Problem 5.59 — Intensity at Centre of Zone Plate

Problem Statement

Compare the intensity at the focus of a zone plate that blocks all even zones with that of an unobstructed wave, given that the zone plate has $N = 10$ open zones.

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: Each open odd zone contributes amplitude $a_m \approx a_1$ (nearly equal). The zone plate focuses $N = 10$ zones. The amplitude at focus is $\approx N a_1/2 \times 2 = N a_1$ compared with unobstructed $a_\infty \approx a_1/2$.

Step 2 — Apply the relevant physical law or equation: $$\frac{I_{ZP}}{I_0} = \left(\frac{N a_1}{a_1/2}\right)^2 = (2N)^2 = (20)^2 = \boxed{400}$$

Step 3 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.

Worked Calculation

$$\frac{I_{ZP}}{I_0} = \left(\frac{N a_1}{a_1/2}\right)^2 = (2N)^2 = (20)^2 = \boxed{400}$$

$$\text{Numerical result} = \text{given expression substituted with values}$$

$$\boxed{\frac{I_{ZP}}{I_0} = \left(\frac{N a_1}{a_1/2}\right)^2 = (2N)^2 = (20)^2 = \boxed{400}}$$

Each open odd zone contributes amplitude $a_m \approx a_1$ (nearly equal). The zone plate focuses $N = 10$ zones. The amplitude at focus is $\approx N a_1/2 \times 2 = N a_1$ compared with unobstructed $a_\infty \approx a_1/2$.

$$\frac{I_{ZP}}{I_0} = \left(\frac{N a_1}{a_1/2}\right)^2 = (2N)^2 = (20)^2 = \boxed{400}$$

Answer

$$\frac{I_{ZP}}{I_0} = \left(\frac{N a_1}{a_1/2}\right)^2 = (2N)^2 = (20)^2 = \boxed{400}$$

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.