One-dimensional photonic crystals with a sawtooth refractive index: another exactly solvable potential

Recently there has been a renewal of interest in the properties of one-dimensional (1D) photonic crystals based on graded index slabs. Rauh et al [1] have considered in some detail optical properties of periodic systems consisting of slabs whose permittivities n²(z) increase linearly with depth z. They also considered periodic systems in which alternating layers have quadratically increasing and decreasing permittivities. Their work pointed out the relevance of such 1D photonic crystals for technological applications. They stressed in particular that to a good approximation all the bandgaps have the same width.

Wu et al [2] considered graded crystals with an isosceles triangle profile of the refractive index on the period. They found that the bandgap structure of such crystals is substantially different from the structure of conventional flat-layered crystals: bands can be wider or narrower, depending on the particular form (upward or downward fold) of the triangle. This evidently widens the application areas of 1D photonic crystals in general. However, their results were only obtained numerically. In subsequent publications [3, 4], the same group considered a sawtooth periodic refractive index, and obtained transmissivities greatly exceeding unity, which must raise some doubts about their method.

Very recently, Guasti and Diamant [5] also considered periodic triangular stacks. They found that reflection characteristics are similar to those of a rugate structure if the proper apodization is chosen. The results also illustrate an approximate analytical method for modelling light propagation in inhomogeneous media with discontinuities in the derivatives of the refractive index, developed by the same authors [6].

In this paper we analyse the properties of sawtooth crystals, i.e. 1D photonic crystals constructed of layers with linearly increasing refractive index n(z) in each period, giving a quadratic increase of the permittivity n²(z). Our approach is substantially analytical and, as a result, possible numerical flaws are avoided.

To be precise, for a slab extending from z = 0 to d we take

$$\begin{equation} n(z) = n_a + (n_b-n_a)\frac{z}{d} \equiv \frac{n_b-n_a}{d}(z-z_0) \end{equation} \tag{ 1 }$$

with z₀ ≡ − n_ad/(n_b − n_a). As seen in figure 1 the refractive index is periodic, increasing linearly from n_a to n_b inside each layer, and falls sharply at the cell boundaries. From data in Adachi [7] one finds that the permittivity of the ternary alloy Al_xGa_1−xAs is approximately linear in the relative content x > 0.4 of Al for wavelengths 10 000 > λ > 6000 Å, so photonic crystals with linearly graded refractive index layers are within present technological possibilities.

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We will express the dispersion equation for the bandgaps of these photonic crystals as well as their reflectance and transmittance in terms of Bessel functions of fractional order. This adds to a group of exactly solvable models of periodic refractive indices which is restricted at the moment to flat-layered periodic structures (binary and ternary photonic crystals) and some specific sinusoidal periodic potentials [8–10]. By means of asymptotic expansions we derive approximate sinusoidal expressions showing that the bandgaps of our sawtooth periodic potential tend to a constant width as the wavelength decreases. For simplicity, as in Rauh's work [1], we restrict our attention to normal propagation.

Description of optical wave propagation of linearly polarized light of circular frequency ω through a periodic structure, with real refractive index n(z + d) = n(z), in case of normal incidence, reduces from Maxwell's equations to the Helmholtz equation

$$\begin{equation} \frac{{\rm d}^2E(z)}{{\rm d}z^2}+k_0^2 n^2(z) E(z) =0, \end{equation} \tag{ 2 }$$

where the total electric and magnetic fields of propagating light are expressed as

$$\begin{equation} \begin{array}{l} {\bf E} = E(z)\exp(-\mathrm{i}\omega t)\,\hat{{\bf y}},\\ {\bf B} = \displaystyle\frac{\rm i}{k_0}\frac{{\rm d}E(z)}{{\rm d}z}\exp(-\mathrm{i}\omega t)\,\hat{{\bf x}}. \end{array} \end{equation} \tag{ 3 }$$

Applying the W transfer matrix method for periodic potentials [11], the field E(z) and its derivative dE(z)/dz ≡ E'(z) at the edge points of the system z = 0 and Nd are related by

$$\begin{equation} \left[ {\begin{array}{@{}c@{}} {E( 0 )} \\ {E'( 0 )} \\ \end{array}} \right] = \left[ {\begin{array}{@{}cc@{}} {v'( d )} & { - v( d )} \\ { - u'( d )} & {u(d)} \\ \end{array}} \right]^{\,N} \left[ {\begin{array}{@{}c@{}} E( Nd ) \\ E'( Nd ) \\ \end{array}} \right], \end{equation} \tag{ 4 }$$

where u(z) and v(z) are the so-called normalized solutions, which satisfy the boundary conditions

$$\begin{equation} \left[ {\begin{array}{@{}c@{}} {u( 0 )} \\ {u'( 0 )} \\ \end{array}} \right] = \left[ {\begin{array}{@{}c@{}} 1 \\ 0 \\ \end{array}} \right], \quad \left[ {\begin{array}{@{}c@{}} {v( 0 )} \\ {v'( 0 )} \\ \end{array}} \right] = \left[ {\begin{array}{@{}c@{}} 0 \\ 1 \\ \end{array}} \right]. \end{equation} \tag{ 5 }$$

The Nth power W^N of a unimodular W-matrix as occurs in equation (4) can be expressed in terms of W and the unit matrix $[\hat {1}]$

$$\begin{equation} W^{N}= \frac{\sin\,N\phi}{\sin\phi}\,\,\,W -\frac{\sin(N-1)\phi}{\sin\phi}\,\,\,[\hat{1}], \end{equation} \tag{ 6 }$$

where the dispersion equation for the Bloch phase ϕ takes the form

$$\begin{equation} \cos\phi \equiv \textstyle\frac{1}{2}\,{\rm Tr}\,W = \textstyle\frac{1}{2}[u(d)+v'(d)]. \end{equation} \tag{ 7 }$$

The advantage of the transfer matrix method is that once we have solved for W of a single layer, the result for N layers is trivial, by use of equation (6).

