$F(t)$ is the external driving force\index{$F(t)$}.
During the non-contact phase of calibration,
$F(t)$ comes from random thermal noise.
-
+%
\nomenclature{$\beta$}{Damped harmonic oscillator drag-acceleration
coefficient $\beta \equiv \gamma/m$}
\nomenclature{$\gamma$}{Damped harmonic oscillator drag coefficient
insignificant ($m \rightarrow 0$). This model is commonly used for
optically trapped beads\citep{bechhoefer02}. Because it is simpler
and solutions are more easily
-available\citep{bechhoefer02,grossman05}, it will serve to outline the
-general approach before we dive into the general case.
+available\citep{bechhoefer02,burnham03,grossman05}, it will serve to
+outline the general approach before we dive into the general case.
Fourier transforming \cref{eq:DHO} with $m=0$ and applying
\cref{eq:four-deriv} we have
Theorem), we can write the one sided thermal power spectral density
per unit time as
\begin{equation}
- \PSD(F, \omega) = G_0
+ G_0 \equiv \PSD(F, \omega)
= \normLimT 2 \magSq{F(\omega)} \;. \label{eq:GOdef} % label O != zero
\end{equation}
+%
+\nomenclature{$G_0$}{The power spectrum of the thermal noise in
+ angular frequency space (\cref{eq:GOdef}).}
Plugging \cref{eq:GOdef} into \cref{eq:ODHO-psd-F} we have
\begin{equation}
\iOInf{\omega}{\PSD(x, \omega)}
= \iOInf{\omega}{\frac{G_0}{\kappa^2 + \gamma^2\omega^2}}
= \frac{G_0}{\gamma}\iOInf{z}{\frac{1}{\kappa^2 + z^2}}
- = \frac{G_0 \pi}{2 \gamma \kappa} \;,
+ = \frac{\pi G_0}{2 \gamma \kappa} \;,
\label{eq:ODHO-psd-int}
\end{align}
-where the integral is solved in \cref{sec:integrals:highly-damped}.
+where we made the simplifying replacement $z\equiv\gamma\omega$, so
+$\dd \omega = \dd z/\gamma$. The integral is solved in
+\cref{sec:integrals:highly-damped}.
Plugging into our corollary to Parseval's theorem (\cref{eq:parseval-var}),
\begin{equation}
- \avg{x(t)^2} = \frac{G_0 \pi}{2 \gamma \kappa} \;. \label{eq:ODHO-var}
+ \avg{x(t)^2} = \frac{\pi G_0}{2 \gamma \kappa} \;. \label{eq:ODHO-var}
\end{equation}
Plugging \cref{eq:ODHO-var} into \cref{eq:equipart} we have
\begin{align}
- \kappa \frac{G_0 \pi}{2 \gamma \kappa} &= k_BT \\
+ \kappa \frac{\pi G_0}{2 \gamma \kappa} &= k_BT \\
G_0 &= \frac{2 \gamma k_BT}{\pi} \;. \label{eq:ODHO-GO}
\end{align}
Combining \cref{eq:ODHO-psd-GO,eq:ODHO-GO}, we expect $x(t)$ to have a
-power spectral density per unit time given by
+power spectral density per unit time given by\footnote{%
+ \cref{eq:ODHO-psd} is Eq.~(A12) from \citet{bechhoefer02} (who's
+ $\tau_0\equiv\gamma/\kappa$), except that they're missing a factor
+ of $1/\pi$.
+ \cref{eq:ODHO-psd} is also Eq.~(8) from \citet{burnham03}, where
+ their damping coefficient $b$ is equivalent to our $\gamma$, their
+ frequency $\nu$ is equivalent to our $f=\omega/2\pi$, and their roll
+ off frequency $\nu_R\equiv k/2\pi b$ is equivalent to our
+ $\kappa/2\pi\gamma$.
+}
\begin{equation}
\PSD(x, \omega) = \frac{2}{\pi}
\cdot
\index{PSD@\PSD}
Plugging \cref{eq:GOdef} into \cref{eq:DHO-psd-F} we have
-\begin{align}
- \PSD(x, \omega) &= \frac{G_0/m^2}{(\omega_0^2-\omega^2)^2 +\beta^2\omega^2}\;,
- \label{eq:model-psd} \\
- &= \frac{G_1}{(\omega_0^2-\omega^2)^2 +\beta^2\omega^2} \;,
- \label{eq:model-psd-Gone}
-\end{align}
-where $G_1\equiv G_0/m^2$ consolidates the unknown fitting parameters
-without loss of generality.
