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Self-focusing in a plasma due to ponderomotive forces and relativistic effects
Volume
16, number
3
SELF-FOCUSING
March 1976
OPTICS COMMUNICATIONS
IN A PLASMA DUE TO PONDEROMOTIVE
AND RELATIVISTIC
FORCES
EFFECTS
M.R. SIEGRIST Department of I:‘n@wering Canberra, A.C. T. Azrstralia
Received
18 November
f’/lvsics. Research
S&o01 of’f’lrvsical
Sciences,
The Australian
Natiorlal
Univcrsit~~.
1975
The propagation of intense laser pulses in a plasma is discussed in terms of a constant shape, paraxial ray approximation. Self-focusing due to ponderomotive forces and relativistic effects is investigated. It is found that the stationary self-focusing behaviour of each mechanism treated separately is similar, with several orders of magnitude difference in critical power. In stationary self-focusing due to the combined mechanisms, complete saturation of ponderomotive self-focusing prevents the occurrence of relativistic effects. Self-focusing lengths and minimum radii are given for a large range of beam powers. A characteristic focal spot radius is found which depends only on the plasma density.
1. Introduction Self-focusing of a laser beam in a plasma has been examined by a number of authors [l-6] Basically two different mechanisms have been proposed: selffocusing due to plasma density variations produced by ponderomotive forces and relativistic self-focusing due to the Lorentz factor of the optical constants in the plasma. It will be shown below that in a first-order approximation the type of focusing and saturation behaviour is the same for the two mechanisms. However. the threshold power and saturation flux, which are the two relevant parameters, differ by several orders of magnitude and the growth rates are different too. The question therefore arises how the two mechanisms interact. If they prove to be independent, then obviously each one influences the behaviour of the beam propagation in a different range of beam powers and a different time domain. Self-focusing is very undesirable in laser fusion applications where it could prevent compression of fuel pellets. On the other hand the self-focusing of a laser beam into a filament would provide an effective method of achieving the high flux densities required to study laser plasma interactions such as the generation of electron-positron pairs. By producing an appropriate nonuniform plasma it might also be possible to taylor the 402
self-focusing in order to preserve a high-quality beam structure. In this paper we shall determine self-focusing distances and focal spot radii for a wide range of beam powers. A constant-shape paraxial ray approximation is used. Absorption and other non-linear effects are neglected.
2. Stationary
beam propagation
in a plasma
Comprehensive studies of the propagation of a laser beam in a medium with nonlinear refractive index have been reported [7] and the basic properties are well understood. We are not at this stage interested in a very accurate description and hence have chosen the very transparent, paraxial ray approximation developed by Wagner et al. [S] Comparison of results with computer studies [9] has shown that many features of the numer. ical solutions are predicted quite accurately. For the radius r of a beam with gaussian intensity distribution as function of distance z, Wagner et al. [S] derive the following differential equation (1)
for a beam of power P. Here E is the envelope of the
Volume
16, number
OPTICS COMMUNICATIONS
3
electric field strength on the beam axis, EI and x are the linear and nonlinear susceptibilities respectively, p is the permeability and k = 27r/X. Expanding the nonlinear part of the susceptibility and keeping only the first nonvanishing term in a saturable form
March 1976
the exponential is developed into a power series up to second order, ~2 is determined as
(8) With /3E2) < 1 which is true as long as the electric field strength is not too close to the saturation value E, = (2/(3)1/2, this can be written as q
eq. (1) can be written as
(9)
PfPU
__=~
(3) (1 +
E2/E;)2
Hereby a critical power PC, and a saturation field Es have been introduced. The former represents the power for which a beam would propagate with a constant cross-section if saturation effects are neglected. The latter is the field strength for which saturation effects become appreciable. In this report we define the nonlinear susceptibility in terms of the time average of the square of the electric field x(E) = c2CE2)
(4)
which differs from Wagner’s notation The critical power then becomes
by a factor of 2.
which is identical with Wagner’s “standard” form [eq. (2)J. For the critical power one obtains PC, = 4nmec3e0kT/e2,
(5)
(10)
where T = i(T, + Ti). It depends only on the temperature and is numerically P,,(W) = 1.7 X 1O4T (eV).
