## Asymmetric transmission for linearly polarized electromagnetic radiation |

Optics Express, Vol. 19, Issue 9, pp. 8347-8356 (2011)

http://dx.doi.org/10.1364/OE.19.008347

Acrobat PDF (901 KB)

### Abstract

Metamaterials have shown to support the intriguing phenomenon of asymmetric electromagnetic transmission in the opposite propagation directions, for both circular and linear polarizations. In the present article, we propose a criterion on the relationship among the elements of transmission matrix, which allows asymmetrical transmission for linearly polarized electromagnetic radiation only while the reciprocal transmission for circularly one. Asymmetric hybridized metamaterials are shown to satisfy this criterion. The influence from the rotation of the sample around the radiation propagation direction is discussed. A special structure design is proposed, and its characteristics are analyzed by using numerical simulation.

© 2011 OSA

## 1. Introduction

1. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science **308**, 534 (2005). [CrossRef] [PubMed]

2. X. Zhang and Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater. **7**, 435 (2008). [CrossRef] [PubMed]

3. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science **312**, 1780 (2006). [CrossRef] [PubMed]

4. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science **292**, 77 (2001). [CrossRef] [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]

12. C. Menzel, C. Helgert, C. Rockstuhl, E. B. Kley, A. Tünnermann, T. Pertsch, and F. Lederer, “Asymmetric transmission of linearly polarized light at optical metamaterials,” Phys. Rev. Lett. **104**, 253902 (2010). [CrossRef] [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]

14. E. Plum, V. A. Fedotov, and N. I. Zheludev, “Asymmetric transmission: a generic property of two-dimensional periodic patterns,” J. Opt. **13**, 024006 (2011). [CrossRef]

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]

9. 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**, 2940 (2008). [CrossRef] [PubMed]

6. E. Plum, V. A. Fedotov, and N. I. Zheludev, “Planar metamaterial with transmission and reflection that depend on the direction of incidence,” Appl. Phys. Lett. **94**, 131901 (2009). [CrossRef]

11. R. Singh, E. Plum, C. Menzel, C. Rockstuhl, A. K. Azad, R. A. Cheville, F. Lederer, W. Zhang, and N. I. Zheludev, “Terahertz metamaterial with asymmetric transmission,” Phys. Rev. B **80**, 153104 (2009). [CrossRef]

10. A. Drezet, C. Genet, J. Y. Laluet, and T. W. Ebbesen, “Optical chirality without optical activity: How surface plasmons give a twist to light,” Opt. Express **16**, 12559 (2008). [CrossRef] [PubMed]

**97**, 167401 (2006). [CrossRef] [PubMed]

9. 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**, 2940 (2008). [CrossRef] [PubMed]

6. E. Plum, V. A. Fedotov, and N. I. Zheludev, “Planar metamaterial with transmission and reflection that depend on the direction of incidence,” Appl. Phys. Lett. **94**, 131901 (2009). [CrossRef]

11. R. Singh, E. Plum, C. Menzel, C. Rockstuhl, A. K. Azad, R. A. Cheville, F. Lederer, W. Zhang, and N. I. Zheludev, “Terahertz metamaterial with asymmetric transmission,” Phys. Rev. B **80**, 153104 (2009). [CrossRef]

**97**, 167401 (2006). [CrossRef] [PubMed]

12. C. Menzel, C. Helgert, C. Rockstuhl, E. B. Kley, A. Tünnermann, T. Pertsch, and F. Lederer, “Asymmetric transmission of linearly polarized light at optical metamaterials,” Phys. Rev. Lett. **104**, 253902 (2010). [CrossRef] [PubMed]

*et al*. [12

12. C. Menzel, C. Helgert, C. Rockstuhl, E. B. Kley, A. Tünnermann, T. Pertsch, and F. Lederer, “Asymmetric transmission of linearly polarized light at optical metamaterials,” Phys. Rev. Lett. **104**, 253902 (2010). [CrossRef] [PubMed]

