Abstract

We have derived an analytic procedure for recording a flat, volume-phase-transmission holographic optical element at a wavelength different from that at which it is to be used. The procedure guarantees that the resulting element will have diffraction-limited aberration performance. Furthermore it guarantees, to a first order, that the Bragg condition for high diffraction efficiency will be satisfied. The technique gives simple analytic expressions for the required object and reference construction-beam phases at the element. In general, the object and reference construction beams must be realized by using computer-generated holograms in conjunction with conventional refractive or reflective optics.

© 1982 Optical Society of America

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References

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  1. B. J. Chang, “Dichromated gelatin holograms and their applications,” Opt. Eng. 19, 642–648 (1980).
    [Crossref]
  2. R. Collier, C. B. Burckhardt, and L. H. Lin, in Optical Holography (Academic, New York, 1971), Chap. 10.
  3. E. B. Champagne, “Nonparaxial imaging, magnification, and aberration properties in holography,” J. Opt. Soc. Am. 57, 51–55 (1967).
    [Crossref]
  4. E. B. Champagne, “A qualitative and quantitative study of holographic imaging,” Ph.D. Dissertation, Ohio State University, catalog no. 67-10876 (University Microfilms, Ann Arbor, Mich., 1967).
  5. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [Crossref]
  6. R. C. Fairchild and J. R. Fienup, “Computer-originated hologram lenses,” Proc. Soc. Photo-Opt. Instrum. Eng. 215, 2–14 (1980).
  7. D. K. Campbell and D. W. Sweeney, “Materials processing with CO2laser holographic scanner systems,” Appl. Opt. 17, 3727–3737 (1978).
    [Crossref] [PubMed]
  8. L. H. Lin and E. T. Doherty, “Efficient and aberration-free wavefront reconstruction from holograms illuminated at wavelengths differing from the forming wavelength,” Appl. Opt. 10, 1314–1318 (1971).
    [Crossref] [PubMed]
  9. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 3.

1980 (2)

B. J. Chang, “Dichromated gelatin holograms and their applications,” Opt. Eng. 19, 642–648 (1980).
[Crossref]

R. C. Fairchild and J. R. Fienup, “Computer-originated hologram lenses,” Proc. Soc. Photo-Opt. Instrum. Eng. 215, 2–14 (1980).

1978 (1)

1971 (1)

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[Crossref]

1967 (1)

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 3.

Burckhardt, C. B.

R. Collier, C. B. Burckhardt, and L. H. Lin, in Optical Holography (Academic, New York, 1971), Chap. 10.

Campbell, D. K.

Champagne, E. B.

E. B. Champagne, “Nonparaxial imaging, magnification, and aberration properties in holography,” J. Opt. Soc. Am. 57, 51–55 (1967).
[Crossref]

E. B. Champagne, “A qualitative and quantitative study of holographic imaging,” Ph.D. Dissertation, Ohio State University, catalog no. 67-10876 (University Microfilms, Ann Arbor, Mich., 1967).

Chang, B. J.

B. J. Chang, “Dichromated gelatin holograms and their applications,” Opt. Eng. 19, 642–648 (1980).
[Crossref]

Collier, R.

R. Collier, C. B. Burckhardt, and L. H. Lin, in Optical Holography (Academic, New York, 1971), Chap. 10.

Doherty, E. T.

Fairchild, R. C.

R. C. Fairchild and J. R. Fienup, “Computer-originated hologram lenses,” Proc. Soc. Photo-Opt. Instrum. Eng. 215, 2–14 (1980).

Fienup, J. R.

R. C. Fairchild and J. R. Fienup, “Computer-originated hologram lenses,” Proc. Soc. Photo-Opt. Instrum. Eng. 215, 2–14 (1980).

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[Crossref]

Lin, L. H.

Sweeney, D. W.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 3.

Appl. Opt. (2)

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

B. J. Chang, “Dichromated gelatin holograms and their applications,” Opt. Eng. 19, 642–648 (1980).
[Crossref]

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

R. C. Fairchild and J. R. Fienup, “Computer-originated hologram lenses,” Proc. Soc. Photo-Opt. Instrum. Eng. 215, 2–14 (1980).

Other (3)

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 3.

R. Collier, C. B. Burckhardt, and L. H. Lin, in Optical Holography (Academic, New York, 1971), Chap. 10.

E. B. Champagne, “A qualitative and quantitative study of holographic imaging,” Ph.D. Dissertation, Ohio State University, catalog no. 67-10876 (University Microfilms, Ann Arbor, Mich., 1967).

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Figures (1)

Fig. 1
Fig. 1

Readout geometry of the aspheric HOE at 0.6328 Å.

