# Laser cooling of ytterbium-doped silica glass ### Determination of the cooling efficiency

The cooling efficiency, ηc, is defined as the net power density (per unit volume) extracted from the material (pnet) per unit total absorbed power density (pabs): ηc = net/pabs. The cooling efficiency can be expressed as5,30 (see Supplementary Note 1 for the derivation)

$${eta }_{{rm{c}}}({lambda }_{{rm{p}}})=frac{{lambda }_{{rm{p}}}}{{lambda }_{{rm{f}}}} {eta }_{{rm{ext}}} {eta }_{{rm{abs}}}-1,$$

(1)

where λf is the mean wavelength of the escaped fluorescence, λp is the laser pump wavelength, ηext is the external quantum efficiency, and ηabs is the absorption efficiency; they are defined as

$${eta }_{{rm{ext}}}=frac{{eta }_{{rm{e}}}{W}_{{rm{r}}}}{{W}_{{rm{tot}}}},quad {W}_{{rm{tot}}}={eta }_{{rm{e}}}{W}_{{rm{r}}}+{W}_{{rm{nr}}},$$

(2)

$${eta }_{{rm{abs}}}({lambda }_{{rm{p}}})=frac{{alpha }_{{rm{r}}}({lambda }_{{rm{p}}})}{{alpha }_{{rm{r}}}({lambda }_{{rm{p}}})+{alpha }_{{rm{b}}}},$$

(3)

where Wr, Wnr, and Wtot are radiative, non-radiative, and total decay rates of the excited state, respectively, and ηe is the fluorescence extraction efficiency. αb is the background absorption coefficient and αr is the resonant absorption coefficient. In practice, both ηext and ηabs must be very close to unity to observe laser cooling, because λp cannot be much longer than λf to keep αr(λp) sufficiently large for a near-unity value of ηabs.

It was recently shown by Mobini et al.16 that it is possible for the Yb excited-state population to have a small non-radiative decay rate in a silica glass host, i.e. WnrWr; therefore, the external quantum efficiency can be near unity. To revisit the arguments presented by Mobini et al.16, note that the non-radiative decay rate, Wnr, can be divided into two separate parts: the multiphonon decay rate (Wmp) and the sum of other non-radiative decay rates (Wi) for those channels that are related to the concentration quenching effect, i.e., Wnr = Wmp + ΣiWi16,31. Using the energy-gap law, we showed that the multiphonon decay rate of silica glass is ({W}_{{rm{mp}}}^{{rm{silica}}}approx 1{0}^{-8} {{rm{s}}}^{-1}), while that of ZBLAN is ({W}_{{rm{mp}}}^{{rm{ZBLAN}}}approx 1{0}^{-4} {{rm{s}}}^{-1}); therefore, as far as the multiphoton non-radiative decay rate is concerned, Yb-doped silica glass is a better material than ZBLAN for optical refrigeration16. Additional information is provided in Supplementary Note 2.

The non-radiative decay channels related to the concentration quenching are mainly due to the dipole–dipole interactions between Yb ions and impurities, which include OH, transition metals, and undesirable RE ions, as well as Yb–Yb interactions in Yb ion clusters. Developing a high-purity Yb-doped silica glass is therefore required to avoid the interactions between the Yb ions and impurities16. Additionally, to ensure that Yb ion clustering is suppressed and to further mitigate Yb-impurity interactions, it is imperative for the Yb ion density to remain below the critical ion concentration31. It is known that the ion solubility of the silica glass is quite low, i.e., for pure silica glass the critical quenching concentration is Nc ≈ 1025 m−3 or lower32. However, by using modifiers such as Al and P, the quenching concentration of silica glass can be increased by an order of magnitude18,19. To prevent concentration quenching and achieve ηext ≈ 1, it is necessary to keep the Yb ion density below Nc. Quite possibly, this issue has been one of the main reasons behind the previously failed attempts in laser cooling of the Yb-doped silica glass17. The Yb-doped silica glass samples that are studied in this paper are all high-purity and are doped with modifiers to increase the Yb ion solubility33. The parasitic background absorption (αb) in these glasses is sufficiently low to ensure that ηabs ≈ 1, as is required to achieve laser cooling.

For the laser cooling experiments, we used three different samples of Yb-doped silica glass optical fiber preforms (see “Methods”). We refer to these preforms as sample A, sample B, and sample C, respectively. These preforms are Yb-doped only in the core and their characteristics are listed in Table 1. The background absorption coefficients were measured for large mode-area fibers drawn from the respective preform samples by means of the well known cut-back technique.

To investigate laser cooling and obtain the cooling efficiency, ηc, of the Yb-doped silica glass preforms as a function of the laser pump wavelength, we perform the laser-induced thermal modulation spectroscopy (LITMoS) test on all three samples5,34 (see Supplementary Notes 3 and 4 for additional information). The LITMoS test setup is shown in Fig. 1. The samples are held by a set of silica fibers inside a vacuum chamber with the pressure of 10−6 Torr to minimize the conductive and convective heat-loads on the samples, so the black body radiation remains the only source of heating from the environment. The samples are pumped by a wavelength-tunable continuous wave (CW) Ti-Sapphire laser (980 nm < λp < 1070 nm) and the laser light passes through each sample twice using an external mirror.

The spectral features of the samples are captured by a spectrometer through a thermally transparent KCl salt window mounted in the chamber. Similarly, the thermal images are recorded by a thermal camera through the KCl salt window and the images are post-processed to determine the changes in the sample temperatures. To calculate the mean fluorescence wavelength, the samples are initially pumped at λp = 1030 nm. The fluorescence emission then is captured with an optical spectrum analyzer. The calculated mean fluorescence wavelengths of the samples A, B, and C are found to be ({lambda }_{{rm{f}}}^{{rm{A}}}) = 1010  nm, ({lambda }_{{rm{f}}}^{{rm{B}}}) = 1008 nm, and ({lambda }_{{rm{f}}}^{{rm{C}}}) = 1008 nm, respectively16 (see Supplementary Note 5).

Figure 2 shows the cooling efficiencies as a function of the laser wavelength for the three samples obtained from the LITMoS test. In each subfigure corresponding to the particular sample A, B, or C, the pump laser wavelength is gradually increased; once it becomes longer than approximately the mean fluorescence wavelength, the anti-Stokes fluorescence begins to extract heat from the sample until the cooling efficiency becomes positive, indicating the net laser cooling. As can be seen in Fig. 2, all three samples have been laser cooled. By fitting Eq. (1) to the experimental results and using the values of αb reported in Table 1, we can find the external quantum efficiency, ηext, of the samples, which are summarized in Table 2. Note that the blue lines in Fig. 2 are the results of the one-parameter fitting—we could have used the fitting procedure to determine the values of αb as well. However, the lack of experimental data for ηc at wavelengths above 1070 nm results in large uncertainties in αb; therefore, we have chosen to use the directly measured values in Table 1, which appear to conform well to our measurements. We note that αb in Table 1 is measured at 1200 nm wavelength to ensure that the resonant absorption is very small; as such, we are implicitly assuming that αb does not change significantly down to  ~1035 nm wavelength.

### Power cooling experiment

The results of the LITMoS tests prove laser cooling in all the Yb-doped silica glass preforms. However, because in the LITMoS test setup, the maximum power of our Ti:Sapphire laser in the cooling wavelength range is less than 900 mW, the signal-to-noise ratio, as can be seen from the error-bars in Fig. 2, is large. Therefore, to enhance and further confirm the laser cooling of our samples, we pumped the preforms with a 10.4 W Nd-doped YLiF4 (Nd:YLF) laser, the wavelength of which at 1053 nm resides in the cooling spectral range of the samples (see Fig. 2). We have shown in Supplementary Note 6 that the optimum wavelength for laser cooling is around 1035 nm in the low absorbance (αrl 1) regime. Our use of the Nd:YLF laser operating at 1053 nm was mainly due to the availability of a high-power source in our laboratory. With multipass pump geometries where Npαrl 1 (Np is the number of passes), optimum cooling wavelength will increase in accordance with the maximum cooling efficiencies shown in Fig. 2. Similar to the LITMoS test, the samples were double-pass pumped by the Nd:YLF laser inside the vacuum chamber and the changes of the temperature were recorded by the thermal camera as a function of the exposure time. Figure 3 shows the thermal images of sample A (a) before and (b) after the exposure to the laser light. Figure 3b was taken after the laser was turned on and the sample temperature was stabilized (~40 min). Note that the heat extraction occurs only in the core of each sample, but the entire sample cools almost uniformly in less than a minute. The cooling is easily recognizable by unaided human eye when the thermal camera image become darker after the exposure to the Nd:YLF laser. The bright regions in the thermal image of the sample in Fig. 3 can be misleading; the reason for these bright regions is that silica glass is not transparent in the thermal window and the bright regions on the sample originate from reflections of the thermal radiation from the side walls of the chamber onto our sample’s cylindrical surface and eventually into the thermal camera.

Figure 4 shows the evolution of the temperature of the samples over time while being exposed to the 10.4 W Nd:YLF laser. In each case, the temperature drop can be fitted to the exponential function

$$Delta T(t)=Delta {T}_{max }({mathrm{e}}^{-t/{tau }_{{rm{c}}}}-1),$$

(4)

where we use the following definitions:

$$Delta {T}_{max }={eta }_{{rm{c}}}frac{{P}_{{rm{abs}}}}{4epsilon sigma {T}_{0}^{3}A},quad quad {tau }_{{rm{c}}}=frac{rho V{c}_{{rm{v}}}}{4epsilon sigma {T}_{0}^{3}A},$$

(5)

where Pabs is the absorbed power, ϵ = 0.85 is the emissivity of the implemented Yb-doped silica glass fiber preforms, σ = 5.67 × 10−8 W m−2 K−4 is the Stefan–Boltzmann constant, T0 is the ambient temperature, l is the sample length, A is the surface area of the sample, V is the volume of the sample, ρ = 2.2 × 103 kg m−3 is the silica glass mass density, and cv = 741 J kg−1 K−1 is the specific heat of the silica glass35,36.

Equations (4) and (5) can be derived by noting that in the vacuum chamber, the convective and conductive heat transfers are negligible; therefore, the temporal behavior of the temperature obeys the following differential equation37 (see Supplementary Note 3 for additional information):

$$rho V{c}_{{rm{v}}}frac{{mathrm{d}}Delta T}{{mathrm{d}}t}approx -{eta }_{{rm{c}}}{P}_{{rm{abs}}}-4epsilon sigma A{T}_{0}^{3}Delta T,$$

(6)

where the absorbed power in the double-pass experiment is given by

$${P}_{{rm{abs}}}={P}_{{rm{in}}}{mathcal{T}}left(right.1-{{mathrm{e}}}^{-{alpha }_{{rm{r}}}({lambda }_{{rm{p}}})l}left)right.left(right.1+{{mathcal{T}}}^{2}{R}_{{rm{m}}},{mathrm{e}}^{-{alpha }_{{rm{r}}}({lambda }_{{rm{p}}})l}left)right..$$

(7)

ΔT = Ts − T0, where Ts is the sample temperature. αr(λp) is the resonant absorption coefficient of the pump laser. We also have ({mathcal{T}}={T}_{{rm{w}}}{T}_{{rm{l}}}{T}_{{rm{g}}}), where Tw = 0.92 is the transmission of the vacuum chamber windows, Tl = 0.998 is the transmission of the lenses, Tg = 0.96 is the transmission of the preforms’ facets, and Rm = 0.998 is the reflection of the mirror. Note that the absorption coefficients of samples A, B, and C were measured to be αr(λp) =  0.43, 0.52, and 0.50 m−1, respectively. The exponential form presented in Eq. (4) is a direct solution to Eq. (6); by fitting Eq. (4) to the measurements in Fig. 4, the values of the two fitting parameters for each sample, i.e., (Delta {T}_{max }) and ηext (via τc) are extracted and reported in Table 2. We note that the slope of the ΔT(t) curve at t = 0 in Eq. (4) gives us the value of the cooling efficiency at 1053 nm wavelength, i.e., ηc = −(ρVcv/Pabs)∂tΔTt=0. For each sample, we also calculate the value of ηc using the two fitted coefficients, and present the results in Table 2; these values all agree well, within the error bars, with the plots of ηc in Fig. 2 obtained using the LITMoS tests. Video clips of the cooling evolution of the samples are presented in Supplementary Note 7.2.