Wood's anomaly

Absorbance of a 1D metallo-dielectric grating with respect to slit width (red) and gold nanowire width (black), which shows the emergence of Wood anomalies.

In optics, Wood's anomaly refers to the rapid variation of light intensities at diffracted spectral orders in metallic gratings. It was first observed by American physicist Robert W. Wood in 1902. ^[2] Initially unexplained by conventional grating theories, the effect was later understood to arise partly from the excitation of surface plasmon polaritons at the grating surface and partly from the coupling of incident light into diffracted orders, one of which becomes evanescent at a grazing angle. The latter effect is also known as Rayleigh anomaly or Rayleigh–Wood anomaly, ^[3] after Lord Rayleigh's 1907 work on gratings. ^[4]

Studies on Wood anomalies acted as a progenitor to the fields of plasmonics and metamaterials. ^{[3] [5] [6]} Wood anomalies were also observed in acoustic gratings, where they were associated both with diffracted waves at a grazing incidence and surface acoustic waves. ^[7]

Background and history

In 1902, Wood studied the spectra of a continuous light source reflected from a metallic surface with periodically-etched grooves. He observed abrupt changes in reflected light intensity under certain conditions, such as sharp drops in reflectance within a range of wavelengths shorter than the distance between the spectral lines of sodium. These anomalous results could not be explained by grating theories of that period. ^[2] Wood reported further studies on this anomaly in 1912 ^[8] and 1935. ^[9]

In 1907, Lord Rayleigh developed a dynamic theory of wave diffraction from a perfectly-conducting grating using Fourier series. He attributed this anomaly to the passing-off of a diffraction order, at which one of the refracted field harmonics emerges at a grazing angle to the grating and becomes evanescent. ^[4] In 1941, Ugo Fano reexamined the effect by incorporating the complex refractive index of the grating, which led to the identification of a surface wave mode contributing to the anomaly. ^[10] This wave was later revealed to be a surface plasmon polariton (SPP). ^[3] The properties of these surface wave modes were later studied by A. Hessel and Arthur A. Oliner in 1965, who provided a comprehensive new theory and identified the contributing leaky wave behavior. ^[11] The anomalies were further studied and identified in two-dimensional periodic structures ^{[12] [13] [14]} and were found to play a part in extraordinary optical transmission phenomenon. ^[3]

Mathematical formulation

Phase matching of surface plasmon polaritons on a sinusoidal grating

A rigorous theory of Wood anomalies can be developed by invoking Bloch's theorem and expanding electromagnetic fields as spatial harmonics, whose boundary conditions are enforced at the grating surface. ^{[4] [10] [11] [12]} Nevertheless, an approximate yet simpler approach can be taken for one-dimensional shallow gratings through the phase matching condition. Assuming that the grating thickness is vanishingly small and thus the SPP mode is not significantly perturbed, the following resonance condition for Wood anomalies can be obtained: ^{[10] [3]}

${\displaystyle {\text{Re}}\left[\beta _{\text{SPP}}(\omega )\right]\approx k_{x,inc}+{\frac {2\pi }{a}}}$

${\displaystyle \beta _{\text{SPP}}(\omega )={\frac {\omega }{c}}{\sqrt {\frac {\varepsilon _{m}(\omega )\varepsilon _{d}}{\varepsilon _{m}(\omega )+\varepsilon _{d}}}}}$

where

• ${\displaystyle \omega }$ is the angular frequency of the light.
• ${\displaystyle \beta _{\text{SPP}}}$ is the SPP propagation constant.
• ${\displaystyle \varepsilon _{m}(\omega )}$ is the frequency-dependent complex dielectric constant of metal.
• ${\displaystyle \varepsilon _{d}}$ is the dielectric constant of the dielectric half-space above the grating.
• ${\displaystyle k_{x,inc}={\frac {\omega }{c}}\sin \theta }$ is the transverse wavevector of a plane wave with an angle of incidence ${\displaystyle \theta }$.
• ${\displaystyle a}$ is the grating period.

In turn, the conditions for Rayleigh anomalies can be represented in a much simpler form, in analogy with the Bragg's law: ^[4]

${\displaystyle k_{x,inc}=m{\frac {2\pi }{a}}}$

where ${\displaystyle m}$ is an integer.

