Broadband one-dimensional photonic crystal wave plate containing single-negative materials

doi:10.1364/OE.18.019920

« Show journal navigation

Broadband one-dimensional photonic crystal wave plate containing single-negative materials

Yihang Chen »View Author Affiliations

Optics Express, Vol. 18, Issue 19, pp. 19920-19929 (2010)
http://dx.doi.org/10.1364/OE.18.019920

View Full Text Article

Acrobat PDF (1840 KB)

Journal Search
Article Lookup

Article Tools

Citations

Alert me when this article is cited
Export Citation/Save

Article Navigation

▸ Article Outline
– Abstract
1. Introduction
2. The model and numerical methods
3. Numerical results and discussion
4. Conclusion
– References and links

Article Tools
Full Text
Article Info
References
Cited By
Figures
Metrics
Download PDF

Viewing Equations in the
HTML Article

Optimal Viewing Recommendations

Full Text
Article Info
References (36)
Cited By
Figures (10)
Metrics

Abstract

The properties of the phase shift of wave reflected from one-dimensional photonic crystals consisting of periodic layers of single-negative (permittivity- or permeability-negative) materials are demonstrated. As the incident angle increases, the reflection phase shift of TE wave decreases, while that of TM wave increases. The phase shifts of both polarized waves vary smoothly as the frequency changes across the photonic crystal stop band. Consequently, the difference between the phase shift of TE and that of TM wave could remain constant in a rather wide frequency range inside the stop band. These properties are useful to design wave plate or retarder which can be used in wide spectral band. In addition, a broadband photonic crystal quarter-wave plate is proposed.

1. Introduction

During the last decade, photonic crystals (PCs) have attracted extensive attention due to their unique physical properties and potential applications in optoelectronics and optical communications [1

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

–3

J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic band gap guidance in optical fibers, ” Science 282(5393), 1476–1478 (1998). [CrossRef] [PubMed]

]. The essential property of PCs is the photonic band gap (PBG), in which the propagation of electromagnetic waves is strongly inhibited [4

K. M. Leung and Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65(21), 2646–2649 (1990). [CrossRef] [PubMed]

A. Z. Genack and N. Garcia, “Observation of photon localization in a three-dimensional disordered system,” Phys. Rev. Lett. 66(16), 2064–2067 (1991). [CrossRef] [PubMed]

]. Conventional PBG originates from the interference of Bragg scattering in a periodical structure with positive-index materials (PIMs). The characteristics of PBG related to the frequency, polarization, and intensity have been studied extensively. Based on such characteristics, many new PC devices have been proposed [6

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and E. F. Schubert, “High Extraction Efficiency of Spontaneous Emission from Slabs of Photonic Crystals,” Phys. Rev. Lett. 78(17), 3294–3297 (1997). [CrossRef]

–9

S. H. Kwon, H. Y. Ryu, G. H. Kim, Y. H. Lee, and S. B. Kim, “Photonic bandedge lasers in two-dimensional square-lattice photonic crystal slabs,” Appl. Phys. Lett. 83(19), 3870–3872 (2003). [CrossRef]

]. On the other hand, the phase properties are as important as frequency, polarization, and intensity for applied physics. Some interesting research has focused on the phase shift of the incident wave in the PC [10

D. R. Solli, C. F. McCormick, R. Y. Chiao, and J. M. Hickmann, “Experimental demonstration of photonic crystal waveplates,” Appl. Phys. Lett. 82(7), 1036–1038 (2003). [CrossRef]

–14

W. F. Zhang, J. H. Liu, W. P. Huang, and W. Zhao, “Self-collimating photonic-crystal wave plates,” Opt. Lett. 34(17), 2676–2678 (2009). [CrossRef] [PubMed]

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 [15

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]

,16

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 was shown that stacking alternating layers of ENG and MNG media leads to a type of PBG corresponding to zero effective phase (denoted as zero-φ _eff gap) [17

L. G. Wang, H. Chen, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals with single-negative materials,” Phys. Rev. B 70(24), 245102 (2004). [CrossRef]

]. Unusual tunneling modes, which are weakly dependent on the incident angle, had been obtained in the zero-φ _eff gap [18

Y. H. Chen, “Defect modes merging in one-dimensional photonic crystals with multiple single-negative material defects,” Appl. Phys. Lett. 92(1), 011925 (2008). [CrossRef]

,19

Y. H. Chen, “Omnidirectional and independently tunable defect modes in fractal photonic crystals containing single-negative materials,” Appl. Phys. B 95(4), 757–761 (2009). [CrossRef]

]. The zero-φ _eff gap and the tunneling modes may be useful for designing omnidirectional filters. However, the properties of the phase shifts of the incident waves in the zero-φ _eff gap have not been reported yet.

In this letter, we investigate the properties of phase shifts upon reflection from one-dimensional (1D) PCs constituted by alternating ENG and MNG materials, including relationship between phase shift of TE and that of TM wave in the zero-φ _eff gap. Based on these properties, we design a broadband quarter-wave plate.

2. The model and numerical methods

Consider 1D PC with the periodic structure of (AB) ^s , where A represents a layer of ENG material with the thickness of d_A and B represents a layer of MNG material with the thickness of d_B , and s is the number of periods. Let a plane wave be injected from vacuum into the considered PC at an angle θ with + z direction, as shown in Fig. 1 . Suppose the wave in the lth layer has a wave vector

{\vec{k}}_{l} = k_{l y} \hat{y} + k_{l z} \hat{z}

, whose magnitude is

ω \sqrt{ε_{l}} \sqrt{μ_{l}} / c

(where c is the speed of light in vacuum). The electric and magnetic fields of adjacent layers can be related via a transfer matrix [17

L. G. Wang, H. Chen, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals with single-negative materials,” Phys. Rev. B 70(24), 245102 (2004). [CrossRef]

,20

W. Li-Gang, L. Nian-Hua, L. Qiang, and Z. Shi-Yao, “Propagation of coherent and partially coherent pulses through one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(1 Pt 2), 016601 (2004). [CrossRef] [PubMed]

]

