## Impedance match of long-wavelength electromagnetic waves incident into magnetic photonic crystals

Optics Express, Vol. 15, Issue 12, pp. 7653-7659 (2007)

http://dx.doi.org/10.1364/OE.15.007653

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### Abstract

- By utilizing an effective-medium method, the effective dielectric constant and effective magnetic permeability of magnetic photonic crystals at long-wavelength limits were calculated. We also examined the impedance ratio when a long-wavelength electromagnetic wave is incident to a magnetic photonic crystal. In this work, we focus on investigating the impact of the magnetic permeability of rods forming magnetic photonic crystals on the impedance ratio. Furthermore, we analyze the dependencies of the incident angle at impedance match on the magnetic permeability and filling factor of rods.

© 2007 Optical Society of America

## 1. Introduction

*H*of EM waves [1

1. R.D. Meade, A.M. Rappe, K.D. Brommer, J.D. Joannopoulos, and O.L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B **48**, 8434–8437 (1993). [CrossRef]

2. S. G. Johnson and J. D. Joannopoulos, ”Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**, 173–190 (2001). [CrossRef] [PubMed]

*B*is taken for finding photonic band structures [3

3. I. Drikis, S.Y. Yang, H.E. Horng, C.-Y. Hong, and H.C. Yang, ”Modified frequency-domain method for simulating the electromagnetic properties in periodic magnetoactive systems,” J. Appl. Phys. **95**, 5876–5881 (2004). [CrossRef]

*ω*versus normalized wave vectors

_{N}*k*. Thus, the phase index

_{N}*n*of an EM wave with a certain frequency

_{p}*ω*and corresponding

_{N}*k*can be calculated via [4

_{N}4. S.Y. Yang and C.T. Chang, “Chromatic dispersion compensators via highly dispersive photonic crystals,” J. Appl. Phys. **98**, 23108–23111 (2005). [CrossRef]

5. T. Matsumoto and T. Baba, “Photonic crystal mmb k-Vector superprism,” J. Lightwave Technol. **22**, 917–922 (2004). [CrossRef]

6. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Photonic crystals for micro lightwave circuits using wavelength-dependent angular beam steering,” Appl. Phys. Lett. **74**, 1370–1372 (1999). [CrossRef]

6. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Photonic crystals for micro lightwave circuits using wavelength-dependent angular beam steering,” Appl. Phys. Lett. **74**, 1370–1372 (1999). [CrossRef]

7. S.Y. Yang and C.T. Chang, “Theoretical analysis for superprisming effect of photonic crystals composed of magnetic material,” J. Appl. Phys. **100**, 831051–831055 (2006). [CrossRef]

8. S.Y. Yang, “Analysis of the contributions of magnetic susceptibility to effective refractive indices of photonic crystals at long-wavelength limits,” Opt. Express **15**, 2669–2676 (2007). [CrossRef] [PubMed]

*ε*, effective magnetic permeability

_{eff}*μ*, and effective phase index of PCs can be obtained. The validity of EMM has been demonstrated for both dielectric and magnetic PCs. It is worthy of note that not only the phase index, but also the dielectric constant and magnetic permeability of a PC, are available at long-wavelength limits through EMM. This creates opportunities to theoretically investigate the impedance ratio between a PC and its surrounding medium, especially for magnetic PCs. According to our previous reports [8

_{eff}8. S.Y. Yang, “Analysis of the contributions of magnetic susceptibility to effective refractive indices of photonic crystals at long-wavelength limits,” Opt. Express **15**, 2669–2676 (2007). [CrossRef] [PubMed]

## 2. Simulated system and simulation formulas

*ε*and

_{rod}*μ*, the effective dielectric constant

_{rod}*ε*, effective magnetic permeability

_{eff}*μ*, and effective phase index

_{eff}*n*of a PC can be obtained using a zero-order EMM for long-wavelength (ω

_{eff}_{N}→ 0) TM (with an

*E*field along the rod) modes via [8

8. S.Y. Yang, “Analysis of the contributions of magnetic susceptibility to effective refractive indices of photonic crystals at long-wavelength limits,” Opt. Express **15**, 2669–2676 (2007). [CrossRef] [PubMed]

*f*denotes the filling factor of the rods, and

*ε*and

_{air}*μ*denote the dielectric constant and magnetic permeability of interstitial air, respectively. We would like to mention that the validity of a zero-order EMM is limited for such cases as the wavelength

