Abstract

Previously, holographic optical elements have been constructed with two spherical waves. Better image quality and efficiency can be achieved by constructing thick HOE's with spherically aberrated wavefronts. We include an exact lens equivalent for a HOE to allow computer optimization of such systems. The effect of changes in the index of refraction and thickness of the emulsion are considered along with the quality of the substrate. Optical systems to construct such HOE's are discussed and analytically described. Many examples are presented.

© 1978 Optical Society of America

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References

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  1. E. B. Champagne, J. Opt. Soc. Am. 57, 51 (1967).
    [CrossRef]
  2. R. W. Meier, J. Opt. Soc. Am. 55, 987 (1965).
    [CrossRef]
  3. J. N. Latta, Appl. Opt. 10, 599 (1971).
    [CrossRef] [PubMed]
  4. A. Offner, J. Opt. Soc. Am. 56, 1509 (1966).
    [CrossRef]
  5. H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
  6. J. N. Latta, R. C. Fairchild, Proc. Soc. Photo-Opt. Instrum. Eng. 39, Aug. (1973).
  7. W. C. Sweatt, J. Opt. Soc. Am. 67, 803 (1977).
    [CrossRef]
  8. W. C. Sweatt, “Designing Holographic Optical Elements” Doctoral Dissertation, U. Arizona, Tucson, (1977).Available from University Microfilms, Ann Arbor Mich., chap. 3
  9. Ref. 8, chap. 4
  10. ACCOS V Users Guide (Scientific Calculations, Inc., Norwalk, Conn., 1970).
  11. M. V. Klein, Optics (Wiley, New York, 1970), pp. 339–342.
  12. A. D. Gara, F. T. S. Yu, Appl. Opt. 10, 1324 (1971).
    [CrossRef] [PubMed]
  13. J. C. Wyant, Optical Testing (Optical Sciences Center, Tucson, Ariz., June1976).
  14. O. Bryngdahl, Appl. Opt. 11, 195 (1972).
    [CrossRef] [PubMed]
  15. R. S. Longhurst, Geometrical and Physical Optics (Jarrold, Norwich, England, 1967), p. 87.
  16. Abbé 3L Reference Manual, catalog. 33-45-58-01 (Bausch & Lomb, Rochester, N.Y.).
  17. A. F. Turner, U. Arizona; private communication (March1977).
  18. Interference Systems, catalog 500-101-13 (Leitz, Rockleigh, N.J., 1973).
  19. A. Offner, Appl. Opt. 2, 153 (1963).
    [CrossRef]
  20. W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966).
  21. R. J. Collier, C. B. Burchart, L. H. Lin, Optical Holography (Academic, New York, 1971), p. 228.

1977 (1)

1973 (1)

J. N. Latta, R. C. Fairchild, Proc. Soc. Photo-Opt. Instrum. Eng. 39, Aug. (1973).

1972 (1)

1971 (2)

1969 (1)

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

1967 (1)

1966 (1)

1965 (1)

1963 (1)

Bryngdahl, O.

Burchart, C. B.

R. J. Collier, C. B. Burchart, L. H. Lin, Optical Holography (Academic, New York, 1971), p. 228.

Champagne, E. B.

Collier, R. J.

R. J. Collier, C. B. Burchart, L. H. Lin, Optical Holography (Academic, New York, 1971), p. 228.

Fairchild, R. C.

J. N. Latta, R. C. Fairchild, Proc. Soc. Photo-Opt. Instrum. Eng. 39, Aug. (1973).

Gara, A. D.

Klein, M. V.

M. V. Klein, Optics (Wiley, New York, 1970), pp. 339–342.

Kogelnik, H.

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

Latta, J. N.

J. N. Latta, R. C. Fairchild, Proc. Soc. Photo-Opt. Instrum. Eng. 39, Aug. (1973).

J. N. Latta, Appl. Opt. 10, 599 (1971).
[CrossRef] [PubMed]

Lin, L. H.

R. J. Collier, C. B. Burchart, L. H. Lin, Optical Holography (Academic, New York, 1971), p. 228.

Longhurst, R. S.

