Abstract

We report on algorithms for the computation of the average phase of a beam over a detector in the near field. The basic idea is to reconstruct the optical field numerically and then use a quadrature algorithm to evaluate the quantity of interest. The various algorithms that employ discrete Fourier transform techniques for the computation of the field are described, and numerical tests that assess the accuracy of these algorithms are presented. No particular algorithm delivers the desired accuracy over the entire range of Fresnel numbers of interest, but each can produce satisfactory results within a particular range. Finally, new methods to evaluate the average phase are introduced, and their efficiency is assessed.

© 2000 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, 1996).
  2. H. H. Hopkins, “The numerical evaluation of frequency response of optical systems,” Proc. Phys. Soc. London Sect. B 70, 1002–1005 (1957).
    [CrossRef]
  3. A. C. Ludwig, “Computation of radiation patterns involving numerical double integration,” IEEE Trans. Antennas Propag. AP-16, 767–769 (1968).
    [CrossRef]
  4. J. J. Stamnes, B. Spjelkavic, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
    [CrossRef]
  5. L. A. D’Arcio, J. J. M. Braat, H. J. Frankena, “Numerical evaluation of diffraction integrals for apertures of complicated shape,” J. Opt. Soc. Am. A 11, 2664–2674 (1994).
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    [CrossRef]
  7. H. G. Kraus, “Finite element area and line integral transforms for generalization of aperture functions and geometry in Kirchhoff scalar diffraction theory,” Opt. Eng. 32, 368–383 (1993).
    [CrossRef]
  8. P. Horwitz, “Modes in misaligned unstable resonators,” Appl. Opt. 15, 167–178 (1976).
    [CrossRef] [PubMed]
  9. E. Carcole, S. Bosch, J. Campos, “Analytical and numerical approximations in Fresnel diffraction. Procedures based on the geometry of the Cornu Spiral,” J. Mod. Opt. 40, 1091–1106 (1993).
    [CrossRef]
  10. E. Sziclas, A. E. Siegman, “Mode calculations in unstable resonators with flowing saturable gain. 2: Fast Fourier transform method,” Appl. Opt. 14, 1874–1889 (1975).
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    [CrossRef]
  13. D. Mas, J. Garcia, C. Ferreira, L. M. Bernado, F. Marinho, “Fast algorithms for free-space diffraction patterns calculations,” Opt. Commun. 164, 233–245 (1999).
    [CrossRef]
  14. J. Stoer, R. Bulirsch, Introduction to Numerical Analysis (Springer-Verlag, New York, 1980).
  15. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).
  16. C. W. Clenshaw, A. R. Curtis, “A method for numericalintegration on an automatic computer,” Numer. Math. 2, 197–205 (1960).
    [CrossRef]
  17. R. K. Littlewood, V. Zakian, “Numerical evaluation of Fourier integrals,” J. Inst. Math. Appl. 18, 331–339 (1976).
    [CrossRef]
  18. D. Levin, “Procedures for computing one- and two-dimensional integrals of functions with rapid irregular oscillations,” Math. Comput. 38, 531–538 (1982).
    [CrossRef]
  19. I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, San Diego, Calif., 1992).
  20. R. Kress, Linear Integral Equations (Springer-Verlag, New York, 1989).
  21. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1977).
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    [CrossRef]
  23. M. Vaez-Iravani, H. K. Wickramasinghe, “Scattering matrix approach of thermal wave propagation in layered structures,” J. Appl. Phys. 58, 122–132 (1983).
    [CrossRef]

1999 (1)

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

1997 (1)

D. Mendlovic, Z. Zalevsky, N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

1994 (1)

1993 (2)

H. G. Kraus, “Finite element area and line integral transforms for generalization of aperture functions and geometry in Kirchhoff scalar diffraction theory,” Opt. Eng. 32, 368–383 (1993).
[CrossRef]

E. Carcole, S. Bosch, J. Campos, “Analytical and numerical approximations in Fresnel diffraction. Procedures based on the geometry of the Cornu Spiral,” J. Mod. Opt. 40, 1091–1106 (1993).
[CrossRef]

1989 (1)

1983 (2)

