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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 12 — Jun. 11, 2007
  • pp: 7653–7659
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Impedance match of long-wavelength electromagnetic waves incident into magnetic photonic crystals

S. Y. Yang  »View Author Affiliations


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

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

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

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]

]. In contrast, for magnetic PCs, the master equation for the magnetic flux density 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]

]. 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 [4

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

]

Fig. 1. Scheme for illustrating the incident angle θi, the refractive angle θr for TM mode incident into a two-dimensional photonic crystal consisting of triangularly-arrayed infinitely long rods surrounded by air. The r and t denote the reflection and the transmission of the electric field E of an incident light at the incident interface (guided by the dashed line). The εeff, μeff, and neff denote the effective dielectric constant, magnetic permeability, and refractive index of a photonic crystal at long-wavelength limits.
np=kNωN
(1)

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

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

,6

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]

], dispersion compensation [6

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

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]

], etc. can be theoretically investigated using the relationship between the phase index and frequency of EM waves.

2. Simulated system and simulation formulas

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 [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]

]

εeff=(1f)εair+fεrod,
(1)
μeff=(1f)μair+f2(μair2μrod+μrod)
(2)
neff=εeffμeff
(3)

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 [10

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

]. 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

r=εeffμeffcosθr+cosθiεeffμeffcosθr+cosθi,
(4)

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.

Z=ZPCZair=μeffεeffcosθicosθr
(5)

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

sinθi,Z=1=μeff(μeffεeff)μeff21
(6)
Fig. 2. Phase diagram for the impedance-non-match region (shadowed) and impedance-match region (downright region) separated by the criterion curve (dashed line) in the coordinates of (εrod, μrod). The criterion curve is obtained from Eq. (7a) or (7b).

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 μrods. 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

nrod2μrod2nrod21,or
(7a)
1εrodnrod42nrod21
(7b)

where nrod=εrodμrod . 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 nrods, 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.

Fig. 3. Impedance-match incident angle θ i,Z versus the magnetic permeability μrod of rods for various nrods. For a given nrod, εrod correspondingly decreases with increasing μrod. Here, the filling factor f of rods in PC is selected as 0.145.

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 μrods 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=1s, 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.

Fig. 4. Impedance-match incident angle θ i,Z=1 as functions of the magnetic permeability μrod of rods for various filling factors f. Here the nrod used is 151/2, and εrod correspondingly decreases with increasing μrod for a given f.

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 μrods 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

t=2cosθicosθi+(εeffμeff)12cosθr
(8)

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 θis corresponding to smaller Zs. However, for any curve in Fig. 5(c), the transmission is close to 1 at θs 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 μrods. 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 μrods. 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.

Fig. 5. Incident angle θi dependent (a) impedance ratio Z, (b) reflection r, and (t) transmission t of the electric field of a TM mode into a PC. Here the nrod used is 101/2, the filling factor f is 0.145. The εrod correspondingly decreases with increasing μrod.
Fig. 6. μrod dependent permitted range Δθi,-3dB of the incident angle to have a high transmission for a TM mode through the air-PC interface. Here the nrod used is 101/2, and εrod correspondingly decreases with increasing μrod for a given filling factor f The gray area is for such case as f being 0.145, while the dotted area is for the filling factor of 0.9.

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.

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 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]

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]

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]

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]

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. 83, 2378–2380 (2003). [CrossRef]

10.

P. Lalanne, “Effective medium theory applied to photonic crystals composed of cubic or square cylinders,” Appl. Opt. 35, 5369–5380 (1996). [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: 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

  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]
  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]
  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]
  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]
  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. 83, 2378-2380 (2003). [CrossRef]

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