含多缺陷光子晶體光學性質之研究

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(1)國立台灣師範大學光電科技研究所碩士論文 Institute of Electro-Optical Science and Technology. National Taiwan Normal University. 含多缺陷光子晶體光學性質之研究 Research on Optical Properties of Photonic Crystals That Contain Multiple Defects 指導教授：吳謙讓研究生：林. 博士. 琦. 中華民國一○三年六月.

(2) Abstract. In this thesis, we shall study the optical properties for the one-dimensional photonic crystals (PCs) containing defects. Two main topics are involved. In the first part, we consider binary defective PCs in the symmetric or asymmetric structures. We investigate the photonic transmission based on the transfer matrix method. The multiple filtering properties can be found in a cascading structure. In the second part, we analyze the transmission properties in a one-dimensional photonic crystal containing twin defects. With the existence of twin defects, a defect mode, i.e., a transmission peak is produced within the photonic band gap in the defect-free photonic crystal. The shape of transmission peak can be controlled by the stack numbers between the twin defects. The dependence of transmission peak on the incident angle is also investigated for both the transverse electric (TE) and the transverse magnetic (TM) waves. The peak becomes narrower as the angle increases in TE wave, whereas it is broadened as a function of the incident angle in TM wave. Discussion on the omnidirectional property in the peak is also given. Keywords：photonic crystals (PCs), dielectric superlattice, defective PC. i.

(3) Acknowledgement First of all, I would like to express my gratitude to all those who have helped me to complete this thesis. I am very indebted to my respected advisor, Professor C.-J. Wu who has constantly encouraged, guided and supported me from the start to the end. He leads me to do and understand the formally scientific study. He teaches me how to write computer software to simulate the optical properties of one-dimensional PCs. Also, I would like to appreciate my classmates, Casey Lin, Allen Hsieh, Sheng-chih Wu at NTNU; Chris Liu at NTUT and Andrea Kang at NTU. They have given me some useful advices on my thesis. Furthermore, I am deeply indebted to my warmhearted and generous supervisors, Chen-lin Chang, Maggie Chang and all the colleagues in Chen-lin Science Institute. Thank you all for training and supporting my full-time teaching job. At last but not least, I would like to thank my parents and brother for their support all the way from the very beginning of my postgraduate study. I am grateful to all my family members for their thoughtfulness and encouragement.. ii.

(4) Contents. Abstract. i. Acknowledgement. ii. Contents. iii. Chapter 1. Introduction. 1-1. Photonic Crystals. 1. 1-2. Motivations and Applications of PCs. 2. 1-3. Thesis Overview. 3. Chapter 2. Theoretical Methods. 2-1. Transfer Matrix Method (TMM). 4. 2-2. Dynamical Matrix of a Medium. 4. 2-3. A Film of Finite Thickness. 7. 2-4. A Multilayer System. 9. 2-5. Transmittance and Reflectance. Reference. Chapter 3 3-1. 11 12. Analysis of Defect Modes in One-Dimensional Defective Photonic Crystals Introduction. 13. 3-2 Theory. 14. 3-3 Design Structures and Numerical Results. 16. iii.

(5) 3-4. Conclusion. 25. Reference. 25. Chapter 4 Transmission properties in a One-dimensional Photonic Crystal Containing Twin Defects 4-1. Introduction. 30. 4-2. Model Structures. 32. 4-3. Numerical results and discussion. 33. 4-4. Conclusion. 38. Reference. Chapter. 5. 39. Conclusions. 41. iv.

(6) Chapter 1 Introduction 1-1 Photonic Crystals. Photonic crystals (PCs) are artificially periodic structures with a period of the order of optical wavelength. Due to the structural periodicity, it has been known that photonic band gaps (PBGs) will be created such that electromagnetic waves with frequencies lying within these cannot allow to propagate. Materials with PBGs have attracted much attention over the past two decades. Depending on the dimension, PCs can be classified as one-, two-, or three-dimensional periodic arrangement, as illustrated in Fig. 1-1. Shown in Fig. 1-2 is the simple 1-D PC.. 1.

(7) Fig. 1-1 The diagrams demonstrating 1-D, 2-D, and 3-D PCs .. Fig. 1-2 A structure of 1-D PC, where a is the lattice constant or period .. 1-2 Motivations and Applications of PCs. The existence of PBGs in PCs can be engineered to achieve many useful optical devices. One of the most useful devices is the narrowband transmission filter which can be achieved by adding the defect layer into the PCs. In a simple 1-D PC, the defective PC can be (AB)ND(AB)N or (AB)ND(BA)N. With the defect layer D, there will be a transmission peak in the PBG. The transmission peak can be sharp by increasing the number of periods N. In this thesis, we shall capitalize on symmetric and asymmetric defective PCs to design the multichanneled transmission filters. The multiple filtering properties can be obtained by cascading two individual single-channel filters. We will also consider the 2.

(8) PC containing twin defects. We investigate the transmission properties in such a defective PC. We shall study the omnidirectional property in this kind of filter.. 1-3 Thesis Overview. This thesis has five Chapters. The physical properties and background of PCs are given in Chapter 1. Chapter 2 deals with theoretical method for the layered media. Chapter 3 and Chapter 4 give the main topics of this thesis. Finally, the conclusion will be given in Chapter 5.. 3.

