一維及二維光子晶體光學性質之計算

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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. 一維及二維光子晶體光學性質之計算 Numerical Studies of Optical Properties of One-Dimensional and Two-Dimensional Photonic Crystals. 指導教授：指導教授：吳謙讓博士研究生：研究生：劉正禮. 中華民國九十八年六月 1.

(2) 摘要本篇論文主要是採用數值模擬的方法，研究一維及二維光子晶體的光學特性及應用。對於一維結構的光子晶體，我們透過轉移矩陣法來計算由超導層、介電層相互交替排列的週期組成，並求得其透射及反射的光學頻譜。利用模擬的結果，我們分析不同的相對週期厚度及入射角度對能帶分佈的影響，進一步歸納出各個變量在整體的結構中，所可能扮演的角色及造成的效應。在處理二維組成的光子晶體時，我們利用平面波展開法探求其能帶結構；並應用時域有限差分法，以光學模擬上常被使用的 FDTD 套裝軟體，進一步透析其電場、磁場、及能量在光子晶體中的傳播方式，以及各個分量在行進中隨時間和空間的變化。最後，我們將探討光通訊系統中，由光子晶體組成的小尺寸光學元件，例如:二維光子晶體波導、耦合共振光學波導、以及指向耦合器，其特有的電磁特性與相關的光學應用。. 2.

(3) Abstract In this thesis, we theoretically study the optical properties and their applications for one-dimensional and two-dimensional photonic crystals. In the one-dimensional photonic crystals (1DPCs) we use the transfer matrix method to calculate the transmittance and reflectance spectra for a superconductor-dielectric photonic crystal (SDPC). Based on the calculated results we investigate the photonic band structures as a function of film thickness and the angle of incidence as well. As for the two-dimensional photonic crystals (2DPCs), the optical properties will be explored by not only the plane wave expansion but FDTD method. In the final part, we design some useful two-dimensional photonic devices such as the photonic crystal waveguides (PCWs), coupled-resonator optical waveguides (CROWs), and director couplers (DCs).. 3.

(4) 誌謝歲月荏苒，光陰似箭，又到了一年一度鳳凰花開的時候。夏天，是一個屬於離別的季節。轉眼間，研究所的備忘錄便要寫到了最終章。一路走來，我並不孤單，因此要感謝的人太多了。承蒙鄭超仁老師在我最迷惘的時候，適時地賜予我最後一根稻草，讓我可以不再猶豫。承蒙昱志學長在 Meeting 時，老師遲來的前幾分鐘裡，還願意盡其所能地幫我把程式修到最完美。承蒙益興同學陪我度過半年多，矜實的實驗室生活，並啟蒙我對日本新番動畫裡的聲優，最初步的體認。緣分是一種很微妙的東西，往往令人嘆為觀止。感謝吳謙讓老師對我在專業及英文方面的不遺餘力；感謝郭文凱老師在 FDTD 光學模擬軟體上的不吝賜教；感謝許恆通老師在 CST 通訊模擬軟體上的傾囊相授；並感謝楊宗哲老師鉅細靡遺地指點迷津。另外，還要感謝柏翰學長教我怎麼找資料、作研究；感謝茂東學長陪我熬夜準備期中考；感謝耀立同學帶我去那家可以盡情吃到飽的羊肉爐，但你的英文名字不叫 YuRi(ゆり:百合)真是太殘念了；再來還要感謝學弟們，雖然我連你們的名字都不知道。但因為有你們的加入，使整個實驗室都充滿了新的活力，雖然我們現在並沒有所謂的實驗室… 我知道能順利畢業是多虧了許多人在背後的默默付出與支持，抱持著感恩的心也讓我深刻體悟到虛心竹尚有低頭葉；傲骨梅豈有仰面花。. 4.

(5) Contents Chapter 1 Introduction 1.1_Foreword………………………………………………………………………...06 1.2_Introduction to Photonic Crystals………………………………………………..07 Reference……………………………………………………………………………..08. Chapter 2 Basic Theories 2.1_ Theory of Photonic Crystals………………………………………...…………..10 2.2_ Plane Wave Expansion Method…………………………………………………11 2.3_ Transfer Matrix Method………………………………………………………...13 2.4_FDTD Method…………………………………………………………………...16 2.5_ An Introduction to Photonic Band Gaps………………………………………..16 2.6_More on Photonic Band Gaps…………………………………………………...20 Reference……………………………………………………………………………..22. Chapter 3 One-Dimensional Photonic Crystals---Using TMM 3.1_ Analysis of Thickness-Dependent Optical Properties in a One-Dimensional Superconducting Photonic Crystal…………………………………...………...24 3.2_ Angle-Dependent Transmittance Spectra in a One- Dimensional Superconducting Photonic Crystal………………………….………………….33. Chapter 4 Two-Dimensional Photonic Crystals---Using FDTD 4.1_Introduction to FDTD……………………………………………………………43 4.2_FDTD Method…………………………………………………..……………….44 4.3_ Cell Size and Time Step……………………………………………...…………48 4.4_ Absorbing Boundary Condition………………………..…………………….49 4.5_ Cases Study………………………………………………………………...50 4.6 Anti-Reflection Film of 1-D PC…………………………………………………72 4.7 Negative Refraction in HTSC PC……………………………………………….75 Reference……………………………………………………………………………..78. Chapter 5 Conclusion and Future Works 5.1_Conclusion…………………………………………………………...…………..79 5.2_Future Works……………………………………………………………...……..79 5.

(6) Chapter 1 Introduction 1.1 Foreword In the past, the investigation of behavior of light can be generally done by the macroscopic and microscopic viewpoints. In the microscopic one, scientists have successfully applied the quantum mechanics to crystalline substance, and have developed the band theory of solids to explain the behaviors of electron in the periodic potential.. Photonic crystals (PCs) are periodic layered structure. The behavior of light in the PCs can be solved by the wave equation derived by the Maxwell’s equations. This wave equation is mathematically equivalent to the Schrodinger equation, the wave equation of electron in solids. It is thus expected the electronic band structure in solids should have its corresponding one for the light in the PCs. That in fact has been confirmed and now is known as the photonic band structure. In addition, there exist some photonic band gaps (PBGs) in some frequency ranges.. When an electromagnetic wave with frequency in the PBG is incident to a PC, it will be reflected totally due to no corresponding propagating modes in the PC. This property thus introduces many interesting applications. For example, one can design a photonic reflective mirror, crate a line defect in the PC to form a PC waveguide (PCW), put a point defect inside to make a PC resonant cavity, and so on. More recently, researchers have also found many strange physical phenomena such as the negative refraction and sub-wavelength imaging as well. In fact, due to the complicated dispersion relation in photonic passband in a PC, which is obviously 6.

(7) totally different from a general homogeneous dielectric material, the anomalous refraction possibly occurs in a PC. Such anomalous behaviors are not seen in the ordinary optic systems which are not based on the PCs.. 1.2 Introduction to Photonic Crystals Photonic crystals are layer-by-layer structured electromagnetic media in which the refractive index function is periodic [1, 2]. According to the Bloch-Floquet theorem and Maxwell equations, we can see the existence of photonic allowed and forbidden bands in the frequency regimes for light propagation which is similar to the electronic bands in the crystalline solids [3]. In the PCs, the length scale of spatial periodicity is on the order of the wavelength of light propagating in the structure [4, 5]. A photonic band gap may allow to suppress spontaneous emission and to achieve localization of light by intentionally introducing defects into the crystal [6]. A linear defect in a PC acts as a channel waveguide for light propagation and may also confine light without loss around sharp bands by doping lower-index media [7, 8]. The linear waveguides made by the PC can thus form a kind of perfect optical “insulator”, and the electromagnetic phenomena is controlled in the form of wavelength-scale [9,10].. Fig. 1.2.1: Schematic arrangements of photonic crystals periodic in one, two, and three directions, respectively, where the periodicity of layer-by-layer is due to the dielectric material structure of the crystal. [From:http://myoops.educities.edu.tw/cocw/mit/Mathematics/ 18-325Fall-2005/CourseHome/index.htm]. 7.

(8) In addition, in the natural world the surface of a butterfly's wing containing a periodic array of holes of a certain size may reflect the specific light and absorb other colors even though the crystal material itself is entirely colorless [11,12]. Because intervals of a crystal array are slightly different from different angles, PCs can lead to shifting shades of iridescent color.. Fig. 1.2.2: A blue butterfly and the two-dimensional FFT power spectra of square areas.. Fig. 1.2.3: A brown butterfly and the two-dimensional FFT power spectra of square areas. [From: http://www.mfa.kfki.hu/int/nano/online/2002_butterfly/index.html]. References 1.. E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. 58, 2059 (1987).. 2.. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486 (1987).. 3.. M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, “Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two-dimensional photonic crystals,” Phys. Rev. B 64, 155113 (2001). 8.

