Abstract

In devices consisting of a large number of coupled resonators, the coupling coefficients are unlikely to be identical throughout the ensemble, owing to statistical fluctuations in the fabrication process, the surrounding environment, or the device operation process. We describe how the frequency spectrum of such a disordered device differs from that of earlier models that assume a perfectly ordered lattice of resonant elements. Based on simulations for a large number of nominally identical resonators perturbed by disorder in the coupling coefficients, we describe the change in the density of modes (resonances) using both Hermitian and non-Hermitian coupling pathways, following recent experimental demonstrations. The band-edge zero-group-velocity state is highly sensitive to disorder, and applications that rely on band-edge effects, such as ultraslow light and group-velocity dispersion compensation, may be strongly impacted.

© 2006 Optical Society of America

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2005 (6)

2004 (5)

2003 (1)

A. Melloni, F. Morichetti, and M. Martinelli, "Optical slow wave structures," Opt. Photonics News 14, 44-48 (2003).
[CrossRef]

2002 (5)

B. Liu, A. Shakouri, and J. E. Bowers, "Wide tunable double ring coupled lasers," IEEE Photon. Technol. Lett. 14, 600-602 (2002).
[CrossRef]

D. Rabus, M. Hamacher, H. Heidrich, and U. Troppenz, "High-Q channel dropping filters using ring resonators with integrated SOAs," IEEE Photon. Technol. Lett. 14, 1442-1444 (2002).
[CrossRef]

S. Mookherjea and A. Yariv, "Kerr-stabilized super-resonant modes in coupled-resonator optical waveguides," Phys. Rev. E 66, 046610 (2002).
[CrossRef]

D. N. Christodoulides and N. K. Efremidis, "Discrete temporal solitons along a chain of nonlinear coupled microcavities embedded in photonic crystals," Opt. Lett. 27, 568-570 (2002).
[CrossRef]

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Mode coupling in traveling-wave resonators," Opt. Lett. 27, 1669-1671 (2002).
[CrossRef]

2001 (2)

2000 (3)

Y. Xu, R. K. Lee, and A. Yariv, "Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide," J. Opt. Soc. Am. B 17, 387-400 (2000).
[CrossRef]

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P.-T. Ho, "Higher order filter response in coupled microring resonators," IEEE Photon. Technol. Lett. 12, 320-322 (2000).
[CrossRef]

M. Bayindir, B. Temelkuran, and E. Ozbay, "Tight-binding description of the coupled defect modes in three-dimensional photonic crystals," Phys. Rev. Lett. 84, 2140-2143 (2000).
[CrossRef] [PubMed]

1999 (4)

J. Feinberg and A. Zee, "Spectral curves of non-hermitian hamiltonians," Nucl. Phys. B 552, 599 (1999).
[CrossRef]

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, "Tight-binding photonic molecule modes of resonant bispheres," Phys. Rev. Lett. 82, 4623-4626 (1999).
[CrossRef]

M. Bayer, I. Gutbrod, A. Forchel, T. Reinecke, P. Knipp, R. Werner, and J. Reithmaier, "Optical demonstration of a crystal band structure formation," Phys. Rev. Lett. 83, 5374 (1999).
[CrossRef]

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, "Coupled-resonator optical waveguide: a proposal and analysis," Opt. Lett. 24, 711-713 (1999).
[CrossRef]

1998 (1)

I. Y. Goldsheid and B. A. Khoruzhenko, "Distribution of eigenvalues in non-Hermitian Anderson models," Phys. Rev. Lett. 80, 2897-2900 (1998).
[CrossRef]

1996 (1)

N. Hatano and D. R. Nelson, "Localized transitions in non-Hermitian quantum mechanics," Phys. Rev. Lett. 77, 570-573 (1996).
[CrossRef] [PubMed]

1995 (1)

1992 (1)

J. B. Pendry and A. MacKinnon, "Calculation of photon dispersion relations," Phys. Rev. Lett. 69, 2772-2775 (1992).
[CrossRef] [PubMed]

