Abstract

We modeled cascade lasing at 1.6 and 1.7 µm and studied how it affects Q-switching at the 2.8 µm in Er:YLF. We showed that enabling 1.7 µm continuous wave (CW) operation in the pre-Q-switch, energy accumulation stage not only reduces heating of the laser media, but also boosts population inversion of the 4I11/24I13/2 laser transition, and increases output pulse energy at 2.8 µm. A cascade, gain-switched operation at 1.62 µm also reduces heating. However, while it does not affect the Q-switched pulse energy, it provides faster de-population of the 4I13/2 level and enables 2.8 µm Q-switching at a higher repetition rate, i.e. increases its average output power. A first cascade assisted, 2.8 µm Q-switched laser based on Er:YLF was experimentally demonstrated in a dual-cavity setup.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Interest in pulsed 3 µm, diode pumped lasers is driven primarily by material processing, medical and environmental applications, spectroscopy and laser technology in general [1,2].

In erbium-doped materials, the most efficient 3 µm laser operation on 4I11/24I13/2 luminescent transitions, see Fig. 1, is achieved under direct, resonant pumping into the upper-state [3]. The biggest drawback of this simple pump-lase scheme is that the lifetime of the bottom laser level is much longer than that of the upper one, which makes laser action self-terminating and inefficient [4–6]. This limitation can be abated by speeding up the de-population of the bottom laser level and can be achieved with two different methods.

 figure: Fig. 1

Fig. 1 Energy levels in Er:YLF, after [5]. Solid arrows denote absorption and laser transitions. Dashed arrows denote energy transfer processes. Broken, arrows denote heat generating non-radiative decay. Also listed on the right are the lifetimes of the levels involved in numerical simulations.

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The first method makes use of high erbium concentration in laser materials to intensify energy transfer up-conversion (ETU) process which depopulates the lower laser level 4I31/2. The majority of the reported CW and Q-switched lasers used this approach [7–9]. However, high Er-doping brings its own issues: (i) deterioration of thermo-optic properties of the host materials, and (ii) inevitable heating of the laser media. These factors reduce the power scaling potential of this pump-lase scheme, though, several attempts were made to optimize erbium concentration to mitigate these drawbacks and increase laser efficiency [10,11].

The second approach enables 3 µm + 1.6 µm cascade lasing: 4I11/24I13/24I15/2 [12], which de-populates 4I13/2 much faster than ETU and spontaneous decay do. Contrary to the first method, high Er-concentration is detrimental to the 4I13/24I15/2 lasing at 1.6 µm because (i) the stronger ground state absorption (GSA) increases its threshold and (ii) the same ETU now depopulates the upper laser level for the 4I13/24I15/2 transition. Thus, any meaningful implementation of the cascade requires low Er3+ content.

The main advantage of cascade schemes with low Er-concentration is in the considerable reduction in heating of the laser media and preservation of their thermo-optic parameters. The obvious drawbacks are relatively weak pump absorption and more complex laser design.

To the best of our knowledge, the 3 + 1.6 µm cascade lasing scheme in Er-doped materials was first suggested in [10] and was implemented in several free running lasers, including those performed at cryogenic temperatures [13–16].

Even more complex, the 3-wavelength cascade process, involving additional 1.7 µm lasing from the 4S3/2 level was demonstrated in Er-doped fibers [17]. This process can benefit 3 um lasing in the Q-switched mode, by re-populating its upper laser 4I11/2 level.

However, as far as we know, there were no reported studies of the cascade lasing in the Q-switched mode in any Er-doped crystalline laser media. Q-switching with 3 µm + 2.1 µm cascade was demonstrated in Ho-doped ZBLAN fiber laser [18]. The laser output consisted of a combination of 3 µm Q-switched and 2.1 µm gain-switched pulses.

In this work we numerically analyzed the impact of the cascade mechanism on the 3 µm, Q-switched lasing and performed the first experimental demonstration using Er:YLF in a dual-cavity setup. We showed that the 1.7 µm lasing helps to increase energy stored in the upper level for the 3 µm transition. The 1.6 µm cascade lasing, manifests itself as gain-switched lasing, increases the average power of the 3 µm, Q-switched operation, but does not affect its pulse energy. Both 1.6 and 1.7 µm cascade lasing reduce heating of the laser media.

2. Energy transfer processes and modeling of cascade lasing

We performed numerical modeling of the Q-switched, cascade operation using Er:LiYF4 (Er:YLF) as an example of laser gain medium. Among the variety of Er-doped materials used to obtain 1.6, 1.7 and 2.6 – 3 µm lasing, we selected this particular crystal for two reasons. First, it happened to be the most studied laser material in this wavelength range with the majority of material and spectroscopic properties, such as, emission and absorption cross-sections, ETU and CR rates, etc. readily available in the literature [2,5,19–25]. Secondly, its 4I11/2 radiation lifetime, ~4.8 msec [21], is long enough to provide adequate energy storage for Q-switched operation. Figure 1 shows an energy level scheme of Er:YLF from its ground state 4I15/2 (level 0) up to the energy level 4F7/2 (level 6). The energy level 2H11/2 is thermally coupled with the 4S3/2 and in the modeling we considered both as one level (level 5). This figure also shows all the radiative (solid arrows) and non-radiative (dashed and broken arrows) processes that were accounted for in the numerical model.

The most important non-radiative process, affecting both 1.6 and 3 um lasing is ETU1: (4I13/2 + 4I13/2) → (4I15/2 + 4I9/2) with the rate of W11.

The next process is ETU2: (4I11/2 + 4I11/2) (4I15/2 + 4F7/2). The 4F7/2 rapidly decays to the 4S3/2 energy level, so its population can be equaled to zero and attributed to the 4S3/2. The exited state absorption (ESA) at the pump wavelengths λP acts with ETU2 in tandem, de-populating the 4I11/2 level. However, for the λP ~972 nm ESA is weak [7] and may be ignored.

The last important process is cross-relaxation CR1: (4S3/24I9/2) + (4I15/24I13/2). It strongly depends on the Er3+concentration, and for the Er-doping exceeding 10%, it becomes the dominant process of depopulating the 4S3/2 level.

The 4I11/2 population loss of can be partially recovered by recycling the energy intermittently stored in 4S3/2 back via 4S3/24I9/2 lasing at 1.7 µm. Without this important process, only a smaller amount of this intermittently stored energy will reach 4I11/2.

We solved a system of rate equations [5] which describe the kinetics of the primary energy levels of Er3+ as well as photon densities corresponding to three laser transitions in the Er:YLF cascade laser: the 4I11/24I13/2, 4I13/24I15/2 and the 4S3/24I9/2 (at the 2.6 - 2.8, 1.62 and the 1.73 µm correspondingly).

Taking into account that the lifetimes of 4F9/2 and 4I9/2 (microseconds) are much shorter than those of 4S3/2, 4I11/2 and 4I13/2 levels, and both of them quickly feed the 4I11/2, we can consider their populations, N3 and N4 equal to zero. As a result, we reduced the total number of rate equations to just six - three for the population densities Ni (i = 1, 2, 5) and three equations for the photon densities φ2.8, φ1.7 and φ1.6 at 2.8, 1.7 and 1.6 µm wavelengths:

dN1dt=N1τ1+β21N2τ2+(β51τ5+W50N0)N52W11N12+RSE2.8RSE1.6;
dN2dt=R02pumpN2τ22W22N22+(β54τ5+W50N0)N5+W11N12RSE2.8+RSE1.7;
dN5dt=N5τ5W50N0N5+W22N22RSE1.7;
dφ2.8dt=RSE2.8ln(R2.81)+G2.82Lcrcφ2.8+N2τ2β21fG2.8;
dφ1.7dt=RSE1.7ln(R1.71)+G1.72Lcrcφ1.7+N5τ5β54fG1.7;
dφ1.6dt=RSE1.6ln(R1.61)+G1.62Lcrcφ1.6+N1τ1β10fG1.6;
The wavelength dependent rates of stimulated emission RSE are given by Eqs. (7) - (9):
RSE2.8=(b24N2g2g1b17N1)cσ2.8φ2.8;
RSE1.6=(b10N1g1g0b08(NN1N2N5))cσ1.6φ1.6;
RSE1.7=b50N5cσ1.7φ1.7;
The pump rate R02pumpgiven by:

R02pump=PpλphcLcrSmod(1exp(σpN0Lcr));

Here we used the following notations:

  • - N is the total Er3+ ions concentration and N0 = N - N1 - N2 - N3 - N5;
  • - Gλ and Rλ (λ = 2.8, 1.7, 1.6) are the wavelength dependent cumulative round trip cavity loss and the out-coupling mirror reflectivity;
  • - Lcr is the Er:YLF crystal length;
  • - fG is the geometrical factor describing the fraction of spontaneous emission into the laser mode [5];
  • - bij are the Boltzmann population coefficients of the 4I15/2, 4I13/2, 4I11/2 and 4S3/2 energy manifolds (where i denotes a manifold and j – its Stark sublevel);
  • - gi is the Kramer`s degeneracy of the i-th Stark level of Er3+ (gi = 2);
  • - σλ is the emission cross-section for the corresponding laser transition;
  • - Pp is the incident pump power; λp is the pump wavelength, in our case - 972 nm;
  • - h is Plank’s constant and c is the speed of light;
  • - Smod is the laser mode cross-section;
  • - σp is the GSA cross-section at the pump wavelength (6∙10−21 cm2 [22]).

The numerical values of the parameters used in simulations are listed in Table 1.

