Luận văn Design and investigation of 1D, 2D photonic crystals for bistable devices

ng the structural parameters of 1D, 2D PhC structures for optical bistable devices which have high quality factor, low optical intensity and time for switching. + Propose and design some integrated photonic structures which have high performance and special characteristic for bistable devices. 3. The main research contents of the thesis + Design and analysis the optical properties of 1D and 2D PhC structures. + Optimization of the structural parameters and resonant spectra of the grating structures to increase the quality factor and reduce the optical intensity for switching. + Investigation of the bistability characteristics of the optimal grating structures. + Design and simulation the narrow high-order resonance linewidth shrinking with multiple coupled resonators in SOH slotted 2D PhCs for reduced optical switching power in bistable devices. Differences and new ones in the research content of the thesis: + Currently in Vietnam, there are very few subjects and thesis mention the PhC structures for application in optical comunication due to the lack of fabrication equipment. This dissertation is considered as the first in computation and simulation of optical bistable devices using 1D and 2D PhC structures in Vietnam. + This dissertation uses the modern and highly accurate calculation and simulation methods to verify the achieved theoretical results, so the dissertation contributes to increase the professional research. This dissertation includes five chapters: Chapter 1. Overview Chapter 2. Calculation and simulation methods Chapter 3. Optimization of quality factor and resonant spectra of grating structures Chapter 4. Optical bistability in slab waveguide gratings Chapter 5. Optical bistability based on interaction between sloted cavities and waveguides in two-dimensional photonic crystals CHAPTER 1: OVERVIEW 1.1. Photonic crystal structures The first concept of PhCs was proposed by Yablonovitch and John in 1987 [7]. PhCs are the periodic structures of the dielectric elements in space. Due to the periodic of the refractive indices, the PhC structures produce the PBGs. Depending on the geometry of the structure, PhCs can be divided into three categories, namely one- dimensional (1D), two-dimensional (2D) and three-dimensional (3D) structures. The examples are shown in Figure 1.1. Figure 1.1. 1D, 2D, and 3D PhC structures (a) 1D PhC, (b) 2D PhC, (c) 3D PhC [27]. 1.2. Optical bistable devices Two features are required for presenting the bistable behavior: nonlinearity and feedback. Both features are available in nonlinear optics. An optical system is shown in Figure 1.33, this system exhibits the bistable behavior: For small inputs (Ivào < 1 ) or large inputs (Ivào > 2 ), each input value has a single response (output). In the intermediate range, 1 < Ivào < 2 , each input value corresponds to two stable output values. CHAPTER 2. CALCULATION AND SIMULATION METHODS 2.1. Coupled mode theory (CMT) Using a simple LC circuit, I have given the dependence of the voltage amplitude on time. This is the method used to calculate the transmission and reflection spectra of the structures. 2.2. Plane wave expansion method (PWE) In order to exploit the extraodinary properties of PhCs, the calculation method is required to accurately determine the PBG. One of the most common methods is the plane wave expansion (PWE). This method allows for solving wave vector equations for electromagnetic fields, calculating the eigen frequency of the PhCs. In addition, it is also used to calculate energy diagram as well as PBG. 2.3. Finite-difference time-domain method (FDTD) The FDTD method is one of the time domain simulation methods based on the mesh generation. Maxwell's equations in differential form are discrete by using the approximation method for differential of the time and space. The finite differential equations will be solved by software according to the leapfrog algorithm. This method aims to provide the mathematic facilities for calculating and simulating the Figure 1.33. Ouput versus input of the bistable device. The dashed line represents an unstable state [85]. device characteristics using PhC structures such as: transmission spectra, energy diagrams, and the characteristics of stability. CHAPTER 3. OPTIMIZATION OF QUALITY FACTOR AND RESONANT SPECTRA OF GRATING STRUCTURES 3.3. Optimization of structural parameters and resonant spectra In this chapter, I will introduce some methods to optimizing the Q- factors and resonant spectra of grating structures. 