Equations (91)

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Φ ^ H ( x , y ) = ϕ ^ out ( x , y , z = 0 ) - ϕ in ( x , y , z = 0 ) .
Φ H ( x , y ) = ϕ obj ( x , y , z = 0 ) - ϕ ref ( x , y , z = 0 ) ,
ϕ out ( x , y , z = 0 ) = Φ H ( x , y ) + ϕ in ( x , y , z = 0 ) .
ϕ out ( x , y , z = 0 ) = ϕ ^ out ( x , y , z = 0 ) ,
Φ H ( x , y ) = Φ ^ H ( x , y ) .
ϕ obj ( x , y , z = 0 ) - ϕ ref ( x , y , z = 0 ) = ϕ ^ out ( x , y , z = 0 ) - ϕ in ( x , y , z = 0 ) .
ϕ out ( x , y , z = 0 ) x = ϕ ^ out ( x , y , z = 0 ) x ,
ϕ out ( x , y , z = 0 ) y = ϕ ^ out ( x , y , z = 0 ) y .
l λ 2 π ϕ x ,
m λ 2 π ϕ y ,
n λ 2 π ϕ z .
ϕ 2 = ( 2 π ν λ ) 2 .
n = ± ( ν 2 - l 2 - m 2 ) 1 / 2 ,
1 - l 2 - m 2 > 0 ,
l out ( x , y , z = 0 ) = l ^ out ( x , y , z = 0 ) ,
m out ( x , y , z = 0 ) = m ^ out ( x , y , z = 0 ) .
constant × exp [ i 2 π λ ( l x + m y + n z ) ] .
| constant 1 × exp [ i 2 π λ ref ( l obj x + m obj y + n obj z ) ] + constant 2 × exp [ i 2 π λ ref ( l ref x + m ref y + n ref z ] | 2
constant 3 + constant 4 × cos [ 2 π K o r · ( x , y , z ) ] ,
K o r = [ ( l obj - l ref ) λ ref , ( m obj - m ref ) λ ref , ( n obj - n ref ) λ ref ] .
K i o = [ ( l out - l in ) λ in , ( m out - m in ) λ in , ( n out - n in ) λ in ] .
K o r = K i o
l obj - l ref = λ ref λ in ( l out - l in ) ,
m obj - m ref = λ ref λ in ( m out - m in ) ,
n obj - n ref = λ ref λ in ( n out - n in ) .
l obj = λ ref λ in ( l ^ out + Δ l ) ,
l ref = λ ref λ in ( l in + Δ l ) ,
m obj = λ ref λ in ( m ^ out + Δ m ) ,
m ref = λ ref λ in ( m in + Δ m ) ,
l out = l ^ out
m out = m ^ out ,
( ν 2 - l obj 2 - m obj 2 ) 1 / 2 - ( ν 2 - l ref 2 - m ref 2 ) 1 / 2 = λ ref λ in [ ( ν 2 - l ^ out 2 - m ^ out 2 ) 1 / 2 - ( ν 2 - l in 2 - m in 2 ) 1 / 2 ] .
( ν 2 - l obj 2 - m obj 2 ) 1 / 2 = ν - l obj 2 + m obj 2 2 ν - ( l obj 2 + m obj 2 ) 2 8 ν 3 - .
( ν 2 - l obj 2 - m obj 2 ) 1 / 2 ν - l obj 2 + m obj 2 2 ν .
( ν - l obj 2 + m obj 2 2 ν ) - ( ν - l ref 2 + m ref 2 2 ν ) λ ref λ in [ ( ν - l ^ out 2 + m ^ out 2 2 ν ) - ( ν - l in 2 + m in 2 2 ν ) ]
- l obj 2 + l ref 2 - m obj 2 + m ref 2 λ ref λ in ( - l ^ out 2 + l in 2 - m ^ out 2 + m in 2 ) .
l in 2 - l ^ out 2 + 2 Δ l ( l in - l ^ out ) + m in 2 - m ^ out 2 + 2 Δ m ( m in - m ^ out ) λ in λ ref ( l in 2 - l ^ out 2 + m in 2 - m ^ out 2 ) .
l in 2 - l ^ out 2 + 2 Δ l ( l in - l ^ out ) = λ in λ ref ( l in 2 - l ^ out 2 ) ,
m in 2 - m ^ out 2 + 2 Δ m ( m in - m ^ out ) = λ in λ ref ( m in 2 - m ^ out 2 ) .
Δ l = 1 2 ( λ in λ ref - 1 ) ( l in + l ^ out ) ,
Δ m = 1 2 ( λ in λ ref - 1 ) ( m in + m ^ out ) .
l obj = 1 2 ( 1 + λ ref λ in ) l ^ out + 1 2 ( 1 - λ ref λ in ) l in ,
l ref = 1 2 ( 1 + λ ref λ in ) l in + 1 2 ( 1 - λ ref λ in ) l ^ out ,
m obj = 1 2 ( 1 + λ ref λ in ) m ^ out + 1 2 ( 1 - λ ref λ in ) m in ,
m ref = 1 2 ( 1 + λ ref λ in ) m in + 1 2 ( 1 - λ ref λ in ) m ^ out .
