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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 17 — Aug. 15, 2011
  • pp: 15652–15668
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Optimization of end-pumped, actively Q-switched quasi-III-level lasers

Jan K. Jabczynski, Lukasz Gorajek, Jacek Kwiatkowski, Mateusz Kaskow, and Waldemar Zendzian  »View Author Affiliations


Optics Express, Vol. 19, Issue 17, pp. 15652-15668 (2011)
http://dx.doi.org/10.1364/OE.19.015652


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Abstract

The new model of end-pumped quasi-III-level laser considering transient pumping processes, ground-state-depletion and up-conversion effects was developed. The model consists of two parts: pumping stage and Q-switched part, which can be separated in a case of active Q-switching regime. For pumping stage the semi-analytical model was developed, enabling the calculations for final occupation of upper laser level for given pump power and duration, spatial profile of pump beam, length and dopant level of gain medium. For quasi-stationary inversion, the optimization procedure of Q-switching regime based on Lagrange multiplier technique was developed. The new approach for optimization of CW regime of quasi-three-level lasers was developed to optimize the Q-switched lasers operating with high repetition rates. Both methods of optimizations enable calculation of optimal absorbance of gain medium and output losses for given pump rate.

© 2011 OSA

1. Introduction

The quasi-III-level (QTL) lasers belong to the most known, explored, wide-spread group of solid-state laser sources (see e.g [1

1. H. Svelto, Principles of Lasers (Plenum Press, 1998).

11

11. M. Eichhorn, “Quasi-three-level solid-state lasers in the near and mid infrared based on trivalent rare earth ions,” Appl. Phys. B 93(2-3), 269–316 (2008). [CrossRef]

].). The main, diversifying property of QTL lasers, namely the occurrence of temperature dependent reabsorption losses [2

2. W. P. Risk, “Modeling of longitudinally pumped solid-state lasers exhibiting reabsorption losses,” J. Opt. Soc. Am. B 5(7), 1412–1423 (1988). [CrossRef]

,3

3. T. Y. Fan, “Optimizing the efficiency and stored energy in quasi-three-level lasers,” IEEE J. Quantum Electron. 28(12), 2692–2697 (1992). [CrossRef]

] is caused by partial occupation of lower laser level. The two main schemes of excitation of QTL gain media can be defined as follows: direct, resonant pumping into upper laser level and indirect pumping. The simplest, the most known and explored example of directly pumped QTL lasers are Yb doped ones [3

3. T. Y. Fan, “Optimizing the efficiency and stored energy in quasi-three-level lasers,” IEEE J. Quantum Electron. 28(12), 2692–2697 (1992). [CrossRef]

11

11. M. Eichhorn, “Quasi-three-level solid-state lasers in the near and mid infrared based on trivalent rare earth ions,” Appl. Phys. B 93(2-3), 269–316 (2008). [CrossRef]

]. Such scheme of pumping characterizes also Er lasers (see e.g [11

11. M. Eichhorn, “Quasi-three-level solid-state lasers in the near and mid infrared based on trivalent rare earth ions,” Appl. Phys. B 93(2-3), 269–316 (2008). [CrossRef]

,12

12. I. Kudryashov, D. Garbuzov, and M. Dubinskii, “Latest developments in resonantly diode-pumped Er:YAG lasers,” in Laser Source Technology for Defence and Security III, Proc. SPIE 6552, 65520K (2007). [CrossRef]

].) operating at 1.6 μm wavelength pumped at ~1.5 μm and Ho lasers resonantly pumped at 1.9 μm (see e.g [13

13. B. M. Walsh, “Review of Tm and Ho Materials; Spectroscopy and Lasers,” Laser Phys. 19(4), 855–866 (2009). [CrossRef]

15

15. J. Kwiatkowski, J. K. Jabczynski, Ł. Gorajek, W. Zendzian, H. Jelínková, J. Sulc, M. Nemec, and P. Koranda, “Resonantly pumped tunable Ho:YAG laser,” Laser Phys. Lett. 6(7), 531–534 (2009). [CrossRef]

].). In a case of indirect excitation, the absorption / pumping band is significantly shifted to shorter wavelengths and several processes of energy transfer (cross-relaxation, migration) have to be exploited to efficiently populate the upper laser level. The classical example of such a case are 2-μm Cr:Tm:Ho:YAG lasers pumped by flash lamps in the visible region (see e.g [16

16. N. P. Barnes, K. E. Murray, and M. G. Jani, “Flash-lamp-pumped Ho:Tm:Cr:YAG and Ho:Tm:Er:YLF lasers: modeling of a single, long pulse length comparison,” Appl. Opt. 36(15), 3363–3374 (1997). [CrossRef] [PubMed]

].). The other important examples of indirect pumping are diode pumped Tm [17

17. E. C. Honea, R. J. Beach, S. B. Sutton, J. A. Speth, S. C. Mitchell, J. A. Skidmore, M. A. Emanuel, and S. A. Payne, “115-W Tm:YAG diode-pumped solid-state laser,” IEEE J. Quantum Electron. 33(9), 1592–1600 (1997). [CrossRef]

25

25. J. K. Jabczynski, L. Gorajek, W. Zendzian, J. Kwiatkowski, H. Jelinkova, J. Sulc, and M. Nemec, “Actively Q-switched thulium lasers,” in Advances in Solid State Lasers: Development and Applications IN-TECH, Vienna, (2010).

] and co-doped Yb,Er and Tm,Ho bulk and fibre oscillators [26

26. S. D. Jackson, “The spectroscopic and energy transfer characteristics of the rare earth ions used for silicate glass fibre lasers operating in the shortwave infrared,” Laser & Photon. Rev. 3(5), 466–482 (2009). [CrossRef]

32

32. G. Rustad and K. Stenersen, “Modeling of laser-pumped Tm and Ho lasers accounting for up conversion and ground-state depletion,” IEEE J. Quantum Electron. 32(9), 1645–1656 (1996). [CrossRef]

].

The goal of our work was to develop the simple model of actively Q-switched, end-pumped laser based on QTL gain medium. Such problem was analyzed previously in many papers (see e.g [3

3. T. Y. Fan, “Optimizing the efficiency and stored energy in quasi-three-level lasers,” IEEE J. Quantum Electron. 28(12), 2692–2697 (1992). [CrossRef]

,4

4. C. D. Nabors, “Q-switched operation of quasi-three-level lasers,” IEEE J. Quantum Electron. 30(12), 2896–2901(1994). [CrossRef]

, 25

25. J. K. Jabczynski, L. Gorajek, W. Zendzian, J. Kwiatkowski, H. Jelinkova, J. Sulc, and M. Nemec, “Actively Q-switched thulium lasers,” in Advances in Solid State Lasers: Development and Applications IN-TECH, Vienna, (2010).

