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Rate equations and solutions of a laser-diode end-pumped passively Q-switched intracavity doubling laser by taking into account intracavity laser spatial distribution
Optics Communications 234 (2004) 321–328 www.elsevier.com/locate/optcom
Rate equations and solutions of a laser-diode end-pumped passively Q-switched intracavity doubling laser by taking into account intracavity laser spatial distribution Guiqiu Li *, Shengzhi Zhao, Hongming Zhao, Kejian Yang, Shuanghong Ding School of Information Science and Engineering, Shandong University, Jinan 250100, China Received 9 November 2003; received in revised form 12 January 2004; accepted 6 February 2004
Abstract The intracavity photon density and the initial population-inversion density are assumed to be Gaussian spatial distributions in the rate equations of a laser-diode end-pumped passively Q-switched intracavity-frequency-doubling Nd:YVO4 /KTP laser with Cr4þ :YAG saturable absorber. These space-dependent rate equations are solved numerically. In the experiment, a laser-diode end-pumped passively Q-switched Nd:YVO4 /KTP laser with Cr4þ :YAG saturable absorber is realized. The dependences of pulse width, pulse repetition rate, single-pulse energy and peak power on incident pump power are measured for the generated-green-light pulses, and the experimental results are consistent with the numerical solutions. Ó 2004 Elsevier B.V. All rights reserved. PACS: 42.55.xi; 42.60.Gd Keywords: Rate equations; Numerical solutions; Gaussian spatial distribution; Intracavity-frequency-doubling; Passively Q-switched; LD-pumped
1. Introduction In recent years, laser-diode (LD) pumped solidstate Q-switched lasers have attracted a great deal of attention because of their high efficiency, simplicity, compactness, good frequency stability. All solid-state Q-switched lasers have wide applica-
*
Corresponding author. Tel.: +86-531-8364840; fax: +86531-8565167. E-mail address: [email protected] (G. Li).
tions in the fields of remote sensing, information storage, coherent telecommunications, medicine, etc. It is easy to obtain high nonlinear conversion efficiency by placing the nonlinear crystal into the laser resonator due to its high fundamental wave power density, which is especially so for a laser with middle or low output power [1–3]. Rate equations are efficient tools for analyzing the performance of a Q-switched laser. By addition of a term to represent the nonlinear loss that is due to second-harmonic generation (SHG) to the photon-density equation, the characteristics of the
0030-4018/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.02.013
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fundamental-frequency and second-harmonic pulses of passively Q-switched intracavity-frequency-doubling lasers have been described [4,5]. However, the rate equations mentioned above are obtained under a plane-wave approximation. Actually, this assumption is not properly satisfied because the fundamental wave is a Gaussian transversal distribution for the TEM00 mode. Recently the transversal distributions of the intracavity photon density and the populationinversion density in the rate equations of passively Q-switched laser [6] and of passively Q-switched self-frequency-doubling laser [7] have been considered and the importance of these distributions has been shown. When the Gaussian distribution of the intracavity laser intensity is taken into account in the rate equations, the theoretical results obtained by numerically solving these rate equations are more close to the experimental results than those obtained under the plane-wave approximation [8,9]. But in [7] the pump source is a pulsed xenon flash-lamp with the pump energy being slightly higher than the threshold pump energy, and thus only one Q-switched pulse is generated in each pump pulse. Besides, the influence of pump rate on the intracavity photon density is not taken into account in [7]. Therefore, for a more accurate theoretical analysis of the pulses from a LD-pumped passively Q-switched intracavity-frequency-doubling laser it is desirable to take into account the influence of pump rate on the intracavity photon density and consider the transversal variations of the intracavity photon density and the population-inversion density in the rate equations. In this paper, we introduce the rate equations of a laser-diode end-pumped passively Q-switched intracavity-frequency-doubling Nd:YVO4 /KTP laser with Cr4þ :YAG saturable absorber, in which the intracavity photon density and the initial population-inversion density are assumed to be Gaussian spatial distributions. These space-dependent rate equations are solved numerically. From the numerical solutions, we obtain the dependences of pulse width, pulse repetition rate, single-pulse energy and peak power on pump power for the generated-green-light pulses. The numerical solutions are consistent with the exper-
imental results obtained from a laser-diode endpumped passively Q-switched Nd:YVO4 /KTP laser.
2. Theoretical calculations 2.1. Nonlinear loss due to harmonic conversion When the rate equations are used to analysis the performance of a Q-switched intracavity-frequency-doubling laser, the second-harmonic conversion can be considered as the nonlinear loss of the fundamental wave. The second-harmonic power for type-II phase-matching KTP crystal is [10] P2x ¼ KN
l2K 2 P ; AK x
ð1Þ
where KN ¼
c3 e
2 x2 deff ; 2x x x 0 ne2 ne2 ne1
lK is the length of KTP; x is the angle frequency of fundamental wave; deff is the effective nonlinear coefficient; c is the velocity of light in vacuum; e0 is x x the dielectric permeability of vacuum; n2x e2 , ne2 , ne1 are harmonic and fundamental wave refractive indices, respectively, and AK ¼ ð1=2Þpw2K is the area of fundamental wave at the position of KTP, where wK is the radius of the TEM00 mode at the position of KTP. According to the relationship of power and photon density: Px ¼ AK hxcð/K =2Þ, where /K is the photon density at the position of KTP, hx is the single-photon energy of the fundamental wave, h is PlanckÕs constant, and the nonlinear loss resulting from harmonic conversion can be expressed as dN ¼
P2x tr KN hxcl2K /K ¼ dK /K ; ¼ hx/K AK Lc 2
ð2Þ
where KN hxcl2K ; 2 Lc is the optical length of the resonator; tr ¼ 2Lc =c is the round-trip time of the resonator. The corresponding parameters for type-II phase-matching KTP crystal are presented in Table 1. dK ¼
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323
Table 1 The parameters of II-type phase-matching KTP crystal
nxe1 nxe2 n2x e2 deff e0
Parameter
Value
Fundamental wave refractive index of o light Fundamental wave refractive index of e light Second-harmonic wave refractive index Effective nonlinear coefficient of KTP Dielectric permeability of vacuum
1.83 1.746 1.79 7.2 pm/V 8.855 1012 C2 /N m2
These data are provided by Coretech Crystal Company, Shandong University, China.
