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In this paper, we investigate the influences of the mode, number, and sequence of unit cell on the production of photonic band gaps (PBGs) in one-dimensional (1D) optical waveguide networks (OWNs) and find that the sufficient condition for producing PBG is related to the mode and number of unit cell, but the sequence of unit cell does not affect the production of PBGs. Only when a 1D OWN contains enough evanescent-mode unit cells can it produce PBGs. Otherwise, no matter how the sequence of unit cell is, the 1D OWN can not produce any PBG. It may deepen people’s knowledge on the mechanism of the production of PBGs in 1D OWNs and may be useful for the designing of PBG materials/devices. On the other hand, according to the classification method of 1D lattices in solid state physics, we classify the unit cells of OWNs as two types: the simple and complex unit cells. This classification method may be useful for investigating OWNs strictly, deeply, and taxonomically.
© 2015 Optical Society of America
1. Introduction
In 1887 Rayleigh [1] studied the propagation of electromagnetic (EM) waves in a one-dimensional (1D) periodic material and found that there exists a small frequency area in this kind of material, where the propagation of EM waves is inhibited. This frequency zone is now called photonic band gap (PBG). After photonic crystals (PCs) were proposed in 1987 [2, 3], people have paid much attention to PBG materials [4–9 ]. It is well known that PBG and photonic localization are two main properties of PBG materials. By use of absolute PBG one can inhibit spontaneous radiation, control and motivate medium radiation, and develop PBG filter for wavelength division multiplexer [5], omnidirectional reflection [6], one-way total reflection [7], photonic frequency converters [8], low threshold lasers [9], and so on. Consequently, what kind of materials/structures can produce PBGs and how to generate PBGs by selected materials/structures have caught people’s eyes in optics, material science, physics, and other related fields.
After PCs were proposed, PCs have been the most important PBG materials/structures that people paid attention to during a long time. It was found that only when the unit cells of a PC are arranged following periodic or quasi-periodic sequence can PBGs be generated. It means that the root cause for producing PBGs in a PC is closely related to the sequence of unit cell.
About 10 years later after PCs were proposed, people raised another kind of PBG structures, the optical waveguide networks (OWNs), which are composed of 1D waveguide segments [10–12 ]. Comparing with PCs, in OWNs only monomode propagation of EM waves needs to be considered and electromagnetic wave functions can be regarded as a linear combination of two opposite traveling plane waves. Figure 1 is the schematic diagram of a 1D OWN containing three unit cells, where d 1 and d 2 are, respectively, the lengths of the two waveguide segments in a unit cell and satisfy the relation of d 1 : d 2 = 1 : 2. Comparing with PCs, OWNs can be realized relatively easier than PCs. The fabrication of a three-dimensional (3D) PC working during visible light region is much more difficult [13]. As structures of OWNs are flexibility, some OWNs can be realized even in university physics laboratory [14]. From the measurement aspect, OWNs are more flexible than PCs. In PCs the measurement of EM waves is mainly on the surface of PCs, but in OWNs people can test the phases and amplitudes of EM waves in arbitrary position, so it brings convenience to experimental studies [11, 12, 14, 15]. From the aspect of generating absolute PBGs in high dimensional structures, OWNs are much easier than PCs. The dimension of the OWN does not affect the generation of absolute PBGs and it asks no special requirement on the dielectric permittivity, even if the waveguide segments composed of one kind of material can also meet the designing request. Our recent studies [15–20 ] also show that the designing of 2D or 3D OWNs for generating large absolute PBGs is not more difficult than that of 1D OWNs. Consequently, OWNs have also become one of the focuses in the field of PBG materials.
Fig. 1 Schematic diagram of a 1D OWN with the waveguide length ratio of d 1 : d 2 = 1 : 2 and containing three unit cells, where EI, ER, and EO represent the incident, reflected, and transmitted EM waves, respectively.
Now, people consider that the root cause for the production of PBGs in PCs is that when EM waves propagate through the structures with spatial translational/rotational symmetry, EM waves are suffered strong Bragg scatterings. Therefore, only the PCs possessing spatial translational/rotational symmetry can generate PBGs. However, A large number of studies show that when the unit cells arrange following periodic [10–12 , 14–16 , 19], quasi-periodic [14], aperiodic [17, 18, 20, 21], and even disordered sequences [11], OWNs can all generate PBGs. This is quite different from PCs. So, what is the sufficient condition for producing PBGs in OWNs?
