Computational model for operation of 2 μm co-doped Tm,Ho solid state lasers

Computational model for operation of 2 μm co-doped Tm,Ho solid state lasers

Oleg A. Louchev1 , Yoshiharu Urata1 , Norihito Saito2 and Satoshi Wada2

1Megaopto Co. Ltd., RIKEN Cooperation Center W414, 2-1 Hirosawa, Wako, Saitama 351-0106, Japan

2Solid-State Optical Science Research Unit, RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan [email protected]

Abstract: A computational model for operation of co-doped Tm,Ho solid-state lasers is developed coupling (i) 8-level rate equations with (ii) TEM00 laser beam distribution, and (iii) complex heat dissipation model. Simulations done for Q-switched ≈0.1 J giant pulse generation by Tm,Ho:YLF laser show that ≈43 % of the 785 nm light diode side-pumped energy is directly transformed into the heat inside the crystal, whereas ≈45 % is the spontaneously emitted radiation from 3F4,

5I7 , 3H4 and 3H5 levels.

In water-cooled operation this radiation is absorbed inside the thermal boundary layer where the heat transfer is dominated by heat conduction. In high-power operation the resulting temperature increase is shown to lead to (i) significant decrease in giant pulse energy and (ii) thermal lensing.

© 2007 Optical Society of America

OCIS codes: (140.3480) Lasers, diode-pumped; (140.3540) Lasers, Q-switched; (140.3580) Lasers, solid-state.

_______________________________________________________________________ References and links

1. J.K. Tyminski, D.M. Franich and M. Kokta, “Gain dynamics of Tm,Ho:YAG pumped in near infrared,” J. Appl. Phys. 65, 3181-3188 (1989).

2. V.A. French, R.R. Petrin, R.C. Powell, and M. Kokta, “Energy-transfer processes in Y3Al5O12:Tm,Ho,” Phys. Rev. B 46, 8018-8026 (1992).

3. R.R. Petrin, M.G. Jani, R.C. Powell and M. Kokta, “Spectral dynamics of laser-pumped Y3Al5O12:Tm,Ho lasers,” Opt. Mater. 1,111-124 (1992).

4. M.G. Jani, R J. Reeves, R.C. Powell, G.J. Quarles and L. Esterovitz, “Alexandrite-laser excitation of a Tm:Ho:Y3Al5O12 laser.” J. Opt. Soc. Am. B 8, 741-746 (1991).

5. M. G. Jani, F. L. Naranjo, N. P. Barnes, K.E. Murray, and G.E. Lockard, ”Diode-pumped long-pulse-length Ho:Tm:YLiF4 laser at 10 Hz,” Opt. Lett. 20, 872-874 (1995)

6. J. Yu, U.N. Singh, N.P. Barnes and M. Petros “125-mJ diode-pumped injection-seeded Ho:Tm:YLF laser,” Opt. Lett. 23, 780-782 (1998).

7. A.N. Alpat'ev, V.A. Smirnov, I.A. Shcherbakov, “Relaxation oscillations of the radiation from a 2-

μm holmium laser with a Cr,Tm,Ho:YSGG crystal,” Quantum Electron. 28, 143-146 (1998). 8. I.F. Elder and M.J.P. Payne, “Lasing in diode-pumped Tm:YAP, Tm,Ho:YAP and Tm,Ho,YLF,”

Opt. Commun. 145, 329-339 (1995). 9. N. P. Barnes, E. D. Filer, C. A. Morrison and C. J. Lee, “Ho:Tm Lasers I: Theoretical,” IEEE J.

Quantum Electron. 32, 92-103 (1996). 10. C. J. Lee, G. Han and N.P. Barnes, ”Ho:Tm Lasers II: Experiments,” IEEE J. Quantum Electron.

