Abstract

It is shown how both the amplitude and the phase of a scattered field may be determined by holography. It is estimated that information about details down to about nine wavelengths of light can be obtained by this technique. The result is of importance for unambiguous determination of the three-dimensional structure of semitransparent objects, such as are frequently encountered, for example, in biology.

© 1970 Optical Society of America

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References

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  1. E. Wolf, Optics Communications 1, No.4, 153 (1969).
    [Crossref]
  2. E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 53, 1377 (1963).
    [Crossref]
  3. If the field distribution U1(x, y) does not satisfy this band-limitation condition, the unwanted spatial-frequency components could be eliminated by using standard optical spatial-filtering techniques. Such spatial-frequency components cannot, of course, be reconstructed at a later stage, i.e., only the filtered field can be reconstructed.
  4. Because of the finite size of the hologram, the intensity distribution IH(x, y) cannot be determined throughout the whole infinite plane z= z1, as formally needed in formula (15). However, because the sinc function in Eq. (15) effectively vanishes when |x− x′| and |y− y′| exceed a moderate multiple of π/ū and π/v¯, respectively, the integration in Eq. (15) needs only be extended over such a finite x′, y′ domain. This result implies, incidently, that information about the scattered field U1 at the point (x, y) is stored in a region of the hologram, centered on the point (x, y), whose linear dimensions are moderate multiples of π/ū and π/v¯ in the x and y directions, respectively.
  5. High-resolution photographic plates are available that would appear, at first sight, to make it profitable to employ reference beams with appreciably larger values of θr. It seems doubtful, however, that this would lead to a significant improvement of the resolution limit of this technique, because of the difficulty of measuring, to sufficient accuracy, the intensity distribution IH(x, y) of the transilluminated hologram.

1969 (1)

E. Wolf, Optics Communications 1, No.4, 153 (1969).
[Crossref]

1963 (1)

J. Opt. Soc. Am. (1)

Optics Communications (1)

E. Wolf, Optics Communications 1, No.4, 153 (1969).
[Crossref]

Other (3)

If the field distribution U1(x, y) does not satisfy this band-limitation condition, the unwanted spatial-frequency components could be eliminated by using standard optical spatial-filtering techniques. Such spatial-frequency components cannot, of course, be reconstructed at a later stage, i.e., only the filtered field can be reconstructed.

Because of the finite size of the hologram, the intensity distribution IH(x, y) cannot be determined throughout the whole infinite plane z= z1, as formally needed in formula (15). However, because the sinc function in Eq. (15) effectively vanishes when |x− x′| and |y− y′| exceed a moderate multiple of π/ū and π/v¯, respectively, the integration in Eq. (15) needs only be extended over such a finite x′, y′ domain. This result implies, incidently, that information about the scattered field U1 at the point (x, y) is stored in a region of the hologram, centered on the point (x, y), whose linear dimensions are moderate multiples of π/ū and π/v¯ in the x and y directions, respectively.

High-resolution photographic plates are available that would appear, at first sight, to make it profitable to employ reference beams with appreciably larger values of θr. It seems doubtful, however, that this would lead to a significant improvement of the resolution limit of this technique, because of the difficulty of measuring, to sufficient accuracy, the intensity distribution IH(x, y) of the transilluminated hologram.

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Figures (4)

Fig. 1
Fig. 1

Illustrating the notation relating to scattering. U(0)(R) = incident field (plane wave propagated in the direction specified by the unit vector s0); U(R) = U(0)(R) + U(s)(R); U(s)(R) = scattered field; r = two-dimensional position vector of a typical point (x, y) in the plane z = z1; U1(r) ≡ U(x, y, z1).

Fig. 2
Fig. 2

Illustrating the notation relating to the formation of the hologram. U(r)(R) = reference field [plane wave propagated in the direction specified by the unit vector sr(pr, qr, mr)]; U(t)(R) = total field incident on the photographic plate in the plane z = z1; U1(r)(r) ≡ U(r)(x, y, z1); U1(t)(r) ≡ U(t)(x, y, z1). Other symbols have same meaning as in Fig. 1.

Fig. 3
Fig. 3

Illustrating the notation relating to the transilluminated hologram: VH(r) represents the distribution of the field on the exit face of the transilluminated hologram, when the hologram is illuminated by a normally incident plane monochromatic wave.

Fig. 4
Fig. 4

The general form of the spatial-frequency spectrum V ˆ H(u) of a (one-dimensional) transilluminated hologram. The scattered field in the plane z = z1 is assumed to be band limited to the range uk |pr|/3. V ˆ H(u) is, in general, complex, but for the purpose of illustration is considered here to be real.

Equations (29)

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U ( R ) = U ( 0 ) ( R ) + U ( s ) ( R )
U ( r ) ( R ) = A ( r ) exp ( i k s r · R )
U ( t ) ( R 1 ) = U ( R 1 ) + U ( r ) ( R 1 ) .
U 1 ( t ) ( r ) = U 1 ( r ) + U 1 ( r ) ( r ) ,
V H ( r ) = V H ( I ) ( r ) + V H ( II ) ( r ) + V H ( III ) ( r ) ,
V H ( I ) ( r ) = C U 1 ( r ) exp [ - i k ( p 1 x + q r y ) ] ,
V H ( II ) ( r ) = C * U 1 * ( r ) exp [ i k ( p r x + q r y ) ] ,
V H ( III ) ( r ) = D + F U 1 * ( r ) U 1 ( r ) .
V H ( r ) = μ ( r ) [ I H ( r ) ] 1 2 ,
V ˆ H ( u , v ) = 1 ( 2 π ) 2 - V H ( x , y ) exp [ - i ( u x + v y ) ] d x d y .
V ˆ H ( u , v ) = V ˆ H ( I ) ( u , v ) + V ˆ H ( II ) ( u , v ) + V ˆ H ( III ) ( u , v ) ,
V ˆ H ( I ) ( u , v ) = C U ˆ 1 ( u + k p r , v + k q r ) ,
V ˆ H ( II ) ( u , v ) = C * U ˆ 1 * ( - u + k p r , - v + k q r ) ,
V ˆ H ( III ) ( u , v ) = D δ ( 2 ) ( u , v ) + F - U ˆ 1 ( u , v ) × U ˆ 1 * ( u - u , v - v ) d u d v ,
U ˆ 1 ( u , v ) = 1 ( 2 π ) 2 - U 1 ( x , y ) exp [ - i ( u x + v y ) ] d x d y
- u ¯ u u ¯ ,             - v ¯ v v ¯ .
u ¯ k ( p r 2 + q r 2 ) 1 2 / 3 2 ,             v ¯ k ( p r 2 + q r 2 ) 1 2 / 3 2 .
V ˆ H ( I ) ( u , v ) = V ˆ H ( u , v ) ˆ ( u , v ) ,
ˆ ( u , v ) = 1 if             - k p r - u ¯ u - k p r + u ¯ ,             - k p r - v ¯ v - k p r + v ¯ , = 0 otherwise . }
U ˆ 1 ( u + k p r , v + k q r ) = ( 1 / C ) V ˆ H ( u , v ) ˆ ( u , v ) .
U 1 ( x , y ) = u ¯ v ¯ π 2 C - [ I H ( x , y ) ] 1 2 × sinc [ u ¯ ( x - x ) π ,             v ¯ ( y - y ) π ] × exp [ i k ( p r x + q r y ) ] d x d y ,
sinc [ ξ , η ] = ( sin π ξ / π ξ ) ( sin π η / π η ) .
U 1 ( x , y ) = k 2 ( p r 2 + q r 2 ) 18 π 2 C - I H ( x , y ) × sinc [ k ( p r 2 + q r 2 ) 1 2 ( x - x ) 3 2 π ,             k ( p r 2 + q r 2 ) 1 2 ( y - y ) 3 2 π ] × exp [ i k ( p r x + q r y ) ] d x d y .
u ¯ = 2 π / Δ x ,             v ¯ = 2 π / Δ y .
p r 2 + q r 2 ( 2 π k ) 2 18 L 2 ,
1 L 2 = 1 ( Δ x ) 2 + 1 ( Δ y ) 2 .
p r 2 + q r 2 = sin 2 θ r ,
L 3 2 λ / sin θ r ,
L 6 2 λ .