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Evolution of the spatial coherence in a copper vapor laser
Optics Communications 92 ( 1992) 50-56 North-Holland
OPTICS C O M MUN ICATIONS
Evolution of the spatial coherence in a copper vapor laser Takashige Omatsu, Tomohiro Takase ~ and Kazuo Kuroda Institute of lndustrial Science, University of Tokyo, Roppongi, Minatoku, Tokyo 106, Japan Received 28 January 1992; revised manuscript received 24 April 1992
We analyzed the pulse shape of a copper vapor laser (CVL) beam by solving the spatio-temporai rate equations, in which the propagation of the laser light in the cavity and the spatial distribution of the population inversion density along the optical axis were taken into consideration. We found that the pulse shape was determined by the fact that the gain saturates and recovers alternatively with the transit time of the laser cavity, 10 ns. We found that the dominant part of the laser pulse changes temporally in the pulse duration with the transit time of 10 ns. Furthermore, we investigated the histories of the parts of the laser pulse which were originated by the spontaneous emission radiated at different time. The temporal change of the spatial coherence can be fully explained by the change of the dominant part in the laser beam. We obtained good agreement between the calculation and the experiment.
1. Introduction The copper vapor laser (CVL) is a high average power and high repetition rate ( ~ 4 k H z ) pulse laser at 510.6 nm and 578.2 n m [1]. The second harmonics o f the CVL beam at 255.3 n m will be a high average power and high repetition rate ultraviolet light [2-5 ]. We have carried out the second harmonic generation o f the CVL using barium borate as a nonlinear crystal and have achieved a conversion efficiency o f 9% [3,4]. In the second harmonic generation, the m a x i m u m conversion efficiency depends on the spatial coherence of a light source [ 6 ]. In general the high power pulse lasers, such as the CVL, radiate partially coherent light [ 7-10 ]. The spatial coherence changes temporally during the pulse oscillation in the CVL. Moreover the pulse length becomes longer as the laser power increases. This causes the saturation o f the conversion efficiency. The dynamics o f the CVL were discussed by rate equations including the discharge circuit [ 11-13]. In the previous papers, however, only total energy had been considered, and little attention was paid to Science University of Tokyo, Kagurazaka, Shinjukuku, Tokyo, 162 Japan. 50
the laser pulse shape and the temporal change o f the spatial coherence. In this paper we report the simulation o f the temporal change o f the spatial coherence of the CVL beam during the pulse oscillation by calculating the spatio-temporal rate equations. Then we discuss the evolution o f the spatial coherence in the laser pulse.
2. Rate equations The electric discharge in the high-repetition rate pulse lasers such as the CVL is divided into two parts: the main discharge and the after-glow. In the main discharge the copper and neon atoms are excited and ionized by the electron impact because the electron temperature is high enough. As a result o f the p u m p processes, the population inversion is produced and the laser starts to oscillate. As the discharge changes to the after-glow, the electron temperature decreases. The atoms are deexcited and the ions are recombined with the electrons rapidly by the electron impact. After the electrons are cooled sufficiently, the a t o m - a t o m collisions and diffusion o f atoms become dominant and all processes such as the relaxation o f the metastable atoms, proceed at slow speed. In our simulation, only the electron-atom colli-
0030-4018/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
Volume 92, number 1,2,3
OPTICS COMMUNICATIONS
sions were considered because our interest is restricted to the optical processes. We ignored the processes in the after-glow, even though the initial conditions of the main discharge, e.g. the population of the residual electrons, electron temperature etc., are determined by the after-glow. The yellow line was neglected for simplicity. The energy level diagrams of Cu and Ne are shown in fig. 1. The rate equations can be written as follows
+ r , 2 N 2 +r13N3 +r14N4] ne,
(l)
ON2/Ot= [ - (r12 +r32 + r42)N2 +rE~N~ +r23N3 +rE4N4] n, - A N 2 - e c ( P + + P - ) (N2 -g3N3/g2) ,
(2)
-- (r,3 +r23 + r43)N3
+r31Nl +r32N2 +r34N4] n~ +AN2 + a c ( e + + e - ) (N: - g 3 N 3 / g 2 ) ,
ON41Ot=[ --
(3)
(r,4 "1-/'24 + r34)N4
+r41Nl +r42N2 +r43N3] n~,
(4)
OMt/Ot= [ - (r'21 +r~l)Mi +r'12M2 +r'13M3] n¢, (5) Cu" (N4)
I. IeV
Ne + (M.)
