1. Introduction

Some years after his official retirement, the writer was invited to collaborate with some Fellows of King’s College, Cambridge, in a scheme to provide improved illumination for orchestras and choirs performing in the Ante-Chapel. Illumination of $500\text{\hspace{0.17em} lx}$ was required over a floor area of $11\mathrm{m}\times 13\mathrm{m}$, via six existing apertures, each about $45\mathrm{mm}$ in diameter, in the stone fan-vault ceiling $24\mathrm{m}$ above.

2. Background

One prototype optical system has been assembled, and this was first used in the Chapel roof space in December 2009. It uses a xenon arc lamp, type PE1500 13F, manufactured by Perkin-Elmer [1, 2], which emits about $100\mathrm{W}$ of visible light and $200\mathrm{W}$ of infrared radiation. A cold mirror was used to reflect the visible light onto the end of a light guide of BK7 glass. The infrared radiation was transmitted by the mirror and was absorbed on the end of a copper rod $50\mathrm{mm}$ in diameter and $120\mathrm{mm}$ long, connected to a substantial heat sink approxi mately $250\mathrm{mm}\times 200\mathrm{mm}\times 40\mathrm{mm}$ supplied by RS Components [3]. This unit delivered $100\mathrm{lx}$, averaged over the $11\mathrm{m}\times 13\mathrm{m}$ area. To achieve this, the illumination of the top of the light guide, $11\mathrm{mm}\times 13\mathrm{mm}$, must be greater than ${10}^8\mathrm{lx}$.

Close examination of the other five holes in the ceiling recently showed that these were too close, either to a brick wall forming the north or south boundary of the roof space or to an adjacent timber roof truss, to allow the same arrangement to be used.

The writer was therefore required to design an optical relay system that would accept light beyond the lamp focus, which was diverging in a cone up to $22°$ from the axis of the lamp, and refocus it at close to unit magnification, on the end of the vertical light guide. Space was needed for a hot mirror, to reflect the infrared flux away from the lamp window and the lenses, and to transmit the visible light. Beyond the lenses, an Amici prism was needed to reflect the horizontal beam vertically downward. The problem was augmented by the diffuse nature of the lamp focus: almost all of the light passed through a patch $12\mathrm{mm}$ in diameter at the lamp focus, but this had no sharp boundary. It was agreed that the source should be modeled by assuming that 50% of the light passed through a point on the lamp axis and at the nominal focus, 20% through each of two points $5\mathrm{mm}$ closer to and farther from the lamp, 4% through points $10\mathrm{mm}$ from the nominal focus, and 1% from points $15\mathrm{mm}$ from the nominal focus.

To minimize cost and possible delays in construction, off-the-shelf items were selected wherever possible from the Comar Instruments catalog [4]. Four designs were investigated. For the purposes of determining the asphericity needed, a wider cone of light was assumed to diverge from the nominal position of the lamp focus and to fill the clear diameter of the aspherized lens surface. A single-lens design was found to suffer spherical aberration so great that both surfaces needed to be given $1.75\mathrm{mm}$ aspheric depth over an aperture of $63.5\mathrm{mm}$. A design with two plano-convex lenses, plane faces outward, needed just under $1\mathrm{mm}$ asphericity on one surface, with an aperture of $50\mathrm{mm}$. Adding a biconvex lens in the center of this system, and using weaker plano-convex lenses, reduced the asphericity needed to $0.29\mathrm{mm}$. A fourth design was examined to see whether the light losses would be tolerable if all surfaces were spherical or plane. This design appeared to be acceptable, so to avoid the delays likely in making relatively deep aspheric surfaces (one for a prototype, five more in total), this design was adopted and the components required for one unit were ordered and received. Nevertheless, in case of unforeseen problems, the writer decided to investigate methods of testing an aspheric lens as needed in the third design (asphericity $0.29\mathrm{mm}$). It was agreed that the lenses would be mounted in the Comar TubeMount system (Section 11 of [4]), which reduces the clear aperture of the lenses to $48.25\mathrm{mm}$. The details of the relay lens design are given in Table 1.

