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Abstract
In this paper, we construct a 1D PT-symmetric Thue-Morse aperiodic optical waveguide network (PTSTMAOWN) and mainly investigate the ultrastrong extraordinary transmission and reflection. We propose an approach to study the photonic modes and solve the problem of calculating photonic modes distributions in aperiodic networks due to the lack of dispersion functions and find that in a PTSTMAOWN there exist more photonic modes and more spontaneous PT-symmetric breaking points, which are quite different from other reported PT-symmetric optical systems. Additionally, we develop a method to sort spontaneous PT-symmetric breaking point zones to seek the strongest extraordinary point and obtain that at this point the strongest extraordinary transmission and reflection arrive at 2.96316 × 105 and 1.32761 × 105, respectively, due to the PT-symmetric coupling resonance and the special symmetry pattern of TM networks. These enormous gains are several orders of magnitude larger than the previous results. This optical system may possess potential in designing optical amplifier, optical logic elements in photon computers and ultrasensitive optical switches with ultrahigh monochromatity.
© 2017 Optical Society of America
1. Introduction
In this decade, people have paid great attention to a kind of novel micro-nano artificial structures, parity-time-symmetric (PT-symmetric) [1–4] optical systems [5–29], whose refractive indices possess even-function real part and odd-function imaginary part and the resonances of the gain and loss coupling effects of electromagnetic (EM) waves are capable of producing unusual optical characteristics, such as noncommutability of EM wave propagation [8, 10, 29], unidirectional invisibility [9], ultrastrong transmissions/reflections [11,29], extraordinary optical modes [17], and so on.
Optical waveguide networks composed of one-dimensional (1D) waveguide segments are another kind of interesting artificial structures [30, 31]. Just like photonic crystals, optical waveguide networks are also a kind of photonic band gap (PBG) materials/structures. However, comparing with the former, the latter are richer and more flexible in symmetry, and the amplitude and phase of EM waves can be conveniently measured anywhere inside their structures [30]. Consequently, optical waveguide networks have also attracted considerable attention over the past twenty years [30–41].
In this paper, combining the optical PT-symmetric concept and waveguide network, we design a kind of new artificial micro-nano material, the 1D PT-symmetric Thue-Morse aperiodic optical waveguide networks (PTSTMAOWNs), whose optical structure is described in detail in Section 2.1. It is well known that being a bridge of linking periodic systems with quasiperiodic ones in a geometrical structure, Thue-Morse (TM) sequence [42,43] is a kind of typical ordered aperiodic one, whose order degree is between those of periodic and quasiperiodic sequences. Due to the specific structural symmetry, a PTSTMAOWN enables EM waves to create richer photonic modes, more spontaneous PT-symmetric breaking points, stronger extraordinary resonant coupling effects, and larger extraordinary transmissions/reflections than those reported yet. Additionally, our designed PTSTMAOWN is aperiodic and unfortunately, the reported methods for calculating spontaneous PT-symmetric breaking points of periodic systems are no longer available. Consequently, in this paper we propose a coupling-resonant-zone approach to obtain the strongest extraordinary transmission and reflection. It may deepen people’s understanding of PT-symmetric optical systems and may provide new selections for the designing of all-optical devices based on strong transmissions/reflections.
2. Model and method
2.1. PTSTMAOWN model
In this paper, we design a kind of new artificial micro-nano structure, the 1D PTSTMAOWNs. As an example, the 2nd generation of 1D PTSTMAOWN is schematically shown in Fig. 1, where each PT-symmetric waveguide segment is composed of three sub-segments and they possess equivalent length but different refractive index (i.e. η(x) = η* (−x)), and the refractive index for each waveguide segment is defined as follows:
where ηR = ηSiO2 = 1.4430 [44, 45] and silicon dioxide is chosen to be the material of the first and third sub-segments, η2 = ηTiO2 = 2.4328 [44, 45] and titanium dioxide is chosen to be the material of the second sub-segment. In fact, making η(x) complex conjugate along the length can be realized by two means: doping gain and loss quantum dots [46] or modulating the density of materials [47]. The parameters aij, bij − aij and lij are, respectively, the lengths of the 1st sub-segment, the 2nd sub-segment, and the total waveguide segment between nodes i and j. On the other hand, in Fig. 1 we set the waveguide lengths in unit cells A and B as d1: d2: d3 = 1 : 1 : 2. In order to measure conveniently, we set the photonic frequency of the strongest extraordinary transmission and reflection to be related to the communication wavelength, i.e., νmax = c/λC. From the result in subsection 3.3 one knows that νmax = 0.245c/d1, and consequently, d1 = 0.245λC = 379.8 nm.
Fig. 1 Schematic diagram of the 2nd generation of PTSTMAOWN with one entrance and one exit, where EI, ER, and EO are the input, reflective, and output EM waves, respectively. Each unit cell contains two PT-symmetric waveguide segments, where the lengths for the upper and lower arms of unit cell A are d1 and d2, respectively, while those of unit cell B are, respectively, d1 and d3.
The unit cells in a 1D PTSTMAOWN is arranged following TM sequence [42, 43]. In this paper, we set the 0th generation of the TM sequence to be B, and the forthcoming generations satisfy the substitution rules B → BA and A → AB. Therefore, TM sequence can be expressed as
where is complementary to Gξ.2.2. Network equation
It is well known that in a 1D waveguide segment, the EM wave function between nodes i and j can be regarded as the following linear combination of two opposite traveling plane waves:
where ku = ωηu/c (u = 1, 2, 3) is the the wave vector of the uth sub-segment. By means of energy flux conservation, one can deduce the following network equation for three-material optical networks: where ψi and ψj are the wave functions at nodes i and j, respectively, andSpecially, for networks composed of vacuum waveguides, there exists η1 = η2 = η3 = 1.0 and Eq. (4) changes to be
This is exactly the network equation of vacuum waveguide networks [32–35,37,38,48].For networks composed of two-dielectric-material waveguides, e.g., η3 = η2 and η1 ≠ η2, then Eq. (4) will be changed into
where M = Φ, Ξ = Γ, U = Π + Θ, Z = −(Π − Θ). It is exactly the network equation of two-segment-connected two-material waveguide network [40].3. Results and discussions
3.1. Photonic mode
In this paper, we mainly investigate the ultrastrong extraordinary optical characteristics in 1D PTSTMAOWNs based on photonic modes.
Generally, obtaining photonic modes in a periodic optical waveguide network needs two steps [33,39,41]. Step I, deduce the dispersion relation by means of network equations [30,32,33] and generalized Floquet-Bloch theorem [33]. Step II, determine photonic modes [33,41] by solving the Bloch wave vector in the dispersion relation. However, our designed 1D PTSTMAOWNs are aperiodic and their dispersion relations can not be analytically deduced. Nonetheless, in terms of symmetry TM sequence is the closest aperiodic model to periodic sequence and each unit cell A (B) is only separated by not more than two Bs (As). It means that TM sequence may be closely related to these two sub-periodic sequences of A and B, and then we calculate the photonic modes in TM networks by combining the solutions of the dispersion equations of these two sub-periodic sequences of A and B with each other.
For 1D periodic optical waveguide network, it is known that the dispersion relation possesses the form of cosK = f (ν, η) [33,41], where K is the dimensionless Bloch wave vector, f is the dispersion function, ν is the frequency of an EM wave, and η is the material refractive index. For an optical waveguide network made up of material with real refractive index (i.e., η is real), it can be demonstrated that f must be real, and obviously, when the absolute value of f is equal to and/or smaller than 1, the dimensionless Bloch wave vector K will only possess real solutions, the EM wave with frequency ν can propagate through the network without attenuation and the photonic mode is a propagation mode (PM); when the absolute value of f is greater than 1, K will possess two complex solutions with positive and negative ImKs at the same time. In mathematical view, the former K corresponds to a loss solution while the latter K denotes a gain solution. However, in physical view, in an optical waveguide network composed of material with real refractive index, there exists none gain mechanism, and consequently, the EM wave with frequency ν will propagate through the network with attenuation and these two photonic modes are all evanescent modes. It means that for an optical waveguide network constructed by material with real refractive index, there only exist one kind of PM and one kind of non-propagation mode (NPM) (i.e., the evanescent NPM), and |f| = 1 is the critical point for judging photonic modes.
For an optical waveguide network composed of PT-symmetric waveguide segments, one can demonstrate that f may be real or complex. When f is complex, K will possess two complex solutions with positive and negative ImKs at the same time. Now, in a PT-symmetric structure, there exists simultaneously loss and gain mechanisms, and consequently, the EM wave with frequency ν will simultaneously possess two kinds of non-propagation photonic modes, the evanescent and crescent modes. It means that for a PT-symmetric optical waveguide network, the evanescent and crescent photonic modes can not be distinguished clearly anywhere inside the system, which is mathematically more complicated than other reported PT-symmetric optical materials, and consequently, we have to use |f| = 1 as the critical point for judging photonic modes in PT-symmetric optical waveguide networks. In detail, we define the four kinds of photonic modes in our designed PTSTMAOWNs as follows: (i) when f is real and |f| ≤ 1, the photonic mode is PM; (ii) when f is real and |f| > 1, the photonic mode is the first kind of NPM, i.e., NPM-I; (iii) when f is complex and |f| ≤ 1, the photonic mode is the second kind of NPM, i.e., NPM-II; (iv) when f is complex and |f| > 1, the photonic mode is the third kind of NPM, i.e., NPM-III. In Fig. 2/3, we plot the distribution of photonic modes in periodic optical waveguide network composed of unit cell A/B. From Figs. 2 and 3 one can see that in these two PT-symmetric optical systems, there exist all of the four kinds of photonic modes. Based on the distribution of photonic modes, one can calculate the spontaneous PT-symmetric breaking points numerically.
