Propagation of Light Among a Fiber

The concept of light propagation, the transmission of light along an optical fiber, can be described by two theories. According to the first theory, light is described as a simple ray. This theory is the ray theory, or geometrical optics, approach. The advantage of the ray approach is that you get a clearer picture of the propagation of light along a fiber. The ray theory is used to approximate the light acceptance and guiding properties of optical fibers. According to the second theory, light is described as an electromagnetic wave. This theory is the mode theory, or wave representation, approach. The mode theory describes the behavior of light within an optical fiber. The mode theory is useful in describing the optical fiber properties of absorption, attenuation, and dispersion. These fiber properties are discussed later in this chapter.

Ray Theory

Two types of rays can propagate along an optical fiber. The first type is called meridional rays. Meridional rays are rays that pass through the axis of the optical fiber. Meridional rays are used to illustrate the basic transmission properties of optical fibers. The second type is called skew rays. Skew rays are rays that travel through an optical fiber without passing through its axis.

MERIDIONAL RAYS.—Meridional rays can be classified as bound or unbound rays. Bound rays remain in the core and propagate along the axis of the fiber. Bound rays propagate through the fiber by total internal reflection. Unbound rays are refracted out of the fiber core. Figure 2-11 shows a possible path taken by bound and unbound rays in a step-index fiber. The core of the step-index fiber has an index of refraction n1. The cladding of a step-index has an index of refraction n2 that is lower than n1. Figure 2-11 assumes the core-cladding interface is perfect. However, imperfections at the core-cladding interface will cause part of the bound rays to be refracted out of the core into the cladding. The light rays refracted into the cladding will eventually escape from the fiber. In general, meridional rays follow the laws of reflection and refraction.

It is known that bound rays propagate in fibers due to total internal reflection, but how do these light rays enter the fiber? Rays that enter the fiber must intersect the core-cladding interface at an angle greater than the critical angle (- c). Only those rays that enter the fiber and strike the interface at these angles will propagate along the fiber.

How a light ray is launched into a fiber is shown in figure 2-12. The incident ray I1 enters the fiber at the angle -a. I1 is refracted upon entering the fiber and is transmitted to the core-cladding interface. The ray then strikes the core-cladding interface at the critical angle (-c). I1 is totally reflected back into the core and continues to propagate along the fiber. The incident ray I2 enters the fiber at an angle greater than -a. Again, I2 is refracted upon entering the fiber and is transmitted to the core-cladding interface. I2 strikes the core-cladding interface at an angle less than the critical angle (-c). I2 is refracted into the cladding and is eventually lost. The light ray incident on the fiber core must be within the acceptance cone defined by the angle -a shown in figure 2-13. Angle -a is defined as the acceptance angle. The acceptance angle (-a) is the maximum angle to the axis of the fiber that light entering the fiber is propagated. The value of the angle of acceptance (-a) depends on fiber properties and transmission conditions.

The acceptance angle is related to the refractive indices of the core, cladding, and medium surrounding the fiber. This relationship is called the numerical aperture of the fiber. The numerical aperture (NA) is a measurement of the ability of an optical fiber to capture light. The NA is also used to define the acceptance cone of an optical fiber.

Figure 2-13 illustrates the relationship between the acceptance angle and the refractive indices. The index of refraction of the fiber core is n1. The index of refraction of the fiber cladding is n2. The index of refraction of the surrounding medium is n0. By using Snell’s law and basic trigonometric relationships, the NA of the fiber is given by:

Since the medium next to the fiber at the launching point is normally air, n0 is equal to 1.00. The NA is then simply equal to sin -a. The NA is a convenient way to measure the light-gathering ability of an optical fiber. It is used to measure source-to-fiber power-coupling efficiencies. A high NA indicates a high source-to-fiber coupling efficiency. Source-to-fiber coupling efficiency is described in chapter 6. Typical values of NA range from 0.20 to 0.29 for glass fibers. Plastic fibers generally have a higher NA. An NA for plastic fibers can be higher than 0.50.

In addition, the NA is commonly used to specify multimode fibers. However, for small core diameters, such as in single mode fibers, the ray theory breaks down. Ray theory describes only the direction a plane wave takes in a fiber. Ray theory eliminates any properties of the plane wave that interfere with the transmission of light along a fiber. In reality, plane waves interfere with each other. Therefore, only certain types of rays are able to propagate in an optical fiber. Optical fibers can support only a specific number of guided modes. In small core fibers, the number of modes supported is one or only a few modes. Mode theory is used to describe the types of plane waves able to propagate along an optical fiber.

SKEW RAYS.—A possible path of propagation of skew rays is shown in figure 2-14. Figure 2-14, view A, provides an angled view and view B provides a front view.Skew rays propagate without passing through the center axis of the fiber. The acceptanceangle for skew rays is larger than the acceptance angle of meridional rays. This conditionexplains why skew rays outnumber meridional rays. Skew rays are often used in thecalculation of light acceptance in an optical fiber. The addition of skew rays increases theamount of light capacity of a fiber. In large NA fibers, the increase may be significant.

Mode Theory

The mode theory, along with the ray theory, is used to describe the propagation of light along an optical fiber. The mode theory is used to describe the properties of light that ray theory is unable to explain. The mode theory uses electromagnetic wave behavior to describe the propagation of light along a fiber. A set of guided electromagnetic waves is called the modes of the fiber.

PLANEWAVES.—The mode theory suggests that a light wave can be represented as a plane wave. A plane wave is described by its direction, amplitude, and wavelength of propagation. A plane wave is a wave whose surfaces of constant phase are infinite parallel planes normal to the direction of propagation. The planes having the same phase are called the wave fronts. The wavelength (λ) of the plane wave is given by: 

where c is the speed of light in a vacuum, f is the frequency of the light, and n is the index of refraction of the plane-wave medium.

