Far-field characterization of the thermal dynamics in lasing microspheres

This work reports the dynamical thermal behavior of lasing microspheres placed on a dielectric substrate while they are homogeneously heated-up by the top-pump laser used to excite the active medium. The lasing modes are collected in the far-field and their temporal spectral traces show characteristic lifetimes of about 2 ms. The latter values scale with the microsphere radius and are independent of the pump power in the studied range. Finite-Element Method simulations reproduce the experimental results, revealing that thermal dynamics is dominated by heat dissipated towards the substrate through the medium surrounding the contact point. The characteristic system scale regarding thermal transport is of few hundreds of nanometers, thus enabling an effective toy model for investigating heat conduction in non-continuum gaseous media and near-field radiative energy transfer.

Standard investigation of dynamical thermal behavior of passive WGM cavities is performed by in-coupling laser light through the evanescent field of a tapered fiber. Under such conditions, the transmitted light is monitored while the WGM cavity is out of thermal equilibrium in response to non-adiabatic heating 11,12 . Conversely, the coupled light can be used both to test the thermal dynamics and as a heating source localized within the WGM volume 10 . However, this approach complicates the dynamics of the system since: i) the heating efficiency is modulated by the shape of the resonance and ii) the cavity takes some time to thermalize internally.
In this work, we present an alternative approach that combines high-power laser emission and narrow spectral linewidths with excitation-detection in the far field, enabling a straightforward investigation of the thermal dynamics of the cavity with all-optical means.
The investigated WGM lasing cavity is a Nd-doped Barium-Titanium-Silicate (BTS) 7 microsphere fabricated with the method reported in ref. 14 (see Methods) placed on top of a 2.7 micron thick SiO 2 layer deposited over a Si substrate. A free space laser is used to resonantly excite the Nd 3+ ions and homogeneously heat the cavity, while the signal of a lasing WGM is collected in the far-field to monitor the thermal dynamics of the cavity. The sphere-plate geometry has been the object of deep investigation during the last years in the context of local thermal transfer between non-planar geometries [15][16][17] mainly driven by their promising application in nanoscale heat stamps 18,19 . We demonstrate that, at ambient conditions, thermal dynamics is governed by heat conduction to the substrate, mediated by the medium surrounding the contact point. Given that the characteristic dimensions of thermal transport are of few hundred nanometers, i.e., substantially lower than the thermal wavelength and comparable to the gas mean free path, we propose lasing microspheres as a simple and versatile scheme to investigate radiative thermal conduction and non-continuum gas thermal behaviors.

Results and Discussion
The measurement setup for the optical experiment is schematically shown in Fig. 1. A continuous wave (CW) laser at 808 nm was focused on a single microsphere (radius between 20 and 25 μ m) by means of a microscope objective 20 , the spot size being large enough to fit the microspheres dimensions. Heat generation can be approximated to be spatially homogeneous within the microspheres since: i) the spatial inhomogeneity of the pump power density provoked by their focusing ability is not very pronounced (the focus lies several microns below the substrate surface), ii) the spatial profile of the PL emission looks homogeneous (see bottom-right panel of Fig. 1) and iii) the internal thermal dynamics is much faster than the dissipation to the surrounding. The PL signal associated to WGMs contained in a plane close to the microsphere equator (parallel to the substrate) and coming out from a lateral edge was collected by a second microscope objective. It was then focused on the input slit of a monochromator and recorded either by a charge-coupled-device (CCD) detector or by a photomultiplier tube (PMT). Time-resolved experiments were made by modulating the pump with a mechanical chopper and by registering the PMT output with a digital oscilloscope (see Methods for further details of the experiment). Dimensions are not to scale. The bottom-right panel illustrates a spatial profile of the emitted PL below lasing threshold, which has been taken by placing the focus of the collection objective at the sphere center and widely opening the pinhole. The sphere contour is highlighted for clarity. The bright spot is probably associated to a scattering region within the sphere. The The Nd 3+ excitation process is described in Fig. 2a. Under external excitation, Nd 3+ ions are promoted from the ground state ( 4 I 9/2 ) up to the 4 F 5/2 and 2 H 9/2 levels, from where a rapid non-radiative thermalization to the 4 F 3/2 transition takes place. Subsequently, another radiative de-excitation occurs from this level down to the 4 I 11/2 state. The final thermalization to the ground state is through non radiative scattering. Nd 3+ effective dynamics can be modeled by a four-level rate equation system, where N 4 , N 3 , N 2 and N 1 correspond to the number of ions excited in transitions 4 F 5/2 , 4 F 3/2 , 4 I 11/2 and 4 I 9/2 , respectively. Laser action at around 1064 nm occurs provided that N 3 > N 2 and internal gain compensates passive losses. Figure 2b shows a 2D color-map displaying the measured PL intensity of a microsphere with radius R = 25 μ m as a function of the observation wavelength (λ c , y-axis scale) and the pump photon flux (Φ p , x-axis scale). Multimode lasing is observed at various wavelengths within the Nd 3+ emission band at around 1064 nm. A typical spectrum taken at the maximum used photon flux (Φ p = 5.5 × 10 23 ph/cm 2 s) is reported on the right hand side of Fig. 2b. The two most intense lasing peaks have been numbered as 1 and 2, respectively.
