# Difference between revisions of "User:Pranav Rathi/Notebook/OT/2012/11/13/Optical Tweezers Calibration"

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− | I use 1st order diffraction beam from AOM to feed the tweezers. NI-DAQ controlled by ''feedback96_main_mx'' labview v7.1 program controls the AOM through analog input voltage to AOM driver. So there is a relationship between the analog input voltage and output laser power in 1st order diffraction beam. Unfortunately this relationship is not linear; it is some odd function (characteristic curve; see the second link). Once I know the curve I can calculate the laser power in the trap at particular input voltage. This information is very useful while calculating the stiffness from the | + | I use 1st order diffraction beam from AOM to feed the tweezers. NI-DAQ controlled by ''feedback96_main_mx'' labview v7.1 program controls the AOM through analog input voltage to AOM driver. So there is a relationship between the analog input voltage and output laser power in 1st order diffraction beam. Unfortunately this relationship is not linear; it is some odd function (characteristic curve; see the second link). Once I know the curve I can calculate the laser power in the trap at particular input voltage. This information is very useful while calculating the stiffness from the corner frequency. |

− | It is absolutely unpractical to measure the laser power in the trap before every power-spectrum data is acquired to calculate the stiffness (for stiffness I need | + | It is absolutely unpractical to measure the laser power in the trap before every power-spectrum data is acquired to calculate the stiffness (for stiffness I need corner frequency from power-spectrum and laser power in the trap at which the spectrum is acquired). So, it is done in advance: I record the laser power after the water-immersion objective for RF-input voltage of 1.3 to 4.9 in .2 volts increments. I put a water droplet on the objective and record the power with a power meter directly. I used Thorlabs sensor: model D10MM (S212A 10W) S/N 0938D08 and detector: PM100 S/N M00229006. Reflection loss at water-air interface is still less than 2%. |

[[Image:AOM RF-input voltage Vs laser power.jpg|right|300x200px|AOM RF_input voltage Vs laser power after objective]] | [[Image:AOM RF-input voltage Vs laser power.jpg|right|300x200px|AOM RF_input voltage Vs laser power after objective]] | ||

− | The Pictures shows the data and characteristic curve for AOM RF-input voltage Vs laser power after the water immersion objective. The curve looks exactly the same as AOM characteristic curve. The data is presented below; I measured the laser power for 10 times each. I use the curve and data in ''OT calibration'' program written in labview V9 to calculate the laser power in the trap from RF-input voltage. Laser power and | + | The Pictures shows the data and characteristic curve for AOM RF-input voltage Vs laser power after the water immersion objective. The curve looks exactly the same as AOM characteristic curve. The data is presented below; I measured the laser power for 10 times each. I use the curve and data in ''OT calibration'' program written in labview V9 to calculate the laser power in the trap from RF-input voltage. Laser power and corner frequency gives me the stiffness. To know corner frequency, I usually do power-spectrum at 1.45 volts; at this voltage power measurement error is 8%. That means calculated stiffness accuracy will not be better than 92%. I will discuss the final number later when i discuss the corner frequency and power-spectrum. |

'''AOM RF-input voltage Vs laser power after the objective:''' | '''AOM RF-input voltage Vs laser power after the objective:''' | ||

Line 88: | Line 88: | ||

2) 2X gain is enough for whole range of AOM RF-signal input voltage from 1.4 to 5 Volts. The relationship is linear between the laser powers in the trap Vs QPD sum signal (see the chart). | 2) 2X gain is enough for whole range of AOM RF-signal input voltage from 1.4 to 5 Volts. The relationship is linear between the laser powers in the trap Vs QPD sum signal (see the chart). | ||

