The exact formula is derived from the “sum over states” (SOS) quantum mechanical model for the frequency dispersion of the nonlinear refractive index coefficient n2 for centrosymmetric molecules in the off-resonance and non-resonant regimes. This expression is characterized by interference between terms from two-photon transitions from the ground state to the even-symmetry excited states and one-photon transitions between the ground state and odd-symmetry excited states. When contributions from the two-photon terms exceed those from the one-photon terms, the non-resonant intensity-dependent refractive index n2>0, and vice versa. Examples of the frequency dispersion for the three-level SOS model are given. Comparison is made with other existing theories.
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There have been ongoing discussions since the early days of nonlinear optics about the dispersion and sign of the non-resonant value of n2 (when the photon energies are all much smaller than the energy to the first excited state), the Kerr nonlinear refractive index coefficient due to transitions between the electronic states of atoms and molecules [1–4]. Atoms and molecules have discrete electronic states and their Kerr nonlinearity has been calculated using various schemes [5–12]. Most common have been two level model approximations involving the ground state and one excited state. They have proven useful for molecules with non-zero permanent dipole moments in the ground and the excited states. Goddard et. al. solved a simple two level system based on charge transfer between the donor and acceptor groups responsible for the permanent dipole moments giving both a second and third order nonlinearity [5,6].
Other researchers have applied the general “sum over states” (SOS) quantum mechanical method to electric dipole allowed transitions between the ground state and the excited states [7–13]. This model contains contributions from both one and two photon transitions and is a powerful tool since it gives the nonlinearity in terms of measurable parameters such as the spectral location of the excited states and the transition dipole moments responsible for transitions from the ground state to the excited states, and the transitions between the excited states. The SOS is generally acknowledged as the fundamentally correct model for dealing with atoms and molecules and its two level version has been remarkably useful in calculating the second order nonlinearity .
Recent experiments on the nonlinear optics of air and its constituent molecules and atoms have stimulated a great deal of interest in the nonlinear optics of atoms and simple molecules such as argon, nitrogen, oxygen etc [15,16]. An “extended Miller formula” based on an anharmonic oscillator has been reported in which the nonlinear response n2 is obtained essentially from the linear susceptibility and contains a phenomenological nonlinear “force” constant . It does not include multi-photon contributions to n2 nor can the magnitude of n2 be calculated from measurable parameters. There is also a model due to Brée et. al which was applied to atomic argon . It includes only two photon “resonance” contributions to n2 and is labeled here as the “two photon resonance model”. Neither of these approaches describe completely the third order nonlinearity of symmetric molecules, nor does the two level SOS model, since such molecules have zero permanent dipole moment.
The SOS model is the only one which includes both one and two photon contributions to n2 [12,13] However, in symmetric molecules with zero permanent dipole moments three levels, including the ground state, are the minimum number needed in order to include both the two photon transitions responsible for two photon absorption and one photon transitions. In symmetric molecules and atoms, the electronic states have wave functions with spatial components which are restricted to have either even or odd symmetry, or equivalently called even or odd parity. (A “bar” over a quantity, e.g. for the wave function, identifies a quantity associated with a single isolated molecule.) The electric dipole operator has odd symmetry so that for the transition dipole moment between two states m and nFig. 1 has been explored previously based on the general SOS formalism of Orr and Ward .
In the three level model,andare the electric dipole transition moments between the ground state 1Ag and the first odd symmetry excited state 1Bu and between that excited state and the dominant even-symmetry excited state mAg respectively [4,11]. Furthermore,andare the energies of the odd symmetry and even symmetry excited states above the ground state, respectively. The contribution to the nonlinearity n2 due to one and two photon transitions are proportional toand respectively. The sign of the non-resonant nonlinearity is determined by the ratio of four parameters, namely 
If the ratio given by Eq. (2) is greater than unity, the sign of the non-resonant nonlinearity is positive and the two photon contributions to the non-resonant n2 exceed those due to one photon transitions, and vice-versa. This model has been successfully applied to the explanation of the nonlinearity, including its sign in the non-resonant regime, for linear organic molecules such as squaraine dyes, CS2, and conjugated polymers [19–22]. The required parametersandcan be obtained from measurements of the linear susceptibility andandfrom two photon absorption measurements. This model has been advanced to the point that exact analytical formulas are available .
In this paper we use the SOS general expressions to extend the results for the three-level model to an arbitrary number of excited states for symmetric molecules or atoms in the off-resonant and non-resonant regimes. This leads to general analytical results for the dispersion with frequency of n2 in terms of electric dipole transition moments and locations of the excited states which for simple atoms and molecules can be calculated from first principles. It will be shown that the relative importance of the contributions of the one- and two-photon transitions still determines the sign of the non-resonant nonlinearity.
