Abstract

This paper discusses the influence of both film size and film resolution on the magnification and resolution achievable in microscopy by wavefront reconstruction. Both Fresnel and Fourier-transform holograms are treated. The derivation of expressions for magnification shows explicitly the two-step nature of the wavefront reconstruction and indicates the conditions for producing Fourier-transform holograms. The limit of resolution depends on the first-stage magnification and the film resolution, when Fresnel holograms are made. Image resolution in Fourier-transform holograms is aperture limited.

© 1967 Optical Society of America

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References

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  1. D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949).
  2. D. Gabor, Proc. Roy. Phys. Soc. (London) B64, 449 (1951).
    [Crossref]
  3. E. Leith and J. Upatnieks, J. Opt. Soc. Am. 54, 1295 (1964).
    [Crossref]
  4. G. W. Stroke and D. G. Falconer, Phys. Letters 13, 306 (1964).
    [Crossref]
  5. G. W. Stroke, Appl. Phys. Letters 6, 201 (1965).
    [Crossref]
  6. J. A. Armstrong, IBM J. Res. Develop. 9, 171 (1965).
    [Crossref]
  7. G. B. Parrent and G. O. Reynolds, J. Soc. Photo–Opt. Instr. Engr. 3(b), 219 (1965).
  8. R. F. van Ligten, J. Opt. Soc. Am. 56, 1 (1966).
    [Crossref]
  9. D. Kelly, J. Opt. Soc. Am. 50, 269 (1960).
    [Crossref]
  10. E. Leith, J. Upatnieks, and K. Haines, J. Opt. Soc. Am. 55, 981 (1965).
    [Crossref]
  11. R. W. Meier, J. Opt. Soc. Am. 55, 1566A (1965).
    [Crossref]
  12. P. M. Morse and H. Feshback, Methods of Theoretical Physics (McGraw–Hill Book Co., New York, 1953).
  13. A. V. Baez, J. Opt. Soc. Am. 42, 156 (1952).
    [Crossref]
  14. G. W. Stroke, D. Brumm, and A. Funkhouser, J. Opt. Soc. Am. 55, 1327 (1965).
    [Crossref]
  15. C. Cook, Proc. IRE 48, 310 (1960).
    [Crossref]

1966 (1)

1965 (6)

G. W. Stroke, Appl. Phys. Letters 6, 201 (1965).
[Crossref]

J. A. Armstrong, IBM J. Res. Develop. 9, 171 (1965).
[Crossref]

G. B. Parrent and G. O. Reynolds, J. Soc. Photo–Opt. Instr. Engr. 3(b), 219 (1965).

E. Leith, J. Upatnieks, and K. Haines, J. Opt. Soc. Am. 55, 981 (1965).
[Crossref]

R. W. Meier, J. Opt. Soc. Am. 55, 1566A (1965).
[Crossref]

G. W. Stroke, D. Brumm, and A. Funkhouser, J. Opt. Soc. Am. 55, 1327 (1965).
[Crossref]

1964 (2)

E. Leith and J. Upatnieks, J. Opt. Soc. Am. 54, 1295 (1964).
[Crossref]

G. W. Stroke and D. G. Falconer, Phys. Letters 13, 306 (1964).
[Crossref]

1960 (2)

1952 (1)

A. V. Baez, J. Opt. Soc. Am. 42, 156 (1952).
[Crossref]

1951 (1)

D. Gabor, Proc. Roy. Phys. Soc. (London) B64, 449 (1951).
[Crossref]

1949 (1)

D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949).

Armstrong, J. A.

J. A. Armstrong, IBM J. Res. Develop. 9, 171 (1965).
[Crossref]

Baez, A. V.

A. V. Baez, J. Opt. Soc. Am. 42, 156 (1952).
[Crossref]

Brumm, D.

Cook, C.

C. Cook, Proc. IRE 48, 310 (1960).
[Crossref]

Falconer, D. G.

G. W. Stroke and D. G. Falconer, Phys. Letters 13, 306 (1964).
[Crossref]

Feshback, H.

P. M. Morse and H. Feshback, Methods of Theoretical Physics (McGraw–Hill Book Co., New York, 1953).

Funkhouser, A.

Gabor, D.

