Light guidance in fountain

Optical fibers are one of the most widely used form of optical waveguide. Unlike other form of waveguides, optical fibers can be manufactured long and bendable, and thus are used for a transmission media for optical fiber communication. Its history is believed to date back to mid 1800s, when an Irish physicist John Tyndall demonstrated light guidance in a fountain. Light is guided in a stream of water even though the water stream is bent by gravity, highlighting two very important features of optical fiber (i.e. long and bendable). Figure 1 shows the schematic of light guidance.

Figure 1: Light guidance in fountain

Total internal reflection in light fountain

In a light fountain, light is guided by a phenomenon referred to as total internal reflection. Figure 2 explains schematically how total internal reflection takes place at the interface of water and air, and light is trapped inside the water stream.


Figure 2: Snell’s law and total internal reflection.

The angle of refraction at the interface of two materials is given by Snell’s law:

\(\mathrm{n_1}\cdot \sin(\theta) = \mathrm{n_2} \cdot \sin(\theta^{\prime}), \)

where n1 and n2 are the refractive indices of the two materials, θ is the incident angle of light, and θ’ is the angle of refraction [see Figure 2(a)]. If we assume that material 1 is water (n1=1.3) and material 2 is air (n2=1.0), θ’ is larger than θ as n1>n2, and θ’ becomes \(\pi\) /2 (i.e. 90 deg) for a certain incident angle θ = θc [Figure 2(b)]. This angle is called the critical angle, and is given by the following formula:

\( \theta_{\mathrm{c}}=\sin^{-1}\left( \frac{\mathrm{n_2}}{\mathrm{n_1}} \right). \)

And at an incident angle θ > θc [Figure 2(c)], the refracted light can no longer exist, and light is totally reflected back to the material 1 (water). This is how light is trapped in a fountain, and this phenomenon is called total internal reflection (TIR).

Light guidance in optical fiber

TIR takes place at any interface between two materials with different refractive indices, not only at the interface of water and air. Figure 3 shows schematically the most basic form of an optical fiber. A core is embedded in a cladding and modifies the refractive index in the transverse (x and y) direction, and the modification is uniform in the longitudinal (z) direction. The refractive-index modification enables the transverse confinement of light in the core, and light is guided longitudinally in the core.

Figure 3: Schematic of optical fiber, and a ray confined in the core by total internal reflection.

Figure 3 shows schematically, using ray optics, the light guided in the core by TIR. The refractive indices of the core and cladding are ncore and nclad, and a ray is travelling in the core with an angle θ. If ncore > nclad and θ is larger than the critical angle θc which is provided by

\( \theta_{\mathrm{c}}=\sin^{-1}\left( \frac{\mathrm{n_{clad}}}{\mathrm{n_{core}}} \right), \)

TIR takes place at the border between the core and cladding, and the ray is confined in the core. On the other hand, the power of a ray in the core rapidly decreases if θ is smaller than θc, because the ray is only partially reflected back at the border. The refractive index of the core has to be higher than that of the cladding, in order to guide light by total internal reflection.

The equation also suggests that a larger refractive-index difference between the core and cladding provides a smaller θc. A small θc allows for launching light into the fiber with a large angle ( \(\pi/2−\theta\) in Figure 1), enabling the launch of higher power density into the core. As a measure of the ability to accept light with a large launch angle, the numerical aperture (NA) of a fiber is defined by

\( \mathrm{NA}=\mathrm{n_{core}}\sin \left( \frac{\pi}{2}-\theta_{\mathrm{c}} \right)=\sqrt{\mathrm{n_{core}^2-n_{clad}^2}}. \)

This equation suggests an optical fiber with a higher NA allows the launch of light with a larger launch angle, therefore enables the launch of higher power density into the core.


Step-index and graded-index fiber

There are two major types of refractive-index profiles – step index (SI) and graded index (GI) – and they are schematically shown in Figure 4. The core of a SI fiber has a uniformly raised refractive index profile, and that of a GI fiber has a continuously raised refractive index profile. The largest difference between these two types of optical fibers lies in the ability to maintain the pulse shape after propagation.


Figure 4: Step-index and graded-index fiber.

When an optical pulse is launched into a SI fiber, the propagation velocity of the pulse is highly dependent on the incident angle. Figure 4(a) intuitively explains that the light launched at a larger incident angle (blue ray in the figure) propagates at a slower velocity, as the path length is longer. This is problematic for high-speed optical communication as the incident pulse shape is degraded at the output.

GI fiber was introduced in order to overcome this limitation of SI-MM fiber. Figure 4(b) shows that, light launched at a small angle (red ray) mainly travels near the center of the core, and light launched at a large angle (blue ray) travels at the outer side of the core. The path length is still longer for the blue ray; however the blue ray travels at a faster speed than the red ray as the refractive index of the outer side of the core is lower than that of the center. The difference in the propagation velocity, therefore, becomes much smaller and degradation in the pulse shape is reduced. It is theoretically possible to eliminate this delay, however in reality, small amount of delay always remains due to small variations in fiber fabrication.

Guided modes in optical fiber

Ray optics gives an intuitive understanding and, in fact, is a good approximation if the size of the core is much larger than the wavelength guided in the optical fiber. The electromagnetic properties of an optical fiber, however, become essential as the size of the waveguide becomes closer to the wavelength. The concept of modes in an optical waveguide is one of the most important concepts when analyzing an optical fiber using electromagnetics.

Figure 5: Schematic of optical fiber and guided modes.