Tables Icon

Table 1. Selected Er(1%):YLF parameters used in numerical simulations.

The system of Eqs. (1) – (10) was solved using the Bulirsch-Stoer algorithm for stiff ordinary differential equations. In the first step, we suggested that during pumping (regime of energy accumulation at the 4I11/2 level), the laser cavity for the 2.8 µm emission has a high loss and the photon density φ2.8 could be initially set to zero.

However, if a free running laser operation at 1.7 µm and 1.6 µm is allowed by designing a laser cavity with a low loss at these wavelengths, then the photon densities φ1.7 and φ1.6 could grow with pumping.

For the 2.8 µm 4I11/24I13/2 transition, the stored energy E2.8ST is given by Eq. (11) [26]:

E2.8ST=α2.8hcSmodσ2.8λ2.8(1b17b17+b24);
Where α2.8 is a coefficient of amplification:
α2.8=(b24N2g2g1b07N1)σ2.8;
The 2.8 µm Q-switched pulse energy is then given by:
E2.8=E2.8stln(R2.81)1αend/α2.8ln(R2.81)+G2.8;
Where αendis the remaining, post Q-switch coefficient of amplification, which is determined by Eq. (14) [26]:
2α2.8αendLcr(ln(R2.81)+G2.8)=ln(α2.8αend);
The output powers Pout1.6 and Pout1.7are calculated using Eq. (15), with i = 1.6 or 1.7:

Pouti=φihλiSmod1Ri1+Riln(Ri1)ln(Ri1)+Gi;

3. Simulations results

Figure 2 shows how populations of the laser levels: 4S3/2, 4I11/2 and 4I13/2, as well as the coefficient of amplification α2.8, change in the pre-Q-switch, energy accumulation stage, when the cavity loss is high and the 2.8 µm lasing is not possible. Here we can compare two distinctive cases: (i) - when the cavity loss is high for all 3 lasing wavelengths of interest, and (ii) - when the loss is high only for the 2.8 µm lasing, while the loss for 1.62 and 1.73 µm lasing is low.

 figure: Fig. 2

Fig. 2 Population dynamics of the 4I11/2, 4I13/2, and 4S3/2 levels, top row, (in red, blue and green correspondingly). Dynamics of the coefficient of amplification α2.8 is shown in the bottom row. Frames (a) & (b) represent the case of high cavity loss for all 3 wavelengths. Frames (c) & (d) represent the case of high cavity loss only for the 2.8 µm emission, while lasing at the 1.73 and 1.62 µm wavelengths is allowed. Cavity parameters: R1.6 = 0.99, R1.7 = 0.97, G1.6 = G1.7 = 0.033, LCR = 15 mm, Pump = 80W.

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One can see that in the case of high losses for all 3 wavelengths, the coefficient of amplification of the 2.8 µm transition α2.8 reaches its maximum of 0.85 cm−1, with about 1 msec delay after pumping began, long before the 4I11/2 lifetime expires, see Fig. 2(b).

But when 1.73 µm lasing is enabled, then the 2.8 µm amplification reaches its maximum much later, at ~1.9 msec, see Fig. 2(d), and at the 30% higher magnitude of 1.1 cm−1. Thus, enabling 1.73 um lasing leads to larger stored energy in the 4I11/2 level and should result in higher energy per pulse for Q-switched operation at the 2.8 µm. The reason for this effect is that the 1.7 µm lasing on the 4S3/2 - 4I9/2 transition, followed by the fast 4I9/24I11/2 decay, populates the upper laser level 4I11/2 of the 2.8 µm laser much faster than it would be populated by CR1 alone, in the absence of 1.7 µm lasing. The speed of depopulation of the 4I11/2 level, however, remains approximately the same for both cases. Eventually, α2.8 gain saturates and stays almost constant, see Fig. 2(d). Therefore, for the Q-switching at the 2.8 um, it is meaningless to extend pump pulsewidth beyond a certain timeframe, which of course varies with pump power, but typically remains noticeably shorter than the 4I11/2 luminescent lifetime (~4.8 msec). In other words, longer pump pulses do not lead to increase in the output pulse energy, but only contribute to the heating of the laser gain media.

Knowing the coefficient of amplification, we can now calculate the expected Q-switched pulse energy using Eqs. (11) - (14), see the results in Fig. 3. Predictably, the Q-switched pulse energy follows the behavior of the coefficient of amplification.

 figure: Fig. 3

Fig. 3 Simulation of the 2.8 µm pulse energy vs time at different pump powers. (a) 2.8 µm lasing only; (b) 2.8 µm lasing with 1.6 + 1.7 µm cascade.. Cavity parameters: R1.6 = 0.99, R1.7 = 0.97, G1.6 = G1.7 = 0.033, LCR = 15 mm.

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When the 2.8 µm gain reaches its maximum, the Q-switch modulator sharply lowers the 2.8 µm cavity loss and the process of the pulse build up begins. It is described by Eqs. (1) – (6) where initial conditions come from the final set of values N1, N2, N5 and φ2.8, φ1.7 and φ1.6, calculated for the last moment of the pre-Q-switch stage. We assumed that the Q-switch operates in a step function manner and the cavity loss remained low only for 5 µsec.

Figure 4 shows the simulated development of the Q-switched and gain-switched pulses at the 2.8 and 1.6 µm correspondingly. The first pulse instantaneously populates 4I13/2 level and creates conditions for the development of the giant, gain-switched pulse at 1.62 μm. It appears a few microseconds later (16 µsec for the particular 80 W pump power). The gain-switched pulse, in its turn, empties the 4I13/2 much faster than ETU1 does, creating preferable conditions for the next 2.8 μm Q-switch pulse faster as well. As a result, with the 2.8 + 1.6 µm cascade lasing, the 2.8 μm Q-switching can operate at a higher pulse repetition rate.

 figure: Fig. 4

Fig. 4 Simulation of the Q-switched + Gain-switched, 4I11/24I13/24I15/2 cascade lasing at the 2.8 and 1.62 µm wavelengths. Cavity parameters: R1.6 = 0.99, R1.7 = 0.97, R2.8 = 0.8, G1.6 = G1.7 = 0.033, LCR = 15 mm, Pump = 80W.

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4. Experimental setup

Until now, the reported 3 + 1.6 µm and 1.7 + 3 + 1.6 µm cascade lasers used a single-cavity design where all three lasing channels shared the same optical path between cavity mirrors, thus, the spatial overlap was automatically achieved, see Fig. 5(a). Fiber lasers belong to this category too. However, a Q-switched laser with cascade requires spatial separation of the 3 µm laser channel/arm from the 1.6 and 1.7 µm arm outside the gain media. This split-channel or dual-cavity design, where both channels overlap only inside the active media, ensures that the Q-switching in the 3 µm arm does not affect cavity loss in the other channel. In other words, when the loss in the ‘3 µm’ cavity is high, the lasing on the 1.6 and 1.7 µm transitions is still possible. An example of such a split-channel optical layout, where 3 and 1.6 + 1.7 µm arms are separated by an intracavity dichroic mirror is shown in Fig. 5(b). This setup requires tighter alignment to provide spatial overlap between 1.6 + 1.7 and 3 um cavity’s arms, which is not automatic anymore.

 figure: Fig. 5

Fig. 5 Simplified schematics of (a) single-cavity and (b) spilt-channel cascade lasers.

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To validate our numerical model we implemented this new, split-channel cavity layout, which more detailed schematic is depicted in Fig. 6.

 figure: Fig. 6

Fig. 6 Er:YLF cascade laser experimental layout. Spectral overlap of the pump emission and Er:YLF absorption is pictured on the left. Pump spectrum was measured with Yokogawa Optical Spectrum Analyzer 3624 and Er:YLF π-absorption spectrum was collected with Cary 600 spectrometer (resolution 0.5 nm).

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A 15 mm long Er(1%):YLF rectangular slab (3 x 6 mm cross-section) was clamped between two copper, water cooled plates (TWATER = 16 CO). The crystal was cut to have its c-axis normal to the optical axis of the crystal. It was end-pumped by a fiber coupled laser diode module (90 W at the 971 nm). The pump was delivered into the crystal using a dichroically coated mirror DCM1 and a pair of collimating and focusing (f = 150 mm) spherical lenses. Spectral matching of the pump and Er:YLF π-polarized absorption was not ideal, as can be seen in Fig. 6(inset). By placing a power meter behind the cavity high reflector HRLAS (HR at 1.6 + 2.8 um, AR at 971 nm) we directly measured that only half of pump was absorbed in a single pass. An unabsorbed portion of pump, that left the cavity was reflected back by a concave mirror HR2 with the radius of curvature RCC = 300 mm. After accounting for all transmission and reflection losses, we conservatively estimated that the total pump absorption was about 70%.

