Rens Breur

Temperature mapping in an optical tweezers setup using fluorescence polarization anisotropy

Student number 11010495
Daily supervisor Joost Geldhof, MSc.
Supervisor Dr. Ir. Iddo Heller
Programme Physics & Astronomy

We build a confocal fluorescence microscope with linearly polarized excitation light measuring emission seperately for polarization directions parallel and perpendicular to the excitation light polarization. Temperature can be quantified using the anisotropy of the emission polarization, calculated using these polarized intensities. We were able to properly align the optical path and performed normalization scans. Calibration was attempted with a thermometer placed near the sample. The fluctuations of the anisotropy over the pixels of the calibration scans were much higher than the difference in anisotropy between the temperatures. The mean of the anisotropy did not change to the extent that is theoretically expected. Further work is necessary to increase the accuracy of the temperature reading so that the instrumentation can be used to analyse heat transfer from optically trapped particles.

Introduction

Single molecule techniques and optical tweezers in particular have allowed an entire new range of experiments in biology over the last decades. The length scale of nanometers and force scale of piconewtons associated with optical tweezers have enabled biological experiments from the single molecule to the single cell level. Examples are the observation of base-pair stepping by RNA polymerase (E. A. Abbondanzieri et al., 2005) or the mechanics of ATP-powered protease (M. E. Abin-Tam et al., 2011).

Optical trapping can also be used to locally introduce a change in temperature. Generally, a trapped particle will dissipate some of the trapping power as heat into the environment, therefore locally adding heat to a system. In contrast, some crystals, such as ytrium lithium fluoride crystals, cool down when trapped due to energy upconversion (P. B. Roder et al., 2015). This encourages the use of optical tweezers for research on a number of temperature dependent processes.

Low-temperature processes would benefit especially from local refridgeration. An example of such a a process is the operation of ice binding proteins. Ice binding proteins prevent damage to organical tissue when temperature reduces to below freezing point, both by preventing ice nucleation to lower the freezing point and by inhibiting ice recrystilization to prevent tissue damage (S. Venketesh, C. Dayananda, 2008). Optical trapping provides a promising tool for studying this behavior on the single molecule level and local temperature control would allow seeding or melting ice without condensation on optical components.

The temperature change induced by laser trapping is local, and the temperature gradient an optical trap creates is only useful if it is quantified. There are several ways to locally quantify temperature. One of these is to use the temperature dependency of the anisotropy of emission light of fluorophores when they are excited with linearly polarized light. When fluorescent particles are excited with linearly polarized light, light polarization after emission conveys information about the amount that the particles have rotated since in the time between excitation and emission, and that in turn depends on the temperature.

In this project we use this phenomenon in combination with confocal microscopy to try and map the temperature near optically trapped particles. Several studies report successful attempts at mapping temperature by measuring fluorescence polarization anisotropy in a confocal setup, attaining an accuracy of 0.1^\circ\mathrm{C} and a 300\mathrm{nm} resolution (G. Baffou et al., 2009). Our goal is to build a setup that can be coupled with a laser trapping path. If successful, this setup would allow analysis of heat transfer from optically trapped particles into the sample and its surroundings, which would help extend the already powerful single molecule toolset with the added ability to locally control temperature.

Theory

Heat transfer from an optically trapped particle

Because we want to build instrumentation to quantify the temperature in a region around an optical trap, it is useful to analyse the theoretical heat transfer in such a system. In any region of a system where no heat is added or removed, heat transfer occurs per conduction or convection. If we approximate the heat transfer in the sample in an optical tweezers setup to be a purely diffusive process, heat transfer follows the heat equation.

The stable, time-independent version of the heat equation is Laplace's equation. Taking the Laplacian of a three dimensional heat map shows where heat is added to the system. Two dimensional maps can be analysed by making assumptions on the symmetry of the system.

Using some simple models we can predict the decay of heat from a heated object to its surroundings. This decay depends on the geometry of the system. If we take the heated object to be round, the system is spherically symmetric and Laplace's equation reduces, in spherical coordinates, to
(1) \frac{1}{r}\frac{\partial^2}{\partial r^2}(rT) = 0 such that the damping of the heat is inversely proportional to the radial distance from the object.

In the case where the heated object is an optically trapped particle, a cylindrical system may be more representative. In this case, Laplace's equation reduces to
(2) \frac{1}{\rho}\frac{\partial}{\partial \rho}\left( r\frac{\partial T}{\partial \rho} \right) = 0 and the damping is logarithmic.

In optical trapping, the sample in which the beads are trapped and the sample stage are not spherically or cylindrically symmetric. If objective and condensor from the most important heat bath, heat will more easily transfer to the top and bottom than to the sides.

Fluorescence polarization anisotropy

The ability of a fluorophore to absorb light depends on the orientation of the fluorophore relative to the excitation light polarization. Linearly polarized light therefore selectively excites fluorophores within an ensemble. The intensity of emission light polarization of a fluorophore similarly depends on its orientation. This causes a relationship between the amount of rotation between absorption and emission and the polarization of emission light.

For particles that are solved in a liquid and are free to rotate this rotation is diffusive and the rate is given by the Debye-Stokes-Einstein relation,
(3) \tau_r = \frac{V \eta (T)}{k_B T}, with V the hydrodynamic molecular volume and \eta(T) the viscosity of the fluid, which may itself depend on the temperature. The temperature dependence of the viscosity is a materialistic property of the solvent.

If fluorescent particles in solution are excited with linearly polarized light, the faster the diffusion, the less the emission light is still polarized in the same direction. We call the emission light to be more isotropic. The anisotropy of the polarization of light emitted by linearly excited fluorophores is defined like (J. R. Lakowicz, 2006) as the difference in light polarization over the total intensity,
(4) a = \frac{I_\parallel - I_\bot}{I_\parallel + 2 \cdot I_\bot} = \frac{I_\parallel / I_\bot - 1}{I_\parallel / I_\bot + 2}, where I_{\parallel} is the intensity of emission light polarized parallel to the excitation light polarization and I_\bot is the intensity of emission light polarized perpendicularly to the excitation light polarization. By symmetry I_\parallel + 2 \cdot I_\bot is proportional to the light emitted by the fluorophores in all directions.

The relationship between anisotropy of the light polarization and the rotational motion of the fluorophores is given by Perrin's equation,
(5) \frac{1}{r} = \frac{1}{r_0} \left ( 1 + \frac{\tau_f}{\tau_r} \right ), where r_0 is the limiting anisotropy that depends on the fluorophore and \tau_f the fluorescent lifetime.

The limiting anisotropy r_0 is determined by the angle between the fluorophores preferred polarization orientations for absorption and emission. In the case that these are equal r_0 is the highest. Because fluorophores are probabilistically excited depending on their orientation with respect to the light polarization, I_\bot is not zero. By the geometry of the situation, the maximal possible anisotropy turns out to be r_0 = 0.4 (J. R. Lakowicz, 2006). In the other extreme the orientations are orthogonal. Then the emitted light will not have any preference for a particular polarization, so I_\parallel = 2 \cdot I_\bot . By equation 4, in this case r_0 = -0.2 .

Any difference in the concentration of fluorescent particles does not affect the anisotropy because concentration will affect the emission intensity in both polarisation directions. This is a great advantage of using anisotropy to quantify temperature.

Confocal scanning microscopy

Unlike in bright-field microscopy, in scanning microscopy only a specific region of the sample is illuminated. The illumination spot is generally formed by the focusing of a point like light source using an objective lens. A collector lens catches light from the sample into a detection system. By scanning the diffraction limited spot in the sample, an image can be formed and recorded to an external device such as a computer.

Figure 1: (T. Wilson, C. Sheppard, 1984) A confocal microscope that images the transmission of light by an object. The dashed and solid lines in the object are in and out of the collector's focal plane respectively.

In a confocal scanning microscope a pinhole is introduced in the detection system. A simple examples of a confocal microscope is shown in figure 1. The pinhole increases the lateral resolution of the image and reduces the intensity of unfocused light in the axial direction.

In the example of figure 1 the illumination spot is shaped by the diffraction of the light by the aperture of the objective lens. This diffraction limited spot has a size on the scale of \lambda/{2\mathit{NA}} . Any point in the illuminated part where the object transmits will form a similar diffraction limited spot on the focal plane of the collector lens, where the detector is. The intensity of the diffraction limited spot at the position of the detector is the detection sensitivity for the point in the object.

