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A population inversion puts more emitters in the upper than in the lower quantum state of a transition in the nondegenerate two-level model. Let $N_2,N_1$ be their number densities. Equal stimulated-transition strengths make the emission rate proportional to $N_2$ and the competing photon absorption rate proportional to $N_1$, so net optical gain is proportional to $N_2-N_1$. For $N_2=6\times 10^{20}{\mathrm{m}}^{-3}$ and $N_1=4\times 10^{20}{\mathrm{m}}^{-3}$, the difference is positive; swapping them makes the medium absorb. Equal populations give zero net stimulated contribution. With degenerate levels, the relevant comparison uses populations per statistical weight of an atomic level, rather than simply $N_2>N_1$.

Why pump through extra levels? Ordinary steady resonant pumping of a nondegenerate two-level system drives both photon absorption and stimulated emission. With equal induced rate $W$ and upper-state decay rate $\Gamma$, its elementary population equation isat a stationary population. Increasing this pump saturates towards $N_2=N_1$; it cannot sustain population inversion in this model. This is a steady rate-equation conclusion, not a universal prohibition on all driven two-level dynamics or all possible laser gain mechanisms. ^[ref].

In a familiar three-level path, pumping excites a higher level which relaxes into the upper lasing level; the lower lasing level is the ground state, so substantial ground-state depletion is needed. A four-level path also provides rapid emptying of the lower lasing level into a separate ground state. Suitable lifetimes then allow accumulation above and depletion below the lasing transition. Pumping still supplies the energy: extra levels provide a route to population inversion, not free energy. ^[ref].

For the field-free, spin-free linear rigid rotor in one fixed internal quantum state, the electric-dipole interaction is $-\mathbf{d}\cdot \mathbf{E}(t)$. A nonzero permanent electric dipole moment along the molecular axis allows pure rotational photon absorption with $\Delta J=+1$ and photon emission with $\Delta J=-1$. The matrix element between initial and final quantum states must be nonzero: the existence of an energy gap alone is insufficient. Relative to a chosen $z$ axis, polarization of an electromagnetic wave along $z$ selects $\Delta M=0$; transverse polarization of an electromagnetic wave permits $\Delta M=\pm 1$.^[ref]

Hydrogen chloride and carbon monoxide are examples with a permanent electric dipole moment. By symmetry, molecular hydrogen and carbon dioxide have no permanent electric dipole moment in the ideal symmetric internal state. They still rotate and have rotational energy eigenvalues, but the simple permanent-electric dipole spectrum described here is absent. This restriction concerns this interaction and these assumptions, not every possible way of detecting molecular rotation.

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.

Arthur Leonard Schawlow's 1981 lecture describes an external-cell arrangement: a beam splitter makes opposed beams, an optical chopper interrupts the pump, and a lock-in amplifier reads the probe detector with the chopper reference. Pump-on reduces absorption and raises the detector signal; scanning the laser finds the narrow resonance. Its first application used iodine; pressure and laser jitter limited the resolution.^[ref]

For an original detector model, let $P_0$ be the probe power reaching a photodetector without sample attenuation and let ${\tau }_{\mathrm{off}},{\tau }_{\mathrm{on}}$ be the sample's probe optical depths. After subtracting a stable detector background, the transmitted powers areA positive synchronous change means less probe photon absorption, provided incident probe power is stable and scattered pump light is excluded. For small optical-depth changes the fractional signal is approximately ${\tau }_{\mathrm{off}}-{\tau }_{\mathrm{on}}$. A lock-in amplifier's voltage also depends on gain, reference phase and chopping waveform; it is not automatically this dimensionless ratio.

