Abstract

It is shown that a system consisting of a polarizer, an Amici prism, and an analyzer may act like a phase and amplitude spatial filter capable of producing four fully distinct images. Two of these are complementary Schlieren or Foucault testlike images, another one is like a phase-edge test pattern, and the fourth is similar to a bright field image. An analysis is made of how the system works when the incident beam is collimated. The formation of the observed images can be explained by assuming that this analysis remains valid for incident beams of small covergence. Some experimental results are presented.

© 1990 Optical Society of America

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References

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  1. A. I. Mahan, “Focal Plane Anomalies in Roof Prisms,” J. Opt. Soc. Am., 35, 623–645 (1945).
    [CrossRef]
  2. A. I. Mahan, E. E. Price, “Diffraction Pattern Deterioration by Roof Prisms,” J. Opt. Soc. Am., 40, 664–686 (1950).
    [CrossRef]
  3. A. M. Tareev, “Effect of Polarization Phenomena in Roof Prisms on the Energy Distribution in an Image,” Sov. J. Opt. Technol., 52, 573–576 (1985).
  4. V. I. Korneev, A. M. Tareev, “Effect of Polarization on the Optical Transfer Function of Roof Prisms,” Sov. J. Opt. Technol., 53, 402–405 (1986).
  5. B. Fuentes-Madariaga, R. Díaz-Uribe, G. Rodríguez-Zurita, “Polarizing Properties of Roof Prisms” in preparation.
  6. R. Barakat, “General Diffraction Theory of Optical Aberration Tests, from the Point of View of Spatial Filtering,” J. Opt. Soc. Am., 59, 1432–1439 (1969).
    [CrossRef]
  7. Rays making an angle >18° from the normal to the entrance face of the prism may not be totally reflected on the roof, and their behavior is not described even in approximate form by the theory of Sec. II.
  8. J. Ojeda-Castaneda, “Foucault, Wire, and Phase Modulation Tests,” in Optical Shop Testing, D. Malacara, Ed., (Wiley, New York, 1978), Chap. 8.
  9. G. Rodríquez-Zurita, R. Díaz-Uribe, “Phase-Edge Effect in Amici Prisms” Proceedings of the SPIE, Vol. 813, 573–574 (1987).
    [CrossRef]

1986 (1)

V. I. Korneev, A. M. Tareev, “Effect of Polarization on the Optical Transfer Function of Roof Prisms,” Sov. J. Opt. Technol., 53, 402–405 (1986).

1985 (1)

A. M. Tareev, “Effect of Polarization Phenomena in Roof Prisms on the Energy Distribution in an Image,” Sov. J. Opt. Technol., 52, 573–576 (1985).

1969 (1)

1950 (1)

1945 (1)

Barakat, R.

Díaz-Uribe, R.

G. Rodríquez-Zurita, R. Díaz-Uribe, “Phase-Edge Effect in Amici Prisms” Proceedings of the SPIE, Vol. 813, 573–574 (1987).
[CrossRef]

B. Fuentes-Madariaga, R. Díaz-Uribe, G. Rodríguez-Zurita, “Polarizing Properties of Roof Prisms” in preparation.

Fuentes-Madariaga, B.

B. Fuentes-Madariaga, R. Díaz-Uribe, G. Rodríguez-Zurita, “Polarizing Properties of Roof Prisms” in preparation.

Korneev, V. I.

V. I. Korneev, A. M. Tareev, “Effect of Polarization on the Optical Transfer Function of Roof Prisms,” Sov. J. Opt. Technol., 53, 402–405 (1986).

Mahan, A. I.

Ojeda-Castaneda, J.

J. Ojeda-Castaneda, “Foucault, Wire, and Phase Modulation Tests,” in Optical Shop Testing, D. Malacara, Ed., (Wiley, New York, 1978), Chap. 8.

Price, E. E.

Rodríguez-Zurita, G.

B. Fuentes-Madariaga, R. Díaz-Uribe, G. Rodríguez-Zurita, “Polarizing Properties of Roof Prisms” in preparation.

Rodríquez-Zurita, G.

