1. Introduction

Photonic crystals have been an interesting field of intense research during the past years because of their potential to be applied in the development of all-optical circuits [1–3] and their interesting properties [4–7]. These systems, which constitute periodic arrays in one, two, and three dimensions of dielectric materials, present characteristic regions in the dispersion plane (frequency versus parallel component of the wave vector) that are called band structures, with bands where the light can propagate (the so-called bulk bands) and bands where light cannot propagate (bandgaps).

Although a photonic crystal represents an infinitely periodic system, in engineering applications they are finite or semi-infinite. When a photonic crystal is truncated, electromagnetic excitations can appear [8–14] at the surface of the photonic crystal in contact with any other bulk medium, such as a dielectric [4,8], a metal [16], or even another photonic crystal [15]. These surface modes are waves that propagate along the surface of the photonic crystal with evanescent fields in the perpendicular direction away from the surface plane, constituting a mechanism for energy loss—via tunneling—of localized bulk excitations.

When observed under reflectance or transmittance, surface modes appear as sharp troughs or peaks, respectively, that are similar to Fabry–Perot guided modes. Although Fabry–Perot filters are useful for solving many important design problems, these photonic-crystal–photonic-crystal bandpass filters deserve a detailed study to explore their properties and potential applications. It is worth mentioning that the knowledge of the existence of these modes is not new, since they were first mentioned by Turner and Baumeister [17], but they considered them more a nuisance than a potentially useful bandpass filter. Then Rusell and others in a patent [18] proposed the application of quarter-wave systems, calling them conjugated filters for reasons that we explain below. Unfortunately, no further studies were published about these systems.

Within this background, the present work proposes the study of other kinds of narrow-bandpass filters that present some advantages over the classical Fabry–Perot multiple-cavity filters. We call these filters multiple-junction photonic-crystal–photonic-crystal (MJPC-PC) filters because they are based on the excitation of surface modes by the coupling of different one-dimensional photonic crystals (1DPCs), as will become evident when the theory of such systems is considered.

2. Theory

In the formalism of the characteristic matrix method of propagation of light through multilayers [19], any simple thin film has an associated 2×2 matrix that relates the total electric and magnetic fields between the boundaries of one medium. This matrix is a function of only the parameters of the corresponding jth film:

It is worth mentioning that in the derivation of this matrix we are using the convention of a positive traveling wave E⃗=E⃗₀exp[i(ωt-k⃗·r⃗)] (in SI units), with ω being the angular frequency, t the time, k⃗ the wave vector, and r⃗ the position vector. Here

represents the phase thickness of the layer and d_j its physical thickness;

stands for its optical admittance (the ratio of the total magnetic to electric fields, H/E), and

is the wave vector’s reduced component perpendicular to interfaces. The constant y represents the admittance of vacuum. Here $\overline{\omega }$=ωΛ/2πc and β̄=βΛ/2π represent the reduced frequency and the reduced parallel component of the wave vector, respectively. The arbitrary constant Λ has the same dimension as the thickness of layers, c is the speed of light in vacuum, and n_j is the refractive index of the medium.

It has been shown elsewhere [8, 19, 20] that a basic multilayer system of three layers that is symmetric with respect to a plane passing through its center of symmetry can be represented mathematically by a single equivalent layer that has an associated equivalent characteristic matrix,

with a given equivalent phase thickness δ_e and optical admittance ηe. Considering the symmetrical period of the three layers as the product m(η_p,δ_p /2) m(η_q,δ_q )m(η_p,δ_p /2), we can determine the elements of the matrix:

where η_p, η_q represent the optical admittance of the layers of the period, δ_p, δ_q their respective phase thickness, and

Although Eqs. (5) and (6)–(9) represent a mathematical equivalence, the equivalent functions η_e, δ_e are physically meaningful and allow us to consider a periodic system as though it were a single thin film with peculiar optical properties [8,10].