OM2/ Ot= [ - ( r'12 +r~2)M2 + r;~ M , +r~3M3] n,,
(6) OM3/Ot= [ - (r13 + r~3)M3 +r~,M, +r~2M2] no,
(7) (O/Ot+cO/Ox)P + =AN2~2 + + trcP + (N2 -g3N3/g2),
21. icy
P
(N,)
(OlOt)(~kT:)=-Zi ~'rijA'ijNj-j ~ ~j-r~jA(iJMJ -(~r4,N,-t-~ra,M~)(~kT,,
I. 411V
S <,,,>
OeV
2m:ri~ 3
2mer~ mcu + ~i--~-N~) [ ~ k ( T e - T g ) ]
!i. IcY
+ W ( t ) l (ne V ) ,
3. h V
D (Na)
(9)
where N,, N2, N3 and N4 are the densities of S, P, D and ionized states of the copper respectively, MI, M2 and M3 are the densities of the ground, metastable and ionized states of the neon respectively, ne is the electron density, and P+ and P - are the photon densities, which propagate through the laser tube to the right and left directions. These variables are functions of both time and space. The coefficients, r~j and r;j are the electron impact excitation rates from the jth level to the ith level of the copper and neon, I2+, /2- are the effective solid angles, A is the Einstein coefficient and a is the cross-section of the stimulated emission. The equation for the electron temperature can be written as
_(~ N ee (M,)
(8)
- (O/Ot- cO/Ox)P=AN212- + acP- (N2 -g3N3/g2 ),
ON, lOt= [ - (r2~ +r3t +r41 )N~
ON3/Ot= [
15 August 1992
,.,
Fig. 1. The energy diagram. The arrow shows the laser transition.
(10)
where W(t) is the input power into the laser medium due to the pulse discharge. The time scale on which the variables change is comparable with the one where the photons propagate through the laser cavity. Therefore we have to take account of the effects of the propagation of the photons and the distribution of the amplification gain. For a numerical computation, we divided the laser cavity, which was composed of two mirrors R1 and R2, into many unit cells. The time interval (At) in the numerical integration of the rate equations was selected to satisfy Ax=cAt, where Ax is the length of the cell. Therefore, at the next step, photons in a cell reach the nearest neighboring cells. The material 51
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OPTICS COMMUNICATIONS
equations arc calculated in each cell. The origin of the x axis is set at the most leftccU in the resonator. If the rcflcctivityof the mirror R~ is r~, the output power I(t) is given by ( l -r~ ) P- (t- At, 0).
15 August 1992
Vc =1 lkV 2 O v-
3. Results C
The parameters used in our calculations are as follows. The output coupling loss for the confocal unstable resonator is approximately l - ( l / M ) 2, where M is a magnification factor. Thus the reflectivities of R~ and R2 are (1/60) 2 and l respectively at the magnification factor 60. The length of the active medium is 72 cm. The input power W(t) was given by the product of the discharge voltage V(t) and current I(t) measured in our laser. The gas temperature depends on the charging voltage. In our experiment, the gas temperatures are 1350°C and 1450°C at the charging voltages of I 1 kV and 13 kV. For the impact cross sections of the S-D and the S-P transitions of copper, we used the results by Trajmar [ 14 ]. For the collisional mixing reaction rate of the D - P transition, we used the equation by Deutsch [ 15 ]. We carried out the calculations using various numbers of cells and found that sufficiently good results could be obtained by dividing the resonator into 100 cells. We calculated the rate equations under these conditions. The results are shown in figs. 2 and 3.
3. I. Pulse shape After about 100 ns from the start of the discharge, the electron temperature reaches the maximum value of 5.5 eV uniformly. Then the laser starts oscillating. When the discharge voltage is 11 kV, the laser pulse has a single peak. The population inversion saturates and scarcely recovers over the whole gain medium. At 13 kV, however, the laser pulse has three peaks. In this case the population inversion becomes spatially nonuniform after is saturates. The population inversion saturates strongly and does not recover at the R~ side of the active medium. On the other hand, near R2, the population inversion saturates weakly and recovers enough in the interval of 20 ns, the transit time of the photon in the cavity. The pulsation of the output power is caused by the saturation 52
g S ov.
m t" °2
2
"5 CL 0 a. C 15
s
1C
I--
100
200
Tme (ha) Fig. 2. The calculation result at the charging voltage of 11 kV. The number in the graph of the population inversion shows the position in the cavity.No. 1 and 4 show the population inversion at the side of Rt and R2 respectively.No. 2 and 3 show the population inversion at the position between R~ and R2.
and recovery of the population inversion. The output coupling loss is very large for the unstableresonator and a small portion of the laserpower is reflected at the output coupler. The photon density is low near the total reflectorbecause the laser power is not amplified sufficientlyin the propagation from the output coupler to the total reflector. O n the other hand, the photon density is very high near the output coupler because the laser power is amplified sufficientlyin the return pass from the total reflectorto the output couplcr. Because of the dif-
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OPTICS COMMUNICATIONS
lo
Vc = 13kV
15 August 1992
~ 10-
==
1 1.5kV v
(O.8W)
x
"~ ~ i-G)
0 0..o l C
o
_L 16o
Time (ns)
~... I(
-=E 12kV (1.5W) ~
g
$
I
0 ¢t
16o
Time (ns)
12.5kV ~: (2W)
,.-
$ 0 ft.
/L 16o
Time (ns) Fig. 4. The experimental results of the laser pulse. '
Time (ns) Fig. 3. The calculation result at the charging voltage of 13 kV.
ference of the laser power, the gain saturates strongly and is hard to recover at the output coupler side, but a weak saturation and quick recovery occur at the total reflector side. This results in a spatial nonuniformity of the population inversion in the active medium. We measured the pulse shapes for our CVL, as shown in fig. 4. At the low discharge voltage the pulse has a single peak. At the high discharge voltage it has two peaks with an interval of about 20 ns. These experimental results agree with our simulation.
3.2. History of laser pulse The spatial coherence depends on how long the photons stay in the laser cavity. The laser pulse is
originated by spontaneous emission and is amplified very strongly while it propagates many times through the cavity. In order to discuss the spatial coherence we must know how many times the emission generated propagates through the laser cavity before it goes out of the cavity. Namely the laser beam must be classified by its history. We add two equations to the rate equations.
(O/Ot+cO/Ox) Q+ (to) =AN2~++~cQ+(to)(N:-g3NJg:),
(ll)
- (O/Ot- cO/Ox) Q - (to)
=AN2~2-+acQ-(to)(N2-g3N3/g2),
(12)
if t to conforms Q + and Q - , we can know the history H(to, t) of the spontaneous emission, which is gen53
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OPTICS COMMUNICATIONS
erated from t = to-At to t = to, by the equation
n(to, t) = [1 - ( I / M ) 2] X [Q+ (to, t, 0) - Q - (to, t, 0) ] .
(13)
Therefore if we calculate eq. ( 11 ) and eq. (12) by varying to, the history of the spontaneous emission generated at time to can be known by eq. ( 13 ). An example of the calculations at the charging voltage of 13 kV is shown in fig. 5. This is the history of the spontaneous emission generated from 81 ns to 87 ns. The spontaneous emission, which propagates in the left direction, is reflected at the output coupler and goes out of the resonator. It forms the first part of the output pulse but the intensity is very low because the amplification is not enough. The spontaneous emission, which propagates in the right direction, is reflected at the concave mirror, goes back into the laser tube and is amplified again. Its outer part goes out of the resonator through the coupler and forms the second part of the output pulse. Since these parts of the output pulse do not propagate via the convex mirror, we call them amplified spontaneous emission (ASE). The inner part of the ASE, which propagates in the left direction, goes through the central hole of the output coupler, is reflected at the convex mirror and goes back to the laser tube. Then it is reflected at the concave mirror, returns to the laser tube again and its outer part finally goes out of the resonator. It be-
15 August 1992
comes the third part of the output pulse power. Similarly the emission, which goes back and forth in the resonator, forms the fourth and later parts of the output pulse successively in every transit time, about 10 ns until the laser amplification terminates. We call them the laser parts. The pulse train in the broken line shown in fig. 5 corresponds with the second, third, fourth, fifth, sixth and seventh parts produced by the spontaneous emission generating from 81 ns to 87ns, respectively. The power of the first part is too small to be seen. Calculating the rate equations by varying to, we can divide the total pulse into the third, fourth, fifth, sixth and seventh part. The result is shown in fig. 6 (a: at l l k V , b: at 13kV). We can find that the energy of odd parts is smaller than that of even parts. Both of the third and fourth parts are reflected once by the output coupler but the third part travels in the active medium shorter than the fourth part. Therefore the energy of the third part is smaller than that of the fourth part. Similarly the energy of the fifth part is smaller than that of the sixth part. At 11 kV, the fifth and the sixth parts start oscillating as soon as the fourth part becomes weaker. On the other hand, at 13 kV, the fifth and the sixth parts are suppressed during the oscillation of the fourth part. After the fourth part stops oscillating, the fifth and the sixth parts start oscillating. Namely the fifth and the sixth parts start oscillating at 13 kV later than at 11 kV. This is caused by the saturation of the population inversion.
M 60
3.3. Temporal change of spatial coherence Vc=13kV
t,=81ns
zxt~6na
I= I l
/,
t,o
9~
,
c
o
0 a.
r '
'
'
100
t
i
200
Time (ns)
Fig. 5. The history of the spontaneous emission generated from 81 ns to 87 ns after the start of the discharge. 54
The spatial coherence of the CVL changes temporally during the pulse duration. This is because the dominant part of the laser pulse changes temporally. The third, fourth, fifth, sixth and seventh parts start to oscillate successively in the pulse duration. Each part of the laser pulse is incoherent with the others. On the assumption that the intensity of the light source is uniform, we can calculate the coherence function as
~(XI,X2)= ~ ~(XI,X2)~, j=3
(13)
Volume 92, number 1,2,3
OPTICS COMMUNICATIONS
15 August 1992
Discharge Voltage 11kV
!
2.5 A
7
E O
o x
¢O ¢-
0.
(a)
3.._.,'
J J
203
Time (ns)
Discharge
Voltage
13kV
10
'E O
¢q
o x
tO O ra.
