Abstract

The dependence of the lasing threshold on the amount of positional disorder in photonic crystal structures is newly studied by means of the finite element method, not of the finite difference time domain method usually used. A two-dimensional model of a photonic crystal consisting of dielectric cylinders arranged on a triangular lattice within a circular region is considered. The cylinders are assumed to be homogeneous and infinitely long. Positional disorder of the cylinders is introduced to the photonic crystals. Optically active medium is introduced to the interspace among the cylinders. The population inversion density of the optically active medium is modeled by the negative imaginary part of dielectric constant. The ratio between radiative power of electromagnetic field without amplification and that with amplification is computed as a function of the frequency and the imaginary part of the dielectric constant, and the threshold of the imaginary part, namely population inversion density for laser action is obtained. These analyses are carried out for various amounts of disorder. The variation of the lasing threshold from photonic-crystal laser to random laser is revealed by systematic computations with numerical method of reliable accuracy for the first time. Moreover, a novel phenomenon, that the lasing threshold have a minimum against the amount of disorder, is found. In order to investigate the properties of the lasing states within the circular system, the distributions of the electric field amplitudes of the states are also calculated.

© 2012 OSA

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  36. R. Ferrini, D. Leuenberger, R. Houdré, H. Benisty, M. Kamp, and A. Forchel, “Disorder-induced losses in planar photonic crystals,” Opt. Lett. 31, 1426–1428 (2006).
    [CrossRef] [PubMed]
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  42. X. Wang and K. Kempa, “Effects of disorder on subwavelength lensing in two-dimensional photonic crystal slabs,” Phys. Rev. B 71, 085101 (2005).
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  43. T. N. Langtry, A. A. Asatryan, and L. C. Botten, “Effects of disorder in two-dimensional photonic crystal waveguides,” Phys. Rev. E 68, 026611 (2003).
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  44. V. N. Astratov, J. P. Franchak, and S. P. Ashili, “Optical coupling and transport phenomena in chains of spherical dielectric microresonators with size disorder,” Appl. Phys. Lett. 85, 5508–5510 (2004).
    [CrossRef]
  45. K. C. Kwan, X. Zhang, Z. Q. Zhang, and C. T. Chan, “Effects due to disorder on photonic crystal-based waveguides,” Appl. Phys. Lett. 82, 4414–4416 (2003).
    [CrossRef]
  46. D. P. Fuell, S. Hughes, and M. M. Dignam, “Effect of disorder strength on the fracture pattern in heterogeneous networks,” Phys. Rev. B 76, 144201 (2008).
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    [CrossRef]
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  49. J. Topolancik and F. Vollmer, “Random high-q cavities in disordered photonic crystal waveguides,” Appl. Phys. Lett. 91, 201102 (2007).
    [CrossRef]
  50. T. A. Leskova, A. A. Maradudin, I. V. Novikov, A. V. Schegrov, and E. R. Méndez, “Design of one-dimensional band-limited uniform diffusers of light,” Appl. Phys. Lett. 73, 1943–1945 (1998).
    [CrossRef]
  51. E. R. Méndez, E. E. García, T. A. Leskova, A. A. Maradudin, J. Muñoz-Lopez, and I. Simonsen, “Design of one-dimensional random surfaces with specified scattering properties,” Appl. Phys. Lett. 81, 798–800 (2002).
    [CrossRef]
  52. E. R. Méndez, T. A. Leskova, A. A. Maradudin, and J. Muñoz-Lopez, “Design of two-dimensional random surfaces with specified scattering properties,” Opt. Lett. 29, 2917–2919 (2004).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  55. K. C. Kwan, X. M. Tao, and G. D. Peng, “Transition of lasing modes in disordered active photonic crystals,” Opt. Lett. 32, 2720–2722 (2007).
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  57. A. Bermúdez, L. Hervella-Nieto, A. Prieto, and R. Rodríguez, “An exact bounded pml for the helmholtz equation,” C. R. Acad, Sci. Paris, Ser. I 339, 803–808 (2004).
    [CrossRef]
  58. A. Bermúdez, L. Hervella-Nieto, A. Prieto, and R. Rodríguez, “Numerical simulation of time-harmonic scattering problems with an optimal PML,” Var. Formul. Mech.:Theory Appl. 13, 58–71 (2006).
  59. A. Bermúdez, L. Hervella-Nieto, A. Prieto, and R. Rodríguez, “An optimal perfectly matched layer with unbounded absorbing function for time-harmonic acoustic scattering problems,” J. Comput. Phys. 223, 469–488 (2007).
    [CrossRef]
  60. A. C. Cangellaris and D. B. Wright, “Analysis of the numerical error caused by the stair-stepped approximation of a conducting boundary in FDTD simulations of electromagnetic phenomena,” IEEE Trans. Antennas Propag. 39, 1518–1525 (1991).
    [CrossRef]
  61. A. Akyurtlu, D. H. Werner, V. Veremey, D. J. Steich, and K. Aydin, “Staircasing errors in FDTD at an air-dielectric interface,” IEEE Microwave Guided Wave Lett. 9, 444–446 (1999).
    [CrossRef]
  62. K. H. Dridi, J. S. Hesthaven, and A. Ditkowski, “Staircase-free finite-difference time-domain formulation for general materials in complex geometries,” IEEE Trans. Antennas Propag. 49, 749–756 (2001).
    [CrossRef]

