Abstract

A numerical scheme for modeling the waveguiding properties of a biased capillary tube infiltrated with nematic liquid crystal is presented. The structure of the liquid crystal under bias is determined by solving the Poisson equation for the electrostatic field and minimizing the elastic free energy of the liquid crystal in a self-consistency procedure. The resulting dielectric tensor is calculated, and the guided modes of the capillary waveguide are found. Results are reported for E7 liquid crystal in a single capillary as well as for a periodic geometry. The influence of the surrounding dielectric structure upon the liquid-crystal structure of an individual capillary tube is found to be minor. The photonic density of states of a square array of biased capillaries is calculated and is found to be highly tunable with respect to both the spectral positions of peaks and bandgaps as well as the widths of the photonic bands.

© 2006 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  4. J. A. Reyes and R. F. Rodriguez, "Guiding of optical fields in a liquid crystal cylindrical fiber," Opt. Commun. 134, 349-361 (1997).
    [CrossRef]
  5. T. T. Larsen, A. Bjarklev, D. S. Hermann, and J. Broeng, "Optical devices based on liquid crystal photonic bandgap fibres," Opt. Express 11, 2589-2596 (2003).
    [CrossRef] [PubMed]
  6. T. T. Alkeskjold, J. Lægsgaard, A. Bjarklev, D. S. Hermann, A. Anawati, J. Broeng, J. Li, and S. Wu, "All-optical modulation in dye-doped nematic liquid crystal photonic bandgap fibers," Opt. Express 12, 5857-5871 (2004).
    [CrossRef] [PubMed]
  7. M. W. Haakestad, T. T. Alkeskjold, M. D. Nielsen, L. Scolari, J. Riishede, H. E. Engan, and A. Bjarklev, "Electrically tunable photonic bandgap guidance in a liquid-crystal-filled photonic crystal fiber," IEEE Photon. Technol. Lett. 17, 819-821 (2005).
    [CrossRef]
  8. L. Scolari, T. Alkeskjold, J. Riishede, A. Bjarklev, D. Hermann, A. Anawati, M. Nielsen, and P. Bassi, "Continuously tunable devices based on electrical control of dual-frequency liquid crystal filled photonic bandgap fibers," Opt. Express 13, 7483-7496 (2005).
    [CrossRef] [PubMed]
  9. F. Du, Y.-Q. Lu, and S.-T. Wu, "Electrically tunable liquid-crystal photonic crystal fiber," Appl. Phys. Lett. 85, 2181-2183 (2004).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  21. J. Lægsgaard, "Gap formation and guided modes in photonic bandgap fibres with high-index rods," J. Opt. 6, 798-804 (2004).
    [CrossRef]

2005 (2)

M. W. Haakestad, T. T. Alkeskjold, M. D. Nielsen, L. Scolari, J. Riishede, H. E. Engan, and A. Bjarklev, "Electrically tunable photonic bandgap guidance in a liquid-crystal-filled photonic crystal fiber," IEEE Photon. Technol. Lett. 17, 819-821 (2005).
[CrossRef]

L. Scolari, T. Alkeskjold, J. Riishede, A. Bjarklev, D. Hermann, A. Anawati, M. Nielsen, and P. Bassi, "Continuously tunable devices based on electrical control of dual-frequency liquid crystal filled photonic bandgap fibers," Opt. Express 13, 7483-7496 (2005).
[CrossRef] [PubMed]

2004 (4)

T. T. Alkeskjold, J. Lægsgaard, A. Bjarklev, D. S. Hermann, A. Anawati, J. Broeng, J. Li, and S. Wu, "All-optical modulation in dye-doped nematic liquid crystal photonic bandgap fibers," Opt. Express 12, 5857-5871 (2004).
[CrossRef] [PubMed]

F. Du, Y.-Q. Lu, and S.-T. Wu, "Electrically tunable liquid-crystal photonic crystal fiber," Appl. Phys. Lett. 85, 2181-2183 (2004).
[CrossRef]

J. Li and S. T. Wu, "Extended Cauchy equations for the refractive indices of liquid crystals," J. Appl. Phys. 95, 896-901 (2004).
[CrossRef]