32, 104-111 (1996). 11. G. Rustad and K. Stenersen, “Modeling of laser-pumped Tm and Ho lasers accounting for

upconversion and ground-state depletion,” IEEE J. Quantum Electron. 32, 1645 -1656 (1996). 12. D. Bruneau, S. Delmonte and J. Pelon, “Modeling of Tm,Ho:YAG and Tm,Ho:YLF 2-μm lasers

and calculation of extractable energies,” Appl. Opt. 37, 8406-8419 (1998). 13. G. L. Bourdet and G. Lescroart, “Theoretical modeling and design of a Tm,Ho:YLiF4 microchip

laser,” Appl. Opt. 38, 3275-3281 (1999). 14. S. D. Jackson and T.A. King, “CW operation of a 1.064-μm pumped Tm-Ho-Doped silica fiber

laser,” IEEE J. of Quantum Electron. 34, 1578-1587 (1998).

#82626 - $15.00 USD Received 2 May 2007; revised 26 Jul 2007; accepted 26 Jul 2007; published 5 Sep 2007

(C) 2007 OSA 17 September 2007 / Vol. 15, No. 19 / OPTICS EXPRESS 11903

15. V. Sudesh and K. Asai, “Spectroscopic and diode-pumped-laser properties of Tm,Ho:YLF; Tm,Ho:LuLF; and Tm,Ho:LuAG crystals: a comparative study,” J. Opt. Soc. Am. B 20, 1829-1837 (2003).

16. A. Sato, K. Asai and K. Mizutani, “Lasing characteristics and optimizations of diode-side-pumped Tm,Ho:GdVO4 laser,” Opt. Lett. 29, 836 –838 (2004).

17. B.M. Walsh, N.P. Barnes, M. Petros, J. Yu and U.N. Singh, “Spectroscopy and modeling of solid state lanthanide lasers: Application to trivalent Tm3+ and Ho3+ in YLiF4 and LuLiF4,” J. Appl. Phys. 95, 3255-3271 (2004).

18. G. Galzerano, E. Sani, A. Toncelli, G. Della Valle, S. Taccheo, M. Tonelli, P. Laporta, “Widely tunable continuous-wave diode-pumped 2-µm Tm-Ho:KYF4 laser,” Opt. Lett. 29, 715-717 (2004).

19. J. Izawa, H. Nakajima, H. Hara, and Y. Arimoto, “Comparison of lasing performance of Tm,Ho:YLF lasers by use of single and double cavities,” Appl. Opt. 39, 2418-2421 (2000).

20. J. Yu, B. C. Trieu, E. A. Modlin, U.N. Singh, M. J. Kavaya, S. Chen, Y. Bai, P. J. Petzar, and M. Petros, “1 J/pulse Q-switched 2 μm solid-state laser,” Opt. Lett. 31, 462-464 (2006).

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

22. O.A. Louchev, Y. Urata, and S. Wada, “Numerical simulation and optimization of Q-switched 2 μm Tm,Ho:YLF laser,” Opt. Express 15, 3940-3947 (2007).

23. P. Černý and D. Burns, “Modeling and experimental investigation of a diode-pumped Tm:YAlO3 laser with a- and b-cut crystal orientations,” IEEE J. of selected topics in quantum electron. 11, 674-681 (2005).

24. V.P. Risk, “Modeling of longitudinally pumped solid-state lasers exhibiting reabsorption losses,” J. Opt. Soc. Am. B 5, 1412-1423 (1988).

25. D. Golla, M. Bode, S. Knoke, W. Schöne, and A. Tünnermann, “62-W cw TEM00 Nd:YAG laser side-pumped by fiber-coupled diode lasers,” Opt. Lett. 21, 210-212 (1996).

26. D. M. Wieliczka, S. Weng, and M. R. Querry, "Wedge shaped cell for highly absorbent liquids: infrared optical constants of water,” Appl. Opt. 28, 1714-1719 (1989).

