Input fields or the ‘right hand side’ vector

After the set of linear equations for a system has been determined, the input fields have to be given by the user. The respective fields are entered into the ‘right hand side’ (RHS) vector of the set of linear equations. The RHS vector consists of complex numbers that specify the amplitude and phase of every input field. Input fields are initially set to zero, and every non-zero entry describes a source. The possible sources in the carrier simulation are lasers and modulators, and in the signal simulation the only sources are signal generators.

Carrier frequencies

Laser

The principal light sources are, of course, the lasers. They are connected to one optical node only. The input power is specified by the user in the input file. For every laser the field amplitude is set as:

\[a_{in} = C \sqrt{(P/\epsilon_c)}~e^{\I \varphi}, \]

with

\[P = C^2|a|^2, \]

as the laser power and \(\varphi\) the specified phase. The constant factor \(C\) depends on the chosen internal scaling for the field amplitudes that can be set in the init file ‘kat.ini’. The default value is \(C=1\). This setting does not yield correct absolute values for light field amplitudes, i.e. when amplitude detectors are used. Instead, one obtains more intuitive numbers from which the respective light power can be easily computed.

Modulators

Modulators produce non-zero entries in the carrier RHS vector for every modulation sideband generated. Depending on the order (\(k\ge 0\)) and the modulation index (\(m\)), the input field amplitude for amplitude modulation is:

\[a_{\rm in} = \frac{m}{4}, \]

and for phase modulation:

\[a_{\rm in} = (-1)^k ~ J_k(m) ~ \mEx{\I\varphi}, \]

with \(\varphi\) given as (Equation (3)):

\[\varphi = \pm k \cdot (\frac{\pi}{2} + \varphi_{\rm s}), \]

where \(\varphi_{\rm s}\) is the user-specified phase from the modulator description. The sign of \(\varphi\) is the same as the sign of the frequency offset of the sideband. For ‘lower’ sidebands (\(f_{\rm mod}<0\)) we get \(\varphi=-\dots\), for ‘upper’ sidebands (\(f_{\rm mod}>0\)) it is \(\varphi=+\dots\).

Signal frequencies

The most complex light fields are the signal sidebands. They can be generated by many different types of modulation inside the interferometer (signal modulation in the following). The components mirror, beam splitter, space, laser and modulator can be used as a source of optical signal sidebands. Primarily, signal sidebands are used as the input signal for computing transfer functions of the optical system. The amplitude, in fact the modulation index, of the signal is assumed to be much smaller than unity so that the effects of the modulation can be described by a linear analysis. If linearity is assumed, however, the computed transfer functions are independent of the signal amplitude; thus, only the relative amplitudes of output and input are important, and the modulation index of the signal modulation can be arbitrarily set to unity in the simulation.

Contrary to previous versions, Finesse 3 includes electronic and mechanical components in the signal simulation, in addition to optics. There is also only one component that can create signals in the RHS vector, the signal generator. This creates an electrical signal which can then be fed into the modulation inputs of various optical components. The output of any other electrical component can also be used in this way to generate optical signals, allowing the creation of feedback loops and other such constructs. Signal frequencies can be ‘applied’ to a number of different components using the command fsig. This will create a signal generator, and attach it to the relevant port of the specified component. The connection of the signal frequency causes the component to in some way modulate the light fields at the component. The frequency, amplitude and phase of the modulation can be specified by fsig.

Finesse always assumes a numerical signal amplitude of 1. The numerical value of 1 has a different meaning for applying signals to different components (see below). The amplitude specified with fsig can be used to define the relative amplitudes of the source when the signal is applied to several components at once. Please note that Finesse does not correct the transfer functions for strange amplitude settings. An amplitude setting of two, for example, will scale the output (a transfer function) by a factor of two.

In order to have a determined number of light fields and electrical and mechanical frequencies, the signal modulation of a signal sideband has to be neglected. This approximation is sensible because in the steady state the signal modulations are expected to be tiny so that second-order effects (signal modulation of the signal modulation fields) can be omitted.

