2. DESCRIPTION OF THE PROPOSED LASER AMPLIFICATION CONCEPT WITH PERFECT REFLECTION AND NO ABSORPTION

The proposed laser amplification concept is explained in Figure 1. A laser pulse L coming from the left is injected into a metallic capillary liner [ell ] over which a large electric current is discharged, imploding the liner by the pinch effect or by ablatively imploding it through the absorption of soft X-rays on its outer surface. As the laser beam bounces back and forth in between the mutually approaching walls of the imploding liner, its frequency rises steadily. Assuming perfect reflection from the liner wall, the number of photons is conserved, and the energy of the laser beam is amplified by its conversion to shorter wave lengths. Because the liner behaves like a waveguide for electromagnetic waves, there is a cut-off frequency ν_c ∼ c/r, where c is the velocity of light and r is the inner liner radius, and no laser pulse with a frequency less than ν_c can propagate inside the liner (Landau & Lifshitz, 1960).

Implosion of a laser beam inside electric pulse power imploded capillary liner by the pinch effect: L laser beam, [ell ] liner, H magnetic field, j pinch current.

The amplification of the laser beam pulse results from the Doppler-effect on a light wave reflected back and forth in between the walls of the imploding liner, leading to an increase in the frequency for each reflection given by

There v is the liner implosion velocity and α is the angle of incidence on the liner wall, with α = 0° for a perpendicular incidence. The factor two results from the twice as large velocity of the light image from the optically reflecting liner wall.

For the described laser amplification process it is important that the implosion time

is shorter than the time

the laser beam is inside the liner of length [ell ], which means the laser pulse must have a length less than cτ_L = [ell ]. Therefore,

For the example v ∼ 3 × 10⁸ cm/s and r ∼ 3 × 10⁻² cm, one has [ell ] [gsim ] 3 cm.

Because metallic conductors become transparent for frequencies larger than their electron plasma frequency ω_p ≃ 10¹⁶ s⁻¹, no amplification is possible for wave lengths shorter than ∼c/ω_p, except by Bragg reflection under a glancing angle of incidence.

Under ideal conditions, the amplification is inversely proportional to the square of the liner radius. Therefore, a 30-fold decrease of the liner radius from 3 × 10⁻² cm down to 10⁻³ cm, would increase the laser energy about 1000-fold, for example, from an infrared laser pulse with a wavelength less than 10⁻³ cm and an energy of ∼100 J, to a 100 kJ laser pulse with a wavelength of ∼10⁻⁶ cm in the far ultraviolet, respectively, the soft X-ray domain. The proposed scheme would accordingly amplify a ∼10⁻¹⁰ s infrared terawatt laser pulse into a 10⁵ J, 10⁻¹⁰ s, soft X-ray petawatt laser pulse, with the energy for the amplification drawn from a Marx generator.

3. LASER BEAM AMPLIFICATION INSIDE THE IMPLODING LINER

As shown in Figure 2, we consider a conical laser beam injected into the liner under angle α. There, the position of constructive interference inside the liner moves to the right with the group velocity

In the time a photon moves from A to B with the velocity of light c, the location of constructive interference is displaced by the distance from A′ to B′ with the group velocity vg = c sin α.

For the following, it is convenient to go into a reference system moving with the velocity v_g. If the energy of the laser pulse in the laboratory rest frame is W and occupies the volume V, the energy and volume in this frame is (β = v_g /c)

With the radiation energy density given by u = W/V, one has the energy density in this system

The advantage of going to this frame is that in this frame the laser light strikes the liner wall under a 90° degree angle of incidence. This makes it easy to compute the radiation pressure in this frame.

The computation of the laser light amplification becomes especially simple in the short wave length limit, where the wave length of the laser light is small compared to the inner diameter of the capillary. Following Linhart (1969), one can use the mechanics of photons, as particles moving with the velocity of light.

