Multimode Input–Output Scattering
Node role: Concept.
Reader task: Understand how an internal dynamical operator, port projections, and a direct path become a scattering matrix.
A port experiment sees both direct transmission and radiation from retained modes. The matrix formula below reorganizes the coupled-mode/input–output picture (Gardiner and Collett 1985; Suh et al. 2004). The time-to-frequency substitution is derived on this page under the assumptions in Convention And Scope; neither source is being credited with every finite-distance specialization below.
The shortest correct summary is
\[ \boxed{ \mathbf S(\omega) =\mathbf S_{\mathrm{dir}}(\omega) +\mathbf D_{\rm port}(\omega)\, \boldsymbol{\chi}_a(\omega)\, \mathbf K_{\rm port}(\omega), \qquad \boldsymbol{\chi}_a(\omega)=\mathbf M^{-1}(\omega). } \]
Read this expression from right to left:
\[ \text{incident port wave} \xrightarrow{\ \mathbf K_{\rm port}\ } \text{internal drive} \xrightarrow{\ \mathbf M^{-1}\ } \text{mode amplitude} \xrightarrow{\ \mathbf D_{\rm port}\ } \text{radiated port wave}, \]
then add the wave that traverses the device without entering the retained internal modes. This page owns that general form.
The physical origin of the port waves and prefactors is derived one layer earlier in From Node and Line Flux to Input–Output Scattering: node and line flux, a physical coupling branch, exact \(S/Y/Z\), mode energy normalization, and bath elimination precede the matrices used here.
- \(\mathbf M\) is an internal dynamical operator.
- \(\boldsymbol{\chi}_a=\mathbf M^{-1}\) is an internal oscillator susceptibility or Green’s function.
- \(\mathbf S\) is a port-to-port observable after direct propagation, port coupling, and reference-plane conventions are included.
In particular, \(\boldsymbol{\chi}_a\) is not an \(S\), \(Y\), or \(Z\) matrix, and a measured \(S_{21}\) is not one entry of \(\boldsymbol{\chi}_a\). The exact second-order flux susceptibility is the different object \(\boldsymbol\chi_\Phi=\boldsymbol{\mathcal D}_{\rm open}^{-1}\).
Convention And Scope
Use the canonical convention defined in Dynamics Conventions and the Flux-to-Mode Symbol Bridge:
\[ x(t)=x(\omega)e^{-i\omega t}. \]
All frequencies, coherent coupling strengths, and decay rates in the equations below are angular rates in \(\mathrm{rad\,s^{-1}}\). A linewidth \(\kappa\) is an energy-decay rate, so the associated amplitude decays as \(e^{-\kappa t/2}\). The conjugated signs apply under an \(e^{+i\omega t}\) convention.
The reusable index contract is
\[ m,n\in\{1,\ldots,N\}, \qquad i,j\in\{1,\ldots,P\}, \qquad \mu\in\{1,\ldots,N_{\mathrm{pole}}\}. \]
\(m,n\) label retained internal coordinates, \(i,j\) label output and input port channels, and \(\mu\) labels a continued complex pole. A pole index is not a mode-composition label: after an avoided crossing, the same \(\mu\) can change which internal coordinate dominates it.
The model is a linear, time-invariant small-signal model. Nonlinearity may first be linearized about an operating point, but a strongly driven nonlinear or frequency-converting system needs a larger harmonic/Floquet scattering model.
This page uses the canonical photon normalization: internal \(a_x\) is dimensionless and \(s_{j,\rm in/out}\) has units \(\mathrm{s^{-1/2}}\). The exact flux bridge may instead use classical RMS power waves in \(\sqrt{\mathrm W}\); the direct crosswalk near carrier frequency \(\omega_0\) is
\[ s_{j,\rm in/out}^{(\mathrm W)} =\sqrt{\hbar\omega_0}\, s_{j,\rm in/out}^{(\mathrm{ph})}. \]
This page abbreviates \(s^{(\mathrm{ph})}\) as \(s\). Scaling both incident and outgoing channel amplitudes consistently leaves the dimensionless scattering matrix unchanged. Do not combine a photon-normalized internal amplitude with an unconverted power-wave drive coefficient.
