Abstract

We present a technique to solve numerically the Fresnel diffraction integral by representing a given complex function as a finite superposition of complex Gaussians. Once an accurate representation of these functions is attained, it is possible to find analytically its diffraction pattern. There are two useful consequences of this representation: first, the analytical results may be used for further theoretical studies and second, it may be used as a versatile and accurate numerical diffraction technique. The use of the technique is illustrated by calculating the intensity distribution in a vicinity of the focal region of an aberrated converging spherical wave emerging from a circular aperture.

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References

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  1. M. Born, and E. Wolf, Principles of Optics 7th ed. (Pergamon, New York, 1980), Chap. 8.
  2. E. H. Linfoot, Recent Advances in Optics (Clarendon Press, Oxford, 1955), Chap. 3.
  3. J. E. A. Landgrave and L. R. Berriel-Valdos, “Sampling expansions for three-dimensional light amplitude distribution in the vicinity of an axial image point,” J. Opt. Soc. Am. A 14(11), 2962–2976 (1997).
    [CrossRef]
  4. J. J. Stamnes and H. Heier, “Scalar and electromagnetic diffraction point-spread functions,” Appl. Opt. 37(17), 3612–3622 (1998).
    [CrossRef]
  5. Y. Li, “Expansions for irradiance distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 23(3), 730–740 (2006).
    [CrossRef]
  6. J. J. Stamnes, “Waves in focal regions,” The Adam Hilger series on Optics and Optoelectronics, (1986).
  7. M. Sypek, “Ligth propagation in the Fresnel region. New numerical approach,” Opt. Commun. 116(1-3), 43–48 (1995).
    [CrossRef]
  8. A. W. Greynolds, “Propagation of generally astigmatic Gaussian beams along skew ray paths,” Proc. SPIE 560, 33–50 (1985).
  9. M. M. Popov, “A new method of computation of wave fields using Gaussian beams,” Wave Motion 4(1), 85–97 (1982).
    [CrossRef]
  10. M. Cywiak, M. Servín, and F. Mendoza-Santoyo, “Wave-front propagation by Gaussian superposition,” Opt. Commun. 195(5-6), 351–359 (2001).
    [CrossRef]
  11. H. P. Hsu, Fourier Analysis (Simon & Schuster, Inc. New York, 1970), Chap. 9.
  12. R. W. Southworth, and S. L. Deleeuw, Digital and computation and numerical methods (McGraw-Hill, N.Y., 1965) Chap. 9.
  13. M. Abramovitz, and I. Stegun, Handbook of mathematical functions in Applied Mathematics series-55, 299 (1972).

2006

2001

M. Cywiak, M. Servín, and F. Mendoza-Santoyo, “Wave-front propagation by Gaussian superposition,” Opt. Commun. 195(5-6), 351–359 (2001).
[CrossRef]

1998

1997

1995

M. Sypek, “Ligth propagation in the Fresnel region. New numerical approach,” Opt. Commun. 116(1-3), 43–48 (1995).
[CrossRef]

1985

A. W. Greynolds, “Propagation of generally astigmatic Gaussian beams along skew ray paths,” Proc. SPIE 560, 33–50 (1985).

1982

M. M. Popov, “A new method of computation of wave fields using Gaussian beams,” Wave Motion 4(1), 85–97 (1982).
[CrossRef]

Berriel-Valdos, L. R.

Cywiak, M.

M. Cywiak, M. Servín, and F. Mendoza-Santoyo, “Wave-front propagation by Gaussian superposition,” Opt. Commun. 195(5-6), 351–359 (2001).
[CrossRef]

Greynolds, A. W.

A. W. Greynolds, “Propagation of generally astigmatic Gaussian beams along skew ray paths,” Proc. SPIE 560, 33–50 (1985).

Heier, H.

Landgrave, J. E. A.

Li, Y.

Mendoza-Santoyo, F.

M. Cywiak, M. Servín, and F. Mendoza-Santoyo, “Wave-front propagation by Gaussian superposition,” Opt. Commun. 195(5-6), 351–359 (2001).
[CrossRef]

Popov, M. M.

M. M. Popov, “A new method of computation of wave fields using Gaussian beams,” Wave Motion 4(1), 85–97 (1982).
[CrossRef]

Servín, M.

M. Cywiak, M. Servín, and F. Mendoza-Santoyo, “Wave-front propagation by Gaussian superposition,” Opt. Commun. 195(5-6), 351–359 (2001).
[CrossRef]

Stamnes, J. J.

Sypek, M.

M. Sypek, “Ligth propagation in the Fresnel region. New numerical approach,” Opt. Commun. 116(1-3), 43–48 (1995).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Commun.

