Transmission properties of defect modes in one-dimensional photonic crystals containing gradient refractive index defects

Transmission properties of defect modes in one-dimensional photonic crystals containing gradient refractive index defects

Optik 126 (2015) 5158–5162 Contents lists available at ScienceDirect Optik journal homepage: www.elsevier.de/ijleo Transmission properties of defec...

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Optik 126 (2015) 5158–5162

Contents lists available at ScienceDirect

Optik journal homepage: www.elsevier.de/ijleo

Transmission properties of defect modes in one-dimensional photonic crystals containing gradient refractive index defects Bobo Xu, Gaige Zheng ∗ , Yigen Wu, Kun Cao School of Physics and Optoelectronic Engineering, Nanjing University of Information Science & Technology, Nanjing 210044, PR China

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Article history: Received 26 November 2014 Accepted 27 September 2015 Keywords: One-dimensional photonic crystal Gradient refractive index FDTD method Defect mode

a b s t r a c t The gradient refractive index (GRIN) defect layer is introduced into one-dimensional photonic crystal (1D PC) to develop the optical filters needed. The transmission properties of 1D PC with GRIN defect layer were investigated based on the finite difference time domain (FDTD) method. Defect mode was obtained according to the GRIN distribution functions. With a GRIN structure in the defect layer, simulation results show that the position and width of the photonic band gap can be effectively modulated by varying the thickness and GRIN distribution functions of defect layer and the parameters of symmetrical layers. The new study provides a certain reference for photonic band gap engineering and designing the photonicbased devices. © 2015 Elsevier GmbH. All rights reserved.

1. Introduction Photonic crystals (PCs), which exhibit photonic band gap (PBG) structure, have received considerable attention for fundamental physics studies as well as for potential applications in photonic devices [1–3] since the initial prediction of Yablonovitch [4] and John [5]. PCs are classified into three categories in terms of the dimensionality of stacks: one-dimensional, two-dimensional and three-dimensional photonic crystals [6]. Different arrangements of one-dimensional photonic crystal (1D PC) structures have attracted extensive studies because it can be easily fabricated by modern experimental techniques. Furthermore, light can be controlled and manipulated by introducing defects into the 1D PCs. The perfect 1D PCs have many applications, but its doped versions can be more useful, as semiconductors doped by impurities are more important than the pure ones [7]. The existence of the PBG and defect modes in the optical spectra of the PCs have leaded to many interesting phenomena and numerous applications [8–12] in improving the performance of optoelectronic and microwave devices, such as high-Q resonators, high-efficiency semiconductor lasers, optical filters and so on [13–15]. In most of the previous works, a conventional 1D PC structure with different arrangements of materials are studied that the refractive indexes of the defect layers are considered to be constant.

∗ Corresponding author. E-mail address: [email protected] (G. Zheng). http://dx.doi.org/10.1016/j.ijleo.2015.09.214 0030-4026/© 2015 Elsevier GmbH. All rights reserved.

For the defective 1D PCs, the transmission properties of gradient refractive index (GRIN) defect mode have almost not been reported. In this paper, GRIN distribution functions are applied to defect layer in the 1D PC structures. The transmission properties of 1D PC with linear GRIN defects were investigated based on the finite difference time domain (FDTD) method. The position and width of the photonic band gap at different GRIN defect modes were analyzed, and the difference between the constant refractive index defect and the GRIN defect under the same incident wavelength was investigated. The aim is to reveal that by introducing the appropriate GRIN defect layer, the wider band gap and desired band position can be obtained with a graded structure, and it is possible to design the optical filters needed.

2. Model and theory 2.1. 1D PC structure with GRIN defect layer Consider a defective 1D PC structure with arrangement of (AB)m C(BA)m , as schematically depicted in Fig. 1 where A and B are two kinds of different materials with constant refractive indexes of nA and nB , respectively, C stands for the GRIN defect layer, which is assumed that the gradient refractive index nC is a linear function of X and Y. And m is the number of couple of layers with high and low refractive index. It is such a structure that defect layer C is sandwiched symmetrically between two same PCs. In this model, we indicate the 1D PC structure as (AB)m C(BA)m where A indicates high dielectric layer with nA = 3.0, B the low dielectric layer with nB = 1.5 and C the GRIN defect layer with

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Fig. 1. Schematic of a defective 1D PC structure, where layers A and B represent the constant refractive index materials with nA = 3.0 and nB = 1.5, respectively and C stands for a GRIN defect layer.