Figure 5 shows schematically a longitudinally uniform optical fiber and examples of modes. Each mode maintains the distribution of the electromagnetic field in the x-y plane (modal-field distribution) as it propagates in the z-direction. Modes are determined once the refractive-index profile and the angular frequency of light \(\omega\) (\(\omega=2\pi C_0/\lambda\) where \(C_0\) is the speed of light in vacuum) are given, the electromagnetic field of the mode at an arbitrary position and time is given as follows:

\( \mathbf{E}_{\beta}(x,y,z,t)=\mathbf{E}_{\beta}(x,y)e^{i(\omega t – \beta z)}, \)
\( \mathbf{H}_{\beta}(x,y,z,t)=\mathbf{H}_{\beta}(x,y)e^{i(\omega t – \beta z)}. \)

The propagation constant \(\beta\) of each mode is provided by solving an eigenvalue problem (which is determined by the refractive-index profile and angular frequency \(\omega\)), and the modal-field distribution \(\mathbf{E}_{\beta}(x,y)\) or \(\mathbf{H}_{\beta}(x,y)\) is the eigenstate for the eigenvalue.

The modal-field distribution is generally either oscillatory or evanescent in the transverse direction. If the modal-field distribution is oscillatory in the core and evanescent in the cladding, the mode is confined in the core as the modal-field distribution rapidly decays in the cladding. These modes are referred to as guided modes. Guided mode is the most important type of modes and is often simply referred to as “mode”.


Multimode and single-mode fiber

A set of guided modes with the same (or very close) \(\beta\) generally show very similar characteristics (e.g. modal-field distribution) except their polarization states; this set of modes is called a mode class. An optical fiber with a large core guides multiple mode classes and the fiber is called a multimode fiber. The number of mode classes reduces as the size of the core decreases, and when the size becomes sufficiently small, only one mode class is allowed. This fiber is called a single-mode fiber.


Fiber material

Low-attenuation silica fiber for optical transmission

Another important feature of optical fiber is its low attenuation. In 1966, Charles Kao predicted that the loss of optical fiber can be reduced significantly if it is made by high-purity glass [1]. The prediction was proven in 1970 by three scientists in Corning – Donald Keck, Robert Maurer, and Peter Schultz [2]. Now the loss of standard telecom optical fiber is well below 0.20 dB/km; light is lost by less than five percent per kilometer. In 2009, Charles Kao was awarded the Novel prize for his contribution to fiber optics.

Today low-loss fibers for telecommunication are all silica-based fibers, and the low losses are enabled by vapor-phase manufacturing methods, for example, MCVD [3, 4], VAD [5], and OVD [6]. In manufacturing silica fiber by vapor-phase methods, SiCl4 and GeCl4 are typically used as starting raw materials. The vapor pressures of these two materials are much higher than those of undesirable contaminants (e.g. FeCl3, PbCl2) remained in the raw material, therefore high-purity raw materials can be delivered to a reaction chamber by vapor-phase delivery. The raw materials delivered to the chamber is then heated and turns into oxide glass, either by oxidation or hydrolysis:

\( \text{SiCl}_{4}/\text{GeCl}_{4} + \text{O}_{2} \longrightarrow \text{SiO}_{2}/\text{GeO}_{2} + \text{2Cl}_{2}, \) (Oxidation)

\( \text{SiCl}_{4}/\text{GeCl}_{4} + \text{4H}_{2} + \text{2O}_{2} \longrightarrow \text{SiO}_{2}/\text{GeO}_{2} + \text{4HCl} + \text{2H}_{2}\text{O}. \) (Hydrolysis)

Silica optical fibers for telecommunication nowadays have negligible amount of contaminants, and the two factors that determine the losses are Rayleigh scattering and infrared absorption.

Non-silica specialty optical fibers

Low loss is arguably the most important characteristic as a transmission media for optical communication, and silica-based glass is by far the most transparent optical fiber material. There are, however, many other areas where fibers are used in a short piece (up to several tens of meters) and other characteristics have more importance. These application-specific fibers are often called as specialty fibers, and non-silica optical fiber materials may be used due to their unique natures.

Fluoride fibers are used for fiber lasers/amplifiers and IR laser power delivery (see our fluoride fiber technology for detail). Chalcogenide fibers are used for optical signal processing due to its high nonliniarity. Plastic fibers are used for cars and consumer electronics, as handling of plastic fibers does not require special skill or training.


FiberLabs’ optical fiber products

FiberLabs manufactures ZBLAN and AlF3-based fluoride fibers in house. Fluoride fibers are mainly used as specialty fibres, and have been used in a variety of industries ranging from telecom to medical. We have been supplying various fluoride fibers, both standard and custom made, to meet customers’ needs. So please feel free to contact us if you would like to use fluoride fibers for your projects.



[1] K. C. Kao and G. A. Hockham, “Dielectric-fibre surface waveguides for optical frequencies,” Proceedings of the Institution of Electrical Engineers, vol. 113, no. 7, pp. 1151–1158, Jul. 1966.
[3] J. B. MacChesney, P. B. O’Connor, and H. M. Presby, “A new technique for the preparation of low-loss and graded-index optical fibers,” Proceedings of the IEEE, vol. 62, no. 9, pp. 1280–1281, 1974.
[4] J. B. MacChesney, “Materials and processes for preform fabrication—Modified chemical vapor deposition and plasma chemical vapor deposition,” Proceedings of the IEEE, vol. 68, no. 10, pp. 1181–1184, 1980.
[5] T. Izawa and N. Inagaki, “Materials and processes for fiber preform fabrication—Vapor-phase axial deposition,” Proceedings of the IEEE, vol. 68, no. 10, pp. 1184–1187, 1980.
[6] P. C. Schultz, “Fabrication of optical waveguides by the outside vapor deposition process,” Proceedings of the IEEE, vol. 68, no. 10, pp. 1187–1190, 1980.