## Optimal synthesis of double-phase computer generated holograms using a phase-only spatial light modulator with grating filter |

Optics Express, Vol. 20, Issue 28, pp. 29844-29853 (2012)

http://dx.doi.org/10.1364/OE.20.029844

Acrobat PDF (2535 KB)

### Abstract

We propose an optical system for synthesizing double-phase complex computer-generated holograms using a phase-only spatial light modulator and a phase grating filter. Two separated areas of the phase-only spatial light modulator are optically superposed by 4-*f* configuration with an optimally designed grating filter to synthesize arbitrary complex optical field distributions. The tolerances related to misalignment factors are analyzed, and the optimal synthesis method of double-phase computer-generated holograms is described.

© 2012 OSA

## 1. Introduction

1. B. R. Brown and A. W. Lohmann, “Computer-generated binary holograms,” IBM J. Res. Develop. **13**(2), 160–168 (1969). [CrossRef]

3. L. G. Neto, D. Roberge, and Y. Sheng, “Full-range, continuous, complex modulation by the use of two coupled-mode liquid-crystal televisions,” Appl. Opt. **35**(23), 4567–4576 (1996). [CrossRef] [PubMed]

4. C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” IEEE. Computer **38**(8), 46–53 (2005). [CrossRef]

3. L. G. Neto, D. Roberge, and Y. Sheng, “Full-range, continuous, complex modulation by the use of two coupled-mode liquid-crystal televisions,” Appl. Opt. **35**(23), 4567–4576 (1996). [CrossRef] [PubMed]

2. C. K. Hsueh and A. A. Sawchuk, “Computer-generated double-phase holograms,” Appl. Opt. **17**(24), 3874–3883 (1978). [CrossRef] [PubMed]

5. S. Reichelt, R. Häussler, G. Fütterer, N. Leister, H. Kato, N. Usukura, and Y. Kanbayashi, “Full-range, complex spatial light modulator for real-time holography,” Opt. Lett. **37**(11), 1955–1957 (2012). [CrossRef] [PubMed]

9. J.-P. Liu, W.-Y. Hsieh, T.-C. Poon, and P. Tsang, “Complex Fresnel Hologram Display using a Single SLM,” Appl. Opt. **50**(34), H128–H135 (2011). [CrossRef] [PubMed]

2. C. K. Hsueh and A. A. Sawchuk, “Computer-generated double-phase holograms,” Appl. Opt. **17**(24), 3874–3883 (1978). [CrossRef] [PubMed]

5. S. Reichelt, R. Häussler, G. Fütterer, N. Leister, H. Kato, N. Usukura, and Y. Kanbayashi, “Full-range, complex spatial light modulator for real-time holography,” Opt. Lett. **37**(11), 1955–1957 (2012). [CrossRef] [PubMed]

7. V. Arrizón, “Complex modulation with a twisted-nematic liquid-crystal spatial light modulator: double-pixel approach,” Opt. Lett. **28**(15), 1359–1361 (2003). [CrossRef] [PubMed]

5. S. Reichelt, R. Häussler, G. Fütterer, N. Leister, H. Kato, N. Usukura, and Y. Kanbayashi, “Full-range, complex spatial light modulator for real-time holography,” Opt. Lett. **37**(11), 1955–1957 (2012). [CrossRef] [PubMed]

9. J.-P. Liu, W.-Y. Hsieh, T.-C. Poon, and P. Tsang, “Complex Fresnel Hologram Display using a Single SLM,” Appl. Opt. **50**(34), H128–H135 (2011). [CrossRef] [PubMed]

## 2. Two double-phase hologram configurations

*f*optical system and a cylindrical 4-

*f*optical system with a grating filter inserted in the Fourier domain, as illustrated in Figs. 1(a) and 1(b), respectively. A plane wave is normally incident on the phase-only SLM in the input plane, where two phase holograms,

*x*-axis cylindrical 4-

*f*system with a periodic grating filter inserted in the Fourier filter plane. The role of the grating filter is bidirectional translation of the SLM image along the ±

*x*-axis directions in the output plane by diffractive beam splitting, which leads to coherent superposition of the upper and lower phase holograms in the overlapped region in the center of the output plane [9

