1. Introduction

Irradiated metallic nanostructures have found a large number of scientific applications, ranging from thermal imaging to cancer therapy [1–3]. For an irradiated structure presence of thermal heating is unavoidable due to electromagnetic absorption. It is shown that by engineering the particle’s shape and size the absorption cross section can be maximized for a given wavelength [4]. Thanks to nanotechnology, today nanoparticles with desired material and shape can be produced. Therefore, irradiating a pre-designed metallic nanostructure could produce a localized nanoscale heat source [5]. One exciting application for such a nanoscale heat source could be photo-thermal cancer therapy, in which, the pre-designed metallic nanoparticles is first functionalised so that they can target the cancerous cells inside a given biological tissue. Once they are accumulated around the cancerous cells irradiating the tissue will selectively damage the cancerous cells. However, in such a delicate application it is crucial not to damage the healthy cells. This can be simply done by controlling the intensity level of the irradiating source while monitoring the temperature distribution around the irradiated nanostructures. Although there have been previous attempts on finding temperature distribution around an irradiated nanostructure, however, a reliable method for measuring temperature in nanoscale is still desired. For instance, the liquid-gel phase transition temperature of lipid membrane [6, 7], melting temperature of ice around an absorbing particle [8], photoacoustic [9], thermal [10] imaging, white light scattering [11], and thermal fluctuation of the trapped particle [12, 13] are used for estimation of the temperature at the surface of an irradiated nanoparticle. In the more recent report the photoluminescence of a laser trapped erbium oxide nanoparticle [14], and changes of re-absorption of fluorescent emission in the mixture of two dyes [15] are used for thermometry at nanoscale. In many of the recent works, the accuracy of the methods is not clearly discussed. Only in [10], it is claimed that the spatial resolution of the technique is diffraction limited, and temperature variations weaker than 1 K can be detected, and also in [15], the accuracy of the temperature is mentioned as the standard deviation of the obtained temperature.

The common feature of the above mentioned methods is that they all need a mediated localized observation which occurs in a very specific condition. For instance, in [6] the temperature can only be correctly estimated at the border of liquid-gel phases. Or in [8] the temperature is only known at the ice-water interface. In the both cases for a given experimental condition the temperature can only be measured at a spatially localized position. In [10], also, the temperature determined as a result of convolution approach from many particles, which have to be precisely arranged on the surface. Therefore, finding a method which allows for measuring temperature profile around a nanoscale heat source is highly desired.

In this report, we introduce a digital holography-based method for measuring temperature profile at submicron scale. We use off-axis digital holographic microscope as an effective tool for quantitative 3D imaging of temperature-induced phase change around a nanoscale hot source. This allows for 3D mapping of temperature distribution around the heat source. The method employs a Mach-Zehnder interferometer implemented in an open setup microscope. Using this method, temperature distribution around a single submicron gold sphere, as well as a silver nano-shell coated micron-sized silica bead, irradiated with laser light were quantified.

2. Theory

Holography technique utilizes the intensity pattern produced by interference between a reference beam with that of reflected from the object of interest. This intensity pattern is then translated into phase difference between the two beams which allows for mapping the object in 3D. In digital holography this pattern is recorded using a digital camera and then transferred to a computer in order to perform the required calculations. Consider that U_r (x′, y′) = A_r (x′, y′) exp[iϕ_r (x′, y′)] and U_o (x′, y′) = A_o (x′, y′) exp[iϕ_o (x′, y′)] are the complex amplitudes of the reference and object waves at the hologram plane, respectively. Here ϕ_r and ϕ_o represent the phase distribution of the reference and object waves. To remove the contributions from the zero-order and the twin-image terms the phase-shift method is applied [16,17]. The recorded intensity distribution can be given by:

The angular spectrum method is applied to the modified hologram in order to calculate the optical field (phase distribution) at the plane that includes the center of microsphere.

Now consider the case where the object beam is passed through a medium containing a nanoparticle (located at the center of the coordinate system). If the nanoparticle is locally irradiated, the absorbed energy would be dissipated as heat and transferred to the surrounding medium. In the steady state, one can assume a continuous temperature distribution around the particle which decays to the ambient temperature at locations very far away from the source. In such a case one can equally assume a continuous Refractive Index (RI) distribution around the irradiated nanoparticle provided that presence of a linear relation between the temperature and the refractive index of the medium. Fig. 1(a) schematically shows such a temperature distribution around the object. Fig. 1(b) and Fig. 1(c), respectively, show a typical RI-temperature dependence (will be explained in more details) and resulted temperature distribution.

 

Fig. 1 (a) Schematic of irradiated microscopic object. (b) Temperature dependency of refractive index for surrounding medium measured by Fresnel diffraction from a phase wedge [18]. (c) Typical refractive index (dashed-blue) and temperature (solid-red) as a function of distance from the center of the irradiated bead.