From Time-Domain Dynamics To The Matrix Formula
Begin with a declared Markov amplitude model and its output boundary (Gardiner and Collett 1985; Suh et al. 2004). Replace the time derivative by \(-i\omega\), solve the internal system, and substitute into the outgoing wave. This is this page’s derivation of the displayed transfer matrix.
The retained internal amplitudes and external port waves are
\[ \mathbf a(t)= \begin{pmatrix} a_1(t)&\cdots&a_N(t) \end{pmatrix}^{T}, \qquad \mathbf s_{\mathrm{in/out}}(t)= \begin{pmatrix} s_{1,\mathrm{in/out}}(t)&\cdots&s_{P,\mathrm{in/out}}(t) \end{pmatrix}^{T}. \]
A Markov coupled-mode model is
\[ \dot{\mathbf a} =\left(-i\mathbf h-\frac{\boldsymbol{\Gamma}}{2}\right)\mathbf a +\mathbf K_{\rm port}\mathbf s_{\mathrm{in}}, \]
\[ \mathbf s_{\mathrm{out}} =\mathbf S_{\mathrm{dir}}\mathbf s_{\mathrm{in}} +\mathbf D_{\rm port}\mathbf a. \]
\(\mathbf h\) contains the retained coherent frequencies and couplings. \(\boldsymbol{\Gamma}\) is the energy-decay matrix whose one-half appears in the amplitude EOM; it need not be diagonal when several modes share a bath. Fourier transformation gives
\[ \underbrace{ \left[ \frac{\boldsymbol{\Gamma}}{2} +i(\mathbf h-\omega\mathbf I) \right]}_{\displaystyle \mathbf M(\omega)} \mathbf a(\omega) =\mathbf K_{\rm port}\mathbf s_{\mathrm{in}}(\omega). \]
Therefore
\[ \mathbf a =\mathbf M^{-1}\mathbf K_{\rm port}\mathbf s_{\mathrm{in}} =\boldsymbol{\chi}_a\mathbf K_{\rm port}\mathbf s_{\mathrm{in}}, \]
and substitution into the output equation produces
\[ \boxed{ \mathbf S =\mathbf S_{\mathrm{dir}} +\mathbf D_{\rm port}\mathbf M^{-1}\mathbf K_{\rm port}. } \]
This is not a curve-fitting trick. It is the frequency-domain solution of the linear equations of motion plus the input–output boundary condition.
For a unitary number-conserving basis rotation \(\mathbf a=\mathbf U\mathbf c\),
\[ \mathbf M_c=\mathbf U^\dagger\mathbf M\mathbf U, \qquad \mathbf K_c=\mathbf U^\dagger\mathbf K_{\rm port}, \qquad \mathbf D_c=\mathbf D_{\rm port}\mathbf U, \]
so \(\mathbf D_c\mathbf M_c^{-1}\mathbf K_c\) is unchanged. A general non-unitary similarity transform can also preserve a transfer function, but it is an abstract state-space realization unless its energy metric is carried; it does not preserve bosonic normalization or the simple passive identities below.
Matrix Recap
| Object | Shape | Units in canonical photon normalization | Physical meaning |
|---|---|---|---|
| \(\mathbf s_{\mathrm{in}}\) | \(P\times1\) | \(\mathrm{s^{-1/2}}\) | Incident photon-flux waves at declared port reference planes. |
| \(\mathbf s_{\mathrm{out}}\) | \(P\times1\) | \(\mathrm{s^{-1/2}}\) | Outgoing photon-flux waves at those same planes. |
| \(\mathbf a\) | \(N\times1\) | dimensionless | Coherent amplitudes of canonically normalized retained oscillators. |
| \(\mathbf h\) | \(N\times N\) | \(\mathrm{rad\,s^{-1}}\) | Conservative diagonal frequencies and coherent off-diagonal couplings. |
| \(\boldsymbol{\Gamma}\) | \(N\times N\) | \(\mathrm{rad\,s^{-1}}\) | Internal and external energy-decay matrix in the retained canonical oscillator basis. |
| \(\mathbf M\) | \(N\times N\) | \(\mathrm{s^{-1}}\) | Inverse dynamical response: detuning, decay, coherent coupling, and any retained self-energy. |
| \(\boldsymbol{\chi}_a=\mathbf M^{-1}\) | \(N\times N\) | \(\mathrm s\) | Internal oscillator susceptibility: amplitude response per normalized oscillator drive. |
| \(\mathbf K_{\rm port}\) | \(N\times P\) | \(\mathrm{s^{-1/2}}\) | Maps an incident photon-flux wave into drives on the retained oscillators. |
| \(\mathbf D_{\rm port}\) | \(P\times N\) | \(\mathrm{s^{-1/2}}\) | Maps internal oscillator amplitudes into outgoing photon-flux waves. |
| \(\mathbf S_{\mathrm{dir}}\) | \(P\times P\) | dimensionless | Scattering that bypasses the retained resonant coordinates. |
| \(\mathbf S\) | \(P\times P\) | dimensionless | Complete device-plane scattering matrix. |
The symbols \(\mathbf K_{\rm port}\) and \(\mathbf D_{\rm port}\) deliberately distinguish port coupling from a circuit capacitance matrix. The symbol \(\mathbf S_{\mathrm{dir}}\) is retained for the direct scattering path so that it cannot be confused with either object.
