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 [56Jump To The Next Citation Point] 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.
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Figure 10: Illustration of an arm cavity of the Virgo gravitational-wave detector [56]: the macroscopic length L of the cavity is approximately 3 km, while the wavelength of the Nd:YAG laser is λ ≈ 1μm. The resonance condition is only affected by the microscopic position of the wave nodes with respect to the mirror surfaces and not by the macroscopic length, i.e., displacement of one mirror by Δx = λ∕2 re-creates exactly the same condition. However, other parameters of the cavity, such as the finesse, only depend on the macroscopic length L and not on the microscopic tuning.

A simple and elegant solution to this problem is to split a distance D between two optical components into two parameters [29Jump To The Next Citation Point]: one is the macroscopic ‘length’ L, defined as the multiple of a constant wavelength λ0 yielding the smallest difference to D. The second parameter is the microscopic tuning T that is defined as the remaining difference between L and D, i.e., D = L + T. Typically, λ 0 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 λ in any equation refers to the actual wavelength at the respective location as λ = λ0∕n with n 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,

E2 = E1 exp (− ikz). (33 )
Thus, the difference in phase between the field at z = z 1 and z = z + D 1 is given as
φ = − kD. (34 )
We recall that k = 2π ∕λ = ω ∕c. We can define ω0 = 2π c∕λ0 and k0 = ω0∕c. For any given wavelength λ we can write the corresponding frequency as a sum of the default frequency and a difference frequency ω = ω0 + Δ ω. Using these definitions, we can rewrite Equation (34View Equation) with length and tuning as
− φ = kD = ω0L- + Δ-ωL- + ω0T- + ΔωT--. (35 ) c c c c
The first term of the sum is always a multiple of 2π, which is equivalent to zero. The last term of the sum is the smallest, approximately of the order −14 Δ ω ⋅ 10. For typical values of L ≈ 1 m, T < 1 μm and Δ ω < 2π ⋅ 100 MHz we find that
ω0L- = 0, Δ-ωL- ≾ 2, ω0T-≾ 6, Δ-ωT- ≾ 2 10−6, (36 ) c c c c
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

ϕ = ω0T ∕c. (37 )
This yields
( ) ( ) ( ω ) ω0 ω ω exp i-T = exp i--T --- = exp i--ϕ . (38 ) c c ω0 ω0
The tuning ϕ is given in radian with 2π referring to a microscopic distance of one wavelength2 λ0.

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

( ) Δ-ωL- -ω- φ = − kD = − c + ϕω , (39 ) 0
or if we neglect the last term from Equation (36View Equation) we can approximate (ω ∕ω0 ≈ 1) to obtain
( ) Δ ωL φ ≈ − --c-- + ϕ . (40 )
This convention provides two parameters L and ϕ, 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 L 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.
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