S L OW- L I G H T PH OTO N I C C RY S TA L S S U P E R V I S O R : D R. N I E L S A S G E R M O R T E N S E N S Ø R E N R A Z A ( S )

S L OW- L I G H T PH OTO N I C C RY S TA L S S U P E R V I S O R : D R. N I E L S A S G E R M O R T E N S E N S Ø R E N R A Z A ( S 0 6 2 0 8 6 ) B A C H E L O R T H E S I S D E P A R T M E N T O F P H O T O N I C S E N G I N E E R I N G T E C H N I C A L U N I V E R S I T Y O F D E N M A R K 2 0 0 9

Abstract This bachelor thesis addresses the phenomenon of slow-light propagation in photonic crystals. Using Bloch theory we determine the dispersion relation and group velocity of the simple 1-D Bragg stack and show that slow-light of less than 5% of the speed of light can be achieved at the band edge. For considerable slower light we examine the coupled-resonator optical waveguide, with and without energy loss. The dispersion relation is derived in both cases, and we show that for the ideal lossless structure slow-light can be achieved by having weak coupling between neighboring resonators. Dissipation is introduced by considering complex dielectric functions, and its implications on the dispersion relation is evident through the dependency on the Q-factor of the single resonators and the absorption loss of the material. The general effect of finite-q resonators on the band structure is studied and shown to alter the band edges, where extreme slow-light is achievable. Absorption losses is shown to narrow the values of allowed wave vectors, and both dissipation sources induce damping, represented by the imaginary part of the wave vector. We also discuss how absorption losses may be avoided in a physical CROW by using a proper dielectric substrate for fabrication. Finally, we simulate a CROW structure from literature, compare and fit the numerical result to the derived theory, and show, in the absence of absorption loss, that the Q-factor of the resonators will limit the minimum attainable group velocity. i

Preface This bachelor thesis was done in the spring semester of 2009 in the group of Structured Electromagnetic Materials at the Department of Photonics Engineering. It is credited 15 ECTS points and is part of the bachelor degree in Physics and Nanotechnology at the Technical University of Denmark. The thesis has been supervised by associate professor Niels Asger Mortensen. Many grateful thanks go out to Niels Asger Mortensen for enlightening discussions, great ideas and in general supreme help. I would also like to thank Ph.D. student Jure Grgić for acquainting me with the simulation program MPB. The front page image is lent from Krauss [1] and illustrates light pulses entering a slowlight regime. Note how the pulses narrow and intensify. Søren Raza, s062086 ii

Contents Abstract Preface Contents List of Figures i ii iii iv 1 Introduction 1 1.1 Objective.......................................... 2 2 Governing equations 3 2.1 Operator properties.................................... 4 3 Bloch s theorem 6 4 Bragg stack 8 4.1 Brief summary....................................... 12 5 Optical resonator theory 14 5.1 Single resonator...................................... 14 5.2 Coupled-resonator optical waveguide (CROW).................... 15 5.3 Analysis of integrals and constants........................... 19 5.4 Energy loss......................................... 19 5.5 Results and discussion.................................. 22 5.6 Brief summary....................................... 26 6 Simulations 27 6.1 Introduction to MPB................................... 27 6.2 MPB results and discussion............................... 28 6.3 Brief summary....................................... 32 7 Conclusion and outlook 34 7.1 Outlook........................................... 35 iii

List of Figures iv Bibliography 36 List of Figures 4.1 Illustration of the Bragg stack. It extends infinitely in the x-y plane, but is periodic in the z-direction when translated a = a 1 + a 2....................... 8 4.2 Plot of the dielectric function. It is clear that ɛ(z + sa) = ɛ(z) for s Z......... 8 4.3 Plot of the photonic band structure of a Bragg stack with a dielectric difference of ɛ 2 ɛ 1 = 12 and layer thickness a 1 = a 2 = a 2. The photonic band gap stretches from approximately 0.15 to 0.25 on the frequency axis..................... 10 4.4 Plot of the frequency ω as function of the imaginary wave vector k z. The dashed lines enclose the photonic band gap area. It is apparent that the imaginary wave vector is only non-zero in the band gap, emphasizing the fact that there are only evanescent states in the gap.................................. 11 4.5 The group velocity of a Bragg stack with a dielectric difference of ɛ 2 ɛ 1 = 12 and layer thickness a 1 = a 2 = a 2. The plot is symmetrical about v g = 0............ 12 5.1 A simple 1-D optical cavity constructed by the use of two identical Bragg stacks. The broadened area (or defect) between the layers creates new extended states in the photonic band gap of the Bragg stacks......................... 14 5.2 An example of 2-D coupled resonators. ɛ 2 is larger than ɛ 1............... 16 5.3 An example of a 2-D single optical cavity. ɛ 2 is larger than ɛ 1.............. 16 5.4 A sketch of the dielectric difference function ɛ(r ), made by subtracting fig. 5.3 from fig. 5.2. White indicates a dielectric value of zero, while green indicates the dielectric difference ɛ 2 ɛ 1................................... 16 5.5 Left panel: Plot of the dispersion relation (5.41) with Q = and κ n = 0.03. Right panel: Plot of the frequency as a function of the imaginary wave vector. Both plots are symmetric about k x = k x = 0............................... 22 5.6 Left panel: Plot of the dispersion relation (5.41) with Q = 1000 and κ n = 0.03. Right panel: Plot of the frequency as a function of the imaginary wave vector. Both plots are symmetric about k x = k x = 0............................... 23 5.7 Left panel: Plot of the dispersion relation (5.41) with Q = 1000 and κ n = 0.03 + i 0.005. Right panel: Plot of the frequency as a function of the imaginary wave vector. Both plots are symmetric about k x = k x = 0...................... 24 5.8 The real (n) and imaginary part (k) of the refractive index of silicon as a function of the wavelength (λ) of light. n(λ) is represented as a solid line, while k(λ) is the dashed line. The graph is from [2].............................. 25 6.1 Image of the MPB unit cell used for simulation. The CROW is made of airholes with radius/b = 0.3 in a dielectric material of ɛ = 7.0225.................... 28