In general, the carrier field at the ‘signal component’ can be written as:

\[E_{in} ~ = ~ E_0 ~ \mEx{\I\w_c\T + \varphi_c}, \]

with \(\w_c\) the carrier frequency, and \(\varphi_c\) the phase of the carrier. In most cases the modulation of the light will be a phase modulation. Then the field after the modulation can be expressed in general as:

\[E_{out} ~ = ~ A ~ E_0 ~ \mEx{\I\w_c\T + \varphi_c + \varphi(t) + B}, \]

with \(A\) as a real amplitude factor, \(B\) a constant phase term and

\[\varphi(t)~=~m\cos{(\w_s\T +\varphi_s)}, \]

with \(m\) the modulation index, \(\w_s\) the signal frequency and \(\varphi_s\) the phase as defined by fsig. The modulation index will in general depend on the signal amplitude \(a_s\) as given by fsig and also other parameters (see below). The simple form for very small modulation indices (\(m \ll 1\)) can be used: only the two sidebands of the first order are taken into account and the Bessel functions can be approximated by:

\[\begin{array}{l} J_0(m) \approx 1,\\ J_{\pm1}(m) \approx \pm\frac{m}{2}. \end{array} \]

Thus, the modulation results in two sidebands (‘upper’ and ‘lower’ with a frequency offset of \(\pm\w_s\) to the carrier) which can be written as:

\[\begin{array}{rcl} E_{sb} &=& \I\frac{m}{2} ~ A ~ E_0 ~ \mEx{\I((\w_c\pm\w_s)\T + \varphi_c + B \pm \varphi_s)}\\ &=&\frac{m}{2} ~ A ~ E_0 ~ \mEx{\I(\w_{sb}\T + \pi / 2 + \varphi_c + B \pm \varphi_s)}. \end{array} \]

Mirror

Fig. 8 Signal applied to a mirror: modulation of the mirror position.

Mirror motion does not change transmitted light. The reflected light will be modulated in phase by any mirror movement. The relevant parameters are shown in Fig. 8. At a reference plane (at the nominal mirror position when the tuning is zero), the field impinging on the mirror is:

\[E_{\rm in} ~ = ~ E_0 ~ \mExB{\I(\w_{\rm c}t + \varphi_{\rm c} - k_{\rm c}x)} ~ = ~ E_0 ~ \mEx{\I\w_{\rm c}t + \varphi_{\rm c}}. \]

If the mirror is detuned by \(x_{\rm t}\) (here given in meters) then the electric field at the mirror is:

\[E_{\rm mir} ~ = ~ E_{\rm in} ~ \mEx{-\I k_{\rm c} x_{\rm t}}. \]

With the given parameters for the signal frequency, the position modulation can be written as \(x_{\rm m} = a_{\rm s}\cos(\w_{\rm s}t + \varphi_{\rm s})\) and thus the reflected field at the mirror is:

\[E_{\rm refl} ~ = ~ r ~ E_{\rm mir} ~ \mEx{\I2k_{\rm c} x_{\rm m}} ~ = ~ r ~ E_{\rm mir} ~ \mExB{\I2k_{\rm c} a_{\rm s}\cos(\w_{\rm s}t + \varphi_{\rm s})}, \]

setting \(m=2k_{\rm c} a_{\rm s}\), this can be expressed as:

\[\begin{array}{rcl} E_{\rm refl} &=& r ~ E_{\rm mir} ~ \Bigl( 1 + \I\frac{m}{2}\mExB{-\I(\w_{\rm s}t + \varphi_{\rm s})} + \I\frac{m}{2}\mExB{\I(\w_{\rm s}t + \varphi_{\rm s})} \Bigr)\\ &=& r ~ E_{\rm mir} ~ \Bigl( 1 + \frac{m}{2}\mExB{-\I(\w_{\rm s}t + \varphi_{\rm s} - \pi/2)} ~ \Bigr)\\ & & + \Bigl( \frac{m}{2}\mExB{\I(\w_{\rm s}t + \varphi_{\rm s} + \pi/2)} \Bigr). \end{array} \]

This gives an amplitude for both sidebands of:

\[a_{\rm sb}~=r~m/2~E_0~=~r~k_{\rm c} a_{\rm s}~E_0. \]

The phase back at the reference plane is:

(4)\[\varphi_{\rm sb} ~ = ~ \varphi_{\rm c} + \frac{\pi}{2} \pm \varphi_{\rm s} - (k_{\rm c} + k_{\rm sb}) ~ x_{\rm t}, \]

where the plus sign refers to the ‘upper’ sideband and the minus sign to the ‘lower’ sideband. As in Finesse the tuning is given in degrees, i.e. the conversion from \(x_{\rm t}\) to \(\Tun\) has to be taken into account:

\[\begin{array}{rcl} \varphi_{\rm sb} &=& \varphi_{\rm c} + \frac{\pi}{2} \pm \varphi_{\rm s} - (\w_{\rm c} + \w_{\rm sb})/c ~ x_{\rm t}\\ &=& \varphi_{\rm c} + \frac{\pi}{2} \pm \varphi_{\rm s} - (\w_{\rm c} + \w_{\rm sb})/c ~ \lambda_0/360 ~ \Tun\\ &=& \varphi_{\rm c} + \frac{\pi}{2} \pm \varphi_{\rm s} - (\w_{\rm c} + \w_{\rm sb})/\w_0 ~ 2\pi/360 ~ \Tun. \end{array} \]

With a nominal signal amplitude of \(a_{\rm s}=1\), the sideband amplitudes become very large. For an input light field at the default wavelength one typically obtains:

\[a_{\rm sb} = r ~ k_{\rm c} ~ E_0 = r ~ \w_{\rm c}/c ~ E_0 = r ~ 2\pi/\lambda_0 ~ E_0 \approx 6 \cdot 10^6. \]

Numerical algorithms have the best accuracy when the various input numbers are of the same order of magnitude, usually set to a number close to one. Therefore, the signal amplitudes for mirrors (and beam splitters) should be scaled: a natural scale is to define the modulation in radians instead of meters. The scaling factor is then \(\w_0/c\), and setting \(a=\w_0/c~a'\) the reflected field at the mirror becomes:

\[\begin{array}{rl} E_{\rm refl} &= ~ r ~ E_{\rm mir} ~ \mEx{\I2 \w_{\rm c}/\w_0 ~ x_{\rm m}}\\ &= ~ r ~ E_{\rm mir} ~ \mExB{\I2 \w_{\rm c}/\w_0 ~ a'_{\rm s}\cos(\w_{\rm s}t + \varphi_{\rm s})}, \end{array} \]

and thus the sideband amplitudes are:

\[a_{\rm sb} ~ = r ~ \w_{\rm c}/\w_0 ~ a'_{\rm s} ~ E_0, \]

with the factor \(\w_{\rm c}/\w_0\) typically being close to one. The units of the computed transfer functions are ‘output unit per radian’; which are neither common nor intuitive. The command scale meter converts the units into the more common ‘Watts per meter’ by applying the inverse scaling factor \(c/\w_0\).

When a light field is reflected at the back surface of the mirror, the sideband amplitudes are computed accordingly. The same formulae as above can be applied with \(x_{\rm m} \rightarrow -x_{\rm m}\) and \(x_{\rm t} \rightarrow -x_{\rm t}\), yielding the same amplitude as for the reflection at the front surface, but with a slightly different phase:

\[\begin{array}{rcl} \varphi_{\rm sb,back} &=& \varphi_{\rm c} + \frac{\pi}{2} \pm (\varphi_{\rm s} + \pi) + (k_{\rm c} + k_{\rm sb}) ~ x_{\rm t}\\ &=& \varphi_{\rm c} + \frac{\pi}{2} \pm (\varphi_{\rm s} + \pi) + (\w_{\rm c} + \w_{\rm sb})/\w_0 ~ 2\pi/360 ~ \Tun. \end{array} \]

Beam splitter

When the signal frequency is applied to the beam splitter, the reflected light is modulated in phase. In fact, the same computations as for mirrors can be used for beam splitters. However, all distances have to be scaled by \(\cos(\alpha)\). Again, only the reflected fields are changed by the modulation and the front side and back side modulation have different phases. The amplitude and phases compute to:

\[\begin{aligned} a_{\rm sb} ~ &= r ~ \w_c/\w_0 ~ a_{\rm s}\cos(\alpha) ~ E_0,\\ \phi_{\rm sb,front} ~ &= ~ \varphi_c + \frac{\pi}{2} \pm \varphi_{\rm s} - (k_{c} + k_{sb}) x_t\cos(\alpha),\\ \phi_{\rm sb,back} ~ &= ~ \varphi_c + \frac{\pi}{2} \pm (\varphi_{\rm s} + \pi) + (k_{c} + k_{sb}) x_t\cos(\alpha). \end{aligned} \]