2.2. Rate equations We consider a LD-pumped passively Q-switched intracavity-frequency-doubling Nd:YVO4 / KTP laser with Cr4þ :YAG saturable absorber, in which Nd:YVO4 works as the gain medium, KTP works as the frequency-doubling crystal, and Cr4þ :YAG works as the passive Q-switch. If the intracavity photon density is assumed to be a Gaussian spatial distribution during the entire formatting process of the LD-pumped passively Qswitched intracavity-frequency-doubling laser pulse, the intracavity photon density /ðr; tÞ for the TEM00 mode can be expressed as 2r2 /ðr; tÞ ¼ /ð0; tÞ exp 2 ; ð3Þ wl where r is the radial coordinate; wl is the average radius of the TEM00 mode, which is mainly determined by the geometry of the resonator; and /ð0; t) is the photon density in the laser axis. Because the radii of the Gaussian beam waists at three positions: Nd:YVO4 crystal, saturable absorber and KTP crystal are not equal, the photon density /g ðr; tÞ; /s ðr; tÞ, and /K ðr; tÞ at three positions: Nd:YVO4 crystal, saturable absorber and KTP crystal can be expressed as ! 2r2 /g ðr; tÞ ¼ /g ð0; tÞ exp 2 ; ð4Þ wg /s ðr; tÞ ¼ /s ð0; tÞ exp
2r2 2 ; ws
/K ðr; tÞ ¼ /K ð0; tÞ exp
2r 2 ; wK
ð5Þ
2
ð6Þ
where wg , ws , and wK are the radii of the TEM00 mode at three positions: Nd:YVO4 crystal, saturable absorber and KTP crystal, respectively; /g ð0; tÞ, /s ð0; tÞ, and /K ð0; tÞ are the photon densities in the laser axis at three positions: Nd:YVO4 crystal, saturable absorber and KTP crystal, reR1 spectively. Because 0 /ðr; tÞ2pr dr can be considered to be independent of the laser longitudinal axis z [7], Eqs. (4)–(6) can be expressed as ! w2l 2r2 /g ðr; tÞ ¼ 2 /ð0; tÞ exp 2 ; ð7Þ wg wg
/s ðr; tÞ ¼
w2l /ð0; tÞ exp w2s
/K ðr; tÞ ¼
w2l /ð0; tÞ exp w2K
2r2 ; w2s
2r2 : w2K
ð8Þ
ð9Þ
So for this laser, if neglecting the spontaneous radiation during the pulse formation, we can obtain the coupling rate equations [11,12] d/ðr; tÞ 1 ¼ 2rnl/g ðr; tÞ 2rg ns1 ls /s ðr; tÞ dt tr 2re ðns0 ns1 Þls /s ðr; tÞ dK /2K ðr; tÞ L/ðr; tÞ ;
ð10Þ
dn n ¼ Rin rcn/g ðr; tÞ ; dt s
ð11Þ
dns1 ns0 ns1 ¼ rg cns1 /s ðr; tÞ; dt ss
ð12Þ
where n is the population-inversion density; ns1 and ns0 are the ground-state and total population
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densities of Cr4þ :YAG saturable absorber, respectively; r and l are the stimulated-emission cross-section and length of Nd:YVO4 gain medium, respectively; rg and re are the ground state and excited state absorption cross-sections of the saturable absorber, respectively; ls is the length of the saturable absorber; tr is the round-trip time of light in the resonator ftr ¼ 2n1 l þ 2n2 ls þ 2n3 lK þ2ðLe l ls lK Þ=cg; n1 ; n2 , and n3 are the refractive indices of Nd:YVO4 gain medium, Cr4þ :YAG saturable absorber, and KTP crystal, respectively; lK is the length of KTP; Le is the cavity length; c is the velocity of light in vacuum; dK is given in Eq. (2); L is the intrinsic loss; s is the stimulated-radiation lifetime of the gain medium; ss is the excited-state lifetime of the saturable absorber; Rin ¼ Pin ½1 expðalÞ=hcp pw2p l is the pump rate, where Pin is the pump power, hcp is the single-photon energy of the pump light, wp is the average radius of the pump beam in the gain medium, a is the absorption coefficient of the gain medium. In the rate equations of a passively Q-switched intracavity-frequency-doubling laser, the spatial variations of the population-inversion density of the gain medium and the ground-state population density of the saturable absorber should also be considered. Besides, the differential equation describing d/ðr; tÞ=dt should be integrated over the beam cross-section to guarantee the beam Gaussian spatial distribution during the entire formatting process of the passively Q-switched intracavity-frequency-doubling laser pulse. In addition, for laser-diode end-pumped lasers, the pump light can be approximated by a Gaussian profile. By making the above modifications to the rate Eqs. (10)–(12), we obtain Z 1 d/ðr; tÞ 2pr dr dt 0 Z 1 1 ¼ 2rnðr; tÞl/g ðr; tÞ 2rg ns1 ðr; tÞls /s ðr; tÞ t r 0 2re ½ns0 ns1 ðr; tÞls /s ðr; tÞ dK /2K ðr; tÞ L/ðr; tÞg2pr dr; dnðr; tÞ nðr; tÞ ¼ Rin ðrÞ rcnðr; tÞ/g ðr; tÞ ; dt s
ð13Þ ð14Þ
dns1 ðr; tÞ ns0 ns1 ðr; tÞ ¼ rg cns1 ðr; tÞ/s ðr; tÞ; dt ss ð15Þ where nðr; tÞ is the average population-inversion density [6]; ns1 ðr; tÞ is the ground-state population density of the saturable absorber; Rin ðrÞ ¼ Rin expð2r2 =w2p Þ ¼ Pin ½1 expðalÞ=hcp pw2p l expð2r2 =w2p Þ is the pump rate. Before the laser action begins, all the saturable-absorber population is situated in the ground state. So the initial value of ns1 ðr; tÞ can be considered to be ns0 . For four-level gain media like Nd:YVO4 , the pumped population is small compared to the total population, the ground state population density can be considered as constant, and the pumped population density should be proportional to the pump light intensity. If the pump light can be approximated by a Gaussian profile, then the pumped population density can also be considered to be a Gaussian distribution. For a single pulse of an LD-endpumped repetitively Q-switched laser, the initial population-inversion density nðr; 0Þ comes from the pumped population density during the interval between the preceding pulse and this pulse if the residual population-inversion density from the preceding Q-switched pulse, which is not a Gaussian distribution, can be neglected. That is to say, under the approximation that no residual population-inversion density from the preceding pulse is included, the initial population-inversion density nðr; 0Þ can be assumed to be a Gaussian spatial distribution [6,7]. Thus the initial conditions of Eqs. (14) and (15) can be written as ns1 ðr; 0Þ ¼ ns0 ; nðr; 0Þ ¼ nð0; 0Þ exp
ð16Þ
2r2 w2p
! ;
ð17Þ
where nð0; 0Þ is the initial population-inversion density in the laser axis. Substituting Eqs. (7), (8), (16) and (17) into Eqs. (14) and (15) and integrating the results over time, we obtain
G. Li et al. / Optics Communications 234 (2004) 321–328
" nðr; tÞ ¼ exp rc
w2l exp w2g
# t /ð0; tÞdt s 0 ( " Z Z
Rin ðrÞ
exp rc 0
325
approximation or by preliminarily solving the equations.
