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Rigrod laser-pumped-laser resonator model: II. Application to thin and optically-dilute laser

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2014 Laser Phys. 24 085003

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1. Introduction

Laser physicists, designers, and engineers often encounter laser design problems in which highly doped materials are needed to obtain good absorption efficiency, and inevitably encounter other problems as a result, including concentration-quenching, upconversion, excited-state absorption, and higher heat fractions. While double and multiple-passing of the gain medium is often used to address increasing the absorption and perhaps using lower doping concentrations to reduce the aforementioned parasitic effects, the additional complexity is often not tolerable, particularly in microlasers where pump-ing directly into the lasing mode is highly desirable to obtain good beam-quality and efficiency. In part I of this article, we presented another approach, involving placing a second thin or dilute lasing crystal intracavity to a pump laser with highly

reflective mirrors, and obtaining good absorption efficiency by the multi-passing of the absorbing laser crystal due to the finite photon lifetime of the pump resonator.

In this paper, we apply the theoretical framework developed in part I to the conceptual design of solid-state laser-pumped laser resonators. We first discuss in section 2 generalized results obtained for a resonator with highly reflective mirrors, typical losses, and a gain element with a gain–length product of 2, 5, 10, 20, and 30. In section 3, we consider the follow-ing operating schemes: a Yb, Er:glass intracavity pumped by a 946 nm Nd:YAG laser, a Yb, Er:glass laser-pumped intracav-ity to a 977 nm diode laser spectrally narrowed using a volume Bragg grating (VBG), an Er:YAG laser-pumped intracavity to a VBG narrowed 1530 nm diode laser, and a Yb:LuAG laser pumped by a VBG narrowed 979 nm diode laser. In sec-tion 4 we discuss the results obtained.

Laser Physics

Rigrod laser-pumped-laser resonator model: II. Application to thin and optically-dilute laser media

D C Brown

Snake Creek Lasers, LLC, Friendsville, PA 18818, USA

E-mail: dbrown@snakecreeklasers.com

Received 17 October 2013, revised 9 April 2014Accepted for publication 1 May 2014Published 15 July 2014

AbstractIn part I of this paper, and to set the foundation for this part II, we derived the resonator equations describing the normalized intensities, output power, gain, and extraction efficiency for a standard resonator incorporating two dielectric mirrors and a gain element. We then generalized the results to include an absorbing region representing a second laser crystal characterized by a small-signal transmission T0. Explicit expressions were found for the output power extracted into absorption by the second laser crystal and the extraction efficiency, and the limits to each were discussed. It was shown that efficient absorption by a thin or dilute second laser crystal can be realized in resonators in which the mirror reflectivities were high and in which the single-pass absorption was low, due to the finite photon lifetime and multi-passing of the absorbing laser element. In this paper, we apply the model derived in part I to thin or dilute laser materials, concentrating on a Yb, Er:glass intracavity pumped by a 946 nm Nd:YAG laser, a Yb, Er:glass laser-pumped intracavity by a 977 nm diode laser, and an Er:YAG laser-pumped intracavity to a 1530 nm diode laser. It is shown that efficient absorption can be obtained in all cases examined.

Keywords: lasers, laser diodes, efficient solid state laser, laser-pumped laser, intracavity-pumped laser, Er laser, Yb laser

(Some figures may appear in colour only in the online journal)

D C Brown

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doi:10.1088/1054-660X/24/8/085003Laser Phys. 24 (2014) 085003 (9pp)

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2. Application to diode-pumped solid-state lasers with intracavity absorbing region

Having reviewed the Rigrod theory and derived expressions for the useful normalized power output, threshold, and extrac-tion efficiency in part I for both non-diode and diode resona-tors, here we apply the results of part I to the optimization of the absorption efficiency in dilute weakly absorbing or thin laser materials.

2.1. Laser-pumped (non-diode) laser resonators

Before discussing the application of the theory developed in part I, we first present some generalized results that are appli-cable to any intracavity laser-pumped laser, with the exception of diode lasers. For the example considered here, we assume the reflectivities R1 and R2 to be 99.5%. The passive loss coef-ficient value is α = 0.001l cm−1, and the reflective loss coeffi-cient =L 0.005R . The crystal length is 0.1 cm, and the value of g L0 varies from 2 to 30.

