Abstract

We propose a new method for the analysis of light propagation through thick phase elements of homogeneous refractive index. The Laue equation is shown to be a generalization of Snell’s law of refraction, yielding correct amplitude and phase changes at a single refracting surface. The application to blazed phase gratings is outlined.

© 1995 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. W. Lohmann, Optical Information Processing Available from A. W. Lohmann at the address on the title page of this paper. (Erlangen, Germany, 1986), Vol. 1, Chap. 22, p. 127.
  2. N. Streibel, U. Nölscher, J. Jahns, S. Walker, “Array generation with lenslet arrays,” Appl. Opt. 30, 2739–2742 (1991).
    [CrossRef]
  3. A. W. Lohmann, J. Schwider, N. Streibl, J. Thomas, “Array illuminator based on phase contrast,” Appl. Opt. 27, 2915–2921 (1988).
    [CrossRef] [PubMed]
  4. H. Kobolla, J. Schmidt, J. T. Sheridan, N. Streibl, R. Völkl, “Optical interconnection based on a light-guiding plate with holographic coupling elements,” J. Mod. Opt. 39, 881–887 (1992).
    [CrossRef]
  5. W. T. Welford, Aberrations of Symmetrical Optical Systems (Academic, London, 1974), Chap. 2, p. 9.
  6. H. Ichikawa, J. Turunen, M. R. Taghizadeh, “Analysis of hybrid holographic gratings by thin-grating decomposition method,” J. Opt. Soc. Am. A 10, 1176–1183 (1993).
    [CrossRef]
  7. M. T. Gale, M. Rossi, H. Schütz, P. Ehbets, H. P. Herzig, D. Prongué, “Continuous-relief diffracted optical elements for two-dimensional array-generation,” Appl. Opt. 32, 2526–2533 (1993).
    [CrossRef] [PubMed]
  8. A. Rubinowicz, Die Beugungswelle in der Kirchhoffschen Theorie der Beugung (Springer-Verlag, Berlin, 1966), Chap. 2, p. 49.
  9. D. S. Jones, Acoustic and Electromagnetic Waves (Oxford U. Press, Oxford, 1986), Chap. 6, p. 321.
  10. E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
    [CrossRef]
  11. A. Sommerfeld, Optik, 3rd ed. (Verlag Harri Deutsch, Frankfurt/Main, Germany, 1978), Chap. 5, p. 162.
  12. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Chap. 4, p. 21.
  13. R. Dändliker, K. Weiss, “Reconstruction of the three-dimensional refractive index from scattered waves,” Opt. Commun. 1, 323–328 (1970).
    [CrossRef]
  14. A. W. Lohmann, “Three-dimensional properties of wave-fields,” Optik 51, 105–117 (1978).
  15. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1966), Chap. 14, p. 64.
  16. J. P. Mathieu, Optics (Pergamon, Oxford, 1975), Chap. 1, p. 9.
  17. E. Merzbacher, Quantum Mechanics (Wiley, New York, 1961), Chap. 1, p. 3.
  18. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 2.4, p. 98.
  19. D. C. Champeney, Fourier Transforms and Their Physical Applications (Academic, New York, 1973), Chap. 2, p. 34.
  20. E. Evans, “Comparison of the diffraction theory of image formation with the three-dimensional, first Born scattering approximation in lens systems,” Opt. Commun. 2, 317–320 (1970).
    [CrossRef]
  21. 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.
  22. D. A. Buralli, G. M. Morris, J. R. Rogers, “Optical performance of holographic kinoforms,” Appl. Opt. 28, 976–983 (1989); W. T. Cathey, Optical Information Processing (Wiley, New York, 1974).
    [CrossRef] [PubMed]
  23. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [CrossRef]
  24. K.-H. Brenner, W. Singer, “Light propagation through microlenses: a new simulation method,” Appl. Opt. 32, 4984–4988 (1993).
    [CrossRef] [PubMed]
  25. J. Saarinen, J. Turunen, J. Huttunen, “Volume diffraction effects in computer-generated guided-wave holography,” Appl. Opt. 33, 1035–1043 (1994).
    [CrossRef] [PubMed]
  26. S. Sinzinger, M. Testorf, W. Singer, “Transition between diffractive and refractive microoptical components,” in Diffractive Optics: Design, Fabrication, and Applications, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 143–146.

