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Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Editor: Franco Gori
  • Vol. 31, Iss. 6 — Jun. 1, 2014
  • pp: 1226–1232
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Jones matrices of perfectly conducting metallic polarizers

Philippe Boyer  »View Author Affiliations


JOSA A, Vol. 31, Issue 6, pp. 1226-1232 (2014)
http://dx.doi.org/10.1364/JOSAA.31.001226


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Abstract

We deduce by the monomode modal method the analytical expressions of the transmission and reflection Jones matrices of an infinitely conducting metallic screen periodically pierced by subwavelength holes. The study is restricted to normal incidence and to the case of neglected evanescent fields (far-field), which covers many common cases. When only one nondegenerate mode propagates in cavities, they take identical forms to those of a polarizer, with Fabry–Perot-like spectral resonant factors depending on the bigrating parameters. The isotropic or birefringent properties are then obtained when holes support two orthogonal polarization modes. This basic formalism is finally applied to design compact and efficient metallic half-wave plates.

© 2014 Optical Society of America

1. INTRODUCTION

Metallic metamaterials made of subwavelength holes are now designed to exhibit new polarization properties [1

1. M. Iwanaga, “Photonic metamaterials: a new class of materials for manipulating light waves,” Sci. Tech. Adv. Mater. 13, 053002 (2012). [CrossRef]

]. A single periodically pierced metallic screen provides a compact linear polarizer [2

2. X.-F. Ren, P. Zhang, G.-P. Guo, Y.-F. Huang, Z.-W. Wang, and G.-C. Guo, “Polarization properties of subwavelength hole arrays consisting of rectangular holes,” Appl. Phys. B 91, 601–604 (2008). [CrossRef]

], double-layer fishnet metamaterials reveal optical activities [3

3. Y.-L. Zhang, W. Jin, X.-Z. Dong, Z.-S. Zhao, and X.-M. Duan, “Asymmetric fishnet metamaterials with strong optical activity,” Opt. Express 20, 10776–10787 (2012). [CrossRef]

], and multilayer structures allow polarization conversion [4

4. J. Xu, T. Li, F. F. Lu, S. M. Wang, and S. N. Zhu, “Manipulating optical polarization by stereo plasmonic structure,” Opt. Express 19, 748–756 (2011). [CrossRef]

]. One topical issue consists of developing efficient theoretical tools to describe with accuracy the polarization properties of a stacked subwavelength metallic bigrating (SMBG).

Fig. 1. Metallic screen periodically pierced by subwavelength holes.

2. ANALYTICAL EXPRESSIONS OF JONES MATRICES

A. Presentation of the Problem

We consider a metallic screen of thickness h pierced periodically by subwavelength holes described in Cartesian coordinate system Oxyz with (ex,ey) as unit vectors in transverse plane (determined by bigrating interfaces) and ez as a longitudinal unit vector (see Fig. 1). We restrict our analysis to biperiodic structures as depicted in Fig. 1 with Ox and Oy as periodic axes, then with dx and dy as periods. The metal is assumed to be perfectly conducting and the refractive index of the hole medium is denoted by n2. The planar object is surrounded by two semi-infinite homogeneous regions (j) of refractive indices nj, j{1,3}. An incident plane wave falls on the SMBG from region (1) or (3) in normal incidence and with φinc as the polarization incident angle. We introduce rotation angle φG with the x axis of the SMBG in the Oxy plane. The far-field approximation consists of neglecting evanescent waves in electromagnetic field description sufficiently distant from the SMBG (half-wavelength about). This hypothesis allows the equivalence between φinc and φG=φinc since the light polarization far from the SMBG is given by polarizations of the specular diffracted waves. That is why only angle φG is used in the following theory (and not φinc) in order to respect the independence of Jones matrices from the incident wave. It is worth noticing that the present theory may be used easily for monoperiodic objects such as subwavelength metallic gratings [7

7. P. Boyer and D. Van Labeke, “Analytical study of resonance conditions in planar resonators,” J. Opt. Soc. Am. A 29, 1659–1666 (2012). [CrossRef]

] and objects under oblique incidence.