What Susceptibility Means
For one uncoupled mode \(x\),
\[ \zeta_x(\omega) =\frac{\kappa_{x,\mathrm{tot}}}{2} +i(\omega_x-\omega) \]
is its inverse bare susceptibility. It has a dissipative part \(\kappa/2\) and a reactive detuning part \(\omega_x-\omega\). The susceptibility itself is
\[ \chi_{x,0}(\omega)=\frac{1}{\zeta_x(\omega)}. \]
Near resonance, the same applied drive produces a larger amplitude because \(|\zeta_x|\) becomes small. This is analogous to a small dynamical impedance, but \(\zeta_x\) is not a physical port impedance and must not be labeled \(Z\).
For coupled modes,
\[ \chi_{ij}(\omega) =\frac{\text{amplitude induced in coordinate }i} {\text{unit generalized drive applied to coordinate }j}. \]
Thus \(\boldsymbol{\chi}_a\) is a Green’s function for the retained canonical oscillator basis declared by the EOM. That basis may be a physically anchored bare-coordinate basis or a transformed hybridized basis, but every other matrix in the equation must use the same choice. Its circuit-to-basis origin is defined by Circuit Models to Bare Coordinates, Open EOM, and Normal Modes. Its off-diagonal elements describe propagation through the internal coupling graph. The matrices \(\mathbf K_{\rm port}\) and \(\mathbf D_{\rm port}\) are required because laboratory ports generally drive and observe linear combinations of those internal coordinates rather than one normalized mode directly.
The following passive constraints reorganize energy-conserving coupled-mode relations (Suh et al. 2004). In a passive Markov realization, radiative damping and port coupling are not independent. With canonically photon-normalized internal amplitudes, a unitary direct path, and the channel gauge used by the fundamental bridge,
\[ \boxed{ \boldsymbol{\Gamma}_{\mathrm{ext}} =\mathbf D_{\rm port}^\dagger\mathbf D_{\rm port}, \qquad \mathbf K_{\rm port}=-\mathbf D_{\rm port}^\dagger\mathbf S_{\mathrm{dir}}. } \]
Since \(\mathbf S_{\mathrm{dir}}\) is unitary, this also gives \(\boldsymbol\Gamma_{\mathrm{ext}}=\mathbf K_{\rm port}\mathbf K_{\rm port}^\dagger\). Energy conservation, reciprocity, time reversal, and the chosen channel gauge constrain the remaining phases and signs. A scalar fit may replace one projected product by a bounded complex residue, but that does not identify independent physical left/right rates or establish a passive multiport model.
How poles, residues, and zeros enter
The response operator becomes singular at the internal open-system poles:
\[ \boxed{\det\mathbf M(\tilde\omega_\mu)=0.} \]
Every entry of \(\boldsymbol{\chi}_a\) can contain those poles, but an observed \(S_{ij}\) contains a pole only when the port projections give it a nonzero residue. A mode may therefore exist physically and remain dark in one trace. Three internal modes do not guarantee three visible dips.
A transfer zero instead comes from cancellation in a named observable. It is not generally a pole, and a zero of \(S_{ij}\) is not interchangeable with a zero of a different \(Y\) or \(Z\) projection. The canonical Poles, Zeros, and Residues node owns the stable-pole convention, residue algebra, dark-mode rules, and observable-specific zero definitions; this page supplies the response operator and port maps used by that analysis.
