Abstract:
An optical apparatus for use in conjunction with spherically shaped concave mirrors for substantially eliminating aberrations in the reflections from such mirror. The mirror cooperates with a light source or a light receiver placed in the vicinity of but off-set relative to the natural focal point of the mirror. A plane parallel, transparent plate is placed between the mirror and the light source or receiver and it has a size so as to intercept light rays between the mirror and the light source or receiver and which respectively result in or are caused by light beams oriented parallel to the optical axis of the mirror. Use of the invention for additionally correcting astigmatism and chromatic aberrations caused by spherical mirrors is disclosed. Also disclosed is the use of the invention in connection with large spherical mirrors forming part of light curtain arrangements.

Description:
BACKGROUND OF THE INVENTION 
     The invention relates to an optical apparatus for correcting the spherical aberration of a spherical concave mirror. 
     Whereas parabolic mirrors provide an absolutely parallel light beam when a punctiform light source is arranged in their focus, in the case of spherical concave mirrors parallelism to the optical axis can only be assumed in the case of light rays close to the axis. In the case of abaxial rays, the parallelism error increases with the distance from the optical axis. As spherical concave mirrors can be manufactured much more economically and accurately than parabolic mirrors, particularly above particular size limits, attempts have been made to eliminate by optical correction elements the parallelism error of a spherical concave mirror, which is also called the spherical abberation. The Schmidt correction plate and the Mangin mirror proposed for this purpose have the disadvantage that their manufacture, particularly when fulfilling the necessary high precision requirements is very expensive. 
     BRIEF SUMMARY OF THE INVENTION 
     The problem of the invention is therefore to remove the spherical aberration of a spherical concave mirror to the extent necessary for practical purposes and at limited cost. 
     According to the invention this problem is solved in that a transparent coplanar plate is positioned between the concave mirror and a distance r/2+d·(n-1/n), whereby r is the radius of the concave mirror, d the thickness of the plate and n the refractive index of the plate, whereby an angular range of &lt;±45° is used, based on the centre of the concave mirror. For the purposes of this definition it is assumed that the coplanar plate is arranged in air. If the correction plate is arranged in a medium whose refractive index is not equal to 1, it is easily possible to carry out a corresponding conversion of the maximum distance (image distance) from the concave mirror measured on the optical axis. The use of a coplanar correction plate makes it possible to substantially eliminate the spherical aberration of rays which are more remote from the optical axis of the concave mirror, in addition to those which are close to said axis, so that by suitable choice of the useful angle range of the concave mirror and the plate parameters, the spherical aberration can be kept below a predetermined absolute value within the complete useful angular range. It is surprising that this is possible through suitable dimensioning and a suitable refractive index of a coplanar plate. It is possible to manufacture economically coplanar plates with an adequate precision level. 
     At least in a plane containing the optical axis, the plate is advantageously positioned at right angles to the optical axis. According to a first advantageous embodiment, the plate is at right angles to the optical axis, whereby the emergent and incident rays pass through the plate. Therefore a light beam which enters or emerges from the correction plate parallel to the optical axis is not influenced by the correction plate. Only those rays which pass from the focus to the concave mirror or from the concave mirror to the focus are subject to the correction within the plate according to the present invention. 
     According to another advantageous embodiment, the plate is only perpendicular to the optical axis in a plane containing said axis, whereby the plate is displaced to such an extent from the optical axis that it does not intercept either the parallel, emergent or, respectively, the incident beam. In this embodiment, the parallel beam not intercepted by the plate and which passes either from or to the concave mirror is at an angle to the optical axis, so that said displacement of the correction plate relative to the optical axis makes it possible for the parallel light beam to pass by the correction plate. As in such an optical arrangement there is a more or less pronounced astigmatism, and more specifically in such an instance the line along which the light beam scans over the mirror becomes curved to an extent which depends on the angle of the parallel beam to the optical axis, a further preferred embodiment of the invention stipulates that the plate is not arranged at right angles to the plane of the incident and emergent beam relative to the projection of the rays passing throught it, but is instead additionally tilted by an angle which eliminates such astigmatism caused by the angle between the optical axis and the parallel beam. Thus, without additional expenditure, the correction plate according to the invention can be used for eliminating a further optical error. 
