Modeling and optimization of actively Q-switched Nd-doped quasi-three-level laser

Optics Communications 305 (2013) 276–281

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Modeling and optimization of actively Q-switched Nd-doped quasi-three-level laser Renpeng Yan a,b,n, Xin Yu a,b, Xudong Li a,b, Deying Chen a,b, Jing Gao c a

National Key Laboratory of Science and Technology on Tunable Laser, Harbin Institute of Technology, Harbin 150080, China Institute of Opto-electronics, Harbin Institute of Technology, Harbin 150080, China c Suzhou Institute of Biomedical Engineering and Technology, Chinese Academy of Sciences, Suzhou 215163, China b

art ic l e i nf o

a b s t r a c t

Article history: Received 3 April 2013 Received in revised form 10 May 2013 Accepted 11 May 2013 Available online 29 May 2013

The energy transfer upconversion and the ground state absorption are considered in solving the rate equations for an active Q-switched quasi-three-level laser. The dependence of output pulse characters on the laser parameters is investigated by solving the rate equations. The influence of the energy transfer upconversion on the pulsed laser performance is illustrated and discussed. By this model, the optimal parameters could be achieved for arbitrary quasi-three-level Q-switched lasers. An acousto-optical Q-switched Nd:YAG 946 nm laser is constructed and the reliability of the theoretical model is demonstrated. & 2013 Elsevier B.V. All rights reserved.

Keywords: Quasi-three-level Q-switched Energy transfer upconversion

1. Introduction Laser diode (LD) end-pumped neodymium-doped quasi-threelevel laser is an efficient and convenient approach to generate light around 900 nm and also wavelengths in blue spectrum region by frequency doubling [1,2]. The blue lasers have a wide range of applications in optical data storage, color displays, submarine communications, and detection. In these applications, high-peakpower pulsed lasers especially at high-repetition-rate are in demand. Since the first demonstration of continuous-wave (cw) neodymium-doped laser in the 4F3/2–4I9/2 transition by Fan and Byer [1], great efforts have been made to generate pulsed lasers around 900 nm and improve their performances [3–7]. However, the output character of pulsed lasers around 900 nm is still far from the laser performance in the four-level transition. The main drawback is attributed to the reabsorption losses at the lasing wavelength combined with a relatively low stimulated emission cross section, which is nearly ten times lower than that in the fourlevel transition [8]. Overcoming these handicaps needs tightfocusing of high-radiance LD pump source, which increases the temperature of laser medium and aggravates the temperaturedependent reabsorption losses and the de-excitation processes.

n Corresponding author at: National Key Laboratory of Science and Technology on Tunable Laser, Harbin Institute of Technology, Harbin 150080, China. fax: +86 45186413161. E-mail addresses: [email protected], [email protected] (R. Yan).

0030-4018/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2013.05.031

In recent years, significant simulation development in understanding the process of quasi-three-level Q-switched lasers has also been achieved [9–13]. Beach derived a set of equations describing the pump excitation and the Q-switched energy extraction of quasi-three-level laser system for performing optimization [9]. The rate equations of actively Q-switched quasi-three-level laser under Gaussian pump distribution were obtained by Zhang et al. with consideration of the spatial distributions of the photon density and the population inversion density [12]. Nevertheless, significant effort is still necessary for determining optimized parameters and achieving high efficiency output in quasi-threelevel pulsed lasers. Besides, the energy transfer upconversion (ETU) should also be taken into account in neodymium-doped quasi-three-level lasers for the great influence on the laser performance, especially in Q-switched operation for its high inversion population density [14–16]. In practical applications, the pump light with a top-hat distribution is more reasonable at high power LD pumping. In this paper, the ETU and the ground state absorption (GSA) effects are considered in the rate equations for quasi-three-level actively Q-switched laser. The reduction of inversion population density by the ETU effects is involved in the pumping stage. The dependence of output pulse on the parameters of pumping, gain medium and resonator is investigated by solving the equations numerically. The influence of ETU effects on the pulse energy and the pulse width under different pumping condition and frequency is analyzed. By using this model, the optimal laser parameters could also be determined under different laser conditions. In experiment, an acousto-optical (A-O) Q-switched Nd:YAG

R. Yan et al. / Optics Communications 305 (2013) 276–281

946 nm laser is constructed to illustrate the utility of the theoretical model.

2. Rate equations and solutions The simulation is based on a geometry of LD end-pumped solid-state laser with a laser rod length of l. The pump source is assumed to be in a top-hat distribution and independent with time. The pumping rate is expressed as following: r p ðr; zÞ ¼

P in αexpð−αzÞΘðω2p −r 2 Þ

Θðω2P −r 2 Þ ¼

hυp 8 <1

r 2 o ω2p

:0

r 2 4 ω2p

ð1Þ

where r is a radial coordinate and z is an axial coordinate, Pin is an incident pump power, α is absorption coefficient, ωp is the pump beam radius in the laser medium, hυp is the pump photon energy. In the simulation, only the fundamental TEM00 laser mode with a Gaussian intensity distribution is considered. In this case, photon density of the cavity mode is given by ! −2r 2 ϕðr; tÞ ¼ ϕð0; tÞexp ð2Þ ω2l ωl is Gaussian beam radius for the laser mode, ϕ(0, t) is photon density along the axis. Length of laser rod is usually much shorter than cavity length in quasi-three-level lasers, so the variation of photon density ϕ(r, t) in laser medium along an axial direction could be neglected. When there is no pumping in laser medium, population density in the lower laser level nl0 is: nl0 ¼ f l N0

In quasi-three-level transitions, the occupation ratio between the lower laser level and the upper laser level fl/fu satisfies the condition fl/fu≪1. So population at the ground state is assumed to be not reduced greatly by pumping. The initial population intensity in the lower laser level nl(r, t) can be assumed to be nl and the initial population intensity of the upper laser level nu(r, t) could be assumed to be proportional to the pumping light distribution. Therefore, the initial conditions of Eqs. (4) and (5) are given by: nl ðr; 0Þ ¼ nl0

ð8Þ

nu ðr; 0Þ ¼ nu ð0; 0ÞΘðω2p −r 2 Þ

ð9Þ

where nu(0, 0) is the initial population density in the upper laser level along the laser axis. For the big amount of population density in the upper level, the ETU process should be taken into account during low Q segment of the cycle. The rate equation for the initial upper laser level population density for the Q-switched segment is written as [18,19]: dnu ð0; 0Þ η P 1 nu ð0; 0Þ ¼ α in − −Unu ð0; 0Þ2 dt τu hvp πω2p l

dnl ðr; tÞ ¼ f l se c½nu ðr; tÞ−nl ðr; tÞϕðr; tÞ dt

ð4Þ

dnu ðr; tÞ ¼ −f u se c½nu ðr; tÞ−nl ðr; tÞϕðr; tÞ dt

ð5Þ

where se is stimulated emission cross section, c is light speed, fu is Boltzmann fraction at upper laser level, nl(r, t) and nu(r, t) are the average population density along the axial coordinate z for lower and upper laser levels, which is given by: Rl ni ðr; z; tÞ dz i ¼ u; l ð6Þ ni ðr; tÞ ¼ 0 l The differential equation describing dϕ(r, t)/dt should be integrated over the beam cross section to guarantee the beam Gaussian distribution during the entire formation process of the Qswitched pulse. It can be written as [17]: Z ∞ Z ∞  dϕðr; tÞ ϕðr; tÞ  2πrdr ¼ 2se ½nu ðr; tÞ−nl ðr; tÞl−lnð1=RÞ−L dt t r 0 0  2πrdr ð7Þ where tr ¼2l′/c is the roundtrip transit time of light in the resonator with an optical length l′, R is the reflectivity of output mirror, and L is the roundtrip dissipated optical losses.