If the total number of photons is N, the fraction N/2 of them move with the velocity of light c toward the wall, and where each of them will change its momentum upon reflection from the wall by 2h/λ = 2hν/c, resulting in a pressure on the wall equal to

and hence,

If V′ decreases by −dV′, the energy W′ increases by

which upon integration yields

Because of (6) and (7), this also means that

With p = W/V, this leads to the equation of state for the photon gas

With V = πr², the liner volume per unit liner length, one has for the pressure and energy density of the photon gas inside the liner

During the compression, the laser energy per unit length, W = πr²u, therefore increases as

With the number N of photons conserved whereby W = Nhν, the laser light frequency increases in the same proportion by

4. THE BUNCHING OF THE LASER PULSE DURING ITS COMPRESSION

During a reflection from the imploding liner wall the direction of the laser beam steepens, reducing the angle of incidence α, and reducing the group velocity v_g = c sin α. For many consecutive reflections, this can lead to a substantial axial bunching of the laser beam, but the angle of incidence can be kept constant if the liner has the shape of a divergent cone. As illustrated in Figure 3, during a reflection the perpendicular wave number component k_⊥ = k sin α is changed by (Sommerfeld, 1950)

where v << c is the radial liner implosion velocity. If δα is the angle by which the angle of incidence α decreases following a reflection from the wall, one has from the sinus law

and hence,

Steepening of the laser beam inside the imploding liner.

To prevent a steepening of the laser beam, taking into account the two reflections from opposite sides of the imploding wall, the opening angle of the capillary liner tube would have to be twice as large as δα. For β << 1, one thus obtains for the opening angle of the conical liner

For the example β = 10⁻² and sin² α ≤ 1, this angle is 2δα ≤ 4 × 10⁻² radians ≃1°, which is quite small.

5. SCATTERING FROM THE LINER WALL

Depending on the smoothness of the liner wall, there will be some scattering of the laser beam into other directions than those which occur under a mirror-like reflection.

In general, one has

where α is the angle of incidence, different from α = 90°, with

the average of the cosine of this angle. For uniform scattering in two dimensions, one has

, and in three dimensions

.

In two dimensions, one thus has p = u/2 and WV^1/2 = const., hence

And in three dimensions, likewise p = u/3 and WV^1/3 = const., hence

With p = u/3 valid for black body radiation, Eq. (24) is also valid for black body radiation inside the imploding capillary.

The photon particle description becomes invalid if the laser light wave length can not be considered small if compared with the diameter r of the capillary. The limit where the laser wave length becomes about equal to the diameter of the capillary, is determined by the uncertainty principle ΔpΔq ≥ ħ, where one has to set Δp ≥ mc, Δq ≥ r, hence W = Nmc² = Nħc/r = const./V^1/2, which leads to (23), in the particle description corresponding to p = u/2.

Assuming that

is in between the case for perfect reflection,

, and two dimensional isotropic scattering,

, the arithmetic average is

. There WV^3/4 = const., and hence

At high radiation intensities where the capillary may become opaque through material ablated from its inner wall, the emerging situation is a mixture of all three cases,

. Taking the arithmetic average one has

and WV^11/18 ≈ WV^0.61 = const., hence

6. ENERGY LOSS BY ABSORPTION

If for perpendicular incidence, the reflection coefficients is R_⊥, it is for non-perpendicular incidence R_⊥ cos α. Upon reflection the fraction of the photon energy absorbed then is (1 − R_⊥)cos α. To have an energy amplifying effect one must have

hence

At an energy of ∼1 eV one has for gold and silver R_⊥ ≈ 0.99. There one must have β > 5 × 10⁻³ or υ > 1.5 × 10⁸ cm/s. For energies larger than 1 eV the reflection coefficient rapidly goes down requiring an even higher implosion velocity.

These are two ways to reduce β:

In the first case, the high refractive material is eventually transformed into a hot plasma. Quite apart from reducing the velocity of light by the factor 1/n, the energy absorbed in the plasma is not lost in the liner wall. The importance of this has been demonstrated by Luther et al. (2004), who showed that a laser beam can with little loss be guided in a highly ionized plasma inside a capillary. For the frequencies of interest, large values of the refractive index occur near resonances, for a magnetized plasma at large magnetic fields, or for high Z non-hydrogen plasmas. A disadvantage is the large absorption near these resonances.

