the third-order nonlinear susceptibility is given as (picture 11) .
Taking the light “2” with the frequency of w2 as the pump beam, and the light “1” with w1 as a probe beam, Eq. (1.55) describes the polarization to generate a signal beam with a frequency of wc = 2w2 - wl. If the two pump beams proceed in the opposite directions, the signal is the phase conjugate wave of the probe beam.
Using X3 of Eq. (1.56) under the degenerate case of w2 = wl, the optical
gain, including the nonlinear effect,and where the power density of the electromagnetic field is given as (picture 22).
thus he third-order optical gain in quantum dots is written as (picture 33) and is the normalized broadening function. Substituting a Gaussian function for
Eq. (1.261), the maximum optical gain, including nonlinear susceptibility, is
written as (picture 33) .This form is often used in laser simulation to avoid negative gain. The third-order nonlinear coefficient is (picture
Despite the above,How can achieve the second-order optical gain in quantum dots ?
You have to break the inversion symmetry:
http://physics.stackexchange.com/questions/127531/lack-of-inversion-symmetry-in-crystal
Then you can generate second harmonic in quantum dots:
http://pubs.acs.org/doi/abs/10.1021/jp111790t
The 2nd (even) order optical gain in QDs is zero. Refer to book of "Self-assembled QDs" by M. Sugawara, Chap 1.
the 2nd order optical nonlinear susceptibilitiy in QDs is zero if the material is an a inversion symmetry But the materials are Noncentro symmetry they have 2nd order optical nonlinear susceptibilitiy thus they have 2nd order optical gain.
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