Space-dependent modeling of diode-pumped actively Q-switched lasers with considering lower level lifetime

Optik 127 (2016) 2545–2550

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Space-dependent modeling of diode-pumped actively Q-switched lasers with considering lower level lifetime Ming Yin ∗ School of Information Science and Technology, Chengdu University of Technology, Chengdu, China

a r t i c l e

i n f o

Article history: Received 7 May 2015 Accepted 16 November 2015 Keywords: Rate equations Diode-pumped actively Q-switched lasers Lower level lifetime

a b s t r a c t With considering both the spatial distributions of the intracavity photon density and population inversion density and the lower level lifetime, the rate equations are obtained and solved numerically for diode-pumped actively Q-switched lasers. The effects of lower level lifetime on pulse rising time, falling time, width, peak power and energy are investigated. The lower level lifetime has no effect on pulse rising, falling and width. A larger value of the lower level time leads to smaller values of the pulse peak power and energy for a fixed pump condition. A more intensive pump leads to relatively lager effect of lower level lifetime on the pulse peak power and energy. The results can give a good understanding of the dependences of the laser pulse on the lower level time and be used to estimate the laser pulse characteristics of diode-pumped actively Q-switched lasers. © 2015 Elsevier GmbH. All rights reserved.

1. Introduction The Modeling and optimization of diode-pumped Q-switched laser has attracted much attention [1–6] due to a wide variety of its applications in civil, military and scientific research fields [7–12]. With the expansion of application range, the more accurate modeling of the laser is required. The spatial distributions of intracavity photon density and population inversion density have been considered for the modeling of diode-pumped continuous-wave [13–15], actively [16] and passively [17] Q-switched lasers, and different results between the assumptions of spatial variation and uniform of intracavity photon density and population inversion density have been shown [13–17]. However, the effect of lower-level lifetime was not considered in [13–17]. In a Q-switched Nd:YAG laser with pulse width of the order of tens of ns, the effect of lower-level lifetime could not be neglected [18]. The effect of lower-level lifetime on the pulse shape and parameters is reported [18–20]. In this paper, the intracavity photon density and initial population inversion density are assumed to be Gaussian distributions, and the lower level lifetime is considered, we obtain the rate equations for diode-pumped actively Q-switched lasers. By solving the rate equations numerically, a group of general curves are generated. These curves show

∗ Corresponding author. E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijleo.2015.11.155 0030-4026/© 2015 Elsevier GmbH. All rights reserved.

the effects of lower level lifetime on laser pulse. The results can be used to estimate the laser pulse characteristics of diode-pumped actively Q-switched lasers.

2. The rate equations for the lasers For an actively Q-switched laser, the rate equations in which the spatial variation of both the intracavity photon density and population inversion density are considered can be written as [16]

∞ 0

d (r, t) 2rdr = dt

∞



 (r, t) 2n (r, t) l − ln tr

  1 R



− L 2rdr

(1)

0

dn (r, t) = −c (r, t) n (r, t) dt

(2)

where r is the radial coordinate, n(r, t) is the population inversion density, (r, t) is the intracavity photon density,  is the stimulated emission cross section, l is length of the gain medium, tr = 2l /c is the round-trip transit time of the light in the resonator of optical length l , c is the light speed in vacuum, R is the reflectivity of the output mirror,  is the inversion reduction factor, and L is the remaining round-trip dissipative optical loss.

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The intracavity photon density and initial population inversion density are assumed to be Gaussian distributions. The intracavity photon density (r, t) for fundamental mode can be written as [16]



 (r, t) =  (0, t) exp

−

2r 2



(3)

ωL2

where ωL is the laser mode radius, (0, t) is the intracavity photon density along the laser axis. The initial population inversion density n(r, 0) can be expressed as [16]



n (r, 0) = n (0, 0) exp

−

2r 2 ωp2



When the lower-level relaxation time t is of the order of pulsewidth, the effect of finite lower-level lifetime should be included in rate equations. The inversion reduction factor is given in [18].  (t) = fb + fa e−t/t



(5)

With the similar method of [16], substituting (3) and (5) into (2) and integrating the result over time, we obtain

 n (r, t) = n (0, 0) exp

(4)

where ωp is the average radius of the pump beam in the gain medium and n(0, 0) is the initial population inversion density in the laser axis.

