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

Interference – Conditions for Observing Two‑Source Fringes (Cambridge IGCSE/A‑Level Physics 9702)

1. Learning Objectives

• Apply the principle of superposition to explain interference, diffraction and standing waves.
• State, justify and quantify all the conditions required for clear, stable two‑source interference fringes.
• Derive the double‑slit fringe‑spacing formula and use it in quantitative calculations.
• Define coherence length, coherence time and fringe visibility; relate them to experimental parameters and calculate visibility from measured intensities.
• Plan, carry out and evaluate a double‑slit interference experiment, including uncertainty analysis and discussion of systematic errors.
• Recognise how interference underpins other syllabus topics (e.g. spectroscopy, wave‑particle duality, the electromagnetic spectrum).

2. Theory Overview (Section 8 – Superposition)

2.1 Wave Superposition

When two or more waves occupy the same region of space the resultant displacement is the algebraic sum of the individual displacements. This principle gives rise to:

• Stationary (standing) waves
• Diffraction (single‑slit, double‑slit, grating)
• Interference (two‑source fringes)

2.2 Stationary Waves

Formed by the superposition of two waves of the same frequency travelling in opposite directions. For a string fixed at both ends the allowed wavelengths satisfy

$$\lambda = \frac{2L}{n}\qquad (n=1,2,3,\dots)$$

Similar relations hold for air columns with the appropriate boundary conditions (open‑open, open‑closed). The node‑antinode spacing is \(\lambda/2\). These results are useful when discussing the wave nature of light (e.g. standing‑wave patterns in lasers).

2.3 Diffraction

Diffraction is the spreading of a wave when it encounters an obstacle or aperture comparable in size to its wavelength.

• Single‑slit diffraction – integrating Huygens’ secondary wavelets across a slit of width \(a\) gives the intensity distribution $$I(\theta)=I_0\left(\frac{\sin\alpha}{\alpha}\right)^2,\qquad \alpha=\frac{\pi a\sin\theta}{\lambda}.$$ The minima occur when \(\alpha = m\pi\;(m=\pm1,\pm2,\dots)\), i.e. \(\displaystyle a\sin\theta = m\lambda\). This “envelope’’ will later modulate the double‑slit pattern.
• Double‑slit interference – two narrow, identical slits separated by centre‑to‑centre distance \(d\) produce an intensity pattern that is the product of the single‑slit envelope and an interference term: $$I(\theta)=I_0\left(\frac{\sin\alpha}{\alpha}\right)^2\cos^2\beta,\qquad \beta=\frac{\pi d\sin\theta}{\lambda}.$$

2.4 Diffraction Grating

For a grating with many equally spaced slits the condition for bright maxima is

$$d\sin\theta = n\lambda\qquad (n=0,\pm1,\pm2,\dots)$$

where \(d\) is the grating spacing (inverse of the line density). This principle is the basis of spectroscopic instruments used later in the syllabus.

3. Conditions for Clear Two‑Source Interference

All of the following must be satisfied simultaneously. Where possible the requirement is expressed quantitatively.

1. Monochromatic (single‑frequency) light
   • Spectral width \(\Delta\lambda\) must be much smaller than the wavelength: \(\Delta\lambda \ll \lambda\). For a typical He‑Ne laser \(\Delta\lambda\approx1\;\text{nm}\) at \(\lambda\approx632\;\text{nm}\).
2. Temporal coherence (constant phase relationship)
   • Coherence time \(\tau_c\) and coherence length \(L_c=c\tau_c\) must exceed the largest path‑difference \(\Delta r_{\max}\) in the set‑up: \(L_c \gg \Delta r_{\max}\).
   • Example: a He‑Ne laser has \(L_c\approx30\;\text{cm}\); for a bench‑top experiment with \(\Delta r_{\max}\le 5\;\text{cm}\) the condition is comfortably met.
3. Identical polarisation
   • The electric‑field vectors of the two beams must be parallel (angular mismatch \(\Delta\theta \lesssim 5^{\circ}\)). Orthogonal polarisations give zero interference (visibility \(V=0\)).
4. Spatial coherence (source size)
   • The angular size of each source \(\theta_s\) must be much smaller than the angular fringe spacing \(\theta_f\approx\lambda/d\): \(\theta_s \ll \lambda/d\).
   • Practically this means using a point source or a narrow pinhole placed sufficiently far from the slits.
5. Path‑difference of the order of a wavelength
   • Constructive interference: \(\Delta r = m\lambda\) (bright fringes).
     Destructive interference: \(\Delta r = (m+\tfrac12)\lambda\) (dark fringes).
6. Mechanical stability
   • During measurement the positions of source, slits and screen must not change by more than \(\sim0.1\;\text{mm}\). Vibrations or thermal drift blur the pattern.

3.1 Fringe Visibility (Contrast)

The visibility quantifies the quality of the pattern:

$$V=\frac{I_{\max}-I_{\min}}{I_{\max}+I_{\min}}$$

For perfectly coherent, monochromatic and identically polarised beams \(V=1\). Any departure from the ideal conditions reduces \(V\) in proportion to the degree of coherence.

Worked example (visibility from measured intensities)

• Measured maximum intensity \(I_{\max}= 12.0\;\text{mW\,m}^{-2}\).
• Measured minimum intensity \(I_{\min}= 2.0\;\text{mW\,m}^{-2}\).
• Visibility \(V = \dfrac{12.0-2.0}{12.0+2.0}= \dfrac{10}{14}=0.71\).
• If a Polaroid sheet is inserted and rotated to make the beams orthogonal, \(I_{\max}=I_{\min}\) and \(V\rightarrow0\), confirming the polarisation condition.

4. Derivation of the Double‑Slit Fringe‑Spacing Formula

1. Two slits \(S_1\) and \(S_2\) are separated by distance \(d\). A point \(P\) on a screen at distance \(D\) (\(D\gg d\)) subtends an angle \(\theta\) with the central axis.
2. Path difference (geometric construction): $$\Delta r = d\sin\theta \approx d\tan\theta = d\frac{y}{D},$$ where \(y\) is the linear displacement of \(P\) from the centre.
3. Constructive condition \(\Delta r = m\lambda\) gives $$d\frac{y_m}{D}=m\lambda\;\;\Longrightarrow\;\;y_m = \frac{m\lambda D}{d}.$$
4. Successive bright fringes are separated by $$\beta = y_{m+1}-y_m = \frac{\lambda D}{d}.$$ This is the standard fringe‑spacing formula (valid for the small‑angle approximation \(\sin\theta\approx\tan\theta\approx\theta\)).

4.1 Uncertainty in Fringe Spacing

Propagation of uncertainties (first‑order) gives

$$\frac{\Delta\beta}{\beta}\approx\frac{\Delta\lambda}{\lambda}+\frac{\Delta D}{D}+\frac{\Delta d}{d}.$$

Typical classroom values:

• \(\Delta\lambda\approx1\;\text{nm}\) (laser)
• \(\Delta D\approx1\;\text{mm}\)
• \(\Delta d\approx0.01\;\text{mm}\)

5. Sample AO2 Problem

Given: He‑Ne laser \(\lambda = 632\;\text{nm}\); slit separation \(d = 0.25\;\text{mm}\); screen distance \(D = 1.20\;\text{m}\).

1. Fringe spacing $$\beta = \frac{\lambda D}{d}= \frac{(632\times10^{-9})(1.20)}{0.25\times10^{-3}} = 3.04\times10^{-3}\;\text{m}=3.04\;\text{mm}.$$
2. Uncertainty (using the values above) $$\frac{\Delta\beta}{\beta}= \frac{1}{632}+ \frac{1}{1200}+ \frac{0.01}{0.25}=0.0016+0.0008+0.04=0.0424,$$ $$\Delta\beta =0.0424\times3.04\;\text{mm}=0.13\;\text{mm}.$$
3. Result \(\displaystyle \beta = 3.04\pm0.13\;\text{mm}\). An experimental measurement within this range confirms the theory.

6. Practical Laboratory Procedure (AO3)

6.1 Equipment

• Laser pointer (λ≈630 nm) or narrow‑band LED with interference filter
• Double‑slit slide (slit width < 0.10 mm, centre‑to‑centre separation \(d\approx0.2–0.5\;\text{mm}\))
• Optical bench with rigid mounts, translation stages for \(D\) and for fine adjustment of the slit slide
• Screen (white matte paper) placed 1–2 m from the slits
• Ruler or vernier caliper, micrometer screw gauge (to measure \(d\))
• Polaroid sheet (optional, to test polarisation)
• Laser‑safety goggles (OD ≥ 3 at λ)

6.2 Safety

• Never look directly into the beam; keep it below eye level.
• Wear appropriate laser‑protective goggles at all times.
• Secure all components to the bench to avoid accidental displacement.
• Turn the laser off while adjusting any component.

6.3 Step‑by‑Step Procedure

1. Mount the laser securely; align the beam horizontally.
2. Insert the double‑slit slide so the beam passes centrally through both slits.
3. Measure the slit separation \(d\) with a microscope reticle or the manufacturer’s specification; record the uncertainty (typically \(\pm0.01\;\text{mm}\)).
4. Place the screen at a measured distance \(D\) from the slits; record \(D\) and its uncertainty (\(\pm1\;\text{mm}\)).
5. Turn the laser on and observe the interference pattern. Mark the positions of several bright fringes (e.g. \(m=-3\) to \(+3\)).
6. Measure the linear displacement \(y\) of each marked fringe from the central maximum. Compute the experimental fringe spacing \(\displaystyle \beta_{\text{exp}} = \frac{\Delta y}{\Delta m}\) (average over all measured orders).
7. Optional polarisation test: place a Polaroid sheet before the slits, rotate it through \(0^{\circ}\)–\(90^{\circ}\) and record the change in visibility \(V\). This demonstrates condition 3.
8. Repeat the measurement for a second screen distance (e.g. double the original \(D\)) to verify the linear relationship \(\beta\propto D\).

