1. Introduction

Radiation-balanced lasers (RBLs) are self-cooling devices in which pump light performs two functions simultaneously. Incident radiation must provide both the gain needed for lasing and the cooling needed to control device temperature. When optical cooling exactly compensates for heating from the lasing process, lasers can operate steadily at a “radiation balance” point without any change in temperature. This radiation balance condition (RBC) was analyzed by Bowman [1], who noted that the elimination of thermal distortions and lensing should be conducive to stable high power laser operation. The first RBL was demonstrated in Yb³⁺:YAG in 2010 [2] and several others have been reported since that time [3,4].

The cooling efficiency of light by the anti-Stokes fluorescence method [5] is limited by the heatload from unintentional background impurities that absorb pump light but do not contribute to refrigeration. Consequently if background absorption is reduced by any means the cooling efficiency improves. Crystal growth with highly purified ingredients is one way to accomplish this. However saturation of the background absorption with the use of intense pump light provides a post-growth method that is also effective. This was pointed out by Cheng [6] who observed an improvement of cooling efficiency in Yb³⁺:LiYF₄ at elevated pump intensities. Cheng also reported laser cooling of Yb³⁺:KYW samples [7] which were candidates for RBL media but had much higher background absorption coefficients than Yb³⁺:YAG or Yb³⁺:LiYF₄. He noted that radiation-balanced lasers operate in the saturation regime where an increase of the background absorption level lowers the pump power required to reach the radiation balance condition. This means saturation could make the difference between whether a solution exists for RBC or not in relatively impure materials like Yb³⁺:KYW.

In this paper two radiation-balanced lasers are described that illustrate the importance of background absorption saturation in RBL analysis. The first utilizes a relatively pure Yb³⁺:YAG crystal to provide a benchmark comparison between theoretical predictions (of laser threshold, wavelength, radiation balance condition, and efficiency) and experimental measurements. Excellent agreement is found with only a single fitting parameter. Thermal imagery was used to confirm that radiation balance was achieved and that the temperature distribution in the gain medium near RBC was uniform and differed from room temperature by less than 0.1 K. The second RBL utilized a relatively impure crystal of Yb³⁺:KYW, a material in which RBL operation has not previously been reported. Due to its high impurity content, no theoretical solution for radiation balance was found for this sample unless background saturation was taken into account. Hence the laboratory demonstration of a Yb³⁺:KYW RBL in this crystal illustrates that saturation permits RBL operation in a broader class of (relatively impure) materials than was previously thought to be suitable for this type of application. Theoretically Yb³⁺:KYW has a higher figure of merit for RBL operation than Yb³⁺:YAG [8]. So while the experimental results reported here are far from optimal they do indicate that, with improved crystal growth efforts, better optical coatings and polishing, an RBL with high efficiency could be realized in Yb³⁺:KYW [9].

2. Theory

The original theory of radiation-balanced laser action treated the active ions as isolated 2-level systems. Background impurity absorption and absorption saturation [10] were ignored. However in this section we show the importance of including the behavior of such impurities due to their significant influence on the cooling efficiency of the gain medium.

The impurity ions are modeled as a second 2-level system coupled to the first (i.e. the active ion population) via internal light fields. Parameters for the background impurities are distinguished by primes. The rate equations for upper state populations of the two systems are

The effective cross sections for active ions and background impurities are denoted by $\frac{{dN_2^{}}}{{dt}} = \frac{{{I_p}{\lambda _p}}}{{hc}}[{\sigma_{a,p}^{}N_1^{} - \sigma_{e,p}^{}N_2^{}} ]+ \frac{{{I_l}{\lambda _p}}}{{hc}}[{\sigma_{a,l}^{}N_1^{} - \sigma_{e,l}^{}N_2^{}} ]- \frac{{N_2^{}}}{{\tau _f^{}}}$ and $\sigma _{j,i}^{\prime} \equiv \sigma _j^{\prime}({{\lambda_i},T} )$ where the index $j = ({a,e} )$ indicates absorption or emission and $i = ({p,l} )$ refers to either the pump or laser field respectively. τ is the excited state decay time.