In the same figure, a red-shift associated to the TO effect is observed for all lasing modes when Φ p is increased. In a first order approximation it is possible to correlate the spectral shift of the lasing mode (Δ λ = λ s − λ ini , where λ s and λ ini are the stationary lasing wavelength for a given temperature and the cold-cavity resonance wavelength respectively) with the temperature difference between the volume where the mode is localized and the surrounding environment as: where a is the TO coefficient, accounting for both physical dilatation and refractive index changes 10 . Figure 2c shows that, for a microsphere with radius R = 25 μ m, while laser emission is inherently nonlinear (superlinear at first and then saturating for high Φ p (left y-axis scale)), Δ λ follows Φ p linearly (right y-axis scale). This indicates that the dominant heating mechanism is partial absorption of the incident pump power radiation ( where σ p , λ p and V are the absorption cross section, pump wavelength and sphere volume respectively), which is to some extent released into heat through the non-radiative decay of the excited ions described above. This result allows us to neglect localized heating within the cavity mode and, therefore, to consider spatially homogeneous heating of the whole sphere. The offset of the linear spectral shift curve corresponds to the cold-cavity resonance (λ ini ), which is λ ini = 1065.0 nm for the measured sphere.
The thermal dynamics of the microsphere can be described by the following heat equation: Intensity (arb. units) which reflects the balance between the incoming heat due to partial absorption of the incident pump power radiation minus the heat dissipated to the surrounding. C p is the heat capacity of the microsphere (C p = ρVc p , ρ and c p being the mass density and the specific heat of the BTS glass respectively) and can be directly calculated, since both the glass intrinsic properties and the dimensions of the spheres are known. γ is the efficiency at which P abs is converted into heat and k is the heat transfer rate per unit temperature. In order to extract an analytic solution of the previous equation we assume that the dynamics governing the ion level population has achieved a stationary regime, making the heat absorbed rate time-independent. Under such condition, and using Eq. 1, the temporal evolution of Δ λ can be written as follows: The exponential fit of the dynamical Δ λ evolution allows extracting the transitory characteristic time This scales linearly with R, since C p and k are proportional to the sphere volume and surface respectively (the heat leaking through the contact point is neglected). Then, the stationary spectral shift is given by: The experimental linear evolution of Δ λ with Φ p reported in Fig. 2c agrees with the theoretical analysis, giving support to the hypothesis of considering k as a temperature-independent parameter. This is confirmed by the large stability of thermal conductivity of air for the particular experimental conditions (50% relative humidity), with a 10% variation over hundred degrees above ambient temperature 21 .
Time-resolved measurements of the laser emission spectral shift were carried out in phase with the modulated pump. Different λ c were collected, accordingly to the monochromator filtering. A transitory signal was detected when the lasing resonance aligned with λ c on its pathway from λ ini towards λ s as a consequence of the thermal-induced red-shift. The red-shift delay time (Δ t) was then extracted taking the chopper as an absolute reference that marks the beginning of the microsphere excitation process. A complete temporal evolution of the lasing wavelength spectral shift was obtained by subsequent recording of Δ t for a complete sweep of λ c from λ ini to λ s in small steps (0.01 nm). Figure 3  captured. This is ascribed to a decrease of the red-shift rate during the measurement, i.e., the resonance wavelength shifts faster when entering into the detection range than when coming out. -Panel c) (λ c ≈ λ s ) shows that a stationary signal appears and persists for the entire excitation period until the pumping is switched off again. This corresponds to a full thermalization of the microsphere, the stationary lasing wavelength being that measured with CW excitation.
