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Step-index profile

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Refractive HRS index distribution in core and cladding with a step-index profile

For an optical fiber, a step-index profile is a refractive index profile characterized by a uniform refractive index within the core and a sharp decrease in refractive index at the core-cladding interface so that the cladding is of a lower refractive index. The step-index profile corresponds to a power-law index profile with the profile parameter approaching infinity. The step-index profile is used in most single-mode fibers^[1] and some multimode fibers.^[2]

A step-index fiber is characterized by the core and cladding refractive indices n₁ and n₂ and the core and cladding radii a and b. Examples of standard core and cladding diameters 2a/2b are 8/125, 50/125, 62.5/125, 85/125, or 100/140 (units of μm). The fractional refractive-index change $\triangle \,={\frac {n_{1}-n_{2}}{n_{1}}}\ll \ 1$ . The value of n₁ is typically between 1.44 and 1.46, and $\triangle$ is typically between 0.001 and 0.02.

Step-index optical fiber is generally made by doping high-purity fused silica glass (SiO₂) with different concentrations of materials like titanium, germanium, or boron.

Modal dispersion in a step index optical fiber is given by

${\text{pulse dispersion}}={\frac {\triangle \ n_{1}\ \ell }{c}}\,\!$

where

\triangle \,\!

is the fractional index of refraction

n_{1}\,\!

is the refractive index of core

\ell \,\!

is the length of the optical fiber under observation

c

is the speed of light.

References

[edit]

^ "SINGLE MODE STEP-INDEX FIBERS". Retrieved 2012-11-23.
^ "Fiber Optics Overview". Archived from the original on 2013-01-17. Retrieved 2012-11-23.

This article incorporates public domain material from Federal Standard 1037C. General Services Administration. Archived from the original on 2022-01-22. (in support of MIL-STD-188).

This optics-related article is a stub. You can help Wikipedia by expanding it.

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from Grokipedia

A step-index profile is a refractive index distribution in optical waveguides, particularly optical fibers, characterized by a uniform, higher refractive index throughout the core and an abrupt transition to a lower, constant refractive index in the surrounding cladding.^[1] This "top-hat" structure enables efficient light guidance via total internal reflection at the core-cladding boundary, serving as the foundational design for both multimode and single-mode fibers used in data transmission and sensing applications.^[2] The development of the step-index profile traces back to the mid-1960s, when Charles K. Kao and George A. Hockham proposed using low-loss glass fibers with a cladding of lower refractive index to confine light propagation, predicting attenuation below 20 dB/km for practical telecommunications.^[3] This breakthrough, for which Kao received the 2009 Nobel Prize in Physics, shifted early experiments from high-loss unclad fibers to structured designs, with initial step-index multimode fibers emerging in the early 1970s as the first viable prototypes for commercial use. By the late 1970s, manufacturing advances like the modified chemical vapor deposition process enabled production of low-loss step-index fibers, revolutionizing long-haul communication by replacing copper cables.^[4] Step-index profiles are classified into multimode and single-mode variants based on core size and numerical aperture. Multimode step-index fibers feature core diameters from 50 μm to 1 mm,^[5] supporting multiple light paths (modes) that propagate at different speeds, leading to modal dispersion but allowing simpler coupling for short-haul links under 2 km, such as in local area networks, video surveillance, and plastic optical fibers for consumer electronics.^[6] Single-mode step-index fibers, with core diameters of 8–10 μm^[7] and numerical apertures around 0.1–0.14,^[8] propagate only the fundamental mode, exhibiting near-zero intermodal dispersion and enabling bandwidths exceeding 100 Gbps over hundreds of kilometers^[9] in submarine cables, metropolitan networks, and high-speed internet backbones. Their simplicity facilitates cost-effective fabrication via methods like outside vapor deposition, though they suffer from chromatic dispersion that requires compensation in wavelength-division multiplexing systems.^[10] Beyond telecommunications, step-index profiles find applications in fiber-optic sensors for temperature, pressure, and strain measurement, leveraging total internal reflection for reliable signal integrity in harsh environments like oil wells and aerospace.^[11] They also enable medical endoscopes and laser delivery in surgery due to high power-handling capacity, and industrial uses in illumination and power transmission where graded-index alternatives may introduce unwanted complexity.^[12] Despite competition from graded-index profiles that reduce dispersion in multimode scenarios, step-index designs remain dominant for single-mode applications owing to their predictable ray optics and mature production ecosystem.^[13]