J. Lægsgaard, "Gap formation and guided modes in photonic bandgap fibres with high-index rods," J. Opt. 6, 798-804 (2004).
[CrossRef]

2003 (1)

2001 (2)

1997 (2)

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, "Erratum: accurate theoretical analysis of photonic band-gap materials," Phys. Rev. B 55, 15942 (1997).
[CrossRef]

J. A. Reyes and R. F. Rodriguez, "Guiding of optical fields in a liquid crystal cylindrical fiber," Opt. Commun. 134, 349-361 (1997).
[CrossRef]

1994 (2)

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]

1992 (1)

1991 (1)

H. Lin, P. Palffy-Muhoray, and M. A. Lee, "Liquid crystalline cores for optical fibers," Mol. Cryst. Liq. Cryst. 204, 189-200 (1991).
[CrossRef]

1986 (1)

Y. Saad and M. H. Schultz, "GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems," SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 7, 856-869 (1986).

1978 (1)

O. Jepsen, J. Madsen, and O. K. Andersen, "Band structure of thin films by the linear augmented-plane-wave method," Phys. Rev. B 18, 605-615 (1978).
[CrossRef]

Alerhand, O. L.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, "Erratum: accurate theoretical analysis of photonic band-gap materials," Phys. Rev. B 55, 15942 (1997).
[CrossRef]

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]

Alkeskjold, T.

Alkeskjold, T. T.

M. W. Haakestad, T. T. Alkeskjold, M. D. Nielsen, L. Scolari, J. Riishede, H. E. Engan, and A. Bjarklev, "Electrically tunable photonic bandgap guidance in a liquid-crystal-filled photonic crystal fiber," IEEE Photon. Technol. Lett. 17, 819-821 (2005).
[CrossRef]

T. T. Alkeskjold, J. Lægsgaard, A. Bjarklev, D. S. Hermann, A. Anawati, J. Broeng, J. Li, and S. Wu, "All-optical modulation in dye-doped nematic liquid crystal photonic bandgap fibers," Opt. Express 12, 5857-5871 (2004).
[CrossRef] [PubMed]

T. T. Alkeskjold, "Optical devices based on liquid crystal photonic bandgap fibers," Ph.D. dissertation (Technical University of Denmark, Lyngby, 2005).

Anawati, A.

Andersen, O. K.

O. Jepsen, J. Madsen, and O. K. Andersen, "Band structure of thin films by the linear augmented-plane-wave method," Phys. Rev. B 18, 605-615 (1978).
[CrossRef]

Bassi, P.

Bjarklev, A.

Broeng, J.

Brommer, K. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, "Erratum: accurate theoretical analysis of photonic band-gap materials," Phys. Rev. B 55, 15942 (1997).
[CrossRef]

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]

Cheng, H. F.

de Gennes, P. G.

P. G. de Gennes, The Physics of Liquid Crystals (Clarendon, 1974).

Du, F.

F. Du, Y.-Q. Lu, and S.-T. Wu, "Electrically tunable liquid-crystal photonic crystal fiber," Appl. Phys. Lett. 85, 2181-2183 (2004).
[CrossRef]

Engan, H. E.

M. W. Haakestad, T. T. Alkeskjold, M. D. Nielsen, L. Scolari, J. Riishede, H. E. Engan, and A. Bjarklev, "Electrically tunable photonic bandgap guidance in a liquid-crystal-filled photonic crystal fiber," IEEE Photon. Technol. Lett. 17, 819-821 (2005).
[CrossRef]

Gao, H. J.

Haakestad, M. W.

M. W. Haakestad, T. T. Alkeskjold, M. D. Nielsen, L. Scolari, J. Riishede, H. E. Engan, and A. Bjarklev, "Electrically tunable photonic bandgap guidance in a liquid-crystal-filled photonic crystal fiber," IEEE Photon. Technol. Lett. 17, 819-821 (2005).
[CrossRef]

Hermann, D.

Hermann, D. S.

Jepsen, O.

O. Jepsen, J. Madsen, and O. K. Andersen, "Band structure of thin films by the linear augmented-plane-wave method," Phys. Rev. B 18, 605-615 (1978).
[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, "Erratum: accurate theoretical analysis of photonic band-gap materials," Phys. Rev. B 55, 15942 (1997).
[CrossRef]

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.

