Preface of this dissertation

In the dissertation, we present the studies of the optical properties of periodic multilayers structures, which include the one-dimension superlattices and the annular Bragg reflectors as well. The materials we investigate for periodic multilayers comprise superconductors, single-negative metamaterials and double-negative metamaterials.

The Chapter 2 depicts basic theory of this dissertation. The PBGs in the superconductor-dielectric superlattices are described in Chapter 3. Optical properties of the SABR are reported in Chapter 4. We propose the photonic band structure of the annular periodic multilayer structure containing the single-negative materials in Chapter 5. Moreover, we discuss the wave properties of the annular periodic multilayer structure with the double-negative materials in Chapter 6. Finally, a conclusion of the results will be included in Chapter 7.

Fig. 1.1. The photonic crystal structures.

Fig. 1.2. An illustration of a multilayered structure with circular cylindrical symmetry.

The figure is imaged from [11]

Chapter 2 Basic theory

2.1 Abeles theory

In order to calculate the transmittance and reflectance for a periodic multilayered structure, the elegant Abeles theorywill be employed [7]. According to this theory, we must, in advance, set up the characteristic matrix corresponding to one period, with the result

cos cos sin sin cos sin sin cos

sin cos cos sin cos cos sin sin

p j j determined by Snell’s law of refraction, are the ray angles in layer 2 and 3, respectively.

Having constructed the matrix in Eq. (2.1.1), the total characteristic matrix for an N-period structure can be obtained, that is

( )

^{11 12}

( )

( ) ( ) ( )

and U are the Chebyshev polynomials of the second kind defined by _N

( ) ( )

¹

Equation (2.1.4) gives the explicit expressions for matrix elementsM , ₁₁ M , ₁₂ M , and ₂₁ M as follows: 22

The reflection and transmission coefficients can be determined and are given by [7]

( ) ( )

for the last medium. Both media here are taken to be free space. The reflectance (reflectivity) R, transmittance (transmissivity) T and r , t are related by

R=  , r2 ² 1

T p t

= p^A  . (2.1.10)

Thus, the transmittance spectrum, T versus ω, can be numerically illustrated, as will be seen in Section III.

2.2 The Floquet (or Bloch) theorem

In the solid state physics, the wave function of a electron which has mass m will satisfied the Schrödinger equation

is lattice transfer vector, then

( ) ( ).

V rG =V rG+RG

(2.2.2) According Bloch theory [34], the solution of Eq. (2.2.1) is

( ) ( ) exp( ),

This theory is also called Floquet theory, because Floquet is the first person to deduce the theory for one-dimensional case.

For the photonic crystal structure, we use Floquet theory to do Furior expantion for the function u. We get that

where ₀ 2

xn x

k k n d

= + π , (2.2.6)

where d is periodic length of the lattice.

Based on the basic assumption of translational symmetry and aided by the Floquet (or Bloch) theorem together with the use of transfer matrix method, one can obtain a transcendental equation determining the band structure, namely [5]

( ) (

2

) (

3

) (

2

) (

3

)

where K is the Bloch wave number,

k_{x r3} sin² ₁ Equation (2.2.7) can be numerically solved for ω as a function of K, yielding the so-called photonic band structure or dispersion relation.

2.3 Theory of annular Bragg reflectors

The structure of a SABR is shown in Fig. 1.2, in which the inner core region has a refractive index of n₀ and a starting radius of ρ0, the layer 1 with index n₁ is assumed to be the superconductor, and layer 2 having index n₂ is the dielectric layer. In addition, the index of refraction of the outer region is denoted by nf. To calculate the reflectance at the first circular boundary, ρ = ρ0, we use the transfer matrix method in the cylindrical waves [11]. The cylindrical wave is assumed to be diverging from the axis of symmetry, ρ = , 0 and then impinges normally on the first circular interface of ρ = ρ0.

Assuming an ^{exp j t}

(

^ω

)

time dependence for the electromagnetic fields, the

source-free two curl Maxwell’s equations are given by

In the case that the propagation of cylindrical wave diverging from or converging to the axis of symmetry ρ= , the derivatives of the fields with 0 respect to z vanish and Eq.(2.3.4) can be reduced to

1 E^z ,

In the circular cylindrical coordinates there are two possible modes, i.e., TE and TM modes. For TE wave, the nonzero fields, E , H_{z φ}, and H_ρin each single layer satisfy the three equations, Eqs. (2.3.7a), (2.3.7b) and (2.3.8c). Solutions for Eqs. (2.3.7c), (2.3.8a) and (2.3.8b) can be obtain for TM wave, which has non-zero components

H , Ez _φ and E_ρ.

