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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 11 — Jun. 2, 2014
  • pp: 13719–13732
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Complete polarimetry on the asymmetric transmission through subwavelength hole arrays

Oriol Arteaga, Ben M. Maoz, Shane Nichols, Gil Markovich, and Bart Kahr  »View Author Affiliations


Optics Express, Vol. 22, Issue 11, pp. 13719-13732 (2014)
http://dx.doi.org/10.1364/OE.22.013719


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Abstract

Dissymmetric, periodically nanostructured metal films can show non-reciprocal transmission of polarized light, in apparent violation of the Lorentz reciprocity theorem. The wave vector dependence of the extraordinary optical transmission in gold films with square and oblique subwavelength hole arrays was examined for the full range of polarized light input states. In normal incidence, the oblique lattice, in contrast to square lattice, showed strong asymmetric, non-reciprocal transmission of circularly polarized light. By analyzing the polarization of the input and the output with a complete Mueller matrix polarimeter the mechanisms that permits asymmetric transmission while preserving the requirement of electromagnetic reciprocity is revealed: the coupling of the linear anisotropies induced by misaligned surface plasmons in the film. The square lattice also shows asymmetric transmission at non-normal incidence, whenever the plane of incidence does not coincide with a mirror line.

© 2014 Optical Society of America

1. Introduction

The Lorentz reciprocity theorem (LRT) [1

1. R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67(5), 717–754 (2004). [CrossRef]

] is a cornerstone of the electromagnetic theory of non-magnetic media that may be formulated as follows: if two incident beams of equal intensity and equal polarization but opposite direction emerge from a medium, the same polarization components of the light must have equal intensities. Arrays of periodic, nanostructured, metallic thin films appear to violate LRT in their nonreciprocal transmission of left and right circularly polarized light (CPL) [2

2. V. A. Fedotov, P. L. Mladyonov, S. L. Prosvirnin, A. V. Rogacheva, Y. Chen, and N. I. Zheludev, “Asymmetric propagation of electromagnetic waves through a planar chiral structure,” Phys. Rev. Lett. 97(16), 167401 (2006). [CrossRef] [PubMed]

7

7. B. Gompf, J. Braun, T. Weiss, H. Giessen, M. Dressel, and U. Hübner, “Periodic nanostructures: spatial dispersion mimics chirality,” Phys. Rev. Lett. 106(18), 185501 (2011). [CrossRef] [PubMed]

]. Such materials are quite distinct from those that exhibit natural optical activity, the parity odd, time even differential refractivity (circular birefringence, CB) or absorption (circular dichroism, CD) of left and right CPL because natural optical activity does not depend on the direction of the wave vector [8

8. N. Berova, P. Polavarapu, K. Nakanishi, and R. W. Woody, eds., Comprehensive Chiroptical Spectroscopy, Volumes 1 & 2, (Wiley VCH, 2012).

]. We presumed that the LRT in thin films must likewise be preserved, but that the mechanism whereby this preservation is achieved is masked in an incomplete polarimetric analysis. CD spectropolarimeters typically modulate the input polarization state of light. Complete polarimeters that also analyze the transmitted light polarization are employed infrequently. Here, we have measured the complete polarimetric response of oblique- and square-lattice metal hole arrays that display extraordinary optical transmission [9

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

] at certain frequencies mediated by the excitation of surface plasmons (SPs) [10

10. P. Lalanne and J. P. Hugonin, “Microscopic theory of the extraordinary optical transmission,” Nat. Phys. 2, 551 (2006). [CrossRef]

,11

11. F. van Beijnum, C. Rétif, C. B. Smiet, H. Liu, P. Lalanne, and M. P. van Exter, “Quasi-cylindrical wave contribution in experiments on extraordinary optical transmission,” Nature 492(7429), 411–414 (2012). [CrossRef] [PubMed]

]. We focus on the changes in the polarization of light upon transmission through the holes so as to assay the full 4 × 4 Mueller scattering matrix, M, of each sample [12

12. D. A. Goldstein, Polarized Light (Marcel Dekker, New York,2003).

]. The reciprocity of optical transmission can be characterized by studying the symmetry properties of the scattering matrix that describes the light-matter interaction. Such a matrix contains all of the information on how the medium behaves upon time reversal (wave vector reversal) [13

13. O. Arteaga, “Number of independent parameters in the Mueller matrix representation of homogeneous depolarizing media,” Opt. Lett. 38(7), 1131–1133 (2013). [CrossRef] [PubMed]

,14

14. A. Schönhofer and H.-G. Kuball, “Symmetry properties of the Mueller matrix,” Chem. Phys. 115(2), 159–167 (1987). [CrossRef]

]. Only in this way can we reckon why LRT is preserved even though the intensity of transmitted light is strikingly wave vector dependent.

2. Experimental details

Gold films (200μm × 200 μm × 250 nm) on fused silica slides were perforated by a focused ion beam (RaithionLiNE 35keV gallium beam, current of 50pA) to make 250 nm diameter holes. Two types of lattices were generated: a square array with lattice parameter a = 530 nm, and a oblique lattice with a = 530 nm, b = 730 nm and φ = 65° (Fig. 1
Fig. 1 Scanning electron michrographs.Square (left) and oblique (right) arrays. The lattice parameters are a = 530 nm, b = 730 nm, d = 250 nm and φ = 65°. The thickness of the gold layer is 250 nm.
). The total array size was 200 × 200 µm2 composed of 50 × 50 µm2 quadrants, the limit on the field of view for writing. The square lattice is mirror symmetric while the oblique lattice is not.

The spectroscopic Ms for the oblique and square lattice samples were measured in transmission at normal incidence using a Mueller matrix polarimeter [15

15. O. Arteaga, J. Freudenthal, B. Wang, and B. Kahr, “Mueller matrix polarimetry with four photoelastic modulators: theory and calibration,” Appl. Opt. 51(28), 6805–6817 (2012). [CrossRef] [PubMed]

]. This instrument uses four different photoelastic modulators (Hinds Instruments) operating at different frequencies, two in the polarization state generator and two in the polarization state analyzer. The fifteen elements of a normalized M are simultaneously measured without moving parts in the instrument. The light source is a 150W Xe Arc lamp and the detector is a photomultiplier tube. To minimize stray light, two synchronized monochromators were used, one after the light source and one before the detector. Ms were measured over the wavelength range 320-840 nm. Collimated light was used for illumination and the transmitted beam was analyzed in the far-field.

The experiments were repeated by turning the backside of the films toward the light source, thus effectively changing the sign of the wave vector. The turn over was done by rotating the sample 180° around the y axis of the instrument system of reference. In the first pass, light was incident on the gold layer (forward configuration) and in the second pass, light entered the sample through the glass substrate (backward configuration). Ms were normalized to their m00 element, the total transmitted intensity, which accounts for reflectivity differences in the two configurations.

2. Analysis of the asymmetric transmission

In the forward configuration, at around 750 nm, the oblique lattice transmits more intensity when it is illuminated with LCPL than RCPL. In the backward configuration, the opposite pertains; the array is more emissive with RCPL illumination, though the emission does not necessarily emerge as RCPL. In agreement with the Lorentz reciprocity theorem, both configurations transmit the same intensity of CPL. In earlier works on non-reciprocal transmission in simple instruments that only modulate the input between LCPL and RCPL, had the investigators placed a circular polarizer in the transmitted beams they would have detected the same intensity of CPL in the forward and backward configurations. This result is counter-intuitive, but it can be described as a polarimetric nonreciprocity that preserves Lorentz reciprocity.

