Abstract

This paper describes a new Fourier propagator for computing the impulse response of an optical system with a curved focal plane array, while including terms ignored in Fresnel and Fraunhofer calculations. The propagator includes a Rayleigh-Sommerfeld diffraction formula calculation from a distant point through the optical system to its image point predicted by geometric optics on a spherical surface. The propagator then approximates the neighboring field points via the traditional binomial approximation of the Taylor series expansion around that field point. This technique results in a propagator that combines the speed of a Fourier transform operation with the accuracy of the Rayleigh-Sommerfeld diffraction formula calculation and extends Fourier optics to cases where the receiver plane is a curved surface. Bounds on the phase error introduced by the approximations are derived, which show it should be more widely applicable than traditional Fresnel propagators. Guidance on how to sample the pupil and detector planes of a simulated imaging system is provided. This report concludes by showing examples of the diffraction patterns computed by the new technique compared to those computed using the Rayleigh-Sommerfeld technique in order to demonstrate the utility of the propagator.

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References

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  1. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  2. D. Voelz, Computational Fourier Optics, A MATLAB Tutorial, (SPIE, 2011).
  3. R. ZEMAX, “ZEMAX User's Manual,” Radiant ZEMAX LLC (RZ), Kirkland, 2011.
  4. C.-Y. Hwang, S. Oh, I.-K. Jeong, and H. Kim, “Stepwise angular spectrum method for curved surface diffraction,” Opt. Express 22(10), 12659–12667 (2014).
    [Crossref]
  5. T. Underwood and D. G. Voelz, “Wave optics approach for incoherent imaging simulation through distributed turbulence,” in Unconventional Imaging and Wavefront Sensing, San Diego, 2013.
  6. J. C. Zingarelli and S. C. Cain, “Phase retrieval and Zernike decomposition using measured intensity data and the estimated electric field,” Appl. Opt. 52(31), 7435–7444 (2013).
    [Crossref]
  7. L. Onural, “Some mathematical properties of the uniformly sampled quadratic phase function and associated issues in digital Fresnel diffraction simulations,” Opt. Eng. 43(11), 2557–2563 (2004).
    [Crossref]
  8. X. Yu, T. Xiahui, Q. Yingxiong, P. Hao, and W. Wei, “Band-limited angular spectrum numerical propagation method with selective scaling of observation window size and sample number,” J. Opt. Soc. Am. A 29(11), 2415–2420 (2012).
    [Crossref]
  9. J. Liang and M. Becker, “Spatial bandwidth analysis of fast backward Fresnel diffraction for precise computer-generated hologram design,” Appl. Opt. 53(27), G84–G94 (2014).
    [Crossref]
  10. S. C. Cain and T. Watts, “Nonparaxial Fourier propagation tool for aberration analysis and point spread function calculation,” Opt. Eng. 55(8), 085104 (2016).
    [Crossref]
  11. P. Davila, H. John Wood, P. D. Atcheson, R. Saunders, J. Sullivan, A. H. Vaughan, and M. Saisse, “Telescope simulators for Hubble: an overview of optical designs,” Appl. Opt. 32(10), 1775–1781 (1993).
    [Crossref]
  12. E. Hecht, Optics, (Addison-Wesley, 1990).

2016 (1)

S. C. Cain and T. Watts, “Nonparaxial Fourier propagation tool for aberration analysis and point spread function calculation,” Opt. Eng. 55(8), 085104 (2016).
[Crossref]

2014 (2)

2013 (1)

2012 (1)

2004 (1)

L. Onural, “Some mathematical properties of the uniformly sampled quadratic phase function and associated issues in digital Fresnel diffraction simulations,” Opt. Eng. 43(11), 2557–2563 (2004).
[Crossref]

1993 (1)

Atcheson, P. D.

Becker, M.

Cain, S. C.

S. C. Cain and T. Watts, “Nonparaxial Fourier propagation tool for aberration analysis and point spread function calculation,” Opt. Eng. 55(8), 085104 (2016).
[Crossref]

J. C. Zingarelli and S. C. Cain, “Phase retrieval and Zernike decomposition using measured intensity data and the estimated electric field,” Appl. Opt. 52(31), 7435–7444 (2013).
[Crossref]

Davila, P.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Hao, P.

Hecht, E.

E. Hecht, Optics, (Addison-Wesley, 1990).

Hwang, C.-Y.

Jeong, I.-K.

John Wood, H.

Kim, H.

Liang, J.

Oh, S.

Onural, L.

L. Onural, “Some mathematical properties of the uniformly sampled quadratic phase function and associated issues in digital Fresnel diffraction simulations,” Opt. Eng. 43(11), 2557–2563 (2004).
[Crossref]

Saisse, M.

Saunders, R.

Sullivan, J.

Underwood, T.

T. Underwood and D. G. Voelz, “Wave optics approach for incoherent imaging simulation through distributed turbulence,” in Unconventional Imaging and Wavefront Sensing, San Diego, 2013.

Vaughan, A. H.

Voelz, D.

D. Voelz, Computational Fourier Optics, A MATLAB Tutorial, (SPIE, 2011).

Voelz, D. G.

