Microcantilever Frequency Noise¶

Detection of Instantaneous Phase. The cantilever signal is

(1)$x(t) = \sqrt{2} \: x_{\text{rms}} \cos{(\omega_0 t + \phi)} + \delta x(t)$

where $$x_{\text{rms}}$$ is the cantilever root mean square amplitude, $$\omega_0$$ is the cantilever frequency, and $$\phi$$ is the cantilever phase. Here $$\delta x(t)$$ is random noise which includes contributions from cantilever thermomechanical fluctuations as well as detector noise.

In order to detect the cantilever frequency we create a quadrature signal by taking the Hilbert transform of the cantilever signal. This procedure gives

(2)$y(t) = \sqrt{2} \: x_{\text{rms}} \sin{(\omega_0 t + \phi)} + \delta y(t)$

where $$\delta y(t)$$ is the Hilbert transform of $$\delta x(t)$$. An expression for $$\delta y(t)$$ can be written down, but it is not instructive. There is a simple relation, however, between $$y$$ and $$x$$ in the Fourier domain:

$\widehat{\delta y}(f) = H(f) \: \widehat{\delta x}(f)$

where $$\widehat{\delta x}(f)$$ indicates the Fourier transform of $$\delta x(t)$$. The function $$H$$ implements the Hilbert transform in Fourier space:

$\begin{split}H(f) = \begin{cases} +\imath & \text{if } f < 0 \\ 0 & \text{if } f = 0 \\ -\imath & \text{if} f > 0 \end{cases}\end{split}$

Since $$H(f) H^{*}(f) = 1$$ (except for the single point at $$f=0$$), it follows that $$\delta y(t)$$ has essentially the same power spectrum as $$\delta x(t)$$.

In our frequency-detection algorithm we measure the instantaneous phase of the cantilever using

(3)$\phi(t) = \arctan{(\frac{y(t)}{x(t)})}$

Substituting equations (1) and (2) into equation (3),

$\phi(t) = \arctan{(\frac{\sqrt{2} \: x_{\text{rms}} \sin{(\omega_0 t + \phi)} + \delta y(t)}{\sqrt{2} \: x_{\text{rms}} \cos{(\omega_0 t + \phi)} + \delta x(t)})}$

Let us now, with the help of Mathematica, expand $$\phi(t)$$ in a Taylor series to first order in both $$\delta y(t)$$ and $$\delta x(t)$$. The result is

$\phi(t) \approx \phi + \omega_0 t - \frac{\delta x(t)}{\sqrt{2} \: x_{\text{rms}}} \sin{(\omega_0 t + \phi)} + \frac{\delta y(t)}{\sqrt{2} \: x_{\text{rms}}} \cos{(\omega_0 t + \phi)}$

We can extract the instantaneous frequency as the slope of the $$\phi(t)$$ versus $$t$$ line. After subtracting away the best-fit line, we are left with phase noise

$\delta \phi(t) = \phi(t) - \omega_0 t - \phi$

given by

(4)$\delta \phi(t) = - \frac{\delta x(t)}{\sqrt{2} \: x_{\text{rms}}} \sin{(\omega_0 t + \phi)} + \frac{\delta y(t)}{\sqrt{2} \: x_{\text{rms}}} \cos{(\omega_0 t + \phi)}$

Phase Noise Power Spectrum. Taking the Fourier transform of $$\delta \phi(t)$$, and switching frequency units

$\begin{split}\begin{gathered} \widehat{\delta \phi}(f) = \frac{1}{\sqrt{2} \: x_{\text{rms}}} \int_{-\infty}^{+\infty} dt \: e^{-2 \pi \imath f t} (- \delta x(t)) \frac{1}{2 \imath} \left( e^{\, 2 \pi \imath f_0 t} e^{\, \imath \, \phi} - e^{-2 \pi \imath f_0 t} e^{-\imath \, \phi} \right) \\ + \frac{1}{\sqrt{2} \: x_{\text{rms}}} \int_{-\infty}^{+\infty} dt \: e^{-2 \pi \imath f t} (\delta y(t)) \frac{1}{2} \left( e^{\, 2 \pi \imath f_0 t} e^{\, \imath \, \phi} + e^{-2 \pi \imath f_0 t} e^{-\imath \, \phi} \right) \end{gathered}\end{split}$