Asymmetric transmission is consequence of M being non-reciprocal. A M is non-reciprocal when it is non Minkowski-symmetric, i.e. when it does not satisfy all the following six relations: m01 = m10, m02 = -m20, m03 = m30, m12 = -m21, m13 = m31 and m23 = -m32. Clearly the forward and backward matrices of Fig. 2 are non-reciprocal because the conditions m03 = m30 and m12 = -m21 are not satisfied. Non-reciprocal Ms are quite usual in optical systems made of sequences of optical elements, because the matrix product in general is not commutative. A classic example of such non-reciprocal response is an optical system composed of a polarizer followed by a retarder that is not equivalent to its reciprocal, a retarder followed by polarizer. However, of interest here is a more subtle manifestation of non-reciprocity, originating in a single medium without need of considering a train of different optical elements. The polarimetric description of light propagation through this medium is well described by differential analysis of its M [17

17. R. Ossikovski, “Differential matrix formalism for depolarizing anisotropic media,” Opt. Lett. 36(12), 2330–2332 (2011). [CrossRef] [PubMed]

,18

18. O. Arteaga and B. Kahr, “Characterization of homogenous depolarizing media based on Mueller matrix differential decomposition,” Opt. Lett. 38(7), 1134–1136 (2013). [CrossRef] [PubMed]

]: L = lnM, where L is the matrix logarithm of M, proportional to the differential Mueller matrix (m) accumulated over a pathlength d: L = dm.

The accumulated matrix L traditionally delivers CD and CB that are defined in terms of the matrix elements as follows: CD = (L03 + L30)/2; CB = (L12-L21)/2where Lij is the element of ith-row and jth-column of L. Other effects such as horizontal linear dichroism [19

19. We use the word “dichroism” to be consistent with published literature. However, it should be reminded that the transmissions peaks observed in CD and LD can be due not only to absorption but also correspond to reflection/scattering peaks for the incoming light.

] and birefringence, LD and LB respectively, and 45° linear dichroism and birefringence, LD’ and LB’, are given by: LD = -(L01 + L10)/2, LB = (L32-L23)/2, LD’ = -(L02 + L20)/2 and LB’ = (L13-L31)/2 [13

13. O. Arteaga, “Number of independent parameters in the Mueller matrix representation of homogeneous depolarizing media,” Opt. Lett. 38(7), 1131–1133 (2013). [CrossRef] [PubMed]

]. These six optical effects are integrated in three complex sets of generalized anisotropies [20

20. J. Schellman and H. P. Jensen, “Optical spectroscopy of oriented molecules,” Chem. Rev. 87(6), 1359–1399 (1987). [CrossRef]

]: horizontal linear anistropy L = LB-iLD, 45° linear anisotropy L’ = LB’-iLD’ and circular anistropy C = CB-iCD. A Mueller matrix is non-reciprocal whenever there is coexistence between two or more of these three sets, in a process where the combination of two sets transforms into an “apparent” (non-reciprocal) contribution of the remaining third. For example, any medium with coexistence of L and L’ will lead to “apparent”, non-reciprocal C. This is the origin of non-reciprocal transmission of CPL in the oblique array of nanoholes. Notably, this non-reciprocity affects M but not L. In L all the optical effects are uncoupled in different matrix elements.

The effects of the combination of L and L’ can be quantified [21

21. O. Arteaga and A. Canillas, “Pseudopolar decomposition of the Jones and Mueller-Jones exponential polarization matrices,” J. Opt. Soc. Am. A 26(4), 783–793 (2009). [CrossRef] [PubMed]

]. It is possible to establish two non-reciprocal parameters, µ and ν, that depend only on L and L’ and that, respectively, determine how strong the asymmetric extinction and refraction of CPL will be.
μ=(LBLD'LB'LD)/2
(2a)
ν=(LBLB'+LD'LD)/2
(2b)
µ is nonvanishing whenever the extinction and refraction eigenmodes do not coincide, i.e. when LD’/LD≠LB’/LB. This optical response is typical for certain low-symmetry crystals (monoclinic and triclinic) or when a crystal is dyed with oriented organic dyes [22

22. K. Claborn, A.-S. Chu, S.-H. Jang, F. Su, W. Kaminsky, and B. Kahr, “Circular extinction imaging: determination of the absolute orientation of embedded chromophores in enantiomorphously twinned LiKSO4 crystals,” Cryst. Growth Des. 5(6), 2117–2123 (2005). [CrossRef]

, 23

23. A. Shtukenberg, Y. Punin, and B. Kahr, Optically anomalous crystals, (Springer, 2007).

]. A nonvanishing ν is much more common effect, as it occurs whenever an eigenmode is at an intermediate position, i.e. not aligned or perpendicular to the horizontal and 45° axes.

