Important phenomena and applications thermal radiation in periodic gratings

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The underlying mechanisms of several phenomena associated with periodic gratings are described below. Whenever possible, the applications for tailoring the radiative properties in energy systems and optoelectronics are discussed.


Surface plasmon polaritons (SPPs) and surface phonon polaritons (SPhPs) in gratings

Plasmons are quasiparticles associated with oscillations of plasma, which is a collection of charged particles such as electrons in a metal or semiconductor [3,36]. Plasmons are longitudinal excitations that can occur either in the bulk or at the interface. The field associated with a plasmon is confined near the surface, while the amplitude decays away from the interface. Such a wave propagates along the surface, and it is called a surface electromagnetic wave. Surface plasmon polaritons (SPPs) can be excited by electromagnetic waves and are important for the study of optical properties of metallic materials, especially near the plasma frequency, which usually lies in the ultraviolet. The associated surface electromagnetic wave in polar materials due to the vibration of ions is called a surface phonon polariton (SPhP). The requirement of evanescent waves on both sides of the interface prohibits the coupling of propagating waves in air to the surface plasmons. Prisms and gratings are commonly used to couple propagating waves in air with surface plasmons. When the plane of incidence is perpendicular to grooves, SPPs or SPhPs can be excited in metallic or polar dielectric gratings due to the evanescent wave nature of some diffraction orders. The Bloch-Floquet condition becomes


where j is the diffraction order. For this reason, the SPP/SPhP dispersion relation can be folded into the region for so that surface polaritons can be excited on a grating surface with propagating waves in air.

Wood’s anomaly

Wood’s anomaly affects radiative properties and causes abrupt changes in the reflectance, transmittance, and absorptance spectra [3,74]. Wood’s anomaly occurs when a diffraction order emerges or disappears at the grazing angle. The transmittance through a 1D slit array can be either enhanced or suppressed due to Wood’s anomaly. Since Wood’s anomaly occurs as a result of diffraction, both polarizations may exhibit such an anomaly. For shallow gratings when the plane of incidence is perpendicular to the grooves, however, Wood’s anomaly is not obvious for TE waves, and thus, initial studies only dealt with the anomaly for TM waves.

Cavity resonance

When cavity resonance occurs, standing waves exist in cavities formed by the grating. At the resonance condition, a strong electromagnetic field exists inside the cavity or slit – the confined and enhanced fields subsequently enhance transmission through the gratings. Therefore, several peaks can be observed from the far-field transmittance shown in Fig. 14 [73]. The cavity resonance in gratings is not the same as a Fabry-Perot resonance, whose resonance condition is given by , where nc and dc are the refractive index and thickness of the medium and m is an integer. Notice that each cavity formed by the grating has two open ends at z = 0 and z = d. However, the boundary condition requires that tangential field components be continuous, enabling electromagnetic waves to be confined inside the cavity. As a matter of fact, the resonance condition strongly depends on the grating’s geometric parameters as well as the boundary conditions [3]. The standing wave in the slit is a combination of all diffracted waves including evanescent waves, and it is not necessary for a single diffraction component to contribute to the cavity resonance. The interferences of all diffracted evanescent waves must be fully considered in order to predict the transmittance peak position. Also, large transmission enhancement and strong field localization can be achieved with nanoscale metallic slit arrays for mid-IR radiation [77].

Figure 14. Transmittance and absorptance of a free-standing Ag grating at normal incidence for (a) TE waves and (b) TM waves [73].

Complex gratings

The concept of complex or aperiodic grating refers to a grating whose surface profile is a superposition of two or more binary or sinusoidal grating profiles. For the application as TPV radiators and infrared detectors, complex gratings may improve simple 1D gratings by reducing the sharpness in the spectral peak and the directional sensitivity [78,79]. The method of forming a complex grating is from a combination of a short-period and a long-period grating. The period of the complex grating is the least common multiple of the periods of two simple gratings. Since the profile of the complex gratings comes from two binary gratings, the normal spectral emittance may have features in common with either of them, whereas the band-folding will form very different emittance features. The emittance/absorptance peaks of the complex grating are wider than those of simple gratings, especially at long wavelengths. This is particularly useful for TPV emitters by enhancing the absorption to frequencies higher than the bandgap of the TPV cell and by suppressing absorption at longer wavelengths [78]. Another example of the application of complex grating is for applications as wavelength-selective absorbers using heavily doped silicon [79]. By properly choosing the carrier concentration and geometry, silicon complex gratings have been shown to exhibit a broadband absorptance peak that is insensitive to the angle of incidence. Furthermore, the peak wavelength can be engineered either with the height of the ridges or the period. Such type of absorptance peak comes from the SPP excitation and is dominated by the first evanescent diffraction order.

Magnetic polaritons

Due to recent development in metamaterials, Localized magnetic polaritons (MPs) have been recently shown as a physical mechanism for extraordinary optical transmission in some artificial nanostructures [80-82]. Some of the phenomena observed previously in periodic gratings can be well described by the excitation of MPs. Diamagnetism is responsible for the magnetic response in split-ring and U-shape structures for metamaterials, as shown in Fig. 15. According to Lenz’s law, when a time-varying magnetic field is introduced perpendicular to the plane of the structure, an oscillating current will be produced in the metal structure that creates an induced magnetic field. Such a diamagnetic response can also occur in a short-strip (or short-rod, short-wire) pair, as shown in Fig. 15(d), since anti-parallel currents are induced in the strips.

