- 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
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 . Regardless of whether dielectric or magnetic PCs are at issue, photonic bands structures are usually shown as normalized frequencies ωN versus normalized wave vectors kN. Thus, the phase index np of an EM wave with a certain frequency ωN and corresponding kN can be calculated via 
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 . Via EMM, analytic expressions for the effective dielectric constant εeff, effective magnetic permeability μeff, 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 , 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.
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 εrod and the magnetic permeability μrod. It is noted that the dielectric constant εrod(ωN) and the magnetic permeability μrod 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 εrod and μrod 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 . Polycarbonate membrane is purely dielectric, whereas permalloy NixFe1-x 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 μrod of rods can be manipulated.
For a given εrod and μrod, the effective dielectric constant εeff, effective magnetic permeability μeff, and effective phase index neff of a PC can be obtained using a zero-order EMM for long-wavelength (ωN → 0) TM (with an E field along the rod) modes via 
where f denotes the filling factor of the rods, and εair and μair 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 λ is much longer than the period d of a PC. Otherwise, higher-order dependencies on εrod, μrod, f, and the period-wavelength ratio d/λ appear in Eqs. (1) and (2). For example, in the case of f= 0.145, εrod = 15, μrod = 1, and d/λ = 0.01, the zero-order εeff is 3.03, while the 2nd -order εeff is about 0.001 . Clearly, the 2nd-order contribution is negligible as compared to the 2nd-order contribution to εeff.
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 θi, and θr 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 Z of PC (ZPC) to air (Zair), i.e.
It is obvious that the reflection vanishes when the impedances of ZPC and Zair match. At impedance match (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 εeff and μeff, which are functions of εrod, μrod, and 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 εrod and μrod keeps constant when μrod is being changed to fix the spatially distributed refractive index for magnetic PCs.
3. Results and discussion
Equations (1) – (6) give the constrains to the values of μrod and εrod to achieve impedance match. Firstly, according to Eq. (2), μeff is always larger than 1 for positive μrod’s. In case, non-trivial θ i,Z=1 in Eq. (6) exists only if μeff is higher than εeff. Secondly, both μrod and εrod are no less than 1. Thirdly, in our discussion, εrod decreases correspondingly with the increasing μrod to keep nrod unchanged. These constrains lead to the requirements for εrod or μrod following
where . 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 (εrod, μrod) (depicted as the dashed line in Fig. 2). For a given nrod, say 151/2, the rods with the coordinates (εrod, μrod), thus lying in the shadowed region, are prohibited from impedance match (shown with the dotted/shadowed segment of the line for nrod = 151/2). In contrast, the rods with the coordinates (εrod, μrod), 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 (μrod = 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 θ i,Z=1 varies with the magnetic permeability μrod of a rod for a given nrod. Figure 3 shows the θ i,Z=1 as functions of μrod for certain values of nrod’s, in which the filling factor f used was 0.145. For a curve of a given nrod in Fig. 3, the εrod correspondingly decreases as μrod increases.
The curves plotted in Fig. 3 exhibit several similar features. First, θ i,Z=1 exists only for a μrod higher than a critical value, which is determined by Eq. (7). Consistent with expectations raised by Eq. (7a), the critical value of μrod increases for a higher nrod. The increase in the critical value of μrod’s with increasing nrod counts for the rightward shift of the θ i,Z=1-μrod curve at higher values of nrod. Secondly, θ i,Z=1 becomes larger for higher values of μrod. This implies that a larger incident angle is required to achieve impedance match when the TM mode is incident to more-magnetic PCs.
Figure 4 shows the effects of the filling factor f of rods on the θ i,Z=1-μrod curve for a given nrod, e.g. 151/2. It is clear that the critical value of μrod for a nonzero θ i,Z=1 is independent of f, as predicted by Eq. (6). But, for a μrod larger than the critical value, the θ i,Z=1-μrod curve moves to the region with larger 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 θi dependent impedance ratio Z is plotted in Fig. 5(a) for a PC having a given refractive index nrod (e.g. nrod = 101/2) and a given filling factor f (e.g. f= 0.145) of rods. The effect of the μrod on the Z-θi curve is also examined.
In Fig. 5(a), the Z decreases from 1 to zero when θi increases from θ i,Z=1 toward 90°. As the magnetism of rods is enhanced, i.e. for higher μrod, the Z-θi curve is shifted rightward. This implies that a higher impedance ratio Z is resulted at a given θi for more-magnetic PCs. Thus, a lower reflection r should be achieved at a given θi for more-magnetic PCs. The r-θi curves for various μrod’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 θi deviates from θ i,Z=1. This means that the transmission of an incident TM mode through the air-PC interface is depressed when θi deviates from θ i,Z=1. The relationship between the transmission t of the electric field of a TM mode and the incident angle θi is expressed as
Figure 5 (c) shows the t-θi curves for μrod ranging from 4.5 to 10 with f being 0.145. It is obvious that the t-θi curve is shifted rightward for larger μrod. Furthermore, a lower transmission is obtained at θi’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 θi 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 Δθi,-3dB of θi 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 μrod’s. For example, in case of μrod = 6 (or εrod = 1.67) and f = 0.145, the Δθi,-3dB is from 38.5° (having t = 1) to 72° (having t = 0.71). The Δθi,-3dB as a function of μrod for nrod = 10 and f= 0.145 is shown with the gray area in Fig. 6. It was found that the permitted range Δθi,-3dB is reduced at higher μrod’s. The effect of the filling factor f on the Δθi,-3dB is also examined. The Δθi,-3dB versus μrod for f= 0.9 and nrod = 10 is plotted with the dotted area in Fig. 6. Through the comparisons between the gray area for f= 0.145 and the dotted area for f= 0.9, the permitted range of Δθi 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 θi for PC’s having higher filling factors.
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.
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.
References and links
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