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Chapter 9: Problem 16

A diffraction-limited laser beam of diameter \(1 \mathrm{~cm}\) is pointed at the moon. What is the diameter of the area illuminated on the moon? (The moon is \(240,000 \mathrm{mi}\) away.) Take the light wavelength to be \(6328 \AA\). Neglect scattering in the earth's atmosphere.

Short Answer

Expert verified
The diameter of the illuminated area on the moon is approximately 29.82 km.

Step by step solution

01

Calculate the Angular Resolution

The angular resolution of a diffraction-limited beam can be calculated using the formula \( \text{angular resolution} = 1.22 \times \frac{\lambda}{D} \), where \( \lambda = 6328 \text{ 脜} = 6328 \times 10^{-10} \text{ m} \) is the wavelength and \( D = 0.01 \text{ m} \) is the beam's diameter. Converting the wavelength, we get \( 6328 \times 10^{-10} \text{ m} \). So, substituting the values, \( \text{angular resolution} = 1.22 \times \frac{6328 \times 10^{-10}}{0.01} \approx 7.72 \times 10^{-5} \text{ radians} \).
02

Calculate the Diameter of the Illuminated Area on the Moon

The diameter of the spot created by the laser beam on the moon can be found using the formula \( \text{diameter} = \text{angular resolution} \times R \), where \( R = 240,000 \text{ miles} \). First, convert \( R \) from miles to meters: \( R = 240,000 \text{ miles} \times 1609.34 \text{ m/mile} \approx 3.8624 \times 10^8 \text{ m} \). Then, the diameter is \( \text{diameter} = 7.72 \times 10^{-5} \times 3.8624 \times 10^8 = 2.982 \times 10^4 \text{ m} \) or approximately 29,820 meters.
03

Convert Diameter to a More Convenient Unit (Kilometers)

Since the resulting diameter \(2.982 \times 10^4\) meters might be easier to comprehend in kilometers, divide by 1000 to convert the units: \( \text{diameter} = \frac{2.982 \times 10^4}{1000} \approx 29.82 \text{ km} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Angular Resolution
Angular resolution is a crucial concept in understanding how well an optical device, like a telescope or a laser beam, can distinguish between two close objects. Simply put, it describes the smallest angle over which you can see two separate points. For a diffraction-limited system, the formula to calculate angular resolution is \[ \text{angular resolution} = 1.22 \times \frac{\lambda}{D} \] where \( \lambda \) is the wavelength of the light used, and \( D \) is the diameter of the lens or the beam.
This formula tells us that the smaller the wavelength or the larger the beam's diameter, the better the angular resolution and the more detailed and clear the image will be. In our example, we calculated that the angular resolution of a laser beam pointed at the moon is approximately \( 7.72 \times 10^{-5} \) radians.
This small angle means the laser can maintain a relatively focused point over vast distances, such as from Earth to the Moon.
Laser Beam
A laser beam is a narrow and focused stream of photons. Lasers are known for their coherence, monochromaticity, and directionality, which means they can travel long distances without spreading much.
When we say a laser is diffraction-limited, we imply that the beam's spread, or divergence, is as minimal as allowed by the laws of physics, given its wavelength and diameter. Let's talk about the parameters of our laser beam shoot to the moon.
  • Diameter: The beam's diameter influences how focused the light remains over distance. A beam with a small diameter tends to spread more quickly.

