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Volumn 80, Issue 1, 2009, Pages

Fano-Hopfield model and photonic band gaps for an arbitrary atomic lattice

Author keywords

[No Author keywords available]

Indexed keywords

ATOMIC DIPOLE; ATOMIC LATTICE; CUBIC LATTICE; GAUSSIANS; HARMONIC OSCILLATORS; HOPFIELD MODELS; LIGHT DISPERSION; LONG-WAVELENGTH LIMITS; LORENTZ; MOMENTUM SPACES; SPECTRAL GAP; ULTRACOLD ATOMS;

EID: 69449100603     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.80.013816     Document Type: Article
Times cited : (53)

References (30)
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    • 36149009557 scopus 로고
    • 10.1103/PhysRev.112.1555
    • J. J. Hopfield, Phys. Rev. 112, 1555 (1958). 10.1103/PhysRev.112.1555
    • (1958) Phys. Rev. , vol.112 , pp. 1555
    • Hopfield, J.J.1
  • 4
    • 36149008368 scopus 로고
    • 10.1103/PhysRev.103.1202
    • U. Fano, Phys. Rev. 103, 1202 (1956). 10.1103/PhysRev.103.1202
    • (1956) Phys. Rev. , vol.103 , pp. 1202
    • Fano, U.1
  • 17
    • 69449108645 scopus 로고    scopus 로고
    • The use of a Gaussian cutoff ensures a rapid convergence of the sums to appear over the reciprocal lattice. Combined with relations 19 25, it allows to avoid Ewald summation tricks, with which it shares however some mathematical features.
    • The use of a Gaussian cutoff ensures a rapid convergence of the sums to appear over the reciprocal lattice. Combined with relations 19 25, it allows to avoid Ewald summation tricks, with which it shares however some mathematical features.
  • 18
    • 0000150031 scopus 로고
    • 10.1063/1.1696973
    • G. D. Mahan, J. Chem. Phys. 43, 1569 (1965). 10.1063/1.1696973
    • (1965) J. Chem. Phys. , vol.43 , pp. 1569
    • Mahan, G.D.1
  • 21
    • 69449101123 scopus 로고    scopus 로고
    • For ω02 to be positive, b should be large enough. For an hydrogenlike atom we may use the estimates ωB ≈ q2 / (4π ε0 a0) and dB ≈ |q| a0, where q is the electron charge and a0 the atomic radius, so that dB2 / ( ε0 ωB) ≈4π a03. The first factor in Eq. 16 is positive if a0 b, which is natural in a dipolar coupling model. Then the second factor is close to unity if (a0 /b)× (ωB a0 /c)2 1. This condition is satisfied, since ωB a0 /c≈α, where α1/1371 is the fine-structure constant.ctron charge and a0 the atomic radius, so that dB2 / (□ ε0 ωB) ≈4π a03. The first factor in Eq. 16 is positive if a0 b, which is natural in a dipolar coupling model. Then the second factor is close to unity if (a0 /b)× (ωB a0 /c)2 1. This condition is satisfied, since ωB a0 /c≈α, where α□1/137□1 is the fine-structure constant.
    • For ω02 to be positive, b should be large enough. For an hydrogenlike atom we may use the estimates ωB ≈ q2 / (4π ε0 a0) and dB ≈
  • 22
    • 69449100577 scopus 로고    scopus 로고
    • The electric field of a free-field solution is a sum of plane waves which interfere destructively in all atomic positions. This sum thus includes at least two plane waves. Let us select arbitrarily two of these plane waves; if one has a wave vector k, the other one has a wave vector necessarily of the form k+K, where K RL-. These two waves have the same frequency ωfree and obey the free space dispersion relation, so that ωfree /c=k= | k+K |. Squaring this last equality gives 2kK+ K2 =0, so that k= k -K/2 where k is orthogonal to K. Then kK/2, which leads to Eq. 21.
    • The electric field of a free-field solution is a sum of plane waves which interfere destructively in all atomic positions. This sum thus includes at least two plane waves. Let us select arbitrarily two of these plane waves; if one has a wave vector k, the other one has a wave vector necessarily of the form k+K, where K RL, because of the Bloch theorem. These two waves have the same frequency ωfree and obey the free space dispersion relation, so that ωfree /c=k= | k+K |. Squaring this last equality gives 2kK+ K2 =0, so that k= k -K/2 where k is orthogonal to K. Then kK/2, which leads to Eq. 21.
  • 23
    • 69449102623 scopus 로고    scopus 로고
    • One introduces Sαβ (k;b) = ∑R L exp (ikR) uαβ (r+R) r, with uαβ (r) = (δαβ -3 rα rβ / r2) / (4π r3) and ... r denotes the average over r with the Gaussian probability distribution exp (- r2 /4 b2). Using Poisson's formula and restricting to k infK RL K, one obtains limb→0 Sαβ (k;b) [kα kβ k2 - δαβ + Jαβ] / VL. The Fourier transform of uαβ is indeed u αβ = kα kβ k2 - 1 3 δαβ. We define the right-hand side of Eq. 32 as limb→0 Sαβ (k;b) k, where ... k stands for the uniform average over the direction of k. We then get Eq. 32.
    • One introduces Sαβ (k;b) = ∑R L exp (ikR) uαβ (r+R) r, with uαβ (r) = (δαβ -3 rα rβ / r2) / (4π r3) and... r denotes the average over r with the Gaussian probability distribution exp (- r2 /4 b2). Using Poisson's formula and restricting to k infK RL K, one obtains limb→0 Sαβ (k;b) [kα kβ k2 - δαβ + Jαβ] / VL. The Fourier transform of uαβ is indeed u αβ = kα kβ k2 - 1 3 δαβ. We define the right-hand side of Eq. 32 as limb→0 Sαβ (k;b) k, where... k stands for the uniform average over the direction of k. We then get Eq. 32.
  • 24
    • 69449104663 scopus 로고    scopus 로고
    • A mathematical question is to know what is the exact minimal free-field frequency in the diamond atomic lattice. Within the two-mode ansatz of one finds a minimal ωfree /c=2π2/a. The general three-mode ansatz can give lower frequency solutions that however form a discrete set. The corresponding minimal value is ωfree /c=3π/ (a2); it is obtained by superimposing with the right amplitudes the three plane waves of wave vectors k= (-3π/2a,3π/2a,0) , k- e 1 and k+ e 2, with a common polarization and the ansatz with four waves or more.
    • A mathematical question is to know what is the exact minimal free-field frequency in the diamond atomic lattice. Within the two-mode ansatz of one finds a minimal ωfree /c=2π2/a. The general three-mode ansatz can give lower frequency solutions that however form a discrete set. The corresponding minimal value is ωfree /c=3π/ (a2); it is obtained by superimposing with the right amplitudes the three plane waves of wave vectors k= (-3π/2a,3π/2a,0) , k- e 1 and k+ e 2, with a common polarization = (1,1,0). We have not explored the ansatz with four waves or more.
  • 30
    • 69449097758 scopus 로고    scopus 로고
    • The locations of the intensity maxima are ± (a/8,a/8,a/8) modulo any vector of the fcc lattice, so that they also form a diamond lattice.
    • The locations of the intensity maxima are ± (a/8,a/8,a/8) modulo any vector of the fcc lattice, so that they also form a diamond lattice.


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.