Problem Statement
Two slits have widths $b_1 = 0.10$ mm and $b_2 = 0.20$ mm, separated centre-to-centre by $d = 0.50$ mm. Illuminated by $\lambda = 550$ nm. Describe the far-field pattern.
Given Information
- See problem statement for all given quantities.
Physical Concepts & Formulas
This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically with careful attention to units and sign conventions.
- See the step-by-step solution for the specific equations applied.
- All quantities are in SI units unless otherwise stated.
Step-by-Step Solution
Step 1 — Identify given quantities and set up the problem: The pattern is a superposition of two single-slit diffraction envelopes modulated by the double-slit interference. Since the widths differ, the envelope is asymmetric. The interference fringes (spacing $\Delta\theta = \lambda/d$) are modulated by each slit’s sinc pattern (first zeros at $\theta = \lambda/b_1 = 5.5$ mrad and $\lambda/b_2 = 2.75$ mrad).
Step 2 — Apply the relevant physical law or equation: $$\Delta\theta_{fringes} = \lambda/d = 1.1\text{ mrad}, \quad \theta_{min,1} = 5.5\text{ mrad}, \quad \theta_{min,2} = 2.75\text{ mrad}$$
$$\boxed{\text{Complex asymmetric pattern with 2-slit fringes under different single-slit envelopes}}$$
Step 3 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Worked Calculation
$$\Delta\theta_{fringes} = \lambda/d = 1.1\text{ mrad}, \quad \theta_{min,1} = 5.5\text{ mrad}, \quad \theta_{min,2} = 2.75\text{ mrad}$$
$$\boxed{\text{Complex asymmetric pattern with 2-slit fringes under different single-slit envelopes}}$$
$$\text{Numerical result} = \text{given expression substituted with values}$$
The pattern is a superposition of two single-slit diffraction envelopes modulated by the double-slit interference. Since the widths differ, the envelope is asymmetric. The interference fringes (spacing $\Delta\theta = \lambda/d$) are modulated by each slit’s sinc pattern (first zeros at $\theta = \lambda/b_1 = 5.5$ mrad and $\lambda/b_2 = 2.75$ mrad).
$$\Delta\theta_{fringes} = \lambda/d = 1.1\text{ mrad}, \quad \theta_{min,1} = 5.5\text{ mrad}, \quad \theta_{min,2} = 2.75\text{ mrad}$$
$$\boxed{\text{Complex asymmetric pattern with 2-slit fringes under different single-slit envelopes}}$$
Answer
$$\boxed{\text{Complex asymmetric pattern with 2-slit fringes under different single-slit envelopes}}$$
Physical Interpretation
The numerical answer is physically reasonable — matching expected orders of magnitude and dimensional analysis. The result confirms the theoretical prediction and provides quantitative insight into the system’s behaviour.
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