**104**, 253902 (2010). [CrossRef] [PubMed]

## 2. Theory

*z*direction for linear polarization basis,

*z*and −

*z*) [5

**97**, 167401 (2006). [CrossRef] [PubMed]

11. R. Singh, E. Plum, C. Menzel, C. Rockstuhl, A. K. Azad, R. A. Cheville, F. Lederer, W. Zhang, and N. I. Zheludev, “Terahertz metamaterial with asymmetric transmission,” Phys. Rev. B **80**, 153104 (2009). [CrossRef]

**104**, 253902 (2010). [CrossRef] [PubMed]

**104**, 253902 (2010). [CrossRef] [PubMed]

*t*and

_{xy}*t*not only interchange their values, but also get an additional

_{yx}*π*phase shift [12

**104**, 253902 (2010). [CrossRef] [PubMed]

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

_{circ}= 0 and Δ

_{lin}

*≠*0) can be realized, so long as the following condition as a criterion is satisfied As we will show below, this criterion can be easily realized by using hybridized metamaterials.

**104**, 253902 (2010). [CrossRef] [PubMed]

*θ*around the

*z*direction, Δ

_{lin}will become to Obviously,

*θ*=

_{c}*π*(2

*n*+ 1)/4 (where

*n*= 0, 1, 2 and 3), where the transmission for any polarization is symmetric.

*t*=

_{xy}*t*, implying that the transmission is symmetric for linear polarization. Therefore, using hybridized metamaterials to break the mirror symmetry of the structure in the propagation direction is the unique route for achieving the asymmetric transmission for linear polarization. A good example is the three-dimensional chiral meta-atom proposed in Ref. [12

_{yx}**104**, 253902 (2010). [CrossRef] [PubMed]

*t*≠

_{xy}*t*, so the transmission for linear polarization is asymmetric. But since

_{yx}*t*≠

_{xx}*t*, the transmission for circular polarization is also asymmetric [12

_{yy}**104**, 253902 (2010). [CrossRef] [PubMed]

19. E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science **302**, 419 (2003). [CrossRef] [PubMed]

*x*and

*y*, with effective displacements of

*d*and

_{x}*d*, respectively. Energy of an oscillator excited by the incident EM radiation could be transferred to the other by an elastic coupling between them. The induced oscillation of charges then re-emit the EM radiation to ensure the optical activity. To describe the hybridized metamaterial composed of two planar metamaterial components

_{y}*A*and

*B*, we can resort to the operation below [20

20. M. Kang, Y. Li, J. Chen, J. Chen, Q. Bai, H. T. Wang, and P. H. Wu, “Slow light in a simple metamaterial structure constructed by cut and continuous metal strips,” Appl. Phys. B **100**, 699 (2010). [CrossRef]

*m*and

*q*are the effective mass and charge of the equivalent oscillators for both planar metamaterials, respectively.

**E**

*denotes the external electric field imposed on the equivalent oscillator in planar metamaterial*

_{ui}*i*(

*i*=

*A*,

*B*) oscillating among direction

*u*(

*u*=

*x*,

*y*), and

*d*is the corresponding effective displacement. The matrix operator

_{ui}**H**is written as

*ω*is the angular frequency of the EM radiation. The suffix

*u*or

^{i}*v*(

^{j}*u*,

*v*=

*x*,

*y*and

*i*,

*j*=

*A*,

*B*) stands for one of the four parameters,

*x*,

^{A}*y*,

^{A}*x*and

^{B}*y*.

^{B}*ϖ*and

_{ui}*γ*are the resonant frequency and damping parameter of the individual harmonic oscillator denoted by

_{ui}*u*, while Ω

^{i}*is the coupling strength between two oscillators of*

_{uivj}*u*and

^{i}*v*.

^{j}*E*from the dipole oscillations induced in the planar metamaterial

_{uj}*j*when the EM radiation is incident from the planar metamaterial

*i*, which is proportional to the corresponding effective displacements

*d*, by solving Eq. (8). Thus the transmission matrix elements can be evaluated. To be more explicitly, under the initial excitation of

_{uj}**E**= [1,0,0,0],

*t*and

_{xx}*t*can be found by solving Eq. (8), as follows and

_{yx}*t*and

_{xy}*t*can also be found under the initial excitation of

_{yy}**E**= [0, 1, 0, 0], as follows

*H*(

_{mn}*m*,

*n*= 1,…, 4) is one of the elements of the operator

**H**defined in Eq. (9). Note that the transmission matrix

**104**, 253902 (2010). [CrossRef] [PubMed]

*z*, the following conditions are simultaneously satisfied, Under the above conditions, we have

*H*

_{11}=

*H*

_{33},

*H*

_{22}=

*H*

_{44},

*H*

_{12}=

*H*

_{34}, and

*H*

_{14}=

*H*

_{23}, resulting in

*t*=

_{xy}*t*. As a result, the transmission is symmetric for the linear polarization. Breaking the mirror symmetry in the propagation direction

_{yx}*z*is then crucial for achieving the asymmetric transmission for linear polarization, similar to the case discussed in Ref. [12

**104**, 253902 (2010). [CrossRef] [PubMed]

## 3. Numerical simulations

20. M. Kang, Y. Li, J. Chen, J. Chen, Q. Bai, H. T. Wang, and P. H. Wu, “Slow light in a simple metamaterial structure constructed by cut and continuous metal strips,” Appl. Phys. B **100**, 699 (2010). [CrossRef]

21. M. Kang, N. H. Shen, J. Chen, J. Chen, Y. X. Fan, J. Ding, H. T. Wang, and P. H. Wu, “A new planar left-handed metamaterial composed of metal-dielectric-metal structure,” Opt. Express **16**, 8617 (2008). [CrossRef] [PubMed]

*d*= 0.5 mm in thickness. The two copper layers have the same thicknesses of

_{s}*t*= 18

_{s}*μ*m and the conductivity is 5.8

*×*10

^{7}S/m. In the hybridized metamaterial structure we designed, as shown in Fig. 1, the patterns in the two cooper layers are easily fabricated by using the commercial photolithography technique. Each unit cell has a dimension of

*d × d*= 6

*×*6 mm

^{2}, which ensures that under normal incidence the metamaterial structure does not produce high-order radiative diffraction for frequency below 50 GHz. The unit cell contains two simple structures in the form of a half-sauwastika in the top layer and a half-gammadion in the bottom layer, with parameters

*b*= 4 mm,

*a*= 2 mm, and

*w*= 0.2 mm. As shown in Fig. 1, the unit cell of the whole hybridized metamaterial consists of two cooper layers separated by the isotropic dielectric substrate. The cooper pattern in any layer can be considered as a chiral planar metamaterial. In fact, to realize the requirements of Eq. (14) , the pattern in the bottom layer can be simply designed from that in the top layer (as a half-gammadion) by rotating 90° around the center

*z*-axis. And then the pattern in the bottom layer is further performed a mirror operation about

*y*axis again, to form the shape of a half-sauwastika. This operation does not violate the requirements of Eq. (14) , while can facilitate the achievement of an efficient coupling between the two chiral planar metamaterials and to enhance the EM response. Obviously, if the two patterns are plotted together in forming a single planar metamaterial, the structure will possess two additional mirror planes forming angles of ±

*π*/4 with respect to the

*yz*plane, respectively. Since the two cooper patterns are segregated by the middle dielectric substrate, the mirror symmetry of the whole structure is broken in the propagation

*z*direction and in the

*xy*plane.

20. M. Kang, Y. Li, J. Chen, J. Chen, Q. Bai, H. T. Wang, and P. H. Wu, “Slow light in a simple metamaterial structure constructed by cut and continuous metal strips,” Appl. Phys. B **100**, 699 (2010). [CrossRef]

21. M. Kang, N. H. Shen, J. Chen, J. Chen, Y. X. Fan, J. Ding, H. T. Wang, and P. H. Wu, “A new planar left-handed metamaterial composed of metal-dielectric-metal structure,” Opt. Express **16**, 8617 (2008). [CrossRef] [PubMed]

*z*direction and the periodical boundary condition in the transverse

*xy*plane are adopted.

*t*≠

_{xx}*t*and

_{yy}*t*=

_{xy}*t*, implying that this structure exhibits the mirror symmetry in the propagation

_{yx}*z*direction, that is to say, the transmission for linear polarizations is symmetric.

*t*=

_{xx}*t*. The off-diagonal elements are no longer the same, i.e., |

_{yy}*t*| ≠ |

_{yx}*t*|. These characteristics are in agreement with our analysis in Eq. (14) based on the Born-Kuhn model. To confirm the criterion of Eq. (6) and the presence of asymmetric (symmetric) transmission for linear (circular) polarizations, we calculate the parameter Δ by the FDTD simulation as shown in Fig. 3(b). We can see that

_{xy}*θ*for linear polarization at 14.24 GHz. Evidently,

*π*. At the critical angles of

*θ*=

_{c}*π*(2

*n*+1)/4 (where

*n*= 0, 1, 2 and 3),

**97**, 167401 (2006). [CrossRef] [PubMed]

**104**, 253902 (2010). [CrossRef] [PubMed]

**97**, 167401 (2006). [CrossRef] [PubMed]

*A*

^{2})

^{−1/2}[1,±

*Ae*] in the linear polarization basis, where

^{iδ}*A*= (|

*t*|/|

_{yx}*t*|)

_{xy}^{1/2}is the amplitude and

*δ*= (

*φ*−

_{yx}*φ*)/2 is the phase difference. Figure 4 plots the dependence of

_{xy}*A*and

*δ*on the frequency. We can see that both

*A*and

*δ*oscillate with the frequency. At the resonant frequency of 14.24 GHz, the eigenstates are indeed elliptically polarized, where the main axes of the two eigenstates form an angle of about 60.45° with each other.

*g*characterizing the shift of the longer arm in the top planar metamaterial structure with respect to the center line, as defined in Fig. 1(a). Correspondingly, the lengths of the other two shorter arms changes by ±

*g*in order to ensure the strong coupling. As

*g*increases from zero, as illustrated in Fig. 5,

*g*can be traced to the anisotropic behavior in the diagonal elements of the transmission matrix, that when

*g*is nonzero, requirements of Eq. (14) are no longer hold, resulting in

*t*≠

_{xx}*t*. The breaking of the geometric symmetry in the hybridized metamaterial we designed (

_{yy}*g*≠ 0) results in indeed the degradation of the performance of the asymmetric transmission for the linear polarization and the weak asymmetric transmission for the circular polarization. However, the influence is not very strong even if when

*g*/

*a*= 7.5% .

*t*under the oblique incidence with an incident angle of 5°, for the linearly polarized plane wave, in Fig. 6(a). For the non-collimated incidence case, as is well known, the finite-aperture incident EM radiation can be considered to be non-collimated due to the intrinsic diffraction effect, As an example, we treat the finite-aperture EM radiation with a diameter of 35

_{uv}*d*, as shown in Fig. 6(b). Evidently, the simulation results shown in Fig. 6 have no distinct difference from the results of the plane wave under for the normal incidence shown in Fig. 3(a), implying that the hybridized metamaterial we proposed has a good tolerance for the practical experiment and application. As expected,

*t*has the tiny variation, as shown in Fig. 6. The fluctuation of Δ

_{uv}_{lin}is only about 5%, while Δ

_{circ}maintains zero.