2. Sum over states for symmetric molecules
The sum over states model assumes at the outset (1) discrete states, see Fig. 2 for an isolated atom or molecule, (2) calculated from quantum mechanics, (3) with the electrons before the application of electromagnetic fields primarily in the ground state and (4) only negligible amounts of electrons in the excited states . An incident electromagnetic field which can contain many different frequency components with potentially different polarizations induces an electric dipole in an atom/molecule which couples the ground state (subscript g) electron to all of the excited states (subscript m). First order perturbation theory is used to calculate the probability for transitions into the excited states m in terms of the transition dipole moments defined in Eq. (1). This yields the induced polarization in each state m by each field component which then gives the linear atomic/molecular susceptibility.
Two interactions with the fields results in a change in the probability of the excitation of a dipole moment in an excited state (n) from the ground state and all of the previously excited states m (due to the first interaction). This leads to the second order atomic/molecular susceptibilitywhich is zero in symmetric molecules and atoms. A third interaction with the applied fields gives the third order atomic/molecular susceptibility in which the additional “hat” superscript signifies that the quantity is a complex number. The subscriptsrefer to the polarizations associated with the incident fields and i with the nonlinear polarization induced. Each of consist of any combination of the input frequencies and is given by a particular combination of the input fields taken three at a time, namely .
This first-order perturbation theory procedure gives the expression shown below for the third order nonlinear susceptibility of atoms or molecules in their frame of reference.
The “hat” over a parameter identifies that parameter as a complex quantity. The summations v, m and n are each over all of the excited states (with the exclusion of the ground state). The frequency terms are and is the time for electrons in the m’th state to decay to the n’th state.
The terms in the first summation correspond to two photon transitions and the second summation to one photon transitions. Note that these transitions are virtual in the sense that the photon energies need not match the energy differences between states. The “pathways” associated with these transitions are shown in Fig. 3 . Note that the general SOS theory allows two photon “pathways” such as 1Ag→6Bu→5Ag→8Bu→1Ag which involve two different odd symmetry (one photon) states which are not allowed in the simple three level model. As a result there are more possible terms for two photon transitions than one photon transitions.
3. Linear symmetric molecules
Equation (3) is now made specific to z-polarized incident fields and, since the interest here is in n2, this restricts the subscripts of the macroscopic third order susceptibilityto z,z,z,z. The further detailed discussion addresses linear molecules since the non-resonant n2 has been measured recently for air and its primary constituents, namely the linear molecules O2 and N2 and its dispersion calculated via the “extended Miller formulas” [15–17]. (Atoms which have spherical symmetry are a special case which will be discussed later.)
Consider a dilute gas (like air) consisting of linear molecules. The dominant third order molecular nonlinearity lies along the inter-atomic axis, specified as for convenience, i.e. only needs to be considered and the molecular re-orientation due to strong fields is neglected. (Typically the reorientation effect is subtracted out in the published data on air molecules [15,16,24].) When the contributions of randomly oriented linear molecules is averaged over all possible angles relative to the z-axis, the net contribution is only 1/5th . For a single incident beam the nonlinearity n2 (defined bywhere I is the intensity) is given byEq. (4) is the average over all possible orientations and is proportional to. The general formula for the individual is
Here N is the density of molecules andis the usual Lorentz-Lorenz local field factorEqs. (3) and (4) contribute to an intensity-dependent refractive index and absorption. However, only the case in which gives the susceptibilityand leads to a summation over the two photon resonances, i.e. peaks at 2ω = whereis an even symmetry state. In fact, these two photon resonances dominate the response due to the two photon transitions near and on these resonances. But, in the non-resonant limit, i.e. the summations in all three susceptibilities of Eq. (4) reduce to the same value, namely
Hence in the molecular case, it is not sufficient to just use the resonance two-photon absorption terms to evaluate n2 in the non-resonant or off resonance limits.
There are a total of 24 terms that need to be evaluated for n2, which is a formidable task, especially near and on the one- and two-photon resonances. However, it has proven possible to obtain closed form solutions for n2 in the off-resonance regime which corresponds simply to neglecting the imaginary parts of the denominators of all of the terms . The result for the contribution to n2 due to both two- and one-photon transitions is given below.
