## 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

Photonic band structures for electromagnetic (EM) waves propagating along dielectric photonic crystals (PCs) can be analyzed by solving a master equation for the magnetic intensity field *H* of EM waves [1,2]. In contrast, for magnetic PCs, the master equation for the magnetic flux density *B* is taken for finding photonic band structures [3]. Regardless of whether dielectric or magnetic PCs are at issue, photonic bands structures are usually shown as normalized frequencies *ω _{N}* versus normalized wave vectors

*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}Therefore, the frequency dependent phase index of guided EM waves along PCs can be obtained. As a result, a number of versatile photonic properties for PCs, such as superprism [5,6], dispersion compensation [6,7], etc. can be theoretically investigated using the relationship between the phase index and frequency of EM waves.

To avoid the complicated simulation processes for photonic band structures, Yang developed the effective-medium method (EMM) to analyze the phase indices of EM waves at long-wavelength limits [8]. Via EMM, analytic expressions for the effective dielectric constant *ε _{eff}*, effective magnetic permeability

*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], the magnetic permeability of PCs demonstrates a greater impact on the photonic properties of TM modes than on the photonic properties of TE modes. Thus, in this work, we theoretically study the transmission properties of long-wavelength TM modes incident into a magnetic PC from an impedance point of view. Furthermore, the magnetic contributions to the impedance ratio from the interaction between a magnetic PC and its surrounding medium are analyzed for long-wavelength TM modes.

_{eff}## 2. Simulated system and simulation formulas

The magnetic PC investigated in this work was a two-dimensional PC consisting of triangularly-arrayed infinitely long rods surrounded by air, as plotted with gray spots in Fig. 1. The rod diameter is represented here as *a*, and the spacing between two nearest neighboring rods as *d*. The rods may be either dielectric or magnetic, with the dielectric constant *ε _{rod}* and the magnetic permeability

*μ*. It is noted that the dielectric constant

_{rod}*ε*(

_{rod}*ω*) and the magnetic permeability

_{N}*μ*of rods are associated with an incident electromagnetic wave, not related to externally static electric or magnetic fields. In case, the rods are not electrically and magnetically polarized. Besides, the

_{rod}*ε*and

_{rod}*μ*are regarded as isotropic. In reality, the material for the rods of the investigated PCs can be the mixture of polycarbonate membrane and permalloy, as used for demonstrating magnetic PCs by Sailb et al. in 2003 [9]. Polycarbonate membrane is purely dielectric, whereas permalloy

_{rod}*Ni*shows soft magnetism and magnetic isotropy under zero magnetic field at frequency range from 100 MHz to 40 GHz. Depending on the amounts of permalloy in the mixture, the magnetic permeability

_{x}Fe_{1-x}*μ*of rods can be manipulated.

_{rod}For a given *ε _{rod}* and

*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]

where *f* denotes the filling factor of the rods, and *ε _{air}* and

*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]. Clearly, the 2

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

^{nd}-order contribution to

*ε*.

_{eff}When a TM mode is incident from air into a magnetic PC, the reflection *r* of the electric field *E* of the TM mode at the air-PC interface can be expressed as

where *θ _{i}*, and

*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}It is obvious that the reflection vanishes when the impedances of *Z _{PC}* and

*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

Equation (6) shows that *θ*
_{i,Z=1} depends on both *ε _{eff}* and

*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

Equations (1) – (6) give the constrains to the values of *μ _{rod}* and

*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}where
${n}_{\mathrm{rod}}=\sqrt{{\epsilon}_{\mathrm{rod}}{\mu}_{\mathrm{rod}}}$
. Equation (7a) or (7b) gives the criterion for the material characteristics of rods to achieve the impedance match for a TM mode incident to a magnetic PC. It is worthy of note that, under the zero-order EMM, this criterion is independent of the filling factor of the rods in a PC and is only dependent on the refractive index of rod. This criterion curve can be plotted in the coordinates of (*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}The curves plotted in Fig. 3 exhibit several similar features. First, *θ*
_{i,Z=1} exists only for a *μ _{rod}* higher than a critical value, which is determined by Eq. (7). Consistent with expectations raised by Eq. (7a), the critical value of

*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}Figure 4 shows the effects of the filling factor *f* of rods on the *θ*
_{i,Z=1}-*μ _{rod}* curve for a given

*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}In Fig. 5(a), the *Z* decreases from 1 to zero when *θ _{i}* increases from

*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}Figure 5 (c) shows the *t-θ _{i}* curves for

*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

Impedance match of a long-wavelength TM mode incident into a PC did not occur unless the PC was made of magnetic rods. Further analysis shows that there exists a range for the magnetic permeability of rods for impedance match to occur under a given rod refractive index. Via using a zero-order effective-medium method, the impedance-match range was found to be independent of the filling factor of the rods, but the impedance-match incident angle became larger when either the filling factor increased or when there was a higher rod magnetic permeability. Moreover, the impedance ratio, the reflection, and the transmission of a TM mode incident to a PC becomes sensitive to the incident angle when the rod is more-magnetic.

## Acknowledgments

I would like to thank Prof. T.J. Yang of National Chiao Tung University for helpful discussion. This work is supported by the National Science Council of Taiwan under Grant Nos. 95-2112-M-003-017-MY2.

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