Abstract

A new type of polygonal holographic scanner that combines a reflection volume hologram with a computer-generated hologram (CGH) is described. The scanner is free from the aberration of field curvature. Such a scanning system can allow for a compact folded version of the scanner and Bragg diffraction into only a single order. The equations expressing the spatial-variable image distance are derived and are fit to the phase function designated by polynomials incorporated into a CGH in terms of the least-squares method. A reflection scanner with field-curvature correction is made by interfering a diffracted wave front from this CGH with a spherical wave front having scanning focal power through a second plane hologram. Experiments demonstrating the feasibility of this scanner are presented. Raster-scan patterns using a multifaceted scanner are shown. Helpful data on the diffraction efficiency and the spectrally diffracted intensity of reflection holograms are also presented.

© 1983 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. O. Bryngdahl, W-H. Lee, Appl. Opt. 15, 183 (1976).
    [CrossRef] [PubMed]
  2. H. Ikeda, M. Ando, T. Inagaki, Appl. Opt. 18, 2166 (1979).
    [CrossRef] [PubMed]
  3. D. K. Campbell, D. W. Sweeney, Appl. Opt. 17, 3727 (1978).
    [CrossRef] [PubMed]
  4. C. J. Kramer, Laser Focus 17, 70 (1981).
  5. C. S. Ih, N. Kong, T. Giriappa, Appl. Opt. 17, 1582 (1978).
    [CrossRef] [PubMed]
  6. S. K. Case, V. Gerbig, Opt. Eng. 19, 711 (1980).
    [CrossRef]
  7. W-H. Lee, Appl. Opt. 16, 1392 (1977).
    [CrossRef] [PubMed]
  8. B. J. Chang, Opt. Eng. 19, 642 (1980).
    [CrossRef]
  9. R. V. Pole, H. W. Werlich, R. J. Krusche, Appl. Opt. 17, 3294 (1978).
    [CrossRef] [PubMed]
  10. R. C. Fairchild, J. R. Fienup, Opt. Eng. 21, 133 (1982).
    [CrossRef]
  11. W-H. Lee, Prog. Opt. 16, 121 (1978).
  12. W. J. Dallas, The Computer in Optical Research (Methods and Applications) (Springer, Berlin, 1980).
  13. H. M. Smith, Principles of Holography (Wiley- Interscience, New York, 1968), Sec. 4.2.
  14. J. Růžek, P. Fiala, Opt. Acta 26, 1257 (1979).
    [CrossRef]
  15. Y. Ishii, K. Murata, Opt.Lett. 7, 230 (1982).
    [CrossRef] [PubMed]
  16. H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
  17. W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, London, 1974), p. 121.
  18. K. J. Birch, F. J. Green, J. Phys. D 5, 1982 (1972).
    [CrossRef]
  19. T. Takahashi, K. Konno, M. Kawai, Jpn. J. Appl. Phys. Suppl. 14-1 14, 247 (1975).
  20. W-H. Lee, Prog. Opt. 16, 155 (1978).
  21. S. Lowenthal, P. Chavel, Appl. Opt. 13, 718 (1974).
    [CrossRef] [PubMed]
  22. Ref. 17, p. 76.

1982 (2)

R. C. Fairchild, J. R. Fienup, Opt. Eng. 21, 133 (1982).
[CrossRef]

Y. Ishii, K. Murata, Opt.Lett. 7, 230 (1982).
[CrossRef] [PubMed]

1981 (1)

C. J. Kramer, Laser Focus 17, 70 (1981).

1980 (2)

S. K. Case, V. Gerbig, Opt. Eng. 19, 711 (1980).
[CrossRef]

B. J. Chang, Opt. Eng. 19, 642 (1980).
[CrossRef]

1979 (2)

1978 (5)

1977 (1)

1976 (1)

1975 (1)

T. Takahashi, K. Konno, M. Kawai, Jpn. J. Appl. Phys. Suppl. 14-1 14, 247 (1975).

1974 (1)

1972 (1)

K. J. Birch, F. J. Green, J. Phys. D 5, 1982 (1972).
[CrossRef]

1969 (1)

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).

