This section contains derivations of analytical formulas for radiation pressure induced modulation of a plane-wave laser beam, in various setups. Test cases simulating these setups can be found in the file plane_wave/test_radiation_pressure.py, in the Finesse validation tests folder.

## Mirrors¶

### Single-Sided Free Mass¶

#### Analytics¶

We start with an amplitude modulated input field, such as produced by connecting a SignalGenerator to a Laser’s amp port in Finesse,

(3)$E_\mathrm{i} = E_0 \cos(\omega_0 t) \left( 1 - \frac{A}{2} \left(1 - \cos(\Omega t)\right) \right)$

where $$A$$ is the modulation index. Writing this in the complex notation used in Finesse gives

\begin{aligned} E_\mathrm{i} &= E_0 e^{i \omega_0 t} \left( 1 - \frac{A}{2} \left(1 - \cos(\Omega t)\right) \right) \\ &= E_0 e^{i \omega_0 t} \left( 1 - \frac{A}{2} + \frac{A}{4}\left(e^{i \Omega t} + e^{-i \Omega t}\right) \right) \end{aligned}

We can then separate this into three fields; the carrier field $$a_0$$, and the upper and lower sidebands $$a_\pm$$, such that

(4)$E_\mathrm{i} = a_0 + a_+ + a_-,$

where

\begin{aligned} a_0 &= \left(1 - \frac{A}{2}\right) E_0 e^{i \omega_0 t} \\ a_+ &= \frac{A}{4} E_0 e^{i (\omega_0 + \Omega) t} \\ a_- &= \frac{A}{4} E_0 e^{i (\omega_0 - \Omega) t}. \end{aligned}

Next, we consider this field when incident on a free-floating mirror. For a free mass, we have

$F(t) = m\ddot{x}(t).$

Taking the Fourier transform, we see that

\begin{aligned} \tilde{F}(\omega) &= -m \omega^2 \tilde{x}(\omega) \\ \therefore \tilde{\chi}(\omega) \equiv \frac{\tilde{x}(\omega)}{\tilde{F}(\omega)} &= -\frac{1}{m \omega^2}. \end{aligned}

For a force of the form $$F_0 \cos(\Omega t)$$, we have

$\tilde{F}(\omega) = \pi F_0 (\delta(\omega + \Omega) + \delta(\omega - \Omega)),$

from which we can find $$x(t)$$:

\begin{aligned} x(t) &= \frac{1}{2 \pi}\int_{-\infty}^{+\infty} \tilde{\chi}(\omega) \tilde{F}(\omega) e^{i \omega t} \mathrm{d}\omega \\ &= -\frac{\pi F_0}{2\pi} \int_{-\infty}^{+\infty} \frac{1}{m \omega^2} e^{i \omega t} (\delta(\omega + \Omega) + \delta(\omega - \Omega)) \mathrm{d}\omega \\ &= -\frac{F_0}{m \Omega^2} \frac{e^{i \Omega t} + e^{-i \Omega t}}{2} \\ &= -\frac{1}{m \Omega^2} F_0 \cos(\Omega t). \end{aligned}

A force of this form arises from the beating of the two amplitude modulation sidebands with the carrier. The force due to radiation pressure of light interacting with a mirror is given by

(5)\begin{aligned} F_\mathrm{rad} &= \cos{\alpha} \frac{P_{1\mathrm{i}} + P_{1\mathrm{o}} - P_{2\mathrm{i}} - P_{2\mathrm{o}}}{c}, \end{aligned}

where $$P_{\{1,2\}\{\mathrm{i,o}\}}$$ is the power of the light on the negative and positive $$x$$ sides of the mirror incoming and outgoing respectively, and $$\alpha$$ is the angle of incidence. In this simple case, $$\alpha = 0$$, and there is no incoming beam from the positive $$x$$ direction. The force can then be calculated to leading order in $$A$$ from Equation (3) as

\begin{aligned} F_\mathrm{rad} &= \frac{|E_\mathrm{i}| ^2 + |E_\mathrm{r}| ^2 - |E_\mathrm{t}| ^2}{c} \\ &= \frac{(1 + R - T)}{c} |E_\mathrm{i}| ^2 \\ &= \frac{(1 + R - T)}{c} \left({E_0}^2 (1 - A) + A {E_0}^2 \cos(\Omega t)\right), \end{aligned}

where $$R$$ is the coefficient of reflectivity of the mirror. Here we are uninterested in the D.C. term as this will not contribute to the mirror’s oscillation, and so have

\begin{aligned} F_0 &= \frac{(1 + R - T) A {E_0}^2}{c}, \\ \therefore x(t) &= -\frac{(1 + R - T) A {E_0}^2}{m c \Omega^2} \cos(\Omega t). \end{aligned}

Fig. 11 shows how an incoming field changes upon reflection from the mirror.