     If chromatic aberrations occur through the use of multicoloured light, according to a further embodiment of the invention the coplanar plate can comprise a plurality of layers, which have different Abbe numbers in such a way that chromatic aberrations are corrected. The various layers must therefore have different dependences of the refractive index from the wave length. 
     According to the invention, the plate parameters are determined in such a way that the displacement of the focal point for areas close to the axis is at least approximately obtained for two further angles within the useful range of the concave mirror. The plate parameters are preferably determined by the Chebyshev approximation, which means that the absolute error in the complete useful angle range is smaller than a predetermined value. 
     If the incident or emergent parallel light beam does not pass through the correction plate but travels alongside it, the plate advantageously has such a trapezoidal shape in the plane of the light beam passing through it that the peripheral rays corresponding to the largest useful angle of the concave mirror are just intercepted at the inclined edges. In other words, the plate dimensions are limited to those areas which are optically necessary, resulting in a considerable saving of space and material. 
     The apparatus according to the invention can be used wherever a spherical concave mirror has hitherto been unusable due to the spherical aberration. Thus, it can be used for example in astronomical reflecting telescopes. 
     The invention can be used with particular advantage in an apparatus with a light ray deflecting device for producing a light curtain. In this case, the coplanar plate is arranged between the concave mirror and the light ray deflecting device and the reflecting surface of said device is, as a function of the plate parameters, arranged at such a distance from half the radius of the concave mirror that the spherical aberration is minimal according to the Chebyshev approximation. In this embodiment, there is somewhat less space for arranging the correction plate according to the invention, because account must also be taken of the movement range of the light ray deflecting device. 
     The invention is used with particular advantage in the case of an apparatus with a spherical concave mirror, a mirror wheel and a light source which illuminates the mirror wheel, whereby the association of the optical elements is such that on rotating the mirror wheel in front of the concave mirror a light curtain is formed comprising a scanning beam displaced parallel to itself. In this case the invention is characterised in that the centre of rotation of the mirror wheel is displaced from the centre of the concave mirror in the direction of said mirror by a distance r/2-a/2+k, 
     whereby r is the concave mirror radius, a the mirror wheel diameter defined by the spacing between two facing mirror surfaces and k a constant, which by means of the Chebyshev approximation is determined in such a way that the spherical aberration is minimal. Even without using the coplanar correction plate, this embodiment is important because in the case of a suitable selection of constant k, the spherical aberration can be made equal to zero not only in the case of rays which are close to the axis but also when they are some distance from said axis, so that in certain circumstances, even without the correction plate according to the invention, the absolute error within the useful angle range can be kept below a predetermined value. The simultaneous use of the correction plate according to the invention is preferred however, because it brings about a further significant decrease of the spherical aberration. 
     A preferred method for the optimization of the setting of the apparatus according to the invention with an incorporated coplanar plate is characterised in that assuming k=O initially for the maximum useful angle of the concave mirror, the spherical aberration caused by the mirror wheel is substantially made to disappear by suitable choice of the parameters of the coplanar plate, whereby subsequently through selecting a suitable value for k, the negative and positive spherical errors within the useful angle range are made substantially the same. Thus, for k initially certain assumptions are made after which the calculation process is repeated to establish in what way the spherical aberration has changed. It is thus possible to approximately find the minimum absolute error. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     Hereinafter, the invention is explained in exemplified manner relative to non-limitative embodiments and with reference to the attached drawings, wherein show: 
     FIG. 1 a part sectional schematic side view of a basic embodiment of the optical apparatus according to the invention. 
