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Lossless reversible propagation preserves the etendue of the same bundle of light rays. Here the model uses stationary isotropic media, specular reflection and refraction, with every selected light ray reaching the output and with no optical gain or scattering of light. An aperture that blocks some incident light rays changes which bundle is counted; the full incident and transmitted etendue need not be equal. ^[ref]

The underlying phase space has four coordinates. In a planar section with constant refractive index, let $(x,y)$ locate a light ray and setThe Jacobian determinant from spherical coordinates givesThus etendue is a four-dimensional volume, preserved by the optical counterpart of Liouville theorem in Hamiltonian mechanics. Individual position and direction extents may change, and correlations between them matter. The Lagrange optical invariant is a related paraxial approximation invariant for pairs of light rays, rather than the full four-dimensional measure.

A Fabry–Pérot interferometer uses repeated specular reflections between two nearly parallel, partially transmitting surfaces. Its transmitted light is the sum of contributions that have made zero, one, two, or more additional round trips. Their complex amplitudes reinforce only near particular wave phases, producing narrow transmission peaks that select wavelength intervals. Scanning the spacing or angle of incidence can build an imaging spectral cube: an image at each selected wavelength.

The surprising feature is that highly reflecting mirrors can together transmit all the incident power at resonance in the ideal symmetric, lossless model. Each individual contribution is weak, but many contributions arrive in phase. Between resonances the same contributions cancel strongly. Increasing mirror reflectivity sharpens the peaks instead of simply suppressing every transmitted signal; finesse quantifies that selectivity. An optical spectrum with nearby spectral lines can therefore produce distinguishable peaks, provided their separation is large enough relative to the instrument's peak width and their orders are identified.

Charles Fabry and Alfred Pérot developed the instrument together; Optica dates its development from 1896 to 1898.^[ref]

Consider two identical, lossless parallel mirrors with intensity reflectivity $0\le R<1$, separation $d$, and a uniform transparent medium of refractive index $n$. Fix one polarization of an electromagnetic wave and an internal angle of incidence $\theta$ measured from the mirror normal vector. Illuminate with a monochromatic wave having planar wavefronts; the emerging contributions must overlap coherently in the detected direction. Define $T=I_{\mathrm{transmitted}}/I_{\mathrm{incident}}$ using the same incident and exit media. Each mirror transmits a fraction $1-R$ of incident power. Its reflected wave amplitude has magnitude $\sqrt{R}$, so two internal specular reflections multiply the circulating complex amplitude by magnitude $R$, not $R^2$.

Let $\lambda$ be the vacuum wavelength and choose the reflection wave phase convention in which the product of the two internal amplitude reflection coefficients is real and positive. The phase difference between successive transmitted contributions, evaluated on a common outgoing wavefront, isWith other mirror coatings or phase conventions, use the total $\delta =4\pi nd\cos \theta /\lambda +{\varphi }_{\mathrm{rt}}$, where ${\varphi }_{\mathrm{rt}}$ is the sum of the two internal reflection wave phases. The stated resonance positions presume ${\varphi }_{\mathrm{rt}}=0$ modulo $2\pi$; a constant nonzero ${\varphi }_{\mathrm{rt}}$ shifts them, while a frequency-dependent one can also change their spacing.

Why does the cosine appear rather than its reciprocal? Inside the gap, the extra two legs have optical path length $2nd/\cos \theta$. But the later emerging contribution starts $2d\tan \theta$ farther along the mirror. Comparing both on one outgoing wavefront subtracts $2nd\tan \theta \sin \theta$ from that path difference, leaving $2nd\cos \theta$. Equivalently, a plane wave's mirror-normal wavevector component is $k_z=(2\pi n/\lambda )\cos \theta$, so the round-trip wave phase is $2k_{z}d$. The diagram uses the same medium outside and inside to show the common-wavefront subtraction directly; at a refracting exit interface, tangential wavevector matching in Snell's law supplies the same result.

