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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 19 — Sep. 10, 2012
  • pp: 21702–21714
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Comparative study of total absorption of light by two-dimensional channel and hole array gratings

Anne-Laure Fehrembach and Evgeny Popov  »View Author Affiliations


Optics Express, Vol. 20, Issue 19, pp. 21702-21714 (2012)
http://dx.doi.org/10.1364/OE.20.021702


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Abstract

A detailed study of light absorption by silver gratings having two-dimensional periodicity is presented for structures constructed either of channels or of holes with subwavelength dimensions. Rigorous numerical modelling shows a systematic difference between the two structures: hole (cavity) gratings can strongly absorb light provided the cavity is sufficiently deep, when compared to the wavelength, whereas very thin channel gratings can induce total absorption. A detailed analysis is given in the limit when the period tends towards zero, and an explanation of the differences in behavior is presented using the properties of effective optical index of the metamaterial layer that substitutes the periodical structure in the limit when the period tend to zero.

© 2012 OSA

1. Introduction

Enhanced light absorption by periodic structures has attracted the attention of scientists and engineers since the first observation of grating anomalies by R. Wood [1

1. R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Phylos. Mag. 4, 396–402 (1902).

]. Resonant excitation of surface plasmon waves, responsible for these anomalies was first proposed by Fano [2

2. U. Fano, “The theory of anomalous diffraction gratings and of quasi-stationary waves on metallic surfaces (Sommerfeld’s waves),” J. Opt. Soc. Am. 31(3), 213–222 (1941). [CrossRef]

] as the explication of this phenomenon. Later, Hutley and Maystre [3

3. M. C. Hutley and D. Maystre, “Total absorption of light by a diffraction grating,” Opt. Commun. 19(3), 431–436 (1976). [CrossRef]

] have predicted and observed total light absorption (called by them Brewster effect) by relatively shallow gratings, with groove height not exceeding 10% of the wavelength. The resonant excitation of surface plasmons gave explanation to surface-enhanced Raman scattering (SERS) [4

4. D. A. Weitz, T. J. Gramila, A. Z. Genack, and J. I. Gersten, “Anomalous low-frequency Raman scattering from rough metal surfaces and the origin of the surface-enhanced Raman scattering,” Phys. Rev. Lett. 45(5), 355–358 (1980). [CrossRef]

] and was largely used in nonlinear optics [5

5. R. Reinisch and M. Nevière, “Electromagnetic theory of diffraction in nonlinear optics and surface enhanced nonlinear optical effects,” Phys. Rev. B 28(4), 1870–1885 (1983). [CrossRef]

].

Since the work of Ebbesen et al. [6

6. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature 391(6668), 667–669 (1998). [CrossRef]

] on the observation of enhanced transmission through metallic hole arrays, surface and volume plasmons became object of so great number of experimental and theoretical studies that gave birth to a new name ‘plasmonics’ of the domain.

Light absorption remains also a separate topic of interest for applications in photovoltaics and microwave isolating. Recently, it has been shown that very shallow periodic structures having one- or two-dimensional periodicity can absorb light totally in relatively large spectral and angular interval [7

7. J. Le Perchec, P. Quémerais, A. Barbara, and T. López-Ríos, “Why metallic surfaces with grooves a few nanometers deep and wide may strongly absorb visible light,” Phys. Rev. Lett. 100(6), 066408 (2008). [CrossRef] [PubMed]

9

9. R.-L. Chern, Y.-T. Chen, and H.-Y. Lin, “Anomalous optical absorption in metallic gratings with subwavelength slits,” Opt. Express 18(19), 19510–19521 (2010). [CrossRef] [PubMed]

]. The advantage of 2D periodicity is the possibility to enhance absorption in unpolarized light, whereas for 1D gratings this is possible by combining surface plasmon and cavity resonances, which imposes tighter constraints to manufacturing.

There are many yet unsolved problems in the theory of periodic structures. A typical example is the study of the similarities and differences between inductive (with continuous perforated metal layer in the grating region) and capacitive grids (where the metal inclusions are separated from each other inside the grating) [10

10. R. C. McPhedran, G. H. Derrick, and L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer, Berlin, 1980).

]. Another problem is the choice of homogetization procedure for structures having very small periods when compared with the wavelength of light, structures known as metamaterials.