M_{l} = (\begin{array}{l} \cos (k_{l z} d_{l}) i \frac{1}{q_{l}} \sin (k_{l z} d_{l}) \\ i q_{l} \sin (k_{l z} d_{l}) \cos (k_{l z} d_{l}) \end{array}),

(1)

where

k_{l z} = ω / c \sqrt{ε_{l}} \sqrt{μ_{l}} \sqrt{1 - (\sin^{2} θ / ε_{l} μ_{l})}

. For TE wave,

q_{l} = \sqrt{ε_{l}} / \sqrt{μ_{l}} \sqrt{1 - (\sin^{2} θ / ε_{l} μ_{l})}

; for TM wave,

q_{l} = \sqrt{μ_{l}} / \sqrt{ε_{l}} \sqrt{1 - (\sin^{2} θ / ε_{l} μ_{l})}

Fig. 1 Schematic representation of the 1D PC constituted by SNG materials. The gray and white regions represent the ENG materials and MIMs, respectively.

Suppose that the matrix connecting the incident end and the exit end is

X_{s} (ω) = \prod_{l = 1}^{s} M_{l}

. The reflection coefficient (r) of the monochromatic plane wave can be obtained as [20

]

r (ω) = \frac{\cos θ (x_{11} - x_{22}) - (\cos^{2} θ x_{12} - x_{21})}{\cos θ (x_{11} + x_{22}) - (\cos^{2} θ x_{12} + x_{21})},

(2)

where x_nm (n,m = 1,2) are the matrix elements of Χ_s (ω). The complex reflection coefficient can be denoted as

r = | r | e^{i Φ}

, where Ф is the phase shift of TE or TM wave reflected from the PC. The reflectance R of the PC can then be calculated from the reflection coefficient.

In addition to the reflectance spectrum, the optical properties for a 1D PC like in Fig. 1 can be directly investigated based on the photonic band structure. According to the Bloch’s theorem, the band structure can be obtained by the characteristic equation given by [21

D. W. Yeh and C. J. Wu, “Analysis of photonic band structure in a one-dimensional photonic crystal containing single-negative materials,” Opt. Express 17(19), 16666–16680 (2009). [CrossRef] [PubMed]

]

\cos β_{z} (d_{A} + d_{B}) = \cos (k_{A z} d_{A}) \cos (k_{B z} d_{B}) - \frac{1}{2} (\frac{k_{A z}}{μ_{1}} \frac{μ_{2}}{k_{B z}} + \frac{k_{B z}}{μ_{2}} \frac{μ_{1}}{k_{A z}}) \sin (k_{A z} d_{A}) \sin (k_{B z} d_{B}),

(3)

where β_z is the z component of Bloch wave vector. The passband can be found when the solution for β_z is real, whereas the forbidden band will be presented if β_z is found to be complex valued.

3. Numerical results and discussion

3.1 Reflection phase properties of zero-φ_eff gap with Drude model dispersion

There are two main approaches to realize SNG materials have been reported: resonant structures made of a periodic array of metallic wires [15

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]

] or split-ring resonators [16

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 structures made of inductors and capacitors [22

A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: Resonance, tunneling and transparency,” IEEE Trans. Antenn. Propag. 51(10), 2558–2571 (2003). [CrossRef]

–25

A. Grbic and G. V. Eleftheriades, “Experimental verification of backward-wave radiation from a negative refractive index metamaterial,” J. Appl. Phys. 92(10), 5930–5935 (2002). [CrossRef]

]. The former can be utilized in three dimensional space and high frequencies, the latter shows advantages of lower loss and wider bandwidth and has already been implemented in various component and microwave applications. According to the equivalent transmission lines models [22

A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: Resonance, tunneling and transparency,” IEEE Trans. Antenn. Propag. 51(10), 2558–2571 (2003). [CrossRef]

], MNG material can be viewed as distributed series and shunt capacitors while ENG material can be viewed as distributed series and shunt inductors. Experimental results show that the dispersion of these SNG materials realized by distributed inductor-capacitor transmission lines can be well described by Drude model [23

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

–25

A. Grbic and G. V. Eleftheriades, “Experimental verification of backward-wave radiation from a negative refractive index metamaterial,” J. Appl. Phys. 92(10), 5930–5935 (2002). [CrossRef]

]. Interesting properties of the SNG PCs were found by numerical simulations using the Drude model [17

L. G. Wang, H. Chen, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals with single-negative materials,” Phys. Rev. B 70(24), 245102 (2004). [CrossRef]

–21

]. Therefore, we first use Drude model to describe such SNG materials, that is,

ε_{1} = ε_{a} - \frac{ω_{e p}^{2}}{ω^{2} - i ω γ_{e}}, μ_{1} = μ_{a}

(4)

in ENG materials and

ε_{2} = ε_{b}, μ_{2} = μ_{b} - \frac{ω_{m p}^{2}}{ω^{2} - i ω γ_{m}}

(5)

in MNG materials. In Eqs. (4) and (5), ω_ep and ω_mp are the electronic plasma frequency and the magnetic plasma frequency, respectively. γ_e and γ_m denote the respective electric and magnetic damping factors that contribute to the absorption and losses. The angular frequency ω is in units of gigahertz. In our calculation, the material parameters are selected as ε_a = μ_b = 1, μ_a = ε_b = 3, ω_ep = ω_mp = 10 GHz, and γ_e = γ_m = 1 × 10⁶ Hz.

Firstly, we consider the PC structure (AB) ^s with d_A = 16 mm, d_B = 8 mm. In Fig. 2 , we plot the band structure for the SNG PC. A zero-φ _eff gap [17

L. G. Wang, H. Chen, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals with single-negative materials,” Phys. Rev. B 70(24), 245102 (2004). [CrossRef]

] appears in frequency region from about 3.8 to 6.3 GHz. The central frequency and the width of the zero-φ _eff gap are insensitive to the incident angle and the light polarizations.

Fig. 2 Photonic band structure as a function of the incident angle for an infinite periodic structure (AB) ^s without considering the losses. The thicknesses are d_A = 16 mm and d_B = 8 mm.