(b) 4b
i
120
O
Time (ns) CVL
........ 3 p a s s
---- 4pass
---5pass
.... 6pass
.... 7pass
Fig. 6. The history of the laser pulse. The total pulse is divided into the 3-7 pass parts. (a): at 11 kV, (b): at 13 kV.
where/j is the intensity of the jth part, ~j(Xl, X2) is the degree of the coherence of theflh part and x i and x2 are the coordinates on the observation plane. By substituting the intensity of the parts shown in fig. 6 into eq. (13), we can calculate the temporal change of the spatial coherence in the pulse duration. We assume that the coherence function is given by a sine function,
~(Xl, X2) ----sinc(~(Xl = sinc(~Ax/Cj),
--X2)/Cj) (14)
sinc(x) = s i n ( x ) / x , where ~x is the distance between xl and x2 and Cj is the coherence width of thejth pan. We defined the coherence width C by the distance at which y(xl, x2) 55
Volume 92, number 1,2,3
E .¢
2C
(9 c° (9
10
OPTICS COMMUNICATIONS
(a)
'it
¢0
o
,
\ \
0
r
'
0
100
200
Time (ns) E
v e-
¢9
o
tO
o
(b)
2C
r--
IC
J'v", 0 0
100
i
i
?'
b 200
Time (ns) Fig. 7. The evolution of the spatial coherence. (a): at 11 kV, (b): at 13 kV.
becomes zero. Thus we calculated the temporal change of the coherence width. In our CVL the diameter of the laser tube is 20 mm and the coherence widths C3, C4 and C5~ C7 are 5 mm, 6 mm and 20 mm, respectively [ 10,16 ]. The result is shown in fig. 7 (a: at 11 kV, b: at 13kV). We found that the coherence width changed temporally like a three-stage step function. The calculation agrees with the experiment very well [ 16 ]. It should be noticed that the coherence width rises about 5 ns later at 13 kV than at 11 kV. This is caused by the fact that the dominant part changes from the fourth part to the fifth and sixth parts, and that this change is delayed about 5 ns at 13 kV compared with the lower voltage discharge. We believe that the saturation of the second harmonic generation efficiency is caused by the delay of the intra-pulse rise of the coherence at higher laser power [3,4].
4. Conclusion We simulated the temporal change of the pulse 56
15 August 1992
shape of a copper vapor laser (CVL) beam with an unstable resonator by solving the spatio-temporal rate equations. The laser cavity was divided into multiple cells. We found that the pulse shape was caused by the gain saturation and recovery in transit time of 20 ns in which the laser pulse went and returned in the laser cavity. The gain saturation and recovery occurred spatially nonuniform. We found that the dominant part of the laser pulse changed temporally during the pulse duration in transit time of 10 ns. This caused the temporal change of the spatial coherence. The agreement is satisfactory between the calculation and the experiment. Moreover the spatial coherence rises later at a high discharge voltage than at a low voltage. This is because the saturation of the population inversion affects the timing at which the highly-coherent part becomes dominant.
References [ 1 ] W.T. Walter, N. Solimene, M. Piltch and G. Gould, IEEE J. Quantum Electron. QE-2 (1966) 474. [ 2 ] A.A. Isaev, G.Yu. Lemmerman and G.L. Malafeeva, Sov. J. Quantum Electron. 10 (1980) 983. [3]K. Kuroda, T. Omatsu, T. Shimura, M. Chihara and I. Ogura, Prec. SPIE 1041 (1989) 60. [4] K. Kuroda, T. Omatsu, T. Shimura, M. Chihara and I. Ogura, Optics Comm. 75 (1990) 42. [5]D.W. Coutts, M.D. Ainsworth and J. Piper, IEEE J. Quantum Electron. QE-25 (1989) 1985. [6] T. Omatsu, K. Kuroda, T. Shimura, M. Chihara, M. Itoh and I. Ogura, Optics Comm. 79 (1990) 129. [7] R.S. Hargrove, R. Grove and T. Kan, IEEE J. Quantum Electron. QE-15 (1979) 1223. [ 8 ] N.A. Lyabin, Sov. J. Quantum Electron. 19 ( 1989 ) 426. [9] M. Amit, S. Lavi, G. Erez and E. Miron, Optics Comm. 62 (1987) 110. [10] T. Omatsu, K. Kuroda, T. Shimura, M. Chihara, M. Itoh and I. Ogura, Opt. Quantum Electron. 23 ( 1991 ) $477. [ 11 ] M.J. Kushner, IEEE J. Quantum Electron. QE-17 ( 1981 ) 1555. [ 12] M.J. Kushncr, J. Appl. Phys. 54 (1983) 2970. [ 13] A.E. Siegraan, Prec. IEEE 53 (1965) 277. [ 14] S. Trajmar, W. Williams and S.K. Srivastava, J. Phys. B 10 (1977) 3233. [ 15] C. Deutsch, J. Appl. Phys. 44 (1973) 1142. [ 16] T. Omatsu, T. Takase and K. Kuroda, Optics Comm. 87 (1992) 278.
