OPTICAL PROPERTIES OF ONE DIMENSIONAL QUATERNARY PHOTONIC CRYSTALS AND ASSOCIATED HETEROSTRUCTURES

The research conducted in this dissertation is the theoretical investigation into the transmission properties of one dimensional inversion symmetric quaternary photonic crystals and heterostructures created by combining quaternary and binary crystals. A photonic crystal is a device constructed from dielectric or conducting scattering elements arranged in a periodic manner. Saying a crystal has inversion symmetry simply means that the individual unit cells do too. Inversion symmetry is needed to ensure the system remains reciprocal and that topological phase of bands in a band structure remains discrete. The terms ”binary” and ”quaternary” refer to the number of layers in a unit cell, in this case, two and four, respectively. Similar to how an ionic lattice manipulates the flow of electrons, the periodicity of a photonic crystal can control the flow of photons. This leads to interesting physical properties in a device, such as photonic band gaps, regions in frequency space where photonic states cannot exist, and interface modes, localized states that can form at the boundary between two different crystals due to some change in material parameters or geometry. In the first manuscript, we investigate interface modes in a heterostructure composed of a single binary photonic crystal and single quaternary photonic crystal. Several papers have already investigated modes in structures solely consisting of binary crystals. Being one dimensional, quaternary crystals are still relatively simple to fabricate while providing more interesting behavior compared with a binary system. In the work, this is done by inserting a tunable layer in between every original layer of the binary crystals. By setting other parameters in the heterostructure constant, it is possible to smoothly transform one binary photonic crystal into another through an intermediate quaternary state. The transfer matrix method is used in the simulation of these crystals and their resultant properties. Identifying topological phase in the band structure for quaternary crystals is discussed and compared to that of binary crystals. Band gap closings are also discussed. Examples of topological interface modes in the transmission spectra of binary/quaternary structures are provided showing that modes only exist in certain band gap when the quaternary crystal is tuned from one binary to another. In the second manuscript, the single binary/quaternary system mentioned previously is generalized to a periodic array of crystals with each crystal still being periodic itself. Interface modes are shown to display different behavior as the tunable layer in the quaternary crystal(s) increases. Topological modes in the band gap are shown to vanish, split in two, or be surrounded by additional modes. Investigations are done for systems consisting entirely of lossless dielectrics and for systems where the tunable layer has a frequency dependent refractive index via the Drude model. Also, examples are given of how modes couple as the number of unit cells in the central crystal of a heterostructure changes. In the third manuscript, effective medium theory is applied to both isolated quaternary photonic crystals and heterostructures. Effective medium theory has been used on photonic crystals and metamaterials in previous works to recover effective parameters, such as refractive index and impedance from transmission and reflection coefficients. We briefly compare analytic and numerical techniques for recovering effective parameters. We then examine how branch point singularities in the real part of the effective refractive index evolve as various parameters in the quaternary crystal, such as tunable layer thickness and loss components of permittivity, change. Construction of the effective refraction index analytically requires knowing the correct branches across the index profile in order to ensure it is both continuous and smooth. Understanding how branch crossings change with other parameters can help prevent abrupt profile changes/discontinuities. We then consider the homogenization of a symmetric heterostructures composed of a binary/quaternary/binary, a possible configuration discussed in the second manuscript. The effective refractive index of the system is shown to display brief regions of normal dispersion in the otherwise anomalous regions which correspond to the interfaces states discussed previously.

erties. Identifying topological phase in the band structure for quaternary crystals is discussed and compared to that of binary crystals. Band gap closings are also discussed. Examples of topological interface modes in the transmission spectra of binary/quaternary structures are provided showing that modes only exist in certain band gap when the quaternary crystal is tuned from one binary to another.
In the second manuscript, the single binary/quaternary system mentioned previously is generalized to a periodic array of crystals with each crystal still being periodic itself. Interface modes are shown to display different behavior as the tunable layer in the quaternary crystal(s) increases. Topological modes in the band gap are shown to vanish, split in two, or be surrounded by additional modes. In the third manuscript, effective medium theory is applied to both isolated quaternary photonic crystals and heterostructures. Effective medium theory has been used on photonic crystals and metamaterials in previous works to recover effective parameters, such as refractive index and impedance from transmission and reflection coefficients. We briefly compare analytic and numerical techniques for recovering effective parameters. We then examine how branch point singularities in the real part of the effective refractive index evolve as various parameters in the quaternary crystal, such as tunable layer thickness and loss components of permittivity, change. Construction of the effective refraction index analytically requires knowing the correct branches across the index profile in order to ensure it is both continuous and smooth. Understanding how branch crossings change with other parameters can help prevent abrupt profile changes/discontinuities. We then consider the homogenization of a symmetric heterostructures composed of a binary/quaternary/binary, a possible configuration discussed in the second manuscript. The effective refractive index of the system is shown to display brief regions of normal dispersion in the otherwise anomalous regions which correspond to the interfaces states discussed previously. This dissertation has been prepared in manuscript form and consists of three separate academic papers which have either been submitted to journals or intend to be submitted. The first manuscript is an investigation into optical properties of one dimensional dielectric inversion symmetric quaternary photonic crystals.
Topology of the band structure for this system is first discussed and compared with that of binary photonic crystals. This is followed by a discussion of the behavior of interface states at the boundary of the heterostructure composed of one binary and one quaternary photonic crystal. This paper is being prepared for submission in OSA Optics Express.
The second paper extends this topic to a heterostructure consisting of an alternationg array of multiple binary and quaternary photonic crystals. The interface states is dependent on whether there is an even or odd number of photonic crystal interfaces in the heterostructure. If there is an odd number, then the properties of the crystals comprising the boundaries will affect the states as well. This paper has been submitted to IOP Journal of Optics.
The third paper examines 1D quaternary photonic crystals and associated heterostructures using effective medium theory. The behavior of branch crossing points in the effective refractive index is examined as different parameters in the crystal change. Effective medium theory is then applied to a heterostructure con-   (a) Transmission spectrum about the 3 rd PBG for inversion symmetric quaternary PC with parameters from Fig. 3(a) as well as C = 3, µ C = 1. There is one region where the PBG closes then reopens, at (ξ, d C ) ≈ (0.9996, 0.1461). Ten unit cells are used. (b) Sign ζ (3) is displayed as described by Eq. (4). Colors are kept consistent with Ref. [26].The impedance switches sign after the gaps reopen. (c) Transmission spectrum for combined system of binary and quaternary PCs. For d C = 0, the system is described by Fig. 2(a). For d C = d max C = 0.341, it is described by Fig. 2(d). For 0 < d C < d max C , the system is described by Fig. 2  This manuscript is being prepared for submission in OSA Optics Express.

