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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

Alternative term: modal index

More general term: refractive index

German: effektiver Brechungsindex

Units: (dimensionless)

Formula symbol: <$n_\textrm{eff}$>

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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 <$n_\rm{eff}$> has the analogous meaning for light propagation in a waveguide with restricted transverse extension: the <$\beta$> value (phase constant) of the waveguide (for some wavelength) is the effective index times the vacuum wavenumber:

$$\beta = {n_{{\rm{eff}}}}\frac{{2\pi }}{\lambda }$$

The mode-dependent and frequency-dependent <$\beta$> values can be calculated with a mode solver software. (Relatively simple calculations are sufficient for LP modes, while a more general mode solver requires a rather sophisticated algorithm.) They depend on the refractive index profile of the waveguide. From frequency-dependent <$\beta$> 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.

Case Studies

## Case Study: Mode Structure of a Multimode Fiber

We explore various properties of guided modes of multimode fibers. We also test how the mode structure of such a fiber reacts to certain changes of the index profile, e.g. to smoothening of the index step.

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.

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## 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 <$4\pi / \lambda$> 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 <$\beta$>.

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

Would it be 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 <$\nu_\textrm{m} = m c / (2 n L)$> where <$m$> is an integer and <$L$> is the laser cavity length, then write <$\Delta \nu_\textrm{m} = c / (2 n_\textrm{eff} L)$>. What is the difference between <$n$> and <$n_\textrm{eff}$>, 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 <$2\pi$>.

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).

2021-09-29

Can the effective refractive index of a waveguide be smaller than the smallest value of the refractive indices of materials comprising the waveguide?

No, I think that cannot happen. But sorry, I have no proof available.

2021-11-12

Is it correct that the effective refractive index difference between two modes increases with the increase of wavelength?

Yes, at least in some typical cases, where the higher-order mode further extends in the leading, particularly for longer wavelengths.

2021-11-25

If the effective refractive index difference of the two modes in the fiber increases with the increase of wavelength, will the group delay difference of the two modes increase or decrease with the increase of wavelength?

Generally, you cannot in fern from refractive indices to group indices, essentially since knowing a function value does not mean that you also know its frequency derivative.

2023-09-25

Can I calculate the numerical aperture using the effective refractive index? I know that normally we use the (n core2 - n cladding2)1/2 to calculate the NA, but in the actual experiment, we cannot get a perfect refractive index profile.

No, that will not work, as you probably do not know the shape of the index profile.

2023-10-19

Can the effective refractive index of a mode in a waveguide be less than that of the cladding for very small widths?

No, it cannot. You would then have an oscillatory field in the cladding, which is not permitted for a guided mode.

2024-02-28

How can I determine the value of the effective refractive index of a polarization-maintaining fiber?

No matter whether the fiber is polarization-maintaining, you will need a mode solver and the refractive index profile as an input for that.

2024-03-18

For Snell's law or Fresnel equations at a waveguide facet, should we use neff instead of n? For time delay calculations of a fiber in OFDR, should we use neff or use ng?

For Snell's law, the normal refractive index <$n$> is relevant.

Time delays are governed by the group index <$n_{\rm g}$>.

In both fields, the effective index has no relevance. It only governs the phase advance inside the waveguide, but in OFDR, e.g., the time delays are relevant.

2024-04-15

For an angled waveguide facet, even for the fundamental mode, some of power is outside the core. Which refractive index should be used in Snell's to calculate the output ray angle?