The present theory is derived from the MMM’s basic equations [7

7. P. Boyer and D. Van Labeke, “Analytical study of resonance conditions in planar resonators,” J. Opt. Soc. Am. A 29, 1659–1666 (2012). [CrossRef]

] extended to biperiodic structures. The electromagnetic fields are described as Fourier–Rayleigh (FR) expansions in homogeneous regions (j), i.e., as sums of Floquet modes. To simplify the notations of FR orders, p orders stands for (n,m) orders with n[N,N], m[N,N] and N is the truncation order of FR expansions. p=0 refers to (0,0) order. Then, we assume that only the nondegenerate fundamental mode (q=1) can propagate in cavities (monomode approximation). This hypothesis restricts the spectral validity domain to [λc,2,λc,1], with λc,q the cut-off wavelength of the qth mode. Futhermore, this condition deals with apertures with Ci (iN), C1v, or C2v cross-section symmetry [15

15. P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides I: summary of results,” IEEE Trans. Microwave Theor. Tech. 23, 421–429 (1975). [CrossRef]

]. Figure 2 depicts common examples gathered in E1 set as rectangular [2

2. X.-F. Ren, P. Zhang, G.-P. Guo, Y.-F. Huang, Z.-W. Wang, and G.-C. Guo, “Polarization properties of subwavelength hole arrays consisting of rectangular holes,” Appl. Phys. B 91, 601–604 (2008). [CrossRef]

,16

16. A. Mary, S. G. Rodrigo, L. Martin-Moreno, and F. J. Garca-Vidal, “Theory of light transmission through an array of rectangular holes,” Phys. Rev. B 76, 195414 (2007). [CrossRef]

], ellipsoidal [3

3. Y.-L. Zhang, W. Jin, X.-Z. Dong, Z.-S. Zhao, and X.-M. Duan, “Asymmetric fishnet metamaterials with strong optical activity,” Opt. Express 20, 10776–10787 (2012). [CrossRef]

], split-ring [17

17. Z. Wei, Y. Cao, Y. Fan, X. Yu, and H. Li, “Broadband polarization transformation via enhanced asymmetric transmission through arrays of twisted complementary split-ring resonators,” Appl. Phys. Lett. 99, 221907 (2011). [CrossRef]

], or chiral [18

18. N. Kanda, K. Konishi, and M. Kuwata-Gonokami, “Terahertz wave polarization rotation with double layered metal grating of complimentary chiral patterns,” Opt. Express 15, 11117 (2007). [CrossRef]

] hole profiles.

Fig. 2. Some examples of common SMBG pattern cross sections considered in the present work. E1 set: one hole with one nondegenerate mode. E2 set: one hole with two degenerate modes. E3 set: two holes, each having a nondegenerate mode.

The Jones matrices are now denoted by JjT and JjR when the incident wave is placed in region (j). In the present case of biperiodic objects, the well-known polarizations (te,tm) or (p,s) are used to describe the electric field of diffracted FR waves. Thus, the analytical terms of JjT,R directly identify to the zero-order (far-field approximation) transmitted and reflected amplitudes given by Eqs. (41) and (42), respectively, in [7

7. P. Boyer and D. Van Labeke, “Analytical study of resonance conditions in planar resonators,” J. Opt. Soc. Am. A 29, 1659–1666 (2012). [CrossRef]

] extended to biperiodic metallic gratings:
JjT,R=f˜jT,R(g˜0,tmg˜0,tm*g˜0,tmg˜0,te*g˜0,teg˜0,tm*g˜0,teg˜0,te*)ξT,R(1001),
(3)
where g˜p,σ are the overlap integrals between FR waves and the cavity modes (expressed below), and σ={te,tm} denotes the polarization of the transverse basis vectors of Floquet mode wave vectors. The factors f˜jT,R are spectral resonant Airy-like functions:
f˜jT=4uη0(j)η˜[C˜(1)+η˜][C˜(3)+η˜]u2[C˜(1)η˜][C˜(3)η˜]
(4)
and
f˜jR=2η0(j){[C˜(j)+η˜]+u2[η˜C˜(j)]}[C˜(1)+η˜][C˜(3)+η˜]u2[C˜(1)η˜][C˜(3)η˜],
(5)
where
C˜(j)=ph˜p(j)·g˜p,
(6)
and j=1 if j=3 and j=3 if j=1. ηp,σ(j) and η˜ are the relative admittances of the pth FR order in region (j) and of the cavity mode, respectively. We use the following notations: η0(j)=η0,te(j)=η0,tm(j)=nj and u=exp(iγ˜h), where γ˜ is the propagation constant of the cavity mode. The terms g˜p,σ and h˜p,σ(j)=ηp,σ(j)g˜p,σ* for σ{te,tm} are components of vectors g˜p and h˜p(j), respectively. It is worth noticing that coefficients C˜(j) are computed for n[N,N] and m[N,N], and so it takes into account coupling between cavity mode and evanescent diffracted waves.

