Etalon pro

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Figure 3: Schematic of a confocal Fabry-Pérot resonator. Mirrors with radius R₁ = R₂ (brown arrows) are separated by a distance L that is equal to the mirror radius. The solid green lines show a ray-trace of an off-axis input beam entering the resonator at a height H. The dashed light green lines represent the beam transmitted through the second mirror; light transmitted through the first mirror is not pictured.

Further Considerations on the Finesse

In fact, a measured finesse has a number of contributing factors: the mirror reflectivity finesse, simply denoted F above, the mirror surface quality finesse F_Q, and the finesse due to the illumination conditions (beam alignment and diameter) of the mirrors F_i. The overall finesse of a system, F_t, is given by the relation 4

Often, the reflectivity finesse, Equation (8), is presented as an effective finesse, which is true for the case when the other contributing factors are negligible. For Thorlabs’ interferometers, the reflectivity finesse dominates when operating with proper illumination. b

The second term in Equation (10) involves, F_Q, which accounts for mirror irregularities that cause a symmetric broadening of the lineshape. The effect of these irregularities is a random position-dependent path length difference that blurs the lineshape. The manufacturing process that is used to produce the cavity mirror substrates always has to ensure that the contribution from F_Q is negligible in comparison to the specified total finesse of a resonator; in other words, the substrates surface figure should never be the limiting factor for the finesse.

The final term in Equation (10), which deals with the illumination finesse, F_i, will reduce the resolution as the beam diameter is increased or as the input beam is offset. When the finesse is limited by the F_i term, the measured lineshape will appear asymmetric. The asymmetry is due to the path length difference between an on-axis beam and an off-axis beam, resulting in different mirror spacings to satisfy the maximum transmission criteria.

To quantify the effects of the variable path length on F_i, consider an ideal monochromatic input, a delta function in wavelength with unit amplitude, entering the Fabry-Pérot cavity coaxial to the optic axis and having a beam radius a. The light entering the interferometer at H = +e, where e is infinitesimally small but not zero, will negligibly contribute to a deviation in the transmitted spectrum. Light entering the cavity at H = +a will cause a shift in the transmitted output spectrum, since the optical path length of the cavity will be less by an approximate distance of a 4 /R 3 . Assuming the input beam has a uniform intensity distribution, the transmitted spectrum will appear uniform in intensity and broader due to the shifts in the optical path length. As a result, the wavelength input delta function will produce an output peak with a FWHM of H 4 /R 3 (Ref. 6).

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Figure 6: The total finesse F_t as a function of beam diameter, 2H, for the SA200 and SA210 Fabry-Pérot Interferometers using Equation (12). The finesse is calculated for a wavelength λ of 633 nm.

Assuming that only F_i contributes significantly to the total finesse, then Eq. (9) can be used to calculate F_i for the idealized input beam. Substituting λ/4 for the FSR, and (H 4 /R 3 ) for FWHM, yields:

The λ/4 substitution for the FSR is understood by considering that the cavity expands by λ/4 to change from one longitudinal mode to the next. For an input beam with a real spectral distribution, the effect of the shift will be a continuous series of shifted lineshapes. It should be noted that the shift is always in one direction, leading to a broadened or asymmetric lineshape due to the over-sized or misaligned beam.

Now, the total finesse for the case of high reflectivity mirrors (r ≈ 1) can be found using Equation (10), which includes significant contributions from both F and F_i (Note: F_q is still considered to have a negligible effect on F_t):

Equation (12) is used to provide an estimate, in general an overestimate, of beam diameter effects on the total finesse of a Fabry-Pérot Interferometer, and several assumptions have been made. The first assumption is that the diameter of the beam is the same as the diameter of the mirror. In practice, the diameter of the beam is typically significantly smaller than that of the mirror, which also helps to reduce spherical aberration. 5 A second assumption is that the light is focused down to an infinitesimally small waist size. Even for monochromatic light, the minimum waist size is limited by diffraction, and in multimode applications the waist size can be quite large at the focus. Figure 6 provides a plot of Equation (12) evaluated at 633 nm for the SA200 and SA210 Fabry-Pérot Interferometers, which have cavity lengths of 50 mm and 7.5 mm respectively. The traces in the plot assume that the reflectivity finesse is equal to 250 for the SA200 and 180 for the SA210, which are typical values obtained for mirrors used in our interferometers.

Cavity Ring-Down Time and Intracavity Power Build-Up

As a light wave travels many round-trips inside the resonator, the light is stored inside for a certain amount of time, and only a small portion of its energy leaks out as it impinges on either the input or the output mirror. In other words, the light wave has a certain life-time inside the resonator. This time is called the cavity ring-down time or cavity storage time, τ_cav, and is given by

This relation can show that τ_cav increases with the cavity finesse, i.e., the higher the finesse and the mirror reflectivity, the longer light is stored inside the resonator. In-line with this, another important quantity is the so-called intracavity power build-up, defined by the ratio of the intracavity intensity, I_c, and the incident intensity according to

which is given by F/π for an impedance-matched cavity (i.e., with a vanishing on-resonance reflection). This relates the intracavity intensity to the finesse by

The fact that the power stored inside the cavity increases with finesse has to be kept in mind, when beams with high incident power are evaluated with a Fabry-Pérot interferometer.

Spectral Resolving Power and Étendue

The spectral resolving power of an interferometer is a metric to quantify the spectral resolution of an interferometer, and is an extension of the Rayleigh criterion. The spectral resolving power, SR, is defined as:

where ν is the frequency of light and λ is its wavelength. It can be shown that for a confocal Fabry-Pérot interferometer, the SR is given by:

where F is the finesse of the interferometer, R is the radius of curvature of the mirrors, and λ is the wavelength. However, to achieve this maximum instrumental profile while the interferometer is in scanning mode, the aperture of the detector would need to be infinitesimally small; as the size of the aperture is increased, the spectral resolving power begins to decrease. The spectral resolving power must be balanced with the étendue of the interferometer. The étendue (U) is the metric for the net light-gathering power of the interferometer. When the light source is a laser beam, the étendue provides a measure of the alignment tolerance between the interferometer and the laser beam. The étendue is defined as the product of the maximum allowed solid angle divergance (Ω) and the maximum allowed aperture area (A). For the confocal system, the étendue is given by:

where F is the finesse of the interferometer, λ is the wavelength, and L is the mirror spacing. The spectral resolving power and étendue need to be balanced for the interferometer to work correctly. The accepted compromise for this balance is to increase the mirror aperture until the the spectral resolving power is decreased by 70% (0.7*SR) (Ref. 4). Under this condition the «ideal» étendue becomes π2λR/F, where R is the mirror’s radius.

For more information on our Fabry-Pérot interferometers, including typical application examples, please see the full web presentation here.

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