R. S. Longhurst, Geometrical and Physical Optics (Jarrold, Norwich, England, 1967), p. 87.

Meier, R. W.

Offner, A.

Smith, W. J.

W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966).

Sweatt, W. C.

W. C. Sweatt, J. Opt. Soc. Am. 67, 803 (1977).
[CrossRef]

W. C. Sweatt, “Designing Holographic Optical Elements” Doctoral Dissertation, U. Arizona, Tucson, (1977).Available from University Microfilms, Ann Arbor Mich., chap. 3

Turner, A. F.

A. F. Turner, U. Arizona; private communication (March1977).

Wyant, J. C.

J. C. Wyant, Optical Testing (Optical Sciences Center, Tucson, Ariz., June1976).

Yu, F. T. S.

Appl. Opt. (4)

Bell Syst. Tech. J. (1)

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

J. Opt. Soc. Am. (4)

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

J. N. Latta, R. C. Fairchild, Proc. Soc. Photo-Opt. Instrum. Eng. 39, Aug. (1973).

Other (11)

J. C. Wyant, Optical Testing (Optical Sciences Center, Tucson, Ariz., June1976).

R. S. Longhurst, Geometrical and Physical Optics (Jarrold, Norwich, England, 1967), p. 87.

Abbé 3L Reference Manual, catalog. 33-45-58-01 (Bausch & Lomb, Rochester, N.Y.).

A. F. Turner, U. Arizona; private communication (March1977).

Interference Systems, catalog 500-101-13 (Leitz, Rockleigh, N.J., 1973).

W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966).

R. J. Collier, C. B. Burchart, L. H. Lin, Optical Holography (Academic, New York, 1971), p. 228.

W. C. Sweatt, “Designing Holographic Optical Elements” Doctoral Dissertation, U. Arizona, Tucson, (1977).Available from University Microfilms, Ann Arbor Mich., chap. 3

Ref. 8, chap. 4

ACCOS V Users Guide (Scientific Calculations, Inc., Norwalk, Conn., 1970).

M. V. Klein, Optics (Wiley, New York, 1970), pp. 339–342.

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Figures (14)

Fig. 1
Fig. 1

HOE-equivalent lens.

Fig. 2
Fig. 2

Simplest HOE constructing optical systems (a) produce thick transmission HOE or any thin HOE; (b) produce thick reflection HOE. SF = spatial filter; BS = beam splitter.

Fig. 3
Fig. 3

Equivalent lenses for constructing wavefronts shown in Figs. 2(a) and 2(b) with nc = 101.

Fig. 4
Fig. 4

Phase HOE showing equivalence between reflected and transmitted images.

Fig. 5
Fig. 5

Aberrated wavefront (dotted line) and reference sphere (solid line).

Fig. 6
Fig. 6

Constructing wavefronts for (a) transmitting HOE; (b) reflecting HOE.

Fig. 7
Fig. 7

Thickness variation of substrate: (a) optical system; (b) example of fringes.

Fig. 8
Fig. 8

Abbé refractometer showing interference between the two reflections.

Fig. 9
Fig. 9

Index matching (a) trench cut in emulsion; (b) Mach-Zehnder interferometer (BS = beam splitter, LG = liquid gate, I = image); (c) example image showing OPD = ±λ/4.

Fig. 10
Fig. 10

Systems producing third order spherical aberration: (a) + Wc and +Zc; (b) −Wc and +Zc; (c) +Wc and −Zc; (d) −Wc and −Zc; S = spatial filter; L1 = aberration generating lens; L2 = imaging lens; O = corrected telescope objective.

Fig. 11
Fig. 11

Sign convention for aberration generation scheme: (a) imaging geometry; (b) lens in contact.

Fig. 12
Fig. 12

Functions f(x) [Eq. (13)] and g(x) for n = 1.52.

Fig. 13
Fig. 13

Flow charts: (a) given a reasonable range of Δu, find the range of lens powers and radii (ϕ and y) that will produce the aberration coefficient Sc. (b) When a specific lens with power ϕ(ϕmin < ϕ < ϕmax) is chosen, is its radius greater than that required (yl)?

Fig. 14
Fig. 14

Constructing a reflecting HOE: (a) reconstruction geometry; (b) equivalent lens (nr = 101); (c) construction geometry; (d) constructing optical system; (e) computer model with HOE equivalent lens (nc = 100λcr + 1).

Equations (19)

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c 1 = 1 / z 1 ( n c 1 ) , c 2 = 1 / z 2 ( n c 1 ) , κ 1 = κ 2 = n c 2 + 2 ,
z 1 = c 1 y 2 / { 1 + [ 1 ( κ 1 + 1 ) c 1 2 y 2 ] 1 / 2 } .
n r = ( n c 1 ) λ r / λ c + 1 .
n m = ( n r 1 ) m + 1 .
ξ ( y ) | n c = [ ( n c 1 ) / λ c ] d / dy [ t ( y ) ] .
W ( y ) = D y 4 + E y 6 + F y 8 + . . . ,
z 1 = c 1 y 2 { 1 + [ 1 ( k 1 + 1 ) c 1 2 y 2 ] 1 / 2 } 1 + AD y 4 + AE y 6 + AF y 8 ,
1 / z c = ( A + B ) / 2 , 1 / z c = ( B A ) / 2 , A = λ c / λ r ( 1 / z r 1 / z r ) , B = n c t r / ( n r t c ) ( 1 / z r + 1 / z r ) .
W c = 1 / 8 S c ( x 2 + y 2 ) 2 / h 0 4 W c = 1 / 8 S c ( x 2 + y 2 ) 2 / h 0 4 } ,
S c = ( E + D ) / 2 S c = ( E D ) / 2 } ,
D = λ c / λ r ( S r S r ) + h 0 4 { λ c / λ r [ ( z r ) 3 ( z r ) 3 ] [ ( z c ) 3 ( z c ) 3 ] } , E = n c t r / ( n r t c ) ( S r + S r ) + h 0 4 ( n c t r / ( n r t c ) { ( 1 1 / 3 n r 2 ) × [ ( z r ) 3 + ( z r ) 3 ] + 1 / 6 n r 2 [ 1 ( t r / t c ) 2 ] · ( 1 / z r + 1 / z r ) 3 } + ( 1 1 / 3 n c 2 ) [ ( z c ) 3 + ( z c ) 3 ] ) .
t c / t r = ( λ c / n c ) / ( λ r / n r ) .
1 / z c . . . 1 / z c = 1 / 2 λ c / λ r ( 1 / z r + 1 / z r ) ± 1 / 2 { 2 ( n c / n r ) 2 [ ( z r ) 2 + ( z r ) 2 ] ( λ c / λ r ) 2 ( 1 / z r + 1 / z r ) 2 } 1 / 2 .
S c + S c = λ c / λ r ( S r + S r ) + h 0 4 { ( z c ) 3 + ( z c ) 3 λ c / λ r [ ( z r ) 3 + ( z r ) 3 ] } ,
W ( x , y ) = ( n 1 ) t ( x , y ) .
Δ t = λ / 2 n .
n d = [ ( 9 n 1 2 n 2 2 ) / 8 ] 1 / 2 , t = λ d [ 2 / ( n 1 2 n 2 2 ) ] 1 / 2 , n f n c = 1 / n d [ λ d ( λ f λ c ) / ( 4 t 2 ) + n 1 ( n f n c ) apparent ] ,
F ( x ) S c h l ( z c / h 0 ) 3 = 2 [ ( G 1 x G 3 ) x + G 6 ] x , x = ϕ h l / [ ( n 1 ) u ] = Δ u / [ ( n 1 ) u ] , G 1 = n 2 ( n 1 ) / 2 , G 3 = ( 3 n + 1 ) ( n 1 ) / 2 , G 6 = ( 3 n + 2 ) ( n 1 ) / 2 n .
t c tan α c = t r tan α r ,

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