With transfer matrix method, the reflectance R and the reflection phase shift Ф are calculated and shown in Fig. 3 . The frequency range of the zero-φ _eff gap in Fig. 3(a) is in accordance with the result in Fig. 2. From Fig. 3(b), we see that the phase shift upon reflection as a function of frequency changes smoothly inside the PBG, while that changes sharply outside it. The dependence of the reflection phase shift Ф on the incident angle is calculated, as shown in Fig. 3(c) – 3(e). It can be seen that, as the incident angle increases, the reflection phase shift of TE wave (Ф _TE) decreases, while that of TM wave (Ф _TM) increases. Moreover, the curves, which represent the dependence of the reflection phase shift on frequency, remain fairly smooth inside the PBG at oblique incidence. We also calculate the difference between the phase shift of TE and that of TM reflected wave, as shown in Fig. 3(c) – 3(e). It is seen that ΔФ ( = Ф _TM - Ф _TE) remains almost constant in a rather wide frequency range inside the zero-φ _eff gap when the incident angle θ is fixed. On the other hand, as the incident angle increases, ΔФ increases gradually. Such variation of ΔФ in Fig. 3 is quite different from the case for ΔФ inside a Bragg gap. In general, ΔФ of the reflected wave inside the Bragg gap of a conventional 1D PC is sensitive to the frequency.

Fig. 3 (a) Reflectance spectrum at normal incidence in structure (AB)⁸ with d_A = 16 mm and d_B = 8. (b)-(e) The reflection phase shift Ф as a function of frequency at different angles of incidence. The dot line and dash line denote Ф for TE and TM wave, respectively. The dash-dot line denotes ΔФ = Ф _TM -Ф _TE.

According to the effective medium theory, the effective permittivity ε _eff and the effective permeability μ _eff of the periodic ENG-MNG layered structures can be written as [26

A. Lakhtakia and C. M. Krowne, “Restricted equivalence of paired epsilon-negative and mu-negative layers to a negative phase-velocity material (alias left-handed material),” Optik (Stuttg.) 114(7), 305–307 (2003).

]

\begin{array}{l} μ_{e f f} = \frac{d_{A}}{d_{A} + d_{B}} μ_{A} + \frac{d_{B}}{d_{A} + d_{B}} μ_{B}, \\ ε_{e f f} = \frac{d_{A}}{d_{A} + d_{B}} ε_{A} + \frac{d_{B}}{d_{A} + d_{B}} ε_{B} - \sin^{2} θ (\frac{d_{A}}{d_{A} + d_{B}} \frac{1}{μ_{A}} + \frac{d_{B}}{d_{A} + d_{B}} \frac{1}{μ_{B}}) \\ + \sin^{2} θ (\frac{1}{\frac{d_{A}}{d_{A} + d_{B}} \frac{1}{μ_{A}} + \frac{d_{B}}{d_{A} + d_{B}} \frac{1}{μ_{B}}}), \end{array}

(6)

for TE wave and

\begin{array}{l} ε_{e f f} = \frac{d_{A}}{d_{A} + d_{B}} ε_{A} + \frac{d_{B}}{d_{A} + d_{B}} ε_{B}, \\ μ_{e f f} = \frac{d_{A}}{d_{A} + d_{B}} μ_{A} + \frac{d_{B}}{d_{A} + d_{B}} μ_{B} - \sin^{2} θ (\frac{d_{A}}{d_{A} + d_{B}} \frac{1}{ε_{A}} + \frac{d_{B}}{d_{A} + d_{B}} \frac{1}{ε_{B}}) \\ + \sin^{2} θ (\frac{1}{\frac{d_{A}}{d_{A} + d_{B}} \frac{1}{ε_{A}} + \frac{d_{B}}{d_{A} + d_{B}} \frac{1}{ε_{B}}}), \end{array}

(7)

for TM wave. It means that the 1D PC structure is equivalent to an anisotropic medium since ε _eff and μ _eff depend on the incident angle θ. The frequency dependence of the values of ε _eff and μ _eff of the considered ENG-MNG structure are plotted in Fig. 4 for both TE and TM waves at incident angle θ = 60°. It is clear shown from Fig. 4 (a)-(c) that the zero-φ_eff gap for TE and TM waves exist in the frequencies where ε _eff of the PC structure is negative and μ _eff is positive.

Fig. 4 (a) The reflectance spectra, the effective permittivity ε _eff and the effective permeability μ _eff for (b) TE and (c) TM wave of structure (AB)⁸ with d_A = 16 mm and d_B = 8 at angle of incidence θ = 60°. (d) The reflection phase of the effective medium slab with parameters the same as in (b) and (c).

The reflection coefficient of the effective medium slab can be obtained from Eq. (2), where the values of the matrix elements x_nm depend on the effective refractive index (

= \sqrt{ε_{e f f}} \sqrt{μ_{e f f}}

), the effective impedance (

= \sqrt{μ_{e f f}} / \sqrt{ε_{e f f}}

) and the incident angle. From the reflection coefficient, the reflection phases of the anisotropic slab can be calculated, as shown in Fig. 4(d). It can be seen that the phase difference ΔФ _eff has little change in frequencies around 5.0 GHz. Such theoretical result agrees with the numerical simulation in Fig. 3(e). For a given incident angle, the reflection coefficient and the reflection phase of the effective medium depend on the values of ε _eff and μ _eff, which deduced from the parameters in Eqs. (4) and (5). So the phenomena that the reflection phase difference stays constant originate from the frequency dispersion of the permittivity and the permeability of the SNG materials. To further confirm our theoretical results, Fig. 5 shows the angle dependence of the reflection phase corresponding to the central frequency (ω = 5.00 GHz) of the zero-φ _eff gap calculated from numerical simulation and effective medium theoretical calculation, respectively. Clearly, the effective medium theory agrees well with the numerical simulation.

Fig. 5 Numerical simulation and effective medium theoretical calculation results of the reflection phase and the phase difference at ω = 5.00 GHz under different incident angles.