Another dichroically coated mirror DCM2 splits the cavity into the two channels – for 1.6 + 1.7 and 2.8 µm. Both arms shared the same flat, dichroically coated high reflector HRLAS. Both cavity arms had the same length ~240 mm and the same radii of curvature of their respective output couplers (OCi) – 250 mm. However, the reflectivity of the OC2 in the arm2 was 99% at the 1.62 µm wavelength and 97% for the 1.73 µm, while the reflectivity of the OC1 (in arm1) was 80% at the 2.8 µm. Such low output coupling in the 1.6 µm channel was chosen intentionally in order to minimize the lasing threshold. No attempts were made to optimize OC1 reflectivity neither in the QCW or Q-switched modes of operation

5. Experimental results

The Er:YLF crystal was first pumped by the fiber coupled laser diode module producing 10 msec long pulses with 20% duty cycle. When losses were low for 2.8 and 1.6 + 1.7 µm cavity channels, a free running QCW cascade lasing was observed with expected performance, see Fig. 7. At the 12W QCW pump power, lasing on the 4I11/24I13/2 transitions began with less than a hundred microseconds delay, see Fig. 7(a), indicating that the threshold was low.

 figure: Fig. 7

Fig. 7 Er(1%):YLF cascade laser performance in the free running, QCW operation. (a) – Oscilloscope traces of pump (grey), 2.8 µm (red) and 1.62 µm lasing (blue). (b) – Wavelength resolved cascade laser output vs the absorbed pump. Inset shows output spectrum consisting of the 2.66, 2.72 and 2.81 µm lines. ΔtPUMP = 10 msec, PRF = 20 Hz.

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After about 1.75 milliseconds, cascade lasing was observed on the 4I13/24I15/2 transition at 1.62 µm. Its boosting impact on 2.8 um was clearly visible in Fig. 7. Similar laser behavior was first reported in [12].

The relatively low slope efficiency of the 2.8 µm lasing could be explained by the high transmission loss introduced by DCM2 (~8% per pass) and un-optimized output coupling provided by OC1.

The spectrum of the QCW laser output consisted of 3 laser lines - 2.66, 2.72 and 2.81 µm, see the inset in Fig. 7(b), as was observed in [27]. A study of the time resolved spectral behavior was outside the scope of this paper.

In the free running mode we did not observe any lasing at 1.73 µm. Indeed, its upper laser level 4S3/2 can be populated only by the ETU2 process, which depends on the population of the 4I11/2. CW lasing at the 2.8 µm keeps the population of the 4I11/2 at the (low) threshold level, denying the “supply” to 4S3/2 and keeping its population below threshold for 1.72 µm lasing.

In the pre-Q-switched stage, a high loss in the 2.8 µm arm prevents lasing on the 4I11/24I13/2 transition. As a result, the population of the 4I11/2 is not “clamped” any more and it keeps increasing with pump until either Q-switching occurs or it reaches some stationary value. As the population of the 4I11/2 grows, so does the population of the 4S3/2 until it reaches a threshold value for lasing on the 4S3/24I9/2 transition at 1.73 µm.

As a Q-switch device, we used a simple mechanical modulator - a fast rotating disk with a slit, similar to that described in [28]. When it was inserted into the 2.8 µm arm of the cavity, we observed the following behavior of the Q-switched cascade laser, see Fig. 8. The 1.73 µm lasing began with about 450 µsec delay and lasted until the Q-switching at 2.8 µm interrupted it sharply. Within 10 to 20 µsec after the appearance of the 2.8 µm pulse, a gain-switched pulse at the 1.62 µm followed.

 figure: Fig. 8

Fig. 8 Er(1%):YLF cascade laser performance (1.6 + 1.7 + 2.8 µm) in the Q-switched mode. Shown are the oscilloscope traces of the pump (grey), 2.8 µm (red), 1.73 µm (green) and 1.62 µm lasing (blue). Pump pulsewidth ΔtPUMP = 1.25 msec, PRF = 160 Hz. Inset shows 1.62 and 1.73 µm laser spectral lines.

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As we expected, fast de-population of the 4I11/2 by Q-switching instantly “chokes” the ETU2 process and ends the 1.73 um lasing. The “giant” 1.6 µm pulse depletes the lower level for the 4I11/24I13/2 laser transition at the 2.8 µm, restoring the above threshold condition faster for the next Q-switching event.

When the 1.6 + 1.7 µm arm of the cavity was blocked, preventing the cascade lasing, the energy per pulse at the 2.81 µm dropped by about 20%.

A significant mechanical jitter of the Q-switch modulator prevented us from making precise measurements of output power. However the goal of this work was to demonstrate a concept of the cascade-assisted Q-switch lasing in Er-doped crystalline material. The existing setup could be considerably improved by using a more stable and much faster Electro- or Acousto-optical Q-switching in lieu of an unstable mechanical one. Output coupler should be optimized and dichroic coatings should be improved as well.

One of the most important factors influencing cascade laser is Erbium doping concentration, which should be a subject of the future performance optimization.

6. Conclusions

We modeled a Q-switched operation at the 2.8 µm, assisted by cascade lasing at the 1.6 and 1.7 µm, in Er(1%):YLF. We showed that enabling CW lasing at the 1.7 µm, in the pre-Q-switch energy accumulation stage, not only reduces heating of the laser media, but also boosts the inversed population of the 4I11/24I13/2 laser transition, and increases output energy of the 2.8 µm pulses. A gain-switched, cascade operation at the 1.62 µm also reduces heating. However, while it does not affect the Q-switched pulse energy, it provides faster de-population of the 4I13/2 level and enables the 2.8 µm Q-switching at a higher repetition rate, i.e. increases its average output power. The first 2.8 µm Q-switched laser with cascade, based on Er-doped crystalline material, was experimentally demonstrated in a dual-cavity setup. We found that cascade operation in Er(1%):YLF increased energy per pulse by 20-25%.

Acknowledgment

The authors wish to thank Dr. Ei Brown and Dr. Zachery Fleishman for helping collecting luminescent spectra of Er:YLF.

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17. M. Pollnau, C. Ghisler, W. Lüthy, H. P. Weber, J. Schneider, and U. B. Unrau, “Three-transition cascade erbium laser at 1.7, 2.7, and 1.6 microm,” Opt. Lett. 22(9), 612–614 (1997). [CrossRef]   [PubMed]  

18. J. Li, T. Hu, and S. D. Jackson, “Q-switched induced gain switching of a two-transition cascade laser,” Opt. Express 20(12), 13123–13128 (2012). [CrossRef]   [PubMed]  

19. C. Li, Y. Guyot, C. Linares, R. Moncorge, and M. F. Joubert, “Radiative transition probabilities of trivalent rare-earth ions in LiYF4,” in “Advanced Solid State Lasers and Compact Blue-Green Lasers” OSA Technical Digest 2, 423–425 (1993).

20. V. Lupei, S. Georgescu, and V. Florea, “On the dynamics of population inversion for 3 µm Er3+ lasers,” IEEE J. Quantum Electron. 29(2), 426–434 (1993). [CrossRef]  

21. H. Chou and H. Jenssen, “Upconversion processes in Er-activated solid state laser materials in Advanced Solid State Lasers, M. Shand and H. Jenssen, eds., Vol. 5 of OSA Proceedings Series (Optical Society of America, 1989), paper DD5.

22. M. Pollnau, W. Lüthy, H. P. Weber, K. Krämer, H. U. Güdel, and R. A. McFarlane, “Excited-state absorption in Er:BaY2F8 and Cs3Er2Br9 and comparison with Er:LiYF4,” Appl. Phys. B 62(4), 339–344 (1996). [CrossRef]  

23. M. Pollnau, R. Spring, S. Wittwer, W. Lüthy, and H. P. Weber, “Investigation on the slope efficiency of a pulsed 2.8-µm Er:LiYF laser,” Opt. Soc. Am. B 14(4), 974–978 (1997). [CrossRef]  

24. C. Labbe, J.-L. Doualan, S. Girard, R. Moncorge, and M. Thuau, “Absolute excited state absorption cross section measurements in Er3+:LiYF4 for laser applications around 2.8 mm and 551 nm,” J. Phys. Condens. Matter 12(30), 6943–6957 (2000). [CrossRef]  

25. S. Hubert, D. Meichenin, B. Zhou, and F. Auzel, “Emission properties, oscillator strengths and laser parameters of Er3+ in LiYF4 at 2.7 mm,” J. Lumin. 50(1), 7–15 (1991). [CrossRef]  

26. A. A. Mak, Yu. A. Anan’ev, and B. A. Ermakov, “Solid State Lasers,” Sov. Phys. Usp. 10(4), 419–452 (1968). [CrossRef]  

27. M. V. Inochkin, V. V. Nazarov, D. Yu. Sachkov, L. V. Khloponin, and V. Yu. Khramov, “Dynamics of the lasing spectrum of a 3-µm Er:YLF laser with semiconductor pumping,” J. Opt. Technol. 76(11), 720–724 (2009). [CrossRef]  

28. X. Ren, Y. Wang, J. Zhang, D. Tang, and D. Shen, “Short-Pulse-Width Repetitively Q-Switched ~2.7-µm Er:Y2O3 Ceramic Laser,” Appl. Sci. (Basel) 7(11), 1201–1207 (2017). [CrossRef]  

