Abstract

Modal phase singularities are identified in linear photonic crystals and the vortex state is explored in detail. Using group theory and phasor geometry, in the vanishing contrast limit, the modal symmetry requirements for the existence of phase singularities are determined. The vortex states are the partner functions of the symmetry groups, and hence one has a qualitative map of these modes in reciprocal space. We find that modes of even rotational symmetry are unable to form vortex states, while modes of odd rotational symmetry may form vortex states. The latter can be further classified into symmetry and accidental vortices. The insights gained using the vanishing contrast approximation are augmented by numerically solving the Maxwell’s equations for the high dielectric lattice forms using the Finite Element method; the general symmetry constraints are confirmed. In addition, symmetry vortices are found to demonstrate form and locational stability over large changes in dielectric contrast, whereas this is not so for the accidental vortices, which are more sensitive to such changes.

© 2007 Optical Society of America

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References

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    [CrossRef]

2006 (3)

2005 (3)

A. Ferrando, M. Zacares, and M. Garcia-March, "Vorticity cutoff in Nonlinear Photonic Crystals," Phys. Rev. Lett. 95, 043901:1-4 (2005).
[CrossRef]

A. Ferrando, "Discrete-symmetry vortices as angular Bloch modes," Phys Rev. E 72, 036612: 1-6 (2005).
[CrossRef]

H. Schriemer, J. Wheeldon, and T. Hall, "Transport Properties of Nonlinear Photonic Crystals," in Photonic Applications in Devices and Communication Systems, Proc. SPIE 5971,10-18 (2005).

2004 (3)

2003 (1)

2001 (1)

J. Masajada and B. Dubik, "Optical vortex generation by three plane wave interference," Opt. Commun. 198, 21-27 (2001).
[CrossRef]

2000 (1)

T. Sondergaard and K. H. Dridi, "Energy flow in photonic crystal waveguides," Phys Rev. B 61, 15688-15695 (2000).
[CrossRef]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

1974 (1)

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. Lond. A 336, 165-190 (1974).
[CrossRef]

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Berry, M. V.

M. V. Berry and M. R. Dennis "Quantum cores of optical phase singularities," J. Opt. A: Pure Appl. Opt. 6, S178-S180 (2004).
[CrossRef]

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. Lond. A 336, 165-190 (1974).
[CrossRef]

Courtial, J.

Dennis, M. R.

Dridi, K. H.

Dubik, B.

J. Masajada and B. Dubik, "Optical vortex generation by three plane wave interference," Opt. Commun. 198, 21-27 (2001).
[CrossRef]

Ferrando, A.

A. Ferrando, "Discrete-symmetry vortices as angular Bloch modes," Phys Rev. E 72, 036612: 1-6 (2005).
[CrossRef]

A. Ferrando, M. Zacares, and M. Garcia-March, "Vorticity cutoff in Nonlinear Photonic Crystals," Phys. Rev. Lett. 95, 043901:1-4 (2005).
[CrossRef]

Garcia-March, M.

A. Ferrando, M. Zacares, and M. Garcia-March, "Vorticity cutoff in Nonlinear Photonic Crystals," Phys. Rev. Lett. 95, 043901:1-4 (2005).
[CrossRef]

Grier, D. G.

Hall, T.

H. Schriemer, J. Wheeldon, and T. Hall, "Transport Properties of Nonlinear Photonic Crystals," in Photonic Applications in Devices and Communication Systems, Proc. SPIE 5971,10-18 (2005).

Ladavac, K.

Masajada, J.

J. Masajada and B. Dubik, "Optical vortex generation by three plane wave interference," Opt. Commun. 198, 21-27 (2001).
[CrossRef]

Musslimnai, Z. H.

Nye, J. F.

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. Lond. A 336, 165-190 (1974).
[CrossRef]

O’Holleran, K.

Padgett, M. J.

Schriemer, H.

H. Schriemer, J. Wheeldon, and T. Hall, "Transport Properties of Nonlinear Photonic Crystals," in Photonic Applications in Devices and Communication Systems, Proc. SPIE 5971,10-18 (2005).