+\begin{equation}
+ \PSD(x, \omega) = \frac{G_0/m^2}{(\omega_0^2-\omega^2)^2 +\beta^2\omega^2}\;,
+ \label{eq:model-psd} \;.
+\end{equation}
Integrating over positive $\omega$ to find the total power per unit
time yields
-\begin{align}
+\begin{equation}
\iOInf{\omega}{\PSD(x, \omega)}
- &= \frac{G_0}{2m^2}
- \iInfInf{\omega}{\frac{1}{(\omega_0^2-\omega^2)^2 + \beta^2\omega^2}}
- = \frac{G_0}{2m^2} \cdot \frac{\pi}{\beta\omega_0^2}
- = \frac{G_0 \pi}{2m^2\beta\omega_0^2}
- = \frac{G_0 \pi}{2m^2\beta \frac{\kappa}{m}} \\
- &= \frac{G_0 \pi}{2m \beta \kappa} \;,
+ = \frac{G_0}{2m^2}
+ \iInfInf{\omega}{\frac{1}{(\omega_0^2-\omega^2)^2 + \beta^2\omega^2}}
+ = \frac{G_0}{2m^2} \cdot \frac{\pi}{\beta\omega_0^2}
+ = \frac{\pi G_0}{2m^2 \beta \omega_0^2}
\label{eq:DHO-psd-int}
-\end{align}
-where the integration is solved in \cref{sec:integrals:general}.
+\end{equation}
+where the integration is solved in \cref{sec:integrals:general}\footnote{
+ Comparing \cref{eq:ODHO-psd-int,eq:DHO-psd-int}, we see
+ \begin{equation}
+ \frac{\pi G_0}{2m^2 \beta \omega_0^2}
+ = \frac{\pi G_0}{2m^2 \frac{\gamma}{m} \frac{k}{m}}
+ = \frac{\pi G_0}{2 \gamma \kappa} \;.
+ \end{equation}
+ This is not a coincidence. Both spectra satisfy the equipartion
+ theorem, so
+ \begin{equation}
+ \iOInf{\omega}{\PSD(x, \omega)} = \avg{x(t)^2} = \frac{k_BT}{\kappa} \;,
+ \end{equation}
+ which is the same for both cases.
+}.
By the corollary to Parseval's theorem (\cref{eq:parseval-var}), we have
-\begin{align}
+\begin{equation}
\avg{x(t)^2}
- &= \frac{G_0 \pi}{2m^2\beta\omega_0^2} \;, \label{eq:DHO-var} \\
- &= \frac{G_1 \pi}{\beta\omega_0^2} \;, \label{eq:DHO-var-Gone}
-\end{align}
+ = \frac{\pi G_0}{2m^2 \beta \omega_0^2} \;. \label{eq:DHO-var}
+\end{equation}
Plugging \cref{eq:DHO-var} into the equipartition theorem
-(\cref{eq:equipart}) we have
+(\cref{eq:equipart}) we can reproduce \cref{eq:ODHO-GO}.