27. W. Koechner, Solid –State Laser Engineering, 6th Edition (New-York, Springer, 2006).

_______________________________________________________________________ 1. Introduction

Co-doped Tm, Ho solid-state lasers present significant interest for a number of advanced applications. In recent years significant progress in understanding the basic phenomena underlying Tm,Ho laser operation as well as in developing high power lasers has been achieved [1-22]. However, many related physical effects which are non-linearly coupled with each other continue to remain unclear. For instance, notwithstanding the importance attributed to thermal effects [23], they are often treated in an excessively simplistic way, which does not allow a correct interpretation of experimental data as well as an adequate laser simulation and optimization. In this Communication we develop a coupled themo-optoical computational model in which specific non-steady-state thermal effects are rigorously coupled with population dynamics and spectroscopic processes involved in the energy transitions, and lasing in Tm,Ho solid state lasers. These effects are found to play a very significant role in high-power high- frequency Q-switched laser operation required for development of coherent-detection lidar systems.

2. Optics model

There are several models describing the electron population dynamics of co-doped Tm,Ho solid state lasers using the main levels involved in the pumping and laser generation shown



ΔEi

ΔE*

i

Fig. 1. Energy transfer processes in co-doped Tm,Ho materials and energy differences used in Eq. (16).

in Fig. 1. These models have included various effects such as end and side pumping, ground state depletion, energy transfer between Tm3+ and Ho3+ ions, and also different types of upconversion processes decreasing the upper laser level population. Several studies have used simplifications allowing model reduction to two-rate equations describing the electron density at the excited levels 3F4 and 5I7, and ground-state levels 3H6 and 5I8. In our study we endeavor to retain all the terms in the rate dynamics model, allowing us to reveal several specific effects. In particular, our analysis is based on computational non-steady state thermo-optical model using 8-level rate equations and related data describing the population dynamics of Tm,Ho lasers from Walsh et al. [17]:

8338166115517227

2222144117718228

2

21 )(

nnpnnpnnpnnp

npnnpnnpnnpn

tRdt

dnp

+−−+

+−−++−=τ , (1)

15517227

2222144117718228

3

3

2

22 22

nnpnnp

npnnpnnpnnpnn

dt

dn

+−

−++−+−=ττ , (2)

83381661

4

4

3

33 nnpnnpnn

dt

dn−++−=

ττ , (3)

22221441

4

44 ),,( npnnpn

rztRdt

dnp +−−=

τ , (4)

15512727

5

55 nnpnnpn

dt

dn−+−=

τ , (5)

38381661

5

5

6

66 nnpnnpnn

dt

dn +−+−=ττ

. (6)

For the upper laser level (5I7):

),()( 887715517227177182286

6

7

77 rtnfnfc

nnpnnpnnpnnpnn

dt

dn se φησ

ττ−−+−−++−= . (7)

For the lower laser level (5I8):



),()( 887783381661177182287

78 rtnfnfc

nnpnnpnnpnnpn

dt

dn se φησ

τ−+−++−= , (8)

where ),( rtni

are the level concentrations, ijp are the probabilities of the optical transitions,

iτ are the level lifetimes, )(tRpis the pumping source, ),( rtφ is the local laser photon

density,seσ is the stimulated emission cross-section, ),( rtfi

are the Boltzmann level

population factors and η is the refractive index of the crystal. All optical transition probabilities and level lifetimes, including characteristic radiative

times, have been considered in detail in Ref. [17]. Although that study neglects some of the possible radiation decays from the upper manifolds which could easily be taken into account in present study, we have not extended the original model of Ref. [17] in view of the good agreement with the known experimental data on Q-switched pulse operation [6].