List of Figures v 6.2 The photonic band structure for the TE modes of a triangular array of air columns in a dielectric substrate (ɛ = 7.0225). The markings Γ, M and K represent the highsymmetry points at the corners of the irreducible Brillouin zone............ 29 6.3 Simulation result of the dispersion relation for the even TE CROW mode (red dots) along with the theoretical fit (solid curve). The curve is symmetric about k x = 0... 29 6.4 The group velocity, given by equation (5.21), as a function of the wave vector with γ = 0.0066 and ω 1a c = 0.6438. Note the sign of the group velocity........... 30 6.5 The black solid line represents the group velocity (5.42) of the dispersion relation (5.46) as a function of the Q-factor. The fitted values γ = 0.0066, γ = 3.705 10 5 and ω 1a c = 0.6438 along with the CROW frequency Ω 1 = ω 1 in the center of the band structure have been used. The red dashed line represents the Taylor expanded approximation (6.10) with the same fitted values....................... 31 6.6 The black solid line represents the group velocity (5.42) of the dispersion relation (5.46) as a function of the Q-factor. The fitted values γ = 0.0066, γ = 3.705 10 5 and ω 1a c = 0.6438 along with the CROW frequency Ω 1 = ω 1 (1 + γ) at the edge of the band structure have been used. The red dashed line represents the Taylor expanded approximation (6.12) with the same fitted values................ 32

1 Introduction The novel field of photonic crystals was first proposed in 1987 simultaneously by Sajeev John and Eli Yablonovitch in two different independent papers [3] [4]. Photonic crystals are periodic arrangements of dielectric media, which allow for manipulation of the propagation of light. The periodic dielectric arrangements affect photons in much the same way as the periodic potential in solid-state physics affects electrons, and therefore photonic crystals are often considered the optical analog of electrical semiconductors. The crystals are usually classified into three categories, 1-dimensional, 2-dimensional and 3-dimensional photonic crystals, and their complexity grows with increasing dimension. The exciting property, that has given the photonic crystals a lot of attention, is their exceptional dispersive character and electromagnetic frequency gaps, also known as photonic band gaps. Inside these frequency gaps no electromagnetic eigenmodes exist. This means that if a lightwave with band gap frequency is incident on the photonic crystal, it will be reflected. Some types of complex photonic crystals even have frequency band gaps regardless of the polarization and angle of the incident light. These gaps are known as complete photonic band gaps. The gaps arise from photon scattering at the multiple dielectric interfaces. Such an intriguing feature gives the means to totally control the propagation of electromagnetic waves, and may be applied to create integrated optical devices for all-optical high-speed computers and energy efficient light-emitting diodes (LEDs) [5] [6]. Another important property is the scalability of photonic crystals. It allows one to tailor which wavelength region one wishes to manipulate, simply by increasing or decreasing the size of the lattice constants of the crystal. For example, if infrared light is to be controlled, the size of the lattice constants should be in the micron region, while control of microwaves requires millimeter dimensions. Due to the previous mentioned unique dispersive properties of photonic crystals, new exciting fields have emanated, such as the study of negative refractive index (also known as metamaterials), negative group velocity, superluminal and slow-light phenomena. This thesis addresses the latter of these new turf by investigating a newly proposed waveguide design consisting of weakly coupled resonators. The coupled-resonator optical waveguide (CROW) was first introduced by Yariv et. al in 1999 [8] and has since attracted a lot of attention. One of the most interesting properties is that light pulses may propagate in the CROW with extremely low group velocities. Slowing light may offer new possibilities in the area of interferometers and optical buffers [9] [10]. As fabrication methods continue to improve, photonic crystals are gradually becoming more and more reality. Today, photonic crystal designs are already used in optical waveguides, but their technological applications in the future are vast, including maybe one day visiblelight cloaking devices [7]. With all these areas of usage photonic crystals may perhaps one day revolutionize our society, as the advances in semiconductor physics have done. 1

CHAPTER 1. INTRODUCTION 2 1.1 Objective The purpose of this thesis is to theoretically examine the phenomena of slow-light in photonic crystals, with emphasis on the coupled-resonator optical waveguide. The theory behind the CROW is first developed in an ideal lossless setting. We then go further and study the implications of having dissipation of energy. We discuss the band structure and group velocity of the CROW, and how energy loss affects this. Finally, we compare the developed theory to numerical simulations, and discuss the limiting factor for minimum attainable group velocity.

2 Governing equations The main equations governing the description of electromagnetic fields in a photonic crystal are Maxwell s four equations. Since a photonic crystal is a specific arrangement of dielectric media, there are no free charges or any free current in such a system, and Maxwell s equations reduce to B(r, t) = 0 D(r, t) = 0 E(r, t) + 1 B(r, t) = 0 c t (2.1) H(r, t) 1 D(r, t) = 0 c t (2.2) A further reduction of Maxwell s equations is done by making some assumptions on D(r, t) and B(r, t). First of all we are only considering small electric field strengths, meaning that we do not look at non-linear effects in this analysis. Next we assume the medium to be macroscopic and isotropic, thus reducing the dielectric function ɛ(r ) to a scalar (instead of a tensor). Thirdly we will ignore any frequency dependency on the dielectric function. With these three assumptions we can write the following relation between the displacement field D(r, t) and the electric field E(r, t) D(r, t) = ɛ(r )E(r, t) (2.3) We will be considering both dielectrics with and without loss, but for now we look at lossless structures, making the dielectric function ɛ(r ) real. It will be highlighted when this is not the case. For most dielectric materials the magnetic permeability is around unity, so we may set B(r, t) = H(r, t) [11]. The last restriction we will impose is that the electromagnetic fields in the crystal are assumed to vary harmonically in time. This means that we can write the magnetic and electric field as a certain spatial field pattern times a complex exponential harmonic mode H(r, t) = H(r )e iωt (2.4) E(r, t) = E(r )e iωt (2.5) With all these assumptions in mind Maxwell s equations reduce to H(r ) = D(r ) = 0 (2.6) E(r ) + iω c H(r ) = 0 (2.7) H(r ) iω c ɛ(r )E(r ) = 0 (2.8) Equation (2.6) is denoted the transversality requirement. It says that the electric and magnetic fields are divergenceless, or in other words, there can be no electric or magnetic point sources. 3

