Abstract

Holographic interferometry is certainly one of the most useful practical applications of holography. This subject was initiated principally by Stetson and Powell with a double exposure technique. In this paper a new technique is exposed which needs only one exposure of the photographic emulsion. This uses two reference beams instead of one as is customary in holography. Depending on the location of the phase object, two configurations are possible, each having some distinct properties. When the object is placed in the central beam, we get, upon reconstruction, an interferogram having twice the sensitivity of one coming from a single-pass interferometer. On the other hand if the object is located in one of the outer beams, the reconstruction step would simultaneously yield an interferogram of the phase object in a first order of diffraction, and the original object wave in a second order. Experimental confirmations are given.

© 1967 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. A. Stetson, R. L. Powell, J. Opt. Soc. Am. 56, 1161 (1966).
    [CrossRef]
  2. G. W. Stroke, A. E. Labeyrie, Phys. Letters 20, 157 (1966).
    [CrossRef]

1966 (2)

G. W. Stroke, A. E. Labeyrie, Phys. Letters 20, 157 (1966).
[CrossRef]

K. A. Stetson, R. L. Powell, J. Opt. Soc. Am. 56, 1161 (1966).
[CrossRef]

Labeyrie, A. E.

G. W. Stroke, A. E. Labeyrie, Phys. Letters 20, 157 (1966).
[CrossRef]

Powell, R. L.

Stetson, K. A.

Stroke, G. W.

G. W. Stroke, A. E. Labeyrie, Phys. Letters 20, 157 (1966).
[CrossRef]

J. Opt. Soc. Am. (1)

Phys. Letters (1)

G. W. Stroke, A. E. Labeyrie, Phys. Letters 20, 157 (1966).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (18)

Fig. 1
Fig. 1

Mach-Zehnder interferometer.

Fig. 2
Fig. 2

Illustrating the two-hologram technique of holographic interferometry.

Fig. 3
Fig. 3

Illustrating the two versions of the three-beam technique of holographic interferometry.

Fig. 4
Fig. 4

Spectrum analysis of a three-beam hologram. (a) General case. The numbers in this diagram refer to the corresponding term in Eq. (5). (b) Three-beam, first version. (c) Three-beam, second version.

Fig. 5
Fig. 5

Reconstruction of a three-beam hologram. (a) Second version. (b) First version.

Fig. 6
Fig. 6

Three-beam holographic interferometry setup.

Fig. 7
Fig. 7

Three-beam interference.

Fig. 8
Fig. 8

Three-beam interference pattern taken prior to alignment.

Fig. 9
Fig. 9

Reconstruction of empty space from a three-beam hologram.

Fig. 10
Fig. 10

Self-aligned setup for three-beam holographic interferometry. The grating G is imaged onto the spherical surface Γ2, and we get its Fourier transform onto the spherical surface Γ1.

Fig. 11
Fig. 11

Two-beam hologram of a linear phase object.

Fig. 12
Fig. 12

Three-beam hologram of a linear phase object.

Fig. 13
Fig. 13

Three-beam hologram of the same linear phase object as in the preceding figure, but now rotated through an angle of 90°.

Fig. 14
Fig. 14

Reconstructed first and second order from a hologram taken with two reference waves conjugate to each other. The second order here is not a nonlinear effect of the emulsion, but a property of the setup. However, the faint fringes in this order result from a nonlinear recording.

Fig. 15
Fig. 15

Reconstructed first and second order from a hologram still taken with two reference waves, but now with one having twice the obliquity of the other to the object wave.

Fig. 16
Fig. 16

Amplitude transmittance vs exposure curve of a typical photographic emulsion.

Fig. 17
Fig. 17

Spectrum of (a) a linearly (b) nonlinearly recorded three-beam hologram of a linear phase object.

Fig. 18
Fig. 18

Spectrum of an actual three-beam hologram of a linear phase object.

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

U 2 = A e i Φ + exp ( i k θ x ) ,
I 2 = ( A 2 + 1 ) + ( A e i Φ ) exp - ( i k θ x ) + ( A e i Φ ) * exp ( i k θ x ) .
I 2 = exp ( i k θ x ) + 1 2 , I 2 = 2 + exp ( i k θ x ) + exp ( - i k θ x ) .
I = I 1 + I 2 = ( A 2 + 3 ) + [ A e i Φ + 1 ] exp ( - i k θ x ) + [ A e - i Φ + 1 ] exp ( i k θ x ) .
T ( x , y ) = k 0 - k 1 u + u 1 + u 2 2 ,
T ( x , y ) = [ k 0 ( 1 ) - k 1 u 2 ( 2 ) - k 1 u 1 2 ( 3 ) - k 1 u 2 2 ( 4 ) ] - k 1 [ u u 1 * ( 5 ) + u u 2 * ( 6 ) + u 1 u 2 * ( 7 ) ] - k 1 [ u * u 1 ( 8 ) + u * u 2 ( 9 ) + u 1 * u 2 ( 10 ) ] ,
u 1 , 2 = exp ( i k θ 1 , 2 x ) ,
I = A e i Φ + exp ( i k θ x ) + exp ( i k 2 θ x ) 2 = ( A 2 + 2 ) + exp ( i k θ x ) [ A e i Φ + 1 ] + [ A e i Φ ] exp ( i k 2 θ x ) + c . c .
U = A e i Φ + exp ( i k θ x ) + exp ( - i k θ x ) ,
I = U U * = ( A 2 + 2 ) + [ A e + i Φ + A e - i Φ ] exp ( i k θ x ) + exp ( i k ( 2 θ ) x ) + c . c .
I = A e - i Φ + A e i Φ 2 = 2 [ A 2 + cos [ ( 2 π / λ ) ( 2 n h ( x , y ) ) ] ] .
h ( x , y ) = K ( λ / 2 n ) with K = 1 , 2 , .
I = A e i Φ + exp ( i k θ x ) exp [ i Ψ 1 ( x , y ) ] + exp ( - i k θ x ) × exp [ i Ψ 2 ( x , y ) ] 2 = ( A 2 + 2 ) + exp [ i k θ x ] [ A e - i Φ exp i Ψ 1 ( x , y ) ] + A e i Φ × exp [ - i Ψ 2 ( x , y ) ] + exp i [ k ( 2 θ ) x ] exp + i [ Ψ 2 ( x , y ) - Ψ 2 ( x , y ) ] + c . c .
exp ( i k θ x ) [ A e - i Φ exp [ i Ψ 1 ( x , y ) ] + A e i Φ exp [ - i Ψ 2 ( x , y ) ] ] = exp ( i k θ x ) exp [ i Ψ ( x , y ) ] [ A e i Φ + A e - i Φ ] ;
Δ I 2 [ cos k θ x + cos k [ ( θ + α ) x + β y ] + cos k [ ( 2 θ + α ) x + β y ] ,
( Δ I ) 2 3 + 2 k θ x + 2 cos 2 k [ ( θ + α ) x + β y ] + 2 cos 2 k [ ( 2 θ + α ) x + β y ] + 2 cos k [ ( 2 θ + α ) x + β y ] + 2 cos k ( α x + β y ) + 2 cos k [ ( 3 θ + α ) x + β y ] + 2 cos k × [ ( α - θ ) x + β y ] + 2 cos k [ ( 3 θ + 2 α ) x + 2 β y ] + 2 cos k θ x .

Metrics