1991 (1)

C. Barnes, T. Wei-chao, and J. B. Pendry, "The localization length and density of states of 1D disordered systems," J. Phys. Condens. Matter 3, 5297-5305 (1991).
[CrossRef]

1988 (1)

1973 (1)

A. Yariv, "Coupled-mode theory for guided-wave optics," IEEE J. Quantum Electron. QE-9, 919-933 (1973).
[CrossRef]

1972 (1)

D. J. Thouless, "A relation between the density of states and range of localization for one dimensional random systems," J. Phys. C 5, 77-81 (1972).
[CrossRef]

1970 (1)

E. R. Smith, "One-dimensional X−Y model with random coupling constants I. Thermodynamics," J. Phys. C 3, 1419-1432 (1970).
[CrossRef]

Absil, P. P.

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P.-T. Ho, "Higher order filter response in coupled microring resonators," IEEE Photon. Technol. Lett. 12, 320-322 (2000).
[CrossRef]

Altug, H.

H. Altug and J. Vuckovic, "Two dimensional coupled photonic crystal resonator arrays," Appl. Phys. Lett. 84, 161-163 (2004).
[CrossRef]

Ashcroft, N. W.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Harcourt, 1976).

Barnes, C.

C. Barnes, T. Wei-chao, and J. B. Pendry, "The localization length and density of states of 1D disordered systems," J. Phys. Condens. Matter 3, 5297-5305 (1991).
[CrossRef]

Bayer, M.

M. Bayer, I. Gutbrod, A. Forchel, T. Reinecke, P. Knipp, R. Werner, and J. Reithmaier, "Optical demonstration of a crystal band structure formation," Phys. Rev. Lett. 83, 5374 (1999).
[CrossRef]

Bayindir, M.

M. Bayindir, B. Temelkuran, and E. Ozbay, "Tight-binding description of the coupled defect modes in three-dimensional photonic crystals," Phys. Rev. Lett. 84, 2140-2143 (2000).
[CrossRef] [PubMed]

Bélanger, N.

Benisty, H.

Blair, S.

Bowers, J. E.

B. Liu, A. Shakouri, and J. E. Bowers, "Wide tunable double ring coupled lasers," IEEE Photon. Technol. Lett. 14, 600-602 (2002).
[CrossRef]

Boyd, R. W.

Chak, P.

Chen, Y.

Christodoulides, D. N.

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics (Wiley, 1977).

Diu, B.

C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics (Wiley, 1977).

Efremidis, N. K.

Etrich, C.

R. Iliew, U. Peschel, C. Etrich, and F. Lederer, "Light propagation via coupled defects in photonic crystals," in Conference on Lasers and Electro-Optics, Vol. 73 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2002), postconference edition, 191-192

Farhang-Boroujeny, B.

Feinberg, J.

J. Feinberg and A. Zee, "Spectral curves of non-hermitian hamiltonians," Nucl. Phys. B 552, 599 (1999).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77 (Cambridge U. Press, 1996).

Forchel, A.

M. Bayer, I. Gutbrod, A. Forchel, T. Reinecke, P. Knipp, R. Werner, and J. Reithmaier, "Optical demonstration of a crystal band structure formation," Phys. Rev. Lett. 83, 5374 (1999).
[CrossRef]

Franklin, J. N.

J. N. Franklin, Matrix Theory (Dover, 2000).

Goldsheid, I. Y.

I. Y. Goldsheid and B. A. Khoruzhenko, "Distribution of eigenvalues in non-Hermitian Anderson models," Phys. Rev. Lett. 80, 2897-2900 (1998).
[CrossRef]

Gonis, A.

A. Gonis, Green Functions for Ordered and Disordered Systems (North-Holland, 1992).

Gredeskul, S. A.

I. M. Lifshits, S. A. Gredeskul, and L. A. Pastur, Introduction to the Theory of Disordered Systems (Wiley, 1988).

Greenberg, M.

Grobe, R.

H. Matsuoka and R. Grobe, "Effect of eigenmodes on the optical transmission through one-dimensional random media," Phys. Rev. E 71, 046606 (2005).
[CrossRef]

Gutbrod, I.

M. Bayer, I. Gutbrod, A. Forchel, T. Reinecke, P. Knipp, R. Werner, and J. Reithmaier, "Optical demonstration of a crystal band structure formation," Phys. Rev. Lett. 83, 5374 (1999).
[CrossRef]

Hamacher, M.

D. Rabus, M. Hamacher, H. Heidrich, and U. Troppenz, "High-Q channel dropping filters using ring resonators with integrated SOAs," IEEE Photon. Technol. Lett. 14, 1442-1444 (2002).
[CrossRef]

Hare, J.

Haroche, S.

Hatano, N.

N. Hatano and D. R. Nelson, "Localized transitions in non-Hermitian quantum mechanics," Phys. Rev. Lett. 77, 570-573 (1996).
[CrossRef] [PubMed]

Haus, H. A.

H. A. Haus, Electromagnetic Noise and Quantum Optical Measurements (Springer, 2000).

Heebner, J. E.

Heidrich, H.

D. Rabus, M. Hamacher, H. Heidrich, and U. Troppenz, "High-Q channel dropping filters using ring resonators with integrated SOAs," IEEE Photon. Technol. Lett. 14, 1442-1444 (2002).
[CrossRef]

Ho, P.-T.

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P.-T. Ho, "Higher order filter response in coupled microring resonators," IEEE Photon. Technol. Lett. 12, 320-322 (2000).
[CrossRef]

Houdré, R.

Hryniewicz, J. V.

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P.-T. Ho, "Higher order filter response in coupled microring resonators," IEEE Photon. Technol. Lett. 12, 320-322 (2000).
[CrossRef]

Iliew, R.

R. Iliew, U. Peschel, C. Etrich, and F. Lederer, "Light propagation via coupled defects in photonic crystals," in Conference on Lasers and Electro-Optics, Vol. 73 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2002), postconference edition, 191-192

Jimba, Y.

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, "Tight-binding photonic molecule modes of resonant bispheres," Phys. Rev. Lett. 82, 4623-4626 (1999).
[CrossRef]

Joseph, R. I.

Khoruzhenko, B. A.

I. Y. Goldsheid and B. A. Khoruzhenko, "Distribution of eigenvalues in non-Hermitian Anderson models," Phys. Rev. Lett. 80, 2897-2900 (1998).
[CrossRef]

Khurgin, J. B.

Kippenberg, T. J.

Knipp, P.

M. Bayer, I. Gutbrod, A. Forchel, T. Reinecke, P. Knipp, R. Werner, and J. Reithmaier, "Optical demonstration of a crystal band structure formation," Phys. Rev. Lett. 83, 5374 (1999).
[CrossRef]

Krauss, T.

Kulishov, M.

Kuwata-Gonokami, M.

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, "Tight-binding photonic molecule modes of resonant bispheres," Phys. Rev. Lett. 82, 4623-4626 (1999).
[CrossRef]

Laloë, F.

C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics (Wiley, 1977).

Laniel, J. M.

Lederer, F.

R. Iliew, U. Peschel, C. Etrich, and F. Lederer, "Light propagation via coupled defects in photonic crystals," in Conference on Lasers and Electro-Optics, Vol. 73 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2002), postconference edition, 191-192

Lee, R. K.

Lefevre-Seguin, V.

Lifshits, I. M.

I. M. Lifshits, S. A. Gredeskul, and L. A. Pastur, Introduction to the Theory of Disordered Systems (Wiley, 1988).

Little, B. E.

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P.-T. Ho, "Higher order filter response in coupled microring resonators," IEEE Photon. Technol. Lett. 12, 320-322 (2000).
[CrossRef]

Liu, B.

