Abstract

A theory of holographic imaging is formulated in terms familiar from conventional optics. The effects of the curvatures and off-axis angles of the reference and read-out waves are described by equivalent thin lenses and prisms. The formation of the true-image wavefield is found to be completely analogous to the conventional imaging of the object wavefield by the equivalent lenses and prisms. To explain the conjugate image, we introduce the concept of time reversal. The conjugate-image wavefield is the time-reversed object wavefield conventionally imaged by equivalent lenses and prisms (and a plane mirror). The finite size and resolution of the photographic plate are taken into account. The size of the plate determines the effective aperture of the equivalent lenses and prisms, it is equivalent to a diaphragm in the hologram plane. The modulation transfer function of the plate has the same effect as a diaphragm inserted in the imaging bundle during the recording (or the reconstruction) with its center at the reference (read-out) point. The two diaphragms limit the field of view and the resolution.

© 1968 Optical Society of America

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References

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  1. D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949); Proc. Phys. Soc. (London) B64, 449 (1951).
  2. E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 52, 1123 (1962); J. Opt. Soc. Am. 53, 1377 (1963); J. Opt. Soc. Am. 54, 1295 (1964).
    [Crossref]
  3. J. A. Armstrong, IBM J. Res. Dev. 9, 171 (1965).
    [Crossref]
  4. E. N. Leith, J. Upatnieks, and K. A. Haines, J. Opt. Soc. Am. 55, 981 (1965).
    [Crossref]
  5. R. W. Meier, J. Opt. Soc. Am. 55, 987 (1965).
    [Crossref]
  6. R. W. Meier, J. Opt. Soc. Am. 56, 219 (1966).
    [Crossref]
  7. That the wavelength ratio between the recording and the reconstructing light in holography is equivalent to the wavelength ratio between the object and the image space in conventional imaging, was first shown by Meier (Ref. 6), by considering both holographic and conventional imaging as projective transformations.
  8. The equivalent lenses of this paper should not be confused with the Fresnel-zone lenses often used to describe holographic imaging [cf. Ref. 1, 2: G. L. Rogers, Nature 166, 237 (1950); Proc. Roy. Soc. (Edinburgh) A63, 193, 313 (1952)]. The Fresnel-zone lenses result from the interaction between the reference wave and the wave issued by an object point. Stroke used the term “equivalent lens” to describe the phase distribution of a spherical wave issued by an object point, which interferes with a plane reference wave [G. W. Stroke, An Introduction to Coherent Optics and Holography (Academic Press Inc., New York, 1966), p. 111]. Our equivalent lenses describe the effect of the curvatures of the reference and the read-out wave, respectively: their properties are independent of the object which they image. This agrees with the basic philosophy of this paper, that the equivalent lenses, prisms, and diaphragms introduced, have exactly the same properties as conventional optical elements.
    [Crossref]
  9. The position and magnification of the image of the object given by the lens formula agree with the results of Refs. 3–6.
  10. H. Kogelnik, Bell System Tech. J. 44, 2451 (1965).
    [Crossref]
  11. E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 56, 523L (1966).
    [Crossref]
  12. F. B. Rotz and A. A. Friesem, Appl. Phys. Letters 8, 146 (1966).
    [Crossref]
  13. H. Frieser, Phot. Korr. 91, 69 (1955); Phot. Korr. 92, 51, 183 (1956).
  14. L. O. Hendeberg, Arkiv Fysik 16, 417, 457 (1960).
  15. Further references are given by H. F. Gilmore, J. Opt. Soc. Am. 57, 75 (1967).
    [Crossref]
  16. R. F. van Ligten, J. Opt. Soc. Am. 56, 1, 1009 (1966).
    [Crossref]
  17. Amplitude transmittance vs exposure curves are given by A. Kozma, J. Opt. Soc. Am. 56, 428 (1966).
    [Crossref]
  18. This can be rigorously proved by describing the propagation of the wavefield by the Green’s-function method used in the Appendix.
  19. From the analogy between holographic and conventional imaging it is obvious that the evanescent (inhomogeneous) waves behind the object, which correspond to very high spatial frequencies, do not contribute to the image [cf. G. C. Sherman, J. Opt. Soc. Am. 57, 1160 (1967].
    [Crossref]
  20. A. Sommerfeld, Vorlesungen über Theoretische Physik, Vol. IV, Optik (Dieterich’sche Verlagsbuchhandlung, Wiesbaden, Germany, 1950), p. 202.