D. Gabor, Proc. Roy. Phys. Soc. (London) B64, 449 (1951).
[Crossref]

D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949).

Haines, K.

Kelly, D.

Leith, E.

Meier, R. W.

R. W. Meier, J. Opt. Soc. Am. 55, 1566A (1965).
[Crossref]

Morse, P. M.

P. M. Morse and H. Feshback, Methods of Theoretical Physics (McGraw–Hill Book Co., New York, 1953).

Parrent, G. B.

G. B. Parrent and G. O. Reynolds, J. Soc. Photo–Opt. Instr. Engr. 3(b), 219 (1965).

Reynolds, G. O.

G. B. Parrent and G. O. Reynolds, J. Soc. Photo–Opt. Instr. Engr. 3(b), 219 (1965).

Stroke, G. W.

G. W. Stroke, Appl. Phys. Letters 6, 201 (1965).
[Crossref]

G. W. Stroke, D. Brumm, and A. Funkhouser, J. Opt. Soc. Am. 55, 1327 (1965).
[Crossref]

G. W. Stroke and D. G. Falconer, Phys. Letters 13, 306 (1964).
[Crossref]

Upatnieks, J.

van Ligten, R. F.

Appl. Phys. Letters (1)

G. W. Stroke, Appl. Phys. Letters 6, 201 (1965).
[Crossref]

IBM J. Res. Develop. (1)

J. A. Armstrong, IBM J. Res. Develop. 9, 171 (1965).
[Crossref]

J. Opt. Soc. Am. (7)

J. Soc. Photo–Opt. Instr. Engr. (1)

G. B. Parrent and G. O. Reynolds, J. Soc. Photo–Opt. Instr. Engr. 3(b), 219 (1965).

Phys. Letters (1)

G. W. Stroke and D. G. Falconer, Phys. Letters 13, 306 (1964).
[Crossref]

Proc. IRE (1)

C. Cook, Proc. IRE 48, 310 (1960).
[Crossref]

Proc. Roy. Phys. Soc. (London) (1)

D. Gabor, Proc. Roy. Phys. Soc. (London) B64, 449 (1951).
[Crossref]

Proc. Roy. Soc. (London) (1)

D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949).

Other (1)

P. M. Morse and H. Feshback, Methods of Theoretical Physics (McGraw–Hill Book Co., New York, 1953).

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Figures (3)

Fig. 1
Fig. 1

Wavefront reconstruction: (a) hologram recording, (b) reconstruction.

Fig. 2
Fig. 2

Factors contributing to the amplitude spectrum of image: A=rect[(fx+x3′2L2′)/(H2L2′)], B=|P(fx)|/|P(0)|, C=|Vd(fxfa)|/|Vd(0)|.

Fig. 3
Fig. 3

Resolution of two point sources: (a) Fresnel holograms, (b) Fourier-transform holograms.

Equations (73)

