Abstract

We consider light propagation in a finite photonic crystal. The transmission and reflection from a one-dimensional system are described in an effective medium theory, which reproduces exactly the results of transfer matrix calculations.We derive simple formulas for the reflection from a semi-infinite crystal, the local density of states in absorbing crystals, and discuss defect modes and negative refraction.

© 2003 Optical Society of America

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References

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  1. E. Merzbacher, Quantum Mechanics, 3rd ed. (Wiley & Sons, New York, 1998).
  2. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, Cambridge, 1999).
  3. D. W. L. Sprung, H. Wu, and J. Martorell, �??Scattering by a finite periodic potential,�?? Am. J. Phys. 61, 1118 (1993).
    [CrossRef]
  4. K. Sakoda, K. Ohtaka, and T. Ueta, �??Low-threshold laser oscillation due to group-velocity anomaly peculiar to two-and three-dimensional photonic crystals,�?? Opt. Express 4, 481 (1999), <a href=" http://www.opticsexpress.org/abstract.cfm?URI=OPEX-4-12-481.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-4-12-481</a>.
    [CrossRef] [PubMed]
  5. D. Y. Jeong, Y. H. Ye, and Q. M. Zhang, �??Effective optical properties associated with wave propagation in photonic crystals of finite length along the propagation direction,�?? J. Appl. Phys. 92, 4194 (2002).
    [CrossRef]
  6. P. Lambropoulos, G. M. Nikolopoulos, T. R. Nielsen, and S. Bay, �??Fundamental quantum optics in structured reservoirs,�?? Rep. Prog. Phys. 63, 455 (2000).
    [CrossRef]
  7. J. M. Bendickson, J. P. Dowling, and M. Scalora, �??Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,�?? Phys. Rev. E 53, 4107 (1996).
    [CrossRef]
  8. A. Moroz, �??Minima and maxima of the local density of states for one-dimensional periodic systems,�?? Europhys. Lett. 46, 419 (1999).
    [CrossRef]
  9. M. Wubs and A. Lagendijk, �??Local optical density of states in finite crystals of plane scatterers,�?? Phys. Rev. E 65, 046612 (2002).
    [CrossRef]
  10. A. Tip, A. Moroz, and J. M. Combes, �??Band structure of absorptive photonic crystals,�?? J. Phys. A: Math. Gen. 33, 6223 (2000).
    [CrossRef]
  11. J. B. Pendry, �??Photonic band structures,�?? J. Mod. Optics 41, 209 (1994).
    [CrossRef]
  12. S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, �??Refraction at media with negative refractive index,�?? Phys. Rev. Lett. 90, 107402 (2003).
    [CrossRef] [PubMed]

Am. J. Phys. (1)

D. W. L. Sprung, H. Wu, and J. Martorell, �??Scattering by a finite periodic potential,�?? Am. J. Phys. 61, 1118 (1993).
[CrossRef]

Europhys. Lett. (1)

A. Moroz, �??Minima and maxima of the local density of states for one-dimensional periodic systems,�?? Europhys. Lett. 46, 419 (1999).
[CrossRef]

J. Appl. Phys. (1)

D. Y. Jeong, Y. H. Ye, and Q. M. Zhang, �??Effective optical properties associated with wave propagation in photonic crystals of finite length along the propagation direction,�?? J. Appl. Phys. 92, 4194 (2002).
[CrossRef]

J. Mod. Optics (1)

J. B. Pendry, �??Photonic band structures,�?? J. Mod. Optics 41, 209 (1994).
[CrossRef]

J. Phys. A: Math. Gen. (1)

A. Tip, A. Moroz, and J. M. Combes, �??Band structure of absorptive photonic crystals,�?? J. Phys. A: Math. Gen. 33, 6223 (2000).
[CrossRef]

Opt. Express (1)

Phys. Rev. E (2)

M. Wubs and A. Lagendijk, �??Local optical density of states in finite crystals of plane scatterers,�?? Phys. Rev. E 65, 046612 (2002).
[CrossRef]

J. M. Bendickson, J. P. Dowling, and M. Scalora, �??Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,�?? Phys. Rev. E 53, 4107 (1996).
[CrossRef]

Phys. Rev. Lett. (1)

S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, �??Refraction at media with negative refractive index,�?? Phys. Rev. Lett. 90, 107402 (2003).
[CrossRef] [PubMed]

Rep. Prog. Phys. (1)

P. Lambropoulos, G. M. Nikolopoulos, T. R. Nielsen, and S. Bay, �??Fundamental quantum optics in structured reservoirs,�?? Rep. Prog. Phys. 63, 455 (2000).
[CrossRef]

Other (2)

E. Merzbacher, Quantum Mechanics, 3rd ed. (Wiley & Sons, New York, 1998).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, Cambridge, 1999).

