Abstract

We describe a phenomenological theory of the phenomenon of binding observed both experimentally and numerically when particles are trapped by an interference system in order to make a structure close to a photonic crystal. This theory leads to a very simple conclusion, which links the binding phenomenon to the bottom of the lowest bandgap of the trapped crystal in a given direction. The phenomenological theory allows one to calculate the period of the trapped crystal by using numerical tools on dispersion diagrams of photonic crystals. It emerges that the agreement of our theory with our rigorous numerical results given in a previous paper [J. Opt A 8, 1059 (2006)] is better than 2% on the crystal period. Furthermore, it is shown that in two-dimensional problems and s polarization, all the optical forces derive from a scalar potential.

© 2007 Optical Society of America

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References

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  1. A. Ashkin, "Acceleration and trapping of particles by radiation pressure," Phys. Rev. Lett. 24, 156-159 (1970).
    [CrossRef]
  2. A. Ashkin, "Optical trapping and manipulation of neutral particles using lasers," Proc. Natl. Acad. Sci. U.S.A. 94, 4853-4860 (1997).
    [CrossRef]
  3. M. Burns, J.-M. Fournier, and J. Golovshenko, "Optical binding," Phys. Rev. Lett. 63, 1233-1236 (1989).
    [CrossRef]
  4. M. Burns, J.-M. Fournier, and J. Golovshenko, "Lateral binding effect, due to particle's optical interaction," Science 289, 749-754 (1990).
    [CrossRef]
  5. J.-M. Fournier, G. Boer, G. Delacrétaz, P. Jacquot, J. Rohner, and R. Salathé, "Building optical matter with binding and trapping forces," Proc. SPIE 5514, 309-317 (2004).
    [CrossRef]
  6. W. Singer, M. Frick, S. Bernet, and M. Ritsch-Marte, "Self-organized array of regularly spaced microbeads in a fiber-optical trap," J. Opt. Soc. Am. B 20, 1568-1574 (2003).
    [CrossRef]
  7. S. Tatarkova, A. Carruthers, and K. Dholakia, "One-dimensional optically bound arrays of microscopic particles," Phys. Rev. Lett. 89, 283901 (2002).
    [CrossRef]
  8. N. Metzger, K. Dholakia, and E. Wright, "Observation of bistability and hysteresis in optical binding of two dielectric spheres," Phys. Rev. Lett. 96, 068102 (2006).
    [CrossRef] [PubMed]
  9. C. Mellor and C. Bain Chem, "Array formation in evanescent waves," ChemPhysChem 7, 329-332 (2006).
    [CrossRef]
  10. N. Metzger, E. Wright, W. Sibbett, and K. Dholakia, "Visualization of optical binding of microparticles using a femtosecond fiber optical trap," Opt. Express 14, 3677-3687 (2006).
    [CrossRef] [PubMed]
  11. P. Chaumet and M. Nieto-Vesperinas, "Time averaged total force on a dipolar sphere in an electromagnetic field," Opt. Lett. 25, 1065-1067 (2000).
    [CrossRef]
  12. T. Grzegorczyk, B. Kemp, and J. Kong, "Trapping and binding of an arbitrary number of cylindrical particles in an in-plane electromagnetic field," J. Opt. Soc. Am. A 23, 2324-2330 (2006).
    [CrossRef]
  13. T. Grzegorczyk, B. Kemp, and J. Kong, "Stable optical trapping based on optical binding forces," Phys. Rev. Lett. 96, 113903 (2006).
    [CrossRef] [PubMed]
  14. M. Povinelli, S. Johnson, M. Lonèar, M. Ibanescu, E. Smythe, F. Capasso, and J. Joannopoulos, "High-Q enhancement of attractive and repulsive optical forces between coupled whispering-gallery-mode resonators," Opt. Express 13, 8286-8295 (2005).
    [CrossRef] [PubMed]
  15. D. McGloin, A. Carruthers, K. Dholakia, and E. Wright, "Optically bound microscopic particles in one dimension," Phys. Rev. E 69, 021403 (2004).
    [CrossRef]
  16. A. Rohrbach and E. Stelzer, "Trapping forces and potentials of dielectric spheres in the presence of spherical aberrations," J. Opt. Soc. Am. A 18, 839-853 (2001).
    [CrossRef]
  17. E. Lidorikis, Q. Li, and C. Soukoulis, "Optical bistability in colloidal crystals," Phys. Rev. E 55, 3613-3618 (1997).
    [CrossRef]
  18. M. Antonoyiannakis and J. Pendry, "Electromagnetic forces in photonic crystals," Phys. Rev. B 60, 2363-2374 (1999).
    [CrossRef]
  19. J. Ng, C. T. Chan, P. Sheng, and Z. Lin, "Strong optical force induced by morphology-dependent resonances," Opt. Lett. 30, 1956-1958 (2005).
    [CrossRef] [PubMed]
  20. D. Maystre and P. Vincent, "Making photonic crystals using trapping and binding optical forces on particles," J. Opt. A, Pure Appl. Opt. 8, 1059-1066 (2006).
    [CrossRef]
  21. J. A. Kong, Maxwell Equations (EMW Publishing, 2002).
  22. J. Van Bladel, Electromagnetic Fields (Mc Graw-Hill, 1964).
  23. D. Maystre, "General study of grating anomalies," in Electromagnetic Surface Modes, A.D.Boardmann, ed. (Wiley, 1982), pp. 661-724.
  24. J.-M. Lourtioz, H. Benisty, V. Berger, J.-M. Gérard, D. Maystre, and A. Tchelnokov, Photonic Crystals (Springer, 2005).
  25. J. Joannopoulos, R. Meade, and J. Winn, Photonic Crystals (Princeton U. Press, 1995).
  26. C. Soukoulis, Photonic Band Gap Materials (Kluwer, 1996).
  27. D. Felbacq and G. Bouchitté, "Homogenization of a set of parallel fibers," Waves Random Media 7, 245-256 (1997).
    [CrossRef]

