Analysis of the contributions of magnetic susceptibility to effective refractive indices of photonic crystals at long-wavelength limits

doi:10.1364/OE.15.002669

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Analysis of the contributions of magnetic susceptibility to effective refractive indices of photonic crystals at long-wavelength limits

S. Y. Yang »View Author Affiliations

Optics Express, Vol. 15, Issue 5, pp. 2669-2676 (2007)
http://dx.doi.org/10.1364/OE.15.002669

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▸ Article Outline
– Abstract
1. Introduction
2. Simulated system
3. Results and discussion
4. Conclusion
– References and links

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Abstract

Since the magnetic susceptibility of materials is significant for low-wavelength regions, we investigated magnetic effects on refractive indices for long-wavelength electromagnetic waves propagating in photonic crystals (PCs). The PCs consisted of triangularly arrayed long rods, and were made of either dielectric or magnetic material, with air as the interstitial medium. According to calculated photonic band structures, the magnetism of rods plays a role in TM modes. Instead of using complicated calculating processes for band structures to find long-wavelength refractive indices, an analytic method was developed to estimate the effective refractive indices of long-wavelength TM modes. The refractive indices obtained through the band structures and the analytic method were consistent with each other. This demonstrates the validity of the analytic method, which we used to further clarify the physical mechanism involving the effects of rod magnetism on the refractive indices of long-wavelength TM modes propagating along magnetic PCs.

1. Introduction

A band-like structure for propagating electromagnetic waves that had forbidden photonic bands was observed, due to Bragg diffraction in a photonic crystal (PC) with a periodic variation in the refractive index. When defects are implanted into PCs, defect modes lying in the forbidden bands are generated. This spectacular property has been widely utilized to develop PC devices, such as cavities [1–3

T. Yoshie, A. Scherer, H. Chen, D. Huffaker, and D. Deppe, “Optical characterization of two-dimensional photonic crystal cavities with indium arsenide quantum dot emitters,” Appl. Phys. Lett. 79, 114 (2001). [CrossRef]

], waveguides [4

E. Chow, S.Y. Lin, J. R. Wendt, S.G. Johnson, and J.D. Joannopoulos, “Quantitative analysis of bending efficiency in photonic-crystal waveguide bends at λ = 1.55 μm wavelengths,” Opt. Lett. 26, 286 (2001). [CrossRef]

], couplers [5

M. Koshiba, “Wavelength Division Multiplexing and Demultiplexing With Photonic Crystal Waveguide Couplers,” J. Lightwave Technol. 19, 1970 (2001). [CrossRef]

], and so on.

In addition to defect-mode engineering, a great deal of attention has been given to investigation into gap-edge modes because of the modes’ unique properties. For example, both PC chromatic super-prisms [6

T. Matsumoto and T. Baba, “Photonic Crystal mmb k-Vector Superprism,” J. Lightwave Technol. 22, 917 (2004). [CrossRef]

S.Y. Yang and C.T. Chang, “Theoretical analysis for superprisming effect of photonic crystals composed of magnetic material,” J. Appl. Phys. 100, 83105 (2006). [CrossRef]

] and dispersion compensators [8–10

P. Halevi, A.A. Krokhin, and J. Arriaga, “Photonic Crystal Optics and Homogenization of 2D Periodic Composites,” Phys. Rev. Lett. 82, 719 (1999). [CrossRef]

] have been explored through the use of the highly-dispersive, frequency-dependent indices of the gap-edge modes. Devices such as birefringence prisms, whose working principle is based on the fact that PCs exhibit a large difference in the phase indices of gap-edge TM and TE modes, have also been intensively developed [9

S.Y. Yang and C.T. Chang, “Chromatic dispersion compensators via highly dispersive photonic crystals,” J. Appl. Phys. 98, 23108 (2005). [CrossRef]

,11

C. Luo, M. Soljačic′, and J.D. Joannopoulos, “Superprism effect based on phase velocities,” Opt. Lett. 29, 745 (2004). [CrossRef] [PubMed]

]. Other PC devices such as super-lenses [12

S. Foteinopoulou and CM. Soukoulis, “Negative refraction and left-handed behavior in two-dimensional photonic crystals,” Phys. Rev. B 67, 235107 (2003). [CrossRef]

,13

S.Y. Yang, Chin-Yih Hong, and H.C. Yang, “Focusing concave lens using photonic crystals involving magnetic materials,” J. Opt. Soc. Am. A 23, 956 (2006). [CrossRef]

], which utilize the negative refraction of the upper gap-edge modes of photonic crystals, have also been the subject of research interest. Therefore, both gap-edge mode and defect mode engineering has enabled a large number of PC components, and both show promise for further applications in lighting, telecommunications, optical storage, etc.

