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Let $\mathcal{R}=|r{|}^2$ be the single-mirror power reflectivity and ${\theta }_g$ the propagation angle inside the gas. Define $\lambda$ as the vacuum wavelength. Successive transmitted complex amplitudes differ by a round-trip factor $\mathcal{R}{e}^{i\delta }$, where, for real dielectric reflection coefficients whose product has positive sign,The cosine arises by comparing successive emerging rays on the same outgoing wavefront; it is not the literal zigzag distance $2d/\cos {\theta }_g$. Equivalently, the round-trip phase is $2k_{z}d$. If the source's $\theta$ is the angle in external air, Snell's law gives $n\sin {\theta }_g=\sin \theta$ andIf $\theta$ denotes the internal angle, simply set ${\theta }_g=\theta$.

For identical lossless mirrors, their power transmission is $1-\mathcal{R}$. In an energy-normalized amplitude convention this is $|t{|}^2=1-\mathcal{R}$; more general interface amplitudes require the corresponding wave impedance factors. The geometric series of transmitted amplitudes givesHenceThis is Fabry–Pérot transmission from multiple reflections.^[ref]

At normal incidence, for nondispersive $n$ and zero net reflection phase modulo $2\pi$, the resonance condition is $\delta =2\pi m$. The resonance frequencies and free spectral range areFor finite $\mathcal{R}<1$ the resonances have finite width, and transmission away from them is nonzero. Thus “only discrete frequencies” is an ideal high-reflectivity description of the selected resonances, not an exact statement about the support of the transmitted spectrum. For example, $\mathcal{R}=0.9$ at $\delta =\pi$ still gives $I_t/I_0=(0.1/1.9{)}^2\simeq 0.00277$.

At half the unit peak, $Q{\sin }^2(\delta /2)=1$. Measuring the phase displacement from a resonance yieldsThe last expression is the high-reflectivity approximation, with finesse $\pi \sqrt{\mathcal{R}}/(1-\mathcal{R})$. At $\mathcal{R}=0.9$, the full phase width is about $0.2109$ radians and $\Delta {\nu }_{\mathrm{FWHM}}/\Delta {\nu }_{\mathrm{FSR}}\simeq 0.03357$. At $\mathcal{R}=0.1$, the minimum transmission is $0.6694$, so the trace never reaches half the peak and this full width at half maximum is not defined. The exact half-maximum condition requires $\mathcal{R}\ge 3-2\sqrt{2}$.

For the absorbing Fabry–Pérot mirror phase and transmission, write $r=\sqrt{\mathcal{R}}{e}^{i{\varphi }_r}$ and let $\mathcal{T}$ be the power transmission of one mirror. Optical absorption makes $\mathcal{R}+\mathcal{T}<1$. The transmitted intensity becomesHere the phase formula is at normal incidence. The peak is reduced below unity, while a change in $\mathcal{R}$ changes the width. The complex reflection phase also moves the resonances: if ${\varphi }_r$ is approximately constant,If either $n$ or ${\varphi }_r$ depends on frequency, this is an implicit condition and the free spectral range need not be constant. No numerical shift is determined without the mirrors' optical response.

Let $\mathbf{n}=\mathbf{r}/a$ be the unit normal from the electrical conductor into the vacuum. The electrostatic boundary conditions at a conductor, together with the vanishing interior electric field, giveWrite $u=\cos \theta$ in spherical coordinates, $D=|\mathbf{r}-{\mathbf{r}}_0|$ on $r=a$, and $z^{\star }=a^2/r_0$. Differentiating the electric potential at fixed angles givesHere the distance identity in the preceding answer was used before simplifying. Hence the induced surface charge density isFor positive $q$ this surface charge density is negative everywhere, with largest magnitude nearest the point charge.

With charge density $\rho$, current density $\mathbf{J}$, and speed of light $c=({ϵ}_0{\mu }_0{)}^{-1/2}$, the four Maxwell equations in vacuum areThe electric potential and magnetic vector potential determine the electric field and magnetic field byThese definitions satisfy Gauss's law for magnetism and Faraday's law identically. The Lorenz gauge condition isSubstituting into Gauss's law and using the Lorenz gauge condition givesFor the Ampère-Maxwell equation, the double-curl identity givesThe two gradient terms cancel by the Lorenz gauge condition. Defining the d'Alembert operator with the sign convention $\square ={\nabla }^2-c^{-2}{\partial }_t^2$, the remaining equations areConversely, if the electric potential and magnetic vector potential satisfy these wave equations and the Lorenz gauge condition, the same substitutions in reverse give Gauss's law and the Ampère-Maxwell equation. The other two Maxwell equations follow identically from the definitions, proving equivalence within the Lorenz gauge.

In photoemission electron microscopy, illuminate a surface with ultraviolet light or X-rays and form an electron-optical image of the emitted electrons. An extraction field accelerates them away from the sample; an immersion objective and projection lenses magnify their spatial distribution onto an electron detector. Unlike a scanned probe, this is a full-field image. Surface preparation and vacuum matter because the emitted low-energy electrons are strongly affected by contamination and local electric fields.

Contrast can come from work-function differences, composition, occupied electronic states, illumination and topography. Near threshold, a small work-function change can produce a large emission-yield change. Therefore brightness is not simply geometrical height. An energy filter can select an emitted-electron energy interval; scanning the selection yields spatially resolved spectra. Sweeping the photon energy across an absorption edge can add element and chemical-state sensitivity. Appropriate polarization-dependent measurements can probe magnetic domains.^[ref]

Resolution depends on electron-lens aberrations, the accepted angular and energy spread, extraction geometry, sample flatness, space charge, mechanical stability and detector sampling. Rejecting electrons improves some aberrations at a cost in signal. An unfiltered photoelectron image and a chemically selected image therefore answer different questions even when they use the same microscope.

Approximate Earth by an isolated, nonrotating, spherically symmetric body. By Birkhoff's theorem, the exterior vacuum metric tensor is the Schwarzschild metric. With $x^0=ct$ it isHere $r$ is the areal radius, $\theta$ and $\varphi$ are spherical angles, $t$ is Schwarzschild time normalized to the proper time of a static observer at infinity, $M$ is Earth's mass, $G$ the gravitational constant, and $c$ the speed of light. The nonzero diagonal metric tensor components are $g_{00}=-f$, $g_{rr}=f^{-1}$, $g_{\theta \theta }=r^2$, $g_{\varphi \varphi }=r^2{\sin }^2\theta$.

The apple is a test particle; neglect its effect on the metric tensor, air resistance, Earth's rotation and nonspherical structure, and external tidal forces. Near Earth's surface, $GM/(R_E{c}^2)\simeq 7\times 10^{-10}$, $h/R_E\ll 1$, and the falling speed remains much smaller than $c$. Thus the Newtonian limit of this metric tensor is appropriate, while the exact radial geodesic is available as a check. A static observer under the tree has proper time $T=\sqrt{f(R_E)}t$, up to an additive constant. Such an observer is supported by the ground and has nonzero proper acceleration; the released apple instead has zero proper acceleration.

In a local inertial frame in vacuum, gravitational waves travel at $c$ in general relativity; there is no ordinary material index of refraction analogous to a transparent dielectric. Ordinary matter couples so weakly that laboratory prisms for gravitational waves are implausible. Propagation through a varying background gravitational field nevertheless changes ray directions and travel times, giving gravitational lensing and Shapiro time delay. In weak static coordinates $d{s}^2\simeq -(1+2{\Phi }_N/c^2)c^{2}d{t}^2+(1-2{\Phi }_N/c^2)d{\mathbf{x}}^2$, the null condition gives an effective coordinate index of refraction $n_{\mathrm{eff}}\simeq 1-2{\Phi }_N/c^2$. This useful description depends on coordinates and does not change the local propagation speed. Thus astronomical gravitational refraction is plausible, while a substantial everyday material effect is not.

The material culture of science comprises the objects, spaces and skilled practices through which inquiry is carried out and made credible. It includes scientific instruments and specimens, but also the workshop which produces them, the arrangements for storing and moving them, the paper on which observations are recorded, and the bodies trained to operate and inspect them. Its history asks how these things helped make a claim possible. It is not merely a catalogue of old apparatus attached to a history whose explanations are otherwise complete.

