Poles, Zeros, and Residues
Node role: Concept.
Reader task: Distinguish mode existence, response visibility, and destructive interference in one declared complex response.
A resonance-like feature can move, disappear, or become asymmetric without the underlying mode ceasing to exist. The reason is that three different analytic objects control three different questions:
| Object | Question it answers |
|---|---|
| Pole | Where is a natural open-system mode? |
| Residue | How strongly does this input/output projection see that pole? |
| Zero | Where do the observed response paths cancel? |
These meanings apply to any scalar entry or projection of the general multimode scattering matrix.
Poles, residues, and zeros are mathematical features of a declared physical response. They do not identify a device mode by themselves. Physical meaning comes from the circuit topology, port/load boundary, response projection, and continuation evidence recorded by Network Trace Views and Extraction Authority.
Start From The Response Matrix
This section reorganizes the direct-path/coupled-mode response idea (Suh et al. 2004). The assumed model is linear and time invariant; port maps and the direct path are already declared. The following analytic manipulations are this page’s derivations for that model, not claims that each device source prints the same matrices.
For a linear internal system,
\[ \boldsymbol{\chi}_a(\omega)=\mathbf M^{-1}(\omega), \]
and a port observable is
\[ \mathbf S(\omega) =\mathbf S_{\mathrm{dir}}(\omega) +\mathbf D_{\rm port}(\omega)\boldsymbol{\chi}_a(\omega) \mathbf K_{\rm port}(\omega). \]
For one selected input \(j\) and output \(i\),
\[ S_{ij} =[\mathbf S_{\mathrm{dir}}]_{ij} +\mathbf d_i^T\mathbf M^{-1}\mathbf k_j. \]
The internal dynamics, the input projection, the output projection, and the direct path are therefore separate layers.
Poles: Natural Open-System Modes
A pole \(\tilde\omega_\mu\) occurs when the internal response operator becomes singular:
\[ \boxed{\det\mathbf M(\tilde\omega_\mu)=0.} \]
Under the canonical convention in Dynamics Conventions and the Flux-to-Mode Symbol Bridge, a stable simple pole is written
\[ \tilde\omega_\mu=\omega_\mu-i\frac{\kappa_\mu}{2}. \]
System-specific pages restore topology, layer, and channel qualifiers; for example, \(\tilde\omega_{\mu,\mathrm H}^{\mathcal T}\) contains \(\kappa_{\mu,\mathrm{tot,H}}^{\mathcal T}\).
\(\omega_\mu\) is the hybridized resonance frequency and \(\kappa_\mu\) is the energy-decay linewidth. Both are properties of the same open-system pole; a hybridized linewidth is not an independent shape parameter to add after the couplings have been fitted.
If \(\mathbf M\) is frequency dependent because of a distributed or delayed environment, the pole equation remains \(\det\mathbf M(\omega)=0\). It need not reduce to an eigenvalue of one constant matrix.
Residues: Visibility Of A Pole
A singular internal operator is only the first step. Multiply its singular part by the input and output maps to determine whether an experiment sees it. This page derives that projection for an analytic operator with a simple isolated root. Rational partial fractions provide the related representation (Gustavsen and Semlyen 1999); they do not establish physical mode identity.
Near separated simple poles, a scalar response has a local expansion
\[ S_{ij}(\omega) =S_{ij,\mathrm{reg}}(\omega) +\sum_\mu\frac{R_{ij,\mu}}{\omega-\tilde\omega_\mu}. \]
\(R_{ij,\mu}\) is the residue of pole \(\mu\) in that selected input/output projection. It combines:
- how strongly input \(j\) excites the mode;
- how the mode is distributed among internal coordinates;
- how strongly the mode radiates toward output \(i\); and
- normalization and phase conventions.
For a simple zero eigenvalue of an analytic \(\mathbf M\), its right and left null vectors satisfy
\[ \mathbf M(\tilde\omega_\mu)\mathbf v_\mu=0, \qquad \mathbf u_\mu^\dagger\mathbf M(\tilde\omega_\mu)=0. \]
Then the singular part of the susceptibility is
\[ \boldsymbol{\chi}_a(\omega) \sim \frac{ \mathbf v_\mu\mathbf u_\mu^\dagger }{ (\omega-\tilde\omega_\mu) \mathbf u_\mu^\dagger \mathbf M'(\tilde\omega_\mu) \mathbf v_\mu }, \]
and the corresponding projected scattering residue is
\[ \boxed{ R_{ij,\mu} = \frac{ (\mathbf d_i^T\mathbf v_\mu) (\mathbf u_\mu^\dagger\mathbf k_j) }{ \mathbf u_\mu^\dagger \mathbf M'(\tilde\omega_\mu) \mathbf v_\mu }. } \]
This equation is the mathematical version of “brightness.” A pole can exist while either the input overlap or output overlap is zero. In an open non-Hermitian system the residue is generally complex; it is not automatically a positive mode-participation fraction.
Degenerate poles, exceptional points, or defective matrices require a higher order/Jordan expansion. The simple-pole formula must not be applied there.
Local residue-normalized open coupling
This is a local response definition made on this page. It uses the exact Schur-reduced operator, not an inferred Hamiltonian. Linearize each selected diagonal about its own simple root; its slope supplies the normalization needed to compare off-diagonal response in frequency units. The starting elimination algebra is in Schur Complement and Kron Reduction.
For declared anchored retained coordinates \(R\) and complete complement \(E\), the exact open-EOM operand is
\[ \mathbf D_R^{E\downarrow}(\omega) =\mathbf D_{RR}(\omega) -\mathbf D_{RE}(\omega)\mathbf D_{EE}^{-1}(\omega) \mathbf D_{ER}(\omega). \]
For selected, simple diagonal root branches \(\widetilde\omega_i\), define
\[ s_i=-\left. \partial_\omega[\mathbf D_R^{E\downarrow}(\omega)]_{ii} \right|_{\widetilde\omega_i}, \qquad [\mathbf D_R^{E\downarrow}(\omega)]_{ii} =-s_i(\omega-\widetilde\omega_i)+O((\omega-\widetilde\omega_i)^2). \]
The pole residue of that scalar local susceptibility is \(-1/s_i\). Thus, at a declared evaluation frequency \(\omega_*\), the local response coupling is
\[ \boxed{ J_{\rm res}(\omega_*) =\frac{[\mathbf D_R^{E\downarrow}(\omega_*)]_{rp}} {\sqrt{s_rs_p}}. } \]
This is the exact open-EOM equation operand at \(\omega_*\), with a declared square-root branch/sign convention. It removes arbitrary retained-coordinate scale factors and makes the local off-diagonal frequency-like; it is not a Hamiltonian coefficient merely because it has frequency units. \(J_{\rm res}(\omega_*)\) may be complex and depends on the selected root branch, anchored basis, phase/gauge convention, physical boundary, and Schur reduction.
Why this is not generally one constant two-mode model
The complete-complement Schur operator is generally frequency dependent. A constant \(2\times2\) model over a finite band is therefore not generally exact, even after residue normalization. Keep the exact \(\mathbf D_R^{E\downarrow}(\omega)\), or use an enlarged realization that retains the relevant eliminated dynamics, when finite-band response is the claim. Pole-residue normalization and a number-conserving RWA are distinct: the former localizes a selected open response; the latter drops pairing in a declared quadratic Hamiltonian.
The comparison \(J_{\rm res}(\omega_*)\approx J_{\rm RWA}\) is a local approximation, not coefficient inheritance. It requires, at minimum:
- isolated positive-frequency, simple root branches with nonzero slopes and no nearby eliminated-sector singularity or exceptional point;
- a narrow comparison band;
- weak pairing/counter-rotating terms and coupling weak relative to the carrier frequencies, so an independently stated number-conserving/RWA reduction is valid;
- a slowly varying, approximately Markov Schur self-energy, with loss and dispersion weak enough for the intended constant response model;
- compatible anchored normalization and phase/gauge/branch conventions; and
- response closure against the declared ports, waves, reference planes, and drive/observation maps.
The fixed-basis full-Hamiltonian coefficient \(J_{\rm ex}=J_{\rm H}\), its pairing companion \(\Lambda\), and the inherited \(J_{\rm RWA}\) retain their declared anchored basis and convention. Symbol Conventions gives concise definitions of the exchange coefficient, its explicitly stated number-conserving/RWA inheritance, and the distinct response couplings. The full quadratic-block derivation, including the pairing block, is not developed in this selected material. The observed pole half-splitting \(J_{\rm split}\) remains a separate quantity.
Dark Modes And Pole Cancellation
Write one projected response as
\[ S_{ij}(\omega) =S_{ij,\mathrm{reg}}(\omega) +\frac{N_{ij}(\omega)}{D(\omega)}. \]
An internal pole \(\tilde\omega_\mu\) satisfies \(D(\tilde\omega_\mu)=0\), but it is dark in this projection when the corresponding numerator or port overlap also vanishes. Equivalently, the residue formula above gives \(R_{ij,\mu}=0\). The same pole may remain bright in another \(S/Y/Z\) projection.
An apparent missing dip is therefore not evidence that a mode or coupling is absent. It may be a small residue, a numerator cancellation, overlap between features, or cancellation with the direct path.
