Introduction to Ray Optics

Introduction to Ray Optics

• The human eye's retina is sensitive to electromagnetic waves within a small range of the spectrum
• Light is electromagnetic radiation with a wavelength of approximately 400 nm to 750 nm
• Light and vision help us understand the world.
• Light travels fast and in a straight line. Value of speed
• Speed of light in a vacuum: c = 2.99792458 × 10^8 m/s
• For practical purposes, c = 3 × 10^8 m/s is used
• The speed of light in a vacuum is nature's highest attainable speed.
• Despite light's straight-line motion notion, light is an electromagnetic wave with wavelengths in the visible spectrum.
• Light's wavelength is small compared to common objects
• Light waves travel in a straight line from one point to another

Rays and Beams

• The path taken by light is called a ray
• A bundle of rays forms a beam of light
• Reflection, refraction, and dispersion are the phenomena considered using ray diagrams.
• Image formation by reflecting and refracting surfaces will be studied, as well as optical instruments, including the human eye.

Reflection by Spherical Mirrors

• Angle of reflection equals the angle of incidence
• Incident ray, reflected ray, and the normal to the reflecting surface all lie in the same plane
• These reflection laws apply to all reflecting surfaces, whether flat or curved
• Discussion is limited to spherical surfaces.
• The normal is the radius along the line connecting the mirror's curvature center to the incidence point.
• The pole refers to a spherical mirror's geometric center
• The optical center refers to a spherical lens's geometric center
• The principal axis is a line connecting the pole and the center of curvature of a spherical mirror
• For spherical lenses, the principal axis connects the optical center to the principal focus

Sign Conventions

• All distances are measured from the pole of the mirror or the optical center of the lens
• Distances measured along the direction of incident light are positive
• Distances measured against the direction of incident light are negative
• Heights measured upwards from the x-axis and normal to the principle axis are positive
• Heights measured downwards are negative
• A single formula for spherical mirrors and lenses applies to all cases.

Focal Length

• Paraxial rays are incident close to the pole P of the mirror with small angles to the principal axis
• Reflected rays converge at a point F on the principal axis in concave mirrors
• Reflected rays appear to diverge from a point F on the principal axis in convex mirrors
• The principal focus (F) is where reflected rays converge or appear to diverge from.
• If a parallel paraxial beam is incident at an angle to the principal axis, reflected rays converge or diverge from a point in a plane through F normal to the axis
• This plane is the focal plane of the mirror

Focal Length Calculation

• Focal length (f) is the distance between the focus F and pole P of the mirror
• Relationship between focal length and radius of curvature: f = R/2, where R is the radius of curvature
• CM is perpendicular to the mirror at M
• θ is the angle of incidence, and MD is the perpendicular from M to principal axis
• For small θ, tan θ ≈ θ and tan 2θ ≈ 2θ
• For small θ, point D is close to point P, FD = f and CD = R
• The equation f = R/2 is derived from this geometry.

Mirror Equation

• The image is real if the rays converge to that point
• The image is virtual if the rays appear to diverge from the point when produced backwards
• An image is thus a point-to-point correspondence with the object established through reflection or refraction
• Obtain the image point by tracing any two rays, in practice any two are chosen from the list below

Ray Tracing Principles

• A ray parallel to the principal axis is reflected through the focus.
• A ray through the center of curvature retraces its path.
• A ray passing through the focus is reflected parallel to the principal axis.
• The ray incident at the pole reflects at the same angle to the x-axis
• An infinite number of rays emanate from any source, in all directions.
• Point A' is the image point of A if every ray from A passes through A' after reflection
• Derive the mirror equation relating object distance (u), image distance (v), and focal length (f)

Mirror Equation Derivation

• Using similar triangles A'B'F and MPF, and the fact that PM = AB results in B'A' / BA = B'F / FP
• Triangles APB and A'PB' are similar because ∠APB = ∠A'PB'
• Formula to note: Β'Α' / BA = Β'P / BP
• Equations to note: B'F / FP = B'P / BP
• Using the sign convention results in: v/f = v/u - 1.
• The mirror equation: 1/v + 1/u = 1/f, this equation uses the distances between objects
• Linear magnification (m) is the ratio of image height (h') to object height (h): m = h'/h
• Magnification calculation: triangles A'B'P and ABP, means B'A'/BA = B'P/BP
• The above triangles with the correct sign convention becomes, m = − v / u
• Derived for real, inverted images from concave mirrors, the mirror equation & magnification formula remains valid for all spherical mirrors with proper sign conventions

Demonstration of reflection

• The reflection is true even if part of the mirror is covered
• An image of the whole object can be seen in a mirror, even if part of it is covered

Refraction

• When a beam of light hits another transparent medium, a part of light gets reflected back into the first medium while the rest enters the other
• A ray of light represents a beam
• The direction of an obliquely incident ray of light changes at the interface of two media, this is called refraction of light
• The incidence, refraction, and normal angles are measured relative to "the normal"
• Snell's law of refraction: sini / sinr = n21 n21=refractive index of the second medium with respect to the first medium
• The equation: n21 > 1, r<i, the refracted bends toward the normal
• Medium 2 is optically denser than media 1
• The equation n21 < 1, r> i, the refracted bends away from the normal
• incident ray is in a rarer medium, incident ray in denser medium will refract into rarer medium

Optical Density Notes and Refractive Index relations

• Optical density is the ratio of the speed of light in two media
• The mass density is the mass per unit volume
• n₁₂ = 1/n₂₁ This is the equation that gives the refractive index of the medium
• The equation: n32 = n31 x n12, where n31 is the refractive index of medium 3 with respect to medium 1. Use this to find medium 3 with respect to 2

Refraction at interfaces

• Follows refraction as well as elementary calculations
• A rectangular slab has two refraction interfaces: air-glass and glass-air
• The emergent ray is parallel to, but shifted from, the incident ray
• Water tanks appear shallower because viewing occurs near the normal direction
• Use the formula: Apparent depth (h₁) = real depth (h₂)/refractive index

Total Internal Reflection

• Light traveling from denser to rarer medium at an interface will be partly reflected and refracted
• Internal reflection occurs because When a ray of light enters from a denser to a rarer medium, it bends away from the normal
• Internal reflection occurs because when light bends away from normal angle of refraction is larger than angle of incidence
• Angle of incidence is increased when the normal does
• Relationship: sin ic = n21, is for angles at which Snells law can be satisfied, hence refraction can be satisfied

Technological applications for internal refraction

• Prisms bend light by 90° or 180°.
• Such prisms invert images without changing size
• For such prisms i is 45°, material must be <45°, which crown and flint glass both satisfy
• Optical fibers transmit audio and video signals over long distances.
• They use optical composite glass
• Fiber makes use of refraction