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Model calculations of non-stationary resonant reflectivity of plasmas
Volume 74A, number 3,4
PHYSICS LErFERS
12 November 1979
MODEL CALCULATIONS OF NON-STATIONARY RESONANT REFLECTIVITY OF PLASMAS K. SAUER and K. BAUMGARTEL Central Institute for Electron Physics, Academy of Sciences of the GDR, Berlin, GDR
and G. AUER Institute for Theoretical Physics, Innsbruck University, Innsbruck, Austria Received 4 May 1979
The reflection coefficient of an overdense plasma with a density plateau or a cavity oscillates and can even reach peak values significantly larger than one (0
Recently in self-consistent calculations of the interaction of strong electromagnetic radiation with inhomogeneous plasmas [1] oscifiations of the reflection coefficient were obtained with maximal values exceeding one (superreflectivity). l’his effect occurs when caviton-like density structures and consequently resonant electromagnetic field patterns are formed temporarily due to the action of the ponderomotive forces. The basic characteristics of these electromagnetic structure resonances can be studied conveniently in a plasma model where the density profile is approximated by homogeneous layers with a stepwise change of the density from one layer to the next [1,2]. Depending on the configuration the superposition of waves propagating forwards and backwards may lead to a strong local increase of the electric field. Resonant states are characterized by a maximum enhancement of this field which occurs for certain values of the physical parameters of the problem (e.g. density proffle, angle of incidence, frequency). Similar phenomena are well known in the theory of coherent plasma optics [3]. We want to emphasize, however, that the temporal
valid for a sufficiently slow variation of the parameters only, i.e. in the adiabatic limit. Otherwise a nonstationary treatment of the problem is necessary, particularly in the case of sharp resonances, i.e. when already a small variation of a parameter leads to a strong change in the electric field pattern. It is the aim of this note to demonstrate this effect of nonstationary resonant plasma reflectivity by means of model calculations. To this end we investigate the reflection of s-polarized waves for two very simple plasma profiles where the dielectric e is a given function ofx and t and is specified in such a way that the configuration passes through an electromagnetic structure resonance. Characteristic oscillations of the reflection coefficient are obtained with a low minimum and a high maximum (0 IR 12 5). We show how the extrema of these oscillations depend on the time rate of change of the parameters. First we consider a three-step density profile cornposed of two overcritical layers (n > n~)separated by an underdense region (caviton model, see fig. la), where the width d2 ofthe undercitical layer increases
evolution of the formation and the decay of the resonance structures is of crucial importance for the reflectivity of the plasma and strongly affects the nonstationary behaviour of the reflection coefficient. The stationary theory used in earlier calculations [2,3] is
linearly with time, d2(t) = d2(O) + u2r. In the special case nof normal incidence and for densities n1 = = 2~c’ 2 =0 (vacuum) a resonant field pattern is obtamed for d = X0/4 under stationary conditions (A0 vacuum wavelength). The spatial behaviour of the am211
Volume 74A, number 3,4
PHYSICS LETFERS
12 November 1979
Crank—Nicholson procedure and gaussian elimination. In figs. 2a—2d the development of both the reflec-
20 15
a)
ui
10
r
5
-
I
i
2
— — — —
I 0 ________________ O 1
tion coefficient jR 12 and the maximum electric field IUlm~is plotted for our two models. The oscifiation of the reflection coefficient can be easily understood from the law of conservation of energy which follows directly from the wave equation (1). In the case of negligible absorption and transmission it reads
(1 2
xj~
d
3
~-~-1fluI2dx+IRI2=l,
1E
iui
(3)
where the amplitude u is normalized with respect to
b)
the amplitude of the incident wave. When the configu-
10
r
2
— — —
n.
