Subject terms: Applied physics, Optical physics, Super-resolution microscopy

Nonlinear structure of the Planck radiation spectrum

While thermal radiation and the notion of a perfectly absorbing blackbody had been explored and partially understood since the pioneering work of Kirchhoff¹¹, Planck proposed in 1901 a keystone heuristic argument, namely the quantification of light-matter interactions, that both initiated the development of quantum mechanics and solved the long-standing mystery called the ultraviolet (UV) catastrophe of blackbody radiation. The so-called Planck law of blackbody radiation¹², which describes the spectrum of the thermal light emitted by a perfectly absorbing surface at thermodynamic equilibrium, can be expressed by the photonic spectral radiance, that is, the number of photons emitted per unit time, unit surface, unit solid angle, and per unit of wavelength (${\#}_{\mathrm{ph}}$ s⁻¹ m⁻² sr⁻¹ μm⁻¹), as:

where$T$ is the equilibrium temperature of the surface in Kelvins,$\lambda$ the wavelength in μm,$h$ andk_B the Planck and Boltzmann constants, and$c$ the speed of light.$\mathcal{S}\left(T,\lambda \right)$ reaches a spectral peak${\mathcal{S}}_{\max }\left(T\right)$ that scales itself as$T^4$, for a wavelength${\lambda }_{\max }\left(T\right)$ that scales as$T^{-1}$ according to Wien’s displacement law.

Let us now analyze the nonlinearities of the photonic spectral radiance$\mathcal{S}\left(T,\lambda \right)$ with respect to$T$ and$\lambda$. These nonlinearities can be characterized locally by two exponents${\nu }_{\lambda }\left(T,\lambda \right)$ and${\nu }_T\left(T,\lambda \right)$ defined by the local scaling$\mathcal{S}\left(T,\lambda \right)\propto T^{{\nu }_T}{\lambda }^{{\nu }_{\lambda }}$. In other words, for two dimensionless scalars$\epsilon$ and$\eta$ with small enough values ($\mid \epsilon \mid ,\mid \eta \mid \ll 1$),$\mathcal{S}\left(\left(1+\epsilon \right)T,\left(1+\eta \right)\lambda \right)\approx \left(1+{\nu }_T\epsilon \right)\left(1+{\nu }_{\lambda }\eta \right)\mathcal{S}\left(T,\lambda \right)$. These two exponents, respectively, represent the ratios of the relative increase of$\mathcal{S}\left(T,\lambda \right)$ to the relative increases of$T$ and$\lambda$. Let us define$\alpha$ as a dimensionless scaling parameter. If the temperature is shifted from$T$ to$\alpha T$, the spectra at those two temperatures are related by the simple scaling relation:

This equation means that the log–log plot representation of$\mathcal{S}\left(\alpha T,\lambda \right)$ is simply obtained from$\mathcal{S}\left(T,\lambda \right)$ by two logarithmic translations$\lambda \to \alpha \lambda$ and$\mathcal{S}\to {\alpha }^4\mathcal{S}$ (Fig. 1 a). This scaling invariance of the radiance is well illustrated in the Supplementary note 2. When combined with the definition of the exponents, that is,$\mathcal{S}\left(T,\lambda \right)\propto T^{{\nu }_T}{\lambda }^{{\nu }_{\lambda }}$, one easily obtains${\nu }_T\left(T,\lambda \right)=4+{\nu }_{\lambda }\left(T,\lambda \right)$. This is an important result, which means that thermal spectra all have the same structure of nonlinearities, that only depends on the ratio$\lambda ∕{\lambda }_{\max }\left(T\right)$. For$\lambda ∕{\lambda }_{\max }\left(T\right)\gg 1$,$\mathcal{S}\propto T{\lambda }^{-3}$, that is,${\nu }_T=1$ and${\nu }_{\lambda }=-3$. Qualitatively, for decreasing wavelengths, the exponent${\nu }_{\lambda }$ increases from −3 to 0 at${\lambda }_{\max }$, and further increases to infinity. The dependence in$T$ has a stronger superlinearity, since${\nu }_T=4$ for${\lambda }_{\max }$, and exhibits a faster divergence toward the UV catastrophe (Fig. 1 a). Quantitatively,${\nu }_{\lambda }$ was numerically assessed from the logarithmic derivative$\frac{\partial \log \left(\mathcal{S}\right)}{\partial \log \left(\lambda \right)}$ (Fig. 1 c). ForT = 300 K and λ = 800 nm, we found the scaling$\mathcal{S}\propto {\lambda }^{\approx 57}$. The practical meaning of exponents is the following. At 800 nm, the spectral radiance increases by 57% when the temperature increases by 1% above the ambient temperature, and it typically doubles for a 2 K temperature increase.