1994 (1)

1993 (3)

1992 (1)

H. Kobolla, J. Schmidt, J. T. Sheridan, N. Streibl, R. Völkl, “Optical interconnection based on a light-guiding plate with holographic coupling elements,” J. Mod. Opt. 39, 881–887 (1992).
[CrossRef]

1991 (1)

1989 (1)

1988 (1)

1981 (1)

1978 (1)

A. W. Lohmann, “Three-dimensional properties of wave-fields,” Optik 51, 105–117 (1978).

1970 (2)

E. Evans, “Comparison of the diffraction theory of image formation with the three-dimensional, first Born scattering approximation in lens systems,” Opt. Commun. 2, 317–320 (1970).
[CrossRef]

R. Dändliker, K. Weiss, “Reconstruction of the three-dimensional refractive index from scattered waves,” Opt. Commun. 1, 323–328 (1970).
[CrossRef]

1969 (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

Born, M.

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

Brenner, K.-H.

Buralli, D. A.

Champeney, D. C.

D. C. Champeney, Fourier Transforms and Their Physical Applications (Academic, New York, 1973), Chap. 2, p. 34.

Dändliker, R.

R. Dändliker, K. Weiss, “Reconstruction of the three-dimensional refractive index from scattered waves,” Opt. Commun. 1, 323–328 (1970).
[CrossRef]

Ehbets, P.

Evans, E.

E. Evans, “Comparison of the diffraction theory of image formation with the three-dimensional, first Born scattering approximation in lens systems,” Opt. Commun. 2, 317–320 (1970).
[CrossRef]

Gale, M. T.

Gaylord, T. K.

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.

Herzig, H. P.

Huttunen, J.

Ichikawa, H.

Jahns, J.

Jones, D. S.

D. S. Jones, Acoustic and Electromagnetic Waves (Oxford U. Press, Oxford, 1986), Chap. 6, p. 321.

Kobolla, H.

H. Kobolla, J. Schmidt, J. T. Sheridan, N. Streibl, R. Völkl, “Optical interconnection based on a light-guiding plate with holographic coupling elements,” J. Mod. Opt. 39, 881–887 (1992).
[CrossRef]

Lohmann, A. W.

A. W. Lohmann, J. Schwider, N. Streibl, J. Thomas, “Array illuminator based on phase contrast,” Appl. Opt. 27, 2915–2921 (1988).
[CrossRef] [PubMed]

A. W. Lohmann, “Three-dimensional properties of wave-fields,” Optik 51, 105–117 (1978).

A. W. Lohmann, Optical Information Processing Available from A. W. Lohmann at the address on the title page of this paper. (Erlangen, Germany, 1986), Vol. 1, Chap. 22, p. 127.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1966), Chap. 14, p. 64.

Mathieu, J. P.

J. P. Mathieu, Optics (Pergamon, Oxford, 1975), Chap. 1, p. 9.

Merzbacher, E.

E. Merzbacher, Quantum Mechanics (Wiley, New York, 1961), Chap. 1, p. 3.

Moharam, M. G.

Morris, G. M.

Nölscher, U.

Prongué, D.

Rogers, J. R.

Rossi, M.

Rubinowicz, A.

A. Rubinowicz, Die Beugungswelle in der Kirchhoffschen Theorie der Beugung (Springer-Verlag, Berlin, 1966), Chap. 2, p. 49.

Saarinen, J.

Schmidt, J.

H. Kobolla, J. Schmidt, J. T. Sheridan, N. Streibl, R. Völkl, “Optical interconnection based on a light-guiding plate with holographic coupling elements,” J. Mod. Opt. 39, 881–887 (1992).
[CrossRef]

Schütz, H.

Schwider, J.

Sheridan, J. T.

H. Kobolla, J. Schmidt, J. T. Sheridan, N. Streibl, R. Völkl, “Optical interconnection based on a light-guiding plate with holographic coupling elements,” J. Mod. Opt. 39, 881–887 (1992).
[CrossRef]

Singer, W.

K.-H. Brenner, W. Singer, “Light propagation through microlenses: a new simulation method,” Appl. Opt. 32, 4984–4988 (1993).
[CrossRef] [PubMed]

S. Sinzinger, M. Testorf, W. Singer, “Transition between diffractive and refractive microoptical components,” in Diffractive Optics: Design, Fabrication, and Applications, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 143–146.

Sinzinger, S.

S. Sinzinger, M. Testorf, W. Singer, “Transition between diffractive and refractive microoptical components,” in Diffractive Optics: Design, Fabrication, and Applications, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 143–146.

Sommerfeld, A.

A. Sommerfeld, Optik, 3rd ed. (Verlag Harri Deutsch, Frankfurt/Main, Germany, 1978), Chap. 5, p. 162.

Stamnes, J. J.

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

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.

Streibel, N.

Streibl, N.

