I cannot give a direct answer to the above question, however can describe how the answer can be obtained. First consider the theoretical discussions in the attached text from Wiley Online Library. A relevant expression for the absorption cross section σa is given in Eq. (A.8), p. 348, of this text. The matrix element Mif (the transition amplitude from the initial state to the final state) is calculated by making use of the time-dependent perturbation theory. For this, I refer you to Sec. 5.6, beginning on page 325, of the book Modern Quantum Mechanics, by J.J. Sakurai, revised edition (Addison-Wesley, Reading, 1994). In Eq. (5.6.1)  cn(t) is expanded in a perturbation series. The explicit expressions for the first three terms in this series are given in Eq. (5.6.17). Taking into account only the first two terms, one arrives at the celebrated Fermi's golden rule for the transition rate, given in Eq. (5.6.34). The answer to your question is to be found by considering the third term in the perturbation series for cn(t). The expression for the associated transition rate is given in Eq. (5.6.37). I note in passing that there is a very important quantum effect accounted for by this expression: one has the absolute value of the sum of two complex quantities [the off-diagonal elements are in general complex], which is crucially distinct from the sum of absolute values of two complex quantities.

For an actual calculation, consider the section with the headline Harmonic Perturbation, beginning on page 333 of the book by Sakurai, which is directly relevant to your question, as well as Sec. 5.7 of the same book, beginning on page 335 (in Eq. (5.7.14) the explicit expression for σa is given, calculated by taking into account only the first two terms in the above-mentioned perturbation series expansion of cn(t)).

http://onlinelibrary.wiley.com/doi/10.1002/9781119998365.app1/pdf

I am afraid that no answer is possible unless you provide more information concerning sigma' and sigma''. Outside optics, for example, whenever abosrpition of neutron beams (in nuclear physics) and electron beams (in plasma phyiscs) people are accustomed to a law like sigma = connst.*E^-1/2, where E is the kinetic energy of the particle (it is just Gamow's limit of s-wave elastic cross section at large energies). For non-relativistic particles E is quadratic in particle velocity, hence in I. In other words, the fastest the particle, the slowest the absorption. As a consequence, fast electrons undergo few collisions, and runaway up to relativistic energies is possible in large lightning bolts. As for photons, which kind of processes you are referring to? Radiologists concerned with X rays assume the absorption to be linear, i.e. sigma' = const, sigma'' = 0 (in oyur formalism). When it comes to visible light, Rayleigh scattering makes sigma to be an increasing function of frequency: this is why a cloudless sky is blue. But different physics leads to different sigmas: after all, cloudy skies are never blue.

@ Andrea Di Vita - What you describe is related to the energy (frequency) dependence of the absorption cross section, while the question is about the dependence of the absorption cross section on the intensity.

@ Sami Birzhandi - Just making a Google search, found the attached article in which they use the expression for the absorption cross section as suggested by you (see their equation (4)).

ps. I think the first "I" in your expression "dI/dz=-n*sigma*I" should be "n".

http://www.maik.ru/full/lasphys_archive/99/1/lasphys1_99p426full.pdf

Behnam Farid

Thanks for article offer.

No this equation describe intensity change of polarized light along medium and when sigma is constant we have

I=I0*exp(-n*sigma*z)

From eqn(2.54) in page 31 of attached article

Thanks again

http://physics.princeton.edu/romalis/papers/Seltzer%20Thesis.pdf

Andrea Di Vita

In my equation sigma' is constant factor of sigma and sigma'' is factor that define intensity effect in sigma. 

You are welcome Sami. Actually, early this morning it downed on me that the relevant expression should be of the form σ = σ΄ + σ΄΄ Ι1/2 + σ΄΄΄ I + ... . Τhis can be immediately seen from the expression in Eq. (5.7.14) in the earlier-mentioned book by Sakurai (p. 337), where σabs is obtained by dividing the transition rate by the intensity of the applied field, which is proportional to the square of the absolute value of  this field -- note the |A0|2 in the denominator of the expression. Since here the transition rate is calculated to second-order in A0, naturally the leading-order contribution to σ is independent of I (note the |A0|2 in the numerator of the same expression). Calculating the transition rate to third-order in A0, unless σ΄΄ is vanishing by symmetry, the next-to-leading-order contribution to σ should be proportional to I1/2. Etc. If σ΄΄ is vanishing by symmetry, then one has σ = σ΄ + σ΄΄΄ I + ... . For the calculation of σ΄΄΄ one has to calculate cn(t) to 4th order in the perturbation, that is one order higher than explicitly described by Sakurai in his book. Of course, if you are doing experiment and the explicit analytic expression for σ΄΄΄ is unimportant from your perspective, then all you will have to do is demonstrating that σ΄΄ is identically vanishing. Once you have done so, then you know for certain that σ = σ΄ + σ΄΄΄ I + ... is the appropriate expression to use.

