## Spatial cross modulation method using a random diffuser and phase-only spatial light modulator for constructing arbitrary complex fields |

Optics Express, Vol. 22, Issue 4, pp. 3968-3982 (2014)

http://dx.doi.org/10.1364/OE.22.003968

Acrobat PDF (5141 KB)

### Abstract

We propose a spatial cross modulation method using a random diffuser and a phase-only spatial light modulator (SLM), by which arbitrary complex-amplitude fields can be generated with higher spatial resolution and diffraction efficiency than off-axis and double-phase computer-generated holograms. Our method encodes the original complex object as a phase-only diffusion image by scattering the complex object using a random diffuser. In addition, all incoming light to the SLM is consumed for a single diffraction order, making a diffraction efficiency of more than 90% possible. This method can be applied for holographic data storage, three-dimensional displays, and other such applications.

© 2014 Optical Society of America

## 1. Introduction

1. A. W. Lohmann and D. P. Paris, “Binary Fraunhofer holograms, generated by computer,” Appl. Opt. **6**(10), 1739–1748 (1967). [CrossRef] [PubMed]

2. A. W. Lohmann and D. P. Paris, “Computer generated spatial filters for coherent optical data processiing,” Appl. Opt. **7**(4), 651–655 (1968). [CrossRef] [PubMed]

3. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. **11**(5), 288–290 (1986). [CrossRef] [PubMed]

5. Y. Sando, M. Itoh, and T. Yatagai, “Holographic three-dimensional display synthesized from three-dimensional Fourier spectra of real existing objects,” Opt. Lett. **28**(24), 2518–2520 (2003). [CrossRef] [PubMed]

6. A. J. MacGovern and J. C. Wyant, “Computer generated holograms for testing optical elements,” Appl. Opt. **10**(3), 619–624 (1971). [CrossRef] [PubMed]

7. V. Arrizón, G. Méndez, and D. Sánchez-de-La-Llave, “Accurate encoding of arbitrary complex fields with amplitude-only liquid crystal spatial light modulators,” Opt. Express **13**(20), 7913–7927 (2005). [CrossRef] [PubMed]

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

10. Z. Göröcs, G. Erdei, T. Sarkadi, F. Ujhelyi, J. Reményi, P. Koppa, and E. Lorincz, “Hybrid multinary modulation using a phase modulating spatial light modulator and a low-pass spatial filter,” Opt. Lett. **32**(16), 2336–2338 (2007). [CrossRef] [PubMed]

11. L. B. Lesem, P. M. Hirch, and J. A. Jordan Jr., “The kinoform: A new wavefront reconstruction device,” IBM J. Res. Develop. **13**(2), 150–155 (1969). [CrossRef]

12. A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE **69**(5), 529–541 (1981). [CrossRef]

13. J. L. Horner and P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. **23**(6), 812–816 (1984). [CrossRef] [PubMed]

14. M. Stanley, M. A. Smith, A. P. Smith, P. J. Watson, S. D. Coomber, C. D. Cameron, C. W. Slinger, and A. Wood, “3D electronic holography display system using a 100Mega-pixel spatial light modulator,” Proc. SPIE **5249**, 297–308 (2004). [CrossRef]

15. E. N. Leith, A. Kozma, J. Upatnieks, J. Marks, and N. Massey, “Holographic data storage in three-dimensional media,” Appl. Opt. **5**(8), 1303–1311 (1966). [CrossRef] [PubMed]

17. M. Takabayashi and A. Okamoto, “Self-referential holography and its applications to data storage and phase-to-intensity conversion,” Opt. Express **21**(3), 3669–3681 (2013). [CrossRef] [PubMed]

18. K. Zukeran, A. Okamoto, M. Takabayashi, A. Shibukawa, K. Sato, and A. Tomita, “Double-referential holography and spatial quadrature amplitude modulation,” Jpn. J. Appl. Phys. **52**(9S2), 09LD13 (2013). [CrossRef]

## 2. Spatial cross modulation

### 2.1. Basic operation

*f*optical system, but this configuration of the SCMM is only one possibility. The SCMM can be also realized by a simpler configuration where the 4

*f*optical system is removed and only the diffuser lies between the input (reconstruction) and output (SLM) planes.

*A*(

*x*,

*y*)exp[

*jφ*(

*x,y*)] is first prepared in an input plane in order to be reconstructed in the subsequent decode step. The complex object is not restricted to either two or three dimensions, but the following explanation proceeds from a two-dimensional (2D) object. Next, the Fourier transformation of the object

*F*{

*A*(

*x*,

*y*)exp[

*jφ*(

*x*,

*y*)]} is calculated with a fast Fourier transform (FFT). Here,

*F*{·} denotes the operator of two-dimensional Fourier transform. Then, the spatial phase distribution of a digital random diffuser (a virtual random diffuser) exp[

*jh*(

_{d}*x*,

*y*)] and the spatial spectrum of the object

*F*{

*A*(

*x*,

*y*)exp[

*jφ*(

*x*,

*y*)]} are multiplied in a spatial Fourier plane. Through an inverse Fourier transformation of the multiplied spectrum distribution

*F*

^{−1}{exp[

*jh*(

*x*,

*y*)]·

*F*{

*A*(

*x*,

*y*)exp[

*jφ*(

*x*,

*y*)]}} by inverse FFT (IFFT), a diffusion image (scattered wavefront) of the original object

*S*(

*x*,

*y*)exp[

*jξ*(

*x*,

*y*)] results within an output plane. Subsequently, amplitude distribution in the diffusion image

*S*(

*x*,

*y*) are uniformed and the phase conjugation exp[-

*jξ*(

*x*,

*y*)] is calculated, which is mathematically equivalent to inverting the sign of the phase distribution in the image. The elimination of the amplitude is based on the assumption that the scattered phase contains more of the important features of the original complex object than the scattered amplitude does. In other words, both the original amplitude

*A*(

*x*,

*y*) and the original phase exp[

*jφ*(

*x*,

*y*)] are spatially cross-modulated into a phase-only diffusion image exp[

*jξ*(

*x*,

*y*)]. This importance of the phase has been demonstrated in a number of different contexts, including optical and acoustical holograms [12

12. A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE **69**(5), 529–541 (1981). [CrossRef]

13. J. L. Horner and P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. **23**(6), 812–816 (1984). [CrossRef] [PubMed]

*jξ*(

*x*,

*y*)] is referred to as cross-modulated image herein. In regard to the complexity of the calculations, the above encoding step includes 2D FFT, 2D IFFT, and the multiplication of 2D distributions. The total complexity is therefore given by 2

*N*

^{2}log

_{2}

*N*+

*N*

^{2}, where the SLM has

*N*

^{2}pixels in the

*x*-

*y*plane. If a three-dimensional (3D) object

*A*(

*x*,

*y*,

*z*)exp[

*jφ*(

*x*,

*y*,

*z*)] is treated with the SCMM, the 3D object

*A*(

*x*,

*y*,

*z*)exp[

*jφ*(

*x*,

*y*,

*z*)] is prepared as an assembly of many 2D images on the input plane. Each 2D image

*A*(

_{n}*x*,

*y*,

*z*)exp[

_{n}*jφ*(

_{n}*x*,

*y*,

*z*)] is changed to a cross-modulated image exp[-

_{n}*jξ*(

_{n}*x*,

*y*)] with the above encode step, and then each cross-modulated image exp[-

*jξ*(

_{n}*x*,

*y*)] is superimposed on the output plane to obtain the final cross-modulated image exp[-

*jξ*(

*x*,

*y*)] to be displayed on the SLM.

*jξ*(

*x*,

*y*)] calculated in step (a) is first displayed onto a phase-only SLM. After incident light falls on the SLM, the modulated light is Fourier-transformed via the first lens. Then, the light

*F*{exp[-

*jξ*(

*x*,

*y*)]} propagates through the optical diffuser with a phase distribution exp[

*jh*(

_{o}*x*,

*y*)], and the reconstructed complex object

*A'*(

*x*,

*y*)exp[

*jφ'*(

*x*,

*y*)] is obtained with a second lens. Note that the reconstructed object

*A'*(

*x*,

*y*)exp[

*jφ'*(

*x*,

*y*)] is extremely similar to the original object

*A*(

*x*,

*y*)exp[

*jφ*(

*x*,

*y*)] but have slightly random noise due to the elimination of the scattered amplitude. The spatial phase distributions exp[

*jh*(

_{d}*x*,

*y*)] and exp[

*jh*(

_{o}*x*,

*y*)] of the digital and optical diffusers must be matched for correctly working the SCMM. Two ways for matching these distributions are possible: one measures the distribution of the optical diffuser by a phase detection method for the decoding and subsequently uses the measured phase distribution as the digital diffuser in the encode step; another uses an electrically addressable phase-only SLM for the optical diffuser. The latter method appears to be more practical.

