A model two-level dipolar molecule, and the rotating wave approximation and perturbation theory, are used to investigate the optimization and the laser intensity dependence of the two-photon excitation rate via the direct permanent dipole mechanism. The rate is proportional to the square of the laser intensity I only for small intensities and times when perturbation theory is applicable. An improvement on perturbation theory is provided by a small time RWA result for the rate which is not proportional to I^{2}; rather it is proportional to the square of an effective intensity I_{eff}. For each laser intensity the optimum RWA excitation rate as a function of time, for low intensities, is proportional to I, not I^{2}, and for high intensities it is proportional to I_{eff}. For a given two-photon transition the laser-molecule coupling optimizes for an intensity I_{max} which, for example, leads to a maximum possible excitation rate as a function of time. The validity of the RWA results of this paper, and the importance of including the effects of virtual excited states, are also discussed briefly.

## I. INTRODUCTION

Molecules with permanent dipoles make good candidates for suitable target molecules for the absorption of two identical photons since they offer two mechanisms for two-photon excitation, the “usual” virtual state mechanism^{1} and the direct permanent dipole mechanism.^{2,3} The latter requires both a transition dipole and a non-zero permanent dipole moment difference between the initial and final states of the two-photon excitation process. The direct permanent dipole mechanism potentially is the stronger of the two excitation mechanisms.^{4–6}

In this paper the optimization of the direct permanent dipole moment two-photon excitation mechanism is discussed through the use of a simple but instructive two-level (giant) dipole molecular model. Continuous wave (CW) laser-molecule interactions are considered as an example and the rotating wave approximation (RWA),^{7,8} the rotating wave perturbation approximation (RWPA), and perturbation theory^{1–3} calculations are employed in the discussion. The relevant expressions for the laser-molecule couplings, the populations of the molecular states, and the rate of excitation are reviewed in Sec. II. The RWPA is obtained from the RWA by replacing the RWA laser-molecule coupling by the perturbation theory coupling.

The optimization of the direct permanent dipole RWA laser-molecule coupling, as a function of laser intensity, for the model two-photon excitation is discussed in Sec. III. Section III A contains discussions of results for the excited state population and the related two-photon excitation rates and cross sections as a function of laser intensity and laser-molecule interaction time. The discussion of the intensity dependence of the excitation rates for intensities and times such that perturbation theory is not applicable is aided by the introduction of an effective laser intensity which is the intensity seen by the dipolar molecule and which is less that the “bare” laser intensity. The discussion is continued in Sec. III B with an emphasis on optimal excitation rates as a function of time for a given laser intensity, including the intensity optimizing the laser-molecule coupling, and with their associated cross sections. In general the laser dependence of the two-photon excitation rate is not quadratic and this has implications on the utility of the use of the concept of two-photon cross sections. Indeed for small to moderate laser intensities the optimum excitation rate is directly proportional to the laser intensity. Some of the more important aspects of this paper are discussed further in Sec. IV.

## II. ROTATING WAVE AND PERTURBATION THEORY EXPRESSIONS

Consider the excitation of a molecule, initially in its ground state 1, to an excited state f by the simultaneous absorption of two identical photons. The time - dependent perturbation representing the interaction of an CW electric field with the molecule is taken to be $V ( t ) =\u2212 \mu \xaf \u22c5 E \xaf ( t ) =\u2212 \mu \xaf \u22c5 e \u02c6 E 0 cos ( \omega t + \delta ) $ where $ \mu \xaf $ is the electric dipole moment vector of the molecule, and $ e \u02c6 $, *E*_{0}, *ω*, δ are the (linear) polarization vector, peak amplitude, circular frequency and phase of the electromagnetic field $ E \xaf $. In what follows the sinusoidal part of the electric field is taken to be a pure cosine field, with *δ* = 0, which is turned on at t = 0. The laser field intensity is $I= ( c / 8 \pi ) E 0 2 $ and c is the speed of light; I(Wcm^{−2}) ≈ 3.509 x 10^{16}(*E*_{0})^{2} where *E*_{0} is given in atomic units (≈ 5.142 x 10^{9} V cm^{−1})^{9}. In what follows the laser intensities will be characterized by an index n such that I = 10^{n} W/cm^{2}.

The original two-level RWA was derived by Rabi^{7} in 1937 for non-polar systems and permits only one-photon excitations from the ground state 1 to the excited state f. After the importance of permanent molecular dipoles in promoting two-photon excitations had been demonstrated^{2,3} by using perturbation theory, an RWA was developed^{8} to include these effects.

### A. The RWA

The original permanent dipole moment RWA was derived for excitations caused by the absorption of N identical photons. The population of the excited state for this excitation is given by^{8,10,11}

Here E_{1f} = E_{f} − E_{1}, E_{i} is the stationary state energy of state i, E = *ħ*ω where *ħ* = h/2π, h is Planck’s constant, P_{1}(t) + P_{f}(t) = 1 and P_{f}(0) = 0. In Eq. (1), C_{d}(N) is the direct permanent dipole laser-molecule coupling for the N- photon 1 → f transition^{8}

where μ_{ii} is the permanent molecular dipole for state i, $ \mu \xaf 1 f $ is the transition dipole connecting states 1 and f, and J_{N}(z) is a Bessel function^{12} of order N and argument z.

In this paper the emphasis is on two-photon excitations. Unless stated otherwise we consider N = 2 and will omit explicit designation of the number of photons involved in the transition; for example C_{d} = C_{d}(2). The perturbation theory expression^{2,3} for C_{d}, which is obtained by keeping the first term (z^{2}/8) in the expansion^{12} of J_{2}(z_{1f}), is given by

The results for N ≠ 2 will be used in part of the discussion in Sec. IV.