Figure 1 is a plot of the required laser threshold gain–length product as a function of the absorbing region small-signal transmission. As expected, the threshold gain–length prod-uct approaches zero as →T 10 , and becomes infinite as →T 0.0 Figure 2 shows how the normalized output power (absorbed power) varies as a function of T .0 Extraction threshold is at about =T 0.150 for a gain–length product of 2. Output power

increases with g0L, and the larger the g0L value the flatter the curves become. It is apparent that for the larger g0L values the performance is nearly independent of T0 over a wide range of values from about 0.2 to 0.9. Figure 3 shows the corresponding extraction efficiencies (total absorption efficiencies), reaching maximum values from 88–96%. As the small-signal transmis-sion →T 10 , the intracavity normalized laser intensity increases, and becomes very large. For zero intracavity passive losses, the intensity would become infinite. This is shown in figure 4 where the normalized positive intensity β2 is plotted as a func-tion of T .0 β2 is the intensity at the location of the interface between the active laser region and the absorbing region and represents the highest intensity in the resonator. The normal-ized intensity for g0L = 2 and =T 0.800 is β = 7.8,2 while for

=T 0.900 , it is β = 18.9.2 It is easy to find the other normalized intensities as well using equations in section 3 of part I. For g0L = 5, =T 0.800 , β = 22.4,2 and at T0 = 0.9, β = 33.5.2

It is obvious from the results presented in figures 1–4 that there is an optimum value of T0 that maximizes the extrac-tion (absorption) efficiency for each value of g0L, analogous to optimizing the outcoupler transmission in a standard laser resonator. For larger values of g L,0 the extraction efficiency increases as expected, the optimum T0 moves towards smaller values (higher small-signal absorption), and the normal-ized output intensity increases. Increasing the normalized intracavity intensity cannot proceed arbitrarily. Optically-induced-damage (OID) places a strict limit on the achievable

Figure 1. Laser threshold gain–length product as a function of absorbing region small-signal transmission.

Figure 2. Normalized output power as a function of absorbing region small-signal transmission for g L0 = 2, 5, 10, 20, and 30 (bottom curve to top).

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intracavity intensity, and in turn places a strict limit on the achievable extraction efficiency. In order to achieve good extraction efficiency, the intracavity CW intensity must be a multiple of the saturation intensity (at least equal to 1 satura-tion fluence, and preferably > 5 times the saturation fluence for good extraction efficiency with TEM00 Gaussian beams). For laser materials with a small saturation intensity, this is not difficult to achieve. Materials that have a saturation fluence close to or in excess of the OID limit, however, will always have limited extraction efficiency. Of particular interest here is the situation where an absorbing lasing element is placed inside a diode laser pump cavity. In that case, the OID of the diode output facet (referred to in the diode laser literature as coefficient of damage or COD) will limit the achievable extraction or absorption efficiency.

3. Laser operating schemes of interest

There are a number of intracavity pumped laser schemes that the Rigrod model presented in this paper can be applied to, and we present an overview of a few such systems here.

3.1. Yb, Er:glass laser-pumped intracavity to a Nd:YAG 946 nm laser

The first scheme we will consider is an 808 nm diode-pumped 946 nm Nd:YAG laser in which a Yb, Er:glass absorbing ele-ment is placed intracavity. Figure 5 shows an 808 nm pump

diode with an fast-axis-collimating (FAC) lens, pumping a Nd:YAG crystal on whose pump and output faces the coat-ings have been designed to suppress the laser transitions at 1064, 1319, and 1338 nm, and optimized for lasing at 946 nm. For good efficiency at 946 nm, the amplified spontaneous emission at 1064 nm should be minimized. The 946 nm reso-nator is formed by the left pumped Nd:YAG crystalline face and the VBG, whose reflectivity (diffraction efficiency) is at least 99% at 946 nm. The VBG is totally transmissive at 1550 nm. A Yb, Er:glass element is placed intra cavity to the 946 nm laser with the appropriate dielectric coatings to be antireflective (AR) at 946 nm on both faces and highly reflective (HR) at 1550 nm for the left face coating, and AR at 1550 nm for the right face coating. A separate outcoupler completes the Yb, Er:glass resonator. The Nd:YAG stimulated-emission

Figure 3. Extraction efficiency (absorption) as a function of absorbing region small-signal transmission for =g L 2,  5,  10,  20,  and 300

(bottom curve to top).