Our aim in this paper is to study light absorption by two different types of 2D gratings, constructed of channels or holes. If the substrate is dielectric, and the grating bulk material is metal, these systems are called capacitive and inductive grids, respectively [10

10. R. C. McPhedran, G. H. Derrick, and L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer, Berlin, 1980).

]. While channel gratings of just several nm depth can absorb light totally, as already shown in [7

7. J. Le Perchec, P. Quémerais, A. Barbara, and T. López-Ríos, “Why metallic surfaces with grooves a few nanometers deep and wide may strongly absorb visible light,” Phys. Rev. Lett. 100(6), 066408 (2008). [CrossRef] [PubMed]

, 8

8. E. Popov, S. Enoch, and N. Bonod, “Absorption of light by extremely shallow metallic gratings: metamaterial behavior,” Opt. Express 17(8), 6770–6781 (2009). [CrossRef] [PubMed]

], the inverted geometry requires much deeper modulation values to obtain similar performance. This difference persists even when the period is reduced to just several nanometers (1/100 of light wavelength). We propose an explanation from metamaterial point of view by testing two approaches for obtaining the formula for the effective dielectric permittivity when the period of the structure tends to zero. A very good agreement is observed between the rigorous electromagnetic modeling and he effective index approach.

2. Comparative study of channel and hole array gratings

The reflection properties of the first system has been studied in [7

7. J. Le Perchec, P. Quémerais, A. Barbara, and T. López-Ríos, “Why metallic surfaces with grooves a few nanometers deep and wide may strongly absorb visible light,” Phys. Rev. Lett. 100(6), 066408 (2008). [CrossRef] [PubMed]

, 8

8. E. Popov, S. Enoch, and N. Bonod, “Absorption of light by extremely shallow metallic gratings: metamaterial behavior,” Opt. Express 17(8), 6770–6781 (2009). [CrossRef] [PubMed]

] and they are characterized by strong absorption in large angular and spectral intervals, even for quite shallow channels. Figure 2(a)
Fig. 2 Reflection of the gratings presented in Figs. 1(a) and 1(b), respectively, as a function of the filing factor f and the channel (hole) depth h (given in µm). Period in x- and y-direction is equal to 250 nm. Wavelength 457 nm in normal incidence.
gives the dependence of the reflectivity on the channel depth h and the filling ratio f:

f=c2d2
(1)

Because of the invariance of the grating region in z-direction, the mode constants are independent of z. As the filling factor is varied, Fig. 3(a) presents the mode constant having the smallest imaginary part for the channel grating.

In the case of 1D perfectly conducting channels, the TEM mode constant is equal to the refractive index of the channels (equal to one in our case). When f = 0, the pillars in Fig. 1(a) have zero width, so that the fundamental harmonic of the field in Eq. (2) is also equal to one. When the bums are growing in size, the interaction between the modes in the parallel and perpendicular channels leads to increase of the losses (reflected in the increase of the imaginary part of γ).

The existence of this fundamental mode can explain the appearance of consecutive minima and maxima in the reflectivity as a function of h, as well as the decrease of the distance between them as f increases. Indeed, a Fabry-Perot resonance within the grating structure due to the fundamental mode has to be quasiperiodic with respect to h, with the period Dh determined by the real part of the mode propagation constant:
Dh=λ2Re(γ)
(4)
For example, when f = 0.6, Dh150 nm in Fig. 2(a), which corresponds to γ = 1.53 in Fig. 3(a).