− | I took this data set with no bead trapped. Laser power in the trap divided by QPD sum signal gives us a calibration parameter which will help us to calculate the laser power in the trap for data analysis. Data analysis program ''Secret peeking software_KL'' written in labview v7.1 needs three calibration parameters (conversion parameters) to determine the force exerted by the trap. These parameters are; detector sensitivity, trap stiffness and laser power from sum. The laser power from sum parameter (mW/mV) calculates the laser power from QPD sum signal. The laser power (mW) and stiffness (pN/nm/W) gives the stiffness at that laser power at which the data was acquired. Now this stiffness and detector sensitivity (mV/nm) gives the force. | + | I took this data set with no bead trapped. Laser power in the trap divided by QPD sum signal gives us a calibration parameter which will help us to calculate the laser power in the trap for data analysis. Data analysis program ''Secret peeking software_KL'' written in labview v7.1 needs three calibration parameters (conversion parameters) to determine the force exerted by the trap. These parameters are; detector sensitivity, trap stiffness and laser power from sum. The laser power from sum parameter (mW/mV) calculates the laser power from QPD sum signal. The laser power (mW) and stiffness (pN/nm/W) gives the stiffness at that laser power at which the data was acquired. Now this stiffness and displacement by detector sensitivity (mV/nm) gives the force. |

Since the relationship is linear I can average this parameter for all the laser powers, which gives me .0700 (mW/mV) with 2.57% error considering the parameter calculation is done using the laser powers which were measured with different errors. So after doing an error analysis the accuracy of the parameter is 97.4%. | Since the relationship is linear I can average this parameter for all the laser powers, which gives me .0700 (mW/mV) with 2.57% error considering the parameter calculation is done using the laser powers which were measured with different errors. So after doing an error analysis the accuracy of the parameter is 97.4%. | ||

Line 94: | Line 94: | ||

Calibration parameter for '''laser power for sum signal is .0700±2.57% mW/mV'''. | Calibration parameter for '''laser power for sum signal is .0700±2.57% mW/mV'''. | ||

− | 3) 3rd data set is taken with a bead trapped with same settings and procedure as 2nd. The purpose of this data set is to decide which method is more accurate. Calibration parameter by this method is .0722 which is only 3.8% different by previous method. If I do error analysis then these two numbers will overlap within the error range, which makes either method correct. But the sum voltage acquired with this method is less for every laser power by 3%. And while doing this experiment I notice a sudden increase in voltage of sum when a second bead falls in the trap, which suggests my notion for beam divergence because of the bead is true. As two bead trapped along the z-axis of light the cone exiting in the case of two beads is smaller in comparison to single bead, so condenser can collect more light, which suggest this method is less accurate because less light is reaching to QPD. | + | 3) 3rd data set is taken with a bead trapped with same settings and procedure as 2nd. The purpose of this data set is to decide which method is more accurate. Calibration parameter by this method is .0722 which is only 3.8% different by previous method. If I do error analysis then these two numbers will overlap within the error range, which makes either method correct. But the sum voltage acquired with this method is less for every laser power by 3%. And while doing this experiment I notice a sudden increase in voltage of sum when a second bead falls in the trap, which suggests my notion for beam divergence because of the bead is true. As two bead trapped along the z-axis of light the cone exiting in the case of two beads is smaller in comparison to single bead, so condenser can collect more light, which suggest this method is less accurate because less light is reaching to QPD. |

==Electronic filters== | ==Electronic filters== | ||

− | I use Krohn-Hite single-ended input 8-pole low-pass Bessel filters with cutoff at 1.5 kHz for x, y, sum and piezo signal. Even though the filters have cutoff at 1.5 kHz but they affect the power-spectrum generated | + | I use Krohn-Hite single-ended input 8-pole low-pass Bessel filters with cutoff at 1.5 kHz for x, y, sum and piezo signal. Even though the filters have cutoff at 1.5 kHz but they affect the power-spectrum generated corner frequency even at 500Hz. I use x-signal from on-track to generate the power-spectrum. |

+ | [[Image:Krohn-Hite filter rolloff.jpg|right|300x200px]] | ||

+ | To study this behavior I generate an interferogram of sound with frequency components from 10 to 1600 Hz with ''whitesoundmain'' program written in labview V9 and produce it at the output of soundcard of one computer. I feed this sound signal into mic-input of another computer and read the interferogram with ''Noise investigator and helper'' program written in labview V9 (I used the same programs and method to study structural resonance of optical setups). See the slide show for more information. | ||