As for the equation "dI/dz=-n*sigma*I", thank you. Yes indeed the expression "I=I0*exp(-n*sigma*z)" is the solution of this differential equation subject to the boundary condition I(z=0) = I0.

Yes you are right. As you said I'm just try to define a practical equation to use for describing my experiment data .

Now I read Demtroder spectroscopy (page 92) doing the same thing for saturation absorption where sigma is defined for one atom and not intensity dependent. instead number of atoms in ground state is intensity dependent so alpa become intensity dependent. Of course the result doesn't change at the end and we have alpha=alpha0+beta*I .

In my opinion it's depend in definition. As you can see in my attached paper (page 13) if  sigma is defined as average absorption cross section of atomic medium we can get sigma intensity dependent. what is your opinion?

As I wrote earlier, calculation of σ involves division of Mif, or wif,  by the intensity (I explicitly referred to Eq. (5.7.14) of the book by Sakurai, where one has the explicit expression for σabs (obtained on the basis of a second-order calculation) and from which one directly sees that the intensity in the denominator cancels that in the numerator). Hence the general series expansion σ = σ΄ + σ΄΄ Ι1/2 + σ΄΄΄ I + ... in my previous response. Now, for symmetry reasons the constant σ΄΄ may be vanishing, resulting in the leading term following the constant σ΄ in the expansion of σ being linear in I. Whatever the case may be, σ is always a function of intensity (as well as frequency). The equality σ = σ΄, that is σ independent of I, therefore amounts to an approximation, which at low intensities could be a very accurate approximation. Compare with the function sin(x) for which one has the Taylor expansion sin(x) = 0 + x - x2/6 + ... . 

With regard to the calculation of σ, theoretically the underlying formalism is the same for atoms and solids. What is different in dealing with these systems, is the nature of the eigenvalues and eigenfunctions that enter into the expression for σ. For atoms, neglecting the scattering states (which in practice can be avoided by placing the relevant atom inside a box and employing a convenient boundary condition) one has to do with discrete energy levels. In solids, atomic wavefunctions hybridize and atomic energy levels transform into energy bands, so that one has to deal with energy bands and associated wavefunctions.

Incidentally, which volume and which edition of the book by Wolfgang Demtröder are you referring to? There are two volumes of Laser Spectroscopy, of which volume 1 has gone into 5th edition. In any case, I have now checked both volumes and concluded that you must have meant volume 2, page 92, since on page 92 of volume 1 (4th edition) the discussion is about line broadening. If you consider Eq. (2.35a) on page 91 of Volume 2 (4th edition), then that expression involves the intensity-dependent function S0 = S0(I,ω0), referred to as 'Lamb depth'. Later in the text one is presented with an expression for S0(I,ω0) which is linear in the intensity, I. This brings us to the expression σ = σ΄ + σ΄΄΄ I. I will have to check the details more carefully, however what is directly evident to me is that use of  S0(I,ω0), instead of S0(I,ω), is certainly an approximation; it is inspired by the fact that the dominant contribution to σs(ω) corresponds to the pole of the expression at ω = ω0 + i γs.

To summarize, my opinion is that when one adopts a specific definition for σ (like any other physical quantity), one has to be consistent; there is no room for allowing arbitrariness, which is only available at the stage of defining σ.

@Farid. You are right. As a matter of principle, the dependence of the cross section on the energy of the incident photon (i.e., on frequency) is known once the Hamiltonian is given, and it is possible to apply quantum mechanics as you have thoroughly explained. But if I ignore what the Hamiltonian is like, I cannot assess which contributions to the cross section are actually relevant. For example, symmetry may lead to selection rules and force some of such contributions to vanish identically, as you have correctly stressed. In its original form, Samis' question is too generic. 

