Lecture note · Fig. 3 · wave optics · wave–particle duality
Two slits, one wave: how a path difference draws light and dark
Numa Koudsie · for school students and first- and second-year undergraduates
01Bright and dark from one source
Stand between two loudspeakers that play the same steady tone and walk slowly across the room. The sound swells and fades, loud, quiet, loud again, although nothing about the speakers has changed.
The loud places are where a crest from one speaker arrives together with a crest from the other; the quiet places are where a crest from one arrives with a trough from the other. Which of the two happens depends only on how much further the sound has come from one speaker than from the other. In air at \(20\ {}^\circ\mathrm{C}\) sound travels at about \(343\ \mathrm{m/s},\) so a tone of \(343\ \mathrm{Hz}\) has a wavelength of \(1\ \mathrm{m},\) and the loud and quiet places lie far enough apart to walk through.
Light does the same on a scale more than a million times finer. The colours of a soap film come from two reflections, one at the front surface of the film and one at the back, which travel slightly different distances; where the extra distance suits one colour, that colour is reinforced and the others are weakened. The rainbow on a CD comes from its tracks, \(1.6\ \mu\mathrm{m}\) apart: the light scattered by neighbouring tracks adds up in step only in certain directions, and those directions are different for each colour.
Between 1801 and 1803 Thomas Young used effects of this kind to argue, in lectures to the Royal Society, that light is a wave. In one experiment he let sunlight in through a small hole in a window shutter and held in the beam a slip of card about \(0.85\ \mathrm{mm}\) wide. Inside the card’s shadow, where light passing the two edges overlaps, he saw a row of bands, with the middle of the shadow always bright. When he blocked the light passing one edge, the bands inside the shadow vanished. Two openings for the light give a pattern; one opening gives none. The double slit is the cleanest form of that experiment, and it is what the figure above draws.
One more step made the experiment famous. Make the light so faint that it reaches the screen one photon at a time, and each photon leaves a single dot. The dots land at random, yet after thousands of them the bands are back. Electrons, atoms and even large molecules do the same. Section 7 returns to this; first we work out where the bands are.
02The model
A flat wave of light of wavelength \(\lambda\) arrives from below and meets a wall with two narrow slits, \(S_1\) and \(S_2,\) a distance \(b\) apart. Each slit lets through a little of the wave and becomes the source of a new circular wave. The same crest reaches both slits at the same moment, so the two new waves start in step. We watch one point \(P\) beyond the wall (Figure 1).
Put the origin at the point \(M\) midway between the slits, with \(x\) along the wall and \(y\) pointing away from it. The point \(P\) lies at distance \(d\) from the wall, in a direction that makes the angle \(\alpha\) with the normal to the wall; as on the page, we take \(P\) on the side of \(S_1.\) With \(r\) the distance from \(M\) to \(P,\)
\[ \begin{aligned} S_1 &= \left(-\tfrac{b}{2},\ 0\right), \qquad S_2 = \left(\tfrac{b}{2},\ 0\right), \\ P &= (-d\tan\alpha,\ d) = (-r\sin\alpha,\ r\cos\alpha), \\ r &= \frac{d}{\cos\alpha} . \end{aligned} \tag{1} \]The model keeps only what decides bright or dark:
- The light has one wavelength and the two slits keep a fixed phase relation: the light is monochromatic and coherent.
- Each slit is a point that sends the same wave in every direction. Section 6 gives the slits a width.
- The two waves reach \(P\) with equal amplitudes; their weakening with distance is ignored.
- The wave is a single number at each point, a scalar wave; the polarisation of light is ignored.
A wave changes its phase by \(2\pi\) over each wavelength, so over a distance \(L\) it changes by \(kL,\) with the wavenumber \(k = 2\pi/\lambda.\) On the page every length is measured in pixels of the drawing: the slider Distance \(d\) runs from \(100\) to \(300\ \mathrm{px},\) Slit separation \(b\) from \(20\) to \(80\ \mathrm{px},\) Wavelength \(\lambda\) from \(8\) to \(40\ \mathrm{px},\) and Angle \(\alpha\) from \(0^\circ\) to \(30^\circ.\) Only the ratios \(b/\lambda\) and \(d/b\) and the angle matter, so the unit drops out of every result (Section 9).