Useful controls are blocking the pump, detuning far from the feature, reversing or adjusting beam alignment, and repeating the scan at several pump light intensities. Frequency calibration, sample pressure, transit through the beams and laser frequency fluctuations must be assessed separately. Nearly opposed beams leave a residual Doppler shift: for wave vectors ${\mathbf{k}}_1,{\mathbf{k}}_2$, the simultaneous-resonance condition gives $2\delta =({\mathbf{k}}_1+{\mathbf{k}}_2)\cdot \mathbf{v}$. Exact opposition removes this transverse sensitivity in the first-order model.

In 1989, William E. Moerner and Lothar Kador reported individual pentacene absorbers in para-terphenyl in Optical detection and spectroscopy of single molecules in a solid by W. E. Moerner and L. Kador (pp. 2535–2538). Different local environments spread photon absorption frequencies; tuning into the sparse wings selected few resonant molecules despite many molecules in the illuminated volume.

Their tunable dye laser focused through a $0.85$ numerical aperture lens onto a $100$–$200\mu \mathrm{m}$ crystal immersed in superfluid helium at $1.6\mathrm{K}$. Frequency modulation accompanied either a $5\mathrm{kHz}$ electric field between coated sapphire disks, detected at $10\mathrm{kHz}$, or a $2\mathrm{MHz}$ quartz shear transducer. An avalanche photodiode measured transmitted-light photon absorption, not fluorescence. Double modulation suppressed residual laser amplitude background.^[ref]

Repeated scans gave the predicted modulated line shape. Blank crystals and off-resonance scans gave background; changing modulation frequency expanded and contracted the feature as predicted. Agreement between both perturbation methods supported the individual-absorber inference. The authors acknowledged out-of-focus molecules and the unlikely alternative of multiple absorbers with exactly coincident frequencies.^[ref]

This is evidence for a selected absorber in a prepared, known chemical substance and host. It does not establish arbitrary unknown molecule identity from one spectral feature, provide an atom-resolved photograph, or count every molecule in the crystal. A measurement sensitive to only a narrow excitation range cannot detect a molecule whose transition lies outside that range. This distinction separates the total particle inventory from the number responding to a particular probe.

A single-molecule experiment resolves a response attributable to an individual molecule, rather than only an average over many molecules. Optical detection need not resolve its constituent chemical atoms in an image: repeated photon absorption by the same molecule can produce a measurable response. One detected photon is not automatically one different molecule.

The measured channel matters. Direct photon absorption detection measures a change in transmitted light. Fluorescence detection collects photons emitted after excitation; a fluorescence excitation spectrum varies the illuminating frequency while observing that emission. Both can probe excitation, but their backgrounds and readouts differ. A labeled particle cartoon explains the interpretation; it supplies no experimental evidence by itself.

Such measurements expose differences that a population average can conceal. Suppose two groups of molecules switch between different internal quantum states. Their average response could vary smoothly even though each molecule changes in discrete steps. Resolving an individual response tests that interpretation directly, subject to the measurement's time resolution, sensitivity, and particle assignment. The following experiment makes that assignment through spectral selection and controlled perturbations.

A form of tomography that reconstructs an internal radioactive source from directional counts of individually emitted photons. Known spatial photon absorption gives the exponential path weight in an attenuated Radon transform. Full directed data and a suitable reconstruction formula recover the source intensity rather than a transmitted exterior-beam attenuation map.

Consider a flat circular disk of radius $a>0$, with its center a perpendicular distance $d>0$ from the viewpoint $O$. The line from $O$ to its center is normal to the disk. Its rim defines a right circular cone with $\tan \theta =a/d$ and $\cos \theta =d/\sqrt{d^2+a^2}$. The solid angle of a circular cone immediately givesFor $a=d$, this is the same $45^{\circ }$ half-opening example: about $1.8403$ steradians. Both the radius and distance enter; doubling both leaves the solid angle unchanged.

A useful local version applies to a small surface patch of area $dA$, at distance $r$ from $O$. If $\psi$ is the angle between its unit normal and the line to $O$, its unsigned geometric contribution isThe numerator is the projected area. For a visible patch, perpendicular projection followed by radial projection onto a small piece of a sphere gives this formula. Integrating counts the directions once provided no direction encounters multiple counted patches. For an opaque detector, use its unoccluded face. At grazing view, the projected area tends to zero.