References

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  1. C. W. Rudy, “Mid-IR Lasers: Power and pulse capability ramp up for mid-IR lasers,” Laser Focus World 50(5), 63–66 (2014).
  2. N. U. Wetter, A. M. Deana, I. M. Ranieri, L. Gomes, and S. L. Baldochi, “Influence of excited-state-energy upconversion on pulse shape in quasi-continuous-wave diode-pumped Er:YLF4 lasers,” IEEE J. Quantum Electron. 46(1), 99–104 (2010).
    [Crossref]
  3. R. C. Stoneman, J. G. Lynn, and L. Esterowitz, “Direct upper-state pumping of the 2.8 µm Er3+:YLF laser,” IEEE J. Quantum Electron. 28(4), 1041–1045 (1992).
    [Crossref]
  4. G. J. Kintz, R. Allen, and L. Esterowitz, “cw and pulsed 2.8 µm laser emission from diode pumped Er3+:LiYF4 at room temperature,” Appl. Phys. Lett. 50(22), 1553–1555 (1987).
    [Crossref]
  5. M. Pollnau, T. Graf, J. E. Balmer, W. Lüthy, and H. P. Weber, “Explanation of the cw operation of the Er3+ 3- microm crystal laser,” Phys. Rev. A 49(5), 3990–3996 (1994).
    [Crossref] [PubMed]
  6. V. Lupei and S. Georgesku, “Erbium 3-µm laser as an upconversion system,” Opt. Eng. 35(5), 1265–1272 (1996).
    [Crossref]
  7. E. V. Zharikov, V. I. Zhekov, I. A. Kulevskii, T. M. Murina, V. V. Osiko, A. M. Prokhorov, A. D. Savelev, V. V. Smirnov, B. P. Starikov, and M. A. Timoshechkin, “Stimulated emission from Er3+ ions in yttrium aluminum garnet crystals at λ=2.94 µm,” Sov. J. Quantum Electron. 4(8), 1039–1040 (1975).
    [Crossref]
  8. M. Messner, A. Heinrich, and K. Unterrainer, “High-energy diode side-pumped Er:LiYF4 laser,” Appl. Opt. 57(6), 1497–1503 (2018).
    [Crossref] [PubMed]
  9. M. Messner, A. Henrich, C. Hagen, and K. Unterrainer, “Acousto-optically Q-switched diode side-pumped Er:YLF laser generating 50-kW peak power in 70-ns pulses,” Proceedings SPIE 10896, Solid State Lasers XXVIII: Technology and Devices; 1089607 (2019).
  10. T. Jensen, A. Diening, G. Huber, and B. H. T. Chai, “Investigation of diode-pumped 2.8-µm Er:LiYF4 lasers with various doping levels,” Opt. Lett. 21(8), 582–584 (1996).
    [Crossref] [PubMed]
  11. H. Uehara, S. Tokita, J. Kawanaka, D. Konishi, M. Murakami, S. Shimizu, and R. Yasuhara, “Optimization of laser emission at 2.8 μm by Er:Lu2O3 ceramics,” Opt. Express 26(3), 3497–3507 (2018).
    [Crossref] [PubMed]
  12. B. Schmaul, G. Huber, R. Clausen, C. Chai, P. LiKamWa, and M. Bass, “Er3+:YLiF4 continuous wave cascade laser operation at 1620 and 2810 nm at room temperature,” Appl. Phys. Lett. 62(6), 541–543 (1993).
    [Crossref]
  13. J. Schneider, “Mid-infrared fluoride fiber lasers in multiple cascade operation,” IEEE Photonics Technol. Lett. 7(4), 354–356 (1995).
    [Crossref]
  14. S. Jackson, M. Pollnau, and J. Li, “Diode-pumped Erbium cascade fiber lasers,” IEEE J. Quantum Electron. 47(4), 471–478 (2011).
    [Crossref]
  15. T. Sanamyan, “Diode-pumped cascade Er:Y2O3 laser,” Laser Phys. Lett. 12(12), 125804 (2015).
    [Crossref]
  16. T. Sanamyan, “Efficient, cryogenic mid-IR and eye-safe Er:YAG laser,” J. Opt. Soc. Am. B 33(11), D1–D6 (2016).
    [Crossref]
  17. M. Pollnau, C. Ghisler, W. Lüthy, H. P. Weber, J. Schneider, and U. B. Unrau, “Three-transition cascade erbium laser at 1.7, 2.7, and 1.6 microm,” Opt. Lett. 22(9), 612–614 (1997).
    [Crossref] [PubMed]
  18. J. Li, T. Hu, and S. D. Jackson, “Q-switched induced gain switching of a two-transition cascade laser,” Opt. Express 20(12), 13123–13128 (2012).
    [Crossref] [PubMed]
  19. C. Li, Y. Guyot, C. Linares, R. Moncorge, and M. F. Joubert, “Radiative transition probabilities of trivalent rare-earth ions in LiYF4,” in “Advanced Solid State Lasers and Compact Blue-Green Lasers” OSA Technical Digest 2, 423–425 (1993).
  20. V. Lupei, S. Georgescu, and V. Florea, “On the dynamics of population inversion for 3 µm Er3+ lasers,” IEEE J. Quantum Electron. 29(2), 426–434 (1993).
    [Crossref]
  21. H. Chou and H. Jenssen, “Upconversion processes in Er-activated solid state laser materials in Advanced Solid State Lasers, M. Shand and H. Jenssen, eds., Vol. 5 of OSA Proceedings Series (Optical Society of America, 1989), paper DD5.
  22. M. Pollnau, W. Lüthy, H. P. Weber, K. Krämer, H. U. Güdel, and R. A. McFarlane, “Excited-state absorption in Er:BaY2F8 and Cs3Er2Br9 and comparison with Er:LiYF4,” Appl. Phys. B 62(4), 339–344 (1996).
    [Crossref]
  23. M. Pollnau, R. Spring, S. Wittwer, W. Lüthy, and H. P. Weber, “Investigation on the slope efficiency of a pulsed 2.8-µm Er:LiYF laser,” Opt. Soc. Am. B 14(4), 974–978 (1997).
    [Crossref]
  24. C. Labbe, J.-L. Doualan, S. Girard, R. Moncorge, and M. Thuau, “Absolute excited state absorption cross section measurements in Er3+:LiYF4 for laser applications around 2.8 mm and 551 nm,” J. Phys. Condens. Matter 12(30), 6943–6957 (2000).
    [Crossref]
  25. S. Hubert, D. Meichenin, B. Zhou, and F. Auzel, “Emission properties, oscillator strengths and laser parameters of Er3+ in LiYF4 at 2.7 mm,” J. Lumin. 50(1), 7–15 (1991).
    [Crossref]
  26. A. A. Mak, Yu. A. Anan’ev, and B. A. Ermakov, “Solid State Lasers,” Sov. Phys. Usp. 10(4), 419–452 (1968).
    [Crossref]
  27. M. V. Inochkin, V. V. Nazarov, D. Yu. Sachkov, L. V. Khloponin, and V. Yu. Khramov, “Dynamics of the lasing spectrum of a 3-µm Er:YLF laser with semiconductor pumping,” J. Opt. Technol. 76(11), 720–724 (2009).
    [Crossref]
  28. X. Ren, Y. Wang, J. Zhang, D. Tang, and D. Shen, “Short-Pulse-Width Repetitively Q-Switched ~2.7-µm Er:Y2O3 Ceramic Laser,” Appl. Sci. (Basel) 7(11), 1201–1207 (2017).
    [Crossref]

2018 (2)

2017 (1)

X. Ren, Y. Wang, J. Zhang, D. Tang, and D. Shen, “Short-Pulse-Width Repetitively Q-Switched ~2.7-µm Er:Y2O3 Ceramic Laser,” Appl. Sci. (Basel) 7(11), 1201–1207 (2017).
[Crossref]

2016 (1)

2015 (1)

T. Sanamyan, “Diode-pumped cascade Er:Y2O3 laser,” Laser Phys. Lett. 12(12), 125804 (2015).
[Crossref]

2014 (1)

C. W. Rudy, “Mid-IR Lasers: Power and pulse capability ramp up for mid-IR lasers,” Laser Focus World 50(5), 63–66 (2014).

2012 (1)

2011 (1)

S. Jackson, M. Pollnau, and J. Li, “Diode-pumped Erbium cascade fiber lasers,” IEEE J. Quantum Electron. 47(4), 471–478 (2011).
[Crossref]

2010 (1)

N. U. Wetter, A. M. Deana, I. M. Ranieri, L. Gomes, and S. L. Baldochi, “Influence of excited-state-energy upconversion on pulse shape in quasi-continuous-wave diode-pumped Er:YLF4 lasers,” IEEE J. Quantum Electron. 46(1), 99–104 (2010).
[Crossref]

2009 (1)

2000 (1)

C. Labbe, J.-L. Doualan, S. Girard, R. Moncorge, and M. Thuau, “Absolute excited state absorption cross section measurements in Er3+:LiYF4 for laser applications around 2.8 mm and 551 nm,” J. Phys. Condens. Matter 12(30), 6943–6957 (2000).
[Crossref]

1997 (2)

M. Pollnau, R. Spring, S. Wittwer, W. Lüthy, and H. P. Weber, “Investigation on the slope efficiency of a pulsed 2.8-µm Er:LiYF laser,” Opt. Soc. Am. B 14(4), 974–978 (1997).
[Crossref]

M. Pollnau, C. Ghisler, W. Lüthy, H. P. Weber, J. Schneider, and U. B. Unrau, “Three-transition cascade erbium laser at 1.7, 2.7, and 1.6 microm,” Opt. Lett. 22(9), 612–614 (1997).
[Crossref] [PubMed]

1996 (3)

T. Jensen, A. Diening, G. Huber, and B. H. T. Chai, “Investigation of diode-pumped 2.8-µm Er:LiYF4 lasers with various doping levels,” Opt. Lett. 21(8), 582–584 (1996).
[Crossref] [PubMed]

V. Lupei and S. Georgesku, “Erbium 3-µm laser as an upconversion system,” Opt. Eng. 35(5), 1265–1272 (1996).
[Crossref]

M. Pollnau, W. Lüthy, H. P. Weber, K. Krämer, H. U. Güdel, and R. A. McFarlane, “Excited-state absorption in Er:BaY2F8 and Cs3Er2Br9 and comparison with Er:LiYF4,” Appl. Phys. B 62(4), 339–344 (1996).
[Crossref]