In what follows the two-photon transition will be considered to be on-resonance in the two-level RWA and so E_{1f} − 2E = 0, E = E_{res} = E_{1f}/2 and the time-dependent populations of the molecular states are given by

where C_{d}(res) is C_{d} evaluated at E = E_{res}. On resonance, for a given laser intensity, the time-dependent population of the excited state f evolves sinusoidally with time, with a (Rabi) period PER(res), and attains a maximum population of unity when t = PER(res)/2;

In the RWA the on-resonance rate of excitation to the state f is given by

For notational convenience in what follows the relevant laser-molecule coupling is to be evaluated at E = E_{res} in expressions for quantities designated by a “res”.

Expanding the sine function in Eq.(5) and (7), and keeping the first term, leads to expressions for the on-resonance population of the excited state and the excitation rate that are valid for “small” times^{4}

The perturbative results for these quantities are then obtained^{4} by expanding the Bessel function in C_{d}, Eq.(2), in powers of z and keeping only the first term varying as z^{2} to obtain

Often it is the time averaged excitation rate over the duration of a laser pulse that is most closely related to experiment; $ R \xaf = ( 1 / t ) \u222b 0 t Rdt$ for a CW laser where t can be regarded as a pulse duration. CW results can be used to model pulsed laser-molecule interactions since a pulse can be regarded as “cut-off” CW laser.^{4,13–15} The time-dependent wave equation for a Gaussian pulsed laser-molecule interaction is the same as that for the CW laser-molecule interaction if the CW electric field amplitude *E*_{0} is replaced by *E*_{0}f(t) where f(t) = exp(−t^{2}/τ^{2}) and τ is the pulse decay constant. Qualitatively^{15–17} for a resonant two-photon excitation the steady state pulsed result for the population of the excited state with τ = (2/*π*)^{1/2}t_{eff} corresponds to the CW population with t = t_{eff}. This connection holds best for relatively weak laser intensities; in the examples of Sec. III for n ≤ 12.

As a function of time, for a given laser intensity, the maximum value of $ R \u0304 f res ( t ) $ occurs at

and using this in Eq.(10) gives the corresponding maximum rate

The two-photon excitation cross section is often of interest. It is commonly defined^{18} to be the excitation rate divided by the square of radiation flux F (= I/E). The excitation cross section defined in this manner, related to the time averaged rates $ R \u0304 f res ( t ) $, $ R \u0304 f , < res ( t ) $ and $ R \u0304 f , pert res ( t ) $, given in Eqs.(10) and (11), are denoted by $ \sigma \u0304 f res ( t ) $, $ \sigma \u0304 f , < res ( t ) $ and $ \sigma \u0304 f , pert res ( t ) $.

That t = PER(res)/2 corresponds to the maximum population of the excited state is related to the condition for a π-pulse;^{19,20} this value of t does not correspond to the maximum excitation rate for a given laser intensity.

### B. Low Intensity RWA

A set of “RWA” results analogous to Eqs.(1), (5)–(7), and (10)–(13) can be defined with the RWA laser-molecule coupling C_{d} replaced by C_{d,pert}. This approach is collectively denoted as the rotating wave perturbative approximation, RWPA, as distinct from the RWA in Sec. II A. The RWPA expression for a given property is designated by adding a C → C_{pert} to the RWA designation for the property; for example $ P f , C \u2192 C pert res ( t ) $, PER_{C→Cpert} and $ R \u0304 f , C \u2192 C pert res ( t ) $ are given by Eqs.(5), (6) and (10) with C_{d}(res) replaced by C_{d,pert}(res). For sufficiently low laser intensities the RWA laser molecule coupling C_{d} can be reliably replaced by C_{d,pert} and the RWPA becomes equivalent to the RWA; in the examples discussed in Sec. III this occurs for laser indices n < 10.5 say. Finally expanding the sine functions in the RWPA results such as $ P f , C \u2192 C pert res ( t ) $ and $ R \u0304 f , C \u2192 C pert res ( t ) $, and keeping only the first term, gives the analogous perturbative results such as $ P f , pert res ( t ) $ and $ R \u0304 f , pert res ( t ) $.

All the results reviewed in this section, and the discussions based on them, neglect excited state life time effects and so are valid for times t considerably less than τ^{∗} where τ^{∗}, in the two-level model, is the lifetime of the excited state f. If appropriate the lifetime of the excited state can be included phenomenologically in the RWA treatment by including a width for the state f into the analysis.^{9,18–20}

## III. MODEL TWO-LEVEL MOLECULE CALCULATIONS

All observables in the RWA considered here depend on the absolute value of the laser-molecule coupling Cd. In the calculations that follow we take the permanent and transition dipoles to be parallel and to make the spherical polar angle θ (0 ≤ θ ≤ π) relative to the positive space fixed z-axis defined by the polarization vector $ e \u02c6 $ of the applied electric field.

In general |C_{d}| can be maximized for given |d_{1f}|, θ and photon energy E = *ħ*ω through the choice of the laser intensity I; in this regard |μ_{1f}| serves as a scale factor. The first and largest maxima^{12} of J_{2}(z) as a function of increasing z is 0.48650 at z = 3.0542 and the absolute value of the laser molecule coupling then maximizes at

when

From Eqs.(14) and (15) it follows that the maximum value of |C_{d}| is independent of θ whereas *E*_{0}(max) varies as ||d_{1f}|cos(θ)|^{−1} while I(max) varies as the square of this quantity. Thus the laser intensities required to maximize |C_{d}| for a given |d_{1f}| increase as *θ* increases from 0 to π/2 and are symmetric in θ about π/2. In the applications that follow we will generally choose E to be the resonance value of the photon energy for the two - photon transition of interest. In passing it is relevant to point out that the perturbative result for the direct permanent dipole laser-molecule coupling, Eq. (4), is generally unreliable for many purposes. For example when z = 3.0542 the lead term z^{2}/8 in the expansion of J_{2}(z), and hence C_{d,pert}, is too high by a factor of 2.4.