03The exact path difference
Pythagoras’ theorem applied to the coordinates (1) gives the two path lengths:
\[ \begin{aligned} L_1 &= \sqrt{\left(d\tan\alpha - \tfrac{b}{2}\right)^2 + d^2}, \\ L_2 &= \sqrt{\left(d\tan\alpha + \tfrac{b}{2}\right)^2 + d^2} . \end{aligned} \tag{2} \]The path difference \(\delta s = L_2 - L_1\) is how much further the wave from \(S_2\) has travelled. Subtracting two square roots is awkward, but subtracting their squares is easy: in \(L_2^2 - L_1^2\) the \(d^2\) and the \(b^2/4\) cancel, and \((d\tan\alpha + \tfrac b2)^2 - (d\tan\alpha - \tfrac b2)^2 = 2bd\tan\alpha.\) Since \(L_2^2 - L_1^2 = (L_2 - L_1)(L_2 + L_1),\)
\[ \begin{aligned} \delta s = L_2 - L_1 &= \frac{L_2^2 - L_1^2}{L_1 + L_2} \\ &= \frac{2\,b\,d\tan\alpha}{L_1 + L_2} . \end{aligned} \tag{3} \]No approximation has been made. The result is zero for \(\alpha = 0,\) where \(P\) lies on the centre line and is equally far from both slits, and positive for \(\alpha > 0,\) where \(S_2\) is the further slit. At the page’s defaults, \(d = 200\ \mathrm{px},\) \(\alpha = 20^\circ\) and \(b = 40\ \mathrm{px},\) it gives \(L_1 = 206.85\ \mathrm{px},\) \(L_2 = 220.48\ \mathrm{px}\) and \(\delta s = 13.6277\ \mathrm{px};\) the readouts Path \(S_1P\), Path \(S_2P\) and \(\delta s\) exact show these numbers rounded.
The page computes \(\delta s\) from the right-hand side of (3), not as \(L_2 - L_1.\) Near the centre line the two lengths are almost equal, and subtracting two nearly equal numbers on a computer throws away significant digits. The form (3) has an exact zero at \(\alpha = 0\) and keeps its relative precision for tiny path differences. Over every combination of slider values the two ways differ by at most \(1.4\times10^{-13}\ \mathrm{px},\) so here the rearrangement is a good habit rather than a visible improvement.
04The far field
Seen from far away, the two slits merge into one point and the two paths run almost parallel.
When \(P\) is far from the slits compared with their separation, both paths are close to \(r,\) so \(L_1 + L_2 \approx 2r = 2d/\cos\alpha.\) Putting this into (3),
\[ \delta s \approx \frac{2\,b\,d\tan\alpha\,\cos\alpha}{2d} = b\sin\alpha . \tag{4} \]The right half of Figure 1 shows the same result without algebra. Drop a perpendicular from \(S_1\) onto the path from \(S_2.\) Beyond the foot of the perpendicular the two paths are equally long, so the extra length is the short side of the right-angled triangle with hypotenuse \(b;\) the perpendicular is turned from the wall by the same angle \(\alpha\) by which the paths are turned from the normal, so that side is \(b\sin\alpha.\) This is the far-field or Fraunhofer approximation, and the page shows it in the readout \(b\sin\alpha\), next to the exact value.
4.1The error shrinks with the square of the distance★
From (1), \(L_{1,2}^2 = r^2 \mp b\,r\sin\alpha + b^2/4,\) so \(L_{1,2} = r\sqrt{1+u}\) with \(u = \mp\varepsilon + \beta,\) \(\varepsilon = b\sin\alpha/r\) and \(\beta = b^2/(4r^2).\) Expand \(\sqrt{1+u} = 1 + \tfrac12 u - \tfrac18 u^2 + \tfrac1{16}u^3 - \dots\) Changing the sign of \(\varepsilon\) swaps the two paths, so only odd powers of \(\varepsilon\) survive in the difference; to third order in \(b/r\) they are \(r\left(\varepsilon - \tfrac12\varepsilon\beta + \tfrac18\varepsilon^3\right),\) which is
\[ \delta s = b\sin\alpha\left[\,1 - \frac{b^2\cos^2\alpha}{8\,r^2} + O\!\left(\frac{b^4}{r^4}\right)\right] . \tag{5} \]The shortcut (4) is always a little too long, by a fraction \(b^2\cos^2\alpha/(8r^2)\): double the distance and the error falls four times. At the defaults the shortcut gives \(13.6808\ \mathrm{px},\) \(0.39\,\%\) more than the exact \(13.6277\ \mathrm{px};\) the corrected form (5) gives \(13.6275\ \mathrm{px}.\) Measured in wavelengths, the error of the shortcut is \(0.0039\,\lambda,\) far too small to change bright into dark. Figure 2 follows the error as \(P\) moves in towards the slits.
