refraction diffraction and intereference

two conditions for sources to be coherent

• waves must have a constant phase difference

• waves must have same frequency

superposition and interference

• when waves meet, they pass through each other / overlap

• at this point, they combine before they move apart - superpose

• the total dispacement at a point is equal to the sum of the individual displacements at that point

• for coherent waves: interference can occur - cancellation (destructive) when crest meets trough (out of phase) and reinforcement (constructive) when crest meets crest (in phase) - occur in fixed positions

• two progressive waves of same speed, frequency and wavelength can superpose to produce a wave with maximum displacement

diffraction

• waves spread out after passing through a gap or around an obstacle - the narrower the gap or the longer the wavelength relative to gap, the more the waves spread out

• diffraction pattern most noticeable when the gap width is approximately equal to the wavelength

• less diffraction / spreading out of waves means that energy is concentrated in smaller area - amplitude increases of central max of interference pattern and side fringes get closer together

refractive index

• $$n=\frac{c}{v}$$

• refractive index of air is approx. 1

• low to high refractie index - bends towards the normal as it slows down

Snell’s law

• n1 sin theta = n2 sin theta

$$n1\sin\theta1=n2\sin\theta2$$

total internal reflection

• theta c = n2/n1

• occurs when ray is incident at greater than the critical angle

• occurs from a higher to lower refractive index boundary

fibre optics

• used in communications to carry light signals and in medical endoscopes to see inside the body

• communication optical fibre allows pulses to light to enter from an transmitter and reach a receiver at the other end - must be highly transparent - minimise absorption which would otherwise reduce the energy and so the amplitude of signal the further it travels

• core: propagates / guides wave by TIR with low absorption

• refractive index of core greater than that of cladding

• cladding; protects core from damage, prevents cross talk between touching fibres, provides clean boundary for TIR

Dispersion problems

• both cause pulse broadening

• Material: different wavelengths have different speeds due to different refractive indices in the core - monochromatic light will reduce this dispersion

  • from highest to lowest refractive index in any material: violet - red. The light with the lower refractive index will have a greater speed and shorter transit time. ( and higher critical angle)

• modal: spreading of pulse / parts of a pulse take different times to travel through the fibre due to entering at diffreent angles - different paths have different lengths so effective time along fibre differs - use narrow core to reduce this dispersion - smaller range of possible paths

• reduce dispersion to increase transmission rate

fibre optic core bending

• angle of incidence on core cladding boundary can decrease and become less than the critical angle (which does not change)

• some light will leave core and be refracted into cladding (partial reflection)

• bending may cause cracks in core / cladding

• amount of internally reflected light changes as a result

path difference

• coherent waves that travel different distances will have a path difference - can cause phase difference

• in phase (phase diff 0) if the path difference is an integer multiple of the wavelength $\operatorname{m}\lambda$

• antiphase (phase diff $\pi$ rad) if path difference is an integer multiple of half the wavelength $\left(\operatorname{m+\frac12\lambda}\right)$

phase difference

• in phase - phase difference is 0 rad

• out of phase any other phase difference that is not in phase

• antiphase refers to a phase difference of $\pi$ rad

single slit diffraction

• a central fringe that is much more intense and twice as large as lower intensity, narrower fringes on either side

• maxima on either side decrease in intensity with distance from centre and are all the same width

• width of central fringe $W=2\lambda D/a$

where a is slit width

• increasing wavelength also increases fringe spacing of subsequent maxima

• range of wavelengths - central maximum unchanged - other maxima gradually getting broader / more spread out for greater order maxima

young’s double slit

• light passes through a single narrow slit first which acts as a single source diffracting light to both slits - the path lengths between the single slit and the double slit are constant / fixed / the same

• this light then passes through two closely spaced double slits which act as cohernt sources of waves

• alternate dark and bright fringes can be seen on a screen which are caused by the interference of light from the two slits

• equally spaced

• width of central diffraction fringe $W=2\lambda D/a$

• fringe spacing $w=\lambda D/s$

• white light interference pattern - intense central white fringe as all wavelengths arrive here - fringes either side each showing range of colours / spectrum - red furthest on outside of fringe and violet closest on inside of fringe to centre - difference wavelengths mean that the two fringe patterns do not overlap exactly

bright and dark fringes

• light from both slits undergoes diffraction

• waves meet and superpose at a point

• at a maxima, they path difference is a whole number of wavelenths so are arriving in phase - reinforce constructively - peaks meet peaks and troughs meet troughs - total amplitude increases

• moving away from the maxima point introduces a path difference that is not a whole number of wavelengths - destructive reinforcement - partial cancellation and amplitude decreases

• At a minima, waves arrive completely out of phase

• moving away, waves move back in phase

diffraction grating

• many closely spaced parallel slits

• monochromatic light incident normally - light transmitted in certain directions only as light passing through each slit is diffracted - interferes constructively in certain directions and destructively in other directions

• central maximum is 0th order

$$d\sin\theta=n\lambda$$

• the maximum possible angle of an order is 90 - total number of maxima given by 2n+

• higher order are less well defined / dimmer - harder to observe than lower orders

diffraction grating uses

• to analyse chemical compositionn of a sample

• to determine wavelength of a light source

• to provide a monochromatic source / select a particular colour of laser light

uncertainty - diffraction grating

• measure across all visible maxima / choose a higher order maxima

• this increases the distance measured, reducing the percentage uncertainty in sin theta

• needed to calculate wavelength so percentage uncertainty in wavelength decreases

or

• increase screen to grating distance

• this increases the fringe spacing

• decreases percentage uncertainty in both

• needed to calculate wavelength so percentage uncertainty in wavelength decreases

• a value that involves fewer readings has a lower percentage uncertainty

• there may be difficulty in locating exact position of centre of each spot / lining up ruler

to ensure a light ray doesn’t devatiate

• same material / same refractive index

• no gaps

effect of refractive index of a material varying by a few percent across it

• variation may be too small to deviate the ray significantly within the material

or

• may cause a summative effect when multiple reflections at boundary occur, changing direction of ray

transmitter and receiver arranged in a line with metal plate parallel to both

• reading at receiver depends on the superposition of waves travelling directly to R and other waves that reach it after reflecting off the metal sheet

• as the metal sheet is moved away, the reading at the receiver continuously fluctuates from decreasing to a minimum and increasing to a maximum

• does not reach 0 as waves reflected at metal plate loose some energy through absorption - intensity decreases - amplitude decreases - combining waves do not have the same amplitude - not complete cancellation even when 180 degrees out of phase

• or if it is not a point source / receiver so imperfect calcellation