Khoo, I. C.

Lægsgaard, J.

Larsen, T. T.

Lee, M. A.

H. Lin, P. Palffy-Muhoray, and M. A. Lee, "Liquid crystalline cores for optical fibers," Mol. Cryst. Liq. Cryst. 204, 189-200 (1991).
[CrossRef]

Li, H.

Li, J.

Liang, Y.

Lin, H.

LoPresti, P. G.

Lu, Y.-Q.

F. Du, Y.-Q. Lu, and S.-T. Wu, "Electrically tunable liquid-crystal photonic crystal fiber," Appl. Phys. Lett. 85, 2181-2183 (2004).
[CrossRef]

Madsen, J.

O. Jepsen, J. Madsen, and O. K. Andersen, "Band structure of thin films by the linear augmented-plane-wave method," Phys. Rev. B 18, 605-615 (1978).
[CrossRef]

Meade, R. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, "Erratum: accurate theoretical analysis of photonic band-gap materials," Phys. Rev. B 55, 15942 (1997).
[CrossRef]

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]

Nielsen, M.

Nielsen, M. D.

M. W. Haakestad, T. T. Alkeskjold, M. D. Nielsen, L. Scolari, J. Riishede, H. E. Engan, and A. Bjarklev, "Electrically tunable photonic bandgap guidance in a liquid-crystal-filled photonic crystal fiber," IEEE Photon. Technol. Lett. 17, 819-821 (2005).
[CrossRef]

Okamoto, K.

K. Okamoto, Fundamentals of Optical Waveguides (Academic, 2000).

Palffy-Muhoray, P.

Rappe, A. M.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, "Erratum: accurate theoretical analysis of photonic band-gap materials," Phys. Rev. B 55, 15942 (1997).
[CrossRef]

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]

Reyes, J. A.

J. A. Reyes and R. F. Rodriguez, "Guiding of optical fields in a liquid crystal cylindrical fiber," Opt. Commun. 134, 349-361 (1997).
[CrossRef]

Riishede, J.

M. W. Haakestad, T. T. Alkeskjold, M. D. Nielsen, L. Scolari, J. Riishede, H. E. Engan, and A. Bjarklev, "Electrically tunable photonic bandgap guidance in a liquid-crystal-filled photonic crystal fiber," IEEE Photon. Technol. Lett. 17, 819-821 (2005).
[CrossRef]

L. Scolari, T. Alkeskjold, J. Riishede, A. Bjarklev, D. Hermann, A. Anawati, M. Nielsen, and P. Bassi, "Continuously tunable devices based on electrical control of dual-frequency liquid crystal filled photonic bandgap fibers," Opt. Express 13, 7483-7496 (2005).
[CrossRef] [PubMed]

Rodriguez, R. F.

J. A. Reyes and R. F. Rodriguez, "Guiding of optical fields in a liquid crystal cylindrical fiber," Opt. Commun. 134, 349-361 (1997).
[CrossRef]

Saad, Y.

Y. Saad and M. H. Schultz, "GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems," SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 7, 856-869 (1986).

Sambles, J. R.

Schultz, M. H.

Y. Saad and M. H. Schultz, "GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems," SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 7, 856-869 (1986).

Scolari, L.

M. W. Haakestad, T. T. Alkeskjold, M. D. Nielsen, L. Scolari, J. Riishede, H. E. Engan, and A. Bjarklev, "Electrically tunable photonic bandgap guidance in a liquid-crystal-filled photonic crystal fiber," IEEE Photon. Technol. Lett. 17, 819-821 (2005).
[CrossRef]

L. Scolari, T. Alkeskjold, J. Riishede, A. Bjarklev, D. Hermann, A. Anawati, M. Nielsen, and P. Bassi, "Continuously tunable devices based on electrical control of dual-frequency liquid crystal filled photonic bandgap fibers," Opt. Express 13, 7483-7496 (2005).
[CrossRef] [PubMed]

Wu, S.

Wu, S. T.