In dimensionless units, equation (2) with the refractive index defined by equation (1), takes the form

$$\begin{equation} \frac{{\rm d}^2E(\tilde{z})}{{\rm d}\tilde{z}^2} + \frac{1}{4}\tilde{z}^2 E(\tilde{z}) = 0, \end{equation} \tag{ 8 }$$

where the dimensionless variable $\tilde {z}$ is

$$\begin{equation} \tilde{z} = \sqrt{\gamma}\,(z-z_0), \quad \gamma \equiv \frac{2 k_0 (n_b-n_a)}{d}. \end{equation} \tag{ 9 }$$

It is straightforward to prove (see appendix A) that the normalized solutions of equation (8) are Bessel functions of order ν = ± 1/4, multiplied by $\tilde {z}^{1/2}$, i.e.

$$\begin{equation} \begin{array}{l} \tilde{u}(\tilde{z}) = {\displaystyle\frac{\pi}{2^{1/4}\Gamma(1/4)}}\,\tilde{z}^{1/2} J_{-1/4}\left(\displaystyle\frac{\tilde{z}^2}{4}\right) \sim 1 - \displaystyle\frac{\tilde{z}^4}{48} \ldots, \\ \tilde{v}(\tilde{z}) = \displaystyle\frac{\pi}{2^{3/4}\Gamma(3/4)}\,\tilde{z}^{1/2} J_{1/4}\left(\displaystyle\frac{\tilde{z}^2}{4}\right) \sim \tilde{z} - \displaystyle\frac{\tilde{z}^5}{80} \ldots . \end{array} \end{equation} \tag{ 10 }$$

Using standard relations among Bessel functions (see chapter 9 in [12]) one finds

$$\begin{equation} \begin{array}{l} \displaystyle\frac{{\rm d}\tilde{u}(\tilde{z})}{{\rm d}\tilde{z}} = -\displaystyle\frac{\pi}{2^{5/4}\Gamma(1/4)} \tilde{z}^{3/2} J_{3/4}\left(\displaystyle\frac{\tilde{z}^2}{4}\right),\\ \displaystyle\frac{{\rm d}\tilde{v}(\tilde{z})}{{\rm d}\tilde{z}} = \displaystyle\frac{\pi}{2^{7/4}\Gamma(3/4)} \tilde{z}^{3/2} J_{-3/4}\left(\displaystyle\frac{\tilde{z}^2}{4}\right). \end{array} \end{equation} \tag{ 11 }$$

If we now change the variable $\tilde {z}$ in equation (10) back to the original variable z, we obtain another set of the fundamental solutions, $\tilde {u}(z)$ and $\tilde {v}(z)$, of equation (2)

$$\begin{equation} \begin{array}{l} \tilde{u}(z) = \displaystyle\frac{\pi}{2^{1/4} \Gamma(1/4)}\,\left[\sqrt\gamma(z-z_0)\right]^{1/2}J_{-1/4}\left[\displaystyle\frac{\gamma(z-z_0)^2}{4}\right], \\ \tilde{v}(z) = \displaystyle\frac{\pi}{2^{3/4}\Gamma(3/4)}\,\left[\sqrt\gamma(z-z_0)\right]^{1/2} J_{1/4}\left[\displaystyle\frac{\gamma(z-z_0)^2}{4}\right]. \end{array} \end{equation} \tag{ 12 }$$

To construct the matrix W of equation (4), one has to express u(z) and v(z) in terms of $\tilde {u}(z)$ and $\tilde {v}(z)$. Taking into account equation (5), one obtains

$$\begin{equation} \begin{array}{l} u(z) =\displaystyle\frac{\tilde{v}'(0)}{\tilde{w}(0)}\,\tilde{u}(z)-\displaystyle\frac{\tilde{u}'(0)}{\tilde{w}(0)}\,\tilde{v}(z),\\ v(z) =-\displaystyle\frac{\tilde{v}(0)}{\tilde{w}(0)}\,\tilde{u}(z)+\displaystyle\frac{\tilde{u}(0)}{\tilde{w}(0)}\,\tilde{v}(z), \end{array} \end{equation} \tag{ 13 }$$

where the Wronskian $\tilde {w}(z)$ of the fundamental solutions $\tilde {u}(z)$ and $\tilde {v}(z)$ of equation (2) at the point z = 0 is

$$\begin{equation} \tilde{w}(0) = \tilde{u}(0)\tilde{v}'(0)-\tilde{v}(0)\tilde{u}'(0)=\sqrt\gamma. \end{equation} \tag{ 14 }$$