t
t
2r2 w2g
!
Z
t 0
2.3. Solutions of the rate equations w2l exp w2g
2r2 w2g
# t /ð0; tÞdt þ dt þ nð0; 0Þ exp s
!
2r2 2 wp
!) ;
ð18Þ
w2 2r2 ns1 ðr; tÞ ¼ exp rg c l2 exp 2 ws ws Z t t /ð0; tÞdt s s 0 Z t ns0 w2l 2r2 exp rg c 2 exp 2 ss 0 w ws s Z t t /ð0; tÞdt þ dt þ ns0 : ð19Þ s s 0 Substituting Eqs. (3),(7)–(9) into Eq. (13), we obtain (
! w2l 2r2 2rnðr; tÞl 2 exp 2 wg wg 0 2 2 w 2r 2rg ns1 ðr; tÞls l2 exp 2 ws ws 2 wl 2r2 2re ½ns0 ns1 ðr; tÞls 2 exp 2 w ws s 2 w4l 4r dK /ð0; tÞ 4 exp 2 wK wK ) 2 2r L exp 2 r dr; ð20Þ wl
d/ð0; tÞ 4/ð0; tÞ ¼ dt w2l tr
Z
1
where nðr; tÞ and ns1 ðr; tÞ are given in Eqs. (18) and (19). Eq. (20) is the basic differential equation describing /ð0; tÞ as a function of t. The initial photons come from spontaneous-emission noise. When Eq. (20) is solved numerically, the initial photon density /ð0; 0Þ can be set at a value much smaller than the peak value of the photon density /m ð0; tÞ (for example, /ð0; 0Þ ¼ 104 /m ð0; tÞÞ. We can estimate /m ð0; tÞ by consulting the results obtained under the plane-wave
Since laser action begins at the moment that the population-inversion density crosses the initial threshold value in a passively Q-switched laser, by setting Eq. (20) equal to zero and t ¼ 0, we obtain the initial population-inversion density in the laser axis
! ln T12 þ L w2g 0 1þ 2 ; ð21Þ nð0; 0Þ ¼ wp 2rl where T0 is the small-signal transmission of the saturable absorber, i.e.,
T0 ¼ exp rg ns0 ls : ð22Þ By numerically solving Eq. (20), we can obtain the relation between /ð0; tÞ and t, and from Eq. (9), we can obtain the relation between /K ð0; tÞ and t. From the relation between /2K ð0; tÞ and t, we can obtain the pulse width (FWHM) W and the pulse repetition rate F of the generated-green-light pulses. According to [4], we can obtain the following expressions for pulse peak power P and single-pulse energy E 1 2 P ¼ nKN AK l2K ðhxcÞ /2Km ; 4
ð23Þ
E ¼ PW ;
ð24Þ
where n is the fraction that coupled out the resonator; KN is given in Eq. (1); /Km is the maximum value of /K ð0; tÞ. The corresponding parameters values of the theoretical calculation are shown in Table 2, in which we mainly adjust the value of L. The smallsignal transmission of Cr4þ :YAG saturable absorber are T0 ¼ 0:91 and T0 ¼ 0:95, respectively. The dotted line in Fig. 1 shows the theoretical pulse shape at T0 ¼ 0:95 and a pump power of 3.05 W. The solid lines in Figs. 2–5 are the theoretical calculation curves for pulse width W , pulse repetition rate F , single-pulse energy E, and peak power P versus pump power, respectively. These solid lines in Figs. 2–5 are obtained by using the following method. First, we calculate pulse width W , pulse
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Table 2 The parameters of the theoretical calculation [4]
r rg re ns0 s ss wl n n1 n2 n3 lK l wp L a wg ws wK
Parameter
Value
Stimulated-emission cross-section of Nd:YVO4 Ground-state absorption cross-section of Cr4þ :YAG excited-state absorption cross-section of Cr4þ :YAG Total population density of Cr4þ :YAG stimulated-radiation lifetime of Nd:YVO4 Excited-state lifetime of Cr4þ :YAG Average radius of the TEM00 mode Coupled coefficient of the resonator Refractive index of Nd:YVO4 Refractive index of Cr4þ :YAG Refractive index of KTP Length of KTP Length of Nd:YVO4 Average radius of the pump beam Intrinsic loss of the resonator Absorption coefficient of Nd:YVO4 Radius of the TEM00 mode at the position of Nd:YVO4 Radius of the TEM00 mode at the position of Cr4þ :YAG Radius of the TEM00 mode at the position of KTP
3.42 1018 cm2 4.3 1018 cm2 8.2 1019 cm2 2.0 1017 cm3 98 ls 3.2 ls 168 lm 0.85 2.183 1.81 1.83 0.7 cm 0.5 cm 330 lm 0.12 5.32 cm1 308 lm 247 lm 125 lm
1.0
350
T0=0.91 T0=0.95
300
Pulse Width(ns)
Normalized Intensity
0.8
0.6
0.4
0.2
0.0
250
200
150 0
500
1000
1500
2000
Time(ns)
1
2
3
4
Pump Power(W)
Fig. 1. Temporal profile of single-pulse with Cr4þ :YAG T0 ¼ 0:95. Solid line, oscilloscope trace; dotted line, calculated result.
