Abstract

The numerical calculation of the Rayleigh–Sommerfeld diffraction integral is investigated. The implementation of a fast-Fourier-transform (FFT) based direct integration (FFT-DI) method is presented, and Simpson's rule is used to improve the calculation accuracy. The sampling interval, the size of the computation window, and their influence on numerical accuracy and on computational complexity are discussed for the FFT-DI and the FFT-based angular spectrum (FFT-AS) methods. The performance of the FFT-DI method is verified by numerical simulation and compared with that of the FFT-AS method.

© 2006 Optical Society of America

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References

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2002

J. Li, Z. Fan, and Y. Fu, "The FFT calculation for Fresnel diffraction and energy conservation criterion of sampling quality," in Lasers in Material Processing and Manufacturing, S. Deng, T. Okada, K. Behler, and X. Wang, eds., Proc. SPIE 4915, 180-186 (2002).
[CrossRef]

2001

1999

A. Dubra and J. A. Ferrari, "Diffracted field by an arbitrary aperture," Am. J. Phys. 67, 87-92 (1999).
[CrossRef]

1998

1991

1989

1984

1981

1979

J. E. Harvey, "Fourier treatment of near-field scalar diffraction theory," Am. J. Phys. 47, 974-980 (1979).
[CrossRef]

1973

1968

1964

1962

1961

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), Chap. 8.

Briggs, W. L.

W. L. Briggs and V. E. Henson, The DFT: An Owner's Manual for the Discrete Fourier Transform (Society for Industrial and Applied Mathematics, 1995).
[CrossRef]

Delen, N.

Dubra, A.

A. Dubra and J. A. Ferrari, "Diffracted field by an arbitrary aperture," Am. J. Phys. 67, 87-92 (1999).
[CrossRef]

Fan, Z.

J. Li, Z. Fan, and Y. Fu, "The FFT calculation for Fresnel diffraction and energy conservation criterion of sampling quality," in Lasers in Material Processing and Manufacturing, S. Deng, T. Okada, K. Behler, and X. Wang, eds., Proc. SPIE 4915, 180-186 (2002).
[CrossRef]

Ferrari, J. A.

A. Dubra and J. A. Ferrari, "Diffracted field by an arbitrary aperture," Am. J. Phys. 67, 87-92 (1999).
[CrossRef]

Fu, Y.

J. Li, Z. Fan, and Y. Fu, "The FFT calculation for Fresnel diffraction and energy conservation criterion of sampling quality," in Lasers in Material Processing and Manufacturing, S. Deng, T. Okada, K. Behler, and X. Wang, eds., Proc. SPIE 4915, 180-186 (2002).
[CrossRef]

Harvey, J. E.

J. E. Harvey, "Fourier treatment of near-field scalar diffraction theory," Am. J. Phys. 47, 974-980 (1979).
[CrossRef]

Hecht, E.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, 1987), Chap. 10.

Henson, V. E.

W. L. Briggs and V. E. Henson, The DFT: An Owner's Manual for the Discrete Fourier Transform (Society for Industrial and Applied Mathematics, 1995).
[CrossRef]

Heurtley, J. C.

Hooker, B.

Hudson, J. A.

Kopp, C.

C. Kopp and P. Meyrueis, "Near-field Fresnel diffraction: improvement of a numerical propagator," Opt. Commun. 158, 7-10 (1998).
[CrossRef]

Lalor, E.

Li, J.

J. Li, Z. Fan, and Y. Fu, "The FFT calculation for Fresnel diffraction and energy conservation criterion of sampling quality," in Lasers in Material Processing and Manufacturing, S. Deng, T. Okada, K. Behler, and X. Wang, eds., Proc. SPIE 4915, 180-186 (2002).
[CrossRef]

Marchand, E. W.

Meyrueis, P.

C. Kopp and P. Meyrueis, "Near-field Fresnel diffraction: improvement of a numerical propagator," Opt. Commun. 158, 7-10 (1998).
[CrossRef]

Mukunda, N.

Osterberg, H.

Pozrikidis, C.

C. Pozrikidis, Numerical Computation in Science and Engineering (Oxford U. Press, 1998), Chap. 7.

Rutt, H. N.

Smith, L. W.

Southwell, W. H.

Stamnes, J. J.

Steane, A. M.

Totzeck, M.

Wolf, E.

Am. J. Phys.

