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Graded-index Fibers

Definition: optical fibers with a continuously varying refractive index in the radial dimension

Alternative term: gradient index fibers

More general term: optical fibers

German: Gradientenindexfasern, Gradientenfasern

Categories: fiber optics and waveguides, lightwave communications


Cite the article using its DOI: https://doi.org/10.61835/21r

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Many multimode fibers are so-called step-index fibers, where the refractive index is a function of the radial position, i.e., it is constant in some regions and exhibits steps (sharp changes) at certain locations. However, there are also so-called graded-index fibers (or gradient index fibers), where the refractive index varies smoothly in the radial direction. That can be achieved with fiber fabrication techniques where the chemical composition of the glass preform varies continuously. For silica fibers, it is common to vary the concentration of germania, but one may in addition use spatially variable fluorine doping.

A typical design of a graded-index fiber contains a parabolic profile from the fiber axis out to a certain radial position; outside that area, there can either be a constant refractive index (the cladding index) or first a depressed index region (trench). Figure 1 shows a simple design without such an index trench.

parabolic graded index profile
Figure 1: Refractive index profile of a graded-index fiber, which is parabolic in the core region. The effective refractive indices of the guided modes (shown as gray lines) for a wavelength of 1.55 μm are approximately equally spaced. For this model, set up with the RP Fiber Power software, a parabolic profile for the GeO2 concentration in a germanosilicate fiber has been assumed, which leads to an approximately parabolic index profile.

The fiber cladding is assumed to consist of pure silica (→ silica fibers).

Precisely speaking, what is meant to be parabolic is the square of the refractive index, i.e., the dielectric constant, rather than the refractive index itself. This detail, however, is quantitatively not that important in most cases, since the refractive index contrast is typically quite small.

A remarkable feature of such parabolic index designs is that the effective refractive indices of the guided modes are equally spaced (only not precisely for the highest-order modes, “seeing” the outer region which deviates from the parabolic shape). Figure 2 shows the effective indices versus mode area; one can see that the effective index essentially depends on the mode index <$l$> only, but hardly on <$m$>. (Note that many horizontal lines in Figure 1 nearly coincide for that reason.)

effective indices of parabolic profile fiber
Figure 2: Effective indices of the fiber versus effective mode area, with the colors of the points depending on the mode index <$l$>.

As shown in Figure 3, the group indices are approximately the same for all modes – except for the highest-order modes. This shows that the intermodal dispersion, which can be characterized by the differential mode delay, is quite small – much smaller than for a step-index fiber.

group indices of parabolic profile fiber
Figure 3: Effective indices of the fiber versus effective mode area, with the colors of the points depending on the mode index <$l$>.

Of course, one can further tailor the refractive index profile, slightly deviating from a parabolic shape, in order to further optimize mode properties. In particular, one may generalize the parabolic profile by using a different exponent; the higher that exponent, the closer the profile would be to a step-index profile. An optimized profile exponent, deviating slightly from 2 (the value for parabolic profile), can be determined for fibers with high index contrast; the optimum value depends on the so-called profile dispersion parameter, describing the relation between group index and refractive index for the chosen material composition. For example, the results of Figure 3, obtained for a parabolic profile, can be significantly enhanced further with a modified profile exponent of ≈1.85.

In an intuitive picture, one may consider hypothetical rays propagating along the fiber. Such rays would perform sinusoidal oscillations around the fiber axis; the index gradient always “bends” them back towards the axis. The strongly reduced intermodal dispersion is sometimes “explained” with the higher velocity of light away from the fiber core, which compensates the longer geometrical path length of the strongly oscillating ray, thus effectively leading to an effective path length per meter fiber which is the same for all rays. This picture is rather crude, however; for example, it suggests that phase delays acquired by the rays are strongly related to time delays, which is actually not true. In fact, the strongly mode-dependent effective indices (see Figure 2) show that the phase delays acquired by different modes are quite different, while the time delays are indeed quite similar according to Figure 3.

Figure 4 shows a simulation (with numerical beam propagation) where a Gaussian input beam has been somewhat displaced against the center of the fiber core. In the fiber, the intensity profile oscillates without fully reaching the edges of the core region. The observed oscillation is somewhat similar as in the ray picture mentioned above, but the perfect periodicity is destroyed by the cut-off parabolic shape. Also note that the transverse size of the oscillating intensity peak varies substantially along the fiber.

beam propagation in a graded-index fiber
Figure 4: Beam propagation in a graded-index fiber, where a Gaussian input beam has been slightly offset against the center of the fiber core.

