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Effect of focusing geometry on the continuous optical discharge properties
Physics Letters A 373 (2009) 3336–3341
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Effect of focusing geometry on the continuous optical discharge properties Ismail Rafatov ∗ Department of Physics, Middle East Technical University, 06531 Ankara, Turkey
a r t i c l e
i n f o
Article history: Received 10 May 2009 Received in revised form 15 June 2009 Accepted 16 July 2009 Available online 21 July 2009 Communicated by F. Porcelli PACS: 52.65.-y 52.80.-s 52.38.-r 52.50.Jm 02.60.Cb
a b s t r a c t We studied the effect of the laser beam focusing geometry on the continuous optical discharge (COD) properties. We used a full two-dimensional radiative gas-dynamic model for the COD, maintained by a vertical CO2 laser beam in free air atmosphere, in the Earth’s gravitational field. The model takes into account all of the factors that are of importance in laser-sustained plasma processes, and uses realistic quasi optics to describe the laser radiation propagation. Results are presented for the optical discharge parameters as functions of applied laser power and degree (f-number) to which the laser beam is focused. © 2009 Elsevier B.V. All rights reserved.
Keywords: Radiative gas dynamics Numerical modeling Continuous optical discharge Laser plasma
1. Introduction In a continuous optical discharge (COD) in air, a low-temperature laser plasma is maintained at a temperature of about 16 to 22 kK by irradiation of a continuous laser power, usually from a CO2 laser. After the production of a low-temperature plasma by a certain method, e.g., by optical or electrical breakdown, in a gas, in free space, or near a surface, the plasma is maintained in the steady state or propagates towards the incident laser by the irradiation of this plasma by laser radiation of a sub-breakdown intensity. This effect was investigated extensively both experimentally [1–6] and theoretically [7–24]. It was found that the shape of the plasma and its position with respect to the focal plane of the laser beam depends on several factors, such as the degree to which the beam is focused, the output power of the laser, and external gas flow conditions. The increase of the laser power and the f-number (ratio of the focus length F of the lens to the beam diameter D) of the optical system [10,11,21,24] results in a shift of the plasma along the laser beam toward the laser. An increase of the convection velocity of the flow propagating co-directionally with the laser radiation, has the opposite effect [6,11,24]. The plasma stabilizes
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in the place where the energy source balances the energy losses due to radiation, thermal conductivity, and convection. If discharge propagates to a point in a beam where the intensity is no longer sufficient to maintain it, it vanishes. In a weakly converging laser beam (large f-number) a stable plasma can only be maintained in a forced convective flow [6,11,12,24]. Convective flow incident to the plasma behaves like a flow over an obstacle (see, e.g. [11,22,24]): it flows around the hot discharge region and only a small fraction of the cold gas enters the discharge. In this Letter, we studied the effect of focusing geometry (f-number) on the COD properties. It is known that for shorter focal lengths, the effect of refraction on the plasma shape is negligible [10,12], however, for larger focal length geometries, the laser refraction process can be of great importance [5,12,21]. Also, most of the existing COD models use geometric optics to trace the laser beam through the optical system, and therefore, are unable to take refraction of the beam in the nonuniform plasma properly into account. Here we applied a full two-dimensional radiative gasdynamic numerical model for the COD, which uses realistic quasi optics and takes refraction of the laser radiation in the plasma properly into consideration in the description of the laser beam propagation. Consequently, the major improvement of the mathematical model in this work, compared with the existing models, is application of a quasi-optical approximation, namely the beam propagation equation in parabolic approximation, to model the
I. Rafatov / Physics Letters A 373 (2009) 3336–3341
3337
laser radiation propagation. We applied the numerical model and corresponding numerical code (realized in fortran 90) of the previous work [24] for COD, free of the external gas flow, and maintained by a vertical CO2 laser beam, in the Earth’s gravitational field. The Letter is organized as follows. Section 2 describes the model of COD. Section 3 presents the results of the calculation of COD characteristics. Finally, Section 4 summarizes the results. 2. Model We consider a steady-state COD sustained by a weakly focused 10.6-μm CO2 laser beam in free space at atmospheric pressure. The beam is focused vertically from below, the discharge is free of the external gas flow, and the free-fall acceleration is acting vertically downward (see Fig. 1). We assume that the gas flow is subsonic and laminar. Moreover, we assume that the discharge plasma is in local thermodynamic equilibrium (LTE). Therefore, the plasma can be described by a single temperature, and its physical properties are only a function of this temperature and pressure. Finally, assuming that all the processes are axially symmetric, we use cylindrical coordinates (r , z).
Fig. 1. Schematic diagram of COD in the Earth’s gravitational field: focusing lens (1), unfocused laser beam (2), COD (3), border of the computational region (4).