Figure 4. Normalized (+) intensity (β2) at interface of laser gain–crystal loss regions as a function of absorbing region small-signal transmission for =g L 2,  5,  10,  20,  and 30 0 (bottom curve to top).

Figure 5. 946 nm Nd:YAG laser intracavity pumping Yb, Er:glass laser.

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cross-section at 946 nm is roughly 11 times smaller than that at 1064 nm. This results in a much larger saturation intensity of ~33 kW cm−2. The saturation fluence is ~8.7 J cm−2. For CW operation, very good extraction efficiency requires that the intracavity 946 nm intensity be at least five times the satu-ration intensity for a Gaussian beam, resulting in an intensity of ~150 kW cm−2. This intensity is well within the OID limits of anti-reflection and high-reflector coatings for CW opera-tion of typically MW cm−2. For Q-switched operation of the Yb, Er:glass laser, typical pump pulsewidths of 2–3 ms are used. The OID threshold for 1 ns FWHM pulses at 1000 nm is typically 5–10 J cm−2. Using the usual pulsewidth to the 1/2 power scaling, OID thresholds at 2 ms can be expected to be in the kJ cm−2 regime, thus it appears that long-pulse 946 nm pumping of Yb, Er:glass lasers for the purpose of Q-switching in which the fluence produced is at least five times the satura-tion fluence is quite feasible with good extraction efficiency.

For the pumping scheme shown in figure 5, the thickness of the Yb, Er:glass element is adjusted to produce maximum absorption efficiency. It should be noted that unlike many laser crystals where the doping density can be adjusted at will, with Yb, Er:YAG the Yb–Er transfer efficiency depends strongly on the doping density of both the Yb and Er ions [1], and sig-nificantly reducing either one will have a deleterious effect on the laser efficiency. As a consequence, using thinned samples of standard laser glass compositions is preferred in this case. For most Yb, Er:glass compositions, it is not difficult to obtain absorption coefficients at 946 nm in the range of 4–6 cm [1]. For =T 0.800 , for example, Yb, Er:glass thicknesses of only 0.4–0.6 mm are needed. Thin Yb, Er:YAG elements are desir-able because of the poor thermal conductivity and large heat fraction, and thin elements can substantially reduce the maxi-mum temperature. While the configuration shown in figure 5 uses a single diode laser and a longitudinal pumping scheme, it is apparent that the single diode may be replaced by an 808 nm bar with much higher output power, and the Yb, Er:glass ele-ment and VBG can be extended transversely so that the absorp-tion takes place along a thin line of ~1 cm in length and with a thickness of 100–200 µm by using an FAC lens. As a conse-quence it is possible to build transversely-pumped Yb, Er:glass lasers with thin elements and much higher output average powers than can be obtained at present. It should be pointed out that the gain–length product g0L for single diode-pumped

Yb, Er:glass lasers is generally small, due primarily to the low (~5 × 10−21 cm−2) emission cross-section of Er, and L is typi-cally in the range of 0.05–0.5. As shown in figure 6, where the extraction (total absorption) efficiency is shown as a function of the small-signal transmission for various g0L values, low gain–length products result in significantly lower extraction efficien-cies. For gain–length products of 0.05 and 0.5, the extraction efficiencies are about 0.29 and 0.63, respectively. Because Yb, Er:glass lasers are usually pumped near 925 nm to enhance the laser temperature operating range, pump powers using single-emitter diodes are currently limited to a few watts. Increasing the single-emitter power to achieve a larger g0L is thus not likely near-term unless OID thresholds are raised for the diode output facet. As previously discussed, the most promising method for achieving large g0L values is to use a 1 cm long Nd:YAG slab to transversely pump a Yb, Er:glass thin slab. VBGs have been used previously to narrow and lock the wavelength of diode bars [2], and can be used here to achieve the same result, pump-ing at 808 nm and lasing at 946 nm with Nd:YAG. Pump powers available from single bars at 808 nm are over 80 W. Using this pumping method, a thin 1 cm long line of excitation results in the Yb, Er:glass element, resulting in a much larger g0L value. The extraction efficiency from such a device can exceed 0.9.