With the aid of newly introduced dimensionless parameters,

$$\begin{equation} \begin{array}{l} z_a = -\sqrt\gamma\,z_0 = \left(\displaystyle\frac{2k_0d\,n_a^2}{n_b-n_a}\right)^{1/2},\\ z_b = \sqrt\gamma\,(d-z_0) = \left(\displaystyle\frac{2k_0d\,n_b^2}{n_b-n_a}\right)^{1/2}, \end{array} \end{equation} \tag{ 15 }$$

the values of $\tilde {u}(0)$, $\tilde {v}(0)$, $\tilde {u}'(0)$, $\tilde {v}'(0)$ take the form

$$\begin{equation} \begin{array}{l} \tilde{u}(0) = \displaystyle\frac{\pi}{2^{1/4}\Gamma(1/4)}z_a^{1/2}J_{-1/4}\left(\displaystyle\frac{z_a^{2}}{4}\right),\\ \tilde{v}(0) = \displaystyle\frac{\pi}{2^{3/4}\Gamma(3/4)}z_a^{1/2}J_{1/4}\left(\displaystyle\frac{z_a^{2}}{4}\right),\\ \tilde{u}'(0) \equiv \left.\displaystyle\frac{{\rm d}\tilde{u}(z)}{{\rm d}z}\right|_{z=0}= -\sqrt\gamma\,\,\displaystyle\frac{\pi}{2^{5/4}\Gamma(1/4)}z_a^{3/2}J_{3/4}\left(\displaystyle\frac{z_a^2}{4}\right),\\ \tilde{v}'(0) \equiv \left.\displaystyle\frac{{\rm d}\tilde{v}(z)}{{\rm d}z}\right|_{z=0}= \sqrt\gamma\,\,\displaystyle\frac{\pi}{2^{7/4}\Gamma(3/4)}z_a^{3/2} J_{-3/4}\left(\displaystyle\frac{z_a^{2}}{4}\right), \end{array} \end{equation} \tag{ 16 }$$

while at z = d, $\tilde {u}(d)$, $\tilde {v}(d)$, $\tilde {u}'(d)$, $\tilde {v}'(d)$ are given by

$$\begin{equation} \begin{array}{l} \tilde{u}(d)= \displaystyle\frac{\pi}{2^{1/4}\Gamma(1/4)}z_b^{1/2}J_{-1/4}\left(\displaystyle\frac{z_b^{2}}{4}\right),\\ \tilde{v}(d) = \displaystyle\frac{\pi}{2^{3/4}\Gamma(3/4)}z_b^{1/2}J_{1/4}\left(\displaystyle\frac{z_b^{2}}{4}\right),\\ \tilde{u}'(d) \equiv \left.\displaystyle\frac{{\rm d}\tilde{u}(z)}{{\rm d}z}\right|_{z=d}= -\sqrt\gamma\,\,\displaystyle\frac{\pi}{2^{5/4}\Gamma(1/4)}z_b^{3/2}J_{3/4}\left(\displaystyle\frac{z_b^{2}}{4}\right),\\ \tilde{v}'(d) \equiv \left.\displaystyle\frac{{\rm d}\tilde{v}(z)}{{\rm d}z}\right|_{z=d}= \sqrt\gamma\,\,\displaystyle\frac{\pi}{2^{7/4}\Gamma(3/4)}z_b^{3/2}J_{-3/4}\left(\displaystyle\frac{z_b^{2}}{4}\right). \end{array} \end{equation} \tag{ 17 }$$

The elements of the W-matrix are then

$$\begin{equation} \fl \hspace{-5.5pt}\begin{array}{l} W_{11} = v'(d)= \displaystyle\frac{\pi}{2^{5/2}}\displaystyle\sqrt{\displaystyle\frac{n_b}{n_a}}z_az_b[J_{1/4}(z_a^2/4)J_{3/4}(z_b^2/4) +J_{-1/4}(z_a^2/4)J_{-3/4}(z_b^2/4)],\\ \displaystyle\frac{W_{21}}{k_0} = - \displaystyle\frac{u'(d)}{k_0}= \displaystyle\frac{\pi}{2^{5/2}}\displaystyle\sqrt{n_an_b}z_az_b [J_{-3/4}(z_a^2/4)J_{3/4}(z_b^2/4) -J_{3/4}(z_a^2/4)J_{-3/4}(z_b^2/4)],\\ k_0W_{12} = - k_0v(d) = \displaystyle\frac{\pi}{2^{5/2}}\displaystyle\sqrt{\displaystyle\frac{1}{n_an_b}}z_az_b [J_{1/4}(z_a^2/4)J_{-1/4}(z_b^2/4) -J_{-1/4}(z_a^2/4)J_{1/4}(z_b^2/4)],\\ W_{22} = u(d) = \displaystyle\frac{\pi}{2^{5/2}}\displaystyle\sqrt{\displaystyle\frac{n_a}{n_b}}z_az_b [J_{-3/4}(z_a^2/4)J_{-1/4}(z_b^2/4) +J_{3/4}(z_a^2/4)J_{1/4}(z_b^2/4)], \end{array} \end{equation} \tag{ 18 }$$

while the dispersion equation (7) takes the form

$$\begin{eqnarray} &&\fl\cos \phi = \frac{\pi}{2^{7/2}}\,\,z_a z_b \left[\sqrt{\frac{n_a}{n_b}}\, J_{-3/4}(z_a^2/4)\,J_{-1/4}(z_b^2/4) + \sqrt{\frac{n_a}{n_b}}\,J_{3/4}(z_a^2/4)\,J_{1/4}(z_b^2/4)\right.\nonumber\\ &&\left. + \sqrt{\frac{n_b}{n_a}}\,J_{1/4}(z_a^2/4)\,J_{3/4}(z_b^2/4) + \sqrt{\frac{n_b}{n_a}}\,J_{-1/4}(z_a^2/4)\,J_{-3/4}(z_b^2/4)\right]. \end{eqnarray} \tag{ 19 }$$

Equation (19) is the exact dispersion relation for our sawtooth 1D photonic crystal. The amplitude reflection r and transmission t coefficients are

$$\begin{equation} \begin{array}{l} r = \displaystyle\frac{W^N_{11}-\frac{n_{\rm ex}}{n_{\rm in}}W^N_{22}+i\left[k_0n_{\rm ex}\,W^N_{12}+\frac{1}{k_0n_{\rm in}}W^N_{21}\right]} {W^N_{11}+\frac{n_{\rm ex}}{n_{\rm in}}W^N_{22}+i\left[k_0n_{\rm ex}W^N_{12}-\frac{1}{k_0n_{\rm in}}W^N_{21}\right]},\\ t = \displaystyle\frac{2}{W^N_{11}+\frac{n_{\rm ex}}{n_{\rm in}}W^N_{22}+i\left[k_0n_{\rm ex}\,W^N_{12} -\frac{1}{k_0n_{\rm in}}W^N_{21}\right]}, \end{array} \end{equation} \tag{ 20 }$$

where the elements of the W-matrix are given in exact form by equation (18) and the elements of the W^N-matrix are obtained by use of equation (6). The indices of refraction in the incident and exit media are n_in and n_ex as seen in figure 1.

The above results can easily be extended to include an isosceles-triangle refractive index profile, as explained in appendix B.

The motivation for exploring those is to have simple analytic approximations for the Bloch phase cos ϕ, as well as for the reflection and transmission coefficients. Using Hankel's asymptotic formula (section 9.2 of [12]) one obtains the elements of the W-matrix in the form

$$\begin{eqnarray} &&\fl W_{11} = \sqrt{\frac{n_b}{n_a}}\left[P_{1/4}(z_a^2/4)\,P_{3/4}(z_b^2/4) + Q_{1/4}(z_a^2/4)\,Q_{3/4}(z_b^2/4)\right]\cos\left(z_a^2/4-z_b^2/4\right) \nonumber\\ &&+ \sqrt{\frac{n_b}{n_a}}\left[P_{1/4}(z_a^2/4)\,Q_{3/4}(z_b^2/4) - Q_{1/4}(z_a^2/4)\,P_{3/4}(z_b^2/4)\right]\sin\left(z_a^2/4-z_b^2/4\right), \nonumber\\ &&\fl \frac{W_{21}}{k_0} = -\sqrt{n_a\,n_b}\left[P_{3/4}(z_a^2/4)\,P_{3/4}(z_b^2/4) + Q_{3/4}(z_a^2/4)\,Q_{3/4}(z_b^2/4)\right]\sin\left(z_a^2/4-z_b^2/4\right) \nonumber\\ &&+ \sqrt{n_a\,n_b}\left[P_{3/4}(z_a^2/4)\,Q_{3/4}(z_b^2/4) - Q_{3/4}(z_a^2/4)\,P_{3/4}(z_b^2/4)\right]\cos\left(z_a^2/4-z_b^2/4\right), \nonumber\\ &&\fl k_0W_{12} = \frac{1}{\sqrt{n_a\,n_b}}\, \left[P_{1/4}(z_a^2/4)\,P_{1/4}(z_b^2/4) + Q_{1/4}(z_a^2/4)\,Q_{1/4}(z_b^2/4)\right]\sin\left(z_a^2/4-z_b^2/4\right) \nonumber\\ &&+ \frac{1}{\sqrt{n_a\,n_b}}\left[P_{1/4}(z_b^2/4)\,Q_{1/4}(z_a^2/4) - P_{1/4}(z_a^2/4)\,Q_{1/4}(z_b^2/4)\right]\cos\left(z_a^2/4-z_b^2/4\right),\nonumber\\ &&\fl W_{22} = \sqrt{\frac{n_a}{n_b}}\left[P_{1/4}(z_b^2/4)\,P_{3/4}(z_a^2/4) + Q_{1/4}(z_b^2/4)\,Q_{3/4}(z_a^2/4)\right]\cos\left(z_a^2/4-z_b^2/4\right) \nonumber\\ &&+ \sqrt{\frac{n_a}{n_b}}\left[P_{3/4}(z_a^2/4)\,Q_{1/4}(z_b^2/4) - P_{1/4}(z_b^2/4)\,Q_{3/4}(z_a^2/4)\right]\sin\left(z_a^2/4-z_b^2/4\right),\nonumber\\ \end{eqnarray} \tag{ 21 }$$