Which can be simplified to

(5)$\begin{split}\begin{gathered} \widehat{\delta \phi}(f) = \frac{1}{\sqrt{2} \: x_{\text{rms}}} \left( -\frac{e^{\, \imath \, \phi}}{2 \imath} \: \widehat{\delta x}(f-f_0) + \frac{e^{-\imath \, \phi}}{2 \imath} \: \widehat{\delta x}(f+f_0) \right. \\ \left. + \frac{e^{\, \imath \, \phi}}{2} \: \widehat{\delta y}(f-f_0) + \frac{e^{-\imath \, \phi}}{2} \: \widehat{\delta y}(f+f_0) \right) \end{gathered}\end{split}$

We can eliminate $$\widehat{\delta y}$$ from equation (5) by recognizing

(6)$\widehat{\delta y}(f+f_0) = \widehat{H}(f+f_0) \: \widehat{\delta x}(f+f_0) = \frac{1}{\imath} \: \widehat{\delta x}(f+f_0)$
(7)$\widehat{\delta y}(f-f_0) = \widehat{H}(f-f_0) \: \widehat{\delta x}(f-f_0) = -\frac{1}{\imath} \: \widehat{\delta x}(f-f_0)$

which holds when $$-f_0 < f < f_0$$; we can arrange for this condition to be met by applying a bandpass filter to the cantilever signal. Substituting equations (6) and (7) into equation (5) gives

(8)$\widehat{\delta \phi}(f) = \frac{1}{\imath} \frac{1}{\sqrt{2} \: x_{\text{rms}}} \left( e^{-\imath \, \phi} \: \widehat{\delta x}(f+f_0) - e^{\, \imath \, \phi} \: \widehat{\delta x}(f-f_0) \right)$

Passing to the power spectrum requires a limiting procedure, as follows. We should consider that $$x(t)$$ is only sampled for a finite amount of time $$T$$, which we can indicate with a subscript: $$x(t) \rightarrow x_{T}(t)$$ where

(9)$\begin{split}x_{T}(t) = \begin{cases} 0 & \text{for } t < 0 \\ x(t) & \text{for } 0 \leq t \leq T \\ 0 & \text{for } T < t \end{cases}\end{split}$

Equation (4) holds with $$\delta x \rightarrow \delta x_T$$, $$\delta x \rightarrow \delta y_T$$, and $$\delta \phi \rightarrow \delta \phi_T$$. Time correlation functions are defined in terms of $$x_T(t)$$, not $$x(t)$$,

$\begin{split}\begin{split} C_x(\tau) & = \lim_{T \rightarrow \infty} \frac{1}{T} \int_{0}^{T} \langle x(t) \: x(t + \tau) \rangle \: dt \\ & = \lim_{T \rightarrow \infty} \frac{1}{T} \int_{-\infty}^{+\infty} \langle x_{T}(t) \: x_{T}(t + \tau) \rangle \: dt \end{split}\end{split}$

where $$\langle \cdots \rangle$$ indicates a statistical average. The manipulations leading to equation (8) are still valid with the $$T$$-subscripted variables, with the result that

(10)$\widehat{\delta \phi}_{T}(f) = \frac{1}{\imath} \frac{1}{\sqrt{2} \: x_{\text{rms}}} \left( e^{-\imath \, \phi} \: \widehat{\delta x}_{T}(f+f_0) - e^{\, \imath \, \phi} \: \widehat{\delta x}_{T}(f-f_0) \right)$

The next step to computing the power spectrum is to calculate

(11)$\begin{split}\begin{gathered} \widehat{\delta \phi}_{T}(f) \: \widehat{\delta \phi}_{T}^{\: *}(f) = \frac{1}{2 \: x_{\text{rms}}^2} \left( e^{-\imath \, \phi} \: \widehat{\delta x}_{T}(f+f_0) - e^{\, \imath \, \phi} \: \widehat{\delta x}_{T}(f-f_0) \right) \\ \left( e^{\, \imath \, \phi} \: \widehat{\delta x}_{T}^{\: *}(f+f_0) - e^{-\imath \, \phi} \: \widehat{\delta x}_{T}^{\: *}(f-f_0) \right) \end{gathered}\end{split}$

We may now pass to the power spectrum by taking the limit

$P_{\delta x}(f) = \lim_{T \rightarrow \infty} \frac{1}{T} \: \widehat{\delta x}_{T}(f) \: \widehat{\delta x}_{T}^{\: *}(f)$

with the power spectrum $$P_{\delta \phi}(f)$$ analogously defined. Carrying out this limiting procedure on both sides of equation (11) yields