M. Vaez-Iravani, H. K. Wickramasinghe, “Scattering matrix approach of thermal wave propagation in layered structures,” J. Appl. Phys. 58, 122–132 (1983).
[CrossRef]

J. J. Stamnes, B. Spjelkavic, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

1982 (1)

D. Levin, “Procedures for computing one- and two-dimensional integrals of functions with rapid irregular oscillations,” Math. Comput. 38, 531–538 (1982).
[CrossRef]

1981 (1)

1977 (1)

1976 (2)

R. K. Littlewood, V. Zakian, “Numerical evaluation of Fourier integrals,” J. Inst. Math. Appl. 18, 331–339 (1976).
[CrossRef]

P. Horwitz, “Modes in misaligned unstable resonators,” Appl. Opt. 15, 167–178 (1976).
[CrossRef] [PubMed]

1975 (1)

1968 (1)

A. C. Ludwig, “Computation of radiation patterns involving numerical double integration,” IEEE Trans. Antennas Propag. AP-16, 767–769 (1968).
[CrossRef]

1960 (1)

C. W. Clenshaw, A. R. Curtis, “A method for numericalintegration on an automatic computer,” Numer. Math. 2, 197–205 (1960).
[CrossRef]

1957 (1)

H. H. Hopkins, “The numerical evaluation of frequency response of optical systems,” Proc. Phys. Soc. London Sect. B 70, 1002–1005 (1957).
[CrossRef]

Bernado, L. M.

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

Bosch, S.

E. Carcole, S. Bosch, J. Campos, “Analytical and numerical approximations in Fresnel diffraction. Procedures based on the geometry of the Cornu Spiral,” J. Mod. Opt. 40, 1091–1106 (1993).
[CrossRef]

Braat, J. J. M.

Bulirsch, R.

J. Stoer, R. Bulirsch, Introduction to Numerical Analysis (Springer-Verlag, New York, 1980).

Campos, J.

E. Carcole, S. Bosch, J. Campos, “Analytical and numerical approximations in Fresnel diffraction. Procedures based on the geometry of the Cornu Spiral,” J. Mod. Opt. 40, 1091–1106 (1993).
[CrossRef]

Carcole, E.

E. Carcole, S. Bosch, J. Campos, “Analytical and numerical approximations in Fresnel diffraction. Procedures based on the geometry of the Cornu Spiral,” J. Mod. Opt. 40, 1091–1106 (1993).
[CrossRef]

Clenshaw, C. W.

C. W. Clenshaw, A. R. Curtis, “A method for numericalintegration on an automatic computer,” Numer. Math. 2, 197–205 (1960).
[CrossRef]

Curtis, A. R.

C. W. Clenshaw, A. R. Curtis, “A method for numericalintegration on an automatic computer,” Numer. Math. 2, 197–205 (1960).
[CrossRef]

D’Arcio, L. A.

Ferreira, C.

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

Frankena, H. J.

Garcia, J.

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

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, San Diego, Calif., 1992).

Hopkins, H. H.

H. H. Hopkins, “The numerical evaluation of frequency response of optical systems,” Proc. Phys. Soc. London Sect. B 70, 1002–1005 (1957).
[CrossRef]

Horwitz, P.

Konforti, N.

D. Mendlovic, Z. Zalevsky, N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

Kraus, H. G.

H. G. Kraus, “Finite element area and line integral transforms for generalization of aperture functions and geometry in Kirchhoff scalar diffraction theory,” Opt. Eng. 32, 368–383 (1993).
[CrossRef]

Kress, R.

R. Kress, Linear Integral Equations (Springer-Verlag, New York, 1989).

Levin, D.

D. Levin, “Procedures for computing one- and two-dimensional integrals of functions with rapid irregular oscillations,” Math. Comput. 38, 531–538 (1982).
[CrossRef]

Littlewood, R. K.

R. K. Littlewood, V. Zakian, “Numerical evaluation of Fourier integrals,” J. Inst. Math. Appl. 18, 331–339 (1976).
[CrossRef]

Ludwig, A. C.

A. C. Ludwig, “Computation of radiation patterns involving numerical double integration,” IEEE Trans. Antennas Propag. AP-16, 767–769 (1968).
[CrossRef]

Mansuripur, M.