, 29

29. J. M. Sousa, J. R. Salcedo, and V. V. Kuzmin, “Simulation of laser dynamics and active Q-switching in Tm,Ho:YAG and Tm:YAG lasers,” Appl. Phys. B 64(1), 25–36 (1996). [CrossRef]

31

31. O. A. Louchev, Y. Urata, and S. Wada, “Numerical simulation and optimization of Q-switched 2 mum Tm,Ho:YLF laser,” Opt. Express 15(7), 3940–3947 (2007). [CrossRef] [PubMed]

].). In our work additional effects, as time dependent ground state depletion (GSD) and Energy Transfer Up-Conversion (ETU) processes [32

32. G. Rustad and K. Stenersen, “Modeling of laser-pumped Tm and Ho lasers accounting for up conversion and ground-state depletion,” IEEE J. Quantum Electron. 32(9), 1645–1656 (1996). [CrossRef]

34

34. Y. F. Chen, Y. P. Lan, and S. C. Wang, “Modeling of diode-end-pumped Q-switched solid-state lasers: influence of energy-transfer upconversion,” J. Opt. Soc. Am. B 19(7), 1558–1563 (2002). [CrossRef]

] have been considered. Moreover we have proposed the new procedure for output energy optimization. The model consists of two stages: excitation /non-generation/ process and Q-switching stage. For the first stage, the semi-analytical model based on solution of averaged rate equation, taking into consideration the transient, time-dependent ground state depletion GSD and ETU processes, was developed. The upper laser level population, dependent on pumping and gain medium parameters, can be determined for different pump durations. Moreover, the quasi-stationary inversion was determined and analyzed for different ETU parameters.

The second part of model describing giant pulse formation and energy extraction process was based on modified for QTL lasers analytical solution of rate equations. Based upon that approach the optimization procedure following Degnan’s theory [35

35. J. Degnan, “Theory of the optimally coupled Q-switched laser,” IEEE J. Quantum Electron. 25(2), 214–220 (1989). [CrossRef]

] was developed for end-pumped oscillators. For quasi-stationary inversion, the worked out, new, optimization procedure enables estimation of maximal energy density extractable for given pump parameters. For comparison with Q-switching regime, the new optimization procedure of CW regime of QTL laser was proposed. The optimal gain medium length and out-coupling losses were determined for wide range of pump and resonator parameters. The effect of ETU on output energy was investigated.

2. Pumping stage

In a case of QTL laser, the additional aperture losses, caused by reabsorption of laser mode on the un-pumped outer regions of gain medium can appear [11

11. M. Eichhorn, “Quasi-three-level solid-state lasers in the near and mid infrared based on trivalent rare earth ions,” Appl. Phys. B 93(2-3), 269–316 (2008). [CrossRef]

,25

25. J. K. Jabczynski, L. Gorajek, W. Zendzian, J. Kwiatkowski, H. Jelinkova, J. Sulc, and M. Nemec, “Actively Q-switched thulium lasers,” in Advances in Solid State Lasers: Development and Applications IN-TECH, Vienna, (2010).

,36

36. T. Y. Fan, “Aperture guiding in quasi-three-level lasers,” Opt. Lett. 19(8), 554–556 (1994). [CrossRef] [PubMed]

]. Thus, to minimize threshold and obtain high efficiency, the optimal mode matching of pumping beam and laser mode is crucial. In further analysis we assumed that laser mode area S 0 is equal to the pump beam area averaged over gain medium.

As it was shown in models of CW generation of QTL laser [5

5. R. J. Beach, “CW Theory of quasi-three level end-pumped laser oscillators,” Opt. Commun. 123(1-3), 385–393 (1996). [CrossRef]

,7

7. G. L. Bourdet, “Theoretical investigation of quasi-three-level longitudinally pumped continuous wave lasers,” Appl. Opt. 39(6), 966–971 (2000). [CrossRef] [PubMed]

11

11. M. Eichhorn, “Quasi-three-level solid-state lasers in the near and mid infrared based on trivalent rare earth ions,” Appl. Phys. B 93(2-3), 269–316 (2008). [CrossRef]

], the absorption efficiency is clamped on the value reached at the threshold, which differs not much from the value of small signal ‘spectroscopic’ absorption efficiency. The model of CW operation of QTL laser with the new optimization procedure is given in p.4.1. The main difference for the pumping at non-lasing stage, comparing to pumping during CW generation is the increasing depletion of ground state, significantly effecting on the available upper level population, inversion and gain for Q-switched lasers. Moreover, for high populations of upper laser level the role of nonlinear relaxation via ETU processes begins to be important in certain gain media [32

32. G. Rustad and K. Stenersen, “Modeling of laser-pumped Tm and Ho lasers accounting for up conversion and ground-state depletion,” IEEE J. Quantum Electron. 32(9), 1645–1656 (1996). [CrossRef]

34

34. Y. F. Chen, Y. P. Lan, and S. C. Wang, “Modeling of diode-end-pumped Q-switched solid-state lasers: influence of energy-transfer upconversion,” J. Opt. Soc. Am. B 19(7), 1558–1563 (2002). [CrossRef]

].

To analyze these effects, let us return to the basics of laser physics i.e. to rate equations set for non-lasing case defined at point z in the crystal and time t. We neglect here transverse spatial distribution of pump beam and operate with averaged across beam pump densities I p(z,t) and population of upper 2-nd level N 2 (z,t). After some transformations, problem can be defined as follows:
{dIp±dz=(NtotγpN2)σpIp±dN2dt=  A  R2N2τkETUN22,
(1)
where: Ip± − pump density in ( + ) forward, (-) backward directions, N2 − population of 2-nd upper laser level, k etu − coefficient of ETU [32

32. G. Rustad and K. Stenersen, “Modeling of laser-pumped Tm and Ho lasers accounting for up conversion and ground-state depletion,” IEEE J. Quantum Electron. 32(9), 1645–1656 (1996). [CrossRef]

,34

34. Y. F. Chen, Y. P. Lan, and S. C. Wang, “Modeling of diode-end-pumped Q-switched solid-state lasers: influence of energy-transfer upconversion,” J. Opt. Soc. Am. B 19(7), 1558–1563 (2002). [CrossRef]

], τ − lifetime of upper laser level. The information about structure of QTL medium is contained in un-dimensional parameters: A, γp (see details [20

20. G. L. Bourdet and G. Lescroart, “Theoretical modeling and design of a Tm:YVO4 microchip lasers,” Opt. Commun. 149(4-6), 404–414 (1998). [CrossRef]

]).