Equations (1) and (2) can be solved for the absorption coefficients of the pump and laser fields. Under steady-state conditions, one obtains

Since the absorption coefficients are intensity-dependent, saturation intensities for both the coolant ions and the background impurities appear in these results. They are defined by

Thermal balance in the gain medium must take into account any heating or cooling by the pump and laser fields. The thermal power density H therefore consists of two terms specifying heating or cooling caused by the absorption of light from these two fields. This function is proportional to the cooling efficiency ${\eta _{c,p}}$ and absorbed power ${P_{abs,p}}$ at the pump wavelength, as well as the cooling efficiency ${\eta _{c,l}}$ and absorbed power ${P_{abs,l}}$ at the lasing wavelength.

The absorption efficiencies at the pump and laser wavelengths are defined respectively by

The radiation balance condition (RBC) is found by setting H equal to zero (H = 0). This determines the point at which heating and cooling exactly balance. The RBC can be written in the form

The RBC in Eq. (16) is central to RBL design. If the variation of the impurity cross sections with wavelength is weak (i.e. β_p’ ≈ β_l’), or the effective absorption cross section is larger than the effective emission cross section (βi′ > 0.5), the factor C is negligible (C ≈ 0). Under these circumstances, Eq. (16) has the same apparent form as in Ref. 1. However the factor R_i (i = p,l) in Eqs. (20)–(21) generally does not vanish, for it depends on the pump and laser intensities, the corresponding saturation intensities, and the small signal absorption coefficients. It also depends on these quantities as they apply to background impurities. When the saturation intensity of active ions is comparable to the saturation intensity of impurities, the factor R becomes important for it alters the RBC significantly, as shown next.

Figure 1 plots curves for intracavity intensities of the laser and pump fields that satisfy the radiation-balance condition in 3% Yb:YAG with an assumed impurity saturation intensity ${I_{sat,b}} = I_{sat,p}^\mathrm{^{\prime}} = I_{sat,l}^\mathrm{^{\prime}} = 2 \times {10^4}$ W/cm² [6]. The curves compare predictions with and without impurity absorption. It can be seen that radiation balance is achievable with an impurity absorption coefficient as high as $2 \times {10^{ - 3}}$ cm^-1. On the other hand, when the absorption coefficient of background impurities drops below $2 \times {10^{ - 4}}$ cm^-1, parasitic heating has a negligible effect on radiation balance in this model.

 

Fig. 1. Intracavity intensities of the laser and pump fields that satisfy the RBC in 3% Yb:YAG for various values of the background impurity absorption coefficient. Fixed parameters for wavelength of the pump, and its absorption and emission cross sections were 1029.3 nm, 1.26 × 10⁻²¹ cm² and 2.24 × 10⁻²⁰ cm² respectively. The wavelength, and absorption and emission cross sections of the laser were 1048 nm, 8.83 × 10⁻²³ cm², and 3.68 × 10⁻²¹ cm². The fluorescence lifetime was set to 0.95 ms. Pump and laser saturation intensities were calculated to be 8.59 × 10³ W/cm² and 5.30 × 10⁴ W/cm². The mean fluorescence wavelength and external quantum efficiency were 1012.1 nm and 99.3%. The impurity saturation intensity was $2 \times {10^4}$ W/cm² [6].

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3. Experiments and results

A well-characterized sample of 3 Molar % Yb³⁺:YAG (Scientific Materials) with dimensions of 1 × 1x10 mm³ and Brewster-angle end faces [7] was inserted as the gain medium into a four mirror, astigmatically-compensated laser cavity. Three of the cavity mirrors were high reflectors at the laser wavelength (R > 99.99%) while the output mirror had a reflectivity specified by the desired RBL efficiency from plots like Fig. 2. The sample contained an unknown impurity that presented a background absorption coefficient of ${\alpha _b} = 2 \times {10^{ - 4}}$ cm^-1 with a saturation intensity of ${I_{sat,b}} = 2 \times {10^4}$ W/cm², taken to be a free parameter with a value close to the experimentally determined value of $2.6 \times {10^4}$ W/cm² in Yb:YLF in an earlier experiment [6]. Saturation intensities at the pump and laser wavelength were calculated from experimental cross sections as described in Ref. 7, yielding ${I_{sat,p}} = 8.59 \times {10^3}$ W/cm² and ${I_{sat,l}} = 5.30 \times {10^4}$ W/cm². The sample was mounted on an aerogel disk for thermal isolation (Aerogel Technologies). All measurements were made at room temperature in air.

