A theoretical analysis on the angle- and thickness-dependent photonic band structure in a one-dimensional photonic crystal containing single-negative (SNG) materials is presented. The photonic crystal consists of two alternating SNG materials, including that one has a negative permittivity (ENG) and the other has a negative permeability (MNG). It is found that there are two types of SNG gaps. The first is the low-frequency gap which is very insensitive to the incident angle in the transversal electric (TE) wave. The second gap, which strongly relies on the incident angle for both TE and transversal magnetic (TM) waves, will close at the zero bandgap frequency at which the impedance match as well as the phase match in the constituent ENG and MNG layers must be simultaneously satisfied. This zero bandgap frequency is also strongly dependent on the incident angle. The band edges and the gap maps are investigated rigorously as a function of the incident angle and the ratio of thickness of the two SNG layers. The analyses are made in the lossless and lossy cases for both TE and TM waves. The inclusion of the loss enables us to further clarify two fundamentally distinct second SNG gaps which are separated by a threshold angle of incidence.
© 2009 OSA
A photonic crystal (PC) is an artificially periodic layered structure that is known to possess some photonic bandgaps (PBGs) at certain frequency ranges in the photonic band structure. Electromagnetic waves with frequencies in the PBGs will be forbidden to propagate in the structure. For a PC made up of conventional dielectrics, the PBGs are generated due to multiple Bragg reflections in the periodic arrangement and are thus referred to as the Bragg gaps. Photonic devices based on the applications of Bragg gaps are now available. A direct example is the Bragg reflector or dielectric mirror which acts as an indispensable part in modern solid state laser system. An omnidirectional Bragg reflector is proven to be achievable in a one-dimensional dielectric PC. Moreover, a defect mode can be created in the PBG by inserting a defect layer in a PC. This defect mode is used to design as a Fabry-Perot resonator or a narrowband transmission filter. All these applications along with further discussion are well described in an excellent book of electromagnetic waves .
Conventional all-dielectric PCs are made of the so-called double-positive (DPS) materials with both positive permittivity and permeability. Recently, a photonic crystal consists of alternating layers of two different single-negative (SNG) materials has attracted much attention [2–12]. There are two kinds of different SNG materials. One is called the epsilon-negative (ENG) material, which has a negative-permittivity and a positive-permeability (ε < 0, μ > 0). The other called mu-negative (MNG) is a material with a negative-permeability and a positive-permittivity (μ < 0, ε > 0). Using a simple metallic microstructure consisting of arrays for thin wires (TWs), an artificial ENG structure operating in the GHz regime has been successfully reported by Pendry et al  in 1996. Later in 1999, they also successfully realized the MNG structure with the help of the split-ring resonators (SRRs) .
For a single ENG-MNG bilayer, it is found that there are some unique properties such as resonance, complete tunneling and transparency . For a PC consisting of periodic ENG-MNG bilayers, the photonic band gaps are called the zero-effective-phase gap  or simply the SNG gap . Unlike the Bragg gaps in the conventional PCs which originate from the interaction of forward and backward propagation waves, the SNG gaps, however, are ascribed to the interaction of forward and backward evanescent waves. The Bragg gap is known to be strongly dependent on the structural periodicity, the angle of incidence, the wave polarization, and the structural disorder, the SNG gap is invariant with the change of the length scale, being very insensitive to the thickness change in the constituent SNG layers. In addition, like the Bragg gap in a one-dimensional all-dielectric PC, the SNG gap is also proven to possess the omindirectional property .
The purpose of this paper is to give a more detailed analysis on the SNG gaps in a one-dimensional SNG PC. First, we shall elucidate that there are, in fact, two types of SNG gaps. The first is called the low-frequency gap which is shown to be robust against to the length scale and the incident angle in TE wave. The second SNG gap, the zero-effective-phase gap, is opened up in the vicinity of the zero bandgap frequency, which is dependent on the incident angle as well as the polarization of the incident wave. Moreover, the center frequency of zero-effective-phase gap is also a strong function of the incident angle for both TE) and TM waves. Next, we shall incorporate the loss in the SNG material to investigate the effects of loss on the gaps that were not considered in the previous reports in SNG PC [2–4].
The present analysis is made based on the reflectance or transmittance spectrum calculated from the transfer matrix method (TMM) in a stratified medium [16,17] together with the band structure calculated based on the Bloch theorem. The format of this paper is as follows: Section 2 describes the theoretical basic equations for the SNG PC. The detailed analyses and discussion are given in Section 3. The final conclusion is drawn in Section 4.