For a perfect point detector, the detection sensitive region then has the same shape as the diffraction limited illumination spot. The confocal volume, determined by a multiplication of the illumination intensity and the detection sensitivity, is then smaller than a simple diffraction limited spot so that the lateral resolution is nearly a third better than that of a one lens system (T. Wilson, C. Sheppard, 1984).

In confocal laser scanning microscopy rather than a point source an objective focuses the collimated light bundle of an excitation laser into the sample. The diffraction of the finitely sized laser beam and other additions, such as the use of a steering mirror, decrease the resolution of the microscope.

In the case of fluorescence microscopy the objective focuses light at the absorption wavelength to excite fluorophores in the sample. Any light that might be fluerescently emitted from the excited volume of the sample is caught through the collector lens and redirected into the detection system.

Method

We combine local temperature quantification by fluorescence polarization anisotropy measurements with confocal microscopy to map the temperature around the focus of an optical trapping laser. For this tempearture quantification we will make use of a solution of green fluorescent protein. The excitation path of our microscope will be linearly polarized. The detection path measures light polarized parallel and perpendicular to the excitation light. We scan the focus of the confocal microscope to create a temperature map.

The design of this setup is first discussed below. The procedure for fine alignment of the excitation and detection path follows. This chapter concludes with the method of using the emission polarization anisotropy for quantifying the temperature, including steps needed for calibration.

Design

Figure 2: Schematic drawing of the optical setup. The rectangle on the top right represents the objective. The direction of excitation light and the direction of emission light that is detected is displayed using arrows.

Figure 2 shows a schematic drawing of the optical setup. The objective lens forms both the objective and collector of a confocal microscope, so it focuses excitation light and collects emission light from the sample. Dichroic mirror 1 decouples the excitation and emission light. The polarizing cube enables measuring light intensity of emission light polarized both parallel and perpendicular to the excitation polarization. Scanning is done by the movement of the steering mirror. A trapping laser can be coupled into the same objective to capture a particle.

The excitation source is a laser with a linearly polarized 488\mathrm{nm} continuous wave output. The orientation of the linear light polarization is adjustable by the half wave plate. This half wave plate is placed before the first mirror so that horizontally or vertically polarized excitation light remains linear.

The excitation light is reflected by dichroic mirror 1, which is a long pass filter. The excitation light is also reflected by the dichroic filters 2-4. These are not used but this array of dichroic mirrors allows the coupling and decoupling of light of different wavelengths in a range of experiments.

Figure 3: A telescope system of a steering mirror, a set of two lenses and the objective. The distance between the lenses is the sum of their focal distances. The dotted line shows the situation in which the mirror is precisely at 45 degrees. In grey the light is shown for the situation in which the mirrors rotation is adjusted to the direction of the arrow.

The focus of the excitation light and the detection region of the confocal setup are moved laterally through the objective's focal plane by the computer-controlled piezo steering mirror and a telescope system of two lenses. To move focus through a lenses focal plane, the angle of incidence on the back focal plane must change. Any change in direction of the steering mirror results results in a different position and angle of incidence of the rays on the first lens of the telescope system, which only causes a different angle of incidence of the rays onto the back focal plane of the objective. The mechanism of the telescope system is illustrated in figure 3.

An external system controls the steering mirror to make a scan and registers the detection signals for every position to a computer. The use of steering mirrors rather than mechanically moving the sample allows the new optical path to make use of the same objective that is also used for a laser trap. Behind the steering mirror a shutter is installed that blocks excitation light when no scan is being performed.

The upward mirror reflects the excitation light into the 1.27\mathit{NA} objective lens. The objective focuses the excitation light into the protein channel of a flow cell that contains a phosphate-buffered saline with green fluorescent protein.

Part of the light emitted by the fluorophores is caught back into the objective. This detection light travels opposite excitation light, through mirror 2, the steering system and dichroic mirrors 4-2. The light is transmitted by dichroic mirror 1. Mirrors 3 and 4 steer the detection beam through an extra filter that prevents any excitation light from leaking into the detection path, through mirror 5 into a set of two lenses and the confocal pinhole.