OPTICS C O M MUN ICATIONS
Evolution of the spatial coherence in a copper vapor laser Takashige Omatsu, Tomohiro Takase ~ and Kazuo Kuroda Institute of lndustrial Science, University of Tokyo, Roppongi, Minatoku, Tokyo 106, Japan Received 28 January 1992; revised manuscript received 24 April 1992
We analyzed the pulse shape of a copper vapor laser (CVL) beam by solving the spatio-temporai rate equations, in which the propagation of the laser light in the cavity and the spatial distribution of the population inversion density along the optical axis were taken into consideration. We found that the pulse shape was determined by the fact that the gain saturates and recovers alternatively with the transit time of the laser cavity, 10 ns. We found that the dominant part of the laser pulse changes temporally in the pulse duration with the transit time of 10 ns. Furthermore, we investigated the histories of the parts of the laser pulse which were originated by the spontaneous emission radiated at different time. The temporal change of the spatial coherence can be fully explained by the change of the dominant part in the laser beam. We obtained good agreement between the calculation and the experiment.
1. Introduction The copper vapor laser (CVL) is a high average power and high repetition rate ( ~ 4 k H z ) pulse laser at 510.6 nm and 578.2 n m [1]. The second harmonics o f the CVL beam at 255.3 n m will be a high average power and high repetition rate ultraviolet light [2-5 ]. We have carried out the second harmonic generation o f the CVL using barium borate as a nonlinear crystal and have achieved a conversion efficiency o f 9% [3,4]. In the second harmonic generation, the m a x i m u m conversion efficiency depends on the spatial coherence of a light source [ 6 ]. In general the high power pulse lasers, such as the CVL, radiate partially coherent light [ 7-10 ]. The spatial coherence changes temporally during the pulse oscillation in the CVL. Moreover the pulse length becomes longer as the laser power increases. This causes the saturation o f the conversion efficiency. The dynamics o f the CVL were discussed by rate equations including the discharge circuit [ 11-13]. In the previous papers, however, only total energy had been considered, and little attention was paid to Science University of Tokyo, Kagurazaka, Shinjukuku, Tokyo, 162 Japan. 50
the laser pulse shape and the temporal change o f the spatial coherence. In this paper we report the simulation o f the temporal change o f the spatial coherence of the CVL beam during the pulse oscillation by calculating the spatio-temporal rate equations. Then we discuss the evolution o f the spatial coherence in the laser pulse.
2. Rate equations The electric discharge in the high-repetition rate pulse lasers such as the CVL is divided into two parts: the main discharge and the after-glow. In the main discharge the copper and neon atoms are excited and ionized by the electron impact because the electron temperature is high enough. As a result o f the p u m p processes, the population inversion is produced and the laser starts to oscillate. As the discharge changes to the after-glow, the electron temperature decreases. The atoms are deexcited and the ions are recombined with the electrons rapidly by the electron impact. After the electrons are cooled sufficiently, the a t o m - a t o m collisions and diffusion o f atoms become dominant and all processes such as the relaxation o f the metastable atoms, proceed at slow speed. In our simulation, only the electron-atom colli-
0030-4018/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
Volume 92, number 1,2,3
OPTICS COMMUNICATIONS
sions were considered because our interest is restricted to the optical processes. We ignored the processes in the after-glow, even though the initial conditions of the main discharge, e.g. the population of the residual electrons, electron temperature etc., are determined by the after-glow. The yellow line was neglected for simplicity. The energy level diagrams of Cu and Ne are shown in fig. 1. The rate equations can be written as follows
+ r , 2 N 2 +r13N3 +r14N4] ne,
(l)
ON2/Ot= [ - (r12 +r32 + r42)N2 +rE~N~ +r23N3 +rE4N4] n, - A N 2 - e c ( P + + P - ) (N2 -g3N3/g2) ,
(2)
-- (r,3 +r23 + r43)N3
+r31Nl +r32N2 +r34N4] n~ +AN2 + a c ( e + + e - ) (N: - g 3 N 3 / g 2 ) ,
ON41Ot=[ --
(3)
(r,4 "1-/'24 + r34)N4
+r41Nl +r42N2 +r43N3] n~,
(4)
OMt/Ot= [ - (r'21 +r~l)Mi +r'12M2 +r'13M3] n¢, (5) Cu" (N4)
I. IeV
Ne + (M.)
