Abstract

A systematic study of the precision required and of the errors committed in testing parabolic mirrors by the longitudinal aberration and by the transversal aberration methods is made, showing that these methods do not possess in general the necessary accuracy of a hundredth of a wave-length per zone. A new method is developed, which consists in determining the caustic or line of the centers of curvature of the surface elements. With this method, a precision of better than a hundredth of a wave-length per zone can easily be obtained. The zones can be comparatively wide. Secondary zones and surface detail can be detected just as well, or even better than by using a plane mirror in conjunction with the parabolic. The use of master flats is thus superfluous. The experimental procedure is explained and results of actual tests are given. By using a thin wire instead of a knife-edge systematic errors are avoided and higher precision is obtained. The caustic method can be applied to the quantitative test of convex surfaces, like Cassegrain mirrors, to the determination of astigmatism, to the study of single lenses, lens systems, Schmidt correcting plates, etc. Its accuracy does not depend on the value of the aberrations, so that surfaces departing considerably from a sphere can be tested just as well as spherical surfaces. It will thus be possible to figure such surfaces with high precision.

© 1939 Optical Society of America

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References

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  1. F. L. O. Wadsworth, Popular Astronomy 10, 337 (1902).
  2. This assumption is made only for the purpose of simplifying the drawings and calculations. In actual practice it is in some cases advantageous to keep the light source fixed. In these cases the aberrations measured are twice the value indicated in the text.
  3. E. Gaviola, J. Opt. Soc. Am. 26, 163 (1936).
    [Crossref]
  4. John Strong, Procedures in Experimental Physics, p. 61–63.

1936 (1)

1902 (1)

F. L. O. Wadsworth, Popular Astronomy 10, 337 (1902).

Gaviola, E.

Strong, John

John Strong, Procedures in Experimental Physics, p. 61–63.

Wadsworth, F. L. O.

F. L. O. Wadsworth, Popular Astronomy 10, 337 (1902).

J. Opt. Soc. Am. (1)

Popular Astronomy (1)

F. L. O. Wadsworth, Popular Astronomy 10, 337 (1902).

Other (2)

This assumption is made only for the purpose of simplifying the drawings and calculations. In actual practice it is in some cases advantageous to keep the light source fixed. In these cases the aberrations measured are twice the value indicated in the text.

John Strong, Procedures in Experimental Physics, p. 61–63.

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Figures (11)

Fig. 1
Fig. 1

Testing along the axis at the so-called center of curvature. When slit and knife-edge are at the crossing of the beams the real images of the slit are farther away.

Fig. 2
Fig. 2

The geometry of testing a parabolic mirror.

Fig. 3
Fig. 3

Photographic determination of the radius of a zone for a given value of the longitudinal aberration using a thin wire instead of a knife-edge.

Fig. 4
Fig. 4

The use of a thin slit instead of a knife-edge for the better determination of the radius of a zone for a given value of the longitudinal aberration.

Fig. 5
Fig. 5

Showing the limitation of the use of a thin slit: at the center the zones become too wide for accurate measurements. The same occurs using a thin wire along the optical axis.

Fig. 6
Fig. 6

Images of a thin slit formed by two symmetrical zones of a parabolic mirror before, at and after the crossing upon the optical axis. The images are sharp beyond the crossing point, at the caustic and not at the axis. Notice the diffraction structure of the images inside and outside focus.

Fig. 7
Fig. 7

Five independent measurements of the curve of shape of a part of the 76-cm Córdoba parabolic mirror, using the caustic method and thin wire. The maximum separation of the curves remains within a hundredth of a wave-length of light.

Fig. 8
Fig. 8

Two high precision measurements of the same large relative aperture parabolic mirror. The systematic errors of the longitudinal aberration method are demonstrated.

Fig. 9a
Fig. 9a

An m=4 parabolic mirror as seen from the “center of curvature” of a peripheral zone. No surface detail is seen. The light fringes are due to diffraction at the edge and at the knife-edge. The picture is a negative.

Fig. 9b
Fig. 9b

The same mirror observed from the real center of curvature—from a point of the caustic—corresponding to the peripheral zone. Two secondary zones are now plainly visible.

Fig. 10
Fig. 10

Surface detail of a bad zone of an m=4 parabolic mirror as seen from the caustic using a thin wire instead of a knife-edge. Notice the two directions in which the polishing tool has moved leaving traces of its cutting edges.

Tables (2)

Tables Icon

Table I Numerical values of the tolerances in the appreciation of the zonal radius for the longitudinal aberration, transversal aberration and caustic method, breadth of the zones, screw tolerance and width of the diffraction images for different values of the relative aperture m, the half-diameter r and the radius of curvature R.

Tables Icon

Table II Numerical calculation of the curve of shape of an m=3 parabolic mirror of 15 cm diameter using the simplified caustic method. The last column gives the results of a measurement of the same mirror by the longitudinal aberration method.