Marinho, F.

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

Mendlovic, D.

D. Mendlovic, Z. Zalevsky, N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

Pedersen, H. M.

J. J. Stamnes, B. Spjelkavic, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, San Diego, Calif., 1992).

Siegman, A. E.

Southwell, W. H.

Spjelkavic, B.

J. J. Stamnes, B. Spjelkavic, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

Stamnes, J. J.

J. J. Stamnes, B. Spjelkavic, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).

Stoer, J.

J. Stoer, R. Bulirsch, Introduction to Numerical Analysis (Springer-Verlag, New York, 1980).

Sziclas, E.

Vaez-Iravani, M.

M. Vaez-Iravani, H. K. Wickramasinghe, “Scattering matrix approach of thermal wave propagation in layered structures,” J. Appl. Phys. 58, 122–132 (1983).
[CrossRef]

Wickramasinghe, H. K.

M. Vaez-Iravani, H. K. Wickramasinghe, “Scattering matrix approach of thermal wave propagation in layered structures,” J. Appl. Phys. 58, 122–132 (1983).
[CrossRef]

Zakian, V.

R. K. Littlewood, V. Zakian, “Numerical evaluation of Fourier integrals,” J. Inst. Math. Appl. 18, 331–339 (1976).
[CrossRef]

Zalevsky, Z.

D. Mendlovic, Z. Zalevsky, N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

Appl. Opt. (2)

IEEE Trans. Antennas Propag. (1)

A. C. Ludwig, “Computation of radiation patterns involving numerical double integration,” IEEE Trans. Antennas Propag. AP-16, 767–769 (1968).
[CrossRef]

J. Appl. Phys. (1)

M. Vaez-Iravani, H. K. Wickramasinghe, “Scattering matrix approach of thermal wave propagation in layered structures,” J. Appl. Phys. 58, 122–132 (1983).
[CrossRef]

J. Inst. Math. Appl. (1)

R. K. Littlewood, V. Zakian, “Numerical evaluation of Fourier integrals,” J. Inst. Math. Appl. 18, 331–339 (1976).
[CrossRef]

J. Mod. Opt. (2)

E. Carcole, S. Bosch, J. Campos, “Analytical and numerical approximations in Fresnel diffraction. Procedures based on the geometry of the Cornu Spiral,” J. Mod. Opt. 40, 1091–1106 (1993).
[CrossRef]

D. Mendlovic, Z. Zalevsky, N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Math. Comput. (1)

D. Levin, “Procedures for computing one- and two-dimensional integrals of functions with rapid irregular oscillations,” Math. Comput. 38, 531–538 (1982).
[CrossRef]

Numer. Math. (1)

C. W. Clenshaw, A. R. Curtis, “A method for numericalintegration on an automatic computer,” Numer. Math. 2, 197–205 (1960).
[CrossRef]

Opt. Acta (1)

J. J. Stamnes, B. Spjelkavic, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

Opt. Commun. (1)

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

Opt. Eng. (1)

H. G. Kraus, “Finite element area and line integral transforms for generalization of aperture functions and geometry in Kirchhoff scalar diffraction theory,” Opt. Eng. 32, 368–383 (1993).
[CrossRef]

Opt. Lett. (1)

Proc. Phys. Soc. London Sect. B (1)

H. H. Hopkins, “The numerical evaluation of frequency response of optical systems,” Proc. Phys. Soc. London Sect. B 70, 1002–1005 (1957).
[CrossRef]

Other (6)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).

J. Stoer, R. Bulirsch, Introduction to Numerical Analysis (Springer-Verlag, New York, 1980).

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, San Diego, Calif., 1992).

R. Kress, Linear Integral Equations (Springer-Verlag, New York, 1989).

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1977).

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Figures (8)

Fig. 1
Fig. 1

Schematic of the test cases.

Fig. 2
Fig. 2

Case A: plot of the optical field at (y=0, z=7 m).

Fig. 3
Fig. 3

Case A: average-phase error of the angular spectrum and direct methods.

Fig. 4
Fig. 4

Case B: plot of the optical field at (y=0, z=10 m).