Set Slit separation \(b\) to \(80\ \mathrm{px},\) Distance \(d\) to \(100\ \mathrm{px}\) and Angle \(\alpha\) to \(30^\circ.\) The readouts show \(\delta s\) exact \(= 38.27\ \mathrm{px}\) against \(b\sin\alpha = 40.00\ \mathrm{px}\): the shortcut is \(4.5\,\%\) too long. Now move Distance \(d\) to \(300\ \mathrm{px}.\) The exact value rises to \(39.80\ \mathrm{px},\) and the error falls from \(1.73\ \mathrm{px}\) to \(0.20\ \mathrm{px},\) an 8.7-fold drop. Equation (5) predicts a nine-fold drop for a three times larger distance; the difference comes from the terms of order \(b^4/r^4\) that (5) leaves out, which are not small when \(b/r = 0.69.\)
05Adding the two waves
A path difference is a length. Turning it into brightness takes one more step: add the two waves, then square.
At \(P\) each wave makes the field oscillate; for light the field is the electric field. Call its amplitude \(E_0\) and its angular frequency \(\omega = 2\pi f.\) The wave from \(S_j\) has travelled \(L_j,\) so it arrives with the phase \(kL_j\): \(E_j = E_0\cos(kL_j - \omega t).\) The two fields simply add, and the identity \(\cos A + \cos B = 2\cos\tfrac{A-B}{2}\cos\tfrac{A+B}{2}\) turns the sum into one oscillation,
\[ \begin{aligned} E &= E_0\cos(kL_1 - \omega t) + E_0\cos(kL_2 - \omega t) \\ &= 2E_0\cos\!\left(\tfrac{\Delta\varphi}{2}\right)\cos\!\left(k\bar L - \omega t\right), \\ \Delta\varphi &= k\,\delta s = \frac{2\pi\,\delta s}{\lambda} , \end{aligned} \tag{6} \]with \(\bar L = (L_1 + L_2)/2.\) The point \(P\) still oscillates at the frequency of the light, but with the amplitude \(2E_0\cos(\Delta\varphi/2),\) which depends only on the phase difference \(\Delta\varphi.\) No eye or camera can follow \(5\times10^{14}\) oscillations a second; they record the intensity, the time average of \(E^2.\) The average of \(\cos^2\) over a period is \(\tfrac12,\) so
\[ \begin{aligned} I &\propto \langle E^2\rangle = 2E_0^2\cos^2\frac{\Delta\varphi}{2}, \\ \frac{I}{I_0} &= \cos^2\!\left(\frac{\pi\,\delta s}{\lambda}\right) . \end{aligned} \tag{7} \]Here \(I_0\) is the intensity where the two waves arrive in step. It is four times what either slit gives alone, \(2E_0^2\) against \(\tfrac12E_0^2.\) Averaged across a fringe, \(\cos^2\) gives \(\tfrac12,\) so on average two slits give twice the light of one: no energy is created, it is moved from the dark places to the bright ones.
The point \(P\) is bright when the path difference is a whole number of wavelengths and dark when it is a whole number and a half:
\[ \begin{aligned} \delta s &= m\lambda & &\text{bright}, \\ \delta s &= \left(m + \tfrac12\right)\lambda & &\text{dark}, \end{aligned} \tag{8} \]Here \(m = 0, 1, 2, \dots\) is the order, the number of extra wavelengths. In the far field \(\delta s = b\sin\theta,\) so the bright directions are \(\sin\theta_m = m\lambda/b.\)
On a flat screen at a large distance \(D,\) the bright lines of small order sit at \(y_m = D\tan\theta_m \approx m\lambda D/b,\) so they are evenly spaced by
\[ \Delta y = \frac{\lambda D}{b} . \tag{9} \]Two approximations were made, in this order: first the far field, \(b \ll r,\) turned \(\delta s\) into \(b\sin\theta;\) only then did small angles allow \(\sin\theta \approx \tan\theta \approx \theta.\) For red light from a helium–neon laser, \(\lambda = 633\ \mathrm{nm},\) slits \(b = 0.25\ \mathrm{mm}\) apart and a screen at \(D = 1\ \mathrm{m},\) equation (9) gives \(\Delta y = 2.53\ \mathrm{mm}.\) At the page’s defaults it gives \(\lambda d/b = 67.5\ \mathrm{px}\) along the observation line.
The order of the steps matters: fields first, then square. Adding the two intensities instead would give \(\tfrac12E_0^2 + \tfrac12E_0^2 = E_0^2\) everywhere, with no fringes at all. That is what happens when the two slits are lit by two unrelated lamps: their phase difference wanders at random, the \(\cos\Delta\varphi\) term averages to zero, and only the sum of intensities is left. Figure 3 shows the addition of fields as arrows. In the language of complex numbers each wave is \(E_0e^{i(kL_j-\omega t)},\) the sum is \(E_0e^{i(kL_1-\omega t)}(1 + e^{i\Delta\varphi}),\) and \(|1 + e^{i\Delta\varphi}|^2 = 2 + 2\cos\Delta\varphi = 4\cos^2(\Delta\varphi/2),\) the same result.
Keep Slit separation \(b = 40\ \mathrm{px},\) Wavelength \(\lambda = 13.5\ \mathrm{px}\) and Distance \(d = 200\ \mathrm{px}.\) Step Angle \(\alpha\) up from \(0^\circ\) and watch the readout \(\delta s/\lambda\) and the verdict under the readouts. Between \(9.5^\circ\) and \(10.0^\circ\) the readout passes from 0.49 to 0.51 and the verdict says Destructive; between \(19.5^\circ\) and \(20.0^\circ\) it passes from 0.99 to 1.01 and the verdict says Constructive · bright fringe, m = 1. In the far field, (8) puts these fringes at \(9.72^\circ\) and \(19.72^\circ;\) the exact path difference at \(d = 200\ \mathrm{px}\) puts them at \(9.76^\circ\) and \(19.81^\circ.\) Then set \(d = 100\ \mathrm{px}\) and find the bright fringe again: it has moved out to \(20.04^\circ.\)
06Slits with a width★
Real slits are not points. A slit of width \(w\) is a row of point sources side by side, and the waves from all of them must be added, just as the two slits were. Measure the position \(x'\) across one slit from its centre. Seen from a far-away point in the direction \(\theta,\) the wave from \(x'\) has a path shorter by \(x'\sin\theta,\) by the argument of (4). Adding all these waves with the complex notation of Section 5,
\[ \begin{aligned} E(\theta) &\propto \int_{-w/2}^{w/2} e^{-ikx'\sin\theta}\,\mathrm{d}x' = w\,\frac{\sin\beta}{\beta}, \\ \beta &= \frac{\pi w\sin\theta}{\lambda} . \end{aligned} \tag{10} \]Both slits carry the same factor, so it comes out of the two-slit sum, which is unchanged; the intensity of a pair of slits of width \(w\) is the two-slit pattern (7) multiplied by a single-slit factor:
\[ \frac{I(\theta)}{I_0} = \cos^2\!\left(\frac{\pi b\sin\theta}{\lambda}\right)\left(\frac{\sin\beta}{\beta}\right)^{2} . \tag{11} \]The second factor is the single-slit envelope. It equals 1 straight ahead and falls to zero where \(w\sin\theta = n\lambda,\) \(n = 1, 2, \dots\) If a zero of the envelope falls on a bright fringe, that fringe is missing. Figure 4 shows this for slits \(20\ \mathrm{px}\) wide at the page’s defaults: the first zero of the envelope, \(\sin\theta = \lambda/w = 0.675,\) coincides with the second bright fringe, \(\sin\theta_2 = 2\lambda/b = 0.675,\) at \(\theta = 42.45^\circ.\) Whenever \(b = 2w,\) every nonzero even order disappears; the central maximum remains because \(\sin\beta/\beta\to1\) at \(\beta=0\).
The page draws each slit as a gap of \(8\ \mathrm{px}\) but treats it as a point. A gap that narrow, narrower than the wavelength, has no zero in its envelope at all: by (11) the envelope falls only to 0.74 at \(30^\circ\) and to 0.26 at \(90^\circ,\) and it would lower the first bright fringe from 1 to 0.88. Within the page’s own plot, which ends at \(30^\circ,\) point slits are a fair picture of the drawing.
07One particle at a time
Turn the light down far enough and the smooth pattern breaks up into dots.