H. Kobolla, J. Schmidt, J. T. Sheridan, N. Streibl, R. Völkl, “Optical interconnection based on a light-guiding plate with holographic coupling elements,” J. Mod. Opt. 39, 881–887 (1992).
[CrossRef]

A. W. Lohmann, J. Schwider, N. Streibl, J. Thomas, “Array illuminator based on phase contrast,” Appl. Opt. 27, 2915–2921 (1988).
[CrossRef] [PubMed]

Taghizadeh, M. R.

Testorf, M.

S. Sinzinger, M. Testorf, W. Singer, “Transition between diffractive and refractive microoptical components,” in Diffractive Optics: Design, Fabrication, and Applications, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 143–146.

Thomas, J.

Turunen, J.

Völkl, R.

H. Kobolla, J. Schmidt, J. T. Sheridan, N. Streibl, R. Völkl, “Optical interconnection based on a light-guiding plate with holographic coupling elements,” J. Mod. Opt. 39, 881–887 (1992).
[CrossRef]

Walker, S.

Weiss, K.

R. Dändliker, K. Weiss, “Reconstruction of the three-dimensional refractive index from scattered waves,” Opt. Commun. 1, 323–328 (1970).
[CrossRef]

Welford, W. T.

W. T. Welford, Aberrations of Symmetrical Optical Systems (Academic, London, 1974), Chap. 2, p. 9.

Wolf, E.

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

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

Appl. Opt. (6)

J. Mod. Opt. (1)

H. Kobolla, J. Schmidt, J. T. Sheridan, N. Streibl, R. Völkl, “Optical interconnection based on a light-guiding plate with holographic coupling elements,” J. Mod. Opt. 39, 881–887 (1992).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (3)

E. Evans, “Comparison of the diffraction theory of image formation with the three-dimensional, first Born scattering approximation in lens systems,” Opt. Commun. 2, 317–320 (1970).
[CrossRef]

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

R. Dändliker, K. Weiss, “Reconstruction of the three-dimensional refractive index from scattered waves,” Opt. Commun. 1, 323–328 (1970).
[CrossRef]

Optik (1)

A. W. Lohmann, “Three-dimensional properties of wave-fields,” Optik 51, 105–117 (1978).

Other (13)

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1966), Chap. 14, p. 64.

J. P. Mathieu, Optics (Pergamon, Oxford, 1975), Chap. 1, p. 9.

E. Merzbacher, Quantum Mechanics (Wiley, New York, 1961), Chap. 1, p. 3.

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

D. C. Champeney, Fourier Transforms and Their Physical Applications (Academic, New York, 1973), Chap. 2, p. 34.

A. Sommerfeld, Optik, 3rd ed. (Verlag Harri Deutsch, Frankfurt/Main, Germany, 1978), Chap. 5, p. 162.

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

A. Rubinowicz, Die Beugungswelle in der Kirchhoffschen Theorie der Beugung (Springer-Verlag, Berlin, 1966), Chap. 2, p. 49.

D. S. Jones, Acoustic and Electromagnetic Waves (Oxford U. Press, Oxford, 1986), Chap. 6, p. 321.

W. T. Welford, Aberrations of Symmetrical Optical Systems (Academic, London, 1974), Chap. 2, p. 9.

A. W. Lohmann, Optical Information Processing Available from A. W. Lohmann at the address on the title page of this paper. (Erlangen, Germany, 1986), Vol. 1, Chap. 22, p. 127.

S. Sinzinger, M. Testorf, W. Singer, “Transition between diffractive and refractive microoptical components,” in Diffractive Optics: Design, Fabrication, and Applications, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 143–146.

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.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (12)

Fig. 1
Fig. 1

Illustration of the Laue equation.

Fig. 2
Fig. 2

Illustration of Snell’s law by the Huygens construction.16

Fig. 3
Fig. 3

Kirchhoff’s screen and related coordinates.

Fig. 4
Fig. 4

Application of the Laue equation to refraction and reflection at a planar interface.

Fig. 5
Fig. 5

Illustration of the independence of refraction and reflection from wavelength changes.

Fig. 6
Fig. 6

Illustration of the transition of the scalar field according to the Laue equation at a prism surface.

Fig. 7
Fig. 7

General interface h(x) between homogeneous media.

Fig. 8
Fig. 8

Frequency domain representation of a thin grating.

Fig. 9
Fig. 9

Geometry of a blazed prism array.

Fig. 10
Fig. 10

Refraction and diffraction at a blazed grating according to the Laue equation.

Fig. 11
Fig. 11

Numerical result showing the incident and the refracted fields for a blazed prism array satisfying the blaze condition.

Fig. 12
Fig. 12

Numerical result showing the incident and the refracted fields for a blazed prism array deviating from the blaze condition.