With Eq. (2.3.3), the governing equation for tangential electric field Ez is given by

The solution of Eq. (2.3.11) can be obtained the method of separation of variables.

Substituting Ez

(

ρ φ^,

)

=V

( ) ( )

ρ Φ φ in Eq. (2.3.7) and using Eq. (2.3.11) we obtain:

where m is a positive or negative integer or zero, called the azimuthal number. Then, we obtain:

which is the Bessel’s differential equation with the solution

where p= ε µ is the intrinsic admittance of the layer, and the primes represent differentiation by the whole argument of the function (not just by ρ).

From Eqs. (2.3.16b) and (2.3.16c) we see that V and U determine the magnetic field components H_ρ and H_φ, respectively. Equations (2.3.8) and (2.3.9) enable us to construct a single layer matrix relating the electric and magnetic fields at its two interfaces. For instance, the matrix for the first layer (with refractive index n and ₁ interfaces at ρ ρ= ₀ and ρ ) is written as [11] ₁ The element of transfer matrix can be found by considering the relations Eqs.

(2.3.20)-(2.3.23) when the vector

(

V

( ) ( )

ρ0 ,U ρ0

)

has the special values

( )

^{1, 0 and}

( )

^0,1 . Solving the equation with the help of the identity

( ) ( ) ( ) ( )

^{2 /}

m m m m

J x Y ^′ x −J′ x Y x = πx, (2.3.24) thus the single layer matrix

11 12 by the ratio of the initial and final radii:

11 12 0 Obviously, the matrix elements are dependent on the radii of the two interfaces.

Similarly, for ith layer the matrix can be obtained by some simple replacements, i.e.,

0 i 1

ρ →ρ₋ , ρ₁→ρ_i , k₁→ =k_i ω µ ε_{i i} , and p₁→ p_i = ε µ_{i i} . In addition, with

structure being periodic, one has

ε ε

_i = ₁ if i=odd, and ε_i = if i = even. For an ε₂ N-period bilayer periodic reflector we have, in total, 2N layers and therefore there should be 2N matrices in order to set up the total system matrix M that relates the first and final interfaces as

11 12

Unlike 1DPC, the analytic expressions for the matrix elements of M for an annular BR cannot be obtained because the elements of each single layer matrix are functions of the radii of the two interfaces. It thus has to be numerically calculated.

Consider an outgoing wave incident on the interface between 0 and 1, which we take to have radius ρ ρ= ₀, and propagating to the medium f , which extends from ρ ρ= _f to

ρ= ∞ . The amplitudes of the electric field and magnetic fields at ρ₀ and ρ_f can be written in terms of the amplitude reflection and transmission coefficients r and _d t and _d are related by the transfer matrix M defined in Eq.(2.3.28) and subsequence discussion:

( )2 ( )1 ¹ ( )²

Equation (2.3.30) enables us to calculate the reflection and transmission coefficients for multilayered structure,

( ) ( ) ( ) Hence, the reflection coefficients is given by

(

( )

)

^{( )}

(

^{( )}

)

Thus the transmission coefficient is given by

( )²

( )

^{( )1}

( ) (

^{( )1 0 0}

)

^{( )2}

(

^{( )1}

)

(2.3.36) and (2.3.39) then leads to the reflectance R and the transmittance T, i.e.,

2 where n and ₀ n are respectively the refractive indices of the starting and the final _f

media. The results for TM wave are also obtainable by simply replacing ε ↔ , and µ j↔ − in the formulas of TE wave. j

2.4 The two-fluid model of superconductors

In 1934, Gorter and Casimir provided the two-fluid model for the electric conduction mechanism of superconductors [12,13]. When T >0 K, the two-fluid model can very successfully explain the character of superconductor device under the

condition of superconductivity conduction. According to this theory, one assumes that the total electron density n can be divided into two kinds of conduction electrons, one of which is normal electrons n and another is superconducting electrons _n n . That is _s

n=n_n+ , (2.4.1) n_s and they have different conductivity

σ σ= _n+σ_s , (2.4.2) where σ_n is the conductivity of normal electrons, and σ_s is the conductivity of paired superconductivity electrons. From the Drude model and London equation, we get