9. J.-P. Liu, W.-Y. Hsieh, T.-C. Poon, and P. Tsang, “Complex Fresnel Hologram Display using a Single SLM,” Appl. Opt. **50**(34), H128–H135 (2011). [CrossRef] [PubMed]

*f*system is constructed for both

*x*- and

*y*-directions. Under the ideal alignment condition, the accurate overlapping imaging of the DPH can be obtained, and as a result, the formation of a nearly perfect complex optical field is expected to be synthesized in the output plane of the 4-

*f*system. We will analyze the influence of the misalignment of the upper and lower phase holograms in the output plane and the vertical shift of the grating filter for the generation of holographic 3D images, and address the conditions for optimal performance of the system.

*f*DPH system with cylindrical lenses shown in Fig. 1(b) is examined comparatively. In the cylindrical 4-

*f*system, the two-dimensional optical transform of the system is separated by

*x*- and

*y*-dependent 1D transforms. For the

*x*-direction, the 1D 4-

*f*system is realized, but for the

*y*-direction, the system looks like 1D free space to the propagating optical field, and as a result, the

*y*-directional 1D Fresnel diffraction image of the DPH is formed in the output plane. In theory, the Fresnel diffraction can be viewed as the multiplication of the quadratic phase term or propagator function to the angular spectrum of the optical field, where the phase profile of the propagator function becomes highly oscillating in proportion to the propagation distance, meaning the Fresnel diffraction makes the phase profile structure of the optical field more complicated.

10. H. Kim and B. Lee, “Analytic design of an anamorphic optical system for generation anisotropic partially coherent Gaussian Schell-model beams,” Opt. Commun. **260**(2), 383–397 (2006). [CrossRef]

*f*systems are represented by the generalized Fresnel transform with separable kernel [10

10. H. Kim and B. Lee, “Analytic design of an anamorphic optical system for generation anisotropic partially coherent Gaussian Schell-model beams,” Opt. Commun. **260**(2), 383–397 (2006). [CrossRef]

*x*-dependent kernel

*x*-dependent inverse kernel

*y*-dependent inverse kernel

*f*system, the transform kernel of the first part is given by the multiplication of Eqs. (1b) and (1c). The optical field in the Fourier domain (grating plane) is multiplied by the transmittance function of the grating filter. The optical field modulated by the grating filter is transmitted to the output plane through the second part, which has the same composition as the first part. In the inverse transform of Eq. (2a), the multiplication of the grating transmittance function should be omitted, where the inverse kernel is composed of Eqs. (2b) and (2c). In the case of the cylindrical 4-

*f*system, the transform kernel of the first and second parts is the multiplication of Eqs. (1b) and (1d), and the corresponding inverse kernel is composed of Eqs. (2b) and (2d). Therefore, the total forward and inverse transformations used in the modeling of the DPH systems are respectively represented asandwhere

*x*-directional width,

*x*-directional shift of the center of the upper area from the origin (

*z*-directional optic axis. In this sense, the grating period,

*f*systems, respectively, and those for ‘SAIT’ are shown in Figs. 2(e) and 2(f), respectively. The second example ‘SAIT’ is dressed with a spherical phase profile to be correctly observed by a virtual camera with a 1cm-radius aperture located at z = 1m. The observation of the accommodation effect of the holographic 3D image ‘SAIT’ is presented and discussed in the next section.

## 3. Comparison of the complex field synthesis by two DPH configurations

*S*and

*N*indicate signal and noise areas in the output plane, respectively. The definition of SNR is appropriate for the evaluation of 2D optical field distribution, but a more sophisticatedly designed measure for quantitatively scoring the quality of holographic 3D image has to be investigated further. Here, the SNR measure of Eq. (7) is only used to evaluate the image quality of the first synthesis example of the ‘SAMSUNG’ 2D flat image in the output plane. As will be discussed in this section, the SNR analysis is used to analyze the influence of the system parameters,

*x*-directional) shift of the grating filter,

*a*) is the sign function that indicates 1 or −1 according to whether the real number

*a*is positive or negative. In practice, the sinusoidal grating profile has a negative and positive gray scale amplitude profile, which requires a composite structure of binary phase and gray scale amplitude modulation layers, while the binary grating can be simply fabricated by a periodic surface relief structure on transparent substrate, so the use of binary phase grating is practical. However, from a theoretical point of view, the comparison of two types of grating filters is valuable to understand the theoretical limit of the synthesis performance.

*f*system with sinusoidal grating filter, (ii) cylindrical 4-

*f*system with sinusoidal grating filter, (iii) spherical 4-

*f*system with binary grating filter, and (iv) cylindrical 4-

*f*system with binary grating filter. The maximum peak SNR indicates the optimal phase compensation parameter for specified translation misalignment of DPH. In the analysis, the SNR versus

*f*configuration, although the diffraction efficiency is relatively low. The use of binary grating is a practical choice, but the 0th order noise and higher-order stray noise are significant in the background of the obtained reconstructed images as indicated in Figs. 3(c) and 3(d) even in the optimal SNR condition. The maximum SNR is smaller than 50% (0.5), which poses a limitation of the image quality achieved using binary phase grating. The diffraction efficiency of the cylindrical 4-

*f*configuration is relatively lower than that of the spherical 4-

*f*system, which is similar to the case of the system with a sinusoidal grating filter.

11. H. Kim, J. Hahn, and B. Lee, “Mathematical modeling of triangle-mesh-modeled three-dimensional surface objects for digital holography,” Appl. Opt. **47**(19), D117–D127 (2008). [CrossRef] [PubMed]

*f*image is comparable to that of the spherical 4-

*f*system with respect to noise distribution, but its diffraction efficiency seems to be lower than that of the spherical 4-

*f*system.

*f*system is advantageous in the minimization and scalability of fabrication in the form of arrays. The configurations shown in Figs. 1(a) and 1(b) are the bulky optical system, where the SLM is supposed to have phase holograms composed of a large number of pixels. Within the total axial length, the

*y*-directional Fresnel diffraction for the cylindrical 4-

*f*system is considerable. For the downsizing of the system to the device level, wherein the SLM component only represents adjacent two pixels on a micrometer scale, the cylindrical 4-

*f*configuration is more acceptable than the spherical 4-

*f*configuration, since a cylindrical lens array is easier to fabricate than a spherical lens array.

## 4. Conclusion

*f*configuration with grating filters, and addressed the optimization method of the system parameters associated with misalignment. It has been shown that two separated phase holograms on a single phase-only SLM can be effectively combined with simple cylindrical 4-

*f*configuration with a binary phase grating filter, to generate the desired complex optical field that can yield correct holographic 3D images. This scheme is the most practical and feasible scheme for device fabrication. The next step is to design a realizable device-level architecture of DPH using cost-effective and scalable elements, such as lens arrays and phase gratings, without using expensive and non-scalable polarization sensitive devices. In the same direction, we will further investigate more simplified device structure without using lenses.

## Acknowledgment

## References and links

1. | B. R. Brown and A. W. Lohmann, “Computer-generated binary holograms,” IBM J. Res. Develop. |

2. | C. K. Hsueh and A. A. Sawchuk, “Computer-generated double-phase holograms,” Appl. Opt. |

3. | L. G. Neto, D. Roberge, and Y. Sheng, “Full-range, continuous, complex modulation by the use of two coupled-mode liquid-crystal televisions,” Appl. Opt. |

4. | C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” IEEE. Computer |

5. | S. Reichelt, R. Häussler, G. Fütterer, N. Leister, H. Kato, N. Usukura, and Y. Kanbayashi, “Full-range, complex spatial light modulator for real-time holography,” Opt. Lett. |

6. | E. Ulusoy, L. Onural, and H. M. Ozaktas, “Full-complex amplitude modulation with binary spatial light modulators,” J. Opt. Soc. Am. A |

7. | V. Arrizón, “Complex modulation with a twisted-nematic liquid-crystal spatial light modulator: double-pixel approach,” Opt. Lett. |

