Reflection and Refraction Equations for Predicting Light’s Direction Reflection and refraction are two processes that change the direction light travels. Using the equations for calculating reflection and refraction, you can predict where rays encountering a surface will go — whether they reflect or refract (bounce off the surface or bend through it) — which is an important concept in the study of optics. The following equations help you determine reflection and refraction angles:.
The law of reflection: The law of reflection shows the relationship between the incident angle and the reflected angle for a ray of light incident on a surface. The angles are measured relative to the surface normal (a line that is perpendicular to the surface), not relative to the surface itself. Here’s the formula:. The index of refraction: This quantity describes the effect of atoms and molecules on the light as it travels through a transparent material. Use this basic formula for the index of refraction:. Snell’s law or the law of refraction: Snell ‘ s law shows the relationship between the incident angle and the transmitted angle (refracted angle) for a ray of light incident on a surface of a transparent material. You can see how Snell’s law works in the following formula:.
The critical angle for total internal reflection: Total internal reflection is the situation where light hits and reflects off the surface of a transparent material without transmitting through the surface. It utilizes the critical angle (the minimum angle of incidence where total internal reflection takes place.). For total internal reflection to occur, the light must start in the material with the higher index. Here’s the formula.
Equations for Optical Imaging Imaging is a key function of optics. Specific optics equations can help you determine the basic characteristics of an image and predict where it will form. Use the following optics equations for your imaging needs:.
Lateral magnification: Lateral magnification is one way you can describe how big the image is compared to the original object. Here are the equations:. Locating images formed by mirrors: An object placed a certain distance away from a mirror will produce an image at a certain distance from the mirror. In some cases where the mirrors are curved, you may be given the focal length of a mirror.
Use these equations:. Location of images formed by a refracting surface: An object placed a certain distance away from a refracting surface will produce an image at a certain distance from the surface. The equation for this is.
The lens maker’s formula: This equation allows you to calculate the focal length of a lens if all you know is the curvature of the two surfaces. Here’s the lens maker’s formula:. The thin lens equation: An object placed a certain distance away from a lens will produce an image at a certain distance from the lens, and the thin lens equation relates the image location to the object distance and focal length. The following is the thin lens equation. Optical Polarization Equations Optical polarization is the orientation of the planes of oscillation of the electric field vectors for many light waves. Optical polarization is often a major consideration in the construction of many optical systems, so equations for working with polarization come in handy.
The following equations highlight some important polarization concepts. The equations listed here allow you to calculate how to make polarized light by reflection and to determine how much light passes through multiple polarizers:.
Polarizing angle or Brewster’s angle: This angle is the angle of incidence where the reflected light is linearly polarized. Here’s the equation:. Malus’ law: This equation allows you to calculate how much polarized light passes through a linear polarizer.
The equation for Malus’ law is. Phase retardation in a birefringent material: A birefringent material has two indexes of refraction.
When you send polarized light into a birefringent material, the two components travel through the material with different speeds. This discrepancy can result in a change in the polarization state or simply rotate the polarization state. Use this equation. Optical Interference Equations Optical interference is just the interaction of two or more light waves. Optical interference is useful in many applications, so you need to understand some basic equations related to this optical phenomenon.
The following equations allow you to calculate various quantities related to optical interference in the two most common interference arrangements. Optical Diffraction Equations Diffraction is light’s response to having something mess with its path, so diffraction occurs only when something blocks part of the wavefront.
Diffraction is the phenomenon where light bends around an obstacle (this bending is not due to refraction, because the material doesn’t change as refraction requires). The following equations cover the most common situations involving diffraction, including resolution. Resolution: Resolution is the minimum angular separation between two objects such that you can tell that there are two distinct objects.
Here’s the equation for determining resolution: The location of the dark fringes produced by diffraction through a single slit: Because a slit has a width larger than the wavelength, light rays from different parts of the slit interfere with each other, creating a fringe pattern. You can relatively easily locate the points where the light destructively interferes by using the following equation:. The location of the different diffraction orders from a diffraction grating: A diffraction grating has a very large number of slits spaced closely together, such that the light from each of these slits interferes with the light from the others.
Geometric Optics Lab Report
You can pretty easily identify where the light constructively interferes by using the following equation. Equations for the Characteristics of Fiber-Optic Fibers Besides imaging, fiber-optic networks are probably the largest application of optics. Fiber optics are very long, thin glass fibers that transfer information-bearing light from one place to another, but that may not be in direct sight of each other.
You need to be aware of a few characteristics of the particular fiber you’re using so that you can ensure the information is accurately transmitted from one end of the fiber to the other. The following equations cover three of the basic parameters necessary for proper use of optical fibers. The maximum acceptance angle for a fiber: This angle is the largest angle of incidence at which light can enter the end of the fiber and be totally internally reflected inside the fiber. Angles of incidence larger than this angle will transmit through the sides of the fiber and not make it to the other end. The equation for this angle is. The numerical aperture for a fiber: The numerical aperture is a measure of the light-gathering power of the fiber.