M. Sypek, “Ligth propagation in the Fresnel region. New numerical approach,” Opt. Commun. 116(1-3), 43–48 (1995).
[CrossRef]

M. Cywiak, M. Servín, and F. Mendoza-Santoyo, “Wave-front propagation by Gaussian superposition,” Opt. Commun. 195(5-6), 351–359 (2001).
[CrossRef]

Proc. SPIE

A. W. Greynolds, “Propagation of generally astigmatic Gaussian beams along skew ray paths,” Proc. SPIE 560, 33–50 (1985).

Wave Motion

M. M. Popov, “A new method of computation of wave fields using Gaussian beams,” Wave Motion 4(1), 85–97 (1982).
[CrossRef]

Other

J. J. Stamnes, “Waves in focal regions,” The Adam Hilger series on Optics and Optoelectronics, (1986).

H. P. Hsu, Fourier Analysis (Simon & Schuster, Inc. New York, 1970), Chap. 9.

R. W. Southworth, and S. L. Deleeuw, Digital and computation and numerical methods (McGraw-Hill, N.Y., 1965) Chap. 9.

M. Abramovitz, and I. Stegun, Handbook of mathematical functions in Applied Mathematics series-55, 299 (1972).

M. Born, and E. Wolf, Principles of Optics 7th ed. (Pergamon, New York, 1980), Chap. 8.

E. H. Linfoot, Recent Advances in Optics (Clarendon Press, Oxford, 1955), Chap. 3.

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Figures (12)

Fig. 1
Fig. 1

Schematic diagram of the physical situation.

Fig. 8
Fig. 8

Amplitude of Eq. (21) after performing the integral compared vs. its representation by superposition of 60 Gaussians for one of the line scans. The solid line corresponds to the integral, the circles to the superposition of Gaussians.

Fig. 2
Fig. 2

(A). Isophotes for an aberration-free focusing wave, φ = 0 ; a = b = c 1 = c 2 = 0. (B). Absolute value of the difference of the isophotes obtained by Gaussian sampling and by Lommel integrals.

Fig. 3
Fig. 3

Isophotes for a tilted focusing wave, φ = 0 , a = π ; b = c 1 = c 2 = 0.

Fig. 4
Fig. 4

Isophotes for a tilted focusing wave. φ = π / 2 , a = π ; b = c 1 = c 2 = 0.

Fig. 5
Fig. 5

Isophotes for a focused wave with coma, φ = 0 , c 1 = π ; a = b = c 2 = 0.

Fig. 6
Fig. 6

Isophotes for a focused wave with coma, φ = π / 4 ;     c 1 = π ; a = b = c 2 = 0 .

Fig. 7
Fig. 7

Isophotes for a focused wave with coma, φ = π / 2 , c 1 = π ; a = b = c 2 = 0 .

Fig. 9
Fig. 9

Plot of the absolute value of the difference between the plots of Fig. 8.

Fig. 10
Fig. 10

Sinusoidal transmittance grating (solid line), superposition of Gaussians (red circles).

Fig. 11
Fig. 11

Intensity distribution at the plane of detection due to the transmittance grating.

Fig. 12
Fig. 12

Intensity distribution at the plane of detection due to the transmittance grating in a region around the first diffraction order.

Equations (32)