2006

N. Metzger, K. Dholakia, and E. Wright, "Observation of bistability and hysteresis in optical binding of two dielectric spheres," Phys. Rev. Lett. 96, 068102 (2006).
[CrossRef] [PubMed]

C. Mellor and C. Bain Chem, "Array formation in evanescent waves," ChemPhysChem 7, 329-332 (2006).
[CrossRef]

T. Grzegorczyk, B. Kemp, and J. Kong, "Stable optical trapping based on optical binding forces," Phys. Rev. Lett. 96, 113903 (2006).
[CrossRef] [PubMed]

D. Maystre and P. Vincent, "Making photonic crystals using trapping and binding optical forces on particles," J. Opt. A, Pure Appl. Opt. 8, 1059-1066 (2006).
[CrossRef]

N. Metzger, E. Wright, W. Sibbett, and K. Dholakia, "Visualization of optical binding of microparticles using a femtosecond fiber optical trap," Opt. Express 14, 3677-3687 (2006).
[CrossRef] [PubMed]

T. Grzegorczyk, B. Kemp, and J. Kong, "Trapping and binding of an arbitrary number of cylindrical particles in an in-plane electromagnetic field," J. Opt. Soc. Am. A 23, 2324-2330 (2006).
[CrossRef]

2005

2004

J.-M. Fournier, G. Boer, G. Delacrétaz, P. Jacquot, J. Rohner, and R. Salathé, "Building optical matter with binding and trapping forces," Proc. SPIE 5514, 309-317 (2004).
[CrossRef]

D. McGloin, A. Carruthers, K. Dholakia, and E. Wright, "Optically bound microscopic particles in one dimension," Phys. Rev. E 69, 021403 (2004).
[CrossRef]

2003

2002

S. Tatarkova, A. Carruthers, and K. Dholakia, "One-dimensional optically bound arrays of microscopic particles," Phys. Rev. Lett. 89, 283901 (2002).
[CrossRef]

2001

2000

1999

M. Antonoyiannakis and J. Pendry, "Electromagnetic forces in photonic crystals," Phys. Rev. B 60, 2363-2374 (1999).
[CrossRef]