3.3.1. Slab waveguide grating structure combining with metallic film Based on the study of waveguide grating structure, so that in order to increase the Q-factor, the grating depth must be reduced, but due to the limitation of manufacturing technology, the grating depth is not too thin of less than 10 nm. Therefore, I have optimized the grating structure by adding a silver (Ag) layer of thickness d (> 50 nm) between the slab waveguide grating and the glass substrate. This thin layer supports a strongly asymmetric resonant profile in the nonlinear slab waveguide grating and reflects the light waves in any direction due to its high reflectivity. These reflected waves will then be coupled into guided-mode resonances in the grating [23]. Figure 3.14. (a) Metallic assisted guided-mode resonance structure with normally incident light. (b) Transmission and reflection spectra for several Ag layer thicknessed d. This results obtained with metallic assisted guided-mode resonance (MaGMR) structure provide the enhancement Q-factor coefficients greater than 1, therefor this structure has a higher Q-factor than grating waveguide structure. Combining with metallic film, the Q-factor has been enhanced. 3.3.2. Coupled grating waveguide structures The second optimal method, coupling two slab waveguide gratings to obtain a higher Q- factor and change the shape of the resonant spectrum. Here, the Q-factor is controlled based on the distance between the two slab waveguide gratings. The schematic of two coupled identical slab waveguide gratings facing each other with a gap-distance of d and horizontal shifted-alignment of s is shown in Fig. 3.18. Each slab waveguide grating supports the Fano resoance, where key structural parameters are defined as the guiding layer made of chalcogenide glass (As2S3, n = 2.38) with a thickness (t) of 220 nm on a thick glass substrate (n=1.5). The grating slit aperture (w) is formed by a rectangular corrugation in As2S3 guided layer with the depth and periodicity of 220 nm and 860 nm, respectively. A normally incident plane wave with transvere electric (TE) polarization is ussed. Figure 3.18. Sketch of coupled slab waveguide gratings. The gap-distance d and horizontal shifted- alignments s are tuned for exciting Fano resonances. Figure 3.19 shows the reflection spectra for various the gap-distances d. With this gap- distance 50 nm ≤d ≤ 300 nm, the resonant wavelengths shifts towards the short wavelenth. The Q-factor increases as the gap- distance d increases due to the long distance of Fabry-Perrot resonantor formed between two slab waveguide gratings. 3.3.3. Multilayer dielectric grating structure Figure 3.21. Multilayer nonlinear dielectric grating structure. The structure consists of N-pair of bilayer As2S3/SiO2 gratings. The structure consits of identically layers of As2S3 and SiO2 with thickness of t = N*(dH + dL), where N are the repetitive identical bilayers of As2S3 and SiO2, and dH và dL are the thickness of As2S3 and SiO2 layers, respectively. In our design, the optical thicknesses of As2S3 and SiO2 layers are chosen to satisfy the quarter-wavelength condition, that mean nH*dH = nL*dL = λ/4, where nH and nL are the refractive indices of As2S3 and SiO2, respectively. In calculations, the center wavelength λcenter = 1550 nm, dH = 162,8 nm và dL = 267,2 nm are used. Figure 3.22 shows the transmission spectra with N = 3 pairs Figure 3.19. Reflection spectra of the coupled slab waveguide gratings depicted in Fig. 3.18. of As2S3/SiO2 layers for several grating widths w from 30 nm to 150 nm. There exits two Fano resonances within the interested wavelength regims, which are associated with the guided-mode resonances in the long and short resonant spectra from 1460 nm to 1610 nm and from 1340 nm to 1480 nm. As it is shown, the increase of grating width w makes the resonance shifts to the short wavelength and the Q-factor decreases. In addition, the spectral resonances show that the side band degrees of Fano lineshapes do not change, it even shows that the linewidths and peaks of resonances change when the grating widths change. Figure 3.22. Transmission spectra of this structure depicted in Fig. 3.18 with N = 3. We investigated and found that the Fano lineshapes were reproducible and readily controlled via the number of layers N and the grating width w, demonstrating the robustness of the suggested structure. With the given grating width w of 70 nm, the resonant peaks and Q-factors of the long and short resonances for several number of layers N were evaluated using Fano lineshapes and plotted in Figure 3.23. When the number of layers N increase, redshifts in resonance, higher Q-factor, and lower sidebands are obtained. Figure 3.23. Resonant peaks and Q-factors of the structure as depicted in Fig. 3.21 for several number of layers N. CHAPTER 4. OPTICAL BISTABILITY IN SLAB WAVEGUIDE GRATINGS After optimizing the Q-factor and resonance spectra of the slab waveguide grating structure as presented in Chapter 3, in this chapter I will examine the bistability characteristics of optimal structures. 