1967 (2)

1966 (5)

1965 (4)

1962 (1)

1960 (1)

L. O. Hendeberg, Arkiv Fysik 16, 417, 457 (1960).

1955 (1)

H. Frieser, Phot. Korr. 91, 69 (1955); Phot. Korr. 92, 51, 183 (1956).

1950 (1)

The equivalent lenses of this paper should not be confused with the Fresnel-zone lenses often used to describe holographic imaging [cf. Ref. 1, 2: G. L. Rogers, Nature 166, 237 (1950); Proc. Roy. Soc. (Edinburgh) A63, 193, 313 (1952)]. The Fresnel-zone lenses result from the interaction between the reference wave and the wave issued by an object point. Stroke used the term “equivalent lens” to describe the phase distribution of a spherical wave issued by an object point, which interferes with a plane reference wave [G. W. Stroke, An Introduction to Coherent Optics and Holography (Academic Press Inc., New York, 1966), p. 111]. Our equivalent lenses describe the effect of the curvatures of the reference and the read-out wave, respectively: their properties are independent of the object which they image. This agrees with the basic philosophy of this paper, that the equivalent lenses, prisms, and diaphragms introduced, have exactly the same properties as conventional optical elements.
[Crossref]

1949 (1)

D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949); Proc. Phys. Soc. (London) B64, 449 (1951).

Armstrong, J. A.

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

Friesem, A. A.

F. B. Rotz and A. A. Friesem, Appl. Phys. Letters 8, 146 (1966).
[Crossref]

Frieser, H.

H. Frieser, Phot. Korr. 91, 69 (1955); Phot. Korr. 92, 51, 183 (1956).

Gabor, D.

D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949); Proc. Phys. Soc. (London) B64, 449 (1951).

Gilmore, H. F.

Haines, K. A.

Hendeberg, L. O.

L. O. Hendeberg, Arkiv Fysik 16, 417, 457 (1960).

Kogelnik, H.

H. Kogelnik, Bell System Tech. J. 44, 2451 (1965).
[Crossref]

Kozma, A.

Leith, E. N.

Meier, R. W.

Rogers, G. L.

The equivalent lenses of this paper should not be confused with the Fresnel-zone lenses often used to describe holographic imaging [cf. Ref. 1, 2: G. L. Rogers, Nature 166, 237 (1950); Proc. Roy. Soc. (Edinburgh) A63, 193, 313 (1952)]. The Fresnel-zone lenses result from the interaction between the reference wave and the wave issued by an object point. Stroke used the term “equivalent lens” to describe the phase distribution of a spherical wave issued by an object point, which interferes with a plane reference wave [G. W. Stroke, An Introduction to Coherent Optics and Holography (Academic Press Inc., New York, 1966), p. 111]. Our equivalent lenses describe the effect of the curvatures of the reference and the read-out wave, respectively: their properties are independent of the object which they image. This agrees with the basic philosophy of this paper, that the equivalent lenses, prisms, and diaphragms introduced, have exactly the same properties as conventional optical elements.
[Crossref]

Rotz, F. B.

F. B. Rotz and A. A. Friesem, Appl. Phys. Letters 8, 146 (1966).
[Crossref]

Sherman, G. C.

Sommerfeld, A.

A. Sommerfeld, Vorlesungen über Theoretische Physik, Vol. IV, Optik (Dieterich’sche Verlagsbuchhandlung, Wiesbaden, Germany, 1950), p. 202.

Upatnieks, J.

van Ligten, R. F.