In this paper we investigate infinite and finite OWNs composed of simple and complex unit cells, which are arranged following periodic, Thue-Morse (TM), Fibonacci (FC), and disordered (DS) sequences. It is found that the sufficient condition for producing PBGs in 1D OWNs is related to the mode and number of unit cell but the sequence of unit cell does not affect the production of PBGs, i.e., only when a 1D OWN possesses enough evanescent-mode unit cells, can it produce PBG(s), otherwise, no matter how the unit cells arrange, it can not create any PBG. This is the fundamental difference for producing PBGs between OWNs and PCs. Our discussions on this problem will deepen people’s understanding on PBG materials/structures and may provide theoretical guidance for the designing of PBG materials/devices.
This paper is organized as follows. In Section 2, we introduce the method for judging the modes of EM wave and unit cell. In Section 3, by use of dispersion relation we analytically study the influence of the mode of unit cell on the production of PBGs. In Section 4, by means of transmission spectra we numerically investigate the influence of the number of unit cell on the production of PBGs. In Section 5, from the examples of some typical OWNs we numerically research the influence of the sequence of unit cell on the production of PBGs. Finally, the conclusions are drawn in Section 6.
2. Modes of EM wave and unit cell
It is well known that EM waves in PBG materials/structures exist two kinds of propagation modes. One is that without considering the absorption effect of materials/structures, EM wave does not evanesce with the increment of propagating distance and it is called propagating EM wave [22–25 ]. The other is that EM wave evanesces with the increment of propagating distance and it is called evanescent EM wave [22–25 ]. On the basis of EM modes we classify the unit cells of OWNs as propagating and evanescent modes. By means of network equations [26] and generalized Floquet-Bloch theorem [15], one can deduce the dispersion relation of a periodic network. Then the modes of EM wave and unit cell can be strictly defined by the absolute value of dispersion relation.
For a periodic OWN, order the dispersion relation be cosK = f(ν), where K is the generalized Bloch wave vector [15], f is the dispersion relation function, and ν is the frequency of a EM wave. If the absolute value of the equation is less than or equal to 1 whenever, obviously, K only possesses real number solutions. EM waves propagating in this network do not evanesce with the increment of propagating distance. This kind of EM waves are propagating ones and we call this kind of unit cell propagating-mode unit cell. However, if the absolute value of the equation can be larger than 1, then K possesses complex number solutions. EM waves propagating in this network evanesce with the increment of propagating distance. This kind of EM waves are evanescent ones and we call this kind of unit cell evanescent-mode unit cell.
For an aperiodic OWN, the mode for each unit cell can be determined by the following two steps. First step, construct a periodic OWN composed of this kind of unit cell and deduce the dispersion relation of this OWN. Second step, determine the mode of the unit cell by the absolute value of the dispersion relation. For example, for the aperiodic OWN with the sequence of ABBCDA one can use the aforementioned two steps to determine the model for each single unit cell (i.e., A, B, C, D) or that for each combined unit cell (e.g., AB, BC, BCDA, etc.). For example, in order to determine the model of the combined unit cell BCDA, firstly, one can use it as a unit cell to construct the periodic OWN with the sequence of ...BCDABCDA..., and then deduce the dispersion relation of this OWN. Secondly, determine the mode of the combined unit cell BCDA by the absolute value of the dispersion relation.
As an example of a 1D OWN, we demonstrate the detailed process for determining the modes of EM wave and unit cell, and show the propagation difference between propagating and evanescent EM waves. For the periodic OWN composed of the unit cells in Fig. 1, using network equations [26] and generalized Floquet-Bloch theorem [15], one can deduce the dispersion relation as follows:
where c is the speed of EM waves in the vacuum. From Eq. (1) one can deduce that when 0 ≤ ν ≤ 0.25c/d 1 and 0.75c/d 1 ≤ ν ≤ c/d 1, |cosK| ≤ 1. On this condition, K only possesses real number solutions. It means that when propagating in this OWN, EM waves in these two frequency zones do not evanesce with the increment of propagating distance. Consequently, these EM waves are all propagating EM waves. However, when 0.25c/d 1 ≤ ν ≤ 0.75c/d 1, |cosK| > 1. On this condition, K only possesses complex number solutions and EM waves in this frequency zone evanesce with the increment of propagating distance. So, these EM waves are all evanescent EM waves. Obviously, both propagating and evanescent EM waves can exist in the 1D periodic OWN composed of the unit cell with the waveguide length ratio of d 1 : d 2 = 1 : 2, consequently, this kind of unit cell is the evanescent-mode unit cell.On the other hand, based on the propagation theory of EM field one knows that the EM wave function between nodes i and j can be expressed as follows [15]:
where ψi and ψj are the wave functions at nodes i and j, respectively, lij is the length of the waveguide segment between nodes i and j, and x is the distance to node i, here we use the traditional local coordinate system. By use of Eq. (2), network equations, and generalized eigenfunction method [27], one can calculate |ψij(x)|2, the intensity of EM waves. The numerical results are shown in Fig. 2.Fig. 2 Intensity map of EM waves propagating in the OWN shown in Fig. 3, where m − di − n (m=1,2,3, i=1,2, and n=2,3,4) represents the waveguide segment between nodes m and n and the length of the segment is di.
Figures 2(a)–2(c) are, respectively, the intensity maps of EM waves propagating in the upper arms of the 1st, 2nd, and 3rd unit cells. Figures 2(d)–2(f) are the intensity maps of EM waves propagating in the lower arms of the 1st, 2nd, and 3rd unit cells, respectively. From Figs. 2(a) and 2(d) one can see that the maximal intensities of EM waves in both upper and lower arms of the 1st unit cell all arrive at 5.1. Although the intensity fluctuates somewhat, the average intensity is large. However, the case of the 2nd unit cell is quite different from that of the 1st unit cell. From Figs. 2(b) and 2(e) one can see that in the frequency zones of 0 ≤ ν ≤ 0.25c/d 1 and 0.75c/d 1 ≤ ν ≤ c/d 1, the maximal intensities of EM waves in both upper and lower arms of the 2nd unit cell all arrive at 5.1, too. Although the intensity fluctuates somewhat, the average intensities of these two zones are still large and are nearly equal to those of the 1st unit cell. In the frequency zone of 0.25c/d 1 ≤ ν ≤ 0.75c/d 1, the maximal intensities of EM waves in both upper and lower arms of the 2nd unit cell only arrive at 0.5 and most of the intensities fluctuate under 0.1, consequently, the average intensity of this zone is about two orders of magnitude smaller than that of the 1st unit cell. From Figs. 2(c) and 2(f) one can see that in the frequency zones of 0 ≤ ν ≤ 0.25c/d 1 and 0.75c/d 1 ≤ ν ≤ c/d 1, the cases of the 3rd unit cell are similar to those of the 1st and 2nd unit cells, i.e., the maximal intensities all arrive at 5.1, too, and the average intensities do not change. But in the frequency zone of 0.25c/d 1 ≤ ν ≤ 0.75c/d 1, the photonic attenuation becomes stronger and stronger. The maximal intensities only arrive at 0.01 and most of the intensities fluctuate under 10−4, consequently, the average intensity of this zone is about two orders of magnitude smaller than that of the 2nd unit cell. In short, from the intensity maps of the three unit cells one can see that in the frequency zones of 0 ≤ ν ≤ 0.25c/d 1 and 0.75c/d 1 ≤ ν ≤ c/d 1, EM waves do not evanesce with the increment of propagating distance, and consequently, they are propagating EM waves. Conversely, in the frequency zone of 0.25c/d 1 ≤ ν ≤ 0.75c/d 1, EM waves evanesce sharply with the increment of propagating distance, and consequently, they are evanescent EM waves. This is the very difference that propagating and evanescent EM waves appear when they are propagating in an OWN.
3. Influence of the mode of unit cell on the production of PBGs
In this section, we propose a precise classification method for the unit cells of OWNs and analytically deduce the dispersion relations of periodic OWNs in categories. Based on these results, we investigate the influence of the mode of unit cell on the production of PBGs.
It is known that in solid state physics, if a unit cell contains only one kind of atoms/ions, then the lattice composed of this kind of unit cell is called Bravais lattice [28, 29]. On the other hand, if a unit cell contains more than one kind of atoms/ions, then the lattice composed of this kind of unit cell is called compound lattice [28, 29]. In this paper, we extend this classification method to OWNs. For 1D OWNs, it is found that the property of unit cell is mainly dependent on node number. Consequently, we call the unit cell containing two nodes simple unit cell and call the unit cell containing more than two nodes complex unit cell, where an intersection connecting more than two waveguide segments with each other is defined as the node of unit cell. Based on this classification method, we divide propagating-mode (evanescent-mode) unit cells into simple and complex propagating-mode (evanescent-mode) unit cells and research the influences of propagating-mode and evanescent-mode unit cells on the production of PBGs in subsections 3.1 and 3.2, respectively.
3.1. Simple unit cell
In this subsection, we study the 1D periodic OWNs composed of simple unit cells. Figure 3 is the schematic diagram of a 1D periodic OWN composed of simple unit cells, where each unit cell contains only two nodes and n waveguide segments. When the lengths of the n waveguide segments are all equal to each other, we call this kind of unit cell the simple unit cell with equal waveguide length ratio; otherwise, we call the unit cell the simple unit cell with unequal waveguide length ratio.
Fig. 3 Schematic diagram of a 1D periodic OWN composed of simple unit cells, where d 1, d 2, ..., and dn are the lengths of the n waveguide segments, respectively.
For a 1D periodic OWN composed of simple unit cells (see Fig. 3), the dispersion relation can be deduced as follows
where cot and csc are the cotangent and cosecant functions, respectively. Equation (3) can be deduced by the following three steps. Step 1, by use of network equation [26] write out the relation of the wave functions of three corresponding nodes for three neighbor unit cells. In Fig. 3, the relation of the wave function of three corresponding nodes, 1, 2, and 3, can be written as Step 2, based on the configuration translation of the OWN, write out the phase relations of the aforementioned three corresponding nodes by generalized Floquet-Bloch theorem [15]. In Fig. 3, the phase relations of nodes 1, 2, and 3 can be expressed as Step 3, deduce the dispersion relation by use of the results obtained in steps 1 and 2. Hence, Eq. (3) can be obtained by means of Eqs. (4) and (5).3.1.1. Simple propagating-mode unit cell
For the simple unit cells with equal waveguide length ratio, there exist di = d 1 (i = 1, 2,..., n) and Eq. (3) can be simplified as
Obviously, the absolute value of Eq. (6) is less than or is equal to 1. Consequently, K only possesses real number solutions and the OWN composed of this kind of simple unit cells can not produce any PBG. The EM waves propagating in this kind of OWN are all propagating EM waves and the unit cell is propagating-mode unit cell. Therefore, the simple unit cell with equal waveguide length ratio must be a simple propagating-mode unit cell, vice versa, a simple propagating-mode unit cell must be the simple unit cell with equal waveguide length ratio. A periodic OWN made up of simple propagating-mode unit cells can not produce any PBG.3.1.2. Simple evanescent-mode unit cell
For the simple unit cells with unequal waveguide length ratio, the length for each waveguide segment does not equal to each other. On this condition, it can be demonstrated that the absolute value of Eq. (3) can be larger than 1 and the unit cell is evanescent-mode unit cell. So, the simple unit cell with unequal waveguide length ratio must be a simple evanescent-mode unit cell; vice versa, i.e., a simple evanescent-mode unit cell must be the simple unit cell with unequal waveguide length ratio. A periodic OWN made up of simple evanescent-mode unit cells can produce PBGs.
3.2. Complex unit cell
Generally, complex unit cells can be regarded as the mixed connection or series connection of simple unit cells, where the mixed connection means the mixture of series connection and parallel connection.
3.2.1. Complex propagating-mode unit cell
So far, we have only found one kind of complex propagating-mode unit cell, i.e., the equilateral polygon consisting of symmetrical mixed connection of simple unit cells with equal waveguide length ratio, where symmetrical mixed connection means that the upper and lower arms of the equilateral polygon contain equal number of simple unit cells, i.e., q up = q low. As an example of complex propagating-mode unit cell, Fig. 4(a) shows a 1D periodic OWN composed of the equilateral decagon consisting of symmetrical mixed connection of simple unit cells with equal waveguide length ratio, where q up = q low = 5 and each waveguide length equals d 1.
Fig. 4 Schematic diagram of a 1D periodic OWN composed of the complex unit cells of equilateral decagon, where each dashed line denotes a simple unit cell with equal waveguide length ratio and each waveguide length equals d 1. (a) Symmetrical mixed connection with q up = q low = 5. (b) Asymmetrical mixed connection with q up = 4 and q low = 6.
Generally, for a 1D periodic OWN composed of complex propagating-mode unit cells, the dispersion relation can be deduced by the steps that are used for deducing Eq. (3). Step 1, by use of network equation write out the relation of the wave functions of three corresponding nodes 1, q up + 1, and 3q up as follows
For example, in Fig. 4(a), the relation of the wave functions of three corresponding nodes, 1, 6, and 15, can be written as −2ψ 6 cos (10πνd 1/c) + (ψ 1 + ψ 15) = 0. Step 2, write out the phase relations of nodes 1, q up + 1, and 3q up as follows Step 3, by use of Eqs. (7) and (8), one can deduce the dispersion relation of the OWN as follows Obviously, the absolute value of Eq. (9) is less than or is equal to 1 and this kind of complex unit cell is propagating-mode unit cell. A periodic OWN made up of the complex propagating-mode unit cells can not produce any PBG.3.2.2. Complex evanescent-mode unit cell
Generally, there are many kinds of complex evanescent-mode unit cells, but they are all the mixed connection or series connection of simple unit cells. As the examples of the mixed connection and series connection of simple unit cells, we only investigate two kinds of complex evanescent-mode unit cells. One is the equilateral polygon consisting of asymmetrical mixed connection of simple unit cells with equal waveguide length ratio (see Fig. 4(b)), and the other one is series connection of two different simple unit cells (see Fig. 5).
Fig. 5 Schematic diagram of a 1D periodic OWN composed of the complex unit cells, which consist of series connection of the simplest and sub-simplest unit cells, where d 1, d 2, and d 3 are, respectively, the lengths of the three kinds of waveguide segments.
For a 1D periodic OWN composed of equilateral polygon consisting of asymmetrical mixed connection of simple unit cells with equal waveguide length ratio (see Fig. 4(b)), the dispersion relation can be deduced by the steps that are used for deducing Eqs. (3) and (9). The result is expressed as follows
Obviously, the absolute value of Eq. (10) can be smaller than and/or larger than 1 and the complex unit cell of equilateral polygon consisting of asymmetrical mixed connections of simple unit cells with equal waveguide length ratio is complex evanescent-mode unit cell. A periodic OWN made up of the complex evanescent-mode unit cells can produce PBGs.For a 1D periodic OWN composed of the series connection of the simplest and sub-simplest unit cells (see Fig. 5), the dispersion relation was deduced and reported in references [15, 30]. One can also deduce it by the steps that are used for deducing Eqs. (3) and (9). The result is expressed as follows:
where L = d 2 + d 3 and ΔL = d 2 − d 3. It can be demonstrated that the absolute value of Eq. (11) can be smaller than and/or larger than 1 and the complex unit cell of the series connection of the simplest and sub-simplest unit cells is complex evanescent-mode unit cell. A periodic OWN made up of the complex evanescent-mode unit cells can produce PBGs. Generally, one can also demonstrate that two different unit cells must be able to construct a complex evanescent-mode unit cell, where these two unit cells can be arbitrary simple/complex propagating-mode/evanescent-mode unit cells.In a word, a periodic OWN made up of propagating-mode unit cells can not produce any PBG while that made up of evanescent-mode unit cells can produce PBGs, where the mode of unit cell can be determined strictly by dispersion relation.
4. Influence of the number of unit cell on the production of PBGs
In this section we investigate the influence of the number of propagating-mode and evanescent-mode unit cells on the production of PBGs in limited OWNs. We use generalized eigenfunction method [27] to calculate transmission spectrum and determine the property of photonic bands by transmission spectrum.
In order to investigate the influence of the number of propagating-mode unit cell on the production of PBGs, we calculate two limited OWNs containing 8 and 32 unit cells, which arrange following the sequence of AO···O, where A and O express evanescent-mode and propagating-mode unit cells, respectively. For the smaller (larger) OWN, there exists 1 evanescent-mode unit cell and 7 (31) propagating-mode unit cells. For simplicity, we select two simple unit cells with d 1 : d 2 = 1 : 2 and d 1 : d 2 = 1 : 1 as the evanescent-mode and propagating-mode unit cells, respectively. The transmission spectra are shown in Fig. 6(a), where the red solid and black dotted lines are the transmissions of the OWNs containing 8 and 32 unit cells, respectively. From Fig. 6(a) one can see that in the zones of [0.331c/d 1, 0.336c/d 1], [0.495c/d 1, 0.505c/d 1], and [0.664c/d 1, 0.669c/d 1], the height of the red solid line is in common with that of the black dotted line. It means that the transmission in these 3 zones dose not change with the increment of unit cell. These are typical passbands. On the other hand, From Fig. 6(a) one can also see that in the remanent zones, the height of the red solid line is sometimes larger than that of the black dotted line and sometimes smaller than that of the black dotted line. It means that transmission in these zones sometimes increases and sometimes decreases with the increment of unit cell. These are also typical passbands. Consequently, there is no stopband in the transmission spectrum and this OWN does not generate any PBG. Increasing propagating-mode unit cells does not produce any PBG. Generally, for a limited OWN containing several kinds of evanescent-mode unit cells and/or several kinds of propagating-mode unit cells, adding propagating-mode unit cells will neither enhance photonic attenuations nor contribute any aid for generating PBGs.
Fig. 6 Transmission spectra of 1D OWNs, where the red solid and black dotted lines are the results for the OWNs containing 8 and 32 unit cells, respectively. (a) OWNs with the sequence of AO···O. (b) OWNs with the sequence of OA···A.
In order to investigate the influence of the number of evanescent-mode unit cell on the production of PBGs, we also calculate two limited OWNs containing 8 and 32 unit cells, respectively, which arrange following the sequence of OA···A. The settings are as the same as those of the aforementioned OWN in this section. The transmission spectra are shown in Fig. 6(b), where the red solid and black dotted lines are the transmissions of the OWNs containing 8 and 32 unit cells, respectively. From Fig. 6(b) one can see that in the zone of [0.25c/d 1, 0.75c/d 1] the height of the red solid line is much larger than that of the black dotted line. It means that the transmission in this zone decreases sharply with the increment of unit cell. This is typical stopband and this OWN generates a PBG. Increasing evanescent-mode unit cells enhances photonic attenuations and produces a PBG. Generally, for a limited OWN containing several kinds of evanescent-mode unit cells and/or several kinds of propagating-mode unit cells, adding evanescent-mode unit cells will either enhance photonic attenuation or contribute aid for generating PBGs.
5. Influence of the sequence of unit cell on the production of PBGs
In this section, we numerically calculate the transmissions of three kinds of typical OWNs to study the influence of the sequence of unit cell on the production of PBGs.
From the aspect of order degree, periodic sequences possess the highest order. TM sequence is the bridge that connects periodic with quasi-periodic sequences and possesses the sub-highest order. FC sequence is the perfect representative of quasi-periodic sequence and possesses lower order than that of TM. DS sequences possess the lowest order. In Section 3 we have investigated the production of PBGs in periodic OWNs in detail. In this section, we research the production of PBGs in TM, FC, and DS OWNs, respectively, where the DS sequence is produced by the intrinsic random function of a computer. For simplicity, we set all of the three kinds of OWNs only contain two kinds of simple propagating-mode unit cells (O 1 and O 2), the simplest and the sub-simplest unit cells, just as the complex unit cell shown in Fig. 5, where the length for each waveguide segment is d 1. In order to investigate the transmission tendency of limited OWNs, we calculate smaller and larger networks for TM, FC, and DS OWNs, respectively. The TM and DS OWNs contain 8 and 32 unit cells, and the FC OWN contains 8 and 34 unit cells.
