Three levels of propagation of the Four-wave mixing signal

Results in Physics 11 (2018) 414–421

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Three levels of propagation of the Four-wave mixing signal J.L. Paz

a,b,⁎

c

a

, Ysaias J Alvarado , Luis Lascano , Cesar Costa-Vera

T

a,d

a

Departamento de Física, Escuela Politécnica Nacional, Ladrón de Guevara, E11-253, 170517, Apdo 17-12-866, Quito, Ecuador Departamento de Química, Universidad Simón Bolívar, Apartado 89000, Caracas 1086, Venezuela Laboratorio de Caracterización Molecular y Biomolecular (LCMB), IVIC, Zulia, Venezuela d Grupo Ecuatoriano para el Estudio Experimental y Teórico de Nanosistemas (GETNano), Diego de Robles y Vía Interoceánica, USFQ, Quito N104-E, Ecuador b c

A R T I C LE I N FO

A B S T R A C T

Keywords: Propagation Four-wave mixing Optical susceptibilities

In this work, we analyze different levels of propagation of the Four-wave mixing signal in a strongly driven twolevel system when the stochastic effects of the thermal bath, are considered. First approximation level, given by an analytical solution valid only for constant pump intensity along the optical path; a second approximation level as an analytical solution valid for a lineal variation of the pump intensity, and finally, a third level as numerical solution, which represent the exact case. In all cases, high dependence of the nonlinear propagation with the chemical concentration, stochastic noise parameters, relaxation times, are studied.

Introduction The study of the interaction between two-level systems and electromagnetic fields has been a very important subject in optics due its applications and usability to explain complex nonlinear phenomena. Under this scheme, in the analysis of the propagation of an electromagnetic signal some phenomena such as the attenuation or amplification of the beam, and the dispersion processes, can be better treated associated with random variables. Particular attention in this framework can be paid to those phenomena that lead to the generation of phonons, and which effects are explained by the notion of white and colored noise correlation functions, and/or Markov processes [1]. In view of the importance of gaining a better understanding in these topics for the propagation of electromagnetic fields in a nonlinear optical medium, we studied the different conditions under which the signal strength of Four-wave mixing FWM is modified by the always occurring effects of absorption and scattering themselves. In this work, the dynamics of such systems are described by the Optical Conventional Bloch equations (OCBE). Still, the strong interaction of the propagating wave with a medium can be subject to multiple collisions and the microscopic nature of the problem becomes complicated [2–8]. To formulate a solution it is necessary to introduce stochastic considerations. In this work, and under this framework, we assume that the systemsolvent interactions induce random shifts in the Bohr frequency and its manifestation should correspond to the broadening of the upper level [9]. Further, the putative effects over the propagation of the fields along the optical path, are analyzed. In this paper, two approximate

⁎

analytical models for the propagation of the FWM signal, and a numerical approximation to a third one, are discussed. The last case is conveniently associated with a transcendental equation. The consistency of the analytical models as dependent on the corresponding approximations of the limiting cases they each describe as compared to the numerical approximation are reviewed. The capability to demonstrate similar correct descriptions of the problem, were established within a 10% tolerance with respect to the more complete numerical model for the analytical solutions, evaluating in this way the extent of applicability of the underlying parameters and particular considerations of each model. The effect of the medium is modeled here with the aid of a stochastic broadening of the molecular energy levels. All the approaches described above are valid in the region 4S / T22 ≪ 1 (S: Saturation parameter, T2 Transversal relaxation time). Since water solutions of Malachite Green chloride satisfy this condition, this system is a good candidate for testing the effects discussed in this work. For this, having analytic expressions for the propagation of the fields allows us understanding the important photonic effects that take place in the interaction. These expressions also help to evaluate which approximations are better suited to study the propagation of the field under different putative experimental conditions. Various works in the same line of research related to the propagation of electromagnetic fields along the optical path have been performed previously. Boyd et al. [10] studied the effect of the propagation of fields in a strongly coupled two-state system, developing a model in which the pump intensity is strictly constant along the optical path. Reif

Corresponding author at: Departamento de Física, Escuela Politécnica Nacional, Ladrón de Guevara, E11-253, 170517, Apdo 17-12-866, Quito, Ecuador. E-mail addresses: [email protected], [email protected] (J.L. Paz).

https://doi.org/10.1016/j.rinp.2018.09.041 Received 3 April 2018; Received in revised form 13 September 2018; Accepted 19 September 2018 Available online 25 September 2018 2211-3797/ © 2018 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).