A telescope is a useful starting point. Galileo Galilei's celestial reports depended on shaped glass, an arrangement of lenses, pointing and focusing, and a practised interpretation of a small, unstable visual field. A potential witness needed reasons to trust that an unfamiliar appearance was celestial rather than an instrumental artefact. His repeated records of the changing positions near Jupiter supplied an evidential pattern: a sequence was more persuasive than a single spectacular image. The printed drawings and account carried an experience beyond the small group able to look through that particular scientific instrument.^[ref]

Microscopy illustrates why seeing is an activity rather than a passive delivery of facts. A specimen has to be selected, prepared, lit and brought into focus. Robert Hooke's Micrographia by Robert Hooke discusses improving sensory capacities and presents observations through text and engraving. Its enlarged structures are reasoned representations intended to communicate salient features; a reader cannot assume that an engraving is an unmediated view identical to every observer's sight. The modern question is not whether every historical drawing could be mistaken for a photograph, but how its production and circulation supported particular claims.^[ref]

The air pump brings construction and maintenance into the explanation. A receiver, piston, connecting passage and valves create a setting in which air can be progressively withdrawn. Leaks, imperfect seals and the size of the pumping chamber affect what a trial actually achieves. In a modern idealized isothermal model with receiver volume $V$ and fresh evacuated pump space $v$, connecting the two changes the receiver pressure by the factor $V/(V+v)$. After $N$ such ideal cycles,This reconstructive model is not being attributed as Robert Boyle's own derivation. It shows why an instruction to use a pump cannot specify an experimental condition without information about the apparatus and procedure. Nor does a practical pump produce an exact vacuum merely because its operator uses that word.^[ref]

The second panel makes the same point about a balance. In a closed vessel, the mass of all contents can remain constant even while a heated metal gains mass by incorporating material from the gas. Weighing only the metal, or opening the vessel before the final weighing, defines a different comparison. Antoine Lavoisier's experimental accounting therefore involved vessels, gases, transfer procedures and the choice of system boundary as well as a numerical reading. Knowing that an investigator owned an accurate balance is insufficient to reconstruct what a particular weighing established.^[ref]

Material culture also makes overlooked contributors visible. Glassworking, engraving, machining, collecting and carrying specimens can be prerequisites for a result whose printed account names only a learned author. Differences in access to workshops, money and audiences affect whose observations can circulate and be repeated. Such dependence does not entail that observations are arbitrary: failure of a seal or disappearance of a specimen can impose a quite definite physical constraint. Social organization and the resistance of materials are both relevant.

Finally, surviving objects have their own source problems. An scientific instrument in a museum may have been repaired, combined with later parts or removed from the room and working routines which made it useful. A printed plate may depict an ideal arrangement rather than an object that existed in exactly that form. Linking an object to notebooks, accounts and published reports can recover the relation between design and use. The explanatory aim is to understand how material arrangements, skilled action and records made claims testable and transportable. It complements intellectual history by showing what an argument required people actually to build, handle and do.

Place the optical lens at $z=0$. For a sufficiently broad incident Gaussian beam, the beam waist positions approach the focal lengths frequency by frequency:Add the group delay at the optical lens to propagation through vacuum:Here an arbitrary common time origin has been omitted. Differentiating both equations with respect to angular frequency yieldsFor $\beta >0$, define $u=2\eta c/\beta$. The velocity graph in the preceding figure has horizontal asymptote zero and a vertical asymptote at $u=-1$. For $\eta >0$, $0<v_f<c$: the higher-frequency components both enter later and travel farther. At $\eta =0$, $v_f=c$. For $-\beta /(2c)<\eta <0$, $v_f>c$: the farther-focusing components leave earlier, partly compensating their extra travel. At $\eta =-\beta /(2c)$ every component focuses at the same time, so the trajectory is not a finite-velocity function of time. For $\eta <-\beta /(2c)$ the farther positions focus first, giving $v_f<0$; it tends to zero from below as $\eta \to -\infty$.

A flying focus is a moving pattern assembled from different portions of an optical pulse. Each portion propagates causally at $c$ in this model. The superluminal or backward motion of the focus therefore does not describe a material object or information overtaking light. The calculation assumes negligible additional frequency-dependent group delay from the optical lens; any such delay must be included in $\psi$.