Behind the pinhole the detection light is split by the polarizing cube into horizontally and vertically polarized light waves, the intensities of which are measured by a pair of avalanche photodiodes. The horizontal D_h and vertical D_v detection signals are used to quantify the temperature of the solution of fluorophores in the confocal volume.

To obtain the highest possible resolution, we require that the scanning step size is smaller than the size of the confocal volume. We therefore choose a scan step size of 80\mathrm{nm} in both directions.

In order to minimize side effects of the excitation light and to reduce photobleaching of the fluorophores, the excitation light intensity and dwell time for each pixel is kept low. The minimal dwell time for the scanning system is 1\mathrm{ms} . The power of the excitation light has to be calibrated to its intput voltage, and is on the order of microwatts.

Mirrors and dichroic mirrors cause a phase shift between light polarisations parallel (p) and perpendicular (s) to their plane of incidence. The beam path of light in this setup is therefore kept at an equal height of 63.5\mathrm{mm} so that for each mirror the s- and p-components of light are the same. The last mirror before the objective reflects the light from the horizontal optical setup up upwards into the objective. For this mirror the s- and p- directions are exactly the p- and s-directions of the other mirrors. The single phase shift that is the result of the reflection of detection light by the mirrors in the detection path will not affect the detection intensities.

When adjusting the half wave plate such that the excitation light afterwards is horizontally polarized, the light will stay horizontally polarized until the upward mirror. The light reflected by the upward mirror will then become polarized in the direction we will call x . This is the direction perpendicular to the plane of incidence of the excitation light on the upward mirror. The detection signal for horizontally polarized detection light D_h is a measure for the intensity of emission light in this direction. We will call the direction of the polarization of vertically polarized excitation light reflected by the upward mirror y . D_v is a measure for the intensity of y -polarized emission light.

If the excitation path light is horizontally polarized such as to create excitation polarized in the x -direction, D_h measures the parallel component I_\parallel of the fluorescence emission light. Signal D_v is a measure for the intensity of perpendicularly polarized light I_\bot . When the half wave plate is set to vertically polarize excitation light instead, the situation is inversed and I_\parallel is measured by D_v and I_\bot by D_h .

Alignment

To maximize the signal, we require that of the part of the sample that is excited by excitation light, most of the possibly fluorescently emitted light can be detected back by our two photo sensors. In order to get most of the emission light detected, we will therefore want to center the excitation and detection regions of the confocal microscope.

Figure 4: Schematic drawing of the optical setup with a beam splitter, mirrors and a photon multiplier tube added. Reflection of excitation light is now detected in the photon multiplier tube rather than the detection photo diodes. Reflection of alignment lasers in the detection fiber outputs can also be imaged on the photon multiplier tube.

To image the excitation region we modify the optical path as shown in figure 4 and replace the flow cell with a prepared microscope slide with reflecting nanobeads. The excitation laser light will illuminate the reflecting particles. Any possibly backward reflected light is caught by the objective and redirected by the pellicle beam splitter and the two added mirrors into a photon multiplier tube.

The sample of reflecting particles we use is a preparation of 50\mathrm{nm} golden beads set in agrose gel. Scanning with the steering mirror makes a raster scan of backward reflection within the sample. The simple is placed in a stage that can be moved up or down by a computer so that the patterns can also be imaged in the z -direction. Because of the small size of the bead, imaging a single bead will image the region in the sample that the excitation laser illuminates.

To determine the center points of the detection regions, we remove the photo sensors from the detection fibers and instead attach alignment lasers. We then use the reflection of the beads in the photon multiplier tube to image the illumination patterns of the alignment laser. This situation is different from the one in figure 4 in that the excitation laser is off, and instead one of the alignment lasers is on, generating light travelling from the detection fiber towards the objective. The regions in the flow cell illuminated by the alignment lasers differ from the detection sensitivity amplitude, especially with the pinhole in place, but the center points are the same.

Centering the patterns that the excitation laser and the two alignment lasers individually illuminate in the sample therefore aligns the excitation and detection paths. Further alignment is done using the original optical setup in figure 2 by slowly adjusting excitation and detection paths until maximal detection signals D_h and D_v are achieved.