*f*system. This feature increases the degree of freedom for designing the optical system, so allows larger objects to be reconstructed as compared to the kinoform based on a Fourier transform lens. In addition, the kinoform adds a random phase into the object to make it diffusive for the smooth Fourier spectrum, and so the reconstructed object includes speckle noise. On contrast, the SCMM places the diffuser with the random phase outside the object, which eliminates most of speckle noises in the reproduced object if the object to be prepared in the computer has homogeneous surface. In common with the kinoform, this method can use all the incoming light energy for the desired complex object, so the resulting diffraction efficiency is much higher than that obtained with conventional CGHs.

### 2.2. Simulation

#### 2.2.1. Models and flows

*f*imaging system. In the simulation flow for the double-phase hologram in Fig. 3(b), the decomposition of the given complex value

*A*(

*x*,

*y*)exp[

*jφ*(

*x,y*)] into two phase

*θ*

_{1}and

*θ*

_{2}is calculated as follows: In the simulation flow for the off-axis amplitude hologram in Fig. 3(c), the amplitude hologram

*I*(

*x*,

*y*) to be displayed on the SLM is given by:where

*I*

_{0}is the bias component to keep the cosine term positive and

*θ*is the incident angle of the off-axis reference plane wave interfering with the object.

_{in}*N*×

_{dx}*N*SLM pixels where

_{dy}*N*and

_{dx}*N*are the number of data pixels along the

_{dy}*x*- and

*y*-axis, respectively. Then, the cross-modulated image is calculated by spreading the original image over a wider region with the diffuser at a diffusion ratio

*N*and composed of

_{diff}*N*×

_{dx}N_{diff}*N*SLM pixels. Here, the diffusion ratio

_{dy}N_{diff}*N*is defined as the ratio of the original and diffusion image (encoded image) sizes and it is calculated as follows:Here, arccosine term represents the maximum angular spectrum of the original complex image,

_{diff}*θ*is the diffusion angle of the diffuser,

_{diff}*L*is the pitch of the data pixels, and

_{dx}*L*is the focal length of the Fourier transform lens. The entire area on the SLM is expressed as

_{f}*N*×

_{dx}N_{0}*N*where

_{dy}N_{0}*N*is zero padding rate for the extra area that spreads out the original complex image. Here, if

_{0}*N*is equal to

_{diff}*N*, the diffusion image (encoded image) occupies the entire area on the SLM. On the SLM plane for the double-phase hologram method in Fig. 4(b), a phase image, similar to a chessboard pattern, is composed of

_{0}*N*×

_{dx}N_{ux}*N*pixels and it is displayed on the SLM. Here,

_{dy}N_{uy}*N*and

_{ux}*N*are the number of subpixels (SLM pixel) along the

_{uy}*x*- and

*y*-axis, respectively. In this method, one data pixel is expressed as an assembly of alternately arranged subpixels (SLM pixel). On the SLM plane for an off-axis amplitude hologram, represented in Fig. 4(c), the amplitude hologram consists of

*N*×

_{dx}N_{s}N_{g}*N*pixels, where

_{dy}N_{s}N_{g}*N*is the number of SLM pixels per grating period along the

_{s}*x-*or

*y*-axis and

*N*is the number of gratings within one data pixel along

_{g}*x-*or

*y*-axis. The region of one data pixel then includes

*N*×

_{s}N_{g}*N*SLM pixels.

_{s}N_{g}*N*= 8. This image shows that the original image spreads across wide area and the original pixel structure totally collapses. Figure 5(b) shows a chessboard-type phase image with

_{diff}*N*×

_{ux}*N*= 8 × 8. In this pattern, it can be observed that two phase values,

_{uy}*θ*

_{1}and

*θ*

_{2}, decomposed from a given complex value, are alternately arranged.

*N*is 4 and

_{s}*N*is 2. It can be seen that one grating period consists of four SLM pixels and within one data pixel there are two gratings including totally 8 × 8 SLM pixels.

