Saturation, Allowed Transitions and Quantum Interference in Laser Cooling of Solids

Abstract

New methods for the rapid cooling of solids with increased efficiency are analyzed and demonstrated experimentally. The advances offered by optical saturation, dipole-allowed transitions, and quantum interference for improved laser cooling of solids are highlighted.

1. Introduction

Optical refrigeration was first achieved using laser-induced anti-Stokes fluorescent (ASF) emission in Yb³⁺:ZBLANP glass in 1995 [1]. Since then, advances in experimental methods and the purification of materials have led to the optical cooling of numerous crystals [2], the reduction of attainable temperatures to the cryogenic range [3], the first optical cryocooler [4], and self-cooling or radiation-balanced lasers (RBLs) [5,6,7]. However, unintended impurities incorporated during sample preparation are still a key limitation to applications of ASF cooling [8]. Saturation of the absorption and reduction in the associated heating caused by background impurities can overcome this limitation under certain conditions. Hence, it is important to understand and characterize such nonlinear effects. In this study, the enhancement of laser cooling due to differential absorption saturation is analyzed and observed by two experimental methods. We also note that the cooling rate is limited by the excited state emission rate of coolant ions as the excited state lifetime determines the number of absorption/emission cycles each ion can undergo per unit time during ASF cooling. Electric-dipole-allowed (ED) transitions have very short upper state lifetimes; so, they can be cycled more rapidly for faster cooling. However, allowed transitions suffer from configuration relaxation and often mediate excited state absorption, both of which contribute to the internal heating of the sample. Consequently, while laser cooling on such transitions has been investigated previously [9], net cooling by this approach has not been reported until now. Finally, quantum interference has not been investigated for possible applications in laser cooling. In this paper, Fano resonance and induced transparency are analyzed for their potential to mediate self-cooled lasing without inversion in a 3-level system with two allowed transitions, modeled after trivalent Cerium.

Several advances are discussed here that enable cooling to lower minimum temperatures or more rapid cooling than has been possible to date with optical refrigeration based on anti-Stokes fluorescence (ASF). The first topic to be analyzed is optical saturation, relevant to attaining lower minimum temperatures by ASF cooling. As the laser cooling of solids is the only known method for refrigeration in space below the thermo-electric limit, any improvements in its capabilities will be important for future technology. Increased cooling efficiency through saturation is also demonstrated experimentally in Yb:LiYF₃ near room temperature. Optical refrigeration of Ti:Al₂O₃ on the allowed ²E-²T₂ transition is realized for the first time in a crystal with an exceptionally high figure of merit. This result demonstrates that cooling on electric-dipole transitions is possible in a bulk solid and should enable more rapid cooling than can be achieved with ASF cooling on forbidden transitions of rare-earth ions. This advance should facilitate the development of imaging arrays with an improved signal-to-noise performance at cryogenic temperatures for sensing applications in outer space. In addition, it should encourage the development of new methods of cooling solids, such as Raman cooling. Analysis of the potential role of quantum interference in self-cooled lasers is also presented. Specific conditions are identified to enable self-cooled lasing without inversion in a Ce:LiCaAlF₆ laser.

2. Theory

In prior experimental research, the forbidden transitions of trivalent rare-earth ions such as Yb³⁺ were employed to demonstrate ASF cooling in insulating crystals and glasses. While many proposals have been advanced to diversify the possible approaches for cooling and to extend the results to a wider class of materials [10,11,12,13,14,15,16], no alternatives to net cooling by anti-Stokes fluorescence have emerged. This failure has impeded the application of laser cooling to the refrigeration of semiconductor circuitry for sensors in space, although an ASF cryocooler for this type of application has been reported [4]. Better speed and efficiency of cooling would be highly advantageous for practical applications. In the following subsections, theoretical advances are described that address the need for the improved performance of optical refrigeration based on anti-Stokes fluorescence cooling through optical saturation and the use of electric-dipole-allowed transitions. The experimental realizations of these proposals are presented in Section 3.

2.1. Improved Cooling Efficiency Using Optical Saturation

The intensity dependence of the absorption coefficients of cooling ions and background impurities can be exploited to improve the cooling efficiency of ASF cooling. The following analysis of differential absorption saturation reveals its impact on laser cooling efficiency when background impurities saturate more easily than cooling ions.

The coolant ions and the background impurities are modeled as 2-level systems [13], with $N_1, N_1^{\prime }$ representing the number of ground state ions and impurities per unit volume. $N_2, N_2^{\prime }$ represent the density of the excited state species, respectively. Absent any internal laser field, the rate equation for the upper state population of the coolant ions is

$$\frac{d{N}_2}{dt}=\frac{I_p{\lambda }_p}{hc}·\left[{\sigma }_a\left({\lambda }_p,T\right)N_1-{\sigma }_e\left({\lambda }_p,T\right)N_2\right]-\frac{N_2}{{\tau }_f}.$$

(1)

For unintended impurities, it is

$$\frac{d{N}_2^{\prime }}{dt}=\frac{{\sigma }^{\prime }{I}_p{\lambda }_p}{hc}·\left[N_1^{\prime }-N_2^{\prime }\right]-\frac{N_2^{\prime }}{{\tau }^{\prime }}.$$

(2)

$I_p$ and ${\lambda }_p$ denote the intensity and wavelength of the pump light. ${\sigma }_a$ and ${\sigma }_a$ are cross-sections for absorption and emission. ${\tau }_f$ and ${\tau }^{\prime }$ are the excited state decay times of the coolant and impurity ions. In this model, the equation for heating power per unit volume is simply

$$H=I_p\left[{\sigma }_a\left({\lambda }_p,T\right)N_1-{\sigma }_e\left({\lambda }_p,T\right)N_2\right]-{\eta }_e\frac{hc}{{\lambda }_f}\frac{N_2}{{\tau }_r}+{\sigma }^{\prime }{I}_p\left[N_1^{\prime }-N_2^{\prime }\right].$$

(3)

Because there is no internal laser field to consider in refrigeration with a single pump beam, the population differences for the system under continuous illumination are given by the steady-state solutions of Equations (1) and (2). These expressions are

$$\mathsf{\Delta }N\equiv N_1-N_2=\frac{N_T}{\frac{I_p}{I_{sat}}+1}$$

(4)

and

$$\mathsf{\Delta }{N}^{\prime }\equiv N_1^{\prime }-N_2^{\prime }=\frac{N_T^{\prime }}{\frac{I_p}{I_{sat}^{\prime }}+1}.$$

