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]]>As an example of embedded LaTeX, several of my posts have included what I call the mode-locking integral

\(\displaystyle\frac{1}{2\pi}\int_0^{2\pi}\frac{d\theta}{a+b\sin\theta}=\frac{1}{\sqrt{a^2-b^2}}\)

The text for this equation is LaTeX, not an image! Very exciting!

You might wonder why I named it the “mode-locking integral”. It comes from being the solution to the mode-locking equation

\(\dot{\Psi}(t)=a+b \sin{\Psi(t)}\)

where \(\Psi\) is the relative phase angle between oscillating modes. The modes can be oppositely running waves in a ring laser, the oscillations of two pendulum clocks near one another on a wall, two nearby tuning forks, etc. For more detailed descriptions, see Section 7.5 Mode Locking in the book *Elements of Quantum Optics*, by Pierre Meystre and yours truly. The variable 𝑎 is the frequency difference between the oscillating modes in the absence of coupling, and the variable 𝑏 is the coupling coefficient. If |𝑎| > |𝑏|, the integral is valid. When |𝑎| ≤ |𝑏|, \(\dot{\Psi}=0\), which has the solution \(\Psi=-\sin^{-1}{\frac{a}{b}}\), and the modes are locked to the same frequency.

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