_{g}19. L. G. Shirley and N. George, “Wide-angle diffuser transmission functions and far-zone speckle,” J. Opt. Soc. Am. A **4**(4), 734–745 (1987). [CrossRef]

*t*(

*x*,

*y*) of the diffuser is represented aswhere

*n*is the refractive index of the diffuser,

_{diff}*λ*is the laser wavelength, and

*h*(

*x*,

*y*) is the diffuser surface profile. Details for calculating the diffuser profile are described in ref [19

19. L. G. Shirley and N. George, “Wide-angle diffuser transmission functions and far-zone speckle,” J. Opt. Soc. Am. A **4**(4), 734–745 (1987). [CrossRef]

#### 2.2.2. Results

*N*. However, note that, when increasing diffusion ratio

_{diff}*N*, the achievable spatial resolution of the image is to be low.

_{diff}*N*. Here, to evaluate the quality of reproduced complex images, we use signal to noise ratio (SNR) defined as follows:where (

_{diff}*mL*,

_{dx}*nL*) denotes the spatial pixel coordinate in

_{dy}*x*-

*y*plane,

*O*(

*mL*,

_{dx}*nL*) denotes amplitude or phase distributions of the original image, and

_{dy}*R*(

*mL*,

_{dx}*nL*) denotes amplitude or phase distribution of the reproduced image. The phase distribution is expressed in the range [0, 2π] and the reproduced phase distribution is not phase-unwrapped. In Fig. 7, it is shown that the SNR of the amplitude image is improved with increasing the diffusion ratio

_{dy}*N*, as expected. The reason for this is that with increasing

_{diff}*N*, more information from the original amplitudes is transferred to the scattered-phase wavefront. It is also interesting to note that the SNR of the phase decreases as

_{diff}*N*reaches 3 and then turns to increase from

_{diff}*N*= 3. We think that this behavior is due to the homogeneity in the scattered amplitude distribution. In particular, while

_{diff}*N*is low, the scattered amplitude distribution is highly inhomogeneous and the resulting gap between the true value in the scattered amplitude and the approximated value of the constant amplitude deteriorates the phase image. On the other hand, while

_{diff}*N*is high, since the inhomogeneity in the scattered amplitude is reduced and the gap described above appears to be small, the SNR of the phase image is improved. For another possible explanation for a the SNR decay, the original phase information may have been transferred to the scattered amplitude by the diffuser. However, if this reason is true, the SNR of the phase should continue to decrease when

_{diff}*N*is greater than 3.0, and so we can rule it out.

_{diff}### 2.3. Application to holographic data storage

18. K. Zukeran, A. Okamoto, M. Takabayashi, A. Shibukawa, K. Sato, and A. Tomita, “Double-referential holography and spatial quadrature amplitude modulation,” Jpn. J. Appl. Phys. **52**(9S2), 09LD13 (2013). [CrossRef]

#### 2.3.1. Basic operation

*A*(

*x*,

*y*)exp[

*jφ*(

*x,y*)] is diffused by a digital random diffuser with the phase distribution exp[

*jh*(

_{d}*x*,

*y*)], as described in Fig. 1(a). Then, the diffused amplitude distribution

*S*(

*x*,

*y*) is made uniform based on the cross modulation effect of the diffuser to obtain the phase-only diffusion (cross-modulated) image exp[-

*jξ*(

*x*,

*y*)]. In the optical hologram recording/reading step in Fig. 11(b), the phase-only image exp[-

*jξ*(

*x*,

*y*)] made during step (a) is projected onto the phase-only SLM and is recorded as a hologram in the same manner as a conventional HDS system. The conventional HDS system directly records the binary amplitude signal in a recording medium, whereas in the new HDS system with the SCMM, the phase-only image exp[-

*jξ*(

*x*,

*y*)] including most of features about the SQAM signal

*A*(

*x*,

*y*)exp[

*jφ*(

*x,y*)] is recorded alternatively in the medium. During reading, cross-modulated image exp[-

*jξ'*(

*x*,

*y*)] is read out from the recorded hologram and is measured through the phase detection method. Here, if we assume that the holographic recording/readout is distortion-free, the image exp[-

*jξ*(

*x*,

*y*)] displayed on the SLM and the image exp[-

*jξ'*(

*x*,

*y*)] measured by the phase detection method will be the same. In the digital phase conjugate reconstruction step (Decode step) in Fig. 11(c), the detected phase-only diffusion image exp[-

*jξ'*(

*x*,

*y*)] is passed through the digital diffuser with the phase distribution exp[

*jh*(

_{d}*x*,

*y*)] again in the phase conjugate optical system of Fig. 11(a). The recovered SQAM signal

*A'*(

*x*,

*y*)exp[

*jφ'*(

*x,y*)] is then obtained from the phase-only image.