B. Overlap Integrals between FR Waves and Cavity Modes

A detailed analysis of g˜p,σ expressions provides the basic and analytic formulation of JjT,R given in Eq. (1) from Eq. (3). To this aim, first we have to pose transverse (in (ex,ey) plane) field expressions of Floquet modes and the ones of the cavity modes.

The transverse field profiles of the well-known Floquet modes in homogeneous regions are given by
{Ep,σ(x,y)=eikp·ρSep,σ,Hp,σ(x,y)=η0ηp,σ(j)ezEp,σ(x,y),
(7)
where ρ=xex+yey, η0=ϵ0/μ0 is vacuum admittance, S is the transverse surface area of the periodic cell, and kp=n2π/dxex+m2π/dyey is the transverse component of the pth FR wave vector. The polarization vectors ep,σ are
ep,tm={kpkpif|kp|0,cosφGex+sinφGeyif|kp|=0,
(8)
and
ep,te={ezkpkpif|kp|0,sinφGex+cosφGeyif|kp|=0.
(9)
Note that |kp|=0 is equivalent to p=0.

Concerning field expressions inside apertures, we note E˜(x,y) the transverse electric field profiles of the fundamental mode. The corresponding transverse magnetic field H˜(x,y) is expressed as in Eq. (7) substituting ηp,σ(j) by η˜ and Ep,σ(x,y) by E˜(x,y).

Thus, the overlap integrals between FR orders and the cavity mode are defined by
g˜p,σ=SEp,σ*(x,y)·E˜(x,y)ds=ep,σ·g˜p.
(10)
The integration is only computed on surface S of the cavity cross section since the fields in the bigrating are different from zero only on S. The g˜p vector is
g˜p=SE˜(x,y)eikp·ρSds=g˜pvp,
(11)
where vp is the unit polarization vector of the overlap integrals g˜p. These overlap vectors cause the linear polarization filtering of the metallic screen, which is described in detail below. We introduce the polarization angle ψp such that vp=cos(ψp)ex+sin(ψp)ey. We easily obtain that
ep,te·vp={mdxcos(ψp)ndysin(ψp)n2dy2+m2dx2if|kp|0,sin(φGψp)if|kp|=0,
(12)
and
ep,tm·vp={ndycos(ψp)+mdxsin(ψp)n2dy2+m2dx2if|kp|0,cos(φGψp)if|kp|=0,
(13)
which is required in Eq. (10) to compute g˜p,σ.

C. Final Expressions of Jones Matrices

Making explicit the zeroth-order overlap integrals finally leads to Eq. (1) where
α˜T,R(λ,ψ)=f˜jT,R(λ,ψ)|g˜0|2.
(14)
The zeroth-order overlap integrals are obtained from Eq. (10) whose ep,σ are expressed in Eqs. (8) and (9) for |kp|=0 (p=0). The square matrix in Eq. (1) is the Jones matrix after |g˜0|2 has been factored out and with θ=ψ0φG. To simplify writing, we introduce the row matrix ψ=(,ψp,) containing all ψp values. The notations here highlight the dependencies of f˜jT,R and α˜T,R on λ and ψ. Note that coefficients α˜T,R(λ,ψ) do not depend on φG since the coupling coefficients C˜(j) do not [but still vary with ψ; see Eqs. (12) and (13)]. In fact, coefficients C˜(j) also take the form
C˜(j)=nj|g˜0|2+p0h˜p(j)·g˜p,
(15)
knowing that |g˜0|2 and the summation for p0 do not depend on φG.

