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

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 2, pp. 11-13.
  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]

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Figures (4)

Fig. 1.
Fig. 1.

(a)First-order Hankel transform g(y) of f(r)=r (0≤r≤1) for a constant C = 20π ; (b) Difference (error) between the analytical transform and the results obtained with the numerical first-order fast hankel transformations.

Fig.2.
Fig.2.

(a)Second-order Hankel transform g(y) of f(r)=r 4 (0≤r≤1) for a constant C = 20π ; (b) Difference (error) between the analytical transform and the results obtained with the numerical second-order fast hankel transformations.

Fig. 3
Fig. 3

Electric field magnitude at the emergent plane of the lens.

Fig. 4
Fig. 4

Electric field magnitude in the focal plane of the lens.

Equations (21)

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g γ ( ρ ) = 0 1 f ( r ) · J γ ( C · ) · rdr ( 0 ρ 1 )
g γ ( ρ ) = H γ { f ( r ) }
r γ + 1 J γ ( r ) = d dr [ r γ + 1 · J γ + 1 ( r ) ]
ξ n ξ n + 1 f ( r ) · J γ ( C · ) · r · dr
= f ( r n ) C · r n γ ρ · [ ξ n + 1 γ + 1 · J γ + 1 ( · ξ n + 1 ) ξ n γ + 1 · J γ + 1 ( · ξ n ) ]
g γ ( ρ m ) = 1 C ρ m F { F { φ n } · F 1 { j ( γ + 1 ) n } }
j ( γ + 1 ) n = J γ + 1 { C r 0 exp [ α ( n + 1 N ) ] }
φ n = { k 0 F 0 exp [ α ( γ + 1 ) ( 1 N ) ] for n = 0 F n exp [ α ( γ + 1 ) ( n + 1 N ) ] for n = 1,2 , , N 1 0 for = N , N + 1 , , 2 N 1
E ρ m ( l · Δ ρ , z 0 ) = ( 1 ) m + 1 2 Δ r 2
× i = 0 M i { [ E r m ( i · Δ r , 0 ) + E ϕ m ( i · Δ r , 0 ) ]
× k 0 2 H m + 1 { J m + 1 ( k · l k 0 Δ ρ ) exp ( j k z z 0 ) }
+ [ E r m ( i · Δ r , 0 ) E ϕ m ( i · Δ r , 0 ) ]
× k 0 2 H m 1 { J m 1 ( k · l k 0 Δ ρ ) exp ( j k z z 0 ) } }
E ϕ m ( l · Δ ρ , z 0 ) = ( 1 ) m + 1 2 Δ r 2
× i = 0 M i { [ E ϕ m ( i · Δ r , 0 ) + E r m ( i · Δ r , 0 ) ]
× k 0 2 H m + 1 { J m + 1 ( k · l k 0 Δ ρ ) exp ( j k z z 0 ) }
+ [ E ϕ m ( i · Δ r , 0 ) E r m ( i · Δ r , 0 ) ]
× k 0 2 H m 1 { J m 1 ( k · l k 0 Δ ρ ) exp ( j k z z 0 ) } }
E z m ( l · Δ ρ , z 0 ) = ( 1 ) m + 1 · ( Δ r ) 2
× i = 0 M i · { E z m ( i · Δ r , 0 )
× k 0 2 H m { J m ( k · l k 0 Δ ρ ) × exp ( j k z · z 0 ) }

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