where a(amplitude) represents the amplitude absorption length in a given material in units of µm-1. By a widely-accepted convention the symbol a = 2a(amplitude) = 2 NLOSS is called the extinction coefficient and describes the exponential decrease in power of a wave traveling in the material.

Following the notation of Pankove, the complex refractive index nc is expressed in terms of its real and imaginary components:

where k0=2p/l. Note the second exponential term on the far right. It describes an exponentially-decaying field amplitude with decay constant equal to kk0. Thus:

In its output file, MODEIG displays the complex refractive index nc (it calls it "RN") which it calculated from NREAL and NLOSS. Thus, in a case where the user begins with nc and converts its complex part (i.e., k) to NLOSS units, the user can check that the conversion was performed properly.

As an example, the complex refractive index of gold is given as nc = 0.188 + 5.39 i at 826.6 nm. From this it may be calculated that NLOSS = 40.97 mm-1 for this wavelength. Using MODEIG/II, the propagation constant b for an (Al,Ga)As waveguide structure coated with gold is calculated. The MODEIG/II layer file shows the structure:

These numbers correspond to the structure given in a MODEIG/II input file containing the following LAYER data (corresponding to a 0.5 mm thick Al0.3Ga0.7As guiding layer embedded between the gold and an Al0.6Ga0.4As cladding layer):

which are respectively the real and imaginary parts of the modal propagation constant b (analogous with the complex index of refraction nc which by comparison describes propagation in a uniform material rather than a waveguide structure):

As mentioned above, the output file shows the correct complex refractive index nc (called RN by MODEIG/II) and an excerpt of the file for the example shows

finding RN = nc as expected.

NLOSS is also commonly used when the background losses in epilayers are included in MODEIG calculations. For example, a cladding layer may have a loss of 10 cm-1, which corresponds to an NLOSS of 0.0005 mm-1 (or NLOSS=0.0005).

Since NLOSS is amplitude loss in mm-1, rather than power loss in cm-1, the use of NLOSS can be confusing. This was not a problem eight or nine years ago when this was coded because there was only one user (G. Evans). As more people use this version of MODEIG, we could change NLOSS to be a power loss in cm-1. If you would like this change, let us know.

On the same subject of defining the index of layers, note that the MODEIG variable PEROPT allows entering either the complex index of refraction nc or the complex dielectric constant e.

To convert between a complex refractive index nc and a complex dielectric constant, e = e1 + i e2 = nc2, the following expressions are appropriate: