A Hertzian dipole is an ideal localized oscillating source much smaller than its radiation wavelength. Its electric version is an infinitesimal current element; its magnetic version is a small current loop with moment $\mathbf{m}=IA\mathbf{n}$. Such elements model the far-field radiation and radiation resistance of electrically small antennas. The approximation requires source dimensions much smaller than $c/\omega$.

The supplied magnetic field has a quasistatic $[m]/r^3$ term, an induction $[\dot{m}]/(c{r}^2)$ term and a transverse radiation $[\ddot{m}]/(c^{2}r)$ term. The electric field has induction and radiation terms. Brackets mean evaluation at the retarded time $t_r=t-r/c$, not a time average. Near terms mainly exchange stored energy; only the $r^{-1}$ fields give a nonvanishing outgoing power integral as $r\to \infty$.

For a fixed dipole axis the radiation fields satisfy $E_{\varphi }=-c{B}_{\theta }$. The radial Poynting flux is thereforeUsing $\int {\sin }^2\theta d\Omega =8\pi /3$ yieldsFor a general changing dipole direction, replace the squared scalar acceleration by $|\ddot{\mathbf{m}}{|}^2$.

Radiation resistance is the equivalent input resistance whose mean power equals radiated power, not ohmic heating. For a fixed small loop carrying $I(t)=I_0\cos \omega t$, $m=AI$, soThis uses peak current $I_0$; with rms current the power is $I_{\mathrm{rms}}^2{R}_{\mathrm{rad}}$.

The satellite loop needs its changing orientation as well as its induced current. Put the spin axis along z, the external field along x, and let $\mathbf{n}(t)=(\cos \omega t,\sin \omega t,0)$. With $A=\pi b^2$ and resistance $R=10^{-4}\Omega$, the resistance-only approximation givesWrite $m_0=B{A}^2\omega /R$. The full magnetic moment isIt has a static component and a rotating component at $2\omega$. Consequently $|\ddot{\mathbf{m}}{|}^2=4m_0^2{\omega }^4$, andHolding the dipole axis fixed while inserting the sinusoidal current would instead give $B^2{A}^4{\omega }^6/(12\pi {ϵ}_0{c}^5{R}^2)$, eight times smaller. That is the fixed-axis-loop result; it does not describe the stated rotating loop. This is frequency doubling of an induction-driven rotating loop; the changing vector direction is essential.

The resistance-only model is conditional: loop inductance is not supplied. If self-inductance L matters, solve $L\dot{I}+RI=BA\omega \sin \omega t$. In the steady state $I_0=BA\omega /\sqrt{R^2+{\omega }^2{L}^2}$. Its phase changes the static moment direction but not the twice-frequency component's magnitude, so the power becomesThe first expression is the requested resistive idealization and an upper estimate within this passive RL correction. Radiation reaction is negligible compared with the stipulated ohmic resistance. The radiated energy comes from rotational kinetic energy: the induced magnetic torque brakes the loop. Most lost rotational energy is Joule heating; a small part escapes as radiation. Maintaining a constant spin would require external mechanical power.

For scale only, take $B=50\mu \mathrm{T}$, a representative near-Earth field rather than a supplied orbital measurement. The terrestrial surface range reported by NOAA is roughly $25$–$65\mu \mathrm{T}$; the actual orbital value depends on position.^[ref] With $b=1\mathrm{m}$ and $\omega =2\pi (500{\mathrm{s}}^{-1})$, the resistance-only model givesInductance would reduce this power. The kilohertz signal is exceptionally weak and a metre-scale receiving antenna is electrically tiny. Ionospheric absorption/reflection and strong natural or instrumental low-frequency backgrounds make ordinary ground reception impractical. This is not a theorem that the ionosphere blocks every low-frequency mode: field-guided propagation and condition-dependent leakage are real, as the VIPER mission's measurements were designed to test.^[ref]

At a separation $r=100\mathrm{km}$, $k_{\mathrm{rad}}r\simeq 2.1$, so the observer is not securely in the far field. The radiation term alone has at most the characteristic magnetic amplitudein this optimistic resistance-only vacuum model. Near and induction fields must be retained for a full prediction; a simple far-field inverse-square power estimate is not an exact received signal here. Another satellite avoids the whole ground ionospheric path, but plasma propagation, receiving sensitivity, bandwidth, orientation, integration time and spacecraft noise remain unspecified. Detection is not forbidden in principle, but routine practical detection is implausible at this scale; a definitive instrument-level yes/no does not follow from the supplied data.