I —
—
—
—
—
ration approaches a resonance, the electric field amplitude and hence the electric field energy blow up. This means that a part of the incident energy is stored in our layers and therefore the reflectedintensity be-
o ___________________________________ 0
comes smaller in that moment (IR 12 assumes a minima! value smaller than one). When afterwards the reson-
Fig. 1. Wave pattern for the resonance state under stationary conditions: electric field amplitude (solid line) versus length measured in vacuum wavelengths. The dashed curve indicates the density profile. 2~c (a) caviton n model with d1 = 0.35 Xo, d2 =0.25 A0, flj = 3, fl2 0,8 = 0 . (b) two-step modetwith d1 = 3.5 A0, d~—~ oo, n1 = O.8nc, fl2 = 2flc, 0 = 25.2
ance structure is destroyed the electric field goes down, and the stored energy is released (JR 12 has a maximum greater than one). In other words, due to the resonances of the electromagnetic structure the plasma acts temporarily as a storage for the electric field energy.
0
2
4
xlA0
6
8
plitude of the resulting field is shown in fig. 1 a. A typical field enhancement in the underdense region is observed, The second model is a two-step profile with an underdense and an overdense layer. Now the angle of incidence is varied with time, 0(t) = 0(0) + w1t. When = 0.8n~,n2 = 2.0 ~ d~= 3.5 A0, d2 -+00 the stationary main resonance occurs for 0 = 25 .25°.The corresponding electric field is plotted in fig. lb. In order to investigate the nonstationary behaviour of the plasma reflectivity near resonance, in the wave equation for the electric field a slow variation in time must be taken into account (modulational representation): 2 a2u/ax2 + c~ e(x, t)u = 0, (l) 2iw0 c e(x, t)au/at = 1 +n/[n~(l + iv/w 20 (2) 0)] sin (w 0 and ~ are the wave frequency and the effective collision frequency, respectively), Eq. (1) was solved numerically by means of a —
—
212
For companson the solution for adiabatic variation, i.e. the solution of eq. (1) without the time derivative, is also plotted in figs. 2a and 2c. The nonstationary treatment leads to a broadening of the resonance and a reduction of the peak amplitude of the electric field. From the foregoing it is clear that the maximum and minimum values of the reflection coefficient and the maximum electric field will depend on the time rate of change of our parameters (length of underdense layer or angle of incidence). This dependence is plotted for the collisionless case as well as for a nonvanishing collision frequency (v = 5 X l0~w0 and 1 X l0-~ w0, respectively) in figs. 3a and 3b. For low time rates the adiabatic results are obtained. In the collisional case the field amplitude is significantly smaller in regime as compared to the coefficient collisionlessis case, andthis also the minimum reflection substantially reduced (in the adiabatic limit I IR I~in equals the maximum absorption coefficient, see also ref. [4]). On the other hand, the maximum reflection coefficient differs little from one. When the speed of parameter variation is increased, the peak electric field
Volume 74A, number 3,4
PHYSICS LETFERS
12 November 1979
20.
____________________________________________________
2C 3.0
10
0
a)
I::
3 2.1O~ 3.1O~~2~t 4~WJ3 C) O_________________________________________________ i’i0
~
0
1
2
if4
~~ v
15~
0 ________________________________________________ o i.iü~ 2~10~ 3.i0~(A,t ~
Ii i~ h nh I I I
1O~
~<
0
bi 10~
2/c
RI2
10
1012 ~.
~
20
15 0 ~10
10 10’
to-’
~o-~
.,,Iu,
~
5
Fig. 3.