The local laser photon density ),( rtφ is represented by the product of (i) the total

number of photons inside the oscillator cavity, )(0 tΦ , depending on t and (ii) the normalized

space distribution function, )(0 rφ . The resulting equation for )(0 tΦ is given by a differential

equation including integration of the stimulated and spontaneous radiation over the crystal volume [17, 23, 24]:

∫∫∫∫ +Φ

−−Φ

=Φ

crcr VcV

se dVnt

dVzrnfnfct

dt

td7

7

008877

00 )(),()(

)()(

τε

τφ

ησ , (9)

where

crV is the crystal volume, and ≈ε 10-7-10-8 is a factor taking into account the proportion

of photons spontaneously emitted within the solid angle of the mirrors, and cτ is the cavity

lifetime given by:

[ ]βτ +−−−=− )1ln(ln2

1out1

optc TR

L

c , (10)

where

crcavopt LLL )1( −+= η is the characteristic optical length, cavL is the cavity length and

crL is the crystal length; lR is the back mirror reflectance,

outT is the output mirror

transmittance and β is the parameter used in our simulations for the optical loss associated with the active Q-switching: β=0 for the open resonator and )1ln(ln 1 outTR −−−>>β for the

closed resonator. For the acousto-optic Q-switch, if the fraction of the main beam diffracted out of the resonator is 0.9, 3.2)9.01ln( =−−=β . We neglect here additional reflectance and scattering loss on crystal and Q-switch. However, these factors can also be included into the round trip optical loss in Eq. (10).

For the case of 100-500 ns pulse generation considered here the cavity length crcav LL >>

and the spatial photon distribution inside the operating crystal can be described by TEM00 fundamental mode as:

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=20

2

20

0

2exp

2)(

w

r

Lw cavπφ r , (11)

where w0 is the beam waist radius of TEM00 mode defined by the resonator parameters.

The solution of the rates equations together with the main oscillator Eq. (9) gives the radial distribution of the output power density (W/m2) at the output mirror as:



⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −×−

Φ=2*

0

2

2*0

00

2exp

2

1

1ln

2),(

w

r

wTL

chrtI

outopt

las

πν , (12)

where *

0w is the modified beam radius outside the resonator (for instance, for the case of a

TEM00 Gaussian beam inside the confocal spherical resonator one has πλ 20 lcavLw =

and0

*0 2ww = at the output mirror).

In this paper we consider a particular case of Tm (6%), Ho(0.4%):YLF operation side pumped by 785 nm LD radiation. For 6 % Tm doped YLF crystal one finds for the absorption coefficient 18.2 −≈= cmNTmaσα [17]. Thus, a 2 mm diameter YLF crystal is able to absorb a

≈−− ))2exp(1( dα 0.67 of the incident beam flux in the case of the double-pass pumping scheme providing high uniformity of the absorbed flux over the crystal volume. In the simulation we assume that the laser rod axis is directed along the c-axis of the YLF crystal and we neglect anisotropy in absorption of the polarized beams of the LD bars. We assume that this anisotropy is not significant for the side-pumped configuration in which three LD bars are arranged around the crystal in threefold symmetry and the internal surface of the tube used for water cooling has a high diffusive reflection [25]. In fact, high incident fluxes are able to deplete the 3H6-level in Tm3+ and to reduce significantly absorption [23]. However, in this study the concentration of the 3H6-level does not fall below 0.9 of the Tm-concentration, the related variations of α do not exceed 5 %, and in the simulations we use:

⎩⎨⎧

Δ>Δ≤

×Δ

≈p

p

ppcr

papp tt

tt

thLd

QtR

,0

,1)( 2 νπ

ηη . (13)

where [ ])2exp(1)1( da αρη −−−= is the absorption efficiency of pumping, ρ is the reflection

factor of the pumping radiation into laser material, pQ is the pumping pulse energy,

ptΔ is

the pumping pulse duration and pη is quantum efficiency.

In Fig. 2 we show a simulated giant pulse (G-pulse) generated by a Tm,Ho:YLF laser producing ≈0.1 J pulses of ≈150 ns duration. In particular, we simulate an active Q-switched laser side-pumped by 0.5 ms LD pulses of 785 nm wavelength for a crystal 2 cm long and 2

mm in diameter placed inside a 1 m long cavity (outT =0.05 and lR =0.98) with a 0.85 mm

1.0x10

6

0.8

0.6

0.4

0.2

0.0

G-p

ulse

pow

er (

W)

1.2010x10-3

1.20061.20041.20021.2000

time (s)

Q-switch open - 1.2 msPumping time - 0.5 ms

Fig. 2. Simulation of G-pulse generation: pulse power versus time.



radius waist in the TEM00 laser beam distribution. The Q-switch is open after a 0.5 ms pumping period with a delay of 0.7 ms to ensure that the G-pulse generation starts after achieving the maximal possible gain. This delay is associated with the delay of excitation transfer from 3H4 to 3F4, and finally towards the lasing 5I7 level [22].