(11)
The oscillation of the electrons in a very intense electromagnetic field has to be treated relativistically. According to Hora [2] and Max [5] the plasma frequency acquires a Lorentz factor ,2
.eY2
saturable
P
= cl_? (1 t -y(&-‘/2,
(12)
PO
with 2e2
In the following it will be shown that both the nonlinear susceptibilities due to ponderomotive forces and due to relativistic terms can be represented in a form similar to eq. (2).
3. Comparison focusing
of ponderomotive
and relativistic self-
The dielectric constant of a plasma due to the timeaveraged force acting on an electron in the field of an electromagnetic wave can be written as [l] * L
(
E(C.!f’)) = 1-
OP
-
exp(-$CE2))
cd2
1
eu,
(13)
y = ,2,2,2 e
and w p0 is the plasma frequency in the absence of the electromagnetic wave. First we study equation (6) with the relativistic term (I 2) but neglecting the ponderomotive force mechanism. This is valid for a pulse which is much shorter than the characteristic time constant for development of a ponderomotive force equilibrium. From (6) one obtains x(E) = (wgd2) [ 1 - (1 + yE2,)-“2]
EO.
(14)
For r(E2) < 1 this can be written as ?
with e2 p=---_ 2mew2k(Te
(7) t Ti)’
where w is the wave and wp the plasma frequency.
The nonlinear susceptibility has again a saturable form. In analogy with the ponderomotive mechanism we obtain a critical power of
If 403
Volume
16, number
3
a2r az2 For a Nd:glass laser this is nutnerically P,(W)
= 2.16 X 1030/~~e(cm
-3
).
Combining eqs. (6) and ( 12) yields the full nonlinear susceptibility in a plasma =
w&i
1 ~_
cd2
expl_:~~ J+ $I+
1
(18)
Substitution into eq. ( 1) and replacement of the axial electric field by the power in the gaussian beam according to (19) results in the following equation for the beam radius r as function of distance of propagation 404
(1
(‘0)
(17)
4. Stationary self-focusing due to the combined mechanisms
I
---1 k2r3
I9 76
with
Comparison of eqs. (11) and (17) shows the large differ. ence in the threshold for the two mechanisms. For a typical laser produced plasma of T = 10 eV and 17, = 1019 cmm3 this amounts to 6 orders of magnitude. Apart from threshold power the two mechanisms have also distinctly different growth rates. This has been investigated in detail by Max et al. [5]. In order to establish a characteristic density profile for ponderemotive self-focusing the beam cross-section has to be partially cleared of particles. For relativistic self-focusing, however, no density gradient is necessary. The change of plasma frequency, as seen by the light wave. is due only to the variation of the mass of the electrons which oscillate with relativistic velocities in the intense light beam. Even with respect to the shortest possible pulses this growth rate is instantaneous. In a discussion of self-focusing due to the combined mechanisms we therefore have to distinguish three different time regimes. They are characterised by the pulse length being much shorter than, comparable to, and longer than the time required for an ion acoustic wave to traverse the filament radius. For a typical laser produced plasma this time constant is of the order of 1 ns. If we consider first pulses of the order of 1O-* s duration, transient effects can obviously be neglected.
X(F)
dlarch
OPTICS COMMUNICATIONS
(31) A term y/20 < 1 has been neglected in eq. (20). Note that P,, is the critical power for the ponderomotive mechanism. To study eq. (20) we introduce dimensionless variables ,!?= PIP cr’
p =,I!?. (‘2)
$ =
z,/kR’,
CY= r/P,
with R = (flPcr /n)“2(p/c
1
)1’4 = 5.314 X 10’ ,r-“2 r
(cgs)
and (Y= 1.4X IO-l4 X2(m2)T(eV)~r,(cm
-3) .
