Abstract

We describe techniques for the quantitative interferometric characterization of wall thickness variations of hollow glass microspheres. By using a combination of techniques, one can rapidly form a picture of the total configuration for most defect combinations. These techniques require only very simple calculations and do not involve detailed ray tracing through the sphere. We also show how ray tracing calculation can be done very simply for careful analysis of the interference phenomenon in a perfect sphere. These calculations show that if both a large illumination angle and a large aperture objective are used, the nonparallel illumination causes degradation of both the spatial and the phase resolution of the interferometer. These problems can be overcome by properly aperturing the illumination.

© 1978 Optical Society of America

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References

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  1. B. W. Weinstein, J. Appl. Phys. 46, 5305 (1975).
    [CrossRef]
  2. We have found that a glass fiber pulled to a very fine tip (~1 μm) works very well for rolling a glass microsphere on a mirror surface. The fine tip keeps the ball from sticking to the fiber. More recently we have developed a manipulator which provides positive control of the microsphere orientation during inspection. See B. W. Weinstein, C. D. Hendricks, C. M. Ward, D. L. Willenborg, Rev. Sci. Instrum. 49, 870, (1978).
    [CrossRef] [PubMed]
  3. R. R. Stone, D. W. Gregg, P. C. Souers, J. Appl. Phys. 46, 2693 (1975).
    [CrossRef]
  4. R. R. Stone, P. C. Souers, G. C. Abell, J. W. Reed, “Light Interference in Hollow Glass Microsphere Laser Targets,” Lawrence Livermore Laboratory, Report UCRL-77487 (1975).
  5. R. R. Stone, P. C. Souers, G. C. Abell, J. W. Reed, Opt. Acta 24, 35 (1977).
    [CrossRef]

1978

We have found that a glass fiber pulled to a very fine tip (~1 μm) works very well for rolling a glass microsphere on a mirror surface. The fine tip keeps the ball from sticking to the fiber. More recently we have developed a manipulator which provides positive control of the microsphere orientation during inspection. See B. W. Weinstein, C. D. Hendricks, C. M. Ward, D. L. Willenborg, Rev. Sci. Instrum. 49, 870, (1978).
[CrossRef] [PubMed]

1977

R. R. Stone, P. C. Souers, G. C. Abell, J. W. Reed, Opt. Acta 24, 35 (1977).
[CrossRef]

1975

B. W. Weinstein, J. Appl. Phys. 46, 5305 (1975).
[CrossRef]

R. R. Stone, D. W. Gregg, P. C. Souers, J. Appl. Phys. 46, 2693 (1975).
[CrossRef]

Abell, G. C.

R. R. Stone, P. C. Souers, G. C. Abell, J. W. Reed, Opt. Acta 24, 35 (1977).
[CrossRef]

R. R. Stone, P. C. Souers, G. C. Abell, J. W. Reed, “Light Interference in Hollow Glass Microsphere Laser Targets,” Lawrence Livermore Laboratory, Report UCRL-77487 (1975).

Gregg, D. W.

R. R. Stone, D. W. Gregg, P. C. Souers, J. Appl. Phys. 46, 2693 (1975).
[CrossRef]

Hendricks, C. D.

We have found that a glass fiber pulled to a very fine tip (~1 μm) works very well for rolling a glass microsphere on a mirror surface. The fine tip keeps the ball from sticking to the fiber. More recently we have developed a manipulator which provides positive control of the microsphere orientation during inspection. See B. W. Weinstein, C. D. Hendricks, C. M. Ward, D. L. Willenborg, Rev. Sci. Instrum. 49, 870, (1978).
[CrossRef] [PubMed]

Reed, J. W.

R. R. Stone, P. C. Souers, G. C. Abell, J. W. Reed, Opt. Acta 24, 35 (1977).
[CrossRef]

R. R. Stone, P. C. Souers, G. C. Abell, J. W. Reed, “Light Interference in Hollow Glass Microsphere Laser Targets,” Lawrence Livermore Laboratory, Report UCRL-77487 (1975).

Souers, P. C.

R. R. Stone, P. C. Souers, G. C. Abell, J. W. Reed, Opt. Acta 24, 35 (1977).
[CrossRef]

R. R. Stone, D. W. Gregg, P. C. Souers, J. Appl. Phys. 46, 2693 (1975).
[CrossRef]

R. R. Stone, P. C. Souers, G. C. Abell, J. W. Reed, “Light Interference in Hollow Glass Microsphere Laser Targets,” Lawrence Livermore Laboratory, Report UCRL-77487 (1975).

Stone, R. R.

R. R. Stone, P. C. Souers, G. C. Abell, J. W. Reed, Opt. Acta 24, 35 (1977).
[CrossRef]

R. R. Stone, D. W. Gregg, P. C. Souers, J. Appl. Phys. 46, 2693 (1975).
[CrossRef]

R. R. Stone, P. C. Souers, G. C. Abell, J. W. Reed, “Light Interference in Hollow Glass Microsphere Laser Targets,” Lawrence Livermore Laboratory, Report UCRL-77487 (1975).

Ward, C. M.

We have found that a glass fiber pulled to a very fine tip (~1 μm) works very well for rolling a glass microsphere on a mirror surface. The fine tip keeps the ball from sticking to the fiber. More recently we have developed a manipulator which provides positive control of the microsphere orientation during inspection. See B. W. Weinstein, C. D. Hendricks, C. M. Ward, D. L. Willenborg, Rev. Sci. Instrum. 49, 870, (1978).
[CrossRef] [PubMed]

Weinstein, B. W.