Next, we turn to investigate the dependence of the reflection phase shift on the ratio of the thicknesses of the two single-negative materials. Figure 6 shows Ф _TE, Ф _TM, and ΔФ as functions of frequency at incident angle θ = 60° in PC structure (AB)⁸ with d_A = 16 mm and different d_B . In our calculation, we found that the bigger the difference between d_A and d_B is, the wider the zero-φ _eff gap will be. Such property of the forbidden gap is in accordance with the variation of the reflection phase shift in Fig. 6. It is seen from Fig. 3 that ΔФ remains invariant in a broader frequency range when the stop band is wider. Moreover, it is found that the changes of the bandwidth of the zero-φ _eff gap would almost not influence the value of ΔФ at frequencies in the middle of the forbidden gap. For example, ΔФ remains 0.43π in the middle of the stop band (around 5.00 GHz) with the changing of the bandwidth of the forbidden gap, as shown in Fig. 6.

Fig. 6 The reflection phase shift Ф as a function of frequency at angle of incidence θ = 60° in PC structure (AB)⁸ with d_A = 16 mm and (a) d_B = 10 mm, (b) d_B = 8 mm, (c) d_B = 6 mm, (d) d_B = 4 mm.

These properties may be useful to achieve broad spectral bandwidth wave plate. Generally, a linear polarized incident wave can be decomposed into TE and TM waves, respectively. Notice that these two waves undergo different phase shifts upon reflection in the stop band of PC. By comparing the difference of reflection phase shift of TE or TM wave, one can obtain the polarization properties of reflection light. In common reflection frequency range, the reflection phase difference between TE and TM wave can be adjusted to a set value such as π/2, meaning that the 1D PC can serve as a quarter-wave plate. More importantly, the reflection phase difference can remain constant in a wide frequency range due to the smooth changes of reflection phase shift within the forbidden gap.

Then, we proposed a broad spectral bandwidth quarter-wave plate based on the 1D PC containing SNG materials. Here the parameters are selected as d_A = 16 mm, d_B = 4 mm and θ = 65°. The corresponding reflectance as a function of frequency is shown in Fig. 7(a) . It can be seen that the reflectance of TE and TM waves is greater than 0.99 in common reflection band from 3.05 to 6.99 GHz. In Fig. 4(b), the reflection phase shifts Ф _TE (dot line), Ф _TM (dash line), and ΔФ = Ф _TM - Ф _TE (dash-dot line) are shown, respectively. According to the accuracy of the usual quarter-wave plate, such as 0.005π for the phase precision, ΔФ is π/2 in the frequency range 4.58–6.03 GHz, as shown in Fig. 4(b), the relative spectral bandwidth Δω/ω is over 27%. In principle, we can achieve other phase difference between TE and TM waves in common reflection band by changing the parameters of the PC.

Fig. 7 (a) Reflectance and (b) the reflection phase shift as functions of frequency at angle of incidence θ = 65° in PC structure (AB)⁸ with d_A = 16 mm and d_B = 4 mm.

It is well known that a simple metallic wall can serve as a broadband reflector. If a plane wave incident to a metal slab, most of it will be reflected. The reflection phase shifts Ф for TE and TM waves are different. However, the phase difference ΔФ between TE and TM waves varies obviously as the frequency of the incident wave changes. So the wave plates based on the metal slabs cannot work efficiently in a broad frequency range. On the other hand, the wave plates based on two-dimensional (2D) metallic photonic crystals, which operate in microwave region [10

D. R. Solli, C. F. McCormick, R. Y. Chiao, and J. M. Hickmann, “Experimental demonstration of photonic crystal waveplates,” Appl. Phys. Lett. 82(7), 1036–1038 (2003). [CrossRef]

,11

F. Miyamaru, T. Kondo, T. Nagashima, and M. Hangyo, “Large polarization change in two-dimensional metallic photonic crystals in subterahertz region,” Appl. Phys. Lett. 82(16), 2568–2570 (2003). [CrossRef]

], have attracted much interested. Our proposed 1D structure provides a convenient way to design wave plates or retarders with broad spectral bandwidth.

3.2 Reflection phase properties of zero-φ_eff gap with Lorentz model dispersion

It was reported that the dispersion of SNG materials made of resonant structures agreed well with Lorentz model [16

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]

,27

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). [CrossRef] [PubMed]

–30

D. Schurig, J. J. Mock, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett. 88(4), 041109 (2006). [CrossRef]

]. So next, let us briefly compare the above results with those obtained when another form of expressions for the permittivity and permeability, which are in the form of the Lorentz model, is used to describe the SNG materials. According to the Lorentz model, the relative permittivity and relative permeability of ENG and MNG layers are respectively expressible as [16

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]

,27

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). [CrossRef] [PubMed]

–30

D. Schurig, J. J. Mock, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett. 88(4), 041109 (2006). [CrossRef]

]

ε_{1} = 1 - \frac{F_{1} ω^{2}}{ω^{2} - ω_{e r}^{2} + i ω γ_{e}}, μ_{1} = 1,

(8)

ε_{2} = 1, μ_{2} = 1 - \frac{F_{2} ω^{2}}{ω^{2} - ω_{m r}^{2} + i ω γ_{m}},

(9)

where ω_er and ω_mr are respectively the electronic resonant frequency and the magnetic resonant frequency, and F ₁ and F ₂ are the structural factors. In the following calculation, we take F ₁ = 0.7, F ₂ = 0.56, ω_er = ω_mr = 4 GHz, and γ_e = γ_m = 1 × 10⁶ Hz. Figure 8 displays the associated photonic bands of structure (AB) ^s with d_A = 4 mm and d_B = 20 mm in the frequency range where the corresponding zero-φ _eff gap is located. As shown in Fig. 8, the zero-φ _eff gap is still insensitive to the incident angle and polarizations although the gap for TM wave is a bit wider at larger incident angle.

Fig. 8 Photonic band structure for the infinite periodic structure (AB) ^s with parameters given by Eqs. (8) and (9) without considering the losses. The thicknesses are d_A = 4 mm and d_B = 20 mm.