4.1. Optical bistability in slab waveguide grating structure combined with metallic film The third-order nonlinear coefficient at a working wavelength of As2S3 is n2 = 3,12x10-18 m2/W (χ (3) = 1,34x10-10). In order to see the optical bistability in MaGMR, we excite the devives with an incident CW source having a suitable working wavelength (frequency) on the surface of the structure. In general, the relation between the working frequency and the resonant frequency requires that [66]:  0 3    (4.1) where, τ is a photon life time, to observe bistability. For our case of an inverse Lorentzian shape, we choose a working wavelength at 80% reflection, which corresponds to a frequency detuning of (ω0 - ω)τ=2 for the Lorentzian shape. In this work, we keep the slab and Ag thickness at 380 nm and 100 nm, respectively. The grating depth δ (< 120 nm) is found close to an optimal value. Table 4.1 shows the trends for the resonant wavelength, the quality factor Q, and the Q-factor enhancement when the grating depth δ changes. As the grating depth increases, the resonant wavelength of MaGMR shifts to shorter wavelengths. It seems that the deeper the grating depth, the more leaky the waveguide mode. The Q- factor enhancement increases as the grating depth increases. For example, a Q-factor enhancement of 5.56 occurs for a grating depth δ of 120 nm. Table 4.1. Linear and nonlinear characteristics of MaGMR gratings with a Ag thickness d = 100 nm for several grating depths. Grating depth, δ (nm) 30 50 80 100 120 Resonant wavelength (nm) 1574,75 1560,61 1524,51 1516,81 1494,55 Q-factor 676,1 506,5 353,9 316,7 293,3 Q-factor enhancement 0,71 1,55 2,97 4,12 5,56 Reduced switching intensity 0,42 2,57 10,7 24,5 45,0 4.2. Optical bistability in coupled grating waveguide structures Figure 4.5 shows the calculated bistable behaviors of the perfect alignment coupled slab wavelength gratings for the gap-distance d of 50 nm, 100 nm, 170 nm, and 300 nm. Bistable behaviors are clearly observed. In each bistable curve, the incident intensity for switching can be estimated as the input intensity for which the reflection increases abruptly in the dotted solid curve. The estimated switching intensities are 1427,1 MW/cm2; 104,1 MW/cm2; 16,2 MW/cm2; và 2,2 MW/cm2; Figure 4.5. Bistability curves of the coupled gratings for various gap-distances d of 50 nm, 100 nm, 170 nm, and 300 nm, respectively. corresponding to the quality factors: Q = 2104, 2543, 3759, và 8522; and asym metric factor q = 1,609; 1,110; 0,835; và 0,655. In contrast to the Lorentzian resonance, these Fano-based results do not follow the 1/Q2 dependence rule of the switching intensity. While the Q- factors increase gradually, the switching intensities dramatically decrease due to a reduction of asymmetric factor q. The Q-factor increases 4.0 times but the switching intensity decreases 648.7 times. 4.3. Optical bistability in multilayer dielectric grating structure Figure 4.9 shows the dependence of transmission (ratio between the transmitted and incident intensities) on the incident intensity of the optical switching/bistability for the long (Fig. 4.9a,b) and short (Fig. 4.9c,d) resonances. For the long resonance, the operating wavelengths are chosen at resonant dip and 10% of transmission as shown in the insets of Fig. 4.9a,b. Figure 4.9a,b with the operating wavelength at 10% of transmission, blue (arrows pointing up/right) and red curves (arrows pointing left/down), it shows that the bistability behaviors and the switching intensities are 0.50 MW∕cm2 and 1178.56 MW/cm2 for grating widths w =30 nm and 150 nm, respectively. Whereas with the operating wavelengths at the resonant dips, the bistability behaviors have not occurred (black curves, on left) even the switching points at 0.04 MW/cm2 and 50.35 MW/cm2 of input intensities and high contrasts are observed for grating widths w = 30 nm and 150 nm, respectively. When the operating wavelength moves away from the resonant dip, the switching intensity is higher and the bistability region is broader. This is attributed to the wavelength detuning, which implies a broader detuning bandwidth and, thus, a higher resonance shift amount is required to change the state. For the short resonance, the operating wavelength is chosen at 1/e transmission as shown in the insets of Figure 4.9c,d. Figure 4.9. Optical switching/bistability behaviors in the nonlinear-pair-grating layers for grating widths w of 30 nm (a and c) and 150 nm (b and d) with operating wavelengths in the long (a and b) and short (c and d) resonances. Figure 4.10 shows the calculated switching intensity for various Q-factors. The fitting equation