Assume equal excess electron and electron hole number densities $N_t$, with no additional doping imbalance. At zero temperature, both occupied carrier spheres have the same Fermi wavevector $k_F$ because their spin counting is the same:For $k<k_F$, the conduction band is occupied and the valence band has electron holes, so $f_c-f_v=1$. Beyond $k_F$ the occupations reverse and the medium absorbs. Thus the upper edge of the positive-optical gain interval is the supremumEquivalently $E_{\mathrm{Fc}}=E_g+{\hslash }^2{k}_F^2/(2m_c^{\ast })$ and $E_{\mathrm{Fv}}=-{\hslash }^2{k}_F^2/(2m_v^{\ast })$. With the supplied effective masses,Using the vacuum wavelength $\lambda =hc/(\hslash \omega )$ reverses the endpoints:These boundaries are idealized and open: the lower-energy edge has zero joint density of states, and the upper edge is the zero-temperature occupation discontinuity, not a finite-width gain peak. The calculation gives positive material optical gain, not the stronger requirement of exceeding the losses at laser threshold.

A mercury barometer balances atmospheric pressure against the hydrostatic head between the open cistern surface and the mercury surface in a closed tube. Writing $p_t$ for the pressure above the column, the ideal relation isThe relevant $h$ is the vertical difference between the two mercury levels. It is not simply the length of filled glass, and the top space need not be an absolutely perfect vacuum.

A cistern barometer needs the following corrections or controlled adjustments:

The observational precautions are also described in the Guide to Meteorological Instruments and Methods of Observation by the World Meteorological Organization.^[ref] The quantitative corrections are separated in Mercury Barometers and Manometers by W. G. Brombacher and others, particularly the cistern, temperature and gravity discussions. Reduction of a weather-station pressure to sea level is a separate atmospheric comparison, not an intrinsic correction to the instrument.^[ref]

At a stated external pressure, the boiling point of a pure liquid is the temperature at which its saturation vapor pressure equals that pressure. A vapour bubble inside the liquid must also withstand the hydrostatic head and any capillary pressure; the simple free-surface boiling-point rule neglects those small local corrections.

Increasing external pressure raises the usual boiling point, while lowering it lowers the boiling point. The Clapeyron equation explains the direction:for ordinary vaporization with positive latent heat $L$ and a larger vapour specific volume. Consequently water can boil below its usual open-vessel temperature at altitude or under partial vacuum, and above it in a sufficiently pressurized vessel.

Composition also matters. A nonvolatile dissolved solute generally lowers the solvent's vapour pressure and produces boiling-point elevation at fixed external pressure. A volatile second component can instead raise or lower the boiling temperature according to the mixture's phase equilibrium; an azeotrope is a useful warning against applying the nonvolatile-solute rule to every mixture.

Finally, distinguish the equilibrium boiling point from the observed onset of bubbles. A smooth clean vessel or a shortage of nucleation sites can permit superheating; agitation or suitable existing sites can make boiling begin nearer equilibrium. Heating power mainly controls the rate of sustained boiling under otherwise fixed conditions, rather than setting an arbitrary new equilibrium boiling point. A thermometer must be placed to measure the intended liquid or saturated vapour region, rather than a locally overheated wall.

The electron-optical column contains an electron gun, an accelerating supply, a condenser lens system, an objective aperture, an electron objective lens, scan coils and alignment/stigmation controls. The electron gun may use thermionic emission or field emission. The condenser lens system sets the incident probe current and source demagnification; the electron objective lens focuses the probe on the specimen. A movable specimen stage controls position, tilt and working distance. The column and specimen chamber need a vacuum system, appropriate pumps and specimen-transfer hardware. Detectors collect secondary electrons, backscattered electrons or characteristic X-rays; amplification and synchronized raster acquisition assign a measured signal to each image pixel. The first two signals provide different imaging information, while characteristic X-rays allow elemental analysis. A source, probe-forming optics, raster scan, specimen stage, detectors, acquisition electronics and vacuum system form the essential chain.^[ref]

Normal evaporative deposition gives approximately horizontal films on the resist tops and exposed substrate, with little coating of vertical walls. A small distant electron-beam-heated source in high vacuum gives nearly line-of-sight arrival. Near-normal rays reach the opening floor, while vertical walls present almost no projected area.

