.. _FreqNoise: Microcantilever Frequency Noise =============================== **Detection of Instantaneous Phase**. The cantilever signal is .. math:: :label: Eq:x x(t) = \sqrt{2} \: x_{\text{rms}} \cos{(\omega_0 t + \phi)} + \delta x(t) where :math:`x_{\text{rms}}` is the cantilever root mean square amplitude, :math:`\omega_0` is the cantilever frequency, and :math:`\phi` is the cantilever phase. Here :math:`\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 .. math:: :label: Eq:y y(t) = \sqrt{2} \: x_{\text{rms}} \sin{(\omega_0 t + \phi)} + \delta y(t) where :math:`\delta y(t)` is the Hilbert transform of :math:`\delta x(t)`. An expression for :math:`\delta y(t)` can be written down, but it is not instructive. There is a simple relation, however, between :math:`y` and :math:`x` in the Fourier domain: .. math:: \widehat{\delta y}(f) = H(f) \: \widehat{\delta x}(f) where :math:`\widehat{\delta x}(f)` indicates the Fourier transform of :math:`\delta x(t)`. The function :math:`H` implements the Hilbert transform in Fourier space: .. math:: H(f) = \begin{cases} +\imath & \text{if } f < 0 \\ 0 & \text{if } f = 0 \\ -\imath & \text{if} f > 0 \end{cases} Since :math:`H(f) H^{*}(f) = 1` (except for the single point at :math:`f=0`), it follows that :math:`\delta y(t)` has essentially the same power spectrum as :math:`\delta x(t)`. In our frequency-detection algorithm we measure the instantaneous phase of the cantilever using .. math:: :label: Eq:phidef \phi(t) = \arctan{(\frac{y(t)}{x(t)})} Substituting equations :eq:`Eq:x` and :eq:`Eq:y` into equation :eq:`Eq:phidef`, .. math:: \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 :math:`\phi(t)` in a Taylor series to first order in *both* :math:`\delta y(t)` and :math:`\delta x(t)`. The result is .. math:: \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 :math:`\phi(t)` versus :math:`t` line. After subtracting away the best-fit line, we are left with phase noise .. math:: \delta \phi(t) = \phi(t) - \omega_0 t - \phi given by .. math:: :label: Eq:dphi \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 :math:`\delta \phi(t)`, and switching frequency units .. math:: \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} Which can be simplified to .. math:: :label: Eq:deltaphiintermediate \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} We can eliminate :math:`\widehat{\delta y}` from equation :eq:`Eq:deltaphiintermediate` by recognizing .. math:: :label: Eq:deltaysimp1 \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) .. math:: :label: Eq:deltaysimp2 \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 :math:`-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 :eq:`Eq:deltaysimp1` and :eq:`Eq:deltaysimp2` into equation :eq:`Eq:deltaphiintermediate` gives .. math:: :label: Eq:FTdeltaphi \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 :math:`x(t)` is only sampled for a finite amount of time :math:`T`, which we can indicate with a subscript: :math:`x(t) \rightarrow x_{T}(t)` where .. math:: :label: Eq:xT 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} Equation :eq:`Eq:dphi` holds with :math:`\delta x \rightarrow \delta x_T`, :math:`\delta x \rightarrow \delta y_T`, and :math:`\delta \phi \rightarrow \delta \phi_T`. Time correlation functions are defined in terms of :math:`x_T(t)`, not :math:`x(t)`, .. math:: \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} where :math:`\langle \cdots \rangle` indicates a statistical average. The manipulations leading to equation :eq:`Eq:FTdeltaphi` are still valid with the :math:`T`-subscripted variables, with the result that .. math:: :label: Eq:FTdeltaphiT \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 .. math:: :label: Eq:PdeltaphiTintermediate \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} We may now pass to the power spectrum by taking the limit .. math:: 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 :math:`P_{\delta \phi}(f)` analogously defined. Carrying out this limiting procedure on both sides of equation :eq:`Eq:PdeltaphiTintermediate` yields .. math:: \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} where :math:`\text{Re} \! \left( \cdots \right)` indicates taking the real part. The last term will not survive statistical averaging over the phase :math:`\phi` since .. math:: \frac{1}{2 \pi} \int_{0}^{2 \pi} e^{-2 \imath \, \phi} \: d\phi = 0 Implicit in this average is the assumption that :math:`\phi` is randomly distributed, that is, there is no correlation between the phase of the cantilever and the cantilever noise. After statistical averaging over :math:`\phi`, the power spectrum of cantilever phase noise becomes .. math:: :label: Eq:Pdeltaphi \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 .. math:: \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 :math:`\delta f_{T}(t)` as in equation :eq:`Eq:xT`. The time-correlation function of the frequency shift is then .. math:: 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 :math:`C_{\delta \phi}` defined likewise. Substituting, and dropping :math:`\langle \cdots \rangle` for notational convenience, .. math:: :label: Eq:Cdeltaf 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 :math:`\delta \dot{\phi}` may be computing from its Fourier transform. With .. math:: \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 .. math:: :label: Eq:deltadotphiT \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 :eq:`Eq:deltadotphiT` into equation :eq:`Eq:Cdeltaf` and use .. math:: \int_{-\infty}^{+\infty} e^{\, 2 \pi \imath (f+f^{\prime}) t} dt = \delta(f+f^{\prime}), where :math:`\delta(t)` is the Kroenecker delta function, then .. math:: 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 :math:`\delta \phi_T(t)` is a real function, .. math:: \widehat{\delta \phi}_{T}(-f) = \widehat{\delta \phi}_{T}^{\: *}(f) The term in braces is thus :math:`P_{\delta \phi}(f)`, the power spectrum of phase fluctuations. We find .. math:: 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 .. math:: C_{\delta f}(\tau) = \int_{-\infty}^{+\infty} P_{\delta f}(f) \: e^{\, 2 \pi \imath f \tau} \: df, we see that .. math:: :label: Eq:PdeltafPdeltaphi P_{\delta f}(f) = f^2 \: P_{\delta \phi}(f) We have used that :math:`P_{\delta \phi}(-f) = P_{\delta \phi}(f)`. Substituting equation :eq:`Eq:PdeltafPdeltaphi` into equation :eq:`Eq:Pdeltaphi` we conclude that position fluctuations lead to frequency noise having a power spectrum .. math:: :label: Eq:Pdeltafresult \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 :eq:`Eq:Pdeltafresult` 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: .. math:: P_{\delta x}(f_0+f) = P_{\delta x}(f_0-f) \equiv P_{\delta x}^{\text{det}} Within this approximation, .. math:: :label: Eq:PdeltaxDet \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 :eq:`Eq:Pdeltafresult` 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 .. math:: 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 :math:`T` is temperature, :math:`k_b` is Boltzmann’s constant, and :math:`f_0`, :math:`\tau_0`, and :math:`\Gamma` are cantilever frequency, ring-down time, and dissipation constant, respectively. For frequency offsets :math:`f \gg f_0 / Q` we find that .. math:: 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 :eq:`Eq:Pdeltafresult` gives .. math:: :label: Eq:PdeltaxTherm \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 :math:`Q^{-2}`, where :math:`Q` is the cantilever quality factor. Using :math:`\Gamma = k /(2 \pi f_0 Q)` we can rewrite the one-sided power spectrum of cantilever frequency fluctuations as .. math:: :label: Eq:PdeltaxTherm2 P_{\delta f}^{\text{therm}}(f) = \frac{k_b T}{k x_{\text{rms}}^2} \frac{1}{2 \pi^2 \tau_0} **Discussion**. Equations :eq:`Eq:PdeltaxDet` and :eq:`Eq:PdeltaxTherm2` agree *exactly* with what Loring and co-workers have derived [#Yazdanian2008jun]_. Together, thermomechanical fluctuations and detector noise lead to cantilever frequency noise with a one-sided power spectrum of .. math:: 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 :math:`f \gg f_0/Q` and assumes for simplicity that detector noise is frequency independent in the vicinity of the cantilever resonance frequency. **References** .. [#Yazdanian2008jun] 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.