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# 3 Theoretical power spectral density for a damped harmonic oscillator

Our cantilever can be approximated as a damped harmonic oscillator (28)

where is the displacement from equilibrium, is the effective mass, is the effective drag coefficient, is the spring constant, and is the external driving force. During the non-contact phase of calibration, comes from random thermal noise.

In the following analysis, we use the unitary, angular frequency Fourier transform normalization (29)

We also use the following theorems (proved elsewhere):  , (30)  , (31)  (Parseval's). (32)

As a corollary to Parseval's theorem, we note that the one sided power spectral density per unit time ( ) defined by  (33)

relates to the variance by  (34)

where is the total time over which data has been aquired.

We also use the Wiener-Khinchin theorem, which relates the two sided power spectral density to the autocorrelation function via  (Wiener-Khinchin), (35)

where is defined in terms of the expectation value  (36)

and represents the complex conjugate of .

## 3.1 Highly damped case

For highly damped systems, the inertial term becomes insignificant ( ). This model is commonly used for optically trapped beads. Because it is simpler and solutions are more easily available, we'll use it outline the general approach before diving into the general case.

Fourier transforming eq. 28 with and applying 31 we have  (37)  (38)

We compute the by plugging eq. 38 into 33 (39)

Because thermal noise is white (not autocorrelated + Wiener-Khinchin Theorem), we can denote the one sided thermal power spectral density per unit time by (40)

Plugging eq. 40 into 39 we have (41)

This is the formula we would use to fit our measured , but let us go a bit farther to find the expected and thermal noise given , and .

Integrating over positive to find the total power per unit time yields  (42) (43) (44)

where the integral is solved in Section 5.

Plugging into our corollary to Parseval's theorem (eq. 34), (45)

Plugging eq. 45 into eqn. 1 we have  (46)  (47)

So we expect to have a power spectral density per unit time given by (48)

## 3.2 General form

The procedure here is exactly the same as the previous section. The integral normalizing just become a little more complicated...

Fourier transforming eq. 28 and applying 31 we have  (49)  (50)  (51)

where is the resonant angular frequency and is the drag-acceleration coefficient.

We compute the by plugging eq. 51 into 33 (52)

Plugging eq. 40 into 52 we have (53)

Integrating over positive to find the total power per unit time yields  (54) (55) (56) (57) (58)

The integration is detailed in Section 5. By our corollary to Parseval's theorem (eq. 34), we have (59)

Plugging eq. 59 into the equipartition theorem (eqn. 1) we have  (60)  (61)

So we expect to have a power spectral density per unit time given by (62)   Next: 4 Contour integration Up: cantilever_calib Previous: 2 Methods