3. Metamaterial analysis of channel and hole arrays

Since the works of Maxwell-Garnett [17

17. J. C. Maxwell-Garnett, “Colors in metal glasses and in. metallic films,” Philos. Trans. R. Soc. London Ser. A 203(359-371), 385–420 (1904). [CrossRef]

] on the homogenization of Maxwell equations for media with inhomogeneous inclusions, there are numerous theoretical, numerical, and experimental works devoted to the determination of an equivalent effective refractive index (or dielectric permittivity and magnetic permeability in case of magnetic properties of the media) that can replace the “alloy” of different substances involved in the structure. It is difficult here to even mention the most important works. One approach is the so called homogenization technique for periodic media, an approach that analyses the optical properties in the quasistatic regime, when the optical frequency tends to zero. While this is interesting and fruitful approach, it is not quite practical in the near-IR, visible and UV domain, where the optical constants depend strongly on the wavelength. To take into account the dispersion, instead of increasing wavelength to infinity, it is possible to diminish the period(s) to zero. If together with that the grating thickness tends to zero, all effects disappear and the limit becomes a reflection from the substrate-superstrate interface. To avoid this, in what follows, we fix the wavelength and the grating thickness h, and decrease the x-y dimensions.

3.1. Numerical results for short periods

The intuitive assumption that the effective permittivity would be the average of the permitivities of the participants fails to be true. The first contra-argument is that it is not clear when to take arithmetic average, when geometric, when some other. The second argument is that the average values for two media are symmetric when the permittivities are mutually permuted, simultaneously with the filling factor.

Numerical results for very small periods, when compared to the wavelength show that the peculiarities in the properties of the gratings, presented in Fig. 3(a) and 3(b) persist, as can be observed in Fig. 5
Fig. 5 As in Fig. 2 but for the period of the structures equal to 3 nm (h is again given in µm).
that shows the dependence of the reflectivity on the filling factor f and grating depth h in the regions of high absorption, obtained by the rigorous numerical method for the two structures of Fig. 1 with period d = 3 nm. In particular, very shallow channel gratings can totally absorb the incident light, which is not the case of hole arrays as seen in Fig. 2(b). Figure 6
Fig. 6 Reflectivity as a function of the filling factor for the channel grating of Fig. 2(a) for two different periods (10 and 3 nm) and for two different forms of the pillars, square and circuler, as described in the legend. Blue curve, the results of Eqs. (23) and (24), cyan line, Eqs. (25) and (26). (a) h = 15 nm, (b) h = 415 nm.
shows the dependence on the filling factor of the reflectivity of the grating of Fig. 2(a) for square pillars with period 10 and 3nm, together with a circular pillar reflectivity having period of 3 nm. The grating thickness h is equal to 15 nm (a) and to 415 nm (b). Results obtained for d = 1 nm practically coincide with the results for d = 3 nm. We see that for h = 15 nm the fill-factor dependence of the reflectivity goes down to less than 0.1%, whereas it remains greater than 94.5% for the gratings consisting of circular or square holes. The blue and cyan lines in the figure describe the effective-index results of the effective-index approach, presented further on.

Figure 7
Fig. 7 Reflectivity as a function of the filling factor for the square hole grating of Fig. 2(b) for two different periods (10 and 3 nm), as described in the legend. Blue curve, the results of Eqs. (23) and (24). (a) h = 15 nm, (b) h = 900 nm.
presents similar results of the reflectivity of a square hole arrays for two different hole depths as a function of the filling factor. Similarly to Fig. 6(b), there is no difference between the gratings with periods 3 and 10 nm.

Two conclusions could be drawn from the results:

  • 1) The behavior of very small pitch gratings shows similar peculiarities, independent on the pitch size; the position of the anomalies tends toward a limit position as the period tends to zero. Channel gratings properties differ significantly from the hole arrays whatever small the period is.
  • 2) There is a small difference in Fig. 6 between the gratings consisting of square and circular pillars, the absorption curve for circular pillars is shifted to larger filling factors, as if they seem to have slightly smaller filling factor than the one given by the simple surface ratio in Eq. (1).

3.2. Equivalent effective-index analysis

In order to obtain a simple model that enables us to better understand the behavior of these short-pitch structures, we use an effective-index approach based on the Maxwell-Garnett approach [17

17. J. C. Maxwell-Garnett, “Colors in metal glasses and in. metallic films,” Philos. Trans. R. Soc. London Ser. A 203(359-371), 385–420 (1904). [CrossRef]

] and Green tensor singularity analysis [18

18. D. Yaghjian, “Electric dyadic Green's functions in the source region,” Proc. IEEE 68(2), 248–263 (1980). [CrossRef]

].

