Question

A laser produces a coherent beam of light that does not spread (diffract) as much in comparison to light from other sources, like an incandescent bulb. Lasers therefore have been used for very accurate measurements of large distances, such as the distance between the Moon and the Earth. In one such experiment, a laser pulse (wavelength of \(633 \mathrm{nm})\) is fired at the Moon. What should be the size of the circular aperture of the laser source in order to produce a central maximum of 1.00 -km diameter on the surface of the Moon? The distance between the Moon and the Earth is \(3.84 \cdot 10^{5} \mathrm{~km}\)

Short answer

Answer: The diameter of the circular aperture required to produce a central maximum of 1.00-km diameter on the surface of the Moon is approximately 0.292 mm.

Step-by-step solution

Find the angular size of the central maximum

Let's use the formula for the angular size of the central maximum in a diffraction pattern produced by a circular aperture: $$\theta = 1.22 \frac{\lambda}{D}$$ where \(\theta\) is the angular size, \(\lambda\) is the wavelength of light, and \(D\) is the diameter of the aperture. We are given the wavelength: \(\lambda = 633 \mathrm{~nm} = 633 \cdot 10^{-9} \mathrm{~m}\). We need to find the angular size corresponding to the diameter of the central maximum of 1.00-km on the surface of the Moon. To do this, we can use the formula that relates the linear size, angular size, and the distance between the observer and the object: $$L = R \theta$$ where \(L\) is the linear size, \(R\) is the distance between the observer and the object, and \(\theta\) is the angular size. In our case, \(L = 1.00 \mathrm{~km} = 1000\mathrm{~m}\) and \(R= 3.84 \cdot 10^{5} \mathrm{~km} = 3.84 \cdot 10^{8} \mathrm{~m}\). Now we can solve for \(\theta\): $$\theta = \frac{L}{R} = \frac{1000\mathrm{~m}}{3.84 \cdot 10^{8} \mathrm{~m}} = 2.60 \cdot 10^{-6}$$

Calculate the diameter of the circular aperture

Now, plug in the values of \(\theta\) and \(\lambda\) into the formula for the angular size of the central maximum: $$2.60 \cdot 10^{-6} = 1.22 \frac{633 \cdot 10^{-9} \mathrm{~m}}{D}$$ To find the diameter \(D\) of the circular aperture, we can solve this equation for \(D\): $$D = 1.22 \frac{633 \cdot 10^{-9} \mathrm{~m}}{2.60 \cdot 10^{-6}} = 2.92 \cdot 10^{-4} \mathrm{~m} $$

Convert the diameter to millimeters and report the answer

Finally, we can convert the diameter of the aperture from meters to millimeters: $$D = 2.92 \cdot 10^{-4} \mathrm{~m} \times \frac{1000\,\mathrm{mm}}{1\,\mathrm{m}} = 0.292\,\mathrm{mm}$$ The diameter of the circular aperture required to produce a central maximum of 1.00-km diameter on the surface of the Moon is approximately 0.292 mm.

Key concepts

Angular Size Calculation

Understanding the concept of angular size is pivotal when discussing the resolution of optical systems. Angular size refers to the angle that a particular object appears to take up in the field of view. In mathematical terms, it's described by the formula:
\[\begin{equation}\theta = \frac{L}{R}\end{equation}\]
where \(L\) is the linear size of the object, \(R\) is the distance to the object, and \(\theta\) is the angular size. This calculation is especially important in astronomy and laser technology where we deal with vast distances and need to pinpoint very small areas. An exercise that often puzzles students involves calculating the right aperture size of a laser source so that the light spreads just enough to fit a specified area at a distance - say a spot on the Moon. Here, knowing how to compute the angular size allows us to reverse-engineer the size of the laser's aperture based on the desired size of the light spot and the distance to the Moon.

Laser Distance Measurement

Laser distance measurement works on the principle of emitting a laser beam and detecting how long it takes for it to reflect back from a target, or by evaluating the spread of the beam over distance. However, for astronomical distances, we use lasers not to time reflections but to create a distinct, highly focused beam. The precision of laser measurement can be affected by many factors including diffraction, which leads to expansion of the beam as it travels. Efficiently minimizing diffraction is crucial in applications like measuring the Earth-Moon distance, where even minor inaccuracies can result in significant errors. The trick lies in calculating an apt aperture that limits diffraction sufficiently for the laser to illuminate a specified area on a distant body, like creating a 1-km wide spot on the Moon, as per the textbook exercise.

Circular Aperture Diffraction

Diffraction occurs when light bends around the edges of an obstacle or passes through an aperture, such as the circular opening of a laser source. It results in the spreading of the light wave, which in case of a circular aperture, leads to a specific pattern known as the Airy disk, central to many optical measurements. Scientists use the formula to predict the angular size of the central bright spot (Airy disk) of diffraction which can be expressed as:
\[\begin{equation}\theta = 1.22 \frac{\lambda}{D}\end{equation}\]
In this formula, \(\lambda\) represents the wavelength of light and \(D\) is the diameter of the circular aperture. It's useful in designing precise optical systems, ensuring the central maximum's size aligns with our requirements for resolution, as in determining what size aperture would create a 1-km spot from Earth to Moon.

Wavelength of Light

The wavelength of light is the distance between two consecutive peaks (or troughs) in a light wave and is typically measured in nanometers (nm) for visible light. The concept of wavelength is integral to understanding not only color but also the behavior of light in different mediums and circumstances, e.g., while passing through a slit or when used in laser applications. The wavelength has an inverse relationship with the energy of a light wave; shorter wavelengths correspond to higher energy and vice versa. For instance, a standard red helium-neon laser has a wavelength of around 633 nm, and this value is crucial when applying the formulae mentioned above for diffraction calculations. Knowledge of the laser's wavelength is instrumental in figuring out how light will spread out after passing through the aperture, directly impacting distance measurements and patterns created at a distance, like the diffraction spot on the Moon.