Abstract

The time resolution of streak cameras is limited by aberrations introduced by the rotating mirror. The effect of geometrical distortions of the mirror, rotating at maximum speed, may be minimized by appropriate choice of mirror material and cross section. The stresses in the rotating mirror and the distortions of its reflecting surface were calculated for various geometries of the mirror. By using a semianalytical point-match method, the same numerical scheme and the same formulas were applicable to all geometries. Sufficient data on the behavior of symmetric and partly symmetric cross sections have been obtained to permit optimization of the mirror design.

© 1966 Optical Society of America

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References

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  1. A. Skinner, J. Sci. Instr. 39, 336 (1962); see also, E. A. Igel, Appl. Opt. 4, 1169 (1965).
    [CrossRef]
  2. B. Brixner, in Proc. 3rd, Intern. Congr. High Speed Photography, (Academic Press, Inc., New York, 1957) p. 289.
  3. T. E. Holland, W. C. Davis, J. Opt. Soc. Am. 48, 365 (1958).
    [CrossRef]
  4. K. K. Stevens, R. E. Miller, J. Soc. Motion Picture Television Engrs. 73, 1032 (1964).
  5. V. V. Novozhilov, Theory of Elasticity (Pergamon Press Inc., New York, 1961), pp. (a) 354, (b) 361, (c) 360.
  6. I. S. Sokolnikoff, Mathematical Theory of Elasticity (McGraw-Hill Book Co., Inc., New York, 1946), p. 358.
  7. S. Timoshenko, J. N. Goodier, Theory of Elasticity (McGraw-Hill Book Co., Inc., New York, 1951), pp. (a) 56, (b) 116.
  8. H. D. Conway, J. Appl. Mech. 82, 275 (1960).
    [CrossRef]
  9. Berylco, Bulletin 2125.

1964 (1)

K. K. Stevens, R. E. Miller, J. Soc. Motion Picture Television Engrs. 73, 1032 (1964).

1962 (1)

A. Skinner, J. Sci. Instr. 39, 336 (1962); see also, E. A. Igel, Appl. Opt. 4, 1169 (1965).
[CrossRef]

1960 (1)

H. D. Conway, J. Appl. Mech. 82, 275 (1960).
[CrossRef]

1958 (1)

Brixner, B.

B. Brixner, in Proc. 3rd, Intern. Congr. High Speed Photography, (Academic Press, Inc., New York, 1957) p. 289.

Conway, H. D.

H. D. Conway, J. Appl. Mech. 82, 275 (1960).
[CrossRef]

Davis, W. C.

Goodier, J. N.

S. Timoshenko, J. N. Goodier, Theory of Elasticity (McGraw-Hill Book Co., Inc., New York, 1951), pp. (a) 56, (b) 116.

Holland, T. E.

Miller, R. E.

K. K. Stevens, R. E. Miller, J. Soc. Motion Picture Television Engrs. 73, 1032 (1964).

Novozhilov, V. V.

V. V. Novozhilov, Theory of Elasticity (Pergamon Press Inc., New York, 1961), pp. (a) 354, (b) 361, (c) 360.

Skinner, A.

A. Skinner, J. Sci. Instr. 39, 336 (1962); see also, E. A. Igel, Appl. Opt. 4, 1169 (1965).
[CrossRef]

Sokolnikoff, I. S.

I. S. Sokolnikoff, Mathematical Theory of Elasticity (McGraw-Hill Book Co., Inc., New York, 1946), p. 358.

Stevens, K. K.

K. K. Stevens, R. E. Miller, J. Soc. Motion Picture Television Engrs. 73, 1032 (1964).

Timoshenko, S.

S. Timoshenko, J. N. Goodier, Theory of Elasticity (McGraw-Hill Book Co., Inc., New York, 1951), pp. (a) 56, (b) 116.

J. Appl. Mech. (1)

H. D. Conway, J. Appl. Mech. 82, 275 (1960).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Sci. Instr. (1)

A. Skinner, J. Sci. Instr. 39, 336 (1962); see also, E. A. Igel, Appl. Opt. 4, 1169 (1965).
[CrossRef]

J. Soc. Motion Picture Television Engrs. (1)

K. K. Stevens, R. E. Miller, J. Soc. Motion Picture Television Engrs. 73, 1032 (1964).

Other (5)

V. V. Novozhilov, Theory of Elasticity (Pergamon Press Inc., New York, 1961), pp. (a) 354, (b) 361, (c) 360.

I. S. Sokolnikoff, Mathematical Theory of Elasticity (McGraw-Hill Book Co., Inc., New York, 1946), p. 358.

S. Timoshenko, J. N. Goodier, Theory of Elasticity (McGraw-Hill Book Co., Inc., New York, 1951), pp. (a) 56, (b) 116.

B. Brixner, in Proc. 3rd, Intern. Congr. High Speed Photography, (Academic Press, Inc., New York, 1957) p. 289.

Berylco, Bulletin 2125.

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Figures (12)

Fig. 1
Fig. 1

Optical layout of the streak camera.

Fig. 2
Fig. 2

Intensity and integrated energy of light as function of α, caused by diffraction.

Fig. 3
Fig. 3

Derivation of ηeff from the displacement ξ of the distorted mirror.