\begin{align}
- \kappa \frac{G_0 \pi}{2m \beta \kappa} &= k_BT \, \\
- G_0 &= \frac{2}{\pi} k_BT m \beta \;, \label{eq:GO} \\
- G_1 &\equiv \frac{G_0}{m^2} = \frac{2}{\pi m} k_BT \beta \;. \label{eq:Gone}
+ \kappa \frac{\pi G_0}{2m^2 \beta \omega_0^2} &= k_BT \, \\
+ G_0 &= \frac{2m^2 \beta \omega_0^2 k_BT}{\pi \kappa}
+ = \frac{2m^2 \beta \frac{\kappa}{m} k_BT}{\pi \kappa}
+ = \frac{2m \beta k_BT}{\pi}
+ = \frac{2m \frac{\gamma}{m} k_BT}{\pi}
+ = \frac{2 \gamma k_BT}{\pi} \;. \label{eq:GO}
\end{align}
Combining \cref{eq:model-psd,eq:GO}, we expect $x(t)$ to have a power
\cref{eq:ODHO-psd} from the general form \cref{eq:DHO-psd}. Plugging
in for $\beta\equiv\gamma/m$ and $\omega_0\equiv\sqrt{\kappa/m}$,
\begin{align}
- \lim_{m\rightarrow 0} \PSD(x, \omega)
- &= \lim_{m\rightarrow 0} \frac{2 k_BT \gamma}
+ \limX{m}{0} \PSD(x, \omega)
+ &= \limX{m}{0} \frac{2 k_BT \gamma}
{ \pi m^2 \p[{\p({\frac{\kappa}{m}-\omega^2})^2 + \frac{\gamma^2}{m^2}\omega^2}] }
- = \lim_{m\rightarrow 0} \frac{2 k_BT \gamma}
+ = \limX{m}{0} \frac{2 k_BT \gamma}
{ \pi \p[{(\kappa-m\omega^2)^2 + \gamma^2\omega^2}] } \\
&= \frac{2}{\pi}
\cdot
+ \omega_0^2 \frac{V_p}{\sigma_p}
&= F(t) \\
\ddt{V_p} + \beta\dt{V_p} + \omega_0^2 V_p
- &= \sigma_p\frac{F(t)}{m} \\
+ &= \sigma_p\frac{F(t)}{m} \label{eq:DHO-ddt-Vp} \\
\ddt{V_p} + \beta\dt{V_p} + \omega_0^2 V_p
- &= \frac{F(t)}{m_p} \;,
+ &= \frac{F_p(t)}{m} \;,
\end{align}
-where $m_p\equiv m/\sigma_p$. This has the same form as
+where $F_p(t)\equiv \sigma_p F(t)$. This has the same form as
\cref{eq:DHO}, which can be rearranged to:
\begin{align}
\ddt{x} + \frac{\gamma}{m} \dt{x} + \frac{\kappa}{m} x &= \frac{F(t)}{m} \\
\ddt{x} + \beta \dt{x} + \omega_0^2 x &= \frac{F(t)}{m} \;,
\end{align}
so the \PSD\ of $V_p(t)$ will be the same as the \PSD\ of $x(t)$,
-after the replacements $x\rightarrow V_p(t)$ and $m\rightarrow m_p$.
-Making these replacements in \cref{eq:model-psd-Gone,eq:DHO-var-Gone},
-we have
+after the replacements $x\rightarrow V_p(t)$, $F\rightarrow F_p$, and
+(because of \cref{eq:GOdef}) $G_0\rightarrow\sigma_p^2G_0$. Making
+these replacements in \cref{eq:model-psd,eq:DHO-var}, we have
\begin{align}
- \PSD(V_p, \omega) &= \frac{G_{1p}}
+ \PSD(V_p, \omega) &= \frac{\sigma_p^2 G_0/m^2}
{ (\omega_0^2-\omega^2)^2 + \beta^2\omega^2 } \\
- \avg{V_p(t)^2} &= \frac{\pi G_{1p}}{2\beta\omega_0^2}
- = \frac{\pi \sigma_p^2 G_{1}}{2\beta\omega_0^2}
- = \sigma_p^2 \avg{x(t)^2} \;,
+ \avg{V_p(t)^2} &= \frac{\pi \sigma_p^2 G_0}{2 m^2 \beta \omega_0^2}
+ = \sigma_p^2 \avg{x(t)^2} \;.
+\end{align}
+The scaling parameters cannot be independently fit though, so lets
+condense the power spectrum of the right hand side of
+\cref{eq:DHO-ddt-Vp} into a single
+\begin{equation}
+ G_1 \equiv \frac{\sigma_p^2 G_0}{m^2} \;. \label{eq:Gone-def}
+\end{equation}
+This gives
+\begin{align}
+ \PSD(V_p, \omega)
+ &= \frac{G_1}{ (\omega_0^2-\omega^2)^2 + \beta^2\omega^2 }
+ \label{eq:psd-Vp-Gone} \\
+ \avg{V_p(t)^2} &= \frac{\pi G_1}{2 \beta \omega_0^2}
+ = \sigma_p^2 \avg{x(t)^2} \;.
+ \label{eq:avg-Vp-Gone}
\end{align}
-where $G_{1p}\equiv G_0/m_p^2=\sigma_p^2 G_1$.
