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Interference

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Topic: Optics > Physical Optics > Interference

Interference

1. Conceptual Definition

Interference is the core of wave optics, referring to the phenomenon where two or more **coherent waves** superimpose in space to form stable patterns of bright and dark fringes.

**Key Exam Point: Coherence Conditions**

To produce stable interference, light sources must satisfy three conditions:

1. **Same Frequency** ($f_1 = f_2$)

2. **Same Vibration Direction**

3. **Constant Phase Difference**

2. Core Principles & Formulas

#### A. Optical Path Difference (OPD)

In physical optics, distance is not just geometry; the refractive index matters.

* **Optical Path (L)**: $L = n \cdot l$ (Refractive Index $\times$ Geometric Path).

* **Optical Path Difference ($\delta$)**: The difference in optical paths between two coherent beams, $\delta = L_2 - L_1$.

#### B. Conditions for Maxima/Minima

* **Constructive Interference (Bright)**: OPD is an integer multiple of the wavelength.

$$\delta = k\lambda \quad (k=0, \pm 1, \pm 2...)$$

* **Destructive Interference (Dark)**: OPD is an odd multiple of half-wavelengths.

$$\delta = (k + 0.5)\lambda \quad (k=0, \pm 1, \pm 2...)$$

#### C. Half-Wave Loss (Phase Change upon Reflection)

**Most Common Trap!**

When light reflects from a **Rarer Medium** ($low \, n$) off a **Denser Medium** ($high \, n$), the phase shifts by $\pi$, effectively adding $\frac{\lambda}{2}$ to the optical path.

* **Rule**: Low $n$ $\to$ High $n$ reflection = Half-Wave Loss.

* **Note**: Transmitted light never suffers half-wave loss.

3. Typical Models

#### Model 1: Young's Double Slit

Interference by Wavefront Splitting.

* **Geometric Relation**: $\delta \approx d \sin\theta \approx d \frac{x}{D}$.

* **Fringe Spacing Formula**:

$$\Delta x = \frac{D\lambda}{d}$$

($D$: Distance to screen, $d$: Slit separation, $\lambda$: Wavelength).

#### Model 2: Thin Film Interference

Interference by Amplitude Splitting (e.g., soap bubbles, anti-reflective coatings).

* **OPD Formula**:

$$\delta = 2n_2 e \pm \frac{\lambda}{2} \quad (\text{Depends on Half-Wave Loss})$$

($e$: Thickness, $n_2$: Refractive index of film).

4. Typical Examples

**Example 1: Young's Double Slit Calculation**

Slit separation $d = 0.50 mm$, screen distance $D = 1.00 m$, fringe spacing $\Delta x = 1.20 mm$. Find wavelength $\lambda$.

* **Solution**:

Rearranging $\Delta x = \frac{D\lambda}{d}$:

$$\lambda = \frac{\Delta x \cdot d}{D} = \frac{(1.20 \times 10^{-3}) \times (0.50 \times 10^{-3})}{1.00} = 6.0 \times 10^{-7} m = 600 nm$$

**Example 2: Anti-Reflective Coating (Trap Question)**

Glass ($n_g = 1.52$) is coated with Magnesium Fluoride ($n_f = 1.38$). Find the minimum thickness for destructive interference of green light ($\lambda = 550 nm$).

* **Analyze Phase Change**:

1. Air ($n=1$) $\to$ Film ($n=1.38$): **Yes**, half-wave loss.

2. Film ($n=1.38$) $\to$ Glass ($n=1.52$): **Yes**, half-wave loss.

3. **Conclusion**: Both reflections shift, so the effects cancel. Total OPD $\delta = 2n_f e$.

* **Calculation**:

Destructive condition $\delta = (k+0.5)\lambda$. Min thickness ($k=0$) implies $2n_f e = 0.5\lambda$.

$$e = \frac{\lambda}{4n_f} = \frac{550}{4 \times 1.38} \approx 99.6 nm$$

**Example 3: Fringe Shift due to Thin Sheet**

A mica sheet ($n=1.58$) covers one slit $S_1$ in a double-slit setup. The central bright fringe shifts by 4 fringe spacings. Find the sheet thickness $t$.

* **Principle**: Inserting the sheet adds optical path. A shift of 4 fringes means the added path difference equals $4\lambda$.

* **Equation**:

Added OPD $\Delta L = (n-1)t = 4\lambda$

$$t = \frac{4\lambda}{n-1} = \frac{4 \times 550 nm}{1.58 - 1} \approx 3793 nm \approx 3.79 \mu m$$