It is interesting to remark that the determinant of JjT given by Eq. (3) is equal to zero, meaning that the metallic array behaves in transmission as a linear polarizer. This result is confirmed by the final expressions of JjT given by Eq. (1). We can also remark that the resonant factor α˜T(λ,ψ) depends on the bigrating parameters (via kp vectors), but the polarization properties given by a polarizer’s Jones matrix Jψ0φG(pol,ex) do not. Indeed, the expressions of Jψ0φG(pol,ex) terms are obtained from Eqs. (12) and (13) for p=0 (|k0|=0). They are thus only depending on ψ0 and φG.

3. CASE OF RECTANGULAR APERTURES: ROLE OF PATTERN ROTATION

We let d=dx=dy. The width and length are ax/d=0.2 and ay/d=0.7, respectively. ψ represents here the angle between the length side and the ey axis. Other parameters are h/d=0.8 and N=5. The geometrical parameters are chosen such that resonant peaks appear for λ/d[1.0,2.0]. In this λ range, the cavities can be effectively assumed to be monomode and the polarization angles of g˜p are given by the linearly polarized electric field direction of the TE01 mode (p, ψp=ψ). Figure 3 depicts the spectra of α˜T,R(λ,ψ) for SMBG patterns rotated by ψ=0°, 15°, 30°, and 45°. The spectra of JjR terms are directly deduced from those of α˜R(λ,ψ) in Eq. (1). Their analysis reveals interesting properties due to the identity matrix. Figure 3 shows two peaks for α˜R(λ,ψ) at resonances. This induces common deep peaks (reflection-like) for the diagonal terms of JjR, whereas the extra-diagonal terms behave as transmission ones (peaks at resonances). This property may cause some special polarization and transmission effects when several SMBGs are piled up.

Fig. 3. Resonant coefficients α˜T,R(λ,ψ) versus wavelength for different values of ψ. The width of the rectangular hole is ax=0.2 and its length is ay=0.7. Other parameters are dx=dy=1, h/d=0.8, n1=n2=n3=1, and N=5.

Fig. 4. Variation of λmax at |α˜T,R(λ,ψ)| maxima according to ψ. See Fig. 3 for parameter values.

4. EXTENSION TO BIMODAL SYSTEMS

For some cavity cross sections and/or frequency ranges, two modes have to be considered in the cavities of the SMBG pattern. The first case deals with one cavity allowing two modes (any cross section a priori). We highlight the particular case of one degenerate mode (two modes with the same effective index). Knowing that the mode field symmetries are independent of the ones of the bigrating lattice only for the studied case of perfectly conducting metals, the degeneracy can be obtained for holes with Civ (i>2) cross-section symmetry [15

15. P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides I: summary of results,” IEEE Trans. Microwave Theor. Tech. 23, 421–429 (1975). [CrossRef]

] such as circular [19

19. R. Ortuno, C. Garcia-Meca, F. J. Rodriguez-Fortuno, J. Marti, and A. Martinez, “Role of surface plasmon polaritons on optical transmission through double layer metallic hole arrays,” Phys. Rev. B 79, 075425 (2009). [CrossRef]

,20

20. T. D. Nguyen, S. Liu, Z. V. Vardeny, and A. Nahata, “Engineering the properties of terahertz filters using multilayer aperture arrays,” Opt. Express 19, 18678–18686 (2011). [CrossRef]

], square [21

21. A. Mary, S. G. Rodrigo, L. Martn-Moreno, and F. J. Garca-Vidal, “Holey metal films: from extraordinary transmission to negative-index behavior,” Phys. Rev. B 80, 165431 (2009). [CrossRef]

,22

22. T. W. H. Oates, B. Dastmalchi, C. Helgert, L. Reissmann, U. Huebner, E.-B. Kley, M. A. Verschuuren, I. Bergmair, T. Pertsch, K. Hinger, and K. Hinrichs, “Optical activity in sub-wavelength metallic grids and fishnet metamaterials in the conical mount,” Opt. Mater. Express 3, 439–451 (2013). [CrossRef]