With transfer matrix method, the dependence of Ф_TE , Ф_TM , and ΔФ on the incident angle in structure (AB)¹² are calculated, as shown in Fig. 9 . We can see that Ф_TE and Ф_TM change smoothly as the frequency varies across the zero-φ _eff gap, and ΔФ remains almost invariant in the central region of the gap. The value of ΔФ can be adjusted by varying the incident angle. We also proposed a broad spectral bandwidth quarter-wave plate based on 1D PC containing SNG materials with parameters given by Eqs. (8) and (9), as shown in Fig. 10 . It can be seen from Fig. 10 that ΔФ is −0.5π ± 0.005π and the reflectance of both TE and TM waves is greater than 0.99 in the frequency range 4.83–5.15 GHz.

Fig. 9 Dependence of Ф_TE , Ф_TM , and ΔФ on the incident angle in periodic structure (AB)¹² with parameters given by Eqs. (8) and (9).

Fig. 10 (a) Reflectance and (b) the reflection phase shift as functions of frequency at angle of incidence θ = 55° in PC structure (AB)¹² with parameters given by Eqs. (8) and (9).

In our calculation, the microwave frequency range is considered since artificial SNG metamaterials were first realized in such frequency range and our proposed 1D PC structures have been investigated experimentally [22

A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: Resonance, tunneling and transparency,” IEEE Trans. Antenn. Propag. 51(10), 2558–2571 (2003). [CrossRef]

–24

L. W. Zhang, Y. W. Zhang, L. He, H. Q. Li, and H. Chen, “Experimental study of photonic crystals consisting of E-negative and μ-negative materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(5 Pt 2), 056615 (2006). [CrossRef]

]. Recently, metamaterials with negative permittivity ε or/and negative permeability μ have been realized in Terahertz frequency range from split-ring resonators [31

J. M. Manceau, N. H. Shen, M. Kafesaki, C. M. Soukoulis, and S. Tzortzakis, “Dynamic response of metamaterials in the terahertz regime: Blueshift tunability and phase modulation,” Appl. Phys. Lett. 96(2), 021111 (2010). [CrossRef]

] and chiral metallic structure [32

O. Paul, C. Imhof, B. Reinhard, R. Zengerle, and R. Beigang, “Negative index bulk metamaterial at terahertz frequencies,” Opt. Express 16(9), 6736–6744 (2008). [CrossRef] [PubMed]

,33

J. F. Dong, J. F. Zhou, T. Koschny, and C. Soukoulis, “Bi-layer cross chiral structure with strong optical activity and negative refractive index,” Opt. Express 17(16), 14172–14179 (2009). [CrossRef] [PubMed]

]. The frequency dependence of the measured values of ε and μ agrees well with the Drude model or Lorentz model. Furthermore, it was demonstrated that nanoscale circuit elements can be obtained using plasmonic and nonplasmonic nanoparticles [34

N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors,” Phys. Rev. Lett. 95(9), 095504 (2005). [CrossRef] [PubMed]

–36

A. Alù and N. Engheta, “All optical metamaterial circuit board at the nanoscale,” Phys. Rev. Lett. 103(14), 143902 (2009). [CrossRef] [PubMed]

]. By properly arranging these nanoscale circuit elements one may form optical nanotransmission lines. It means that the dispersions of Drude model in Eq. (4) and (5) may occur in optical frequency and 1D PC structures made of transmission lines may be realized in the optical domain, similarly to what was proposed in the microwave range [23

,24

]. So our results can be extended to other frequency ranges.

