Super-resolution provided by the arbitrarily strong superlinearity of the blackbody radiation

Blackbody radiation is a fundamental phenomenon in nature, and its explanation by Planck marks a cornerstone in the history of Physics. In this theoretical work, we show that the spectral radiance given by Planck’s law is strongly superlinear with temperature, with an arbitrarily large local exponent for decreasing wavelengths. From that scaling analysis, we propose a new concept of super-resolved detection and imaging: if a focused beam of energy is scanned over an object that absorbs and linearly converts that energy into heat, a highly nonlinear thermal radiation response is generated, and its point spread function can be made arbitrarily smaller than the excitation beam focus. Based on a few practical scenarios, we propose to extend the notion of super-resolution beyond its current niche in microscopy to various kinds of excitation beams, a wide range of spatial scales, and a broader diversity of target objects.

Supplementary note 1: Derivation of the psf compression factor µ The purpose of this annex is to derive the formula we used to directly compute the compression factor µ induced by the superlinearity of the blackbody radiation spectrum. As mentioned in the main text, the conditions for such a compression are as follow: a focused excitation beam with a 2D transverse spatial intensity profile I(x, y) = I maxĨ (x, y), illuminates an object smaller than the beam waist. At each relative position (x, y), we assume that the object uniformly experiences the intensity I(x, y) that produces a proportional temperature increase ∆T (x, y) relative to the background temperature T ref . The object then emits a thermal signal determined by the photonic spectral radiance, i.e. the number of photons emitted per unit time, unit surface, unit solid angle, and per unit of wavelength, as: S(T, λ) = 2.10 8 c λ 4 exp where T is the equilibrium temperature of the surface in Kelvins, λ the wavelength in microns, h and k B the Planck and Boltzmann constants, and c the speed of light. The spectral integral of the spectral radiance over the wavelength detection window [0, Λ] is called the photonic radianceand and reads: But T is a function of the position (x, y) of the target, and the spatial profile of the response P(T (x, y), Λ) can be compared to the illumination profile I(x, y). We consider a Gaussian illumination and define the waist of these two profiles by the distance between their center the point where they reach 1 e 2 of their maximum value. The compression factor µ is then defined as the ratio of the illumination waist to the thermal response waist. Other definitions of the waist using the 1 e or 1 2 levels could be considered, with slightly different results.
By definition of the waist of the thermal response, it comes that: By definition of the compression µ and for the Gaussian illumination, we have: Following the main text, let's consider α = Tmax T ref and α 0 = T thermal waist T ref .
It comes that α 0 = 1 + e −2/µ 2 (α − 1), and we can then compute µ from the following equation: Using the relation P(αT, Λ) = α 3 P(T, αΛ) (see equation (4) in main text) we obtain: For known values of T ref , T max and Λ, the second member of the latter equation is determined, and the equation can be numerically solved for α 0 . The compression factor µ is then explicitly computed from equation (5) and reads : For large values of the compression ratio, T thermal waist ≈ T max and α 0 ≈ α. In such circumstances, assuming P(T ref , α 0 Λ) ≈ P(T ref , αΛ), equation (6) can be written an simplified as: For T ref = 300K, T max = 400K and Λ = 12µm, we find that µ ≈ 18. The approximation α 0 ≈ α comes with a 0.15% error, while the approximation P(T ref , α 0 Λ) ≈ P(T ref , αΛ) is better than 1%. In such circumstances, the compression factor can be explicitly computed from the following equation: And the quality of the approximation P(T ref , α 0 Λ) ≈ P(T ref , αΛ) can be checked using equation 6.

Supplementary note 2: Scaling Invariance of the Planck's law
This annex provides a tilted double logarithmic representation that best shows the scaling invariance of the Planck's radiation spectrum. When the temperature of an object relaxes with time from a maximum T max to an equilibrium temperature T min , the superlinearity of the spectral radiance with temperature makes the former relax faster than the latter. This temporal compression has two practical consequences, when using a pulsed illumination. If the expected signal to noise ratio is an issue, gated acquisition will be the best approach, and we can use acquisition times that are shorter than heat relaxation times. On the contrary, if noise is not an issue, the strong superlinearity leads to a potentially very low noise levels, and the continous acquisition of the thermal radiation signal will be dominated by the radiation pulses. (bottom) Detection scheme of an induced thermal signal. An object undergoes a series of periodical heat pulses followed by thermal relaxations at τ Rep intervals. The thermal radiation pulse can be detected by a synchronously gated detection of period τ Rep and duration τ Gate .

Supplementary note 4: Glossary
The symbol˜(tilde) accounts for the dimensionless normalized variables.
• c: Speed of light • P: Photonic radiance (# ph s −1 m −2 sr −1 ), obtained from the spectral integral of the photonic spectral radiance (S) • R: Spatial profile (x,y) of the thermal photonic radiance above background.
• C: Effective thermal signal (# ph s −1 ) obtained from the product of the Etendue and the photonic radiance; C ref : Thermal signal due to the background temperature (# ph s −1 ); ∆C: Difference between the thermal signal C and its background level C ref (# ph s −1 ) • ν λ , ν T : Local scaling exponents (as a function of T and λ) characterizing the nonlinearities of the photonic spectral radiance (S) with respect to the wavelength λ and the temperature T • ω T : Exponent characterizing the nonlinearities of the photonic radiance (P) with respect to the temperature T and defined as ω T = 3 + ν λ • µ: Compression factor of the point spread function (psf ) • τ emission : Temporal width of the thermal emission pulses