For a Diverging lens, derive a formula for the output angle with respect to the refractive indexes and input angle. Assume paraxial approximation and thin lens.
For a Diverging lens, construct a derivation of Newton’s lens equation x_o*x_i = f^2.
Previously featured was an article that derived a matrix formation of an equation for a thick lens. This matrix equation, it was said can be used to build a variety of optical systems. This will be undertaken using MATLAB. One of the great parts of using a matrix formula in MATLAB is that essentially any known parameter in the optical system can not only be altered directly, but a parameter sweep can be used to see how the parameter will effect the system. Parameters that can be altered include radius of curvature in the lens, thickness of the lens or distance between two lenses, wavelength, incidence angle, refractive indexes and more. You could also have MATLAB solve for a parameter such as the radius of curvature, given a desired angle. All of these parameters can be varied and the results can be plotted.
Matrix Formation for Thick Lens Equation
The matrix equation for the thick lens is modeled below:
- nt2 is the refractive index beyond surface 2
- αt2 is the angle of the exiting or transmitted ray
- Yt2 is the height of the transmitted ray
- D2 is the power of curvature of surface 2
- D1 is the power of curvature of surface 1
- R1 is the radius of curvature of surface 1
- R2 is the radius of curvature of surface 2
- d1 is the thickness of the lens or distance between surface 1 and 2
- ni is the refractive index before surface 1
- αi is the angle of the incident ray
- Yi1 is the height of the incident ray
The following plots show a parameter sweep on an number of these variables. The following attachment includes the code that was used for these calculations and plots: optics1hw
The following set of notes presents first a trigonometric derivation of the thick lens equation using principles such as Snell’s law and the paraxial approximation. A final formula for the thick lens equation is rather unwieldy. A matrix form is much more usable, we will find. Moreover, a matrix form allows for one to add a number of lenses together in series with ease. Parameters of the lenses can be altered as well. Soon, the matrix formation of these equations will be used in MATLAB to demonstrate the ease at which an optical system can be built using matrix formations. The matrix formation of the thick lens equation can be summarized as three matrices multiplied, for the first curved surface, the separation between the next curved surface and the final curved surface. By altering the radius of curvature, the refractive indexes at each position, distances between them using these matrices, a new lens can also be made, such as a convex thin lens by inverting the curvature of the lens and reducing the thickness on the lens. A second lens can be added in series. Once a matrix formation is made handy, there are numerous applications that then become simple.
The following ray tracing examples all utilize Fermat’s principle in examining ray traces incident at a mirror.
Example 1. Draw a ray trace for a ray angled at a convex mirror.
The ray makes a 40 degree angle with the normal of the mirror at the point of incidence. In accordance with the law of reflection (Fermat’s Principle), the ray will exit at 40 degrees on the other side of the normal.
The above example shows a single ray at an angle. Often, rays are drawn together in a group of parrallel rays. This example shows how an incident set of parallel rays will no longer be parallel when reflected by a non-uniform (not flat) mirror surface.
This example brings up an important concept that happens especially with concave mirrors. Two rays drawn seem to be directed towards the same point, known as the focal point. A focal point however is only consistent for smaller angles. The third ray at the bottom makes a 55 degree incident angle with the normal of the surface. The reflected ray is also 55 degrees separated from the normal but is directed to the other side of the normal. The ray does not converge at the focal point as the others do. This effect is known as an aberration and may be discussed further at length in a later article.
This example makes use of the above concept of focal point. An object placed at the focal point will not make an image at the focal point. This is useful if for instance, some type of lense or collecter should be placed at the focus of the mirror. This can be done without worry for it causing disturbances to the image that is formed at the focal point by the reflected rays.
In geometrical optics, light is treated as rays, typically drawn as lines that propagate in a straight line from one point to another. Ray tracing is a method of determining how a ray will react to a surface or mirror. Rays are understood to propagate always in a straight line, however when entering an angled surface, rebounding from an angled surface or propagating through a different medium, there are a few techniques that are needed to reliably determine the direction and path of a light ray. The following properties are the basis for ray tracing.
The refractive index is a property intrinsic to a medium that describes how fast or slow light propagates in the medium. Light speed in a vacuum is 3*10^8 m/s. Light speed will only get slower in real mediums. The formula for refractive index is the speed of light c devided by the velocity of light in the medium.
The refractive index of air is approximately 1. The refractive index of glass for instance is about 1.5. This has implications on how light will propage when changing from one medium to another.
Snell’s law uses the angle of incidence (incoming ray), the angle of refraction (exiting ray) and the refractive indexes of each medium at a boundary to determine the path of propagation. Consider the example below:
Snell’s Law: η1*sin(θ1) = η2*sin(θ2)
The angle of incidence and the angle of refraction are both with respect to the normal of the surface!
Fermat’s Principle is also demonstrated in the above figure. Fermat’s Principle states that the angle of incidence of a ray will be equal to the angle of reflection, but exiting from the other side of the normal of the surface.
Using these principles alone, many optical instruments and technologies can be designed and built that manipulate the direction of light rays.
Of the four ways of manipulating light, these examples employ shaping of a lens and the refractive index to change the path of a ray.
Guiding light rays with multiple lenses or mirrors is possible, however this may result in great loss of optical power due to refraction in a system if there are many lenses or mirrors. Using total internal reflection however, rays may be transmitted over long distances without these losses. Glass fibers are the primary choice for guiding light in this manner using total internal reflection. Glass fibers consist of a glass wire with a cladding. The refractive index of the outer cladding will be smaller than the glass core. This allows for a consistent total internal reflection throughout the wire.
A graded-index material (GRIN) has a refractive index that varies throughout the material. When a ray moves through a graded-index material, the variance in refractive index causes the ray to bend and curve according to how the graded index is laid out.
The path of an optical ray in graded-index material is determined by Fermat’s principle, which states that the path of a ray is the integral of the refractive index (a function of position on the material) between two points, equated to zero. The ray equation can solve this problem, however for simplification, a paraxial approach is taken to give the paraxial ray equation.
Paraxial Ray Equation:
A graded index glass fiber is modeled below:
A paraxial ray is described by a coordinate and angle. Using this approximation, the output paraxial ray going through a system can be written in matrix form:
An optical system can be modeled using the 2×2 ABCD matrix. Matrices of systems may also be cascaded to describe the effect of many systems on a ray.
Refraction is an important effect in ray optics. The refractive index of a material influences how rays react when entering or leaving a boundary. For instance, if the ray is exiting a medium of smaller refractive index and entering a medium with a higher refractive index, the angle will tend towards being perpendicular to the boundary line. The angle of refraction is also greater than the angle of incidence. This case is called external refraction (n1 < n2) and (θ1 > θ2). If the ray is exiting a medium of higher refractive index into a medium with a lower refractive index, the rays will tend towards being closer to parallel with the medium boundary. This case is referred to as internal refraction (n1 > n2) and (θ2 > θ1). Both of these situations are governed by Snell’s Law:
n1*sin(θ1) = n2*sin(θ2)
When the rays are paraxial, the relation between θ1 and θ2 is linear (n1*θ1 = n2*θ2).
The critical angle occurs when n1*sin(θ1) = n2*sin(pi/2) = n2. θ1 in this case is then equal to the critical angle. If θ1 is greater than the critical angle θC, refraction cannot occur and the situation is characterized by a phenomenon known as total internal reflection (TIR). Total internal reflection is the basis for many optical systems and devices. Systems with total internal reflection are understood to be highly efficient even under more rigorous approaches to optics such as electromagnetic optics.
Prisms are common applications of refraction. A prism of apex angle α and refractive index n deflects a ray incident at an angle of θ:
This is taken by using Snell’s law twice along two planar boundaries.
A beamsplitter is an optical component that divides a ray into a reflected and refracted (or transmitted) ray. The proportions of reflected to refractive light is a problem dealt with in electromagnetic optics. Beamsplitters are also used to combine two rays.
Beam directors apply Snell’s law and the rules governing refraction to direct rays in different directions. Three methods of directing waves are the biprism, the Fresnel biprism and the axicon.
The main types of mirrors used as simple optical components are planar mirrors, paraboloidal mirrors, spherical mirrors and elliptical mirrors.
Planar Mirrors reflect rays in a manner that the apparent object location reflects rays from a position that forms a reflected angle (Snell’s law) with the angle between the point of reference and the mirror.
Paraboloidal Mirrors focus all incident rays to a single point, the focus or focal point. The distance from the end of the paraboloidal mirror to the focal point is the focal length. Paraboloidal mirrors are used in telescopes to collect light. Paraboloidal mirrors are also used in flashlight bulbs and light-emitting diodes to direct rays in one direction from a source of light.
Elliptical Mirrors reflect all rays from one source point to another point. Hero’s principle concludes that any path traveled from either point to another will be equal in distance, no matter the direction.
Spherical Mirrors will direct all rays in varying directions. Spherical mirrors may be concave and convex. A spherical mirror acts like a paraboloidal mirror of focal length f = radius/2.
Rays that make small angles with the mirrors axis are called paraxial rays. For paraxial rays, a spherical mirror exhibits a focusing property similar to an elliptical mirror and an imaging property as present in elliptical mirrors. The paraxial approximation considers only paraxial rays and therefore allows spherical mirrors to be considered for the above tendencies. Paraxial Optics is an approach to optics which operates under a set of rules derived from the paraxial approximation. Paraxial Optics is also referred to as first-order optics or Gaussian optics.
In spherical mirrors, considering the paraxial approximation, a focal point is assigned for each source point. All rays that are emitted from a a very far distance (approaching infinite distance) are focused to a point at distance f = (-R)/2.
The following is an example of a use of a paraxial approximation for an image formation using a spherical mirror:
Images are credit of Fundamentals of Photonics, Wiley Series in Pure and Applied Optics
The following principles of ray optics may be used to describe many optical systems. The numbering system is of no significance.
1. Light travels in the form of a ray. This means that light will travel from a source and is observed when reaching a detector.
2. Optical rays are vector which point in the direction of energy flow.
3. An optical medium is characterized by a refractive index, n = c0 / c, where c0 is the speed of light in free space and c is the speed of light in the medium. The time taken by light to travel a distance d is d/c = nd/c0. The optical pathlength is n*d.
4. In an inhomogeneous medium, the refractive index n(r) is a function of the position r(x,y,z). The optical pathlength along a path between A and B is the integral of A to B of n(r)*ds.
5. Fermat’s Principle states that optical rays travel from A to B following the path that requires the least amount of travel time.
6. Hero’s Principle states that light travels in straight lines in a homogeneous medium. A homogeneous medium means that the refractive index is consistent throughout.
7. Light reflects from mirrors in accordance with the law of reflection: The angle of reflection equals the angle of incidence and the reflected ray lies in the plane of incidence. This may be proven using Hero’s principle.
8. At a boundary between two mediums of different refracting indexes, a ray is split in two. One resulted ray is a reflected ray and the other is a refracted or transmitted ray. The reflected ray is shown in figure (b) above as vector C, while the refracted ray is C’.
9. The refracted ray lies in the place of incidence. The angle of refraction is related to the angle of incidence by Snell’s Law:
10. The proportion of reflected light to refracted light is not dealt with in ray optics.
In describing the nature of light, numerous theories have been described. One of the oldest and most simplest of explanations of the nature of light is Ray Optics. In variable contrast to Wave Optics, Electromagnetic Optics or Quantum Optics, the theory of Ray Optics describes light as obeying a set of geometrical rules. Ray Optics assumes that the wavelength of light is infinitesimally smaller than the objects that light “rays” interact with. Ray Optics is also referred to as Geometrical Optics due to the geometrical nature of the understanding of the theory and the manner of calculations involved.
Ray Optics has limitations and does not describe many phenomenon. However, Ray Optics or Geometrical Optics is is useful in determining the conditions in which light travels and is guided within various mediums, such as in relation to a lens, mirror or glass fiber. Optical rays may also be described as vectors which point in the direction of travel of a light ray.
The above diagram describes the relationship between Ray Optics to other important theories regarding the nature of light. Electromagnetic Optics describes light as an electromagnetic wave phenomenon and therefore assesses light using concepts applied to electromagnetic radiation, such as the form of electric field waves and magnetic field waves coupled. Wave Optics approximates this wave phenomenon as a scalar wave. Electromagnetic Optics, Wave Optics and Ray Optics encompass what is known as Classical Optics. To describe the nature of light in a manner consistent with quantum mechanics, the theory of Quantum Optics meets these purposes.