1997

D. Felbacq and G. Bouchitté, "Homogenization of a set of parallel fibers," Waves Random Media 7, 245-256 (1997).
[CrossRef]

E. Lidorikis, Q. Li, and C. Soukoulis, "Optical bistability in colloidal crystals," Phys. Rev. E 55, 3613-3618 (1997).
[CrossRef]

A. Ashkin, "Optical trapping and manipulation of neutral particles using lasers," Proc. Natl. Acad. Sci. U.S.A. 94, 4853-4860 (1997).
[CrossRef]

1990

M. Burns, J.-M. Fournier, and J. Golovshenko, "Lateral binding effect, due to particle's optical interaction," Science 289, 749-754 (1990).
[CrossRef]

1989

M. Burns, J.-M. Fournier, and J. Golovshenko, "Optical binding," Phys. Rev. Lett. 63, 1233-1236 (1989).
[CrossRef]

1970

A. Ashkin, "Acceleration and trapping of particles by radiation pressure," Phys. Rev. Lett. 24, 156-159 (1970).
[CrossRef]

Antonoyiannakis, M.

M. Antonoyiannakis and J. Pendry, "Electromagnetic forces in photonic crystals," Phys. Rev. B 60, 2363-2374 (1999).
[CrossRef]

Ashkin, A.

A. Ashkin, "Optical trapping and manipulation of neutral particles using lasers," Proc. Natl. Acad. Sci. U.S.A. 94, 4853-4860 (1997).
[CrossRef]

A. Ashkin, "Acceleration and trapping of particles by radiation pressure," Phys. Rev. Lett. 24, 156-159 (1970).
[CrossRef]

Bain Chem, C.

C. Mellor and C. Bain Chem, "Array formation in evanescent waves," ChemPhysChem 7, 329-332 (2006).
[CrossRef]

Benisty, H.

J.-M. Lourtioz, H. Benisty, V. Berger, J.-M. Gérard, D. Maystre, and A. Tchelnokov, Photonic Crystals (Springer, 2005).

Berger, V.

J.-M. Lourtioz, H. Benisty, V. Berger, J.-M. Gérard, D. Maystre, and A. Tchelnokov, Photonic Crystals (Springer, 2005).

Bernet, S.

Boer, G.

J.-M. Fournier, G. Boer, G. Delacrétaz, P. Jacquot, J. Rohner, and R. Salathé, "Building optical matter with binding and trapping forces," Proc. SPIE 5514, 309-317 (2004).
[CrossRef]

Bouchitté, G.

D. Felbacq and G. Bouchitté, "Homogenization of a set of parallel fibers," Waves Random Media 7, 245-256 (1997).
[CrossRef]

Burns, M.

M. Burns, J.-M. Fournier, and J. Golovshenko, "Lateral binding effect, due to particle's optical interaction," Science 289, 749-754 (1990).
[CrossRef]

M. Burns, J.-M. Fournier, and J. Golovshenko, "Optical binding," Phys. Rev. Lett. 63, 1233-1236 (1989).
[CrossRef]

Capasso, F.

Carruthers, A.

D. McGloin, A. Carruthers, K. Dholakia, and E. Wright, "Optically bound microscopic particles in one dimension," Phys. Rev. E 69, 021403 (2004).
[CrossRef]

S. Tatarkova, A. Carruthers, and K. Dholakia, "One-dimensional optically bound arrays of microscopic particles," Phys. Rev. Lett. 89, 283901 (2002).
[CrossRef]

Chan, C. T.

Chaumet, P.

Delacrétaz, G.

J.-M. Fournier, G. Boer, G. Delacrétaz, P. Jacquot, J. Rohner, and R. Salathé, "Building optical matter with binding and trapping forces," Proc. SPIE 5514, 309-317 (2004).
[CrossRef]

Dholakia, K.