6.4 Data‑Analysis Checklist

Task	What to Do	Typical Pitfalls
Unit conversion	Express all lengths in metres before calculations.	Mixing mm and m introduces systematic error.
Calculate \(\beta_{\text{exp}}\)	Use several fringe orders; report the mean and the standard deviation as \(\pm\Delta\beta_{\text{exp}}\).	Using only two points under‑estimates random error.
Compare with theory	Compute \(\beta_{\text{theory}}=\lambda D/d\) and its uncertainty; check whether \(|\beta_{\text{exp}}-\beta_{\text{theory}}|\le\sqrt{\Delta\beta_{\text{exp}}^{2}+\Delta\beta_{\text{theory}}^{2}}\).	Ignoring uncertainties can lead to false conclusions.
Plot \(\beta\) versus \(D\)	Linear fit; slope should equal \(\lambda/d\). Evaluate \(R^{2}\) and residuals.	Non‑linear behaviour often signals mis‑alignment or excessive source size.
Visibility from intensities	Measure \(I_{\max}\) and \(I_{\min}\) (light‑meter or calibrated photograph) and compute \(V\). Compare with the expected value from the degree of coherence.	Neglecting background light inflates \(V\).
Systematic error discussion	Consider slit width (single‑slit envelope), finite spectral bandwidth, vibrations, inaccurate \(D\) or \(d\).	Omitting these reduces the quality of the AO3 evaluation.

7. Summary Table of Conditions

Condition	Quantitative Requirement	Consequence if Not Met
Monochromatic light	\(\Delta\lambda \ll \lambda\) (e.g. \(\Delta\lambda\le1\;\text{nm}\) for \(\lambda\approx600\;\text{nm}\))	Fringe washing out; reduced visibility.
Temporal coherence	Coherence length \(L_c = c\tau_c \gg \Delta r_{\max}\) (e.g. \(L_c>30\;\text{cm}\) for bench‑top set‑ups)	Phase drift → blurred fringes, visibility falls with order \(m\).
Identical polarisation	Polarisation vectors parallel; angular mismatch \(\Delta\theta\lesssim5^{\circ}\)	Visibility drops to zero for orthogonal polarisations.
Spatial coherence (source size)	Angular source size \(\theta_s \ll \lambda/d\) (e.g. \(\theta_s<0.1\lambda/d\))	Fringe contrast reduced; pattern may disappear.
Path‑difference ≈ λ	\(\Delta r = m\lambda\) for bright, \(\Delta r = (m+\tfrac12)\lambda\) for dark	Only intensity variation, no distinct fringes.
Mechanical stability	Positional changes \(<0.1\;\text{mm}\) during measurement	Moving fringes, larger random error.

8. Quick Reference – Key Equations

• Path difference: \(\displaystyle \Delta r = d\sin\theta\)
• Constructive interference: \(\displaystyle \Delta r = m\lambda\)
• Destructive interference: \(\displaystyle \Delta r = (m+\tfrac12)\lambda\)
• Fringe spacing (small‑angle): \(\displaystyle \beta = \frac{\lambda D}{d}\)
• Single‑slit envelope: \(\displaystyle I(\theta)=I_0\left(\frac{\sin\alpha}{\alpha}\right)^2,\; \alpha=\frac{\pi a\sin\theta}{\lambda}\)
• Double‑slit intensity: \(\displaystyle I(\theta)=I_0\left(\frac{\sin\alpha}{\alpha}\right)^2\cos^2\beta\)
• Grating condition: \(\displaystyle d\sin\theta = n\lambda\)
• Visibility: \(\displaystyle V=\frac{I_{\max}-I_{\min}}{I_{\max}+I_{\min}}\)
• Uncertainty in fringe spacing: \(\displaystyle \frac{\Delta\beta}{\beta}\approx\frac{\Delta\lambda}{\lambda}+\frac{\Delta D}{D}+\frac{\Delta d}{d}\)

9. Links to Other Syllabus Areas

• Spectroscopy – Diffraction gratings use the same interference condition \(d\sin\theta=n\lambda\) to separate wavelengths, a technique examined in the “Atomic spectra” sub‑topic.
• Wave‑particle duality – Interference patterns provide direct evidence for the wave nature of light; the same principles are applied to electron diffraction in later A‑Level modules.
• Electromagnetic spectrum – The requirement for monochromatic light highlights the need for narrow‑band sources (lasers, filtered LEDs) across the spectrum.