## Anomalous Faraday Effect of a system with extraordinary optical transmittance

Optics Express, Vol. 15, Issue 11, pp. 6612-6622 (2007)

http://dx.doi.org/10.1364/OE.15.006612

Acrobat PDF (1567 KB)

### Abstract

It is shown theoretically that the Faraday rotation becomes anomalously large and exhibits extraordinary behavior near the frequencies of the extraordinary optical transmittance through optically thick perforated metal film with holes filled with a magneto-optically active material. This phenomenon is explained as result of strong confinement of the evanescent electromagnetic field within magnetic material, which occurs due to excitation of the coupled plasmon-polaritons on the opposite surfaces of the film.

© 2007 Optical Society of America

## 1. Introduction

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

2. J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking Surface Plasmons with
Structured Surfaces,” Science **305**, 847–848
(2004). [CrossRef] [PubMed]

3. V. I. Belotelov, L. L. Doskolovich, and A. K. Zvezdin, “Extraordinary magneto-optical
effects and transmission through Metal-Dielectric Plasmonic
Systems,” Phys. Rev. Lett. **98**,
077401–077404(2007). [CrossRef] [PubMed]

4. Y. M. Strelniker and D. J. Bergman, “Optical transmission through metal
films with a subwavelength hole array in the presence of a magnetic
field,” Phys. Rev. B **59**, R12763–R12766
(1999). [CrossRef]

5. L. E. Helseth, “Tunable plasma response of a
metal/ferromagnetic composite material,”
Phys. Rev. B **72**, 033409–033411
(2005). [CrossRef]

6. Z. Wang and S. Fan, “Optical circulators in
two-dimensional magneto-optical photonic
crystals,” Opt. Lett. **30**, 1989–1991
(2005). [CrossRef] [PubMed]

9. M. Inoue, R. Fujikawa, A. Baryshev, A. Khanikaev, P. B. Lim, H. Uchida, O. Aktsipetrov, A. Fedyanin, T. Murzina, and A. Granovsky, “Magnetophotonic
crystals,” J. Phys. D: Appl. Phys. **39**, R151–R161
(2006). [CrossRef]

10. A. A. Fedyanin, O. Aktsipetrov, D. Kobayashi, K. Nishimura, H. Uchida, and M. Inoue, “Enhanced Faraday and nonlinear
magneto-optical Kerr effects in magnetophotonic
crystals,” J. Magn. Magn. Mater. **282**, 256–259
(2004). [CrossRef]

11. M. Inoue, K. I. Arai, T. Fujii, and M. Abe, “Magneto-optical properties of
one-dimensional photonic crystals composed of magnetic and dielectric
layers,” J. Appl. Phys. **83**, 6768–6770
(1998). [CrossRef]

12. A. B. Khanikaev, A. V. Baryshev, M. Inoue, A. B. Granovsky, and A. P. Vinogradov, “Two-dimensional magnetophotonic
crystal: Exactly solvable model,” Phys.
Rev. B **72**, 035123–035131
(2005). [CrossRef]

8. P. M. Hui and D. Stroud, “Theory of Faraday rotation by dilute
suspensions of small particles,” Appl.
Phys. Lett. **50**, 950–952
(1987). [CrossRef]

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

2. J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking Surface Plasmons with
Structured Surfaces,” Science **305**, 847–848
(2004). [CrossRef] [PubMed]

## 2. Model description

13. 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**, 1114–1117
(2001). [CrossRef] [PubMed]

*x*and

*y*directions to allow extraordinary transmittance to appear for two orthogonal polarizations at the same frequency. Holes are chosen to be square because, as it will be demonstrated later, this gives maximal mode conversion.

13. 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**, 1114–1117
(2001). [CrossRef] [PubMed]

*ρ*

^{12}

_{αf},

*τ*

^{21}

_{αf}and

*τ*

^{12}

_{αf}can be found in Ref. 13, and

*ê*is the propagation matrix of the waveguide, which describes evolution of the phase and polarization state when guided mode is propagating along the hole and is considered later.

_{α}*ε̂*are usually very small and can be treated as a perturbation to the diagonal permittivity tensor

*ε̂*

_{0}of the non-magnetic case. I.e. modes of the magnetic waveguide can be found starting from the known solution of the non-magnetic problem by application of the coupled mode theory (CMT) [14, 15]. However, before applying CMT to our problem we would shortly remind its main expressions and limitations to demonstrate its applicability to the evanescent fields.

