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The linear limb-darkening law parametrizes angular specific intensity by a stipulated limb-darkening coefficient $u$:Here $I$ denotes specific intensity at one frequency, or radiance integrated over a specified interval of frequency or wavelength; it does not denote radiant intensity. Within this stipulated range, $u=0$ gives a uniform disk and $I(0)/I_c=1-u$. The linear limb-darkening law is linear in $\mu$, while its radial profile $I(r)=I_c[1-u+u\sqrt{1-(r/R_{\star }{)}^2}]$ is curved. In the linear source function example above, $u=b_{\nu }/(a_{\nu }+b_{\nu })$ when $a_{\nu },b_{\nu }\ge 0$ and $a_{\nu }+b_{\nu }>0$.

For the illustrative choice $u=0.6$, the specific intensity at $r/R_{\star }=0$, $0.8$, and the limiting edge is respectively $I_c$, $0.76I_c$, and $0.4I_c$. These are synthetic values, not measurements of the Sun. Real optical spectra and stellar atmospheres need not obey a linear limb-darkening law; more flexible profiles can be necessary for accurate exoplanet transit photometry. ^[ref]

For a diatomic molecule, let $m_1,m_2$ be the two masses and $r$ their fixed separation. The distances from the centre of mass are $r_1=m_{2}r/(m_1+m_2)$ and $r_2=m_{1}r/(m_1+m_2)$. A perpendicular axis through the centre of mass therefore has moment of inertiaThe reduced mass $\mu$ here is not the electric dipole moment, for which we use $\mathbf{d}$. For a longer linear molecule, replace $\mu r^2$ by ${\sum }_i{m}_i{\rho }_i^2$, where ${\rho }_i$ is each distance along the axis from the centre of mass. In the point-mass idealization, the axial moment of inertia is zero; turning about that axis changes no nuclear positions, so no independent axial rotational degree of freedom is included.

For the field-free linear rigid rotor, assume a fixed internal quantum state with no electronic angular momentum or electronic spin, and omit couplings to nuclear spin. The rotational Hamiltonian is ${\hat{H}}_{\mathrm{rot}}={\hat{\mathbf{L}}}^2/(2I)$. Its orientation wavefunctions are spherical harmonics $Y_{JM}$, and its angular momentum eigenstates satisfywhere $J=0,1,2,\dots$ and $M=-J,-J+1,\dots ,J$. Each $E_J$ belongs to $2J+1$ angular momentum eigenstates distinguished by the magnetic quantum number $M$, an instance of degenerate energy levels. The Planck constant is $h=2\pi \hslash$; $B$ is the frequency-valued rotational constant. Notice that the energy eigenvalues scale as $J(J+1)$, not $J^2$.

Subtracting consecutive energy eigenvalues givesThus increasing level gaps produce a regularly spaced series of lines at $2B,4B,6B,\dots$, with spacing $2B$. Allowed positions do not guarantee that every line is visible: electric-dipole selection rules, occupation of the initial quantum state, and instrument response also matter.

At normal incidence, let $z$ measure distance from the reflector into a uniform transparent layer with refractive index $n$. For a monochromatic wave of vacuum wavelength ${\lambda }_0$, the layer wavenumber is $k=2\pi n/{\lambda }_0$. Represent the incident and returning electric fields by complex wave amplitudes with time factor $e^{-i\omega t}$ suppressed:The time-averaged squared electric field, normalized to the incident contribution, isThis electric exposure pattern has maxima separated byThe wave phase of the reflector shifts the maxima's positions but leaves their spacing unchanged in this uniform-layer model. The electric pattern, rather than the zero net energy flux of an ideal standing wave, is the relevant exposure proxy. An imperfect mirror reduces its contrast: $(I_{\max }-I_{\min })/(I_{\max }+I_{\min })=2|r|/(1+|r{|}^2)$, where $I$ denotes this exposure proxy.

Choose ${\lambda }_0=600\mathrm{nm}$ and $n=1.5$. Then the medium wavelength is $400\mathrm{nm}$ and $d=200\mathrm{nm}$. For an ideal phase-reversing reflector, $r=-1$ and the normalized pattern is $4{\sin }^2(kz)$: its first two maxima are at $z=100$ and $300\mathrm{nm}$. These are chosen calculations, not measured properties of a historical photographic emulsion or the optical response of Mercury.