Appl. Phys. Letters (1)

F. B. Rotz and A. A. Friesem, Appl. Phys. Letters 8, 146 (1966).
[Crossref]

Arkiv Fysik (1)

L. O. Hendeberg, Arkiv Fysik 16, 417, 457 (1960).

Bell System Tech. J. (1)

H. Kogelnik, Bell System Tech. J. 44, 2451 (1965).
[Crossref]

IBM J. Res. Dev. (1)

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

J. Opt. Soc. Am. (9)

Nature (1)

The equivalent lenses of this paper should not be confused with the Fresnel-zone lenses often used to describe holographic imaging [cf. Ref. 1, 2: G. L. Rogers, Nature 166, 237 (1950); Proc. Roy. Soc. (Edinburgh) A63, 193, 313 (1952)]. The Fresnel-zone lenses result from the interaction between the reference wave and the wave issued by an object point. Stroke used the term “equivalent lens” to describe the phase distribution of a spherical wave issued by an object point, which interferes with a plane reference wave [G. W. Stroke, An Introduction to Coherent Optics and Holography (Academic Press Inc., New York, 1966), p. 111]. Our equivalent lenses describe the effect of the curvatures of the reference and the read-out wave, respectively: their properties are independent of the object which they image. This agrees with the basic philosophy of this paper, that the equivalent lenses, prisms, and diaphragms introduced, have exactly the same properties as conventional optical elements.
[Crossref]

Phot. Korr. (1)

H. Frieser, Phot. Korr. 91, 69 (1955); Phot. Korr. 92, 51, 183 (1956).

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

D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949); Proc. Phys. Soc. (London) B64, 449 (1951).

Other (4)

The position and magnification of the image of the object given by the lens formula agree with the results of Refs. 3–6.

That the wavelength ratio between the recording and the reconstructing light in holography is equivalent to the wavelength ratio between the object and the image space in conventional imaging, was first shown by Meier (Ref. 6), by considering both holographic and conventional imaging as projective transformations.

This can be rigorously proved by describing the propagation of the wavefield by the Green’s-function method used in the Appendix.

A. Sommerfeld, Vorlesungen über Theoretische Physik, Vol. IV, Optik (Dieterich’sche Verlagsbuchhandlung, Wiesbaden, Germany, 1950), p. 202.

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Figures (5)

Fig. 1
Fig. 1

Recording process. O object; H photographic plate; R reference point. The effects of the curvature and the off-axis angle of the (a) divergent, and (b) convergent reference wave on the formation of the true image are described by the equivalent lens L1 and the prism P1, respectively.

Fig. 2
Fig. 2

First imaging step (corresponding to reconstruction with a plane read-out wave traveling in the +z direction). (a) object wave field: The object O, with transmittance T, is illuminated by the light source S. The imaging by L1 and P1 yields (b) the true-image wavefield: The image O′ of O, with transmittance T′ = T, is illuminated by the image S′ of S. (c) the conjugate-image wavefield is the time-reversed true-image wavefield, reflected at the hologram plane: The image Ō′, with transmittance T ¯ = T*, is illuminated by the wave converging to S ¯ . Ō′, S ¯ mirror images of O′ and S′, respectively. The field (c) is produced by the lens L ¯ 1 and the prism P ¯ 1 from (d) the time-reversed object wavefield, reflected at the hologram plane: The mirror image Ō of O, with transmittance T ¯ = T*, is illuminated by the wave converging to the mirror image S ¯ of S.

Fig. 3
Fig. 3

Reconstruction process. H hologram plane; C read-out point. The effect of the curvature and the off-axis angle of the (a) divergent, and (b) convergent read-out wave are described by the equivalent lens L2 and the prism P2. L2, P2 image, in the second step, the true and the conjugate image of the first step.

Fig. 4
Fig. 4

Equivalent to the influence of the modulation transfer function of the photographic plate is the insertion of (a) a diaphragm e, with its center at the reference point R, during the recording, or (b) a diaphragm e′, with its center at the read-out point C, during the reconstruction. The diaphragm H in the hologram plane describes the effect of the finite size (DH) of the plate.

Fig. 5
Fig. 5

The diaphragms H and e limit the cross sections of the imaging bundles and thus determine the field of view and the resolution. The geometrical-optical construction shown yields the boundaries ( josa-58-8-1084-i001 and josa-58-8-1084-i002) of the part of the object space imaged with nonzero resolution. (The real object space is to the left of H.) The reference wave is (a) divergent, and (b) convergent.

