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Throughput

Throughput is one name for the optical invariant that is used for the product of the pupil area and the solid angle subtended at this pupil by the window area. This means that the interaction between entrance pupil and the entrance window are the same as for the exit pupil and exit window. Stated mathematically,

In this expression, EP and EP' as well as EW and EW' are the areas of the respective pupils and windows. S is the spacing between EP and EW, and S' is the separation of EP' from EW'. Figures 1 and 2 show the arrangement for our two-lens example.

Figure 1 Throughput invariance.

Figure 2 A three-dimensional look at throughput invariance.

Throughput has been called by many different names, optical extent, light gathering power, and area-solid-angle product. Etendue is the French term, and “geometrical light flux” is the translation of the German term Geometrischer Lichtstrom.^1,2

What is really important is the choice that is available with this invariance. One can calculate the throughput using information from the object side or from the image side. When the image is at infinity, as in the case of a collimator, dimensions referring to the entrance pupil and entrance window can be used because S is a finite quantity, while S' is infinite. Similarly, when the object is at infinity, values for EP', EW', and S' from the image side can be used.

It must be added that pupils and windows are not always circular. In fact, for infrared systems, exit windows are mostly rectangular in shape. A typical example is the staring focal plane array. In a Cassegrain telescope, the pupils are doughnut-shaped, due to the central obstruction of the secondary mirror. What counts is not the shape but the areas of the pupils and the windows.

References

1. W. Steel, “Luminosity, throughput, or etendue?,” Applied Optics 13, (1974), page 704.
2. K. Räntsch, Die Optik in der Feinmesstechnik, Carl Hanser Verlag (1949), page 53.