3. Principles

It is well known that a paraboloidal mirror, when tested with a point source of light placed close beside its mean center of curvature, shows a moderate amount of spherical aberration. This aberration can be compensated by a thick plano-convex lens, which relays the imperfect image formed by the mirror to a second focus at which the Seidel, third-order, spherical aberration of the mirror is corrected. A small field lens is required at the imperfect focus to fine tune the correction of higher orders of spherical aberration. This test, described by Offner [5], was a great improvement compared with all previous null tests that did not require a plane mirror as large as the paraboloid. Zonal errors and other irregularities can be revealed by a knife-edge test, wavefront shearing interferometer, or other test procedure.

In the test to be described here, we replace the paraboloidal mirror by a spherical mirror of known high quality, used well away from unit magnification (where it would produce no spherical aberration). The plano-convex lens in the Offner test is replaced here by the aspheric lens to be tested, in its water tank. The field lens remains, as in the Offner test.

Light from a small source at a position between the center of curvature of the mirror and its focus for paraxial rays will be refocused at the long conjugate focus, where rays which fall near to the edge of the mirror will be brought to a focus closer to the mirror than the rays which fall on it near to the axis. The lens has the greatest depth of material removed at a zone at 70% of its aperture, with none removed on axis or at the edge. Therefore, the curvature of the aspheric surface is greatest near to its axis, and the focus for these rays will be shorter than that of rays that pass through its outer zones. So the aberrations of the lens and mirror will tend to cancel each other, but it remains to test whether the spherical aberration of a mirror of reasonable size is great enough to compensate for that of the lens.

4. Quantitative Fitting

The writer has available a spherical mirror nearly $305\mathrm{mm}$ ($12\mathrm{in}\mathrm{.}$) in diameter and with a radius of curvature of $-696\mathrm{mm}$, which he made more than 20 years ago for tests of the convex secondary mirror of a three-mirror telescope of $0.5\mathrm{m}$ aperture [6, 7]. Ray-tracing calculations initially failed to find any arrangement that could cancel the spherical aberration of the lens when this was tested in air. The asphericity of the lens was too great.

As an experiment, two plane-parallel windows were added to the design (item 63 GQ 00, in Section 6.9 of [4]), and the air between these and the aspheric lens ($n=1.523$) was replaced by water ($n=1.3330$, $\nu =56$ at $20°\mathrm{C}$, and $\lambda =589.3\mathrm{nm}$) [8]. In some trial calculations, the final focus moved to an inconveniently large distance, because the water reduced the focusing effect of the lens, as well as its asphericity. One of the windows was briefly replaced by a lens of $63\mathrm{mm}$ aperture and $1000\mathrm{mm}$ focal length (item 1000 PQ 63, in Section 1.1 of [4]), but it has not been necessary to keep this.

A systematic search was then carried out to find if there was any pair of conjugate foci of the mirror that would create an amount of spherical aberration equal and opposite to that of the aspheric lens, now immersed in water. In the computer program, a cone of rays was generated from a parallel bundle that fell on a small paraboloidal mirror; the diameter of the parallel bundle was adjusted so that, starting from the shorter conjugate focus, the divergent rays exactly covered a diameter of $300\mathrm{mm}$ on the $\mathrm{12}\mathrm{in.}$ mirror. The spherical face of the plano-convex field lens was placed at the focus of the marginal rays from the mirror, so that when the radius of curvature of that surface was changed to make fine adjustments to the test, the rays from the margin of the mirror should not deviate and the area of the aspheric lens that was illuminated should not change. The distance of the lens, in its water cell, was chosen so that the outer rays always met its aspheric surface at $25.0\mathrm{mm}$ from the optical axis. The radius of curvature of the field lens was then adjusted to find the best null test that could be obtained with that selected pair of conjugate foci.