(23)
Eq. (20) then becomes
“2P_ 1 at2 p3
( 1
__
p
exp(--dp2 -. 1 vGs7
1
(74)
Since Q is the ratio of the two critical powers it is always very small. For the term pa/p2 under the square root to become significant, the beam power has to be very high. But at this stage the exponential is so small that the self-focusing term in eq. (24) vanishes. Hence the saturation behaviour is always completely governed by the ponderomotive mechanism. Considerable rarefaction of the plasma within the filament saturates both ponderomotive and relativistic self-focusing and hence prevents relativistic effects altogether. This could only be avoided by increasing cysome orders of magnitude. According to eq. (23) this can be achieved ,by increasing X, T or ne. Since beam propagation is only possible below the plasma cut-off frequency an upper limit exists for the product r$
< 4#e0’neje~.
So, if at all, relativistic self-focusing in this time regime can only be observed in very hot plastnas (some keV), but even then (Yremains small. Fig. la shows the self-focusing length as function of the ratio of power to critical power for different initial beam diameters as obtained by numerically solving eq.
Volume
16, number
3
OPTICS COMMUNICATIONS
March 1976
and remains constant from there on. If the initial beam radius PO and the power ratio p satisfy the condition p20 =plln P
BEAM
POWER/CRITICAL
h__/ /I’,’
POWER
,,,,’
f-----p~~l ;
I
IOf
IO0 BEAM
POWER
102 /CRITICAL
10)
IO'
--__-
IO5
POWER
Fig. 1. The self-focusing distance (a) and minimum radius of the beam (b) as function of beam power for ponderomotive self-focusing in dimensionless units. The radius of the parallel incoming beam is PO.
(24) (for (Y= 0). The incident beam is assumed to be a plane wave. It can be seen that for increasing power the self-focusing length decreases until it reaches a minimum. Saturation effects are now noticeable and slow down the focusing process. All the curves then show a kink from where on they continue with an almost constant slope in the log-log plot. The nature of this kink becomes obvious from fig. lb where the minimum beam radius is shown for the same conditions. The focal radius is fairly constant over a large range of powers. It then increases until it abruptly reaches a maximum corresponding to the initial radius of the beam
(25)
the term in brackets in eq. (24) vanishes. The beam propagates with constant cross-section. For each value of p. > de * 1.65, eq. (25) can be satisfied with two different p values: 1 < per < e and ps > e. The lower value per is the factor with which the critical power given by eq. (10) has to be corrected in order to take the saturation term into account. This makes it slightly dependent on the initial beam radius. The upper value characterises the power for which the focusing mechanism saturates. It provides also an estimate for the rarefaction in the focal region where the plasma density drops by a factor of -l/p,. For a power above critical but srnaller than ps the beam contracts to a focus. From there on the beam radius oscillates between the initial beam radius and the radius of the focal spot. If p is larger than ps, the beam initially diverges. The self-focusing mechanism is so heavily saturated that diffraction is more effective. With expansion, however, both diffraction and saturation decrease. The beam reaches a maximum diameter and starts to contract again. The resulting oscillation is similar to the case p < p,, except that the initial radius is now equal to the minimum radius. The oscillatory behaviour is a property of the differential equation as has already been pointed out by Wagner [8] . Physically it is unlikely to be significant since (a) computer studies [9] have shown that the constant-shape approximation breaks down in the focal spot where a ring structure is formed and (b) other nonlinear mechanisms and absorption have to be taken into account. It is interesting to note that the radius of the focal spot is not only fairly constant (p z 1) over a wide range of powers between per and ps, but also practically independent of the initial beam radius pg. This suggests that R defined by eq. (23) is a focal spot radius which is characteristic for ponderomotive self-focusing. It is independent of the beam parameters and varies only with the plasma density.
405
Volume
16, number
5. Self-focusing
3
OPTICS COMMUNICATIONS
March 1976
of short pulses
Consider first pulses of picosecond duration. They are obviously so short that no density profile can be established within the beam cross-section and hence no ponderomotive self-focusing takes place. Relativistic self-focusing can occur, but only if the beam power is above the critical value given by ey. (16) which is of the order of 10” watts. In this high-power regime the focusing behaviour can then be described in a way similar to the ponderomotive self-focusing of long pulses. The equation for the beam radius vs distance of propagation in dimensionless units can be expressed in analogy with eq. (24).
IO' IO"
IO' BEAM
a2p --_=
-I
406
POWER/
CRITICAL
IO‘
IO'
POWER
( _____r,-
1 (26) (1 + p/p2)3’2 ’i P3 The definitions of < and p are those given by eq. (22) and p is now the ratio of beam power to critical power for relativistic self-focusing. Self-focusing lengths and minimum beam radii as functions of input beam power are shown in fig. 2a and 2b for different initial beam radii. These curves are in every respect fairly similar to those of fig. 1. The only significant difference is that the ratio of saturation power to critical power is much higher for relativistic self-focusing. There is again a characteristic focal spot radius R which is in fact the same as for ponderomotive self-focusing. With respect to self-focusing in a plasma we therefore conclude that short powerful pulses (ps, 2 10” W) behave very similar to long and weak pulses (2 10 ns, 2 lo6 W). A study of self-focusing in the intermediate time regime (” 1 ns) has to include transient effects. The nonlinear refractive index in a given cross-section at a given time is now not only a function of the instantaneous beam parameters, but depends on the time history as well. A detailed investigation might produce some interesting results, but is beyond the scope of this paper. Some general remarks can, however, be made. Due to the vastly different power ranges relevant for the two mechanisms, self-focusing at any instant in time is always either due to ponderomotive forces or relativistic effects, apart from the influence of the density profile which has already been established. There is no overlap of the two mechanisms but their self-focusing properties are comparable. The beam diameter is thus expected to acquire a similar horn-shaped form as has been reported for non-stationary self-focusing in media due
at2
IO3
IO'
0.3 100
IO’ BEAM
Fig. 2. The self-focusing the beam (b) as function focusing in dimensionless coming beam is po.
to one mechanism that the minimum stationary case.
IO]
IO’ POWER
/
CRITICAL
IO4
105
POWER
distance (a) and minimum radius of of beam power for relativistic selfunits. The radius of the parallel in-
only [ 10-121. It seems unlikely diameter deviates much from the
6. Conclusions Self-focusing of laser pulses in a plasma can be due to ponderomotive forces or relativistic effects. The two mechanisms operate in vastly different ranges of beam power and pulse length, but exhibit very similar properties. A parallel incoming beam with a power above critical and below saturation is focused to a spot of an almost constant diameter, dependent only on the plasma density. Pulses of some ten ns length and a power between
Volume 16, number 3
OPTICS COMMUNICATIONS
- IO5 to lo8 W can be focused to about 1 pm radius, resulting in power densities from 1012 to 1O1’ W/cm2. Pulses of ps duration and a power exceeding 1O’l W, focus to the same spot size, yielding power densities >1018 W/cm2. The latter ones are mainly interesting to study nonlinear beam plasma interactions with very high thresholds, such as electron-positron pair production. Here it is essential that self-focusing takes place without expulsion of plasma from the beam cross-section. On the other hand there are large ranges of pulse parameters to avoid self-focusing where it is undesirable, such as in laser fusion.
Acknowledgements The author would like to thank Professor H. Hora and Dr. J.L. Hughes for initiating the project and helpful discussions.
March 1976
References
111P. Kaw, G. Schmidt, T. Wilcox, Physics of Fluids 16 (1973) 1522.
121H. Hora, J. of Opt. Sot. Am. 65 (1975) 882. I31 P. Kaw, Appl. Phys. Lett. 15 (1969) 16. [41 H. Hora, Z. Physik 226 (I 969) 156. [51 C.E. Max, J. Arons, A.B. Langdon, Phys. Rev. L&t. 33 (1974) 209. 161 C.E. Max, to be publ. in Phys. of Fluids. 171 0. Svelto, Progress in Optics XII, ed. E. Wolf (NorthHolland, 1974) pp. 1-51 and references given therein. 181 W.G. Wagner, H.A. Haus, J.H. Marburger, Phys. Rev. 175 (1968) 256. PI EL. Dawes, J.H. Marburger, Phys. Rev. 179 (1969) 862. [lOI S.A. Akhmanov, A.P. Sukhorukov and R.V. Khokhlov, Sov. Phys. JETP 24 (1967) 198. 1111 F. Shimizu, E. Courtens, Fundamental and Applied Laser Physics, Proc. of the Esfahan Symp. (John Wiley, N.Y., 1973) pp. 67-79. 1121 J.A. Fleck, R.L. Carman, Appl. Phys. Lett. 20 (1972) 290.