The incident field will be multiplied by the reflection coefficient of the mirror, $$r$$. The reflected light will also be phase shifted due to propagation through space by a total amount $$e^{-2i \phi}$$, where

\begin{aligned} \phi &= \frac{2 \pi x(t)}{\lambda} \\ &= -\frac{2 \pi (1 + R - T) A {E_0}^2}{\lambda m c \Omega^2} \cos(\Omega t) \\ &= -\frac{(1 + R - T) A {E_0}^2 \omega_0}{m c^2 \Omega^2} \cos(\Omega t). \end{aligned}

This results in the incoming beam being phase modulated by the mirror motion,

$E_\mathrm{r} = r E_\mathrm{i} \exp(i B \cos(\Omega t)),$

where

$B = \frac{2 (1 + R - T) A {E_0}^2 \omega_0}{m c^2 \Omega^2},$

which to first order gives

\begin{aligned} E_\mathrm{r} &= r E_\mathrm{i} \left( 1 + i\frac{B}{2}\left(e^{i \Omega t} + e^{-i \Omega t}\right) \right) \\ &= r(b_0 + b_+ + b_-), \end{aligned}

where

\begin{aligned} b_0 &= E_\mathrm{i} \\ b_+ &= i\frac{B}{2} E_\mathrm{i} e^{i \Omega t} \\ b_- &= i\frac{B}{2} E_\mathrm{i} e^{-i \Omega t}. \end{aligned}

Putting Equation (4) into this gives

\begin{aligned} b_0 &= a_0 + a_+ + a_- \\ b_+ &= i\frac{B}{2} E_0 e^{i (\omega_0 + \Omega) t} \\ b_- &= i\frac{B}{2} E_0 e^{i (\omega_0 - \Omega) t}, \end{aligned}

to leading order, where we can once again see an input field and an upper and lower sideband. The full expression for the reflected field can then be written as

\begin{aligned} E_\mathrm{r} &= r (a_0 + a_+ + a_- + b_+ + b_-) \\ &= r E_0 e^{i \omega_0 t} \left( 1 + \left(\frac{A + 2i B}{4}\right) \left(e^{i \Omega t} + e^{-i \Omega t}\right) \right) \\ &= r E_0 e^{i \omega_0 t} \left( 1 + \left( \frac{A}{4} + i\frac{(1 + R - T) A {E_0}^2 \omega_0}{m c^2 \Omega^2} \right) \left(e^{i \Omega t} + e^{-i \Omega t}\right) \right) \end{aligned}

#### Finesse¶

The analytics above are compared to the sidebands produced by the following Finesse file:

l l1 P=3
s s1 l1.p1 m1.p1 L=5
m m1 R=0.5 T=0.4
free_mass m1_sus m1.mech 0.5e-3

sgen sig l1.amp.i 0.1
fsig(0.1)

xaxis(sig.f, log, 0.1, 1e7, 400)


The results are shown in Fig. 12.

### Double-Sided Free Mass, One Modulation¶

#### Analytics¶

This is similar to the previous test, however there are no amplitude modulation sidebands present in the input field. As a result, we only expect to see the output containing the phase modulation sidebands. The second laser will not contribute to the oscillation of the mirror, as it is a purely D.C. force. The sidebands will also be $$180^\circ$$ out of phase, as the mirror’s motion is inverted when looking from the other side. In addition, we assume that the mirror is opaque i.e. $$T = 0$$. For a given input field

$E_\mathrm{i} = E_1 \cos(\omega_0 t),$

we therefore have

$E_\mathrm{r} = r E_1 \left( 1 - i\frac{(1 + R) A {E_0}^2 \omega_0}{m c^2 \Omega^2} \left(e^{i \Omega t} + e^{-i \Omega t}\right) \right).$

#### Finesse¶

The analytics above are compared to the sidebands produced by the following Finesse file:

l l1 P=3
s s1 l1.p1 m1.p1 L=5
m m1 R=1 T=0
s s2 m1.p2 l2.p1 L=5
l l2 P=2

free_mass m1_sus m1.mech 0.5e-3

sgen sig l1.amp.i 0.1
fsig(0.1)

xaxis(sig.f, log, 0.1, 1e7, 400)


The results are shown in Fig. 14.

### Double-Sided Free Mass, Two Modulations¶

#### Analytics¶

As the second beam is now also amplitude modulated, it will contribute to the mirror’s motion, so we must recalculate the radiation pressure force from Equation (5). For the left and right input fields,

\begin{aligned} E_\mathrm{Left} &= E_0 \cos(\omega_0 t) \left( 1 - \frac{A_0}{2} \left(1 - \cos(\Omega t)\right) \right) \\ E_\mathrm{Right} &= E_1 \cos(\omega_0 t) \left( 1 - \frac{A_1}{2} \left(1 - \cos(\Omega t - \psi)\right) \right), \end{aligned}

we then have

$F_\mathrm{rad} = \frac{1 + R}{c} \left( {E_0}^2 (1 - A_0) + A_0 {E_0}^2 \cos(\Omega t) - {E_1}^2 (1 - A_1) - A_1 {E_1}^2 \cos(\Omega t - \psi) \right),$

or ignoring the D.C. terms,

$F_\mathrm{rad} = \frac{1 + R}{c} \left( A_0 {E_0}^2 \cos(\Omega t) - A_1 {E_1}^2 \cos(\Omega t - \psi) \right).$

To write this as a single cosine, we use some trigonometry:

\begin{aligned} F_\mathrm{rad} &= \frac{1 + R}{c} \left[ A_0 {E_0}^2 \cos(\Omega t) - A_1 {E_1}^2 (\cos(\Omega t)\cos(\psi) + \sin(\Omega t)\sin(\psi)) \right] \\ &= \frac{1 + R}{c} \left[ (A_0 {E_0}^2 - A_1 {E_1}^2 \cos(\psi)) \cos(\Omega t) - A_1 {E_1}^2 \sin(\psi) \sin(\Omega t) \right]. \end{aligned}

Writing the term in brackets in the form $$D \cos(x + \theta)$$, we get

\begin{aligned} D^2 &= {(A_0 {E_0}^2 - A_1 {E_1}^2\cos(\psi))}^2 + {(A_1 {E_1}^2\sin(\psi))}^2 \\ D &= \sqrt{ {A_0}^2 {E_0}^4 - 2 A_0 A_1 {E_0}^2 {E_1}^2 \cos(\psi) + {A_1}^2 {E_1}^4 } \end{aligned}

and

$\theta = \tan^{-1} \left( \frac{ A_1 {E_1}^2 \sin(\psi) }{ A_0 {E_0}^2 - A_1 {E_1}^2 \cos(\psi) } \right).$

We can now proceed in the same way as in Single-Sided Free Mass, to obtain an expression for the reflected field on the left hand side of the mirror,

$E_\mathrm{r} = r E_{0} e^{i \omega_0 t} \left[ 1 + \left( \frac{A_0}{4} \left(e^{i\Omega t} + e^{-i\Omega t}\right) + i\frac{(1 + R) D \omega_0}{m c^2 \Omega^2} \left( e^{i (\Omega t + \theta)} + e^{-i (\Omega t + \theta)} \right) \right) \right].$

To get the field on the right hand side of the mirror, we can apply the following (remembering that $$\theta$$ will also be affected):

\begin{aligned} E_0 &\Longleftrightarrow E_1 \\ A_0 &\Longleftrightarrow A_1 \\ \psi &\Longrightarrow -\psi. \end{aligned}

#### Finesse¶

The analytics above are compared to the sidebands produced by the following Finesse file:

l l1 P=3
s s1 l1.p1 m1.p1 L=5
m m1 R=1 T=0
s s2 m1.p2 l2.p1 L=5
l l2 P=4

free_mass m1_sus m1.mech 0.5e-3

fsig(0.1)
sgen sig1 l1.amp.i 0.1
sgen sig2 l2.amp.i 0.1 phase=29

xaxis(fsig.f, log, 0.1, 1e7, 400)


For comparison with the analytical results, we must also include an extra factor of $$e^{\pm i\psi}$$ in the upper / lower sidebands respectively when looking at the right hand side field. This is because Finesse measures phase relative to the first laser in the file, which in this case is the left hand side laser. The results are shown in Fig. 16 & Fig. 17.

## Beamsplitters¶

The beamsplitter behaves similarly to the mirror, with two key differences. Firstly, the angle of incidence $$\alpha$$ in Equation (5) is no longer necessarily 0. This also affects the phase change on reflection due to the beamsplitter tuning, $$\phi$$. Secondly, there are now 4 ports at which light can enter and exit, rather than two. Here we repeat similar experiments to the previous section, taking these factors into account.

### Single-Sided Free Mass¶

#### Analytics¶

The steps here are much the same as in the equivalent mirror case, with the only changes being the inclusion of $$\alpha$$ in the radiation pressure force calculation, and a change in the phase shift due to mirror tuning $$\phi$$ — the total phase shift of reflected light will now be $$e^{-2i\phi\cos{\alpha}}$$, as the tuning $$\phi$$ is defined relative to the surface normal.