Geometric Optics Quiz
It has a maximum value of 1 (all the light remains trapped inside the fiber) and a minimum value of 0 (only light incident at an angle of 0 degrees on the end of the fiber remains trapped in the fiber). Use this equation:. Intermodal dispersion in a fiber: This characteristic measures the difference in time that different fiber modes take to reach the end of the fiber. The larger this time difference, the shorter the fiber has to be so that the information on this light doesn’t turn into junk. Here’s the equation.
Main article: Geometrical optics, or ray optics, describes in terms of. The ray in geometric is an useful for approximating the paths along which light propagates under certain circumstances. The simplifying assumptions of geometrical optics include that light rays:. propagate in straight-line paths as they travel in a medium. bend, and in particular circumstances may split in two, at the between two dissimilar. follow curved paths in a medium in which the changes. may be absorbed or reflected.
Geometrical optics does not account for certain optical effects such as and. This simplification is useful in practice; it is an excellent approximation when the wavelength is small compared to the size of structures with which the light interacts.
The techniques are particularly useful in describing geometrical aspects of, including. As light travels through space, it in. In this image, each maximum amplitude is marked with a to illustrate the. The is the arrow to these surfaces.
A light ray is a or that is to the light's (and is therefore with the ). A slightly more rigorous definition of a light ray follows from, which states that the path taken between two points by a ray of light is the path that can be traversed in the least time. Geometrical optics is often simplified by making the, or 'small angle approximation.' The mathematical behavior then becomes, allowing optical components and systems to be described by simple matrices. This leads to the techniques of and paraxial, which are used to find basic properties of optical systems, such as approximate and object positions and. Reflection. Diagram of Glossy surfaces such as reflect light in a simple, predictable way.
This allows for production of reflected images that can be associated with an actual or extrapolated location in space. With such surfaces, the direction of the reflected ray is determined by the angle the incident ray makes with the, a line perpendicular to the surface at the point where the ray hits. The incident and reflected rays lie in a single plane, and the angle between the reflected ray and the surface normal is the same as that between the incident ray and the normal. This is known as the. For, the law of reflection implies that images of objects are upright and the same distance behind the mirror as the objects are in front of the mirror. The image size is the same as the object size.
(The of a flat mirror is equal to one.) The law also implies that are, which is perceived as a left-right inversion. Can be modeled by and using the law of reflection at each point on the surface. For, parallel rays incident on the mirror produce reflected rays that converge at a common.
Other curved surfaces may also focus light, but with aberrations due to the diverging shape causing the focus to be smeared out in space. In particular, spherical mirrors exhibit. Curved mirrors can form images with magnification greater than or less than one, and the image can be upright or inverted. An upright image formed by reflection in a mirror is always virtual, while an inverted image is real and can be projected onto a screen. Refraction. Incoming parallel rays are focused by a convex lens into an inverted real image one focal length from the lens, on the far side of the lens Rays from an object at finite distance are focused further from the lens than the focal distance; the closer the object is to the lens, the further the image is from the lens.
With concave lenses, incoming parallel rays diverge after going through the lens, in such a way that they seem to have originated at an upright virtual image one focal length from the lens, on the same side of the lens that the parallel rays are approaching on. With concave lenses, incoming parallel rays diverge after going through the lens, in such a way that they seem to have originated at an upright virtual image one focal length from the lens, on the same side of the lens that the parallel rays are approaching on.
Rays from an object at finite distance are associated with a virtual image that is closer to the lens than the focal length, and on the same side of the lens as the object. The closer the object is to the lens, the closer the virtual image is to the lens., An Introduction to the Theory of Optics, London: Edward Arnold, 1904. Greivenkamp, John E. Field Guide to Geometrical Optics. SPIE Field Guides. Young (1992).
University Physics 8e. Marchand, Gradient Index Optics, New York, NY, Academic Press, 1978. ^ Hecht, Eugene (1987). Optics (2nd ed.). Addison Wesley. Chapters 5 & 6. Sommerfeld, A., & Runge, J.
Anwendung der Vektorrechnung auf die Grundlagen der geometrischen Optik. Annalen der Physik, 340(7), 277-298. Born, M., & Wolf, E.
Principles of optics: electromagnetic theory of propagation, interference and diffraction of light. Borowitz, S.
Fundamentals of quantum mechanics, particles, waves, and wave mechanics. D., & Lifshitz, E. The classical theory of fields. Further reading. (1900) from.
is a manuscript, in Arabic, about geometrical optics, dating from the 16th century. Hamilton in Transactions of the Royal Irish Academy, Vol. English translations of some early books and papers. External links.