In Fig. 1, the state of Yb3+ can be described as eigenstates of the crystal field potential, and is composed of |J, J
z> = (J
−)m
Y
33(θ, ϕ)χ
↑ (m = 0 to 7), where Y
lm
(θ, ϕ), J
− and χ
↑ are spherical harmonics, the ladder operator of total angular momentum and the wave function of spin with +0.5 ħ, respectively. The crystal field potentials for the crystals were calculated using the Stevens equivalent operator as in Equation (2)19. We can approximate the radius of the electron cloud in the direction of angle (θ, ϕ) by the azimuthal component to the 1.5th power. Electron clouds in this viewgraph were drawn larger than the real size to improve visibility.
Here we show how to estimate Δχ of RE3+-doped microdomains. In order to calculate the effective total angular momentum of doped RE3+-ions, we have to introduce two new multipliers. One describes Coulomb shielding, and it is a ratio of the calculated V
cryst to the spectroscopically measured V
cryst. Another describes Néel rotation of doped RE3+-ions32, which can be estimated from the ratio of the calculated Δχ to the measured Δχ in the other host crystal where the same RE3+-ions are doped.
Using a superconducting quantum interference device magnetometer under 7.0 T (MPMS-7, Quantum Design Inc., San Diego, CA, USA) in the temperature range from 80 to 300 K, we measured Δχ of Yb3+-doped and undoped yttrium orthovanadate (YVO4) single crystals with dimensions of 4.0 × 4.0 × 3.0 mm, and S-FAP with dimensions of 3.5 × 3.5 × 3.5 mm (Fig. 2). Yb3+ are substituted into 4a sites in YVO4 and 6 h sites in apatite crystals. According to Equation (2), V
cryst in YVO4, S-FAP and FAP were evaluated using lattice sums that were calculated with over 30,000 nearest ions around the doped sites for the crystals. V
cryst was multiplied by a constant to correct the crystal field splitting modulated by Coulomb shielding of RE3+. Mainly because of Néel rotation of the magnetic moments, the measured Δχ became smaller than the theoretical value determined by Equation (3). We estimated this factor to be 20% from the difference between calculated and experimental values for Yb:YVO4. Generally, undoped crystals show non-zero Δχ because of Larmor diamagnetism; Δχ was measured to be 3.9 × 10−6 for the YVO4 crystal. We estimated that Δχ for undoped S-FAP was 2.4 × 10−6 from the temperature dependence of the calculated and measured Δχ of Yb:S-FAP.
While the main crystal axis (c-axis) of Yb:S-FAP and Yb:FAP is the hard magnetization axis (Δχ < 0), the main crystal axis of Yb:YVO4 is the easy magnetization axis (Δχ > 0). This difference arises from the sign of their lattice sum. Similarly, the main crystal axis of Nd3+-doped apatite crystals is the easy magnetization axis because of the different signs of θ
J
33.
In uniaxial materials, refractive index n(θ) is given by
where n
m and n
s are refractive indices of the microdomains for polarizations that are parallel and perpendicular to the controlled axis, respectively. The mean refractive index of all domains n
mean and the difference of refractive index between the averaged matrix and each microdomain Δn are calculated from the distribution function F(θ) by
In the case of Yb:FAP, wavevector k of the irradiated light is parallel to the c-axis. Consequently, n
m and n
s are 1.622 and 1.620, respectively23. F(θ) can be derived from Equation (1) as
where Erfi(z) gives the imaginary error function, and B
min is a parameter that indicates the lower limit of the magnetic flux density for orientation control20. B
min is proportional to the square root of RE3+ concentration. C
sca arising from this birefringent scattering is expressed by16
where λ is the wavelength of the incident light.
Although Equation (9) gives C
sca of 0.12 cm−1 for D of 30 μm, spectroscopic measurements indicated C
sca was 1.0 cm−1, which is eight times larger than the calculated value21. This difference suggests that there are other scattering sources in our Yb:FAP as well as birefringent scattering.