2012

G. Fujii, T. Matsumoto, T. Takahashi, and T. Ueta, “A study on the effect of filling factor for laser action in dielectric random media,” (2012). Appl. Phys. A DOI: .
[CrossRef]

2010

G. Fujii, T. Matsumoto, T. Takahashi, and T. Ueta, “A study on optical properties of photonic crystals consisting of hollow rods,” IOP Conf. Ser.: Mater. Sci. Eng. 10, 012072 (2010).
[CrossRef]

G. Fujii, T. Matsumoto, T. Takahashi, and T. Ueta, “Finite element analysis for laser oscillation in random system consisting of heterogeneous dielectric materials,” Trans. Jpn. Soc. Comput. Methods Eng. 10, 117–122 (2010).

2009

C. Vanneste and P. Sebbah, “Complexity of two-dimensional quasimodes at the transition from weak scattering to anderson localization,” Phys. Rev. A 79, 041802 (2009).
[CrossRef]

M. Patterson, S. Hughes, S. Combrié, N.-V.-Q. Tran, A. D. Rossi, R. Gabet, and Y. Jaouën, “Disorder-induced coherent scattering in slow-light photonic crystal waveguides,” Phys. Rev. Lett. 102, 253903 (2009).
[CrossRef] [PubMed]

2008

D. P. Fuell, S. Hughes, and M. M. Dignam, “Effect of disorder strength on the fracture pattern in heterogeneous networks,” Phys. Rev. B 76, 144201 (2008).

D. S. Wiersma, “The physics and applications of random lasers,” Nat. Phys. 4, 359–367 (2008).
[CrossRef]

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef] [PubMed]

2007

T. Prasad, V. L. Colvin, and D. M. Mittleman, “The effect of structural disorder on guided resonances in photonic crystal slabs studied with terahertz time-domain spectroscopy,” Opt. Express 15, 16954–16965 (2007).
[CrossRef] [PubMed]

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007).
[CrossRef] [PubMed]

C. Vanneste, P. Sebbah, and H. Cao, “Lasing with resonant feedback in weakly scattering random systems,” Phys. Rev. Lett. 98, 143902 (2007).
[CrossRef] [PubMed]

J. Topolancik and F. Vollmer, “Random high-q cavities in disordered photonic crystal waveguides,” Appl. Phys. Lett. 91, 201102 (2007).
[CrossRef]

L. O. Faolain, T. P. White, D. O. Brien, X. Yuan, M. D. Settle, and T. F. Krauss, “Dependence of extrinsic loss on group velocity in photonic crystal waveguides,” Opt. Express 15, 13129–13138 (2007).
[CrossRef]

K. C. Kwan, X. M. Tao, and G. D. Peng, “Transition of lasing modes in disordered active photonic crystals,” Opt. Lett. 32, 2720–2722 (2007).
[CrossRef] [PubMed]

A. Bermúdez, L. Hervella-Nieto, A. Prieto, and R. Rodríguez, “An optimal perfectly matched layer with unbounded absorbing function for time-harmonic acoustic scattering problems,” J. Comput. Phys. 223, 469–488 (2007).
[CrossRef]

2006

A. Bermúdez, L. Hervella-Nieto, A. Prieto, and R. Rodríguez, “Numerical simulation of time-harmonic scattering problems with an optimal PML,” Var. Formul. Mech.:Theory Appl. 13, 58–71 (2006).

R. Ferrini, D. Leuenberger, R. Houdré, H. Benisty, M. Kamp, and A. Forchel, “Disorder-induced losses in planar photonic crystals,” Opt. Lett. 31, 1426–1428 (2006).
[CrossRef] [PubMed]

P. Sebbah, B. Hu, J. K. Klosner, and A. Z. Genack, “Extended quasimodes within nominally localized random waveguides,” Phys. Rev. Lett. 96, 183902 (2006).
[CrossRef] [PubMed]

2005

C. Vanneste and P. Sebbah, “Localized modes in random arrays of cylinders,” Phys. Rev. E 71, 026612 (2005).
[CrossRef]

A. Rodriguez, M. Ibanescu, J. D. Joannopoulos, and S. G. Johnson, “Disorder-immune confinement of light in photonic-crystal cavities,” Opt. Lett. 30, 3192–3194 (2005).
[CrossRef] [PubMed]

X. Wang and K. Kempa, “Effects of disorder on subwavelength lensing in two-dimensional photonic crystal slabs,” Phys. Rev. B 71, 085101 (2005).
[CrossRef]

D. Gerace and L. C. Andreani, “Effects of disorder on propagation losses and cavity q-factors in photonic crystal slabs,” Photon. Nanostruct. Fundam. Appl. 3, 120–128 (2005).
[CrossRef]

2004

V. N. Astratov, J. P. Franchak, and S. P. Ashili, “Optical coupling and transport phenomena in chains of spherical dielectric microresonators with size disorder,” Appl. Phys. Lett. 85, 5508–5510 (2004).
[CrossRef]

W. R. Frei and H. T. Johnson, “Finite-element analysis of disorder effects in photonic crystals,” Phys. Rev. B 70, 165116 (2004).
[CrossRef]

A. Bermúdez, L. Hervella-Nieto, A. Prieto, and R. Rodríguez, “An exact bounded pml for the helmholtz equation,” C. R. Acad, Sci. Paris, Ser. I 339, 803–808 (2004).
[CrossRef]

E. R. Méndez, T. A. Leskova, A. A. Maradudin, and J. Muñoz-Lopez, “Design of two-dimensional random surfaces with specified scattering properties,” Opt. Lett. 29, 2917–2919 (2004).
[CrossRef]

S. Mujumdar, M. Ricci, R. Torre, and D. S. Wiersma, “Amplified extended modes in random lasers,” Phys. Rev. Lett. 93, 053903 (2004).
[CrossRef] [PubMed]

2003

T. N. Langtry, A. A. Asatryan, and L. C. Botten, “Effects of disorder in two-dimensional photonic crystal waveguides,” Phys. Rev. E 68, 026611 (2003).
[CrossRef]

K. C. Kwan, X. Zhang, Z. Q. Zhang, and C. T. Chan, “Effects due to disorder on photonic crystal-based waveguides,” Appl. Phys. Lett. 82, 4414–4416 (2003).
[CrossRef]

2002

E. R. Méndez, E. E. García, T. A. Leskova, A. A. Maradudin, J. Muñoz-Lopez, and I. Simonsen, “Design of one-dimensional random surfaces with specified scattering properties,” Appl. Phys. Lett. 81, 798–800 (2002).
[CrossRef]

M. A. Kaliteevski, J. M. Martinez, D. Cassagne, and J. P. Albert, “Disorder-induced modification of the transmission of light in a two-dimensional photonic crystal,” Phys. Rev. B 66, 113101 (2002).
[CrossRef]

P. Sebbah and C. Vanneste, “Random laser in the localized regime,” Phys. Rev. B 66, 144202 (2002).
[CrossRef]

2001

C. Vanneste and P. Sebbah, “Selective excitation of localized modes in active random media,” Phys. Rev. Lett. 87, 183903 (2001).
[CrossRef]

K. H. Dridi, J. S. Hesthaven, and A. Ditkowski, “Staircase-free finite-difference time-domain formulation for general materials in complex geometries,” IEEE Trans. Antennas Propag. 49, 749–756 (2001).
[CrossRef]

2000

A. A. Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, and C. M. de Sterke, “Effects of geometric and refractive index disorder on wave propagation in two-dimensional photonic crystals,” Phys. Rev. E 62, 5711–5720 (2000).
[CrossRef]

H. Li, H. Chen, and X. Qiu, “Band-gap extension of disordered 1d binary photonic crystals,” Physica B 279, 164–167 (2000).
[CrossRef]

P. Sebbah, R. Pnini, and A. Z. Genack, “Field and intensity correlation in random media,” Phys. Rev. E 62, 7348–7352 (2000).
[CrossRef]

H. Cao, J. Y. Xu, D. Z. Zhang, S. H. Chang, S. T. Ho, E. W. Seeling, X. Liu, and R. P. H. Chang, “Spatial confinement of laser light in active random media,” Phys. Rev. Lett. 84, 5584–5587 (2000).
[CrossRef] [PubMed]

H. Cao, J. Y. Xu, S. H. Chang, and S. T. Ho, “Transition from amplified spontaneous emission to laser action in strongly scattering media,” Phys. Rev. E 61, 1985–1989 (2000).
[CrossRef]

Z.-Y. Li and Z.-Q. Zhang, “Fragility of photonic band gaps in inverse-opal photonic crystals,” Phys. Rev. B 62, 1516–1519 (2000).
[CrossRef]

E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Gap deformation and classical wave localization in disordered two-dimensional photonic-band-gap materials,” Phys. Rev. B 61, 13458–13464 (2000).
[CrossRef]

1999

K. Ohtaka, “Density of states of slab photonic crystals and the laser oscillation in photonic crystals,” J. Lightwave Technol. 17, 2161–2169 (1999).
[CrossRef]

K. Sakoda, “Enhanced light amplification due to group-velocity anomaly peculiar to two- and three-dimensional photonic crystals,” Opt. Express 4, 167–176 (1999).
[CrossRef] [PubMed]

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, 481–489 (1999).
[CrossRef] [PubMed]

Y. A. Vlasov, M. A. Kaliteevski, and V. V. Nikolaev, “Different regimes of light localization in disordered photonic crystal,” Phys. Rev. B 60, 1555–1562 (1999).
[CrossRef]

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82, 2278–2281 (1999).
[CrossRef]

A. Golshani, H. Pier, E. Kapon, and M. Moser, “Photon mode localization in disordered arrays of vertical cavity surface emitting lasers,” Appl. Phys. Lett. 85, 2454–2456 (1999).

