Interference

Coherent sources emit light waves of the same frequency (and wavelength) with a constant phase difference that does not change with time.

They are necessary because, if the phase difference between the sources fluctuates randomly (non-coherent sources), the positions of the bright and dark regions keep shifting very rapidly over the screen. Averaged over the observation time, this gives uniform illumination instead of distinct bright and dark fringes.

The path difference \(\Delta x\) is the difference between the distances travelled by two light waves from their sources to a common point where they superpose (for example a point on the screen). It decides the relative phase with which the waves meet.

Constructive interference: the waves meet in phase and reinforce each other, giving maximum intensity (a bright fringe). This happens when the path difference is a whole number of wavelengths:

Destructive interference: the waves meet out of phase and cancel each other, giving minimum intensity (a dark fringe). This happens when the path difference is an odd multiple of half a wavelength:

When two or more waves travel through the same medium at the same time, the net displacement of the medium at any point and instant is the vector sum of the displacements produced by the individual waves at that point:

For a stable pattern that can be observed over time (sustained interference):

A crest of one wave exactly overlaps a trough of the other, so the interference is destructive. If the amplitudes are equal the waves cancel completely and a dark fringe forms at that point.

Sustained interference is an interference pattern (bright and dark bands) that stays fixed in position on the screen without fluctuating, so it can be observed easily by the eye.

The necessary conditions are those listed in Question 2: the sources must be strictly coherent and continuous, emit light of the same frequency (monochromatic), preferably have the same amplitude, be placed close together, and have the same state of polarization.

Interference is a distribution of varying light energy in space. A screen intercepts the light and scatters it towards our eyes, so the variation in intensity (the bright and dark fringes) can be seen and measured.

Two independent sources (such as two separate bulbs) can never be coherent. Their atoms emit light in random, uncoordinated bursts of about \(10^{-8}\ \mathrm{s}\), so the phase difference between the sources changes millions of times a second. This washes out any pattern and gives uniform illumination.

Constructive interference: overlapping waves are in phase, their displacements add and the resultant has a larger amplitude and maximum intensity (brightness).

Destructive interference: overlapping waves are out of phase, their displacements subtract and the resultant has minimum or zero amplitude and minimum intensity (darkness).

Light from an ordinary bulb comes from billions of independent atoms that emit spontaneously. Each emission is a short wave train with a completely random phase. The phase difference between two independent bulbs therefore varies randomly within tiny fractions of a second, so they cannot keep the constant phase difference required for coherence.

Figure: The completed diagram of Young’s double-slit experiment

Light from the coherent sources \(S_1\) and \(S_2\) reaches a point \(P\) on the screen. Constructive interference (a bright fringe) occurs when the path difference is a whole number of wavelengths:

The path difference must be an odd multiple of half a wavelength:

Let \(S_1\) and \(S_2\) be two coherent sources separated by a distance \(d\), and let \(D\) be the distance from the slits to the screen. For a point \(P\) at a distance \(y\) from the central point \(O\), the path difference is \(\Delta x=S_2P-S_1P\).

Drawing a perpendicular from \(S_1\) to \(S_2P\), for small angles \(\theta\):

From the large triangle formed by the slits, the screen and the central axis, \(\tan\theta=\dfrac yD\). For small angles \(\sin\theta\approx\tan\theta\approx\dfrac yD\), so

For the \(n\)th bright fringe, \(\dfrac{y_nd}{D}=n\lambda\):

The fringe width is the distance between two consecutive bright fringes:

Position of the central fringe: the central fringe corresponds to zero path difference (\(n=0\), \(y=0\)). It lies neither upwards nor downwards; it lies exactly at the geometric centre \(O\) of the screen, on the perpendicular bisector of \(S_1S_2\).

Interference of light is the phenomenon in which two or more light waves from coherent sources superpose to form a resultant wave, so that the light energy is redistributed in space. It gives alternating regions of maximum intensity (bright fringes) and minimum intensity (dark fringes).

Coherent sources emit light waves of the same frequency that keep a strictly constant phase difference over time.

Example: the two slits \(S_1\) and \(S_2\) illuminated by the same primary source, or two beams split from a single laser by a beam splitter.

The experiment is based on division of the wavefront and superposition. A single wavefront is divided into two coherent wavefronts by two closely spaced slits. These act as coherent sources, and their superposition produces the interference pattern.

Conditions: monochromatic light, strictly coherent sources, narrow slits, small slit separation \(d\), large screen distance \(D\), and equal amplitudes for perfect contrast.

