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Focal-length measurements of deep parabolic mirrors
Volume
11, number
OPTICS
3
FOCAL-LENGTH
July 1974
COMMUNICATIONS
MEASUREMENTS
OF DEEP PARABOLIC
MIRRORS
M. MALi’ and B. NABCLEK Dpt. of Applied
Optics, Institute of Physics, Czechoslovak Academy Prague, Czechoslovakia
Revised
Received 11 February 1974 manuscript received 27 March
of Sciences,
1974
The focal length of deep parabolic mirrors is determined by using the shadows of two fibres placed in front of the mirror and illuminated by collimated light. By double reflexion on the deep mirror two shadow images of each fibre are formed and if all these images are in contact, the fibre distance is 45
The methods for testing concave mirrors commonly used in practice are applicable only for small relative openings, when the aperture angle of the rays is considerably smaller than 90”. The aperture angle is much greater than 90” in the case of “deep mirrors” with a very great relative opening, if the depth of the parabolic segment is greater than the focal length. Because of the impossibility to locate the measuring or indicating device in the neighborhood of the focal point it is impossible to use the known measuring methods. This difficulty can be avoided by the use of the autocollimating shadow method according to Maly [ 11, which exploits two successive reflections of a parallel beam of rays falling on the deep parabolic mirror under examination. The mirror Z (fig. 1) is illuminated by a parallel beam of rays coming from the collimating lens K. The rays are reflected from the surface of the paraboloid, then they go through the focus F and are again reflected. The reflected beam remains parallel in the case of a perfect paraboloid. If a straight fibre V is placed between the paraboloid and the collimator perpendicularly to the optical axis a complex shadow image of the fibre is formed on the screen S by the lens Q (fig. 2). The black straight line is the shadow of one fibre, illuminated by the light coming from the paraboloid, falling into the entrance pupil of the collimator. The black circle is the transformed image of the fibre, illuminated by the light coming from the collimator,
a
b
Fig. 1. Measuring the focal length of a deep parabolic mirror by autocollimation. a) Optical diagram; Z - measured mirror (F - its focus). K - collimating lens. R - semireflecting illuminating mirror. L - light source. V - a fibre perpendicular to the optical axis; the displacement between the fibre and axis is adjustable (ttvo such fibres parallel to each other are used in the actual device). Q - lens projecting the image of the collimator entrance pupil on to the observation screen S. b) Two shadow image of the fibre V on the screen S; the straight one is the shadow cast into the entrance pupil by the fibre V, illuminated by mirror Z; the circular one is the shadow of the same fibre (illuminated by L) transformed by two reflections on Z.
after two reflections on the paraboloid. The ray reflected on the central part of the mirror Z misses the margin of a mirror of a finite diameter and is lost for the second reflection; in this way the central dark area on the photographs figs. 2 and 3 has been formed. One can obtain two shadow images on the screen by inserting a second straight fibre, parallel to the first one, into the beam of the parallel rays falling on the parabolic mirror. 321
Volume
11, number
3
OPTICS COMMUNICATIONS
Fig. 2. Photograph
of the two shadow images of the fibre V as seen on the screen S (for optical diagram see fig. 1). For any position of the fibre V its circular shadow (if prolongated into the central dark area) intersects the optical axis. The paraboloid dimensions: 135 mm dia, 27 mm focal length, zonal errors less than + 1 urn.
The position of the fibres can be altered independently as to bring the straight image of the first fibre in contact with the circular image of the second fibre and vice versa, as seen in fig. 3. If the two fibres are placed symmetrically in respect to the axis of the paraboloid and the straight and the circular fibre images are in contact, the fibre distance is a minimum and equals 4J In the simplest case, if the fibre distance is measured on a millimetre scale without any aid, the focal length
322
July
1974
Fig. 3. Photograph of shadow images of two parallel fibres as seen on the screen S if the fibre positions are adjusted for the focal length measurement; the distance of both straight lines is 4fin this case. The same paraboloid as in fig. 2.
may be obtained within an accuracy of k 0.03 mm. Zonal deformations of the parabolic mirror may be easily indicated by deformations of the straight and circular shadows.
Reference [l]
M. Mal$, Czech.
Patent
No. 14 84 46 (18.10.1972).
