This is a problem in Goldman's 'Introduction to Fourier Optics', Chapter 2.

Please show that zero-order Bessel function $J_0(2\pi\rho_0r$) is eigenfunction of an invariant linear system with circularly symmetric response.

I have tried two methods. One is introduced on linear time-invariant system by Wikipedia. It proves that complex exponential function is eigenfunction of any invariant linear system. The methods depends on the communitive property of convolution. It works in a pretty clear way in $\exp(ikx)$ but not in $J_0(kx)$.

Another one is to transform the convolution in spatial domain into multiplication in frequency domain. The transformation of $J_0$ is shown here: Fourier transform of Bessel functions. Not very simple though. The fourier transformation of circularly symmetric impulse response involves $J_0$ in an integral again. I cannot show that the latter is scalar.

Already tried for one week on this problem. Any feedback is strongly appreciated!


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