The Direct Path And Propagation Phase
For a two-port line whose reference planes have been de-embedded to one ideal loading point, the zero-phase, matched, lossless direct path is
\[ \mathbf S_{\mathrm{dir}} = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}. \]
This says an incoming wave at either port leaves through the other port when the retained resonant coordinates are absent. It is a special reference-plane choice, not the general definition of an ideal line.
The transmission-line field derivation, including voltage/current signs and power-wave normalization, is owned by From Node and Line Flux to Input–Output Scattering. Here only its device-plane consequence is needed.
Propagation between reference planes
The line-field derivation and propagation sign are owned by the node/line-flux bridge. For this matrix model, only its reference-plane result is needed. A reciprocal matched line with equal real reference impedances has
\[ t_0(\omega)=e^{-\alpha(\omega)\ell}e^{i\beta(\omega)\ell} =A_{\mathrm{dir}}(\omega)e^{i\phi_{\mathrm{dir}}(\omega)}, \qquad \mathbf S_{\mathrm{dir}}(\omega) = \begin{pmatrix}0&t_0(\omega)\\t_0(\omega)&0\end{pmatrix}. \]
For a nondispersive delay \(\tau=\ell/v_p\), the present \(e^{-i\omega t}\) convention gives \(t_0=e^{+i\omega\tau}\); the opposite phasor convention conjugates this factor. A phase function is dimensionless, whereas a symbol multiplying \(\omega\) in an exponent has units of time. Constant phase, delay, attenuation, or dispersion belongs in \(\mathbf S_{\mathrm{dir}}\) only when it lies between the declared device reference planes.
Input-capacitor placement and source approximation
Heinsoo et al. place their Purcell-filter couplings near voltage antinodes, approximately \(n\lambda/2\) from a series input capacitor. In that reported device, \(C_{\mathrm{in}}=40\,\mathrm{fF}\) and the boundary model directs about \(98\%\) of emitted readout photons toward the output side (Heinsoo et al. 2018, secs. II–III). These numbers describe that paper’s device, not default design values or measurement validation for another device. The directionality is reciprocal interference, not nonreciprocity.
Appendix C neglects dispersion between the T-junction and the input capacitor when deriving its effective linewidth expressions (Heinsoo et al. 2018, Appendix C). Those expressions are the source’s zero-distance boundary approximation. The half-wavelength placement argument does not authorize extending every channel or total-linewidth expression to finite distance by inserting only a phase-rotated reflection coefficient. The retained total and channel linewidth derivation and the paper-specific source record keep those meanings separate.
Series input capacitor and finite-distance feedback
The capacitor two-port below is this page’s circuit algebra from a reciprocal series element and equal real-positive power-wave references (Kurokawa 1965). Its conjugate source-native form appears in (Heinsoo et al. 2018, Appendix C). The later propagation/cascade treatment is a network construction here, not an equation attributed wholesale to that paper.
For a reciprocal ideal series capacitor \(C_{\mathrm{in}}\) between ports with the same positive real reference impedance \(Z_0\), the present \(e^{-i\omega t}\) convention gives
\[ \Gamma_C(\omega) =\frac{1}{1-2i\omega Z_0C_{\mathrm{in}}}, \qquad \mathbf S_C(\omega) = \begin{pmatrix} \Gamma_C&1-\Gamma_C\\ 1-\Gamma_C&\Gamma_C \end{pmatrix}. \]
Here \(Y_C=-i\omega C_{\mathrm{in}}\) and \(Z_C=i/(\omega C_{\mathrm{in}})\), so \(\Gamma_C=Z_C/(2Z_0+Z_C)\). Heinsoo et al. print the complex-conjugate plus-sign expression in their source-native opposite phasor convention; its real part and magnitude are unchanged, but its phase must not be imported without converting conventions. Explicitly, the source-native coefficient is \(\Gamma=(1+2i\omega Z_0C_{\mathrm{in}})^{-1}\) (Heinsoo et al. 2018, Appendix C). For this declared boundary, \(C_{\mathrm{in}}\to\infty\) gives direct continuation and \(\Gamma\to0\), while \(C_{\mathrm{in}}\to0\) opens the capacitive path and gives \(\Gamma\to1\). Saying another topology has “no input capacitor” selects neither limit by itself.
