Nonexistence of smooth electromagnetic fields in nonlinear dielectrics. 2: Shock development in a half-space

If over the distance travelled by the beam into the half-space, until shock development, max epsilon approximately max epsilon 0 then we have the (admittedly crude but, nonetheless, informative) estimate (3.8) S max equal C0 max epsilon 0) the minus 3/2 (max/epsilon prime 0/)3 over 2 to the minus 1 where Co = one over the square root of 2 micron one half power(0) over chi to the 3 power(1) is a characteristic material coefficient which may be associated with a particular nonlinear dielectric substance. For most common nonlinear dielectric substance, e.g., index matched KDP, C0 will be a very large number, something of the order of magnitude 10 to the 21th power. This (3.8) indicates that even for an incident high intensity beam of the order of magnitude of 10 to the 9th power volts/meter a steep gradient on the incident beam will be needed so that shock development may occur within distances obtainable under laboratory conditions.