## Zone plate method for electronic holographic display using resolution redistribution technique |

Optics Express, Vol. 19, Issue 15, pp. 14707-14719 (2011)

http://dx.doi.org/10.1364/OE.19.014707

Acrobat PDF (1309 KB)

### Abstract

The resolution redistribution (RR) technique can increase the horizontal viewing-zone angle and screen size of electronic holographic display. The present study developed a zone plate method that would reduce hologram calculation time for the RR technique. This method enables calculation of an image displayed on a spatial light modulator by performing additions of the zone plates, while the previous calculation method required performing the Fourier transform twice. The derivation and modeling of the zone plate are shown. In addition, the look-up table approach was introduced for further reduction in computation time. Experimental verification using a holographic display module based on the RR technique is presented.

© 2011 OSA

## 1. Introduction

1. Y. Takaki and Y. Hayashi, “Increased horizontal viewing zone angle of a hologram by resolution redistribution of a spatial light modulator,” Appl. Opt. **47**(19), D6–D11 (2008). [CrossRef] [PubMed]

2. Y. Takaki and Y. Hayashi, “Elimination of conjugate image for holograms using a resolution redistribution optical system,” Appl. Opt. **47**(24), 4302–4308 (2008). [CrossRef] [PubMed]

3. Y. Takaki and Y. Tanemoto, “Modified resolution redistribution system for frameless hologram display module,” Opt. Express **18**(10), 10294–10300 (2010). [CrossRef] [PubMed]

^{−1}(λ/2

*p*), where

*p*is the pixel pitch of the SLM and λ is the wavelength of light. An extremely high resolution is required for an SLM to obtain a large screen size; the screen size is given by

*N*×

_{x}p*N*, where

_{y}p*N*×

_{x}*N*is the resolution of the SLM. Unlike two-dimensional (2D) displays, pixel pitch cannot be enlarged in order to increase the screen size. For instance, a holographic display with a viewing-zone angle of 30° and a screen size of 10 inches requires an SLM with a pixel pitch of 1 μm and a resolution of 203,000 × 152,000 pixels.

_{y}1. Y. Takaki and Y. Hayashi, “Increased horizontal viewing zone angle of a hologram by resolution redistribution of a spatial light modulator,” Appl. Opt. **47**(19), D6–D11 (2008). [CrossRef] [PubMed]

2. Y. Takaki and Y. Hayashi, “Elimination of conjugate image for holograms using a resolution redistribution optical system,” Appl. Opt. **47**(24), 4302–4308 (2008). [CrossRef] [PubMed]

3. Y. Takaki and Y. Tanemoto, “Modified resolution redistribution system for frameless hologram display module,” Opt. Express **18**(10), 10294–10300 (2010). [CrossRef] [PubMed]

1. Y. Takaki and Y. Hayashi, “Increased horizontal viewing zone angle of a hologram by resolution redistribution of a spatial light modulator,” Appl. Opt. **47**(19), D6–D11 (2008). [CrossRef] [PubMed]

2. Y. Takaki and Y. Hayashi, “Elimination of conjugate image for holograms using a resolution redistribution optical system,” Appl. Opt. **47**(24), 4302–4308 (2008). [CrossRef] [PubMed]

19. J. P. Waters, “Holographic image synthesis utilizing theoretical methods,” Appl. Phys. Lett. **9**(11), 405–407 (1966). [CrossRef]

20. G. L. Rogers, “Gabor diffraction microscopy: the hologram as a generalized zone-plate,” Nature **166**(4214), 237 (1950). [CrossRef] [PubMed]

21. W. J. Siemens-Wapniarski and M. P. Givens, “The experimental production of synthetic holograms,” Appl. Opt. **7**(3), 535–538 (1968). [CrossRef] [PubMed]

*f*optical system with a single-sideband filter made it possible to eliminate the conjugate image [6

6. T. Mishina, F. Okano, and I. Yuyama, “Time-alternating method based on single-sideband holography with half-zone-plate processing for the enlargement of viewing zones,” Appl. Opt. **38**(17), 3703–3713 (1999). [CrossRef] [PubMed]

