We can express plane wave in terms of a spherical waves in 3D as: \begin{equation} e^{ikz}=\frac{2\pi i}{k}\delta(\vec{n}-\vec{e}_z)\frac{e^{ikr}}{r}+\frac{2\pi i}{k}\delta(\vec{n}+\vec{e}_z)\frac{e^{-ikr}}{r} \end{equation}
Where $\vec{n}$: vector in radial direction
and $\vec{e}_z$: basis vector in z direction
I want to know what is it's analogue in 2D ?
The expression you wrote is not right, here is the correct expansion: using spherical coordinates $(r,\theta, \phi)$, there is something called partial wave decomposition, and for the wave $e^{i\vec k\cdot\vec r}$ it gives
$$e^{i\vec k\cdot\vec r} =\sum_{l=0}^{+\infty}i^l(2l+1)j_l(kr)P_l(\cos \theta)$$ where $\theta$ is the angle between the vertors $\vec k$ (the direction of propagation) and $\vec r$, $P_l(x)$ are the Legendre polynomials https://en.wikipedia.org/wiki/Legendre_polynomials and the $j_l(x)$ are the modified Bessel functions of the first kind https://en.wikipedia.org/wiki/Bessel_function#Modified_Bessel_functions.
In the particular case of a wave propagating in the $z$ direction, i.e $\theta=0$, $\cos\theta=1$, the decomposition semplifies because $P_l(1)=1 \quad\forall l$.
The most interesting aspect though, is that these Bessel functions are oscillating functions with a decreasing amplitude, and for large $r=|\vec r|$ it can be shown that is valid the approximation
$$j_l(kr)\approx \dfrac{1}{kr}\sin\left( kr-l\frac{\pi}{2} \right)= \dfrac{1}{2ikr}\left(e^{ikr}e^{-il\pi/2}-e^{-ikr}e^{il\pi/2}\right)=\\ \dfrac{1}{2ikr}\left((-i)^le^{ikr}-i^le^{-ikr}\right)$$
So, for large $r$ it is clear that we have a decomposition in outgoing and ingoing spherical waves. I don't know your background, but in quantum mechanics each term is the component of a free particle with angular momentum $l$, and this is relevant in scattering theory.
Similarly to the 3D case it can be shown in 2D, using polar coordinates, that it is valid the expansion $$e^{i\vec k\cdot\vec r}=e^{ikr\cos\theta}=\sum_{m=0}^{+\infty}\epsilon_mi^m\cos(m\theta)J_m(kr)$$ where $\epsilon_0=1$ and $\epsilon_m=2\forall m>0$, where the functions $J_m(x)$ are another kind of Bessel functions (cylindrical Bessel functions) that have a similar approximation for large $r$.