ϕ obj ( x , y , z = 0 ) = 1 2 ( 1 + λ in λ ref ) ϕ ^ out ( x , y , z = 0 ) + 1 2 ( λ in λ ref - 1 ) ϕ in ( x , y , z = 0 ) ,
ϕ ref ( x , y , z = 0 ) = 1 2 ( 1 + λ in λ ref ) ϕ in ( x , y , z = 0 ) + 1 2 ( λ in λ ref - 1 ) ϕ ^ out ( x , y , z = 0 ) .
l obj 2 + m obj 2 < 1
l ref 2 + m ref 2 < 1.
ϕ in ( x , y , z = 0 ) = - 2 π λ in [ x 2 + y 2 + ( 0.4 ) 2 ] 1 / 2 ,
ϕ ^ out ( x , y , z = 0 ) = 2 π λ in [ x 2 + ( y - 0.4 ) 2 + ( 0.3 ) 2 ] 1 / 2 .
ϕ obj ( x , y , z = 0 ) = 1 2 ( 1 + λ in λ ref ) ϕ ^ out ( x , y , z = 0 ) + 1 2 ( λ in λ ref - 1 ) ϕ in ( x , y , z = 0 ) = 1.115 2 π λ in [ x 2 + ( y - 0.4 ) 2 + ( 0.3 ) 2 ] 1 / 2 - 0.115 2 π λ in [ x 2 + y 2 + ( 0.4 ) 2 ] 1 / 2 ,
ϕ ref ( x , y , z = 0 ) = 1 2 ( 1 + λ in λ ref ) ϕ in ( x , y , z = 0 ) + 1 2 ( λ in λ ref - 1 ) ϕ ^ out ( x , y , z = 0 ) = - 1.115 2 π λ in [ x 2 + y 2 + ( 0.4 ) 2 ] 1 / 2 + 0.115 2 π λ in [ x 2 + ( y - 0.4 ) 2 + ( 0.3 ) 2 ] 1 / 2 .
ϕ obj ( x , y , z = 0 ) - ϕ ref ( x , y , z = 0 ) = ϕ ^ out ( x , y , z = 0 ) - ϕ in ( x , y , z = 0 ) ,
K i o = [ ( l obj - l ref ) λ ref , ( m obj - m ref ) λ ref , ( n out - n in ) λ in ] .
K o r = [ ( l obj - l ref ) λ ref , ( m obj - m ref ) λ ref , ( n obj - n ref ) λ ref ] .
K o r = K i o .
K o r K o r · K i o K i o = 1 ,
K o r K o r · K i o K i o 1 ,
K o r K i o .
cos - 1 ( K o r K o r · K i o K i o ) ,
K o r K o r · K i o K i o 0.999982.
OUT ( l out , m out , n out ) ,
IN ( l in , m in , n in ) ,
OBJ ( l obj , m obj , n obj ) ,
REF ( l ref , m ref , n ref ) .
( OUT - IN ) = λ in λ ref ( OBJ - REF ) .
OUT · ( OUT - IN ) = OUT · OUT - OUT · IN = ν 2 - OUT · IN .
IN · ( OUT - IN ) = IN · OUT - IN · IN = IN · OUT - ν 2 .
OUT · ( OBJ - REF ) = - IN · ( OBJ - REF )
( OUT - IN ) OUT - IN · ( OBJ - REF ) OBJ - REF = 1.
λ ref λ in OUT - IN = OBJ - REF .
2 λ ref OBJ - REF ( - IN η ) ( OBJ - REF ) OBJ - REF = λ in ν .
( OBJ - REF ) OBJ - REF
IN ν
cos θ = ( - IN ν ) · ( OBJ - REF ) OBJ - REF ,
d = λ ref OBJ - REF .
2 d cos θ = λ in ν .
δ = cos - 1 [ ( - IN ν ) · ( OBJ - REF ) OBJ - REF ] - θ β .
l obj 1 2 ( 1 + λ ref λ in ) l ^ out + 1 2 ( 1 - λ ref λ in ) l in ,
m obj 1 2 ( 1 + λ ref λ in ) m ^ out + 1 2 ( 1 - λ ref λ in ) m in ,
λ in > λ ref > 0 ,
l ^ out 2 + m ^ out 2 < 1 ,
l in 2 + m in 2 < 1 ,
l obj 2 + m obj 2 < 1.
a = 1 2 ( 1 + λ ref λ in ) ,             b = 1 2 ( 1 - λ ref λ in ) .
l obj 2 + m obj 2 = a 2 ( l ^ out 2 + m ^ out 2 ) + b 2 ( l in 2 + m in 2 ) = 2 a b ( l ^ out l in + m ^ out m in ) + a 2 ( l ^ out 2 + m ^ out 2 ) + b 2 ( l in 2 + m in 2 ) + 2 a b ( l ^ out , m ^ out ) · ( l in , m in ) a 2 + b 2 + 2 a b × ( l ^ out , m ^ out ) · ( l in , m in ) .
( l ^ out , m ^ out ) · ( l in , m in ) ( l ^ out , m ^ out ) · ( l in , m in ) = ( l ^ out 2 + m ^ out 2 ) 1 / 2 ( l in 2 + m in 2 ) 1 / 2 < 1.
l obj 2 + m obj 2 < a 2 + b 2 + 2 a b .
a > 0 ,             b > 0 ,             a + b = 1
l obj 2 + m obj 2 < a 2 + b 2 + 2 a b < ( a + b ) 2 < 1.