In this report we demonstrate the transformation of the photon density curves in operation of a three modes random laser when the gain coefficients of modes depending on time.Supposing gain coefficients having Gaussian profile and using Matlab language, we have solved the basic equations describing the operation of a three modes random laser. The received results give us more real knowledge about the appearance of modes in random laser

Anh, Le Ngoc

Hoang, Dinh Van

Phu, Nguyen Van

Vietnam Academy of Science and Technology: Journals Online

Communications in Physics, Vol. 22, No. 4 (2012), pp. 349-355
INFLUENCE OF TIME DEPENDENCE OF DYNAMIC LASER
PARAMETER ON THE THREE MODES RANDOM LASER
OPERATION
DINH VAN HOANG AND LE NGOC ANH
Hanoi National University
NGUYEN VAN PHU
Vinh University
Abstract. In this report we demonstrate the influence of the photon density on the operation of a
three modes random laser when the gain coefficients of modes depending on time. Supposing gain
coefficients had Gaussian profile and using Matlab language, we have solved the basic equations
describing the operation of the three modes random laser. The received results give us more real
knowledge about the appearance of modes in random laser.
I. INTRODUCTION
Random laser appeared in the early 1990’s since Lawandy et al. [1] reported stimu-
lated emission from dye solution which contains micro-particles. After this report, many
experimental and theoretical studies on random laser have been published [2-8]. In fact,
the mechanism of light amplification in disordered media up to now is not clear. Therefore
one has proposed hypothesis, supposition and realized many experimental or theoretical
studies for explaining the operation of random laser.
With this aim, in the article [9] we have studied characteristics of modes generated
by two and three modes random laser. The transformation of these mode characteristics
depends mainly on the change of laser parameters like the gain and loss coefficients or
the field coupling and photon hopping coefficients. However, in [9] that transformation
corresponded only to the variation of the values of coefficients indicated above, while each
coefficient, in fact, depended on time in the process of laser operation, and this time
dependence was not disscussed.
Starting from this situation, in this work we examine the case in which the gain
coefficient depends on time and we hope that the received results will give us a general
and nearly real scenario about mechanism of random laser.
II. BASIC EQUATIONS AND METHOD OF RESOLUTION
Supposing random laser generated three modes with photon densities n1, n2, n3,
respectively, and where n2 belongs to the middle mode. Following [4], equations describing
the variation on time of modes are written, as follows:
dn1
dt
= α1n1 − β1n
2
1 − θ12n1n2 + γ12n2 (1)
350 DYNAMIC LASER PARAMETER ON THE THREE MODES RANDOM LASER OPERATION ...
dn2
dt
= α2n2 − β2n
2
2 − (θ21n1 + θ23n3)n2 + γ21n1 + γ23n3 (2)
dn3
dt
= α3n3 − β3n
2
3 − θ32n2n3 + γ32n2 (3)
Here αi, βi (i = 1, 2, 3) are gain and loss coefficients and θij, γij (j = 1, 2, 3) are field
coupling and photon hopping coefficients, respectively.
In case the gain coefficients depend on time, we choose for αi:
αi = α0ie
−∆T t2 (4)
this αi is corresponding to Gaussian profile of generated pulse mode.
For solving this system of equations with αi following (4), we use numerical method
on the base of Matlab language. Initial conditions are given:
n1 (0) = n2 (0) = n3 (0) = 0.2 (in a.u)
∆T1 = ∆T2 = ∆T3 = 2 (ns)
Other values of coefficients α0i, βi, θij, γij are given in:
α01 = 0.9 s
−1; β1 = 0.4 cm
3.s−1; θ12 = θ21 = 0.4 cm
3.s−1; γ12 = 0.8s
−1.
α02 = 1.3 s
−1; β2 = 0.3 cm
3.s−1; θ23 = 0.4 cm