(9) 4.. M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696 (2000).. 5.. Introduction to Photonic Crystals: Bloch’s Theorem, Band Diagrams, and Gaps(But No Defects)Steven G. Johnson and J. D. Joannopoulos, MIT 3rd (February 2003).. 6.. Alexei A. Erchak, Daniel J. Ripin, Shanhui Fan, Peter Rakich, John D. Joannopoulos, Erich P. Ippen, Gale S. Petrich and Leslie A. Kolodziejski, “Enhanced coupling to vertical radiation using a two-dimensional photonic crystal in a semiconductor light-emitting diode,” Appl. Phys. Lett. 78, 563 (2001).. 7.. J. C. Knight, J. Broeng, T. A. Birks, and P. St. J.Russell, “Photonic band gap-guidance in opticalfibers,” Sicience, 282, 1476 (1998).. 8.. Attila Mekis, J. C. Chen, I. Kurland, Shanhui Fan, Pierre R. Villeneuve, and J. D.Joannopoulos, “High Transmission through Sharp Bends in Photonic Crystal Waveguides,” Phys. Rev. Lett. 77, 3787 (1996).. 9.. M. Agio and C. M. Soukoulis, “Ministop bands in single-defect photonic crystal waveguides,” Phys. Rev. E 64, 055603R (2001).. 10. P. G. Luan, and K. D. Chang, “Transmission characteristics of finite periodic dielectric waveguides,” Opt. Express 14, 3263 (2006). 11. 欒丕綱、陳啓昌, 光子晶體—從蝴蝶翅膀到奈米光子學, 五南出版社 (2005). 12. Biró, L., P.; Bálint, Zs.; Kertész, K.; Vértesy, Z.; Márk, G., I.; Horváth, Z., E.; Balázs, J.; Méhn, D.; Kiricsi, I.; Lousse, V.; Vigneron, J.-P.: Role of photonic-crystal-type structures in the thermal regulation of a lycaenid butterfly sister species pair; Phys. Rev. E 67, 021907-1(2003).. 9.

(10) Chapter 2 Basic Theories 2.1 Theory of Photonic Crystals A photonic crystal is a periodic structure of dielectrics in 1D, 2D or 3D. Due to the . . . periodicity of the lattice, we have ε (r + R) = ε (r ) . Furthermore, ε (−r ) = ε (r ) and any. vector r ′ in space can be written as r ′ = r + R , where R is the translational vector . . . . in space defined by R = α1a1 + α 2 a2 + α 3 a3 , where α1, α2, α3 = 0, ±1, ±2, ±3,…and a1, a2,. a3 are so called as the primitive vectors (Fig. 2.1.1).. Fig. 2.1.1: The translation vector R and letting. a1 = a , a2 = b , a3 = c here.. [From: Chapter 7 Dielectric Waveguides and Some Selected Topics for Photonics]. Fig. 2.1.2: The Wigner-Seitz cell for an arbitrary position of points. [From: Study of EM waves in Periodic Structures with addenda: “Study of EM waves in Periodic Structures (mathematical details)”]. 10.

(11) Wigner-Seitz cell is constructed by joining the center element to its closest neighbors and drawing perpendicular lines from to the center of these segments [13]. The polygon thus created is the smallest repeatable cell of the periodic lattice, and is defined as the Wigner-Seitz cell (Fig. 2.1.2) which is the first Brillouin Zone (BZ). A detailed description of the BZ of a solid will be given in the section 2.5 later. The calculation of photonic band structures and modes in the 2DPC are appropriate to be treated by the plane-wave expansion (PWE) method [14, 15]. If the electromagnetic field is periodic then we can write the electromagnetic properties as the sum of sinusoidal functions and the Maxwell equations can be simplified to Helmholtz problems. Usually, the PWE method is used to calculate the dispersion relation of photonic crystals. We describe PWE method in Sec. 2.2. The calculation of optical properties in the 1DPC like the reflection and transmission can be made by the transfer matrix method (TMM) [16] to be given in Sec. 2.3. Numerical simulation of the propagation of an EM wave in the real PCs will be made by using the Finite-Difference Time-Domain (FDTD) method [17]. The FDTD method to be described in Sec. 2.4 is used to calculate the extraction efficiency of periodic lattices. This technique is based on a real-space discretization and is able to model photonic crystal defects of any geometry.. 2.2 Plane Wave Expansion Method. The plane wave expansion method may be used to calculate the band structures using an eigen-formulation from the Bloch's theorem and Maxwell's equations [18]. After solving for the eigen-problem of the wave vectors, it is possible to directly obtain the dispersion diagram. Electric field strength values can also be calculated over the. 11.

(12) spatial domain by solving the eigen-vectors of the fields. A Bloch wave is the wavefunction of a particle placed in a periodic potential. It consists of the product of a plane wave envelope function and a periodic function. u ( r + Λ ) = u ( r ) where Λ is the period, and which is the same periodicity as the potential:. ψ k (r ) = eik ⋅r uk (r ) ,. (1). where k is known as the Bloch wave number. The result that the eigenfunctions can be written in this form for a periodic system is called Bloch's theorem. The PWE method is commonly used to calculate the dispersion relation, the photonic band structure in a photonic crystal. In the source-free (J = 0 and ρ = 0) the two curl Maxwell’s equations are: ∂H ∂E , , ∇× E = µ ∇× H = ε ∂t ∂t. (2). which together with Faraday’s law and Ampere’s law give the following master equation for H-field  ω2  1 ∇× ∇× H(r) = 2 H(r) .  ε(r)  c. (3). The Fourier coefficients of periodical permittivity is . . . ε (r ) = ∑ ε (G )e iG⋅r ,. (4). G. where G is translation vector in the reciprocal lattice [19]. Also, an expansion of the inverse of the permittivity function of the crystal is required iG ⋅r 1 −1 = ∑ ε (G ) e . ε (r ) G. (5). A Fourier expansion of the magnetic field over the reciprocal space yields 2 ⌢ H ( r ) = ∑ u j H j ( k ) e ik ⋅ r ,. (6). j =1. ⌢ where u j is the unit vector of the jth coordinate basis. With the periodic lattice and the. 12.

(13) reciprocal lattice, we know H ( r ) = H (r + R ) ,. (7). and H j (k )eik ⋅r = H j (k + G)ei (k +G)⋅r .. (8). Substituting these two expressions into Eq. (3) leads to the following equation,. ∑ε G ', j. −1. ω2 ⌢ (G'−G) (k + G' ) × [(k + G) H G ', j × u j ] + 2 c. ∑H. G, j. ⌢ uj = 0. j. (9). The equation is a linear eigenvalue problem where ω2/c2 is the eigenvalue and HG is . the eigenvector given in terms of a plane wave e i ( k +G )⋅r . In 2-dimensional PCs, Eq. (9) can be decomposed into two parts [20]: TM mode:. . . . G'. TE mode:. . . . ∑ ε −1 (G − G' )(k + G) ⋅ (k + G' ) H G ',1 = ∑ε G'. −1. ω2 c2. H G ,1 ,. ω2 (G − G ' ) k + G ⋅ k + G ' H G ', 2 = 2 H G , 2 . c. (10) (11). 2.3 Transfer Matrix Method. According to Maxwell's equations and the continuous boundary conditions, the electromagnetic reflection and transmission of a multilayer structure can be investigated by the so-called transfer matrix method (TMM) [21]. In TMM, the field at the end of the layer can be derived from the field at the beginning through the simple matrix operation of the system matrix represented from a stack of layers, which is the product of the individual layer matrices. We now briefly introduce TMM as follows: Consider a light-wave propagating along x-direction in a media characterized by a refractive index n homogeneous in the yz plane but possibly x-dependent. Referring to Fig.2.3.1 as a simple case, we have a homogeneous and isotropic layer structure 13.

(14) with three different indices of refraction  n1 , x < 0    n(x) = n2 , 0 < x < d  ,   n3 , d < x . (12). and we define the propagation matrix as. Fig. 2.3.1 A thin homogeneous layer of dielectric material..  e ikd 0 Pd =  − ikd  0 e.  . . (13). Here k is the component of wave vector in the tangent direction inward a medium, and we can write it as k =. ω c. n cos θ .. (14). for the refractive index of a medium as n, incidence of a beam as θ, velocity of light in vacuum c, and angular frequency ω. Then we will have  A 2 '  A2   B ' = Pd  B  .  2   2. (15). The boundary conditions on the tangential components of the electric and magnetic fields require that Ez, Ey, Hz, and Hy, be continuous at the interface x = 0, and we have the dynamical matrix for the medium i (i =1, 2) as  E1  E2  D1   = D 2  ,  E 1 '  E 2 '. 14. (16).