Most PCs are made of semiconductors due to the well-established lithographical techniques available for semiconductors, having periodic variation in dielectric constants and no variation in magnetic permeability (≈ 1). Recently, non-unit and negative magnetic permeabilities [14

S. O’Brien and John B. Pendry, “Photonic band-gap effects and magnetic activity in dielectric composites,” J. Phys.: Condens. Matter 14, 4035 (2002). [CrossRef]

,15

V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges,” J. Phys. D: Condens. Matter 17, 3717 (2005). [CrossRef]

], together with negative refraction, have observed in PCs made of non-magnetic metamaterials. Magnetic PCs can also be realized with magnetic materials having positive non-unit magnetic permeability. Several interesting phenomena have been reported for magnetic PCs [16–20

Y. Saado, M. Golosovsky, D. Davidov, and A. Frenkel, “Tunable photonic band gap in self-assembled clusters of floating magnetic particles,” Phys. Rev. B 66, 195108 (2002). [CrossRef]

]. For example, in such magnetic PCs, the band-gap size can be reduced to achieve single-mode waveguides or cavities [20

C.-Y. Hong, S.Y. Yang, H.E. Horng, and H.C. Yang, “Slab-thickness dependent band gap size of two-dimensional photonic crystals with triangular-arrayed dielectric or magnetic rods,” J. Appl. Phys. 94, 2188 (2003). [CrossRef]

]. The TE-mode resonance can be enhanced in a magnetic-rod photonic-crystal cavity [19

S.Y. Yang, C.-Y. Hong, I. Drikis, H.E. Horng, and H.C. Yang, “Resonant electromagnetism in photonic crystals composed of triangular-arrayed rods with both dielectric constant and magnetic permeability functions,” J. Opt. Soc. Am. B 21, 413 (2004). [CrossRef]

]. These phenomena are significant for electromagnetic waves at a frequency lower than optical frequency, due to the larger inertia associated with magnetic dipole moments as compared with the electric dipole moment associated with the shifting of electron clouds. As a result of these properties, magnetic PCs made of magnetic materials can only operated at a wavelength long compared with optical wavelength.

Although magnetic effects on PC properties have been reported, magnetic effects on refractive indices have not been so clearly analyzed. In this work, we derive an analytic form for the long-wavelength refractive index to explore the physical mechanism associated with the dependence of the refractive index on magnetic susceptibility of PCs. After demonstrating the validity of the derived analytic form, the role of the magnetic susceptibility of PCs in its refractive index is clarified.

2. Simulated system

The PC investigated here was a two-dimensional PC consisting of triangularly-arrayed, infinitely-long rods surrounded by air, as illustrated in Fig. 1. The ratio of the rod diameter a to the rod spacing d was 0.4. Generally, the rods may be either dielectric or magnetic, with the dielectric constant ε_rod and the magnetic permeability μ_rod In our examples, we set (ε_rod , μ_rod ) to be (15, 1) and (10, 1.5) to investigate the magnetic effects on photonic band structures for TM (with E field along the rod) and TE (with H field along the rod) modes. It is worth noting that the refractive index of dielectric rods equaled that of the magnetic rods. This means that the contrast in the refractive index over PCs was equivalent for the dielectric PCs and magnetic PCs investigated here. However, as indicated later, the effective refractive index of dielectric PCs may deviate from that of magnetic PCs.

Fig. 1. Scheme of a two-dimensional photonic crystal composed of triangularly-arrayed, infinitely-long rods surrounded by air. The ratio of the rod diameter a to the rod spacing d is 0.4. The dielectric constant and magnetic permeability of the rods have been denoted with ε_rod and μ_rod , respectively.