Fig. 1. Normalized photon density ˚(0, ) for different values of   and N(0, 0) in the case of ωp  ωL . (a) N(0, 0) = 1.5, (b) N(0, 0) = 3, (c) N(0, 0) = 5.

⎡ exp ⎣−c

t 0

−



2r 2



ωp2

fb + fa e

−t/t 

  (0, t) dt exp

−

2r 2 ωL2



⎤ ⎦

(6)

Fig. 2. Normalized cavity photon density ˚(0, ) for different values of   and N(0, 0) in the case of ωp = ωL . (a) N(0, 0) = 1.5, (b) N(0, 0) = 3, (c) N(0, 0) = 5.

M. Yin / Optik 127 (2016) 2545–2550

Fig. 3. Normalized photon density ˚(0, ) for different values of   and N(0, 0) in the case of ωp  ωL . (a) N(0, 0) = 1.5, (b) N(0, 0) = 3, (c) N(0, 0) = 5.

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Fig. 4. 1/ r as a function of N(0, 0) for different   and ωp /ωL . (a) ωp  ωL , (b) ωp = ωL , (c) ωp  ωL .

Substituting (6) into (1) yields 4l d(0, t) = 2  (0, t) n (0, 0) dt ωL tr

∞ × 0

⎡

exp⎣−c





exp −2r 2

t



fb + fa e

0

1 ωp2

+

1

−t/t 



ωL2



  (0, t) dt exp

−

2r 2 ωL2

 1

 (0, t) 2rdr − ln tr

R

⎤  ⎦

+L



nth (0, 0) =



ln 1/R + L 2l



1+

ωL2 ωp2

=

t t = tc tr

 =

t tc



(8)

 1 ln

R

+L



(9) (10)

 (r, )

˚ (r, ) =

 

N (r, ) =

n (r, ) nth (0, 0)

(7)

By setting (7) equal to zero and t = 0, we can obtain the threshold of the initial population inversion density in the laser axis



We introduce the normalized time , normalized lower-level relaxation time   , normalized photon density ˚(r, ), and normalized population inversion density N(r, ) as





ln 1/R + L /2l

(11)

(12)

where tc is the photon decay time. N(0, 0) is the ratio of the initial population inversion density in the laser axis to the threshold nth (0, 0). For four-level gain media like Nd:YAG and Nd:YVO4 , the pumped population density is proportional to the pump beam intensity and

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M. Yin / Optik 127 (2016) 2545–2550

Fig. 5. 1/ f as a function of N(0, 0) for different   and ωp /ωL . (a) ωp  ωL , (b) ωp = ωL , (c) ωp  ωL .

N(0, 0) equals the ratio of the pump power to the threshold pump power when the pulse repetition rate is fixed [16]. Substituting (9)–(12) into (7) yields



×

exp

−



case of ωp  ωL , ωp = ωL , and ωp  ωL , simpler equations can be derived from (13) d˚(0, ) 1 − exp (−D) = ˚ (0, ) N (0, 0) − ˚ (0, ) , D d

d˚ (0, ) = ˚ (0, ) N (0, 0) d

1

Fig. 6. ˚m as a function of N(0, 0) for different   and ωp /ωL . (a) ωp  ωL , (b) ωp = ωL , (c) ωp  ωL .



fb + fa e

−/ 



 ˚(0, )dg



1/1+

2 ω2 /ωp L

 dg− ˚ (0, )

d˚ (0, ) 2 = ˚ (0, ) N (0, 0) D d

0

− ˚ (0, ) , for ωp = ωL

0

(13) where



g = exp −2r 2



1 ωL2

+

1 ωP2

d˚(0,) d

(17) ωL where





(14)

Eq. (13) is the differential equation describing the normalized photon density in the laser axis versus the normalized time. In the

 1 − exp

D= 0



D

(−D)

for ωp  ωL

(15)



− exp (−D)

(16)

= ˚ (0, ) N (0, 0) exp (−D) − ˚ (0, ) , for ωp 

fb + fa e−/





˚ (0, ) d

(18)

M. Yin / Optik 127 (2016) 2545–2550

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For the same value of N(0, 0) and   , ˚(0, ) is different for different values of ωp /ωL . The intracavity photon density (0, ) is proportional to the normalized photon density ˚(0, ) in the laser axis. The pulse shape is different for different values of lower level lifetime for the same pump condition. A more intensive pump leads to relatively lager effect of lower level lifetime on the pulse shape.  r ,  f , ˚m and ˚e can be obtained from the shape of ˚ (0, ).

∞

Here ˚m is the maximum value of ˚ (0, ),˚e =

˚ (0, ) d,  r is 0

the normalized pulse rising time,  f is the normalized pulse falling time. Figs. 4 and 5 show 1/ r and 1/ f as a function of N(0, 0) for different values of   in the case of ωp  ωL , ωp = ωL , and ωp  ωL , respectively.   has no effect on 1/ r and 1/ f for a given ωp /ωL . 1/ r and 1/ f increases with increasing N(0, 0) for a given ωp /ωL . A larger value of ωp /ωL leads to a smaller value of 1/ r and 1/ f for a given N(0, 0). The pulse width is the sum of  r and  f . As can be seen from Figs. 4 and 5, the lower level lifetime has no effect on pulse rising, falling and width. Figs. 6–7 show ˚m and ˚e as a function of N(0, 0) for different values of   in the case of ωp  ωL , ωp = ωL , and ωp  ωL , respectively. A larger value of   leads to relatively smaller ˚m and ˚e for a given ωp /ωL and N(0, 0). The effect of   on ˚m and ˚e increases with N(0, 0) increasing. ˚m and ˚e increases with N(0, 0) increasing for a given ωp /ωL . A larger value of ωp /ωL leads to a larger value of ˚m and ˚e for a given N(0, 0). ˚m and ˚e are proportional to pulse peak power and energy. A larger value of the lower level time leads to smaller values of the pulse peak power and energy for a fixed pump condition. A more intensive pump leads to relatively lager effect of lower level lifetime on the pulse peak power and energy. 4. Conclusion

Fig. 7. ˚e as a function of N(0, 0) for different   and ωp /ωL . (a) ωp  ωL , (b) ωp = ωL , (c) ωp  ωL .

In this paper, the rate equations are obtained and solved numerically for diode-pumped actively Q-switched lasers with considering both the spatial distributions of intracavity photon density and population inversion density and the lower level lifetime. The effects of lower level lifetime on pulse rising time, falling time, width, peak power and energy are investigated. The lower level lifetime has no effect on pulse rising, falling and width. A larger value of the lower level time leads to smaller values of the pulse peak power and energy for a fixed pump condition. A more intensive pump leads to relatively lager effect of lower level lifetime on the pulse peak power and energy. The results can give a good understanding of the dependences of the laser pulse on the lower level time and be used to estimate the laser pulse characteristics of diode-pumped actively Q-switched lasers.

3. Simulation results and discussion As an example, the pulse parameters of the 1.064 ␮m diodepumped actively Q-switched Nd:YAG laser are calculated. In Nd:YAG, fa = 0.19 and fb = 0.40 at room temperature [19]. In the case of ωp  ωL , ωp = ωL , and ωp  ωL , ˚(0, ), can be obtained by numerically solving (15)–(18). Figs 1–3 show the normalized photon density ˚(0, ) for different values of   and N(0, 0) in the case of ωp  ωL , ωp = ωL , and ωp  ωL , respectively. For the same N(0, 0) and ωp /ωL , ˚(0, ) is different for different values of   . A larger value of N(0, 0) leads to relatively lager effect of   on ˚(0, ) for a given ωp /ωL . For example, in Fig. 1, in the case of N(0, 0) = 1.5, the ˚(0, ) is the same for   = 0, 1 and 3; in the case of N(0, 0) = 3, the ˚(0, ) is the same for   = 0 and 1 while the ˚(0, ) is different between   = 3 and 1; in the case of N (0, 0) = 5, the ˚(0, ) is different between   = 0, 1 and 3.

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