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.

Carriers

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:

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

with

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

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:

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

and for phase modulation:

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

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 \(\varphi=+\dots\).

Signals

Signals in Finesse refer to very small, linearised modulations of the optical fields, or mechanical, or electrical states. Optical signal sidebands come in pairs of ‘upper’ and ‘lower’ sidebands, and are equally spaced around each carrier optical frequency by a frequency \(\Omega = 2\pi f_{\text{sig}}\). In general, at any optical node the optical field near each carrier will be described by:

\[E_{c} [1 + E_{+}\mEx(\I\Omega t) + E_{-} \mEx(-\I\Omega t)] \mEx(\omega t)\]

They are used to compute linear time invariant transfer functions in Finesse. These sidebands describe the modulation of the carrier light field with respect to exciting some other state in the simulation, such as a mechanical or electrical state. Signal sidebands are always generated at a single frequency, this is the ‘signal frequency’ or what is seen in the Finesse code as fsig in many locations, which is \(f_{\text{sig}}\) as mentioned above. fsig is in units of Hertz. For mechanical and electrical states only the ‘upper’ sideband is directly calculated to compute the modulation around the DC value of the state. Finesse does not handle multiple mechanical or electrical carrier frequencies like with the optical calculations.

As with the carrier field calculations a matrix system is constructed to describe how all the signal sidebands interact with each other, modulate the carrier frequencies, and propagate through the model. This is called the signal matrix. By solving this matrix system you can ask how much signal at one node is generated by exciting another signal node in the system. This output divided by the input computes the transfer function. As the system is linearised, the input signal is often just a unitary value. Therefore it computes how much optical field modulation you might see got one meter of motion of a mirror. Obviously one cannot modulate a mirror position by a meter in an interferometer and the system still work. Experimentally this modulation is very small, small enough so that the linearisation assumptions are true.

Note

Signal sidebands can also be referred to as ‘audio’ sidebands in literature. Signal is used in Finesse as it is more general and the frequency range is not technically limited to audio frequencies. The term audio sidebands is mostly a carry over from gravitational wave detectors, where the signals are often in the audio frequency range.

It is critical to understand that the ampltiude of signal sidebands are not simply in units of optical field, mechanical motion, or electrical signal. They are the amplitude induced due to some excitation, such as an optic being moved or a voltage being applied. The units of a signal sideband field must be carefully considered as they depend on how you setup your simulation. An optical signal field will have units of \(\sqrt{\text{W}}/X\) where X depends on what you excite. If you excite a mirror position, then X is in meters. If you excite a voltage that induces some change in an optical field, then X is in Volts.

Warning

You should be careful not to excite too many signals at once which do not make sense, such as exciting a voltage and a mirror position at the same. The units of that resulting transfer function will be more complicated.

In practice Finesse actually needs to compute the conjugate of the lower optical signal sidebands. Thus internally the solver computes:

\[E_{c} [1 + E_{+}\mEx(\I\Omega t) + E_{-}^{\ast} \mEx(\I\Omega t)] \mEx(\omega t)\]

The reason for this is that this allows us to compute non-linear effects such as radiation pressure and photodiode detection inside the matrix solver. In practice the user should not need to ever worry about this and Finesse will return \(E_{-}\). However, if you delve into the internals or write your own elements then you should keep this in mind.

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 optical 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’.

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:

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

Beamsplitter

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:

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

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]):

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

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:

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

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

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

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:

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

• 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:

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