where

$$\begin{equation} \fl \hspace{-6pt}\begin{array}{l} P_{\nu}(y) = 1 -\displaystyle\frac{(4\nu^2-1)(4\nu^2-9)}{2!(8y)^2}- \frac{(4\nu^2-1)(4\nu^2-9)(4\nu^2-25)(4\nu^2-49)}{4!(8y)^4} \ldots ,\\ Q_{\nu}(y) = \displaystyle\frac{4\nu^2-1}{8y} -\frac{(4\nu^2-1)(4\nu^2-9)(4\nu^2-25)}{3!(8y)^3} \ldots \end{array} \end{equation} \tag{ 22 }$$

and the dispersion equation in the form

$$\begin{eqnarray} &&\fl \cos\,\phi = \frac{1}{2}\sqrt{\frac{n_b}{n_a}}\left[P_{1/4}(z_a^2/4)\,P_{3/4}(z_b^2/4)+ Q_{1/4}(z_a^2/4)\,Q_{3/4}(z_b^2/4)\right]\cos\left(z_a^2/4-z_b^2/4\right) \nonumber\\ &&+\frac{1}{2}\sqrt{\frac{n_a}{n_b}}\left[P_{1/4}(z_b^2/4)\,P_{3/4}(z_a^2/4)+ Q_{1/4}(z_b^2/4)\,Q_{3/4}(z_a^2/4)\right]\cos\left(z_a^2/4-z_b^2/4\right) \nonumber\\ &&+ \frac{1}{2} \sqrt{\frac{n_b}{n_a}}\left[P_{1/4}(z_a^2/4)\,Q_{3/4}(z_b^2/4) - Q_{1/4}(z_a^2/4)\,P_{3/4}(z_b^2/4)\right]\sin\left(z_a^2/4-z_b^2/4\right)\nonumber\\ &&+ \frac{1}{2} \sqrt{\frac{n_a}{n_b}}\left[P_{3/4}(z_a^2/4)\,Q_{1/4}(z_b^2/4) - P_{1/4}(z_b^2/4)\,Q_{3/4}(z_a^2/4)\right]\sin\left(z_a^2/4-z_b^2/4\right).\nonumber\\ \end{eqnarray} \tag{ 23 }$$

Equations (21)–(23) are the expressions we sought. By truncating the terms P_ν and Q_ν various asymptotic approximations are obtained. In particular, after some trigonometric simplifications, the matrix elements become in the first approximation (P = 1, Q = 0)

$$\begin{equation} \begin{array}{l} W_{11} = \displaystyle\sqrt{\displaystyle\frac{n_b}{n_a}}\,\cos\left(z_a^2/4-z_b^2/4\right),\\ \displaystyle\frac{W_{21}}{k_0} = -\displaystyle\sqrt{n_a\,n_b}\sin\left(z_a^2/4-z_b^2/4\right),\\ k_0W_{12} = \displaystyle\frac{1}{\displaystyle\sqrt{n_a\,n_b}}\sin\left(z_a^2/4-z_b^2/4\right),\\ W_{22} = \displaystyle\sqrt{\displaystyle\frac{n_a}{n_b}}\cos\left(z_a^2/4-z_b^2/4\right), \end{array} \end{equation} \tag{ 24 }$$

in the second (P = 1, Q = (ν² − 0.25)/2z)

$$\begin{equation} \fl \hspace{-5.5pt}\begin{array}{l} W_{11} = \displaystyle \sqrt{\frac{n_b}{n_a}}\left[\cos\left(z_a^2/4-z_b^2/4\right) +\left(\displaystyle\frac{3}{8z_a^2}+\displaystyle\frac{5}{8z_b^2}\right)\sin\left(z_a^2/4-z_b^2/4\right)\right],\\ \displaystyle\frac{W_{21}}{k_0} = -\displaystyle\sqrt{n_a\,n_b}\left[\sin\left(z_a^2/4-z_b^2/4\right) +\left(\displaystyle\frac{5}{8z_a^2}-\displaystyle\frac{5}{8z_b^2}\right)\cos\left(z_a^2/4-z_b^2/4\right)\right],\\ k_0W_{12} = \displaystyle\frac{1}{\sqrt{n_a\,n_b}}\left[\sin\left(z_a^2/4-z_b^2/4\right) -\left(\displaystyle\frac{3}{8z_a^2}-\displaystyle\frac{3}{8z_b^2}\right)\cos\left(z_a^2/4-z_b^2/4\right)\right],\\ W_{22} = \sqrt{\displaystyle\frac{n_a}{n_b}}\left[\cos\left(z_a^2/4-z_b^2/4\right) -\left(\displaystyle\frac{5}{8z_a^2}+\displaystyle\frac{3}{8z_b^2}\right)\sin\left(z_a^2/4-z_b^2/4\right)\right], \end{array} \end{equation} \tag{ 25 }$$

