Optimization of the passively Q-switched laser with a novel central semiconductor saturable absorption mirror (C-SESAM)

Optik 122 (2011) 1558–1564

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Optimization of the passively Q-switched laser with a novel central semiconductor saturable absorption mirror (C-SESAM) Wenjing Tang, Dechun Li ∗ , Shengzhi Zhao, Guiqiu Li, Kejian Yang School of Information Science and Engineering, Shandong University, Jinan 250100, PR China

a r t i c l e

i n f o

Article history: Received 1 April 2010 Accepted 1 September 2010

Keywords: Optimization C-SESAM Q-switched Rate equation Numerical solutions

a b s t r a c t The diode-pumped passively Q-switched laser, realized with a novel central semiconductor saturable absorption mirror (C-SESAM), can obtain much higher repetition rate and shorter pulse duration which have been proved theoretically and experimentally. Therefore, it is essential to optimize this kind of lasers to achieve many of the desired properties. The normalized rate equations are solved numerically, when the direct band-gap absorption of the InGaAs thin layer, the single-photon absorption (SPA) and two-photon absorption (TPA) processes of GaAs substrate are simultaneously considered. Some new normalized parameters are introduced and the key parameters of an optimally passively Q-switched laser are determined, including the optimal normalized coupling parameter and the optimal normalized saturable absorber parameters, which can maximize the output energy. A group of general curves are generated, and sample calculations for a diode-pumped Nd3+ :YVO4 laser with a novel central semiconductor saturable absorption mirror (C-SESAM) are presented to demonstrate the use of the curves and the relevant formulas. © 2010 Elsevier GmbH. All rights reserved.

1. Introduction Passively Q-switched all-solid-state lasers are of great interest because of their potential applications in remote sensing, ranging, micromachining, and nonlinear wavelength conversion. Most of the work on passively Q-switched solid-state lasers has been performed with Cr4+ :YAG crystals as saturable absorbers. Recently, the semiconductor saturable absorption mirror (SESAM) has been successfully used in passively Q-switched lasers to generate pulses [1,2], and offers a distinct range of parameters not available with other approaches. Different designs of semiconductor saturable absorber mirrors (SESAMs) can be used to achieve many of the desired properties. In 1992, Keller et al. realized the self-starting passive modelocking for the Nd3+ :YLF laser with a pulse width of 3.3 ps by using the anti-resonance Fabry–Perot saturable absorber (A-FPSA) [3]. In 1999, Spu¨hler et al. obtained the shortest Q-switched pulse width (pulse width of 37 ps) by using SESAM in the Nd3+ :YVO4 sheet laser till now [4,5]. However, different structures of SESAMs are required in Q-switched lasers and mode-locked lasers, and the thickness of the absorption layer of SESAMs used in Q-switched lasers (a few hundred nanometers) is much thicker than that used in modelocked lasers (10–20 nm) [5]. In 2005, Wang et al. developed a novel

∗ Corresponding author. Tel.: +86 531 88361581: fax: +86 531 88364613. E-mail address: [email protected] (D. Li). 0030-4026/$ – see front matter © 2010 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2010.09.040

central semiconductor saturable absorption mirror (C-SESAM) [6], and studied the passive Q-switching properties of the C-SESAM in the passively Q-switched Nd3+ :YVO4 lasers. They obtained the pulse width shorter than 10 ns and the repetition rate up to 2 MHz [7]. The C-SESAM [7] is manufactured by growing the InGaAs nonlinear absorption layer on the GaAs substrate of about 500 ␮m directly, polishing the back of the substrate, and then plating medium antireflection coatings on both sides of the wafer, the reflectivity of the coatings at 1064 nm is less than 0.1%. This kind of SESAM is no longer required to produce the Bragg reflector, and also do not need to follow the anti-resonant cavity or resonant cavity structure, having advantages of simplicity and low cost. When the C-SESAM is used as Q-switching components, its Q-switching properties are different to the SESAMs with a Bragg reflector [8], and different to the GaAs saturable absorber [9–12], either. As far as we know, several optimization theories of single Q-switched lasers have been put forward [13–19]. A single Qswitched laser can be optimized for a certain pump condition and dissipative optical loss through the proper choice of the reflectivity of an output mirror [18,19]. However, those results cannot be applied to this kind of passively Q-switched lasers with a C-SESAM. To our knowledge, the optimization of a passively Q-switched laser with a C-SESAM has not been reported. In this paper, the normalized rate equations describing the doubly Q-switched lasers [2,20–23] are introduced to the diode-pumped passively Q-switched laser with a C-SESAM. The key parameters of an optimally passively

W. Tang et al. / Optik 122 (2011) 1558–1564

Q-switched laser are determined, and a group of general curves are generated. Sample calculations for a diode-pumped Nd3+ :YVO4 laser with a C-SESAM are presented to demonstrate the use of the curves and the relevant formulas.

density of ground-state of In0.25 Ga0.75 As, ntot is the total population density of In0.25 Ga0.75 As;  g and  e are the absorption cross sections of the ground-state and excited state of In0.25 Ga0.75 As, respectively; ls is the thickness of In0.25 Ga0.75 As;  and l are the stimulated-emission cross section and length of the gain medium, respectively;  0 and  + are the absorption cross sections of EL20 and EL2+ , respectively; d is the length of the GaAs saturable substrate; B = 6ˇG hc(ω0 /ωq )2 is the coupling coefficient of TPA in GaAs, where ˇG is the absorption coefficient of two photons, ω0 and ωq are the spot size of the beam in the gain medium and GaAs substrate, h is the photon energy, respectively; tr = 2L /c is the round-trip transit time of light in the resonator of optical length l , c is the light speed in vacuum; L is the remaining round-trip dissipative optical loss;  is the inversion-reduction factor, which corresponds to the net reduction in the population inversion resulting from the stimulated emission of a single photon; SN and SI are the beam cross section in the gain medium and in the In0.25 Ga0.75 As absorption area, respectively. For a passively Q-switched laser with a C-SESAM, assuming the intracavity photon density is a Gaussian distribution during the entire process of the Q-switched pulse formation, by defining the initial conditions of Eqs. (1)–(4) and integrating the results over time, we can obtain the expressions of n(r,t), ng (r,t) and n+ (r,t). Then substituting these expressions into Eq. (1) and regrouping the result yields:

2. Rate equations The C-SESAM includes two saturable absorption layers, one is the thin-layer absorber, which grows at low temperature and usually has its thickness about several tens of nanometers; the other part is the semi-insulating GaAs substrate whose thickness is usually around 500 ␮m. It is inevitable that the non-linear effects of semi-insulating GaAs substrate will be introduced when In0.25 Ga0.75 As is used as the thin-layer absorber in passively Q-switched lasers [7]. The two absorption mechanisms can be controlled separately, which can help us to obtain ideal Q-parameters. Therefore, by considering the single-photon absorption (SPA) and two-photon absorption (TPA) processes of GaAs substrate and the direct band-gap absorption of the In0.25 Ga0.75 As thin layer, and considering the horizontal spatial distribution of the initial population-inversion density, the ground-state population density of In0.25 Ga0.75 As thin layer and the population density of the deep

d(0, t) 4n(0, 0)l(0, t) = dt ωL2 tr −



ωL2 tr −

4(0, t)d



2  t ωL tr

×

∞



0



exp −c exp



∞

exp −



−

2r 2 ωL2



2r dr −

0

d(r, t) 2r dr = dt



0

∞

(r, t) tr







ωL2

ln

R

ϕ(0, t) dt exp

 +

−

0

e ln g

1





+L −B

T02



1

+

2r 2 2

ωL

exp[−c 0 (1 + ı) exp



1

2rdr

2 2  ωp ωL



t

2

−

2r dr

2r 2



ωL2

(5)

2

[(0, t)] d 2tr

where ı =  + / 0 and T0 is the small-signal transmission of In0.25 Ga0.75 As; absorption area which can be expressed as: T0 = exp(−g ntot ls )

(6)

3. Normalizations of the rate equations 2n(r, t)l − 2g ng (r, t)ls

−2e [ntot − ng (r, t)]ls − 2 0 [n0 − n+ (r, t)]d −2 + n+ (r, t)d − ln



1

TG0 = exp{−[ 0 (n0 − n+ ) +  + n+ ]d} is the small-signal trans0 0 mission of the GaAs saturable substrate, and by making Eq. (5) equal zero we can obtain: n(0, 0) =

R

−L − B(r, t)d 2r dr

(1)

dn(r, t) = −c(r, t)n(r, t) dt

(2)

dn+ (r, t) = { 0 [n0 − n+ (r, t)] −  + n+ (r, t)}c(r, t) dt

(3)

dng (r, t) SN = − g c(r, t)ng (r, t) SI dt

−

0 2r 2

   1 (0, t)

level EL2 of GaAs, we get the following rate equations (the pump and spontaneous emission processes are neglected during the process of the pulse formation) [20–22]: ∞



tr



ϕ(0, t) dt exp −2r

ωL2

SN cg exp SI



t

2ı 0 n0  n0 −  0 (1 − ı) n+ − 0 1+ı 1+ı

0



−



2r 2



0

exp

(0, t) dt



0

4(g − e )ntot ls (0, t)



∞

1559

(4)

The intracavity photon density (r, t) for the TEM00 mode is a Gaussian spatial distribution, which can be expressed as (r, t) = (0, t) exp(−(2r 2 )/(ωL2 )); (0, t) is the photon density in the laser axis; ωL is the average radius of the TEM00 mode; n(r,t) is the population-inversion density, where r is the radial coordinate and t is the time; n0 is the total population density of the EL2 defect level (including EL20 and EL2+ ) of GaAs substrate; n+ (r,t) is the population density of positively charged EL2+ ; ng (r,t) is the population

2 )+L ln(1/R) + ln(1/T02 ) + ln(1/TG0

2l



1+

ωL2 ωp2



(7)

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. Then the normalized time = (t/tr )[2εn(0, 0)l], the normalized photon density ˚(r, ) = (r, )(2l /(2εn(0, 0)l)) and the new parameters x, z, y1 , y2 are introduced as follows [19–22]): x=

ln(1/R) 2εn(0, 0)l

(8)

z=

L 2εn(0, 0)l

(9)

y1 = y2 =

ln(1/T02 ) 2εn(0, 0)l 2 ) ln(1/TG0

2εn(0, 0)l

where ε = 1/[1 + (ωL /ωp )2 ].

(10)

(11)

1560

W. Tang et al. / Optik 122 (2011) 1558–1564

From Eqs. (7) and (8)–(11), we can obtain: x + y1 + y2 + z = 1

(12)

Substituting these normalized parameters into Eq. (5), using Eq. (12) and regrouping the result, the differential equation describing the normalized photon density in the laser axis versus the normalized time can be obtained as follows:



1

1 − exp[−˛1 A( )] ˛1 A( )

exp{−A( )uε } du + Y1 ˚(0, ) 1 −

−1 y2 ˚(0, ) − 2 y2 ˚(0, )

1 − exp[−˛2 A( )] − (1 − y2 )˚(0, ) ˛2 A( )

B 2 −  3 y2 ˛2 [˚(0, )] 8L

where

u = exp[−2r 2 ((1/ωL2 ) + (1/ωp2 ))],

A( ) =

0

˚(0, ) d ,

(t) dt = 0

ωL2 h 4

ln

1 R

˚int

1 R

˚m

(15) (16)

where ˚ int is the integral of ˚(0, ) over from zero to infinity, i.e.,

∞ ˚int = 0 ˚(0, ) d and ˚m is the maximum value of ˚(0, ). Then we define the normalized parameters e = ((4)/(ωL2 h))(1/(2ˇn(0, 0)l))E, p= (1/((2ˇn(0, 0)l) ))((4tr )/(ωL2 h))Pm , and ((2ˇn(0, 0)l)/tr )W , which can be expressed as:

w=

e = x˚int

(17)

p = x˚m

(18)

e p

0.26 0.24 0.22 0.20

0.1

0.65 0.60 0.55

a

0.50

b

0.45

c d

0.40 0.35 0.30 0.25 0.20 0.15

z

0.1

Fig. 2. Dependence of y1opt on z for different y2 in the case of ωL /ωp = 1, ˛1 = 20 and ˛2 = 30 when the laser is an energy-maximized passively Q-switched laser. (a) y2 = 0.05, (b) y2 = 0.1, (c) y2 = 0.15, and (d) y2 = 0.2.

(14)

˚int E tr W≈ = Pm 2ˇn(0, 0)l ˚m

w≈

0.28

0.01

ωL2 h hAl ln(1/R) int = [2εn(0, 0)l] ln tr 4tr

2

a

Fig. 1. Dependence of xopt on z for different y2 in the case of ωL /ωp = 1, ˛1 = 20 and ˛2 = 30 when the laser is an energy-maximized passively Q-switched laser. (a) y2 = 0.05, (b) y2 = 0.1, (c) y2 = 0.15, and (d) y2 = 0.2.

emax=4σγE/[2εσn(0,0)l]hνπωL2

Pm =

∞

0.30

0.10

The pulse energy E, the peak power Pm and pulse width W can be expressed as follows [22]:



0.32

d c b

0.16 0.01

4. Maximization of output energy

hAl ln(1/R) tr

0.34

z

˛1 = ( g SN )/(SI ), ˛2 = (( 0 (1 + ı))/), here ˛1 is a mark indicating how easily the In0.25 Ga0.75 As saturble absorber is bleached. The larger ˛1 is, the more easily the In0.25 Ga0.75 As, saturable absorber is bleached. ˛2 is not only a mark indicating how easily the process of SPA is saturated, but also a parameter showing how strongly of the TPA process of GaAs. 1 = (2n0 ı)/((1 + ı)[n0 − (1 − ı)n+ ]), 0 )/(n0 − (1 − ı)n+ )), 3 = 2 = ((1 − ı)/(1 + ı))((n0 − (1 + ı)n+ 0 0 (1/( 0 (1 + ı)[ 0 n0 −  0 (1 − ı)n+ ])), L = l /(2εn(0, 0)l), 1 , 2 , and 0 3 are constants for the GaAs substrate and L is the normalized optical length of resonator. Y1 = (1 −  e / g )y1 . Eq. (13) shows us how the In0.25 Ga0.75 As saturble absorber and the GaAs substrate working together.

E=

0.36

0.18

(13)

2

0



y1opt=ln(1/T 0)/[2εσn(0,0)l]

d˚(0, ) = ˚(0, ) d



Xopt=ln(1/R)/[2εσn(0,0)l]

0.38

1.0 0.8 0.6

a b c d

0.4 0.2 0.0 0.01

0.1

z Fig. 3. Dependence of emax on z for different y2 in the case of ωL /ωp = 1, ˛1 = 20 and ˛2 = 30 when the laser is an energy-maximized passively Q-switched laser. (a) y2 = 0.05, (b) y2 = 0.1, (c) y2 = 0.15, and (d) y2 = 0.2.

(19)

What we want to do is to maximize the output pulse energy by selecting the optimal ln(1/R) and ln(1/T02 ) for a given amplifying medium, a given C-SESAM ( 0 ,  + ,  g ,  e and TG0 is fixed, T0 is changeable, here the thickness of GaAs is supposed to be fixed and the thickness of In0.25 Ga0.75 As is changeable), and a given pump level (i.e., a given n(0, 0) and a given ωp /ωL ). This means maximizing e by selecting the optimal x and Y1 under the condition of (12). By numerically solving Eqs. (12), (13) and (17), a series of numerical solutions of the normalized output pulse energy e versus x and Y1 for given z, ˛1 , ˛2 , ωp /ωL and y2 can be derived, then we can maximize it and determine the optimal x, Y1 and the corresponding p, w.

5. Results and discussion The results are shown in Figs. 1–15, respectively. Figs. 1–5 shows the dependence of the normalized parameters on z for different y2 in the case of ωp /ωL = 1, ˛1 = 20, ˛2 = 30, when the laser is an energy-maximized passively Q-switched laser. From Fig. 1, it can be seen that the xopt decreases monotonically with increasing z, and when z is small, xopt increases monotonically with increasing y2 , but in order to guarantee laser’s oscillation the xopt decreases with increasing y2 when z is larger. From curves of Fig. 2 we can see that the parameters y1opt decreases monotonically with increasing y2 and z. Figs. 3–5 show the variations of the normalized parameters emax , popt and w as functions of z for different y2

40 38 36 34 32 30 28 26 24 22 20 18 16 14 12

Xopt=ln(1/R)/[2εσn(0,0)l]

w=W[2βσn(0,0)l]/tr

W. Tang et al. / Optik 122 (2011) 1558–1564

d c b a

0.01

a

0.06

0.04 0.03

c b a

z

0.6

2

b c d

0.5

a

0.4

b

0.3

c

0.2 0.1

d

0.02

0.0 0.01

0.01

z

0.00 0.01

0.1

Fig. 6. Dependence of xopt on z for different Y1 in the case of ˛1 = 16, ˛2 = 36 and ωL /ωp = 1 when the laser is an energy-maximized passively Q-switched laser. (a) Y1 = 0.01, (b) Y1 = 0.1, (c) Y1 = 0.2, and (d) Y1 = 0.3.

y2opt=ln(1/T 0)/[2εσn(0,0)l]

2

popt=4σγ tr Pm/[2εσn(0,0)l] hνπω0

2

0.08

0.05

d

0.1

z

Fig. 4. Dependence of w on z for different y2 in the case of ωL /ωp = 1, ˛1 = 20 and ˛2 = 30 when the laser is an energy-maximized passively Q-switched laser. (a) y2 = 0.05, (b) y2 = 0.1, (c) y2 = 0.15, and (d) y2 = 0.2.

0.07

0.62 0.60 0.58 0.56 0.54 0.52 0.50 0.48 0.46 0.44 0.42 0.40 0.38 0.36 0.34 0.32 0.01

1561

z

0.1

Fig. 7. Dependence of y2opt on z for different Y1 in the case of ˛1 = 16, ˛2 = 36 and ωL /ωp = 1 when the laser is an energy-maximized passively Q-switched laser. (a) Y1 = 0.01, (b) Y1 = 0.1, (c) Y1 = 0.2, and (d) Y1 = 0.3.

emax=4σγ E/[2εσn(0,0)l]hνπωL

2

Fig. 5. Dependence of popt on z for different y2 in the case of ωL /ωp = 1, ˛1 = 20 and ˛2 = 30 when the laser is an energy-maximized passively Q-switched laser. (a) y2 = 0.05, (b) y2 = 0.1, (c) y2 = 0.15, and (d) y2 = 0.2.

when the laser is an energy-maximized passively Q-switched laser. Figs. 3 and 5 show that the emax and the popt decreases monotonically with increasing y2 and z. From Fig. 4 we can obtain that the pulse width w increases monotonically with increasing y2 and z. Figs. 6–10 show the dependence of the normalized parameters on z for different Y1 in the case of ˛1 = 16, ˛2 = 36 and ωL /ωp = 1 when the laser is an energy-maximized passively Q-switched laser. Fig. 6 shows us that the xopt increases monotonically with increasing z and it also decreases monotonically with increasing Y1 . From Fig. 7, it can be seen that y2opt decreases monotonically with increasing z and Y1 , it means that the less the T0 of In0.25 Ga0.75 As, the more the TG0 . And from Figs. 8–10 it can be obtained that the optimal output energy emax , the pulse width w and the pulse peak power popt have the same variations with which we found from Figs. 3–5 when z increases monotonically. But the variations of emax , w and popt versus Y1 are completely different from the variations versus y2 which we found from Figs. 3–5. From Fig. 8 we can see that the emax increases monotonically with increasing Y1 . Figs. 9 and 10 show the variations of the w and the popt versus z for different Y1 . The pulse width w decreases monotonically with increasing Y1 and the peak power popt increases monotonically with increasing Y1 . Then we can conclude that we need a larger Y1 and a smaller y2 to maximize the output pulse energy, it means that T0 should be smaller and TG0 should be larger. Figs. 11–15 show the dependence of the normalized parameters on ␻L /ωp for different y2 in the case of ˛1 = 20, ˛2 = 30 and z = 0.05 when the laser is an energy-maximized passively Q-switched laser.

0.1

0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0.01

d c b a

z

0.1

Fig. 8. Dependence of emax on z for different Y1 in the case of ˛1 = 16, ˛2 = 36 and ωL /ωp = 1 when the laser is an energy-maximized passively Q-switched laser. (a) Y1 = 0.01, (b) Y1 = 0.1, (c) Y1 = 0.2, and (d) Y1 = 0.3.

First, it can be seen that the variations of these normalized parameters in Figs. 11–15 versus y2 are all same with the variations shown in Figs. 1–5. The xopt and w increases monotonically with increasing y2 , the y1opt , emax and popt decreases monotonically with increasing y2 . What we want to discuss is the dependence of the normalized parameters on ωL /ωp . As we know, the laser mode is a Gaussian spatial distributions, the interaction between the central part of the mode and the inversion population is strong, but the interaction between the external part of the mode and the inversion population is weak. Therefore, if ωL /ωp is comparatively small(ωL /ωp < 1), the external part of the inversion population will go on interact

1562

W. Tang et al. / Optik 122 (2011) 1558–1564 0.65

60

y1opt=ln(1/T 0)/[2εσn(0,0)l]

50 45

2

w=W[2βσn(0,0)l]/tr

55

40

a b c

35 30

b 0.50

c

0.45

d

0.40

0.1

0.1

z

Fig. 9. Dependence of w on z for different Y1 in the case of ˛1 = 16, ˛2 = 36 and ωL /ωp = 1 when the laser is an energy-maximized passively Q-switched laser. (a) Y1 = 0.01, (b) Y1 = 0.1, (c) Y1 = 0.2, and (d) Y1 = 0.3.

wL/wP

1

10

Fig. 12. Dependence of y1opt on ωL /ωp for different y2 in the case of ˛1 = 20, ˛2 = 30 and z = 0.05 when the laser is an energy-maximized passively Q-switched laser. (a) y2 = 0.05, (b) y2 = 0.1, (c) y2 = 0.15, and (d) y2 = 0.2.

1.4

a

2

0.026 0.024 0.022 0.020 0.018 0.016 0.014 0.012 0.010 0.008 0.006 0.004 0.002 0.000 0.01

emax=4σγ E/[2εσn(0,0)l]hνπωL

2

0.55

0.35

0.01

2

a

d

25

popt=4σγ tr Pm/[2εσn(0,0)l] hνπω0

0.60

d

c b a

1.2 1.0

b c

0.8

d

0.6 0.4

0.1

0.1

z

1

10

wL/wP

Fig. 10. Dependence of popt on z for different Y1 in the case of ˛1 = 16, ˛2 = 36 and ωL /ωp = 1 when the laser is an energy-maximized passively Q-switched laser. (a) Y1 = 0.01, (b) Y1 = 0.1, (c) Y1 = 0.2, and (d) Y1 = 0.3.

Fig. 13. Dependence of emax on ωL /ωp for different y2 in the case of ˛1 = 20, ˛2 = 30 and z = 0.05 when the laser is an energy-maximized passively Q-switched laser. (a) y2 = 0.05, (b) y2 = 0.1, (c) y2 = 0.15, and (d) y2 = 0.2.

Xopt=ln(1/R)/[2εσn(0,0)l]

0.36 0.35 0.34 0.33

d c b

0.32 0.31 0.30

a 0.1

1

10

wL/wP Fig. 11. Dependence of xopt on ωL /ωp for different y2 in the case of ˛1 = 20, ˛2 = 30 and z = 0.05 when the laser is an energy-maximized passively Q-switched laser. (a) y2 = 0.05, (b) y2 = 0.1, (c) y2 = 0.15, and (d) y2 = 0.2.

with the laser mode after the central part of the inversion population exhausted, then the output energy will be very high. But when ωL /ωp is too small(ωL /ωp  1), the interaction between the laser mode and the inversion population only occur in the center, and when ωL /ωp is too big(ωL /ωp  1), the inversion population will be exhausted thoroughly. From Fig. 11, it can be seen that xopt increases monotonically with increasing ωL /ωp when ωL /ωp is smaller (ωL /ωp < 1). But when ωL /ωp is larger (ωL /ωp > 1), xopt decreases with increasing ωL /ωp , this is because the interaction between the laser mode and the population-inversion density is weaker, and the gain become smaller, so it need to reduce the out-

Fig. 14. Dependence of w on ωL /ωp for different y2 in the case of ˛1 = 20, ˛2 = 30 and z = 0.05 when the laser is an energy-maximized passively Q-switched laser. (a) y2 = 0.05, (b) y2 = 0.1, (c) y2 = 0.15, and (d) y2 = 0.2.

put loss. From Fig. 12 it can be seen that y1opt has minor changes around ωL /ωp = 1 when y2 is fixed. Fig. 13 show us that for a fixed y2 the emax decreases monotonically with increasing ωL /ωp , but the change is only obvious around ωL /ωp = 1. Figs. 14 and 15 show us the variations of the corresponding normalized pulse width w and the peak power popt versus ωL /ωp for different y2 . We can see that the change of the pulse width w versus ωL /ωp for a fixed y2 mainly occurs when ωL /ωp near 1, the w become small obviously around ωL /ωp = 1, , this is because the interaction between the external part of the mode and the inversion population will become weaker with

W. Tang et al. / Optik 122 (2011) 1558–1564 Table 1 The relevant parameters of GaAs substrate [19,22].

2

popt=4σγ tr Pm/[2εσn(0,0)l] hνπω0

2

0.10 0.09

a

0.08 0.07

b

0.06 0.05 0.04

c

0.02 0.1

1

5

w L/w P Fig. 15. Dependence of popt on ωL /ωp for different y2 in the case of ˛1 = 20, ˛2 = 30 and z = 0.05 when the laser is an energy-maximized passively Q-switched laser. (a) y2 = 0.05, (b) y2 = 0.1, (c) y2 = 0.15, and (d) y2 = 0.2.

increasing ωL /ωp . We also can obtain that the popt decreases monotonically with increasing ωL /ωp . For a fixed y2 , the change is only obvious around ωL /ωp = 1, too. 6. Applications Figs. 1–15 can help us to understand the relationship between the normalized parameters of the passively Q-switched laser and the parameters of the C-SESAM. It also can be used to predict the pulse characteristics and perform the design of an optimally passively Q-switched laser. We can treat the problem as follows. First, calculate ωL /ωp , ˛1 and ˛2 ; second, by solving Eqs. (13) and (17), we can obtain y2opt , xopt and emax for a fixed Y1 or obtain Y1opt , xopt and emax for a fixed y2 ; third, determine the corresponding, p and ω according to Eq. (12) and Eqs. (18) and (19). Fourth, obtain Ropt , T0opt and TG0opt , and the real maximum output energy Emax , the corresponding real peak power Pm , the real pulse width W. As an illustration of the technique described in our paper, we optimized a passively Q-switched Nd3+ :YVO4 laser with a C-SESAM based on maximizing output energy. The sketch of the experimental set-up [7] is shown in Fig. 16, it mainly includes pump source (808 nm), optical coupling system which can couple the pump laser to the Nd3+ :YVO4 crystal, the C-SESAM and the output mirror. In this example, the length of the resonator is 10 cm; the gain medium length l is 5 mm; the dissipative loss of the laser L is measured to be 0.15. In the case of ωp = 0.22 mm and the pump power Pp = 6 W, the laser-mode radius ωL is approximately equal to ωp . The stimulated emission cross section of the gain medium is  = 3.42 × 10−18 cm2 , doped with 1.0 at.% Nd3+ ions.  = fa + fb , where fa , fb are the Boltzmann occupation factors of upper and lower laser levels of the gain medium, respectively. For a Nd3+ :YVO4 gain medium at room temperature, fa = 0.43, and fb = 0.23. The absorption cross section of In0.25 Ga0.75 As is  g = 1.1 × 10−16 cm2 , and ntot = 4.2 × 1020 cm−3 . The relevant parameters of GaAs [22] are shown in Table 1.

Nd3+:YVO4 Pump source

Parameters

Values

Parameters

Values

0 + n0 n+ 0

1.0 × 10−16 cm2 2.3 × 10−17 cm2 1.2 × 1016 cm−3 1.4 × 1015 cm−3

ω0 ωq ˇG d

252 ␮m 156 ␮m 2.6 × 10−8 cm W−1 500 ␮m

d

0.03

0.01

1563

Coupling system

C-SESAM Output mirror

Fig. 16. The sketch of the experimental set-up which contains pump source, coupling system, Nd3+ : YVO4 , C-SESAM and output mirror.

Thus ωL /ωp , ˛1 and ˛2 are calculated to be 1, 32, 36, y2 = 0.07 when TG0 is approximately equal to 94.7 %. Then we can derive Y1opt = 0.53, xopt = 0.35, and we can obtain emax = 0.81897, p = 0.0608, w = 13.465. So we can obtain Ropt = 58.07 %, T0opt = 63.6 %, Emax = 40.11 ␮J, Pm = 6.95 kW, W = 5.77 ns, ls = 98 nm, respectively, when the laser is an energy-maximized passively Q-switched laser. 7. Conclusions The coupled rate equations are used to describe the passively Q-switched laser with a novel C-SESAM. These coupled rate equations are solved numerically. The key parameters of an optimally passively Q-switched laser are determined, and a group of general curves are generated. Sample calculations for a diode-pumped Nd3+ :YVO4 laser with a C-SESAM are presented to demonstrate the use of the curves and the relevant formulas. The C-SESAM is the combination of the In0.25 Ga0.75 As absorption layer and the GaAs substrate, therefore, the Q-switched pulse properties also formatted by the combined effect of the two(which have be shown in our figures). A significant advantage of the C-SESAM is that the two absorption mechanisms can be controlled separately, which can help us to obtain the ideal Q-parameters. Acknowledgements This work was partially supported by the National Science Foundation of China (60876056) and the Natural Science Foundation of Shandong Province (Y2007G17). References [1] Y. Wang, X. Ma, Passive Q-switching and modelocking of Nd-doped crystal lasers with semiconductor saturable absorption mirror, Prog. Laser Optoelectron. 40 (12) (2003) 18–20. [2] H. Liang, J. Huang, K. Su, H. Lai, Passively Q-switched Yb3:YCa4O(BO3)3 laser with InGaAs quantum wells as saturable absorbers, Appl. Opt. 46 (12) (2007) 2292–2296. [3] U. Keller, D.A.B. Miller, G.D. Boyd, Solid-state low-loss intracavity saturable absorber for Nd:YLF lasers: an A-FPSA, Opt. Lett. 17 (7) (1992) 505–508. [4] G.J. Spu¨hler, R. Paschotta, R. Fluck, Experimentally confirmed design guidelines for passively Q-switched microchip lasers using semiconductor saturable absorbers, J. Opt. Soc. Am. B 16 (3) (1999) 376–388. [5] D. Li, Y. Wang, X. Ma, The development of semiconductor saturable absorption mirror as passive Q-switching absorber, Appl. Opt. 26 (2) (2005) 7–9. [6] Y. Wang, X. Ma, Y. Liu, Passive mode-locked Nd:YV04 laser with central semiconductor saturable absorption mirror, Chin. J. Lasers 32 (8) (2005) 1034–1035. [7] Y. Wang, J. Peng, H. Tan, Passive Q-switched Nd3+ :YVO4 laser with central semiconductor saturable absorption mirror, Acta Photon. Sinica 36 (3) (2007) 401–404. [8] Y. Wang, X. Ma, K. Zheng, The research of long wavelength semiconductor saturable absorption and it’s application in passive modelocking and Q-switching of the solid-state lasers, Laser J. 13 (6) (2003) 7–8. [9] T.T. Kajava, A.L. Gaeta, Q-switching of a diode-pumped Nd:YAG laser with GaAs, Opt. Lett. 21 (1996) 1244–1246. [10] T.T. Kajava, A.L. Gaeta, Intra-cavity frequency-doubling of a Nd:YAG laser passively Q-switched with GaAs, Opt. Commun. 137 (1997) 93–97. [11] L. Chen, S. Zhao, H. Zhao, Passively Q-switching of a laser-diode-pumped intracavity-frequency-doubling Nd:NYW/KTP laser with GaAs saturable absorber, Opt. Laser Technol. 35 (7) (2003) 563–567. [12] S. Zhao, X. Zhang, Passively Q-switched self-frequency-doubling Nd3+ :GdCa4 O(BO3 )3 laser with GaAs saturable absorber, Opt. Eng. 41 (2002) 559–560. [13] J.J. Degnan, Theory of the optimally coupled Q-switched lasers, IEEE J. Quantum Electron. 25 (1989) 214–220.

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