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f ( x ) = exp ( ( x + A / 2 ) 2 σ 2 ) + exp ( ( x ) 2 σ 2 ) + exp ( ( x A / 2 ) 2 σ 2 ) .
f ( x ) = exp ( ( x + 2 A / 4 ) 2 σ 2 ) + exp ( ( x + A / 4 ) 2 σ 2 ) + exp ( ( x ) 2 σ 2 ) + exp ( ( x A / 4 ) 2 σ 2 ) + exp ( ( x 2 A / 4 ) 2 σ 2 ) ,
f ( x ) = n = N 1 2 N 1 2 exp ( ( x n A N 1 ) 2 ( A N 1 ) 2 ) ,
F ( u ) = f ( x ) exp ( i 2 π u x ) d x ,
F ( u ) = π A N 1 exp ( ( π A u N 1 ) 2 ) n = N 1 2 N 1 2 exp ( i 2 π n A u N 1 ) .
F ( u ) = π A N 1 exp ( ( π A u N 1 ) 2 ) sin ( N N 1 π A u ) sin ( π A u N 1 ) .
F ( u ) = π A sin ( π A u ) π A u .
f s ( x ) = n = f ( n Δ ) g ( x n Δ ) ,
F s ( u ) = G ( u ) n = f ( n Δ ) exp ( i 2 π n Δ u ) ,
F s ( u ) = 1 Δ G ( u ) n = F ( u n / Δ ) .
g ( x ) = A exp ( x 2 σ 2 ) ,
G ( u ) = A π σ exp ( π 2 σ 2 u 2 ) .
F s ( u ) = A Δ π σ exp ( π 2 σ 2 u 2 ) n = F ( u n / Δ ) .
1 / π σ = K U M     and       1 / Δ U M = K ( 1 / π σ )       with       K > 1 ,
Ψ I N ( x , y ) = c i r c ( r , a ) g ( x , y ) exp [ i π λ f ( x 2 + y 2 ) ] ,
ψ ( ξ , μ ) = exp ( i 2 π λ z ) i λ z ψ I N ( x , y ) exp [ i π λ z { ( x ξ ) 2 + ( y η ) 2 } ] d x d y .
x = r cos ( θ ) ,     y = r sin ( θ ) ;     ξ = ρ cos ( φ ) , η = ρ sin ( φ ) ,
Ψ ( ρ , φ ) = a 2 exp ( i 2 π λ ( f + Δ z ) ) i λ ( f + Δ z ) exp ( i π λ ( f + Δ z ) ρ 2 ) × 0 1 0 2 π g ( s , θ ) exp [ i π a 2 Δ z λ f ( f + Δ z ) s 2 ] exp [ i 2 π a λ ( f + Δ z ) s ρ cos ( θ φ ) ] s d s d θ .
u = 2 π a 2 Δ z λ f 2 ;     v = 2 π a ρ λ f .
Ψ ( ρ , φ ) = a 2 exp ( i 2 π λ ( f + Δ z ) ) i λ f exp ( i π λ f ρ 2 ) × 0 1 0 2 π g ( s , θ ) exp [ i u s 2 2 ] exp [ i v s cos ( θ φ ) ] s d s d θ .
s 0 2 π g ( s , θ ) exp [ i v s cos ( θ φ ) ]     d θ = n = 0 N G A n exp ( ( s s n ) 2 σ 2 ) ,
Ψ ( ρ , φ ) = a 2 exp ( i 2 π λ ( f + Δ z ) ) i λ f exp ( i π λ f ρ 2 ) × n = 0 N G A n 0 1 exp [ i u s 2 2 ] exp ( ( s s n ) 2 σ 2 ) d s .
Ψ ( ρ , φ ) = n = 0 N G A n exp ( s n 2 σ 2 ) exp ( 2 s n 2 σ 2 ( i σ 2 u + 2 ) ) × 0 1 exp [ ( i σ 2 u + 2 2 σ 2 ) ( s 2 s n i σ 2 u + 2 ) 2 ] d s .
p = s 2 s n i σ 2 u + 2 .
Ψ ( ρ , φ ) = n = 0 N G A n exp ( ( i σ 2 u + 2 ) s n 2 + 2 s n 2 σ 2 ( i σ 2 u + 2 ) ) ( 2 s n i σ 2 u + 2 ) 1 2 s n i σ 2 u + 2 exp [ i σ 2 u + 2 2 σ 2 p 2 ] d p .
q = i σ 2 u + 2 2 σ 2 p ,
Ψ ( ρ , φ ) = n = 0 N G A n exp ( i u s n 2 i σ 2 u + 2 ) 2 σ 2 i σ 2 u + 2 ( 2 s n i σ 2 u + 2 ) i σ 2 u + 2 2 σ 2 1 2 s n i σ 2 u + 2 i σ 2 u + 2 2 σ 2 exp [ q 2 ] d q .
Ψ ( ρ , φ ) = π 2 2 σ 2 i σ 2 u + 2 n = 0 N G A n exp ( i u s n 2 ( i σ 2 u + 2 ) ) × { e r f [ ( 1 2 s n i σ 2 u + 2 ) i σ 2 u + 2 2 σ 2 ] + e r f [ 2 s n i σ 2 u + 2 i σ 2 u + 2 2 σ 2 ] } .
e r f ( x + i y ) = e r f ( x ) + exp ( x 2 ) 2 π x [ 1 cos ( 2 x y ) + i sin ( 2 x y ) + 2 π exp ( x 2 ) n = 1 exp ( n 2 4 ) n 2 + 4 x 2 [ f ( n , x , y ) + i g ( n , x , y ) ] ] ,
f ( n , x , y ) = 2 x 2 x cosh ( n y ) cos ( 2 x y ) + n sinh ( n y ) sin ( 2 x y ) ,
g ( n , x , y ) = 2 x cosh ( n y ) sin ( 2 x y ) + n sinh ( n y ) cos ( 2 x y ) .
g ( s , θ ) = exp [ i a s cos ( θ ) + i b s sin ( θ ) + i c 1 s 3 cos ( θ ) + i c 2 s 3 sin ( θ ) ]     ,

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