Spectrally Resolved Reflection Matrix

• Spectrally resolved reflection matrices are frequency-dependent operators that map incident channels to reflected modes, capturing the full spectral variation of the reflection process.
• They are computed using techniques such as Fast Fourier Transforms and temporal coupled-mode theory, enabling precise modeling in waveguiding, imaging, and scattering applications.
• Their formulation varies across fields—from coherent field measurements in optics to flux-based descriptions in radiative transfer—providing insights into phenomena like lasing thresholds and polarization effects.

A spectrally resolved reflection matrix is a frequency- or wavelength-dependent operator that maps incident channels to reflected channels at each spectral point. Its specific meaning depends on the underlying wave model: in guided-wave optics it is the matrix $r_{mn}(\omega)$ or $r_{mn}(\lambda)$ that converts an incident guided mode $n$ into a reflected mode $m$; in radiative-transfer models it is a reflection submatrix acting on specular and diffuse flux components; in wavefront-shaping and reflection matrix microscopy it is the field operator $R(\omega)$ or $R(\lambda)$ linking controlled input modes to measured reflected modes; in focused ultrasonic imaging it is a frequency-resolved matrix between virtual sources and receivers; and in coherence scanning interferometry it can appear as a measurement matrix relating a reflection spectrum to an interferometric scan (Svendsen et al., 2010, Slovick et al., 2017, Yu et al., 2015, Lambert et al., 2019, Chen et al., 15 Jun 2025). Across these settings, “spectral resolution” means that the reflection operator is not treated as a single monochromatic object but as a family of matrices indexed by $\omega$ or $\lambda$.

1. Definitions and channel spaces

The common structure is a linear map evaluated at each spectral variable. In field-based formulations this is typically written as

$$\mathbf{E}_{\text{out}}(\omega)=R(\omega)\,\mathbf{E}_{\text{in}}(\omega),$$

where the channel basis may be spatial modes, guided modes, angular modes, or polarization states (Yu et al., 2015). In flux-based radiative transfer, the reflected state is written as

$$\mathbf{F}_{\text{refl}}^{(0)}(\lambda)=\mathbf{R}(\lambda)\,\mathbf{F}_{\text{in}}^{(0)}(\lambda),$$

with channels corresponding to forward/backward specular and diffuse components (Slovick et al., 2017). In coherence scanning interferometry, the matrix may instead map a spectral reflectivity vector to the measured interferometric signal,

$r_{mn}(\lambda)$0

so the “reflection matrix” is a system-specific spectral measurement operator rather than the sample’s intrinsic field-reflection operator (Chen et al., 15 Jun 2025).

This variation in meaning is substantive. Some formulations act on complex amplitudes and preserve phase information; others act on fluxes or intensities; others are inverse-problem operators assembled from system optics, source spectrum, and calibration. The term therefore denotes a family of formally analogous objects rather than a single universal matrix.

Context	Channels	Matrix quantity
Waveguide end facet	Guided modes and plane-wave modes	$r_{mn}(\lambda)$1, $r_{mn}(\lambda)$2
Four-flux radiative transfer	Specular and diffuse flux components	Reflection subblock of $r_{mn}(\lambda)$3
Turbid-media optics	Controlled input modes and measured output modes	$r_{mn}(\lambda)$4
Focused ultrasonic imaging	Virtual focal points in transmit and receive	$r_{mn}(\lambda)$5
CSI reflectivity reconstruction	Wavenumber samples or channels	$r_{mn}(\lambda)$6, $r_{mn}(\lambda)$7

A related but narrower use appears in interferometric systems with spectrally marked paths. There the frequency-dependent transfer coefficient $r_{mn}(\lambda)$8, or equivalently a spectrally resolved scattering matrix, governs how different spectral modes encode path information, and direct spectral detection recovers behavior consistent with standard unitary quantum mechanics and the Englert–Greenberger–Yasin complementarity relation (Bula et al., 2016).

