3.2 refraction, diffraction, interference

• Coherent waves - wave sources of the same type and a single frequency

  • Ensures they maintain a constant phase relationship

  Path difference - the difference in length of the paths travelled by different waves
• Interference 

  • Interaction of different waves
  • Results in altered waves

  Constructive interference

  • In phase
  • Path difference is n x wavelength

   Destructive interference

  • 180° out of phase
  • Path difference is (2n+1) x wavelength / 2

  Standing waves 

  • Stationary and oscillating
  • Always in phase
  • Constant change

  
• Node - point of no displacement

  • Has no energy

  Antinode - point of maximum displacement

  Phase difference between two particles

  • 0 is separated by even number of nodes
  • 180 (pi radians) if separated by odd number of nodes

  
• • To observe a clear diffraction pattern, a monochromatic, coherent light source is needed
  • E.g. lasers 
  • Monochromatic - all light has same wavelength and frequency so is same colour
  • If the wavelength of light is equal to the aperture, a diffraction pattern appears
  • The central fringe is bright (central maximum) with alternating dark and bright fringes on either side
  • Caused by destructive and constructive interference 

  
• Young’s double slit experiment

  • Can use two separate coherent light sources or shine laser through two slits
  • Laser light is monochromatic and coherent
  • Both slits have to be approximately the same size as the wavelength so it is diffracted
  • The light from the slits act as two coherent point sources
  • Get a pattern of light and dark fringes depending on constructive or destructive interference

  
• Practical considerations

  • It is easier to measure centre point of dark fringes than light 
  • Measure across several fringes (centre to centre) to reduce error
  • Wavelengths of light written in nanometers (x10-9m)
  • Ensure all figures are returned to meters for calculations
• Fringe spacing

  • Fringe spacing - distance from the centre of one minimum to the centre of the next minimum
  • The wavelength of light is so small a high D/s ratio is needed so the fringe spacing is big enough to see
  • Measure across several fringes and divide by the number of fringes to find the average
  • More accurate result

  Superposition - constructive and destructive interference 

  sin(x)= w/D     sin(x) = (width of central fringe) / (distance of screen from slits)

  sin(x) = wavelength / distance between slits

  • w/D = wavelength / s     w = (wavelength x D) / s
• Production of an interference pattern

  • White light is a mixture of all colours, each with a different wavelengths
  • All are diffracted by different amounts
  • Get a spectra of colour

  
• Laser safety precautions

  • Never shine the laser towards someone
  • Wear safety goggles
  • Avoid shining laser at reflective surface
  • Have a warning sign on display
  • Turn laser off when not needed
• Interference of sound and EM waves

  Sound waves

  • Two-source interference of waves with a measurable wavelength
  • Two coherent sources driven by the same oscillator

  Electromagnetic waves

  • End of 17th century, two theories of light were published
  • Newton - suggested light was made up of particles called corpuscles
  • Huygens - light as waves
  • Corpuscular theory - explains reflection and refraction but diffraction and interference are uniquely wave properties
  • Youngs double slit experiment provides evidence showing light can diffract and interfere
• Diffraction pattern from a single slit

  Diffraction pattern

  • The slit is the same width as the wavelength
  • The bright central maximum with fringes

  Intensity - power per unit area Wm-2

  • Monochromatic - all photons have the same energy

  
• Huygens construction

  • Infinite sources of the wave
  • wavelength > gap size

  Width of central fringe maximum

  • Bigger slit width then less diffraction then narrower but more intense central maximum
  • The central fringe has the greatest intensity
  • The intensity of the central fringe is significantly greater than the outer fringe
  • The central fringe has twice the width of subsequent fringes 
  • w = (2Dxwavelength) / a  
  • width of central fringe = (2 x distance from slit to screen x wavelength) / width of single slit

  
• Transmission diffraction grating

  • For monochromatic light, all maxima are sharp lines
  • Line of maximum brightness at the centre - zero order line
  • Lines on either side are first-order lines

