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Back-focal-plane interferometry


There exist several ways of measuring the position of a trapped bead. The most common method consists of analyzing the light exiting the trap with a quadrant photodiode QPD. These photons are collected with the same condenser that is used to illuminate the sample.




A dicrhoic mirror is placed after the condenser. It reflects the light coming from the trap, yet it permits the illumination light go through. The light pattern at the back-focal-plane BFP of the condenser is projected onto the detector by an auxiliar lens. Then, the signal from the QPD allows us to calculate the position of the sphere. In order to obtain an intense signal, a high numerical aperture condenser has to be used. This is the way to collect as much photons as possible and, hence, to record a good signal from the detector.


Light pattern at the back-focal-plane of the condenser

When the size of the bead is smaller than the wavelength of the laser, the sample behaves as a dipole. Hence, the light passing through the microsphere is scattered, creating a new spherical wave. Then the light exiting the trap has two different contributions:

            1. The light scattered by the sample.

            2. The ligth that do not interact with the bead.

Both terms are collected by the condenser and interfere at its back focal plane.



The movie shows the interference pattern generated by a 1 um bead when moving along the x axis. The sphere was stuck on the slide and moved with a piezoelectric stage.



The images were recorded with a CCD camera placed at the BFP of the condenser. The interference clearly changes when the sample moves in the 'x' direction.

When a large bead (~several microns) is used instead, there is no interference. What one observes then, is a spot that changes its position when the microsphere is displaced form the center of the trap. The following movie shows the light pattern for an 8 um trapped bead. The particle was moved by the piezostage, which was generating a peridoic flow around the trapped sample.



The light pattern acquired with a QPD produces a signal that permits us to calculate the position of the bead within the trap.


Why do we use the light from the BFP to detect the position of the sample?


The scalar field at the back-focal-plane of any lens depicts the Fourier transform of the object before the lens (see figure). This has some advantages when detecting the position of a sample at the specimen plane.



One of the interesting properties of this mathematical operation is that any movement of the object just involves a change in the phase of the field at the BFP of the lens, which vanishes when calculating the intensity of the light pattern. Hence, at the back focal plane of the condenser we do not have any information about the absolut position of the trap within the sample, but just information about the relative distance between the microsphere and the laser beam. This is the main reason why this plane is used to obtain a signal from the QPD. Furthermore, since the mathematical operation that relates both planes is well-known one can compute the intensity pattern at the BFP and, therefore, it is possible to estimate the signal from the QPD.



Photodetector

A quadrant photodiode is a silicon detector whose surface is divided in four quadrants. Everyone of these behaves as an independent detector, providing a voltage Vi that depends on the amount of light. The QPD generates three different signals Sx, Sy and Sz which are linear combinations of those four voltages.



Sx and Sy depend linearly on the position of the light spot on the surface of the detector for small displacements. Therefore, these two signals provide a measure of the position (x,y) of the beam. On the other hand, Sz gives a signal that is proportional to the intensity of the beam. The main feature of this kind of detectors is that the sensibility (the slope of the linear region) can be increased as much as we nee, just by reducing the size of the spot.



Other kinds of silicon diodes, such as the position sensing detector PSD, have been used with the same purpose taking advantage of their wide linear range. Nevertheless, the QPD is widely used because of its high spatial resolution.


QPD signal-position curve

Now, we do not consider a single spot of light but the interference pattern that we have already described. Then, the signal Sx from the QPD when moving a bead through the laser beam along the direction 'x' is given by the following curves. We show the signal for different effective numerical apertures of the condenser using a 5 um sphere, and this reveals that the sensitivity increases when NA is higher. This explains why it is desirable the use of a high-numerical aperture condenser to detect the position using this back-focal-plane interferometry technique.



Therefore, the signal is proportional to the position of the microsphere, for a small displacements (Gittes et al.):
The constant can be obtained by computing the interference pattern at the BFP as the Fourier transform of the field at the specimen plane. Then, calculating the signal which would generate the QPD with this distribution of light, one gets the relation:
Now, we must experimentally measure two constants, b and k, in order to calculate the force. This will be given from the signal Sx of the detector by the equation: