1. Introduction

Fluorescent-based temperature probing techniques are of great importance because of their non-contact and non-invasive advantages [1–11]. These techniques typically measure the thermal dependence of phosphor fluorescence from fluorescent materials, many of them inorganic fluorescent molecules. The temperature dependence can be categorized into two types. (i) Temperature changes the radiating behavior of a fluorescent molecule by modifying the electronic structure and energy distribution of electrons. Such techniques include the thermometries exploiting fluorescence lifetime, spectrum and intensity. (ii) Temperature can also change the way a fluorescent molecule interacting with its surroundings. For example, because the rotational speed of a molecule in solution typically increases with temperature, the molecule orientation is expected to deviate faster from its initial orientation. Baffou et.al. demonstrated that molecular diffusion based fluorescence polarization anisotropy can be used to map local temperature with an accuracy of 0.1 °C [3, 6]. Instead of slowly scanning each pixel that is typically needed in lifetime and spectrum measurements, they used CCD cameras to map two fluorescence images of orthogonal polarizations upon excitation by linearly polarized light. Then they correlated temperature with the ratiometric parameter polarization anisotropy, rather than the fluorescence intensity which is susceptible to photobleaching of fluorescent molecules. However, anisotropy measurement is inapplicable to polarizing materials or near metal surfaces since it requires polarizations of excitation and emission fields are well-defined and unaffected by the measuring environment.

In this paper, we report a new two-dimensional temperature mapping technique based on fluorescence thermometry in liquid solution that promises video-rate readout speeds with desired reliability against photobleaching and illumination fluctuations. Essentially we simultaneously capture two fluorescence images of different collection cones, corresponding to different numerical apertures (NAs), to resolve the local temperature through rotational thermal diffusion of fluorescent molecules. This technique shares the same advantages of fluorescence polarization anisotropy method because the images are captured directly using CCD cameras so the operation speed can be as high as the readout rates of cameras. This fast read-out rate provides specific advantage to capture the transient processes at microscale which typically have thermal response at millisecond time scale. Also, it uses the intensity ratio of two simultaneously collected images and thus is insensitive to the variations in fluorescence intensity caused by photobleaching of molecules, fluctuations of illumination and inhomogeneity of molecule concentration. Different from the fluorescence anisotropy, this new technique measures the directional fluorescent emission that is robust against polarization change, therefore it is applicable to measure temperature inside polarizing environment and near a metal surface. Similar to other optical thermometry approaches, this new technique also requires to perform calibration to quantify accurate temperature values. Here, we also develop a theoretical model to predict the trend of signal dependence on temperature which can be used for the general calibration purpose. Possible measurement errors resulted by strong scattering or background are also discussed. Besides the applications in temperature mapping, we expect this 2NA technique can be used to measure other quantities related to thermal diffusion, such as viscosity of the solution and rotational coefficient of the molecules. Moreover, its dark-field illumination scheme is compatible with total internal reflection fluorescence (TIRF) microscopy, so it may be integrated to optical microscopes to provide additional information that is not easily available using traditional setups.

2. Experimental platform

Fluorescein (free acid) is used in the experiment. It is dissolved in glycerol-water (4:1) mixture to achieve a concentration about 10⁻³ M, a region where energy transfer between molecules is negligible [12]. A 25 μm thick layer of solution is sandwiched between two coverslips made of borosilicate glass. Borosilicate glass has a low thermal conductivity (1.14 W/(m·K) at 300 K), providing good thermal insulation to build up temperature gradient while applying current to the 750 nm thick aluminum heater that deposited on the top coverslip using photolithography [13]. The fluorescent molecules are excited by a p-polarized beam using a 450-nm blue light-emitting diode (LED) (Thorlabs M470L3) and the power density is around 10 mW/cm², which is well below the intensity for ground-state depletion (around 10 MW/cm² for fluorescein in moderate concentration) [14]. The fluorescence is collected using a 20X dry objective lens with NA 0.4 (20X Mitutoyo Plan Apo NUV) with 1.5X intermediate magnification selected. Under this configuration, the total depth of imaging field (along z) is about 7 μm in solution determined by the optical depths in the solution and the finite size of CCD pixel. And the lateral resolving power (along x & y) of the imaging system is about 1 μm. The gathered fluorescence is then filtered by a long-pass emission filter (Chroma HQ485lp) and sent to two CCDs (CCD1: Infinity 2–3C in gray mode; CCD2: Cascade 1K) after manipulating imaging NAs using limiting apertures. The two limiting apertures are placed at the conjugate location of the entrance pupil of the microscope objective which provides equivalent NA reductions. The choice of the NA combination can provide the tenability of mapping sensitive over a temperature region (see appendix). The two CCDs are synchronized with heating and illumination using a homemade LabVIEW FPGA program. The captured gray scale images are then post-processed using MATLAB. The detailed sample configuration and heating and illumination methods are detailed in the XXXXX section.

The aluminum heater is fabricated using a standard photolithography procedure [13]. A 750 nm thick aluminum film is deposited on a cleaned glass slide using electron beam evaporation. A positive photoresist layer of S1805 with a thickness of 1 μm is then spin-coated onto Al film, followed by 60 s softbake at a temperature of 115 °C. The whole sample is subsequently exposed for 10 s under a premade photo mask with desired patterns. The MJB3 mask aligner is operated under a constant exposure power of 10 mW with a 3 inch opening in the mask holder. After development, the unprotected Al film is removed using aluminum etchant. Electric contacts are connected using silver paste afterwards.

3. Physical principles

The experimental configuration is shown in Fig. 1(a). A microscale thin-film heater is used to generate a temperature gradient across the fluorescent solution sandwiched between two coverslips. A blue beam is used to excite fluorescent molecules and enters the solution at a large incident angle. The beam direction is nearly parallel to the coverslip so that reflected beam leaves without entering the imaging system, which is known as dark-field illumination. By choosing a p-polarized beam with a dominant off-plane (along z-axis direction) electric field (E), we preferably excite fluorescent molecules and induce dipole transitions along the off-plane orientation. At this original orientation, the relaxation of excited molecules emits green photons mainly in the in-plane directions (x–y plane) which again fall outside of the imaging system. However, at finite temperatures, the excited molecules are able to rotate away from their original orientation driven by thermal motion therefore a larger portion of fluorescent emission can enter the imaging system at higher temperatures. This rotational diffusion of the molecules can serve as a thermal indicator for the local environment: the higher the surrounding temperature is, the faster the molecule rotates, leaving its emission dipole a larger deviation from the excitation direction, as shown in Fig. 1(b). Detailed description can be found in theoretical analysis in the appendix. As illustrated in Fig. 1(c), an infinity-corrected microscope system with two NAs is used to quantitatively capture this temperature information through measuring the orientations of emission dipoles. For the emission dipoles located at the front focal plane of the objective lens, their fluorescence emissions are collected by an objective and sent toward the subsequent tube lens to form an intermediate image. Between the objective and the tube lens, a longpass filter (not shown in the schematic) blocks unwanted excitation and background light. The objective lens and the tube lens are assembled in a commercial microscope, so a second tube lens is used for extending the beam path outside the microscope body to further engineer the imaging NA. The second tube lens is put 2 f after the intermediate image plane to keep the original magnification of the infinity corrected optical system. A 50:50 beamsplitter is positioned after the second tube lens to form images on two separate CCD cameras. In one beam path, a limiting aperture is placed at the conjugate location of the objective entrance pupil to reduce the effective NA. A computer system controls these two CCD cameras to simultaneously capture images and display two-dimensional (2D) temperature maps from the ratio between two fluorescence intensities. A Nikon inverted microscope (Eclipse Ti-U model) is used as base microscopy system and modified externally in this work.

 

Fig. 1 Schematic of working principle and experimental setup of 2NA technique. (a) Excitation, thermal rotation and emission of fluorescent molecules in solution. Molecules in the solution are excited using dark field illumination and rotate before emitting fluorescence under a temperature gradient generated by a heater. CS: coverslip; HT: heater; SL: solution. (b) Effect of thermal rotation on emission: at low/large temperature, rotation angle of the molecule is small/large before emission. τ_F is the lifetime of transition dipole moment. (c) 2NA setup: experimental configuration for simultaneous collection of fluorescence using two different numerical apertures (NAs). One beam path is limited by a limiting aperture at the conjugate plane of pupil plane to have a small imaging NA. Temperature map is calculated based the ratio between fluorescence intensities collected by two CCDs. OBJ: objective; M: mirror; TL: tube lens; BS: beamsplitter; LA: limiting aperture; CCD: charge-coupled device (camera); PP: pupil plane; CPP: conjugate pupil plane.

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To characterize its directional properties of fluorescent emission, we first use a rotational half-wave plate to vary the excitation polarization of a 450-nm laser beam (Osram PLTB450) and use a compact CCD spectrometer (Thorlabs CCS200) to collect fluorescence intensity at a specific detection angle, as shown in Fig. 2(a). In this part, we use the glycerol solution containing common fluorescein (2(3,6-dihydroxyxanthyl)-benzoic acid), as shown in the inset of Fig. 2(a), a xanthene-type chromophore having a large extinction coefficient, a good photostability and a close-to-unity quantum yield [15]. Figure 2(b) shows a typical emission spectrum under this arrangement which has a peak measured at 518 nm. Initially, the fast axis of the half-wave plate is aligned with the polarization direction of the illumination light and perpendicular to the detection plane. As the rotation angle of the fast axis (θ_r) varies from 0 ° to 180 °, the detection intensity varies by two cycles with form of cos²(2θ_r) when the detection angle is 90 ° as shown in Fig. 2(c). As a comparison, the periodic fluctuation diminishes at a detection angle less than 20 ° since the detection direction is always nearly perpendicular to the direction of emission dipole. These two representative cases imply that the overall emission dipoles of fluorescent molecules are well correlated to the absorption dipoles which are along the direction of excitation electric field. This physical picture is consistent with the fact that the excited molecules emit photons before significant rotation occurs at room temperature where glycerol is viscous [6]. In the following experiments, only the fluorescence with a wavelength longer than 500 nm is collected to measure temperature since a longpass filter is used to remove excitation background.

 

Fig. 2 Optical characterization of fluorescein in glycerol. (a) Experimental configuration to study emission spectra of fluorescein in glycerol. Polarization of incident light is controlled by a half-wave plate and the emission is collected using a spectrometer. Inset: chemical structure of fluorescein. (b) Excitation and emission spectra of fluorescein. (c) Spectrum peak intensity as a function of rotation angle of the half-wave plate. When the detection angle is at 90 °, the peak intensity has a sinusoidal oscillation. While the peak intensity is a constant when the detection angle is at an angle smaller than 20 °.

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4. Results and discussion

To generate a local temperature gradient in the fluorescent solution, we fabricate an aluminum heater with a width of 25 μm on a cover slide. Figure 3(a) shows the scanning electron microscope (SEM) image of the heater, which is composed of a long stripe and four side markers. These markers are used for the alignment between two CCD images. The heater has well-defined edges, shown in Fig. 3(b) and its thickness is measured to be about 750 nm using atomic force microscopy (AFM), as shown in Fig. 3(c). The edge region shows a height of 800 nm because of the residual of photoresist from patterning process. With these geometry parameters, the thermal response of the heater is simulated using COMSOL, a commercial multiphysics software package implementing finite element method. The boundaries in the simulation are set to be far away from the heated zone at ambient condition. The schematic of the simulation model is shown in Fig. 3(d), where glycerol solution is sandwiched between two glass substrates and an aluminum heater is attached to the bottom of the top glass substrate. In the model, resistive heating of the metal film serves as a heat source and only heat diffusion is considered [16]. The transient thermal response is represented along a cutline right underneath the heater, as shown in Fig. 3(e). At time zero, the system stays at room temperature and the temperature increases rapidly in the first five milliseconds. Then the growth becomes smaller and smaller within the same time interval of five milliseconds. At time instant 35 ms, a temperature gradient of about 10 degrees per micrometer is achieved along horizontal direction (along y), as shown in Fig. 3(f). Along the off-plane direction (along z), Fig. 3(f) also indicates that the temperature rise mainly occurs near the surface of the heater which locates within the depth of imaging field. The simulation results provide a rough estimation of the temperature field inside the solution since the accurate temperature distribution is hard to get [16].

 

Fig. 3 Characterization of fabricated heater. (a) Scanning electron microscope (SEM) image of aluminum heater on glass substrate. Scale bar: 20 μm. (b) Enlarged view of the edge of the heater. Scale bar: 1 μm. (c) Height profile across the edge of the heater measured using atomic force microscopy (AFM). (d) Schematic of cross-sectional view of the simulation model (blocks are not to scale). The dotted line indicates the line of symmetry. (e) Temperature profiles along a line right underneath the heater at different time instants. (f) Temperature distribution inside solution at time instant 35 ms.