The floor film and top film can remain disconnected, which helps a later lift-off process. A finite source size, tilted incidence, scattering or sample rotation changes that idealization and may coat one wall or shadow part of the floor. The stated $200\mathrm{nm}$ thickness is well below the $1\mu \mathrm{m}$ resist height, but this alone does not guarantee a clean break at every real edge.

Four useful interactions are dispersion attraction, electrostatic interaction, capillary attraction and short-range repulsion. Their ranges are relative to atomic contact and depend on geometry and environment.

Van der Waals forces provide an attractive contribution extending beyond direct atomic overlap; for an ideal sphere and plane it scales as $F=-AR/(6z^2)$ in a simple nonretarded model. Electrostatic forces can also be long-ranged: biased conducting surfaces are commonly attractive, whereas charged insulating surfaces can attract or repel depending on their charge and screening.

A capillary meniscus in humid air can pull tip and sample together. It acts over the formation and rupture range of the liquid bridge and can greatly exceed atomically localized forces; it is not a universal interaction in a dry vacuum. At the shortest separations, overlap-related Pauli repulsion provides a steep repulsive contact contribution.

Atomic-scale contact contrast relies on strongly localized short-range tip–surface forces, especially the rapidly varying repulsive interaction in the simple contact-mode picture. Long-range attraction mainly adds a background. Real atomic-resolution imaging also depends on apex structure, stability and operating mode; a generic long-range force alone does not resolve individual surface atoms.

Deposit a dilute dispersion onto a clean, electron-transparent TEM support and let the ethanol evaporate. A practical sequence is to redisperse gently, dilute enough to reduce overlap, place a small drop on a holey or lacey support grid, wick away excess liquid and dry thoroughly. Choose support chemistry compatible with wetting by ethanol.

Look for flakes spanning support holes, where no support film obscures the light-element lattice. Adjacent supported regions help locate and stabilize the flakes. Dilution reduces accidental multilayer stacks, while a clean grid and clean solvent reduce hydrocarbon background. Gentle agitation may be useful; prolonged aggressive sonication can itself change the defects one intends to measure.

Drying prevents a volatile liquid from entering the microscope vacuum. Additional cleaning must be chosen cautiously because heating or energetic plasma treatment can alter oxygen groups and defects in reduced graphene oxide. Record a low-dose survey before atomic-resolution imaging, and distinguish pre-existing defects from changes induced during acquisition.

Use multilayer reflective optics and a reflective patterned mask. Near the usual extreme ultraviolet wavelength of 13.5 nm, ordinary lenses and thick transmissive masks absorb too strongly. Multilayer mirrors focus and project the image, while an absorber pattern on a multilayer-coated mask defines the exposed regions.

The optical path operates in vacuum to avoid absorption by air. The key distinction is reflective mask projection, rather than a transparent chromium-on-glass mask or a scanned electron probe.

The electric potential and magnetic vector potential represent the fields through $\mathbf{B}=\nabla \times \mathbf{A}$ and $\mathbf{E}=-\nabla \varphi -{\partial }_t\mathbf{A}$. They are not unique: a differentiable function $\chi$ giveswithout changing either field. Gauge fixing restricts this freedom to suit the sources, boundaries or symmetry; it does not alter observables.

The Lorenz gauge $\nabla \cdot \mathbf{A}+c^{-2}{\partial }_t\varphi =0$ is Lorentz covariant. Maxwell's equations then reduce to sourced wave equations,Retarded solutions implement causal radiation conditions. This gauge is useful for radiation and relativistic calculations; residual transformations obey the homogeneous wave equation.^[ref]

The Coulomb gauge $\nabla \cdot \mathbf{A}=0$ instead gives ${\nabla }^2\varphi =-\rho /{ϵ}_0$. Its electric potential is an instantaneous Poisson solution, but the accompanying magnetic vector potential makes the full physical electric field causal. It is useful in separating longitudinal electric fields from transverse radiation. In a source-free vacuum region, suitable radiation boundary conditions permit radiation gauge $\varphi =0$, $\nabla \cdot \mathbf{A}=0$, leaving two transverse plane-wave polarizations. These extra conditions are not a gauge choice applicable unchanged to arbitrary nonzero charge distributions.

