Abstract

A general relation is derived by which small displacements of a diffusely reflecting surface may be determined using holographic interferometry. It is shown how a single hologram method of analysis, utilizing parallax and fringe counting, and a multiple hologram method, using interference order assignment are related. Both methods may be viewed in a unified manner. The requirements for application of both methods are discussed.

© 1969 Optical Society of America

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References

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  1. D. Gabor, Nature 161, 777 (1948);Proc. Roy. Soc. (London) A197, 454 (1949).
    [Crossref] [PubMed]
  2. E. Leith, J. Upatnieks, J. Opt. Soc. Amer. 52, 1123 (1962).
    [Crossref]
  3. E. Archbold, J. M. Burch, A. E. Ennos, J. Sci. Instrum. 44, 489 (1967).
    [Crossref]
  4. M. H. Horman, Appl. Opt. 4, 333 (1965).
    [Crossref]
  5. B. P. Hildebrand, K. A. Haines, Appl. Opt. 5, 172 (1966).
    [Crossref] [PubMed]
  6. E. B. Aleksandrov, A. M. Bonch-Bruevich, Soviet Phys.-Tech. Phys. 12, 258 (1967).
  7. K. A. Haines, B. P. Hildebrand, Appl. Opt. 5, 595 (1966).
    [Crossref] [PubMed]
  8. A. E. Ennos, J. Sci. Instrum. (J. Phys. E) Ser. 2 1, 731 (1968).
    [Crossref]
  9. L. W. Orr, S. W. Tehon, N. E. Barnett, Appl. Opt. 7, 202 (1968).
    [Crossref]

1968 (2)

A. E. Ennos, J. Sci. Instrum. (J. Phys. E) Ser. 2 1, 731 (1968).
[Crossref]

L. W. Orr, S. W. Tehon, N. E. Barnett, Appl. Opt. 7, 202 (1968).
[Crossref]

1967 (2)

E. B. Aleksandrov, A. M. Bonch-Bruevich, Soviet Phys.-Tech. Phys. 12, 258 (1967).

E. Archbold, J. M. Burch, A. E. Ennos, J. Sci. Instrum. 44, 489 (1967).
[Crossref]

1966 (2)

1965 (1)

1962 (1)

E. Leith, J. Upatnieks, J. Opt. Soc. Amer. 52, 1123 (1962).
[Crossref]

1948 (1)

D. Gabor, Nature 161, 777 (1948);Proc. Roy. Soc. (London) A197, 454 (1949).
[Crossref] [PubMed]

Aleksandrov, E. B.

E. B. Aleksandrov, A. M. Bonch-Bruevich, Soviet Phys.-Tech. Phys. 12, 258 (1967).

Archbold, E.

E. Archbold, J. M. Burch, A. E. Ennos, J. Sci. Instrum. 44, 489 (1967).
[Crossref]

Barnett, N. E.

Bonch-Bruevich, A. M.

E. B. Aleksandrov, A. M. Bonch-Bruevich, Soviet Phys.-Tech. Phys. 12, 258 (1967).

Burch, J. M.

E. Archbold, J. M. Burch, A. E. Ennos, J. Sci. Instrum. 44, 489 (1967).
[Crossref]

Ennos, A. E.

A. E. Ennos, J. Sci. Instrum. (J. Phys. E) Ser. 2 1, 731 (1968).
[Crossref]

E. Archbold, J. M. Burch, A. E. Ennos, J. Sci. Instrum. 44, 489 (1967).
[Crossref]

Gabor, D.

D. Gabor, Nature 161, 777 (1948);Proc. Roy. Soc. (London) A197, 454 (1949).
[Crossref] [PubMed]

Haines, K. A.

Hildebrand, B. P.

Horman, M. H.

Leith, E.

E. Leith, J. Upatnieks, J. Opt. Soc. Amer. 52, 1123 (1962).
[Crossref]

Orr, L. W.

Tehon, S. W.

Upatnieks, J.