\(\beta=\dfrac{\lambda D}{d}\). If \(d\) increases, \(\beta\) decreases: the fringes become thinner and more crowded.

\(D\) increases and \(\beta\propto D\), so the fringes become wider and farther apart.

In a medium of refractive index \(\mu>1\) the wavelength becomes \(\lambda’=\dfrac{\lambda}{\mu}\). Since \(\beta\propto\lambda\), the fringe width decreases.

At a point \(P\) at a distance \(y\) from the centre, the path difference is \(\Delta x=\dfrac{yd}{D}\). For consecutive bright fringes \(n\) and \(n+1\):

So \(\beta\propto\lambda\) and \(\beta\propto\dfrac1d\).

Path difference is the difference between the distances travelled by two waves from their respective sources before they meet at a common point of superposition.

The optical path is the distance light would travel in vacuum in the same time that it takes to travel a given geometrical distance \(x\) in a medium of refractive index \(\mu\):

The geometrical path is just the physical distance \(x\), whatever the medium. The optical path takes into account the slower speed of light in a denser medium.

The conditions are: coherent sources, monochromatic light, the same polarization and nearly equal amplitudes.

Temporal coherence ensures that the phase of a wave at a given point does not jump randomly with time, so the fringes stay stable.

Spatial coherence ensures that different points of the wavefronts from the sources keep a fixed phase relation, so a clear fringe pattern extends across the screen.

If the frequencies differ, the phase difference between the waves changes steadily with time. The positions of the bright and dark fringes then shift rapidly across the screen (like beats), and to the eye the screen looks uniformly illuminated.

If the amplitudes are unequal (\(a_1\ne a_2\)), the intensity at a dark fringe, \(I_{\min}\propto(a_1-a_2)^2\), is no longer zero. The dark fringes receive some light, so the visibility (contrast) of the fringes falls, although their positions stay the same.

For bright fringes the waves must arrive in phase, so the path difference is a whole number of wavelengths:

From the geometry of the double slit, \(\Delta x\approx\dfrac{yd}{D}\). Equating:

where \(y_n\) is the position of the \(n\)th bright fringe.

In glass (refractive index \(\mu>1\)) the wavelength changes from \(\lambda\) to \(\lambda’=\dfrac\lambda\mu\), so

All the fringes move closer to the centre by a factor \(\dfrac1\mu\).

Fringe visibility (contrast) measures how clearly the fringes can be distinguished:

where \(I_{\max}\) is the intensity of the bright fringes and \(I_{\min}\) that of the dark fringes. High visibility means crisp dark minima and bright maxima.

A dark fringe appears when the interference is destructive, that is, when the path difference is an odd multiple of half a wavelength:

For perfect darkness (\(I_{\min}=0\)), the amplitudes of the two waves must be equal (\(a_1=a_2\)).

Monochromatic light has a single wavelength. Since the fringe positions depend on the wavelength (\(y_n\propto\lambda\)), a single wavelength gives sharp, well-defined fringes.

With white light, every wavelength forms its own pattern with a different fringe width. The central fringe is white (all wavelengths interfere constructively at \(\Delta x=0\)), but the fringes farther out turn into coloured bands and, after a few orders, the overlapping maxima and minima wash out into nearly uniform white light.

An extended source emits light from many independent points, which would give overlapping, uncoordinated patterns. A narrow single slit acts as a point (line) source. By Huygens’ principle, the cylindrical wavefronts from this slit reach \(S_1\) and \(S_2\) together, so \(S_1\) and \(S_2\) are derived from the same wavefront. This keeps them coherent and locks their relative phase.

(i) Fringe width

(ii) Position of the 4th bright fringe

Consider slits \(S_1\) and \(S_2\) separated by \(d\) and a screen at a distance \(D\). Let \(O\) be the central point of the screen. For a point \(P\) at a distance \(y\) from \(O\), the path difference is \(\Delta x=S_2P-S_1P\).

Drawing a perpendicular from \(S_1\) to \(S_2P\), for \(D\gg d\) we have \(\Delta x=d\sin\theta\). Also \(\tan\theta=\dfrac yD\), and for small angles \(\sin\theta\approx\tan\theta\), so

For the \(n\)th and \((n+1)\)th bright fringes:

Bright fringes form where the waves from the two slits arrive in phase (\(\Delta x=n\lambda\)) and interfere constructively. Dark fringes form where they arrive out of phase (\(\Delta x=(n+\tfrac12)\lambda\)) and interfere destructively.

Since \(\beta=\dfrac{\lambda D}{d}\), increasing the slit distance \(d\) makes the fringes narrower (tighter). Changing to a smaller wavelength decreases the fringe width in proportion.

