Question

The fact that a laser's resonant cavity so effectively sharpens the wavelength can lead to the output of several closely spaced laser wavelengths, called longitudinal modes. Here we see how. Suppose the spontaneous emission serving as the seed for stimulated emission is of wavelength $633\mathrm{nm}$, but somewhat fuzzy, with a line width of roughly $0.001\mathrm{nm}$ either side of the central value. The resonant cavity is exactly $60\mathrm{cm}$ kng. (a) How many wavelengths fit the standing-wave condition? (b) If only u single wavelength were desired. would changing the length of the cavity help? Explain.

Short answer

Approximately $947,900$ wavelengths fit the standing-wave condition in the cavity. Adjusting the length of the cavity would not effectively isolate a single wavelength because the spontaneous emission of wavelength is somewhat fuzzy around 633 nm.

Step-by-step solution

Determine the Number of Wavelengths

First, convert the length of the cavity from centimeters to nanometers to match the units for the wavelength. This calculation will yield the cavity length in nanometers = $60cm\ast 1\times {10}^{7}nm/cm=6\times {10}^{8}nm$. The number of wavelengths that fit in the cavity is given by the length of the cavity divided by the wavelength. Therefore, to calculate the number of wavelengths, divide the length of the cavity in nanometers by the given wavelength. The number of wavelengths = $6\times {10}^{8}nm/633nm=9.479\times {10}^5$. Thus, approximately $947,900$ wavelengths fit the standing-wave condition.

Determine if changing the length of the cavity would help

In this part, we are asked about the impact on the output if the cavity is modified for a single wavelength. Even though we have counted how many wavelengths fit in the cavity, we have to remember that the initial emission is not at an exact wavelength but varies slightly around 633 nm. This means there's a range around 633 nm that fits into the resonant cavity. If we adjust for just one mode (a single wavelength), other wavelengths from this range will still fulfill the resonant condition. Therefore, adjusting the cavity length would not effectively isolate a single wavelength due to the fuzzy line width of the initial emission.

Key concepts

Longitudinal Modes

In a laser system, longitudinal modes represent distinct wavelengths of light that resonate within the laser's cavity. These modes occur because the laser cavity only supports waves that satisfy the standing-wave condition, meaning the waves must fit exactly within the length of the cavity with an integral number of half-wavelengths.

Think of longitudinal modes like the distinct notes produced by a guitar string. Just as only certain vibrations produce clear tones on a string, only specific wavelengths form stable standing waves in a laser cavity.

• When conditions are right for stimulated emission, numerous wavelengths close to each other can be amplified.
• However, this 'fuzziness' can lead to multiple wavelengths being emitted simultaneously.
• The quantity of these modes is influenced by both the physical length of the cavity and the wavelength of light.

In the provided exercise, the number of wavelengths that fit within a 60 cm resonant cavity for a 633 nm emission is approximately 947,900. This illustrates the potential for a large number of longitudinal modes.

Stimulated Emission

The concept of stimulated emission is at the heart of how lasers work. When an electron in an atom is excited to a higher energy level, it can fall back to a lower level by releasing a photon of light. If this happens naturally, it's called spontaneous emission. But in stimulated emission, an incoming photon can interact with the excited electron to trigger the release of another photon.

This second photon is identical to the incoming one: it has the same wavelength, phase, direction, and polarization. This process is what allows a laser to emit a concentrated beam of light.

• To achieve laser action, a population inversion is required, where more electrons are in an excited state than a lower one.
• The resonant cavity of a laser helps maintain and amplify these emissions through repeated reflections back and forth.
• As seen in the exercise, a 'fuzzy' line width means that several wavelengths around the target value can be equally stimulated, leading to multiple longitudinal modes.

Effective laser operation relies on managing stimulated emissions to maintain a desired purity of the light output.

Standing-Wave Condition

The standing-wave condition is a requirement for the stable operation of a laser resonant cavity, dictating that only certain wavelengths of light will be amplified to become part of the laser's output. In a fixed cavity length, only the wavelengths that are an integral multiple of half the cavity's length will be reinforced and sustained.

In our analogy, the guitar string vibrates to produce standing waves at specific frequencies corresponding to musical notes. Similarly, the laser's cavity only 'rings' with certain wavelengths that complete whole 'loops' from one end to the other.

• Altering the length of the cavity can change the wavelengths that fulfill the standing-wave condition.
• For lasers with adjustable cavities, tuning the length can refine the output to fewer longitudinal modes.
• However, as the exercise suggests, due to the initial fuzzy emission width, a simple adjustment in cavity length will not completely isolate a single wavelength.

In essence, the standing-wave condition governs the selection of longitudinal modes, shaping the laser's spectral output.