3.2. Yb, Er:glass laser-pumped intracavity to a 977 nm diode laser

Yb, Er:glass laser materials have a peak zero- phonon absorption line near 977 nm. In addition, powerful

Figure 6. Extraction (total absorption) efficiency as a function of small-signal transmission for g0L values of 0.05, 0.25, 0.5, 1, and 2 (lowest to highest peak value). The highest extraction efficiency of about 0.83 for a small-signal transmission of ~0.9 corresponds to g0L = 2. A reflectivity of 0.99 was assumed for the VBG, and 0.995 for the high-reflector.

Figure 7. 977 nm diode laser intracavity pumping Yb, Er:glass laser.

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InGaAs–InGaP–InGAs based diode lasers with high effi-ciency are available. Pumping at 977 nm results in about a 37% quantum defect, while pumping at 946 nm results in a 39% quantum defect. In figure 7, we show a pumping scheme similar to figure 5, but with the 946 nm Nd:YAG laser replaced with a direct diode laser. The 977 nm diode laser is injection-locked with a VBG. A collimated beam is obtained from the diode laser using an FAC lens. While schemes like this have been recently demonstrated to nar-row and lock diode lasers, VBG reflectivities of ~20% are typically used with diode facet reflectivities of 2% [2]. Here we assume a 99% reflective VBG. As in the previous case, we place a Yb, Er:glass thin element in the resonator to absorb the 977 nm pump light, and form a 1550 nm reso-nator between the left face of the Yb, Er:glass element and the outcoupler or a second grating in the VBG.

Unlike the previous case where the beam size is more or less uniform throughout the structure, here the FAC lens is used to expand and collimate the diode beam in the fast-axis direction from ~1 μm to 100–200 μm diameter. In the slow axis, a slow-axis collimating (SAC) lens may or may not be used, depending on the exact configuration chosen. As a result, the highest intensities achieved in the system are at the diode output facet and the diode HR facet. Since the diode output facet is usually the most susceptible to damage, the OID limit of the system is located there. COD values for diode facets vary widely, typically from 1 to 20 MW cm−2, the exact

value depending upon the diode material system, optical coat-ing, and facet passivation techniques [3], although COD val-ues have continued to rise since that study. A typical value for the saturation intensity of a GaAs semiconductor laser is 0.5 MW cm−2 [4]. The maximum COD threshold for InGaAs/InGaP/GaAs lasers operating near 977 nm is ~20 MW cm−2. Consequently, to avoid OID the normalized intensity at the facet should not exceed a value of about 40.

We have used the Schindler model presented in part I to simulate the performance of a diode resonator formed as shown in figure 7 between the diode HR facet and the external VBG. We assume a broad area design, and idealize the extract-ing laser beam as flat-topped. The gain–length product g L0 typically takes a value in the range of 10–20 for a single diode with a 1 mm cavity length. Here we assume the loss coefficient loss αL to be 1 cm, resulting in a loss–length product of 0.1 for a 1 mm cavity length. Figure 8 shows the obtained extraction efficiency as a function of the small signal transmission. For g0L = 10 (lower curve), T0 optimizes at about 0.6, whereas for g0L = 20, the optimum T0 is about 0.30. It should be noted that for g0L = 10, a T0 value as high as 0.7 still results in about 0.80 extraction efficiency. For g0L = 20, a T0 value as high as 0.75, the extraction efficiency can be as high as 0.80.

In figure 9, the normalized intensityβ2 at the diode output facet is shown as a function of the small-signal transmission; it can be seen that for a T0 value of ~0.80, the normalized power reaches a value of about 40 for g0L = 20, or the limit imposed

Figure 8. Extraction efficiency as a function of small-signal transmission for a diode laser with gain–length products of 10 (lower curve) and 20 (top curve), an HR facet reflectivity of 99.5%, a loss–length product of 0.1, and a VBG reflectivity of 99%.

Figure 9. Normalized (+) intensity β2, as a function of small-signal transmission, for diode gain–length products of 10 (bottom curve) and 20 (top curve), and αLL = 0.1. Also shown (horizontal curve) is the normalized COD value assumed as 40.

