Abstract

At the exit surface of a photonic crystal, the intensity of the diffracted wave can be periodically modulated, showing a maximum in the “positive” (forward diffracted) or in the “negative” (diffracted) direction, depending on the slab thickness. This thickness dependence is a direct result of the so-called Pendellösung phenomenon, consisting of the periodic exchange inside the crystal of the energy between direct and diffracted beams. We report the experimental observation of this effect in the microwave region at about 14GHz by irradiating 2D photonic crystal slabs of different thickness and detecting the intensity distribution of the electromagnetic field at the exit surface and inside the crystal itself.

© 2008 Optical Society of America

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References

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  1. P. P. Ewald, "Zur Theorie der Interferenzen der R¨ontgenstrahlen," Physik Z. 14, 465 (1913).
  2. P. P. Ewald, "Crystal optics for visible light and X rays," Rev. Mod. Phys. 37, 46 (1965).
    [CrossRef]
  3. A. Authier, Dynamical Theory of X-ray Diffraction (Oxford University Press,Oxford, 2001).
  4. N. Kato and A. R. Lang, "The projection topograph: a new method in X-ray diffraction microradiography," Acta Cryst. 12, 249 (1959).
    [CrossRef]
  5. D. Sippel, K. Kleinstu, and G. E. R. Schulze, "Pendellösungs-interferenzen mit thermischen neutronen an Sieinkristallen," Phys. Lett. 14, 174 (1965).
    [CrossRef]
  6. C. G. Shull, "Observation of Pendellösung Fringe Structure in Neutron Diffraction," Phys. Rev. Lett. 21, 1585 (1968).
    [CrossRef]
  7. V. Mocella, J. Hrtwig, J. Baruchel, and A. Mazuelas, "Influence of the transverse and longitudinal coherence in dynamical theory of X-ray diffraction," J. Phys. D: Appl. Phys. 32, A88 (1999).
    [CrossRef]
  8. M. Born, Athomtheorie des festen Zustandes (Springer, Berlin, 1923).
  9. V. Mocella, "Negative refraction in Photonic Crystals: thickness dependence and Pendell¨osung phenomenon," Opt. Express 13, 1361 (2005).
    [CrossRef] [PubMed]
  10. A. Balestreri, L. C. Andreani, and M. Agio, "Optical properties and diffraction effects in opal photonic crystals," Phys. Rev. E 74, 036.603 (2006).
    [CrossRef]
  11. O. Francescangeli, S. Melone, and R. D. Leo, "Dynamical diffraction of microwaves by periodic dielectric media," Phys. Rev. A 40, 4988 (1989).
    [CrossRef] [PubMed]
  12. M. L. Calvo, P. Cheben, O. Martnez-Matos, F. del Monte, and J. A. Rodrigo, "Experimental Detection of the Optical Pendell¨osung Effect," Phys. Rev. Lett. 97, 084,801 (2006).
    [CrossRef]
  13. V. Mocella, P. Dardano, L. Moretti, and I. Rendina, "A polarizing beam splitter using negative refraction of photonic crystals," Opt. Express 13, 7699 (2005).
    [CrossRef] [PubMed]
  14. N. Kato, "The determination of structure factors by means of Pendell¨osung fringes," Acta Cryst. A 25, 119 (1969)

2006 (1)

M. L. Calvo, P. Cheben, O. Martnez-Matos, F. del Monte, and J. A. Rodrigo, "Experimental Detection of the Optical Pendell¨osung Effect," Phys. Rev. Lett. 97, 084,801 (2006).
[CrossRef]

2005 (2)

1999 (1)

V. Mocella, J. Hrtwig, J. Baruchel, and A. Mazuelas, "Influence of the transverse and longitudinal coherence in dynamical theory of X-ray diffraction," J. Phys. D: Appl. Phys. 32, A88 (1999).
[CrossRef]

1989 (1)

O. Francescangeli, S. Melone, and R. D. Leo, "Dynamical diffraction of microwaves by periodic dielectric media," Phys. Rev. A 40, 4988 (1989).
[CrossRef] [PubMed]

1969 (1)

N. Kato, "The determination of structure factors by means of Pendell¨osung fringes," Acta Cryst. A 25, 119 (1969)

1968 (1)

C. G. Shull, "Observation of Pendellösung Fringe Structure in Neutron Diffraction," Phys. Rev. Lett. 21, 1585 (1968).
[CrossRef]

1965 (2)

P. P. Ewald, "Crystal optics for visible light and X rays," Rev. Mod. Phys. 37, 46 (1965).
[CrossRef]

D. Sippel, K. Kleinstu, and G. E. R. Schulze, "Pendellösungs-interferenzen mit thermischen neutronen an Sieinkristallen," Phys. Lett. 14, 174 (1965).
[CrossRef]

1959 (1)

N. Kato and A. R. Lang, "The projection topograph: a new method in X-ray diffraction microradiography," Acta Cryst. 12, 249 (1959).
[CrossRef]

1913 (1)

P. P. Ewald, "Zur Theorie der Interferenzen der R¨ontgenstrahlen," Physik Z. 14, 465 (1913).

Agio, M.

