How does the grid on the microwave oven window prevent microwave radiation from coming out? If I look through the microwave window I can see through, which means visible radiation can get out. We know also that there is a mesh on the microwave window which prevents microwave from coming out.
My question is how does this work? how come making stripes or mesh of metals can attenuate microwave radiation yet allow visible radiation?
Looks like an electrodynamics problem to me with periodic boundary conditions (because of the partitions on the microwave oven window). Is it discussed in any textbook?
 A: As John and others have said, the wavelength of the microwaves is very large compared to the size of the holes in the screen which allows the screen to act as a solid.  Visible light has much smaller wavelengths and can pass through the holes unobstructed.  It isn't possible to see (resolve) objects and features smaller than the wavelength of light (electromagnetic radiation) used so this is why the mesh works.  See http://hyperphysics.phy-astr.gsu.edu/hbase/waves/mwoven.html for more details.
A:    The metal mesh, or 'cage' around a microwave's oven cavity acts as a Faraday cage (see Wikipedia article on Faraday cage here), although a 'true' Faraday cage is grounded, and a microwave cage is not.
   A cellular phone inside a Faraday cage will be protected from outside EM transmissions, just as conversely, the transmissions of the phone inside the cage will be blocked from reaching outside the cage.
   According to Wikipedia, a Faraday cage can be thought of as an approximation to an ideal hollow conductor; When an external electrical field is applied to the cage, the electrons in the metal move towards the side of the metal that is closest to the source of the transmission, giving it a negative charge, while the remaining unbalanced charge of the nuclei give the other side a positive charge. These induced charges create an opposing electric field that cancels the external electric field throughout the box.
   However you wish to visualize the principals that govern how a Faraday cage with holes works. It is well established that to block a transmission of a particular frequency, size of the largest hole in the Faraday cage must be AT MOST 1/2 the wavelength of the frequency of the undesired transmission. 
   According to an online calculator, the wavelength of 2.45 GHz (the frequency of most domestic microwave ovens), the wavelength will be approximately 12.24 cm, or 4.82 inches. Taking half of that, we learn that the holes on a microwave oven could be 2.41 inches at most, although I'm not sure I would be very comfortable with that!
   Now in theory, light could be blocked on the same principle, but considering that visible light has a wavelength somewhere around 390 to 700 nm, you can see now why visible light passes through the mesh of a microwave door, where as microwaves do not; the gaps would have to be on the order of 200-350 nanometers to block light.
   So why, then, are the holes in the microwave's viewing window so small when they could be a lot bigger? Well, in reality, the EM waves do travel outside past the hole, beyond the boundaries of the metal cage for a bit. The distance drops off at some exponential rate, depending on the inverse proportion of the gap versus the wavelength, and having a gap less than half the wavelength only assures us that the wave will [eventually] terminate [at some distance] beyond the cage, and I speculate that manufacturers choose the size of the gap that they did out of an abundance of caution. Also, if you mistakenly put metal into a microwave, you would get arcing and re-transmission of EM energy, possibly at different frequencies.
   Also note here that to get microwave leakage, a gap only needs to be larger than 1/2 the wavelength ALONG ONE AXIS. That is, a gap that is 6.12 cm long but only 0.5 mm wide, will leak significant microwave radiation. That's why most microwaves' doors have direct metal-to-metal contact, a conductive bushing, or a metallic lip lining the doorway. This is why its important to replace your microwave if the conductive bushing (door seal gasket) is falling off or has gaps, for those who's microwaves have such a seal.
   I looked at my microwave door, and the holes on the front are approximately 1 mm wide. Working backwards from length, a 2 mm wavelength would be at a frequency of 150 GHz. Since most antenna are some fraction of a wavelength, typically 1/2 or 1/4, a 1 mm antenna could produce a signal of this frequency, but I am unsure as to whether such an antenna would be very efficient at re-transmitting a 2.45 GHz signal, and what frequency that would be at; I am not much for antenna theory. I do feel, however, it would be very efficient at arcing and vaporizing itself rather quickly.
