Modern mirrors and beam splitters that make use of dielectric coatings are complex optical systems, see Figure 8 whose reflectivity and transmission depend on the multiple interference inside the coating layers and thus on microscopic parameters. The phase change upon transmission or reflection depends on the details of the applied coating and is typically not known. In any case, the knowledge of an absolute value of a phase change is typically not of interest in laser interferometers because the absolute positions of the optical components are not known to sub-wavelength precision. Instead the relative phase between the incoming and outgoing beams is of importance. In the following we demonstrate how constraints on these relative phases, i.e., the phase relation between the beams, can be derived from the fundamental principle of power conservation. To do this we consider a Michelson interferometer, as shown in Figure 9, with perfectly-reflecting mirrors. The beam splitter of the Michelson interferometer is the object under test. We assume that the magnitude of the reflection and transmission are known. The phase changes upon transmission and reflection are unknown. Due to symmetry we can say that the phase change upon transmission should be the same in both directions. However, the phase change on reflection might be different for either direction, thus, we write for the reflection at the front and for the reflection at the back of the beam splitter.

Then the electric fields can be computed as

We do not know the length of the interferometer arms. Thus, we introduce two further unknown phases: for the total phase accumulated by the field in the vertical arm and for the total phase accumulated in the horizontal arm. The fields impinging on the beam splitter compute as The outgoing fields are computed as the sums of the reflected and transmitted components: with and .It will be convenient to separate the phase factors into common and differential ones. We can write

with and similarly with For simplicity we now limit the discussion to a 50:50 beam splitter with , for which we can simplify the field expressions even further: Conservation of energy requires that , which in turn requires which is only true if with as in integer (positive, negative or zero). This gives the following constraint on the phase factors One can show that exactly the same condition results in the case of arbitrary (lossless) reflectivity of the beam splitter [48].We can test whether two known examples fulfill this condition. If the beam-splitting surface is the front of a glass plate we know that , , , which conforms with Equation (28). A second example is the two-mirror resonator, see Section 2.2. If we consider the cavity as an optical ‘black box’, it also splits any incoming beam into a reflected and transmitted component, like a mirror or beam splitter. Further we know that a symmetric resonator must give the same results for fields injected from the left or from the right. Thus, the phase factors upon reflection must be equal . The reflection and transmission coefficients are given by Equations (7) and (8) as

and We demonstrate a simple case by putting the cavity on resonance (). This yields with being purely real and imaginary and thus and which also agrees with Equation (28).In most cases we neither know nor care about the exact phase factors. Instead we can pick any set which fulfills Equation (28). For this document we have chosen to use phase factors equal to those of the cavity, i.e., and , which is why we write the reflection and transmission at a mirror or beam splitter as

In this definition and are positive real numbers satisfying for the lossless case.Please note that we only have the freedom to chose convenient phase factors when we do not know or do not care about the details of the optical system, which performs the beam splitting. If instead the details are important, for example when computing the properties of a thin coating layer, such as anti-reflex coatings, the proper phase factors for the respective interfaces must be computed and used.

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