## Abstract

We demonstrate that a series of one-dimensional photonic crystals made of any dielectric materials, with the periods are distributed in a geometrical progression of a common ratio, r<rc (θ,P), where rc is a structural parameter that depends on the angle of incidence, θ, and polarization, P, is capable of blocking light of any spectral range. If an omni-directional reflection is desired for all polarizations and for all incident angles smaller than θo, then r<rc (θo,p), where p is the polarization with the electric field parallel to the plane of incidence. We present simple and formula like expressions for rc, width of the bandgap, and minimum number of photonic crystals to achieve a perfect light reflection.

© 2009 Optical Society of America

Light reflection for all angles of incidence and all polarizations is useful for variety of applications. Achieving light reflection in all angles and for all polarizations requires a photonic crystal (PC) with a huge refractive index contrast, Δ*n* [1–5]. For instance, in one-dimensional (1D) PCs, to block light with frequencies covering the entire visible range and the typical ultraviolet (UV) spectrum from the sun at a normal incidence, Δ*n* of 1.6 and 0.8 are required, respectively [6,7]. For omni-directional reflections, the requirements on Δ*n* are even higher [3,4]. Unfortunately, such large Δ*n* values cannot be realized for applications of light filtering with UV and visible frequencies, as there are no pair of materials that is non-absorptive (in visible or UV) and at the same time exhibiting large Δ*n* [8]. The demand on the Δ*n* in the PC is the limiting factor for successful technological applications. Previously, there were several works on how to enlarge the spectral range for omni-directional reflection [5,9]. In particular, hetero-structures of 1D PCs with each PC possessing omni-directional bandgaps – the spectral range of reflection [5], and each PC possessing bandgaps for different range of incident angles [9] were proposed. In this paper, we demonstrate remarkably simple and practical design principles to achieve light reflection in all angles without any material constraints. We show that a series of 1D PCs of any Δ*n*, with the periods distributed in a geometrical progression of a common ratio, *r*, *r*<*r _{c}* (where

*r*is a structural parameter we introduced), is capable of reflecting light of any spectral range, at any part of the electromagnetic spectrum. The 1D PCs in the series can be fabricated using matured techniques and non-absorptive materials of any refractive indices and, therefore, are of low fabrication cost.

_{c}Firstly we will examine how to create a large bandgap for a particular polarization (*P*) and incident angle (*θ*). Thereafter, omni-directional bandgaps will be considered.

Figure 1 shows a PC hetero-structure formed by a series of *m* number of 1D PCs with the period for the *k*-th PC is *p _{k}* (

*p*

_{k}_{-1}<

*pk*<

*p*

_{k+1}). For a maximum bandgap, let us assume each of these PCs contains two alternating materials of refractive indices,

*n*

_{1}and

*n*

_{2}with quarter wavelength thicknesses,

*p*/(

_{k}n_{2}*n*

_{1}+

*n*

_{2}) and

*p*/(

_{k}n_{1}*n*

_{1}+

*n*

_{2}), respectively. The center wavelength of the bandgap of the

*k*-th PC is

*λ*=

_{k}*p*, where

_{k}/ω_{c}*ω*is the normalized frequency of the bandgap center. The ratio of the width of the bandgap frequencies, Δ

_{c}*ω*, to

*ω*is independent of

_{c}*p*and can be denoted as

_{k}*g*. Both

_{n}*g*and

_{n}*ω*are dependent on

_{c}*P*and

*θ*, and the values can be calculated using the plane wave expansion methodology [10,11], for a PC with an infinite number of unit cells. For

*θ*=0, we have exact analytical equations given by the following expressions [6,7],

Note that for *θ*=0, *g _{n}* and

*ω*are polarization independent. The lower and upper bandgap edges of the

_{c}*k*-th PC can be denoted as

*λ-,*and

_{k}*λ+,*, respectively. These edges can be written as,

_{k}When *g _{n}* is small,

*λ*±,

_{k}≈

*pk*(1±

*g*/2)/

_{n}*ω*.