CHAPTER 2. GOVERNING EQUATIONS 4 Isolating E(r ) in (2.8) and inserting it in (2.7) yields the main equation ( ) 1 ( ω ) 2 ɛ(r ) H(r ) = H(r ) (2.9) c which is a linear differential equation for H(r ). We can rewrite (2.9) by defining the operator Ô(r ) = 1 ɛ(r ) to ( ω ) 2 Ô(r )H(r ) = H(r ) (2.10) c which emphasizes the fact that (2.9) is an eigenvalue equation with H(r ) being the eigenfunctions and ( ) ω 2 c being the eigenvalue. In [11] it is shown that the operator Ô(r ) is Hermitian. Once the H-field is determined one can find the electric field from equation (2.8). One may also isolate H(r ) in (2.7) and insert it in to (2.8). This will yield a linear differential equation for the electric field ( ω ) 2 E(r ) = ɛ(r )E(r ) (2.11) c This is now a generalized eigenvalue equation, since there is not only an operator on the left hand side of the equation but also on the right hand side. We now define the two operators Û 1 and Û 2 as Û 1 = (2.12) ( ω ) 2 Û 2 (r ) = ɛ(r ) (2.13) c We will show in the next section that these operators also are Hermitian. In this report we will be using both equation (2.9) and (2.11) in our analysis of different photonic crystal structures. For some applications one formulation is more convenient than another, which is the reason for introducing both. 2.1 Operator properties We define the inner product of two vector fields E 1 (r ) and E 2 (r ) as E 1 E 2 = E1 E 2dr (2.14) where the integration is over all r. The star denotes a complex conjugation. An operator Û is Hermitian if it satisfies the condition E 1 ÛE 2 = ÛE 1 E 2 for any E 1 and E 2 [11]. We first examine the operator Û 1 from (2.12) E 1 Û 1 E 2 = E1 ( E 2 )dr = ( E 1 ) ( E 2 )dr (2.15) = ( E 1 ) E 2 dr = Û 1 E 1 E 2 (2.16) In the above derivation integration by parts has been used, and the surface terms have been neglected since these terms involve values of the fields at the boundaries of integration. Either the fields decay to zero at large distances or the fields are periodic in the integration region, causing the surface terms to vanish.

CHAPTER 2. GOVERNING EQUATIONS 5 Now operator Û 2 (r ) from (2.13) is examined E 1 Û 2 E 2 = ( (ω E1 c ) 2 ɛ(r )E2 ) dr = ( (ω ) ) 2 ɛ(r )E1 E 2 dr = Û 2 E 1 E 2 (2.17) Note that the above derivation is only possible because we assume that ɛ(r ) is real. We have now proved that both the operator Û 1 and Û 2 are Hermitian. This property is important since it can be shown that this will lead to a real ω and the orthogonality relation E i (r ) ɛ(r )E j (r ) = δ i j with δ i j being the Kronecker delta. c

3 Bloch s theorem Bloch s theorem will be needed for coming analysis and is therefore derived in this chapter. The theorem originates from the periodic potential problem in solid state physics. We will now prove the theorem 1 when dealing with a similar problem in the field of photonic crystals, that is when we have a periodic dielectric function ɛ(r + R) = ɛ(r ) (3.1) where R is a Bravais lattice vector. This implies that the operator Û 2 (r ) defined in equation (2.13) also is periodic Û 2 (r + R) = Û 2 (r ) (3.2) First of all we define the translation operator ˆT R which, when operated on a function f (r ), shifts the argument by R, as ˆT R f (r ) = f (r + R) (3.3) We now apply the translation operator on the governing equation (2.11) ˆT R Û 1 E(r ) = ˆT R Û 2 (r )E(r ) = Û 2 (r + R)E(r + R) = Û 2 (r ) ˆT R E(r ) = Û 1 (r ) ˆT R E r (3.4) Since the above relation holds for any E(r ), we have that ˆT R Û 1 = Û 1 ˆT R (3.5) This relation is essential, since it means that the translation operator ˆT R and the operator Û 1 commute for all lattice vectors R. Using quantum mechanical theory we know that commuting operators have simultaneous eigenfunctions, so eigenfunctions E(r ) satisfying the translation equation ˆT R E(r ) = c(r)e(r ) (3.6) will also satisfy the governing equation (2.11). We wish now to determine the eigenvalues c(r), so by applying two successive translations and noting that the order does not matter, we find But it also follows that 2 ˆT R ˆT R E(r ) = ˆT R c(r )E(r ) = c(r)c(r )E(r ) (3.7) ˆT R ˆT R E(r ) = ˆT R+R E(r ) = c(r + R )E(r ) (3.8) 1 We employ an approach similar to that of [12]. 2 This is valid for all Bravais lattice vectors R and R, since ˆT R ˆT R E(r ) = ˆT R ˆT R E(r ) = ˆT R+R E(r ) = E(r + R + R ) 6

CHAPTER 3. BLOCH S THEOREM 7 The eigenvalues must satisfy the relation c(r)c(r ) = c(r + R ) (3.9) which indicates that the constant c(r) must be some sort of exponential function. Let a i (i = 1,2,3) be the three primitive vectors for the Bravais lattice. We can now write the lattice vector R in terms of the primitive vectors as R = n 1 a 1 + n 2 a 2 + n 3 a 3 (3.10) for some integers n 1, n 2 and n 3. From equation (3.9) we see that c(nr) = c(r) N, so c(r) can be written as c(r) = c(a 1 ) n 1 c(a 2 ) n 2 c(a 3 ) n 3 (3.11) For convenience we write the c(a i ) in the following exponential form c(a i ) = e 2πi x i (3.12) for some x i. Using equation (3.12) we can rewrite equation (3.11) to c(r) = e i 2π(n 1x 1 +n 2 x 2 +n 3 x 3 ) = e ik R (3.13) where k = x 1 b 1 + x 2 b 2 + x 3 b 3 (3.14) and the b i are the reciprocal lattice vectors satisfying the condition b i a j = 2πδ i j. We have shown that if we have a dielectric function, which is periodic when translated by R, then the eigenfunctions E(r ) of equation (2.11) will have the important Bloch property ˆT R E(r ) = E(r + R) = e ik R E(r ) (3.15) This means that the electric field only changes phase when it translates by R. The same Bloch property is valid for the magnetic field H(r ), when considering periodic dielectrics, as proven in [11].