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v 2 ( x 2 ) = j λ 1 r 1 v 1 ( x 1 ) exp [ j k 1 r 1 ] d x 1 ,
v 0 = exp [ j k 1 r 2 ] ,
r 1 = [ ( x 2 - x 1 ) 2 + L 1 2 ] 1 2 L 1 + ( 1 / 2 L 1 ) ( x 2 - x 1 ) 2 ,
r 2 = [ ( x 2 - x 0 ) 2 + R 1 2 ] 1 2 R 1 + ( 1 / 2 R 1 ) ( x 2 - x 0 ) 2 .
τ ( x 2 ) = - [ v 0 + v 2 ( x 2 ) ] × [ v 0 + v 2 ( x 2 ) ] * p ( x 2 - x 2 ) d x 2 .
h R ( x 2 ) = rect ( x 2 H ) - I 2 R ( x 2 ) p ( x 2 - x 2 ) d x 2 ,
I 2 R ( x 2 ) = exp [ ( j π / λ 1 R 1 ) x 2 2 - j 2 π f a x 2 ] · { - v 1 * ( x 1 ) exp [ - j π λ 1 L 1 ( x 2 - x 1 ) 2 ] d x 1 }
f a = x 0 / λ 1 R 1 .
rect ( x 2 H ) = { 1 , x 2 H / 2 0 , x 2 > H / 2.
v R ( x 3 ) = - h R ( x 2 ) exp [ ( j π / λ 2 R 2 ) x 2 2 + ( j π / λ 2 R 2 ) ( x 3 - x 2 ) 2 ] d x 2 ,
r 3 ( 1 / L 2 ) + ( x 3 - x 2 ) 2 / 2 L 2
v R ( x 3 ) = - v 1 D * ( x 2 ) exp [ ( j π / λ 1 R 1 ) x 2 2 ] · { χ ( x 2 , x 3 ) exp [ - j π f a x 2 ] } d x 2 ,
v 1 D ( x 2 ) = - v 1 ( x 1 ) exp [ ( j π / λ 1 L 1 ) ( x 2 - x 1 ) 2 ] d x 1
χ ( x 2 , x 3 ) = - rect ( x 2 / H ) p ( x 2 - x 2 ) · { exp [ ( j π / λ 2 R 2 ) x 2 2 + ( j π / λ 2 L 2 ) × ( x 3 - x 2 ) 2 ] } d x 2 .
v R ( x 3 ) = exp { j π [ ( L 2 / R 2 ) / λ 2 L 2 ] x 3 2 + j π λ 1 L 1 f a 2 } · - V D * ( f x - f a ) P * ( f x ) × { exp [ j π ( λ 1 L 1 - λ 2 L 2 ) f x 2 - j 2 π f x ( x 3 + λ 1 L 1 f a ) ] } × rect [ f x + x 3 / λ 2 L 2 H / λ 2 L 2 ] d f x ,
L 1 = L 1 / ( 1 - L 1 / R 1 )
L 2 = L 2 / ( 1 - L 2 / R 2 )
x 3 = x 3 / ( 1 + L 2 / R 2 )
V D ( f x ) = - v 1 ( x 1 ) exp [ - j π L 1 / R 1 1 - L 1 / R 1 x 1 2 λ 1 L 1 - j 2 π f x x 1 1 - L 1 / R 1 ] d x 1 .
v R ( x 3 ) = - V D * ( f x - f a ) exp [ j π L 2 / R 2 x 3 2 λ 2 L 2 + j π ( λ 1 L 1 - λ 2 L 2 ) f x 2 - j 2 π f x ( x 3 + λ 1 L 1 f a ) ] d f x .
λ 1 L 1 - λ 2 L 2 = 0 ,
v R ( x 3 ) = v 1 * ( x 3 / M + λ 1 L 1 f a ) exp [ j ϕ ( x 3 ) ] ,
ϕ ( x 3 ) = [ ( π R 1 / L 1 ) / λ 1 L 1 ] x 3 2 - 2 π ( 1 - L 1 / R 1 ) f a x 3 + π ( 2 - L 1 / R 1 ) λ 1 L 1 f a 2
M = ( 1 + L 2 / R 2 ) / ( 1 - L 1 / R 1 ) .
1 / M = 1 - ( L 1 / R 1 ) - λ 1 L 1 / λ 2 R 2 .
v 1 D ( x 2 ) = exp [ j π λ 1 L 1 x 2 2 ] - v 1 ( x 1 ) × exp [ ( j π / λ 1 L 1 ) x 1 2 - ( 2 j π / λ 1 L 1 ) x 1 x 2 ] d x 1 ,
v R ( x 3 ) = - V s ( x 2 ) χ ( x 2 , x 3 ) exp [ - j 2 π f a x 2 ] d x 2 ,
