If you google strain in silicon waveguides you'll see that by using strain to break the symmetry it is possible to obtain Khi2 and hence SHG or a Pockels effect.
Note that both of the responses so far- and both are correct- refer to situations where that central symmetry is broken; near a surface, applying non-centrosymmetric stress. Other ways: near dislocations, thin bicrystals, etc. ( Just thinking aloud.)
The most trivial (and well know in nonlinear optics) trick is to apply a DC electrical field (bias). There are plenty of examples of that. In some systems it occurs in natural way; for example, SHG in plasma described by (the late) Boris Stoicheff -- due to DC field generated by separated charges in irradiated plasma. SHG even in vacuum could be generated by a very powerful laser provided an inhomogenious DC magnetic field is applied
http://psi.ece.jhu.edu/~kaplan/PUBL/AEK.pubs/109.pdf
(but this is rather exotic)
SHG is possible in centrosymmetric crystals, where inversion center of symmetry is broken. Otherwise in small amount of SHG is possible due to surface polarization, but it is very weak signal.
For symmetry reasons dipolar second harmonic generation is null, but this is not the case for quadrupolar and higher orders terms. These are very small signals though with no practical use in wavelength doubling.
I introduce you a book about it: nonlinear optics ( Robert Boyd )
Second harmonic generation (SHG) is a very powerful technique for checking the centrosymmetry or non-centrosymmetry in a crystal. Any break of centrosymmetry will lead to the second harmonic generation. So this technique has to be used very carefully. Since the structure of the surface and skin of crystals is often different from the bulk and since SHG is essentially a surface probe then SHG may give no information about the bulk unless your sample is truly transparent to the radiation used.
I agree with Khalil. That book answers the question under discussion and many more with sufficient theoretical detail.
To go after Tapan. There will always be SHG at least from the surface of a crystal. The exciting wavelength has to be chosen as to explore the greatest possible volume in order to minimize the surface SHG. The best would be to record multiple wavelengths SHG yields in order to separate the surface SHG.
It is possible to measure SHG from the sample.I suppose that you consider the case when the high intensity laser field incide on metal sample( the grazing incidence). In the reflected light from sample you should have the frequencies which are the multplikations of the w (w-frequency of laser) i.e. 2w, 3w etc. The signal is weak few order weaker than first harmonic w but it is possible to measure that signal by separation of the expected
frequency. You can precisly choose the duration of laser luse as well as frequency of laser and power ( below the damage of materials). Measurement of signal SHG using photomultiplayer
Ity is very possible, from the surface there is symmetry breaking so SHG can be generate there. You also have to consider bulk quadrupolar effects from within the material.
The challenge is to be able to determine the strenth of each contribution especially if surface atomic orientation is the same as in the bulk. Please check our work which is very relevant to your question:
http://iopscience.iop.org/article/10.1088/2040-8978/18/3/035501/pdf
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