(ne is in cm−3 , T in K, and μ in cm−1 ), ne is the equilibrium electron number density determined from the Saha equation, and nc is the critical plasma density at which its dielectric permittivity vanishes,
nc =
0me ω2 e2
2.1. Equations We determine the thermal and gas-dynamic structure of COD by solving the set of equations consisting of the continuity equation
∇ · (ρ v) = 0,
(1)
the Navier–Stokes equations,
2 3
The quantity Q L expresses the specific power of the energy release within the plasma caused by the laser radiation absorption,
QL = μ
∇·
3χm
∇ U m = χm (U m − U eq,m ) (m = 1, 2, . . . , Nm ),
(4)
and the amplitude equation of a weakly focused laser beam propagating in positive z direction in parabolic approximation of quasioptics [24,25],
2ik
∂E 1 ∂ ∂E + k2 (c − 1)E . = r ∂z r ∂r ∂r
(5)
Here, v( v r = v , v z = u ) is the mean mass velocity, T is the temperature, p is the deviation of the pressure from p 0 = 105 Pa, g (0, 0, g z = − g ) is the gravity acceleration, S˙ = 12 (∇ v + ∇ v T ) is the deformation rate tensor. Parameters ρ , C p , η and λ are the density, heat capacity, viscosity, and thermal conductivity, respectively. E is the complex amplitude of the laser radiation field,
E (r , z) = E (r , z) exp(−ikz + iωt ),
(6)
k = ω/c, where ω is angular frequency of laser radiation, c is the speed of light. The complex dielectric permittivity of the plasma, c , is defined as
c = 1 −
ne nc
−i
μ k
(7)
,
where μ is the volume coefficient of the laser radiation absorption [15],
μ=
2.82 × 10−29n2e T 3/2
lg
2.17 × 103 T 1/3
ne
(10)
(3)
1
c = μ J,
QR =
the equation for the selective thermal radiation transport in the multigroup diffusion approximation [21,24,25],
2
(2)
the equation of energy balance,
ρ v · C p ∇ T = ∇ · (λ∇ T ) + Q L − Q R ,
0 |E |2
where J is laser radiation intensity, and 0 is permittivity of vacuum. The power density of the heat sources associated with thermal radiation transport is
ρ (v · ∇)v = −∇ p + η∇ · v + 2∇ · (η S˙ ) + (ρ0 − ρ )g,
(9)
.
(8)
Nm
c χm (U eq,m − U m ),
(11)
m =1
where χm , U m , and U eq,m are the group values of the volume absorption coefficient and of the radiation density of the plasma and ideal blackbody, averaged over each of the N m spectral intervals [21,25]. The data on the temperature dependence of the thermodynamic, transport, and optical properties of air, which are ρ ( T ) at p = p 0 , C p ( T ), η( T ), and χm ( T ) (m = 1, 2, . . . , N m ) were taken from [26–28]. In the calculations we used N m = 37 spectral intervals [26]. 2.2. Boundary conditions The boundary conditions at the rectangular contour around the computational region describe the COD in an unbounded free space. The inflow and outflow of a cold gas at atmospheric pressure are normal to the lower boundary and the outer cylindrical surface bounding the computation domain, through which no thermal radiation fluxes enter the domain from the outside. Along the lower boundary (z = 0, 0 < r < R) we set
v = 0,
p = 0,
T = T 0,
U m = U eq,m ,
E (r , 0) = E0 (r ), (12)
where m = 1, 2, . . . , N m , and E0 (r ) is given function. Along the outer cylindrical surface (0 < z < L , r = R) we set
u = 0,
p = 0,
T = T 0,
U m = U eq,m ,
E (r , 0) = 0.
(13)
We impose symmetry conditions at the z-axis (r = 0, 0 < z < L),
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I. Rafatov / Physics Letters A 373 (2009) 3336–3341
Fig. 2. Axial distribution of plasma temperature T in a COD in air for p 0 = 1 atm and laser powers P L = 2, 3, 4, 5 and 6 kW for f / D = 3.5 (a) and 5.0 (b).
Fig. 3. Axial distributions of plasma temperature T (a), laser radiation intensity I (b), axial component of velocity u (c), deviation from the atmospheric pressure p (d), and electric field amplitude | E | (e) in a COD in air for laser power P L = 3 kW and numbers f / D = 3.5, 5.0, 6.0, and 8.5.
