Class 12/Notes/Wave Optics
Diffraction of Light at a Single Slit: Experimental Setup & Working Principle
Light Source & Lenses:
A monochromatic source S is placed at the principal focus of a convex lens L1, rendering the light rays parallel before they fall on a slit AB of width d.
Wavefront Propagation: A plane wavefront WW’ strikes the slit. Every point on the exposed portion of the wavefront acts as a source of secondary wavelets according to Huygens’ Principle.
Focusing Lens: A second convex lens L2 focuses these diffracted parallel rays onto a screen located at a focal distance D from the slit.
Path Difference: For wavelets traveling at an angle relative to the central axis, the path difference x between rays originating from the upper edge A and lower edge B of the slit is given by:
x=d × sin θ
What is diffraction?
Diffraction of light is the bending and spreading of light waves when they pass through a narrow aperture or around the edge of an obstacle whose size is comparable to the wavelength of light.
For a single narrow slit, the light does not simply form a geometrical image of the slit. Instead, the light spreads into a characteristic pattern consisting of:
- a broad and bright central maximum,
- dark regions called minima on either side,
- weaker secondary maxima between successive minima.
The most important feature is that the central maximum is twice as wide as each secondary maximum.
Distinguish between interference and diffraction.
| Feature | Interference | Diffraction |
| Source | Two or more separate coherent sources | Secondary wavelets from same wavefront |
| Fringes | All fringes are of equal width and intensity | Central max is brightest and widest; secondary maxima decrease in intensity |
| Minima | Perfectly dark | Not perfectly dark (except approximate) |
Summary Table
| Quantity | Formula |
| Condition for minima | a sin(θ) = nλ |
| Condition for secondary maxima | a sin(θ) = (2n+1)λ/2 |
| Position of nth minimum | y_n = nλD / d |
| Position of nth secondary maximum | y_n = (2n+1)λD / 2d |
| Angular width of central maximum | 2λ / d |
Important Observations
- The central maximum is twice as wide as any secondary maximum.
- The intensity of the central maximum is maximum; secondary maxima have much lower intensity.
- Narrower slit → wider diffraction pattern (inverse relationship).
- Longer wavelength → wider diffraction pattern.
- The diffraction pattern is symmetric about the center.
- If slit width d < λ, the central maximum covers the entire screen (no minima observed).
Why Do We Get Maxima and Minima?
Each point on the slit emits secondary wavelets. These wavelets travel different distances to reach a point on the screen, causing path differences. Depending on the path difference:
- Constructive interference → Bright fringe (Maximum)
- Destructive interference → Dark fringe (Minimum)
Key Formula
The path difference between waves from the top and bottom of the slit is:
Path Difference = d × sin(θ)
where:
- d = slit width
- θ = angle of diffraction
The slit can be imagined as divided into N parts. The condition for minima and maxima depends on how many such parts produce cancelling pairs.
Positions of Minima (Dark Fringes)
Condition for Minima:
The slit is divided into 2n equal parts (where n = 1, 2, 3, …). Each pair of parts cancels each other.
d × sin(θ) = n × λ
where:
- n = 1, 2, 3, … (order of minimum)
- λ = wavelength of light
For small angles (θ small):
sin(θ) ≈ θ ≈ y/D
So,
y_n = (n × λ × D) / d
where:
- y_n = distance of nth minimum from center
- D = distance between slit and screen
Positions of Secondary Maxima (Bright Fringes)
Condition for Secondary Maxima:
The slit is divided into (2n + 1) parts. One part remains unpaired, giving a bright fringe.
d × sin(θ) = (2n + 1) × (λ / 2)
where:
- n = 1, 2, 3, … (order of secondary maximum)
For small angles:
y_n = ((2n + 1) × λ × D) / (2d)
Important Points:
- The central maximum (n = 0) is the brightest.
- Secondary maxima are less intense than the central maximum.
- Intensity decreases as we move away from the center.
Angular Width of Central Maximum
The central maximum lies between the first minima on either side.
Condition for first minima:
d × sin(θ) = λ
For small angles:
θ = λ / d
Angular width of central maximum:
The angular width is the angle between the two first-order minima (one on each side).
Angular Width = 2θ = 2λ / d
Linear Width of Central Maximum
The linear width is the actual distance on the screen between the two first-order minima.
Distance of first minimum from center:
y₁ = (λ × D) / d
Linear width of central maximum:
Linear Width = 2y₁ = 2λD / d
Linear Width of a Secondary Maximum
A secondary maximum lies between two consecutive minima.
For example, the first secondary maximum lies between:
- 1st minimum (n = 1): y = λD / d
- 2nd minimum (n = 2): y = 2λD / d
Linear width of first secondary maximum:
Width = (2λD / d) − (λD / d) = λD / d
In general, the linear width of any secondary maximum is:
Width of Secondary Maximum = λD / d
Note: The width of the secondary maxima is half the width of the central maximum.
Summary Table
| Physical quantity / Feature | Expression / Description |
|---|
| 1 | Slit width | d |
| 2 | Wavelength of light | λ |
| 3 | Condition for minima (dark fringes) | d sin θ = nλ, where n = ±1, ±2, ±3, … |
| 4 | Position of first minima | sin θ = ±λ/d |
| 5 | Position of nth minima | yₙ = nλD/d |
| 6 | Angular position of nth minima | θₙ ≈ nλ/d (in radians) |
| 7 | Angular width of central maximum | 2λ/d |
| 8 | Linear width of central maximum | 2λD/d |
| 9 | Linear width of secondary maxima (approximately) | λD/d |
| 10 | Position of secondary maxima (approximate) | d sin θ ≈ (n + 1/2)λ, for n = 1, 2, 3, … on either side |
| 11 | Intensity distribution | I = I₀ (sin β / β)², where β = πd sin θ / λ |
| 12 | Intensity at the central maximum | I = I₀ (maximum intensity) |
| 13 | Intensity at minima | I = 0 |
| 14 | Nature of central maximum | Brightest and widest fringe |
| 15 | Nature of secondary maxima | Bright fringes of decreasing intensity |
| 16 | Relative intensity of first secondary maximum | Approximately 4.7% of the central maximum |
| 17 | Effect of increasing slit width | Central maximum becomes narrower |
| 18 | Effect of increasing wavelength | Diffraction pattern becomes wider |
| 19 | Effect of increasing screen distance | Linear width of the diffraction pattern increases |
| 20 | Number of secondary maxima | Infinitely many in the ideal mathematical model |
Conclusion
Single-slit diffraction clearly demonstrates the wave nature of light. When light passes through a narrow slit of width d, it spreads into the region behind the slit and produces a characteristic pattern of a broad central maximum surrounded by weaker secondary maxima and dark minima. The condition for minima is given by d sin θ = nλ, showing that the diffraction pattern depends directly on the wavelength of light and inversely on the slit width.
The most important result is that the angular width of the central maximum is 2λ/d, while its linear width on a screen at distance D is 2λD/d. Thus, a smaller slit or a larger wavelength produces greater diffraction. The decreasing intensity of successive secondary maxima also distinguishes diffraction from an ordinary interference pattern.
In essence, single-slit diffraction provides powerful experimental evidence for the wave nature of light and establishes how the spreading of light is governed by the relationship between wavelength and slit width.
Link to this page
Leave a Reply