2. Generalized Fresnel matrices for waveguide end facets

A particularly explicit definition appears in the nanowire end-facet formalism of Svendsen, Weman, and Skaar. For a waveguide of arbitrary cross section terminated at $r_{mn}(\lambda)$9, the field on the waveguide side is expanded in true waveguide modes $n$0, while the field in the ambient is expanded in plane-wave modes of a homogeneous medium. The end-facet boundary conditions for the transverse fields are

$n$1

$n$2

Here $n$3 is the reflection coefficient from incoming guided mode $n$4 into reflected guided mode $n$5, and $n$6 is the transmission coefficient into ambient plane-wave mode $n$7. The spectrally resolved reflection matrix is simply the matrix $n$8, or equivalently $n$9, whose entries are the complex reflected amplitudes as functions of frequency or wavelength (Svendsen et al., 2010).

Projecting the boundary conditions onto plane-wave modes yields overlap matrices $m$0 and $m$1, and eliminating the transmission coefficients gives the generalized Fresnel equation

$m$2

with $m$3 the Moore–Penrose pseudoinverse. This is a direct generalization of the classical Fresnel equations to arbitrary cross section. In the limit where the “mode” is a single plane wave in a homogeneous medium, the matrix equation reduces to the usual scalar Fresnel coefficient. The spectral dependence enters through the modal fields, the propagation constants $m$4, and the ambient plane-wave wavenumbers $m$5 (Svendsen et al., 2010).

The computation is organized around Fourier decomposition of the guided modes. The overlaps $m$6 and $m$7 can be written in terms of the Fourier transforms of the transverse modal fields, so the reflection can be conveniently computed using Fast Fourier Transforms. Operationally, one computes the guided modes at a given $m$8, samples the transverse fields on a grid, applies 2D FFTs, assembles $m$9 and $R(\omega)$0, solves the matrix Fresnel equation, and repeats over the spectral range of interest. This produces a spectrally resolved family $R(\omega)$1 and, by back-substitution, $R(\omega)$2 (Svendsen et al., 2010).

The same work emphasizes that the reflection is qualitatively described by two main parameters: the modal field confinement and the average Fresnel reflection of the plane waves constituting the waveguide mode. The confinement factor

$R(\omega)$3

measures the fraction of modal power in the core, while the average Fresnel reflection captures the mode’s angular spectrum in Fourier space. As $R(\omega)$4 varies, both $R(\omega)$5 and the angular spectrum change, so the diagonal elements $R(\omega)$6 and the off-diagonal coupling terms vary spectrally. For nanowire optics, this spectrally varying reflection matrix is used to predict lasing thresholds, mode competition, and far-field emission as functions of wavelength (Svendsen et al., 2010).

3. Scattering, transfer, and resonance formulations

In finite-channel scattering theory, the spectrally resolved reflection matrix is embedded in the full scattering matrix. For a 2D periodic structure supporting two propagating radiation channels, the scattering problem is written as

$R(\omega)$7

Here the diagonal entries are the reflection coefficients for left and right incidence, so the reflection matrix is encoded in the diagonal part of $R(\omega)$8. Near an isolated nondegenerate resonance $R(\omega)$9, the paper shows that the spectrally resolved reflection and transmission spectra can be approximated using only the scattering matrix at $R(\lambda)$0, the complex resonant frequency, and the radiation coefficients of the resonant mode. The central approximation is

$R(\lambda)$1

where $R(\lambda)$2, $R(\lambda)$3 is the normalized radiation vector, and $R(\lambda)$4 for real $R(\lambda)$5. A revised temporal coupled-mode theory produces the same approximate formulas for the transmission and reflection spectra (Wu et al., 2022).

A conceptually different but structurally analogous formulation appears in four-flux radiative transfer. There the state vector is

$R(\lambda)$6

representing forward/backward collimated and diffuse fluxes. A homogeneous layer has a $R(\lambda)$7 film transfer matrix $R(\lambda)$8, interfaces have transfer matrices $R(\lambda)$9, and a multilayer slab is described by

$\omega$0

After S↔M conversion, the global S-matrix $\omega$1 maps incoming fluxes to outgoing fluxes, and the entrance-face reflection submatrix defines the spectrally resolved reflection matrix $\omega$2. In this context the matrix distinguishes specular-for-specular reflection, diffuse reflection produced by collimated incidence, and the corresponding quantities for diffuse incidence. The spectral dependence arises from wavelength-dependent Fresnel coefficients, angle-averaged diffuse coefficients, Mie cross sections, anisotropy parameter $\omega$3, and attenuation exponents in the layer matrices (Slovick et al., 2017).