The complete polarimetric analysis derived from L reveals that the oblique hole array has a remarkable natural activity, manifested through CD and CB [Figs. 4(a)
Fig. 4 Reciprocal (a,b) and non-reciprocal (c,d) contributions to the chiroptical response of the oblique array of nanoholes.(a) CD, (b) CB, (c) µ and (d) ν are compared in the forward (red, from gold to substrate) and backward (blue) directions. (a,b) emphasize that the film with an oblique array displays some natural, wave vector invariant, optical activity, as would any 3D enantiomorph. (c,d) emphasize the “apparent” non-reciprocal chiroptical contributions, that reverse sign as the wave vector is reversed. These contributions are retrieved from the Mueller calculus as combinations of L and L’.
and 4(b)]. Obviously, these effects remain unchanged for forward and backward propagation. At the same time, the sample is rich in linear effects that, when combined together, lead to a non-reciprocal contribution [Figs. 4(c) and 4(d)]. The achiral square lattice, measured analogously as a control, did not yield appreciable CD, CB, µ or ν nor any linear anisotropies, confirming its anticipated, isotropic response when the electric vector of the light is parallel to the sample plane.

3. Discussion

The non-reciprocaltransmission of CPL light in planar 2D nanostructures has been previously studied in pioneering papers by Zheludev and associates [2

2. V. A. Fedotov, P. L. Mladyonov, S. L. Prosvirnin, A. V. Rogacheva, Y. Chen, and N. I. Zheludev, “Asymmetric propagation of electromagnetic waves through a planar chiral structure,” Phys. Rev. Lett. 97(16), 167401 (2006). [CrossRef] [PubMed]

4

4. A. S. Schwanecke, V. A. Fedotov, V. V. Khardikov, S. L. Prosvirnin, Y. Chen, and N. I. Zheludev, “Nanostructured metal film with asymmetric optical transmission,” Nano Lett. 8(9), 2940–2943 (2008). [CrossRef] [PubMed]

, 24

24. A. Papakostas, A. Potts, D. M. Bagnall, S. L. Prosvirnin, H. J. Coles, and N. I. Zheludev, “Optical manifestations of planar chirality,” Phys. Rev. Lett. 90(10), 107404 (2003). [CrossRef] [PubMed]

]. They correctly asserted that non-reciprocal transmission does not violate the LRT but they did not solve the complete polarimetric analysis. In our experiments, symmetric, wave-vector invariant CB and CD coexist with non-reciprocal contributions that arise due to combinations of linear effects [25

25. L. Wu, Z. Y. Yang, Y. Cheng, Z. Lu, P. Zhang, M. Zhao, R. Gong, X. Yuan, Y. Zheng, and J. Duan, “Electromagnetic manifestation of chirality in layer-by-layer chiral metamaterials,” Opt. Express 21(5), 5239–5246 (2013). [CrossRef] [PubMed]

]. Complete polarimetric analysis is essential here because it can distinguish CD from µ and, ultimately, reveal the mechanism that permits the oblique array to asymmetrically transmit CPL without breaking the LRT (Fig. 5
Fig. 5 Asymmetric transmission of CPL. Cartoon of the transmission of CPL through an ideal 2D oblique array. The solid black arrows represent the portion of the transmitted light that has a polarization different from the incoming. In the limit of a 2D material, the intensity of RCPL and LCPL transmitted through the sample is the same (because no optical activity is possible) and independent of the sense of wavevector. The different amounts of other forms of polarized light generated for LCPL and RCPL in forward and backward configurations give rise to the asymmetric transmission.
). Likewise, it is worth to remind that, although this plasmonic nanomaterial is showing rather large non-reciprocal “effects”, such type of optical response is not a consequence of any new optical phenomena and it is a quite ubiquitous in absorbing media with low symmetry.

Extraordinary optical transmission in nanohole arrays is a process whereby incident light launches SPs that propagate in the array until scattered at the far interface as photons with characteristic polarizations [26

26. C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature 445(7123), 39–46 (2007). [CrossRef] [PubMed]

]. The SPs are longitudinal, compressive electron density waves with directions of propagation in the gold interfaces parallel to the electric field of the exciting photons. In the far field, a plane wave is reconstructed from the scattered radiation from each hole. Both the propagation velocity [27

27. V. V. Temnov, U. Woggon, J. Dintinger, E. Devaux, and T. W. Ebbesen, “Surface plasmon interferometry: measuring group velocity of surface plasmons,” Opt. Lett. 32(10), 1235–1237 (2007). [CrossRef] [PubMed]

] and radiative decay rates depend on the plasmon momentum. CPL, as opposed to linearly polarized light, can excite SPs in any in-plane direction. At either metal-dielectric interface, the square array supports degenerate SPs with a relative orientation of 90° that will be excited with equal probability by CPL [Fig. 6(a)
Fig. 6 SPs in hole arrays with plane group symmetries p4mm (a) and p2 (b).Colored double-headed arrows represent plasmons of unique energies. In (a) at normal incidence, there is perfect polarization preservation; M is proportional to the identity matrix. For (b), the optically response is anisotropic, with non-zero linear dichroism and linear birefringence. For SP modes associated with similar lattice constants (w and v), the plasmon moments are similar, and the transmitted is affected by the scattering of distinct, misaligned SPs. Reversing the sense of the wavector initially propagating along positive Z is equivalent to rotating 180° around Y the sample (c). In the laboratory/instrument system of reference, a SP mode initially oriented at 135° has an orientation of 45° after the sample turn.
]. Because the modes have equal momenta, and identical temporal evolutions, the superpositon of the radiation by this degenerate pair of SPs is coherent and reconstructs the input light polarization. The response is isotropic for light at normal incidence [28

28. E. Altewischer, M. P. van Exter, and J. P. Woerdman, “Polarization analysis of propagating surface plasmons in a subwavelength hole arrays,” J. Opt. Soc. Am. B 20(9), 1927 (2003). [CrossRef]

].

The oblique metallic nanohole array can hold two sets of SP modes, one at the air-metal interface and another at a glass-metal interface. By turning over the sample, the excitation of modes by the incoming light (in-coupling) and the radiation (out-coupling) switch from one interface to the other. It could be argued that the observed asymmetric transmission is a direct consequence of exchanging the in-coupling and out-coupling interfaces. However, this serial excitation, which also takes place for the square lattice, is not responsible of the non-reciprocal optical effect. An interpretation can be secured in the lattice geometry of one interface, as we have described above. The physical process must be understood as the coupled action of two linear plasmon modes that creates a “mixed” circular eigenmode for light propagation regardless of the interface in which they originate. In agreement with this description, µ and ν predominate between a 650 nm and 800 nm, the spectral region where there is an asymmetric non-reciprocal transmission of light in the range of the characteristic length parameters of the two coupled plasmon modes shown in Fig. 6(b).

4. Further results of the square lattice at oblique incidence

The case of oblique incidence in lattices of nanoholes has rarely been treated experimentally. In this configuration there is an in plane component of the incident wave vector that provides additional momentum for SP coupling in this direction. Even at relatively small angles of incidence the polarimetric response of the square lattice becomes drastically different from normal incidence, and in fact when tilted around certain lattice direction it can show asymmetric transmission of CPL in a similar way the oblique lattice does.

For these experiments we have varied the angle of incidence (θ) between 0° and 30° and the azimuthal tilt angle (α) between 0° and 360° [Fig. 7(a)
Fig. 7 Oblique incidence measurements through the square nanohole array. Angle of incidence θ and azimuthal orientation α (a). Lattice projections as function of α and θ (b). Polar plots measured at 615 nm and 640 nm showing the evolution of µ and CD with α and θ (c). Asymmetric transmission (a direct consequence of µ) is only non-zero for values of α in which the square lattice has an oblique projection.
] thereby characterizing the polarimetric response through the measurement of the M of the square lattice in a sizable region of k-space at several wavelengths (see appendix A). The results of these measurements have been plotted in polar coordinates, where α is the polar angle and the radius is the angle of incidence θ [Fig. 7(c)]. Although the polarimetric response of the nanohole array changes from one wavelength to another, the polar symmetry of every element of M is preserved in all the spectrums, as it depends only on the symmetry of the lattice of holes and the orientation of the sample.

At a given θ, angle α determines the projection of the lattice sampled by the optical wave [Fig. 7(b)]. Therefore, the experiments could be described approximately as rectangular, rhombic, or oblique 2D lattices of nanoholes excited at normal incidence. As anticipated from the results above, asymmetric transmission of CPL (i.e. a nonvanishing µ) through the square array is only possible for oblique projections as it is shown in the polar plots of Fig. 7(c); that is when plane of incidence does not correspond with a mirror line [8

8. N. Berova, P. Polavarapu, K. Nakanishi, and R. W. Woody, eds., Comprehensive Chiroptical Spectroscopy, Volumes 1 & 2, (Wiley VCH, 2012).

]. When α = 0°, 90° and when α = 45°, 135° the square lattice projects, respectively, rectangular and rhombic arrays that contain mirror lines thus giving a completely reciprocal polarimetric response. Opposing sides of any mirror line must show opposite signs of CD and µ because the projected lattices are enantiomorphous, identified as D-oblique and L-oblique in Fig. 7(b).

5. Conclusion

By studying the extraordinary asymmetric transmission of light in sub-wavelength hole arrays with a complete polarimeter for the first time, we have discovered mechanisms that enable non-reciprocal transmission while preserving Lorentz reciprocity. This brings new possibilities for the control of the polarization information in SPs. We have shown that engineered, plasmonic nanostructured materials based on arrays of holes distributed over an oblique lattice can be useful to encode and decode information-rich SPs, since they are not only sensitive to the intensity, frequency, and polarization of the optical signal but also to the k-vector direction. Understanding the intrinsic differences in the plasmonic response for 2D lattices is a first step in the development of plasmonic crystallography to localize and control light in sub-wavelength regions.

Appendix A: k-space mapping of the square array at different wavelengths

This appendix shows in Figs. 8
Fig. 8 Normalized Mueller matrix mapping at 495 nm.
, 9
Fig. 9 Normalized Mueller matrix mapping at 570 nm.
, 10
Fig. 10 Normalized Mueller matrix mapping at 615 nm.
, 11
Fig. 11 Normalized Mueller matrix mapping at 640 nm.
, 12
Fig. 12 Normalized Mueller matrix mapping at 687 nm.
, and 13
Fig. 13 Normalized Mueller matrix mapping at 750 nm.
the results of the Mueller matrix k-space mapping at different wavelengths. To elaborate these polar (θ, α) plots the angle of incidence (θ) has been varied between 0° and 30° in steps of 1° and the azimuthal tilt angle (α) between 0° and 360° in steps of 2°.

Acknowledgments

This work was supported by the US National Science Foundation (CHE-0845526, DMR-1105000), by the Israel Science Foundation grant no. 172/10 and by the James Frank program on light-matter interaction. O. A. acknowledges a Marie Curie Fellowship within the European Union's Seventh Framework Programme (FP7/2007-2013) under REA grant agreement n° [330513]. The authors are grateful to Dr. A. Tzukernik, Dr. Y.Lilach, M. Eitan and A. Hazzan for sample fabrication. B.M.M. was supported by The Tel Aviv University Center for Nanoscience and Nanotechnology.

References and links

1.

R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67(5), 717–754 (2004). [CrossRef]

2.

V. A. Fedotov, P. L. Mladyonov, S. L. Prosvirnin, A. V. Rogacheva, Y. Chen, and N. I. Zheludev, “Asymmetric propagation of electromagnetic waves through a planar chiral structure,” Phys. Rev. Lett. 97(16), 167401 (2006). [CrossRef] [PubMed]

3.