− | + | I record the data; when there was no filter between the output and input (signal goes straight from one computer to another) and when there is a filter between (signal goes through the filter before input). The result is as expected: Graph is a power spectrum of input interferogram; without filter (Red curve) it is a square function over 1600Hz, with filter (white curve) frequency starts rolling off after 360Hz. This means with filter on power-spectrum taken at frequencies higher than 360Hz will report false corner frequency (low corner frequency will report low stiffness). | |

+ | Fix of this problem is simple; remove the filter when doing power-spectrum. | ||

+ | |||

+ | =Calibration= | ||

+ | Now we have collected all the essential information about the active components of our setup, it is time to proceed for calibration. Optical tweezers calibration is a tedious task, but it is done only once, if no major changes are made to the optical path or components downstream from the objective. | ||

+ | [[Image:OT calibration schematics.jpg|right|300x200px]] | ||

+ | |||

+ | Optical tweezers calibration schematics in image shows all the parameters and programs I use. To determine the force exerted by the trap I need to know two parameters: Stiffness and displacement. | ||

+ | |||

+ | ==Calibration of Trap Stiffness== | ||

+ | The stiffness of the trap is proportional to the corner frequency of the overdamped trapped free bead’s Brownian motion. Basically trap stiffness depends on two parameters: Corner frequency and Stokes drag coefficient. There are multiple ways to determine the trap stiffness but we do it Power-spectrum way. | ||

+ | |||

+ | ===Background=== | ||

+ | When a bead is trapped the mechanics of Brownian motion in a harmonic potential can be exploited to determine the stiffness of the trap. We can assume that the trap exerts a linear spring force on the bead. The one-dimensional equation of motion of a trapped bead is that of a damped driven oscillator: | ||

+ | |||

+ | :<math> | ||

+ | m\ddot{x}(t)+{\beta}\dot{x}(t)+Kx(t)=F(t) | ||

+ | </math> | ||

+ | Where x is position of bead, m is the mass of the bead, β=6πηr is the stokes drag coefficient for a sphere of radius r in a fluid of dynamic viscosity η, K is the stiffness along x-axis. and F(t) is the Brownian forcing term, whose power spectrum is white. | ||

+ | |||

+ | I use 520nm radius polystyrene bead which has a mass of 6.24e-13 gm. Also Reynolds number in the experiments is no greater than 10e-2. For this mass I can ignore the first term and rewrite after Fourier transformation (frequency representation): | ||

− | [ | + | :<math> |

+ | \mathbf | ||

+ | X(f)[K-i2{\pi}f\beta]=F(f) | ||

+ | </math> | ||

+ | here Brownian noise have amplitude of: | ||

− | + | :<math> | |

+ | \mathbf | ||

+ | |F(f)|^2=4\beta k_B T | ||

+ | </math> | ||

+ | where k<sub>B</sub>T is the thermal energy. Square both sides and rearrange: | ||

+ | :<math> | ||

+ | |X(f)|^2=\frac{k_BT}{\pi^2\beta[(\frac{K}{2\pi\beta})^2+f^2]} | ||

+ | </math> | ||

− | + | This is the power-spectrum equation for the bead's motion. This has a Lorentzian roll-off with corner frequency of: | |

+ | :<math> | ||

+ | f_c=\frac{K}{2\pi\beta} | ||

+ | </math> | ||

+ | |||

+ | Stiffness: | ||

+ | |||

+ | :<math> | ||

+ | \mathbf | ||

+ | K=2\pi\beta f_c | ||

+ | </math> | ||

+ | Stiffness depends on the corner frequency f<sub>c</sub> and drag coefficient. Drag coefficent plays an important role | ||

[[Category:Construction of Optical Tweezers and devices]] | [[Category:Construction of Optical Tweezers and devices]] |

## Revision as of 20:22, 21 November 2012

{{#widget:Picasa |user=101108414393941264686 |album=5810773395749044641 |width=600 |height=400 |captions=1 |autoplay=1 }}

To perform accurate measurements with optical tweezers, it is necessary to calibrate some experimental parameters and even before that study the active components of the optical tweezers setup. The parameters I want to calibrate: Stiffness; which measures the force that is exerts on the trapped bead, and Sensitivity; which measures the relative position of bead within the trap. The components I want to characterize: Laser, AOM (acoustooptic module), microscope objective, z and x piezo, lens and mirrors, QPD (quadrant photo diode), and electronic filters. It is necessary to characterize these components first because not only they affect the experimental parameters but also reduce the chances to see something unexpected in data later on.