( )

If the relaxation time τ → ∞, the conductivity is a complex number, so under the condition of f ≠ , we have 0 In order to satisfy the approximation of the conductivity imaginary part, we assumed the condition is low frequency, so

Therefore according to the two-fluid model the electromagnetic response of a

superconductor can be described in terms of the complex conductivity, σ σ= ₁− jσ₂, where the real part, σ₁, indicating the loss, is contributed by the normal electrons, whereas the imaginary part, σ₂, is due to the superelectrons.

Under the low temperature condition,the imaginary part is expressed as [12,13]

_{2 2}

0

1

L

σ ⁼ωµ λ , (2.4.7)

where the temperature-dependent penetration depth is given by

( )

We shall consider the lossless case, meaning that the real part of the complex conductivity of the superconductor can be neglected and consequently it becomes

2 2 The conditions for a lossless superconductor are well described in Ref. [10,11]. With Eq.

(2.4.4), the relative permittivity as well as its associated index of refraction can be obtained, namely

Chapter 3

Photonic band structure for a

superconductor-dielectric superlattice

3.1 Introduction

It is well known that photonic crystals have photonic band gaps (PBGs) in the photonic dispersion relation. In the PBGs, optical waves with certain frequencies are not allowed to propagate through the crystal [1, 2].The PBGs are analogous to the electronic band gaps in a solid and their physical origin can be ascribed to the Bragg diffraction in a periodic multilayer structure. A simple one-dimensional photonic crystal is, in general, made of alternating layers of material with different permittivities, forming a superlattice with infinite periods. The band structure for a dielectric-dielectric photonic crystal shows that the PBG between the first and second bands widens considerably as the difference in dielectric permittivity is increased [3]. In addition, no low-frequency band gap below the first (lowest) band can be found. In a metallic photonic made of a normal metal and a dielectric, it is however found that a low-frequency (or metallicity) gap may exist.

Contrary to a PBG, this metallicity gap which does not depend on the periodicity, is of the order of the plasma frequency and thus is regarded as a modified effective plasma frequency [35-37].

On the other hand, studies of photonic crystals consisting of a superconducting material and a dielectric have also been reported recently [10-12].The electromagnetic properties of Abrkosov vortex lattice as a photonic crystal were investigated by changing the Ginzburg-Landau parameter and static magnetic field [18]. In addition to a low-frequency band gap below the first band, they also obtained the PBGs for a

superconductor in the presence of vortices. In fact, the issue of a superconducting photonic crystal was first investigated by a group in Singapore [19,20]. They considered a one-dimensional superconductor-dielectric superlattcie. By making use of the transfer matrix method accompanied by the Bloch theorem [19], a low-frequency band gap was seen for both transversal magnetic (TM) and transversal electric (TE) modes. This band gap was found to be about one third of the threshold frequency of a bulk superconducting material. The physical information from this work for TE mode however is quite limited because only the first band is given. As for the other higher bands in addition to the possible PBGs cannot be obtained there. In other words, a full band structure for this one-dimensional superconducting photonic crystal remains unavailable thus far.

A full band structure is a basic and important means for understanding the fundamental physics about electromagnetic wave propagation characteristics in a photonic

crystal. This information is not only of fundamental but also of technical use for a superconducting material. Motivated by this, in this dissertation we shall extend the

work of Ref. [20]. We would like to present the full photonic band structure for TE mode in a superconductor-dielectric photonic crystal. Firstly, we use the Abeles theory for a stratified media to calculate the frequency-dependent transmittance [7]. From the transmittance spectrum, we can clearly learn the locations of all possible pass bands and stop bands. With these in hand, one is able to calculate the band structure from the transcendental equation based on the transfer matrix method together with the Bloch theorem. Then a comparison between the transmittance spectrum and full band structure will be made.

The format of this work is as follows: Section II describes the theoretical approaches to be used in the calculation. The calculated transmittance spectrum and band structure will be given in Section III. Discussion on the PGBs will also be made in Section III. A summary will be addressed in Section 4.