$\begin{split}\begin{split} P_{\delta \phi}(f) & = \frac{1}{2 x_{\text{rms}}^2} \left( P_{\delta x}(f+f_0) + P_{\delta x}(f-f_0) \right) \\ & - \frac{1}{2 x_{\text{rms}}^2} \lim_{T \rightarrow \infty} \frac{1}{T} \text{Re} \! \left\{ \widehat{\delta x}_{T}(f+f_0) \: \widehat{\delta x}_{T}^{\: *} (f-f_0) \: e^{-2 \imath \, \phi} \right\} \end{split}\end{split}$

where $$\text{Re} \! \left( \cdots \right)$$ indicates taking the real part. The last term will not survive statistical averaging over the phase $$\phi$$ since

$\frac{1}{2 \pi} \int_{0}^{2 \pi} e^{-2 \imath \, \phi} \: d\phi = 0$

Implicit in this average is the assumption that $$\phi$$ is randomly distributed, that is, there is no correlation between the phase of the cantilever and the cantilever noise. After statistical averaging over $$\phi$$, the power spectrum of cantilever phase noise becomes

(12)$\boxed{P_{\delta \phi}(f) = \dfrac{1}{2 x_{\text{rms}}^2} \left( P_{\delta x}(f+f_0) + P_{\delta x}(f-f_0) \right)}$

Frequency Shift Power Spectrum. Let us define the instantaneous frequency shift as

$\delta f(t) = \frac{1}{2 \pi} \frac{d}{d t} \: \delta \phi(t) = \frac{1}{2 \pi} \delta \dot{\phi}$

and compute the power spectrum of the instantaneous frequency shift. Let us define $$\delta f_{T}(t)$$ as in equation (9). The time-correlation function of the frequency shift is then

$C_{\delta f}(\tau) = \lim_{T \rightarrow \infty} \: \frac{1}{T} \int_{-\infty}^{+\infty} \langle \delta f_{T}(t) \: \delta f_{T}(t+\tau) \rangle \: dt$

with $$C_{\delta \phi}$$ defined likewise. Substituting, and dropping $$\langle \cdots \rangle$$ for notational convenience,

(13)$C_{\delta f}(\tau) = \frac{1}{4 \pi^2} \lim_{T \rightarrow \infty} \: \frac{1}{T} \int_{-\infty}^{+\infty} \langle \delta \dot{\phi}_{T}(t) \: \delta \dot{\phi}_{T}(t+\tau) \rangle \: dt$

The time derivative $$\delta \dot{\phi}$$ may be computing from its Fourier transform. With

$\delta \phi_T(t) = \int_{-\infty}^{+\infty} \widehat{\delta \phi}_{T}(f) \: e^{\, 2 \pi \imath f t} \: df$

we can compute the time derivative of the instantaneous phase shift as

(14)$\delta \dot{\phi}_T(t) = \int_{-\infty}^{+\infty} \widehat{\delta \phi}_{T}(f) \: (2 \pi \imath f) \: e^{\, 2 \pi \imath f t} \: df$

If we substitute equation (14) into equation (13) and use

$\int_{-\infty}^{+\infty} e^{\, 2 \pi \imath (f+f^{\prime}) t} dt = \delta(f+f^{\prime}),$

where $$\delta(t)$$ is the Kroenecker delta function, then

$C_{\delta f}(\tau) = \int_{-\infty}^{+\infty} f^2 \left\{ \lim_{T \rightarrow \infty} \: \frac{1}{T} \: \widehat{\delta \phi}_{T}(f) \: \widehat{\delta \phi}_{T}(-f) \right\} \: e^{-2 \pi \imath f \tau} \: df$

where we have passed the limit into the integral. Because $$\delta \phi_T(t)$$ is a real function,

$\widehat{\delta \phi}_{T}(-f) = \widehat{\delta \phi}_{T}^{\: *}(f)$

The term in braces is thus $$P_{\delta \phi}(f)$$, the power spectrum of phase fluctuations. We find

$C_{\delta f}(\tau) = \int_{-\infty}^{+\infty} f^2 \: P_{\delta \phi}(f) \: e^{-2 \pi \imath f \tau} \: df$

