Difficulty: Intermediate | Prerequisites: Parts 1 and 2 of these notes (wave optics fundamentals, double-slit interference, diffraction gratings).
The previous sections dealt with interference from multiple separate slits. Here the focus shifts to what happens when light passes through a single opening: the slit itself acts as a source of many wavelets that interfere with each other, producing a characteristic pattern of a broad central bright band flanked by weaker secondary maxima. Circular apertures (round holes, camera lenses, telescope mirrors, the pupil of your eye) produce the same effect in a circular pattern, and this ultimately limits how fine a detail any optical instrument can resolve. These concepts tie together every earlier topic in the chapter and connect directly to the practical limits of telescopes, microscopes, and cameras.
A single narrow slit produces a diffraction pattern with a wide, bright central maximum and progressively weaker secondary maxima on either side. A circular aperture produces the same kind of pattern but with circular symmetry. The narrower the slit (or the smaller the aperture), the wider the diffraction pattern spreads. The width of the central maximum and the angle to the first dark fringe are the key quantities to calculate.
Huygens' principle
Every point on a wavefront acts as the source of a tiny spherical wavelet that spreads out at the wave speed. The new wavefront, a moment later, is the surface tangent to all those wavelets. This principle explains why waves bend around obstacles and spread through openings.
Single-slit diffraction
The spreading and interference pattern produced when light passes through a single narrow slit of width a. In simple terms, the slit itself generates many wavelets that interfere with each other, creating a pattern of bright and dark bands.
Central maximum
The broad, bright band at the centre of a single-slit or circular-aperture diffraction pattern. It is the brightest and widest feature, and its width is defined as the distance between the first dark fringes on either side.
Secondary maxima
The weaker bright bands that appear on either side of the central maximum. Each successive secondary maximum is dimmer and narrower than the one before it.
Circular-aperture diffraction
The diffraction pattern produced when light passes through a round opening (a hole, a lens, or a pupil). It consists of a central bright disc (the Airy disc) surrounded by concentric bright and dark rings.
Airy disc
The central bright spot in a circular diffraction pattern. Its angular radius to the first dark ring is theta_1 = 1.22 lambda / D. Think of it as the smallest spot of light that any circular lens or mirror can produce, no matter how perfectly it is made.
Huygens' principle applied
Each point on the wavefront arriving at the slit acts as a source of a spherical wavelet. All these wavelets overlap and interfere on the far side of the slit.
At the centre of the screen (theta = 0), all wavelets travel the same distance and arrive in phase, producing the central maximum.
Dark fringes
To find the dark fringes, pair every point on the wavefront with another point exactly a/2 away (half the slit width). Each pair produces destructive interference at the same angle.
Condition for dark fringes: a sin(theta_p) = p lambda, where p = 1, 2, 3, ... Note: p = 0 is excluded because theta = 0 is the central maximum, not a dark fringe.
Positions of dark fringes on a screen at distance L: y_p = p lambda L / a.
Central maximum width
The central maximum extends between the p = 1 dark fringes on either side. Its width is w = 2 lambda L / a.
A narrower slit (smaller a) produces a wider central maximum. This is the opposite of what your intuition might suggest: squeezing light through a smaller gap makes it spread out more.
Secondary maxima
Between the dark fringes are weaker bright bands (secondary maxima). Each one is dimmer and narrower than the central maximum.
Any wave passing through an opening of any shape will diffract. A circular opening produces a circular diffraction pattern: a central bright disc surrounded by concentric dark and bright rings.
First minimum
The angle to the first dark ring (first minimum) is: theta_1 = 1.22 lambda / D, where D is the diameter of the circular aperture. This uses the small-angle approximation (theta in radians).
The factor 1.22 comes from the circular geometry. For a slit the corresponding factor is 1; for a circle it is 1.22.
Width of the central maximum
The width of the central bright disc on a screen at distance L from the aperture is: w = 2.44 lambda L / D.
This is twice the distance from the centre to the first minimum (the "2.44" is simply 2 x 1.22).
Key behaviours
A larger aperture D produces a narrower, tighter diffraction pattern.
A smaller aperture produces a wider pattern: the light spreads out more.
The diffraction pattern grows linearly with the distance L to the screen.