Sondergaard, T.

T. Sondergaard and K. H. Dridi, "Energy flow in photonic crystal waveguides," Phys Rev. B 61, 15688-15695 (2000).
[CrossRef]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Vasnetsov, M.

Wheeldon, J.

H. Schriemer, J. Wheeldon, and T. Hall, "Transport Properties of Nonlinear Photonic Crystals," in Photonic Applications in Devices and Communication Systems, Proc. SPIE 5971,10-18 (2005).

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Yang, J.

Zacares, M.

A. Ferrando, M. Zacares, and M. Garcia-March, "Vorticity cutoff in Nonlinear Photonic Crystals," Phys. Rev. Lett. 95, 043901:1-4 (2005).
[CrossRef]

Zambrini, R.

J. Opt. A: Pure Appl. Opt. (1)

M. V. Berry and M. R. Dennis "Quantum cores of optical phase singularities," J. Opt. A: Pure Appl. Opt. 6, S178-S180 (2004).
[CrossRef]

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

Nature (1)

D. G. Grier, "A revolution in optical manipulation," Nature 424, 21-27 (2006).

Opt. Commun. (1)

J. Masajada and B. Dubik, "Optical vortex generation by three plane wave interference," Opt. Commun. 198, 21-27 (2001).
[CrossRef]

Opt. Express (3)

Opt. Lett. (1)

Phys Rev. B (1)

T. Sondergaard and K. H. Dridi, "Energy flow in photonic crystal waveguides," Phys Rev. B 61, 15688-15695 (2000).
[CrossRef]

Phys Rev. E (1)

A. Ferrando, "Discrete-symmetry vortices as angular Bloch modes," Phys Rev. E 72, 036612: 1-6 (2005).
[CrossRef]

Phys. Rev. A (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

A. Ferrando, M. Zacares, and M. Garcia-March, "Vorticity cutoff in Nonlinear Photonic Crystals," Phys. Rev. Lett. 95, 043901:1-4 (2005).
[CrossRef]

Proc. R. Soc. Lond. A (1)

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. Lond. A 336, 165-190 (1974).
[CrossRef]

Proc. SPIE (1)

H. Schriemer, J. Wheeldon, and T. Hall, "Transport Properties of Nonlinear Photonic Crystals," in Photonic Applications in Devices and Communication Systems, Proc. SPIE 5971,10-18 (2005).

Other (2)

K. Sakoda, Optical Properties of Photonic Crystals (Springer, New York, 2005).

M. Tinkham, Group Theory and Quantum Mechanics (McGraw-Hill, New York, 1964).

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Figures (7)

Fig. 1.
Fig. 1.

Square lattice (a) the unit cell in real space (b) the irreducible Brillouin zone and (b) the band structure with the locations of the accidental singularity and the modes which are associated with the A 1, B 2 and E irreducible representations of the C 4v point group.

Fig. 2.
Fig. 2.

The hexagonal lattice: (a) real-space unit cell, (b) the Brillouin zone, and (c) the band structure with the locations of the modes associated with the A 1 and E irreducible representations of the C 3v point group.

Fig. 3.
Fig. 3.

Phasor geometries for the Bloch mode of the A 1 representation, where (a) is υ = 1 and (b) is υ = -1 .

Fig. 4.
Fig. 4.

Phasor geometries for the Bloch mode of the first E representation, where (a) is υ = 1 and (b) is υ = -1 .

Fig. 5.
Fig. 5.

Phase (grey scale from -π to π) and optical flux (arrows) distributions in the hexagonal unit cells. Symmetry phase singularities for dielectric contrasts of (a) 1.2 and (b) 12.

Fig. 6.
Fig. 6.

Phase (grey scale from -π to π) and optical flux (arrows) distributions in the square-lattice unit cells. Conjugate plane waves for dielectric contrasts of (a) 1.2 and (b) 12.

Fig. 7.
Fig. 7.

Phase (grey scale from -π to π) and optical flux (arrows) distributions in the square-lattice unit cells. Accidental phase singularities for dielectric contrasts of (a) 1.2 and (b) 12.