(15) from which we finally relate the amplitudes A1, B1 and A3′, B3′of the single layer by  A1   A 3 ' −1 −1  B  = D 1 D 2 P2 D 2 D 3  B ' .  1  3 . (17). Here the dynamical matrix D is defined by    n Dα =   α  . 1 1  cos θ α − n α cos θ α  ,  cos θ α cos θ α   n − n α  α . s − wave p − wave. (18). We now consider the case of multilayer structures below (Fig. 2.3.2), where the solutions can be written as.  A0 e −ik0 x ( x− x0 ) + B0 eik0 x ( x− x0 ) , x < x0   −ik ( x− x )  E( x) =  Al e lx l + Bl eiklx ( x− xl ) , xl −1 < x < xl  ,   −ik ( x− x ) ik ( x− x )  As ' e sx N + Bs ' e sx N , xN < x . (19). Fig. 2.3.2 The multilayer structure of thin homogeneous layers of dielectric material.. where E + (-) ( x) is the complex amplitude of light-wave propagating in positive (negative) direction, and l = 0,1,2,…N. The relationship between A0, B0 and A s′, B s′ can be thus be established as follows:  A0   A3   A1   A2  −1 −1 −1 −1 −1 −1   = D 0 D 1   = D 0 D 1 P1 D 1 D 2   = D 0 D 1 P1 D 1 D 2 P2 D 2 D 3   , (20) B B B  1  2  0  B3 . Then 15.

(16)  A 0   M 11 M 12   A s ' ,  B  = M   0   21 M 22   B s '. (21). with the M-matrix given by N  M 11 M 12  −1  −1    = D 0  ∏ D l Pl D l  D s . l = 1 M M   22    21. (22). Equation (20) can be summarized as the following figure (Fig. 2.3.3).. Fig. 2.3.3: Expression for the layer-by-layer in M-matrix for the multilayer structure.. 2.4 FDTD Method. The FDTD method, based on the two curl Maxwell’s equations, can compute the time-dependent behaviors of the electromagnetic waves. This method can also compute the radiation field in open space by using appropriate boundary conditions. Therefore, the FDTD method is widely used in the study of the optical properties of the photonic crystal structures, and we will introduce it in details in the next chapter.. 2.5 An Introduction to Photonic Band Gaps. A PC is a periodic dielectric structure whose lattice spacing is comparable to the wavelength of light. It is known that a PC has some photonic band gaps (PBGs), in which the light will be localized in the defect when a defect is introduced into a PC.. 16.

(17) Therefore, the PBG allows us to control the propagation of the electromagnetic wave in PC. The PBGs depend upon the arrangement of constituent filling factor and dielectric contrasts of the two media that form a PC. Taking into account in the convenience of studying band gap graphs forward, we will introduce the concept of Brillouin zones [22]. In solid-state physics, the . reciprocal lattice of a crystal structure is the set of all vectors K such that e iK ⋅R = 1 for all lattice point position vectors R [23]. This reciprocal lattice is the Fourier domain of original lattice by optical diffraction, and the reciprocal of the reciprocal lattice is the original lattice. The reciprocal space lattice is a set of imaginary points constructed in such a way that the direction of a vector from one point to another is orthogonal to the real space planes and the absolute value of the vector is equal to the reciprocal of the real interplanar distance. It is convenient to let the reciprocal lattice vector be 2π times the reciprocal of the interplanar distance, and this convention converts the units from periods per unit length to radians per unit length. A reciprocal lattice of a crystal structure is a set of vectors denoted by K that. satisfies the constructive interference condition, K ⋅ R = 2πm . Hence, if (k − k ′) was equal to any vector in the set K, the set of K obeying this requirement can be written as: K = β1b1 + β 2 b2 + β 3b3 , where β1, β2, β3=0, ±1, ±2, ±3,…. ∵ ε(r + R) =ε(r) , . . . . . . . ∴ ε (r ) = ∑ ε (G)e iG⋅r = ε (r + R) = ∑ ε (G )e iG⋅( r + R ) . Consideration of Bragg’s diffraction G. G. law of G ⋅ R = 2nπ , we get b j ⋅ ai = 2πδ ij , and let b1 = k (a2 × a3 ) , b1 ⋅ a1 = 2π then. 2π k= a1 ⋅ (a 2 × a3 ). a ×a b3 = 2π 1 2 , a3 ⋅ (a1 × a2 ). and. 2π ( a 2 × a 3 ) b1 = a1 ⋅ ( a 2 × a 3 ). .. Whereas,. a ×a b2 = 2π 3 1 a 2 ⋅ ( a3 × a1 ). and. where a1, a2, a3 are the primitive lattice vectors describing the. structure of the crystal.. 17.

(18) Fig. 2.5.1 Direct square lattice and corresponding reciprocal lattice. [From: Study of EM waves in Periodic Structures with addenda: “Study of EM waves in Periodic Structures (mathematical details)”]. The first Brillouin zone is a uniquely defined primitive cell of the reciprocal lattice in the frequency domain. Another definition is as the set of points in k-space that can be reached from the origin without crossing any Bragg plane. The importance of the Brillouin zone stems from the Bloch wave description of waves in a periodic medium, in which it is found that the solutions can be completely characterized by their behavior in a single Brillouin zone.. Fig. 2.5.2 First Brillouin zones of (a) square lattice and (b) hexagonal lattice. [From: http://en.wikipedia.org/wiki/Brillouin_zone]. Now we would like to know this convention of definition of the reciprocal lattice vector in First Brillouin zones further for Square lattice and Hexagonal lattice:. 18.

(19) Fig. 2.5.3: Direct square lattice and corresponding reciprocal lattice with Brillouin zone. The reciprocal. π. π. a. a. lattice vector: Γ → ( k x = k y = 0) , X →(kx = , ky = 0) , M → (kx = k y = ). Fig. 2.5.4: Direct hexagonal lattice and corresponding reciprocal lattice with Brillouin zone. The reciprocal lattice vector: Γ →(kx = ky = 0) , M →(kx = 0, ky =. 2π , 2π 2π ) K →(kx = , ky = ) 3a 3a 3a. [From: Chapter 7 Dielectric Waveguides and Some Selected Topics for Photonics]. Figure 2.5.5: Band diagrams and photonic band gaps for hexagonal lattices of air holes (right). The radius is 0.3α, where α is defined by center-center periodicity in dielectric (left). The frequencies are plotted at longitudinal axis by normalization with ωα/2πc around the boundary of the irreducible Brillouin zone, with solid-red/dashed-blue lines denoting TE/TM polarization (electric field parallel/perpendicular to plane of periodicity). [From: Introduction to Photonic Crystals:Bloch’s Theorem, Band Diagrams, and Gaps(But No Defects)Steven G. Johnson and J. D. Joannopoulos, MIT 3rd February 2003]. 19.

(20) In this example for 2-dimensional hexagonal photonic crystals lattice of air-hole structures in TE modes (which electric fields are polarized parallel to the plane of a photonic crystal slab), the dielectric constant of material is chosen to be 12 in the calculation, and there is a large band gap for the TE polarization in this slab below.. 2.6 More on Photonic Band Gaps. The main feature of photonic crystals is the periodic modulation of dielectric constant in space. In a composite formed by two dielectrics, if the interference causes that some frequencies are not allowed to propagate, the forbidden and allowed bands will appear [24]. In a medium of a photonic crystal, a spatial modulation of the dielectric constant ε (r ) = ε (r + a ) exists periodically, the eigenvalues of this equation are also periodic functions with period a. The dispersion relationship derived will present a forbidden band for all frequencies ω which have imaginary values. The photon propagation is governed by the classical wave equation for the magnetic field H(r): 2 1  ωn (k )    Hn,k (r) (∇+ ik ) × (∇+ ik ) × Hn,k (r) =   ε (r)  c . Fig. 2.6.1 Energy dispersion relations for a free photon and a photon in a photonic crystal, it is shown how gaps are developed in a PBG material. [From: http://luxrerum.icmm.csic.es/?q=node/research/PCintro]. Dispersion relation (band diagram) is frequency versus wavenumber, and the 20.

