## Unusual transmission bands of one-dimensional photonic crystals containing single-negative materials

Optics Express, Vol. 17, Issue 22, pp. 20333-20341 (2009)

http://dx.doi.org/10.1364/OE.17.020333

Acrobat PDF (276 KB)

### Abstract

Unusual transmission bands are found in one-dimensional photonic crystals composed of alternating layers of positive-index materials and single-negative (permittivity- or permeability-negative) materials. By varying the thicknesses of the positive-index material layers, the number and central frequencies of these transmission bands can be tuned. On the other hand, by varying the thicknesses of the single-negative material layers, only the bandwidths of these transmission bands will change. Furthermore, omnidirectional transmission bands for TE or TM polarization can be realized from these periodic photonic structures.

© 2009 OSA

## 1. Introduction

1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. **58**(20), 2059–2062 (1987). [CrossRef] [PubMed]

4. P. Lodahl, A. Floris Van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature **430**(7000), 654–657 (2004). [CrossRef] [PubMed]

5. S. Fan, P. R. Villeneuve, J. D. Joanopulos, and H. A. Haus, “Channel Drop Tunneling through Localized States,” Phys. Rev. Lett. **80**(5), 960–963 (1998). [CrossRef]

6. S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature **407**(6804), 608–610 (2000). [CrossRef] [PubMed]

7. Z. S. Wang, L. Wang, Y. G. Wu, L. Y. Chen, X. S. Chen, and W. Lu, “Multiple channeled phenomena in heterostructures with defects mode,” Appl. Phys. Lett. **84**(10), 1629–1631 (2004). [CrossRef]

8. Y. H. Chen, “Independent modulation of defect modes in fractal photonic crystals with multiple defect layers,” J. Opt. Soc. Am. B **26**(4), 854–857 (2009). [CrossRef]

*ε*) and permeability (

*μ*) is negative, has been realized [9

9. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs I, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. **76**(25), 4773–4776 (1996). [CrossRef] [PubMed]

10. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. **47**(11), 2075–2084 (1999). [CrossRef]

*ε*but positive

*μ*and the mu-negative (MNG) media with negative

*μ*but positive

*ε*. It is well known that the EM waves in the SNG materials are decaying since their wave vectors are imaginary. However, tunneling modes can be obtained in 1D photonic structures composed of alternate MNG and ENG materials [11

11. G. S. Guan, H. T. Jiang, H. Q. Li, Y. W. Zhang, H. Chen, and S. Y. Zhu, “Tunneling modes of photonic heterostructures consisting of single-negative materials,” Appl. Phys. Lett. **88**(21), 211112 (2006). [CrossRef]

13. Y. H. Chen, “Frequency response of resonance modes in heterostructures composed of single-negative materials,” J. Opt. Soc. Am. B **25**(11), 1794–1799 (2008). [CrossRef]

## 2. The model

*AB*)

*and (*

^{s}*AC*)

*, where*

^{s}*A*represents a layer of PIM with the thickness of

*d*and

_{A}*B*(

*C*) represents a layer of ENG (MNG) material with the thickness of

*d*(

_{B}*d*), and

_{C}*s*is the number of periods. We assume that the relative permittivity and permeability take the forms ofin ENG materials andin MNG materials, where

*ω*and

_{ep}*ω*

_{0}are, respectively, the electronic plasma frequency and the magnetic resonance frequency. These kinds of dispersion for

*ε*and

_{B}*μ*can be realized in special metamaterials [14

_{C}14. T. Fujishige, C. Caloz, and T. Itoh, “Experimental demonstration of transparency in ENG-MNG pair in a CRLH transmission-line implementation,” Microw. Opt. Technol. Lett. **46**(5), 476–481 (2005). [CrossRef]

17. H. Zhang, X. Chen, Y. Q. Li, Y. Fu, and N. Yuan, “The Bragg gap vanishing phenomena in one-dimensional photonic crystals,” Opt. Express **17**(10), 7800–7806 (2009). [CrossRef] [PubMed]

*μ*=

_{A}*ε*= 1,

_{A}*μ*=

_{b}*ε*= 1, and

_{c}*ω*=

_{ep}*πc*/

*d*,

*ω*

_{0}= 4

*c*/

*d*,

*F*= 0.56.

