# Two-Photon Absorption

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.
 dI = -αN IN dx

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
 dI = -α2 I2 dx
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
 dP = -α2 P2 dx A
For constant beam area and initial power P0, this has the solution
 P(x) = P0 1 + α2 x P0 ⁄ A

For a focused beam with a focal point at xf, the area changes with x as
 A(x) = A0 ( xf-x )² xf
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)]

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
 s = xf - x . a xf
Therefore
 xf = x 1 - s⁄a
 xf - x = x s⁄a 1 - s⁄a
Collecting results
 P(x) = P0 1 + (4 ⁄ π)(α2 P0 ⁄ [a s]) x

The rapidly increasing intensity as you approach the focus means that if
 4 α2 P0 x > 1 π a s
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 are difficult to find. However, based on typical two photon cross sections of other atoms and molecules the quantity (4/π) α2 for sea level air on Earth is likely to be somewhere in the range of 10-7 cm/W to 10-10 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 (4/π) α2 = 10-9 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.