3. Thermal model

The heat absorbed inside the crystal leads to a temperature increase over the crystal volume. For high power operation this temperature shift is able to change the local values of the Boltzmann population factors of the upper and lower lasing levels:

[ ]

[ ]∑ −−

=

jBjj

Biii tTkEg

tTkEgtf

),(exp

),(exp),(

rr

r , (14)

where

Bk is the Boltzmann constant, ig is the degeneracy of the i-level, and ),( rtT is the

local temperature. Generally, the operating crystal is heated via lattice vibrations due to non-radiative decay of electrons from all levels involved in the excitations. The local heat source is defined by:

inriicr nEtq τ/),(

7

2i∑

=

Δ=r , (15)

where

iEΔ is the energy difference between the i-manifold and the next lower manifold into

which the electron makes the transition (Fig. 1) and inrτ are the non-radiative times inversely

proportional to the non-radiative transition probabilities. In order to avoid difficulties in defining the probabilities of non-radiative transitions, an estimate of the heat source can be made via the difference between the pumped energy and the energy of stimulated and spontaneous radiation leaving the crystal [12]. This approach is mainly used for the CW mode or as an averaged estimate for high-repetition pulsed mode. However, we use this approach for normal or Q-switched mode operation by introducing a

modification which takes into account the rate, dtdnE ii /7

2i

*∑

=

Δ , at which the pumped energy is

stored inside Tm3+ and Ho3+ ions as:

dtdnEnEtnfnfhchtRtq iiiriilseppcr / / ),()()(),(7

2i

*7

2i8877

1∑∑

==

− Δ−Δ−−−= τφνησν rr , (16)

where in addition to iEΔ we introduce the energy difference between the i-manifold and the

ground state *iEΔ (Fig. 1), and then

irτ are the corresponding radiative times [17].

The calculation of Eq. (16) for the Tm,Ho:YLF laser reveals several effects significant for energy extraction by lasing pulse. First, Fig. 3(a) shows the energy balance integrated over the crystal volume versus time. It reveals a very significant extension of the heat release period as compared with the pumping period. Fig. 3(a) shows that the heat is released inside



3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

ener

gy (

J)

20x10-3

151050time (s)

pumping heat source laser ouput optical loss

(a)

1.0

0.8

0.6

0.4

0.2

0.0

opti

cal

loss

(J)

20x10-3

151050time (s)

3F4

3H5

3H4

5I7

total optical loss(b)

Fig. 3. Energy balance versus time during laser operation: (a) energy pumping and release and (b) optical loss by spontaneous radiation from different levels.

the crystal over a period of ≈10 ms, whereas the pumping period is 0.5 ms during which only ≈30 % of heat is released. A two-time lower resulting temperature increase is achieved in the crystal prior to G-pulse generation (1.2 ms) as follows from an estimate neglecting the thermal conductivity effect:

∫≈Δt

crcr dttqc

tT0

),(1

),( rrρ

. (17)

Second, Fig. 3(a) also shows that in the final energy balance ≈0.12 J corresponds to the G-pulse energy, ≈0.75 J corresponds to the heat released inside the crystal and ≈0.84 J corresponds to the energy lost by spontaneous emission. Thus, about 43 % of the pumped energy is directly converted into heat. We should note that the estimates of heat release based on 2-level rate equations treat this value as the difference between the pumped energy and the optical energy of the laser pulse and the spontaneous emission from two levels, 3F4 and 5I7 [12]. The energy spontaneously emitted by other levels, i.e. 3H5,