_{c}Now we assume that each PC in Fig. 1 consists of a large number of alternating layers so that it can perfectly reflect light with all wavelengths that fall within the bandgap. If the bandgap of the adjacent PCs in the hetero-structure overlaps, a large bandgap can be created. This condition can be written as,

When *λ*
_{+,k-1}=*λ*
_{-,k,} from Eq. (3) it can be shown that the periods, *p*
_{1}, *p*
_{2}, *p*
_{3}, …, must obey a geometrical progression with a common ratio, *r _{c}*, given by,

If *g*
_{n} is small, then for *θ*=0, it can be shown that *r _{c}*(

*θ*=0)≈(9

*n*

_{2}+2

*n*

_{1})/(2

*n*

_{2}+9

*n*

_{1}) (with

*n*

_{2}>

*n*

_{1}). Using Eq. (5) and assuming the periods follow a geometrical progression of a common ratio,

*r*, the condition on the wavelength (Eq. (4) can be written as

*r*<

*r*. It is important to remember that the condition,

_{c}*r*<

*r*, is valid only for a particular set of (

_{c}*θ,P*). The question on how the common ratio should be, in order to reflect light in all angles, will be answered in the later part of the paper.

When the PC has a finite number of unit cells, the band edge wavelengths will be slightly different from Eq. (3) and, thus, the common ratio value in Eq. (5) will be different too. For PCs with the finite number of unit cells, Δ*ω/ω _{c}* is slightly bigger than the

*g*of a PC with an infinite number of unit cells [6,7] and, therefore, the common ratio for a PC with

_{n}*N*number of unit cells (see Fig. 1),

*r*, obeys

_{N}*r*>

_{N}*r*. Consequently, the condition

_{c}*r*<

*r*, for creation of the large bandgap, is valid even for the PC with the finite number of unit cells.

_{c}For a hetero-structure with *r*<*r _{c}*, the overall bandgap is only determined by the lower edge of the first PC,

*λ*

_{-,1}, and the upper edge of the last PC,

*λ*

_{+,m}. Using Eq. (3), we can show that the bandgap width to the bandgap center of the hetero-structure,

*g*, to be exactly,

where *g _{p}* is the ratio of the variation in the period, |

*p*-

_{m}*p*

_{1}|, to the average period, (

*p*+

_{m}*p*

_{1})/2. Note that if

*g*/4≪1,

_{p}g_{n}*g*≈

*g*+

_{p}*g*. Equation (6) summarizes two important conclusions. The first is that, even if the refractive index modulation is very small (i.e.,

_{n}*g*≈0), a large bandgap is still possible, if the variation in the period (i.e.,

_{n}*g*) is large. The second is well known [2–4], where in the absence of the variation in the period (i.e., a constant period,

_{p}*g*=0), a large bandgap can be created, if the refractive index modulation (i.e.,

_{p}*g*) is large.

_{n}Assuming *r*<*r _{c}* for a given pair of materials,

*θ*, and

*P*, we can create a bandgap of any size and at any part of the electromagnetic spectrum by just controlling the modulation in the period. So, what is the variation in the period and how many PCs are required to achieve the bandgap for an arbitrary spectral region,

*λ*

_{a}to

*λ*

_{b}? Firstly we set

*λ*-,

_{1}=

*λ*

_{a}and

*λ*

_{+,m}=

*λ*

_{b}in Eq. (3), and find the periods of the first and the last PCs of the hetero-structure. Then, assuming the in between periods follow a geometrical progression with the first and last terms are

*p*

_{1}and

*p*, respectively, the common ratio,

_{m}*r*=(

*p*)

_{m}/p_{1}^{1/m-1}, can be found. The condition in Eq. (3) requires

*r*<

*r*and, therefore, the number of required PCs must satisfy,

_{c}Equation (7) describes the minimum number of required PCs to achieve the bandgap in the spectral region, *λ*
_{a} to *λ _{b}*. When

*g*is very small, the hetero-structure must have a large number of PCs and the period variation must be more continuous [i.e.,

_{n}*g*→0,

_{n}*m*→∞ and hence

*r*=(

*p*)

_{m}/p_{1}^{1/m-1}→1].

Figure 2(a) shows the transmission spectrum (i.e., absolute value of transmission coefficient [6] versus wavelength) at *θ*=0, for the PC hetero-structure (blue curve) with *n*
_{1}=1.45, *n*
_{2}=1.8, *p*
_{1}=125 nm, *θ*=0, *N*=12, and *m*=6. For a comparison, the transmission spectrum of a uniform PC (i.e., a constant period), with the bandgap is designed to be the mid of the visible range is plotted as a green curve in Figs. 2(a). All spectrums are obtained using the transfer matrix method [13] by assuming the ambient medium to be air. As we can see from Fig. 2(a), the hetero-structure produces bandgap in the desired spectral range (i.e., 380nm–780 nm). However, the bandgap region of the hetero-structure (blue curve) exhibits narrow spikes, which are absent in the transmission spectrum of the uniform PC (green curve). The transmission of the narrow spikes can be reduced by increasing *m*, *N*, or the refractive index modulation (i.e., Δ*n*). For example, Figs. 2(b) and 2(c) shows the transmission spectrums for hetero-structures with the specifications as in Fig. 2(a), but *m* is increased from 6 to 8 and 16, respectively. As we can readily verify from the figures, the transmission of the narrow spikes in the bandgap region when *m*=8 [Fig. 2(b)] is smaller than those of *m*=6 [Fig. 2(a)]. When *m*=16 [Fig. 2(c)], the narrow spikes cannot be seen in the bandgap region.