We have found that a glass fiber pulled to a very fine tip (~1 μm) works very well for rolling a glass microsphere on a mirror surface. The fine tip keeps the ball from sticking to the fiber. More recently we have developed a manipulator which provides positive control of the microsphere orientation during inspection. See B. W. Weinstein, C. D. Hendricks, C. M. Ward, D. L. Willenborg, Rev. Sci. Instrum. 49, 870, (1978).
[CrossRef] [PubMed]

B. W. Weinstein, J. Appl. Phys. 46, 5305 (1975).
[CrossRef]

Willenborg, D. L.

We have found that a glass fiber pulled to a very fine tip (~1 μm) works very well for rolling a glass microsphere on a mirror surface. The fine tip keeps the ball from sticking to the fiber. More recently we have developed a manipulator which provides positive control of the microsphere orientation during inspection. See B. W. Weinstein, C. D. Hendricks, C. M. Ward, D. L. Willenborg, Rev. Sci. Instrum. 49, 870, (1978).
[CrossRef] [PubMed]

J. Appl. Phys.

B. W. Weinstein, J. Appl. Phys. 46, 5305 (1975).
[CrossRef]

R. R. Stone, D. W. Gregg, P. C. Souers, J. Appl. Phys. 46, 2693 (1975).
[CrossRef]

Opt. Acta

R. R. Stone, P. C. Souers, G. C. Abell, J. W. Reed, Opt. Acta 24, 35 (1977).
[CrossRef]

Rev. Sci. Instrum.

We have found that a glass fiber pulled to a very fine tip (~1 μm) works very well for rolling a glass microsphere on a mirror surface. The fine tip keeps the ball from sticking to the fiber. More recently we have developed a manipulator which provides positive control of the microsphere orientation during inspection. See B. W. Weinstein, C. D. Hendricks, C. M. Ward, D. L. Willenborg, Rev. Sci. Instrum. 49, 870, (1978).
[CrossRef] [PubMed]

Other

R. R. Stone, P. C. Souers, G. C. Abell, J. W. Reed, “Light Interference in Hollow Glass Microsphere Laser Targets,” Lawrence Livermore Laboratory, Report UCRL-77487 (1975).

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Figures (7)

Fig. 1
Fig. 1

Double-pass Twyman-Green interferometer used for wall thickness and uniformity measurements.

Fig. 2
Fig. 2

Interferometric photographs illustrating the measurement of several different defects in a hollow glass microsphere: (a) Fringe pattern showing nonconcentricity. (b) The same sphere and orientation as (a), but with a linear phase gradient compensating for the nonconcentricity. A second more subtle defect is now apparent. (c) The same ball rolled 90° to the right. The nonconcentricity is now invisible, and a small thick spot has come into view at the 7 o’clock position.

Fig. 3
Fig. 3

Diagram of a nonconcentric hollow sphere showing the relevant parameters. The average thickness t0 is r0ri.

Fig. 4
Fig. 4

Parameters used in tracing a ray through a hollow microsphere.

Fig. 5
Fig. 5

Diagram of the reflection of a ray from the background mirror.

Fig. 6
Fig. 6

Blurring of lower wall caused by nonparallel illumination. The amount of overlap or blurring is plotted as a function of illumination half-angle for three values of the ratio of wall thickness t0 to radius r0.

Fig. 7
Fig. 7

Variations in optical path of rays as a function of the angle between the incident ray and the optic axis. The curves are for different ratios of thickness to radius (aspect ratio): (a) t0/r0 = 0.02, (b) t0/r0 = 0.1, (c) t0/r0 = 0.25. The optical path difference is normalized by 4t0(n − 1), the pathlength for the ray incident on the center of the ball parallel to the optic axis. The double valued nature of (c) results from the fact that different rays with the same incidence angle but which strike the ball at different points can both appear at the same point in the image. (They also exit the ball at different points and with different angles.)

Equations (19)

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t = x λ / 4 ( n 1 ) ,
t = 1 ( n 1 ) [ x λ 4 r i ( n 1 ) ] ,
t ( θ ) = t 0 ( 1 + Δ t 1 t 0 cos θ ) .
t ( z ) = t 0 ( 1 + Δ t 1 t 0 r i z ) ,
Δ t 1 t 0 = r i tan θ t 2 ( n 1 ) t 0 ,
Δ t 1 t 0 = r i tan θ t 2 ( n 1 ) t 0 cos θ 1 .
Δ t = ( Δ x λ ) 2 ( n 1 ) ,
θ 1 = sin 1 ( n 0 n 1 sin θ I 1 ) ,
Δ = sin 1 ( r 0 r i sin θ 1 ) θ 1 ,
θ 2 = sin 1 ( n 0 n 2 r 0 r i sin θ I 1 ) ,
θ T = π + 2 θ 2 2 Δ .
Δ P 1 = 2 r i ( n 1 sin Δ sin θ 1 + n 2 cos θ 2 ) .
ψ E 1 = ψ I 1 + θ T .
ψ I 2 = π + γ ,
θ I 2 = ϕ + γ ,
γ = sin 1 [ t cos ϕ ( 1 t 2 ) 1 / 2 sin ϕ ]
t = 2 sin ϕ + sin θ I
ϕ = ψ E 1 π θ I 1 .
Δ P 2 = n 0 r 0 cos ϕ ( 2 + cos ψ E 1 + cos ψ I 2 ) ,

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