Abstract
The existence of topological interface states is investigated in a heterostructure consisting of a binary photonic crystal and a quaternary photonic crystal. The individual crystals possess inversion symmetric unit cells. In this work, a photonic crystal is made quaternary by inserting a tunable third layer into two different locations in the unit cell. Conditions are established that describe where the quaternary crystal can exist in parameter space subject to constraints. The closing of band gaps is discussed for different optical path ratios. When the binary and quaternary crystals share an interface, optical states appear at the interface when the two crystals have different signs for the surface impedance. The evolution of the states is displayed as the geometry of the quaternary crystal changes.

Introduction
A photonic crystal (PC) is an periodic array of dielectrics and/or conductors able to scatter electromagnetic (EM) fields, where the scattering elements and incident wavelengths are of similar size [1]. This periodicity implies that a PC possesses discrete translational symmetry. Therefore, as in solid state lattices, EM waves are described using Bloch's Theorem [2], and the wave equation can be solved for the modes allowed in the PC. Destructive interference due to multiple scattering inside the PC produces frequency ranges in which no mode is allowed to propagate through the crystal regardless of crystal momentum (i.e. Bloch wavevector). These regions of reduced transmission are known as photonic band gaps (PBG), although in 1-D systems they are also called stop bands. While investigations about 1-D photonic systems began with Lord Rayleigh [3], PBG research in two and three dimensions did not accelerate until the works of Yablonovitch [4,5] and John [6] about a century later. Applications of PBGs in PCs are numerous, including dielectric mirrors [1], channeling EM modes through photonic slab waveguides [7] and optical fibers [8], and construction of defect states [9,10]. PBGs are also not just limited to PCs; photonic aperiodic structures [11,12,13,14] and certain types of disordered hyperuniform [15] media can also support band gaps, due to isotropy [16,17,18].
When multiple PCs are joined together, a superlattice can be constructed that can possess optical properties not observed in an isolated crystal. Among these properties is the ability to localize EM fields at an interface. This was theoretically demonstrated by Kavokin et al. [19], using two adjacent lossless PCs with different periods. The interface states that developed at the boundary of the individual crystals were called optical Tamm states (OTS), due to similarities with electronic surface modes discovered by Tamm [20], and were found to be strongly dependent on the order of the individual layers in the crystals. Follow up investigations by   [21] demonstrated that these OTSs require the normal wavevector to decrease with distance from the interface on both sides. Since there is a PC on both sides, the only way this can happen is if the wave is trying to propagate in a PBG. This implies that a necessary condition for the formation of interface modes is that PBGs of the individual PCs in the superlattice must overlap. These states were experimentally verified soon after and appear as a sharp peak in transmission spectra [22]. Kang  show that OTSs would appear if impedance matching was satisfied at the interface and confirmed the idea that the order of layers in a unit cell mattered for the appearance of states [19]; however,no physical connection between the appearance of states and the layer order was found [23]. While OTSs were mostly studied in asymmetric unit cell configurations, Vinogradov et al. (2010) [24] showed that symmetric unit cells were also valid. Interface states with symmetric unit cells were classified as optical Shockley [25] states (OSSs); however, it was determined that the underlying physical mechanism that produced OTSs and OSSs was the same, thus all optical states are referred to as Tamm states, although many papers simply call them interface states. While the bulk band structure for an infinite crystal will be unaffected by the symmetry of the unit cell, the exact location of the interface state will shift slightly [24].
Utilizing a symmetric unit cell, a special type of OTS called a topological interface state can be studied. As stated before, the existence of interface states in a superlattice is strongly dependent on the order of the layers comprising the PCs.
Xiao et al. [26] was able to explain the surface impedance of a PC in terms of a topological invariant known as the Zak phase [27]. A Zak phase is assigned to each isolated bulk band in the band structure. It was shown that interface states emerge in the band gaps if these phases change value via a topological phase transition.
One way these phase transitions can occur is if the order of layers in unit cells on one side of the interface is reversed while on the other side it is not. The electric field will acquire a Zak phase (for each band) as the Bloch wavevector, κ, travels on a closed path around the 1 st Brillouin zone. Since the system is 1D, this path is a ring. If the PC is D-dimensional, the trajectory that κ traces in momentum space would exist on the surface of a D-torus. For a PC whose unit cell possesses inversion symmetry, the Zak phase is a convenient measure of topological phase of the band structure as it is constrained to 0 or π, depending on the inversion center [27]. For EM systems with period Λ, the Zak phase can be written as [26]: where i u n,κ ∂ κ u n,κ = i unit cell u * n,κ (z) (z) ∂ κ u n,κ (z) dz represents the Berry connection. In Eq. (2), (z) is the relative permittivity across the unit cell, and u n,κ (z) is the periodic function from Bloch's Theorem, E n,κ (z) = exp(i κ z)u n,κ (z), where E n,κ (z) is the electric field. The label n specifies the isolated band.
A topological interface state will appear if the surface impedances of the PCs on both sides of the interface sum to zero [26]: Since the impedance in a PBG is imaginary, it can be written as Z/Z 0 = iζ, where: with Z 0 being the vacuum impedance. The variable l indicates the number of band crossings (Dirac points) in the band structure below gap n. In a binary PC, a Dirac point will occur if the ratio of the optical path lengths of the two layers is a rational number [26,28].
After Xiao's paper, research in this field began to rapidly expand. Experimental measurements of Zak phase in a 1-D SiO 2 − TiO 2 composite structure were conducted by first measuring reflection phase [29,30]. Other experiments have measured Zak phases through direct observation of interface states [31]. Recent theoretical work has shown that by manipulating the unit cell inversion centers, a superlattice can be designed that supports topological states in every PBG [32].
The robustness of topological states in photonic systems with a finite number of layers has also been examined as the unit cell number varies [33]. Some photonic systems simultaneously support topological and Fano resonances [34]. Other works have extended these ideas to include PCs including metallic layers [35,36,37].
While the concept of interface states at the boundary of two inversion symmetric binary PCs is well understood, the literature is sparse about what happens when an additional layer is added to one of the crystals [38,39,40]. More specifically, we consider inserting an additional layer inbetween every original layer of the binary PC. The binary crystal is displayed in Fig. 1(a) while the new crystal is shown in Fig. 1(b). Despite being composed of three different materials, this new PC is not ternary. In order to keep the unit cell inversion symmetric, two layer C's must be included, thus giving it four layers and being referred to as a quaternary PC. The primes are used to indicate that the thicknesses of layers A and B can be different between the two PCs, even if the respective layers are composed of the same material.
Section III discusses the geometry of the PCs used in this work. With a constant optical path length, as a new layer (C) is introduced to a binary PC, the original layers will grow or shrink, depending on their respective refractive indices.
If this new layer is large enough, the now quaternary PC will become binary again, but with a different periodic configuration. A parameter space with dimensions of the refractive index and thickness of the new layer is also constructed. As properties of the PC are altered, the acceptable region within this parameter space where the PC can exist will change. Section IV discusses topological phase and band crossing conditions for a quaternary PC and contrasts them with the more familar binary PC. Transfer matrix elements are derived for a quaternary PC and crossing conditions are written in terms of a ratio between two propagation phase terms. As this ratio becomes irrational, the first band crossing occurs at higher and higher bands. Next, several examples are given of interface state behavior in a binary-quaternary structure as the geometry changes. It is shown, especially at higher frequencies, that when an additional layer is introduced to make half of the structure quaternary, an interface state can appear but then disappear again before the quaternary PC becomes another different binary crystal. Section V concludes this work.

