Lecture 8 Chapter 4 The Propagation of Light: Transmission Reflection Refraction Macroscopic manifestations of scattering and interference occurring at the atomic level
Reflection
Reflection Inside the dense substance each of the scattering molecules has a pair that is /4 away and scatters backward wave that is out-of-phase - complete destructive interference On an interface between two substances there are many unpaired molecules, or even if there is a pair scattering efficiency or phase will be different for different atoms or molecules Backwards scattering close to interface (~ /4) will not experience complete destructive interference: reflection /4 /4
Reflection Scattering properties are reflected in in index of refraction n Reflection occurs on the surface between two materials with different n Depending on the interface type: external reflection internal reflection n i < n t n i > n t Example: reflection of beam falling on glass (beam coming from air) - ~4% reflected Example: reflection of beam coming from under water from water-air interface
Reflection: microscopic view When an incident plane wave front strikes the surface at some angle it does not reach all the atoms along the surface simultaneously. Each consequent atom will scatter at slightly different phase, while the spherical wave created by previous atom had a chance to move away some distance. The resulting reflected wave front created as a superposition of all scattered wavelets will emerge also at an angle to the surface.
Scattered spherical waves often combine to form plane waves. A plane wave impinging on a surface (that is, lots of very small closely spaced scatterers!) will produce a reflected plane wave because all the spherical wavelets interfere constructively along a flat surface.
We ll check the interference one direction at a time, usually far away. This way we can approximate spherical waves by plane waves in that direction, vastly simplifying the math. Far away, spherical wave-fronts are almost flat Usually, coherent constructive interference will occur in one direction, and destructive interference will occur in all others. If incoherent interference occurs, it is usually omni-directional.
The mathematics of scattering The math of light scattering is analogous to that of light sources. If the phases aren t random, we add the fields: Coherent E total = E 1 + E 2 + + E n I I I... I c Re EE EE... E E * * * total 1 2 N 1 2 1 3 N 1 N I 1, I 2, I n are the irradiances of the various beamlets. They re all positive real numbers and add. E i E j* are cross terms, which have the phase factors: exp[i( i - j )]. When the s are not random, they don t cancel out! If the phases are random, we add the irradiances: Incoherent I total = I 1 + I 2 + + I n
To understand scattering in a given situation, we compute phase delays. Because the phase is constant along a wavefront, we compute the phase delay from one wave-front to another potential wave-front. Wave-fronts Scatterer L 1 L 2 L3 L 4 kl i Potential wave-front i If the phase delay for all scattered waves is the same (modulo 2), then the scattering is constructive and coherent. If it varies uniformly from 0 to 2, then it s destructive and coherent. If it s random (perhaps due to random motion), then it s incoherent.
Coherent constructive scattering: Reflection from a smooth surface when angle of incidence equals angle of reflection A beam can only remain a plane wave if there s a direction for which coherent constructive interference occurs. The wave-fronts are perpendicular to the k-vectors. i r Consider the different phase delays for different paths. Coherent constructive interference occurs for a reflected beam if the angle of incidence = the angle of reflection: i = r.
Coherent destructive scattering: Reflection from a smooth surface when the angle of incidence is not the angle of reflection Imagine that the reflection angle is too big. The symmetry is now gone, and the phases are now all different. = ka sin( too big ) i too big = ka sin( i ) a Potential wave front Coherent destructive interference occurs for a reflected beam direction if the angle of incidence the angle of reflection: i r.
Coherent scattering occurs in one (or a few) directions, with coherent destructive scattering occurring in all others. A smooth surface scatters light coherently and constructively only in the direction whose angle of reflection equals the angle of incidence. Looking from any other direction, you ll see no light at all due to coherent destructive interference.
Reflection: constructive interference For constructive interference spherical waves created by the atoms on the surface must arrive in-phase. Let us consider two atoms on the surface. Wave function depends only on xt-t: E f ( k r t) Wave phase along incident wavefront is the same: E E f ( t) The scattered wavefront CD: points C and D must be at the same phase: EC f ( k AC t) AC BD E f ( k BD t) D phase shift on scattering atom A B
Triangles ABD and ACD: The angle of reflection AC AD B BD AD C 90 The Law of Reflection (1st part): The angle-of-incidence equals the angle-of-reflection i = r
Rays and the Law of Reflection A ray is a line drawn in space along the direction of flow of radiant energy. Rays are straight and they are perpendicular to the wavefront Conventionally talk about rays instead of wavefronts The Law of Reflection 1. The angle-of-incidence equals the angle-of-reflection ( i = r ) 2. The incident ray, the perpendicular to the surface and the reflected ray all lie in a plane (plane-of-incidence)
Incoherent scattering: reflection from a rough surface No matter which direction we look at it, each scattered wave from a rough surface has a different phase. So scattering is incoherent, and we ll see weak light in all directions. This is why rough surfaces look different from smooth surfaces and mirrors.
Specular and diffuse reflection Smooth surface: specular reflection Rough surface: diffuse reflection
Example: stealth technology Flat surfaces: microwaves do not reflect back
Lecture 9 Chapter 4 The Propagation of Light: Transmission Reflection Refraction Macroscopic manifestations of scattering and interference occurring at the atomic level
What about light that scatters on transmission through a surface? Again, a beam can remain a plane wave if there is a direction for which constructive interference occurs. Huygens Principle Constructive interference will occur for a transmitted beam if Snell's Law is obeyed.
Refraction Incident beams are bent when they enter a substance with different index of refraction - refraction. The phase difference between wavefronts AB and ED must be the same - it must take the same time for the wavefront to cover distance BD and AE: BD AE v t i v t t AD sini AD sin t v v i t sini sin t cnt n c i sini sin t n i sin n i t sin t
The Law of Refraction 1. (Snell s law): n i sin i n t sin t 2. The incident, reflected and refracted rays all lie in the plane of incidence The laws of refraction and reflection are reversible Willebrord Snel van Royen (1580-1626)
Magic: see over edge See over the edge cup no water Apparent (virtual) image The cup seems to be shallow Add water
Spoon bending
Refraction and wavelength The wavelength changes when light enters a substance: vit i vtt t ct n i ct n i t t n n i i t t If 0 is wavelength in vacuum (n=1): n 0 vacuum In the media with n>1 wavelength decreases: = 0 /n speed decreases: v = c/n frequency does not change.
Dispersion Speed of light in matter depends on frequency (or wavelength) Refraction index depends on wavelength Amount of bending depends on wavelength Rainbows: white light is separated into colors due to dispersion in water droplets
Rainbows Skier will see red at the top of the rainbow, and blue at bottom. Rainbows are one of the most beautiful examples of dispersion in nature.
Secondary rainbow In the secondary rainbow pattern is reversed There are 2 total internal reflections in water droplets that form the second rainbow