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Fiber Optics: Theory and Applications

TABLE OF CONTENTS Fiber Optics Fundamentals Total Internal Reflection Numerical Aperture Depth of Focus Contrast Versus EMA Resolution Fiber Optics Configurations 1 1 1 3 4 4 FIBER NON-IMAGING CONFIGURATIONS Single Fiber Light Guides Plastic Fibers 5 5 5 5 6 FIBER IMAGING CONFIGURATIONS Flexible Imagescopes Rigid Combiner/Duplicators 6 6 6 FUSED IMAGING CONFIGURATION Image Conduit Faceplates Tapers Inverters Fibreye 7 7 8 11 11 12

Fiber Optics Fundamentals The science of fiber optics deals with the transmission or guidance of light (rays or waveguide modes in the optical region of the spectrum) along transparent fibers of glass, plastic, or a similar medium. The phenomenon responsible for the fiber or light-pipe performance is the law of total internal reflection. Total Internal Reflection A ray of light, incident upon the interface between two transparent optical materials having different indices of refraction, will be totally internally reflected (rather than refracted) if (1) the ray is incident upon the interface from the direction of the more dense material and (2) the angle made by the ray with the normal to the interface is greater than some critical angle, the latter being dependent only on the indices of refraction of the media (see Figure 1). Rays may be classified as meridional and skew. Meridional rays are those that pass through the axis of a fiber while being internally reflected. Skew rays are those that never intersect the fiber axis although their Figure 1 Refraction, Reflection, and Numerical Aperture behavior patterns resemble those of meridional rays in all other respects. For convenience, this discussion will deal only with the geometric optics of meridional ray tracing. An off-axis ray of light traversing a fiber 50 microns in diameter may be reflected 3000 times per foot of fiber length. This number increases in direct proportion to diameter decrease. Total internal reflection between two transparent optical media results in a loss of less than 0.001 percent per reflection; thus a useful quantity of illumination can be transported. This spectral reflectance differs significantly from that of aluminum (shown graphically in Figure 2). An aluminum mirror cladding on a glass fiber core would sustain a loss of approximately 10 percent per reflection, a level that could not be tolerated in practical fiber optics. As indicated in Figure 1, the angle of reflection is equal to the angle of incidence. (By definition, the angle is that measured between the ray and the normal to the interface at the point of reflection.) Light is transmitted down the length of a fiber at a constant angle with the fiber axis. Scattering from the Figure 2 Aluminum Mirror Reflectance true geometric path can occur, however, as a result of (1) imperfections in the bulk of the fiber; (2) irregularities in the core/clad interface of the fiber; and (3) surface scattering upon entry. In the first two instances, light will be scattered in proportion to fiber length, depending upon the angle of incidence. To be functional, therefore, long fibers must have an optical quality superior to that of short fibers. Surface scattering occurs readily if optical polishing has not produced a surface that is perpendicular to the axis of the fiber; pits, scratches, and scuffs diffuse light very rapidly. The speed of light in matter is less than the speed of light in air, and the change in velocity that occurs when light passes from one medium to another results in refraction. It should be noted that a portion of the light incident on a boundary surface is not transmitted but is instead reflected back into the air. That portion that is transmitted is totally reflected from the sides, assuming that the angle is less than the critical angle (see Figure 1). The relationship between the angle of incidence I and the angle of refraction R is expressed by Snell’s law as N1 sin I = N2 sin R where N1 is the index of refraction of air and N2 the index of refraction of the core. Since N1 = 1 for all practical purposes, the refractive index of the core becomes N2 = sin I/sin R Numerical Aperture Numerical aperture (abbreviated N.A. in this paper) is a basic descriptive characteristic of specific fibers. It can be thought of as representing the size or “degree of openness” of the input acceptance cone (Figure 3). Mathematically, numerical aperture is defined as the

azimuthal angle on emergence varies so rapidly with θ, the length and diameter of the fiber, etc. that the emerging ray spreads to fill an annulus of a cone twice Figure 3 Numerical Aperture sine of the half-angle of the acceptance cone (sin θ). The light-gathering power or flux-carrying capacity of a fiber is numerically equal to the square of the numerical aperture, which is the ratio between the area of a unit sphere within the acceptance cone* and the area of a hemisphere (2π solid angle). A fiber with a numerical aperture of 0.66 has 43 percent as much fluxcarrying capacity as a fiber with a numerical aperture of l.0; i.e., (0.66)2/(1.0)2 = 0.43. Snell’s law can be used to calculate the maximum angle within which light will be accepted into and conducted through a fiber (see Figure 1): Figure 4 Fiber Light Transmission angle θ, as shown in Figure 4. In a two-lens system, where the output light is fed from a fiber annulus back into a second lens (Figure 5), the light will be distributed over a 360-degree angle and only a small fraction of it will strike the second lens. This effect is important in fiber-optic system design: in N1 sin θ max(N22 - N23)1/2 In this equation, sin θ max is the numerical aperture, N1 the refractive index of air (1.00), N2 the refractive index of the core, and N3 the refractive index of the clad. As light emerges from the more dense medium (glass) into a less dense medium such as air, it is again refracted. The angle of refraction is greater than the angle of incidence (R > I) at emergence only; and because R is by necessity 90 degrees, there must be a limiting value of I beyond which no incident ray would be refracted and emerge into the air. This becomes the critical angle, and rays that strike at a greater angle are totally reflected. It should be noted that this formula for the calculation of numerical aperture does not take into account striae, surface irregularities, and diffraction, all of which tend to decollimate the beam bundle. Source Distributions A lambertian source plane is one that looks equally bright from all directions. It emits a flux proportional to the cosine of the angle from the normal. Matte white paper and phosphors are approximate examples of such source planes, and the light diffused by opal glass is a close approximation for most measurements. In the lambertian case, the flux contained out to an angle θ is proportional to sin2 θ. Input-Output Phenomena If a ray is incident at angle θ, it will ideally emerge from a fiber at angle θ. In practice, however, the Figure 5 Projection System Transmission general, fiber optics should be utilized as image or light transporters rather than as focusing components. The exit end of a fiber will act as a prism if it is not cut perpendicular to the fiber axis. A bias cut will tip the exit cone as shown in Figure 6. Thus β = sin-1 [(N2/N1)sin α] - α ≅[(N2 /N1) – 1] a (for small angles) where β is the axis of the deflected ray and α is the cut angle to the normal of the fiber. The preservation of angle θ on exit is only an approximation. Diffraction at the ends, bending, striae, Figure 6 Biased End Cut and surface roughness will cause decollimation or

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