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I am currently studying the textbook Laser Systems Engineering by Keith Kasunic. Chapter 1.2.1 Temporal coherence says the following:

Even a frequency-stabilized SLM laser has a linewidth – an unavoidable consequence of random spontaneous-emission phase variations resulting from quantum fluctuations in excited-state energy levels. For such lasers, the ideal single-mode (Schawlow-Townes) linewidth $\Delta \nu_{\text{ST}}$ for a laser cavity $\Delta \nu_{\text{R}}$ (not its etalon) is given by Refs 9 and 12 as $$\Delta \nu_{\text{ST}} = \dfrac{\pi h \nu \cdot (2 \pi \Delta \nu_{\text{R}})}{P} \ \ \ \text{[Hz]} \tag{1.8}$$ where $P$ is the output power of the given axial mode. Physically, the higher power has a larger fraction of amplified in-phase photons in comparison with the incoherent, out-of-phase photons from spontaneous emission, reducing $\Delta \nu_{\text{R}}$ of the laser's axial modes [also Eq. (1.7)] given the laser-cavity reflectivity $R$.

Equation (1.8) represents the ultimate lower limit on linewidth, with no commercial products able to achieve this level of frequency stability ($< 1 \ \text{Hz})$. In practice, some SLM fibre lasers have a linewidth smaller than $10 \ \text{kHz}$ – limited by "technical noise" (fluctuations in optical and optomechanical parameters, e.g.), and resulting in a coherence length $d_c \ge 15 \ \text{km}$.

For details, the Wikipedia article regarding Eq. (1.8) can be found here.

I'm curious about this part:

Equation (1.8) represents the ultimate lower limit on linewidth, with no commercial products able to achieve this level of frequency stability ($< 1 \ \text{Hz})$. In practice, some SLM fibre lasers have a linewidth smaller than $10 \ \text{kHz}$ – limited by "technical noise" (fluctuations in optical and optomechanical parameters, e.g.), and resulting in a coherence length $d_c \ge 15 \ \text{km}$.

Why can no commercial products achieve a frequency stability of $< 1 \ \text{Hz}$, and how does (1.8) show this?

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In the meanwhile there are commercial laser systems with stability below 1 Hz.[1] However, this is still very challenging. Your equation $(1.8)$ is the Schawlow-Townes linewidth, which can be understood as the fundamental physical lower limit for laser linewidth. In reality other effects are dominating. If the optical length of the laser resonator changes due to

  • temperature
  • mechanical vibrations
  • changes in the composition of the gas in the resonator
its resonance frequency and therefore the frequency of emission changes. Consider a resonator with a round-trip length of $L = 1 \, \text{m}$ emitting at a frequency of $f = 500 \, \text{THz}$. Then the length change $\Delta L$ which changes the resonance frequency by $\Delta f = 1 \, \text{Hz}$ is $$\Delta L = L \frac{\Delta f}{f} = 2 \, \text{fm} \text{.}$$ To achieve such a stability one needs to eliminate all fluctuations in the environment carefully and isolate the system well. Additionally one needs to actively regulate the cavity length, for example by a mirror mounted on a piezo crystal. As frequency reference one uses monolithic well-isolated ultrastable external cavity. This is more stable than the laser cavity, because it doesn't have any active elements besides temperature stabilization. Even better frequency references can be optical transitions in atoms[2], but I don't know of any commercial system yet.
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  • $\begingroup$ Thanks for the answer. There's still one thing I'm confused about: If the Schawlow-Townes linewidth is the fundamental physical lower limit for laser linewidth, then how can we get below $1 \ \text{Hz}$, which the author says is physically impossible due to the same equation? Is the equation actually invalid somehow? Or was one of the original Schawlow-Townes assumptions incorrect (since, as your article states, the equation was discovered before even the demonstration of the first laser)? $\endgroup$ Jan 17, 2021 at 22:03
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    $\begingroup$ @ThePointer As far as I understand the quote they mean that there was no commercial laser system with a linewidth below $1 \, \text{Hz}$ at the time of writing. But this is not a typical value for the Schawlow-Townes limit. In the case described in my answer with the $L = 1 \, \text{m}$ resonator assuming a finesse of $100$ and an output power of $P = 1 \, \text{W}$ the Schawlow-Townes limit is $\Delta f_{\text{ST}} = 18.7 \, \text{µHz}$. $\endgroup$
    – A. P.
    Jan 17, 2021 at 22:19
  • $\begingroup$ Ahh, ok; in that case, I misinterpreted what the author was saying. Thanks for taking the time to clarify this. $\endgroup$ Jan 17, 2021 at 22:42

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