**E**

^{0}

_{m}(

*x*,

*y*)exp(

*iq*

_{m}*z*-

*iωt*), supported by the dielectric medium characterized by the unperturbed permittivity tensor

*ε̂*

_{0}, satisfies unperturbed wave equation

*ε̂*causes an energy exchange between unperturbed modes. In order to take into account influence of the perturbation one expresses the electric field vector of the electromagnetic wave as an expansion in the normal modes of the unperturbed dielectric structure, where the expansion coefficients depend on

*z*

*A*(

_{m}*z*) is slow and parabolic approximation

*A*(

_{m}*z*)

*N*are the normalization constants and

_{m}*M*are overlap integrals determined by expressions

_{ml}*ε̂*is small and attenuation rate determined by the imaginary part of

*q*is big enough. However, CMT will fail if absolute value of the

_{m}*q*is small. In our problem this happens when we are near cut off frequency of the waveguide. This limitation should be kept in mind when applying CMT for both propagating and evanescent regimes.

_{m}*i*) the most slowly decaying waveguided modes and (

*ii*) the first-order diffraction. Consideration of the higher order diffraction and waveguided modes introduces insignificant correction to intensity and frequency shift to the transmission spectra [13

13. 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**, 1114–1117
(2001). [CrossRef] [PubMed]

*M*that occur in CMT. On the other hand the overlap integral

_{ml}*M*takes maximal values when orthogonally polarized modes are degenerate due to symmetry with respect to 90 degrees rotation. Thus, the maximal mode conversion is expected, for example, for the square or circular apertures. Therefore, we choose the holes to be square with degenerate

_{ml}*x*- and

*y*-polarized modes.

*x*- and

*y*- polarized most slowly decaying modes of the non-magnetic square waveguide are

*q*=(

_{z}*ε*

_{11}((

*ω*/

*c*)

^{2}-(

*π*/

*a*)

^{2})

^{1/2}. Then, utilizing CMT we look for solution in the form

*A*

^{0}

_{10}

*A*

^{0}

_{01})

^{T}defines initial “polarization state” of the guided wave.

*α*∣<<∣

*q*∣ for this problem. In all the subsequent calculations we choose holes’ diameter small enough so that the cutoff frequency is far from frequency range of our interest and this condition is satisfied with very high precision.

_{z}*ê*can be obtained from (11), (12) and (13); it has the following form

*h*is the film thickness.

*t̂*

_{00}allows us to evaluate the Faraday rotation angle of the transmitted wave [14].

## 3. Calculation results and discussion

11. M. Inoue, K. I. Arai, T. Fujii, and M. Abe, “Magneto-optical properties of
one-dimensional photonic crystals composed of magnetic and dielectric
layers,” J. Appl. Phys. **83**, 6768–6770
(1998). [CrossRef]

*L*. Transmittance spectra are thoroughly studied and explained in Ref. 13. Therefore, we concentrate on the MO properties and only briefly remind the points of that work which are the most relevant for their explanation. The extraordinary optical transmittance occurs at frequencies where the denominator in Eq. (15) vanishes. It becomes possible only owing to the evanescent character of the modes inside the holes, which allows for removal of the restrictions on the real part or the modulus of

*ρ*. Two peaks in transmission spectra correspond to the two different resonances and two different values of (

*ρ̂*

^{21}

*ê*)

^{2}and are positioned around 703 and 709 nm. Strong enhancement of the Faraday rotation at frequencies of the peaks of high optical transmittance is evident from Fig. 2. Comparison with the case of homogeneous Bi:YIG film of the same thickness as the structure under study, reveals about 18 and 8 times rotation enhancement for the left and right peaks, respectively.

16. V. Gasparian, M. Ortuno, J. Ruiz, and E. Cuevas, “Faraday Rotation and Complex-Valued
Traversal Time for Classical Light Waves,”
Phys. Rev. Lett. **75**, 2312–2315
(1995). [CrossRef] [PubMed]

17. R. Rosenberg, C. B. Rubinstein, and D. R. Herriott, “Resonant Optical Faraday
Rotator,” Appl. Opt. **3**, 1079–1083,
(1964). [CrossRef]

18. H. Y. Ling, “Theoretical investigation of
transmission through a Faraday-active Fabry-Perot
etalon,” J. Opt. Soc. Am. A **11**, 754–758,
(1994). [CrossRef]

*q*results in imaginary mode conversion parameter

_{z}*α*and changes the nature of the sine and cosine functions in (16). This causes dramatic changes in the underlying physics of the phenomenon of the polarization transformation. Firstly, it is well-known that the Faraday rotation is determined by real part of the ratio

*t̂*

^{21}

_{00}/

*t̂*

^{11}

_{00}. In the case of homogeneous non-absorbing MO material it can be represented as

*α*in this case is also real. In the case of the structure under study this ratio can be approximately expressed for the small