], and annular [23

23. F. I. Baida and D. Van Labeke, “Light transmission by subwavelength annular aperture arrays in metallic films,” Opt. Commun. 209, 17–22 (2002). [CrossRef]

,24

24. Z. Chen, C. Wang, Y. Lou, B. Cao, and X. Li, “Quarter-wave plate with subwavelength rectangular annular arrays,” Opt. Commun. 297, 198–203 (2013). [CrossRef]

] hole cross sections (E2 set in Fig. 2), or can occur accidentally. The second case deals with two monomode cavities as for any combination of geometries in E1 set as example. The working spectral range determines the monomode or bimode regime of apertures. However, we focus on the particular case of noncoupled modes via evanescent waves, which implies the basic expressions of Jones matrices.

Similarly to Eq. (3) for the monomode SMBG described in Section 2, Eqs. (30), (31), (33), and (34) in [7

7. P. Boyer and D. Van Labeke, “Analytical study of resonance conditions in planar resonators,” J. Opt. Soc. Am. A 29, 1659–1666 (2012). [CrossRef]

] applied to bimode holes (q{1,2}) lead to semi-analytical Jones matrices after tedious calculations:
{J1T=2[(ug˜)tM11+g˜tM21]g˜*η(1),J1R=2[g˜tM11+(ug˜)tM21]g˜*η(1)Id,J3T=2[(ug˜)tM12+g˜tM22]g˜*η(3),J3R=2[g˜tM21+(ug˜)tM22]g˜*η(3)Id,
(16)
where superscript t stands for transpose and u is a 2×2 matrix such that (u)q,q=uqδq,q, with uq=exp(iγ˜qh) (q and q{1,2}); then (η(j))σ,σ=ησ(j)δσ,σ and (g˜)q,σ=g˜q,σ, with g˜q,σ the overlap integral between the FR (0,σ) order and the qth mode. To simplify notations, we consider here that σ{tm,te} (and σ) stands for the subscript (0,σ), i.e., for p=0 (and (0,σ)). The 2×2 matrices Mχ,χ, with χ and χ{1,2}, are 2×2 blocks of the 4×4 matrix M=M1 with
M=(C˜1,1(1)+η˜1C˜1,2(1)[C˜1,1(1)η˜1]u1C˜1,2(1)u2C˜2,1(1)C˜2,2(1)+η˜2C˜2,1(1)u1[C˜2,2(1)η˜2]u2[C˜1,1(3)η˜1]u1C˜1,2(3)u2C˜1,1(3)+η˜1C˜1,2(3)C˜2,1(3)u1[C˜2,2(3)η˜2]u2C˜2,1(3)C˜2,2(3)+η˜2).
(17)
This matrix linking the field amplitudes of both cavity modes depends on the cross-coupling coefficients C˜q,q(j) between the qth and qth modes via FR orders in the (j) region. These coefficients are defined similarly as in Eq. (6):
C˜q,q(j)=ph˜p,q(j)·g˜p,q,
(18)
with p=(n,m), n[N,N], and m[N,N]; then q and q{1,2}. The numerical inversion of M makes the theory semi-analytical for most cases.

We now have to clarify the noncoupling conditions of modes. A tedious analysis of the C˜q,q(j) terms from Eqs. (12) and (13) shows that two modes are not coupled via evanescent diffracted waves when vp,1·vp,2=0p, and when vp,1 and vp,2 vectors coincide with ex and ey, respectively. Moreover, the SMBG pattern’s cross section must respect C1v symmetry (E3 set in Fig. 2). These assertions reduce to ψ0,1=0 and ψ0,2=π/2 when the mode fields are linearly polarized for which ψ(p)ψp (rectangular or square apertures as example). Other geometries inducing C˜q,q(j)=0 may exist but remain difficult to obtain.

