Abstract

We describe in detail the partial pixel architecture that permits the realization of three-dimensional (3-D) displays that are functionally equivalent to a real-time holographic stereogram. This architecture permits the simultaneous presentation of multiple stereoscopic images so that motion parallax is discernible in the resultant 3-D scene. The key innovation of the architecture is that each pixel is subdivided into partial pixels, which in turn can be implemented as individual diffraction gratings. We describe a static display that exhibits a 3-D image with one-dimensional motion parallax, thereby demonstrating key features of the architecture. A variety of partial pixel implementations are discussed that can operate at video frame rates. These include voltage-controlled liquid crystal gratings and binary optic gratings integrated with conventional liquid crystal amplitude modulators. In addition, we describe how the partial pixel architecture can be generalized for the implementation of full-color displays and displays having two-dimensional motion parallax.

© 1995 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. St. Hilaire, S. A. Benton, M. Lucente, M. L. Jepson, J. Kollin, H. Yoshikawa, J. Underkoffler, “Electronic display system for computational holography,” in Practical Holography IV, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1212, 174–182 (1990).
    [CrossRef]
  2. P. St. Hilaire, S. A. Benton, M. Lucente, H. Yoshikawa, J. Underkoffler, “Real-time holographic display: improvements using a multichannel acousto-optic modulator and holographic optical elements,” in Practical Holography V, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1461, 256–261 (1991).
  3. S. A. Benton, “Experiments in holographic video imaging,” in Industrial Applications of Holographic and Speckle Measuring Techniques, W. P. Jueptner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1508, 247–267 (1991).
  4. M. Lucente, “Optimization of hologram computation for real-time display,” in Practical Holography VI, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1667, 32–43 (1992).
    [CrossRef]
  5. P. St. Hilaire, S. A. Benton, M. Lucente, “Synthetic aperture holography: a novel approach to three-dimensional displays,” J. Opt. Soc. Am. A 9, 1969–1977 (1992).
    [CrossRef]
  6. G. Saxby, Practical Holography (Prentice-Hall, Englewood Cliffs, N.J., 1988).
    [CrossRef]
  7. S. A. Benton, “Survey of holographic stereograms,” in Processing and Display of Three-Dimensional Data, J. J. Pearson, ed., Proc. Soc. Photo-Opt. Instrum. Eng.367, 15–19 (1982).
    [CrossRef]
  8. S. A. Benton, “The mathematical optics of white light transmission holograms,” in Proceedings of the International Symposium on Display Holography (Holography Workshops, Lake Forest College, Lake Forest, Ill., 1982).
  9. J. Walker, “In-situcolor control for reflection holograms,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1987).
  10. D. B. Diner, D. H. Fender, Human Engineering in Stereoscopic Viewing Devices (Plenum, New York, 1993).
  11. D. Leseberg, O. Bryngdahl, “Computer-generated rainbow holograms,” Appl. Opt. 23, 2441–2447 (1984).
    [CrossRef] [PubMed]
  12. D. Leseberg, “Computer-generated holograms: display using one-dimensional transforms,” J. Opt. Soc. Am. A 3, 1846–1851 (1986).
    [CrossRef]
  13. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).
  14. G. P. Nordin, J. H. Kulick, M. Jones, P. Nasiatka, R. G. Lindquist, S. T. Kowel, “Demonstration of a novel 3-D autostereoscopic display,” Opt. Lett. 19, 901–903 (1994).
    [CrossRef] [PubMed]
  15. R. G. Lindquist, J. H. Kulick, G. P. Nordin, J. M. Jarem, S. T. Kowel, M. Friends, T. M. Leslie, “High-resolution liquid crystal phase grating formed by fringing fields from interdigitated electrodes,” Opt. Lett. 19, 670–672 (1994).
    [CrossRef] [PubMed]
  16. J. Kulick, S. T. Kowel, T. Leslie, R. Ciliax, “ICVision—a VLSI based holographic display system,” in Practical Holography VII: Imaging and Materials, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1914, 219–229 (1993).
    [CrossRef]
  17. J. H. Kulick, S. T. Kowel, G. P. Nordin, A. Parker, R. Lindquist, P. Nasiatka, M. Jones, “ICVision—a VLSI-based diffractive display for real-time display of holographic stereograms,” in Practical Holography VIII, Stephen Bentson, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2176, 2–11 (1994).
    [CrossRef]
  18. O. Bryngdahl, F. Wyrowski, “Digital holography—computer-generated holograms,” in Progress in Optics, E. Wolf, ed. (North-Holland, New York, 1990), Vol. 28, pp. 3–85.
    [CrossRef]
  19. J. M. Younse, “Mirrors on a chip,” IEEE Spectrum 30(11), 27–31 (1993).
    [CrossRef]
  20. O. Solgaard, F. S. A. Sandejas, D. M. Bloom, “Deformable grating optical modulator,” Opt. Lett. 17, 688–690 (1992).
    [CrossRef] [PubMed]

1994

1993

J. M. Younse, “Mirrors on a chip,” IEEE Spectrum 30(11), 27–31 (1993).
[CrossRef]

1992

1986

1984

Benton, S. A.

P. St. Hilaire, S. A. Benton, M. Lucente, “Synthetic aperture holography: a novel approach to three-dimensional displays,” J. Opt. Soc. Am. A 9, 1969–1977 (1992).
[CrossRef]

S. A. Benton, “The mathematical optics of white light transmission holograms,” in Proceedings of the International Symposium on Display Holography (Holography Workshops, Lake Forest College, Lake Forest, Ill., 1982).

P. St. Hilaire, S. A. Benton, M. Lucente, H. Yoshikawa, J. Underkoffler, “Real-time holographic display: improvements using a multichannel acousto-optic modulator and holographic optical elements,” in Practical Holography V, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1461, 256–261 (1991).

S. A. Benton, “Experiments in holographic video imaging,” in Industrial Applications of Holographic and Speckle Measuring Techniques, W. P. Jueptner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1508, 247–267 (1991).

S. A. Benton, “Survey of holographic stereograms,” in Processing and Display of Three-Dimensional Data, J. J. Pearson, ed., Proc. Soc. Photo-Opt. Instrum. Eng.367, 15–19 (1982).
[CrossRef]

P. St. Hilaire, S. A. Benton, M. Lucente, M. L. Jepson, J. Kollin, H. Yoshikawa, J. Underkoffler, “Electronic display system for computational holography,” in Practical Holography IV, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1212, 174–182 (1990).
[CrossRef]

Bloom, D. M.

Bryngdahl, O.

D. Leseberg, O. Bryngdahl, “Computer-generated rainbow holograms,” Appl. Opt. 23, 2441–2447 (1984).
[CrossRef] [PubMed]

O. Bryngdahl, F. Wyrowski, “Digital holography—computer-generated holograms,” in Progress in Optics, E. Wolf, ed. (North-Holland, New York, 1990), Vol. 28, pp. 3–85.
[CrossRef]

Ciliax, R.

J. Kulick, S. T. Kowel, T. Leslie, R. Ciliax, “ICVision—a VLSI based holographic display system,” in Practical Holography VII: Imaging and Materials, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1914, 219–229 (1993).
[CrossRef]

Diner, D. B.

D. B. Diner, D. H. Fender, Human Engineering in Stereoscopic Viewing Devices (Plenum, New York, 1993).

Fender, D. H.

D. B. Diner, D. H. Fender, Human Engineering in Stereoscopic Viewing Devices (Plenum, New York, 1993).

Friends, M.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

Hilaire, P. St.

P. St. Hilaire, S. A. Benton, M. Lucente, “Synthetic aperture holography: a novel approach to three-dimensional displays,” J. Opt. Soc. Am. A 9, 1969–1977 (1992).
[CrossRef]

P. St. Hilaire, S. A. Benton, M. Lucente, H. Yoshikawa, J. Underkoffler, “Real-time holographic display: improvements using a multichannel acousto-optic modulator and holographic optical elements,” in Practical Holography V, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1461, 256–261 (1991).

P. St. Hilaire, S. A. Benton, M. Lucente, M. L. Jepson, J. Kollin, H. Yoshikawa, J. Underkoffler, “Electronic display system for computational holography,” in Practical Holography IV, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1212, 174–182 (1990).
[CrossRef]

Jarem, J. M.

Jepson, M. L.

P. St. Hilaire, S. A. Benton, M. Lucente, M. L. Jepson, J. Kollin, H. Yoshikawa, J. Underkoffler, “Electronic display system for computational holography,” in Practical Holography IV, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1212, 174–182 (1990).
[CrossRef]

Jones, M.

G. P. Nordin, J. H. Kulick, M. Jones, P. Nasiatka, R. G. Lindquist, S. T. Kowel, “Demonstration of a novel 3-D autostereoscopic display,” Opt. Lett. 19, 901–903 (1994).
[CrossRef] [PubMed]

J. H. Kulick, S. T. Kowel, G. P. Nordin, A. Parker, R. Lindquist, P. Nasiatka, M. Jones, “ICVision—a VLSI-based diffractive display for real-time display of holographic stereograms,” in Practical Holography VIII, Stephen Bentson, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2176, 2–11 (1994).
[CrossRef]

Kollin, J.

P. St. Hilaire, S. A. Benton, M. Lucente, M. L. Jepson, J. Kollin, H. Yoshikawa, J. Underkoffler, “Electronic display system for computational holography,” in Practical Holography IV, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1212, 174–182 (1990).
[CrossRef]

Kowel, S. T.

G. P. Nordin, J. H. Kulick, M. Jones, P. Nasiatka, R. G. Lindquist, S. T. Kowel, “Demonstration of a novel 3-D autostereoscopic display,” Opt. Lett. 19, 901–903 (1994).
[CrossRef] [PubMed]

R. G. Lindquist, J. H. Kulick, G. P. Nordin, J. M. Jarem, S. T. Kowel, M. Friends, T. M. Leslie, “High-resolution liquid crystal phase grating formed by fringing fields from interdigitated electrodes,” Opt. Lett. 19, 670–672 (1994).
[CrossRef] [PubMed]

J. H. Kulick, S. T. Kowel, G. P. Nordin, A. Parker, R. Lindquist, P. Nasiatka, M. Jones, “ICVision—a VLSI-based diffractive display for real-time display of holographic stereograms,” in Practical Holography VIII, Stephen Bentson, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2176, 2–11 (1994).
[CrossRef]

J. Kulick, S. T. Kowel, T. Leslie, R. Ciliax, “ICVision—a VLSI based holographic display system,” in Practical Holography VII: Imaging and Materials, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1914, 219–229 (1993).
[CrossRef]

Kulick, J.

J. Kulick, S. T. Kowel, T. Leslie, R. Ciliax, “ICVision—a VLSI based holographic display system,” in Practical Holography VII: Imaging and Materials, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1914, 219–229 (1993).
[CrossRef]

Kulick, J. H.

Leseberg, D.

Leslie, T.

J. Kulick, S. T. Kowel, T. Leslie, R. Ciliax, “ICVision—a VLSI based holographic display system,” in Practical Holography VII: Imaging and Materials, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1914, 219–229 (1993).
[CrossRef]

Leslie, T. M.

Lindquist, R.

J. H. Kulick, S. T. Kowel, G. P. Nordin, A. Parker, R. Lindquist, P. Nasiatka, M. Jones, “ICVision—a VLSI-based diffractive display for real-time display of holographic stereograms,” in Practical Holography VIII, Stephen Bentson, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2176, 2–11 (1994).
[CrossRef]

Lindquist, R. G.

Lucente, M.

P. St. Hilaire, S. A. Benton, M. Lucente, “Synthetic aperture holography: a novel approach to three-dimensional displays,” J. Opt. Soc. Am. A 9, 1969–1977 (1992).
[CrossRef]

P. St. Hilaire, S. A. Benton, M. Lucente, H. Yoshikawa, J. Underkoffler, “Real-time holographic display: improvements using a multichannel acousto-optic modulator and holographic optical elements,” in Practical Holography V, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1461, 256–261 (1991).

M. Lucente, “Optimization of hologram computation for real-time display,” in Practical Holography VI, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1667, 32–43 (1992).
[CrossRef]

P. St. Hilaire, S. A. Benton, M. Lucente, M. L. Jepson, J. Kollin, H. Yoshikawa, J. Underkoffler, “Electronic display system for computational holography,” in Practical Holography IV, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1212, 174–182 (1990).
[CrossRef]

Nasiatka, P.

G. P. Nordin, J. H. Kulick, M. Jones, P. Nasiatka, R. G. Lindquist, S. T. Kowel, “Demonstration of a novel 3-D autostereoscopic display,” Opt. Lett. 19, 901–903 (1994).
[CrossRef] [PubMed]

J. H. Kulick, S. T. Kowel, G. P. Nordin, A. Parker, R. Lindquist, P. Nasiatka, M. Jones, “ICVision—a VLSI-based diffractive display for real-time display of holographic stereograms,” in Practical Holography VIII, Stephen Bentson, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2176, 2–11 (1994).
[CrossRef]

Nordin, G. P.

Parker, A.

J. H. Kulick, S. T. Kowel, G. P. Nordin, A. Parker, R. Lindquist, P. Nasiatka, M. Jones, “ICVision—a VLSI-based diffractive display for real-time display of holographic stereograms,” in Practical Holography VIII, Stephen Bentson, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2176, 2–11 (1994).
[CrossRef]

Sandejas, F. S. A.

Saxby, G.

G. Saxby, Practical Holography (Prentice-Hall, Englewood Cliffs, N.J., 1988).
[CrossRef]

Solgaard, O.

Underkoffler, J.

P. St. Hilaire, S. A. Benton, M. Lucente, M. L. Jepson, J. Kollin, H. Yoshikawa, J. Underkoffler, “Electronic display system for computational holography,” in Practical Holography IV, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1212, 174–182 (1990).
[CrossRef]

P. St. Hilaire, S. A. Benton, M. Lucente, H. Yoshikawa, J. Underkoffler, “Real-time holographic display: improvements using a multichannel acousto-optic modulator and holographic optical elements,” in Practical Holography V, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1461, 256–261 (1991).

Walker, J.

J. Walker, “In-situcolor control for reflection holograms,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1987).

Wyrowski, F.

O. Bryngdahl, F. Wyrowski, “Digital holography—computer-generated holograms,” in Progress in Optics, E. Wolf, ed. (North-Holland, New York, 1990), Vol. 28, pp. 3–85.
[CrossRef]

Yoshikawa, H.

P. St. Hilaire, S. A. Benton, M. Lucente, M. L. Jepson, J. Kollin, H. Yoshikawa, J. Underkoffler, “Electronic display system for computational holography,” in Practical Holography IV, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1212, 174–182 (1990).
[CrossRef]

P. St. Hilaire, S. A. Benton, M. Lucente, H. Yoshikawa, J. Underkoffler, “Real-time holographic display: improvements using a multichannel acousto-optic modulator and holographic optical elements,” in Practical Holography V, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1461, 256–261 (1991).

Younse, J. M.

J. M. Younse, “Mirrors on a chip,” IEEE Spectrum 30(11), 27–31 (1993).
[CrossRef]

Appl. Opt.

IEEE Spectrum

J. M. Younse, “Mirrors on a chip,” IEEE Spectrum 30(11), 27–31 (1993).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Lett.

Other

P. St. Hilaire, S. A. Benton, M. Lucente, M. L. Jepson, J. Kollin, H. Yoshikawa, J. Underkoffler, “Electronic display system for computational holography,” in Practical Holography IV, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1212, 174–182 (1990).
[CrossRef]

P. St. Hilaire, S. A. Benton, M. Lucente, H. Yoshikawa, J. Underkoffler, “Real-time holographic display: improvements using a multichannel acousto-optic modulator and holographic optical elements,” in Practical Holography V, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1461, 256–261 (1991).

S. A. Benton, “Experiments in holographic video imaging,” in Industrial Applications of Holographic and Speckle Measuring Techniques, W. P. Jueptner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1508, 247–267 (1991).

M. Lucente, “Optimization of hologram computation for real-time display,” in Practical Holography VI, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1667, 32–43 (1992).
[CrossRef]

G. Saxby, Practical Holography (Prentice-Hall, Englewood Cliffs, N.J., 1988).
[CrossRef]

S. A. Benton, “Survey of holographic stereograms,” in Processing and Display of Three-Dimensional Data, J. J. Pearson, ed., Proc. Soc. Photo-Opt. Instrum. Eng.367, 15–19 (1982).
[CrossRef]

S. A. Benton, “The mathematical optics of white light transmission holograms,” in Proceedings of the International Symposium on Display Holography (Holography Workshops, Lake Forest College, Lake Forest, Ill., 1982).

J. Walker, “In-situcolor control for reflection holograms,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1987).

D. B. Diner, D. H. Fender, Human Engineering in Stereoscopic Viewing Devices (Plenum, New York, 1993).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

J. Kulick, S. T. Kowel, T. Leslie, R. Ciliax, “ICVision—a VLSI based holographic display system,” in Practical Holography VII: Imaging and Materials, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1914, 219–229 (1993).
[CrossRef]

J. H. Kulick, S. T. Kowel, G. P. Nordin, A. Parker, R. Lindquist, P. Nasiatka, M. Jones, “ICVision—a VLSI-based diffractive display for real-time display of holographic stereograms,” in Practical Holography VIII, Stephen Bentson, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2176, 2–11 (1994).
[CrossRef]

O. Bryngdahl, F. Wyrowski, “Digital holography—computer-generated holograms,” in Progress in Optics, E. Wolf, ed. (North-Holland, New York, 1990), Vol. 28, pp. 3–85.
[CrossRef]

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 (17)

Fig. 1
Fig. 1

Schematic diagram of a holographic stereogram that is viewed from a region that consists of virtual viewing slits. A single 2-D image is visible in the plane of the display when an observer’s eye looks through any particular virtual viewing slit. In general, a different 2-D image can be seen through each virtual viewing slit. When each eye sees the appropriate image of a given pair of stereoscopic images, the observer perceives a 3-D scene. The 2-D images shown in (a) and (b) are intended to illustrate such a pair of stereoscopic images. If an appropriate set of stereopair images is displayed through the virtual viewing slits, 1-D motion parallax is perceived as an observer moves horizontally in the viewing region.