For multiple mutually incoherent wavelengths, time averaging adds exposure contributions rather than their instantaneous electric fields. In a linearized recording model,Here $S$ is incident spectral exposure density, $q$ a wavelength-dependent recording sensitivity and $D$ a model exposure response. An optical spectrum therefore writes overlapping depth oscillations, not necessarily visibly separated sheets. This is a useful connection to the Fourier transform: approximately, depth is paired with optical spatial frequency $2n/{\lambda }_0$. Finite recording depth, varying sensitivity and nonlinear photographic processing prevent treating this expression as an exact inversion rule for an actual plate.

Connect a passive electrical load of electrical resistance $R_L>0$ from the junction to the reference. In the steady direct current model, $R_L$ and $R_2$ share the same electric potential difference and are connected in parallel. Their equivalent electrical resistance isThis is the replacement required when a voltmeter is connected across $R_2$. The inspected teaching guide explicitly models the meter this way.^[ref]

The Kirchhoff's current law proof makes the changed assumption visible:The upper resistor carries the sum of the lower and load electric currents, so the two original resistors no longer have equal electric currents. In the stipulated $R_L=10\mathrm{k\Omega }$ example, $R_p=5\mathrm{k\Omega }$, the upper current is $2/3\mathrm{mA}$, and each lower branch carries $1/3\mathrm{mA}$.

For a digital multimeter used as a voltmeter, let its input electrical resistance on the selected voltage range be $R_m$. With no other load, set $R_L=R_m$; if an intended load is also present, use $R_L\parallel R_m$ in place of $R_L$. Two instruments connected simultaneously add two branches. Their ideal indicated value equals the voltage of this altered circuit; voltmeter loading can exist even with perfect measurement calibration. Real readout additionally has measurement uncertainty, offsets, and finite resolution. Consult the actual range and input specifications rather than assigning every digital multimeter the same $R_m$.

For an oscilloscope, the resistance-only model applies to settled direct current with a compatible input coupling setting. The oscilloscope input and cable also have capacitance; changing signals can therefore have loading that depends on frequency. A probe changes the impedance seen at its tip and the relation between tip voltage and displayed voltage; use the probe-and-input combination's specification and attenuation setting. NJIT's guide models the parallel electrical resistance and capacitance and adjusts probe compensation with a square-wave readout.^[ref]

A lock-in amplifier measures the component of an input signal synchronized with a reference at a chosen frequency. It multiplies the input by a reference sinusoid, then uses a low-pass filter to retain the slowly varying product. Two references in phase quadrature recover both amplitude and phase difference. A shared timing source keeps the relative wave phase stable: this is coherent detection.^[ref]

The useful trick is to move the wanted frequency to zero before averaging. An electronic amplifier alone enlarges both the wanted signal and unwanted noise; a lock-in amplifier instead selects a component by its timing. A photodiode's electric current can first be converted to an electric potential difference, so the same calculation can measure weak modulated light. The chopped-light lock-in experiment makes this selection tangible.

Consider this original constructed input electric potential difference, with $t$ in seconds and $f_0=1\mathrm{Hz}$:The unwanted component has three times the reference frequency and more than twice the wanted peak amplitude. Nevertheless, the lock-in quadratures over $T=4/f_0=4\mathrm{s}$ are exactlyso $R=0.4\mathrm{V}$ and $atan2(Y,X)=\pi /3=60^{\circ }$. This is a peak amplitude calculation, not an SR830 display prediction.

To see why the larger component vanishes, multiply $B\cos (3\omega t+\psi )$ by each reference:Each sinusoid has zero average value of a function over a whole reference cycle. More generally, a component $B\cos (n\omega t+\psi )$ with integer $n\ge 2$ gives products at $(n-1)\omega$ and $(n+1)\omega$, whose integrals vanish over any integer number of reference cycles. This is the orthogonality behind selecting a Fourier coefficient with sinusoidal references. A nonsinusoidal reference can contain additional frequencies and need not have the same rejection property.