+%
+\nomenclature{$G_1$}{The scaled power spectrum of the thermal noise in
+ angular frequency space (\cref{eq:Gone-def}).}
+
Plugging into the equipartition theorem (\cref{eq:equipart_k}) yields
\begin{align}
\kappa &= \frac{\sigma_p^2 k_BT}{\avg{V_p(t)^2}}
- = \frac{2 \beta\omega_0^2 \sigma_p^2 k_BT}{\pi G_{1p}} \;.
+ = \frac{2 \beta \omega_0^2 \sigma_p^2 k_BT}{\pi G_1} \;.
\end{align}
-
-From \cref{eq:Gone}, we find the expected value of $G_{1p}$ to be
+Shifting this around, we can find the expected value of $G_1$.
\begin{equation}
- G_{1p} \equiv \sigma_p^2 G_1 = \frac{2}{\pi m} \sigma_p^2 k_BT \beta \;.
- \label{eq:Gone-p}
+ G_1 = \frac{2 \beta \omega_0^2 \sigma_p^2 k_BT}{\pi \kappa}
+ = \frac{2 \beta \frac{\kappa}{m} \sigma_p^2 k_BT}{\pi \kappa}
+ = \frac{2 \beta \sigma_p^2 k_BT}{\pi m}
+ \label{eq:Gone}
\end{equation}
\subsection{Fitting deflection voltage in frequency space}
= \iOInf{f}{\frac{1}{2\pi}\PSD_f(x,f)2\pi\cdot}
= \iOInf{f}{\PSD_f(x,f)} \;.
\end{align}
-We can now extract \cref{eq:psd-Vp,eq:Vp-from-freq-fit}.
+We can now extract \cref{eq:psd-Vp,eq:Vp-from-freq-fit} from
+\cref{eq:psd-Vp-Gone,eq:avg-Vp-Gone}.
\begin{align}
\begin{split}
- \PSD_f(V_p, f) &= 2\pi\PSD(V_p,\omega)
- = \frac{2\pi G_{1p}}{(4\pi f_0^2-4\pi^2f^2)^2 + \beta^2 4\pi^2f^2}
- = \frac{2\pi G_{1p}}{16\pi^4(f_0^2-f^2)^2 + \beta^2 4\pi^2f^2} \\
- &= \frac{G_{1p}/8\pi^3}{(f_0^2-f^2)^2 + \frac{\beta^2 f^2}{4\pi^2}}
+ \PSD_f(V_p, f) &= 2\pi\PSD(V_p, \omega)
+ = \frac{2\pi G_1}{(4\pi^2f_0^2-4\pi^2f^2)^2 + \beta^2 4\pi^2f^2}
+ = \frac{2\pi G_1}{16\pi^4(f_0^2-f^2)^2 + \beta^2 4\pi^2f^2} \\
+ &= \frac{G_1/8\pi^3}{(f_0^2-f^2)^2 + \frac{\beta^2 f^2}{4\pi^2}}
= \frac{G_{1f}}{(f_0^2-f^2)^2 + \beta_f^2 f^2}
\end{split} \\
- \avg{V_p(t)^2} &= \frac{\pi G_{1f}}{2\beta_f f_0^2} \;.
-% = \frac{\pi G_{1p} / (2\pi)^3}{2\beta/(2\pi) \omega_0^2/(2\pi)^2}
-% = \frac{\pi G_{1p}}{2\beta\omega_0^2} = \avg{V_p(t)^2} % check!
+ \avg{V_p(t)^2}
+ &= \frac{\pi \frac{G_1}{(2\pi)^3}}
+ {2 \frac{\beta}{2\pi} \p({\frac{\omega_0}{2\pi}})^2}
+ = \frac{\pi G_{1f}}{2 \beta_f f_0^2} \;.
\end{align}
where $f_0\equiv\omega_0/2\pi$, $\beta_f\equiv\beta/2\pi$, and
-$G_{1f}\equiv G_{1p}/8\pi^3$. Finally, we can generate
-\cref{eq:kappa}.
