- Email: [email protected]
Numerical calculations of intracavity dye Q-switched ruby laser
ARTICLE IN PRESS
Optics and Lasers in Engineering 41 (2004) 177–187
Numerical calculations of intracavity dye Q-switched ruby laser M. Soukieh, B.Abdul Ghani*, M. Hammadi Atomic Energy Commission, P.O. Box 6091, Damascus, Syria Received 13 November 2001; accepted 27 August 2002
Abstract A mathematical model describing the dynamic emission of the Q-switched ruby laser has been adapted. The suggested model allows the investigation of the effects of a dye cell on the mode characteristics of the ruby laser and, moreover, the study of the effect of the laser input parameters on the output laser pulse. This model simulates the nonlinear effects of dye pulse modulation on the laser emission. In addition, a numerical solution of a nonlinear rate equation system of the adapted model is discussed. The solution estimates the density of the emitted radiation, population inversion and energy transfer processes of the ruby laser rod and dye cell for different emission regimes (one pulse regime, free running pulses, repetition periodic pulses). The estimated results of the laser output pulse characteristics are in a good agreement with the other calculated and experimental results. r 2002 Published by Elsevier Science Ltd. Keywords: Modeling; Ruby laser; Dye cell
1. Introduction The first successfully operated laser was the ruby laser. It consists of a ruby rod whose ends are flat, at the two ends of the resonator cavity there are reflecting mirrors (see Fig. 1a). Ruby consists of aluminum oxide Al2O3 doped with 0.05% Cr+3; however, higher concentrations have also been used. The energy levels of the *Corresponding author. Fax: +963-11-611-2289. E-mail address: [email protected] (B.A. Ghani). 0143-8166/04/$ - see front matter r 2002 Published by Elsevier Science Ltd. PII: S 0 1 4 3 - 8 1 6 6 ( 0 2 ) 0 0 1 4 9 - 5
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L2
L1 rFr
Ruby rod
L*
L
r1
Dye cell
rFr
R1
R2
r2
(a)
P13
E3
S1 E2 P21
E1
B21
U
S0
(b) Fig. 1. (a) Schematic diagram of the experimental setup and (b) the scheme of transition between different energy levels for Cr+3 ions and dye cell.
chromium are shown in Fig. 1b , the state in the band E3 has a short lifetime (p109 s), whereas the metastable state E2 has a much longer lifetime (milliseconds). A part of energy is absorbed by the Cr+3 ions in the ground state E1 resulting in their excitation to an energy level inside the band E3 : The Cr+3 ions make very fast nonradiative transition from the excited state to the E2 state. Once a state of population inversion is achieved, lasing action (l ¼ 694 nm) is triggered by spontaneously emitted photons (free running generation) and it can be operated using dye Q-switching (such as: DDI, Cryptocyanine, Chlorophyll, Methly DOTCI, Rhodamine, Styryl-15, and many other Phthalocyanines [1,2]). The dye molecules get excited in laser cavity from ground state S0 to higher vibrational rotational levels of the next electronic state S1 : Radiation is emitted when the molecules decay from rotational vibrational sublevels of higher electronic state S1 to any rotational vibrational sublevels of electronic ground state S0 [3]. Of course, the low-intensity emissions vanish due to absorption process in the dye cell. When the dye is saturated it becomes transparent, and then by increasing amplification, the emitted photons can pass through it to be subsequently reflected by the rear mirror. Consequently, one, several, or a train of giant pulses are generated. The dynamics of the nonlinear absorption coefficient of most dye materials used in Q-switching laser operation is not well studied. This work will concentrate on the effect of the ruby laser and dye cell parameters on the output characteristics (mainly studying the effect of variation nonlinear absorption coefficient on the line width of the generated giant output pulses).
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2. Mathematical model The following equation describes the variation of relative population inversion as a function of time [4]: dY ¼ ðP13 P21 Þ ðP13 þ P21 ÞY 2B21 ðuÞUY : dt
ð1Þ
The time evolution of the volume density of the field intensity in the cavity is given by the equation dU aK L vmU0 ¼ vm wY Kloss : ð2Þ Uþ dt L 2L The time evolution of the absorption coefficient of a dye cell is determined as follows: dK ¼ A21 ðK0 K Þ sUK : dt
ð3Þ
The intensity of laser beam will be given by the relation IðtÞ ¼ cUðtÞ; where P13 is the pumping rate (the probability of Cr+3 ions transition from the E1 state to the excited state E3 by affecting the light emission of flash lamp), P21 is the probability of Cr+3 ions spontaneous transition from the upper laser level E2 to the state E 1 (see Fig. 1b), Y ¼ ðN2 N1 Þ=N is the relative population inversion, Ni ði ¼ 1; 2Þ is the population densities of the lower and upper laser level, N is the total density of Cr+3 ions, B21 is the probability of Cr+3 ions stimulated emission from the upper laser level E2 to the state E1 ; U is the radiation density of laser beam in unit volume, v ¼ c=Z is the light velocity in the ruby rod, c is the light velocity, Z is the refractive index of the ruby rod, m ¼ ðLZ þ Ld Zd Þ=ðL1 þ L2 þ LZ þ L Zd Þ is the filling factor of laser resonator, L is the effective length of laser rod, L1 and L2 are the distance between the ends of laser rod and the reflecting mirrors, Ln the dye cell length, Z the refractive index of the dye cell, w is the maximum value of amplification coefficient in the ruby rod (when the total number of Cr+3 ions are excited), Kloss ¼ r þ ðrn Ln =LÞ þ ð1=2LÞLnð1=R1 R2 Þ is the loss factor of laser resonator in general, r is the loss factor by absorption, scattering, etc., R1 and R2 are the reflecting coefficient of the complex resonator mirrors (consisting of reflecting mirrors and rod faces), a ¼ ð1=K n Ln ÞLnðR1 R2 ÞK ¼0 =ðR1 R2 ÞK a0 is a constant amount, cd ¼ c=Zd is the light 0 0 velocity in the dye cell. The following relation estimates the effective value of reflection coefficient Ri ði ¼ 1; 2Þ: Ri ¼ ðrFr 2t2 rFr ri þ t2 ri Þ=ð1 t2 rFr ri Þ; where t is the transmitivity coefficient of the dye cell, rFr is the Frenel reflection coefficient of rod surface, and r1 ; r2 reflecting coefficient of resonator mirrors (see Fig. 1a).
