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• # Chapter 4 - Lenses and Mirrors

In optics we can generally claim that the fundamental differences between lenses and mirrors concern the ability to transmit or reflect light.

• The lenses are affected by chromatic aberration: their behavior depends by the crossing radiation wavelength. The mirrors are achromatic: they reflect in the same way each wavelength within the range of the spectral reflectivity of the surface treatment.
• Even if corrected, lenses always suffer from mono-chromatic aberrations (other than those chromatic). On the contrary mirrors can be completely free from aberration effects. However the absence of aberrations can be achieved only under certain conditions: a parabolic mirror can perfectly collimate a point source in axis, placed directly on the focal plane or it may converge in a point of the focal plane, the radiation, coming from a source at infinite distance and propagating parallel to its optical axis. An ellipsoid mirror can form a perfect image of a point source coincident with one of its foci. A spherical mirror can reflect, without aberrations, a point source placed on the curvature center on itself (whatever the diameter of the mirror is). Note that the real sources are never point sources. In order to keep negligible the aberration effects, the size of the object must be small compared to the mirror curvature radius and the angle at which the mirror collects the radiation must be small as well, the effective mirror diameter must be small with respect to all its other dimensions.
• In order to correct aberration effects, it is possible to project a lens playing with the following parameters:
2. the thickness;
3. the shape, eventually aspheric, of its optical surfaces;
4. the refractive index (in case of isotropic glass);
5. the gradient of the refractive index (along axis and along radius);
• The refractive index of the lenses glass can uncontrollably vary both along radius (GRIN) and along axis (AGRIN). The radial refractive gradient forces rays to travel a curved path, this effect is mainly exploited in optical fibers (but also in small cylinders, employed for transport of laser beams or small images – selfoc). The axial gradient, achievable by melting layers of different glasses, improves the aberration correction (lenti Gradium) by allowing differences between the optical paths of peripheral and central rays.
• A lens is transparent within the transparency range of the material. Similarly, a mirror reflects radiation within the reflectivity spectral range of the reflecting treatment of its surface.
• The most common material of which are made the lenses for visible radiation is borosilicate glass, transparent in the interval ranging from 0.4 to 2.5 µm. There are very few materials transparent to the infrared (IR) and ultraviolet (UV) spectral ranges and often they are very expensive and not transparent to visible light. Moreover there are not glues transparent to IR radiation: two infrared lenses cannot be glued as it is possible for visible lenses.

• The total transparency of lenses is limited, not only by their material, but also by the reflection of their surfaces that, for a refractive index of about 1.5 (i.e. for a common glass), is roughly 4% for each surface (for normal incidence and more for larger angles). Germanium has a refractive index of 4 in the infrared range and each surface reflects 36% of the incident radiation. These losses can be, also strongly, reduced by using anti-reflex treatments (can be reduced to less than 0.1%).

The mirror reflectivity is limited by the reflectivity of their surface treatment.

Dielectric reflective treatments can exceed the 99% and have very low absorption losses, but they are angularly and spectrally strongly limited; conversely metallic treatments are less reflective and can be strongly absorbing, but they are angularly unlimited and capable to reflect over a spread spectral ranges. A metal coating that allows to split in equal percentages the reflected and the transmitted beam, usually presents an absorption of roughly 20%. Metallic coating that better reflect in the medium IR range are made of gold. The reflectivity that can be obtained by a surface treatment is, generally, higher than the reflectivity obtainable by polishing the solid material.

All these considerations must be taken into account in choosing the most proper lenses and mirrors for the different applications.