Here \(S_{ij}\) is the outgoing wave at port \(i\) due to an incident wave at port \(j\). The capacitor plane, the filter–feedline coupling or T-junction plane, and the external matched-port planes must be named separately. Electrical distance \(\ell\) is the path length between the declared capacitor and coupling reference planes, interpreted through the line propagation constant; it is not an unspecified layout-center distance.
For the intervening line,
\[ u(\omega)=e^{-\alpha(\omega)\ell}e^{i\beta(\omega)\ell}. \]
In the diagnostic boundary of a matched line with no other discontinuity, the capacitor reflection seen at the coupling plane contains the round trip, \(\Gamma_{C,\mathrm{seen}}=u^2\Gamma_C\). The complete finite-distance response is not obtained by inserting that phase-rotated scalar into every zero-distance formula. Cascade the full capacitor two-port, propagation section, T-junction or coupling network, resonant device, and matched external ports. Keep every reflection coefficient so the cascade or equivalent global network retains repeated round trips.
An integer-half-wavelength anchor position \(x_n\) is meaningful only after declaring the frequency and reference planes:
\[ \ell(x_n)=n\,\frac{\lambda_g(\omega_\star)}{2}. \]
For an anchored filter coordinate \(p\), a dimensionless anchored-bare brightness diagnostic may be defined as
\[ B_p(x) =\frac{\kappa_{p,\mathrm{AB}}(x)} {\kappa_{p,\mathrm{AB}}(x_n)}. \]
\(\kappa_{p,\mathrm{AB}}/(2\pi)=-2\operatorname{Im} \widetilde f_{p,\mathrm{AB}}\) is the anchored filter-coordinate diagonal-root linewidth diagnostic from the same Direct open-system model, branch continuation, port boundary, and reference-plane convention at both positions. The subscript \(p\) labels the retained coordinate, not a scattering port, and this total diagnostic has no port decomposition. The ratio is unavailable when the reference value is not a defined nonzero linewidth. \(\sqrt{B_p}\) is only an amplitude-like retention diagnostic; without an additional normalization it is not a field amplitude, an \(S\)-parameter magnitude, or a power ratio.
Do not interchange the anchored-bare DirectSolve filter-coordinate diagonal-root linewidth diagnostic with an HB fit linewidth, a hybridized full-system pole linewidth, or the \(-3\) dB width of an interfering \(S_{21}\) trace. Their exact linewidth conventions are collected in Resonator Decay, Linewidth, and Quality Factor. The zero-distance approximation and qualified source-device context are retained above.
Direct and pump-off harmonic-balance results may be compared as numerical and model-agreement evidence only when they use the same network, candidate, ports, reference planes, \(Z_0\), phasor convention, and exact frequency points. Compare the full complex entries of \(\mathbf S\), not magnitude alone. No such comparison establishes measurement validation, fabrication closure, nonreciprocity, an optimum, a tolerance, or scientific-result acceptance. A GIF derived from the sealed complex response and sweep coordinates is a presentation view; the source arrays and their provenance remain the primary evidence.
Direct device path is not measurement background
The scalar background below is an explicitly limited engineering approximation, not a universal calibration model. Complex resonator fitting motivates retaining amplitude and phase (Probst et al. 2015), but the error-box separation and complete cascade are choices for the declared reference planes in this page’s model.
The observable has three distinct layers:
\[ \underbrace{\mathbf S_{\mathrm{dir}}}_{ \substack{\text{device-plane path that bypasses}\\ \text{the retained resonant coordinates}}} + \underbrace{\mathbf D_{\rm port}\boldsymbol\chi\mathbf K_{\rm port}}_{ \substack{\text{resonant radiation through}\\ \text{the retained internal system}}} \quad\longrightarrow\quad \underbrace{\text{external error boxes / measurement chain}}_{ \text{outside the declared device reference planes}}. \]
Propagation inside the declared device reference planes belongs to \(\mathbf S_{\mathrm{dir}}\). Cable, connector, gain, attenuation, and delay outside those planes belong to calibration or measurement background. Moving a reference plane moves phase between these descriptions, so the reference-plane location is part of the parameter definition.