Figure 2-15 shows the direction and wave fronts of plane-wave propagation. Plane waves, or wave fronts, propagate along the fiber similar to light rays. However, not all wave fronts incident on the fiber at angles less than or equal to the critical angle of light acceptance propagate along the fiber. Wave fronts may undergo a change in phase that prevents the successful transfer of light along the fiber.

Wave fronts are required to remain in phase for light to be transmitted along the fiber. Consider the wave front incident on the core of an optical fiber as shown in figure 2-15. Only those wave fronts incident on the fiber at angles less than or equal to thecritical angle may propagate along the fiber. The wave front undergoes a gradual phasechange as it travels down the fiber. Phase changes also occur when the wave front isreflected. The wave front must remain in phase after the wave front transverses the fibertwice and is reflected twice. The distance transversed is shown between point A and pointB on figure 2-16. The reflected waves at point A and point B are in phase if the totalamount of phase collected is an integer multiple of 2π radian. If propagating wave frontsare not in phase, they eventually disappear. Wave fronts disappear because of destructive interference. The wave fronts that are in phase interfere with the wave fronts that are out of phase. This interference is the reason why only a finite number of modes can propagate along the fiber.

The plane waves repeat as they travel along the fiber axis. The direction the plane wave’s travel is assumed to be the z direction as shown in figure 2-16. The plane waves repeat at a distance equal to λ/sin- . Plane waves also repeat at a periodic frequency β = 2π sin -/λ. The quantity β is defined as the propagation constant along the fiber axis. As the wavelength (λ) changes, the value of the propagation constant must also change. For a given mode, a change in wavelength can prevent the mode from propagating along the fiber. The mode is no longer bound to the fiber. The mode is said to be cut off. Modes that are bound at one wavelength may not exist at longer wavelengths. The wavelength at which a mode ceases to be bound is called the cutoff wavelength for that mode. However, an optical fiber is always able to propagate at least one mode. This mode is referred to as the fundamental mode of the fiber. The fundamental mode can never be cut off. The wavelength that prevents the next higher mode from propagating is called the cutoff wavelength of the fiber. An optical fiber that operates above the cutoff wavelength (at a longer wavelength) is called a single mode fiber. An optical fiber that operates below the cutoff wavelength is called a multimode fiber. Single mode and multimode optical fibers are discussed later in this chapter.

In a fiber, the propagation constant of a plane wave is a function of the wave’s wavelength and mode. The change in the propagation constant for different waves is called dispersion. The change in the propagation constant for different wavelengths is called chromatic dispersion. The change in propagation constant for different modes is called modal dispersion. These dispersions cause the light pulse to spread as it goes down the fiber, see figure 2-17. Some dispersion occurs in all types of fibers. Dispersion is discussed later in this chapter.

The TE mode field patterns shown in figure 2-18 indicate the order of each mode. The order of each mode is indicated by the number of field maxima within the core of the fiber. For example, TE0 has one field maxima. The electric field is a maximum at the center of the waveguide and decays toward the core cladding boundary. TE0 is considered the fundamental mode or the lowest order standing wave. As the number of field maxima increases, the order of the mode is higher. Generally, modes with more than a few (5-10) field maxima are referred to as high-order modes.

The order of the mode is also determined by the angle the wave front makes with the axis of the fiber. Figure 2-19 illustrates light rays as they travel down the fiber. These light rays indicate the direction of the wave fronts. High-order modes cross the axis of the fiber at steeper angles. Low-order and high-order modes are shown in figure 2-19.

Before we progress, let us refer back to figure 2-18. Notice that the modes are not confined to the core of the fiber. The modes extend partially into the cladding material. Low-order modes penetrate the cladding only slightly. In low-order modes, the electric and magnetic fields are concentrated near the center of the fiber. Low-order modes take parallel or modestly transverse paths. However, high-order modes penetrate further into the cladding material and take considerably more transverse paths. In high-order modes, the electrical and magnetic fields are distributed more toward the outer edges of the fiber.

This penetration of low-order and high-order modes into the cladding region indicates that some portion is refracted out of the core. The refracted modes may become trapped in the cladding due to the dimension of the cladding region. The modes trapped in the cladding region are called cladding modes. As the core and the cladding modes travel along the fiber, mode coupling occurs. Mode coupling is the exchange of power between two modes. Mode coupling to the cladding results in the loss of power from the core modes.

In addition to bound and refracted modes, there are leaky modes. Leaky modes are similar to leaky rays. Leaky modes lose power as they propagate along the fiber. For a mode to remain within the core, the mode must meet certain boundary conditions. A mode remains bound if the propagation constant β meets the following boundary condition:

where n1 and n2 are the index of refraction for the core and the cladding, respectively. When the propagation constant becomes smaller than 2πn2/λ, power leaks out of the core and into the cladding. Generally, modes leaked into the cladding are lost in a few centimeters. However, leaky modes can carry a large amount of power in short fibers.

NORMALIZED FREQUENCY.—Electromagnetic waves bound to an optical fiber are described by the fiber’s normalized frequency. The normalized frequency determines how many modes a fiber can support. Normalized frequency is a dimensionless quantity. Normalized frequency is also related to the fiber’s cutoff wavelength. Normalized frequency (V) is defined as:

where n1 is the core index of refraction, n2 is the cladding index of refraction, a is the core diameter, and λ is the wavelength of light in air.

The number of modes that can exist in a fiber is a function of V. As the value of V increases, the number of modes supported by the fiber increases. Optical fibers, single mode and multimode, can support a different number of modes. The number of modes supported by single mode and multimode fiber types is discussed later in this chapter.