緊束縛理論在光子晶體波導的應用

Abstract

本論文利用緊束縛理論來研究脈衝在單一且非線性光子晶體波導或共振耦合波導以及電磁波在對稱或非對稱線性光子晶體耦合波導的傳播情形。從考慮缺陷間耦合的緊束縛理論，電場在光子晶體和共振耦合波導的振幅可以寫成一個解析演化方程式，此方程式稱為延伸離散非線性薛丁格方程式。在光子晶體波導或共振耦合波導中，藉由解這個方程式我們可以得到光調制不穩定區域及在不同的平面波向量(p)和不同擾動波向量(q)的增益係數G(p,q)的解析解形式。在共振耦合波導中，光調變不穩定的區域只能出現在pa大於钉/2或者小於钉/2中。這裡a指的是缺陷間的距離。而光調變不穩定區域的位置會由缺陷間界電柱數目以及克爾係數的正負號所決定。然而，在光子晶體波導中，光調變不穩定區域中的pa可以超過钉/2。當平面波的相位pa超過钉/2，在固定pa的情況下，光調變不穩定區域的增益曲線G(q)會有兩個最大值，這和非線性光纖中的情況有很大的不同。 另外一方面，我們也成功地利用延伸離散非線性薛丁格方程式來描述光固子在含有克爾介質的非線性光子晶體波導及共振耦合波導中的傳播及其傳播的條件。從這個條件，我們得到了光固子在不同數目的間隔界電柱及不同自相位調變強度下的穩定傳播區域，這和光調變不穩定區域是吻合的。光子晶體波導中，在低頻或者低波向量的脈衝傳播時，需要加入正的克爾係數的物質到波導中，反之亦然。由於光子晶體波導和共振耦合波導的耦合係數大小不同，導致共振耦合波導的群速度、色散和支持光固子傳播的自聚焦強度比光子晶體波導小。對於一個長脈衝在光固子傳播條件下，他的脈衝擴散由於大於二階的的最低階色散係數所造成。當脈衝變窄，其他高階項需要被考慮，這導致光固子最小擴散的自聚焦強度會比導出來的傳播條件小，尤其在三階色散趨近於0的時候更加明顯。 當另一個相同的波導刻入光子晶體中且與原波導相隔一排或數排介電柱，我們可以得到一個光子晶體耦合器。耦合器的色散曲線會有一個交點，稱為不耦合點。在這個點上，能量不能耦合到另一個波導。所以我們可以利用調變不耦合點來改變耦合器的性質。從考慮到兩個波導間的耦合到次鄰近缺陷的緊束縛理論，我們發現如果平行於耦合器的方向去移動缺陷柱，不耦合點頻率在正方晶格會有藍移的現象，但在三角晶格則有紅移的現象。如果讓缺陷互相靠近，由於耦合強度變強，導致耦合器的傳播頻率及兩條色散曲線間的頻率差都有增加的趨勢。當我們利用平面波展開法及時域有限差分法來作模擬時，發現其結果跟我們的理論非常吻合。 如果光子晶體中的兩條波導不一樣，耦合器變的不對稱。利用考慮到兩個波導間的耦合到次鄰近缺陷的緊束縛理論，我們解釋了幾個非對稱耦合波導的物理性質:(1)在某個特定點時，耦合波導的色散曲線會退化到單一波導的色散曲線且電場只會侷限在單一個波導中，此點我們稱為不耦合點；(2)即使色散曲線沒交叉，本徵模態的宇稱仍會在不耦合點交換；(3)即使本徵模態交換，高頻色散曲線的電場會主要分佈在擁有較高本徵模態的單一波導，反之亦然。當一個單頻光射入耦合器中，能量的轉換也可以用解析解的形式表達。由於色散曲線沒交叉，所以在非對稱的耦合波導的耦合長度並非無限大，但是在不耦合點兩個波導間有最低的能量轉換。

Tight binding theory (TBT) is used to study the pulse propagation in singe photonic crystal waveguides (PCWs) and coupled resonant optical waveguides (CROWs) with nonlinear media as well as an electromagnetic (EM) wave propagation in the symmetric and asymmetric photonic crystal (PC) coupler. From the TBT and considering the coupling between the defects, the amplitude of the electric field in the PCWs or CROWs can be expressed as an analytic evolution equation and we termed it the extended discrete nonlinear Schrödinger (EDNLS) equation. By solving this equation for CROWs and PCWs, we obtained the modulation instability (MI) region and the MI gains, G(p,q), for different wavevectors of the incident plane wave (p) and perturbation (q) analytically. In CROWs, the MI region, in which solitons can be formed, can only occur for pa being located either before or after 钉/2, where a is the separation of the cavities. The location of the MI region is determined by the number of the separation rods between defects and the sign of the Kerr coefficient. However, in the PCWs, pa in the MI region can exceed the 钉/2. For those wavevectors close to 钉/2, the MI profile, G(q), can possess two gain maxima at fixed pa. It is quite different from the results of the nonlinear CROWs and optical fibers. We also successfully used the EDNLS equation to describe the soliton propagation and to obtain the soliton propagation criteria (SPC) in the nonlinear PCWs and CROWs containing Kerr media. From these criteria, we obtained the soliton propagating region of CROWs in different numbers of separated rods and strengths of self-phase modulation which coincides with the region of MI of the CROWs. In the PCWs, the positive Kerr coefficient medium needs to be added to support the pulse propagation in the low frequency or wave vector region of the dispersion relation and vice versa. Due to the different magnitudes of coupling coefficients in CROWs and PCWs, the group velocity, dispersion and self phase modulation strength to support soliton propagation in CROWs are smaller than those in PCWs. For a long pulse, only the lowest nonzero dispersion coefficients, 刍n with n > 2 needs to take into consideration for pulse broadening at the SPC. However, as decreasing the pulse width, even higher order dispersion should be taken into account that makes the self phase modulation strength smaller than the criteria when the third order dispersion is almost zero. As the other identical waveguides is inserted into the PC with one or several partition rods, the PC coupler is created. The dispersion relation curves of the coupler could be crossing. The crossing point is named as the decupling point. At this point, the energy cannot be transfer into the other waveguide. Controlling the decoupling point can modify the properties of the coupler. From the TBT that includes coupling of the guiding mode field up to the next nearest-neighbor defects, we find there is a blue shift in the frequency of the decoupling point in the square lattice and red shift in the triangular lattice by translating the defect rods along the axis of the coupler. By moving defects of the coupler close to each other transversely, not only the eigenfrequencies of the coupler but also separations of dispersion curves increase due to the stronger coupling between defect rods. From the simulation results of the plane wave expansion method and finite difference time domain method, the theoretical analysis of TBT gets a great agreement with the numerical ones. If the other waveguide inserted into the PC is different from the original one, the coupler will be asymmetric. By considering the next nearest-neighbor defects between two PCWs, analytic formulas derived by the TBT, we will explain the physical properties of the asymmetric directional coupler made of two coupled PCWs: (1) The dispersion curves of a PC coupler will decouple into the dispersion curves of a single line defect, and the electric field would only be localized in one waveguide of the coupler at a particular point that we name the decoupling point; (2) The parities of the eigenmode switch at the decoupling point, even though the dispersion curves are not crossing; (3) The eigenfield at a higher (lower) dispersion curve is always mainly localized in the waveguides that have higher (lower) eigenfrequencies of single line defects, even though the eigenmodes are switched. As a given frequency is incident into the coupler, the energy transfer between two waveguides and the coupling length can be expressed analytically. Due to no dispersion curve crossing, the coupling length is no longer infinite at the decoupling point in asymmetric PCWs, but it still possesses the minimal energy transfer between two waveguides when the frequency of the incident wave is close to the decoupling point.