In a research paper that I'm currently going through, spectrally modulated light is being described with the optical probe beam written as: $$E_0 e^{i(\omega_0 t+M sin(\omega_mt))}$$ Where $\omega_0$ is the carrier frequency, $\omega_m$ is the wavelength modulation, and $M$ is the modulation index (although the conceptual significance of these variables is not extremely relevant to the question which I'm posing).
The paper then utilizes Fourier Series to remove the periodic nature of the $sin(\omega_m t)$ in the exponent, which then gives the following equation: $$E_0 e^{i(\omega_0 t+M sin(\omega_mt))}=E_0 e^{i \omega_0 t} \sum_{n=-\infty}^{+\infty}{J_n(M) e^{i n \omega_m t}}$$
How does the infinite sum of the Bessel Function from $-\infty$ to $\infty$, multipled obviously by $e^{i n \omega_m t}$, mathematically replace the $e^{sin(\omega_m t)}$ in the original equation?
The exponential term $\mathrm e^{iM\sin\omega_mt}$ is a periodic signal with period $\frac{2\pi}{\omega_m}$ and can be expanded by the exponential Fourier series: $$ \mathrm e^{iM\sin\omega_mt}=\sum_{n=-\infty}^\infty C_n\mathrm e^{in\omega_mt} $$ where
$$ C_n=\frac{\omega_m}{2\pi}\int_{-\pi/\omega_m}^{\pi/\omega_m}\mathrm e^{iM\sin\omega_mt}\,\mathrm e^{-in\omega_mt}\mathrm d t $$ By changing variables $\omega_mt= x$, we get $$ C_n=\frac{1}{2\pi}\int_{-\pi}^{\pi}\mathrm e^{i(M\sin x-nx)}\,\mathrm d t=J_n(M) $$ where the integral is the integral representation of the Bessel function $J_n(M)$ of the first kind and order $n$