while in the third

$$\begin{eqnarray} &&\fl W_{11} = \sqrt{\frac{n_b}{n_a}}\left[\left(1-\frac{105}{128z_a^4}+\frac{135}{128z_b^4}-\frac{15}{64z_a^2z_b^2}\right) \cos\left(z_a^2/4-z_b^2/4\right)\right.\nonumber\\ &&\left. +\left(\frac{3}{8z_a^2}+\frac{5}{8z_b^2}\right)\sin\left(z_a^2/4-z_b^2/4\right)\right],\nonumber\\ &&\fl \frac{W_{21}}{k_0} = -\sqrt{n_a\,n_b}\left[\left(1+\frac{135}{128z_a^4}+\frac{135}{128z_b^4}+\frac{25}{64z_a^2z_b^2}\right)\sin\left(z_a^2/4-z_b^2/4\right)\right.\nonumber\\ &&\left. +\left(\frac{5}{8z_a^2}-\frac{5}{8z_b^2}\right)\cos\left(z_a^2/4-z_b^2/4\right)\right],\nonumber\\ &&\fl k_0W_{12} = \frac{1}{\sqrt{n_a\,n_b}}\left[\left(1-\frac{105}{128z_a^4}-\frac{105}{128z_b^4}+\frac{9}{64z_a^2z_b^2}\right)\sin\left(z_a^2/4-z_b^2/4\right)\right.\nonumber\\ &&\left. -\left(\frac{3}{8z_a^2}-\frac{3}{8z_b^2}\right)\cos\left(z_a^2/4-z_b^2/4\right)\right],\nonumber\\ &&\fl W_{22} = \sqrt{\frac{n_a}{n_b}}\left[\left(1+\frac{135}{128z_a^4}-\frac{105}{128z_b^4}-\frac{15}{64z_a^2z_b^2}\right) \cos\left(z_a^2/4-z_b^2/4\right)\right.\nonumber\\ &&\left. -\left(\frac{5}{8z_a^2}+\frac{3}{8z_b^2}\right)\sin\left(z_a^2/4-z_b^2/4\right)\right]. \end{eqnarray} \tag{ 26 }$$

As one can see, the dispersion equation $\cos \phi =\frac {1}{2}(W_{11}+W_{22})$ takes a particularly simple form in the first approximation

$$\begin{equation} \cos\,\phi_1 = \frac{1}{2}\bigg(\sqrt{\frac{n_a}{n_b}}+\sqrt{\frac{n_b}{n_a}}\bigg)\,\cos\left(z_a^2/4-z_b^2/4\right). \end{equation} \tag{ 27 }$$

Due to its simplicity equation (27) may be particularly useful in photonic crystal design. Let us estimate its range of validity. Using expressions for W₁₁ and W₂₂ in equations (24) and (25), we find that cos ϕ in the second approximation can be written as

$$\begin{equation} \cos\,\phi_2 = \cos\,\phi_1 + \left|\epsilon\right|\sin\left(z_a^2/4-z_b^2/4\right), \end{equation} \tag{ 28 }$$

where

$$\begin{eqnarray} \epsilon& =& \frac{1}{2}\sqrt{\frac{n_b}{n_a}}\left(\frac{3}{8z_a^2}+\frac{5}{8z_b^2}\right)\, -\frac{1}{2}\sqrt{\frac{n_a}{n_b}}\left(\frac{5}{8z_a^2}+\frac{3}{8z_b^2}\right) \nonumber\\ & < & \frac{1}{2}\left(\frac{5}{8}\sqrt{\frac{n_b}{n_a}}-\frac{3}{8}\sqrt{\frac{n_a}{n_b}}\right)\left(\frac{1}{z_a^2}+\frac{1}{z_b^2}\right)\nonumber\\ &=&\frac{1}{2}\left(\frac{5}{8}\sqrt{\frac{n_b}{n_a}}-\frac{3}{8}\sqrt{\frac{n_a}{n_b}}\right)\! \left(\frac{1}{n_a^2}+\frac{1}{n_b^2}\right)\,\frac{n_b-n_a}{2k_0d}\nonumber\\ & \leqslant & \frac{143\,{\sqrt 5}}{250}\,\frac{1}{4}\,\frac{n_b-n_a}{k_0d} < \frac{1}{3}\,\frac{n_b-n_a}{k_0d}. \end{eqnarray} \tag{ 29 }$$

At the last line we took into account that for typical dielectrics the refractive indices are in the range n_a,b = 1...5. As a result, it is safe to use equation (27) for design purposes in the range of wavenumbers k₀ such as

$$\begin{equation} k_0d > \frac{|n_b-n_a|}{3}. \end{equation} \tag{ 30 }$$

Figures 2–4 illustrate all of the above approximations. In the second and third approximations, cos ϕ can also be written as

$$\begin{equation} \cos \phi_{\,2,3}=A_{2,3} \cos (z_a^2/4-z_b^2/4 + \delta_{\,2,3}), \end{equation} \tag{ 31 }$$

which allows for a shift of both the band centres and their widths. For our choice of n_b/n_a = 2, the amplitude in the first approximation A₁ = 1.06 accounts for the narrow bandgaps observed in the drawings.

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In appendix C we also compare the exact results to approximations provided by binary photonic crystals (replacing the ramp by discrete steps).