Fig. 3. The transmission spectra of defective 1D PCs with different refractive index defect modes are illustrated, where the constant refractive index of defect layer is n = 2 and the GRIN is n = 2 +0.3X + Y.

layer with high refractive index. That is, in the (AB)m C(BA)m structure, the layer A is high refractive index layer, and the layer B is low refractive index layer. In theory, as long as increasing the layer numbers, the transmittance can reach closely to 100%. However, due to the absorption and scattering loss of membrane layers, the layer numbers could not be infinite. Here, we select m = 4 as the object for research. In this section, we present our simulation results of 1D PC containing GRIN defect layer. The transmission properties of defective 1D PC depend on both GRIN defects and crystal structure. Fig. 2. Three-dimensional diagram of GRIN distribution function, where nC varies linearly along with X, Y.

nC = nC (X,Y). Different GRIN distribution functions can be applied to defect layer C in the structure of (AB)m C(BA)m . Here, we choose the linear GRIN distribution function which has a simple manufacturing process. The applied GRIN distribution function is depicted in Fig. 2. 2.2. Finite difference time domain (FDTD) method Different analytical and numerical methods are proposed to investigate the optical transmission properties of the PC structures [16–18]. In this paper, the finite difference time domain (FDTD) method for computing the PBG structure of 1D PC containing gradient refractive index defects is presented. It is a state-of-the-art method for solving Maxwell’s equations [19] in complex geometries. The technique is discrete in both space and time. Maxwell’s equations are solved discretely in time, where the time step used is related to the mesh size through the speed of light [20]. Frequency domain information at any spatial point may be obtained through the Fourier transform of the time domain information at that point. The band structures of defective 1D PC to be simulated can have a wide variety of electromagnetic properties. The FDTD program is self made code using Fortran language. The calculation is done with limited resource, e.g. a normal PC. 3. Results and discussion As the optical coating theory shows, for membrane stack with high and low refractive index alternating, the transmittance will be reduced and the absorption loss will be increased when the last layer is the low refractive index layer. Therefore, in order to get higher transmittance and lower loss, we choose the outermost

3.1. Influence of GRIN defect layer on the transmission spectra Fig. 3 shows the transmission spectrum of the defective 1D PCs with different refractive index defect modes. In order to have an explicit understanding of the introduced GRIN defect, it is useful to compare the transmission spectra of 1D PCs with constant refractive index (n = 2) defect layer and GRIN (n = 2 + 0.3X + Y ) defect layer. In Fig. 3, X- and Y-axis stand for the incident wavelength and the transmittance, respectively. Our simulation results show that the photonic band gap width is becoming wider with the introduction of the GRIN defect. Thus, the band gap width can effectively be tuned by introducing GRIN defect. In practical design, we can change the GRIN distribution function of defect layer to find suitable parameters of the band gap width which meet the manufacturing requirement. 3.2. Influence of the thickness of defect layer on the transmission spectra Consider a defective 1D PC structure with arrangement of (AB)4 C(BA)4 , where layers A and B represent the constant refractive index materials with nA = 3.0 and nB = 1.5, respectively. C stands for the GRIN defect layer, which is assumed a linear function of X and Y. We explore the influence of the thickness of defect layer on the transmittance of 1D PC with GRIN defect layer. Originally, dC = 0.234 ␮m. Now, we respectively increase and reduce the thickness ( = 0.047 ␮m) of the defect layer. As one can see from Figs. 4 and 5, the transmission peaks red-shift and transmittance decreases in a certain extent with the increase of the thickness of GRIN defect layer; the transmission peaks blue-shift and transmittance increases in a certain extent with the decrease of the thickness of GRIN defect layer. However, the band gap width is a constant even with different thickness of GRIN defect layer. In practical design, according to the specific

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Fig. 4. The transmission spectra in defective 1D PC structure when we increase the thickness (dC = 0.234 ␮m, 0.281 ␮m, 0.328 ␮m) of GRIN defect layer. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 6. The structures (a) (AB)m CD(BA)m and (b) (AB)m DC(BA)m of defective 1D PC.

Fig. 5. The transmission spectra in defective 1D PC structure when we reduce the thickness (dC = 0.234 ␮m, 0.187 ␮m, 0.140 ␮m) of GRIN defect layer. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

manufacturing requirements, when the wavelength has a little change, the peaks can move to the required position through changing the thickness of the GRIN defect layer.

Fig. 7. The transmission spectra in defective 1D PC structure with different GRIN defect layers.

3.4. Influence of different parameters of crystal structure on the transmission spectra 3.3. Influence of different GRIN structures on the transmission spectra The influence of defect layer on the transmission properties of 1D PC is mainly reflected in the changes of the thickness and refractive index. Two different GRIN defect layers are inserted in the structure of (AB)4 C(BA)4 . Fig. 6, respectively, shows the structures (AB)m CD(BA)m and (AB)m DC(BA)m of defective 1D PC, where C stands for the parabolic gradient refractive index distribution function (nC = 3sin(1 − (X2 + Y2 )/3.5)), and D the linear GRIN distribution function (nD = 2 +0.3X + Y). By considering the different GRIN defect structures, the transmission spectra are plotted in Fig. 7. In Fig. 7, we can clearly observe the original positions of transmission peaks have changed. However, the band gap width is still invariant in the defective 1D PC structure even with different GRIN defect layers. So, some desired filters can be obtained by designing 1D PC structures using the proposed defect layers.

Until now, we have considered the thickness and structure of GRIN defect layer. However, the transmission properties of defect mode depend on both GRIN defects and crystal structure. Thus, we will focus on the parameters of crystal structure. Originally, nA :nB = 2, dA :dB = 0.58, m = 4.We consider a defective 1D PC structure with arrangement of (AB)m C(BA)m , where C stands for the constant refractive index defect layer and the GRIN defect layer, respectively. Next, we will respectively change the refractive index ratio nA :nB , the thickness ratio dA :dB and the periodic number m. In Fig. 8(a) and (c), the band gap width is getting wider with larger nA :nB . Meanwhile, the GRIN defect mode has the wider band gap width than refractive index defect mode under the same nA :nB . In Fig. 8(b) and (d), the band gap width is getting narrower with smaller nA :nB . Meanwhile, the GRIN defect mode has the narrower band gap width than refractive index defect mode under the same nA :nB . It is shown in Fig. 8 that band gaps appearing in transmission spectra can experience significant changes as we change the

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Fig. 8. Band gap width changing caused by the variations of nA :nB but the same dA :dB . The transmission spectra of defective 1D PC with constant refractive index defect layer are shown in (a) and (b), respectively, and the transmission spectra of defective 1D PC with GRIN defect layer are shown in (c) and (d) respectively. (a)–(c) nA is constant (nA = 3), and nB is 1.5, 1.2 and 1.0, respectively. Namely nA :nB = 2 (black), nA :nB = 2.5 (red), nA :nB = 3 (blue); (b)–(d) nA is constant (nA = 3), and nB is 1.5, 2.0 and 2.5, respectively. Namely nA :nB = 2 (black), nA :nB = 1.5 (red), nA :nB = 1.2 (blue). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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Fig. 9. Transmission peaks shifting caused by the variations of dA :dB but the same nA :nB . The transmission spectra of defective 1D PC with constant refractive index defect layer are shown in (a) and (b), respectively, and the transmission spectra of defective 1D PC with GRIN defect layer are shown in (c) and (d), respectively. (a and c) dA is constant (dA = 0.137 ␮m), and dB is 0.234 ␮m, 0.187 ␮m and 0.140 ␮m, respectively. Namely dA :dB = 0.58 (black), dA :dB = 0.73 (red), dA :dB = 1 (blue); (b and d) dA is constant (dA = 0.137 ␮m), and dB is 0.234 ␮m, 0.281 ␮m and 0.328 ␮m, respectively. Namely dA :dB = 0.58 (black), dA :dB = 0.48 (red), dA :dB = 0.41 (blue). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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significant changes as we adjust the gradation profiles of gradedindex defect layer and other structure parameters. This study introduced the capability of the 1D PC structure with GRIN defects to modulate the position and width of the photonic band gap. We investigate the transmission properties of defect modes in 1D PCs by varying the parameters of the symmetrical layers and GRIN defect layer. For such a GRIN defect structure, with appropriate structure parameters, a desired band gap and a band position can be obtained, which has the great potential as the optical filters and the plane wave modulators or works as the sensing materials. This study could thus have a great impact on the effective modulation of spectral response and the further development of photonic band gap engineering. Moreover, with the guidance of our research results, the tenability in our proposed structure is easy to implement in the experiment. Fig. 10. The transmission spectra in constant refractive index defect structure with different m are illustrated, where the periodic number of A and B is 3, 5 and 7, respectively.

Acknowledgments This work is partially supported by the National Natural Science Foundation of China (grant no. 61203211), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (grant no. 13KJB140006) and the Natural Science Foundation of the Jiangsu Province (grant no. BK20141483). References

Fig. 11. The transmission spectra in GRIN defect structure with different m are illustrated, where the periodic number of A and B is 3, 5 and 7, respectively.

refractive index ratio nA :nB . So, it is possible to modulate the filtering channel width by choosing different nA :nB with GRIN defect mode. In Fig. 9(a) and (c), the transmission peaks move towards short wave direction when the thickness ratio dA :dB is closing to 1.0. However, the band gap width has no obvious change. In Fig. 9(b) and (d), the transmission peaks move towards long wave direction when the thickness ratio dA :dB is closing to 0.4. Meanwhile, the GRIN defect mode has greater movement than refractive index defect mode under the same dA :dB . We analyze the transmission spectrum of the liner GRIN defect mode versus the constant refractive index defect mode when we change the periodic number m. At this time, nA :nB and dA :dB are unvaried. With a variation of periodic number m in the defective 1D PC structure, the simulation results are shown in Figs. 10 and 11. When the periodic number changes from 3, 5 to 7, the band gap width will become larger. The most important is that GRIN defect mode has the wider band gap width than refractive index defect mode under the same m. 4. Conclusions We studied 1D PC structure with arrangement of (AB)4 C(BA)4 , that A and B represent the constant refractive index materials and C stands for a GRIN defect layer. The linear and parabolic distribution functions, which practically are utilizable, are proposed for defect layer C. Band gaps appearing in transmission spectra can experience

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