(5)

In these expressions, we have defined the saturation intensities [17] for the coolant and impurity ions, given by

$$I_{sat}=\frac{hc}{{\mathsf{\lambda }}_p{\tau }_f\left[{\sigma }_a\left({\mathsf{\lambda }}_p,T\right)+{\sigma }_e\left({\mathsf{\lambda }}_p,T\right)\right]} \mathrm{and} I_{sat}^{\prime }=\frac{hc}{2{\sigma }^{\prime }{\tau }^{\prime }{\mathsf{\lambda }}_p}.$$

The total numbers of coolant and impurity species per unit volume are

$$N_T=N_1+N_2$$

(6)

and

$$N_T^{\prime }=N_1^{\prime }+N_2^{\prime }.$$

(7)

The steady-state heating power/volume obtained from Equation (3) is

$$H=I_p\left[{\sigma }_a\left({\mathsf{\lambda }}_p,T\right)\mathsf{\Delta }N+{\sigma }^{\prime }\mathsf{\Delta }{N}^{\prime }\right]·\left[1-\frac{{\sigma }_a\left({\mathsf{\lambda }}_p,T\right)\mathsf{\Delta }N}{{\sigma }_a\left({\mathsf{\lambda }}_p,T\right)\mathsf{\Delta }N+{\sigma }^{\prime }\mathsf{\Delta }{N}^{\prime }}{\eta }_{ext}\frac{{\mathsf{\lambda }}_p}{{\mathsf{\lambda }}_f}\right].$$

(8)

The final expression for cooling power in the presence of the optical saturation of the active ions and background impurities is therefore

$$P_c=-P_{abs}\left[1-\frac{{\sigma }_a\left({\mathsf{\lambda }}_p,T\right)\mathsf{\Delta }N}{{\sigma }_a\left({\mathsf{\lambda }}_p,T\right)\mathsf{\Delta }N+{\sigma }^{\prime }\mathsf{\Delta }{N}^{\prime }}{\eta }_{ext}\frac{{\mathsf{\lambda }}_p}{{\mathsf{\lambda }}_f}\right].$$

(9)

The cooling efficiency ${\eta }_c$ may be derived from Equation (9) as the ratio of the cooling power, $P_c$, to absorbed power, $P_{abs}$. This yields

$${\eta }_c=\frac{ P_c}{ P_{abs}}={\eta }_{ext}\frac{\mathsf{\lambda }}{{\mathsf{\lambda }}_f}\left[\frac{1}{1+{\alpha }_b\left(I\right)/{\alpha }_r\left(I\right)}\right]-1.$$

(10)

Here, the external quantum efficiency ${\eta }_{ext}$ is the product of the fluorescence escape efficiency ${\eta }_e$ and the internal quantum efficiency ${\eta }_{QE}$ given by the ratio of the fluorescence and radiative lifetimes of the upper state. $\mathsf{\lambda }$ and $I$ are the wavelength and intensity of the pump laser. The polarization-averaged mean fluorescence wavelength ${\mathsf{\lambda }}_f$ is determined from emission spectra. The absorption coefficients of the coolant ions, ${\alpha }_r$, and background impurities, ${\alpha }_b$, were assumed to be intensity-dependent. The form of this dependence [17] was taken to be $\alpha \left(I\right)=\alpha \left(0\right)/\left(1+I/I_{sat}\right)$, where the saturation intensity $I_{sat}$ was assigned the values $I_r$ or $I_b$ for the coolant or background impurity ions, respectively.

The saturation intensity of the coolant ions varies with wavelength through its dependence on absorption and emission cross-sections. It is defined by the expression

$$I_r=\frac{hc}{\mathsf{\lambda }{\tau }_f\left[{\sigma }_a\left(\mathsf{\lambda }\right)+{\sigma }_e\left(\mathsf{\lambda }\right)\right]},$$

(11)

where ${\tau }_f$ is the fluorescence lifetime. The effective absorption cross-section ${\sigma }_a\left(\mathsf{\lambda }\right)$ was measured directly with a spectrophotometer (SHIMADZU UV-3600) and corrected for instrument response. The effective emission cross-section ${\sigma }_e\left(\mathsf{\lambda }\right)$ was then calculated using the McCumber relation [18]. This procedure ensured that the multiple Stark levels of Yb³⁺ and the Boltzmann population distribution among them were taken into account. Figure 1 displays the effective absorption and emission cross-sections of the 10% Yb:YLF crystal studied in our experiments and shows the position of the polarization-averaged mean fluorescence wavelength at 997.6 nm. Based on these cross-sections and a literature value for the Yb³⁺ fluorescence lifetime of 2.2 ms [19], the theoretical saturation intensity of the Yb³⁺ coolant ions was determined using Equation (11) and plotted as the continuous curve in Figure 2 versus wavelength. Surmising that an impurity such as Fe³⁺ is the main source of background absorption [9] and that it has a broadband spectrum centered around 1 micron [20], the saturation intensity $I_b$ of the background may be presumed to be only weakly dependent on wavelength over the range of our experiments. With this one assumption, the experimental value of $I_b$ can be estimated using the empirical approach outlined next and illustrated as a dashed line in Figure 2.

