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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 17 — Aug. 18, 2008
  • pp: 12559–12570
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Optical chirality without optical activity: How surface plasmons give a twist to light

Aurélien Drezet, Cyriaque Genet, Jean-Yves Laluet, and Thomas W. Ebbesen  »View Author Affiliations


Optics Express, Vol. 16, Issue 17, pp. 12559-12570 (2008)
http://dx.doi.org/10.1364/OE.16.012559


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Abstract

Light interacts differently with left and right handed three dimensional chiral objects, like helices, and this leads to the phenomenon known as optical activity. Here, by applying a polarization tomography, we show experimentally, for the first time in the visible domain, that chirality has a different optical manifestation for twisted planar nanostructured metallic objects acting as isolated chiral metaobjects. Our analysis demonstrate how surface plasmons, which are lossy bidimensional electromagnetic waves propagating on top of the structure, can delocalize light information in the just precise way for giving rise to this subtle effect.

© 2008 Optical Society of America

1. Introduction

Since the historical work of Arago [1

1. C. -F. M. Arago, “Mémoire sur une modification remarquable qu’éprouvent les rayons lumineux dans leur passage à travers certains corps diaphanes, et sur quelques autres nouveaux phénomènes d’optique,” Mém. Inst. France , Part I 12 (1811).

] and Pasteur [2

2. L. Pasteur, “Mémoire sur la relation qui peut exister entre la forme cristalline et la composition chimique, et sur la cause de la polarization rotatoire,” C. R. Acad. Sci. Paris 26, 535–539 (1848).

], chirality (the handedness of nature) has generally been associated with optical activity, that is the rotation of the plane of polarisation of light passing through a medium lacking mirror symmetry [3

3. E. Hecht, Optics 2nd ed. (Addison-Wesley, Massachusetts, 1987). [PubMed]

, 4

4. L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of continuous media 2nd ed. (Pergamon, New York, 1984). [PubMed]

]. Optical activity is nowadays a very powerful probes of structural chirality in varieties of system. However, two-dimensional chiral structures, such as planar molecules, were not expected to display any chiral characteristics since simply turning the object around leads to the opposite handedness (we remind that a planar structure is chiral if it can not be brought into congruence with its mirror image unless it is lifted from the plane). This fundamental notion was recently challenged in a pioneering study where it was shown that chirality has a distinct signature from optical activity when electromagnetic waves interact with a 2D chiral structure and that the handedness can be recognized [5

5. 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, 167401 (2006). [CrossRef] [PubMed]

]. While the experimental demonstration was achieved in the giga-Hertz (mm) range for extended 2D structures, the question remained whether this could be achieved in the optical range since the optical properties of materials are not simply scalable when downsizing to the nanometer level. However, theoretical work has been done which suggests ways to overcome this difficulty by using localized plasmon modes excited at the scale of the nanostructures [6

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

]. Here we report genuine optical planar chirality for a single subwavelength hole surrounded by left and right handed Archimedian spirals milled in a metallic film. Key to this finding is the involvement of surface plasmons, lossy electromagnetic waves at the metal surfaces, and the associated planar spatial dispersion [7

7. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824 (2003). [CrossRef] [PubMed]

, 8

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

]. Our results reveal how, in a stringent and unusual way, this optical phenomenon connects concepts of chirality, reciprocity and broken time symmetry.

We remind that partly boosted by practical motivations, such as the quest of negative refractive lenses [9

9. J. B. Pendry, “A chiral route to negative refraction,” Science 306, 1353–1355 (2004). [CrossRef] [PubMed]

] or the possibility to obtain giant optical activity for applications in optoelectronics, there is currently a renewed interest [9

9. J. B. Pendry, “A chiral route to negative refraction,” Science 306, 1353–1355 (2004). [CrossRef] [PubMed]

, 10

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

, 11

11. A. S. Schwanecke, A. Krasavin, D. M. Bagnall, A. Potts, A. V. Zayats, and N. I. Zheludev, “Broken time symmetry of light interaction with planar chiral nanostructures,” Phys. Rev. Lett. 91, 247404 (2003). [CrossRef] [PubMed]

, 12

12. T. Vallius, K. Jefimovs, J. Turunen, P. Vahimaa, and Y. Svirko, “Optical activity in subwalength-period arrays of chiral metallic particles,” Appl. Phys. Let. 83, 234–236 (2003). [CrossRef]

, 13

13. M. Kuwata-Gonokami, N. Saito, Y. Ino, M. Kauranen, K. Jefimovs, T. Vallius, J. Turunen, and Y. Svirko, “Giant optical activity in quasi-two-dimensional planar nanostructures,” Phys. Rev. Lett. 95, 227401 (2005). [CrossRef] [PubMed]

, 14

14. B. K. Canfield, S. Kujala1, K. Jefimovs, J. Turunen, and M. Kauranen, “Linear and nonlinear optical responses influenced by broken symmetry in an array of gold nanoparticles,” Opt. Express 12, 5418–5423 (2004). [CrossRef] [PubMed]

, 15

15. B. K. Canfield, S. Kujala1, K. Laiho1, K. Jefimovs, J. Turunen, and M. Kauranen, “Remarkable polarization sensitivity of gold nanoparticle arrays,” Opt. Express 12, 5418–5423 (2004). [CrossRef] [PubMed]

, 16

16. W. Zhang, A. Potts, A. Papakostas, and D. M. Bagnall, “Intensity modulation and polarization rotation of visible light by dielectric planar chiral materials,” Appl. Phys. Lett. 86, 231905 (2005). [CrossRef]

