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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 22 — Oct. 26, 2009
  • pp: 20333–20341
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Unusual transmission bands of one-dimensional photonic crystals containing single-negative materials

Yihang Chen  »View Author Affiliations


Optics Express, Vol. 17, Issue 22, pp. 20333-20341 (2009)
http://dx.doi.org/10.1364/OE.17.020333


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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

During the last decade, the study of electromagnetic (EM) properties of photonic crystals (PCs) has been the intriguing subject of great attention [1

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

4

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]

]. Conventional photonic band gap (PBG) originates from the interference of Bragg scattering in a periodical structure with positive-index materials (PIMs). Such a Bragg gap depends strongly on the details of the interference process. When the structural periodicity is broken by introducing defects into a PC, defect modes will appear inside the PBG [5

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

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]

]. One-dimensional (1D) PCs with defect layers have been used for filters that possess narrow passbands with high transmittance [7

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

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]

]. Since the defect modes originate from the interference, its frequencies will blue shift as the incident angle increases. Furthermore, the bandwidths of the passbands corresponding to the defect modes are difficult to tune. So the applications of the conventional filters are restricted.

Recently, a new type of artificial composites, in which only one of the two material parameters permittivity (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

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]

]. These single-negative (SNG) materials include epsilon-negative (ENG) media with negative 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

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]

]. The transmission bands corresponding to the tunneling modes are insensitive to the incident angle, but their bandwidths are still hard to control.

In this letter, 1D PCs consisting of alternating layers of PIMs and SNG (ENG or MNG) materials are demonstrated. Such PCs can produce a number of transmission bands, whose number, corresponding central frequencies and bandwidths can be controlled by changing its structural parameters. Moreover, omnidirectional transmission bands for the TE or TM wave can be realized. Such properties will lead to potential applications.

2. The model

Consider 1D PCs with the periodic structures of (AB)s and (AC)s, where A represents a layer of PIM with the thickness of dA and B (C) represents a layer of ENG (MNG) material with the thickness of dB (dC), and s is the number of periods. We assume that the relative permittivity and permeability take the forms of
εB=1ωep2ω2,   μB=μb
(1)
in ENG materials and
εC=εc,   μC=1Fω2ω2ω02
(2)
in MNG materials, where ωep and ω 0 are, respectively, the electronic plasma frequency and the magnetic resonance frequency. These kinds of dispersion for εB and μC can be realized in special metamaterials [14

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

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]

]. In the following calculation, we choose μA = εA = 1, μb = εc = 1, and ωep = πc/d, ω 0 = 4c/d, F = 0.56.

Let a plane wave be injected from vacuum into the considered PC at an angle θ with + z direction, as show in Fig. 1
Fig. 1 Schematic representation of the PC constituted by SNG materials and PIMs. The gray and white regions represent the SNG materials and PIMs, respectively.
. 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 (s), according to Bloch’s theorem, the dispersion at any incident angle follows the relation [18

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]

]
cosβz(dA+dB)=cos(kAzdA)cos(kBzdB)12(qAqB+qBqA)sin(kAzdA)sin(kBzdB),
(3)
where βz is the z component of Bloch wave vector, kiz=ω/cεiμi1(sin2θ/εiμi) (i = A, B) is the z component of the wave vector, and c is the light speed in the vacuum. For TE wave, qi=εi/μi1(sin2θ/εiμi); for TM wave, qi=μi/εi1(sin2θ/εiμi). The EM fields in the consider system can be propagating or evanescent, corresponding to real or imaginary Bloch wave numbers, respectively. The solution of Eq. (3) defines the band structure for the infinite system (AB)s. Similarly, the dispersion relation for infinite structure (AC)s can be obtained by replacing dB, kBz and qB of Eq. (3) by dC, kCz and qC, respectively.

3. Numerical results and discussion

2kAzdA=(2t+1)π.
(4)

In case of normal incidence, substitute μA and εA into Eq. (4), we obtain

2ωcdA=(2t+1)π.
(5)

For example, when dA = 2d, it can be got from Eq. (5) that ω/ωep = 1/4, 3/4 corresponding to t = 0, 1. The two corresponding transmission bands exist in Fig. 2 when dA = 2d, although the frequencies of them are a little different from the approximately theoretical values. For another example, when dA = 3d, it can be got from Eq. (5) that ω/ωep = 1/6, 3/6, 5/6 corresponding to t = 0, 1, 2. It can be seen from Fig. 2 that the three corresponding transmission bands exist when dA = 3d. Furthermore, from Eq. (5) we can obtain, as dA increases, ω decreases and the transmission band shifts to lower frequency, in accordance with Fig. 2.

Next, we study the dependence of the transmission bands on the ratio dB/dA when dA is fixed. In Fig. 3
Fig. 3 Dependence of the transmission bands on the ratio dB/dA in infinite structure (AB)s with dA = 2.5d at normal incidence.
, we choose dA = 2.5d. As shown in Fig. 3, three transmission bands appear, their central frequencies are 0.32ωep, 0.63ωep, and 0.91ωep, respectively. As dB 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 dA, not dB. The properties of Figs. 2 and 3 can be used to design filters with multiple transmission channels, whose bandwidths can be varied conveniently.

The dependence of the transmission spectra on dB in finite structure (AB)12 is shown in Fig. 4
Fig. 4 Dependence of the transmission bands on dB in finite structure (AB)12 with dA = 2.5d.
. It can be seen from Fig. 4 that the transmission bands will be narrowed with the increasing of dB, the same as the results in Fig. 3.

Then, we turn to investigate the dependence of the transmission bands on the incident angle. In Fig. 5
Fig. 5 Photonic band structure as a function of the incident angle in infinite structure (AB)s with dA = 2.5d and dB = d.
we show the dispersion relations of the transmission bands with the thicknesses dA = 2.5d and dB = 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

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]

]. On the other hand, we note from Fig. 5 that the frequency shift of band I is relatively small for TM wave. So we then study the dependence of the dispersion relation of transmission band I on dA, as shown in Fig. 6
Fig. 6 Dependence of the dispersion relations of the transmission bands on dA for infinite structure (AB)s with dB = 1.5d.
. 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 dA changes from 0.5d to 0.82d and onward to 1.5d. 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.

Moreover, the properties of the transmission bands in structure (AC)s containing MNG materials are also studied. It is found that the properties of the transmission bands in (AC)s for TE (TM) polarization are similar to those in (AB)s for TM (TE) polarization. So the properties of the transmission bands of structure (AC)s can be easily obtained from those of (AB)s.

Figure 7
Fig. 7 Dispersion relation of the transmission band in infinite structure (AC)s with dA = 0.36d and dC = 0.85d.
shows the dispersion relation of the transmission band in infinite structure (AC)s with dA = 0.36d and dC = 0.85d. 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
Fig. 8 Dependence of the transmission band on incident angle θ for TE wave in finite structure (AC)12 with dA = 0.36d and dC = 0.85d.
. As the incident angle θ varies, the central frequency of the transmission band remains almost invariant.

In practice, two main approaches to realize SNG materials have been reported: resonant structures made of the array of wires and split-ring resonators [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

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]

] and nonresonant transmission line (TL) structures made of lumped inductors and lumped capacitors [14

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

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]

]. The TL approach towards metamaterial with left-handed and right-handed attributes, known as composite right/left-handed transmission line (CRLH TL), which presents the advantage of lower losses over a broader bandwidth and has already been demonstrated in various component and coupler applications. 1D PCs containing SNG materials have been fabricated using the CRLH TL by periodically loading lumped-element series capacitors and shunt inductors [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

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]

]. The unit cell of the CRLH TL is shown in Fig. 9(a)
Fig. 9 (a) The schematic and circuit model of a CRLH TL unit with the loading lumped element series capacitors (Ci) and shunt inductors (Li). (b) The calculated relative permittivity (ε) and permeability (μ) of the ENG TL and PIM TL.
. The structure consists of a host TL medium with the distributed parameters L 0 and C 0 periodically loaded with discrete lumped element components, Li and Ci. Such structure exhibits a macroscopic behavior rigorously expressed with the constitutive parameters 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

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]

]. The effective relative permittivity and permeability are given by the following approximate expressions,
εi[C01(ω2iγeω)Lidi]/(ε0p),   μip[L01(ω2iγmω)Cidi]/μ0,
(6)
where p is a structure constant and i = 1, 2 denotes the different type of CRLH TL. γe and γm 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

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]

]. Consider ENG TL and PIM TL both made on a FR-4 substrate with a thickness of 1.6mm, relative permittivity εr = 4.75, and relative permeability μr = 1.0. In this situation, 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.

In the following calculation, we use the relatively large damping factors γe = γm = 1 × 107 Hz. Firstly, we consider the dependence of the transmission band of the (PIM5ENG4)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
Fig. 10 Dependence of the transmission band of the (PIM5ENG4)2 TL on the incident angle.
, 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.

Figure 12
Fig. 12 The simulated transmission of the structures (PIM10ENG4)4, (PIM10ENG5)4, and (PIM10ENG6)4.
shows the transmission spectra of the structures (PIM10ENG4)4, (PIM10ENG5)4, and (PIM10ENG6)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

In conclusion, we showed that 1D PCs stacking of PIMs and SNG materials can produce a number of transmission bands. The bandwidths of such transmission bands can be tuned conveniently by adjusting the thicknesses of the SNG layers. On the other hand, the number and central frequencies of these bands depend on the thicknesses of the PIM layers. Omnidirectional transmission bands for the TE or TM wave are, respectively, obtained from such PC structures containing MNG or ENG materials. Our results will lead to further applications in optical guided transmission devices, such as multichannel filters and omnidirectional filters.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant No. 10704027), and the Natural Science Foundation of Guangdong Province of China (Grant Nos. 9151063101000040 and 07300205).

References and links

1.

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

2.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58(23), 2486–2489 (1987). [CrossRef] [PubMed]

3.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: Putting a new twist on light,” Nature 386(6621), 143–149 (1997). [CrossRef]

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]

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]

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]

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. 89(14), 141101 (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]

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]

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]

16.

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]

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]

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]

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]

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


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References

  1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987). [CrossRef] [PubMed]
  2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58(23), 2486–2489 (1987). [CrossRef] [PubMed]
  3. J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: Putting a new twist on light,” Nature 386(6621), 143–149 (1997). [CrossRef]
  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]
  9. 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]
  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]
  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]
  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. 89(14), 141101 (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]
  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]
  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]
  16. 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]
  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]
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