For this on-axis disk, a ring of radius $\rho$ has $r=\sqrt{d^2+{\rho }^2}$ and $|\cos \psi |=d/r$. Its polar coordinates gives $dA=\rho d\rho d\varphi$, so the same answer follows directly from the surface integral:The changing distance and obliquity are both already accounted for; an extra detector-wide cosine factor would count the projection twice.

For $d\gg a$, the Taylor series of $(1+a^2/d^2{)}^{-1/2}$ givesThus the accepted fraction of an ideal point emitter's isotropic output is approximately $A/(4\pi d^2)$, where $A=\pi a^2$. This is the geometry behind the inverse-square law for a small detector. At fixed $a$, however, taking $d\to 0^{+}$ gives $\Omega \to 2\pi$, so the accepted fraction approaches $1/2$ rather than diverging.

For a chosen numerical example, let $a=1\mathrm{cm}$ and $d=5\mathrm{cm}$. The exact solid angle is approximately $0.122015$ steradians, giving an acceptance fraction $0.00970966$. An ideal point emitter with isotropy and total radiant flux $1\mathrm{mW}$ would therefore send about $9.71\mu \mathrm{W}$ onto the disk. The distant-source approximation gives $10.0\mu \mathrm{W}$, about $3.0\%$ high. This calculation assumes unobstructed propagation and counts incident radiant flux; photon absorption, detector response, and any optical attenuator require separate factors.

Stimulated emission is emission induced by incident light, in contrast with the unprompted decay in spontaneous emission. It supplies the amplification mechanism of a laser. ^[ref].

For an emitter with upper and lower energy values $E_2>E_1$, the resonant photon has frequency $\nu$ satisfyingwhere $h$ is the Planck constant. In the elementary photon picture, an incident photon stimulates an emitter in the upper quantum state to emit an additional photon into the same optical normal mode, with the same frequency, propagation direction and photon polarization. The emitter supplies the extra energy; the stimulating photon is not consumed. This builds the existing field and its optical coherence. An emitter in the lower quantum state can instead undergo photon absorption, so stimulated emission alone does not establish net optical gain.

For example, a $2.0\mathrm{eV}$ gap corresponds to $\nu \approx 4.8360\times 10^{14}\mathrm{Hz}$ and a vacuum wavelength of about $620\mathrm{nm}$. One ideal stimulated emission event converts one incoming photon into two outgoing photons in the selected optical normal mode, adding $2.0\mathrm{eV}$ of light energy while lowering the emitter's energy by that amount.

A three-level laser pumps emitters out of their ground state into a higher pump state, which rapidly relaxes into the upper lasing state. The lower lasing state is the ground state itself. Photon absorption on the lasing transition therefore competes with stimulated emission until enough emitters have been removed from the ground state. This is the ground state depletion problem implicit in population inversion.

The model below treats nondegenerate lasing energy levels with equal absorption and emission cross-sections. Its inversion condition is $N_2>N_1$. Real materials require the relevant sublevel populations, degeneracies and transition strengths; the simple half-population condition should not be imposed on every optical transition. Extra levels let the pump and the emitted light act on different transitions, avoiding the steady two-level saturation limit discussed in population inversion.

Choose a closed two-level system, equal stimulated absorption and emission strengths, a monochromatic pump, and no extra dephasing. Let $p_g+p_e=1$ be the lower and upper populations, and let spontaneous emission remove upper population at rate $\Gamma =1/{\tau }_e$. In this convention the natural linewidth is full width at half maximum $\Gamma$ in angular frequency, its half width at half maximum is $\Gamma /2$, and its full width at half maximum in hertz is $\Gamma /(2\pi )$. All $\Delta$ below are angular detunings.