1995 (1)

J. Schneider, “Mid-infrared fluoride fiber lasers in multiple cascade operation,” IEEE Photonics Technol. Lett. 7(4), 354–356 (1995).
[Crossref]

1994 (1)

M. Pollnau, T. Graf, J. E. Balmer, W. Lüthy, and H. P. Weber, “Explanation of the cw operation of the Er3+ 3- microm crystal laser,” Phys. Rev. A 49(5), 3990–3996 (1994).
[Crossref] [PubMed]

1993 (2)

B. Schmaul, G. Huber, R. Clausen, C. Chai, P. LiKamWa, and M. Bass, “Er3+:YLiF4 continuous wave cascade laser operation at 1620 and 2810 nm at room temperature,” Appl. Phys. Lett. 62(6), 541–543 (1993).
[Crossref]

V. Lupei, S. Georgescu, and V. Florea, “On the dynamics of population inversion for 3 µm Er3+ lasers,” IEEE J. Quantum Electron. 29(2), 426–434 (1993).
[Crossref]

1992 (1)

R. C. Stoneman, J. G. Lynn, and L. Esterowitz, “Direct upper-state pumping of the 2.8 µm Er3+:YLF laser,” IEEE J. Quantum Electron. 28(4), 1041–1045 (1992).
[Crossref]

1991 (1)

S. Hubert, D. Meichenin, B. Zhou, and F. Auzel, “Emission properties, oscillator strengths and laser parameters of Er3+ in LiYF4 at 2.7 mm,” J. Lumin. 50(1), 7–15 (1991).
[Crossref]

1987 (1)

G. J. Kintz, R. Allen, and L. Esterowitz, “cw and pulsed 2.8 µm laser emission from diode pumped Er3+:LiYF4 at room temperature,” Appl. Phys. Lett. 50(22), 1553–1555 (1987).
[Crossref]

1975 (1)

E. V. Zharikov, V. I. Zhekov, I. A. Kulevskii, T. M. Murina, V. V. Osiko, A. M. Prokhorov, A. D. Savelev, V. V. Smirnov, B. P. Starikov, and M. A. Timoshechkin, “Stimulated emission from Er3+ ions in yttrium aluminum garnet crystals at λ=2.94 µm,” Sov. J. Quantum Electron. 4(8), 1039–1040 (1975).
[Crossref]

1968 (1)

A. A. Mak, Yu. A. Anan’ev, and B. A. Ermakov, “Solid State Lasers,” Sov. Phys. Usp. 10(4), 419–452 (1968).
[Crossref]

Allen, R.

G. J. Kintz, R. Allen, and L. Esterowitz, “cw and pulsed 2.8 µm laser emission from diode pumped Er3+:LiYF4 at room temperature,” Appl. Phys. Lett. 50(22), 1553–1555 (1987).
[Crossref]

Anan’ev, Yu. A.

A. A. Mak, Yu. A. Anan’ev, and B. A. Ermakov, “Solid State Lasers,” Sov. Phys. Usp. 10(4), 419–452 (1968).
[Crossref]

Auzel, F.

S. Hubert, D. Meichenin, B. Zhou, and F. Auzel, “Emission properties, oscillator strengths and laser parameters of Er3+ in LiYF4 at 2.7 mm,” J. Lumin. 50(1), 7–15 (1991).
[Crossref]

Baldochi, S. L.

N. U. Wetter, A. M. Deana, I. M. Ranieri, L. Gomes, and S. L. Baldochi, “Influence of excited-state-energy upconversion on pulse shape in quasi-continuous-wave diode-pumped Er:YLF4 lasers,” IEEE J. Quantum Electron. 46(1), 99–104 (2010).
[Crossref]

Balmer, J. E.

M. Pollnau, T. Graf, J. E. Balmer, W. Lüthy, and H. P. Weber, “Explanation of the cw operation of the Er3+ 3- microm crystal laser,” Phys. Rev. A 49(5), 3990–3996 (1994).
[Crossref] [PubMed]

Bass, M.

B. Schmaul, G. Huber, R. Clausen, C. Chai, P. LiKamWa, and M. Bass, “Er3+:YLiF4 continuous wave cascade laser operation at 1620 and 2810 nm at room temperature,” Appl. Phys. Lett. 62(6), 541–543 (1993).
[Crossref]

Chai, B. H. T.

Chai, C.

B. Schmaul, G. Huber, R. Clausen, C. Chai, P. LiKamWa, and M. Bass, “Er3+:YLiF4 continuous wave cascade laser operation at 1620 and 2810 nm at room temperature,” Appl. Phys. Lett. 62(6), 541–543 (1993).
[Crossref]

Clausen, R.

B. Schmaul, G. Huber, R. Clausen, C. Chai, P. LiKamWa, and M. Bass, “Er3+:YLiF4 continuous wave cascade laser operation at 1620 and 2810 nm at room temperature,” Appl. Phys. Lett. 62(6), 541–543 (1993).
[Crossref]

Deana, A. M.

N. U. Wetter, A. M. Deana, I. M. Ranieri, L. Gomes, and S. L. Baldochi, “Influence of excited-state-energy upconversion on pulse shape in quasi-continuous-wave diode-pumped Er:YLF4 lasers,” IEEE J. Quantum Electron. 46(1), 99–104 (2010).
[Crossref]

Diening, A.

Doualan, J.-L.

C. Labbe, J.-L. Doualan, S. Girard, R. Moncorge, and M. Thuau, “Absolute excited state absorption cross section measurements in Er3+:LiYF4 for laser applications around 2.8 mm and 551 nm,” J. Phys. Condens. Matter 12(30), 6943–6957 (2000).
[Crossref]

Ermakov, B. A.

A. A. Mak, Yu. A. Anan’ev, and B. A. Ermakov, “Solid State Lasers,” Sov. Phys. Usp. 10(4), 419–452 (1968).
[Crossref]

Esterowitz, L.

R. C. Stoneman, J. G. Lynn, and L. Esterowitz, “Direct upper-state pumping of the 2.8 µm Er3+:YLF laser,” IEEE J. Quantum Electron. 28(4), 1041–1045 (1992).
[Crossref]

G. J. Kintz, R. Allen, and L. Esterowitz, “cw and pulsed 2.8 µm laser emission from diode pumped Er3+:LiYF4 at room temperature,” Appl. Phys. Lett. 50(22), 1553–1555 (1987).
[Crossref]

Florea, V.

V. Lupei, S. Georgescu, and V. Florea, “On the dynamics of population inversion for 3 µm Er3+ lasers,” IEEE J. Quantum Electron. 29(2), 426–434 (1993).
[Crossref]

Georgescu, S.

V. Lupei, S. Georgescu, and V. Florea, “On the dynamics of population inversion for 3 µm Er3+ lasers,” IEEE J. Quantum Electron. 29(2), 426–434 (1993).
[Crossref]

Georgesku, S.

V. Lupei and S. Georgesku, “Erbium 3-µm laser as an upconversion system,” Opt. Eng. 35(5), 1265–1272 (1996).
[Crossref]

Ghisler, C.

Girard, S.

C. Labbe, J.-L. Doualan, S. Girard, R. Moncorge, and M. Thuau, “Absolute excited state absorption cross section measurements in Er3+:LiYF4 for laser applications around 2.8 mm and 551 nm,” J. Phys. Condens. Matter 12(30), 6943–6957 (2000).
[Crossref]

Gomes, L.

N. U. Wetter, A. M. Deana, I. M. Ranieri, L. Gomes, and S. L. Baldochi, “Influence of excited-state-energy upconversion on pulse shape in quasi-continuous-wave diode-pumped Er:YLF4 lasers,” IEEE J. Quantum Electron. 46(1), 99–104 (2010).
[Crossref]

Graf, T.

M. Pollnau, T. Graf, J. E. Balmer, W. Lüthy, and H. P. Weber, “Explanation of the cw operation of the Er3+ 3- microm crystal laser,” Phys. Rev. A 49(5), 3990–3996 (1994).
[Crossref] [PubMed]

Güdel, H. U.

M. Pollnau, W. Lüthy, H. P. Weber, K. Krämer, H. U. Güdel, and R. A. McFarlane, “Excited-state absorption in Er:BaY2F8 and Cs3Er2Br9 and comparison with Er:LiYF4,” Appl. Phys. B 62(4), 339–344 (1996).
[Crossref]

Heinrich, A.

Hu, T.

Huber, G.

T. Jensen, A. Diening, G. Huber, and B. H. T. Chai, “Investigation of diode-pumped 2.8-µm Er:LiYF4 lasers with various doping levels,” Opt. Lett. 21(8), 582–584 (1996).
[Crossref] [PubMed]

B. Schmaul, G. Huber, R. Clausen, C. Chai, P. LiKamWa, and M. Bass, “Er3+:YLiF4 continuous wave cascade laser operation at 1620 and 2810 nm at room temperature,” Appl. Phys. Lett. 62(6), 541–543 (1993).
[Crossref]

Hubert, S.

S. Hubert, D. Meichenin, B. Zhou, and F. Auzel, “Emission properties, oscillator strengths and laser parameters of Er3+ in LiYF4 at 2.7 mm,” J. Lumin. 50(1), 7–15 (1991).
[Crossref]

Inochkin, M. V.

Jackson, S.

S. Jackson, M. Pollnau, and J. Li, “Diode-pumped Erbium cascade fiber lasers,” IEEE J. Quantum Electron. 47(4), 471–478 (2011).
[Crossref]

Jackson, S. D.