εeff,zz=εeff,e=ε1ε2fε1+(1f)ε2
(26)

There are two arguments to choose the first approach (the singularity of infinitely long cylinder. First, when the unit cell cross-section tends to zero with the reduction of the periods and keeping the thickness h fixed, the cross-section diameter can become infinitely smaller than h. Second, Eqs. (25) and (26) are symmetrical when permuting ε2andε1 simultaneously with fand(1f), i.e., channel and hole gratings would be equivalent by interchanging f to 1-f, which is not the case.

If we go back to Fig. 6, the results using the two approaches are presented in blue (for infinitely long cylinders) and cyan (infinitely thin disks). As can be expected, the first approach, Eqs. (23) and (24), gives much better results, which are quite close to the numerical values for circular pillars.

Similar correspondence between the results of the effective-index approach and the numerical method are observed in Fig. 7 for cavity resonances.

The main disadvantage of this simplified approach is that the results depend only on the filling factor, while not taking into account the form of the cross-section of the pillars or holes, while numerical results given in Fig. 6 show that there is a slight difference in the position of the reflectivity dip for circular and square cross-sections. In order to take into account the cross-section form, it is necessary to use more sophisticated approaches that have been developed during the last 120 years. An interested reader can find a detailed work on homogenization theory in a recent review by G. W. Milton [19

19. G. W. Milton, The Theory of Composites (Cambridge Univ. Press, 2002).

]. The specific form can be taken into account by the concept of polarisability, and the necessity to use the average values not only of the fields but of the modulus square of the electric field (see ch.16 of [19

19. G. W. Milton, The Theory of Composites (Cambridge Univ. Press, 2002).

] and ref [20

20. G. W. Milton and K. Golden, “Representations for the Conductivity Functions of Multicomponent Composites,” Commun. Pure Appl. Math. 43(5), 647–671 (1990). [CrossRef]

].). However, we are not able to directly apply this approach, because it is developed in the quasistatic limit as the frequency tends to zero, while in our case the frequency is finite, and the periods in the grating plane tend to zero.

4. Single-mode model and effective index behavior

In normal incidence, the normalized propagation constant of the waves propagating in direction to +/− z is equal to the effective refractive index in direction x for the waves polarized in y. The values of neff,xx are given in Fig. 9
Fig. 9 Dependence on the filling factor f of the real and imaginary parts of the normalized propagation constant of the wave propagating in direction of z, equal to neff,xx=εeff,xx/ε0. (a) channel grating, (b) hole array.
. For the channel gratings, as f increases, the real part of the effective index (and thus the real part of the normalized propagation constant of the vertical mode) increases, which explains why the distance in h of the resonances in Fig. 8(a) decreases with h (see Eq. (4)). In fact, for f < 0.7, the metamaterial behaves like a lossy dielectric with large optical index, having real part that varies from 1 to 6. The same behavior was observed for d = 250 nm in sec.1. As f approaches 0.7, both real and imaginary part grow significantly, thus a very thin optical layer (several nanometers thick) can totally absorb the incident light.

When the channel width decreases (f > 0.7), the effective medium is transferred into a lossy metal, with optical proparties tending towards the properties of the metal of the pillar.

Let us consider now the case of hole gratings. When the holes cover completely the grating region (f = 1), the modulated layer is air. Reducing the hole size, the real part of the effective index decreases, in contrast with Fig. 9(a). The other important fact is the small increase of its imaginary part, as compared with Fig. 9(a). Close to the cut-off of the mode near f = 0.7, both the real and the imaginary parts remain small, which fact explains why for small values of h there is no strong absorption, as seen in Fig. 8(b).