Geometry
Before discussing the results of this investigation, it is benifical to establish some dimensionless quantities that will make scaling more natural. Since any periodic PCs are constructed of identical unit cells, we need only consider a single unit cell. The period and optical path length of the unit cell are Λ = l A + l B + 2l C and Γ = n A l A + n B l B + 2n C l C , respectively. The individual layers have thicknesses l i and refractive indices n i , for i = {A, B, C}. In this paper, Λ and Γ are held constant. Therefore, we define a parameter, γ ≡ Γ/Λ ≥ 1. The layer thicknesses are made dimensionless with d i ≡ l i /Λ.With this information: In Eqs. (5) and (6), d C is an independent variable. As d C changes, d A and d B increase or decrease depending on the relative sizes of γ, n A , and n B . An example of this behavior is shown in Fig. 3. For the given parameters in Fig. 3 and d C can exist only on the blue, red, and black surfaces, respectively, shown in (RHS of Fig. 2(c)). Note that in Fig. 3(c), since n B = n C , d A is flat 1 . Also, as n C → γ, Eqs. (5) and (6) It is important to reiterate that in the example described in Fig. 3, n A and n B are held constant while n C is free to vary. The surface boundaries in Fig. 3 are shaped by the conditions that the layer thicknesses must be non-negative and their collective sum must be the period (i.e d A + d B + 2d C = 1). In other words, we can imagine viewing Fig. 3(b) from a "top-down" perspective, projecting the surfaces into a 2-D parmeter space of coordinates (d C ,n C ) In this space, setting Eqs. (5) and (6) to 0 and solving each for n C yields: Eqs. (7) and (8) are displayed in Fig. 4 and represent the boundaries in parameter space. The blue curve is Eq. (7) and the red curve is Eq. (8). The shaded area is the region where the PC is allowed to exist. Here γ < n A and both constants are given the same values they had in Fig. 3(a); however, now n B is allowed to change. In Fig. 4(a), since n B < γ, the shaded region is unbounded as n C → ∞, although Eq. (7) asymptotically approaches d C = 0. Fig. 4(a) represents the parameter space for Fig. 3(b). As n B → γ, the shaded area transforms into  difference is that n C = γ serves as a lower bound rather than an upper bound.