For indirect pumping with very fast relaxation and negligible population of higher than 2-nd levels (N2N3,N4 and N1NtotN2 see Fig. 1
Fig. 1 Scheme of indirect pumping.
) γp = 1 and A is the measure of quantum efficiency (including for example cross-relaxation) of pumping process between N1N2. For a case of direct pumping A = f a’, γp=1+fb'/fa'where f a’, f b’ are fractional occupations of (a’) lower and (b’) upper levels of pumping process (see e.g [5

5. R. J. Beach, “CW Theory of quasi-three level end-pumped laser oscillators,” Opt. Commun. 123(1-3), 385–393 (1996). [CrossRef]

,8

8. G. L. Bourdet, “New evaluation of ytterbium-doped materials for CW laser applications,” Opt. Commun. 198(4-6), 411–417 (2001). [CrossRef]

,9

9. C. Lim and Y. Izawa, “Modeling of end-pumped CW quasi-three-level lasers,” IEEE J. Quantum Electron. 38(3), 306–311 (2002). [CrossRef]

].), R 2 – pumping rate into 2-nd upper level defined as follows:
R2=N1σpIp++Iphνp=(NtotγpN2)σpIp++Iphνp,
(2)
where: h – Planck constant, ν p = c/λ p – frequency of pump wave, c – light speed, λ p – pump wavelength. Let us introduce the un-dimensional variables t’, ζ and functions x 2, ip± defined as follows:
t'=t/τ;ζ=α0z;x2=N2/Ntot;ip±=Ip±/Ip,sat,
(3)
where: α 0 – absorption coefficient, I p,sat – saturation density for pump wavelength defined for indirect and direct pumping as follows:

  • a) for indirect pumping
    α0=σpNtot;Ip,sat=hνpA​  σpτ   ;γp=1;A=η122,
    (4a)
  • b) for direct pumping
    α0=fa'σpNtot;Ip,sat=hνpfa'​  σpτ   ;γp=1+fb'/fa';A=1.
    (4b)

After a few transformations, substituting (2), (3), (4) into (1) we obtain the following equation set:
{dip±dξ=(1γpx2)ip±dx2dt'=(1γpx2)(ip++ip)x2kx22,
(5)
where:

k=kETUτNtot.
(6)

Further, we perform averaging along rod length (integration in the range (0, a 0) where a 0 = αp l 0 –absorbance of gain medium. Let us notice that:

0a0(1γpx2)(ip++ip)dξ=0a0dip0a0dip+=ip+(0)ip+(a0)+ip(a0)ip(0).
(7)

From 2-nd equation of (5) we derived final pumping equation describing dynamics of y 2 averaged relative population of 2-nd laser level as follows:
dy2dt'=rp(y2)y2ky22,
(8)
where: y 2 – averaged relative population of 2-nd laser level defined as follows:

y2=x2¯=a010a0x2dξ.
(9)

The function rp(y2) is the relative averaged pump rate. Let us consider two cases: i) pumping from 1 (left) end of rod with R p ≥ 0 – reflection coefficient of the latter one, ii) pumping from both ends with R p = 0.

Taking into consideration (7) we have derived the following formulae on pumping rate function:
rp(y2)=ip,0a0{η0(a0(1γpy2))[1+Rp(1η0(a0(1γpy2)))]for1endpump2η0(a0(1γpy2))               for2endpump,
(10)
where: i p,0 – relative pump density, η 0(x) – initial absorption efficiency defined as follows:

ip,0=P0SpIp,sat;η0(x)=1exp(x).
(11)

P0 –pump power, Sp – averaged area of pump beam.

2.1. Semi-analytical solution of pumping equation

Preliminary analysis of (8) shows, that with increase in pumping time t’, y 2 increases, causing depletion of ground state and decrease in pump rate rp(y2). The 1-st order differential Eq. (8) does not have analytical solution for function rp(y2). However, for constant coefficients, it is the nonlinear differential equation of Ricatti type [37

37. I. N. Bronstein, K. A. Semendjajew, G. Musiol, and H. Muehling, Taschenbuch der Mathematik, (Verlag Harri Deutsch 2001)

] and has the following analytical solution:
y2(Δt')=β1(β2ky2,0)exp(β1Δt')β2(β1ky2,0)exp(β2Δt')κ[(β2ky2,0)exp(β1Δt')(β1ky2,0)exp(β2Δt')],
(12)
where: y 2,0 = y 2(t 0’) – initial value of y 2 for time t 0’ ; Δt’ = t’- t 0’,

β1=12(1+1+4kp0),β2=12(11+4kp0),p0=rp(y2,0).
(13)

For sufficiently long pump duration for given parameters of pump and gain medium we can achieve quasi-stationary state of 2-nd level population y 2,st. In such a case pump rate is balanced by linear and nonlinear (via ETU effects) relaxation. The value of y 2,st can be determined as a solution of transcendent equation:

rp(y2,st)y2,stky2,st2=0.
(15)

The quasi-stationary population y 2,st determines the maximal inversion, gain and stored energy for given set of pumping parameters.

3. Analytical model of Q-switching regime for QTL laser

After finishing of pumping stage with temporal scale of parts of lifetime τ, the high loss of a cavity is immediately switched off to low value, enabling the giant pulse formation. For analysis of such a case, let us start from the modified for QTL medium averaged rate equation set:
{dϕldt=Φltrt(2σel0γl(N¯2βlNtot)xOCδpas)dN¯2dt=AR¯2N¯2τkETUN¯22Φltrtlcl0(2σel0γl(N¯2βlNtot)),
(16)
where:
γl=(fa+fb)/fb;βl=fa/(fa+fb)=(γl1)/γl,
(17)
fa, fb – are fractional occupations of (a) lower and (b) upper levels of laser transition (see e.g [3

3. T. Y. Fan, “Optimizing the efficiency and stored energy in quasi-three-level lasers,” IEEE J. Quantum Electron. 28(12), 2692–2697 (1992). [CrossRef]

5

5. R. J. Beach, “CW Theory of quasi-three level end-pumped laser oscillators,” Opt. Commun. 123(1-3), 385–393 (1996). [CrossRef]

].), lc – cavity length, trt = 2lc/c – round trip time of cavity, σe – emission cross section, l0 – gain medium length, Ntot – concentration of dopant, N2¯ – averaged over gain medium occupation of upper level, Φl – laser field density, xOC=ln(1TOC) – out-coupling losses, T oc – output transmission, δ pas – passive losses of cavity.

We assume here, that processes of stimulated emission are much faster than pumping rate and linear and nonlinear relaxation. Let us change to un-dimensional variables and quantities as follows:

θ=t/trt   ;   Δy2=   (N¯2βlNtot)/Ntot=y2βl;   ϕl=Φl/Ntot.
(18)

After substitution of (25) to (23) and neglecting the 3 first slowly varying components in the 2-nd Eq. (23) we obtain the well-known form of simplified rate equation set (see e.g [1

1. H. Svelto, Principles of Lasers (Plenum Press, 1998).

,11

11. M. Eichhorn, “Quasi-three-level solid-state lasers in the near and mid infrared based on trivalent rare earth ions,” Appl. Phys. B 93(2-3), 269–316 (2008). [CrossRef]

,16

16. N. P. Barnes, K. E. Murray, and M. G. Jani, “Flash-lamp-pumped Ho:Tm:Cr:YAG and Ho:Tm:Er:YLF lasers: modeling of a single, long pulse length comparison,” Appl. Opt. 36(15), 3363–3374 (1997). [CrossRef] [PubMed]

,36

36. T. Y. Fan, “Aperture guiding in quasi-three-level lasers,” Opt. Lett. 19(8), 554–556 (1994). [CrossRef] [PubMed]

]:
{dϕldθ=2g0l0(Δy2Δy2,t)ϕldΔy2dθ=lcl02g0l0Δy2ϕl,
(19)
where:
g0=γlσeNtot;Δy2,t=(xoc+δpas)/2g0l0,
(20)
with the following initial conditions for θ = 0 ; Δy 2 = Δy2,i, ϕi = 0.