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Figures (4)

Fig. 1.
Fig. 1.

(Left panel) complex eigenvalue λ+=eika vs. frequency for the Kronig-Penney model, Eq. (9), with point scatterers of polarizability α=(0.2+0i)a. (Right panel) band structure ω(k) of the infinite crystal. The thick dispersion curves correspond to the physical Bloch momentum fixed by the requirements of causality and energy conservation. Even bands exhibit negative refraction (k<0).

Fig. 2.
Fig. 2.

(Left) reflectance |rN | for the Kronig-Penney model with α=(0.2+0i)a, N=4 (blue line). The envelope, Eq. (6), is shown as well. Solid green line: half-space approximation |r|, Eq. (11). Dashed dark green line: |r| for absorbing scatterers (α=(0.2+0.02i)a). (Right) reflection coefficient |r i| for Bloch waves reflected from the end face of a semi-infinite crystal. Solid line: our result, Eq. (13); dashed line: proposed by Sakoda [4]. Kronig-Penney model with α=(0.2+0i)a.

Fig. 3.
Fig. 3.

LDOS, Eq. (15), in an infinite crystal with scatterers at x=…,-a/2,a/2,… (Left) no absorption (α=0.2a). (Right) nonzero absorption (α=(0.2+0.05i)a).

Fig. 4.
Fig. 4.

(Left) reflectance from two N=6 layer crystals (α=0.2a) with a defect in between (dR =dL =a/2). Green line with peak: active defect with scattering strength α d/a≈-0.62-0.04i, given by Eq. (16) for ωa/2πc≈0.462. The same structure with a passive defect (α d≈-0.62a) gives the green line with the transmission dip. Blue line: reflectance for α d=α. (Right) defect frequency in the first band gap vs. real part of scattering strength α d.

Equations (16)

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E n ( x ) = a n e i ω ( x na ) + b n e i ω ( x na ) ,
( a n + 1 b n + 1 ) = T ( a n b n ) .
T N = M ( e ikaN 0 0 e ikaN ) M 1 , M = ( N + N N + c + N c ) .
c ± = e ± ika T 11 T 12 = T 21 e ± ika T 22 .
r N = c + c ( e ikaN e ikaN ) c + e ikaN c e ikaN .
0 r N 2 c + c c + + c ,
d 2 E ( x ) d x 2 + ω 2 ( 1 + n = 1 N α n δ ( x ( n 1 2 ) a ) ) E ( x ) = 0
T = ( ( 1 + i 2 ω α ) e i ω a i 2 ω α i 2 ω α ( 1 i 2 ω α ) e i ω a )
λ ± = cos ( ω a ) ω α 2 sin ( ω a ) ± i sin ( ω a ) 1 + ω α cot ( ω a ) 1 4 ω 2 α 2
( t 0 ) = M 1 ( 1 r ) .
r = c + = e iωa e ika e iωa e ika 1 ,
( r i 1 ) = M 1 ( 0 t )
r i = N N + = N + c + N c = c + .
r N , FP = r + t r i t e 2 ikaN 1 r 1 2 e 2 ikaN ,
ρ ( x , ω ) = Re ( r L ( ω ) e 2 i d L ω + r R ( ω ) e 2 i d R ω + 2 1 r L ( ω ) r R ( ω ) e 2 i ( d L + d R ) ω 1 ) , d L = d R = x .
i ω α d = r R ( ω ) e 2 i ω d R 1 r R ( ω ) e 2 i ω d R + 1 + r L ( ω ) e 2 i ω d L 1 r L ( ω ) e 2 i ω d L + 1 ,

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