Download Fundamentals of Light and Waves: An In-depth Analysis - Prof. Charles Sackett and more Study notes Optics in PDF only on Docsity! Phys 531 Lecture 1 25 August 2005 What is Light? Historical discussion: Hecht Ch. 1 Simplest theory: light consists of a stream of particles. The particles • are emitted by a source (ie, a lamp), • bounce off an object (ie, a book), • and enter your eye. Usually the particles travel in straight lines called rays. 1 Optical elements like lenses and mirrors work by deflecting the rays: Source Focus Rays Lens 2 Particle theory is called ray optics or geometrical optics. • The nature of the “particles” is not specified, so focus more on trajectories = rays. Ray optics is useful for many problems in optics, including most imaging and illumination applications. It fails to explain phenomena like interference, diffraction • require wave optics 3 Wave Optics (Hecht Ch. 2) Say light consists of a wave = disturbance in a medium Just like water waves, sound waves For light, medium is “electromagnetic field” • not very well defined, but doesn’t matter Wave optics is very accurate: Treat wave optics as the “true” theory for most of this course But ray optics is easier, use it when possible! 4 Wave Equation How can we tell if a function is a sum of travelling waves? Any function f(x− vt) has ∂f∂x = − 1 v ∂f ∂t . Any function f(x+ vt) has ∂f∂x = + 1 v ∂f ∂t . So if ψ(x, t) is a sum of travelling waves, must have ( ∂ ∂x − 1 v ∂ ∂t ) ( ∂ ∂x + 1 v ∂ ∂t ) ψ(x, t) = 0 or ∂2ψ ∂x2 − 1 v2 ∂2ψ ∂t2 = 0 Call this the wave equation. Say that function describes a wave if and only if it satisfies the wave equation. 9 Generalize to 3D: ∂2ψ ∂x2 + ∂2ψ ∂y2 + ∂2ψ ∂z2 − 1 v2 ∂2ψ ∂t2 = 0 Recognize Laplacian operator ∇2 ≡ ∂2 ∂x2 + ∂ 2 ∂y2 + ∂ 2 ∂z2 so write 3D wave equation as ∇2ψ(r, t) = 1 v2 ∂2ψ ∂t2 Question: Which of these would you consider a wave? Dye spreading in a pool of water. A line of dominos falling over. A superposition of two travelling waves with different speeds. Which do you think satisfies the wave equation? 10 Harmonic Waves (Hecht 2.2) Most important solutions of wave equation are harmonic waves: ψ(x, t) = A cos(kx− ωt+ φ) where A ≡ amplitude k ≡ wave number (units rad/m, usually just m−1) ω ≡ frequency (units rad/s) φ ≡ phase (units rad) Also have λ = 2π/k ≡ wave length (units m) τ = 2π/ω ≡ period (units s) ν = 1/τ = ω/2π ≡ frequency (units cycles/s or Hz) 11 Also, the wave equation requires that ω = vk. Harmonic waves are periodic in both space and time: ψ(x+ λ, t) = ψ(x, t+ τ) = ψ(x, t) These definitions and relationships are very important, so you should memorize them! Question: What are the units of A? 12 The 3D version of a harmonic wave is called a plane wave: ψ(r, t) = A cos(k · r − ωt+ φ) Here k is the wave vector, while k = |k| is the wave number. We still have k = ω/v = 2π/λ. The condition of spatial periodicity becomes ψ(r + λk̂, t) = ψ(r, t) where k̂ = k/|k| is the propagation direction of the wave. 13 Complex Representation (Hecht 2.5, handout) Harmonics waves are useful, but trig functions get tedious. Instead represent with complex functions. Complex numbers: form z = x+ iy, where i = √ −1. Define x ≡ real part, write Re z y ≡ imaginary part Im z. Complex numbers follow the normal rules of algebra: (x1 + iy1) + (x2 + iy2) = (x1 + x2) + i(y1 + y2) (x1 + iy1)(x2 + iy2) = x1(x2 + iy2) + iy1(x2 + iy2) = x1x2 + ix1y2 + iy1x2 − y1y2 = (x1x2 − y1y2) + i(x1y2 + y1x2) 14
