doPhysics · Optics
Interference and Young's Double Slit
Light is a wave. Huygens' wavelets and the laws of reflection and refraction, the Doppler shift of light and the expanding universe, coherent sources and the intensity of two waves, Young's double slit with its fringe width and intensity pattern, two colours and water, a slab over one slit, and white-light fringes.
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Why this chapter
Why this chapter exists
Rays explain mirrors, lenses and prisms, but they cannot explain darkness made of light. Shine one laser through two fine slits, and the wall shows a row of bright and dark bands. This chapter treats light as what it is, an electromagnetic wave. Huygens' wavelets rebuild the laws of reflection and refraction and show why a lens focuses. A moving source crowds its waves, and the shift of a galaxy's spectrum tells its speed. Two coherent sources add their waves into bright and dark fringes, and Young's double slit measures a wavelength with a ruler. A thin slab over one slit, and white light, complete the picture.
You will learn
By the end of this chapter you will
- Describe light as a transverse electromagnetic wave, and draw wavefronts and rays for point, line and distant sources.
- State Huygens' principle and use it to derive the laws of reflection and refraction, with \(\mu = v_1/v_2\).
- Use \(\Delta\lambda/\lambda = v/c\) for \(v \ll c\) to find the speed of a star, a galaxy or a car. Tell a red shift from a blue shift.
- Explain what makes two sources coherent, and why two bulbs never make fringes.
- Use \(I = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos\phi\), find \(I_{\max}\) and \(I_{\min}\), and show that energy is conserved on average.
- Derive the path difference yd/D in Young's experiment and the positions of the bright and dark fringes.
- Use \(\beta = \lambda D/d\) and the angular fringe width \(\lambda/d\), and measure a wavelength with the fringes.
- Find the intensity at any point, the coincidences of two wavelengths, and the fringes under water.
- Find the shift \((\mu - 1)tD/d\) of the pattern when a slab covers one slit, and explain white-light fringes.
Before you begin
Before you begin
From the chapters on waves, you should know \(\Delta\phi = 2\pi\Delta x/\lambda\), coherent sources and the sum of two waves, \(2A\cos(\Delta\phi/2)\). You should also know the wavefronts of a moving source. From the chapter on refraction, you should know \(\mu = c/v\), Snell's law and the optical path \(\mu d\). You should be comfortable with the cosine rule and small-angle approximations.
Lessons
1
Light as a Wave: Huygens' Principle
Light is a transverse electromagnetic wave. Wavefronts and rays, Huygens' secondary wavelets and their envelope, the laws of reflection and refraction built from wavefronts, \(\mu = v_1/v_2\), and why a convex lens turns a plane wavefront into a converging sphere.
Light is a waveWavefronts and raysHuygens principleReflection and refraction by wavefronts
2
The Doppler Effect for Light
A moving source crowds its wavefronts ahead and spreads them behind. For light only the relative speed matters, and \(\Delta\lambda/\lambda = v/c\) for \(v \ll c\). Red shift and blue shift, the expanding universe, and the traffic police speed gun.
A moving source crowds its wavefrontsRed shift and blue shiftThe expanding universe
3
Coherent Sources and the Intensity of Two Waves
A path difference \(\Delta x\) gives a phase difference \(\Delta\phi = 2\pi\Delta x/\lambda\). Why two torches never make fringes and what makes sources coherent. The resultant intensity \(I = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos\phi\), its maximum and minimum, and why the average is still \(I_1 + I_2\).
Phase difference from path differenceCoherent sourcesResultant intensity of two waves
4
Young's Double Slit Experiment
One slit lights two, and the two coherent slits paint equally spaced bright and dark fringes on a screen. The path difference \(yd/D\), the positions \(n\lambda D/d\) and \((2n - 1)\lambda D/2d\), the always-bright centre, the fringe width \(\beta = \lambda D/d\) and the angular fringe width \(\lambda/d\).
Youngs arrangementPath difference at a point on the screenPositions of bright and dark fringesFringe width
5
Reading the Fringe Pattern
The intensity across the screen, \(I = I_{\max}\cos^2(\pi y/\beta)\), and where it falls to half. Two wavelengths on one screen and where their bright fringes meet, the pattern under water, and what unequal slits, or one slit closed, do to the dark fringes.
Intensity across the screenTwo colours on one screenFringes under waterUnequal slits
6
A Slab in One Beam, and White Light
Optical path \(\mu t\) and the extra path \((\mu - 1)t\) of a thin slab. A slab over one slit moves the whole pattern by \((\mu - 1)tD/d\) towards that slit while \(\beta\) stays the same. White-light fringes, the colours that are bright or missing at a point, and why the white fringe finds the shift.
Optical pathA slab shifts the whole patternWhite light fringesFinding the shift with white light
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- angular fringe width
- The angle that one fringe subtends at the slits, \(\beta/D = \lambda/d\). It does not depend on the distance of the screen.
- blue shift
- A shortening of the wavelengths of light from a source moving towards the observer, which moves spectral lines towards the violet.
- central fringe
- The fringe at the point of zero path difference, bright for every wavelength. With white light it is the only white fringe.
- coherent sources
- Sources that emit waves of the same frequency with a phase difference that stays constant in time. Only they give a steady interference pattern.
- constructive interference
- Interference in which the waves arrive in phase, crest on crest, and their amplitudes add. It needs a path difference of a whole number of wavelengths.
- destructive interference
- Interference in which the waves arrive in opposite phase, crest on trough, and their amplitudes subtract. It needs a path difference of an odd number of half wavelengths.
- Doppler effect
- The change in the wavelength and frequency received when the source and the observer move towards or away from each other.
- electromagnetic wave
- A wave of electric and magnetic fields that vary together, at right angles to each other and to the direction of travel. Light is one; it needs no medium.
- fringe
- One of the bright or dark bands of an interference pattern.
- fringe width
- The distance \(\beta\) between two neighbouring bright, or two neighbouring dark, fringes. In Young's experiment \(\beta = \lambda D/d\).
- Hubble's law
- The speed at which a distant galaxy moves away is proportional to its distance, \(v = H_0 d\), with \(H_0\) about 70 km/s per megaparsec.
- Huygens' principle
- Every point of a wavefront is a source of secondary wavelets, and the new wavefront a time t later is their forward envelope.
- intensity
- The power carried by a wave through unit area at right angles to its travel, in W/m². It grows as the square of the amplitude, and it is what the eye sees as brightness.
- interference
- The combining of two or more waves by the principle of superposition. Coherent waves give a steady pattern of bright and dark places.
- optical path
- The distance \(\mu t\) that light travels in air in the time it takes to cross a thickness \(t\) of a medium of index \(\mu\).
- order
- The whole number \(n\) that counts a bright fringe from the centre, where the path difference is \(n\lambda\). The central fringe has order zero.
- path difference
- The difference \(\Delta x\) between the distances that two waves travel from their sources to a point.
- phase difference
- The difference between the phases of two waves at a point. For sources in phase, a path difference \(\Delta x\) gives \(\Delta\phi = 2\pi\Delta x/\lambda\).
- plane wavefront
- A flat wavefront, as from a very distant source such as the Sun or a star. Its rays are parallel.
- red shift
- A lengthening of the wavelengths of light from a source moving away from the observer, which moves spectral lines towards the red.
- secondary wavelet
- In Huygens' construction, the small wave that spreads from each point of a wavefront at the speed of the wave in that medium.
- wavefront
- A surface through all the points that a wave reaches in the same time, so that all of them vibrate in the same phase.