6.4 Michelson length sensing

Similarly to the two-mirror cavity, we can start to understand the length-sensing capabilities of the Michelson interferometer by looking at the output light power as a function of a mirror movement, as shown in Figure 29View Image. The power changes as sine squared with the maximum slope at the point when the output power (in what we call the South port) is half the input power. The slope of the output power, which is the optical gain of the instrument for detecting a differential arm-length change ΔL with a photo detector in the South port can be written as
( ) -dS--= 2πP0-sin 4π-ΔL (106 ) dΔL λ λ
and is shown in Figure 36View Image. The most notable difference of the optical gain of the Michelson interferometer with respect to the Fabry–Pérot interferometer (see Figure 33View Image) is the wider, more smooth distribution of the gain. This is due to the fact that the cavity example is based on a high-finesse cavity in which the optical resonance effect is dominant. In a basic Michelson interferometer such resonance enhancement is not present.
View Image

Figure 36: Power and slope of a Michelson interferometer. The upper plot shows the output power of a Michelson interferometer as detected in the South port (as already shown in Figure 29View Image). The lower plot shows the optical gain of the instrument as given by the slope of the upper plot.

However, the main difference is that the measurement is made differentially by comparing two lengths. This allows one to separate a larger number of possible noise contributions, for example noise in the laser light source, such as amplitude or frequency noise. This is why the main instrument for gravitational-wave measurements is a Michelson interferometer. However, the resonant enhancement of light power can be added to the Michelson, for example, by using Fabry–Pérot cavities within the Michelson. This construction of new topologies by combining Michelson and Fabry–Pérot interferometers will be described in detail in a future version of this review.

The Michelson interferometer has two longitudinal degrees of freedom. These can be represented by the positions (along the optical axes) of the end mirrors. However, it is more efficient to use proper linear combinations of these and describe the Michelson interferometer length or position information by the common and differential arm length, as introduced in Equation (97View Equation):

¯L = LN+LE- ΔL = L2 − L . N E

The Michelson interferometer is intrinsically insensitive to the common arm length ¯L.

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