Some of the Ca2+ in FAP microcrystals were substituted by Yb3+ by reacting commercial FAP powder (10 g) and Yb(NO3)3 (1.5 g) in an aqueous solution (120 mL) containing 0.012 mol of hydrochloric acid and 0.012 mol of sodium nitrate for 5 min at room temperature. After drying, Yb:FAP microcrystals were dispersed in distilled water as a slurry for casting. As described in Fig. 2, slurry in a gypsum mold was rotated at 17 rpm, and slip-cast under a magnetic field of 1.4 T generated by an electric magnet (JER-3XG, JEOL Ltd., Tokyo, Japan). Cast powder compacts were pre-sintered for 2 h at 1600 °C in air, and hot isostatic pressed at 1600 °C and 190 MPa in Ar for 1 h (System 20 J, Kobe Steel, Ltd., Tokyo, Japan). The Yb:FAP sample was cut and polished to dimensions of 3.4 × 3.0 × 0.48 mm (Lambda Precision Co., Atsugi, Japan).
Quantum control under a magnetic field was a stochastic process that could be reproduced statistically if isolated microcrystals were well stabilized in the slurry used for the slip-casting process. A rough indication of mean deflection angle <θ> is given by
In the orientation control of Yb:FAP domains, the alignment of one hard magnetization axis is equivalent to the alignment of two easy magnetization axes. Consequently, B
min for Yb:FAP domains should be
which is √2 times compared to the orientation control of one easy magnetization axis20. Equation (4) can be directly derived from Equations (10) and (11). However, we only managed to synthesize two samples of polycrystalline Yb:FAP with QC-MDs that displayed laser oscillations. The consistency of material processes in this work relies on the reproducibility of current laser ceramic technologies such as casting, sintering and polishing.
Figure 4(a) was taken under irradiation with white light to emphasize the optical scattering of the sample. To prevent charge-up on the surface of uncoated Yb:FAP, a back-scattered electron image was obtained by SEM (SU6600, Hitachi Ltd., Tokyo, Japan) under a vacuum of 50 Pa with a 7.0-kV accelerating voltage and 400× magnification (Fig. 4(b)). There were almost no structures in a SEM secondary electron image. Observation of the image in Fig. 4(b) indicates that the C
sca of 1.0 cm−1 in ref. 21 should be caused by both birefringence and the secondary amorphous phase.
Two surfaces of a Yb:FAP sample with an area of 3.4 × 3.0 mm were coated with coating apertures of 2.5 × 2.5 mm (Showa Optronics Co. Ltd., Tokyo, Japan): one side had reflection below 1% at 905 nm and above 99.9% around 1 µm (A-side), and the other side had reflection above 99.5% at 905 nm and below 0.5% around 1 µm (B-side). The laser resonator with a cavity length of 4.5 mm was composed of the A-side of Yb:FAP as a total reflection mirror and plane mirror with a reflection R
OC of 95.8% as an output coupler. The B-side of Yb:FAP was placed on a copper holder, and fixed from the A-side by a copper plate. A Cr4+:YAG film with a thickness of 1 mm, initial transmittance T
0 of 80% and covered on both surfaces with anti-reflection coating (Scientific Materials Co., Boseman, MT, USA) was inserted as a saturable absorber between the output coupler and B-side of Yb:FAP. The pump source was a fiber-coupled laser diode with an output power of 112 W at 905 nm, duration of 800 µs and rise time of 20 µs (LE0379, Hamamatsu Photonics K.K., Hamamatsu, Japan). This pump source was collimated by an aspheric condenser lens with a focal length of 18 mm, and focused onto the A-side of Yb:FAP by an aspheric condenser lens with a focal length of 12 mm. Generated laser output passed across the pump-cut filter with 86.3% transmittance around 1 µm and was detected by an energy sensor (PD10-C, Ophir Optics LLC, North Andover, MA, USA) or photodetector (ET-3500F, Electro-Optic Technology Inc., Traverse City, MI, USA) and spectrum analyzer (AQ6763, Yokogawa Meters & Instruments Co., Tokyo, Japan).