     FIG. 2 a somewhat modified embodiment compared with FIG. 1 in a partly broken away view along the line II--II of FIG. 1. 
     FIG. 3 similar view to FIG. 1 of a further embodiment of the optical apparatus according to the invention. 
     FIG. 4 a part sectional diagrammatical side view of a measuring light curtain equipped with the apparatus according to the invention. 
     FIG. 5 a diagram showing the spherical aberration ε as a function of the angular distance φ of the considered point on the concave mirror. 
     FIG. 6 an identical diagram to FIG. 5 for a different practical case. 
     FIG. 7 a diagrammatic view of a similar measuring light curtain to FIG. 4 along the line VII--VII of FIG. 4. 
     FIG. 8 a view of the object of FIG. 7 along the line VIII--VIII of FIG. 7. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     FIG. 1 shows a spherical concave mirror 11 with a useful angle range φ max , whereby the said range is measured from the centre M of concave mirror 11. The coplanar correction plate 12 according to the invention is arranged perpendicularly to the optical axis 13 of the concave mirror directly in front of the latter. For reasons of clarity of the parallel light beam 14 parallel to the optical axis 13 only one ray close to the axis and a further ray remote from the axis are shown. Both rays pass through the coplanar plate 12, initially without any displacement because they strike it in perpendicular manner. After reflection at the surface of the spherical concave mirror 11, the emergent rays 15, as represented, are displaced in a more or less parallel manner from coplanar plate 12. Taking account of the thickness d of the coplanar plate 12 and its reflective index n, the following amount is obtained for the focus displacement z o  of rays close to the axis: 
     
         z.sub.o =d(n-1/n)                                          (1) 
    
     Thus, due to the arrangement of correction plate 12, the focus for rays close to the axis is no longer at r/2, but at r/2+z o . This point, designated by F must now be considered as the focus of the concave mirror/correction plate system. 
     The reflected rays 15 of the abaxial incident rays 14 generally do not precisely intercept the optical axis 13 at point F, but instead are displaced by an amount Δz relative to F on the optical axis 13. As a function of the angle φ of the particular incident ray 14 the relationship between Δz and φ is as follows: ##EQU1## It is now possible, for example, by means of the Chebyshev approximation, to fix an absolute error below a predetermined value for the entire useful angle range 2·φ max . 
     It is stressed that the considerations based on FIG. 1 apply to an incident parallel beam. However, the corrective action of the coplanar plate 12 also applies for the reverse direction of the rays, i.e. 15 are rays coming from a central light source and 14 is a parallel light beam emanating from correction plate 12. 
     FIG. 2 shows a view in the direction of the line II--II of FIG. 1, whereby however a slight modification has been made in that the parallel beam now no longer strikes concave mirror 11 parallel to optical axis 13, but now does so at an angle thereto. The coplanar plate 12 is displaced relative to optical axis 13 in such a way that the parallel beam 14 reaches concave mirror 11, without traversing the same. Only the reflected beam 15 passes through the coplanar plate 12. 
     According to the preferred embodiment shown diagrammatically in FIG. 2, the emergent beams 15 in the projection of FIG. 2 no longer strike the coplanar sides of plate 12 in a perpendicular manner, but at an angle φ of &gt;45°. By suitably choosing the angle ψ the formation of a curved scanning line on concave mirror 11 resulting from the inclination of parallel beam 14 can be substantially eliminated. The direction of the rays can also be reversed without difficulty in the case of FIG. 2, without impairing the corrective action of plate 12. 
