Electromagnetic normal modes and Casimir effects in layered structures

We derive a general procedure for finding the electromagnetic normal modes in layered structures. We apply this procedure to planar, spherical, and cylindrical structures. These normal modes are important in a variety of applications. They are the only input needed in calculations of Casimir interactions. We present an explicit expression for the condition for modes and Casimir energy for a large number of specific geometries. The layers are allowed to be two-dimensional so graphene and graphenelike sheets as well as two-dimensional electron gases can be handled within the formalism. Also, forces on atoms in layered structures are obtained. One side result is the van der Waals and Casimir-Polder interaction between two atoms.

Figure 1

Integration contour in the complex $z$ plane suited for zero-temperature calculations. Crosses and circles are poles and zeros, respectively, of the function $f\left(z\right)$. The radius of the circle is let to go to infinity.

Figure 2

Integration contour in the complex $z$ plane suited for finite temperature calculations. Crosses and circles are poles and zeros, respectively, of the function $f\left(z\right)$. The small semicircles are centered at the poles of the coth function in the integrand. The radius of the large semicircle is let to go to infinity.

Figure 3

Schematic illustration of the layered structure. The numbering of the $N+2$ media are indicated at the bottom of the figure and the numbering of the $N+1$ interfaces at the top. In the general solution of Maxwell's equations, there are one wave moving towards the right and one towards the left inside each medium. If a normal mode is excited, there is no wave moving to the right in medium 0, which is the ambient medium; if the medium number $N+1$ is unlimited there is no wave moving to the left in that medium. See the text for more details.

Figure 4

The geometry of a single planar interface.

Figure 5

The one planar layer geometry.

Figure 6

The two planar layer geometry. This geometry is used to find the atom substrate interaction as illustrated by the cartoon in the lower right corner. We start from the gas layer substrate geometry in the upper right corner.

Figure 7

The three planar layer geometry.

Figure 8

The geometry of a thin gas layer at a distance $d$ from a thin film. This geometry is used to find the interaction between an atom and a thin film.

Figure 9

The amplitudes of the waves at the interface number $n$.

Figure 10

The geometry of a solid sphere or a solid cylinder in the nonretarded treatment.

Figure 11

The geometry of a coated sphere or cylinder in the nonretarded treatment.

Figure 12

The geometry of a thin gas layer the distance $d$ from a sphere or cylinder of radius $a$ in the nonretarded treatment.

Figure 13

The geometry of a thin gas layer at radius $d$ inside a spherical or cylindrical cavity of radius $a$ in the nonretarded treatment.

Figure 14

The geometry of a thin gas layer the distance $d$ from thin spherical or cylindrical shell of radius $a$ in the nonretarded treatment.

Figure 15

The geometry of a thin gas layer at radius $d$ inside a thin spherical or cylindrical shell of radius $a$ in the nonretarded treatment.

Figure 16

The geometry of a solid sphere or cylinder of radius $a$ in the fully retarded treatment.

Figure 17

The geometry of a coated sphere or cylinder of radius $a$ in the fully retarded treatment.

Figure 18

The geometry of a thin gas layer the distance $d$ from a sphere or cylinder of radius $a$ in the fully retarded treatment.

Figure 19

The geometry of a thin gas layer at radius $d$ inside a spherical or cylindrical cavity of radius $a$ in the fully retarded treatment.

Figure 20

The geometry of a thin gas layer at distance $d$ from a thin spherical or cylindrical shell of radius $a$ in the fully retarded treatment.

Figure 21

The geometry of a thin gas layer at radius $d$ inside a thin spherical or cylindrical shell of radius $a$ in the fully retarded treatment.