In the second possibility, where one makes use of the “scissors effect” by giving the liner a non-circular cross section, the effective implosion velocity can be increased by at least one order of magnitude.

If the liner has a lenticular cross section as shown in Figure 4a, idealized in Figure 4b by a diamond shape with a major and minor semi-axis a and b, and if the implosion velocity along the minor semi axis is υ_○, the implosion velocity along the major semi-axis is by the “scissors effect” enlarged from v₀ to

With a ≥ 20b which is technically feasible, this would mean a more than 20-fold increase of the implosion velocity in one direction. By giving the liner a “star-like” cross section with a sufficient number of spikes, the implosion velocity can be increased by the same factor in all directions.

Lenticular liner cross section.

To prevent jetting from the corners of the scissors, the opening angle of the corners must be less than 10°, as experiments with shape charges have shown. This means that a/b ≥ 20 or υ ≥ 20υ_○. On the other hand, one must have υ = (a/b)υ_o < c for the photons to be “squeezed out” from the corners, respectively, reflected from the mirror formed by the scissors. For example, υ_○ = 3 × 10⁷ cm/s, one would have υ ≥ 6 × 10⁸ cm/s and υ/c ≈ 2 × 10⁻² << 1.

The foregoing considerations are not valid for high photon intensities inside the capillary. There the wall of the capillary is heated up to high temperatures by photon absorption. But there the wall also becomes the source of intense black body photon reemission into the capillary. For an imploding capillary, we have the situation which exists in an imploding dynamic hohlraum (Winterberg, 1980). If the wall temperature is T ° K, the degree of ionization of the wall material can be estimated from the formula

Knowing the value of T, one can then compute the value of Z and the opacity coefficient (Schwarzschild, 1958)

From there one obtains the photon path length

where ρ is the density of the wall material. Knowing the photon path length, one can compute the velocity v_d by which the photons diffuse the distance x into the wall

It is reasonable to assume a wall temperature 5 × 10⁶° K < T < 10⁷° K. For ρ = 18 g/cm⁻³ (gold capillary), one obtains 5 × 10⁻⁶ cm < λ_p < 5 × 10⁻⁵ cm. Assuming that x ≃ 10⁻³ cm, about the same length as the capillary radius at maximum compression, one finds that 3 × 10⁷ cm/s < v_d < 3 × 10⁸ cm/s. The velocity v_d must be smaller than the implosion velocity. In case of the scissors effect, the effective implosion velocity must be taken, because it is this implosion velocity that determines the time taken by the implosion.

This example shows that the condition v_d < v_imp can be reasonably well satisfied.

7. CONNECTION TO DYNAMIC HOHLRAUM CONFIGURATIONS

In indirect drive hohlraum configurations, one or more laser beams dissipate their energy inside a cavity, transforming the non-absorbed laser energy into soft X-rays. And in the dynamic hohlraum configurations the produced soft X-ray radiation is subsequently imploded, increasing its energy inside the hohlraum.

The conversion of the incoming laser light into soft X-rays can simply be understood as a conversion of the laser energy into black body radiation, by the absorption and reemission of the laser light energy from the inner wall of the hohlraum. Because this is an irreversible process, the conversion takes place with an efficiency η < 1. Experiments suggest that by order of magnitude η ≈ 0.2.