Let us notice, that generation of giant pulse can begin, if the initial occupation of upper level y2,i satisfies the threshold condition:

y2,i>y2,t(l0,xOC,δpas)=(xoc+δpas)/2g0l0+βl.
(21)

The solution of equation set (19) is given in the following form:

ϕl(Δy2)=l0lcΔy2,iΔy2xΔy2,txdx=l0lc(Δy2,iΔy2Δy2,tln(Δy2,iΔy2)).
(22)

After passage of certain time (usually few dozens of roundtrips) the laser signal falls sharply to initial value for which the final occupation Δy2,f > 0 satisfies the following transcendent equation:

Δy2,iΔy2,fΔy2,tln(Δy2,iΔy2,f)=0.
(23)

After solving this equation we can calculate output energy and peak power. It can be shown (see details [1

1. H. Svelto, Principles of Lasers (Plenum Press, 1998).

,36

36. T. Y. Fan, “Aperture guiding in quasi-three-level lasers,” Opt. Lett. 19(8), 554–556 (1994). [CrossRef] [PubMed]

]) that the pulse energy density is given by the following formula:
Jout=J0xocxoc+δpas(Δy2,iΔy2,f),
(24)
where:

J0=hνll0Ntot.
(25)

4. Optimization of output of QTL laser

As it was shown in section 3, the parameters of Q-switched generation are unambiguously determined by the cavity parameters {lc, x oc, δ pas}, gain medium parameters {α p, l 0, σ p, σ e, τ, γ p, A, γl, βl, k ETU} and pump parameters {y2,0 , tpump, P0, Sp}. In general case of optimization of Q-switched laser the merit function is output energy, control variables of optimization are output coupling losses x oc and gain medium length l 0. The rest of parameters are constant, given data. For definite initial gain (i.e. definite values of pump density, pump duration and gain medium length) the modified Degnan’s method can be adopted [11

11. M. Eichhorn, “Quasi-three-level solid-state lasers in the near and mid infrared based on trivalent rare earth ions,” Appl. Phys. B 93(2-3), 269–316 (2008). [CrossRef]

,25

25. J. K. Jabczynski, L. Gorajek, W. Zendzian, J. Kwiatkowski, H. Jelinkova, J. Sulc, and M. Nemec, “Actively Q-switched thulium lasers,” in Advances in Solid State Lasers: Development and Applications IN-TECH, Vienna, (2010).

,36

36. T. Y. Fan, “Aperture guiding in quasi-three-level lasers,” Opt. Lett. 19(8), 554–556 (1994). [CrossRef] [PubMed]

] for optimization. Let us notice, that initial population Δy2,i is a function of pump power P 0 and pump duration tpump. For repetitively Q-switching, the final inversion has to be considered because it affects on the gain reached in the end of the following period of pumping. Thus, directly Degnan’s approach is not adequate for such a case because we cannot separate pumping and Q-switching processes.

In the presented work we intend to analyze the problem of maximum extractable energy density for a given pump density. The optimization procedure enabling the calculation of optimal gain medium length and OC losses will be derived. It is reasonable to compare results of optimization in Q-switched regime to the CW regime, thus we have worked out the adequate procedure of optimization in CW regime, firstly.

4.1. Procedure of optimization of CW regime of QTL laser

In case of CW generation for 1-end pumping with R p = 0, the output laser density of QTL laser is given by:
il(l0,xOC;ip,0)=ηslope,i(xOC,δpas)ηabs,t(l0,xOC,δpas)(ip,0it,cw(l0,xOC,δpas)),
(26)
where:

ηslope(xOC,δpas)=g0α0(1ROC)Rpas(1ROCRpas)(ROC+Rpas).
(27)

ηabs,t(l0,xOC,δpas)=1exp[α0l0(1γpy2,t(l0,xOC,δpas))].
(28)

The threshold pump intensity was directly determined from the Eq. (15) as follows:

it,cw(l0,xOC,δpas)=α0l0y2,t(l0,xOC,δpas)(1+ky2,t(l0,xOC,δpas))1exp(α0l0(1γpy2,t(l0,xOC,δpas))).
(29)

The optimal gain medium length for CW regime can be determined from the following equation:

ip,0ddl0ηabs,t(l0,xOC,δpas)=ddl0α0l0[1+ky2,t(l0,xOC,δpas)]y2,t(l0,xOC,δpas).
(30)

After a few simple transformations, we obtained the final formula on optimal absorbance of gain medium a 0,opt,cw as follows:
a0,opt,cw=α0lopt,cw=11γpβl[ln(ip,0(1+γpβl)βl(1+βlk))+α0γp(xOC+δpas)2g0],
(31)
for relative pump density i p,0 > i t,cw. It is easy to show, that optimal length l opt,cw increases with pump density and OC losses. The effect of ETU is contrary; increase in k causes the decrease in optimal length.

The optimal OC losses x OC,opt,cw for CW case can be found as a solution of the following equation:

ddxOCil(lopt,cw(xOC;ip,0),xOC;ip,0)=0.
(32)

In such a case x OC,opt,cw, l opt,cw are coupled pair of parameters, because the value of x OC effects on the absorption efficiency in a case of CW generation, unlike in the case of Q-switching regime, where the inversion is determined by incident pump density and duration and does not depend on out-coupling losses of cavity. The above presented procedure can be applied for optimization of Q-switched lasers operating with high repetition rates.

4.2. Procedure of optimization of Q-switching regime for quasi-stationary inversion

Preliminary intuitive analysis of Q-switching generation shows that gain medium length should be longer in comparison to CW case, because the higher inversion and deeper ground state depletion are reached in pumping process as a rule. Let us notice, that change of l0 results in change of averaged pump area, thus we should consider this combined effect in calculation of incident pump density, quasi-stationary inversion etc. To simplify approach, in analysis presented below, we do not consider optimization or change of pump area (which depends on gain medium length) and assume the same pump and mode area S 0. Such approximation is valid for quasi-coherent pumping, for which the Rayleigh range of pumping beam is much longer than gain medium length and absorption depth.

