2.4 Phase relation at a mirror or beam splitter

The magnitude and phase of reflection at a single optical surface can be derived from Maxwell’s equations and the electromagnetic boundary conditions at the surface, and in particular the condition that the field amplitudes tangential to the optical surface must be continuous. The results are called Fresnel’s equations [33]. Thus, for a field impinging on an optical surface under normal incidence we can give the reflection coefficient as
n1-−-n2- r = n1 + n2, (17 )
with n1 and n2 the indices of refraction of the first and second medium, respectively. The transmission coefficient for a lossless surface can be computed as t2 = 1 − r2. We note that the phase change upon reflection is either 0 or 180°, depending on whether the second medium is optically thinner or thicker than the first. It is not shown here but Fresnel’s equations can also be used to show that the phase change for the transmitted light at a lossless surface is zero. This contrasts with the definitions given in Section 2.1 (see Figure (3View Image)ff.), where the phase shift upon any reflection is defined as zero and the transmitted light experiences a phase shift of π∕2. The following section explains the motivation for the latter definition having been adopted as the common notation for the analysis of modern optical systems.

2.4.1 Composite optical surfaces

Modern mirrors and beam splitters that make use of dielectric coatings are complex optical systems, see Figure 8View Image 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 9View Image, with perfectly-reflecting mirrors. The beam splitter of the Michelson interferometer is the object under test. We assume that the magnitude of the reflection r and transmission t are known. The phase changes upon transmission and reflection are unknown. Due to symmetry we can say that the phase change upon transmission φt should be the same in both directions. However, the phase change on reflection might be different for either direction, thus, we write φ r1 for the reflection at the front and φ r2 for the reflection at the back of the beam splitter.

View Image

Figure 8: This sketch shows a mirror or beam splitter component with dielectric coatings and the photograph shows some typical commercially available examples [45]. Most mirrors and beam splitters used in optical experiments are of this type: a substrate made from glass, quartz or fused silica is coated on both sides. The reflective coating defines the overall reflectivity of the component (anything between R ≈ 1 and R ≈ 0, while the anti-reflective coating is used to reduce the reflection at the second optical surface as much as possible so that this surface does not influence the light. Please note that the drawing is not to scale, the coatings are typically only a few microns thick on a several millimetre to centimetre thick substrate.
View Image

Figure 9: The relation between the phase of the light field amplitudes at a beam splitter can be computed assuming a Michelson interferometer, with arbitrary arm length but perfectly-reflecting mirrors. The incoming field E0 is split into two fields E1 and E2 which are reflected atthe end mirrors and return to the beam splitter, as E3 and E4, to be recombined into two outgoing fields. These outgoing fields E 5 and E 6 are depicted by two arrows to highlight that these are the sum of the transmitted and reflected components of the returning fields. We can derive constraints for the phase of E1 and E2 with respect to the input field E0 from the conservation of energy: |E0|2 = |E5|2 + |E6 |2.

Then the electric fields can be computed as

E1 = r E0 eiφr1 ; E2 = t E0 eiφt. (18 )
We do not know the length of the interferometer arms. Thus, we introduce two further unknown phases: Φ1 for the total phase accumulated by the field in the vertical arm and Φ2 for the total phase accumulated in the horizontal arm. The fields impinging on the beam splitter compute as
i(φr1+Φ1) i(φt+Φ2) E3 = r E0 e ; E4 = t E0 e . (19 )
The outgoing fields are computed as the sums of the reflected and transmitted components:
( i(2φr1+Φ1) i(2φt+Φ2)) E5 = E0 R e + T e (20 ) E = E rt(ei(φt+φr1+Φ1) + ei(φt+φr2+Φ2)), 6 0
with R = r2 and T = t2.

It will be convenient to separate the phase factors into common and differential ones. We can write

( ) E5 = E0 eiα+ R eiα− + T e−iα− , (21 )
1- 1- α+ = φr1 + φt + 2 (Φ1 + Φ2) ; α− = φr1 − φt + 2 (Φ1 − Φ2 ), (22 )
and similarly
E = E rt eiβ+ 2 cos(β ), (23 ) 6 0 −
β+ = φt + 1(φr1 + φr2 + Φ1 + Φ2 ) ; β − = 1-(φr1 − φr2 + Φ1 − Φ2) . (24 ) 2 2
For simplicity we now limit the discussion to a 50:50 beam splitter with √ -- r = t = 1∕ 2, for which we can simplify the field expressions even further:
iα+ iβ+ E5 = E0 e cos(α− ) ; E6 = E0 e cos(β− ). (25 )
Conservation of energy requires that |E0 |2 = |E5 |2 + |E6|2, which in turn requires
cos2(α− ) + cos2(β− ) = 1, (26 )
which is only true if
π- α− − β− = (2N + 1)2, (27 )
with N as in integer (positive, negative or zero). This gives the following constraint on the phase factors
1- π- 2 (φr1 + φr2) − φt = (2N + 1)2. (28 )
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 φt = 0, φr1 = π, φr2 = 0, which conforms with Equation (28View Equation). 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 φr = φr1 = φr2. The reflection and transmission coefficients are given by Equations (7View Equation) and (8View Equation) as

( r2t2exp(− i2kD ) ) rcav = r1 − -----1-------------- , (29 ) 1 − r1r2exp(− i2kD )
--− t1t2exp-(− ikD-) tcav = 1 − r1r2exp(− i2kD ). (30 )
We demonstrate a simple case by putting the cavity on resonance (kD = N π). This yields
( 2 ) r = r − --r2t1--- ; t = --i t1t2-, (31 ) cav 1 1 − r1r2 cav 1 − r1r2
with rcav being purely real and tcav imaginary and thus φt = π ∕2 and φr = 0 which also agrees with Equation (28View Equation).

In most cases we neither know nor care about the exact phase factors. Instead we can pick any set which fulfills Equation (28View Equation). For this document we have chosen to use phase factors equal to those of the cavity, i.e., φt = π∕2 and φr = 0, which is why we write the reflection and transmission at a mirror or beam splitter as

Erefl = r E0 and Etrans = i t E0. (32 )
In this definition r and t are positive real numbers satisfying 2 2 r + t = 1 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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