## Fast calculation method for computer-generated cylindrical hologram based on wave propagation in spectral domain |

Optics Express, Vol. 18, Issue 25, pp. 25546-25555 (2010)

http://dx.doi.org/10.1364/OE.18.025546

Acrobat PDF (1636 KB)

### Abstract

A fast calculation method for computer generation of cylindrical holograms is proposed. The calculation method is based on wave propagation in spectral domain and in cylindrical co-ordinates, which is otherwise similar to the angular spectrum of plane waves in cartesian co-ordinates. The calculation requires only two FFT operations and hence is much faster. The theoretical background of the calculation method, sampling conditions and simulation results are presented. The generated cylindrical hologram has been tested for reconstruction in different view angles and also in plane surfaces.

© 2010 Optical Society of America

## 1. Introduction

1. Tung H. Jeong, “Cylindrical holography and some proposed applications,” J. Opt. Soc. Am.57, 31396–1398 (1967), http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-57-11-1396. [CrossRef]

2. O. D. D. Soares and J. C. A. Fernandes, “Cylindrical hologram of 360° field of view,” Appl. Opt.21,3194–3196 (1982), http://www.opticsinfobase.org/abstract.cfm?URI=ao-21-17-3194. [CrossRef] [PubMed]

3. A. W. Lohmann and D. P. Paris, “Binary Fraunhofer holograms, generated by computer,” Appl. Opt.6, 739–1748 (1967), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-6-10-1739. [CrossRef]

5. T. Tommasi and B. Bianco, “Computer-generated holograms of tilted planes by a spatial frequency approach,” J. Opt. Soc. Am. A10, 299–305 (1993), http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-10-2-299. [CrossRef]

7. A. Kashiwagi and Y. Sakamoto, “A Fast calculation method of cylindrical computer-generated holograms which perform image reconstruction of volume data,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM, OSA Technical Digest (CD) (Optical Society of America, 2007), paper DWB7, http://www.opticsinfobase.org/abstract.cfm?URI=DH-2007-DWB7. [PubMed]

8. T. Yamaguchi, T. Fujii, and H. Yoshikawa, “Fast calculation method for computer-generated cylindrical holograms,” Appl. Opt.47, D63–D70 (2008), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-47-19-D63. [CrossRef] [PubMed]

10. Y. Sando, M. Itoh, and T. Yatagai, “Fast calculation method for cylindrical computer-generated holograms,” Appl. Opt.13, 1418–1423 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-5-1418.

## 2. Theoretical background

*p*is given by where

*k*is the wave number, given by

*k*= 2

*π/λ*, and

*p*denotes the complex amplitude of the wavefront as a function of spatial coordintes. The solution to the Helmholtz equation in cylindrical co-ordinates, as shown in Fig. 1 can be represented as shown in Eq. (2) where

*R*(

*r*)Φ,(

*φ*) and

*Y*(

*y*) represent the radial, angular and vertical components, respectively

*A*(

_{n}*k*) and

_{y}*B*(

_{n}*k*) are the arbitrary constants.

_{y}- Let all the radiating sources be within the circular cylinder of radius ‘
*a*’,*i.e*,*the radiating source is cylindrical surface of radius ‘a’* - The complex amplitude is measured only at positions outside the cylindrical radius ‘
*a’ ie, we consider only outward propagation*. Let the measuring position be denoted by ‘*r*’.

*P*(

_{n}*r, k*) to be the two-dimensional Fourier transform in

_{y}*φ*and

*y*in cylindrical coordinates The inverse relation for Eq. (5) is given by where

*n*can take only integer values because the cylindrical surface is a closed one in the circumferential direction. Comparing Eq. (6) at

*r*=

*a*with Eq. (4) we get Using Eq. (7) to eliminate

*A*in Eq. (4) yields Eq. (8) calculates the complex amplitude at any position

_{n}*p*(

*r, φ, y*), for ((

*r > a*)), when the complex amplitude in a cylindrical surface of radius ‘

*a*’ is given. The spectral soulution in Eq. (8) is similar in form to the angular spectrum of plane waves which is defined as

*r*’ in Eq. (8) is kept constant, ie, the measurement plane (hologram surface) is also a cylinder, then the system is shift invariant. Hence we can use FFT to evaluate Eq. (8) and hence fast calculation. Deriving out the analytical expression for the Transfer Function for propagation from cylindrical surface as shown in Eq. (11), is the most important step in this work.

## 3. Computational Procedure

*a*and

*r*respectively. The height of the cylindrical surfaces were

*y*=

*h*. The object of size

*N*×

*N*was chosen for simulation which is graphically represented in Fig. 3. The object is illuminated virtually by a point source form the center which also serves as the reference.

### 3.1. Sampling Conditions

*π*when

*n*=

*N/*2, where

*N*= [–

*N/*2...0...

*N*/2]. From the analysis of Eq. (11) the spatial rate of change of

*k*is higher than that of

_{r}r*k*. Hence as long as the sampling condition for

_{r}a*k*is satisfied, the entire Transfer function also meets the sampling condition approximately. Accordingly the Nyquist sampling condition can be expressed by the inequality as shown in below from the above inequality with conditions,

_{r}r*L*

_{0}=

*h*is the height of the cylinder) we can obtain which again reduces to as

*a*= 1

*cm*and

*r*= 10

*cm*respectively. The height of the cylindrical sufrace was assumed to be

*y*= 10

*cm*. To avoid harsh sampling requirements, the wavelength

*λ*was assumed to be large i.e,

*λ*= 180

*μm*. When all these dimensions were substituted in the sampling condition, given by Eq. (15), the required number of samples turned out to be

*N*≃ 512. Hence the object and transfer functinon were generated as 512 × 512 matrices.

### 3.2. Hologram generation

- The transfer function(TF) was generated according to Eq. (11).
- The complex amplitudes of object and reference were generated as 512 × 512 matrices. The generated object is graphically shown in Fig. 3
- The Fourier spectrum of object and reference wavefield was computed (using FFT) according to Eq. (5) which gives the corresponding complex amplitude in spectral domain at the object surface.
- The calculated spectrum (object and reference) is multiplied with the transfer function which gives the corresponding complex amplitude in spectral domain at the hologram surface.
- The complex amplitude in spectral domain is inverse Fourier transformed according to Eq. (6) to get the complex amplitude in real space at the hologram surface
- The complex amplitudes due to object and reference are added(superposed) at the hologram surface and their intensity is calculated.

*Hologram*is the resulting 2D image(matrix) that holds the intensity pattern of hologram,

*TF*is the transfer function and

*FFT*and

*IFFT*are the forward and inverse discrete fast Fourier transforms respectively. The calculated intensity pattern is the hologram and is graphically shown in Fig. 4

### 3.3. Simulated reconstruction

*Reconstruction*is the 2D image(matrix) that holds the reconstructed intensity distribution,

*TF*is the transfer function and

*FFT*and

*IFFT*are the forward and inverse discrete fast Fourier transforms respectively. The reconstructed image pattern is shown in Fig. 5 which exactly resembles the original object choosen.

## 4. Testing the hologram

### 4.1. Reconstruction in plane surface

*z*= 1 and

*z*= −1 is simulated. The schematic and geometry of this setup are shown in Fig. 6. The width and height of the reconstruction plane were set to be 3 cm and 10cm respectively. Each reconstruction plane coincides with the original cylindrical object only at one position, ie at θ = 0° and at θ = 180°. Hence during reconstruction, only these two vicinities(of the object) are expected to be reconstructed in focus and other areas unfocused. But now, these two reconstruction planes nomore maintain any shift invariance relation to the hologram surface and hence FFT cannot be used for reconstruction. Hence the reconstruction was done using direct integration according to Eq. (16) shown below

*z*= 1 and

*z*= −1 are shown in Fig. 7 and Fig. 8 respectively. As seen from the Fig. 7 only the letter ‘P’ that lies in the vicinity of θ= 0° is in focus. While Fig. 8 shows the letter ‘O’ which is in the vicinity of θ = 180° reconstruced sharply. Hence the cylindrical hologram could reconstruct the stored data properly in any surface.

### 4.2. Reconstruction for variable viewing angles

## 5. Concluding Remarks

## References and links

1. | Tung H. Jeong, “Cylindrical holography and some proposed applications,” J. Opt. Soc. Am.57, 31396–1398 (1967), http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-57-11-1396. [CrossRef] |

2. | O. D. D. Soares and J. C. A. Fernandes, “Cylindrical hologram of 360° field of view,” Appl. Opt.21,3194–3196 (1982), http://www.opticsinfobase.org/abstract.cfm?URI=ao-21-17-3194. [CrossRef] [PubMed] |

3. | A. W. Lohmann and D. P. Paris, “Binary Fraunhofer holograms, generated by computer,” Appl. Opt.6, 739–1748 (1967), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-6-10-1739. [CrossRef] |

4. | J. W. Goodman, |

5. | T. Tommasi and B. Bianco, “Computer-generated holograms of tilted planes by a spatial frequency approach,” J. Opt. Soc. Am. A10, 299–305 (1993), http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-10-2-299. [CrossRef] |

6. | Y. Sakamoto and M. Tobise, “Computer generated cylindrical
hologram,” in Practical Holography XIX:
Materials and Applications, Tung H. Jeong and Hans I. Bjelkhagen, eds., |

7. | A. Kashiwagi and Y. Sakamoto, “A Fast calculation method of cylindrical computer-generated holograms which perform image reconstruction of volume data,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM, OSA Technical Digest (CD) (Optical Society of America, 2007), paper DWB7, http://www.opticsinfobase.org/abstract.cfm?URI=DH-2007-DWB7. [PubMed] |

8. | T. Yamaguchi, T. Fujii, and H. Yoshikawa, “Fast calculation method for computer-generated cylindrical holograms,” Appl. Opt.47, D63–D70 (2008), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-47-19-D63. [CrossRef] [PubMed] |

9. | T. Yamaguchi, T. Fujii, and H. Yoshikawa, “Computer-generated cylindrical rainbow
hologram,” in Practical Holography XXII:
Materials and Applications, Hans I. Bjelkhagen and Raymond K. Kostuk, eds., |

10. | Y. Sando, M. Itoh, and T. Yatagai, “Fast calculation method for cylindrical computer-generated holograms,” Appl. Opt.13, 1418–1423 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-5-1418. |

11. | G. E. Williams, |

12. | N. N. Lebedev, |

13. | G. B. Arfken and H. J. Weber, |

**OCIS Codes**

(050.1960) Diffraction and gratings : Diffraction theory

(090.1760) Holography : Computer holography

(090.2870) Holography : Holographic display

**ToC Category:**

Holography

**History**

Original Manuscript: September 8, 2010

Revised Manuscript: October 11, 2010

Manuscript Accepted: October 17, 2010

Published: November 22, 2010

**Citation**

Boaz Jessie Jackin and Toyohiko Yatagai, "Fast calculation method for computer-generated cylindrical hologram based on wave propagation in spectral domain," Opt. Express **18**, 25546-25555 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-25-25546

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### References

- T. H. Jeong, “Cylindrical holography and some proposed applications,” J. Opt. Soc. Am. 57, 31396–31398 (1967), http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-57-11-1396. [CrossRef]
- O. D. D. Soares and J. C. A. Fernandes, “Cylindrical hologram of 360o field of view,” Appl. Opt. 21, 3194–3196 (1982), http://www.opticsinfobase.org/abstract.cfm?URI=ao-21-17-3194. [CrossRef] [PubMed]
- A. W. Lohmann and D. P. Paris, “Binary Fraunhofer holograms, generated by computer,” Appl. Opt. 6, 739–1748 (1967), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-6-10-1739. [CrossRef]
- J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
- T. Tommasi and B. Bianco, “Computer-generated holograms of tilted planes by a spatial frequency approach,” J. Opt. Soc. Am. A 10, 299–305 (1993), http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-10-2-299. [CrossRef]
- Y. Sakamoto and M. Tobise, “Computer generated cylindrical hologram,” in Practical Holography XIX: Materials and Applications, Tung H. Jeong and Hans I. Bjelkhagen, eds., Proc.SPIE 5742, 267–274 (2005).
- A. Kashiwagi and Y. Sakamoto, “A Fast calculation method of cylindrical computer-generated holograms which perform image reconstruction of volume data,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM, OSA Technical Digest (CD) (Optical Society of America, 2007), paper DWB7, http://www.opticsinfobase.org/abstract.cfm?URI=DH-2007-DWB7. [PubMed]
- T. Yamaguchi, T. Fujii, and H. Yoshikawa, “Fast calculation method for computer-generated cylindrical holograms,” Appl. Opt. 47, D63–D70 (2008), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-47-19-D63. [CrossRef] [PubMed]
- T. Yamaguchi, T. Fujii, and H. Yoshikawa, “Computer-generated cylindrical rainbow hologram,” in Practical Holography XXII: Materials and Applications, Hans I. Bjelkhagen and Raymond K. Kostuk, eds., Proc.SPIE 6912, 69121C (2009).
- Y. Sando, M. Itoh, and T. Yatagai, “Fast calculation method for cylindrical computer-generated holograms,” Appl. Opt. 13, 1418–1423 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-5-1418.
- G. E. Williams, Fourier Acoustics, Sound Radiation, and Near-field Acoustical Holography (Academic Press, 1999).
- N. N. Lebedev, Special Functions and Their Applications (Prentice Hall, 1965), pp. 98–160.
- G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic Press, 2001), pp. 702–705.

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