4 Bragg stack We start by considering the simplest photonic crystal, the 1-D photonic crystal consisting of an infinite number of slabs with alternating dielectric constant and thickness, see figure 4.1. This photonic crystal is also known as a Bragg stack or a multilayerfilm, and is a well-known system. Our goal is to determine the dispersion relation using Bloch theory, so we can examine the band structure and group velocity. Figure 4.1: Illustration of the Bragg stack. It extends infinitely in the x-y plane, but is periodic in the z-direction when translated a = a 1 + a 2. Figure 4.2: Plot of the dielectric function. It is clear that ɛ(z + sa) = ɛ(z) for s Z. We consider light propagating only in the z-direction, so the wave vector is given by k = k z ẑ. Due to the high symmetry of the stack, it is obvious that the polarization of the incident light is irrelevant, as long as it is in the x-y plane, therefore we choose for simplicity that H(r ) = H x (r ) ˆx (4.1) With these assumptions in mind and using the transversality requirement (2.6), the governing equation (2.9) reduces to d 2 ω ) 2 dz 2 H x(z) = ( ɛn H x (z) (4.2) c where n = 1,2. In the region 0 < z < a 1, see figure 4.2, the solution to the differential equation is H x1 (z) = Ae iκz + Be iκz (4.3) where κ = ω c ɛ1 and A,B are arbitrary complex constants. Similarly in the region a 2 < z < 0 we find the solution H x2 (z) = Ce iqz + De iqz (4.4) 8

CHAPTER 4. BRAGG STACK 9 where Q = ω c ɛ2 and C,D are arbitrary complex constants. Due to the discrete translational symmetry in the z-direction, when translated R = (a 1 + a 2 )ẑ, we know that the H-field must have the important Bloch property (3.15) H x (z + a) = e ik z a H x (z) (4.5) where a = a 1 + a 2. To find the dispersion relation we must now apply Maxwell s electromagnetic boundary conditions, when going from one dielectric medium to another. We need 4 boundary conditions to determine the 4 arbitrary constans A,B,C and D. The first 2 boundary conditions are that the parallel components of the H-field must be continuous across the boundaries z = 0 and z = a 1. This means that These two conditions along with equation (4.5) lead to H x1 (0) = H x2 (0) (4.6) H x1 (a 1 ) = H x2 (a 1 ) (4.7) A + B = C + D ( (4.8) ) Ae iκa 1 + Be iκa 1 = e ik z a Ce iqa 2 + De iqa 2 (4.9) Now the final 2 boundary conditions apply for the electric field E, which we can determine using equation (2.8) E n (r ) = ic ωɛ n H xn (r ) (4.10) We find the electric fields to be E 1 (z) = E 2 (z) = ɛ1 ( Ae iκz Be iκz) ŷ (4.11) ɛ 1 ɛ2 ( Ce iqz De iqz) ŷ (4.12) ɛ 2 The boundary conditions for the electric fields are the same as for the magnetic fields, the parallel components must be continuous across the boundaries z = 0 and z = a 1. This leads to the final 2 conditions ɛ1 ɛ 1 (A B) = ɛ1 ɛ 1 ( Ae iκa 1 Be iκa 1 ) = ɛ2 (C D) (4.13) ɛ 2 ( ) e ik z a Ce iqa 2 De iqa 2 (4.14) ɛ 2 The equations (4.8), (4.9), (4.13) and (4.14) can be written in matrix form, as ɛ2 Mb = 0 (4.15) where the vector b = (A,B,C,D). Exploiting that the matrix M must be singular and doing a lot of simple, but lengthy algebra, we arrive at the result cos(k z a) = cos ( ω ) ( c ɛ2 a 2 cos ω ) ɛ 1 + ɛ 2 c ɛ1 a 1 2 sin ( ω ) ( ɛ 1 ɛ2 c ɛ2 a 2 sin ω ) c ɛ1 a 1 (4.16) which is the dispersion relation for the Bragg stack. Because the primitive lattice vector of the Bragg stack is a = aẑ, the reciprocal vector is given as b = 2π a ẑ. Thus the Brillouin zone is set to π a < k z π a.

CHAPTER 4. BRAGG STACK 10 A plot of the dispersion relation (4.16) is seen on figure 4.3 with a 1 = a 2 = a/2, ɛ 2 = 13 and ɛ 1 = 1. The gap on the frequency axis is known as the photonic band gap, since light with frequency within the gap cannot propagate through the multilayerfilm. Instead it is reflected, so the structure works as a mirror for distinct frequencies that can be tuned depending on the values of parameters such as layer thickness and dielectric values. The size of the gap is especially dependent on the dielectric difference ɛ 2 ɛ 1, where a larger difference produces a bigger frequency gap. The frequencies below the gap constitute the first band, also named the dielectric band, while frequencies above the gap formate the second band, or sometimes known as the air band. Figure 4.3: Plot of the photonic band structure of a Bragg stack with a dielectric difference of ɛ 2 ɛ 1 = 12 and layer thickness a 1 = a 2 = a 2. The photonic band gap stretches from approximately 0.15 to 0.25 on the frequency axis. We would like to understand a bit more about what happens when light, consisting of frequencies in the photonic band gap, enters the structure. This examination is done in [11] by approximating the second band near the gap with a Taylor expansion to lowest order. The result is that whenever the frequency is in the gap, the wave vector becomes complex, inducing evanescent electromagnetic waves. This can be detailed by writing the wave vector k z as a complex number with a real and imaginary part The electromagnetic wave has the form H(r ) e ik z z ˆx = k z = k z + ik z (4.17) (e ik z z e k z z) ˆx (4.18) We see that if the wave vector is complex it introduces an exponentially decaying term, which size is determined by the imaginary part of the wave vector. This means that light with frequency within the band gap will quickly decay to zero and stop propagating in the structure.

CHAPTER 4. BRAGG STACK 11 In other words, we may say that there are no extended modes with frequency in the photonic band gap, only evanescent modes. Figure 4.4: Plot of the frequency ω as function of the imaginary wave vector k z. The dashed lines enclose the photonic band gap area. It is apparent that the imaginary wave vector is only non-zero in the band gap, emphasizing the fact that there are only evanescent states in the gap. From these considerations, it must be equally important to examine the imaginary part of the wave vector. Therefore we have plotted the frequency as a function of k z on figure 4.4. This is done by writing k z in equation (4.16) as a complex number, and then setting the imaginary part of each side of the equation equal to each other. As expected from our previous analysis, the imaginary part of the wave vector is zero in the bands, but non-zero in the gap. We observe that k z is largest in the middle of the gap, causing the quickest decay of waves with that frequency. Now that we know the form of the electromagnetic wave in the band gap, we could estimate the amount of dielectric double-layers N needed to give a satisfactory reflection of light with frequency in the band gap. So far we have only considered the ideal Bragg stack with infinite dielectric layers giving a reflection of 100%. Obviously, an infinite number of layers is impossible, so setting a less stringent requirement on the reflection would lead to a practical number of layers. From equation (4.18) the intensity of light with band gap frequency is I (z) e ik z z 2 = e i (k z +ik z )z 2 = e 2k z z (4.19) The transmittance of light through the finite Bragg stack is given as I (z = Na) T = I (0) = e 2k z Na (4.20)