*θ*with +

*z*direction, as show in Fig. 1 . For the transverse electric (TE) [or transverse magnetic (TM)] wave, the electric field [or the magnetic field] is in the

*x*direction. For an infinite periodic structure (

*AB*)

*(*

^{s}18. M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **60**(44 Pt B), 4891–4898 (1999). [CrossRef]

*β*is the

_{z}*z*component of Bloch wave vector,

*i*=

*A*,

*B*) is the

*z*component of the wave vector, and

*c*is the light speed in the vacuum. For TE wave,

*AB*)

*. Similarly, the dispersion relation for infinite structure (*

^{s}*AC*)

*can be obtained by replacing*

^{s}*d*,

_{B}*k*and

_{Bz}*q*of Eq. (3) by

_{B}*d*,

_{C}*k*and

_{Cz}*q*, respectively.

_{C}## 3. Numerical results and discussion

*AB*)

*containing ENG materials. Figure 2 shows the dependence of the band structure on the ratio of the thicknesses*

^{s}*d*/

_{A}*d*under

_{B}*d*=

_{B}*d*at normal incidence. The gray areas represent the regions of propagating states, whereas the white areas represent regions containing evanescent states. It can be seen from Fig. 2 that transmission bands appear in the ENG frequency range 0 <

*ω*/

*ω*< 1. As

_{ep}*d*increases, more and more transmission bands emerge, and these transmission bands all shift to lower frequencies. The physical mechanism of the transmission bands is different from the tunneling mechanism of the ENG-MNG multilayered periodic structure [11

_{A}11. G. S. Guan, H. T. Jiang, H. Q. Li, Y. W. Zhang, H. Chen, and S. Y. Zhu, “Tunneling modes of photonic heterostructures consisting of single-negative materials,” Appl. Phys. Lett. **88**(21), 211112 (2006). [CrossRef]

*k*. If the phase difference is equal to odd times of

_{Az}d_{A}*π*, these reflected waves will cancel each other and thus the transmission bands form. Therefore, the frequencies of the transmission bands should be close to the values of

*ω*satisfying

*d*= 2

_{A}*d*, it can be got from Eq. (5) that

*ω*/

*ω*= 1/4, 3/4 corresponding to

_{ep}*t*= 0, 1. The two corresponding transmission bands exist in Fig. 2 when

*d*= 2

_{A}*d*, although the frequencies of them are a little different from the approximately theoretical values. For another example, when

*d*= 3

_{A}*d*, it can be got from Eq. (5) that

*ω*/

*ω*= 1/6, 3/6, 5/6 corresponding to

_{ep}*t*= 0, 1, 2. It can be seen from Fig. 2 that the three corresponding transmission bands exist when

*d*= 3

_{A}*d*. Furthermore, from Eq. (5) we can obtain, as

*d*increases,

_{A}*ω*decreases and the transmission band shifts to lower frequency, in accordance with Fig. 2.

*d*/

_{B}*d*when

_{A}*d*is fixed. In Fig. 3 , we choose

_{A}*d*= 2.5

_{A}*d*. As shown in Fig. 3, three transmission bands appear, their central frequencies are 0.32

*ω*, 0.63

_{ep}*ω*, and 0.91

_{ep}*ω*, respectively. As

_{ep}*d*increases, the central frequencies of these bands remain invariant while the bandwidths of the transmission bands decrease. Such properties can also be understood from Eq. (5), the central frequencies of the transmission bands depend on

_{B}*d*, not

_{A}*d*. The properties of Figs. 2 and 3 can be used to design filters with multiple transmission channels, whose bandwidths can be varied conveniently.

_{B}*d*in finite structure (

_{B}*AB*)

^{12}is shown in Fig. 4 . It can be seen from Fig. 4 that the transmission bands will be narrowed with the increasing of

*d*, the same as the results in Fig. 3.

_{B}*d*= 2.5

_{A}*d*and

*d*=

_{B}*d*. It is seen that the transmission bands are sensitive to the incident angle. The transmission bands will shift to higher frequencies as the incident angle increases. It means that the dispersion relations of such transmission bands are all positive type [19

19. K. Y. Xu, X. G. Zheng, and W. L. She, “Properties of defect modes in one-dimensional photonic crystals containing a defect layer with a negative refractive index,” Appl. Phys. Lett. **85**(25), 6089–6091 (2004). [CrossRef]

20. Y. H. Chen, J. W. Dong, and H. Z. Wang, “Conditions of near-zero dispersion of defect modes in one-dimensional photonic crystals containing negative-index materials,” J. Opt. Soc. Am. B **23**(4), 776–781 (2006). [CrossRef]

*d*, as shown in Fig. 6 . It can be seen from Fig. 6(a)-6(c) that the dispersion relation of band I for TM wave changes from negative to near-zero then to positive type as

_{A}*d*changes from 0.5

_{A}*d*to 0.82

*d*and onward to 1.5

*d*. As shown in Fig. 6(b), the transmission band remains almost invariant as the incident angle varies for TM wave. The transmission band with weak incident angle dependence may be useful in applications, such as omnidirectional filters.

