A Note on Sign Conventions
When describing the propagation of a plane electromagnetic wave, different references will employ different sign conventions. The choice of sign convention can have far-reaching implications for the interpretation of phase delays and material properties, among other things.
A plane electromagnetic wave is sometimes defined as
(3-1)
and at other times as
(3-2)
where
•
E (SI unit: V/m) is the electric field along the ray,
•
E0 (SI unit: V/m) is the electric field amplitude at r = 0 and t = 0,
•
k (SI unit: rad/m) is the wave vector,
•
r (SI unit: m) is the position vector,
•
ω (SI unit: rad/s) is the angular frequency, and
•
t (SI unit: s) is time.
For example, Refs. 1 and 6 use the sign convention in Equation 3-1, whereas Refs. 7, 9, 12, 13, and 16 use the sign convention in Equation 3-2.
Suppose that the wave propagates in the positive x-direction. Then the wave vector may be defined as
where n (dimensionless) is the refractive index of the medium, assumed to be isotropic. For an absorbing or gain medium, n is complex-valued. Using Equation 3-1, the electric field along a plane wave in an absorbing or gain medium is
which has an exponentially decaying magnitude (absorbing medium) if the imaginary part of n is positive, and an exponentially growing magnitude (gain medium) if the imaginary part of n is negative. In contrast, when starting from Equation 3-2, the electric field is
So the medium is absorbing if the imaginary part of n is negative, or a gain medium if the imaginary part of n is positive.
The Geometrical Optics interface uses the convention of Equation 3-2. Therefore the complex-valued refractive index may be written as n - iκ, where both n and κ are real-valued. The variable κ, sometimes called the extinction coefficient, should then be positive when defining an absorbing medium, for example when specifying the Refractive index, imaginary part in the Medium Properties node.