A. Balestreri, L. C. Andreani, and M. Agio, "Optical properties and diffraction effects in opal photonic crystals," Phys. Rev. E 74, 036.603 (2006).
[CrossRef]

Andreani, L. C.

A. Balestreri, L. C. Andreani, and M. Agio, "Optical properties and diffraction effects in opal photonic crystals," Phys. Rev. E 74, 036.603 (2006).
[CrossRef]

Balestreri, A.

A. Balestreri, L. C. Andreani, and M. Agio, "Optical properties and diffraction effects in opal photonic crystals," Phys. Rev. E 74, 036.603 (2006).
[CrossRef]

Baruchel, J.

V. Mocella, J. Hrtwig, J. Baruchel, and A. Mazuelas, "Influence of the transverse and longitudinal coherence in dynamical theory of X-ray diffraction," J. Phys. D: Appl. Phys. 32, A88 (1999).
[CrossRef]

Calvo, M. L.

M. L. Calvo, P. Cheben, O. Martnez-Matos, F. del Monte, and J. A. Rodrigo, "Experimental Detection of the Optical Pendell¨osung Effect," Phys. Rev. Lett. 97, 084,801 (2006).
[CrossRef]

Cheben, P.

M. L. Calvo, P. Cheben, O. Martnez-Matos, F. del Monte, and J. A. Rodrigo, "Experimental Detection of the Optical Pendell¨osung Effect," Phys. Rev. Lett. 97, 084,801 (2006).
[CrossRef]

Dardano, P.

del Monte, F.

M. L. Calvo, P. Cheben, O. Martnez-Matos, F. del Monte, and J. A. Rodrigo, "Experimental Detection of the Optical Pendell¨osung Effect," Phys. Rev. Lett. 97, 084,801 (2006).
[CrossRef]

Ewald, P. P.

P. P. Ewald, "Crystal optics for visible light and X rays," Rev. Mod. Phys. 37, 46 (1965).
[CrossRef]

P. P. Ewald, "Zur Theorie der Interferenzen der R¨ontgenstrahlen," Physik Z. 14, 465 (1913).

Francescangeli, O.

O. Francescangeli, S. Melone, and R. D. Leo, "Dynamical diffraction of microwaves by periodic dielectric media," Phys. Rev. A 40, 4988 (1989).
[CrossRef] [PubMed]

Hrtwig, J.

V. Mocella, J. Hrtwig, J. Baruchel, and A. Mazuelas, "Influence of the transverse and longitudinal coherence in dynamical theory of X-ray diffraction," J. Phys. D: Appl. Phys. 32, A88 (1999).
[CrossRef]

Kato, N.

N. Kato, "The determination of structure factors by means of Pendell¨osung fringes," Acta Cryst. A 25, 119 (1969)

N. Kato and A. R. Lang, "The projection topograph: a new method in X-ray diffraction microradiography," Acta Cryst. 12, 249 (1959).
[CrossRef]

Kleinstu, K.

D. Sippel, K. Kleinstu, and G. E. R. Schulze, "Pendellösungs-interferenzen mit thermischen neutronen an Sieinkristallen," Phys. Lett. 14, 174 (1965).
[CrossRef]

Lang, A. R.

N. Kato and A. R. Lang, "The projection topograph: a new method in X-ray diffraction microradiography," Acta Cryst. 12, 249 (1959).
[CrossRef]

Leo, R. D.

O. Francescangeli, S. Melone, and R. D. Leo, "Dynamical diffraction of microwaves by periodic dielectric media," Phys. Rev. A 40, 4988 (1989).
[CrossRef] [PubMed]

Martnez-Matos, O.

M. L. Calvo, P. Cheben, O. Martnez-Matos, F. del Monte, and J. A. Rodrigo, "Experimental Detection of the Optical Pendell¨osung Effect," Phys. Rev. Lett. 97, 084,801 (2006).
[CrossRef]

Mazuelas, A.

V. Mocella, J. Hrtwig, J. Baruchel, and A. Mazuelas, "Influence of the transverse and longitudinal coherence in dynamical theory of X-ray diffraction," J. Phys. D: Appl. Phys. 32, A88 (1999).
[CrossRef]

Melone, S.

O. Francescangeli, S. Melone, and R. D. Leo, "Dynamical diffraction of microwaves by periodic dielectric media," Phys. Rev. A 40, 4988 (1989).
[CrossRef] [PubMed]

Mocella, V.

Moretti, L.

Rendina, I.

Rodrigo, J. A.

M. L. Calvo, P. Cheben, O. Martnez-Matos, F. del Monte, and J. A. Rodrigo, "Experimental Detection of the Optical Pendell¨osung Effect," Phys. Rev. Lett. 97, 084,801 (2006).
[CrossRef]

Schulze, G. E. R.

D. Sippel, K. Kleinstu, and G. E. R. Schulze, "Pendellösungs-interferenzen mit thermischen neutronen an Sieinkristallen," Phys. Lett. 14, 174 (1965).
[CrossRef]

Shull, C. G.

C. G. Shull, "Observation of Pendellösung Fringe Structure in Neutron Diffraction," Phys. Rev. Lett. 21, 1585 (1968).
[CrossRef]

Sippel, D.

D. Sippel, K. Kleinstu, and G. E. R. Schulze, "Pendellösungs-interferenzen mit thermischen neutronen an Sieinkristallen," Phys. Lett. 14, 174 (1965).
[CrossRef]

Acta Cryst. (1)

N. Kato and A. R. Lang, "The projection topograph: a new method in X-ray diffraction microradiography," Acta Cryst. 12, 249 (1959).
[CrossRef]

Acta Cryst. A (1)

N. Kato, "The determination of structure factors by means of Pendell¨osung fringes," Acta Cryst. A 25, 119 (1969)

J. Phys. D: Appl. Phys. (1)

V. Mocella, J. Hrtwig, J. Baruchel, and A. Mazuelas, "Influence of the transverse and longitudinal coherence in dynamical theory of X-ray diffraction," J. Phys. D: Appl. Phys. 32, A88 (1999).
[CrossRef]

Opt. Express (2)

Phys. Lett. (1)

D. Sippel, K. Kleinstu, and G. E. R. Schulze, "Pendellösungs-interferenzen mit thermischen neutronen an Sieinkristallen," Phys. Lett. 14, 174 (1965).
[CrossRef]

Phys. Rev. A (1)

O. Francescangeli, S. Melone, and R. D. Leo, "Dynamical diffraction of microwaves by periodic dielectric media," Phys. Rev. A 40, 4988 (1989).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

M. L. Calvo, P. Cheben, O. Martnez-Matos, F. del Monte, and J. A. Rodrigo, "Experimental Detection of the Optical Pendell¨osung Effect," Phys. Rev. Lett. 97, 084,801 (2006).
[CrossRef]

C. G. Shull, "Observation of Pendellösung Fringe Structure in Neutron Diffraction," Phys. Rev. Lett. 21, 1585 (1968).
[CrossRef]

Physik Z. (1)

P. P. Ewald, "Zur Theorie der Interferenzen der R¨ontgenstrahlen," Physik Z. 14, 465 (1913).

Rev. Mod. Phys. (1)

P. P. Ewald, "Crystal optics for visible light and X rays," Rev. Mod. Phys. 37, 46 (1965).
[CrossRef]

Other (3)

A. Authier, Dynamical Theory of X-ray Diffraction (Oxford University Press,Oxford, 2001).

A. Balestreri, L. C. Andreani, and M. Agio, "Optical properties and diffraction effects in opal photonic crystals," Phys. Rev. E 74, 036.603 (2006).
[CrossRef]

M. Born, Athomtheorie des festen Zustandes (Springer, Berlin, 1923).

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Figures (6)

Fig. 1.
Fig. 1.

The band structure of the square-lattice PhC for the TE polarization. The red line represents the normalized frequency ωn =0.722 at which the Pendellösung effect takes place (colour online).

Fig. 2.
Fig. 2.

The reciprocal space with the first Brillouin zone (dotted line) and symmetry points for the square-lattice PhC. The contours for the normalized frequency ωn =0.722 are plotted. Arrows indicate the directions of group velocity νg , whereas shows the normal to the incident surface (colour online).

Fig. 3.
Fig. 3.

Schematic layout of the experiment carried out on the square-lattice PhC slab having 25 rods columns and a number of rows n varying from 10 to 1. The dashed line box represents the scanned area during the measurements (colour online).

Fig. 4.
Fig. 4.

(a)–(e): mapping of the measured electric field (real part) in a normalized scale for even n; (f)–(j): mapping of the measured electric field (real part) in a normalized scale for odd n (colour online).

Fig. 5.
Fig. 5.

The measured electric field intensity ratio I -/I + for all the crystal configurations considered. The case of 10 rows corresponds to the maximum thickness t=(10a+2r)=16.4cm.

Fig. 6.
Fig. 6.

(a) FDTD simulation of the propagation pattern inside a crystal consisting of 10 rows of the 13.784 GHz plane wave modulated by a rectangular profile and incident at an angle of 43.8° across the XM interface; (b) & (c): electric field intensity distribution (blue dashed lines) along line 1 and 2 respectively compared with the experimental results (red solid lines) (colour online).

Equations (2)

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t = 2 m Λ 2 max ( I + )
t = ( 2 m 1 ) Λ 2 max ( I )

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