It is well known that any symmetric system can be represented by a repetition of the basic symmetric three-layer system p/2q p/2 repeated σ times (a truncated 1DPC of σ periods or, equivalently, 2σ+1 layers; see Fig. 1). With this kind of system Λ=2(n_pd_p +n_qd_q ) represents the double optical thickness of a period if we assume that the layers of the symmetric period have the thicknesses d_p /2 and d_q , respectively. In general any arbitrary symmetric system can be considered a single layer with an equivalent phase thickness σδ_e and the same equivalent optical admittance of the basic three-layer period. According to the equivalence theorem [19, 20] and from Eqs. (5), (6), and (8), the equivalent functions δ_e and η_e can be determined:

It has been shown elsewhere [8] that in regions where the equivalent optical admittance η_e is purely imaginary there exists a bandgap in photonic crystal terms or a stop band in thin-film design nomenclature. As can be appreciated from Fig. 2 (upper part), η_e is always real and positive in the bulk bands or the bandpass region. On the other hand, the equivalent phase thickness δ_e also provides important information about the band structure, since the imaginary part of δ_e is nonzero only within the bandgaps, and in those regions Re(δ_e )=mπ, with m being the order of the corresponding bandgap. The real part of δ_e is approximately equal to the phase thickness of the period δ_p +δ_q outside the bandgaps, as is evident from Fig. 2 (lower graph).

 

Fig. 1. Periodic multilayer of symmetric periods. η ₀ and η_s stand for the optical admittance of the incidence and the transmission media, respectively. In this multilayer the indicated symmetric period p/2q p/2 indicates that layers of a material with the refractive index n_p have the physical thickness d_p /2.

Download Full Size | PDF

Thus either of these two functions, η_e or δ_e , can be used as a criteria to determine the limits of stop bands on a 1DPC.

 

Fig. 2. Equivalent functions versus reduced frequency under normal incidence (β̄=0). The system considered is n_p =2.1, n_q =1.444, and n_pd_p =n_qd_q =λ ₀/4, with λ ₀=1516.2 nm. The imaginary part of the equivalent phase thickness given in the lower graph was multiplied by a factor of eight to make the function more clearly visible on the same scale as the real part of the same function. The real part of δ_e is given in units of π radians.

Download Full Size | PDF

3. Conjugated photonic crystals

Let us consider two similar 1DPCs of σ periods: (p/2q p/2)^σ with the equivalent functions η_e, δ_e , and (q/2p q/2)^σ with the equivalent functions $\tilde{\eta }$_e, $\tilde{\delta }$_e. We can consider these systems almost identical, since both have the same period (considering that there are two different constituent alternate materials and not properly a symmetric period), and as a consequence they have the same band structure. The only difference between them is in the most external layers, which are constituted of different materials. The reason to call these systems conjugated becomes clear from Fig. 3, where we have overlapped graphs of the imaginary parts of the equivalent admittance of both systems. As we are mainly interested in the regions within the stop bands, we can appreciate that in these regions the functions are purely imaginary and opposite in sign. Although these functions are not rigorously complex conjugated, we informally call them conjugated 1DPCs (C-PCs).

To emphasize two important properties of these systems that will be useful for studying the MJPC-PC filters, we formulate two theorems.

 

Fig. 3. Imaginary part of equivalent functions of two C-PCs. The parameters are the same as in previous example.

Download Full Size | PDF

Theorem I: The equivalent phase thickness of C-PCs constitute identical functions:

To demonstrate this property, let us consider the equivalent phase thickness of the C-PC (q/2p q/2)^σ :

The right-hand side of Eqs. (12) and (15) are identical, so that

with k being an integer. This means that both functions are equal but displaced by a constant quantity. To find the value of k, let us consider the following physical argument. Since Eqs. (12), (15), and (16) are valid in general, let us consider the limiting case in which n_p =n_q . In this case we can show from Eqs. (12) and (15) that δ_e =δ_p +δ_q =δ̃_e and k=0. As the functions δ_e and δ̃_e are continuous, it follows that in general k=0.

Theorem II: If δ_p =δ_q , the equivalent optical admittance of C-PCs satisfy the relation

If we consider the element M̃₁₂ of the characteristic matrix M̃ of the conjugated C-PC (q/2p q/2)^σ when δ_p =δ_q ,

Rearranging Eqs. (8) and (18) makes it possible to find that

Since, from Eq. (5),

by substituting Eqs. (20) and (21) into Eq. (19) and using Theorem I, we can derive Eq. (17).

