1. Introduction
Recent developments in resonantly-pumped, cryo-cooled, eye-safe Er:YAG lasers have demonstrated great progress in reducing the thermal load and improving the laser efficiency [
11. M. Dubinskii, N. Ter-Gabrielyan, G. A. Newburgh, and L. D. Merkle, “Cryogenically cooled Er:YAG laser.” in CLEO/Europe and IQEC 2007 conference Digest, (OSA 2007), paper CA6–5.
–
33. S. D. Setzler, M. J. Shaw, M. J. Kukla, J. R. Unternahrer, K. M. Dinndorf, J. A. Beattie, and E. P. Chicklis, “A 400 W cryogenic Er:YAG slab laser at 1645 nm,” Proc. SPIE 7686, 76860C (2010).
]. In spite of the perceived complexity from the engineering point of view, cryogenic cooling offers numerous benefits. Along with improved thermal and thermo-optical properties of the host materials [
44. R. L. Aggarwal, D. J. Ripin, J. R. Ochoa, and T. Y. Fan, “Measurement of thermo-optic properties of Y_{3}Al_{5}O_{12}, Lu_{3}Al_{5}O_{12}, YALO_{3}, LiLuF_{4}, BaY_{2}F_{8}, KGd(WO_{4})_{2}, and KY(WO_{4})_{2} laser crystals in the 80-300 K temperature range,” J. Appl. Phys. 98, 103514 (2005). [CrossRef]
], most transitions in Er:YAG become narrower and stronger. The reduction of the thermal population of terminal laser energy levels enables the lowest quantum defect (QD) operating schemes that are either inefficient or impossible at room temperature.
The
^{4}I_{15/2} →
^{4}I_{13/2} absorption spectrum of the Er in YAG consists of two distinct groups of transitions in the wavelength ranges of 1450–1490 nm and 1520–1550 nm. This clear distinction arises from the large split in the energy levels of the
^{4}I_{13/2} manifold, see
Fig. 1
Fig. 1 a. The energy level diagram with major pump and laser transitions. b. Absorption spectrum of Er(0.5%):YAG at 77K taken with Cary 6000i spectrophotometer (0.05 nm resolution).
.
Both groups originate from the four lowest Stark sublevels of the
^{4}I_{15/2} manifold. The transitions of the first group terminate at four upper Stark sublevels of the
^{4}I_{13/2} manifold and, they remain broad (around 1nm), even at cryogenic temperatures. Recently, a record ~400W laser output at 1645 nm from a cryogenically cooled Er:YAG laser was reported with pumping into the 1452.85 nm absorption line using spectrally-narrowed diode lasers with a bandwidth of approximately 1.2 nm full width at half maximum (FWHM) [
33. S. D. Setzler, M. J. Shaw, M. J. Kukla, J. R. Unternahrer, K. M. Dinndorf, J. A. Beattie, and E. P. Chicklis, “A 400 W cryogenic Er:YAG slab laser at 1645 nm,” Proc. SPIE 7686, 76860C (2010).
]. However, pumping into any of the 14XX nm lines does not offer the lowest QD, defined as
QD=λlas/λp−1. Meanwhile, QD defines the theoretical limit of laser optical-to-optical efficiency – the smaller the QD, the higher the laser efficiency, and the lower the thermal load which can be potentially achieved. The second group of lines, which terminate at the four lowest Stark sublevels of the
^{4}I_{13/2} manifold, enables laser schemes with much lower QD. Unfortunately, at cryogenic temperatures these absorption lines become extremely narrow and require even more advanced pump sources targeting the 15XX nm wavelength range. With continuing technological advances, it has become possible to develop and implement diode pump lasers with the output bandwidth as narrow as 0.2 nm [
55. V. Fromzel, N. Ter-Gabrielyan, M. Dubinskii, G. Venus, I. Divliansky, L. Glebov, O. Mokhun, and V. Smirnov, “Cryo-cooled Er:YAG laser resonantly pumped by a fiber coupled ultra-spectrally-narrowed diode source,” presented at Solid State and Diode Laser Technology Review, 15–18 June 2010, Broomfield, Colorado.
] around 1532 nm, but moving in this direction is very costly and may not be very practical.