If the incoming laser light has an energy W₀, the energy converted into soft X-rays then is W_x = ηW₀, with the hohlraum black body radiation temperature T_h given by the Stefan-Boltzmann law

and with the average frequency ν of this radiation given by Wien's law (k Boltzmann constant):

We have to apply these results to a capillary hohlraum of initial radius r₀ and length l, with the soft X-ray black body radiation in it generated by a laser beam shot into the capillary, and with the capillary imploded from its initial radius r₀, to a smaller radius r < r₀. If the implosion is fast enough, with the time for the implosion shorter than the time the radiation is lost by absorption in the wall of the capillary, the black body radiation behaves more like a grey body radiation, provided the photon mean free path is large compared to [ell ]. If n is the number density of atoms vaporized from the wall of the capillary, and if σ_p ≈ 10⁻¹⁸ cm² (

cm, Bohr radius) is the absorption cross section of the photons, one has for the photon mean free path λ_p = 1/nσ_p ∼ (10¹⁸/n)cm, and hence n < (10¹⁸/[ell ])cm⁻³, for grey body radiation. For a 10⁻¹⁰ s laser pulse one has [ell ] ≈ 3 cm, and hence n < 3 × 10¹⁷ cm⁻³, a comparatively small number density.

According to (26), the energy of the soft X-ray photon gas rises in proportion to r^−1.22, and one has:

For the energy W(r) and power P(r), with W₀ and P₀ being the initial laser energy and power, one then has:

For example, η = 0.2, r₀ /r = 30, one finds that W(r)/W₀ = P(r)/P₀ ≈ 10, that is a 10-fold amplification.

If the soft X-ray pulse released from the capillary following it's implosive compression lasts 10⁻¹⁰ s, and if this energy is W_x(r) ≈ 10⁵ J, as it is required for the fast ignition of 10³-fold compressed liquid DT target, one would need W₀ ∼ 10⁴ J, for the incoming laser energy of photons in the optical region, amplified 10-fold into soft X-ray photons. If the laser energy is supplied in ∼10⁻¹⁰ s, the laser power would be 10¹⁴ W, amplified into a 10¹⁵ W soft X-ray pulse.

This example shows, that the proposed technique, can under rather pessimistic assumptions eliminate the need for an expensive 10⁵ J petawatt laser by at least one order of magnitude, otherwise required for fast ignition.

8. “DIRECT DRIVE” LINER IMPLOSION BY THE PINCH EFFECT

If m is the mass per liner length and H is the magnetic field at its surface set up by a pinch current I in Ampere whereby H = I/5r, the equation of motion of the liner is

With H² = H₀²(r₀ /r)², this becomes

and with dv/dt = (½)dv²/dr and integration one has

For a thin liner of thickness δ and density ρ we can set m ≃ 2πr₀ ρδ whereby (41) can be written as follows

where

is the Alfvén velocity at r = r₀. For thin liners we may put δ ∼ r₀. This shows that the implosion velocity is determined by the Alfvén velocity. As an example, we choose I = 5 × 10⁶ A, r₀ = 3 × 10⁻³ cm, and 4πρ ∼ 10²g/cm³. We obtain v_A ≃ 3 × 10⁷ cm/s. The implosion time then is τ_imp ∼ r₀ /v_A ∼ 10⁻¹⁰ s. With I = 5 × 10⁷ A, the implosion velocity with v_A ≈ 3 × 10⁸ cm/s would be 10 times larger.

To obtain the maximum possible laser amplification, one simply has to equate (½)mυ² given by (41) with W. One obtains

This has to be combined with (16) and (17), respectively. Taking the example I = 5 × 10⁶ A, r₀ = 3 × 10⁻³ cm, r = 10⁻⁴ cm, ν₀ = c/r₀ = 10⁻¹³ s⁻¹, one finds W ≃ 10¹² erg/cm = 10⁵ J/cm, W/W₀ = 10³, ν = 10¹⁶ s⁻¹, W₀ = 100 J. This means a cm long capillary liner would under ideal conditions with no absorption by the liner wall amplify an infrared 100 J terawatt laser pulse into an ultraviolet 10⁵ J petawatt laser pulse. With absorption losses the amplification is, of course, much less.

