Calculating Light Spot Size on the Moon for a Diffraction-Limited Laser Beam

[RJLiberator]

Homework Statement

A laser is a light source that emits a diffraction-limited beam (like waves diffracting through a wide slit) of diameter 2 mm. Ignoring any scattering due to the Earth's atmosphere, calculate how big a light spot would be produced on the surface of the moon, 240,000 miles away. Assume a wavelength of approximately 600 nm.

Homework Equations

d = 2mm
L = 240,000 miles
λ = 600nm

The Attempt at a Solution

I am using a small angle approximation where Θ = λ/d from dsinΘ = λ

And so, converting the proper units, we have
Θ = (6e-7)/0.002 = 0.0003

Angle is a dimensionless unit, so this seems to be correct.

Now, if I wanted to find how big the light spot is, do I simple do
tanΘ = x/240,000 => 240,000*tan(0.0003) = x = 72 miles

Looking good?

 

[BvU]

I am using a small angle approximation where Θ = λ/d from dsinΘ = λ

Is that the angle for the width of the central spot ?

 

[RJLiberator]

Yes, I believe that is true. It is the angle for the width of the central maximum.

 

[BvU]

These guys think differently. Depends on how you define the width, I suppose. I had in mind this is the expression for the angle for the first minimum and was inclined to multiply by 2.

 

[RJLiberator]

So, the 240,000 miles plays no part in this question then? eh?

 

[RJLiberator]

Er, my answer seems to be incorrect. A laser would create a 72 mile bright spot on the moon? That doesn't seem reasonable. mmm.

 

[BvU]

So, the 240,000 miles plays no part in this question then? eh?

It does play a role and you did that correctly. My hunch is they want the distance between the two minima on either side. And a 144 miles spot isn't all that big when seen from Earth (namely a viewing angle of $\arctan 0.0006$ ).

 

[RJLiberator]

BvU, you were absolutely correct. He accepted both answers, but he did mention the factor of 2.

Thank you for the help.