22. O. Bryngdahl and A. Lohmann, “Single-sideband holography,” J. Opt. Soc. Am. **58**(5), 620–624 (1968). [CrossRef]

23. Y. Takaki and Y. Tanemoto, “Band-limited zone plates for single-sideband holography,” Appl. Opt. **48**(34), H64–H70 (2009). [CrossRef] [PubMed]

## 2. RR technique for holographic display

3. Y. Takaki and Y. Tanemoto, “Modified resolution redistribution system for frameless hologram display module,” Opt. Express **18**(10), 10294–10300 (2010). [CrossRef] [PubMed]

**47**(24), 4302–4308 (2008). [CrossRef] [PubMed]

**18**(10), 10294–10300 (2010). [CrossRef] [PubMed]

*K*, and the size of the Fourier transformed images is denoted by

*w*×

*w*. The multiple point light sources are arranged such that the Fourier transformed images are aligned with a horizontal pitch of

*w*and a vertical pitch of

*w*/2

*K*, as shown in Fig. 2. A horizontal slit with a height of

*w*/2

*K*is placed in the Fourier plane. The distribution passed through the horizontal slit has a width of

*Kw*and a height of

*w*/2

*K*. The reshaped distribution consists of different regions of the original Fourier transformed image so that the complex-amplitude of the reshaped distribution can be controlled arbitrarily. The width of the Fourier transformed image increases by

*K*times, and its height decreases by 2

*K*times in the Fourier plane. Therefore, the horizontal resolution in the image plane increases by

*K*times, and the vertical resolution decreases by 2

*K*times. When the pixel pitch of the SLM is denoted by

*p*and the magnification of the imaging system is denoted by

*M*, the horizontal pixel pitch in the image plane is given by

*Mp*/

*K*and the vertical pixel pitch is given by 2

*KMp*. By proper determination of the parameters

*K*and

*M*, both horizontal pixel pitch reduction and image size enlargement can be achieved. Therefore, both the horizontal viewing-zone angle and the screen size of electronic holographic display can be increased. Because the vertical pixel pitch increases, the RR technique can be applied to horizontal-parallax-only (HPO) holography [24

24. D. Leseberg, “Computer-generated holograms: display using one-dimensional transforms,” J. Opt. Soc. Am. A **3**(11), 1846–1851 (1986). [CrossRef]

**47**(24), 4302–4308 (2008). [CrossRef] [PubMed]

*Mp*/

*K*and a vertical pitch of 2

*KMp*. Then, the inverse Fourier transform is performed. The inverse-Fourier transformed image, which has a width of

*Kw*and a height of

*w*/2

*K*, is split into

*K*regions in the horizontal direction. The split distributions are rearranged in the vertical direction and the complex conjugate and symmetric distribution of the rearranged distribution is added. The synthesized distribution, with size

*w*×

*w*, is the Fourier transform of the distribution displayed on the SLM. Finally, the second inverse Fourier transform is performed to obtain the distribution displayed on the SLM. The addition of the complex conjugate and symmetric distribution in the Fourier plane makes the SLM image a real-valued distribution. A constant distribution is added to the obtained SLM image to make it positive real so that the image can be displayed on the amplitude-modulated SLM. For electronic holographic displays that use the RR technique, Fourier transform must be performed twice to calculate the distribution displayed on the SLM, in addition to calculating the object wave.

## 3. Theory

### 3.1. Zone plate method

19. J. P. Waters, “Holographic image synthesis utilizing theoretical methods,” Appl. Phys. Lett. **9**(11), 405–407 (1966). [CrossRef]

21. W. J. Siemens-Wapniarski and M. P. Givens, “The experimental production of synthetic holograms,” Appl. Opt. **7**(3), 535–538 (1968). [CrossRef] [PubMed]

*f*optical system with a single-sideband filter [22

22. O. Bryngdahl and A. Lohmann, “Single-sideband holography,” J. Opt. Soc. Am. **58**(5), 620–624 (1968). [CrossRef]

6. T. Mishina, F. Okano, and I. Yuyama, “Time-alternating method based on single-sideband holography with half-zone-plate processing for the enlargement of viewing zones,” Appl. Opt. **38**(17), 3703–3713 (1999). [CrossRef] [PubMed]