G. Rodríquez-Zurita, R. Díaz-Uribe, “Phase-Edge Effect in Amici Prisms” Proceedings of the SPIE, Vol. 813, 573–574 (1987).
[CrossRef]

Tareev, A. M.

V. I. Korneev, A. M. Tareev, “Effect of Polarization on the Optical Transfer Function of Roof Prisms,” Sov. J. Opt. Technol., 53, 402–405 (1986).

A. M. Tareev, “Effect of Polarization Phenomena in Roof Prisms on the Energy Distribution in an Image,” Sov. J. Opt. Technol., 52, 573–576 (1985).

J. Opt. Soc. Am. (3)

Sov. J. Opt. Technol. (2)

A. M. Tareev, “Effect of Polarization Phenomena in Roof Prisms on the Energy Distribution in an Image,” Sov. J. Opt. Technol., 52, 573–576 (1985).

V. I. Korneev, A. M. Tareev, “Effect of Polarization on the Optical Transfer Function of Roof Prisms,” Sov. J. Opt. Technol., 53, 402–405 (1986).

Other (4)

B. Fuentes-Madariaga, R. Díaz-Uribe, G. Rodríguez-Zurita, “Polarizing Properties of Roof Prisms” in preparation.

Rays making an angle >18° from the normal to the entrance face of the prism may not be totally reflected on the roof, and their behavior is not described even in approximate form by the theory of Sec. II.

J. Ojeda-Castaneda, “Foucault, Wire, and Phase Modulation Tests,” in Optical Shop Testing, D. Malacara, Ed., (Wiley, New York, 1978), Chap. 8.

G. Rodríquez-Zurita, R. Díaz-Uribe, “Phase-Edge Effect in Amici Prisms” Proceedings of the SPIE, Vol. 813, 573–574 (1987).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Sketch of an Amici prism. A and B are the entrance and exit faces of the prism. R1 and R2 are the two faces of the roof. r1 and r2 are rays of type one and two, respectively.

Fig. 2
Fig. 2

Diagram of an Amici prism placed between two linear polarizers. E0 is the formerly incident natural light (unpolarized) on the polarizer P1. E is the field incident on the prism; it is linearly polarized along a plane making an angle θ with the X-axis. E 1 and E 2 are the fields of the rays of type one and two, respectively, that exits the prism. They are elliptically polarized. After passing through the analyzer P2, the amplitudes and phases of the final emergent linearly polarized fields E 1 and E 2 are different in general.

Fig. 3
Fig. 3

Ellipses described by the fields E 1 and E 2 , for the case θ = 0°. Both ellipses have the semimajor axis length a = 0.931, the semiminor axis length b = 0.215 and the angle of inclination σ = 33.04°.

Fig. 4
Fig. 4

Amplitude variation of the emergent fields ‖ E 1 , 2 ; η ‖ as a function of the angle θ. a) Case η = θ, both types of rays. b) Case η = θ ± 90°, both types of rays. Case η = θ + 45°: c) rays of type one, d) rays of type two. Case η = θ − 45°: e) rays of type one, f) rays of type two.

Fig. 5
Fig. 5

Phase difference between rays of type one and two for the emergent fields, as a function of the angle θ. a) Case η = θ. b) Case η = θ ± 90°. c) Case η = θ ± 45°. d) Case η = θ − 45°.

Fig. 6
Fig. 6

Emerging fields schematically represented with their relative amplitudes and phases (not in scale). a) η = θ. b) θ = 0° or θ = 90°, and η = θ ± 90°. c) η = θ ± 90°, 0° < θ < 90°. d) η = θ + 45°. e) η = θ − 45°.

Fig. 7
Fig. 7

Experimental setup to use an Amici prism as a phase or amplitude knife edge. Description in the text.

Fig. 8
Fig. 8

Simulated effect of the polarizer–prism–analyzer system, on the spatial frequency spectrum of an object placed against the lens L1 of the setup of Fig. 7 (the object may be the lens itself). a) Is the simulated incident spatial spectrum of an object. After being reflected on the roof, the spectrum is transformed in b) when η = θ; c) η = θ + 90°; d) η = θ + 45°; and e) η = θ − 45°. In those figures, e is the Fourier transform of the complex amplitude of the field, whereas f represents the spatial frequencies. Rays of type one belongs to negative frequencies.