M. M. Sigalas, C. M. Soukoulis, C. T. Chan, R. Biswas, and K. M. Ho, “Effect of disorder on photonic band gaps,” Phys. Rev. B 59, 12767–12770 (1999).
[CrossRef]

A. A. Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, and C. M. de Sterke, “Effects of disorder on wave propagation in two-dimensional photonic crystals,” Phys. Rev. E 60, 6118–6127 (1999).
[CrossRef]

A. Akyurtlu, D. H. Werner, V. Veremey, D. J. Steich, and K. Aydin, “Staircasing errors in FDTD at an air-dielectric interface,” IEEE Microwave Guided Wave Lett. 9, 444–446 (1999).
[CrossRef]

1998

T. A. Leskova, A. A. Maradudin, I. V. Novikov, A. V. Schegrov, and E. R. Méndez, “Design of one-dimensional band-limited uniform diffusers of light,” Appl. Phys. Lett. 73, 1943–1945 (1998).
[CrossRef]

1997

H. Li, B. Cheng, and D. Zhang, “Two-dimensional disordered photonic crystals with an average periodic lattice,” Phys. Rev. B 56, 10734–10736 (1997).
[CrossRef]

1995

D. S. Wiersma, M. P. van Albada, and A. Lagendijk, “Random laser ?” Nature 373, 203–204 (1995).
[CrossRef]

S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Theoretical investigation of fabrication-related disorder on the properties of photonic crystals,” J. Appl. Phys. 78, 1415–1418 (1995).
[CrossRef]

1994

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

N. M. Lawandy, R. M. Balachandra, A. S. L. Gomez, and E. Sauvain, “Laser action in strongly scattering media,” Nature 368, 436–438 (1994).
[CrossRef]

1993

1991

A. C. Cangellaris and D. B. Wright, “Analysis of the numerical error caused by the stair-stepped approximation of a conducting boundary in FDTD simulations of electromagnetic phenomena,” IEEE Trans. Antennas Propag. 39, 1518–1525 (1991).
[CrossRef]

1989

E. Yablonovitch and T. J. Gmitter, “Photonic band structure: the face-centered-cubic case,” Phys. Rev. Lett. 63, 1950–1953 (1989).
[CrossRef] [PubMed]

1987

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
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Figures (16)

Fig. 1
Fig. 1

Concept of random media. (a) A random structure model. (b) An illustration of whole random system.

Fig. 2
Fig. 2

Parameterization of disorder by Δxr.

Fig. 3
Fig. 3

Analysis models. (a) |Δxr|max/a = 0.00. (b) |Δxr|max/a = 0.250 (c) |Δxr|max/a = 0.500 (d) |Δxr|max/a = 1.00 (e) |Δxr|max/a = 2.00. (f) |Δxr|max/a = 4.00.

Fig. 4
Fig. 4

Radial distribution functions. (a) |Δxr|max/a = 0.00. (b) |Δxr|max/a = 0.250. (c) |Δxr|max/a = 0.500. (d) |Δxr|max/a = 1.00. (e) |Δxr|max/a = 2.00. (f) |Δxr|max/a = 4.00.

Fig. 5
Fig. 5

Finite elements. (a) Mesh in cylinder. (b) Mesh in PML.

Fig. 6
Fig. 6

Amplification factor versus normalized frequency ωa/2πc for γ = 0.002.

Fig. 7
Fig. 7

Dispersion relation of light waves in the periodic structure corresponding to |Δxr|max/a = 0.00.

Fig. 8
Fig. 8

Electric intensity distributions corresponding to the wave vector, k = (3π/4a, 3 π / 4 a ) in the unit cell of the periodic structure. (a) The unit cell of the periodic structure. (b) Lowest frequency band. (c) 2nd lowest frequency band. (d) 3rd lowest frequency band. (e) 4th lowest frequency band. (f) 5th lowest frequency band.

Fig. 9
Fig. 9

Amplification factor A versus normalized frequency ωa/2πc and γ for each |Δxr|max/a. (a) |Δxr|max/a = 0.00 (periodic structure). (b) |Δxr|max/a = 0.0625. (c) |Δxr|max/a = 0.125. (d) |Δxr|max/a = 0.250. (e) |Δxr|max/a = 0.500. (f) |Δxr|max/a = 1.00. (g) |Δxr|max/a = 2.00. (h) |Δxr|max/a = 4.00. (i) A peak at the bottom edge frequency ωa/2πc = 0.22196531 of the 4th band for |Δxr|max/a = 0.00 (periodic structure).