Results
Now that the conditions for the existence of an inversion-symmetric 4-layer PC and its connection to the binary crystal have been shown, it is easier to discuss the conditions in which a topological state can form. By using the transfer matrix method (See Appendix), we can extend the familiar transfer matrix elements from a binary unit cell [26,41]: to a symmetric 4 layer unit cell: frequency and c 0 is the speed of light in vacuum. The impedance mismatch terms are defined as: for relative impedance, z i . It is easy to show that Eq. (11) reduces to Eq. (9) and Eq. (12) reduces to Eq. (10) when either will also work, but is a bit more subtle; in order for this case to simplify, the substitution z A → z C must be made. Adding Eq. (11) with its complex conjugate yields the dispersion relation (See Appendix): With Eqs. (11) and (12), we can use the expression for surface impedance in Ref. [26], Note that Eq. (16) where, The value n denotes the band gap number and x represents a decay factor.
As in Ref. [26], calculating the Zak phase for each isolated band, from Eq. (15), requires finding the set of frequencies, ξ, in which Im(t 12 exp(iφ A )) = 0, assuming the center of layer A is chosen as the center of inversion. If such a value of ξ intersects a band n > 0, then for that band, θ zak n = π; for all bands not intersected, θ zak n = 0. For the binary PC, the ξ and thus the Zak phases can be found analytically for all bands. This is done using Eq. (10) [26]: For the 0 th band: For all other bands, sin φ B = 0. A similar procedure can be done for the 4-layer unit cell, using Eq. (12); however, one quickly realizes that now ξ cannot be found analytically. Furthermore, the situation is complicated by the fact that θ zak 0 cannot be separately calculated. While it is still true that θ zak n>0 = π for bands intersected by ξ, this rule does not appear to consistently hold for the 0 th band. In addition, for bands n > 0, there may be instances where two different ξ values intersect the same band. If this happens, that band has θ zak n>0 = 0; two crossings are treated as no crossing. An example of this behavior is displayed in Fig. 6. Note that in Analytic results for band crossings can be obtained if the constraint, M φ C = φ B , is applied, for M ∈ Q, assuming that all layer widths remain non-negative.
Applying this condition to Eqs. (5) and (6) yields: It is easy to check that as M → ∞, d C → 0 while d A and d B reduce to their respective binary expressions. This constraint allows for the following band crossing condition to hold [39,40] 3 : for {m 1 , m 2 , m 3 } ∈ N. Therefore, bands l(m 1 +m 2 +m 3 ) and l(m 1 +m 2 +m 3 )−1 will cross at frequency ξ cross = l(m 1 + m 2 + m 3 )/(2γ), where l ∈ N + . It is productive to illustrate these crossing with examples. In Fig. 7, four examples of band crossings are shown, each with a different M value. To ensure that a crossing exists, all refractive indices are rational numbers. Therefore, Eq. (23) can be written as a trio of non-negative integers. In Fig 7(a), when one of the terms is 0, the PC becomes binary and the first crossing occurs at a low frequency. In fact, for the particular refractive index values used in this example, only the 0 th band is isolated.   Fig. 11. For the quaternary PC let the parameters for layer C be C = 3, µ C = 1 for Fig. 9 and Fig. 10, and C = 1, µ C = 2.25 for Fig. 11. Let us first consider Fig. 9. A transmission map about the 3 rd PBG of the isolated quaternary PC is displayed in Fig. 9(a), showing two transmission deserts. At d C ≈ 0.1461, a Dirac point occurs. In Fig. 9(b), it is shown that this crossing produces a change in the sign of the surface impedance of the gap, thus producing a change in topological phase of the band structure. As in Ref. [26], cyan is negative impedance and magenta is positive; however, the topology changes due to change in d C rather than changes in i or µ i . Ideally, the cyan and magenta parts of the impedance map should meet at a point. The reason why they do not is because the map was created using the transmission map. Everywhere the transmission from Fig. 9(a) was less than some selected percentage (say 0.05), that value would be placed in Fig. 9(b) and assigned the correct color according to Eq. (4). Now if the binary PC is placed next to the quaternary crystal, the new transmission is shown in Fig. 9(c). A topological state can be easily seen in the upper half of the map. For the binary PC, d C = 0, so the impedance in the 3 rd PBG is always negative. As d C increases in the other crystal, its impedance eventually flips sign.
Therefore, in the region of d C values above the transition, Eq. (3) holds and thus a state appears. This is also clearly shown in Fig. 9(d), in which the imaginary part of Eq. (3) is directly plotted. The state can be seen starting from the crossing point.
In a similar manner to Fig. 9(a), Fig. 10(a) displays the transmission map for the 10 th PBG. The main difference now is that there are two points of band gap closure. Unlike the previous case, the gap width undergoes somewhat oscillatory behavior. It can be seen in Fig. 10(b) that the second closing causes the sign of the surface impedance to revert back to the sign it had when d C = 0. This means that the topological state produced at the interface between the binary and quaternary PCs will vanish before d C = d max C . That state is seen in the transmission map in Fig. 10(c). Lastly, Fig. 10 Fig. 2(b). It is not present for the PC configurations corresponding to the extreme values of d C (i.e. Fig. 2(a) for d C = 0 and Fig. 2 . This is in constrast to Fig. 9(d), where the state persisted for d max C .
In Fig. 11(a) and Fig. 11 states are more clearly defined in Fig. 11(d).

Conclusion
The optical properties of inversion symmetric quaternary PCs have been investigated along with the interface states that appear at the boundary of binary and quaternary PCs. First, the geometry of a quaternary PC was discussed. By inserting two tunable layers into a binary PC to make it quaternary, the rate at which all layers in the unit cell change width with respect to this tunable layer was examined. Depending on the refractive indices, as the tunable layer got large enough, the quaternary PC achieved one of three final configurations. Next, inequalites among n A , n B , and γ were established that displayed the quaternary PC in a parameter space consisting of dimensions of the layer thickness and refractive index of the tunable layer. As an extension to binary crystals, it was shown that while it was still possible to determine whether the Zak phase of an isolated band is 0 or π for a quaternary PC by finding the zeros of t 12 , this must be done numerically; however, unlike for binary PCs, a band can have two frequency crossings, which indicates a Zak phase of 0 for that band. Band crossings were also exam-ined in terms of a ratio between two phase coefficents, in this case φ B /φ C . As this ratio became irrational, the first band crossing occured at increasingly higher band numbers. There also appeared to be two different types of band crossings: one, as stated in previous works, that is described by a simple ratio between the individual optical lengths in the unit cell, but also another class that seems to only     (c) Transmission spectrum for combined system of binary and quaternary PCs. For d C = 0, the system is described by Fig. 2(a). For d C = d max C = 0.341, it is described by Fig. 2(d). For 0 < d C < d max C , the system is described by Fig. 2(b). Five unit cells are used for each PC. Note the topological state after the gap reopens. (d) The state is clearly shown by plotting the implicit equation, Z left (ξ) + Z right (ξ, d C ) = 0, where each term is described by Eq. (16).  This manuscript has been submitted to IOP Journal of Optics.

Abstract
Interface states in a 1-D photonic crystal heterostructure with multiple interfaces are examined. The heterostructure is a periodic network consisting of two different photonic crystals. In addition, the two crystals themselves are periodic, with one being made of alternating binary layers and the other being a quaternary crystal with a tunable layer. The second crystal can thus be smoothly transformed from one binary crystal to another. All individual photonic crystals in the superstructure have symmetric unit cells, as well as identical periods and optical path lengths. Therefore, as the tunable layer in the quaternary crystal expands, other layers will shrink. It is found that the behavior of the localized modes in the band gaps is dependent on whether there is an even or odd number of interfaces in the heterostructure. With certain sequences of all dielectric photonic crystals, topological states are shown to split in two, whereas for other heterostructures they are shown to vanish. Additional resonant modes appear depending on how many crystals are in the heterostructure. If the tunable layer is frequency dependent, the band gap can still support topological/resonant modes with some band gaps even supporting two separate groups.