The calculations were repeated with different pairs of conjugate foci, which are listed in Table 2. Here the field lens remains plano-convex, although, in some preliminary calculations, there was a small improvement to be obtained by making its second surface slightly convex (radius $12\mathrm{m}$). This modification of the field lens, which is strictly plano-convex in the original Offner test, was suggested to the writer by the late D. S. Brown [9] and used in proposed tests of the primary mirror of a large three-mirror telescope [6]. Changing the shape of the “imaging” or compensating lens, as was suggested by Offner [5] as a way of improving the test, would change its Seidel spherical aberration, rather than reduce the residual higher-order aberration.

The unusually small residual zonal spherical aberration shown in the last five lines of Table 2 will be illusory unless the optical components used are of very good quality. The plane-parallel windows supplied by Comar Instruments should be flat to $\lambda /4$ (He–Ne laser light, $633\mathrm{nm}$) over 90% of their diam eter. If these can be tested against a master flat, and if one surface is found to be significantly better than the other, then the better one should face outward, and the worse one be in contact with the water. The weak lens used in some preliminary calculations is expected only to have surfaces true to one wavelength [10]. Modesty prevents the writer from making any claims for the quality of the spherical mirror.

In the real world, the writer would place the field lens at the correct distance from the spherical mirror, and the water cell at the correct distance from the field lens, then place the light source at such a distance from the mirror as to cause the marginal rays to be focused on the field lens, and also to illuminate the whole $50\mathrm{mm}$ aperture of the aspheric lens.

Complete details of the arrangement of the test are given in Table 3 and illustrated in Fig. 1. An enlarged view of a possible design of the water tank and the aspheric lens in it is shown in Fig. 2.

If plane-parallel windows larger than the lens to be tested, and of sufficient surface accuracy, are not available, the lens might be tested in a long trough, using the field lens as the entrance window and a small plane-parallel window close to the final focus. The arrangement of this test is given in Table 4. The distance from the field lens to the final focus is greater than in the first version of the test (Table 3), but the conjugate foci of the mirror are almost unchanged. Because of the substantial path length of water used in this version of the test, it may be desirable to stir the water before testing the lens to ensure a uniform temperature.

5. Discussion

The writer has, perhaps, been rather fortunate that the existing spherical mirror was not large enough, or deep enough, to be able to compensate for the asphericity of the lens when that was in air, but was able to do so when it was immersed in water. If this had been a purely theoretical exercise, it would have been too easy to make the mirror large enough to create a null test with the lens in air.

A paraboloid tangent to the sphere at its center and intersecting it at the full aperture of $300\mathrm{mm}$ has the greatest aspheric height of $0.0486\mathrm{mm}$ at $106\mathrm{mm}$ from the center. Allowing for the doubling by reflection, the greatest wavefront distortion this spherical mirror could create would be $0.0972\mathrm{mm}$, and this would require the longer conjugate focus to be at an inconveniently large distance. The asphericity of the lens, $0.29\mathrm{mm}$, results in a wavefront distortion that is reduced by the factor $(n-1)=0.523$ to $0.15167\mathrm{mm}$ when it is in air, but by $(n_g-n_w)=(1.523-1.333)=0.190$ to $0.0551\mathrm{mm}$ when the lens is immersed in water.

Other liquids, with higher refractive indices, such as paraffin oil ($n=1.44$) or glycerol ($n=1.47$) [8], might be used to extend the test to lenses with even greater asphericity, or to allow the use of a smaller spherical mirror to test this lens.

This increased versatility of the test using a water bath comes at a cost of reduced sensitivity to small surface errors, but in the application described here, i.e., a set of lenses to relay an image that is already very ill defined, this is of little importance.

A good null test can also be obtained with the aspheric lens in air with the same spherical mirror if this is used in double-pass mode. This may be arranged with the light source and a beam-splitting cube (item 16 JQ 01 in Section 5.9 of [4]) near its long conjugate focus and a small spherical mirror at the short focus, as described in Table 5, but the final focus is less than $100\mathrm{mm}$ from the lens, which might not be convenient.