A two - level molecular model will be used to illustrate the maximization of |C_{d}| and some of the resulting effects. The parameters characterizing the model are based on dipolar molecules that have been used previously to discuss the effects permanent dipoles in two - photon excitations.^{4,5,21–24} μ_{1f} = 10D, d_{1f} = 30D and E_{f1}/2 = 0.05 Eh. These correspond qualitatively to the two-photon resonance for the transition from the ground to first excited state of the molecule 1-[p-(N,N-dimethylamino)phenyl]-4-(p-nitophenyl)-1,3-buatdiene. Here the Hartree, E_{h} = 27.211 eV, is used as the unit of energy, and the Debye (D) is chosen as the unit of electric dipole.

Plots of |C_{d}(res)| and |C_{d,pert}(res)| versus I (i.e. n) for the model molecule are shown in Fig. 1 for θ = 0, 2.5π, 0.4π. The common maxima |C_{d}(res)| = 0.03243 E_{h} is independent of θ and occurs at n = n_{max} = 12.769, 13.070 and 13.789 respectively; as θ → π/2 , n_{max} → ∞. |C_{d,pert}(res)| can overestimate the absolute value of the RWA laser-molecule coupling severely as n increases. In what follows the (res) will be omitted in quantities like Cd(res), PER(res), …for notational convenience.

### A. Results for two-photon excitation populations, rates and cross sections

As an illustrative example the molecule is taken to be aligned with the laser electric field (θ = 0). The maximum two-photon laser-molecule coupling then occurs for the smallest value of laser intensity (n = 12.769). Analogous results to those discussed below apply for all θ with n_{max} increasing from 12.769 as θ increases to π/2.

The RWA on-resonance population of the excited state $ P f res ( t ) $, and its small time approximation $ P f , < res ( t ) $, are shown in Fig. 2 as a function of time and for values of the laser intensity characterized by n values surrounding n = 12.769. Figure 3 shows the time-averaged RWA excitation rate, and its small time approximation, as a function of time for the laser intensities of Fig. 2. The various maxima $ R \u0304 f res ( t op ) $ in Fig. 3, and their location in time t_{op}, are in agreement with Eqs.(13) and (12) and are included in Table I.

. | RWA . | RWPA . | ||||
---|---|---|---|---|---|---|

n . | $ R \u0304 f res ( t op ) $ . | t_{op}
. | $ \sigma \u0304 f res ( t op ) $ . | $ R \u0304 f , C d \u2192 C d , pert res ( t op \u2032 ) $ . | $ t op \u2032 $ . | $ \sigma \u0304 f , C d \u2192 C d , pert res ( t op \u2032 ) $ . |

13.5 | 2.930(0) | 2.882(0) | 1.392(0) | 6.267(2) | 1.347(-1) | 2.978(1) |

13 | 3.674(0) | 2.299(0) | 1.746(1) | 1.982(1) | 4.261(-1) | 9.418(1) |

12.769 | 4.858(0) | 1.739(0) | 6.688(1) | 1.164(1) | 7.254(-1) | 1.603(2) |

12.5 | 4.024(0) | 2.099(0) | 1.912(2) | 6.267(0) | 1.348(0) | 2.978(2) |

12 | 1.732(0) | 4.875(0) | 8.232(2) | 1.982(0) | 4.261(0) | 9.418(2) |

11.5 | 6.009(-1) | 1.405(1) | 2.855(3) | 6.267(-1) | 1.348(1) | 2.978(3) |

11 | 1.956(-1) | 4.318(1) | 9.294(3) | 1.982(-1) | 4.261(1) | 9.418(3) |

10.5 | 6.241(-2) | 1.353(2) | 2.966(4) | 6.267(-2) | 1.348(2) | 2.978(4) |

10 | 1.979(-2) | 4.267(2) | 9.405(4) | 1.982(-2) | 4.261(2) | 9.418(4) |

9.5 | 6.265(-3) | 1.348(3) | 2.977(5) | 6.267(-3) | 1.348(3) | 2.978(5) |

9 | 1.982(-3) | 4.262(3) | 9.416(5) | 1.982(-3) | 4.261(3) | 9.418(5) |

. | RWA . | RWPA . | ||||
---|---|---|---|---|---|---|

n . | $ R \u0304 f res ( t op ) $ . | t_{op}
. | $ \sigma \u0304 f res ( t op ) $ . | $ R \u0304 f , C d \u2192 C d , pert res ( t op \u2032 ) $ . | $ t op \u2032 $ . | $ \sigma \u0304 f , C d \u2192 C d , pert res ( t op \u2032 ) $ . |

