Abstract

In this paper an autocollimation method of refractive index gradient measurement is described. The autocollimation method, in contrast to those applied up-to-date, makes it possible to measure the linear changes of refractive index. Small nonlinear changes of the refractive index in the region under test are approximated to the corresponding linear changes. From the theoretical considerations as well as by the measurements made it follows that the measurement accuracy is of the order of 10−7 cm−1. The method can be applied in the quality control of the applied glass.

© 1965 Optical Society of America

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References

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  1. Karlheinz Rau, 60 Jahre Quarzglas—25 Jahre Hochvacuumtechnik (W. C. Heraeus, Hanau), pp. 77–104.
  2. Z. Bodnar, Pomiary Automatyka Kontrola I, 35 (1963).

1963 (1)

Z. Bodnar, Pomiary Automatyka Kontrola I, 35 (1963).

Bodnar, Z.

Z. Bodnar, Pomiary Automatyka Kontrola I, 35 (1963).

Rau, Karlheinz

Karlheinz Rau, 60 Jahre Quarzglas—25 Jahre Hochvacuumtechnik (W. C. Heraeus, Hanau), pp. 77–104.

Pomiary Automatyka Kontrola (1)

Z. Bodnar, Pomiary Automatyka Kontrola I, 35 (1963).

Other (1)

Karlheinz Rau, 60 Jahre Quarzglas—25 Jahre Hochvacuumtechnik (W. C. Heraeus, Hanau), pp. 77–104.

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Figures (11)

Fig. 1
Fig. 1

Glass heterogeneity in an arbitrarily chosen direction AB expressed in terms of its linear and nonlinear component.

Fig. 2
Fig. 2

A ray falling on an isorefractive surface at the point A changes its direction under the influence of the normal component of the grad nA.

Fig. 3
Fig. 3

A ray falling on a glass block under the angle 90-α.

Fig. 4
Fig. 4

Glass block cross section in the incidence plane of the ray.

Fig. 5
Fig. 5

Decomposition of the grad n into its components the x and y direction, respectively.

Fig. 6
Fig. 6

The ray trace in a heterogeneous glass block: – – –the ray trace under the influence of grad yn only, —the ray trace when influenced by both the gradient components.

Fig. 7
Fig. 7

The ray path illustration in the glass block examined by means of an autocollimation method. Determination of the angle θ″.

Fig. 8
Fig. 8

Optical schema of the arrangement used for measurement of the angle θ.

Fig. 9
Fig. 9

The vector of cross-hair displacement due to glass heterogeneity represented as a difference between the measured displacement θ and the displacement θ′ caused by the wedge between the A and B sides of the examined glass block.

Fig. 10
Fig. 10

Illustration of the change in the direction of the ray after its passing through a heterogeneous glass block on the base of Huygens Principle.

Fig. 11
Fig. 11

A simplified manner for the determination of the ray trace in the heterogeneous glass.

Equations (41)

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grad n A = d n d l
sin α 0 sin α i = n i n 0 .
sin α i = n 0 sin α 0 n i .
grad y n = n i - n 0 y .
sin α i = n 0 sin α 0 y grad y n + n 0 .
Δ x i Δ y i = tan α i = sin α i 1 - sin 2 α i .
Δ x i Δ y i = n 0 sin α 0 y 2 grad y 2 n + 2 y n 0 grad y n + n 0 2 cos 2 α 0 .
d x = n 0 sin α 0 y 2 grad y 2 n + 2 y n 0 grad y n + n 0 2 cos 2 α 0 d y .
x = n 0 sin α 0 grad y n ln 2 ( grad y 2 n ( y 2 grad y 2 n + 2 y n 0 grad y n + n 0 2 cos 2 α 0 ) + y grad y 2 n + n 0 grad y n ) + C .
C = - n 0 sin α 0 grad y n ln 2 n 0 grad y n ( 1 + cos α 0 ) ,
x = n 0 sin α 0 grad y n In ( y 2 grad y 2 n n 0 2 ( 1 + cos α 0 ) 2 + 2 y grad y n n 0 ( 1 + cos d 0 ) 2 + cos 2 α 0 ( 1 + cos α 0 ) 2 + y grad y n n 0 ( 1 + cos α 0 ) + 1 ( 1 + cos α 0 ) ) .
x = n 0 grad y n ln ( 2 y grad y n n 0 + 1 ) .
y = n 0 2 grad y n [ exp ( x grad y n / n 0 ) - 1 ] 2 .
sin β = sin ( 90 - α i ) = 1 - sin 2 α i ,
sin β = 2 y n 0 grad y n .
sin β = exp ( x grad y n / n 0 ) - 1.
sin β β ,
grad y n = n 0 x ln ( 1 + β ) .
θ = n β .
θ = 2 n φ .
θ = θ + θ .
θ y = θ y - θ y = θ y - 2 n φ y = n β y ,
β y = θ y - 2 n φ y n ,
β z = θ z - 2 n φ z n .
grad y n = n 0 2 x ln ( 1 + θ y - 2 n φ y n ) , grad z n = n 0 2 x ln ( 1 + θ z - 2 n φ z n ) ,
n 2 D E = n 1 A B + B C ,
A B = D E = x .
B C = x ( n 2 - n 1 )
sin γ γ = B C y = x ( n 2 - n 1 ) y γ = x grad n .
θ = 2 γ = 2 x grad n ,
grad n = θ 2 x .
grad n = θ - 2 n φ 2 x .
grad y n = θ y - 2 n φ y 2 x , grad z n = θ z - 2 n φ z 2 x .
tan β β = Δ y Δ x = c x , d y = c x d x .
y = c 2 x 2 + c .
θ n = β = c x .
c = θ n x = x grad n x n = grad n n .
y = grad n 2 n x 2 .
Δ grad n = | Δ θ 2 x | + | φ Δ n x | + | n Δ φ x | + | θ - 2 n φ 2 x 2 Δ x | .
θ = 9 0 Δ θ = 1 x = 6 cm Δ x = 0.01 cm φ = 3 0 Δ φ = 1 n = 1.56 Δ n = 1 × 10 - 4 .
Δ grad n = 1 × 10 - 6 cm - 1 .

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