Fig. 2
Fig. 2

Schematic illustration showing two display pixels illuminating each of the virtual viewing slits. In general, all the display pixels are able to illuminate each virtual viewing slit simultaneously.

Fig. 3
Fig. 3

In the composite-pixel approach multiple holograms are superimposed within the area of any given pixel.

Fig. 4
Fig. 4

Schematic representation of a single pixel that is composed of multiple partial pixels, each of which directs light to a single virtual viewing slit.

Fig. 5
Fig. 5

Arbitrarily oriented collimated readout beam (represented by its k vector) incident upon a grating (also represented by its k vector) that is in the xy plane.

Fig. 6
Fig. 6

(a) Labeling of the vectors that represent the center of each pixel and each partial pixel. (b) Vectors specifying the direction (vijm) in which the +1-order diffracted beam must travel to propagate from partial pixel ijm to the mth virtual viewing slit.

Fig. 7
Fig. 7

Ideal and actual (i.e., diffraction-induced) intensity profiles at a single virtual viewing slit that are due to a single partial pixel.

Fig. 8
Fig. 8

3-D object displayed by the chrome mask. Each letter is located in a different plane (all of which are parallel to the xy plane), and each is composed of a series of dots. The top of the U and the bottom of the H are at the xz plane.

Fig. 9
Fig. 9

Schematic illustration of a single pixel that is composed of an array of partial pixels.

Fig. 10
Fig. 10

(a) Side view of the chrome mask readout geometry. (b) Top view of the readout geometry showing the virtual viewing slits (i.e., slits −16 to −7 and 6 to 15) from which images can be observed on the mask.

Fig. 11
Fig. 11

Photomicrograph of a single pixel in which there are diffraction gratings in 20 partial pixels (after Ref. 14).

Fig. 12
Fig. 12

Photographs of stereoscopic images seen through slits (a) −7 and (b) 15.

Fig. 13
Fig. 13

Schematic diagram of a liquid crystal film on interdigitated electrodes (a) without an applied voltage and (b) with an applied voltage. The details are discussed in the text. ITO, indium tin oxide.

Fig. 14
Fig. 14

Schematic diagram of the interdigitated electrodes in a single partial pixel.

Fig. 15
Fig. 15

The diagram in the upper left-hand corner represents a display composed of tessellated silicon dies. The other diagrams are (in clockwise order) a single die on which multiple pixels are implemented, a single pixel composed of an array of partial pixels, and the interdigitated electrodes in a single partial pixel.

Fig. 16
Fig. 16

Schematic representation of a binary optic grating partial pixel.

Fig. 17
Fig. 17

Display geometry for 2-D motion parallax.

Equations (15)

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

k = k ( α x ^ + β y ^ + γ z ^ ) ,
K G = K G [ ( cos θ G ) x ^ + ( sin θ G ) y ^ ] ,
U ( r ) = q = - J q ( v ) exp ( i k q · r ) ,
k q = k ( α q x ^ + β q y ^ + γ q z ^ ) ,
α q = α + q λ Λ cos θ G , β q = β + q λ Λ sin θ G , γ q = ( 1 - α q 2 - β q 2 ) 1 / 2 .
s i j m = p i j + q m ,
v i j m = u m - s i j m ,
v i j m = v i j m ( α i j m x ^ + β i j m y ^ + γ i j m z ^ ) .
Λ i j m = λ [ ( α i j m - α ) 2 + ( β i j m - β ) 2 ] 1 / 2 ,
θ G i j m = tan - 1 ( β i j m - β α i j m - α ) .
T aperture ( x , y ) = rect ( x / l x ) rect ( y / l y ) ,
rect ( ξ ) = { 1 if ξ 1 / 2 0 otherwise ,
I slit ( x , y ) = I 0 sinc 2 ( k l x α i j m x / 2 v i j m ) sinc 2 ( k l y β i j m y / 2 v i j m ) ,
sinc x ( sin x ) / x .
Δ x = 2 λ v i j m / l x .

Metrics