understand the conditions required if two-source interference fringes are to be observed

Published by Patrick Mutisya · 14 days ago

Interference – Conditions for Observing Two‑Source Fringes

Interference

Objective

To understand the conditions required for the observation of interference fringes produced by two coherent sources.

Key Concepts

  • Interference occurs when two or more waves superpose and produce a resultant intensity pattern.

  • For clear, stable fringes the sources must be coherent.

Conditions for Two‑Source Interference

  1. Same Frequency (Monochromatic)

    The two waves must have the same wavelength \$\lambda\$ (or a very narrow range of wavelengths).

  2. Constant Phase Relationship

    The phase difference \$\Delta\phi\$ between the sources must remain constant in time.

  3. Identical Polarisation

    Both waves should be polarised in the same direction; orthogonal polarisations do not interfere.

  4. Point‑like Sources (or Small Angular Size)

    The effective size of each source must be much smaller than the fringe spacing, otherwise fringe contrast is reduced.

  5. Path‑Difference Comparable to Wavelength

    Constructive interference occurs when the path difference \$\Delta r\$ satisfies \$\Delta r = m\lambda \quad (m = 0, \pm1, \pm2,\dots)\$ and destructive interference when \$\Delta r = \left(m+\tfrac12\right)\lambda.\$

  6. Stable Geometry

    The relative positions of the sources and the observation screen must remain fixed during the measurement.

  7. Sufficient Coherence Length

    The distance over which the wave maintains a well‑defined phase (coherence length \$L_c\$) must exceed the maximum path difference in the experiment.

Typical Experimental Arrangement

Suggested diagram: Two narrow slits (S₁ and S₂) illuminated by a monochromatic source, with a screen at distance \$D\$ showing bright and dark fringes.

Fringe Spacing Formula

For the classic double‑slit geometry, the fringe spacing \$ \beta \$ on a screen a distance \$D\$ from the slits (separation \$d\$) is given by

\$\beta = \frac{\lambda D}{d}\$

where \$d \ll D\$ so that the small‑angle approximation holds.

Table: Summary of Conditions

ConditionRequirementConsequence if Not Satisfied
Same frequencyMonochromatic light (single \$\lambda\$)Fringe washing out due to varying \$\lambda\$
Constant phaseCoherent sources (fixed \$\Delta\phi\$)Temporal variation → blurred fringes
Identical polarisationSame polarisation directionNo interference if orthogonal
Small source sizeAngular size \$\ll\$ fringe angular spacingReduced visibility (contrast)
Path‑difference \$\sim\lambda\$\$\Delta r\$ within a few \$\lambda\$Only intensity variations, no distinct fringes
Stable geometryFixed \$d\$, \$D\$, and alignmentMoving fringes, measurement error
Coherence length \$L_c\$\$L_c \gg\$ maximum \$\Delta r\$Fringe contrast falls off with distance

Practical Tips for A‑Level Experiments

  • Use a laser pointer or a narrow‑band LED with a filter to ensure monochromatic light.

  • Employ a double‑slit mask with slit width \$<0.1\$ mm and separation \$d\$ of a few millimetres.

  • Mount the slits and screen on a rigid optical bench to minimise vibrations.

  • Check polarisation with a Polaroid sheet; align both beams parallel.

  • Measure fringe spacing with a ruler and compare with \$\beta = \lambda D/d\$ to verify the condition.