8. | M. M. M. Makowski, A. S. A. Siemion, I. D. I. Ducin, K. K. K. Kakarenko, M. S. M. Sypek, A. M. S. A. M. Siemion, J. S. J. Suszek, D. W. D. Wojnowski, Z. J. Z. Jaroszewicz, and A. K. A. Kolodziejczyk, “Complex light modulation for lensless image projection,” Chin. Opt. Lett. |

9. | J.-P. Liu, W.-Y. Hsieh, T.-C. Poon, and P. Tsang, “Complex Fresnel Hologram Display using a Single SLM,” Appl. Opt. |

10. | H. Kim and B. Lee, “Analytic design of an anamorphic optical system for generation anisotropic partially coherent Gaussian Schell-model beams,” Opt. Commun. |

11. | H. Kim, J. Hahn, and B. Lee, “Mathematical modeling of triangle-mesh-modeled three-dimensional surface objects for digital holography,” Appl. Opt. |

**OCIS Codes**

(090.2870) Holography : Holographic display

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

(230.3720) Optical devices : Liquid-crystal devices

(230.6120) Optical devices : Spatial light modulators

**ToC Category:**

Holography

**History**

Original Manuscript: November 12, 2012

Revised Manuscript: December 16, 2012

Manuscript Accepted: December 16, 2012

Published: December 21, 2012

**Citation**

Hoon Song, Geeyoung Sung, Sujin Choi, Kanghee Won, Hong-Seok Lee, and Hwi Kim, "Optimal synthesis of double-phase computer generated holograms using a phase-only spatial light modulator with grating filter," Opt. Express **20**, 29844-29853 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-28-29844

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### References

- B. R. Brown and A. W. Lohmann, “Computer-generated binary holograms,” IBM J. Res. Develop.13(2), 160–168 (1969). [CrossRef]
- C. K. Hsueh and A. A. Sawchuk, “Computer-generated double-phase holograms,” Appl. Opt.17(24), 3874–3883 (1978). [CrossRef] [PubMed]
- L. G. Neto, D. Roberge, and Y. Sheng, “Full-range, continuous, complex modulation by the use of two coupled-mode liquid-crystal televisions,” Appl. Opt.35(23), 4567–4576 (1996). [CrossRef] [PubMed]
- C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” IEEE. Computer38(8), 46–53 (2005). [CrossRef]
- S. Reichelt, R. Häussler, G. Fütterer, N. Leister, H. Kato, N. Usukura, and Y. Kanbayashi, “Full-range, complex spatial light modulator for real-time holography,” Opt. Lett.37(11), 1955–1957 (2012). [CrossRef] [PubMed]
- E. Ulusoy, L. Onural, and H. M. Ozaktas, “Full-complex amplitude modulation with binary spatial light modulators,” J. Opt. Soc. Am. A28(11), 2310–2321 (2011). [CrossRef] [PubMed]
- V. Arrizón, “Complex modulation with a twisted-nematic liquid-crystal spatial light modulator: double-pixel approach,” Opt. Lett.28(15), 1359–1361 (2003). [CrossRef] [PubMed]
- M. M. M. Makowski, A. S. A. Siemion, I. D. I. Ducin, K. K. K. Kakarenko, M. S. M. Sypek, A. M. S. A. M. Siemion, J. S. J. Suszek, D. W. D. Wojnowski, Z. J. Z. Jaroszewicz, and A. K. A. Kolodziejczyk, “Complex light modulation for lensless image projection,” Chin. Opt. Lett.9(12), 120008 (2011). [CrossRef]
- J.-P. Liu, W.-Y. Hsieh, T.-C. Poon, and P. Tsang, “Complex Fresnel Hologram Display using a Single SLM,” Appl. Opt.50(34), H128–H135 (2011). [CrossRef] [PubMed]
- H. Kim and B. Lee, “Analytic design of an anamorphic optical system for generation anisotropic partially coherent Gaussian Schell-model beams,” Opt. Commun.260(2), 383–397 (2006). [CrossRef]
- H. Kim, J. Hahn, and B. Lee, “Mathematical modeling of triangle-mesh-modeled three-dimensional surface objects for digital holography,” Appl. Opt.47(19), D117–D127 (2008). [CrossRef] [PubMed]

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