Conclusions

As already observed in [7

7. J. Le Perchec, P. Quémerais, A. Barbara, and T. López-Ríos, “Why metallic surfaces with grooves a few nanometers deep and wide may strongly absorb visible light,” Phys. Rev. Lett. 100(6), 066408 (2008). [CrossRef] [PubMed]

9

9. R.-L. Chern, Y.-T. Chen, and H.-Y. Lin, “Anomalous optical absorption in metallic gratings with subwavelength slits,” Opt. Express 18(19), 19510–19521 (2010). [CrossRef] [PubMed]

], the reflection properties of subwavelength two-dimensional grating depend significantly on the form of the structure. When it consists of pillars separated by channels, there exists a fundamental mode that propagates in vertical direction due to the field interaction through the channels, even for narrow channels, with width less than 5% of the light wavelength. Due to the existence of this mode, the grating structure behaves like a lossy high-index anisotropic dielectric, with characteristic Fabry-Perot resonances observed as a function of its thickness. For channels narrower than 5%, this fundamental mode becomes evanescent and the structure behaves like lossy metal. Close to the cut-off, due to the very high losses and large real part of the effective index, one can observe total light absorption by the layer having thickness of the order of 10 nm.

The corresponding structure that consists of hole instead of pillars is characterized by a vertical mode that has a cut-off (fact well known for single apertures and periodical hole arrays). When propagating, its real part is smaller than unity; this is why strong light absorption and Fabry-Perot resonances appear for thicknesses comparable to the wavelength of light.

These conclusions are also valid in the limit as the structure periods in both x- and y-directions tend toward zero, as shown numerically. A simple Maxwell-Garnett homogenization approach gives very good coincidence with the rigorous numerical results. An open question remains to explain the small difference between the numerical results for circular and square-form pillars, difference that cannot be explained using effective-index approximation that depends only on the filling factor.

References and links

1.

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Phylos. Mag. 4, 396–402 (1902).

2.

U. Fano, “The theory of anomalous diffraction gratings and of quasi-stationary waves on metallic surfaces (Sommerfeld’s waves),” J. Opt. Soc. Am. 31(3), 213–222 (1941). [CrossRef]

3.

M. C. Hutley and D. Maystre, “Total absorption of light by a diffraction grating,” Opt. Commun. 19(3), 431–436 (1976). [CrossRef]

4.

D. A. Weitz, T. J. Gramila, A. Z. Genack, and J. I. Gersten, “Anomalous low-frequency Raman scattering from rough metal surfaces and the origin of the surface-enhanced Raman scattering,” Phys. Rev. Lett. 45(5), 355–358 (1980). [CrossRef]

5.

R. Reinisch and M. Nevière, “Electromagnetic theory of diffraction in nonlinear optics and surface enhanced nonlinear optical effects,” Phys. Rev. B 28(4), 1870–1885 (1983). [CrossRef]

6.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature 391(6668), 667–669 (1998). [CrossRef]

7.

J. Le Perchec, P. Quémerais, A. Barbara, and T. López-Ríos, “Why metallic surfaces with grooves a few nanometers deep and wide may strongly absorb visible light,” Phys. Rev. Lett. 100(6), 066408 (2008). [CrossRef] [PubMed]

8.

E. Popov, S. Enoch, and N. Bonod, “Absorption of light by extremely shallow metallic gratings: metamaterial behavior,” Opt. Express 17(8), 6770–6781 (2009). [CrossRef] [PubMed]

9.

R.-L. Chern, Y.-T. Chen, and H.-Y. Lin, “Anomalous optical absorption in metallic gratings with subwavelength slits,” Opt. Express 18(19), 19510–19521 (2010). [CrossRef] [PubMed]

10.

R. C. McPhedran, G. H. Derrick, and L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer, Berlin, 1980).

11.

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996). [CrossRef]

12.

L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14(10), 2758–2767 (1997). [CrossRef]

13.

M. Nevière and E. Popov, “Crossed gratings,” in Light Propagation in Periodic Media, Differential Theory and Design (Marcel Dekker, New York, 2003) Chap. 9.

14.

M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1780–1787 (1986). [CrossRef]

15.

P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13(4), 779–784 (1996). [CrossRef]

16.

G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13(5), 1019–1023 (1996). [CrossRef]

17.

J. C. Maxwell-Garnett, “Colors in metal glasses and in. metallic films,” Philos. Trans. R. Soc. London Ser. A 203(359-371), 385–420 (1904). [CrossRef]

18.