4. Conclusion

In conclusion, we studied the properties of the phase shift of the reflected wave in the zero-φ _eff gap of the 1D PCs stacking with ENG and MNG materials. The phase difference between TE and TM reflected wave remains invariant in a wide frequency region. According to such property, broadband 1D PC wave plate and retarder can be conveniently designed.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (NNSFC) (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. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic band gap guidance in optical fibers, ” Science 282(5393), 1476–1478 (1998). [CrossRef] [PubMed]
4.	K. M. Leung and Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65(21), 2646–2649 (1990). [CrossRef] [PubMed]
5.	A. Z. Genack and N. Garcia, “Observation of photon localization in a three-dimensional disordered system,” Phys. Rev. Lett. 66(16), 2064–2067 (1991). [CrossRef] [PubMed]
6.	S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and E. F. Schubert, “High Extraction Efficiency of Spontaneous Emission from Slabs of Photonic Crystals,” Phys. Rev. Lett. 78(17), 3294–3297 (1997). [CrossRef]
7.	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]
8.	D. G. Ouzounov, F. R. Ahmad, D. Müller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, “Generation of megawatt optical solitons in hollow-core photonic band-gap fibers,” Science 301(5640), 1702–1704 (2003). [CrossRef] [PubMed]
9.	S. H. Kwon, H. Y. Ryu, G. H. Kim, Y. H. Lee, and S. B. Kim, “Photonic bandedge lasers in two-dimensional square-lattice photonic crystal slabs,” Appl. Phys. Lett. 83(19), 3870–3872 (2003). [CrossRef]
10.	D. R. Solli, C. F. McCormick, R. Y. Chiao, and J. M. Hickmann, “Experimental demonstration of photonic crystal waveplates,” Appl. Phys. Lett. 82(7), 1036–1038 (2003). [CrossRef]
11.	F. Miyamaru, T. Kondo, T. Nagashima, and M. Hangyo, “Large polarization change in two-dimensional metallic photonic crystals in subterahertz region,” Appl. Phys. Lett. 82(16), 2568–2570 (2003). [CrossRef]
12.	E. Istrate and E. H. Sargent, “Measurement of the phase shift upon reflection from photonic crystals,” Appl. Phys. Lett. 86(15), 151112 (2005). [CrossRef]
13.	Q. F. Dai, Y. W. Li, and H. Z. Wang, “Broadband two-dimensional photonic crystal wave plate,” Appl. Phys. Lett. 89(6), 061121 (2006). [CrossRef]
14.	W. F. Zhang, J. H. Liu, W. P. Huang, and W. Zhao, “Self-collimating photonic-crystal wave plates,” Opt. Lett. 34(17), 2676–2678 (2009). [CrossRef] [PubMed]
15.	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]
16.	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]
17.	L. G. Wang, H. Chen, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals with single-negative materials,” Phys. Rev. B 70(24), 245102 (2004). [CrossRef]
18.	Y. H. Chen, “Defect modes merging in one-dimensional photonic crystals with multiple single-negative material defects,” Appl. Phys. Lett. 92(1), 011925 (2008). [CrossRef]
19.	Y. H. Chen, “Omnidirectional and independently tunable defect modes in fractal photonic crystals containing single-negative materials,” Appl. Phys. B 95(4), 757–761 (2009). [CrossRef]
20.	W. Li-Gang, L. Nian-Hua, L. Qiang, and Z. Shi-Yao, “Propagation of coherent and partially coherent pulses through one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(1 Pt 2), 016601 (2004). [CrossRef] [PubMed]
21.	D. W. Yeh and C. J. Wu, “Analysis of photonic band structure in a one-dimensional photonic crystal containing single-negative materials,” Opt. Express 17(19), 16666–16680 (2009). [CrossRef] [PubMed]
22.	A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: Resonance, tunneling and transparency,” IEEE Trans. Antenn. Propag. 51(10), 2558–2571 (2003). [CrossRef]
23.	T. Fujishige, C. Caloz, and T. Itoh, “Experimental demonstration of transparency in the ENG-MNG pair in a CRLH transmission-line implementation,” Microw. Opt. Technol. Lett. 46(5), 476–481 (2005). [CrossRef]
24.	L. W. Zhang, Y. W. Zhang, L. He, H. Q. Li, and H. Chen, “Experimental study of photonic crystals consisting of E-negative and μ-negative materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(5 Pt 2), 056615 (2006). [CrossRef]
25.	A. Grbic and G. V. Eleftheriades, “Experimental verification of backward-wave radiation from a negative refractive index metamaterial,” J. Appl. Phys. 92(10), 5930–5935 (2002). [CrossRef]
26.	A. Lakhtakia and C. M. Krowne, “Restricted equivalence of paired epsilon-negative and mu-negative layers to a negative phase-velocity material (alias left-handed material),” Optik (Stuttg.) 114(7), 305–307 (2003).
27.	R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). [CrossRef] [PubMed]
28.	T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(6 Pt 2), 065602 (2003). [CrossRef]
29.	R. P. Liu, T. J. Cui, D. Huang, B. Zhao, and D. R. Smith, “Description and explanation of electromagnetic behaviors in artificial metamaterials based on effective medium theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(2 Pt 2), 026606 (2007). [CrossRef] [PubMed]
30.	D. Schurig, J. J. Mock, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett. 88(4), 041109 (2006). [CrossRef]
31.	J. M. Manceau, N. H. Shen, M. Kafesaki, C. M. Soukoulis, and S. Tzortzakis, “Dynamic response of metamaterials in the terahertz regime: Blueshift tunability and phase modulation,” Appl. Phys. Lett. 96(2), 021111 (2010). [CrossRef]
32.	O. Paul, C. Imhof, B. Reinhard, R. Zengerle, and R. Beigang, “Negative index bulk metamaterial at terahertz frequencies,” Opt. Express 16(9), 6736–6744 (2008). [CrossRef] [PubMed]
33.	J. F. Dong, J. F. Zhou, T. Koschny, and C. Soukoulis, “Bi-layer cross chiral structure with strong optical activity and negative refractive index,” Opt. Express 17(16), 14172–14179 (2009). [CrossRef] [PubMed]
34.	N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors,” Phys. Rev. Lett. 95(9), 095504 (2005). [CrossRef] [PubMed]
35.	N. Engheta, “Circuits with light at nanoscales: optical nanocircuits inspired by metamaterials,” Science 317(5845), 1698–1702 (2007). [CrossRef] [PubMed]
36.	A. Alù and N. Engheta, “All optical metamaterial circuit board at the nanoscale,” Phys. Rev. Lett. 103(14), 143902 (2009). [CrossRef] [PubMed]

OCIS Codes
(260.2110) Physical optics : Electromagnetic optics
(160.3918) Materials : Metamaterials
(050.5298) Diffraction and gratings : Photonic crystals

ToC Category:
Metamaterials

History
Original Manuscript: June 18, 2010
Revised Manuscript: August 16, 2010
Manuscript Accepted: August 23, 2010
Published: September 3, 2010

Citation
Yihang Chen, "Broadband one-dimensional photonic crystal wave plate containing single-negative materials," Opt. Express 18, 19920-19929 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-19-19920