We thus deduce from Eq. (19) that JjT,R is a diagonal matrix for φG=0:
JjT,R=(α˜R,T(λ,ψ(1))ξT,R00α˜R,T(λ,ψ(2))ξT,R).
(20)
Thus, such metallic plates can be divided into two sets:
  • i. α˜R,T(λ,ψ(1))=α˜R,T(λ,ψ(2)) for the case of one cavity in SMBG pattern with one degenerate mode (E2 set). Consequently, the SMBG behaves as a Fabry–Perot-like isotropic resonator in transmission.
  • ii. α˜R,T(λ,ψ(1))α˜R,T(λ,ψ(2)) for other cases, i.e., for patterns made of one cavity allowing two nondegenerate modes (E1 set) or two monomode cavities (E3 set). Consequently, the SMBG behaves as a Fabry–Perot-like birefringent resonator in transmission.

5. APPLICATION TO METALLIC HALF-WAVE PLATES

Fig. 5. Coupling analysis between the modes of each cavity for L-shape pattern made of two orthogonal rectangular apertures: τq,q|q(1) and |C˜q,q| versus λ/d. The transmission spectrum is plotted in gray color (scale not mentioned).
Fig. 6. PD between txx and tyy versus λ/d and ay/d for a SMBG with C1v pattern made of two orthogonal rectangular apertures. The white contour plots show the couples (λ, ay) that correspond to |txx|=|tyy|. The black line corresponds to PD=π, the blue line to PD=π/2, and the red lines to |txx| maxima. Point A represents the case of a half-wave plate. The parameters are h/d=0.83, ax/d=0.73, cx/d=0.067, ay/d=0.58, by/d=0.2, and cy/d=0.45.

We now propose to take advantage of our very efficient analytical model to improve the performance of such metallic wave plates. Our goal is to achieve a more compact system (of lower thickness) with better transmission. To this aim, the object designed must satisfy the following three conditions simultaneously:
{L1=arg(txx/tyy)/π1=0:PD condition,L2=|txx||tyy|=0:identical transmission moduli condition,L3=|txx|1=0:total transmission condition,
(23)
which are gathered in the following global condition:
L=l=13|Ll|.
(24)

The value of h/d is changed from 0.5 to 0.85. For each value of h, we determine point A (and so the values of λmax/d and ay/d) as in Fig. 6. The variations of L1 and L1toL3 according to h/d are shown in Fig. 7. We see that the conditions are satisfied for many values of h/d (hollow peaks). Then, the discontinuities close to h/d=0.55 correspond to ay/d=0.5, i.e., when the cut-off wavelength of one cavity mode (position of the first |tyy| peak) is equal to the Rayleigh wavelengths. Thus, a half-wave plate cannot be designed for h/d less than about 0.55. We also remark that L1=0 and L2=0 cannot occur for the same value of h/d (see subfigure), and L3=0 never occurs. So we deduce that perfect half-wave plates cannot be obtained in general with metallic plates made of subwavelength rectangular holes. The variations of ay/d, Tmax, λmax/d, and PD at each minimum of L are plotted in Fig. 8. In order to achieve our goal, we have chosen the most compact system: h/d=0.5484 for one minimum of L such that L10 and L3 reaches one of the minima. Finally, the transmission of the retained metallic plate is Tmax=96.16% at λmax/d=1.073 and with PD=3.1324rad and ay/d=0.5008. To complete the analysis, the corresponding transmission and PD spectra are plotted in Fig. 9.

Fig. 7. Design of one optimized half-wave plate: L1L3 and L computed at point A (see Fig. 6) as functions of h/d. The gray lines refer to equivalent Baida’s waveplate (h/d=0.83) [13] and to the optimized one (h/d=0.5484).
Fig. 8. Variations of ay/d, Tmax, λmax/d, and PD at each minimum of L depicted in Fig. 7.
Fig. 9. Transmission spectra of the retained metallic plate (h/d=0.5484 and ay/d=0.5008).

6. CONCLUSION

We have provided an efficient theoretical tool to analyze the polarization features of subwavelength metallic bigratings in monomode and different bimode regimes. The geometries considered cover a wide part of applications studied in the literature. This model has especially been used to optimize thin metallic half-wave plates with high transmission (patterns with two orthogonal rectangular apertures). The analytical Jones matrices for one metallic plate and the scattering-matrix propagation algorithm can be combined in an analytical recurrence way. This basic process allows the computation of the global Jones matrices of stacked structures and forms an extended Jones-like formalism for metallic plates. Further works are in progress to show with the help of this new formalism that efficient polarization conversion with total transmission occurs for stacked twisted metallic polarizers.

ACKNOWLEDGMENTS

I would like to thank Lifeng Li of the Department of Precision Instruments (Tsinghua University, China) and Daniel Van Labeke of the FEMTO-ST Institute (Besançon, France) for their helpful advice.

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M. Boutria, R. Oussaid, D. Van Labeke, and F. I. Baida, “Tunable artificial chirality with extraordinary transmission metamaterials,” Phys. Rev. B 86, 155428 (2012). [CrossRef]

15.

P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides I: summary of results,” IEEE Trans. Microwave Theor. Tech. 23, 421–429 (1975). [CrossRef]

16.

A. Mary, S. G. Rodrigo, L. Martin-Moreno, and F. J. Garca-Vidal, “Theory of light transmission through an array of rectangular holes,” Phys. Rev. B 76, 195414 (2007). [CrossRef]

17.

Z. Wei, Y. Cao, Y. Fan, X. Yu, and H. Li, “Broadband polarization transformation via enhanced asymmetric transmission through arrays of twisted complementary split-ring resonators,” Appl. Phys. Lett. 99, 221907 (2011). [CrossRef]

18.

N. Kanda, K. Konishi, and M. Kuwata-Gonokami, “Terahertz wave polarization rotation with double layered metal grating of complimentary chiral patterns,” Opt. Express 15, 11117 (2007). [CrossRef]

19.

R. Ortuno, C. Garcia-Meca, F. J. Rodriguez-Fortuno, J. Marti, and A. Martinez, “Role of surface plasmon polaritons on optical transmission through double layer metallic hole arrays,” Phys. Rev. B 79, 075425 (2009). [CrossRef]

20.

T. D. Nguyen, S. Liu, Z. V. Vardeny, and A. Nahata, “Engineering the properties of terahertz filters using multilayer aperture arrays,” Opt. Express 19, 18678–18686 (2011). [CrossRef]

21.

A. Mary, S. G. Rodrigo, L. Martn-Moreno, and F. J. Garca-Vidal, “Holey metal films: from extraordinary transmission to negative-index behavior,” Phys. Rev. B 80, 165431 (2009). [CrossRef]

22.

T. W. H. Oates, B. Dastmalchi, C. Helgert, L. Reissmann, U. Huebner, E.-B. Kley, M. A. Verschuuren, I. Bergmair, T. Pertsch, K. Hinger, and K. Hinrichs, “Optical activity in sub-wavelength metallic grids and fishnet metamaterials in the conical mount,” Opt. Mater. Express 3, 439–451 (2013). [CrossRef]

23.

F. I. Baida and D. Van Labeke, “Light transmission by subwavelength annular aperture arrays in metallic films,” Opt. Commun. 209, 17–22 (2002). [CrossRef]

24.

Z. Chen, C. Wang, Y. Lou, B. Cao, and X. Li, “Quarter-wave plate with subwavelength rectangular annular arrays,” Opt. Commun. 297, 198–203 (2013). [CrossRef]

OCIS Codes
(050.1960) Diffraction and gratings : Diffraction theory
(230.5440) Optical devices : Polarization-selective devices
(260.1960) Physical optics : Diffraction theory
(160.3918) Materials : Metamaterials
(050.6624) Diffraction and gratings : Subwavelength structures

ToC Category:
Diffraction and Gratings

History
Original Manuscript: March 7, 2014
Revised Manuscript: April 12, 2014
Manuscript Accepted: April 14, 2014
Published: May 14, 2014

Citation
Philippe Boyer, "Jones matrices of perfectly conducting metallic polarizers," J. Opt. Soc. Am. A 31, 1226-1232 (2014)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-31-6-1226