13.5 | 2.930(0) | 2.882(0) | 1.392(0) | 6.267(2) | 1.347(-1) | 2.978(1) |

13 | 3.674(0) | 2.299(0) | 1.746(1) | 1.982(1) | 4.261(-1) | 9.418(1) |

12.769 | 4.858(0) | 1.739(0) | 6.688(1) | 1.164(1) | 7.254(-1) | 1.603(2) |

12.5 | 4.024(0) | 2.099(0) | 1.912(2) | 6.267(0) | 1.348(0) | 2.978(2) |

12 | 1.732(0) | 4.875(0) | 8.232(2) | 1.982(0) | 4.261(0) | 9.418(2) |

11.5 | 6.009(-1) | 1.405(1) | 2.855(3) | 6.267(-1) | 1.348(1) | 2.978(3) |

11 | 1.956(-1) | 4.318(1) | 9.294(3) | 1.982(-1) | 4.261(1) | 9.418(3) |

10.5 | 6.241(-2) | 1.353(2) | 2.966(4) | 6.267(-2) | 1.348(2) | 2.978(4) |

10 | 1.979(-2) | 4.267(2) | 9.405(4) | 1.982(-2) | 4.261(2) | 9.418(4) |

9.5 | 6.265(-3) | 1.348(3) | 2.977(5) | 6.267(-3) | 1.348(3) | 2.978(5) |

9 | 1.982(-3) | 4.262(3) | 9.416(5) | 1.982(-3) | 4.261(3) | 9.418(5) |

The variation of many important results, as a function of n, follows that of |C_{d}|; see Fig. 1 with θ = 0. The period of the excited state population PER and t_{op} vary as |C_{d}|^{−1} [Eqs.(6) and (12)] while $ R \u0304 f res ( t op ) $ varies as |C_{d}| [Eq.(13) ]; $ P f , < res ( t ) $ and $ R \u0304 f , < res ( t ) $ vary as (C_{d})^{2} [ Eqs.(8) and (11) ]. Hence PER and t_{op} minimize, while $ P f , < res ( t ) $ and $ R \u0304 f , < res ( t ) $, for a given time, and $ R \u0304 f res ( t op ) $ maximize as a function of increasing n at n_{max} = 12.769. The following results for the period PER will be useful in the discussions that follow: PER = 11487.7, 1150.14, 364.75, 116.39, 13.14, 5.66, 4.69, 6.20, 7.77 fs respectively for n = 9, 10, 10.5, 11, 12, 12.5, 12.769, 13, 13.5; the corresponding values of t_{op} = 0.3710 PER are included in Table I.

The time averaged RWA rate of excitation as a function of time is closely related to the temporal evolution of the excited state $ P f res ( t ) $; indeed the former is simply (1/t) times the later. Thus for a given laser intensity the rate is zero when t = PER and maximizes when t = t_{op} with a value of $ R \u0304 f res ( t op ) =0.3623| C d |/\u0127$. The population of the excited state is unity when t = PER/2 and $ R \u0304 f res ( PER / 2 ) =0.3183| C d |/\u0127$ while $ P f res ( t op ) =0.8446$ for all n. As a function laser intensity the minimum values of PER, t_{op} and PER/2 are 4.69 fs , 1.74 fs and 2.35fs and the maximum values of $ R \u0304 f res ( t op ) $ and $ R \u0304 f res ( PER / 2 ) $ are 4.86x10^{14}s^{−1} and 4.27x10^{14}s^{−1} corresponding to n = 12.769.

Let $ Q f res ( t ) $ represent either $ P f res ( t ) $ or $ R \u0304 f res ( t ) $. The percent disagreement DIS(n) between $ Q f , pert res ( t ) $ and $ Q f , < res ( t ) $ is independent of time and given by

which increases with increasing n following the behaviour of |C_{d,pert}| versus |C_{d}| as a function of n (see Fig. 1). For example the perturbative results are greater than the small time RWA results by DIS(n) = 0.27, 2.69, 30.9, 142.6, 474.5 and 2810.6 % for n = 10, 11, 12, 12.5, 12.769, 13 respectively. They severely overestimate the small time RWA results for n > 11 and do not reflect the changes in behavior as n increases through n = 12.769 observed for the RWA results.

The percent differences between $ Q f , pert res ( t ) $ or $ Q f , < res ( t ) $ and the full RWA result $ Q f res ( t ) $, denoted by $\Delta Q f , pert res ( t ) $ or $\Delta Q f , < res ( t ) $ respectively, are related by

Clearly $ Q f , pert res ( t ) $ gives a poorer representation of the full RWA results than does $ Q f , < res ( t ) $, more so as n increases.

The small time RWA result $ Q f , < res ( t ) $ can give quite reliable approximations for $ Q f res ( t ) $ for significant values of time. For a given n the error $\Delta Q f , < res ( t ) $ is greater than κ% for t > t(κ) = N(κ)PER; t(κ) depends on n through PER. For example, N(κ) = 0.05497, 0.07751, 0.1215 and 0.1694 for κ = 1, 2, 5, 10. The error $\Delta Q f , pert res ( t ) $ is essentially κ% + DIS(n) where DIS(n) can increase significantly relative to κ% for n >10, see above. For example the small time RWA result is reliable to 1% for t = 63.22, 20.05, 6.40, 2.08, 0.722 and 0.311 fs for n = 10, 10.5, 11, 11.5, 12 and 12.5 respectively whereas the perturbative results are reliable to 1.27, 1.85, 3.71, 9.86, 32.20 and 145.04 % for these times and intensity values.

The two cross sections $ \sigma \u0304 f , < res ( t ) = R \u0304 f , < res ( t ) / F 2 $ and $ \sigma \u0304 f , pert res ( t ) = R \u0304 f , pert res ( t ) / F 2 $ are compared, as a function of time, in Fig. 4 for various values of the laser index n including those used in previous figures; the unit of two-photon excitation cross section used here is 10^{−50}cm^{4}s = 1 GM = 1 Goeppert Meyer. The maximum values of $ \sigma \u0304 f res ( t ) = R \u0304 f res ( t ) / F 2 $ for various n, which occur for t = t_{op} (n) and are denoted by $ \sigma \u0304 f res ( t op ) $, are included in Table I.

The perturbative cross section is independent of laser intensity and yields the perturbative results for the rate, which steadily increase with increasing intensity, when multiplied by F^{2} = (I/E)^{2}. An important result of Fig. 4 is that the small time RWA cross section is not generally independent of laser intensity; they steadily decrease as n increases. As intensity decreases the RWA small time cross section becomes equal to the intensity independent perturbative cross section; on the scale of Fig. 4 this occurs for n < 10.5.

When perturbation theory is applicable only one cross section for a particular molecular excitation is required to provide excitation rates as a function of laser intensity. The usefulness of this is clear. Tables of cross sections can be provided which nicely summarize excitation rate information; the size of the rate correlates with the cross section on multiplication by F^{2}. Clearly in general the cross sections $ \sigma \u0304 f res ( t ) $ and $ \sigma \u0304 f , < res ( t ) $ do not have this property. In general these would have to be provided as a function of laser intensity and would be no more convenient than providing the corresponding rate data.