E. Leith, J. Upatnieks, J. Opt. Soc. Amer. 52, 1123 (1962).
[Crossref]

Appl. Opt. (4)

J. Opt. Soc. Amer. (1)

E. Leith, J. Upatnieks, J. Opt. Soc. Amer. 52, 1123 (1962).
[Crossref]

J. Sci. Instrum. (1)

E. Archbold, J. M. Burch, A. E. Ennos, J. Sci. Instrum. 44, 489 (1967).
[Crossref]

J. Sci. Instrum. (J. Phys. E) (1)

A. E. Ennos, J. Sci. Instrum. (J. Phys. E) Ser. 2 1, 731 (1968).
[Crossref]

Nature (1)

D. Gabor, Nature 161, 777 (1948);Proc. Roy. Soc. (London) A197, 454 (1949).
[Crossref] [PubMed]

Soviet Phys.-Tech. Phys. (1)

E. B. Aleksandrov, A. M. Bonch-Bruevich, Soviet Phys.-Tech. Phys. 12, 258 (1967).

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Figures (12)

Fig. 1
Fig. 1

After Fig. 6 of Aleksandrov and Bonch-Bruevich.6 Formation of fringes in rotation of the object. (a) Axis of rotation in the plane of the object; (b) axis normal to the plane.

Fig. 2
Fig. 2

The most general case. Light with propagation vector k1 is incident on a point P located by position vector r1 from an arbitrary origin at 0. The light is scattered with propagation vector k2 and detected at S which is located by R. The point P translates to point Q. Then light with propagation vector k3 is incident, and is scattered into k4 which is also detected at S.

Fig. 3
Fig. 3

The two-dimensional case. The displacement vector d, which arises from motion of a point at P to point Q, lies in the plane of the propagation vectors k1 and k2. Angles θ1 and θ2 are measured from k1 and k2, respectively, with respect to d. The angle θ bisects the angle between k1 and k2, and the angle ψ is measured from the bisector to the displacement vector d.

Fig. 4
Fig. 4

This is the same as Fig. 3. Now, however, all the angles are measured with respect to a line in the plane of displacements.

Fig. 5
Fig. 5

The two-dimensional case, α3 = α2. The components of d on k3 + k2, and k3k2 are sought.

Fig. 6
Fig. 6

The two-dimensional case, but k1 is now allowed to wander out of the plane of k1 and k2. The angle ϑ is the azimuthal angle in a system of spherical coordinates where α1 is the polar angle. The angle ϑ is a measure of how far out of the k2, k3 plane k1 is.

Fig. 7
Fig. 7

The three-dimensional case. The propagation vector k1 describes the incident light, and k2, …, k5 are propagation vectors of light scattered into these directions. The angles θ1, …,θ6 are defined in the figure. The difference vectors K1, K2, K3, which may be used as basis vectors provided k1 is known, are also shown.

Fig. 8
Fig. 8

This shows the basis vectors, K1, K2, and K3 and the unknown displacement vector d, as taken from Fig. 7, showing clearly how the angles α, β, and γ are defined.

Fig. 9
Fig. 9

This shows explicitly the six difference vectors A1, …, A6 any three of which, provided they are not coplanar, provide a basis. For the example in the text A1, A2, and A3 have been chosen as basis. The angles 2χ1, 2χ2,and 2χ3 are shown. These determine the magnitudes of A1, A2, and A3, respectively, viz., Aj = 2k sinχj. From Fig. 7 we see that 2χ1 = θ5 and 2χ3 = θ6.

Fig. 10
Fig. 10

This shows basis vectors A1, A2, and A3 and the unknown displacement vector d, as taken from Fig. 9, showing clearly how the angles a, b, and c are defined.

Fig. 11
Fig. 11

This is a diagram of a typical holographic setup consisting of a single hologram H, positioned with its normal through the image object point 0. The dihedral angle α, between the reference planes ADC and ABC shown, is about 168°.

Fig. 12
Fig. 12

The relation between the generalized coordinate system k ̂ 1, k ̂ 2, and k ̂ 3, and the orthogonal, rectilinear system x ̂, ỳ,ẑ is given by the angles a, b, and c. The angle a = cos 1 x ̂ · k ̂ 2, b = cos 1 · k ̂ 3, and c is the angle between the planes of k ̂ 1 k ̂ 3, and x ̂ k ̂ 2. The angles α1, β1, and γ1 define the direction cosines in the orthogonal system and the angles α1, β2, and γ2 define the direction cosines in the generalized system.

Equations (65)

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δ 1 = k 1 · r 1 + k 2 · ( R r 1 ) + δ r ,
δ 2 = k 3 · r 3 + k 4 · ( R r 3 ) + δ r ,
k 3 = k 1 + Δ k 1 ,
k 4 = k 2 + Δ k 2 .
δ = δ 1 δ 2 = ( k 1 k 2 ) · ( r 1 r 3 ) Δ k 1 · r 3 Δ k 2 · ( R r 3 ) .
| r 1 | | r 3 | | d | = | r 3 r 1 | ,
δ = ( k 1 k 2 ) · ( r 1 r 3 ) .
δ = ( k 1 k 2 ) · d ,
δ = kd ( cos θ 1 + cos θ 2 )
θ = 1 2 ( θ 1 + θ 2 ) ,
ψ = 1 2 ( θ 1 θ 2 ) ,
δ = 2 kd cos θ 1 cos ψ .
θ 12 = 1 2 ( α 1 α 2 ) ,
ψ 12 = π ( α 2 + θ 12 ) ϕ ,
θ 13 = 1 2 ( π α 1 α 3 ) ,
ψ 13 = π ( α 1 + θ 13 ) ϕ ,
δ 12 = 2 kd cos 1 2 ( α 2 α 1 ) cos [ ϕ + 1 2 ( α 2 + α 1 ) ] ,
δ 13 = 2 kd sin 1 2 ( α 3 + α 1 ) sin [ ϕ 1 2 ( α 3 α 1 ) ] ,
cos 1 2 ( α 2 + α 1 ) cos 1 2 ( α 2 α 1 ) = 1 2 ( cos α 2 + cos α 1 ) ,
sin 1 2 ( α 2 + α 1 ) cos 1 2 ( α 2 α 1 ) = 1 2 ( sin α 2 + sin α 1 ) ,
sin 1 2 ( α 2 α 1 ) sin 1 2 ( α 2 + α 1 ) = 1 2 ( cos α 1 cos α 2 ) ,
d cos ϕ = [ k ( cos α 2 + cos α 1 ) ] 1 [ 1 2 δ 13 ( 1 + sin α 2 sin α 1 ) δ 12 ] = x ,
d sin ϕ = [ k ( sin α 2 + sin α 1 ) ] 1 [ 1 2 δ 13 ( 1 + sin α 2 sin α 1 ) ] = δ 13 2 k sin α 1 = y .
ϕ = tan 1 y / x ,
d = ( x 2 + y 2 ) 1 2 .
d cos ϕ = ( δ 13 δ 12 ) / 2 k cos α 2 ,
d sin ϕ = δ 13 [ 1 + ( cos α 1 / cos α 2 ) ] + δ 12 [ 1 ( cos α 1 / cos α 2 ) ] 2 k ( sin α 2 + sin α 1 ) ,
δ 12 = ( k 2 k 1 ) · d ,
δ 13 = ( k 3 k 1 ) · d ,
δ 13 δ 12 = ( k 3 k 2 ) · d ,
δ 13 + δ 12 = ( k 3 + k 2 2 k 1 ) · d .
( k 3 k 2 ) · d = 2 kd cos ϕ cos α 2 ,
( k 3 + k 2 2 k 1 ) · d = kd [ cos ( ϕ α 2 ) cos ( ϕ + α 2 ) + 2 cos χ ] ,
cos χ = cos ϕ cos α 1 + sin ϕ sin α 1 cos ϑ .
δ 13 + δ 12 = 2 kd [ sin ϕ ( sin α 2 + sin α 1 ) cos ϕ cos α 1 ] .
δ 13 + δ 12 = kd [ cos ( ϕ α 2 ) cos ( ϕ + α 2 ) 2 cos ϕ cos α 1 ] = 2 kd [ sin ϕ sin α 2 cos ϕ cos α 1 ] .
d cos ϕ = δ 13 δ 12 2 k cos α 2 , d sin ϕ = δ 13 + δ 12 2 k sin α 2 ,
tan ϕ = ( δ 13 + δ 12 δ 13 δ 12 ) cos α 2 ,
d 2 = 1 4 k 2 [ ( δ 13 δ 12 ) 2 cos 2 α 2 + ( δ 13 + δ 12 ) 2 sin 2 α 2 ] .
δ 1 = ( k 2 k 1 ) · d = K 1 · d ,
δ 2 = ( k 3 k 1 ) · d = K 2 · d ,
δ 3 = ( k 4 k 1 ) · d = K 3 · d ,
δ 4 = ( k 5 k 1 ) · d = K 4 · d .
δ 1 = 2 k sin θ 1 d cos α ,
δ 2 = 2 k sin θ 2 d cos β ,
δ 3 = 2 k sin θ 3 d cos γ .
d = ( K 1 · d ) K 1 K 1 2 + ( K 2 · d ) K 2 K 2 2 + ( K 3 · d ) K 3 K 3 2
d 2 = ( K 1 · d ) 2 K 1 2 + 2 ( K 1 · d ) ( K 2 · d ) K 1 2 K 2 2 ( K 1 · K 2 ) + ( K 2 · d ) 2 K 2 2 + 2 ( K 2 · d ) ( K 3 · d ) K 2 2 K 3 2 ( K 2 · K 3 ) + ( K 3 · d ) 2 K 3 2 + 2 ( K 3 · d ) ( K 1 · d ) K 3 2 K 1 2 ( K 3 · K 1 ) .
K 1 · K 2 = ( k 2 k 1 ) · ( k 3 k 1 ) = k 2 ( cos θ 4 cos 2 θ 2 cos 2 θ 1 + 1 ) ,
K 2 · K 3 = ( k 3 k 1 ) · ( k 4 k 1 ) = k 2 ( cos θ 5 cos 2 θ 3 cos 2 θ 2 + 1 ) ,
K 3 · K 1 = ( k 4 k 1 ) · ( k 2 k 1 ) = k 2 ( cos θ 6 cos 2 θ 1 cos 2 θ 3 + 1 ) ,
cos θ 6 = cos θ 5 cos θ 4 + sin θ 5 sin θ 4 cos ( A + B ) ,
cos A = cos 2 θ 3 cos θ 5 cos 2 θ 2 sin θ 5 sin 2 θ 2
cos B = cos 2 θ 1 cos 2 θ 2 cos θ 4 sin 2 θ 2 sin θ 4 .
f 1 = δ 2 δ 3 = ( k 3 k 4 ) · d = A 1 · d ,
f 2 = δ 4 δ 3 = ( k 5 k 4 ) · d = A 2 · d ,
f 3 = δ 1 δ 3 = ( k 2 k 4 ) · d = A 3 · d .
f 1 = δ 2 δ 3 = 2 π N 34 .
f 1 = 2 k sin χ 1 d cos a ,
f 2 = 2 k sin χ 2 d cos b ,
f 3 = 2 k sin χ 3 d cos c .
d cos α 1 = d ( cos α 2 cos a + cos β 2 sin b cos c ) , d cos β 1 = d cos β 2 sin b cos c , d cos γ 1 = d ( cos γ 2 cos α 2 sin a + cos β 2 cos b ) .
d cos γ 1 = d cos γ 2 ,
cos α 2 sin a = cos β 2 cos b .
Δ 13 = 2 d sin [ ϕ + 1 2 ( α 3 α 1 ) ] sin 1 2 ( α 3 + α 1 )

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