Light is an electromagnetic wave with extremely high frequency, and optical fiber itself is a dielectric waveguide; therefore, the theory of light propagation in optical fibers is extremely complex. A comprehensive understanding requires knowledge of electromagnetic field theory, wave optics theory, and even quantum field theory.
To facilitate understanding, this textbook discusses the light-guiding principle of optical fibers from the perspective of geometric optics, which is more intuitive, visual, and easier to comprehend. Moreover, for multimode optical fibers, since their geometric dimensions are much larger than the wavelength of light, the light wave can be treated as a single ray, which is the fundamental starting point for geometric optics.
Total internal reflection principle
"When light propagates in a uniform medium, it travels in a straight line direction, but when it reaches the interface between two different media, reflection and refraction phenomena occur. The reflection and refraction of light are shown in Figure 2-4.
According to the law of reflection, the angle of reflection equals the angle of incidence; according to the law of refraction, n₁sinθ₁ = n₂sinθ₂. Where n₁ is the refractive index of the fiber core; n₂ is the refractive index of the cladding.
Obviously, if n₁ > n₂, then θ₂ > θ₁. If the ratio of n₁ to n₂ increases to a certain extent, the refraction angle θ₂ ≥ 90°, and the refracted light will no longer enter the cladding, but will be refracted along the interface between the fiber core and cladding (when θ₂ = 90°), or return back into the fiber core for propagation (when θ₂ > 90°). This phenomenon is called total internal reflection of light. As shown in Figure 2-5."
The angle of incidence corresponding to a refraction angle θ₂ = 90° is called the critical angle (θ₀), which can be easily obtained.
It's easy to understand that when total internal reflection occurs in an optical fiber, since almost all the light propagates within the fiber core, and no light escapes into the cladding, the fiber's attenuation is greatly reduced. Early step-index optical fibers were designed based on this concept.
Propagation of light in step-index optical fiber
(1) Propagation of Light Rays in Optical Fibers To facilitate understanding, we will first use the ray method theory to give a simple description of the propagation of light waves in optical fibers. When a beam of light is coupled into the optical fiber from the end face, there may be different forms of light rays in the fiber: meridional rays and oblique rays. Figure 2-6a shows a ray that always propagates in a plane containing the central axis 00' of the optical fiber, and intersects the central axis twice in one propagation cycle. This type of ray is called a meridional ray, and the plane containing the central axis of the optical fiber is called the meridional plane. Figure 2-6a shows a meridional plane MN. Another type is where the trajectory of the light ray during propagation is not in the same plane and does not intersect the central axis of the optical fiber. This type of ray is called an oblique ray, as shown in Figure 2-6b. The analysis of oblique rays is quite complicated even using the ray method theory. This is because the propagation of oblique rays is not in a plane like that of meridional rays, but rather in a spiral pattern within a three-dimensional space, as shown in Figure 2-6b. Analysis requires the use of three-dimensional coordinates, which is somewhat abstract, but its basic light-guiding principle is the same as the meridian method, so a detailed analysis is not provided.
(2) Meridian propagation in step-index fiber The propagation of the meridian in a step-index fiber is shown in Figure 2-7. A step-index fiber consists of a core with a refractive index of n2 and a cladding with a refractive index of n1, where n1and n2 are constants, and n1> n2.
"When light O enters from air (n₀ = 1) into the optical fiber end surface at angle φ₁, a portion of the light will enter the optical fiber. At this time, according to Snell's law n₀sinφ₁ = n₁sinθ₁, and since the fiber core refractive index n₁ > n₀ (air refractive index), the refraction angle θ₁ < φ₁, and the light continues to propagate, incident at angle θᵢ = 90° - θ₁ to the interface between the fiber core and cladding. If θᵢ is smaller than the critical angle θc = arcsin(n₂/n₁) at the fiber core and cladding interface, then part of the light will be refracted into the cladding and lost, while another part reflects back into the fiber core. In this way, after several reflections and refractions, this light ray will quickly be attenuated. If φ₁ decreases to φ₀ (as in light ray ②), then θᵢ also decreases, while θᵢ = 90° - θ₁ increases. If φ₁ increases to exceed the critical angle θc, then this light ray will undergo total internal reflection at the fiber core and cladding interface, with all energy reflected back into the fiber core. When it continues to propagate and encounters the fiber core and cladding interface again, total internal reflection occurs again. Repeating this process, the light can be transmitted from one end along a zigzag path to the other end.
