Understanding single mode laser dynamics
In this post, I explore the intricacies of single mode lasers, with a particular emphasis on the Helium-Neon laser. As a continuous wave (CW) laser, the Helium-Neon laser operates in a steady state within a resonant cavity, allowing me to derive and discuss the number of photons, their distribution, and the resultant implications on measurement precision.
Single mode laser setup
A single mode laser consists of a resonant cavity formed by mirrors, where light amplifies by repeatedly bouncing between these reflective surfaces. The cavity is usually cylindrical, and in a Helium-Neon laser, it contains a gas mixture of helium and neon. The length of the cavity, denoted as L_{\text{cav}}, is pivotal because it determines the resonant frequencies of the cavity. The mirrors at the ends, M_0 and M_1, are highly reflective, while the semi-transparent mirror M_S allows a fraction of the light to transmit outside the cavity.
The transmission coefficient T and reflection coefficient R of M_S are crucial for understanding the power dynamics within the cavity:
P_{\text{out}} = T \cdot P_{\text{cav}}
where P_{\text{out}} is the power output and P_{\text{cav}} is the power circulating within the cavity.
Quantizing the radiation in the cavity
The radiation in the cavity can be described using a quasi-classical approach, where the state of the light is represented by a large complex number, implying a high photon count:
\langle n_{\text{cav}} \rangle \hbar \omega_\lambda = P_{\text{cav}} \frac{L_{\text{cav}}}{c}
Here, \hbar \omega_\lambda is the energy per photon, with \omega_\lambda being the angular frequency of the light.
Calculating photon distribution
For a typical Helium-Neon laser with an output power of 1 milliwatt and a wavelength of 633 nm, the detailed calculations yield:
\langle n_{\text{cav}} \rangle = \frac{P_{\text{out}}}{T} \frac{L_{\text{cav}}}{c} \cdot \frac{1}{\hbar\omega_\lambda}
Substituting the values:
\langle n_{\text{cav}} \rangle = \frac{1}{1.0545718 \times 10^{-34} \cdot 2.98 \times 10^{15}} \times \frac{10^{-3}}{10^{-2}} \times \frac{0.6}{3 \times 10^8} \approx 6.4 \times 10^8
Implications of photon distribution
The high number of photons results in a low relative dispersion:
\frac{\Delta N_{\text{cav}}}{\langle N_{\text{cav}} \rangle} = \frac{1}{\sqrt{\langle N_{\text{cav}} \rangle}} \approx 4 \times 10^{-5}
This small dispersion underpins the laser’s precision and its utility in scientific measurements, where stability and accuracy are paramount. The phenomenon, known as shot noise, is critical in determining the standard quantum limit of measurements.
Conclusion
The physics of single mode lasers, especially CW lasers like the Helium-Neon, is fundamental in various applications, from scientific research to industrial uses. Understanding the photon dynamics within the resonant cavity not only helps in optimizing the laser’s performance but also enhances the precision in applications like metrology and spectroscopy.
For more insights into this topic, you can find the details here.