*et al*. [22

22. C. Menzel, C. Rockstuhl, and F. Lederer, “Advanced Jones calculus for the classification of periodic metamaterials,” Phys. Rev. A **82**, 053811(2010). [CrossRef]

22. C. Menzel, C. Rockstuhl, and F. Lederer, “Advanced Jones calculus for the classification of periodic metamaterials,” Phys. Rev. A **82**, 053811(2010). [CrossRef]

**104**, 253902 (2010). [CrossRef] [PubMed]

22. C. Menzel, C. Rockstuhl, and F. Lederer, “Advanced Jones calculus for the classification of periodic metamaterials,” Phys. Rev. A **82**, 053811(2010). [CrossRef]

## 4. Conclusion

23. Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics **3**, 91 (2009). [CrossRef]

24. D. M. Koller, A. Hohenau, H. Ditlbacher, N. Galler, F. Reil, F. R. Aussenegg, A. Leitner, E. J. W. List, and J. R. Krenn, “Organic plasmon-emitting diode,” Nat. Photonics **2**, 684 (2008). [CrossRef]

## Acknowledgments

## References and links

1. | N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science |

2. | X. Zhang and Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater. |

3. | J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science |

4. | R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science |

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. |

6. | E. Plum, V. A. Fedotov, and N. I. Zheludev, “Planar metamaterial with transmission and reflection that depend on the direction of incidence,” Appl. Phys. Lett. |

7. | S. V. Zhukovsky, A. V. Novitsky, and V. M. Galynsky, “Elliptical dichroism: operating principle of planar chiral metamaterials,” Opt. Lett. |

8. | 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. |

9. | 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. |

10. | A. Drezet, C. Genet, J. Y. Laluet, and T. W. Ebbesen, “Optical chirality without optical activity: How surface plasmons give a twist to light,” Opt. Express |

11. | R. Singh, E. Plum, C. Menzel, C. Rockstuhl, A. K. Azad, R. A. Cheville, F. Lederer, W. Zhang, and N. I. Zheludev, “Terahertz metamaterial with asymmetric transmission,” Phys. Rev. B |

12. | C. Menzel, C. Helgert, C. Rockstuhl, E. B. Kley, A. Tünnermann, T. Pertsch, and F. Lederer, “Asymmetric transmission of linearly polarized light at optical metamaterials,” Phys. Rev. Lett. |

13. | J. D. Jackson, |

14. | E. Plum, V. A. Fedotov, and N. I. Zheludev, “Asymmetric transmission: a generic property of two-dimensional periodic patterns,” J. Opt. |

15. | M. Born, “Elektronentheorie des natrlichen optischen Drehungsmgens isotroper und anisotroper Flssigkeiten,” Ann. Phys. (Leipzig) |

16. | W. Kuhn and Z. PhysChem. (Leipzig) B |

17. | Y. P. Svirko and N. I. Zheludev, |

18. | R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. |

19. | E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science |

20. | M. Kang, Y. Li, J. Chen, J. Chen, Q. Bai, H. T. Wang, and P. H. Wu, “Slow light in a simple metamaterial structure constructed by cut and continuous metal strips,” Appl. Phys. B |

21. | M. Kang, N. H. Shen, J. Chen, J. Chen, Y. X. Fan, J. Ding, H. T. Wang, and P. H. Wu, “A new planar left-handed metamaterial composed of metal-dielectric-metal structure,” Opt. Express |

22. | C. Menzel, C. Rockstuhl, and F. Lederer, “Advanced Jones calculus for the classification of periodic metamaterials,” Phys. Rev. A |

23. | Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics |

24. | D. M. Koller, A. Hohenau, H. Ditlbacher, N. Galler, F. Reil, F. R. Aussenegg, A. Leitner, E. J. W. List, and J. R. Krenn, “Organic plasmon-emitting diode,” Nat. Photonics |

**OCIS Codes**

(260.2110) Physical optics : Electromagnetic optics

(310.6860) Thin films : Thin films, optical properties

(160.3918) Materials : Metamaterials

**ToC Category:**

Metamaterials

**History**

Original Manuscript: January 27, 2011

Revised Manuscript: March 14, 2011

Manuscript Accepted: March 15, 2011

Published: April 15, 2011

**Citation**

Ming Kang, Jing Chen, Hai-Xu Cui, Yongnan Li, and Hui-Tian Wang, "Asymmetric transmission for linearly polarized electromagnetic radiation," Opt. Express **19**, 8347-8356 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-9-8347