See also

• Bragg's law
• Electromagnetic metasurface
• Extraordinary optical transmission
• Frequency selective surface
• Photonic crystal
• Spoof surface plasmon

References

1. Darweesh, Ahmad A.; Bauman, Stephen J.; Debu, Desalegn T.; Herzog, Joseph B. (2018). "The Role of Rayleigh-Wood Anomalies and Surface Plasmons in Optical Enhancement for Nano-Gratings". Nanomaterials . 8 (10): 809. doi: 10.3390/nano8100809 . PMC   6215216 . PMID   30304809.
2. Wood, R. W. (1902). "XLII. On a remarkable case of uneven distribution of light in a diffraction grating spectrum". Philosophical Magazine . 4 (12): 396–402. doi:10.1080/14786440209462857.
3. Benisty, Henri; Greffet, Jean-Jacques; Lalanne, Philippe (2022). Introduction to Nanophotonics. Oxford University Press. ISBN   9780198786139.
4. Strutt, John William (1907). "On the dynamical theory of gratings". Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character . 79 (532): 399–416. Bibcode:1907RSPSA..79..399R. doi:10.1098/rspa.1907.0051.
5. Barnes, William L.; Dereux, Alain; Ebbesen, Thomas W. (2003). "Surface plasmon subwavelength optics". Nature . 424 (6950): 824–830. Bibcode:2003Natur.424..824B. doi:10.1038/nature01937. PMID   12917696.
6. Gao, Zhen; Wu, Lin; Gao, Fei; Luo, Yu; Zhang, Baile (2018). "Spoof Plasmonics: From Metamaterial Concept to Topological Description". Advanced Materials . 30 (31): 1706683. Bibcode:2018AdM....3006683G. doi:10.1002/adma.201706683. PMID   29782662.
7. Liu, Jingfei; Declercq, Nico F. (2015). "Investigation of the origin of acoustic Wood anomaly". Journal of the Acoustical Society of America . 138 (2): 1168–1179. Bibcode:2015ASAJ..138.1168L. doi:10.1121/1.4926903.
8. Wood, R. W. (1912). "XXVII. Diffraction gratings with controlled groove form and abnormal distribution of intensity". Philosophical Magazine . 23 (6): 310–317. doi:10.1080/14786440208637224.
9. Wood, R. W. (1935). "Anomalous Diffraction Gratings". Physical Review . 48 (12): 928–936. Bibcode:1935PhRv...48..928W. doi:10.1103/PhysRev.48.928.
10. Fano, U. (1941). "The Theory of Anomalous Diffraction Gratings and of Quasi-Stationary Waves on Metallic Surfaces (Sommerfeld's Waves)". Journal of the Optical Society of America. 31 (3): 213–222. Bibcode:1941JOSA...31..213F. doi:10.1364/JOSA.31.000213.
11. Hessel, A.; Oliner, A. A. (1965). "A New Theory of Wood's Anomalies on Optical Gratings". Applied Optics. 4 (10): 1275–1297. Bibcode:1965ApOpt...4.1275H. doi:10.1364/AO.4.001275.
12. Sarrazin, Michaël; Vigneron, Jean-Pol; Vigoureux, Jean-Marie (2003). "Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes". Physical Review B . 67 (8) 085415. arXiv: physics/0311013 . Bibcode:2003PhRvB..67h5415S. doi:10.1103/PhysRevB.67.085415.
13. Kim, Tae Jin; Thio, Tineke; Ebbesen, T. W.; Grupp, D. E.; Lezec, H. J. (1999). "Control of optical transmission through metals perforated with subwavelength hole arrays". Optics Letters . 24 (4): 256–258. Bibcode:1999OptL...24..256K. doi:10.1364/OL.24.000256. PMID   18071472.
14. Gao, H.; McMahon, J. M.; Lee, M. H.; Henzie, J.; Gray, S. K.; Schatz, G. C.; Odom, T. W. (2009). "Rayleigh anomaly-surface plasmon polariton resonances in palladium and gold subwavelength hole arrays". Optics Express . 17 (4): 2334–2340. Bibcode:2009OExpr..17.2334G. doi:10.1364/OE.17.002334.