Confocal FPI

Fig.4.48a-d. Trajectories of rays in a confocal FPI (a) incident beam parallel to the FPI axis (b) inclined incident beam (c) perspective view for illustrating the skew angle (d) projection of the skewed rays onto the mirror surfaces... 

Because of spherical aberration, rays with different distances p from the axis will not all go through F but will intersect the axis at different positions F depending on p and 0. Also, each ray will not exactly reach the entrance point Pi after four passages through the confocal FPI since it is slightly shifted at successive passages. However, it can be shown [4.36,4.39] that for sufficiently small angles 9, all rays intersect at a distance p p, 9) from the axis in the vicinity of the two points P and P located in the central plane of the confocal FPI (Fig. 4.48b). 

An incident light beam with diameter D = 2pi therefore produces, in the central plane of a confocal FPI, an interference pattern of concentric rings. Analogous of the treatment in Sect. 4.2.5, the intensity 7(p, X) is obtained by adding all amplitudes with their correct phases S = So(lit/X) As. According to (4.50) we get... 

This large dispersion can be used for high-resolution spectroscopy of narrow line profiles with a scanning confocal FPI and photoelectric recording (Fig. 4.49). 

If the central plane of the near-confocal FPI is imaged by a lens onto a circular aperture with sufficiently small radius b < (Ar ) / only the central interference order is transmitted to the detector while all other orders are stopped. Because of the large radial dispersion for small p one obtains a high spectral resolving power. With this arrangement not only spectral line profiles but also the instrumental bandwidth can be measured, when an incident monochromatic wave (from a stabilized single-mode laser) is used. The mirror separation d = r - is varied by the small amount e and the power... 

Fig. 4.49. Photoelectric recording of the spectral light power transmitted of a scanning confocal FPI...

The total finesse of the confocal FPI is, in general, higher than that of a plane FPI for the following reasons ... 

The alignment of spherical mirrors is far less critical than that of plane mirrors, because tilting of the spherical mirrors does not change (to a first approximation) the optical path length 4r through the confocal FPI, which remains approximately the same for all incident rays (Fig. 4.50). For the plane FPI, however, the path length increases for rays below the interferometer axis, but decreases for rays above the axis. 

The total finesse of a confocal FPI is therefore mainly determined by the reflectivity R of the mirrors. For R = 0.99, a finesse F = n R/( — R) 300 can be achieved, which is much higher than that obtainable with a plane FPI, where other factors decrease F. With the mirror separation r = d = 3 cm, the free spectral range is 3 = 2.5 GHz and the spectral resolution is Au = 7.5 MHz at the finesse F = 300. This is sufficient to measure the natural linewidth of many optical transitions. With modem high-reflection coatings, values of F = 0.9995 can be obtained and confocal FPI with a finesse F > 10" have been realized [4.41]. 

For a given finesse F, the etendue of the confocal FPI increases with the mirror separation d — r. The spectral resolving power... 

While the spectral resolving power is proportional to U for the confocal FPI, it is inversely proportional to U for the plane FPL This is because the etendue increases with the mirror separation d for the confocal FPI but decreases proportional to I/d for the plane FPI. For a mirror radius r > /Ad, the etendue of the confocal FPI is larger than that of a plane FPI with equal spectral resolution. 

This example shows that for a given light-gathering power, the confocal FPI can have a much higher spectral resolving power than the plane FPI. 

While method (a) is often used for high-resolution fluorescence spectroscopy with slow scan rates or for tuning pulsed dye lasers, method (b) is realized in a scanning confocal FPI (used as an optical spectrum analyzer) for monitoring the mode structure of lasers. 

In Sect. 4.2.10 we saw that for a given resolving power the spherical FPI has a larger etendue for mirror separations r > /Ad. For Example 4.19 with D — 5 cm, d = 1 cm, the confocal FPI therefore gives the largest product RU of all interferometers for r > 6 cm. Because of the higher total finesse, however, the confocal FPI may be superior to all other instruments even for smaller mirror separations. 

A confocal FPI shall be used as optical spectrum analyzer, with a free spectral range of 3 GHz. Calculate the mirror separation d and the finesse that is necessary to resolve spectral features in the laser output within 10 MHz. What is the minimum reflectivity R of the mirrors, if the surface finesse is 500 ... 

An incident light beam with diameter D = 2pi therefore produces, in the central plane of a confocal FPI, an interference pattern of concentric rings. Analogous of... 

The free spectral range 8v, i.e., the frequency separation between successive interference maxima, is for the near-confocal FPI with p d... 

If the central plane of the near-confocal FPI is imaged by a lens onto a circular aperture with sufficiently small radius b < only the central interference... 

A confocal FPI with r = d = 5 cm has for A = 500 nm the etendue U = (2.47 X 10 /F ) cm /sr. This is the same etendue as that of a plane FPI with d = 5cm. and D = 10cm. However, the diameter of the spherical mirrors can be much smaller (less than 5 mm). With a hnesse F = 100, the dtendue is, U = 2.5x10 [cm sr] and the spectral resolving power is v/Av = 4 X 10. With this dtendue the resolving power of the plane FPI is 6x10 , provided the whole plane mirror surface has a surface quality to allow a surface finesse of F > 100. In practice, this is difficult to achieve for a flat plane with D = 10 cm diameter, while for the small spherical mirrors even F > lO is feasible. 

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