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- Fundamentals of Optics - An Introduction for Beginners
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- Fundamentals of Optics - An Introduction for Beginners - Vision Systems Design

As described above, refraction is related to the differing speed of light in various media.

The Law of Refraction can also be stated as:. Take, as an example, air to be medium 1. In accordance with Equation 5 , the propagating speed of light in medium 2 is:. Equation 6 infers that the speed of light in an optical medium more dense than air is smaller than air. Only in a vacuum or in a medium less dense than that of air is speed of light greater than in air. As is known trigonometrically, the sine function has a maximum value of 1. The Law of Refraction suggests that light passes from one media to the next.

The limiting angle for total reflection from medium 1 to air as medium 2 yields the following expression of the Law of Refraction:. Total reflection is a prerequisite for guided waves in optical fibers and light guides. The optically denser material is the core usually glass, quartz, or plastic and the cladding is an optically less dense material of glass, quartz, or plastic.

The specific selection of core and cladding materials will result in a specific numerical aperture or acceptance angle for the fiber. Optical Image Formation and Optical Systems Certainly, the concept of optical phenomena was first realized from observing nature, i. In an effort to reproduce these effects, attempts were made to construct experiments, which could use these effects. The scientific approach was also employed, which made use of the many observations.

Formulating laws was attempted through sketching diagrams and experimentally varying parameters. For more than years the Law of Refraction was not known in its current mathematical form, in spite of the fact that there are clear indications that people knew how to work with transparent crystals and aided vision with enlarging optics. The predecessor of today's geometrical optics was the telescope builder of the 16th and 17th century, above all Galileo Galilei. They searched for homogeneous materials without inclusions or spatial variations in the index of refraction.

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Attention was given to work methods, which yielded good surface quality and polishes, so as not to affect image formation by way of scattering effects. It is noteworthy that Fresnel experimented with various natural crystals. A type of lens, the Fresnel lens, with many ground prismatic rings was named after him. This point propagates by way of a diverging image bundle homocentric bundle through an optical system to a corresponding point in the image plane image point.

At this point the rays from the point source converge again Fig. A particularly good image results when all rays within a region i. If the image lies behind the optical system, where the emitted object rays define the direction, the image is called a real image. If the image lies in front of the optical system, the image is called a virtual image. A virtual image can not be acquired by any sensor behind the optical system without additional optical elements. Sensors can only acquire real images.

In Figure 6, the constructed ray which creates O' obviously doesn't exist at infinity.

The lens in our eye can, however, focus divergent rays from a virtual image onto the retina and simulate the existence of a virtual image at point O'. Lenses are transparent elements for spectrums of interest made usually of glass. The entrance and exiting surfaces are usually spherically ground and polished.

We differentiate between converging lenses with positive refractive power and diverging lenses with negative refractive power.

## Fundamentals of Optics - An Introduction for Beginners

Parallel rays falling on a converging lens will be refracted to the focal point located behind the lens. The same experiment performed on a divergent lens will disperse light. Rays will not converge behind the lens, but will yield the projection of a virtual image point in the object space. Rotationally symmetric lenses are generally used for optical systems, meaning that the refractive surfaces are spherical or conical and a useful outer diameter is concentrically ground.

Some non-rotationally symmetric lenses are, for example, cylindrical or toric lenses. These lenses have differing vertical and horizontal cross sections. Currently, more and more compact video camera systems are using rotationally symmetric aspherical lens systems. Imaging errors can be reduced which occur mainly from slight conical form spherical aberrations of spherical lenses Fig. The lens is placed a certain distance from the object Object distance a.

The direction of propagation is defined from the object to lens for this system. The region before the lens is referred to as the object space. In the even of a real image, the image space lies behind the lens. The distance between the lens and the image is referred to as the image distance a'. Figure 10 has sufficient parameters to calculate the most important dimensional quantities of an imaging system, such as linear magnification, acceptance angle, and field of view FOV. If a virtual image lies in the object space, a' negative is inserted.

These are called paraxial rays.

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One of the most important equations characterizing optical imaging is the conjugate distance equation. For a given lens of focal length f, the expression relates the object and image distance. This expression is based on the assumption that the lens thickness t is small relative to a and a', so that the lens is considered thin. This may deviate from expected magnification due to effects such as magnification distortion or deviations from paraxial assumptions.

For small angles of FOV, both quotients should be identical. A negative magnification implies the image is inverted from that of the object upside-down image. By combining Equations 8 , 9 and 10 , the following useful relationships apply to paraxial systems:. Paraxial systems with thin lenses are a simplified configuration for crudely analyzing optical systems. They present a special case. Thick lenses have two refracting surfaces to which the thin lens parameters can be applied. They are referred to as principle planes object side h, image side h'. The corresponding intersection points with the optical axes are principle or nodal points H and H'.

Equation 8 is used by replacing object and image distances with nodal points. Here, z and z' are the distance to the object and image from their respective focal points. In general, the object principle plane is nearer the object than the image principle plane, but this can be reversed with thick lenses and unique surface configurations. The intersection of the polished lens surface with the optical axes is called the object vertex, S, and the image vertex, S'.

The corresponding distances between vertex and object or image are referred to as the back focus, s and s'. Optical systems usually contain converging and diverging lenses made of different materials. Such systems are known as compound lens systems. A compound lens system can also produce an object and image principle plane. The intersection of the outer most lens surface with the optical axes defines the vertex and associated back focus. Lenses in an optical system can have different shapes or bends with radii r 1 and r 2. Note that for most compound systems rotationally symmetric , the center point of all refractive surfaces lies on the optical axes.

If the radius is to the left of the lens toward the object , it is negative by convention and to the right of the lens it is positive. By way of this lens bending, the location of the principle plane can be influenced. The lens collapses to a single principle plane. In choosing a meniscus lens b , as opposed to a symmetrical lens a , the object related back focus can be shortened, thus maintaining the same focal length. For understanding optical configurations and instruments, the choice of the principle plane location is an important characteristic. For example, large focal length lenses with short back foci can be conceptualized.

The arrangement of several single lens systems into a compound lens system will combine the different bendings, glasses, thicknesses of individual lenses, and lens distances into a single system. A single object side and image side principle plane describe such a system.

This is especially true of systems whose object or image back focus is infinite. This is usually true of photographic lenses where the object is at infinity and the image side back focus is very short. The included angle between the outermost ray and the optical axis is related to the aperture of a lens. According to Equation 22 , systems with large aperture have small aperture numbers.

The aperture numbers of most lenses are assigned a number according to the diaphragm range. This range is defined such that a step in aperture number reflects a factor of 2 change in the light flux. The diaphragm range is as follows:. Optical professionals sometimes refer to the numerical aperture of a lens system. This is defined as:. From Equation 22 the relationship between numerical aperture and aperture number can be readily seen. Frequently, lens systems project an object onto an image sensor in the image space that is made of photosensitive material.

If one concretely wants to know the effective aperture number, Feff, of an imaging system magnification m , it can be derived from Equations 8 , 10 , and For single lens systems, the lens diameter can act as the iris at the principle plane. Irises, also called aperture stops, are precise mechanical apertures, which restrict light rays transmitted through a lens.

This restricting is obviously related to the aperture number. The free diameter in Equation 22 can be replaced with the aperture diameter. The iris defines the amount of light flux transmitted by an optical system. Figure 16 schematically shows the rays transmitted through an optical system with an iris. Typically, the middle ray of a ray bundle principle ray intersects the optical axis at the iris plane. The corresponding peripheral ray is restricted by the iris.

Depending on the lens system, the position of the iris may be in the middle of system or shifted in front of, or behind, the precise center. In general, by shifting the location of the iris, the oblique aberrations of a lens system such as coma, astigmatism, distortion, lateral color can be influenced and minimized. A lens stack will, in most cases, create a virtual image of the iris in image planes in front of, and behind the iris.

The iris image appears as a typical limiting ray diameter when viewing the lens.

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These iris images are also called pupils. The front and back images of the iris are called entrance pupil EP and exit pupil XP , respectively. Figure 17 is a schematic diagram of the pupils. Ray bundles emitted from the object or in the reverse direction from the image are restricted by their respective pupils, and their propagation direction is defined.

The magnitude is given by the following expression.

## Fundamentals of Optics - An Introduction for Beginners - Vision Systems Design

Equation 26 is only valid for the imaging of light emitting points lying on the optical axis. When the luminous flux of light emitting points transmitted through optics is not lying on the optical axis, then the optical axis is smaller than Equation Equation 26 is reduced by a factor cos 4 alpha , thereby formulating the cos 4 - Law for off-axis points. The sharpness region increases as the aperture number decreases. For object distance a, focal length f, and aperture number F, the limits of the object distance av and ah are related by.

Object distance for theoretically sharp imaging Fig. Lens focal length; F Circle of confusion diameter. The permissible circle of confusion diameter, p, for an image point refers to the dimension of the image, being approximately one thousandth of the image diagonal.

The depth of field, a h - a v , is calculated by subtracting Equation 28 from Equation 29 , and also using Equation It can be seen from Equation 31 the depth of field can be increased by choosing larger aperture numbers smaller apertures , and smaller magnifications. As a basic theoretical requirement, two optical systems must have matched numerical apertures at their point of coupling. Several things can jeopardize light transport through fibers.

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Added to Your Shopping Cart. Description The easy way to shed light on Optics In general terms, optics is the science of light. Tracks a typical undergraduate optics course Detailed explanations of concepts and summaries of equations Valuable tips for study from college professors If you're taking an optics course for your major in physics or engineering, let Optics For Dummies shed light on the subject and help you succeed! About the Author Galen Duree, Jr. Table of contents Introduction 1 Part I: Introducing Optics, the Science of Light 9 Chapter 2: A Little Light Study: Reviewing Light Basics 31 Chapter 4: Bouncing Many Rays Around 69 Chapter 7: Using the Light Wave 95 Chapter 8: Changing Optical Polarization Chapter