Jensen, T.

Kawanaka, J.

Khloponin, L. V.

Khramov, V. Yu.

Kintz, G. J.

G. J. Kintz, R. Allen, and L. Esterowitz, “cw and pulsed 2.8 µm laser emission from diode pumped Er3+:LiYF4 at room temperature,” Appl. Phys. Lett. 50(22), 1553–1555 (1987).
[Crossref]

Konishi, D.

Krämer, K.

M. Pollnau, W. Lüthy, H. P. Weber, K. Krämer, H. U. Güdel, and R. A. McFarlane, “Excited-state absorption in Er:BaY2F8 and Cs3Er2Br9 and comparison with Er:LiYF4,” Appl. Phys. B 62(4), 339–344 (1996).
[Crossref]

Kulevskii, I. A.

E. V. Zharikov, V. I. Zhekov, I. A. Kulevskii, T. M. Murina, V. V. Osiko, A. M. Prokhorov, A. D. Savelev, V. V. Smirnov, B. P. Starikov, and M. A. Timoshechkin, “Stimulated emission from Er3+ ions in yttrium aluminum garnet crystals at λ=2.94 µm,” Sov. J. Quantum Electron. 4(8), 1039–1040 (1975).
[Crossref]

Labbe, C.

C. Labbe, J.-L. Doualan, S. Girard, R. Moncorge, and M. Thuau, “Absolute excited state absorption cross section measurements in Er3+:LiYF4 for laser applications around 2.8 mm and 551 nm,” J. Phys. Condens. Matter 12(30), 6943–6957 (2000).
[Crossref]

Li, J.

J. Li, T. Hu, and S. D. Jackson, “Q-switched induced gain switching of a two-transition cascade laser,” Opt. Express 20(12), 13123–13128 (2012).
[Crossref] [PubMed]

S. Jackson, M. Pollnau, and J. Li, “Diode-pumped Erbium cascade fiber lasers,” IEEE J. Quantum Electron. 47(4), 471–478 (2011).
[Crossref]

LiKamWa, P.

B. Schmaul, G. Huber, R. Clausen, C. Chai, P. LiKamWa, and M. Bass, “Er3+:YLiF4 continuous wave cascade laser operation at 1620 and 2810 nm at room temperature,” Appl. Phys. Lett. 62(6), 541–543 (1993).
[Crossref]

Lupei, V.

V. Lupei and S. Georgesku, “Erbium 3-µm laser as an upconversion system,” Opt. Eng. 35(5), 1265–1272 (1996).
[Crossref]

V. Lupei, S. Georgescu, and V. Florea, “On the dynamics of population inversion for 3 µm Er3+ lasers,” IEEE J. Quantum Electron. 29(2), 426–434 (1993).
[Crossref]

Lüthy, W.

M. Pollnau, C. Ghisler, W. Lüthy, H. P. Weber, J. Schneider, and U. B. Unrau, “Three-transition cascade erbium laser at 1.7, 2.7, and 1.6 microm,” Opt. Lett. 22(9), 612–614 (1997).
[Crossref] [PubMed]

M. Pollnau, R. Spring, S. Wittwer, W. Lüthy, and H. P. Weber, “Investigation on the slope efficiency of a pulsed 2.8-µm Er:LiYF laser,” Opt. Soc. Am. B 14(4), 974–978 (1997).
[Crossref]

M. Pollnau, W. Lüthy, H. P. Weber, K. Krämer, H. U. Güdel, and R. A. McFarlane, “Excited-state absorption in Er:BaY2F8 and Cs3Er2Br9 and comparison with Er:LiYF4,” Appl. Phys. B 62(4), 339–344 (1996).
[Crossref]

M. Pollnau, T. Graf, J. E. Balmer, W. Lüthy, and H. P. Weber, “Explanation of the cw operation of the Er3+ 3- microm crystal laser,” Phys. Rev. A 49(5), 3990–3996 (1994).
[Crossref] [PubMed]

Lynn, J. G.

R. C. Stoneman, J. G. Lynn, and L. Esterowitz, “Direct upper-state pumping of the 2.8 µm Er3+:YLF laser,” IEEE J. Quantum Electron. 28(4), 1041–1045 (1992).
[Crossref]

Mak, A. A.

A. A. Mak, Yu. A. Anan’ev, and B. A. Ermakov, “Solid State Lasers,” Sov. Phys. Usp. 10(4), 419–452 (1968).
[Crossref]

McFarlane, R. A.

M. Pollnau, W. Lüthy, H. P. Weber, K. Krämer, H. U. Güdel, and R. A. McFarlane, “Excited-state absorption in Er:BaY2F8 and Cs3Er2Br9 and comparison with Er:LiYF4,” Appl. Phys. B 62(4), 339–344 (1996).
[Crossref]

Meichenin, D.

S. Hubert, D. Meichenin, B. Zhou, and F. Auzel, “Emission properties, oscillator strengths and laser parameters of Er3+ in LiYF4 at 2.7 mm,” J. Lumin. 50(1), 7–15 (1991).
[Crossref]

Messner, M.

Moncorge, R.

C. Labbe, J.-L. Doualan, S. Girard, R. Moncorge, and M. Thuau, “Absolute excited state absorption cross section measurements in Er3+:LiYF4 for laser applications around 2.8 mm and 551 nm,” J. Phys. Condens. Matter 12(30), 6943–6957 (2000).
[Crossref]

Murakami, M.

Murina, T. M.

E. V. Zharikov, V. I. Zhekov, I. A. Kulevskii, T. M. Murina, V. V. Osiko, A. M. Prokhorov, A. D. Savelev, V. V. Smirnov, B. P. Starikov, and M. A. Timoshechkin, “Stimulated emission from Er3+ ions in yttrium aluminum garnet crystals at λ=2.94 µm,” Sov. J. Quantum Electron. 4(8), 1039–1040 (1975).
[Crossref]

Nazarov, V. V.

Osiko, V. V.

E. V. Zharikov, V. I. Zhekov, I. A. Kulevskii, T. M. Murina, V. V. Osiko, A. M. Prokhorov, A. D. Savelev, V. V. Smirnov, B. P. Starikov, and M. A. Timoshechkin, “Stimulated emission from Er3+ ions in yttrium aluminum garnet crystals at λ=2.94 µm,” Sov. J. Quantum Electron. 4(8), 1039–1040 (1975).
[Crossref]

Pollnau, M.

S. Jackson, M. Pollnau, and J. Li, “Diode-pumped Erbium cascade fiber lasers,” IEEE J. Quantum Electron. 47(4), 471–478 (2011).
[Crossref]

M. Pollnau, C. Ghisler, W. Lüthy, H. P. Weber, J. Schneider, and U. B. Unrau, “Three-transition cascade erbium laser at 1.7, 2.7, and 1.6 microm,” Opt. Lett. 22(9), 612–614 (1997).
[Crossref] [PubMed]

M. Pollnau, R. Spring, S. Wittwer, W. Lüthy, and H. P. Weber, “Investigation on the slope efficiency of a pulsed 2.8-µm Er:LiYF laser,” Opt. Soc. Am. B 14(4), 974–978 (1997).
[Crossref]

M. Pollnau, W. Lüthy, H. P. Weber, K. Krämer, H. U. Güdel, and R. A. McFarlane, “Excited-state absorption in Er:BaY2F8 and Cs3Er2Br9 and comparison with Er:LiYF4,” Appl. Phys. B 62(4), 339–344 (1996).
[Crossref]

M. Pollnau, T. Graf, J. E. Balmer, W. Lüthy, and H. P. Weber, “Explanation of the cw operation of the Er3+ 3- microm crystal laser,” Phys. Rev. A 49(5), 3990–3996 (1994).
[Crossref] [PubMed]

Prokhorov, A. M.

E. V. Zharikov, V. I. Zhekov, I. A. Kulevskii, T. M. Murina, V. V. Osiko, A. M. Prokhorov, A. D. Savelev, V. V. Smirnov, B. P. Starikov, and M. A. Timoshechkin, “Stimulated emission from Er3+ ions in yttrium aluminum garnet crystals at λ=2.94 µm,” Sov. J. Quantum Electron. 4(8), 1039–1040 (1975).
[Crossref]

Ranieri, I. M.

N. U. Wetter, A. M. Deana, I. M. Ranieri, L. Gomes, and S. L. Baldochi, “Influence of excited-state-energy upconversion on pulse shape in quasi-continuous-wave diode-pumped Er:YLF4 lasers,” IEEE J. Quantum Electron. 46(1), 99–104 (2010).
[Crossref]

Ren, X.

X. Ren, Y. Wang, J. Zhang, D. Tang, and D. Shen, “Short-Pulse-Width Repetitively Q-Switched ~2.7-µm Er:Y2O3 Ceramic Laser,” Appl. Sci. (Basel) 7(11), 1201–1207 (2017).
[Crossref]

Rudy, C. W.

C. W. Rudy, “Mid-IR Lasers: Power and pulse capability ramp up for mid-IR lasers,” Laser Focus World 50(5), 63–66 (2014).

Sachkov, D. Yu.

Sanamyan, T.

T. Sanamyan, “Efficient, cryogenic mid-IR and eye-safe Er:YAG laser,” J. Opt. Soc. Am. B 33(11), D1–D6 (2016).
[Crossref]

T. Sanamyan, “Diode-pumped cascade Er:Y2O3 laser,” Laser Phys. Lett. 12(12), 125804 (2015).
[Crossref]

Savelev, A. D.

