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Optical systems are characterized by two fundamental angles (shown in Figures 7 and 8).
Fig. 7  Field of view F.O.V. (Field Of View)
Fig. 7 shows the field of view of a simple optical system. The two rays shown are the "principal rays", namely those rays that, starting from the edges of the object, after crossing the center of the entrance pupil of the optical system (and, thus, the center of the output pupil), contribute to form on the images plane, the images of the points from which they started.
The image size collected by the optical system is firstly limited by the detector size (photographic film, CCD, photomultiplier, other optical instrument, etc.).
The field of view can be also mechanically limited, such as for example, by a hood. By increasing the field of view, the aberration effects usually grow; thus the edges can be much more poor than the centre of the image.
Typically the human eye is not able to directly focus the image plane (unless it is sufficiently far away), hence it needs of an auxiliary optics to move far away enough the image from the eye (eyepiece or ocular lens).
Fig. 8  Object at infinite distance, on axis.
Fig. 9  Object at finite distance, at the edge of the field of view.
Fig. 10  Object at finite distance, on axis.
In Fig. 8, 9 and 10 is shown the angle corresponding to the F # or Fno or Fnumber. Fig. 8 represents what happens when the object is positioned at infinite distance (i.e. at very large distance with respect to the lens focal length); Fig. 9 represents what happens to a point at the edge of the field of view at finite distance; Fig. 10 shows the case of a point at the center of the field of view at finite distance.
When the object is at infinite distance (or very large), the image is formed on the focal plane and at a distance, from the principal axis of the optics (and not from the last lens), equal to the focal length. In this case we can properly speak of F #, or Fno. The value usually indicated in this case, is the ratio between the focal length and the diameter of the entrance pupil. Since the amount of the collected energy is proportional to the area of the entrance pupil (and therefore to its diameter) and the image size is proportional to the focal length, the ratio between the focal length and the entrance pupil diameter is proportional to the energy density of each point of the image.
When the object is not placed at an infinite, that is at very large distance, we cannot speak of F# (as defined above). However, the ratio between the distance from the principal plane of the lens to the plane where the image is formed, and the useful optics diameter, is still an indication of the energy density of the image points. When the object is placed at a finite distance, the distance from the optics principal axis and the image plane is larger than the focal length. Consequently the angle between the peripheral rays and the optical axis is smaller, such angle must be considered in the coupling with other optical systems. If an optical system, meant to work at infinite distance with a certain F#, is used under unitary magnification conditions, then its effective F#doubles (i.e. the image distance from the optics principal axis becomes twice the focal length).
The F # indicated on most commercial objectives, refers to an object at infinite distance. In practice, when the object distance is not enough large, the real F # is greater (i.e. less bright) with respect to the value indicated for an infinite distance.