  • Distance: Even with a small angular resolution, when a laser travels a massive distance (like 240,000 miles to the moon), the spot it creates inevitably becomes larger.
Wavelength
The wavelength of light is another essential factor when considering diffraction and resolution. It is the distance between successive peaks in a wave of light. This determines the color in the visible spectrum and affects how light interacts with objects and openings.In our exercise, the wavelength is specified as 6328 \( \text{脜} \), or 632.8 nm, which falls within the red part of the visible spectrum. This particular wavelength is common in helium-neon lasers, often used in scientific demonstrations.
Wavelength impacts the angular resolution since longer wavelengths result in poorer resolution in diffraction-limited systems, making it key to stay in the range that provides sufficient clarity for your needs.
Diffraction-limited
In optics, being "diffraction-limited" refers to a system whose performance is reduced only by the physical law of diffraction, and not by imperfections in the optics. Essentially, a diffraction-limited laser or telescope operates at its maximum theoretical resolution. How does diffraction affect a laser beam directed at the moon? Using our calculated angular resolution, the laser beam, although narrow, spreads into a large circle as it travels toward its target.
The diameter of the illuminated area we calculated, about 29.82 km, is the best achievable result given the beam's size and light wavelength we are using. For an ideal diffraction-limited system, this spot size is as refined as possible, emphasizing the influence of physics over design.

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Most popular questions from this chapter

A "corner reflector" consists of three plane mirrors joined so as to form an inside corner of a rectangular box. Show that a light beam that strikes a corner reflector is directed back at 180 deg to its original direction, independent of the angle of incidence, as long as it hits all three surfaces.

A baby-food jar full of air and immersed in water is a diverging lens. Use a fish tank with glass sides, or use an ordinary pan with a mirror to change a vertically downward flashlight beam into a horizontal beam. Put a little milk in the water so you can see the beam. A good pencil-sized beam is obtained from a flashlight covered by an opaque piece of cardboard with an off-center hole. (The flashlight bulb is usually irregular at the tip. Also, you don't want the direct light from the bulb, which falls off as the inverse square of distance, but the parallel beam from the parabolic reflector.) You can study lenses of air and mineral oil and glass using a suspension of milk in water to see the beam.

Burn a piece of toilet paper and look at it through your diffraction grating (held, as always, close in front of one eye). Notice the beautifully clear "first-order flame." This shows that the soft yellow light is almost monochromatic, with very little "white light" color spectrum due to hot carbon. The yellow that you see is the by now familiar (we hope) sodium doublet of wavelengths 5890 and \(5896 \AA\). Now that you recognize "sodium yellow," light an ordinary paper match and look at it with your grating. Most of the light is "hot carbon yellow," which is not really yellow but a complete "white" color spectrum. But look closely! In the yellow part of the hot carbon spectrum, down low next to the cardboard, where the flame is "blue" looking-below the blindingly bright hot carbon spectrum-do you see a crisp, clear little monochromatic match flame? If you don't, try again! Now burn other things and look. You may well conclude that everything is made of salt or is at least contaminated by it.

The world's cheapest broad, almost monochro- matic light source is obtained by burning a wad of toilet paper. You can use this source to see Fabry-Perot fringes. Burn the paper. (The room should be dark-perhaps also you should have some water handy!) Look through the flame at the image of the flame at near- normal incidence in a piece of glass-a microscope slide or picture- frame glass. You will see finger-print-like fringes. If the glass is optically flat, the fringes will be circles centered on your eyeballs; in any case, you can see them easily. If you have a gas stove or bunsen burner, you can get a brighter monochromatic sodium source by sprinkling salt on a wet knife and immersing it in the flame. Then you can see the Fabry-Perot fringes even in the daytime. For a nice, steady, broad monochromatic source with which to look at the fringes, use the G.E. neon bulb NE-34.

Eye-pupil size and mental activity. If someone shows you a picture of a good- looking individual of the opposite sex, your eye-pupil diameter may increase by as much as \(30 \%\), according to Eckhard \(\mathrm{H}\). Hess, Scientific American p. 46 (April, 1965 ). This large a change is very easy to detect in your own pupil by using a pinhole in a piece of aluminum foil that covers one eye, with a bright source illuminating the pinhole, as discussed in Sec. 9.7. Perhaps by just thinking, you can vary your pupil size, depending on what you think about. Have someone read to you. (Concentrate on listening, not on the pupil size.)
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