## 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

Recently, the study of electromagnetic properties of photonic crystals has attracted a great deal of attention. The emergence of left-handed materials (LHMs) [1–2] introduces many unique photonic band-gaps (PBG) to photonic crystals. LHMs, a class of artificially designed and structured materials with simultaneously negative permittivity and permeability, exhibit many different physical properties [3, 4] from the conventional right-handed materials (RHMs). The PBG of photonic crystals containing LHMs or RHMs generally originates from the Bragg scattering mechanism [5]. The Bragg gaps can be identified through the index m in the Bragg conditions

where we suppose that photonic crystals are staked by alternating layers of two materials. The parameter *ψ _{peroid}* denotes the phase shift of a period,

*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.

In this paper, we study the Bragg gap vanishing phenomena in one-dimensional photonic crystals containing LHMs or RHMs. If the Bragg gap is expected to appear, the vanishing conditions should be avoided. However, in some cases the Bragg gap is needed to be closed up to propagate the electromagnetic wave, for example, in the multilayered radome for high antenna gain [6] and the leaky wave antenna [7]. Thus, the Bragg gap vanishing conditions are essential guidelines on how to design Bragg gaps. Two photonic crystals containing LHMs and RHMs are fabricated to demonstrate the *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], is a special Bragg gap, because it is independent of the lattice constant. However, when the vanishing conditions are satisfied, the *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

Consider a 1-D infinite periodic structure with alternating layers of two materials, as shown in Fig. 1. The parameters *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, 9] can be determined using the Bloch-Floquet theorem:

with ${\alpha}_{i}=\genfrac{}{}{0.1ex}{}{\omega \sqrt{{\omega}_{i}{\mu}_{i}}}{c}\mathrm{cos}{\theta}_{i}$, ${F}_{i}=\sqrt{\genfrac{}{}{0.1ex}{}{{\epsilon}_{i}}{{\mu}_{i}}}\mathrm{cos}{\theta}_{i}$, $\mathrm{cos}{\theta}_{i}=\sqrt{1-\genfrac{}{}{0.1ex}{}{1}{{\epsilon}_{i}{\mu}_{i}}\left(\genfrac{}{}{0.1ex}{}{{k}_{\parallel}c}{\omega}\right)}\left(i=\mathrm{1,2}\right)$ for *s*-polarized waves and ${\alpha}_{i}=\genfrac{}{}{0.1ex}{}{\omega \sqrt{{\omega}_{i}{\mu}_{i}}}{c}\mathrm{cos}{\theta}_{i}$, ${F}_{i}=\sqrt{\genfrac{}{}{0.1ex}{}{{\mu}_{i}}{{\epsilon}_{i}}}\mathrm{cos}{\theta}_{i}$, $\mathrm{cos}{\theta}_{i}=\sqrt{1-\genfrac{}{}{0.1ex}{}{1}{{\epsilon}_{i}{\mu}_{i}}\left(\genfrac{}{}{0.1ex}{}{{k}_{\parallel}c}{\omega}\right)}\left(i=\mathrm{1,2}\right)$ for *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 _{i}* represent effective characteristic impedances of two media.

We suppose that the *m*th mode of Bragg gap occur at the frequency *ω*
_{0}, then

In the case of matched impedance (*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

where the frequency *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

which make the dispersion relation in Eq. (4) valid around *ω*
_{0}. This behavior exhibits the disappearance of the *m*th gap.

In a mismatched impedance case (*F*
_{1} ≠ *F*
_{2}), Eq. (2) at *ω*
_{0} becomes

Except for the discrete solutions *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]. For the dispersive materials, the discrete solutions make ∣cos(*βa*)∣*ω*
_{0}∣ = 1 valid and the approximate relation at *ω* is obtained

$${\alpha}_{2}{d}_{2}{\mid}_{\omega}\approx \genfrac{}{}{0.1ex}{}{d{\alpha}_{2}}{\mathrm{d\omega}}{\mid}_{{\omega}_{0}}{d}_{2}\left(\omega -{\omega}_{0}\right)+\left(m-l\right)\pi \triangleq B+\left(m-l\right)\pi $$

where *A* and *B* are infinitesimal for *ω* very close to *ω*
_{0}. Thus the dispersion relation in Eq. (2) becomes

which implies that the Bloch propagation constant *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).

Consequently, if the constitutive materials of photonic crystals have different impedances, Bragg gaps will vanish as long as the discrete modes appear in photonic crystals containing dispersive materials, while for matched impedance case, Bragg gaps will always disappear.

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

To experimentally demonstrate the Bragg gap vanishing phenomena, we respectively fabricate two photonic crystals containing LHMs and RHMs for mismatched/matched impedance cases. LHMs do not exist in nature, but the artificial structures have been realized by using periodic structures. Two main approaches to realize LHMs have been reported: resonant structures made of the array of wires and split-ring resonators [2] and non-resonant transmission line (TL) structures made of lumped elements [11, 12]. In particular, the TL method to LHMs, which is known as composite right/left-handed transmission line (CRLH-TL), presents the advantage of lower losses over a broader bandwidth [13]. The two photonic crystals, which are implemented by TL method, are both made on a Teflon substrate of the thickness *h*=0.5 mm, relative permittivity *ε _{r}* =2.65 and relative permeability

*μ*= 1.

_{r}One photonic crystal for the mismatched impedance case is implemented by the proposed CRLH-TL and composite right-handed transmission line (CRH-TL). The equivalent circuit model of CRLH-TL is shown in Fig. 2 (a). The host microstrip transmission line of the CRLH-TL is designed with the width *w _{1}*=1.37 mm for

*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.

In the lossless case, the ABCD matrix of the CRLH-TL and the CRH-TL can be obtained from the equivalent circuit models in Fig. 2 and the resulting dispersion relations [11, 12] are determined

where *k _{i}* and

*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}According to Eqs. (9) and (10), the calculated dispersion characteristics of the CRLH-TL and the CRH-TL are shown in Fig. 3. The CRLH-TL exhibits the LH passband in the lower frequency range, while the RH transmission properties are dominant in the upper frequency range. The LH property is attributed to shunt inductors *L _{L}* and series capacitors

*C*. As the balanced condition ${Z}_{1}=\sqrt{{L}_{L}/{C}_{L}}=50\Omega $ is satisfied, there are no gaps between the LH passband and the RH passband, and the transition frequency is about 2.25 GHz. The dispersion diagram of the CRH-TL only shows a RH passband in the lower frequency range.

_{L}In Fig. 3, the LH passband of the CRLH-TL intersects the RH passband of the CRH-TL at 0.84 GHz with the Bloch phase shift 60°. Therefore, by cascading three CRLH-TL cells for *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 _{g}* in the certain range of frequencies).

The photonic crystal is simulated by using the advanced design systems (ADS) of agilent and measured by the vector network analyzer. Figure 5(b) draws the simulated phase delay of effective media CRLH_{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.

Two neighboring Bragg gaps are also observed in both the transmission spectrum (Fig. 5) and the dispersion curve (Fig. 6). The Bragg gap at the frequency 0.59 GHz is in the effective left-handed passband, corresponding to the phase shift *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.

Another photonic crystal is a periodic structure in the matched impedance case, as shown in Fig. 7(a). When the balanced CRLH-TL is cascaded with a matched RH-TL in a period, the transmission line can be adjusted together for RHMs, while lumped elements (series capacitors and shunt inductors) can be approximately considered as purely LHMs. The circuit model of the LH unit cell and RH-TL are provided in Figs. 7(b) and 7(c). The RH-TL is a section of microstrip transmission line with the width *w _{L}*=137 mm for

*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 ${Z}_{1}=\sqrt{L/C}=50\Omega $. Therefore, the two materials are with matched impedances but inverse refractive index, and LHMs are dispersive.

_{L}Figure 8 shows the simulated and measured transmission properties of the photonic crystal (LH/RH)_{12}, as well as the simulated phase delay of a period (-*ψ _{peroid}*). It is seen that there is no gap around the frequencies 2.25 GHz with the phase shift

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

_{peroid}## 4. Conclusion

To summarize, we have studied the Bragg gap vanishing phenomenon in detail. The vanishing conditions, as a guideline to design Bragg gaps, are applicable to one-dimensional photonic crystals of two materials which include LHMs or RHMs. Although the Bragg gap vanishing phenomena are experimentally observed in microwave frequency, the vanishing conditions are also suitable to infrared or optic bands. For the mismatched cases, as long as discrete modes of the Bragg gap appear in photonic crystals containing dispersive materials, the gap will vanish, while for the matched impedance cases, Bragg gaps will always vanish.

## Acknowledgments

This work was partially supported by the National Science Foundation of China (Grant No.60871069) and Aeronautical Science Foundation of China (Grant No.20070188002).

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