An unwanted component at the reference frequency survives. If $u(t)=B\cos (\omega t+\psi )$ is added to the input, its coordinate vector adds to the wanted one:For example, adding $B=0.1\mathrm{V}$ with $\psi =0$ to the wanted component gives $X=0.3\mathrm{V}$, $Y=0.2\sqrt{3}\mathrm{V}$, $R=\sqrt{0.21}\mathrm{V}\approx 0.458258\mathrm{V}$, and phase difference $atan2(0.2\sqrt{3},0.3)\approx 49.1066^{\circ }$. Longer averaging cannot separate these two synchronized contributions.

A normal mode whose longitudinal phase satisfies the resonator's round-trip condition. Holding the transverse mode fixed, an empty nondispersive two-mirror cavity of length L has adjacent optical frequencies separated by $c/(2L)$. An offset from mirror or transverse phases changes the position of the mode sequence without changing this ideal spacing. A dispersive cavity instead uses its round-trip group delay.

A spectral line profile with the shape of a Cauchy distribution. In the displayed peak-normalized convention, $\nu$ is frequency, ${\nu }_0$ is the line centre, and $\gamma >0$ is the half width at half maximum: $L({\nu }_0\pm \gamma )=1/2$, so the full width is $2\gamma$. The normalization used for a probability density function differs from this unit-peak convention.

The photon phase-space occupation is a Lorentz scalar. For radiation propagating through vacuum, this makes specific intensity divided by cubed frequency invariant under a change of inertial observer. With Doppler factor $D$, $I_{\nu }(\nu )=D^3{I}_{{\nu }^{\prime }}^{\prime }(\nu /D)$. This describes corresponding ray directions; it does not assert that an angularly integrated radiation field is isotropic in both frames.

A loudspeaker converts an electrical signal into sound. It can drive an acoustic wave whose frequency is controlled independently of the spatial measurement.

For a slow uniform layer over a faster half-space, the nth Love wave branch reaches the half-space shear wave speed at the displayed threshold. Its half-space decay exponent then vanishes, so the threshold is a limiting nonlocalized state, not a trapped finite-energy mode. The fundamental has zero threshold and exists at every positive frequency; overtones require frequency strictly above their positive threshold. At high frequency each fixed branch approaches the layer shear wave speed from above.

A low-pass filter preferentially transmits low frequencies and attenuates higher frequencies. It can smooth rapid noise in a slowly varying signal, but it also suppresses wanted variations outside the useful signal bandwidth. An RC low-pass filter gives a concrete continuous-time example; a moving average implements related smoothing on a discrete-time signal.

The useful question is how much change the measurement can tolerate. Preserving the low frequencies of a temperature record while reducing a fast unwanted oscillation can improve readability. Preserving the sharp onset of a pulse requires higher frequencies, so excessive smoothing changes its timing and shape. The RC low-pass frequency response and RC low-pass step response make this signal bandwidth versus settling time tradeoff quantitative.

Luminosity is emitted radiative power: the energy carried away by electromagnetic radiation per unit time, measured in watts. Bolometric luminosity includes all wavelengths; a measurement through one photometric passband captures only part of it. Spectral luminosity records how that power is distributed over frequency. Received observed radiative flux has different units, watts per square metre, because it measures what crosses an observer's collecting area. ^[ref].

For example, an ideal source with luminosity $10^{26}\mathrm{W}$ emits $10^{26}\mathrm{J}$ each second regardless of whether it is viewed from $20$ or $40$ parsecs. Its observed radiative flux changes with distance, while its luminosity remains fixed. Combining stellar parallax with observed radiative flux therefore lets us compare the actual outputs of stars rather than their apparent brightnesses.

Stig Lundquist, the author printed as S. Lundquist, reported an early laboratory test in 1949. The apparatus used mercury in a stainless-steel cylinder, a vibrating disk, and a floating mirror to indicate the motion at the liquid surface. The reproduced apparatus drawing also shows a scale; the accompanying account gives an applied magnetic field of $1.3\mathrm{T}$.^[ref]

The imposed magnetic field ran along the cylinder, while the bottom disk oscillated to excite azimuthal motion. Varying the drive frequency sought a standing wave resonance. A later authored experimental account describes this as an early hint: the resonance peak was not reached, and magnetic damping was substantial.^[ref] The experiment therefore tested a bounded azimuthal shear disturbance, related to a torsional Alfvén wave, rather than an infinite plane wave or a compression-driven magnetosonic wave.

The magnetic field provides the controlled physical parameter: the ideal Alfvén wave prediction makes a propagation speed proportional to its strength, while a resonance also depends on the vessel's boundary conditions. A quantitative test must compare the driven response with finite-electrical conductivity theory and account for mechanical coupling and wall losses. An observed surface motion alone cannot distinguish all these mechanisms. The reproduced apparatus account gives neither a numerical drive frequency nor a field-off comparison. Lundquist's paper itself calls the experiment preliminary and proposes further investigations.^[ref]

The 1985 spectroscopy experiment used niobium-based tunnel Josephson junctions on silicon, thermally anchored with final bias low-pass filters to a dilution refrigerator. Calibrated germanium and cobalt thermometers measured temperature. A filtered coaxial cable injected microwaves; repeated switches to nonzero electric potential difference measured escape with irradiation on and off. Bias scans at fixed microwave frequency resolved separate enhancements matching calculated unequal energy level spacings. Raising temperature merged the structure toward classical response. Junction parameters were calibrated in the classical regime; reducing the superconducting critical current with a magnetic field checked thermal readout. Parameter uncertainties limited absolute resonance positions.^[ref]

The companion escape experiment mounted a tunnel Josephson junction on an attenuating coaxial cable, cold-anchored its bias filters, ramped the electric current, and counted switches to nonzero electric potential difference. Rapid current shutoff limited heating. Classical escape and microwave resonance calibrated the superconducting critical current, capacitance and environmental loading. Low-temperature escape agreed with the quantum prediction using these calibrated parameters. A magnetic field reduced the superconducting critical current to keep the same apparatus thermally activated at lower temperature, testing excess noise. Frequency-dependent loading contributed uncertainty; the predicted damping correction was smaller than combined errors, so the experiment did not resolve that correction.^[ref]

Together these tests connect spectral structure and escape of the same kind of collective coordinate. They do not constitute a demonstration of coherent gate operations on a qubit, or prove that every macroscopic coordinate retains quantum behavior under arbitrary environmental coupling. The attraction is more concrete: a fabricated electrical circuit admits both an independently measurable classical model and quantitative quantum tests of its phase dynamics.

A maser uses stimulated emission to amplify microwaves. With population inversion and feedback it can sustain a coherent wave field as an oscillator. The name abbreviates "microwave amplification by stimulated emission of radiation".^[ref]

Why make molecules supply an oscillator? Their energy levels offer a reproducible frequency reference, while stimulated emission lets successive molecules reinforce one coherent wave field. In his 1964 lecture, Charles Hard Townes traced his motivation to microwave spectroscopy: resolving molecular structure required useful coherent sources.^[ref] The interesting balance is between molecular amplification and escaping energy: even a beam with population inversion fails to oscillate if cavity losses win. This is the microwave counterpart of laser threshold.

Ammonia gas entered a vacuum chamber through fine channels; the selected beam entered a cavity with input and output waveguides.^[ref] With external microwaves swept across a molecular resonance, the 1954 experiment read an increase in transmitted power; with sufficient beam emission, it sustained oscillation.^[ref]

The 1955 readout mixed two maser signals with a local oscillator in a crystal detector, amplified them at an intermediate frequency, and combined them in a diode detector. An oscilloscope showed their beat frequency. Changing cavity tuning, source pressure or focuser field tested sensitivity; excessive detuning stopped oscillation.^[ref] The beat tests relative frequency stability; it does not by itself establish an absolute frequency or an intrinsic linewidth independent of detector noise. Detuning also changes the field–molecule interaction, so the threshold model's constant $G$ cannot describe arbitrary tuning scans.

Multiplying the upper-level and photon equations by $h{\nu }_0$ makes the cancellation of internal transfer visible:Incoming excitation becomes unused excitation carried away, internal dissipation, useful output, or a change in stored energy. Translational kinetic energy, the source's other internal states and preparation losses lie outside this two-level budget. The output cannot exceed the incoming excitation power $h{\nu }_{0}F$.

Choose $G=10^{-4}{\mathrm{s}}^{-1}$ per unit inversion, $\gamma =10^4{\mathrm{s}}^{-1}$, ${\kappa }_{\mathrm{int}}={\kappa }_{\mathrm{out}}=10^6{\mathrm{s}}^{-1}$, and $F=4\times 10^{14}{\mathrm{s}}^{-1}$. These are stipulated teaching parameters. Then $N_0=4\times 10^{10}$, $F_{\mathrm{th}}=2\times 10^{14}{\mathrm{s}}^{-1}$, $N_{\mathrm{ss}}=2\times 10^{10}$ and $n_{\mathrm{ss}}=5\times 10^7$. Using the rounded ammonia inversion transition frequency, the loaded quality factor is approximately $7.50\times 10^4$. The incoming excitation power is $6.327\mathrm{nW}$: $4.745\mathrm{nW}$ leaves unused, $0.791\mathrm{nW}$ becomes internal dissipation and $0.791\mathrm{nW}$ is useful output, with rounding in these displayed values.

There is an instructive extraction optimum within this model. With the same $F$, $G$ and $\gamma$, put $\rho ={\kappa }_{\mathrm{out}}/{\kappa }_{\mathrm{int}}$ and $p=FG/(\gamma {\kappa }_{\mathrm{int}})=4$. For $0\le \rho <p-1$,Differentiation gives ${\rho }_{\mathrm{opt}}=\sqrt{p}-1=1$. At zero coupling no power is extracted; at $\rho =3$ the device reaches onset and the nonzero branch disappears. The optimum balances extraction against the inversion required to compensate that extraction. Equal residence rates and loss of molecular phase memory are simplifying assumptions; the calculated efficiency is not a universal bound for beam masers.

The following original model isolates amplification by inverted electric dipoles. Let $N=N_u-N_l$ be the excess number in the upper member of one equally degenerate resonant pair. Let $a$ be a slowly varying cavity amplitude with photon number $n=|a{|}^2$, and let $p$ describe the collective molecular dipole component in the amplifying phase convention. On exact resonance, stipulate the linear equationsThe dipole relaxation rate is $\Gamma$, and $g_0=|d_{ul}{\mathcal{E}}_{\mathrm{vac}}|/\hslash$ is a single-molecule coupling in the chosen mode normalization. Here $d_{ul}$ is an electric dipole moment matrix element, ${\mathcal{E}}_{\mathrm{vac}}$ is the field's zero-point amplitude, and $\hslash =h/(2\pi )$. Field overlap and dipole orientation enter this coupling. The characteristic polynomial for growth rate $\lambda$ isFor $N>0$, one root becomes positive precisely when $g_0^{2}N>\kappa \Gamma /2$. If dipole relaxation is fast enough to eliminate $p$, then $p\approx g_{0}Na/\Gamma$ andThus positive population inversion alone is insufficient: the gain rate $GN$ must exceed the passive decay rate. This reduction describes the onset near resonance; it does not predict output phase noise, actual transit dynamics or an exact historical threshold. Nikolay Basov and Aleksandr Prokhorov's molecular-generator theory likewise treats self-excitation as competition between molecular gain and resonator-plus-load losses.^[ref]

To include replenishment, suppose upper-state molecules enter at rate $F$ and both levels leave with residence rate $\gamma$. A deliberately simple ordinary differential equation model isOne downward transition removes two units of the difference $N_u-N_l$, so $\dot{N}=F-\gamma N-2GNn$. Below onset the unsaturated inversion is $N_0=F/\gamma$, givingFor fixed mode volume $V$, the equivalent number density condition is $N_0/V>\kappa /(GV)$. Raising selected beam intensity raises gain; increasing quality factor lowers the loss rate at fixed frequency. These statements hold with $G$, residence time and mode overlap fixed.

Above onset, the nonzero steady state hasThe inversion is clamped while excess supply supports output. A finite initial seed is understood: the noiseless equations leave $n=0$ exactly unchanged even above onset. Actual startup has spontaneous emission and field fluctuations.

For fixed spring constant $k$ and linear damping coefficient $b$, replacing mass $m$ by $\lambda m$ changes the exponential envelope rate from $\gamma =b/(2m)$ to $\gamma /\lambda$. The damped angular frequency becomes ${\omega }_{d,\lambda }^2=({\omega }_d^2+{\gamma }^2)/\lambda -{\gamma }^2/{\lambda }^2$. Thus envelope decay times scale exactly as mass, while the familiar inverse-square-root frequency scaling is a weak-damping approximation. A constant gravitational force shifts equilibrium without changing either rate.