407
16, number
3
SELF-FOCUSING
March 1976
OPTICS COMMUNICATIONS
IN A PLASMA DUE TO PONDEROMOTIVE
AND RELATIVISTIC
FORCES
EFFECTS
M.R. SIEGRIST Department of I:‘n@wering Canberra, A.C. T. Azrstralia
Received
18 November
f’/lvsics. Research
S&o01 of’f’lrvsical
Sciences,
The Australian
Natiorlal
Univcrsit~~.
1975
The propagation of intense laser pulses in a plasma is discussed in terms of a constant shape, paraxial ray approximation. Self-focusing due to ponderomotive forces and relativistic effects is investigated. It is found that the stationary self-focusing behaviour of each mechanism treated separately is similar, with several orders of magnitude difference in critical power. In stationary self-focusing due to the combined mechanisms, complete saturation of ponderomotive self-focusing prevents the occurrence of relativistic effects. Self-focusing lengths and minimum radii are given for a large range of beam powers. A characteristic focal spot radius is found which depends only on the plasma density.
1. Introduction Self-focusing of a laser beam in a plasma has been examined by a number of authors [l-6] Basically two different mechanisms have been proposed: selffocusing due to plasma density variations produced by ponderomotive forces and relativistic self-focusing due to the Lorentz factor of the optical constants in the plasma. It will be shown below that in a first-order approximation the type of focusing and saturation behaviour is the same for the two mechanisms. However. the threshold power and saturation flux, which are the two relevant parameters, differ by several orders of magnitude and the growth rates are different too. The question therefore arises how the two mechanisms interact. If they prove to be independent, then obviously each one influences the behaviour of the beam propagation in a different range of beam powers and a different time domain. Self-focusing is very undesirable in laser fusion applications where it could prevent compression of fuel pellets. On the other hand the self-focusing of a laser beam into a filament would provide an effective method of achieving the high flux densities required to study laser plasma interactions such as the generation of electron-positron pairs. By producing an appropriate nonuniform plasma it might also be possible to taylor the 402
self-focusing in order to preserve a high-quality beam structure. In this paper we shall determine self-focusing distances and focal spot radii for a wide range of beam powers. A constant-shape paraxial ray approximation is used. Absorption and other non-linear effects are neglected.
2. Stationary
beam propagation
in a plasma
Comprehensive studies of the propagation of a laser beam in a medium with nonlinear refractive index have been reported [7] and the basic properties are well understood. We are not at this stage interested in a very accurate description and hence have chosen the very transparent, paraxial ray approximation developed by Wagner et al. [S] Comparison of results with computer studies [9] has shown that many features of the numer. ical solutions are predicted quite accurately. For the radius r of a beam with gaussian intensity distribution as function of distance z, Wagner et al. [S] derive the following differential equation (1)
for a beam of power P. Here E is the envelope of the
Volume
16, number
OPTICS COMMUNICATIONS
3
electric field strength on the beam axis, EI and x are the linear and nonlinear susceptibilities respectively, p is the permeability and k = 27r/X. Expanding the nonlinear part of the susceptibility and keeping only the first nonvanishing term in a saturable form
March 1976
the exponential is developed into a power series up to second order, ~2 is determined as
(8) With /3E2) < 1 which is true as long as the electric field strength is not too close to the saturation value E, = (2/(3)1/2, this can be written as q
eq. (1) can be written as
(9)
PfPU
__=~
(3) (1 +
E2/E;)2
Hereby a critical power PC, and a saturation field Es have been introduced. The former represents the power for which a beam would propagate with a constant cross-section if saturation effects are neglected. The latter is the field strength for which saturation effects become appreciable. In this report we define the nonlinear susceptibility in terms of the time average of the square of the electric field x(E) = c2CE2)
(4)
which differs from Wagner’s notation The critical power then becomes
by a factor of 2.
which is identical with Wagner’s “standard” form [eq. (2)J. For the critical power one obtains PC, = 4nmec3e0kT/e2,
(5)
(10)
where T = i(T, + Ti). It depends only on the temperature and is numerically P,,(W) = 1.7 X 1O4T (eV).