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by the assumed COD threshold. This value is not reached in this case for g0L = 10.

If we operate the diode laser with T0 ~0.70, pumping at 977 nm results in much decreased Yb, Er:glass thicknesses, since the absorption coefficient at 977 nm is ~5 times that at 946 nm. The absorption coefficients at 977 nm can then be typically 18–28 cm. For a small signal absorption of 0.7, the Yb, Er:glass absorbing elements can then have a thickness in the range of 127–200 μm. If we were to operate the diode laser with T0 = 0.30, then the thickness increases to a range of 430–670 µm. From equation (129) of part I, the average number of photon absorption round-trips is about 5.

In the case discussed here, we adjust the absorption by using thin Yb, Er:glass elements, a development that prom-ises to make thermal effects in very low conductivity laser materials more manageable, and result in higher-average power operation. It is clear that in order to enhance the perfor-mance even further, larger COD values than that assumed here (20 MW cm−2) must be obtained. In one recent publication, for example, a single-emitter 915 nm laser was demonstrated to have a COD value >25.5 MW cm−2 [5], thus it seems that there is considerable room for improvement. Larger COD values will also have a very beneficial effect on the use of dilute laser materials where the density of the ion may be reduced, result-ing in larger T0 values.

3.3. Er:YAG pumped-laser intracavity to a 1530 nm diode laser

Near-infrared lasers such as Er:YAG and Ho:YAG may also be pumped using the intracavity scheme discussed in this paper. Direct pumping into the 4I13/2 upper laser level in Er:YAG has been demonstrated at room [6] and cryogenic temperatures [7]. In [7], pumping was achieved using 1530 nm InGaAsP/InP diode bars, with a 9 nm FWHM spectral output. The 4I15/2 to 4I13/2 absorption transitions from the ground-state manifold directly into the upper laser manifold have weak cross-sec-tions (~1 × 10–20 cm−2) with FWHM absorption bandwidths at room temperature of a few nm, becoming <1 nm at 77 K. Er:YAG crystals with cm lengths are needed to obtain rea-sonable absorption, with double-passing often used. The use of a VBG to narrow the operation of the pump diode, as well as the multi-passing inherent in the method discussed in this paper to increase absorption, can greatly improve the

performance of Er lasers at any temperature. We consider the case where Er:YAG is used at room temperature. We further assume a low Er doping density of 0.5 at.% (6.925 × 1019 ions cm−3). Figure  10 shows the room-temperature absorp-tion cross-sections of Er:YAG from 1400–1700 nm. To obtain a low heat fraction, we assume the same pump wavelength of about 1530 nm as [7], and pump directly into the 4I13/2 level from the 4I15/2 ground-state level. This corresponds to the absorption wavelength of 1533.6 nm in figure 9, with a peak absorption cross-section of 1.30 × 10–20 cm2. The absorp-tion coefficient is then 0.90 cm–1 for monochromatic pump light. An external VBG or a fiber Bragg grating can be used to narrow down the diode emission FWHM bandwidth from a typical 9–10 nm to ~1 nm [8], allowing the efficient pumping of the 1533.6 nm absorption line, which has a FWHM absorp-tion of about 3.8 nm. Diode lasers with an internal grating [9] have also been demonstrated with FWHM bandwidths <1 nm at 1532 nm. If we assume monochromatic light for now, 1 mm and 2 mm thick absorbing Er:YAG crystals then have T0 val-ues of 0.91 and 0.84, respectively.

In figure 11 we show the pumping scheme considered here. An InP diode laser operating at 1530 nm has an FAC lens attached to collimate the diode output in the fast-axis direc-tion. A VBG is then placed a short distance away to form a diode resonator with narrow band output. An Er:YAG crystal

Figure 10. Absorption cross-sections of Er:YAG as a function of wavelength at room temperature for the spectral region 1400–1700 nm.

Figure 11. Intracavity-transverse-pumped Er:YAG laser at room temperature using a volume Bragg grating.