9. “INDIRECT DRIVE” ABLATIVE LINER IMPLOSION

The liner implosion by the pinch effect is not the only one possible. As in inertial confinement fusion, one can also ablatively implode the liner. If this is done with electric pulse power, the energy must first be converted into a burst of soft X-ray black body radiation inside a hohlraum, which thereafter ablatively implodes the liner. As it has been demonstrated by the Sandia National Laboratories, this can be done by discharging a large current over a cylindrical wire array (Sanford et al., 1997). This scheme has the advantage that it leads to a pulse shortening by about one order of magnitude. The indirect drive has the additional benefit in that it promises a more uniform implosion, reducing the growth of the Rayleigh-Taylor instability.

The indirect drive liner implosion suggests an integrated soft X-ray compression fast ignition inertial confinement fusion concept, illustrated in Figure 5.

Integrated fast ignition concept where both the thermonuclear target T and the capillary liner [ell ], are compressed inside a hohlraum filled with soft X-rays produced from electric pulse power.

It consists of a spherical inertial confinement fusion target connected to one end of the laser beam amplifying liner, with both the liner and the target imploded inside one hohlraum by the soft X-ray from the electric pulse power. A low energy laser beam entering from the other side of the liner and after being greatly amplified, is then used for the fast ignition of the target.

10. COUPLING OF THE ELECTRIC PULSE POWER TO THE LINER IMPLOSION

The energy flux required to implode the liner to an implosion velocity, v ∼ v_A is determined by the Poynting vector

With E = (v_A /c)H one has

For a liner of radius r and length [ell ], the power which has to be delivered is

which has to be equal to the electric pulse power (I current, V voltage)

Assuming that I remains constant, one has (in Gaussian units)

where the self inductance, L, of a wire of length [ell ] and radius r is

Hence,

and thus,

With H = 2I/rc (in Gaussian units) one then obtains from (50) and (51)

the same as in (46). Therefore, the impedance must be (in Gaussian units)

Conversion into practical units, achieved by multiplying (53) with 10⁻⁹ c², results in

If for example, v_A = 3 × 10⁷ cm/s, 2[ell ]/r = 10³, one has Z ≃ 30Ω. Assuming a high energy transfer efficiency from the electric pulse power into the laser pulse by putting W ≃ P one has

We compare what can be achieved with electric pulse power with the extreme example of a high efficient petawatt laser where W ≃ P = 10¹⁵ Watt. Electric pulse power of this magnitude is, in theory, attainable with magnetic insulation (Winterberg, 1968). We find that I ≃ 5.8 × 10⁶ Ampere, V = 1.7 × 10⁸ Volt. It is the high voltage which implies a short discharge time, τ, given by ¼ of the Thomson time (in Gaussian units)

where L is given by (49) and the capacitance C by

with R the radius of the return current conductor, hence,

For the example [ell ] ∼ 10 cm one has τ ≃ 3 × 10⁻⁹ s.

With spark gap switches closing within 10⁻⁹ s, the discharge time under these conditions cannot be shorter than 10⁻⁹ s. This requires a lower voltage which leads to a reduction in the energy transfer efficiency, going in the inverse proportion to the square of the voltage. It is for this reason that it is desirable to employ a pulse power compression scheme. To some degree, it already exists in the mechanism of the liner implosion, where electric energy is cumulated into kinetic energy of the liner. It has been realized in the indirect ablative implosion mode where as mentioned above, a ∼10-fold pulse power compression is possible (Sanford et al., 1997).

A potentially more efficient way for pulse power compression is by the implosion of an array of concentric shell, as shown in Figure 6. There, unlike as in a convergent cylindrical shock wave, a much larger rise in the kinetic energy toward the center of convergence seems possible (Winterberg, 1981). If a ∼ 100-fold pulse power compression can be achieved in this way, which seems theoretically possible, then an electric pulse power input of 10¹³ Watt would be sufficient to amplify a laser beam to 10¹⁵ Watt.

Implosion of multiple concentric cylindrical shells with a laser amplifying liner in the center.

11. ENERGY CONSIDERATIONS

If e is the laser beam energy following its amplification in the liner of length [ell ], one has

where

One therefore has

which means that [ell ] scales as 1/W, leading to large liner lengths for a small laser power W at the same total laser energy e.

Also, since

one has

Setting, as before, W ≃ P to have a high electric pulse power laser energy transfer efficiency, and using Eq. (46) for P, one has

and hence, with (60) that

or that

This result shows that at a given energy e the implosion velocity v ∼ v_A scales as W^1/2.

As an example, we take P ≃ W ∼ 10¹² Watt, τ ≃ 10⁻⁸ s, and e ≃ 10⁵ J. We find that [ell ] ≃ 30 cm and v ≃ 3 × 10⁶ cm/s. A velocity of 3 × 10⁶ cm/s was estimated by Linhart (1969) as the lower bound to obtain microwave amplification by compression.

The large length of the liner [ell ] ≃ 30 cm, suggests to deform it into a helix with a radius R >> r. In such a configuration, shown in Figure 7, the length of the wound up helical transmission line is

where N is the number of helical turns. If n is the number of turns per unit length, the height h of the helix is

hence

Helical liner to axially compactify long liner of length [ell ] >> h, into a configuration which can be placed in a hohlraum of axial length h << [ell ].

As an example, we take n ≃ 10 cm⁻¹ and 2R ≃ 0.3 cm, resulting in h/[ell ] ≃ 0.1. In this example, the 30 cm long liner would be axially compactified to 3 cm. The direct pinch effect drive is not here feasible, but the indirect soft X-ray induced implosion drive by placing the helix in a small hohlraum seems feasible.

For liners smaller than [ell ] ≃ cτ ≃ (c/v)r, the energy conversion efficiency, electric pulse power into laser energy, is smaller in the same proportion, but it would still be of interest for the conversion into laser radiation with smaller energy but shorter wavelengths.

12. PREVENTION OF SHOCK FORMATION AND LINER WALL ABLATION

While the effective liner implosion velocity can be increased about 10-fold from a few 10⁷ cm/s to more than 10⁸ cm/s (as it is required by (28)) by the “scissors effect,” this leaves open the question how during the implosion the formation of shocks inside the liner can be prevented. In reaching the inner surface of the capillary liner, a shock would be followed by a rarefaction wave, filling and closing the capillary with material from its wall. There are three means to prevent this from happening, all of which or any combination of them may be used. They are:

1. Imploding the capillary with a special current-time profile, similar as for the shock-less isentropic compression to reach high densities.
2. Placing the liner inside a large (∼10⁵ G) axial magnetic field which during the implosion of the liner is amplified to high field strengths (∼10⁸ G), acting like a cushion, preventing the occurrence of a rarefaction wave and preventing the capillary from being filled with material from its inner wall.
3. Twisting the non-circular cross section of the capillary (Fig. 4) into a helix, leading in the course of its implosion to a rapid azimuthal rotation, with the centrifugal force field preventing the occurrence of a rarefaction wave and the wall material to fill the capillary.

To 1: As it was shown by the author (Winterberg, 1978), the occurrence of shocks during the implosion of a pinch discharge can be prevented by a current-time dependence of the form

For this time dependence, all the simple waves (Mach lines) arrive simultaneously at r = 0. Eq (70) was derived for a pinch discharge, not for a hollow liner, but must be there quite similar, in as much as during its isentropic implosion the distribution of matter approaches the form of a cylindrical shell, with the material in its outer periphery more compressed that toward its center.

To 2: An axial magnetic field H ∼ 10⁵ G placed inside the capillary would due to Hr² = const., for an implosion ratio r₀ /r ≃ 30, be amplified ∼10³ fold to H ∼ 10⁸ G, with a magnetic pressure of the order ∼10¹⁵ dyn/cm². This pressure is strong enough to balance the stagnation pressure ρv² of the capillary with an inward velocity of v ∼10⁷ cm/s.

To 3: This possibility had been previously proposed by the author (Winterberg, 1999) as a means to stabilize a linear pinch discharge by axial and rotational shear flow, and by the centrifugal force from a rapid azimuthal rotation.