## Shifted angular spectrum method for off-axis numerical propagation |

Optics Express, Vol. 18, Issue 17, pp. 18453-18463 (2010)

http://dx.doi.org/10.1364/OE.18.018453

Acrobat PDF (1781 KB)

### Abstract

A novel method is proposed for simulating free-space propagation from an input source field to a destination sampling window laterally shifted from that in the source field. This off-axis type numerical propagation is realized using the shifted-Fresnel method (Shift-FR) and is very useful for calculating non-paraxial and large-scale fields. However, the Shift-FR is prone to a serious problem, in that it causes strong aliasing errors in short distance propagation. The proposed method, based on the angular spectrum method, resolves this problem. Numerical examples as well as the formulation are presented.

© 2010 OSA

## 1. Introduction

9. K. Matsushima, H. Schimmel, and F. Wyrowski, “Fast calculation method for optical diffraction on tilted planes by use of the angular spectrum of plane waves,” J. Opt. Soc. Am. A **20**(9), 1755–1762 (2003), http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-20-9-1755. [CrossRef]

12. S. J. Jeong and C. K. Hong, “Pixel-size-maintained image reconstruction of digital holograms on arbitrarily tilted planes by the angular spectrum method,” Appl. Opt. **47**(16), 3064–3071 (2008). [CrossRef] [PubMed]

9. K. Matsushima, H. Schimmel, and F. Wyrowski, “Fast calculation method for optical diffraction on tilted planes by use of the angular spectrum of plane waves,” J. Opt. Soc. Am. A **20**(9), 1755–1762 (2003), http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-20-9-1755. [CrossRef]

10. K. Matsushima, “Formulation of the rotational transformation of wave fields and their application to digital holography,” Appl. Opt. **47**(19), D110–D116 (2008), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-47-19-D110. [CrossRef] [PubMed]

10. K. Matsushima, “Formulation of the rotational transformation of wave fields and their application to digital holography,” Appl. Opt. **47**(19), D110–D116 (2008), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-47-19-D110. [CrossRef] [PubMed]

13. S. De Nicola, A. Finizio, G. Pierattini, P. Ferraro, and D. Alfieri, “Angular spectrum method with correction of anamorphism for numerical reconstruction of digital holograms on tilted planes,” Opt. Express **13**(24), 9935–9940 (2005). [CrossRef] [PubMed]

14. K. Matsushima, “Computer-generated holograms for three-dimensional surface objects with shade and texture,” Appl. Opt. **44**(22), 4607–4614 (2005), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-44-22-4607. [CrossRef] [PubMed]

15. K. Matsushima and S. Nakahara, “Extremely high-definition full-parallax computer-generated hologram created by the polygon-based method,” Appl. Opt. **48**(34), H54–H63 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-22-19662. [CrossRef] [PubMed]

5. R. P. Muffoletto, J. M. Tyler, and J. E. Tohline, “Shifted Fresnel diffraction for computational holography,” Opt. Express **15**(9), 5631–5640 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-9-5631. [CrossRef] [PubMed]

8. K. Matsushima and T. Shimobaba, “Band-limited angular spectrum method for numerical simulation of free-space propagation in far and near fields,” Opt. Express **17**(22), 19662–19673 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-22-19662. [CrossRef] [PubMed]

## 2. Off-axis numerical propagation

- Step 1. Extend the source sampling window so that the sampling window includes the region of interest after numerical propagation.
- Step 2. Pad the extended source sampling window with zeros.
- Step 3. Numerically propagate the source wave field onto the destination plane.
- Step 4. Cut out the region of interest from the destination sampling window.

*off-axis numerical propagation*in this paper. The most notable method for off-axis numerical propagation is the Shift-FR [5

5. R. P. Muffoletto, J. M. Tyler, and J. E. Tohline, “Shifted Fresnel diffraction for computational holography,” Opt. Express **15**(9), 5631–5640 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-9-5631. [CrossRef] [PubMed]

## 3. Formulation of the shifted angular spectrum method

### 3.1 The angular spectrum method for shifted coordinates

8. K. Matsushima and T. Shimobaba, “Band-limited angular spectrum method for numerical simulation of free-space propagation in far and near fields,” Opt. Express **17**(22), 19662–19673 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-22-19662. [CrossRef] [PubMed]

*u*and

*v*are Fourier frequencies with respect to

*x*and

*y*, respectively. The spectrum of the source field is given as:

*F*represents the Fourier transform. The transfer function

*λ*denotes the wavelength.

### 3.2 One-dimensional wave fields

*x*are discussed in this section. In this case, the transfer function of Eq. (7) is redefined as follows:

*u*. Therefore, supposing that the transfer function is sampled at intervals of

8. K. Matsushima and T. Shimobaba, “Band-limited angular spectrum method for numerical simulation of free-space propagation in far and near fields,” Opt. Express **17**(22), 19662–19673 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-22-19662. [CrossRef] [PubMed]

**17**(22), 19662–19673 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-22-19662. [CrossRef] [PubMed]

### 3.3 Two-dimensional wave fields

*S*is the size of the source sampling window in the

_{y}*y*direction. Note that the sampling window is also doubled in the

*y*direction to linearize the discrete convolution.

*x*and

*u*to

*y*and

*v*in the above relations, respectively. Note that the region under the conditions

**17**(22), 19662–19673 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-22-19662. [CrossRef] [PubMed]

*λ*

^{−1}in the (

*u*,

*v*) plane, while the relations for shifting

### 3.4 Approximated rectangular region to avoid aliasing errors

## 4. Numerical verification of the shifted angular spectrum method

*θ*. Supposing that the plane wave is diffracted by a circular aperture, we calculate the diffracted field in the plane at position

*x-*and

*y*-axes so that the diffracted field is included within the destination sampling window. In addition, the approximated transfer function given by Eq. (23) is used for calculating the Shift-AS.

*z*

_{0}. The shift of the destination sampling window is

*x*

_{0}= 0.4 cm both for

*z*

_{0}= 5 cm in (a) and 10 cm in (b), whereas

*x*

_{0}= 1 cm for

*z*

_{0}= 40 cm in (c). In the case of

*z*

_{0}= 5 cm in (a), the position of the destination field is shifted slightly along the

*x*-axis because of the slanted plane wave. In this case, the field calculated by the Shift-AS agrees with that by the BL-AS, whereas the field by the Shift-FR disagrees with these fields because of the aliasing error. The field calculated by the Shift-FR is also not normal in

*z*

_{0}= 10 cm in (b), while the BL-AS and Shift-AS again give similar fields.

*z*

_{0}= 40 cm in (c), the destination field lies outside the extended sampling area of the BL-AS. However, the Shift-AS and Shift-FR can calculate the field within the sampling area owing to their off-axis property. The Shift-FR, in this case, gives the same result as that using the Shift-AS, due to the long distance of the propagation.

## 5. Discussion on the limit frequency

## 6. Conclusion

## Acknowledgments

## References and links

1. | T. M. Kreis, M. Adams, and W. P. O. Jüptner, “Methods of digital holography: A comparison,” Proc. SPIE |

2. | F. Zhang, I. Yamaguchi, and L. P. Yaroslavsky, “Algorithm for reconstruction of digital holograms with adjustable magnification,” Opt. Lett. |

3. | L. Yu and M. K. Kim, “Pixel resolution control in numerical reconstruction of digital holography,” Opt. Lett. |

4. | D. Wang, J. Zhao, F. Zhang, G. Pedrini, and W. Osten, “High-fidelity numerical realization of multiple-step Fresnel propagation for the reconstruction of digital holograms,” Appl. Opt. |

5. | R. P. Muffoletto, J. M. Tyler, and J. E. Tohline, “Shifted Fresnel diffraction for computational holography,” Opt. Express |

6. | M. Sypek, “Light propagation in the Fresnel region. New numerical approach,” Opt. Commun. |

7. | J. W. Goodman, |

8. | K. Matsushima and T. Shimobaba, “Band-limited angular spectrum method for numerical simulation of free-space propagation in far and near fields,” Opt. Express |

9. | K. Matsushima, H. Schimmel, and F. Wyrowski, “Fast calculation method for optical diffraction on tilted planes by use of the angular spectrum of plane waves,” J. Opt. Soc. Am. A |

10. | K. Matsushima, “Formulation of the rotational transformation of wave fields and their application to digital holography,” Appl. Opt. |

11. | N. Delen and B. Hooker, “Free-space beam propagation between arbitrarily oriented planes based on full diffraction theory: a fast Fourier transform approach,” J. Opt. Soc. Am. A |

12. | S. J. Jeong and C. K. Hong, “Pixel-size-maintained image reconstruction of digital holograms on arbitrarily tilted planes by the angular spectrum method,” Appl. Opt. |

13. | S. De Nicola, A. Finizio, G. Pierattini, P. Ferraro, and D. Alfieri, “Angular spectrum method with correction of anamorphism for numerical reconstruction of digital holograms on tilted planes,” Opt. Express |

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

15. | K. Matsushima and S. Nakahara, “Extremely high-definition full-parallax computer-generated hologram created by the polygon-based method,” Appl. Opt. |

16. | J. W. Goodman, |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

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

(090.1760) Holography : Computer holography

**ToC Category:**

Physical Optics

**History**

Original Manuscript: July 12, 2010

Manuscript Accepted: August 8, 2010

Published: August 13, 2010

**Citation**

Kyoji Matsushima, "Shifted angular spectrum method for off-axis numerical propagation," Opt. Express **18**, 18453-18463 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-17-18453

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### References

- T. M. Kreis, M. Adams, and W. P. O. Jüptner, “Methods of digital holography: A comparison,” Proc. SPIE 3098, 224–233 (1997). [CrossRef]
- F. Zhang, I. Yamaguchi, and L. P. Yaroslavsky, “Algorithm for reconstruction of digital holograms with adjustable magnification,” Opt. Lett. 29(14), 1668–1670 (2004). [CrossRef] [PubMed]
- L. Yu and M. K. Kim, “Pixel resolution control in numerical reconstruction of digital holography,” Opt. Lett. 31(7), 897–899 (2006). [CrossRef] [PubMed]
- D. Wang, J. Zhao, F. Zhang, G. Pedrini, and W. Osten, “High-fidelity numerical realization of multiple-step Fresnel propagation for the reconstruction of digital holograms,” Appl. Opt. 47(19), D12–D20 (2008). [CrossRef] [PubMed]
- R. P. Muffoletto, J. M. Tyler, and J. E. Tohline, “Shifted Fresnel diffraction for computational holography,” Opt. Express 15(9), 5631–5640 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-9-5631 . [CrossRef] [PubMed]
- M. Sypek, “Light propagation in the Fresnel region. New numerical approach,” Opt. Commun. 116(1-3), 43–48 (1995). [CrossRef]
- J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), chap. 3.10.
- K. Matsushima and T. Shimobaba, “Band-limited angular spectrum method for numerical simulation of free-space propagation in far and near fields,” Opt. Express 17(22), 19662–19673 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-22-19662 . [CrossRef] [PubMed]
- K. Matsushima, H. Schimmel, and F. Wyrowski, “Fast calculation method for optical diffraction on tilted planes by use of the angular spectrum of plane waves,” J. Opt. Soc. Am. A 20(9), 1755–1762 (2003), http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-20-9-1755 . [CrossRef]
- K. Matsushima, “Formulation of the rotational transformation of wave fields and their application to digital holography,” Appl. Opt. 47(19), D110–D116 (2008), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-47-19-D110 . [CrossRef] [PubMed]
- N. Delen and B. Hooker, “Free-space beam propagation between arbitrarily oriented planes based on full diffraction theory: a fast Fourier transform approach,” J. Opt. Soc. Am. A 15(4), 857–867 (1998). [CrossRef]
- S. J. Jeong and C. K. Hong, “Pixel-size-maintained image reconstruction of digital holograms on arbitrarily tilted planes by the angular spectrum method,” Appl. Opt. 47(16), 3064–3071 (2008). [CrossRef] [PubMed]
- S. De Nicola, A. Finizio, G. Pierattini, P. Ferraro, and D. Alfieri, “Angular spectrum method with correction of anamorphism for numerical reconstruction of digital holograms on tilted planes,” Opt. Express 13(24), 9935–9940 (2005). [CrossRef] [PubMed]
- K. Matsushima, “Computer-generated holograms for three-dimensional surface objects with shade and texture,” Appl. Opt. 44(22), 4607–4614 (2005), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-44-22-4607 . [CrossRef] [PubMed]
- K. Matsushima and S. Nakahara, “Extremely high-definition full-parallax computer-generated hologram created by the polygon-based method,” Appl. Opt. 48(34), H54–H63 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-22-19662 . [CrossRef] [PubMed]
- J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), chap. 2.2.

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