Lasers, Mirrors and Star Pyramids

Lasers can hit targets at extreme ranges, at the fastest speed possible. They are ideal weapons for space warfare. 

However, everyone knows that lasers bounce off mirrors... does this make lasers useless?

The post is inspired by the discussion that arose from the conclusions stated by Kyle Hill (@sci_phile) in 'The Truth about Space Warfare' video for Because Science.

Many commenters asked me about how mirrors could be used to counter lasers. This is the response I promised. 

Common assumptions and common answers

An electrically pumped VECSEL, which promises high efficiency and powerful beams. 

The usual understanding of laser weapons is that they produce a bright light that slices through armor. Mirrors can bounce away this beam, rendering it harmless.

A laser defeated by a mirror?

The realistic science fiction community, centered around the efforts of the Atomic Rockets website, did their research with regards to the effectiveness of mirrors against lasers, and found that no mirror could be 100% reflective. It would always absorb some of the laser's energy, which would result in it heating up, melting and therefore losing its reflective properties. Because laser weaponry produce several kilowatts to hundreds of megawatts of power, the transition from 'cold and reflecting' to 'molten and ineffective' would happen very quickly.

As a result, the common answer to 'what about mirrors?' is that they are ineffective.

ToughSF will revisit the concept, as it has done with other 'established' truths of science fiction.

How do lasers deal damage?

A laser guide star for a telescope's adaptive optics. 

To first understand the effectiveness of a defense, we must first understand the type of damage it is trying to mitigate.

Lasers produce an intense spot of light on the surface of the target. As a directed energy weapon, its primary purpose is to heat up that surface until it deals damage. 

Continuous lasers produce a constant stream of energy. Pulsed lasers produce short bursts of much higher power, but much shorter duration, spaced by intervals as small as a microsecond.

Pulsed laser effects at different pulse lengths. 

One measure of a laser's destructive capability is its intensity at the target, in watts per square meter (W/m^2).

Sunlight has an intensity of 1kW/m^2 on Earth's surface, lightbulb filaments produce an intensity of 1MW/m^2 at their surface and lightning strikes flash at an impressive 10GW/m^2 in some cases. 

When a laser strikes a surface, it raises the temperature of that surface's material. At a certain temperature, it melts and moves out of the way. Other materials vaporize, turning into a gas that expands rapidly. At very high intensities, such as those produced by a pulsed laser, the vaporization gases expand so quickly that they overcome the mechanical strength of the surrounding material and shear off chunks or drag along debris out into space.

The best armor materials are those that take the most energy to heat up (high heat capacity), the most energy to destroy (high melting or vaporization energy) and have the highest mechanical strength (tensile strength). In space, every kilogram is important, so we also want the armor to have good characteristics for its weight.

For these reasons, steel is a bad armor material while graphite is an excellent armor material.

Steel cutting.

Steel has a low heat capacity (0.5kJ/kg/K), a low melting energy (240kJ/kg at 1640K) and good strength (600MPa). 

Vaporizing graphite to produce carbon nanotubes require extreme temperatures. 

Graphite has a great heat capacity (0.72kJ/kg/K), an extreme vaporization energy (59.5MJ/kg at 4000K) and poor strength (30MPa). 

Steel can only absorb 923kJ/kg before it is destroyed. Graphite absorbs up to 62MJ/kg, meaning each kilogram of graphite is worth 67kg of steel armor. 

What does this all mean in practice?

Laser heating up and vaporizing the surface of an asteroid. 

Well, when no mirrors are being used, lasers can burn through significant depths of armor from long distances, and can keep scratching away at armor from even farther away. To find the penetration rate of a laser, we divide the laser intensity at the target by the armor's ability to absorb energy before being destroyed. 

Here's a worked example:

100kW MBDA laser.

Imagine a laser producing 10 MegaWatts of power. It has a wavelength of 450 nanometers, which is great at travelling through our atmosphere. The focusing mirror is 10 meters wide, about half as wide as the one the James Webb telescope uses. 

At 100 kilometers, this laser produces a beam 11mm wide with an intensity of 105.6GW/m^2 at the target. It can melt away 10.32 kg of steel per second, or vaporize 154 grams of graphite. This translates into a penetration rate of 13.5m/s and 0.7m/s respectively.

We'll assume a perfect weapon that produces diffraction-limited beams (as focused as possible).

At 1000km, the beam spreads to 110mm wide intensity drops to 1.05GW/m^2. The penetration rate falls to 7mm/s in graphite.

At 10,000km, the penetration rate falls to 0.07mm/s. At 20,000km, it is 0.017mm/s, and so on.

With each increase in distance, the penetration rate falls by the square of that increase. These numbers might not seem to be impressive at the distances usually discussed when talking about space travel (millimeters?!) but they do add up over time. If the distances are great, they take a long time to cross. During that time, a huge amount of armor can be removed. 

When talking lasers, we are handling distances long enough to make the Earth look small. 