A reflecting bead also reflects part of the light rays focused behind it. It is therefore important to center the front most parts of the images.

Temperature quantification

For temperature quantification we use a solution of green fluorescent protein in phosphate buffered saline. The viscosity of phosphate buffered saline the same as that of distilled water (P. B. Roder et al., 2015). Phosphate buffered saline stabilizes the pH and therefore has good resemblence to a biological environment.

Green fluorescent protein has a fluorescent lifetime \tau_f of around 3.0\mathrm{ns} (G. Striker et al., 1999). Its hydrodynamic volume V has been determined by previous experiments at 17.1\mathrm{nm}^3 (N. A. Busch et al., 2000), making use of a diffusion measurement and inversely applying the Stokes-Einstein relation.

The rotational correlation time of green fluorescent protein depends on the viscosity of the solvent. Phosphate-buffered saline has the same temperature dependent viscosity as distilled water, which may be given by a phenomenological expression such as (G. Tammann, W. Hesse, 1926)
(6) \eta (T) = \eta_\infty e^{\frac{A}{T-T_{VF}}} where \eta_\infty , A and T_{VF} are variables. Experimental data shows distilled water to fit this expression as (P. B. Roder et al., 2015)
(7) \eta (T) = ( 2.664 \cdot 10^{-5} \mathrm{Pa} \cdot \mathrm{s} ) \, e^{\frac{536.5 \mathrm{K}}{T-145.5 \mathrm{K}}}.

Figure 5: Theoretical relationship between temperature ( x -axis) and anisotropy ( y -axis) for green fluorescent protein in phosphate buffered saline, using the unfitted literature parameters given in the text.

At room temperature therefore, the rotational correlation time \tau_r and the fluorescent lifetime \tau_f of green fluorescent protein are of the same order of magnitude, which will give a high sensitivity of anisotropy r on temperature (see equation 5). The literature values of \tau_f , V and r_0 = 0.4 give the typical relationship between temperature and the emission anisotropy of linearly polarized green fluorescent protein in phosphate-buffered saline shown in figure 5.

Figure 6: (H. E. Seward, C. R. Bagshaw, 2009) Absorption and emission spectra for wild-type green fluorescent protein. Absorption peaks are at 395\mathrm{\hspace{0.25mm}nm} and 475\mathrm{\hspace{0.25mm}nm} , the emission peaks at 510\mathrm{\hspace{0.25mm}nm} .

The instrumentation is built for use with enhanced green fluorescent protein, which has an emission spectrum with a single peak at 488\mathrm{nm} (H. E. Seward, C. R. Bagshaw, 2009), the wavelength of our excitation laser. For measurements we will use the less expensive wild-type green fluorescent protein, the absorption and emission spectra of which are shown in figure 6. The wavelengths of the absorption peaks are at 395\mathrm{\hspace{0.25mm}nm} and 475\mathrm{\hspace{0.25mm}nm} , the emission peaks at 510\mathrm{\hspace{0.25mm}nm} .

There are two steps to calibrating the measured detection signal to a temperature. The first step consists of normalizing the detection signals using unpolarized light. The second step measuring the anisotropies for different flow cell flow cell temperatures. The resulting data is then fit with parameters \tau_f/V , r_0 and the variables of equation 6. The literature values of \tau_F and hydrodynamic volume V give \tau_F/V = 1.76\cdot 10^{17} .

For these calibration steps a fixed microscope slide is used rather than the flow cell. This will require less gfp for a higher signal, and give the same calibration as in both situations the fluorescent particles are free to rotate.

Optical components affect the emission light polarization. Dichroic filters or monochromators have different transmission intensities depending on the incident light polarization. The further the incident light is from the normal, the more this effect is amplified. Normalizing the signals will account for this. Normalizing will also correct in linear order for a difference in detector sensitivities or slight misalignment.

Normalization is commonly done by making the excitation light polarized in such a way as to remove the preference of emission polarization toward parallel or perpendicular orientation, so that the fluorophores emit light for which I_\parallel=I_\bot . If the measured detection signals D_h and D_v are not the same, this difference must be caused by the optical paths for parallel and perpendicular detection. The detection ratio will then give the normalization of the system.