Absorb the common first-pass wave phase into the incident complex amplitude $E_{\mathrm{in}}$. Successive transmitted contributions are $(1-R)E_{\mathrm{in}}$, $(1-R)E_{\mathrm{in}}R{e}^{i\delta }$, and so on. The superposition principle and the convergent geometric series giveThe last step expands the squared complex modulus as $1+R^2-2R\cos \delta$. This transmission profile is often called the Airy transmission function; it is distinct from the Airy disk, a diffraction image of a circular aperture. The University of Washington's lab handout gives the same transmission profile and distinguishes mirror reflectivity from other sources of peak broadening.^[ref]

For an original example, set $R=3/4$. At $\delta =0$, the amplitudes add to $(1/4)/(1-3/4)=1$, so $T=1$. At $\delta =\pi$, they alternate in sign and add to $(1/4)/(1+3/4)=1/7$, so $T=1/49$. At $\delta =\pi /2$, the amplitude is $4/25+3i/25$ and $T=1/25$. Adding the individual intensities instead would give $(1-R{)}^2{\sum }_j{R}^{2j}=(1-R)/(1+R)=1/7$, independent of wave phase; that discards the wave interference responsible for the peaks.

In general the maximum is $T_{\max }=1$ at $\delta =2\pi m$, with integer $m$, and the minimum is $T_{\min }=[(1-R)/(1+R){]}^2$ halfway between peaks. Perfect resonant transmission requires identical lossless mirrors, coherent illumination, and sufficient time to establish the steady field. Absorption, unequal mirror transmission, finite beams that fail to overlap, and fluctuations of $d$ can reduce it. The exactly reflecting case $R=1$ is excluded: no incident power enters through a perfectly reflecting mirror, and the resonant limit $R\to 1^{-}$ is singular.

Geometrical optics describes propagation by rays and Snell's law, neglecting diffraction on scales much larger than the wavelength.

A light ray is a path in an approximation, not a physical thread of zero width. In a transparent, homogeneous, isotropic medium it is straight and perpendicular to a wavefront; boundaries redirect it by the law of reflection or Snell's law. The approximation is useful when the wavelength is small compared with relevant apertures, surface curvature scales, and distances over which the refractive index changes. For an illustrative wavelength of $500\mathrm{nm}$ and a $10\mathrm{mm}$ aperture, their ratio is $5\times 10^{-5}$. That small ratio supports ray tracing for image location, but does not eliminate diffraction when calculating the fine structure of a focused spot. At a wave caustic, geometrical optics can fail despite large optical elements. ^[ref].

The mechanism of refraction is a change in phase velocity. For transparent isotropic media at the chosen frequency, the refractive index iswhere $c$ is the vacuum speed of light and the angles are measured from the interface normal vector. At a stationary planar interface the frequency and tangential wavevector match across the boundary. Since the wavenumber is $k=n\omega /c$, that matching gives Snell's law. The transmitted wavelength changes while the frequency stays fixed. Here $n$ uses phase velocity, which need not equal group velocity. ^[ref]. ^[ref].

Choose model air with $n_1=1$ and model glass with $n_2=1.5$, without claiming these are exact values for every sample or wavelength. At ${\theta }_1=45^{\circ }$, Snell's law givesThe transmitted light ray bends toward the normal vector; its phase velocity is $c/1.5$, and its wavelength is two thirds of the air value. By reversibility of an optical ray, incidence from glass at $28.1255^{\circ }$ returns the outgoing air angle $45^{\circ }$.

Incidence from glass at $45^{\circ }$ is a different experiment. It would require $\sin {\theta }_2=1.5\sin 45^{\circ }\approx 1.06066$, so there is no real transmitted angle. The critical angle for total internal reflection isAbove this angle, an ideal lossless glass-air boundary produces total internal reflection. The physical optics description still permits an evanescent wave in the air, although no propagating transmitted light ray crosses the boundary. ^[ref]. ^[ref].

The usefulness of geometrical optics is that complicated instruments can be assembled from these local rules. Applying Snell's law at successive curved surfaces changes a spreading bundle into a converging one. The Fermat principle unifies the construction through stationary optical path length; stationarity does not require a global minimum. ^[ref]. Fang's MIT notes explicitly connect this principle to specular reflection, refraction, and thin lens imaging, providing a route from individual paths to complete instruments. ^[ref].