(9) Chapter 2 Transfer Matrix Method in Layered Structure. 2-1 Transfer Matrix Method (TMM). In this thesis, we shall use the transfer matrix method (TMM) to study the optical properties of the defective PC. The TMM developed Yeh will be used to calculate the optical transmittance and reflectance for a layered structure. In the following we briefly describe this method. [1]. 2-2 Dynamical Matrix of a Medium. In the beginning we state the transmission and reflection problem for a single interface as shown in Fig. 2.1, in which the incident wave is Ei exp[i(t  ki  r )] , the. reflected. wave. is. Er exp[i(t  kr  r )] ,. and. the. refracted. wave. is. Et exp[i(t  kt  r )] . The phase continuity at the boundary x = 0 gives n1 sin i  n1 sin r  n2 sin t ,. (2.1). which in turn leads to the Snell’s law of reflection, i.e. i   r and the Snell’s law of refraction, 4.

(10) n1 sin i  n2 sin t .. (2.2). Fig. 2-1 Reflection and refraction of a plane wave at a boundary between two dielectric media.. In order to treat multilayer structure in the future, Fig. 2-1 can be generalized to Fig. 2-2, where a TE wave is considered, and the plane of incidence is in the xz plane. The electric field in each medium can be expressed as  ik r  ik  r it   ( E1e 1  E1e 1 )e , x  0 E   ik2 r  E2eik2 r )eit , x  0  ( E2e. (2.3). Then the magnetic field can be calculated via the Faraday’s law, H. 1. . k E ,. (2.4). ˆ , and the intrinsic impedance     . With boundary conditions at where k  kk x = 0, we have the matrix equation 1    1 cos  1    1. 1     E1s        1 cos 1   E1s   2 cos  2 1   2 1. 5.    E2 s  2  .  cos  2   E2 s  2  1. (2.5).

(11) Fig. 2-2. Reflection and Refraction of s Wave (TE).. If we define the dynamical matrix of the s wave for the medium i (i =1, 2) as 1   Ds (i )    i   cos i  i.   i  cos i  i  1. (2.6). Similarly, for the TM configuration in Fig. 2-3, we have the dynamical matrix of the p wave for the medium i (i =1, 2)  cos i  D p (i )    i   i . cos i   i   i . Fig. 2-3 Reflection and Refraction of p Wave (TM) 6. (2.7).

(12) 2-3 A Film of Finite Thickness. Referring to Fig. 2-4, we have a homogeneous and isotropic layer structure with three different refractive indices. x0 0 xd..  n1 ,  n( x)  n2 , n ,  3. (2.8). dx. The electric field E ( x) consisting of a right-traveling wave and left-traveling wave and can be written as E ( x ) R  eikx x  L  eikx x  A x( ) B x ( ). (2.9). where we have assumed the electric field vector is s-polarized, and 1/ 2.  n  2  kix   i    2   c  . .  c. n1 cos i ,. i  1,2,3. (2.10). where  i is the ray angle measured from the x axis and R and L are constants in each layer. Let A(x) represent the amplitude of the right-traveling wave and B(x) be that of the left-traveling one. To illustrate the matrix method, we define A1  A(0 ) , B1  B(0 ) , A2  A(0 ) , B2  B(0 ) ,. A2  A(d  ) , B2  B(d  ) A3  A(d  ) B3  B(d  ). 7. (2.11).

(13) Fig. 2-4 A thin layer of dielectric material If we represent the two amplitudes of E (x) as column vectors, then we have  A1   A2   A2  1    D1 D2     D12    ,  B1   B2   B2   A2   A2   ei2     P2      B2   B2   0. 0   A2    , ei2   B2 .  A3   A3   A2  1    D2 D3     D23    ,  B2   B3   B3 . (2.12). (2.13). (2-14). where D1 , D2 , and D3 are the dynamical matrices introduced in Section 2-1, and P2 in Eq. (13) is the called propagation matrix, which accounts for propagation through the thickness of the slab, and 2 is given by. 2  k2 x d. (2.15). The matrix D12 and D23 may be regarded as transmission matrices that links the amplitudes of the waves on the two sides of the interfaces and are given by. 8.

(14) 1 k2 x  1  k1x 2 D12  D11 D2    1 1  k2 x 2 k1x  .      . k2 x   1 1   2 k1x    k2 x   1 1   2 k1x  . (2.16). Similarly, for p wave:.  1  n2 2 k1x   1  2   2  n1 k2 x  D12   2  1 1  n2 k1x  2 2   n1 k2 x . 1  n2 2 k1x   1   2  n12 k2 x    1  n2 2 k1x   1    2  n12 k2 x  . (2.17). From Eqs. (2-12)-(2-14), the amplitudes A1 , B1 and A3 , B3 of the single layer are related by  A3   A1  1 1    D1 D2 P2 D2 D3     B1   B3 . (2.18). 2-4 A Multilayer System. Referring to Fig. 2-6, we now consider the case of multilayer structures. The dielectric constants in the structure are described by n0, n1, n2…, and ns, respectively. The thickness of the lth layer is. dl  xl  xl 1 where the electric field distribution E  x  can be written as. 9. (2.25).