N. Metzger, E. Wright, W. Sibbett, and K. Dholakia, "Visualization of optical binding of microparticles using a femtosecond fiber optical trap," Opt. Express 14, 3677-3687 (2006).
[CrossRef] [PubMed]

N. Metzger, K. Dholakia, and E. Wright, "Observation of bistability and hysteresis in optical binding of two dielectric spheres," Phys. Rev. Lett. 96, 068102 (2006).
[CrossRef] [PubMed]

D. McGloin, A. Carruthers, K. Dholakia, and E. Wright, "Optically bound microscopic particles in one dimension," Phys. Rev. E 69, 021403 (2004).
[CrossRef]

S. Tatarkova, A. Carruthers, and K. Dholakia, "One-dimensional optically bound arrays of microscopic particles," Phys. Rev. Lett. 89, 283901 (2002).
[CrossRef]

Felbacq, D.

D. Felbacq and G. Bouchitté, "Homogenization of a set of parallel fibers," Waves Random Media 7, 245-256 (1997).
[CrossRef]

Fournier, J.-M.

J.-M. Fournier, G. Boer, G. Delacrétaz, P. Jacquot, J. Rohner, and R. Salathé, "Building optical matter with binding and trapping forces," Proc. SPIE 5514, 309-317 (2004).
[CrossRef]

M. Burns, J.-M. Fournier, and J. Golovshenko, "Lateral binding effect, due to particle's optical interaction," Science 289, 749-754 (1990).
[CrossRef]

M. Burns, J.-M. Fournier, and J. Golovshenko, "Optical binding," Phys. Rev. Lett. 63, 1233-1236 (1989).
[CrossRef]

Frick, M.

Gérard, J.-M.

J.-M. Lourtioz, H. Benisty, V. Berger, J.-M. Gérard, D. Maystre, and A. Tchelnokov, Photonic Crystals (Springer, 2005).

Golovshenko, J.

M. Burns, J.-M. Fournier, and J. Golovshenko, "Lateral binding effect, due to particle's optical interaction," Science 289, 749-754 (1990).
[CrossRef]

M. Burns, J.-M. Fournier, and J. Golovshenko, "Optical binding," Phys. Rev. Lett. 63, 1233-1236 (1989).
[CrossRef]

Grzegorczyk, T.

Ibanescu, M.

Jacquot, P.

J.-M. Fournier, G. Boer, G. Delacrétaz, P. Jacquot, J. Rohner, and R. Salathé, "Building optical matter with binding and trapping forces," Proc. SPIE 5514, 309-317 (2004).
[CrossRef]

Joannopoulos, J.

Johnson, S.

Kemp, B.

Kong, J.

Kong, J. A.

J. A. Kong, Maxwell Equations (EMW Publishing, 2002).

Li, Q.

E. Lidorikis, Q. Li, and C. Soukoulis, "Optical bistability in colloidal crystals," Phys. Rev. E 55, 3613-3618 (1997).
[CrossRef]

Lidorikis, E.

E. Lidorikis, Q. Li, and C. Soukoulis, "Optical bistability in colloidal crystals," Phys. Rev. E 55, 3613-3618 (1997).
[CrossRef]

Lin, Z.

Lonèar, M.

Lourtioz, J.-M.

J.-M. Lourtioz, H. Benisty, V. Berger, J.-M. Gérard, D. Maystre, and A. Tchelnokov, Photonic Crystals (Springer, 2005).

Maystre, D.

D. Maystre and P. Vincent, "Making photonic crystals using trapping and binding optical forces on particles," J. Opt. A, Pure Appl. Opt. 8, 1059-1066 (2006).
[CrossRef]

D. Maystre, "General study of grating anomalies," in Electromagnetic Surface Modes, A.D.Boardmann, ed. (Wiley, 1982), pp. 661-724.

J.-M. Lourtioz, H. Benisty, V. Berger, J.-M. Gérard, D. Maystre, and A. Tchelnokov, Photonic Crystals (Springer, 2005).

McGloin, D.

D. McGloin, A. Carruthers, K. Dholakia, and E. Wright, "Optically bound microscopic particles in one dimension," Phys. Rev. E 69, 021403 (2004).
[CrossRef]

Meade, R.

J. Joannopoulos, R. Meade, and J. Winn, Photonic Crystals (Princeton U. Press, 1995).

Mellor, C.