Dear Andrea, experimentally one can deduce the dependence of σ, which is naturally frequency dependent, on the intensity I. To this end, one has to produce a log-log plot of |σ(ω)|, at some peak frequency, as a function of I. But the question posed on this page is not about this; it is about the theoretical expectation with regard to the dependence of σ = σ(ω) on I. To answer this question, one will have to assume a model Hamiltonian, or one will be speculating. Once this Hamiltonian has been assumed, its symmetries are directly inferred.

Incidentally, and the question posed on this question may be a consequence of this fact, that for producing a log-log plot of |σ(ω)| capable of yielding the sought-after power of I (whether 1, or 1/2, or whatever), one has to have data for σ(ω) corresponding to I extending over a relatively broad interval. This may be experimentally problematical, since either one is limited in the range that I can vary, or for large I the sample being investigated may be damaged or destroyed. It is here that a theoretical understanding of how σ(ω) varies as a function of I becomes indispensable.

Behnam Farid

Sorry for delay and thanks for your answer

I just like to add this to your answer

In p77 of  volume 2 (4th edition) of spectroscopy by Demtroder and in p16 of nonlinear optics by Boyd for nonlinear absorption is solved in two way with different assumption of sigma which leads to same answer at the end.  

Thank you Sami. No problem with the delay. I have now checked both references. If we consider Eq. (1.2.18) of the book by Boyd, or the equation following Eq. (2.2) in the book by Domtröder (Vol. 2), taking into account that S(ω) is proportional to the intensity I, the conclusion seems to be that for sufficiently small I (considering Eq. (1.2.18) of the book by Boyd, for I < Is) the absorption coefficient is describable as a power series in I, this on account of the expansion 1/(1+x) = 1 - x + x2 - x3 + ..., which applies for |x| < 1 (the condition I < Is can certainly be brought into contact with the condition for which the perturbation series expansion for the function cn(t), discussed earlier, is applicable). From this perspective, it would appear that your original suggestion is correct, that there is no term proportional to I1/2 to be expected in the expression for the absorption coefficient. I am however not fully convinced that this is a general result, although as I have indicated earlier, for symmetry reasons (for instance, inversion symmetry and time-reversal symmetry) indeed the term in the absorption coefficient proportional to I1/2 can be vanishing. I think this should be looked into, if not by you, by someone who specialises in the theoretical aspects of quantum optics.

Domtröder very explicitly arrives at his result, to which I have referred above. The kind of model he uses (and no doubt this model is widely used in the community) is a two-level system which technically is a zero-dimensional model. One will have to verify (and it is very likely that someone has verified this already) whether absence of terms proportional to In/2, where n is an odd integer, is not a peculiarity of zero-dimensional models. I do not know the answer and cannot therefore rule out existence of terms proportional to In/2, with n = odd integer, in the expression for the absorption coefficient. Undoubtedly, for symmetry reasons such terms are identically vanishing in some systems, but the question to be answered is whether they are vanishing for all systems. 

Just one general remark. With reference to my previous comment on this page, note that for I > Is one is in the non-perturbative regime. Martin Wegener's textbook Extreme Nonlinear Optics (Springer, Berlin, 2005) touches on this issue.

As you mentioned I think absorption cross section vanishing In/2 relate to sigma nature. 

In sec nonlinear optics (21) of  Fundamentals Of Photonics by Saleh Polarization only depend on real part of electric field. Maybe it's the solution to why odd term vanished for 

No, what is relevant is the symmetry of the problem (this symmetry determines what modes couple to what other modes of the unperturbed system as a result of a perturbation of a certain symmetry). For this, I refer you to the last paragraph on page 2, to be continued on page 3, of the book by Boyd (Nonlinear Optics, 3rd edition). Here it is briefly described in what kind of systems second-order non-linear optical interactions take place and in what kind of systems they don't. Actually, one does not need to speculate; writing down the relevant expressions, one can immediately infer (using some basic elements of the group theory, including the Wigner-Eckart theorem) the conditions under which certain contributions are identically vanishing.