Electronic detection: Quadrant photodiode

Detecting the position of a trapped particle electronically with a quadrant photo- diode (QPD) is very fast with acquisition rates up to 50kHz. This circumvents the inherent low-pass filtering of video detection which conceals detailed information about the fast moving metal nanoparticles. Therefore we decided to use a QPD for position detection and trap stiffness measurements of our tweezing experiments described in more detail in Chapter 4. Contrary to our Laguerre-Gaussian trap experiment, we aimed to measure the precise position of the trapped particle. The particle was confined to a much smaller space in our tweezing experiments that would have been even harder to detect with video analysis. A QPD is a photo detector made up of four individual photodiodes shaped as quadrants that form a circle. Figure 3.18 shows a picture of the InGaAs quadrant photodiode (infrared sensitive) we used in our experiment.

Implementing a QPD in our setup requires an additional optical train for detec- tion. After passing the sample chamber, the trapping laser is collected by a second objective (Nikon Plan Apo, 40x,NA 0.65). A telescope relays the condenser back focal plane on the QPD (the QPD is in a position optically conjugate to the con- denser back focal plane) and matches the size of the back aperture to the diameter of the photo detector. This technique is called back focal plane interferometry and used in most nanoparticle detection experiments [13, 14, 22, 38].

Back focal plane interferometry images the far field interference pattern of the trapping laser and the scattered laser light from the trapped object on the QPD. This method measures the particle’s position relative to the trap. As the trapping laser is also the detection laser, large temporal bandwidths are achievable. This is one of the key advantages compared to video detection of trapped particles. Metal nanoparticles with diameters of 80-100nm move very rapidly due to their Brownian motion. Using only one laser to trap and to detect leaves the detection system automatically aligned with the trap. The position of the trap can be anywhere in

U - UC D U - UA B U - UA C U - UB D x y I V A B D C 10mm laser line mirror

silver mirror 50/50 beamsplitter

microscope objective

microscope

objective lens 2 lens 3

lens 1 digital colour CCD camera QPD sample laser f1 f1 f3 f3 f2 f2

Figure 3.18: We expanded our previously described tweezing setup with a quadrant

photodiode (QPD) for electronic detection of particle motion. The diagram on the left shows this addition to the setup. The image in the middle is a photograph of the InGaAs QPD we used. The graph on the right side illustrates the functionality of the photo detector (see also Eq. (3.1)-(3.3)).

the field of view of the detection objective as the interference pattern is a far field effect. All light passing through the focus is imaged onto the QPD.

The QPD converts the output current from each quadrant photodiode to a volt- age. The signal of the four quadrants gives precise information about the position of the interference pattern on the QPD. The intensity of the interference pattern is compared in each half (consisting of two quadrants) of the detector, vertically and horizontally. If the interference pattern is perfectly symmetrical and exactly posi- tioned at the centre of the QPD, the x and y values (see Eq. (3.1)-(3.3)) are zero. Displacements of the trapped particle in x and y result in lateral intensity shifts in the interference pattern. Therefore we obtain the position data of the trapped particle by adding and subtracting the individual voltages as follows:

Ux = (UA+ UC)−(UB+ UD) UA+ UB+ UC+ UD (3.1) Uy = (UA+ UB)−(UC+ UD) UA+ UB+ UC+ UD (3.2) Uz = UA+ UB+ UC+ UD (3.3)

The x and y position values are normalised with the total voltage of all four quadrants (z coordinate). The sign of x and y indicates the direction of the dis- placement. The QPD measurement is precise for position detection provided the voltage response of the QPD is linear to the intensity. It is possible to detect the movement in z direction, which may require additional adjustment, depending on the system [68]. We acquired the voltage data for all three coordinates with our own LABVIEW programme. The subsequent data analysis is discussed in detail in

Chapter 4. Calibrating the QPD enabled us to assign real physical units of length to the position data which is delivered in volts by the QDP.

We would like to point out some specific challenges connected with nanoparticle position detection. The amount of light interacting with the nanoparticle is a lot smaller than for a large micron sized bead. The scattered and refracted light is the laser light that interferes with the unperturbed trapping beam and hence gives the information about the position of the trapped object. As the interacting laser light for particles of nanometre dimensions is only a fraction of the unperturbed beam, the signal containing the relevant position information is thus only a fraction of the laser light impinging on the QPD. A good signal to noise ratio is therefore essential to pick up this relatively weak signal containing valuable information.

The z position of the trap influences the detection of the trapped particle’s move- ment and subsequent data analysis. The deeper the trapping position is inside the sample chamber, the more aberrations occur at the trapping focus. Therefore the trap deep inside the sample is weaker compared to just below the top coverslip. However, trapping too close to the sample coverslip leads to interference with back reflections from the coverslip and a significant change in the particle’s viscosity [69]. As the aberration of the focus and thus the deterioration of the trap can be corrected by using the appropriate immersion oil for the trapping objective [57] we were able to trap 10µm away from the top glass coverslip. The particle’s diameter (80-100nm) was thus negligible in comparison to this distance and we did not need to include hydrodynamic corrections (Faxen’s law) in our data analysis. We aimed to trap the particle at the same depth at all times to obtain comparable measurements.

We made an effort to eliminate noise from the system in order to achieve the best possible signal to noise ratio. We took a dark spectrum (power spectrum of the position data with the laser switched off) to measure the electronic noise of our system. This turned out to be negligible. However, electronic noise is not the only noise in the system. Mechanical (and even acoustical) noise is transmitted via the sample solution onto the sphere and can only be seen in the final motion of the trapped particle. We tried to circumvent mechanical noise sources by floating the optical table and taking measurements out of normal working hours when the building was quiet.

The largest source of noise to our system turned out to be the laser itself. We measured the light spectrum (power spectrum of the position data), with only the laser beam shining on the photo detector without a particle in the trap, to check for beam pointing instabilities. Significant contributions in the low frequency regime (<20Hz) indicated low pointing stability. By replacing the QPD with a camera for a test we were able to see movements of the entire interference pattern and hence the laser beam in space from time to time. As the beam remained stable for seconds in between movements, this was not a problem for the trap stiffness measurement (see Chapter 4) because the data acquisition only took 5s at the longest. However, to calibrate our QPD with a nanopositioning stage turned out to be impossible as

20 40 60 20 40 60 3 x (pixels) y (pixels) 20 40 60 20 40 60 x (pixels) y (pixels)

Figure 3.19:We analysed a section (left) of an imaged grid ruler (Comar, 11GG76) with

our particle tracking programme to allocate a physical length scale to each pixel. The graph on the right shows a summary after analysing several different images to minimise errors. In this example 1 pixel equals 200nm.

this requires the beam to remain stationary for at least a couple of minutes (see the following section).