Tables (1)

Tables Icon

Table 1 (a). C2v Character Table χ

Equations (26)

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= 2 πυ .
ψ ( r ) = Σ s S F s e s ( r )
F S F 1 + F 2 + + F S 1 .
ψ k ( r ) = Σ G′ U ( k G′ ) e i ( k G′ ) r ,
ω = ν ph q = ν ph ( k G ) ( k G ) = ν ph ( k G′ ) ( k G′ ) = ν ph ( k G″ ) ( k G″ ) = ,
ψ k ( r ) U ( k G ) e i ( k G ) r + U ( k G′ ) e i ( k G′ ) r + U ( k G″ ) e i ( k G″ ) r + .
ψ k ( r ) U ( k 0 ) e i ( k 0 ) r + U ( k ( 2 π a ) x ̂ ) e i ( k ( 2 π a ) x ̂ ) . r ,
Θ [ ε ( r ) , k ] ψ ( r ) = λ ψ ( r ) ,
R Θ [ ε ( r ) , k ] R 1 = Θ [ ε ( r ) , k ] .
R ψ Ai ( r ) = Σ i l D Aij ( R ) ψ Aj ( r ) .
χ RA = Tr [ D A ( R ) ] = Σ i l D Aii ( R ) .
{ A 1 } : ψ k ( r ) = U ( k 0 ) [ e iπx a + e iπx a ] ,
{ B 1 } : ψ k ( r ) = U ( k 0 ) [ e iπx a e iπx a ] .
{ A 1 } : ψ 1 , k ( r ) = U ( k 0 ) ( 1 + e i 2 π ( x a y ( 3 a ) ) + e i 2 π ( x a + y ( 3 a ) ) ) e ix ( 4 π 3 a ) ,
{ E } : ψ 2 , k ( r ) = U ( k 0 ) ( 2 e i 2 π ( x a y ( 3 a ) ) e i 2 π ( x a + y ( 3 a ) ) ) e ix ( 4 π 3 a ) , and
{ E } : ψ 3 , k ( r ) = U ( k 0 ) 3 ( e i 2 π ( x a y ( 3 a ) ) + e i 2 π ( x a + y ( 3 a ) ) ) e ix ( 4 π 3 a ) ,
[ 2 π a 2 π 3 a 2 π a 2 π 3 a ] [ x y ] = [ 4 π 3 2 π 3 ] and [ 2 π a 2 π 3 a 2 π a 2 π 3 a ] [ x y ] = [ 2 π 3 4 π 3 ] .
[ 2 π a 2 π 3 a 2 π a 2 π 3 a ] [ x y ] = [ π π ] ,
ψ k ( r ) = ψ 2 , k ( r ) + ( A + iB ) ψ 3 , k ( r ) ,
ψ k ( r ) = ( 2 ( A + 1 + iB ) e i 2 π ( x a y ( 3 a ) ) + ( A 1 + iB ) e i 2 π ( x a + y ( 3 a ) ) ) e ix ( 4 π 3 a ) .
2 ( A + 1 ) 2 + B 2 + ( A 1 ) 2 + B 2
ψ sum , k ( r ) = U ( k + ( 2 π a ) x ̂ ) ( e i 2 π x a + e i 2 πy a + e i 2 π ( x a + y a ) ) e i 2 π ( x 3 a + y 3 a ) .
{ A 1 } : ψ k ( r ) = U ( k 0 ) ( e ( x a + y a ) + e ( x a + y a ) + e ( x a y a ) + e ( x a y a ) ) ,
{ B 2 } : ψ k ( r ) = U ( k 0 ) ( e ( x a + y a ) e ( x a + y a ) e ( x a y a ) + e ( x a y a ) ) ,
{ E } : ψ k ( r ) = U ( k 0 ) ( e ( x a + y a ) e ( x a y a ) ) , and
{ E } : ψ k ( r ) = U ( k 0 ) ( e ( x a + y a ) e ( x a y a ) ) .

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