### 2.5 Lengths and tunings: numerical accuracy of distances

The resonance condition inside an optical cavity and the operating point of an interferometer depends on the optical path lengths modulo the laser wavelength, i.e., for light from an Nd:YAG laser length differences of less than 1 µm are of interest, not the full magnitude of the distances between optics. On the other hand, several parameters describing the general properties of an optical system, like the finesse or free spectral range of a cavity (see Section 5.1) depend on the macroscopic distance and do not change significantly when the distance is changed on the order of a wavelength. This illustrates that the distance between optical components might not be the best parameter to use for the analysis of optical systems. Furthermore, it turns out that in numerical algorithms the distance may suffer from rounding errors. Let us use the Virgo [56] arm cavities as an example to illustrate this. The cavity length is approximately 3 km, the wavelength is on the order of 1 µm, the mirror positions are actively controlled with a precision of 1 pm and the detector sensitivity can be as good as 10–18 m, measured on  10 ms timescales (i.e., many samples of the data acquisition rate). The floating point accuracy of common, fast numerical algorithms is typically not better than 10–15. If we were to store the distance between the cavity mirrors as such a floating point number, the accuracy would be limited to 3 pm, which does not even cover the accuracy of the control systems, let alone the sensitivity.

A simple and elegant solution to this problem is to split a distance between two optical components into two parameters [29]: one is the macroscopic ‘length’ , defined as the multiple of a constant wavelength yielding the smallest difference to . The second parameter is the microscopic tuning that is defined as the remaining difference between and , i.e., . Typically, can be understood as the wavelength of the laser in vacuum, however, if the laser frequency changes during the experiment or multiple light fields with different frequencies are used simultaneously, a default constant wavelength must be chosen arbitrarily. Please note that usually the term in any equation refers to the actual wavelength at the respective location as with the index of refraction at the local medium.

We have seen in Section 2.1 that distances appear in the expressions for electromagnetic waves in connection with the wave number, for example,

Thus, the difference in phase between the field at and is given as
We recall that . We can define and . For any given wavelength we can write the corresponding frequency as a sum of the default frequency and a difference frequency . Using these definitions, we can rewrite Equation (34) with length and tuning as
The first term of the sum is always a multiple of , which is equivalent to zero. The last term of the sum is the smallest, approximately of the order . For typical values of , and we find that
which shows that the last term can often be ignored.

We can also write the tuning directly as a phase. We define as the dimensionless tuning

This yields
The tuning is given in radian with referring to a microscopic distance of one wavelength .

Finally, we can write the following expression for the phase difference between the light field taken at the end points of a distance :

or if we neglect the last term from Equation (36) we can approximate () to obtain
This convention provides two parameters and , that can describe distances with a markedly improved numerical accuracy. In addition, this definition often allows simplification of the algebraic notation of interferometer signals. By convention we associate a length with the propagation through free space, whereas the tuning will be treated as a parameter of the optical components. Effectively the tuning then represents a microscopic displacement of the respective component. If, for example, a cavity is to be resonant to the laser light, the tunings of the mirrors have to be the same whereas the length of the space in between can be arbitrary.