The distinction between these two settings is fundamental. In the resonant scattering formulation the matrix acts on coherent complex amplitudes and its pole-zero structure governs Fano line shapes, Lorentzian peaks, and exact or approximate reflection zeros. In the four-flux formulation the matrix acts on incoherent flux components within the diffusion approximation and neglects coherent interference. The two uses share matrix language and spectral indexing, but they encode different physical observables (Wu et al., 2022, Slovick et al., 2017).

4. Polarization, bi-anisotropy, and constitutive complexity

For anisotropic, bi-anisotropic, magneto-electric, and chiral media, the spectrally resolved reflection matrix is naturally expressed as a polarization matrix derived from Berreman’s $\omega$4 formalism. Maxwell’s equations with constitutive relations

$\omega$5

are rewritten as a first-order propagation equation for the transverse field vector

$\omega$6

Solving the eigenproblem for $\omega$7, constructing the layer matrix $\omega$8, and enforcing boundary conditions yields the $\omega$9 Jones reflection matrix

$\lambda$0

Its entries may be diagonal in symmetric cases or fully populated when p/s conversion is induced by magneto-electric or chiral coupling (Rogers et al., 2011).

The spectral dependence is inherited from dispersive tensor elements. The paper uses Lorentz oscillator models for $\lambda$1 and $\lambda$2, and also considers dispersive $\lambda$3 and $\lambda$4. Consequently the reflection matrix captures electric, magnetic, magneto-electric, and chiral resonances in both magnitude and phase. In simple cases this reproduces Fresnel-like coefficients; in more general bi-anisotropic settings it yields cross-polarized terms $\lambda$5 and $\lambda$6 with explicitly dispersive structure (Rogers et al., 2011).

A central result is the Adjusted Oscillator Strength Matching condition for hybrid electric- and magnetic-dipole resonances. For a semi-infinite isotropic medium at normal incidence, cancellation of a hybrid mode in reflection occurs when

$\lambda$7

At oblique incidence the condition becomes

$\lambda$8

This means that a resonant feature may disappear from the reflection spectrum while remaining visible in transmission. In thin films the same structure persists, with the reflection adjusted oscillator strength proportional to $\lambda$9 and the transmission strength proportional to $$\mathbf{E}_{\text{out}}(\omega)=R(\omega)\,\mathbf{E}_{\text{in}}(\omega),$$0 (Rogers et al., 2011).

The Jones reflection matrix also underlies the construction of Mueller matrices. Because Mueller elements are bilinear functions of the Jones coefficients, spectrally resolved Mueller matrix spectroscopy provides a polarization-rich view of the same reflection operator. In the paper, magneto-electric excitations produce off-diagonal Mueller elements $$\mathbf{E}_{\text{out}}(\omega)=R(\omega)\,\mathbf{E}_{\text{in}}(\omega),$$1 and $$\mathbf{E}_{\text{out}}(\omega)=R(\omega)\,\mathbf{E}_{\text{in}}(\omega),$$2 of opposite sign, whereas chiral excitations produce the same sign. This establishes a spectrally resolved matrix signature for distinguishing magneto-electric from chiral responses (Rogers et al., 2011).

5. Measurement and imaging in complex media

In turbid optics, the reflection matrix is measured experimentally as the linear operator linking controlled incident optical modes to reflected spatial modes. Using Hadamard phase patterns on a phase-only spatial light modulator and full-field Michelson interferometry, a large optical reflection matrix of a ZnO scattering slab was measured at a single wavelength. The matrix relation is

$$\mathbf{E}_{\text{out}}(\omega)=R(\omega)\,\mathbf{E}_{\text{in}}(\omega),$$3

with the input basis formed by Hadamard phase patterns and the output basis given by CCD camera pixels. Calibration uses a mirror reference and a regularized inverse

$$\mathbf{E}_{\text{out}}(\omega)=R(\omega)\,\mathbf{E}_{\text{in}}(\omega),$$4

The measured experiment is monochromatic at $$\mathbf{E}_{\text{out}}(\omega)=R(\omega)\,\mathbf{E}_{\text{in}}(\omega),$$5, but the paper explicitly identifies the natural extension to a spectrally resolved family $$\mathbf{E}_{\text{out}}(\omega)=R(\omega)\,\mathbf{E}_{\text{in}}(\omega),$$6 or a continuous operator $$\mathbf{E}_{\text{out}}(\omega)=R(\omega)\,\mathbf{E}_{\text{in}}(\omega),$$7. It also shows that finite numerical aperture makes the measured RM a filtered random matrix and modifies the eigenvalue distribution, so apparent open or closed channels must be interpreted cautiously under incomplete channel control (Yu et al., 2015).

Reflection matrix microscopy treats the reflection matrix as an imaging operator. If $$\mathbf{E}_{\text{out}}(\omega)=R(\omega)\,\mathbf{E}_{\text{in}}(\omega),$$8 and $$\mathbf{E}_{\text{out}}(\omega)=R(\omega)\,\mathbf{E}_{\text{in}}(\omega),$$9 are vectorized stacks of incident and reflected fields, then

$$\mathbf{F}_{\text{refl}}^{(0)}(\lambda)=\mathbf{R}(\lambda)\,\mathbf{F}_{\text{in}}^{(0)}(\lambda),$$0

After Fourier transformation to the spatial-frequency basis,

$$\mathbf{F}_{\text{refl}}^{(0)}(\lambda)=\mathbf{R}(\lambda)\,\mathbf{F}_{\text{in}}^{(0)}(\lambda),$$1

the single-scattering kernel of a thin layer object is modeled as

$$\mathbf{F}_{\text{refl}}^{(0)}(\lambda)=\mathbf{R}(\lambda)\,\mathbf{F}_{\text{in}}^{(0)}(\lambda),$$2

Synthetic aperture imaging is then obtained by remapping $$\mathbf{F}_{\text{refl}}^{(0)}(\lambda)=\mathbf{R}(\lambda)\,\mathbf{F}_{\text{in}}^{(0)}(\lambda),$$3 to $$\mathbf{F}_{\text{refl}}^{(0)}(\lambda)=\mathbf{R}(\lambda)\,\mathbf{F}_{\text{in}}^{(0)}(\lambda),$$4. The 2024 algorithm paper introduces logical indexing, power iterations, and low-frequency blocking to accelerate aperture synthesis, 3D image reconstruction, and aberration correction. Although the experiments are single-wavelength, the paper explicitly connects them to time-gated reflection matrices and wavelength scanning for spatio-spectral reflection matrices and volumetric responses, so the formalism extends directly to $$\mathbf{F}_{\text{refl}}^{(0)}(\lambda)=\mathbf{R}(\lambda)\,\mathbf{F}_{\text{in}}^{(0)}(\lambda),$$5, $$\mathbf{F}_{\text{refl}}^{(0)}(\lambda)=\mathbf{R}(\lambda)\,\mathbf{F}_{\text{in}}^{(0)}(\lambda),$$6, and wavelength-dependent propagation operators $$\mathbf{F}_{\text{refl}}^{(0)}(\lambda)=\mathbf{R}(\lambda)\,\mathbf{F}_{\text{in}}^{(0)}(\lambda),$$7 (Kang et al., 2024).

In focused ultrasonic imaging, the reflection matrix is frequency resolved from the outset. Raw data are stored as

$$\mathbf{F}_{\text{refl}}^{(0)}(\lambda)=\mathbf{R}(\lambda)\,\mathbf{F}_{\text{in}}^{(0)}(\lambda),$$8

and Fourier transformed to obtain

$$\mathbf{F}_{\text{refl}}^{(0)}(\lambda)=\mathbf{R}(\lambda)\,\mathbf{F}_{\text{in}}^{(0)}(\lambda),$$9

A model-based double focusing operation produces the focused reflection matrix

$r_{mn}(\lambda)$00

whose entries are monochromatic responses between virtual transducers located at focal points inside the medium. Bandwidth integration yields a time-gated broadband matrix $r_{mn}(\lambda)$01, but the monochromatic $r_{mn}(\lambda)$02 remains the primary spectrally resolved object. From its diagonal and off-diagonal structure the paper derives local focusing criteria, sound-speed maps, and spatially resolved measures of multiple scattering such as $r_{mn}(\lambda)$03 and $r_{mn}(\lambda)$04 (Lambert et al., 2019).