3.s−1; γ21 = γ23 = 0.7 s
−1.
α03 = 0.9 s
−1; β3 = 0.5 cm
3.s−1; θ32 = 0.4 cm
3.s−1; γ32 = 0.8 s
−1.
The distribution of mode photon densities ni (t) is presented in Fig. 1
0 5 10 15 20 25 30 35 40 45 50
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
n
1
n
2
n3
E
m
is
s
io
n
I
n
te
n
s
it
y
(a
.u
)
Time (ns)
Fig. 1. The curves of ni(t) versus time.
From Fig. 1, we see that the profile of curves ni (t) in case the gain coefficient
depending on time has same form as the gain coefficients are constant.
DINH VAN HOANG, NGUYEN VAN PHU, LE NGOC ANH 351
For examining the evaluation of photon density ni (t) when the profile of gain coef-
ficients αi varies, we note the three parameters of these curves as following:
The ∆ωi is the pulse width of the i
th mode.
The Imaxi is the maximum photon density of i
th mode.
The Tmaxi is the time for reaching maximum photon density of i
th mode.
III. INFLUENCE OF THE VARIATION OF GAIN COEFFICIENTS
III.1. The influence of ∆T
Starting with the formula (4), we can vary one of two parameters ∆T and α0i.
Giving different values of ∆T and remaining invariable all values of coefficients presented
above, we can receive curves ni(t) presented in figures 2, 3 and 4. From these figures we
determine pulses mode parameters. The results are given in Table 1.
0 5 10 15 20 25 30 35 40 45 50
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
n
1
n
2
n
3
E
m
is
s
io
n
I
n
te
n
s
it
y
(a
.u
)
Time (ns)
Fig. 2. The curves of ni(t) with ∆T = 2 ns.
Table 1. Computation results of the influence of ∆T on the characteristics of
modes of random laser
∆T Tmax1 I
max
1 ∆ω1(ns) T
max
2 I
max
2 ∆ω2 T
max
3 I
max
3 (ns)
(ns) (ns) (a.u) (ns) (ns) (a.u) (ns) (ns) (a.u) (ns)
1 2.3848 0.6927 4.3548 2.2465 1.1050 4.8378 2.2465 0.6547 4.0092
2 3.4101 0.8284 9.4009 3.4908 1.5038 9.5391 3.2258 0.7751 9.2166
3 3.9174 0.8526 26.7281 6.1060 1.6854 29.9539 3.8018 0.7965 26.0368
4 6.4516 0.8620 94.0092 10.0230 1.7556 97.2458 3.8018 0.7965 87.5576
6 →∞ →∞ →∞
352 DYNAMIC LASER PARAMETER ON THE THREE MODES RANDOM LASER OPERATION ...
0 5 10 15 20 25 30 35 40 45 50
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
n
1
n
2
n
3
E
m
is
s
io
n
I
n
te
n
s
it
y
(a
.u
)
Time (ns)
Fig. 3. The curves of ni(t) with ∆T = 4 ns.
0 5 10 15 20 25 30 35 40 45 50
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
E
m
is
s
io
n
In
te
n
s
it
y
(a
.u
)
Time (ns)
n
1
n
2
n
3
E
m
is
s
io
n
In
te
n
s
it
y
(a
.u
)
Time (ns)
Fig. 4. The curves of ni(t) with ∆T = 6 ns.
From received results, we see that: The more the values of ∆T are small, the
quickly the time for reaching maximum photon density of mode attains and analogously
the smaller the values of maximum photon density and pulse width have.
DINH VAN HOANG, NGUYEN VAN PHU, LE NGOC ANH 353
III.2. Influence of initial values of gain coefficients α0i
By the same method of calculation, here we change in turn the values of α0i (i = 1,
2, 3). Obtained results are given in Table 2, 3, 4.
Table 2. Computation results of the influence of α01 on the characteristics of
modes of random laser
α01 T
max
1
Imax
1
∆ω1 T
max
2
Imax
2
∆ω2 T
max
3
Imax
3
) ∆ω3
(s−1) (ns) (a.u) (ns) Tmax
2
(ns) Imax
2
(a.u) (ns) (ns) (a.u) (ns)
0.1 4.0438 0.4658 7.8111 3.9747 1.8000 8.7788 2.6613 0.7553 8.8479
0.3 3.9747 0.5205 8.9171 3.9055 1.7509 9.0553 2.7304 0.7591 9.2627
0.5 3.2834 0.5906 9.1244 3.8364 1.6854 9.3318 2.8687 0.7684 9.4700
0.7 3.2143 0.6936 9.7465 3.7673 1.6012 9.8847 2.9378 0.7778 10.0921
0.9 3.1452 0.8365 10.3686 3.4908 1.5079 10.1612 3.0530 0.7833 10.1785
1.1 3.3525 1.0289 10.9908 3.0069 1.4015 10.4378 3.2834 0.7956 10.2304
1.3 3.5599 1.2950 11.6820 2.7304 1.2991 11.3364 3.6290 0.8120 10.7834
1.5 3.9747 1.6620 10.7834 2.5230 1.2082 9.6774 3.9747 0.8339 9.3318