It is thought that the use of a water cell or trough for optical testing as described here may be original. Others have used fluids in optics, for example, an inclined concave spherical mirror was used in a water bath as a monochromator for the ultraviolet by Harrison [11], and powdered glass in a mixture of carbon disulphide and benzene is the basis of the Christiansen filter [12, 13]. The examination of roughly shaped pieces of glass for internal strain or for striae or other imperfections can be aided by immersion in a fluid having the same refractive index as the glass [14]. There are no references in Malacara [15] to the immersion in water or any other fluid of an optical component the surfaces of which are to be tested.

I thank Wyn Evans for suggesting my name, during a lunch-time conversation with Tom White, as a possible collaborator in the scheme to improve the lighting in King’s College Chapel; George Efstathiou, while Director of the Institute of Astronomy, for forwarding that invitation; and Tom White for many stimulating discussions.

Table 1. Relay Lens Design^a

View Table | View all tables in this article

Table 2. Convergence of Null Test Arrangement^a

View Table | View all tables in this article

Table 3. Optical Arrangement of the Null Test for an Aspheric Lens^a

View Table | View all tables in this article

Table 4. Optical Arrangement of the Null Test for the Aspheric Lens Using a Water Trough^a

View Table | View all tables in this article

Table 5. Optical Arrangement of the Null Test for the Aspheric Lens in Air, Using the Spherical Mirror in Double Pass^a

View Table | View all tables in this article

 

Fig. 1 Scale drawing of the null test. M, spherical mirror; S, light source; C, center of curvature of the spherical mirror; F, field lens; W, water tank containing the aspheric lens; I, final image. Only two rays are drawn: one at the margin of the mirror M, and the other near to the edge of the illuminated part of the field lens.

Download Full Size | PDF

 

Fig. 2 Cross section of the water tank and aspheric lens in the vertical plane. V, air vent, above; D, water inlet and drain, below. O rings are needed to make the tank watertight, and to prevent damage to the glass windows when the components of the tank are held together by screws (not shown). The dotted line at the upper left side of the lens shows a sphere that departs from the lens surface by a maximum of $0.29\mathrm{mm}$.

Download Full Size | PDF

1. http://www.perkinelmer.com.

2. http://www.excelitas.com.

3. http://www.rs-components.com.

4. http://www.comaroptics.com.

5. A. Offner, “A null corrector for paraboloidal mirrors,” Appl. Opt. 2, 153–155 (1963). [CrossRef]  

6. R. V. Willstrop, “The flat-field Mersenne-Schmidt,” Mon. Not. R. Astron. Soc. 216, 411–427 (1985).

7. R. V. Willstrop, “Atmospheric dispersion compensator for a wide-field three-mirror telescope,” Mon. Not. R. Astron. Soc. 225, 187–198 (1987).

8. G. W. C. Kaye and T. H. Laby, “Radiation and optics,” in Tables of Physical and Chemical Constants, 14th ed., A. E. Bailey, ed. (Longman, 1973), p. 95.

9. D. S. Brown, Grubb-Parsons, Walkergate, Newcastle-upon-Tyne, UK (personal communication, 1984).

10. J. Vos, Comar Instruments, 70 Hartington Grove, Cambridge CB1 7UH, UK (personal communication, 14 March 2011).

11. G. R. Harrison, “Simply constructed ultraviolet monochromators for large area illumination,” Rev. Sci. Instrum. 5, 149–152 (1934). [CrossRef]  

12. C. Christiansen, “Untersuchungen uber die optischen Eigenschaften von fein vertheilten Korpen. Erste theil,” Ann. Phys. u. Chem. 23, 298–306 (1884). [CrossRef]  

13. C. Christiansen, “Untersuchungen uber die optischen Eigenschaften von fein vertheilten Korpen. Zweiter theil,” Ann. Phys. u. Chem. 24, 439–446 (1885). [CrossRef]  

14. J. Strong, “Optics: light sources, filters and optical instruments,” in Procedures in Experimental Physics (Prentice-Hall, 1939).

15. D. Malacara, “Appendix 2: Some useful null testing configurations,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 1992).