The total dipole moment induced by an electric field is usually written as a Taylor series, where the third term is directly related to the intensity of second harmonic generation (SHG).

This term precisely contains the hyperpolarizability tensor (a "molecular" parameter)

which is also related to the second nonlinear susceptibility tensor (a bulk property) and

the density of molecules which generate your non-linear field (such as collagen fibers).

Maybe these papers could help you in this topic:

http://www.nature.com/nbt/journal/v21/n11/abs/nbt894.html

http://biomedicaloptics.spiedigitallibrary.org/article.aspx?articleid=1183111

Good luck

P= X0E + X2E2+X3E3

E=E0e -jWt , Applied field

P(w)= X0e-jwt E0 + X2  e-j2wtE02 , Induced polarization

Where X0 and X2 are linear and second order susceptibility

SHG: X0= X2 E0  This happens using sufficiently intense laser field !

For more information you can read nonlinear optics of Robert Boyd

I study Boyd *s BooK and other papers. But I dont undrestand realationShip between intensity of generation of the second optical harmonic and d(ijk)?

dijk is simply another definition for susceptibility Xijk representing the type of crystal.

On the other hand, X2 E02 is proportional to SHG intensity I(2W) too.

Dear Marzieh,

I assume you are familiar with the couple mode equations derived from the Maxwell equation (inhomogenous wave equation with nonlinear polarization source) , e.g. in Boyds book they are given in Eq. 2.2.12 (Third ed):

http://prntscr.com/5ddoql

The SHG or here the SFG field can be obtained rather easily if the amplitude of the SHG field (E_2w) is only slowly varying and is given by

http://prntscr.com/5ddp3r

where the sinc function contains phase matching condition, an efficient energy transfer from the linear to SHG for a depth beyond the coherence length of the optical field occurs if  the phase velocity of the nonlinear polarization and the generated SHG field is nearly equal.

The intensity is obtained by multiplying the SHG field with its conjugate partner. And because E_2w is proportional to d_ijk then the intensity is proportional to the square of  nonlinear SHG susceptibility d_SHG_ijk of (d_SHG_ijk)^2  (NOT linearly proportional to d_SHG_ijk!) as is given in Eq. 2.2.17:

http://prntscr.com/5ddpx9

Note that the nonlinear SHG susceptibility also contains the square of the linear susceptibility:

http://prntscr.com/5ddsxu

therefore the SHG intensity is also proportional to (d_jk)^4!

The SHG field can also be obtained microscopically, using the antenna picture or direct dipole summation. There is a beginners book for NLO researchers written by Jeffrey New, Introduction to NLO.

Good luck on your research!

1- Downvote to my comments above is downvote to Boyd's book of

NLO !! and it is not reasonable.

Furthermore, it is much better to emphasize on the efficiency of SHG that is written as below:

SHG - efficiency =2 (mu0/eps0)3/2 w2d2/ n12n2 (sinc(delta k. L))2 L2 I1

or simply:

SHG efficiency= 2 (mu0/eps0)3/2 (W2d2 L2 / n3 ) I1

 where I1 is the irradiance of primary beam. Therefore, the conversion efficiency is proportional to d2 and high conversion ratio requires high peak power only available  from pulsed lasers!

Hi professors

First of all i have to say thank you so much for your guidance, I'm really happy to know you in this place.

but now i have some question of you again:

1. Dear Dr.parvin in your description  where is the reference of these equations

(SHG - efficiency =2 (mu0/eps0)3/2 w2d2/ n12n2 (sinc(delta k. L))2 L2 I1

or simply:

SHG efficiency= 2 (mu0/eps0)3/2 (W2d2 L2 / n3 ) I1)?

because i need to drive these equations.

2. Dear Hendradi Hardhienata

In your description, would you please give me more description specially in this  part(The intensity is obtained by multiplying the SHG field with its conjugate partner. And because E_2w is proportional to d_ijk then the intensity is proportional to the square of  nonlinear SHG susceptibility d_SHG_ijk of (d_SHG_ijk)^2  (NOT linearly proportional to d_SHG_ijk!) as is given in Eq. 2.2.17:

) of your offers?

Refs :

1- A.Yarif, Optical electronics, 4th edn, Saunders college Publishing, 1991, Section 8.2.

2- P.A. Franken, A.E.Hill, C.W.Peters,  "Generation of Optical Harmonics", Phys. Rev .Lett , 7, 118-119,1961

thank you dear Dr.Parvin

Unfortunately I couldn't download this book  would you please send for me the pdf file of that ?

with best regard

Dear Marzieh,

In physics every physical observable must be described by a REAL quantity. Because the electric field (both the laser or the generated linear and nonlinear field) is a physical observable (e.g. you can measure a field directly such as using a capacitor set up) it can be described by a plane wave $\cos$ function (real) or exponential form with its conjugate partner.

In the cosinus form the phase of the field is mixed inside the cosinus with the frequency e.g. the laser field at fundamental frecueny $\omega$ is given by

\begin{equation} E=E_0 \cos(wt-\phi)\end{equation}

where $\phi$ is the phase of the field

in the exponential form this phase shift is included in the amplitude term, this is what Boyds uses in the Lorentz (oscilator) model and has the advantage that the phase dependence does not occur explicitly. There are also other literatures where the cos term is used for the field, see e.g. Berkeley physics course mechanics book....where you can see that the polarization source and the generated field phase are described. This phase is important when deriving the dispersion relation where the imaginary part of the polarization or susceptibility has a phase difference of $\pi/2$ with the real part resulting in the well known linear dispersion relation.  The fundamental field and generated field are written in exponential form and various authors usually omitting its conjugate partner because of equaliy, the conjugate term can be ommited (it will produce $d^*_{SHG}$ but with a star denoting the conjugate form, $-i$ of $d_{SHG}$  Because you omit the conjugate field the obtained field is complex and when you want to get the intensity you must multiply it with the conjugate partner and NOT by squaring complex fields!

When you derive $\chi_{ijk}$ or $d_{SHG}$ using the anharmonic Lorentz model you will see that it depends on the linear susceptibility. This is characteristics of perturbative models also arising in quantum mechanics (Schroedinger-Rayleigh models) where the first order solution is inserted into the second order. 

http://prntscr.com/5eorwl

therefore it follows logically that the SHG intensity depends on $d^4_{SHG}$

best regards

Dear Henderadi

I(2W)= efficiency* I0

and as states above eff is proportional to d2

Then, SHG intensity depends on d2 , in contradiction with d4.

Dear Prof. Parvis,

You are right! It was a typho on my part. Thank you for your correction  it should be (d_linear)^4

 Dear Dr.ParvinUnfortunately I couldn't download this book (1- A.Yarif, Optical electronics, 4th edn, Saunders college Publishing, 1991). would you please send for me the pdf file of that ?

with best regard