To make use of the superlinearity of the blackbody radiation with the temperature, photons should ideally be collected over a spectral region$0\le \lambda \le \mathrm{\Lambda }$ with high nonlinearity exponents, that is, with$\mathrm{\Lambda }\le {\lambda }_{\max }$. We therefore introduce the notion of partial photonic radiance$\mathcal{P}\left(T,\mathrm{\Lambda }\right)$ defined by the spectral integration up to the boundary$\mathrm{\Lambda }$:

which easily leads to a new scaling relation:

When$\mathrm{\Lambda }\to \infty$, Eq. (4) indicates that the total number of thermal photons is superlinear in$T$ as it scales as$T^3$, which is in agreement with the$T^4$ scaling given by the well-known Stefan–Boltzmann law for the total energy. However, for any finite integration boundary$\mathrm{\Lambda }$ on the UV side of the Planck spectrum, that is,$\mathrm{\Lambda }\le {\lambda }_{\max }$, the increasing superlinearity of$\mathcal{S}$ for decreasing wavelengths leads to the fact that the spectral integral in Eq. (3) is largely dominated by the contribution at the upper boundary$\mathcal{S}\left(T,\mathrm{\Lambda }\right)$. As a consequence, the superlinearity of the spectral integral is characterized by a new scaling relation,$\mathcal{P}\left(\alpha T,\mathrm{\Lambda }\right)={\alpha }^{{\omega }_T}\mathcal{P}\left(T,\mathrm{\Lambda }\right)$, in which the scaling exponent can be approximated as${\omega }_T\approx 3+{\nu }_{\lambda }\left(T,\mathrm{\Lambda }\right)$. Numerically, this nonlinearity exponent${\omega }_T$ was assessed from the logarithmic derivative$\frac{\partial \log \left(\mathcal{P}\right)}{\partial \log \left(T\right)}$ (Fig. 1 d). ForT = 300 K and$\mathrm{\Lambda }=$ 800 nm, we found that$\mathcal{P}\propto T^{\approx 60}$, in agreement with${\omega }_T\approx 3+{\nu }_{\lambda \left(T,\mathrm{\Lambda }\right)}$.

Compression of the point spread function

Let us now examine why and the circumstances under which the superlinearity described above can lead to super-resolved detection and imaging. If an extended object is locally heated up by a focused beam of energy characterized by a transverse intensity profile$I\left(x,y\right)$, the surface temperature will locally rise accordingly, but heat will be driven away from where it is initially created, by heat diffusion within the surface and perpendicular to it. However, if energy is delivered by short enough pulses, heat accumulation can be made faster than the relevant diffusion times, and the resulting thermal radiation can be captured from the focus before its spatial profile is broadened by diffusion. Pulsed illumination and/or fast beam scanning is therefore the preferred option for experimental realizations, and the following analysis will be performed under the assumption that heat does not diffuse away. In addition, we will assume that the spectral photonic radiance is determined by the local thermodynamic temperature of the surface and Planck’s law, despite the surface being far from equilibrium. This is a valid assumption for time-scales larger than the phonon equilibration time (1–10 ps). For the sake of simplicity, we make the assumption that the temperature locally rises proportionally to the intensity$I\left(x,y\right)$, that is, with no change of heat capacity and no phase transition. Also, the spectral emissivity will be considered to be invariant with temperature. Finally, a change of the target temperature may cause a variation of the spectral emissivity and thermal expansion. The emissivity comes as a wavelength-dependent pre-factor of the Planck spectrum, which possibly varies with temperature, although very smoothly. Emissivity was therefore assumed to be unity at all wavelengths. For most solid materials, expansion coefficients are typically$10^{-5}-10^{-6}$, and rarely exceed$10^{-4}$. In the worst-case scenario, a 100 K increase would typically degrade the point spread function (psf) by a factor 1.01, and thermal expansion can be safely neglected.

Under the above assumptions, energy absorption, heat production, and temperature increase operate locally and linearly, and the notion of psf becomes meaningful for the following analysis. Let us consider a focused beam with a two-dimensional spatial intensity profile$I\left(x,y\right)=I_{\max }Ĩ\left(x,y\right)$ such as a Gaussian or Airy shape, and an object smaller than the beam section. When located at position$\left(x,y\right)$, we assume that the object experiences uniform intensity$I\left(x,y\right)$ that produces a proportional temperature increase$\mathrm{\Delta }T\left(x,y\right)=T\left(x,y\right)-T_{\mathrm{ref}}$, with$T_{\mathrm{ref}}\le T\left(x,y\right)\le T_{\max }$. Thermal detection over a given spectral domain$\lambda \le \mathrm{\Lambda }$ can be described using the photonic radiance response, and we define the effective response as$\mathcal{R}\left(x,y\right)$ defined as the difference:

