## Magneto-optical enhancement through gyrotropic gratings

Optics Express, Vol. 16, Issue 8, pp. 5378-5384 (2008)

http://dx.doi.org/10.1364/OE.16.005378

Acrobat PDF (151 KB)

### Abstract

Diffracted magneto-optical (MO) effects are numerically investigated for one-dimensional lossy gyrotropic gratings in the zeroth and the first orders for the polar magnetization by utilizing the rigorous coupled-wave approach implemented as an Airy-like internal-reflection series. The simulated Kerr spectra agree well with the experimental ones. The dependence of the MO Kerr enhancement on the grating depth in the first-order diffraction, compared with that in the zeroth one, is illustrated, and the diffracted MO Faraday effect is theoretically investigated as well. Such a MO enhancement through the gyrotropic gratings is superior to the conventional MO devices and magneto-photonic crystals. The potential applications are also suggested.

© 2008 Optical Society of America

## 1. Introduction

1. M. Inoue, K. Arai, T. Fujii, and M. Abe, “One-dimensional magnetophotonic crystals,” J. Appl. Phys. **85**, 5768–5770 (1999). [CrossRef]

2. E. Takeda, N. Todoroki, Y. Kitamoto, M. Abe, M. Inoue, T. Fujii, and K. Arai, “Faraday effect enhancement in Co-ferrite layer incorporated into one-dimensional photonic crystal working as a Fabry-Pérot resonator,” J. Appl. Phys. **87**, 6782–6784 (2000). [CrossRef]

3. M. J. Steel, M. Levy, and R. M. Osgood, “High transmission enhanced Faraday rotation in one-dimensional photonic crystals with defects,” IEEE Photon. Technol. Lett. **12**, 1171–1173 (2000). [CrossRef]

4. M. J. Steel, M. Levy, and R. M. Osgood, “Large magnetooptical Kerr rotation with high reflectivity from photonic bandgap structures with defects,” J. Lightwave Technol. **18**, 1289–1296 (2000). [CrossRef]

5. M. J. Steel, M. Levy, and R. M. Osgood, “Photonic bandgaps with defects and the enhancement of Faraday rotation,” J. Lightwave Technol. **18**, 1297–1308 (2000). [CrossRef]

6. I. L. Lyubchanskii, N. N. Dadoenkova, M. I. Lyubchanskii, E. A. Shapovalov, and Th. Rasing, “Magnetic photonic crystals,” J. Phys. D: Appl. Phys. **36**, R277–R287 (2003). [CrossRef]

7. 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]

6. I. L. Lyubchanskii, N. N. Dadoenkova, M. I. Lyubchanskii, E. A. Shapovalov, and Th. Rasing, “Magnetic photonic crystals,” J. Phys. D: Appl. Phys. **36**, R277–R287 (2003). [CrossRef]

8. A. Figotin and I. Vitebskiy, “Electromagnetic unidirectionality in magnetic photonic crystals,” Phys. Rev. B **67**, 165210 (2003). [CrossRef]

9. A. K. Zvezdin and V. A. Kotov, *Modern Magnetooptics and Magnetooptical Materials* (Institute of Physics Publishing, London, 1997). [CrossRef]

10. S. Fan, M. F. Yanik, Z. Wang, S. Sandhu, and M. L. Povinelli, “Advances in theory of photonic crystals,” J. Lightwave Technol. **24**, 4493–4501 (2006). [CrossRef]

4. M. J. Steel, M. Levy, and R. M. Osgood, “Large magnetooptical Kerr rotation with high reflectivity from photonic bandgap structures with defects,” J. Lightwave Technol. **18**, 1289–1296 (2000). [CrossRef]

## 2. Theoretical approach

21. R. Antos, J. Postora, J. Mistrik, T. Yamaguchi, S. Yamaguchi, M. Horie, S. Visnovsky, and Y. Otani, “Convergence properties of critical dimension measurements by spectroscopic ellipsometry on gratings made of various materials,” J. Appl. Phys. **100**, 054906 (2006). [CrossRef]

*k*

_{0}are transformed to dimensionless ones by setting

**r̄**=

*k*

_{0}

**r**:

**=**

*H̃**cµ*

_{0}

*and*

**H**. c*µ*

_{0}represent light velocity in vacuum and magnetic permeability in vacuum, respectively.

*λ*,

*d*, and

*θ*denote the wavelength of the incident beam, the period of the grating, and the incident angle, respectively.

_{i}*z*coordinate, the propagation eigenmodes of electromagnetic fields are expressed in terms of pseudo-Fourier series:

*k*denotes

*x*or

*y*. The permittivity is a periodic function of only one lateral coordinate

*y*with periodicity

*d*:

*ε*] and [

_{w}*ε*] are the permittivity in wires with the width of

_{b}*w*and spaces between them, respectively. For the wires made of magnetic materials, [

*ε*] is a 3×3 tensor with off-diagonal elements, whose positions are dependent on the direction of magnetization with respect to the incident plane. In the case of polar magnetization, the permittivity is written as below:

_{w}*y*-dependent permittivity can be expressed in terms of Fourier series:

*α*and

**β**represent any of

*x*,

*y*and

*z*. Substituting Eqs. (5), (6) and (9) into Maxwell’s equations, a system composed of ordinary differential equations are obtained as

*ε*] is a Toeplitz matrix consisting of Fourier coefficients

_{αβ}*ε*

_{αβ,n}and the tangential column vectors are denoted to

**f**

_{x},

**f**

_{y},

**g**

_{x}, and

**g**

_{y}composed of

*f*and

_{k}*g*in Eqs. (5) and (6). Here

_{k}**q**is a diagonal matrix in the form of

22. H. Kato, T. Matsushita, A. Takayama, M. Egawa, K. Nishimura, and M. Inoue, “Theoretical analysis of optical and magneto-optical properties of one-dimensional magnetophotonic crystals,” J. Appl. Phys. **93**, 3906–3911 (2003). [CrossRef]

*n*th order as follows,

*χ*

^{(n)}is estimated by the ratio of reflection amplitudes

*f*

^{r}

_{y,n}/

*f*

^{r}

_{x,n}. The procedure of modeling diffracted MO Faraday effect is almost identical to the case of the diffracted MO Kerr effect except for replacing the ratio of reflection amplitudes

*f*

^{r}

_{y,n}/

*f*

^{r}

_{x,n}with that of transmission amplitudes

*f*

^{t}

_{y,n}/

*f*

_{t}

_{x,n}.

*n*is chosen carefully to guarantee a sufficient convergence.

_{max}_{2}O

_{3}capping layer, Ni

_{81}Fe

_{19}permalloy layer, and SiO

_{2}oxide layer on the substrates are 2, 12, and 3 nm, respectively. All the wavelength-dependent optical constants are taken from the literature [23

23. G. Neuber, P. Rauer, J. Kunze, T. Korn, C. Pels, G. Meier, U. Merkt, J. Backstrom, and M. Rubhausen, “Temperature-dependent spectral generalized magneto-optical ellipsometry,” Appl. Phys. Lett. **83**, 4509–4511 (2003). [CrossRef]

24. P. Hones, M. Diserens, and F. Levy, “Characterization of sputter-deposited chromium oxide thin films,” Surf. Coat. Technol. **120**, 277–283 (1999). [CrossRef]

## 3. Results and discussion

19. R. Antos, J. Mistik, T. Yamaguchi, S. Visnovsky, S. O. Demokritov, and B. Hillebrands, “Evidence of native oxides on the capping and substrate of Permalloy gratings by magneto-optical spectroscopy in the zeroth- and first-diffraction orders,” Appl. Phys. Lett. **86**, 231101 (2005). [CrossRef]

19. R. Antos, J. Mistik, T. Yamaguchi, S. Visnovsky, S. O. Demokritov, and B. Hillebrands, “Evidence of native oxides on the capping and substrate of Permalloy gratings by magneto-optical spectroscopy in the zeroth- and first-diffraction orders,” Appl. Phys. Lett. **86**, 231101 (2005). [CrossRef]

21. R. Antos, J. Postora, J. Mistrik, T. Yamaguchi, S. Yamaguchi, M. Horie, S. Visnovsky, and Y. Otani, “Convergence properties of critical dimension measurements by spectroscopic ellipsometry on gratings made of various materials,” J. Appl. Phys. **100**, 054906 (2006). [CrossRef]

*M*as below:

*θ*

^{(-1)}

_{Ks}and

*θ*

^{(0)}

*are the Kerr rotation of the first- and the zeroth-order diffraction with*

_{Ks}*s*-polarized incidence, respectively. It is possible to observe the MO responses in the zeroth- and the first-order diffraction according to the grating equation [26], for both transmission and reflection, as below:

*θ*and

_{D}*N*represent the diffraction angle and order, respectively. In the following calculations, the effects of the capping and the oxide layers are excluded to investigate the diffracted MO effects from pure gyrotropic gratings. The simulated Kerr rotation is shown in Fig. 3 as a function of the grating depth varying from 10 to 500 nm. The magnitudes of the Kerr rotation in two diffraction orders vary in opposite ways with the increase in the depth even though a fluctuation is found in the depth greater than 100 nm. Obviously, we can suppose that the diffracted MO enhancement through gyrotropic gratings comes out only when the grating is thin compared with the grating period and the width of gyrotropic rod, as presented in Fig. 4, where

*M*monotonously decreases down to zero with depths. It is thought that the exaltation of the internal diffraction-edge effects in thick gratings suppresses the MO effect in high orders. The maximum Kerr rotation is even up to ~36°/

*µ*m for the 10 nm grating in the first-order diffraction. The thickness is reduced from the micrometer level down to the nanometer level compared with conventional MPCs to achieve such a huge MO effect. Also in the form of the single layer with the same thickness the estimated Kerr rotation is merely ~5°/

*µ*m by the propagation-matrix method [27

27. I. Abdulhalim, “Analytic propagation matrix method for linear optics of arbitrary biaxial layered media,” J. Opt. A: Pure Appl. Opt. **1**, 646–653 (1999). [CrossRef]

16. J. B. Kim, G. J. Lee, Y. P. Lee, J. Y. Rhee, K. W. Kim, and C. S. Yoon, “One-dimensional magnetic grating structure made easy,” Appl. Phys. Lett. **89**, 151111 (2006). [CrossRef]

## 4. Conclusions

## Acknowledgments

## References and links

1. | M. Inoue, K. Arai, T. Fujii, and M. Abe, “One-dimensional magnetophotonic crystals,” J. Appl. Phys. |

2. | E. Takeda, N. Todoroki, Y. Kitamoto, M. Abe, M. Inoue, T. Fujii, and K. Arai, “Faraday effect enhancement in Co-ferrite layer incorporated into one-dimensional photonic crystal working as a Fabry-Pérot resonator,” J. Appl. Phys. |

3. | M. J. Steel, M. Levy, and R. M. Osgood, “High transmission enhanced Faraday rotation in one-dimensional photonic crystals with defects,” IEEE Photon. Technol. Lett. |

4. | M. J. Steel, M. Levy, and R. M. Osgood, “Large magnetooptical Kerr rotation with high reflectivity from photonic bandgap structures with defects,” J. Lightwave Technol. |

5. | M. J. Steel, M. Levy, and R. M. Osgood, “Photonic bandgaps with defects and the enhancement of Faraday rotation,” J. Lightwave Technol. |

6. | I. L. Lyubchanskii, N. N. Dadoenkova, M. I. Lyubchanskii, E. A. Shapovalov, and Th. Rasing, “Magnetic photonic crystals,” J. Phys. D: Appl. Phys. |

7. | 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. |

8. | A. Figotin and I. Vitebskiy, “Electromagnetic unidirectionality in magnetic photonic crystals,” Phys. Rev. B |

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

10. | S. Fan, M. F. Yanik, Z. Wang, S. Sandhu, and M. L. Povinelli, “Advances in theory of photonic crystals,” J. Lightwave Technol. |

11. | M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. |

12. | P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A |

13. | K. Watanabe, “Study of the differential theory of lamellar gratings made of highly conducting materials,” J. Opt. Soc. Am. A |

14. | M. Grimsditch and P. Vavassori, “The diffracted magneto-optical Kerr effect: what does it tell you?” J. Phys. : Condens. Matter |

15. | A. Westphalen, A. Schumann, A. Remhof, H. Zabel, T. Last, and U. Kunze, “Magnetization reversal of equilateral Fe triangles,” Phys. Rev. B |

16. | J. B. Kim, G. J. Lee, Y. P. Lee, J. Y. Rhee, K. W. Kim, and C. S. Yoon, “One-dimensional magnetic grating structure made easy,” Appl. Phys. Lett. |

17. | F. Jonsson and C. Flytzanis, “Nonlinear magneto-optical Bragg gratings,” Phys. Rev. Lett. |

18. | J. B. Kim, G. J. Lee, Y. P. Lee, J. Y. Rhee, and C. S. Yoon, “Enhancement of magneto-optical properties of a magnetic grating,” J. Appl. Phys. |

19. | R. Antos, J. Mistik, T. Yamaguchi, S. Visnovsky, S. O. Demokritov, and B. Hillebrands, “Evidence of native oxides on the capping and substrate of Permalloy gratings by magneto-optical spectroscopy in the zeroth- and first-diffraction orders,” Appl. Phys. Lett. |

20. | R. Antos, J. Mistrik, T. Yamaguchi, S. Visnovsky, S. O. Demokritov, and B. Hillebrands, “Evaluation of the quality of Permalloy gratings by diffracted magneto-optical spectroscopy,” Opt. Express |

21. | R. Antos, J. Postora, J. Mistrik, T. Yamaguchi, S. Yamaguchi, M. Horie, S. Visnovsky, and Y. Otani, “Convergence properties of critical dimension measurements by spectroscopic ellipsometry on gratings made of various materials,” J. Appl. Phys. |

22. | H. Kato, T. Matsushita, A. Takayama, M. Egawa, K. Nishimura, and M. Inoue, “Theoretical analysis of optical and magneto-optical properties of one-dimensional magnetophotonic crystals,” J. Appl. Phys. |

23. | G. Neuber, P. Rauer, J. Kunze, T. Korn, C. Pels, G. Meier, U. Merkt, J. Backstrom, and M. Rubhausen, “Temperature-dependent spectral generalized magneto-optical ellipsometry,” Appl. Phys. Lett. |

24. | P. Hones, M. Diserens, and F. Levy, “Characterization of sputter-deposited chromium oxide thin films,” Surf. Coat. Technol. |

25. | D. F. Edwards, “Silicon (Si)” in |

26. | E. Hecht, |

27. | I. Abdulhalim, “Analytic propagation matrix method for linear optics of arbitrary biaxial layered media,” J. Opt. A: Pure Appl. Opt. |

**OCIS Codes**

(050.1950) Diffraction and gratings : Diffraction gratings

(050.1960) Diffraction and gratings : Diffraction theory

(160.3820) Materials : Magneto-optical materials

(050.5298) Diffraction and gratings : Photonic crystals

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: January 2, 2008

Revised Manuscript: March 24, 2008

Manuscript Accepted: March 29, 2008

Published: April 3, 2008

**Citation**

Y. H. Lu, M. H. Cho, J. B. Kim, G. J. Lee, Y. P. Lee, and J. Y. Rhee, "Magneto-optical enhancement through gyrotropic gratings," Opt. Express **16**, 5378-5384 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-8-5378