  
• Derivation of nth order maximum equation 

  • dsin(x) = n x wavelength
  • The angle between the incident beam and the nth-order maximum
  • First-order maximum happens at an angle when waves from one slit line up with waves from the next and are exactly one wavelength behind
  • Difference worked out using trigonometry
  • Other maxima occur when the path difference is 'n'
• Conclusions 

  • If the wavelength is greater, the angle is greater - the larger the wavelength, the more the pattern will spread out
  • If the slit spacing is greater, the angle is smaller - the coarser the grating, the less the pattern will spread out
  • Values of sin greater than 1 are impossible - for certain n you get a result of more than one so that order cannot exist
• Identifying elements 

  • White light is a mixture of colours - diffracting white light through a grating gives a spectrum of colour
  • Each order in the pattern becomes a spectrum with red on the outside and violet on the inside
  • The zero-order maximum stays white because all wavelengths pass-through
  • The wavelength of x-rays is a similar scale to the spacing between atoms in crystalline solids - x-rays form a diffraction pattern when directed at a thin crystal 
  • The crystal acts as a diffraction grating and spacing between atoms (slit width) can be found in pattern - x-ray crystallography
• Refractive index

  • Light travels at different speeds in different materials
  • Refractive index (n) - the ratio of the speed of light in a vacuum to the speed of light in the material
  • n = sin(i) / sin(r)    n = c1 / c2    
  • sin(i) / sin(r) = n2 / n1 = c1 / c2
  • The relative refractive index of air is 1
  • Relative refractive index - the ratio of the speed of light in two different materials
  • 1n2 = c1 / c2    1n2 = n2 / n1 
  • Property of the interface between two materials - different for every pair
  • The absolute refractive index of a material is a property of that material only 
• Snell’s law

  • Law of refraction
  • n1sin(x)1 = n2sin(x)2
  • Incident ray = reflection ray
  • The angle of incidence - the angle incoming light makes to the normal
  • The angle of refraction - the angle the refracted ray makes with the normal
  • When light enters an optically denser medium it is refracted towards the normal
• Total internal reflection (TIR)

  • Only from more dense material to less dense material
  • Only when the refractive index of the first material is greater than that of the second 
  • ni > nr
  • E.g. air to glass - bends towards the normal
• Refraction vs. TIR 

  • Light leaving an optically denser material is refracted away from the normal
  • Increasing the angle of incidence brings the angle of refraction closer to 90°
  • When the critical angle is reached, light is refracted along the boundary
  • sin(x)c = n2 / n1 =1n2
  • Beyond the critical angle, refraction is impossible and all the light is reflected into the material - TIR

  
• Fibre optics and cladding 

  Optical fibre - thin, flexible tube of glass/plastic fibre

  • Carries light signals over long distances and round corners
  • E.g. step-index optical fibres

  Step-index optical fibres

  • Have a high refractive index
  • Surrounded by cladding with a lower refractive index
  • Allows for total internal reflection (TIR)
  • Protects fibre from scratches letting light escape
  • Light shone in at one end
  • Extremely narrow fibre
  • Light hits the boundary between fibre and cladding at an angle > critical angle
  • All light is TIR from boundary to boundary until it reaches the other end

  
• Dispersion and absorption

  A signal (stream of pulses of light) travelling down an optical fibre can be degraded 

  • Can cause loss of information

  Absorption - causes loss in amplitude

  • Energy is lost through absorption by the material of fibre
  • Reduces the amplitude of the signal
• Dispersion - causes pulse broadening

  • Modal dispersion - light rays enter fibre at different angles so they take different paths
  • Rays take a longer path than those down the middle of the fibre
  • Single-mode fibre only lets light take one path reducing modal dispersion

  
• • Material dispersion - light consists of different wavelengths travelling at different speeds
  • Some wavelengths reach the end of the fibre faster than others
  • Monochromatic light stops material dispersion

  Both types of dispersion lead to pulse broadening - the signal sent down the fibre is broader at the other end

  • Broadened pulses can overlap each other, confusing the signal

  Optical fibre repeater - boosts and regenerates signal every so often reducing signal degradation