J. E. Harvey, "Fourier treatment of near-field scalar diffraction theory," Am. J. Phys. 47, 974-980 (1979).
[CrossRef]

A. Dubra and J. A. Ferrari, "Diffracted field by an arbitrary aperture," Am. J. Phys. 67, 87-92 (1999).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

C. Kopp and P. Meyrueis, "Near-field Fresnel diffraction: improvement of a numerical propagator," Opt. Commun. 158, 7-10 (1998).
[CrossRef]

Proc. SPIE

J. Li, Z. Fan, and Y. Fu, "The FFT calculation for Fresnel diffraction and energy conservation criterion of sampling quality," in Lasers in Material Processing and Manufacturing, S. Deng, T. Okada, K. Behler, and X. Wang, eds., Proc. SPIE 4915, 180-186 (2002).
[CrossRef]

Other

C. Pozrikidis, Numerical Computation in Science and Engineering (Oxford U. Press, 1998), Chap. 7.

W. L. Briggs and V. E. Henson, The DFT: An Owner's Manual for the Discrete Fourier Transform (Society for Industrial and Applied Mathematics, 1995).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), Chap. 8.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, 1987), Chap. 10.

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Figures (8)

Fig. 1
Fig. 1

Illustration of the coordinate system of the Rayleigh–Sommerfeld diffraction theory.

Fig. 2
Fig. 2

Magnitude of g ( x , y , z ) at several observation planes.

Fig. 3
Fig. 3

Oscillating period of g ( x , y , z ) at several observation planes.

Fig. 4
Fig. 4

Axial intensity distribution behind a circular aperture.

Fig. 5
Fig. 5

Calculation errors of the FFT-AS method and the FFT-DI method: (a) FFT-AS and (b) FFT-DI methods.

Fig. 6
Fig. 6

Calculation errors of the FFT-AS method for the sampling numbers and computation window sizes shown.

Fig. 7
Fig. 7

Calculation errors of the FFT-DI method for several sampling numbers.

Fig. 8
Fig. 8

Simulation results of diffraction pattern of a single slit.

Tables (1)

Tables Icon

Table 1 Comparison of Calculation Speeds

Equations (162)