Fig. 2. Pulse width versus pump power for Cr4þ :YAG saturable absorbers of T0 ¼ 0:91 and 0.95.
repetition rate F , single-pulse energy E, and peak power P under certain pump power. Then using these theoretical results, we do the nonlinear curve fitting and obtain the solid lines in Figs. 2–5.
by Semiconductor Institute, Chinese Academic, maximum output power 5 W) which works at the maximum absorption wavelength (808 nm) of the Nd:YVO4 crystal. The mirror M1 with 150-mm curvature radius is high antireflection coated at 808 nm and high reflection coated at 1064 nm and 532 nm. The Nd:YVO4 crystal doped with 1.0 at.% Nd3þ ions is 4 4 5 mm3 and its absorption coefficient at 808 nm is 5.32 cm1 . Its first surface is antireflection coated at 808 nm and the other
3. Experimental setup and results The experimental setup is shown in Fig. 6. The pump source is a fiber-coupled laser-diode (made
G. Li et al. / Optics Communications 234 (2004) 321–328
327
12
T0=0.91 T0=0.95
250
T0=0.91 T0=0.95
10
200
Peak Power(W)
Pulse Repetition Rate(kHz)
300
150
100
8
6
50
4 0 1
2
3
4
5
2 1
Pump Power(W)
2
3
4
Pump Power(W)
Fig. 3. Pulse repetition rate versus pump power for Cr4þ :YAG saturable absorbers of T0 ¼ 0:91 and 0.95.
1.8
Fig. 5. Pulse peak power versus pump power for Cr4þ :YAG saturable absorbers of T0 ¼ 0:91 and 0.95.
T0=0.91 T0=0.95
M1
Nd:YVO4
Cr4+:YAG
M3
1.6
Energy (µJ)
Filter 1.4
Laser-diode 1.2
KTP
Focusing Optics
M2
Fig. 6. Schematic of the experimental setup.
1.0
1
2
3
4
Pump Power(W)
Fig. 4. Single-pulse energy versus pump power for Cr4þ :YAG saturable absorbers of T0 ¼ 0:91 and 0.95.
surface is high antireflection coated at 1064 nm. The small-signal transmission of the Cr4þ :YAG saturable absorber is T0 ¼ 0:91 and T0 ¼ 0:95, respectively. The mirror M3 with 100-mm curvature radius is also used as the output mirror of the generated green light. The KTP crystal cut for type-II phase matching (made by Crystal Material Institute of Shandong University, China) is 3 3 7 mm3 and both of its surfaces are antireflection coated at 1064 and 532 nm. The temperature of the Nd:YVO4 crystal and the KTP crystal is controlled at 20 and 22 °C by means of a temperature controller, respectively. M2 is a plane mirror and its surface is high reflection coated at 1064 and 532 nm. The distance between M1 and M3
is about 22 cm and the distance between M2 and M3 is about 8 cm. So from the cavity configuration, we calculate that the radii of the Gaussian beam waists at three positions: Nd:YVO4 crystal, saturable absorber and KTP crystal are 0.308, 0.247and 0.125 mm, respectively. The values of the three radii are given in Table 2. The filter is used for separating the 532-nm wave from the remainder 1064-nm fundamental wave leaking out from the resonator. A LPE-1B power (Institute of Physics, Chinese Academy of Science) is used to measure the generated-green-light power and a TED620B digital oscilloscope (Tektronix Inc., USA) is used to measure the generated-green-light pulse width and pulse repetition rate. A single-pulse temporal profile of an oscilloscope trace at T0 ¼ 0:95 and a pump power of 3.05 W is shown by the solid line in Fig. 1. Fig. 7 shows the average output power PA of the generatedgreen-light pulses versus pump power. The
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G. Li et al. / Optics Communications 234 (2004) 321–328
dependences of pulse width, pulse repetition rate, single-pulse energy and peak power on pump power for the generated-green-light pulses. The theoretical calculations of the numerical solutions agree with the experimental results obtained from a laser-diode end-pumped passively Q-switched Nd:YVO4 /KTP laser.
500
T0=0.91 T0=0.95
Output Power(mW)
400
300
200
100
Acknowledgements 0 1
2
3
4
Pump Power(W)
Fig. 7. Average output power versus pump power for Cr4þ :YAG saturable absorbers of T0 ¼ 0:91 and 0.95.
dependences of pulse width W and pulse repetition rate F of the generated-green-light pulses on pump power are shown by scattered dots in Figs. 2 and 3. Using the equations E ¼ PA =F and P ¼ E=W , we obtain the single-pulse energy E and the peak power P , which are shown by scattered dots in Figs. 4 and 5. From Figs. 1–5, we can see that the experimental results are in agreement with the theoretical calculations.
4. Conclusions We have assumed the intracavity photon density and the initial population-inversion density to be Gaussian spatial distributions in the rate equations of a laser-diode end-pumped passively Q-switched intracavity-frequency-doubling Nd:YVO4 /KTP laser with Cr4þ :YAG saturable absorber. These space-dependent rate equations are solved numerically. From the numerical solutions, we obtain the
This work is partially supported by the Science and Technology Development Program of Shandong Province and the Research Fund for the Doctoral Program of Higher Education.
