B. Investigation of coupling effect of higher bands
3. Numerical Results and Discussion
In the following numerical results the typical superconducting system, niobium (Nb, with Tc = 9.2 K and λ0 = 83 nm), is taken as the superconducting layer and the operating temperature at 4.2 K is used. In addition, the dielectric layer with a relative permittivity of 10 and N = 200 are used. In Fig. 2 we plot the frequency response of the transmittance for the TE wave under three distinct angles of incidence 150, 300, and 450, respectively. In the upper figure we take d2 = 160 nm and d3 = 80 nm. It is seen that the first passband is weakly dependent on the incident angle. However, the second and third passbands are very sensitive to the incident angle, moving to the higher frequency regions as the incident angle increases. Similar results can also be seen when the thicknesses of two layers are interchanged, as illustrated in the lower figure of Fig. 2. Another feature in Fig.2 is that the more passbands are seen when the dielectric layer is thicker than the superconducting layer. The results in Fig. 2 indicate that the number of passbands can be primarily controlled by the thickness of the dielectric layer.
Fig. 2 Calculated frequency-dependent transmittance for the TE wave at three different angles of incidence, where the upper one is for d2 = 160 nm and d3 = 80 nm, and the lower one is for d2 = 80 nm and d3 = 160 nm, respectively.
Figure 3 depicts the transmittance for the TM wave for the same conditions in Fig. 2. Again, higher passbands are sensitive to the incident angle. The vertical dashed line marks the threshold frequency of 560 GHz. For the TM wave, it is noted that there exist narrow passbands between 560-600 GHz for the oblique incidence, which actually are not present in the cases of the normal incidence and the TE wave as well.
Such narrow passbands called the superpolariton bands come from the existence of the normal component of the electric field in the TM wave. The normal component of the electric field causes the superelectrons to be polarized and thus polaritons are formed.
Fig. 3 Calculated frequency-dependent transmittance for the TM wave at three different angles of incidence, where the upper one is for d2 = 160 nm and d3 = 80 nm, and the lower one is for d2 = 160 nm and d3 = 80 nm, respectively.
In Fig. 4, the calculated transmittance is limited to the case where the superconducting layer is thicker than the dielectric layer. We take d2 = 160 nm, and d3 is varied as 80 nm (top), 93 nm (middle), and 108 nm (bottom), respectively. In the top figure, at normal incidence (the red curve), the second passband is located at frequency higher than the threshold frequency (indicated by the vertical dashed line).
It then splits into two bands at oblique incidence, a narrow band (near 600 THz) and a wide band (See the black and blue curves). The gap between these two split bands is largely increased with increasing the incident angle. In addition, both these two bands remain higher than the threshold frequency. If now the dielectric thickness is increased to 93 nm (middle), then the threshold frequency can be arranged to fall near the center of the second band at 00. The second band is again split into two bands at 300 and 450. However, the split two bands are separated by the threshold frequency. If the dielectric thickness is further increased to 108 nm, then the second band falls below the threshold frequency at 00. In this case, the second band at 300 or 450 will also be split into two bands separated by the threshold frequency. The results shown in Fig. 4 reveal that the band shift is mainly controlled by the change in the thickness of the dielectric layer. Another important feature in the oblique incidence is that the threshold frequency must be located within the photonic forbidden band, as marked by the arrows.
Fig. 4 Calculated frequency-dependent transmittance for TM wave at three different dielectric thicknesses, d3 = 80, 93, and 108 nm and superconductor layer has a fixed thickness of d2 = 160 nm.
Each figure includes three different angles of incidence.
Fig. 5 Calculated frequency-dependent transmittance for TM wave at five different dielectric thicknesses, d3 = 70, 78, 87, 97, and 106 nm and superconductor layer has a fixed thickness of d2 = 160 nm. The angle of incidence is fixed at 450.
In Fig. 5, taking a fixed angle of incidence at 450 we plot the transmittance at various dielectric thicknesses d3 = 70, 78, 87, 97, and 105 nm for a fixed thickness of superconductor d2 = 160 nm. It is seen that the second passband will be shifted to the left when d3 increases. Meanwhile, its bandwidth is also narrowed down substantially.
It is of interest to note that this passband disappears at d3 = 87 nm. The disappearance is due to the fact that at this thickness the passband almost shrunk to a single frequency just at the threshold frequency. Because the threshold frequency must be within the forbidden band, the passband at this frequency is thus smeared out and disappears. If we continue to increase d3 over 87 nm, then the passband reappears in the left side of the threshold frequency and its bandwidth is increased as the d3
increases. Conclusively, we can find a critical thickness d3 = 87 nm such that a much wider stop band can be obtained. That is, a wider superconducting Bragg reflector ranging from 240 – 750 THz is achievable by controlling the dielectric thickness in such a superconducting photonic crystal, as depicted in the middle of Fig. 5. This region obviously includes all the visible frequencies.
Fig. 6 Calculated frequency-dependent transmittance for the TE and TM waves at three different superconductor thicknesses, d2 = 100, 150, and 200 nm and dielectric layer has a fixed thickness of d3 = 100 nm. The angle of incidence is fixed at 300.
In Fig. 6, we investigate the effect due to the change in the superconductor thickness. Here the incident angle is fixed at 300 and dielectric thickness d3 = 100 nm is taken. For the TE wave, the passbands are enhanced as the superconductor thickness decreases. However, the band center for the lowest passaband remains unchanged, whereas the shift in the other passbands is apparently seen. Similar behaviors are also seen in the TM wave. Thus, in order to have a wider passbands, it is preferable to use a thinner superconductor in a superconducting photonic crystal.
4. Conclusion
The angle- and thickness-dependent photonic band structures in a superconducting 1DPC have been theoretically investigated. Numerical results illustrate that the bandwidth of the lowest passband can be enhanced by using a thinner superconducting film. The band shift and numbers can be primarily controlled by the thickness of the dielectric layer. As for the angular dependence, it is found that, in the TM wave, the passband near the threshold frequency will be split into two bands with
a gap being controlled by the angle of incidence. The positions of the two split passbands are also proven to be adjustable by the variation of the thickness of the dielectric layer.
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