Figure 3 presents a theoretical plot of the cooling efficiencies given by Equation (10) at two pump intensities, namely $I=3\times {10}^2 \mathrm{W}/{\mathrm{cm}}^2$ and $I=3\times {10}^4 \mathrm{W}/{\mathrm{cm}}^2.$ The first thing to notice is that at long wavelengths (>1 μm), the cooling efficiency increases at the higher pump intensity because the background impurities are easier to saturate than Yb³⁺ near their own broad resonance. Such long wavelengths are highly detuned from the absorption peak of Yb³⁺ (Figure 2). Naturally, saturation of the background reduces parasitic heating, leading to a significantly higher cooling efficiency in the long wavelength range. Closer to the absorption resonances of Yb³⁺, at wavelengths of less than ~1 μm, the saturation intensity of the coolant ions drops, falling to a value below that of the background because the detuning is small. This raises the heat load and lowers the cooling efficiency at short wavelengths. Between these opposing trends, there is an intersection point in the plots versus wavelength of the saturation intensities (Figure 2) on the “red” side of the Yb³⁺ absorption band. The same is true of the intensity-dependent cooling efficiency plot (Figure 3). In Figure 3, this crossing point can be seen around 1010 nm, identifying a wavelength ${\mathsf{\lambda }}_{cr}$, where the saturation intensities of the coolant ion and the background impurities are equal. The efficiency curves for all the input intensities cross at ${\mathsf{\lambda }}_{cr}$, so the cooling efficiency is intensity-independent there. As the experimental saturation intensity of the background impurities equals that of Yb³⁺ at ${\mathsf{\lambda }}_{cr}$, it can be determined by inserting ${\mathsf{\lambda }}_{cr}$ into Equation (11), yielding the result $I_b=I_r({\mathsf{\lambda }}_{cr})$.

2.2. Electric-Dipole-Allowed Transitions for Rapid Cooling

The efficiency of laser cooling by the ASF method is independent of the transition rate of the coolant atoms. This is clear because the upper state lifetime of these atoms is absent from Equation (10). On the other hand, the number of transitions per unit time does determine the speed of cooling. Hence, the transition rate of coolant atoms has an important impact on how quickly bulk systems can be cooled with light. This is confirmed in the expression for heating power density $H$, where the cooling term (second term on the right side of Equation (3) depends on the upper state radiative lifetime ${\tau }_r$. From this equation, it is evident that the net heating or cooling power varies with the transition rate of the coolant atoms and improves significantly when the radiative lifetime on allowed transitions is short. Another way to think about the cooling term is that $N_2/{\tau }_r$ is proportional to the absorption cross-section. A medium 2, therefore, improves on the cooling power density of a medium 1 by the factor ${\tau }_{r1}/{\tau }_{r2}={\sigma }_2/{\sigma }_1$, making it advantageous to use electric-dipole-allowed transitions for laser cooling, rather than forbidden transitions, because of their larger cross-sections. As most ED transitions have upper state decay times of less than 100 ns and forbidden transitions have lifetimes of ~1 ms, the cooling power can theoretically be increased by a factor of roughly 10⁴ by using allowed transitions. Thus, this could result in a substantial improvement of the cool-down times.

Despite the high transition rate afforded by dipole-allowed transitions in solids, they normally incur heating due to non-radiative configuration relaxation. The mechanism is illustrated in Figure 4. The polarizability of excited state orbitals on allowed transitions in solids is susceptible to interactions with neighboring atoms. This shifts the minimum of the excited state potential to a greater distance from the nucleus than that of the ground state. According to the Franck–Condon principle, the absorption transition takes place rapidly enough so that the configuration coordinate Q does not change. Thus, the electron undergoes a vertical transition in Figure 4. which places it far from the equilibrium point at the bottom of the excited state well. This necessitates relaxation to the potential minimum, which is shifted to a larger value of the configuration coordinate Q. This relaxation, and a similar one that follows the vertical radiative transition to the ground state potential, is mediated by the emission of the vibrational quanta of the atomic cluster, consisting of the atom itself together with its neighbors in its coordination sphere. Fortunately, the emission of phonons and consequent heating that takes place during configuration relaxation can be mitigated to achieve net cooling in the same way as for systems with forbidden transitions, such as Yb³⁺. Theoretically, one simply tunes the pump wavelength to a value exceeding the average Stokes-shifted emission wavelength of the coolant species. In Section 3.2, this strategy is shown to work on an ED transition in Ti³⁺:Al₂O₃.

In the context of laser cooling, one thing which differentiates optical from acoustic phonons is their higher frequency and the limited number of allowed modes determined by crystal symmetry. Transitions involving optical modes have narrow linewidths and are well-defined in frequency. Hence, they exhibit quantum effects more readily than acoustic modes, which have a continuous density of states versus frequency. One consequence of this is illustrated in Figure 5, where the red arrow indicates excitation from the lowest sub-level of a vibrational ladder in the ground state (v = 1) to the lowest sub-level of the excited state (v’ = 0).

Such a transition avoids configuration relaxation because it takes place at a wavelength longer than the mean fluorescence wavelength. When this is the case, the quanta of the incident field have less energy than required to generate both the electronic transition and a phonon. Absorption is followed by anti-Stokes emission, shown as the downward blue arrow connecting v’ = 0 to v = 0. The red and blue transitions jointly remove a vibrational quantum of the mode in question, thereby contributing to refrigeration. Other quantized absorption transitions are possible at longer wavelengths from the occupied sub-levels of other modes and other electronic sub-levels of the ground state. Figure 6 depicts a more detailed picture of the vibrational and electronic sub-levels of the ²T₂ ground state of Ti³⁺:Al₂O₃ [21,22].

The absorptive transitions that contribute to cooling were all assumed to terminate in the lowest v’ = 0 sub-level of the electronic excited state. This is necessary to avoid the generation of optical phonons. Because of the common final state, the cooling resonances are distinguishable by their initial states alone and form a discrete spectrum as a function of wavelength because all the electronic and optical mode energies exceed ~400 cm⁻¹ and have low damping parameters [21].

2.3. Quantum Interference for Self-Cooled Lasing without Inversion

In this section, Fano resonance mediated by excited state mixing and electromagnetically induced transparency (without any state mixing) are investigated in the system depicted in Figure 7, modeled after Ce³⁺ ions in the host crystal LiCaAlF₆. The Ce³⁺ dopant ions are characterized by having two allowed transitions, from states 1 to 2 and from 2 to 3. These are located in the ultraviolet spectral region. The transition from 1 to 3 is assumed to be forbidden by the usual rule of Laporte, and the lifetime of state 3 is correspondingly long.