## Contents

# Characterization

I characterized most of the active components time to time in the setup. The characterization also helped me in designing this tweezers.

## Laser

I use two lasers: 1064nm 2W ND:YAG crystalaser for tweezing and 633nm 2mW He-Ne for surface detection and alignment purposes. It is important to know some details about the source. IR-laser is more important because it is tweezing laser. I did not do a rigorous study (because it was not necessary), just specified few parameters like power output, beam waist and its location, polarization, beam mode profile and beam propagation factor. I used this information to design the tweezers expansion optics. The laser specifications are given in the following link:

User:Pranav_Rathi/Notebook/OT/2010/08/18/CrystaLaser_specifications

## AOM

Acoustooptic modulator is used to modulate the laser intensity in the trap. So it is second most important component of the tweezers. AOM has two components: AOM driver and AOM module, I characterized both. The specifications are given in the link:

- AOM driver

User:Pranav_Rathi/Notebook/OT/2010/01/29/Characterization_of_AOM_driver

- AOM module

User:Pranav_Rathi/Notebook/OT/2010/05/12/Characterization_of_AOM_module

I use 1st order diffraction beam from AOM to feed the tweezers. NI-DAQ controlled by *feedback96_main_mx* labview v7.1 program controls the AOM through analog input voltage to AOM driver. So there is a relationship between the analog input voltage and output laser power in 1st order diffraction beam. Unfortunately this relationship is not linear; it is some odd function (characteristic curve; see the second link). Once I know the curve I can calculate the laser power in the trap at particular input voltage. This information is very useful while calculating the stiffness from the corner frequency.

It is absolutely unpractical to measure the laser power in the trap before every power-spectrum data is acquired to calculate the stiffness (for stiffness I need corner frequency from power-spectrum and laser power in the trap at which the spectrum is acquired). So, it is done in advance: I record the laser power after the water-immersion objective for RF-input voltage of 1.3 to 4.9 in .2 volts increments. I put a water droplet on the objective and record the power with a power meter directly. I used Thorlabs sensor: model D10MM (S212A 10W) S/N 0938D08 and detector: PM100 S/N M00229006. Reflection loss at water-air interface is still less than 2%.

The Pictures shows the data and characteristic curve for AOM RF-input voltage Vs laser power after the water immersion objective. The curve looks exactly the same as AOM characteristic curve. The data is presented below; I measured the laser power for 10 times each. I use the curve and data in *OT calibration* program written in labview V9 to calculate the laser power in the trap from RF-input voltage. Laser power and corner frequency gives me the stiffness. To know corner frequency, I usually do power-spectrum at 1.45 volts; at this voltage power measurement error is 8%. That means calculated stiffness accuracy will not be better than 92%. I will discuss the final number later when i discuss the corner frequency and power-spectrum.

**AOM RF-input voltage Vs laser power after the objective:**

{{#widget:Google Spreadsheet |key=0ApjWjFYiQdkfdElUNmQ5bE1wSktkbzlIdlJOZE9JQnc |width=750 |height=200 }}

*Note: the data is good until no change is made to the optical path/components downstream from the objective that may change the laser power at the objective.*

## Microscope objective

Objective is another very important part of the tweezers. I am using Olympus UPLANSAPO (UIS 2) water immersion IR objective. The objective gives a maximum spot size of 760nm with a Rayleigh range of 567nm. The objective has 55% transmittance (55% of the input laser power makes through) and it has a collar for cover glass correction (spherical aberration gets worse with the depth in the sample, but i do not have to worry about it while doing power spectrum or DOG). The full details of the objective are available here:

User:Pranav_Rathi/Notebook/OT/2010/12/10/Olympus_Water_Immersion_Specs

## X & Z-piezo

We use "Mad city lab" nano position systems; Nano-OP30 for x-piezo and Nano-F25HS for z-piezo. None of the stages required any calibration. Characterization shows that x-piezo sensor output ground from drive has noise of 60Hz and its multiples, but very low.