∂u = 0, ∂r
v = 0,
∂p = 0, ∂r
∂T = 0, ∂r
∂ Um = 0, ∂r
∂E = 0. ∂r (14)
The conditions at the outflow boundary (z = L , 0 < r < R) correspond to a one-dimensional gas flow,
∂ρu = 0, ∂z
v = 0,
p = 0,
∂T = 0, ∂z
∂ Um = 0. ∂z
(15)
2.3. Solution method We used the numerical model and corresponding numerical code (realized in fortran 90) of the previous work [24]. ‘Hydrodynamic’ part of the system, which consists of coupled mass, momentum, energy conservation and radiation transport equations (1)–(3), (4), was solved by Patankar’s pressure-correction SIMPLE algorithm [29] with application of SIP iterations [30]. For the
I. Rafatov / Physics Letters A 373 (2009) 3336–3341
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Fig. 4. Temperature field (the increment between adjacent contours is 2 kK) and gas streamlines G /G max in COD under the same conditions as in Fig. 3: laser power P L = 3 kW and f / D = 3.5 (a), 5.0 (b), 6.0 (c), and 8.5 (d). Panels at the right column illustrate a close-up view of the data plotted on the left. Dashed lines indicate the laser beam channel border (at the level corresponding to e −1 of the laser radiation intensity) in the plasma and solid lines in the absence of plasma.
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I. Rafatov / Physics Letters A 373 (2009) 3336–3341
laser beam propagation equation (5) we employed the ‘method of lines’ [31]. The computational domain was a cylinder of length 10 cm along the axis and 4 cm in radius. The discharge was initially located in the middle of this region: the distance between the ‘geometrical’ focal point and bottom of the chamber was zF = 5 cm. The laser beam was of Gaussian shape in the initial section, z = 0,
kr 2 r2 . E0 (r ) = E0 exp − + i 2 2w
2zF
(16)
Focusing was controlled by spot size of the laser beam in a vacuum w 0 (where radius of the beam is taken at the e −1 laser radiation intensity level) at z F = 5 cm fixed. The beam radius at the initial section (w in Eq. (16)) was calculated as w = w 0 1 + ( zF /kw 20 )2 .
3. Numerical results We calculated parameters of a COD in a vertical CO2 laser beam, free of external forced gas flow, as functions of the laser power P L = 2–6 kW at p 0 = 105 Pa and T 0 = 300 K, for different focusing geometries (f-numbers) f / D = 3.5, 5.0, 6.0, and 8.5 (here the beam diameter is taken at the e −2 intensity level). The values f / D = 5.0 and 6.0 correspond to the conditions of experiments [3] and [4], where the COD was produced in a free air atmosphere by a carbon-dioxide laser of diameters of 2.5 cm and 4 cm, operated at 10.6-μm, focused by a 15 cm and 20 cm focal-length lens along the vertical axis. Fig. 2 illustrates the effect of increase in the laser power P L = 2–6 kW on the plasma temperature in the steady state optical discharge, maintained by weaker and stronger focused beams, namely by the beams with f / D = 3.5 (panel (a)) and f / D = 5.0 (panel (b)). In the case of the stronger focused beam, f / D = 3.5, the discharge’s peak temperature remains fixed in space as the laser power increases, the discharge plasma expands (Fig. 2(a)), and fraction of the laser power dissipated within the plasma grows steadily from 40% at P L = 2 kW to 72% at P L = 6 kW. For a weaker focused beam, f / D = 5.0, the increase in the laser power causes the discharge to shift towards the laser beam (Fig. 2(b)). The discharge cools down and contracts as it displaces from the ‘geometrical’ focal point, dissipated fraction of the laser power first increases from 50% at P L = 2 kW to 53% at P L = 3 kW and then it steadily decreases to 32% at P L = 6 kW. Fig. 3 presents comparison of the COD characteristics (axial distributions of the plasma temperature T (a), laser radiation intensity I (b), deviation from the atmospheric pressure p (c), axial component of velocity u (d), and electric field amplitude |E | (e)), maintained by the laser power P L = 3 kW, for different f-numbers. For greater f-numbers (weaker degree to which beam is focused), discharge stabilizes further away from the ‘geometrical’ focal point (Fig. 3(a)), the laser radiation intensity decays and its profile flattens (b), and the discharge shrinks in size and temperature in it decreases (a). Fraction of the laser power, dissipated within the plasma, decreases as 54%, 53%, 44%, and 22% with increase in f / D = 3.5, 5.0, 6.0, and 8.5. Figs. 4(a)–(d) show the temperature and the mass flux vector fields in the discharge plasma corresponding to the same model solutions demonstrated in Fig. 3, i.e., for a laser power P L = 3 kW and f-numbers 3.5, 5.0, 6.0, and 8.5. Figs. 4(a2 )–(d2 ) present closeup views of the same plasma and gas-dynamic flow from the corresponding left panels (a1 )–(d1 ). These figures also contain the laser beam channel borders in the plasma as well as in the absence of plasma. (We determined the beam radius as a radius of a circular cross section in the transverse plane, so that the fraction of the laser power, passing across it, was equal to 1 − 1/e = 0.63
of the total passing power. This is the level corresponding to e −1 of the laser radiation intensity for the case of a Gaussian beam.) As it can be seen from these figures, the effect of refraction of the laser radiation in a nonuniform plasma, and consequently, the divergence of the laser beam in the plasma, is less noticeable in the case of a strongly focused beam (smaller f-number) (panel (a)). The refraction effect becomes significant for weakly focused beams (panels (b) and (c)). Values of the laser beam waists, corresponding to numbers f / D = 3.5, 5.0, 6.0, and 8.5, are 0.09, 0.35, 0.45, and 0.50 mm, which are about one order as large as to the corresponding ‘undisturbed’ values in a vacuum, which are w 0 = 0.0169, 0.024, 0.0284, and 0.040 mm. We should also notice that the discharge plasma, displaced from the initial focal region, contrary to the statement, e.g., in [21–23], does not necessary localize in a region of the maximal laser radiation intensity. For weakly focused beams and larger value of laser power, the model might have a converging solution for a steady state plasma localized in the place (which is rather a local maximum of the laser radiation intensity) below the beam waist, where the beam intensity is still sufficient to maintain it. 4. Conclusions We considered the effect of focusing geometry on the continuous optical discharge (COD) properties. We used a full two-dimensional radiative gas-dynamic model for the COD, maintained by a vertical CO2 laser beam in free air atmosphere, free of the external gas flow, in the Earth’s gravitational field. The model takes into account all of the important factors that are of influence in laser-sustained plasma processes, and, as a further development of the existing radiative gas-dynamic models of COD, uses realistic quasi optics, namely the beam propagation equation in parabolic approximation, to model the laser radiation propagation. The results presented here describe thermal and gas-dynamic properties of the COD as functions of applied laser power and degree to which laser beam is focused. The model solutions improve results of previous modelling approaches and it performs well in calculating the size and temperature distribution of laboratory plasmas as well as their dynamics in the converged laser beams. For strongly focused beams, the discharge localizes at the ‘geometric’ focal point, dissipates most of the applied laser power, heats up, and expands with increase in the laser power. For weakly focused beams, the discharge displaces from the ‘geometric’ focal point toward the laser beam, shrinks and cools down. Depending on the focusing geometry and applied laser power, it either vanishes or stabilizes in the place where the energy source is balanced by the energy losses due to radiation, thermal conductivity and convection. Acknowledgement The work was supported by the research grant 106T705 from The Scientific and Technical Research Council of Turkey (TUBITAK). Author is grateful to Prof. S.T. Surzhikov (Institute of Problems of Mechanics, Russian Academy of Sciences) for providing the mean absorption coefficients of air. References [1] Yu.P. Raizer, Gas Discharge Physics, Springer-Verlag, Berlin, 1991. [2] A.A. Vvedenov, G.G. Gladush, Physical Processes in Laser Treatment of Materials, Energoatomizdat, Moscow, 1985. [3] N.A. Generalov, A.M. Zakharov, V.D. Kosynkin, M.Yu. Yakimov, Fiz. Goreniya Vzryva 2 (1986) 91. [4] D.R. Keefer, B.B. Henriksen, F. Braerman, J. Appl. Phys. 46 (1975) 1080. [5] M.C. Fowler, D.C. Smith, J. Appl. Phys. 46 (1975) 138.
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[6] M.V. Gerasimenko, G.I. Kozlov, V.A. Kuznetsov, Sov. J. Quantum Electron. 13 (1983) 438. [7] S.-M. Jeng, D.R. Keefer, J. Propul. Power 3 (1987) 255. [8] S.-M. Jeng, D.R. Keefer, R.P. Welle, C. Peters, AIAA J. 25 (1987) 1224. [9] S.-M. Jeng, D.R. Keefer, J. Appl. Phys. 60 (1986) 2272. [10] Z. Szymanski, J. Phys. D: Appl. Phys. 25 (1992) 413. [11] Z. Szymanski, Z. Peradzynski, J. Kurzyna, J. Phys. D: Appl. Phys. 27 (1994) 2074. [12] J.M. Girard, A. Lebehot, R. Campargue, J. Phys. D: Appl. Phys. 27 (1994) 253. [13] S.T. Surzhikov, A.A. Chentsov, Plasma Phys. Rep. 22 (1996) 957. [14] S.T. Surzhikov, Sov. J. Quantum Electron. 30 (2000) 416. [15] S.T. Surzhikov, Fluid Dyn. 3 (2005) 446. [16] E.B. Kulumbaev, V.M. Lelevkin, Plasma Phys. Rep. 6 (1999) 517. [17] E.B. Kulumbaev, V.M. Lelevkin, Plasma Phys. Rep. 7 (2000) 617. [18] E.B. Kulumbaev, V.M. Lelevkin, Plasma Phys. Rep. 7 (2000) 621. [19] S.T. Surzhikov, High Temp. 4 (2002) 546. [20] S.T. Surzhikov, Dokl. Phys. 6 (2008) 305. [21] Yu.P. Raizer, A.Yu. Silant’ev, Sov. J. Quantum Electron. 3 (1986) 384.