Let $s=I/I_{\mathrm{sat}}$ be the on-resonance saturation parameter. An effective stimulated transition rate that reproduces the monochromatic steady response isHere $R$ describes each stimulated direction, while $\Gamma$ describes irreversible spontaneous emission. The rate equation gives the steady populations; it is not a description of transient Rabi oscillations. Setting ${\dot{p}}_e=0$ givesand the population difference relevant to net weak-probe photon absorption isThe pump depletes the lower population by $p_e$, but reduces the population difference by $q=2p_e$: stimulated emission contributes as well as missing ground-state absorbers. On resonance, $s=1$ gives $p_e=1/4$, $p_g=3/4$, $d=1/2$; $s=9$ gives $p_e=0.45$, $p_g=0.55$, $d=0.10$. As $s\to \infty$, the two populations approach $1/2$, rather than producing a population inversion. Far off resonance, $p_e\to 0$ and $d\to 1$.

For moving atoms use $\Delta =\delta -kv$ in these pump-prepared populations. A weak counterpropagating probe interrogates them near $\delta +kv=0$. A simple population-only approximation to its absorption coefficient averages the weak-probe Lorentzian line shape times $d(\delta -kv)$ over the Maxwell-Boltzmann velocity distribution. It neglects interference between the beams, changes of velocity during illumination, and multilevel optical pumping. Those restrictions matter for real alkali chemical atoms.

Ultraviolet is electromagnetic radiation with wavelengths shorter than the violet end of ordinarily visible light. The CIE convention uses $100$–$400\mathrm{nm}$, subdivided into UV-A at $315$–$400\mathrm{nm}$, UV-B at $280$–$315\mathrm{nm}$ and UV-C at $100$–$280\mathrm{nm}$. These divisions name intervals; they do not mark universal changes in every material's response. The boundary with visible light is not sharp, and other fields use near-, far-, vacuum- and extreme-ultraviolet divisions. ^[ref]. For example, NASA's Solar Dynamics Observatory observes extreme-ultraviolet bands below $100\mathrm{nm}$; the CIE interval is therefore not the full extent of astronomical ultraviolet usage. ^[ref].

The CIE ultraviolet interval corresponds to frequencies about $7.49\times 10^{14}$–$3.00\times 10^{15}\mathrm{Hz}$ and photon energies $4.97\times 10^{-19}$–$1.99\times 10^{-18}\mathrm{J}$, using vacuum wavelength and $E=hc/\lambda$ with Planck constant $h$ and speed of light $c$. Shorter wavelength means larger energy per photon; larger irradiance means more incident power per area. These are different comparisons. At the same incident power, a $254\mathrm{nm}$ ultraviolet beam has fewer photons per second than a $940\mathrm{nm}$ infrared beam, even though each ultraviolet photon has more energy.

A common ultraviolet mechanism is photon absorption promoting an electron from a ground state to an excited state of a chemical atom or molecule. By contrast, many infrared absorption bands involve stretching or bending chemical bonds. For either mechanism, both the energy difference and the coupling to the incident electromagnetic field matter. One concrete example is the $217\mathrm{nm}$ absorption discussed for butadiene: an electron moves between molecular orbitals, and the position of the absorption reveals their energy separation. Ultraviolet absorption can therefore probe molecular orbitals, while infrared spectroscopy probes many molecular motions. ^[ref].

Subsequent chemical reactions depend on the absorbing chemical substance and its environment; the label ultraviolet alone does not predict a reaction rate. At sufficiently short wavelength, a photon can cause photoionization if its energy exceeds the relevant ionization energy. There is no single wavelength that separates all ionizing from all nonionizing behavior for every chemical substance. Likewise, eventual heating after photon absorption is possible in both ultraviolet and infrared. The distinction is useful because different optical spectra expose different processes, not because the two ranges obey different laws of electromagnetism.

