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, a1a_{1} impinging on the mirror on the front and a2a_{2} on the back surface. Two output fields leave the mirror, b1b_{1} and b2b_{2}.

../../_images/flat_surface.svg

Fig. 5 Schematic for two coherent beams falling on the front and back surface of a mirror, a1a_{1} and a2a_{2}, along with the outcoming beams reflected off either surface, b1b_{1} and b2b_{2}.

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

(b1b2)=(M11M12M21M22)(a1a2)\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}

Tuning

We define the tuning ϕ0\phi_{0} of a surface as the change in the mirror position expressed in radiants (with respect to the reference plane). A tuning of ϕ0=2π\phi_{0}=2\pi translates the mirror by one vacuum wavelength (default wavelength set in Finesse): x=λ0x=\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 xx, then the corresponding tuning computes as follows:

ϕ0=k0x=2πλ0x=ω0cx\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. To take that into account, ϕ\phi can be generalised as follows:

ϕ=ϕ0ωω0\phi = \phi_{0} \frac{\omega}{\omega_{0}}
../../_images/flat_surface_tuning.svg

Fig. 6 Tuning of a mirror under normal incidence. The solid line is the reference untuned mirror, the dashed one is tuned mirror.


We define the light-mirror coupling coefficients as:

M11=reiφ11(ϕ)M22=reiφ22(ϕ)M12=M21=teiφ12(ϕ)=teiφ21(ϕ)M_{11} = r e^{i\varphi_{11}(\phi)}\\ M_{22} = r e^{i\varphi_{22}(\phi)}\\ M_{12} = M_{21} = t e^{i\varphi_{12}(\phi)} = t e^{i\varphi_{21}(\phi)}

where rr is the amplitude reflectance of the mirror, tt the mirror transmittance and φij(ϕ)\varphi_{ij}(\phi) the phase jump at the surface for the impinging light, which depends on the surface tuning. We assume φ12(ϕ)\varphi_{12}(\phi) to be equal to φ21(ϕ)\varphi_{21}(\phi) not to introduce a preferred direction of propagation. For a loss-less surface we can compute conditions for φij(ϕ)\varphi_{ij}(\phi) from energy conservation:

b12+b22=a12+a22|b_1|^{2} + |b_2|^{2} = a_1^{2} + a_2^{2}

each term in the equation above is defined as:

b12=r2a12+t2a22+2rta1a2cos(φ12φ11)b22=t2a12+r2a22+2rta1a2cos(φ12φ22)|b_{1}|^{2} = r^{2} a_{1}^{2} + t^{2} a_{2}^{2} + 2rt\, a_{1} a_{2}\, \cos(\varphi_{12}-\varphi_{11})\\ |b_{2}|^{2} = t^{2} a_{1}^{2} + r^{2} a_{2}^{2} + 2rt\, a_{1} a_{2}\, \cos(\varphi_{12}-\varphi_{22})

where we temporarily forget the dependencies. Energy conservation requires:

cos(φ12φ11)=cos(φ12φ22)\cos(\varphi_{12}-\varphi_{11}) = -\cos(\varphi_{12}-\varphi_{22})

which in turn requires:

φ12φ11=(2N+1)π(φ12φ22)\varphi_{12}-\varphi_{11} = (2N+1)\pi - (\varphi_{12}-\varphi_{22})

where NN is an integer. After some simple algebraic steps, we obtain:

φ12=(2N+1)π2+(φ11+φ22)2\varphi_{12} = (2N+1)\frac{\pi}{2} + \frac{(\varphi_{11}+\varphi_{22})}{2}

We arbitrarily set N=0N=0 and we will follow this convention throughout this modeling. In general, the conditions for φij(ϕ)\varphi_{ij}(\phi) are given by the following equation:

φ12(ϕ)=π2+(φ11(ϕ)+φ22(ϕ))2\varphi_{12}(\phi) = \frac{\pi}{2} + \frac{(\varphi_{11}(\phi)+\varphi_{22}(\phi))}{2}