 

Fig. 2. Predicted design requirements for radiation-balanced lasing in (a) 3% Yb³⁺:YAG, (b) 1% Yb³⁺:KYW, and (c) 2% Yb³⁺:KYW with and without saturation of background impurity absorption. The dashed black curves plot the radiation balance condition ignoring background absorption (${\alpha _b} = 0$). Background absorption coefficients for the three crystals were ${\alpha _b} = 2 \times {10^{ - 4}}$ cm^-1 for 3% Yb³⁺:YAG, ${\alpha _b} = 1.7 \times {10^{ - 3}}$ cm^-1 for 1% Yb³⁺:KYW and ${\alpha _b} = 9.4 \times {10^{ - 4}}$ cm^-1 for 1% Yb³⁺:KYW. Blue curves assumed the background absorption was not saturable (${I_b} = 0$) while the red curve assumed saturability (${I_b} = 2 \times {10^4}$ W/cm²). Note that in KYW no radiation balance is possible without saturation. (d), (e), (f) Laser efficiency under radiation-balanced conditions versus reflectance of the input and output couplers in 1% Yb³⁺:YAG (${\lambda _p} = 1029.3$ nm and ${\lambda _l} = 1048$ nm), in 1% Yb³⁺:KYW (${\lambda _p} = 1022.5$ nm and ${\lambda _l} = 1039.6$ nm), and in 2% Yb³⁺:KYW (${\lambda _p} = 1022.5$ nm and ${\lambda _l} = 1050$ nm).

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In the case of Yb³⁺:KYW, several samples from different vendors were available for experimentation. A sample with 1% doping (Altechna) was shown in prior experiments [7] to have the highest cooling efficiency, so mirrors for the RBL experiment were selected for pumping at 1022.5 nm and lasing at 1040 nm to optimize conditions for this sample. However, no lasing was observed in this crystal due to surface damage at one end. Consequently a sample with 2 Molar % doping (from FEE) and lower cooling efficiency was investigated instead [7]. The sample had dimensions of 0.9 × 1.2 × 10 mm³ with Brewster-angle end faces and the Nm axis was parallel to the shortest edge. The background impurity absorption coefficient was ${\alpha _b} = 9.4 \times {10^{ - 4}}$ cm^-1 and the saturation intensity was assigned the value ${I_{sat,b}} = 2 \times {10^4}$ W/cm². Saturation intensities at the pump and laser wavelength were again calculated from experimental cross sections as described in Ref. 7, yielding the values $1.66 \times {10^4}$W/cm² and $8.81 \times {10^4}$W/cm². The 2% sample called for dichroic cavity optics with tight tolerances. The design specifications were met only approximately using an angled 1064 nm laser line mirror (Edmund 11-075) as the output coupler. The pump wavelength was maintained at 1022.5 nm although this was not optimal for cooling the higher concentration sample. These factors forced the laser to operate at a longer wavelength and prevented complete absorption of the pump light. Consequently its efficiency was severely reduced. Nevertheless the 2% crystal did support radiation-balanced laser action. Crystals having higher concentrations of Yb had lower cooling efficiencies and were not investigated for operation as RBLs.

In previous research on RBL operation in bulk crystals [2,3] researchers used schemes in which the pump and laser fields took different paths within the cavity. Naturally this approach reduces laser efficiency because pump and laser mode volumes cannot overlap perfectly. An alternative approach capable of achieving total pump absorption and modal overlap is to lock the frequency of the RBL cavity to the pump laser and match the pump and laser modes, as in the present work. With mode-matching and frequency-locking the pump laser can circulate within the cavity until absorption is complete and pump/laser overlap can be fully optimized. In our experimental setup (Fig. 3), frequency-locking was achieved by imposing radio frequency (rf) sidebands on the input light with an electro-optic modulator. After separating the total pump field from laser light transmitted through mirror M4 with spectral filtering, the rf beat signal between the carrier and sidebands was detected with a photodiode to implement Drever-Pound-Hall locking. This error signal was applied to the current driving the diode seed laser (Innovative Photonic Solutions) to correct for frequency excursions. Seed light of variable intensity was amplified using a fiber amplifier (IPG YAR-10K-1020-LP-SF) to provide frequency-locked input to the RBL cavity.

