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  • mbenkerumass 9:00 am on January 14, 2020 Permalink | Reply
    Tags: Geometrical Optics,   

    Ray Optics – Graded-Index Fibers, Matrix Optics 

    Graded-Index Fibers

    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.

    fiber

    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.

    grinparax

    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.

    Ray Equation:

    rayeq

    Paraxial Ray Equation:

    rayeqpar

    A graded index glass fiber is modeled below:

    gribfiber


     

    Matrix Optics

    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:

    abcd          ,            abcd1

    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.

    abcdmatricies

     
  • mbenkerumass 9:00 am on January 7, 2020 Permalink | Reply
    Tags: Geometrical Optics,   

    Planar Boundaries, Total Internal Reflection, Beamsplitters 

    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).

    refraction

    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.

    tir

     

    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:

    prism2

    This is taken by using Snell’s law twice along two planar boundaries.

    prism1

     

    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.

    beamsplit

    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.

     

     

     
  • mbenkerumass 9:00 am on December 31, 2019 Permalink | Reply
    Tags: Geometrical Optics,   

    Mirrors in Geometrical Optics, Paraxial Approximation 

    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.

    mirror1

    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.

    ellipticalmirror

    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.

    spmirror

    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.

    spmir2

    The following is an example of a use of a paraxial approximation for an image formation using a spherical mirror:

    sp

    Images are credit of Fundamentals of Photonics, Wiley Series in Pure and Applied Optics

     
  • mbenkerumass 9:00 am on December 24, 2019 Permalink | Reply
    Tags: Geometrical Optics,   

    Postulates of Ray 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.

    planeofincidence

    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:

    snell

    10. The proportion of reflected light to refracted light is not dealt with in ray optics.

    ray1

     
  • mbenkerumass 9:00 am on December 19, 2019 Permalink | Reply
    Tags: Geometrical Optics,   

    Ray Optics & Geometrical Optics (Introduction) 

    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.

    2

    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.

    optics1

    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.

     

     
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