Phase jumps in the untuned case

At the reference position of the mirror, we arbitrarily set φ11(0)=φ22(0)=0\varphi_{11}(0)=\varphi_{22}(0)=0 and we will follow this convention for the rest of the modeling. The phase gain for a beam transmitted through the mirror at the reference position is:

φ12(0)=φ21(0)=π2\varphi_{12}(0) = \varphi_{21}(0) = \frac{\pi}{2}

Phase jumps in the tuned case

As the mirror is tuned, the phase of a beam reflected off the surface φii(ϕ)\varphi_{ii}(\phi) is given as:

φ11(ϕ)=2n1ϕφ22(ϕ)=2n2ϕ\varphi_{11}(\phi) = 2n_{1}\phi\\ \varphi_{22}(\phi) = -2n_{2}\phi

where n1n_{1} and n2n_{2} are the indices of refraction of the media on either side of the surface. Substituting φ11(ϕ)\varphi_{11}(\phi) and φ22(ϕ)\varphi_{22}(\phi): in the equation for φij(ϕ)\varphi_{ij}(\phi), we obtain φ21(ϕ)\varphi_{21}(\phi):

φ12(ϕ)=φ21(ϕ)=π2+(n1n2)ϕ\varphi_{12}(\phi) = \varphi_{21}(\phi) = \frac{\pi}{2} + (n_{1}-n_{2})\phi

Beam splitter

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

Reflection

../../_images/flat_surface_tilted_Refl.svg

Fig. 7 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=xcos(α)b=acos(2α)c=xcos(β)d=ccos(2β)a = \frac{x}{\cos(\alpha)}\\[10pt] b = a\cos(2\alpha)\\[10pt] c = \frac{x}{\cos(\beta)}\\[10pt] d = c\cos(2\beta)\\[10pt]

where β\beta is the refraction angle given by the Snell’s law:

n1sin(α)=n2sin(β)n_{1}\sin(\alpha) = n_{2}\sin(\beta)

The phase change for a beam reflected on one side of a beam splitter is:

(ωω0k0)n1a+b=2n1ϕcosα\left(\frac{\omega}{\omega_{0}}k_{0}\right)\, n_{1}|a + b| = 2 n_{1}\phi \cos{\alpha}\\[10pt]

As for a beam reflected on the other side, the phase change is:

(ωω0k0)n2c+d=2n2ϕcosβ\left(\frac{\omega}{\omega_{0}}k_{0}\right)\, n_{2}|c + d| = 2 n_{2}\phi \cos{\beta}\\[10pt]

As was done for the mirror, we model the beam splitter-light coupling via linear coefficients MijM_{ij}:

M11=reiφ11(ϕ)M22=reiφ22(ϕ)M12=M21=teiφ12(ϕ)=teiφ21(ϕ)M_{11} = r e^{i\varphi_{11}(\phi)}\\ M_{22} = r e^{i\varphi_{22}(\phi)}\\ M_{12} = M_{21} = t e^{i\varphi_{12}(\phi)} = t e^{i\varphi_{21}(\phi)}

The conditions for φij(ϕ)\varphi_{ij}(\phi) are given by:

φ12(ϕ)=π2+(φ11(ϕ)+φ22(ϕ))2\varphi_{12}(\phi) = \frac{\pi}{2} + \frac{(\varphi_{11}(\phi)+\varphi_{22}(\phi))}{2}

The phase change for a beam reflected off either side of the beam splitter is given as:

φ11(ϕ)=2n1ϕcosαφ22(ϕ)=2n2ϕcosβ\varphi_{11}(\phi) = 2n_{1}\phi\cos\alpha\\ \varphi_{22}(\phi) = -2n_{2}\phi\cos\beta

Substituting φ11(ϕ)\varphi_{11}(\phi) and φ22(ϕ)\varphi_{22}(\phi): in the equation for φij(ϕ)\varphi_{ij}(\phi), we obtain φ21(ϕ)\varphi_{21}(\phi):

φ12(ϕ)=φ21(ϕ)=π2+(n1cosαn2cosβ)ϕ\varphi_{12}(\phi) = \varphi_{21}(\phi) = \frac{\pi}{2} + (n_{1}\cos\alpha-n_{2}\cos\beta)\phi