_{air}*λ*is much longer than the period

*d*of a PC. Otherwise, higher-order dependencies on

*ε*,

_{rod}*μ*,

_{rod}*f*, and the period-wavelength ratio

*d*/

*λ*appear in Eqs. (1) and (2). For example, in the case of

*f*= 0.145,

*ε*= 15,

_{rod}*μ*= 1, and

_{rod}*d*/

*λ*= 0.01, the zero-order

*ε*is 3.03, while the 2

_{eff}^{nd}-order

*ε*is about 0.001 [10

_{eff}10. P. Lalanne, “Effective medium theory applied to photonic crystals composed of cubic or square cylinders,” Appl. Opt. **35**, 5369–5380 (1996). [CrossRef] [PubMed]

^{nd}-order contribution is negligible as compared to the 2

^{nd}-order contribution to

*ε*.

_{eff}*r*of the electric field

*E*of the TM mode at the air-PC interface can be expressed as

*θ*, and

_{i}*θ*are the incident and refractive angles (illustrated in Fig. 1). From the impedance point of view, the ratio of the two terms in the numerator in Eq. (4) is the impedance ratio

_{r}*Z*of PC (

*Z*) to air (

_{PC}*Z*), i.e.

_{air}*Z*and

_{PC}*Z*match. At impedance match (

_{air}*Z*= 1), the incident angle

*θ*

_{i,Z=1}(hereinafter the impedance-match incident angle) can be expressed as

*θ*

_{i,Z=1}depends on both

*ε*and

_{eff}*μ*, which are functions of

_{eff}*ε*,

_{rod}*μ*, and

_{rod}*f*Through Eqs. (1)–(6), the dependencies of

*θ*

_{i,Z=1}on the magnetic permeability of rods can be investigated. In this work, the product of

*ε*and

_{rod}*μ*keeps constant when

_{rod}*μ*is being changed to fix the spatially distributed refractive index for magnetic PCs.

_{rod}## 3. Results and discussion

*μ*and

_{rod}*ε*to achieve impedance match. Firstly, according to Eq. (2),

_{rod}*μ*is always larger than 1 for positive

_{eff}*μ*’

_{rod}*s*. In case, non-trivial

*θ*

_{i,Z=1}in Eq. (6) exists only if

*μ*is higher than

_{eff}*ε*. Secondly, both

_{eff}*μ*and

_{rod}*ε*are no less than 1. Thirdly, in our discussion,

_{rod}*ε*decreases correspondingly with the increasing

_{rod}*μ*to keep

_{rod}*n*unchanged. These constrains lead to the requirements for

_{rod}*ε*or

_{rod}*μ*following

_{rod}*ε*,

_{rod}*μ*) (depicted as the dashed line in Fig. 2). For a given

_{rod}*n*, say 15

_{rod}^{1/2}, the rods with the coordinates (

*ε*,

_{rod}*μ*), thus lying in the shadowed region, are prohibited from impedance match (shown with the dotted/shadowed segment of the line for

_{rod}*n*= 15

_{rod}^{1/2}). In contrast, the rods with the coordinates (

*ε*,

_{rod}*μ*), lying in the solid segment, can have impedance match. One remarkable result that can be inferred from Fig. 2 is that the purely dielectric PC (

_{rod}*μ*= 1) does not exhibit impedance match for the long-wavelength TM mode. Impedance match only occurs for a TM mode incident into magnetic PCs. Furthermore, according to Eqs. (1), (2), and (6), the

_{rod}*θ*

_{i,Z=1}varies with the magnetic permeability

*μ*of a rod for a given

_{rod}*n*. Figure 3 shows the

_{rod}*θ*

_{i,Z=1}as functions of

*μ*for certain values of

_{rod}*n*’

_{rod}*s*, in which the filling factor

*f*used was 0.145. For a curve of a given

*n*in Fig. 3, the

_{rod}*ε*correspondingly decreases as

_{rod}*μ*increases.

_{rod}*θ*

_{i,Z=1}exists only for a

*μ*higher than a critical value, which is determined by Eq. (7). Consistent with expectations raised by Eq. (7a), the critical value of

_{rod}*μ*increases for a higher

_{rod}*n*. The increase in the critical value of

_{rod}*μ*’

_{rod}*s*with increasing

*n*counts for the rightward shift of the

_{rod}*θ*

_{i,Z=1}-

*μ*curve at higher values of

_{rod}*n*. Secondly,

_{rod}*θ*

_{i,Z=1}becomes larger for higher values of

*μ*. This implies that a larger incident angle is required to achieve impedance match when the TM mode is incident to more-magnetic PCs.

_{rod}*f*of rods on the

*θ*

_{i,Z=1}-

*μ*curve for a given

_{rod}*n*, e.g. 15

_{rod}^{1/2}. It is clear that the critical value of

*μ*for a nonzero

_{rod}*θ*

_{i,Z=1}is independent of

*f*, as predicted by Eq. (6). But, for a

*μ*larger than the critical value, the

_{rod}*θ*

_{i,Z=1}-

*μ*curve moves to the region with larger

_{rod}*θ*

_{i,Z=1}’

*s*, as

*f*increases. This reveals that a TM mode has to be along a larger incident angle into a magnetic PC with more rods-occupied space to achieve impedance match. But, how is the variation in the impedance ratio

*Z*when a TM mode is not incident into a PC along

*θ*

_{i,Z=1}? This can be investigated according to Eq. (5). The incident angle

*θ*dependent impedance ratio

_{i}*Z*is plotted in Fig. 5(a) for a PC having a given refractive index

*n*(e.g.

_{rod}*n*= 10

_{rod}^{1/2}) and a given filling factor

*f*(e.g.

*f*= 0.145) of rods. The effect of the

*μ*on the

_{rod}*Z*-

*θ*curve is also examined.

_{i}*Z*decreases from 1 to zero when

*θ*increases from

_{i}*θ*

_{i,Z=1}toward 90°. As the magnetism of rods is enhanced, i.e. for higher

*μ*, the

_{rod}*Z-θ*curve is shifted rightward. This implies that a higher impedance ratio

_{i}*Z*is resulted at a given

*θ*for more-magnetic PCs. Thus, a lower reflection

_{i}*r*should be achieved at a given

*θ*for more-magnetic PCs. The

_{i}*r-θ*curves for various

_{i}*μ*’

_{rod}*s*are calculated using Eq. (4) and are plotted in Fig. 5(b). The curves in Fig. 5(b) exhibit similar behaviors as those in Fig. 5(a). A reasonable characteristics found in Fig. 5(b) is that the reflection r is enhanced when

*θ*deviates from

_{i}*θ*

_{i,Z=1}. This means that the transmission of an incident TM mode through the air-PC interface is depressed when

*θ*deviates from

_{i}*θ*

_{i,Z=1}. The relationship between the transmission

*t*of the electric field of a TM mode and the incident angle

*θ*is expressed as

_{i}*t-θ*curves for

_{i}*μ*ranging from 4.5 to 10 with

_{rod}*f*being 0.145. It is obvious that the

*t-θ*curve is shifted rightward for larger

_{i}*μ*. Furthermore, a lower transmission is obtained at

_{rod}*θ*’

_{i}*s*corresponding to smaller

*Z*’

*s*. However, for any curve in Fig. 5(c), the transmission is close to 1 at

*θ*’

*s*around

*θ*

_{i,Z=1}. This reveals that a certain range for

*θ*is permitted to achieve a high transmission for an incident TM mode into PC’s as if the material characteristics satisfies the requirement expressed as Eq. 7(a) or 7(b). With the data shown in Fig. 5(c), the permitted range

_{i}*Δθ*of

_{i,-3dB}*θ*to have the transmission higher than 0.71,which gives -3 dB (= 50 %) for the intensity transmittance of a TM mode through an air-PC interface, is investigated for various

_{i}*μ*’

_{rod}*s*. For example, in case of

*μ*= 6 (or

_{rod}*ε*= 1.67) and

_{rod}*f*= 0.145, the

*Δθ*is from 38.5° (having

_{i,-3dB}*t*= 1) to 72° (having

*t*= 0.71). The

*Δθ*as a function of

_{i,-3dB}*μ*for

_{rod}*n*= 10 and

_{rod}*f*= 0.145 is shown with the gray area in Fig. 6. It was found that the permitted range

*Δθ*is reduced at higher

_{i,-3dB}*μ*’

_{rod}*s*. The effect of the filling factor

*f*on the

*Δθ*is also examined. The

_{i,-3dB}*Δθ*versus

_{i,-3dB}*μ*for

_{rod}*f*= 0.9 and

*n*= 10 is plotted with the dotted area in Fig. 6. Through the comparisons between the gray area for

_{rod}*f*= 0.145 and the dotted area for

*f*= 0.9, the permitted range of

*Δθ*to achieve a high transmission for a TM mode at the air-PC interface is significantly reduced at high filling factors. This implies that the transmission becomes more sensitive to

_{i}*θ*for PC’s having higher filling factors.

_{i}## 4. Conclusion

## Acknowledgments

## References and links

1. | R.D. Meade, A.M. Rappe, K.D. Brommer, J.D. Joannopoulos, and O.L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B |

2. | S. G. Johnson and J. D. Joannopoulos, ”Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express |

3. | I. Drikis, S.Y. Yang, H.E. Horng, C.-Y. Hong, and H.C. Yang, ”Modified frequency-domain method for simulating the electromagnetic properties in periodic magnetoactive systems,” J. Appl. Phys. |

4. | S.Y. Yang and C.T. Chang, “Chromatic dispersion compensators via highly dispersive photonic crystals,” J. Appl. Phys. |

5. | T. Matsumoto and T. Baba, “Photonic crystal mmb k-Vector superprism,” J. Lightwave Technol. |

6. | H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Photonic crystals for micro lightwave circuits using wavelength-dependent angular beam steering,” Appl. Phys. Lett. |

7. | S.Y. Yang and C.T. Chang, “Theoretical analysis for superprisming effect of photonic crystals composed of magnetic material,” J. Appl. Phys. |

8. | S.Y. Yang, “Analysis of the contributions of magnetic susceptibility to effective refractive indices of photonic crystals at long-wavelength limits,” Opt. Express |

9. | A. Sailb, D. Vanhoenacker-Janvier, I. Huynen, A. Encinas, L. Piraux, E. Ferain, and R. Legras, “Magnetic photonic band-gap material at microwave frequencies based on ferromagnetic nanowires,” Appl. Phys. Lett. |

10. | P. Lalanne, “Effective medium theory applied to photonic crystals composed of cubic or square cylinders,” Appl. Opt. |

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(230.3990) Optical devices : Micro-optical devices

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: April 4, 2007

Revised Manuscript: May 4, 2007

Manuscript Accepted: May 30, 2007

Published: June 6, 2007

**Citation**

S. Y. Yang, "Impedance match of long-wavelength electromagnetic waves incident into magnetic photonic crystals," Opt. Express **15**, 7653-7659 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-12-7653

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### References

- R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, "Accurate theoretical analysis of photonic band-gap materials," Phys. Rev. B 48, 8434-8437 (1993). [CrossRef]
- S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis," Opt. Express 8, 173-190 (2001). [CrossRef] [PubMed]
- I. Drikis, S. Y. Yang, H. E. Horng, C.-Y. Hong, and H. C. Yang, "Modified frequency-domain method for simulating the electromagnetic properties in periodic magnetoactive systems," J. Appl. Phys. 95, 5876-5881 (2004). [CrossRef]
- S. Y. Yang and C. T. Chang, "Chromatic dispersion compensators via highly dispersive photonic crystals," J. Appl. Phys. 98, 23108-23111 (2005). [CrossRef]
- T. Matsumoto and T. Baba, "Photonic crystal mmb k-Vector superprism," J. Lightwave Technol. 22, 917-922 (2004). [CrossRef]
- H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Photonic crystals for micro lightwave circuits using wavelength-dependent angular beam steering," Appl. Phys. Lett. 74, 1370-1372 (1999). [CrossRef]
- S. Y. Yang and C. T. Chang, "Theoretical analysis for superprisming effect of photonic crystals composed of magnetic material," J. Appl. Phys. 100, 831051-831055 (2006). [CrossRef]
- S. Y. Yang, "Analysis of the contributions of magnetic susceptibility to effective refractive indices of photonic crystals at long-wavelength limits," Opt. Express 15, 2669-2676 (2007). [CrossRef] [PubMed]
- A. Sailb, D. Vanhoenacker-Janvier, I. Huynen, A. Encinas, L. Piraux, E. Ferain, and R. Legras, "Magnetic photonic band-gap material at microwave frequencies based on ferromagnetic nanowires," Appl. Phys. Lett. 83, 2378-2380 (2003). [CrossRef]

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