It is relevant to compare the laser intensity dependence of the two-photon excitation rate $ R \u0304 f , < res ( t ) $ versus $ R \u0304 f , pert res ( t ) $. The laser field strength E_{0} is proportional to I^{1/2} and so, from Eqs. (9), (4) and (3), the perturbation theory result for the excitation rate is proportional to (*E*_{0})^{4} or I^{2}. This I^{2} intensity dependence of the rate is often taken to be characteristic of two-photon excitation and leads to a two-photon excitation cross section $ \sigma \u0304 f , pert res ( t ) $ that is independent of laser intensity. The small time RWA two-photon excitation rate $ R \u0304 f , < res ( t ) $ does not behave as I^{2} in general. The RWA laser-molecule coupling given by Eq.(2) can be written in terms of the perturbative laser-molecule coupling Eq.(4) as $ C d = ( 8 / z 1 f 2 ) J 2 ( z 1 f ) C d,pert $. Thus the excitation rate $ R \u0304 f , < res ( t ) $, in Eq.(8), can be written as $ R \u0304 f , < res ( t ) = [ ( 8 / z 1 f 2 ) J 2 ( z 1 f ) ] 2 R \u0304 f , pert res ( t ) $ and so the laser intensity dependence of $ R \u0304 f , < res ( t ) $ is not given by I^{2} but by the square of the effective laser intensity I_{eff};

where z_{1f}, Eq.(3), depends on I. The ratios $ R \u0304 f , < res ( t ) / F eff 2 $, F_{eff} = I_{eff}/E, and $ R \u0304 f , pert res ( t ) / F 2 $ are the same and so the perturbative cross section can be regarded as an effective cross section for the rate $ R \u0304 f , < res ( t ) $ which can be rewritten as $ R \u0304 f , < res ( t ) = \sigma \u0304 f , pert res ( t ) F eff 2 $. The “n” - dependence of the effective laser intensity and of I(n) = 10^{n} are compared in Fig. 5 with d_{1f} = 10, 20, 30, 40, 50D. The effective intensity is independent of the transition dipole μ_{1f} and becomes equal to the laser intensity as the permanent dipole difference d1f tends to zero. The effective laser intensity is the intensity seen by the dipolar molecule and it is generally less than the laser intensity I.

### B. Optimal excitation rates and related cross sections

The optimal excitation rates were discussed briefly in Sec. III A, see also Table I, and the details of their laser intensity dependence is of interest.

To begin consider the intensity dependence of the time averaged excitation rate in the RWPA (see Sec. II B). This expression for the rate is denoted $ R \u0304 f , C \u2192 C pert res ( t ) $ and the corresponding cross section by $ \sigma \u0304 f , C \u2192 C pert res ( t ) $. The maxima are denoted $ R \u0304 f , C \u2192 C pert res ( t op \u2032 ) $ and $ \sigma \u0304 f , C \u2192 C pert res ( t op \u2032 ) $; their position in time is $ t op \u2032 = t op ( C d \u2192 C pert ) $. These results are compared with the full RWA analogues in Table I.

The agreement with the full RWA results is poor until n becomes reasonably small where C_{d,pert} becomes a good approximation for Cd. For example $ R \u0304 f , C \u2192 C pert res ( t op \u2032 ) $ (and $ \sigma \u0304 f , C \u2192 C pert res ( t op \u2032 ) $) are about 55.8%, 14.4%, 4.3% , 1.3%, 0.42%, 0.13% higher than $ R \u0304 f res ( t op ) $ (and $ \sigma \u0304 f res ( t op ) $) for n = 12.5, 12 ,11.5, 11, 10.5 and 10 respectively; the corresponding t_{op}(C → C_{pert}) values are 35.8%, 12.6%, 4.1%, 1.3%, 0.42% and 0.13% lower than t_{op}.

The rate $ R \u0304 f res ( t op ) $ maximizes, while t_{op} minimizes, as a function of intensity at n = 12.769, following the behaviour of C_{d} as a function of n discussed previously. On the other hand $ R \u0304 f , C \u2192 C pert res ( t op \u2032 ) $ steadily increases, while $ t op \u2032 = t op ( C \u2192 C pert ) $ steadily decreases, with increasing n following the behaviour of C_{d,pert} as a function of n. The two sets of results merge when C_{d} and C_{d,pert} merge as a function of n. The cross sections $ \sigma \u0304 f res ( t op ) $ and $ \sigma \u0304 f , C \u2192 C pert res ( t op \u2032 ) $ steadily decrease with increasing laser intensity due to the factor I^{2} in their denominators. They tend to agree for low intensity but $ \sigma \u0304 f , C \u2192 C pert res ( t op \u2032 ) $ becomes much larger than $ \sigma \u0304 f res ( t op ) $ as n increases. It is interesting to note that the cross sections $ \sigma \u0304 f res ( t op ) $ and $ \sigma \u0304 f , C \u2192 C pert res ( t op \u2032 ) $ are not useful for tabulation in the sense discussed previously even for low n. Even though they agree well as n decreases, these cross sections do not become independent of intensity since the corresponding rates are not proportional to I^{2} for low n. This arises since keeping only the first term in the expansion of the sine function occurring in $ R \xaf $, which is what leads to an intensity independent cross section, is not justified for the times of interest in Table I. For example $ R \u0304 f , pert res ( t op \u2032 ) $ is higher than $ R \u0304 f , C \u2192 C pert res ( t op \u2032 ) $ by 60.85% for all n; this can be shown using Eqs. (11)–(13) with C_{d} replaced by C_{d,pert} in the last two equations.

