Wave Optics – Interference, Young’s experiment, thin‑film & interferometers

1. Principle of Superposition

When two or more waves occupy the same region of space, the resultant displacement $y$ at any point is the algebraic sum of the individual displacements:

$$y_{\text{total}}=y_1+y_2+\cdots +y_n$$

If the waves are sinusoidal with the same angular frequency $\omega$ and wave‑number $k$,

$$y_1=A_1\cos (kx-\omega t+{\varphi }_1),y_2=A_2\cos (kx-\omega t+{\varphi }_2)$$

the resultant amplitude $A_R$ follows

$$A_R=\sqrt{A_1^2+A_2^2+2A_1{A}_2\cos \Delta \varphi },$$

where $\Delta \varphi ={\varphi }_2-{\varphi }_1$ is the phase difference.

• Constructive interference: $\Delta \varphi =2m\pi$ → $A_R=A_1+A_2$.
• Destructive interference: $\Delta \varphi =(2m+1)\pi$ → $A_R=|A_1-A_2|$.

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2. Conditions for Stable Interference

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3. Young’s Double‑Slit Experiment

Setup description
A monochromatic source illuminates a single narrow slit (to ensure spatial coherence). The emerging wave passes through two parallel slits separated by distance $d$. On a screen placed at distance $D$ from the slits, an interference pattern of bright and dark fringes appears.

Mathematical derivation

Path difference between the two rays reaching a point $P$ on the screen at transverse coordinate $y$:

$$\Delta L=d\sin \theta \approx d\frac{y}{D}(\theta \text{small})$$

Constructive condition: $\Delta L=m\lambda$ →

$$y_m=\frac{m\lambda D}{d}(m=0,\pm 1,\pm 2,\dots )$$

Fringe spacing (distance between successive bright fringes):

$$\beta =y_{m+1}-y_m=\frac{\lambda D}{d}$$

Worked example – Determining the wavelength of a red laser

A red diode laser ($\lambda$ unknown) shines on a double‑slit plate with $d=0.25\text{mm}$. The screen is $D=1.20\text{m}$ away. Measured distance between the 3rd‑order bright fringe ($m=3$) and the central maximum is $y_3=7.2\text{mm}$.

$$\lambda =\frac{y_{3}d}{mD}=\frac{7.2\times 10^{-3}\text{m}\times 2.5\times 10^{-4}\text{m}}{3\times 1.20\text{m}}=5.0\times 10^{-7}\text{m}=500\text{nm}$$

Thus the laser emits greenish light, confirming the instrument’s calibration.

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4. Multiple‑Slit (Diffraction Grating)

When many equally spaced slits (grating constant $d$) are used, the condition for principal maxima becomes

$$d\sin \theta =m\lambda ,m=0,\pm 1,\pm 2,\dots$$

The intensity distribution sharpens with increasing number of slits $N$; the angular width of each principal maximum is approximately $\Delta \theta \approx \lambda /(Nd\cos \theta )$. Diffraction gratings are the heart of spectrometers and optical communication wavelength multiplexers.

Close‑up of a ruled diffraction grating used in spectrometers (Image: Robert Thompson, CC BY-SA 4.0, via Wikimedia Commons)

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5. Thin‑Film Interference

A thin transparent film (thickness $t$, refractive index $n$) placed on a substrate produces reflected beams from the front and back surfaces. The optical path difference (OPD) between the two reflected rays is

$$\text{OPD}=2nt\cos r+\frac{\lambda }{2},$$

where the extra $\lambda /2$ term accounts for a phase reversal when reflection occurs from a medium of higher refractive index. Constructive interference (bright reflected colour) occurs when

$$2nt\cos r=(m+\frac{1}{2})\lambda ,$$

and destructive interference (dark reflected colour) when

$$2nt\cos r=m\lambda .$$

Real‑world example – anti‑reflective (AR) coating on smartphone screens. The coating thickness is chosen so that reflected waves from the air‑coating and coating‑glass interfaces destructively interfere for visible wavelengths, reducing glare.

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6. Interferometers

6.1 Michelson Interferometer

Two perpendicular arms reflect light from a beam‑splitter. By moving one mirror by a distance $\Delta x$, the optical path difference changes by $2\Delta x$. Each shift of one fringe corresponds to a path change of one wavelength:

$$2\Delta x=\lambda \Rightarrow \Delta x=\frac{\lambda }{2}$$

Michelson interferometers are used for precise distance measurement, refractive‑index determination, and in gravitational‑wave detectors (LIGO).

6.2 Fabry‑Perot Interferometer

Consists of two partially reflecting parallel plates separated by distance $L$. Multiple reflections produce sharp transmission peaks when

$$2nL\cos \theta =m\lambda .$$

High‑resolution spectroscopy and laser cavity design rely on this principle.

Classic Michelson interferometer used in laboratory optics (Image: Shivam Nilesh Shingade, CC BY-SA 4.0, via Wikimedia Commons)

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7. Applications Overview

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8. Worked Example – Thin‑Film Coating for a Solar Panel

A solar panel glass is to be coated with an AR layer to minimise reflection at $\lambda =550\text{nm}$ (peak solar spectrum). The coating material has $n=1.38$. Required thickness $t$ for destructive interference at normal incidence:

$$t=\frac{\lambda }{4n}=\frac{550\text{nm}}{4\times 1.38}=99.6\text{nm}\approx 100\text{nm}$$

Thus a 100 nm SiO₂ coating will reduce reflection by ≈ 90 % at 550 nm, increasing panel efficiency.

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9. Comparison of Interference Set‑ups

| Feature                | Young’s Double‑Slit | Diffraction Grating | Thin‑Film Coating | Michelson Interferometer |
|------------------------|---------------------|---------------------|-------------------|--------------------------|
| Primary use            | Measuring λ, teaching concept | Spectroscopy, wavelength multiplexing | Reducing/controlling reflection | Precise distance/refractive index measurement |
| Number of apertures    | 2                   | 10⁴–10⁵ (grooves)   | 2 surfaces        | 2 arms (mirrors)         |
| Fringe visibility      | Moderate (depends on coherence) | Very sharp (high N) | Colour patterns (angle‑dependent) | High contrast, adjustable |
| Typical λ range        | Visible–UV          | UV–IR               | Visible           | Visible–IR               |
| Sensitivity to λ      | Linear (β ∝ λ)      | Linear (θ ∝ λ)      | Non‑linear (phase shift) | Direct (Δx = λ/2 per fringe) |

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10. In the real world

1. Smartphone screen (e.g., iPhone, Samsung) – The glossy screen is coated with a multilayer thin‑film stack. Each layer’s thickness is engineered so that reflected light from the air‑film and film‑glass interfaces undergoes destructive interference for most visible wavelengths, giving a near‑zero‑glare display. The principle is exactly the thin‑film interference formula discussed above.

2. NEPSE (Nepal Stock Exchange) live‑ticker display – The high‑speed LED panels use diffraction‑grating based backlights to spread white light into a uniform white illumination. The grating’s constructive‑interference angles are chosen so that the dominant wavelengths overlap, producing a bright, colour‑balanced screen.

3. LIGO (Laser Interferometer Gravitational‑Wave Observatory) – Although not a Nepali institution, LIGO’s Michelson interferometer detects spacetime ripples by measuring minute changes in fringe position (as small as $10^{-19}$ m). The same interferometric principle is taught in the lab for measuring tiny displacements, illustrating the extreme sensitivity achievable with interference.

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11. Exam tip

• Memorise the three conditions for stable interference (monochromatic, coherent, spatially coherent) – they appear in many short‑answer questions.
• Derive the fringe‑spacing formula $\beta =\lambda D/d$ and be ready to plug numbers quickly; the exam often gives $D,d,$ and $\beta$ and asks for $\lambda$.
• Thin‑film phase‑change rule: always add a $\lambda /2$ term when the reflected ray comes from a higher‑index medium; forgetting this leads to sign errors.
• Interferometer fringe shift: remember that moving a mirror by $\lambda /2$ shifts one fringe. This is a favourite quantitative problem.
• Diagram credit: draw a clean labelled diagram for Young’s experiment or Michelson interferometer; even a brief sketch earns marks for “illustration of principle”.

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