Abstract

We reveal the fundamental relation between linear photonic crystal symmetries and the local polarization states of its Bloch modes, in particular the location and nature of polarization singularities as established by rigorous group theoretic analysis, encompassing the full system symmetry. This is illustrated with the fundamental transverse electric mode of a two-dimensional hexagonal photonic crystal, in the vanishing contrast limit and at the K point. For general Wyckoff positions within the fundamental domain, the transformation of a local polarization state is determined by the nature of the symmetry operations that map to members of its crystallographic orbit. In particular, the site symmetries that correspond to specific Wyckoff positions constrain the local polarization state to singular character — circular, linear or disclination. Moreover, through the application of a local symmetry transformation relation, and the group’s character table, the precise natures of the singularities may be determined from self-consistency arguments.

© 2009 Optical Society of America

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References

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  1. J. V. Hajnal, "Singularities in the transverse fields of electromagnetic waves. I. Theory," Proc. R. Soc. Lond. A 414, 433-446 (1987).
    [CrossRef]
  2. M. V. Berry and M. R. Dennis, "Polarization singularities in isotropic random vector waves," Proc. R. Soc. Lond. A 457, 141-155 (2001).
    [CrossRef]
  3. I. Freund, "Polarization singularities in optical lattices," Opt. Let. 29, 875-877 (2004).
    [CrossRef]
  4. O. Toader, S. John, and K. Busch, "Optical trapping, Field enhancement and Laser cooling in photonic crystals," Opt. Express 8, 217-222 (2001).
    [CrossRef] [PubMed]
  5. M. Born and E. Wolf, Principles of optics: Electromagnetic theory of propagation, interference and diffraction of light, 7th edition (Cambridge, Cambridge University Press, 2003).
    [PubMed]
  6. J. F. Nye, Natural focusing and fine structure of light (Bath, IOP Publishing Ltd, 1999).
  7. J. F. Wheeldon, T. Hall and H. Schriemer, "Symmetry constraints and the existence of Bloch mode vortices in linear photonic crystals", Opt. Express 15, 3531-3542 (2007).
    [CrossRef] [PubMed]
  8. K. Sakoda, Optical Properties of Photonic Crystals (Springer, New York, 2005).
  9. M. Tinkham, Group Theory and Quantum Mechanics (McGraw-Hill, New York, 1964).
  10. J. D. Jackson, Classical Electrodynamics, 3rd ed. (USA, John Wiley and Sons, Inc., 1999).
  11. J. F. Wheeldon and H. Schriemer, "Symmetry and the local field response in photonic crystals," to appear in Proceedings of the NATO-ASI in Laser Control & Monitoring in New Materials, Biomedicine, Environment Security and Defense, Ottawa (Canada), November 24 - December 5 (2008),
  12. R. A. Evarestov and V. P. Smirnov, Site symmetry in crystals theory and applications, Springer series in solid-state sciences 108, (Springer-Verlag, New York, 1993).
  13. J. Wheeldon, Group Theoretic Expressions of Optical Singularities in Photonic Crystals (PhD Thesis, University of Ottawa, Ottawa, Canada, 2008).

2007 (1)

2004 (1)

I. Freund, "Polarization singularities in optical lattices," Opt. Let. 29, 875-877 (2004).
[CrossRef]

2001 (2)

O. Toader, S. John, and K. Busch, "Optical trapping, Field enhancement and Laser cooling in photonic crystals," Opt. Express 8, 217-222 (2001).
[CrossRef] [PubMed]

M. V. Berry and M. R. Dennis, "Polarization singularities in isotropic random vector waves," Proc. R. Soc. Lond. A 457, 141-155 (2001).
[CrossRef]

1987 (1)

J. V. Hajnal, "Singularities in the transverse fields of electromagnetic waves. I. Theory," Proc. R. Soc. Lond. A 414, 433-446 (1987).
[CrossRef]

Berry, M. V.

M. V. Berry and M. R. Dennis, "Polarization singularities in isotropic random vector waves," Proc. R. Soc. Lond. A 457, 141-155 (2001).
[CrossRef]

Busch, K.

Dennis, M. R.

M. V. Berry and M. R. Dennis, "Polarization singularities in isotropic random vector waves," Proc. R. Soc. Lond. A 457, 141-155 (2001).
[CrossRef]

Freund, I.

I. Freund, "Polarization singularities in optical lattices," Opt. Let. 29, 875-877 (2004).
[CrossRef]

Hajnal, J. V.

J. V. Hajnal, "Singularities in the transverse fields of electromagnetic waves. I. Theory," Proc. R. Soc. Lond. A 414, 433-446 (1987).
[CrossRef]

Hall, T.

John, S.

Schriemer, H.

Toader, O.

Wheeldon, J. F.

