Single-Slit Diffraction
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Direct answer
Light passing through a single narrow slit of width a spreads beyond geometrical shadow, with dark fringes at a sin(theta) = n lambda for n = 1, 2, 3..., flanking a central maximum of angular width 2 lambda/a — twice as wide as any other fringe and far the brightest, since the first secondary maximum is under 5% of the central. On a screen at distance D, the central maximum spans 2 lambda D/a between its first minima. Diffraction distinguishes itself from interference by the unequal fringe spacing and steeply falling intensity; combined with a double slit, it engraves missing orders at d/a ratio integers, and it sets every optical instrument's resolving limit through the Rayleigh criterion.
What you must remember
- Minima condition: a sin(theta) = n lambda, n = 1, 2, ...; secondary maxima sit roughly halfway between minima at a sin(theta) = (2n + 1) lambda/2 — approximately, not exactly.
- Central maximum width: angular width 2 lambda/a; linear width on a distant screen 2 lambda D/a; every other fringe is about half as wide.
- Intensity distribution: central maximum strongest; first secondary maximum about 4.5% of central intensity; falling rapidly — the diffraction signature.
- Against interference: interference fringes are equally spaced with equal maxima (two slits idealised), diffraction fringes are unequal and fading (single aperture real).
- Missing orders in double slit: interference maxima at d sin(theta) = n lambda vanish when a diffraction minimum coincides — order n = d/a times an integer missing; the envelope modulates the fringes.
- Fresnel versus Fraunhofer: the JEE treatment is Fraunhofer (parallel rays, slit-lens-screen); Fresnel (near field, zones) appears only qualitatively.
- Rayleigh criterion: two point sources are just resolved when one's maximum falls on the other's first minimum; telescope limit 1.22 lambda/D (aperture D), microscope limit d(min) = 0.61 lambda/(mu sin theta).
- Pattern note: Main asks slit-width-from-fringe-width numericals; Advanced asks missing orders and resolution comparisons.
Widths, orders and a missing order
A slit of width 0.2 mm lit by 600 nm sodium light throws its pattern on a screen 1 m away. First minimum: sin(theta) = 600 × 10^-9/(0.2 × 10^-3) = 3 × 10^-3, so the half-width of the central patch on screen is D tan(theta) ≈ 3 mm and its full width 6 mm. Halve the slit and the patch doubles — the inverse relation between aperture and spread is the quantitative heart of the chapter.
Now make it a double slit with centres separated d = 1 mm and the same slit width a = 0.2 mm: interference maxima at d sin(theta) = n lambda would appear, but wherever a diffraction minimum lands — a sin(theta) = m lambda — the maximum is annihilated; the ratio d/a = 5 means orders 5, 10, 15... go missing from the pattern. Reading the d/a ratio off a photograph and naming the missing orders is the Advanced version of the same physics.
Where students slip
The condition a sin(theta) = n lambda locates minima, not maxima — the single commonest slip; only the central maximum sits at theta = 0, and the secondary maxima are offset from the halfway points by a small amount. Second, the central maximum's width is 2 lambda D/a while other fringes run about lambda D/a; questions asking "width of the central maximum" versus "fringe width" expect exactly this factor of two. Third, in the double-slit composite, fringe spacing is governed by d but the envelope by a — two independent lengths, and mixing them scrambles which orders disappear. Fourth, resolution: Rayleigh is "maximum on the first minimum", not "any visible dip"; the telescope formula carries 1.22 for a circular aperture, the slit-based 1.22 does not apply to a rectangular one. And increasing aperture D improves resolution toward smaller angles — saying a bigger telescope resolves less is a sign the whole idea inverted.
Frequently asked questions
What condition gives the diffraction minima of a single slit?
a sin(theta) = n lambda with n = 1, 2, 3... — the path difference between the slit's two edges is a whole number of wavelengths, and the slit halves cancel pairwise.
How wide is the central maximum on a screen?
Angular width 2 lambda/a, linear width 2 lambda D/a at distance D — double the width of the other fringes and far brighter than all of them.
How does a diffraction pattern differ from an interference pattern?
Diffraction fringes are unequally spaced with rapidly fading intensity from one aperture; interference fringes from narrow slits are equally spaced with comparable maxima.
What are missing orders in a double-slit experiment?
Interference maxima that coincide with single-slit diffraction minima are suppressed — orders that are integer multiples of d/a disappear from the pattern.
What does the Rayleigh criterion state?
Two images are just resolved when one diffraction maximum falls on the other's first minimum; for a circular aperture of diameter D, the limit angle is 1.22 lambda/D.