J. Li and S. T. Wu, "Extended Cauchy equations for the refractive indices of liquid crystals," J. Appl. Phys. 95, 896-901 (2004).
[CrossRef]

Wu, S.-T.

F. Du, Y.-Q. Lu, and S.-T. Wu, "Electrically tunable liquid-crystal photonic crystal fiber," Appl. Phys. Lett. 85, 2181-2183 (2004).
[CrossRef]

Yang, F. Z.

Appl. Phys. Lett. (1)

F. Du, Y.-Q. Lu, and S.-T. Wu, "Electrically tunable liquid-crystal photonic crystal fiber," Appl. Phys. Lett. 85, 2181-2183 (2004).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

M. W. Haakestad, T. T. Alkeskjold, M. D. Nielsen, L. Scolari, J. Riishede, H. E. Engan, and A. Bjarklev, "Electrically tunable photonic bandgap guidance in a liquid-crystal-filled photonic crystal fiber," IEEE Photon. Technol. Lett. 17, 819-821 (2005).
[CrossRef]

J. Appl. Phys. (1)

J. Li and S. T. Wu, "Extended Cauchy equations for the refractive indices of liquid crystals," J. Appl. Phys. 95, 896-901 (2004).
[CrossRef]

J. Opt. (1)

J. Lægsgaard, "Gap formation and guided modes in photonic bandgap fibres with high-index rods," J. Opt. 6, 798-804 (2004).
[CrossRef]

J. Opt. Soc. Am. B (1)

Mol. Cryst. Liq. Cryst. (1)

H. Lin, P. Palffy-Muhoray, and M. A. Lee, "Liquid crystalline cores for optical fibers," Mol. Cryst. Liq. Cryst. 204, 189-200 (1991).
[CrossRef]

Opt. Commun. (1)

J. A. Reyes and R. F. Rodriguez, "Guiding of optical fields in a liquid crystal cylindrical fiber," Opt. Commun. 134, 349-361 (1997).
[CrossRef]

Opt. Express (4)

Opt. Lett. (3)

Phys. Rev. B (3)

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]

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, "Erratum: accurate theoretical analysis of photonic band-gap materials," Phys. Rev. B 55, 15942 (1997).
[CrossRef]

O. Jepsen, J. Madsen, and O. K. Andersen, "Band structure of thin films by the linear augmented-plane-wave method," Phys. Rev. B 18, 605-615 (1978).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. (1)

Y. Saad and M. H. Schultz, "GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems," SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 7, 856-869 (1986).

Other (3)

T. T. Alkeskjold, "Optical devices based on liquid crystal photonic bandgap fibers," Ph.D. dissertation (Technical University of Denmark, Lyngby, 2005).

K. Okamoto, Fundamentals of Optical Waveguides (Academic, 2000).

P. G. de Gennes, The Physics of Liquid Crystals (Clarendon, 1974).

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Figures (11)

Fig. 1
Fig. 1

Illustration of the deformations described by the LC elastic constants.

Fig. 2
Fig. 2

(a) Geometry and structural parameters for a single biased capillary. In principle, the structure extends laterally to infinity; however, only the domain of width W y is included in the finite-difference calculation. (b) Geometry and structural parameters for a square array of capillaries. In this case an infinite domain can be modeled due to the quasi-periodic boundary conditions on the electrostatic potential.

Fig. 3
Fig. 3

Symmetry lines (dashed) for the problem of an isolated biased capillary. The symmetries of the electrostatic potential [ Ψ ( x , y ) = Ψ ( x , y ) and Ψ ( x , y ) = Ψ ( x , y ) ] are indicated.

Fig. 4
Fig. 4

Distribution of the transverse components of the LC director field n for an isolated capillary at three different values of the bias voltage. The electrode separation ( W x in Fig. 2) is two times the capillary diameter. The darker color indicates a higher value of the n component in question. The LC is perpendicularly anchored to the capillary walls.

Fig. 5
Fig. 5

x component of the electrostatic field in a capillary with an 8 V bias infiltrated with a perpendicularly anchored E7 LC. Only the upper left quadrant of the calculational domain shown in Fig. 2a is shown. The dark quarter-circle indicates the capillary boundary. The unit on the color scale is in volts per micrometer, for W x = 3 μ m .

