## The Bragg gap vanishing phenomena in one-dimensional photonic crystals

Optics Express, Vol. 17, Issue 10, pp. 7800-7806 (2009)

http://dx.doi.org/10.1364/OE.17.007800

Acrobat PDF (176 KB)

### Abstract

We theoretically deduce the Bragg gap vanishing conditions in one-dimensional photonic crystals and experimentally demonstrate the *m*=0 band-gap vanishing phenomena at microwave frequencies. In the case of mismatched impedance, the Bragg gap will vanish as long as the discrete modes appear in photonic crystals containing dispersive materials, while for the matched impedance cases, Bragg gaps will always disappear. The experimental results and the simulations agree extremely well with the theoretical expectation.

© 2009 Optical Society of America

## 1. Introduction

1. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of permittivity and permeability,” Sov. Phys. Usp. **10**, 509–514 (1968). [CrossRef]

4. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science **292**, 77–79 (2001). [CrossRef] [PubMed]

5. M. W. Feise, I. V. Shadrivov, and Y. S. Kivshar, “Tunable transmission and bistability in left-handed band-gap structures,” Appl. Phys. Lett. **85**, 1451 (2004). [CrossRef]

*ψ*denotes the phase shift of a period,

_{peroid}*ψ*

_{1}and

*ψ*

_{2}respectively denote the phase shifts of two inclusion layers in a period, and

*m*is the band-gap index which is integer, including zero, negative and positive numbers. The photonic crystals stacked by RHMs will only lead to the positive modes (

*m*>0) of Bragg gaps, while the multilayered LHMs can just possess the negative modes (

*m*<0) of Bragg gaps. However, the zero modes (

*m*=0) of Bragg gaps can just exist in the alternating layers of LHMs and RHMs.

6. S. Enoch, G. Tayeb, P. Sabouroux, N. Guerin, and P. Vincent, “A Metamaterial for Directive Emission,” Phys. Rev. Lett. **89**, 213902 (2002). [CrossRef] [PubMed]

7. L. Sungjoon, C. Caloz, and T. Itoh, “Metamaterial-based electronically controlled transmission-line structure as a novel leaky-wave antenna with tunable radiation angle and beamwidth,” IEEE Trans. Microwave Theory Tech. **53**, 161–172 (2005). [CrossRef]

*m*=0 band-gap vanishing phenomena at microwave frequencies. The

*m*=0 band-gap, also called the zero averaged refractive index gap (the zero-

*n*̄ band-gap) [8–10

8. J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. **90**, 083901 (2003). [CrossRef] [PubMed]

*m*=0 band-gap is completely closed up in experiment. The experimental results and the simulations agree extremely well with the theoretical expectation.

## 2. Bragg gap vanishing conditions

*d*

_{1}and

*d*

_{2}are the widths of the two inclusion layers respectively, and

*a*=

*d*

_{1}+

*d*

_{2}is the lattice constant. The dispersion relation [8

8. J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. **90**, 083901 (2003). [CrossRef] [PubMed]

9. Y. Weng, Z. G. Wang, and H. Chen, “Band structures of one-dimensional subwavelength photonic crystals containing metamaterials,” Phys. Rev. E. **75**, 046601 (2007). [CrossRef]

*s*-polarized waves and

*p*-polarized waves. Here,

*θ*

_{1,2}are the angles between the propagating waves and normal to the interface in the two media, respectively. The parameter

*k*

_{∥}is the parallel part of the wave vector and

*β*is the Bloch propagation constant in the first Brillouin zone. In addition, the parameters

*F*represent effective characteristic impedances of two media.

_{i}*m*th mode of Bragg gap occur at the frequency

*ω*

_{0}, then

*F*

_{1}=

*F*

_{2}), ∣cos(

*βa*)∣

*ω*

_{0}∣ = 1 is obtained. If the constitutive materials are both nondispersive, the dispersion relation in Eq. (2) is given by

*ω*is very close

*ω*

_{0}. Equation (4) indicates that the Bloch propagation constant

*β*is real around

*ω*

_{0}, thus the mth gap is closed up. If one of the constitutive materials or both are dispersive, then

*ω*

_{0}. This behavior exhibits the disappearance of the

*m*th gap.

*ψ*

_{1}=

*α*

_{1}

*d*

_{1}=

*lπ*and

*ψ*

_{2}=

*α*

_{2}

*d*

_{2}= (

*m*-

*l*)

*π*, where

*l*and

*m*are integral numbers, Eq. (6) indicates that the Bloch propagation constant

*β*has no real solutions around

*ω*

_{0}, which renders the appearance of the

*m*th gap. These discrete solutions for the non-dispersive media will exhibit singular frequency propagations with no sidelobes in the Bragg gaps [8

8. J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. **90**, 083901 (2003). [CrossRef] [PubMed]

*βa*)∣

*ω*

_{0}∣ = 1 valid and the approximate relation at

*ω*is obtained

*A*and

*B*are infinitesimal for

*ω*very close to

*ω*

_{0}. Thus the dispersion relation in Eq. (2) becomes

*β*is real around

*ω*

_{0}for ∣cos(

*β*)∣

_{a}*ω*∣<1. In other words, if one of two materials or both are dispersive, the passband of discrete modes will be enlarged to cover the Bragg gaps (the band-gaps entirely disappear).

