$\mathbb{T}$ Operator Bounds on Angle-Integrated Absorption and Thermal Radiation for Arbitrary Objects

We derive fundamental per-channel bounds on angle-integrated absorption and thermal radiation for arbitrarily structured bodies—for any given material susceptibility and bounding region—that simultaneously encode both the per-volume limit on polarization set by passivity and geometric constraints on radiative efficiencies set by finite object sizes through the scattering $\mathbb{T}$ operator. We then analyze these bounds in two practical settings, comparing against prior limits as well as near optimal structures discovered through topology optimization. Principally, we show that the bounds properly capture the physically observed transition from the volume scaling of absorptivity seen in deeply subwavelength objects (nanoparticle radius or thin film thickness) to the area scaling of absorptivity seen in ray optics (blackbody limits).

Figure 1

Bounds on angle-integrated absorption and thermal radiation for arbitrarily structured bodies enclosed within compact and extended domains. Absorptivity ($\mathrm{\Phi }$ normalized by area $A$) bounds ${\mathrm{\Phi }}_{\mathrm{opt}}$ (orange lines) and ${\mathrm{\Phi }}_{\mathrm{qs}}$ (purple lines), for a range of $\zeta =|\chi {|}^2/\mathrm{Im}[\chi ]$ at a fixed wavelength $\lambda$. These quantities are shown as a function of the wavelength normalized radius $R$ of an enclosing sphere (a) and thickness $h$ of a semi-infinite film (b). Schematics of each setting are included as insets. Even for small characteristic lengths ($\{R,h\}\le 0.1\lambda$), ${\mathrm{\Phi }}_{\mathrm{opt}}$ is orders of magnitude smaller than ${\mathrm{\Phi }}_{\mathrm{qs}}$.

Figure 2

Comparison of bounds with geometries discovered by inverse design. Absorptivity ($\mathrm{\Phi }$ over area $A$) of structures discovered using gradient topology optimization for a variety of metallic (a) and dielectric (b) materials characterized by the material figure of merit $\zeta =|\chi {|}^2/\mathrm{Im}[\chi ]$. (See text for more information.) For comparison, the bounds ${\mathrm{\Phi }}_{\mathrm{opt}}$ [Eq. (6)] and ${\mathrm{\Phi }}_{\mathrm{qs}}$ [Eq. (7)] are also depicted. In (a), all structures are bound by a ball of radius $R=0.05\lambda$. For (b), the confining domain is a ball of $R=0.5\lambda$. The inset provides a visualization of the structure (exterior and planar cut) for the rightmost green square. The observation that optimized structures come within factors of unity of ${\mathrm{\Phi }}_{\mathrm{opt}}$ provides case evidence of the tightness of Eq. (6).