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Rate equations and solutions of a laser-diode end-pumped passively Q-switched intracavity doubling laser by taking into account intracavity laser spatial distribution Guiqiu Li *, Shengzhi Zhao, Hongming Zhao, Kejian Yang, Shuanghong Ding School of Information Science and Engineering, Shandong University, Jinan 250100, China Received 9 November 2003; received in revised form 12 January 2004; accepted 6 February 2004
Abstract The intracavity photon density and the initial population-inversion density are assumed to be Gaussian spatial distributions in the rate equations of a laser-diode end-pumped passively Q-switched intracavity-frequency-doubling Nd:YVO4 /KTP laser with Cr4þ :YAG saturable absorber. These space-dependent rate equations are solved numerically. In the experiment, a laser-diode end-pumped passively Q-switched Nd:YVO4 /KTP laser with Cr4þ :YAG saturable absorber is realized. The dependences of pulse width, pulse repetition rate, single-pulse energy and peak power on incident pump power are measured for the generated-green-light pulses, and the experimental results are consistent with the numerical solutions. Ó 2004 Elsevier B.V. All rights reserved. PACS: 42.55.xi; 42.60.Gd Keywords: Rate equations; Numerical solutions; Gaussian spatial distribution; Intracavity-frequency-doubling; Passively Q-switched; LD-pumped
1. Introduction In recent years, laser-diode (LD) pumped solidstate Q-switched lasers have attracted a great deal of attention because of their high efficiency, simplicity, compactness, good frequency stability. All solid-state Q-switched lasers have wide applica-
*
Corresponding author. Tel.: +86-531-8364840; fax: +86531-8565167. E-mail address: [email protected] (G. Li).
tions in the fields of remote sensing, information storage, coherent telecommunications, medicine, etc. It is easy to obtain high nonlinear conversion efficiency by placing the nonlinear crystal into the laser resonator due to its high fundamental wave power density, which is especially so for a laser with middle or low output power [1–3]. Rate equations are efficient tools for analyzing the performance of a Q-switched laser. By addition of a term to represent the nonlinear loss that is due to second-harmonic generation (SHG) to the photon-density equation, the characteristics of the
0030-4018/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.02.013
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G. Li et al. / Optics Communications 234 (2004) 321–328
fundamental-frequency and second-harmonic pulses of passively Q-switched intracavity-frequency-doubling lasers have been described [4,5]. However, the rate equations mentioned above are obtained under a plane-wave approximation. Actually, this assumption is not properly satisfied because the fundamental wave is a Gaussian transversal distribution for the TEM00 mode. Recently the transversal distributions of the intracavity photon density and the populationinversion density in the rate equations of passively Q-switched laser [6] and of passively Q-switched self-frequency-doubling laser [7] have been considered and the importance of these distributions has been shown. When the Gaussian distribution of the intracavity laser intensity is taken into account in the rate equations, the theoretical results obtained by numerically solving these rate equations are more close to the experimental results than those obtained under the plane-wave approximation [8,9]. But in [7] the pump source is a pulsed xenon flash-lamp with the pump energy being slightly higher than the threshold pump energy, and thus only one Q-switched pulse is generated in each pump pulse. Besides, the influence of pump rate on the intracavity photon density is not taken into account in [7]. Therefore, for a more accurate theoretical analysis of the pulses from a LD-pumped passively Q-switched intracavity-frequency-doubling laser it is desirable to take into account the influence of pump rate on the intracavity photon density and consider the transversal variations of the intracavity photon density and the population-inversion density in the rate equations. In this paper, we introduce the rate equations of a laser-diode end-pumped passively Q-switched intracavity-frequency-doubling Nd:YVO4 /KTP laser with Cr4þ :YAG saturable absorber, in which the intracavity photon density and the initial population-inversion density are assumed to be Gaussian spatial distributions. These space-dependent rate equations are solved numerically. From the numerical solutions, we obtain the dependences of pulse width, pulse repetition rate, single-pulse energy and peak power on pump power for the generated-green-light pulses. The numerical solutions are consistent with the exper-
imental results obtained from a laser-diode endpumped passively Q-switched Nd:YVO4 /KTP laser.
2. Theoretical calculations 2.1. Nonlinear loss due to harmonic conversion When the rate equations are used to analysis the performance of a Q-switched intracavity-frequency-doubling laser, the second-harmonic conversion can be considered as the nonlinear loss of the fundamental wave. The second-harmonic power for type-II phase-matching KTP crystal is [10] P2x ¼ KN
l2K 2 P ; AK x
ð1Þ
where KN ¼
c3 e
2 x2 deff ; 2x x x 0 ne2 ne2 ne1
lK is the length of KTP; x is the angle frequency of fundamental wave; deff is the effective nonlinear coefficient; c is the velocity of light in vacuum; e0 is x x the dielectric permeability of vacuum; n2x e2 , ne2 , ne1 are harmonic and fundamental wave refractive indices, respectively, and AK ¼ ð1=2Þpw2K is the area of fundamental wave at the position of KTP, where wK is the radius of the TEM00 mode at the position of KTP. According to the relationship of power and photon density: Px ¼ AK hxcð/K =2Þ, where /K is the photon density at the position of KTP, hx is the single-photon energy of the fundamental wave, h is PlanckÕs constant, and the nonlinear loss resulting from harmonic conversion can be expressed as dN ¼
P2x tr KN hxcl2K /K ¼ dK /K ; ¼ hx/K AK Lc 2
ð2Þ
where KN hxcl2K ; 2 Lc is the optical length of the resonator; tr ¼ 2Lc =c is the round-trip time of the resonator. The corresponding parameters for type-II phase-matching KTP crystal are presented in Table 1. dK ¼
G. Li et al. / Optics Communications 234 (2004) 321–328
323
Table 1 The parameters of II-type phase-matching KTP crystal
nxe1 nxe2 n2x e2 deff e0
Parameter
Value
Fundamental wave refractive index of o light Fundamental wave refractive index of e light Second-harmonic wave refractive index Effective nonlinear coefficient of KTP Dielectric permeability of vacuum
1.83 1.746 1.79 7.2 pm/V 8.855 1012 C2 /N m2
These data are provided by Coretech Crystal Company, Shandong University, China.