CHAPTER 4. BRAGG STACK 12 Requiring the transmittance to be T = e 4 1.8%, we find that the amount of double-layers N is given by the simple relation N = 2 k (4.21) If we consider light with frequency right in the center of the band gap, where the reflection is highest, we need N = 3 double-layers to obey the transmittance demand, while at a frequency near the band edge N = 32 double-layers are needed. Another aspect of the Bragg stack worth examining is the group velocity. Due to unique dispersive property of the Bragg stack, or photonic crystals in general, the group velocity can be somewhat exceptional. We may calculate the group velocity from the dispersion relation (4.16) using the relation v g = dω dk = 1 (4.22) dk dω The result is plotted in figure 4.5. It is clear that the light is slowed down with lowest velocity z a Figure 4.5: The group velocity of a Bragg stack with a dielectric difference of ɛ 2 ɛ 1 = 12 and layer thickness a 1 = a 2 = a 2. The plot is symmetrical about v g = 0. at the band edge. In the band gap the group velocity is zero as expected, since no electromagnetic modes exist here. Thus a simple 1-D photonic crystal as the Bragg stack can support very slow light waves, much slower than structures in which only a single dielectric media is present. This slowdown is due to the scattering processes that happens at each dielectric interface of the multilayerfilm. 4.1 Brief summary We have examined the infinite layer Bragg stack by determining the dispersion relation (4.16), using Maxwell s equations and Bloch theory. The band structure is plotted in figure 4.3 and

CHAPTER 4. BRAGG STACK 13 the characteristic photonic band gap is examined and understood in terms of the imaginary wave vector k z. Writing the wave vector as a complex number illustrates that the only modes allowed in the band gap are evanescent, see figure 4.4. A plot of the group velocity is also presented in figure 4.5 and the slowdown of light around the band edges is observed. Lastly, a quick study has been done on how many double-layers N that are actually needed for sufficient reflection in the band gap (instead of infinite layers). It is found that N is inversely proportional to the k z, see equation (4.21), and therefore more layers are needed at band edge frequencies than at band center frequency.

5 Optical resonator theory We will now discuss and examine more advanced photonic crystal structures, in particular a recently proposed waveguide structure which displays the phenomenon of extreme slow-light propagation. The waveguide consists of coupled optical cavities, and therefore we start by discussing the physics behind optical cavities exhibiting resonance, and how such an optical resonator could be designed using photonic crystals. 5.1 Single resonator In chapter 4 we have detailed the mirror-like properties of the Bragg stack. We may create an optical resonator in which light with certain frequency or wavelength will be trapped by using two identical Bragg stacks, see figure 5.1. Figure 5.1: A simple 1-D optical cavity constructed by the use of two identical Bragg stacks. The broadened area (or defect) between the layers creates new extended states in the photonic band gap of the Bragg stacks. We know that the Bragg layers have a photonic band gap, in which light can not propagate. Imagine sending in light with that specific frequency; it will not be supported in the Bragg layers, but it might be able to exist in the broadened area between the layers. This broadened area creates new extended states at certain frequencies in the otherwise forbidden band gap. If light is sent in with one of these frequencies, it can be localized between the Bragg layers. This is an optical resonator, since the cavity exhibits resonance, which is the increasing of wave amplitude at certain frequencies. How well the light is localized depends on a number of parameters. If the resonator in figure 5.1 is fabricated with ideal impurity-free infinite Bragg stacks and with no light attenua- 14

CHAPTER 5. OPTICAL RESONATOR THEORY 15 tion, the electromagnetic waves would be trapped in the cavity forever. One essential parameter for describing the quality of a resonator is the Q-factor, which is defined as Stored energy Q = 2π Loss of energy (5.1) For a perfect resonator where there is no loss of energy the Q-factor is infinite. Energy loss can be due to several factors, for example if the fabrication process is imprecise, creating an imperfect geometrical form or it could be that the material used absorbs some of the photons. These are different physical effects, which influence the value of Q. There is another, but equivalent, definition of the Q-factor, which will be useful in later discussions. The definition comes from the frequency distribution of the energy, E(ω) 2, where E(ω) is determined by the Fourier integral The Q-factor can be found by the following ratio E(ω) = 1 E(t)e iωt dt (5.2) 2π 0 Q = ω 0 ω (5.3) where ω 0 is the maximum (or resonance) value for E(ω) 2 and ω is the full width at halfmaximum (FWHM). From this definition, a perfect resonator of Q = leads to a FWHM of zero, meaning that the frequency distribution is a delta function E(ω) 2 δ(ω ω 0 ). So far we have only considered 1-D photonic crystals, with and without defects. On figure 5.3 is shown a 2-D triangular lattice photonic crystal with infinitely long rods of dielectric value ɛ 1 embedded in a material of higher dielectric value ɛ 2. A defect has been induced by removing one of the rods to create an optical cavity. Analogous to the Bragg stack, this photonic crystal without the defect has a photonic band gap. For some specific rod sizes, this dielectric structure even has a complete band gap for both TE and TM polarized light. As before the defect creates extended states in the gap, allowing for light to be localized. By having an array of cavities in a 2-D photonic crystal, see figure 5.2, one can create a novel type of waveguide, which is the topic of study for the next sections. The light can propagate by coupling to neighboring resonators. 5.2 Coupled-resonator optical waveguide (CROW) In this section we will employ a formalism familiar to the tight-binding model in solid-state physics, see for example [12], to approximate the dispersion relation of the first band of an optical waveguide made of weakly coupled resonators. For now we assume that there is no energy loss in the structure, so ɛ(r ) is real, and that the resonators are perfect so Q =. We will be working with the governing equation (2.11), written in a more convenient form as Û 1 E(r ) = Ω2 n c 2 [ɛ sc(r ) + ɛ(r )]E(r ) (5.4) where Ω n is the nth mode frequency of the coupled resonators, and E(r ) is the electric field in the waveguide. Here the dielectric function is split in to two contributions: one from a single resonator cavity ɛ sc (r ), see figure 5.3, and another part ɛ(r ), which contains the rest of the dielectric structure, see figure 5.4.