5.1. Sequences
For TM sequence, the successive substitutions are defined as O 1 → O 1 O 2 and O 2 → O 2 O 1. Choosing O 1 as the starting block, one can obtain the 4th (containing 8 unit cells) and 6th (containing 32 unit cells) TM sequences as follows:
For FC sequence, the successive substitutions are defined as O 1 → O 2 and O 2 → O 2 O 1. Choosing O 1 as the starting block, one can obtain the 6th (containing 8 unit cells) and 9th (containing 34 unit cells) FC sequence as follows:
For the DS sequence, we produce it by the intrinsic random function of a computer. The two DS sequences containing 8 and 32 unit cells are as follows:
where the beginning 8 UCs of DS32 are as the same as those of DS8 in order to keep the relationship between DS8 and DS32.5.2. Results
The transmission spectra for TM, FC, and DS OWNs are shown in Fig. 7, where the red solid and black dotted lines are the results of the smaller and larger OWNs, respectively. From Fig. 7 one can see that for not only TM but also FC and DS OWNs, the height of the red solid line in some zones is much larger than that of the black dotted line. This means that the transmission in these zones decreases sharply with the increment of unit cell. These are typical stopbands. Consequently, TM, FC, and DS OWNs composed of two kinds of propagating-mode unit cells can all produce PBGs. It denotes that increasing both O 1 and O 2 propagating-mode unit cells at the same time, but not increasing O 1 or O 2 propagating-mode unit cells alone, may enhance photonic attenuations and then produce PBGs. However, our results in Sections 3 and 4 demonstrate that only evanescent-mode unit cells, but not propagating-mode unit cells, can produce photonic attenuations and then generate PBGs, but in these three kinds of OWNs there exist only propagating-mode but no evanescent-mode unit cells, and PBGs are still generated. What is the reason of the production of PBGs? In subsection 3.2.2, we have demonstrated that a complex unit cell composed of series connection of different unit cells must be a complex evanescent-mode unit cell. Therefore, in these OWNs each pair of O 1 and O 2 propagating-mode unit cells form a complex evanescent-mode unit cell and then, these OWNs are all made up of many indirect evanescent-mode unit cells. This is the very reason that the OWNs can produce PBGs. In a word, PBGs are still generated by evanescent-mode unit cells but not propagating-mode unit cells. Consequently, any kind of OWN must possess enough direct or indirect evanescent-mode unit cells and then, can certainly produce PBGs. Here the production of PBGs is caused by evanescent-mode unit cells but not sequence.
Fig. 7 Transmission spectra of three kinds of 1D OWNs, where the red solid and black dotted lines are the results of the smaller and larger networks, respectively. (a) TM OWNs. (b) FC OWNs. (c) DS OWNs.
In conclusion, one can see that whether OWNs are periodic or aperiodic, and even whether OWNs are ordered or disordered, some OWNs can produce PBGs and some OWNs can not produce PBGs. The sufficient condition for producing PBGs is nether the sequence of unit cell, nor the order degree of a OWN, but the mode of unit cell. Only the OWNs containing enough direct and/or indirect evanescent-mode unit cells can produce PBGs. Therefore, the request for producing PBGs in OWNs is lower than that in PCs. It means that OWNs can generate PBGs easier than PCs do.
6. Conclusion
In this paper, we analytically deduce dispersion relations of periodic OWNs, numerically calculate the transmissions of limited OWNs, systematically investigate the influences of the mode, number, and sequence of unit cell on the production of PBGs.
It is found that by use of dispersion relation one can not only divide EM waves into propagating and evanescent EM waves, but also classify unit cells as propagating-mode and evanescent-mode unit cells. With the increment of unit cell, the intensity of propagating EM wave does not decrease while that of evanescent EM wave decreases sharply. This is as the same as the characteristic of EM wave propagating through PCs.
By use of dispersion relation we investigate the influence of the mode of unit cell on the production of PBGs. It is found that the periodic OWNs composed of propagating-mode unit cells can not produce PBGs, only the periodic OWNs composed of evanescent-mode unit cells can produce PBGs. On the other hand, we also find that simple propagating-mode unit cells must be simple unit cells with equal waveguide length ratio and simple evanescent-mode unit cells must be simple unit cells with unequal waveguide length ratio. So far, we have only found one kind of complex propagating-mode unit cells, i.e., complex unit cells consisting of symmetric mixed connections of simple unit cells with equal waveguide length ratio.
By means of transmission spectra we study the influence of the number of unit cell on the production of PBGs. It is found that adding propagating-mode unit cells will neither enhance photonic attenuations nor improve any contribution of producing PBGs, but adding evanescent-mode unit cells can either enhance photonic attenuations or improve contribution of producing PBGs.
On the other hand, by means of transmission spectra we research the influence of the sequence of unit cell on the production of PBGs. It is found that in 1D OWNs the sufficient condition for producing PBGs is neither the sequence of unit cell, nor the order degree of OWNs, but the mode of unit cell. Only the OWNs containing enough direct and/or indirect evanescent-mode unit cells can produce PBGs.
In summary, we analytically and numerically investigate large numbers of typical 1D OWNs, propose a strict and clear classification method for the modes of EM waves and unit cells, and obtain the sufficient condition for producing PBGs in 1D OWNs. It may deepen people’s knowledge on the mechanism of the production of PBGs in 1D OWNs and may be useful for the designing of PBG materials/devices.
Acknowledgments
This work was supported by the National Natural Science Foundation of China, Grant Nos. 11374107 and 11374108, and the Natural Science Foundation of Guangdong Province, Grant No. 2015A030313374.
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