Introduction
A photonic crystal (PC) is a periodic array of dielectrics and/or conductors used to scatter light [1,2]. In a similar manner to how semiconductors control the passage of electrons, PCs possess passbands which allow photons in certain frequency ranges to propagate through the crystal and photonic band gaps (PBGs), which inhibit photon flow, producing regions of suppressed transmission. The existence of these pass and stop bands are governed by Bloch's Thereom. Photonic heterostructure devices are comprised of multiple periodic components that can produce transmission properties and field localization not seen in isolated crystals [3,4]. Heterostructures with a single PC interface have been extensively studied.
Examples of localized behavior are the surface or interface modes, also known as optical Tamm [5] states (OTSs). These modes can exist at a boundary only if their field amplitudes decay away as the distance from the boundary increases in either direction. This means the wavevectors must be imaginary. In the case of a PC, this occurs if the mode is trying to travel through a PBG. These modes have been found in a variety of photonic structures including 1-D [8,6,7,9] and 2-D [10] PC interfaces, air-PC surfaces at oblique angles [11], and PCs bordering media with a graded refractive index [12]. Tamm states have also been investigated in various systems containing a PC with a tunable cap layer adjacent to a uniform medium. Examples include PCs containing superconducting layers [13], systems containing metamaterials, both the PC layers [14,15] and the uniform medium [16], and systems with liquid crystal [17] and chiral [18] cap layers. Note that in Ref. [11], despite the PC being adjacent to a uniform medium with positive dielectric constant, surface modes can still form due to total internal reflection.
The component of the wavevector parallel to the boundary, k , is large enough to cause the normal component, k ⊥ to become imaginary.
A varient of OTSs are the Tamm plasmon-polaritons (TPP) formed at a boundary between a metal and a PC [19,20,21]. In order for a TPP to form, the condition, must be satisfied. The reflection coefficent r metal describes the amplitude of the electric field, incident from the PC side of the interface, reflecting off the metallic surface. In the same manner, r P C describes the electric field amplitude from a wave incident from the metallic side reflecting off the PC surface of the interface.
In the case described in Ref. [19], the TPP is excited at a frequency below the plasma frequency of the metal, implying that r metal = −1. Therefore, to ensure that Eq. 25 remains satisifed, r P C = −1, implying that the higher index material in the PC should be adjacent to the metal. In Ref. [20], the plasmon is produced above the plasma frequency. Since the permittivity of the metal is now positive, r metal flips sign. For the state to exist now, the sign of r P C must also flip, meaning that, in the PC, the low index material is adjacent to the material. Similar to Ref. [11], the state is supported on the metallic side by total internal reflection.
If an interface is generated between two PCs with symmetric unit cells, localized states at the boundary can form that are governed by the bulk band structure of the two crystals. These states are referred to as topological interface states.
Xiao et.al. [22] showed that their existence in a PBG can be predicted by ensuring PCs, the topological state splits. If more layers are added in this scenario, keeping the two ends quaternary, the split state is joined by resonant states.

Methods
Our work was conducted using transfer matrix method (TMM) [33]. Keeping The primes indicate that the individual layer lengths are different from those of the quaternary PC. Note that the PC is capped on both sides with a half-width of layer A, making the unit cells symmetric. The top diagram shows the quaternary PC, which is represented in the middle picture as the light blue regions. In the quaternary PC, layers A and B are the same material as in the binary crystal. Layer C is orange. As in the binary case, the symmetric unit cell is displayed by the double arrow.
length scale, we set γ = Γ/Λ. For the quaternary PC, shown at the top of Fig. 12, the widths of layers A, in green, and B, in blue, can be expressed in terms of a free parameter, the width of the introduced layer, d C , in orange, [32], Note that d C can only take on values in which both Eqs. 27 and 28 are nonnegative. When d C reaches its maximum, the quaternary PC will become binary again, but with configuration, ...CBCBC..., if d A tends to zero, or, ...ACACA..., if d B tends to zero. For the special case, γ = n C , both d A and d B will be zero when d C reaches its maximum; this will result in a uniform layer C. The lengths of the layers in the binary PC, displayed at the bottom of Fig. 12, are simply Eqs. 27 and 28 but with d C = 0 and thus do not change. The index of refraction of layer j is n 2 j = j µ j , where j and µ j are the (relative) permittivites and permeabilites.
In the binary and quaternary crystals, the n A 's are the same and the n B 's are the same, although n A = n B [32].
For the system described in Fig. 12, we only consider an electric field incident from the left, E 1+ . The reflected field is E 1− and the field that is transmitted through the entire structure is E (N +1)+ . To compute the transmission spectra for the system, first we must construct the transfer matrix, M, from the individual interface matrices, I j , and propagation matrices, P j , where the index, j, specifies the layer in question [34], where r j and τ j are the reflection and transmission coefficients, respectively, τ j = 2µ j+1 n j µ j+1 n j + µ j n j+1 (32) In scaled variables, the phase argument, ik j l j becomes 2πin j d j ξ. The frequency, f , becomes ξ = f Λ/c 0 , where c 0 is the speed of light in vacuum. The incident and scattered field are related by, where, The transmitted power is calculated via, Since there is an even number of interfaces in the heterostructure, a single topological peak (Fig. 13(g)) is absent, even though Eq. 26 states that there is a change in the sign of surface impedance between the binary and quaternary components as d C increases from 0 to 0.341. For the transmission maps in the top row, the heterostructure has the form bqb, bqbqb and bqbqbqb. It can be seen that the state in a single interface system splits into two sets of resonances, with one set below the original frequency and the other above. At d C = 0, all these states exist as pass band modes; however, as d C increases, they begin to wander into the band gap. As When the heterostructure changes from bqb to bqbqb, the two states themselves split into pairs such that these pairs ( Fig. 13(b)) each have a higher and lower frequency state relative to their respective states in Fig. 13(a). This splitting is illustrated in Fig. 14. A horizontal slice of Fig. 13(b) at d C = 0.25 is considered, except now the states are plotted for varying thickness of the middle cystral. Each binary and quaternary represent 4 unit cells; b/2 represents 2 unit cells. In Fig. 14(a), we see two distant edge states (blue) in the absence of a middle b: bqqb. To see the two interface states, though, we must zoom into the cluttered middle region. These interface states are clearly seen in Fig. 14(b). When two binary unit cells are inserted in the center of the structure (bq(b/2)qb), there is now strong coupling between the two central states and the two edge states. The edge states rapidly move toward the central region. Inserting another two binary unit cells produces the familiar structure bqbqb and the black transmission profile.