*AC*)

*containing MNG materials are also studied. It is found that the properties of the transmission bands in (*

^{s}*AC*)

*for TE (TM) polarization are similar to those in (*

^{s}*AB*)

*for TM (TE) polarization. So the properties of the transmission bands of structure (*

^{s}*AC*)

*can be easily obtained from those of (*

^{s}*AB*)

*.*

^{s}*AC*)

*with*

^{s}*d*= 0.36

_{A}*d*and

*d*= 0.85

_{C}*d*. The MNG frequency range is 1 <

*ω*/

*ω*

_{0}< 1.5. It is seen from Fig. 7 that the transmission band is insensitive to the incident angle for TE wave. The dependence of the transmission band on incident angle in finite structure (

*AC*)

^{12}is also calculated, as shown in Fig. 8 . As the incident angle

*θ*varies, the central frequency of the transmission band remains almost invariant.

9. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs I, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. **76**(25), 4773–4776 (1996). [CrossRef] [PubMed]

10. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. **47**(11), 2075–2084 (1999). [CrossRef]

14. T. Fujishige, C. Caloz, and T. Itoh, “Experimental demonstration of transparency in ENG-MNG pair in a CRLH transmission-line implementation,” Microw. Opt. Technol. Lett. **46**(5), 476–481 (2005). [CrossRef]

17. H. Zhang, X. Chen, Y. Q. Li, Y. Fu, and N. Yuan, “The Bragg gap vanishing phenomena in one-dimensional photonic crystals,” Opt. Express **17**(10), 7800–7806 (2009). [CrossRef] [PubMed]

15. L. W. Zhang, Y. W. Zhang, L. He, H. Q. Li, and H. Chen, “Experimental study of photonic crystals consisting of ε-negative and μ-negative materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **74**(5), 056615 (2006). [CrossRef]

17. H. Zhang, X. Chen, Y. Q. Li, Y. Fu, and N. Yuan, “The Bragg gap vanishing phenomena in one-dimensional photonic crystals,” Opt. Express **17**(10), 7800–7806 (2009). [CrossRef] [PubMed]

*L*

_{0}and

*C*

_{0}periodically loaded with discrete lumped element components,

*L*and

_{i}*C*. Such structure exhibits a macroscopic behavior rigorously expressed with the constitutive parameters

_{i}*ε*and

*μ*. The CRLH TL fabricated by cascading the unit cells of Fig. 9(a) periodically is effectively homogeneous in a certain range of frequencies [15

15. L. W. Zhang, Y. W. Zhang, L. He, H. Q. Li, and H. Chen, “Experimental study of photonic crystals consisting of ε-negative and μ-negative materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **74**(5), 056615 (2006). [CrossRef]

**17**(10), 7800–7806 (2009). [CrossRef] [PubMed]

*p*is a structure constant and

*i*= 1, 2 denotes the different type of CRLH TL.

*γ*and

_{e}*γ*denote the respective electric and magnetic damping factors that contribute to the absorption and losses. Here we use the material parameters the same as those in Ref [15

_{m}15. L. W. Zhang, Y. W. Zhang, L. He, H. Q. Li, and H. Chen, “Experimental study of photonic crystals consisting of ε-negative and μ-negative materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **74**(5), 056615 (2006). [CrossRef]

*ε*= 4.75, and relative permeability

_{r}*μ*= 1.0. In this situation,

_{r}*p*= 4.05. We choose

*L*

_{1}= 13 nH,

*C*

_{1}= 8.6 pF and

*d*

_{1}= 5mm for the ENG TL, and

*L*

_{2}= 111 nH,

*C*

_{2}= 8.6 pF and

*d*

_{2}= 5mm for the PIM TL. The calculated relative permittivity and permeability of the ENG TL and PIM TL according to Eq. (6) in the lossless case are shown in Fig. 9(b). It is clearly shown that

*ε*and

*μ*are dependent on the frequency. As shown in Fig. 9(b), the frequency range, where

*ε*<0 and

*μ*>0, for ENG TL is 0.817-1.961 GHz, and the frequency range, where

*ε*>0 and

*μ*>0, for PIM TL is over 0.817 GHz.