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with a thickness of 1–2 mm is placed intracavity and has a single-pass transmission T0. The Er:YAG faces orthogonal to the pump beam are AR-coated at 1530 nm. In this example, lasing is in a direction orthogonal to the pump beam, hence this is an intracavity side-pumped configuration. Because the doping is very low, transverse gain and heat inhomogenei-ties are minimized, particularly since the absorbing crystal is multiply-passed in both absorbing directions. The end faces of the Er:YAG crystal are dielectrically-coated. One flat end is HR-coated at 1645 nm, while the other is AR-coated, allowing the use of an external outcoupler whose reflectivity at 1645 is optimized to maximize output power. Because Er:YAG is a quasi-three-level kinetics system, the transverse extent of the pump beam should be matched to the Er:YAG crystal length to avoid absorbing regions.

For a dilutely-doped Er:YAG crystal pumped at 1530 nm and lasing around 1645 nm, the quantum defect, which is in this case the heat fraction, takes the value of 7%, which is very low and comparable to typical Yb lasers. Using an orthogo-nal pump and laser beams also solves the problems associ-ated with fabricating good dielectric coatings on the resonator optics with pump and lasing wavelengths that are spectrally very close to one another by allowing the use of completely separated optics.

Diodes built using the InP/GaInAsP material system, las-ing around 1540 nm, are not as well developed as the InGaAs system (920–980 nm) used for fabricating high efficiency (>70% wallplug efficiency) single-emitters and bars, but can now achieve >50% wallplug efficiency and have the highest COD values reported [10]. It is thus likely that very efficient dilute Er lasers can be demonstrated using the intracavity diode-pumping scheme discussed here.

3.4. InGaAs (920–980 nm) diode-pumped Yb lasers

Our last example is InGaAs diode-pumped Yb lasers. Yb has been shown to be a nearly ideal laser material because of the absence of concentration-quenching, upconversion, and excited-state absorption [11]. At room temperature, the effec-tive stimulated-emission cross-section for Yb:YAG is small, ~1.9 × 10–20 cm−2, and the saturation fluence is large, about 9.9 J cm−2. At the cryogenic temperature of 77 K, however, the cross-section increases to about 1.2 × 10–19 cm−2, about

6.2 times increase, and the saturation fluence is reduced to 1.6 J cm−2 [12]. Yb:YAG is normally diode-pumped in the broad absorption band near 940 nm at both room and cryo-genic temperatures; the zero-phonon line near 968 nm has a bandwidth of about 2.8 nm at room temperature, amenable to pumping with VBG-narrowed diodes, but at 77 K the band-width narrows to <0.1 nm [13], so narrow that even VBG nar-rowed diodes cannot pump the band efficiently. Nevertheless, if a method can be found to pump the zero-phonon line at 968 nm, the quantum defect is reduced from 0.081 to 0.059, and a substantial decrease in the heat load is obtained. Investigation of other Yb-based laser materials, for example, has revealed a number that may be pumped at the zero- phonon line. For Yb:LuAG, for example [14], whose absorption cross-sections at 82 and 295 K are shown in figure 12, where the line occurs at about 969.5–969.7 nm, at 295 K the FWHM bandwidth is 2.75 nm while at 82 K the bandwidth is reduced to 1.1 nm. For Yb:Lu2O3, the bandwidth at 295 K was mea-sured to be about 4 nm [15]. It thus appears quite feasible to efficiently pump the zero-phonon line for both Yb:LuAG and Yb:Lu2O3, using VBG-narrowed InGaAs diode sources. In particular, it appears that the type of intracavity pumping scheme examined in this article will work quite well and result in efficient diode-pumped laser sources.

Figure 13 shows one configuration that can be used to dem-onstrate very high wallplug efficiency CW Yb:LuAG lasers. A diode bar operating at about 969.6 nm and whose output is collimated using an FAC lens is used to transversely pump a Yb:LuAG crystal sandwiched between two high thermal-conductivity plates like sapphire, which has a very good index match to Yb:LuAG. For room-temperature operation, the length of the Yb:LuAG crystal is matched to the ~1 cm length of the diode bar to avoid absorbing regions. A VBG, whose reflection coefficient (diffraction efficiency) exceeds 0.99 is used to form the pump resonator; the rear diode bar facets are also highly reflecting while the output facets are AR coated. The laser pump resonator thus formed is used to intracavity pump Yb:LuAG, which lases transversely to the pump resona-tor. The laser resonator is formed using a flat HR reflector at one end and a partial reflector (outcoupler or OC) at the oppo-site end. The reflectivity of the OC is adjusted to obtain opti-mum extraction efficiency from the Yb:LuAG crystal. Other similar configurations are possible, which use much longer

Figure 12. Yb:LuAG absorption cross-sections as a function of wavelength from 900–1100 nm at temperatures of 82 (green) and 295 K (blue).