V. A. Fedotov, A. S. Schwanecke, N. I. Zheludev, V. V. Khardikov, and S. L. Prosvirnin, “Asymmetric transmission of light and enantiomerically sensitive plasmon resonance in planar chiral nanostructures,” Nano Lett. 7(7), 1996–1999 (2007). [CrossRef]

4.

A. S. Schwanecke, V. A. Fedotov, V. V. Khardikov, S. L. Prosvirnin, Y. Chen, and N. I. Zheludev, “Nanostructured metal film with asymmetric optical transmission,” Nano Lett. 8(9), 2940–2943 (2008). [CrossRef] [PubMed]

5.

R. Zhao, L. Zhang, J. Zhou, T. Kochny, and C. M. Soukoulis, “Conjugated gammadion chiral metamaterial with uniaxial optical activity and negative refractive index,” Phys. Rev. B 83(3), 035105 (2011). [CrossRef]

6.

E. Plum, X. X. Liu, V. A. Fedotov, Y. Chen, D. P. Tsai, and N. I. Zheludev, “Metamaterials: optical activity without chirality,” Phys. Rev. Lett. 102(11), 113902 (2009). [CrossRef] [PubMed]

7.

B. Gompf, J. Braun, T. Weiss, H. Giessen, M. Dressel, and U. Hübner, “Periodic nanostructures: spatial dispersion mimics chirality,” Phys. Rev. Lett. 106(18), 185501 (2011). [CrossRef] [PubMed]

8.

N. Berova, P. Polavarapu, K. Nakanishi, and R. W. Woody, eds., Comprehensive Chiroptical Spectroscopy, Volumes 1 & 2, (Wiley VCH, 2012).

9.

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

10.

P. Lalanne and J. P. Hugonin, “Microscopic theory of the extraordinary optical transmission,” Nat. Phys. 2, 551 (2006). [CrossRef]

11.

F. van Beijnum, C. Rétif, C. B. Smiet, H. Liu, P. Lalanne, and M. P. van Exter, “Quasi-cylindrical wave contribution in experiments on extraordinary optical transmission,” Nature 492(7429), 411–414 (2012). [CrossRef] [PubMed]

12.

D. A. Goldstein, Polarized Light (Marcel Dekker, New York,2003).

13.

O. Arteaga, “Number of independent parameters in the Mueller matrix representation of homogeneous depolarizing media,” Opt. Lett. 38(7), 1131–1133 (2013). [CrossRef] [PubMed]

14.

A. Schönhofer and H.-G. Kuball, “Symmetry properties of the Mueller matrix,” Chem. Phys. 115(2), 159–167 (1987). [CrossRef]

15.

O. Arteaga, J. Freudenthal, B. Wang, and B. Kahr, “Mueller matrix polarimetry with four photoelastic modulators: theory and calibration,” Appl. Opt. 51(28), 6805–6817 (2012). [CrossRef] [PubMed]

16.

To determine the intensity of transmitted RCPL and LCPL it is necessary to multiply the output Stokes vector by the Mueller matrix of a right and left circular polarizer. This operation is equivalent to multiply the first Stokes parameters by the last (negated in the case of LCPL).

17.

R. Ossikovski, “Differential matrix formalism for depolarizing anisotropic media,” Opt. Lett. 36(12), 2330–2332 (2011). [CrossRef] [PubMed]

18.

O. Arteaga and B. Kahr, “Characterization of homogenous depolarizing media based on Mueller matrix differential decomposition,” Opt. Lett. 38(7), 1134–1136 (2013). [CrossRef] [PubMed]

19.

We use the word “dichroism” to be consistent with published literature. However, it should be reminded that the transmissions peaks observed in CD and LD can be due not only to absorption but also correspond to reflection/scattering peaks for the incoming light.

20.

J. Schellman and H. P. Jensen, “Optical spectroscopy of oriented molecules,” Chem. Rev. 87(6), 1359–1399 (1987). [CrossRef]

21.