## Lens and mirrors

All the optics is rated for 1064nm. Some power measurements are as follow:

- Laser power before the AOM: 1.8W
- Laser power after the AOM: 1.7W, 6% transmission loss
- Power in 1st order diffraction beam: 1.238W, 27% transfer loss
- Laser power before the 1st lens of steering assembly: 1.18W, 5% transmission loss through expansion optics and mirrors
- Maximum laser power achievable in the trap: 610mW, 48% transmission loss through lens, dichroic mirror and objective.

## QPD

Infrared-sensitive quadrant photo diode is used as a detector in optical tweezers. Bead position relative to trap center in x-direction is measured by the deflection of the laser beam. This deflection is imaged at the detector-plane by condenser and imaging lens in front of the QPD-detector. I use Hamamatsu S5981 QPD with analog amplifier On Track OT 301. Amplifier measures the current from the four quadrants of the diode and produces normalized position signals and sum signal in voltage.

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle X= \frac{q1-q2-q3+q4}{q1+q2+q3+q4}; Y= \frac{q1+q2-q3-q4}{q1+q2+q3+q4}; Sum= q1+q2+q3+q4 }**

This voltage signal goes through low-pass 8-pole Bessel filters cutoff at 1.5 kHz by Krohn-Hite. The signal is sampled at 13-20 kHz rate by data acquisition box BNC-2111 through PCI-card PCI-6052E by national Instruments.

Calibration data: power-spectrum and detector sensitivity is produced in X, Y and sum voltages by QPD at some laser power. On track amplifier can introduce a gain from 1 to 6 to this signal before it is sampled. So there is a need of characterization of QPD for laser power in the trap Vs voltage signal for various laser powers at different gains. This will help me choosing the right laser power in the trap (at right AOM RF-input voltage) and gain for power-spectrum.

My goal is to get a data set for Laser power in the trap (mW) Vs QPD sum signal (mV) at 1x and 2x gain (3x and above saturates the DAQ; maximum detection voltage is 10V). I will do this for two methods:

- Laser power in trap Vs QPD sum signal without trapped bead.
- Laser power in trap Vs QPD sum signal with trapped bead.

These two methods will help me decide which method is more accurate. In my experience when a bead is trapped, the beam through the trap diverges (expands) much faster. So some of the light is not entirely collected by the condenser and there is a reflection loss too because some of the light is going through total internal reflection inside the bead. The two can drop the voltage signal making this method more inaccurate.

**Laser power in trap (mW) Vs QPD sum signal**:
{{#widget:Google Spreadsheet
|key=0ApjWjFYiQdkfdGlwX3dNNTlja2tZM2l1NHV5WkJ3Z3c
|width=750
|height=200
}}

I took 3 data sets with 1.04 μm(r=520nm) bead sample in water. I put the sample on the tweezers and start recording the sum signal for 1.3 to 4.9 volts in .2 volts increment.

1) 1X gain data set is not useful; a voltage under .1 volt is not registered accurately by DAQ, which makes power-spectrum inaccurate for laser powers below 45mW (AOM RF-signal input 1.5V). See the data in spread sheet.

2) 2X gain is enough for whole range of AOM RF-signal input voltage from 1.4 to 5 Volts. The relationship is linear between the laser powers in the trap Vs QPD sum signal (see the chart).

I took this data set with no bead trapped. Laser power in the trap divided by QPD sum signal gives us a calibration parameter which will help us to calculate the laser power in the trap for data analysis. Data analysis program *Secret peeking software_KL* written in labview v7.1 needs three calibration parameters (conversion parameters) to determine the force exerted by the trap. These parameters are; detector sensitivity, trap stiffness and laser power from sum. The laser power from sum parameter (mW/mV) calculates the laser power from QPD sum signal. The laser power (mW) and stiffness (pN/nm/W) gives the stiffness at that laser power at which the data was acquired. Now this stiffness and displacement by detector sensitivity (mV/nm) gives the force.

Since the relationship is linear I can average this parameter for all the laser powers, which gives me .0700 (mW/mV) with 2.57% error considering the parameter calculation is done using the laser powers which were measured with different errors. So after doing an error analysis the accuracy of the parameter is 97.4%.