*α*from (15) as

3. V. I. Belotelov, L. L. Doskolovich, and A. K. Zvezdin, “Extraordinary magneto-optical
effects and transmission through Metal-Dielectric Plasmonic
Systems,” Phys. Rev. Lett. **98**,
077401–077404(2007). [CrossRef] [PubMed]

8. P. M. Hui and D. Stroud, “Theory of Faraday rotation by dilute
suspensions of small particles,” Appl.
Phys. Lett. **50**, 950–952
(1987). [CrossRef]

*q*[Eq. (4)], in the case of the holes filled with dielectric of high dielectric constant, significant extraordinary transmittance exists even for very thick films (Fig. 3). With increase of the thickness both transmittance and Faraday rotation gradually decrease because of the increase of absorption in MO holes’ filler. At the same time, two peaks in transmittance are getting closer to each other in the same manner as was shown in Ref. [13

_{z}**86**, 1114–1117
(2001). [CrossRef] [PubMed]

*θ*∣ ∙√

_{F}*T*) [14]. Figure 4 shows corresponding dependencies for the left and right resonances. One can see that the envelope of the FOM at the left peak gradually decreases with increase of the thickness, while at the right peak it has maximum at thickness of 290 nm. When thickness of the structure is small the FOM at the left peak is considerably higher than at the right peak because of huge rotation angles. At the same time the left peak is much narrower than the right one demonstrating that this difference originates in difference of quality of the resonances. Finally, FOM calculated for the homogeneous Bi:YIG film is up to two orders of magnitude smaller then for the structure under study. When the thickness is increased the left peak is getting wider and FOM decreases and takes values close to those corresponding to the right peak and found in the homogeneous MO film.

*a*is presented in Fig. 5. The upper limit for this parameter is determined by the criterion of validity of the couple mode approximation and was estimated to be approximately equal to

*a*≈147 nm . At bigger values the cutoff frequency of the guided modes approaches the frequency range of the extraordinary transmittance giving rise to inapplicability of CMT.

*a*should be appropriately chosen. As expected, the maximal values of FOM correspond to bigger holes. Nevertheless at smaller values of

*a*this dependence is also rather unusual and nontrivial and exhibits strongly undulating behavior caused by the interference of the evanescent guided modes. Appreciable transmittance and Faraday rotation exist and FOM riches the local maxima for some critical values of

*a*. Note that in general these values do not coincide for the left and right peaks. Summarizing we therefore conclude that the most optimal structure should have as big holes as possible while maintaining the evanescent character of the guided modes.

## 4. Conclusion

## Acknowledgments

## References and links

1. | T. W. Ebbssen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission
through sub-wavelength hole arrays,”
Nature |

2. | J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking Surface Plasmons with
Structured Surfaces,” Science |

3. | V. I. Belotelov, L. L. Doskolovich, and A. K. Zvezdin, “Extraordinary magneto-optical
effects and transmission through Metal-Dielectric Plasmonic
Systems,” Phys. Rev. Lett. |

4. | Y. M. Strelniker and D. J. Bergman, “Optical transmission through metal
films with a subwavelength hole array in the presence of a magnetic
field,” Phys. Rev. B |

5. | L. E. Helseth, “Tunable plasma response of a
metal/ferromagnetic composite material,”
Phys. Rev. B |

6. | Z. Wang and S. Fan, “Optical circulators in
two-dimensional magneto-optical photonic
crystals,” Opt. Lett. |

7. | K. Ando, “Nonreciprocal devices for integrated
optics,” Proc. SPIE |

8. | P. M. Hui and D. Stroud, “Theory of Faraday rotation by dilute
suspensions of small particles,” Appl.
Phys. Lett. |

9. | M. Inoue, R. Fujikawa, A. Baryshev, A. Khanikaev, P. B. Lim, H. Uchida, O. Aktsipetrov, A. Fedyanin, T. Murzina, and A. Granovsky, “Magnetophotonic
crystals,” J. Phys. D: Appl. Phys. |

10. | A. A. Fedyanin, O. Aktsipetrov, D. Kobayashi, K. Nishimura, H. Uchida, and M. Inoue, “Enhanced Faraday and nonlinear
magneto-optical Kerr effects in magnetophotonic
crystals,” J. Magn. Magn. Mater. |

11. | M. Inoue, K. I. Arai, T. Fujii, and M. Abe, “Magneto-optical properties of
one-dimensional photonic crystals composed of magnetic and dielectric
layers,” J. Appl. Phys. |

12. | A. B. Khanikaev, A. V. Baryshev, M. Inoue, A. B. Granovsky, and A. P. Vinogradov, “Two-dimensional magnetophotonic
crystal: Exactly solvable model,” Phys.
Rev. B |

