Light as Waves and Particles - Reflection, Refraction, Dispersion

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This is a topic from Higher Physics 1B

Contents

Introduction to the Dual Nature of Light

During the 19th and 20th centuries there was a lot of debate about the true nature of light. Some scientists argued that light was an electromagnetic wave because it refracts and interferes and because of Maxwell's wave equations which were confirmed by Hertz. Others believed light to be a stream of particles, supported by observations of the photoelectric effect. It was Einstein who proposed an explanation of the photoelectric effect that used the idea of quantisation whereby light is both a wave and particles.


Light, according to Einstein, is made up of particles (quanta) of light known as photons, the energy of which is given by:

E = hf

Where E is the energy of the photon in Joules (J)

h is Planck's constant (6.63 x 10-34
f is the frequency of the light wave in Hertz (Hz)


This combination of light and wave characteristics is known as the wave-particle duality. At any given time we can comprehend light as a stream of particles, or as a wave, but it is difficult to comprehend it as being both simultaneously. We choose the method of analysing light which is most suitable to the situation and its scale. Typically the wave model is more useful on a macroscopic scale and the particle model is more useful on a microscopic scale.

The Ray Approximation

In geometric optics, the assumption is made that light travels in a straight-line path in a uniform medium which can be approximated as a ray, where a ray is a straight line perpendicular to the wave fronts. This straight-line path comes from Fermat's principle, that the path light takes between two points is the one which requires the smallest time interval (hence a straight path).

Reflection

  • When a ray of light travelling through one medium meets a second medium part of the light bounces back into the first medium. This is known as reflection
  • Specular reflection is the reflection from a smooth surface
  • Diffuse reflection is the reflection from a rough surface
  • All reflection in the course is assumed to be specular unless told otherwise
  • The normal is the vector perpendicular to the surface of reflection
  • The Law of Reflection states that when an incident ray makes an angle θ with the normal, the reflected ray also makes an angle of θ with the normal - the angle of incidence equals the angle of reflection

Refraction

  • When a ray of light travelling through one transparent medium meets a second transparent medium part of the light is reflected, while the part of the light which is bent as it passes into the second medium is said to have been refracted
  • The angle of refraction is dependant upon the materials that the medium is made of and the angle of incidence, as given in the following equation in which the subscripts 1 and 2 refer to mediums 1 and 2 respectively
Sin θ2/Sin θ1 = v2/v1 = constant
  • The path of light upon refraction is reversible (In figure (a) below, if a ray went down path AB when originating at A it would follow the same path in the opposite direction when originating at B)

Screen Shot 2012-09-10 at 2.53.40 PM.png

  • When light moves from a medium of slower speed to a medium of higher speed the ray bends away from the normal (θ2 > θ1)
  • When light moves from a medium of faster speed to a medium of slower speed the ray bends towards the normal (θ2 < θ1)

Index of Refraction

  • The reason light travels at different speeds in different mediums is that when light strikes an electron within a medium that electron absorbs the light, oscillates and then re-radiates the light, a process which reduces the average speed of the light. The more of these collisions which occur in a given medium the slower the speed of light in that medium
  • The index of refraction, n, of a medium is defined by Snell's Law as follows:
Screen Shot 2012-09-10 at 3.01.34 PM.png
  • For a vacuum, n=1 (we assume n=1 for air as well)
  • For other media, n>1, where n is not necessarily an integer
  • As light travels from one medium to another its frequency remains constant which its speed and wavelength change in a manner governed by v = fλ
  • This leads to the following relationship:
Screen Shot 2012-09-10 at 3.18.39 PM.png
  • From this, and the assumption that the refractive index for air is n=1, we get the following:
Screen Shot 2012-09-10 at 3.19.25 PM.png

Optical Path Length

Optical path length, or OPL is a quantity invented to allow us to account for the changes in wavelength in different mediums that can cause one wave to have a different phase to a wave in another medium which has travelled the same distance.