3. Results and discussion

For magnetoactive PCs, the photonic band structure can be simulated by solving the master equation [18

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 (2004). [CrossRef]

]

\frac{1}{μ} \nabla x \frac{1}{ε} \nabla x \frac{1}{μ} B = {(\frac{ω}{c})}^{2} \frac{1}{μ} B,

(1)

for the magnetic flux density B of electromagnetic waves propagating in a PC via the frequency-domain plane-wave expansion method, where ε is the dielectric constant, μ is the magnetic permeability, and ω is the frequency of a propagation mode.

With ∇ ∙ B = 0, the field B can be expanded in a basis of transverse plane waves

B = \exp (i k ∙ r) \sum_{G} (b_{G}^{u} e_{G}^{u} + b_{G}^{v} e_{G}^{v}) \exp (i G ∙ r),

(2)

where k is the Bloch vector, r denotes a position vector, G refers to a vector in a reciprocal lattice, e^u _G and e^v _G are unit vectors perpendicular to wave vector k + G, and the column vector b = (⋯,b^u _G,b^v _G,⋯)^T represents a magnetic field in the Fourier space of transverse plane wave basis. It is noted that the longitudinal components of B in Eq. (1) are not independent but can be expressed in terms of the transversal components. Here, only the transversal components in Eq. (1) are solved. As a result, the master equation becomes

P_{⊥} \frac{1}{μ} \nabla x \frac{1}{ε} \nabla x \frac{1}{μ} B = {(\frac{ω}{c})}^{2} P_{⊥} \frac{1}{μ} B,

(3)

where P⊥ denotes a projection operator to transverse basis. Remarkably, the master Eq. (3) can be expressed as the generalized eigenproblem with the general eigenequation

A b = {(\frac{ω}{c})}^{2} C b,

(4)

where b is the column vector in Eq. (2). The products of A b and C b for the trivial eigenvector are calculated according to the pseudospectral algorithm described in Ref. 21: the curl operators in A are treated in wave vector space, but the multiplication by ε^-1 and ε^-1 is done in the spatial domain after a Fourier transform. Also, both µ^-1 and μ^-1 are represented as smoothed effective dielectric and magnetic tensors near the material interfaces. Since both A and C are Hermitian operators in the present case, the eigenvalues and eigenvectors of the generalized eigenproblem of Eq. (4) can be solved by using the block Rayleigh-quotient method [20

]. Adequately providing a preconditioning operator Ã^-1 that denotes the inverse of approximated and which can be quickly computable significantly accelerates the convergence of this method. The Ã, approximate A for the dielectric photonic crystals was given in Ref. 22. For a system with both periodically distributed dielectric and magnetic permeability functions, the Ã can be selected as

Ã B = P_{⊥} \frac{1}{μ} \nabla x P_{⊥} \frac{1}{ε} \nabla x P_{⊥} \frac{1}{μ} B

(5)

Other details about solving the master equation (1) using this modified frequency-domain method to find photonic band structures are discussed in Ref. 18.

With photonic band structures shown as dimensionless frequency ω_N (= d/λ) vs. dimensionless wave vector k_N (= kd/2π = n_pd/λ), the phase index n_p of a propagating wave is then obtained from the photonic band structure via

n_{p} = \frac{k_{N}}{ω_{N}}

(6)

The simulated photonic band structure for a TM mode propagating in a PC of dielectric rods (ε_rod = 15, μ_rod = 1) was plotted with dashed lines in Fig. 2(a). The first forbidden gap occurred at ω_N from 0.25 to 0.4. Using Eq. (6), the phase index at the long-wavelength limit for the TM mode propagating in a dielectric PC was evaluated as 1.746, as listed in Table I.

For a TM mode propagating in the PC of magnetic rods (ε_rod = 10, μ_rod = 1.5), the dispersion relations are shown with solid lines in Fig. 2(a). It is clear that the ω_N -k_N curve of a certain band shifts upward for a photonic crystal consisted of magnetic rods. Thus, the forbidden frequencies become higher for a magnetic PC with respect to a dielectric PC. Meanwhile, the phase index for a TM mode propagating in the magnetic PC was definitely reduced. For example, the phase index at the long-wavelength limit decreased to be 1.567 in this case.