Equations (46)

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- h = y sin φ d / ( R + y cos φ ) = y r d / R ( R + y ) .
- h = y r d R - 2 .
δ r = δ h · R 3 / r 2 d .
δ r = δ h · 4800 m 2 = 2.7 m 2 · 10 - 3 cm .
δ y = δ h / c = δ h R 2 / r d = 1200 m δ h = 6.7 m · 10 - 4 cm
δ r = δ h R 2 / y d = δ h · 2 R 3 / r 2 d ,
δ h = ( h / R ) δ R = 2 y r d R - 3 δ R
δ R = 1.1 m 3 · 10 - 2 cm .
δ s = 3.1 m 3 · 10 - 2 cm .
2 r 1 2 m 4 cm .
- h = ( x cos φ + y 0 sin φ ) d / ( R + y 0 cos φ ) .
- h = ( x + y 0 r / R ) d / ( R + y 0 ) .
x = r 3 / 2 R 2 - y 0 r / R .
δ h = δ r ( x / r ) d / ( R + y 0 ) .
δ r = δ h 2 R 2 ( R + y 0 ) / ( 3 r 2 - 2 y 0 R ) d .
δ r = 2.15 m 2 · 10 - 3 cm .
δ x = δ h ( R + y 0 ) d - 1 = 1.7 · 10 - 4 cm ,
- h = ( y sin φ + x cos φ ) d / ( R + y cos φ ) ,
- h = ( y r + x R ) d / R ( R + y ) .
δ h = ( h / r ) δ r = ( a y / r + b x / r ) δ r .
η = s - r ( 1 + r 2 ) / r ,                         ξ = r + ( 1 + r 2 ) / r ,
η = R + 3 r 2 / 2 R ,                         ξ = - r 3 / R 2 .
y = 3 r 2 / 2 R ,             x = - r 3 / R 2 .
δ h = ( 3 a r / R - 3 b r 2 / R 2 ) δ r = 0 ,
δ 2 h = 1 2 ( 2 h / r 2 ) δ r 2 = 1 2 ( a 2 y / r 2 + b 2 x / r 2 ) δ r 2 .
δ 2 h = δ r 2 3 r d / 2 R 2 ( R + y )
δ r = 2.1 ( m R ) 1 2 10 - 2 cm .
δ y = δ r 3 r / R + δ r 2 3 / 2 R + , δ x = - δ r 3 r 2 / R 2 - δ r 2 3 r / R 2 - .
± δ r = ± Δ r ( i - 1 2 ) / 2 n ,             i = 1 , 2 , n .
δ y = ± Δ r ( i - 1 2 ) 3 r / 2 n R + ( Δ r ( i - 1 2 ) / 2 n ) 2 3 / 2 R ± , δ x = Δ r ( i - 1 2 ) 3 r 2 / 2 n R 2 - ( Δ r ( i - 1 2 ) / 2 n ) 2 3 r / R 2 .
δ y ¯ = δ y c = ( 3 Δ r 2 / 8 R ) 1 n 3 1 n ( i - 1 2 ) 2 , δ x ¯ = δ x c = - ( 3 r Δ r 2 / 4 R 2 ) 1 n 3 ( i - 1 2 ) 2 .
1 n 3 1 n ( i - 1 2 ) 2 = ( 1 + 1 / n ) ( 2 + 1 / n ) / 6 - ( 1 + 1 / n ) / n - 1 / 4 n 2 = lim n             1 3 .
δ y c = Δ r 2 / 8 R ,             δ x c = - Δ r 2 r / 4 R 2
y c = y + δ y c = 3 r 2 / 2 R + Δ r 2 / 8 R , x c = x + δ x c = - r 3 / R 2 - Δ r 2 r / 4 R 2 .
- h = d δ r [ ( δ y - δ y c ) r + ( δ x - δ x c ) R ] / R ( R + y ) .
h = ( r / 2 R 2 ( R + y ) ) ( 3 δ r 2 - Δ r 2 / 4 ) d δ r ,
h = ( r / 2 R 2 ( R + y ) ) ( δ r 3 - δ r Δ r 2 / 4 ) .
δ r = ± Δ r / 2 3.
h max = r Δ r 3 / 24 3 R 2 ( R + y ) .
Δ r 3 = 24 3 · 5.6 · 10 - 7 R 2 ( R + y ) / r .
Δ r = 4.53 ( m R 2 ) 1 3 10 - 2 cm .
a = 2.44 R λ / Δ r cm .
- h = ( ( y + δ R ) r + x R ) d / R 2
- h = ( y r + ( x + δ R r / R ) R ) d / R 2 .
- h = x d R - 1 ,
y t = 3 e 2 r 2 / 2 R ,             x t = - e 2 r 3 / R 2 ,