Fig. 6
Fig. 6

Intensity profiles for the two guided modes of highest effective index H 1 and H 2 in the LC structures shown in Fig. 4. H 1 is mainly x polarized, while H 2 is mainly y polarized. The thick solid circle indicates the boundary of the LC-filled capillary.

Fig. 7
Fig. 7

Convergence with W y of an LC structure for an isolated capillary. The reported quantity Δ n eff is the difference in effective index at a 1.55 μ m wavelength of the x-polarized mode with an LC structure calculated in the geometry of Fig. 2a and that calculated with periodic boundary conditions, as in Fig. 2b. The capillary diameter d c was 1.5 μ m , and d c W x = 0.9 . The applied bias was V 0 = 3.5 V .

Fig. 8
Fig. 8

Difference in effective index at 1.55 μ m for the fundamental (x-polarized) mode of a capillary waveguide, with the LC director structure calculated in either an isolated capillary, Fig. 2a, or periodic, Fig. 2b, geometry. The LC is in the escape-radial configuration, and the capillary radius is discretized in 150 mesh points when minimizing the LC free energy. The capillary diameter d c was 1.5 μ m , and d c W x = 0.9 .

Fig. 9
Fig. 9

Convergence with discretization of the LC director structure. n eff is the effective index of the x-polarized guided mode in the capillary, while N FD is the number of finite-difference discretization points across the capillary radius used to minimize the LC free energy. The Poisson equation was solved with periodic boundary conditions. Other parameters are as in Fig. 7.

Fig. 10
Fig. 10

Effective indices of the fundamental x- and y-polarized mode of the LC-infiltrated capillary as a function of bias voltage. (a) Results calculated for a parallel anchored LC and (b) results calculated for a perpendicular anchored crystal. The capillary diameter is d c = 1.5 μ m , and W x = W y = 3 μ m . To solve the Poisson equation, the upper quadrant of the calculational domain was discretized in 150 × 150 grid points, whereas the resolution to minimize the LC free energy was 150 points over the capillary radius.

Fig. 11
Fig. 11

Photonic density of states (PDOS) at a wavelength of 1.55 μ m for a square array of capillary waveguides having d c Λ = 0.5 . (a) Results for a parallel anchored LC with Λ = 2.5 μ m and (b) results for an LC in the escape-radial configuration with Λ = 3 μ m .

Tables (1)

Tables Icon

Table 1 Elastic and Static Dielectric Constants Used for the E7 LC in the Present Work a

Equations (10)

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

f el 0 = K 1 2 ( n ̂ ) 2 + K 2 2 n ̂ ( × n ̂ ) 2 + K 3 2 n ̂ × ( × n ̂ ) 2 .
f el = f el 0 1 2 E D .
ϵ ̿ S = ϵ S I ̿ + Δ ϵ S [ sin 2 ( θ ) cos 2 ( ϕ ) sin 2 ( θ ) sin ( ϕ ) cos ( ϕ ) sin ( θ ) cos ( θ ) cos ( ϕ ) sin 2 ( θ ) sin ( ϕ ) cos ( ϕ ) sin 2 ( θ ) sin 2 ( ϕ ) sin ( θ ) cos ( θ ) sin ( ϕ ) sin ( θ ) cos ( θ ) cos ( ϕ ) sin ( θ ) cos ( θ ) sin ( ϕ ) cos 2 ( θ ) ] .
F el = d r f el ( r ) .
[ ϵ ̿ S ( r ) Ψ ( r ) ] = 0 ,
R 0 2 f el = K 1 2 ( u n ̂ ) 2 + K 2 2 n ̂ ( u × n ̂ ) 2 + K 3 2 n ̂ × ( u × n ̂ ) 2 1 2 u Ψ ϵ ̿ S u Ψ ,
Ψ ( r + R x ) = Ψ ( r ) + V 0 , Ψ ( r + R y ) = Ψ ( r ) .
× ϵ ̿ 1 ( r ) × H ( r ) = ( ω c ) 2 H ( r )
H ( r ) = H ( r ) exp ( i β z )
β β 0 + ω ω 0 v g ,

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