Figure 2(b) shows the photonic band structures for a TE mode propagating in PCs of dielectric rods (dashed lines) and magnetic rods (solid lines). In contrast to the clear shift in the ω_N -k_N curve for the TM mode, a slightly lower value was found for ω_N at a given k_N in the magnetic situation. Thus, a slightly larger phase index was achieved for a TE mode propagating in a magnetic PC. Here, we gave an example, with the long-wavelength-limit phase indices n_p listed in Table I for the TE mode. From the example, we can see that np increased from 1.145 to 1.174 when the dielectric rods (ε_rod = 15, μ_rod = 1) were replaced with magnetic rods (ε_rod = 10, μ_rod = 1.5).

Fig. 2. Photonic band structure of the photonic crystal made of triangular-arrayed magnetic (solid lines) or dielectric (dashed lines) rods for (a) TM and (b) TE modes. The magnetic rods were (10,1.5) for (ε_rod , μ_rod ), whereas dielectric rods were (15, 1). The ratio of the rod radius to the rod spacing a/2d was 0.2.

Table I. Phase index n_p at the long-wavelength limit (ω∼ 0) for TM- and TE-polarized light propagating photonic crystals consisting of triangular arrayed dielectric (ε_rod = 15, μ_rod = 1) or magnetic (ε_rod = 10, μ_rod = 1.5) rods in air.

Polarization	TM		TE
Rod material	Dielectric	Magnetic	Dielectric	Magnetic
n_p at ω∼ 0	1.746	1.567	1.145	1.174

Although the long-wavelength refractive index of guided modes propagating along PCs can be obtained from photonic band structures, the simulation processes for photonic band structures are usually complicated. Moreover, via the band structure calculation, it is difficult to realize why the magnetic susceptibility of rods manipulates the refractive index of guided modes. As a result, a method that is able to easily calculate long-wavelength refractive indices of guided modes and that is able to reflect clearly the impact of magnetism to long-wavelength refractive indices of guided modes is required. As can be seen from the results shown in Figs. 2(a) and (b), the magnetism of the rods is crucial to the refractive index for TM modes, the magnetism hardly modulates the refractive index of TE modes. Thus, we focused on developing an effective analytical method that met the demands of both situations for calculating the long-wavelength refractive indices of TM modes propagating along magnetic PCs. Via the effective resultant method, the effective dielectric constant ε_eff and the effective magnetic permeability μ_eff of magnetic PCs are calculated first. Next, the effective phase index n_p of long-wavelength TM modes propagating in magnetic PCs is found via

n_{p} = \sqrt{ε_{eff} μ_{eff}}

(7)

At the long-wavelength limit, the spatial variations in the electric field E and the magnetic field H over the rod spacing of a PC are negligible. Besides, for a TM mode, the electric field is along the axis of a rod and is tangential to the interface between the rod and interstitial air. Hereafter, we use E _∥ for the electric field of a TM mode, where the subscript ∥ refers to the tangential direction of the rod-air interface. According to the boundary condition, E _∥ is continuous at the rod-air interface. Therefore, E _∥ is almost constant over a unit cell of the ordered structure of a PC at the long-wavelength limit for a TM mode. Thus, the electric energy density W_E of a TM mode in a PC at the long-wavelength limit can be expressed as

W_{E} = \frac{1}{4} [(1 - f) ε_{o} ε_{air} E_{∥}^{2} + {fε}_{o} ε_{rod} E_{∥}^{2}] = \frac{1}{4} ε_{o} [(1 - f) ε_{air} + {fε}_{rod}] E_{∥}^{2}

(8)

where ε_air (= 1) is the dielectric constant of the interstitial air, and f denotes the filling factor of rods and equals to πa ²/(2√3d ²) = 0.145 here. Comparing Eq. (8) with

W_{E} = \frac{1}{4} ε_{0} ε_{e f f} E_{∥}^{2}

, the long-wavelength-limit effective dielectric constant ε _eff for a TM mode propagating in a PC is