V s ( x 2 ) = - v 1 ( x 1 ) exp [ ( j π / λ 1 L 1 ) x 1 2 - ( j 2 π / λ 1 L 1 ) x 1 x 2 ] d x 1 .
v s ( f x ) = λ 1 L 1 v 1 ( - f x ) exp [ j π λ 1 L 1 f x 2 ] ,
v R ( x 3 ) = - λ 1 L 1 v 1 * ( - f x + f a ) × exp [ - j π λ 1 L 1 ( f x - f a ) 2 ] X ( f x , x 3 ) d f x ,
v R ( x 3 ) = λ 1 L 1 exp { [ ( j π L 2 / R 2 ) / L 2 2 ] x 3 2 - j π λ 1 L 1 f 2 } · { - v 1 * ( - f x + f a ) exp [ - j π ( λ 1 L 1 - λ 2 L 2 ) f x - 2 π ( x 3 + λ 1 L 1 f a ) f x ] d f x } .
λ 1 L 1 - λ 2 L 2 = 0 ,
v R ( x 3 ) = λ 1 L 1 exp { [ ( j π L 1 / R 1 ) / λ 2 L 2 ] x 3 2 - j π λ 1 L 1 f a 2 } · { v 1 * ( - f x + f a ) × exp [ - j 2 π ( x 3 + λ 1 L 1 f a ) f x ] d f x } .
v R ( x 3 ) = λ 1 L 1 V 1 * ( x 3 / M + λ 1 L 1 f x ) exp [ - j ϕ ( x 3 ) ] ,
M = 1 + L 2 / R 2 ,
v R ( x 3 ) = - exp [ - j π λ 1 L 1 ( 1 - L 1 R 1 ) x 2 2 ] × V s * ( x 2 ) χ ( x 2 , x 3 ) exp [ - j 2 π f a x 2 ] d x 2 ,
( π / λ 1 L 1 ) 1 - ( L 1 / R 1 ) x 2 max 2 < π / 4 ,
1 - L 1 / R 1 < λ 1 L 1 / 4 x 2 max 2 .
v 1 ( x 1 ) = δ ( x 1 - x 0 ) ,
v 1 D ( x 2 ) = exp [ ( j π / λ 1 L 1 ) ( x 2 - x 0 ) 2 ] .
k R ( x 3 , x 0 ) = exp [ ( - j π / λ 1 L 1 ) x 0 2 ] × - χ ( x 2 , x 3 ) · { exp [ ( - j π / λ 1 L 1 ) x 2 2 - j 2 π ( f a - x 0 / λ 1 L 1 ) x 2 ) ] } d x 2 .
k R ( x 3 , x 0 ) = A 0 - P * ( f x ) rect [ f x + x 3 / λ 2 L 2 H / λ 2 L 2 ] × exp [ j π L 2 / R 2 λ 2 L 2 x 3 2 - j π λ 2 L 2 f x 2 - j 2 π f x x 3 + j 2 π λ 1 L 1 ( f x 2 - 2 f a f x + f a 2 ) ] d f x ,
f a = f a - x 0 / λ 1 L 1
λ 1 L 1 = λ 2 L 2 ,
k R ( x 3 , x 0 ) = A 1 exp [ - j π λ 1 L 1 f a 2 - j π ( L 2 / R 2 ) x 3 2 ] p ( y ) ,
y = ( 1 + L 2 / R 2 ) - 1 [ x 3 + M λ 1 L 1 f a - M x 0 ] .
d min = ( 1 - L 1 / R 1 ) a .
k R ( x 3 , x 0 ) = exp { [ ( - j π / λ 1 L 1 ) x 0 2 ] - χ ( x 2 , x 3 ) } × exp [ - j 2 π ( f a - x 0 / λ 1 L 1 ) ] d x 2 .
X ( f x , x 3 ) = P * ( f x ) rect [ f x + x 3 / λ 2 L 2 H / λ 2 L 2 ] × exp [ j π R 2 / L 2 λ 2 L 2 x 3 2 - j π λ 2 L 2 f x 2 - j 2 π f x x 3 ] ,
k R ( x 3 , x 0 ) = A 0 rect [ x 3 + f a λ 2 L 2 - ( x 0 λ 2 L 2 / λ 1 L 1 ) H ( 1 + L 2 / R 2 ) ] · { exp [ j π x 3 2 λ 2 ( R 2 + L 2 ) + j 2 π x 0 λ 1 L 1 x 3 ] } ,
d min = λ 1 L 1 / 2 H ,