The 2D color-map plot of Fig. 4a illustrates a particular case obtained by pumping a sphere of R = 25 μ m with Φ p = 5.5 × 10 23 ph/cm 2 s. The spectrum covers the full temporal trace of lasing mode 1 and the last part of that of lasing mode 2 (see the corresponding CW spectrum in Fig. 1b). Both curves follow the same exponential law, reaching λ s after about 10 ms. From this point on, stable thermal equilibrium is maintained. Figure 4b shows the temporal traces of the spectral position of lasing mode 1 upon low and high Φ p (2.5 × 10 23 ph/cm 2 s and 5.5 × 10 23 ph/cm 2 s, respectively). As expected, Δ λ max scales with Φ p , in accordance with Eq. 4 and with the previously reported values in Fig. 2c (marked with dashed lines). When normalizing Δ λ to Δ λ max for the specific value of Φ p , the obtained curves overlap (Fig. 4c). In fact, a characteristic lifetime of τ (R = 25 μm) = 2.4 ms is extracted for both curves, which, according to Eq. 4, allows us to conclude that also C p can be assumed independent of the temperature of the cavity. Indeed, it has been previously demonstrated that although Δ T can reach a few hundreds of degrees in extreme cases 22 , C p in BTS is almost unresponsive to temperature variations 23 . The pure exponential behavior of the thermal dynamics can be further inspected in Fig. 4d, given that, when represented in log scale, 1 − Δ λ /Δ λ max shows a linear behavior with negative slope.
Finally, in Fig. 5a we have compared the thermal dynamics of a sphere of R = 25 μ m (τ (R = 25 μm) = 2.4 ms, black dots), with that of a smaller radius (R = 20 μ m, green dots). As expected, the characteristic lifetime scales with R, the value obtained for the latter case being τ (R = 20 μm) = 1.8 ms. It is worth noting that an equivalent trend was observed in several lasing spheres of similar radii, denoting high consistency and reproducibility.
In order to get more insight on the origin of the observed thermal dynamics we have performed Finite-Element Method (FEM) simulations on a system consisting of a BTS glass microsphere that is heated up by a homogenous source distributed along the sphere volume. The used FEM module solves Fourier's heat conduction law just by considering conductive heat transfer mechanisms through  purely diffusive processes. We have neglected convection on the basis of the small dimensions of the spheres 24 and radiation since it only becomes significant with respect to conduction at gas pressures below 10 −2 mb 17 . On the basis of the high relative humidity of the ambient, we have considered a temperature independent air thermal conductivity, as discussed previously. The temperature has been monitored within the sphere with two probes. The first is the whole sphere domain, while the second is a much smaller spherical region (radius = 1 μ m). The normalized thermal dynamics extracted from the two probes are equivalent, regardless the position of the small one. We have first solved the problem of a sphere fully surrounded by air (orange solid curve of Fig. 5a). The obtained result is a spatially homogenous output heat flux along the sphere surface, but with a much slower thermal dynamics than that observed experimentally. The second case of study was that of a sphere under vacuum deposited on a substrate such as that of the experiment, which leads to an even slower dynamics (cyan solid curve of Fig. 5a). This is an expected result, as the released heat can only be dissipated through the contact point between the sphere and the substrate. The direct combination of both cases, i.e., microsphere surrounded by air on top of a substrate, successfully reproduces the experimental results for the two microsphere radii under analysis (R 1 = 20 μ m and R 2 = 25 μ m). It is worth noting the absence of free parameters in these simulations, the input values being only the geometric dimensions and the intrinsic properties of the materials involved. Figure 5b shows the corresponding stationary Δ T profiles along the horizontal (r, parallel to the substrate surface) and vertical (z, perpendicular to the substrate surface) directions passing through the center of the sphere, i.e. r = 0 and z = 0. The inset shows the bidimensional distribution along a cross section of the whole system geometry. In the horizontal direction (along the r-axis) the temperature profile (black curve of Fig. 5b) is isotropic. Starting from r = 0, a subtle decrease in Δ T is observed when approaching the surface of the sphere, followed by a steep decay for r values beyond the radius of the sphere (r > 25 μ m in this case). On the contrary, along the vertical direction, the profile is strongly affected by the substrate, providing a highly asymmetric feature. In the upper semisphere, Δ T is much higher while it gets very close to zero at the contact point with the substrate. This spatial distribution is better understood by analyzing the output heat flux across the microsphere surface (Fig. 5c), which reveals that most of the heat dissipates through its lower part. It is thus possible to conclude that the main heat loss mechanism is thermal conduction towards the substrate through the region surrounding the substrate contact point. In fact, temperature gradients increase towards it, given that the substrate remains very close to ambient temperature. As described above, we monitor lasing modes that come out from one of the lateral edges of a microsphere, i.e., they circulate on planes close to the equator of the microsphere. Thus, given that there is an azimuthal symmetry in the temperature spatial distribution and that the mode is confined in the outer part of the sphere, it is a good approximation to assume that the temperature along the volume mode is spatially homogeneous. The characteristic length scale of the system, which is the distance between the lower part of the sphere surface around the contact point and the substrate (Δ z = R(1 − cosθ ), where θ is the angle with respect to the z axis pointing downwards), remains below 400 nm in the region dissipating 90% of the heat (being θ < 11°).