In our first investigation
Doubling the central region causes coupling of the states in each pair to weaken due to the increased distance bewteen the interface pairs bqb. This is seen in the magenta curve as the four peaks mostly merge into two, recovering Fig. 13(a) at d C = 0.25. There is also a new pair of edge states in Fig. 14(a).
An important change occurs in the transmission behavior as d C increases if the sandwiching layers are quaternary. In Fig. 13(d interfaces and reversing the order of the components (bq → qb) will not change the transmission. Fig. 13(g) is the familiar single topological state from heterostruture bq. For Fig. 13(h) & 13(i), the addition of bq layers produces resonant states that behave like those discussed previously.
In our second investigation, layer C is given a permittivity with frequency dependence, in accordance with the Drude model of dispersion, where ξ p and g are the dimensionless plasma and collision frequencies. Eq. 36 is plotted in Fig. 15 with plasma frequency ξ p = 2 and negligible collision frequency g = 10 −10 . Therefore, layer C acts as a metal. Layers A and B remain unchanged.
Since the optical path in metal is not constant with frequency, the layer width defined in Eqs. 27 & 28 are given simplier forms, Now, γ is only relevant when defining the layer widths before the metal is introduced. As d C increases, the band gap closing points are skewed towards higher frequencies due to the behavior of Eqs. 37 & 38. As a concequence of this, toppological states in a single interface bq system do not start and terminate at the closing points nor are they positioned near the center of the gap. An example of this behavior is shown in Fig. 16. In Fig. 16(a), the transmission map is plotted around ξ = 2. The metallic layer, d C , follows the behavior in Fig. 15. Note that we can have a case where one gap (top center) can support two states. The left state is much sharper than the right one. Also worth noting is that the two center states appear to cross the plasma frequency of the metallic layer without anything unusual happening. This is acceptable because the effective plasma frequency of the entire heterostructure is much lower than the plasma frequency of the metallic inclusion, so the effective permittivity of the heterosructure is positive in the region of these states [35]. This means that all visible gaps in Fig. 16(a) are classifed as PBGs. There are also two distinct groups of Fabrey-Perot resonances.
The brighter, more slanted triplets that largely encase the PBGs are caused by coupling among the 3 interfaces of the four unit cells in the quaternary PC. There is also a fainter vertical triplet of resonances between about 1.72 < ξ < 1.9, that is caused by the three interfaces in the binary PC. As d C increases, the leftmost topological state eventually appears to turn into one of these resonances and the rightmost of these states breaks away to become the top-center topological state.
The equation Im(Z b + Z q ) = 0 is plotted in Fig. 16(b), showing the exact location of those five topological states.
As in the all dielectric case, when there are multiple binary/quaternary interfaces, topological states can split; however, the split states are much closer together, meaning that they are more difficult to resolve. Transmission maps for the qbq and bqb configurations are displayed in Fig. 17. While they look very similar to each other and to the single interface system, some subtleties can be pointed out.
Resonances in the qbq system are much sharper compared to those in bqb. Also the splitting can be seen, although it is more pronounced in qbq. Cross sections of the lower center topological state for d C = 0.1 are shown in both structures in Fig. 18 as the number of interface increases. In Fig. 18(a), the transmission is shown for a heterostructures sandwiched between two quaternary PCs. As the number of interfaces increases, each split state itself divides such that the total number equals the number of interfaces. Fig. 18(b) zooms into the left cluster of states. If the heterostructure is bounded by binary PCs, shown in Fig. 18(c), the two central split states appear much closer together. As the number of interfaces increases these two eventually merge and the resultant peak decreases. In the plot, this occurs for six interfaces (bqbqbqb).This makes it appear that there is a missing state; however, similar to the all dielectric heterostructures qb...bq To help understand what is happening within the heterostructure, it is beneficial to compare the optical system to the more familar 1D coupled harmonic oscillator, shown in Fig. 19. The interfaces between the individual PCs act as identical masses and the PCs themselves can be thought of as the spring constants. Since there are two different PCs, two distinct spring constants are used.
In this example, the constant k corresponds to the binary PC while κ corresponds to the quaternary PC or vice virsa. The topological state will split into a number of states corresponding to the number of interfaces. With an even number of interfaces, the central state vanishes and splits such that half are above the original frequency and half are below. Using this analogy with two interfaces, the lower of the two states is the symmetric state while the higher one is the antisymmetric state [36]. With an odd number of interfaces, the central state still splits as in the even case except now the original state remains.This splitting is shown in Fig. 20.
Overall, the original and split frequencies can be related by an average, where N is the number of PCs in the entire structure and the index, i, is summed through all frequencies after the splitting.

Conclusion
We The fabrication of such a structure could be useful for filtering applications.

Abstract
Effective medium theory is used to model a one dimensional lossy dielectric quaternary photonic crystal as a homogeneus slab. The unit cell of the original crystal is inversion symmetric with layer sequence ACBCA. The behavior of the branch frequency singularities in the effective refractive index is investigated as parameters in the layered structure change. A heterostructure composed of multiple photonic crystals is also modeled with effective medium theory. It is shown that the effective refractive index of such a structure possesses regions of normal dispersion that corresponding to localized interface states within the overall regions of anaomalous dispersion corresponding to the photonic band gap.