Fig. 2 Distribution of photonic modes in the periodic optical waveguide network composed of unit cell A, where PM denotes propagation mode, NPM means non-propagation mode, BP indicates breaking point, and GBP represents generalized breaking point.
Fig. 3 Distribution of photonic modes in the periodic optical waveguide network composed of unit cell B, where PM denotes propagation mode, NPM means non-propagation mode, BP indicates breaking point, and GBP represents generalized breaking point.
3.2. Spontaneous PT-symmetric breaking point
For a PT-symmetric optical system, it is known that extraordinary optical characteristics occur at/near the spontaneous PT-symmetric breaking points [5]. Generally, for the EM waves with certain frequency ν0, if the imaginary part of the refractive index increases with the photonic mode changing from PM (NPM) to NPM (PM), the spontaneous breaking point can be defined as the imaginary part of the refractive index (i.e., ηI) on the boundary between PM and NPM. However, in this paper there exist three kinds of NPMs, so we modify the definition of spontaneous breaking point and call the extreme value of ηI on the boundary between PM and any kind of NPM where its curve is non-differentiable the “breaking point” (BP), call the extreme value of ηI on the boundary between one kind of NPM and another kind of NPM where its curve is non-differentiable the “generalized breaking point” (GBP), as marked by circles in Figs. 2 and 3.
From Fig. 2 one can see that there exist one BP (i.e., BP-I) and one GBP (i.e., GBP-IV) for the periodic optical waveguide network composed of unit cell A. Fig. 3 shows that there exist two BPs (i.e., BP-II, BP-III) and three GBPs (i.e., GBP-I, GBP-II, GBP-III) for the periodic optical waveguide network composed of unit cell B. Obviously, from Figs. 2 and 3 one can see that comparing with usual PT-symmetric waveguide [17], a PTSTMAOWN generates richer photonic modes and more spontaneous PT-symmetric breaking points. In order to investigate the distribution of spontaneous PT-symmetric breaking points of the 1D PTSTMAOWN and obtain the exact ηI corresponding to the strongest extraordinary transmission and reflectioin, we arrange all of the BPs and GBPs following the value of ηI in Fig. 4. On the other hand, we also add some coupling resonant (CR) zones on the ηI axis, during which the transmission and reflection peaks totally overlap. It means that transmission and reflection EM waves with the frequencies in these CR zones will create sympathetic vibrations and produce maximal transmission and reflection at the same frequency.
Fig. 4 Distribution of spontaneous PT-symmetric breaking points and coupling resonant zones of the 1D PTSTMAOWN, where CR means coupling resonant, BP denotes breaking point, GBP indicates generalized breaking point, respectively. Orange (green) circles represent the BPs and/or GBPs produced by the periodic optical waveguide network composed of unit cell B (A), the colour of a semicircle indicates the photonic mode which is defined in Figs. 2 and 3.
From Fig. 4 one can see that BP-I and GBP-IV construct zone A, which is the zone of the spontaneous PT-symmetric breaking points for the periodic optical waveguide network composed of unit cell A, and similarly, BP-II, BP-III, GBP-I, GBP-II, and GBP-III construct zone B, which is the zone of the spontaneous PT-symmetric breaking points for the periodic optical waveguide network composed of unit cell B. We call the total range of zones A and B zone TM, which is the zone of the spontaneous PT-symmetric breaking points for PTSTMAOWN. From Fig. 4 one can also see that on the left-hand side of, in the middle of, and on the right-hand side of zone TM, there exist one CR zone, respectively. In fact, we have found more than ten CR zones by simulation calculations and most of them are on the right-hand side of zone TM. For simplicity, we have not plotted them in detail.