When external reflections are negligible over one narrow fit window, a scalar approximation may be written
\[ S_{21}^{\mathrm{meas}}(\omega) =B_{21}^{\mathrm{meas}}(\omega) S_{21}^{\mathrm{device}}(\omega), \]
\[ B_{21}^{\mathrm{meas}}(\omega) =A_B(\omega) e^{i[\phi_{B,0}+\omega\tau_B+\phi_{B,\mathrm{disp}}(\omega)]}. \]
This scalar multiplication is not the general mismatch model. When the external chain reflects appreciably, use complete two-port error boxes and cascade them with the device; repeated reflections then matter.
| Quantity | Units | Meaning |
|---|---|---|
| \([\mathbf S_{\mathrm{dir}}]_{21}=t_0\) | dimensionless complex | Direct device-plane transmission that bypasses retained modes. |
| \(A_{\mathrm{dir}}\) | dimensionless | Direct-path attenuation within the device reference planes. |
| \(\phi_{\mathrm{dir},0}\) | rad | Constant device direct-path/reference-plane phase. |
| \(\alpha\) | \(\mathrm{Np\,m^{-1}}\) | Distributed amplitude-attenuation constant. |
| \(\beta\) | \(\mathrm{rad\,m^{-1}}\) | Distributed phase constant. |
| \(\ell\) | m | Separation of the relevant reference planes. |
| \(\tau\) or \(\tau_g\) | s | Propagation or local group delay. |
| \(B_{21}^{\mathrm{meas}}\) | dimensionless complex | External scalar measurement-chain approximation, not an internal mode parameter. |
| \(A_B,\phi_{B,0},\tau_B\) | mixed as above | External gain/attenuation, phase, and delay nuisance parameters. |
For a general imperfect two-port environment,
\[ \mathbf S_{\mathrm{dir}}(\omega) = \begin{pmatrix} r_1(\omega)&t_{1\leftarrow2}(\omega)\\ t_{2\leftarrow1}(\omega)&r_2(\omega) \end{pmatrix}. \]
The matrix-entry convention is
\[ t_{i\leftarrow j}(\omega)\equiv S_{ij}(\omega), \qquad r_j(\omega)\equiv S_{jj}(\omega). \]
Reciprocity constrains the two transmissions only after the port impedances, power-wave normalization, and reference planes have been made consistent.
Mismatch changes this matrix and may also feed energy back into the internal system. No direct-path or measurement-background parameter is permitted to move the physical poles of the declared internal model.
The algebraic pattern survives; the matrices become more complete.
| Physical change | What remains | What changes |
|---|---|---|
| Unequal or complex port coupling | \(\mathbf S=\mathbf S_{\mathrm{dir}}+\mathbf D_{\rm port}\mathbf M^{-1}\mathbf K_{\rm port}\) | Use the full complex \(\mathbf K_{\rm port}\) and \(\mathbf D_{\rm port}\); do not infer independent Port-1/Port-2 rates from one scalar trace. |
| Feedline mismatch | Internal conservative matrix may remain | \(\mathbf S_{\mathrm{dir}}\) gains reflection; feedback can also modify \(\mathbf M\). |
| Feedline directly couples to \(q\) or \(r\) | Same matrix formula | Additional rows of \(\mathbf K_{\rm port}\) and columns of \(\mathbf D_{\rm port}\) become nonzero, so many \(\chi_{ij}\) contribute. |
| Shared radiative bath | Same retained coordinates | \(\boldsymbol{\Gamma}\) gains off-diagonal correlated decay and usually a coherent bath-induced shift. |
| Frequency-dependent environment | Same input–output idea | Replace \(\mathbf M_0\) by \(\mathbf M_0+\boldsymbol{\Sigma}_{a,\mathrm{env}}(\omega)\) and retain causality. |
| Propagation delay comparable to mode lifetime | Port waves and local responses remain meaningful | A constant Markov matrix may fail; retain explicit propagation or a frequency-dependent self-energy. |
In the frequency-dependent case,
\[ \mathbf M(\omega) =\mathbf M_0(\omega)+\boldsymbol{\Sigma}_{a,\mathrm{env}}(\omega), \]
where diagonal entries of \(\boldsymbol{\Sigma}_{a,\mathrm{env}}\) shift and damp individual coordinates, while off-diagonal entries create environment-mediated coherent coupling or correlated decay.