Equations (28)

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

Δ u + n ¯ 2 k 0 2 u = - [ 2 δ n ( r ) n ¯ + δ n 2 ( r ) ] k 0 2 u = f ( r ) k 0 2 u ,
u = u i + u s ,
u s ( r ) = k 0 2 v f ( r ) u ( r ) G ( r ; r ) d 3 r .
G ( r , r ) = i 2 π - - exp { i [ k x ( x - x ) + k y ( y - y ) + k z ( z - z ) ] } k z × d k x d k y .
u ˜ s ( k x , k y ; z ) = i k 0 2 k z x , y , z f ( r ) u i ( r ) × exp { - i [ k x x + k y y + k z ( z - z ) ] } × d x d y d z = i k 0 2 k z exp ( i k z z ) g f ˜ ( g ) u ˜ i ( k - g ) d 3 g ( 2 π ) 3 .
u ˜ s [ k x , k y , k z = ( k 2 - k x 2 - k y 2 ) 1 / 2 ; z ] = i k 0 2 k z exp ( i k z z ) f ˜ ( g ) u ˜ i ( g ) .
u ˜ i ( g ) = δ ( g - k i ) ,
u ˜ s [ k x , k y , k z = ( k 2 - k x 2 - k y 2 ) 1 / 2 ; z ] = i k 0 2 k z exp ( i k z z ) f ˜ ( k - k i ) .
g = k - k i
P 1 - P 0 = Γ N , Γ = ( P 1 · N ) - ( P 0 · N ) = n 1 cos α 1 - n 0 cos α 0 .
P = ƛ k = λ 2 π k = λ ν .
k - k i = 2 π λ Γ N = k 0 Γ N .
Δ G ( r - r ) + n 1 2 k 0 2 G ( r - r ) = - δ ( r - r ) right half - space , Δ G ( r - r ) + n 0 2 k 0 2 G ( r - r ) = - δ ( r - r ) left half - space .
u ˜ ( k x , k y ; z ) = i χ k 0 2 k z exp ( i k z z ) { Δ ˜ ( g x , g y , g z ) u ˜ i [ k x , k y , k z = ( k 2 - k x 2 - k y 2 ) 1 / 2 ] } ,
Δ ( r ) = { 1 r right half - space 0 r left half - space
u ˜ ( k x , k y , k z ; z = 0 ) = ( n 1 2 - n 0 2 ) k 0 2 k z δ ( k x - k x i ) δ ( k y - k y i ) k z - k z i .
k t / r - k i = g t / r ,
g t / r = k 0 ( n t / r cos α t / r - n i cos α i ) .
u ˜ t = n 1 2 - n 0 2 n 1 cos α t δ ( k t = g t + k i ) ( n 1 cos α t - n 0 cos α i ) ,
Δ ˜ ( g x , g z ) x [ δ ( g z ) + i g z ] exp [ - i h ( x ) g z ] × exp ( - i g x x ) d x .
u ( x , z ) = χ k 0 2 4 π ( 2 π ) 2 x , z x g x , g z k x exp [ - i h ( x ) g z ] g z × u i ( x , z ) exp { i [ g x ( x - x ) + g z z ] } × exp { i [ k x ( x - x ) + k z ( z - z ) ] } k z × d k x d x d x d z d g x d g z .
u ( x , z ) = χ k 0 2 2 ( 2 π ) 2 x , k x k z i exp [ - i h ( x ) ( k z - k z i ) ] k z ( k z - k z i ) × u ˜ i ( x , k z i ) exp { i [ k x ( x - x ) + k z z ] } d k x d x d k z i .
T ( x ) = exp [ i ϕ ( x ) ] = exp [ i 2 π λ Δ n h ( x ) ] .
u ( x , z = 0 ) T ( x ) u i ( x , z = 0 ) .
F { rect ( x / b ) Δ ( x , z ) comb ( x / b ) } = i g z sinc [ b 2 π ( g x - g z tan β ) ] comb ( b 2 π g x ) ,
u ˜ s ( k x ; z = 0 ) = - i k 0 2 2 k z f ˜ ( g x = k x - k i sin ϕ , g z = k z - k i cos ϕ ) = χ k 0 2 2 k z ( k z - k i cos ϕ ) × sinc { b 2 π [ k x - k i sin ϕ - ( k z - k i cos ϕ ) tan β ] } × comb [ b 2 π ( k x - k i sin ϕ ) ] .
k x - k i sin ϕ = m ( 2 π / b ) .
m ( 2 π / b ) = ( k z - k i cos ϕ ) tan β .

Metrics