A Framework for Formulating Polychromatic Theories of Emission

TL;DR

This work presents a Hilbert-space framework for polychromatic electromagnetic emission that treats emission as trains of coherent polychromatic pulses derived from the natural resonances of a finite object. By decomposing the absorption operator $Q$ into orthogonal subspaces via a frequency-diagonal $T$-matrix and performing a spectral decomposition, it defines orthogonal absorption modes and, from their poles, constructs single-photon emission modes $|sp^{em}\rangle$ tied to resonances with normalized weights $e_s^2(k)=q_s^2(k)$. The total emission is the superposition of many such pulses with phase and delay statistics, ensuring energy conservation across subspaces and enabling emissions at frequencies not present in the illumination, with potential applications to luminescence. An explicit example for a silicon carbide sphere demonstrates the method, highlighting differences from Kirchhoff-based monochromatic theory, such as extended tails and mode-by-mode contributions, and motivating further development of phase-delay statistics and temperature-dependent material responses. The framework thus provides a versatile, generalizable approach to modeling thermal and non-thermal radiative processes in the Hilbert space, with implications for radiative transfer limits and luminescence descriptions.

Abstract

The emission of energy as electromagnetic radiation is ubiquitous, in particular because objects release thermal energy in the form of photons. Most theories of thermal radiation assume that the thermal emissions originate from a continuum of elementary monochromatic sources, uncorrelated to each other. The universality of thermal radiation motivates the consideration of theories that allow for more general kinds of elementary emissions. In here, we introduce a framework for formulating polychromatic theories of emission in the electromagnetic Hilbert space, whose computational side is based on the transition matrix, or T-matrix. Each photon is emitted as a coherent polychromatic pulse. The spectra of the different emitted pulses are derived using the natural resonance frequencies of the given finite-size object. Each resonance belongs to one of the orthogonal subspaces which decompose the absorption operator according to the symmetries of the object. Energy conservation in the steady-state is ensured by equalizing the absorption and emission of energy at each individual subspace. The framework can accommodate general illuminations, and produce emissions with frequencies that are much suppressed in or even absent from the illumination, resulting in different rates of emission and absorption of photons. This makes the framework suitable for describing other kinds of emissions, such as luminescence, in the Hilbert space.

Figures (6)

• Figure 1: Conceptual illustration of the polychromatic framework for electromagnetic emission. An incident field with a known spectrum is decomposed into the orthogonal absorption subspaces of a given object, each of which absorbs a fraction of the incident power. The energy absorbed into each subspace is re-emitted through polychromatic modes connected to the same subspace, which are obtained using the natural resonance (complex) frequencies of the object. The total emission spectrum is produced by the combination of all the polychromatic emissions.
• Figure 2: Spectral photon density (a) and time dependence (b) corresponding to the $\ket{sp}^\text{em}$ modes and the common Lorentzian, as defined in Eq. (\ref{['eq:sp2']}) and Eq. (\ref{['eq:lorentzian']}), respectively. The value $\omega_{sp}=2-\mathrm{i}0.5$ is taken. The Lorentzian spectrum is asymmetric regarding $k\rightarrow -k$, while for $\ket{sp}^\text{em}$, all the information is already contained in $k>0$. As a consequence, the time dependence obtained in (b) is complex for the Lorentzian and real for the $\ket{sp}^\text{em}$ states [Eq. (\ref{['eq:time']})]. The latter is the behavior expected for physical electromagnetic fields. The total emitted energy for the Lorentzian, obtained by integrating $({{$-1mu'26$}\mkern-1mu\mkern-8mu\mathrm{h}}\mathrm{c_0} k)L(k)$ diverges. While this divergence can be avoided by assuming the Lorentzian lineshape for the energy density, this then causes the corresponding photon density to diverge due to the behavior of the integrand as $|k|\rightarrow 0$, as seen in the inset in (a).
• Figure 3: Properties of a 2µm SiC sphere, as a function of the wavenumber $k/(2\pi)$. Panel (a) shows the real and imaginary parts of the permittivity (blue and orange lines, respectively). Panel (b) shows the absorption cross section (blue line) and the Planckian energy density (orange line) at 600K. Panel (c) shows the thermal emission spectrum (precisely, the spectral density of photon emission rate) for the poly- and monochromatic theories (blue and orange lines, respectively). Panel (d) shows the data of panel (c) in linear scale and for a reduced range of frequencies.
• Figure 4: The contribution to the polychromatic-theory emission spectrum [see Fig. \ref{['fig:sicsphere1']}(c,d)] by different $q$-resonances. Each resonance's contribution is plotted with a line of a different color. Contributions by the electric dipole and quadrupole are shown in panels (a) and (b), while those of the magnetic dipole and quadrupole are shown in panels (c) and (d), respectively.
• Figure 5: Temperature-dependent emission by the 2µm SiC sphere. Panel (a) shows the emitted power as a function of temperature; both the polychromatic theory (blue line) and monochromatic theory (orange line) give the same result. Panel (b) shows the photon emission rate as a function of temperature, with the polychromatic and monochromatic results (blue and orange lines, respectively) on the left y-axis. The black curve shows, for the polychromatic theory, the ratio between emitted and absorbed photons. For the monochromatic theory, this ratio is always unity.