Material Response to Peak Intensity

Where high peak pulse power really starts to have a significant effect is when the intensity is high enough to flash a thin surface layer to plasma. As previously described, a dense enough plasma will absorb the light incident upon it. When initially formed, the plasma will be the same density as the solid or liquid from which it was produced. Condensed matter tends to have between 1×1029 and 2×1030 ionizable (valence) electrons per cubic meter. This corresponds to a transparency cutoff at wavelengths of between 2×10-8 meters and 4×10-9 meters. All wavelengths longer than this will be absorbed by solid density plasmas. These very short wavelengths are in the soft x-ray range - any light to which the atmosphere is transparent, and any light which can be conveniently focused by mirrors or lenses, will be absorbed by the solid density plasma produced by sufficiently intense pulses.

How intense is sufficiently intense? A general rule of thumb is that the threshold for plasma production occurs at between 1013 W/m2 and 1014 W/m2.

Plasma is simply an ionized gas. It is a fluid that expands to fill the available volume. A solid density plasma is under intense pressure - the same pressure as a gas would be if you squeezed it down to the same density as water or rock and then heated it up to over 100,000 K. We are talking hundereds of thousands of atmospheres. This has two effects. The first is that the plasma immediately starts to expand. The expanding plasma continues to absorb the incident light from the beam. In some cases, the energy of the beam drives the expansion of the plasma, and the plasma shields the target from further damage.

The second effect occurs from the intense pressure of the plasma. This is far above the material strength of any matter held together by chemical bonds. The expanding plasma simply pushes away the intervening matter, which flows aside as if it were a fluid. The plasma is expanding much faster than the speed of sound in the target, thus the surrounding matter is pushed away faster than the speed of sound. This creates a shock wave that propagates through the matter. We are familiar with the effect of shock waves from detonating high explosives. It will gouge out a crater, drive cracks through the material, send fragments flying, and may cause spallation (spallation is when the back surface of the target breaks away and flies off at high speeds after the shock wave hits it).

The problem for a laser weapons engineer is to adjust the parameters of the beam such that the maximum energy is delivered to the material through the plasma. The blast from the plasma will have a similar effect to the detonation of a high explosive with a similar energy yeild to the pulse energy that is in direct contact with the target. The energy yeild of TNT is about 4 MJ/kg, so a pulse of energy E will have a blast effect similar to (E / 4 MJ) kg of TNT.

	The beam is initially incident on the target. A thin layer of target material is flashed to plasma, and the plasma absorbs the rest of the pulse. At this point, the plasma is at extremely high pressure, much greater pressure than the strength of the target material.
	The expanding plasma launches a blast wave. A shell of hot vapor blasts out into the air, while a shell of super-hot, highly compressed material propagates into the target, accumulating target material into the blast front. Early on, the blast front into the target is moving as fast as the blast bubble into the air. Pressures are so high the target material flows like a fluid.
	As more material is accumulated into the blast front, the blast front slows down. This is almost entirely due to the increased mass of moving material, the dynamic pressures are still so much higher than material strengths that the material essentially behaves like a fluid.
	Eventually, the blast wave has done so much work pushing through the target material and has spread out so much that the pressure is no longer far in excess of the material strength. The target material starts resisting compression and starts behaving like a rigid solid. The flowing blast wave, no longer able to bull its way through the target material, is instead re-directed along the crater walls and splashes out into the air.
	Once the fluidized material that made up the blast wave has squirted out of the crater along the crater sides and into the air, what is left is a permanent crater, surrounded by a layer of compressed and deformed material, and beyond that the mostly undamaged target material.

A death ray will probably want to drill deeply in order to reach vital organs or components. It can do this with less energy by blasting out new crater inside of previous ones. You can imagine that early in the process, adding new energy to the energetic blast wave will not have too much difference compared to just hitting the target with a single pulse of greater energy, and you get a unidirectional expansion rather than drilling deep. It is therefore preferable to wait until the blast wave is no longer much greater in pressure than the material strength.

Here is a calculator for estimating the effects of pulsed laser blasts on a target.

Here are the details of how the calculator does its calculations.

A rough idea of the size of a crater can be obtained by considering the following: Pressure is the energy per unit volume, P = dE/dV ≈ E/V, where E is the energy of the pulse delivered to the material and V is the volume of the plasma (or the gas that condenses from the plasma). If the pressure is above the strength of the material K, the material give way as it is pushed aside by the plasma. We can therefore find the approximate volume of the excavated crater by setting E/V = K. Assuming the crater is a sphere, the radius of the crater will be approximately
R_c = (3 E/(4 π K))^1/3.
If delivered to the surface, the crater will actually be a hemisphere, but the blast is spherical and energy going away from the surface is lost and does not contribute to excavating a crater. Therefore, we will approximate the radius as what would be blasted out if it were fully inside the target material.

So what value do you use for the strength of the material. Intuitively, you want the ultimate compressive strength of the material, which is the pressure required to strain the material to the point of failure. This value can give the distance to which the target is permanently changed by the blast - perhaps dimpled or distorted or cracked or torn. However, it generally overestimates the crater size significantly, and ductile materials may not display any noticable damage at these distances. In practice, using a strength some 3 to 4 times higher than the compressive strength works well. Theoretically, the strength appropriate to gouging a crater is K_crater = (2/3) K_comp × (1+ln(2 G/K_comp)) where K_comp is the compressive strength and G is the shear modulus. An upper limit to the pressure at which the material flows like a fluid, and the strength needed to resist that fate, is to consider the energy neccessary to disorder the molecular bonds holding the solid together. We are familar with this happening when heat is applied, it is called melting. The energy necessary to turn a solid at its melting temperature into a liquid at its melting temperature, divided by the mass of the solid being melted, is called the heat of fusion. Presumably, if this much energy is applied as a pressure, the bonds will also disorder and the material can flow. Multiplying the heat of fusion by the material's density gives you the pressure that can do enough work to fluidize the material.

When estimating material strengths, the ultimate compresive strength is close to the ultimate tensile strength, which is more commonly measured. Either are slightly larger than the yield strength, which is the pressure at which the material permanently deforms but before it gives way. To the accuracy of this approximation, any of these can be used.

A complication can arise if the spot size S is much larger than the depth of the crater. In this case the laser blast is pushing the fluidized material in front of it rather than out of the way. Since it is more difficult for the material to escape to the sides, a shallow crater may just mostly bounce back once the material becomes solid again and the elastic parts rebound. In this case, the only material actually removed from the spot will be what was directly vaporized by the crater, leading to a crater depth of (E/(H_v ρ (π S²/4)^1/3.

By delivering pulses rapidly, you can blast out each subsequent crater inside the previous one, drilling a deep hole into your target. It takes a time T roughly equal to the crater radius divided by the speed of sound c in the material for the crater to fully form,
T = R_c/c.
If a second pulse is incident on the crater while it is still forming, it merely adds to the energy of the fluidized material pushing outward, and makes the crater larger in all dimensions. In our approximation, the blast cavities expand as fast as the speed of sound in the target material. Thus, the depth of the hole at the time of the last pulse is
D_last = (N_p-1) c τ
where τ is the time between pulses and N_p is the number of pulses.

The total energy E_t delivered to the target is
E_t = N_p E.
If the hole has a radius of r, the total volume excavated is
E/K = (4/3) π r³ + π r² D_last.
where again we assume the effective volume excavated is as for a cavity blasted entirely inside the target material even though one end is at the surface. This requires solving a cubic equation, which has analytical solutions. The total depth D is then
D = D_last + r.

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