This is why telescopes need large mirrors or lenses: a bigger aperture gives a smaller Airy disc, which means finer angular resolution.
a\sin\theta_p = p\lambda \quad (\text{single-slit dark fringes, } p = 1, 2, 3, \ldots)y_p = \frac{p\lambda L}{a} \quad (\text{positions of single-slit dark fringes})w = \frac{2\lambda L}{a} \quad (\text{width of central maximum, single slit})\theta_1 = \frac{1.22\lambda}{D} \quad (\text{first minimum, circular aperture})w = \frac{2.44\lambda L}{D} \quad (\text{width of central maximum, circular aperture})Circular-aperture diffraction sets the fundamental resolution limit of every optical instrument. The pupil of your eye, a camera lens, and a telescope mirror are all circular apertures, and the Airy disc determines the smallest angular separation at which two point sources (two stars, two features on a chip) can be told apart. This is why the Hubble Space Telescope has a 2.4-metre mirror and the James Webb Space Telescope has a 6.5-metre mirror: larger aperture, smaller Airy disc, finer detail.
Students often confuse single-slit dark fringes with double-slit bright fringes. In single-slit diffraction, the formula a sin(theta) = p lambda gives the positions of the dark fringes, not the bright ones. In double-slit interference, d sin(theta) = m lambda gives the bright fringes.
A common error is including p = 0 in the single-slit dark-fringe formula. p = 0 would mean theta = 0, which is the central maximum (the brightest point), not a dark fringe.
Students sometimes think a narrower slit produces a narrower diffraction pattern. The opposite is true: a narrower slit causes more spreading.
For circular apertures, the factor 1.22 is frequently forgotten or confused with 1. It applies specifically to circular openings; rectangular slits use the factor 1.
Single-slit problems typically ask you to find the width of the central maximum or the position of a specific dark fringe. Make sure you use a (slit width), not d (slit spacing).
Circular-aperture questions often ask for the angular size of the Airy disc or the width of the central maximum on a screen. Remember the 1.22 factor.
Conceptual questions may ask you to explain why a narrower slit produces a wider pattern, or to compare single-slit diffraction with double-slit interference.
Be ready for problems that combine single-slit diffraction with double-slit interference: in a real double-slit experiment, each slit has a finite width, so the overall pattern is a double-slit interference pattern modulated by a single-slit diffraction envelope.
True or false: In single-slit diffraction, p = 0 corresponds to a dark fringe. (False, p = 0 is the central maximum)
Fill in the blank: Halving the slit width a causes the central maximum width to ___. (Double)
True or false: The factor 1.22 in the circular-aperture formula comes from the circular geometry. (True)
Fill in the blank: The width of the central maximum for a circular aperture is w = ___. (2.44 lambda L / D)
True or false: A larger telescope mirror produces a larger Airy disc. (False, a larger mirror produces a smaller Airy disc)
Q: Light of wavelength 500 nm passes through a single slit of width 0.10 mm. A screen is 2.0 m behind the slit. What is the width of the central maximum?
A: w = 2 lambda L / a = 2(500 x 10^-9)(2.0) / (0.10 x 10^-3) = 0.020 m = 2.0 cm.
Q: If the slit in the previous problem is narrowed to 0.050 mm, what happens to the central maximum?
A: w = 2(500 x 10^-9)(2.0) / (0.050 x 10^-3) = 0.040 m = 4.0 cm. The central maximum doubles in width because halving the slit width doubles w.
Q: A circular aperture of diameter 1.0 cm is illuminated with light of wavelength 600 nm. What is the angle (in radians) to the first dark ring?
A: theta_1 = 1.22 lambda / D = 1.22(600 x 10^-9) / (1.0 x 10^-2) = 7.3 x 10^-5 rad.
Q: For the same aperture, what is the width of the central maximum on a screen 5.0 m away?
A: w = 2.44 lambda L / D = 2.44(600 x 10^-9)(5.0) / (1.0 x 10^-2) = 7.3 x 10^-4 m = 0.73 mm.
Q: Why do stars viewed through a small telescope appear as discs rather than points?
A: The telescope's circular aperture diffracts the incoming light, producing an Airy disc. The angular size of this disc (theta_1 = 1.22 lambda / D) is larger than the true angular size of the star, so what you see is the diffraction pattern, not the star itself. A larger telescope aperture shrinks the Airy disc and brings the image closer to a true point.
Single-slit diffraction explains the finite width of each bright fringe in a diffraction-grating pattern (Part 2), since each slit in the grating is itself a single slit. Circular-aperture diffraction connects directly to the Rayleigh criterion for resolving two point sources, which appears in optics and astronomy courses. Huygens' principle, introduced here to explain single-slit diffraction, is also the basis for understanding refraction (Snell's law) from a wave perspective.
Single-slit diffraction, single slit, Huygens principle, Huygens' construction, wavelet, central maximum, secondary maxima, dark fringes single slit, slit width a, p = 1 2 3, circular aperture diffraction, Airy disc, Airy pattern, 1.22 lambda over D, angular resolution, Rayleigh criterion, diffraction limit, telescope resolution, camera resolution, aperture diameter, w = 2 lambda L / a, w = 2.44 lambda L / D, Chapter 17, General Physics, wave optics