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2 U x 2 + 2 U y 2 + 2 U z 2 + k 2 U = 0 ,
A ( α , β , z ) = A ( α , β , 0 ) G ( α , β , z ) ,
A ( α , β , z )
A ( α , β , z ) = F { U ( x , y , z ) } = U ( x , y , z ) exp ( j α x j β y ) d x d y ,
G ( α , β , z ) = exp ( j k 2 α 2 β 2 z )
U ( x , y , z )
A ( α , β , z )
U ( x , y , z ) = F 1 { A ( α , β , z ) }
= 1 4 π 2 A ( α , β , 0 ) exp ( j α x + j β y + j k 2 α 2 β 2 z ) d α .
U ( x , y , z )
Q = I F F T 2 { F F T 2 { U ( x m , y n , 0 ) } ⋅ ×   G ( α m , β n , z ) } ,
U ( x m , y n , 0 )
G ( α m , β n , z )
U ( x , y , 0 )
G ( α , β , z )
G ( α , β , z )
g ( x , y , z ) = 1 4 π 2 G ( α , β , z ) exp ( j α x + j β y ) d α d β
= 1 2 π exp ( j k r ) r z r ( 1 r j k ) ,
r = x 2 + y 2 + z 2
U ( x , y , z )
U ( x , y , 0 )
g ( x , y , z )
U ( x , y , z ) = A U ( ς , η , 0 ) g ( x ς , y η , z ) d ς d η
= A U ( ς , η , 0 ) exp ( j k r ) 2 π r z r ( 1 r j k ) dςd η ,
r = ( x ς ) 2 + ( y η ) 2 + z 2
U ( ς , η , 0 )
N × N
( x m , y n , z )
U ( x m , y n , z ) = i = 1 N j = 1 N U ( ς i , η j , 0 ) g ( x m ς i , y n η j , z ) Δ ς Δ η ,
Δς
Δ η
U ( ς i , η j , 0 )
g ( x m , y n , z )
N 2
S = I F F T 2 [ F F T 2 ( U ) ⋅ ×   F F T 2 ( H ) ] Δ ς Δ η ,
U = [ U 0 0 0 0 ] ( 2 N - 1 ) × ( 2 N - 1 ) = [ U ( ς 1 , η 1 , 0 ) U ( ς 1 , η N , 0 ) | | 0 N × ( N - 1 ) U ( ς N , η 1 , 0 ) U ( ς N , η N , 0 ) | - - - | - 0 ( N - 1 ) × N | 0 ( N - 1 ) × ( N - 1 ) ] ,
H = [ g ( X 1 , Y 1 , z ) g ( X 1 , Y 2 N 1 , z ) g ( X 2 N 1 , Y 1 , z ) g ( X 2 N 1 , Y 2 N 1 , z ) ] ( 2 N - 1 ) × ( 2 N - 1 ) ,
X j = { x 1 ς N + 1 j j = 1 , , N 1 x j N + 1 ς 1 j = N , , 2 N 1 ,
Y j = { y 1 η N + 1 j j = 1 , , N 1 y j N + 1 η 1 j = N , , 2 N 1 .
U ( ς i , η j , 0 )
( 2 N 1 ) × ( 2 N 1 )
( 2 N 1 ) × ( 2 N 1 )
N × N
U ( x m , y n , z ) = S m + N , n + N .
E = O ( Δ ς 2 ) + O ( Δ η 2 ) .
U = [ W  ⋅ × U 0 O O O ] ( 2 N - 1 ) × ( 2 N - 1 ) ,
W = B T B ,
B = 1 / 3 [ 1 4 2 4 2 2 4 1 ]
E = O ( Δ ς 4 ) + O ( Δ η 4 ) ,
g ( x , y , z )
U ( ς , η , 0 )
U ( ς , η , 0 )
A ( α , β , 0 ) | | α | > α M   or   | β | > β M = 0
( 2 α M ) 1
( 2 β M ) 1
g ( x , y , z )
g ( x , y , z )
g ( x , y , z )
g ( x , y , z )
g ( x , y , z )
z = 0.5 , 1 , 2.5 , 5 , 10 , 25 , 50 , 100 μ m
g ( x , y , z )
Δ ρ
2 π
k ( ρ + Δρ ) 2 + z 2 k ρ 2 + z 2 = 2 π ,
ρ = x 2 + y 2
Δ ρ = λ 2 + ρ 2 + 2 λ ρ 2 + z 2 ρ .
Δρ
g ( x , y , z )
g ( x , y , z )
g ( x , y , z )
g ( x , y , z )
Δ ρ
g ( x , y , z )
z min
ρ max
Δ ρ min
Δ ρ min / 2
Q ( x m , y n , z )
U ( x , y , z )
U ( x , y , z )
Q ( x m , y n , z ) = i = j = U ( x m + i X , y n + j Y , z ) ,
g ( x , y , z )
| g ( x , y , z ) | < ε | g ( 0 , 0 , z ) | ,
| 1 r 2 ( 1 r j k ) | < ε | 1 z 2 ( 1 z j k ) | ,
r = ρ 2 + z 2
1 r 2 < ε 1 z 2 ,
ρ > z [ ( 1 / ε ) 1 ] 1 / 2 .
( 2 ρ + a ) × ( 2 ρ + b )
a × b
A = F F T ( U ) × G
N × N
C 1 = C a + C b + C c + C d = O ( N 2 log 2 N ) + O ( N 2 ) + O ( N 2 ) + O ( N 2 log 2 N ) .
N 2
O ( N 4 )
O ( N 2 )
2 m × 2 m
N F
C 2 = C a + C b + C c + C d + C e
= O ( N F       2 log 2 N F ) + O ( N F       2 ) + O ( N F       2 log 2 N F ) + O ( N F       2 ) + O ( N F       2 log 2 N F ) ,
O ( N 4 )
N × N
2 N × 2 N
2 × 2
2 × 2
N × N = ( N 1 / 2 ) × ( N 1 / 2 )
N 1
4 × O ( N 2 log 2 N ) = O [ N 1     2 ( log 2 N 1 1 ) ] ,
O ( N 1       2 log 2 N 1 )
h ( x , z ) = j k z 2 r H 1     ( 1 ) ( k r ) ,
H 1     ( 1 ) ( k r )
r = x 2 + z 2
U ( x , 0 )
h ( x , z )
U ( x , z ) = A U ( ς , 0 ) h ( x ς , z ) d ς = A U ( ς , 0 ) j k z 2 r H 1     ( 1 ) ( k r ) d ς ,
r = ( x ς ) 2 + z 2
S = I F F T [ F F T ( U ) ⋅ ×   F F T ( H ) ] Δ ς ,
U = [ U ( ς 1 , 0 ) U ( ς N , 0 ) 0 0 ] 2 N 1 ,
H = j k z 2 [ H 1     ( 1 ) ( k r 1 ) r 1 H 1     ( 1 ) ( k r 2 N 1 ) r 2 N 1 ] 2 N 1 ,
r j = { ( x 1 ς N + 1 j ) 2 + z 2 , j = 1 , , N 1 ( x j N + 1 ς 1 ) 2 + z 2 , j = N , , 2 N 1 ,
S ( N : 2 N 1 )
U ( z ) = U 0 z [ exp ( j k z ) z exp ( j k z 2 + a 2 ) z 2 + a 2 ] ,
U 0
λ = 0.5 μ m
a = 10 λ = 5 μ m
0.1 λ
2 × 2
| U U AS | 2
| U U DI | 2
U AS
U DI
128 × 128
512 × 512
2 × 2
2 × 2
6 × 6
128 × 128
| U U DI | 2
z = 8 μ m
128 × 128
1024 × 1024
0.5 μ m
10 μ m
80 μ m
z = 5 , 50 , 250 μ m
( 5 μ m )
( 50 μ m )
( 250 μ m )
1.4   GHz
512   Mbytes
10 μ m × 10 μ m
512 × 512
3.36   s
1024 × 1024
16 × 16
10.80   s
2048 × 2048
1024 × 1024
8   Mbytes
16   Mbytes
g ( x , y , z )
g ( x , y , z )

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