Analytical Investigation of In-plane Focusing Surface Plasmon Modes by a Dielectric Lens

A dielectric lens placed on a metal-dielectric interface (Fig. 1) can be used for in-plane focusing surface plasmon (SP) modes. The propagation behind the lens is usually studied using scalar diffraction theory which is not... [ view full abstract ]

A dielectric lens placed on a metal-dielectric interface (Fig. 1) can be used for in-plane focusing surface plasmon (SP) modes. The propagation behind the lens is usually studied using scalar diffraction theory which is not accurate in nanometer regime and cannot explain the propagation and energy flow of SP modes. Here, we have used vectorial derivation of Huygens-Fresnel principle [1] to analytically calculate the diffracted fields behind the lens to study the energy flow of SP modes in a focusing system.

In a plasmonic structure, a dielectric lens cannot be considered as a thin lens. In the first step, we have derived the phase delay of a SP mode considering the lens thickness. This phase delay in conjunction with the vectorial form of Huygens-Fresnel principle is then used to calculate the diffraction pattern of a SP mode. The resulted electric field is

E(r)=(1/2π)∫E(k_y)exp(i√(K²-k_y²)+ik_yy+ik_xx)dk_y,

where k_x and k_y are the components of wavevector in the x and y directions, respectively. The wave propagates along the z direction and K indicates the tangential element of the wavevector. The above equation can be considered as the Fourier transform of the product of the phase function of SP and its complex amplitude. This relation can be expressed in the form of a convolution integral and its solution provides the diffracted fields behind the lens. The resulted fields have the form of higher order SP modes [2] which means that passing through a dielectric lens has excited higher order modes. Using the Poynting vector, one can calculate the energy flow behind the lens. The results are shown in Fig. 2 for different points on a metal-dielectric interface. Using the vectorial Huygens-Fresnel principle and accurate SP phase delay, we have shown that higher order SP modes are excited behind a lens in an in-plane focusing system. This is equivalent to more energy loss from the surface since the higher SP modes are less bounded to the surface than the fundamental mode.

References

[1]        A. Archambault, et al. Phys. Rev. B 79, 2009.

[2]        M. Kordi, et al. Phys. Rev. A 95, 2017.