Two types of quantum interference can manifest themselves in 3-level systems driven by a single pump wave. If there is no mixing between the two excited states of the system, quantum interference is found to produce induced transparency on the 12 transition above a threshold corresponding to the optical saturation of the transition. If the two excited states are mixed or coupled by a configuration interaction of strength $V$ for example, transparency appears due to Fano resonance on the 13 transition. Interestingly, in each of these cases, gain appears even though the populations of states 1 and 3 always exceed that of state 2, and no special tuning requirement needs to be met [23]. Thus, lasing without inversion takes place at all frequencies close to either ${\omega }_{12}$ or ${\omega }_{13}$ depending on whether there is significant excited state mixing or not. In addition, the 12 transition in particular has gain over a range of red-detuned frequencies near the ${\omega }_{12}$ resonance. Therefore, self-cooled lasing without inversion may be possible.

To examine the behavior of this system, when irradiated by a single pump wave of frequency $\omega$ tuned near the 12 transition frequency, the imaginary part of the susceptibility must be calculated. To do this, one can determine the density matrix elements ${\rho }_{ij}$ between states $i$ and $j$ in the expression [23]

$${\chi }^{″}\left(\omega \right)=\mathrm{Im}\left\{\left(\frac{2N}{{\epsilon }_0{E}_0}\right)\left({\mu }_{12}{\tilde{\rho }}_{21}+{\mu }_{13}{\tilde{\rho }}_{31}^{(-)}+{\mu }_{31}{\tilde{\rho }}_{13}^{(-)}+{\mu }_{32}{\tilde{\rho }}_{23}\right)\right\}.$$

(12)

Optical absorption is proportional to ${\chi }^{″}\left(\omega \right)$. Hence, when the expression in Equation (12) is evaluated, the absorption and gain spectra can be plotted. When the optical frequency is close to ${\omega }_{12}$ or ${\omega }_{13}$, the rotating wave approximation (RWA) can be made for the corresponding off-diagonal matrix elements, but not for the 13 transition. This accounts for the co- and counter-rotating terms (${\tilde{\rho }}_{31}^{(-)}$ and ${\tilde{\rho }}_{13}^{(-)}$, respectively) in Equation (12). To plot the response near the 13 transition frequency, the RWA can be applied to the 13 transition but not to the other two. These requirements arise because of the relative placement of the levels assumed in Figure 7, which reflect those of Ce³⁺. Steady-state expressions for the various matrix elements obtained by third-order perturbation theory are given in Appendix A. The absorption spectra showing the effects of the two types of quantum interference described above are presented in Section 3.

3. Methods and Results

3.1. Improved Cooling via Saturation of Background Absorption in Yb:LiYF₄

Laser cooling was investigated in a crystal of 10% Yb³⁺:YLF, and its efficiency was measured by two different methods, namely Differential Luminescence Thermometry (DLT) and Thermal Lens Spectroscopy (TLS). The Yb³⁺:YLF sample had dimensions of $3.4\times 5.1\times 5.6 {\mathrm{mm}}^3$ and was supported by an aerogel disk (Classic Silica Disk, Aerogel Technologies) to minimize the conductive thermal load. A tunable Ti:Sapphire laser (M Squared SolsTiS) was used as the excitation source. To study the intensity dependence, the pump beam radius at the sample was varied by using different focal length lenses and measuring the spot size with a beam profiler (Thorlabs BC106-VIS). When the pump beam was introduced into the sample, temperature changes and thermal lensing were measured by the DLT and TLS methods, respectively. Cooling efficiency was then computed based on these measurements and found to be intensity-dependent.

3.1.1. Differential Luminescence Thermometry (DLT)

DLT was employed to deduce the crystal temperature from variations of the fluorescence line shape [3]. The infrared fluorescence was collected with a multimode optical fiber (Ocean Optics QP600-2-VIS-NIR; NA = 0.4) connected to a 0.25 m grating spectrometer (Oriel 74100) equipped with a CCD detector (Andor DU491A-1.7). The difference between the measurement and the reference was spectrally integrated to calculate the DLT signals, which were pre-calibrated by the direct heating or cooling of the sample using a temperature controller (Quantum Northwest Flash 300) with an accuracy of ±0.01 K. A thermal camera (FLIR A655sc) was used to monitor the crystal surface temperature and cross-check the temperature measured by DLT [24]. Figure 8 and Figure 9 plot the crystal temperature change versus the time measured by DLT at wavelengths of 1015 nm and 920 nm, showing the representative signals for the cooling and heating ranges.

The cooling efficiency can be computed directly from the temperature evolution measurements. Assuming a crystal of thermal emissivity $\epsilon$ and a surface area $A_{surf}$ enclosed by a blackbody, the thermal balance equation relating the various contributions to heating power at a crystal temperature of T [25,26] is:

$$cM\frac{dT}{dt}=-{\eta }_c{P}_{abs}+A_{surf}ϵ{\sigma }_B\left(T_0^4-T^4\right)+\left(h_c{A}_{surf}+k\frac{A}{L}\right)\left(T_0-T\right),$$

(13)

where ${\sigma }_B$, $T_0$, $h_c$, $k$, $A$, and $L$ are the Stefan–Boltzmann constant, environmental temperature, convective heat transfer coefficient, thermal conductivity of the link, cross-sectional area, and sample length, respectively. The specific heat $c$ of 10% Yb:YLF is 0.79 J/(g K) at room temperature [27]. The crystal mass ($M=0.408 \mathrm{g}$) was determined using a precision balance (QUINTIX213-1S) with an accuracy of 0.001 g.

The first term on the right side of Equation (13) is the laser-induced heat density. The second term is the power exchange between the crystal and its surroundings through blackbody radiation. The last term is the conductive heat load, which takes into account the contact with the air and sample support. The blackbody radiation and conductive heat load depend on the temperature difference between the crystal and the environment. As indicated by the data of Figure 8 and Figure 9, the temperature change is linear at very short times, reflecting the negligible difference between $T$ and $T_0$ in the second and third terms on the right of Equation (13). During the first minute after the pump beam enters the sample, only the first term on the right of Equation (13) is responsible for the temperature changes of the crystal. This term is numerically equal to the cooling power; so, the cooling power can be determined directly from the initial slope of the sample temperature versus time. At long times, the difference between $T$ and $T_0$ increases. Consequently, the last two terms in Equation (13) become significant, and the temperature dependence on time becomes nonlinear. By restricting the data analysis to the early times (or to temperature excursions $dT<1 \mathrm{K}$) and the measuring power absorbed by the sample, the blackbody radiation and conduction terms can be ignored, permitting the cooling efficiency to be determined directly from Equation (13) in a simple way.