CHAPTER 5. OPTICAL RESONATOR THEORY 16 Figure 5.2: An example of 2-D coupled resonators. ɛ 2 is larger than ɛ 1. Figure 5.3: An example of a 2-D single optical cavity. ɛ 2 is larger than ɛ 1. Figure 5.4: A sketch of the dielectric difference function ɛ(r ), made by subtracting fig. 5.3 from fig. 5.2. White indicates a dielectric value of zero, while green indicates the dielectric difference ɛ 2 ɛ 1. In the vicinity of one resonator cavity we assume that equation (5.4), which describes the whole structure, can be reduced to Û 1 E n (r ) = ω2 n c 2 ɛ sc(r )E n (r ) (5.5) where ω n is the nth resonator frequency of a single cavity and E n (r ) is the electric field of the nth resonator frequency, normalized according to the orthogonality relation ɛ sc (r )E m (r ) E n(r )dr = δ mn (5.6) Now E n (r ) would also satisfy (5.4) if ɛ(r ) = 0 whenever E n (r ) is non-zero and vice versa. This is however not likely, so instead we assume that ɛ(r ) is small whenever E n (r ) is appreciable

CHAPTER 5. OPTICAL RESONATOR THEORY 17 and vice versa. Therefore we guess for a solution for E(r ) as a linear combination of some function φ(r ) E(r ) = c j φ(r ) (5.7) j From figure 5.2 we see that there is discrete translational symmetry in the x-direction indicating that the field E(r ) must have the form of a Bloch wave. This leads to E(r ) = R e ik R φ(r R) (5.8) where k ranges through the values in the first Brillouin zone, and R is the Bravais lattice vector. We now show that (5.8) preserves the Bloch description E(r + R ) = e ik R φ(r + R R) R = e ik R e ik (R R ) φ(r (R R )) R = e ik R e ik R φ(r R) = e ik R E(r ) (5.9) R This result is in accordance with (3.15). We expect that the functions φ(r ) will not differ too much from E n (r ), so we expand φ(r ) in terms of the single resonator electric field φ(r ) = n a n E n (r ) (5.10) Combining (5.10) and (5.8) leads to E(r ) = R,n a n e ik R E n (r R) (5.11) We now multiply the equation (5.4) with Em (r ) and spatial integrate E m (r ) [Û 1 E(r )]dr = Ω2 n c 2 ( ɛ sc (r )E m (r ) E(r )dr + ɛ(r )E m (r ) E(r )dr ) (5.12) But since Û 1 is a Hermitian operator, as proven in section 2.1, we use that Em (r ) [Û 1 E(r )]dr = [Û 1 E m (r )] E(r )dr = ω2 m c 2 ɛ sc (r )Em (r ) E(r )dr (5.13) Combining (5.12) and (5.13) with the expression for E(r ) from (5.11) and then isolating for Ω 2 n gives us the result we have been searching for Ω 2 n (k) = ω2 m [ ] R,n a n e ik R β(r) R,n a n e ik R β(r) + R,n a n e ik R γ(r) (5.14) where we have introduced β(r) and γ(r) as β(r) = ɛ sc (r )Em (r ) E n(r R)dr (5.15) γ(r) = ɛ(r )Em (r ) E n(r R)dr (5.16)

CHAPTER 5. OPTICAL RESONATOR THEORY 18 Equation (5.14) is the general result for the dispersion relation of a coupled-resonator structure. To simplify things we will investigate further the coupling of the lowest mode frequency, setting m = n = 1. This will give us just one equation, since the sums over n disappear, and thereby enable us to determine the first band of the CROW. So equation (5.14) reduces to [ ] Ω 2 1 (k) = ω2 R e ik R β(r) 1 R e ik R β(r) + R e ik R (5.17) γ(r) We now make our first serious approximation; the resonators are weakly coupled, so the sum over R is reduced to only the nearest neighbors. This essentially means that a photon in a resonator cavity is assumed to only hop to neighboring resonators, either backwards or forwards, thereby neglecting the possibility that the photon could hop from one cavity to the third or fourth. From figure 5.2, this means that R ranges through a ˆx, 0, a ˆx. Since there is inversion symmetry in the CROW structure the following identities must apply: β( a ˆx) = β(a ˆx) and γ( a ˆx) = γ(a ˆx). With these considerations in mind equation (5.17) is reduced to Ω 1 (k x ) = ω 1 [ 1 + 2cos(k x a)β(a ˆx) 1 + γ + 2cos(k x a)[β(a ˆx) + γ(a ˆx)] ] 1 2 (5.18) where k x is the x-component of the wave vector k and γ is γ = ɛ(r ) E 1 (r ) 2 dr (5.19) Here E 1 (r ) is the lowest mode electric field of a single resonator cavity. It is fair to assume that E 1 (r ) is real. Assuming that γ(a ˆx), β(a ˆx) and γ are small (this assumption will be justified in section 5.3), we Taylor expand the square root in equation (5.18) and arrive at our final result ( Ω 1 (k x ) = ω 1 1 γ ) 2 γcos(k xa) (5.20) where we have omitted the argument of γ. The group velocity is determined as v g (k x ) = dω 1(k x ) dk x = ω 1 γa sin(k x a) (5.21) Using the identity sin(k x a) = 1 cos 2 (k x a) we may rewrite the group velocity as a function of the frequency Ω 1 ( v g (Ω 1 ) = ω 1 a γ 2 1 γ 2 Ω ) 2 1 (5.22) ω 1 The results in equations (5.20), (5.21) and (5.22) are in accordance with results achieved by [8], who have used a slightly different approach. The results also match those of which a completely different approach has been used, see [13]. The constant γ is labeled the coupling factor, since the size of γ determines the amount of coupling between neighboring resonators. If γ is very small, it will eliminate the cosine-term in equation (5.20) and thereby remove the wave vector dependency on the dispersion relation. It is also seen from the expression for the group velocity that a coupling factor of about zero will stop the light from moving between resonators. However, when this is said, a small coupling factor could be preferred since this will give slowlight propagation in the CROW. If we look closer at the definition of γ γ = γ(a ˆx) = ɛ(r )E 1 (r ) E 1 (r a ˆx)dr (5.23)