(11)
The oscillation of the electrons in a very intense electromagnetic field has to be treated relativistically. According to Hora [2] and Max [5] the plasma frequency acquires a Lorentz factor ,2
.eY2
saturable
P
= cl_? (1 t -y(&-‘/2,
(12)
PO
with 2e2
In the following it will be shown that both the nonlinear susceptibilities due to ponderomotive forces and due to relativistic terms can be represented in a form similar to eq. (2).
3. Comparison focusing
of ponderomotive
and relativistic self-
The dielectric constant of a plasma due to the timeaveraged force acting on an electron in the field of an electromagnetic wave can be written as [l] * L
(
E(C.!f’)) = 1-
OP
-
exp(-$CE2))
cd2
1
eu,
(13)
y = ,2,2,2 e
and w p0 is the plasma frequency in the absence of the electromagnetic wave. First we study equation (6) with the relativistic term (I 2) but neglecting the ponderomotive force mechanism. This is valid for a pulse which is much shorter than the characteristic time constant for development of a ponderomotive force equilibrium. From (6) one obtains x(E) = (wgd2) [ 1 - (1 + yE2,)-“2]
EO.
(14)
For r(E2) < 1 this can be written as ?
with e2 p=---_ 2mew2k(Te
(7) t Ti)’
where w is the wave and wp the plasma frequency.
The nonlinear susceptibility has again a saturable form. In analogy with the ponderomotive mechanism we obtain a critical power of
If 403
Volume
16, number
3
a2r az2 For a Nd:glass laser this is nutnerically P,(W)
= 2.16 X 1030/~~e(cm
-3
).
Combining eqs. (6) and ( 12) yields the full nonlinear susceptibility in a plasma =
w&i
1 ~_
cd2
expl_:~~ J+ $I+
1
(18)
Substitution into eq. ( 1) and replacement of the axial electric field by the power in the gaussian beam according to (19) results in the following equation for the beam radius r as function of distance of propagation 404
(1
(‘0)
(17)
4. Stationary self-focusing due to the combined mechanisms
I
---1 k2r3
I9 76
with
Comparison of eqs. (11) and (17) shows the large differ. ence in the threshold for the two mechanisms. For a typical laser produced plasma of T = 10 eV and 17, = 1019 cmm3 this amounts to 6 orders of magnitude. Apart from threshold power the two mechanisms have also distinctly different growth rates. This has been investigated in detail by Max et al. [5]. In order to establish a characteristic density profile for ponderemotive self-focusing the beam cross-section has to be partially cleared of particles. For relativistic self-focusing, however, no density gradient is necessary. The change of plasma frequency, as seen by the light wave. is due only to the variation of the mass of the electrons which oscillate with relativistic velocities in the intense light beam. Even with respect to the shortest possible pulses this growth rate is instantaneous. In a discussion of self-focusing due to the combined mechanisms we therefore have to distinguish three different time regimes. They are characterised by the pulse length being much shorter than, comparable to, and longer than the time required for an ion acoustic wave to traverse the filament radius. For a typical laser produced plasma this time constant is of the order of 1 ns. If we consider first pulses of the order of 1O-* s duration, transient effects can obviously be neglected.
X(F)
dlarch
OPTICS COMMUNICATIONS
(31) A term y/20 < 1 has been neglected in eq. (20). Note that P,, is the critical power for the ponderomotive mechanism. To study eq. (20) we introduce dimensionless variables ,!?= PIP cr’
p =,I!?. (‘2)
$ =
z,/kR’,
CY= r/P,
with R = (flPcr /n)“2(p/c
1
)1’4 = 5.314 X 10’ ,r-“2 r
(cgs)
and (Y= 1.4X IO-l4 X2(m2)T(eV)~r,(cm
-3) .
(23)
Eq. (20) then becomes
“2P_ 1 at2 p3
( 1
__
p
exp(--dp2 -. 1 vGs7
1
(74)
Since Q is the ratio of the two critical powers it is always very small. For the term pa/p2 under the square root to become significant, the beam power has to be very high. But at this stage the exponential is so small that the self-focusing term in eq. (24) vanishes. Hence the saturation behaviour is always completely governed by the ponderomotive mechanism. Considerable rarefaction of the plasma within the filament saturates both ponderomotive and relativistic self-focusing and hence prevents relativistic effects altogether. This could only be avoided by increasing cysome orders of magnitude. According to eq. (23) this can be achieved ,by increasing X, T or ne. Since beam propagation is only possible below the plasma cut-off frequency an upper limit exists for the product r$
< 4#e0’neje~.