(21) band of a physical periodic dielectric variation has been opened by splitting the degeneracy at the. k =±. π. as Brillouin-zone boundaries. A complete photonic band gap. a. is a range of frequency in which there are no real solutions of propagating wavevector in 2 1  ωn (k )   Hn,k (r) . (∇ + ik ) × (∇ + ik ) × Hn,k (r ) =   c ε(r)  . There are also incomplete gaps, which only exist over a subset of all possible wavevectors in specific polarizations or symmetries. In a one-dimensional system of ε = 1 , (this ε must have trivial periodicity a for any a ≠ 0 ) the eigensolutions of plane wave is ω (k ) = ck . When a = 0 , it will give the usual unbounded dispersion relation as Figure 1.7.1 (left). If we label the states in terms of Bloch envelope functions and wavevectors for some a ≠ 0 , in which case the bands for. k=−. π. k >. π. will be folded into the first Brillouin zone. In this description, the. a. mode lies at an equivalent wavevector to the. a. k=. π. mode at the same. a. frequency, and this appearing degeneracy here is caused by the periodicity chosen.. Fig. 2.6.2 Schematic origin of the band gap in one dimension. The degenerate k = ±π / a plane waves of a uniform medium are split into e( x ) and o( x ) standing waves by a dielectric periodicity, forming the lower and upper edges of the band gap, respectively. [From: Introduction to Photonic Crystals:Bloch’s Theorem, Band Diagrams, and Gaps(But No Defects)Steven G. Johnson and J. D. Joannopoulos, MIT 3rd February 2003]. 21.

(22) Now. we. π  o(x ) = sin  x  a . equivalently. write. linear. combinations. π  e(x ) = cos x  a . and. to substitute for the exponential express of electric fields as shown in. Fig. 2.6.2. If the dielectric constant is nontrivially periodic with period a in a photonic crystal, the accidental degeneracy between e( x ) and o( x ) is broken. The field e( x ) is more concentrated in the higher dielectric regions than o( x ) such as electric-field peaks in the higher dielectric, and so lies at a lower frequency, and this opposite shifting of the bands creates a band gap. The frequency dispersion relation for a wave in vacuum is parabolic with no gaps. A periodic dielectric medium will present frequency regions where propagating waves are not allowed and will find it impossible to travel the crystal. Since waves are vectors, polarization must be taken into account. This finally results in much more restrictive conditions for gap appearance. The wave equation is scalable, hence if a photonic crystal presents a given periodicity length, it will show photonic bands in certain range of frequency and a scaling. The optical features of a photonic crystal will depend on the type of symmetry of the structure, dielectric constant contrast ( ε 1 / ε 1 ), and the ratio between the volume occupied by each dielectric component.. References. 13. C. Kittel, Introduction to Solid State Physics, 8th Ed., John Wiley & Sons, USA, (2005). 14. J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals—Molding the Flow of Light, Princeton University Press (1995). 15. K. Sakoda, Optical Properties of Photonic Crystals, Springer, Berlin (2001). 16. A. Yariv, P. Yeh, Optical Waves in Crystals, John Wiley & Sons, New York, 22.

(23) (1984). 17. A. Taflove, Computational Electrodynamics, Artech House, Boston, (1995). 18. D. K. Cheng, Field and Wave Electromagnetics 2nd, Addison-Wesley, New York, (1989). 19. George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists, 5th Ed, Academic Press, (2000). 20. I. S. Gradshteyn, I. M. Ryzhik, Alan Jeffrey, Table of Integrals, Series, and Products, 5th Ed., Academic Press, (1994). 21. J. D. Jackson, Classical electrodynamics 3rd, John Wiley&Sons, New York, (1999). 22. L. Brillouin, Wave propagation in periodic structure 2nd, Mineola, New York, (1946). 23. J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals—Molding the Flow of Light, Princeton University Press (1995). 24. Kazuaki Sakoda, Noriko Kawai, and Takunori Ito, “Photonic bands of metallic systems. I. Principle of calculation and accuracy,” Phys. Rev. B 64, 045116 (2001).. 23.

(24) Chapter 3 One-Dimensional Photonic Crystals---Using TMM 3.1 Analysis of Thickness-Dependent Optical Properties in a One-Dimensional Superconducting Photonic Crystal (J. Electromagnetic Waves and Applications, vol. 23, 1113-1122, 2009). 1. INTRODUCTION. A photonic crystal (PC) is a periodic layered structure consisting of alternating two dielectric layers with different refractive indices. It is known that there exist some photonic bandgaps (PBGs) within which electromagnetic waves cannot propagate through the layered structure. The existence of a PBG originates from the multiple Bragg reflections in a periodic dielectric structure [1-2]. A variety of applications of PBG media have enriched modern photonic science and technology. For a typical one, a PBG medium can be designed as a high-reflectance mirror called the Bragg reflector [3]. For a one-dimensional all-dielectric PC, to have an omnidirectional reflector is also obtainable [4]. In addition, by inserting a defect layer inside the PC, it can be further used to design a narrowband transmission filter or Fabry-Perot resonator [5]. A metallic photonic crystal consisting of alternating metallic and dielectric materials is also an interesting subject and it indeed has attracted much attention in the past decade [1]. With the highly dispersive behavior in the refractive index for a metal, the periodic layered arrangement causes the structure to have a variety of filtering responses at distinct frequency regimes [6]. The use of metal however may introduce some inherent loss issue in a photonic device. The loss issue can be reduced greatly by using the superconducting material instead of the metallic ones. In fact, there have been many reports on the superconducting PC [7-11]. The use of superconducting material in the PCs has some advantages. For instance, optical features can be tunable in a superconductor-based PC because of the temperature-dependent London penetration length. They can further be tuned by the external magnetic field when the superconductor in the mixed state. 24.

(25) In this paper, we investigate the fundamental optical properties for a one-dimensional superconducting photonic crystal stacked alternately by the superconducting and dielectric materials. The transmittance spectrum will be calculated by using the transfer matrix method (TMM). We pay attention to the effects of thickness changes in the constituent two materials. The results indicate that the transmission bands can be enhanced, shifted and even merged due to the thickness variation. The roles play by the thicknesses of superconductor and dielectric layers will be numerically elucidated.. 2. SUPERCONDUCTOR ELECTROMAGNETICS AND ABELES THEORY Let us consider a one-dimensional superconducting photonic crystal which is modeled as a periodic layered structure of 1/(23)N/1, where layer 1 is taken to be the free space, layer 2 is the superconductor, layer 3 is the dielectric, and N is the number of periods. The thicknesses of the superconducting and dielectric layers are denoted by d2 and d3, respectively, and the spatial periodicity is Λ = d2 + d3. A unit amplitude optical wave is normally incident on the plane boundary between layer 1 and layer 2. The reflectance and transmittance are denoted by R and T, respectively. Our goal is to calculate the optical transmittance spectrum for such a periodic layered structure. With the calculated spectrum, the thickness dependence of the optical properties can be explored.. Fig. 1 The one-dimensional superconducting photonic crystal, where layer 2 with refractive index n2 is the superconductor, layer 3 with n3 is the dielectric, and region 1 is assumed to be the free space with n1 = 1.. The transmittance can be analytically calculated by making use of the Abeles theory that is known as an elegant theory in dealing with a stratified medium. In describing the formulation of current problem we have, in advance, to know the refractive indices for the dielectric and superconducting materials. The refractive 25.

(26) index of a dielectric material is simply given by n3 = ε r 3 , where ε r 3 is the relative permittivity of the dielectric. The refractive index of the superconductor is denoted by n 2 which can be obtained as follows. If we assume that the temporal part is. exp ( jω t ) for all fields, then following the superconducting two-fluid model together. with Maxwell’s equations, one can obtain the wave equation for electric field, i.e., ∇ 2 E + k s2 E = 0 ,. (1). where the wave number in the superconductor is given by 12.  ω2  k s =  2 − jωµ0σ  c . where c = 1. 12.  ω2 1  = 2 − 2  , λL  c. (2). µ0ε 0 is the speed of light in free space and the expression for. superconducting conductivity is used, i.e., [12]. σ (ω ) = − j. 1. ωµ0λL2. ,. (3). where the temperature-dependent London penetration length of a superconductor is given by. λL (T ) =. λL ( 0 ) T  1−    Tc . 4. ,. (4). where λL ( 0 ) is the London penetration length at T = 0 K. In Eq. (3) the conductivity is purely imaginary because we have neglect the contribution of the normal-fluid. In other words, we are interested in the lossless superconductor that is valid for temperature well below transition temperature. A detailed discussion on the validity of a lossless superconductor can be found in Ref. [13]. With Eq. (2) in hand, the refractive index of a superconductor can be readily determined to be 12.  c2  n2 = k s = 1 − 2 2  . ω  ω λL  c. (5). It can be seen from Eq. (3) that the zero-index can be found at a frequency of. ωλ = c λL . Such a frequency is thus referred to as a singular frequency or the so-called threshold frequency. For a considered periodic superconductor-dielectric layered structure, the 2. transmittance is defined by T = t with t being the transmission coefficient.. 26.

