Abstract

A method for the calculation of the transmitted and reflected scalar fields at arbitrary dielectric surfaces is presented. The method is based on an evaluation of the Born series expansion and is of high accuracy as multiple reflections and refractions are taken into account. We show by comparison with Fresnel formulas that with the algorithm the ratio of transmitted and reflected field amplitudes can be calculated exactly. Results obtained by our algorithm are compared with results from rigorous diffraction calculations for a dielectric cylinder. We also demonstrate the application of the method to a more complicated surface geometry. Furthermore, advantages and restrictions of this algorithm are identified.

© 1998 Optical Society of America

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  1. M. Hausmann, J. Dölle, A. Arnold, B. Stepanow, B. Wickert, J. Boscher, P. C. Popescu, C. Cremer, “Development of a two-parameter slit-scan flow cytometer for screening of normal and aberrant chromosomes: application to a karyotype of Sus scrofa domestica (pig),” Opt. Eng. 31, 1463–1469 (1992).
    [CrossRef]
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  6. E. Zimmermann, R. Dändliker, N. Souli, B. Krattiger, “Scattering of an off-axis Gaussian beam by a dielectric cylinder compared with a rigorous electromagnetic approach,” J. Opt. Soc. Am. A 12, 398–403 (1995).
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  8. P. Török, P. Varga, Z. Laczik, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12, 325–332 (1995).
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    [CrossRef]
  13. H. Weyl, “Ausbreitung elektromagnetischer Wellen über einem ebenen Leiter,” Ann. Phys. (Leipzig) 60, 481–500 (1919).
    [CrossRef]
  14. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980), Chap. 2.4, p. 98.
  15. G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, San Diego, Calif., 1995), pp. 510–516.
  16. E. Lalor, “Conditions for the validity of the angular spectrum of plane waves,” J. Opt. Soc. Am. 58, 1235–1237 (1968).
    [CrossRef]
  17. W. Freude, G. K. Grau, “Rayleigh Sommerfeld and Helmholtz Kirchhoff integrals: application to the scalar and vectorial theory of wave propagation and diffraction,” J. Lightwave Technol. 13, 24–32 (1995).
    [CrossRef]
  18. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Chap. 4, p. 21.
  19. J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
    [CrossRef]
  20. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980), Chap. 1.5.2, p. 40.
  21. M. Totzeck, “Validity of scalar Kirchhoff and Rayleigh–Sommerfeld diffraction theories in the near field of small phase objects,” J. Opt. Soc. Am. A 8, 27–32 (1991).
    [CrossRef]
  22. W. Freude, G. K. Grau, “Rayleigh–Sommerfeld and Helmholtz–Kirchhoff integrals: application to the scalar and vectorial theory of wave propagation and diffraction,” J. Lightwave Technol. 13, 30 (1995).
    [CrossRef]

1997 (1)

M. Totzeck, H. J. Tiziani, “Interference microscopy of sub-lambda structures: a rigorous computation method and measurements,” Opt. Commun. 136, 61–74 (1997).
[CrossRef]

1995 (5)

1993 (2)

1992 (1)

M. Hausmann, J. Dölle, A. Arnold, B. Stepanow, B. Wickert, J. Boscher, P. C. Popescu, C. Cremer, “Development of a two-parameter slit-scan flow cytometer for screening of normal and aberrant chromosomes: application to a karyotype of Sus scrofa domestica (pig),” Opt. Eng. 31, 1463–1469 (1992).
[CrossRef]

1991 (2)

1981 (1)

1979 (1)

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

1972 (1)

1968 (1)

1919 (1)

H. Weyl, “Ausbreitung elektromagnetischer Wellen über einem ebenen Leiter,” Ann. Phys. (Leipzig) 60, 481–500 (1919).
[CrossRef]

Arfken, G. B.

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, San Diego, Calif., 1995), pp. 510–516.

Arnold, A.

M. Hausmann, J. Dölle, A. Arnold, B. Stepanow, B. Wickert, J. Boscher, P. C. Popescu, C. Cremer, “Development of a two-parameter slit-scan flow cytometer for screening of normal and aberrant chromosomes: application to a karyotype of Sus scrofa domestica (pig),” Opt. Eng. 31, 1463–1469 (1992).
[CrossRef]

Booker, G. R.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980), Chap. 2.4, p. 98.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980), Chap. 1.5.2, p. 40.

Boscher, J.