Maximum electric field amplitude, maximum and mini-
mum reflection coefficient versus the time rates of change
2
3w i~respectively. (a) caviton model (solid line: v = 0, dashed line: p = 5 X 10 0) (b) two-step model (solid line:
and ~
0 O
2.10~
3
4~10~
640
b)0t
3~o).
v
1
RI2
=
0, dashed line: v = i0~
decreases continuously, whereas the maximum and minimum reflectivity for a certain time rate pass through a maximum and a minimum, respectively. For a high time rate the coffisional results approach the collisionless (since in this case the time derivative term in eq. (1) dominates over the colisional contribution
2
0
u
_____________________________________________
0
2.1o~
~.io3
61o3 ~,,t
840~
Fig. 2. Maximum electric field amplitude (a), (c) and reflection coefficient (b), (d) versus time when the wave structure passes through a resonance state. (a),. (b) caviton model with
e). The effect of superreflectivity (1R12 > i)is suppressed when the structure resonance is passed sufficiently fast. In the caviton model the region of significant deviation from the stationary result (JR 12 = 1 without collisions) lies in the range of 103c to 104c in
d 2(r) = 0.22 A0 + iO’~Ct. (c) (d) two-step model with 0(t) = 5 x 10~ c.j0t. The other parameters are the same as in fig. 1. Curve 0: adiabatic variation, curve 1: nonstationary 240 +
result.
(see fig. 3a), which corresponds to the ion sound velocity of a 0.2 to 2 keV deuterium plasma. This is consistent with our self-consistent calculations [1] in 213
Volume 74A, number 3,4
PHYSICS LETFERS
which a similar caviton density profile was produced by the ponderomotive force. The model calculations presented in this letter provide a basic understanding of nonstationary resonant reflection and absorption, an important phenomenon when considering self-consistent laser—plasma interactions. Moreover, these effects may also occur in other areas where similar structure resonances exist. We wish to thank Prof. G. Wallis and Dr. D. Sunder for fruitful discussions.
214
12 November 1979
References [1] K. Sauer, N.E. Andreev and K. Baumgartel, 7th Intern. Conf. on Plasma phys. contr. nucL fusion res. (Innsbruck, 1978) Ext. Syn. p. 69. [21 F.J. Mayer, R.K. Osborn, D.W. Daniels and J.F. McGrath, Phys. Rev. Lett. 40 (1978) 30. [3] L.M. Brekhovskikh, Waves in layered media (Academic Press, New York, 1960). [41K. Sauer, Phys. Lett. 66A (1978) 37.
PHYSICS LErFERS
12 November 1979
MODEL CALCULATIONS OF NON-STATIONARY RESONANT REFLECTIVITY OF PLASMAS K. SAUER and K. BAUMGARTEL Central Institute for Electron Physics, Academy of Sciences of the GDR, Berlin, GDR
and G. AUER Institute for Theoretical Physics, Innsbruck University, Innsbruck, Austria Received 4 May 1979
The reflection coefficient of an overdense plasma with a density plateau or a cavity oscillates and can even reach peak values significantly larger than one (0
Recently in self-consistent calculations of the interaction of strong electromagnetic radiation with inhomogeneous plasmas [1] oscifiations of the reflection coefficient were obtained with maximal values exceeding one (superreflectivity). l’his effect occurs when caviton-like density structures and consequently resonant electromagnetic field patterns are formed temporarily due to the action of the ponderomotive forces. The basic characteristics of these electromagnetic structure resonances can be studied conveniently in a plasma model where the density profile is approximated by homogeneous layers with a stepwise change of the density from one layer to the next [1,2]. Depending on the configuration the superposition of waves propagating forwards and backwards may lead to a strong local increase of the electric field. Resonant states are characterized by a maximum enhancement of this field which occurs for certain values of the physical parameters of the problem (e.g. density proffle, angle of incidence, frequency). Similar phenomena are well known in the theory of coherent plasma optics [3]. We want to emphasize, however, that the temporal
valid for a sufficiently slow variation of the parameters only, i.e. in the adiabatic limit. Otherwise a nonstationary treatment of the problem is necessary, particularly in the case of sharp resonances, i.e. when already a small variation of a parameter leads to a strong change in the electric field pattern. It is the aim of this note to demonstrate this effect of nonstationary resonant plasma reflectivity by means of model calculations. To this end we investigate the reflection of s-polarized waves for two very simple plasma profiles where the dielectric e is a given function ofx and t and is specified in such a way that the configuration passes through an electromagnetic structure resonance. Characteristic oscillations of the reflection coefficient are obtained with a low minimum and a high maximum (0 IR 12 5). We show how the extrema of these oscillations depend on the time rate of change of the parameters. First we consider a three-step density profile cornposed of two overcritical layers (n > n~)separated by an underdense region (caviton model, see fig. la), where the width d2 ofthe undercitical layer increases
