Radiometric properties of an optical instrument

In most electro-optical devices, the optical sensor does not consist in a detector alone, but is made up of some (collecting) optics associated with a detector : this association makes possible to select a particular zone in space and to eliminate the rest, whereas the use of a detector alone does not allow to discriminate the incident the rays with respect to their incidence angle inside the entire half-space in front of the detector. The association of a lens with a detector also allows to optimize the flux collecting capability of the sensor because the area of a lens may be made much larger than that of a detector.

The association « lens + detector » operates a double selection among incident rays. Firstly, it is clear that the photons from the source that do not enter the lens will never reach the detector. The optical diaphragm that is responsible for this first selection is called the aperture stop. It is conventionally defined for the beam that propagates along the axis of the sensor (on axis point source). The entrance pupil, that defines the size of the beam entering the lens, is the image of the aperture stop in the object space of the optics. The exit pupil, or image of the aperture stop in the image space, defines the size of the beam at the outset of the lens.

In most applications, the second selection among the rays is done by the detector itself : among all the photons having gone through the lens, only a fraction succeeds in reaching the sensitive area of the detector : these are the ones originating from the part of the object space which is called the object field of view of the sensor. It is why, in this configuration, the sensitive area of the detector is called the field stop of the sensor.

If the object, or the source, is at a finite distance from the sensor, the field of view is usually specified or measured in terms of linear dimensions at the object plane. For example, the field of view of a microscope will be said to be 1 mm in diameter if that is the size of the area being observed. If the object (or the source) is far away, the field of view will be preferably specified, or measured, in terms of angular dimensions, usually along two axes, horizontal and vertical ; if that is the case, one will say, for instance, that the field of view of a camera is 9° by 16°. The half angle θ of the field of view along either of these two axes is given by the following relationship, where a is half the detector size along the corresponding axis, and f' the focal length of the lens:

In some cases, there may happen that the detector also collects some light from regions that are theoretically outside the field of view of the sensor, because of scattering by particles along the path, or from reflections by mechanical structures or by optical surfaces inside the sensor. The resulting light must be taken into account when evaluating the incident flux upon the detector (Figure 12), not as part of the useful signal, but as stray light.

The shape of a beam, which is defined by the field and aperture stops of the sensor, leads to a basic quantity called the geometrical extent of the sensor. In order to seize what it means, let us consider an E-O sensor that is observing a far away, on-axis, source inside a small solid angle : for this purpose, it is made up of a small detector, of sensitive area Sdét, at the back focus of a lens, of focal length f. In the object space, the geometrical extent of the sensor may be defined, as that of the pencil of light propagating along the sensor axis inside the solid angle Ωdét =Sd/f2   and passing through the entrance pupil (of area Sopt ) : in the image space, the geometrical extent is that of the beam passing through the exit pupil and converging onto the detector, so that :

or, since

where N is the aperture number of the lens (N = f'/De )

By applying geometrical optics rules, one shows that the input and output geometrical extents, G1 G2 of a pencil of light traversing an optical component without being limited by it, are related to each other by :

where n1 and n2 are the refractive indices of the initial and final media.

Whenever the initial and final media of the lens are identical (which happens in a vast majority of EO sensors, the common medium being air), the input and output pencils of light have got the same geometrical extent. One notable exception is underwater imaging systems, where the object medium is water and the image medium is air. Furthermore, if the lens is lossless, input and output fluxes are identical, which means that a perfect optical system that does not limit the geometrical extent of an entering beam delivers an output beam of same extent and radiance as the input beam if the initial and final media are the same. This is called the radiance conservation theorem by optical systems (beware : this theorem holds only for perfect systems, without loss and having identical input and output media). If the lens is not perfectly transparent, then the ouput radiance is the product of the input radiance by the transmittance of the lens.

This theorem applies for example to simple components such as mirrors (or specular surfaces) for which, by definition, the final medium is the same as the initial one: the radiance of the reflected beam along the direction of geometrical optics (θ=-θ', et φ=φ') is the product of the input radiance Linc (λ,θ',φ') by the reflectance of the mirror :

The BRDF (defined in part 3.a) of a mirror or, more generally, of a specular surface, is a Dirac distribution : it is naught in all directions of observation, except along the direction that correspond to geometrical optics, i.e. that is symetrical of the direction of illumination with respect to the normal.