(27) According to the Abeles theory, the transmission coefficient t is determined by the matrix elements of the characteristic matrix of the entire N-period layered system M and given by [14] 2Y0 n1 , ( M11 + M12Y0 n1 ) Y0 n1 + ( M 21 + M 22Y0 n1 ). t=. (6). where Y0 = ε 0 µ0 is the free-space admittance and the matrix elements of matrix M are given as follows:.   p M 11 =  cos β 2 cos β3 − 3 sin β 2 sin β 3  U N −1 (φ ) − U N − 2 (φ ) , p2  . (7a).  1  1 M 12 = j  cos β 2 sin β3 + sin β 2 cos β3  U N −1 (φ ) , p2  p3 . (7b). M 21 = j ( p2 sin β 2 cos β3 + p3 cos β 2 sin β3 ) U N −1 (φ ) ,. (7c).   p M 22 =  cos β 2 cos β 3 − 2 sin β 2 sin β3  U N −1 (φ ) − U N − 2 (φ ) , p3  . (7d). where 1  p2 p3  +  sin β 2 sin β 3 , 2  p3 p2 . φ = cos β 2 cos β3 − . (8). and U N are the Chebyshev polynomials of the second kind defined by. U N (φ ) =. sin ( N + 1) cos −1 φ  1−φ 2. .. (9). If all the media are nonmagnetic, i.e., µ = µ0 , then the parameters in Eqs. (7) and (8) are defined by. β2 =. 2π. λ. n2 d 2 cos θ 2 ,. p2 = Y0 n2 cos θ 2 ,. β3 =. 2π. λ. n3d3 cos θ3 ,. p3 = Y0 n3 cos θ3 ,. where λ = 2π c ω is the wavelength of the incident electromagnetic wave.. 27. (10) (11).

(28) 3. NUMERICAL RESULTS AND DISCUSION A. Investigation of band enhancement and shift. In what follows, the conventional superconducting material, Nb, will be used in our numerical calculations. The transition temperature is 9.2 K and the London penetration length λL ( 0 ) = 83 nm is taken [15]. In the following calculations, the temperature T = 4.2 K is taken. At this temperature, the threshold frequency is calculated to be f λ = ωλ / 2π = 559.7 THz. The calculated transmittance spectra for different thicknesses are shown in Fig. 2. The upper one is plotted at distinct thicknesses of the superconducting layer d2 = 100 nm (blue), 150 nm (green), and 200 nm (red) for a fixed dielectric thickness d3 = 100 nm. For convenience of discussion, we mark the bandedges as 1, 2, 3, 4, and 5 for the case of d2 = 100 nm. It is seen that the bandwidths are apparently enhanced as d2 is decreased for the first two passbands. For the first passband, the enhancement is due to the shift in both the left and right bandedges. The enhancement in the second band is mostly ascribed to the shift in the right bandedge. The third passband is, however, not enhanced but reduced appreciably in addition to a salient shift. To discuss the shift behavior, we define the thickness ratio as ρ = d 2 / d3 . The point 1 indicates the so-called low cutoff frequency (coming from the combined effect of introduction of superconducting material together with the structural periodicity), which reduces to zero frequency for an all-dielectric photonic, i.e., ρ → 0. Thus, the increase in ρ (or in d2) will increase the low cutoff frequency, as indicated in Fig. 2, in which point 1 is shifted to the right as d2 increases. The band gap between points 2 and 3 is called the fundamental band gap and, based on the theory of Bragg reflector, is. inversely. proportional. to. the. average. index. defined. by. n = ( n2 d 2 + n3d3 ) / ( d 2 + d 3 ) ≅ n3 d3 / ( d 2 + d 3 ) , where we have used the fact that. 28.

(29) n2 ≪ n3 . Thus, n is inverse proportional to d2 when d3 is fixed, which in turn the gap between points 2 and 3 should be proportional to d2. As a result, point 2 must be shifted to the left to enlarge the bandgap, as illustrated in Fig. 2.. 1. 2. 3. 4. 5. Fig. 2 The calculated transmittance as a function of the frequency at a fixed thickness of the dielectric layer (upper) and a fixed thickness of the superconducting layer (lower).. If now the thickness of the superconductor layer is keep fixed at 100 nm and then the dielectric thickness d3 is changed as 100 nm (blue), 150 nm (green), and 200 nm (red). The transmittance spectra are depicted in the lower one in Fig. 2. The increase in the dielectric thickness has two strong effects and the spectrum. The transmission bands are not only shifted to left but also reduced in bandwidth as the thickness of dielectric is increased. The shifting behavior is more pronounced for the higher bands. Since the lower band is of practical interest, we can control the pass bandwidth by changing the thickness of the superconducting layer. Moreover, changing the thickness of the dielectric layer has a pronounced shift in the band structure. The roles played by the thickness changes for both superconducting and dielectric layers are thus illustrated. 29.

(30) B. Investigation of coupling effect of higher bands. The results in Fig. 2 suggest that the first passband can be strongly narrowed down when d2 > d3. This can be seen in the top figure of Fig. 3, where we taken d2 = 230 nm and d3 = 60 nm. Then keeping d3 fixed and increasing d2 to 250.4 nm, the first passband is largely suppressed and almost negligible, as illustrated in the middle of Fig. 3. In addition, the gap between the higher second and third bands is substantially reduced. If we continue to increase d2 to 268.6 nm, then the bandgap between the second and third bands is closed up, indicating the two higher bands are coupled as a wider second band. The first band, however, reappears when the coupling effect occurs between the second and third bands. On the other hand, In Fig. 4, we now keep d2 fixed and increase d3 from 45 nm to 56 nm. It is seen that the merging of the higher two bands will happen at a critical thickness of d3 = 50.7 nm. The band merging then disappears again when d3 is larger than this critical thickness. The higher band merging effect due to the thickness change can be qualitatively ascribed to the phase matching between the superconducting and dielectric layers. It can be seen from Figs. 3 and 4 that the higher bands are located at frequency higher than the bulk threshold frequency, which in turn leads to ε2 > 0.. Fig. 3 The calculated transmittance as a function of the frequency at a fixed d3 and three different values in d2.. 30.

(31) The phase matching means that k2d2 = k3d3 → d2 ε 2 = d3 ε 3 . In this case, the stop band (band gap) shrinks to zero and thus the band merging occurs there. The merging behavior cannot be found for the lower band because it is seen that the lower passband is located below threshold frequency. In this region, the permittivity of superconductor is negative. Thus, the condition of phase matching will never be satisfied and therefore the merging effect is not seen. If the matching is broken due the variation in the thickness change, then the gap is opened up again, as illustrated in the bottom of Fig. 4.. Fig. 4 The calculated transmittance as a function of the frequency at a fixed d2 and three different values in d3.. 4. CONCLUSION The. thickness-dependent. photonic. band. structures. in. a. one-dimensional. superconducting photonic crystal have been theoretically investigated. As far as the lower fundamental passband is concerned, the bandwidth enhancement can be controlled by the thickness of the superconducting layer, while the band shift is more pronounced due to the change in the thickness of the dielectric layer. By suitably choosing the thickness of the superconducting or dielectric, it is possible to obtain a single second wider bandwidth due to the merging of the higher second and third bands. REFERENCES. 31.

