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.