Reciprocal scattering of spatial modes

Summary: A solved conservative wave model turns a two-sided spatial overlap into reflection and transmission, separating protected polarization mismatch from narrow resonances and coherent transparency.

From an allowed overlap to an actual encounter

A nonzero geometric vertex does not specify how much of an incident wave scatters. The target responds, and that response changes the wave which drives it. This note solves that reciprocal problem in a conservative mode benchmark, including both incident polarizations and all retained target shape modes.

The variables are modes of the one field on the nested macroscopic spatial manifold. The propagating modes represent a chosen physical propagation direction, with their bound transverse profiles already included in their normalization. The target variables represent localized spatial changes of the same field. They are not additional mediator particles or additional physical spatial dimensions.

The model is conditional on a self-adjoint first-order mode reduction. A general Dirac linearization retains the ordinary and conjugate blocks in Dirac source linearization; reducing it to the form below requires justification. The lattice dispersion below is an exact solvable benchmark, not a proposed photon dispersion law.

One reciprocal quadratic energy

Let $a_n\in\mathbb C^p$, $n\in\mathbb Z$, describe $p$ propagating polarization amplitudes on a line, and let $b\in\mathbb C^m$ describe the target response modes. Take

$$E=-J\sum_n\left(a_n^\dagger a_{n+1}+a_{n+1}^\dagger a_n\right) +b^\dagger H_b b+b^\dagger Va_0+a_0^\dagger V^\dagger b, \tag{1}$$

with $J>0$, $H_b=H_b^\dagger$ and an $m\times p$ spatial vertex $V$. The two preparations enter through the target profiles which determine $V$ and through the incident polarization. Their canonical normalization must come from the field action. A shared frequency offset can be added to all modes if positive absolute frequencies are required; the calculation uses frequencies relative to that offset and sets $\hbar=1$.

Variation yields

$$i\dot a_n=-J(a_{n-1}+a_{n+1})+\delta_{n0}V^\dagger b, \qquad i\dot b=H_bb+Va_0. \tag{2}$$

The conserved mode norm is $\sum_n\|a_n\|^2+\|b\|^2$. Both appearances of $V$ in (2) are fixed by the same energy. No independently chosen one-way response coefficient is allowed.

The scalar version belongs to the exactly solvable family of discrete wave resonance models studied by Miroshnichenko and Kivshar. The following matrix formulas are derived directly from (2).

Solve the incoming and outgoing wave equations

Choose $0<k<\pi$. A free wave has relative frequency and positive group velocity

$$\varepsilon=-2J\cos k,\qquad v=2J\sin k. \tag{3}$$

For an incident polarization $c$, use the stationary amplitudes

$$a_n=e^{ikn}c+e^{-ikn}r\quad(n\leq0), \qquad a_n=e^{ikn}t\quad(n\geq0). \tag{4}$$

The value at the shared site gives $t=c+r$. Substitution into its original difference equation gives $iv r=V^\dagger b$. Combining that equation with the target equation gives

$$D(\varepsilon)=\varepsilon I-H_b+\frac{i}{v}VV^\dagger, \qquad \boxed{b=D^{-1}Vc,\quad r=\mathsf r c=-\frac{i}{v}V^\dagger D^{-1}Vc,\quad t=\mathsf t c=(I+\mathsf r)c.} \tag{5}$$

The inverse is evaluated with outgoing boundary conditions. If an exact decoupled target eigenmode makes it singular, use the limiting resolvent on the coupled subspace; an unexcited dark bound mode does not spoil the scattering solution.

The term $iVV^\dagger/v$ is the response into the propagating modes of the same field. Its size and sign have been derived from the spatial junction equation, not inserted as a phenomenological loss rate.

Exact flux conservation

Because $D-D^\dagger=2iVV^\dagger/v$, the inverse identity gives

$$D^{-1}-D^{-\dagger} =-\frac{2i}{v}D^{-\dagger}VV^\dagger D^{-1}.$$

Consequently

$$\mathsf r+\mathsf r^\dagger=-2\mathsf r^\dagger\mathsf r, \qquad \boxed{\mathsf r^\dagger\mathsf r+\mathsf t^\dagger\mathsf t=I.} \tag{6}$$

The two-port scattering matrix and its second unitarity identity are

$$S=\begin{pmatrix}\mathsf r&\mathsf t\\\mathsf t&\mathsf r\end{pmatrix}, \qquad \mathsf r^\dagger\mathsf t+\mathsf t^\dagger\mathsf r=0, \qquad S^\dagger S=I. \tag{7}$$

All propagating polarizations here have the same group velocity. If their dispersions differ, the flux metric must be included. Complex polarized targets need not give $S=S^T$ while that preparation is held fixed: time reversal transforms the target and probe preparations as well. Adjoint terms in one energy establish conservative mutual coupling; they do not erase those polarization conditions.

One shape channel: exact mismatch and a narrow resonance

Let there be one target mode of frequency $\Omega$, and write $V=gq^\dagger$, where $q^\dagger q=1$ is the normalized spatial coupling direction in the two incident polarizations. Then

$$P=q q^\dagger,\qquad \delta=\varepsilon-\Omega, \qquad \Gamma(\varepsilon)=\frac{g^2}{v(\varepsilon)}, \qquad \mathsf r=-\frac{i\Gamma}{\delta+i\Gamma}P. \tag{8}$$

For $c^\dagger c=1$, the reflected fraction is

$$\boxed{\mathcal R(\varepsilon,c) =\frac{\Gamma^2}{\delta^2+\Gamma^2}|q^\dagger c|^2.} \tag{9}$$

This separates two effects with different physical meanings:

• If $q^\dagger c=0$, the target is never driven: $b=0$, $r=0$, $t=c$. This is a geometric polarization mismatch for the specified channel.
• For the matched polarization $c=q$, any nonzero $g$ gives complete reflection at $\delta=0$. A small coupling makes the resonance narrow; it does not make its peak small.

Away from a band edge and at weak coupling, $v$ varies little across the resonance. Its full width at half maximum is approximately $2\Gamma$, and its integrated reflected spectral weight is approximately $\pi\Gamma$. The exact expression remains (9) with its energy-dependent velocity. A broad incident spectrum can therefore scatter weakly even when a finely tuned monochromatic wave scatters strongly.

For a quadrupole channel with $g=g_0\cos(2\theta)$ from transition-density coupling, the width scales as $\cos^2(2\theta)$. At exactly zero overlap the channel closes. For arbitrarily small nonzero overlap its ideal resonant peak can remain large. The limits of zero coupling and exact resonant frequency do not commute. Real finite-time preparations must be evaluated with their actual bandwidth.

Coherent transparency can retain target motion

Take one incident polarization and two target modes,

$$H_b=\begin{pmatrix}-\Omega&0\\0&\Omega\end{pmatrix}, \qquad V=\begin{pmatrix}g\\g\end{pmatrix},\qquad\Omega\ne0.$$

At the central relative frequency $\varepsilon=0$, (5) has

$$b=\frac g\Omega\begin{pmatrix}1\\-1\end{pmatrix}c, \qquad V^\dagger b=0,\qquad r=0,\qquad t=c. \tag{10}$$

The two target responses cancel in the outgoing source while both are nonzero. This is a frequency-specific transparency, unlike the invariant input nullspace $\ker V$. It illustrates why observing transmission alone does not establish that the incoming pattern never engaged the target.

Independent spatial wavepacket check

The diagnostic evolves (2) directly on 501 sites with two polarizations and one target mode. Before evaluating it, the benchmark fixes $J=1$, $g=0.6$, $\Omega=0$, carrier $k=\pi/2$, an incident Gaussian intensity width 12 centred at site $-80$, and final time 100. The boundaries are at $\pm250$, outside the relevant encounter and outgoing packets. The target direction is $q=(\cos0.43,e^{0.71i}\sin0.43)^T$; its complex polarization is retained.

Integration of (9) over the incoming packet spectrum predicts matched reflection $0.8590726693$. Direct evolution gives:

Reflected and transmitted packets are counted outside 25 sites from the target. Their values agree with the integrated stationary prediction to within $1.2\times10^{-6}$; a small remainder is still near the target at the finite stopping time. The full evolving norm and energy are conserved. The dark input leaves the target unexcited throughout the sampled evolution.

These are model units and mode norms. They are not measured particle cross sections or an assignment of a lattice to physical space.

Consequence for the coupling mechanism

Shape and polarization enter the spatial vertex. Reciprocal field dynamics then determine its resonant enhancement, interference and outgoing flux. An exact rotation-neutral source can close a specified vertex entirely. A small but nonzero overlap instead requires a bandwidth and spectrum calculation before it can explain rare scattering.

To apply this result across the particle hierarchy, derive $V$, $H_b$, the continuum dispersion and the background preparation from the same field action. The higher spatial geometry may determine the full vertex; the two photon polarizations specify the incident channel on which it acts.

Verification and related

spatial_coupling_mechanism_checks.py checks the unreduced spatial equations, multichannel unitarity, a protected dark input, resonant reflection, coherent transparency and full wavepacket evolution. The propagating lattice is solved directly for the final check; the scattering formula is not used to evolve it.

• Transition-density coupling — supplies spatial vertices with both shape preparations retained.
• Polarized response spectrum — gives finite-exposure response before completing the reciprocal scattering problem.
• Coherent spatial overlap — includes spatial phase and interference in the input source profiles.