The normalized spatial profile of that response,$\tilde{\mathcal{R}}\left(x,y\right)=\mathcal{R}\left(x,y\right){\mathcal{R}}_{\max }^{-1}$, then appear as a function of the dimensionless excitation$Ĩ\left(x,y\right)$ parametrized by$T_{\mathrm{ref}}$,$\mathrm{\Delta }{T}_{\max }$ and$\mathrm{\Lambda }$ as:

This normalized dimensionless response$\tilde{\mathcal{R}}\left(Ĩ\left(x,y\right)\right)$ is a complex non-algebraic function that exhibits a strong superlinearity (Fig. 2a1, a2) with apparent exponents that only depend on the parameters$T_{\mathrm{ref}}$,$\mathrm{\Delta }{T}_{\max }$, and$\mathrm{\Lambda }$. As a consequence, the spatial extension of$\tilde{\mathcal{R}}$ as a function of$\left(x,y\right)$ is contracted compared to$Ĩ\left(x,y\right)$, as schematically illustrated on Fig. 2b. The extent of psf contraction is assessed by the compression factor$\mu$ between these two profiles measured from their full width at$1∕e^2$, which is computed numerically from Eq. (6), or by directly inverting that equation (see Supplementary Note 1). Numerical computations with$\mathrm{\Lambda }$ = 1.5 μm indicate that the thermal radiation psf$\tilde{\mathcal{R}}\left(x,y\right)$ can be made 10-fold smaller than the excitation psf (Fig. 2d).

The above theoretical analysis shows that the superlinearity of the blackbody radiation response can be made arbitrarily high by moving the thermal detection to smaller wavelengths, and this leads to an arbitrarily high spatial compression factor between the cross-section of the excitation beam and the thermal radiation signal. Practically however, superlinearity is best achieved when$\mathrm{\Lambda }\ll {\lambda }_{\max }\left(T_{\mathrm{ref}}\right)$, that is, when the spectral radiance becomes dramatically weak. Therefore, the photon budget must be assessed to optimize the trade-off between the detection sensitivity and the resolution.

Photon budget and suggested conditions for super-resolved detection

For the sake of the present argument, we consider a beam scanning excitation scheme and a target object that contains details smaller than the beam cross-section. The thermal signal is not imaged, but is collected instead with a point detector, for each position$\left(x,y\right)$ of the target relative to the beam center (Fig. 3a). The signal is a number of photons per second$\mathcal{C}\left(x,y\right)$ given by:

where the étendue$\mathcal{G}$ of the optical system is proportional to the apparent cross-section of the object seen by the detector and the solid angle of detection. While the object is scanned, the signal of interest is collected as a temporal profile$\mathcal{C}\left(t\right)$, in which the target should produce a detectable variation from${\mathcal{C}}_{\mathrm{ref}}$ due to the background temperature to$\mathcal{C}\left(T_{\max }\right)$ (Fig. 3b). The relevant signal is therefore the differential count rate$\mathrm{\Delta }\mathcal{C}=\mathcal{C}-{\mathcal{C}}_{\mathrm{ref}}=\mathcal{G}\mathcal{R}$, and the number of signal photons is the integral${\int }_{\mathrm{scan}}\mathrm{\Delta }\mathcal{C}\mathrm{d}t$, which can be approximated by the product of the maximal rate$\mathrm{\Delta }{\mathcal{C}}_{\max }$ by the typical time width${\tau }_{\mathrm{emission}}$ of the peak emission. Meanwhile, the resolution will be limited by the psf compression factor$\mu$ introduced above. As expected,$\mathrm{\Delta }{\mathcal{C}}_{\max }$ and$\mu$ vary in opposite ways with the temperature and the integration wavelength$\mathrm{\Lambda }$ (Fig. 3c1, c2, d), and a trade-off must be arbitrated between the benefits of the spatial compression and the need of large enough photon rates (Fig. 3e).

Obviously, the integration time matters tremendously, as well as the optical étendue$\mathcal{G}$. Because the actual photon budget can only be meaningfully discussed if all parameters are considered, a few practical situations are examined at various scales (microscopic, macroscopic, mid-range, and long range), and for three different temperature amplitudes ($\mathrm{\Delta }{T}_{\max }=$ 50, 100, and 150 K) above ambient temperature ($T_{\mathrm{ref}}=$300 K) in Table 1. A compression factor$\mu$ between 5 and 10 is obtained, but the major role played by the optical étendue$\mathcal{G}$ is such that lower values of$\mathcal{G}$ must be compensated for by relatively larger detection times to achieve super-resolution. At any rate, the present analysis can serve the purpose of designing experimental tests of the new concept proposed here.

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Supplementary Materials

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.