Equations (54)

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u 0 ( x , y , z ; t ) = u 0 ( x , y , z ) exp ( - i ω t ) ,
u r ( x , y , z ; t ) = u r ( x , y , z ) exp ( - i ω t )
I ( x , y ) = [ u r ( x , y , z = 0 ) + u 0 ( x , y , z = 0 ) ] * × [ u r ( x , y , z = 0 ) + u 0 ( x , y , z = 0 ) ] = u r ( x , y , z = 0 ) 2 + u 0 ( x , y , z = 0 ) 2 + u r * ( x , y , z = 0 ) u 0 ( x , y , z = 0 ) + u r ( x , y , z = 0 ) u 0 * ( x , y , z = 0 ) .
u c ( x , y , z ; t ) = u c ( x , y , z ) exp ( - i ω t ) ,
u 0 ( x , y , z = 0 ; t ) = u c ( x , y , z = 0 ) u r * ( x , y , z = 0 ) × u 0 ( x , y , z = 0 ) exp ( - i ω t ) ,
ū 0 ( x , y , z = 0 ; t ) = u c ( x , y , z = 0 ) u r ( x , y , z = 0 ) × u 0 * ( x , y , z = 0 ) exp ( - i ω t ) ,
n = λ / λ = ω / ω .
u r ( x , y , z = 0 ) = u r exp { ± i k [ ( x - x r ) 2 + y 2 + z r 2 ] 1 2 } ,
u r ( x , y , z = 0 ) = u r exp { ± i k [ r 0 - x sin α r + 1 2 ( x 2 + y 2 ) / r 0 ] } ,
L ( x , y ; f k ) = L exp [ - i k ( x 2 + y 2 ) / 2 f ] ,
f = n f .
P ( x ; α k ) = P exp ( i k x sin α ) ,
n sin α = sin α .
u r * ( x , y , z = 0 ) = const L ( x , y ; f 1 k ) P ( x ; α 1 k ) ,
f 1 = ± ( x r 2 + z r 2 ) 1 2 ,             α 1 = ± α r ,
f 1 = n f 1 ,             n sin α 1 = sin α 1 .
f / z - f / z = 1 , x = ( x + z sin α ) z / n z = x z / n z + z sin α , y = ( y + z sin β ) z / n z = y z / n z + z sin β .
T ( x , y ) = T ( x , y ) ,
u c ( x , y , z = 0 ) = u c exp { ± i k [ ( x - x c ) 2 + ( y - y c ) 2 + z c 2 ] 1 2 } ,
u c ( x , y , z = 0 ) = u c exp { ± i k [ r c - x sin α c - y sin β c + ( x 2 + y 2 ) / 2 r c ] } ,
sin α c = x c / r c ,             sin β c = y c / r c .
u c ( x , y , z = 0 ) = const P ( x ; α 2 k ) × P ( y ; β 2 k ) L ( x , y ; f 2 k ) ,
f 2 = [ x c 2 + y c 2 + z c 2 ] 1 2 ,             α 2 = α c ,             β 2 = β c .
f 2 = f 2 / n ,             sin α 2 = n sin α 2 ,             sin β 2 = n sin β 2 .
1 / f 1 , 2 = 1 / f 1 + 1 / f 2 , sin α 1 , 2 = sin α 1 + sin α 2 , sin β 1 , 2 = sin β 2 .
Δ u - ( n / c ) 2 2 u / t 2 = 0
ũ ( x , y , z ; t ) = u ( x , y , z ; - t )
u ( x , y , z ; t ) = Re { u ( x , y , z ) exp ( - i ω t ) } ,
u ( x , y , z ) = ( 2 π ) - 2 - + u ( k x , k y ) × exp [ i ( k x x + k y y + k z z ) ] d k x d k y ,
k z = ± [ k 2 - k x 2 - k y 2 ] 1 2 ,