Sort: Author | 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. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic band gap guidance in optical fibers, ” Science 282(5393), 1476–1478 (1998). [CrossRef] [PubMed]
K. M. Leung and Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65(21), 2646–2649 (1990). [CrossRef] [PubMed]
A. Z. Genack and N. Garcia, “Observation of photon localization in a three-dimensional disordered system,” Phys. Rev. Lett. 66(16), 2064–2067 (1991). [CrossRef] [PubMed]
S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and E. F. Schubert, “High Extraction Efficiency of Spontaneous Emission from Slabs of Photonic Crystals,” Phys. Rev. Lett. 78(17), 3294–3297 (1997). [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]
D. G. Ouzounov, F. R. Ahmad, D. Müller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, “Generation of megawatt optical solitons in hollow-core photonic band-gap fibers,” Science 301(5640), 1702–1704 (2003). [CrossRef] [PubMed]
S. H. Kwon, H. Y. Ryu, G. H. Kim, Y. H. Lee, and S. B. Kim, “Photonic bandedge lasers in two-dimensional square-lattice photonic crystal slabs,” Appl. Phys. Lett. 83(19), 3870–3872 (2003). [CrossRef]
D. R. Solli, C. F. McCormick, R. Y. Chiao, and J. M. Hickmann, “Experimental demonstration of photonic crystal waveplates,” Appl. Phys. Lett. 82(7), 1036–1038 (2003). [CrossRef]
F. Miyamaru, T. Kondo, T. Nagashima, and M. Hangyo, “Large polarization change in two-dimensional metallic photonic crystals in subterahertz region,” Appl. Phys. Lett. 82(16), 2568–2570 (2003). [CrossRef]
E. Istrate and E. H. Sargent, “Measurement of the phase shift upon reflection from photonic crystals,” Appl. Phys. Lett. 86(15), 151112 (2005). [CrossRef]
Q. F. Dai, Y. W. Li, and H. Z. Wang, “Broadband two-dimensional photonic crystal wave plate,” Appl. Phys. Lett. 89(6), 061121 (2006). [CrossRef]
W. F. Zhang, J. H. Liu, W. P. Huang, and W. Zhao, “Self-collimating photonic-crystal wave plates,” Opt. Lett. 34(17), 2676–2678 (2009). [CrossRef] [PubMed]
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]
L. G. Wang, H. Chen, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals with single-negative materials,” Phys. Rev. B 70(24), 245102 (2004). [CrossRef]
Y. H. Chen, “Defect modes merging in one-dimensional photonic crystals with multiple single-negative material defects,” Appl. Phys. Lett. 92(1), 011925 (2008). [CrossRef]
Y. H. Chen, “Omnidirectional and independently tunable defect modes in fractal photonic crystals containing single-negative materials,” Appl. Phys. B 95(4), 757–761 (2009). [CrossRef]
W. Li-Gang, L. Nian-Hua, L. Qiang, and Z. Shi-Yao, “Propagation of coherent and partially coherent pulses through one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(1 Pt 2), 016601 (2004). [CrossRef] [PubMed]
D. W. Yeh and C. J. Wu, “Analysis of photonic band structure in a one-dimensional photonic crystal containing single-negative materials,” Opt. Express 17(19), 16666–16680 (2009). [CrossRef] [PubMed]
A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: Resonance, tunneling and transparency,” IEEE Trans. Antenn. Propag. 51(10), 2558–2571 (2003). [CrossRef]
T. Fujishige, C. Caloz, and T. Itoh, “Experimental demonstration of transparency in the 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 E-negative and μ-negative materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(5 Pt 2), 056615 (2006). [CrossRef]
A. Grbic and G. V. Eleftheriades, “Experimental verification of backward-wave radiation from a negative refractive index metamaterial,” J. Appl. Phys. 92(10), 5930–5935 (2002). [CrossRef]
A. Lakhtakia and C. M. Krowne, “Restricted equivalence of paired epsilon-negative and mu-negative layers to a negative phase-velocity material (alias left-handed material),” Optik (Stuttg.) 114(7), 305–307 (2003).
R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). [CrossRef] [PubMed]
T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(6 Pt 2), 065602 (2003). [CrossRef]
R. P. Liu, T. J. Cui, D. Huang, B. Zhao, and D. R. Smith, “Description and explanation of electromagnetic behaviors in artificial metamaterials based on effective medium theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(2 Pt 2), 026606 (2007). [CrossRef] [PubMed]
D. Schurig, J. J. Mock, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett. 88(4), 041109 (2006). [CrossRef]
J. M. Manceau, N. H. Shen, M. Kafesaki, C. M. Soukoulis, and S. Tzortzakis, “Dynamic response of metamaterials in the terahertz regime: Blueshift tunability and phase modulation,” Appl. Phys. Lett. 96(2), 021111 (2010). [CrossRef]
O. Paul, C. Imhof, B. Reinhard, R. Zengerle, and R. Beigang, “Negative index bulk metamaterial at terahertz frequencies,” Opt. Express 16(9), 6736–6744 (2008). [CrossRef] [PubMed]
J. F. Dong, J. F. Zhou, T. Koschny, and C. Soukoulis, “Bi-layer cross chiral structure with strong optical activity and negative refractive index,” Opt. Express 17(16), 14172–14179 (2009). [CrossRef] [PubMed]
N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors,” Phys. Rev. Lett. 95(9), 095504 (2005). [CrossRef] [PubMed]
N. Engheta, “Circuits with light at nanoscales: optical nanocircuits inspired by metamaterials,” Science 317(5845), 1698–1702 (2007). [CrossRef] [PubMed]
A. Alù and N. Engheta, “All optical metamaterial circuit board at the nanoscale,” Phys. Rev. Lett. 103(14), 143902 (2009). [CrossRef] [PubMed]

Ahmad, F. R.
Alù, A.
Beigang, R.
Birks, T. A.
Broeng, J.
Caloz, C.
Chen, H.
Chen, Y. H.
Chiao, R. Y.
Chutinan, A.
Cui, T. J.
Dai, Q. F.
Dong, J. F.
Eleftheriades, G. V.
Engheta, N.
Fan, S.
Fujishige, T.
Gaeta, A. L.
Gallagher, M. T.
Garcia, N.
Genack, A. Z.
Grbic, A.
Hangyo, M.
He, L.
Hickmann, J. M.
Holden, A. J.
Huang, D.
Huang, W. P.
Imada, M.
Imhof, C.
Istrate, E.
Itoh, T.
Joannopoulos, J. D.
John, S.
Kafesaki, M.
Kim, G. H.
Kim, S. B.
Knight, J. C.
Koch, K. W.
Kondo, T.
Koschny, T.
Krowne, C. M.
Kwon, S. H.
Lakhtakia, A.
Lee, Y. H.
Leung, K. M.
Li, H. Q.
Li, Y. W.
Li-Gang, W.
Liu, J. H.
Liu, R. P.
Liu, Y. F.
Manceau, J. M.
Markos, P.
McCormick, C. F.
Miyamaru, F.
Mock, J. J.
Müller, D.
Nagashima, T.
Nian-Hua, L.
Noda, S.
Ouzounov, D. G.
Paul, O.
Pendry, J. B.
Qiang, L.
Reinhard, B.
Robbins, D. J.
Russell, P. S. J.
Ryu, H. Y.
Salandrino, A.
Sargent, E. H.
Schubert, E. F.
Schultz, S.
Schurig, D.
Shelby, R. A.
Shen, N. H.
Shi-Yao, Z.
Silcox, J.
Smith, D. R.
Solli, D. R.
Soukoulis, C.
Soukoulis, C. M.
Stewart, W. J.
Thomas, M. G.
Tzortzakis, S.
Venkataraman, N.
Villeneuve, P. R.
Wang, H. Z.
Wang, L. G.
Wu, C. J.
Yablonovitch, E.
Yeh, D. W.
Youngs, I.
Zengerle, R.
Zhang, L. W.
Zhang, W. F.
Zhang, Y. W.
Zhao, B.
Zhao, W.
Zhou, J. F.
Zhu, S. Y.

