Radiation Pressure at a Mirror

When light is incident on a mirror’s surface, it generates a force pushing on the mirror. In the steady state, we may assume that any constant light field is accounted for by control systems, such that the only effect will be due to the beats between different frequency light fields e.g. a carrier and its signal sidebands.

The coupling relations relevant to model radiation-pressure effects at a mirror are given below.

Coupling of radiation pressure force to mirror motion

Any electromagnetic field will exert a force on a suspended optic, the frequency spectrum of such a force from one beam is given as,

\[F(f)=\frac{P(f)\cos(\alpha)}{c} \]

where \(P(f)\) is the fluctuation of the power in the laser at frequency \(f\), \(\alpha\) is the angle of incidence and \(c`\) the speed of light. Finesse models only consider forces with \(f > 0\), in other words, we assume that the DC radiation pressure force is compensated for by some control system that keep the mirror in the same position.

Any fluctuation in the light power at a given signal frequency \(f_s\) can be described by a positive and negative part in the frequency domain,

\[P(f_s)=P_s^+ e^{\mathrm{i}2\pi f_s t} + P_s^- e^{-\mathrm{i}2\pi f_s t}. \]

As \(P_s^+=P_s^{-\ast}\) there is no need to explicitly solve \(P_s^{-\ast}\) in any of our simulations, instead we only solve the positive frequency components. Assuming that all carrier fields possess both an upper \(a^+_{s,jnm}\) and a lower \(a^-_{s,jnm}\) sideband of mode \(\mathrm{TEM}_{nm}\) and which belongs to the jth carrier field, the fluctuating power is then given by

\[P_s^+=\sum_j\sum_{n,m}\left(a_{s,jnm}^{+}a_{c,jnm}^{\ast}+a_{s,jnm}^{-\ast}a_{c,jnm}\right). \]

It should be noted that this power fluctuation calculation depends on both the upper sideband and the conjugate of the lower sideband - conjugation is not a linear operation so cannot be represented in a matrix form of the equations. By choosing to model only the positive frequencies we must now propagate the conjugate of the lower signal sidebands in the interferometer matrices that we construct. Although internally this requires constructing the matrices differently, any output from Finesse will always output the non-conjugated lower sideband, such as if you use the ad detectors.

The total force applied to a suspended mirror is then the sum of all forces due to each incoming and outgoing beams present at the optical component; for a mirror this is four beams and at a beamsplitter it is eight,

\[F_{\mathrm{total}}(f) &= -F_{1i}(f) - F_{1o}(f) + F_{2i}(f) + F_{2o}(f) \\ &= \frac{2\cos(\alpha)}{c}\left(-P_{1i}(f) - P_{1o}(f) + P_{2i}(f) + P_{2o}(f)\right) \]

The \(\pm\) for each \(P_s\) is determined by which side of the mirror or beamsplitter the beam is on: The positive direction of motion, the surface normal, for the mirror is defined as the direction of the beam reflected by the side of the first node. Thus an incoming or outgoing beam imparts a negative momentum on the side of the first node and a positive on the side of the last.

As we only compute the output for a single value of \(f_s`\) in each step we set \(F_s=F_{\text{total}}(f)\) and \(P_x(f)= P_{s,x}\) leaving,

\[F_{s}=\frac{2\cos(\alpha)}{c}\left(-P_{1i} - P_{1o} + P_{2i} + P_{2o}\right) \]

Variables subscripted with an s are typically frequency dependent terms which are being computed at an assumed signal frequency \(f_s\). The longitudinal motion at the signal frequency is then found using,

\[Z_{s}=H_{s}\sum_{n}^{N_{F}}F_{s,n}\,, \]

where \(H_{s}\) is the transfer function for the mechanical response of the suspended optic due to a force being applied to it in the \(z\) direction, i.e. in the direction of beam propagation. The sum of forces can also contain other external forces being applied, such as from actuators.

Mirror motion to optical phase change

Evaluating above equation for \(Z_s\) determines the amplitude and phase of the mirror oscillations. When carrier fields and RF modulation sidebands are reflected from a moving mirror, this oscillation will produce phase modulation signal sidebands, \(a^{\pm}\), around these fields. The variation in height of a mirror’s surface at a frequency \(f_s\) is described by \(z_{s}(x; y)\). This function is a normalised description of the general motion; its amplitude is given by \(A_s\), where \(A^{+}_{s}=A_{s}\) and \(A^{-}_{s}=A_{s}^{\ast}\) as lower sideband computation requires the conjugate of the motion amplitude. For example, longitudinal motion is described by \(z_{s}(x; y)=1\) and thus \(A_s=Z_s\) is in meters. Rotational motion is defined by \(z_s(x; y)=x\) or \(y\) with \(A_s = \Theta_{x/y}\) in radians. Making the assumption that \(|A_s| \ll 1064nm\), the creation of signal sidebands of a carrier field \(a_{c,j}\) is linearised [36],

\[a_{s,jnm}^\pm = i r k Z_s^\pm\sum_{n',m'}a_{c,jn'm'}K_{nmn'm'}^0 \]

where \(Z_s^+ \equiv Z_s\) and \(Z_s^- \equiv Z_s^*\) are the surface motion along the beam axis, and \(K_{nmn'm'}^0\) is the HOM coupling due to a static mirror distortion. Note that this is for the side of the mirror defined as “positive” i.e. the port1 (p1) side. For the opposite side (p2), the longitudinal motion is negative, so an extra 180° phase is added.

In Finesse, we must also propagate the carrier field to and the generated field away from the mirror’s surface by the mirror’s static tuning \(\phi\). The overall couplings for a carrier field \(a_{c,j}\) with frequency offset \(f_{c,j}\) and signal field \(a_{s,j}\) with frequency offset \(f_{s,j}\) are then

\[a_{s,jnm}^\pm = e^{i\phi(1 + \frac{f_{s,j}}{f_0} + \frac{f_{c,j}}{f_0})} i r k Z_s^\pm\sum_{n',m'}a_{c,jn'm'}K_{nmn'm'}^0 \]

for the positive side and

\[a_{s,jnm}^\pm = -e^{-i\phi(1 + \frac{f_{s,j}}{f_0} + \frac{f_{c,j}}{f_0})} i r k Z_s^\pm\sum_{n',m'}a_{c,jn'm'}K_{nmn'm'}^0 \]

for the negative, where \(f_0\) is the simulation’s default frequency.