So, if at all, relativistic self-focusing in this time regime can only be observed in very hot plastnas (some keV), but even then (Yremains small. Fig. la shows the self-focusing length as function of the ratio of power to critical power for different initial beam diameters as obtained by numerically solving eq.
Volume
16, number
3
OPTICS COMMUNICATIONS
March 1976
and remains constant from there on. If the initial beam radius PO and the power ratio p satisfy the condition p20 =plln P
BEAM
POWER/CRITICAL
h__/ /I’,’
POWER
,,,,’
f-----p~~l ;
I
IOf
IO0 BEAM
POWER
102 /CRITICAL
10)
IO'
--__-
IO5
POWER
Fig. 1. The self-focusing distance (a) and minimum radius of the beam (b) as function of beam power for ponderomotive self-focusing in dimensionless units. The radius of the parallel incoming beam is PO.
(24) (for (Y= 0). The incident beam is assumed to be a plane wave. It can be seen that for increasing power the self-focusing length decreases until it reaches a minimum. Saturation effects are now noticeable and slow down the focusing process. All the curves then show a kink from where on they continue with an almost constant slope in the log-log plot. The nature of this kink becomes obvious from fig. lb where the minimum beam radius is shown for the same conditions. The focal radius is fairly constant over a large range of powers. It then increases until it abruptly reaches a maximum corresponding to the initial radius of the beam
(25)
the term in brackets in eq. (24) vanishes. The beam propagates with constant cross-section. For each value of p. > de * 1.65, eq. (25) can be satisfied with two different p values: 1 < per < e and ps > e. The lower value per is the factor with which the critical power given by eq. (10) has to be corrected in order to take the saturation term into account. This makes it slightly dependent on the initial beam radius. The upper value characterises the power for which the focusing mechanism saturates. It provides also an estimate for the rarefaction in the focal region where the plasma density drops by a factor of -l/p,. For a power above critical but srnaller than ps the beam contracts to a focus. From there on the beam radius oscillates between the initial beam radius and the radius of the focal spot. If p is larger than ps, the beam initially diverges. The self-focusing mechanism is so heavily saturated that diffraction is more effective. With expansion, however, both diffraction and saturation decrease. The beam reaches a maximum diameter and starts to contract again. The resulting oscillation is similar to the case p < p,, except that the initial radius is now equal to the minimum radius. The oscillatory behaviour is a property of the differential equation as has already been pointed out by Wagner [8] . Physically it is unlikely to be significant since (a) computer studies [9] have shown that the constant-shape approximation breaks down in the focal spot where a ring structure is formed and (b) other nonlinear mechanisms and absorption have to be taken into account. It is interesting to note that the radius of the focal spot is not only fairly constant (p z 1) over a wide range of powers between per and ps, but also practically independent of the initial beam radius pg. This suggests that R defined by eq. (23) is a focal spot radius which is characteristic for ponderomotive self-focusing. It is independent of the beam parameters and varies only with the plasma density.
405
Volume
16, number
5. Self-focusing
3
OPTICS COMMUNICATIONS
March 1976
of short pulses
Consider first pulses of picosecond duration. They are obviously so short that no density profile can be established within the beam cross-section and hence no ponderomotive self-focusing takes place. Relativistic self-focusing can occur, but only if the beam power is above the critical value given by ey. (16) which is of the order of 10” watts. In this high-power regime the focusing behaviour can then be described in a way similar to the ponderomotive self-focusing of long pulses. The equation for the beam radius vs distance of propagation in dimensionless units can be expressed in analogy with eq. (24).
IO' IO"
IO' BEAM
a2p --_=
-I
406
POWER/
CRITICAL
IO‘
IO'
POWER
( _____r,-
1 (26) (1 + p/p2)3’2 ’i P3 The definitions of < and p are those given by eq. (22) and p is now the ratio of beam power to critical power for relativistic self-focusing. Self-focusing lengths and minimum beam radii as functions of input beam power are shown in fig. 2a and 2b for different initial beam radii. These curves are in every respect fairly similar to those of fig. 1. The only significant difference is that the ratio of saturation power to critical power is much higher for relativistic self-focusing. There is again a characteristic focal spot radius R which is in fact the same as for ponderomotive self-focusing. With respect to self-focusing in a plasma we therefore conclude that short powerful pulses (ps, 2 10” W) behave very similar to long and weak pulses (2 10 ns, 2 lo6 W). A study of self-focusing in the intermediate time regime (” 1 ns) has to include transient effects. The nonlinear refractive index in a given cross-section at a given time is now not only a function of the instantaneous beam parameters, but depends on the time history as well. A detailed investigation might produce some interesting results, but is beyond the scope of this paper. Some general remarks can, however, be made. Due to the vastly different power ranges relevant for the two mechanisms, self-focusing at any instant in time is always either due to ponderomotive forces or relativistic effects, apart from the influence of the density profile which has already been established. There is no overlap of the two mechanisms but their self-focusing properties are comparable. The beam diameter is thus expected to acquire a similar horn-shaped form as has been reported for non-stationary self-focusing in media due
at2
IO3
IO'
0.3 100
IO’ BEAM
Fig. 2. The self-focusing the beam (b) as function focusing in dimensionless coming beam is po.
to one mechanism that the minimum stationary case.
IO]
IO’ POWER
/
CRITICAL
IO4
105
POWER
distance (a) and minimum radius of of beam power for relativistic selfunits. The radius of the parallel in-
only [ 10-121. It seems unlikely diameter deviates much from the
6. Conclusions Self-focusing of laser pulses in a plasma can be due to ponderomotive forces or relativistic effects. The two mechanisms operate in vastly different ranges of beam power and pulse length, but exhibit very similar properties. A parallel incoming beam with a power above critical and below saturation is focused to a spot of an almost constant diameter, dependent only on the plasma density. Pulses of some ten ns length and a power between
Volume 16, number 3
OPTICS COMMUNICATIONS
- IO5 to lo8 W can be focused to about 1 pm radius, resulting in power densities from 1012 to 1O1’ W/cm2. Pulses of ps duration and a power exceeding 1O’l W, focus to the same spot size, yielding power densities >1018 W/cm2. The latter ones are mainly interesting to study nonlinear beam plasma interactions with very high thresholds, such as electron-positron pair production. Here it is essential that self-focusing takes place without expulsion of plasma from the beam cross-section. On the other hand there are large ranges of pulse parameters to avoid self-focusing where it is undesirable, such as in laser fusion.
Acknowledgements The author would like to thank Professor H. Hora and Dr. J.L. Hughes for initiating the project and helpful discussions.
March 1976
References
111P. Kaw, G. Schmidt, T. Wilcox, Physics of Fluids 16 (1973) 1522.
121H. Hora, J. of Opt. Sot. Am. 65 (1975) 882. I31 P. Kaw, Appl. Phys. Lett. 15 (1969) 16. [41 H. Hora, Z. Physik 226 (I 969) 156. [51 C.E. Max, J. Arons, A.B. Langdon, Phys. Rev. L&t. 33 (1974) 209. 161 C.E. Max, to be publ. in Phys. of Fluids. 171 0. Svelto, Progress in Optics XII, ed. E. Wolf (NorthHolland, 1974) pp. 1-51 and references given therein. 181 W.G. Wagner, H.A. Haus, J.H. Marburger, Phys. Rev. 175 (1968) 256. PI EL. Dawes, J.H. Marburger, Phys. Rev. 179 (1969) 862. [lOI S.A. Akhmanov, A.P. Sukhorukov and R.V. Khokhlov, Sov. Phys. JETP 24 (1967) 198. 1111 F. Shimizu, E. Courtens, Fundamental and Applied Laser Physics, Proc. of the Esfahan Symp. (John Wiley, N.Y., 1973) pp. 67-79. 1121 J.A. Fleck, R.L. Carman, Appl. Phys. Lett. 20 (1972) 290.
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