Beam expansion standard concepts re-interpreted Ulrike Fuchs (Ph.D.), Sven R. Kiontke asphericon GmbH Stockholmer Str. 9 07743 Jena, Germany Tel: +49-3641-3100500 Introduction Everyday work in an optics laboratory would be unthinkable without beam expansion systems 1. They serve to optimally adjust the beam cross sections between the light sources (e.g. lasers) and the subsequent optical elements. Precise illumination of the optically effective surfaces is essential, especially for beam shaping and focusing with high numerical apertures. The most widespread basic principles of afocal beam expansion systems are based on the telescopes of Kepler and Galileo. In the following article, the various modes of action are revealed in more detail and new approaches based on the use of aspherical surfaces are presented and discussed. Basic principles of beam expansion The simplest principle of a beam expansion system is the combination of two convergent lenses with different focal lengths, which corresponds to the configuration of a Kepler telescope. The enlargement or reduction of the beam cross section results from the relationship between the focal lengths as per =. (1) The total length of this optical system is primarily determined by the distance between the lenses, which can be estimated using the sum of the focal lengths, = +. (2) If the construction length needs to be shortened while retaining the enlargement, the first convergent lens (left in fig. 1a) can be replaced with a divergent lens, resulting in a construction similar to that of a Galileo telescope. Fig. 1: Illustration of the course of a beam for a Kepler (a) and a Galileo (b) telescope for ten times enlargement, realized with a convergent lens with a long focal length (f 2 =200mm) and a convergent (f 1 =20mm)(a) or divergent lens (f 1 =-20mm)(b) with a short focal length. 1 For the benefit of clarity, this article only makes reference to beam expansion and enlargement. All aspects discussed also apply to the inversion of the course of the beam for beam reduction and diminishment.
In addition, the sign of the enlargement changes, whereby the beam profile is no longer subject to point reflection during enlargement. Figure 1 shows both a Kepler (a) and a Galileo (b) telescope for ten times enlargement, realized with a convergent lens with a long focal length (f 2 =200mm) and a convergent (f 1 =20mm) or divergent lens (f 1 =-20mm) with a short focal length. Different implementations of a Galileo telescope In order to be able to further shorten the construction length while retaining the enlargement, according to equations (1) and (2), the focal lengths f 1 and f 2 should be scaled correspondingly with a factor. The same enlargement of M=10, for instance, can also be achieved with the focal lengths f 1 =150mm and f 2 =15mm. If only spherical lenses are used in this kind of configuration, spherical aberrations soon become noticeable, which reduce the quality of the wavefront for the expanded beam. Figure 2(a) shows the resultant wavefront as a cross section. This comparison clearly shows the compromise between the achievable wavefront quality and minimal achievable construction length when using spherical lenses with a certain minimal requirement on the quality of the wavefront, it is not possible to go below a certain construction length using this mode of action. Fig. 2: Illustration of three different implementations of a Galileo telescope for the enlargement M=10, realized using the focal lengths f 1 =150mm and f 2 =15mm, with a (a) spherical convergent lens, (b) achromatic doublet, (c) aspherical convergent lens. The gradual reduction of the wavefront aberrations by several orders of magnitude is very easy to understand using the cross-section images on the right. The use of achromatic groups counteracts this effect, as can be seen in figure 2(b). This shows a Galileo telescope for the same enlargement M=10, in which the convergent lens has been replaced by a doublet with f 2 =150mm. The spherical aberrations can thus be reduced significantly while retaining the construction length. This effect can be increased further by adding additional optical surfaces. An alternative approach is to aspherize one of the lens surfaces, whereby the spherical aberrations can be reduced to a minimum in line with the working principle. The shape of the lens surface differs from that of a sphere and can be described by
zh= + A h (3) In figure 2(c), such a beam expansion system for M=10 and f 2 =150mm is shown in comparison to the approach using an achromatic lens. In the selected example, the last optical surface (on the right) has an aspherical design. If this kind of system is used for beam expansion, it is usually optimized for one, or, in the case of using a doublet, for two, wavelengths as regards the shape of the optical surfaces and the lens spacing. Using the system for another wavelength thus inevitably leads to a greater wavefront error and an additional divergence. By slightly altering the spacing of the lenses, the afocal system can be reproduced and thus the residual divergence and the spherical aberrations can be minimized. Occasionally, a targeted misalignment of the formerly afocal system is also desirable in order to produce or compensate for a defined residual divergence. Symmetrical zoom systems All beam expansion system versions discussed so far are designed for a set enlargement, which is determined by the relationship between the focal lengths (equation 1). If additional flexibility with regard to the enlargement level that can be set is required, an additional optical group must be added to the system. The simplest system is thus a symmetrical zoom system, consisting of three individual lenses (convergent lens - divergent lens - convergent lens[1]), of which two have to be able to change their position in order to ensure an afocal beam course for every enlargement level. The fundamental configuration of such an optical system is shown schematically in figure 3. Fig. 3: Example to show the mode of action of an afocal zoom system consisting of convergent lens - divergent lens - convergent lens. To alter the beam expansion, two of the lenses must always be moved, as shown by the arrows. The third is fixed in position. The resultant wavefront aberrations also vary with the respective position of the lenses in the system. The illustration on the right is purely intended for placement. The previously discussed methods of improving the achievable wavefront quality also apply here - the increase in the number of optical surfaces e.g. by using a doublet or the aspherization of one of the optical surfaces. Depending on the requirement, such an optical system can be as complex as one likes, whereby in turn the requirements on the mechanical fitting also increase, particularly if movable optical groups are included.