We therefore have

\begin{aligned} x(t) &= -\frac{(1 + R - T) A {E_0}^2}{m c \Omega^2} \cos(\Omega t) \cos{\alpha}.\\ \therefore \phi &= -\frac{(1 + R - T) A {E_0}^2 \omega_0}{m c^2 \Omega^2} \cos(\Omega t) \cos{\alpha}.\end{aligned}

This results in the incoming beam being phase modulated by the mirror motion,

$E_\mathrm{r} = r E_\mathrm{i} \exp(i B \cos(\Omega t)),$

where

$B = \frac{2 (1 + R - T) A {E_0}^2 \omega_0}{m c^2 \Omega^2} \cos^2{\alpha},$

Following the same steps as in the mirror case, the full expression for the reflected field can then be written as

\begin{aligned} E_\mathrm{r} &= r E_0 e^{i \omega_0 t} \left( 1 + \left(\frac{A + 2i B}{4}\right) \left(e^{i \Omega t} + e^{-i \Omega t}\right) \right) \\ &= r E_0 e^{i \omega_0 t} \left( 1 + \left( \frac{A}{4} + i\frac{(1 + R - T) A {E_0}^2 \omega_0}{m c^2 \Omega^2} \cos^2{\alpha} \right) \left(e^{i \Omega t} + e^{-i \Omega t}\right) \right) \end{aligned}

#### Finesse¶

The analytics above are compared to the sidebands produced by the following simple Finesse file:

l l1 P=3
s s1 l1.p1 bs1.p1 L=5
bs bs1 R=0.5 T=0.4 alpha=37
s s2 bs1.p2 out.p1 L=5
nothing out

free_mass bs1_sus bs1.mech 0.5e-3

fsig(0.1)
sgen sig l1.amp.i 0.1

xaxis(fsig.f, log, 0.1, 1e7, 400)


The results are shown in Fig. 19.

### Double-Sided Free Mass, One Modulation¶

#### Analytics¶

This is similar to the equivalent mirror case, with the inclusion of the $$\cos^2{\alpha}$$ term from the previous section. For a given input field

$E_\mathrm{i} = E_1 \cos(\omega_0 t),$

we therefore have

$E_\mathrm{r} = r E_1 \left( 1 - i\frac{(1 + R) A {E_0}^2 \omega_0}{m c^2 \Omega^2} \cos^2{\alpha} \left(e^{i \Omega t} + e^{-i \Omega t}\right) \right).$

#### Finesse¶

The analytics above are compared to the sidebands produced by the following Finesse file:

l l1 P=3
s s1 l1.p1 bs1.p1 L=5

l l2 P=2
s s2 l2.p1 bs1.p4 L=5

bs bs1 R=1 T=0 alpha=37
s sout bs1.p3 out.p1 L=5
nothing out

free_mass bs1_sus bs1.mech 0.5e-3

fsig(0.1)
sgen sig l1.amp.i 0.1

xaxis(fsig.f, log, 0.1, 1e7, 400)


The results are shown in Fig. 21

### Double-Sided Free Mass, Two Modulations¶

#### Analytics¶

Once again, this is similar to the procedure in the equivalent mirror case, with the addition of a $$\cos^2{\alpha}$$ term on the amplitude of the generated sidebands. The output at the top detector will therefore be

\begin{aligned} E_\mathrm{r} &= r E_{0} e^{i \omega_0 t} \left[ 1 + \left( \frac{A_0}{4} \left(e^{i\Omega t} + e^{-i\Omega t}\right) + i\frac{(1 + R) D \omega_0}{m c^2 \Omega^2}\cos^2{\alpha} \left( e^{i (\Omega t + \theta)} + e^{-i (\Omega t + \theta)} \right) \right) \right], \end{aligned}

with $$D$$ defined as in the mirror case. The results for the right detector can be obtained with the same transformations as before.

#### Finesse¶

The analytics above were compared to the sidebands produced by the following Finesse file:

l l1 P=3
s s1 l1.p1 bs1.p1 L=5

l l2 P=4
s s2 l2.p1 bs1.p4 L=5

bs bs1 R=1 T=0 alpha=37

s sout_left bs1.p2 out_left.p1 L=5
nothing out_left

s sout_right bs1.p3 out_right.p1 L=5
nothing out_right

free_mass bs1_sus bs1.mech 0.5e-3


As in the mirror case, for comparison with the analytical results, we must also include an extra factor of $$e^{\pm i\psi}$$ in the upper / lower sidebands respectively when looking at the right hand side field. This is because Finesse measures phase relative to the first laser in the file, which in this case is the left hand side laser. The results are shown in Fig. 23 & Fig. 24.