For example, a spaceship travelling from the Moon (400,000km away) in a straight line towards the Earth at a rapid rate (10km/s) while facing the 10MW laser described above would lose a full 3358 meters of graphite armor before it even reaches Low Earth orbits (200km)! It would be very impractical if all spaceships had to cover themselves in several kilometers of armor to survive crossing the relatively short Earth-Moon distance!

Hot Surfaces

At high temperatures, surfaces emit light and heat. 

Is the scenario described above actually realistic? Can lasers scrape away meters of armor if left unchecked?

Like many SF writers and thinkers so far, we ignored the fact that as the armor heats up, it radiates away energy like any hot surface does. Blackbody radiation is proportionate to the 4th power of temperature and can be measured in Watts of heat radiated per square meter (W/m^2).

Thermal radiators on all spacecraft are needed to radiate away waste heat.

This blackbody radiation is directly comparable to the laser intensity that reaches the armor.

If the blackbody radiation is equal to the laser intensity, then the armor is dissipating as much heat as it is absorbing. The temperature therefore cannot increase to the point where melting or vaporization happens!

Planets achieve thermal equilibrium with their star.

So, at what distance does the laser beam become weak enough that the armor's heat dissipation properties prevent any damage from occurring? Let's work this out for steel and graphite, using the same laser as before.

Steel melts at 1640K. The blackbody radiation rate at that temperature is 420.2kW/m^2. For the laser intensity to match this value, it needs its 10MW of power to be spread out over a 23.8m^2 surface area. This happens at a distance of... 50,000km. 

Graphite vaporizes at 4000K. It can dissipate up to 14.5MW/m^2, meaning it can withstand the laser beam from a distance of 8506km. 

What does this mean in practice?

Well, for steel, no damage occurs at distances further than 50,000km. For graphite, this distance is as little as 8506km. The armor only accumulates damage at distances shorter than that number of kilometers.

Furthermore, as the spaceship approaches the laser, you get to subtract the radiated energy from the laser energy to get the amount of heat absorbed and used to vaporize the armor. 

So, for example, if the spaceship closed in to 8000km, the laser intensity reaches 16.5MW/m^2. We subtract the heat radiated (14.5MW/m^2) and obtain a 'net' intensity of 2MW/m^2. This 'net' amount only vaporizes about 20 grams of graphite per second. 

The Mirrors

What if we used mirrors?

A mirror with 90% reflectivity at the wavelength the laser uses would reduce the heat absorbed by a factor 10. 

However, they are not good blackbody surfaces. A 90% reflective mirror would radiate 10 times less heat through blackbody radiation than a black surface such a graphite. Alone, if would not provide any benefit and it would just melt or burn up.

Low emissivity (reflective) surfaces don't lose a lot of heat through blackbody radiation.

Mirrors must be paired with active cooling.

Mechanisms such as those used to keep rocket nozzles cool or jet turbines functional can be applied to mirror armor to remove massive amounts of heat in a short amount of time. It involves moving a large amount of coolant at high pressure and velocity through the narrow channels of a heat exchanger to transfer heat from the component into the coolant. 

Again, we can describe the effectiveness of active cooling in terms directly comparable to laser intensity: W/m^2. 

If the laser intensity equals the active cooling rate, the mirror surface will not heat up.

Thermal receivers for concentrated solar power face many of the same challenges.

Let's continue the previous example and equip a spaceship with actively cooled mirrors that reflect 90% of the laser energy away. To use a real world example, we will use the numbers from the cooling system that absorbs intense, concentrated sunlight in solar thermal receivers. This example of a volumetric gas tube absorbs over 2MW/m^2. Another example from the solar energy industry is the High-Flux Solar Furnace that handles up to 11MW/m^2.  

Liquid salts coolants allow for higher heat fluxes to be absorbed.

We work out that the actively cooled mirror can survive a laser intensity of 11*10: 110MW/m^2. 

The laser would have to be firing from a distance of 3500km to overwhelm the set-up so that it can start damaging the mirror and then the armor underneath. 

3500km is of course, much shorter than the distances cited so far, and no damage takes place at longer distances.  

Active cooling systems that can handle even more impressive heat fluxes, and mirrors that reflect a greater percentage of the laser energy, will shorten the lasers' effective range even further. 

Sloped surfaces and Fresnel Reflection

There is more that can be done to reduce the combat distances imposed by powerful space lasers. 

Sloped surfaces are able to spread a laser beam's energy over a larger surface area. This decreases its intensity and the damage that it can do. We can work it out using cos(slope angle). 

At 45 degrees, the surface area is increased by 41% and the laser intensity decreases by 29%.

At 60 degrees, the surface area is increased by 100% and the laser intensity decreases by 50%.

At 80 degrees, the the surface area is increased by 476% and the laser intensity decreases by 83%.

If graphite armor is sloped at 80 degrees, it can withstand a laser of 5.76 times the intensity it could have without sloping. This reduces the 'no damage' distance from the laser by a factor 2.4.