Calibration parameter for **laser power for sum signal is .0700±2.57% mW/mV**.

3) 3rd data set is taken with a bead trapped with same settings and procedure as 2nd. The purpose of this data set is to decide which method is more accurate. Calibration parameter by this method is .0722 which is only 3.8% different by previous method. If I do error analysis then these two numbers will overlap within the error range, which makes either method correct. But the sum voltage acquired with this method is less for every laser power by 3%. And while doing this experiment I notice a sudden increase in voltage of sum when a second bead falls in the trap, which suggests my notion for beam divergence because of the bead is true. As two bead trapped along the z-axis of light the cone exiting in the case of two beads is smaller in comparison to single bead, so condenser can collect more light, which suggest this method is less accurate because less light is reaching to QPD.

## Electronic filters

I use Krohn-Hite single-ended input 8-pole low-pass Bessel filters with cutoff at 1.5 kHz for x, y, sum and piezo signal. Even though the filters have cutoff at 1.5 kHz but they affect the power-spectrum generated corner frequency even at 500Hz. I use x-signal from on-track to generate the power-spectrum.

To study this behavior I generate an interferogram of sound with frequency components from 10 to 1600 Hz with *whitesoundmain* program written in labview V9 and produce it at the output of soundcard of one computer. I feed this sound signal into mic-input of another computer and read the interferogram with *Noise investigator and helper* program written in labview V9 (I used the same programs and method to study structural resonance of optical setups). See the slide show for more information.

I record the data; when there was no filter between the output and input (signal goes straight from one computer to another) and when there is a filter between (signal goes through the filter before input). The result is as expected: Graph is a power spectrum of input interferogram; without filter (Red curve) it is a square function over 1600Hz, with filter (white curve) frequency starts rolling off after 360Hz. This means with filter on power-spectrum taken at frequencies higher than 360Hz will report false corner frequency (low corner frequency will report low stiffness).

Fix of this problem is simple; remove the filter when doing power-spectrum.

# Calibration

Now we have collected all the essential information about the active components of our setup, it is time to proceed for calibration. Optical tweezers calibration is a tedious task, but it is done only once, if no major changes are made to the optical path or components downstream from the objective.

Optical tweezers calibration schematics in image shows all the parameters and programs I use. To determine the force exerted by the trap I need to know two parameters: Stiffness and displacement.

## Calibration of Trap Stiffness

The stiffness of the trap is proportional to the corner frequency of the overdamped trapped free bead’s Brownian motion. Basically trap stiffness depends on two parameters: Corner frequency and Stokes drag coefficient. There are multiple ways to determine the trap stiffness but we do it Power-spectrum way.

### Background

When a bead is trapped the mechanics of Brownian motion in a harmonic potential can be exploited to determine the stiffness of the trap. We can assume that the trap exerts a linear spring force on the bead. The one-dimensional equation of motion of a trapped bead is that of a damped driven oscillator:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle m\ddot{x}(t)+{\beta}\dot{x}(t)+Kx(t)=F(t) }**

Where x is position of bead, m is the mass of the bead, β=6πηr is the stokes drag coefficient for a sphere of radius r in a fluid of dynamic viscosity η, K is the stiffness along x-axis. and F(t) is the Brownian forcing term, whose power spectrum is white.

I use 520nm radius polystyrene bead which has a mass of 6.24e-13 gm. Also Reynolds number in the experiments is no greater than 10e-2. For this mass I can ignore the first term and rewrite after Fourier transformation (frequency representation):

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathbf X(f)[K-i2{\pi}f\beta]=F(f) }**

here Brownian noise have amplitude of:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathbf |F(f)|^2=4\beta k_B T }**

where k_{B}T is the thermal energy. Square both sides and rearrange:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle |X(f)|^2=\frac{k_BT}{\pi^2\beta[(\frac{K}{2\pi\beta})^2+f^2]} }**

This is the power-spectrum equation for the bead's motion. This has a Lorentzian roll-off with corner frequency of:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle f_c=\frac{K}{2\pi\beta} }**

Stiffness:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathbf K=2\pi\beta f_c }**

Stiffness depends on the corner frequency f_{c} and drag coefficient. Drag coefficent plays an important role