E. V. Zharikov, V. I. Zhekov, I. A. Kulevskii, T. M. Murina, V. V. Osiko, A. M. Prokhorov, A. D. Savelev, V. V. Smirnov, B. P. Starikov, and M. A. Timoshechkin, “Stimulated emission from Er3+ ions in yttrium aluminum garnet crystals at λ=2.94 µm,” Sov. J. Quantum Electron. 4(8), 1039–1040 (1975).
[Crossref]

Schmaul, B.

B. Schmaul, G. Huber, R. Clausen, C. Chai, P. LiKamWa, and M. Bass, “Er3+:YLiF4 continuous wave cascade laser operation at 1620 and 2810 nm at room temperature,” Appl. Phys. Lett. 62(6), 541–543 (1993).
[Crossref]

Schneider, J.

Shen, D.

X. Ren, Y. Wang, J. Zhang, D. Tang, and D. Shen, “Short-Pulse-Width Repetitively Q-Switched ~2.7-µm Er:Y2O3 Ceramic Laser,” Appl. Sci. (Basel) 7(11), 1201–1207 (2017).
[Crossref]

Shimizu, S.

Smirnov, V. V.

E. V. Zharikov, V. I. Zhekov, I. A. Kulevskii, T. M. Murina, V. V. Osiko, A. M. Prokhorov, A. D. Savelev, V. V. Smirnov, B. P. Starikov, and M. A. Timoshechkin, “Stimulated emission from Er3+ ions in yttrium aluminum garnet crystals at λ=2.94 µm,” Sov. J. Quantum Electron. 4(8), 1039–1040 (1975).
[Crossref]

Spring, R.

M. Pollnau, R. Spring, S. Wittwer, W. Lüthy, and H. P. Weber, “Investigation on the slope efficiency of a pulsed 2.8-µm Er:LiYF laser,” Opt. Soc. Am. B 14(4), 974–978 (1997).
[Crossref]

Starikov, B. P.

E. V. Zharikov, V. I. Zhekov, I. A. Kulevskii, T. M. Murina, V. V. Osiko, A. M. Prokhorov, A. D. Savelev, V. V. Smirnov, B. P. Starikov, and M. A. Timoshechkin, “Stimulated emission from Er3+ ions in yttrium aluminum garnet crystals at λ=2.94 µm,” Sov. J. Quantum Electron. 4(8), 1039–1040 (1975).
[Crossref]

Stoneman, R. C.

R. C. Stoneman, J. G. Lynn, and L. Esterowitz, “Direct upper-state pumping of the 2.8 µm Er3+:YLF laser,” IEEE J. Quantum Electron. 28(4), 1041–1045 (1992).
[Crossref]

Tang, D.

X. Ren, Y. Wang, J. Zhang, D. Tang, and D. Shen, “Short-Pulse-Width Repetitively Q-Switched ~2.7-µm Er:Y2O3 Ceramic Laser,” Appl. Sci. (Basel) 7(11), 1201–1207 (2017).
[Crossref]

Thuau, M.

C. Labbe, J.-L. Doualan, S. Girard, R. Moncorge, and M. Thuau, “Absolute excited state absorption cross section measurements in Er3+:LiYF4 for laser applications around 2.8 mm and 551 nm,” J. Phys. Condens. Matter 12(30), 6943–6957 (2000).
[Crossref]

Timoshechkin, M. A.

E. V. Zharikov, V. I. Zhekov, I. A. Kulevskii, T. M. Murina, V. V. Osiko, A. M. Prokhorov, A. D. Savelev, V. V. Smirnov, B. P. Starikov, and M. A. Timoshechkin, “Stimulated emission from Er3+ ions in yttrium aluminum garnet crystals at λ=2.94 µm,” Sov. J. Quantum Electron. 4(8), 1039–1040 (1975).
[Crossref]

Tokita, S.

Uehara, H.

Unrau, U. B.

Unterrainer, K.

Wang, Y.

X. Ren, Y. Wang, J. Zhang, D. Tang, and D. Shen, “Short-Pulse-Width Repetitively Q-Switched ~2.7-µm Er:Y2O3 Ceramic Laser,” Appl. Sci. (Basel) 7(11), 1201–1207 (2017).
[Crossref]

Weber, H. P.

M. Pollnau, R. Spring, S. Wittwer, W. Lüthy, and H. P. Weber, “Investigation on the slope efficiency of a pulsed 2.8-µm Er:LiYF laser,” Opt. Soc. Am. B 14(4), 974–978 (1997).
[Crossref]

M. Pollnau, C. Ghisler, W. Lüthy, H. P. Weber, J. Schneider, and U. B. Unrau, “Three-transition cascade erbium laser at 1.7, 2.7, and 1.6 microm,” Opt. Lett. 22(9), 612–614 (1997).
[Crossref] [PubMed]

M. Pollnau, W. Lüthy, H. P. Weber, K. Krämer, H. U. Güdel, and R. A. McFarlane, “Excited-state absorption in Er:BaY2F8 and Cs3Er2Br9 and comparison with Er:LiYF4,” Appl. Phys. B 62(4), 339–344 (1996).
[Crossref]

M. Pollnau, T. Graf, J. E. Balmer, W. Lüthy, and H. P. Weber, “Explanation of the cw operation of the Er3+ 3- microm crystal laser,” Phys. Rev. A 49(5), 3990–3996 (1994).
[Crossref] [PubMed]

Wetter, N. U.

N. U. Wetter, A. M. Deana, I. M. Ranieri, L. Gomes, and S. L. Baldochi, “Influence of excited-state-energy upconversion on pulse shape in quasi-continuous-wave diode-pumped Er:YLF4 lasers,” IEEE J. Quantum Electron. 46(1), 99–104 (2010).
[Crossref]

Wittwer, S.

M. Pollnau, R. Spring, S. Wittwer, W. Lüthy, and H. P. Weber, “Investigation on the slope efficiency of a pulsed 2.8-µm Er:LiYF laser,” Opt. Soc. Am. B 14(4), 974–978 (1997).
[Crossref]

Yasuhara, R.

Zhang, J.

X. Ren, Y. Wang, J. Zhang, D. Tang, and D. Shen, “Short-Pulse-Width Repetitively Q-Switched ~2.7-µm Er:Y2O3 Ceramic Laser,” Appl. Sci. (Basel) 7(11), 1201–1207 (2017).
[Crossref]

Zharikov, E. V.

E. V. Zharikov, V. I. Zhekov, I. A. Kulevskii, T. M. Murina, V. V. Osiko, A. M. Prokhorov, A. D. Savelev, V. V. Smirnov, B. P. Starikov, and M. A. Timoshechkin, “Stimulated emission from Er3+ ions in yttrium aluminum garnet crystals at λ=2.94 µm,” Sov. J. Quantum Electron. 4(8), 1039–1040 (1975).
[Crossref]

Zhekov, V. I.

E. V. Zharikov, V. I. Zhekov, I. A. Kulevskii, T. M. Murina, V. V. Osiko, A. M. Prokhorov, A. D. Savelev, V. V. Smirnov, B. P. Starikov, and M. A. Timoshechkin, “Stimulated emission from Er3+ ions in yttrium aluminum garnet crystals at λ=2.94 µm,” Sov. J. Quantum Electron. 4(8), 1039–1040 (1975).
[Crossref]

Zhou, B.

S. Hubert, D. Meichenin, B. Zhou, and F. Auzel, “Emission properties, oscillator strengths and laser parameters of Er3+ in LiYF4 at 2.7 mm,” J. Lumin. 50(1), 7–15 (1991).
[Crossref]

Appl. Opt. (1)

Appl. Phys. B (1)

M. Pollnau, W. Lüthy, H. P. Weber, K. Krämer, H. U. Güdel, and R. A. McFarlane, “Excited-state absorption in Er:BaY2F8 and Cs3Er2Br9 and comparison with Er:LiYF4,” Appl. Phys. B 62(4), 339–344 (1996).
[Crossref]

Appl. Phys. Lett. (2)

G. J. Kintz, R. Allen, and L. Esterowitz, “cw and pulsed 2.8 µm laser emission from diode pumped Er3+:LiYF4 at room temperature,” Appl. Phys. Lett. 50(22), 1553–1555 (1987).
[Crossref]

B. Schmaul, G. Huber, R. Clausen, C. Chai, P. LiKamWa, and M. Bass, “Er3+:YLiF4 continuous wave cascade laser operation at 1620 and 2810 nm at room temperature,” Appl. Phys. Lett. 62(6), 541–543 (1993).
[Crossref]

Appl. Sci. (Basel) (1)

X. Ren, Y. Wang, J. Zhang, D. Tang, and D. Shen, “Short-Pulse-Width Repetitively Q-Switched ~2.7-µm Er:Y2O3 Ceramic Laser,” Appl. Sci. (Basel) 7(11), 1201–1207 (2017).
[Crossref]

IEEE J. Quantum Electron. (4)

V. Lupei, S. Georgescu, and V. Florea, “On the dynamics of population inversion for 3 µm Er3+ lasers,” IEEE J. Quantum Electron. 29(2), 426–434 (1993).
[Crossref]

S. Jackson, M. Pollnau, and J. Li, “Diode-pumped Erbium cascade fiber lasers,” IEEE J. Quantum Electron. 47(4), 471–478 (2011).
[Crossref]

N. U. Wetter, A. M. Deana, I. M. Ranieri, L. Gomes, and S. L. Baldochi, “Influence of excited-state-energy upconversion on pulse shape in quasi-continuous-wave diode-pumped Er:YLF4 lasers,” IEEE J. Quantum Electron. 46(1), 99–104 (2010).
[Crossref]

R. C. Stoneman, J. G. Lynn, and L. Esterowitz, “Direct upper-state pumping of the 2.8 µm Er3+:YLF laser,” IEEE J. Quantum Electron. 28(4), 1041–1045 (1992).
[Crossref]

IEEE Photonics Technol. Lett. (1)

J. Schneider, “Mid-infrared fluoride fiber lasers in multiple cascade operation,” IEEE Photonics Technol. Lett. 7(4), 354–356 (1995).
[Crossref]

J. Lumin. (1)

S. Hubert, D. Meichenin, B. Zhou, and F. Auzel, “Emission properties, oscillator strengths and laser parameters of Er3+ in LiYF4 at 2.7 mm,” J. Lumin. 50(1), 7–15 (1991).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Opt. Technol. (1)

J. Phys. Condens. Matter (1)

C. Labbe, J.-L. Doualan, S. Girard, R. Moncorge, and M. Thuau, “Absolute excited state absorption cross section measurements in Er3+:LiYF4 for laser applications around 2.8 mm and 551 nm,” J. Phys. Condens. Matter 12(30), 6943–6957 (2000).
[Crossref]

Laser Focus World (1)

C. W. Rudy, “Mid-IR Lasers: Power and pulse capability ramp up for mid-IR lasers,” Laser Focus World 50(5), 63–66 (2014).

Laser Phys. Lett. (1)

T. Sanamyan, “Diode-pumped cascade Er:Y2O3 laser,” Laser Phys. Lett. 12(12), 125804 (2015).
[Crossref]

Opt. Eng. (1)

V. Lupei and S. Georgesku, “Erbium 3-µm laser as an upconversion system,” Opt. Eng. 35(5), 1265–1272 (1996).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Opt. Soc. Am. B (1)

M. Pollnau, R. Spring, S. Wittwer, W. Lüthy, and H. P. Weber, “Investigation on the slope efficiency of a pulsed 2.8-µm Er:LiYF laser,” Opt. Soc. Am. B 14(4), 974–978 (1997).
[Crossref]

Phys. Rev. A (1)

M. Pollnau, T. Graf, J. E. Balmer, W. Lüthy, and H. P. Weber, “Explanation of the cw operation of the Er3+ 3- microm crystal laser,” Phys. Rev. A 49(5), 3990–3996 (1994).
[Crossref] [PubMed]

Sov. J. Quantum Electron. (1)

E. V. Zharikov, V. I. Zhekov, I. A. Kulevskii, T. M. Murina, V. V. Osiko, A. M. Prokhorov, A. D. Savelev, V. V. Smirnov, B. P. Starikov, and M. A. Timoshechkin, “Stimulated emission from Er3+ ions in yttrium aluminum garnet crystals at λ=2.94 µm,” Sov. J. Quantum Electron. 4(8), 1039–1040 (1975).
[Crossref]

Sov. Phys. Usp. (1)

A. A. Mak, Yu. A. Anan’ev, and B. A. Ermakov, “Solid State Lasers,” Sov. Phys. Usp. 10(4), 419–452 (1968).
[Crossref]

Other (3)

C. Li, Y. Guyot, C. Linares, R. Moncorge, and M. F. Joubert, “Radiative transition probabilities of trivalent rare-earth ions in LiYF4,” in “Advanced Solid State Lasers and Compact Blue-Green Lasers” OSA Technical Digest 2, 423–425 (1993).

H. Chou and H. Jenssen, “Upconversion processes in Er-activated solid state laser materials in Advanced Solid State Lasers, M. Shand and H. Jenssen, eds., Vol. 5 of OSA Proceedings Series (Optical Society of America, 1989), paper DD5.

M. Messner, A. Henrich, C. Hagen, and K. Unterrainer, “Acousto-optically Q-switched diode side-pumped Er:YLF laser generating 50-kW peak power in 70-ns pulses,” Proceedings SPIE 10896, Solid State Lasers XXVIII: Technology and Devices; 1089607 (2019).

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Figures (8)

Fig. 1
Fig. 1 Energy levels in Er:YLF, after [5]. Solid arrows denote absorption and laser transitions. Dashed arrows denote energy transfer processes. Broken, arrows denote heat generating non-radiative decay. Also listed on the right are the lifetimes of the levels involved in numerical simulations.
Fig. 2
Fig. 2 Population dynamics of the 4I11/2, 4I13/2, and 4S3/2 levels, top row, (in red, blue and green correspondingly). Dynamics of the coefficient of amplification α2.8 is shown in the bottom row. Frames (a) & (b) represent the case of high cavity loss for all 3 wavelengths. Frames (c) & (d) represent the case of high cavity loss only for the 2.8 µm emission, while lasing at the 1.73 and 1.62 µm wavelengths is allowed. Cavity parameters: R1.6 = 0.99, R1.7 = 0.97, G1.6 = G1.7 = 0.033, LCR = 15 mm, Pump = 80W.
Fig. 3
Fig. 3 Simulation of the 2.8 µm pulse energy vs time at different pump powers. (a) 2.8 µm lasing only; (b) 2.8 µm lasing with 1.6 + 1.7 µm cascade.. Cavity parameters: R1.6 = 0.99, R1.7 = 0.97, G1.6 = G1.7 = 0.033, LCR = 15 mm.
Fig. 4
Fig. 4 Simulation of the Q-switched + Gain-switched, 4I11/24I13/24I15/2 cascade lasing at the 2.8 and 1.62 µm wavelengths. Cavity parameters: R1.6 = 0.99, R1.7 = 0.97, R2.8 = 0.8, G1.6 = G1.7 = 0.033, LCR = 15 mm, Pump = 80W.
Fig. 5
Fig. 5 Simplified schematics of (a) single-cavity and (b) spilt-channel cascade lasers.
Fig. 6
Fig. 6 Er:YLF cascade laser experimental layout. Spectral overlap of the pump emission and Er:YLF absorption is pictured on the left. Pump spectrum was measured with Yokogawa Optical Spectrum Analyzer 3624 and Er:YLF π-absorption spectrum was collected with Cary 600 spectrometer (resolution 0.5 nm).
Fig. 7
Fig. 7 Er(1%):YLF cascade laser performance in the free running, QCW operation. (a) – Oscilloscope traces of pump (grey), 2.8 µm (red) and 1.62 µm lasing (blue). (b) – Wavelength resolved cascade laser output vs the absorbed pump. Inset shows output spectrum consisting of the 2.66, 2.72 and 2.81 µm lines. ΔtPUMP = 10 msec, PRF = 20 Hz.
Fig. 8
Fig. 8 Er(1%):YLF cascade laser performance (1.6 + 1.7 + 2.8 µm) in the Q-switched mode. Shown are the oscilloscope traces of the pump (grey), 2.8 µm (red), 1.73 µm (green) and 1.62 µm lasing (blue). Pump pulsewidth ΔtPUMP = 1.25 msec, PRF = 160 Hz. Inset shows 1.62 and 1.73 µm laser spectral lines.

Tables (1)

Tables Icon

Table 1 Selected Er(1%):YLF parameters used in numerical simulations.

Equations (15)

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d N 1 dt = N 1 τ 1 + β 21 N 2 τ 2 +( β 51 τ 5 + W 50 N 0 ) N 5 2 W 11 N 1 2 + R SE 2.8 R SE 1.6 ;
d N 2 dt = R 02 pump N 2 τ 2 2 W 22 N 2 2 +( β 54 τ 5 + W 50 N 0 ) N 5 + W 11 N 1 2 R SE 2.8 + R SE 1.7 ;
d N 5 dt = N 5 τ 5 W 50 N 0 N 5 + W 22 N 2 2 R SE 1.7 ;
d φ 2.8 dt = R SE 2.8 ln( R 2.8 1 )+ G 2.8 2 L cr c φ 2.8 + N 2 τ 2 β 21 f G 2.8 ;
d φ 1.7 dt = R SE 1.7 ln( R 1.7 1 )+ G 1.7 2 L cr c φ 1.7 + N 5 τ 5 β 54 f G 1.7 ;
d φ 1.6 dt = R SE 1.6 ln( R 1.6 1 )+ G 1.6 2 L cr c φ 1.6 + N 1 τ 1 β 10 f G 1.6 ;
R SE 2.8 =( b 24 N 2 g 2 g 1 b 17 N 1 )c σ 2.8 φ 2.8 ;
R SE 1.6 =( b 10 N 1 g 1 g 0 b 08 ( N N 1 N 2 N 5 ) )c σ 1.6 φ 1.6 ;
R SE 1.7 = b 50 N 5 c σ 1.7 φ 1.7 ;
R 02 pump = P p λ p hc L cr S mod ( 1exp( σ p N 0 L cr ) );
E 2.8 ST = α 2.8 hc S mod σ 2.8 λ 2.8 ( 1 b 17 b 17 + b 24 );
α 2.8 =( b 24 N 2 g 2 g 1 b 07 N 1 ) σ 2.8 ;
E 2.8 = E 2.8 st ln( R 2.8 1 ) 1 α end / α 2.8 ln( R 2.8 1 )+ G 2.8 ;
2 α 2.8 α end L cr ( ln( R 2.8 1 )+ G 2.8 ) =ln( α 2.8 α end );
P out i = φ i h λ i S mod 1 R i 1+ R i ln( R i 1 ) ln( R i 1 )+ G i ;

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