evolution of the formation and the decay of the resonance structures is of crucial importance for the reflectivity of the plasma and strongly affects the nonstationary behaviour of the reflection coefficient. The stationary theory used in earlier calculations [2,3] is
linearly with time, d2(t) = d2(O) + u2r. In the special case nof normal incidence and for densities n1 = = 2~c’ 2 =0 (vacuum) a resonant field pattern is obtamed for d = X0/4 under stationary conditions (A0 vacuum wavelength). The spatial behaviour of the am211
Volume 74A, number 3,4
PHYSICS LETFERS
12 November 1979
Crank—Nicholson procedure and gaussian elimination. In figs. 2a—2d the development of both the reflec-
20 15
a)
ui
10
r
5
-
I
i
2
— — — —
I 0 ________________ O 1
tion coefficient jR 12 and the maximum electric field IUlm~is plotted for our two models. The oscifiation of the reflection coefficient can be easily understood from the law of conservation of energy which follows directly from the wave equation (1). In the case of negligible absorption and transmission it reads
(1 2
xj~
d
3
~-~-1fluI2dx+IRI2=l,
1E
iui
(3)
where the amplitude u is normalized with respect to
b)
the amplitude of the incident wave. When the configu-
10
r
2
— — —
n.
I —
—
—
—
—
ration approaches a resonance, the electric field amplitude and hence the electric field energy blow up. This means that a part of the incident energy is stored in our layers and therefore the reflectedintensity be-
o ___________________________________ 0
comes smaller in that moment (IR 12 assumes a minima! value smaller than one). When afterwards the reson-
Fig. 1. Wave pattern for the resonance state under stationary conditions: electric field amplitude (solid line) versus length measured in vacuum wavelengths. The dashed curve indicates the density profile. 2~c (a) caviton n model with d1 = 0.35 Xo, d2 =0.25 A0, flj = 3, fl2 0,8 = 0 . (b) two-step modetwith d1 = 3.5 A0, d~—~ oo, n1 = O.8nc, fl2 = 2flc, 0 = 25.2
ance structure is destroyed the electric field goes down, and the stored energy is released (JR 12 has a maximum greater than one). In other words, due to the resonances of the electromagnetic structure the plasma acts temporarily as a storage for the electric field energy.
0
2
4
xlA0
6
8
plitude of the resulting field is shown in fig. 1 a. A typical field enhancement in the underdense region is observed, The second model is a two-step profile with an underdense and an overdense layer. Now the angle of incidence is varied with time, 0(t) = 0(0) + w1t. When = 0.8n~,n2 = 2.0 ~ d~= 3.5 A0, d2 -+00 the stationary main resonance occurs for 0 = 25 .25°.The corresponding electric field is plotted in fig. lb. In order to investigate the nonstationary behaviour of the plasma reflectivity near resonance, in the wave equation for the electric field a slow variation in time must be taken into account (modulational representation): 2 a2u/ax2 + c~ e(x, t)u = 0, (l) 2iw0 c e(x, t)au/at = 1 +n/[n~(l + iv/w 20 (2) 0)] sin (w 0 and ~ are the wave frequency and the effective collision frequency, respectively), Eq. (1) was solved numerically by means of a —
—
212
For companson the solution for adiabatic variation, i.e. the solution of eq. (1) without the time derivative, is also plotted in figs. 2a and 2c. The nonstationary treatment leads to a broadening of the resonance and a reduction of the peak amplitude of the electric field. From the foregoing it is clear that the maximum and minimum values of the reflection coefficient and the maximum electric field will depend on the time rate of change of our parameters (length of underdense layer or angle of incidence). This dependence is plotted for the collisionless case as well as for a nonvanishing collision frequency (v = 5 X l0~w0 and 1 X l0-~ w0, respectively) in figs. 3a and 3b. For low time rates the adiabatic results are obtained. In the collisional case the field amplitude is significantly smaller in regime as compared to the coefficient collisionlessis case, andthis also the minimum reflection substantially reduced (in the adiabatic limit I IR I~in equals the maximum absorption coefficient, see also ref. [4]). On the other hand, the maximum reflection coefficient differs little from one. When the speed of parameter variation is increased, the peak electric field
Volume 74A, number 3,4
PHYSICS LETFERS
12 November 1979
20.
____________________________________________________
2C 3.0
10
0
a)
I::
3 2.1O~ 3.1O~~2~t 4~WJ3 C) O_________________________________________________ i’i0
~
0
1
2
if4
~~ v
15~
0 ________________________________________________ o i.iü~ 2~10~ 3.i0~(A,t ~
Ii i~ h nh I I I
1O~
~<
0
bi 10~
2/c
RI2
10
1012 ~.
~
20
15 0 ~10
10 10’
to-’
~o-~
.,,Iu,
~
5
Fig. 3.
Maximum electric field amplitude, maximum and mini-
mum reflection coefficient versus the time rates of change
2
3w i~respectively. (a) caviton model (solid line: v = 0, dashed line: p = 5 X 10 0) (b) two-step model (solid line:
and ~
0 O
2.10~
3
4~10~
640
b)0t
3~o).
v
1
RI2
=
0, dashed line: v = i0~
decreases continuously, whereas the maximum and minimum reflectivity for a certain time rate pass through a maximum and a minimum, respectively. For a high time rate the coffisional results approach the collisionless (since in this case the time derivative term in eq. (1) dominates over the colisional contribution
2
0
u
_____________________________________________
0
2.1o~
~.io3
61o3 ~,,t
840~
Fig. 2. Maximum electric field amplitude (a), (c) and reflection coefficient (b), (d) versus time when the wave structure passes through a resonance state. (a),. (b) caviton model with
e). The effect of superreflectivity (1R12 > i)is suppressed when the structure resonance is passed sufficiently fast. In the caviton model the region of significant deviation from the stationary result (JR 12 = 1 without collisions) lies in the range of 103c to 104c in
d 2(r) = 0.22 A0 + iO’~Ct. (c) (d) two-step model with 0(t) = 5 x 10~ c.j0t. The other parameters are the same as in fig. 1. Curve 0: adiabatic variation, curve 1: nonstationary 240 +
result.
(see fig. 3a), which corresponds to the ion sound velocity of a 0.2 to 2 keV deuterium plasma. This is consistent with our self-consistent calculations [1] in 213
Volume 74A, number 3,4
PHYSICS LETFERS
which a similar caviton density profile was produced by the ponderomotive force. The model calculations presented in this letter provide a basic understanding of nonstationary resonant reflection and absorption, an important phenomenon when considering self-consistent laser—plasma interactions. Moreover, these effects may also occur in other areas where similar structure resonances exist. We wish to thank Prof. G. Wallis and Dr. D. Sunder for fruitful discussions.
214
12 November 1979
References [1] K. Sauer, N.E. Andreev and K. Baumgartel, 7th Intern. Conf. on Plasma phys. contr. nucL fusion res. (Innsbruck, 1978) Ext. Syn. p. 69. [21 F.J. Mayer, R.K. Osborn, D.W. Daniels and J.F. McGrath, Phys. Rev. Lett. 40 (1978) 30. [3] L.M. Brekhovskikh, Waves in layered media (Academic Press, New York, 1960). [41K. Sauer, Phys. Lett. 66A (1978) 37.