.. |MLA (TM)| unicode:: MLA U+2122
.. _glossary:
Glossary
========
Multi-frequency lockin measurement and intermodulation spectroscopy are developing fields
and the language used to describe the concepts changes as our understanding
becomes more refined. Mathematical
and programing symbols are part of this language.
We provide this glossary to establish definitions of
the terms used in this manual.
Table of symbols used in this manual
------------------------------------
+------------------------+---------------------+------------------------------+
| Term | Programming | Math |
| | | |
+========================+=====================+==============================+
| Base tone | Df = 1/T | :math:`\Delta f = 1/T` |
+------------------------+---------------------+------------------------------+
| Waveform period | T = 1/Df | :math:`T = 1/\Delta f` |
+------------------------+---------------------+------------------------------+
| Measurement time | Tm = 1/df | :math:`T_m = 1 / \delta f` |
+------------------------+---------------------+------------------------------+
| Measurement bandwidth | df = 1/Tm | :math:`\delta f = 1 / T_m` |
+------------------------+---------------------+------------------------------+
| Tone index | idx | :math:`i` |
+------------------------+---------------------+------------------------------+
| Tone-integer array | narray | :math:`n_i` |
+------------------------+---------------------+------------------------------+
| Frequency array | freqs = narray*df | :math:`f_i = n_i * df` |
+------------------------+---------------------+------------------------------+
.. _amplitude:
amplitude
---------
The word amplitude is used in different ways and its exact definition depends on the context. Consider a signal consisting of one pure tone.
:math:`V(t) = A \cos(\omega t + \phi)`.
For this single-frequency signal, the amplitude :math:`A` is just the peak value of the signal. This amplitude is related to the two quadratures :math:`I` and :math:`Q` by the following expression,
:math:`A = 2 \sqrt{I^2 + Q^2}`
as described in the section on :ref:`lockin-measurement-label`.
We may also refer to the peak-to-peak amplitude of the signal,
:math:`V_\mathrm{pp} = 2A`
or the root-mean-square (rms) amplitude of the signal,
:math:`V_\mathrm{RMS} = \sqrt{ \frac{1}{T_m}\int_0^{T_m} V^2(t) dt} = \frac{1}{\sqrt{2}}A`
when the :ref:`measurement_time_window` :math:`T_m` is an integer multiple of the period :math:`T=2\pi /\omega`.
The table below summarizes how these various amplitudes are related.
+----------------------------+--------------+---------------------------------------------------------+
| Name | Symbol | Mathematical relationship |
| | | |
+============================+==============+=========================================================+
| amplitude | A | :math:`A` |
+----------------------------+--------------+---------------------------------------------------------+
| quadrature amplitude | Aquad |:math:`\sqrt{I^2 + Q^2} = A/2` :sup:`*` |
+----------------------------+--------------+---------------------------------------------------------+
| peak-to-peak amplitude | Vpp | :math:`V_{pp} = 2A` |
+----------------------------+--------------+---------------------------------------------------------+
| root-mean-square amplitude | Vrms |:math:`V_{rms}=\tfrac{1}{\sqrt{2}}A` :sup:`*` |
+----------------------------+--------------+---------------------------------------------------------+
:sup:`*` :math:`=A` at :math:`\omega=0` or DC
Note that when we build a multifrequency waveform consisting of a superposition (sum) of :math:`N` sinusoidal oscillations,
:math:`V(t) = \sum_{i=0}^N{A_i \cos(\omega_i t + \phi_i)}`
it is not trivial to determine the peak, or maximum voltage of the signal. Each component oscillates with a different frequency and arbitrary phase and one can not generally say how all signals superpose to create the resultant signal.
.. _base_tone:
base tone
---------
The frequency of the base tone is the inverse of the waveform period.
Any periodic waveform is represented on a comb in the frequency-domain,
where all tones in the comb have frequencies that
are integer multiples of the frequency of the base tone. The base tone is the greatest common divisor of all frequencies in the comb.
programing symbol - Df = 1/T
math symbol :math:`\Delta f = 1/T`
.. _frequency_comb:
frequency comb
--------------
We use frequency comb to mean **any** superposition of sine and cosine
oscillations from a discrete set frequencies which are integer multiples of a :ref:`base_tone` :math:`\Delta f`.
:math:`f_i = k_i \Delta f`,
where :math:`k_i` are a set of integers.
The term 'comb' is used because the spectrum of all such tones with equal amplitude,
looks like the instrument used to straighten your hair. The Fourier components of a comb
are referred to as :ref:`tones`.
In optics, frequency combs generated from nonlinear media often have an offset frequency.
:math:`f_i = f_\mathrm{offset} + k_i \Delta f`
The |MLA (TM)| digitally synthesis a comb having a maximum of 40 tones, with :math:`f_\mathrm{offset}=0`, and measures the response these frequencies.
.. _measurement_time_window:
measurement time window
-----------------------
The time window over which Fourier sums, or Lockin calculations are preformed.
For optimal Fourier analysis, all tones of a multifrequency waveform should
have a period which is an integer fraction of the measurement time window.
The measurement time is the inverse of the measurement bandwidth.
programing symbol: - Tm = 1/df
math symbol :math:`T_m = 1 / \delta f`
.. _measurement_bandwidth:
measurement bandwidth
---------------------
The inverse of the time window. Noise power is inversely proportional to
the measurement bandwidth.
programing symbol - df = 1/Tm
math symbol :math:`\delta f = 1 / T_m`
Do not use and depreciate: delta_f, Delta_f, :math:`\Delta f`
.. _pixel:
pixel
-----
A fundamental unit of multifrequency lockin data containing the quadrature
amplitudes at all measured frequencies. Pixel can also refer to the
particular measurement time window on which this lockin data was calculated.
This term comes from scanning probe microscopy, where each mutifrequency
lockin measurement corresponds to one image pixel.
.. _tones:
tones
-----
The components of a multifrequency waveform, or components of a frequency comb.
Tones have frequency, amplitude and phase, or alternatively,
frequency and two quadrature amplitudes.
Sometimes tones are loosely referred to 'frequencies', but strictly speaking, the frequency
is only one of three attributes of a tone.
The tones in the |MLA (TM)|,
are specified by a **tone index** :math:`i = 0,1,2,\dots,N-1`,
where :math:`N` is the number of tones in the |MLA (TM)|.
The **frequency array** maps the tone index to frequency.
programming symbols: freqs = narray * df
- freqs : np.array(dtype=float) [Hz] frequencies of tones.
- narray : np.array(dtype=int) [-] frequencies of tones as multiples of df.
- df : float [Hz] measurement bandwidth
math symbols: :math:`f_i = n_i \delta f`
.. _tuning:
tuning
------
The process of choosing the frequency of all measurement and excitation tones,
as well as the measurement bandwidth, such that all have an integer relation to one base tone.
In mathematical terms, tuning enforces the orthogonality of the drive tones and all
their intermodulation products, on the finite interval of time over which the signal is
integrated.
.. _units:
units
-----
The |MLA (TM)| is a digital instrument that works with discrete numbers.
We use the word 'units' in this context to distinguish which number we refer to:
- PCU - phase counter units, integer between 0 and 2**42.
- ADU - analog to digital units, integer steps of AD converter.
- DAU - digital to analog units, integer steps of DA converter.
The documentation should give physical units in hard brackets e.g. [Hz], unless
the quantity is obviously non-dimensional (i.e. unit-less), in which case one could also write [-].
.. _waveform_period:
waveform period
---------------
The period of a multifrequency signal, for example the period of the drive
waveform or period of the response waveform.
programing symbol - T = 1/Df
math symbol :math:`T=1/\Delta f`