Results in Physics 11 (2018) 414–421

J.L. Paz et al.

et al. [11] developed a propagation model considering three levels of approximation, but not necessarily making the pump beam constant. Both models are totally deterministic. Paz and Recamier [12] introduced a propagation model to analyze two analytical solutions for the study of the propagation of the FWM signal in a strongly driven twolevel system, when the stochastic effects of the solvent are explicitly considered. In that work, the pump field was treated at all orders but the probe and signal fields at first order only. In contrast to this, in the present work, the probe beam is treated to the second order thus exploring important effects at this level. Thus, considering perturbationally the probe beam, photonic processes that generate energy at the frequency of the pump-beam are extracted. Unlike previous models, the pump beam was attenuated along the optical path and its intensity was never restored. As demonstrated in this article, most important in this regard for the description of the signal propagation, are the saturation parameter and the concentration of the analyte. The most relevant aspect of this work is its contribution to the understanding of the putative generation of nonlinear multiphotonic processes induced by the probe beam in the FWM scheme.

defines the intensity of the coupling between matter and radiation, μba , (disregarding in connected only to the transition dipole moments → → → → this case, permanent dipole moments). Here, is defined E = E1 + E2 , → → → → with Ej (t ) = Ej0 exp(i kj . r −ωj t ) (j = 1,2), and ξt = iξ (t ) + T2−1; T1 and T2 are defined as longitudinal and transversal relaxation times, respectively. In this methodology, we consider that the system-solvent interactions induce random shifts in the Bohr frequency, given by ξ (t ) = ω0 + σ (t ) , where ω0 is the Bohr-frequency for the isolated twolevel molecular system; σ (t ) integrates all the stochasticity of the problem. Our study is localized in the Four-wave mixing FWM spectroscopy, where the most general process involves the interaction of three → → → laser fields with wave vector k1, k2 and k3 and frequencies ω1, ω2 and → ω3 , respectively, with a nonlinear medium. Here, kS and ωS are given by any linear combination of the applied wave-vectors and frequencies. Solving Eq. (2) for a two-level molecular system, the Fourier components of the coherences at the indicated frequencies, are given by:

Theoretical considerations

with m = 1, 2, 3 and where λ ξ is given by:

In this work, we describe the time-dependent process of interaction of molecular systems with a total external electromagnetic field (radiative process) and with a thermal bath (non-radiative process) using the Liouville-Von Newmann treatment in the semiclassical approximation. The reduced density matrix equation describing the dynamics behavior, is given by:

λξ = −

i ∂t ρ ̂ = − [H ,̂ ρ ]̂ −Γρ̂ ,̂ ħ

ρba (ωm) = {iΩm + iλ ξ [Ω1δm,3 + Ω2 δm,1] + iλξ∗ [Ω1δm,2 + Ω3 δm,1 ]}

Lξ , m

, (3)

2 (Ω1Ω∗2 J1 + Ω1∗ Ω3 υ3, −1 J2−Ω1Ω∗3 J3), |J2 |2

(4)

where we have defined:

J1 = Γ1∗ (Δ12) + 2 |Ω1|2 υ1, −2 υ2, −3 , J2 = Γ1∗ (Δ12) + 2 |Ω1|2 υ2, −3 +

(1)

(4.a)

|2

2 |Ω2 , Lξ∗,1

J3 = 2 |Ω2 |2 υ1, −1 υ1, −3 .