OM2/ Ot= [ - ( r'12 +r~2)M2 + r;~ M , +r~3M3] n,,
(6) OM3/Ot= [ - (r13 + r~3)M3 +r~,M, +r~2M2] no,
(7) (O/Ot+cO/Ox)P + =AN2~2 + + trcP + (N2 -g3N3/g2),
21. icy
P
(N,)
(OlOt)(~kT:)=-Zi ~'rijA'ijNj-j ~ ~j-r~jA(iJMJ -(~r4,N,-t-~ra,M~)(~kT,,
I. 411V
S <,,,>
OeV
2m:ri~ 3
2mer~ mcu + ~i--~-N~) [ ~ k ( T e - T g ) ]
!i. IcY
+ W ( t ) l (ne V ) ,
3. h V
D (Na)
(9)
where N,, N2, N3 and N4 are the densities of S, P, D and ionized states of the copper respectively, MI, M2 and M3 are the densities of the ground, metastable and ionized states of the neon respectively, ne is the electron density, and P+ and P - are the photon densities, which propagate through the laser tube to the right and left directions. These variables are functions of both time and space. The coefficients, r~j and r;j are the electron impact excitation rates from the jth level to the ith level of the copper and neon, I2+, /2- are the effective solid angles, A is the Einstein coefficient and a is the cross-section of the stimulated emission. The equation for the electron temperature can be written as
_(~ N ee (M,)
(8)
- (O/Ot- cO/Ox)P=AN212- + acP- (N2 -g3N3/g2 ),
ON, lOt= [ - (r2~ +r3t +r41 )N~
ON3/Ot= [
15 August 1992
,.,
Fig. 1. The energy diagram. The arrow shows the laser transition.
(10)
where W(t) is the input power into the laser medium due to the pulse discharge. The time scale on which the variables change is comparable with the one where the photons propagate through the laser cavity. Therefore we have to take account of the effects of the propagation of the photons and the distribution of the amplification gain. For a numerical computation, we divided the laser cavity, which was composed of two mirrors R1 and R2, into many unit cells. The time interval (At) in the numerical integration of the rate equations was selected to satisfy Ax=cAt, where Ax is the length of the cell. Therefore, at the next step, photons in a cell reach the nearest neighboring cells. The material 51
Volume 92, number 1,2,3
OPTICS COMMUNICATIONS
equations arc calculated in each cell. The origin of the x axis is set at the most leftccU in the resonator. If the rcflcctivityof the mirror R~ is r~, the output power I(t) is given by ( l -r~ ) P- (t- At, 0).
15 August 1992
Vc =1 lkV 2 O v-
3. Results C
The parameters used in our calculations are as follows. The output coupling loss for the confocal unstable resonator is approximately l - ( l / M ) 2, where M is a magnification factor. Thus the reflectivities of R~ and R2 are (1/60) 2 and l respectively at the magnification factor 60. The length of the active medium is 72 cm. The input power W(t) was given by the product of the discharge voltage V(t) and current I(t) measured in our laser. The gas temperature depends on the charging voltage. In our experiment, the gas temperatures are 1350°C and 1450°C at the charging voltages of I 1 kV and 13 kV. For the impact cross sections of the S-D and the S-P transitions of copper, we used the results by Trajmar [ 14 ]. For the collisional mixing reaction rate of the D - P transition, we used the equation by Deutsch [ 15 ]. We carried out the calculations using various numbers of cells and found that sufficiently good results could be obtained by dividing the resonator into 100 cells. We calculated the rate equations under these conditions. The results are shown in figs. 2 and 3.
3. I. Pulse shape After about 100 ns from the start of the discharge, the electron temperature reaches the maximum value of 5.5 eV uniformly. Then the laser starts oscillating. When the discharge voltage is 11 kV, the laser pulse has a single peak. The population inversion saturates and scarcely recovers over the whole gain medium. At 13 kV, however, the laser pulse has three peaks. In this case the population inversion becomes spatially nonuniform after is saturates. The population inversion saturates strongly and does not recover at the R~ side of the active medium. On the other hand, near R2, the population inversion saturates weakly and recovers enough in the interval of 20 ns, the transit time of the photon in the cavity. The pulsation of the output power is caused by the saturation 52
g S ov.
m t" °2
2
"5 CL 0 a. C 15
s
1C
I--
100
200
Tme (ha) Fig. 2. The calculation result at the charging voltage of 11 kV. The number in the graph of the population inversion shows the position in the cavity.No. 1 and 4 show the population inversion at the side of Rt and R2 respectively.No. 2 and 3 show the population inversion at the position between R~ and R2.
and recovery of the population inversion. The output coupling loss is very large for the unstableresonator and a small portion of the laserpower is reflected at the output coupler. The photon density is low near the total reflectorbecause the laser power is not amplified sufficientlyin the propagation from the output coupler to the total reflector. O n the other hand, the photon density is very high near the output coupler because the laser power is amplified sufficientlyin the return pass from the total reflectorto the output couplcr. Because of the dif-
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OPTICS COMMUNICATIONS
lo
Vc = 13kV
15 August 1992
~ 10-
==
1 1.5kV v
(O.8W)
x
"~ ~ i-G)
0 0..o l C
o
_L 16o
Time (ns)
~... I(
-=E 12kV (1.5W) ~
g
$
I
0 ¢t
16o
Time (ns)
12.5kV ~: (2W)
,.-
$ 0 ft.