## 3. Experiments of the m=0 band-gap vanishing phenomena

2. D. R. Smith, W. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “A composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. **84**, 4184–4187 (2000). [CrossRef] [PubMed]

11. C. Caloz and T. Itoh, “Transmission line approach of Left-Handed (LH) materials and microstrip implementation of an artificial LH transmission Line,” IEEE Trans. Antennas Propag. **52**, 1159 (2004). [CrossRef]

12. G. V. Eleftheriades, A.K. Iyer, and P.C. Kremer,”Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. Microwave Theory Tech. **50**, 2702–2712 (2002). [CrossRef]

13. A. Sanada, C. Caloz, and T. Itoh, “Characteristics of the composite right/left-handed transmission lines,” IEEE Microwave Wireless Components Lett. **14**, 68–71, 2004. [CrossRef]

*h*=0.5 mm, relative permittivity

*ε*=2.65 and relative permeability

_{r}*μ*= 1.

_{r}*w*=1.37 mm for

_{1}*Z*

_{1}= 50Ω and the length

*d*=6 mm, and the lumped-element components are chosen as

_{1}*C*=3.3 pF and

_{L}*L*= 8.2 nH. In Fig. 2 (b) , the host transmission line of the CRH-TL are designed with the width

_{L}*w*= 5mm for

_{2}*Z*

_{2}= 18Ω and the length

*d*=7 mm, and the lumped-element components are chosen as

_{2}*C*=9.1 pF and

_{R}*L*=2.7 nH. The CRH-TL structures are used here to realize RH phase shift

_{R}*ψ*

_{2}=

*π*with extraordinary shorter length of the transmission line.

11. C. Caloz and T. Itoh, “Transmission line approach of Left-Handed (LH) materials and microstrip implementation of an artificial LH transmission Line,” IEEE Trans. Antennas Propag. **52**, 1159 (2004). [CrossRef]

12. G. V. Eleftheriades, A.K. Iyer, and P.C. Kremer,”Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. Microwave Theory Tech. **50**, 2702–2712 (2002). [CrossRef]

*k*and

_{i}*Z*; (i.e.

_{i}*i*=1, 2) are the wave numbers and the characteristic impedances of the host transmission lines for the CRLH-TL and the CRH-TL, respectively. The parameters

*β*and

_{CRLH}*β*are the Bloch propagation constants of the CRLH-TL and the CRH-TL, respectively.

_{CRH}*L*and series capacitors

_{L}*C*. As the balanced condition

_{L}*ψ*

_{1}= -

*π*and three CRH-TL cells for

*ψ*

_{2}=

*π*as a period (CRLH

_{3}CRH

_{3}), the periodic structure will exhibit a discrete mode in the

*m*=0 gap at 0.84 GHz. A photograph of the fabricated (CRLH

_{3}CRH

_{3})

_{7}using real lumped-element components is illustrated in Fig. 4, where the subscript “7” represents the period number. This periodic structure is a typical photonic crystal, because they are made of two different effectively homogeneous media (the average electrical length of a unit cell CRLH or CRH is much smaller than the guide wavelength

*λ*in the certain range of frequencies).

_{g}_{3}(-

*ψ*

_{1}) and CRH

_{3}(-

*ψ*

_{2}) compared with a period CRLH

_{3}CRH

_{3}(-

*ψ*). The zero phase shifts of CRLH-TL and CRH-TL occur respectively at the balanced point (2.25 GHz) and zero frequency point (0 GHz). Therefore, at the frequency 0.84 GHz, the phase shift of unit cell CRLH

_{3}happens to be

*ψ*

_{1}=-

*π*, while CRH

_{3}is

*ψ*

_{2}=

*π*, corresponding to the phase shift of a period CRLH

_{3}CRH

_{3}

*ψ*to be zero. Figure 5(a) shows the simulated and measured transmission properties of the photonic crystals (CRLH

_{3}CRH

_{3})

_{7}. Both the experimental results and the simulation show that there is a broad passband around the frequency of 0.84 GHz. The calculated dispersion relation of infinite periodic structure CRLH

_{3}CRH

_{3}-TL is provided in Fig. 6 and the

*m*=0 band-gap is entirely closed up. This behaviors can be explained by that the dispersive property of CRLH-TL and CRH-TL enlarge the passband of discrete modes to cover the

*m*=0 band-gap.

*ψ*= -

*π*for the band-gap index

*m*=-1, while the band-gap at the frequency 1.20 GHz is in the effective right-handed passband, corresponding to the phase shift

*ψ*=

*π*for the index

*m*=+1.