Ritter's 1801 observation used a chemical readout rather than a thermometer. In Caltech/IPAC's account, sunlight passed through a glass optical prism and silver chloride was placed in different parts of the optical spectrum. Darkening was weak toward red, stronger toward violet, and pronounced beyond the visible violet edge. The measured evidence was a change in the chemical substance's appearance at a spatial location with no visible color, establishing radiation beyond the eye's ordinary range. It was not a calibrated measurement of a universal ultraviolet optical spectrum. ^[ref].

A controlled modern interpretation of such a chemical test would compare equal exposure durations and matched fresh samples, keep an unexposed sample in darkness, monitor temperature, and distinguish scattered visible light from the intended short-wavelength beam with characterized filtering. These are proposed controls, not an assertion that Ritter performed them. Unequal sample preparation or heating can change a reaction rate. The optical prism, terrestrial atmosphere and sample response restrict which wavelengths the test detects. Darkening is a detector-weighted response, so its spatial maximum is not automatically the maximum of incoming radiant flux or of photon energy.

Thomas Larason and Jeanne Houston's NIST report documents a quantitative ultraviolet detector comparison. An argon source feeds a double-diffraction grating spectrometer; a shutter and mirrors deliver the selected band to a working-standard or test photodiode, successively at the same position. A beam splitter supplies a monitor photodiode. Amplifiers and voltmeters record detector and monitor simultaneously; shutter-closed readings provide dark baselines. The $200$–$500\mathrm{nm}$ scan, typically with $4\mathrm{nm}$ bandpass, measures spectral responsivity. Monitor normalization suppresses source drift; known spectral lines calibrate wavelength. The report assesses temperature effects and unwanted light outside the band, weighted by detector response. The latter need not cancel between different spectral responsivities. ^[ref].

This determines electrical response without visual perception. Bandpass and spatial nonuniformity limit the calibration; aging and changed temperature can affect later use. The scan includes visible light and ultraviolet; no response below $200\mathrm{nm}$ follows from it.

For a narrow beam, the idealized substitution calculation isHere $R$ is spectral responsivity; $I$ and $M$ are dark-subtracted detector and monitor electric currents. This follows by writing $I=RP$ for each detector and using the monitor to track changes in radiant flux. It assumes linear response, unchanged beam splitting and collection geometry, and a sufficiently narrow band. For an illustrative standard with $R_{\mathrm{std}}=0.15\mathrm{A}{\mathrm{W}}^{-1}$, ratios $I_{\mathrm{std}}/M_{\mathrm{std}}=2.0$ and $I_{\mathrm{test}}/M_{\mathrm{test}}=1.6$ give $R_{\mathrm{test}}=0.12\mathrm{A}{\mathrm{W}}^{-1}$. These invented arithmetic inputs illustrate the method; they are not NIST measurements.

An independent photon-counting model connects spectral responsivity to quantum efficiency. If $\eta (\lambda )$ is the number of collected electrons per incident photon, there is no internal multiplication, and $q$ is the elementary charge, thenAt illustrative constant $\eta =0.60$ and $P=1\mu \mathrm{W}$, this gives about $0.123\mu \mathrm{A}$ at $254\mathrm{nm}$ and $0.455\mu \mathrm{A}$ at $940\mathrm{nm}$. The ultraviolet readout is smaller in this model because there are fewer incident photons, not because their individual energies are smaller. Real quantum efficiency depends on wavelength, windows, absorption and carrier collection; a detector's band gap and construction determine its usable range. Constant $\eta$ across these two wavelengths is an analytical comparison, not a claimed specification of a real photodiode.

The curve is the calculated relation $E=hc/\lambda$, and the shaded ranges use the stated CIE conventions. The lower diagram abstracts the detector comparison rather than reproducing an instrument drawing. Together with the infrared heat-balance illustration, it shows why a calibrated detector response, rather than what looks bright to an observer, is the bridge between invisible radiation and a quantitative measurement.