/L 16o
Time (ns) Fig. 4. The experimental results of the laser pulse. '
Time (ns) Fig. 3. The calculation result at the charging voltage of 13 kV.
ference of the laser power, the gain saturates strongly and is hard to recover at the output coupler side, but a weak saturation and quick recovery occur at the total reflector side. This results in a spatial nonuniformity of the population inversion in the active medium. We measured the pulse shapes for our CVL, as shown in fig. 4. At the low discharge voltage the pulse has a single peak. At the high discharge voltage it has two peaks with an interval of about 20 ns. These experimental results agree with our simulation.
3.2. History of laser pulse The spatial coherence depends on how long the photons stay in the laser cavity. The laser pulse is
originated by spontaneous emission and is amplified very strongly while it propagates many times through the cavity. In order to discuss the spatial coherence we must know how many times the emission generated propagates through the laser cavity before it goes out of the cavity. Namely the laser beam must be classified by its history. We add two equations to the rate equations.
(O/Ot+cO/Ox) Q+ (to) =AN2~++~cQ+(to)(N:-g3NJg:),
(ll)
- (O/Ot- cO/Ox) Q - (to)
=AN2~2-+acQ-(to)(N2-g3N3/g2),
(12)
if t
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OPTICS COMMUNICATIONS
erated from t = to-At to t = to, by the equation
n(to, t) = [1 - ( I / M ) 2] X [Q+ (to, t, 0) - Q - (to, t, 0) ] .
(13)
Therefore if we calculate eq. ( 11 ) and eq. (12) by varying to, the history of the spontaneous emission generated at time to can be known by eq. ( 13 ). An example of the calculations at the charging voltage of 13 kV is shown in fig. 5. This is the history of the spontaneous emission generated from 81 ns to 87 ns. The spontaneous emission, which propagates in the left direction, is reflected at the output coupler and goes out of the resonator. It forms the first part of the output pulse but the intensity is very low because the amplification is not enough. The spontaneous emission, which propagates in the right direction, is reflected at the concave mirror, goes back into the laser tube and is amplified again. Its outer part goes out of the resonator through the coupler and forms the second part of the output pulse. Since these parts of the output pulse do not propagate via the convex mirror, we call them amplified spontaneous emission (ASE). The inner part of the ASE, which propagates in the left direction, goes through the central hole of the output coupler, is reflected at the convex mirror and goes back to the laser tube. Then it is reflected at the concave mirror, returns to the laser tube again and its outer part finally goes out of the resonator. It be-
15 August 1992
comes the third part of the output pulse power. Similarly the emission, which goes back and forth in the resonator, forms the fourth and later parts of the output pulse successively in every transit time, about 10 ns until the laser amplification terminates. We call them the laser parts. The pulse train in the broken line shown in fig. 5 corresponds with the second, third, fourth, fifth, sixth and seventh parts produced by the spontaneous emission generating from 81 ns to 87ns, respectively. The power of the first part is too small to be seen. Calculating the rate equations by varying to, we can divide the total pulse into the third, fourth, fifth, sixth and seventh part. The result is shown in fig. 6 (a: at l l k V , b: at 13kV). We can find that the energy of odd parts is smaller than that of even parts. Both of the third and fourth parts are reflected once by the output coupler but the third part travels in the active medium shorter than the fourth part. Therefore the energy of the third part is smaller than that of the fourth part. Similarly the energy of the fifth part is smaller than that of the sixth part. At 11 kV, the fifth and the sixth parts start oscillating as soon as the fourth part becomes weaker. On the other hand, at 13 kV, the fifth and the sixth parts are suppressed during the oscillation of the fourth part. After the fourth part stops oscillating, the fifth and the sixth parts start oscillating. Namely the fifth and the sixth parts start oscillating at 13 kV later than at 11 kV. This is caused by the saturation of the population inversion.
M 60
3.3. Temporal change of spatial coherence Vc=13kV
t,=81ns
zxt~6na
I= I l
/,
t,o
9~
,
c
o
0 a.
r '
'
'
100
t
i
200
Time (ns)
Fig. 5. The history of the spontaneous emission generated from 81 ns to 87 ns after the start of the discharge. 54
The spatial coherence of the CVL changes temporally during the pulse duration. This is because the dominant part of the laser pulse changes temporally. The third, fourth, fifth, sixth and seventh parts start to oscillate successively in the pulse duration. Each part of the laser pulse is incoherent with the others. On the assumption that the intensity of the light source is uniform, we can calculate the coherence function as
~(XI,X2)= ~ ~(XI,X2)~, j=3
(13)
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15 August 1992
Discharge Voltage 11kV
!
2.5 A
7
E O
o x
¢O ¢-
0.
(a)
3.._.,'
J J
203
Time (ns)
Discharge
Voltage
13kV
10
'E O
¢q
o x
tO O ra.