4. Electromagnetic surface modes in conjugated photonic crystals

It is well known that, when a 1DPC is truncated, electromagnetic surface modes can appear within the bandgaps when the outermost periods are complete or incomplete [4, 8]. Let us consider an air–1DPC–C-PC system. For simplicity we are assuming that the 1DPC and C-PC are semi-infinite C-PCs, so if the number of periods of the C-PC goes to infinity (σ̃→∞) we can consider the C-PC a bulk material; we are always in the spectral regions within the bandgaps. In this configuration the dispersion relation that determines the existence of surface modes will be given by

which results from assuming a null incident wave in the Fresnel reflection coefficient of the system. In this equation η ₀ represents the optical admittance of the incidence medium. Equation (22) is valid for any number of periods (σ≥2), and it is useful to determine the surface modes of a finite truncated 1DPC that may be present on each one of its surfaces in spite of its having only few periods. In this equation η_e and η̃_e stand for the equivalent optical admittance of the two different photonic crystals, and σ represents the number of periods of the 1DPC. When the phase thickness of the finite crystal goes to infinity (σ→∞), Eq. (22) simplifies to the conditions

In this case Eq. (23) gives the condition for the existence of modes between the incidence medium and the 1DPC; if the incidence medium is a dielectric, this is a condition for the existence of the well-known photonic crystal–bulk dielectric surface modes [10–11]. Equation (24) establishes the condition for the existence of surface modes at the boundary 1DPC-C-PC. According to the analysis given elsewhere [15], the regions in which the optical admittances of the 1DPCs satisfy this condition lie within their corresponding bandgaps, so that the 1DPC-C-PC modes exist within the regions of overlapping bandgaps.

By means of the result of Theorem I and the condition that δ≡δ_p =δ_q , the relation given in Eq. (24) can be expressed as

This equation is valid even under nonnormal incidence. In the particular case in which both layers have the same quarter-wave optical thickness under normal incidence, it is customary to use the subscripts H (for a high-refractive-index layer) and L (for a low-index layer) instead of p and q. As Eq. (25) is independent of the order of the materials, it does not matter whether we take the basic period (H/2 L H/2) or (L/2 H L/2); the position of the modes is unaltered. Since in this case n_Hd_H =n_Ld_L =λ ₀/4,

or

Then, by substituting Eq. (26) into Eq. (25) and solving for the wavelength that denotes the position of the mode λ ₁, we obtain

In this equation we used the subscript 1 for λ to indicate the bandgap number where the mode appears (first bandgap). It is worth mentioning that, since the argument of the cosine function is between zero and one, the denominator of Eq. (28) is less than (or equal to) unity; thus the wavelength where the mode appears, λ ₁, is larger than the design wavelength, λ ₀.

Equation (25) is multivalued in wavelength (or frequency). With the help of Fig. 4 it is possible to find a relation to determine the phase values corresponding to the position of modes at the subsequent bandgaps:

where m=1, 2,… represents the order of the bandgap. By definition δ_m represents the phase thickness at various wavelengths where the modes appear:

Therefore, from Eqs. (28) and (30), the general relation that determines the position of the modes will be

Considering the conjugated systems of Figs. (2) and (3) with nine periods each, joined to form the PC–PC junction, it is possible to see the modes as dips in the regions of stop bands in the reflectance curve in Fig. 5. As before, λ ₀=1516.2 nm, and by using Eq. (31) we can find that λ ₁=1550 nm, λ ₂=501.8 nm, λ ₃=304.6 nm, λ ₄=215.9 nm, etc.

 

Fig. 4. Multiple values of the cosine function of the equivalent phase function.

Download Full Size | PDF

 

Fig. 5. Reflectance as a function of reduced frequency for single-junction system with a C-PC of six periods. The integer numbers in the ω̄ axis represent the order where the bandgaps are centered.

Download Full Size | PDF

In the infrared region around λ ₁=1550 nm it is possible to solve Eq. (31) for λ0 to determine the physical thickness of each layer of the period. If we consider the typical materials (tantalum pentoxide) n_H =2.1 and (silicon dioxide) n_L =1.444, then λ ₀=1516.2 nm, d_H =180.5 nm, and d_L =262.5 nm. In Fig. 6 (solid curve) we have the transmittance of this single mode in the first bandgap for the conjugated 1DPCs of six periods. The peak is not positioned exactly at λ ₁ because the 1DPCs are finite. However, as we increase the number of periods, the position of the peak converges toward λ ₁. Another important feature that deserves some discussion is the width of the peak. In much the same way as in Fabry–Perot modes, the width is also function of the number of periods. Ideally when both C-PCs are infinite the peak should be like a Dirac delta function. This becomes evident from Fig. 6 (dashed curve), where we have graphed the mode for a system with 1DPCs of 14 periods.