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### References

- M. Inoue, K. Arai, T. Fujii, and M. Abe, "One-dimensional magnetophotonic crystals," J. Appl. Phys. 85, 5768-5770 (1999). [CrossRef]
- E. Takeda, N. Todoroki, Y. Kitamoto, M. Abe, M. Inoue, T. Fujii, and K. Arai, "Faraday effect enhancement in Co-ferrite layer incorporated into one-dimensional photonic crystal working as a Fabry-Pérot resonator," J. Appl. Phys. 87, 6782-6784 (2000). [CrossRef]
- M. J. Steel, M. Levy, and R. M. Osgood, "High transmission enhanced Faraday rotation in one-dimensional photonic crystals with defects," IEEE Photon. Technol. Lett. 12, 1171-1173 (2000). [CrossRef]
- M. J. Steel, M. Levy, and R. M. Osgood, "Large magnetooptical Kerr rotation with high reflectivity from photonic bandgap structures with defects," J. Lightwave Technol. 18, 1289-1296 (2000). [CrossRef]
- M. J. Steel, M. Levy, and R. M. Osgood, "Photonic bandgaps with defects and the enhancement of Faraday rotation," J. Lightwave Technol. 18, 1297-1308 (2000). [CrossRef]
- I. L. Lyubchanskii, N. N. Dadoenkova, M. I. Lyubchanskii, E. A. Shapovalov, and Th. Rasing, "Magnetic photonic crystals," J. Phys. D: Appl. Phys. 36, R277-R287 (2003). [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. Figotin and I. Vitebskiy, "Electromagnetic unidirectionality in magnetic photonic crystals," Phys. Rev. B 67, 165210 (2003). [CrossRef]
- A. K. Zvezdin and V. A. Kotov, Modern Magnetooptics and Magnetooptical Materials (Institute of Physics Publishing, London, 1997). [CrossRef]
- S. Fan, M. F. Yanik, Z. Wang, S. Sandhu, and M. L. Povinelli, "Advances in theory of photonic crystals," J. Lightwave Technol. 24, 4493-4501 (2006). [CrossRef]
- M. G. Moharam and T. K. Gaylord, "Rigorous coupled-wave analysis of planar-grating diffraction," J. Opt. Soc. Am. 71, 811-818 (1981). [CrossRef]
- P. Lalanne and G. M. Morris, "Highly improved convergence of the coupled-wave method for TM polarization," J. Opt. Soc. Am. A 13, 779-784 (1996). [CrossRef]
- K. Watanabe, "Study of the differential theory of lamellar gratings made of highly conducting materials," J. Opt. Soc. Am. A 23, 69-72 (2006). [CrossRef]
- M. Grimsditch and P. Vavassori, "The diffracted magneto-optical Kerr effect: what does it tell you?" J. Phys.: Condens. Matter 16, R275-R294 (2004).
- A. Westphalen, A. Schumann, A. Remhof, H. Zabel, T. Last, and U. Kunze, "Magnetization reversal of equilateral Fe triangles," Phys. Rev. B 74, 104417 (2006). [CrossRef]
- J. B. Kim, G. J. Lee, Y. P. Lee, J. Y. Rhee, K. W. Kim, and C. S. Yoon, "One-dimensional magnetic grating structure made easy," Appl. Phys. Lett. 89, 151111 (2006). [CrossRef]
- F. Jonsson and C. Flytzanis, "Nonlinear magneto-optical Bragg gratings," Phys. Rev. Lett. 96, 063902 (2006). [CrossRef] [PubMed]
- J. B. Kim, G. J. Lee, Y. P. Lee, J. Y. Rhee, and C. S. Yoon, "Enhancement of magneto-optical properties of a magnetic grating," J. Appl. Phys. 101, 09C518 (2007). [CrossRef]
- R. Antos, J. Mistik, T. Yamaguchi, S. Visnovsky, S. O. Demokritov, and B. Hillebrands, "Evidence of native oxides on the capping and substrate of Permalloy gratings by magneto-optical spectroscopy in the zeroth- and first-diffraction orders," Appl. Phys. Lett. 86, 231101 (2005). [CrossRef]
- R. Antos, J. Mistrik, T. Yamaguchi, S. Visnovsky, S. O. Demokritov, and B. Hillebrands, "Evaluation of the quality of Permalloy gratings by diffracted magneto-optical spectroscopy," Opt. Express 13, 4651-4656 (2005). [CrossRef] [PubMed]
- R. Antos, J. Postora, J. Mistrik, T. Yamaguchi, S. Yamaguchi, M. Horie, S. Visnovsky, and Y. Otani, "Convergence properties of critical dimension measurements by spectroscopic ellipsometry on gratings made of various materials," J. Appl. Phys. 100, 054906 (2006). [CrossRef]
- H. Kato, T. Matsushita, A. Takayama, M. Egawa, K. Nishimura, and M. Inoue, "Theoretical analysis of optical and magneto-optical properties of one-dimensional magnetophotonic crystals," J. Appl. Phys. 93, 3906-3911 (2003). [CrossRef]
- G. Neuber, P. Rauer, J. Kunze, T. Korn, C. Pels, G. Meier, U. Merkt, J. Backstrom, and M. Rubhausen, "Temperature-dependent spectral generalized magneto-optical ellipsometry," Appl. Phys. Lett. 83, 4509-4511 (2003). [CrossRef]
- P. Hones, M. Diserens, and F. Levy, "Characterization of sputter-deposited chromium oxide thin films," Surf. Coat. Technol. 120, 277-283 (1999). [CrossRef]
- D. F. Edwards, "Silicon (Si)" in Handbook of Optical Constants of Solids, edited by E. D. Palik (Academic, New York, 1998); H. R. Philipp, "Silicon Dioxide (SiO2) (Glass)," ibid.
- E. Hecht, Optics (Addison-Wesley, New York, 1998).
- I. Abdulhalim, "Analytic propagation matrix method for linear optics of arbitrary biaxial layered media," J. Opt. A: Pure Appl. Opt. 1, 646-653 (1999). [CrossRef]

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