7. T. Mishina, M. Okui, and F. Okano, “Viewing-zone enlargement method for sampled hologram that uses high-order diffraction,” Appl. Opt. **41**(8), 1489–1499 (2002). [CrossRef] [PubMed]

**47**(19), D6–D11 (2008). [CrossRef] [PubMed]

**18**(10), 10294–10300 (2010). [CrossRef] [PubMed]

25. K. Matsushima, “Computer-generated holograms for three-dimensional surface objects with shade and texture,” Appl. Opt. **44**(22), 4607–4614 (2005). [CrossRef] [PubMed]

## 3.2. Derivation of zone plate

*f*(

*x*,

_{i}*y*), where

_{i}*x*and

_{i}*y*are the coordinates of the image plane. The inverse-Fourier transformed image of

_{i}*f*(

*x*,

_{i}*y*) is represented by

_{i}*F*(

*u*,

*v*), where

*u*and

*v*are the spatial frequencies in the Fourier plane. As shown in Fig. 3, the inverse-Fourier transformed image is horizontally split into

*K*rectangular regions, which are then rearranged vertically. Figure 5 depicts the realignment of the

*j*-th rectangular region (−

*K*/2 ≤

*j*≤

*K*/2 − 1). The positions of the rectangular regions before and after the realignment are designated by

*u*= (

_{s}*j*+ 1/2)

*w*and

*v*= (

_{d}*K*/2 −

*j −*1/2)(

*w*/2

*K*). The distribution after the realignment of the

*j*-th rectangular region is given by

*G*(

_{j}*u*,

*v*) gives the SLM display pattern

*g*(

_{j}*x*,

*y*), which generates the

*j*-th rectangular region of the Fourier plane, where

*x*=

*x*/

_{i}*M*and

*y = y*/

_{i}*M*are the coordinates of the SLM plane:

*w*= 1/

*p*, sinc[

*w*(

*x*−

*ξ*)] = sinc[(

*x*−

*ξ*)/

*p*], which becomes zero with an interval of

*p*except at

*x*=

*ξ*. Therefore, sinc[

*w*(

*x*−

*ξ*)] ≃

*p δ*(

*x*−

*ξ*) at the SLM pixel positions. With this approximation, Eq. (2) becomes

*f*(

*x*,

_{i}*y*) is a 1D spherical wave because the RR technique is applied to HPO holography. Therefore, the 1D spherical wave that generates an object point at the distance of

_{i}*z*is given by

*u*and

_{s}*v*are represented by using the region number

_{d}*j*:

*x*-direction and that of an inclined plane wave in the

*y*-direction. The horizontal phase distribution

*ϕ*= −

_{x}*π*[

*Mx*+ λ

*z*(

*j*+ 1/2)/

*Mp*]

^{2}/λ

*z*needs to satisfy the sampling theorem, i.e.,

*j*-th region of the Fourier plane has a finite width in the SLM display plane.

*z*/

*M*

^{2}

*p*. To generate all rectangular regions of the Fourier plane,

*K*distributions with a width of λ

*z*/

*M*

^{2}

*p*are aligned with a pitch of λ

*z*/

*M*

^{2}

*p*in the horizontal direction in the SLM display plane. Finally, the zone plate for the RR technique is given by the sum of all distributions:

*K*regions. Each region has a spherical phase distribution

*ϕ*in the horizontal direction, whose center of curvature lies at the center of each region. Each region has a phase distribution of an inclined plane wave in the vertical direction,

_{x}*ϕ*=

_{y}*π*(

*K*/2−

*j*−1/2)

*y*/

*Kp*, whose inclination angle depends on the region number

*j*. The amplitude of the derived zone plate, represented by sinc (

*y*/2

*Kp*), varies in the vertical direction. Figure 7(a) shows an example of the derived zone plate when

*K*= 4. A constant amplitude was added to make the distribution non-negative.

