Optics Concepts

• Physics 323 Lecture Notes cover Optics
• Author: A. A. Louro

Nature of Light

• Both experimental evidence supports that Light is a wave and a stream of particles, although contradictory.

What is a Wave?

• A perturbation with a periodic spatial pattern propagates or travels in space.
• Sound waves: perturbed quantity is the pressure oscillating around atmospheric pressure.
• Water surface waves: perturbed quantity is the height of the surface oscillating around its stationary level.
• Wave fronts visualize waves using crests.

Evidence for Wave Properties of Light

• Interference is exclusive to waves.
• Ripples in a pond caused by two pebbles exhibit interference: crests overlap and reinforce each other, crest and trough coincide and cancel.
• Thomas Young's experiment in the early 1800s: light as a wave produces alternating bright/dark bands on a screen.
• Problem: no medium for light waves (unlike sound/surface waves).
• The "luminiferous aether" was the medium name.
• James Clerk Maxwell's (1831-1879) theory of electromagnetism: light is a wave in combined electric and magnetic fields needing no material medium.
• Coherent waves: two sources oscillate in step.

Evidence for Light as a Stream of Particles

• Isaac Newton supported the idea of light as a stream of particles.
• Photoelectric effect: light striking a metal dislodges electrons, generating current.
• Albert Einstein won the Nobel prize for explaining the photoelectric effect with photons.
• Light's behavior depends on the context of the light
• Wave theory is adequate for understanding optical instruments.

Features of a Wave

• Wavelength: the distance between successive wave fronts.
• Propagation speed (v): dx/dt.
• Stationary observer sees periodic oscillation ("cycles").
• Period: duration of each cycle.
• Frequency (f): the number of cycles measured each second (SI unit: Hertz (Hz), equivalent to s⁻¹).
• Formula for the relationship between wavelength, frequency and propagation speed: v = fλ
• Electromagnetic waves in vacuum propagate at c = 3.0 × 10⁸ m/s.
• Visible light: wavelengths of 400 - 700 nm (perceived as color).
• Red: longest visible wavelengths.
• Violet: shortest visible wavelengths.
• Wavelengths longer than visible are Infrared, microwave, and radio.
• Wavelengths shorter than visible are Ultraviolet, X rays, and gamma rays.

Propagation of Light

Huygens' Principle

• Christian Huygens proposed in the 1670's
• Each point on a wavefront acts as the new wave source.
• Envelopes of secondary waves are the new wave front.
• Huygens' Principle describes a property of waves, useful to explain refraction.

Refraction

• Light slows when it propagates in transparent material medium, compared to the speed in vacuum c.
• Light changes direction when passing from one medium to another, it is called refraction.
• Plane wavefront approaches the interface between two media.
• One end's new wavefront: propagates outward, interface reached in time t (radius v₁t).
• Other end's new wavefront: propagates slower into medium 2 (radius v2t in time t)
• Angle of incidence θi
• Angle of refraction θr
• sin θi / sin θr = v₁ / v₂
• Index of refraction is defined as n = c/v, where c is the speed of light in a vacuum and v is the speed of light in the medium.
• Snell's law formula n₁ sin θ₁ = n₂ sin θr
• Refractive indices > 1 (vacuum = 1).
• Water's index of refraction is 1.33.
• Diamond's index of refraction is ~1.5.
• Optical density measures material's index of refraction.

Total Internal Reflection

• One consequence of Snell's Law.
• Light propagates from denser to less dense medium, and n₁ > n2, then sin θr > sin θᵢ .
• Maximum angle of incidence for refraction, sin θᵢ < n₂/n₁.
• Incident ray does not cross the interface it is reflected back.
• Fiber optics is a consequence of light propagating inside a fiber with higher refractive index than air.

Images

• An optical system creates an image from an object
• Examples include projectors
• Two types: real and virtual.
• Extended objects are collections of point sources of light.

Real Images

• A point object emits light rays in all directions.
• Optical elements redirect the light which converges to a point image
• If a screen is placed at the same point, the image may be seen as the light concentrated there.

Virtual Images

• Reflection from a plane mirror creates a virtual image.
• Reflected rays appear to originate from a point behind the mirror.
• Rays entering an observer's eye or a camera objective lens are seen as coming from a point.
• Placing a screen behind the mirror will not project the virtual image.

Curved Mirrors

• Curved mirrors are telescope components
• Parabolic cross-sections for focusing properties although spherical is a good approximation if the curvature is low.
• Mirrors focus beam parallel to the optical axis to the focal point.

Ray Tracing with Mirrors

• Locate images via auxiliary rays from the object.
• Ray (1) can be parallel to the optical axis, passing through the focal length.
• Ray (2) passes through the focal point is reflected parallel to the optical axis (reversibility of light).
• Ray (3) at the mirror's vertex: reflected ray is symmetrical with respect to the axis of the mirror.
• Intersection of rays means this is the location of the image.
• Two rays forms an image if the object is extended and image can be: real, inverted, enlarged.

The Mirror Equation

• Location of the image by object position and mirror's focal point and curvature.
• Object distance p: along the axis from the mirror's vertex.
• Image distance i
• Focal length f measured from the object
• Mirror Equation: Curvature of the mirror should be very low, as well as the object being close to optical axis.
• Derivation from similar triangles to determine equal ratios, triangles △OPF and △FQI.
• Simplified: 1/p + 1/i = 1/f

Lenses

Introduction

• Mirrors use reflection; lenses use refraction. Lenses are used in optical systems and refracting telescopes.
• An image may be formed as light pass through air/glass with point images located after a single spherical interface.

Refraction at a Spherical Interface

• Interface curvature should be represented as a flat surface to simplify.
• Point object at O emits a ray along optical axis and one other point to reach the interface with another to form the image at I.
• Radius of curvature is r, with Object distance designated as p, but there are sign conversions.
• Sign convention differs: object distances are positive when the object is in front of the interface; image distances are positive when the image is formed behind the interface.
• Exterior angle theorem: ∠PCO = θ − α and ∠PIC = θ − α − φ.
• Snell's Law n₁θ = n₂φ
• The formula derived is n1/p+n2/i =(n2-n1)/r

A Lens

• Two refractions one air to glass then glass to air. Formula derived n1/p + n2/i' = (n2 - n1)/rr
• n1/L-i' + n2/i = (n1-n2)/r for the thin lends approximation L -> 0
• This then reduces to n1/i+n2/i'=n2-n1/r

Locating the Image

• There are two consecutive refractions air to glass, and glass back to air.
• n1/p+n2/i' = n2-n1/r1
• Where r1 is the curvatur radius of the first surfacer, distance is L-i
• n2/L- i' + n1/i= n1-n2/r2 , the thin less approximation is L->0
• Can be reduced into an image object of 2 lenses from two equations
• n1/p + n1/i = (n2 - n1)(1/r1 -1/r2)

The Lensmaker's Equation

• Usually outside is air n=1 with glass index of depending
• 1/p + 1/i = (R-1)(1/r1 - 1/i) -Right had side only affects the lens features measured at a point in space at length
• Incident rays are parallel, the formed image equals to the inverse on the right side.
• Lens focal Length
• 1/f= (n-1)(1/r1 - 1/r2) -Lensmaker tells the curvatur needed for the perfect focal length and index type

The Thin Lens Equation

• When to get the location of the image by lems

Converging and Diverging Lenses

• If positive then the lens is converging
• IF negatie, then the rasy diverge after appearing infront from source infornt of the lends

Ray Tracing with Lenses

• Each Lens has two focal points. -primary focal point on the light source side
• Converging Lens has it source on the left.

Real and Virtual Images

Lateral magnification

• y Axis has been marked off as it shows the height of the inage and object related.

Optical Instruments Using Lenses

Single-Lens Systems

• Analytic toos and devleop of the system

A Magnifying Glass

• What is perceived to us as an object is the angel of the feild vision.

Single-Lens Systems

• Analytic tools devwloped

Wave Phenomenon

Introduction

• Optics understood by instruments with the power of light or interference and diffreaction.

Interference

• In 1773-1829 to show cases of light props like beams and sources by effectively splitting the beams of light
• wavefronts are used to split into interference combining space that enhances other.

Diffraction

• More important is we observe the dark band.

Diffraction gratings

• One important for the wave on the applications based on the interface producing very thin slits and spacing

Summary of formulas in this chapter

Two-slit Wave Interference and Diffraction

derivations

• derivation for module 7 equation to light in Snell and media and triangle
• deriving for mirror and young experiment

Small angle approximation

• Function with values of 0 and consider 0 if measure in radians.

Explore fundamental principles within optics, including Huygens' Principle and refraction. Investigate image formation using lenses and mirrors, and define real versus virtual images. Learn about calculations, total internal reflection, and optical fibers.