(15) Fig. 2-6 A multilayer structure.  A0 eik0 x ( x0  x )  B0 eik0 x ( x0  x ) , x  x0  E   Al eiklx ( xl  x )  Bl eiklx ( xl  x ) , xl 1  x  xl ,  iksx ( x  xN )  Bseiksx ( x  xN ) , xN  x  Ase. (2.26). where klx is the x component of the wave vector 1/ 2.    2  klx    nl    2   c    . , l  0,1,2,3......N , s. (2.27). and is related to the ray angle  i by klx  nl.  c. cos i. (2.28). According to Eqs. (2-26) and (2-27), Al and Bl represent the amplitude of the plane waves at interface, x  xl . Thus, using the same argument as in Section 2-2, we can write  A0   A1  1    D0 D1   ,  B1   B0   Al   Al 1  1    PD  , l  1,2,3......N , l l Dl 1   Bl   Bl 1  10. (2.29).

(16) where N + 1 represents s, AN 1  As , BN 1  Bs . where. l  klx dl . 2 dl. . nl cos l .. (2.33). The relation between A0 , B0 and As , Bs can thus be written as  A0   M1 1 M 1 2 As       B0   M 2 1 M 22 Bs . (2.34). with the M-matrix given by  M11   M 21. N M12  1  1    D0  Dl PD l l  Ds M 22   l 1 . (2.35). From the matrix elements we can calculate the reflectance and transmittance.. 2-5 Transmittance and Reflectance. Using the 2 x 2 matrix method, we now discuss the reflectance and transmittance of monochromatic plane waves through a multilayer dielectric structure. If the light is incident from medium 0, the reflection and transmission coefficients are defined as B  . r  0   A0  Bs 0. (2.36). and A  t  s  .  A0  Bs 0 11. (2.37).

(17) Using the above matrix relation it is easy to have the reflectance (which is meaningful only when medium 0 is nonabsorbing), namely 2. M R  r  21 , M 11 2. (2.40). and the transmittance is given by n cos  s T s t n0 cos 0. 2. 2. n cos  s 1 .  s n0 cos 0 M11. (2.41). Reference. [1] Pochi Yeh, Optical Waves in Layered Media, John Wiley & Sons, Singapore, 1991.. 12.

(18) Chapter 3 Analysis of Defect Modes in One-Dimensional Defective Photonic Crystals 3-1 Introduction A one-dimensional binary dielectric superlattice is a periodic structure formed by two distinct dielectrics with different refractive indices. A periodic bilayer system is now known as a one-dimensional photonic crystal (1DPC), which has been a hot topic in optical physics over the past two decades [5-16]. In a PC, there will be photonic band gaps (PBGs) created due to the spatial periodicity. By capitalizing on the existence of the PBGs in a PC, one of the important and useful applications in photonics is to utilize the PBG to design a multilayer narrowband transmission filter also called a Fabry-Perot resonator (FPR). An FPR can be simply realized by introducing a defect layer into the PC to break the spatial periodicity. This defect layer behaves, in principle, like a cavity resonator when the resonant condition is satisfied. The defect modes will be located within the PBG, which are much similar to the defect states generated in the forbidden band in a doped semiconductor. Materials for the defect layer usually call for dielectrics [1-4]. In a 1DPC, adding a defect layer C in it will lead to the system denoted as V/(HL)NC(LH)N/S where V means a vacuum medium which refractive index is nV = 13.

(19) 1.00,the same as the usual air. N is the number of periods for the periodic bilayers, and S denotes the substrate. In addition, H and L are the high- and low-index layers with their thicknesses usually set to be quarter-wavelength i.e., the optical thickness is 1H = 1L = nHdH = nLdL = 0 / 4, where nH, nL, and dH, dL are the refractive indices and physical thicknesses of H and L, respectively, and 0 is the designed wavelength. The purpose of this chapter is to investigate the properties of defect modes in these two defective PCs. We specifically investigate the defect modes as a function of the angle of incidence for both TE and TM waves for the symmetric defective PCs. As we shall show later, in the symmetric PC, we find there is only one defect mode within the PBG. However, there will be two defect modes in the symmetric PC. The presence of two defect modes may be more efficient in utilizing the PBG compared to the single mode in the asymmetric one [25,26]. Moreover, the feature of two-mode PC makes the structure more useful in the signal processing especially in optical communication. For instance, it can be used as a frequency-selective filter (or wavelength multiplexer) that can extract two signals with fairly different frequencies. The analysis on the properties of defect modes gives some useful information for the design of narrowband transmission filter based on the one-dimensional PCs.. 3-2 Theory Let us first consider the symmetric defective PC with a structure of V/(HL)ND(LH)N/S, 14.

(20) as depicted in Fig. 3-1. The transmittance T is related to the transmission coefficient t by nS cos  S 2 t , nA cos  A. T. (3.1). where t is given by t. 1 , M 11. (3.2). according to the transfer matrix method [2], where M11 is one of the matrix elements of the total system matrix M written by M M   11  M 21. M12  . M 22 . (3.3). Figure 3-1. Two defective PCs, in which the upper panel is a symmetric PC, whereas the asymmetric one is in the lower panel.. The total system matrix is given by N. N. M sym  DV1 DH PH DH1DL PL DL1  DC PC DC1  DL PL DL1DH PH DH1  DS ,(3.4) where the dynamical matrix in Eq. (3.4) for medium i is given by.  1 Di    ni cos i for TE wave and 15. 1  , ni cos i . (3.5).