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[22] Yu.P. Raizer, A.Yu. Silant’ev, S.T. Surzhikov, High Temp. 25 (1987) 454. [23] R. Conrad, Yu.P. Raizer, S.T. Surzhikov, AIAA J. 8 (1996) 1584. [24] I.R. Rafatov, B. Yedierler, E.B. Kulumbaev, J. Phys. D: Appl. Phys. 42 (2009) 055212. [25] V. Mazhukin, I. Smurov, G. Flamant, J. Comput. Phys. 112 (1994) 78. [26] S.T. Surzhikov, Optical Properties of Gases and Plasmas, MGTU, Moscow, 2004. [27] A.S. Predvoditelev, et al., Thermodynamic Functions of Air, Akad. Nauk SSSR, Moscow, 1960. [28] J. Bacri, S. Raffanel, Plasma Chem. Plasma Process. 9 (1989) 133. [29] S. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corp., New York, 1980. [30] J.H. Ferziger, M. Peric, Computational Methods for Fluid Dynamics, Springer, New York, 2002. [31] W. Hundsdorfer, J.G. Verwer, Numerical Solution of Time-Dependent Advection–Diffusion-Reaction Equations, Springer Series in Computational Mathematics, vol. 33, Springer, Berlin, 2003.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Effect of focusing geometry on the continuous optical discharge properties Ismail Rafatov ∗ Department of Physics, Middle East Technical University, 06531 Ankara, Turkey
a r t i c l e
i n f o
Article history: Received 10 May 2009 Received in revised form 15 June 2009 Accepted 16 July 2009 Available online 21 July 2009 Communicated by F. Porcelli PACS: 52.65.-y 52.80.-s 52.38.-r 52.50.Jm 02.60.Cb
a b s t r a c t We studied the effect of the laser beam focusing geometry on the continuous optical discharge (COD) properties. We used a full two-dimensional radiative gas-dynamic model for the COD, maintained by a vertical CO2 laser beam in free air atmosphere, in the Earth’s gravitational field. The model takes into account all of the factors that are of importance in laser-sustained plasma processes, and uses realistic quasi optics to describe the laser radiation propagation. Results are presented for the optical discharge parameters as functions of applied laser power and degree (f-number) to which the laser beam is focused. © 2009 Elsevier B.V. All rights reserved.
Keywords: Radiative gas dynamics Numerical modeling Continuous optical discharge Laser plasma
1. Introduction In a continuous optical discharge (COD) in air, a low-temperature laser plasma is maintained at a temperature of about 16 to 22 kK by irradiation of a continuous laser power, usually from a CO2 laser. After the production of a low-temperature plasma by a certain method, e.g., by optical or electrical breakdown, in a gas, in free space, or near a surface, the plasma is maintained in the steady state or propagates towards the incident laser by the irradiation of this plasma by laser radiation of a sub-breakdown intensity. This effect was investigated extensively both experimentally [1–6] and theoretically [7–24]. It was found that the shape of the plasma and its position with respect to the focal plane of the laser beam depends on several factors, such as the degree to which the beam is focused, the output power of the laser, and external gas flow conditions. The increase of the laser power and the f-number (ratio of the focus length F of the lens to the beam diameter D) of the optical system [10,11,21,24] results in a shift of the plasma along the laser beam toward the laser. An increase of the convection velocity of the flow propagating co-directionally with the laser radiation, has the opposite effect [6,11,24]. The plasma stabilizes
*
Tel.: +90 312 2103254; fax: +90 312 2105099. E-mail address: [email protected].
