## Abstract

We present a simple, fast and accurate technique to characterize intracavity losses in broadband quasi-three-level lasers based on spectroscopic gain analysis. The technique is based on spectral gain measurement and potentially can be used at any laser output power levels, thus allowing a dynamic optimization of laser performance. Successful experimental demonstration was carried out with a diode-pumped Yb:KGW continuous wave oscillator. A comparison with traditional Findlay-Clay analysis and numerical modeling was also made.

© 2014 Optical Society of America

## 1. Introduction

Intrinsic losses introduced by various laser cavity components (or intracavity losses) play a fundamental role in limiting efficiency of laser performance. Currently, however, their accurate characterization is only available for the four-level laser systems [1] and only at the threshold of operation. At the same time, over the past decade quasi-three-level laser systems based on Yb-ion doped materials have become technologically important. Ytterbium ion solid-state [2] and fiber [3] lasers are revolutionizing industrial and scientific applications such as micromachining [4] and multiphoton microscopy [5]. Despite this fact, determination of parasitic intracavity losses in such laser systems remained a technological and scientific challenge. Here we demonstrate a simple, fast and accurate analysis of intracavity losses of quasi-three-level lasers based on spectral gain measurement. The technique overcomes limitations of the four-level laser model [1], enabling dynamic characterization of loss, and hence, providing a potential for real-time optimization of laser performance at any working point above the threshold.

Intracavity losses are important factor in determining the overall performance of a laser. Because minimization of such parasitic losses is an obvious task, practical laser development often requires their accurate characterization. Findlay and Clay addressed this problem by developing the well-known method for a 4-level laser system [1]. This model is based on the assumption that the peak gain for a particular laser transition is spectrally independent from loss, which results in a linear plot between the logarithm of the output coupler reflectivity and the threshold pump power. Several measurements with different output couplers are needed in order to extract the intracavity loss. While perfectly valid for a typical 4-level laser system (e.g. 1064 nm transition of Nd:YAG crystal), this assumption no longer holds in case of a quasi-three-level laser transition which also exhibits reabsorption loss. Unfortunately, in case of broadband quasi-three-level transitions such as those of the Yb-ions in solid matrices, the situation becomes even much more complicated because the gain and reabsorption losses are wavelength dependent. This leads to a nonlinear relationship between the threshold pump power and the introduced loss and to erroneous analysis. Another serious limitation of the traditional Findlay-Clay analysis is that it determines the intracavity losses only at the *threshold* of laser operation. Knowledge of the intracavity losses *during* the laser operation would be particularly valuable for the quasi-three-level lasers where thermal lens induced mismatch between the pump and laser mode volumes at high pump power levels can introduce reabsorption losses in the unpumped regions of the laser mode. Such additional losses would effectively contribute to the total intracavity loss. Therefore, the ability of dynamic intracavity loss characterization can be used to ensure that a laser always operates in optimized configuration.

The intracavity loss can be determined from the loss dependent peak gain coefficient in an Yb-ion gain medium. It is well known that in a steady state operation the overall cavity loss is balanced by the gain,

where*L*is the round-trip intracavity loss,

_{in}*R*is the reflectivity of the output coupler, and

_{oc}*g(λ*is the wavelength-dependent effective gain coefficient which takes into account the reabsorption loss. It follows from Eq. (1) that if the gain coefficient is known, then the intracavity loss can be found. Since the gain coefficient determines the oscillating wavelength of a laser, it can be easily calculated by measuring the lasing spectrum and using the following equation from [6]:

_{L})*N*is the doping concentration,

_{0}*l*is the crystal length,

_{c}*σ*and

_{em}(λ_{L})*σ*are the emission and absorption cross sections at lasing wavelength

_{abs}(λ_{L})*λ*. The factor of 2 accounts for the double pass of the laser radiation in the gain medium. The value of

_{L}*β*, which is the fractional population inversion, controls the shape and the peak wavelength of the gain spectrum and therefore is defined by the experimentally measured lasing wavelength. The sources of original data on emission and absorption spectra of a particular gain medium can be usually found in the literature.