Let us analyze how small φ₁ must be to transmit light from one end of the optical fiber to the other end.
Assuming φ₁ = φ₀, then θc = θc₀, θᵢ = θc, n₀ = 1, we have: n₀sinφ₀ = sinφ₀ = n₁sinθ₀ = n₁sin(90° - θc) = n₁cosθc
Thus we have: sinφ₀ = n₁cosθc = n₁√(1 - sin²θc) = n₁√(1 - (n₂/n₁)²) = n₁√(2Δ) = √(n₁² - n₂²)
In the equation, Δ is the relative refractive index difference of the optical fiber, Δ = (n₁² - n₂²)/(2n₁²) ≈ (n₁ - n₂)/n₁.
From this it can be seen that as long as the incident angle φ₁ ≤ φ₀ at the optical fiber end surface, light can be transmitted through total internal reflection in the fiber core. φ₀ is called the maximum incident angle of the optical fiber end surface, and 2φ₀ is the maximum acceptance angle of the optical fiber for light."
(Figure 2-7 Meridian propagation in a step-index optical fiber)
"(3) Numerical Aperture: Since the difference between n₁ and n₂ is small, the sine of the maximum incident angle at the optical fiber end surface when total internal reflection occurs in the optical fiber is sinφ₀ ≈ φ₀, which is called the numerical aperture of the optical fiber, generally denoted as NA (Numerical Aperture), i.e.:
NA = sinφ₀ = n₁√2Δ = √(n₁² - n₂²)
This equation expresses the light-gathering ability of the optical fiber. Any incident light rays with an incident angle smaller than φ₀ can satisfy the total internal reflection condition and will be confined within the fiber core to propagate along the axial direction. It can be seen that the numerical aperture of the optical fiber is directly proportional to the square root of the relative refractive index difference. In other words, the greater the refractive index difference between the fiber core and cladding, the larger the numerical aperture of the optical fiber, and the stronger its light-gathering ability."
Propagation of light in graded-color optical fiber
The refractive index of the core of a graded-index fiber is not constant; it gradually decreases with increasing fiber radius until it equals the refractive index of the cladding, as shown in Figure 2-8. To analyze the propagation of light in a graded-index fiber, a method similar to the "integral definition" in mathematics can be used. First, the fiber core is divided into numerous concentric thin cylindrical layers. Each layer is very thin, and its refractive index is approximately constant within each layer. There is a small step difference in refractive index between adjacent layers.
The meridional plane and layering of graded-index optical fiber are shown in Figure 2-8. The refractive indices of each layer satisfy the following relationship: n(rO) > n(r1)>n(r2)>n(r4)>…>n(r),When a ray of light is incident from the end face of an optical fiber at a median angle, its propagation in a multilayered optical fiber with varying refractive indexes is shown in Figure 2-8. When the ray strikes the interface between layers 1 and 2 at an incident angle of θ, since the ray is traveling from a denser medium to a less dense medium, its angle of refraction θ will be larger than θ. As shown in the figure, this ray will then refract at the interface between layers 2 and 3 with a new incident angle of θ, and so on. Since light always propagates from a denser medium to a less dense medium, its angle of incidence gradually increases, i.e., θ<θ<θ<θ4<θ5", until at a certain interface (interface u in the diagram), the angle of incidence exceeds the critical angle, at which point total internal reflection occurs. Afterward, the light travels along a perfectly symmetrical trajectory, layer by layer, from less dense to denser, towards the central axis. At this point, the angle of incidence decreases as the light propagates towards the center due to the increasing refractive index of each layer, and the light crosses the central axis. Since the refractive index distribution below the central axis is exactly the same as above, after passing the central axis, the light is essentially propagating from a denser medium to a less dense medium again, and its angle of incidence gradually increases, subsequently undergoing total internal reflection and returning to the central axis. Then, it again enters the interface of layers 1 and 2 at an angle θ, and the cycle repeats. In this way, light can be transmitted from one end to the other.
(Figure 2-8 Meridian plane and layering of graded-ratio optical fiber)