Fig. 5
Fig. 5

Case B: average-phase error of the angular spectrum and direct methods.

Fig. 6
Fig. 6

Case B: average-phase error of the two methods as a function of Fresnel number. z=0.1 ,, 2.2 m.

Fig. 7
Fig. 7

Case C: plot of the optical field at (y=0, z=10 m).

Fig. 8
Fig. 8

Case C: numerical convergence of the two methods. N is the number of grid points at each direction in the aperture plane, and z=11.65 m.

Equations (41)

Equations on this page are rendered with MathJax. Learn more.

2x2+2y2+2z2+k2U(x, y, z)=0,
(x, y, z)Ω,
U(x, y, 0)=U(ξ, η)if(ξ, η)Γ0otherwise,
limRR UR-jkUR=0.
U(x, y, z)=-+F ( fx, fy, 0)exp{ jkz[1-(λfx)2-(λfy)2]1/2}exp[2π(xfx+yfy)]d fxd fy.
F ( fx, fy, 0)=F(U(x, y, 0))
U(x, y, z)=12πΓU(ξ, η) zr0121r01-jk×exp(jkr01)dξdη,
U(x, y, z)=exp(jkz)u˜(x, y, z),
2x2+2y2+j2k zu˜(x, y, z)=0,(x, y, z)Ω.
U(x, y, z)=exp(jkz)-+F ( fx, fy, 0)×exp[-jπλz( fx2+fy2)]×exp[j2π( fxx+fyy)]d fxd fy,
U(x, y, z)=exp(jkz)jλzexpjk2z (x2+y2)×-+ U(ξ, η)expjk2z (ξ2+η2)×exp-j2πλz (xξ+yη)d ξd η,
Fr=w2λz,
ϕtan-1Im(I)Re(I),
IΔU(x, y, z)dxdy.
G(ξ, η)=-+A(x-ξ, y-η)U(x, y)d x d y,
G(ξ, η)=F -1(F(A)F(U)).
U(x, y, z)=exp(jkz)2j×[C(a2)-C(α1)+j(S(α2)-S(α1))]×[C(β2)-C(β1)+j(S(β2)-S(β1))],
α1=-(w+x)2/λz,α2=(w-x)2/λz,
β1=-(w+y)2/λz,β2=(w-y)2/λz.
λ=0.7 μm,2w=3.0cm,
2W=6.0cm,z=7.0 ,, 9.0m,
λ=1.3μm,2w=5.0mm,
2W=10.0mm,z=10.0 ,, 12.0m.
λ=1.3μm,2w=3.0cm,
2W=6.0cm,z=10.0 ,, 12.0m.
I(kz)=-+dxdyΓU(ξ, η) zjλr012exp(jkr01)d ξdη.
I(kz)=Γdξdη02πdθ0r˜U(ξ, η)×z exp[jk(r˜2+z2)1/2]jλ(r˜2+z2)dr˜,
r˜=[(x-ξ)2+(y-η)2]1/2=(r012-z2)1/2.
I(kz)=ΓU(ξ, η)d ξdηkzj1exp(jkzσ)σdσ.
I(kz)=jkzIΓEi(jkz),
IΓΓU(ξ, η)dξdη,
I(kz)IΓexp(jkz)1+1jkz-2(kz)2+,
kz1.
I(kz)=jkzIΓγ+log(kz)-jπ2+n=1(jkz)nn×n!,
U(x, y, z)=ΓU(ξ, η)K(x, y, z; ξ; η)dξdη,
I=ΓU(ξ, η)ΔK(x, y, z; ξ, η)d xd ydξdη.
U(ξ, η)=U(r, 0)=1ifr=(x2+y2)1/2w0otherwise.
U(r, z)=2π0ρJ0(2πrρ)H(U(r, 0))×exp{ jkz[1-(λρ)2]1/2}d ρ,
H(G(r))=2π0rG(r)J0(2πrρ)d r.
H(U(r, 0))=wJ1(2πρw),
U(r, z)=2πw0J1(2πρw)J0(2πrρ)×exp{ jkz[1-(λρ)2]1/2}dρ.

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