D. Yaghjian, “Electric dyadic Green's functions in the source region,” Proc. IEEE 68(2), 248–263 (1980). [CrossRef]

19.

G. W. Milton, The Theory of Composites (Cambridge Univ. Press, 2002).

20.

G. W. Milton and K. Golden, “Representations for the Conductivity Functions of Multicomponent Composites,” Commun. Pure Appl. Math. 43(5), 647–671 (1990). [CrossRef]

OCIS Codes
(050.1950) Diffraction and gratings : Diffraction gratings
(240.6680) Optics at surfaces : Surface plasmons
(260.3910) Physical optics : Metal optics
(050.5745) Diffraction and gratings : Resonance domain

ToC Category:
Diffraction and Gratings

History
Original Manuscript: July 11, 2012
Revised Manuscript: August 21, 2012
Manuscript Accepted: August 21, 2012
Published: September 6, 2012

Citation
Anne-Laure Fehrembach and Evgeny Popov, "Comparative study of total absorption of light by two-dimensional channel and hole array gratings," Opt. Express 20, 21702-21714 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-19-21702


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References

  1. R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Phylos. Mag.4, 396–402 (1902).
  2. U. Fano, “The theory of anomalous diffraction gratings and of quasi-stationary waves on metallic surfaces (Sommerfeld’s waves),” J. Opt. Soc. Am.31(3), 213–222 (1941). [CrossRef]
  3. M. C. Hutley and D. Maystre, “Total absorption of light by a diffraction grating,” Opt. Commun.19(3), 431–436 (1976). [CrossRef]
  4. D. A. Weitz, T. J. Gramila, A. Z. Genack, and J. I. Gersten, “Anomalous low-frequency Raman scattering from rough metal surfaces and the origin of the surface-enhanced Raman scattering,” Phys. Rev. Lett.45(5), 355–358 (1980). [CrossRef]
  5. R. Reinisch and M. Nevière, “Electromagnetic theory of diffraction in nonlinear optics and surface enhanced nonlinear optical effects,” Phys. Rev. B28(4), 1870–1885 (1983). [CrossRef]
  6. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature391(6668), 667–669 (1998). [CrossRef]
  7. J. Le Perchec, P. Quémerais, A. Barbara, and T. López-Ríos, “Why metallic surfaces with grooves a few nanometers deep and wide may strongly absorb visible light,” Phys. Rev. Lett.100(6), 066408 (2008). [CrossRef] [PubMed]
  8. E. Popov, S. Enoch, and N. Bonod, “Absorption of light by extremely shallow metallic gratings: metamaterial behavior,” Opt. Express17(8), 6770–6781 (2009). [CrossRef] [PubMed]
  9. R.-L. Chern, Y.-T. Chen, and H.-Y. Lin, “Anomalous optical absorption in metallic gratings with subwavelength slits,” Opt. Express18(19), 19510–19521 (2010). [CrossRef] [PubMed]
  10. R. C. McPhedran, G. H. Derrick, and L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer, Berlin, 1980).
  11. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A13, 1870–1876 (1996). [CrossRef]
  12. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A14(10), 2758–2767 (1997). [CrossRef]
  13. M. Nevière and E. Popov, “Crossed gratings,” in Light Propagation in Periodic Media, Differential Theory and Design (Marcel Dekker, New York, 2003) Chap. 9.
  14. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am.72, 1780–1787 (1986). [CrossRef]
  15. P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A13(4), 779–784 (1996). [CrossRef]
  16. G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A13(5), 1019–1023 (1996). [CrossRef]
  17. J. C. Maxwell-Garnett, “Colors in metal glasses and in. metallic films,” Philos. Trans. R. Soc. London Ser. A203(359-371), 385–420 (1904). [CrossRef]
  18. D. Yaghjian, “Electric dyadic Green's functions in the source region,” Proc. IEEE68(2), 248–263 (1980). [CrossRef]
  19. G. W. Milton, The Theory of Composites (Cambridge Univ. Press, 2002).
  20. G. W. Milton and K. Golden, “Representations for the Conductivity Functions of Multicomponent Composites,” Commun. Pure Appl. Math.43(5), 647–671 (1990). [CrossRef]

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