3.2 Theory

A one-dimensional nonmagnetic superconductor-dielectric photonic crystal will be modeled as a periodic superconductor-dielectric multilayer structure with a large number of periods, N >>1. Such an N-period superlattice is shown in Fig. 3.1, where

2 3

a=a + is the spatial periodicity, wherea a₂ is the thickness of the superconducting layer and a denotes the thickness of the dielectric layer. We consider that a TE wave is ₃ incident at an angle θ₁ from the top medium which is taken to be free space with a refractive index, n₁= . The index of refraction of the lossless dielectric is given by 1

3 r3

n = ε , whereε_r3 is its relative permittivity. For the superconductor, the index of refraction can be described on the basis of the conventional two-fluid model [20].

According to the two-fluid model the relative permittivity as well as its associated index of refraction can be obtained, namely According to the Abeles theory, the reflection and transmission coefficients can be determined and are given by¹¹

for the last medium. Both media here are taken to be free space. The reflectance (reflectivity) R, transmittance (transmissivity) T and r , t are related by

R=  , r2 ² 1

T p t

= p^A  . (3.5)

Next, we are going to briefly describe the method used in Ref. [20] for a direct calculation of the band structure in a periodic superconductor-dielectric medium. Based on the basic assumption of translational symmetry and aided by the Floquet (or Bloch) theorem together with the use of transfer matrix method, one can obtain a transcendental equation determining the band structure, namely [8,20]

where K is the Bloch wave number,

k_{x r3} sin² ₁

Equation (3.17) can be numerically solved for ω as a function of K, yielding the so-called photonic band structure or dispersion relation. In Ref. [20], only the first band is given and thus it is not sufficient to explore the whole optical properties in a photonic crystal. In the next section, we shall give other possible higher bands. The higher bands then enable us to study the PGBs.

Before presenting the numerical results we mention that the above theoretical formulations are based on the flat interface model. This is legitimate and widely used to theoretically study the fundamental optical properties in a photonic crystal [3, 19, 20]. In the actual material, some interface issues such as interface roughness, lattice imperfection, and surface discontinuity may arise due to the process of a film growth. To

study surface effect on a photonic crystal, other method such as the plane-wave expansion may be employed and some works are available [38]. A study of interface effect on the photonic crystal is not our interest here.

3.3 Numerical Results and Discussion

3.3.1 Superconductor-dielectric superlattice

Let us now present the numerical results according to the aforementioned equations. Two dimensionless quantities such as Ω ω= a 2πc and Λ=a 2πλ_L will be used as usual in the analysis of photonic bands. We also define the dielectric thickness ratio as r=a a₃ . Figure 3.2 displays the calculated transmittance spectrum (right) and the band structure (left) for the conditions of θ₁=45⁰, ε_r3 =15, Λ=0.05, r=1 3, and N =500. It is seen that the calculated transmission spectrum is in fairly good agreement with that of the calculated band structure. For the sake of convenience, the first six cutoff frequencies (at which T = 0) are denoted by Ω_i, 1, 2, 3 ~ 6i= , as shown in Fig. 3.2. The first band gap, denoted by ∆₁, is equal to Ω₁ =0.017. The first band gap is referred to as the low-frequency (LF) gap [20], which is not seen in the dielectric-dielectric superlattice. This gap size is nearly equal to one third of the cutoff frequency Ω_c for a bulk superconductor which is in value of 0.05 here. Thus, its origin can be regarded as a combined effect of the spatial periodicity and of the addition of dielectric material [20]. The dimensionless bulk cutoff frequency Ω_c =0.05 is equal to a real frequency of ω_c =c λ_L ~ 10¹⁵ s^-1, which is of the same order of plasma frequency for most alkali metals. The dispersion relation for a bulk superconductor is thus recognized as an analogy to the plasma dispersion in metals.⁸ On the other hand, ∆₁ also appears in a metallic photonic crystal but its size is near the plasma cutoff frequency, meaning that it does not depend on the periodicity [18].Thus in the metallic superlattice

∆1 is not a real PBG, whereas it is a true PBG in the superconductor superlattice because

∆1 is indeed related to the periodicity.

