Research ArticleCLASSICAL MECHANICS

Electromagnetic stress at the boundary: Photon pressure or tension?

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Science Advances  11 Mar 2016:
Vol. 2, no. 3, e1501485
DOI: 10.1126/sciadv.1501485
  • Fig. 1 Boundary optical stress in the microscopic model system.

    (A) Boundary formed by two different homogeneous media described by effective parameters. (B) For metamaterials, the effective medium has an underlying microscopic structure, taken to be a square lattice in this example. (C) The effective-medium (A) is a good representation of the microscopic system (B) if the fields calculated for them are nearly the same as shown in this panel, where we compare the calculated electric fields (Ez). (D) Force distribution in the model system with εeff,1 = μeff,1 = − εeff,2 = −μeff,2 = −2 and λ = 100a, with a being the size of the unit cell. d is the relative shift between two sublattices along the y direction. The inserted panel shows the total force (boundary stress) acting on the boundary, which is the sum of the nonvanishing force values in the blue shadowed region.

  • Fig. 2 Boundary optical stress in impedance-matched lattice systems.

    (A to D) Boundary stress in the square-square (A), triangle-triangle (B), square-triangle (C), and triangle-square (D) systems. We fix the refractive index on the left neff,1 = −1 or neff,1 = 1 (corresponding to air) and vary the refractive index neff,2 on the RHS. The dots correspond to the boundary stresses of the microscopic systems calculated with multiple scattering theory, whereas the lines denote the analytical values obtained using the Helmholtz stress tensor. The insets show the configurations under consideration. (E) Four types of lattice systems correspond to the same effective-medium system (neff,1 = −1 on the left, neff,2 = 4 on the right) but have entirely different boundary stresses. The direction of the stresses is shown in the panels by an arrow.

  • Fig. 3 Boundary optical stress in impedance-mismatched lattice systems.

    (A and C) Boundary stress as a function of μeff,2, εeff,2 in the “air/square-lattice” configuration, respectively. (B and D) Boundary stress as a function of μeff,2, εeff,2 in the “air/triangular-lattice” configuration, respectively. In (A) and (B), we fix εeff,2 = −1, whereas in (C) and (D), we fix μeff,2 = −1.

  • Fig. 4 Optical force distribution in finite slab and total reflecting systems.

    Optical force distribution in finite square-lattice (A and D) and triangle-lattice (B and E) slab systems. For (A) and (B), the impedances of the lattice systems match that of air, whereas for (D) and (E), they do not. The solid and dashed lines in (C) and (F) denote the accumulated force counting from the first layer, and the final value (as marked by a blue dot) corresponds to the total force exerting on the slab. We note that whereas the forces on the boundary depend strongly on the microscopic details, the total force acting on the slab is exactly the same independent of the microscopic details as long as the macroscopic εeff and μeff are the same. Optical force distribution for a semi-infinite square (G) [triangle (H)] lattice with εeff,2 = −4, μeff,2 = 1 attached to a semi-infinite square (triangle) lattice with εeff,1 = μeff,1 = −1 or air with εeff,1 = μeff,1 = 1. Here, x denotes the position of the unit, with a being the column layer distance.

  • Fig. 5 Boundary optical stress in amorphous-lattice system.

    (A) Part of an amorphous-lattice sample drawn to scale. (B) Comparison between the boundary stress of a square lattice and an amorphous lattice. Circles denote the stress calculated numerically for the microscopic lattice systems. Solid lines denote the stress calculated analytically using the Helmholtz stress tensor in the effective-medium system. Dashed lines are the average of the values represented by circles. δ is the random amplitude defined in the main text.

  • Fig. 6 Boundary optical stress in a metamaterial system.

    (A) Metamaterial system formed by core-shell cylinders and normal dielectric cylinders. (B) Photonic band diagram for the core-shell square lattice. (C) Retrieved effective permittivity and permeability based on S parameters of the core-shell structure. (D) Boundary stresses of the microscopic model system (dots) and the corresponding effective-medium system (lines). (E) Magnetic field distribution in the neff,2 = 1 (air) case.

Supplementary Materials

  • Supplementary Materials

    This PDF file includes:

    • Fig. S1. Boundary stress evaluation by Maxwell stress tensor method in the amorphous-lattice system.
    • Fig. S2. Boundary stress in square-lattice system under H-z polarization.

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