Abstract

Analyses of prism anamorphic optical systems with single and multiple prisms in plane and spherical wave illumination geometries are presented. A theoretical evaluation and experimental demonstration of a new technique for recording holographic anamorphic elements by using prism anamorphic optical systems is discussed. It is seen that this technique offers the advantage of generating elements with low f/number. Another advantage of this recording is the easy generation of elements with different anamorphic factors by using a single configuration.

© 1992 Optical Society of America

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References

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  1. T. Szoplik, H. H. Arsenault, “Shift and scale-invariant anamorphic Fourier correlator using multiple circular harmonic filters,” Appl. Opt. 24, 3179–3184 (1985).
    [CrossRef] [PubMed]
  2. T. Szoplik, “Shift- and scale-invariant anamorphic Fourier correlator,” J. Opt. Soc. Am. A 2, 1419–1423 (1985).
    [CrossRef]
  3. T. Szoplik, H. H. Arsenault, “Nonsymmetrical Fourier transform hologram,” J. Opt. Soc. Am. A 1, 1203–1205 (1984).
  4. E. Bonet, C. Ferreira, P. Andres, A. Pons, “Nonsymmetrical Fourier correlator to increase angular discriminant in character recognition,” Opt. Commun. 58, 161–166 (1986).
    [CrossRef]
  5. T. Szoplik, K. Chalasinska-Macukow, J. Kosek, “Accuracy of angular spectral analysis with an anamorphic Fourier transformer,” Appl. Opt. 25, 188–192 (1986).
    [CrossRef] [PubMed]
  6. K. Tatsuno, R. Drenten, C. J. van der Poel, J. Opschoor, G. A. Acket, “Diffraction-limited circular single spot from phased array lasers,” Appl. Opt. 28, 4560–4568 (1989).
    [CrossRef] [PubMed]
  7. B. Cuony, “Calcul de l’Activité optique Vibrationnelle Raman,” Ph.D. dissertation (University of Fribourg, Fribourg, Switzerland, 1981).
  8. T. Szoplik, W. Kosek, C. Ferreira, “Nonsymmetrical Fourier transforming with an anamorphic system,” Appl. Opt. 23, 905–909 (1984).
    [CrossRef] [PubMed]
  9. D. H. Close, “Holographic optical elements,” Opt. Eng. 14, 408–419 (1975).
  10. B. Chang, K. A. Winick, “Holographic optical elements,” in Advances in Laser Scanning Technology, L. Beiser, ed., Proc. Soc. Photo-Opt. Instrum. Eng.299, 157–162 (1981).
  11. R. R. A. Syms, L. Solymar, “Analysis of volume holographic cylindrical lenses,” J. Opt. Soc. Am. 72, 179–186 (1982).
    [CrossRef]
  12. R. Kingslake, “Dispersing prism,” in Applied Optics and Optical Engineering; R. Kingslake, ed. (Academic, New York, 1969).
  13. G. A. Boutry, Instrumental Optics (Hilger & Watts, London, 1961).
  14. G. W. Stroke, “Lensless Fourier transform method for optical holography,” Appl. Phys. Lett. 6, 201–203 (1966).
    [CrossRef]
  15. M. Miler, C. W. Slinger, J. M. Heaton, “Off axis zone plates recorded and reconstructed by cylindrical wavefronts,” Opt. Acta. 31, 745–758 (1984).
    [CrossRef]

1989 (1)

1986 (2)

E. Bonet, C. Ferreira, P. Andres, A. Pons, “Nonsymmetrical Fourier correlator to increase angular discriminant in character recognition,” Opt. Commun. 58, 161–166 (1986).
[CrossRef]

T. Szoplik, K. Chalasinska-Macukow, J. Kosek, “Accuracy of angular spectral analysis with an anamorphic Fourier transformer,” Appl. Opt. 25, 188–192 (1986).
[CrossRef] [PubMed]

1985 (2)

1984 (3)

1982 (1)

1975 (1)

D. H. Close, “Holographic optical elements,” Opt. Eng. 14, 408–419 (1975).

1966 (1)

G. W. Stroke, “Lensless Fourier transform method for optical holography,” Appl. Phys. Lett. 6, 201–203 (1966).
[CrossRef]

Acket, G. A.

Andres, P.

E. Bonet, C. Ferreira, P. Andres, A. Pons, “Nonsymmetrical Fourier correlator to increase angular discriminant in character recognition,” Opt. Commun. 58, 161–166 (1986).
[CrossRef]

Arsenault, H. H.

Bonet, E.

E. Bonet, C. Ferreira, P. Andres, A. Pons, “Nonsymmetrical Fourier correlator to increase angular discriminant in character recognition,” Opt. Commun. 58, 161–166 (1986).
[CrossRef]

Boutry, G. A.

G. A. Boutry, Instrumental Optics (Hilger & Watts, London, 1961).

Chalasinska-Macukow, K.

Chang, B.

B. Chang, K. A. Winick, “Holographic optical elements,” in Advances in Laser Scanning Technology, L. Beiser, ed., Proc. Soc. Photo-Opt. Instrum. Eng.299, 157–162 (1981).

Close, D. H.

D. H. Close, “Holographic optical elements,” Opt. Eng. 14, 408–419 (1975).

Cuony, B.

B. Cuony, “Calcul de l’Activité optique Vibrationnelle Raman,” Ph.D. dissertation (University of Fribourg, Fribourg, Switzerland, 1981).

Drenten, R.

Ferreira, C.

E. Bonet, C. Ferreira, P. Andres, A. Pons, “Nonsymmetrical Fourier correlator to increase angular discriminant in character recognition,” Opt. Commun. 58, 161–166 (1986).
[CrossRef]

T. Szoplik, W. Kosek, C. Ferreira, “Nonsymmetrical Fourier transforming with an anamorphic system,” Appl. Opt. 23, 905–909 (1984).
[CrossRef] [PubMed]

Heaton, J. M.

M. Miler, C. W. Slinger, J. M. Heaton, “Off axis zone plates recorded and reconstructed by cylindrical wavefronts,” Opt. Acta. 31, 745–758 (1984).
[CrossRef]

Kingslake, R.

R. Kingslake, “Dispersing prism,” in Applied Optics and Optical Engineering; R. Kingslake, ed. (Academic, New York, 1969).

Kosek, J.

Kosek, W.

Miler, M.

M. Miler, C. W. Slinger, J. M. Heaton, “Off axis zone plates recorded and reconstructed by cylindrical wavefronts,” Opt. Acta. 31, 745–758 (1984).
[CrossRef]

Opschoor, J.

Pons, A.

E. Bonet, C. Ferreira, P. Andres, A. Pons, “Nonsymmetrical Fourier correlator to increase angular discriminant in character recognition,” Opt. Commun. 58, 161–166 (1986).
[CrossRef]

Slinger, C. W.

M. Miler, C. W. Slinger, J. M. Heaton, “Off axis zone plates recorded and reconstructed by cylindrical wavefronts,” Opt. Acta. 31, 745–758 (1984).
[CrossRef]

Solymar, L.

Stroke, G. W.

G. W. Stroke, “Lensless Fourier transform method for optical holography,” Appl. Phys. Lett. 6, 201–203 (1966).
[CrossRef]

Syms, R. R. A.

Szoplik, T.

Tatsuno, K.

van der Poel, C. J.

Winick, K. A.

B. Chang, K. A. Winick, “Holographic optical elements,” in Advances in Laser Scanning Technology, L. Beiser, ed., Proc. Soc. Photo-Opt. Instrum. Eng.299, 157–162 (1981).

Appl. Opt. (4)

Appl. Phys. Lett. (1)

G. W. Stroke, “Lensless Fourier transform method for optical holography,” Appl. Phys. Lett. 6, 201–203 (1966).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Acta. (1)

M. Miler, C. W. Slinger, J. M. Heaton, “Off axis zone plates recorded and reconstructed by cylindrical wavefronts,” Opt. Acta. 31, 745–758 (1984).
[CrossRef]

Opt. Commun. (1)

E. Bonet, C. Ferreira, P. Andres, A. Pons, “Nonsymmetrical Fourier correlator to increase angular discriminant in character recognition,” Opt. Commun. 58, 161–166 (1986).
[CrossRef]

Opt. Eng. (1)

D. H. Close, “Holographic optical elements,” Opt. Eng. 14, 408–419 (1975).

Other (4)

B. Chang, K. A. Winick, “Holographic optical elements,” in Advances in Laser Scanning Technology, L. Beiser, ed., Proc. Soc. Photo-Opt. Instrum. Eng.299, 157–162 (1981).

B. Cuony, “Calcul de l’Activité optique Vibrationnelle Raman,” Ph.D. dissertation (University of Fribourg, Fribourg, Switzerland, 1981).

R. Kingslake, “Dispersing prism,” in Applied Optics and Optical Engineering; R. Kingslake, ed. (Academic, New York, 1969).

G. A. Boutry, Instrumental Optics (Hilger & Watts, London, 1961).

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Figures (8)

Fig. 1
Fig. 1

Propagation of plane waves through a prism with refractive index n placed in air.

Fig. 2
Fig. 2

Variation of the anamorphic factor with incident angle for an equilateral prism, A = 60 deg; right-angle prism, A = 45 deg; Littrow prism, A = 34.35 deg (the refractive index of the material of the prism is 1.648 at 632.8 nm).

Fig. 3
Fig. 3

Variation of the anamorphic factor with a refractive index of the prism for an equilateral prism, A = 60 deg, right-angle prism, A = 45 deg, Littrow prism, A = 34.35 deg (angle of incidence is 75 deg).

Fig. 4
Fig. 4

Plane-wave propagation through two prisms [refractive index is n(j), j = 1, 2] in a general configuration. L is the distance between the two prisms.

Fig. 5
Fig. 5

Spherical wave propagation through a prism; the distances are SO = Zi, OO = e, and OH = Zh.

Fig. 6
Fig. 6

Schematic of the arrangement used to record HAE’s: BS, beam splitter; M1, M2 mirrors; MO1, MO2, microscope objectives (10×, 0.3 numerical aperture); PR, equilateral prism (n = 1.648 at 632.8 nm); HP, hologram recording plane.

Fig. 7
Fig. 7

Photographs of reconstructions under conjugate illumination from (a) HAE I, vertical focus; (b) HAE II, horizontal focus.

Fig. 8
Fig. 8

Photograph of the curved focal line reconstructed from HAE’s (conjugate illumination) when recording conditions are not satisfied (see text for explanation).

Tables (1)

Tables Icon

Table I Wavelength Dependence of Focal Lengths (Longitudinal Chromatic Aberration) of HAE I Recorded at 632.8 nm and Reconstructed at Different Wavelengths

Equations (34)

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M = cos ( i 1 ) cos ( i 3 ) cos ( i 2 ) cos ( i 4 ) .
[ h 4 V 4 ] = [ 1 / M b 0 M ] [ h 1 V 1 ] ,
[ h o V o ] = [ 1 / ( M 1 M 2 ) ( b 2 M 1 M 2 + L M 1 + b 1 ) / M 2 0 ( M 1 M 2 ) ] [ h e V e ] ,
M j = cos i 1 ( j ) cos i 3 ( j ) / [ cos i 2 ( j ) cos i 4 ( j ) ] , b j = d ( j ) sin A ( j ) cos i 1 ( j ) cos i 4 ( j ) / [ n cos 2 i 3 ( j ) cos i 2 ( j ) ] ,
[ h o V o ] = [ 1 / M p b p 0 M p ] [ 1 L ( p 1 ) p 0 1 ] [ 1 / M p 1 b p 1 0 M p 1 ] × [ 1 L ( p 2 ) ( p 1 ) 0 1 ] [ 1 / M 2 b 2 0 M 2 ] [ 1 L 12 0 1 ] [ 1 / M 1 b 1 0 M 1 ] [ h e V e ] ,
R s = Z i ( e / n ) Z h ,
R t = ( Z i / M 2 ) ( e / n ) ( cos 2 i 4 / cos 2 i 3 ) Z h ,
A G = Z i ( 1 / M 2 1 ) + ( e / n ) [ 1 ( cos 2 i 4 / cos 2 i 3 ) ] ,
A MD = ( e / n ) [ 1 ( cos 2 i 4 / cos 2 i 3 ) ] .
R s C = R s ( 1 ) + R s ( 2 ) Z h = Z i [ e ( 1 ) / n ( 1 ) ] [ e ( 2 ) / n ( 2 ) ] Z h Z 12 ,
R t C = R t ( 1 ) R t ( 2 ) Z h = [ Z i / M 2 ( 1 ) ] [ e ( 1 ) / n ( 1 ) ] [ cos 2 i 4 ( 1 ) / cos 2 i 3 ( 1 ) ] + [ Z 12 / M 2 ( 2 ) ] + [ e ( 2 ) / n ( 2 ) ] [ cos 2 i 4 ( 2 ) / cos 2 i 3 ( 2 ) ] Z h .
A G C = Z i { [ 1 / M 2 ( 1 ) ] 1 } + Z 12 { [ 1 / M 2 ( 2 ) ] 1 } + [ e ( 1 ) / n ( 1 ) ] { 1 [ cos 2 i 4 ( 1 ) / cos 2 i 3 ( 1 ) ] } + [ e ( 2 ) / n ( 2 ) ] { 1 [ cos 2 i 4 ( 2 ) / cos 2 i 3 ( 2 ) ] } ,
R s A = R s ( 1 ) + R s ( 2 ) Z h = Z i [ e ( 1 ) / n ( 1 ) ] [ e ( 2 ) / n ( 2 ) ] Z h Z 12 ,
R t A = R t ( 1 ) + R t ( 2 ) Z h = [ Z i / M 2 ( 1 ) ] [ e ( 1 ) / n ( 1 ) ] [ cos 2 i 4 ( 1 ) / cos 2 i 3 ( 1 ) ] [ Z 12 / M 2 ( 2 ) ] [ e ( 2 ) / n ( 2 ) ] [ cos 2 i 4 ( 2 ) / cos 2 i 3 ( 2 ) ] Z h .
A G A = Z i { [ 1 / M 2 ( 1 ) ] 1 } Z 12 { [ 1 / M 2 ( 2 ) ] 1 } + [ e ( 1 ) / n ( 1 ) ] { 1 [ cos 2 i 4 ( 1 ) / cos 2 i 3 ( 1 ) ] } [ e ( 2 ) / n ( 2 ) ] { 1 [ cos 2 i 4 ( 2 ) / cos 2 i 3 ( 2 ) ] } ,
R s = R s ( 1 ) + R s ( 2 ) + R s ( 3 ) + + R s ( p ) Z h ,
R t = R t ( 1 ) ± R t ( 2 ) + R t ( 3 ) + + R t ( p ) Z h ,
r o = R t + x 2 / ( 2 R s ) + y 2 / ( 2 R t ) ,
r o = R t + x 2 / ( 2 R s ) + ( y 2 cos 2 θ 1 ) / ( 2 R t ) y sin θ 1 .
r r = R s + x 2 / ( 2 R s ) + y 2 / ( 2 R s ) .
r r = R s + x 2 / ( 2 R s ) + [ y 2 cos 2 θ 2 / ( 2 R s ) ] y sin θ 2 .
ϕ o = ( 2 π / λ 1 ) ( r o ) ,
ϕ r = ( 2 π / λ 1 ) ( r r ) ,
ϕ o ϕ r = ( 2 π / λ 1 ) { ( R s R t ) + ( y 2 / 2 ) × [ ( cos 2 θ 2 / R s ) ( cos 2 θ 1 / R t ) ] + y ( sin θ 2 + sin θ 1 ) } .
ϕ w = ϕ c ± ( ϕ o ϕ r ) ,
ϕ c = ( 2 π / λ 2 ) ( y sin θ 3 ) .
ϕ w = ( 2 π / λ 2 ) y sin θ 3 ± ( 2 π / λ 1 ) { ( R s R t ) + ( y 2 / 2 ) [ ( cos 2 θ 2 / R s ) ( cos 2 θ 1 / R t ) ] + y ( sin θ 2 + sin θ 1 ) } .
ϕ ω = 2 π y [ ( sin θ 3 / λ 2 ) + ( sin θ 2 / λ 1 ) + ( sin θ 1 / λ 1 ) ] + ( 2 π / λ 1 ) { ( R s R t ) + ( y 2 / 2 ) [ ( cos 2 θ 2 / R s ) ( cos 2 θ 1 / R t ) ] } .
d y = ( sin θ 1 + sin θ 2 ) / λ 1 .
d y = ( sin θ 3 + sin θ 4 ) / λ 2 .
d y = ( sin θ 1 + sin θ 2 ) / λ 1 = ( sin θ 3 + sin θ 4 ) / λ 2 .
ϕ w = ( 2 π / λ 2 ) { 2 y sin θ 3 + y sin θ 4 + N A + [ N y 2 / ( 2 B ) ] } ,
ϕ w = ϕ c ( ϕ o ϕ r ) .
ϕ w = ( 2 π / λ 2 ) [ y sin θ 4 A N ( y 2 / 2 B ) ] .

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