*γ*=

_{e}*γ*= 1 × 10

_{m}^{7}Hz. Firstly, we consider the dependence of the transmission band of the (PIM

_{5}ENG

_{4})

^{2}TL on the incident angle, where the subscripts “5” and “4” represent the number of PIM TL and ENG TL units in one period, respectively, and the superscript “2” represents the number of periods. As shown in Fig. 10 , the central frequency of the transmission band is insensitive to the incident angle for TM wave. Such property is in accord with that of infinite structure in Fig. 6(b). The losses will only slightly decrease the transmittance.

_{5}ENG

_{4})

^{2}, (PIM

_{7}ENG

_{4})

^{2}, and (PIM

_{10}ENG

_{4})

^{2}, as shown in Fig. 11(a) . It is shown from Fig. 11(a) that, as the lengths of PIM TLs increase, the transmission bands shift to lower frequencies, the same as the property in Fig. 2. Figure 11(b) shows the simulated spectra of (PIM

_{10}ENG

_{4})

^{2}, (PIM

_{10}ENG

_{4})

^{3}, and (PIM

_{10}ENG

_{4})

^{4}based on CRLH TLs. It can be seen from in Fig. 11(b) that there are

*S*– 1 peaks in each transmission band in the structure (PIM-ENG)

*. Such properties can be explained as follows. From our previous analysis, we know that reflected wave exists at each interface from PIM to SNG layer. So there are*

^{S}*S*reflected waves exist in structure (PIM-ENG)

*. According to the wave optics, for*

^{S}*S*light beams with the same amplitude

*A*

_{0}and phase difference Δ

*ϕ*between two neighboring beams, if they superpose, the total amplitude is equal to

*A*

_{0}[sin(

*S*Δ

*ϕ*/2)/ sin(Δ

*ϕ*/2)]. From Δ

*ϕ*= 0 to 2π, the total amplitude has

*S*-1 minimum values, corresponding to the

*S*– 1 transmission peaks. For the current structure with period number

*S*, there are

*S*reflection beams with equal phase difference but different intensities between two neighboring beams. Although the different intensities may change the frequencies of the peaks, there still exist

*S*–1 transmission peaks in each transmission band.

_{10}ENG

_{4})

^{4}, (PIM

_{10}ENG

_{5})

^{4}, and (PIM

_{10}ENG

_{6})

^{4}. As the lengths of ENG TLs increase, the transmission bands become narrower and narrower, while the central frequencies of the two bands are unchanged. Such result is in accordance with that of Fig. 3.

## 4. Conclusion

## Acknowledgements

## References and links

1. | E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. |

2. | S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. |

3. | J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: Putting a new twist on light,” Nature |

4. | P. Lodahl, A. Floris Van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature |

5. | S. Fan, P. R. Villeneuve, J. D. Joanopulos, and H. A. Haus, “Channel Drop Tunneling through Localized States,” Phys. Rev. Lett. |

6. | S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature |

7. | Z. S. Wang, L. Wang, Y. G. Wu, L. Y. Chen, X. S. Chen, and W. Lu, “Multiple channeled phenomena in heterostructures with defects mode,” Appl. Phys. Lett. |

8. | Y. H. Chen, “Independent modulation of defect modes in fractal photonic crystals with multiple defect layers,” J. Opt. Soc. Am. B |

9. | J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs I, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. |

10. | J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. |

11. | G. S. Guan, H. T. Jiang, H. Q. Li, Y. W. Zhang, H. Chen, and S. Y. Zhu, “Tunneling modes of photonic heterostructures consisting of single-negative materials,” Appl. Phys. Lett. |

12. | Y. H. Chen, J. W. Dong, and H. Z. Wang, “Twin defect modes in one-dimensional photonic crystals with a single-negative material defect,” Appl. Phys. Lett. |

13. | Y. H. Chen, “Frequency response of resonance modes in heterostructures composed of single-negative materials,” J. Opt. Soc. Am. B |

14. | T. Fujishige, C. Caloz, and T. Itoh, “Experimental demonstration of transparency in ENG-MNG pair in a CRLH transmission-line implementation,” Microw. Opt. Technol. Lett. |

15. | L. W. Zhang, Y. W. Zhang, L. He, H. Q. Li, and H. Chen, “Experimental study of photonic crystals consisting of ε-negative and μ-negative materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

16. | L. W. Zhang, Y. W. Zhang, L. He, H. Q. Li, and H. Chen, “Experimental investigation on zero-ϕ |

17. | H. Zhang, X. Chen, Y. Q. Li, Y. Fu, and N. Yuan, “The Bragg gap vanishing phenomena in one-dimensional photonic crystals,” Opt. Express |

