Evaluation of the flux incident upon the detector of an electro-optical sensor

As was mentionned above, one of the major tasks in electro-optical system design is evaluating the detector output, and before that, evaluating the flux that is incident on the detector. Computation procedures vary from one application to another, depending upon the goal of the sensor and its configuration (free space propagation, fiber optic sensor, extended or point source, image forming sensor or flux collector,...). The scope of this course is limited to two configurations in free space propagation : image forming sensors and flux collectors.

Image forming sensors are such that they spatially or angularly resolve the light source they are looking at. They are generally comprising a lens and an array of elementary detectors (picture elements or pixels) in the image plane. The field of view of each one of these is much smaller than the object. The number of elementary detectors in the array may vary from a few (coarse image) to several millions (megapixel imagery). The geometrical extent Gcapteur of the beam incident on each element is then not defined by the whole object, but by the association of the lens with each one of the elemental detectors. Following the relationship of § 2.2, the spectral flux incident upon the sensitive area of each pixel is the product of the geometrical extent of the beam along the direction seen by the pixel by the apparent spectral radiance of the scene along that direction, so that :

and hence, if the object is at infinity :

where N and Top(λ) are respectively the aperture number (ratio between the back focal length and the diameter of the entrance pupil) of the lens (in imaging applications, the lens is supposed to be « aplanetic », i.e. devoid of major aberrations) and its spectral transmittance. Ad is the sensitive area of one elemental detector.

From this point, one will compute the electrical output of the detector by means of the procedure which will be described in the next part.

The main difference between an image forming sensor and a flux collector is the fact that flux collectors do not resolve the source, while image forming sensors do : the field of view of a flux collector is larger than the source, so that it collects light from the entire source. In these conditions, the most adequate parameter for specifying the source is its intensity : the flux that is incident upon the detector is the product of the apparent intensity of the source, Iapp (evaluated at the sensor entrance pupil) by the transmittance of the lens and by the solid angle under which the entrance pupil is being seen from the source :

Since the field of view of a flux collector is larger than the source, that means that the sensor is also observing some part of the surrounding background. The background being an extended object (its solid angle is much larger than the field of view of the sensor), a flux collector behaves on the background like an image forming sensor : hence, the background must be specified by its apparent spectral radiance Lapp,fond , inside the field of view of the sensor. If the source is very tiny as compared to the sensor field of view, and if the object is far away (« at infinity »), the total spectral flux incident upon the detector is the following :

Let us mention a third configuration, which occurs quite often in astronomy, and in which the sensor is an image forming sensor that observes point sources (stars) with an array of elemental detectors. Even though the objects are point sources and their images tiny (the optical instrument is diffraction limited), the flux from a star that enters the lens may not be collected entirely by only one detector, but spread over several of them.

In that case, one must evaluate the percentage of flux from a star which, having gone through the lens, is incident upon each of the corresponding detectors. The specification of the lens answering that problem is its Point Spread Function (PSF), which is the relative irradiance distribution in the image plane whenever the source is a point. As for the flux contribution from the background, it remains unchanged with respect to the previous result.