For a narrow beam of monoenergetic photons crossing homogeneous material of thickness $x$, the uncollided wave intensity obeys $I(x)=I_0{e}^{-\mu x}$. Here $\mu$ is the linear attenuation coefficient, counting removal from the incident beam by photon absorption and scattering of light. It depends on photon energy and material; it is not universally independent of energy.^[ref]

An original derivation makes the model explicit. If each thin layer removes fraction $\mu dx$ of the surviving beam, with the same $\mu$ throughout, thenThus equal added thicknesses multiply transmission by the same factor. The ordinary differential equation describes uncollided beam survival. To test it with a detector, keep source output, distance, selected photon energy band and detector response fixed. Use a collimator and suitable detector acceptance to exclude scattering of light into the detector. Subtract background, measure an empty-path reference, and keep the detector in its linear operating range. A changing source distance adds geometric dilution and is not an attenuation measurement at fixed geometry.

For a stipulated example, take $\mu =0.40\mathrm{m}{\mathrm{m}}^{-1}$ and $x=2.5\mathrm{mm}$. Transmission is $e^{-1}\simeq 0.368$. If the empty-path detector rate is $1010{\mathrm{s}}^{-1}$, including background $b=10{\mathrm{s}}^{-1}$, the model predicts sample rate $10+1000e^{-1}\simeq 378{\mathrm{s}}^{-1}$. Recover transmission as $(R-b)/(R_0-b)$; the raw ratio $R/R_0\simeq 0.374$ is biased. These chosen values are a calculation, not measurements of a particular material or a historical apparatus.

X-ray emission spectroscopy studies the frequency spectrum of emitted X-rays. The spectral line positions identify electronic energy differences; their strengths also depend on excitation, photon absorption inside the sample, geometry and detector response. An X-ray tube may excite a sample with electrons, while X-ray fluorescence uses incident photons. The distinction matters when choosing what the experiment measures: identifying a chemical element from a line position is different from determining its abundance from a line strength.

In August 1896, Pieter Zeeman observed broadened sodium spectral lines at Leiden, initially without resolved components. His lecture describes a slit and concave diffraction grating, then compares the broadening with the sodium doublet separation: about one thirtieth, roughly $0.2$ ångström. This supplied a spectral reference scale, not an absolute magnetic field calibration. Later cadmium photographs displayed resolved transverse triplets and axial doublets.^[ref]

The fuller report specifies a concave diffraction grating of ten-foot radius with 14,438 lines per inch, an adjustable slit and a micrometer eyepiece with vertical crosswire. Salt in a flame between electromagnet poles supplied the sodium spectral lines. Repeated switching broadened and restored them. To test changes in flame temperature or vapour density, Pieter Zeeman examined photon absorption through a heated porcelain tube with water-cooled glass windows. The arc lamp was displaced from the magnet and its light redirected by a mirror to avoid disturbing the arc. Rotating a narrower tube made the absorption spectral lines equally broad at different heights; the magnetic field still broadened their whole length.^[ref]

For the axial test, pierced electromagnet poles let the light travel along the magnetic field. A quarter-wave plate and polarizer selected one broadened edge; rotating the analyser by $90^{\circ }$ or reversing the magnet current exchanged the visible edge. The crosswire made its displacement detectable. This tested electromagnetic polarization as well as line width. For its rough estimate the report compared each side's widening with the sodium doublet separation, obtaining about one fortieth, and stated an approximate magnetic field intensity in historical units. These passages give no absolute field-calibration procedure; the inferred charge-to-mass ratio was explicitly a first approximation.^[ref]

Hendrik Antoon Lorentz's charged-oscillator theory connected the observed splitting to moving electric charge. In modern language, a line-position measurement needs a wavelength or frequency calibration and an independent magnetic field calibration before it determines a magnetic dipole moment. Resolving the individual components adds the transition information used in the worked example below.