(32) 1.. Joannopoulos, J. D., R. D. Meade, and J. N. Winn, Photonic Crystals: Molding. the Flow of Light, Princeton University Press, Princeton, NJ, 1995. 2.. Soukoulis, C., Photonic Band Gap Materials, Kluwer Academic, Dordrecht, 1996.. 3.. Yeh, P., Optical Wave in Layered Media, John Wiley & Sons, New York, 1998.. 4.. Fink, Y., J. N. Winn, S. Fan, C. Chen, J. Michael, and J. D. Joannopoulos, “A dielectric omnidirectional reflector,” Science, Vol. 282, 1679-1682, 1998.. 5.. Orfanidis, S. J., Electromagnetic Waves and Antennas, Rutger University, 2008, www.ece.rutgers.edu/~orfanidi/ewa.. 6.. Contopanagos, H., E. Yablonovitch, and N. G. Alexopoulos, “Electromagnetic properties of periodic multilayers of ultrathin metallic films from dc to ultraviolet frequencies”, J. Opt. Soc. Am. A, Vol. 16, 2294-2306, 1999.. 7.. Pei, T.-H., Y.-T. Huang, “A temperature modulation photonic crystal Mach-Zehnder interferometer composed of copper oxide high-temperature superconductor”, J. Appl. Phys., Vol. 101, 084502, 2007.. 8.. Bermann, O. L., Y. E. Lozovik, S. L. Eiderman, and R. D. Coalson, “Superconducting photonic crystals”, Phys. Rev. B, Vol. 74, 092505, 2006.. 9.. Takeda, H., K. Yoshino, “Tunable photonic band schemes in two-dimensional photonic crystals composed of copper oxide high-temperature superconductors”,. Phys. Rev. B, Vol. 67, 245109, 2005. 10. Raymond Ooi, C. H., T. C. Au Yeung, C. H. Kam, and T. K. Lim, “Photonic band gap in a superconductor-dielectric superlattice”, Phys. Rev. B, Vol. 61, 5920-5926, 2000. 11. Wu, C.-J., “Transmission and reflection in a periodic superconductor/dielectric film multilayer structure,” Journal of Electromagnetic Waves and Applications, Vol. 19, 1991-1996, 2006. 12. Van Duzer, T., C. W. Turner, Principles of Superconductive Devices and Circuits, Edward Arnold, London, 1981. 13. Raymond Ooi, C. H., and T. C. Au Yeung, “Polariton gap in a superconductor-dielectric superlattice”, Phys. Lett. A, Vol. 259, 413-419, 1999. 14. Born, M., E. Wolf, Principles of Optics, Cambridge, London, 1999. 15. Fosshein, K., Superconducting Technology: 10 Case Studies, World Scientific, Singapore, 1991.. 32.

(33) 3.2 Angle-Dependent Transmittance Spectra in a One- Dimensional Superconducting Photonic Crystal. 1. Introduction The photonic crystals (PCs), artificially periodic multilayer media consisting of alternating two or more materials with different refractive indices, have attracted much attention over the past decades [1, 2]. The most fundamental characteristic in the PCs is the existence of the forbidden bands or photonic band gaps (PBGs) in some certain frequency regimes. Electromagnetic waves with frequencies falling within PBGs are prohibited to propagate through the PCs. The presence of the PCs has provided many useful and important applications in modern photonic engineering, including the optoelectronics and optical communications [3]. For a simple one-dimensional periodic multilayer structure or one dimensional PC (1DPC), there have been many reports on the interesting and useful optical properties. For instance, an all-dielectric 1DPC is omnidirectional, i.e., it can totally reflect both the transversal electric (TE) and the transversal magnetic (TM) lights at all angles within the PBGs [4-6]. A 1DPC containing metallic films can be used to act as a optical filter operated in distinct frequency regimes because of the highly dispersive behavior in the refractive index of the metal [1,7]. The application of a metallic PC may be inevitable to face the inherent loss issue arising from the metallic extinction coefficient. To remedy this loss problem, it is possible to use the superconducting materials in place of the metals. Indeed, 1DPCs containing the superconductors have been reported [8-13]. In addition to reducing the loss, a superconductor-based PC has some advantages such as its PBG can be tuned because of the temperature-dependent London penetration length.. 33.

(34) In this paper, we investigate the photonic band structure for a superconducting 1DPC as a function of the polarization of the incident wave and the thicknesses of the constituent superconducting and dielectric layers. The photonic band structure is investigated through the frequency-dependent transmittance spectrum calculated by using the transfer matrix method (TMM). The roles play by the thicknesses of superconductor and dielectric layers and by the incident angle will be numerically elucidated.. 2. Basic Equations In Fig. 1, we model the superconducting 1DPC as a superconductor-dielectric periodic structure. Here the space is divided by three regions. The region 1 with refractive index n1 is the free space, the superconductor film with index n2 and thickness d2 occupies the region 2, and the dielectric film with index n3 and thickness. d3 is in the region 3. The lattice period is Λ = d2 + d3 and the number of periods is denoted by N. An electromagnetic wave with unit amplitude impinges obliquely on the plane boundary between layer 1 and layer 2 at an angle of incidence θ. The optical reflectance and transmittance are denoted by R and T, respectively.. Fig. 1 A one-dimensional superconducting/dielectric superlattice, where layer 2 is the superconductor with index n2 and thickness d2, layer 3 is the dielectric with index n3 and thickness d3, and spatial periodicity is Λ = d2 + d3.. The optical transmittance can be calculated by making use of the transfer matrix method (TMM) [14]. To formulate the matrix method for the electromagnetic problem in Fig. 1, we have, in advance, to describe the superconductor refractive index n2. For the time-harmonic fields with the temporal part of exp ( jωt ) , Maxwell’s equations 34.

(35) and the London electrodynamics of superconductors give the following governing wave equation for the electric field [8], ∇ 2ψ + ks2ψ = 0 ,. (1). where ψ may be E or H, the electric field or magnetic field, respectively, and the wave number in the superconductor is given by. k s = ω 2 µ0ε 0 − λL−2 ,. (2). where the temperature-dependent London penetration length is given by. λL (T ) =. λL ( 0 ) 1 − (T Tc ). 4. ,. (3). where λL ( 0 ) is the London penetration length at T = 0 K. Equation (2) then can be used to define the refractive index of a superconductor, i.e., 12.  k  1 , n2 = s = 1 − 2 2  k0  ω µ0ε 0 λL . (4). where k0 = ω µ0ε 0 is the free-space wave number. It can be seen from Eq. (4) that the refractive index is equal to zero when the angular frequency is at the threshold. (. ). frequency of ωth = 1 λL µ0ε 0 . For a bulk superconductor, the electromagnetic wave with frequency higher than ωth can propagate in the superconductor like a usual dielectric because in this case n2 is real. However, when the frequency is less than ωth,. n2 becomes imaginary, and then the wave will be evanescent wave which cannot propagate in it. The threshold frequency thus characterizes the wave properties for a bulk superconductor. Another point is worth mentioning. In obtaining the refractive index in Eq. (4), we neglect the contribution of the normal-fluid because we are interested in the lossless superconductor. For a detailed description on the validity of a lossless superconductor we mention Ref. [8]. 35.

(36) 2. For an N-period superconducting 1DPC, the transmittance is T = t , where t is the transmission coefficient. According to the TMM, the transmission coefficient t is determined by the matrix elements of the characteristic matrix of the total system, i.e., t=. 1 , M11. (5). where M11 is one of the the matrix elements of the total transfer matrix M given by. M M =  11  M 21. M 12  −1 N  = D1 M Λ D1 , M 22 . (6). and MΛ is the transfer matrix of a single period,. M Λ = D2 P2 D2 −1 D3 P3 D3−1 .. (7). The propagation matrix P in Eq. (7) of the individual layer is.  e iφℓ Pℓ =   0. 0  , e − iφℓ . (8). nℓ cos θ ℓ (ℓ = 2,3) .. (9). where. φ ℓ = k ℓx d ℓ =. 2π d ℓ. λ. The dynamical matrix in each layer is defined by. 1  1  Dℓ =  ,  nℓ cosθℓ −nℓ cosθℓ . (10). for the TE wave, and.  cosθℓ cosθℓ  Dℓ =  , −nℓ   nℓ for the TM wave, respectively. The ray angle of the. (11). ℓ th layer is denoted by θ ℓ ,. where ℓ = 1, 2, and 3 with θ1 = θ being the incident angle. The ray angle in layer 2 and 3 can be calculated by the Snell’s law of refraction, i.e., 36.

(37) n1 sin θ1 = n2 sin θ2 = n3 sin θ3 .. (12). 3. Numerical Results and Discussion In the following numerical results the typical superconducting system, niobium (Nb, with Tc = 9.2 K and λ0 = 83 nm), is taken as the superconducting layer and the operating temperature at 4.2 K is used. In addition, the dielectric layer with a relative permittivity of 10 and N = 200 are used. In Fig. 2 we plot the frequency response of the transmittance for the TE wave under three distinct angles of incidence 150, 300, and 450, respectively. In the upper figure we take d2 = 160 nm and d3 = 80 nm. It is seen that the first passband is weakly dependent on the incident angle. However, the second and third passbands are very sensitive to the incident angle, moving to the higher frequency regions as the incident angle increases. Similar results can also be seen when the thicknesses of two layers are interchanged, as illustrated in the lower figure of Fig. 2. Another feature in Fig.2 is that the more passbands are seen when the dielectric layer is thicker than the superconducting layer. The results in Fig. 2 indicate that the number of passbands can be primarily controlled by the thickness of the dielectric layer.. Fig. 2 Calculated frequency-dependent transmittance for the TE wave at three different angles of incidence, where the upper one is for d2 = 160 nm and d3 = 80 nm, and the lower one is for d2 = 80 nm and d3 = 160 nm, respectively.. 37.