Fig. 9
Fig. 9

Experimental filtered images of an f/4 simple lens, in all the figures θ = 0°. a) Bright field image, η = 0°. b) Phase edge filter, η = 90°. c) and d) Two complementary Foucault testlike images, η = ± 45°.

Fig. 10
Fig. 10

Patterns of the same lens used for Fig. 9. In this case η = θ = 0° for the first picture. The others were taken with θ = 0°, and successive steps of 15° in η.

Fig. 11
Fig. 11

Amici-like prism to avoid introducing additional spherical aberration to a converging beam. S and S′ are the entrance and exit faces; both faces are spherical surfaces with a common center of curvature C.

Equations (20)

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E 1 = M 1 E E 2 = M 2 E .
M 1 = [ A e i α B e i β - B e i β A e i γ ] M 2 = [ A e i α - B e i β B e i β A e i γ ] ,
E = ( cos θ sin θ ) .
E 1 , 2 = ( A exp ( i α ) cos θ ± B exp ( i β ) sin θ B exp ( i β ) cos θ + A exp ( i γ ) sin θ ) ,
E 1 , 2 ; η = ( A e i α cos θ ± B e i β sin θ ) cos η + ( B e i β cos θ + A e i γ sin θ ) sin η .
E 1 , 2 ; η = θ = A ( exp ( i α ) cos 2 θ + exp ( i γ ) sin 2 θ ) .
E 1 , 2 ; η = θ = A [ 1 - sin 2 ( α - γ 2 ) sin 2 ( 2 θ ) ] 1 / 2 .
E 1 , 2 ; η = θ ± 90 ° = A sin θ cos θ [ - exp ( i α ) + exp ( i γ ) ] B exp ( i β ) .
E 1 , 2 ; η = θ ± 90 ° = - exp ( i β ) [ i A sin ( 2 θ ) sin Δ ± B ] ,
E 1 , 2 ; η = θ ± 90 ° = [ B 2 + A 2 sin 2 Δ sin 2 ( 2 θ ) ] 1 / 2 .
δ ϕ = 180 ° + 2 tan - 1 [ A B sin Δ sin ( 2 θ ) ] .
E 1 , 2 ; η = θ ± 45 ° = K M 1 , 2 exp ( i μ 1 , 2 )
K = sin 45 ° = cos 45 ° , M 1 , 2 2 = A 1 , 2 2 + B 1 , 2 2 , μ 1 , 2 = tan - 1 ( A 1 , 2 B 1 , 2 ) , }
A 1 , 2 = A [ ½ sin ( 2 θ ) ( cos γ - cos α ) + cos α cos 2 θ + cos γ sin 2 θ ] B cos β B 1 , 2 = A [ ½ sin ( 2 θ ) ( sin γ - sin α ) + sin α cos 2 θ + sin γ sin 2 θ ] B sin β
E 1 , 2 ; η = 45 ° θ = 0 ° = K [ A e i α B e i β ] = K e i β [ A e i Δ B ] ,
E 1 , 2 ; η = 45 ° θ = 0 ° = K [ A 2 B 2 2 A B cos Δ ] 1 / 2 .
E 1 , η = 45 ° θ = 0 ° = 0.972             E 2 , η = 45 ° θ = 0 ° = 0.237.
A 1 , 2 = A [ ½ sin ( 2 θ ) ( cos α - cos γ ) + cos α cos 2 θ + cos γ sin 2 θ ] ± B cos β B 1 , 2 = A [ ½ sin ( 2 θ ) ( sin α - sin γ ) + sin α cos 2 θ + sin γ sin 2 θ ] ± B sin β
E 1 , 2 ; η = - 45 ° θ = 0 ° = K e i β [ A e i Δ ± B ] .
E 1 , 2 ; η = 45 ° θ = 0 ° = K [ A 2 + B 2 ± 2 A B cos Δ ] 1 / 2 .

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