M. Hausmann, J. Dölle, A. Arnold, B. Stepanow, B. Wickert, J. Boscher, P. C. Popescu, C. Cremer, “Development of a two-parameter slit-scan flow cytometer for screening of normal and aberrant chromosomes: application to a karyotype of Sus scrofa domestica (pig),” Opt. Eng. 31, 1463–1469 (1992).
[CrossRef]

Brenner, K.-H.

Bruno, A. E.

Cremer, C.

M. Hausmann, J. Dölle, A. Arnold, B. Stepanow, B. Wickert, J. Boscher, P. C. Popescu, C. Cremer, “Development of a two-parameter slit-scan flow cytometer for screening of normal and aberrant chromosomes: application to a karyotype of Sus scrofa domestica (pig),” Opt. Eng. 31, 1463–1469 (1992).
[CrossRef]

Dandliker, R.

Dändliker, R.

DeSanto, J. A.

Dölle, J.

M. Hausmann, J. Dölle, A. Arnold, B. Stepanow, B. Wickert, J. Boscher, P. C. Popescu, C. Cremer, “Development of a two-parameter slit-scan flow cytometer for screening of normal and aberrant chromosomes: application to a karyotype of Sus scrofa domestica (pig),” Opt. Eng. 31, 1463–1469 (1992).
[CrossRef]

Freude, W.

W. Freude, G. K. Grau, “Rayleigh Sommerfeld and Helmholtz Kirchhoff integrals: application to the scalar and vectorial theory of wave propagation and diffraction,” J. Lightwave Technol. 13, 24–32 (1995).
[CrossRef]

W. Freude, G. K. Grau, “Rayleigh–Sommerfeld and Helmholtz–Kirchhoff integrals: application to the scalar and vectorial theory of wave propagation and diffraction,” J. Lightwave Technol. 13, 30 (1995).
[CrossRef]

Geiser, M.

Gelius, L.-J.

L.-J. Gelius, J. J. Stamnes, “Diffraction tomography: potentials and problems,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (Elsevier, Amsterdam, 1990), pp. 91–109.

Grau, G. K.

W. Freude, G. K. Grau, “Rayleigh–Sommerfeld and Helmholtz–Kirchhoff integrals: application to the scalar and vectorial theory of wave propagation and diffraction,” J. Lightwave Technol. 13, 30 (1995).
[CrossRef]

W. Freude, G. K. Grau, “Rayleigh Sommerfeld and Helmholtz Kirchhoff integrals: application to the scalar and vectorial theory of wave propagation and diffraction,” J. Lightwave Technol. 13, 24–32 (1995).
[CrossRef]

Harvey, J. E.

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

Hausmann, M.

M. Hausmann, J. Dölle, A. Arnold, B. Stepanow, B. Wickert, J. Boscher, P. C. Popescu, C. Cremer, “Development of a two-parameter slit-scan flow cytometer for screening of normal and aberrant chromosomes: application to a karyotype of Sus scrofa domestica (pig),” Opt. Eng. 31, 1463–1469 (1992).
[CrossRef]

Krattiger, B.

Krattinger, B.

Laczik, Z.

Lagasse, P. E.

Lalor, E.

Popescu, P. C.

M. Hausmann, J. Dölle, A. Arnold, B. Stepanow, B. Wickert, J. Boscher, P. C. Popescu, C. Cremer, “Development of a two-parameter slit-scan flow cytometer for screening of normal and aberrant chromosomes: application to a karyotype of Sus scrofa domestica (pig),” Opt. Eng. 31, 1463–1469 (1992).
[CrossRef]

Singer, W.

Souli, N.

Stamnes, J. J.

L.-J. Gelius, J. J. Stamnes, “Diffraction tomography: potentials and problems,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (Elsevier, Amsterdam, 1990), pp. 91–109.

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Chap. 4, p. 21.

Stepanow, B.

M. Hausmann, J. Dölle, A. Arnold, B. Stepanow, B. Wickert, J. Boscher, P. C. Popescu, C. Cremer, “Development of a two-parameter slit-scan flow cytometer for screening of normal and aberrant chromosomes: application to a karyotype of Sus scrofa domestica (pig),” Opt. Eng. 31, 1463–1469 (1992).
[CrossRef]

Tiziani, H. J.

M. Totzeck, H. J. Tiziani, “Interference microscopy of sub-lambda structures: a rigorous computation method and measurements,” Opt. Commun. 136, 61–74 (1997).
[CrossRef]

Török, P.

Totzeck, M.

M. Totzeck, H. J. Tiziani, “Interference microscopy of sub-lambda structures: a rigorous computation method and measurements,” Opt. Commun. 136, 61–74 (1997).
[CrossRef]

M. Totzeck, “Validity of scalar Kirchhoff and Rayleigh–Sommerfeld diffraction theories in the near field of small phase objects,” J. Opt. Soc. Am. A 8, 27–32 (1991).
[CrossRef]

Turunen, J.

J. Turunen, “Diffraction theory of microrelief gratings,” in Microoptics: Elements, Systems and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1977), pp. 31–52.

van der Donk, J.

Van Roey, J.

Varga, P.

Weber, H. J.

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, San Diego, Calif., 1995), pp. 510–516.

Weyl, H.

H. Weyl, “Ausbreitung elektromagnetischer Wellen über einem ebenen Leiter,” Ann. Phys. (Leipzig) 60, 481–500 (1919).
[CrossRef]

Wickert, B.

M. Hausmann, J. Dölle, A. Arnold, B. Stepanow, B. Wickert, J. Boscher, P. C. Popescu, C. Cremer, “Development of a two-parameter slit-scan flow cytometer for screening of normal and aberrant chromosomes: application to a karyotype of Sus scrofa domestica (pig),” Opt. Eng. 31, 1463–1469 (1992).
[CrossRef]

Widmer, H. M.

Wolf, E.

E. Lalor, E. Wolf, “Exact solution of the equations of molecular optics for refraction and reflection of an electromagnetic wave on a semi-infinite dielectric,” J. Opt. Soc. Am. 62, 1165–1174 (1972).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980), Chap. 2.4, p. 98.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980), Chap. 1.5.2, p. 40.

Wombell, R. J.

Zimmermann, E.

Am. J. Phys. (1)

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

Ann. Phys. (Leipzig) (1)

H. Weyl, “Ausbreitung elektromagnetischer Wellen über einem ebenen Leiter,” Ann. Phys. (Leipzig) 60, 481–500 (1919).
[CrossRef]

Appl. Opt. (2)

J. Lightwave Technol. (2)

W. Freude, G. K. Grau, “Rayleigh Sommerfeld and Helmholtz Kirchhoff integrals: application to the scalar and vectorial theory of wave propagation and diffraction,” J. Lightwave Technol. 13, 24–32 (1995).
[CrossRef]

W. Freude, G. K. Grau, “Rayleigh–Sommerfeld and Helmholtz–Kirchhoff integrals: application to the scalar and vectorial theory of wave propagation and diffraction,” J. Lightwave Technol. 13, 30 (1995).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (5)

Opt. Commun. (1)

M. Totzeck, H. J. Tiziani, “Interference microscopy of sub-lambda structures: a rigorous computation method and measurements,” Opt. Commun. 136, 61–74 (1997).
[CrossRef]

Opt. Eng. (1)

M. Hausmann, J. Dölle, A. Arnold, B. Stepanow, B. Wickert, J. Boscher, P. C. Popescu, C. Cremer, “Development of a two-parameter slit-scan flow cytometer for screening of normal and aberrant chromosomes: application to a karyotype of Sus scrofa domestica (pig),” Opt. Eng. 31, 1463–1469 (1992).
[CrossRef]

Other (6)

J. Turunen, “Diffraction theory of microrelief gratings,” in Microoptics: Elements, Systems and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1977), pp. 31–52.

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Chap. 4, p. 21.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980), Chap. 2.4, p. 98.

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, San Diego, Calif., 1995), pp. 510–516.

L.-J. Gelius, J. J. Stamnes, “Diffraction tomography: potentials and problems,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (Elsevier, Amsterdam, 1990), pp. 91–109.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980), Chap. 1.5.2, p. 40.

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Figures (10)

Fig. 1
Fig. 1

Transmission and reflection of a plane wave at a planar interface according to the Born approximation.

Fig. 2
Fig. 2

Reflection, refraction, and diffraction at an interface with a cosine-modulated edge.

Fig. 3
Fig. 3

Coordinate system with two halves of the spherical surface and scattered fields according to a second-order Born series expansion.

Fig. 4
Fig. 4

The angle-dependent factor b for both transitions with nt/ni=1.5 (factor b1) and nt/ni=1/1.5 (factor b2) normalized to k02.

Fig. 5
Fig. 5

Amplitude distribution of a plane wave scattered at a dielectric cylinder with index 1.5 and diameter 21 λ (image extension 63×42 λ).

Fig. 6
Fig. 6

Intensity distribution corresponding to Fig. 5.

Fig. 7
Fig. 7

Comparison of intensities in the x direction directly behind the cylinder.

Fig. 8
Fig. 8

Comparison of intensities along the optical axis in the z direction.

Fig. 9
Fig. 9

Intensity distribution of the scattering of a plane wave at a deformed dielectric cylinder with index 1.5.

Fig. 10
Fig. 10

Illustration of the angular restriction in both space and frequency domains.

Equations (37)

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n(x, y, z)=n0+(n1-n0)·step[z-h(x, y)],
Δu(r)+n02·k02·u(r)=V(r),
V(r)=f(r)·k02·u(r)=(n02-n12)·step[z-h(x, y)]·k02·u(r),
[u(r)·LG(r, r)-G(r, r)·Lu(r)]d3r=[u(r)nG(r, r)-G(r, r)nu(r)]dS.
u(r)=V(r)·G(r, r)·d3r+u0(r)nG(r, r)-G(r, r)nu0(r)dS=us(r)+ui(r).
ui(r)=[u0(r)nG(r, r)-G(r, r)nu0(r)]dS=u0(r)·(/z-ik0z)·G(r, r)dxdy.
H(kx, ky, z)=Fxy[(/z-ik0z)G(x, y, z)]=12π(1+k0z/kgz)×exp(ikxx+ikyy+ikzz).
us(r)=V(r)·G(r, r)·d3r=ur(r)+ut(r).
G(r, r)=Gr(r, r, kr=n0·k0)+Gt(r, r, kt=n1k0).
u˜s(kx, ky, z)=k02f(r)·ui(r)·1kzexp[-i(kxx+kyy+kz(z-z)]dxdydz=k02(2π)3exp(ikzz)kzkif˜(k-ki)·u˜i(ki)d3ki.
kz=±(k2-kx2-ky2)1/2.
u˜mt(k, dm)=(ik0)2m(2π)3mexp(ikzmdm)kzm·f˜mexp[ikzm-1d(m-1)]kzm-1·f˜m-1exp(ikz1d1)kz1(f˜1u˜0).
u˜s(ksx, ksz)=(ni2-nt2)[k02·δ(ksx-kix, ksz-kiz)+i·k02·δ(ksx-kix)/(ksz-kiz)].
t=u˜t/u˜r=f˜t/f˜r=(krz-kiz)/(ktz-kiz).
ui[x, h(x)]exp{i[kixx+kizh(x)]}=b0·ut[x, h(x)]exp{i[ktxx+ktzh(x)]}-b0·ur[x, h(x)]exp{i[kixx-kizh(x)]}.
b=u˜t-u˜ru˜i=k02ktz+kiz(ktz-kiz)2kiz.
t=2·kiz/(kiz-ktz)=2·kiz/(kiz-i·κ)=2·kiz·(kiz+i·κ)/(kiz2+κ2),
G(x, z, k)=- exp(ikr)rdy=iπ·H0(1)(kρ)=exp(ikzz)kzexp[i(kxx)]dkx.
up+1(r)=k02f1/2(r)·up(r)·G(r-r, ka/b)d2r,
uˆ1=ut0+ur1-+ut3+ut5+ut7+,
onEwaldcircle:kaz=±(ka2-kx2)1/2,
uˆ2=ut1+ur2+ur3+ur4+ur5+ur6+,
onEwaldcircle:kbz=±(kb2-kx2)1/2,
uˆ3=ut0+ur1++ut2+ut4+ut6+ut8+,
onEwaldcircle:kaz=±(ka2-kx2)1/2,
|kxs|kzx/hm=ks/(1+hm2)1/2.
|tan(αs)|<1/hm=|gmx/gmz|,
f(x, z)=Δn·step[z-h(x)]-Δn[z-h(x0)]/N,
f˜(gx, gz)=Δn·step{±[z-h(x)]}×exp[-i(xgx+zgz)]dxdz=Δn·δ(gz)±igz×exp[ih(x)gz]exp(-ixgx)dx.
f˜(gx, gz)=Δn·δ(gx, gz)+Δnsin[h(x)gz]gz+i cos[h(x)gz]gzexp(-ixgx)dx.
u˜s[kx, kz=(ks2-kx2)1/2]=i·k02/kz·f˜(kx-kix, kz-kiz),
u˜s[kx, kz=(ks2-kx2)1/2+kiz]=i·k02/kz·f˜(gx, gz),
u˜s(kx, kz, z)=i·k02 exp(ikzz)kz×kif˜ [kx-kix, kz-(ki2-kix2)1/2].
u(x)=u(x)·A(x),
A(x)=exp[-κ·(x-x0)]ifx>x0,A(x)=1ifx<x0.
kxr(z)=ki sin[2αN(z)],
withsin(2αN)=2 sin(αN)cos(αN)=2h-1(x)(R-x)/R2.

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