For propagation along z, write $\mathbf{E}=\mathrm{Re}\{(E_x\hat{\mathbf{x}}+E_y\hat{\mathbf{y}})e^{i(kz-\omega t)}\}$. A Jones vector contains the complex transverse components. Equal phase modulo $\pi$ gives linear polarization; equal magnitudes with phase difference $\pm \pi /2$ give circular polarization. A common overall phase has no effect on the polarization ellipse. A deterministic optical element acts by a Jones matrix, with successive matrices ordered from right to left along the light path.

A circular polarizer uses a linear polarizer transmitting x and a quarter-wave plate whose principal axes are at $\pm 45^{\circ }$. Up to an overall phase, convenient matrices in fixed physical x/y axes areThe plate is lossless, $Q{Q}^{†}=I_2$, and $Q(1,0{)}^T=(1,i{)}^T/\sqrt{2}$ is circular in the supplied convention. Put P on side A and Q on side B: forward passage is $J_{A\to B}=QP$.

Unpolarized light has no single Jones vector. Represent its intensity by the coherency matrix $C_{\mathrm{in}}=(I_0/2)I_2$ and propagate it as $C_{\mathrm{out}}=J{C}_{\mathrm{in}}{J}^{†}$. Forward transmission givesThus the transmitted light is circular with intensity $I_0/2$. The initial polarizer removes one incoherent linear component; the plate changes phase without restoring the lost intensity.

Reverse passage encounters Q before P and has $J_{B\to A}=PQ$ in the same fixed transverse axes. It givesThe output is linearly polarized along x and again has intensity $I_0/2$. A retarder alone leaves an unpolarized coherency matrix unchanged. Consequently an unpolarized scene appears darker from either side, although the transmitted polarization differs. No polarization-dependent response of the observer is needed to explain this reduction.

For the mirror calculations below, use an ideal near-normal isotropic reflector, $M=-I_2$ up to a common phase. In fixed physical transverse axes reflection leaves the polarization ellipse unchanged; it reverses circular helicity because the propagation direction reverses. A reciprocal plate has the same relative phase delay on the return traversal in this fixed-axis convention. In particular $Q^2=i\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$: two passes exchange x and y linear states.

An optical squeezed vacuum has zero mean field, nonzero mean photon number ${\sinh }^{2}r$, and unequal noise in two principal optical quadratures. With $X_1=(a+a^{†})/2$ and $X_2=(a-a^{†})/(2i)$, the vacuum variance is $1/4$ and $[X_1,X_2]=i/2$. In axes aligned with the squeezing,The ideal pure state saturates $\Delta X_1\Delta X_2=1/4$. A below-threshold degenerate optical parametric amplifier or oscillator can generate it: a coherent pump drives pair creation into a mode, and the pump phase selects the squeezed quadrature. Loss mixes in vacuum noise, while technical phase fluctuations mix the squeezed and antisqueezed axes.

For the stated vacuum–one-photon superposition, normalization requires $|ϵ|\le 1$. Direct Fock-basis matrix elements giveUsing $a{a}^{†}=a^{†}a+1$,Consequently there is no squeezing of the specified quadrature:It equals the vacuum uncertainty only at $ϵ=0$ and reaches $\sqrt{3}/2$ at $ϵ=\pm 1$. The curve is even in $ϵ$. The orientation matters: the same real superposition has $(\Delta X_1{)}^2=1/4-{ϵ}^2/2+{ϵ}^4$, which is below $1/4$ for $0<{ϵ}^2<1/2$. This does not turn it into a Gaussian squeezed vacuum; it simply illustrates quadrature squeezing of a vacuum–one-photon superposition.

For the printed beam splitter convention, expansion gives the operator identitiesThuswhere different input modes commute. The source's equality mixing an expectation on one side with an operator on the other is understood as this operator identity, or with expectation brackets on both sides. The mean difference reveals phase-sensitive mutual coherence between inputs; it is not by itself a photon-number correlation measurement of the usual Hanbury Brown–Twiss type. Repeated fluctuations, rather than one mean value, are needed to infer quadrature uncertainty.

For the independent coherent local oscillator, replace $b,b^{†}$ by $\beta ,{\beta }^{\ast }$ in the mean. The effective signal operator isTherefore $\varphi =0$ measures $X_2$, and $\varphi =-\pi /2$ measures $X_1$, each multiplied by $2|\beta |$. Choosing $\varphi =\pi /2$ measures $-X_1$. These phase choices follow the specified complex beam-splitter matrix; importing a different beam-splitter convention would shift them.

This is balanced homodyne detection. A stable strong local oscillator supplies the phase reference, the signal and oscillator meet at a 50:50 beam splitter, and matched photodetector currents are subtracted. Varying the reference phase selects different quadrature projections. Recording their distributions supplies more information than their means and supports optical homodyne tomography.

The strong-reference limit also needs a noise qualification. Define $X_{\vartheta }=X_1\cos \vartheta +X_2\sin \vartheta$, with $\vartheta =\varphi +\pi /2$. For independent signal and coherent reference, an exact ideal-detector calculation givesOne derivation writes $b=\beta +\delta b$, where $\delta b$ is in vacuum. The reference-amplitude term gives the first contribution; the remaining vacuum term contributes $⟨a^{†}a⟩$. Thus the normalized variance approaches the signal quadrature variance when the reference is sufficiently strong relative to the signal. A finite reference is not exactly a classical noiseless field.

A single detector can also carry quadrature information, but its mean alone still does not determine the variance. It measures $n_C=(n_a+n_b+I_{CD})/2$, retaining the large reference background and its fluctuations. For fixed signal moments and a large coherent reference,The additional vacuum-level term and the uncancelled reference power must be calibrated. In practice the reference may also have much larger classical intensity noise. Balanced subtraction cancels the common total-power term and strongly rejects common-mode technical noise while retaining the phase-sensitive difference. Detector imbalance, unequal efficiencies and reference phase noise limit that advantage. It is therefore the more direct and usually more sensitive quadrature-noise measurement, rather than a claim that a single-port measurement contains no useful information.

The process is sum-frequency generation, with ${\omega }_3={\omega }_1+{\omega }_2$. The same reversible three-wave mixing interaction permits the reverse conversion when the higher-frequency field supplies the pump. The second-order nonlinear polarization couples the three waves; its effective coefficient includes the crystal orientation and polarizations.

One efficiency limit is phase mismatch: contributions generated at different positions acquire different phases and can cancel. A second practical limit is spatial walk-off, which reduces the overlap of finite beams in a birefringent crystal. A beam radius $w$ and relative walk-off angle $\rho$ give an overlap length of order $w/|\tan \rho |$. Absorption and scattering provide further losses. Group-velocity mismatch matters for pulses, and pump depletion eventually invalidates an undepleted-pump calculation; none of these effects should be silently included in the ideal lossless plane-wave equations.

Define ${\kappa }_j=2d_{\mathrm{eff}}{\omega }_j/(n_{j}c)$ and hold $A_1,A_2$ constant. Direct integration of the driven equation giveswhere $\mathrm{sinc}x=\sin x/x$. The second expression has a continuous phase-matched limit. If the generated field is initially absent,For $\Delta k\ne 0$ this is proportional to ${\sin }^2(\Delta kz/2)$: the first maximum is at the nonlinear optical coherence length ${\ell }_c=\pi /|\Delta k|$, and the first subsequent zero is at $2{\ell }_c$. A seed $A_3(0)\ne 0$ adds interference with the generated amplitude and can change that pattern. At exact phase matching, the unseeded result grows as $z^2$ only while pump depletion remains negligible.

To retain depletion consistently, differentiate each intensity before inserting the coupled equations:For real $d_{\mathrm{eff}}$ in the lossless convention, put $Q=A_1{A}_2{A}_3^{\ast }{e}^{i\Delta kz}$. The result isThe signs follow the printed amplitude equations. In initial up-conversion, the newly generated $A_3$ makes $\mathrm{Im}Q<0$, so the two input intensities decrease while the sum-frequency intensity increases. Adding the derivatives gives zero because ${\omega }_1+{\omega }_2-{\omega }_3=0$. The total intensity is conserved, even though the three individual intensities change.

The Manley-Rowe relations areDividing an intensity by $\hslash {\omega }_j$ gives photon flux density. A conversion event removes one photon from each lower-frequency mode and adds one to the upper mode, or does the reverse. Therefore $C_1/\hslash$ and $C_2/\hslash$ count conserved sums of photon fluxes, while $C_3/\hslash$ counts the conserved lower-frequency difference. They are not three independent invariants: $C_3=C_2-C_1$. Energy conservation is also recovered as $I={\omega }_1{C}_2+{\omega }_2{C}_1$.

The unweighted combinations generally fail because the exchanged photons have different energies. Explicitly,The qualification “generally” matters: $D_3$ is conserved for equal lower frequencies in a distinct-mode degenerate case, and every derivative vanishes when there is no exchange. The equations establish nonconservation for generic unequal-frequency interaction, not for all special solutions.

An ideal cross-Kerr interaction between two single modes is $H_{\mathrm{int}}=\hslash \chi n_1{n}_2$, with $n_j=a_j^{†}{a}_j$. The number operators commute with the interaction, so this coupling changes phases without transferring photons between the modes. To implement the specified positive conditional phase, usewhere $n_{2V}=|V⟩⟨V|$ within the one-photon polarization subspace. The horizontal branch has zero interacting occupation. Since $e^{i\Theta n}|\alpha ⟩=|e^{i\Theta }\alpha ⟩$, the specified conditional rotation follows. This is an ideal lossless single-mode model; an actual travelling-wave medium's bandwidth, loss and noise are not established by writing this Hamiltonian.

Use $X_1=(a+a^{†})/2$ and $X_2=(a-a^{†})/(2i)$. A coherent state has centre $(\mathrm{Re}\alpha ,\mathrm{Im}\alpha )$ and variance $1/4$ in either optical quadrature. Multiplication of $\alpha$ by $e^{i\Theta }$ rotates that centre anticlockwise through $\Theta$, keeping the coherent state noise disk unchanged. For real positive $\alpha$ the two centres are $(\alpha ,0)$ and $(\alpha \cos \Theta ,\alpha \sin \Theta )$.

Normalize the initial polarization of an electromagnetic wave qubit. The output of the medium isThe joint state is generally an entangled state. The overlap magnitude is $|⟨\alpha |\beta ⟩|=e^{-2|\alpha {|}^2{\sin }^2(\Theta /2)}$. The state factorizes when the amplitudes coincide, while nearly distinguishable amplitudes make the polarization nearly maximally entangled with the optical mode. Before any polarization conditioning, tracing out mode 2 gives the mixture $(|\alpha ⟩⟨\alpha |+|\beta ⟩⟨\beta |)/2$, with no interference between branches.

Include the normalization factors in the half-wave plate map: $|H⟩\mapsto (|H⟩+|V⟩)/\sqrt{2}$ and $|V⟩\mapsto (-|H⟩+|V⟩)/\sqrt{2}$. The transformed joint state isConditioning on a vertical result and setting $\Theta =\pi$ therefore preparesThis pure optical cat state is the conditional output; it is not the unconditioned reduced state immediately after the Kerr medium. The measurement erases the polarization label, allowing the two amplitudes to interfere. The calculated optical Wigner functions below display the resulting fringes; their negative regions are quasiprobability, not negative detection probabilities. These illustrative plots choose $\alpha =1.15$: the rotation panel uses $\Theta =2\pi /3$, while the cat panels use the conditional Kerr rotation $\Theta =\pi$ and three different relative phases $\phi$.

The coherent-state overlap follows by expanding in the Fock basis:Thus $s=⟨\alpha |-\alpha ⟩=e^{-2|\alpha {|}^2}$ andThe postselection probability for Kerr-cat preparation includes both the normalized input qubit and the wave plate. At $\alpha =0,\varphi =\pi$ the vertical branch has zero probability and no normalized conditional state. For the remaining calculation take $\alpha$ real, as required.

Using $a|\pm \alpha ⟩=\pm \alpha |\pm \alpha ⟩$, put $D=1+s\cos \varphi$. Directly evaluating diagonal and cross terms givesThe last quantity is the mean photon number in the conditional cat, which differs from the input coherent state mean ${\alpha }^2$. Since $a{a}^{†}=n+1$,However, its first moment iswhich is generally nonzero. The original PDF's assertion of a zero mean for arbitrary $\varphi$ is inconsistent with its stated pure state. Consequently its displayed expression is the second moment, not the general variance. The phase-dependent cat-state quadrature variance isThe printed result does become the variance for $\sin \varphi =0$, notably the even and odd cat states. It could also describe an ensemble mixing equal weights of $+\varphi$ and $-\varphi$: that different state's mean vanishes while its second moment is unchanged. Such phase averaging is not part of the specified pure-state preparation.

For nonzero finite $\alpha$, $0<s<1$ and $D>0$. The correct quadrature squeezing condition in this quadrature is thereforeFor a representative phase in $-\pi <\varphi \le \pi$, this is $|\varphi |<\arccos (-e^{-2{\alpha }^2})$. If instead one applies the printed second moment as a variance, the resulting condition is $\cos \varphi >0$, or $|\varphi |<\pi /2$ modulo $2\pi$. That is sufficient, but not necessary, for the actual pure state's squeezing. For example, at $\varphi =\pi /2$, $⟨X_2^2⟩=1/4$ while $\mathrm{Var}{X}_2=1/4-{\alpha }^2{e}^{-4{\alpha }^2}<1/4$. This explicitly tests the difference.

For a fixed phase other than $\pi$ modulo $2\pi$, expandAs the input mean number ${\alpha }^2\to 0$, the normalized state tends to vacuum, $⟨X_2⟩\sim -\alpha \tan (\varphi /2)$ andThis expansion is nonuniform near the odd-cat phase. At exactly $\varphi =\pi$ the vacuum terms cancel and the normalized odd coherent-state superposition tends to $|1⟩$, giving $\mathrm{Var}{X}_2\to 3/4$, while its postselection probability tends to zero. The even coherent-state superposition at $\varphi =0$ instead contains only even photon numbers and tends smoothly to vacuum.

At large input mean number, $s\to 0$, the two coherent state branches become nearly orthogonal, and $\mathrm{Var}{X}_2\to 1/4$ for every fixed phase. The mean tends to zero as well. This does not make the state an incoherent mixture: phase-space interference still distinguishes the ideal quantum superposition, although its contribution to this low-order moment is exponentially small. The loss of coherence in an optical cat state arises when ordinary loss reveals the branch to the environment and suppresses that interference, with factor $e^{-2(1-\eta ){\alpha }^2}$ for transmissivity $\eta$.

For the especially simple even cat,It starts at $1/4$, falls to about $0.1108$ at $|\alpha |\simeq 0.7995$, and returns towards $1/4$ from below. The odd-cat curve decreases from $3/4$ towards $1/4$ from above. Other fixed-phase curves have the small- and large-amplitude limits derived above; for negative $\cos \varphi$, squeezing ends when $e^{-2{\alpha }^2}=-\cos \varphi$. In balanced homodyne detection, centre each quadrature distribution on its measured mean before inferring its variance.