A diffraction-limited laser of length l and aperture diameter d generates light of wavelength `lambda`. If the beam is directed at the surface of the Moon a distance D away, the radius of the illuminated area on the Moon is approximately

Options

A. `dD // l`
B. `d D // lambda`
C. `D lambda // l`
D. D

Answer

To find the radius of the illuminated area on the Moon when a diffraction-limited laser beam is directed at it, we can follow these steps:

### Step 1: Understand the diffraction limit
The diffraction limit for a laser beam is determined by the angle θ at which the beam spreads out. This angle can be approximated using the formula:
\[ \theta \approx \frac{1.22 \lambda}{d} \]
where:
- \( \lambda \) is the wavelength of the light,
- \( d \) is the diameter of the aperture of the laser.

### Step 2: Calculate the angle θ
Using the formula from Step 1, we can express the angle θ:
\[ \theta = \frac{1.22 \lambda}{d} \]

### Step 3: Relate the angle to the radius on the Moon
The radius \( R \) of the illuminated area on the Moon can be found using the relationship between the angle θ and the distance \( D \) to the Moon. The radius can be approximated as:
\[ R \approx \frac{1}{2} D \cdot \theta \]

### Step 4: Substitute θ into the radius formula
Now, substituting the expression for θ into the radius formula gives:
\[ R \approx \frac{1}{2} D \cdot \frac{1.22 \lambda}{d} \]

### Step 5: Simplify the expression
By simplifying the equation, we can express the radius \( R \) as:
\[ R \approx \frac{0.61 \lambda D}{d} \]

### Final Answer
Thus, the radius of the illuminated area on the Moon is approximately:
\[ R \approx \frac{0.61 \lambda D}{d} \]

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