The horizontal gray lines indicate the edges of the core.

For comparison, Figure 5 shows the same for a step-index design with the same core radius and maximum refractive index. The result looks quite different; one obtains a complicated evolution of the intensity profile.

beam propagation in a step-index fiber
Figure 5: Same as Figure 7, but for a step-index profile.

Fabrication of Graded-index Fibers

Optical fibers are usually fabricated by drawing them from a preform. (That applies to glass fibers as well as to plastic optical fibers.) The preform then needs to be made with a graded-index profile, so that the fiber exhibits the same profile, just on a different scale.

Different processes can be used to make such preforms:

  • In the case of glass preforms, the core material is usually generated with some method of chemical vapor deposition. That process needs to be controlled such that the refractive index is varied over time by changing the composition of the deposited material.
  • In the case of polymers, the refractive index can be increased by doping with high-index substances having a larger molecular weight. These also tend to exhibit lower diffusion constants. That can be exploited to maintain a concentration gradient until the polymerization is complete, and the gradient is “frozen in”. Alternatively, one may use a mixture of monomers with different density and refractive indices, and obtain a concentration gradient e.g. using centrifugal forces until polymerization is complete.

Applications of Graded-index Fibers

The above example has been for a germanosilicate fiber, i.e., a glass fiber. Similar designs, with typical core diameters being 50 μm and 62.5 μm, are used for multimode telecom fibers in fiber-optic links with a transmission distance of a few hundred meters, for example. The fiber designs and high-precision fabrication techniques have been refined more and more for obtaining a minimum differential mode delay, in that way maximizing the modal bandwidth and thus the transmission capacity of such links. Early standards for named OM1 and OM2; the optimization later lead to OM3 and OM4 fibers, allowing for substantially higher performance.

Typically, a small differential group delay and thus a high modal bandwidth is achieved only in a relatively limited wavelength region e.g. around 850 nm; the performance is already seriously degraded for a wavelength deviation of only 30 nm, for example. However, special wideband multimode fibers have been developed which offer relatively low intermodal dispersion over a broader wavelength range (e.g. 100 μm).

In the future, further substantially increases of transmission capacity may be achieved with mode division multiplexing, which can be realized with so-called multiple input multiple output (MIMO) techniques. For certain practical reasons, it is then still important to achieve a rather small differential group delay.

Graded-index designs are also sometimes used with other types of glass fibers, e.g. for mid-infrared fibers [8].

There are also plastic optical fibers (POF) having graded index profiles. They are often used in the same way, i.e., with the goal of minimizing intermodal dispersion effects in fiber-optic links.

Graded-index fibers are not only used for telecom purposes, but also e.g. for laser power transmission (power over fiber), where one may profit from the better output beam profile. For such applications, fibers with much larger core diameters of e.g. 100, 200, 400 or even 600 μm are available. Another application is using short pieces of such fibers as mode field adapters [13]. Some graded-index fibers are used in fiber-optic sensors, some are developed as large mode area fibers [10], and there are even versions for guiding terahertz radiation [17].

Graded-index Single-mode Fibers

Usually, the term graded-index fibers is used for multimode fibers only. However, single-mode fibers also often have a graded (but normally non-parabolic) refractive index profile. This may sometimes just result from certain fabrication conditions, while in other cases it is actively tailored for obtaining certain mode properties – mostly concerning chromatic dispersion. For example, triangular, trapezoidal or Gaussian profiles, possibly equipped with additional features, are used for dispersion-shifted fibers.

Case Study

The following case study is available:


[1]R. Olshansky, “Mode coupling effects in graded-index optical fibers”, Appl. Opt. 14 (4), 935 (1975); https://doi.org/10.1364/AO.14.000935
[2]L. Jacomme, “Modal dispersion in multimode graded-index fibers”, Appl. Opt. 14 (11), 2578 (1975); https://doi.org/10.1364/AO.14.002578
[3]D. Marcuse, “Gaussian approximation of the fundamental modes of graded-index fibers”, J. Opt. Soc. Am. 68 (1), 103 (1978); https://doi.org/10.1364/JOSA.68.000103
[4]M. Horiguchi, Y. Ohmori and H. Takata, “Profile dispersion characteristics in high-bandwidth graded-index optical fibers”,“Appl. Opt. 19 (18), 3159 (1980)”]
[5]W. Emkey nd C. Jack, “Analysis and evaluation of graded-index fiber lenses”, J. Lightwave Technol. 5 (9), 1156 (1987); https://doi.org/10.1109/JLT.1987.1075651
[6]M.-S. Chung and C.-M. Kim, “Analysis of optical fibers with graded-index profile by a combination of modified Airy functions and WKB solutions”, J. Lightwave Technol. 17 (12), 2534 (1999); https://doi.org/10.1109/50.809674
[7]G. Yabre, “Comprehensive theory of dispersion in graded-index optical fibers”, J. Lightwave Technol. 18 (2), 166 (2000); https://doi.org/10.1109/50.822789
[8]B.-Z. Dekel and A. Katzir, “Graded-index silver chlorobromide fibers for the mid-infrared”, Appl. Opt. 44 (16), 3343 (2005); https://doi.org/10.1364/AO.44.003343
[9]A. Kondo and T. Ishigure, “Fabrication process and optical properties of perdeuterated graded-index polymer optical fiber”, J. Lightwave Technol. 23 (8), 2443 (2005); https://doi.org/10.1109/JLT.2005.852021
[10]J. M. Fini, “Bend-resistant design of conventional and microstructure fibers with very large mode area”, Opt. Express 14 (1), 69 (2006); https://doi.org/10.1364/OPEX.14.000069
[11]M. B. Shemirani et al., “Principle modes in graded-index multimode fiber in presence of spatial- and polarization-mode coupling”, J. Lightwave Technol. 27 (10), 1248 (2009)
[12]Y. Akimoto et al., “Poly(styrene)-based graded-index plastic optical fiber for home networks”, Opt. Lett. 37 (11), 1853 (2012); https://doi.org/10.1364/OL.37.001853
[13]P. Hofmann et al., “Detailed investigation of mode-field adapters utilizing multimode-interference in graded index fibers”, J. Lightwave Technol. 30 (14), 2289 (2012); https://doi.org/10.1109/JLT.2012.2196406
[14]J. Sun et al., “Novel bending-resistant design of two-layer low-index trench fiber with parabolic-profile core”, Opt. Express 22 (15), 18036 (2014); https://doi.org/10.1364/OE.22.018036
[15]B. Ung et al., “Few-mode fiber with inverse-parabolic graded-index profile for transmission of OAM-carrying modes”, Opt. Express 22 (15), 18044 (2014); https://doi.org/10.1364/OE.22.018044
[16]R. Ryf et al., “Mode-multiplexed transmission over conventional graded-index multimode fibers”, Opt. Express 23 (1), 235 (2015); https://doi.org/10.1364/OE.23.000235
[17]T. Ma et al., “Graded index porous optical fibers – dispersion management in terahertz range”, Opt. Express 23 (6), 7856 (2015); https://doi.org/10.1364/OE.23.007856
[18]X. Zheng et al., “Bending losses of trench-assisted few-mode optical fibers”, Appl. Opt. 55 (10), 2639 (2016); https://doi.org/10.1364/AO.55.002639
[19]A. Inoue and Y. Koike, “Low-noise graded-index plastic optical fiber for significantly stable and robust data transmission”, J. Lightwave Technol. 36 (24), 5887 (2018)
[20]S. Savović et al., “Power flow in graded-index plastic optical fibers”, J. Lightwave Technol. 37 (19), 4985 (2019)
[21]A. Simović et al., “Investigation of bandwidth in multimode graded-index plastic optical fibers”, Opt. Express 29 (19), 29587 (2021); https://doi.org/10.1364/OE.433481

(Suggest additional literature!)

See also: fibers, telecom fibers, optical fiber communications, multimode fibers, intermodal dispersion, modal bandwidth, dispersion-shifted fibers

Questions and Comments from Users


Why don't we use single-mode graded-index fibers?

The author's answer:

We actually do – most single-mode fibers have an approximately smooth refractive index profile. Of course, the effect discussed in the context of multimode fibers are not relevant here. In particular, we have no intermodal dispersion anyway.


Why does a multimode gradient index fiber need a glass clad?

The author's answer:

You would not need that cladding for the optical function, but the preforms for these fibers are usually fabricated by depositing material on the inner side of a glass tube, and the material of that glass tube will form the cladding.


Do you know a tool with which the mode fields of a GRIN fiber can be calculated, i.e. a publication providing the mathematical model for mode field calculation for GRIN fibers? How do the GRIN mode fields differ from the well-known LP modes of a step-index fiber?

The author's answer:

A powerful tool for such things is our software RP Fiber Power. If you want to do it yourself, get one of the textbooks on fiber optics to find the required equations.

Your last question is quite vague; I am not sure what exactly you mean.

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