Next we answer the important question; how to make the PC hetero-structure such that it is able to reflect light of all polarizations and all *θ, θ*<*θ*
_{o}? To answer this, we first write *r _{c}* as

*r*(

_{c}*θ,*), and

_{P}*g*as

_{n}*g*(

_{n}*θ,P*). The polarization

*P*can be either

*s*(electric field is perpendicular to the plane of incidence)-polarization or

*p*(electric field is parallel to the plane of incidence)-polarization. In order to achieve a common bandgap for all

*θ*≤

*θ*

_{o}and all polarizations,

*r*should be smaller than the extreme minimum of the function,

*r*(

_{c}*θ,P*). The minimum value can be obtained by finding the first derivative of

*r*(

_{c}*θ,P*) [Eq. (5) with respect to

*g*(

_{n}*θ,P*) as,

The derivative is always positive as |*g _{n}*(

*θ,P*)| is always less than 1. Therefore, the minimum of

*r*(

_{c}*θ,P*) occurs when

*g*(

_{n}*θ,P*) is minimum. The minimum of

*g*(

_{n}*θ,P*) is

*g*(

_{n}*θ*) [determined by the band structure of the

_{o},p*p*-polarization] [3,4]. If the light line is above the point where the bandgap of the

*p*-polarization becomes zero, then, there is no condition on

*θ*. On the other hand, if the corresponding point is above the light line, then,

_{o}*θ*<Brewster angle,

_{o}*θ*[4]. In this case, when

_{B}*θ*

_{o}=

*θ*, we have

_{B}*g*(

_{n}*θ*)=0.

_{o},pHence, an important condition for existence of omni directional bandgaps for both polarizations of light and 0≤*θ*≤*θ*
_{o} is *r*<*r _{c}*(

*θo,p*). With

*r*<

*r*(

_{c}*θo, p*), the band edges for all polarizations and all angles in the range 0<

*θ*<

*θ*will satisfy Eq. (3) and, therefore, the PC hetero-structure will possess a large bandgap at each value of

_{o}*θ*. Once

*r*is fixed, the next thing to do is to appropriately choose the first and the last periods of the hetero-structure based upon the desired spectral range. The first and the last periods of the hetero-structure can be found using Eq. (3) and noting that the overall common bandgap for

*θ*<

*θ*

_{o}is only determined by the band edges of the first and the last PCs when

*θ*=0 and

*θ*=

*θ*, respectively. In most cases,

_{o}*λ*-,

_{1}(

*θ*=0)>

*λ*-,

_{1}(

*θ*=

*θ*

_{o}) and

*λ+*,

_{m}(

*θ*=

_{0})>

*λ*+,

*(*

_{m}*θ*=

*θ*

_{o}) and, hence, the

*p*

_{1}and

*p*can be found using

_{m}*λ*

_{-,1}(

*θ*=0) and

*λ*

_{+,m}(

*θ*=

*θ*

_{o}), respectively [Eq. (3). Once the periods are found, we can follow the same arguments that lead to Eq. (7), to show that

*m*>ln(

*p*)/ln[

_{m}/p_{1}*r*(

_{c}*θo,*)]≈[

_{p}*g*(

_{p}/g_{n}*θ*)]+1.

_{o,p}In order to give an illustration to the approach, we have tabulated the values of *r _{c}*(

*θ*

_{o}=90°,

*p*),

*p*, and the smallest integer values of

_{1}, p_{m}*m*for few materials systems in Tables 1 and 2, for omni-directional light blocking, with UV wavelengths on earth surface (200 nm–400 nm) and visible light (380 nm–780 nm), respectively. Optical and UV materials with the refractive indices shown in Tables 1 and 2 are typical materials and readily available [8,12,14,15]. As we can see from the tables, there is no way to ensure a broad band omni-directional light blocking using a single unit of PC, with the constraint of the available optical and UV materials. We picked the material system with

*n*

_{1}=1.45 and

*n*

_{2}=2.40 in Table 2, to calculate the transmission spectrums with air as the ambient medium, a substrate with refractive index, 1.45, and

*N*=12. The spectrums are shown in Fig. 3 for

*θ*=0, 30°, 45°, 60°, 75°, and 85°. As we can see from the figure, the omni-directional bandgap exists for the visible range of the electromagnetic spectrum.

