Interference & Diffraction Estimates (IPhO)

This lesson is about “scale thinking”: before doing detailed algebra, you should know what sets the characteristic angle, spacing, or envelope width, and you should be able to estimate a number in one or two lines.

Mind-stretcher feedback

• In the focal plane of a lens, angle maps to position as y ≈ fθ, so replace the free-propagation distance L by focal length f: for example the single-slit central width is approximately 2fλ/a.

1. Definitions (Must Know)

• Wavelength and wavenumber: λ and k = 2π/λ.
• Path difference: Δ (a length). Phase difference: Δφ = (2π/λ)Δ.
• Angle and screen coordinate: for a screen distance L, small angles give y ≈ Lθ.
• Slit width vs separation: a is the width of one slit, d is the center-to-center separation of two slits (or grating spacing).
• Order number: m labels principal maxima (integer).
• Envelope vs fringes: a broad diffraction envelope (set by a) that modulates fine interference fringes (set by d).

2. Key Ideas (What Earns Marks)

• Start from phase: write Δφ = (2π/λ)Δ and get Δ from geometry.
• Identify the controlling length scale: a sets the envelope width, d sets fringe spacing.
• Use small-angle estimates only after you check the problem is near the axis (angles in radians are small enough that sin θ and θ are close).
• Convert grating specs early: “N lines per mm” means d = 1/N mm.
• Do an order-of-magnitude pass first, then refine with the exact factor (like 2 or 1.22).

3. Detailed Explanations

3.1 Double slit (two point-like slits)

Path difference for two slits separated by d:

Constructive interference occurs when

Near the axis, the angular fringe spacing is

so the screen fringe spacing is

3.2 Single slit (diffraction envelope)

For slit width a, the first minimum is set by

So the characteristic angular half-width of the central maximum is

and the central maximum width on a screen is

3.3 Double slit with finite slit width (envelope plus fringes)

If each slit has width a and separation d, then:

• the fringe spacing is still set by d,
• the envelope zeros are set by a.

An estimate for how many bright fringes fit in the central envelope is

3.4 Diffraction grating (many equally spaced slits)

Principal maxima occur at the same condition

but they get much sharper as the number of slits increases. A useful estimate for the angular width of a principal maximum is

where N is the number of illuminated slits. Near the axis, this is roughly δθ∼ λ/(Nd).

3.5 Circular aperture (diffraction limit)

For aperture diameter D, the first dark ring is at

This is the scale behind the Rayleigh resolution criterion.

4. Common Mistakes

• Swapping a and d (envelope versus fringe spacing).
• Unit slips: nm, μm, mm, m, and “lines per mm”.
• Treating y and θ as the same quantity (always connect them via y ≈ Lθ).
• Applying small-angle formulas far from the axis without checking whether the computed angle is actually small.
• Forgetting that in a medium λ changes; the phase uses the wavelength in the medium.

5. Exam Tips

• Write one line for the scale before the full derivation, for example “Δ y set by λ L/d”.
• When you compute sin θ = mλ/d, check it is physically allowed by remembering sin θ cannot exceed one in magnitude.
• Keep a “typical visible” number ready: λ∼ 5 × 10⁻⁷ m.
• If you are unsure about a factor, get the scale right first; many IPhO marks are for setting up the right dependence.

6. Worked Examples

7. Mind Stretchers

• A laser beam hits a slit of width a and then a lens focuses the Fraunhofer pattern onto a screen. How does the “screen distance” L get replaced by the lens focal length in the scaling laws?
• Two wavelengths λ and λ + Δλ go through the same grating. Estimate the angular separation of their first-order maxima and relate it to spectral resolution.
• A finite-sized source reduces fringe visibility. Model the source as many incoherent point sources and estimate how large the source can be before fringes wash out.

8. Practice