Space

For interferometric gravitational wave detectors, the ‘free space’ is an important source for a signal frequency: a passing gravitational wave modulates the length of the space (i.e. the distance between two test masses). A light field passing this length will thus be modulated in phase. The phase change \(\phi(t)\) which is accumulated over the full length is (see, for example, [28]):

\[\phi(t) ~ = ~ \frac{\w_c ~ n ~ L}{c} + \frac{a_s}{2}\frac{\w_c}{\w_s} \sin{\left(\w_s\frac{n ~ L}{c}\right)} \cos{\left(\w_s\left(t - \frac{n ~ L}{c}\right)\right)}, \]

with \(L\) the length of the space, \(n\) the index of refraction and \(a_s\) the signal amplitude given in strain (h). This results in a signal sideband amplitude of:

\[\begin{aligned} a_{\rm sb} ~ &= ~ \frac{1}{4} ~ \frac{\w_c}{\w_s} \sin{\left(\w_s\frac{n ~ L}{c}\right)} a_{\rm s} ~ E_0,\\ \phi_{\rm sb} ~ &= ~ \varphi_c + \frac{\pi}{2} \pm \varphi_s - (\w_c + \w_s)\frac{nL}{c}. \end{aligned} \]

Laser

A laser has three types of signal modulation available: phase, frequency and amplitude. These are treated as in Modulation of light fields.

Frequency

Frequency modulation gives a field

\[E ~ = ~ E_0 ~ e^{\I (\w_c\,t + a_s/\w_s\cos(\w_s\,t + \varphi_s) + \varphi_c)}, \]

with sideband amplitude and phase

\[\begin{array}{l} a_{sb} = \frac{a_{\rm s}}{2 \w_{\rm s}} ~ E_0,\\ \phi_{\rm sb} = \varphi_c + \frac{\pi}{2} \pm \varphi_s. \end{array} \]

Phase

With phase modulation, the laser produces a field

\[E = E_0 ~ \exp\Bigl( \mathrm{i}\,(\omega_0\,t + a_{\rm s}\cos{\left(\omega_s\,t\right)}) \Bigr), \]

giving sidebands with

\[\begin{array}{l} a_{sb} = \frac{a_{\rm s}}{2} ~ E_0,\\ \phi_{\rm sb} = \varphi_c \pm \frac{\pi}{2} \pm \varphi_s. \end{array} \]

Amplitude

Amplitude modulation gives us

\[\begin{array}{lcl} E &=& E_0 ~ \mEx{\mathrm{i}\,\omega_0\,t} ~ \left( 1 - \frac{m}{2}\Bigl(1 - \cos{\left(\omega_m \,t\right)}\Bigr) \right), \end{array} \]

with sidebands

\[\begin{array}{l} a_{sb} = \frac{a_{\rm s}}{4} ~ E_0,\\ \phi_{\rm sb} = \varphi_c \pm \varphi_s. \end{array} \]

Modulator

Signal frequencies at a modulator are treated as ‘phase noise’. The electric field that leaves the modulator can be written as:

\[\begin{array}{rl} E ~ = ~ E_0 & e^{\I(\w_0 \T + \varphi_0)} ~ \sum_{k=-order}^{order} i^{\,k} ~ J_k(m) ~ e^{\I k(\w_m\T + \varphi_m)}\\[5pt] & \times \left( 1 + \I\frac{k ~ m_2 ~ a_s}{2} ~ e^{-i(\w_s\T + \varphi_s)} + \I\frac{k ~ m_2 ~ a_s}{2} ~ e^{\I(\w_s\T + \varphi_s)} + O((km_2)^2) \right), \end{array} \]

with

\[\begin{array}{l} E_{\rm mod} ~ = ~ E_0 ~ J_k(m),\\ \varphi_{\rm mod} = \varphi_0 + k\frac{\pi}{2} + k\varphi_m. \end{array} \]

The sideband amplitudes are:

\[\begin{array}{l} a_{sb} = \frac{a_{\rm s}k m_2}{2} ~ E_{\rm mod},\\ \phi_{\rm sb} = \varphi_{\rm mod} + \frac{\pi}{2} \pm \varphi_s. \end{array} \]

TODO

• Swap from using \(a_s\) to \(a_e\) (?), and mention electrical inputs rather than fsig everywhere

• Update each components signal modulation text as we go

• Insert something like the following, maybe not in this section:

  • As we must generate a conjugate pair of sidebands, one may think that we need to carry two electrical fields throughout the simulation. To get around this, Finesse instead carries the conjugate of the lower optical sideband everywhere, so we have for e.g. amplitude modulation of a laser:

    \[\begin{array}{l} a_+ = C\frac{E_0}{4} a_e \\[5pt] a_-^* = C\frac{E_0^*}{4} a_e \end{array} \]