes**

(070.0070) Fourier optics and signal processing : Fourier optics and signal processing

(110.3000) Imaging systems : Image quality assessment

(110.4100) Imaging systems : Modulation transfer function

(120.4820) Instrumentation, measurement, and metrology : Optical systems

(120.5060) Instrumentation, measurement, and metrology : Phase modulation

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: January 30, 2008

Revised Manuscript: March 19, 2008

Manuscript Accepted: March 27, 2008

Published: April 2, 2008

**Citation**

Darwin Palima and Jesper Glückstad, "Comparison of generalized phase contrast and computer generated holography for laser image projection," Opt. Express **16**, 5338-5349 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-8-5338

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### References

- C. David, J. Wei, T. Lippert, and A. Wokaun, "Diffractive grey-tone phase masks for laser ablation lithography," Microelectron. Eng. 57-8, 453-460 (2001). [CrossRef]
- M. G. L. Gustafsson, "Nonlinear structured-illumination microscopy: Wide-field fluorescence imaging with theoretically unlimited resolution," P. Natl. Acad. Sci. USA 102, 13081-13086 (2005). [CrossRef]
- M. P. MacDonald, G. C. Spalding, and K. Dholakia, "Microfluidic sorting in an optical lattice," Nature 426, 421-4 (2003). [CrossRef] [PubMed]
- M. Klosner and K. Jain, "Massively parallel, large-area maskless lithography," Appl. Phys. Lett. 84, 2880-2882 (2004). [CrossRef]
- A. W. Lohmann and D. P. Paris, "Binary Fraunhofer holograms, generated by computer," Appl. Opt. 6, 1739-1748 (1967). [CrossRef] [PubMed]
- J. Glückstad, "Phase contrast image synthesis," Opt. Commun. 130, 225-230 (1996). [CrossRef]
- J. Glückstad and P. C. Mogensen, "Optimal Phase Contrast in Common-Path Interferometry," Appl. Opt. 40, 268-282 (2001). [CrossRef]
- D. Gabor, "A new microscopic principle," Nature 161, 777- 778 (1948). [CrossRef] [PubMed]
- F. Zernike, "How I discovered phase contrast," Science 121, 345-349 (1955). [CrossRef] [PubMed]
- J. Glückstad, L. Lading, H. Toyoda, and T. Hara, "Lossless light projection," Opt. Lett. 22, 1373-1375 (1997). [CrossRef]
- P. J. Rodrigo, V. R. Daria, and J. Glückstad, "Real-time three-dimensional optical micromanipulation of multiple particles and living cells," Opt. Lett. 29 2270-2272 (2004). [CrossRef] [PubMed]
- P. Rodrigo, V. Daria, and J. Glückstad, "Dynamically reconfigurable optical lattices," Opt. Express 13, 1384-1394 (2005). [CrossRef] [PubMed]
- C. A. Alonzo, P. J. Rodrigo, and J. Glückstad, "Photon-efficient grey-level image projection by the generalized phase contrast method," New J. Phys. 9, 132 (2007). [CrossRef]
- J. Glückstad, D. Palima, P. J. Rodrigo, and C. A. Alonzo, "Laser projection using generalized phase contrast," Opt. Lett. 32, 3281-3283 (2007). [CrossRef] [PubMed]
- 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]
- V. Arrizon and M. Testorf, "Efficiency limit of spatially quantized Fourier array illuminators," Opt. Lett. 22, 197-199 (1997). [CrossRef] [PubMed]
- W. Lukosz, "Optical systems with resolving powers exceeding classical limit," J. Opt. Soc. Am. 56, 1463-1472 (1966). [CrossRef]
- I. J. Cox and C. J. R. Sheppard, "Information capacity and resolution in an optical system," J. Opt. Soc. Am. A 3, 1152-1158 (1986). [CrossRef]
- P. B. Fellgett and E. H. Linfoot, "On the assessment of optical images," Phil. Trans. R. Soc. London Ser. A 247, 369-407 (1955). [CrossRef]
- F. Wyrowski, "Upper bound of the diffraction efficiency of diffractive phase elements," Opt. Lett. 16, 1915-1917 (1991). [CrossRef] [PubMed]
- D. Palima and V. R. Daria, "Effect of spurious diffraction orders in arbitrary multifoci patterns produced via phase-only holograms," Appl. Opt. 45, 6689-6693 (2006). [CrossRef] [PubMed]
- A. J. Waddie and M. R. Taghizadeh, "Interference Effects in Far-Field Diffractive Optical Elements," Appl. Opt. 38, 5915-5919 (1999). [CrossRef]
- F. Wyrowski, "Diffractive optical elements: iterative calculation of quantized, blazed phase structures," J. Opt. Soc. Am. A 7, 961-969 (1990). [CrossRef]
- P. Senthilkumaran, F. Wyrowski, and H. Schimmel, "Vortex Stagnation problem in iterative Fourier transform algorithms," Opt. Laser Eng. 43, 43-56 (2005). [CrossRef]
- H. Aagedal, M. Schmid, T. Beth, S. Teiwes, and F. Wyrowski, "Theory of speckles in diffractive optics and its application to beam shaping," J. Mod. Opt. 43, 1409-1421 (1996). [CrossRef]
- R. W. Cohn, "Fundamental properties of spatial light modulators for the approximate optical computation of Fourier transforms: a review," Opt. Eng. 40, 2452-2463 (2001). [CrossRef]
- D. Palima and V. R. Daria, "Holographic projection of arbitrary light patterns with a suppressed zero-order beam," Appl. Opt. 46, 4197-4201 (2007). [CrossRef] [PubMed]
- V. Arrizón, E. Carreón, and M. Testorf, "Implementation of Fourier array illuminators using pixelated SLM: efficiency limitations," Opt. Commun. 160, 207-213 (1999). [CrossRef]
- M. L. Hsieh, K. Y. Hsu, E. G. Paek, and C. L. Wilson, "Modulation transfer function of a liquid crystal spatial light modulator," Opt. Commun. 170, 221-227 (1999). [CrossRef]
- E. Hällstig, T. Martin, L. Sjöqvist, and M. Lindgren, "Polarization properties of a nematic liquid-crystal spatial light modulator for phase modulation," J. Opt. Soc. Am. A 22, 177-184 (2005). [CrossRef]
- G. Moddel and L. Wang, "Resolution limits from charge transport in optically addressed spatial light modulators," J. Appl. Phys. 78, 6923-6935 (1995). [CrossRef]
- M. Duelli, L. Ge, and R. W. Cohn, "Nonlinear effects of phase blurring on Fourier transform holograms," J. Opt. Soc. Am. A 17, 1594-1605 (2000). [CrossRef]
- A. Márquez, C. Iemmi, I. Moreno, J. Campos, and M. Yzuel, "Anamorphic and spatial frequency dependent phase modulation on liquid crystal displays. Optimization of the modulation diffraction efficiency," Opt. Express 13, 2111-2119 (2005). [CrossRef] [PubMed]

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