While the intensity dependence of the rates is not quadratic, it is in fact linear for the rate $ R \u0304 f , C \u2192 C pert res ( t op \u2032 ) $ for all n as can be seen from the tabular data in Table I. This can be shown explicitly using Eqs. (13), with C_{d} replaced by C_{d,pert}, and Eqs. (4) and (3) and (*E*_{0})^{2} ∝ I. It can similarly be shown that $ \sigma \u0304 f , C \u2192 C pert res ( t op \u2032 ) $ and $ t op \u2032 = t op ( C \u2192 C pert ) $ vary inversely with I in agreement with the corresponding numerical data of Table I. These results lead to a more appropriate definition of an “optimum” two-photon cross section, associated with $ R \u0304 f , C \u2192 C pert res ( t op \u2032 ) $, that is intensity independent. Namely $ [ \sigma \u0304 f , C \u2192 C pert res ( t op \u2032 ) ] redefine = R \u0304 f , C \u2192 C pert res ( t op \u2032 ) /F$ with units of cm^{2} which are units usually associated with one-photon excitation cross sections. For the model transition considered here $ [ \sigma \u0304 f , C \u2192 C pert res ( t op \u2032 ) ] redefine =$ 4.32025x10^{−17} cm^{2} and t_{op}(C → C_{pert}) = T/I where T = 4.26150/x10^{−3} s (W/cm^{2}). Using the equality $ R \u0304 f , C \u2192 C pert res ( t op \u2032 ) = \sigma \u0304 f , C \u2192 C pert res ( t op \u2032 ) F 2 = [ \sigma \u0304 f , C \u2192 C pert res ( t op \u2032 ) ] redefine $ F the explicit result for the original cross section for the model transition is $ \sigma \u0304 f , C \u2192 C pert res ( t op \u2032 ) =9.4176x1 0 \u2212 ( n + 36 ) cm 4 s$. Both cross sections correspond to a rate $ R \u0304 f , C \u2192 C pert res ( t op \u2032 ) =1.9819x1 0 n + 2 s \u2212 1 $.

The optimized RWA rate, Eq.(13), can be written in terms of the effective laser intensity defined by Eq.(18). Namely $ R \u0304 f res ( t op ) = ( 8 / z 1 f 2 ) | J 2 z 1 f | R \u0304 f , C \u2192 C pert res ( t op \u2032 ) $ and its intensity dependence is I_{eff} and not I. The appropriate effective intensity independent cross section is then given by $ [ \sigma \u0304 f res ( t op ) ] redefine = R \u0304 f res ( t op ) / ( F eff ) $ which is identical to $ [ \sigma \u0304 f , C \u2192 C pert res ( t op \u2032 ) ] redefine = R \u0304 f , C \u2192 C pert res ( t op \u2032 ) /F$. Finally the time for which the RWA rate optimizes can also be written in terms of the effective intensity I_{eff}; t_{op} = T/I_{eff} where T is identical to that occurring in $ t op \u2032 = t op ( C \u2192 C pert ) $ above. For the model transition $ R \u0304 f res ( t op ) =1.9819x1 0 2 I eff s \u2212 1 $ and $ \sigma \u0304 f res ( t op ) =9.4176x1 0 \u2212 ( 2 n + 36 ) I eff cm 4 s$.

The calculations of the populations of the molecular states and the excitation rates in this paper are for θ = 0, that is when the molecular dipole is aligned with the electric field vector. Often the target molecules are randomly orientated with respect to the laser field direction. Averaging over the field - molecule orientations reduces the strength of the laser-molecule interaction. For example^{3,25} in the perturbative limit the two-photon excitation rate or cross section is reduced by a factor of five.

## IV. CONCLUDING REMARKS

The RWA laser-molecule coupling C_{d} for two-photon excitation, via the direct permanent dipole excitation mechanism, can be maximized as a function of laser intensity (I = 10^{n}) as discussed at the beginning of Sec. III. This is illustrated for a model two-level giant dipole molecule, on two-photon resonance, and for several choices of the orientation θ of $ d \xaf 1 f = \mu \xaf ff \u2212 \mu \xaf 11 $ relative to the laser field polarization vector $ e \u02c6 $. The intensity maximizing C_{d} is characterized by n = n_{max} which increases with increasing θ over 0 ≤ θ ≤ π/2; C_{d}(max) is independent of θ (Fig. 1).

The remainder of Sec. III is concerned with discussions of RWA and perturbative calculations of the temporal behaviour of the population of the excited state $ P f res ( t ) $, and the time averaged rate of two-photon excitation $ R \u0304 f res ( t ) $ and related cross sections, for the model resonant two-photon transition. The effects of varying the intensity index n from weak to strong values through the “optimum” value n_{max} are illustrated; θ = 0 is chosen as a specific example.

The period PER of $ P f res ( t ) $ and the time t_{op} required to obtain the maximum in $ R \u0304 f res ( t ) $ minimize, and the small time RWA results $ P f , < res ( t ) $ and $ R \u0304 f , < res ( t ) $ at a given time t and the $ R \u0304 f res ( t op ) $ maximize, as n varies from small to large values, at n = n_{max} = 12.769. For example $ R \u0304 f res ( t op ) $ and t_{op} vary from 1.982 x10^{11}s^{−1}, through 4.858x10^{14}s^{−1}, to 2.930x10^{14}s^{−1} and from 4262 fs, through 1.739 fs, to 2.882 fs respectively as n changes om 9, through 12.769, to 13.5. The perturbation theory results $ P f , pert res ( t ) $ and $ R \u0304 f , pert res ( t ) $ do not show this laser intensity breathing effect.

The small time RWA results generally give a considerably better representation of the full RWA results than does perturbation theory as a function of increasing time. For example $ P f , < res ( t ) $ and $ R \u0304 f , < res ( t ) $ are reliable to within 2% until t > t2% = .07751 PER = 0.2089 t_{op}; typical values of t_{op} and hence PER are in Table I. This result applies for perturbation theory as well for small n but as n increases the failure of the perturbative results becomes clear; they are in error by greater than 2.270, 2.858, 4.740, 10.95, 33.51 and 147.46 % , instead of 2%, for n = 10, 10.5, 11,11.5, 12 and, 12.5 respectively when t2% = 89.14, 28.27, 9.021, 2.936, 1.019, 0.439 fs. An analogous discussion based on a 1% error is given in Sec. III A.