There are two distinct physical reductions. They must not be mixed silently.
Separated two-port scatterers
This page derives the denominator below by summing repeated round trips: each additional trip contributes \(r_1^Rr_2^Lu^2\). The construction assumes complete local two-ports and a declared intervening line; it is not the same reduction as one shared bath.
If each local system has already been reduced to a complete two-port scattering matrix and the systems communicate only through the intervening line, cascade the two-ports including propagation and repeated reflections. A three-mode local system \(m\) first contributes
\[ \mathbf S^{(m)} =\mathbf S_{\mathrm{dir}}^{(m)} +\mathbf D_{\rm port}^{(m)} [\mathbf M^{(m)}]^{-1} \mathbf K_{\rm port}^{(m)}. \]
Use the complete \(t_m\), \(r_m^L\), and \(r_m^R\) entries of that matrix. For two systems,
\[ t_{\mathrm{total}} =\frac{t_2\,u\,t_1} {1-r_1^{R}r_2^{L}u^2}, \qquad u(\omega)=e^{-\alpha\ell}e^{i\beta\ell}. \]
\(r_1^R\) is the reflection of System 1 seen from its right, and \(r_2^L\) is the reflection of System 2 seen from its left. The denominator sums all round trips. For many systems, convert each full two-port to a consistently oriented transfer matrix and multiply local systems and propagation sections in physical order.
Multiplying only the scalar transmissions, \(t_{\mathrm{total}}\approx t_N\cdots t_1\), is valid only after demonstrating that the omitted reflections and round trips are negligible.
Evidence Contract
The following engineering record explains what makes this page’s model interpretable. It supplies no universal fit window, sampling requirement, uncertainty cutoff, or numerical acceptance threshold.
A reported scattering fit is interpretable only when it records:
- phasor convention, angular-frequency or hertz units, and amplitude/energy linewidth convention;
- internal coordinate order and normalization;
- port order, wave normalization, reference impedances, and reference planes;
- which entries of \(\mathbf K_{\rm port}\) and \(\mathbf D_{\rm port}\) are physically allowed;
- whether \(\mathbf S_{\mathrm{dir}}\) is ideal, calibrated, fitted, or embedded in a larger network;
- whether damping is diagonal or contains shared-bath terms;
- complex residuals, frequency windows, resolution, branch continuation, parameter bounds, and uncertainty or stability evidence; and
- which parameters are physical and which describe the measurement chain.
The following are invalid shortcuts:
- treating a fitted complex residue as proof of separate physical channel linewidths;
- identifying every dip with a pole or every pole with a visible dip;
- hiding propagation phase in a coupling strength;
- using \(\boldsymbol{\chi}_a\) as though it were a port \(Y\) or \(Z\) matrix;
- combining local scattering blocks without their reflections or spacing; and
- assigning independent linewidths to hybridized poles instead of deriving them from the complex roots of the same \(\mathbf M\).
Connections
- Finite-Order Port-Response Models and Chain Realizations owns the finite-dimensional artifact built from this matrix formula and the distinction between a physically anchored basis and a later Chain basis.
- From Node and Line Flux to Input–Output Scattering owns the physical node/line boundary and derives the prefactors and passive constraints used by this general matrix form.
- Poles, Zeros, and Residues owns analytic feature and visibility semantics.
- Resonator Decay, Linewidth, and Quality Factor owns decay and linewidth units.
- Network Trace Views owns how physical \(S/Y/Z\) observables are selected and promoted.
- Port Reference Impedance Semantics owns the wave/reference-impedance boundary.
- Harmonic Balance owns the nonlinear periodic extension.
References
- Gardiner and Collett, Input and output in damped quantum systems (1985).
- Suh, Wang, and Fan, Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities (2004).
- Spring et al., Appendix E building block.
| Field | Value |
|---|---|
| Status | Source-backed |
| Review state | Rewritten from the Human-directed general-form contract; pending Human review of physics, readability, and rendered equations. |
| Human review gate | Can a reader explain every matrix in \(\mathbf S=\mathbf S_{\mathrm{dir}}+\mathbf D_{\rm port}\mathbf M^{-1}\mathbf K_{\rm port}\), distinguish poles/residues/zeros, and choose between a cascade and a shared-bath global matrix? |