Figure 15. Common structures that serve as “magnetic atoms”. (a) The split-ring resonator, (b) single split-ring resonator, (c) U-shape resonator, (d) short-wire, short-rod, or short-strip pairs, (e) fishnet structure.

Even in a 1D deep grating, when the magnetic resonance condition is satisfied, strong absorption and enhanced transmission can occur at specific frequencies. Figure 16 illustrates the effect of MP for a slit array. The induced current flow, shown as red arrows, in the 1D grating can be modeled by an equivalent LC circuit model shown in Fig. 16(b). The contour plot of 1R, or the sum of the transmittance T and absorptance , as a function of  and kx is shown in Fig. 16(c). The radiative properties of considered structure are calculated with RCWA and the predicted resonance frequency from the LC model for MP1 is illustrated as triangles. Excellent agreement between the LC model and the RCWA results further confirms the mechanism of magnetic resonance. The bright bands indicate usually a strong transmission, but can also be associated with a strong absorption, due to the resonance behavior of SPPs or MPs. The inclined line close to the light line, which is then folded due to the Bloch-Floquet condition in the gratings, is associated with the excitation of SPP at the Ag-vacuum interface. The SPP dispersion relation and the effect of folding by gratings have been discussed in the previous section. Several relatively flat dispersion curves correspond to the fundamental, second, and third modes of MPs and are marked as MP1, MP2 and MP3 in the figure. The frequency of higher order MPs is approximately an integer times the fundamental resonance frequency of MP1. The bending and truncation of MP2 is due to the interaction with the SPP. The flatness of MP dispersion curves indicates their unique feature as directional independence. The directional independence of MPs can be understood by the diamagnetic response, as the oscillating magnetic field is always along the y-direction no matter what incident angles is for TM waves. It should be noted that, the cavity-like resonance or coupled SPPs were previously proposed to explain the resonance phenomenon in simple gratings, but only MPs can quantitatively account for the geometric effects on the resonance conditions [80].

Figure 16. Effect of magnetic polaritons (MPs) on the radiative properties of a single grating (slit array): (a) Schematic of a deep grating; (b) the equivalent LC circuit model; (c) Contour plots of the sum of absorptance and transmittance (i.e., 1–R) for a Ag grating with period = 500 nm, h = 400 nm, and b = 50 nm. Triangle marks indicate the frequency of the fundamental mode predicted by the LC circuit model [80].

Another potential application of the MPs is the construction of coherent thermal emission sources. It has been demonstrated that a nanostructure consisting of a periodic metallic strips separated by a thin dielectric layer over an opaque metal film [81,82]. The coupling of the metallic strips and the film induces a magnetic response that is characterized by a negative permeability and positive permittivity. On the other hand, the metallic film intrinsically exhibits a negative permittivity and positive permeability in the near infrared. This artificial structure is equivalent to a pair of single-negative materials. By exciting surface magnetic polaritons, large emissivity peaks can be achieved at the resonance frequencies and are almost independent of the emission angle. The resonance frequency of the magnetic response can be predicted by an analogy to an inductor and capacitor circuit. Furthermore, phonon-assisted MPs have also been predicted and to exhibit similar features as metallic gratings or slit arrays [83].


[3] Zhang, Z. M., 2007, Nano/Microscale Heat Transfer, McGraw-Hill, New York.

[36] Raether, H., 1988, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, Springer, Berlin.

[74] Chen, Y.-B., Zhang, Z. M., and Timans, P. J., 2007, “Radiative Properties of Pattered Wafers with Nanoscale Linewidth,” Journal of Heat Transfer, 129, pp. 79-90.

[73] Lee, B. J., Chen, Y.-B., and Zhang, Z. M., 2008, “Transmission Enhancement through Nanoscale Metallic Slit Arrays from the Visible to Mid-Infrared,” Journal of Computational and Theoretical Nanoscience, 5, pp. 201-213.

[77] Lee, B. J., Chen, Y.-B., and Zhang, Z. M., 2008, “Confinement of Infrared Radiation to Nanometer Scales through Metallic Slit Arrays,” Journal of Quantitative Spectroscopy and Radiative Transfer, 109, pp. 608-619.

[78] Chen, Y.-B., and Zhang, Z. M., 2007, “Design of Tungsten Complex Gratings for Thermophotovoltaic Radiators,” Optics Communications, 269, pp. 411-417.

[79] Chen, Y.-B., and Zhang, Z. M., 2008, “Heavily Doped Silicon Complex Gratings as Wavelength-Selective Absorbing Surfaces,” Journal of Physics D: Applied Physics, 41, p. 095406.

[80] Wang, L. P., and Zhang, Z. M., 2009, “Resonance Transmission or Absorption in Deep Gratings Explained by Magnetic Polaritons,” Applied Physics Letters, 95, p. 111904.

[81] Lee, B. J., Wang, L. P., and Zhang, Z. M., 2008, “Coherent Thermal Emission by Excitation of Magnetic Polaritons between Periodic Strips and a Metallic Film,” Optics Express, 16, pp. 11328-11336

[82] Zhang, Z., Park, K., and Lee, B. J., 2011, “Surface and Magnetic Polaritons on Two-Dimensional Nanoslab-Aligned Multilayer Structure,” Optics Express, 19, p. 16375.

[83] Wang, L. P., and Zhang, Z. M., 2011, “Phonon-Mediated Magnetic Polaritons in the Infrared Region,” Optics Express, 19, pp. A126–A135.