+$G_{1f}\equiv G_1/8\pi^3$. Finally, we can generate
+\cref{eq:kappa} from \cref{eq:equipart_k,eq:x-from-Vp}.
\begin{align}
\kappa &= \frac{\sigma_p^2 k_BT}{\avg{V_p(t)^2}}
= \frac{2 \beta_f f_0^2 \sigma_p^2 k_BT}{\pi G_{1f}} \;.
\end{align}
-From \cref{eq:Gone-p}, we expect $G_{1f}$ to be
+Shifting this around, we can find the expected value of $G_{1f}$.
\begin{equation}
- G_{1f} = \frac{G_{1p}}{8\pi^3}
- = \frac{\frac{2}{\pi m} \sigma_p^2 k_BT \beta}{8\pi^3}
- = \frac{\sigma_p^2 k_BT \beta}{4\pi^4 m} \;.
+ G_1 = \frac{2 \beta_f f_0^2 \sigma_p^2 k_BT}{\pi \kappa}
+ = \frac{2 \beta_f \frac{\kappa}{4\pi^2 m} \sigma_p^2 k_BT}{\pi \kappa}
+ = \frac{\beta_f \sigma_p^2 k_BT}{2\pi^3 m}
\label{eq:Gone-f}
\end{equation}
+Plugging \cref{eq:Gone-f} into \cref{eq:psd-Vp}, we have
+\begin{equation}
+ \PSD_f(V_p, f) = \frac{\sigma_p^2 k_BT \beta_f}{2\pi^3 m} \cdot
+ \frac{1}{(f_0^2-f^2)^2 + \beta_f^2 f^2}
+\end{equation}
+From which we can recover \citet{burnham03}'s Eq.~(6).
+\begin{align}
+ \PSD_f(x, f) &= \frac{\PSD_f(V_p, f)}{\sigma_p^2}
+ = \frac{k_BT \colA{\beta_f}}{2\pi^3 m} \cdot
+ \frac{1}{(f_0^2-f^2)^2 + \colA{\beta_f^2} f^2} \\
+ &= \frac{k_BT \colAB{f_0}}{2\pi^3 m \colA{Q}} \cdot
+ \frac{1}{\colB{(f_0^2}-f^2)^2 + \frac{\colAB{f_0^2}f^2}{\colA{Q^2}}}
+ = \frac{k_BT}{2\pi^3 m Q \colAB{f_0^3}} \cdot
+ \frac{1}{(\colB{1}-\frac{f^2}{\colB{f_0^2}})^2 +
+ \frac{f^2}{\colB{f_0^2}Q^2}} \\
+ &= \frac{k_BT}{\colB{2\pi^3} m Q \colAB{\p({\frac{\omega_0}{2\pi}})^3}}
+ \cdot
+ \frac{1}{(1-\frac{f^2}{f_0^2})^2 + \frac{f^2}{f_0^2Q^2}}
+ = \frac{\colB{4}k_BT}{m Q \colB{\omega_0}\colAB{\omega_0^2}} \cdot
+ \frac{1}{(1-\frac{f^2}{f_0^2})^2 + \frac{f^2}{f_0^2Q^2}} \\
+ &= \frac{4k_BT}{\colB{m} Q \omega_0\frac{\colA{\kappa}}{\colAB{m}}} \cdot
+ \frac{1}{(1-\frac{f^2}{f_0^2})^2 + \frac{f^2}{f_0^2Q^2}}
+ = \frac{4 k_BT}{\omega_0 Q \kappa}
+ \frac{1}{(1-\frac{f^2}{f_0^2})^2 + \frac{f^2}{f_0^2Q^2}} \;,
+ \label{eq:psd-f-x}
+\end{align}
+where $Q$ is the quality factor\citep{burnham03}
+\begin{equation}
+ Q \equiv \frac{\sqrt{\kappa m}}{\gamma}
+ = \sqrt{\frac{\kappa}{m}}\frac{m}{\gamma}
+ = \frac{\omega_0}{\beta}
+ = \frac{2\pi f_0}{2\pi\beta_f}
+ = \frac{f_0}{\beta_f} \;.
+ \label{eq:Q}
+\end{equation}
+%
+\nomenclature{$Q$}{Quality factor of a damped harmonic oscillator.
+ $Q\equiv \frac{\sqrt{\kappa m}}{\gamma}$ (\cref{eq:Q}).}
% TODO: re-integrate the following
projects / thesis.git / commitdiff