Abstract

Accurate simulation of scalar optical diffraction requires consideration of the sampling requirement for the phase chirp function that appears in the Fresnel diffraction expression. We describe three sampling regimes for FFT-based propagation approaches: ideally sampled, oversampled, and undersampled. Ideal sampling, where the chirp and its FFT both have values that match analytic chirp expressions, usually provides the most accurate results but can be difficult to realize in practical simulations. Under- or oversampling leads to a reduction in the available source plane support size, the available source bandwidth, or the available observation support size, depending on the approach and simulation scenario. We discuss three Fresnel propagation approaches: the impulse response/transfer function (angular spectrum) method, the single FFT (direct) method, and the two-step method. With illustrations and simulation examples we show the form of the sampled chirp functions and their discrete transforms, common relationships between the three methods under ideal sampling conditions, and define conditions and consequences to be considered when using nonideal sampling. The analysis is extended to describe the sampling limitations for the more exact Rayleigh–Sommerfeld diffraction solution.

© 2009 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts and Company, 2005).
  2. A. Papoulis, “Pulse compression, fiber communications and diffraction, a unified approach,” J. Opt. Soc. Am. A 11, 3-13(1994).
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  3. F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293-297 (1981).
    [CrossRef]
  4. D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculations,” Opt. Commun. 164, 233-245 (1999).
    [CrossRef]
  5. L. Onural, “Sampling of the diffraction field,” Appl. Opt. 39, 5929-5935 (2000).
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  6. R. Rao, “Statistics of the fractal structure and phase singularity of a plane light wave propagation in atmospheric turbulence,” Appl. Opt. 47, 269-276 (2008).
    [CrossRef] [PubMed]
  7. D. P. Kelly and B. M. Hennelly, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
    [CrossRef]
  8. A. Stern and B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. 43, 239-250(2004).
    [CrossRef]
  9. A. Brodzik, “On the Fourier transform of finite chirps,” IEEE Signal Process. Lett. 13, 541-544 (2006).
    [CrossRef]
  10. S. M. Flatte and J. S. Gerber, “Irradiance-variance behavior by numerical simulation for plane-wave and spherical-wave optical propagation through strong turbulence,” J. Opt. Soc. Am. A 17, 1092-1097 (2000).
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  11. A. Belmonte, “Feasibilty study for the simulation of beam propagation: consideration of coherent lidar performance,” Appl. Opt. 39, 5426-5445 (2000).
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    [CrossRef]
  13. Computation time for the FFT algorithm is minimized when N is a power of 2, although depending on the value, other lengths can be computed nearly as fast. The value of N=250 was chosen here to simplify our illustration.
  14. The size is chosen so an odd number of samples spans the aperture and one sample can be placed on the optical axis. In this case 51 samples×0.2 cm=10.2cm width.
  15. The analytic Fourier transform of the chirp q(x,y)=exp(j(k/2z)(x2+y2)) is Q(fX,fY)=jλz exp(−jπλ(fX2+fY2)). In starting with a sampled version of Q, the inverse FFT of Q typically needs to be multiplied by N2ΔfX2 to reproduce the correct magnitude for the sampled version of q (assumes 1/N2 scaling on inverse 2D FFT).
  16. C. Rydberg and J. Bengtsson, “Efficient numerical representation of the optical field for the propagation of partially coherent radiaction with a specified spatial and temporal coherence function,” J. Opt. Soc. Am. A 23, 1616-1625 (2006).
    [CrossRef]
  17. X. Deng, B. Bihari, J. Gan, F. Zhao, and R. T. Chen, “Fast algorithm for chirp transforms with zooming-in ability and its applications,” J. Opt. Soc. Am. A 17, 762-771 (2000).
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  18. L. Onural, “Exact analysis of the effects of sampling of the scalar diffraction field,” J. Opt. Soc. Am. A 24, 359-367 (2007).
    [CrossRef]

2008 (1)

2007 (1)

2006 (3)

C. Rydberg and J. Bengtsson, “Efficient numerical representation of the optical field for the propagation of partially coherent radiaction with a specified spatial and temporal coherence function,” J. Opt. Soc. Am. A 23, 1616-1625 (2006).
[CrossRef]

D. P. Kelly and B. M. Hennelly, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

A. Brodzik, “On the Fourier transform of finite chirps,” IEEE Signal Process. Lett. 13, 541-544 (2006).
[CrossRef]

2005 (1)

2004 (1)

A. Stern and B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. 43, 239-250(2004).
[CrossRef]

2000 (4)

1999 (1)

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

1994 (1)

1981 (1)

F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293-297 (1981).
[CrossRef]

Belmonte, A.

Bengtsson, J.

Bernardo, L. M.

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

Bihari, B.

Brodzik, A.

A. Brodzik, “On the Fourier transform of finite chirps,” IEEE Signal Process. Lett. 13, 541-544 (2006).
[CrossRef]

Chen, R. T.

Deng, X.

Ferreira, C.

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

Flatte, S. M.

Gan, J.

Garcia, J.

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

Gerber, J. S.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts and Company, 2005).

Gori, F.

F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293-297 (1981).
[CrossRef]

Hennelly, B. M.

Javidi, B.

A. Stern and B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. 43, 239-250(2004).
[CrossRef]

Kelly, D. P.

D. P. Kelly and B. M. Hennelly, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

Marinho, F.

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

Mas, D.

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

Onural, L.

Papoulis, A.

Rao, R.

Rydberg, C.

Sheridan, J. T.

Stern, A.

A. Stern and B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. 43, 239-250(2004).
[CrossRef]

Zhao, F.

Appl. Opt. (3)

IEEE Signal Process. Lett. (1)

A. Brodzik, “On the Fourier transform of finite chirps,” IEEE Signal Process. Lett. 13, 541-544 (2006).
[CrossRef]

J. Opt. Soc. Am. A (6)

Opt. Commun. (2)

F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293-297 (1981).
[CrossRef]

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

Opt. Eng. (2)

D. P. Kelly and B. M. Hennelly, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

A. Stern and B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. 43, 239-250(2004).
[CrossRef]

Other (4)

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts and Company, 2005).

Computation time for the FFT algorithm is minimized when N is a power of 2, although depending on the value, other lengths can be computed nearly as fast. The value of N=250 was chosen here to simplify our illustration.

The size is chosen so an odd number of samples spans the aperture and one sample can be placed on the optical axis. In this case 51 samples×0.2 cm=10.2cm width.

The analytic Fourier transform of the chirp q(x,y)=exp(j(k/2z)(x2+y2)) is Q(fX,fY)=jλz exp(−jπλ(fX2+fY2)). In starting with a sampled version of Q, the inverse FFT of Q typically needs to be multiplied by N2ΔfX2 to reproduce the correct magnitude for the sampled version of q (assumes 1/N2 scaling on inverse 2D FFT).

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Figures (7)

Fig. 1
Fig. 1

Sampled phase profiles (solid) for (a)  h X undersampled by a factor of three ( λ z / Δ x L = 0.33 ) and (b)  H X undersampled by a factor of two ( λ z / Δ x L = 2 ). Analytic phase profiles (dashed) shown for comparison.

Fig. 2
Fig. 2

Magnitude and phase profiles (solid curves) for H X DFT , the discrete Fourier transform of h X , where h X is (a) slightly undersampled ( λ z / Δ x L = 0.75 ), (b) ideally sampled ( λ z / Δ x L = 1 ), and (c) oversampled by a factor of 2 ( λ z / Δ x L = 2 ). Analytic transform profiles (dashed) shown for comparison. Phase profiles are the downward concave curves.

Fig. 3
Fig. 3

Irradiance profiles for IR [(a), (c), (e), and (g)] and TF [(b), (d), (f), and (h)] simulations corresponding to cases listed in Table 2.

Fig. 4
Fig. 4

(a) Field magnitude and (b) phase profiles for numerical integration result (dashed) and TF simulation approach (solid) where z = 1000 m ( λ z / Δ x L = 0.5 ).

Fig. 5
Fig. 5

(a) Field magnitude and (b) phase profiles for numerical integration result (dashed) and IR simulation approach (solid) where z = 2000 m ( λ z / Δ x L = 1 , ideal sampling).

Fig. 6
Fig. 6

(a) Field magnitude and (b) phase profiles for numerical integration result (dashed) and IR simulation approach (solid) where z = 20 , 000 m ( λ z / Δ x L = 10 ).

Fig. 7
Fig. 7

(a) Irradiance, (b) field magnitude, and (c) phase profiles for numerical integration result (dashed) and two-step simulation approach (solid) where z = 20 , 000 m and L 1 = 0.5 m , L 2 = 0.8 m .

Tables (2)

Tables Icon

Table 1 Parameters and Sampling Regimes for IR and TF Simulation Example

Tables Icon

Table 2 Parameters for Two-Step Simulation Example

Equations (36)

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U 2 ( x , y ) = U 1 ( x , y ) h ( x , y ) ,
h ( x , y ) = e j k z j λ z exp ( j k 2 z ( x 2 + y 2 ) ) ,
U 2 ( x , y ) = F 1 { F { U 1 ( x , y ) } F { h ( x , y ) } } ,
H ( f X , f Y ) = e j k z exp ( j π λ z ( f X 2 + f Y 2 ) ) .
U 2 ( x , y ) = F 1 { F { U 1 ( x , y ) } H ( f X , f Y ) } .
ϕ h ( x , y ) = k 2 z ( x 2 + y 2 ) .
Δ x | ϕ h x | max π ,
Δ x λ z 2 | x max | .
Δ x λ z L .
D Undersampled h = λ z Δ x .
ϕ H ( f X , f Y ) = π λ z ( f X 2 + f Y 2 ) ,
Δ f X | ϕ H f X | max π ,
Δ f X 1 λ z 2 | f X max | ,
Δ x λ z L .
B Undersampled H = L λ z .
Δ x λ z L .
h X ( x ) = 1 j λ z exp { j k 2 z x 2 } ,
H X DFT ( p Δ f X ) = n = N / 2 N / 2 1 h X ( n Δ x ) exp ( j 2 π p Δ f X n Δ x ) Δ x ,
B DFT { Oversampled h } L λ z .
D DFT 1 { Oversampled H } λ z Δ x ,
B 1 L λ z .
D 2 D 1 + λ z Δ x .
U 2 , X ( x 2 ) = 1 j λ z n U 1 , X ( n Δ x 1 ) exp ( j k 2 z ( n Δ x 1 x 2 ) 2 ) Δ x 1 ,
U 2 ( x 2 , y 2 ) = e j k z j λ z exp ( j k 2 z ( x 2 2 + y 2 2 ) ) F { U 1 ( x 1 , y 1 ) exp ( j k 2 z ( x 1 2 + y 1 2 ) ) } .
Δ x 1 λ z L 1 .
Δ x 1 λ z L 1 .
Δ x 1 λ z L 1 .
U 2 ( x 2 , y 2 ) = z 2 z 1 e j k ( z 1 z 2 ) exp ( j k 2 z 2 ( x 2 2 + y 2 2 ) ) × F 1 { exp ( j k 2 ( 1 z 1 1 z 2 ) ( x d 2 + y d 2 ) ) F { U 1 ( y 1 , x 1 ) exp ( j k 2 z 1 ( x 1 2 + y 1 2 ) ) } } .
Δ x 1 Δ x 2 = L 1 L 2 = z 1 z 2 ,
z 1 = z ( L 1 L 1 L 2 ) , z 2 = z ( L 2 L 1 L 2 ) .
Δ x 1 λ z | L 1 L 2 | ,
Δ x 1 L 1 L 2 λ z | L 1 L 2 | .
Δ x 1 λ z L 2 .
h ( x , y ) = z j λ exp ( j k z 2 + x 2 + y 2 ) z 2 + x 2 + y 2 .
H ( f X , f Y ) = exp ( j 2 π z λ 1 ( λ f X ) 2 ( λ f Y ) 2 ) ,
Δ x λ ( z 2 + ( L / 2 ) 2 ) 1 / 2 L .

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