3H4, 5I5 and 5I6, are

implicitly included into the heat released inside the crystal [12]. The 8-level model used here shows that the contribution of the 5I5 and 5I6 levels into the spontaneous emission loss is negligibly small, whereas the contribution of 3H4 and 3H5 appears to be quite significant, ≈0.1 and ≈0.4 J, respectively (see Fig. 3(b)). Adding these values to the heat release of ≈0.75 J gives ≈70 %, similar to the result from the 2-level model [12]. Thus, only ≈43 % of the pumped energy is released directly as heat inside the crystal, whereas ≈45 % is spontaneously emitted radiation from the crystal at wavelengths: λ2=1.93 μm, λ3=4.32 μm, λ4=2.46 μm and λ7=2.07 μm. These wavelengths are within the transparency range of the crystal and are therefore able to leave the crystal. Fig. 3(b) shows

the values of the total optical loss, irii nE τ/

7

2i∑

=

Δ , and also the losses emitted from all levels,

irii nE τ/Δ integrated over the crystal volume. These radiation fluxes leaving the crystal are

absorbed by the water flow typically used for crystal cooling. The water absorption coefficients for these wavelengths are [26]: α2=124 cm-1, α3=300 cm-1, α4=63.5 cm-1 and α7=31 cm-1. That is, the spontaneously emitted fluxes leaving the crystal are absorbed within lengths of 1−≈ iα , i.e. within 80, 33, 157 and 320 μm from the surface, respectively. The

absorption of these fluxes in the vicinity of the crystal surface can significantly inhibit heat dissipation from the crystal. The heat transfer to the water flow depends on the Reynolds number, Re, defining the level of the flow turbulency dependent on the water flow rate through the channel inside which the operating crystal is set up. Numerical estimates show



that for the typical coaxial crystal in a tube water channel geometry and typical flow rates, the value of the heat transfer coefficient is h=103-105 W/m2 K [27]. The main thermal resistance to the heat flow from the crystal surface is due to the thermal boundary layer,

Tδ , within which the heat conductance dominates over the convective transport. The estimate of

Tδ follows from the equivalency of )( ∞−=∂∂−=∂∂− wsurcrsurwwsurcrcr TThrTkrTk ,where kcr≈6

and kw≈0.6 W/m K are the thermal conductivity of crystal and water, respectively. That is, using

Twsurcrsurw TTrT δ/)( ∞−−≈∂∂ one finally obtains for h=103-105 W/m2 K:

≈≈ hkwT /δ 6-600 μm . (18)

Thus, the spontaneous IR fluxes are absorbed by water within a distance where the heat transfer is dominated by the thermal conductivity. Hence, the absorption of these fluxes is able to significantly inhibit the heat dissipation from the crystal. In order to consider the thermal effect we simulate the complex heat transfer non-steady state, two-dimensional problem by coupling the above optical model with the heat generation and heat transport through the operating crystal, and the water boundary layer inside which the absorption of spontaneously emitted IR radiation takes place. The radially symmetric temperature distribution inside the cylindrical crystal, ),( rtTcr

, and the thermal boundary layer in water,

),( rtTw, are defined by:

)r,()( tqTktTC iiiiii +∇∇=∂∂ρ , (19)

for crystal (i=cr) and water (i=w) with the boundary condition

∞= ww TT at TRr δ+= 0

,

where ( )[ ]1exp 00 −= hRkR wTδ takes into account the radial curvature.

