Abstract

Pseudoscopic (inverted depth) images that keep a continuous parallax were shown to be possible by use of a double diffraction process intermediated by a slit. One diffraction grating directing light to the slit acts as a wavelength encoder of views, while a second diffraction grating decodes the projected image. The process results in the enlargement of the image under common white light illumination up to infinite magnification at a critical point. We show that this point corresponds to another simple-symmetry object–observer system. Our treatment allows us to explain the experience by just dealing with main ray directions.

© 2006 Optical Society of America

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References

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  1. F. Okano and J. Arai, "Optical shifter for a three-dimensional image by use of a gradient-index lens array," Appl. Opt. 41, 4140-4147 (2002).
    [CrossRef] [PubMed]
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    [PubMed]
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    [CrossRef]
  4. L. S. Pedrotti and F. L. Pedrotti, "Optics of the eye," in Optic and Vision (Prentice Hall, 1998), pp. 202-203.
  5. J. M. Simon and M. A. Gil, "Diffraction grating and optical aberrations: a new and exact formulation," Appl. Opt. 24, 2956-2958 (1985).
  6. E. N. Hogert, M. A. Rebollo, and N. G. Gaggioli, "Align-ment and/or tilting measurement by means of conical diffraction phenomena," Opt. Laser Technol. 23, 341-344 (1991).
    [CrossRef]

2002 (2)

1999 (1)

1998 (1)

L. S. Pedrotti and F. L. Pedrotti, "Optics of the eye," in Optic and Vision (Prentice Hall, 1998), pp. 202-203.

1991 (1)

E. N. Hogert, M. A. Rebollo, and N. G. Gaggioli, "Align-ment and/or tilting measurement by means of conical diffraction phenomena," Opt. Laser Technol. 23, 341-344 (1991).
[CrossRef]

1985 (1)

Arai, J.

Gaggioli, N. G.

E. N. Hogert, M. A. Rebollo, and N. G. Gaggioli, "Align-ment and/or tilting measurement by means of conical diffraction phenomena," Opt. Laser Technol. 23, 341-344 (1991).
[CrossRef]

Gil, M. A.

Hogert, E. N.

E. N. Hogert, M. A. Rebollo, and N. G. Gaggioli, "Align-ment and/or tilting measurement by means of conical diffraction phenomena," Opt. Laser Technol. 23, 341-344 (1991).
[CrossRef]

Hyde, R. A.

Lunazzi, J. J.

Okano, F.

Pedrotti, F. L.

L. S. Pedrotti and F. L. Pedrotti, "Optics of the eye," in Optic and Vision (Prentice Hall, 1998), pp. 202-203.

Pedrotti, L. S.

L. S. Pedrotti and F. L. Pedrotti, "Optics of the eye," in Optic and Vision (Prentice Hall, 1998), pp. 202-203.

Rebollo, M. A.

E. N. Hogert, M. A. Rebollo, and N. G. Gaggioli, "Align-ment and/or tilting measurement by means of conical diffraction phenomena," Opt. Laser Technol. 23, 341-344 (1991).
[CrossRef]

Rivera, N. I.

Simon, J. M.

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Figures (13)

Fig. 1
Fig. 1

Ray-tracing scheme for the depth inverted image.

Fig. 2
Fig. 2

Schematic ray tracing showing the limit of sharpness due to the width of the slit.

Fig. 3
Fig. 3

Schematic perspective view of the setup including ray tracing that generates vertical astigmatism. A, point object; A, astigmatic image.

Fig. 4
Fig. 4

Ray-tracing scheme for calculating the angular field of view.

Fig. 5
Fig. 5

Ray-tracing scheme showing object field of view. Object in three positions around the critical point: O 1 , in front of the critical point; O 2 , at the critical point; O 3 , behind the critical point.

Fig. 6
Fig. 6

Single ray path for a vertically displaced observer, O 2 .

Fig. 7
Fig. 7

Observer’s position that keeps the critical point invariant when the second grating is displaced.

Fig. 8
Fig. 8

Ray-tracing scheme showing magnification and observer’s position when the second grating’s distance is doubled.

Fig. 9
Fig. 9

Photographic evidence of the pseudoscopy of the image: right-hand-side image (blue wavelength) and left-hand-side image (red wavelength).

Fig. 10
Fig. 10

Top, unfiltered red-wavelength view of the filament; middle, same view after filtering to 634 640 nm bandwidth; bottom, same view as top image but filtered to 643 657 nm bandwidth.

Fig. 11
Fig. 11

Magnification by displacement of the second grating: (a) undisplaced, (b) displaced 300 mm .

Fig. 12
Fig. 12

Schematic view of the experiment performed to show the properties of the critical point.

Fig. 13
Fig. 13

Sequence of object positions at increasing distances showing the inversion of the image: (a) between the first grating and the critical point, (b) at the critical position, and (c) farther from the critical position, showing lateral inversion.

Tables (1)

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Table 1 Experimental Checking of the Angular Field a

Equations (8)

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sin θ i sin θ d = λ ν ,
θ i = arcsin [ ( X X i ) ( X X i ) 2 + Z 2 ] ,
( X X i ) ( X X i ) 2 + Z 2 X i X i 2 + Z R 2 = λ ν .
θ i = θ i ( X , Z , Z R , ν , λ ) ,
Δ θ = θ i ( X , Z , X B , Z B , Z R , ν , λ M ) θ i ( X , Z , Z A , Z A , Z R , ν , λ m ) .
θ i = arcsin λ ν ,
Δ θ = arcsin λ M ν arcsin λ m ν
sin θ i sin θ d = λ ν / 1 φ 2 .

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