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The page on linear absorption & scattering describes the effects when an individual photon has the right energy to excite atmospheric molecules, and single photon ionization is about when individual photons have enough energy to knock an electron off an atom.
You can also get multi-photon absorption and ionization, where the simulatneous absorption of many photons provides enough energy to create the excitation.
Because all the photons must participate at once, it requires much higher light intensities. In general, for absorption of N photons at once, the rate of absorption will be proportional to intensity raised to the Nth power.
As the intensity of the death ray rises, the first such non-linear effect to occur is two photon absorption. This gives the differential equation
The power P of a beam is equal to the beam area A times its intensity I, the beam energy E is equal to its power times its duration δt. This gives us
For constant beam area and initial power P0, this has the solution
| P(x) =
| P0
| 1 + α2 x P0 ⁄ A
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For a focused beam with a focal point at xf, the area changes with x as
until the beam get close to its diffraction-limited area. As a rough approximation, the beam can be considered to converge according to the above formula until it reaches its diffraction limited area Ad, and then propagate with its area equal to its diffraction limited area for a distance of xd = Ad/λ, where λ is the beam's wavelength.
For focused beams, while the beam is still converging the differential equation for power loss gives
| P(x) =
| P0
| 1 + (α2 xf P0 ⁄ A0)[x ⁄ (xf - x)]
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Let's get this into something a bit more useful.
We are going to focus our beam to a spot size s on the target, from an initial aperture diameter of a (note that this means that A0 = π a2 / 4).
The spot size s can be as small as the diffraction limited spot size, but might be larger.
Geometry of similar triangles, with a common point at the focal point and one with the opposite face of length a and the other of length s, tells us that
Therefore
Collecting results
| P(x) =
| P0
| 1 + (4 ⁄ π)(α2 P0 ⁄ [a s]) x
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The rapidly increasing intensity as you approach the focus means that if
the beam will rapidly lose power to two-photon absorption and very little of the original power can be expected to be incident on the target.
The actual two-photon absorption cross sections of oxygen and nitrogen molecules are difficult to find. There is data for water across much of the near ultraviolet, and if this is typical of light elements in small molecules you might expect that at a molecular density similar to sea level air on Earth you might have α2 somewhere in the range of 10-14 cm/W to 10-12 cm/W.
If you are building a real-life ultraviolet laser death ray, you will need to get the correct value for the wavelength you are using. However, for the purposes of fiction just choosing α2 = 10-13 cm/W can at least give some idea of how close you are to having your beam absorbed before it gets anywhere near its target.
This α2 will be proportional to the atmospheric density.
Reference
- David N. Nikogosyan and Dmitrij A. Angelov, "Formation of free radicals in water under high-power laser uv irradiation", Chemical Physics Letters Volume 77, Issue 1, 1 January 1981, Pages 208-210.
- Adrian Dragomir, John G. McInerney, and David N. Nikogosyan, "Femtosecond measurements of two-photon absorption coefficients at λ = 264 nm in glasses, crystals, and liquids", Applied Optics Vol. 41, Issue 21, pp. 4365-4376 (2002)
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