(b) 4b
i
120
O
Time (ns) CVL
........ 3 p a s s
---- 4pass
---5pass
.... 6pass
.... 7pass
Fig. 6. The history of the laser pulse. The total pulse is divided into the 3-7 pass parts. (a): at 11 kV, (b): at 13 kV.
where/j is the intensity of the jth part, ~j(Xl, X2) is the degree of the coherence of theflh part and x i and x2 are the coordinates on the observation plane. By substituting the intensity of the parts shown in fig. 6 into eq. (13), we can calculate the temporal change of the spatial coherence in the pulse duration. We assume that the coherence function is given by a sine function,
~(Xl, X2) ----sinc(~(Xl = sinc(~Ax/Cj),
--X2)/Cj) (14)
sinc(x) = s i n ( x ) / x , where ~x is the distance between xl and x2 and Cj is the coherence width of thejth pan. We defined the coherence width C by the distance at which y(xl, x2) 55
Volume 92, number 1,2,3
E .¢
2C
(9 c° (9
10
OPTICS COMMUNICATIONS
(a)
'it
¢0
o
,
\ \
0
r
'
0
100
200
Time (ns) E
v e-
¢9
o
tO
o
(b)
2C
r--
IC
J'v", 0 0
100
i
i
?'
b 200
Time (ns) Fig. 7. The evolution of the spatial coherence. (a): at 11 kV, (b): at 13 kV.
becomes zero. Thus we calculated the temporal change of the coherence width. In our CVL the diameter of the laser tube is 20 mm and the coherence widths C3, C4 and C5~ C7 are 5 mm, 6 mm and 20 mm, respectively [ 10,16 ]. The result is shown in fig. 7 (a: at 11 kV, b: at 13kV). We found that the coherence width changed temporally like a three-stage step function. The calculation agrees with the experiment very well [ 16 ]. It should be noticed that the coherence width rises about 5 ns later at 13 kV than at 11 kV. This is caused by the fact that the dominant part changes from the fourth part to the fifth and sixth parts, and that this change is delayed about 5 ns at 13 kV compared with the lower voltage discharge. We believe that the saturation of the second harmonic generation efficiency is caused by the delay of the intra-pulse rise of the coherence at higher laser power [3,4].
4. Conclusion We simulated the temporal change of the pulse 56
15 August 1992
shape of a copper vapor laser (CVL) beam with an unstable resonator by solving the spatio-temporal rate equations. The laser cavity was divided into multiple cells. We found that the pulse shape was caused by the gain saturation and recovery in transit time of 20 ns in which the laser pulse went and returned in the laser cavity. The gain saturation and recovery occurred spatially nonuniform. We found that the dominant part of the laser pulse changed temporally during the pulse duration in transit time of 10 ns. This caused the temporal change of the spatial coherence. The agreement is satisfactory between the calculation and the experiment. Moreover the spatial coherence rises later at a high discharge voltage than at a low voltage. This is because the saturation of the population inversion affects the timing at which the highly-coherent part becomes dominant.
References [ 1 ] W.T. Walter, N. Solimene, M. Piltch and G. Gould, IEEE J. Quantum Electron. QE-2 (1966) 474. [ 2 ] A.A. Isaev, G.Yu. Lemmerman and G.L. Malafeeva, Sov. J. Quantum Electron. 10 (1980) 983. [3]K. Kuroda, T. Omatsu, T. Shimura, M. Chihara and I. Ogura, Prec. SPIE 1041 (1989) 60. [4] K. Kuroda, T. Omatsu, T. Shimura, M. Chihara and I. Ogura, Optics Comm. 75 (1990) 42. [5]D.W. Coutts, M.D. Ainsworth and J. Piper, IEEE J. Quantum Electron. QE-25 (1989) 1985. [6] T. Omatsu, K. Kuroda, T. Shimura, M. Chihara, M. Itoh and I. Ogura, Optics Comm. 79 (1990) 129. [7] R.S. Hargrove, R. Grove and T. Kan, IEEE J. Quantum Electron. QE-15 (1979) 1223. [ 8 ] N.A. Lyabin, Sov. J. Quantum Electron. 19 ( 1989 ) 426. [9] M. Amit, S. Lavi, G. Erez and E. Miron, Optics Comm. 62 (1987) 110. [10] T. Omatsu, K. Kuroda, T. Shimura, M. Chihara, M. Itoh and I. Ogura, Opt. Quantum Electron. 23 ( 1991 ) $477. [ 11 ] M.J. Kushner, IEEE J. Quantum Electron. QE-17 ( 1981 ) 1555. [ 12] M.J. Kushncr, J. Appl. Phys. 54 (1983) 2970. [ 13] A.E. Siegraan, Prec. IEEE 53 (1965) 277. [ 14] S. Trajmar, W. Williams and S.K. Srivastava, J. Phys. B 10 (1977) 3233. [ 15] C. Deutsch, J. Appl. Phys. 44 (1973) 1142. [ 16] T. Omatsu, T. Takase and K. Kuroda, Optics Comm. 87 (1992) 278.