Mirror and beamsplitter

As from Fresnel equations, passive optical components, such as mirrors, beam splitters and lenses, can be described as flat thin layers linearly coupling with the incident light. When light impinges on that surface, both reflection and refraction of the light may occur. A coupling coefficient is the ratio of the reflected, or transmitted, light to the incident light. These ratios are generally complex, describing not only the relative amplitudes but also the phase shifts at the interface.

We define a mirror as the contact plane between two media. We derive the coupling coefficients under normal incidence. To do that, we calculate the phase jump for the transmitted and reflected beam. Finally, we obtain the phase for the transmitted beam as a function pf the position of the mirror. We repeat this study for a beam splitter, which is similar to a mirror except for an inclination angle.

Mirror

The coupling of light field amplitudes with a mirror under normal incidence can be described as follows: there are two coherent input fields, \(a_{1}\) impinging on the mirror on the front and \(a_{2}\) on the back surface. Two output fields leave the mirror, \(b_{1}\) and \(b_{2}\).

Fig. 3 Schematic for two coherent beams falling on the front and back surface of a mirror, \(a_{1}\) and \(a_{2}\), along with the outcoming beams reflected off either surface, \(b_{1}\) and \(b_{2}\).

The following linear equations can be used to describe the coupling:

\[\begin{pmatrix} b_{1}\\ b_{2} \end{pmatrix} = \begin{pmatrix} M_{11} & M_{12}\\ M_{21} & M_{22} \end{pmatrix} \begin{pmatrix} a_{1} \\ a_{2} \end{pmatrix} \]

Where \(M_{ij}\) is a coupling coefficent. We define te coupling coefficients as:

\[M_{11} = r e^{i\rho_{1}}\\ M_{22} = r e^{i\rho_{2}}\\ M_{12} = M_{21} = t e^{i\tau} \]

where \(r\) is the amplitude reflectance of the mirror, \(t\) the mirror transmittance, \(\rho_{1}\), \(\rho_{2}\) and \(\tau\) are constant phase jumps for a beam reflected off the front surfece, reflected off the back surface and transmitted through the mirror, respectively. We assume \(\tau\) to be the same both for \(M_{12}\) and \(M_{21}\) not to introduce a preferred direction of propagation.

Phase jumps at the surface

For a loss-less surface we can compute conditions for \(\rho\) and \(\tau\) from energy conservation:

\[|b_1|^{2} + |b_2|^{2} = a_1^{2} + a_2^{2} \]

each term in the equation above is defined as:

\[|b_{1}|^{2} = r^{2} a_{1}^{2} + t^{2} a_{2}^{2} + 2rt\, a_{1} a_{2}\, \cos(\tau-\rho_{1})\\ |b_{2}|^{2} = t^{2} a_{1}^{2} + r^{2} a_{2}^{2} + 2rt\, a_{1} a_{2}\, \cos(\tau-\rho_{2})\]

Energy conservation requires:

\[\cos(\tau-\rho_{1}) = -\cos(\tau-\rho_{2}) \]

which in turn requires:

\[\tau-\rho_{1} = (2N+1)\pi - (\tau-\rho_{2}) \]

where \(N\) is an integer. After some simple algebraic steps, we obtain:

\[\tau = (2N+1)\frac{\pi}{2} + \frac{(\rho_{1}+\rho_{2})}{2} \]

We arbitrarily set \(\rho_{1}=\rho_{2}=0\) and \(N=0\) and we will follow this convention throughout this modelling. Thus:

\[\tau =\frac{\pi}{2} \]

which is the total phase gain for a beam transmitted by the surface.

Tuning

The tuning \(\phi_{0}\) gives the change in the mirror position expressed in radiants (with respect to the reference plane). A tuning of \(\phi_{0}=2\pi\) translates the mirror by one vacuum wavelength (default wavelength set in Finesse): \(x=\lambda_{0}\). The direction of the change is defined to be in the direction of the normal vector on the front surface.

