Reflective Optics Chapter 25

Reflective Optics  Wavefronts and Rays  Law of Reflection  Kinds of Reflection  Image Formation  Images and Flat Mirrors  Images and Spherical Mirrors  The Paraxial Approximation and Aberrations

Wavefronts and Rays A wave is the propagation of a condition or disturbance. A wavefront is a surface over which the value of that condition is constant.

Wavefronts and Rays The direction of motion is always locally normal to the wavefront. A line drawn in the direction of advance is called a ray.

Wavefronts and Rays The directional distribution of these rays depends on the nature and geometry of the source of the waves.

Wavefronts and Rays As distance from the point source increases, the radii of the spherical wavefronts becomes larger, until the wavefronts approximate planes. Waves from an infinitely-distant source are sometimes called plane waves.

Law of Reflection When light encounters the surface of a material, three things happen:  reflection  transmission  absorption

Law of Reflection In reflection, the light “bounces” off the surface. The bounce occurs according to the law of reflection:

Law of Reflection Notice that:  Both angles are measured from the surface normal  The incident ray, the reflected ray, and the surface normal all lie in a single plane: the plane of incidence

Law of Reflection Notice that:  If the surface normal is rotated through an angle  within the plane of incidence, and the incident direction is constant, the reflected ray rotates through twice the angle (2  )  If the plane of incidence rotates, the reflected ray rotates with it (“one for one”)

Kinds of Reflection We distinguish between two sorts of reflection:  Specular (from smooth surfaces)  mirror  polished metal  calm liquid  Diffuse (from rough or irregular surfaces)  white paper  projection screen  clouds or snow

Kinds of Reflection: Specular A surface producing specular reflection has a constant, or a “well-behaved” (slowly and continuously changing) normal direction. For a constant incident direction, the reflected direction is either constant or changes continuously: “organized.”

Kinds of Reflection: Diffuse A surface producing diffuse reflection has random surface normal directions that change chaotically with location on the surface. The law of reflection is everywhere obeyed: but with random results.

Image Formation Consider an object that either produces light, or that scatters light from its surroundings. Each point on its surface acts as a spherically-symmetric source (“point source”), sending out rays in many directions.

Image Formation If something acts on some of the rays that originate at one point on the object, and causes them to converge at a point somewhere else, or to diverge from a point somewhere else, then it has formed an image of that object.

Image Formation: Two Kinds of Image Images may be sorted into two categories:  virtual images: formed when the rays never physically come back to one point, but instead diverge as if they came from one point. The place they appear to have come from is the image.  real images: formed when the rays converge, so that they physically arrive at the same point. That point of physical reconvergence is the image.

Image Formation: Real Image The rays here physically converge: real image.

Image Formation: Virtual Image The rays here diverge as if they came from an image point: virtual image.

Image Formation by a Flat Mirror The image formed by a flat mirror:  virtual  upright  same size  same distance on the other side of the mirror

Image Formation by a Flat Mirror Two flat mirrors: the image formed by one mirror acts as an object for the second mirror.

Image Formation by Spherical Mirrors A spherical mirror is one whose surface is a portion of a sphere. The radius of the sphere at the mirror’s center is called the optical axis. The center of the sphere is the center of curvature.

Image Formation by Spherical Mirrors Necessary terms:

Spherical Mirrors: Special Rays Chief ray: a ray striking the vertex reflects symmetrically about the axis.

Spherical Mirrors: Special Rays Axial ray: a ray parallel to the axis passes through the focal point after being reflected (or appears to have)

Spherical Mirrors: Special Rays This means that the image (real or virtual) of an infinitely-distant object is formed at the focal point.

Spherical Mirrors: Special Rays A ray passing through the center of curvature passes through it again after reflection.

Finding Images by Ray Tracing We can use these special ray properties to find the locations where images are formed. We can also find out:  the image size  the image orientation  whether the image is real or virtual

Finding Images by Ray Tracing Example: concave mirror, object outside the center of curvature Image: real, inverted, between focal point and center of curvature

Finding Images by Ray Tracing Concave mirror, object at the center of curvature Image: real, inverted, at center of curvature

Finding Images by Ray Tracing Object between center of curvature and focal point: Image: real, inverted, outside center of curvature

Finding Images by Ray Tracing Object at the focal point: Image: real, inverted, located at infinity

Finding Images by Ray Tracing Object inside the focal point: Image: virtual, upright

Image Formation Mathematics Trace a single chief ray from object to image: magnification: d o and d i are the conjugate distances

Image Formation Mathematics Trace a single axial ray:

Image Formation Mathematics

Mirror equation: The mirror equation relates the conjugate distances and the focal length. With the definition of magnification it can be used generally to characterize images formed by mirrors.

Image Formation Mathematics In solving problems, we must keep a standard set of sign conventions in mind. In the picture, all dimensions shown are positive except for h i, which is negative.

Image Formation Mathematics Sign convention summary:  f : + for a concave mirror; - for a convex mirror  conjugate distances (d o and d i ): + if object or image is in front of mirror … - if behind  magnification, m : + if image is upright; - if image is inverted

The Paraxial Approximation Did you notice a “stolen base?” f, which is the distance from the focal point to the vertex, isn’t quite the base of the green triangle.

The Paraxial Approximation The larger the height of the axial ray, the more difference there is between f and the length of the triangle’s base.

The Paraxial Approximation The mirror equation is valid only as a paraxial approximation. It applies to a threadlike cylinder of infinitesimal diameter, centered on the axis. The difference between the paraxial approximation and the consequences of exact spherical geometry cause what is called spherical aberration. The larger the height of an axial ray, the closer to the vertex it passes through the axis.

Spherical Aberration

The Paraxial Approximation So: what good is the mirror equation, since it is only an approximation? Optical system design:  paraxial layout (approximation)  computer modeling and optimization (exact)