Interferometer finesse

Frequency noise. If the two interferometer arms are not exactly symmetric (if they have different lengths / 0 or finesses()F 0), then the laser frequency noise Ujy can mimic a gravitational wave signal... 

At 10Hz in a typical Nd-YAG laser 1000Hz/- /Hz, and the typical finesse asymmetry is of the order of one percent. In order to detect a gw signal the laser frequency noise has to be lowered by six orders of magnitudes (compared to the noise of a free running laser), and the two arms made as identical as possible. In order to achieve this complex frequency stabilization methods are employed in all interferometric detectors, and in order to insure the perfect symmetry of the interferometer, all pairs of Virgo optical components are coated during the same run (both Fabry-Perot input mirrors then both end mirrors are coated simultaneously). 

We have undertaken an experiment to try to improve the performance of pulse amplifier experiments. The system is shown schematically in figure 2. It consisted of a continuous-wave C102 dye laser amplified in three stages by a frequency tripled Q-switched NdtYAG laser. The output energy was approximately 2.0 mJ in a 150 MHz linewidth and was up-shifted from the continuous-wave laser by 60 MHz caused by the frequency chirp. This light was then spectrally filtered in a confocal interferometer with a finesse of 40 and a free spectral range of 300 MHz. The linewidth of the filtered radiation was approximately 16 MHz. 

There are two important parameters that characterize Fabry-Perot interferometers, the free spectral range (FSR) and the finesse. The FSR is essentially the frequency spacing between adjacent cavity modes. For a flat-plate interferometer, it is given by... 

A second independent method of direct absorption spectroscopy has been recently applied to clusters cavity ring down (CRD) spectroscopy. This method, where a sample is introduced into the cavity of a high finesse Fabry-Perot interferometer, and is shown schematically in Fig. 3. 

Because of their capability to separate very closely spaced and narrow spectral features (down to a few megahertz with high-finesse devices), FP etalons and interferometers are often used in high-resolution spectroscopy. Such devices are common tools for the analysis of laser radiation, for example (i) to determine the line width of a laser source, (ii) to characterize the mode composition or to ascertain that it is operating in a single longitudinal mode, or (iii) to provide an accurate, calibrated frequency scale when a laser is scanned in wavelength. The radiation analysed by the FP device is coupled to it either in collimated or focused beams. 

Fig. 4.37. Transmittance of an absorption-free multiple-beam interferometer as a function of the phase difference 0 for different values of the finesse F ...

The ratio 8v/Av of free spectral range 8v to the halfwidth Av of the transmission maxima is called ht finesse F of the interferometer. From (4.51b) and (4.52c) we obtain for the finesse... 

The finesse is a measure for the effective number of interfering partial waves in the interferometer. This means that the maximum path difference between interfering waves is A5max = F 2nd. 

Since we have assumed an ideal plane-parallel plate with a perfect surface quality, the finesse (4.53a) is determined only by the reflectivity R of the surfaces. In practice, however, deviations of the surfaces from an ideal plane and slight inclinations of the two surfaces cause imperfect superposition of the interfering waves. This results in a decrease and a broadening of the transmission maxima, which decreases the total finesse. If, for instance, a reflecting surface deviates by the amount X/q from an ideal plane, the finesse cannot be larger than q. One can define the total finesse F of an interferometer by... 

The spectral resolution, v/Av or A./AA., of an interferometer is determined by the free spectral range 8v and by the finesse F. Two incident waves with frequencies v and V2 = v - - Av can still be resolved if their frequency separation Av is larger than <5i /F, which means that their peak separation should be larger than their full halfwidth. 

The resolving power of an interferometer is the product of finesse F and optical path difference As/X in units of the wavelength X. 

The optimum choice for the radius of the aperture is based on a compromise between spectral resolution and transmitted intensity. When the interferometer has the finesse F, the spectral halfwidth of the transmission peak is Sv/F, see (4.53b), and the maximum spectral resolving power becomes F A /A (4.56). For the radius b = (Px/F y of the aperture, which is just (F )1/4 iiYnes the radius p of a fringe with p = 1 in (4.77), the spectral resolving power is reduced to about 70% of its maximum value. This can be verified by inserting this value of b into (4.79) and calculating the halfwidth of the transmission peak P(X, F, 6). 

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. 

Since the laser resonator is a Fabry-Perot interferometer, the spectral distribution of the transmitted intensity follows the Airy formula (4.57). According to (4.53b), the halfwidth Avr of the resonances, expressed in terms of the free spectral range 5y, is Avr = Sy/F. If diffraction losses can be neglected, the finesse F is mainly determined by the reflectivity R of the mirrors, therefore the halfwidth of the resonance becomes... 

Figure 4.39 Finesse of a Fabry-Perot interferometer as a function of the mirror reflectivity R...

This results in a decrease and a broadening of the transmission maxima, which decreases the total finesse. If, for instance, a reflecting surface deviates by the amount Xjq from an ideal plane, the finesse cannot be larger than q. One can define the total finesse F of an interferometer by... 

To achieve a finesse F with photoelectric recording, this variation of the path length for the different rays through the interferometer should not exceed A/F, which restricts the solid angle Q = acceptable by the detector to Q < X/ d F ). The 6tendue is therefore... 

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