ε_{eff} = (1 - f) ε_{air} + {f ε}_{rod}

(9)

We then estimated the effective magnetic permeability of a TM mode in a PC on the long-wavelength limit. The magnetic field H of a TM mode lies in a plane perpendicular to the rods. This field exhibits the normal H⊥ and the tangential H _∥ components with respect to the interface between a rod and interstitial air. At the long-wavelength limit, and according to the boundary condition, the H _∥ of a TM mode is almost constant over a unit cell of the ordered structure of a PC, whereas the H⊥ is discontinuous across the rod-air interface. Thus, the magnetic energy density W _H of a TM mode in a PC at the long-wavelength limit can be written as

W_{H} = \frac{1}{4} μ_{o} {[(1 - f) μ_{air} H_{air, ⊥}^{2} + {fμ}_{rod} H_{rod, ⊥}^{2} + [(1 - f) μ_{air} + {fμ}_{rod}] H_{∥}^{2}},

(10)

where μ_air (= 1) is the magnetic permeability of the interstitial air, and H_air ⊥ and H_rod ⊥ denote the normal components of magnetic fields in the rod and the air regions, respectively. By substituting H_rod ⊥ with μ_air H_air ⊥/μ_rod , Eq. (10) becomes

W_{H} = \frac{1}{4} μ_{o} {[(1 - f) μ_{air} + f \frac{1}{μ_{rod}}] H_{air, ⊥}^{2} + [(1 - f) μ_{air} + {fμ}_{rod}] H_{∥}^{2}}

= \frac{1}{4} μ_{o} [(1 - f) μ_{air} + \frac{f}{2} (\frac{μ_{air}^{2}}{μ_{rod}} + μ_{rod}) H^{2}],

(11)

where H ² _air,⊥ =H ²/2 and H ² _∥=H ²/2. Comparing Eq. (11) with W_H =¼μ_o μ_eff H ², the long-wavelength-limit effective magnetic permeability μ _eff for a TM mode propagating in a PC is

μ_{eff} = (1 - f) μ_{air} + \frac{f}{2} (\frac{μ_{air}^{2}}{μ_{rod}} + μ_{rod})

(12)

For an asymptotic case in which the PC consists completely of rod material, μ_air would be replaced with μ_rod in Eq. (12), and ε_air would be substituted with ε_rod in Eq. (9). Thus, ε_eff and μ_eff would become ε_rod and μ_rod respectively. When the PC is empty, then the equation would be modified similarly (i.e. f= 0, ε_eff and μ_eff in Eqs. (9) and (12) would be ε_air and μ_air ). Through the use of this method, reasonable results can be obtained for ε_eff and μ_eff of PCs in even these two asymptotic cases. In addition, for a PC consisting of dielectric rods with (ε_rod , μ_rod ) being (15, 1), the (ε_eff , μ_eff ) of a propagating TM mode would be evaluated via Eqs. (9) and (12) to be (3.030, 1), as listed in Table II. This would lead to a value of 1.741 for the effective phase index n_p via Eq. (7). For a magnetic PC having (ε_rod , μ_rod ) as (10, 1.5), the (ε_eff , μ_eff ) obtained would be (2.305, 1.012), and in turn, the n_p would be 1.527. These two n_p ’s obtained via the effective method are very close to those calculated from the PC band structures, as listed in Table I. This agreement demonstrates the validity of using the effective method to find the effective refractive index of long-wavelength TM modes along magnetic and dielectric PCs.

Table II. Effective dielectric constant ε_eff , magnetic permeability μ_eff , and phase index n_p for long-wavelength TM mode propagating along dielectric PC (ε_rod = 15, μ_rod = 1) and magnetic PC (ε_rod = 10, μ_rod = 1.5) having a filling factor of 0.145. These values are calculated using the effective method.