, 17

17. M. Decker, M. W. Klein, M. Wegener, and S. Linden, “Circular dichroism of planar chiral magnetic metamaterials,” Opt. Lett. 32, 856–858 (2007). [CrossRef] [PubMed]

, 18

18. E. Plum, V. A. Fedotov, A. S. Schwanecke, N. I. Zheludev, and Y. Chen “Giant optical gyrotropy due to electromagnetic coupling,” Appl. Phys. Lett. 90, 223113 (2007). [CrossRef]

, 19

19. A. V. Rogacheva, V. A. Fedotov, A. S. Schwanecke, and N. I. Zheludev, “Giant gyrotropy due to electromagnetic-field coupling in a bilayered chiral structure.” Phys. Rev. Lett.97, 177401 (2006). [CrossRef] [PubMed]

] in the optical activity in artificial photonic media with planar chiral structures. It was shown for instance that planar gammadionic structures, which have by definition no axis of reflection but a four-fold rotational invariance [10

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

, 12

12. T. Vallius, K. Jefimovs, J. Turunen, P. Vahimaa, and Y. Svirko, “Optical activity in subwalength-period arrays of chiral metallic particles,” Appl. Phys. Let. 83, 234–236 (2003). [CrossRef]

], can generate optical activity with giant gyrotropic factors [13

13. M. Kuwata-Gonokami, N. Saito, Y. Ino, M. Kauranen, K. Jefimovs, T. Vallius, J. Turunen, and Y. Svirko, “Giant optical activity in quasi-two-dimensional planar nanostructures,” Phys. Rev. Lett. 95, 227401 (2005). [CrossRef] [PubMed]

, 19

19. A. V. Rogacheva, V. A. Fedotov, A. S. Schwanecke, and N. I. Zheludev, “Giant gyrotropy due to electromagnetic-field coupling in a bilayered chiral structure.” Phys. Rev. Lett.97, 177401 (2006). [CrossRef] [PubMed]

, 18

18. E. Plum, V. A. Fedotov, A. S. Schwanecke, N. I. Zheludev, and Y. Chen “Giant optical gyrotropy due to electromagnetic coupling,” Appl. Phys. Lett. 90, 223113 (2007). [CrossRef]

, 17

17. M. Decker, M. W. Klein, M. Wegener, and S. Linden, “Circular dichroism of planar chiral magnetic metamaterials,” Opt. Lett. 32, 856–858 (2007). [CrossRef] [PubMed]

]. Importantly, and in contrast to the usual three dimensional (3D) chiral medium (like quartz and its helicoidal structure [3

3. E. Hecht, Optics 2nd ed. (Addison-Wesley, Massachusetts, 1987). [PubMed]

, 20

20. J. C. Bose, “On the rotation of plane of polarization of electric waves by a twisted structure.” Proc. R. Soc. London A 63, 146–152 (1898). [CrossRef]

]), planar chiral structures change their observed handedness when the direction of light is reversed through the system [10

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

, 21

21. L. Hecht and L. D. Barron, “Rayleigh and Raman optical activity from chiral surfaces,” Chem. Phys. Lett. 225, 525–530 (1994). [CrossRef]

]. This challenged Lorentz principle of reciprocity [4

4. L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of continuous media 2nd ed. (Pergamon, New York, 1984). [PubMed]

] (which is known to hold for any linear non magneto-optical media) and stirred up considerable debate [10

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

, 11

11. A. S. Schwanecke, A. Krasavin, D. M. Bagnall, A. Potts, A. V. Zayats, and N. I. Zheludev, “Broken time symmetry of light interaction with planar chiral nanostructures,” Phys. Rev. Lett. 91, 247404 (2003). [CrossRef] [PubMed]

, 13

13. M. Kuwata-Gonokami, N. Saito, Y. Ino, M. Kauranen, K. Jefimovs, T. Vallius, J. Turunen, and Y. Svirko, “Giant optical activity in quasi-two-dimensional planar nanostructures,” Phys. Rev. Lett. 95, 227401 (2005). [CrossRef] [PubMed]

, 22

22. L. D. Barron, “Parity and optical activity,” Nature 238, 17–19 (1972). [CrossRef] [PubMed]

] which came to the conclusion that optical activity cannot be a purely 2D effect and always requires a small dissymmetry between the two sides of the system [13

13. M. Kuwata-Gonokami, N. Saito, Y. Ino, M. Kauranen, K. Jefimovs, T. Vallius, J. Turunen, and Y. Svirko, “Giant optical activity in quasi-two-dimensional planar nanostructures,” Phys. Rev. Lett. 95, 227401 (2005). [CrossRef] [PubMed]

, 19

19. A. V. Rogacheva, V. A. Fedotov, A. S. Schwanecke, and N. I. Zheludev, “Giant gyrotropy due to electromagnetic-field coupling in a bilayered chiral structure.” Phys. Rev. Lett.97, 177401 (2006). [CrossRef] [PubMed]

, 18

18. E. Plum, V. A. Fedotov, A. S. Schwanecke, N. I. Zheludev, and Y. Chen “Giant optical gyrotropy due to electromagnetic coupling,” Appl. Phys. Lett. 90, 223113 (2007). [CrossRef]

, 17

17. M. Decker, M. W. Klein, M. Wegener, and S. Linden, “Circular dichroism of planar chiral magnetic metamaterials,” Opt. Lett. 32, 856–858 (2007). [CrossRef] [PubMed]

]. Nevertheless Zheludev and colleagues did demonstrate in the GHz spectrum that a pure 2D chiral structure lacking rotational symmetry can have an optical signature which is distinct from optical activity [5

5. 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, 167401 (2006). [CrossRef] [PubMed]

]. They went on to predict that it should be possible to observe the same phenomena in the optical range by scaling down their fish-scale structure and playing on localized plasmons [6

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

]. Following a different strategy, we show here that SP waves propagating on a 2D metal chiral grating resonantly excited by light provide an elegant solution to generate planar optical chirality in the visible.