Introduction
Effective medium theory (EMT) is the modeling of scattering parameters, such as reflection and transmission, from a complex heterogeneous structure by replacing it with a partially or fully homogenized material, in an effort to simplify a problem. While any field of wave mechanics can make use of EMT, it has seen significant applications in electromagnetism (EM). In 1956, Rytov [1] proposed that a one dimensional Bragg grating composed of two different isotropic layers could be thought of as a uniform slab with anisotropic effective permittivity and permeability. Since then, EMT has been used to model both photonic crystals [2,3], which have scattering elements on the order of the incident wavelength, and metamaterials [4], where wavelength is several orders of magnitude larger. In the long wavelength regime, EMT was applied to 2D [5,6,7] and 3D [8,9] periodic photonic crystals (PCs) by approximating them as 1D Bragg gratings. Reflection from a 1D Bragg grating at oblique angles using EMT was examined [10,11].
EMT has also been applied the photonic systems for higher frequencies. In 1976, Yariv and Yeh [12] used an effective index profile representation of a binary PC whose layers have the same optical path to achieve nonlinear phase matching.
It was shown that when a PC is homogenized, the resultant slab behaves as a single negative material inside the photonic band gaps (PBGs) [13]. Later, it was found that a bi-anisotropic parameter (magnetoelectric coupling) is present when homogenizing unit cells without inversion symmetry [14,15]. These works also showed that power expansions in frequency for effective parameters becomes invalid at the start of the first PBG. This breakdown was due to the presence of branch points (i.e. singularites) [14]. Liu [14] also showed that effective index and impedance can be modeled with the Kramers-Kroing relations. Various concepts, such as density of states [16], reflection phase [17,18], and defects [19] have been studied in PCs using EMT.
In this work, we base the design of the laminated PCs from Ref. [20]. The individual layers in the symmetric unit cell are denoted as A, B, and C with refractive indices n A , n B , and n C . The permittivity, and permeability, µ as defined as n 2 i = i µ i The period, Λ and optical path, Γ, across a unit cell are given as constants, where γ = Γ/Λ. As a result, the scaled layer lengths are, From the given and µ in the PC, the transfer matrix method [21,22] is

Methods
While in most scattering problems the objective is to find reflection and transmission spectra based on the given material parameters (i.e. & µ or n & z), with effective medium theory the inverse problem can be performed. In this case, a system has a known or measured spectrum. With this information, a more uniform object can be considered that produces the same scattering. Essentially, this means that for the uniform material to return the same scattering information as the heterogeneous one, the object must act as though it has frequency dependent dispersion rather than spatial dispersion.
Smith et. al [23] provided an analytical method for extracting the effective refractive index and impedance from the transmission and reflection coefficents, t and r respectively. Matching electric and magnetic boundary conditions for a single slab yields, where φ = n ef f k 0 L for freespace wavevector, k 0 = ω/c, and slab length, L. Using dimensionless variables, φ = 2πn ef f ξN . The length must be an integer multiple, N , of the unit cell period, Λ, and ξ = f Λ/c is the scaled frequency. Equations 42 and 43 can then be inverted to produce, Finding the permittivity and permeability is now trivial, = n/z and µ = nz.
It is assumed that the dielectrics in the original unit cell can only have lossy components. Therefore, the positive root is chosen for the impedance in Eq. 44, since the materials are passive [23]. While symmetric unit cells produce well defined, unique values of impedance, it was found that when homogenizing an asymmetric cell, impedance will depend on the order of the layers. This discrepancy is due to the diagonal scattering matrix elements, S 11 and S 22 having different amounts of phase advance [24]. Calculating the effective refractive index is more difficult due to the presence of the cosine function. Mathematically, there are inifinite values of n ef f that satisfy Eq. 45, but, physically, there should only be one.
As before, to avoid phase ambiguity, it is best to consider symmetric unit cells, although n ef f dependence on unit cell asymmetry is less pronounced compared with z ef f [24]. Since the original system is periodic, the thickness of the slab should be taken as small as possible to minimize the number of roots in cos φ; however, since L = N Λ, the smallest value of L is a single unit cell (N = 1) [23,24,25].
If we let n ef f = n R + in I and X = X R + iX I , Eq. 45 can be separated into [23], Re(cos −1 (X(ξ))) + 2πm (46) and where m is an integer that denotes the current branch. Figure 22 shows an example of the effective parameters for a photonic crystal with inversion-symmetric quaternary unit cells. Only 1 unit cell is used. Figure 22a shows the effective impedance using Eq. 44. In Fig. 22b, which displays Eq. 46, branch crossing points, denoted by black dots, only occur in regions of anomalous dispersion, where dn I dξ < 0. Figure 22c displays the real part but without the extra branches. Some early attempts at plotting Re(n ef f ) have resulted in erroneous solutions, such as incorrect branch transitions [26], and incorrect or missing anomalous behavior [27,28]. The corresponding crossing points are shown in the imaginary part in Fig. 22d. The positions of these points can be found by setting X I (ξ) = 0 and are independent of N . This equation will be used in the next section to explore a rich supply of behaviors of the branch crossing points. Increasing N , however, would cause additional crossing points to appear in the regions of normal dispersion, where dn R dξ > 0, with the number of crossing points in these region equal to N − 1.
An alternative method to find n R and n I is to calculate them numerically. Starting from Eq. 45, we can separate out the real and imaginary equations [21], sin(2πN ξn R ) sinh(2πN ξn I ) = −X I (ξ) Note that these equations are both coupled and nonlinear, making solutions challenging to find. As before, the imaginary component is easier to obtain since it is only located in hyperbolic functions, meaning that there is only a finite number of possible values it can have; however, we see again that the real component only appears in circular trigonometric functions, meaning we must be careful to select the correct branch. To proceed, it is best to start at zero frequency or at least very close to it. Here all non-physical solutions will diverge. The frequency ξ is run through a list of values and both components are calculated at each step, gradually building an array of values. The main difficulty is selecting an appropriate initial starting value each interation so that (n R , n I ) converges to the correct solution.
While the numerical method can still be used at higher frequencies (or higher N ), its practicality starts to falter as possible solutions become closer and closer together, making selecting the correct one more of a challenge. Figure 23 illustrates this problem by logarithmically plotting the difference between the analytical and numerical transmission spectra for 1 and 3 uinits cells. Crystal parameters are the same as in Fig. 22. In Fig. 23a, a single unit cell is used, allowing the arguments of the sine and cosine funtions in Eqs. 48 and 49 to remain small enough to allow for proper convergence of n R and n I for all frequencies. In Fig. 23b, three unit cells are used, and it can be clearly seen that there are three regions, marked by arrows, in which agreement is poor. This is a result of Eqs. 48 and 49 converging to an incorrect solution for those frequency ranges. To remedy this problem for higher ξ (or N ), the same initial (n R , n I ) should not be used throughout the simulation. If it appears that (n R , n I ) will converge to non-physical values for a certain frequency range, then the initial point can be altered for that range.