(21)  cos i Di    ni. cos i  , ni . (3.6). for TM wave, and i = A, H, L, and S, respectively. The translational matrix in layer H, L, or D is expressed as 0  exp  ji   Pi   , 0 exp   ji   . (3.7). where the phase is i = 2ni di /, where  is the wavelength of the incident wave. In the above formulations, ni, i, and di are the refractive index, the ray angle, and the thickness of layer i, respectively. For i = A, the ray angle in the air is then equal to the incident angle denoted by    A . All the ray angles are related by the Snell’s law of refraction. As for the asymmetric defective PC shown in the bottom of Fig. 3-1, the total system matrix is given by N. N. Masy  DV1 DH PH DH1DL PL DL1  DC PC DC1  DH PH DH1DL PL DL1  DS , (3.8) From Eq. (3.8), we can obtain the matrix element M11, the transmittance T in this case is determined again by Eqs. (3.1) and (3.2).. 3-3 Design Structures and Numerical Results The defective PCs to be considered are shown in Fig. 3-2, where A, B are composed of (x1Hx1L)N2xH(x2Lx2H)N, respectively, where xi is the coefficient of optical thickness and N is the period number of unit. TiO2 and SiO2 are chosen as high and 16.

(22) low refractive index media, and their refractive indices are. nH = 2.32 and nL = 1.46,. respectively.. A’. A. B’. B. Figure 3-2.. Four structures of symmetric defective PCs, in which the upper ones are: A =. 4. (1.0H1.0L) 2.0H(1.0L1.0H)4 4. 4. and ’. A’. (1.0L1.0H)42.0H(1.0H1.0L)4,. = 4. whereas. B. =. 4. (1.4H1.4L) 2.8H(1.4L1.4H) and B = (1.4L1.4H) 2.8H(1.4H1.4L) are depicted in the lower ones.. Without loss of generality, the defect modes are investigated for the quarter-wavelength stack, i.e., nHdH = nLdL = 0 / 4 with a designed wavelength, where a wavelength in infrared region, 0 = 520 nm, is taken. The use of the quarter-wavelength stack enables us to have the analytical expression for the transmittance and then some physical phenomena can be gained insight directly. The number of periods for each PC is N = 4. The substrate S is simply assumed to be the same with air with nS = nV = 1.. 17.

(23) Figure 3-3.. The upper half ones are referred to as calculated TE-wave transmittance for the. symmetric filters as a function of the wavelength of normal incidence, in which the left is referred to as A structure, whereas the right is referred to as A’ structure. The lower ones are referred to as the similar symmetric but different thickness filters as a function of the wavelength of normal incidence, in which the left is referred to as B structure, whereas the right is referred to as B’ structure.. In Fig. 3-2, we have plotted four structures which are under consideration, namely A = (1.0H1.0L)42.0H(1.0L1.0H)4, B = (1.4H1.4L)42.8H(1.4L1.4H)4 and A’ = (1.0L1.0H)42.0H(1.0H1.0L)4, B’ = (1.4L1.4H)42.8H(1.4H1.4L)4. The calculated transmittance spectra for these four structures are shown in Fig. 3-3, where the normal incidence is considered. Although the ways of piling up these thin films are different, the transmittance of A and A’ are very similar to each other among the visible spectrum. The sharp peak of transmittance function of A is much steeper than A’ . 18.

(24) Similarly, the sharp peak of transmittance function of B is much steeper than B’ . In this design, we have seen that the peak positions for A and B are different. With these different peak positions, we are in a position to design a multichanneled filter by combining A, A’, B, and B’ as a cascaded structure. In Fig. 3-4, we have designed four possible structures with the help of the coupling layer C, where C = 1.0L.. c. A. B. c. A’. B’. c. B’. A. c. A’. Figure 3-4.. B. Structures of symmetric and asymmetric PCs, in which the upper half ones are referred to. as symmetric PCs A-C-B and A’-C-B’, whereas asymmetric PCs A-C-B’ and A’-C-B are depicted in the lower half ones.. In Fig. 3-4, we have A-C-B, A’-C-B’, A-C-B’ and A’-C-B. According to this, we can clearly find that A-C-B and A’-C-B’ are symmetric PCs while A-C-B’ and A’-C-B are asymmetric ones. All these four structures are investigated as follows:. A. Defect Modes in A-C-B and A’-C-B’ In Fig. 3-5, we have plotted the wavelength-dependent transmittance for the 19.

(25) symmetric PC structure A-C-B in TE wave at different angles of incidence, 00, 150, 300, 450, 600, and 750, respectively. In this case, there are two defect modes, as indicated by the two resonant peaks 1 and 2 within the PBG. The presence of two defect modes can be ascribed to the structural reflection symmetry [27,28]. This feature of symmetry will give rise to two possible field solutions in the defect layer at resonance. One solution proportional to cosine function belongs to an even symmetry which corresponds to the lower energy state, as marked by 2. The other proportional to sine function is an odd symmetry and is indicated by 1. The dependence of defect modes 1 and 2 on the angle of incidence is plotted in Fig. 3-6. We could see that the left peak locates at the wavelength less than 0 whereas right peak wavelength is much larger than 0. If the incident angle increases, the two defect modes are shifted to the left, and peak heights are changed irregularly.. Figure 3-5. The transmittance spectra of photonic quantum-well symmetric structure A-C-B , in TE wave, at different angles of incidence 0o, 15o, 30o, 45o, 60o and 75o, respectively. 20.