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crystals with a plurality of pump bars and VBGs. In addition, it is quite possible to produce longitudinally-pumped configu-rations as well. The lasing crystal can be a straight-through or zig-zag slab to average any transverse inhomogeneties and increase the extraction efficiency, and the slab may be oper-ated at room or cryogenic temperatures.

As we have seen previously in connection with section 2, figures 8 and 9, InGaAs lasers can be used in intracav-ity pumping schemes and very high extraction efficiencies obtained over a broad range of T0 values, due to the low pas-sive losses as well as relatively large COD values. Using the peak absorption cross-section value at ~969 nm as shown in figure 12, ~2.5 x 10–20 cm−2, the absorption coefficients for T0 values of 0.3 and 0.7 are 12.04 cm–1 and 3.57 cm–1, respectively (absorption depth is equal to 1 mm). If we use an FAC lens that results in about a 1 mm 1/e2 spot diameter, then a 1 mm absorption depth can be obtained with mod-est 3.38 and 1.0 at.% Yb concentrations for T0 values of 0.3 aned 0.7 respectively. Unlike other laser materials based on Nd, for example, that display concentration-quenching and upconversion, for Yb the concentration can be varied at will to obtain the T0 value desired, at least up to concentrations (~10 at.% for Yb:YAG) where anecdotal evidence indicates that the presence of impurities can become important and deleteriously affect performance. To obtain absorption that is as uniform as possible in the transverse direction, however, larger T0 values are preferred due to multi-passing of the pump beam. Low T0 values result in less uniform transverse absorp-tion profiles since only one or two passes occur. In recent years, diode bars operating around 976 nm have produced the largest wallplug efficiencies, 0.76 at room temperature [14]. 946 nm Nd:YAG lasers have been used to produce a Yb:YAG cryogenic laser operating near 77 K, which has been shown to be capable of a photon–photon efficiency of 1.0 and very high slope efficiency, limited only by the quantum defect [16]. An optical–optical efficiency of >0.84 was demonstrated. As a consequence, in optimized transverse diode-pumped systems

such as the one shown in figure 13, an ultimate wallplug effi-ciency of >62% can be obtained. This of course assumes no further improvements in the efficiency of the diode lasers, and that present diode lasers can be further optimized to oper-ate in the type of configuration shown in figure 13. It is worth pointing out that the transverse pumping scheme of figure 13 is inherently scalable since resonators can be extended in the transverse direction to accommodate more diode bars, and other configurations can be demonstrated that use the angular multiplexing of N diodes around the perimeter of the laser crystal as well.

4. Discussion

In this article, we have presented modeling of absorbing laser crystals within or intracavity to a laser resonator, using the Rigrod approach. We have shown that it is possible to obtain excellent absorption efficiency using thin dilute absorbing laser elements, due to the finite photon lifetime of the reso-nator. The application of the method presented here to the pumping of thin Yb, Er:glass discs can result in high absorp-tion efficiency as well as more manageable thermal effects. This will likely make higher repetition rate and higher aver-age power Yb, Er:glass lasers possible. The highest efficiency laser-pumped laser devices that can be demonstrated at pres-ent are likely Yb devices intracavity-pumped at the zero-phonon line at 976–979 nm with highly efficient diode lasers. High efficiency Er:YAG and Ho:YAG devices are likely in the near future as well. In this article, we have presented idealized results and have for brevity ignored other important techni-cal issues and approaches. Nevertheless, we believe that the optimized intracavity pumping of solid-state lasers has a very bright future, and further accelerates the integration of diode and solid-state laser technology to produce new devices that will have high wallplug efficiency as well as scalability to high average and peak powers.

Figure 13. Transverse intracavity diode-pumped CW Yb:LuAG laser using high power ~970 nm diode bar and volume Bragg grating.

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References

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Laser Phys. 24 (2014) 085003