(38) Figure 3 depicts the transmittance for the TM wave for the same conditions in Fig. 2. Again, higher passbands are sensitive to the incident angle. The vertical dashed line marks the threshold frequency of 560 GHz. For the TM wave, it is noted that there exist narrow passbands between 560-600 GHz for the oblique incidence, which actually are not present in the cases of the normal incidence and the TE wave as well. Such narrow passbands called the superpolariton bands come from the existence of the normal component of the electric field in the TM wave. The normal component of the electric field causes the superelectrons to be polarized and thus polaritons are formed.. Fig. 3 Calculated frequency-dependent transmittance for the TM wave at three different angles of incidence, where the upper one is for d2 = 160 nm and d3 = 80 nm, and the lower one is for d2 = 160 nm and d3 = 80 nm, respectively.. In Fig. 4, the calculated transmittance is limited to the case where the superconducting layer is thicker than the dielectric layer. We take d2 = 160 nm, and d3 is varied as 80 nm (top), 93 nm (middle), and 108 nm (bottom), respectively. In the top figure, at normal incidence (the red curve), the second passband is located at frequency higher than the threshold frequency (indicated by the vertical dashed line). 38.

(39) It then splits into two bands at oblique incidence, a narrow band (near 600 THz) and a wide band (See the black and blue curves). The gap between these two split bands is largely increased with increasing the incident angle. In addition, both these two bands remain higher than the threshold frequency. If now the dielectric thickness is increased to 93 nm (middle), then the threshold frequency can be arranged to fall near the center of the second band at 00. The second band is again split into two bands at 300 and 450. However, the split two bands are separated by the threshold frequency. If the dielectric thickness is further increased to 108 nm, then the second band falls below the threshold frequency at 00. In this case, the second band at 300 or 450 will also be split into two bands separated by the threshold frequency. The results shown in Fig. 4 reveal that the band shift is mainly controlled by the change in the thickness of the dielectric layer. Another important feature in the oblique incidence is that the threshold frequency must be located within the photonic forbidden band, as marked by the arrows.. Fig. 4 Calculated frequency-dependent transmittance for TM wave at three different dielectric thicknesses, d3 = 80, 93, and 108 nm and superconductor layer has a fixed thickness of d2 = 160 nm. Each figure includes three different angles of incidence.. 39.

(40) Fig. 5 Calculated frequency-dependent transmittance for TM wave at five different dielectric thicknesses, d3 = 70, 78, 87, 97, and 106 nm and superconductor layer has a fixed thickness of d2 = 160 nm. The angle of incidence is fixed at 450.. In Fig. 5, taking a fixed angle of incidence at 450 we plot the transmittance at various dielectric thicknesses d3 = 70, 78, 87, 97, and 105 nm for a fixed thickness of superconductor d2 = 160 nm. It is seen that the second passband will be shifted to the left when d3 increases. Meanwhile, its bandwidth is also narrowed down substantially. It is of interest to note that this passband disappears at d3 = 87 nm. The disappearance is due to the fact that at this thickness the passband almost shrunk to a single frequency just at the threshold frequency. Because the threshold frequency must be within the forbidden band, the passband at this frequency is thus smeared out and disappears. If we continue to increase d3 over 87 nm, then the passband reappears in the left side of the threshold frequency and its bandwidth is increased as the d3 increases. Conclusively, we can find a critical thickness d3 = 87 nm such that a much wider stop band can be obtained. That is, a wider superconducting Bragg reflector ranging from 240 – 750 THz is achievable by controlling the dielectric thickness in such a superconducting photonic crystal, as depicted in the middle of Fig. 5. This region obviously includes all the visible frequencies. 40.

(41) Fig. 6 Calculated frequency-dependent transmittance for the TE and TM waves at three different superconductor thicknesses, d2 = 100, 150, and 200 nm and dielectric layer has a fixed thickness of d3 = 100 nm. The angle of incidence is fixed at 300.. In Fig. 6, we investigate the effect due to the change in the superconductor thickness. Here the incident angle is fixed at 300 and dielectric thickness d3 = 100 nm is taken. For the TE wave, the passbands are enhanced as the superconductor thickness decreases. However, the band center for the lowest passaband remains unchanged, whereas the shift in the other passbands is apparently seen. Similar behaviors are also seen in the TM wave. Thus, in order to have a wider passbands, it is preferable to use a thinner superconductor in a superconducting photonic crystal.. 4. Conclusion The angle- and thickness-dependent photonic band structures in a superconducting 1DPC have been theoretically investigated. Numerical results illustrate that the bandwidth of the lowest passband can be enhanced by using a thinner superconducting film. The band shift and numbers can be primarily controlled by the thickness of the dielectric layer. As for the angular dependence, it is found that, in the TM wave, the passband near the threshold frequency will be split into two bands with 41.

(42) a gap being controlled by the angle of incidence. The positions of the two split passbands are also proven to be adjustable by the variation of the thickness of the dielectric layer. REFERENCES. [1] J. D. Joannopoulos, R. Meade, and J. Winn, Photonic Crystals, Princeton University Press, Princeton, 1995. [2] E. Yablonovitch, Phys. Rev. Lett. 58, 2059-2062, 1987. [3] C. M. Soukoulis, Photonic Band Gap Materials, Kluwer Academic, Dordrecht, 1996. [4] Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science, 282, 1679-1682, 1998. [5] D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, Appl. Phys. A: Mater. Sci. Precess., 68, 25-28, 1999. [6] E. Yablonovitch, J. Opt. Soc. Amer. B, 10, 283-295, 1993. [7] H. Contopanagos, E. Yablonovitch, and N. G. Alexopoulos, J. Opt. Soc. Am. A, Vol. 16, 2294-2306, 1999. [8] C. H. Raymond Ooi, and T. C. Au Yeung, Phys. Lett. A, 259, 413-419, 1999. [9] C. H. Raymond Ooi, T. C. Au Yeung, C. H. Kam, and T. K. Lim, Phys. Rev. B, 61, 5920-5926, 2000. [10] C.-J. Wu, M.-S. Chen, and T.-J. Yang, Physica C 432, 133-138, 2005. [11] C.-J. Wu, J. Electromagn. Waves Appl., 19, 1991-1996, 2006. [12] O. L. Berman, Y. E. Lozovik, S. L. Eiderman, and R. D. Coalson, Phys. Rev. B 74, 092505, 2006. [13] Arafa H. Aly, H.-T. Hsu, T.-J. Yang, C.-J. Wu, C. K. Hwangbo, J. Appl. Phys. 105, 083917, 2009. [14] P. Yeh, Optical Waves in Layered Media, John Wiley & Sons, New York, 1998. 42.

(43) Chapter 4 Two-Dimensional Photonic Crystals---Using FDTD 4.1 Introduction to FDTD Electromagnetic fields can be implemented by experiment or can be calculated by computer numerical simulation. By modeling of electromagnetic processes on the computer, we can rapidly predict the work using the computer-aided design (CAD). In Maxwell’s equations, we could see the electromagnetic field is time-dependent and it is appropriate to deal a transient problem with the homogeneous finite difference time domain method (FDTD). The FDTD method is now popular in the numerical calculation for the electromagnetic problem. This method encloses the divided domains of interest in an interlocking lattice of cubes. On each cube, the electric and magnetic field components have their own values at corresponding particular points. The FDTD method is held by the general class of grid-based differential time-domain numerical modeling methods [25]. Not like in many electromagnetic simulation techniques applied in the frequency-domain, FDTD solves Maxwell’s equations in the time-domain with discrete central-difference approximations to the space and time partial derivatives. The resulting finite-difference equations are progressed at discrete steps and the electric field vector components in a volume of space are solved at a given instant in time; then the corresponding magnetic fields are solved at the next instant [26]. The process goes on until the wanted temporary resonance or the electromagnetic field behavior is grown into steady-state fully. In the FDTD approach, both space and time are divided into discrete segments, the algorithm is traced back to a paper by Kane Yee in IEEE Transactions on 43.