As can be seen, the presented technique eliminates repetitive measurements (only one output coupler is needed) thus providing a simple, fast and accurate characterization of the intracavity loss. Moreover, by taking into account dependence of the emission and absorption spectra on the temperature, the estimation of losses potentially can be done at any output power level; therefore the measurements provide the current value of the intracavity loss and can be used for dynamic monitoring of the lasing performance. In addition to loss, the laser gain coefficient is also found.

## 2. Experimental setup

The experiments were carried out with a diode-pumped continuous wave Yb:KGW oscillator with a standard delta cavity shown in Fig. 1 which could support two polarizations, N_{m} and N_{p}. The cavity was designed to have a good overlap between the pump and cavity modes. The Yb:KGW crystal slab (Eksma) with 5 mm × 8 mm × 1.2 mm dimensions and 1.5 at.% doping was cut along the N_{g}-axis and antireflection (AR) coated. A BK7 glass plate was used at Brewster’s angle to select oscillation with polarization either along the N_{m}-axis or N_{p}-axis [7]. The pump beam from a fiber coupled laser diode was delivered through the two AR coated achromatic doublets, forming a beam spot with 375 μm diameter in the crystal. The pump absorption in the crystal was measured to be 50% under non-lasing conditions. The cavity optics (Laseroptik GmbH) were configured to provide a cavity mode size of 320 μm in diameter inside the crystal. This relatively small beam size took into account the effect of thermal lensing and ensured that the cavity mode was always within the pumped region. The output power was measured by a calibrated thermopile power meter and the lasing spectrum was recorded by a spectrum analyzer (Anritsu MS2687B) with a wavelength resolution of 0.07 nm.

## 3. Results and discussion

The laser performance was evaluated with a series of output couplers with transmission ranging from 0.4% to 10%. The best performance for both polarizations (N_{m} and N_{p}) was achieved with a 5% output coupler as shown in Fig. 2. The output power was 3.3 W for the N_{m}-polarization and 3.5 W for the N_{p}-polarization at the pump power of 21 W (~10 W absorbed), corresponding to a slope efficiency of more than 70% with respect to the absorbed pump power. The output power plots exhibited a nonlinear behavior since the pump wavelength shifted to a longer one as the diode’s drive current was increased as well as varying overlap between the cavity and pump modes as a result of thermal lensing.

The lasing spectra of the laser and the corresponding calculated gain spectra for the N_{m}-polarization and the N_{p}-polarization are shown in Fig. 3. For all output couplers the lasing spectra were recorded at the pump power of 17.5 W, and no shifts in wavelength for each output coupler were observed as the pump power was increased up to the maximum of 22 W. The lasing wavelength *λ _{L}*, the fitted population inversion

*β*, the gain coefficient

*g(λ*at the lasing wavelength, and the calculated intracavity losses

_{L})*L*are listed in Table 1. The room-temperature data on emission and absorption spectra of the Yb:KGW crystal that were used in these calculations were published in [8] and kindly provided to us by S. R. Bowman.

_{in}The intracavity losses for the N_{m}- and N_{p}-polarizations have consistent values of ~2.1% and ~3.0% for the 3-10% and 3-5% output couplers, respectively. There are a number of factors that could have contributed to the uncertainty in these obtained values of intracavity loss. The difference in intracavity losses between both polarizations can be attributed to a slight misalignment of the Brewster plate during realignment procedure, which was used (in both cases) to select the N_{m}- or N_{p}-polarization. The difference in intracavity losses between the individual output couplers (for a particular polarization) could have also resulted from some residual misalignment that was left after their changing and from variation of the cavity mirror’s reflectivity (including that of the output coupler) in the 1023-1043 nm wavelength range. More generally, systemic errors came from the wavelength resolution of the spectrum analyzer (0.07 nm) and of the available emission and absorption spectra (0.1 nm) of the Yb:KGW crystal. In addition, the dependence of the emission and absorption spectra on the temperature and the actual level of doping concentration also can provide a degree of uncertainty. While all used highly reflective cavity optics were specified to be 99.9% reflective in the 1010-1100 nm range and had less than 0.01% change in reflectivity in the observed lasing wavelength range, we were unable to find any temperature dependent data on emission and absorption cross sections of Yb:KGW. For this reason the room-temperature data set from [8] was used in our calculations. The fact that for every output coupler no wavelength shift was observed as the pump power was increased points out that in our case the effect of temperature at least on the gain shape was negligible. We believe that a cumulative error in the estimated intracavity loss in our case did not exceed 1%. A simple estimation supports this statement: taking manufacturer’s specified loss of 0.1% for highly reflecting, dichroic and crystal’s AR coatings the total intracavity loss per roundtrip can be put at 1.3%. Considering the scattering introduced by the dust particles on optics (the laser was not operated in a clean room and did not have a cover) the actual loss will be closer to the experimentally inferred values.

On the other hand, the data show a large error with 0.4% output coupler for both polarizations. This is caused by a relatively flat gain spectrum at low level of loss. This leads to a difficulty in assignment of the peak gain wavelength because of multiple wavelengths in the spectrum (N_{m}-polarization) or very vague peaks in the calculated gain spectrum (N_{p}-polarization). Another possible limitation could be due to spectroscopic data of the crystal since reabsorption at the longer wavelengths is fairly weak thus making its accurate measurement difficult. In addition, the lasing wavelength of the N_{p}-polarization (see Fig. 3(b)) had a shorter tuning range and was clamped at 1037 nm as the cavity loss exceeded a certain level. Thus with the 10% output coupler, the population inversion and the corresponding gain spectrum could not be reliably determined. Therefore, the intracavity losses estimated by this spectroscopic method could have large errors when the output couplers induce too large or too small losses. In order to determine the range of applicable losses, the calculated lasing wavelength from Eq. (1) was plotted in Fig. 4 as a function of the total cavity loss (i.e. *R _{oc}* +

*L*). At low level of loss the curves are relatively steep, corresponding to a spectrally flat gain spectrum and at high level of loss they become flat, resulting in an invariable lasing wavelength. It can be seen that the optimum ranges of total cavity losses that can be used for correct intracavity loss estimation are 2.5% - 19% for the N

_{in}_{m}-polarization and 3.5% - 8.5% for the N

_{p}-polarization.

For comparison the Findlay-Clay analysis [1] was also performed and the data are displayed in Fig. 5. The main equation of this method can be written in a simplified form as

where*K*is a constant and

*δ*= 2

_{reabs}*N*is the round-trip reabsorption loss (in our case it is wavelength-dependent through

_{0}l_{c}σ_{abs}(λ_{L})*σ*). For the analysis the threshold pump power could not be determined by linear fitting of the output power plots because of their obvious nonlinearity (shown in Fig. 2). Therefore, in this work, the threshold pump power was recorded when the lasing ceased while decreasing the pump diode’s drive current. A noticeable deviation from linear behavior of the measured relationship between the

_{abs}(λ_{L})*P*and ln(

_{th}*R*) was observed for the N

_{oc}_{m}-polarization (see Fig. 5). For the N

_{p}-polarization, owing to a narrower wavelength tuning range and almost constant level of the reabsorption loss, the relationship is closer to a linear. According to the Findlay-Clay analysis, the intercept of the linear data fit with the horizontal axis gives the overall cavity loss (excluding the loss of the output coupler). By subtracting the reabsorption loss of about 5%, the estimated intracavity loss for the N

_{p}-polarization is 1.8% and is close to our previous measurements. For the N

_{m}-polarization, however, although the linear fit points to an overall cavity loss of about 10%, it is still impossible to estimate the intracavity loss because the reabsorption loss varied from 3% to 20% (see Table 2) for the observed oscillating wavelength range. This clearly shows the limitation of the Findlay-Clay analysis. It is worth noting that similar to the work of Findlay and Clay, Caird’s analysis [9] also assumes that the lasing wavelength is spectrally independent from loss and, therefore, cannot be used for analysis of intracavity losses in broadband quasi-three-level gain media.