(26) Figure 3-6. The angle-dependent two peak wavelengths for the defect mode in symmetric filter A-C-B in TE wave. In Fig. 3-7, the two defect modes in TM wave for the A-C-B are shown. The positions of two defect modes are also moved to shorter wavelengths as the incident angle increases. Moreover, the separation of two peaks is decreased as the incident angle increases.. Figure 3-7. The transmittance spectra of photonic quantum-well symmetric structure A-C-B , in TM wave, at different angles of incidence 0o, 15o, 30o, 45o, 60o and 75o, respectively. 21.

(27) Figure 3-8. The angle-dependent two peak wavelengths for the defect mode in symmetric filter A-C-B in TM wave.. In Fig. 3-8, we plot the peak wavelengths for the two defects as a function of the angle of incidence. It can be seen both peak wavelengths are a decreasing function of the angle of incidence. The presence of two defect modes of the TE wave can be compared to the TM wave from Fig. 3-5 and Fig. 3-7. When the wavelength is larger than 750 nm, the transmittance oscillates, especially in the TM wave. Finally, in Figs. 3-9 and 3-10, we plot the defect modes for both A-C-B and A’-C-B’, respectively. It can be seen that the defect modes in symmetric PCs A-C-B are much steeper than A’-C-B’.. 22.

(28) Figure 3-9. The transmittance spectrum of A-C-B under normal incidence.. Figure 3-10. The transmittance spectrum of A’-C-B’ under normal incidence.. B. Defect Modes in A-C-B’ And A’-C-B. Figure 3-11. The transmittance spectrum of the asymmetric structure A-C-B’ under normal incidence. 23.

(29) Figure 3-12. The transmittance spectrum of the asymmetric structure A’-C-B under normal incidence.. In Fig. 3-11 and 3-12, we have plotted the wavelength-dependent transmittance for the asymmetric PC in the case of normal incidence. Some features are of note. In this case, there are three resonant peaks within the PBG. A resonant peak locates near the designed wavelength of 0 = 520 nm. The peak shape is not sharp in these two asymmetric structures. The resonant peak height of the PC structure A-C-B’ is up to 0.91. The transmittance for these four structures is shown in Fig. 3-13 for the purpose of comparison.. Figure 3-13. The transmittance spectra of the four structures under consideration. 24.

(30) 3-4 Conclusion In this chapter, we have studied the high transmittance for the cascaded defective photonic crystal. Based on the two single defective PCs with different resonant wavelengths, a multichannel filter can be achieved by cascading these two defective PC. Four possible structures including two symmetric and two asymmetric ones have been investigated. The positions of peaks as a function of the angle of incidence are also demonstrated.. References. 1. Orfanidis, S. J., Electromagnetic Waves and Antennas, Chapter 7, Rutger University, 2008, www.ece.rutgers.edu/~orfanidi/ewa. 2. Yeh, P., Optical Waves in Layered Media, John Wiley & Sons, Singapore, 1991. 3. Born, M., E. Wolf, Principles of Optics, Cambridge, London, 1999. 4. Hecht, E., Optics, Addison Wesley, New York, 2002, Ch. 9. 5. Yablonovitch, E., “Inhibited spontaneous emission in solid state physics and electronics,” Phys. Rev. Lett., Vol. 58, 2059-2062, 1987. 6. John, S., “Strong localization of photons in certain disordered lattices,” Phys. Rev. 25.

(31) Lett., Vol. 58, 2486-2489, 1987. 7. Joannopoulos, J. D., R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, Princeton University Press, Princeton, NJ, 1995. 8. Yariv, A., and P. Yeh, Photonics, Oxford University Press, New York, 2007. 9. Srivastava, R., K. B. Thapa, S. Pati, and S. P. Ojha, "Omni-direction reflection in one dimensional photonic crystal," Progress In Electromagnetics Research B, Vol. 7, 133-143, 2008. 10. Srivastava, R., S. Pati, and S. P. Ojha, “Enhancement of omnidirectional reflection in photonic crystal heterostructures,” Progress In Electromagnetics Research B, Vol. 1, 197-208, 2008. 11. Awasthi, S. K., U. Malaviya, S. P. Ojha, N. K. Mishra, and B. Singh, "Design of a tunable polarizer using a one–dimensional nano sized photonic bandgap structure," Progress In Electromagnetics Research B, Vol. 5, 133-152, 2008. 12. Golmohammadi, S., Y. Rouhani, K. Abbasian, and A. Rostami, “Photonic bandgaps in quasiperiodic multilayer using Fourier transform of the refractive index profile,” Progress In Electromagnetics Research B, Vol. 18, 311-325, 2009. 13. Banerjee, A.,”Tunable polarizer using a one–dimensional nano sized photonic bandgap structure," Progress In Electromagnetics Research B, Vol. 5, 133-152, 2008. 26.