When the second and third terms on the right side of Equation (13) can be ignored, the thermal balance equation reduces to ${\eta }_c=P_{abs}^{-1}cMdT/dt$. The temperature derivative with time can then be determined by applying a linear fit to the data (for temperature changes of less than 1 Kelvin or at times of less than 1 min, whichever occurs first). Examples of such fits are plotted in the insets of Figure 8 and Figure 9. By measuring the injected, reflected, and transmitted pump power, the absorbed power can also be determined. Thus, the cooling efficiency can be measured directly from the temperature evolution measurements.

3.1.2. Thermal Lens Spectroscopy (TLS)

TLS was used to measure the thermal lensing signal by probing the refractive index variation of the crystal induced by a pump beam. The tunable Ti:Sapphire laser was used to pump the crystal and, by chopping its beam, synchronous detection was enabled. In general, two lensing effects are induced by the pump—a fast transient from photoinduced population transfer between the ground and excited states and a slower transient from thermal diffusion. Both of these lensing effects change the refractive index of the crystal and must be accurately modeled to account for their separate contributions [27,28]. The TLS signal was obtained by measuring the intensity variation of a co-propagating He-Ne probe laser (Melles Griot 25-LHP-991-249), which sensed the crystal refractive index changes. Two normalized TLS signals and fits accounting for population and thermal effects [29] are shown in Figure 10. These best fits provided quantitative values for the thermal lens strength and the corresponding cooling efficiency ${\eta }_c$ derived from it.

The cooling efficiency determined at different intensities is plotted versus wavelength in Figure 11 and Figure 12, based on the TLS and DLT data, respectively. The measured pump radii and intensities are shown in

Table 1. Pump beam radius (half-width at 1/e² intensity) and intensity values used for the measurements presented in Figure 11 and Figure 12. Entries indicate the small variations of intensity at multiple wavelengths for each intensity category.

	Beam Radius (µm)	Intensity (W/cm2)
DLT High	32.4–39.3	(1.6–2.4) $\times$ 104
DLT Low	284–318	(2.0–3.3) $\times$ 102
TLS High	316–364	(1.1–1.3) $\times$ 103
TLS Low	316–364	(1.1–1.3) $\times$ 102

. The main result to note is a steepening of the cooling efficiency curve over most of the range as intensity is increased. This behavior arises from the different variations of the saturation intensity of the coolant and the impurity ions versus the wavelength, as discussed further below and in Ref. [30]. This effect, referred to here as differential absorption saturation, was observed in both the DLT and the TLS experiments and can be explained using the analysis of Section 2.1.

The experimental cooling efficiency data coalesce at 1000 nm. There, all the curves pass through the same point, as predicted by the theory of Section 2. At such a crossing point, the cooling efficiency is clearly independent of intensity, indicating the saturation intensity of the background equals the saturation intensity of the coolant ions, which is $~2.6\times {10}^4 \mathrm{W}/{\mathrm{cm}}^2$. At longer wavelengths in the cooling range, a significantly higher cooling efficiency is observed at high pump intensity in both sets of data. The results are therefore consistent with a saturation intensity for background impurities that is less than that of the coolant ions, leading to reduced parasitic heating and higher cooling efficiency. At wavelengths shorter than 1000 nm, closer to the main absorption peak of Yb³⁺, the cooling efficiency drops at high intensity, consistent with a drop in the Yb³⁺ saturation intensity as resonance is approached. The separation of the experimental curves is somewhat larger than the theoretical efficiency curves of Figure 3. In the short wavelength range, this may be attributable to the easy saturation of Yb³⁺, which would maintain the maximum population density in the excited state, thereby enhancing the well-known energy transfer from Ytterbium to other rare-earth ions such as Erbium [31,32]. The characteristic green upconversion emission of Erbium was in fact visible to the eye. Enhanced energy transfer to the Erbium present as an unintended impurity would elevate the heating load, leading to a lower cooling efficiency at high intensity.

3.2. Laser Cooling on an Electric-Dipole-Allowed Transition in Ti:Al₂O₃

A Ti³⁺:Al₂O₃ sample (GT Advanced Technologies) was grown using the heat exchange method (HEM) known to produce high-quality crystals with exceptionally high figures of merit. In this material, the figure of merit (FOM) is defined as the ratio of the absorption coefficients at two specific wavelengths (${\alpha }_{532\mathrm{nm}}/{\alpha }_{800\mathrm{nm}}$) and is used as a measure of crystal quality and laser performance. The GTAT sample had a quoted FOM of 844 and was Brewster-cut with the dimensions of $4\times 5\times 20 {\mathrm{mm}}^3$. The normalized absorption and emission spectra of this sample are shown in Figure 13. Two other laser-grade samples with lower FOMs were included in the cooling tests.

Thermal lens spectroscopy (TLS) in a mode-mismatched configuration [28] was used to investigate the thermal characteristics of the three sapphire samples for pump light tuned close to the absorption peak of Ti³⁺ and in the absorption tail. An experimental approach that corrects for intensity variations of the pump laser [33] was applied in experiments on Yb:YLF but not in the GTAT sapphire sample because weak fluorescence prevented any reliable correction. A weak helium-neon laser (${\mathsf{\lambda }}_p=633$ nm) was used as the probe laser. A test was performed with a thermal camera to ensure that the probe power was low enough to avoid any detectable heating. The GTAT sample was first pumped with 532 nm light (Coherent Verdi V6) to show heating. The TLS transient signal for this excitation wavelength (Figure 14a) shows the positive slope expected for a material with $ds/dT>0$ with increasing temperature. To pump at red-detuned wavelengths in the absorption tail, the tunable Ti:Sapphire laser was used. For some wavelengths longer than the mean fluorescence wavelength, the slope of the TLS signal then became negative, as shown in Figure 14b, which was indicative of sample cooling within the pumping region. Using the known material parameters and the measured experimental values, in both cases, the data fit very well with TLS theory. The TLS signals in the other samples showed only positive sloping curves, such as that in Figure 14a, at all wavelengths.

