Open Access
Add to BibSonomy
Abstract
A kind of one-dimensional (1D) complete-connected network (CCN) is designed and its extraordinary optical property for producing an ultrawide photonic band gap (PBG) is investigated. The gap-midgap ratio formulaes of the largest PBGs created by CCNs are analytically derived, and the results indicate that with the increment of the node number in a unit cell, the number of the loops that can produce antiresonances increases fleetly, and consequently the gap-midgap ratio of the PBG produced by CCNs enlarges rapidly and tends rapidly to the limit at 200%. Moreover, the general transmission formula for 1D CCNs is analytically determined. Due to the periodicity, two types of transmission resonance peaks are generated, and the condition is analytically obtained from the transmission formula. This kind of CCN may have wide applications to design superwide band optical filters, optical devices with large PBGs and strong photonic attenuations, and other related optical communication and optical increment processing devices.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Periodic dielectric structures on a wavelength scale exhibit photonic band gaps (PBGs), where the propagation of electromagnetic (EM) waves is forbidden [1,2]. Many significant applications [3–7] of PBG structures are based on absolute PBGs and the larger the width of the absolute PBG is, the better the characteristic will be. Early work produced PBGs by photonic crystals (PCs) [8–10], and offer the possibility to confine and control the propagation of EM waves. Since 1987 PCs have attracted people’s much attention and have been applied to, e.g.,all-optical microchips [11], all-optical switching [12], polarization beam splitter [13,14] and all-optical RAM [15], etc. Many significant applications of PCs are based on absolute PBGs and the larger the width of the absolute PBG is, the better the characteristic will be. So people made attempts to design PCs producing larger absolute PBGs. However, as the lattice constant of a PC is on the scale of the wavelength of an EM wave, the shorter the work wavelength is, the smaller the characteristic length of a PC will be and then the more difficult to fabricate the structure will be, especially for a three-dimensional (3D) PC structure. Meanwhile, PCs generally require materials with large dielectric contrasts.
Fortunately, another kind of PBG networks composed of 1D waveguides [16–19] overcome these shortcomings and can be experimentally easily realized. These structures are flexible and the phase, amplitude and wave function at each node can be measured anywhere inside the systems. On the other hand, strong scattering can be easily introduced in a unit cell to generate large full gaps in any dimension and thus, these systems do not require a material with a large dielectric constant [16]. Compared to PCs used in producing PBG, OWNs are capable of producing ultrawide PBG by introducing the loops that can produce antiresonances into the system. Meanwhile, many studies [16,19,20] show that the most effective loop is the ring with the waveguide length ratio of 1:2. The larger the proportion of this loops in a network is, the wider the PBG will be. The presently reported the maximum gap-midgap ratio of photonic band gap produced 1D tetrahedral network is 146% and the average attenuation of the photonic band gap arrives at $10^{-106}$ [21]. Therefore, whether can a new OWN that can generate wider PBG be carried out?
Compared to the tetrahedral network [21], in this study, the kind of complete-connected networks (CCNs) with high proportional loops were designed and their extraordinary optical characteristic for producing ultrawide photonic band gap is investigated. For example, the schematic diagram of CCNs is shown in Fig. 1, which is a periodic structure with 1D arranging unit cells. The results of this study indicated that with the increment of the node number in a unit cell, not only the number of loops increases rapidly, but also the proportion of loops increases monotonously. Consequently, the gap-midgap ratio of the PBG produced by these networks enlarges rapidly and quickly tends to the limit the gap-midgap ratio to 200% . The PBG generated by CCNs is one order of magnitude larger than that of the widest absolute PBG created by PCs [22–24] and attains the limit of the widest PBG yielded by waveguide network systems. Our designed CCNs provide a kind of new PBG structures with high efficiency and may be useful for the designing of superwide band optical filters, optical devices with large PBGs and strong photonic attenuations, and other related optical communication and optical increment processing devices.
Fig. 1. Schematic diagram of a kind of CCNs. The CCN with two unit cells connecting with one entrance and one exit, where each dashed line denotes $l$ segments with the lengths of $d_1$, $d_2$ and $d_l$, and each solid line indicates one segment of 1D waveguide with the length of $d_1$.(a) The CCN with each unit cell possesses five nodes. (a) The CCN with each unit cell possesses $n$ nodes.
This paper is organized as follow. In Section 2, we introduce the main theory and method for analytical deducing and numerical calculation. In Section 3, we deduce the dispersion relation and transmission coefficient for CCNs. Simultaneously, the analytical and numerical results of the PBG properties and transmission characteristics are given in this section. Finally, a conclusion of this paper is drawn in Section 4.
2. Theory and methods
2.1 Network equation
The networks studied in this paper are formed by segments of 1D vacuum waveguides, where 1D waveguide means the radius of the waveguide cross-section should be at least one order of magnitude smaller than waveguide length and the length of propagation of EM waves. Based on the propagation theory by EM field, the waveguide satisfying the above conditions only single-mode propagation of EM waves needs to be considered. EM wave function with angular frequency $\omega$ in the segment between nodes $i$ and $j$ can be regarded as a linear combination of two opposite traveling plane waves
where $k=\omega /c$ and $c$ is speed of the EM wave in vacuum. For the waveguide networks composed of the dielectric waveguides with the refractive index $n_e$, the speed of EM waves should be changed into $c/n_e$. By use of the energy flux conservation one can deduce the network equation as follows [16,18–20] where $\psi _{i}$ and $\psi _{j}$ are the wave functions at nodes $i$ and $j$, respectively, the summation of $j$ is over all nodes linked to node, $l_{ij}$ is the length of the segment between nodes $i$ and $j$. Zhang and co-workers [16] used this network equation to investigate the transmission of EM waves through a 3D network in a diamond structure and the calculations are accurately confirmed by experimental results. Making use of this network equation and the dimensionless Floquet-Bloch theorem proposed by us [19], we deduced the dispersion relation of a serial loop waveguide network and the result we obtained is exactly that reported in Ref. [25], where the simulations of the transmission coefficients are accordant with experimental values.2.2 Generalized Floquet-Bloch theorem
In Ref. [24] we develop a dimensionless Floquet-Bloch theorem to study the band structures and the attenuation behavior of different EM modes inside the PBGs of square networks where nearest neighbor nodes are connected by more than one segment. We found that as long as the configure period is defined, the band gap can be obtained. We have conducted that, for a configuration-periodic network, there is the following relation for the Bloch function when a discrete configuration translation $\vec {T}$ is made:
where $\vec {N}$, $\vec {T}$ and $\vec {K}$ are all dimensionless and their values depend on the configuration of the network. When $\vec {K}$ is real, the wave is a propagation mode and can travel through the network without attenuation, whereas in the PBGs wave is a evanescent mode with complex $\vec {K}$. In Fig. 1, we can obtain the equation following,2.3 Generalized eigenfunction method
In this paper, we use the generalized eigenfunction method to calculate transmissivity and wave function, where wave transfer equations are changed into a transfer matrix, and transmission is regarded as generalized wave function.
3. Properties of photonic bands
3.1 Photonic band structure
For a CCN with infinite unit cells, taking into account the network equation and the dimensionless Floquet-Bloch theorem [19] one can obtain the dispersion relation. In Fig. 1, from network equation, one can write out the relation of the wave functions as following.
From the equation, one can deduce
From Eq. (4) and (7), one can obtain
From the dispersion relation, it shows that if the absolute value of the right hand of Eq. (8) is smaller than or equal to 1.0, real solutions for $K$ can be found, the wave with this frequency is a propagation mode and can travel through the network without attenuation. Otherwise, $K$ will be a complex vector, the wave with this frequency is an evanescent mode and then a PBG appears. In a PBG, the wave is Bragg scattered and the attenuation within a gap is due to the destructive interference of scattered waves. By use of the Eq. (8) their band structures are shown in Fig. 2 for one-segment-connected (1SC) CCN and two-segment-connected (2SC) CCN. As shown, a large PBG and two wide photonic pass bands can be seen. Obviously, with the increment of nodes $n$, the width of PBG rapidly increases.
Fig. 2. The spectra of the CCNs with 1SC and 2SC. (a) and (c) Photonic band structures, where $\Gamma$ and $X$ correspond to the high symmetric points $K=0$ and $\pi$, respectively. (b) and (d) Dispersion spectrum, where the two black vertical solid lines with $\cos K=\pm 1$ are the marker lines for distinguishing passbands from stopbands.