C. Mellor and C. Bain Chem, "Array formation in evanescent waves," ChemPhysChem 7, 329-332 (2006).
[CrossRef]

Metzger, N.

N. Metzger, E. Wright, W. Sibbett, and K. Dholakia, "Visualization of optical binding of microparticles using a femtosecond fiber optical trap," Opt. Express 14, 3677-3687 (2006).
[CrossRef] [PubMed]

N. Metzger, K. Dholakia, and E. Wright, "Observation of bistability and hysteresis in optical binding of two dielectric spheres," Phys. Rev. Lett. 96, 068102 (2006).
[CrossRef] [PubMed]

Ng, J.

Nieto-Vesperinas, M.

Pendry, J.

M. Antonoyiannakis and J. Pendry, "Electromagnetic forces in photonic crystals," Phys. Rev. B 60, 2363-2374 (1999).
[CrossRef]

Povinelli, M.

Ritsch-Marte, M.

Rohner, J.

J.-M. Fournier, G. Boer, G. Delacrétaz, P. Jacquot, J. Rohner, and R. Salathé, "Building optical matter with binding and trapping forces," Proc. SPIE 5514, 309-317 (2004).
[CrossRef]

Rohrbach, A.

Salathé, R.

J.-M. Fournier, G. Boer, G. Delacrétaz, P. Jacquot, J. Rohner, and R. Salathé, "Building optical matter with binding and trapping forces," Proc. SPIE 5514, 309-317 (2004).
[CrossRef]

Sheng, P.

Sibbett, W.

Singer, W.

Smythe, E.

Soukoulis, C.

E. Lidorikis, Q. Li, and C. Soukoulis, "Optical bistability in colloidal crystals," Phys. Rev. E 55, 3613-3618 (1997).
[CrossRef]

C. Soukoulis, Photonic Band Gap Materials (Kluwer, 1996).

Stelzer, E.

Tatarkova, S.

S. Tatarkova, A. Carruthers, and K. Dholakia, "One-dimensional optically bound arrays of microscopic particles," Phys. Rev. Lett. 89, 283901 (2002).
[CrossRef]

Tchelnokov, A.

J.-M. Lourtioz, H. Benisty, V. Berger, J.-M. Gérard, D. Maystre, and A. Tchelnokov, Photonic Crystals (Springer, 2005).

Van Bladel, J.

J. Van Bladel, Electromagnetic Fields (Mc Graw-Hill, 1964).

Vincent, P.

D. Maystre and P. Vincent, "Making photonic crystals using trapping and binding optical forces on particles," J. Opt. A, Pure Appl. Opt. 8, 1059-1066 (2006).
[CrossRef]

Winn, J.

J. Joannopoulos, R. Meade, and J. Winn, Photonic Crystals (Princeton U. Press, 1995).

Wright, E.

N. Metzger, E. Wright, W. Sibbett, and K. Dholakia, "Visualization of optical binding of microparticles using a femtosecond fiber optical trap," Opt. Express 14, 3677-3687 (2006).
[CrossRef] [PubMed]

N. Metzger, K. Dholakia, and E. Wright, "Observation of bistability and hysteresis in optical binding of two dielectric spheres," Phys. Rev. Lett. 96, 068102 (2006).
[CrossRef] [PubMed]

D. McGloin, A. Carruthers, K. Dholakia, and E. Wright, "Optically bound microscopic particles in one dimension," Phys. Rev. E 69, 021403 (2004).
[CrossRef]

ChemPhysChem

C. Mellor and C. Bain Chem, "Array formation in evanescent waves," ChemPhysChem 7, 329-332 (2006).
[CrossRef]

J. Opt. A, Pure Appl. Opt.

D. Maystre and P. Vincent, "Making photonic crystals using trapping and binding optical forces on particles," J. Opt. A, Pure Appl. Opt. 8, 1059-1066 (2006).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Express

Opt. Lett.

Phys. Rev. B

M. Antonoyiannakis and J. Pendry, "Electromagnetic forces in photonic crystals," Phys. Rev. B 60, 2363-2374 (1999).
[CrossRef]

Phys. Rev. E

D. McGloin, A. Carruthers, K. Dholakia, and E. Wright, "Optically bound microscopic particles in one dimension," Phys. Rev. E 69, 021403 (2004).
[CrossRef]

E. Lidorikis, Q. Li, and C. Soukoulis, "Optical bistability in colloidal crystals," Phys. Rev. E 55, 3613-3618 (1997).
[CrossRef]

Phys. Rev. Lett.

T. Grzegorczyk, B. Kemp, and J. Kong, "Stable optical trapping based on optical binding forces," Phys. Rev. Lett. 96, 113903 (2006).
[CrossRef] [PubMed]

S. Tatarkova, A. Carruthers, and K. Dholakia, "One-dimensional optically bound arrays of microscopic particles," Phys. Rev. Lett. 89, 283901 (2002).
[CrossRef]

N. Metzger, K. Dholakia, and E. Wright, "Observation of bistability and hysteresis in optical binding of two dielectric spheres," Phys. Rev. Lett. 96, 068102 (2006).
[CrossRef] [PubMed]

A. Ashkin, "Acceleration and trapping of particles by radiation pressure," Phys. Rev. Lett. 24, 156-159 (1970).
[CrossRef]

M. Burns, J.-M. Fournier, and J. Golovshenko, "Optical binding," Phys. Rev. Lett. 63, 1233-1236 (1989).
[CrossRef]

Proc. Natl. Acad. Sci. U.S.A.

A. Ashkin, "Optical trapping and manipulation of neutral particles using lasers," Proc. Natl. Acad. Sci. U.S.A. 94, 4853-4860 (1997).
[CrossRef]

Proc. SPIE

J.-M. Fournier, G. Boer, G. Delacrétaz, P. Jacquot, J. Rohner, and R. Salathé, "Building optical matter with binding and trapping forces," Proc. SPIE 5514, 309-317 (2004).
[CrossRef]

Science

M. Burns, J.-M. Fournier, and J. Golovshenko, "Lateral binding effect, due to particle's optical interaction," Science 289, 749-754 (1990).
[CrossRef]

Waves Random Media

D. Felbacq and G. Bouchitté, "Homogenization of a set of parallel fibers," Waves Random Media 7, 245-256 (1997).
[CrossRef]

Other

J. A. Kong, Maxwell Equations (EMW Publishing, 2002).

J. Van Bladel, Electromagnetic Fields (Mc Graw-Hill, 1964).

D. Maystre, "General study of grating anomalies," in Electromagnetic Surface Modes, A.D.Boardmann, ed. (Wiley, 1982), pp. 661-724.

J.-M. Lourtioz, H. Benisty, V. Berger, J.-M. Gérard, D. Maystre, and A. Tchelnokov, Photonic Crystals (Springer, 2005).

J. Joannopoulos, R. Meade, and J. Winn, Photonic Crystals (Princeton U. Press, 1995).

C. Soukoulis, Photonic Band Gap Materials (Kluwer, 1996).

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Figures (9)

Fig. 1
Fig. 1

System of interference generated by four plane waves. The origin of the system of coordinates is the center of the figure. The gray circles show the optical traps, and the sign of the field at these points is shown inside each circle.

Fig. 2
Fig. 2

Bloch wave vector inside the Brillouin zone (in gray) for a trapped crystal with square symmetry. The circles show the extremities of other wave vectors of the plane-wave expansion described by Eq. (37), which are located at the other corners of the Brillouin zone (circles).

Fig. 3
Fig. 3

Scheme of the band diagram of a dielectric crystal with square symmetry in the direction Γ M . Δ denotes the tangent to the dispersion curve at the origin. Its slope is given by the rules of homogenization.

Fig. 4
Fig. 4

Comparison of the rigorous numerical results with those obtained from the phenomenological theory for square symmetry. The radius R of the particles is equal to 0.0819 μ m , the wavelength in vacuum is equal to 0.546 μ m , and the external medium has an index 1.3 in such a way that the wavelength in the external medium is equal to 0.42 μ m . The dotted curve (squares) represents the rigorous numerical results. The solid curves show the results of the phenomenological theory: triangles, approximation deduced from the theory of homogenization [Eq. (42)]; circles, phenomenological theory using rigorous dispersion curves of photonic crystals.

Fig. 5
Fig. 5

System of interference generated by three plane waves. The origin of the system of coordinates is the center of the figure. The circles show the points where the three terms of the right-hand member of Eq. (44) have the same phase, the radii being related to this phase. The modulus of the total field is equal to 3 at the center of each circle.

Fig. 6
Fig. 6

Direct and reciprocal lattices.

Fig. 7
Fig. 7

Bloch wave vector inside the Brillouin zone (in gray) for a trapped crystal with hexagonal symmetry. The circles show the extremities of other wave vectors of the plane waves included in the Bloch mode.

Fig. 8
Fig. 8

Scheme of the dispersion diagram of a dielectric crystal with hexagonal symmetry in the direction Γ K . Δ denotes the tangent to the dispersion curve at the origin. Its slope is given by the rules of homogenization.

Fig. 9
Fig. 9

Comparison of the rigorous numerical results with those obtained from the phenomenological theory for hexagonal symmetry. The radius R of the particles is equal to 0.0819 μ m , the wavelength in vacuum is equal to 0.546 μ m , and the external medium has an index 1.3, in such a way that the wavelength in the external medium is equal to 0.42 μ m . The dotted curve (squares) represent the rigorous numerical results. The solid curves show the results of the phenomenological theory: triangles, approximation deduced from the theory of homogenization [Eq. (65)]; circles, phenomenological theory using rigorous dispersion curves of photonic crystals.

Equations (67)

