Mirror¶

mirror¶ m Mirror

The mirror component represents a thin dielectric surface with associated properties such as reflectivity, tuning, and radius of curvature. Mirror components are nominally at normal incidence to the beams. It has two optical ports p1 and p2 which describes the two beams incident on either side of this surface. The surface normal points out of the mirror on the p1 side. A mirror also has a mechanical port mech which has nodes for longitudinal, yaw, and pitch motions. These mechanical nodes are purely for exciting small signal oscillations of the mirror. Static offsets in longitudinal displacements are set by the phi parameter (in units of degrees), yaw by the xbeta parameter, and pitch the ybeta parameter.

Syntax

m name R=none T=none L=none phi=0 Rc=inf xbeta=0 ybeta=0 misaligned=false

Required

name: Name of newly created mirror.

Optional

R: Reflectivity of the mirror; defaults to 0.5.

T: Transmittance of the mirror; defaults to 0.5.

L: Loss of the mirror; defaults to 0.0.

phi: Tuning of the mirror (in degrees); defaults to 0.0.

Rc: The radius of curvature of the mirror (in metres); defaults to a flat mirror (Rc=np.inf). Astigmatic mirrors can also be set with Rc=(Rcx, Rcy). A positive value results in a concave mirror on the p1 side of the mirror.

xbeta, ybeta: Misalignment of the mirror in yaw and pitch in units of radians

misaligned: When True the mirror will be significantly misaligned and assumes any reflected beam is dumped. Transmissions will still occur.

Phase relationship¶

See Mirror and beamsplitter phase relationships for more details on the phase relationship to ensure energy conservation that is used in |Finesse|.

Parameters¶

Listed below are the parameters of the mirror component. Certain parameters can be changed during a simulation and some cannot, which is highlighted in the can change during simulation column. These changeable parameters can be used by actions such as xaxis or change. Those that cannot must be changed before a simulation is run.

Name	Description	Units	Data type	Can change during simualation
R	Reflectivity	None	float	✓
T	Transmission	None	float	✓
L	Loss	None	float	✓
phi	Microscopic tuning (360 degrees = 1 default wavelength)	degrees	float	✓
Rcx	Radius of curvature (x)	m	float	✓
Rcy	Radius of curvature (y)	m	float	✓
xbeta	Yaw misalignment	radians	float	✓
ybeta	Pitch misalignment	radians	float	✓
misaligned	Misaligns mirror reflection (R=0 when True)	None	bool	✓

Coordinate systems¶

There are various coordinate systems involved in modelling the mirror component. Each mechanical and optical node has an associated coordinate system. Shown in the figure below are the coordinate systems for the optical p1 (Port 1) and the mechanical mode mech.

The optical nodes, as for all optical nodes for any component, are in a left-handed coordinate system, shown in blue. The incoming and output going nodes represent the beam in the incident plane. For a mirror the angle of incidence is fixed to zero, \(\alpha=0\). A beamsplitter component is exactly the same as a mirror component, except that the angle of incidence can be non-zero. The outgoing node is a reflection of the incoming in the z and x coordinates.

The mechanical node coordiante system is shown in black and is a right-handed coordinate system. A positive z motion is the surface normal of mirror surface going in the port 1 direction. Yaw and pitch are right-handed rotations around the y and x axes respectively.

Signals¶

Signals can be excited by injecting a signal into the relevant electrical or mechanical node at a component. The mirror component only has mechanical signals.

Longitudinal motions¶

Small longitudinal oscillations can be excited directly by using the mirror.mech.z (whose units are meters) or by applying a force mirror.mech.F_z (units of Newtons). If no suspension is attached to the mirror then applying a force will not move the mirror.

When the longitudinal motion is driven any carrier light reflected from the mirror surface will be phase modulated. This creates upper and lower signal sidebands around each carrier. Mathematically, the amount of sideband generated, \(a_{\pm}\), is linearly proportional to induced motion, \(z\). The higher order mode vector for the upper or lower sideband generated at the outgoing node of port 1 will be

\[\hat{a}_{\pm} = -\I k_{\pm} r z^{\pm} \hat{E}_{c}^{i} \exp\left(\I\frac{f_{\mathrm{sig}}}{f_0}\phi\right)\]

Where \(k_{\pm}\) is the sideband wavenumber, \(f_0\) is the default optical frequency, \(\hat{E}_{c}^{i}\) is a vector of carrier higher order modes, \(r\) the mirror reflectivity, and \(\phi\) the mirror tuning in radians. \(z^{\pm}\) is defined as \(z^+ \equiv z\) and \(z^- \equiv z^\ast\). Sidebands generated from reflection on the port 2 side will have a 180 degree phase shift relative to the port 1.

Yaw and pitch rotations¶

Small rotational oscillations can be excited by using the mirror.mech.yaw and mirror.mech.pitch both in units of radians. Torques can also be applied by using the mirror.mech.F_yaw or mirror.mech.F_pitch. If no suspension is attached to the mirror then applying a force will not rotate the mirror.

When either yaw or pitch is driven any carrier light reflected from the surfaces has a linear phase shifted applied to its wavefront, thus tilting it.

\[\hat{a}_{\pm} = -\I k_{\pm} r \beta^{\pm} \mathbf{K}_{y/p} \hat{E}_{c}^{i} \exp\left(\I\frac{f_{\mathrm{sig}}}{f_0}\phi\right)\]

The terms are the same as for the longitudinal equation except for \(\beta^\pm\) which is the misalignment equivalent of \(z^\pm\) for either pitch or yaw, and \(\mathbf{K}_{y/p}\) which is the mode scattering matrix for pitch or yaw for some linearised unit radian alignment change.