Definition and Structure

Refractive Index Profile

The step-index profile in optical fibers is characterized by a uniform refractive index within the core that abruptly decreases at the core-cladding interface. Specifically, the core maintains a constant refractive index

n_1

up to the core radius

a

, beyond which the refractive index sharply drops to

n_2

in the cladding, with

n_1 > n_2

. This discontinuous change enables total internal reflection at the boundary, confining light propagation within the core.^[14]^[1] Mathematically, the refractive index profile

n(r)

as a function of radial distance

r

from the fiber axis is expressed as:

n(r) = \begin{cases} n_1 & \text{for } r \leq a \\ n_2 & \text{for } r > a \end{cases}

This piecewise function represents the "step" nature of the profile, distinguishing it from graded-index designs where the index varies continuously.^[14] Graphically, the step-index profile appears as a rectangular plot when refractive index is graphed against radial distance: a flat plateau at

n_1

from

r = 0

r = a

, followed by an immediate vertical drop to

n_2

that remains constant thereafter. This top-hat shape illustrates the abrupt transition, often depicted in fiber optics literature to highlight the simplicity of the design.^[1] The step-index profile originated in early optical fiber designs during the 1960s, pioneered by Charles K. Kao and George A. Hockham in their seminal 1966 paper, which proposed low-loss dielectric waveguides with a higher-index core surrounded by lower-index cladding to enable practical light transmission over long distances.^[15]^[16]

Core and Cladding Composition

The core of a step-index optical fiber is the central cylindrical region with a uniform higher refractive index that serves as the primary conduit for light propagation. This region confines optical signals through total internal reflection at its boundary with the surrounding cladding. Typical core diameters are approximately 8–10 μm in single-mode step-index fibers, allowing for efficient single-path transmission over long distances, while multimode variants feature larger cores of 50 to 1000 μm to support multiple simultaneous light paths for shorter-range applications.^[17]^[5] The cladding forms the outer layer encasing the core, possessing a lower refractive index to facilitate light confinement and provide mechanical stability to the fiber structure. With a standard thickness resulting in a 125 μm diameter for most silica-based fibers, the cladding not only protects the delicate core but also ensures compatibility with industry-standard connectors and handling procedures.^[18] Structurally, the step-index fiber exhibits a coaxial cylindrical geometry, where the core radius

a

precisely delineates the abrupt refractive index transition to the cladding, defining the step boundary essential for waveguiding. Total internal reflection at this core-cladding interface is enabled by the index contrast, with the boundary conditions dictated by Snell's law, ensuring rays exceeding the critical angle remain trapped within the core.^[1]^[19]

Types of Step-Index Fibers

Multimode Step-Index Fibers

Multimode step-index fibers are optical fibers characterized by a core diameter sufficiently large, typically greater than 50 μm (such as 50 μm or 100 μm), to support the simultaneous propagation of multiple transverse guided modes at a given wavelength and polarization.^[20]^[21] This design contrasts with single-mode fibers by enabling a higher light-gathering capacity, though it introduces complexities in signal transmission due to mode interactions. The numerical aperture (NA) plays a key role in determining the range of angles for light acceptance into the core, influencing the diversity of modes excited.^[22] The number of guided modes in these fibers is governed by the V-number, defined as

V = \frac{2\pi a}{\lambda} \cdot NA

, where

a

is the core radius,

\lambda

is the wavelength, and NA is the numerical aperture. For multimode operation,

V > 2.405

, the cutoff value for the fundamental mode, resulting in hundreds to thousands of modes; for example, a step-index fiber with a 100 μm core diameter and NA of 0.2 supports approximately 994 modes.^[22]^[23] The approximate total number of modes

M

for large

V

(typically

V > 20

) follows

M \approx \frac{V^2}{2}

. In multimode step-index fibers, light rays propagate along either meridional paths, which repeatedly cross the fiber axis in a plane containing the axis, or skew paths, which zigzag helically around the axis without crossing it.^[24] These differing path lengths cause intermodal dispersion, where modes travel at varying group velocities, leading to pulse broadening and signal distortion over distance.^[25] This effect is most pronounced in step-index profiles compared to graded-index alternatives.^[26] Due to intermodal dispersion, multimode step-index fibers exhibit bandwidth limitations, with typical bandwidth-distance products ranging from 20 to 30 MHz·km, restricting their use to shorter distances and lower data rates.^[27] For instance, uncompensated step-index designs often achieve only 20 to 30 MHz·km.^[27]

Single-Mode Step-Index Fibers

Single-mode step-index fibers are optical waveguides engineered to support only a single propagating mode, achieved through a precise core diameter of approximately 8-10 μm and a normalized frequency parameter