ð10Þ

where ηα is the pump absorbed efficiency ηα ¼1−exp(−αl), U is the ETU rate constant, and τu is the lifetime of the upper laser level. When pump power exceeds threshold, the accumulation population density at the end of low-Q segment are given as:   ð1 þ Bnf þ AÞ þ ð1 þ Bnf −AÞexpð−A=τu f Þ 1 A −1 ð11Þ nu ð0; 0Þ ¼ B ð1 þ Bnf þ AÞ−ð1 þ Bnf −AÞexpð−A=τu f Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi η P 1 A ¼ 1 þ 4τ2u U α in 2 hvp πωp l

ð12Þ

B ¼ 2τu U

ð13Þ

ð3Þ

where N0 is an total population density in laser rod, and fl is Boltzmann occupation at lower laser level. In the rapid Q-switch process, the spontaneous relaxation and the pumping can be neglected during the development of a shortduration output pulse. Therefore, the rate equations in quasithree-level transition could be given by:

277

nf ¼

Δnf þ f l N 0 γ

Δnf ¼

L 2se l

ð14Þ ð15Þ

where f is frequency rate, nf is the resident population density in the upper laser level from previous pulse procedure, γ¼ fu+fl is the inversion reduction factor corresponding to the net reduction in the population inversion resulting from the stimulated emission of a single photon, Δnf is the residual population inversion density from the preceding pulse. By using Eqs. (4), (5), (8), (9), the inversion population density is obtained: h i nu ðr; tÞ−nl ðr; tÞ ¼ nu ð0; 0ÞΘðω2p −r 2 Þ−nl0 " !# Z t −2r 2 ϕð0; tÞdtexp ð16Þ  exp −γse c ω2l 0 Substituting Eqs. (2) and (16) into Eq. (7) and performing the integration over time, one obtains: R∞ dϕð0;tÞ 2 2 ¼ 4seωlϕð0;tÞ 2t 0 ½nu ð0; 0ÞΘðωp −r Þ−nl0  dt l r " !# ! Z t −2r 2 −2r 2 ϕð0; tÞdtexp exp −γse c  exp 2rdr ω2l ω2l 0 −

 ϕð0; tÞ  lnð1=RÞ þ L tr

ð17Þ

This equation is the basic differential equation describing ϕ(r, t) as a function of t in Q-switched quasi-three-level lasers. When Eq. (17) equals to zero at t¼0, the threshold condition for upper

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laser level population density in the laser axis is expressed as: nut h ð0; 0Þ ¼

lnð1=RÞ þ L þ 2se nl0 l

ω2 2se l 1−exp − ωp2 l

¼ nut h1 ð0; 0Þ þ nut h2 ð0; 0Þ nut h1 ð0; 0Þ ¼

ð18Þ

lnð1=RÞ þ L

ω2 2se l 1−exp − ωp2

ð19Þ

l

nut h2 ð0; 0Þ ¼

nl0

ω2 1−exp − ωp2

ð20Þ

l

The threshold inversion population density nu_th(0, 0) can be divided into two parts: nu_th1(0, 0) comes from intracavity losses including the output mirror transmission and the round-trip dissipative optical losses; nu_th2(0, 0) represents the reabsorption losses in quasi-three-level system. It is convenient to define a normalized time τ, a normalized photon density Φ(r, τ), and three new parameters M, N, β as τ¼

 t  lnð1=RÞ þ L tr

Φðr; τÞ ¼ ϕðr; τÞ

2γse l′ lnð1=RÞ þ L

ð21Þ ð22Þ

N¼

nu ð0; 0Þ nut h1 ð0; 0Þ

ð23Þ

M¼

nut h2 ð0; 0Þ nut h1 ð0; 0Þ

ð24Þ

β¼

ω2p

ð25Þ

ω2l

where N is the ratio between the total inversion population to the threshold inversion population, which is proportional to the pump power density, and M is the proportion of the reabsorption losses in quasi-three-level transition to the intrinsic intracavity losses. Substituting Eqs. (21)–(25) into Eq. (17), it yields: dΦð0; τÞ expð−AðτÞexpð−βÞÞ−expð−AðτÞÞ ¼ NΦð0; τÞ dt AðτÞð1−expð−βÞÞ 1−expð−AðτÞÞ −Φð0; τÞ −MΦð0; τÞ AðτÞ where AðτÞ ¼

Z

τ 0

Φð0; τÞ dτ

ð26Þ

ð27Þ

Eq. (26) describes the normalized photon density along the laser axis versus the normalized time. By numerically solving Eqs. (26), (27) with Runge–Kutta method [20], the normalized photon density Φ(0, τ) in different conditions can be obtained. Then normalized pulse duration Δτ, full width at half maximum (FWHM) of Φ(0, τ), are also possible to achieve. Φinteg, the integration of Φ(0, τ) over τ from 0 to ∞, is written as: Z ∞ Φinteg ¼ Φð0; τÞ dτ ð28Þ 0

From Eq. (26), it could be seen that the output characters of pulsed laser are mainly in relation with M and β at a fixed pump power, N. Fig. 1 shows the plots of Φinteg and Δτ as a function of β with different M when N equals to 2. The parameter M has a negative influence on the laser performance of quasi-three-level lasers. The influence becomes more serious at a smaller value of β. It is reasonable by the population density participating in the laser operation. When ωp oωl, nearly all the population participates in

Fig. 1. Φinteg (a) and Δτ (b) as a function of β for different M when N equals to 2.

the laser operation. When ωp 4ωl, only the central part of the inversion population interacts with the laser mode. Besides, a smaller β also leads to higher intracavity diffraction losses in quasi-three-level lasers [21]. According to theory, pulse energy E, pulse peak power Pm and pulse width W can be expressed as [22]: E¼

πω2l hυl lnð1=RÞΦinteg 4se γ

ð29Þ

Pm ¼

 πω2l hυl  lnð1=RÞ þ L lnð1=RÞΦm 4se γt r

ð30Þ

W¼

Δτ  t r lnð1=RÞ þ L

ð31Þ

where Φm is the maximum value of Φ(0, τ), hυl is the laser photon energy. With the previous model, the influence of ETU effects on the performance of Q-switched quasi-three-level laser is possible to be investigated. Let's consider a LD end-pumped Q-switched Nd:YAG quasi-three-level 946 nm laser for simulation and the parameters used are fully specified in Ref. [13]. Fig. 2 presents the pulse energy and the pulse width versus incident pump power for different U in Nd:YAG 946 nm pulsed laser at 10 kHz. The pulse energy increases

R. Yan et al. / Optics Communications 305 (2013) 276–281

Fig. 2. Pulse energy (a) and pulse width (b) versus incident pump power for different U in 10 kHz Nd:YAG 946 nm laser.

and the pulse width decreases with the incident pump power. It could also be seen that the pulse energy decreases and the pulse width increases with the enhancement of U values at the same pump power. It is also noted from Fig. 2 that the influence of the ETU effects on the pulse energy becomes bigger as the incident pump power increases. Fig. 3 plots the pulse energy and the pulse width as a function of pulse repetition rate for different U at the incident pump power of 40 W. With the reduction of the frequency, the influence of ETU effects on the pulse energy and the pulse width becomes more obvious. When the repetition rate is lower than 5 kHz, the laser output character remains nearly constant due to the existence of ETU effects (U≠0). When the ETU effects are taken into consideration (U¼ 2  10−16 cm3/s for Nd:YAG crystal [23]), optimization of laser rod parameters in Q-switched Nd:YAG 946 nm pulsed laser can be conducted. The output character of 10 kHz 946 nm laser versus doping concentration of the laser rod is presented in Fig. 4. In the concentration optimization, the absorption coefficient is calculated by the product of the doping concentration and the absorbed cross section at the pump wavelength. It could be seen that the output pulse character becomes nearly constant and then fails when the doping concentration is higher than 0.5 at%. It could be ascribed to the absorption saturation in the laser rod at high doping concentration and the inversion population depletion induced by the ETU and the GSA effects. The optimal doping concentration of 0.5 at% is

279

Fig. 3. Pulse energy (a) and pulse width (b) versus frequency for different U in Nd: YAG 946 nm laser at the incident pump power of 40 W.

Fig. 4. Profile of the pulse energy and the pulse width versus the doping concentration in Nd:YAG Pin ¼ 30 W, l ¼ 5 mm, T ¼ 9%.

much lower than those often used in the Nd:YAG quasi-three-level laser operations [2,24]. If the thermal effects induced by these effects are considered in the simulation, the lower concentration

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would be preferred. The profile of the pulse energy and the pulse width versus the transmissivity of output mirror is shown in Fig. 5 when the doping concentration is set to be 0.3 at%. The optimized transmissivity of output mirror is about 10% for the pulsed Nd:YAG 946 nm laser. It is higher than that used in cw quasi-three-level laser operation but lower than that used in four-level laser operation. It is necessary to mention that the optimal parameter also has relation with the pump condition.

3. Experiments and results The schematic of LD end-pumped Nd:YAG pulsed 946 nm laser by an A-O Q-switch is presented in Fig. 6. The pump source is a 50 W fiber-coupled LD with a diameter of 400 μm and a numerical aperture (N.A.) of 0.22. The output radiation is re-imaged into the laser rod by a coupling system with a ratio of 1:1. The doping concentration of Nd:YAG rod is chosen to be 0.3 at% to minimize the thermal effects induced by the high doping concentration [25]. The laser rod is wrapped by a 0.05 mm thick indium foil, mounted in a micro-channel copper heat sink, and kept at 286 70.2 K by water cooling. Both facets of laser rod are coated with high transmissivity at 946 nm (T 499%) as well as antireflection coated at 1064 nm (R o5%) and 1342 nm (Ro10%). The A-O Q-switch is a 15 mm long quartz crystal with a repetition rate range from 1 kHz to 200 kHz. The laser cavity is a plane–plane cavity with a cavity length of 50 mm. Fig. 7 plots the output pulse energy and the pulse width versus absorbed pump power in Q-switched 946 nm laser at 10 kHz. At the absorbed pump power of 16.8 W, a pulse energy of 0.16 mJ and a pulse width of 23 ns are achieved at 10 kHz. The simulation results with and without ETU effects are also associated in Fig. 7. It could be noted that the simulation results without ETU effects overestimate the output performance in the pulsed 946 nm laser. In contrast, the experimental results could be described properly with the simulation results with ETU effects. The difference

Fig. 7. Output pulse energy (a) and pulse width versus (b) absorbed pump power in Q-switched Nd:YAG 946 nm laser at 10 kHz.

between the experimental and theoretical results is due to the multimode oscillation and the thermal effects in the practical operation. The temporal profile of output single pulse in 10 kHz Nd:YAG 946 nm laser is shown in Fig. 8, in which good agreement between experimental data and theoretical calculation could be found.

4. Conclusion Fig. 5. Profile of the pulse energy and the pulse width versus the transmissivity of output mirror Pin ¼30 W, l¼ 5 mm.

Fig. 6. Schematic setup of LD end-pumped Nd:YAG 946 nm pulsed laser.

We have obtained the rate equations for quasi-three-level actively Q-switched laser under a top-hat pump beam distribution by considering GSA and ETU effects, the spatial distributions of the intracavity photon density and the population inversion density. The influence of pumping, gain medium and resonator on the laser performance is analyzed by solving the rate equations. The ETU effects are involved in the initial population accumulation and the influence is investigated under different conditions. With the theoretical model the parameter optimization is conducted and meaningful results could be achieved. The experimental results in an A-O Q-switched Nd:YAG 946 nm laser have good agreement with the theoretical results.

R. Yan et al. / Optics Communications 305 (2013) 276–281

Fig. 8. Experimental and theoretical temporal profile of output single pulse at 10 kHz.

Acknowledgements This work was partially supported by the National Scientific Foundation of China (Grant No. 61275127), the Fundamental Research Funds for the Central Universities (Grant No. HIT. NSRIF. 201165), and the Natural Science Foundation of Jiangsu Province (Grant No. BK2011330). References [1] T.Y. Fan, R.L. Byer, IEEE Journal of Quantum Electronics 23 (1987) 605.

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