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In order to achieve repeatable temperature gradients in the solution at microscale, we use a pulsed heating pattern within each measurement cycle, as shown in Fig. 4(a). The heater operates in pulsed mode with a pulse width of 35 ms out of a period of 60 ms. The peak heating current is 180 mA. In the first four measurement cycles, the heating magnitude remains zero therefore the solution stays near room temperature. In the following five measurement cycles, a finite heating magnitude is used to generate temperature gradient. Then the heating magnitude goes back to zero for another five measurement cycles. Because the heat generation is highly confined at microscale, the pulsed heating can rapidly heat up the nearby solution within a few 10s of milliseconds. Despite of the high transient temperature generated near the heater line, the total amount of heat generated remains negligibly small compared to the overall heat capacity of the sample, therefore this microscale heated zone can rapidly cool down after the heating pulse ends and returns to near room temperature without noticeably heat accumulations. Meanwhile each illumination pulse is synchronized with each heating pulse to excite a snapshot of fluorescence image only at the presence of the generated temperature field. The two CCDs are also coordinated with laser pulses and measure fluoresces at a frequency of 1 Hz and a duty cycle of 30%. The choices of above parameters can give a temperature gradient over heat diffusion length ∼ 100 μm [17] and avoid awakeness of strong convections inside the solution. Figure 4(b) shows the fluorescent intensities near the heater are simultaneously measured by two CCD cameras. It can be seen that the intensities of fluorescence captured by both cameras change even though temperature field remains unchanged. These intensity changes are mainly caused by photobleaching of fluorescent molecules, thermal agitation of solution and illumination fluctuation of the LED source. It is also worth noting that even at different temperatures the measured fluorescence intensity could be the same, depending on the imaging NA of the optical system, as shown by the data from CCD2 in Fig. 4(b). And the same temperature doesn’t necessarily give the same fluorescence intensity. This implies that measured fluorescence intensity itself as thermal indicator could be ambiguous. By contrast, the ratio between collected intensities is less affected by intensity variations as shown in Fig. 4(c). It is stable under the same temperature and can well distinguish different temperatures, demonstrating a good quality of being a thermal indicator. Moreover, comparison between intensities captured by two CCDs also reveals that 2NA configuration is most sensitive to the off-plane emissions. It can automatically filter out the in-plane background which emitted by non-rotated dipoles. This is consistent with our theoretical analysis that is detailed in the appendix. The inset of Fig. 4(c) shows the location of a 4 μm × 10 μm area that is used to extract the intensity information. This location is on the metallic heater surface, highlighted by a solid rectangular box whose center is about 4 μm away from the right edge of the heater. Locations on the glass portion of the surface give similar conclusions. The illumination light shines from left to right, so the left edge of the 750-nm-tall heater is bright due to light scattering while the right one casts a shadow on its right.

 

Fig. 4 Comparison between ratiometric measurement and intensity measurement. (a) Time sequences of heating, illumination and CCD exposure. Each measurement cycle contains five heating pulses, five illumination pulses and one CCD exposure. Insets: enlarged view of time sequence of heating (upper panel) and illumination (middle panel) within one CCD exposure. (b) Simultaneously measured intensities from CCD1 (left axis) and CCD2 (right axis) under conditions indicated in panel a. (c) Calculated ratio between simultaneously collected fluorescence intensities shown in panel b. Inset: A fluorescence image. The solid rectangular highlights the area on the heater (25 in width)that is used to extract the information of intensity and ratio.