In his 1888 reflection experiments, Heinrich Hertz excited a straight conductor with a central spark gap driven by an induction coil. The receiver was a wire circle of $35\mathrm{cm}$ radius with an adjustable small spark gap; tiny sparks were observed in darkness. There was no connecting transmission wire. ^[ref]

He removed hanging metal fixtures, installed a $4\mathrm{m}$ by $2\mathrm{m}$ zinc reflecting sheet, and moved and rotated the receiver. Spark strength alternated with position and gap orientation, supporting specular reflection and wave interference. He checked observations from farther away because the observer's body perturbed the response. Changing oscillator dimensions changed the spatial pattern. ^[ref]

These tests supported propagating electromagnetic waves, rather than mere detection of nearby induction. They did not establish unlimited reception distance. His wire-versus-air wave interference experiment addressed finite velocity; he later judged its quantitative velocity comparison doubtful because of room effects. ^[ref]

The physical inference combines transmission with a spatially structured response to a reflected disturbance. In an ideal monochromatic incident-and-reflected model, standing wave nodes are separated by $\lambda /2$; combining a reliable wavelength with independently known frequency gives phase velocity $v=\nu \lambda$. A spark is a threshold event, however: its presence or length is not a calibrated wave intensity or a direct arrival-time measurement. The actual loop also responds to its orientation and finite size, so the simple standing-wave model is an explanation of the test's logic, rather than a reconstruction of Hertz's measured points. Poynting theorem explains how source energy can reach and drive the receiver's discharge.

Take $0<{\theta }_0<\pi$, put $a=k_0\cos {\theta }_0$ and $b=k_0\sin {\theta }_0>0$, and suppress the common factor $e^{i\omega t}$. The upper-half-plane incident acoustic plane wave is $e^{iax+iby}$. A rigid zero-thickness screen imposes the same normal-derivative cancellation on both faces; the scattered velocity potential is consequently odd in $y$. Across the open part $x>0$ it is continuous, so its trace there is zero. These complementary half-line data are what make the Wiener-Hopf method applicable.

Use the Fourier transform $\hat{\varphi }(\kappa ,y)={\int }_{\mathbb{R}}\varphi (x,y)e^{i\kappa x}dx$, with inverse exponential $e^{-i\kappa x}$. The outgoing Helmholtz equation solution is $\hat{\varphi }=F^{-}(\kappa )e^{-\gamma y}$ for $y>0$, where $\gamma =\sqrt{{\kappa }^2-k_0^2}$ has $\gamma =i\sqrt{k_0^2-{\kappa }^2}$ on the propagating interval and positive real part on evanescent components. The limiting absorption principle supplies the contour prescription. The transform $F^{-}$ of the trace supported on $x<0$ is analytic below the contour. Let $G^{+}$ be the analytic upper transform of the unknown derivative on $x>0$. The prescribed derivative on $x<0$ has transformSince the upper normal derivative is $-\gamma F^{-}$, the Wiener-Hopf equation isFor the Wiener-Hopf factorization, choose ${\gamma }_{+}=(\kappa -k_0{)}^{1/2}$ analytic above and ${\gamma }_{-}=(\kappa +k_0{)}^{1/2}$ analytic below. Their branch cuts go into the opposite half-planes, with ${\gamma }_{+}(-a)=i\sqrt{k_0+a}$. Dividing by ${\gamma }_{+}$ and performing pole subtraction in a Wiener-Hopf equation givesThe forcing pole is allocated to the lower analytic part; the second quotient has a removable pole and is upper analytic. Subtracting these parts leaves a common entire function. The finite-energy edge condition excludes an inverse-square-root singularity in the potential: its odd scattered trace is $O(\sqrt{|x|})$, so $F^{-}=O(|\kappa {|}^{-3/2})$. The normal derivative on the open side can be $O(x^{-1/2})$, giving $G^{+}=O(|\kappa {|}^{-1/2})$. Both sides of the entire remainder tend to zero. The Liouville theorem sets that remainder to zero, giving the explicit Wiener-Hopf solution of rigid half-plane diffraction:The scattered field on either side of the screen is thereforeIts forcing-pole residue at $\kappa =-a$ is $-i$. Deforming to the outgoing steepest descent contour crosses this pole precisely when $|\theta |>\pi -{\theta }_0$, where $x=r\cos \theta$, $y=r\sin \theta$ and $-\pi <\theta <\pi$. The residue contributes $\mathrm{sgn}(y)e^{iax-ib|y|}$. Thus, away from transition boundaries, the scattered geometrical optics field isAdding the incident acoustic plane wave gives incident plus unit specular reflection in the upper reflected sector, just the incident wave in the remaining illuminated sectors, and zero in the lower shadow sector. The geometrical shadow boundaries also follow directly by tracing a straight ray to $y=0$: the reflected ray hits the screen at $x+y\cot {\theta }_0<0$, and the transmitted incident ray crosses the open part at $x-y\cot {\theta }_0>0$.

