Local density of optical states in the band gap of a finite one-dimensional photonic crystal

We study the local density of states (LDOS) in a finite photonic crystal, in particular in the frequency range of the band gap. We propose an original point of view on the band gap, which we consider to be the result of vacuum fluctuations in free space that tunnel in the forbidden range in the crystal. As a result, we arrive at a model for the LDOS that is in two major items modified compared to the well-known expression for infinite crystals. First, we modify the Dirac $\delta$ functions to become Lorentzian with a width set by the crystal size. Second, building on characterization of the fields versus frequency and position we calculated the fields in the band gap. We start from the fields at the band edges, interpolated in space and position, and incorporating the exponential damping in the band gap. We compare our proposed model to exact calculations in one dimension using the transfer matrix method and find very good agreement. Notably, we find that in finite crystals, the LDOS inside the band gap depends on frequency, on position, and on crystal size, in contrast to the well-known results for infinite crystals where the LDOS is zero, independent of frequency and position.

Figure 1

Schematic drawing of a light source (orange sphere) embedded in a finite (1D) photonic band gap crystal with the size $L$ (photonic crystal slab). The source is located at a depth $z=\Delta z$. The interface between the crystal and free space is indicated by the vertical dashed line. The shading in the crystal lightens away from the interface to illustrate the decreasing LDOS with depth into the crystal $\Delta z$.

Figure 2

Schematic drawing of a one-dimensional periodic structure. (a) Infinite structure and (b) finite Bragg stack. One period $a$ consists of two different media with width $a_1$ and $a_2$ and dielectric permittivities of ${\epsilon }_1$ and ${\epsilon }_2$, respectively. The crystal is taken to be one dimensional and the other directions are used for illustration purposes.

Figure 3

Calculated band structure of the one-dimensional periodic structure depicted at Fig. 2, where $a_1=0.2a$ and $a_2=0.8a$ and ${\epsilon }_1$ and ${\epsilon }_2$ are 13 and 1, respectively. The bands are shown with dark blue lines where the wave vector is real (left ordinate). The first two photonic band gaps are shown with yellow bars. In the band gaps the wave vector becomes complex $\mathbf{K}={\mathbf{K}}^{\prime }+i{\mathbf{K}}^{\prime \prime }$, where the real part is clamped at the Brillouin zone edge wave vector $(K^{\prime }=\pi /a)$ and the imaginary part $K^{\prime \prime }$ strongly depends on the frequency as is shown by red dashed line (right ordinate).

Figure 4

Absolute value of the electric field associated with (a) reduced frequency $\omega a/2\pi c=0.2$, (b) reduced frequency $\omega a/2\pi c=0.3$, (c) reduced frequency $\omega a/2\pi c=0.45$, and (d) reduced frequency $\omega a/2\pi c=0.47$ of the band structure plotted in Fig. 3. The main maximum of the field amplitude is indicated with a black arrow.

Figure 5

(a) Behavior of the first field maximum in the frequency range of the first band gap. The ordinate between 0 and $0.2$ is the high index material (indicated by the orange hatched bar) and from $0.2$ to 1 is the low index material (indicated by the gray hatched bar). The red solid line is the exact behavior of the field and the black dashed line is the linear field interpolation with the position. (b) Comparison of the exact maximum field amplitude (red curve) and the model (black dashed curve) as a function of reduced frequency. In the model a linear interpolation is used for the position of the field maximum [see (a)]. The model shows a very good agreement with the exact calculations.

Figure 6

(a) Absolute value of the electric field at a frequency $\omega a/2\pi c=0.2$ for a finite structure with $m=10$ periods. The environment of the crystal is assumed to be air. (b) Absolute value of the electric field at a frequency $\omega a/2\pi c=0.3$ inside the photonic band gap (see Fig. 3). At this frequency the field inside the structure is damped. The field in the finite structure decays with the same damping as the field in the infinite structure at the same frequency.

Figure 7

The LDOS calculated using the ansatz Eq. (5) (blue) and the exact LDOS calculated for a finite periodic structure at two different reduced frequencies $\omega a/2\pi c=0.33$ and $\omega a/2\pi c=0.43$ within the photonic band gap (logarithmic axis). At frequency $\omega a/2\pi c=0.33$ there is a small shift in the unit cell modulations of the LDOS. At frequency $\omega a/2\pi c=0.43$ our model matches perfectly with the exact calculation. This small deviation between our model (blue curve) and the exact calculations of the LDOS using the field (red curve) and the Green function (green curve) is due to our simple interpolation and can be already seen in Fig. 5 at the same frequency.

Figure 8

Local density of optical states (LDOS) in the frequency range of the band gap for (a) a finite crystal consisting of $m=10$ periods and (b) a finite crystal consisting of $m=20$ periods. In the band gap the envelope of the LDOS decays exponentially inside the structure with a minimum at the center of the structure. The attenuation of the LDOS is strongest at the center of the band gap because the imaginary part of the $K^{\prime \prime }$ vector is maximal. Moreover the summation over all $F(\omega -{\omega }_{i,j})$ is minimal at frequencies in the center of the band gap.