11, number
OPTICS
3
FOCAL-LENGTH
July 1974
COMMUNICATIONS
MEASUREMENTS
OF DEEP PARABOLIC
MIRRORS
M. MALi’ and B. NABCLEK Dpt. of Applied
Optics, Institute of Physics, Czechoslovak Academy Prague, Czechoslovakia
Revised
Received 11 February 1974 manuscript received 27 March
of Sciences,
1974
The focal length of deep parabolic mirrors is determined by using the shadows of two fibres placed in front of the mirror and illuminated by collimated light. By double reflexion on the deep mirror two shadow images of each fibre are formed and if all these images are in contact, the fibre distance is 45
The methods for testing concave mirrors commonly used in practice are applicable only for small relative openings, when the aperture angle of the rays is considerably smaller than 90”. The aperture angle is much greater than 90” in the case of “deep mirrors” with a very great relative opening, if the depth of the parabolic segment is greater than the focal length. Because of the impossibility to locate the measuring or indicating device in the neighborhood of the focal point it is impossible to use the known measuring methods. This difficulty can be avoided by the use of the autocollimating shadow method according to Maly [ 11, which exploits two successive reflections of a parallel beam of rays falling on the deep parabolic mirror under examination. The mirror Z (fig. 1) is illuminated by a parallel beam of rays coming from the collimating lens K. The rays are reflected from the surface of the paraboloid, then they go through the focus F and are again reflected. The reflected beam remains parallel in the case of a perfect paraboloid. If a straight fibre V is placed between the paraboloid and the collimator perpendicularly to the optical axis a complex shadow image of the fibre is formed on the screen S by the lens Q (fig. 2). The black straight line is the shadow of one fibre, illuminated by the light coming from the paraboloid, falling into the entrance pupil of the collimator. The black circle is the transformed image of the fibre, illuminated by the light coming from the collimator,
a
b
Fig. 1. Measuring the focal length of a deep parabolic mirror by autocollimation. a) Optical diagram; Z - measured mirror (F - its focus). K - collimating lens. R - semireflecting illuminating mirror. L - light source. V - a fibre perpendicular to the optical axis; the displacement between the fibre and axis is adjustable (ttvo such fibres parallel to each other are used in the actual device). Q - lens projecting the image of the collimator entrance pupil on to the observation screen S. b) Two shadow image of the fibre V on the screen S; the straight one is the shadow cast into the entrance pupil by the fibre V, illuminated by mirror Z; the circular one is the shadow of the same fibre (illuminated by L) transformed by two reflections on Z.
after two reflections on the paraboloid. The ray reflected on the central part of the mirror Z misses the margin of a mirror of a finite diameter and is lost for the second reflection; in this way the central dark area on the photographs figs. 2 and 3 has been formed. One can obtain two shadow images on the screen by inserting a second straight fibre, parallel to the first one, into the beam of the parallel rays falling on the parabolic mirror. 321
Volume
11, number
3
OPTICS COMMUNICATIONS
Fig. 2. Photograph
of the two shadow images of the fibre V as seen on the screen S (for optical diagram see fig. 1). For any position of the fibre V its circular shadow (if prolongated into the central dark area) intersects the optical axis. The paraboloid dimensions: 135 mm dia, 27 mm focal length, zonal errors less than + 1 urn.
The position of the fibres can be altered independently as to bring the straight image of the first fibre in contact with the circular image of the second fibre and vice versa, as seen in fig. 3. If the two fibres are placed symmetrically in respect to the axis of the paraboloid and the straight and the circular fibre images are in contact, the fibre distance is a minimum and equals 4J In the simplest case, if the fibre distance is measured on a millimetre scale without any aid, the focal length
322
July
1974
Fig. 3. Photograph of shadow images of two parallel fibres as seen on the screen S if the fibre positions are adjusted for the focal length measurement; the distance of both straight lines is 4fin this case. The same paraboloid as in fig. 2.
may be obtained within an accuracy of k 0.03 mm. Zonal deformations of the parabolic mirror may be easily indicated by deformations of the straight and circular shadows.
Reference [l]
M. Mal$, Czech.
Patent
No. 14 84 46 (18.10.1972).