(44) Antennas and Propagation (1966). Space is segmented into box-shaped “cells”, which are small compared to the wavelength. The descriptor "Finite-difference time-domain" and its corresponding "FDTD" acronym were originated by Allen Taflove in a 1980 paper in IEEE Transactions on Electromagnetic Compatibility . Time is quantized into many small time-steps, and the step represents the time of EM wave traveling the field from one cell to the next. Gotten the offset of the magnetic fields in space from the coordinating electric fields, the field with respect to time is offset, too. The values of the electric and magnetic fields are constantly updated using a leapfrog scheme where the electric fields are computed firstly, then the magnetic are next in time at following each time-step.. 4.2 FDTD method From the statements of Maxwell's differential equations, one can see that the change in the E-field in time as the time derivative is dependent on the change in the H-field cross space by the curl operator. This incurs the basic FDTD time-stepping relation at any point in space, then the updated value of the E-field in time is dependent on the stored value of the E-field and the numerical curl of the local distribution of the H-field in space. The H-field is time-stepped in a similar manner. At any point in space, the updated value of the H-field in time is dependent on the stored value of the H-field and the numerical curl of the local distribution of the E-field in space. The cell size, the dimensions of the box, with respect to the discrete distribution of the E-field or H-field in space, is the most important. The constraint in the FDTD simulation for it determines not only the step size in time, but also the exact calculation range of upper limit in frequency. In general, we set the minimum resolution at ten cells per wavelength, and thus we can get the upper frequency limit. If an input source be brought into the FDTD simulation by supporting a sampled 44.

(45) waveform to the field, the program will update equation at every location. Iterating the E-field and H-field updates results in a marching-in-time process wherein sampled-data analogs of the continuous electromagnetic waves under consideration propagate in a numerical grid stored in the computer memory. At each step in time, the value of the information of the wave pass that time period is overlapped into the field value. The surrounding fields will propagate the induced waveform throughout the FDTD grid appropriately, depending on the features of each cell in the structure. The iterating must continue until a state of convergence has been approached. This typically expresses that every field value has decayed to nearly zero or a steady-state condition has been arrived. A geometrical structure can be formed within the FDTD grid such as 1-D, 2-D, and 3-D models, these descriptions are able to holds true for FDTD techniques. When multiple dimensions are considered, evaluation of the numerical curl will become complex. According to Kane Yee's paper, the spatially existing vector components of the E-field and H-field within rectangular unit cells of a Cartesian computational grid can be explain that each E-field vector component is located midway between a pair of H-field vector components, and contrariwise. About any presented calculation of the geometry belonging to the structure being simulated, FDTD will set the cell edges at specific locations to the given material. All of the simulated geometry space is generally called the grid or the mesh, and a three dimensional block of these cells is formed. This scheme, as known as a Yee lattice, has proven to be very robust, and remains at the core of many current FDTD software constructs (Yee 1966) [27].. 45.

(46) n+. Hx. 1 2. (i , j + 12 , k + 12 ) n+. 1. H y 2 (i + 12 , j , k + 12 ). Ezn (i , j , k + 12 ). n+. Hz. E yn (i , j + 12 , k ). 1 2. (i + 12 , j + 12 , k ). Exn (i + 12 , j , k ) Illustration of a standard Cartesian Yee cell used for FDTD, where the electric and magnetic field vector components are pointed out. (Yee 1966). The electric field components are marked on the edges of the cube, and the magnetic field components on the normals to the faces of the cube. A three-dimensional space lattice is comprised of a multiplicity of such Yee cells. [From: Implementation of an Electromagnetic Perfectly Matched Layer in Head Model using FDTD Terapass Jariyanorawiss* and Nuttaka Homsup**]. Here, Yee presented a leapfrog scheme which processes in time wherein the E-field updates are conducted midway during each time-step between successive H-field updates, and conversely (Yee 1966). By Maxwell’s equations in the source-free region: ∂E , ∂H ∇ × H = σE + ε ∇× E = µ ∂t ∂t. ⇒ σE x + ε ∂E x = ∂H z − ∂H y , σE y + ε ∂t. µ. ∂y. ∂z. ∂E y ∂t. =. ∂H y ∂H x ∂H x ∂H z , ∂E , − σE z + ε z = − ∂z ∂x ∂t ∂x ∂y. ∂H ∂E ∂E ∂H z ∂E x ∂E y ∂H x ∂E y ∂E z , µ y = z− x, µ = − = − ∂t ∂x ∂z ∂t ∂y ∂x ∂t ∂z ∂y. By defining the following notations:. E(i△x,j△y,k△z,n△t)≡E(i,j,k,n) , so. ∂E x ≈ ∂t. E x (i +. 1 1 , j , k , n + 1) − E x (i + , j, k , n) 2 2 , ∆t. 1 1 1 1 E y (i, j + , k , n + 1) − E y (i, j + , k , n) E z (i, j , k + , n + 1) − E z (i, j , k + , n) ∂E z 2 2 2 2 , , ≈ ≈ ∂t ∆t ∂t ∆t. ∂E y. and 46.

(47) ∂E x ≈ ∂y. E x (i +. 1 1 1 1 E x (i + , j, k , n) − E x (i + , j, k − 1, n) , j , k , n ) − E x (i + , j − 1, k , n ) ∂E 2 2 2 2 , x ≈ , ∂z ∆z ∆y. 1 1 1 1 E y (i, j + , k , n) − E y (i − 1, j + , k , n) ∂E E y (i, j + , k , n) − E y (i, j + , k − 1, n) y 2 2 2 2 , , ≈ ≈ ∂x ∆x ∂z ∆z. ∂E y. ∂E z ≈ ∂x. 1 1 1 1 E z (i, j , k + , n) − E z (i − 1, j, k + , n) E z (i, j , k + , n ) − E z (i, j − 1, k + , n) E ∂ 2 2 z 2 2 , , ≈ ∆x ∂y ∆y. H(i△x,j△y,k△z,n△t)≡H(i,j,k,n) , so. ∂H x ≈ ∂t. 1 1 1 1 1 1 H x (i, j + , k + , n + ) − H x (i, j + , k + , n − ) 2 2 2 2 2 2 , ∆t. 1 1 1 1 1 1 1 1 1 1 1 1 Hy (i + , j, k + , n + ) − H y (i + , j, k + , n − ) H z (i + , j + , k, n + ) − Hz (i + , j + , k, n − ) ∂ H 2 2 2 2 2 2 , z ≈ 2 2 2 2 2 2 , ≈ ∂t ∆t ∂t ∆t. ∂Hy. and ∂H x ≈ ∂y. 1 1 1 1 1 1 1 1 1 1 1 1 H x (i, j + , k + , n − ) − H x (i, j + , k − , n − ) Hx (i, j + , k + , n − ) − H x (i, j − , k + , n − ) ∂H 2 2 2 2 2 2 , 2 2 2 2 2 2 , x≈ ∂z ∆z ∆y. 1 1 1 1 1 1 1 1 1 1 1 1 Hy (i + , j, k + , n − ) − Hy (i − , j, k + , n − ) ∂H Hy (i + , j, k + , n − ) − Hy (i + , j, k − , n − ) y 2 2 2 2 2 2 2 2 2 2 2 2 , , ≈ ≈ ∂x ∆x ∂z ∆z. ∂Hy. ∂Hz ≈ ∂x. 1 1 1 1 1 1 1 1 1 1 1 1 Hz (i + , j + , k, n − ) − Hz (i − , j + , k, n − ) H (i + , j + , k, n − ) − H z (i + , j − , k, n − ) 2 2 2 2 2 2 , ∂H z ≈ z 2 2 2 2 2 2 , ∆x ∂y ∆y. then 1 1 2ε (i + , j, k ) − σ (i + , j , k )∆t 1 2∆t 2 2 E .E xn (i + , j , k ) + ⋅ 1 1 1 1 2 2ε (i + , j , k ) + σ (i + , j, k )∆t 2ε (i + , j , k ) + σ (i + , j, k )∆t , 2 2 2 2 1 1 1 n− n− n−  n − 12  1 1 1 1 1 1 1 1  H z (i + , j + , k ) − H z 2 (i + , j − , k ) H y 2 (i + , j, k + ) − H y 2 (i + , j, k − )  2 2 2 2 2 2 2 2   − ∆y ∆z     1 1 2ε (i, j + , k ) − σ (i, j + , k )∆t 1 1 2∆t n +1 2 2 E y (i, j + , k ) = .E yn (i, j + , k ) + ⋅ 1 1 1 1 2 2 2ε (i, j + , k ) + σ (i, j + , k )∆t 2ε (i, j + , k ) + σ (i, j + , k )∆t , 2 2 2 2 1 1 1  n − 12 n − n − n −  1 1 1 1 1 1 1 1  H x (i, j + , k + ) − H x 2 (i, j + , k − ) H z 2 (i + , j + , k ) − H z 2 (i − , j + , k )  2 2 2 2 − 2 2 2 2   ∆z ∆x     n +1 x. 1 (i + , j, k ) = 2. 47.