The presence or absence of inversion symmetry in a medium has a direct impact on the types of nonlinear interactions that it can support; specifically, media which do have inversion symmetry cannot support nonlinear effects of even order.

The reason for this is that adding an even harmonic to the fundamental will yield an asymmetric dependence of the electric field, and this is only possible if the medium itself is asymmetric.

"Inversion symmetry" is the property that the material remain the same when you change the position rj of each particle j to its 'inverse', −rj. Because we can typically move materials around, this is equivalent to saying that the medium is identical to its mirror image. This is true, for example, for a gas (if the rj are random, then the −rj will also be random), or for crystals like body-centred cubic lattices [http://i.stack.imgur.com/B5xHc.png].

On the other hand, certain lattices do not have this symmetry, like you get when you displace the atom in the centre towards one of the faces of the cubic unit cell [http://i.stack.imgur.com/McMZi.png]. In this case the symmetry is broken, and if you invert all the coordinates with respect to (say) the middle atom, you no longer recover the original lattice.

It is fairly easy to see why this asymmetry is necessary for second-harmonic generation. Suppose that the second harmonic has a maximum in the +z direction at the same time as the fundamental, so that they add constructively. If you wait for half a period of the fundamental, its electric field will be in the −z direction, but the second harmonic will have undergone a full period and will be pointing towards +z, so that the two interfere destructively [http://i.stack.imgur.com/iv8A1.png].

This means that the maximum of the total field is stronger in the +z direction than it is in the −z direction. This is actually quite remarkable! In particular, the medium itself needs to be asymmetric to "know" which direction the stronger fields need to go towards. If the medium has inversion symmetry, then the +z and −z directions are equivalent, and an asymmetric output like this is impossible.

Consider, on the other hand, a process with odd order like third harmonic generation. Here a half-integral period of the fundamental is also a half-integral period of the harmonic, which means that they add in the same direction in each half-cycle, and the output is symmetric [http://i.stack.imgur.com/1FPZG.png].

[by Emilio Pisanty]

http://physics.stackexchange.com/questions/127531/lack-of-inversion-symmetry-in-crystal

Nonlinearity of group of atoms (ions) is a sum of nonlinearities of each interatomic bond (with account for the phases of nonlinear polarization). Symmetry, e.g., of crystal unit cell, means certain order in the atoms' positions and, correspondingly, in the bonds. Zero components correspond to completely destructive interference of contributions from all bonds. Naturally, it is governed by the symmetry of unit cell.

The presence or absence of inversion symmetry in a medium has a direct impact on the types of nonlinear interactions that it can support; specifically, media which do have inversion symmetry cannot support nonlinear effects of even order.

The reason for this is that adding an even harmonic to the fundamental will yield an asymmetric dependence of the electric field, and this is only possible if the medium itself is asymmetric.

"Inversion symmetry" is the property that the material remain the same when you change the position rj of each particle j to its 'inverse', −rj. Because we can typically move materials around, this is equivalent to saying that the medium is identical to its mirror image. This is true, for example, for a gas (if the rj are random, then the −rj will also be random), or for crystals like body-centred cubic lattices [http://i.stack.imgur.com/B5xHc.png].

On the other hand, certain lattices do not have this symmetry, like you get when you displace the atom in the centre towards one of the faces of the cubic unit cell [http://i.stack.imgur.com/McMZi.png]. In this case the symmetry is broken, and if you invert all the coordinates with respect to (say) the middle atom, you no longer recover the original lattice.

It is fairly easy to see why this asymmetry is necessary for second-harmonic generation. Suppose that the second harmonic has a maximum in the +z direction at the same time as the fundamental, so that they add constructively. If you wait for half a period of the fundamental, its electric field will be in the −z direction, but the second harmonic will have undergone a full period and will be pointing towards +z, so that the two interfere destructively [http://i.stack.imgur.com/iv8A1.png].

This means that the maximum of the total field is stronger in the +z direction than it is in the −z direction. This is actually quite remarkable! In particular, the medium itself needs to be asymmetric to "know" which direction the stronger fields need to go towards. If the medium has inversion symmetry, then the +z and −z directions are equivalent, and an asymmetric output like this is impossible.

Consider, on the other hand, a process with odd order like third harmonic generation. Here a half-integral period of the fundamental is also a half-integral period of the harmonic, which means that they add in the same direction in each half-cycle, and the output is symmetric [http://i.stack.imgur.com/1FPZG.png].

[by Emilio Pisanty]

http://physics.stackexchange.com/questions/127531/lack-of-inversion-symmetry-in-crystal

Parity conservation is the key principle. The simplest cases to consider are those that only involve the symmetry of the photon, which has odd parity since it represents a vector field. When the material medium is of even parity, like a gas which is isotropic, it can absorb N  photons and emit one (actually more than one, but this is a higher-order process which I think has never been observed). So, the initial state is N photons plus isotropic medium: parity (-1)^N.  The final state is one photon plus isotropic medium: parity -1.  This is the situation with "four-wave mixing" in gases: three photons - parity (-1)^3 = -1 - absorbed, and one photon is emitted  - parity = -1 .  If the medium has odd parity, 

Vladimir: Let me again be very simple here: The belief is that all optical properties of materials can be explained by charge distribution within its unit cell (I am using unit cell in very general meaning, which can also be a molecule or an atom). Now charge distribution should have same symmetry properties as as the unit cell itself  (there is no reason to think otherwise) . Susceptibilities (including nonlinear ones) are subject for same spatial (symmetry) transformations as the unit cell, so that should define there properties.

Let me bring an example.

Suppose you have a molecule (molecular crystal) having say C3 symmetry transformation - molecule look exactly in the same way if you rotate it by 120 degrees, it means that ALL measurable properties should be invariant under rotation by 120 degrees.That defines what of the tensor elements of susceptibilities do not vanish under transformation. Check also term of Kleinman symmetry

Let me recommend you a book of R. W. Boyed as an excellent and rather comprehensive reference to the entire field.

@Yuri,

Most of the my knowledge about nonlinear optics is from book of the Boyd.

I am "understand" what happening in the case of the 2nd order nonlinearity, but when I try apply same concepts on the higher order non-linearity I don't really shure if I do that correct, and I not really understand Kleinman's symmetry in the case of high (for example third or forth) order nonlinearity. Can somebody explain it in the case of the 3rd order nonlinearity? Can somebody explain how mach reduces the number of elements in the high-order nonlinear susceptibility?