This result is exact within the stated approximations. Clearly the dispersion with frequency of n2 can be quite complicated, including multiple sign changes. In the non-resonant limit (ω→0),
Note that the second term due to one-photon transitions is always negative and that the first term due to two photon transitions is positive. If the non-resonant nonlinearity is measured to be positive for a given symmetric molecule, then the two-photon transitions dominate the one photon ones, and vice-versa. For air molecules, which are not exactly one-dimensional, other tensor components contribute. However, many of the additional dipole terms in the expression for n2 will still vanish because of centrosymmetry and the expression will again separate into positive and negative terms similar in form to the one given by Eq. (9). Thus, a positive value of n2 will always be associated with two-photon transitions.
Without additional knowledge of the details of the different transition moments and the locations of the excited states, no further information can be gotten from this general analysis.
4. Three-level model for linear symmetric molecules
As discussed in the introduction, there are molecular systems which exhibit only one strong one photon and one two photon absorption peak, for which the pertinent parameters in Eq. (3) have either been calculated or measured and for which the three level model has worked well. The detailed analytical solution for n2 is shown below in the limit that the decay times are small, i.e. and ,23]. The nonlinearity is simpler than for the general case for off resonance,Eq. (2).
It is instructive to use this simple three-level model to probe the dispersion in n2 for a few scenarios in which both one- and two-photon transitions are important. Specifically n2 was calculated for the ratiofor both , and the results are shown in Fig. 4 . In both cases the nonlinearity n2 is negative in the frequency range between the one and two photon resonances. There is a pronounced dispersion resonance in the nonlinearity near the normalized frequency of the two photon absorption peak which in this case appears at . Furthermore, for, n2 changes sign twice in the vicinity of this resonance. This occurs over a narrow range ofand for smaller values of this parameter n2 remains negative since the one photon transitions dominate. For values of, the nonlinearity remains positive right down to ω = 0 and the two photon transitions dominate.
5. Spherically symmetric molecules and atoms
This case is simpler than that needed for linear molecules since the nonlinearityis a scalar quantity and no angular averaging is needed. This leads to Eqs. (5)–(9) multiplied by 5 to remove the averaging factor for linear molecules.
6. Comparison with other models of n2
It is clear that none of the other models contain the correct ingredients to reproduce the SOS model. The two level SOS model gives the same contributions as in the three level model due to one-photon transitions since it is based on a single one photon excited state. However, because the permanent dipole moments are zero for symmetric molecules, hence the two level model does not contain the two photon contributions.
The “two photon resonance” model contains exactly what the name implies and is approximately valid in the spectral vicinity of the two photon peaks [25,26] but does not contain all of the two photon contributions in the off resonance or non-resonant regimes . Nor does it include the one photon transition contributions.
The “extended Miller formulas” model fails to capture the essential elements of molecular nonlinear optics, i.e. one and two photon transitions between discrete states. It is based on an anharmonic oscillator and the third order susceptibility responsible for n2 is given in terms of a nonlinear force constant parameter Q(3), namely Eq. (22) in Eqs. (5) and (8) derived from the SOS model. Equation (13) does not reproduce the one photon contributions of the SOS model. Furthermore, in  the authors claim that for arbitrary “s”, n2s would be proportional to and thus predict a monotonic decrease of n2s with decreasing frequency whereas the SOS model predicts low frequency resonances in the dispersion and potentially multiple sign changes. For example for n8, multiphoton resonances will occur at, , . Unfortunately these “extended Miller formulas” results have been used in subsequent publications [27,28], and could thus result in an erroneous analysis.
7. Concluding remarks
We have used the widely accepted “sum over states” model for molecular nonlinear optics to calculate a general formula for the third-order nonlinearity n2 in the off-resonance and non-resonant regimes. The net result is that the non-resonant sign of n2 can be used to determine whether one-photon or two-photon transitions dominate the nonlinear response of molecules. The frequency dispersion of n2 is complicated and none of the previous theories discussed here have captured completely the essential physics. Using a three-level model, it has been shown that the sign of n2 can change at least twice, depending on the details of the molecular properties.
Although at present our results due to the lack of precise values for the transition moments and the exact locations of the excited states cannot be used to shed light in the current controversy regarding the interpretation of the experimental results on the higher order Kerr effect [15,16], based on the sign of the non-resonant electronic nonlinearity measured in air, nitrogen, oxygen and argon gases, it is concluded that two-photon transitions dominate the optical nonlinearity.
This work was supported by the European Union (EU) Marie Curie Excellence Grant “MULTIRAD” MEXT-CT-2006–042683 and partially by the EU FP7 project “ENSEMBLE”. MGK thanks the National Science Foundation (NSF) (ECCS-0756936) and the Air Force Office of Scientific Research (Grant No: FA9550-10-1-0286) for their generous support.
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