Radiation pressure and photon flux

Maxwell's equations give a plane wave a momentum density g = S/c², where S is the Poynting vector. Put a perfectly absorbing surface of area A in the beam and ask what arrives in a time Δt: the field in a column of length cΔt and cross-section A, carrying momentum (I/c²)(A c Δt) = I A Δt/c. Divide by Δt for the force and by A for the pressure, and radiation pressure is p = I/c. That is an intensity in W m⁻² divided by a number of order 10⁸, which is why every answer in this lesson lands in micropascals. Nothing quantum has entered anywhere in that argument. The ratio of momentum to energy carried by a wave is fixed at 1/c by the field equations alone — and the same 1/c will come back out of a photon count two sections from now, with h cancelling on the way.

Three cases, and the differences are pure geometry. A black surface at normal incidence absorbs the momentum: p = I/c. A perfect mirror reverses it, so twice as much is transferred: p = 2I/c — the mirror feels more force while absorbing no energy at all, which is the first sign that force tracks momentum, not energy. Tilt the surface by θ from its normal and two separate cosines appear. First, a surface of area A now intercepts only a cross-section A cosθ of the beam, so the power landing per unit of surface is I cosθ. Second, specular reflection reverses only the normal component of each photon's momentum and leaves the tangential component untouched, so the transfer per photon is 2(hf/c)cosθ, directed along the surface normal rather than along the beam. Multiply the two: p⊥ = 2I cos²θ/c, half its face-on value at 45° and a quarter at 60°. For reflectance r the normal pressure is (1 + r)I cos²θ/c, and a shear (1 − r)I cosθ sinθ/c appears beside it.

Now count the same beam. Each photon carries energy hf and momentum hf/c = h/λ, so a power P delivers R = P/hf photons per second. A black surface stops each one: F = R(hf/c) = P/c. A mirror sends each one back: F = 2P/c. Identical to the wave answers, and h has cancelled — it entered once through the energy and once through the momentum. That cancellation is a historical fact worth holding on to. Lebedev, and then Nichols and Hull, weighed radiation pressure on torsion balances around 1901 and confirmed Maxwell two decades before anyone accepted photons, precisely because the mean force cannot tell the two pictures apart. Work one case both ways: a 10 W beam at 1064 nm on a black target gives P/c = 3.34 × 10⁻⁸ N; counted, it is 5.36 × 10¹⁹ photons per second, each carrying h/λ = 6.23 × 10⁻²⁸ kg m s⁻¹, and the product is the same 3.34 × 10⁻⁸ N.

Microscopic is not the same as irrelevant; it depends on what the force acts on and for how long. On a bench a few micropascals is hopeless — Nichols and Hull had to work in a partial vacuum because residual gas heated by the beam produced radiometric forces larger than the effect they were after. In orbit nothing competes. A perfect mirror at 1 AU collects 2I/c = 9.08 μPa, that is 9.08 μN of thrust for every square metre, while the Sun's gravity there pulls at GM/r² = 5.93 × 10⁻³ m s⁻²; the two balance when sail plus payload masses 1.5 g per square metre, and since both fall off as 1/r² a craft that beats gravity at 1 AU beats it at every radius. At atomic scale the ratio is better still. A sodium atom absorbing a 589 nm photon takes a velocity kick h/mλ = 2.9 cm s⁻¹, and at saturation it scatters about 3 × 10⁷ photons per second, giving some 9 × 10⁵ m s⁻² — around 90 000 g. The same pressure that cannot lift a speck of dust is what stops an atomic beam and holds a bead in optical tweezers.

So far the two pictures agree exactly. They stop agreeing at the second moment. A classical wave of intensity I deposits exactly I A τ of energy into every gate of length τ: the prediction carries no spread at all. The photon picture says arrivals are discrete events, and for an ideal laser — a coherent state — they are Poissonian and independent, P(N) = e^(−⟨N⟩)⟨N⟩^N / N!, with Var N = ⟨N⟩ and σ_N = √⟨N⟩. The absolute noise therefore grows as √N while the relative noise falls as 1/√N. Take 1.0 mW at 633 nm for a full second: ⟨N⟩ = 3.19 × 10¹⁵ and σ = 5.6 × 10⁷, a fractional wobble of 1.8 × 10⁻⁸ that no bench measurement would ever resolve. Attenuate the same beam to 1.0 nW and gate it for 1 μs and the identical law gives ⟨N⟩ = 3187 with σ = 56 — a 1.8% scatter that now dominates the measurement. The graininess was always there; only its fractional size changed.

That fluctuation belongs to the light, not to the detector, and reading it correctly needs three facts. First, imperfect detection can only make things worse. A detector of efficiency η keeps each photon independently with probability η, and thinning a Poisson stream that way leaves a Poisson stream, of mean ηN — so the relative noise climbs to 1/√(ηN), and no amount of gain afterwards recovers what was discarded. Second, in a photodiode the same statistics appear as current noise, iᵣₘₛ = √(2 e i_dc B): for a 1.0 mA photocurrent in a 1.0 MHz bandwidth that is 18 nA, and cooling does not touch it, because it is the granularity of the beam rather than thermal agitation in the load. Third, the variance is diagnostic where the mean is blind. The second-order correlation g⁽²⁾(0) is 1 for a coherent beam, 2 for thermal light of the same mean power, which bunches, and tends to 0 for a single-atom source, which cannot emit two photons at once. Three sources, one radiation pressure, three different noise spectra. In LIGO the same number fluctuation becomes a fluctuating force on the mirrors — quantum radiation-pressure noise, the low-frequency partner of the shot noise on the readout.