Interestingly, the cooling behavior at wavelengths longer than the mean fluorescence wavelength was not uniform. Instead, sharp resonances were observed at which heating abruptly switched to cooling. This behavior is shown as TLS signals with negative polarity at discrete wavelengths in Figure 15, consistent with the discussion of Section 2.2 and Figure 5. In the case of well-defined local or host optical phonons, the cooling transitions should be discrete. The spectrum of infrared-active modes is well-known for sapphire [21], but the frequencies of local modes exhibiting site-dependent perturbations with respect to host modes are not. Nevertheless, the cooling resonances observed in Figure 15a,b are well accounted for by assignments that assume four mildly perturbed sites for Ti³⁺ ions in sapphire, as listed in

Table 2. Assignments of cooling resonances in Ti:Al₂O₃ for $\pi$-polarization (mean fluorescence wavelength 760 nm). Wavenumbers for observed and calculated resonances are subtracted for comparison in the last column on the right. The average discrepancy is given in the bottom row.

λobs (nm)	kobs (cm−1)	Initial Level	Site	kcalc (cm−1)	kobs − kcalc (cm−1)
812.55	12,306.93	(2A(0), vd)	3	12,276.79	−30.14
817.68	12,229.72	(2A(0), vd)	4	12,276.79	47.07
822.85	12,152.88	(2A(1), v = 0)	1	11,957.89	−194.99
827.43	12,085.61	(2A(1), v = 0)	2	11,957.89	−127.72
829.41	12,056.76	(2A(1), v = 0)	3	11,957.89	−98.87
831.39	12,028.05	(2A(1), v = 0)	4	11,957.89	−70.15
906.92	11,026.33	(2A(1), vd)	2	11,076.89	50.56
930.47	10,747.26	(2A(1), vd)	3	11,076.89	329.64
937.19	10,670.19	(2A(1), vd)	4	11,076.89	406.70
944.04	10,592.77	(2A(2), v = 0)	1	10,547.89	−44.88
950.08	10,525.43	(2A(2), v = 0)	2	10,547.89	22.47
953.61	10,486.47	(2A(2), v = 0)	3	10,547.89	61.43
955.36	10,467.26	(2A(2), v = 0)	4	10,547.89	80.64
958.87	10,428.94	(2A(2), va)	1	10,149.89	−279.05
961.51	10,400.31	(2A(2), va)	2	10,149.89	−250.41
				Average:	−1.21

and

Table 3. Assignments of cooling resonances in Ti:Al₂O₃ for $\sigma$-polarization (mean fluorescence wavelength 763 nm). Wavenumbers for observed and calculated resonances are subtracted for comparison in the last column on the right. The average discrepancy is given in the bottom row.

λobs (nm)	kobs (cm−1)	Initial Level	Site	kcalc (cm−1)	kobs − kcalc (cm−1)
815.80	12,257.91	(2A(0), vd)	4	12,199.56	−58.35
832.14	12,017.21	(2A(1), v = 0)	4	11,906.16	−111.05
899.11	11,122.11	(2A(1), vd)	1	10,999.56	−122.55
906.95	11,025.97	(2A(1), vd)	2	10,999.56	−26.41
928.00	10,775.86	(2A(1), vd)	3	10,999.56	223.70
933.84	10,708.47	(2A(1), vd)	4	10,999.56	291.09
944.03	10,592.88	(2A(2), v = 0)	1	10,496.16	−96.72
950.08	10,525.43	(2A(2), v = 0)	2	10,496.16	−29.27
954.42	10,477.57	(2A(2), v = 0)	3	10,496.16	18.59
957.90	10,439.50	(2A(2), v = 0)	4	10,496.16	56.66
				Average:	13.24

.

Although the observed resonances were grouped into sets of four perturbed site transitions for the purpose of making transition assignments, the tabulations were based purely on unperturbed host vibrational modes and electronic splitting of the ground state. Absorption transitions at wavelengths on the low energy side of the mean fluorescence wavelength 760 nm were calculated by subtracting the vibrational mode frequencies [21] or electronic splitting [22] (in cm⁻¹) for the listed transitions from the mean fluorescence wavenumber 13,158 cm⁻¹ ($\pi$-polarization only). The mean fluorescence wavelength for $\sigma$-polarization was 763 nm. Despite the lack of information from the literature on site perturbations, agreement between the observed and the calculated resonant wavelengths in the tables was found to be remarkably good. The width of the resonances was close to the step-size tuning of the laser, consistent with the low damping parameters of the optical modes in sapphire [21]. The number of observed cooling resonances in the range of 750–790 nm also decreased when the optical polarization was rotated from $\pi$ to $\sigma$, consistent with higher losses from the impurity background absorption for $\sigma$-polarization [34]. This quenching of resonances in $\sigma$-polarization was therefore ascribed to increased background heating over much of the spectral range.

3.3. Fano Resonance and Induced Transparency in Laser Cooling

The 3-level system of Ce³⁺ can be viewed either as a Λ-system or a V-system with a large detuning from at least one of the three transitions. The validity of these two perspectives explains why two forms of quantum interference are possible in this system, namely Fano resonance [35] and induced transparency [36,37]. The former requires mixing of the two distinct excited states by some interaction (V-picture), and the latter does not (Λ-picture). The two features that result from interference are the negative peaks displayed in the plots of Figure 16a,b respectively, which display results calculated from Equation (12) using input parameters [38] that specialize the problem to Ce³⁺: LiCaAlF6.

In Figure 16a, the coupling frequency was assumed to have a value of $V/\hslash ={10}^8$ rad/s. This is considerably less than known optical dephasing and decay parameters of trivalent Ce ions in LiCaAlF6 [37]: ${\mathsf{\Gamma }}_{12}={10}^{12} \mathrm{rad}/\mathrm{s}$, ${\mathsf{\Gamma }}_{23}={10}^{12} \mathrm{rad}/\mathrm{s}$, ${\mathsf{\gamma }}_{12}=3.3\times {10}^7{ \mathrm{s}}^{-1}$, ${\mathsf{\gamma }}_{13}=1.0\times {10}^4 {\mathrm{s}}^{-1}$, and ${\mathsf{\gamma }}_{23}=1.1\times {10}^7 {\mathrm{s}}^{-1}$. In Figure 16b, its value was set to $V/\hslash =0 \mathrm{rad}/\mathrm{s}$. The negative absorption (gain) peak in Figure 16a disappeared when the coupling frequency was set equal to zero. In addition, this feature was found to appear at arbitrary intensity of the pump wave. This identifies it as a Fano resonance that results in gain when the system is pumped at frequency ${\omega }_{13}$. At the low pump intensity assumed in the plot ($I~{10}^8 \mathrm{W}/{\mathrm{m}}^2$) it may also readily be confirmed that the upper state 3 of the transition is less populated than the ground state 1, showing that lasing without inversion can take place under these circumstances. For example, the steady-state solution for the ratio of populations in states 2 and 3 is ${\rho }_{22}/{\rho }_{33}={\gamma }_{13}/{\gamma }_{12}$, which is always small when state 3 is metastable. Near the ${\omega }_{12}$ resonance in Figure 16b, negative absorption (gain) is also induced, but only above a threshold value of pump intensity. The apparent need to meet a threshold requirement for the appearance of gain, in this case, indicates an induced transparency origin of the interference.

Figure 17 presents the theoretical cooling efficiency of Ce³⁺ ions in the wavelength range where the absorption and emission curves overlap. Net cooling is possible from 293–310 nm on the red side of the ${\omega }_{12}$ resonance when the quantum efficiency is one. This range encroaches on the shaded region of Figure 16b, where lasing without inversion can take place. Hence, self-cooled lasing without inversion may be possible in this system. While no experiments have been performed to date to test this prediction, this intriguing possibility indicates that quantum interference could play a role in future schemes for the laser cooling of solids.

4. Discussion

Laser cooling efficiencies in 10% Yb:YLF, measured with the DLT and TLS methods at various intensities and wavelengths, were in satisfactory mutual agreement. Absorption saturation of the background impurity was observed at high pump intensity, leading to a significantly higher cooling efficiency, as predicted by the differential absorption saturation theory. A method for measuring the saturation intensity of background impurities was devised based on identifying the cross-over point in the intensity-dependent cooling efficiency plot. Our experiments were in close agreement with analysis incorporating saturation effects, revealing that because background absorption can be reduced at high pump intensity, higher pump power reduced parasitic heating and enhanced cooling efficiency. This finding may allow the attainment of lower minimum temperatures and will promote the use of cooling materials that are not well developed and suffer from higher background impurity levels, such as KYW [24]. In addition, this result should enable laser refrigeration of some impure materials that could not be cooled previously at all.

The method used to determine the saturation intensity of background absorption consisted of systematic measurements of laser cooling efficiency by Differential Luminescence Thermometry (DLT) and Thermal Lens Spectroscopy (TLS) versus wavelength and incident intensity. The slope of cooling efficiency versus wavelength exhibited an intensity dependence with a common intersection point at $\mathsf{\lambda }=1000 \mathrm{nm}$ in the 10% Yb³⁺:YLF sample, indicating that at this wavelength, the saturation intensities of the coolant ions and background impurities had the same value ($~2.6\times {10}^4 \mathrm{W}/{\mathrm{cm}}^2$). For wavelengths longer than 1000 nm, higher pump intensities improved the cooling efficiency significantly, agreeing well with the differential saturation model using the measured saturation intensities.

Although major efforts have been made to improve crystal purity in hosts such as YLF to reach low temperatures by conventional ASF, the present results on improved cooling efficiency through optical saturation establish a new route for lowering the minimum achievable temperatures below those previously possible at a given background absorption level. However, it should be noted that room temperature measurements are insufficient to predict outcomes at cryogenic temperatures. The absorption coefficients of coolant ions and background impurities decrease with temperature [39], resulting in temperature-dependent saturation intensities. It is only in the case when the background absorption saturates first, at an intensity lower than the coolant ions, that lower minimum temperatures can be realized by increasing pump intensity.

The intensity dependence of the cooling efficiency observed in this work is surprising because the saturation intensity of the background impurity absorption is unexpectedly low. This result must stem from a dominant impurity of particular valence and site symmetry. Iron is thought to be the main source of background absorption in Yb:YLF crystals [40]. This impurity is likely a substituent at the trivalent sites (Y³⁺) in YLF, which would render it predominantly trivalent. It is therefore of possible interest to note that trivalent iron in a material such as Fe:YAG has a low-energy excited state transition from ⁶A_1g to ⁴T_1g, which is forbidden by both spin and parity selection rules [41]. Associated with the forbidden transition is a long-lived excited state with a fluorescence lifetime of 170 µs [42], making the transition easy to saturate. The resonance wavelength of this forbidden transition is 903 nm in YAG, with an absorption cross-section of ~$1.2\times {10}^{-20}{ \mathrm{cm}}^2$ at 1000 nm [19]. Unfortunately, the emission cross-section in YAG and both the absorption and emission cross-sections in LiYF₄ are currently unknown. However, based on the spectroscopy of Fe:YAG, it is reasonable to surmise that the background saturation intensity of trivalent iron in LiYF₄ could be extremely low in the same wavelength range. Using solely the value of the cross-section given above, the absorption saturation intensity of Fe³⁺ is estimated to be 20% lower than that of Yb³⁺ at 1000 nm, which is in quite good agreement with the experimental background saturation intensity reported here (albeit in a different material). For iron in other valence states, such as Fe²⁺ or Fe⁴⁺ ions, the transition near 1000 nm is allowed [43]. This would lead to a short-lived excited state and a much higher saturation intensity compared with Fe³⁺. Thus, the divalent and tetravalent ions would not be expected to saturate at the low intensity measured in our experiments. However, these valence states may be less prevalent in YLF than the trivalent state. Hence, one could reasonably conclude that iron impurities may saturate easily and that the cooling efficiency should improve substantially if a sizeable fraction of background impurities become non-absorbing. The properties of trivalent iron seem consistent with the low-absorption saturation intensity determined in our experiments.