Light delivers its energy in grains, photons, each carrying the energy \(hf,\) where \(f\) is the frequency and \(h = 6.626\times10^{-34}\ \mathrm{J\,s}\) is Planck’s constant. A sensitive camera behind the slits records each photon as one dot at one place. Where the next dot will land cannot be predicted. Where dots are likely to land can: the probability of a dot near a direction \(\theta\) is proportional to the intensity (7) there,
\[ p(\theta)\,\mathrm{d}\theta \propto \cos^2\!\left(\frac{\pi b\sin\theta}{\lambda}\right)\mathrm{d}\theta . \tag{12} \]Figure 5 draws dots at random according to this rule. Twenty dots look like noise; two thousand show the fringes. Of those 2000 dots, 140 land within \(1^\circ\) of the centre and only 3 within \(1^\circ\) of the two dark directions at \(\pm 9.72^\circ;\) the rule (12) expects 131 and 2.2.
The pattern needs both slits for every single photon. Close one slit, or place a detector that records which slit each photon passed, and the cross term \(\cos\Delta\varphi\) disappears: the dots then pile up as the sum of two single-slit patterns, without fringes. So the photons are not a crowd interfering with one another. Each photon, on its own, has two ways of reaching the screen, and as long as nothing in the apparatus records which way it took, the two ways add the way waves add. Quantum mechanics gives each way a complex number, a probability amplitude, and the probability of a dot at \(\theta\) is the squared modulus of their sum. The field \(E\) of Section 5 predicts the same fringes, because its two parts add with the same path difference and the same cross term; but \(E\) is not that amplitude. It is the classical account of light made of very many photons, and it gets the single-photon pattern right here because, for two slits, both sums have the same shape.
One more care with the dots. Figure 5 draws them with a probability per unit angle, the rule (12). A flat screen at distance \(D\) counts dots per unit length instead, and that is a different measure: a step \(\mathrm{d}\theta\) in angle covers a step \(\mathrm{d}y = D\sec^2\theta\,\mathrm{d}\theta\) on the screen, longer out to the sides, so the density per unit length is \(p(\theta)/(D\sec^2\theta)\), smaller than straight ahead by a factor \(3/4\) at \(30^\circ\). The animation shows how a fringe pattern builds up dot by dot; it is not a full model of a counting experiment.
Matter behaves the same way. In 1924 Louis de Broglie proposed that a particle of momentum \(p\) has the wavelength \(\lambda = h/p.\) An electron accelerated from rest through a voltage \(V\) has \(p = \sqrt{2m_e eV},\) where \(m_e\) is its mass and \(e\) its charge, so
\[ \begin{aligned} \lambda &= \frac{h}{\sqrt{2\,m_e\,e\,V}}, \\ V = 100\ \mathrm{V}:\quad \lambda &= 0.123\ \mathrm{nm} . \end{aligned} \tag{13} \]That is about 4500 times shorter than green light of \(550\ \mathrm{nm},\) so the slits for electrons must be correspondingly finer. With suitable slits, electrons, neutrons, atoms and molecules of sixty carbon atoms all show the same fringes, arriving one at a time and building the pattern of Figure 5 dot by dot.
08How the page computes
Nothing on this page is stepped forward in time. Every quantity has a closed formula and is recomputed whenever a slider moves:
- The slits and \(P\) are placed by (1); \(L_1\) and \(L_2\) are the distances from the slits to \(P;\) \(\delta s\) comes from (3). The point \(Q\) of Figure 1 is \(P + L_1\hat u,\) with \(\hat u\) the unit vector from \(P\) towards \(S_2.\)
- The readouts show \(L_1,\) \(L_2,\) the exact \(\delta s,\) the shortcut \(b\sin\alpha,\) the order \(\delta s/\lambda\) and the phase \(\Delta\varphi\) in units of \(\pi.\)
- The verdict uses the fractional part \(f = \delta s/\lambda - \lfloor\delta s/\lambda\rfloor\): Constructive when \(f < 0.08\) or \(f > 0.92,\) Destructive when \(|f - \tfrac12| < 0.08,\) otherwise Partial with \(I/I_0\) from (7). A tolerance of 0.08 wavelengths means that Constructive covers intensities from 0.94 to 1 and Destructive from 0 to 0.06.
- The intensity plot under the figure samples 301 directions from \(0^\circ\) to \(30^\circ,\) one every \(0.1^\circ.\) The burgundy curve is the far-field law \(\cos^2(\pi b\sin\theta/\lambda).\) The dashed curve uses the exact path difference (3) for the point of the observation line in the direction \(\theta,\) at \(x = d\tan\theta;\) where the two curves part, the far field is failing. Even for the finest fringes the sliders allow, \(b = 80\ \mathrm{px}\) and \(\lambda = 8\ \mathrm{px},\) one fringe spans 57 samples.