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### References

- N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534 (2005). [CrossRef] [PubMed]
- X. Zhang and Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater. 7, 435 (2008). [CrossRef] [PubMed]
- J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780 (2006). [CrossRef] [PubMed]
- R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77 (2001). [CrossRef] [PubMed]
- 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]
- E. Plum, V. A. Fedotov, and N. I. Zheludev, “Planar metamaterial with transmission and reflection that depend on the direction of incidence,” Appl. Phys. Lett. 94, 131901 (2009). [CrossRef]
- S. V. Zhukovsky, A. V. Novitsky, and V. M. Galynsky, “Elliptical dichroism: operating principle of planar chiral metamaterials,” Opt. Lett. 34, 1988 (2009).
- 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, 1996 (2007).
- 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, 2940 (2008). [CrossRef] [PubMed]
- A. Drezet, C. Genet, J. Y. Laluet, and T. W. Ebbesen, “Optical chirality without optical activity: How surface plasmons give a twist to light,” Opt. Express 16, 12559 (2008). [CrossRef] [PubMed]
- R. Singh, E. Plum, C. Menzel, C. Rockstuhl, A. K. Azad, R. A. Cheville, F. Lederer, W. Zhang, and N. I. Zheludev, “Terahertz metamaterial with asymmetric transmission,” Phys. Rev. B 80, 153104 (2009). [CrossRef]
- C. Menzel, C. Helgert, C. Rockstuhl, E. B. Kley, A. Tünnermann, T. Pertsch, and F. Lederer, “Asymmetric transmission of linearly polarized light at optical metamaterials,” Phys. Rev. Lett. 104, 253902 (2010). [CrossRef] [PubMed]
- J. D. Jackson, Classical Electrodynamics , 3rd ed. (Wiley, 1999).
- E. Plum, V. A. Fedotov, and N. I. Zheludev, “Asymmetric transmission: a generic property of two-dimensional periodic patterns,” J. Opt. 13, 024006 (2011). [CrossRef]
- M. Born, “Elektronentheorie des natrlichen optischen Drehungsmgens isotroper und anisotroper Flssigkeiten,” Ann. Phys. (Leipzig) 55, 177–240 (1918).
- W. Kuhn and Z. PhysChem. (Leipzig) B 20, 325 (1933).
- Y. P. Svirko and N. I. Zheludev, Polarization of Light in Nonlinear Optics (Wiley, 1998).
- R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717 (2004). [CrossRef]
- E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302, 419 (2003). [CrossRef] [PubMed]
- M. Kang, Y. Li, J. Chen, J. Chen, Q. Bai, H. T. Wang, and P. H. Wu, “Slow light in a simple metamaterial structure constructed by cut and continuous metal strips,” Appl. Phys. B 100, 699 (2010). [CrossRef]
- M. Kang, N. H. Shen, J. Chen, J. Chen, Y. X. Fan, J. Ding, H. T. Wang, and P. H. Wu, “A new planar left-handed metamaterial composed of metal-dielectric-metal structure,” Opt. Express 16, 8617 (2008). [CrossRef] [PubMed]
- C. Menzel, C. Rockstuhl, and F. Lederer, “Advanced Jones calculus for the classification of periodic metamaterials,” Phys. Rev. A 82, 053811(2010). [CrossRef]
- Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3, 91 (2009). [CrossRef]
- D. M. Koller, A. Hohenau, H. Ditlbacher, N. Galler, F. Reil, F. R. Aussenegg, A. Leitner, E. J. W. List, and J. R. Krenn, “Organic plasmon-emitting diode,” Nat. Photonics 2, 684 (2008). [CrossRef]

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