O. Arteaga and A. Canillas, “Pseudopolar decomposition of the Jones and Mueller-Jones exponential polarization matrices,” J. Opt. Soc. Am. A 26(4), 783–793 (2009). [CrossRef] [PubMed]

22.

K. Claborn, A.-S. Chu, S.-H. Jang, F. Su, W. Kaminsky, and B. Kahr, “Circular extinction imaging: determination of the absolute orientation of embedded chromophores in enantiomorphously twinned LiKSO4 crystals,” Cryst. Growth Des. 5(6), 2117–2123 (2005). [CrossRef]

23.

A. Shtukenberg, Y. Punin, and B. Kahr, Optically anomalous crystals, (Springer, 2007).

24.

A. Papakostas, A. Potts, D. M. Bagnall, S. L. Prosvirnin, H. J. Coles, and N. I. Zheludev, “Optical manifestations of planar chirality,” Phys. Rev. Lett. 90(10), 107404 (2003). [CrossRef] [PubMed]

25.

L. Wu, Z. Y. Yang, Y. Cheng, Z. Lu, P. Zhang, M. Zhao, R. Gong, X. Yuan, Y. Zheng, and J. Duan, “Electromagnetic manifestation of chirality in layer-by-layer chiral metamaterials,” Opt. Express 21(5), 5239–5246 (2013). [CrossRef] [PubMed]

26.

C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature 445(7123), 39–46 (2007). [CrossRef] [PubMed]

27.

V. V. Temnov, U. Woggon, J. Dintinger, E. Devaux, and T. W. Ebbesen, “Surface plasmon interferometry: measuring group velocity of surface plasmons,” Opt. Lett. 32(10), 1235–1237 (2007). [CrossRef] [PubMed]

28.

E. Altewischer, M. P. van Exter, and J. P. Woerdman, “Polarization analysis of propagating surface plasmons in a subwavelength hole arrays,” J. Opt. Soc. Am. B 20(9), 1927 (2003). [CrossRef]

29.

K. L. van der Molen, F. B. Segerink, N. F. van Hulst, and L. Kuipers, “Influence of hole size on theextraordinary transmission through subwavelength hole arrays,” Appl. Phys. Lett. 85(19), 4316–4318 (2004). [CrossRef]

30.

L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86(6), 1114–1117 (2001). [CrossRef] [PubMed]

OCIS Codes
(120.5410) Instrumentation, measurement, and metrology : Polarimetry
(240.6680) Optics at surfaces : Surface plasmons
(160.1585) Materials : Chiral media

ToC Category:
Plasmonics

History
Original Manuscript: February 3, 2014
Revised Manuscript: March 29, 2014
Manuscript Accepted: March 31, 2014
Published: May 30, 2014

Citation
Oriol Arteaga, Ben M. Maoz, Shane Nichols, Gil Markovich, and Bart Kahr, "Complete polarimetry on the asymmetric transmission through subwavelength hole arrays," Opt. Express 22, 13719-13732 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-11-13719


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References

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  16. To determine the intensity of transmitted RCPL and LCPL it is necessary to multiply the output Stokes vector by the Mueller matrix of a right and left circular polarizer. This operation is equivalent to multiply the first Stokes parameters by the last (negated in the case of LCPL).
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  19. We use the word “dichroism” to be consistent with published literature. However, it should be reminded that the transmissions peaks observed in CD and LD can be due not only to absorption but also correspond to reflection/scattering peaks for the incoming light.
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  28. E. Altewischer, M. P. van Exter, J. P. Woerdman, “Polarization analysis of propagating surface plasmons in a subwavelength hole arrays,” J. Opt. Soc. Am. B 20(9), 1927 (2003). [CrossRef]
  29. K. L. van der Molen, F. B. Segerink, N. F. van Hulst, L. Kuipers, “Influence of hole size on theextraordinary transmission through subwavelength hole arrays,” Appl. Phys. Lett. 85(19), 4316–4318 (2004). [CrossRef]
  30. L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86(6), 1114–1117 (2001). [CrossRef] [PubMed]

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