### Basics of conoscopy I

Figure 1: The incoming elementary parallel beams are focused to form spots in the back focal plane of the lens. The location of these spots is a direct function of the direction of propagation of the incoming beam (i.e. angle of inclination and and azimuth).

The raytracing of Figure 1 illustrates the basic concept of conoscopy: transformation of a directional distribution in the front focal plane into a lateral distribution (named: directions image) appearing in the back focal plane (which is curved in most of the cases). The incoming elementary parallel beams (illustrated by the colors blue, green and red) are converging in the back focal plane of the lens with the distance of their focal point from the optical axis being a (monotonous) function of the angle of beam inclination.

This transformation can easily be deduced from two simples rules for the thin lens:
• the rays through the center of the lens remain unchanged,
• the rays through the front focal point are transformed into parallel rays.
In the above illustration all rays hitting the lens with the same direction of propagation are focused to the same spot in the back focal plane. It is obvious however that rays with different directions of propagation may emerge from different locations in the fron focal plane.

The object of measurement is placed into the front focal plane of the lens. In order to select a specific area of interest on the object (i.e. definition of a measuring spot) an aperture is placed on top of the object. In this configuration only rays from the measuring spot (= aperture) hit the lens.

The image of the aperture is projected to infinity while the image of the directional distribution of the light passing through the aperture (directions image) is generated in the back focal plane of the lens.

Figure 2: Placement of an aperture in the front focal plane ensures that all elementary parallel beams emerge from the same area on the object.

When it is not considered appropriate to place an aperture into the front focal plane of the lens, the selection of the measuring spot can also be achieved by using a second lens.

Figure 3: Selection of the measuring spot by an aperture placed in the image of the object.

An image of the object (located in the fron focal plane of the first lens) is generated in the back focal plane of the second lens. The magnification of this imaging is given by the ratio of the focal lengths of the lenses L1 and L2, M = f2 / f1.

Figure 4: Schematic raytracing of a conoscopic system: formation of the directions image and imaging of the object.

A third lens transforms the rays passing through the aperture (located in the plane of the object-image) into a second directions image which may be analyzed by an image sensor (electronic camera).

The functional sequence is as follows:
• the first lens forms the directions image (transformation of directions into locations),
• the second lens together with the first projects an image of the object,
• the aperture allows selection of the area of interest (measuring spot) on the object,
• the third lens together with the second images the directions image on a 2-dimensional optical sensor.

This simple arrangement is the basis for all conoscopic devices. It is not straight forward hoewever to design lenses that meet the following requirements at the same time:
• maximum angle of light incidence from object as high as possible (e.g. 80°),
• diameter of measuring spot in the range of some millimeters,
• achromatic performance for all angles of light incidence,
• minimum effect of polarization of incident light.
Design and manufacturing of such lens systems requires assistance by numerical modelling and a sophisticated manufacturing process.