CHAPTER 5. OPTICAL RESONATOR THEORY 19 we see that the size of the integral is determined by ɛ(r ), which contains all information of the dielectric structure of the coupled resonators, and the electric field overlap between neighboring resonators. Thus the size, distance and amount of cavities determines the properties of the waveguide, as reported in [14]. A few words should be said about γ as well. Since this factor is non-negative (see analysis in section 5.3), it will reduce the coupled-resonator frequency Ω 1, compared to the single cavity resonance frequency ω 1. If the coupling factor is zero, this reduction becomes clear from equation (5.20). 5.3 Analysis of integrals and constants In this section we examine closer the assumption of γ, β(a ˆx) and γ being small. Let us first start with γ which is given by equation (5.19). There are only two functions determining the size of the integral; E 1 (r ), which is the lowest mode electric field of a single resonator, and ɛ(r ) which contains the difference between the coupled-resonator structure and the single resonator structure. An illustration of ɛ(r ) is given in figure 5.4 and we remark that the function is non-negative for all r, making the integrand in equation (5.19) nonnegative for all r. This means that the integral can only be positive or zero. The field E 1 (r ) is primarily concentrated in the region of the single resonator. Thus the overlap between the electric field squared and the dielectric difference function is extremely small, making the integral almost negligible. This justifies our earlier assumption. Now we move on to examine the constant β(a ˆx) given by the integral β(a ˆx) = ɛ sc (r )E 1 (r ) E 1 (r a ˆx)dr (5.24) A sketch of the single cavity dielectric function ɛ sc (r ) is given in figure 5.3. As mentioned before E 1 (r ) is mainly concentrated in the region of the cavity, so the product of ɛ sc (r ) and E 1 (r ) is not necessarily a small quantity. But E 1 (r a ˆx) is confined to a region that is shifted a distance a away. Since we assume weakly coupled resonators, the overlap of the two relatively shifted fields will be small enough to justify our assumption that β(a ˆx) in fact is small. The last constant is γ given by the integral in equation (5.23). The same considerations apply here as for the two previous constants. The product of E 1 (r ) and ɛ(r ) is almost zero, as with the constant γ, but E 1 (r a ˆx) contributes to the integral since its argument is shifted. Thus it is safe to say that this constant is also small. 5.4 Energy loss So far we have considered lossless coupled-resonator structures. We will now investigate what happens when energy loss is taking into account, by considering complex dielectric functions, ɛ sc (r ) and ɛ(r ). Tilde will be used to emphasize that the functions or constants no longer can be assumed to be real. The governing equation of the CROW structure is now Û 1 Ẽ(r ) = Ω2 n c 2 [ ɛ sc(r ) + ɛ(r )]Ẽ(r ) (5.25) where Ẽ(r ) is the electric field in the lossy waveguide. While in the vicinity of a single resonator we assume that the equation (5.25) reduces to Û 1 Ẽ n (r ) = ω2 n c 2 ɛ sc(r )Ẽ n (r ) (5.26)

CHAPTER 5. OPTICAL RESONATOR THEORY 20 where Ẽ n (r ) is the nth mode electric field of the single resonator with loss. We notice that the operators on the right hand side of equations (5.25) and (5.26) are no longer Hermitian, since the dielectric functions are complex. The operators were only Hermitian as long as the dielectric functions were real, as we noted in section 2.1. This has two very important consequences. The first is that the single cavity resonance frequency ω n is not necessarily real anymore and second, we cannot employ the exact same procedure as before to determine the dispersion relation. Instead we insert our qualified guess Ẽ(r ) from equation (5.8) with k replaced by k and with n = 1 (we only want the first band, so this simplifies calculations) into equation (5.25) Û 1 Ẽ(r ) = Û 1 R a 1 e i k R Ẽ 1 (r R) = Ω2 1 c 2 [ ɛ sc(r ) + ɛ(r )] R a 1 e i k R Ẽ 1 (r R) (5.27) = R a 1 e i k R Û 1 Ẽ 1 (r R) = ω2 1 c 2 ɛ sc(r R) R a 1 e i k R Ẽ 1 (r R) (5.28) We multiply these results with Ẽ1 (r ), spatially integrate and solve for Ω2 1 [ Ω 2 1 = ω2 R e i k ] R α(r) 1 R e i k R β(r) + R e i k R γ(r) (5.29) where α(r), β(r) and γ(r) are given by α(r) = ɛ sc (r R)Ẽ1 (r ) Ẽ 1 (r R)dr (5.30) β(r) = ɛ sc (r )Ẽ1 (r ) Ẽ 1 (r R)dr (5.31) γ(r) = ɛ(r )Ẽ1 (r ) Ẽ 1 (r R)dr (5.32) As before, we employ the nearest-neighbor approximation, the inversion symmetry considerations and assume that β(a ˆx), γ(a ˆx), α(a ˆx) and γ = Ẽ 1 (r ) 2 ɛ(r )dr are small. This reduces equation (5.29) to Ω 1 ( k x ) = ω 1 ( 1 γ β(a ˆx) α(a ˆx) + γ(a ˆx) 2 β(0) β(0) ) cos( k x a) (5.33) where k x is the x-component of k. We can further reduce this expression by noting that β(a ˆx) α(a ˆx) + γ(a ˆx) = [ ɛ sc (r ) + ɛ(r ) ɛ sc (r a ˆx)]Ẽ 1 (r ) Ẽ 1 (r a ˆx)dr (5.34) = ɛ(r a ˆx)Ẽ 1 (r ) Ẽ 1 (r a ˆx)dr κ (5.35) where we have used the property of the periodic CROW structure: ɛ sc (r )+ ɛ(r ) = ɛ sc (r a ˆx)+ ɛ(r a ˆx). We end up with ( Ω 1 ( k x ) = ω 1 1 γ κ 2 β(0) ) ( β(0) cos( k x a) ω 1 1 γ n 2 κ n cos( k x a) ) (5.36) Equation (5.36) represents the dispersion relation of the CROW, where the possibility of dissipation of energy is taken into account. This result is somewhat similar to (5.20), though with