3.2 Photonic band gaps
3.2.1 Width
For researching the variation trend of PBG with the number of nodes, taking advantage of Eq. (8), one can deduce the gap-midgap ratio for the largest PBG generated by a one-segment-connected (1SC) CCN as follows:
In a similar way, the gap-midgap for a two-segment-connected (2SC) CCN with $d_1:d_2=2:1$ can be written as
Comparing Eqs. (9) with (10) one can see that the efficiency of generating giant PBGs of 2SC-2:1 CCNs is much higher than that of 1SC CCNs. For example, when $n=2$, $({\Delta} \omega /\omega _\textrm {{C}})_{2\rm {SC-2:1}}=100\%$. If $n=6$, $({\Delta} \omega /\omega _\textrm {{C}})_{2\rm {SC-2:1}}=159\%$. The results are ploted in Figs. 2(c) and (d). It means that to produce two giant PBGs with the same width, the 1SC CCN will be nearly twice complicated of the 2SC-2:1 CCN. Comparing with the examples of 1SC CCNs, the node numbers of a unit cell of 2SC-2:1 CCNs are much smaller than those of 1SC CCNs. This is also because of the number and density of triangular loops in 2SC-2:1 CCNs. Between each pair of nodes in a unit cell of 2SC-2:1 CCNs there exist two segments of 1D waveguides with the lengths of $2d_1$ and $d_1$, respectively. Comparing with 1SC CCNs, these two segments compose new equilateral triangle loops and make the number of triangular loops in 2SC-2:1 CCNs improve approximately twice. This is the very reason that makes the efficiency of producing PBGs by the latter much higher than the former.
3.2.2 Average attenuation
By use of Eq. (1) one can obtain the normalized wave functions of the entrance (nodes 0 and 1) and the exit (nodes 9 and 10) in Fig. 1(a) as follow:
From Eqs. (11) and (12), one can write the following equation,
By use of Eqs. (2) and (12), one can deduce that
Furthermore, for CCN with each unit cell possesses $n$ nodes and $m$ unit cells, one can obtain the following equations by the same method.
Where the node with number $f(m)$ is the public node of the two cells with the number $m$ and $m+1$, and nodes with number between $f(m-1)$ and $f(m)$ belong to the cell with the number $m$.
By means of Eqs. (7) and (15), one can deduce the transmission coefficient $T$ for a CCN with $m$ unit cells as follows:
whereMaking use of Eq. (16) we calculate the average attenuations of the largest PBGs resulted in some 1SC and 2SC-2:1 CCNs and list the results in Table 1. From each row of Table 1 one can see that for the 1SC or 2SC-2:1 CCNs with identical node number of a unit cell, photonic attenuations increase rapidly with the increment of the number of unit cells. From each column of Table 1 one can see that for the 1SC or 2SC-2:1 CCNs with same number of unit cells, photonic attenuations increase slowly with the increment of the node number of a unit cell. Further, for two CCNs with same number of unit cells, the average attenuation of a 2SC-2:1 CCN with $n$ nodes in a unit cell is approximately equal to that of a 1SC CCN with $2n$ nodes in a unit cell. For example, when $m=3$, the average attenuation for a 2SC-2:1 (1SC) CCN with $n=12(n=24)$ is -106.3dB (-107.4dB). If $m=4$, the average attenuation for a 2SC-2:1 (1SC) CCN with $n=24 (n=48)$ is -166.4dB (-167.0dB). Obviously, the latter is nearly twice complex of the former.
Table 1. Average attenuations of the largest PBGs generated by 1SC and 2SC CCNs.
3.3 Photonic pass bands
By use Eq. (16), we plot in Fig. 3 the transmission of CCNs with $d_1:d_2=2:1$. Transmission spectra calculated by the generalized eigenfunction method are also shown in Fig. 3. From Fig. 3, we observe that the results calculated by the Eq. (16) (green dotted line) are identical with those obtained by the generalized eigenfunction method (black solid line).
Fig. 3. Transmission spectra of CCNs with $d_1:d_2=1:2$, where the black and blue solid lines are obtained by the Eq. (16), the green and red dotted lines are obtained by the generalized eigenfunction method. (a) and (d) one unit cell. (b) and (e) two unit cells. (c) and (f) three unit cells.
Type I transmission resonance peak. The peaks at $\pi c/d_1$ and $2\pi c/d_1$ are the type I transmission resonance peak and are labeled as I [27]. In brief, such peaks are located at $\omega _{I}=\alpha c/d_1$, where $\alpha$ is an integer.