2. Experiments and results

This is a challenging issue as it leads to two fundamental points which are apparently incompatible. On the one hand, finding such a 2D chiral effect in the optical domain is not equivalent to a simple rescaling of the problem from the GHz to the visible part of the spectrum. Indeed, losses in metal become predominant at the nanometer scale so that the penetration length of light through any chiral structure will become comparable to the thickness of the structure. In-depth spatial dispersion along the propagation direction of light will hence be induced, corresponding to the usual 3D optical activity [12

12. T. Vallius, K. Jefimovs, J. Turunen, P. Vahimaa, and Y. Svirko, “Optical activity in subwalength-period arrays of chiral metallic particles,” Appl. Phys. Let. 83, 234–236 (2003). [CrossRef]

, 13

13. M. Kuwata-Gonokami, N. Saito, Y. Ino, M. Kauranen, K. Jefimovs, T. Vallius, J. Turunen, and Y. Svirko, “Giant optical activity in quasi-two-dimensional planar nanostructures,” Phys. Rev. Lett. 95, 227401 (2005). [CrossRef] [PubMed]

, 14

14. B. K. Canfield, S. Kujala1, K. Jefimovs, J. Turunen, and M. Kauranen, “Linear and nonlinear optical responses influenced by broken symmetry in an array of gold nanoparticles,” Opt. Express 12, 5418–5423 (2004). [CrossRef] [PubMed]

, 15

15. B. K. Canfield, S. Kujala1, K. Laiho1, K. Jefimovs, J. Turunen, and M. Kauranen, “Remarkable polarization sensitivity of gold nanoparticle arrays,” Opt. Express 12, 5418–5423 (2004). [CrossRef] [PubMed]

, 16

16. W. Zhang, A. Potts, A. Papakostas, and D. M. Bagnall, “Intensity modulation and polarization rotation of visible light by dielectric planar chiral materials,” Appl. Phys. Lett. 86, 231905 (2005). [CrossRef]

, 17

17. M. Decker, M. W. Klein, M. Wegener, and S. Linden, “Circular dichroism of planar chiral magnetic metamaterials,” Opt. Lett. 32, 856–858 (2007). [CrossRef] [PubMed]

, 18

18. E. Plum, V. A. Fedotov, A. S. Schwanecke, N. I. Zheludev, and Y. Chen “Giant optical gyrotropy due to electromagnetic coupling,” Appl. Phys. Lett. 90, 223113 (2007). [CrossRef]

, 19

19. A. V. Rogacheva, V. A. Fedotov, A. S. Schwanecke, and N. I. Zheludev, “Giant gyrotropy due to electromagnetic-field coupling in a bilayered chiral structure.” Phys. Rev. Lett.97, 177401 (2006). [CrossRef] [PubMed]

]. One thus expects optical activity, through the losses, to be a more favorable channel than 2D optical chirality. On the other hand, losses (i.e., broken time invariance at the macroscopic scale) are necessary to guarantee planar chiral behavior [5

5. 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, 167401 (2006). [CrossRef] [PubMed]

, 6

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

]. It is also important to emphasize that in order to observe a signature of planar chirality the structure considered must necessarily lack the rotational invariance. Indeed, rotational invariance together with chirality necessarily imply the existence of circular birefringence (i.e., gyrotropy and optical activity). This can be proven from group theory arguments and is also consistent with theoretical analysis of transmission through gammadion holes [23

23. S. L. Prosvirnin and N. I. Zheludev, “Polarization effects in the diffraction of light by planar chiral structure”, Phys. Rev. E 71, 037603 (2005). [CrossRef]

, 24

24. A. Krasavin, A. S. Schwanecke, and N. I. Zheludev, J. Opt. A: Pure Appl. Opt. 8, S98–S105 (2006). [CrossRef]

, 25

25. M. Reichelt, S. W. Koch, A. Krasavin, J. V. Moloney, A. S. Schwanecke, T. Stroucken, E. M. Wright, and N. I. Zheludev, “Broken enantiomeric symmetry for electromagnetic waves interacting with planar chiral nanostructures”, Appl. Phys. B 84, 97–101 (2006). [CrossRef]

] (compare with the analysis in ref. [6

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

]). With this in mind, we chose to make single SP structures such as a single hole in an optically thick metal film surrounded by an Archimedian spirals (Fig. 1) which can provide all the necessary ingredients for observing 2D optical chirality. It is a 2D structure lacking point symmetry, that is rotational and mirror invariances. At the same time, it resonates due to coupling to surface plasmons which, as lossy waves, represent a natural way for delocalizing information along a planar interface, moving in-depth losses to the surface. Importantly, the thickness of the metal film optically decouples both interfaces [26

26. A. Degiron and T. W. Ebbesen, “Analysis of the transmission process through a single aperture surrounded by periodic corrugations,” Opt. Express 12, 3694–3700 (2004). [CrossRef] [PubMed]

], and consequently only the structured chiral side is involved in the 2D optical chiral effect reported here. Finally, the structures gives rise to enhanced transmission [8

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

] enabling high optical throughput for all the characterization measurements.

Using focus ion beam (FIB), we milled in an opaque gold film a clockwise (right 𝓡) or anticlockwise (left 𝓛) Archimedian spiral grooves around a central subwavelength hole. The polar equation (ρ,θ) of the left handed Archimedian spiral is ρ=P·θ/(2π), and the right handed enantiomeric spiral is obtained by reflection across the y axis (see Fig. 1). The geometrical parameter P is the radial grating period and we take its value equal to the SP wavelength λSPP≃760 nm for an excitation at λ≃780 nm (by analogy with what occurs for bull’s eye circular antennas [8

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

] this conditions PλSPP is required in order to obtain a resonant excitation of SPs on the structured metal film interface). We recorded optical transmission spectra at normal incidence with unpolarized light for both isolated structures (Fig. 1). As it can be seen, both enantiomers behave like resonant antennas with quasi identical transmission properties. This resonant behaviour is a direct indication of the SP excitation by the grating similarly to what is observed for circular antennas [27

27. H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “beaming light from a subwavelength aperture,” Science 297, 820–822 (2002). [CrossRef] [PubMed]

].

Fig. 1. Chiral plasmonic metamolecules. On the top panel: scanning electron micrographs of the left (L) and right (R) handed enantiomer (mirror image) planar chiral structures investigated. The scale bar is 3 µm long. The parameters characterizing the structure are the following: hole diameter d=350 nm, film thickness h=310 nm, grating period P=760 nm, groove width w=370 nm, and groove depth s=80 nm. Values for h,w, and s were chosen from the known optimal resonant geometrical properties of circular SP antennas [8]. The structures are milled, with a focus ion beam, in a gold film deposited on a glass substrate. On the bottom panel: transmission spectra at normal incidence of individual left (blue curve) and right handed (red curve) Archimede spirals illuminated from the air side.

To observe and fully characterize the optical signature of planar chirality we perform a full polarization tomography [28

28. F. Le Roy-Brehonnet and B. Le Jeune, “Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties,” Prog. Quant. Electr. 21, 109–151 (1997). [CrossRef]

, 29

29. E. Altewisher, C. Genet, M. P. van Exter, J. P. Woerdman, P. F. A. Alkemade, A. van Zuuk, and E. W. J. M. van der drift, “Polarization tomography of metallic nanohole arrays.” Opt. Lett. 30, 90–92 (2005). [CrossRef]

] (see appendix A and Fig. 4) with the aim of determining the 4×4 Mueller matrix 𝓜 associated with each enantiomer. Experimental results 𝓜𝓛 exp., and 𝓜𝓡 exp. respectively obtained for left and right handed spirals are given in appendixes A and B. Here the important point is that the degree of purity F of the Mueller matrices [28

28. F. Le Roy-Brehonnet and B. Le Jeune, “Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties,” Prog. Quant. Electr. 21, 109–151 (1997). [CrossRef]

] is near unity with F (𝓜𝓛 exp.)≃0.967 and F(𝓜𝓡 exp.)≃0.939. This shows that the coherence in polarization is not degraded by the structure and that we can therefore restrict our discussion to Jones matrices [28

28. F. Le Roy-Brehonnet and B. Le Jeune, “Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties,” Prog. Quant. Electr. 21, 109–151 (1997). [CrossRef]

, 3

3. E. Hecht, Optics 2nd ed. (Addison-Wesley, Massachusetts, 1987). [PubMed]

]. In the convenient left |L〉 and right |R〉 circular polarization basis, these Jones matrices tie the excitation [E in L,E in R] to the transmitted [E out L,E out R] electric fields. In the case of planar chiral structures displaying 2D chiral activity, they have the following form [5

5. 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, 167401 (2006). [CrossRef] [PubMed]

, 6

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

]:

𝒥𝓛th.=(ABCA),𝒥𝓛th.=(ACBA),
(1)

where A, B and C are complex valued numbers such that |B|≠|C| (in references [5

5. 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, 167401 (2006). [CrossRef] [PubMed]

, 6

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

] these constants were written α,β and γ respectively). This inequality accounts for chirality. Being non diagonal, these matrices correspond to polarization converter elements with no rotational invariance around the z axis (Fig. 1). They are thus fundamentally different from Jones matrices associated with optical activity, e.g., gammadions. Importantly the conditions |B|≠|C| implies the non unitarity of 𝓙th. 𝓛,𝓡 which means that losses are necessarily present. Since losses implies (macroscopic) time irreversibility, the condition |B|≠|C| also means that reversing the light path through the chiral structures is not mathematically equivalent to simply reversing the time. This explains why the simple argument based on time invariance given in ref. [13

13. M. Kuwata-Gonokami, N. Saito, Y. Ino, M. Kauranen, K. Jefimovs, T. Vallius, J. Turunen, and Y. Svirko, “Giant optical activity in quasi-two-dimensional planar nanostructures,” Phys. Rev. Lett. 95, 227401 (2005). [CrossRef] [PubMed]

], to exclude planar optical activity, does not here lead to contradiction with our study. From Eq. (1) we deduce the associated theoretical forms for the Mueller matrices 𝓜𝓛 th., 𝓜𝓡 th. (see appendix C). The determination of the coefficients A, B and C (see Eq. (1)) are obtained by comparing the experimental results 𝓜𝓛,𝓡 exp. with 𝓜𝓛,𝓡 th.. After normalization by A we deduce

𝒥𝓛fit=(1.0000.166+i0.2210.131+i0.0991.000),
𝒥𝓡fit=(1.0000.129+i0.0980.170+i0.2301.000).
(2)

These matrices indeed satisfy the chirality criteria of Eq. (1) within the ~1% uncertainty evaluated from the degree of purity of the Mueller matrix of the empty setup.

3. Discussion and conclusion

To illustrate the polarization conversion properties of our chiral structures, we compare in Fig. 2 theory and experiment when the input state is linearly polarized and when the output transmitted intensity is analyzed along different orthogonal directions. A good agreement between the measurements and the theoretical predictions deduced from the Jones matrices (see appendix D) is clearly seen, together with the mirror symmetries between the two enantiomers. This agreement shows that our theoretical hypothesis about the form of the matrices 𝓙𝓛,𝓡(see Eq. (1)) is experimentally justified. Importantly, the observed symmetries also imply that for unpolarized light, and in complete consistency with Fig. 1, the total intensity transmitted by the structures is independent of the chosen enantiomer. Furthermore, the conversion of polarization is well (geometrically) illustrated by using the Poincaré sphere representation [28

28. F. Le Roy-Brehonnet and B. Le Jeune, “Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties,” Prog. Quant. Electr. 21, 109–151 (1997). [CrossRef]

]. Indeed, as shown in Fig. 3, the Mueller matrix defines a geometrical transformation which projects the unit Poincaré sphere, drawn by the input Stokes vector, on an output closed surface with typical radius F(𝓜exp.)≃1 in agreement with the absence of net depolarization as already noticed (from theory, F (𝓜th.)=1 exactly). Data shown on Fig. 2 are also plotted on this sphere. The input state draws a circle in the equator plane while the output state (for each enantiomer) draws a circle in a different plane, which center is not located at the center of the sphere. This is a direct manifestation of planar chirality (see appendix E). There is clearly an antisymmetrical behaviour between both enantiomers. The good agreement between the experiment and the prediction of Eqs. (1), (2) shows the sensitivity of the polarization tomography method and the high reliability of the FIB fabrication.

Fig. 2. Analysis of the polarization states for an input light with variable linear polarization for both the left (left panel) and right handed (right panel) individual chiral structures of Fig. 1. The data points (acquired with a laser light at λ=780 nm) are compared to the predictions from Eq. (2) (continuous curves) for respectively the transmitted intensity analyzed along the direction: |x〉(green), |y〉 (yellow), |+45°〉 (cyan), |-45°〉 (magenta), |L〉 (red), and |R〉 (blue). The total transmitted intensity is also shown (black). The symmetries between both panel expected from group theory (see appendix D) are observed experimentally. The insets show in each panel the ellipses of polarization and the handedness (arrow) associated with the two corotating eingenstates associated with the Jones matrix 𝓙𝓛 (blue) and 𝓙𝓡 (red).
Fig. 3. Full polarization tomography. Poincaré sphere of unity radius associated with the input state represented by the Stokes vector X [3, 28]. Also shown are the results of Fig. 2 for the left (blue) and right handed (red) structures if the linearly polarized incident state draw the black circle in the (X 1, X 2) equator plane of the input sphere. Data points are compared with the predictions from 𝓜𝓛,𝓡 exp. (continuous curves) and of Eq. (2) (dashed curves).

4. Appendix A: Polarization tomography setup.

We apply a procedure similar to the one considered in [29

29. E. Altewisher, C. Genet, M. P. van Exter, J. P. Woerdman, P. F. A. Alkemade, A. van Zuuk, and E. W. J. M. van der drift, “Polarization tomography of metallic nanohole arrays.” Opt. Lett. 30, 90–92 (2005). [CrossRef]

, 30

30. C. Genet, E. Altewischer, M. P. van Exter, and J. P. Woerdman, “Optical depolarization induced by arrays of subwavelength metal holes,” Phys. Rev. B. 71, 033409 (2005). [CrossRef]

] in order to record the Mueller matrix: a collimated laser beam at λ=785 nm is focussed normally on the structure by using an objective L 1 (×50, numerical aperture=0.55). The transmitted light is collected and recollimated by using a second objective L 2 (×40, numerical aperture=0.6). The input and output states of polarization are respectively prepared and analyzed in the collimated part of the light path by using polarizers, half wave plates and quarter waveplates. A sketch of the setup is provided below (see Fig. 4).

Fig. 4. Principle of the polarization experiment. (a), Sketch of the optical set up described in the text. The images are recorded by using a CMOS camera. (b), A typical image of the transmitting nanohole showing the Airy spot associated with diffraction by the optical microscope. The scale bar is 2 µm long. (c), Crosscut of the intensity profile along the yellow dotted line shown in (b).

The Mueller matrix is built by applying an experimental algorithm equivalent to the one described in [28

28. F. Le Roy-Brehonnet and B. Le Jeune, “Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties,” Prog. Quant. Electr. 21, 109–151 (1997). [CrossRef]

]. More precisely, in order to write down the full Mueller matrix, we measured here 6×6 intensity projections corresponding to the 6 unit vectors |x〉, |y〉, |+45°〉, |-45°〉, |L〉, and |R〉 for the input and the output polarizations. Actually only 16 measures are needed to determine 𝓜 [28

28. F. Le Roy-Brehonnet and B. Le Jeune, “Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties,” Prog. Quant. Electr. 21, 109–151 (1997). [CrossRef]

]. Our actual procedure is thus more than sufficient to obtain 𝓜.

The isotropy of the setup was first checked by measuring the Mueller matrix 𝓜glass with a glass substrate. Up to a normalization constant, we deduced that 𝓜glass is practically identical to the identity matrix 𝓘 with individuals elements deviating by no more than 0.02. More precisely, the optical depolarization (i. e, the losses in polarization coherence) can be precisely quantified through the degree of purity of the Mueller matrix defined by [28

28. F. Le Roy-Brehonnet and B. Le Jeune, “Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties,” Prog. Quant. Electr. 21, 109–151 (1997). [CrossRef]

] F(𝓜)=(Tr[𝓜𝓜]𝓜0023𝓜002)121 Here we measured F(𝓜glass)= 0.9851. It implies that the light is not depolarized when going through the setup and that consequently we can rely on our measurement procedure for obtaining 𝓜.

5. Appendix B: Experimental Mueller matrices.

The experimental Mueller matrices deduced from the polarization tomography are after normalization of 𝓜exp. 00:

𝓜𝓛exp.=(1.0000.0310.1070.0290.0290.9580.0440.2510.1050.0370.9530.2870.0290.2610.2820.809),
𝓜𝓡exp.=(1.0000.0350.1110.0230.0270.9490.0510.2460.0960.0340.9430.2670.0110.2520.2770.745).
(3)

We have F(𝓜𝓛 exp.)≃0.967 and F(𝓜𝓡 exp.)≃0.939.

We must also note that the normalization used here neglects a small additional coefficient of proportionality 𝓜𝓛ijexp.𝓜𝓡ijexp.0.954 imputed to experimental errors and uncertainties. We also recorded the Mueller matrix of the set up with the glass substrate only. Up to a normalization factor we deduced

𝓜glass=(1.0000̲0.00600.00400.00700.00300.9851̲0.00100.00200.00200.00200.9965̲0.00300.00500.00400.00300.9821̲)
(4)

which satisfies 𝓜glass≃𝓘 with 𝓘 the identity matrix (ideally one should have 𝓜glass=𝓘). It implies that the optical set up do not induce depolarization and that consequently we can rely on our measurement procedure for obtaining 𝓜.

6. Appendix C: Theoretical Mueller matrices.

𝓜𝓛th.=(𝓜00th.𝓜01th.𝓜02th.𝓜03th.𝓜01th.𝓜11th.𝓜12th.𝓜13th.𝓜02th.𝓜12th.𝓜22th.𝓜23th.𝓜03th.𝓜13th.𝓜23th.𝓜33th.).
(5)

with 𝓜th. 00=(2|A|2+|B|2+|C|2)/2,𝓜th. 01=Re[BA*+AC*],𝓜th. 02=Im[AB*+CA*],𝓜th. 03=(|C|2-|B|2)/2, 𝓜th. 11=|A|2+Re[B*C], 𝓜th. 12=Im[B*C],𝓜th. 13=Re[AC*-BA*],𝓜th. 22=|A|2-Re[B*C], 𝓜th. 23=Re[CA*-AB*], 𝓜th. 33=(2|A|2-|B|2-|C|2)/2. Similar formula are obtained for 𝓜𝓡 th. after permuting B and C.

BA=𝓜01th.𝓜13th.𝓜00th.+𝓜33th.+i𝓜23th.𝓜02th.𝓜00th.+𝓜33th.
CA=𝓜01th.𝓜13th.𝓜00th.+𝓜33th.+i𝓜23th.+𝓜02th.𝓜00th.+𝓜33th..
(6)

Together with Eq. (3) Eq. (6) allow us to fit B/A and C/A if we replace 𝓜th. by 𝓜𝓛 exp. (a similar procedure is applicable to 𝓜𝓡 exp. after permuting B and C).

The best fit we obtained (see Eq. (2)) are:

𝓜𝓛fit=(1.0000.0330.1160.0230.0330.9510.0430.2820.1160.0430.9510.3040.0230.2820.3040.902),
𝓜𝓡fit=(1.0000.03590.1250.0260.0390.9490.0440.2830.1250.0440.9480.3110.0260.2830.3110.897).
(7)

From theory we can deduce that F(𝓜𝓛,𝓡 th.)=1 (i.e., after normalization by 𝓜th. 00). We have thus F(𝓜𝓛,𝓡 fit)=1

7. Appendix D: Symmetries due to chirality [interpreting Fig. 2].

Let |Ψin〉=Ex|x〉+Ey|y〉 and |Ψout〉=Ex|x〉+Ey|y〉 be respectively the incident and transmitted electric fields when we consider the left handed planar chiral structure. We have

Ψout=𝓙̂𝓛Ψin
(8)

where 𝓙^ 𝓛 is the operator associated with the Jones matrix 𝓙𝓛. The mathematical definition of planar chirality is that whatever the mirror symmetry operation Π^ in the plane X-Y we have 𝓙^ Π^ -Π^ 𝓙^ ≠=0. It equivalently states that Π^ 𝓙^ Π^ -1𝓙^ . If we consider for example the mirror reflection through the Y axis (see Fig. 1) we have the matrix representation (in the cartesian basis) Π=Π1=(1001) and consequently

Π̂𝒥̂𝓛Π̂1=𝒥̂𝓡𝒥̂𝓛
(9)

which agrees with Eq. (1) and constitutes an other optical definition of chirality.

Π̂Ψout=𝓙̂𝓡Π̂Ψin.
(10)

The input state considered in Fig. 2 is a linearly polarized light |θ〉=sin(θ)|x〉+cos(θ)|y〉 (the angle is measured relatively to the Y axis) and the transmitted intensity projected along a direction of analysis |i〉 (i.e, |x〉, |y〉, |+45°〉, |-45°〉, |L〉, and |R〉) is written Ii (Left)(θ)=|〈iout〉|2=|〈i|𝓙^ 𝓛|θ〉|2. Similarly we also write Ii (Right)(θ)=|〈i|𝓙^ 𝓡|θ〉|2. From Eq. (10) we deduce:

i𝓙̂𝓛θ=i𝓙̂𝓡θ,
(11)

where we used |i′〉=Π^ -1|i〉=Π^ |i〉 and |-θ〉=Π^ |θ〉. We consequently have:

Itotal(Left)(θ)=Itotal(Right)(θ),
Ix,y(Left)(θ)=Ix,y(Right)(θ),
I±45°(Left)(θ)=I45°(Right)(θ),
IL,R(Left)(θ)=IR,L(Right)(θ),
(12)

Such symmetries are clearly visible in Fig. 2 and correspond to a direct signature of optical chirality in the planar systems considered.

8. Appendix E: Planar chirality on the Poincaré Sphere [interpreting Fig. 3]

We remind that the Stokes parameters associated with a polarization state of light |Ψ〉 are defined by

S0=Ix+Iy,S1=IxIy
S2=I+45°I45°,S3=ILIR,
(13)

where Ii are projection measurement along the direction i, i.e, Ii=|〈i|Ψ〉|2. The Stokes vector X is a convenient representation of such a state. We have X=X 1 x 1+X 2 x 2+X 3 x 3 with X 1=S 1/S 0, X 2=S 2/S 0, X 3=S 3/S 0 and with (x 1, x 2, x 3) a cartesian orthogonal and normalized vector basis.

The coherent input state satisfies the normalization [28

28. F. Le Roy-Brehonnet and B. Le Jeune, “Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties,” Prog. Quant. Electr. 21, 109–151 (1997). [CrossRef]

] |X|=1, that is the vector draw a Poincaré sphere of unit radius in the space X 1,X 2,X 3. The transmitted output state after interaction with the left or right handed structure is defined by the relation

(S𝓛,𝓡;0S𝓛,𝓡;1S𝓛,𝓡;2S𝓛,𝓡;3)=𝓜𝓛,𝓡(S0S1S2S3).
(14)

The output state defines a Stokes vector X 𝓛,𝓡 such that |X 𝓛,𝓡|≤1. A typical value for this radius is given by F(𝓜𝓛,𝓡).

If the input state is linearly polarized the input Stokes vector is:

Xin(θ)=(cos(2θ)sin(2θ)0),
(15)

and draw a circle (∑in) along the equator contained in the plane X 1,X 2 of the unit radius Poincaré sphere. Using Eq. 14 the output Stokes vector is now a function of θ: X 𝓛,𝓡(θ) drawing a closed curve (∑𝓛,𝓡) (see Fig. 3) which is the image, through the Mueller matrix transformation, of the equator circle (∑in) above mentioned. Importantly, since the Mueller matrix 𝓜 given by Eq. (5) represents a linear relation connecting X in to X out, we conclude that the image of the incident polarization state contained in the equator plane X 1,X 2 through 𝓜 must also be contained in a plane in the space X 1,X 2,X 3 (in other words, this curve (∑𝓛,𝓡) is the circle arising from the section of the sphere by a plane).

To analyze this point more in details we consider the normalized Vector product

n𝓛,𝓡=(X𝓛,𝓡(0)X𝓛,𝓡(2π3))×(X𝓛,𝓡(0)X𝓛,𝓡(π2))(X𝓛,𝓡(0)X𝓛,𝓡(2π3))×(X𝓛,𝓡(0)X𝓛,𝓡(π2))
(16)

and we write it

n𝓛,𝓡=(U𝓛,𝓡V𝓛,𝓡W𝓛,𝓡),
(17)

with |U 𝓛,𝓡|2+|V 𝓛,𝓡|2+|W 𝓛,𝓡|2=1. It represents a normal to the closed curve (∑𝓛,𝓡) (the vectorial products of 2 non colinear vectors connecting three points contained in a plane is indeed normal to this plane). From Eqs. 7,14 and 15 we obtain

n𝓛=(0.28450.30650.9084),n𝓡=(0.28610.31390.95053).
(18)

Actually, if each curve (∑𝓛,𝓡) is contained in a (different) plane P 𝓛,𝓡 we must have

n𝓛,𝓡·(X𝓛,𝓡(θ)X𝓛,𝓡(0))=0
(19)

for every θ. This was indeed checked numerically up to a precision of 10-11. It was also checked that |X 𝓛,𝓡(θ)|=1 up to the same precision. This also proves that each curve (∑𝓛,𝓡) must be a circle (indeed the intersection of a plane with a sphere is a circle). The equations of the two planes P 𝓛,𝓡 are given by n 𝓛,𝓡·(X-X 𝓛,𝓡(0))=0 where X is the Stokes vector associated with a running point belonging to each plane. We write

U𝓛,𝓡X1+V𝓛,𝓡X2+W𝓛,𝓡X3+D𝓛,𝓡=0
(20)

with D 𝓛=-0.0237 and D 𝓡=-0.0266. |D 𝓛,𝓡| represents the distance separating the center of the circle (∑𝓛,𝓡) to the origin of the poincaré sphere. This proves that the planes are not going through the center of the sphere. It was checked after lengthy calculations that if |B|=|C| in the Jones matrix (see Eq. (1) then D=0. This shows that the property |D 𝓛,𝓡|≠0 is a characteristic of planar chirality (i.e, the condition |B|≠|C|). The radius of each circle (∑𝓛,𝓡) is given by r𝓛,𝓡=(1D𝓛,R2) and we have r 𝓛=0.9997 and r 𝓛=0.9996 which are slightly smaller than r=1 in agreement with the fact that P 𝓛,𝓡 are not going through the center of the sphere.

References and links

1.

C. -F. M. Arago, “Mémoire sur une modification remarquable qu’éprouvent les rayons lumineux dans leur passage à travers certains corps diaphanes, et sur quelques autres nouveaux phénomènes d’optique,” Mém. Inst. France , Part I 12 (1811).

2.

L. Pasteur, “Mémoire sur la relation qui peut exister entre la forme cristalline et la composition chimique, et sur la cause de la polarization rotatoire,” C. R. Acad. Sci. Paris 26, 535–539 (1848).

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E. Hecht, Optics 2nd ed. (Addison-Wesley, Massachusetts, 1987). [PubMed]

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L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of continuous media 2nd ed. (Pergamon, New York, 1984). [PubMed]

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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, 167401 (2006). [CrossRef] [PubMed]

6.

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

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W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824 (2003). [CrossRef] [PubMed]

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

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J. B. Pendry, “A chiral route to negative refraction,” Science 306, 1353–1355 (2004). [CrossRef] [PubMed]

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

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A. S. Schwanecke, A. Krasavin, D. M. Bagnall, A. Potts, A. V. Zayats, and N. I. Zheludev, “Broken time symmetry of light interaction with planar chiral nanostructures,” Phys. Rev. Lett. 91, 247404 (2003). [CrossRef] [PubMed]

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T. Vallius, K. Jefimovs, J. Turunen, P. Vahimaa, and Y. Svirko, “Optical activity in subwalength-period arrays of chiral metallic particles,” Appl. Phys. Let. 83, 234–236 (2003). [CrossRef]

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M. Kuwata-Gonokami, N. Saito, Y. Ino, M. Kauranen, K. Jefimovs, T. Vallius, J. Turunen, and Y. Svirko, “Giant optical activity in quasi-two-dimensional planar nanostructures,” Phys. Rev. Lett. 95, 227401 (2005). [CrossRef] [PubMed]

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B. K. Canfield, S. Kujala1, K. Jefimovs, J. Turunen, and M. Kauranen, “Linear and nonlinear optical responses influenced by broken symmetry in an array of gold nanoparticles,” Opt. Express 12, 5418–5423 (2004). [CrossRef] [PubMed]

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B. K. Canfield, S. Kujala1, K. Laiho1, K. Jefimovs, J. Turunen, and M. Kauranen, “Remarkable polarization sensitivity of gold nanoparticle arrays,” Opt. Express 12, 5418–5423 (2004). [CrossRef] [PubMed]

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W. Zhang, A. Potts, A. Papakostas, and D. M. Bagnall, “Intensity modulation and polarization rotation of visible light by dielectric planar chiral materials,” Appl. Phys. Lett. 86, 231905 (2005). [CrossRef]

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28.

F. Le Roy-Brehonnet and B. Le Jeune, “Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties,” Prog. Quant. Electr. 21, 109–151 (1997). [CrossRef]

29.

E. Altewisher, C. Genet, M. P. van Exter, J. P. Woerdman, P. F. A. Alkemade, A. van Zuuk, and E. W. J. M. van der drift, “Polarization tomography of metallic nanohole arrays.” Opt. Lett. 30, 90–92 (2005). [CrossRef]

30.

C. Genet, E. Altewischer, M. P. van Exter, and J. P. Woerdman, “Optical depolarization induced by arrays of subwavelength metal holes,” Phys. Rev. B. 71, 033409 (2005). [CrossRef]

OCIS Codes
(240.6680) Optics at surfaces : Surface plasmons
(160.1585) Materials : Chiral media
(310.6628) Thin films : Subwavelength structures, nanostructures
(240.5440) Optics at surfaces : Polarization-selective devices

ToC Category:
Optics at Surfaces

History
Original Manuscript: June 12, 2008
Revised Manuscript: July 16, 2008
Manuscript Accepted: July 16, 2008
Published: August 5, 2008

Citation
Aurelien Drezet, Cyriaque Genet, Jean-Yves Laluet, and Thomas W. Ebbesen, "Optical chirality without optical activity: How surface plasmons give a twist to light," Opt. Express 16, 12559-12570 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-12559


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References

  1. D. -F.M. Arago, "Memoire sur une modification remarquable qu�??eprouvent les rayons lumineux dans leur passage a travers certains corps diaphanes, et sur quelques autres nouveaux phenomenes d�??optique," Mem. Inst. France Part I, 12 (1811).
  2. L. Pasteur, "Memoire sur la relation qui peut exister entre la forme cristalline et la composition chimique, et sur la cause de la polarization rotatoire," C. R. Acad. Sci. Paris 26, 535-539 (1848).
  3. E. Hecht, Optics 2nd ed. (Addison-Wesley, Massachusetts, 1987). [PubMed]
  4. L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of continuous media, 2nd ed. (Pergamon, New York, 1984). [PubMed]
  5. 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, 167401 (2006). [CrossRef] [PubMed]
  6. V. A. Fedotov, A. S. Schwanecke, N. I. Zheludev, V. V. Khardikov and S. L. Prosvirnin, "Asymmetric transmission of light and enantiomerically sensistive plasmon resonance in planar chiral nanostructures," Nano Lett. 7, 1996-1999 (2007). [CrossRef]
  7. W. L. Barnes, A. Dereux, T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824 (2003). [CrossRef] [PubMed]
  8. C. Genet, T. W. Ebbesen, "Light in tiny holes," Nature 445, 39-46 (2007). [CrossRef] [PubMed]
  9. J. B. Pendry, "A chiral route to negative refraction," Science 306, 1353-1355 (2004). [CrossRef] [PubMed]
  10. A. Papakostas, A. Potts, D. M. Bagnall, S. L. Prosvirnin, H. J. Coles and N. I. Zheludev, "Optical manisfestation of planar chirality," Phys. Rev. Lett. 90, 107404 (2003). [CrossRef] [PubMed]
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