Results and Discussions
The question now is, how do the crossing frequencies in Fig. 22b evolve as different parameters in the quaternary PC change? As mentioned in the previous section, constructing the real part of the effective refractive index requires knowing the appropriate branches such that the index is both continuous and smooth. By understanding how properties such as material loss can influence the location of these crossings, the construction of effective index functions can be improved, while reducing the possibility of connecting adjacent branches incorrectly.
We consider a single inversion symmetric quaternary PC made of three different dielectrics, as in Fig. 21. The optical path to unit cell period ratio is γ = Γ/Λ = 1.5. Only one unit cell is used so that the only crossing points occur in the band gaps. We will consider three different systems and examine the lowest three crossings in each case.
In the first study, Im (  to some saturation level, ξ ∞ , marked by horizontal dashed lines. This is because as Im( A ) grows without bound, d A → 0, in accordance with Eq. 40. In all three figures, as d C increases, there is general trend not only for the saturation frequency to be lower, but also for the initial starting value to be closer to it. This makes sense because as d C increases, d A will go to zero, for the chosen n A , n B , and  Figure 25: Behavior of (a) 1 st (b)2 nd (c) 3 rd crossing point as Im( C ) changes convergence, especially when d C = 0. Also in Fig. 24c, note that when d C = 0, the rate of increase temporarily slows around Im( A ) = 10 before increasing again.
In the second study, Im( C ) is allowed to change, keeping all other parameters constant. All other parameters are unchanged from the first study and A is now lossless. Results are shown in Fig. 25 for six different values of d C . Crossing frequencies asymptotically approach constants as Im( C ) → 0. Note that for larger d C , the graphs abruptly end. This is due to the behavior of the layer thicknesses of the corresponding unit cell, given by Eqs. 40 Fig. 25b and Fig. 25c the frequencies are shown to reach a maximum, at least for smaller d C , before sharply declining.
In the third study , d C is allowed to change. Other parameters are: A = 6 + 10 −8 i, µ A = 1, B = 1, µ B = 1, Re( C ) = 3, µ C = 1. The tiny imaginary part of A produces noticeable behavior in the crossing frequency values for d C ∼ 10 −7 and below. If Im( A ) were zero, then all values would asymptotically approach 0.31682 in Fig. 26a, 0.70029 in Fig. 26b, and 0.95837 in Fig. 26c; however, nonzero Im( A ) elevates the crossing point values in odd band gaps and depresses them in even gaps as d C → 0. Common behavior among all three plots are that as Im( C ) becomes larger, the crossing frequency terminates at lower d C . This is only significant for Im( C ) ≥ 1 and, as in the second study, is due to Eq. 40 eventually becoming negative, which is unphysical. For larger d C , behavior of first, second, and third crossing frequencies is quite different. In Fig. 26a, the frequency rises relatively rapidly, more so for larger Im( C ), before leveling off as d C reaches its maximum value. In Fig. 26b, the frequency achieves a maximum before sharply falling. The maximum for Im( C ) < 1 is present but much smaller. In Fig. 26c, the crossing point frequencies undergo undulatory behavior.
We now consider a heterostructure consisting of a quaternary PC sandwiched between two identical binary PCs ( Fig. 21a but with no C layer) [29]. All crystals in the structure are inversion symmetric and so is the entire structure. A system with a binary PC surrounding by two quaternary PCs would also be valid; however, a system with an equal number of both cystals would not (e.g.quaternary-binary).
This is because such system, as a whole is not inversion symmetric depsite the crystals composing it to being so. We can apply the techniques in Section 3.3 to this system and find the overall effective refractive index just as for an individual PC. Now though, we do not have the liberty of setting the effective slab thickness equal to a unit cell since now periodicity is broken. We must set the slab thickness equal to the total length of the heterostructure. Since both crystals have the same unit cell period, L = (2m + 1)N Λ, where m is an integer. The effective index profile is similar to that described in Ref. [12], except now spatially dispersive behavior, caused by the breaking of periodicity via the three different PCs, is superimposed onto the overall spatial dispersion described previously. In Ref. [12], material dispersion was superimposesd onto spatial dispersion.
Coupling between the two PC interfaces results in two distinct interface states, Re(n ef f ) is displayed in Fig. 27b. The boxed region is displayed in Fig. 27c.
Normally, the band gap is represented with a negative derivative in Re(n ef f ).
Note, however, the two regions marked by arrows that show brief instances of normal dispersion. These parts correspond to the two transmssion peaks. Small loss terms were added to some of the permittivities to make the branches smooth across the transition points. As in the case for an isolated crystal, the band gaps are represented by a large decrease in the index; however, now each branch crossing contains a small resonance, since the heterostructure, as a whole, is not periodic. This break in periodicity also means the exact behavior of the index is now dependent on the number of unit cells in each PC. As mentioned above, this break in periodicity forces us to use the full length of the structure in Eqs. 46 and 47, leading to a higher number density of branches. It is still possible to apply to analytical and numerical techniques disccssed in Section 3.3 to this system, but it may be more tedious.

Conclusion
We have used effective medium theory to describe some properites of quaternary photonic crystals, both isolated and within heterostructures. For an isolated quaternary PC, the behavior of branch crossing points within the anomalous dispersion region was discussed as various parameters changed. As Im( A ) increases, crossing points eventually reach a saturation frequency, that becomes lower for larger d C . In the case of varying Im( C ), keeping all else constant, the crossing points are constant for small Im( C ) but display different behavior depending on the point in question. The highest defined crossing point becomes smaller as d C increases. The roles of Im( C ) and d C are then reversed with d C now as the independent parameter. It is found that asymptotic behavior as d C → 0 is strongly dependent on Im( A ). The effective index is then examined for a binary/quaternary/binary heterostructure. It was found that interface states within the band gap are represented as rapid increases in an otherwise decreasing region of the real part of the refractive index.

APPENDIX Transfer Matrix Method for general quaternary unit cell
In this appendix, the dispersion relation for a PC with a general 4-layer unit cell is derived. The derivation follows a similar format to the binary unit cell, found in [1]. The direction of propagation is normal to the interfaces in the +ẑ direction, with E = E xx and H = H yŷ , where: