1. Introduction

Recently, metamaterials, first proposed by Veselago [1], have attracted people’s eyes due to their unusual electromagnetic (EM) properties and potential applications. They include negative-index materials (NIMs) [2–4], which are up to now artificial composites with simultaneously negative permittivity ε and permeability μ. When EM waves propagate in them, the corresponding wave vector k, electric field E, and magnetic field H form a left-handed orthogonal set, contrary to all known natural materials where the three vectors are right handed. Besides, metamaterials include mu-negative materials (MNMs) with ε > 0 and μ < 0 and epsilon-negative materials (ENMs) with ε < 0 and μ > 0. The MNMs and ENMs are all called single-negative materials (SNMs) [5,6], which are different from NIMs and may exist in nature.

It is well known that a variety of new devices have been developed based on semiconductor quantum wells (QWs) [7], which have great impacts on modern sciences and technologies. Naturally, their photonic counterparts, photonic QWs, have also attracted intensive attentions [8–11]. Qiao et al [11]. proposed a kind of photonic QWs by assembling different photonic crystals (PhCs) with positive-index materials (PIMs). Through the so called photonic QWs, the photonic bandgaps (PBGs) can be enlarged effectively, and narrow multichannel filters can be obtained when the constituent PBGs are designed properly. For these structures, however, the resonance modes as well as the PBGs vary noticeably with different incident angles and polarizations. Consequently, this kind of structures can only be used for filters mounted at restrictive angles.

Structures consisting of two different SNMs have been proposed then to obtain omnidirectional and multi-channel filtering [12,13]. When the average permittivity and average permeability of the whole structure equal zero, there will be a number of resonance modes insensitive to incident angles. But the number of resonance modes in these structures cannot be accurately controlled by adjusting the number of periods in the constituent PhCs, and the resonance modes usually appear in pairs.

In this paper, we propose a type of photonic QWs made of two different PhCs with NIMs and PIMs. When the PBGs of the constituent PhCs are designed properly, omnidirectional and multi-channel filtering can be achieved. These photonic QWs share the advantages of both the photonic QWs with PIMs and photonic QWs with SNMs: the resonance modes are insensitive to the incident angle from 0 to 85 degrees and the scaling of the barrier PhCs at a certain range; and the number of resonance modes is equal to the number of periods of the well PhC. The structures investigated in this paper distinguish themselves from others in that their volume-averaged effective index [3] of the whole structure is approximately zero.

2. Computational model and numerical method

Suppose the NIMs are isotropic and dispersive, with effective permittivity ε_NIM and μ_NIM and permeability μ_NIM given by [3,4]

where ω is the frequency measured in GHz, ε and μ represent permittivity and permeability at infinite frequencies, α and β are circuit parameters and can be modulated L and capacitance C. Recent experiments demonstrated that this type of NIM has attractive properties in the microwave region, such as low loss and broad bandwidth [14].

We consider the photonic QW with the form of (AB)ⁿ (CD)^m (AB)ⁿ, where A and C are NIMs, B and D are PIMs, and m and n are the corresponding number of periods. The thicknesses of A, B, C and D are expressed as d_A, d_B, d_C and d_D, respectively. The left and right AB)ⁿ in the structure serve as photonic barriers, and (CD)^m works as a well.

In the following, without loss of generality, we consider A and C as the same NIM, and B and D as the same PIM. The difference of A and C or B and D is in their thicknesses. In the following simulations, for practical applications, we set ε = 1.21, μ = 1, α = β = 100, d_A = 6mm, d_B = 12mm for the NIMs, and ε_B = ε_D = 16 , μ_B = μ_D = 1 for the PIMs. According to the dispersion properties of the well PhC, the following typical thicknesses of materials in the well PhC are considered: d _C1 = 6mm and d _D1 = 50mm for the first set of parameters in investigations; and d _C2 = 40mm and d _D2 = 6mm for the second set of parameters.

For convenience in studying the influence of size scaling of the structures, we define a scaling factor ρ as follows

where d ₀ indicates the thickness before scaling and d indicates the thickness after scaling.

Let a wave be incident from a vacuum at an angle θ onto a periodical layered structure containing NIM and PIM. For the TE wave, amplitudes of the forward and backward waves on the two surfaces of a layer are related via a transfer matrix:

where $k_{\mathrm{zj}}=(\omega /c)\sqrt{{\epsilon }_j}\sqrt{{\mu }_j}\sqrt{1-{\left({\epsilon }_j{\mu }_j\right)}^{-1}{\sin }^2\theta }$ is the component of the wave vector in the jth layer along the z direction. c is the light speed in the vacuum and Δz is the thickness of the layer. For TE and TM waves, we have $q_j=\sqrt{{\epsilon }_j}/\sqrt{{\mu }_j}\sqrt{1-{\left({\epsilon }_j{\mu }_j\right)}^{-1}{\sin }^2\theta }$ and $q_j=\sqrt{{\mu }_j}/\sqrt{{\epsilon }_j}\sqrt{1-{\left({\epsilon }_j{\mu }_j\right)}^{-1}{\sin }^2\theta }$, respectively. Therefore, the amplitudes of the transmitted and reflected waves can be connected with that of the incident wave by the matrix X _N = Π^2N _j=1 M _j(d_j, ω). The transmission coefficient of the wave passing through this structure into vacuum can be written as [15]:

where x_ij (i, j = 1,2) are the matrix elements of X _N.

3. Numerical results and discussions

Referring to the definition of quantum wells in solid-state physics, to operate as a photonic quantum well, the passband of the well PhC(CD) should be within the PBG of the barrier PhC(AB), so that resonance modes could exist in the well and be confined by the barriers around the well. Figure 1 shows the PBGs for the PhC(AB) and PhC(CD) with infinite periods calculated according to the dispersion relation in Ref. [16]. From Fig. 1 we see that the parameters are chosen for at least one passband of the well PhC is inside the PBG of the barrier PhC: in Fig. 1(a), the passband ω=2.8~4.8GHz of the well PhC(CD) is located inside the PBG of the barrier PhC(AB); in Fig. 1(b), the passbands ω=2~3GHz and ω=3.6~5.6GHz of the well PhC(CD) is located inside the PBG of the barrier PhC(AB).

It should be pointed out that the PBGs of PhC(AB) and PhC(CD) investigated are all zero volume-averaged refractive index (zero- n̄) gaps which are independent on the incident angles and polarizations [4].

 

Fig. 1. The PBGs of PhC(AB) and PhC(CD) for the first set of parameters (a) and that for the second set of parameters (b). The solid lines and dashed lines are for PhC(AB) and PhC(CD), respectively. The gray areas are the PBGs of PhC(AB).

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Fig. 2. Transmittance spectra of the photonic QW structures (AB)ⁿ (CD)^m(AB)ⁿ (m = 1 ~ 4) for the first set of parameters (a) and that for the second set of parameters (b).

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Now, we move to study the transmittance spectra of the entire photonic QW (AB)ⁿ (CD)^m (AB)ⁿ .

The transmittance spectra of (AB)ⁿ (CD)^m (AB)ⁿ for m = 1 ~ 4 are shown in Fig. 2, from which we can see easily that, for each passband of the well PhC in the PBG of the barrier PhC, the number of resonance modes is just equal to the number of periods m. This is easily understood by noting that the structure resembles a Fabry-Perot cavity with (CD)^m being the cavity length, so that m resonant modes can exist in the cavity. This feature is useful for obtaining as many resonance modes as desired simply by adjusting m. It should be noted that the modes are all in the regions ω = 2.8 ~ 4.8GHz in Fig. 2(a) and ω = 2 ~ 3GHz, ω = 3.6 ~ 5.6GHz in Fig. 2(b), which are required to form the quantum wells.

For a further step, we go to investigate the influences of the incident angle and the scaling of the barrier PhCs on the resonance modes. The results are indicated in Figs. 3 and 4.

 

Fig. 3. Influence of incident angle θ on the resonance modes for the first set of parameters (a) and that for the second set of parameters (b), where I, II, III, and IV refer to the modes indicated in Fig. 2.

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Figure 3 shows the influence of the incident angle on the resonance modes, where ω ₀ is the frequency of the resonance mode at normal incidence, ω is the frequency of the resonance mode at an angle θ, Δω = ω - ω ₀ is the frequency shift due to changes in the incident angle, and Δω/ ω ₀ is the relative frequency shift. For the first set of parameters, we see from Fig. 3(a) that the relative frequency shift ∣Δω/ω ₀∣ is under 0.002, which is 1/10 the value in photonic QWs with just SNMs [13], for incident angles from 0 ° to 85 °. The relative frequency shift is no more than 0.0000027 for incident angles from 0 ° to 20 °, for both TE and TM polarizations, i.e., the resonance modes are almost independent on the incident angles.

For the second set of parameters, the relative frequency shift ∣Δω/ω ₀∣ is under 0.007 as shown in Fig. 3(b), which can also be regarded as independent on the incident angles. Furthermore, from Fig. 3 we can see that the frequency shifts are always positive for the first set of parameters and negative for the second set of parameters. This can be understood by noting that the equivalent period of the constituent PhCs increases as the incident angle θ increases. It means the optical-path phase kd will also increase, i.e., for the first set of parameters, the allowed frequency ω will increase with the increase of kd as shown in Fig. 1(a), corresponding to that the frequency shift Δω is always positive at different θ. Similarly, for the second set of parameters, the frequency ω will decrease with the increase of kd as shown in Fig. 1(b), corresponding to that the frequency shift Δω is always negative at different θ.

Figure 4 shows the influence of the scaling factor ρ of the barrier PhC (AB)ⁿ on the relative resonance-frequency shift Δω/ω ₀, where ω ₀ is the frequency of the resonance mode at ρ = 1, ω is the frequency of the resonance mode at other ρ, and Δω = ω - ω ₀ is the frequency shift. The incident angle is set to be 30°. The thicknesses of A and B are d_A = 6ρmm and d_B =12pmm, respectively.

From Fig. 4, we can see that the resonance modes are also insensitive to the scaling of the barrier PhCs: when ρ varies from 0.5 to 1.5 for both TE and TM polarizations, the absolute relative frequency shift ∣Δω/ω ₀∣ is less than 0.006 for the first set of parameters and under 0.004 for the second set of parameters. This is understandable from Fig. 1 that the bandgap of the barrier PhC may vary at a certain range but the requirements for forming the photonic quantum well can still be satisfied. Yet, if the scaling factor is very large, the PBG of the barrier PhC will move to regions outside the passband of the well PhC, so that the condition for forming the quantum well will be unsatisfied and the resonance tunneling modes will disappear, i.e., large scaling factors may destroy the quantum well and should be avoided.

 

Fig. 4. Influence of the scaling factor ρ of the barrier PhCs on the relative resonance-frequency shift Δω/ω ₀ under θ = 30° for the first set of parameters (a) and that for the second set of parameters (b), where I, II, III, and IV refer to the modes indicated in Fig. 2.

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We can also see from Fig. 4 that the frequency shift increases as ρ increases for the first set of parameters and decreases for the second set of parameters. The explanation is similar to the discussion about the influence of incident angles on the frequency shift.

Nevertheless, the frequency shifts in Figs. 3 and 4 are very small and may be neglected. This is reasonable since we choose n̄ = 0 under which the PhCs have zero- n̄ gaps and the influences of both the incident angle and the scaling factor of the barrier PhC on the resonance frequency should be approximately zero.

It should be pointed out that our theory is limited to one-dimensional structures with infinite extent in the transverse direction, which is parallel to the contact surfaces of the dielectric layers. However, it is a reasonable approximation for practical structures whose transverse sizes are many times of the wavelength of the operating wave.

4. Conclusions

In summary, we have proposed and demonstrated a type of photonic QWs made of two different PhCs with NIMs and PIMs. Omnidirectional and multi-channel filtering can be achieved. These photonic QWs share the advantages of both the photonic QWs with PIMs and photonic QWs with SNMs: the resonance modes are insensitive both to the incident angle from 0 to 85 degrees and to the scaling of the barrier PhCs at a certain range; and for each passband of the well PhC in the PBG of the barrier PhC, the number of resonance modes in our structure is equal to the number of periods of the well PhC. Such structures are useful for all-direction receiving, sending, or linking-up of multi-channel signals for avoiding blind angles or cones in communication networks. And they can be applied in signal-detection systems to enhance signal-detection sensitivity.