When the laser intensity is relatively small, n < 10.5 for the model calculations of this paper, C_{d} and C_{d,pert} agree well and the RWPA and RWA results for the observables agree; see for example Table I. Even so $ P f , pert res ( t ) \u2248 P f , < res ( t ) $ and $ R \u0304 f , pert res ( t ) \u2248 R \u0304 f , < res ( t ) $ do not represent the full RWPA or RWA results at all well for important times associated with PER/2 or t_{op}.

Consider n = 9 as an example; the RWPA and RWA results are essentially identical. For t = t_{op} = 4262.06 fs, $ R \u0304 f res =1.9816$ x 10^{11} s^{−1} and $ P f res =0.84457$ while for t = PER/2 = 5743.71 fs, $ R \u0304 f res =1.7410$ x 10^{11} s^{−1} and $ P f res =1$. The analogous perturbation theory (or small time RWA) results are $ R \u0304 f , pert res ( t op ) =3.1883$ x10^{11}s^{−1}, $ P f , pert res ( t op ) =1.3589$, $ R \u0304 f , pert res ( PER / 2 ) =4.2968$ x10^{11}s^{−1},$ P f , pert res ( PER / 2 ) =2.4681$ and are much too high. The perturbative results are in serious error due to the omission of the higher order terms in the expansion of the sine function when obtaining them from the RWA or RWPA results for $ P f res ( t ) $ and $ R \u0304 f res ( t ) $; see the relevant discussions in Sec. III B versus that in Sec. III A. For n = 9 the perturbation theory (or small time RWA) results are reliable to 1%, 2% and 5% for times t < 631.5, 890.4 and 1395.6 fs which are far below t_{op}.

Choosing n = n_{max} = 12.769 is of particular interest since this corresponds to the maximum value of the time averaged rate of excitation $ R \u0304 f res ( t op ) $, 4.8658x10^{14} s^{−1}, and the minimum values of t_{op} = 1.739 fs and PER/2 = 2.343 fs ; $ P f res ( t op ) =0.8446$. The small time RWA results for the rate and population are inadequate for these laser-molecule interaction times. For example they are too high by 1%, 2%, 5% and 10% at t = 0.258, 0.363, 0.569 and 0.794 fs respectively while at these times the analogous perturbative results are too high by 480% or more.

The small time RWA and perturbative results for the state populations and rates are reliable only for relatively small laser-molecule interaction times which depend on the laser intensity. The two examples discussed directly above furnish complimentary examples for low versus high laser intensities. As the laser intensity increases, so that the RWPA and the RWA disagree significantly, the small time RWA results are preferred relative to the perturbation theory results for $ P f res ( t ) $ and $ R \u0304 f res ( t ) $.

For n = 9 the small time RWA and perturbative results agree well and when t = t2% = 890.38 fs, $ P f , < res =0.0593$ and $ R \u0304 f , < res =6.66x1 0 10 s \u2212 1 $ compared to $ P f res =0.0581$ and $ R \u0304 f res =6.53$ x 10^{10} s^{−1}. On the other hand when n = 12.769 the small time RWA and perturbative results do not agree at all well and when t = t2% = 0.3632 fs, $ P f , < res =0.0593$ and $ R \u0304 f , < res =1.632$ x10^{14} s^{−1} while $ P f , pert res =0.341$ and $ R \u0304 f , pert res =9.377$ x 10^{14} s^{−1} versus $ P f res =0.0581$ and $ R \u0304 f res =1.600$ x 10^{14} s^{−1}. In both examples while the small time RWA results are not useful for times much greater than t2% they are useful for applications involving “short” pulses or a sequence of short pulses separated by a suitable repetition interval.^{5,26–28} This comment is only applicable for the perturbative results for relatively weak laser intensities.

The Gaussian pulse durations, see Sec. II A, required to obtain an on-resonance steady state population of 0.0581, 0.8446 and unity are 0.2898 fs, 1.388 fs and 1.869 fs respectively when n = 12.769 and 710.422 fs, 3400.64 fs and 4582.82 fs respectively for n = 9.

As discussed in Sec. III A a two-photon excitation cross section based on the definition Rate/(F)^{2} where F = I/E, which has units of cm^{4}s, is only useful in the perturbative limit; $ \sigma \u0304 f , pert res ( t ) = R \u0304 f , pert res ( t ) / F 2 $. The RWPA results for the time averaged excitation rate leads to a maximized two-photon excitation rate $ R \u0304 f , C \u2192 C pert res ( t op \u2032 ) $, for each laser intensity n, which occurs at $t= t op \u2032 = t op $ (C_{d} → C_{d,pert}). The laser intensity dependence of this rate is not I^{2}, rather it is I (Sec. III B). This leads to a natural choice, assuming the RWPA is valid, for a maximized (as a function of time) laser intensity independent two-photon excitation cross section, namely $ [ \u2009 \sigma \u0304 f , C \u2192 C pert res ( t op \u2032 ) ] redefine = R \u0304 f , C \u2192 C pert res ( t op \u2032 ) /F$, which has units, cm^{2}, usually associated with one-photon excitations.

Analogous results can be written, through the use of the effective laser intensity I_{eff} defined by Eq.(18), for the corresponding RWA rate expressions. These are obtained by replacing I by I_{eff}. The excitation cross sections $ \sigma \u0304 f , pert res ( t ) $ and $ [ \u2009 \sigma \u0304 f , C \u2192 C pert res ( t op \u2032 ) ] redefine $ can then be used to obtain the associated RWA excitation rates as a function of laser intensity via $ R \u0304 f , < res ( t ) = \sigma \u0304 f , pert res ( t ) F eff 2 $ and $ R \u0304 f res ( t op ) = [ \u2009 \sigma \u0304 f , C \u2192 C pert res ( t op \u2032 ) ] redefine F eff $ where F_{eff} = I_{eff}/*E*. The effective intensity I_{eff} is the laser intensity seen by the dipolar molecule and is generally less than the bare laser intensity I; see Sec. III and Fig. 5. The notion of effective laser intensities connects two-photon excitation rates based on the perturbative laser-molecule coupling, including the RWPA, to the analogous RWA results. In general the two-photon excitation rate is not proportional to I^{2}.