Appl. Phys. B

Y. H. Chen, “Omnidirectional and independently tunable defect modes in fractal photonic crystals containing single-negative materials,” Appl. Phys. B 95(4), 757–761 (2009). [CrossRef]

Appl. Phys. Lett.

Y. H. Chen, “Defect modes merging in one-dimensional photonic crystals with multiple single-negative material defects,” Appl. Phys. Lett. 92(1), 011925 (2008). [CrossRef]
S. H. Kwon, H. Y. Ryu, G. H. Kim, Y. H. Lee, and S. B. Kim, “Photonic bandedge lasers in two-dimensional square-lattice photonic crystal slabs,” Appl. Phys. Lett. 83(19), 3870–3872 (2003). [CrossRef]
D. R. Solli, C. F. McCormick, R. Y. Chiao, and J. M. Hickmann, “Experimental demonstration of photonic crystal waveplates,” Appl. Phys. Lett. 82(7), 1036–1038 (2003). [CrossRef]
F. Miyamaru, T. Kondo, T. Nagashima, and M. Hangyo, “Large polarization change in two-dimensional metallic photonic crystals in subterahertz region,” Appl. Phys. Lett. 82(16), 2568–2570 (2003). [CrossRef]
E. Istrate and E. H. Sargent, “Measurement of the phase shift upon reflection from photonic crystals,” Appl. Phys. Lett. 86(15), 151112 (2005). [CrossRef]
Q. F. Dai, Y. W. Li, and H. Z. Wang, “Broadband two-dimensional photonic crystal wave plate,” Appl. Phys. Lett. 89(6), 061121 (2006). [CrossRef]
D. Schurig, J. J. Mock, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett. 88(4), 041109 (2006). [CrossRef]
J. M. Manceau, N. H. Shen, M. Kafesaki, C. M. Soukoulis, and S. Tzortzakis, “Dynamic response of metamaterials in the terahertz regime: Blueshift tunability and phase modulation,” Appl. Phys. Lett. 96(2), 021111 (2010). [CrossRef]

IEEE Trans. Antenn. Propag.

A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: Resonance, tunneling and transparency,” IEEE Trans. Antenn. Propag. 51(10), 2558–2571 (2003). [CrossRef]

IEEE Trans. Microw. Theory Tech.

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]

J. Appl. Phys.

A. Grbic and G. V. Eleftheriades, “Experimental verification of backward-wave radiation from a negative refractive index metamaterial,” J. Appl. Phys. 92(10), 5930–5935 (2002). [CrossRef]

Microw. Opt. Technol. Lett.

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

Nature

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]

Opt. Express

Opt. Lett.

W. F. Zhang, J. H. Liu, W. P. Huang, and W. Zhao, “Self-collimating photonic-crystal wave plates,” Opt. Lett. 34(17), 2676–2678 (2009). [CrossRef] [PubMed]

Optik (Stuttg.)

A. Lakhtakia and C. M. Krowne, “Restricted equivalence of paired epsilon-negative and mu-negative layers to a negative phase-velocity material (alias left-handed material),” Optik (Stuttg.) 114(7), 305–307 (2003).

Phys. Rev. B

L. G. Wang, H. Chen, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals with single-negative materials,” Phys. Rev. B 70(24), 245102 (2004). [CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

W. Li-Gang, L. Nian-Hua, L. Qiang, and Z. Shi-Yao, “Propagation of coherent and partially coherent pulses through one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(1 Pt 2), 016601 (2004). [CrossRef] [PubMed]
L. W. Zhang, Y. W. Zhang, L. He, H. Q. Li, and H. Chen, “Experimental study of photonic crystals consisting of E-negative and μ-negative materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(5 Pt 2), 056615 (2006). [CrossRef]
T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(6 Pt 2), 065602 (2003). [CrossRef]
R. P. Liu, T. J. Cui, D. Huang, B. Zhao, and D. R. Smith, “Description and explanation of electromagnetic behaviors in artificial metamaterials based on effective medium theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(2 Pt 2), 026606 (2007). [CrossRef] [PubMed]

Phys. Rev. Lett.

A. Alù and N. Engheta, “All optical metamaterial circuit board at the nanoscale,” Phys. Rev. Lett. 103(14), 143902 (2009). [CrossRef] [PubMed]
N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors,” Phys. Rev. Lett. 95(9), 095504 (2005). [CrossRef] [PubMed]
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]
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]
K. M. Leung and Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65(21), 2646–2649 (1990). [CrossRef] [PubMed]
A. Z. Genack and N. Garcia, “Observation of photon localization in a three-dimensional disordered system,” Phys. Rev. Lett. 66(16), 2064–2067 (1991). [CrossRef] [PubMed]
S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and E. F. Schubert, “High Extraction Efficiency of Spontaneous Emission from Slabs of Photonic Crystals,” Phys. Rev. Lett. 78(17), 3294–3297 (1997). [CrossRef]

Science

J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic band gap guidance in optical fibers, ” Science 282(5393), 1476–1478 (1998). [CrossRef] [PubMed]
D. G. Ouzounov, F. R. Ahmad, D. Müller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, “Generation of megawatt optical solitons in hollow-core photonic band-gap fibers,” Science 301(5640), 1702–1704 (2003). [CrossRef] [PubMed]
N. Engheta, “Circuits with light at nanoscales: optical nanocircuits inspired by metamaterials,” Science 317(5845), 1698–1702 (2007). [CrossRef] [PubMed]
R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). [CrossRef] [PubMed]

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.

Click here to see a list of articles that cite this paper