#### 2.3.2. Experimental setup and results

*N*= 4.0, is shown in Fig. 14(a).It is displayed on a phase-only LCoS-SLM in Fig. 12. The diffusion image, modulated by the SLM, propagates through a recording medium. The phase-only diffusion image is then detected by a holographic diversity interferometer (HDI) composed of two CCDs [20

_{diff}20. A. Okamoto, K. Kunori, M. Takabayashi, A. Tomita, and K. Sato, “Holographic diversity interferometry for optical storage,” Opt. Express **19**(14), 13436–13444 (2011). [CrossRef] [PubMed]

*N*. This result is similar to the simulated result in Fig. 7.

_{diff}## 3. Conclusion

## References and links

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

2. | A. W. Lohmann and D. P. Paris, “Computer generated spatial filters for coherent optical data processiing,” Appl. Opt. |

3. | A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. |

4. | V. Bagnoud and J. D. Zuegel, “Independent phase and amplitude control of a laser beam by use of a single-phase-only spatial light modulator,” Opt. Lett. |

5. | Y. Sando, M. Itoh, and T. Yatagai, “Holographic three-dimensional display synthesized from three-dimensional Fourier spectra of real existing objects,” Opt. Lett. |

6. | A. J. MacGovern and J. C. Wyant, “Computer generated holograms for testing optical elements,” Appl. Opt. |

7. | V. Arrizón, G. Méndez, and D. Sánchez-de-La-Llave, “Accurate encoding of arbitrary complex fields with amplitude-only liquid crystal spatial light modulators,” Opt. Express |

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

9. | J. M. Florence and R. D. Juday, “Full complex spatial filtering with a phase mostly DMD,” Proc. SPIE |

10. | Z. Göröcs, G. Erdei, T. Sarkadi, F. Ujhelyi, J. Reményi, P. Koppa, and E. Lorincz, “Hybrid multinary modulation using a phase modulating spatial light modulator and a low-pass spatial filter,” Opt. Lett. |

11. | L. B. Lesem, P. M. Hirch, and J. A. Jordan Jr., “The kinoform: A new wavefront reconstruction device,” IBM J. Res. Develop. |

12. | A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE |

13. | J. L. Horner and P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. |

14. | M. Stanley, M. A. Smith, A. P. Smith, P. J. Watson, S. D. Coomber, C. D. Cameron, C. W. Slinger, and A. Wood, “3D electronic holography display system using a 100Mega-pixel spatial light modulator,” Proc. SPIE |

15. | E. N. Leith, A. Kozma, J. Upatnieks, J. Marks, and N. Massey, “Holographic data storage in three-dimensional media,” Appl. Opt. |

16. | K. Anderson and K. Curtis, “Polytopic multiplexing,” Opt. Lett. |

17. | M. Takabayashi and A. Okamoto, “Self-referential holography and its applications to data storage and phase-to-intensity conversion,” Opt. Express |

18. | K. Zukeran, A. Okamoto, M. Takabayashi, A. Shibukawa, K. Sato, and A. Tomita, “Double-referential holography and spatial quadrature amplitude modulation,” Jpn. J. Appl. Phys. |

19. | L. G. Shirley and N. George, “Wide-angle diffuser transmission functions and far-zone speckle,” J. Opt. Soc. Am. A |

20. | A. Okamoto, K. Kunori, M. Takabayashi, A. Tomita, and K. Sato, “Holographic diversity interferometry for optical storage,” Opt. Express |

**OCIS Codes**

(090.1760) Holography : Computer holography

(090.1970) Holography : Diffractive optics

(210.2860) Optical data storage : Holographic and volume memories

(110.7348) Imaging systems : Wavefront encoding

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: December 26, 2013

Revised Manuscript: February 7, 2014

Manuscript Accepted: February 7, 2014

Published: February 12, 2014

**Citation**

Atsushi Shibukawa, Atsushi Okamoto, Masanori Takabayashi, and Akihisa Tomita, "Spatial cross modulation method using a random diffuser and phase-only spatial light modulator for constructing arbitrary complex fields," Opt. Express **22**, 3968-3982 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-4-3968