In Fig. 6, National Ignition Facility (NIF), Laser Mégajoule (LMJ), PETawatt Aquitaine Laser (PETAL), LUCH (ISKRA-6), OMEGA, GEKKO, Z-Beamlet, Petawatt High Energy Laser for Heavy Ion Experiments (PHELIX), TEXAS, SLAC National Accelerator Laboratory (SLAC), Prague Asterix Laser System (PALS), Petawatt Field Synthesizer (PFS), JAEA Kansai Advanced Relativistic ENgineering (J-KAREN), Apollon, Qiangguang, SuperIntense Laser for Experiments on eXtremes (SILEX), Xtreme Light III (XL-III), Astra-Gemini, Laserix, Advanced Photonics Research Institute (APRI), Petawatt Optical Laser Amplifier for Radiation Intensive experimentS (POLARIS), Petawatt, Energy-Efficient Laser for Optical Plasma Experiments (PEnELOPE) and Mercury are petawatt-class laser facilities34. PEARL35 is the current laser system at Exawatt Center for Extreme Light Studies (XCELS)34. ISKRA-536 is an iodine laser that is the previous generation of LUCH. ELECTRA, NIKE, and Super-ASHURA are high-power excimer lasers37. LUCIA and Diode Pumped Optical Laser Experiment (DiPOLE) are Yb:YAG ceramic laser systems that were developed as a part of the HiLASE project27. Generation of ENergetic Beam Ultimate (GENBU)28 and High Average power Laser for Nuclear fusion Application (HALNA)38 are categorized as slab laser systems that have high thermal performance.
In these laser facilities, effective cooling of laser gain media is quite important, and the advantage of MCL technology has been proved by the possibility of face cooling of thin laser gain media in Mercury, LUCIA, DiPOLE, HALNA, GENBU and SLAC. Consequently, MCLs such as MW-UV39, multi-point laser ignitor40, and is-TPG41 can provide comparable optical energy densities to the above high-power devices aiming to realize inertial confinement fusion.
The averaged pump intensity in effective mode volume I
p inside Yb:FAP was 278 kW/cm2 in our setup, where the number density of Yb3+
N
tot was 1.86 × 1020/cm3 = 1.62 at. % and the pump saturation intensity I
s was 13.9 kW/cm2. The time evolution of ξ, which is the ratio of N to N
tot, is given by
where f
1, f
2 and f
3 are the fractional population ratios of the ground state, laser upper level and pumping levels, respectively. We estimated that f
1, f
2 and f
3 were 0.827, 0.909 and 0.0128, respectively23. Because the effective pumping time was 235 µs, we calculated that ξ was up to ξ
i = 0.785. Therefore, the initial population inversion N
i in this work was calculated to be 1.46 × 1020/cm3. The number of Yb3+ that were totally excited into the upper manifold N
U can be expressed by
where 2* is the bottleneck factor and equals f
1 + f
2. In Yb:FAP, 2* is calculated to be 1.74. Thus, we can conclude that 92.5% of Yb3+ were excited. According to Equation (12), the repetition rate of 8.1 kHz indicates that the final population inversion after pulsing N
f was 1.16 × 1020/cm3.
We can estimate the stimulated emission cross section σ
e of our Yb:FAP sample. From the balance between loss and gain, with a C
sca of 0.120 cm−1, the initial gain coefficient g
i of Yb:FAP can be calculated to be
Therefore, σ
e is given by
This value can be confirmed by the extraction energy density. Total extracted photon density ΔN is
Therefore, the photon extraction efficiency η
ext and threshold inversion N
th are calculated by
Finally, we can obtain the output coupling efficiency η
OC and energy extraction density E
out by
The fact that the experimental E
out of 0.34 J/cm3 coincided with the calculated value proves the validity of the estimated σ
e. In addition, C
sca of 0.12 cm−1 and σ
e of 3.6 × 10−20 cm2 were also confirmed experimentally from the strong relaxation oscillation observed under the condition without Cr4+:YAG using recently developed methods42.
The dependence of E
out on N
tot can be calculated using η
ext as
From Equations (14) and (18), ξ
i and η
ext are given as functions of N
tot by
where Ω(x) is the omega function (the inverse function of xe
x). The red dashed line in Fig. 6 indicates the power scaling calculated by Equation (23) assuming that R
OC had the same ξ
i as in our experiment of 0.785.