For PT-symmetric mechanical systems [1–4], it is known that, when potential function is below (smaller than) the spontaneous PT-symmetric breaking point, the Hamiltonian eigenvalues are all real; and when potential function is at (equal to) the spontaneous PT-symmetric breaking point, the Hamiltonian eigenvalues occur complex solution(s). Similarly, for PT-symmetric optical systems [5–9], when imaginary part of the refractive index is below (smaller than) the spontaneous PT-symmetric breaking point, the photonic modes are all ordinary; and when imaginary part of the refractive index is at (equal to) the spontaneous PT-symmetric breaking point, the photonic modes occur extraordinary one(s). Consequently, extraordinary optical properties can only be generated when imaginary part of the refractive index is at (equal to) and/or near the spontaneous PT-symmetric breaking point. It is known that strong optical properties is generally produced when EM waves create sympathetic vibrations in the system. So, in order to obtain the strongest extraordinary transmission and reflection, we try to seek the exact ηI of EM waves in the CR zone on the right-hand side of zone TM and nearest to zone TM at the same time. By means of this approach we obtain the exact ηI in CR Zone III, which is on the right-hand side of zone TM and nearest to zone TM at the same time. The exact value of ηI at this point is 4.145836 × 10−3. In Fig. 4, we mark the strongest extraordinary point in CR Zone III.
3.3. Ultrastrong extraordinary transmission and reflection
After determine the value of ηI for the strongest extraordinary point, by means of generalized eigenfunction method [49] one can calculate the transmission and reflection spectra. On the other hand, we find that the strongest extraordinary transmission and reflection are related not only to the value of ηI for the strongest extraordinary point but also to the generation number of network. In PT-symmetric waveguide networks, there exists a “coupling resonance & attenuation” balance. With the increment of the number of TM unit cell, the coupling resonances will increase. Meanwhile, the light attenuation will also increase. However, with the generation number increases, the number of TM unit cells is in an exponential explosive growth (2n), which causes attenuation to increase more quickly than coupling resonances. Therefore, when the two factors (coupling resonance and attenuation) reach an optimal balance, one can locate the “optimal generation number” and acquire the best performance, which in our designed PTSTMAOWN, the optimal generation number is six. In Fig. 5, we plot the transmission and reflection spectra for the 6th generation of PTSTMAOWN at the strongest extraordinary point ηI = 4.145836 × 10−3, where TL and RL denotes the transmissivity and reflectivity for the EM waves propagating from left to right, respectively, and TR and RR denotes the transmissivity and reflectivity for the EM waves propagating from right to left, respectively.
Fig. 5 Transmission and reflection spectra of the 6th generation of PTSTMAOWN at the strongest extraordinary point, where ηI = 4.145836 × 10−3.
From Fig. 5 one can see that at the resonant frequency ν = 0.245c/d1, ultrastrong extraordinary transmission and reflection are generated and the peak values are, respectively, TL = 2.96316×105 and RL = 1.32761 × 105. Our results show giant gains and are several orders of magnitude larger than the results reported previously [9,11,29], which proves our new approach correct and efficient. In order to investigate the reason of producing ultrastrong gains, we compare the transmission of our designed PTSTMAOWN with those of other two TM networks. The results are ploted in Fig. 6, where the blue thin solid line expresses the results of our designed PTSTMAOWN (η1 = 1.4430 + 0.004145836ι, , PT-symmetric network), the green thick dashed line denotes the results of the network constructed by the materials with positive imaginary part of the refractive index (η1 = η3 = 1.4430 + 0.004145836ι, pure loss network), and the red thick dotted line represents the results of the network constructed by the materials with negative imaginary part of the refractive index (η1 = η3 = 1.4430 − 0.004145836ι, pure gain network).
Fig. 6 Transmission spectra of the sixth-generation Thue-Morse waveguide networks composed of three different kinds of materials, where EM waves all propagate from left to right.
From Fig. 6 one can see that the peak transmissivity of PT-symmetric network is 2.96316 ×105, that of the pure gain network arives at 1.78843 × 103, and that of the pure loss network is only 0.99006. The results show that pure loss network is composed of loss materials and the accumulated loss effects make the total transmissivity be smaller than 1.0. The results also demonstrate that pure gain network is composed of gain materials and the accumulated gain effects make the total transmissivity be much larger than 1.0. However, PT-symmetric network is composed of loss and gain materials simultaneously, but the total transmissivity is not caused by accumulated loss and gain effects, otherwise the transmissivity of PT-symmetric network would be larger than that of pure loss network and smaller than that of pure gain network. The ultrastrong extraordinary transmission must be created by some resonant effects. Based on its symmetric structure, we think it may be produced by the accumulated couplings of two kinds of photonic resonant modes, the evanescent and crescent modes. These coupling resonances make giant gain amplifications, which indicates high efficiency for a PTSTMAOWN to convert the energy of pump light into output, unveiling great potential in further related experiments.