'OPL = nl

Where n represents the refractive index

l represents the distance travelled

Total Internal Reflection

  • As previously stated, light going from a more dense medium to a less dense medium bends away from the normal
  • In such a circumstance there is an angle of incidence known as the critical anglec) which results in an angle of refraction of 90°
  • For all angles greater than this critical angle the ray will be completely reflected at the boundary between the mediums (there is no refraction)
  • One should be familiar with the idea that total internal reflection is sued in telecommunications technologies such as fibre optic cables.

Dispersion

  • The index of refraction n for a medium varies depending on the wavelength and frequency of the incident light, leading to a phenomenon known as dispersion
  • Dispersion is the splitting of incident light into various strata through the process of refraction
  • For white light for example, the various components of the light have different wavelengths, different refractive indices and so different refracted angles. As a result of this the refracted light is split into the component colours of white light, all leaving the medium at slightly different angles
  • Due to this variation in refracted angle a parallel incident beam is refracted non-parallel in different directions, hence the term dispersion
  • The difference in angles of refraction of different components of light is called the angle of deviation and it is dependant on the incident wavelength
  • For white light the red component deviates the least while violet deviates the most
  • Refraction and reflection of sunlight through raindrops is the cause of rainbows, while the second rainbow which makes up a double rainbow is caused by some light internally reflecting twice within a raindrop before exiting

Huygen's Principle

  • Huygen's principle is a geometric tool which allows us to determine the position of the wavefront at a point in time based on previous knowledge of the wavefront
  • The principle considers light as waves, not as particles
  • The principle involves taking all points on known wavefront as point sources for new spherical secondary waves called wavelets which propagate at the speed of waves in the given medium
  • After some time interval the new wavefront is given by the surface tangent to all of the wavelets

Consider the following diagrams which show Huygen's principle applied to two different wavefronts:
Screen Shot 2012-09-17 at 11.07.28 AM.png Screen Shot 2012-09-17 at 11.07.41 AM.png

  • The black dots represents a sample of the infinitely many points on the original wavefront taken to be point sources (for the purposes of explanation only several points on the wavefront are considered)
  • The red arcs represent arcs of the spherical wavelets which propagate from the point sources
  • The blue lines, as labelled, represent the original wavefront, and the final wavefront constructed as a surface tangent to the red arcs

Deriving the Law of Reflection Using Huygen's Principle

Consider the following diagram in which two parallel rays of light strike a reflective surface:
Screen Shot 2012-09-17 at 11.19.24 AM.png

  • AB represents a wavefront of the incident light
  • By Huygen's principle we imagine a wavelet from point A towards D and from B towards C such that we have a wavefront of reflected light, DC

Screen Shot 2012-09-17 at 11.27.38 AM.png

  • Examining the triangles ABC and ADC in isolation we see that they are congruent, and so Υ = Υ'
  • This shows us that the angle of incidence equals the angle of reflection, which is the law of reflection

Deriving the Law of Refraction Using Huygen's Principle

Consider the following diagram in which two parallel rays of light strike a semi-transparent surface:

Screen Shot 2012-09-17 at 11.32.15 AM.png
  • AB represents z wavefront of the incident light
  • By Huygen's principle we imagine a wavelet from point A towards D and from B towards C such that we have a wavefront of refracted light, DC
  • In the new denser medium light travels more slowly and so the wavelet path AD is shorter than BC, causing the wavefront to bend as seen
  • Examining triangles ABC and ADC we see that they are right-angled triangles and that AD and BC can be given in terms of the time interval Δt and the speed of light in their respective media:
Sin θ1 = BC/AC = v1Δt/AC
and Sin θ2 = AD/AC = v2Δt/AC
  • Dividing one equation by the other gives:
Sin θ1/Sin θ2 = v1/v2
  • Expressing v1 and v2 in terms of the speed of light and their respective indices of refraction gives us Snell's law

Phase Change Due to Reflection

  • When an EM wave is reflected from a boundary between its medium and one of a higher refractive index is undergoes a phase change of 180 degrees analogous to a pulse on a string reflected off a rigid support
  • When an EM wave is reflected from a boundary between its medium and one of a lower refractive index it undergoes no phase change, analogous to a pulse on a string reflected off a free support
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