Further, to consider the maximum extractable energy density available for given pump density, we assume that quasi stationary inversion is reached. Applying Degnan’s method [36

36. T. Y. Fan, “Aperture guiding in quasi-three-level lasers,” Opt. Lett. 19(8), 554–556 (1994). [CrossRef] [PubMed]

] the formula on optimal OC losses is following:
xoc,optQsw(l0;ip,0)=δpaszst(l0;ip,0)1ln[zst(l0;ip,0)]ln[zst(l0;ip,0)],
(33)
where: zst – initial gain ratio for quasi-stationary (long pulse) pumping
zst(l0;ip,0)=2g0l0δpas[y2,st(l0;ip,0)βl],
(34)
y2,st - quasi-stationary population of 2-nd level determined as a solution of the Eq. (15). Let us notice, that unlike in a case of CW regime, now we have to solve (15) with respect to population y 2,st whereas pump density i p,0 is a given parameter. The Eq. (33) defines the extremal curve in the 2D space of (x oc, l 0) tracing the route to the global maximum on the 2D surface of output energy density. The output energy density determined on the extremal curve (33) is given by the following formula:
Jmax(l0;ip,0)=δpasJsat,e2γl(zst(l0;ip,0)1ln[zst(l0;ip,0)]),
(35)
where J sat,e = hνle – saturation energy density.

Thus, to find the optimal gain medium length, we have to solve the following equation:

ddl0Jmax(l0;ip,0)=δpasJsat,e2γl(zst(l0;ip,0)1zst(l0;ip,0))dzst(l0;ip,0)dl0=0.
(36)

Let us notice, that to optimize output energy density we have to find maximum of initial gain ratio, which is not a function of out-coupling loss.

The problem of optimization of initial gain ratio was solved applying Lagrange multiplier technique (see details in Appendix). In result we have obtained transcendent equation on optimal population of 2-nd level y opt of the following form:
ip,0exp(f1(yopt)(1yopt1))=ip0f1(yopt)(1+kyopt),
(37)
where:

f1(y)=11+ky(ip,0+1+2ky1+2kyky21βl).
(38)

Optimal absorbance a opt,Qsw = α0 l 0,opt,qsw is given by:
aopt,Qsw=f1(yopt)/yopt,
(39)
and resulting maximal gain ratio z max:

zmax=2g0α0δpasaopt,Qsw(yoptβl).
(40)

In a case of k = 0 (i.e. for medium with negligible ETU processes), we have obtained explicté formula on optimal absorbance:
aopt,0=ip,0βl1βl+ln(ip,0(1βl)βl),
(41)
yopt,0=ip,0βl1βlip,0βl1βll+ln(ip,0(1βl)βl),
(42)
zmax,0=2g0α0δpas[ip,0(1βl)+βlln(ip,0(1βl)βle)],
(43)
where: a opt,0, y opt,0, z max,0 - optimal absorbance, 2-nd level population and maximal gain ratio respectively.

As was shown, the optimization procedure of Q-switching regime for quasi-stationary inversion consists of two steps:

  • i/ the search for optimal absorbance of gain medium with respect to maximum of initial gain ratio according to formulae (37)(40)
  • ii/ the determination of optimal OC losses according to formula (33).

Let us notice, that optimization of initial gain ratio is realized during first pumping step, thus we do not need any information about cavity and losses. In the second step of optimization, for definite optimal gain medium length, we can use well known procedure for optimization of OC losses in Q-switching regime. The proposed mathematical procedure of optimization reflects well the physics of Q-switching and experimental practice. The l0, opt,qsw is here control parameter dependent on pumping parameters set {i p,0, α0, k }, whereas the optimal OC loss x OC,opt,qsw is the variable dependent on l0, opt,qsw and other parameters.

4.3. Role of gain medium absorbance and passive losses

In designing and optimization of Q-switched lasers for given pump rate, we can play with absorbance and out-coupling losses. As was shown in p. 4.2. the independent parameter of optimization is absorbance; out-coupling losses depends on it. The ETU parameter characterizes the gain medium and cannot optimized. The last, but not least parameter in optimization procedure is the passive loss, which, as a rule, should be as small as possible.

We have calculated these dependencies for two most important cases: gain medium with negligible ETU effect (e.g. case of Er doped lasers) and with significant ETU parameter (i.e. case of Tm doped lasers) for Q-switched and CW cases (see Fig. 3
Fig. 3 Optimal absorbance vs. relative pump density for ETU parameter k = 8 (blue curves) and ETU parameter k = 0 (red curves) for Q-switched (continuous curves) and CW (dashed curves).
).

As was shown, optimal absorbance increases with pump density, moreover for higher pump densities the difference between optimal absorbance for Q-switched lasers and CW lasers increases as well. Thus, we can conclude, that QTL laser optimized for CW regime is not optimized as a rule for Q-switched regime for elevated pump densities (i p,0 > 1).

Output energy density depends significantly on ETU parameter (see Fig. 4a
Fig. 4 Output energy density vs. absorbance for different relative pump densities i p,0 = 1,2,4; ETU parameters k = 0 (plot a)) and k = 8 (plot b)) .
, 4b). For negligible ETU effects (Fig. 4a) the relative tolerance in absorbance change is greater i.e. gain medium can be longer than optimal case without significant drop in output. For gain media with high ETU (Fig. 4b) the length of medium has to be precisely determined especially for moderate pump rates. Quite similar properties can be observed for CW lasers but the decrease in efficiency is lower because of much lower inversions reached in CW regime.

The passive loss does not cause the significant drop in output for negligible ETU case (Fig. 5a
Fig. 5 (a) Normalized output energy density vs. absorbance for different passive losses; ETU parameters k = 0, relative pump density i p,0 = 2. (b) Normalized output energy density vs. absorbance for different passive losses δpas; ETU parameters k = 8, relative pump density i p,0 = 2.
, 6
Fig. 6 Normalized output energy density vs. passive losses for ETU parameter k = 0 (continuous curves) and ETU parameter k = 8 (dotted curves) and several relative pump density: i p,0 = 2 (red curves), i p,0 = 4 (blue curves).
). Comparing to the ideal case of vanishing passive losses the relatively high passive loss of 10% causes only 20% decrease in output.

The contrary situation is in a case of significant ETU effects (Fig. 5b, 6). Thus, we can interpret the ETU effect as the additional mechanism of losses (not for laser field but for inversion) which influences dramatically on the extractable energy of Q-switched laser. The relatively lower role of ETU effects in a case of CW regime is caused by much lower relative population clamped at threshold of QTL lasers.

In practical design of QTL lasers additional, very important factors have to be considered: the average temperature increase in gain medium, aperture losses caused by not perfect mode matching between laser mode and pump beam, thermal-optical aberrations, thermally induced birefringence and of course the surface and volume damage threshold of laser elements.

5. Conclusions

We have proposed the new approach for optimization of actively Q-switched end-pumped QTL lasers. The role of transient, time-dependent ground state depletion and ETU processes was considered in the model. The new optimization procedure, applying Lagrange’s multiplier technique was derived, which enable the optimization of gain medium and cavity parameters for given pump rate. Final explicte formulae for ETU-less (k = 0) gain medium were derived. The novel method of optimization of QTL laser in CW regime was proposed and applied to compare results of absorbance optimization for both regimes. The procedure for CW regime optimization can be applied for actively Q-switched lasers operating with high repetition rates. Analysis of role of ETU parameter and passive losses was performed for two important for practice cases: negligible ETU (k = 0) and significant ETU (k = 8) cases.

