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# Effective Refractive Index

Definition: a number quantifying the phase delay per unit length in a waveguide, relative to the phase delay in vacuum

More general term: refractive index

German: effektiver Brechungsindex

Formula symbol: neff

Units: (dimensionless)

Author:

For plane waves in homogeneous transparent media, 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 but also (for multimode waveguides) on the mode in which the light propagates. For this 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 quantity. In that case, the imaginary part is related to gain or loss – see the article on refractive index for more details.

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. This 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.

## Questions and Comments from Users

2020-05-01

Does this mean, that when simulation software gives the effective refractive index, the gain or loss can be calculated? If so, what formula should be used?

Only if the complex refractive index is computed. Its imaginary part times 4π / λ gives you the exponential intensity absorption or gain coefficient – with signs depending on conventions.

2020-06-02

If I want to calculate the effective refractive index in the process of coupling light into a microring to form an optical frequency comb, do I need to take into account the different effective refractive indices in the waveguide and the microring?

Each device (waveguide and microring) will have its own effective refractive index – more precisely, one such value for each guided mode, if it is not single-mode.

2020-06-03

Can we use RP Fiber Power to directly calculate the effective refractive index in microrings?

Unfortunately, this might work only with limited accuracy if you have a large refractive index contrast. There might also be a problem with the limitation that the mode solver cannot take into account the strong curvature.

2020-08-25

Imagine two cladding materials with the same real refractive index, but one has absorption while the other does not. Is the real part of the propagation constant of the waveguide, beta, the same for both scenarios?

They will generally not have the same real part of β.

2020-10-23

Can the effective index of modes be used in Snell's law or Fresnel's equations?

No, that won't work. These laws assume plane waves, while the effective refractive index is calculated for a waveguide mode.

2020-12-26

Is it possible to calculate the effective index of a 3D structure, e.g. a dielectric nanorod antenna superimposed on a dielectric waveguide?

I don't think that in such a case the effective refractive index can be defined in a meaningful way.

2021-04-09

Why does a waveguide's effective index decreases for longer wavelengths?

This is not always so, but it is the typical behavior that the refractive index, and also the effective refractive index, decreases following a wavelengths within the transparency region. That behavior can be understood based on Kramers–Kronig relations, considering the absorption bands on both sides of the transparency region.

2021-05-22

Why is effective index transverse mode-dependent?

We can consider the field distribution corresponding to a mode as a superposition of plane waves with different propagation directions. That superposition will be different for each mode, and the phase delay is reduced for modes traveling in directions deviating from the waveguide axis.

2021-05-28

I find that some researchers write νm = m c / (2 n L) where m is an integer and L is the laser cavity length, then write Δνm = c / (2 neffL). What is the difference between n and neff, and how can I get the second equation from the first one?

The first equation applies to the mode frequencies of a linear optical resonator which is filled with the material having that refractive index. It is also assumed that additional phase changes by beam divergence or for reflection at the end mirrors are negligible. It is then correct, but one should keep in mind that the refractive index is frequency-dependent; that leads to non-equidistant mode frequencies, and to a mode spacing which also depends on the frequency derivative of the refractive index.

You can take the second equation as the definition of the effective refractive index. So you can calculate that quantity from the mode spacing, but this takes a little more work. You best use the condition that from one mode to the next one the round-trip phase shift changes by 2π.

2021-09-14

For a 3D structure, if I know the electromagnetic modes. How can I define the phase constant or propagation constant? And what is the relationship between the propagation constant and the effective refractive index?

That is defined only for propagation in a specific direction. If your 3D structure is a waveguide, you can easily calculate the phase constant of the modes, but otherwise it may not be defined. The same applies to the effective refractive index, which is simply related to the phase constant as explained in the article (see the first equation).

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