Making the parallel and perpendicular light intensities of emission light equal I_\parallel=I_\bot is generally achieved either by polarizing excitation linearly perpendicular to the objective, or by making the linear excitation light circularly polarized. The first option requires excitation and detection to occur in different directions, therefore requiring the use of separate objective and collector lenses. The second option requires a quarter wave plate to polarize the excitation light for which there is no place. A quarter wave plate should be placed after the very last mirror in the excitation path because any mirrors will introduce an additional phase shift in the light polarization.

To normalize the detection signals, we therefore illuminate the fluorophores with x and y -polarized excitation light seperately, and sum the measured values D_v and D_h . The average detection values of a scan gives the normalization as a factor by which the D_h should be multiplied to get the proper polarization anisotropy,
(8) F_\mathrm{sum} = \frac{[D_v]_y+[D_v]_x}{[D_h]_y+[D_h]_x} where [D_j]_i is the reflected detection intensity polarized in direction j for i -polarized excitation light. The resulting factor would be the same if we had excited the sample with light polarized with light half of which is linear in p-direction and half of which is linear in s- direction.

Another option is to adjust the half wave plate, which is placed before the first excitation path mirror directly after the excitation laser fiber output, such that the light is polarized at an angle of 45 degrees before it hits the first mirror. Any mirrors in the excitation path will alter the phase shift between horizontal or vertical polarizations, such that the light will become elliptically polarized, but the individual components in the excitation light will still be equal. In this case the normalization factor is simply
(9) F_\mathrm{elliptical} = \left [ \frac{D_v}{D_h} \right ]_\mathrm{elliptical}.

A third way to normalize is to use the fact that, whether the excitation light is polarized linearly in x or y direction, the ratio I_\parallel/I_\bot should be the same, and therefore also the ratio of normalized detection signals. We use a calibration constant G defined by \left [\frac{D_v}{GD_h} \right ]_y = \left [\frac{GD_h}{D_v} \right ]_x, or similarly, by equalling the anisotropies, \left [\frac{D_v-GD_h}{D_v+2GD_h}\right ]_y=\left [\frac{GD_h - D_v}{GD_h + 2D_v}\right ]_x, which both reduce to
(10) G^2=\frac{[D_v]_y [D_v]_x}{[D_h]_y [D_h]_x}.

For temperature calibration we change the flow cell temperature using collars around the objective and around a condensor, that are connected to a heat pump. The condensor is a lens above the flow cell, that may be used for force acquisition on the optical trap but that is not used for temperature mapping. The heat pump can be adjusted in the range of 5^\circ\mathrm{C} to 20^\circ\mathrm{C} . A thermometer attached to the objective lens collar using a piece of tape.

Results

Excitation and detection regions

Figure 7: Final point spreading functions of the system with alignment lasers at the fiber exit of respectively excitation, detection transmission and detection reflection paths in the x - y plane. The pinhole is included in the detection path.

Figure 7 shows images of the reflection of the excitation laser and of the alignment lasers at the detection fiber outputs of a bead in the photon multiplier tube, after alignment of the paths. The images were made by scanning the steering mirror and therefore show the regions the lasers illuminate in the sample in the lateral ( x - y ) plane. The scanned frame is the same for the three images to show that the brightest parts are aligned. This alignment was performed mainly by tuning the orientation of the polarizing cube, mirror 2 and dichroic mirror 1.

Figure 8: Final point spreading functions of respectively excitation, detection transmission and detection reflection paths in the x - z plane. The lower part of the central spot is at the same position in all images.

Similar measurements show the reflection images in the x - z plane as shown in figure 8. They are translationally aligned, using the lower part of the patterns. Movement in the z -direction was done by moving the lens collecting the light from the alignment laser from the fiber exit in the direction of the light, therefore changing the collimation of the light. Reflection of light rays behind the focus point explain why the images of figure 7 with a pinhole narrow the lateral diffraction pattern more than expected by an increase of lateral resolution.

Because the illuminated patterns are three dimensional, the cross-section images above were all made in the middle of the brightest part in the perpendicular direction. That is, for the x - y scans, an x - z scan was made, and then in the middle of the z -axis another x - y scan was made.

Figure 9: The left image shows a distorted diffraction pattern in x - z plane, which was solved in two steps (middle, right) by adjusting the flow cell orientation.

Earlier measurements, such as the one in figure 9, show a less symmetric diffraction pattern, which we could fix by slightly changing the orientation of the flow cell. When the direction of the incoming laser light is offset from normal of the flow cell, the different places of refraction by the flow cell distorts the diffraction pattern.

Normalization measurements

Figure 10: Scan of 25 by 25 (80nm)^2 pixels of green fluorescent protein in solution for horizontally D_h (left) and vertically polarized detection light D_v (middle) and the ratio D_v/D_h (right), under vertically polarized excitation.
Excitation polarizationAverage D_h (photons)Average D_v (photons)
y 22.32\pm 4.76 136.66\pm 11.9
x 3.918\pm 2.02 5.764\pm 2.43
elliptical 12.27\pm 3.52 63.73\pm 8.35
Table 1: Average detection values with standard deviation for the 25 by 25 pixels of (80nm)^2 of a green fluorescent protein solution scan using a pixel dwell time of 1\mathrm{ms} .

Three scans were performed for normalization using a buffer with 3.6\mu\mathrm{M} green fluorescent protein at room temperature, for x -, y - and elliptically polarized excitation light. The resulting images of the scan under y -polarized excitation light are shown in figure 10 as an example together with the ratio of D_v/D_h . The average pixel values of all the scans and the variance are listed in table 1. The dwell time per pixel used was 1\mathrm{ms} . The y -polarized excitation light intensity was measured at 176\mathrm{\mu W} at the back focal plane of the objective.

The scan under x -polarized excitation light and the scan under y -polarized excitation light both show an average D_v signal that is higher than D_h . For the y -polarized excitation scan this ratio is higher.

The first two normalization constants are F_\mathrm{sum} = \frac{[D_v]_y+[D_v]_x}{[D_h]_y+[D_h]_x} = 5.43 and F_\mathrm{elliptical} = \frac{D_v}{D_h} = 5.20. A repeated normalization scan with elliptically polarized excitation light, the data of which is not shown in the table, gives F_\mathrm{elliptical} = 5.32 .

Applying F=5.3 to the averages of the values in table 1 gives an extremely low anisotropy of 0.049 for vertically polarized excitation light and an undefined anisotropy of 0.46 for horizontally polarized excitation light. Note that the relationship between temperature and anisotropy is only roughly linear between 0.15 and 0.35.

The behavior of dichroic mirrors is commonly polarization dependent. The normalization procedure accounts for this in the detection path, but it turns out to also affect the excitation path, with a total factor 10 difference between the intensities of x - and y -polarized excitation. The transmission of p-polarized light incident on dichroic mirror 1 is a factor 8 smaller than transmission of s-polarized light.

Apart from normalizing the detection path, normalization constant G also corrects for this, G' = \sqrt{\frac{10 [D_v]_v [D_v]_h}{10 [D_h]_v [D_h]_h}} = G = 3.00.

The normalization constants F_\mathrm{sum} and F_\mathrm{elliptical} do not, but it is possible to correct F_\mathrm{sum} to F_\mathrm{sum}' = \frac{[D_v]_v+10 [D_v]_h}{[D_h]_v+10 [D_h]_h} = 3,2.

Using normalization constant G , the anisotropy for the pixels of the y -polarized excitation scan is 0.265 on average, which corresponds to a typical temperature of 282\mathrm{K} . The fluctuation of this value is 0.088 , corresponding to a typical temperature difference of 33\mathrm{K} . For the x -polarized excitation scan the detected intensities result in an anisotropy fluctuation of 0.265 .

The detection intensities normalized with normalization constant F_\mathrm{sum}' give anisotropies 0.277 and 0.238 respectively for x - and y -polarized excitation. The corresponding temperatures are 278\mathrm{K} and 292\mathrm{K} .

We checked the difference in reflection efficiency for a single beam steering mirror between s- and p-polarized incident light. Such a single mirror reduces the intensity of p-polarized incident light 5% more than the intensity of s-polarized incident light.

Temperature and emission polarization