Jacobi-Anger Expansion

The Jacobi-Anger expansion expresses a complex exponential with sinusoidal argument as an infinite series of Bessel functions:

$$ e^{jz\sin\theta} = \sum_{n=-\infty}^{\infty} J_n(z) \, e^{jn\theta} $$

where $J_n(z)$ is the Bessel function of the first kind of order $n$.

Equivalent Forms

Cosine argument:

$$ e^{jz\cos\theta} = \sum_{n=-\infty}^{\infty} j^n J_n(z) \, e^{jn\theta} $$

Real-valued form (using Euler’s formula):

$$ \cos(z\sin\theta) = J_0(z) + 2\sum_{n=1}^{\infty} J_{2n}(z)\cos(2n\theta) $$$$ \sin(z\sin\theta) = 2\sum_{n=0}^{\infty} J_{2n+1}(z)\sin((2n+1)\theta) $$

Application to Phase Gratings

For a sinusoidal phase grating with transmittance $t(\xi) = e^{j(m/2)\sin(2\pi f_0 \xi)}$, the Jacobi-Anger expansion decomposes the field into diffraction orders:

$$ e^{j(m/2)\sin(2\pi f_0 \xi)} = \sum_{q=-\infty}^{\infty} J_q(m/2) \, e^{j2\pi q f_0 \xi} $$

Each term $J_q(m/2) \, e^{j2\pi q f_0 \xi}$ represents the $q$th diffraction order, with amplitude $J_q(m/2)$ determined by the modulation depth $m$.

Connection to FM Modulation

The expansion is mathematically identical to the sideband structure of a frequency-modulated signal, where the modulation index plays the role of $z$ and the Bessel functions determine the sideband amplitudes.