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To demonstrate this 2NA technique can capture two-dimensional temperature distribution, we calculate the pixelwise ratio change and characterize the corresponding temperature field. Figure 5 plots the 2D temperature mapping for a 12.5 μm × 30 μm area containing both glass and metal surfaces. The upper panel of Fig. 5(a) shows typical fluorescent images captured by two CCDs with shown pixel size ∼ 500 nm. Boundaries of three regions are indicated by dotted lines, which are the glass, edge and metal regions respectively. The glass and metal regions show different fluorescence intensity where in the metal region is brighter because of the higher surface reflections. The edge region shows some intensity variations caused by the light scattering near the heater edge. The upper panel of Fig. 5(c) shows the 2D map of ratio change at a heating current of 205 mA, directly calculated using the simultaneously captured images under two NAs. Here the ratio is defined as intensity of larger NA divided by the one of smaller NA. Comparing with the metal region, the glass region shows relatively larger ratio change at similar temperatures. This difference is caused by the optical properties of different surfaces and can be calibrated out, which will be detailed in the theoretical analysis in the appendix. Figure 5(b) shows the correlations between the intensity ratio change and the surface temperature. The form of ratio dependence on the temperature is chosen based on theoretical prediction as 1/Δr = A/(Δg)^α + B. In this correlation, A, B and α are constants to be obtained from calibrations, Δr is the ratio change from measurements, and Δg = (g(T) − g(T₀)) where Δg is a known function of temperature. The three calibration constants can be experimentally determined by uniformly heating up the sample to known temperatures. Here, A, B and α are we determined as 82.06, 0.36 and 0.53 for glass, and 245.15, −22.31 and 0.40 for metal, respectively.

 

Fig. 5 Map of ratio change and the corresponding map of temperature. (a) 2D map of intensity of two CCDs. Two dotted lines divide each panel into three regions: glass (I), edge (II) and metal (III). Scale bar: 3 μm. (b) Calibration curves of ratio change as a function of temperature for glass and metal surfaces. (c) 2D experimental map of ratio change and temperature converted from panel a based on the calibration curves in panel b. (d) Temperature profiles on glass along the direction perpendicular to the boundary between glass and metal film. Length is measured from left to right. Trend lines are plotted to aid visualization.

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Detailed derivation for the form of the correlation is shown in the appendix. The solid lines are the trends of theoretical predictions and the markers are experimental measurements with error bars. The error bars are calculated using standard deviation in each measurement. The heater temperature is controlled by the amplitude of heating current. Since performance optimization is not the focus of this work, the maximum temperature of the solution is estimated using the change of electric resistance of the heater by assuming solution near the heater shares the same temperature with the heater. The ratio change on the metal can be directly obtained from the metal region while the ratio change on the glass is from location very close to the heater. To reduce the noise caused by random air flow in the extended beam path outside the microscope, a CCD binning factor of two has been used and a total of twenty pairs of CCD images are averaged to calculate the map of ratio change. The lower panel of Fig. 5(c) shows the obtained temperature map at a 205 mA heating current after applying the correlation curves for glass and metal surfaces separately. The correlation curve for metal surface is also applied to the edge region here. The obtained temperature measurements show good trends in the glass and metal regions. In the edge region, the temperature is underestimated because the light scattering is not considered in the analysis. Figure 5(d) shows the measured temperature profiles in the glass region at different heating currents. The measurements are obtained by applying the calibration data directly into measured ratio maps. As we can see from the trend lines of the temperature measurements at the different heater powers, the overall temperature and gradient both increase with heater power correspondingly, demonstrating good reliability of this 2NA technique.

5. Conclusions

We have demonstrated a new 2NA technique that is capable of reliable mapping of temperature. Its advantage is threefold: (i) it is robust, because calculated ratio change from two simultaneously captured images is insensitive to variations in fluorescence intensity; (ii) it is applicable to measure temperature near polarizing materials including metal surfaces, because it measures directional emission patterns of fluorescent dipoles regardless the polarization of collected fluorescence; (iii) it possesses a natural filter against the signal background that falls out the collection cone of the objective when the excitation filed along the direction of optical axis. This new technique is built upon optical microscopes which can provide additional information that is not easily available before. Moreover, this 2NA method has the potential of fast readout rate since 2D temperature images are directly achievable from captures of two CCDs, at the spatial resolution of the microscopy system.

Appendix

To gain insight into the experimental results, we develop a theoretical model to quantify this 2NA method.

A. Fluorescence emission of a single dipole

Assume that a fluorescent dipole, whose orientation is characterized by (θ′, ϕ′), is located at the intersection point between the optical axis of the optical system and the plane z = z₀, where z is the vertical distance from the dipole to the interface between the solution and the heated coverslip, illustrated in Fig. 6(a). An observation point (r, θ, ϕ), where r ≫ z₀, together with the optical axis defines an observation plane, which can be used to define a right-handed spz coordinate system. As shown in Fig. 6(a), s is perpendicular to the observation plane, p is in the plane and perpendicular to the optical axis and z is along the optical axis. Thus the emission dipole vector can be decomposed as

(1)$$\mu =\mu \text{cos}{\theta }^{\prime }\widehat{z}+\mu \text{sin}{\theta }^{\prime }\text{cos}(\varphi -{\varphi }^{\prime })\widehat{p}+\mu \text{sin}{\theta }^{\prime }(\varphi -{\varphi }^{\prime })\widehat{s}$$

where μ = |μ|, ẑ, p̂ and ŝ are unit vectors along z, p and s directions, respectively. Considering the random orientations of emission dipoles and the rotational symmetry along the optical axis of the optical system, the directional emission (magnitude of the average Poynting flux) of the fluorescent dipole is given by [18]

(2)$$I=\frac{n}{2Z_0}\left[{\text{cos}}^2{\theta }^{\prime }{\left|{\mathbf{E}}^{{\mu }_z}\right|}^2+\frac{1}{2}{\text{sin}}^2{\theta }^{\prime }\left({\left|{\mathbf{E}}^{{\mu }_p}\right|}^2+{\left|{\mathbf{E}}^{{\mu }_s}\right|}^2\right)\right]$$

where n is the refractive index of surrounding medium and Z₀ is the impedance of free space. E^μ_i (i = z, p, s) is the generated electric field assuming μ is along i direction. It is a function of r, μ, θ, angular frequency of dipole oscillation ω and the boundary conditions that the dipole is subjected to. Given that the total emitted power of a dipole is equal to its rate of energy absorption under steady illumination, E^μ_i should be calculated using a fixed-power oscillator [18]. One way of quantifying the fixed-power emission is to normalize the dipole emission intensities calculated from a dipole of fixed amplitude using its total emitted power. Here the total emitted power of a dipole is defined as 𝕇 = ω/2𝕀m(μ^* · E), including both radiating and non-radiating energies. Figure 6(b) plots the total emitted power by fixed-amplitude dipoles as a function of z₀, normalized using their same asymptotic value when z₀ → ∞. The nominal emission wavelength is chosen to be 518 nm. The refractive indices of glycerol and glass are 1.47 and 1.52, respectively. For simplicity, only the reflection interface between solution and the heated coverslip is considered in the calculation. When the dipole is 0.2 μm away from the interface, the fixed-amplitude model gives nearly a constant dissipated power for dipoles oriented both parallel and perpendicular to the interface, as shown in Fig. 6(b). This implies that the fixed amplitude model works essentially the same as the fixed-power model when dipoles are more than 0.2 μm away from the interface. Thus fixed-amplitude approximation is valid in our model since the thickness of solution between two coverslips is typically tens of micrometers in our experiments. Note that fluorescence emission has a finite lifetime τ_F, i.e., I = I₀ exp (−t/τ_F), where I₀ = I(t = 0). So integrating Eq. 2 over a solid angle of a cone corresponding to a specific collecting NA yields the collected fluorescence

(3)$$F=\left[{\text{cos}}^2{\theta }^{\prime }{\eta }_z+\frac{1}{2}{\text{sin}}^2{\theta }^{\prime }({\eta }_p+{\eta }_s)\right]\text{exp}\left(-\frac{t}{{\tau }_F}\right)$$

where η_i is the collection coefficient of |E^μ_i (θ, ϕ)|², defined as ${\eta }_i=n/(2Z_0){\iint }_{\mathit{NA}}{\left|{\mathbf{E}}_{t=0}^{{\mu }_i}\right|}^{2}d\mathrm{\Omega }$ (i = z, p, s). When multiple interfaces or metallic surfaces are considered, only the expressions of |E^μ_i (θ, ϕ)|² need to be modified.

 

Fig. 6 Theoretical study of 2NA method. (a) Orientation annotation: observation plane (θ, ϕ); emission dipole plane (θ′, ϕ′); rotational cone of emission dipole (θ″). z₀ is the height of dipole μ measured from the bottom surface of the heated coverslip. (b) The normalized total emitted power of a dipole as a function of height z₀ in fixed-amplitude model at emission wavelength equaling to 518 nm. The normalization value is the same asymptotic value as z₀ → ∞ for dipoles oriented both parallel (horizontal dipole) and perpendicularly (vertical dipole) to the interface. (c) The ratio change as a function of temperature for a dipole in free space. Vertical excitation means excitation field E is along interface normal while tilted excitation corresponds to the case where the angle between E and interface normal is π/6.