This suggests that sharp cones are very effective at reducing laser effectiveness, and would be the optimal shape for armor on a spaceship. 

Space warship with conical frontal section, by Grokodaemon.

Polygonal shapes such as pyramids with triangular, square, hexagonal or other bases could be more effective than a rounded cone by creating a compound angle (vertical and horizontal sloping) against the laser.

The 75.04 degree vertical angle compounds with the 45 degree horizontal angle to create a 10.55 compound angle.

An even better arrangement would be a pyramid with a star-shaped base, or 'star pyramid'. 

Drawn in Geogebra.

The vertical angle can be compounded by a very sharp horizontal angle. For example, an octagram 'star' can form the base of a pyramid 10 meters wide and 57 meters long, and achieve an 80 degree vertical angle and a horizontal slope of 22.5 degrees. The compounded angle becomes just 3.81 degrees, allowing for a 15.05x decrease in laser intensity. 

Another benefit from extreme sloping is that Fresnel Reflection becomes significant. 

At low angles, even dark surfaces become shiny. 

We first have to find the refractive index of the armor material with regards to the wavelength the laser is using. A database such as this one is excellent for this purpose.

We find that graphite has an refractive index n = 1.5179 in 450nm light. 

That index, along with the slope angle, can be used to calculate how much of the laser's energy is reflected away. At 45 degrees, it is an insignificant 5.45%. An extreme compounded slope such as the one produced by the star pyramid as described above would allow for 68.9% of the laser energy to bounce off harmlessly. 

A graphite octagram star pyramid with 80 degrees of vertical slope and 67.5 degrees of horizontal slope would survive the 10MW laser at a distance of 2010km. 

Dielectric mirrors, when sloped, lose effectiveness against lasers of certain polarizations. Omnidirectional mirrors are needed. 

If we used an extremely sloped pyramid (80 degrees) in addition to a star-base with a very large number of sides (18 sides for 80 degrees), omnidirectional dielectric mirrors (99% reflectivity) and active cooling (11MW/m^2), we could expect to increase the damaging intensity threshold to 33.16*100*11:  36.5GW/m^2. 

The effective range of the 10MW laser falls to a mere 1700km.

Further damage reduction techniques and conclusion

With a bit of imagination, we can come up with even more ways to reduce laser damage.

By rotating the armor, we would force the laser beam to wander over fresh, unheated surface area. The maximum effectiveness of spinning is to divide the circumference of the armor by the width of the laser beam.

For example, a 4 meter wide cylinder would have a circumference of 12.56 meters. At 2000km, the laser example we have used so far would produce a beam 0.195 meters wide. Spinning the armor would therefore spread the laser by a maximum of 12.56/0.195: 64.4 times. 

The effectiveness of armor rotation increases as distances become shorter. The laser beam becomes narrower, so we divide the circumference by a smaller number to get a bigger reduction in intensity. At 1000km, the beam thins to 0.0976m, increasing the spread to a factor 128.7.

Or, instead of just spinning in circles, armor can be split into segmented bands (like tank tracks) that move diagonally up a spaceship's hull. This would spread the laser both along the circumference of the spaceship and along the band's length. 

Another option would be to stop thinking of armor as solid blocks of material. A forest of metal wires could intercept a laser and absorb its energy, but it would have a much better surface area to mass ratio - like a hot surface of graphite, it could passively radiate away a lot of energy.

If we using several of these techniques together, we can reasonably conclude that lasers will always be powerful, long-ranged weapons, but they actually have an effective range that can be reduced to a few hundreds kilometers, instead of the tens of thousands of kilometers assumed so far.

Why bother with all this?

Well, at shorter distances, other weapons can come into play. Railguns would be able to hit targets before they can dodge out of the way, and missiles don't have to grow to obscene sizes so that they may pack enough propellant to drive themselves up to immense speeds. 

Particle beam weapons, generally shunned for their short effective range when compared to lasers, can bypass reflective surfaces and deal damage directly. You'd even want different types of laser, to shoot at wavelengths that mirrors struggle to handle, or in pulses intense enough to overwhelm active cooling. 

More weapons means more options in combat, and sci-fi writers have more tools to shake up their action scenes and make them more interesting.

Maneuvering can become important again too. At extreme distances, a spaceship accelerating as hard as it can will only shift its position in its enemy's sky by a few degrees, and it cannot close or open the distance meaningfully before the battle is over.

At shorter ranges, flanking maneuvers become possible. Tactics such as rushing the target or accelerating away are suddenly more effective. The importance of pre-positioning before a battle become less important than the maneuvers taken during a battle. 

"Very, very far away..."

In other words, shorter ranges allow for more dynamic battles where tactical decisions matter and human actors would have a role to play.

Without all this, space warfare would solely be determined by which side brought the most spaceships with the biggest lasers into the fight. Fleets would know whether to engage or not deterministically, and even if they do lock themselves into battle, it will be a drawn-out sequence of spaceships' armor being burned away over the course of days to weeks. That's boring.