*w*=137 mm for

_{L}*Z*

_{1}= 50Ω and length

*d*=6 mm, whereas the LH unit cell is designed with series capacitors

_{1}*C*=3.3 pF and shunt inductors

_{L}*L*=8.2 nH for the balanced condition

_{L}_{12}, as well as the simulated phase delay of a period (-

*ψ*). It is seen that there is no gap around the frequencies 2.25 GHz with the phase shift

_{peroid}*ψ*= 0. The experimental results and the simulations agree extremely well with the theoretical expectation for the matched impedance case.

_{peroid}## 4. Conclusion

## Acknowledgments

## References and links

1. | V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of permittivity and permeability,” Sov. Phys. Usp. |

2. | D. R. Smith, W. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “A composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. |

3. | J. B. Pendry, “Electromagnetic materials enter the negative age,” Physics World. |

4. | R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science |

5. | M. W. Feise, I. V. Shadrivov, and Y. S. Kivshar, “Tunable transmission and bistability in left-handed band-gap structures,” Appl. Phys. Lett. |

6. | S. Enoch, G. Tayeb, P. Sabouroux, N. Guerin, and P. Vincent, “A Metamaterial for Directive Emission,” Phys. Rev. Lett. |

7. | L. Sungjoon, C. Caloz, and T. Itoh, “Metamaterial-based electronically controlled transmission-line structure as a novel leaky-wave antenna with tunable radiation angle and beamwidth,” IEEE Trans. Microwave Theory Tech. |

8. | J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. |

9. | Y. Weng, Z. G. Wang, and H. Chen, “Band structures of one-dimensional subwavelength photonic crystals containing metamaterials,” Phys. Rev. E. |

10. | H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative-index materials,” Appl. Phys. Lett. |

11. | C. Caloz and T. Itoh, “Transmission line approach of Left-Handed (LH) materials and microstrip implementation of an artificial LH transmission Line,” IEEE Trans. Antennas Propag. |

12. | G. V. Eleftheriades, A.K. Iyer, and P.C. Kremer,”Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. Microwave Theory Tech. |

13. | A. Sanada, C. Caloz, and T. Itoh, “Characteristics of the composite right/left-handed transmission lines,” IEEE Microwave Wireless Components Lett. |

**OCIS Codes**

(230.1480) Optical devices : Bragg reflectors

(350.3618) Other areas of optics : Left-handed materials

(160.5298) Materials : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: March 12, 2009

Revised Manuscript: April 10, 2009

Manuscript Accepted: April 16, 2009

Published: April 27, 2009

**Citation**

Hui Zhang, Xi Chen, Youquan Li, Yunqi Fu, and Naichang Yuan, "The Bragg gap vanishing phenomena in one-dimensional photonic crystals," Opt. Express **17**, 7800-7806 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-10-7800

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### References

- V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of permittivity and permeability," Sov. Phys. Usp. 10, 509-514 (1968). [CrossRef]
- D. R. Smith, W. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "A composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000). [CrossRef] [PubMed]
- J. B. Pendry, "Electromagnetic materials enter the negative age," Physics World. 14, 47 (2001).
- R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental Verification of a Negative Index of Refraction," Science 292, 77-79 (2001). [CrossRef] [PubMed]
- M. W. Feise, I. V. Shadrivov, and Y. S. Kivshar, "Tunable transmission and bistability in left-handed band-gap structures," Appl. Phys. Lett. 85, 1451 (2004). [CrossRef]
- S. Enoch, G. Tayeb, P. Sabouroux, N. Guerin, and P. Vincent, "A Metamaterial for Directive Emission," Phys. Rev. Lett. 89, 213902 (2002). [CrossRef] [PubMed]
- L. Sungjoon; C. Caloz, T. Itoh, "Metamaterial-based electronically controlled transmission-line structure as a novel leaky-wave antenna with tunable radiation angle and beamwidth," IEEE Trans. Microwave Theory Tech. 53, 161-172 (2005). [CrossRef]
- J. Li, L. Zhou, C. T. Chan, and P. Sheng, "Photonic band gap from a stack of positive and negative index materials," Phys. Rev. Lett. 90, 083901 (2003). [CrossRef] [PubMed]
- Y. Weng, Z. G. Wang, and H. Chen, "Band structures of one-dimensional subwavelength photonic crystals containing metamaterials," Phys. Rev. E. 75, 046601 (2007). [CrossRef]
- H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, and S. Y. Zhu, "Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative-index materials," Appl. Phys. Lett. 83, 5386 (2003). [CrossRef]
- C. Caloz and T. Itoh, "Transmission line approach of Left-Handed (LH) materials and microstrip implementation of an artificial LH transmission Line," IEEE Trans. Antennas Propag. 52, 1159 (2004). [CrossRef]
- G. V. Eleftheriades, A.K. Iyer and P.C. Kremer,"Planar negative refractive index media using periodically L-C loaded transmission lines," IEEE Trans. Microwave Theory Tech. 50, 2702-2712 (2002). [CrossRef]
- A. Sanada, C. Caloz, and T. Itoh, "Characteristics of the composite right/left-handed transmission lines," IEEE Microwave Wireless Components Lett. 14, 68-71, 2004. [CrossRef]

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