Appendix

The problem of optimization of initial gain ratio can be solved applying Lagrange multiplier technique. We have to find max[f(x,y)], where: f(x,y) - normalized gain ratio (40) determined in the 2D space of x = a − absorbance, y = yst. – averaged relative stationary population of 2-nd level, assuming the boundary condition G(x,y) = 0, which is pumping equation for quasi-stationary inversion (15):
f(x,y)=x(yβl)G(x,y)=rp(x,y)y(1+ky)=0
(A1)
where:
rp(x,y)=ip,0(1exp(xyx))x-1
(A2)
i p,0 – incident relative pump density, βl – lower laser level relative population factor defined by (17), k – relative parameter of ETU defined by (6). Let us introduce the multiplier Λ and a new function F(x,y) as follows:

F(x,y)=f(x,y)+ΛG(x,y)
(A3)

According to Lagrange method [37

37. I. N. Bronstein, K. A. Semendjajew, G. Musiol, and H. Muehling, Taschenbuch der Mathematik, (Verlag Harri Deutsch 2001)

], we have to solve the following set of equations for three unknowns (x, y, Λ):

{F(x,y,Λ)x=0F(x,y,Λ)y=0G(x,y)=0
(A4)

After substitution of (A1,A2) into (A4) and a few transformations we obtain the following set of coupled nonlinear algebraic equations:

{yβ+Λ(ip,0(1y)exp(xyx)y(1+ky))=0xΛ(ip,0xexp(xyx)x(12ky))=0ip,0(1-exp(xyx))xy(1ky)=0
(A5)

From the 2-nd equation (A5) we can calculate Λ:

Λ=xip,0exp(xyx)12ky
(A6)

After substituting (A6) to (A5) we obtain final set of 2 coupled nonlinear equations:

{(1βl)(ip,0exp(xyx)+1+2κy)=1+2κyκy2ip,0exp(xyx)=ip,0xy(1+κy)
(A7)

Further, combining both equations of (A7) we can eliminate x variable in the following way:

x=f1(y)y-1
(A8)

where:

f1(y)=11+ky(ip,0+1+2κy1+2kyky21βl)
(A9)

Substituting (A8) to the 2-nd equation of (A7) we obtain the final transcendent equation on optimal population y opt:

ip,0exp(f1(yopt)(1yopt1))=ip,0f1(yopt)(1+yopt)
(A10)

The optimal absorbance is determined by formula (A8), and maximal gain ratio z max,st is given by:

zmax,st=2g0α0δpas(1βlyopt1)f1(yopt)
(A11)

In a case of negligible ETU effects (k0) we can directly determine optimal absorbance a opt,0, 2-nd level population y opt,0 and maximal gain ratio z max,0 as follows:

aopt,0=ip,0βl1βl+ln(ip,0(1βl)βl)
(A12)
yopt,0=ip,0βl1βlip,0βl1βll+ln(ip,0(1βl)βl)
(A13)
zmax,0=2g0α0δpas[ip,0(1βl)+βlln(ip,0(1βl)βle)]
(A14)

Acknowledgments

This research has been supported by Ministry of Science and Higher Education of Poland under projects NN 515 345 036. We would like to thanks Dr. M. Skorczakowski for fruitful discussion.

References and links

1.

H. Svelto, Principles of Lasers (Plenum Press, 1998).

2.

W. P. Risk, “Modeling of longitudinally pumped solid-state lasers exhibiting reabsorption losses,” J. Opt. Soc. Am. B 5(7), 1412–1423 (1988). [CrossRef]

3.

T. Y. Fan, “Optimizing the efficiency and stored energy in quasi-three-level lasers,” IEEE J. Quantum Electron. 28(12), 2692–2697 (1992). [CrossRef]

4.

C. D. Nabors, “Q-switched operation of quasi-three-level lasers,” IEEE J. Quantum Electron. 30(12), 2896–2901(1994). [CrossRef]

5.

R. J. Beach, “CW Theory of quasi-three level end-pumped laser oscillators,” Opt. Commun. 123(1-3), 385–393 (1996). [CrossRef]

6.

C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hugel, “A 1-kW CW thin disk laser,” IEEE J. Sel. Top. Quantum Electron. 6, 650–657 (2000). [CrossRef]

7.

G. L. Bourdet, “Theoretical investigation of quasi-three-level longitudinally pumped continuous wave lasers,” Appl. Opt. 39(6), 966–971 (2000). [CrossRef] [PubMed]

8.

G. L. Bourdet, “New evaluation of ytterbium-doped materials for CW laser applications,” Opt. Commun. 198(4-6), 411–417 (2001). [CrossRef]

9.

C. Lim and Y. Izawa, “Modeling of end-pumped CW quasi-three-level lasers,” IEEE J. Quantum Electron. 38(3), 306–311 (2002). [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.

M. Eichhorn, “Quasi-three-level solid-state lasers in the near and mid infrared based on trivalent rare earth ions,” Appl. Phys. B 93(2-3), 269–316 (2008). [CrossRef]

12.

I. Kudryashov, D. Garbuzov, and M. Dubinskii, “Latest developments in resonantly diode-pumped Er:YAG lasers,” in Laser Source Technology for Defence and Security III, Proc. SPIE 6552, 65520K (2007). [CrossRef]

13.

B. M. Walsh, “Review of Tm and Ho Materials; Spectroscopy and Lasers,” Laser Phys. 19(4), 855–866 (2009). [CrossRef]

14.

E. P. Chicklis, J. R. Mosto, M. L. Lemons, and P. A. Budni, “High-Power/High-Brightness Diode-Pumped 1.9-μm Thulium and resonantly Pumped 2.1-μm Holmium Lasers,” IEEE J. Sel. Top. Quantum Electron. 6(4), 629–635 (2000). [CrossRef]

15.

J. Kwiatkowski, J. K. Jabczynski, Ł. Gorajek, W. Zendzian, H. Jelínková, J. Sulc, M. Nemec, and P. Koranda, “Resonantly pumped tunable Ho:YAG laser,” Laser Phys. Lett. 6(7), 531–534 (2009). [CrossRef]

16.

N. P. Barnes, K. E. Murray, and M. G. Jani, “Flash-lamp-pumped Ho:Tm:Cr:YAG and Ho:Tm:Er:YLF lasers: modeling of a single, long pulse length comparison,” Appl. Opt. 36(15), 3363–3374 (1997). [CrossRef] [PubMed]

17.

E. C. Honea, R. J. Beach, S. B. Sutton, J. A. Speth, S. C. Mitchell, J. A. Skidmore, M. A. Emanuel, and S. A. Payne, “115-W Tm:YAG diode-pumped solid-state laser,” IEEE J. Quantum Electron. 33(9), 1592–1600 (1997). [CrossRef]

18.

P. Peterson, M. P. Sharma, and A. Gavrielides, “Extraction efficiency and thermal lensing in Tm:YAG lasers,” Opt. Quantum Electron. 28(6), 695–707 (1996). [CrossRef]

19.