Ando, M.

Birch, K. J.

K. J. Birch, F. J. Green, J. Phys. D 5, 1982 (1972).
[CrossRef]

Bryngdahl, O.

Campbell, D. K.

Case, S. K.

S. K. Case, V. Gerbig, Opt. Eng. 19, 711 (1980).
[CrossRef]

Chang, B. J.

B. J. Chang, Opt. Eng. 19, 642 (1980).
[CrossRef]

Chavel, P.

Dallas, W. J.

W. J. Dallas, The Computer in Optical Research (Methods and Applications) (Springer, Berlin, 1980).

Fairchild, R. C.

R. C. Fairchild, J. R. Fienup, Opt. Eng. 21, 133 (1982).
[CrossRef]

Fiala, P.

J. Růžek, P. Fiala, Opt. Acta 26, 1257 (1979).
[CrossRef]

Fienup, J. R.

R. C. Fairchild, J. R. Fienup, Opt. Eng. 21, 133 (1982).
[CrossRef]

Gerbig, V.

S. K. Case, V. Gerbig, Opt. Eng. 19, 711 (1980).
[CrossRef]

Giriappa, T.

Green, F. J.

K. J. Birch, F. J. Green, J. Phys. D 5, 1982 (1972).
[CrossRef]

Ih, C. S.

Ikeda, H.

Inagaki, T.

Ishii, Y.

Y. Ishii, K. Murata, Opt.Lett. 7, 230 (1982).
[CrossRef] [PubMed]

Kawai, M.

T. Takahashi, K. Konno, M. Kawai, Jpn. J. Appl. Phys. Suppl. 14-1 14, 247 (1975).

Kogelnik, H.

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).

Kong, N.

Konno, K.

T. Takahashi, K. Konno, M. Kawai, Jpn. J. Appl. Phys. Suppl. 14-1 14, 247 (1975).

Kramer, C. J.

C. J. Kramer, Laser Focus 17, 70 (1981).

Krusche, R. J.

Lee, W-H.

Lowenthal, S.

Murata, K.

Y. Ishii, K. Murata, Opt.Lett. 7, 230 (1982).
[CrossRef] [PubMed]

Pole, R. V.

Ružek, J.

J. Růžek, P. Fiala, Opt. Acta 26, 1257 (1979).
[CrossRef]

Smith, H. M.

H. M. Smith, Principles of Holography (Wiley- Interscience, New York, 1968), Sec. 4.2.

Sweeney, D. W.

Takahashi, T.

T. Takahashi, K. Konno, M. Kawai, Jpn. J. Appl. Phys. Suppl. 14-1 14, 247 (1975).

Welford, W. T.

W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, London, 1974), p. 121.

Werlich, H. W.

Appl. Opt. (7)

Bell Syst. Tech. J. (1)

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).

J. Phys. D (1)

K. J. Birch, F. J. Green, J. Phys. D 5, 1982 (1972).
[CrossRef]

Jpn. J. Appl. Phys. (1)

T. Takahashi, K. Konno, M. Kawai, Jpn. J. Appl. Phys. Suppl. 14-1 14, 247 (1975).

Laser Focus (1)

C. J. Kramer, Laser Focus 17, 70 (1981).

Opt. Acta (1)

J. Růžek, P. Fiala, Opt. Acta 26, 1257 (1979).
[CrossRef]

Opt. Eng. (3)

S. K. Case, V. Gerbig, Opt. Eng. 19, 711 (1980).
[CrossRef]

B. J. Chang, Opt. Eng. 19, 642 (1980).
[CrossRef]

R. C. Fairchild, J. R. Fienup, Opt. Eng. 21, 133 (1982).
[CrossRef]

Opt.Lett. (1)

Y. Ishii, K. Murata, Opt.Lett. 7, 230 (1982).
[CrossRef] [PubMed]

Prog. Opt. (2)

W-H. Lee, Prog. Opt. 16, 121 (1978).