For the diffracted contribution, the outgoing phase $-i\kappa r\cos \theta -\gamma r|\sin \theta |$ has saddle ${\kappa }_s=k_0\cos \theta$, value $-i{k}_{0}r$, and second derivative $ir/(k_0{\sin }^2\theta )$. Stationary phase therefore supplies $\sqrt{2\pi k_0/r}|\sin \theta |e^{-i{k}_{0}r+i\pi /4}$. Substituting $F^{-}({\kappa }_s)$, and using $\mathrm{sgn}(y)|\sin \theta |/\sqrt{1+\cos \theta }=\sqrt{2}\sin (\theta /2)$, givesThe full asymptotic scattered field is ${\varphi }_{\mathrm{GO}}+{\varphi }_d$. The angular edge-wave expression is not uniform on the geometrical boundaries. There is also a sign inconsistency in the supplied saddle formula: with its printed inverse exponential $e^{+i\kappa x}$, the outgoing saddle is $-k_0\cos \theta$, not $+k_0\cos \theta$. The consistent inverse convention used above gives the requested outgoing field without that ambiguity.

Write the unit-amplitude incident plane wave as ${\psi }_i=e^{i{p}_{i}x-i{q}_{i}z}$, with $q_i=\sqrt{k^2-p_i^2}>0$. The flat Dirichlet reflected field is ${\psi }_s^{[0]}=-e^{i{p}_{i}x+i{q}_{i}z}$, so the flat total field vanishes at $z=0$ and has derivative ${\partial }_z{\psi }^{[0]}(x,0)=-2i{q}_i{e}^{i{p}_{i}x}$.

For a controlled height expansion take $f=\epsilon h$ with fixed regular bounded shape $h$. Taylor expansion at the moving boundary yieldsThus the first-order scattered trace is ${\psi }_s^{[1]}(x,0)=2i{q}_{i}f(x)e^{i{p}_{i}x}$, and its transform follows by the Translation property of the Fourier transform:The first formula is the roughness correction; the second includes the zeroth-order reflection. The small-height assumption controls the Taylor expansion for a fixed profile; arbitrarily fine surface scales can require additional derivative control, not merely a small height at fixed $k$.

If the mean roughness height is a constant $\bar{f}$, as for stationary roughness, then $⟨\hat{f}(\nu -p_i)⟩=2\pi \bar{f}\delta (\nu -p_i)$. Consequently the specular first-order mean reflection isIts tangential wavenumber is unchanged and its normal wavenumber is reversed relative to incidence, which is precisely specular reflection. For zero mean height the first-order mean correction is zero, but the full mean reflected field is not zero: it still contains the flat reflection.

Small height alone does not imply the statistical conclusion. For example, deterministic mean height $\bar{f}(x)=a\cos Kx$ with $|ka|\ll 1$ produces first-order mean sidebands at $p_i\pm K$ and is not purely specular. Constant mean is sufficient at first order; stationarity supplies it and, with suitable translation-invariant statistics, constrains higher-order coherent reflection as well.

At $X=P(u,v)$, orient the unit normal vector $n=(P_u\times P_v)/\Vert P_u\times P_v\Vert$ towards the eye. Define the unit directions from the reflecting point to the source and eye byThe incident propagation direction is $-a$. The law of specular reflection is $b=-a+2(a\cdot n)n$, so $a+b$ is normal to the surface. Conversely, if the tangential components of $a+b$ vanish and $a\cdot n,b\cdot n>0$, its unit-length summands have equal positive normal components and obey the reflection law. Therefore solve the two scalar equationsThe stated orientation condition gives $b\cdot n>0$. It removes the back-facing ambiguity, but not multiple images or caustics. Exclude $Q(t)=P(u,v)$ and $E=P(u,v)$, where these directions are undefined.

At a root where the $2\times 3$ Jacobian matrix $DF$ has rank two, the implicit function theorem makes the solution set a local curve in parameter space. Compute a unit tangent $w$ in its nullspace; equivalently take the normalized cross product of the two rows of $DF$. A continuation method predicts $x_p=x+hw$, where $x=(u,v,t)$, then corrects with the Newton method applied to the three equationsThe last equation fixes a transverse section and allows tracing through a turning point of $t$, where solving only for $(u,v)$ at successive fixed source parameters would fail. The corresponding space point is $R=P(u,v)$, with derivative $R^{\prime }=P_u{u}^{\prime }+P_v{v}^{\prime }$ along the traced parameter-space curve.

Find initial roots by subdividing the parameter cube, using interval arithmetic bounds on $F$ to discard impossible boxes and validated local solves in the remaining boxes. Trace in both directions, stop at the boundary or a recognized closed loop, and retain a record of traced boxes to avoid duplicates. Search the remaining boxes as well: starting from just one seed would miss disconnected closed reflection curves. Where $DF$ loses rank, reduce the step and subdivide locally to find the outgoing branches instead of continuing a single guessed tangent. If physical visibility is required, discard roots for which either open segment $XE$ or $XQ(t)$ is occluded; test these segments against the surface or any supplied scene geometry.

Control the geometric approximation rather than just the parameter step. For any regular traced arc $R(s)$ with a bound $\Vert R^{\prime \prime }(s)\Vert \le M$, the chord error bound gives an arc-to-chord Hausdorff distance at most $M(\Delta s{)}^2/8$. Budget, for example, half of $ϵ$ for corrected-vertex error and half for chord error; accept a segment only whenBound the vertex error through the inverse transverse Jacobian matrix and a bound on $DP$, or use validated root enclosures. If analytic second-derivative bounds are unavailable, recursively subdivide and enclose each arc in a geometric tube of radius at most the remaining tolerance around its chord. A midpoint test alone is a useful heuristic but cannot certify the requested error for an arbitrary oscillatory curve.

The returned vertices $P(u_k,v_k)$ and their ordered connecting segments give the reflection curve to the prescribed tolerance on every resolved branch. Finite certified termination requires regular evaluable patches, derivative or enclosure bounds, and finitely resolvable branches; eye-facing normals alone do not guarantee these properties. Singular components or an identically vanishing reflection constraint must be handled as such rather than silently reported as an ordinary curve.

Use the limit subdivision surface and limit subdivision curve, rather than treating their initial control meshes as the reflecting geometry. Build adaptive refinement hierarchies for both. Each surface cell supplies a limit-position evaluator, local tangent vectors and a normal vector; each curve interval supplies a limit-position evaluator and tangent. Regular regions may be evaluated as local splines, while extraordinary neighborhoods require the evaluation machinery of the specified subdivision rules. A piecewise smooth surface is handled patch by patch, with explicit boundary or crease joins.

A useful initial polygonal construction refines the surface to small triangles and the source curve to small line segments. For a candidate mirror triangle in a plane $n\cdot (X-X_0)=0$, reflect the eye across that plane:For a source edge $Y(t)=(1-t)Y_0+t{Y}_1$, a physical reflection point is the plane intersection of the line $E^{\ast }Y(t)$:Keep only $0\le t\le 1$, $0<\lambda <1$, points inside the mirror triangle, and eye/source points on the appropriate reflecting side. Unfolding the reflected path across the mirror proves the specular reflection condition. As $Y$ moves along a straight edge, $X$ lies on the intersection of the mirror plane with the plane through $E^{\ast },Y_0,Y_1$, so the retained locus is a straight segment, a point or the empty set. Plane and triangle clipping computes its endpoints. If these defining points are collinear, or a denominator vanishes, use the corresponding limiting geometric case rather than a generic plane formula. A bounding volume hierarchy accelerates candidate search where geometric bounds exclude pairs; refined facet segments provide seeds and approximate connectivity.