u ( k x , k y ) = - + u ( x , y , z = 0 ) × exp [ - i ( k x x + k y y ) ] d x d y ,
ū ( x , y , z ; t ) = Re { ū ( x , y , z ) exp ( - i ω t ) } ,
ū ( x , y , z = 0 ) = u * ( x , y , z = 0 ) ,
ū ( k x , k y ) = u * ( - k x , - k y ) .
ū ( x , y , z ) = ( 2 π ) - 2 - + ū ( k x , k y ) × exp [ i ( k x x + k y y + k z z ) ] d k x d k y ,
k z = + [ k 2 - k x 2 - k y 2 ] 1 2 .
ū ( x , y , z ) = u * ( x , y , - z ) ,
ū ( x , y , z ; t ) = u ( x , y , - z ; - t ) .
ū 0 ( x , y , z ; t ) = u 0 ( x , y , - z ; - t ) .
f ¯ 1 = - f 1 ,             α ¯ 1 = - α 1 ,
1 / f ¯ 1 , 2 = 1 / f ¯ 1 + 1 / f 2 = - 1 / f 1 + 1 / f 2 , sin α ¯ 1 , 2 = sin α ¯ 1 + sin α 2 = - sin α 1 + sin α 2 , sin β ¯ 1 , 2 = sin β 2 .
I eff ( x , y ) = - + I ( x ¯ , y ¯ ) E ( x - x ¯ ) , ( y - y ¯ ) d x ¯ d y ¯ = I ( x , y ) * E ( x , y ) ,
i eff ( k x , k y ) = i ( k x , k y ) e ( k x , k y ) ,
e ( k x , k y ) = - + E ( x , y ) exp [ - i ( k x x + k y y ) ] d x d y .
[ u r * ( x , y , z = 0 ) u 0 ( x , y , z = 0 ) + u r ( x , y , z = 0 ) u 0 * ( x , y , z = 0 ) ] * E ( x , y ) .
u 0 ( x , y , z = 0 ; t ) = { [ u r * ( x , y , z = 0 ) u 0 ( x , y , z = 0 ) ] * E ( x , y ) } × u c ( x , y , z = 0 ) exp ( - i ω t ) = { [ L ( x , y ; f 1 k ) P ( x ; α 1 k ) u 0 ( x , y , z = 0 ) ] * E ( x , y ) } × L ( x , y ; f 2 k ) P ( x ; α 2 k ) × P ( y ; β 2 k ) exp ( - i ω t ) ,
ū 0 ( x , y , z = 0 ; t ) = { [ u r ( x , y , z = 0 ) u 0 * ( x , y , z = 0 ) ] * E ( x , y ) } × u c ( x , y , z = 0 ) exp ( - i ω t ) = { [ L ( x , y ; f ¯ 1 k ) P ( x ; α ¯ 1 k ) u 0 * ( x , y , z = 0 ) ] * E ( x , y ) } × L ( x , y ; f 2 k ) P ( x ; α 2 k ) × P ( y ; β 2 k ) exp ( - i ω t ) ,
x - x c = f 2 k x / k ,             y - y c = f 2 k y / k .
D e = 2 f 1 k ˆ / k ,             and             D e = 2 f 2 k ˆ / k ,
u ( x , y , z 1 ) = - + u ( x ¯ , y ¯ , z 0 ) G ( x - x ¯ , y - y ¯ ; z 1 - z 0 k ) d x ¯ d y ¯ = u ( x , y , z 0 ) * G ( x , y ; z 1 - z 0 k ) ,
G ( x , y ; z k ) = ( i λ z ) - 1 exp { i k [ z + 1 2 ( x 2 + y 2 ) / z ] } ,
u 0 ( x , y , z ) = const G ( x - x s , y - y s ; z - z s k ) T ( x , y ) .
u 0 ( x , y , z ) = { [ u 0 ( x , y , z ) * G ( x , y ; - z k ) ] L ( x , y ; f k ) × P ( x ; α k ) P ( y ; β k ) } * G ( x , y ; z k )
u 0 ( x , y , z ) = cons t T ( x , y ) × G ( x - x s , y - y s ; z - z s k ) ,