Abstract

One dimensional photonic crystals combining positive and negative index layers have shown to present a photonic band gap insensitive to the period scaling when the volume average index vanishes. Defect modes lying in this zero- gap can in addition be obtained without locally breaking the symmetry of the crystal lattice. In this work, index dispersion is shown to broaden the resonant frequencies creating then a conduction band lying inside the zero- gap. Self-collimation and focusing effects are in addition demonstrated in zero-average index metamaterials supporting defect modes. This beam shaping is explained in the framework of a beam propagation model by introducing an harmonic average index parameter.

© 2011 Optical Society of America

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References

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  1. J. Joannopoulos, S. Johnson, D. Winn, and R. Meade, Photonic crystals: molding the flow of light, 2nd edition, (Princeton University Press 2008).
  2. I. Nefedov, and S. Tretyakov, “Photonic band gap structure containing metamaterial with negative permittivity and permeability,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66, 036611 (2002).
    [CrossRef]
  3. J. Li, L. Zhou, C. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 83901 (2003).
    [CrossRef]
  4. D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. Vigneron, E. E. Boudouti, and A. Nougaoui, “Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69, 066613 (2004).
    [CrossRef]
  5. Y. Yuan, L. Ran, J. Huangfu, H. Chen, L. Shen, and J. Kong, “Experimental verification of zero order bandgap in a layered stack of left-handed and right-handed materials,” Opt. Express 14, 2220–2227 (2006).
    [CrossRef] [PubMed]
  6. S. Kocaman, R. Chatterjee, N. Panoiu, J. Mcmillan, M. Yu, R. Osgood, D. Kwong, and C. Wong, “Observation of zeroth-order band gaps in negative-refraction photonic crystal superlattices at near-infrared frequencies,” Phys. Rev. Lett. 102, 203905 (2009).
    [CrossRef] [PubMed]
  7. V. Mocella, S. Cabrini, A. Chang, P. Dardano, L. Moretti, I. Rendina, D. Olynick, B. Harteneck, and S. Dhuey, “Self-collimation of light over millimeter-scale distance in a quasi-zero-average-index metamaterial,” Phys. Rev. Lett. 102, 133902 (2009).
    [CrossRef] [PubMed]
  8. N. Panoiu, J. Osgood, S. Zhang, and S. Brueck, “Zero-n bandgap in photonic crystal superlattices,” J. Opt. Soc. Am. B 23, 506–513 (2006).
    [CrossRef]
  9. I. Shadrivov, A. Sukhorukov, and Y. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2009).
    [CrossRef]
  10. F. Krayzel, R. Poll`es, A. Moreau, and M. Mihailovic, “Simulation and analysis of exotic non-specular phenomena,” J. Europ. Opt. Soc.: Rap. Pub. 5, 10025 (2010).
    [CrossRef]
  11. E. Silvestre, R. Depine, M. Mart’ınez-Ricci, and J. Monsoriu, “Role of dispersion on zero-average-index bandgaps,” J. Opt. Soc. Am. B 26, 581–586 (2009).
    [CrossRef]
  12. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
    [CrossRef]
  13. Throughout this paper, the inverse Fourier Transform is defined by TF−1(f (α)) = R ∞ −∞ f (x)exp(iαx)dα.
  14. M. Born, E. Wolf, and A. B. Bhatia, Principles of optics: Electromagnetic theory of propagation, interference and diffraction of light (Cambridge University press, Cambridge 2000).
    [PubMed]

2010 (1)

F. Krayzel, R. Poll`es, A. Moreau, and M. Mihailovic, “Simulation and analysis of exotic non-specular phenomena,” J. Europ. Opt. Soc.: Rap. Pub. 5, 10025 (2010).
[CrossRef]

2009 (4)

S. Kocaman, R. Chatterjee, N. Panoiu, J. Mcmillan, M. Yu, R. Osgood, D. Kwong, and C. Wong, “Observation of zeroth-order band gaps in negative-refraction photonic crystal superlattices at near-infrared frequencies,” Phys. Rev. Lett. 102, 203905 (2009).
[CrossRef] [PubMed]