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B. Thermal rotation of an emission dipole

Then the effect of dipole rotation can be considered. Assume that the excitation electric field E has a polar angle of θ_e so the induced absorption dipole is also along θ_e, as shown in Fig. 6(a). Strictly speaking, this induced absorption dipole is a hypothetical concept to simplify the physical picture and it only indicates the possibility for the intrinsic absorption dipole getting excited. It implies that the input power for the dipole moment is |μ_i · E|² [18,19], where μ_i is the intrinsic absorption dipole. If the difference between the orientations of absorption and emission dipoles θ″, shown in Fig. 6(a), is only introduced by rotational molecular diffusion, θ″ satisfies the rotational diffusion equation considering the isotropic property of the solution

(4)$$\frac{\partial P}{\partial t}=D_r\left[\frac{1}{\text{sin}{\theta }^{″}}\frac{\partial }{\partial {\theta }^{″}}\left(\text{sin}{\theta }^{″}\frac{\partial P}{\partial {\theta }^{″}}\right)\right]$$

where P(θ″; t) is the normalized probability that μ is along θ″ at time instant t, satisfying $2\pi {\int }_0^{\pi }P\text{sin}{\theta }^{″}d{\theta }^{″}=1$. D_r is the diffusion coefficient by simplifying the molecule as a sphere. It is related to the local temperature T, solution dynamic viscosity ν(T) and the hydrodynamic volume of molecule V by the Debye-Stokes-Einstein equation [6]

(5)$$6D_r=\frac{k_{B}T}{V\nu (T)}\equiv \frac{1}{{\tau }_R}$$

where k_B is the Boltzmann constant and τ_R is the rotational correlation time. Based on Eq. 4, it can be shown 〈cos² θ″〉 ≡ ∫ cos² θ″P(θ″)dΩ″ = [1 + 2 exp (−t/τ_R)]/3 [20]. In the notations, we use 〈X〉 to denote the average of X with respect to orientation. 〈cos² θ′〉 is related to 〈cos² θ″〉 by a rotation of the coordinate system

(6)$$〈{\text{cos}}^2{\theta }^{\prime }〉={\text{cos}}^2{\theta }_e〈{\text{cos}}^2{\theta }^{″}〉+\frac{1}{2}{\text{sin}}^2{\theta }_e〈{\text{sin}}^2{\theta }^{″}〉$$

Substituting Eq. 6 into Eq. 3 and simple rearrangement gives the expected value of the collected dipole emission at a particular time instant t

(7)$$〈F(z_0,t)〉=\frac{\left(3{\text{cos}}^2{\theta }_e-1\right)}{2}\frac{(2{\eta }_z-{\eta }_p-{\eta }_s)}{3}\text{exp}\left[-\left(\frac{1}{{\tau }_R}+\frac{1}{{\tau }_F}\right)t\right]+\frac{({\eta }_p+{\eta }_s+{\eta }_z)}{3}\text{exp}\left(-\frac{t}{{\tau }_F}\right)$$

Since the integration time of CCDs (∼ ms) is much longer compared to both τ_R (∼ ns to μs) and τ_F (∼ ns), the expected fluorescence contributed by a single dipole at z₀ on average is given by

(8)$$〈\overline{F}(z_0)〉=\left[\frac{\left(3{\text{cos}}^2{\theta }_e-1\right)}{2}\frac{\left(2{\eta }_z-{\eta }_p-{\eta }_s\right)}{3}\frac{1}{1+{\tau }_F/{\tau }_R}+\frac{({\eta }_p+{\eta }_s+{\eta }_z)}{3}\right]{\tau }_F$$

where τ_F is assumed to be temperature independent [6]. And the only strong temperature-dependence goes to the term τ_F/τ_R. In principle, τ_R can be engineered to control the temperature region where the measurement is most sensitive to temperature [6]. Note that when θ_e = 0, corresponding to the vertical excitation in Fig. 6(c), cos² θ_e becomes the largest, so 〈F̄(z₀)〉 has the strongest dependence on temperature. Under this excitation configuration, the polarization anisotropy becomes an invariant zero because the fluorescence becomes unpolarized measured from the direction of optical axis due to the rotational symmetry. This also suggests that a separate fitting may be needed to get accurate temperature estimation at the edge region of the heater where the excitation electric field may not along the off-plane direction. In addition, when the excitation direction is off-plane, the collection efficiency of fluorescence is the smallest, which essentially filters out the fluorescence background, considering the donut-shaped pattern of directional emission of a free dipole. Here we use X̄ to denote the average of X with respect to time. Since z₀ ≪ r and the excitation electric field E is not strongly z-dependent, it is reasonable to assume that under constant temperature field the overall fluorescence collected by the objective is just 〈F̄(z₀)〉 multiplied by a prefactor C which accounts for the concentration and thickness of the solution. This conclusion is also based on the fact that the solution concentration is not high in our experiment so that the interactions between dipoles are negligible [12].