W-H. Lee, Prog. Opt. 16, 155 (1978).

Other (4)

Ref. 17, p. 76.

W. J. Dallas, The Computer in Optical Research (Methods and Applications) (Springer, Berlin, 1980).

H. M. Smith, Principles of Holography (Wiley- Interscience, New York, 1968), Sec. 4.2.

W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, London, 1974), p. 121.

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 (13)

Fig. 1
Fig. 1

Diffraction efficiency of reflection holographic gratings as a function of exposure at 514.5- and 488.0-nm wavelengths.

Fig. 2
Fig. 2

Spectral diffraction intensity of reflection volume gratings corresponding to the data indicated by arrows in Fig. 1. The reconstructing wavelengths are shown in the figure.

Fig. 3
Fig. 3

Reflection volume holographic scanner for one facet: (a) top view and (b) side view of experimental setup. Recording geometry is shown by solid lines and readout geometry by dashed lines.

Fig. 4
Fig. 4

Photographs of magnified scan spots on the (a) flat scanning plane and (b) curved scanning plane, at different scan angles, (c) Raster-scan pattern on the curved scanning plane generated by four reflection volume hologram facets.

Fig. 5
Fig. 5

Schematic illustration for computing the spatially variable image distance fx(x) while the scanner is rotating about the center O.

Fig. 6
Fig. 6

Reciprocal of space-variant image distance 1/fx (x) vs a hologram coordinate x with F = 470 mm and r = 80 mm.

Fig. 7
Fig. 7

Schematic diagram for calculating a spatially variable image distance i(x) using Eq. (20). Two rays diffracted at points M and N fall on an image point S.

Fig. 8
Fig. 8

(a) Magnified portion of the CGH used as the field-curvature correction. (b) Interferogram at half-plane is observed with ± 1st-order diffracted waves from the CGH. (c) Computer-predicted interferogram which agrees with the measured one shown in (b).

Fig. 9
Fig. 9

Top view of the geometry for making a reflection holographic scanner with field-curvature correction. The diffracted wave from the CGH is imaged onto the second hologram plane after magnification by the telescope system. The diffracted, divergent wave from the second hologram together with the divergent reference wave interferes to form the reflection hologram.

Fig. 10
Fig. 10

(a) Holographic interferogram obtained by illuminating the second hologram with both the divergent spherical wave emerging from point source P in Fig. 9 and the plane wave. (b) Computer-predicted interferogram which agrees with the interferogram shown in (a).

Fig. 11
Fig. 11

Magnified scan spots (b) from the corrected reflection holographic scanner together with those (a) from the uncorrected scanner similar to those in Fig. 4(a). A significant improvement in the scan spots over 8° can be seen. The focused spots are displayed on half of the flat scanning plane, for simplicity.

Fig. 12
Fig. 12

Formation of raster-scan pattern on (A) the curved scanning plane similar to the one shown in Fig. 4(c), (B) the flat plane from four uncorrected reflection hologram facets, and (C) the flat plane from one corrected facet. Magnified portions of scan lines corresponding to the areas marked (a)–(c) of (A)–(C) at 13.6° scan angle are shown in (D). A significant improvement in (c) can be seen.

Fig. 13
Fig. 13

(a) Ray trace sample aperture used for calculations of transverse ray aberrations. Ray aberration plots vs input heights in the (b) x and (c) y coordinates, respectively.

