# Mode Radius

Acronym: MFD = mode field diameter = twice the mode radius

Definition: a measure of the transverse extent of a laser mode or laser beam

Alternative term: mode field radius

German: Modenradius

Category: fiber optics and waveguides

Units: m

Author: Dr. Rüdiger Paschotta

The transverse extent of the optical intensity distribution of a mode (e.g. of an optical cavity or a waveguide) is usually specified as a *mode radius* (or *mode field radius*). It is mostly of interest in the context of single-mode fibers, or for the fundamental mode of multimode fibers (mostly few-mode fibers). It can be defined in different ways, as described in the following section.

The *mode diameter* (or *mode field diameter*, MFD) is simply twice the mode radius.

It is not recommended to use the common terms *mode size* or *spot size* in a quantitative sense because it is then not clear whether the radius or the diameter is meant.

## Near-Field Based Definitions

Traditionally, the mode field radius (or diameter) was defined as for Gaussian beams, where the beam radius is the radius where the intensity has dropped to <$1/e^2$> of the intensity on the beam axis. This is a suitable definition as long as the mode intensity profile has a close to Gaussian shape. For better accuracy, one can then use a Gaussian fit and take its radius as the result, rather than directly applying the <$1/e^2$> criterion to the mode intensity profile.

For modes with clearly non-Gaussian profiles, as occur for some fibers (e.g., many dispersion-shifted fibers), such methods are not suitable. A more accurate way of obtaining the near-field beam radius is then to apply the D4σ method (ISO Standard 11146) to the measured near-field beam profile. This technique is explained in the article on beam radius. It requires a high dynamic range and can profit from special smoothing techniques.

In any case, near-field measurement methods are used. For example, one may image the near field (i.e., the field distribution at a fiber end) onto a somewhat larger spot and scan that spot with a small photodetector.

## Far-Field Based Definitions

It is quite common to measure the far field profile, i.e., the beam profile outside the fiber in a distance which is much larger than the Rayleigh length. This can be done e.g. with a camera mounted in some distance to the fiber end, or by scanning the far field with a small photodetector.

For the fundamental mode (i.e., not for higher-order modes), one can then calculate a near field mode radius (or diameter) from the measured angular width in the far field. The nowadays most common method is using the “Petermann II mode field diameter definition” [2, 3], assuming a radially symmetric fiber mode:

$${\rm{MFD}} = \frac{\lambda }{\pi }\sqrt {\frac{{2\int\limits_0^{\pi /2} {I(\theta )\sin \theta \cos \theta \;{\rm{d}}\theta } }}{{\int\limits_0^{\pi /2} {I(\theta ){{\sin }^3}\theta \cos \theta \;{\rm{d}}\theta } }}} $$(There is also a Petermann I definition, which applies directly to the near field and agrees with the common D4σ definition of the beam radius.)

The mode *radius* is of course half that mode diameter.

For fibers with close to Gaussian mode profiles, the near-field and far-field mode field diameters agree quite well, whereas in other cases the far-field values can be significantly smaller. Both values are relevant for coupling losses at fiber joints: the far-field mode radius is relevant for losses related to a transverse core offset, whereas the near-field mode radius is relevant concerning angular misalignment.

## Indirect Measurement Techniques

There are also various indirect techniques for measuring the mode field diameter (or radius). For example, the *transverse offset technique* is based on measuring coupling losses between two pieces of single-mode fiber as a function of a transverse core offset. These are related to the far-field beam radius. Other techniques are based on variable aperture.

## Marcuse Formula for the Mode Radius of a Step-index Single-mode Fiber

For step-index single-mode fibers, the mode radius (based on the <$1/e^2$> criterion) may be estimated from the core radius <$a$> and the *V* number, using Marcuse's equation [1]:

This shows that the mode radius becomes smaller for higher frequencies, which have higher <$V$> values. The equation is fairly accurate for <$V$> values above 1. In the multimode range (<$V > 2.405$>), it applies to the fundamental mode.

There is also a modified formula for approximating the Petermann II mode field radius [4]:

$$\frac{w}{a} \approx 0.65 + \frac{{1.619}}{{{V^{3/2}}}} + \frac{{2.879}}{{{V^6}}} - (0.016 + 1.561\;{V^{ - 7}})$$This is accurate within 1% for <$V$> between 1.5, and 2.5, the range of highest practical interest.

For a given numerical aperture, one may expect that a smaller core radius will lead to a smaller mode radius. However, for core radii below a certain value, the mode radius rises again (see Figure 2). In that regime, the mode field extends far into the cladding.

## Relation of Mode Field Radius and Effective Mode Area

It appears natural to relate the mode field radius to the mode area according to

$$A_{\rm{eff}} = \pi w^2$$and to call this the *mode field area* (MFA). However, the effective mode area is actually defined differently:

Depending on the mode shape, the values from the two definitions can deviate substantially from each other. For calculating the effective mode area from the mode field diameter, one can thus apply a correction factor, which depends on the wavelength [6].

For judging the strength of fiber nonlinearities, the latter definition (effective mode area) is clearly more appropriate.

## Relevance of the Mode Field Diameter

The mode field diameter (or radius) is relevant e.g. at fiber joints, where a mismatch of mode sizes and angular deviations can lead to substantial coupling losses. Note, however, that efficient coupling requires not only matching the mode field diameters, but of the complete mode profiles.

As mentioned above, the strength of fiber nonlinearities is determined by the effective mode area, which is not directly related to the mode field diameter according to the usual definitions.

### Bibliography

[1] | D. Marcuse, “Loss analysis of single-mode fiber splices”, Bell Syst. Tech. J. 56, 703 (1977), DOI:10.1002/j.1538-7305.1977.tb00534.x |

[2] | K. Petermann, “Constraints for fundamental mode spot size for broadband dispersion-compensated single-mode fibers”, Electron. Lett. 19, 712 (1983), DOI:10.1049/el:19830485 |

[3] | C. Pask, “Physical interpretation of Petermann's strange spot size for single-mode fibres”, Electron. Lett. 20 (3), 144 (1984), DOI:10.1049/el:19840097 |

[4] | C. D. Hussey and F. Martinez, “Approximate analytical forms for the propagation characteristics of single-mode optical fibres”, Electron. Lett. 21 (23), 1103 (1985), DOI:10.1049/el:19850783 |

[5] | M. Artiglia et al., “Mode field diameter measurements in single-mode optical fibers”, J. Lightwave Technol. 7 (8), 1139 (1989) |

[6] | Y. Namihira, “Relationship between nonlinear effective area and modefield diameter for dispersion shifted fibres”, Electron. Lett. 30 (3), 262 (1994), DOI:10.1049/el:19940149 |

See also: modes, Gaussian beams, beam radius, effective mode area, fiber joints, mode field converters, fibers, single-mode fibers, dispersion-shifted fibers, spotlight 2014-07-28

## Questions and Comments from Users

2020-11-25

Several vendors of PANDA980-type fiber claim core NA = 0.12, core diameter 5.5 μm, and MFD 6.6 μm at 980 nm wavelength. Mode field calculations however, show that these numbers are not mutually consistent: for NA = 0.12, the MFD at 980 nm should be 5.4 μm, i.e. about 20% less than claimed, which isn't negligible in some applications. The quoted MFD of 6.6 μm would be about correct for NA = 0.10, instead. Is this just a typo in the spec sheets or something more?

The author's answer:

Presumably, your calculations are based on the assumption of a step-index fiber. However, many single-mode fibers substantially deviate from that: the refractive index varies substantially within the core region. In that situation, it is of course questionable what exactly is meant with the given NA. I would then recommend just to work with the mode size specified by the manufacturer. The problem is still often to know what exactly they mean with their mode size specification.

2021-02-01

How does the wavelength affect the mode size of a single mode fiber?

The author's answer:

Tentatively, shorter wavelengths mean smaller modes.

2021-11-25

If I were to launch blue light into a single mode fiber designed for telecom wavelengths (i.e. fiber will get few-moded) how does the MFD change if compared to the same wavelength but launched into an optical fiber designed for the visible?

The author's answer:

That depends on the fiber designs, therefore one cannot generally answer that question.

2022-11-24

If I have the simulated E_{f}ield beam profile in a PCF fiber. Is the MFD (in the context of fiber simulations) better defined via a Gaussian 2D fit and the D4sigma method or via <$A_\rm{eff}$> and then A_{eff} = pi * (MFD/2)^{2}?

The author's answer:

That depends on what you want to use the value for. The effective mode area <$A_\rm{eff}$> is a good measure for how strong nonlinear effects will be, while a D4σ value may be more relevant in other situations.

2023-02-09

How to calculate bent fiber mode radius with constant bending radius? Is there any analytical formula including the bending radius?

The author's answer:

I am not aware of such an equation, and I am afraid the situation is too complicated for such a simple solution. The dependence of the mode radius on the beam radius will generally depend on various factors.

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2020-08-19

Since MFD value for single mode fiber does not really exceed 12 microns typically, why is the cladding diameter still 125 microns? Is this due to optical reasons or more to do with mechanical strength and processing methods?

The author's answer:

Considering only the optical performance, the fiber diameter could indeed be much smaller. However, such extremely thin fibers would indeed not have a sufficiently high mechanical stability. One may use such things, however, in special situations.