 

Fig. 3. Schematic of the experimental setup for radiation-balanced lasing in a mode-matched, frequency-locked optical cavity. A frequency-programmable gate array (FPGA) generated 12.5 MHz rf sidebands via an electro-optic modulator (EOM) and processed the error signal to modulate the current driver of the diode seed laser. Abbreviations are as follows, PBS: polarizing beam splitter, EOM: electro-optic modulator, PM: power meter, BD: beam dump, PD: photodetector.

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Sample temperature was monitored with a thermal camera (FLIR A655sc) equipped with a high resolution lens (T198059) that achieved 50 µm/pixel spatial resolution in the range 7.5-14 µm. Calibration was effected by placing the sample in a temperature-controlled oven (Quantum Northwest Flash 300) which had an accuracy and stability of ±0.01 K. Thermal image intensities were typically recorded at ten temperatures between 10-50 C so that emissivity could be determined as the value that returned matching temperatures calculated from the Stefan-Boltzmann law by FLIR’s ResearchIR software. Real-time temperature measurements were then possible with averaging over selected portions of the image at a frame rate of 1 Hz.

To measure absorption and emission cross sections, the crystals were pumped with a tunable Ti:Sapphire laser (M Squared SolsTiS, 750–1050 nm). Infrared luminescence 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). A polarizer was placed between the crystal and the collecting fiber to ensure the measured spectrum corresponded to the same polarization as the pump laser. Absorption cross sections were calculated from the measured fluorescence data using the McCumber relation [11] after correction for instrumental response. To permit scaling of the absolute value of absorption and emission cross-sections, the absorption coefficients were determined by tuning the Ti:Sapphire laser to 1030 nm (Yb:YAG) or 1025 nm (Yb:KYW) and measuring the incident, reflected and transmitted laser powers. Figure 4 displays the resulting calibrated absorption and emission spectra.

 

Fig. 4. (a) Absorption and emission cross sections for 3% Yb:YAG. (b) Absorption and emission cross sections for 1% Yb:KYW (polarization E||Nm).

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To demonstrate RBL operation, pump light of variable intensity was focused into the cavity using a mode-matching lens combination. At selected input intensities, the temperature evolution and output intensities were then recorded. The results are shown in Fig. 5, 6, 7 and 8. In Yb:YAG, the optics available for experimentation met all design specifications closely. The laser reached radiation balance at 1.8 W of input power and had a gradient of less than 0.1 K temperature along its length. The threshold, observed RBC point, and laser efficiency all agreed with theory to a few percent with the single fitting parameter ${I_{sat,b}}$, providing compelling support for the model. In Yb:KYW, the measured background impurity absorption was very high and many factors contributed to additive losses that shifted the RBC and reduced laser efficiency dramatically. These additional losses were treated as an extra output coupling loss of 2.2% in order to plot the solid black curve in Fig. 8. With this one adjustable parameter, the laser threshold, observed RBC point, and laser efficiency then all agreed very closely with experiment. The laser reached radiation balance at an input power of 1.92 W as predicted and displayed a gradient of only 0.6 K along its length under radiation-balanced conditions. The temperature gradient in Yb:KYW is larger than Yb:YAG because KYW crystal has a lower thermal conductivity, which limits the heat transfer between cooled and heated ends of the crystal.

 

Fig. 5. (a) Longitudinal temperature distribution of 3% Yb:YAG crystal at steady state with an input pump power of 1.2 W, which is at the balanced condition. Inset shows the thermal image and pump beam was introduced from the left. (b) Measured temperature change of 3% Yb:YAG crystal versus time at a series of input pump powers. The pump was introduced at ∼ 10 s and was blocked after ∼ 500 s. (Insert) The temperature was averaged over the red rectangular region shown in the image. RBC corresponds to the purple curve recorded at an input pump power of 1.20 W.

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Fig. 6. (a) Longitudinal temperature distribution of 2% Yb:KYW crystal at steady state RBC with an input pump power of 1.92 W at 1022.5 nm. Inset shows the thermal camera image for a pump beam introduced from the left. (b) Measured temperature change of 2% Yb:KYW crystal versus time at a series of input pump powers. The pump was introduced at ∼ 10 s and was blocked after ∼ 600 s. (Insert) Temperature was averaged over the red rectangular region shown in the image. RBC corresponds to the purple curve recorded at an input pump power of 1.92 W.

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Fig. 7. The data points represent the measured output laser power versus input pump power in 3% Yb:YAG. Blue circles are in the cooling regime (ΔT < 0), red triangles are in the heating regime (ΔT > 0) and the black square is the RBL point (ΔT ∼ 0). The black curve, background color and red curve are the theoretical predicted output power versus input power, thermal intensity and RBC with 88.3% and 97.8% reflectors at 1029.3 nm and 1048 nm. Fixed parameters for wavelength of the pump, and its absorption and emission cross sections were 1029.3 nm, 1.26 × 10⁻²¹ cm² and 2.24 × 10⁻²⁰ cm² respectively. The wavelength, and absorption and emission cross sections of the laser were 1048 nm, 8.83 × 10⁻²³ cm², and 3.68 × 10⁻²¹ cm². The fluorescence lifetime was set to 0.95 ms. Pump and laser saturation intensities were calculated to be 8.59 × 10³ W/cm² and 5.30 × 10⁴ W/cm² together with I_p,min= 1.23I_sat,p and I_l,min= 0.99I_sat,l. The mean fluorescence wavelength and external quantum efficiency are 1012.1 nm and 99.3%. The background absorption coefficient and impurity saturation intensity are 2 × 10⁻⁴cm⁻¹ and 2 × 10⁴ W/cm².

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Fig. 8. Output versus input power in 2% Yb:KYW with 81.3% and 98.9% reflectors at 1022.5 nm and 1050 nm. The background absorption coefficient and impurity saturation intensity are 9.4 × 10⁻⁴cm⁻¹ and 2 × 10⁴ W/cm². Fixed parameters for wavelength of the pump, and its absorption and emission cross sections were 1022.5 nm, 3.90 × 10⁻²¹ cm² and 3.51 × 10⁻²⁰ cm² respectively. The wavelength, and absorption and emission cross sections of the laser were 1050 nm, 2.19 × 10⁻²² cm², and 6.94 × 10⁻²¹ cm². The fluorescence lifetime was taken to be 0.3 ms. Pump and laser saturation intensities were calculated to be 1.66 × 10⁴ W/cm² and 8.81 × 10⁴ W/cm². The mean fluorescence wavelength and external quantum efficiency are 1002.1 nm and 98.3%. The black curve includes an added 2.2% loss due to non-optimal optical properties.

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4. Discussion and conclusions

Although radiation-balanced lasing has been reported previously in Yb:YAG, the present experimental results in 3% Yb:YAG closely matched the predictions of the extended model for laser threshold, RBC point, and laser efficiency with only one free parameter. This provided strong validation for analysis that includes impurity absorption and saturation. Indeed, impurity absorption cannot be ignored because, although it is often small, it is the limiting factor in laser cooling by the anti-Stokes fluorescence method and RBLs operate in the saturation regime.

The other results reported here in 2% Yb:KYW constitute the first report of RBL operation in the tungstate host. They also agreed quantitatively with predictions once uncertainties in output reflectance, non-parallel Brewster surfaces, and radiation trapping were lumped into an effective contribution to the total output coupling loss. The mere demonstration of an RBL in 2% Yb³⁺:KYW with a background impurity absorption of ${\sim} {10^{ - 3}}$ cm^-1 illustrates that the saturation of impurities can have a major impact on self-cooled lasers. RBLs can be operated in a broader class of (relatively impure) materials than was previously thought possible, provided the impurities saturate before the coolant ions [6].