Taken together, these measurement frameworks show that spectral resolution can be added in two ways. One can either repeat a monochromatic matrix measurement across wavelengths, as in optical RM and RMM, or one can start from broadband data and retain the full $r_{mn}(\lambda)$05-dependence before time gating or frequency integration, as in focused ultrasonic imaging (Yu et al., 2015, Kang et al., 2024, Lambert et al., 2019).

6. Conditioning, approximations, and interpretive limits

The practical value of a spectrally resolved reflection matrix depends on how faithfully the matrix matches the true physics and how stably it can be inverted or interpreted. In coherence scanning interferometry, the 2025 high-NA formulation makes this dependence explicit. For a single pixel, the CSI signal is written as

$r_{mn}(\lambda)$06

where $r_{mn}(\lambda)$07, and equivalently

$r_{mn}(\lambda)$08

Here $r_{mn}(\lambda)$09 is the spectrum measurement matrix and $r_{mn}(\lambda)$10 the reflection spectrum measurement matrix. Because low-intensity regions of the source spectrum attenuate some columns, $r_{mn}(\lambda)$11 is more ill-conditioned than $r_{mn}(\lambda)$12. The conditioned counterpart $r_{mn}(\lambda)$13 is formed by summing columns within wavenumber channels, which improves conditioning at the cost of spectral resolution (Chen et al., 15 Jun 2025).

This formalism also makes the dominant error mechanisms explicit. Under the paper’s assumptions, the framework is restricted to NA $r_{mn}(\lambda)$14 and angle-independent sample and reference reflection spectra. Even in that regime, the condition number of the matrix governs sensitivity to perturbations, and phase change on reflection is the dominant factor influencing the reconstruction of the reflection coefficient spectrum. If phase change on reflection is completely neglected while using ideal height, the relative Frobenius error of the matrix can be as large as $r_{mn}(\lambda)$15; when height is reconstructed from the CSI signals, it drops but remains about $r_{mn}(\lambda)$16. NA inaccuracy, pupil apodization, height reconstruction error, and spectral inaccuracies of the light source also contribute materially to reconstruction error (Chen et al., 15 Jun 2025).

Comparable caveats recur in other formulations. In optical RM measurements, finite numerical aperture and incomplete channel control mean that the measured matrix is a filtered random matrix; for smaller $r_{mn}(\lambda)$17, the probability of observing eigenvalues very close to $r_{mn}(\lambda)$18 vanishes, and modes with $r_{mn}(\lambda)$19 may appear dark because reflected energy escapes outside the NA or into another polarization rather than because they are genuinely open channels (Yu et al., 2015). In four-flux radiative transfer, the reflection matrix is built within a diffusion approximation, a plane-parallel geometry, and an assumption of no specular–diffuse coupling at interfaces, so it is an efficient reduced-order description rather than a coherent wave-optics operator (Slovick et al., 2017). In near-resonance scattering theory, the rational approximation to $r_{mn}(\lambda)$20 is valid only for a single isolated nondegenerate resonance and fails in the presence of multiple resonances, overlapping poles, or degeneracies (Wu et al., 2022).

A persistent source of confusion is the assumption that all spectrally resolved reflection matrices are directly comparable. They are not. A waveguide end-facet matrix $r_{mn}(\lambda)$21, a Jones matrix $r_{mn}(\lambda)$22, a four-flux reflection submatrix $r_{mn}(\lambda)$23, a focused reflection matrix $r_{mn}(\lambda)$24, and a CSI measurement matrix $r_{mn}(\lambda)$25 each resolve reflection spectrally, but they act on different state spaces and encode different observables. The unifying idea is not a unique algebraic form; it is the elevation of reflection from a scalar or monochromatic quantity to a frequency-indexed operator whose channel structure is chosen to match the physics of the problem (Svendsen et al., 2010, Rogers et al., 2011, Lambert et al., 2019, Chen et al., 15 Jun 2025).