(32) 14. Banerjee, A., “Enhanced temperature sensing by using one-dimensional ternary photonic band gap structures,” Progress In Electromagnetics Research Letters, Vol. 11, 129-137, 2009. 15. Wu, C.-J., B.-H. Chu, M.-T. Weng, H.-L. Lee, “Enhancement of bandwidth in a chirped quarter-wave dielectric mirror,” J. Electromagnetic Waves and Applications, Vol. 23, No. 4, 437-447, 2009. 16. Wu, C.-J., B.-H. Chu, M.-T. Weng, “Analysis of optical reflection in a chirped distributed Bragg reflector,” J. Electromagnetic Waves and Applications, Vol. 23, No. 1, 129-138, 2009. 17. Veselago, V. G., “The electrodynamics of substances with simultaneously negative values of permittivity and permeability”, Sov. Phys. Usp., Vol. 10, 509-514, 1968. 18. Hsu, H.-T., and C.-J. Wu, "Design rules for a Fabry-Perot narrow band transmission filter containing a metamaterial negative-index defect," Progress In Electromagnetics Research Letters, Vol. 9, 101-107, 2009. 19. Wang, Z.-Y., X.-M. Cheng, X.-Q. He, S.-L. Fan, and W.-Z. Yan, "Photonic crystal narrow filters with negative refractive index structural defects," Progress In Electromagnetics Research, PIER 80, 421-430, 2008. 20. Boedecker, G., and C. Henkle, “All-frequency effective medium theory of a photonic crystal,” Optics Express, Vol. 13, 1590-1595, 2003. 27.

(33) 21. Ha, Y. K., Y. C. Yang, J. E. Kim, and H. Y. Park, “Tunable omnidirectional reflection bands and defect modes of a one-dimensional photonic band gap structure with liquid crystals,” Appl. Phys. Lett., Vol. 79, 15-17, 2001. 22. Lu, Y. Q., and J. J. Zheng, “Frequency tuning of optical parametric generator in periodically poled optical superlattice LiNbO3 by electro-optic effect,” Appl. Phys. Lett., Vol. 74, 123-125, 1999. 23. Zhu, Q., and Y. Zhang, “Defect modes and wavelength tuning of one-dimensional photonic crystal with lithium niobate,” Optik, Vol. 120, 195-198, 2009. 24. Wu, C.-J., J.-J. Liao, and T. W. Chang, “Tunable multilayer Fabry-Perot resonator using electro-optical defect layer,” J. Electromagnetic Waves and Applications, Vol. 24, 531-542, 2010. 25. Hu, X., Q. Gong, S. Feng, B. Cheng, and D. Zhang, “Tunable multichannel filter in nonlinear ferroelectric photonic crystal,” Optics Communication, Vol. 253, 138-144, 2005. 26. Liu, J., J. Sun, C. Huang, W. Hu, and D. Huang, “Optimizing the spectral efficiency of photonic quantum well structures”, Optik, Vol. 120, 35-39, 2009. 27. Smith, D. R., R. Dalichaouch, N. Kroll, S. Schultz, S. L. McCall, and P. M. Platzman, “Photonic band structure without and with defect in one-dimensional photonic crystal,” J. Opt. Soc. Am. B: Optical Physics, Vol. 10, 314-321, 1993. 28.

(34) 28. Qiao, F., C. Zhang, and J. Wan, “Photonic quantum-well structure: Multiple channeled filtering phenomena,” Appl. Phys. Lett., Vol. 77, 3698-3700, 2000.. 29.

(35) Chapter 4 Transmission Properties in A One-dimensional Photonic Crystal Containing Twin Defects. 4-1 Introduction Over the past two decades, the field of photonic crystals (PCs) has attracted much attention in the communities of condensed matter physics, optics and photonics, and material science and engineering. The emergence of PCs was initialized by two pioneering works by Yablonovitch and John in 1987 [1,2]. With some novel and unique electromagnetic properties in PCs, researches on their applications and fundamental issues have been extensively reported thus far [3-8]. Depending on the structural dimension, PCs can be classified as one- , two- and three-dimensional. In a PC, the most important and fundamental feature is the existence of the photonic band gap (PBG), which arises from the interference of Bragg scattering in a periodically stacked structure. A PC can be applied to function as an optical reflector in some frequency regions where PBGs are covered. In a simple finite one-dimensional (1D) PC, the structure is (HL)N, where H and L are layers with high and low refractive indices, respectively, and N is the number of periods. This structure is commonly referred to as a distributed Bragg reflector (DBR). In addition 30.

(36) to being used as a DBR, 1D PC can usually be designed as a narrowband transmission filter which can be made by adding a defect layer to break the periodic structure, i.e., (HL)ND(HL)N, where D is the defect layer. With the addition of defect, a transmission peak within the desired PBG can be created. Narrowband transmission filters also called the multilayer Fabry-Perot resonators (FPRs) are often designed based on the DBR containing quarter-wavelength stack [9-11]. In this paper, we would like to investigate a modified version of multilayer FPR containing twin defects, namely a structure of air/(HL)PD1(HL)ND2(HL)Q/air, in which the twin defects D1 and D2 are sandwiched by three Bragg mirrors with stack numbers of P, N, and Q, respectively. Without loss of generality, we assume that H and L are high- and low-index quarter-wavelength layers, i.e., nHdH = nLdL = 0/4, where nH, nL and dH, dL are their refractive indices and thickness, respectively, and 0 is the design wavelength. In this twin-defect filter, the stack numbers P, N, Q, are taken to satisfy the relation of N = P + Q. All constituents H, L, D1, and D2 are taken to be lossless dielectrics. In addition, the refractive index of air is na = 1. We shall analyze the transmission properties for this filter. The roles played by the stack numbers P, N, Q, in the transmission peak will be illustrated. The dependence of angle of incidence in both the TE and TM waves will also be elucidated.. 31.