CHAPTER 5. OPTICAL RESONATOR THEORY 21 modified constants and arguments. β(0) is not unity as before, since we cannot impose the normalization condition ɛ sc (r ) Ẽ 1 (r ) 2 dr = 1. It is a lot more difficult to estimate the sizes of the constants γ n and κ n in equation (5.36), since we can not assume that the electric field Ẽ 1 (r ) is real. But nevertheless we can try and understand their physical meaning and effect on the CROW structure. The complex coupling factor κ n is given as ɛ(r a ˆx)Ẽ 1 κ n = (r ) Ẽ 1 (r a ˆx)dr [ ɛ (r a ˆx) + iɛ (r a ˆx)]Ẽ1 ɛsc (r ) Ẽ 1 (r ) 2 = (r ) Ẽ 1 (r a ˆx)dr dr [ɛ sc (r ) + iɛ sc(r )] Ẽ 1 (r ) 2 dr (5.37) where ɛ(r ) = ɛ (r ) + i ɛ (r ) and ɛ sc (r ) = ɛ sc (r ) + iɛ sc (r ). The size of κ n only depends on the dielectric functions and the single resonator electric field. This means that the main way this factor introduces energy loss in the dispersion relation is through photon absorption of the material, and not like the Q-factor, which could also be due to fabrication imperfections. Of course fabrication imperfections changes the single resonator field Ẽ 1 (r ), but this change is negligible. Therefore, if there is no energy loss due to absorption (meaning that Re( ɛ(r )) Im( ɛ(r ))), we may safely approximate the complex coupling factor to the usual coupling factor from equation (5.20), i.e. κ n γ. The same considerations apply for the complex constant γ n. Before we continue to plotting the dispersion relation of the CROW, we wish first to relate the frequency of the single resonator ω 1 to the Q-factor, since this value characterizes resonators in the literature. The resultant electrical field of the single resonator is Ẽ 1 (r, t) = Ẽ 1 (r )e i ω 1t = Ẽ 1 (r )e ω 1 t e iω 1 t (5.38) where we have defined ω 1 = ω 1 + iω 1. According to [15], this leads to a frequency distribution for the energy in the resonator having a resonant line shape E(ω) 2 1 (ω ω 1 )2 + (ω 1 )2 (5.39) The resonance shape (5.39) has a full width at half-maximum (FWHM) equal to 2ω 1 and maximum (or resonance) at ω 1. From equation (5.3) the Q-factor is With this result we can write ω 1 as ω 1 = ω 1 terms of Q ( Ω 1 ( k x ) = ω 1 1 1 + i 2Q Q = ω 1 2ω 1 ( 1 + i 1 2Q (5.40) ), and then reformulate equation (5.36) in )( 1 γ ) n 2 κ n cos( k x a) (5.41) This is the form of the dispersion relation, which we will examine in the following section. Due to the complex constants in (5.41) it is not possible to find an analytical expression for the group velocity, but it may be computed numerically using the definition v g = 1 Re( d kx dω ) (5.42) where we have assumed that the angular frequency Ω 1 is real, while the wave vector k x is complex.

CHAPTER 5. OPTICAL RESONATOR THEORY 22 5.5 Results and discussion We will now plot the dispersion relation of the CROW structure for various values of κ n and Q. This will enable us to understand the effect of the sizes of the constants on the light propagating in the waveguide. In the following considerations, we set γ n = 0 and define k x as k x = k x + ik x (5.43) Once again, the primitive lattice vector is a = a ˆx so the reciprocal lattice vector is given as b = 2π π a ˆx, setting the Brillouin zone to a < k x π a. The symmetry of the CROW about the y-axis, see figure 5.2, restricts the k x values further to 0 < k x π a. We first reproduce and examine the results of [8] by considering equation (5.41) with Q = and κ n = 0.03. This is a CROW of perfect resonators with no absorption. The dispersion relation and the frequency as a function of the imaginary part of the wave vector is plotted in figure 5.5. We remark that the real and imaginary part of the wave vector are never present Figure 5.5: Left panel: Plot of the dispersion relation (5.41) with Q = and κ n = 0.03. Right panel: Plot of the frequency as a function of the imaginary wave vector. Both plots are symmetric about k x = k x = 0. simultaneously. The imaginary part of the wave vector is zero when the real part is nonzero. This confirms that the extended states have no energy loss, which we expected. Outside the frequency range of the dispersion relation, only the imaginary part of the wave vector is present, giving rise to reflection of the light. This is the ideal CROW structure, which is very difficult to fabricate. Therefore we now examine a more realistic CROW structure, still with no absorption, but consisting of resonators with a finite Q-factor of 1000. Figure 5.6 shows the band structure of this waveguide. The main difference here, compared to the previous figure, is that the imaginary wave vector is non-zero for all frequencies, giving rise to an exponential damping factor. The damping is however very small in the frequency range of the extended states

CHAPTER 5. OPTICAL RESONATOR THEORY 23 Figure 5.6: Left panel: Plot of the dispersion relation (5.41) with Q = 1000 and κ n = 0.03. Right panel: Plot of the frequency as a function of the imaginary wave vector. Both plots are symmetric about k x = k x = 0. (0.97 < Ω 1 < 1.03). The effect of low quality resonators in the CROW is clear; the light is not ω 1 sufficiently well-reflected in the cavities, so it loses intensity as it propagates. This is of course an undesirable property, so high Q-factor resonators are essential for stable light propagation. The smooth transitions around the band edges k x = 0 and k x = π a compared to the sharp cutoffs on figure 5.5 should also be noted. The transitions show that there are more extended modes available. The allowed frequency range for the light has broadened. Speaking in terms of density of states, one may say that the number of states decays smoothly around the band edges, and not abruptly as seen on figure 5.5, making more states accessible. This is understandable, since a finite Q means that we no longer represent the frequency distribution of the energy of each resonator by a delta function, but by a more smoothed out Lorentzian function (see equation (5.39)). Thus the coupling of these finite Q resonators must lead to a more smooth dispersion relation as we also see. In the ideal CROW on figure 5.5 the group velocity could be zero at the band edges, because the slope is zero. This is no longer the case when a finite Q is present, because the slope is never zero on figure 5.6. Though at the band edges the group velocity should be very low. Let us now look closer at what happens when we add absorption by introducing a small imaginary part to the complex coupling factor κ n. On figure 5.7 the band structure can be seen with κ n = 0.03 + i 0.005 and Q = 1000. Many of the same effects appear as when we just had a finite Q; the imaginary part of the wave vector is almost non-zero everywhere and the transitions around the band edges have been smoothed out. This is somewhat not surprising, since a finite Q-factor means that there is a loss of energy in the resonators, and these energy losses could be due to photon absorption by the material, as discussed in section 5.1. There is though a new big change around the band edges. In figure 5.6 the bands ended at k x = 0 and k x = π a, but now the k x values have narrowed. This means that not all k x values are supported