Equations (41)

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

Δ λ = λ 2 2 n π T sin ψ B ξ 0 ,
ϕ i = ϕ c ± ( ϕ o ϕ r ) ,
ϕ = 2 π λ ( l x + m y + n z ) ,
l i = λ c 2 π ϕ i x = l c + λ c λ o ( l o l r ) ,
m i = λ c 2 π ϕ i y = m c + λ c λ o ( m o m r ) ,
n i = λ c 2 π ϕ i z = n c + λ c λ o ( n o n r ) ,
cos ( K ox x + K oy y + K oz z ) ,
K 0 = ( 2 π λ o ( l o l r ) , 2 π λ o ( m o m r ) , 2 π λ o β ( n o n r ) ] .
K c = [ 2 π λ c ( l i l c ) , 2 π λ c ( m i m c ) , 2 π λ c ( n i n c ) ] .
K o = K c ,
D = 2.44 λ F d .
N = 2 ( F + r ) θ max D ,
θ max = tan 1 ( l d 2 r ) .
( r + F ) 1 cos θ cos θ 2 λ ( F d ) 2 .
( x r + z ) = ( cos θ sin θ sin θ cos θ ) ( h ( θ ) r + F ) ,
x = 0 ,
z = ( r + F ) ( tan θ sin θ + cos θ ) r ,
1 f x ( x ) = r F r 2 + x 2 + r ( r 2 + x 2 r ) ,
ϕ 1 ( x ) = 2 π λ ( F 2 + x 2 F + a x 2 + b x 3 + c x 4 ) ,
1 i ( x ) = 1 ρ [ tan α ( x + ρ ) tan α ( x ) ] .
1 i ( x ) = 1 ρ cos [ α ( x + ρ ) α ( x ) 2 ] [ sin α ( x + ρ ) sin α ( x ) ] cos [ α ( x + ρ ) + α ( x ) 2 ] cos α ( x + ρ ) cos α ( x ) 1 ρ 1 cos 3 α ( x ) [ sin α ( x + ρ ) sin α ( x ) ] ,
1 i ( x ) 1 ρ 1 cos 3 α ( x ) [ λ 2 π ( ϕ 1 x ) x + ρ λ 2 π ( ϕ 1 x ) x ] ,
1 i ( x ) 1 ρ F 3 ( F 2 + x 2 ) 3 / 2 [ x + ρ F 2 + ( x + ρ ) 2 x F 2 + x 2 + A + B x + C x 2 ] ,
cos α ( x ) F ( F 2 + x 2 ) 1 / 2 .
i = 1 n [ 1 f x ( x i ) 1 i ( x i ) ] 2 = minimum ,
a = 3.22 × 10 7 / mm , b = 1.21 × 10 8 / mm 2 , c = 1.41 × 10 8 / mm 3 ,
ϕ 2 ( x , y ) = 2 π λ [ F 2 + x 2 + y 2 F + a x 2 + b x 3 + c x 4 + 1 2 f y ( x ) y 2 ] ϕ S ( x , y ) + ϕ C ( x , y ) ,
ϕ S ( x , y ) = 2 π λ ( F 2 + x 2 + y 2 F )
ϕ C ( x , y ) = 2 π λ [ a x 2 + b x 3 + c x 4 + 1 2 f y ( x ) y 2 ] .
1 f y ( x ) = 1 f x ( x ) 1 F = ( F + r ) ( r r 2 + x 2 ) F ( F + r ) r 2 + x 2 F r 2 .
g ( x , y ) = y Λ + 1 2 π ϕ C ( x , y ) = μ ,
g ( x , y ) y = 1 Λ + 1 2 π ϕ C ( x , y ) y .
1 Λ = M 2 × 0.16 L 1.5 B ,
B = 2 × 1 2 π ( ϕ C y ) x = L y = 0.16 L .
B = [ 2 λ f y ( x ) y ] x = L y = 0.16 L ,
X k = x k + l i n i z x ,
Y k = y k + m i n i z y ,
n i = 1 l i 2 m i 2
l i = λ 2 π ( ϕ 2 x ) x = d / 2 + r tan θ y = 0 ,
m i = λ 2 π ( ϕ 2 y ) x = r tan θ y = d / 2 ,
W x ( x k ) = 1 z x 0 x k X k dx ,

Metrics