Thermal Blooming

If the beam is tracking a moving target, or if the wind is blowing cooler air across the beam, you end up with the hot air mostly on one side of the beam. This makes the air act as a lens to bend the beam toward the cooler, denser air, which will throw off your aim.

Thermal blooming can be difficult to adaptively correct. An adaptive optics system will tend to try to compensate for the increased divergence of the beam by focusing the beam even more. This makes the light more concentrated, which heats the air even more, which makes the thermal blooming even worse! Adaptive optics systems may be able to be designed to handle blooming - or they may not, we do not yet know.

A similar effect occurs with beam power. If thermal blooming keeps you from delivering enough inetnsity to your target and you try to compensate by increasing the beam power, the extra power heats the air even more and can end up decreasing your intensity on the target.

There are ways to avoid thermal blooming. The most straightforwad is to use short, high power pulses. Hot air takes time to expand into low density air, so if the beam pulse is done propagating before the air expands, it will not be affected. The time it takes before the hot air expands depends on the beam width, with wider beams having more time before the onset of blooming. Meter wide beams do not suffer significant blooming for perhaps a couple of hudredths of a second, centimeter wide beams for perhaps a few tens of microseconds. This means that practical death rays that are pulsed for a few tens of microseconds or less need not concern themselves with thermal blooming.

Even if you want to put a continuous ray of heat on the target, you can mitigate the effects of thermal blooming. The closer the range, the less the blooming. The less your wavelength of light is absorbed, the less the blooming. The wider your beam, the less blooming it causes. The faster the wind (or the faster your beam slews to follow a target) the less blooming the beam will experience. Finally, it is obvious that the less powerful the beam the less blooming you will experience, but as death ray designers we do not want to go too far in this direction!

We can estimate when thermal blooming will become important. We compute a thermal distortion factor called N. If N is more than one, we need to start taking thermal blooming into account. Roughly speaking, the beam intensity on target will be reduced by a factor of N compared to if thermal blooming was not present. we will use the following definitions (values in parentheses are for typical temperate sea-level conditions on Earth)

n	index of refraction of the medium the beam is going through (n = 1.000273)
dn ⁄ dT	the rate of change of the index of refraction with increasing temperature (dn ⁄ dT = -9.48×10-7 / K)
ρ	mass density of the medium the beam is going through (ρ = 1.2 kg / m³)
T	initial temperature of the medium the beam is going through (T = 288 K)
CP	heat capacity per unit mass at constant pressure of the medium the beam is going through (CP = 1.004×103 J/(kg K))
Ra	absorption length of the medium the beam is going through
R0	attenuation length of the medium the beam is going through
cs	speed of sound in the medium the beam is going through (cs = 330 m/s)
g	acceleration due to gravity (g = 9.8 m/s²)
E	energy of the beam
P	power of the beam
Δt	beam pulse duration
R	range to target
v	windspeed across the beam at the aperture
vt	windspeed across the beam at the target
S	beam spot size at target
D	diameter of focusing aperture
θ	beam diffraction angle θ = 1.2 λ ⁄ D
φ	beam angular slew rate
π	circle constant π ≈ 3.14159
N	thermal distortion factor
β	thermal coefficient β = (-dn ⁄ dT) . (β = 7.87×10-10 m³/J)     ρ CP

With these, the thermal distortion factor is
N = N_c f q s

Nc =	16 √(2)	β P R2
	π	Ra v D2

f = (2/N_E²) (N_E - 1 + exp[-N_E])
f = 1 if N_E = 0
q = (2 N_F² / (N_F - 1)) [1 - ln[N_F]/(N_F - 1)]
q = 1 if N_F = 1
s = (2/N_W²) [(N_W+1) ln[N_W+1] - N_W]
or
s = (2/(N_V-1)²) [N_V ln[N_V] - (N_V-1)]
s = 1 if N_W = 0 or N_V = 1
N_E = R/R₀
N_F = (π/4) n D/S
N_W = φ R / v
N_V = v_t / v

If S ≪ D (the beam is focused to a much smaller spot than the aperture), then it can be can be convenient to use

q Nc ≈ 8 √(2)	β n P R2
	Ra v D S

For still air, there will still be air motion across the beam driven by convection as the beam heats the air which expands and rises out of the beam because of its lower density, sucking cooler denser air into the beam from below.
v_convection = [2 P g / (ρ C_P R_a T) ]^1/3
If the wind speed is lower than the convection speed, use the convection speed instead.

If the pulse duration is shorter than the time it takes for the wind to blow across the beam (Δt < S/v_t and Δt < D/v), then the dynamics of the thermal blooming will be dominated by the beam essentially warming up stationary air rather than wind blowing across the beam. If the pulse happens so quickly that the air cannot move out of the beam to equalize pressure (Δt < S/c_s and Δt < D/c_s) then
N = N_s = (1/(30 π) (β E R² (Δt c_s)²)/(R_a D² S⁴)
On the other hand, if the pulse duration is long enough that the air can equalize pressure (Δt > S/c_s and Δt > D/c_s)
N = N_l = (1/(6 π) (β E R²)/(R_a D² S²)
(N_l differs by a factor of 2 from T_lc and N_s by a factor of 4 from T_sc in H. Kleiman and R. W. O'Neil "Thermal blooming of pulsed laser radiation", Appl. Phys. Lett. 23, 43 (1973) because those give the time-dependent change in the laser spot and we are interested in the total fluence dumped on the target during the time the spot is on. So the presented results are for time-averaging the T parameters.)

If N is significantly less than one, you don't have to worry about thermal blooming. At around one, you can probably correct for blooming with linear adaptive optics. If N is much larger than one, either you need much more sophisticated non-linear controls for your adpative optics or you will need to adjust your beam power, spot size, range, pulse duration, or slew rate.

References

1. Philip E. Nielsen, "Effects of Directed Energy Weapons"
2. Frederick Gebhardt, "Twenty-five years of thermal blooming: an overview", Proc. SPIE 1221, Propagation of High-Energy Laser Beams Through the Earth's Atmosphere, (1 May 1990); doi: 10.1117/12.18326
3. H. Kleiman and R. W. O'Neil "Thermal blooming of pulsed laser radiation", Appl. Phys. Lett. 23, 43 (1973)

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