Monolithical beam expansion systems Monolithical beam expansion systems take a slightly different approach. In terms of the mode of action, they correspond to the Galileo telescope, although they consist of only one optical element - a meniscus lens, which means both of the optically effective surfaces possess a common center of curvature. The principle behind has already been known for some time, although they produce severe spherical aberrations in their original design with two spherical surfaces, and can thus only be used for very small incoming beam diameters and very small enlargements. Figure 4 shows an example of such a lens. Fig. 4: Two monolithic beam expansion systems are shown, a) with spherical surfaces and (b) with a (convex) aspherical surface, for an enlargement of M=2. The incoming beam diameter is 5mm in (a) and 10mm in (b). Nevertheless, the resulting wavefront errors for the aspherical solution are three times smaller. These optical elements become very interesting when one of the two surfaces is aspherized. In line with the mode of action, this enables the spherical aberrations to be corrected and an afocal system to be realized, even for large incoming beam diameters. The improvement in the optical properties is clearly visible in the comparison in figure 4. The enlargement corresponds in both cases to M=2, whereby the incoming beam diameter is decreased by a factor of 2. One of the most exciting questions in this context is, of course: how large is the maximum enlargement that can be achieved with such an individual element? This is estimated using the paraxial enlargement =1+! # #, (4) whereby n is the refractive index of the glass, r the radius of the concave side and d the center thickness. If glass is chosen as the material, the refractive index in VIS is 1.4 < n < 2.1 and thus the corresponding factor approximately between 0.3 and 0.5. Accordingly, the contribution of the two summands is significantly determined by the relationship between the center thickness d and radius of curvature r of the concave surface. Of course, in principle, a very large center thickness could be chosen, although this does not make practical sense. As a result, a limit is set on an aspect ratio of center thickness to diameter of 1 (e.g. center
thickness = diameter = 25mm) for this estimation.the concave radius also has a lower limit of approx. 8mm; if one chooses a smaller radius, one places significant unnecessary limitations on the free aperture of the lens. This produces an optimum in the maximum individual element enlargement of M=2. If these considerations are extended to semiconducting materials for use in IR, enlargements of up to M=3.5 are possible. Cascade systems for beam expansion based on monolithic systems As shown, the individual element enlargements of monolithic Galileo telescopes are relatively small due to the limitation in center thickness. However, as these are afocal beam expansion systems, they can be connected in series to successively enlarge the incoming beam one after the other in the beam course (figure 5). Fig. 5: Illustration shows cascade systems for beam expansion based on monolithic individual systems, (a) 10.5-times enlargement (b) 21-times enlargement, (c) 9.3-times enlargement. The systems (a) and (b) differ by the additional element with M=2. When transferring from (b) to (c), the orientation of the last element with M=1.5 has been inverted. This opens up completely new opportunities. With just three of these elements, one can enlarge by 8 times, with five elements even 32 times. If only individual elements with M=2 are used, the increments of the possible enlargements are very approximate at M=2, 4, 8, 16, 32... This is an optimal solution if one requires strong enlargement with minimal space used and a high wavefront quality. If, however, finer increments between the enlargement levels are desired, it is necessary to introduce other individual element enlargements, which also lie very close together. Two lower levels are offered here at M=1.5 and M=1.75 especially for the version in glass. Due to the afocal dimensioning of the individual elements, the meniscus lenses can be oriented in the course of the beam however one likes, as shown in figure 5(c). This means, when combined, there are not only three but actually six individual element enlargements available, which significantly increases the combinatorics level. If, for instance, there is one element available for each basic enlargement, this produces 13 possibilities for the overall enlargement with just these three meniscus lenses. If one pushes these
combinatorics further with additional elements, it can be seen that an above-average number of combination options are opened up by certain lens groupings. What is common to all these groups is the presence of one element each with M=1.5 and M=1.75 and an increasing number of elements with M=2. Figure 5(b) shows an example of an overall enlargement of M=21 consisting of five individual elements. Using the specific group shown there (1x M=1.5; 1x M=1.75; 3x M=2), it is possible to realize 62 different enlargement levels with the maximum at M=21. The use of monolithic beam expansion systems in a cascade construction, as shown in figure 5, involves significantly more optical surfaces than the optical systems shown in figure 3. What s more, every other surface is aspherical! In order to be able to implement such a cascade system for flexible beam expansion in practice, very high surface qualities for the individual elements are required. To prevent any restrictions on combinatorics for later use, each individual element must be significantly better over the whole free aperture than the diffraction-limited requirement, i.e. wavefront error RMS< λ/14. For the Ti:sapphire laser wavelength of 780nm, this means, for instance, RMS < 55nm, for λ=532nm it is even just RMS < 32nm. If the center thickness and the decentration of the surfaces are also produced very precisely for these requirements, this system is completely adjustment-free, as all adjustment degrees of freedom in the conventional system in figure 3 are already set at an optimum as a monolithic element during manufacture. This means the attachment of additional monolithic elements to change the enlargement level also takes place completely adjustment-free and is thus quick and easy. Summary Standard afocal beam expansion systems, which are based on the basic principles of Kepler and Galileo telescopes, can be improved in terms of wavefront quality by using additional optical surfaces or aspherizing them. Due to their mode of action, it is not possible to shorten the construction length of such a system however one likes while also retaining the wavefront quality. A new approach, based on the use of monolithic beam expansion elements, which can be used as a cascade, achieves construction lengths much smaller than those of conventional systems. Furthermore, this system is diffraction-limited, and offers a high level of flexibility and continuous extensibility using new elements. Literature [1] H. Gross, Handbook of Optical Systems, volume 4, p.470f., 2008