With the mirror position given in meters \(x\), then the corresponding tuning computes as follows:

\[\phi_{0} = k_{0} x = \frac{2\pi}{\lambda_{0}} x = \frac{\omega_{0}}{c} x \]

A certain displacement results in different changes in the optical path for light fields with different frequencies. In order to take that into account, \(\phi\) can be generalised as follows:

\[\phi = \phi_{0} \frac{\omega}{\omega_{0}} \]

Fig. 4 Tuning of a mirror under normal incidence. The solid line is the reference untuned mirror, the dashed one is tuned mirror.

To account for the effect of tuning, we re-define the coupling coefficients as:

\[M_{11} = r e^{2in_{1}\phi}\\ M_{22} = r e^{-2in_{2}\phi}\\ M_{12} = M_{21} = te^{i(\tau+\varphi)} \]

where \(\varphi\) is the chenge in the optical path for a beam transmitted through the mirror, \(n_{1}\) and \(n_{2}\) are the indices of refraction of the media on either side of the surface. As was done in the previous section, we compute the conditions for \(\tau\) from energy conservation:

\[|b_1|^{2} + |b_2|^{2} = a_1^{2} + a_2^{2} \]

each term in the equation above is defined as:

\[|b_{1}|^{2} = r^{2} a_{1}^{2} + t^{2} a_{2}^{2} + 2rt\, a_{1} a_{2}\, \cos(\tau-\rho_{1}-2n_{1}\phi+\varphi)\\ |b_{2}|^{2} = t^{2} a_{1}^{2} + r^{2} a_{2}^{2} + 2rt\, a_{1} a_{2}\, \cos(\tau-\rho_{2}+2n_{2}\phi+\varphi)\]

Conservation of energy requires:

\[\cos(\tau-\rho_{1}-2n_{1}\phi+\varphi) = -\cos(\tau-\rho_{2}+2n_{2}\phi+\varphi) \]

Which in turn requires:

\[\tau-2n_{1}\phi+\varphi = \pi - (\tau+2n_{2}\phi+\varphi) \]

After some simple algebraic steps, we obtain:

\[\tau = \frac{\pi}{2} -\varphi + (n_{1}-n_{2})\phi \]

Therefore, the total phase gain for a beam transmitted by a mirror in the case of tuning is given by:

\[\arg(M_{12}) = \arg(M_{21}) = \frac{\pi}{2}+(n_{1}-n_{2})\phi \]

which is a function of tuning.

Beam splitter

A beam splitter is similar to a mirror except for the extra parameter \(\alpha\) which indicates the tilt angle relative to the incoming beams.

Reflection

Fig. 5 Schematic for the beam reflected off a beam splitter in the reference (solid lines) and tuned case (dashed lines).

Referring to the figure above, we define the following geometrical paths as:

\[a = \frac{x}{\cos(\alpha)}\\[10pt] b = a\cos(2\alpha)\]

The change in the optical path due to tuning for a reflected beam is:

\[\left(\frac{\omega}{\omega_{0}}k_{0}\right)\, n_{i}|a + b| = 2 n_{i}\phi \cos{\alpha} \]

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We define the coupling coefficients for a beam splitter as:

\[M_{11} = r e^{2in_{1}\phi\cos{\alpha}}\\ M_{22} = r e^{-2in_{2}\phi\cos{\alpha}}\\ M_{12} = M_{21} = te^{i(\tau+\varphi)} \]

The only difference relative to the mirror case is the following replacement:

\[\phi \rightarrow \phi\cos(\alpha) \]

In a similar way to what was done for the mirror component in the case of tuning, the conditions for \(\tau\) are given as:

\[\tau = \frac{\pi}{2} - \varphi + (n_{1}-n_{2})\phi\cos(\alpha) \]

Therefore, the total phase gain for a beam transmitted by the beam splitter in the case of tuning is given by:

\[\arg(M_{12}) = \arg(M_{21}) = \frac{\pi}{2} + (n_{1}-n_{2})\phi\cos(\alpha) \]