= Radiation pressure of a directed beam
{title2=$p_\perp=(a+2R)I\cos^2\theta/c$}
For normal incidence in <vacuum>, a uniform beam delivers <energy> $IA\,\Delta t$ to an illuminated <surface area> $A$ during time $\Delta t$. A perfect absorber receives <momentum> $IA\,\Delta t/c$, so <force> divided by <surface area> gives $p_{\rm abs}=I/c$. A perfect stationary mirror sends the <photons> back with reversed <momentum>, so <momentum conservation> gives twice the normal transfer, $p_{\rm refl}=2I/c$. These are time-averaged <pressures>; the microscopic arrivals need not be uniform.
\Image[/radiation-pressure-beam-photon-transfer.png]
{title=<Photon> <momentum> transfer to an absorber and a mirror}
{description=Original idealized normal-incidence diagram. Blue arrows show the incoming <photon> <momentum>; orange shows the reflected <photon> <momentum>. The green arrow is the <momentum> delivered to the receiver, not another outgoing <photon>. The mirror reverses the incoming <momentum> and therefore receives twice as much.}
For a stipulated model <light intensity> $I=1000\,\mathrm{W\,m^{-2}}$ and the exact <SI> <speed of light> $c=299792458\,\mathrm{m\,s^{-1}}$,
$$
p_{\rm abs}=3.335640952\times10^{-6}\,\mathrm{Pa}\simeq3.3356\,\mu\mathrm{Pa},\qquad
p_{\rm refl}=6.671281904\times10^{-6}\,\mathrm{Pa}\simeq6.6713\,\mu\mathrm{Pa}.
$$
These are computed predictions, not measured data or a universal sunlight value. If the beam fully covers a normally oriented <surface area> $A=0.010\,\mathrm{m^2}$, the intercepted <radiant flux> is $\Phi=IA=10\,\mathrm W$ and the absorber's <force> is $\Phi/c\simeq33.356\,\mathrm{nN}$. Using the total source <power> would be wrong if only part of its beam reaches the receiver. The absorption prediction assumes complete <photon absorption>, while the reflection prediction assumes complete <specular reflection>; transmission, <scattering of light> and directed thermal <photon emission> require their own outgoing <momentum> accounting.
Measure the <angle of incidence> $\theta$ between the incoming path and the surface's normal line, with $0\leq\theta<\pi/2$. There are two separate <cosine> factors. A fixed actual <surface area> $A$ intercepts only its <projected area> $A\cos\theta$, so $\Phi=IA\cos\theta$. Each incoming <photon> supplies normal <momentum> $(E_\gamma/c)\cos\theta$ on absorption, or twice that on <specular reflection>. Thus, for <radiometric absorptance> $a$ and reflected <power> fraction $R$,
$$
F_\perp=(a+2R)\frac{\Phi}{c}\cos\theta,\qquad
p_\perp=\frac{F_\perp}{A}=(a+2R)\frac{I}{c}\cos^2\theta.
$$
This model assumes transmitted <light> continues straight in the same medium and carries away its original <momentum>. It omits thermal recoil and assumes a stationary receiver. At $\theta=60^\circ$, fixed beam <light intensity> therefore gives one quarter of the normal-incidence <pressure>. For fixed intercepted <radiant flux> $\Phi$, however, the normal <force> has only one <cosine> factor. Equivalently, if the given quantity is receiving-surface <irradiance> $E_{\rm inc}=I\cos\theta$, then $p_\perp=(a+2R)E_{\rm inc}\cos\theta/c$. The distinction matters when interpreting an experiment's angle scan. <Use of radiation pressure for measurement of high-power laser emission by Paul A. Williams and others>[Williams and collaborators, Use of radiation pressure for measurement of high-power laser emission, equation (1): force expressed using intercepted optical power and the incidence-angle cosine]{ref}
Absorption also transfers a tangential <momentum> component at oblique incidence. It contributes a tangential <force>, rather than normal <pressure>. Ideal <specular reflection> preserves the tangential <momentum> component, so it transfers only normal <momentum>.
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