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As it is shown in Fig. 1(a), let’s divide the surrounding medium into a series of concentric annular rings with thickness of δρ and RI of n_i. The thickness of the ring is so small that the temperature is assumed to be constant over each ring. By this consideration and using Abel equation, the phase information can be derived as, $\mathrm{\Delta }\varphi (x,z)=(4\pi /\lambda ){\displaystyle {\int }_x^R[n(x,y,z)-n_0]/\sqrt{{\rho }^2-x^2}\rho d\rho .}$ Here, n₀ and n(x, y, z) are the RI of the medium far away from the heat source and at position with coordinates of x, y, and z, respectively [19]. R is the radius of the irradiated object, and λ is the wavelength. To reach the RI gradient from the measured phase, the inverse transformation of Abel equation can be used:

Once RI distribution is determined by above mentioned calculations, the temperature profile can be immediately determined using the pre-known RI-temperature relation.

In the absence of phase transition, a quantitative description of the temperature increase around a hot sphere placed in an infinite medium can be given by heat transfer equation [20,21]: ρ_m(r)c(r)∂T (r, t)/∂t = ∇k(r)∇T (r, t) + Q(r, t), where, T (r, t) and Q(r, t) denotes the spatial and temporal distribution of temperature and local heat, respectively. ρ_m(r), c(r), and k(r) are mass density, specific heat, and thermal conductivity of the surrounding medium, respectively. By solving the heat transfer equation in a steady state situation, temperature profile around the sphere can be written as [21]:

3. Experiment

Our experimental setup is based on a Mach-Zehnder interferometer implemented in a custom-designed open setup microscope, as schematically shown in Fig. 2(a). A collimated He:Ne laser beam with λ_p = 632.8 nm (hereafter would be called probe beam) is first divided into two separate beams with equal intensities using a 50:50 beam splitter (BS₁).

 

Fig. 2 (a) Schematic of the setup: A Mach-Zehnder interferometer implemented into a custom-designed microscope. The heating and probe beams are shown in green and red, respectively. The mirror M₂ is mounted on a PZT with nanometer resolution. The interference patterns are recorded using a CMOS camera. (b) and (c) show typical recorded holograms for a 3.2 μm silver coated silica bead, respectively, in absence and presence of the heating beam.

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The object beam reflects from the mirror M₁ after which passes through a condenser (Olympus, 10×, NA= 0.3, ∞), sample chamber (S), and an objective (Bresser, 100×, NA= 1.25, 160) before reaching the CCD. After reflection from the mirror M₂, the reference beam is passes through another 100× objective (MO₂) in order to balance the wavefront curvatures of the reference and object beams. A Nd:YAG laser (λ_h = 532 nm) - to be used as heating source - is also introduced into the optical path of the microscope from the condenser side. The object and reference beams are combined using another beam splitter (BS₂) and their interference pattern is recorded using a CMOS camera (DCC1545M, Thorlabs, 8 bit, 5.2 μm pixel pitch). A laser line filter (LLF) positioned before the camera blocks the heating laser beam. The mirror M₂ is mounted on a Piezo Transducer (PZT) with nanometer resolution in order to introduce very accurate phase-shifts between the two beams. The sample chamber was made using a coverglass and a microscope slide spaced by two strips of double sided scotch tape with a thickness of ~80 μm.

Gold nanospheres with mean diameter of 400 nm (Sigma-Aldrich) and also spherical silica beads with mean diameters of D = 1.2 μm, 2.0 μm, and 3.2 μm coated with a thin layer of silver were used as absorptive objects in this research. The coating on the silica beads were made using wet chemical coating method [22] with a thickness of ~100 nm.

The beads were embedded in silicone oil by ultrasonic agitation∼before being loaded into the sample chamber. Due to high viscosity of the silicone oil, it took few hours for the solution to be loaded into the chamber. It is worth mentioning that the RI of silicone oil is a steeper function of temperature compared to water which increases the sensitivity of the measurement. Another advantage of silicone oil is that due to its large viscosity convection around the heated bead can be neglected.

In each measurement, first, the particle of interest was accurately positioned under the focal spot of the heating laser beam. Then, the interference patterns were recorded with and without presence of the heating beam - will be referred to as object and reference holograms, respectively. The recorded reference hologram allows to remove the heating effect of the probe beam and also to remove any optical aberrations of the system. Figures 2(b) and 2(c), respectively, show typical reference and object holograms for a 3.2 μm silver coated bead.