18. | M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics |

19. | K. Y. Xu, X. G. Zheng, and W. L. She, “Properties of defect modes in one-dimensional photonic crystals containing a defect layer with a negative refractive index,” Appl. Phys. Lett. |

20. | Y. H. Chen, J. W. Dong, and H. Z. Wang, “Conditions of near-zero dispersion of defect modes in one-dimensional photonic crystals containing negative-index materials,” J. Opt. Soc. Am. B |

**OCIS Codes**

(260.2110) Physical optics : Electromagnetic optics

(160.3918) Materials : Metamaterials

(050.5298) Diffraction and gratings : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: August 5, 2009

Revised Manuscript: October 9, 2009

Manuscript Accepted: October 9, 2009

Published: October 23, 2009

**Citation**

Yihang Chen, "Unusual transmission bands of one-dimensional photonic crystals containing single-negative materials," Opt. Express **17**, 20333-20341 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-22-20333

Sort: Year | Journal | Reset

### References

- E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987). [CrossRef] [PubMed]
- S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58(23), 2486–2489 (1987). [CrossRef] [PubMed]
- J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: Putting a new twist on light,” Nature 386(6621), 143–149 (1997). [CrossRef]
- P. Lodahl, A. Floris Van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430(7000), 654–657 (2004). [CrossRef] [PubMed]
- S. Fan, P. R. Villeneuve, J. D. Joanopulos, and H. A. Haus, “Channel Drop Tunneling through Localized States,” Phys. Rev. Lett. 80(5), 960–963 (1998). [CrossRef]
- S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature 407(6804), 608–610 (2000). [CrossRef] [PubMed]
- Z. S. Wang, L. Wang, Y. G. Wu, L. Y. Chen, X. S. Chen, and W. Lu, “Multiple channeled phenomena in heterostructures with defects mode,” Appl. Phys. Lett. 84(10), 1629–1631 (2004). [CrossRef]
- Y. H. Chen, “Independent modulation of defect modes in fractal photonic crystals with multiple defect layers,” J. Opt. Soc. Am. B 26(4), 854–857 (2009). [CrossRef]
- J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76(25), 4773–4776 (1996). [CrossRef] [PubMed]
- J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999). [CrossRef]
- G. S. Guan, H. T. Jiang, H. Q. Li, Y. W. Zhang, H. Chen, and S. Y. Zhu, “Tunneling modes of photonic heterostructures consisting of single-negative materials,” Appl. Phys. Lett. 88(21), 211112 (2006). [CrossRef]
- Y. H. Chen, J. W. Dong, and H. Z. Wang, “Twin defect modes in one-dimensional photonic crystals with a single-negative material defect,” Appl. Phys. Lett. 89(14), 141101 (2006). [CrossRef]
- Y. H. Chen, “Frequency response of resonance modes in heterostructures composed of single-negative materials,” J. Opt. Soc. Am. B 25(11), 1794–1799 (2008). [CrossRef]
- T. Fujishige, C. Caloz, and T. Itoh, “Experimental demonstration of transparency in ENG-MNG pair in a CRLH transmission-line implementation,” Microw. Opt. Technol. Lett. 46(5), 476–481 (2005). [CrossRef]
- L. W. Zhang, Y. W. Zhang, L. He, H. Q. Li, and H. Chen, “Experimental study of photonic crystals consisting of ε-negative and μ-negative materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(5), 056615 (2006). [CrossRef]
- L. W. Zhang, Y. W. Zhang, L. He, H. Q. Li, and H. Chen, “Experimental investigation on zero-ϕeff gaps of photonic crystals containing single-negative materials,” Eur. Phys. J. B 62(1), 1–6 (2008). [CrossRef]
- H. Zhang, X. Chen, Y. Q. Li, Y. Fu, and N. Yuan, “The Bragg gap vanishing phenomena in one-dimensional photonic crystals,” Opt. Express 17(10), 7800–7806 (2009). [CrossRef] [PubMed]
- M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(44 Pt B), 4891–4898 (1999). [CrossRef]
- K. Y. Xu, X. G. Zheng, and W. L. She, “Properties of defect modes in one-dimensional photonic crystals containing a defect layer with a negative refractive index,” Appl. Phys. Lett. 85(25), 6089–6091 (2004). [CrossRef]
- Y. H. Chen, J. W. Dong, and H. Z. Wang, “Conditions of near-zero dispersion of defect modes in one-dimensional photonic crystals containing negative-index materials,” J. Opt. Soc. Am. B 23(4), 776–781 (2006). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.