V. Mocella, S. Cabrini, A. Chang, P. Dardano, L. Moretti, I. Rendina, D. Olynick, B. Harteneck, and S. Dhuey, “Self-collimation of light over millimeter-scale distance in a quasi-zero-average-index metamaterial,” Phys. Rev. Lett. 102, 133902 (2009).
[CrossRef] [PubMed]

I. Shadrivov, A. Sukhorukov, and Y. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2009).
[CrossRef]

E. Silvestre, R. Depine, M. Mart’ınez-Ricci, and J. Monsoriu, “Role of dispersion on zero-average-index bandgaps,” J. Opt. Soc. Am. B 26, 581–586 (2009).
[CrossRef]

2006 (2)

2004 (1)

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. Vigneron, E. E. Boudouti, and A. Nougaoui, “Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69, 066613 (2004).
[CrossRef]

2003 (1)

J. Li, L. Zhou, C. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 83901 (2003).
[CrossRef]

2002 (1)

I. Nefedov, and S. Tretyakov, “Photonic band gap structure containing metamaterial with negative permittivity and permeability,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66, 036611 (2002).
[CrossRef]

1999 (1)

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[CrossRef]

Akjouj, A.

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. Vigneron, E. E. Boudouti, and A. Nougaoui, “Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69, 066613 (2004).
[CrossRef]

Boudouti, E. E.

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. Vigneron, E. E. Boudouti, and A. Nougaoui, “Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69, 066613 (2004).
[CrossRef]

Bria, D.

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. Vigneron, E. E. Boudouti, and A. Nougaoui, “Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69, 066613 (2004).
[CrossRef]

Brueck, S.

Cabrini, S.

V. Mocella, S. Cabrini, A. Chang, P. Dardano, L. Moretti, I. Rendina, D. Olynick, B. Harteneck, and S. Dhuey, “Self-collimation of light over millimeter-scale distance in a quasi-zero-average-index metamaterial,” Phys. Rev. Lett. 102, 133902 (2009).
[CrossRef] [PubMed]

Chan, C.

J. Li, L. Zhou, C. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 83901 (2003).
[CrossRef]

Chang, A.

V. Mocella, S. Cabrini, A. Chang, P. Dardano, L. Moretti, I. Rendina, D. Olynick, B. Harteneck, and S. Dhuey, “Self-collimation of light over millimeter-scale distance in a quasi-zero-average-index metamaterial,” Phys. Rev. Lett. 102, 133902 (2009).
[CrossRef] [PubMed]

Chatterjee, R.

S. Kocaman, R. Chatterjee, N. Panoiu, J. Mcmillan, M. Yu, R. Osgood, D. Kwong, and C. Wong, “Observation of zeroth-order band gaps in negative-refraction photonic crystal superlattices at near-infrared frequencies,” Phys. Rev. Lett. 102, 203905 (2009).
[CrossRef] [PubMed]

Chen, H.

Dardano, P.

V. Mocella, S. Cabrini, A. Chang, P. Dardano, L. Moretti, I. Rendina, D. Olynick, B. Harteneck, and S. Dhuey, “Self-collimation of light over millimeter-scale distance in a quasi-zero-average-index metamaterial,” Phys. Rev. Lett. 102, 133902 (2009).
[CrossRef] [PubMed]

Depine, R.

Dhuey, S.

V. Mocella, S. Cabrini, A. Chang, P. Dardano, L. Moretti, I. Rendina, D. Olynick, B. Harteneck, and S. Dhuey, “Self-collimation of light over millimeter-scale distance in a quasi-zero-average-index metamaterial,” Phys. Rev. Lett. 102, 133902 (2009).
[CrossRef] [PubMed]

Djafari-Rouhani, B.

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. Vigneron, E. E. Boudouti, and A. Nougaoui, “Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69, 066613 (2004).
[CrossRef]

Dobrzynski, L.

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. Vigneron, E. E. Boudouti, and A. Nougaoui, “Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69, 066613 (2004).
[CrossRef]

Harteneck, B.