Zeros: Destructive Interference
The direct and resonant paths add as complex amplitudes (Suh et al. 2004). This page’s cancellation equation below therefore depends on both magnitude and phase. A magnitude minimum can approach a zero without reaching one, and changing the observable can change the cancellation condition.
A zero of the selected transfer function satisfies
\[ \boxed{S_{ij}(\omega_z)=0.} \]
Using the input–output form,
\[ [\mathbf S_{\mathrm{dir}}]_{ij} +\mathbf d_i^T\mathbf M^{-1}(\omega_z)\mathbf k_j=0. \]
Thus a zero normally describes cancellation between a direct path and one or more resonant paths, or cancellation among resonant paths. It is not generally a pole, and it is not generally a zero of one susceptibility numerator.
The observable name matters. These are different statements:
\[ S_{21}(\omega_z)=0, \qquad Z_{21}^{\mathrm{PTC}}(\omega_n)=0, \qquad Y_{\mathrm{diff}}(\omega_y)=0. \]
They use different port variables and boundary conditions. A zero in one view does not imply that another view contains the same zero.
Pole, Dip, Peak, And Zero Are Not Synonyms
| Observation | What can cause it | What it does not prove |
|---|---|---|
| Pole of a fitted rational model | A natural mode represented in that response projection. | Correct physical mode identity without stability and continuation evidence. |
| Minimum of \(|S_{21}|\) | Direct/resonant interference, linewidth, background, and nearby modes. | Pole frequency or transfer-zero frequency. |
| Maximum of \(|S_{21}|\) | Resonant enhancement and background. | An isolated mode or positive residue. |
| Real-axis phase crossing | Response geometry and reference-plane phase. | A unique pole without convention and continuation checks. |
| Complex response zero | Exact cancellation of both real and imaginary parts. | A natural mode. |
Engineering Use
Splitting is not a universal coupling
For two reported hybridized frequencies, the observed half-splitting \(J_{\rm split}=(\omega_+-\omega_-)/2\) is a response-derived feature. It does not by itself identify a fixed-basis Hamiltonian exchange coefficient or a local residue-normalized coupling: detuning, loss, pairing, additional modes, and frequency-dependent loading can all alter the pole separation. See Which coupling is this? before assigning a coupling name.
Use poles for hybridized frequencies and total linewidths after the full physical response model is fixed. Use residues for observability, branch continuation, bright/dark classification, and comparison of port projections. Use zeros for matching, rejection, notch, and interference design.
Vector fitting approximates poles and residues of a sampled response (Gustavsen and Semlyen 1999). Pole sensitivity, resolution, continuation, and complex residuals are engineering evidence for interpreting that approximation; this page supplies no numerical promotion thresholds. It still does not know the physically declared anchored coordinates, their Schur-reduced diagonal roots, or coherent couplings. A physical coupled-mode fit adds the shared topology and parameter structure that relates several poles and residues across a sweep. Comparing its derived complex poles with the eligible VF poles tests whether that physical decomposition reproduces the hybridized response performance.
- Is the phasor convention stated?
- Is the feature a pole of \(\mathbf M^{-1}\), a zero of a named observable, or a sampled extremum?
- Are the input/output projections and direct path explicit?
- Are pole frequency and linewidth derived from the same complex pole?
- Are residues continued across sweep points rather than sorted only by frequency?
- Are nearly cancelling poles and zeros reported?
- Is a dark mode checked in another physically valid observable?
- Is a real-axis complex zero gated on both real and imaginary residuals?
Limits And Failure Modes
- Magnitude-only plots can hide pole/zero cancellation and reference-plane phase.
- A dense pole cluster makes isolated partial-fraction labels unstable.
- A slowly varying background can mimic a broad pole when the fit window is too narrow.
- A flexible background can cancel a physical pole and create false confidence.
- Sorting poles only by real frequency can exchange branch identity at an avoided crossing.
- The simple residue formula fails at repeated or defective poles.
Connections
- Multimode Input–Output Scattering owns the general response matrix.
- Vector Fitting and Passivity explains rational approximation and the distinction between scalar and full-network evidence.
- Network Trace Views owns the observable and reference-plane contract.
- Resonator Decay, Linewidth, and Quality Factor owns the pole-to-linewidth conversion.
References
- Gustavsen and Semlyen, Rational approximation of frequency domain responses by vector fitting (1999).
- Suh, Wang, and Fan, Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities (2004).
- Spring et al. paper context.
| Field | Value |
|---|---|
| Status | Source-backed |
| Review state | Rewritten to separate existence, visibility, and interference; pending Human review. |
| Human review gate | Can a reader explain why three internal poles may produce one visible dip and why a transfer zero is not a pole? |