### 3.3. Modeling of zone plate

*π*times that of the main lobe. The height of the modeled zone plate becomes 4

*Kp*. Figure 7(b) shows an example of the modeled zone plate when

*K*= 4.

*R*, the size of the look-up table becomes

*R*times larger.

## 4. Experiments

### 4.1. Newly developed hologram display module

*K*= 4). A polarization beam splitter (PBS) reflects light from the optical fiber array. A combination lens works as a condenser lens as well as an imaging lens. Multiple plane waves illuminate a reflection-type SLM. A liquid-crystal-on-silicon SLM with a resolution of 4,096 × 2,400 and a pixel pitch of 4.8 μm was used. The SLM performed amplitude modulation. The light reflected and modulated by the SLM passes through the combined lens and the PBS to form an image on the screen lens. The focal length of the combined lens was 38.4 mm and the magnification of the imaging system was

*M*= −2.88. The central 3,200 × 1,800 pixels of the SLM were used considering the imaging properties of the combined lens so that the screen size of the module was 2.0 inches. The Fourier plane coincided with one surface of the PBS, to which a horizontal slit with a height of 0.60 mm was attached. A plano-convex lens with a focal length of 100.0 mm was used as the screen lens and placed in the image plane. A lenticular lens was used as the vertical diffuser and attached to the screen lens. A fiber-coupled laser diode with a wavelength of 635 nm was used as the light source. A fiber coupler was used to split the laser light into four optical fibers, which were connected to the optical fiber array. The RR technique changed the resolution to 12,800 × 225 in the image plane. The horizontal pixel pitch became 3.46 μm and the horizontal viewing-zone angle was enlarged to 10.5°.

**18**(10), 10294–10300 (2010). [CrossRef] [PubMed]

### 4.2. Experimental results

## 5. Discussion

*sinc*function in the vertical direction and light is diffused in the vertical direction by the vertical diffuser placed on the screen.

## 6. Conclusion

## Acknowledgments

## References and links

1. | Y. Takaki and Y. Hayashi, “Increased horizontal viewing zone angle of a hologram by resolution redistribution of a spatial light modulator,” Appl. Opt. |

2. | Y. Takaki and Y. Hayashi, “Elimination of conjugate image for holograms using a resolution redistribution optical system,” Appl. Opt. |

3. | Y. Takaki and Y. Tanemoto, “Modified resolution redistribution system for frameless hologram display module,” Opt. Express |

4. | P. St. Hilaire, S. A. Benton, and M. Lucente, “Synthetic aperture hologram: a novel approach to three-dimensional display,” J. Opt. Soc. Am. |

5. | M. Lucente and T. A. Galyean, “Rendering interactive holographic images,” in |

6. | T. Mishina, F. Okano, and I. Yuyama, “Time-alternating method based on single-sideband holography with half-zone-plate processing for the enlargement of viewing zones,” Appl. Opt. |

7. | T. Mishina, M. Okui, and F. Okano, “Viewing-zone enlargement method for sampled hologram that uses high-order diffraction,” Appl. Opt. |

8. | M. Stanley, R. W. Bannister, C. D. Cameron, S. D. Coomber, I. G. Cresswell, J. R. Hughes, V. Hui, P. O. Jackson, K. A. Milham, R. J. Miller, D. A. Payne, J. Quarrel, D. C. Scattergood, A. P. Smith, M. A. G. Smith, D. L. Tipton, P. J. Watson, P. J. Webber, and C. W. Slinger, “100-megapixel computer-generated holographic images from Active Tiling: a dynamic and scalable electro-optic modulator system,” Proc. SPIE |

9. | K. Maeno, N. Fukaya, O. Nishikawa, K. Sato, and T. Honda, “Electro-holographic display using 15 mega pixels LCD,” Proc. SPIE |

10. | J. Hahn, H. Kim, Y. Lim, G. Park, and B. Lee, “Wide viewing angle dynamic holographic stereogram with a curved array of spatial light modulators,” Opt. Express |

11. | T. Senoh, T. Mishina, K. Yamamoto, R. Oi, and T. Kurita, “Viewing-zone-angle-expanded color electronic holography system using ultra-high-definition liquid-crystal displays with undesirable light elimination,” J. Display Technol. (to be published). |

12. | Y. Takaki and N. Okada, “Hologram generation by horizontal scanning of a high-speed spatial light modulator,” Appl. Opt. |

13. | Y. Takaki and N. Okada, “Reduction of image blurring of horizontally scanning holographic display,” Opt. Express |

14. | Y. Takaki, M. Yokouchi, and N. Okada, “Improvement of grayscale representation of the horizontally scanning holographic display,” Opt. Express |

15. | S. Tay, P. A. Blanche, R. Voorakaranam, A. V. Tunç, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. St Hilaire, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic three-dimensional display,” Nature |

16. | P.-A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W.-Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature |

17. | A. Schwerdtner, N. Leister, and R. Häussler, “A new approach to electro-holography for TV and projection displays,” in |

18. | R. Häussler, A. Schwerdtner, and N. Leister, “Large holographic displays as an alternative to stereoscopic displays,” Proc. SPIE |

19. | J. P. Waters, “Holographic image synthesis utilizing theoretical methods,” Appl. Phys. Lett. |

20. | G. L. Rogers, “Gabor diffraction microscopy: the hologram as a generalized zone-plate,” Nature |

21. | W. J. Siemens-Wapniarski and M. P. Givens, “The experimental production of synthetic holograms,” Appl. Opt. |

22. | O. Bryngdahl and A. Lohmann, “Single-sideband holography,” J. Opt. Soc. Am. |

23. | Y. Takaki and Y. Tanemoto, “Band-limited zone plates for single-sideband holography,” Appl. Opt. |

24. | D. Leseberg, “Computer-generated holograms: display using one-dimensional transforms,” J. Opt. Soc. Am. A |

25. | K. Matsushima, “Computer-generated holograms for three-dimensional surface objects with shade and texture,” Appl. Opt. |

**OCIS Codes**

(090.1760) Holography : Computer holography

(090.2870) Holography : Holographic display

(120.2040) Instrumentation, measurement, and metrology : Displays

**ToC Category:**

Holography

**History**

Original Manuscript: May 23, 2011

Revised Manuscript: June 22, 2011

Manuscript Accepted: June 25, 2011

Published: July 15, 2011

**Citation**

Yasuhiro Takaki and Junya Nakamura, "Zone plate method for electronic holographic display using resolution redistribution technique," Opt. Express **19**, 14707-14719 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-15-14707