Rod material	ε_eff	μ_eff	n_eff
Dielectric rods	3.030	1	1.741
Magnetic rods	2.305	1.012	1.527

Notably, by using this effective method, not only the effective refractive index but also the effective dielectric constant ε_eff and the magnetic permeability μ_eff of PCs are available for long-wavelength TM modes. Since the dielectric constant and the magnetic permeability,

Fig. 3. Effective (a) dielectric constant ε_eff (b) magnetic permeability μ_eff , and (c) refractive index n_p of the PC, shown in Fig. 1, as functions of rod magnetic permeability μ_rod for various filling factors f. The rod dielectric constant ε_rod correspondingly decreases from 15 to 7.5 when μ_rod increases from 1 to 1.5, to keep the product of ε_rod and μ_rod constant (= 15 here)

instead of refractive index, are fundamentally optical parameters for propagating electromagnetic waves along a medium, the feasibility of finding ε_eff and μ_eff makes it possible to investigate the physical mechanism of μ_rod -modulated n_p of magnetic PCs. It is worth noting that the validity of the effective resultant method is not limited to cases where (ε_rod , μ_eff ) are (15, 1) and (10, 1.5), but that it is also applicable to scenarios using positive ε_rod and μ_rod .

As (ε_rod , μ_rod ) changes from (15, 1) to (10, 1.5), ε_eff decreases by 23.9 %, which is defined as [ε_eff (ε_rod =15) - ε_eff (ε_rod =10)]/ε_eff (ε_rod =15), and μ_eff increases only by 1.2 %, defined as [μ_eff (μ_rod =1.5) - μ_eff (μ_rod =1)]/μ_eff (μ_rod =1.5), for long-wavelength TM modes. The decrease in ε_eff is much larger than the increase in μ_eff . Through a careful inspection show in Eq. (12), it can be seen that when the μ_rod increases from 1 to 1.5, the μ_eff becomes larger because of the term f μ_rod /2, whereas it is reduced by the term fμ² _air/2μ_rod. But according to Eq. (9), the ε_eff is linearly reduced with decreasing ε_rod . Hence, an announced decrease in the ε_eff results and dominates the reduction in the effective phase index n_p . This also accounts for the upward shift in the ω_N -k_n curve for the TM modes shown in Fig. 2(a).

It is obvious that variations in ε_eff , μ_eff , and n_p with varying ε_rod and μ_rod , depend on the filling factor f Figures 3(a)–(c) plot the dependencies of ε_eff , μ_eff , and n_p with μ_rod increasing from 1 to 2 for f being 0.073, 0.145, and 0.290. The ε_rod correspondingly decreases from 15 to 7.5 when μ_rod increases from 1 to 1.5, to keep the product of ε_rod and μ_rod constant (= 15 here). For a given f both ε_eff and n_p decrease as μ_rod increases, while μ_eff becomes larger. As f increases, each of the ε_eff -μ_rod , μ_eff -μ_rod and n_p -μ_rod curves in Figs. 3(a) – (b) moves upwards. Additionally, the variations in ε_eff , μ_eff , and n_p due to increasing μ_rod are enhanced at a larger f. This implies that the contribution of optical properties of rods to the PC is enhanced as the rods occupy more space of PC.

4. Conclusion

Magnetic susceptibility of PCs plays a role in the photonic properties of guided TM modes. Through the effective medium method, analytic expressions of effective dielectric constant and effective magnetic permeability of PCs at long-wavelength limit are obtained for TM modes. These analytic expressions provide quantitative variations in effective dielectric constant and effective magnetic permeability of PCs caused by the magnetic susceptibility of PCs. According to these variations, the physical mechanism relevant to the effect of magnetic permeability of PCs on TM-mode effective refractive index is clarified.

Acknowledgments

I would like to thank Prof. C.T. Chang of San Diego State 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

