Diffraction lens diameter Problem

In summary, the conversation discusses the ability of a spy satellite to measure the size of an aircraft's air intake from 150 km above the Earth's surface. Diffraction calculations are used to determine the effective lens diameter needed for this task, with a final result of approximately 0.27 m.
  • #1
purduegrad
11
0
Tried this problem for about 2 hrs and no dice...

A spy satellite orbiting at 150 km above the Earth's surface has a lens with a focal length of 3.5 m and can resolve objects on the ground as small as 36 cm; it can easily measure the size of an aircraft's air intake. What is the effective lens diameter, determined by diffraction consideration alone? Assume = 527 nm.

I don't think the 150km matters...
i just did .36 = f*theta and i got theta = .36/3.5


then i did theta = wavelength /d (diameter) and i get like 5.12e-6M

totally wrong...help

btw as soon as someone replies i will prolly reply back since its kinda urgent
 
Last edited:
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  • #2
Consider the Rayleigh criterion for a circular aperture:

[tex]\sin \theta_R = 1.22 \frac{\lambda}{d}[/tex]

In this case, [itex]\theta_R[/itex] is the angle subtended by a 36 cm feature from 150 km away. Hint: draw a right triangle. The angle subtended is

[tex]\begin{align*}
\theta_R &= 2 \tan^{-1} \left( \frac{0.18}{150 \cdot 10^3} \right)\\
& \approx \tan^{-1} \left( \frac{0.36}{150 \cdot 10^3} \right)\\
& \approx 2.4 \cdot 10^{-6}\ \text{rad}
\end{align*}
[/tex]

The Rayleigh criterion then gives

[tex]\begin{align*}
d &= \frac{1.22 \lambda}{\sin \theta_R}\\
& \approx 0.27\ m
\end{align*}
[/tex]

Does this make sense?

- Warren
 
  • #3


First of all, I would like to commend you for putting in the effort to solve this problem for 2 hours. Diffraction problems can be tricky, so don't get discouraged if you haven't found the correct answer yet.

To solve this problem, we need to use the diffraction limit formula: θ = 1.22λ/D, where θ is the angular resolution, λ is the wavelength, and D is the diameter of the lens.

Since we are given the focal length of the lens (3.5 m) and the smallest resolvable object (36 cm), we can use the formula θ = 1.22λ/f to find the angular resolution. Plugging in the values, we get:

θ = 1.22 * 527 nm / 3.5 m = 1.83e-4 radians

Now, we can use the formula θ = λ/D to find the effective lens diameter. Rearranging the formula, we get:

D = λ/θ = 527 nm / 1.83e-4 = 2.88 m

Therefore, the effective lens diameter, determined by diffraction consideration alone, is approximately 2.88 meters. This is larger than the actual lens diameter of 3.5 m, which means that diffraction is not the only factor limiting the resolution of the spy satellite.

I hope this helps you understand the problem better. Keep practicing and don't give up, you will eventually get the hang of diffraction problems!
 

1. What is the diffraction lens diameter problem?

The diffraction lens diameter problem refers to the phenomenon where the diameter of a lens affects the quality of an image produced by that lens. This is due to the bending of light waves as they pass through the edges of the lens, causing aberrations and distortions in the image.

2. How does the diffraction lens diameter problem affect image quality?

The diffraction lens diameter problem can cause a decrease in sharpness and clarity of an image, as well as introduce color fringing and other distortions. This is especially noticeable when using high magnification or zoom levels.

3. Can the diffraction lens diameter problem be avoided?

While it cannot be completely avoided, the effects of the diffraction lens diameter problem can be minimized by using high-quality lenses with larger diameters and avoiding extreme magnification levels. Additionally, post-processing techniques can be used to correct some of the distortions.

4. How do I choose the right lens diameter to avoid the diffraction lens diameter problem?

The right lens diameter will depend on your specific needs and the type of photography you are doing. Generally, using a lens with a larger diameter can help reduce the effects of the diffraction lens diameter problem, but it is also important to consider other factors such as focal length and aperture.

5. Are there any other factors besides lens diameter that can contribute to the diffraction lens diameter problem?

Yes, other factors such as the quality and design of the lens, the focal length, and the aperture can also play a role in the diffraction lens diameter problem. It is important to consider all of these factors when choosing a lens to achieve the best possible image quality.

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