T. Underwood and D. G. Voelz, “Wave optics approach for incoherent imaging simulation through distributed turbulence,” in Unconventional Imaging and Wavefront Sensing, San Diego, 2013.

Watts, T.

S. C. Cain and T. Watts, “Nonparaxial Fourier propagation tool for aberration analysis and point spread function calculation,” Opt. Eng. 55(8), 085104 (2016).
[Crossref]

Wei, W.

Xiahui, T.

Yingxiong, Q.

Yu, X.

Zingarelli, J. C.

Appl. Opt. (3)

J. Opt. Soc. Am. A (1)

Opt. Eng. (2)

S. C. Cain and T. Watts, “Nonparaxial Fourier propagation tool for aberration analysis and point spread function calculation,” Opt. Eng. 55(8), 085104 (2016).
[Crossref]

L. Onural, “Some mathematical properties of the uniformly sampled quadratic phase function and associated issues in digital Fresnel diffraction simulations,” Opt. Eng. 43(11), 2557–2563 (2004).
[Crossref]

Opt. Express (1)

Other (5)

T. Underwood and D. G. Voelz, “Wave optics approach for incoherent imaging simulation through distributed turbulence,” in Unconventional Imaging and Wavefront Sensing, San Diego, 2013.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

D. Voelz, Computational Fourier Optics, A MATLAB Tutorial, (SPIE, 2011).

R. ZEMAX, “ZEMAX User's Manual,” Radiant ZEMAX LLC (RZ), Kirkland, 2011.

E. Hecht, Optics, (Addison-Wesley, 1990).

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Figures (5)

Fig. 1.
Fig. 1. Diagram showing how the z coordinate in the curved detector (solid arc at the top) is computed from its (x’,y’) coordinates in Eq. (3). S is the radius of curvature of the curved detector whose surface lies on a sphere centered at (0,0,zo -S). The oval region represents the pupil plane containing the center of the coordinate system but not necessarily the center of curvature of the detector unless the radius of curvature is equal to zo which is the vertical distance between the aperture and the center of the detector.
Fig. 2.
Fig. 2. Optical arrangement used to produce a spherical aberration.
Fig. 3.
Fig. 3. Image of the PSF simulated using the Rayleigh-Sommerfeld diffraction formula for the arrangement shown in Fig. 2.
Fig. 4.
Fig. 4. Image of the PSF simulated using the new Fourier Propagator.
Fig. 5.
Fig. 5. Plot of the normalized PSF amplitude through the middle from the Rayleigh-Sommerfeld propagator as well as the new propagator. The PSFs were normalized to sum to one.

Equations (26)

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U ( x , y , z ) = z j λ x y U ( x , y , 0 ) e j 2 π R ( x , y , x , y ) / λ R 2 ( x , y , x , y ) d x d y
R ( x , y , x , y ) = ( x x ) 2 + ( y y ) 2 + z ( x , y ) 2
z ( x , y ) = ( S ) 2 x 2 y 2 ( S z o ) ,
R ( x , y , x , y ) = ( S ) 2 + x 2 + y 2 2 x x 2 y y
R o 2 ( x , y ) = ( S ) 2 + x 2 + y 2
R ( x , y , x , y ) = R o ( x , y ) 1 + 2 x x 2 y y R o 2 ( x , y )
R ( x , y , x , y ) R o ( x , y ) ( 1 + 2 x x 2 y y 2 R o 2 ( x , y ) )
U ( x , y , z o ) = 1 j λ ( S ) x y U ( x , y , 0 ) e j 2 π R o ( x , y ) / λ e j 2 π ( x x + y y ) λ R o ( x , y ) d x d y
2 π λ ( ( x x + y y ) R o ( x , y ) ( x x + y y ) S )
2 π ( x x + y y ) λ ( 1 R o ( x , y ) 1 S )
θ e r r 2 π ( x x + y y ) λ ( S ) 2 | R o ( x 2 , y 2 ) S |
R o ( x , y ) ( S ) ( 1 + x 2 + y 2 2 ( S ) 2 )
| 2 π ( x x + y y ) λ | | x 2 + y 2 2 ( S ) 3 | << 1
π n max D 2 4 ( S ) 2 << 1
n max << ( F # ) 2
U r ( n , m ) = Δ s 2 j λ ( S ) k , l U s ( k , l ) e j 2 π R o ( k , l ) λ e j 2 π ( n k + m l ) N
1 + b 1 + b / 2
2 π b 2 / 8 λ = 2 π ( 2 x x + 2 y y ) 2 8 λ R o 4 ( x , y ) << 1
π n max 2 λ ( x + y ) 2 4 D 2 ( S ) << 1
( F # ) 4 λ << ( S )
π D 4 16 λ << z 3
( F # ) << ( S / λ ) 1 / 4
( π D 16 λ ) 1 / 3 << ( F # )
θ t o t ( k , l ) = θ l e n s ( k , l ) + 2 π R o ( k , l ) / λ ,
Δ r = λ S 2 L ,
% E r r o r = x = 1 N y = 1 N ( I R S ( x , y ) I F T ( x , y ) ) 2 x = 1 N y = 1 N I R S 2 ( x , y ) × 100

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