     Relative to FIG. 3 it is shown what is meant by the spherical aberration ε. If it is assumed that a light ray passes from focus F of the system to a particular point on concave mirror 11, which in FIG. 3 is assumed as the extreme outermost point corresponding to the maximum useful angle φ max , then ε designates the angle of emergent light beam 15 formed by the latter with a direction parallel to optical axis 13. On the basis of this consideration, it is clear that the displacement of the focus Δz defined relative to FIG. 1 corresponds with the spherical aberration ε. In other words, an incident light beam 14 directed parallel to the optical axis 15 and striking a particular point of concave mirror 11 undergoes a displacement of the focus Δz, whereas a light ray passing from focus F to the same point on concave mirror 11 after reflection forms an angle ε with the direction of the optical axis 13. 
     Compared with the embodiment of FIG. 1, FIG. 3 shows two further modifications. Firstly, the coplanar plate 12 comprises a total of four layers 12a, 12b, 12c, 12d, having different Abbe numbers in order to eliminate any chromatic aberrations. For reasons of simplicity, the layers are shown by straight lines, but in actual fact the boundary layers within the correction plate are convex. 
     In addition, the coplanar plate 12 according to FIG. 3 has inclined edges 16, so that overall a trapezoidal shape is obtained in the view of FIG. 3. The arrangement and direction of the inclined edges 16 are such that the outermost rays of the useful angular range 2φ max  are just within the coplanar plate 12. 
     It can be gathered from the preceeding description that the maximum useful angle range 2φ max  which is possible according to the invention is 90°, because at an angle of φ max  &gt;45° the parallel rays reflected on the concave mirror would no longer be reflected into correction plate 12. When dimensioning the coplanar plate 12, it must be ensured that the light rays 15 which have entered the plate after reflection at concave mirror 11 pass out of plate 12 on the same side of the optical axis 13 as that on which they entered it. 
     The optical apparatus according to the invention is preferably used with a measuring light curtain 22 according to FIG. 4. According to FIG. 4, a light source 18, which preferably comprises a laser, illuminates via an optical system 23, a mirror wheel 17 on which is projected an image of the light source. As can be gathered particularly clearly from FIGS. 7 and 8, the correction plate 12 according to the invention is arranged between mirror wheel 17 and the spherical concave mirror 11. Plate 12 has the trapezoidal shape explained relative to FIG. 3 and in the view of FIG. 7 is tilted at an angle of ψ&gt;90° relative to the light rays passing through it in order to eliminate the above-mentioned astigmatism. Without this tilting of the correction plate, on rotating the mirror wheel 17 by means of a motor 24 a curved scanning line 25 would form on concave mirror 11 and is indicated by dotted lines in FIG. 8. By a suitable choice of angle ψ a straight scanning line 26 can without difficulty be formed on concave mirror 11. 
     Concave mirror 1 reflects the light rays passing through correction plate 12 and directs them as a parallel emergent beam, for example onto a retroreflector 19 which can be constructed as a triple reflector. In this case, the light is reflected back onto itself and is diverted to a photoelectric receiver 21 by means of concave mirror 11, coplanar plate 12, mirror wheel 17 and a beam splitter 20. 
     In light curtain 22 is performed the measurement of objects located there by the scanning beam technique, i.e. a scanning beam preferably produced by a laser 18 is moved backwards and forwards parallel to itself over light curtain 22, as indicated by arrow 27. If it is assumed that the migration rate of scanning beam 15 is known, photoelectric receiver 21 supplies a signal, whose duration has a known relationship with the path displacement of the light ray. If an obstacle is placed in the measuring circuit, the measurement ray is interrupted for a short time, which is dependent on the thickness of the object. On the basis of this time, it is possible to determine the thickness of the obstacle, taking account of the migration rate of the scanning beam. In addition to this analogue evaluation, a digital evaluation is also possible, by using in per se known manner a timing scale and a further photoelectric cell, which are not shown. This not shown timing scale is preferably located at the light exit of the apparatus and is scanned by part of the scanning beam. By means of further optical aids and a second photoelectric cell, it supplies electrical pulses, which are proportional to the path displacement of the scanning beam. If an obstacle is located in the measuring area, the scanning beam is interrupted in accordance with the thickness of the obstacle, whereby the number of pulses during the interruption of the scanning beam is a measure of the width of the obstacle. 
     In the case of such measuring light curtains, the light ray must always be precisely parallel to the optical axis. This is particularly important if the object to be measured can have a random position between the vector and apparatus. To achieve this with a spherical concave mirror, the position of the mirror wheel 17 and the parameters of the coplanar plate 12 are according to the invention coordinated with one another. 
     Firstly, the centre of rotation of the mirror wheel is displaced relative to the centre M of concave mirror 11 in the direction of said mirror by a distance 
     
         r/2-a/2+k 
    
     Constant k can be selected in such a way that a spherical aberration ε=0 is obtained not only for rays which are close to the axis but also for those which are at a certain distance from the optical axis. The error curve I in FIGS. 5 and 6 show how the constant k can be selected in an optimum manner. For the complete useful angle range φ max  the absolute error should not for example exceed 10 -1  (FIG. 5) or 10 -2  (FIG. 6). FIG. 5 shows an example for a relatively large useful angle range of 13° and FIG. 6 for a relatively small aperture of about 6°. Both diagrams only give half the useful angle range, the other half being completely symmetrical. 
     Through the additional arrangement of the correction plate according to the invention and by suitable choice of the parameters of said plate 12, it is possible according to FIGS. 5 and 6 to completely eliminate the spherical aberration not only for rays close to the axis but also for two further rays remote from the axis. By the Chebyshev approximation, the individual parameters are varied until the absolute error remains below a predetermined value, which according to FIG. 5 is 10 -3  and according to FIG. 6 10 -5 . 
     As can be gathered from FIGS. 5 and 6, the spherical abberration ε is reduced by two or three decimal powers when using the coplanar plate according to the invention compared with using a mirror wheel without a correction plate. It should be noted that the spherical aberration ε in FIGS. 5 and 6 is shown in a logarithmic scale. By means of the correction plate according to the invention, a gradual and considerable improvement is made in connection with the spherical aberration of a spherical concave mirror. 
     When rotating the mirror wheel, the reflection point which reflects the incident light beam must always be located at such a point that the ray reflected towards the concave mirror always emanates from it in a parallel manner. However, as the mirror wheel cannot be made randomly large, even with the optimum arrangement at a distance r/2-a/2+k from the 
     centre of the concave mirror, there is still a slight deviation Δx from the ideal reflection point at any time and this is represented mathematically as follows: 
     
         Δx=(r/2-a/2+k)·(1/cos φ-1)              (3) 
    
     By equating Δx and Δz and by varying the various parameters it is possible to form symmetrical error curves II according to FIGS. 5 and 6. 
     A particularly important finding of the Applicant is that for practical purposes, it is completely adequate if only two or three clearly defined points of the spherical aberration are reduced to zero, whereas it is sufficient if between these ideal points certain absolute values are not exceeded. 
     Whereas in the embodiment according to FIG. 4, a tetrahedral mirror wheel is shown, particular preference is given to mirror wheels with twelve mirrors, i.e. an angle of 30° per mirror. For manufacturing reasons, they have a maximum half useful angle range of ψ=15°. However, they must always have an even number of faces. 
     The above statements clearly show that the spherical aberration can be substantially eliminated with the correction plate according to the invention alone. A mirror wheel with a larger diameter than zero leads to a considerable further improvement. In theory, extremely large a values would be advantageous, the diameters of mirror wheels having speeds of 3000 to 30000 r.p.m. are limited to 10 to 60 mm and an absolute maximum of 200 mm due to the maximum material strength. 
     As can be gathered more particularly from FIG. 1, the thickness d of the coplanar plate 12 is firstly determined by the position of the system focus F and secondly by the maximum useful angle φ max . 
     The invention is not limited to the embodiments described and represented hereinbefore and various modifications can be made thereto without passing beyond the scope of the invention.