To obtain the reflection curve of the actual limits, correct these seeds. In a local surface chart write $S(u,v)$, and on a source interval write $C(t)$. SetThe exact limit reflection locus is $F=0$ together with the front-side conditions. For a two-sided mirror, orient $n$ locally so that $b\cdot n>0$ and require $a\cdot n>0$; for a one-sided mirror use its prescribed orientation and discard the wrong side. At grazing incidence this orientation condition ceases to be strict and the case needs separate handling. Unlike the eye-facing assumption in the parametric question, no global orientation condition is supplied here.

Apply a continuation method: at a rank-two root of $DF$, predict in a nullspace tangent direction and correct by the Newton method on a transverse section. Map corrected parameters to $S(u,v)$, crossing patch and curve-interval boundaries through their hierarchy adjacency. Trace all branches, recognizing closed components and endpoints. Supplement facet seeds by recursive searches of the surface-cell/source-interval product, with interval arithmetic or other certified enclosures of $F$; otherwise a small loop absent from the initial polygonal approximation could be missed. Split singular neighborhoods where the Jacobian matrix loses rank, and test the open source-to-mirror and mirror-to-eye segments for occlusion when selecting actually visible components.

Allocate a geometric error budget to limit evaluation, root correction and polygonal chords. For example, require combined position-evaluation and corrected-root error at most $ϵ/2$ and, on every regular arc, require the chord error bound $M(\Delta s{)}^2/8\le ϵ/2$. Use subdivision position and normal bounds for evaluations: a tiny position error alone is insufficient, since a wrong normal vector changes the reflected ray. Away from singularities an inverse transverse Jacobian matrix bound converts reflection residual into parameter error and a bound on $DS$ converts this to space error. Refine until these bounds hold, or use validated enclosures of entire arcs and their distances to the chords.

Near a caustic the inverse bound can become large, so the hierarchy must refine more strongly or isolate the singular branch geometry. Mesh flatness alone does not certify a reflection tolerance. With regular convergent evaluators and finite resolvable components, the resulting ordered vertices and chords satisfyThe problem supplies no particular subdivision rules or smoothness guarantees, so this construction specifies what their evaluators and bounds must provide; an arbitrary nonsmooth or degenerate limit need not have an ordinary finite reflection curve. Refine for seeds, enforce reflection on the actual limit geometry, then refine the traced arcs to the required spatial tolerance.

For the overlapping monochromatic wave components, let $d$ be the thickness and ${\lambda }_0$ the vacuum wavelength. Define $\Delta n=n_s-n_f>0$ using the slow and fast refractive indices. With complex amplitude convention $e^{i(kz-\omega t)}$, the accumulated slow-minus-fast phase difference isThis is the optical path difference $\Delta nd$ expressed as a wave phase. It is a signed accumulation convention; with the opposite time convention the sign reverses. The outgoing polarization of an electromagnetic wave depends on $\delta$ modulo $2\pi$. Here the positive slow-minus-fast $\Delta n$ differs from the signed principal-index $n_e-n_o$ used in the parent birefringence entry when $n_e<n_o$.

For an original worked example, stipulate $\Delta n=0.0100$ at ${\lambda }_0=600\mathrm{nm}$ and a uniform thickness $d=15.0\mu \mathrm{m}$. These are design assumptions, not measured material data. The optical path difference is $150\mathrm{nm}$, soIt is a quarter-wave plate. At the same wavelength and refractive indices, doubling $d$ to $30.0\mu \mathrm{m}$ gives $\delta =\pi$, a half-wave plate. A $1.0\mu \mathrm{m}$ thickness increase changes $\delta$ by $\pi /30$, or $6^{\circ }$; retaining the same birefringence at $500\mathrm{nm}$ would instead give $\delta =3\pi /5$ for the $15.0\mu \mathrm{m}$ plate. Real refractive indices also vary with wavelength, so this last calculation explicitly neglects optical dispersion.

For these equal transmitted wave amplitudes, orthogonal projection onto a polarizer crossed with the initial $45^{\circ }$ linear polarization gives a simple independent readout. After removing a common wave phase, the component complex amplitudes are proportional to $(1,e^{i\delta })/\sqrt{2}$; orthogonal projection onto $(1,-1)/\sqrt{2}$ gives $(1-e^{i\delta })/2$. Squaring its modulus yields $I_{\perp }/I_{\mathrm{in}}={\sin }^2(\delta /2)$: zero at $d=0$, one half at $15\mu \mathrm{m}$, and one at $30\mu \mathrm{m}$. This assumes ideal polarizers, negligible specular reflection and wave absorption, overlapping components and stable relative wave phase. It measures a periodic function of phase retardation, so one brightness measurement cannot determine the thickness uniquely.