and the line for the switching intensity are also noted. It is clearly seen that the switching intensities decrease roughly as 1∕Q2.4 and 1/Q2.3 for bistability and switching behaviors, Figure 4.10. Optical incident intensity for the switching of optical switching/bistability devices based on 3-pair-grating layers for various Q-factors. respectively. It is well known that the switching intensities of an established Lorentzian lineshape optical bistable device in photonic crystal slabs or slab waveguide gratings scale as 1/Q2, where Q = λo/Δλ, λo and Δλ are the resonant wavelength and full-width at half- maximum, respectively. This implies that the switching intensity based on Fano resonances decrease faster than that of the Lorentzian lineshapes. If the nonlinear characteristics of the Fano resonances are similar to that of a Lorentzian lineshape, the normalized switching intensity should be proportional to the 1/(Δλ)2. CHAPTER 5. OPTICAL BISTABILITY BASED ON INTERACTION BETWEEN SLOTED CAVITIES AND WAVEGUIDES IN TWO-DIMENSIONAL PHOTONIC CRYSTALS 5.1. Photonic devices and two-dimensional photonic crystal structure using silicon photonic material Silicon-on-insulator has become the foundation of silicon photonic materials due to a number of advantages [128,129]: (i) promote the strengths of the technology of electronic components which have been perfect on crystalline silicon, (ii) the material cost is relatively cheap, durable in operation and proactively sizing the components down to a few tens of nanometers and (iii) the high refractive index difference between crystalline silicon and silicon oxide, is very effective in propagating of light. Silicon photonic material promises to fabricate photonic integrated circuits (PICs) on the same large SOI plate. Integrating the materials on a large SOI plate is essential, for example minimizing the effect of the free carrier in the optical sensors. The SOI plate fabrication technique is compatible with the complementary metal oxide semiconductor (CMOS) technology, thus achieving high accuracy. 5.2. Slot waveguide and cavity To reduce the calculation time without reducing the accuracy of the simulation results, I use the method to estimate the effective refractive index of the PhCs plate to bring the structure of the PhCs plate to 2D PhCs structure. The parameters of SOH (Silicon organic hybrid - SOH) plate are given as follows: refractive index of silicon plate nSi = 3.48; silicon thickness d = 220 nm and refractive index of organic material DDMEBT nDDMEBT = 1.8, I found that the effective refractive index of SOH plate is n = 2.9812. Figure 5.8a shows the structure of a slot waveguide with a narrow width d = 50 nm with the following structural parameters: the effective refractive index of the SOH n = 2.9812, the lattice constant a = 380 nm, the air hole radius r = 0.3a, and the refractive index of the organic material DDMEBT nDDMEBT = 1.8, filled in the holes. Figure 5.11a shows a cavity with the slot width at the center d = 50 nm, the central of slot length L. The slot width gradually increases at equal intervals of 10 nm/a until the wall width is reached prevent electromagnetic waves d = 120 nm. Figure 5.11b shows the distribution of the electric field inside the cavity with a slot length L = Figure 5.5. Relationship diagram between the transmission coefficient and the effective refractive index of the structure. 1a. With this resonator structure, I obtained the quality factor Q = 2403. Similar in Figure 5.11c and Figure 5.11d is the distribution of the electric field inside the cavity with slot length L = 3a and L = 5a corresponds to the quality factor Q = 6161 and Q = 9163. 5.3. Interaction between resonator and slot waveguide 5.3.1.1 Theoretical model Figure 5.12 shows a schematic diagram of n identical resonators through the bus waveguide. Figure 5.12. Schematic of coupled n identical resonators through the bus waveguide. Figure 5.8. (a) Slot waveguide channel with narrow width d, (b) Energy diagram of the waveguide channel, (c) Distribution of electric field within the waveguide channel. Figure 5.11. (a) The slot cavity, (b, c, d) is the electric field distribution within the cavity with slot length L = 1a, 3a and 5a, respectively. It is assumed that each resonator has a resonant frequency ωo and large enough Q factor so that the direct coupling between resonators is negligible and the resonators interact through the bus waveguide. 