(37) 4-2 Model Structure The transmission properties in the above-mentioned twin-defect filter will be analyzed through the transmittance calculated by the transfer matrix method (TMM) [12]. In TMM, the transmittance can be determined by the matrix elements of the total system matrix M written by.  M11 M12  1 1 1 P 1 1 1 N    Da  DH PH DH DL PL DL  DD1 PD1 DD1  DH PH DH DL PL DL   M 21 M 22 . . D. . . P DD2 1  DH PH DH 1DL PL DL 1  Da .. D2 D2. Q. (4.1). The transmittance T, which is related to the transmission coefficient t, is calculated by the element M11, with the result 2. 1 Tt  . M 11 2. (4.2). With the temporal part of exp  jt  for all fields, the propagation matrix Pi (i =H, L, D1 and D2) in Eq. (1) for each layer i is written by.  exp  jki di   0 Pi   , 0 exp  jk d   i i  . (4.3). where ki  ni cos i c is the wave number, where i is the ray angle. The dynamical matrix Dq (q = a, H, L, D1 and D2) in Eq. (1) is dependent on the polarization of the incident wave, namely. 1  Dq    nq cos  q. 1  , nq cos  q  32. (4.4).

(38) for TE wave, and.  cos  q Dq    nq. cos  q  , nq . (4.5). for TM wave, respectively, where q = a is for air with na = 1 and 0   a is defined as the angle of incidence.. 4-3 Numerical Results and Discussion In what follows, we adopt the following refractive indices for our design. The refractive indices for layers H and L are nH = 3.61 (GaAs), nL = 2.78 (ZnTe). The twin defects are taken to be the same material D1 = D2 = D with nD1  nD2  nD  1.36 (MgF2) [9]. The design wavelength 0 = 9.2 m is taken. The corresponding design angular frequency is thus given by 0 = 2c/0 = 2.05 x 1014 rad/s. The twin defects are also taken to be quarter-wavelength layers.. Figure 4-1. The calculated normal-incidence transmittance spectrum for an ideal defect-free PC of (HL)N at N = 50. There is a photonic band gap with gap center near the design frequency. 33.

(39) In Fig. 4-1, we plot the transmittance spectrum for the ideal defect-free PC of air/(HL)N/air at N = 50 under normal incidence. It is seen that there is a PBG with gap center near the design frequency 0. This gap is known as the Bragg gap. In an ideal defect-free PC, it is known that the Bragg gap can be widened in the oblique incidence for TE wave. However, in TM wave, it will be shrunk as the incident angle increases [9]. Let us now consider the twin-defect filter air/(HL)PD(HL)ND(HL)Q/air. The calculated normal-incidence transmittance spectrum inside the PBG of Fig. 4-1 is shown in Fig. 4-2, in which different values of (P, N, Q) are used. It is seen that there is a transmission peak (the defect mode) with peak frequency that is equal to the design frequency. Such transmission peak indicates that this twin-defect structure can actually work as a narrowband transmission filter. The transmission peak shape,. Figure 4-2.. The calculated normal-incidence transmittance spectrum for air/(HL) PD(HL)ND(HL)Q/air. at four different values of (P, N, Q). 34.

(40) which in turn determines the quality factor of the filter, is mainly controlled by N, the stack number of the middle Bragg mirror. It can become narrower at a large number of N. For a sufficiently large N, a very sharp peak can be obtained. The dependence of (P, N, Q) can be explained as follows: With the condition of N = P + Q, the whole structure can be viewed as a system that is cascaded by two individual multilayer FPRs, i.e., (HL)PD(HL)P and (HL)QD(HL)Q. These two FPRs have the same peak frequency but different bandwidths if P  Q . The peak shape will be narrower when the two FPRs are stacked in series, because the two transmission spectra must be multiplied to yield the resultant spectrum for the cascaded system. The narrowing effect will be more salient when N is large, as illustrated in Fig. 4-2. In Fig. 4-3, we plot transmittance spectra for this filter at four different angles of incidence, 00, 300, 450, and 600, respectively. Here, (P, N, Q) are fixed at (5, 7, 2), and the results of TE- and TM-wave are shown in the upper and lower panels, respectively. It can be seen from both figures that the transmission peak will be blue-shifted, that is, shifted to the higher frequency as the incident angle increases for both TE and TM waves. The distinction between TE and TM waves is the change in the peak shape. In TE wave, the peak will be narrowed when the angle increases. However, the peak is broadened at a large angle of incidence in TM wave. The results shown in Fig. 4-3 suggest that it is preferable to increase the quality factor of this narrowband filter at a 35.

(41) larger angle of incidence in TE wave.. Figure 4-3. The calculated transmittance spectrum for air/(HL) PD(HL)ND(HL)Q/air at (P, N, Q) = (5, 7, 2) for TE wave (upper panel) and TM wave (lower panel).. In Fig. 4-4, we plot the transmittance at  = 0 versus angle of incidence at four different values of (P, N, Q), and the results of TE and TM waves are given in the upper and lower panels, respectively. According to Fig. 4-3, we see that the resonant transmission at 0 holds for the normal incidence. In the oblique incidence, the 36.

(42) resonance transmission no longer occurs at 0. The increase in the angle of incidence will then decrease the transmittance at 0, as illustrated in Fig. 4-4. It is also seen that the decreasing curve is broadened when N increases. In addition, at N = 7, T = 1 for both TE and TM waves holds at angle 00-60. However, at N = 4, the range of angle of T = 1 for both TE and TM waves is extended to about 200. This indicates the resonance transmission cannot be expected to be omnidirectional, especially for large N. At N = 4, we can say that the resonance transmission is partially omnidirectional with a range of ~200. That is, the partially omnidirectional feature in the transmission peak can be more pronounced at a small N. However, a small N will obviously reduce the quality factor as depicted in Fig. 4-2. Conclusively, it is hard to have a strongly narrowband filter which is omnidirectional. In fact, it is known that an omnidirectional PBG in a defect-free dielectric PC is much easier to have [13].. 37.

(43) Figure 4-4.. The calculated transmittance at  = 0 as a function of the angle of incidence for. twin-defect filter of air/(HL)PD(HL)ND(HL)Q/air at four different values of (P, N, Q) for TE wave (upper panel) and TM wave (lower panel).. 4-4 Conclusion Based on the use of the twin-defect in a PC, we have proposed a design of narrowband transmission filter. The transmission properties of such filter have been investigated and some conclusions can be drawn. First, the narrowing and widening in the transmission peak can be mainly controlled by the stack number of the central Bragg mirror. Second, in the oblique incidence, the resonance frequency is shifted to a higher frequency as the angle of incidence increases. In addition, the narrowing effect in the peak shape is seen in the TE wave whereas a widening effect in it is found in the TM wave. Finally, in view of the design of a narrowband filter, the omnidirectional peak is hard to obtain, which is in sharp contrast to the defect-free PC where an omnidirectional PBG is easy to obtain. 38.

(44) REFERENCES 1.. Yablonovitch, E., “Inhibited spontaneous emission in solid state physics and electronics,” Phys. Rev. Lett., Vol. 58, 2059-2062, 1987.. 2.. John, S, “Strong localization of photons in certain disordered lattices,” Phys. Rev. Lett., Vol. 58, 2486-2489, 1987.. 3.. Bowden, C. M., J. P. Dowling, and H. O. Everitt, “Development and applications of materials exhibiting photonic band gaps: introduction,” J. Opt. Soc. Am. B: Optical Physics, Vol. 10, 280-413, 1993.. 4.. Joannopoulos, J. D., R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, Princeton University Press, Princeton, NJ, 1995.. 5.. Knight, J. C., J. Broeng, T. A. Birks, and P. St. J. Russell, “Photonic band gap guidance in optical fibers,” Science, Vol. 282, 1476-1478, 1998.. 6.. Yariv, A. and P. Yeh, Photonics, Oxford University Press, New York, 2007.. 7.. Yeh, D.-W. and C.-J. Wu, "Analysis of photonic band structure in a one-dimensional photonic crystal containing single-negative material," Optics Express, Vol. 17, 16666-16680, 2009.. 8.. Rahimi, H., A. Namdar, S. R. Entezar, and H. Tajalli, "Photonic transmission spectra. in. one-dimensional. Fibonacci. multilayer. structures. containing. single-negative metamaterials," Progress In Electromagnetics Research, Vol. 102, 39.

(45) 15-30, 2010. 9.. Orfanidis, S. J., Electromagnetic Waves and Antennas, Rutger University, 2008, www.ece.rutgers.edu/~orfanidi/ewa, Ch.7.. 10. Smith, D. R., R. Dalichaouch, N. Kroll, S. Schultz, S. L. McCall, and P. M. Platzman, “Photonic band structure without and with defect in one-dimensional photonic crystal,” J. Opt. Soc. Am. B: Optical Physics, Vol. 10, 314-321, 1993. 11. Wu, C.-J. and Z.-H. Wang, "Properties of defect modes in one-dimensional photonic crystals," Progress In Electromagnetics Research, Vol. 103, 169-184, 2010. 12. Yeh, P., Optical Waves in Layered Media, John Wiley & Sons, Singapore, 1991. 13. Chigrin, D. N., A. V. Lavrinenko, D. A. Yarotsky, and S.V. Gaponenko, “Observation of total omnidirectional reflection from a one-dimensional dielectric lattice,” Appl. Phys. A: Mater. Sci. Process, Vol. 68, 25-28, 1999.. 40.

(46) Chapter 5 Conclusions In this thesis, we have studied the optical properties in a defective PC. In Chapter 3, we use the specific materials, i.e., TiO2 and SiO2, to investigate the defect modes for the symmetric and asymmetric narrowband transmission filters. In the symmetric structures, there are two modes in the visible region. In addition, defect modes in the A-C-B structure are much shaper than those in A’-C-B’. In the asymmetric structures, more than two modes in the visible region can be produced, but the magnitudes in transmittance of these modes are lower than the symmetric structures. Moreover, we could use some suitable modifications in these symmetric and asymmetric structures to optimize the narrowband filters at the designed wavelength. In Chapter 4, three specific coating materials GaAs, ZnTe and MgF2 are used. Firstly, the narrowed and widened transmission peaks can be mainly controlled by the stack numbers of the central Bragg mirror. Secondly, in the oblique incidence, the resonance frequency is shifted to a higher frequency as the angle of incidence increases. It is difficult to obtain the omnidirectional peak from the designed narrowband filters, which is in sharp contrast to the defect-free PC where an omnidirectional PBG is easy to obtain.. 41.

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