C. Characterization of 2NA method

The ratio between two intensities collected by 2NA setup is defined as r ≡ 〈F̄₁〉/〈F̄₂〉, so ratio change Δr ≡ r₀ − r is given by

(9)$$\frac{1}{\mathrm{\Delta }r}=A\frac{1}{\mathrm{\Delta }g}+B$$

where A = [a₂g(T₀) + b₂]²/(a₁b₂ − a₂b₁), B = a₂[a₂g(T₀) + b₂]/(a₁b₂ − a₂b₁) and Δg = (g(T) − g(T₀)), with a ≡ (η_p + η_s + η_z)/3, b ≡ (η_p + η_s + η_z)/3 + (3 cos² θ_e − 1)(2η_z − η_p − η_s)/6 and g(T) ≡ τ_F/τ_R. The subscript “0” indicates room temperature. Figure 6(c) gives two examples of the change of intensity ratio of free dipoles as a function of temperature where τ_F = 3.7 ns, V = 0.41 nm³, T₀ = 20°C, NA₁ = 0.6 and NA₂ = 0.5. The dynamic viscosity is calculated using the empirical formula for glycerol-water mixture given by Cheng [21]. In Cheng’s paper, temperature only extends up to 100 °C but the error of estimated viscosity is verified to be less than 14% at a high temperature of 167 °C for glycerol [22]. For a dipole in free space, ${\left|{\mathbf{E}}_{t=0}^{{\mu }_s}\right|}^2=(1/2\pi )\left({\mu }^2{\omega }^4/r^2{c}^4\right)$, ${\left|{\mathbf{E}}_{t=0}^{{\mu }_z}\right|}^2={\left|{\mathbf{E}}_{t=0}^{{\mu }_s}\right|}^2{\text{sin}}^2\theta$ and ${\left|{\mathbf{E}}_{t=0}^{{\mu }_p}\right|}^2={\left|{\mathbf{E}}_{t=0}^{{\mu }_s}\right|}^2{\text{cos}}^2\theta$. As shown in Fig. 6(c), Δr increases monotonically with temperature, which agrees well with Fig. 5(b), making it a good indicator for thermometers. By comparing the points where Δr changes most rapidly with temperature in two curves indicated by dotted lines, it can be concluded that the excitation angle θ_e offers another degree of freedom to tune the sensitive window of the 2NA method besides the thermal correlation time τ_R.

Besides parameters A and B in Eq. 9, we introduced another fitting parameter α in Fig. 5(b), to take into account some non-ideal factors that could modify the effective collecting NA: (1) τ_F is actually temperature dependent [23] though the dependence is not strong compared to τ_R; (2) some fluorescence signals are collected from dipoles that are out of focus considering the depth of focus of the objective; (3) the decay of fluorescence intensity is more accurately described by several time constants [24]; (4) the orientations of the absorption dipole and the emission dipole in the derivation are simplified compared to real system [25]; (5) the measured temperature is an averaged quantify with respect to time t and height z. While these factors alter calibration in a systematic way and can be compensated using fitting parameters to some extent, temperature gradient may change calibration relationship locally since the induced non-uniformity in the refractive index of the solution can deviate fluorescence beam. However, this temperature gradient effect is not strong. The maximum angle deviation is estimated to be less than 0.6° for NA= 0.4 by assuming that dn/dT = −6 × 10⁻⁵/°C, [26] dT/dy = 10 °C/μm and the thickness of solution is 25 μm.

Funding

National Science Foundation (NSF) (Grant No. CMMI-1405078, Grant No. CMMI-1554189 and Grant No. CMMI-1634832).

Acknowledgments

The authors appreciate Professor Amy Marconnet’s help in performing thermal characterization of aluminum heaters using infrared thermal microscope. The authors are also grateful for the awarded Purdue Research Foundation (PRF) Grant.

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