Figure 2 shows that the dispersion equation for cos ϕ in the first approximation is already in quite good agreement with the exact results. We now discuss some of the predictions for the bandgaps that follow from it. The bandgap edges are defined by the condition cos ϕ = (− 1)^q, where the index q = 1,2,... numbers the bandgaps. This condition applied to equation (27) leads to expressions for the right k^r₀ and left k^l₀ bandedges in the first approximation:

$$\begin{equation} k_0^{\rm r}\,d = \frac{q\pi +\alpha}{n_{\rm av}}, \quad k_0^{\rm l}\,d = \frac{q\pi-\alpha}{n_{\rm av}}, \end{equation} \tag{ 32 }$$

where

$$\begin{equation} \alpha = \arccos \left(\frac{2}{\sqrt{n_a/n_b+n_b/n_a}}\right), \quad n_{\rm av} =\frac{n_a+n_b}{2}. \end{equation} \tag{ 33 }$$

Therefore, the centres of the bandgaps k^c₀ and their widths w are given in the first approximation by

$$\begin{equation} k_0^{\rm c}\,d = q\,\frac{\pi}{n_{\rm av}}, \quad wd = \frac{2\alpha}{n_{\rm av}}. \end{equation} \tag{ 34 }$$

This is qualitatively similar to the asymptotic results obtained by Rauh et al [1]. In figure 5 we compare the accuracy of the three approximations for the bandgap centres, and in figure 6 for their widths. As one can see from equations (24)–(26), each approximation incorporates new corrections of orders 1/z²_a,b and their squares, respectively. With the exception of the first bandgap (but that is inherent in use of an asymptotic series), all three approximations provide one with quite accurate estimates and the second one balances accuracy and simplicity.

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We have derived the exact dispersion equation, reflection and transmission coefficients, in terms of Bessel functions, for our one-dimensional photonic crystal with a sawtooth refractive index profile. Using the Hankel expansion, we worked out the first three asymptotic approximations to those coefficients. Even the first approximation, which has a particularly simple analytic form, provides one with very reasonable estimates. Using the results of appendix A, one can easily work out solutions for any monomial behaviour (k₀n(z))² ∼ (z/L)ⁿ of the permittitivity within a single cell of the periodic system. They always involve Bessel functions J_±m(z) of order m = 1/(n + 2).

We acknowledge support by FIS2011-24154 and FIS2009-SGR1289 (JM), and by NSERC discovery grant no. RGPIN-3198 (DWLS). We thank one of the referees for suggesting the addition of appendix B.

While discussing the Wentzel–Kramers–Brillouin approximation, Schiff [13, p 272] points out that when k²(z) = (z/L)ⁿ, an exact solution of the stationary state wave equation

$$\begin{equation} 0 = \psi^{\prime\prime}(z) + k^2(z)\psi(z), \end{equation} \tag{ A.1 }$$

is given by

$$\begin{equation} \begin{array}{l} \psi(z) \equiv A(z) J_m(\xi), \quad A(z) \equiv \sqrt{\xi(z)/k(z)},\\ \xi(z) \equiv \displaystyle\int_0^z k(t) \mathrm{d}t, \quad m = \displaystyle\frac{1}{n+2}. \end{array} \end{equation} \tag{ A.2 }$$

The task is to show that J_m(ξ) is a generic Bessel function of order m, a result given originally by Langer [14]. The most used case is a linear turning point with n = 1⇒m = 1/3, where the solution coincides with the Airy function. The general result makes it easy to work out the transfer matrix for a 1D photonic crystal whose refractive index profile has any monomial 'sawtooth shape' in each layer. To begin, we note that for our assumed k²(z)

$$\begin{equation} \begin{array}{l} \xi(z) = \displaystyle\frac{2L}{n+2} \left(\displaystyle\frac{z}{L}\right)^{(n+2)/2}, \\ k^\prime(z) = \displaystyle\frac{n}{2L} \left(\displaystyle\frac{z}{L}\right)^{(n-2)/2},\\ A^2(z) = \displaystyle\frac{\xi}{k} = 2\,m z,\\ 2 A(z) A^\prime(z) = 2\,m = {\rm const} ,\\ A^3(z) A^{\prime\prime}(z) = -[A(z) A^\prime ]^2 = -m^2. \end{array} \end{equation} \tag{ A.3 }$$

From here we work out ψ' and ψ'', and substitute into the wave equation. On functions of z, the prime means d/dz, but on the Bessel function it means d/dξ. We have

$$\begin{equation} \begin{array}{l} \psi^\prime = A^\prime(z) J_m(\xi) + A(z) k(z) J_m^\prime(\xi),\\ \psi^{\prime\prime} = A^{\prime\prime}(z) J_m(\xi) + A(z) k^2(z) J_m^{\prime\prime}(\xi) + [2 A^\prime(z) k(z) + A(z) k^\prime(z)] J^\prime_m(\xi) . \end{array} \end{equation} \tag{ A.4 }$$

Substituting into the wave equation and removing the factor Ak², we find that the coefficient of J'_m(ξ) is

$$\begin{equation} \frac{2 A^\prime k + Ak^\prime }{A k^2} = \frac{1}{\xi}, \end{equation} \tag{ A.5 }$$

see equation (A.8) for details, while the coefficient of J_m(ξ) is 1 + [A''/Ak²], where the variable portion is

$$\begin{equation} \frac{A^{\prime\prime}}{A k^2} = \frac{A^3(z) A^{\prime\prime} }{A^4 k^2} = - \frac{ (A^{\prime})^2 A^2 }{A^4 k^2} = - \frac{m^2}{\xi^2}. \end{equation} \tag{ A.6 }$$