P. Cemy and D. Burns, “Modeling and experimental investigation of a diode-pumped Tm:YAlO3 laser with a- and b-cut orientation,” IEEE J. Sel. Top. Quantum Electron. 11(3), 674–681 (2005). [CrossRef]

20.

G. L. Bourdet and G. Lescroart, “Theoretical modeling and design of a Tm:YVO4 microchip lasers,” Opt. Commun. 149(4-6), 404–414 (1998). [CrossRef]

21.

S. So, J. I. Mackenzie, D. P. Shepherd, W. A. Clarkson, J. G. Betterton, and E. K. Gorton, “A power-scaling strategy for longitudinally diode-pumped Tm:YLF lasers,” Appl. Phys. B 84(3), 389–393 (2006). [CrossRef]

22.

M. Schellhorn, M. Eichhorn, C. Kieleck, and A. Hirth, “High repetition rate mid-infrared laser source,” C. R. Phys. 8(10), 1151–1161 (2007). [CrossRef]

23.

J. K. Jabczynski, W. Zendzian, J. Kwiatkowski, H. Jelínková, J. Šulc, and M. Němec, “Actively Q-switched diode pumped thulium laser,” Laser Phys. Lett. 4(12), 863–867 (2007). [CrossRef]

24.

N. G. Zakharov, O. L. Antipov, A. P. Savikin, V. V. Sharkov, O. N. Eremeikin, Y. N. Frolov, G. M. Mishchenko, and S. D. Velikanov, “Efficient emission at 1908 nm in a diode-pumped Tm:YLF laser,” Quantum Electron. 39(5), 410–414 (2009). [CrossRef]

25.

J. K. Jabczynski, L. Gorajek, W. Zendzian, J. Kwiatkowski, H. Jelinkova, J. Sulc, and M. Nemec, “Actively Q-switched thulium lasers,” in Advances in Solid State Lasers: Development and Applications IN-TECH, Vienna, (2010).

26.

S. D. Jackson, “The spectroscopic and energy transfer characteristics of the rare earth ions used for silicate glass fibre lasers operating in the shortwave infrared,” Laser & Photon. Rev. 3(5), 466–482 (2009). [CrossRef]

27.

G. L. Bourdet and G. Lescroart, “Theoretical modeling and design of a Tm:Ho:YLiF4 microchip laser,” Appl. Opt. 38(15), 3275–3281 (1999). [CrossRef] [PubMed]

28.

G. L. Bourdet, “Gain and absorption saturation coupling in end pumped Tm:YVO4 and Tm:Ho:YLF amplifiers,” Opt. Commun. 173(1-6), 333–340 (2000). [CrossRef]

29.

J. M. Sousa, J. R. Salcedo, and V. V. Kuzmin, “Simulation of laser dynamics and active Q-switching in Tm,Ho:YAG and Tm:YAG lasers,” Appl. Phys. B 64(1), 25–36 (1996). [CrossRef]

30.

X. Zhang, Y. Ju, and Y. Wang, “Theoretical and experimental investigation of actively Q-switched Tm,Ho:YLF lasers,” Opt. Express 14(17), 7745–7750 (2006). [CrossRef] [PubMed]

31.

O. A. Louchev, Y. Urata, and S. Wada, “Numerical simulation and optimization of Q-switched 2 mum Tm,Ho:YLF laser,” Opt. Express 15(7), 3940–3947 (2007). [CrossRef] [PubMed]

32.

G. Rustad and K. Stenersen, “Modeling of laser-pumped Tm and Ho lasers accounting for up conversion and ground-state depletion,” IEEE J. Quantum Electron. 32(9), 1645–1656 (1996). [CrossRef]

33.

L. B. Shaw, R. S. F. Chang, and N. Djeu, “Measurement of up-conversion energy-transfer probabilities in Ho:Y3Al5O12 and Tm:Y3Al5O12,” Phys. Rev. B 50, 6009–6019 (1996).

34.

Y. F. Chen, Y. P. Lan, and S. C. Wang, “Modeling of diode-end-pumped Q-switched solid-state lasers: influence of energy-transfer upconversion,” J. Opt. Soc. Am. B 19(7), 1558–1563 (2002). [CrossRef]

35.

J. Degnan, “Theory of the optimally coupled Q-switched laser,” IEEE J. Quantum Electron. 25(2), 214–220 (1989). [CrossRef]

36.

T. Y. Fan, “Aperture guiding in quasi-three-level lasers,” Opt. Lett. 19(8), 554–556 (1994). [CrossRef] [PubMed]

37.

I. N. Bronstein, K. A. Semendjajew, G. Musiol, and H. Muehling, Taschenbuch der Mathematik, (Verlag Harri Deutsch 2001)

OCIS Codes
(140.3460) Lasers and laser optics : Lasers
(140.3540) Lasers and laser optics : Lasers, Q-switched
(140.3580) Lasers and laser optics : Lasers, solid-state
(140.5560) Lasers and laser optics : Pumping
(140.5680) Lasers and laser optics : Rare earth and transition metal solid-state lasers

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: March 29, 2011
Revised Manuscript: June 14, 2011
Manuscript Accepted: June 23, 2011
Published: August 1, 2011

Citation
Jan K. Jabczynski, Lukasz Gorajek, Jacek Kwiatkowski, Mateusz Kaskow, and Waldemar Zendzian, "Optimization of end-pumped, actively Q-switched quasi-III-level lasers," Opt. Express 19, 15652-15668 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-17-15652