Comparing this to the usual relation between the correlation function and the power spectrum

$C_{\delta f}(\tau) = \int_{-\infty}^{+\infty} P_{\delta f}(f) \: e^{\, 2 \pi \imath f \tau} \: df,$

we see that

(15)$P_{\delta f}(f) = f^2 \: P_{\delta \phi}(f)$

We have used that $$P_{\delta \phi}(-f) = P_{\delta \phi}(f)$$. Substituting equation (15) into equation (12) we conclude that position fluctuations lead to frequency noise having a power spectrum

(16)$\boxed{P_{\delta f}(f) = \dfrac{f^2}{2 x_{\text{rms}}^2} \left( P_{\delta x}(f_0+f) + P_{\delta x}(f_0-f) \right)}$

Instrument Noise. Equation (16) is a general relation between the position-fluctuation power spectrum and the frequency-fluctuation power spectrum. The power spectrum of detector noise is typically flat:

$P_{\delta x}(f_0+f) = P_{\delta x}(f_0-f) \equiv P_{\delta x}^{\text{det}}$

Within this approximation,

(17)$\boxed{P_{\delta f}^{\text{det}}(f) = \dfrac{f^2 \: P_{\delta x}^{\text{det}}}{x_{\text{rms}}^2} \: \sim \: [\dfrac{\text{Hz}^2}{\text{Hz}}] }$

This relation holds whether the power spectra are defined as one-sided or two-sided, as long as the power spectrum is computed consistently on both sides of equation. We typically work up data using a one-sided power spectrum. The more general equation (16) can be used when the detector noise spectrum is not independent of frequency.

Cantilever Thermomechanical Fluctuations. We have previously shown that the (one sided) power spectrum of cantilever position fluctuation is

$P_{\delta z}^{\text{therm}}(f) = \dfrac{k_b T \tau_0^2}{\Gamma} \dfrac{1}{(\pi \tau_0)^4(f_0^2 - f^2)^2 + (\pi \tau_0)^2 f^2}$

where $$T$$ is temperature, $$k_b$$ is Boltzmann’s constant, and $$f_0$$, $$\tau_0$$, and $$\Gamma$$ are cantilever frequency, ring-down time, and dissipation constant, respectively. For frequency offsets $$f \gg f_0 / Q$$ we find that

$P_{\delta z}^{\text{therm}}(f_0 \pm f) \approx \dfrac{k_b T \tau_0^2}{\Gamma} \times \frac{1}{(\pi \tau_0)^4 \: 4 f_0^2 f^2}$

Substituting this result into equation (16) gives

(18)$\boxed{ P_{\delta f}^{\text{therm}}(f) = \dfrac{k_b T}{\Gamma x_{\text{rms}}^2} \dfrac{1}{4 \pi^2} \dfrac{1}{(\pi \tau_0 f_0)^2} \: \sim \: [\dfrac{\text{Hz}^2}{\text{Hz}}] }$

The last term equals $$Q^{-2}$$, where $$Q$$ is the cantilever quality factor. Using $$\Gamma = k /(2 \pi f_0 Q)$$ we can rewrite the one-sided power spectrum of cantilever frequency fluctuations as

(19)$P_{\delta f}^{\text{therm}}(f) = \frac{k_b T}{k x_{\text{rms}}^2} \frac{1}{2 \pi^2 \tau_0}$

Discussion. Equations (17) and (19) agree exactly with what Loring and co-workers have derived [1]. Together, thermomechanical fluctuations and detector noise lead to cantilever frequency noise with a one-sided power spectrum of

$P_{\delta f}(f) = \frac{1}{x_{\mathrm{rms}}^2} \left( \frac{1}{4 \pi^2} \frac{k_b T}{\Gamma} \frac{1}{(\pi \tau_0 f_0)^2} + f^2 P_{\delta x}^{\mathrm{det}} \right)$

This equation is valid for offset frequencies $$f \gg f_0/Q$$ and assumes for simplicity that detector noise is frequency independent in the vicinity of the cantilever resonance frequency.

References

 [1] Yazdanian, S. M.; Marohn, J. A. & Loring, R. F. Dielectric Fluctuations in Force Microscopy: Noncontact Friction and Frequency Jitter. J. Chem. Phys., 2008, 128: 224706 [http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2674627/] [http://dx.doi.org/10.1063/1.2932254] . See equations 6.7 through 6.9.