In conclusion, we have performed an experimental characterization of the thermal dynamics of Nd-doped BTS laser microspheres deposited on a glass substrate when they are homogeneously heated up by a pump laser. By using a lasing cavity, we overcome the stringent requirements for characterizing passive WGM cavities, i.e., near-field excitation of the optical modes, hence allowing for far-field pumping and detection. Heating characteristic lifetimes of about 2 ms were determined by tracing the dynamic spectral behavior of the lasing modes. Those were dominated by the heat exchange towards the substrate through the medium in the close surrounding of the contact point (moist air in our case).
Provided that the heat transfer occurs on distance scales on the hundreds of nanometers, we believe that this system and technique could become an interesting tool for studying thermal transport dynamics in non-continuum regimes. For the same reason, it would be possible to shine some light in the yet not fully understood enhancement of the radiative heat transfer 17 by performing a similar characterization below 10 −3 mb, in which it becomes dominant over thermal conduction 16 .

Methods
For the optical measurements, a continuous wave laser diode at 808 nm is modulated at a frequency of 11 Hz through a mechanical chopper. After spatial filtering by means of a pinhole it is shined into an infrared objective (Mitutoyo 20X, NA = 0.4). The resulting focused beam has a spot with an area of 1.5 × 10 −5 cm 2 , where the power is ensured to be within 90% of the maximum value. A single microsphere can be aligned with the pump beam using a xyz micro-precision stage. The PL is collected by an objective placed in front of the microsphere. A long-pass filter with a cut-off wavelength of 950 nm is used to filter out the laser beam excitation. The output beam is directed towards a second pinhole that removes the scattered signal. The diameter of the microspheres is far greater than the spatial resolution of the collection system (about 5 μ m) 20 , hence the collected PL emission comes out of localized volumes within the microsphere. In the current work we align the collection objective focal plane with the center of the microsphere and collect the signal from a region around one of its lateral edges, where the signal associated to WGMs is well-defined and not masked by the Nd 3+ bulk emission 20 . Moreover, those are the modes experiencing less radiative losses due to the presence of the substrate. Collected light is then focused at the entrance slit of a 750 mm monochromator. For signal detection, a CCD is placed at one of the monochromator output ports and directly connected to a computer. An infrared PMT (H10330-25) is placed at the other output port of the monochromator, and used in time-resolved measurements at various wavelengths. In the latter case, the signal is monitored by a digital oscilloscope using the TTL signal of the chopper as a reference. The best attainable temporal resolution is roughly 1 μ s, and the maximum spectral resolution is 0.01 nm. The dimensions of the microspheres have been quantified using the collection objective and forming the device image in the CCD. Using a structure of known size as calibration, we achieved a spatial resolution of about 1 μ m 20 .
The microspheres were fabricated from BTS glass doped with Nd 3+ ions with the composition of 40%BaO-20%TiO 2 -40%SiO 2 and doped with 1.5% Nd 2 O 3 (in the molar ratio). The glass is reduced to dust by means of a mortar and is heated up to its fusion temperature, which is around 900 °C. Most of the splinters melt and, when the temperature decreases, solidify in a spherical shape of several micrometer radii 14 .