Use $\mathbf{E}(z,t)=\mathrm{Re}[\mathbf{J}{e}^{i(kz-\omega t)}]$. A Jones vector $\mathbf{J}=(J_x,J_y{)}^{\mathsf{T}}$ records the two transverse complex amplitudes. The amplitude ratio and relative phase determine the polarization ellipse; a common nonzero scalar does not change the polarization. A fixed Jones vector describes fully polarized monochromatic light, not an unpolarized statistical mixture.

Linear birefringence gives orthogonal principal polarizations different refractive indices. Put $n_y-n_x=\Delta n>0$, so $x$ is the fast axis. A plate of thickness $d$ gives slow-minus-fast phase retardationThe thinnest quarter-wave plate hasHere ${\lambda }_0=2\pi c/{\omega }_0$ is the vacuum wavelength. For linear input at angle $\theta$ to the fast axis, the output at design frequency is $(\cos \theta ,i\sin \theta {)}^{\mathsf{T}}$ up to common phase. Axis-aligned input remains linear; equal nonzero components at $45^{\circ }$ modulo $90^{\circ }$ give circular polarization; other input angles give elliptical polarization. There is no need to spatially separate the two components.^[ref]

For waveplate retardance detuning, defineNeglecting material dispersion reduces this to $ϵ=(\pi /2)(\omega -{\omega }_0)/{\omega }_0$. Remove the common fast-axis phase. Then $Q=\mathrm{diag}(1,i{e}^{iϵ})$. A similar half-wave plate of twice the thickness has twice the retardation, giving $H=\mathrm{diag}(1,-e^{2iϵ})$. The shared error parameter presumes the same material and matched thickness ratio.

For the half-wave and quarter-wave achromatic circularizer, the first fast-axis angle is $\alpha =15^{\circ }$ and the second is $\gamma =\alpha +\beta =75^{\circ }$. With the stated passive rotation convention,An explicit expression valid for the given shared error is obtained by putting $z=e^{iϵ}$ and definingAt $ϵ=0$, the half-wave plate rotates the linear input to $30^{\circ }$, which is $-45^{\circ }$ relative to the second fast axis. ThusThis is circular polarization. With our time convention its fixed-point electric vector is proportional, after a time-origin shift, to $(\cos \omega t,-\sin \omega t)$; this specifies the rotation without relying on conflicting handedness conventions.

Expanding the exact product to first order givesThese coefficients are rounded to three significant figures. Before rounding, the derivative is exactly ${\mathbf{J}}_0^{\prime }=i\kappa {\mathbf{J}}_0$, with $\kappa =(3-\sqrt{3})/2$. The Jones-component ratio sensitivity therefore vanishes:Only common phase changes at first order. This proves compensation for the prescribed input near the design frequency, not exact achromatic performance throughout the visible spectrum for arbitrary input polarization.

wave phase labels the stage of a physical oscillation at a spacetime event. Two observers comparing the same event must identify the same crest, trough or intermediate stage: a change of coordinates cannot change the number of completed cycles between physically specified wavefronts. The continuous wave phase can therefore be treated as a Lorentz scalar, up to a common choice of wave phase origin. This does not assert invariance of the measured angular frequency, which is the rate of change of that wave phase along the observer's trajectory.

For Lorentz invariance of plane-wave phase, writeusing the metric with signature $(+,-,-,-)$. The gradient $K_{\mu }={\partial }_{\mu }\Phi$ transforms as a covector because $\Phi$ is a scalar. Raising its index therefore gives the four-wavevector. Explicitly, substituting the inverse boost $t=\gamma (t^{\prime }+v{x}^{\prime }/c^2)$, $x=\gamma (x^{\prime }+v{t}^{\prime })$ givesThese are exactly the Lorentz transformation laws for $(\omega /c,\mathbf{k})$. In vacuum, $K_{\mu }{K}^{\mu }=0$; the Doppler effect and aberration of light are two consequences of transforming the same four-vector.

Polarization under a boost. Lorentz invariance of normalized plane-wave polarization preserves the polarization ellipse's shape, handedness and degree of polarization when its transverse basis is carried to the aberrated ray. The propagation direction changes; equality of a fixed laboratory Cartesian pair of field components is not the appropriate comparison.

Rotate the $yz$ axes so that a general incident vacuum plane wave has unit direction $\hat{\mathbf{n}}=(\cos \vartheta ,\sin \vartheta ,0)$. Choose the transverse basis ${\mathbf{e}}_p=(-\sin \vartheta ,\cos \vartheta ,0)$, ${\mathbf{e}}_s=(0,0,1)$, and arbitrary complex amplitudes $E_p,E_s$. ThenPutting $\beta =v/c$, the given transformation givesThe four-wavevector transforms with Doppler factor $\mathcal{D}=\gamma (1-\beta \cos \vartheta )>0$, givingIn the corresponding basis ${\mathbf{e}}_p^{\prime }=(-\sin {\vartheta }^{\prime },\cos {\vartheta }^{\prime },0)$, ${\mathbf{e}}_s^{\prime }={\mathbf{e}}_s$, the transformed field isBoth components of the Jones vector acquire the same positive real multiplier. Their amplitude ratio and relative wave phase are unchanged, proving preservation of arbitrary linear, circular or elliptical polarization. All Stokes parameters scale by ${\mathcal{D}}^2$, so their normalized values and the degree of polarization are unchanged. An unpolarized light ensemble remains unpolarized under this single-ray boost. A mixture of different ray directions requires separate transformations and is outside this statement.

Reflection from the moving slab. Take the upper interface as $z=0$, the material below it, and the incident ray along $-\hat{\mathbf{z}}$. This fixes the sign left implicit by “along the $z$-axis”; choosing the opposite surface convention changes no result below. Treat $n$ as the real refractive index in the slab's rest frame, with unit relative magnetic permeability. In the laboratory,Transform to the slab rest frame:The incidence is now oblique, in the $xz$ plane, at angle $i$ to the surface normal:The incoming ensemble remains unpolarized light. In this frame the ordinary stationary-interface Fresnel equations give the amplitude coefficients:For generic oblique incidence, $R_s=|r_s{|}^2$ and $R_p=|r_p{|}^2$ differ. Equal incident $s$ and $p$ intensities therefore produce unequal reflected intensities: the reflection is partially polarized. Its degree of linear polarization isThe inverse boost preserves this degree, so the same conclusion holds in the laboratory even though the laboratory incident ray is normal to the surface. The moving material cannot be treated as an isotropic stationary dielectric in that frame.

The reflected rest-frame wavevector is $(\omega /c)(\gamma ,-\gamma \beta ,0,+1)$. Boosting back gives $(\omega /c)(1,0,0,+1)$: the reflected laboratory ray is normal to the surface and has the incident frequency. There is no contradiction with the oblique rest-frame geometry. Translation tangential to a uniform interface does not itself introduce a moving normal mirror.

Complete linear polarization. For Brewster reflection from a tangentially moving dielectric, at Brewster's angle, $r_p=0$ and $\tan i_B=n$. Combining this with the aberration relation givesOnly the $s$ component is reflected. The plane of incidence in the slab frame is $xz$, so the reflected electric field is along $y$, perpendicular to the slab velocity. Boosting this $s$ field back keeps its electric direction along $y$. For example, the reflected rest-frame $s$ field has $B_z^{\prime }=-\beta E_y^{\prime }/c$; hence $E_y=\gamma (E_y^{\prime }+v{B}_z^{\prime })=E_y^{\prime }/\gamma$ and $E_x=E_z=0$.

This exact extinction assumes a transparent nonmagnetic dielectric with $n\ne 1$. At $v=0$ the reflected intensities are equal and the reflected beam is unpolarized. If $n=1$, both reflection coefficients vanish and there is no reflected beam whose polarization can be assigned. With dispersion, evaluate $n$ at the shifted frequency ${\omega }^{\prime }=\gamma \omega$; the Brewster-speed equation is then implicit. absorption generally prevents a real incidence angle from making the complex $r_p$ exactly zero.