Of course, a low value of background saturation intensity could arise from impurities other than iron. Several elements are significantly correlated with the background absorption in crystalline Yb:YLF [40], such as vanadium and chromium. Some of these elements could also lead to a low ionic saturation intensity near 1000 nm. For example, when the transition metal chromium is in the quadrivalent state in a YAG host, the saturation intensity of Cr⁴⁺ is estimated to be only 10% of Yb³⁺ at 1000 nm [44]. Consequently, the absorption of a variety of background ionic species could saturate similarly at relatively low intensities, contributing to the overall reduction in parasitic heating and enhanced cooling at elevated pump powers. Although we cannot state for certain what impurities contribute to heightened efficiency with increasing pump intensity in Yb:YLF, it seems evident that the dominant transition metal ion and others present in this crystal have properties that are consistent with the low saturation intensity we determined at ${\mathsf{\lambda }}_{cr}=1000 \mathrm{nm}$.

Our TLS experiments also indicate that laser cooling can take place on dipole-allowed transitions. The curves in Figure 10 and Figure 14, which show heating and cooling at wavelengths shorter and longer than the mean fluorescence wavelengths of the corresponding ions, respectively, are qualitatively the same for Yb³⁺:LiYF₄ (dipole-forbidden) and Ti³⁺:Al₂O₃ (dipole-allowed). The cooling behavior observed in Ti³⁺:Al₂O₃ reveals that configuration relaxation can be avoided by tuning to wavelengths longer than the mean fluorescence wavelength. This wavelength condition also determines the cooling range of rare-earth ions such as Yb³⁺. However, the allowed transition of Ti³⁺ has a much faster decay rate; so, faster cool-down times could be enabled for semiconductor sensor circuitry grown on Ti³⁺:Al₂O₃ substrates and cooled optically to cryogenic temperatures. Sapphire is an excellent substrate for GaN, Al_xGa_1-xN or In_xGa_1-xN, and ZnO epitaxy [21,45,46]; so, if cooling were to be demonstrated with laser diodes, the improved performance of imaging arrays in outer space could be achieved in the near future.

In view of the general agreement between calculated and observed wavelengths in Table 2 and Table 3, it can be concluded that the discrete resonances in cooling response in Ti:Al₂O₃ (Figure 15) arise from absorptions on all-electronic transitions or optical phonon sidebands. The discrepancies between the calculated and observed wavenumbers were less than $~400 {\mathrm{cm}}^{-1}$ in all cases ($<3\%$) and better than $1.2 {\mathrm{cm}}^{-1}$ ($<0.01\%$) on average for fifteen π resonances and $60 {\mathrm{cm}}^{-1}$ ($<5.6\%$) on average for ten σ resonances. Experimentally, more resonances were observed for $\pi$- than $\sigma$-polarization, even though $\sigma$-polarization excites more optical modes in sapphire [21]. Whether or not a particular absorption transition gives rise to observable cooling in Ti³⁺:Al₂O₃ seems to be dictated by the level of background impurity absorption. The background absorption coefficient for $\sigma$-polarization has a peak in the 750–790 nm range, with a value nearly twice that for $\pi$-polarization [34]. So, more resonances may be observed in this range for $\pi$- than for $\sigma$-polarization despite the greater number of infrared active $E_u$ modes [21]. The relative efficiencies of cooling transitions also depend on the occupation of the various levels in question, determined by the Boltzmann distribution, and the nature of the transitions, which in general is of mixed electronic and vibrational character. Due to these complexities, initial and final states were assigned to each resonance, but no attempt was made to analyze the relative intensities of the cooling resonances.

Because the dipole-allowed transition of Ti³⁺ has proven useful in this work for the laser cooling of sapphire, the analysis of quantum interference in the 3-level system of Ce³⁺, which has two such transitions, acquires heightened interest. Trivalent Cerium can undergo quantum interference effects of several kinds. This was evident from the calculations in Figure 16a,b, where gain was induced on the two dipole-allowed absorption transitions by a single incident pump wave. Both figures indicate that lasing without inversion can take place on these transitions as the two lower states are always more occupied than the upper level (given that state 3 is long-lived in Ce³⁺). Gain on the 13 transition in Figure 16a is due to Fano resonance as it has no threshold and requires the mixing of the excited states ($V\ne 0$). Gain on the 12 transition in Figure 16b is induced only at high pump powers and does not require excited state mixing ($V=0$). Hence, quantum interference that originates in two distinct ways is possible in this system. At the same time, laser cooling can be sustained theoretically on the red side of the transition at ${\omega }_{13}$. Therefore, the intriguing possibility emerges of accomplishing self-cooled lasing without inversion in materials with allowed transitions such as Ce³⁺:LiCaAlF₆.

5. Conclusions

Steady-state analysis of ASF laser cooling revealed that improved cooling efficiency becomes possible at elevated pump intensities whenever the saturation intensity of the background absorption is less than that of the coolant ions. Experimental results (Figure 11 and Figure 12) showed that this situation obtains at wavelengths longer than 1000 nm in 10% Yb:LiYF₄. In this crystal, the saturation intensity of the background absorption was determined experimentally to be $I_b~2.6\times {10}^4 \mathrm{W}/{\mathrm{cm}}^2$, a value low enough to be reached with the continuous-wave excitation used in our experiments. Above this intensity, the cooling efficiency doubled at 1035 nm when input power was increased to the maximum available. This affirmed that in some materials, ASF cooling efficiency could improve with higher pump intensity.

Cooling of bulk sapphire doped with Ti³⁺ ions was also demonstrated on the allowed ²E→²T₂ transition of the dopant. The polarity of the TLS signals reversed when the wavelength of excitation was tuned from near resonance at 532 nm to wavelengths longer than the mean fluorescence value. The use of Brewster-cut sapphire that was highly purified and lightly doped was essential to minimize losses from the crystal coatings and parasitic absorptions from valence-modified titanium pairs or iron impurities [47,48,49]. Samples with FOM values less than 800 did not show net laser cooling at any wavelength. Hence, although configuration relaxation on allowed transitions normally contributes to strong heating in optical interactions, cooling is achievable in high-quality Ti³⁺:Al₂O₃ at long wavelengths in the absorption tail. Cooling in sapphire occurred at discrete absorption resonances which were assigned to electronic transitions and optical phonon sidebands. This behavior differed from cooling on forbidden transitions in rare-earth systems where continuous tuning is observed due to the continuous density of states spectrum for the acoustic phonons which mediate refrigeration. Overall, the present results provide accelerated cooling to lower minimum temperatures based on electric-dipole-allowed transitions and optical saturation. These findings should enable Raman cooling, self-cooled lasing without inversion, and other advancements of the field.