For further comparison we also estimated intracavity loss using a relationship derived from a numerical modeling of the quasi-three-level lasers presented in [10]:

*hυ*= 2 × 10

_{p}^{−19}J is the pump photon energy and

*τ*= 300 µs is the lifetime of the upper energy manifold. The

*f*= 0.75 and

_{u}*f*= 0.04 are the fractional population in the upper and lower energy levels which were calculated based on the energy structure of the Yb:KGW crystal [11]. The cavity mode radius

_{l}*w*of 165 µm was used when considering a weak thermal lens (~1 diopter) at the lasing threshold. Taking the pump beam quality factor (M

_{L}^{2}) of 35 into consideration, an averaged pump mode radius

*w*= 205 µm was used [12].

_{ap}*T*is the transmission of the output coupler. Other parameters, including the reabsorption loss

_{oc}*δ*and the emission and absorption cross sections (

_{reabs}*σ*and

_{em}(λ_{L})*σ*), are all wavelength-dependent. Their values are listed in Table 2. With the measured absorbed pump power (and its spectrum) at lasing threshold, the intracavity losses were calculated for each output coupler (see Table 2).

_{abs}(λ_{L})The data in Table 2 indicate that this numerical model also has large errors. This is caused by the uncertainties and approximations that were involved in the measurements and calculations. Firstly, the threshold pump power was determined in the same way as for the Findlay-Clay analysis, although the power meter itself has a finite accuracy. Secondly and most important, the effect of the overlap between the pump mode and the cavity mode had a large uncertainty since the effect of thermal lensing (even at threshold) would change the shape of the cavity mode. Most likely this is reflected in the growing intracavity losses as the output coupling was increased.

As can be concluded from the above discussion, both conventional methods have serious limitations and result in large errors. More importantly, they estimate the intracavity loss only at the lasing threshold, a condition that is very different from the normal operating regime. The newly proposed method, however, was shown to be more accurate and consistent, and potentially allows one to estimate the intracavity losses at any output power levels. It can be further improved by taking into account temperature-dependent emission and absorption cross sections, spatially varying level of inversion in the crystal [13,14] as well as inhomogeneity of pumping. We anticipate that this method will be also applicable to other broadband gain media such as, for example, Yb:CaF, Yb:glass, and Yb:CALGO. While questionable, its application to narrowband materials like Yb:YAG is not completely impossible and can be assessed by making calculations similar to the ones presented in Fig. 4 to identify a suitable range of losses.

To demonstrate dynamic application of our technique we adjusted the cavity to provide a lasing mode that would have a poor overlap with the pump mode at low pump power and a better overlap at high pump power as a result of thermal lensing. Figure 6 shows the lasing spectra as the pump power was increased from the lasing threshold at 11.7 W to 24.7 W. It can be seen that the lasing wavelength shifted from 1029 nm to 1030.4 nm, corresponding to a reduction of intracavity loss from 4.6% to 2.3% as better overlap between the modes was achieved. While optimization of laser performance can be done by simply monitoring of its output power, our approach allows one to do the same in the spectral domain. It is worth noting that in many cases simple monitoring of the output power is not always objective enough criterion because the lasing efficiency may experience a loss-induced degradation despite the continuous growth of the output power (e.g. in case of thermal roll off).

## 4. Conclusion

In conclusion, a technique to estimate the intracavity losses based on the spectroscopic gain analysis was presented. This technique can be applied to the laser oscillators based on the quasi-three-level energy structure with broad emission spectrum, where the traditional Findlay-Clay analysis is not applicable. It provides an accurate estimation of the intracavity loss based on a single measurement. More importantly, the estimation is not limited only to the lasing threshold, thus allowing one to adjust laser cavity parameters according to the dynamically varying intracavity loss. A comparison with the traditional Findlay-Clay analysis and numerical modeling method was made and limitations were discussed. An experimental test with a diode-pumped Yb:KGW laser oscillator was demonstrated. We believe that this technique will be equally important for the continuous wave as well as ultrafast lasers based on broadband quasi-three-level gain media.

## Acknowledgments

The authors gratefully acknowledge S. R. Bowman for providing data on Yb:KGW absorption and emission. This work was supported by the Natural Sciences and Engineering Research Council of Canada, University of Manitoba, and Western Economic Diversification Canada.

## References and links

**1. **D. Findlay and R. A. Clay, “The measurement of internal losses in 4-level lasers,” Phys. Lett. **20**(3), 277–278 (1966). [CrossRef]

**2. **T. Südmeyer, S. V. Marchese, S. Hashimoto, C. R. E. Baer, G. Gingras, B. Witzel, and U. Keller, “Femtosecond laser oscillators for high-field science,” Nat. Photon. **2**(10), 599–604 (2008). [CrossRef]

**3. **C. Jauregui, J. Limpert, and A. Tunnermann, “High-power fibre lasers,” Nat. Photon. **7**(11), 861–867 (2013). [CrossRef]

**4. **R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photon. **2**(4), 219–225 (2008). [CrossRef]

**5. **C. Xu and F. W. Wise, “Recent advances in fibre lasers for nonlinear microscopy,” Nat. Photon. **7**(11), 875–882 (2013). [CrossRef] [PubMed]

**6. **C. Hönninger, R. Paschotta, M. Graf, F. Morier-Genoud, G. Zhang, M. Moser, S. Biswal, J. Nees, A. Braun, G. A. Mourou, I. Johannsen, A. Giesen, W. Seeber, and U. Keller, “Ultrafast ytterbium-doped bulk lasers and laser amplifiers,” Appl. Phys. B **69**(1), 3–17 (1999). [CrossRef]

**7. **A. Major, D. Sandkuijl, and V. Barzda, “A diode-pumped continuous-wave Yb:KGW laser with N_{g}-axis polarized output,” Laser Phys. Lett. **6**(11), 779–781 (2009). [CrossRef]

**8. **S. R. Bowman, S. P. O’Connor, and S. Biswal, “Ytterbium laser with reduced thermal loading,” IEEE J. Quantum Electron. **41**(12), 1510–1517 (2005). [CrossRef]

**9. **J. A. Caird, S. A. Payne, P. R. Staber, A. J. Ramponi, L. L. Chase, and W. F. Krupke, “Quantum electronic properties of the Na_{3}Ga_{2}Li_{3}F_{12}:Cr^{3+} laser,” IEEE J. Quantum Electron. **24**(6), 1077–1099 (1988). [CrossRef]

**10. **T. Taira, W. M. Tulloch, and R. L. Byer, “Modeling of quasi-three-level lasers and operation of cw Yb:YAG lasers,” Appl. Opt. **36**(9), 1867–1874 (1997). [CrossRef] [PubMed]

**11. **N. V. Kuleshov, A. A. Lagatsky, A. V. Podlipensky, V. P. Mikhailov, and G. Huber, “Pulsed laser operation of Yb-doped KY(WO_{4})_{2} and KGd(WO_{4})_{2.},” Opt. Lett. **22**(17), 1317–1319 (1997). [CrossRef] [PubMed]

**12. **I. D. Lindsay and M. Ebrahimzadeh, “Efficient continuous-wave and Q-switched operation of a 946-nm Nd:YAG laser pumped by an injection-locked broad-area diode laser,” Appl. Opt. **37**(18), 3961–3970 (1998). [CrossRef] [PubMed]

**13. **S. Yiou, F. Balembois, and P. Georges, “Numerical modelling of a continuous-wave Yb-doped bulk crystal laser emitting on a three-level laser transition near 980 nm,” J. Opt. Soc. Am. B **22**(3), 572–581 (2005). [CrossRef]

**14. **B. Jacobsson, J. E. Hellström, V. Pasiskevicius, and F. Laurell, “Widely tunable Yb:KYW laser with a volume Bragg grating,” Opt. Express **15**(3), 1003–1010 (2007). [CrossRef] [PubMed]