In addition to ∆₁, along with the first band from Ω₁=0.017 to Ω₂ =0.165, other higher bands as well as PBGs are also displayed in Fig. 2. The second PBG is denoted by ∆₂ equal to Ω Ω₃− ₂ =0.368 0.165− =0.203 . That is almost twelve times larger than ∆₁. The second band is located from Ω₃=0.368 to Ω₄ =0.455. The third PBG, ∆₃, is Ω Ω₅− ₄ =0.712 0.455− =0.257 in magnitude and is greater than ∆₂ appreciably. The third band is then above Ω₅ and under Ω₆. From the results in Fig. 2 we can deduce that the photonic band structure for a one-dimensional superconducting photonic crystal is quite reminscent of the electronic band structure. Moreover, it has multiple PGBs, instead of having just one lowest band gap as reported in Ref. [20].

Figure 3.3 shows the first five cutoff frequencies and PBGs as a function of penetration depth at the conditions of θ =45⁰, ε_r3 =15, r=1 3, and N=500. The first one, Ω₁, being equal to ∆₁, increases with increasing Λ. The dependence of ∆₂ on Λ is similar to that of ∆₃. Both also increase as Λ increases. The variations in ∆₂ and ∆₃, however, are not as salinet as ∆₁, especially at small values of Λ. Figure 3.4 shows the calculated frequencies and PBGs as a function of angle of incidence at

3 15

εr = , Λ=0.05, r=1 3, and N =500. It is seen that gap ∆₁ essentially does not change with the variation in the angle of incidence, indicating an omnidirectional feature.

In addition, ∆₂ changes slightly as a function of angle of incidence. The change in the third gap size, ∆₃ is appreciable for θ₁ smaller than 20⁰ and becomes nearly linear between 20⁰ and 60⁰. It then approaches a saturation value of about 0.32. In Fig. 3.5, we have plotted the cutoff frequencies versus dielectic constant of dielectric layer for

450

θ = , Λ=0.05, r=1 3, and N =500. All the cutoff frequencies, in general, decrease with increasing dielectric constant. The corresponding first three gap sizes are

depicted in Fig. 3.6, where ∆₁ decreases slowly with increasing dielectric constant. A peak vaule in ∆₂ is attained for ε_r3 = , and then decreases as the dielectric constant 5 increases. As for ∆₃, it also attains a maximum when ε_r3 =10, and ∆₃ is equal to ∆₁ for ε_r3 = . 3

In the above numerical results, the calculated frequency for a superconductor photonic crystal (SPC) is normalized in 1 λ₀ , the sole material parameter of a superconductor involved in the formulation. This indicates that the results are valid for all the possible superconductors described by the two-fluid model [20]. Most high-T_c cuprates have a value of λ₀ ≈200 300− nm, corresponding to infrared region. As for the conventional superconductor such as a typical A15 compound superconductor with Tc above 10 K, λ₀ ≈60 90− nm, it then can work in the yellow to violet region. The feasibility of a SPC has been well discussed by Feng et al [39].

3.3.2 Extraordinary optical properties in near-zero-permittivity operation range

Let us investigate the reflectance in the vicinity of superconductor threshold wavelength at various angle of incidence. To calculate the reflection response, the layer 1 is taken to be the typical high-T_c superconductor, YBa₂Cu₃O₇ (YBCO) with T_c = 92 K and λ₀ =140 nm [40], and the layer 2 is MgO with ε_2r =10 . The operating temperature is T = 77 K in our simulation. The penetration depth λ_L and the permittivity ε_1r of YBCO can be calculated according to Eqs. (3.16) and (3.17). With these material parameters, the threshold wavelength of YBCO is calculated to be

λth=1245 nm. In addition, the superconductor-dielectric superlattice is immersed in free space, i.e, n =₀ n =1. _f

We first consider the conditions of that the thicknesses of YBCO and MgO layers

are set to be d₁=130 nm and d₂ =80 nm, respectively, and the number of periods is wavelength λ_th is contained. Such a PBG is referred to as a near-zero-ngap because within this gap the refractive index of superconductor is much less one and very close to

are set to be d₁=130 nm and d₂ =80 nm, respectively, and the number of periods is wavelength λ_th is contained. Such a PBG is referred to as a near-zero-ngap because within this gap the refractive index of superconductor is much less one and very close to

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