An Analog of electrically induced transparency via surface delocalized modes

We demonstrate theoretically and experimentally an interesting opaque state, which is based on an analog of electromagnetically induced transparency (EIT) in mechanism, in a metal hole array of the dimer lattice. By introducing a small difference to the dimer holes of each unit cell, the surface delocalized modes launching out from the dimer holes can have destructive interferences. Consequently, a narrow opaque window in the transparent background can be observed in the transmission spectrum. This surface-mode-induced opacity (SMIO) state is very sensitive to the difference of the dimer holes, which will promise various applications.

(1) In above expression, we have: which gives the amplitudes of the electromagnetic field at upper holes in the unit cell for a given mode labeled by l, s, and σ. (3) which gives the amplitudes of the electromagnetic field at lower holes in the unit cell for a given mode labeled by l, s, and σ.
The z-component wave vector for a given waveguide modes is given by The admittances of a mode of given polarization are The details of the ket expressions are given in the appendix.
In subwavelength region, we can only consider the fundamental TE mode for the longer sides of the holes. Thus, the equation (1) Then we compare the result of the full calculation (2 waveguide modes is enough to make the result converge.) and the single waveguide mode result. It can be seen that the single mode approximation does not derivate very much from the full calculation, and all features of the transmission spectra have been captured by the single mode approximation.
By defining the following correlation functions: The equation (4) can be re-written as: From the expressions of the ket, one can find that (also proved by numerical calculation): Then the equation (11) One can solve out the expression for the field amplitudes of waveguide modes at the output interface analytically, and they are: Where φ (θ) is angle of the upper (lower) hole tilting from y axis. Since   , we can further simplify (54) as: 2 2 2 2 2 2 22 0,1,1 0,1,1 0 0,1,1 0,1,1 0 0,1,1 0,1,1 0 Since the shape of the hole are the same, 12 I I I  , which can further simplify the expressions.
A. The case without difference in dielectric constants: To feeling the power of the results above, we firstly consider the case that the two holes in the unit cell fills with nothing.
As a result, we have a very simple expression for the transmission:  One can observe that one term in the denominator cancels the same term in the numerator. We can expect that when the small dielectric difference is taken into account, the term would not be cancelled, because of the different deviations in denominator and numerator. On the other hand, we can also expect that the deviations should be small, since the dielectric difference is small. Therefore, it would be useful for us to plot  Indeed the shift of the 0 points of the numerator induces the change of the transmission profile from Fano to anti-Fano. The amplitude of 0,1,1  is always opposite to that of 0,1,1  in the frequency region between the 0 points of the two numerators. Fig. 7 the plot of the absolute values, real parts and imaginary parts of the numerators of 0,1,1  and 0,1,1  From Fig. 7, It can be seen that the real parts of them are always in opposite signs, and the imaginary parts are in opposite signs in the region between two 0 points.
Then we can expect that the cross point of the two absolute values should be identical to the SMIO frequency. To see it, we will plot them in Fig. 8. In conclusion, the SMIO phenomenon can be viewed in the following way: the introduction of the difference in the dielectric constants of helps the 0 points of the field amplitudes at the two holes to join into the physical picture. As a consequence, the dielectric constant difference shifts the 0 points of the field amplitudes of the two holes from each other, and thus one of them gives a Fano transmission profile and the other gives anti-Fano. In the frequency region between the two 0 points, the amplitudes of the two fields are opposite in signs, which provides the cancellation and gives the transmission dip. All these eigenmodes are bounded on the metal surface and thus are surface modes. Therefore, the transmission dip is indeed due to the interaction between two surface modes launching out from the two dimer holes, which are different in phase.
III. The effect of angle and dielectric constant difference: In the part, we will check whether the understanding above can be applied to different tilting angles and different dielectric constant differences. We will plot the absolute value of the numerators of the field amplitudes at the two holes ( 0,1,1  and 0,1,1  ), and comparing the results with Fig. 3 of the manuscript. We will find the position of SMIO fits well with the 0 points of the numerators. The results are summarized in Fig. 9 below.