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### References

- A. W. Lohmann, D. P. Paris, “Binary Fraunhofer holograms, generated by computer,” Appl. Opt. 6(10), 1739–1748 (1967). [CrossRef] [PubMed]
- A. W. Lohmann, D. P. Paris, “Computer generated spatial filters for coherent optical data processiing,” Appl. Opt. 7(4), 651–655 (1968). [CrossRef] [PubMed]
- A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11(5), 288–290 (1986). [CrossRef] [PubMed]
- V. Bagnoud, J. D. Zuegel, “Independent phase and amplitude control of a laser beam by use of a single-phase-only spatial light modulator,” Opt. Lett. 29(3), 295–297 (2004). [CrossRef] [PubMed]
- Y. Sando, M. Itoh, T. Yatagai, “Holographic three-dimensional display synthesized from three-dimensional Fourier spectra of real existing objects,” Opt. Lett. 28(24), 2518–2520 (2003). [CrossRef] [PubMed]
- A. J. MacGovern, J. C. Wyant, “Computer generated holograms for testing optical elements,” Appl. Opt. 10(3), 619–624 (1971). [CrossRef] [PubMed]
- V. Arrizón, G. Méndez, D. Sánchez-de-La-Llave, “Accurate encoding of arbitrary complex fields with amplitude-only liquid crystal spatial light modulators,” Opt. Express 13(20), 7913–7927 (2005). [CrossRef] [PubMed]
- C. K. Hsueh, A. A. Sawchuk, “Computer-generated double-phase holograms,” Appl. Opt. 17(24), 3874–3883 (1978). [CrossRef] [PubMed]
- J. M. Florence, R. D. Juday, “Full complex spatial filtering with a phase mostly DMD,” Proc. SPIE 1558, 487–498 (1991). [CrossRef]
- Z. Göröcs, G. Erdei, T. Sarkadi, F. Ujhelyi, J. Reményi, P. Koppa, E. Lorincz, “Hybrid multinary modulation using a phase modulating spatial light modulator and a low-pass spatial filter,” Opt. Lett. 32(16), 2336–2338 (2007). [CrossRef] [PubMed]
- L. B. Lesem, P. M. Hirch, J. A. Jordan., “The kinoform: A new wavefront reconstruction device,” IBM J. Res. Develop. 13(2), 150–155 (1969). [CrossRef]
- A. V. Oppenheim, J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69(5), 529–541 (1981). [CrossRef]
- J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23(6), 812–816 (1984). [CrossRef] [PubMed]
- M. Stanley, M. A. Smith, A. P. Smith, P. J. Watson, S. D. Coomber, C. D. Cameron, C. W. Slinger, A. Wood, “3D electronic holography display system using a 100Mega-pixel spatial light modulator,” Proc. SPIE 5249, 297–308 (2004). [CrossRef]
- E. N. Leith, A. Kozma, J. Upatnieks, J. Marks, N. Massey, “Holographic data storage in three-dimensional media,” Appl. Opt. 5(8), 1303–1311 (1966). [CrossRef] [PubMed]
- K. Anderson, K. Curtis, “Polytopic multiplexing,” Opt. Lett. 29(12), 1402–1404 (2004). [CrossRef] [PubMed]
- M. Takabayashi, A. Okamoto, “Self-referential holography and its applications to data storage and phase-to-intensity conversion,” Opt. Express 21(3), 3669–3681 (2013). [CrossRef] [PubMed]
- K. Zukeran, A. Okamoto, M. Takabayashi, A. Shibukawa, K. Sato, A. Tomita, “Double-referential holography and spatial quadrature amplitude modulation,” Jpn. J. Appl. Phys. 52(9S2), 09LD13 (2013). [CrossRef]
- L. G. Shirley, N. George, “Wide-angle diffuser transmission functions and far-zone speckle,” J. Opt. Soc. Am. A 4(4), 734–745 (1987). [CrossRef]
- A. Okamoto, K. Kunori, M. Takabayashi, A. Tomita, K. Sato, “Holographic diversity interferometry for optical storage,” Opt. Express 19(14), 13436–13444 (2011). [CrossRef] [PubMed]

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