V. Mocella, S. Cabrini, A. Chang, P. Dardano, L. Moretti, I. Rendina, D. Olynick, B. Harteneck, and S. Dhuey, “Self-collimation of light over millimeter-scale distance in a quasi-zero-average-index metamaterial,” Phys. Rev. Lett. 102, 133902 (2009).
[CrossRef] [PubMed]

Huangfu, J.

Kawakami, S.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[CrossRef]

Kawashima, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[CrossRef]

Kivshar, Y.

I. Shadrivov, A. Sukhorukov, and Y. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2009).
[CrossRef]

Kocaman, S.

S. Kocaman, R. Chatterjee, N. Panoiu, J. Mcmillan, M. Yu, R. Osgood, D. Kwong, and C. Wong, “Observation of zeroth-order band gaps in negative-refraction photonic crystal superlattices at near-infrared frequencies,” Phys. Rev. Lett. 102, 203905 (2009).
[CrossRef] [PubMed]

Kong, J.

Kosaka, H.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[CrossRef]

Krayzel, F.

F. Krayzel, R. Poll`es, A. Moreau, and M. Mihailovic, “Simulation and analysis of exotic non-specular phenomena,” J. Europ. Opt. Soc.: Rap. Pub. 5, 10025 (2010).
[CrossRef]

Kwong, D.

S. Kocaman, R. Chatterjee, N. Panoiu, J. Mcmillan, M. Yu, R. Osgood, D. Kwong, and C. Wong, “Observation of zeroth-order band gaps in negative-refraction photonic crystal superlattices at near-infrared frequencies,” Phys. Rev. Lett. 102, 203905 (2009).
[CrossRef] [PubMed]

Li, J.

J. Li, L. Zhou, C. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 83901 (2003).
[CrossRef]

Mart’inez-Ricci, M.

Mcmillan, J.

S. Kocaman, R. Chatterjee, N. Panoiu, J. Mcmillan, M. Yu, R. Osgood, D. Kwong, and C. Wong, “Observation of zeroth-order band gaps in negative-refraction photonic crystal superlattices at near-infrared frequencies,” Phys. Rev. Lett. 102, 203905 (2009).
[CrossRef] [PubMed]

Mihailovic, M.

F. Krayzel, R. Poll`es, A. Moreau, and M. Mihailovic, “Simulation and analysis of exotic non-specular phenomena,” J. Europ. Opt. Soc.: Rap. Pub. 5, 10025 (2010).
[CrossRef]

Mocella, V.

V. Mocella, S. Cabrini, A. Chang, P. Dardano, L. Moretti, I. Rendina, D. Olynick, B. Harteneck, and S. Dhuey, “Self-collimation of light over millimeter-scale distance in a quasi-zero-average-index metamaterial,” Phys. Rev. Lett. 102, 133902 (2009).
[CrossRef] [PubMed]

Monsoriu, J.

Moreau, A.

F. Krayzel, R. Poll`es, A. Moreau, and M. Mihailovic, “Simulation and analysis of exotic non-specular phenomena,” J. Europ. Opt. Soc.: Rap. Pub. 5, 10025 (2010).
[CrossRef]

Moretti, L.

V. Mocella, S. Cabrini, A. Chang, P. Dardano, L. Moretti, I. Rendina, D. Olynick, B. Harteneck, and S. Dhuey, “Self-collimation of light over millimeter-scale distance in a quasi-zero-average-index metamaterial,” Phys. Rev. Lett. 102, 133902 (2009).
[CrossRef] [PubMed]

Nefedov, I.

I. Nefedov, and S. Tretyakov, “Photonic band gap structure containing metamaterial with negative permittivity and permeability,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66, 036611 (2002).
[CrossRef]

Notomi, M.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[CrossRef]

Nougaoui, A.

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. Vigneron, E. E. Boudouti, and A. Nougaoui, “Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69, 066613 (2004).
[CrossRef]

Olynick, D.

V. Mocella, S. Cabrini, A. Chang, P. Dardano, L. Moretti, I. Rendina, D. Olynick, B. Harteneck, and S. Dhuey, “Self-collimation of light over millimeter-scale distance in a quasi-zero-average-index metamaterial,” Phys. Rev. Lett. 102, 133902 (2009).
[CrossRef] [PubMed]

Osgood, J.