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

E ( x , y ) = e i k 1 r + e i k 2 r + e i k 3 r + e i k 4 r .
x = n λ 2 , y = m λ 2 .
f = 1 4 R e Ω + n ( ε E E * + μ 0 H H * ) d s + 1 2 R e Ω [ ε E ( n E * ) + μ 0 H ( n H * ) ] d s ,
J = μ 0 4 Ω [ H ( n H * ) + H * ( n H ) ] d s .
Ω ( a ( n b ) + b ( n a ) n ( a b ) ) d s = Ω ( a b + b a a ( × b ) b × ( × a ) ) d S ,
J = μ 0 4 Ω n ( H H * ) d s μ 0 4 Ω ( H × ( × H * ) + H * × ( × H ) ) d S .
J = μ 0 4 Ω n ( H H * ) d s + ε r 4 Ω ( E ( E * ) + E * ( E ) ) d S ,
= μ 0 4 Ω n ( H H * ) d s + ε r 4 Ω ( E E * ) d S .
Ω n ( E E * ) d s = Ω ( E E * ) d S ,
f = ( ε r ε ) 4 Ω n ( E E * ) d s = ( ε r ε ) 4 Ω ( E E * ) d S .
f = Ω V d S ,
V = ( ε r ε ) 4 E E * .
V = μ 0 ( ε r ε ) 4 ε r H H * ,
F ( x , y ) = E ( x + c , y + d ) E ( c , d ) .
Symmetry with respect to the lines x = n p 0 and y = m p 0 ,
Symmetry with respect to the lines x + y = n p 0 and x y = m p 0 , n and m relative integers .
Symmetry with respect to the lines x = n p c and y = m p c
Symmetry with respect to the lines x + y = n p c and x y = m p c , n and m relative integers .
F 1 ( x , y ) = E 1 ( x + c , y + d ) E 1 ( c , d ) .
U ( x , y ) = E 1 ( x , y ) ,
V ( x , y ) = [ 1 ( i ω μ 0 ) ] × [ U ( x , y ) z ̂ ] ;
E 2 ( x , y ) = U ( x , y ) = E 1 ( x , y )
F 2 ( x , y ) = E 2 ( x c , y + d ) E 2 ( c , d ) .
E 1 ( c x , y + d ) = E 1 ( c + x , y + d ) ,
E ( x , y ) = exp ( i k B r ) n = + m = + b n , m exp [ i ( n G x + m G y ) ] ,
G 2 < k B x G 2 , G 2 < k B y G 2 .
E = exp ( i k 2 ( x y ) ) + exp ( i k 2 ( + x y ) ) + exp ( i k 2 ( + x + y ) ) + exp ( i k 2 ( x + y ) ) .
E = exp ( i G 0 2 ( x y ) ) + exp ( i G 0 2 ( + x y ) ) + exp ( i G 0 2 ( + x + y ) ) + exp ( i G 0 2 ( x + y ) ) ,
E = exp ( i G 0 2 ( x + y ) ) [ exp ( i G 0 ( x + y ) ) + exp ( i G 0 y ) + 1 + exp ( i G 0 x ) ] .
k B = ( G 0 2 , G 0 2 ) .
( x , y ) , F ( x , y ) = α 2 α 1 F ( x p c , y ) = γ F ( x p c , y ) ,
( x , y ) , F ( x + p c , y ) = γ F ( x , y ) = γ 2 F ( x p c , y ) ,
( x , y ) , F ( x + p c , y ) = F ( x p c , y ) ,
γ 2 = 1 .
( x , y ) , F ( x + p c , y ) = F ( x , y ) .
( x , y ) , E ( x + p c , y ) = E ( x , y ) .
( x , y ) , E ( x , y + p c ) = E ( x , y ) .
E ( x , y ) = exp ( i G 2 ( x + y ) ) n = + m = + b n , m exp [ i ( n G x + m G y ) ] .
k B = ( G 2 , G 2 ) .
ν e = ν 2 + π ( ν r 2 ν 2 ) R 2 p c 2 .
p c λ 0 ν e 2 ν ν e p 0 .
p c 1 1 + π ( ν r 2 ν 2 1 ) R 2 p c 2 p 0 , p 0 = λ 2 .
p c p 0 2 π R 2 ( ν r 2 ν 2 1 ) .
R < p 0 2 1 + π 4 ( ν r 2 ν 2 1 ) .
E ( x , y ) = e i k 1 r + e i k 2 r + e i k 3 r .
a i G j = 2 π δ i , j , i , j ( 1 , 2 ) .
a 1 = p c ( 3 2 j x ̂ + 1 2 y ̂ ) ,
a 2 = p c ( 3 2 x ̂ 1 2 y ̂ ) ,
G 1 = 4 π p c 3 ( 1 2 x ̂ + 3 2 y ̂ ) ,
G 2 = 4 π p c 3 ( 1 2 x ̂ 3 2 y ̂ ) .
E ( x , y ) = exp ( i k B r ) n = + m = + b n , m exp [ i ( n G 1 + m G 2 ) r ] ,
k q 1 , q 2 = q 1 G 1 + q 2 G 2 , q 1 and q 2 relative integers .
E ( x , y ) = exp ( i k B r ) n = + m = + b n , m exp [ i 2 π p c ( ( n + m ) x 3 + ( n m ) y ) ] .
symmetry with respect to the y axis and the two axes deduced from it by rotations of 2 π 3 around the origin ,
symmetry with respect to the axes deduced from these three axes by translations of n a 1 + m a 2 , n and m relative integers .
F ( r ) = E ( r + t ) E ( t ) , t = OO .
F ( r ) = α 2 α 1 F ( r a 1 ) = γ F ( r a 1 ) .
F ( r ) = γ F ( r + a 2 ) ,
F ( r ) = γ F ( r + a 1 a 2 ) .
F ( r + a 1 a 2 ) = γ 2 F ( r ) .
k B x = 0 ,
k B y = k B = 4 π 3 p c .
p c ν ν e p 0 .
ν e = ν 2 + 2 π 3 ( ν r 2 ν 2 ) R 2 p 2 ,
p c 1 1 + 2 π 3 ( ν r 2 ν 2 1 ) R 2 p c 2 p 0 , p 0 = 2 λ 3 .
p c p 0 2 2 π 3 ( ν r 2 ν 2 1 ) R 2 .
R < p 0 2 1 + π 2 3 ( ν r 2 ν 2 1 ) .

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