## Fast Hankel transform and its application for studying the propagation of cylindrical electromagnetic fields

Optics Express, Vol. 10, Issue 12, pp. 521-525 (2002)

http://dx.doi.org/10.1364/OE.10.000521

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### Abstract

We present a fast Hankel transform (FHTn) method for direct numerical evaluation of electromagnetic (EM) field propagation through an axially symmetric system. Comparing with the vector-based plane-wave spectrum (VPWS) method, we present an alternative approach to implement the fast Hankel transform which does not require an additional coordinate transformation for Fourier transform. The proposed FHTn method is an efficient approach for numerical evaluation of an arbitrary integer order of the Hankel transform (HT). As an example to demonstrate the effectiveness of the proposed method, we apply the FHTn technique to the analysis of cylindrical EM field propagation through a diffractive microlens.

© 2002 Optical Society of America

## 1. Introduction

2. Shouyuan Shi and Dennis W. Prather, “Vector-based plane-wave spectrum method for the propagation of cylindrical electromagnetic fields,”Opt. Lett. **24**, 1445 (1999). [CrossRef]

3. Vittorio Magni, Giulio Cerullo, and Sandro De Silvestri, “High-accuracy fast Hankel transform for optical beam propagation,”J. Opt. Soc. Am. A. **9**, 2031–2033 (1992). [CrossRef]

4. José A. Ferrari, Daniel Perciante, and Alfredo Dubra, “Fast Hankel transform of n-th order,”J. Opt. Soc. Am. A. **16**, 2581–2582 (1999). [CrossRef]

5. Dennis W. Prather and Shouyuan Shi, “Formulation and application of the finite-difference time-domain method for the analysis of axially symmetric diffractive optical elements,”J. Opt. Soc. Am. A. **16**, 1131–1142 (1999). [CrossRef]

## 2. Theory

*J*

_{γ}is the γth-order Bessel function and

*C*is a constant. We use

*H*

_{γ}{

*f*(

*r*)} to represent the γth-order HT of a function

*f*(

*r*), so Eq. (1) can be re-written as

*ξ*

_{i}<

*r*<

*ξ*

_{i+1}, the value of

*f*(

*r*)/

*r*

^{γ}can be determined by

*f*(

*r*

_{n})/

*r*

_{n}. Thus the integral over each interval is given by

3. Vittorio Magni, Giulio Cerullo, and Sandro De Silvestri, “High-accuracy fast Hankel transform for optical beam propagation,”J. Opt. Soc. Am. A. **9**, 2031–2033 (1992). [CrossRef]

*F*{} represents the Fourier transform (FT) and

*F*

^{-1}{}the inverse FT. In Eq. (5), the functions

*j*

_{(γ+1)n}and

*φ*

_{n}are defined as

*n*= 0, 1,…,2N-1,and

*f*(

*r*

_{N}) ≡ 0,

*F*

_{n}=

*f*(

*r*

_{n})/

*f*(

*r*

_{n+l})/

*k*

_{0}= [2exp(

*α*)+exp(2

*α*)]/{[1+exp(

*α*)]

^{2}[1-exp(-2

*α*)]} , and

*N*is the total number of sampling points. It is seen that the FHTn can be reduced to the FHATHA for the case of

*γ*=0 in Eq. (7). In numerical calculation we only need to reserve the values of

*g*

_{γ}(

*ρ*

_{m}) from

*m*= 0 to

*N*-1 in Eq. (5) because we pattern the value of

*φ*

_{n}to be zero for

*n*=

*N*,

*N*+1,…,2

*N*-1 according to Eq. (7), therefore the values of

*g*

_{γ}(

*ρ*

_{m}) from

*N*to 2

*N*-1 are insignificant in mathematics. The parameters

*r*

_{0}and

*α*can be chosen arbitrarily in theory, however, for better numerical results, we take the same values of

*r*

_{0}=[1+exp(

*α*)]exp(-

*αN*)/2 and

*α*= -ln[1-exp(-

*α*)](

*N*-1) as used in Ref [3

3. Vittorio Magni, Giulio Cerullo, and Sandro De Silvestri, “High-accuracy fast Hankel transform for optical beam propagation,”J. Opt. Soc. Am. A. **9**, 2031–2033 (1992). [CrossRef]

## 3. Numerical experiments

*H*

_{1}{

*r*} and

*H*

_{2}{

*r*

^{4}} with a constant

*C*= 20

*π*. The numerical calculations were completed within 0.01 second on a 450MHz PC with 128MByte RAM for 1024 sampling points. Figures 1(a) and 2(a) show the plots of both numerical results

*g*_{1}(

*ρ*)and

*g*_{2}(

*ρ*). The errors between the numerical results and the analytical solutions are plotted in Figures 1(b) and 2(b). The good agreement between the two results show that the FHTn method is highly accurate, and it can thus be used as a direct Hankel transform approach for studying the propagation through an axially symmetric system.

## 4. Application for VPWS

2. Shouyuan Shi and Dennis W. Prather, “Vector-based plane-wave spectrum method for the propagation of cylindrical electromagnetic fields,”Opt. Lett. **24**, 1445 (1999). [CrossRef]

*z*=

*z*

_{0}can be determined with those in the near field

*z*=

*0*as follows

*j*= √-1 ,

*k*

_{0}= 2

*π*/

*λ*, 0 ≤

*k*’ ≤ 1, Δ

*r*and Δ

*ρ*are the sampling lengths in the original and propagated planes, respectively, and λ is the wavelength of illumination light. The index

*i*and

*l*represent the values of the

*i*-th sampling point in the near field and the

*l*-th in the focal plane, respectively.

*M*is the total number of sampling points in the near field, and the constant

*C*in the process of HT is equal to

*Nk*

_{0}Δ

*r*.

*m*= 1 mode is needed. Thus, only zero-, first-, and second-order HTs are used in the calculation. For oblique incidence, higher-order HT will be involved in the VPWS calculation, and in this case, the FHTn method can still be used because it is a generic formula for an arbitrary integer order HT.

5. Dennis W. Prather and Shouyuan Shi, “Formulation and application of the finite-difference time-domain method for the analysis of axially symmetric diffractive optical elements,”J. Opt. Soc. Am. A. **16**, 1131–1142 (1999). [CrossRef]

_{r}= 2.25; wavelength λ = 1.0 μm; diameter = 102.47 μm;

*f*-number = 0.78; and 15 zones. Firstly, we employed our axially symmetric finite-difference time-domain (FDTD) program to obtain the near electric field and then used Eqs. (8) to (10) to calculate the electric field propagated onto the focal plane. Figure 3 shows the near electric field magnitude at the emergent plane of the lens as determined by the axially symmetric FDTD method. Figure 4 shows the numerical result of the electric field distribution at the focal plane of the lens, which is the same as that determined with the EM method presented in Ref. [5

5. Dennis W. Prather and Shouyuan Shi, “Formulation and application of the finite-difference time-domain method for the analysis of axially symmetric diffractive optical elements,”J. Opt. Soc. Am. A. **16**, 1131–1142 (1999). [CrossRef]

## 5. Summary

## References and Links

1. | J. W. Goodman, |

2. | Shouyuan Shi and Dennis W. Prather, “Vector-based plane-wave spectrum method for the propagation of cylindrical electromagnetic fields,”Opt. Lett. |

3. | Vittorio Magni, Giulio Cerullo, and Sandro De Silvestri, “High-accuracy fast Hankel transform for optical beam propagation,”J. Opt. Soc. Am. A. |

4. | José A. Ferrari, Daniel Perciante, and Alfredo Dubra, “Fast Hankel transform of n-th order,”J. Opt. Soc. Am. A. |

5. | Dennis W. Prather and Shouyuan Shi, “Formulation and application of the finite-difference time-domain method for the analysis of axially symmetric diffractive optical elements,”J. Opt. Soc. Am. A. |

**OCIS Codes**

(050.1970) Diffraction and gratings : Diffractive optics

(070.2580) Fourier optics and signal processing : Paraxial wave optics

(090.1970) Holography : Diffractive optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: May 24, 2002

Revised Manuscript: June 11, 2002

Published: June 17, 2002

**Citation**

D. Zhang, X. Yuan, N. Ngo, and P. Shum, "Fast Hankel transform and its application for studying the propagation of cylindrical electromagnetic fields," Opt. Express **10**, 521-525 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-12-521

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### References

- J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 2, pp. 11-13.
- Shouyuan Shi and Dennis W. Prather, �??Vector-based plane-wave spectrum method for the propagation of cylindrical electromagnetic fields,�?? Opt. Lett. 24, 1445 (1999). [CrossRef]
- Vittorio Magni, Giulio Cerullo, and Sandro De Silvestri, �??High-accuracy fast Hankel transform for optical beam propagation,�?? J. Opt. Soc. Am. A 9, 2031-2033 (1992). [CrossRef]
- José A. Ferrari, Daniel Perciante, and Alfredo Dubra, �??Fast Hankel transform of n-th order,�?? J. Opt. Soc. Am. A. 16, 2581-2582 (1999). [CrossRef]
- Dennis W. Prather and Shouyuan Shi, �??Formulation and application of the finite-difference time-domain method for the analysis of axially symmetric diffractive optical elements,�?? J. Opt. Soc. Am. A 16, 1131-1142 (1999). [CrossRef]

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