3.4. Noncommutability
Just as other PT-symmetric optical systems [8, 10, 29], our designed 1D PTSTMAOWN also exhibits asymmetric incident-direction-dependent noncommutability of EM wave propagation.
Figure 5 shows the transmission and reflection spectra for the 6th generation of PTSTMAOWN at the strongest extraordinary point ηI = 0.004145836, where TL (red thin solid line) and RL (green thick dotted line) denotes the transmissivity and reflectivity for the EM waves propagating from left to right, respectively, and TR (blue thick solid line) and RR (orange thick dashed line) denotes the transmissivity and reflectivity for the EM waves propagating from right to left, respectively.
From Fig. 5 one can see that the red thin solid line is all above the the thick blue solid line. It means that for the frequency within the range of 0.243c/d1 to 0.247c/d1, the transmissivities for the EM waves propagating from left to right are all larger than those for the EM waves propagating from right to left. Figure 5 also demonstrates that the green thick dotted line is all above the the thick orange dashed line. It shows that for the frequency within the range of 0.243c/d1 to 0.247c/d1, the reflectivities for the EM waves propagating from left to right are all larger than those for the EM waves propagating from right to left. This is a typical noncommutability of EM wave propagation which can be utilized to develop optical logic gates of photon computers.
3.5. Photonic band gap
In order to investigate the PBG properties of our designed PTSTMAOWNs, in Fig. 7 we draw the transmission spectra of the 1st, 2nd, and 3rd generations of the 1D PTSTMAOWNs at the strongest extraordinary point ηI = 0.004145836 for the EM waves with the frequency within the range of 0.25c/d1 to 0.40c/d1.
From Fig. 7 one can see that within the range of 0.273c/d1 to 0.390c/d1, the transmission attenuates seriously with the increment of generation number/unit cell number. This is a typical PBG and its gap-midgap ratio is Δν/νC = 35.3%. It means that our designed 1D PTSTMAOWNs are also capable of producing PBGs and are a kind of PBG structures. Based on our previous works [33–41], one can demonstrate that changing the parameters of waveguide lengths (d1, d2, d3) and materials (η1, η2, η3), the frequency position, width, number, and attenuation depth of PBGs can be all adjusted flexibly.
4. Conclusions
In this paper, we construct a 1D PTSTMAOWN and mainly investigate the ultrastrong extraordinary transmission and reflection, noncommutability of EM waves propagation, and PBG properties.
For the aperiodic network, we propose a new approach to study the photonic modes, where the dispersion relations for two periodic waveguide sub-networks are used for calculating the distributions of photonic modes and spontaneous PT-symmetric breaking points. It is found that in this aperiodic PT-symmetric network there exist one kind of propagation mode and three kinds of non-propagation modes, which are quite different from those of previous PT-symmetric optical systems. Consequently, spontaneous PT-symmetric breaking points in our designed network are also different from those reported previously. In order to obtain the strongest extraordinary transmission and reflection generated in 1D PTSTMAOWN, we develop an approach to sort all of the zones of spontaneous PT-symmetric breaking points, seek the strongest extraordinary point, and determine the value of ηI at this point. The strongest extraordinary transmission and reflection we obtained are generated in the 6th generation of 1D PTSTMAOWN and arrive at 2.96316 × 105 and 1.32761 × 105, respectively, at the same frequency of 0.245c/d1, which are several orders of magnitude larger than the results reported previously. The 1D PTSTMAOWN may be used for designing optical amplifier, optical communication relays, and ultrasensitive optical switches with ultrahigh monochromatity.
Additionally, we research the noncommutability of EM waves propagating in a 1D PTSTMAOWN and find that this network also exhibits incident-direction-dependent noncommutability in both transmission and reflection spectrum. It may possess potential in developing optical arithmetic elements and photonic logic gates in photon computers.
Finally, the PBG properties are also investigated. It is found that the 1D PTSTMAOWN is also capable of producing PBGs and changing the parameters of waveguide lengths (d1, d2, d3) and materials (η1, η2, η3), the frequency position, width, number, and attenuation depth of PBGs can be all adjusted flexibly. Combining this “traditional” property with the “new” extraordinary optical characteristics aforementioned, one may use it to create some complex optical filters with special purposes.
Funding
National Natural Science Foundation of China (NSFC) Grant Nos. 11674107, 11374107, 11374108, 11775083, 61774062; Natural Science Foundation of Guangdong Province, Grant No. 2015A030313374.
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