2.2. Rate equations We consider a LD-pumped passively Q-switched intracavity-frequency-doubling Nd:YVO4 / KTP laser with Cr4þ :YAG saturable absorber, in which Nd:YVO4 works as the gain medium, KTP works as the frequency-doubling crystal, and Cr4þ :YAG works as the passive Q-switch. If the intracavity photon density is assumed to be a Gaussian spatial distribution during the entire formatting process of the LD-pumped passively Qswitched intracavity-frequency-doubling laser pulse, the intracavity photon density /ðr; tÞ for the TEM00 mode can be expressed as 2r2 /ðr; tÞ ¼ /ð0; tÞ exp 2 ; ð3Þ wl where r is the radial coordinate; wl is the average radius of the TEM00 mode, which is mainly determined by the geometry of the resonator; and /ð0; t) is the photon density in the laser axis. Because the radii of the Gaussian beam waists at three positions: Nd:YVO4 crystal, saturable absorber and KTP crystal are not equal, the photon density /g ðr; tÞ; /s ðr; tÞ, and /K ðr; tÞ at three positions: Nd:YVO4 crystal, saturable absorber and KTP crystal can be expressed as ! 2r2 /g ðr; tÞ ¼ /g ð0; tÞ exp 2 ; ð4Þ wg /s ðr; tÞ ¼ /s ð0; tÞ exp
2r2 2 ; ws
/K ðr; tÞ ¼ /K ð0; tÞ exp
2r 2 ; wK
ð5Þ
2
ð6Þ
where wg , ws , and wK are the radii of the TEM00 mode at three positions: Nd:YVO4 crystal, saturable absorber and KTP crystal, respectively; /g ð0; tÞ, /s ð0; tÞ, and /K ð0; tÞ are the photon densities in the laser axis at three positions: Nd:YVO4 crystal, saturable absorber and KTP crystal, reR1 spectively. Because 0 /ðr; tÞ2pr dr can be considered to be independent of the laser longitudinal axis z [7], Eqs. (4)–(6) can be expressed as ! w2l 2r2 /g ðr; tÞ ¼ 2 /ð0; tÞ exp 2 ; ð7Þ wg wg
/s ðr; tÞ ¼
w2l /ð0; tÞ exp w2s
/K ðr; tÞ ¼
w2l /ð0; tÞ exp w2K
2r2 ; w2s
2r2 : w2K
ð8Þ
ð9Þ
So for this laser, if neglecting the spontaneous radiation during the pulse formation, we can obtain the coupling rate equations [11,12] d/ðr; tÞ 1 ¼ 2rnl/g ðr; tÞ 2rg ns1 ls /s ðr; tÞ dt tr 2re ðns0 ns1 Þls /s ðr; tÞ dK /2K ðr; tÞ L/ðr; tÞ ;
ð10Þ
dn n ¼ Rin rcn/g ðr; tÞ ; dt s
ð11Þ
dns1 ns0 ns1 ¼ rg cns1 /s ðr; tÞ; dt ss
ð12Þ
where n is the population-inversion density; ns1 and ns0 are the ground-state and total population
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G. Li et al. / Optics Communications 234 (2004) 321–328
densities of Cr4þ :YAG saturable absorber, respectively; r and l are the stimulated-emission cross-section and length of Nd:YVO4 gain medium, respectively; rg and re are the ground state and excited state absorption cross-sections of the saturable absorber, respectively; ls is the length of the saturable absorber; tr is the round-trip time of light in the resonator ftr ¼ 2n1 l þ 2n2 ls þ 2n3 lK þ2ðLe l ls lK Þ=cg; n1 ; n2 , and n3 are the refractive indices of Nd:YVO4 gain medium, Cr4þ :YAG saturable absorber, and KTP crystal, respectively; lK is the length of KTP; Le is the cavity length; c is the velocity of light in vacuum; dK is given in Eq. (2); L is the intrinsic loss; s is the stimulated-radiation lifetime of the gain medium; ss is the excited-state lifetime of the saturable absorber; Rin ¼ Pin ½1 expðalÞ=hcp pw2p l is the pump rate, where Pin is the pump power, hcp is the single-photon energy of the pump light, wp is the average radius of the pump beam in the gain medium, a is the absorption coefficient of the gain medium. In the rate equations of a passively Q-switched intracavity-frequency-doubling laser, the spatial variations of the population-inversion density of the gain medium and the ground-state population density of the saturable absorber should also be considered. Besides, the differential equation describing d/ðr; tÞ=dt should be integrated over the beam cross-section to guarantee the beam Gaussian spatial distribution during the entire formatting process of the passively Q-switched intracavity-frequency-doubling laser pulse. In addition, for laser-diode end-pumped lasers, the pump light can be approximated by a Gaussian profile. By making the above modifications to the rate Eqs. (10)–(12), we obtain Z 1 d/ðr; tÞ 2pr dr dt 0 Z 1 1 ¼ 2rnðr; tÞl/g ðr; tÞ 2rg ns1 ðr; tÞls /s ðr; tÞ t r 0 2re ½ns0 ns1 ðr; tÞls /s ðr; tÞ dK /2K ðr; tÞ L/ðr; tÞg2pr dr; dnðr; tÞ nðr; tÞ ¼ Rin ðrÞ rcnðr; tÞ/g ðr; tÞ ; dt s
ð13Þ ð14Þ
dns1 ðr; tÞ ns0 ns1 ðr; tÞ ¼ rg cns1 ðr; tÞ/s ðr; tÞ; dt ss ð15Þ where nðr; tÞ is the average population-inversion density [6]; ns1 ðr; tÞ is the ground-state population density of the saturable absorber; Rin ðrÞ ¼ Rin expð2r2 =w2p Þ ¼ Pin ½1 expðalÞ=hcp pw2p l expð2r2 =w2p Þ is the pump rate. Before the laser action begins, all the saturable-absorber population is situated in the ground state. So the initial value of ns1 ðr; tÞ can be considered to be ns0 . For four-level gain media like Nd:YVO4 , the pumped population is small compared to the total population, the ground state population density can be considered as constant, and the pumped population density should be proportional to the pump light intensity. If the pump light can be approximated by a Gaussian profile, then the pumped population density can also be considered to be a Gaussian distribution. For a single pulse of an LD-endpumped repetitively Q-switched laser, the initial population-inversion density nðr; 0Þ comes from the pumped population density during the interval between the preceding pulse and this pulse if the residual population-inversion density from the preceding Q-switched pulse, which is not a Gaussian distribution, can be neglected. That is to say, under the approximation that no residual population-inversion density from the preceding pulse is included, the initial population-inversion density nðr; 0Þ can be assumed to be a Gaussian spatial distribution [6,7]. Thus the initial conditions of Eqs. (14) and (15) can be written as ns1 ðr; 0Þ ¼ ns0 ; nðr; 0Þ ¼ nð0; 0Þ exp
ð16Þ
2r2 w2p
! ;
ð17Þ
where nð0; 0Þ is the initial population-inversion density in the laser axis. Substituting Eqs. (7), (8), (16) and (17) into Eqs. (14) and (15) and integrating the results over time, we obtain
G. Li et al. / Optics Communications 234 (2004) 321–328
" nðr; tÞ ¼ exp rc
w2l exp w2g
# t /ð0; tÞdt s 0 ( " Z Z
Rin ðrÞ
exp rc 0
325
approximation or by preliminarily solving the equations.
t
t
2r2 w2g
!
Z
t 0
2.3. Solutions of the rate equations w2l exp w2g
2r2 w2g
# t /ð0; tÞdt þ dt þ nð0; 0Þ exp s
!
2r2 2 wp
!) ;
ð18Þ
w2 2r2 ns1 ðr; tÞ ¼ exp rg c l2 exp 2 ws ws Z t t /ð0; tÞdt s s 0 Z t ns0 w2l 2r2 exp rg c 2 exp 2 ss 0 w ws s Z t t /ð0; tÞdt þ dt þ ns0 : ð19Þ s s 0 Substituting Eqs. (3),(7)–(9) into Eq. (13), we obtain (
! w2l 2r2 2rnðr; tÞl 2 exp 2 wg wg 0 2 2 w 2r 2rg ns1 ðr; tÞls l2 exp 2 ws ws 2 wl 2r2 2re ½ns0 ns1 ðr; tÞls 2 exp 2 w ws s 2 w4l 4r dK /ð0; tÞ 4 exp 2 wK wK ) 2 2r L exp 2 r dr; ð20Þ wl
d/ð0; tÞ 4/ð0; tÞ ¼ dt w2l tr
Z
1
where nðr; tÞ and ns1 ðr; tÞ are given in Eqs. (18) and (19). Eq. (20) is the basic differential equation describing /ð0; tÞ as a function of t. The initial photons come from spontaneous-emission noise. When Eq. (20) is solved numerically, the initial photon density /ð0; 0Þ can be set at a value much smaller than the peak value of the photon density /m ð0; tÞ (for example, /ð0; 0Þ ¼ 104 /m ð0; tÞÞ. We can estimate /m ð0; tÞ by consulting the results obtained under the plane-wave
Since laser action begins at the moment that the population-inversion density crosses the initial threshold value in a passively Q-switched laser, by setting Eq. (20) equal to zero and t ¼ 0, we obtain the initial population-inversion density in the laser axis
! ln T12 þ L w2g 0 1þ 2 ; ð21Þ nð0; 0Þ ¼ wp 2rl where T0 is the small-signal transmission of the saturable absorber, i.e.,
T0 ¼ exp rg ns0 ls : ð22Þ By numerically solving Eq. (20), we can obtain the relation between /ð0; tÞ and t, and from Eq. (9), we can obtain the relation between /K ð0; tÞ and t. From the relation between /2K ð0; tÞ and t, we can obtain the pulse width (FWHM) W and the pulse repetition rate F of the generated-green-light pulses. According to [4], we can obtain the following expressions for pulse peak power P and single-pulse energy E 1 2 P ¼ nKN AK l2K ðhxcÞ /2Km ; 4
ð23Þ
E ¼ PW ;
ð24Þ
where n is the fraction that coupled out the resonator; KN is given in Eq. (1); /Km is the maximum value of /K ð0; tÞ. The corresponding parameters values of the theoretical calculation are shown in Table 2, in which we mainly adjust the value of L. The smallsignal transmission of Cr4þ :YAG saturable absorber are T0 ¼ 0:91 and T0 ¼ 0:95, respectively. The dotted line in Fig. 1 shows the theoretical pulse shape at T0 ¼ 0:95 and a pump power of 3.05 W. The solid lines in Figs. 2–5 are the theoretical calculation curves for pulse width W , pulse repetition rate F , single-pulse energy E, and peak power P versus pump power, respectively. These solid lines in Figs. 2–5 are obtained by using the following method. First, we calculate pulse width W , pulse
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G. Li et al. / Optics Communications 234 (2004) 321–328
Table 2 The parameters of the theoretical calculation [4]
r rg re ns0 s ss wl n n1 n2 n3 lK l wp L a wg ws wK
Parameter
Value
Stimulated-emission cross-section of Nd:YVO4 Ground-state absorption cross-section of Cr4þ :YAG excited-state absorption cross-section of Cr4þ :YAG Total population density of Cr4þ :YAG stimulated-radiation lifetime of Nd:YVO4 Excited-state lifetime of Cr4þ :YAG Average radius of the TEM00 mode Coupled coefficient of the resonator Refractive index of Nd:YVO4 Refractive index of Cr4þ :YAG Refractive index of KTP Length of KTP Length of Nd:YVO4 Average radius of the pump beam Intrinsic loss of the resonator Absorption coefficient of Nd:YVO4 Radius of the TEM00 mode at the position of Nd:YVO4 Radius of the TEM00 mode at the position of Cr4þ :YAG Radius of the TEM00 mode at the position of KTP
3.42 1018 cm2 4.3 1018 cm2 8.2 1019 cm2 2.0 1017 cm3 98 ls 3.2 ls 168 lm 0.85 2.183 1.81 1.83 0.7 cm 0.5 cm 330 lm 0.12 5.32 cm1 308 lm 247 lm 125 lm
1.0
350
T0=0.91 T0=0.95
300
Pulse Width(ns)
Normalized Intensity
0.8
0.6
0.4
0.2
0.0
250
200
150 0
500
1000
1500
2000
Time(ns)
1
2
3
4
Pump Power(W)
Fig. 1. Temporal profile of single-pulse with Cr4þ :YAG T0 ¼ 0:95. Solid line, oscilloscope trace; dotted line, calculated result.
Fig. 2. Pulse width versus pump power for Cr4þ :YAG saturable absorbers of T0 ¼ 0:91 and 0.95.
repetition rate F , single-pulse energy E, and peak power P under certain pump power. Then using these theoretical results, we do the nonlinear curve fitting and obtain the solid lines in Figs. 2–5.
by Semiconductor Institute, Chinese Academic, maximum output power 5 W) which works at the maximum absorption wavelength (808 nm) of the Nd:YVO4 crystal. The mirror M1 with 150-mm curvature radius is high antireflection coated at 808 nm and high reflection coated at 1064 nm and 532 nm. The Nd:YVO4 crystal doped with 1.0 at.% Nd3þ ions is 4 4 5 mm3 and its absorption coefficient at 808 nm is 5.32 cm1 . Its first surface is antireflection coated at 808 nm and the other
3. Experimental setup and results The experimental setup is shown in Fig. 6. The pump source is a fiber-coupled laser-diode (made
G. Li et al. / Optics Communications 234 (2004) 321–328
327
12
T0=0.91 T0=0.95
250
T0=0.91 T0=0.95
10
200
Peak Power(W)
Pulse Repetition Rate(kHz)
300
150
100
8
6
50
4 0 1
2
3
4
5
2 1
Pump Power(W)
2
3
4
Pump Power(W)
Fig. 3. Pulse repetition rate versus pump power for Cr4þ :YAG saturable absorbers of T0 ¼ 0:91 and 0.95.
1.8
Fig. 5. Pulse peak power versus pump power for Cr4þ :YAG saturable absorbers of T0 ¼ 0:91 and 0.95.
T0=0.91 T0=0.95
M1
Nd:YVO4
Cr4+:YAG
M3
1.6
Energy (µJ)
Filter 1.4
Laser-diode 1.2
KTP
Focusing Optics
M2
Fig. 6. Schematic of the experimental setup.
1.0
1
2
3
4
Pump Power(W)
Fig. 4. Single-pulse energy versus pump power for Cr4þ :YAG saturable absorbers of T0 ¼ 0:91 and 0.95.
surface is high antireflection coated at 1064 nm. The small-signal transmission of the Cr4þ :YAG saturable absorber is T0 ¼ 0:91 and T0 ¼ 0:95, respectively. The mirror M3 with 100-mm curvature radius is also used as the output mirror of the generated green light. The KTP crystal cut for type-II phase matching (made by Crystal Material Institute of Shandong University, China) is 3 3 7 mm3 and both of its surfaces are antireflection coated at 1064 and 532 nm. The temperature of the Nd:YVO4 crystal and the KTP crystal is controlled at 20 and 22 °C by means of a temperature controller, respectively. M2 is a plane mirror and its surface is high reflection coated at 1064 and 532 nm. The distance between M1 and M3
is about 22 cm and the distance between M2 and M3 is about 8 cm. So from the cavity configuration, we calculate that the radii of the Gaussian beam waists at three positions: Nd:YVO4 crystal, saturable absorber and KTP crystal are 0.308, 0.247and 0.125 mm, respectively. The values of the three radii are given in Table 2. The filter is used for separating the 532-nm wave from the remainder 1064-nm fundamental wave leaking out from the resonator. A LPE-1B power (Institute of Physics, Chinese Academy of Science) is used to measure the generated-green-light power and a TED620B digital oscilloscope (Tektronix Inc., USA) is used to measure the generated-green-light pulse width and pulse repetition rate. A single-pulse temporal profile of an oscilloscope trace at T0 ¼ 0:95 and a pump power of 3.05 W is shown by the solid line in Fig. 1. Fig. 7 shows the average output power PA of the generatedgreen-light pulses versus pump power. The
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G. Li et al. / Optics Communications 234 (2004) 321–328
dependences of pulse width, pulse repetition rate, single-pulse energy and peak power on pump power for the generated-green-light pulses. The theoretical calculations of the numerical solutions agree with the experimental results obtained from a laser-diode end-pumped passively Q-switched Nd:YVO4 /KTP laser.
500
T0=0.91 T0=0.95
Output Power(mW)
400
300
200
100
Acknowledgements 0 1
2
3
4
Pump Power(W)
Fig. 7. Average output power versus pump power for Cr4þ :YAG saturable absorbers of T0 ¼ 0:91 and 0.95.
dependences of pulse width W and pulse repetition rate F of the generated-green-light pulses on pump power are shown by scattered dots in Figs. 2 and 3. Using the equations E ¼ PA =F and P ¼ E=W , we obtain the single-pulse energy E and the peak power P , which are shown by scattered dots in Figs. 4 and 5. From Figs. 1–5, we can see that the experimental results are in agreement with the theoretical calculations.
4. Conclusions We have assumed the intracavity photon density and the initial population-inversion density to be Gaussian spatial distributions in the rate equations of a laser-diode end-pumped passively Q-switched intracavity-frequency-doubling Nd:YVO4 /KTP laser with Cr4þ :YAG saturable absorber. These space-dependent rate equations are solved numerically. From the numerical solutions, we obtain the
This work is partially supported by the Science and Technology Development Program of Shandong Province and the Research Fund for the Doctoral Program of Higher Education.
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