So far in the analysis, we have neglected the variation of the refractive index with respect to the wavelength of the light (i.e., refractive index dispersion). In reality, every material will exhibit a refractive index dispersion. This dispersion and the material loss are negligible, if the desired spectral range of reflection [at one direction or omni-directions] is located far from the absorption edge wavelengths of the materials.

Let us assume the materials with refractive indices 1.45, 1.8, and 2.4 used in Figs. 2 and 3 to be silica [8], amorphous silicon nitride (a-Si_{1-x}N* _{x}*) with

*x*=0.56 [15], and diamond [14], respectively. The absorption edge wavelengths for silica, amorphous silicon nitride, and diamond are 180 nm, 300 nm, and 240 nm, respectively [8,14,15]. The refractive index dispersions of these materials can be obtained using Sellmeier Eqs. (8),14–15], and they are shown in Fig. 4(a). For an extensive listing of Sellmeier equations, for optical and UV materials, refer to Ref. 8. In the visible range, the refractive index of the diamond changes about 0.07 [from 2.47 (violet) to 2.40 (red)]. The similar change for silica and amorphous silicon nitride are about 0.02 and 0.08, respectively [Fig. 4(a)]. Figure 4(b) shows transmission spectrums for

*θ*=0 with the same parameters as in Fig. 3, but with the dispersions of diamond and silica [Fig. 4(a)] are included (dark green curve) and excluded (red curve). As we can readily verify from Fig. 4(b), both spectrums share good agreements, except at the wavelengths closer to the lower bandgap edge (i.e., wavelengths closer to the absorption edge wavelength of the diamond).

The overall omni-directional bandgap is only determined by the lower bandgap edge wavelength when *θ*=0, and the upper bandgap edge wavelength of the *p*-polarization when *θ*=90° (see Fig. 3). To analyze the effect of the refractive index dispersions [Fig. 4(a)] to the omni-directional bandgap of the hetero-structure with parameters as in Fig. 3, let assume the bandgap region as a region with the transmissions below 0.3. When *θ*=0, the lower bandgap edge wavelength increases from 377.6 nm (with the dispersion excluded) to 383.9 nm (with the dispersion included). For *θ*=90°, the upper bandgap edge wavelength of the *p*-polarization increases from 773.7 nm (with the dispersion excluded) to 776.0 nm (with the dispersion included). Therefore, the width of the omni-directional bandgap reduces from 396.1 nm (with the dispersion excluded) to 392.1 nm (with the dispersion included).

A compact picture of the omni-directional reflection can be obtained using an angle-averaged reflectance (AAR) spectrum [16,17]. The overall effect of the refractive index dispersions to the omni-directional reflection also can be justified using the AAR spectrum. The calculated AAR spectrums according to [17], for a hetero-structure with the parameters as in Fig. 3, are shown in Figs. 5(a) and 5(b) for the *s* and *p*-polarizations, respectively. The blue curves in these figures represent AAR spectrums with the refractive index dispersions are neglected, while the red curves in these figures represent AAR spectrums with the dispersions in Fig. 4(a) included. As we can see from Figs. 5(a) and 5(b), the spectral range of the omni-directional reflection for the *s*-polarization is wider than the corresponding range for the *p*-polarization. This is consistent with the transmission spectrums shown in Fig. 3, where we can find that the bandgap of the *s*-polarization is always wider than the bandgap of the *p*-polarization for all *θ*=0, 30°, 45°, 60°, 75°, and 85°. The AAR spectrums also clearly indicate that the effect of the dispersion is larger for the UV wavelengths compared to the infra-red wavelengths.

In summary, we have presented the general formulation of light reflection using a series of 1D PCs with different periods, made of non-absorptive materials of any refractive indices. In order to have a large bandgap – the spectral range of reflection, the periods of 1D PCs must be distributed in a geometrical progression with a common ratio, *r*, smaller than a maximum value of *r _{c}*. The paper have presented exact expressions for

*r*, bandgap to mid gap ratio of the PC hetero-structure, and the minimum number of PCs to achieve the desired range of bandgap in single and all angles of incidence. The proposed method can be used to design filters for vast range of applications such as UV filters (i.e., sunglasses, eye safety glasses, UV photography filters) and visible light filters.

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