White light gives a white central fringe surrounded by coloured fringes. White light is a mixture of all visible wavelengths, each forming its own pattern of width \(\beta\propto\lambda\): red has the widest fringes and violet the narrowest. These slightly shifted patterns overlap and wash out into white light after a few orders.

Implication: the sources are still coherent, but because the light is not monochromatic, useful fringes are seen only near the centre.

Coherent sources (same frequency, constant phase), the same state of polarization, monochromatic light (to avoid overlapping coloured patterns), \(D\gg d\) (to make the fringes wide enough to resolve), and equal amplitudes (to make the dark fringes perfectly dark and the contrast maximum).

The condition is

where \(\Delta x\) is the path difference. It fixes exactly where the crests of one wave overlap the crests of the other, giving a local maximum of energy on the screen (a bright fringe).

The path difference at a vertical position \(y\) is \(\dfrac{yd}{x}\). For consecutive bright fringes \(n\) and \(n+1\):

The path difference at a height \(y\) on the screen is \(\Delta x\approx\dfrac{yd}{D}\). The \(n\)th maximum is at \(y_n=\dfrac{n\lambda D}{d}\) and the next at \(y_{n+1}=\dfrac{(n+1)\lambda D}{d}\). Hence

The fringe width is directly proportional to the wavelength \(\lambda\) and the screen distance \(D\), and inversely proportional to the slit separation \(d\).

Since \(\beta=\dfrac{\lambda D}{d}\propto\dfrac1d\), doubling \(d\) gives

The fringe width is halved, so the pattern becomes twice as dense.

Constructive: \(\Delta x=n\lambda\). The crests of both waves arrive together, the amplitudes add and a region of high intensity (a bright fringe) forms.

Destructive: \(\Delta x=\left(n+\tfrac12\right)\lambda\). The crest of one wave arrives with the trough of the other, the amplitudes cancel and a region of nearly zero intensity (a dark fringe) forms.

Interference is the redistribution of energy in space caused by the superposition of two or more coherent light waves.

Conditions: coherent sources (the most important), monochromatic light, the same polarization and suitably narrow slits.

Difference: constructive interference gives maximum intensity (bright bands) by in-phase addition, \(\Delta x=n\lambda\); destructive interference gives minimum intensity (dark bands) by out-of-phase cancellation, \(\Delta x=(n+\tfrac12)\lambda\).

Because \(\beta\propto\lambda\), increasing the wavelength from \(500\ \mathrm{nm}\) to \(600\ \mathrm{nm}\) (a 20% increase) increases the fringe width by the same proportion, from \(3.75\ \mathrm{mm}\) to \(4.5\ \mathrm{mm}\).

The fringe width \(\beta\) is the distance between the centres of any two consecutive bright fringes or any two consecutive dark fringes.

The path difference is \(\Delta x=\dfrac{yd}{D}\). For maximum intensity \(\dfrac{y_nd}{D}=n\lambda\), so \(y_n=\dfrac{n\lambda D}{d}\), and

Factors: \(\beta\) increases directly with the wavelength \(\lambda\) and the screen distance \(D\), and decreases as the slit separation \(d\) increases.

Since \(y_n\propto\lambda\), a larger wavelength pushes all the fringes outwards, farther from the central maximum.

Interference is the superposition of waves that modifies the intensity. It needs coherent sources and monochromatic light. Constructive interference adds the amplitudes to give maximum brightness (\(\Delta x=n\lambda\)); destructive interference subtracts them to give darkness (\(\Delta x=(n+\tfrac12)\lambda\)). See Question 12 for details.

\(\beta=\dfrac{\lambda D}{d}\). If \(d\) is halved to \(\dfrac d2\):

The pattern expands and the fringe width becomes twice as large.

The distance between any two consecutive dark fringes equals one fringe width \(\beta\).

The separation between adjacent maxima is the fringe width \(\beta=\dfrac{\lambda D}{d}\):

The point P is the central maximum on the axis, symmetrical between the slits. The path difference for waves reaching P from the two slits is zero, so

(For any other bright fringe the phase difference is \(2n\pi\).)

Slits far apart: the fringe width is inversely proportional to the slit separation (\(\beta\propto\dfrac1d\)). When the slits are far apart, the fringes become extremely narrow and crowded, beyond the resolving power of the eye or the detector, so the pattern blurs into a continuous spread of light.

Incoherent source: the phase difference changes rapidly and randomly, and it fixes the positions of the maxima and minima. The fringes therefore shift millions of times a second, and averaged over the observation time the pattern is washed out completely.