This verifies that the wave equation has been reduced to Bessel's equation for the fractional order m,

$$\begin{equation} J^{\prime\prime}_m(\xi) + \frac{1}{\xi} J^\prime_m(\xi) + \left[ 1 - \frac{m^2}{\xi^2} \right] J_m(\xi) = 0. \end{equation} \tag{ A.7 }$$

The regular solution J_m(ξ) will vanish at the origin when m > 0, so the irregular solution J_−m(ξ) is also required if a non-zero initial value is in order. In equation (A.5) we used the following relations:

$$\begin{equation} \begin{array}{l} A^2(z) = \xi/k, \quad 2 A A^\prime = 1 - \xi \displaystyle\frac{k^\prime}{k^2}, \\ \displaystyle\frac{2 A^\prime k + Ak^\prime }{A k^2} = \displaystyle\frac{2 A A^\prime }{A^2 k} + \displaystyle\frac{k^\prime}{k^2} = \displaystyle\frac{1}{\xi}. \end{array} \end{equation} \tag{ A.8 }$$

Our purpose here is to show how the results of the section 2 can be used to describe wave propagation through a periodic structure with a triangular refractive index profile, see figure B.1. Since the refractive index of the period with −d ⩽ z ⩽ d is an even function, the normalized solutions u and v, relating the field E(z) and its derivative dE(z)/dz at the edge points z = ± d, can be taken to be the even and odd functions, respectively. As a result, we have

$$\begin{equation} \left[ {\begin{array}{@{}c@{}} {E\left( {-d} \right)} \\ {E'\left({-d} \right)} \\ \end{array}} \right] = W_t \left[ {\begin{array}{@{}c@{}} {E( d )} \\ {E'( d )} \\ \end{array}} \right], \end{equation} \tag{ B.1 }$$

where

$$\begin{eqnarray} W_t &=&\left[ {\begin{array}{@{}cc@{}} {u\left( -d \right)} & { v\left(- d \right)} \\ { u'\left( -d \right)} & {v'(-d)} \\ \end{array}} \right]\, \left[ {\begin{array}{@{}cc@{}} {v'( d )} & { - v( d )} \\ { - u'( d )} & {u(d)} \\ \end{array}} \right] \nonumber\\ &=&\left[ {\begin{array}{@{}cc@{}} {u \left(d \right)} & { -v( d )} \\ {-u'\left(d \right)} & {v'(d)} \\ \end{array}} \right]\, \left[ \begin{array}{@{}cc@{}} {v'( d )} & { - v( d )} \\ { - u'( d )} & {u(d)} \\ \end{array} \right]. \end{eqnarray} \tag{ B.2 }$$

The dispersion equation takes the form

$$\begin{equation} \cos\phi = u(d)\,v'(d)+v(d)\,u'(d). \end{equation} \tag{ B.3 }$$

Expressions for the elements u(d), u'(d), v(d) and v'(d) are given by equation (18).

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Reversing the order of the two matrices whose product gives W_t in equation (B.2) would give the transfer matrix for the inverted V triangle pattern. That is to say, if figure B.1 is a pattern DUDUDU, the new pattern would be UD UD UD. These are examples of biperiodic superlattices [16], and some interesting physics arises if one of the two components U or D is given a slightly different width d_U ≠ d_D.

We consider a binary photonic crystal composed of layers of constant refractive indices n₁, n₂, with corresponding thicknesses d₁, d₂. The elements of a binary W-matrix are well-known [11] and given by

$$\begin{equation} \begin{array}{l} W_{11} = \cos(k_0n_1d_1)\cos(k_0n_2d_2) - \displaystyle\frac{n_2}{n_1}\sin(k_0n_1d_1)\sin(k_0n_2d_2),\\ \displaystyle\frac{W_{21}}{k_0} = n_1\sin(k_0n_1d_1)\cos(k_0n_2d_2) + n_2\cos(k_0n_1d_1)\sin(k_0n_2d_2),\\ k_0W_{12} = -\displaystyle\frac{1}{n_1}\sin(k_0n_1d_1)\cos(k_0n_2d_2) - \displaystyle\frac{1}{n_2}\cos(k_0n_1d_1)\sin(k_0n_2d_2),\\ W_{22} = \cos(k_0n_1d_1)\cos(k_0n_2d_2) - \displaystyle\frac{n_1}{n_2}\sin(k_0n_1d_1)\sin(k_0n_2d_2). \end{array} \end{equation} \tag{ C.1 }$$

The simplest binary approximation to the sawtooth profile discussed above is given by n₁ = n_a, n₂ = n_b and d₁ = d₂ = d/2. Figure C.1 compares the exact results for the sawtooth potential with parameters as in figure 2 to that of the simplest binary approximation. As one can see there are some similarities, but the bandgap widths differ strongly; for example, the third bandgap of the sawtooth potential disappears entirely. A better binary approximation, see figure C.2, is obtained if one follows the recipe of Kalotas and Lee [15], taking the layer indices to be n₁ = (n_a + 3n_b)/4, n₂ = (3n_a + n_b)/4 and widths d₁ = d₂ = d/2. (These values give the correct average wave number in each layer.)

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