Fig. 10
Fig. 10

Relation between normalized disorder index |Δxr|max/a and the average value of lasing thresholds, namely, lowest γ with error bar.

Fig. 11
Fig. 11

Electric amplitude distributions of lasing states with tight confinement (sample 1) (a) |Δxr|max/a = 0.00, ωa/2πc = 0.22196531, γ = 0.0004965. (b) |Δxr|max/a = 0.0625, ωa/2πc = 0.2221812, γ = 0.0007350. (c) |Δxr|max/a = 0.125, ωa/2πc = 0.2223220, γ = 0.001140. (d) |Δxr|max/a = 0.250, ωa/2πc = 0.2222716, γ = 0.002490.

Fig. 12
Fig. 12

Polar plots of the group velocity of tight-binding model. (a) ωa/2πc = 0.3650. (b) ωa/2πc = 0.3675.

Fig. 13
Fig. 13

Electric amplitude distributions of lasing states with spatial extension (sample 1). (a) |Δxr|max/a = 0.00, ωa/2πc = 0.2310212, γ = 0.001380. (b) |Δxr|max/a = 0.125, ωa/2πc = 0.2308372, γ = 0.001470. (c) |Δxr|max/a = 0.250, ωa/2πc = 0.2300848, γ = 0.001995. (d) |Δxr|max/a = 0.500, ωa/2πc = 0.2297800, γ = 0.001785. (e) |Δxr|max/a = 0.750, ωa/2πc = 0.2280060, γ = 0.001515. (f) |Δxr|max/a = 1.00, ωa/2πc = 0.2238840, γ = 0.001080. (g) |Δxr|max/a = 1.25, ωa/2πc = 0.2276240, γ = 0.001575. (h) |Δxr|max/a = 4.00, ωa/2πc = 0.2321304, γ = 0.002430.

Fig. 14
Fig. 14

Mode volume versus normalized disorder index |Δxr|max/a.

Fig. 15
Fig. 15

Average values of | Δ x r | | Δ x r | max for each |Δxr|max.

Fig. 16
Fig. 16

Lasing thresholds versus normalized disorder index |Δxr|max/a (5 samples with another positioning algorithm).

Tables (1)

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Table 1 Model parameters.

Equations (16)

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x r ( n , m ) = x p ( n , m ) + Δ x r ,
x p ( n , m ) = n r 1 + m r 2 ,
r 1 = [ 3.47735 a × 3 / 2 , 3.47735 a × 1 / 2 ] T ,
r 2 = [ 0.0 , 3.47735 a ] T ,
0 | Δ x r | | Δ x r | max
× [ × E ( x ) ] ω 2 c 2 ε ( x ) E ( x ) = ω 2 ε 0 c 2 D d δ ( x x 0 ) ,
E ( x ) = E s ( x ) + E i ( x ) ,
× [ × E i ( x ) ] ω 2 c 2 ε i E i ( x ) = ω 2 ε 0 c 2 D d δ ( x x 0 ) ,
× [ × E s ( x ) ] ω 2 c 2 ε ( x ) E s ( x ) = ω 2 c 2 [ ε ( x ) ε i ] E i ( x ) .
E i ( x ) = ω 2 ε 0 c 2 D d i 4 H 0 ( 1 ) ( ω c ε i | x x 0 | ) ,
ε ( x ) = { 1.0 + i ( γ ) x Ω act 4.0 x Ω cylinder 1.0 x Ω out .
S = Re ( E × H * 2 )
A = C out S n out dl | γ 0 C out S n out dl | γ = 0 .
Ω act + Ω cylinder ε ( x ) | E ( x ) | 2 d Ω max [ ε ( x ) | E ( x ) | 2 ] λ 2 ,
( m σ ) | Δ x r | max < 0.73867 a < ( m + σ ) | Δ x r | max ,
0.93660 < | Δ x r | max / a < 3.49544.

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