v R ( x 3 ) = - v 1 D * ( x 2 ) exp [ j π λ 1 R 1 x 2 2 ] χ ( x 2 , x 3 ) × exp [ - j 2 π f a x 2 ] d x 2 ,
v 1 D ( x 2 ) = - v 1 ( x 1 ) exp [ j π λ 1 L 1 ( x 2 - x 1 ) 2 ] d x 1
χ ( x 2 , x 3 ) = - rect ( x 2 / H ) p ( x 2 - x 2 ) × exp [ ( j π / λ 2 R 2 ) x 2 2 + ( j π / λ 2 L 2 ) ( x 3 - x 2 ) 2 ] d x 2 .
χ ( x 2 , x 3 ) = - P * ( f x ) exp [ j 2 π f x x 2 ] I ( f x ) d f x ,
I ( f x ) = - exp [ j π λ 2 L 2 ( x 3 - x 2 ) 2 + ( j π / λ 2 R 2 ) x 2 2 - j 2 π f x x 2 ] d x 2 ,
I ( f x ) = exp [ j π λ 2 L 2 x 3 2 ] × - H / 2 H / 2 exp [ j π λ 2 L 2 x 2 2 ( 1 + L 2 R 2 ) - 2 x 2 ( x 3 + λ 2 L 2 f x ) ] d x 2 .
L 2 = L 2 / ( 1 + R 2 / L 2 )
x 3 = x 3 / ( 1 + R 2 / L 2 ) ,
I ( f x ) = - H / 2 H / 2 exp [ j π λ 2 L 2 [ x 2 - ( x 3 + f x λ 2 L 2 ) ] 2 × ( - j π / λ 2 L 2 ) ( x 3 + f x λ 2 L 2 ) 2 + ( j π / λ 2 L 2 ) ( 1 + L 2 / R 2 ) x 3 2 ] d x 2 .
ω 2 / 2 = ( 1 / λ 2 L 2 ) [ x 2 - ( x 3 + f x λ 2 L 2 ) ] 2 ,
I ( f x ) = [ λ 2 L 2 / 2 ] 1 2 exp [ ( j π / λ 2 L 2 ) ( 1 + L 2 / R 2 ) x 3 2 - ( j π / λ 2 L 2 ) ( x 3 + λ 2 L 2 f x ) 2 ] · { ϕ 1 ϕ 2 exp [ j π ω 2 2 ] d ω ,
ϕ 1 = [ H 2 λ 2 L 2 ] 1 2 [ 1 - λ 2 L 2 H ( 2 f x + 2 x 3 λ 2 L 2 ) ]
ϕ 2 = - [ H 2 λ 2 L 2 ] 1 2 [ 1 + λ 2 L 2 H ( 2 f x + 2 x 3 λ 2 L 2 ) ] .
Z * ( ϕ 2 ) - Z * ( ϕ 1 ) rect [ f x + x 3 / λ 2 L 2 H / λ 2 L 2 ] ,
Z ( ϕ ) = 0 ϕ exp [ - j π 2 ω 2 ] d ω .
χ ( x 2 , x 3 ) = a 0 ( λ 2 L 2 ) 1 2 × - P * ( f x ) rect [ f x + x 3 / λ 2 L 2 H / λ 2 L 2 ] · { exp [ j π L 2 / R 2 x 3 2 λ 2 L 2 - j π λ 2 L 2 f x 2 + j 2 π f x ( x 2 - x 3 ) ] } d f x ,
X ( f x , x 3 ) = a 0 ( λ 2 L 2 2 ) 1 2 P * ( f x ) × rect [ f x + x 3 / λ 2 L 2 H / λ 2 L 2 ] · { exp [ j π L 2 / R 2 λ 2 L 2 x 3 2 - j π λ 2 L 2 f x 2 - j 2 π f x x 3 ] }
v R ( x 3 ) = - X ( f x , x 3 ) V * ( f x - f a ) d f x ,
V ( f x - f a ) = - v 1 D ( x 2 ) exp [ ( - j π / λ 1 R 1 ) x 2 2 + j 2 π f a x 2 - j 2 π f x x 2 ] d x 2 .
L 1 = L 1 / ( 1 - R 1 / L 1 ) ,
v R ( x 3 ) = exp { [ ( j π L 2 / R 2 ) / λ 2 L 2 ] x 3 2 + j π λ 1 L 1 f a 2 } · - V D * ( f x - f a ) P * ( f x ) × exp [ j π ( λ 1 L 1 - λ 2 L 2 ) f x - j 2 π f x ( x 3 + λ 1 L 1 f a ) ] × rect [ f x + x 3 / λ 2 L 2 H / λ 2 L 2 ] d f x ,
V D ( f x ) = - v 1 ( x 1 ) [ - j π L 1 / R 1 1 - L 1 / R 1 x 1 2 λ 1 L 1 - j 2 π f x x 1 1 - L 1 / R 1 ] d x 1 .