Opt. Express (2)

Opt. Let. (1)

I. Freund, "Polarization singularities in optical lattices," Opt. Let. 29, 875-877 (2004).
[CrossRef]

Proc. R. Soc. Lond. A (2)

J. V. Hajnal, "Singularities in the transverse fields of electromagnetic waves. I. Theory," Proc. R. Soc. Lond. A 414, 433-446 (1987).
[CrossRef]

M. V. Berry and M. R. Dennis, "Polarization singularities in isotropic random vector waves," Proc. R. Soc. Lond. A 457, 141-155 (2001).
[CrossRef]

Other (8)

M. Born and E. Wolf, Principles of optics: Electromagnetic theory of propagation, interference and diffraction of light, 7th edition (Cambridge, Cambridge University Press, 2003).
[PubMed]

J. F. Nye, Natural focusing and fine structure of light (Bath, IOP Publishing Ltd, 1999).

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

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

J. D. Jackson, Classical Electrodynamics, 3rd ed. (USA, John Wiley and Sons, Inc., 1999).

J. F. Wheeldon and H. Schriemer, "Symmetry and the local field response in photonic crystals," to appear in Proceedings of the NATO-ASI in Laser Control & Monitoring in New Materials, Biomedicine, Environment Security and Defense, Ottawa (Canada), November 24 - December 5 (2008),

R. A. Evarestov and V. P. Smirnov, Site symmetry in crystals theory and applications, Springer series in solid-state sciences 108, (Springer-Verlag, New York, 1993).

J. Wheeldon, Group Theoretic Expressions of Optical Singularities in Photonic Crystals (PhD Thesis, University of Ottawa, Ottawa, Canada, 2008).

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Figures (2)

Fig. 1.
Fig. 1.

Two-dimensional hexagonal lattice photonic crystal; (a) composed of a periodic array of dielectric cylinders, which have periodic symmetry in the x-y plane and infinite translational symmetry in the z direction; (b) top view of the dielectric cylinders and the symmetry operators R which leave the dielectric profile invariant; (c) Brillouin zone and equivalent K points. [8]

Fig. 2.
Fig. 2.

Polarization response for the A 2 irreducible representation of C . (a) calculated local polarization states: polarization ellipses (blue), dielectric columns (closed grey circles), linear polarization (black lines), circular polarization (red circles) disclinations (green circles); the primitive unit cell is delineated by the dashed red lines and the fundamental domain is shown in gold; (b) field magnitude, ∣E(r)∣ (arbitrary units); (c) the ẑ component of P(rQ(r) (arbitrary units; note that all other vector components are zero); (d) unit cell of the hexagonal lattice (dashed line), fundamental domain (orange region), reflection planes (lines), centers of C 3 rotational symmetry (red triangles), and the orbit of a general Wyckoff position (black dots).

Tables (1)

Tables Icon

Table 1. The C 3ν Character Table

Equations (14)

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R ψ i ( Γ ) = j l D ij ( Γ ) ( R ) ψ j ( Γ ) ,
χ R ( Γ ) = Tr [ D ( Γ ) ( R ) ] = j l D jj ( Γ ) ( R ) .
R F ( r ) = ( R F ) ( R 1 r ) = F ( r ) .
R F ( Γ ) ( r ) = χ R ( Γ ) F ( Γ ) ( r ) .
E ( r ) = 4 π 3 ε 0 ε [ ( 0 2 ) e i 4 πx / 3 a + ( 3 1 ) e i 2 π ( x / 3 a y / ( 3 a ) ) + ( 3 1 ) e i 2 π ( x / 3 a + y / ( 3 a ) ) ] .
E ( r ) = P ( r ) + i Q ( r ) ,
tan [ 2 τ ( r ) ] = 2 P ( r ) · Q ( r ) / [ P 2 ( r ) Q 2 ( r ) ]
R E ( r 1 ) = R E ( R 1 r 1 ) = E ( r 2 ) ,
R P ( R 1 r 1 ) × R Q ( R 1 r 1 ) = P ( r 2 ) × Q ( r 2 ) = det ( R ) P ( r 1 ) × Q ( r 1 ) .
R E ( r 1 ) = χ R ( r 1 ) E ( r 1 ) ,
E ( r 1 ) = P ( r 1 ) + i Q ( r 1 ) = ( P x ( r 1 ) + i Q x ( r 1 ) P y ( r 1 ) + i Q y ( r 1 ) ) = Ω ( r 1 ) ( D ( r 1 ) + iB ( r 1 ) C ( r 1 ) ) ;
[ 1 0 0 1 ] ( D + iB C ) = ( D + iB C ) = χ σ y ( r 1 ) ( D + iB C ) .
[ cos ( 2 π / n ) sin ( 2 π / n ) sin ( 2 π / n ) cos ( 2 π / n ) ] ( D + iB C ) = χ C n ( r 1 ) ( D + iB C ) .
1 2 ( ( D + iB ) C 3 ( D + iB ) 3 C ) = χ C 3 ( r 1 ) ( D + iB C ) .

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