V

less than 2.405, which restricts light to the fundamental linearly polarized LP

_{01}

mode.^[7]^[28]^[29] The abrupt refractive index change at the core-cladding interface in the step-index profile facilitates tight confinement of this mode, minimizing leakage into the cladding.^[8] The transition to single-mode operation is governed by the cutoff wavelength

\lambda_c

, defined as the longest wavelength at which higher-order modes can propagate; for wavelengths longer than

\lambda_c

, only the LP

_{01}

mode is supported. This cutoff is calculated using the formula:

\lambda_c = \frac{2\pi a}{V_c} \sqrt{n_1^2 - n_2^2}

λc=Vc2πan12−n22

where $V_c = 2.405$ is the critical normalized frequency for the LP $_{11}$ mode cutoff, $a$ is the core radius, and $n_1$ and $n_2$ are the refractive indices of the core and cladding, respectively.^[30] This parameter determines the operational wavelength range, typically ensuring single-mode behavior in the near-infrared spectrum for telecommunications applications.^[31] Due to the absence of multiple modes, single-mode step-index fibers exhibit minimal intermodal dispersion, with total dispersion dominated by material dispersion—arising from wavelength-dependent refractive index variations in the silica core—and waveguide dispersion, which stems from the fiber's geometric and index profile effects.^[32]^[28] These contributions can be balanced to achieve near-zero dispersion at specific wavelengths, such as around 1310 nm, enhancing signal integrity over long distances.^[33] Standardization of single-mode step-index fibers is outlined in the ITU-T G.652 recommendation, which defines key attributes for telecommunications use, including a core-cladding refractive index difference $\Delta \approx 0.3\%$ to optimize mode confinement and low attenuation.^[34]^[35] This standard ensures compatibility across global networks, with variants like G.652.D incorporating reduced water peak losses for extended wavelength bands.^[36]

Optical Characteristics

Numerical Aperture and Light Acceptance

The numerical aperture (NA) of a step-index optical fiber is a dimensionless parameter that quantifies the fiber's capacity to accept light from an external source, specifically the maximum angle at which incident light rays can enter the core and undergo total internal reflection at the core-cladding interface.^[37] It is defined by the formula

NA = \sqrt{n_1^2 - n_2^2},

NA=n12−n22

, where $n_1$ is the refractive index of the core and $n_2$ is the refractive index of the cladding, with $n_1 > n_2$ .^[37]^[38] This expression arises from the condition for total internal reflection, where the index contrast between core and cladding determines the light-gathering efficiency.^[39] The derivation of the NA formula relies on Snell's law applied sequentially at the air-core interface and the core-cladding boundary. Consider a ray incident from air (refractive index $n_0 \approx 1$ ) at an angle $\theta_{\max}$ to the fiber axis, refracting into the core at angle $\theta_1$ . For the ray to propagate without loss, it must reach the core-cladding interface at the critical angle $\theta_c = \arcsin(n_2 / n_1)$ . Applying Snell's law at the air-core interface gives $n_0 \sin \theta_{\max} = n_1 \sin \theta_1$ , and at the core-cladding interface, $n_1 \sin(90^\circ - \theta_1) = n_2 \sin 90^\circ$ , leading to $\sin \theta_{\max} = \sqrt{n_1^2 - n_2^2} / n_0 \approx NA$

/n0≈NA for air.^[37]^[38]^[39] This maximum acceptance angle $\theta_{\max} = \arcsin(NA)$ defines the half-angle of the acceptance cone, a conical region of directions from which light rays can efficiently couple into the fiber and be guided by total internal reflection.^[37]^[39] The acceptance cone is crucial for coupling efficiency in applications such as laser-to-fiber connections, as rays outside this cone will refract into the cladding and experience higher attenuation.^[38] In practice, NA values for step-index multimode fibers typically range from 0.1 to 0.3, reflecting a balance between light collection and propagation quality.^[37]^[39] A higher NA enhances the fiber's ability to accept a wider range of incident angles, supporting more guided modes and improving coupling from divergent sources, but it can also increase losses due to greater sensitivity to bending, scattering, and leaky modes.^[37]^[38] The NA contributes to the V-number calculation, which ultimately governs the total number of propagating modes in multimode fibers.^[37]

Mode Propagation and Dispersion

In step-index optical fibers, light propagation can be modeled using ray optics, where meridional rays travel in planes containing the fiber axis and bounce axially via total internal reflection at the core-cladding interface, while skew rays in multimode fibers follow helical paths that do not intersect the axis, contributing to azimuthal propagation components.^[40] This ray model illustrates the basic guidance mechanism but overlooks wave interference effects. A more precise description employs waveguide theory, where modes are solutions to the Helmholtz equation under the weakly guiding approximation, assuming a small refractive index difference between core and cladding. These modes are linearly polarized (LP) modes, denoted as LP_{lm}, with l and m as azimuthal and radial indices, respectively, and each characterized by a propagation constant β that determines the axial phase velocity along the fiber.^[41] The propagation constant β lies between the cladding wavenumber nk_2 and the core wavenumber nk_1, where k = 2π/λ is the free-space wavenumber, enabling guided wave solutions.^[42] Dispersion in step-index fibers manifests in three primary forms, leading to temporal broadening of optical pulses. Intermodal dispersion occurs in multimode fibers due to differences in path lengths among various ray trajectories or modes, causing higher-order modes to arrive later than axial ones. Material dispersion arises from the wavelength dependence of the refractive index, where different spectral components of the pulse travel at varying group velocities. Waveguide dispersion results from the mode-dependent effective index, as the fiber geometry influences the fraction of light guided in the core versus cladding, altering propagation speeds across wavelengths.^[43] Attenuation during mode propagation is predominantly caused by Rayleigh scattering from microscopic density fluctuations in the glass, with a typical intrinsic loss coefficient of approximately 0.2 dB/km at the 1550 nm wavelength used in long-haul communications.^[44] The numerical aperture affects the initial excitation of these modes by determining the range of input angles that couple light into guided paths.^[41]

Fabrication and Materials

Manufacturing Processes

The manufacturing of step-index optical fibers centers on fabricating a preform with an abrupt refractive index transition between core and cladding, followed by drawing the fiber while maintaining this profile. Primary techniques for preform creation include modified chemical vapor deposition (MCVD) and outside vapor deposition (OVD), both leveraging vapor-phase reactions to deposit layered glass structures. These methods enable precise control over the index step, essential for the fiber's guiding properties. In the MCVD process, gaseous silicon, germanium, and other precursors flow into a rotating fused silica substrate tube, where an external oxyhydrogen torch heats the inner wall to induce chemical reactions that deposit thin soot layers of doped oxide glass. For step-index profiles, multiple deposition passes build a central core region with higher-index material, surrounded by lower-index cladding layers, after which the tube is heated to collapse into a solid, cylindrical preform. This inside-deposition approach ensures uniformity and is widely used for producing preforms up to 1 meter long.^[45]^[46] The OVD process deposits soot externally on a rotating target rod (often ceramic) via flame hydrolysis of vaporized chloride precursors in an oxyhydrogen flame, forming porous layers that are later consolidated. Core layers with elevated refractive index are deposited first on the rod, followed by cladding layers to create the step-index structure, allowing flexible control over profile dimensions; the soot body is then sintered at high temperature (~1400°C) in a chlorine atmosphere to form a clear glass preform, with the rod removed chemically. This outside method excels in scalability for large preforms and precise index tailoring.^[47]^[45] Fiber drawing transforms the preform into a continuous strand by feeding it vertically into a drawing tower, where the tip is heated to ~2000°C in a resistance or induction furnace to soften the glass without devitrification. The molten end is pulled downward by a capstan or tractor at speeds of 10-20 m/s, reducing the diameter from centimeters to 125 μm while preserving the core-cladding structure through controlled tension and temperature gradients; the drawn fiber is then passed through a coating applicator for dual-layer polymer protection (typically 250 μm total diameter) and cured via UV light.^[48] Post-drawing quality control verifies the index profile integrity using the refracted near-field (RNF) technique, which scans the near-field light pattern refracted from the fiber end immersed in index-matching liquids of varying refractive indices, enabling reconstruction of the radial profile. This non-destructive method offers a precision of

4 \times 10^{-5}

in index measurements and detects deviations from the ideal step profile, ensuring compliance with specifications.^[49]

Common Materials Used

In step-index optical fibers, the core is typically composed of silica glass doped with germanium dioxide (GeO₂) to achieve a refractive index of approximately 1.46, enabling light confinement through total internal reflection at the core-cladding interface.^[50]^[51] For dispersion-shifted variants, phosphorus doping is employed in the silica core to modify the refractive index profile and optimize chromatic dispersion characteristics.^[52] The cladding is commonly made from undoped silica with a refractive index of about 1.444, providing the necessary index contrast for guiding.^[51] Fluorine doping in the silica cladding lowers the refractive index further, allowing for a larger refractive index difference (Δn) between core and cladding to support higher numerical apertures in multimode applications.^[53] For plastic optical fibers (POF), polymethyl methacrylate (PMMA) serves as the core material with a refractive index around 1.49, paired with a fluorinated polymer cladding of approximately 1.40 refractive index.^[54]^[55] Material selection prioritizes low optical absorption in the key transmission windows of 850 nm, 1310 nm, and 1550 nm to minimize signal loss, alongside thermal stability for silica-based fibers up to 1000°C to withstand processing and operational conditions.^[7]^[56]

Applications and Comparisons

Practical Uses in Optics

Step-index fibers, particularly single-mode variants, are widely employed in telecommunications for long-haul transmission due to their low attenuation and ability to support high-bit-rate signals over extended distances. For instance, standard single-mode step-index fibers conforming to ITU-T G.652 specifications are integral to submarine cable systems, enabling data transmission across transoceanic distances exceeding 10,000 km with minimal signal loss.^[57] As of 2024, ITU-T G.652 remains the standard, supporting enhanced low-loss variants.^[58] In contrast, multimode step-index fibers find application in short-reach local area networks (LANs), where distances under 500 m are common, leveraging their simpler coupling requirements for cost-effective connectivity in enterprise environments.^[21] In optical sensing, step-index fibers serve as the host medium for Fiber Bragg Gratings (FBGs), which are inscribed into the core to enable precise measurements of environmental parameters. These gratings in single-mode step-index fibers exhibit distinct wavelength shifts in response to temperature variations (typically 10-14 pm/°C) and mechanical strain (around 1.2 pm/µε), allowing for multiplexed sensor arrays in structural health monitoring and aerospace applications.^[59] For medical procedures, multimode step-index fibers are utilized in endoscopy probes to deliver illumination and collect imaging signals, benefiting from their flexibility and high numerical aperture (NA up to 0.5) for efficient light collection in confined spaces.^[60] In industrial settings, these fibers facilitate high-power laser delivery for processes like welding, where their robust step-index design supports NA values greater than 0.2 to couple intense beams (up to several kW) while maintaining beam integrity over short distances.^[61] Emerging applications in quantum communication leverage low-loss single-mode step-index fibers to transmit entangled photons over fiber networks, achieving quantum key distribution with fidelities above 90% in metropolitan-scale deployments.^[62]

Differences from Graded-Index Profiles

The step-index profile features an abrupt change in refractive index at the core-cladding interface, with a uniform index

n_1

in the core and a lower constant index

n_2

in the cladding, whereas the graded-index profile exhibits a gradual variation, typically parabolic, described by the formula

n(r) = n_1 \sqrt{1 - 2\Delta (r/a)^\alpha}

n(r)=n11−2Δ(r/a)α

, where $\Delta = (n_1^2 - n_2^2)/(2n_1^2)$ is the relative index difference, $r$ is the radial distance from the center, $a$ is the core radius, and $\alpha = 2$ for the standard parabolic case.^[63]^[64] This sharp discontinuity in step-index fibers leads to straightforward ray paths that reflect at the boundary via total internal reflection, while the graded-index design causes rays to continuously refract inward, equalizing travel times across modes.^[7] In terms of dispersion, step-index multimode fibers suffer from higher intermodal dispersion due to significant differences in propagation velocities among modes, typically on the order of 50 ns/km, which broadens optical pulses and limits signal integrity over distance.^[65] Graded-index profiles mitigate this by optimizing the index gradient to nearly equalize path lengths and group velocities for different modes, reducing intermodal dispersion to around 0.5–1 ns/km and enabling longer transmission distances with less pulse spreading.^[63]^[7] This difference in mode propagation arises because step-index fibers propagate modes with distinct axial and radial velocities, whereas graded-index fibers focus rays periodically to minimize differential delays.^[66] Bandwidth performance reflects these dispersion characteristics, with step-index multimode fibers offering lower modal bandwidth, typically around 50 MHz·km at 850 nm, suitable only for short-haul applications.^[7] In contrast, graded-index multimode fibers achieve significantly higher bandwidths, up to 4.7 GHz·km (4700 MHz·km) in optimized designs like OM4 or OM5, supporting data rates of 10 Gbit/s over several hundred meters. For multimode, OM5 fibers enable 400 Gbps over 100 m in data centers as of 2025.^[67]^[68] Step-index profiles are generally simpler and less costly to fabricate, involving uniform doping of the core material during processes like chemical vapor deposition, making them ideal for plastic optical fibers (POF) used in low-cost, short-distance links.^[7]^[69] Graded-index profiles, however, require more complex manufacturing to achieve the precise radial index variation, often using modified chemical vapor deposition with graded dopant concentrations, and are thus reserved for high-performance glass fibers in telecommunications.^[63]^[69]

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