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Table 1 The ruby and dye parameters Constant P13 P21 B21 v m a w N
Value 1500–7000 300 150 1.7 1010 0.5 1 0.25 1 1019
Unit 1
s s1 cm3/erg s cm/s — — cm1 cm3
Constant
Value
Unit
Kloss L A21 s L r1 r2 Z
0.02–0.1 12 108 103–107 1 100% 45% 1.76
cm1 cm s1 cm3/erg s cm — — —
The following relation estimates the total extracted energy at single pulse emission: Z A 1 R2 Kloss þ R2 Kloss t IðtÞ dt; Eout ¼ ln½R2 ð1 Kloss Þ 2 1 R2 ð1 Kloss Þ 0 where A is the cross-section of the laser mode.
3. Numerical solution of rate equations The rate Eqs. (1)–(3) represent a system of stiff ordinary nonlinear differential equations. These equations describe the dynamic emission in both laser media, ruby rod and gain switched cell. A FORTRAN computer program, based on the Runga–Kutta method, was used to solve these equations. The integration of the equations was achieved using an error criterion ep104 : This program allows the investigation of the effects of a dye cell on mode characteristics of a ruby laser. It also studies the effect of the laser input parameters on the output laser pulse. The physical constants of Eqs. (1)–(3) and the geometrical dimensions of the laser cavity are given in Table 1 [4]. The initial values of the rate equations were chosen as follows: P13 P21 w L Kloss Y0 ¼ ; U0 ¼ 200 erg=cm3 ; K0n ¼ n Y0 : L P13 þ P21 w The initial values of K depend on the value of Kloss ; it means, there are different initial values for different values of Kloss :
4. Results and discussion Fig. 2 shows the relation between the laser density and the relative population inversion as a function of time. From this figure it can be seen that the pulse duration is very short (approximately 30 ns). Initially, the photon density is low while the laser
ARTICLE IN PRESS M. Soukieh et al. / Optics and Lasers in Engineering 41 (2004) 177–187
1.0
5.E+6 1
3.E+6
0.8 0.6
3 Y
U (erg/cm3)
4.E+6
0.4
2.E+6
1
2 1.E+6
3 2
4
0.2
4
0.0
0.E+0 0 (a)
181
10
20 t (ns)
30
0 (b)
10
20
30
t (ns)
Fig. 2. (a) The laser energy density as a function of time and (b) the relative population density as a function of time (for P13 ¼ 1500 s1 (curves 3, 4), P13 ¼ 7000 s1 (curves 1, 2), and Kloss ¼ 0:02 cm1(curves 1, 3), Kloss ¼ 0:1 cm1 (curves 2, 4)).
is being pumped where the cavity losses are higher than the amplification coefficient, then the losses suddenly decrease, because the photon density rises from initial value, reaches a peak value many order of magnitude higher than the initial value. The pulse duration, output power and relative population inversion increase with increasing pumping rate of radiation (P13 ) and decrease with decreasing loss coefficient Kloss : The population inversion at single pulse emission decreases and reaches the stabilized minimum value Ymin at the end of the pulse duration. This minimum value increases with increasing loss coefficient. The pulse width of curves 1, 3 (Fig. 2a) reaches 6.67 and 7.78 ns and reaches 4.44 and 8.89 ns for curves 2, 4, respectively. The pulse width increases with decreasing pumping rate. The delay time increases with increasing loss coefficient and decreases with increasing pumping rate. Fig. 3 was obtained using pumping rate of P13 ¼ 150027000 (s1), Kloss ¼ 0:0220:1 (cm1), not null absorption coefficient of dye cell, nonlinear factor of the dye solution s ¼ 107 ðcm3 =erg sÞ and t > 0: It can be noticed from Figs. 3a and b that the energy radiation density and the pulse duration decrease with decreasing pumping rate and with increasing loss coefficient. It also can be seen from Figs. 3a and b that with increasing loss coefficient, the delay time increases and the pulse width decreases until the loss coefficient reaches the value Kloss ¼ 0:08 cm1. When the loss coefficient reaches its maximum value (0.1 cm1), the pulse width begin to increase. The population inversion process will be delayed with decreasing pumping rate and increasing the loss coefficient (Figs. 3c and d). From Figs. 4a and b it can be seen that the dye cell becomes transparent through nearly 1 ns (the absorption coefficient of the dye cell tends to zero). Figs. 5a and b give the effect of nonlinear coefficient of the dye solution on the characteristics of laser emission, which was studied for different values; s ¼ 107 ; 105 ; 104 ðcm3 =erg sÞ; and for different values of pumping rate and loss coefficient. From Figs. 5a and b it can be noticed that the radiation density did not change the peak of giant pulse for the following values of nonlinear coefficient of the dye solution s ¼ 107 ; 105 ðcm3 =erg sÞ at limited values of pumping rate (see curves 1, 2).
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182
5.E+6
4.E+6
U (erg/cm3)
4.E+6
U (erg/cm3)
1 2
3.E+6
3
2.E+6
4 5
1.E+6
3.E+6
1
2.E+6
2 3
1.E+6
4
0.E+0
0.E+0 0
10
(a)
20
30
t (ns)
0
5
10
20
30
20
30
t (ns)
(b) 1.0
0.8
0.8
0.6
0.6
Y
Y
1.0
0.4
0.4
0.2
0.2
12 3 4 5
1 23 4
5
0.0
0.0 0
10
20 t (ns)
(c)
30
10
0
t (ns)
(d)
5 4 3 2.0 12 1.5
3.0
3.0
2.5
2.5 K* (cm-1)
K* (cm-1)
Fig. 3. (a) The laser energy density as a function of time (P13 ¼ 7000 s1), (b) the laser energy density as a function of time (P13 ¼ 1500 s1), (c) the relative population inversion as a function of time (P13 ¼ 7000 s1) and (d) the relative population inversion as a function of time (P13 ¼ 1500 s1) (for Kloss ¼ 0:02 cm1 (1), 0.04 (2), 0.06 (3), 0.08 (4), 0.1 (5) and ) K0 ¼ ðXL=L Þ½Y0 ðKloss =X Þ:
1.0
0.5 0.0
0.5 0.0 0 (a)
5 4 3 1.5 2 1.0 1
2.0
5
10 t (ns)
0 15
20
25
(b)
5
10 t (ns)
15
20
25
Fig. 4. (a) The absorption coefficient of dye cell as a function of time (P13 ¼ 7000 s1) and (b) the absorption coefficient of dye cell as a function of time (P13 ¼ 1500 s1) (for Kloss (cm1)=0.02 (1), 0.04 (2), 0.06 (3), 0.08 (4),0.1 (5)).
When the nonlinear coefficient of the dye solution decreases, the emission of the giant pulse is delayed at the value of s ¼ 104 ðcm3 =erg sÞ until the delay time is about 50 ns. This delay is due to the decrement of the solution transparency in the dye cell,
ARTICLE IN PRESS M. Soukieh et al. / Optics and Lasers in Engineering 41 (2004) 177–187
5.E+6
1.6E+6
1
2
1
4.E+6
2
1.2E+6 4
3.E+6
U (erg/cm3)
U (erg/cm3)
183
5
2.E+6
8.0E+5 4
5
4.0E+5
1.E+6 3.6
0.E+0 0
10
(a)
20 30 t (ns)
40
3.6
0.0E+0
50
0
10
(b)
20 30 t (ns)
40
50
Fig. 5. (a) The laser energy density as a function of time (Kloss ¼ 0:02 cm1) and (b) the laser energy density as a function of time (Kloss ¼ 0:1 cm1) (for P13 ¼ 7000 s1 (1–3); P13 ¼ 1500 s1(4–6) and sðcm3 =erg:sÞ=107 (1, 4), 105 (2, 5), 104 (3, 6)).
Table 2 Figure # 5a 5a 5b 5b
Curve # 1 5 1 5
Pulse width (ns) 7.27 8.18 4.27 7.27
sðcm3 =erg sÞ 7
10 105 107 105
Delay time (ns) 3 17 5 26
and it requires irradiating the solution for a longer period in order to become more transparent for the laser beam. It can also be seen from Figs. 5a and b that the pulse width decreases with increasing nonlinear coefficient of the dye solution. The increment of this coefficient and the pumping rate leads to a shorter time delay, and the decrement of pumping rate leads to an increased pulse width as shown in Table 2. The relative population inversion decreases monotonously with time starting at initial inversion and ending at the final inversion (see Figs. 6a and b, curves 1, 2, 4, 5). On the other hand, the relative population inversion remains constant during the same duration for s ¼ 104 ðcm3 =erg sÞ (Figs. 6a and b, curves 3, 6). From Figs. 7a and b it can be seen that the used dye cell solution renews its properties by the end of the giant pulse especially in the case of high losses (approximately after 30–50 ns from the beginning of the giant laser pulse). The ruby laser could be operated in the periodic regime when the relative population inversion reaches the initial value Y0 by influence of pumping radiation after giant pulse emission in order to produce another pulse. In this case, the population inversion and laser emission could be repeated periodically. Eqs. (1)–(3) were solved considering the following constants P13 ¼ 7000 s1, Kloss ¼ 0:08 cm1 and K ¼ 1:77 cm1.
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184
1.0
1.0 2
0.6
3
Y
5
6
6
0.6
4
0.4
3
2
1
0.8
Y
1
0.8
4
0.4
0.2
5
0.2
0.0
0.0 0
10
20
(a)
30 t (ns)
40
50
0
10
20
(b)
30 t (ns)
40
50
Fig. 6. (a) The relative population density as a function of time (Kloss ¼ 0:02 cm1) and (b) the relative population density as a function of time (Kloss ¼ 0:1 cm1) (for P13 ¼ 7000 s1 (1–3); P13 ¼ 1500 s1(4–6), sðcm3 =erg:sÞ=107 (1, 4), 105 (2, 5), 104 (3, 6)).
1.6
3.0 2
2.0 1.5
6
5
1.0
1 2
0.8 1
1 0.5 4
0.4
0.0
0.0 0
2
1.2 K* (cm-1)
K* (cm-1)
2.5
(a)
3
3
5
6 4 5
4 10
20 30 t (ns)
40
0
50 (b)
10
20 30 t (ns)
40
50
Fig. 7. (a) The absorption coefficient of dye cell as a function of time (Kloss ¼ 0:02 cm1) and (b) the absorption coefficient of dye cell as a function of time (Kloss ¼ 0:1 cm1) (for P13 ¼ 7000 s1 (1–3); P13 ¼ 1500 s1(4–6), sðcm3 =erg:sÞ=107 (1, 4), 105 (2, 5), 104 (3, 6)).
Fig. 8 shows the laser energy density as a function of time in the periodic emissions regime. From this figure it can be noticed that the period of emission reaches nearly 40 ns. The period and the density of laser emission could be affected by the losses and nonlinear solution factor. The laser energy density of the ruby laser in free running regime as a function of time (for P13 ¼ 7000 s1 and Kloss ¼ 0:02 cm1) is shown in Fig. 9. It can also be noticed that the free running regime lasts for about 90 ms. The laser energy density is damped (the emission of free running pulses is chaotic regime). This may be due to the multimode and linear characteristic emission (each line is amplified independently). It can be noticed from Fig. 10 that there is a good quantitative and qualitative agreement between the calculated results in this work and the theoretical results given in [4]. For example, the calculated FWHM of the output pulse is about 14.5 ns, while it is about 15 ns in Ref. [4].
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185
2.5E+6
U (erg/cm3)
2.0E+6 1.5E+6 1.0E+6 5.0E+5 0.0E+0 0
100
200
300
400
500
600
700
800
t (ns) Fig. 8. The energy density as a function of time in ruby laser with dye cell during the periodic emission regime (P13 ¼ 7000 s1, Kloss ¼ 0:08 cm1 and sðcm3 =erg:sÞ=107)
U (erg/cm3)
325
300
275
250 20
40
60
80
t (µs) Fig. 9. The energy density as a function of time in ruby laser during the free-running emission regime (P13 ¼ 7000 s1, Kloss ¼ 0:02 cm1).
Fig. 11 shows a good agreement between the calculated radiation intensity as a function of time in the dye Q-switched ruby laser and the experimental result reported in [1]. The calculated results show a 5.6 ns pulse width and 0.76 J of total energy power, while the experimental results were 5.3 ns and 0.74 J, respectively. The above numerical results were obtained at a single frequency. If the active material is pumped considerably above threshold then the gain drops quickly in a short cavity transit time. The results show that a decrease in nonlinear solution coefficient causes high losses and consequently a longer period is needed to exceed these losses (coupling losses, incidental losses such as scattering, diffraction and absorption, cavity losses introduced by inserting the dye cell). This increase in the duration between laser pulses leads to higher values of population inversion and higher maximum of peak power. Therefore, the stored energy in the pulse is emitted in a shorter laser pulse. Also, it is possible to ensure operation on lowest order transverse mode by taking advantage of the higher losses suffered by higher order mode.
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Fig. 10. The energy density (1), the relative population density (2), and the absorption coefficient of dye cell (3) as a function of time in ruby laser; (a) calculated in this work and (b) calculated in [4] (for P13 ¼ 1500 s1, Kloss ¼ 0:11 cm1, sðcm3 =erg:sÞ=6000, A21 ¼ 107 s1).
Fig. 11. The radiation intensity as a function of time in the dye Q-switched ruby laser: (a) calculated results for: L ¼ 10 cm, m=0.4, Kloss ¼ 0:04; sðcm3 =erg:sÞ=107, P13 ¼ 7000 s1 and (b) experimental results (pulse width=5.3 ns, pulse energy=0.74 J) [1].
5. Conclusion In this work, a mathematical model has been adapted to predict different characteristics of a Q-switched ruby laser pulse. The adapted approach depends on using a general three-equation model given by Eqs. (1)–(3), that describes the dynamic emission in gain switched ruby laser. The adapted mathematical model allows the investigation of the effects of nonlinear coefficient of the Q-switched dye as a selective absorber for obtaining giant laser pulse, and the laser input parameters on the output laser pulse. This model can be applied to any solid-state laser such as Nd-glass, Nd-YAG with different dye materials, and it can adapted to describe the gas lasers such as CO2 laser with different saturable absorbers as well [5].
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Acknowledgements The authors would like to express their thanks to the Director General of AECS Prof. I. Othman for his continuous encouragement, guidance and support. They also thank Dr. B. Masarani and Dr. F. Awad for their revision.
References [1] [2] [3] [4] [5]
Koechner W. Solid state laser engineering. Berlin: Springer; 1988. Brackmann U. Lambdachrome laser dyes. Datasheets, 1994. Thyagarajan K, Ghatak AK. Lasers theory and applications. New York: Macmillan; 1987. Stepanov BI. Calculation methods of optical quantum generators II. Minsk: Nayka, 1968 (in Russian). Kulikov VV, Kuntsevich BF, Chigevski VN, Churakov VV. The effect of refraction index variation of investigation material on the output parameters of CO2 laser. J Appl Spectrosc 1992;57(5-6):464–71 (in Russian).
Optics and Lasers in Engineering 41 (2004) 177–187
Numerical calculations of intracavity dye Q-switched ruby laser M. Soukieh, B.Abdul Ghani*, M. Hammadi Atomic Energy Commission, P.O. Box 6091, Damascus, Syria Received 13 November 2001; accepted 27 August 2002
Abstract A mathematical model describing the dynamic emission of the Q-switched ruby laser has been adapted. The suggested model allows the investigation of the effects of a dye cell on the mode characteristics of the ruby laser and, moreover, the study of the effect of the laser input parameters on the output laser pulse. This model simulates the nonlinear effects of dye pulse modulation on the laser emission. In addition, a numerical solution of a nonlinear rate equation system of the adapted model is discussed. The solution estimates the density of the emitted radiation, population inversion and energy transfer processes of the ruby laser rod and dye cell for different emission regimes (one pulse regime, free running pulses, repetition periodic pulses). The estimated results of the laser output pulse characteristics are in a good agreement with the other calculated and experimental results. r 2002 Published by Elsevier Science Ltd. Keywords: Modeling; Ruby laser; Dye cell
1. Introduction The first successfully operated laser was the ruby laser. It consists of a ruby rod whose ends are flat, at the two ends of the resonator cavity there are reflecting mirrors (see Fig. 1a). Ruby consists of aluminum oxide Al2O3 doped with 0.05% Cr+3; however, higher concentrations have also been used. The energy levels of the *Corresponding author. Fax: +963-11-611-2289. E-mail address: [email protected] (B.A. Ghani). 0143-8166/04/$ - see front matter r 2002 Published by Elsevier Science Ltd. PII: S 0 1 4 3 - 8 1 6 6 ( 0 2 ) 0 0 1 4 9 - 5
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L2
L1 rFr
Ruby rod
L*
L
r1
Dye cell
rFr
R1
R2
r2
(a)
P13
E3
S1 E2 P21
E1
B21
U
S0
(b) Fig. 1. (a) Schematic diagram of the experimental setup and (b) the scheme of transition between different energy levels for Cr+3 ions and dye cell.
chromium are shown in Fig. 1b , the state in the band E3 has a short lifetime (p109 s), whereas the metastable state E2 has a much longer lifetime (milliseconds). A part of energy is absorbed by the Cr+3 ions in the ground state E1 resulting in their excitation to an energy level inside the band E3 : The Cr+3 ions make very fast nonradiative transition from the excited state to the E2 state. Once a state of population inversion is achieved, lasing action (l ¼ 694 nm) is triggered by spontaneously emitted photons (free running generation) and it can be operated using dye Q-switching (such as: DDI, Cryptocyanine, Chlorophyll, Methly DOTCI, Rhodamine, Styryl-15, and many other Phthalocyanines [1,2]). The dye molecules get excited in laser cavity from ground state S0 to higher vibrational rotational levels of the next electronic state S1 : Radiation is emitted when the molecules decay from rotational vibrational sublevels of higher electronic state S1 to any rotational vibrational sublevels of electronic ground state S0 [3]. Of course, the low-intensity emissions vanish due to absorption process in the dye cell. When the dye is saturated it becomes transparent, and then by increasing amplification, the emitted photons can pass through it to be subsequently reflected by the rear mirror. Consequently, one, several, or a train of giant pulses are generated. The dynamics of the nonlinear absorption coefficient of most dye materials used in Q-switching laser operation is not well studied. This work will concentrate on the effect of the ruby laser and dye cell parameters on the output characteristics (mainly studying the effect of variation nonlinear absorption coefficient on the line width of the generated giant output pulses).
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2. Mathematical model The following equation describes the variation of relative population inversion as a function of time [4]: dY ¼ ðP13 P21 Þ ðP13 þ P21 ÞY 2B21 ðuÞUY : dt
ð1Þ
The time evolution of the volume density of the field intensity in the cavity is given by the equation dU aK L vmU0 ¼ vm wY Kloss : ð2Þ Uþ dt L 2L The time evolution of the absorption coefficient of a dye cell is determined as follows: dK ¼ A21 ðK0 K Þ sUK : dt
ð3Þ
The intensity of laser beam will be given by the relation IðtÞ ¼ cUðtÞ; where P13 is the pumping rate (the probability of Cr+3 ions transition from the E1 state to the excited state E3 by affecting the light emission of flash lamp), P21 is the probability of Cr+3 ions spontaneous transition from the upper laser level E2 to the state E 1 (see Fig. 1b), Y ¼ ðN2 N1 Þ=N is the relative population inversion, Ni ði ¼ 1; 2Þ is the population densities of the lower and upper laser level, N is the total density of Cr+3 ions, B21 is the probability of Cr+3 ions stimulated emission from the upper laser level E2 to the state E1 ; U is the radiation density of laser beam in unit volume, v ¼ c=Z is the light velocity in the ruby rod, c is the light velocity, Z is the refractive index of the ruby rod, m ¼ ðLZ þ Ld Zd Þ=ðL1 þ L2 þ LZ þ L Zd Þ is the filling factor of laser resonator, L is the effective length of laser rod, L1 and L2 are the distance between the ends of laser rod and the reflecting mirrors, Ln the dye cell length, Z the refractive index of the dye cell, w is the maximum value of amplification coefficient in the ruby rod (when the total number of Cr+3 ions are excited), Kloss ¼ r þ ðrn Ln =LÞ þ ð1=2LÞLnð1=R1 R2 Þ is the loss factor of laser resonator in general, r is the loss factor by absorption, scattering, etc., R1 and R2 are the reflecting coefficient of the complex resonator mirrors (consisting of reflecting mirrors and rod faces), a ¼ ð1=K n Ln ÞLnðR1 R2 ÞK ¼0 =ðR1 R2 ÞK a0 is a constant amount, cd ¼ c=Zd is the light 0 0 velocity in the dye cell. The following relation estimates the effective value of reflection coefficient Ri ði ¼ 1; 2Þ: Ri ¼ ðrFr 2t2 rFr ri þ t2 ri Þ=ð1 t2 rFr ri Þ; where t is the transmitivity coefficient of the dye cell, rFr is the Frenel reflection coefficient of rod surface, and r1 ; r2 reflecting coefficient of resonator mirrors (see Fig. 1a).
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Table 1 The ruby and dye parameters Constant P13 P21 B21 v m a w N
Value 1500–7000 300 150 1.7 1010 0.5 1 0.25 1 1019
Unit 1
s s1 cm3/erg s cm/s — — cm1 cm3
Constant
Value
Unit
Kloss L A21 s L r1 r2 Z
0.02–0.1 12 108 103–107 1 100% 45% 1.76
cm1 cm s1 cm3/erg s cm — — —
The following relation estimates the total extracted energy at single pulse emission: Z A 1 R2 Kloss þ R2 Kloss t IðtÞ dt; Eout ¼ ln½R2 ð1 Kloss Þ 2 1 R2 ð1 Kloss Þ 0 where A is the cross-section of the laser mode.
3. Numerical solution of rate equations The rate Eqs. (1)–(3) represent a system of stiff ordinary nonlinear differential equations. These equations describe the dynamic emission in both laser media, ruby rod and gain switched cell. A FORTRAN computer program, based on the Runga–Kutta method, was used to solve these equations. The integration of the equations was achieved using an error criterion ep104 : This program allows the investigation of the effects of a dye cell on mode characteristics of a ruby laser. It also studies the effect of the laser input parameters on the output laser pulse. The physical constants of Eqs. (1)–(3) and the geometrical dimensions of the laser cavity are given in Table 1 [4]. The initial values of the rate equations were chosen as follows: P13 P21 w L Kloss Y0 ¼ ; U0 ¼ 200 erg=cm3 ; K0n ¼ n Y0 : L P13 þ P21 w The initial values of K depend on the value of Kloss ; it means, there are different initial values for different values of Kloss :
4. Results and discussion Fig. 2 shows the relation between the laser density and the relative population inversion as a function of time. From this figure it can be seen that the pulse duration is very short (approximately 30 ns). Initially, the photon density is low while the laser
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1.0
5.E+6 1
3.E+6
0.8 0.6
3 Y
U (erg/cm3)
4.E+6
0.4
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1
2 1.E+6
3 2
4
0.2
4
0.0
0.E+0 0 (a)
181
10
20 t (ns)
30
0 (b)
10
20
30
t (ns)
Fig. 2. (a) The laser energy density as a function of time and (b) the relative population density as a function of time (for P13 ¼ 1500 s1 (curves 3, 4), P13 ¼ 7000 s1 (curves 1, 2), and Kloss ¼ 0:02 cm1(curves 1, 3), Kloss ¼ 0:1 cm1 (curves 2, 4)).
is being pumped where the cavity losses are higher than the amplification coefficient, then the losses suddenly decrease, because the photon density rises from initial value, reaches a peak value many order of magnitude higher than the initial value. The pulse duration, output power and relative population inversion increase with increasing pumping rate of radiation (P13 ) and decrease with decreasing loss coefficient Kloss : The population inversion at single pulse emission decreases and reaches the stabilized minimum value Ymin at the end of the pulse duration. This minimum value increases with increasing loss coefficient. The pulse width of curves 1, 3 (Fig. 2a) reaches 6.67 and 7.78 ns and reaches 4.44 and 8.89 ns for curves 2, 4, respectively. The pulse width increases with decreasing pumping rate. The delay time increases with increasing loss coefficient and decreases with increasing pumping rate. Fig. 3 was obtained using pumping rate of P13 ¼ 150027000 (s1), Kloss ¼ 0:0220:1 (cm1), not null absorption coefficient of dye cell, nonlinear factor of the dye solution s ¼ 107 ðcm3 =erg sÞ and t > 0: It can be noticed from Figs. 3a and b that the energy radiation density and the pulse duration decrease with decreasing pumping rate and with increasing loss coefficient. It also can be seen from Figs. 3a and b that with increasing loss coefficient, the delay time increases and the pulse width decreases until the loss coefficient reaches the value Kloss ¼ 0:08 cm1. When the loss coefficient reaches its maximum value (0.1 cm1), the pulse width begin to increase. The population inversion process will be delayed with decreasing pumping rate and increasing the loss coefficient (Figs. 3c and d). From Figs. 4a and b it can be seen that the dye cell becomes transparent through nearly 1 ns (the absorption coefficient of the dye cell tends to zero). Figs. 5a and b give the effect of nonlinear coefficient of the dye solution on the characteristics of laser emission, which was studied for different values; s ¼ 107 ; 105 ; 104 ðcm3 =erg sÞ; and for different values of pumping rate and loss coefficient. From Figs. 5a and b it can be noticed that the radiation density did not change the peak of giant pulse for the following values of nonlinear coefficient of the dye solution s ¼ 107 ; 105 ðcm3 =erg sÞ at limited values of pumping rate (see curves 1, 2).
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2.E+6
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30
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Y
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12 3 4 5
1 23 4
5
0.0
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(c)
30
10
0
t (ns)
(d)
5 4 3 2.0 12 1.5
3.0
3.0
2.5
2.5 K* (cm-1)
K* (cm-1)
Fig. 3. (a) The laser energy density as a function of time (P13 ¼ 7000 s1), (b) the laser energy density as a function of time (P13 ¼ 1500 s1), (c) the relative population inversion as a function of time (P13 ¼ 7000 s1) and (d) the relative population inversion as a function of time (P13 ¼ 1500 s1) (for Kloss ¼ 0:02 cm1 (1), 0.04 (2), 0.06 (3), 0.08 (4), 0.1 (5) and ) K0 ¼ ðXL=L Þ½Y0 ðKloss =X Þ:
1.0
0.5 0.0
0.5 0.0 0 (a)
5 4 3 1.5 2 1.0 1
2.0
5
10 t (ns)
0 15
20
25
(b)
5
10 t (ns)
15
20
25
Fig. 4. (a) The absorption coefficient of dye cell as a function of time (P13 ¼ 7000 s1) and (b) the absorption coefficient of dye cell as a function of time (P13 ¼ 1500 s1) (for Kloss (cm1)=0.02 (1), 0.04 (2), 0.06 (3), 0.08 (4),0.1 (5)).
When the nonlinear coefficient of the dye solution decreases, the emission of the giant pulse is delayed at the value of s ¼ 104 ðcm3 =erg sÞ until the delay time is about 50 ns. This delay is due to the decrement of the solution transparency in the dye cell,
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2
1
4.E+6
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5
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8.0E+5 4
5
4.0E+5
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10
(a)
20 30 t (ns)
40
3.6
0.0E+0
50
0
10
(b)
20 30 t (ns)
40
50
Fig. 5. (a) The laser energy density as a function of time (Kloss ¼ 0:02 cm1) and (b) the laser energy density as a function of time (Kloss ¼ 0:1 cm1) (for P13 ¼ 7000 s1 (1–3); P13 ¼ 1500 s1(4–6) and sðcm3 =erg:sÞ=107 (1, 4), 105 (2, 5), 104 (3, 6)).
Table 2 Figure # 5a 5a 5b 5b
Curve # 1 5 1 5
Pulse width (ns) 7.27 8.18 4.27 7.27
sðcm3 =erg sÞ 7
10 105 107 105
Delay time (ns) 3 17 5 26
and it requires irradiating the solution for a longer period in order to become more transparent for the laser beam. It can also be seen from Figs. 5a and b that the pulse width decreases with increasing nonlinear coefficient of the dye solution. The increment of this coefficient and the pumping rate leads to a shorter time delay, and the decrement of pumping rate leads to an increased pulse width as shown in Table 2. The relative population inversion decreases monotonously with time starting at initial inversion and ending at the final inversion (see Figs. 6a and b, curves 1, 2, 4, 5). On the other hand, the relative population inversion remains constant during the same duration for s ¼ 104 ðcm3 =erg sÞ (Figs. 6a and b, curves 3, 6). From Figs. 7a and b it can be seen that the used dye cell solution renews its properties by the end of the giant pulse especially in the case of high losses (approximately after 30–50 ns from the beginning of the giant laser pulse). The ruby laser could be operated in the periodic regime when the relative population inversion reaches the initial value Y0 by influence of pumping radiation after giant pulse emission in order to produce another pulse. In this case, the population inversion and laser emission could be repeated periodically. Eqs. (1)–(3) were solved considering the following constants P13 ¼ 7000 s1, Kloss ¼ 0:08 cm1 and K ¼ 1:77 cm1.
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10
20
(a)
30 t (ns)
40
50
0
10
20
(b)
30 t (ns)
40
50
Fig. 6. (a) The relative population density as a function of time (Kloss ¼ 0:02 cm1) and (b) the relative population density as a function of time (Kloss ¼ 0:1 cm1) (for P13 ¼ 7000 s1 (1–3); P13 ¼ 1500 s1(4–6), sðcm3 =erg:sÞ=107 (1, 4), 105 (2, 5), 104 (3, 6)).
1.6
3.0 2
2.0 1.5
6
5
1.0
1 2
0.8 1
1 0.5 4
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K* (cm-1)
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(a)
3
3
5
6 4 5
4 10
20 30 t (ns)
40
0
50 (b)
10
20 30 t (ns)
40
50
Fig. 7. (a) The absorption coefficient of dye cell as a function of time (Kloss ¼ 0:02 cm1) and (b) the absorption coefficient of dye cell as a function of time (Kloss ¼ 0:1 cm1) (for P13 ¼ 7000 s1 (1–3); P13 ¼ 1500 s1(4–6), sðcm3 =erg:sÞ=107 (1, 4), 105 (2, 5), 104 (3, 6)).
Fig. 8 shows the laser energy density as a function of time in the periodic emissions regime. From this figure it can be noticed that the period of emission reaches nearly 40 ns. The period and the density of laser emission could be affected by the losses and nonlinear solution factor. The laser energy density of the ruby laser in free running regime as a function of time (for P13 ¼ 7000 s1 and Kloss ¼ 0:02 cm1) is shown in Fig. 9. It can also be noticed that the free running regime lasts for about 90 ms. The laser energy density is damped (the emission of free running pulses is chaotic regime). This may be due to the multimode and linear characteristic emission (each line is amplified independently). It can be noticed from Fig. 10 that there is a good quantitative and qualitative agreement between the calculated results in this work and the theoretical results given in [4]. For example, the calculated FWHM of the output pulse is about 14.5 ns, while it is about 15 ns in Ref. [4].
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2.5E+6
U (erg/cm3)
2.0E+6 1.5E+6 1.0E+6 5.0E+5 0.0E+0 0
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200
300
400
500
600
700
800
t (ns) Fig. 8. The energy density as a function of time in ruby laser with dye cell during the periodic emission regime (P13 ¼ 7000 s1, Kloss ¼ 0:08 cm1 and sðcm3 =erg:sÞ=107)
U (erg/cm3)
325
300
275
250 20
40
60
80
t (µs) Fig. 9. The energy density as a function of time in ruby laser during the free-running emission regime (P13 ¼ 7000 s1, Kloss ¼ 0:02 cm1).
Fig. 11 shows a good agreement between the calculated radiation intensity as a function of time in the dye Q-switched ruby laser and the experimental result reported in [1]. The calculated results show a 5.6 ns pulse width and 0.76 J of total energy power, while the experimental results were 5.3 ns and 0.74 J, respectively. The above numerical results were obtained at a single frequency. If the active material is pumped considerably above threshold then the gain drops quickly in a short cavity transit time. The results show that a decrease in nonlinear solution coefficient causes high losses and consequently a longer period is needed to exceed these losses (coupling losses, incidental losses such as scattering, diffraction and absorption, cavity losses introduced by inserting the dye cell). This increase in the duration between laser pulses leads to higher values of population inversion and higher maximum of peak power. Therefore, the stored energy in the pulse is emitted in a shorter laser pulse. Also, it is possible to ensure operation on lowest order transverse mode by taking advantage of the higher losses suffered by higher order mode.
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Fig. 10. The energy density (1), the relative population density (2), and the absorption coefficient of dye cell (3) as a function of time in ruby laser; (a) calculated in this work and (b) calculated in [4] (for P13 ¼ 1500 s1, Kloss ¼ 0:11 cm1, sðcm3 =erg:sÞ=6000, A21 ¼ 107 s1).
Fig. 11. The radiation intensity as a function of time in the dye Q-switched ruby laser: (a) calculated results for: L ¼ 10 cm, m=0.4, Kloss ¼ 0:04; sðcm3 =erg:sÞ=107, P13 ¼ 7000 s1 and (b) experimental results (pulse width=5.3 ns, pulse energy=0.74 J) [1].
5. Conclusion In this work, a mathematical model has been adapted to predict different characteristics of a Q-switched ruby laser pulse. The adapted approach depends on using a general three-equation model given by Eqs. (1)–(3), that describes the dynamic emission in gain switched ruby laser. The adapted mathematical model allows the investigation of the effects of nonlinear coefficient of the Q-switched dye as a selective absorber for obtaining giant laser pulse, and the laser input parameters on the output laser pulse. This model can be applied to any solid-state laser such as Nd-glass, Nd-YAG with different dye materials, and it can adapted to describe the gas lasers such as CO2 laser with different saturable absorbers as well [5].
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Acknowledgements The authors would like to express their thanks to the Director General of AECS Prof. I. Othman for his continuous encouragement, guidance and support. They also thank Dr. B. Masarani and Dr. F. Awad for their revision.
References [1] [2] [3] [4] [5]
Koechner W. Solid state laser engineering. Berlin: Springer; 1988. Brackmann U. Lambdachrome laser dyes. Datasheets, 1994. Thyagarajan K, Ghatak AK. Lasers theory and applications. New York: Macmillan; 1987. Stepanov BI. Calculation methods of optical quantum generators II. Minsk: Nayka, 1968 (in Russian). Kulikov VV, Kuntsevich BF, Chigevski VN, Churakov VV. The effect of refraction index variation of investigation material on the output parameters of CO2 laser. J Appl Spectrosc 1992;57(5-6):464–71 (in Russian).