13. | 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. |

14. | A. K. Zvezdin and V.A. Kotov, |

15. | P. Yeh, |

16. | V. Gasparian, M. Ortuno, J. Ruiz, and E. Cuevas, “Faraday Rotation and Complex-Valued
Traversal Time for Classical Light Waves,”
Phys. Rev. Lett. |

17. | R. Rosenberg, C. B. Rubinstein, and D. R. Herriott, “Resonant Optical Faraday
Rotator,” Appl. Opt. |

18. | H. Y. Ling, “Theoretical investigation of
transmission through a Faraday-active Fabry-Perot
etalon,” J. Opt. Soc. Am. A |

**OCIS Codes**

(050.1950) Diffraction and gratings : Diffraction gratings

(230.2240) Optical devices : Faraday effect

(240.6680) Optics at surfaces : Surface plasmons

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: March 5, 2007

Revised Manuscript: May 10, 2007

Manuscript Accepted: May 10, 2007

Published: May 15, 2007

**Citation**

Alexander B. Khanikaev, Alexander V. Baryshev, Andrey A. Fedyanin, Alexander B. Granovsky, and Mitsuteru Inoue, "Anomalous Faraday effect of a system with
extraordinary optical transmittance," Opt. Express **15**, 6612-6622 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-11-6612

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### References

- T. W. Ebbssen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998). [CrossRef]
- J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, "Mimicking Surface Plasmons with structured surfaces," Science 305, 847-848 (2004). [CrossRef] [PubMed]
- V. I. Belotelov, L. L. Doskolovich, and A. K. Zvezdin, "Extraordinary magneto-optical effects and transmission through Metal-Dielectric Plasmonic Systems," Phys. Rev. Lett. 98, 077401-077404 (2007). [CrossRef] [PubMed]
- Y. M. Strelniker and D. J. Bergman, "Optical transmission through metal films with a subwavelength hole array in the presence of a magnetic field," Phys. Rev. B 59, R12763-R12766 (1999). [CrossRef]
- L. E. Helseth, "Tunable plasma response of a metal/ferromagnetic composite material," Phys. Rev. B 72, 033409-033411 (2005). [CrossRef]
- Z. Wang and S. Fan, "Optical circulators in two-dimensional magneto-optical photonic crystals," Opt. Lett. 30, 1989-1991 (2005). [CrossRef] [PubMed]
- K. Ando, "Nonreciprocal devices for integrated optics," Proc. SPIE 1126, 58-65 (1989).
- P. M. Hui and D. Stroud, "Theory of Faraday rotation by dilute suspensions of small particles," Appl. Phys. Lett. 50, 950-952 (1987). [CrossRef]
- M. Inoue, R. Fujikawa, A. Baryshev, A. Khanikaev, P. B. Lim, H. Uchida, O. Aktsipetrov, A. Fedyanin, T. Murzina, and A. Granovsky, "Magnetophotonic crystals," J. Phys. D: Appl. Phys. 39, R151-R161 (2006). [CrossRef]
- A. A. Fedyanin, O. Aktsipetrov, D. Kobayashi, K. Nishimura, H. Uchida and M. Inoue, "Enhanced Faraday and nonlinear magneto-optical Kerr effects in magnetophotonic crystals," J. Magn. Magn. Mater. 282, 256-259 (2004). [CrossRef]
- M. Inoue, K. I. Arai, T. Fujii, M. Abe, "Magneto-optical properties of one-dimensional photonic crystals composed of magnetic and dielectric layers," J. Appl. Phys. 83, 6768-6770 (1998). [CrossRef]
- A. B. Khanikaev, A. V. Baryshev, M. Inoue, A. B. Granovsky, and A. P. Vinogradov, "Two-dimensional magnetophotonic crystal: Exactly solvable model," Phys. Rev. B 72, 035123-035131 (2005). [CrossRef]
- 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, 1114-1117 (2001). [CrossRef] [PubMed]
- A. K. Zvezdin and V.A. Kotov, Modern Magnetooptics and Magnetooptical Materials (Taylor & Francis, New York, 1997).
- P. Yeh, Optical Waves in Layered Media (John Wiley & Sons, New York, 1988).
- V. Gasparian, M. Ortuno, J. Ruiz, and E. Cuevas, "Faraday Rotation and Complex-Valued Traversal Time for Classical Light Waves," Phys. Rev. Lett. 75, 2312-2315 (1995). [CrossRef] [PubMed]
- R. Rosenberg, C. B. Rubinstein, and D. R. Herriott, "Resonant Optical Faraday Rotator," Appl. Opt. 3, 1079-1083, (1964). [CrossRef]
- H. Y. Ling, "Theoretical investigation of transmission through a Faraday-active Fabry-Perot etalon," J. Opt. Soc. Am. A 11, 754-758, (1994). [CrossRef]

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