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Diffraction

CSCA Diffraction study guide organized around the publicly available CSCA syllabus. Practice Physics questions on aicsca.com.

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Topic: Optics > Physical Optics > Diffraction

Diffraction

1. Conceptual Definition

Diffraction is the phenomenon where light bends around the edges of an **obstacle** or passes through a small **aperture**, entering the geometric shadow. It is direct evidence of the **Wave Nature** of light.

**Key Exam Point: Conditions for Diffraction**

* Diffraction is most pronounced when the size of the aperture/obstacle $a$ is **comparable to** or **smaller than** the wavelength $\lambda$ ($a \approx \lambda$).

* When $a \gg \lambda$, diffraction is negligible, and light travels in straight lines (Geometrical Optics).

2. Core Models & Formulas

#### A. Single Slit Fraunhofer Diffraction

Parallel light hits a single slit normally, creating a pattern on a screen.

* **Pattern Characteristics**:

* **Central Maximum**: Located at the center, widest (approx. 2x width of others), and brightest.

* **Secondary Maxima**: Located on sides, intensity decreases rapidly.

* **Condition for Minima (Dark Fringes) —— ⚠️ Memorize**

Using the "Half-Period Zone" method, minima occur when the path difference between edges is an integer multiple of wavelength:

$$ a \sin\theta = k\lambda \quad (k = \pm 1, \pm 2, \pm 3...) $$

* **Note**: Here $k\lambda$ corresponds to **DARK** fringes, which is the opposite of double-slit interference (where $k\lambda$ is Bright)!

* **Width of Central Maximum**:

$$ \Delta x \approx \frac{2f\lambda}{a} $$

($f$ is focal length, $a$ is slit width). Narrower slit ($a$ decreases) $\to$ Wider pattern.

#### B. Circular Aperture & Rayleigh Criterion

Light passing through a circular hole forms an "Airy Disc".

* **Resolution Limit**: The resolving power of optical instruments (telescopes, microscopes) is limited by diffraction.

* **Rayleigh Criterion**: Two point sources are considered **just resolved** when the center of the diffraction pattern of one lies exactly on the first dark ring of the other.

* **Minimum Angle of Resolution**:

$$ \theta_{min} = 1.22 \frac{\lambda}{D} $$

($D$ is aperture diameter). Smaller $\theta_{min}$ means higher resolving power.

#### C. Diffraction Grating

Consists of many equally spaced, identical slits.

* **Features**: Compared to double-slit, grating fringes are **sharper, brighter, and more widely spaced**.

* **Grating Equation (Maxima Condition)**:

$$ d \sin\theta = k\lambda \quad (k = 0, \pm 1, \pm 2...) $$

($d$ is the grating constant, i.e., spacing between slits).

3. Comparison: Double-Slit vs. Single-Slit

| Feature | Double-Slit Interference | Single-Slit Diffraction |

| :--- | :--- | :--- |

| **Cause** | Superposition of 2 beams | Superposition of infinite wavelets from 1 wavefront |

| **Fringe Width** | Uniform width | Central max is widest |

| **Intensity** | Relatively uniform | Central max is brightest, decays fast |

| **Formula ($k\lambda$)** | $d\sin\theta = k\lambda$ (**Bright**) | $a\sin\theta = k\lambda$ (**Dark**) |

**Advanced Note**: Real double-slit patterns are **modulated by the single-slit diffraction envelope**. If an interference maximum coincides with a diffraction minimum, that interference fringe disappears. This is called a **Missing Order**.

4. Typical Examples

**Example 1: Single Slit Calculation**

Parallel light ($\lambda = 500 nm$) hits a slit ($a = 0.5 mm$). Screen is at focal plane of lens ($f = 1 m$). Find the width of the central bright fringe.

* **Solution**:

Central width = distance between first dark fringes ($k=\pm 1$).

Angle of 1st dark fringe: $\sin\theta \approx \theta = \frac{\lambda}{a}$.

Distance on screen: $x_1 = f \cdot \tan\theta \approx f \cdot \frac{\lambda}{a}$.

Width $\Delta x = 2x_1 = \frac{2f\lambda}{a}$.

$$ \Delta x = \frac{2 \times 1.0 \times 500 \times 10^{-9}}{0.5 \times 10^{-3}} = 2 \times 10^{-3} m = 2 mm $$

**Example 2: Resolving Power**

Hubble Telescope ($D=2.4m$) observes a star ($\lambda=550nm$). Find the minimum angular resolution.

* **Solution**:

$$ \theta_{min} = 1.22 \frac{\lambda}{D} = 1.22 \times \frac{5.5 \times 10^{-7}}{2.4} \approx 2.8 \times 10^{-7} rad $$