## Angular spectrum calculations for arbitrary focal length with a scaled convolution |

Optics Express, Vol. 19, Issue 15, pp. 14268-14276 (2011)

http://dx.doi.org/10.1364/OE.19.014268

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### Abstract

Nyquist sampling theorem in an image calculation with angular spectrum method restricts a propagation distance and a focal length of a lens. In order to avoid these restrictions, we studied suitable expressions for the image computations depending on their conditions. Additionally, a lateral scale in an observation plane can be magnified freely by using a scaled convolution in each expression.

© 2011 OSA

## 1. Introduction

5. M. Mansuripur, “Certain computational aspects of vector diffraction problems,” J. Opt. Soc. Am. A **6**(6), 786–805 (1989). [CrossRef]

6. D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. **164**(4-6), 233–245 (1999). [CrossRef]

7. S. Nishiwaki, “Calculations of optical field by fast Fourier transform analysis,” Appl. Opt. **27**(16), 3518–3521 (1988). [CrossRef] [PubMed]

## 2. Aberration-free lens model

*f*. A monochromatic plane wave field parallel to the optical axis z transmits an aperture in

*ξ*-

*η*plane. A numerical aperture (NA) of the lens depends on the diameter

*d*for a lens aperture as follows

*f*, the larger NA if

*d*is constant. The wave field

**and**

*ξ***denote vectors in**

*x**ξ*-

*η*(image plane) and

*x*-

*y*plane (observation plane), respectively.

**is a spatial-frequency vector against**

*u***vector. The function**

*ξ**T*over the aperture is defined by

4. F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula,” Appl. Opt. **45**(6), 1102–1110 (2006). [CrossRef] [PubMed]

*Δξ*is a sampling interval on the aperture plane, the area of the sampling window is defined by

*Δξ = L/M*) . Here we assumed that the phase of

4. F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula,” Appl. Opt. **45**(6), 1102–1110 (2006). [CrossRef] [PubMed]

5. M. Mansuripur, “Certain computational aspects of vector diffraction problems,” J. Opt. Soc. Am. A **6**(6), 786–805 (1989). [CrossRef]

4. F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula,” Appl. Opt. **45**(6), 1102–1110 (2006). [CrossRef] [PubMed]

**45**(6), 1102–1110 (2006). [CrossRef] [PubMed]

*Δξ*on the aperture plane is not sufficiently small. In this case, it is necessary to rewrite Eq. (2) as a convolution representation. Specifically, we derived such formulas:

**is a spatial-frequency vector against**

*p***vector. Here we utilized Weyl’s representation of a spherical wave (See appendix B for details). When computing these formulas by the FFT (e.g., the convolution theorem), one must expand the sampling area of**

*ξ**f*≥

*M*(Δ

*ξ)*

^{2}/

*λ*,

*z*≤

*M*(Δ

*ξ*)

^{2}/

*λ*,

*z*≥

*M*(Δ

*ξ*)

^{2}/

*λ*,

*f*≤

*M*(Δ

*ξ*)

^{2}/

*λ*, respectively). Thus, one can make it possible to compute the wave field

## 3. Scaled convolution

*Δξ = Δx*) if one performs the AS computation by means of the FFT. Therefore, such method is not adequate for practical computation of the spot image near the focal plane. Let us modify the formulas of the propagations so that the lateral scale of the image can be magnified by reducing the sampling interval

*Δx*in the observation plane. Equation (6) of the convolution representation for long distance propagation is directly expressed as

**m**and

**q**are two-dimensional vectors of integers,

8. R. P. Muffoletto, J. M. Tyler, and J. E. Tohline, “Shifted Fresnel diffraction for computational holography,” Opt. Express **15**(9), 5631–5640 (2007). [CrossRef] [PubMed]

*γ*indicates the ability to vary the total number of data for the Discrete Fourier Transform in one period. The parameter

**u**after computing the scaled convolution of Eq. (3). Thus, one can obtain the scale-down image of the scalar wave field in the observation plane.

*z*and the focal length

*f*. It also indicates original data requiring for the zero-padding in each scaled convolution.

## 4. Numerical verifications

*z*and focal length

*f*should take almost the same values. According to Table 1, one should use the Type-II or-III for this computation. Figure 2 shows the magnifications of the Airy disc images using Type-II and -III. We computed the Airy disc images of the aberration-free lenses having NA 0.2 and 0.9 at the focal planes by using Type-II and -III, respectively. The images where the scale convolutions are not applied are shown for comparison (1x magnification). As expected, we can clearly confirm the magnifications of these Airy disc images. Furthermore, we can hardly see the fine features of Airy disc images, when we expand the images of 1x magnifications by digitally scale-up. Thus, we verified the capability of the scale convolution for the magnification.

*f*. When we consider an aberration-free lens having

*f*= 11.456 mm, NA = 0.4,

*L*= 10 mm and

*D*= 9.999 mm in

*λ*= 0.6 um with

*M*= 2

^{14}, threshold value

*f*= 0.128 mm, NA = 0.9,

*L*= 0.778 mm and

*D*= 0.529 mm in

*λ*= 0.6 um with

*M*= 2

^{10}, threshold value

## 5. Summary and Discussion

5. M. Mansuripur, “Certain computational aspects of vector diffraction problems,” J. Opt. Soc. Am. A **6**(6), 786–805 (1989). [CrossRef]

## Appendix A

*d*Ω:

*θ, ϕ*) [9]. This solid angle can be represented in tangent (

*ξ, η*) and sine (

*α, β*) coordinates as

## Appendix B

*k*, we obtain

## Acknowledgments

## References and links

1. | L. Mandel and E. Wolf, |

2. | J. W. Goodman, |

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

4. | F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula,” Appl. Opt. |

5. | M. Mansuripur, “Certain computational aspects of vector diffraction problems,” J. Opt. Soc. Am. A |

6. | D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. |

7. | S. Nishiwaki, “Calculations of optical field by fast Fourier transform analysis,” Appl. Opt. |

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

9. | M. Nakahara, |

**OCIS Codes**

(220.2560) Optical design and fabrication : Propagating methods

(260.1960) Physical optics : Diffraction theory

(070.7345) Fourier optics and signal processing : Wave propagation

**ToC Category:**

Physical Optics

**History**

Original Manuscript: May 27, 2011

Revised Manuscript: June 24, 2011

Manuscript Accepted: June 25, 2011

Published: July 11, 2011

**Citation**

Satoru Odate, Chiaki Koike, Hidemitsu Toba, Tetsuya Koike, Ayako Sugaya, Katsumi Sugisaki, Katsura Otaki, and Kiyoshi Uchikawa, "Angular spectrum calculations for arbitrary focal length with a scaled convolution," Opt. Express **19**, 14268-14276 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-15-14268

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### References

- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
- J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
- 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). [CrossRef] [PubMed]
- F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula,” Appl. Opt. 45(6), 1102–1110 (2006). [CrossRef] [PubMed]
- M. Mansuripur, “Certain computational aspects of vector diffraction problems,” J. Opt. Soc. Am. A 6(6), 786–805 (1989). [CrossRef]
- D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164(4-6), 233–245 (1999). [CrossRef]
- S. Nishiwaki, “Calculations of optical field by fast Fourier transform analysis,” Appl. Opt. 27(16), 3518–3521 (1988). [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). [CrossRef] [PubMed]
- M. Nakahara, Geometry, Topology and Physics, 2nd ed. (Taylor & Francis, 2003).

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