The closer scrutiny of the fringes in the vicinity of the bead in Fig. 2 reveals that irradiating the bead [2(c)] deflects the fringes from that of the non-irradiated one [2(c)]. The reason for this deflection is that RI of the medium is decreased due to increase in the temperature of the surrounding medium which in turn causes a phase deflection in the wavefront of the object beam. Three phase-shifted images were recorded. In practice, the phase shift was applied by translating the PZT mirror (M₂) by a constant distance for each recorded hologram. Eq. (2) was used for the recorded holograms to obtain the modified hologram without the zero-order and twin-image terms. Finally, angular spectrum method was used to propagate the modified hologram to an appropriate distance, ~23 mm, so that the phase distribution at the plane which consist the center of microsphere is determined.

4. Results

Figure 3 shows typical resulted spatial phase distribution around the irradiated silver-coated silica bead with sizes of 1.2μm (a) and 2.0μm (b–d). On can see that: 1) There is a region around the bead for which increase in the temperature can be recognized. 2) The size of this region depends on the irradiating power, absorption cross-section of the irradiated bead, and resolution of the detection method. We noticed that at a given irradiating laser power the larger beads became hotter, which is due to their larger absorption cross- section. 3) Another advantage of the current method is that the whole temperature field can be determine at once and there is no need to change any parameter of the irradiating experiment (such as the irradiating power). 4) Using the measured phase distribution and Eq. (3), the RI profile around the irradiated bead can be obtained, in which the deviation from circular symmetry was less than 5%.

 

Fig. 3 Two-dimensional phase distribution around the silver-coated beads. Particles size are 1.2 μm and 2.0 μm for (a) and (b)–(d), respectively. Δϕ axis and colors show the phase in radian. (b)–(d) Phase distribution in the situation at three different intensities of heating beam (I_h). The colorbar represents phase values in radian for right column.

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As it was mentioned earlier, in order to correctly determine the temperature profile around the irradiated bead one has to accurately determine the RI-temperature dependence for the medium. In our case, this was done using the high resolution optical refractometry based on Fresnel diffraction from a phase wedge [18] at different temperatures, results of which are shown Fig. 1(b). The solid line shows linear fit to the data point with a slope of (35.75 ± 3.25) × 10⁻⁵/°C. Dividing the phase values to the current conversion factor will provide temperature distribution around the irradiated bead.

Figures 4(a)–4(d) show radial temperature profile (averaged over the angular direction) around the particle under different irradiation. As it was mentioned earlier, the temperature around the irradiated bead should drop with inverse of the distance from the center of the bead. Therefore, the resulted data were fitted to the function of T = a/r + T₀ with a and T₀ being the free parameters, results of which are shown as solid graphs in Figs. 4(a)–4(d). It is evident that the fitted curves nicely follow the data points, which has a coefficient of determination (RMS) below 1 °C. By knowing the fit parameters and the particle’s radius, one can calculate the tem perature on the surface of the particle. Resulted temperatures are summarized in Fig. 4(e). As it was expected, the temperature of the gold sphere is significantly higher than silica beads at the same irradiating laser power [23]. Temperature on the surface of the particle could also be extracted directly from Figs. 4(a) to 4(d) right at the boarder of the bead in the image results of which are shown in Fig. 4(f). It is worth mentioning that the results are averaged over at least 15 measurements (5 different beads with 3 measurements for each), whit the standard deviation less than 1 °C. For the case of 400 nm gold sphere this method was not applicable as its size was below the imaging resolution of the microscope. Fit to a linear function of T = αI_h + β with α and β being the free parameters are shown with solid lines which confirms that temperature on the surface of the irradiated bead is a linear function of the power of irradiating laser.

 

Fig. 4 (Top row:) Radial temperature profile around the irradiated beads: (a)–(c) Silver-coated silica beads with mean diameter of 3.2 μm, 2.0 μm, and 1.2 μm, respectively, and (d) Gold sphere with mean diameter of 400 nm. Solid lines show fit to the equation T = a/r + T₀. (Bottom row:) Temperature measured on the surface of the beads resulted by (e) fit parameters of temperature profile around the particle, and (f) directly taken from the data points on the vicinity of the particle’s radius in plots (a)–(d).

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Comparison of Fig. 4(e) and Fig. 4(f) shows that surface temperature resulted from the two methods are in a very good agreement. Though both methods may seem similar at first look, however, the second method has the advantage that the temperature at a given position can be directly extracted from the local (the containing pixel) information whereas for the first method one needs the information (temperature) is a reasonable length scale. In other words our method is able to determine the local temperature based on the local information.

5. Conclusion

We propose a method for thermometry in sub-micron scale. The method utilizes the well-known Mach-Zender interferometer coupled into a diffraction limited microscope. This method allowed us to map the temperature field around an irradiated (sub-)micron beads. Experimental results perfectly revealed the reciprocal temperate decay with the distance from the surface of the hot particle, T ∝ 1/r.