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References

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  3. Y.-L. Zhang, W. Jin, X.-Z. Dong, Z.-S. Zhao, and X.-M. Duan, “Asymmetric fishnet metamaterials with strong optical activity,” Opt. Express 20, 10776–10787 (2012). [CrossRef]
  4. J. Xu, T. Li, F. F. Lu, S. M. Wang, and S. N. Zhu, “Manipulating optical polarization by stereo plasmonic structure,” Opt. Express 19, 748–756 (2011). [CrossRef]
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  6. F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010). [CrossRef]
  7. P. Boyer and D. Van Labeke, “Analytical study of resonance conditions in planar resonators,” J. Opt. Soc. Am. A 29, 1659–1666 (2012). [CrossRef]
  8. R. C. McPhedran and D. Maystre, “On the theory and solar application of inductive grids,” Appl. Phys. 14, 1–20 (1977). [CrossRef]
  9. R. Petit, Electromagnetic Theory of Gratings, Topics in Current Physics (Springer-Verlag, 1980).
  10. R. Ulrich, K. F. Renk, and L. Genzel, “Tunable submillimeter interferometers of the Fabry–Perot type,” IEEE Trans. Microwave Theor. Tech. 11, 363–371 (1963). [CrossRef]
  11. R. Ulrich, “Far-infrared properties of metallic mesh and its complementary structure,” Infrared Phys. 7, 37–55 (1967). [CrossRef]
  12. T. Li, S. M. Wang, J. X. Cao, H. Liu, and S. N. Zhu, “Cavity-involved plasmonic metamaterial for optical polarization conversion,” Appl. Phys. Lett. 97, 261113 (2010). [CrossRef]
  13. F. I. Baida, M. Boutria, R. Oussaid, and D. Van Labeke, “Enhanced-transmission metamaterials as anisotropic plates,” Phys. Rev. B 84, 035107 (2011). [CrossRef]
  14. M. Boutria, R. Oussaid, D. Van Labeke, and F. I. Baida, “Tunable artificial chirality with extraordinary transmission metamaterials,” Phys. Rev. B 86, 155428 (2012). [CrossRef]
  15. P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides I: summary of results,” IEEE Trans. Microwave Theor. Tech. 23, 421–429 (1975). [CrossRef]
  16. A. Mary, S. G. Rodrigo, L. Martin-Moreno, and F. J. Garca-Vidal, “Theory of light transmission through an array of rectangular holes,” Phys. Rev. B 76, 195414 (2007). [CrossRef]
  17. Z. Wei, Y. Cao, Y. Fan, X. Yu, and H. Li, “Broadband polarization transformation via enhanced asymmetric transmission through arrays of twisted complementary split-ring resonators,” Appl. Phys. Lett. 99, 221907 (2011). [CrossRef]
  18. N. Kanda, K. Konishi, and M. Kuwata-Gonokami, “Terahertz wave polarization rotation with double layered metal grating of complimentary chiral patterns,” Opt. Express 15, 11117 (2007). [CrossRef]
  19. R. Ortuno, C. Garcia-Meca, F. J. Rodriguez-Fortuno, J. Marti, and A. Martinez, “Role of surface plasmon polaritons on optical transmission through double layer metallic hole arrays,” Phys. Rev. B 79, 075425 (2009). [CrossRef]
  20. T. D. Nguyen, S. Liu, Z. V. Vardeny, and A. Nahata, “Engineering the properties of terahertz filters using multilayer aperture arrays,” Opt. Express 19, 18678–18686 (2011). [CrossRef]
  21. A. Mary, S. G. Rodrigo, L. Martn-Moreno, and F. J. Garca-Vidal, “Holey metal films: from extraordinary transmission to negative-index behavior,” Phys. Rev. B 80, 165431 (2009). [CrossRef]
  22. T. W. H. Oates, B. Dastmalchi, C. Helgert, L. Reissmann, U. Huebner, E.-B. Kley, M. A. Verschuuren, I. Bergmair, T. Pertsch, K. Hinger, and K. Hinrichs, “Optical activity in sub-wavelength metallic grids and fishnet metamaterials in the conical mount,” Opt. Mater. Express 3, 439–451 (2013). [CrossRef]
  23. F. I. Baida and D. Van Labeke, “Light transmission by subwavelength annular aperture arrays in metallic films,” Opt. Commun. 209, 17–22 (2002). [CrossRef]
  24. Z. Chen, C. Wang, Y. Lou, B. Cao, and X. Li, “Quarter-wave plate with subwavelength rectangular annular arrays,” Opt. Commun. 297, 198–203 (2013). [CrossRef]

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