In three of the Apollo lunar experiments, astronauts left...

In three of the Apollo lunar experiments, astronauts left reflector panels on the moon so that laser beams from the earth could be reflected off the panels and back to earth. Lasers with $\lambda=532 \mathrm{~nm}$ send pulses of $0.3 \mathrm{~J}$ to the moon, and the round-trip time is measured within $\delta t=0.4 \mathrm{~ns}$. In this way the one-way distance to the reflector on the moon is determined regularly within $\delta l=c \delta t / 2 \approx 6 \mathrm{~cm}$. The beams from the lasers have a diameter of $5 \mathrm{~km}$ on the moon. (This is the result of sprcading due to diffraction and to atmosphoric turbulence.) The reflector panels contain 300 mirrors, each of diameter $4 \mathrm{~cm}$. (a) Find the number of photons sent from the earth in a single pulse. (b) Find the fraction of these photons that strike any one of the small mirrors. (c) Find the angular divergence $\delta \theta \approx \lambda / d$ of the reflected beam from this mirror (where $d$ is the mirror's diameter). (d) What is the diameter of the reflected beam when it returns to earth? (e) The return light is measured by a photomultiplier at the focus of a telescope. What fraction of the return light is captured by the telescope, whose diameter is $1 \mathrm{~m}$ ? (f) For a single pulse, find the total number of photons captured by the telescope from all 300 mirrors (The actual number is somewhat smaller because of losses in the atmosphere and in the telescope.) This experiment monitors the moon's orbit within a few centimeters (compared to the earth-moon distance of $4 \times 10^{\mathrm{s}} \mathrm{m}$ ), allowing stringent tests of competing theories of gravity.

In three of the Apollo lunar experiments, astronauts left reflector panels on the moon so that laser beams from the earth could be reflected off the panels and back to earth. Lasers with $\lambda=532 \mathrm{~nm}$ send pulses of $0.3 \mathrm{~J}$ to the moon, and the round-trip time is measured within $\delta t=0.4 \mathrm{~ns}$. In this way the one-way distance to the reflector on the moon is determined regularly within $\delta l=c \delta t / 2 \approx 6 \mathrm{~cm}$. The beams from the lasers have a diameter of $5 \mathrm{~km}$ on the moon. (This is the result of sprcading due to diffraction and to atmosphoric turbulence.) The reflector panels contain 300 mirrors, each of diameter $4 \mathrm{~cm}$. (a) Find the number of photons sent from the earth in a single pulse. (b) Find the fraction of these photons that strike any one of the small mirrors.
(c) Find the angular divergence $\delta \theta \approx \lambda / d$ of the reflected beam from this mirror (where $d$ is the mirror's diameter). (d) What is the diameter of the reflected beam when it returns to earth? (e) The return light is measured by a photomultiplier at the focus of a telescope. What fraction of the return light is captured by the telescope, whose diameter is $1 \mathrm{~m}$ ? (f) For a single pulse, find the total number of photons captured by the telescope from all 300 mirrors (The actual number is somewhat smaller because of losses in the atmosphere and in the telescope.) This experiment monitors the moon's orbit within a few centimeters (compared to the earth-moon distance of $4 \times 10^{\mathrm{s}} \mathrm{m}$ ), allowing stringent tests of competing theories of gravity.

Key Concepts

Time of Flight and Distance Measurement

The time of flight concept involves measuring the duration it takes for a light pulse to travel to a target and back. Using the speed of light, one determines the distance to the target by applying the formula distance = (c × ?t) / 2. This method is a cornerstone in precise distance measurements and ranging experiments.

Optical System Collection Efficiency

Collection efficiency in optical systems is determined by the ratio of the detector or telescope aperture area to the area of the incoming light beam. This concept is critical in the design of telescopes and photodetectors, as it defines how much of the reflected or scattered light is actually captured for measurement.

Reflector Cross-section and Photon Capture Efficiency

This concept deals with evaluating the fraction of the light beam intercepted by a target, based on the ratio of the target’s effective area to the total beam area. It is important in scenarios where a small reflective surface, such as a mirror, needs to capture a portion of a widely spread laser beam for effective signal return.

Diffraction and Angular Divergence

Diffraction is the phenomenon where light waves spread when they encounter an aperture or an obstacle. The angular divergence of a beam, which is influenced by diffraction, is approximately given by ?? ? ?/d, where d is the effective diameter of the light source or mirror. This principle is essential for understanding how beams expand over long distances and affects beam quality in laser experiments.

Photon Energy and Photon Count

This concept involves calculating the energy carried by individual photons using the relation E = hc/?, where h is Planck’s constant, c is the speed of light, and ? is the wavelength. By determining the energy per photon, one can estimate the total number of photons in a light pulse by dividing the pulse’s energy by the energy of one photon—a fundamental idea in quantum mechanics and photonics.

Beam Spreading and Spot Size

Beam spreading refers to the increase in the cross-sectional area of a laser beam as it propagates, primarily due to diffraction and atmospheric effects. The resulting spot size at a distant target is crucial in determining how much of the light actually interacts with a small target area, which impacts the efficiency of optical systems in remote sensing and ranging.

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