Osgood, R.

S. Kocaman, R. Chatterjee, N. Panoiu, J. Mcmillan, M. Yu, R. Osgood, D. Kwong, and C. Wong, “Observation of zeroth-order band gaps in negative-refraction photonic crystal superlattices at near-infrared frequencies,” Phys. Rev. Lett. 102, 203905 (2009).
[CrossRef] [PubMed]

Panoiu, N.

S. Kocaman, R. Chatterjee, N. Panoiu, J. Mcmillan, M. Yu, R. Osgood, D. Kwong, and C. Wong, “Observation of zeroth-order band gaps in negative-refraction photonic crystal superlattices at near-infrared frequencies,” Phys. Rev. Lett. 102, 203905 (2009).
[CrossRef] [PubMed]

N. Panoiu, J. Osgood, S. Zhang, and S. Brueck, “Zero-n bandgap in photonic crystal superlattices,” J. Opt. Soc. Am. B 23, 506–513 (2006).
[CrossRef]

Poll`es, R.

F. Krayzel, R. Poll`es, A. Moreau, and M. Mihailovic, “Simulation and analysis of exotic non-specular phenomena,” J. Europ. Opt. Soc.: Rap. Pub. 5, 10025 (2010).
[CrossRef]

Ran, L.

Rendina, I.

V. Mocella, S. Cabrini, A. Chang, P. Dardano, L. Moretti, I. Rendina, D. Olynick, B. Harteneck, and S. Dhuey, “Self-collimation of light over millimeter-scale distance in a quasi-zero-average-index metamaterial,” Phys. Rev. Lett. 102, 133902 (2009).
[CrossRef] [PubMed]

Sato, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[CrossRef]

Shadrivov, I.

I. Shadrivov, A. Sukhorukov, and Y. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2009).
[CrossRef]

Shen, L.

Sheng, P.

J. Li, L. Zhou, C. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 83901 (2003).
[CrossRef]

Silvestre, E.

Sukhorukov, A.

I. Shadrivov, A. Sukhorukov, and Y. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2009).
[CrossRef]

Tamamura, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[CrossRef]

Tomita, A.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[CrossRef]

Tretyakov, S.

I. Nefedov, and S. Tretyakov, “Photonic band gap structure containing metamaterial with negative permittivity and permeability,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66, 036611 (2002).
[CrossRef]

Vigneron, J.

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. Vigneron, E. E. Boudouti, and A. Nougaoui, “Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69, 066613 (2004).
[CrossRef]

Wong, C.

S. Kocaman, R. Chatterjee, N. Panoiu, J. Mcmillan, M. Yu, R. Osgood, D. Kwong, and C. Wong, “Observation of zeroth-order band gaps in negative-refraction photonic crystal superlattices at near-infrared frequencies,” Phys. Rev. Lett. 102, 203905 (2009).
[CrossRef] [PubMed]

Yu, M.

S. Kocaman, R. Chatterjee, N. Panoiu, J. Mcmillan, M. Yu, R. Osgood, D. Kwong, and C. Wong, “Observation of zeroth-order band gaps in negative-refraction photonic crystal superlattices at near-infrared frequencies,” Phys. Rev. Lett. 102, 203905 (2009).
[CrossRef] [PubMed]

Yuan, Y.

Zhang, S.

Zhou, L.

J. Li, L. Zhou, C. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 83901 (2003).
[CrossRef]

Appl. Phys. Lett. (2)

I. Shadrivov, A. Sukhorukov, and Y. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2009).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[CrossRef]

J. Europ. Opt. Soc.: Rap. Pub. (1)

F. Krayzel, R. Poll`es, A. Moreau, and M. Mihailovic, “Simulation and analysis of exotic non-specular phenomena,” J. Europ. Opt. Soc.: Rap. Pub. 5, 10025 (2010).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Express (1)

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (2)

I. Nefedov, and S. Tretyakov, “Photonic band gap structure containing metamaterial with negative permittivity and permeability,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66, 036611 (2002).
[CrossRef]

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. Vigneron, E. E. Boudouti, and A. Nougaoui, “Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69, 066613 (2004).
[CrossRef]

Phys. Rev. Lett. (3)

J. Li, L. Zhou, C. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 83901 (2003).
[CrossRef]

S. Kocaman, R. Chatterjee, N. Panoiu, J. Mcmillan, M. Yu, R. Osgood, D. Kwong, and C. Wong, “Observation of zeroth-order band gaps in negative-refraction photonic crystal superlattices at near-infrared frequencies,” Phys. Rev. Lett. 102, 203905 (2009).
[CrossRef] [PubMed]

V. Mocella, S. Cabrini, A. Chang, P. Dardano, L. Moretti, I. Rendina, D. Olynick, B. Harteneck, and S. Dhuey, “Self-collimation of light over millimeter-scale distance in a quasi-zero-average-index metamaterial,” Phys. Rev. Lett. 102, 133902 (2009).
[CrossRef] [PubMed]

Other (3)

J. Joannopoulos, S. Johnson, D. Winn, and R. Meade, Photonic crystals: molding the flow of light, 2nd edition, (Princeton University Press 2008).

Throughout this paper, the inverse Fourier Transform is defined by TF−1(f (α)) = R ∞ −∞ f (x)exp(iαx)dα.

M. Born, E. Wolf, and A. B. Bhatia, Principles of optics: Electromagnetic theory of propagation, interference and diffraction of light (Cambridge University press, Cambridge 2000).
[PubMed]

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Figures (2)

Fig. 1
Fig. 1

(a) and (c): dispersion relation given by Eq. (1) (black curve) and its Taylor expansions (Eq. (2)) (red curve). (b) and (d): reflectance of the structure composed of N = 50 periods. Black curves correspond to the nondispersive case and red curves to the dispersive case. (a) and (b): the structure is characterized by n1 = 1, n 2 0 = 2, η1 = 1, η2 = 0.5, d1 = 2D/3, d2 = D/3 and D/λ0 = 13/8 (which corresponds to D = 1.625λ0). (c) and (d): the parameters are the same but D/λ0 = 3/2.

Fig. 2
Fig. 2

Modulus of the field when a PBGM, embedded in a air-medium (n0 = 1), is illuminated by a beam. (a) The parameters of the PBGM are N = 200, d1 = d2 = D/2, n1 = 2, n 2 0 = 2, η1 = 1, η2 = 0.5 and D = λ0. (b) The parameters of the PBGM are N = 400, d1 = D/3, d2 = 2D/3, n1 = 1, n 0 2 = 0.5, η1 = 1, η2 = 0.5 and D = 3λ0.

Equations (11)

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

cos ( κ D ) = cos ( n ¯ k D ) + ( η 1 η 2 ) 2 2 η 1 η 2 sin ( n 1 k d 1 ) sin ( | n 2 | k d 2 ) ,
cos ( κ D ) 1 + Γ x m 2 + Γ Δ n 2 ( λ ) k d 2 x m ( Δ n 2 ( λ ) k d 2 ) 2 / 2 ,
Δ m λ = m A ( η 2 2 η 1 2 ) ( λ 0 Λ m ) A 2 η 2 η 1 m ( A + m ) ( η 2 η 1 ) 2 ,
U ( x , L ) = T F 1 { U i ( α ) ( P 1 ( α , d 1 ) P 2 ( α , d 2 ) ) N } .
U ( x , L ) = T F 1 { U i ( α ) P ˜ ( α , D ) N } .
U ( x , L ) = W 0 ω ¯ ( L ) e ( x W ( L ) ) 2 e i ω c n L e i φ ( x , L ) ,
W ( L ) = W 0 1 + θ 0 2 ( N D ) 2 1 n 2 ,
d 1 n 1 + d 2 n 2 = 0 .
U ( x , L ) = T F 1 { U i ( α ) P 0 ( α , f ) P ˜ ( α , D ) N P 0 ( α , f ) } ,
W ( f ) = W 0 1 + θ 0 2 ( f n 0 + N D 1 n + f n 0 ) 2 .
f + f = N D 1 n n 0 .

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