The Plane FPI as a Transmission Filter

In laser spectroscopy etalons are mainly used as wavelength-selective transmission filters within the laser resonator to narrow the laser bandwidth (Sect. 5.4). The wavelength Aj or frequency for the transmission maximum of m order, where the optical path between successive beams is As = mA, can be deduced from (4.47a) and Fig. 4.36 to be 

however, that this is only true for A 1 and infinitely extended plane waves where the different reflected partial waves completely overlap. If the incident wave is a laser beam with the finite diameter D, the different reflected partial beams do not completely overlap because they are laterally shifted by b = 2dtany5cosa (Fig.4.40). For a rectangular intensity profile of the laser beam the fraction b/D of the reflected partial amplitudes does not overlap and cannot interfere destructively. This means that even for maximum transmission the reflected intensity is not zero but a background reflection remains, which is missing in the transmitted light. For small angles a one obtains for the intensity loss per transit due to reflection [4.30a] for a rectangular beam profile 

For a Gaussian beam profile the calculation is more difficult, and the solution can only be obtained numerically. The result for a Gaussian beam with the radius w (Sect.5.3) is [4.30b] 

A parallel light beam with the diameter D passing a plane parallel plate with the angle of incidence a therefore suffers, besides eventual absorption losses, reflection losses which increase with and which are proportional to the ratio (d/D)2 of the etalon thickness d and the beam diameter D (walk-off losses). 

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