In this paper the direct permanent dipole moment mechanism for two-photon excitation has been considered. This requires a two-level molecular model and the discussion here has been based on the RWA and perturbative results related to the two-photon RWA. The validity of the RWA, both with and without permanent dipole moments, as a representation of exact two-level calculations has often been discussed in the literature.

The RWA can be expected to represent the two-photon excitation only if it is well separated from neighboring multi-photon resonances within the RWA. Using the RWA expressions^{29} for the N-photon resonance profiles, which are given^{30,31} by the long-time average of Eq.(1), the N = 1 and N = 3 resonances do not seriously overlap with that for N = 2 for most of the laser intensities considered here.

There are certain parameters^{10,20,29,30} which are useful in discussing the validity of the multi-photon RWA with respect to exact two-level calculations. Two of these are β(N) = NC_{d}(res, N)/E_{1f} and $b=| e \u02c6 \u22c5 \mu \xaf 1 f | E 0 / E 1 f $ where C_{d}(res, N) is C_{d} (N), Eq.(2), evaluated at the N-photon resonance energy E = E_{1f}/N. The parameter b is the strength parameter^{20,30} associated with atomic transitions (d_{1f} = 0), *β* (N) is the molecular (d_{1f} → 0) analogue.

The two-photon resonance is reasonably well separated from the one-photon and three-photon resonances if the sum of the half widths for adjacent resonance profiles is less that the difference between the corresponding resonance energies. This leads to the conditions V(1) = [β(1) + (1/4)β(2)] < 1/2 and V(2) = (1/4)β(2) + (1/9)β(3) < 1/6 where the RHS of these equations are (E_{res,r}(1) – E_{res,r}(2)) and (E_{res,r}(2) – E_{res,r}(3)) where E_{res,r}(N) = 1/N is the RWA N-photon resonance energy is units of E_{f1}. Expressions for the widths of the multi-photon resonances used in the derivation of these expressions are in the literature.^{29}

As a specific example consider the laser intensity n = n_{max} = 12.769 which corresponds to the maximum two-photon laser molecule coupling for the model transition considered in Sec. III. Here b(1) = 0.509, β(N) = 0.374, 0.649, 0.837 for N = 1, 2, 3 while V(1) = 0.537 and V(2) = 0.255. The decrease in β(1) relative to b(1) indicates a modest reduction in the one-photon laser-molecule coupling due to the presence of permanent dipoles and the values of V(1) and V(2) indicate a sizeable overlap of the two-photon resonance with its neighbouring resonances. For n = 12.5, V(1) = 0.452 and V(2) = 0.216 while for n = 13 they are 0.507 and 0.150 respectively indicating that the two-photon resonance will be more noticeably separate from the neighbouring resonances for these intensities than for n = 12.769. Part of the reduction of the V’s, as n increases from n = 12.679 to 13, is due to the corresponding decrease in C_{d}(2). As n decreases from n = 12.5 the V’s decrease significantly indicating, as expected, that the multi-photon resonances will become progressively more separated as the laser intensity decreases; for example V(1) = 0.26 and 0.14 while V(2) = 0.083 and 0.025 for n = 12 and 11.5 respectively. Generally there will be some overlap of the resonances in exact two-level calculations but the RWA results, especially for the two-photon resonances considered in this paper, will be qualitatively reasonable even for n = n_{nax}. For weaker laser intensities the RWA will be a (much) better representation of the exact two-level results.

The ideas used in the last paragraph have been tested against exact resonance profiles for CW lasers interacting with a variety of two-level dipolar molecules.^{8,10,29,32} In addition to misrepresenting the overlap of the resonances as the β’s and V’s increase the RWA also does not represent the dynamic background^{29,30} in the resonance profiles or the shift^{29,30,33} in the resonance positions from the RWA predictions; the latter can be either to low or the high energy for dipolar molecules.^{29} Nevertheless, as has been emphasized previously,^{23,34} even if the laser field is too large for RWA predictions to be quantitatively reliable they are often qualitatively useful for interpreting strong CW or pulsed laser-molecule interactions.

The focus of this paper has been on the optimization of the permanent dipole mechanism for two-photon excitation and on the intensity dependence of the excitation rate within this mechanism. This requires a two- level molecular system and our model is specified in Secs. II and III. The other mechanism for two-photon excitation involves virtual states. It can be considered analytically, together with the permanent dipole mechanism, through the use of the many-level analytical rotating wave approximation (AGRWA).^{6,35} Permanent dipole effects are included for the virtual state mechanism in the AGRWA. Other techniques are also available for these types of problems including the generalized rotating wave approximation (GRWA), the precursor of the AGRWA, and the generalization of the two-level RWA to many-level molecules.^{36–39} One approach for maximizing two-photon excitations is to seek molecules that lead to optimal excitation rates for the excitation from the ground state 1 to the excited state f via the permanent dipole excitation mechanism and then consider molecules with the desired or similar (1,f) framework and excited molecular states that add in the effects of the virtual state mechanism optimally and so it constructively^{4,5,24,40} augments the contribution of the permanent dipole mechanism.

The direct permanent dipole two-photon excitation mechanism is not available for non-polar molecules (C_{d} = 0). In this situation only the virtual state mechanism is operational and the treatment of the problem requires a model with at least three molecular states (1, f and a virtual state v say). The inclusion of virtual states is also important if the permanent dipole mechanism is weak relative to the contribution of the virtual sates.

## ACKNOWLEDGEMENTS

The support of the Natural Sciences and Engineering Research Council of Canada is acknowledged.