1.	T. Yoshie, A. Scherer, H. Chen, D. Huffaker, and D. Deppe, “Optical characterization of two-dimensional photonic crystal cavities with indium arsenide quantum dot emitters,” Appl. Phys. Lett. 79, 114 (2001). [CrossRef]
2.	H.Y. Ryu, M. Notomi, E. Kuramoti, and T. Segawa, “Large spontaneous emission factor (>0.1) in the photonic crystal monopole-mode laser,” Appl. Phys. Lett. 84, 1067 (2004). [CrossRef]
3.	S.Y. Yang, H.E. Horng, Y.T. Shiao, C.-Y. Hong, and H.C. Yang, “Photonic-crystal Resonant Effect Using Self-assembly Ordered Structures in Magnetic Fluid Films under External Magnetic Fields,” J. Magn. Magn. Mater. 307, 43 (2006). [CrossRef]
4.	E. Chow, S.Y. Lin, J. R. Wendt, S.G. Johnson, and J.D. Joannopoulos, “Quantitative analysis of bending efficiency in photonic-crystal waveguide bends at λ = 1.55 μm wavelengths,” Opt. Lett. 26, 286 (2001). [CrossRef]
5.	M. Koshiba, “Wavelength Division Multiplexing and Demultiplexing With Photonic Crystal Waveguide Couplers,” J. Lightwave Technol. 19, 1970 (2001). [CrossRef]
6.	T. Matsumoto and T. Baba, “Photonic Crystal mmb k-Vector Superprism,” J. Lightwave Technol. 22, 917 (2004). [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, 83105 (2006). [CrossRef]
8.	P. Halevi, A.A. Krokhin, and J. Arriaga, “Photonic Crystal Optics and Homogenization of 2D Periodic Composites,” Phys. Rev. Lett. 82, 719 (1999). [CrossRef]
9.	S.Y. Yang and C.T. Chang, “Chromatic dispersion compensators via highly dispersive photonic crystals,” J. Appl. Phys. 98, 23108 (2005). [CrossRef]
10.	S.Y. Lin, V.M. Hietala, L. Wang, and E.D. Jones, “Highly dispersive photonic band-gap prism,” Opt. Lett. 21, 1771 (1996). [CrossRef] [PubMed]
11.	C. Luo, M. Soljačic′, and J.D. Joannopoulos, “Superprism effect based on phase velocities,” Opt. Lett. 29, 745 (2004). [CrossRef] [PubMed]
12.	S. Foteinopoulou and CM. Soukoulis, “Negative refraction and left-handed behavior in two-dimensional photonic crystals,” Phys. Rev. B 67, 235107 (2003). [CrossRef]
13.	S.Y. Yang, Chin-Yih Hong, and H.C. Yang, “Focusing concave lens using photonic crystals involving magnetic materials,” J. Opt. Soc. Am. A 23, 956 (2006). [CrossRef]
14.	S. O’Brien and John B. Pendry, “Photonic band-gap effects and magnetic activity in dielectric composites,” J. Phys.: Condens. Matter 14, 4035 (2002). [CrossRef]
15.	V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges,” J. Phys. D: Condens. Matter 17, 3717 (2005). [CrossRef]
16.	Y. Saado, M. Golosovsky, D. Davidov, and A. Frenkel, “Tunable photonic band gap in self-assembled clusters of floating magnetic particles,” Phys. Rev. B 66, 195108 (2002). [CrossRef]
17.	A. Figotin and I. Vitebskiy, “Electromagnetic unidirectionality in magnetic photonic crystals,” Phys. Rev. B 67, 165210 (2003). [CrossRef]
18.	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 (2004). [CrossRef]
19.	S.Y. Yang, C.-Y. Hong, I. Drikis, H.E. Horng, and H.C. Yang, “Resonant electromagnetism in photonic crystals composed of triangular-arrayed rods with both dielectric constant and magnetic permeability functions,” J. Opt. Soc. Am. B 21, 413 (2004). [CrossRef]
20.	C.-Y. Hong, S.Y. Yang, H.E. Horng, and H.C. Yang, “Slab-thickness dependent band gap size of two-dimensional photonic crystals with triangular-arrayed dielectric or magnetic rods,” J. Appl. Phys. 94, 2188 (2003). [CrossRef]
21.	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 (1993). [CrossRef]
22.	S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell′s equations in a planewave basis,” Opt. Express 8, 173 (2001). [CrossRef] [PubMed]

OCIS Codes
(000.3860) General : Mathematical methods in physics
(230.3990) Optical devices : Micro-optical devices

ToC Category:
Photonic Crystals

History
Original Manuscript: January 4, 2007
Revised Manuscript: February 13, 2007
Manuscript Accepted: February 13, 2007
Published: March 5, 2007

Citation
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)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-5-2669

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References

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