- The calculations use JavaScript’s double-precision numbers, about 16 significant digits; Section 3 shows that this is far more than the drawing needs.
Only the wavelets move. At each frame of the browser, the phase of the animation advances by \(0.8\,\Delta t\) periods, with \(\Delta t\) the time since the previous frame, and the arcs are redrawn; the geometry and the readouts do not change. The page’s data panel stores every number named here at the line that computes it, including the direct difference \(L_2 - L_1\) and the length of the drawn bar \(S_2Q,\) so the three ways of getting \(\delta s\) can be compared side by side.
09From lengths to pixels
The figure is a vector drawing, \(600 \times 440\) units, that scales with the page, so lines stay sharp at any size; positions are written to two decimals. The wall lies at \(y = 370\) with the slit midpoint at \(x = 300,\) and screen \(y\) points down, so \(P\) is drawn at \((300 - d\tan\alpha,\ 370 - d)\): equation (1) flipped and shifted. The wall is a 5-unit-thick line with an \(8\)-unit gap at each slit. The two paths are drawn in ink, the bar \(S_2Q\) in burgundy, the construction dashed; the dashed arc and the bar appear only when \(\delta s > 0.05\ \mathrm{px},\) and the arc that marks \(\alpha\) only when \(\alpha > 0.2^\circ.\)
The rose arcs are the wavelets: semicircles of radius \(r_n = (n + \phi)\lambda\) about each slit, out to \(430\ \mathrm{px},\) where \(\phi\) is the animation phase, a fraction of a period. Their opacity falls as \(0.55\,(1 - r/430)\) to suggest the weakening of each wavelet. Below the wall, straight rose lines one wavelength apart are the incoming plane wave, moving in step with the arcs at \(0.8\lambda\) per second, \(10.8\ \mathrm{px/s}\) at the defaults. Real green light oscillates \(5.5\times10^{14}\) times a second, so the animation is slowed down by a factor of about \(7\times10^{14}.\)
Because only ratios matter, the pixel is no real unit of length. The default \(b = 40\ \mathrm{px}\) with \(\lambda = 13.5\ \mathrm{px}\) is a slit pair 2.96 wavelengths apart, which for red laser light would mean slits \(1.88\ \mu\mathrm{m}\) apart. Laboratory slits are usually wider apart, \(0.25\ \mathrm{mm}\) or 395 wavelengths, and their first bright fringe lies only \(0.145^\circ\) from the centre. The page enlarges the wavelength so that the angles can be seen. In the intensity plot, \(\theta\) runs from \(0^\circ\) to \(30^\circ\) over 500 units, and \(I/I_0\) from 0 to 1 over 118 units; the dot marks the exact intensity at \(\alpha,\) and the labels \(m = 0, 1, \dots\) mark the far-field bright directions.
10What the model leaves out
- Slits have a width, which multiplies the fringes by the envelope of Section 6.
- The amplitudes are taken as equal. In the plane of the drawing, the wave from a long slit weakens as \(1/\sqrt L.\) At the dark fringe for the defaults, \(L_1 = 200.52\ \mathrm{px}\) and \(L_2 = 207.27\ \mathrm{px},\) so the amplitudes differ by \(1.6\,\%\) and the dark fringe keeps \(7\times10^{-5}\) of the peak intensity: invisible.
- Polarisation and the vector nature of light are ignored. For small angles this changes nothing.
- The light is perfectly coherent and of one colour. White light, as in Young’s experiment, gives a white central fringe and coloured fringes that wash out after a few orders, because each colour has its own spacing (9).
- The drawing is a two-dimensional slice through slits that are long in the third direction.
- The speed of the wavelets is illustrative only.
10.1Further reading
- E. Hecht, Optics, 5th edition, Pearson, 2017: chapters 9 (interference) and 10 (diffraction).
- D. Halliday, R. Resnick and J. Walker, Fundamentals of Physics, 10th edition, Wiley, 2014: chapters 35 (interference) and 36 (diffraction).
- R. P. Feynman, R. B. Leighton and M. Sands, The Feynman Lectures on Physics, Addison-Wesley, 1963–1965: volume I, chapters 29 (interference) and 30 (diffraction); volume III, chapter 1 (quantum behaviour).