It can be shown from the characteristic matrix of the system incidence-medium–1DPC–1DPC*–transmission-medium [8],

that the condition for maximum transmittance in the mode is given by

 

Fig. 6. Transmittance as a function of wavelength. The peak indicated by a solid curve corresponds to 1DPCs of 10 periods each. The peak on the dotted curve illustrates a mode resulting from a system with C-PCs of 14 periods.

Download Full Size | PDF

If for simplicity we assume that the incident and the transmission media are the same (η ₀=η_s ), then by find the matrix product of Eq. (32), substituting into Eq. (33), and using the results of Eqs. (14) and (24) to simplify the expression, we have

In general η_s ≠η_e everywhere in the spectrum. This is also the situation at the wavelength where the mode appears; thus the conclusion is that transmittance is 100% at the position of the mode whenever the C-PCs have the same number of periods (σ=σ̃).

5. Surface modes with nonnormal incidence

When the system considered in the previous section is analyzed for nonnormal incidence (β̄≠0), it is more convenient to use dispersion diagrams (Figs. 7 and 8, both polarizations) that show the band structure for an ideal photonic crystal with an infinite number of periods. In Figs. 7 and 8 shaded zones represent the bulk bands where light can propagate, and white regions represent the bandgaps or high-reflectance zones, except for the lower region in which any electromagnetic wave is evanescent in all media that constitute the system. Surface modes are indicated by dashed lines, and it can be observed that they are located at the same relative position within the bandgaps for both polarizations. In the insets of Figs. 7 and 8 two example of modes are shown, corresponding to the positions marked by diamonds in the dispersion diagrams for each polarization. To determine these modes, we considered the system (H/2 L H/2)³ (L/2 H L/2)³ embedded in a medium with refractive index n ₀=2.0. The angles of incidence were θ_i =46.7° for the TE case and θ_i =50.07° for the TM case.

 

Fig. 7. Band structure for the TE case. The nonshaded bands represent the regions of high reflectance where the surface modes can be present. Surface electromagnetic excitations are indicated by dotted curves. One example of a surface mode is shown in the inset, corresponding to the point indicated by a diamond for β̄=2.5, ω̄=1.717. If we consider an incident medium with refractive index n ₀=2.0, then the mode appears at θ ₀=46.7°. The dotted-dashed line represents the light line for vacuum.

Download Full Size | PDF

 

Fig. 8. Band structure for the TM case. As in Fig. 7, the surface modes are indicated by dashed curves. One example of surface mode under this polarization is shown in the inset corresponding to the point indicated by a diamond for β̄=4, ω̄=2.608. If we consider the incident medium with a refractive index n ₀=2.0, then the mode appears at θ ₀=50.07°. The points where the bandgaps narrow and close represent the Brester points (spheres).

Download Full Size | PDF

6. Multiple-junction filters: coupling of modes

Let us now consider a system (H/2 L H/2)¹² (L/2 H L/2)²⁵ (H/2 L H/2)¹² with two junctions of C-PCs. When the crystals have a number of periods that guarantees a total optical thickness that is greater than a certain decay length, then it seems that only a single mode exists (Fig. 9, dotted curve).

 

Fig. 9. Transmittance as a function of wavelength. The peak on the dotted curve illustrates coupled modes resulting from a system (H/2 L H/2)¹² (L/2 H L/2)²⁵(H/2 L H/2)¹². In this case the peaks are almost overlapping. The double peak indicated by the solid curve corresponds to the system (H/2 L H/2)¹² (L/2 H L/2)¹⁵(H/2 L H/2)¹².

Download Full Size | PDF

In Fig. 10 we graph the amplitude of the electric field for λ ₁=1550 nm. It is worth observing that the amplitude is greatly amplified at the interfaces corresponding to both junctions and decays exponentially as we move away from these surfaces in the perpendicular direction. Indeed, this characteristic, defines a surface mode. When the crystals are finite, the modes become degenerate or coupled to a variable degree. This effect produces a separation or splitting of modes (in the frequency domain) that among other factors is function of the number of periods of the photonic crystal in the middle. This effect can be explained in terms of the strength of the interaction of modes present in each junction. In Fig. 7 (solid curve) the two peaks are clearly separated when the number of periods of 1DPC2, in the middle, is reduced as in the system (H/2 L H/2)¹² (L/2 H L/2)¹⁵(H/2 L H/2)¹².

It becomes evident that the strength of the interaction depends strongly on the number of periods of 1DPC2 and consequently on the degree of separation of the tails of the electric fields of each interface (Fig. 11). In this interaction the decaying length is an important parameter that is inversely proportional to the imaginary part of the equivalent phase thickness of 1DPC2. In other words, the envelope of the electric field is determined by the exponential $e^{-\mid {\delta }_e\mid }$. Another important factor that determines the mode separation is the relative position of the mode within the bandgap, since, as can be observed in the lower graph of Fig. 2, Im(δ_e ) is almost zero at the edges of bandgaps and becomes maximum at the center [21].

 

Fig. 10. Electric field magnitude as a function of the optical thickness normalized to λ ₀/4. This field corresponds to the system (H/2 L H/2)¹² (L/2 H L/2)²⁵(H/2 L H/2)¹² with vacuum as incidence and transmission media.

Download Full Size | PDF

 

Fig. 11. Electric field magnitude as a function of the optical thickness normalized to λ ₀/4. This field corresponds to the system (H/2 L H/2)¹² (L/2 H L/2)¹⁵(H/2 L H/2)¹² with vacuum as incidence and transmission media.

Download Full Size | PDF

It is possible to derive an analytic expression to determine the separation of coupled modes as a function of the number of periods in the case of a double junction with C-PC. The dispersion equation for coupled modes can be considered by analogy to Eq. (22),

This equation results from assuming that the external C-PCs with equivalent admittance η̃_e are of infinite thickness and can be transformed, by use of the result of Theorem II [Eq. (17)], to

where

is valid within the bandgaps. Equation (36) is a quartic equation that has four possible solutions. However, two of them can be discarded by considering the limiting case when σ→∞, so we have

The two solutions of this equation (${\eta }_e^{+}$ and ${\eta }_e^{-}$) determine the values of the equivalent admittance function where the coupled modes will appear in each bandgap. It is worth mentioning that, because we are looking for the frequencies where the coupled modes appear, it is not possible to solve Eq. (38) explicitly for ω̄ as a function of the number of periods σ unless we determine δ_e (ω̄) and ${\eta }_e^{\pm }$(ω̄). If we assume a linear dependence on frequency for

where ω̄₊, ω̄_- are the frequencies at which the coupled modes appear, ω ₁ is below the frequency where the single nonperturbed mode is located, and $\alpha ={[\partial {\eta }_e⁄\partial \varpi ]}_{{\varpi }_1}$. The function α can be determined to good approximation by substituting Eq. (25) into Eq. (13) and deriving

where

In a similar way we can derive a relation for δ_e (ω̄) by substituting Eq. (25) into Eq. (12),

where

Then, by substituting Eq. (42) into Eq. (38),

So, with Eq. (44), the solutions for ${\eta }_e^{\pm }$ in Eq. (38) can be determined explicitly. Finally, if we solve Eqs. (39) for ω ₊ and ω _-,

Although relation (45) is an approximation, it gives rather exact results even for a small number of periods. In Fig. 12 we have graphed the separation of two coupled modes as a function of the number of periods σ of the internal 1DPC. The positions of modes for a five-period internal 1DPC departs from exact values for just a fraction of a nanometer.

Although the behavior of systems with three or more junctions can be explained in a similar way, Eq. (45) is not valid for these cases. However, this calculation gives good insight into the coupling of modes in general.

 

Fig. 12. Separation of two coupled modes as a function of the number of periods of the middle 1DPC in the system.

Download Full Size | PDF

Multiple coupled modes will constitute a band that is similar to supermodes in other systems, i.e. guided modes in periodic multilayers [4,10]. In other words the spectral distribution of these coupled modes is a consequence of the orthogonality of the system modes.

7. Multiple-junction systems for DWDM filters

Because of strong phase dispersion at the bandpass frequencies, the design of multiple-junction narrow-bandpass optical filters has some limitations if we seek applications for the telecommunications industry. Let us consider the triple-junction system (H/2 L H/2)⁹ (L/2 H L/2)¹⁸(H/2 L H/2)¹⁸ (L/2 H L/2)⁹, with n_H =2.1 and n_L =1.444 as before, embedded in glass with a refractive index 1.52 (n ₀=n_s ) at normal incidence. Although this condition does not avoid considering admittance, matching antireflection layers produces a symmetric bandpass. This system represents a triple-junction filter with three coupled surface modes. The electric field profile is shown in Fig. 13, and the transmittance is given in Fig. 14.

In this case the multilayers of 18 periods determine the strength of interaction of these 3 modes and consequently the separation of the bandpass peaks. The number of periods of external multilayers determine the width of the peaks.

At first glance this filter looks promising for dense wavelength division and multiplexing (DWDM) technology [22]. However, the group-delay dispersion (second derivative of the phase retardation under transmission, ∂² φ_t /∂ω ²) is an important feature to be considered with such filters, and in analyzing it we find some limitations (see Fig. 15). It is stipulated that this function should be small and constant near the center of the wavelength where the filter is centered, let us say within the interval [1549.65, 1550.35 nm] (gray dotted lines). This behavior is far from the merit function required to guarantee nondistortion of the pulses passing through these optical elements, and some future research could be interesting to analyze this problem.

 

Fig. 13. Electric field magnitude for a triple-junction filter (H/2 L H/2)⁹ (L/2 H L/2)¹⁸(H/2 L H/2)¹⁸(L/2 H L/2)⁹. The materials are those used in previous examples, and the design wavelength was weakly modified to 1516.18 nm to center the peak at 1550 nm.

Download Full Size | PDF

 

Fig. 14. Transmittance versus wavelength of the triple-junction filter given in Fig. 13.

Download Full Size | PDF

One advantage of MJPC-PC over multiple-cavity Fabry–Perot filters is that the total thickness of a similar multiple-junction system is smaller for a filter in the first bandgap (see Fig. 5), because the mode always appears at a wavelength longer than the design wavelength. In the example considered here the design wavelength was λ ₀=1516.18 nm. This means that the physical thickness of quarter-wave layers are d_H =180.5 nm and d_L =262.5 nm. In the case of a multiple-cavity Fabry–Perot filter the physical thickness of quarter-wave layers will be d_H =184.5nm, and d_L =268.4 nm. This difference of some nanometers in thickness can be important when a large number of layers is considered.

It is worth mentioning the possibility of designing an even thinner system by managing the thickness of the outermost layers of each photonic crystal. In this case the modes will appear more shifted within the bandgaps than the modes we have analyzed here.

In summary, the advantages and disadvantages of these PC–PC modes relative to Fabry–Perot guided modes will become apparent as applications are developed and experimental results accumulate. However, we can be assured in general that the physical and chemical properties of these multilayers and their stability should be very similar, since the number of layers needed for similar applications are almost the same when the same materials are used as building blocks for both kind of multilayers. Important differences can become evident from a design point of view. If we think of the problem of designing a bandpass at a given frequency by using PC–PC systems and Fabry–Perot filters, the resulting multilayers will be quite different, although with approximately the same number of layers. Other important differences will reside in the group-delay dispersion function. It is also possible that the sensitivity to thickness errors in different layers in each system during manufacturing will differ, depending on the specific problem.

 

Fig. 15. Group-delay dispersion as a function of wavelength of the system given in Fig. 13. The vertical axis is given in squared picoseconds.

Download Full Size | PDF

8. Conclusions

The properties of single- and multiple-junction bandpass filters were studied in equivalent systems. In the case of simple junction filters the conditions for existence were stated by our giving one analytical expression for the positions of the modes within the stop bands when the optical thicknesses of the layers of a period are quarter waves. The width of the bandpass was considered qualitatively.

In the case of multiple-junction filters it was possible to derive an analytic expression to determine the separation of the modes in the spectrum as a function of the number of periods of the internal photonic crystal. This was explained in terms of the strength of interaction of both modes simultaneously present in the system.

A multiple-junction system was considered that presents some interesting features to be applied as filters in DWDM technology. The design deserves future research, perhaps adding some layers to improve performance and considering arbitrary truncation of the outermost layers of each photonic crystal.

It is worth mentioning that although for simplicity this study was done by considering one-dimensional photonic crystals, these ideas can be applied to systems of two and three dimensions.