Table 3. The influence of α02 on the characteristics of modes of random laser
α02 T
max
1 I
max
1 ∆ω1 T
max
2 I
max
2 ∆ω2 T
max
3 I
max
3 ω3
(s−1) (ns) (a.u) (ns) (ns) (a.u) (ns) (ns) (a.u) (ns)
0.2 2.8687 0.8105 9.6774 3.9055 1.7556 8.9171 3.6982 0.4784 7.6037
0.4 3.0069 0.8199 9.8848 3.8364 1.6947 9.2627 3.5599 0.5345 8.2949
0.6 3.0760 0.8246 9.9539 3.7673 1.6339 9.3318 3.4908 0.6140 9.3318
0.8 3.1452 0.8325 10.2304 3.6290 1.5488 9.5318 2.9397 0.7178 10.0230
1.0 3.3525 0.8447 10.3329 3.4217 1.4629 10.2304 3.0069 0.8570 10.6452
1.2 3.4908 0.8611 10.6451 3.1452 1.3687 10.4378 3.1452 1.0412 10.0230
1.4 3.8364 0.8816 10.9907 2.7304 1.2746 11.4746 3.2143 1.2827 10.0922
1.6 4.1820 0.9102 11.6129 2.5230 1.1927 11.6820 3.4217 1.5898 10.2995
Table 4. The influence of α03 on the characteristics of modes of random laser
α03 T
max
1 I
max
1 ∆ω1 T
max
2 I
max
2 ∆ω2 T
max
3 I
max
3 ∆ω3
(s−1) (ns) (a.u) (ns) (ns) (a.u) (ns) (ns) (a.u) (ns)
0.5 4.3894 1.1079 9.4009 3.1452 0.6898 9.2627 4.1820 0.9792 9.4009
0.7 4.3203 1.0465 9.4700 3.1665 0.8096 9.5391 4.1129 0.9383 9.6774
0.9 4.1820 0.9778 9.6082 3.1943 0.9731 9.7465 3.8364 0.8865 9.7466
1.1 3.6982 0.9032 9.7330 3.2143 1.1965 9.8847 3.4908 0.8330 9.8157
1.3 3.0760 0.8325 9.8848 3.4908 1.5038 9.8156 2.9378 0.7792 9.9539
1.5 2.7304 0.7763 10.0230 3.6982 1.9289 9.8848 2.6613 0.7395 10.0230
1.7 2.2465 0.7346 10.1613 3.7673 2.4598 10.0230 2.3848 0.7054 10.2304
2.0 1.9700 0.6908 10.0231 3.6290 3.3823 9.8856 2.0392 0.6601 10.5069
354 DYNAMIC LASER PARAMETER ON THE THREE MODES RANDOM LASER OPERATION ...
From tables 2, 3 and 4, we see:
- The variation of α01 and α03 give the same effect for the evaluation of pulse mode
parameters Tmaxi , I
max
i , ∆ωi.
- When α02 augments, the photon density of 2
th mode increases quickly. This point
demonstrates that the more value of α02 is large, the quicker the photon hopping from
near modes to middle mode grows.
IV. CONCLUSIONS
We have simulated the influence of the dynamic laser parameters (such as variation
of gain coefficients) on the characteristics of three modes of random laser by rate equations.
We demonstrate the influence of the photon density curves on the operation of a three
modes random laser when the gain coefficients of modes depend on time. Especially, when
the gain coefficient α0i of any mode becomes largest, this mode will be brought in relief
and laser generating three modes becomes laser generating single mode.
The numerical method used here in the case of three modes random laser, will be
applied completely for the case of multimode random laser.
0 5 10 15 20 25 30 35 40 45 50
0
2
4
6
8
10
12
P
ho
to
n
de
n
si
ty
(a
.u
)
Time (ns)
n
1
n2
n
3
P
h
o
to
n
d
e
n
s
it
y
(a
.u
)
Time s)
Fig. 5. The curves of ni(t) with α02 = 4 s
−1.
DINH VAN HOANG, NGUYEN VAN PHU, LE NGOC ANH 355
REFERENCES
[1] N. M. Lawandy et al., Nature 368 (1994) 436.
[2] S. John and G. Pang, Phys. Rev. A54 (1996) 3642.
[3] G. A. Berger et al., Phys. Rev. E56 (1997) 6118.
[4] G. Zacharakis et al., Opt. Lett. 25 (2001) 923.
[5] A. L. Burin et al., Phys. Rev. Lett. 88 (2002) 3904.
[6] X. Jiang et al., Phys. Rev. B69 (2004) 104202.
[7] H. Cao et al., Appl. Phys. Lett. 73 (2005) 3656.
[8] D. V. Hoang, M. H. Hanh, VNU J. of Science Maths-Phys. 23(3) (2007) 139.
[9] Dinh Van Hoang, Nguyen Thi Phuong, Nguyen Van Phu, Computational Methods in Science and
Technology, Special Issue 2, 47-53, Poland (2010).
Received 25 April 2012.


Influence of Time Dependence of Dynamic Laser Parameter on the Three Modes Random Laser Operation

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Influence of Time Dependence of Dynamic Laser Parameter on the Three Modes Random Laser Operation

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