Why are light rays able to cross each other?

Note: this answer was in response to the original question:

My question is that Why the light rays able to cross each other weather water waves and air could not cross each other

Other waves pass through each other just as with light. This is easy to test. Place four people at the corner of a large room. Have two of them, at adjacent corners talk to the person at a diagonal corner. Use a cone such as a cheerleader might use to somewhat channel the sound. You may be a bit distracted by the other voice but you will clearly hear the voice from the opposite corner.

Here's a standard demo in a high school science class. Have two students hold each end of a moderately stretched slinky resting on a smooth floor. Have each student give the slinky a sharp snap to their right. Since the students are facing each other, the pulses will be opposite one another as they travel toward opposite ends. When the two pulses meet in the middle the slinky will appear relatively straight but only for an instant. The two pulses will continue to travel past one another as if they never had met.

Waves of the same kind traveling through one another maintain their original identity after the encounter. This is a basic property of waves, you can read about it in any introductory Physics text.

I believe it's because you are thinking of the light as particles (little solid balls) that it seems a bit odd that "they can pass through each other." I think thinking in terms of (quantum) fields gets rid of this. As someone mentioned, photons are bosons and so there is no Pauli exclusion principle that applies to them. That is to say, they can be at the same place at the same time....so the fact that they can "pass through" each other seems less odd, knowing that.

Said more simply, the information of the photons spreads out, and there is nothing to prohibit this information from allowing the photons to cross the same point in space at the same time. This is easier to see if you know math and look at the equations explaining photons.

*Note: We are neglecting any interactions between photons in this sort of explanation.

You seem to be looking for a more mathematically oriented answer, so let's try that. You cannot prove that light rays are able to cross each other because that is false: it only works in the approximation of very dim light. If you were to take your source of light and increase the frequency or amplitude of the electric field, you'd observe a non-linear behaviour: rays would very clearly influence each other as the pass through each other.

The key aspect of the superposition principle is linearity, that is, the fact that Maxwell's equations are linear. If you consider electromagnetic radiation in a certain material, it is well-known that for intense enough radiation the polarisation becomes non-linear, and you enter the realm of nonlinear optics. Here, the polarisation $\boldsymbol P$ becomes a (non-linear) function of $\boldsymbol E$, and therefore the wave equation becomes $$ \left(\nabla\times\nabla\times+\frac{n^2}{c^2}\frac{\partial^2}{\partial t^2}\right)\boldsymbol E=\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\boldsymbol P(\boldsymbol E) $$ which is a non-linear equation for $\boldsymbol E$. For example, a very strong source of light (travelling through the air) the Kerr effect or other non-linear effects kick in. This is due to quadratic (and higher) terms in the polarisation tensor $\boldsymbol P\sim\chi_1\boldsymbol E+\chi_2\boldsymbol E\otimes\boldsymbol E+\cdots$.

The polarisation of realistic materials is always non-linear, and therefore light rays always influence each other. Only in the limit $\chi_2E^2\ll \chi_1E$ the wave equation becomes linear; in other words, only in the limit of very weak radiation are light rays oblivious to other light rays.

Even in vacuum you can observe non-linear behaviour. The Maxwell Lagrangian, corrected by quantum-mechanical effects, becomes $$ \mathcal L=\frac12(\boldsymbol E^2-\boldsymbol B^2)+\frac{2\alpha}{45m^4}\left[(\boldsymbol E^2-\boldsymbol B^2)^2+7(\boldsymbol E\cdot\boldsymbol B)^2\right] $$ where $\alpha\sim 1/137$ is the fine-structure constant, and $m$ is the mass of the electron. As $\mathcal L$ is non-linear in $\boldsymbol E,\boldsymbol B$, the propagation of electromangetic waves is no longer linear, and the superposition principle ceases to hold. Only when you can neglect the non-linear terms the superposition principle becomes valid.

In a nutshell, your question is based on a false premise. Light rays seem to be able to cross each other and continue their path unaffected, but this is because you are not using sensitive enough instruments (e.g., your eyes). If you were to measure the effect of light rays on each other with a very good piece of equipment, you'd observe that they do affect each other, both in a material and in vacuum.