Heat source density inside the crystal is defined by Eq. (16) whereas the heat source density due to the absorption of spontaneously emitted IR fluxes in water is defined by:

[ ]∑ −−=i

iiiw RrtJr

Rrtq )(exp)(),( 00

0 αα , (20)

where )(0 tJ i

are the IR flux densities isotropically leaving the crystal given by:

dVtnES

tJV

iriicr

i ∫ Δ= τ/)r,(1

)(0. (21)

The effect of IR radiation absorption is negligibly small for h>105 W/m2 K and

Tδ <6 μm,

when 1−<< iT αδ . However, for h≈104 W/m2 K (Tδ ≈60 μm ) this effect is very significant,

and can lead to the onset of an inverted temperature distribution inside the crystal when the temperature inside the boundary layer is higher than that inside the crystal. This effect is shown to take place in a coupled thermo-optical simulation performed using a conservative 50x100 conservative finite-difference approximation. The main results of this simulation given in Fig. 4 at three times show that a significant temperature increase has occured (≈2 K) by the start of G-pulse generation (1.2 ms), which leads to a pulse energy decrease due to the decrease in

8877 nfnf − over the crystal volume. This figure also shows that an inverted

temperature distribution inside the crystal is present at period of time of ≈10 ms.



300

298

296

294

292

290

T (

K)

1.4x10-3

1.00.80.60.40.20.0

radius (m)

operating crystal

boundary layer in water 60 μm

t=1.2 ms

t=5 ms

t=50 ms

Fig. 4. Temperature distribution inside the operating Tm,Ho:YLF crystal and thermal boundary

layer for single G-pulse generation for h=104 W/m2 K (δT≈60 μm, water temperature ∞wT =290 K).

This effect, associated with the strong absorption of emitted IR radiation in water, produces a different result in high repetition mode. In particular, the simulation of 20 and 50 Hz G-pulse repetition mode given in Fig. 5 shows that in contrast with first few pulses after pulsed operation stabilization the temperature inside the crystal becomes higher than that at the crystal surface due to the onset of a quasi-steady state gradient. Finally, Fig. 5(a) shows that this thermal effect leads to ≈10-25 % reduction of G-pulse energy combined with a strong thermal lensing effect known to be detrimental for laser beam quality.

4. Summary

A complex thermo-optical model for Tm,Ho solid state lasers has been developed based on an 8-level rate dynamics model for the excitation transfer to Ho3+ ions from LD pumped Tm3+ ions integrated together with the equation for the total number of stimulated photons inside the cavity. This model is also coupled with a two-dimensional time dependent heat transfer model including absorption, heat release and heat transfer inside the operating crystal as well as the absorption and the thermal effect of infrared radiation fluxes spontaneously emitted by the operating crystal. In the case of water cooled laser operation the thermal effect is shown to

1.0x106

0.8

0.6

0.4

0.2

0.0

G-p

ulse

pow

er (

W)

1.00.80.60.40.20.0time (s)

(a) 20 Hz 50Hz

380

360

340

320

300

T (

K)

1.00.80.60.40.20.0

time (s)

20 Hz 50 Hz

(b)

axis

surface

axis

surface

Fig. 5. 20 and 50 Hz G-pulse laser operation: (a) G-pulse power modification with time and (b) temperature increase in the operating crystal versus time for h=104 W/m2 K for crystal axis and surface.

be split into two simultaneously occurring processes: (i) direct heat release inside the crystal and (ii) infrared spontaneously emitted radiation fully absorbed in water over a distance of several hundreds of microns, which corresponds to a typical value of boundary layer thickness. In particular, the simulations show that only ≈43 % of the pumped energy is



transformed into heat directly inside the crystal, whereas ≈45 % is IR radiation spontaneously emitted by 3H4,

3H5, 3F4 and 5I7 levels and absorbed in the vicinity of the

crystal surface. The absorption taking place within the boundary layer provides an additional strong thermal effect, inhibiting the dissipation of the heat from the crystal and significantly increasing crystal temperature. The resulting temperature increase is shown to reduce significantly G-pulse energy.

Acknowledgments

We would like to acknowledge the financial support from the National Institute of Information and Communications Technology (Japan). We would also like to thank Dr. J. Hester from the Australian Nuclear Science and Technology Organization for careful reading of this paper and valuable comments.



Documents

Computational model for operation of 2 μm co-doped Tm,Ho solid state lasers