B. Liu, A. Shakouri, and J. E. Bowers, "Wide tunable double ring coupled lasers," IEEE Photon. Technol. Lett. 14, 600-602 (2002).
[CrossRef]

Louisell, W. H.

W. H. Louisell, Coupled Mode and Parametric Electronics (Wiley, 1960).

MacKinnon, A.

J. B. Pendry and A. MacKinnon, "Calculation of photon dispersion relations," Phys. Rev. Lett. 69, 2772-2775 (1992).
[CrossRef] [PubMed]

Martin, R. M.

R. M. Martin, Electronic Structure (Cambridge U. Press, 2004).

Martinelli, M.

A. Melloni, F. Morichetti, and M. Martinelli, "Optical slow wave structures," Opt. Photonics News 14, 44-48 (2003).
[CrossRef]

Matsuoka, H.

H. Matsuoka and R. Grobe, "Effect of eigenmodes on the optical transmission through one-dimensional random media," Phys. Rev. E 71, 046606 (2005).
[CrossRef]

Melloni, A.

A. Melloni, F. Morichetti, and M. Martinelli, "Optical slow wave structures," Opt. Photonics News 14, 44-48 (2003).
[CrossRef]

Mermin, N. D.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Harcourt, 1976).

Miyazaki, H.

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, "Tight-binding photonic molecule modes of resonant bispheres," Phys. Rev. Lett. 82, 4623-4626 (1999).
[CrossRef]

Mookherjea, S.

S. Mookherjea, "Mode cycling in microring optical resonators," Opt. Lett. 30, 2751-2753 (2005).
[CrossRef] [PubMed]

S. Mookherjea, "Dispersion characteristics of coupled-resonator optical waveguides," Opt. Lett. 30, 2406-2408 (2005).
[CrossRef] [PubMed]

S. Mookherjea and A. Yariv, "Kerr-stabilized super-resonant modes in coupled-resonator optical waveguides," Phys. Rev. E 66, 046610 (2002).
[CrossRef]

S. Mookherjea and A. Yariv, "Optical pulse propagation in the tight-binding approximation," Opt. Express 9, 91-96 (2001).
[CrossRef] [PubMed]

S. Mookherjea, "Coupled-resonator optical waveguides and multiplexed solitons," Ph.D. thesis (California Institute of Technology, 2003).

S. Mookherjea, "Principles and applications of coupled microring optical resonators," in Workshop on Fibres and Optical Passive Components 2005: 4th IEEE/LEOS Workshop on Fibers and Optical Passive Components (IEEE, Piscataway, NJ, 2005), pp. 51-57 .

Morichetti, F.

A. Melloni, F. Morichetti, and M. Martinelli, "Optical slow wave structures," Opt. Photonics News 14, 44-48 (2003).
[CrossRef]

Mukaiyama, T.

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, "Tight-binding photonic molecule modes of resonant bispheres," Phys. Rev. Lett. 82, 4623-4626 (1999).
[CrossRef]

Nelson, D. R.

N. Hatano and D. R. Nelson, "Localized transitions in non-Hermitian quantum mechanics," Phys. Rev. Lett. 77, 570-573 (1996).
[CrossRef] [PubMed]

Oesterlé, U.

Olivier, S.

Orenstein, M.

Ozbay, E.

M. Bayindir, B. Temelkuran, and E. Ozbay, "Tight-binding description of the coupled defect modes in three-dimensional photonic crystals," Phys. Rev. Lett. 84, 2140-2143 (2000).
[CrossRef] [PubMed]

Park, Q.-H.

Pasrija, G.

Pastur, L. A.

I. M. Lifshits, S. A. Gredeskul, and L. A. Pastur, Introduction to the Theory of Disordered Systems (Wiley, 1988).

Pendry, J. B.

J. B. Pendry and A. MacKinnon, "Calculation of photon dispersion relations," Phys. Rev. Lett. 69, 2772-2775 (1992).
[CrossRef] [PubMed]

C. Barnes, T. Wei-chao, and J. B. Pendry, "The localization length and density of states of 1D disordered systems," J. Phys. Condens. Matter 3, 5297-5305 (1991).
[CrossRef]

Pereira, S.

Peschel, U.

R. Iliew, U. Peschel, C. Etrich, and F. Lederer, "Light propagation via coupled defects in photonic crystals," in Conference on Lasers and Electro-Optics, Vol. 73 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2002), postconference edition, 191-192

Plant, D. V.

Poon, J. K. S.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77 (Cambridge U. Press, 1996).

Rabus, D.

D. Rabus, M. Hamacher, H. Heidrich, and U. Troppenz, "High-Q channel dropping filters using ring resonators with integrated SOAs," IEEE Photon. Technol. Lett. 14, 1442-1444 (2002).
[CrossRef]

Raimond, J.-M.

Rattier, M.

Reinecke, T.

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Figures (5)

Fig. 1
Fig. 1

Schematic of a coupled-resonator waveguide, where each resonator has multiple degenerate eigenmodes. κ labels the bidirectional coupling coefficients between adjacent resonators, and γ labels the unidirectional coupling coefficients. (b) Implementation using coupled microring resonators, where κ is the coupling coefficient for the direct ring-to-ring coupling, which couples both cw and ccw modes, and γ is the coupling coefficient for the ring-waveguide-ring coupler, which is selective between cw and ccw modes of the rings according to the layout of the waveguides.

Fig. 2
Fig. 2

Eigenvalue spectrum [a slice through the density of states distribution, ρ ( ω x , ω y ) ]. The peaks correspond to the location of eigenvalues (resonance frequencies) ω = ω x + i ω y in the complex plane, and all the eigenvalues are real in both cases, even though the coupling matrix is non-Hermitian. The real-ω axis is measured in megahertz as a detuning from ω 0 , the resonance frequency of an uncoupled resonator. (a) Ordered, κ = 10 MHz ; the resonance frequencies lie within the interval δ ω < 2 κ and are more densely distributed at the edges. (b) Disordered, κ = 10 ± 5 MHz ; eigenvalues distributed more uniformly along the interval, and the width of the band increases slightly. Changing either the mean or variance of γ has no effect on the eigenvalue spectrum.

Fig. 3
Fig. 3

Density of states ρ ( ω ) for the ordered case, where ω is measured in megahertz from zero detuning, showing that ρ ( ω ) diverges at the band edges. The apparent thickness of the curve at the band center is an artifact arising from the discretization of the ω axis. Normalization, d ω ρ ( ω ) = 2 .

Fig. 4
Fig. 4

Density of states ρ ( ω ) for the ordered case, where ω is measured in megahertz from zero detuning. Compared with Fig. 3, the band edge is no longer a singularity. The inset shows that the tail of ρ ( ω ) is exponential and hence linear on a logarithmic scale. The density of states away from the band edges is relatively unaffected.

Fig. 5
Fig. 5

Eigenvalue spectrum for on-diagonal (isotopic) disorder, where ω 0 has random variations ± 10 MHz .

Equations (37)

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u = ( u 1 u 2 u N ) ,
E ( r , t ) = l = 1 N 1 2 [ e ̂ a l ( r ) a l ( t ) exp ( i ω a l t ) + e ̂ b l ( r ) b l ( t ) exp ( i ω b l t ) + + e ̂ m l ( r ) m l ( t ) exp ( i ω m l t ) ] + c . c . ,
i d dt u = M u ,
M = ( Ω 1 C 12 0 0 C 21 Ω 2 C 23 0 0 C 32 Ω 3 C 34 0 0 C 43 Ω 4 ) ,
d d t ( a 1 b 1 ) = ( i Δ ω 1 2 τ i 2 γ i 2 γ i Δ ω 1 2 τ ) ( a 1 b 1 ) ,
M ω I = 0 ,
T 1 M T = Λ = diag { ω 1 , ω 2 } ,
δ ω i max i ( j p i j ) .
M I = ( ω 1 κ 1 0 0 κ 1 * ω 2 κ 2 0 0 κ 2 * ω 3 κ 3 0 0 κ 3 * ω 4 ) ,
ϕ 0 ( ω ) = 1 ,
ϕ 1 ( ω ) = ω 1 ω ,
ϕ n ( ω ) = ( ω n ω ) ϕ n 1 κ n 1 2 ϕ n 2 ( n 2 ) ,
M II = ( Ω C 12 C 13 C 14 0 Ω C 23 C 24 0 0 Ω C 34 0 0 0 Ω )
ϕ ( ω ) = [ det ( Ω ω I ) ] n ,
d d t [ a 1 b 1 a 2 b 2 ] = ( i ω 1 0 0 κ 12 0 i ω 2 κ 21 γ 22 γ 11 κ 12 i ω 1 0 κ 21 0 0 i ω 2 ) [ a 1 b 1 a 2 b 2 ] .
ω = ω 1 + ω 2 2 ± [ ( ω 1 ω 2 2 ) 2 + κ 12 κ 21 ] 1 2 .
M = ( Ω C 12 0 0 C 21 Ω C 23 0 0 C 32 Ω C 34 0 0 C 43 Ω ) ,
Ω = ( Δ ω 1 0 0 Δ ω 2 ) , C l + 1 , l = ( γ l + 1 , l κ l + 1 , l ( 1 ) κ l + 1 , l ( 2 ) 0 ) ,
C l , l + 1 = ( 0 κ l + 1 , l ( 2 ) * κ l + 1 , l ( 1 ) * γ l + 1 , l * ) .
ω m i i j i m i j .
δ ω i = ( δ M v i , v ̃ i ) ( v i , v ̃ i ) ,
a 1 = m 2 ( v 1 + i ω 1 x 1 ) ,
a 2 = m 2 ( v 1 + i ω 2 x 2 ) ,
d d t ( a 1 a 2 a 1 * a 2 * ) = C ( a 1 a 2 a 1 * a 2 * ) ,
c 11 = c 33 = i ω 1 ( 1 + k 2 m ω 1 2 ) ,
c 22 = c 44 = i ω 1 ( 1 + k 2 m ω 2 2 ) ,
c 13 = c 21 = c 23 = c 31 = c 41 = c 43 = i k 2 m ω 1 ,
c 12 = c 14 = c 24 = c 32 = c 34 = c 42 = i k 2 m ω 2 .
E = a 1 2 + a 1 * 2 + a 2 2 + a 2 * 2 k 2 m ( a 1 a 1 * ω 1 a 2 a 2 * ω 2 ) 2 .
k 2 m ω 1 , 2 2 .
2 E ( r , t ) μ ϵ ( r ) 2 t 2 E ( r , t ) = μ 2 t 2 P ( r , t ) ,
ϵ ( r ) l = 1 N ω a l [ i d a l d t ] exp ( i ω a l t ) e ̂ a l ( r ) + + c . c . Δ ϵ ( r , t ) l = 1 N 1 2 [ ω a l 2 e ̂ a l ( r ) a l ( t ) exp ( i ω a l t ) + + c . c . , ]
i d dt a l ( t ) = ω a l 2 e ̂ a l Δ ϵ e ̂ a l a l + ω b l 2 2 ω a l e ̂ a l Δ ϵ e ̂ b l exp [ i ( ω b l ω a l ) t ] b l +     ;
e ̂ a l ϵ e ̂ a l d 3 r ϵ ( r ) e ̂ a l ( r ) 2 = 1.
ω ( K ) = ω 0 + Δ ω cos ( K R ) .
δ K = 2 π N R .
δ ω ( δ K ) R sin ( K R ) .

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