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

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  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]

2007 (1)

2006 (1)

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]

2005 (1)

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

2004 (2)

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

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]

2003 (1)

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]

2001 (1)

1999 (1)

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]

1993 (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]

Alerhand, O. L.

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]

Baba, T.

Brommer, K. D.

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]

Chang, C. T.

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 and C. T. Chang, "Chromatic dispersion compensators via highly dispersive photonic crystals," J. Appl. Phys. 98, 23108-23111 (2005).
[CrossRef]

Drikis, I.

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]

Encinas, A.

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]

Ferain, E.

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]

Hong, C.-Y.

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]

Horng, H. E.

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]

Huynen, I.

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]

Joannopoulos, J. D.

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]

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]

Johnson, S. G.

Kawakami, S.

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]

Kawashima, T.

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]

Kosaka, H.

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]

Legras, R.

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]

Matsumoto, T.

Meade, R. D.

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]

Notomi, M.

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]

Piraux, L.

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]

Rappe, A. M.

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]

Sailb, A.

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]

Sato, T.

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]

Tamamura, T.

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]

Tomita, A.

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]

Vanhoenacker-Janvier, D.

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]

Yang, H. C.

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]

Yang, S. Y.

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]

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 and C. T. Chang, "Chromatic dispersion compensators via highly dispersive photonic crystals," J. Appl. Phys. 98, 23108-23111 (2005).
[CrossRef]

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]

Appl. Phys. Lett. (2)

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]

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]

J. Appl. Phys. (3)

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]

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]

J. Lightwave Technol. (1)

Opt. Express (2)

Phys. Rev. B (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]

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Figures (6)

Fig. 1.
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.

Fig. 2.
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).

Fig. 3.
Fig. 3.

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

Fig. 4.
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.

Fig. 5.
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.
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.

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

n p = k N ω N
ε eff = ( 1 f ) ε air + f ε rod ,
μ eff = ( 1 f ) μ air + f 2 ( μ air 2 μ rod + μ rod )
n eff = ε eff μ eff
r = ε eff μ eff cos θ r + cos θ i ε eff μ eff cos θ r + cos θ i ,
Z = Z PC Z air = μ eff ε eff cos θ i cos θ r
sin θ i , Z = 1 = μ eff ( μ eff ε eff ) μ eff 2 1
n rod 2 μ rod 2 n rod 2 1 , or
1 ε rod n rod 4 2 n rod 2 1
t = 2 cos θ i cos θ i + ( ε eff μ eff ) 1 2 cos θ r

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