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| The sign convention for the imaginary part of the refractive index | Parent topic |
Propagation of electromagnetic waves in a medium is governed by the Maxwell equations. For a medium with electrical conductivity, σ, magnetic permeability, μ, and electric susceptibility, χ, the relevant Maxwell equations for the electric field, E, and magnetic field, B, can be written as follows:
| curl E = - ∂B / ∂t | (1) |
| curl B = µσE + µε0( 1 + χ)∂E / ∂t | (2) |
where ε0 is the electric permittivity of free space, also referred to as the electric constant. The curl operator is defined, for example, in MathWorld. By combining Eqs. 1 and 2 we obtain
| curl curl E + [ µσ ∂/∂t + µε0( 1 + χ) ∂2/∂t2 ] E = 0 | (3) |
Now consider an electric field of the electromagnetic wave expressed as follows (note the plus sign of the exponent of the time-dependent part):
| E = E0(r) exp(iωt) | (4) |
where r is the position vector. By substituting that expression into Eq. 3, we obtain
| curl curl E0 - ω2 εµ E0 = 0 | (5) |
where ε is the complex electric permittivity of the medium:
| ε = ε0( 1 + χ ) - ( σ / ω ) i | (6) |
By using a relationship between the refractive index, electric permittivity, ε, and magnetic permeability, μ we have:
| m2 | = με | |
| = με0( 1 + χ ) - ( μσ / ω ) i | ||
| = ( m' - m"i )2 | (7) |
where the last line is just the definition of the complex refractive index but the minus sign in that line is a consequence of the minus sign of the imaginary part of the electric permittivity (Eq. 6). Incidentally, it follows from Eq. 7 that:
| m' 2 - m" 2 = με0( 1 + χ ) | (8) |
| 2m'm" = μσ / ω | (9) |
| CITATION: Jonasz M. 2006. Refractive index (www.tpdsci.com/Tpc/RI.php). In: Top. Part. Disp. Sci. (www.tpdsci.com). |
HISTORY: Published: 04-May-2006 Modified: 04-May-2006 Peer-reviewed: PENDING |
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