Radiation pressure is the normal momentum flux transferred by electromagnetic radiation to a surface. A photon travelling in vacuum carries momentum $p_{\gamma }=E_{\gamma }/c=h\nu /c$, where $E_{\gamma }$ is its energy, $\nu$ its frequency, $h$ the Planck constant and $c$ the speed of light. Photon absorption transfers that momentum; specular reflection reverses its normal component. This lets light exert a force even when almost no energy becomes heat in the receiver.

Two useful formulas describe different preparations. A directed beam normally incident on a stationary perfect absorber gives pressure $I/c$, and on a stationary perfect mirror gives $2I/c$. Here $I$ is light intensity, the incident power per beam cross-sectional area perpendicular to propagation, in $\mathrm{W}{\mathrm{m}}^{-2}$. An isotropic photon gas has pressure $u/3$, where $u=U/V$ is its radiation internal energy per volume, in $\mathrm{J}{\mathrm{m}}^{-3}$. Stored energy density and travelling energy flux must be distinguished: an isotropic field can have zero net energy flux and positive radiation pressure.

The physical appeal is that the same momentum conservation connects a single photon collision, a mirror's measurable force, and a radiation-filled cavity's equation of state. The directional calculation below complements radiation pressure on a perfectly absorbing surface; the equilibrium spectrum and its temperature dependence belong to blackbody radiation and the photon gas.

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 $p_{\mathrm{abs}}=I/c$. A perfect stationary mirror sends the photons back with reversed momentum, so momentum conservation gives twice the normal transfer, $p_{\mathrm{refl}}=2I/c$. These are time-averaged pressures; the microscopic arrivals need not be uniform.

For a stipulated model light intensity $I=1000\mathrm{W}{\mathrm{m}}^{-2}$ and the exact SI speed of light $c=299792458\mathrm{m}{\mathrm{s}}^{-1}$,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 $\Phi /c\simeq 33.356\mathrm{nN}$. 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 $0\le \theta <\pi /2$. 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 $R$,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 $E_{\mathrm{inc}}=I\cos \theta$, then $p_{\perp }=(a+2R)E_{\mathrm{inc}}\cos \theta /c$. The distinction matters when interpreting an experiment's angle scan. ^[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.

At a smooth mirror with unit normal vector $n$, an incoming unit direction $i$ has outgoing direction $i-2(i\cdot n)n$. Thus the incident and reflected rays make equal angles with the normal.

More precisely, specular reflection redirects each incident light ray into the direction prescribed by the law of reflection, rather than spreading it over many outgoing directions. The incident light ray, reflected light ray, and local normal vector lie in one plane. The surface must be sufficiently smooth on the scale of the wavelength for this model to describe its reflected beam; a visibly flat surface can still scatter light if its microscopic roughness is appreciable. ^[ref].

The displayed formula uses unit vectors $\mathbf{i},\mathbf{n},\mathbf{r}$ and the dot product. It preserves the component parallel to the mirror and reverses the normal component. For a mirror along $y=0$, take $\mathbf{n}=(0,1)$ and $\mathbf{i}=(3/5,-4/5)$. Then $\mathbf{i}\cdot \mathbf{n}=-4/5$ and $\mathbf{r}=(3/5,4/5)$: both directions have vector norm one, and both make an angle $\arccos (4/5)\approx 36.87^{\circ }$ with the relevant normal direction. Replacing $\mathbf{n}$ by $-\mathbf{n}$ leaves the result unchanged.

This same local rule explains a flat mirror's virtual image. For a source at $(0,a)$ above $y=0$, backward extensions of reflected light rays intersect at $(0,-a)$. The light remains above the mirror; its apparent origin lies below it. A curved mirror changes its normal vector from point to point and can instead make reflected light rays converge. The geometry of image formation therefore starts with one simple direction rule. ^[ref].

// Target: optics.bigb