## 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.

© 2010 OSA

## 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 [1–3]. The essential property of PCs is the photonic band gap (PBG), in which the propagation of electromagnetic waves is strongly inhibited [4,5]. 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 [6–9]. 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 [10–14].

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 [15,16]. 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]. Unusual tunneling modes, which are weakly dependent on the incident angle, had been obtained in the zero-*φ*
_{eff} gap [18,19]. 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*and

_{A}*B*represents a layer of MNG material with the thickness of

*d*, and

_{B}*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

*l*th layer has a wave vector ${\overrightarrow{k}}_{l}={k}_{ly}\widehat{y}+{k}_{lz}\widehat{z}$, whose magnitude is $\omega \sqrt{{\epsilon}_{l}}\sqrt{{\mu}_{l}}/c$ (where

*c*is the speed of light in vacuum). The electric and magnetic fields of adjacent layers can be related via a transfer matrix [17,20]

Suppose that the matrix connecting the incident end and the exit end is ${X}_{s}(\omega )={\displaystyle \prod _{l=1}^{s}{M}_{l}}$. The reflection coefficient (*r*) of the monochromatic plane wave can be obtained as [20]

*x*(

_{nm}*n*,

*m*= 1,2) are the matrix elements of

*Χ*(

_{s}*ω*). The complex reflection coefficient can be denoted as $r=\left|r\right|{e}^{i\mathrm{\Phi}}$, 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]

*β*is the

_{z}*z*component of Bloch wave vector. The passband can be found when the solution for

*β*is real, whereas the forbidden band will be presented if

_{z}*β*is found to be complex valued.

_{z}## 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] or split-ring resonators [16] and nonresonant transmission line structures made of inductors and capacitors [22–25]. 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], 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 [23–25]. Interesting properties of the SNG PCs were found by numerical simulations using the Drude model [17–21]. Therefore, we first use Drude model to describe such SNG materials, that is,

*ω*and

_{ep}*ω*are the electronic plasma frequency and the magnetic plasma frequency, respectively.

_{mp}*γ*and

_{e}*γ*denote the respective electric and magnetic damping factors that contribute to the absorption and losses. The angular frequency

_{m}*ω*is in units of gigahertz. In our calculation, the material parameters are selected as

*ε*=

_{a}*μ*= 1,

_{b}*μ*=

_{a}*ε*= 3,

_{b}*ω*=

_{ep}*ω*= 10 GHz, and

_{mp}*γ*=

_{e}*γ*= 1 × 10

_{m}^{6}Hz.

Firstly, we consider the PC structure (*AB*)* ^{s}* with

*d*= 16 mm,

_{A}*d*= 8 mm. In Fig. 2 , we plot the band structure for the SNG PC. A zero-

_{B}*φ*

_{eff}gap [17] 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.

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 Δ*Ф* ( = *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, Δ*Ф* increases gradually. Such variation of Δ*Ф* in Fig. 3 is quite different from the case for Δ*Ф* inside a Bragg gap. In general, Δ*Ф* of the reflected wave inside the Bragg gap of a conventional 1D PC is sensitive to the frequency.

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]

*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.

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 ($=\sqrt{{\epsilon}_{eff}}\sqrt{{\mu}_{eff}}$), the effective impedance ($=\sqrt{{\mu}_{eff}}/\sqrt{{\epsilon}_{eff}}$) 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 Δ

*Ф*

_{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.

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 Δ*Ф* as functions of frequency at incident angle *θ* = 60° in PC structure (*AB*)^{8} with *d _{A}* = 16 mm and different

*d*. In our calculation, we found that the bigger the difference between

_{B}*d*and

_{A}*d*is, the wider the zero-

_{B}*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 Δ

*Ф*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 Δ

*Ф*at frequencies in the middle of the forbidden gap. For example, Δ

*Ф*remains 0.43

*π*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.

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*= 4 mm and

_{B}*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 Δ

*Ф*=

*Ф*

_{TM}

*- Ф*

_{TE}(dash-dot line) are shown, respectively. According to the accuracy of the usual quarter-wave plate, such as 0.005

*π*for the phase precision, Δ

*Ф*is

*π*/2 in the frequency range 4.58–6.03 GHz, as shown in Fig. 4(b), the relative spectral bandwidth Δ

*ω*/

*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.

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 Δ*Ф* 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 [10,11], 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 [16,27–30]. 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 [16,27–30]

*ω*and

_{er}*ω*are respectively the electronic resonant frequency and the magnetic resonant frequency, and

_{mr}*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*= 4 GHz, and

_{mr}*γ*=

_{e}*γ*= 1 × 10

_{m}^{6}Hz. Figure 8 displays the associated photonic bands of structure (

*AB*)

*with*

^{s}*d*= 4 mm and

_{A}*d*= 20 mm in the frequency range where the corresponding zero-

_{B}*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.

With transfer matrix method, the dependence of *Ф _{TE}*,

*Ф*, and Δ

_{TM}*Ф*on the incident angle in structure (

*AB*)

^{12}are calculated, as shown in Fig. 9 . We can see that

*Ф*and

_{TE}*Ф*change smoothly as the frequency varies across the zero-

_{TM}*φ*

_{eff}gap, and Δ

*Ф*remains almost invariant in the central region of the gap. The value of Δ

*Ф*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 Δ

*Ф*is −0.5

*π*± 0.005

*π*and the reflectance of both TE and TM waves is greater than 0.99 in the frequency range 4.83–5.15 GHz.

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 [22–24]. Recently, metamaterials with negative permittivity *ε* or/and negative permeability *μ* have been realized in Terahertz frequency range from split-ring resonators [31] and chiral metallic structure [32,33]. 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 [34–36]. 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).

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