Effective Refractive Index
For plane waves of light in homogeneous transparent media (e.g. in optical materials), the refractive index n can be used to quantify the increase in the wavenumber (phase change per unit length) caused by the medium: the wavenumber is n times higher than it would be in vacuum. The effective refractive index neff has the analogous meaning for light propagation in a waveguide with restricted transverse extension: the β value (phase constant) of the waveguide (for some wavelength) is the effective index times the vacuum wavenumber:
The mode-dependent and frequency-dependent β values can be calculated with a mode solver software. They depend on the refractive index profile of the waveguide. From frequency-dependent β values, one may also calculate chromatic dispersion, e.g. by numerical differentiation.
Note that the effective refractive index depends not only on the wavelength (or optical frequency) but also (for multimode waveguides) on the mode in which the light propagates. For that reason, it is also called modal index. Obviously, the effective index is not just a material property, but depends on the whole waveguide design. Its value can be obtained with numerical mode calculations, for example. It can vary substantially near a mode cut-off.
The effective index may be a complex (rather than purely real) quantity. In that case, the imaginary part is related to gain or loss – see the article on refractive index for more details. In fiber amplifiers, for example, the imaginary part of the effective index is always far smaller than the real part – which implies that gain or loss does not have substantial effects on a length scale where diffraction and waveguiding start to become relevant.
The effective refractive index contains information on the phase velocity of light, but not on the group velocity; for the latter, one can similarly define an effective group index in analogy to the group index for plane waves in a homogeneous medium.
A common but wrong belief is that the effective refractive index is a kind of weighted average of the refractive index of core and cladding of the waveguide, with the weight factors determined by the fractions of the optical power propagating in the core and cladding. That impression may result from the common observation that higher-order modes, e.g. of a fiber, have a lower effective index and also a lower mode overlap with the core. However, consider e.g. a step-index multimode waveguide with a high numerical aperture and large core diameter. Here, all modes overlap to nearly 100% with the core (i.e. the mode overlaps are very similar), whereas the effective indices differ substantially. That demonstrates that the effective refractive index cannot be interpreted as a weighted average.
Questions and Comments from Users
Here you can submit questions and comments. As far as they get accepted by the author, they will appear above this paragraph together with the author’s answer. The author will decide on acceptance based on certain criteria. Essentially, the issue must be of sufficiently broad interest.
Please do not enter personal data here; we would otherwise delete it soon. (See also our privacy declaration.) If you wish to receive personal feedback or consultancy from the author, please contact him e.g. via e-mail.
By submitting the information, you give your consent to the potential publication of your inputs on our website according to our rules. (If you later retract your consent, we will delete those inputs.) As your inputs are first reviewed by the author, they may be published with some delay.