Propagation Constant
Author: the photonics expert Dr. Rüdiger Paschotta (RP)
Definition: a mode- and frequency-dependent quantity describing the propagation of light in a medium or waveguide
Alternative term: wave propagation constant
Category:
- fiber properties
- acceptance angle
- bend losses
- cut-off wavelength
- differential mode delay
- effective mode area
- effective refractive index
- group velocity dispersion
- intermodal dispersion
- modal bandwidth
- mode radius
- polarization beat length
- propagation constant
- propagation losses
- V-number
- waveguide dispersion
- zero dispersion wavelength
- (more topics)
Related: modes effective refractive index waveguides fibers group delay dispersion propagation losses
Units: rad/m
Formula symbol: ($k$), ($\beta$), ($\gamma$)
Page views in 12 months: 3851
DOI: 10.61835/x1u Cite the article: BibTex BibLaTex plain text HTML Link to this page! LinkedIn
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Propagation Constant in Physics
There are substantially different meanings of the term propagation constant; we begin with the common meaning in physics. The conventions outlined in this section are generally used in this encyclopedia.
We consider propagation of a plane wave in ($+z$) direction, described with a complex amplitude:
$$A(z) = A(0) \; e^{i k z}$$where we also use physicists' sign conventions. The temporal evolution is then governed by a term ($\exp(-i \omega t)$) with the angular frequency ($\omega$), which is (2ドル\pi$) times the optical frequency.
We consider ($k$) as the propagation constant. In lossless media, ($k$) is real. (It is also called wavenumber.) In a medium with propagation losses due to absorption, we can either add a term with an absorption coefficient or add an imaginary part to ($k$). If we split ($k$) into a real and an imaginary part according to
$$k = k_\textrm{r} + i k_\textrm{i}$$we can identify the absorption coefficient ($\alpha$) with (2ドル k_\textrm{i}$). (Note that the absorption coefficient is usually specified for optical intensity, which is proportional to ($|A(z)|^2$).) For an amplifying medium, e.g. with laser gain, ($k_\textrm{i}$) can become negative. The real part ($k_\textrm{r}$) can also be called the phase constant, as it determines the evolution of optical phase.
The propagation constant depends on the optical frequency (or wavelength) of the light. The frequency dependence of its real part determines the group velocity and the group velocity dispersion; see the article on chromatic dispersion.
For modes of waveguides, one usually has the symbol ($\beta$) instead of ($k$), but can otherwise use the same kinds of equations. Note that while in a homogeneous medium one can have a k vector oriented in an arbitrary direction, a waveguide fixes a certain propagation direction, and ($\beta$) is a scalar quantity. Again, one may either consider ($\beta$) as purely real and treat attenuation separately, or allow it to become complex, thus including attenuation or gain.
Normalized Propagation Constant
For propagation in waveguides such as optical fibers, one sometimes introduces a normalized propagation constant which can only vary between 0 and 1. Here, the value zero corresponds to the wavenumber of a plane wave in the cladding material, and 1 to that for the core. Modes which are mostly propagating in the cladding will have a value close to 0.
Calculating Propagation Constants
For calculating propagation constants, one requires a mode solver. The RP Fiber Power software is an ideal tool for such work, allowing such calculations very conveniently and with great flexibility.
Propagation Constant in Electrical Engineering
In electrical engineering, the field evolution is often described in the form
$$A(z) = A(0) \; e^{-\gamma z}$$where ($A(z)$) is again the complex field amplitude, and ($\gamma$) is called the propagation constant. A purely real ($\gamma$) value can be identified with ($\alpha / 2$), if the absorption coefficient ($\alpha$) is defined for the intensity rather than amplitude. Its imaginary part is related to the phase evolution.
Also note that the usual sign convention in engineering is that waves propagating in ($+z$) direction are described with
$$A(z) = A(0) \; e^{-j k z}$$so that we can identify ($\gamma$) with ($j k$). The temporal evolution is then governed by a term ($\exp(+j \omega t)$). Note that the imaginary constant is usually named ($j$) rather than ($i$) in electrical engineering.
In lossless media, ($\gamma$) is purely imaginary; we have ($\gamma = j k$) with the (real) phase constant ($k$) (or ($\beta$) in cases with waveguides). Generally we have:
$$\gamma = \frac{\alpha}{2} + j k$$Frequently Asked Questions
This FAQ section was generated with AI based on the article content and has been reviewed by the article’s author (RP).
What is the propagation constant in physics?
In physics, the propagation constant ($k$) describes how a plane wave's complex amplitude evolves in space, according to ($A(z) = A(0) \; e^{i k z}$). While it is real in lossless media, it can be a complex number to account for optical gain or loss.
What is the difference between the propagation constant and the phase constant?
The propagation constant ($k$) can be a complex number, written as ($k = k_\textrm{r} + i k_\textrm{i}$). Its real part ($k_\textrm{r}$) is the phase constant, which governs the change in optical phase, while its imaginary part ($k_\textrm{i}$) is related to absorption or gain.
How does the definition of the propagation constant differ in electrical engineering?
In electrical engineering, the propagation constant ($\gamma$) is often defined via ($A(z) = A(0) \; e^{-\gamma z}$), using a different sign convention than in physics. In this convention, the real part of ($\gamma$) relates to attenuation, and the imaginary part to the phase change.
What is a normalized propagation constant?
For waveguides like optical fibers, the normalized propagation constant is a value scaled to be between 0 and 1. A value of 0 corresponds to the wavenumber in the cladding material, and 1 corresponds to that in the core material.
Questions and Comments from Users
2020年08月13日
How can the effective propagation constant be written in terms of relative permittivity and permeability?
The author's answer:
For propagation in a homogeneous medium, it is (2ドル\pi / \lambda$) = (2ドル\pi n / \lambda_0$), where the refractive index ($n$) can be written as the square root of ($\epsilon \mu$), the product of the relative permittivity and permeability.
2021年12月06日
How can I calculate the propagation constant for a fiber given the values of the specsheet?
The author's answer:
You would need to know the transverse refractive index profile of the fiber, which is unfortunately not usually available on specification sheets, and take that as an input for some mode solver software.
2025年06月02日
Is the weighted average of the refractive index multiplied by E field norm squared for a VCSEL like structure provide the effective index for the mode inside?
The author's answer:
Although it may seem plausible, this is not the case. See the article on effective refractive index.
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