Type II transmission resonance peak. Following Eq. (16), when $\cos mK=1$ (i.e., $mK=N\pi$, $N=1, 2, \ldots , m-1$), $T\equiv 1(m>1)$.It indicates that the transmission spectrum of a network with $m$ unit cells will from another $m-1$ transmission resonance peak because of the periodicity. These peaks are the type II transmission resonance peak and labeled as II. All these transmission resonances are Bragg resonances, which are due to the augment of the number of unit cell. According to Eq. (8), these peaks will appear at frequency
As illustrated in Fig. 3, the transmission spectrum of CCNs with one unit cell cannot produce the type II transmission resonance peaks [Fig. 3(a) and (d)]. Nevertheless, when $m>1$, there generate $m-1$ transmission resonance peaks in every monotonic interval of the dispersive function. From Eq. (18), these type II transmission resonance peaks in Fig. 3(b) are locate at $0.8335\pi c/d_1$, $1.8335\pi c/d_1$ and $2.1655\pi c/d_1$. Homoplastically, the frequency of the type II transmission resonance peaks in Figs. 3(c), (e) and (f) can also calculate by using Eq. (18).
4. Conclusions
In conclusion, a kind of 1D CCNs containing high proportional loops are proposed. With the increment of the node number in a unit cell, not only the number of loops increases rapidly, but also the proportion of loops increases monotonously. Consequently, the gap-midgap ratio of the PBG produced by these networks enlarges rapidly and quickly tends to the limit the gap-midgap ratio to 200% . In addition, for a CCN with infinite unit cells, the gap-midgap ratio formulaes of the largest PBGs created by 1SC and 2SC-2:1 CCNs were deduced accurately. The results show that even if the simplest 1SC and 2SC-2:1 CCNs can yield huge PBGs. When the node number of a unit cell tends to be infinite, the ratio of ${\Delta} \omega /\omega _\textrm {{C}}$ attains the limit of the widest PBG reproduced by waveguide networks, 200%, which is about one order of magnitude larger than that of the widest absolute PBG resulted in PCs. For a CCN with finite unit cells, the number of unit cells influences the attenuations of the largest PBGs significantly. To yield a huge PBG with same attenuation, the 1SC CCN has to be nearly twice complex of the 2SC-2:1 one. So, a 2SC-2:1 CCN is now the best selectable system for the designing of optical devices with giant PBGs and strong attenuations. Finally, the transmission formula is analytically determined for any 1D CCNs. Due to the periodicity, two types of transmission resonance peaks can be produced. The condition for producing this phenomena are analytically obtained from the transmission formula. This study may be useful for the designing of superwide band optical filters, optical devices with large PBGs and strong photonic attenuations, and other related optical communication and optical increment processing devices.
Funding
National Natural Science Foundation of China (11674107); Foundation for Distinguished Young Talents in Higher Education of Guangdong (2018KQNCX308).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987). [CrossRef]
2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58(23), 2486–2489 (1987). [CrossRef]
3. J. Foresi, P. R. Villeneuve, J. Ferrera, E. Thoen, G. Steinmeyer, S. Fan, J. Joannopoulos, L. Kimerling, H. I. Smith, and E. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390(6656), 143–145 (1997). [CrossRef]
4. S. Noda, K. Tomoda, N. Yamamoto, and A. Chutinan, “Full three-dimensional photonic bandgap crystals at near-infrared wavelengths,” Science 289(5479), 604–606 (2000). [CrossRef]
5. S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics 1(8), 449–458 (2007). [CrossRef]
6. X. Hu, P. Jiang, C. Ding, H. Yang, and Q. Gong, “Picosecond and low-power all-optical switching based on an organic photonic-bandgap microcavity,” Nat. Photonics 2(3), 185–189 (2008). [CrossRef]
7. A. Tandaechanurat, S. Ishida, D. Guimard, M. Nomura, S. Iwamoto, and Y. Arakawa, “Lasing oscillation in a three-dimensional photonic crystal nanocavity with a complete bandgap,” Nat. Photonics 5(2), 91–94 (2011). [CrossRef]
8. E. Yablonovitch and T. Gmitter, “Photonic band structure: The face-centered-cubic case,” Phys. Rev. Lett. 63(18), 1950–1953 (1989). [CrossRef]
9. C. M. Anderson and K. P. Giapis, “Larger two-dimensional photonic band gaps,” Phys. Rev. Lett. 77(14), 2949–2952 (1996). [CrossRef]
10. Y. Chan, C. T. Chan, and Z. Liu, “Photonic band gaps in two dimensional photonic quasicrystals,” Phys. Rev. Lett. 80(5), 956–959 (1998). [CrossRef]
11. A. Chutinan, S. John, and O. Toader, “Diffractionless flow of light in all-optical microchips,” Phys. Rev. Lett. 90(12), 123901 (2003). [CrossRef]
12. H. Sun, Y. Hu, Y. Tang, J. You, J. Zhou, H. Liu, and X. Zheng, “Ultrafast polarization-dependent all-optical switching of germanium-based metaphotonic devices,” Photonics Res. 8(3), 263–270 (2020). [CrossRef]
13. F. Gan, C. Sun, H. Li, Q. Gong, and J. Chen, “On-chip polarization splitter based on a multimode plasmonic waveguide,” Photonics Res. 6(1), 47–53 (2018). [CrossRef]
14. C. Sun, Y. Yu, Y. Ding, Z. Li, W. Qi, and X. Zhang, “Integrated mode-transparent polarization beam splitter supporting thirteen data channels,” Photonics Res. 8(6), 978–985 (2020). [CrossRef]
15. K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, and M. Notomi, “Ultralow-power all-optical ram based on nanocavities,” Nat. Photonics 6(4), 248–252 (2012). [CrossRef]
16. Z. Zhang, C. Wong, K. Fung, Y. Ho, W. Chan, S. Kan, T. Chan, and N. Cheung, “Observation of localized electromagnetic waves in three-dimensional networks of waveguides,” Phys. Rev. Lett. 81(25), 5540–5543 (1998). [CrossRef]
17. L. Dobrzynski, A. Akjouj, B. Djafari-Rouhani, J. Vasseur, and J. Zemmouri, “Giant gaps in photonic band structures,” Phys. Rev. B 57(16), R9388–R9391 (1998). [CrossRef]
18. S.-K. Cheung, T.-L. Chan, Z.-Q. Zhang, and C. T. Chan, “Large photonic band gaps in certain periodic and quasiperiodic networks in two and three dimensions,” Phys. Rev. B 70(12), 125104 (2004). [CrossRef]
19. Z.-Y. Wang and X. Yang, “Strong attenuation within the photonic band gaps of multiconnected networks,” Phys. Rev. B 76(23), 235104 (2007). [CrossRef]
20. Z.-Q. Zhang and P. Sheng, “Wave localization in random networks,” Phys. Rev. B 49(1), 83–89 (1994). [CrossRef]
21. Q. Xiao, X. Yang, J. Lu, and C. Liu, “Huge photonic band gaps with strong attenuations resulted from quasi-one-dimensional waveguide networks composed of triangular fundamental loops,” Opt. Commun. 285(18), 3775–3780 (2012). [CrossRef]
22. R. Chern, C. C. Chang, C. C. Chang, and R. Hwang, “Large full band gaps for photonic crystals in two dimensions computed by an inverse method with multigrid acceleration,” Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 68(2), 026704 (2003). [CrossRef]
23. O. Toader, T. Y. Chan, and S. John, “Photonic band gap architectures for holographic lithography,” Phys. Rev. Lett. 92(4), 043905 (2004). [CrossRef]
24. Y.-F. Chau, F.-L. Wu, Z.-H. Jiang, and H.-Y. Li, “Evolution of the complete photonic bandgap of two-dimensional photonic crystal,” Opt. Express 19(6), 4862–4867 (2011). [CrossRef]
25. A. Mir, A. Akjouj, J. Vasseur, B. Djafari-Rouhani, N. Fettouhi, E. El Boudouti, L. Dobrzynski, and J. Zemmouri, “Observation of large photonic band gaps and defect modes in one-dimensional networked waveguides,” J. Phys.: Condens. Matter 15(10), 1593–1598 (2003). [CrossRef]
26. E. Lidorikis, M. Sigalas, E. N. Economou, and C. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. 81(7), 1405–1408 (1998). [CrossRef]
27. J. Zheng, X. Yang, X. Zhang, D. Deng, and H. Liu, “Transmission characteristics of one-dimensional periodic optical waveguide networks,” Phys. Rev. A 99(2), 023809 (2019). [CrossRef]