1/τ is the decay rate into the bus waveguide from each resonator. s+1 and s-1 are the amplitudes of the incoming and the outgoing waves in the first resonator; s+n and s-n are defined similar for the n resonator. The temporal changes of the mode amplitudes of the resonators a1, a2,,an are derived: 1 0 1 2 1 0 1 1 0 1 1 [ ( ) ] ( 2 ) 1 [ ( ) ] i i i i n n n n da j a a k s dt da j a a a dt da j a a k s dt                                             với 1 < i < n where 2je    is the coupling coefficient from the waveguide to the resonator. δ and µ represent the shifted resonant frequency and direct coupling coefficient, respectively, which are given by δ = cotφ/τ and µ = -jcscφ/τ. φ is the phase shift between two adjacent resonators through the bus waveguide. |a|2 and |s|2 are the energy stored in the resonator and the wave power, respectively. The transmission spectra of the coupling n identical resonators can be calculated in the frequency domain with s+n = 0 can be given by:   22 * 21 2 ( ) 1/ n n i i s k k T f s j f                (5.11) With     1 1 1 1/ 1 / n n n i i i i a f a j a f a j f                     (5.12)  is the frequency detuned from the resonant frequency,  =  - o. Eq. (5.11) gives us the transmission spectra as show in Fig. 5.13. As shown in Fig. 513a, for φ = π/2, the transmission spectra are symmetric and their center resonant frequencies remain stationary with nearly flattop at unity and a little fluctuation. With deviation of the phase shift   /2, the symmetry of transmission spectra will be broken and the depth of valley will increase with an increase of the detuning frequency. Another characteristic is that the transmission spectrum shifted to a lower frequency (or higher frequency) with increase (or decrease) of the phase shift . Fig. 5.13d shows the fifth-order filter for several phase shifts . As can be seen, the different resonant peaks located on both sides of the central resonant frequency depending on the phase shift larger or smaller than /2. The linewidth and depth of valley of the Figure 5.13. Theoretical transmission spectra of the structure depicted in Fig. 5.12 for several numbers of resonators by using the CMT with: (a)  = π/2, (b)  = π/3, (c)  = 2π/3 và (d) The transmission spectra of the fifth-order filters for several phase-shifts resonances far from the center resonance tend to become narrower and deeper, respectively. The induced resonance linewidth shrinking of the right- (or left-) most peak with multiple coupled resonator for phase shift   /2 shows better than that with phase shift  = /2. For switching/bistability applications, the choice of phase shift  = /2 is easy comparison among transmission spectra for different number of resonators n by maintaining the same center frequency, even phase shift  = /2 is not the best choice in terms of switching power. 5.3.1.2 Simulation results The SOI slotted 2D PhC structure under consideration is started from the 260 nm thick silicon film of the SOI wafer, and then made of a 2D triangular lattice PhC structure of holes with lattice constant a = 380 nm. The holes are inscribed through the silicon film with radius of r = 0.30a = 114 nm. The slotted waveguide is formed by removing one center row of holes along  - K direction and inscribing with the rectangular slot of width d through the silicon film. The SOH slotted PhC waveguide is obtained by first producing the SOI slotted PhC waveguide and then filling and covering it with DDMEBT of refractive index of 1.8 as shown in Fig. 5.14a. Figs. 5.14b, c show the 2D PhC structures of coupled five identical slotted cavities together through the waveguide and two identical side-coupled slotted cavities and slotted waveguide, respectively. Each cavity is formed by gradually changing the slotted width from 50 nm at the center to 120 nm of both sides as shown in Fig. 5.13d. The increasing step of slotted width of cavity is 10 nm for each periodicity, whereas the slotted waveguide widths at the input and output ports are kept at 50 nm. Figure 5.14. (a) Sketch of SOH slotted PhC waveguide. (b) and (c) are the designs of the fifth- order filter and two side-coupled resonators and waveguide, respectively. The details of the one resonator in a slotted 2D-PhCs are shown in (d). Fig. 5.15a shows the transmittance characteristics for n from 1 to 5 by using the FDTD method. The center wavelength which is the resonant peak of the single resonator at λ1 = 1555.28 nm and it is used for optical communication. The Q-factor of single resonator is estimated at 4462. The transmission spectra of the higher-order filters (n > 1) have n resonant lineshapes and the linewidths of the rightmost resonant peak in the transmission spectra, which are used for bistable switching operations, are estimated. Since their full-width at half- maximum (FWHM) cannot be defined as seen in Fig. 5.15a (note that the first dip on the left side of the right-most peak does not go below 50%), so that the linewidth and Q-factor can be estimated as fitting the right-most resonance to the Fano lineshape as follows [160]: 2 2 ( ) ( ) 1 q R F       (5