Fig. 4
Fig. 4

Maximum absolute value of slope against Poisson’s ratio. (1) triangle, (2) square, (3) pentagon, (4) hexagon, (5) octagon, (6) rectangle b/a = ½, (7) rectangle b/a = ¼, (8) rectangle b/a = 1/6.5, (9) rectangle b/a = 1/10.

Fig. 5
Fig. 5

Displacements of rectangular cross section (ν = 0.027).

Fig. 6
Fig. 6

Slope of rectangular cross section (ν = 0.027).

Fig. 7
Fig. 7

Displacements of rectangular cross section (ν = 0.29).

Fig. 8
Fig. 8

Slope of rectangular cross section (ν = 0.29).

Fig. 9
Fig. 9

Displacements of regular polygons (ν = 0.027).

Fig. 10
Fig. 10

Displacements of regular polygons (ν = 0.29).

Fig. 11
Fig. 11

Displacements of triangular cross section with chipped edges (ν = 0.027).

Fig. 12
Fig. 12

Slope of triangular cross section with chipped edges(ν = 0.027).

Tables (1)

Tables Icon

Table I Summary of Results

Equations (24)

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t = φ / 2 ω ,
I ~ sin 2 α / α 2 ,
α eff = 5.4 rad , φ eff = 5.4 λ / π a cos γ ,
t eff = φ / 2 ω = 0.137 λ / a f cos γ ,
η = d ξ / d l .
e ~ 0 η Σ d l / d η d η .
t eff = φ eff / 2 ω = 2 η eff / ω .
η ~ a 2 ω 2 ρ / E .
λ * = 2 λ μ / ( λ + 2 μ ) , μ * = μ ,
E * = E ( 1 + 2 ν ) / ( 1 + ν ) 2 , ν * = ν / ( 1 + ν ) .
σ r r r + 1 r σ r θ θ + 1 r ( σ r r - σ θ θ ) = - ρ ω 2 r σ r θ r + 1 r σ θ θ θ + 2 r σ r θ = 0
2 σ r r + 1 1 + ν 2 σ r 2 = 0 2 σ θ θ + 1 1 + ν ( 1 r σ r + 1 r 2 2 σ θ 2 ) = 0 2 σ r θ + 1 1 + ν ( 1 r 2 σ r θ - 1 r 2 σ θ ) = 0 σ = σ r r + σ θ θ + σ z z = ( 1 + ν ) ( σ r r + σ θ θ ) .
( σ r r ) h = 1 r 2 2 ϕ h θ 2 + 1 r ϕ h r ( σ r θ ) h = - 1 r 2 ϕ h r θ + 1 r 2 ϕ h d θ ( σ θ θ ) h = 2 ϕ h r 2 .
4 ϕ h = 0.
( σ r r ) p = 1 r 2 2 ϕ p θ 2 + 1 r ϕ p r - 1 2 ρ ω 2 r 2 ( σ r θ ) p = - 1 r 2 ϕ p r θ + 1 r 2 ϕ p θ ( σ θ θ ) p = 2 ϕ p r 2 - 1 2 ρ ω 2 r 2 .
4 ϕ p = 2 [ ( 1 - 2 ν ) / ( 1 - ν ) ] ρ ω 2 .
ϕ p             = ( 1 / 32 ) [ ( 1 - 2 ν ) / ( 1 - ν ) ] ρ ω 2 r 4 ,
( σ r r ) p = - { ( 3 - 2 ν ) / [ 8 ( 1 - ν ) ] } ρ ω 2 r 2 ( σ r θ ) p = 0 ( σ θ θ ) y = - { ( 1 + 2 ν ) / [ 8 ( 1 - ν ) ] } ρ ω 2 r 2 .
( u r ) p = - 1 8 ( 1 + ν ) ( 1 - 2 ν ) E ( 1 - ν ) ρ ω 2 r 3 ( u θ ) p = 0.
ϕ h = A 0 r 2 + n = 1 [ A n r s n + C n r s n + 2 ] cos ( s n θ ) ,
( σ r r ) h = 2 A 0 + n = 1 { A n s n ( 1 - s n ) r s n - 2 + C n [ 2 + s n ( 1 - s n ) ] r s n } cos ( s n θ ) ( σ r θ ) h =             n = 1 [ A n s n ( s n - 1 ) r s n - 2 + C n s n ( 1 + s n ) r s n ] sin ( s n θ ) ( σ θ θ ) h = 2 A 0 + n = 1 { A n s n ( s n - 1 ) r s n - 2 + C n [ 2 + s n ( 3 + s n ) ] r s n } cos ( s n θ ) ;
( u r ) h = 2 A 0 ( 1 + ν ) ( 1 - 2 ν ) E r + 1 + ν E n = 1 { - A n s n r s n - 1 + C n [ 4 ( 1 - ν ) - ( 2 + s n ) ] r s n + 1 } cos ( s n θ ) ( u θ ) n = 1 + ν E n = 1 { A n s n r s n - 1 + [ C n 4 ( 1 - ν ) + s n ] r s n + 1 } sin ( s n θ ) .
σ r r cos ( θ - β ) - σ r θ sin ( θ - β ) = 0 σ r θ cos ( θ - β ) - σ θ θ sin ( θ - β ) = 0 ,
σ r r = σ r θ = σ θ θ = 0.

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