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### References

- Y. Takaki and Y. Hayashi, “Increased horizontal viewing zone angle of a hologram by resolution redistribution of a spatial light modulator,” Appl. Opt. 47(19), D6–D11 (2008). [CrossRef] [PubMed]
- Y. Takaki and Y. Hayashi, “Elimination of conjugate image for holograms using a resolution redistribution optical system,” Appl. Opt. 47(24), 4302–4308 (2008). [CrossRef] [PubMed]
- Y. Takaki and Y. Tanemoto, “Modified resolution redistribution system for frameless hologram display module,” Opt. Express 18(10), 10294–10300 (2010). [CrossRef] [PubMed]
- P. St. Hilaire, S. A. Benton, and M. Lucente, “Synthetic aperture hologram: a novel approach to three-dimensional display,” J. Opt. Soc. Am. 9(11), 1969–1977 (1992). [CrossRef]
- M. Lucente and T. A. Galyean, “Rendering interactive holographic images,” in SIGGRAPH '95 Proceedings of the 22nd Annual Conference on Computer Graphics and Interactive Techniques (ACM, 1995), pp. 387–394.
- T. Mishina, F. Okano, and I. Yuyama, “Time-alternating method based on single-sideband holography with half-zone-plate processing for the enlargement of viewing zones,” Appl. Opt. 38(17), 3703–3713 (1999). [CrossRef] [PubMed]
- T. Mishina, M. Okui, and F. Okano, “Viewing-zone enlargement method for sampled hologram that uses high-order diffraction,” Appl. Opt. 41(8), 1489–1499 (2002). [CrossRef] [PubMed]
- M. Stanley, R. W. Bannister, C. D. Cameron, S. D. Coomber, I. G. Cresswell, J. R. Hughes, V. Hui, P. O. Jackson, K. A. Milham, R. J. Miller, D. A. Payne, J. Quarrel, D. C. Scattergood, A. P. Smith, M. A. G. Smith, D. L. Tipton, P. J. Watson, P. J. Webber, and C. W. Slinger, “100-megapixel computer-generated holographic images from Active Tiling: a dynamic and scalable electro-optic modulator system,” Proc. SPIE 5005, 247–258 (2003). [CrossRef]
- K. Maeno, N. Fukaya, O. Nishikawa, K. Sato, and T. Honda, “Electro-holographic display using 15 mega pixels LCD,” Proc. SPIE 2652, 15–23 (1996). [CrossRef]
- J. Hahn, H. Kim, Y. Lim, G. Park, and B. Lee, “Wide viewing angle dynamic holographic stereogram with a curved array of spatial light modulators,” Opt. Express 16(16), 12372–12386 (2008). [CrossRef] [PubMed]
- T. Senoh, T. Mishina, K. Yamamoto, R. Oi, and T. Kurita, “Viewing-zone-angle-expanded color electronic holography system using ultra-high-definition liquid-crystal displays with undesirable light elimination,” J. Display Technol. (to be published).
- Y. Takaki and N. Okada, “Hologram generation by horizontal scanning of a high-speed spatial light modulator,” Appl. Opt. 48(17), 3255–3260 (2009). [CrossRef] [PubMed]
- Y. Takaki and N. Okada, “Reduction of image blurring of horizontally scanning holographic display,” Opt. Express 18(11), 11327–11334 (2010). [CrossRef] [PubMed]
- Y. Takaki, M. Yokouchi, and N. Okada, “Improvement of grayscale representation of the horizontally scanning holographic display,” Opt. Express 18(24), 24926–24936 (2010). [CrossRef] [PubMed]
- S. Tay, P. A. Blanche, R. Voorakaranam, A. V. Tunç, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. St Hilaire, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic three-dimensional display,” Nature 451(7179), 694–698 (2008). [CrossRef] [PubMed]
- P.-A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W.-Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature 468(7320), 80–83 (2010). [CrossRef] [PubMed]
- A. Schwerdtner, N. Leister, and R. Häussler, “A new approach to electro-holography for TV and projection displays,” in SID Symposium Digest of Technical Papers (Society for Information Display, 2007), pp. 1224–1227.
- R. Häussler, A. Schwerdtner, and N. Leister, “Large holographic displays as an alternative to stereoscopic displays,” Proc. SPIE 6803, 68030M, 68030M-9 (2008). [CrossRef]
- J. P. Waters, “Holographic image synthesis utilizing theoretical methods,” Appl. Phys. Lett. 9(11), 405–407 (1966). [CrossRef]
- G. L. Rogers, “Gabor diffraction microscopy: the hologram as a generalized zone-plate,” Nature 166(4214), 237 (1950). [CrossRef] [PubMed]
- W. J. Siemens-Wapniarski and M. P. Givens, “The experimental production of synthetic holograms,” Appl. Opt. 7(3), 535–538 (1968). [CrossRef] [PubMed]
- O. Bryngdahl and A. Lohmann, “Single-sideband holography,” J. Opt. Soc. Am. 58(5), 620–624 (1968). [CrossRef]
- Y. Takaki and Y. Tanemoto, “Band-limited zone plates for single-sideband holography,” Appl. Opt. 48(34), H64–H70 (2009). [CrossRef] [PubMed]
- D. Leseberg, “Computer-generated holograms: display using one-dimensional transforms,” J. Opt. Soc. Am. A 3(11), 1846–1851 (1986). [CrossRef]
- K. Matsushima, “Computer-generated holograms for three-dimensional surface objects with shade and texture,” Appl. Opt. 44(22), 4607–4614 (2005). [CrossRef] [PubMed]

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