Previous attempts to operate Er:YAG lasers in the lowest-QD mode relied heavily on pumping into the 1532 nm absorption band. It has generally been assumed that the strongest
Z_{2}→Y_{1} absorption line centered at ~1532.3 nm was the one mostly responsible for the efficient resonant excitation of Er
^{3+} ions into
^{4}I_{13/2} manifold, while the impact of all other adjacent transitions was minimal. While this assumption is satisfactory for lasers operating at room temperature, experimental evidence suggests that cryogenic operation requires a more detailed analysis of Er:YAG fine spectral features. For example, we recently found that the widths and strengths of absorption lines in the 1520–1550 nm range measured by standard spectrophotometry were not adequately defined due to their extremely narrow bandwidths [
66. N. E. Ter-Gabrielyan, L. D. Merkle, V.Fromzel, J.O.White, J. McElhenny and A. Michael, “Cryogenic Er:YAG lasers: aspects of diode pumping,” Solid State and Diode Laser Technology Review, Technical Digest, pp. 110–113, 15–18 June 2010, Broomfield, Colorado.
]. At 77 K, the absorption cross-sections of these transitions are very strong (~10
^{−18}–10
^{−19} cm
^{2}) and have a bandwidth of only ~0.02–0.05 nm. For comparison, a typical line-narrowed pump diode spectrum has a bandwidth of 0.5–1 nm. Also, with such strong absorption, significant pump saturation should take place. Indeed, for the strongest transition at 1532 nm with an absorption coefficient of ~50 cm
^{−1} at 1% of Er concentration, the pump saturation fluence is only 86 W/cm
^{2}.
In this paper, we present the investigative results of a cryogenic Er:YAG laser utilizing different pumping transitions which can be used for the optimization of the lowest QD operation. In the first step, we carried out a detailed spectroscopic study of an Er:YAG absorption spectrum in the 1520–1550 nm wavelength range, including an investigation of the fine structure of the strongest 1532 nm absorption band. In the second step, we numerically simulated the laser operation. In order to anchor the modeling results, we also carried out laser experiments with a cryogenically-cooled Er:YAG resonantly in-band pumped at 1532-, 1534- and 1546 nm and achieved a laser operation with very high efficiency (~75%).
2. Spectroscopy
The necessity of detailed spectroscopic studies of the Er^{3+} in YAG at cryogenic temperatures was caused by insufficient knowledge of its fine spectral features. We focused our attention on the 1520–1550 nm wavelength range which is the most suitable for pumping aimed at achieving the lowest QD operation.
High resolution absorption spectra were derived from spectral scanning through a thin Er:YAG sample with the collimated output of a tunable narrow-band (~800KHz) diode laser (Santec model TSL-210). The transmitted power was measured with a Germanium detector (New Focus model 2033). The AR-coated samples were mounted on a temperature controlled “cold finger” inside a standard liquid nitrogen cryostat. In order to properly address the dynamic range issue, we used Er:YAG samples with different Er^{3+} concentrations and with different thicknesses depending on the absorption strength of a particular absorption transition. For the strongest absorption line, around 1532 nm, we used 1-mm thick Er:YAG samples with the lowest Er-concentrations of 0.15 and 0.2%. Weaker absorption lines were studied with 2.5 mm thick Er(1%):YAG crystals. All samples were acquired from Scientific Materials, Inc.
The resulting absorption spectrum of Er:YAG at 77 K is shown in
Fig. 2a
Fig. 2 a. Fragment of the ^{4}I_{15/2} → ^{4}I_{13/2} absorption cross section spectrum of Er^{3+}:YAG crystal at 77K taken with the 1 pm spectral resolution; b. Close-up of the Lorentzian fit simulation of the 1532 nm absorption band at 77 K.
. It can be seen that along with the 1532 nm absorption band, which is commonly used for pumping, there are also two absorption transitions centered at ~1534- and ~1546 nm. Their utilization for effective pumping at cryogenic temperatures has not been investigated previously, though pumping into the 1549 nm absorption line at room temperature (the same as 1546 nm line at 77 K) was successfully demonstrated in [
77. K. Spariosu and M. Birnbaum, “Intracavity 1549-um pumped 1634-mm Er:YAG lasers at 300 K,” IEEE J. Quantum Electron. 30(4), 1044–1049 (1994). [CrossRef]
].
The two most intense and well-defined transitions, 1532.3 nm (Z_{2}→Y_{1} major peak) and ~1532.5 nm (Z_{4}→Y_{3} satellite), are displayed in the table in bold. The origin of the other transitions is not clear; they may be associated with minority sites, clusters or vibronics, but their precise identification is beyond the scope of this paper.
Figure 3
Fig. 3 Temperature behavior of the 1532.3 and 1534 nm (inset) absorption bands.
shows the temperature evolution of the 1532 nm absorption band. The most striking behavior was observed for the major 1532.3 nm ultra-narrow absorption peak. It remains quite narrow (~20–40 pm) in the entire 77–100 K temperature range and its maximum cross-section (~7.6∙10
^{−19} cm
^{2}), measured with adequate resolution, is much higher than reported earlier [
66. N. E. Ter-Gabrielyan, L. D. Merkle, V.Fromzel, J.O.White, J. McElhenny and A. Michael, “Cryogenic Er:YAG lasers: aspects of diode pumping,” Solid State and Diode Laser Technology Review, Technical Digest, pp. 110–113, 15–18 June 2010, Broomfield, Colorado.
]. Its temperature dependence is consistent with thermal broadening due to phonon scattering, with a residual (temperature independent) contribution less than 0.02 nm. It is best fit by a Debye temperature of approximately 450 K, reasonably consistent with that reported for another transition in this material by Beghi et al [
88. M. G. Beghi, C. E. Bottani, and V. Russo, “Debye temperature of erbium-doped yttrium aluminum garnet from luminescence and Brillouin scattering data,” J. Appl. Phys. 87(4), 1769–1774 (2000). [CrossRef]
].
The 1534- and the 1546 nm absorption transitions both originate at the
Z_{4} Stark sublevel (79 cm
^{−1}) of the
^{4}I_{15/2} manifold and terminate at the
Y_{2} (6598 cm
^{−1}) and the
Y_{1} (6548 cm
^{−1}) Stark sublevels of the
^{4}I_{13/2} manifold, respectively (see
Fig. 1a). Their intensities and bandwidths are very similar to those of the wide satellite transition at ~1532.5 nm (
Z_{4}→Y_{3}). Therefore, one can expect that the performance of the cryogenically-cooled Er:YAG laser resonantly pumped into the 1532-, 1534- or 1546 nm lines will be very similar. Thetemperature behavior of the 1534 nm absorption line is shown in the inset of
Fig. 3. Its cross-section remains nearly constant within the 77–150K temperature range (~2.1∙10
^{−20} cm
^{2}), while its bandwidth (~180 pm FWHM at 77 K) changes consistently with the theory of thermal broadening of a homogeneously broadened line. The 1546 nm absorption behaves similarly, but has a much lower cross-section (~7∙10
^{−21} cm
^{2}) and a much wider bandwidth (~700 pm).
3. Modeling
Let
N_{2}(λ_{p,}z) be the local population density of the upper energy manifold
^{4}I_{13/2} and
λ_{p} be the pumping wavelength corresponding to the
Z_{i}→
Y_{j} transition between Stark levels of
^{4}I_{13/2} and
^{4}I_{15/2} manifold, see
Fig. 1a. When only
^{4}I_{13/2} and
^{4}I_{15/2} manifolds are involved and the laser transition occurs between the
Y_{2} and the
Z_{5} Stark levels (
λ_{las} = 1618 nm), the local population density in the steady state can be derived as:
Herein:
N_{0} is the total Er
^{3+} concentration;
λ_{p} is the wavelength of the peak of the absorption line;
σ_{p}(λ) is the absorption cross-section such that
σp(λ)=σ(λp)⋅g(λ), where
σ(λ_{p}) is the peak absorption cross-section and
g(λ) is the spectral shape of the absorption line with the bandwidth of
Δλ_{abs} normalized as
∫Δλabsg(λ)⋅dλ=1;
λ_{las} is the laser wavelength;
σ_{las} is the absorption cross-section, its wavelength dependence can be ignored since laser emission has a very narrow spectrum;
τ is the fluorescence lifetime;
f_{Zi}, f_{Yj}, f_{Y2} and
f_{Z5} are the Bolzmann occupation factors of the corresponding Stark levels. The local pump intensity -
I_{p}(λ, z) and the local laser intensity -
I_{las} are defined as:
and
Here:
P_{p}(z) is the pump power along laser axis
z;
G_{p}(λ) is the spectral distribution of the pump power and normalized as
Δλ_{pump} is the bandwidth of the pump spectrum;
d_{p} and
d_{las} are diameters of the pump and laser beams in the crystal;
P_{las} is the laser output power.
For simplicity, we assume that the laser operates at a single wavelength and the transverse distributions of the pump and the laser beams have top hat profiles. A small dependence of
P_{las} on
z inside the laser medium was neglected. Following [
99. Y. Sato and T. Taira, “Saturation factors of pump absorption in solid-state lasers,” IEEE J. Quantum Electron. 40(3), 270–280 (2004). [CrossRef]
], the pump saturated absorption coefficient can be expressed as:
where
ISP(λ)=h⋅cσp(λ)⋅λp⋅τ⋅fZiand
ISlas=h⋅cσlas⋅λlas⋅τ⋅fYk are the pump and the laser saturation intensities;
α_{0}(λ) is the unsaturated pump absorption coefficient:
Pump intensity
I_{p}(λ, z) decreases with pump propagation along axis
z in accordance with the saturated absorption coefficient (
Eq. (4)), and in the absence of lasing (
I_{las} = 0), i.e. before the laser threshold is reached, it can be defined at any point
z from the numerical solution of the transcendental equation:
If
Ip(λ,0)=Ip0(λ) is the incident pump intensity, then by using
Eq. (6) one can calculate the pump intensity at every point
z along the laser axis. According to
Eq. (2), the difference between the calculated values of the pump intensities
ΔIp(λ,z)=Ip(λ,z+dz)−Ip(λ,z) defines the absorbed pump power in every elementary volume
dV=π⋅dp2⋅dz/4. By integrating
ΔI_{p}(λ,z) over the length of the active medium and then over the wavelengths under the absorption contour and converting pump intensity into absorbed power, one can finally calculate the averaged population density
N_{2}(P_{p}) and the total absorbed pump power
P_{abs}(P_{p}):
where
l_{a} is the length of the active medium.
It should be emphasized that, according to definition
(2), both
P_{abs} and
N_{2} in
Eqs. (7) and
(8) are implicit functions of the incident pump power
P_{p}. Then, the laser gain coefficient,
α_{g}(P_{p}), which also varies with
P_{p}, can be expressed by:
The threshold pump power,
P_{th}, is defined as the incident pump
P_{p}, for which the laser gain determined by expression
(9) equals the total laser cavity losses,
α_{loss}:
where
R_{out} and
R_{HR} = 1 are the reflection coefficients of the laser resonator mirrors and
L is the passive round-trip resonator loss.
Further calculations are carried out in two approximation steps. At first, we assumed that the absorption is independent of the intensity of the laser emission
I_{las} even after the pump exceeds the laser threshold. In this approximation the absorption saturation is caused only by the pump interaction with the laser medium and the intracavity laser emission does not affect the absorption. Then, the laser output power
P_{out,1} can be expressed by:
Here
Kabs(Pth)=Pabs(Pth)/Pth is the absorption of the laser medium when pumping reaches the threshold and, in the first order of approximation, the absorption coefficient remains constant after the threshold.
In the second step, we take into account the influence of the laser emission on the absorption coefficient which, according to
Eq. (4), causes partial or full “restoration” of the saturated absorption. With
I_{las} > 0, the saturated absorption coefficient (4) can be expressed as:
where
and
Here,
K_{abs}(P_{p,min}) is the unsaturated absorption of the laser medium, when the incident pump power is very low and thus the saturation can be neglected. One can see that if
I_{las} = 0, the
Eq. (12) reduces to
α_{abs.las} =
α_{abs}. Using
Eq. (12) with
I_{las} ≠ 0, the laser output power can be finally determined by:
For Er:YAG with Er
^{3+} concentration less than 1%, up-conversion losses are low and can be neglected [
1010. J. W. Kim, J. I. Mackenzie, and W. A. Clarkson, “Influence of energy-transfer-upconversion on threshold pump power in quasi-three-level solid-state lasers,” Opt. Express 17(14), 11935–11943 (2009). [CrossRef] [PubMed]
]. For higher Er
^{3+} concentrations, the influence of the up-conversion becomes more noticeable. These losses can be described by introducing an additional term in
Eq. (1) -
w_{up}N_{2}^{2}, where
w_{up} is the up-conversion coefficient. For a continuous wave (CW) Er:YAG laser, the population of the upper
^{4}I_{13/2} manifold -
N_{2} is relatively low (
N_{2} <<
N_{0}) and it remains constant after the laser threshold is reached. Thus,
w_{up}N_{2} can be interpreted as a modification of the decay time the same way as
τ enters the term
N_{2}/τ in
Eq. (1). Let us introduce the effective lifetime of the upper laser level,
τ_{eff}:
Expression
(17) shows that the up-conversion essentially shortens the upper level lifetime; hence, it mainly impacts the laser threshold. The up-conversion coefficient
w_{up} for cryogenically-cooled Er:YAG is unknown. By using a smaller value of
τ than its fluorescence value of
τ ~10 ms [
22. N. Ter-Gabrielyan, M. Dubinskii, G. A. Newburgh, A. Michael, and L. D. Merkle, “Temperature dependence of a diode-pumped cryogenic Er:YAG laser,” Opt. Express 17(9), 7159–7169 (2009). [CrossRef] [PubMed]
], it is possible to estimate the up-conversion parameter. A good fit between experimental and calculated data can serve as a criterion for the accuracy of this approach. In our case all spectroscopic and laser parameters employed in the laser modeling (except
τ_{eff}) were independently measured. Then, by varying the value of
τ_{eff} only, one can find the best fit between the experimental and the calculated laser outputs described by
Eq. (16). The results of the above approach and modeling were validated by the experimental data and will be presented below.
5. Results and discussion
Employing the model described above, we simulated the performance of the cryogenically cooled Er(2%):YAG laser resonantly pumped in several absorption transitions. In the modeling, we used independently measured spectroscopic parameters of the crystal: σ_{p}, σ_{g}, Δλ_{abs}, g(λ), τ, N_{0}, f_{Zi}, f_{Yj}, f_{Z5}, f_{Y2}, parameters of the pump source - G_{p}(λ), Δλ_{pump} and parameters of the laser - l_{a}, R_{out}, R_{HR}, L, d_{p}, d_{las}.
For example, for the 1534 nm pumping, these parameters were taken as:
σ_{p} = 2.2∙10
^{−20} cm
^{2},
σ_{g} = 1.1∙10
^{−20} cm
^{2},
Δλ_{abs} = 180 pm (FWHM),
g(λ) – Lorentzian shape,
τ = 10 ms,
N_{0} = 2.76∙10
^{20} cm
^{−3},
f_{Zi} =
f_{Z4} = 0.103,
f_{Z5} = 0.00018,
f_{Y2} = 0.222,
f_{Yj} =
f_{Y2} = 0.222,
Δλ_{pump} = 300 pm,
G_{p}(λ) – Gaussian shape,
l_{a} = 5- and 10 mm,
R_{out} = 0.85,
L = 0.05,
d_{p} =
d_{las} = 0.47 mm. By varying
τ, we determined that the best fit between the experimental data and modeling occurs when
τ_{eff} = 3.7 ms for 5 mm long and
τ_{eff} = 5.5 ms for 10 mm long laser crystals, respectively. This effective lifetime is much shorter than the fluorescence time
τ ~10 ms measured at 77K directly [
22. N. Ter-Gabrielyan, M. Dubinskii, G. A. Newburgh, A. Michael, and L. D. Merkle, “Temperature dependence of a diode-pumped cryogenic Er:YAG laser,” Opt. Express 17(9), 7159–7169 (2009). [CrossRef] [PubMed]
]. Using
Eq. (17) with the averaged population of the upper laser manifold
N_{2} ~8.5∙10
^{19} cm
^{−3} for the 5 mm long laser crystal and
N_{2} ~4.4∙10
^{19} cm
^{−3} for the 10 mm long one, we estimated
w_{up}~2∙10
^{−18} cm
^{3}/s. This value is close to the indirect estimation of the
w_{up} at 77K made in [
1111. J. O. White, M. Dubinskii, L. D. Merkle, I. Kudryashov, and D. Garbuzov, “Resonant pumping and upconversion in 1.6 um Er^{3+} lasers,” J. Opt. Soc. Am. B 24(9), 2454–2460 (2007). [CrossRef]
] and nearly twice lower than the up-conversion parameter estimated for room temperature (3.5∙10
^{−18} cm
^{3}/s for 1% of Er in YAG [
1212. M. O. Iskandarov, A. A. Nikitichev, and A. I. Stepanov, “Quasi-two-level Er^{3+}:Y_{3}Al_{5}O_{12} laser for 1.6 µm range,” J. Opt. Technol. 68(12), 885–888 (2001). [CrossRef]
]).
The results of the numerical modeling are shown in
Fig. 5 by solid lines. One can see a good fit between the experimental and the calculated data. All calculated curves correspond to the same up-conversion parameter. Thus, in the case of the 1532 nm resonant pumping, the modeling shows that the major contribution to the laser operation comes not from the strongest and extremely narrow peak corresponding to
Z_{1}→Y_{2} line (1532.3 nm), but from the much weaker and broader satellite transition
Z_{4}→Y_{3} centered at 1532.5 nm (see
Fig. 2b, olive bell-curve) forming a long-wavelength shoulder of the 1532 nm absorption band. The predominance of the satellite contribution to the absorbed pump power remains even when bandwidth-narrowed diodes (0.2–0.4 nm) are used. In the case of pumping in the weaker 1534 nm and 1546 nm absorption lines, modeling explains why such pumping is as effective as pumping into the 1532 nm band and provides nearly the same laser efficiency: this is simply because spectroscopic features of 1534- and 1546 nm lines are very similar to those of the satellite transition
Z_{4} →Y_{3} at 1532.5 nm