2. Basic Equations
A one-dimensional SNG periodic multilayer is modeled as a periodic structure that is made of the ENG (layer 1) and the MNG (layer 2) materials as depicted in Fig. 1 , in which the number of periods is denoted by N and the spatial periodicity is Λ = d 1 + d 2, where d 1 and d 2 are respectively the thicknesses of the ENG and MNG layers. Using the temporal part proportional to exp(-iωt) for all fields, the relative permittivity and relative permeability of ENG and MNG layers are respectively expressible as 18]. Many previous studies have been focused on the lossless case with Γ = 0 [2–4,15]. With Eqs. (1) and (2) for ε 1 and μ 2 and using ωp = 10 GHz, ω 0 = 4 GHz and F = 0.56 , the frequency ranges for both ENG and MNG are plotted in Fig. 2 , where the validity for ε 1 < 0 and μ 2 < 0 lies in the typical microwave region of 4.2-6.0 GHz. In addition, the refractive index, (j = 1, 2), must be imaginary for each SNG material.
To analyze the photonic band structure shown in Fig. 1, the optical reflectance or transmittance spectrum will be employed together with the dispersion diagram, the band structure. For the reflectance and transmittance spectra, they can be calculated by making use of the standard transfer matrix method (TMM) . According to TMM, one has to first calculate the total transfer matrix for this 2N-layer system, which connects the electric and magnetic fields at the incident boundary and those at the transmitted boundary. The matrix can be written as,16]Eq. (3) for the SNG layer (j = 1, 2) is expressed as
Having obtained the total matrix in Eq. (3), the transmittance, T and reflectance, R can thus be calculated from its matrix elements, i.e.,
In addition to the reflectance or transmittance spectrum, the optical properties for a one-dimensional PC like in Fig. 1 can be directly investigated based on the dispersion relation, the photonic band structure. For the number of periods being sufficiently large, aided by the Bloch theorem the band structure can be obtained by the so-called characteristic equation given by 
3. Numerical results and discussion
3.1 Band structure in normal incidence
In Fig. 3 , we plot the normal-incidence (a) transmittance spectrum and (b) band structure for the SNG PC shown in Fig. 1. Here d1 = 10 mm, d2 = 20 mm and three different numbers of periods N = 1, 2, and 20 are used for the calculation of transmittance spectra. The high-reflectance (T = 0) band, i.e., the photonic bandgap, can be formed when N is sufficiently large, say N = 20. The band structure, in general, has two passbands and two bandgaps and can be best described in terms of four band edges denoted by ω 1, ωL, ω H, and ω 2 as well. The two passbands are located in the interval between ω 1 and ωL, and between ω H and ω 2, respectively. The two bandgaps have different physical insights. The first one called the low-frequency gap is located below the band edge ω 1, the cutoff frequency. It can be seen from Fig. 3(a) that ω 1 exists even for N = 1, and 2, indicating that it is indeed not related to the periodic stack. This cutoff frequency comes from the composite structure made up of the ENG and MNG materials. An electromagnetic wave with frequency higher than ω 1 is permitted to propagate through the structure containing the ENG-MNG bilayer, whereas it will be totally reflected at frequencies lower than ω 1. Thus, this low-frequency gap is not a real photonic bandgap. The similar low-frequency gap is also seen in a superconductor-dielectric PC because the permittivity of superconducting material is also of the similar form as Eq. (1) [20–22]. Moreover, the superconductor is an ENG material at frequencies where the low-frequency gap is shown up. It is thus to be believed that there should be the low-frequency gap when the two constituent layers in PC are not all DPS dielectrics.
The second SNG gap is a real PBG that has a size of 0.42 GHz because its two band edges are found to be at ωL = 5.1 GHz and ωH = 5.52 GHz, as illustrated in Fig. 3 (b). As stated previously in Section 1, this SNG gap originates from the interaction of forward and backward evanescent waves and its nature is different from the usual Bragg gap [2,4]. In Ref , this SNG gap is called the zero-effective-phase gap because it will vanish at the phase match condition, i.e., the zero-effective-phase delay point. The gap size is insensitive to the change of the spatial periodicity but it is dependent on the change of the thickness ratio of the ENG and MNG layers [2,9].
3.2 Condition of vanishing zero-effective-phase gap in normal incidence
As shown in Fig. 3, there is usually an SNG zero-effective-phase gap falls between ωL and ωH. Now we try to find a condition such that this gap will vanish there. Let us start from the case of normal incidence. The impedance match condition requires that the intrinsic impedances of both ENG and MNG layers are equal, i.e.,Equation (10) then directly leads to an another equivalent expression for the impedance match condition, namelyEqs. (1) and (2) can be used to determine the impedance match frequency ω 0. Moreover, with the impedance match condition in Eq. (11), the characteristic equation in Eq. (9) becomesEq. (12) can be found only when
Equation (13) obviously indicates that the phases of layer 1 and 2 are equal and consequently it is called the phase match condition . In this case, the solution results in a vanishing zero-effective-phase gap. In summary, to cause the SNG zero-effective-phase gap to be vanishing, the impedance match condition in Eq. (10) or (11), and the phase match condition in Eq. (13) must be simultaneously satisfied. All the discussion here is illustrated in Fig. 4 , where we plot the calculated reflectance spectrum (left) and the calculated K versus frequency, the band structure (right). It is seen that the zero-effective-phase gap is closed at the impedance match frequency ω 0 = 5.2 GHz, consistent with the frequency obtained by Eq. (10). With this impedance match frequency, we can first calculate the wave number ratio and then the thickness ratio can be determined according to Eq. (13). This value of (in fact, d1 = 10 mm and d2 = 27 mm) has been used to obtain the results of Fig. 4.
Moreover, Eq. (12) shows that the zero-effective-phase gap will be opened up in the vicinity of ω 0 when . There will be no real solutions for K within the gap, that is, the solution for K must be complex-valued and thus the nonpropagating mode is achieved. Conclusively, the condition for the presence of the second SNG gap, zero-effective-phase gap, should be read as (14)
3.3 Condition of vanishing zero-effective-phase gap in oblique incidence
To discuss the angular dependence of the photonic band structure in both TE and TM waves, let us extend the condition of zero SNG gap in Sec. 3.2 to the case of oblique incidence. In the oblique incidence, the wave number in each layer is given by1]Eqs. (16) and (17) are also directly obtainable by using Eq. (11) with kj being replaced by Eq. (15).
We have known that the vanishing zero-effective-phase gap can happen when the impedance match, Eq. (16) or (17), and the phase match, Eq. (13), are simultaneously satisfied. Both conditions are consistent with previous results given in Ref . However, it should be noted that these two match conditions in Ref . are valid only for a single bilayer structure made of ENG and MNG slabs. The conditions can be derived based on the zero reflectance calculated by making use of the successive impedance transform method (ITM) for a two-layer system . However, using ITM to calculate the reflectance for a multilayer structure more than two layers such as a photonic crystal is a formidable task and is not easy to conclude the two conditions. In fact, in a single bilayer we can also derive these two conditions directly from Eqs. (3)-(8) with N = 1. However, the photonic band structure or bandgap is obviously meaningless for a single ENG-MNG bilayer. Therefore, to investigate the conditions of the vanishing of the zero-effective-phase gap in an SNG PC, it is better to start from the impedance match and then to get the phase match condition from the characteristic equation, Eq. (9), obtained based on the Bloch theorem.
Equations (16) and (17) can be used to calculate the angle-dependent impedance match frequency ω 0. The results are plotted in Fig. 5 , in which ω 0 as a function of the incident angle for both TE and TM waves are given. For TE wave, ω 0 decreases as the incident angle increases, whereas it increases with the increase in the incident angle in TM wave.
With Fig. 5, we can then calculate the wave number ratio in the impedance match condition defined as ρ = k 2 / k 1 for each incident angle according to Eq. (15). The results in ρ for both polarizations are plotted in Fig. 6 . We can see that the value in ρ increases with the increase in the incident angle for both TE and TM waves. However, the ρ-value in TE wave is larger than that of TM wave, i.e., , except at θ = 0°. In fact, varies from 0.370 (θ = 0°) to 0.566 (θ = 50°), whereas is changed from 0.37 (θ = 0°) to 0.5 (θ = 50°), It can be seen that the SNG zero-effective-phase gap can be vanishing simultaneously for both TE and TM waves only at the extreme case of θ = 0°, the normal incidence.
Moreover, the angle-dependent ρ under the impedance match condition shown in Fig. 6 can also be converted to the angle-dependent thickness ratio by making use of the phase match condition given in Eq. (13) with kj being replaced by Eq. (15). Accordingly, at a fixed angle of , will be different each other for TE and TM waves. That is, if is taken to be equal to then it will be different from at the same incident angle. In this case, there will be a zero bandgap for TE wave and a nonzero bandgap for TM wave. In the similar manner, if now is equal to then it will be different from , and therefore the bandgap will be missing for TM wave and be present for TE wave. A further look at the band diagram for the case of the zero-effective-phase gap in TE wave is shown in Fig. 7 . The agreement with Fig. 5 is obviously seen because ω 0 decreases with increasing the incident angle. Another feature is of note in Fig. 7. It is seen that the low-frequency SNG gap (the shaded horizontal strips) is enlarged as the angle increases when the zero-effective-phase gap, the second SNG gap is closed.
3.4 Angular dependence of gap map
We now investigate the angular dependence of the gap map for this SNG PC. The gap map will be studied by plotting the four band edge frequencies ω 1, ωL, ωH, and ω 2 (defined in Fig. 3) as a function of the incident angle for both TE and TM waves. According to the result of shown in Fig. 6, for convenience of discussion, we now take three different values of the thickness ratio to investigate the gap maps for both TE and TM waves. In case I, will be taken. It is larger than the ρ−value at θ = 50° ( = 0.566, and = 0.5). In case II, we choose as the value of () at some angle between θ = 0° and θ = 50°, say 30°. Case III is that which is smaller than = 0.37, the value at θ = 0°.
The gap map of case I with = 0.6 for both TE (a) and TM (b) is plotted in Fig. 8 , where d 2 = 20 mm is taken and the shaded areas are the gaps. In this case where the phase match condition is not satisfied, the zero-effective-phase gaps in TE and TM waves are evidently seen in the figure. In TE wave, the gap size, ωH – ωL, of zero-effective-phase gap tends to reduce as the incident angle increases. In addition, we find that the low-frequency gap (below ω 1) is nearly unchanged as the incident angle increases in TE wave. In TM wave, the gap size, ωH – ωL, is also reduced with the increase in the incident angle. However, the low-frequency gap will be enhanced as the angle increases in TM wave.
In case II, d 1 / d 2 is fixed at 0.483 (0.422) which is equal to the ρ TE (ρ TM) value at 30° (See Fig. 6). The gap maps for both polarizations are plotted in Fig. 9, where again d 2 = 20 mm is used. We see that the zero-effective-phase gap first reduces as the angle increases for both TE (a) and TM (b). Both gaps are then close at 30° at which the phase match condition is satisfied for the taken d 1 / d 2. Then the gaps are reopened up after 30° and their sizes are increased as the incident angle increases.
Finally, for case III, d 1/d 2 is fixed at 0.3, which again the phase match condition is invalid for all incident angles. The gap maps are now shown in Fig. 10 , in which we again take d 2 = 20 mm. It is seen that the zero-effective-phase gaps are enhanced as the incident angle increases for both TE (a) and TM (b). In addition, the low-frequency gap is more sensitive to the incident angle for TM wave compared to TE wave.
Conclusively, the results of these three cases can be summarized as follows: If the value in the difference of or is decreased as the incident angle increases, then the zero-effective-phase gap will consequently be decreased, as illustrated in case I. Conversely, if or is increased with increasing the incident angle, the zero-effective-phase gap will be enhanced as a function of the incident angle, as demonstrated in case III. Specifically, when or is equal to zero, the zero-effective-phase gap will be closed because the phase match condition is satisfied which is seen in case II.
Next, we investigate how the center frequency of the zero-effective-phase gap (defined as ωc = (ωL + ωH) / 2) is affected by the incident angle. Following the above three cases, the center frequency ωc as a function of the incident angle for both TE (left) and TM (right) waves are plotted in Fig. 11 , where the solid curve is for the impedance match frequency ω 0 and the other three curves are ωc’s for the above three conditions. The angular dependence for ωc and ω 0 are clearly demonstrated for both polarizations.
3.5 Effects of losses in SNG gap
In the above discussion, we have assumed that the SNG materials are lossless. We now turn our attention to the loss parameter Γ given in Eq. (2) for the MNG layer. To investigate the effects of loss, Γ = 2π × 107 rad/s will be taken in our calculation. Let us first consider the case where there is no zero-effective-phase gap. Following Fig. 6 we take the incident angle at 30°, at which the thickness ratio d 1 / d 2 = k 2 / k 1 is equal to 0.483, 0.422 for TE and TM waves, respectively. In this case, the reflectance spectra with Γ = 2π × 107 rad/s are shown in Fig. 12 . We see that the effect of loss is pronounced only in the oscillating passband, in which the oscillating dips (T = 1, R = 0) in the lossless case shown in Fig. 4 are now strongly raised up. This is because some of the energy is dissipated in the lossy MNG layers and thus the power absorptance exists, i.e. , which in turn makes T = 1 no longer valid according to the power balanced equation, R + T + A = 1.
Now let consider the case where the zero-effective-phase gap opens, i.e., the angle is taken to be smaller than 30° while d 1 / d 2 = 0.483 remains unchanged in TE wave. The TE-wave reflectance spectra for two incident angles, 0° and 20°, are shown in Fig. 13 . It is seen that the band edges ωL and ωH become smooth due to the presence of the loss. Moreover, the oscillating passband is raised up as a whole just like in Fig. 12.
Similarly, the zero-effective-phase gap can also open when incident angle is changed to be larger than 30° at d 1 / d 2 = 0.483. The TE-wave reflectance spectra are plotted in Fig. 14 , where two incident angles, 40° and 50°, are taken. Surprisingly, a deep notch near ωc is present in the zero-effective-phase gap due to the inclusion of loss. The presence of the notch leads to the missing of the second SNG zero-effective-phase gap. The above features in Figs. 13 and 14 are also seen in TM wave and thus are not presented.
The results in Figs. 13 and 14 clearly illustrate that the two zero-effective-phase gaps separated by the angle of 30° in Fig. 9 are fundamentally different. The first gap located below 30° is insensitive to the loss, while the second gap above 30° is strongly affected by the inclusion loss. This second gap will no more exist due to the presence the notch. Thus, as far as the loss is concerned, the zero-gap angle at 30° in Fig. 9 can be characterized as a new threshold angle θc. The loss effect in the zero-effective-phase gap will be weak and only smooth the band edges when θ < θc. On the other hand, a strong influence will be seen, i.e., zero-effective-phase gap will disappear at θ > θc.
Although the photonic band structure of an SNG PC has been reported previously by Jiang et al. , and Wang et al. , there are some novelties in our present study. First, in Ref , the study is only limited to the case of the normal incidence in the second SNG gap that is described by two band edges ωL and ωH. However, here we have shown that the SNG gaps should be best demonstrated by four band edges, i.e., ω 1, ωL, ωH, and w 2, as illustrated in Fig. 3. In addition, we have extended to the case of the oblique incidence, in which we introduce the ρ-parameter to investigate the angle-dependent band structure and we further find that the nature of the two possible second SNG gaps separated by the threshold angle (Fig. 9) can be clearly distinguished when the loss is introduced (Figs. 13 and 14). These features are obviously not seen in Ref . On the other hand, in Ref , the authors point out the fundamental difference between the omnidirectional SNG gap and Bragg gap, They also show the same difference in the defect mode located in these two different gaps. In fact, we also elucidate that the first SNG gap is omnidirectional, as shown in Figs. 8-10. However, for the second SNG gap, the appearance of omnidirectional feature can only occur at some special condition such as shown in Fig. 10, which is not pointed out in Ref .
The PBG structure for an SNG PC made of ENG-MNG layers has been analyzed in a systematical way. For an SNG PC, there are two so called SNG gaps. The first one is called the low-frequency gap which is shown to be insensitive to the incident angle in TE wave, but sensitive in TM wave. The second SNG gap is referred to as the zero-effective-phase gap which will close at the conditions of both impedance match as well as the phase match. The angle-dependent band edges and SNG bandgaps with center frequencies have been rigorously investigated. Finally, the effects of loss can be salient when the incident angle is great than the threshold one. Due to the presence of loss the zero-effective-phase gap becomes a notch and consequently the gap vanishes. In addition, all the oscillating passbands are enhanced well above zero in the reflectance spectra.
C.-J. Wu wishes to acknowledge the financial support from the National Science Council of the Republic of China under grant No. NSC-97-2112-M-003-013-MY3.
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