0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.07.043
in the place where the energy source balances the energy losses due to radiation, thermal conductivity, and convection. If discharge propagates to a point in a beam where the intensity is no longer sufficient to maintain it, it vanishes. In a weakly converging laser beam (large f-number) a stable plasma can only be maintained in a forced convective flow [6,11,12,24]. Convective flow incident to the plasma behaves like a flow over an obstacle (see, e.g. [11,22,24]): it flows around the hot discharge region and only a small fraction of the cold gas enters the discharge. In this Letter, we studied the effect of focusing geometry (f-number) on the COD properties. It is known that for shorter focal lengths, the effect of refraction on the plasma shape is negligible [10,12], however, for larger focal length geometries, the laser refraction process can be of great importance [5,12,21]. Also, most of the existing COD models use geometric optics to trace the laser beam through the optical system, and therefore, are unable to take refraction of the beam in the nonuniform plasma properly into account. Here we applied a full two-dimensional radiative gasdynamic numerical model for the COD, which uses realistic quasi optics and takes refraction of the laser radiation in the plasma properly into consideration in the description of the laser beam propagation. Consequently, the major improvement of the mathematical model in this work, compared with the existing models, is application of a quasi-optical approximation, namely the beam propagation equation in parabolic approximation, to model the
I. Rafatov / Physics Letters A 373 (2009) 3336–3341
3337
laser radiation propagation. We applied the numerical model and corresponding numerical code (realized in fortran 90) of the previous work [24] for COD, free of the external gas flow, and maintained by a vertical CO2 laser beam, in the Earth’s gravitational field. The Letter is organized as follows. Section 2 describes the model of COD. Section 3 presents the results of the calculation of COD characteristics. Finally, Section 4 summarizes the results. 2. Model We consider a steady-state COD sustained by a weakly focused 10.6-μm CO2 laser beam in free space at atmospheric pressure. The beam is focused vertically from below, the discharge is free of the external gas flow, and the free-fall acceleration is acting vertically downward (see Fig. 1). We assume that the gas flow is subsonic and laminar. Moreover, we assume that the discharge plasma is in local thermodynamic equilibrium (LTE). Therefore, the plasma can be described by a single temperature, and its physical properties are only a function of this temperature and pressure. Finally, assuming that all the processes are axially symmetric, we use cylindrical coordinates (r , z).
Fig. 1. Schematic diagram of COD in the Earth’s gravitational field: focusing lens (1), unfocused laser beam (2), COD (3), border of the computational region (4).
(ne is in cm−3 , T in K, and μ in cm−1 ), ne is the equilibrium electron number density determined from the Saha equation, and nc is the critical plasma density at which its dielectric permittivity vanishes,
nc =
0me ω2 e2
2.1. Equations We determine the thermal and gas-dynamic structure of COD by solving the set of equations consisting of the continuity equation
∇ · (ρ v) = 0,
(1)
the Navier–Stokes equations,
2 3
The quantity Q L expresses the specific power of the energy release within the plasma caused by the laser radiation absorption,
QL = μ
∇·
3χm
∇ U m = χm (U m − U eq,m ) (m = 1, 2, . . . , Nm ),
(4)
and the amplitude equation of a weakly focused laser beam propagating in positive z direction in parabolic approximation of quasioptics [24,25],
2ik
∂E 1 ∂ ∂E + k2 (c − 1)E . = r ∂z r ∂r ∂r
(5)
Here, v( v r = v , v z = u ) is the mean mass velocity, T is the temperature, p is the deviation of the pressure from p 0 = 105 Pa, g (0, 0, g z = − g ) is the gravity acceleration, S˙ = 12 (∇ v + ∇ v T ) is the deformation rate tensor. Parameters ρ , C p , η and λ are the density, heat capacity, viscosity, and thermal conductivity, respectively. E is the complex amplitude of the laser radiation field,
E (r , z) = E (r , z) exp(−ikz + iωt ),
(6)
k = ω/c, where ω is angular frequency of laser radiation, c is the speed of light. The complex dielectric permittivity of the plasma, c , is defined as
c = 1 −
ne nc
−i
μ k
(7)
,
where μ is the volume coefficient of the laser radiation absorption [15],
μ=
2.82 × 10−29n2e T 3/2
lg
2.17 × 103 T 1/3
ne
(10)
(3)
1
c = μ J,
QR =
the equation for the selective thermal radiation transport in the multigroup diffusion approximation [21,24,25],
2
(2)
the equation of energy balance,
ρ v · C p ∇ T = ∇ · (λ∇ T ) + Q L − Q R ,
0 |E |2
where J is laser radiation intensity, and 0 is permittivity of vacuum. The power density of the heat sources associated with thermal radiation transport is
ρ (v · ∇)v = −∇ p + η∇ · v + 2∇ · (η S˙ ) + (ρ0 − ρ )g,
(9)
.
(8)
Nm
c χm (U eq,m − U m ),
(11)
m =1
where χm , U m , and U eq,m are the group values of the volume absorption coefficient and of the radiation density of the plasma and ideal blackbody, averaged over each of the N m spectral intervals [21,25]. The data on the temperature dependence of the thermodynamic, transport, and optical properties of air, which are ρ ( T ) at p = p 0 , C p ( T ), η( T ), and χm ( T ) (m = 1, 2, . . . , N m ) were taken from [26–28]. In the calculations we used N m = 37 spectral intervals [26]. 2.2. Boundary conditions The boundary conditions at the rectangular contour around the computational region describe the COD in an unbounded free space. The inflow and outflow of a cold gas at atmospheric pressure are normal to the lower boundary and the outer cylindrical surface bounding the computation domain, through which no thermal radiation fluxes enter the domain from the outside. Along the lower boundary (z = 0, 0 < r < R) we set
v = 0,
p = 0,
T = T 0,
U m = U eq,m ,
E (r , 0) = E0 (r ), (12)
where m = 1, 2, . . . , N m , and E0 (r ) is given function. Along the outer cylindrical surface (0 < z < L , r = R) we set
u = 0,
p = 0,
T = T 0,
U m = U eq,m ,
E (r , 0) = 0.
(13)
We impose symmetry conditions at the z-axis (r = 0, 0 < z < L),
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Fig. 2. Axial distribution of plasma temperature T in a COD in air for p 0 = 1 atm and laser powers P L = 2, 3, 4, 5 and 6 kW for f / D = 3.5 (a) and 5.0 (b).
Fig. 3. Axial distributions of plasma temperature T (a), laser radiation intensity I (b), axial component of velocity u (c), deviation from the atmospheric pressure p (d), and electric field amplitude | E | (e) in a COD in air for laser power P L = 3 kW and numbers f / D = 3.5, 5.0, 6.0, and 8.5.
∂u = 0, ∂r
v = 0,
∂p = 0, ∂r
∂T = 0, ∂r
∂ Um = 0, ∂r
∂E = 0. ∂r (14)
The conditions at the outflow boundary (z = L , 0 < r < R) correspond to a one-dimensional gas flow,
∂ρu = 0, ∂z
v = 0,
p = 0,
∂T = 0, ∂z
∂ Um = 0. ∂z
(15)
2.3. Solution method We used the numerical model and corresponding numerical code (realized in fortran 90) of the previous work [24]. ‘Hydrodynamic’ part of the system, which consists of coupled mass, momentum, energy conservation and radiation transport equations (1)–(3), (4), was solved by Patankar’s pressure-correction SIMPLE algorithm [29] with application of SIP iterations [30]. For the
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Fig. 4. Temperature field (the increment between adjacent contours is 2 kK) and gas streamlines G /G max in COD under the same conditions as in Fig. 3: laser power P L = 3 kW and f / D = 3.5 (a), 5.0 (b), 6.0 (c), and 8.5 (d). Panels at the right column illustrate a close-up view of the data plotted on the left. Dashed lines indicate the laser beam channel border (at the level corresponding to e −1 of the laser radiation intensity) in the plasma and solid lines in the absence of plasma.
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laser beam propagation equation (5) we employed the ‘method of lines’ [31]. The computational domain was a cylinder of length 10 cm along the axis and 4 cm in radius. The discharge was initially located in the middle of this region: the distance between the ‘geometrical’ focal point and bottom of the chamber was zF = 5 cm. The laser beam was of Gaussian shape in the initial section, z = 0,
kr 2 r2 . E0 (r ) = E0 exp − + i 2 2w
2zF
(16)
Focusing was controlled by spot size of the laser beam in a vacuum w 0 (where radius of the beam is taken at the e −1 laser radiation intensity level) at z F = 5 cm fixed. The beam radius at the initial section (w in Eq. (16)) was calculated as w = w 0 1 + ( zF /kw 20 )2 .
3. Numerical results We calculated parameters of a COD in a vertical CO2 laser beam, free of external forced gas flow, as functions of the laser power P L = 2–6 kW at p 0 = 105 Pa and T 0 = 300 K, for different focusing geometries (f-numbers) f / D = 3.5, 5.0, 6.0, and 8.5 (here the beam diameter is taken at the e −2 intensity level). The values f / D = 5.0 and 6.0 correspond to the conditions of experiments [3] and [4], where the COD was produced in a free air atmosphere by a carbon-dioxide laser of diameters of 2.5 cm and 4 cm, operated at 10.6-μm, focused by a 15 cm and 20 cm focal-length lens along the vertical axis. Fig. 2 illustrates the effect of increase in the laser power P L = 2–6 kW on the plasma temperature in the steady state optical discharge, maintained by weaker and stronger focused beams, namely by the beams with f / D = 3.5 (panel (a)) and f / D = 5.0 (panel (b)). In the case of the stronger focused beam, f / D = 3.5, the discharge’s peak temperature remains fixed in space as the laser power increases, the discharge plasma expands (Fig. 2(a)), and fraction of the laser power dissipated within the plasma grows steadily from 40% at P L = 2 kW to 72% at P L = 6 kW. For a weaker focused beam, f / D = 5.0, the increase in the laser power causes the discharge to shift towards the laser beam (Fig. 2(b)). The discharge cools down and contracts as it displaces from the ‘geometrical’ focal point, dissipated fraction of the laser power first increases from 50% at P L = 2 kW to 53% at P L = 3 kW and then it steadily decreases to 32% at P L = 6 kW. Fig. 3 presents comparison of the COD characteristics (axial distributions of the plasma temperature T (a), laser radiation intensity I (b), deviation from the atmospheric pressure p (c), axial component of velocity u (d), and electric field amplitude |E | (e)), maintained by the laser power P L = 3 kW, for different f-numbers. For greater f-numbers (weaker degree to which beam is focused), discharge stabilizes further away from the ‘geometrical’ focal point (Fig. 3(a)), the laser radiation intensity decays and its profile flattens (b), and the discharge shrinks in size and temperature in it decreases (a). Fraction of the laser power, dissipated within the plasma, decreases as 54%, 53%, 44%, and 22% with increase in f / D = 3.5, 5.0, 6.0, and 8.5. Figs. 4(a)–(d) show the temperature and the mass flux vector fields in the discharge plasma corresponding to the same model solutions demonstrated in Fig. 3, i.e., for a laser power P L = 3 kW and f-numbers 3.5, 5.0, 6.0, and 8.5. Figs. 4(a2 )–(d2 ) present closeup views of the same plasma and gas-dynamic flow from the corresponding left panels (a1 )–(d1 ). These figures also contain the laser beam channel borders in the plasma as well as in the absence of plasma. (We determined the beam radius as a radius of a circular cross section in the transverse plane, so that the fraction of the laser power, passing across it, was equal to 1 − 1/e = 0.63
of the total passing power. This is the level corresponding to e −1 of the laser radiation intensity for the case of a Gaussian beam.) As it can be seen from these figures, the effect of refraction of the laser radiation in a nonuniform plasma, and consequently, the divergence of the laser beam in the plasma, is less noticeable in the case of a strongly focused beam (smaller f-number) (panel (a)). The refraction effect becomes significant for weakly focused beams (panels (b) and (c)). Values of the laser beam waists, corresponding to numbers f / D = 3.5, 5.0, 6.0, and 8.5, are 0.09, 0.35, 0.45, and 0.50 mm, which are about one order as large as to the corresponding ‘undisturbed’ values in a vacuum, which are w 0 = 0.0169, 0.024, 0.0284, and 0.040 mm. We should also notice that the discharge plasma, displaced from the initial focal region, contrary to the statement, e.g., in [21–23], does not necessary localize in a region of the maximal laser radiation intensity. For weakly focused beams and larger value of laser power, the model might have a converging solution for a steady state plasma localized in the place (which is rather a local maximum of the laser radiation intensity) below the beam waist, where the beam intensity is still sufficient to maintain it. 4. Conclusions We considered the effect of focusing geometry on the continuous optical discharge (COD) properties. We used a full two-dimensional radiative gas-dynamic model for the COD, maintained by a vertical CO2 laser beam in free air atmosphere, free of the external gas flow, in the Earth’s gravitational field. The model takes into account all of the important factors that are of influence in laser-sustained plasma processes, and, as a further development of the existing radiative gas-dynamic models of COD, uses realistic quasi optics, namely the beam propagation equation in parabolic approximation, to model the laser radiation propagation. The results presented here describe thermal and gas-dynamic properties of the COD as functions of applied laser power and degree to which laser beam is focused. The model solutions improve results of previous modelling approaches and it performs well in calculating the size and temperature distribution of laboratory plasmas as well as their dynamics in the converged laser beams. For strongly focused beams, the discharge localizes at the ‘geometric’ focal point, dissipates most of the applied laser power, heats up, and expands with increase in the laser power. For weakly focused beams, the discharge displaces from the ‘geometric’ focal point toward the laser beam, shrinks and cools down. Depending on the focusing geometry and applied laser power, it either vanishes or stabilizes in the place where the energy source is balanced by the energy losses due to radiation, thermal conductivity and convection. Acknowledgement The work was supported by the research grant 106T705 from The Scientific and Technical Research Council of Turkey (TUBITAK). Author is grateful to Prof. S.T. Surzhikov (Institute of Problems of Mechanics, Russian Academy of Sciences) for providing the mean absorption coefficients of air. References [1] Yu.P. Raizer, Gas Discharge Physics, Springer-Verlag, Berlin, 1991. [2] A.A. Vvedenov, G.G. Gladush, Physical Processes in Laser Treatment of Materials, Energoatomizdat, Moscow, 1985. [3] N.A. Generalov, A.M. Zakharov, V.D. Kosynkin, M.Yu. Yakimov, Fiz. Goreniya Vzryva 2 (1986) 91. [4] D.R. Keefer, B.B. Henriksen, F. Braerman, J. Appl. Phys. 46 (1975) 1080. [5] M.C. Fowler, D.C. Smith, J. Appl. Phys. 46 (1975) 138.
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