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References

  1. H. Svelto, Principles of Lasers (Plenum Press, 1998).
  2. W. P. Risk, “Modeling of longitudinally pumped solid-state lasers exhibiting reabsorption losses,” J. Opt. Soc. Am. B 5(7), 1412–1423 (1988). [CrossRef]
  3. T. Y. Fan, “Optimizing the efficiency and stored energy in quasi-three-level lasers,” IEEE J. Quantum Electron. 28(12), 2692–2697 (1992). [CrossRef]
  4. C. D. Nabors, “Q-switched operation of quasi-three-level lasers,” IEEE J. Quantum Electron. 30(12), 2896–2901(1994). [CrossRef]
  5. R. J. Beach, “CW Theory of quasi-three level end-pumped laser oscillators,” Opt. Commun. 123(1-3), 385–393 (1996). [CrossRef]
  6. C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hugel, “A 1-kW CW thin disk laser,” IEEE J. Sel. Top. Quantum Electron. 6, 650–657 (2000). [CrossRef]
  7. G. L. Bourdet, “Theoretical investigation of quasi-three-level longitudinally pumped continuous wave lasers,” Appl. Opt. 39(6), 966–971 (2000). [CrossRef] [PubMed]
  8. G. L. Bourdet, “New evaluation of ytterbium-doped materials for CW laser applications,” Opt. Commun. 198(4-6), 411–417 (2001). [CrossRef]
  9. C. Lim and Y. Izawa, “Modeling of end-pumped CW quasi-three-level lasers,” IEEE J. Quantum Electron. 38(3), 306–311 (2002). [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. M. Eichhorn, “Quasi-three-level solid-state lasers in the near and mid infrared based on trivalent rare earth ions,” Appl. Phys. B 93(2-3), 269–316 (2008). [CrossRef]
  12. I. Kudryashov, D. Garbuzov, and M. Dubinskii, “Latest developments in resonantly diode-pumped Er:YAG lasers,” in Laser Source Technology for Defence and Security III, Proc. SPIE 6552, 65520K (2007). [CrossRef]
  13. B. M. Walsh, “Review of Tm and Ho Materials; Spectroscopy and Lasers,” Laser Phys. 19(4), 855–866 (2009). [CrossRef]
  14. E. P. Chicklis, J. R. Mosto, M. L. Lemons, and P. A. Budni, “High-Power/High-Brightness Diode-Pumped 1.9-μm Thulium and resonantly Pumped 2.1-μm Holmium Lasers,” IEEE J. Sel. Top. Quantum Electron. 6(4), 629–635 (2000). [CrossRef]
  15. J. Kwiatkowski, J. K. Jabczynski, Ł. Gorajek, W. Zendzian, H. Jelínková, J. Sulc, M. Nemec, and P. Koranda, “Resonantly pumped tunable Ho:YAG laser,” Laser Phys. Lett. 6(7), 531–534 (2009). [CrossRef]
  16. N. P. Barnes, K. E. Murray, and M. G. Jani, “Flash-lamp-pumped Ho:Tm:Cr:YAG and Ho:Tm:Er:YLF lasers: modeling of a single, long pulse length comparison,” Appl. Opt. 36(15), 3363–3374 (1997). [CrossRef] [PubMed]
  17. E. C. Honea, R. J. Beach, S. B. Sutton, J. A. Speth, S. C. Mitchell, J. A. Skidmore, M. A. Emanuel, and S. A. Payne, “115-W Tm:YAG diode-pumped solid-state laser,” IEEE J. Quantum Electron. 33(9), 1592–1600 (1997). [CrossRef]
  18. P. Peterson, M. P. Sharma, and A. Gavrielides, “Extraction efficiency and thermal lensing in Tm:YAG lasers,” Opt. Quantum Electron. 28(6), 695–707 (1996). [CrossRef]
  19. P. Cemy and D. Burns, “Modeling and experimental investigation of a diode-pumped Tm:YAlO3 laser with a- and b-cut orientation,” IEEE J. Sel. Top. Quantum Electron. 11(3), 674–681 (2005). [CrossRef]
  20. G. L. Bourdet and G. Lescroart, “Theoretical modeling and design of a Tm:YVO4 microchip lasers,” Opt. Commun. 149(4-6), 404–414 (1998). [CrossRef]
  21. S. So, J. I. Mackenzie, D. P. Shepherd, W. A. Clarkson, J. G. Betterton, and E. K. Gorton, “A power-scaling strategy for longitudinally diode-pumped Tm:YLF lasers,” Appl. Phys. B 84(3), 389–393 (2006). [CrossRef]
  22. M. Schellhorn, M. Eichhorn, C. Kieleck, and A. Hirth, “High repetition rate mid-infrared laser source,” C. R. Phys. 8(10), 1151–1161 (2007). [CrossRef]
  23. J. K. Jabczynski, W. Zendzian, J. Kwiatkowski, H. Jelínková, J. Šulc, and M. Němec, “Actively Q-switched diode pumped thulium laser,” Laser Phys. Lett. 4(12), 863–867 (2007). [CrossRef]
  24. N. G. Zakharov, O. L. Antipov, A. P. Savikin, V. V. Sharkov, O. N. Eremeikin, Y. N. Frolov, G. M. Mishchenko, and S. D. Velikanov, “Efficient emission at 1908 nm in a diode-pumped Tm:YLF laser,” Quantum Electron. 39(5), 410–414 (2009). [CrossRef]
  25. J. K. Jabczynski, L. Gorajek, W. Zendzian, J. Kwiatkowski, H. Jelinkova, J. Sulc, and M. Nemec, “Actively Q-switched thulium lasers,” in Advances in Solid State Lasers: Development and Applications IN-TECH, Vienna, (2010).
  26. S. D. Jackson, “The spectroscopic and energy transfer characteristics of the rare earth ions used for silicate glass fibre lasers operating in the shortwave infrared,” Laser & Photon. Rev. 3(5), 466–482 (2009). [CrossRef]
  27. G. L. Bourdet and G. Lescroart, “Theoretical modeling and design of a Tm:Ho:YLiF4 microchip laser,” Appl. Opt. 38(15), 3275–3281 (1999). [CrossRef] [PubMed]
  28. G. L. Bourdet, “Gain and absorption saturation coupling in end pumped Tm:YVO4 and Tm:Ho:YLF amplifiers,” Opt. Commun. 173(1-6), 333–340 (2000). [CrossRef]
  29. J. M. Sousa, J. R. Salcedo, and V. V. Kuzmin, “Simulation of laser dynamics and active Q-switching in Tm,Ho:YAG and Tm:YAG lasers,” Appl. Phys. B 64(1), 25–36 (1996). [CrossRef]
  30. X. Zhang, Y. Ju, and Y. Wang, “Theoretical and experimental investigation of actively Q-switched Tm,Ho:YLF lasers,” Opt. Express 14(17), 7745–7750 (2006). [CrossRef] [PubMed]
  31. O. A. Louchev, Y. Urata, and S. Wada, “Numerical simulation and optimization of Q-switched 2 mum Tm,Ho:YLF laser,” Opt. Express 15(7), 3940–3947 (2007). [CrossRef] [PubMed]
  32. G. Rustad and K. Stenersen, “Modeling of laser-pumped Tm and Ho lasers accounting for up conversion and ground-state depletion,” IEEE J. Quantum Electron. 32(9), 1645–1656 (1996). [CrossRef]
  33. L. B. Shaw, R. S. F. Chang, and N. Djeu, “Measurement of up-conversion energy-transfer probabilities in Ho:Y3Al5O12 and Tm:Y3Al5O12,” Phys. Rev. B 50, 6009–6019 (1996).
  34. Y. F. Chen, Y. P. Lan, and S. C. Wang, “Modeling of diode-end-pumped Q-switched solid-state lasers: influence of energy-transfer upconversion,” J. Opt. Soc. Am. B 19(7), 1558–1563 (2002). [CrossRef]
  35. J. Degnan, “Theory of the optimally coupled Q-switched laser,” IEEE J. Quantum Electron. 25(2), 214–220 (1989). [CrossRef]
  36. T. Y. Fan, “Aperture guiding in quasi-three-level lasers,” Opt. Lett. 19(8), 554–556 (1994). [CrossRef] [PubMed]
  37. I. N. Bronstein, K. A. Semendjajew, G. Musiol, and H. Muehling, Taschenbuch der Mathematik, (Verlag Harri Deutsch 2001)

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