5.2 Michelson interferometer

We came across the Michelson interferometer in Section 2.4 when we discussed the phase relation at a beam splitter. The typical optical layout of the Michelson interferometer is shown again in Figure 28View Image: a laser beam is split by a beam splitter and send along two perpendicular interferometer arms. The four directions seen from the beam splitter are called North, East, West and South. The ends of these arms (North and East) are marked by highly reflective end mirrors, which reflect the beams back into themselves so that they can be recombined by the beam splitter. Generally, the Michelson interferometer has two outputs, namely the so far unused beam splitter port (South) and the input port (West). Both output ports can be used to obtain interferometer signals, however, most setups are designed such that the signals with high signal-to-noise ratios are detected in the South port.
View Image

Figure 28: Typical optical layout of a Michelson interferometer: a laser beam is split into two and sent along two perpendicular interferometer arms. We will label the directions in a Michelson interferometer as North, East, West and South in the following. The end mirrors reflect the beams such that they are recombined at the beam splitter. The South and West ports of the beam splitter are possible output port, however, in many cases, only the South port is used.

The Michelson interferometer output is determined by the laser wavelength λ, the reflectivity and transmittance of the beam splitter and the end mirrors, and the length of the interferometer arms. In many cases the end mirrors are highly reflective and the beam splitter ideally a 50:50 beam splitter. In that case, we can compute the output for a monochromatic field as shown in Section 2.4. Using Equation (20View Equation) we can write the field in the South port as

i( i2kL i2kL ) ES = E0 2-e N + e E . (96 )
We define the common arm length and the arm-length difference as
¯ LN+LE- L = 2 ΔL = LN − LE, (97 )
which yield ¯ 2LN = 2L + ΔL and ¯ 2LE = 2L − ΔL. Thus, we can further simplify to get
i-i2k¯L( ikΔL − ikΔL ) i2k¯L ES = E0 2e e + e = E0 ie cos(kΔL ). (98 )
The photo detector then produces a signal proportional to
∗ 2 2 S = ESE S = P0 cos (k ΔL ) = P0 cos (2 πΔL ∕λ ). (99 )
This signal is depicted in Figure 29View Image; it shows that the power in the South port changes between zero and the input power with a period of ΔL ∕λ = 0.5. The tuning at which the output power drops to zero is called the dark fringe. Current interferometric gravitational-wave detectors operate their Michelson interferometer at or near the dark fringe.
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Figure 29: Power in the South port of a symmetric Michelson interferometer as a function of the arm length difference ΔL.

The above seems to indicate that the macroscopic arm-length difference plays no role in the Michelson output signal. However, this is only correct for a monochromatic laser beam with infinite coherence length. In real interferometers care must be taken that the arm-length difference is well below the coherence length of the light source. In gravitational-wave detectors the macroscopic arm-length difference is an important design feature; it is kept very small in order to reduce coupling of laser noise into the output but needs to retain a finite size to allow the transfer of phase modulation sidebands from the input to the output port; this is illustrated in the Finesse example below and will be covered in detail in Section 6.4.

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