where H ̂ = H0̂ + V ,̂ with H0̂ the Hamiltonian for the isolated system → and V ̂ the perturbation, given by V ̂ = −→ μ . E ; Γ ̂ represents the relaxation matrix that includes the relaxation rates between two-states considered 1/ T1 and for the induces coherences 1/ T2 . The decay rate of the excited state population and the dephasing rate of the optical transition must also be included in the analysis whenever the frequency ω1 becomes comparable to or smaller than this rates (decay of the levels, T1 effects, and of optical transitions, T2 effects). In this work, we use the Optical Stochstic Bloch equations (OSBE) to model the dynamical behavior of the system, which is given by [13]:

∂t ρ (t ) = Mξ (t ) ρ (t ) + R,

ρDdc

(4.b) (4.c)

With Γ1 (Δ12) = 1/ T1−iΔ12 ; υn, −m =

(2T2−1) + iΔn, m Lξ , n Lξ∗, m

where Δnm = ωn−ωm (for

the indicated values of n and m). The lorentzian Lξ , m ≡ Lξ ,2n + 1 = T2−1 + i [ξ (t )−(ω1 + nΔ12)], and where considered values of n = 0, −1, 1 for pump, probe and FWM signal, respectively. For simplicity, we define Lξ , −1 ≡ Lξ ,2 for the probe. The zero-frequency Fourier component is given by:

ρDdc =

(2)

ρD(0) (1−fξ ) +

4S1 T22 | Lξ ,1 |2

. (5)

|Ω1|2 T1T2

where Mξ (t ) and R are the radiative and non-radiative matrices, respectively. The density matrix for the two-level system, is defined as:

S1 = is defined as the saturation parameter, associated to pump beam. Next, a perturbative method at all orders in the pump- beam, second order in the probe-beam, and at first order in the generated FWM signal, is used to solve the equations. The function fξ is given by:

⎛ ρba (t ) ⎞ ρ (t ) = ⎜ ρab (t ) ⎟, ⎜ ρ (t ) ⎟ ⎝ D ⎠

υ1, −2 υ1, −2 υ2, −1 υ2, −1 ⎞ ⎡ fξ = 4T1 ⎢|Ω1|2 |Ω2 |2 ⎛⎜ + ⎟ J2∗ J2 ⎝ ⎠ ⎣

(2.a)

υ2, −1 υ3, −1 υ1, −2 ⎞ ⎤ + (Ω1∗)2Ω2 Ω3 ⎛⎜ + ⎟ . ⎥ L J J2 ,1 2 ξ ⎝ ⎠⎦

Mξ (t ) is a matrix containing strictly all of the matter–radiation interaction details and is defined as: 0 iΩ ⎞ ⎛ − ξt ∗ Mξ (t ) = ⎜ 0 − ξt − iΩ∗ ⎟, ⎜ ⎟ ∗ ⎝ 2iΩ − 2iΩ − 1/ T1⎠

Given that the coherences, Eq. (3), are dependent of the stochastic variable ξ (t ) (through of Lξ , j ), it is necessary to establish an average over all realizations of the random variables as 〈ρba (ωk ) 〉ξ . In order to carry out the mentioned above averages, Van Kampen [14] has proposed a method where he formally solves the stochastic differential equation assuming it to be deterministic and then takes an average over the realizations of the stochastic variable. A different approach consists in taking the same average before solving the Optical Bloch equations OBE. In the latter case, the set of differential equations obtained can be described as an Ornstein-Uhlenbeck process (OUP), the set of equations solved and one obtains an equation for the average of ρba (ωk ) . In the present work, we solve OBE as if they were deterministic and then, acknowledging the fact that ρba (ωk ) depends upon the realizations of

(2.b)

and, R is defined as the relaxation matrix associated with the equilibrium condition, given by:

⎛ 0 ⎞ R = ⎜ 0 ⎟, ⎜ ρ (0) / T ⎟ ⎝ D 1⎠

(2.c)

ρD(0)

(6)

(0) ρgg −ρee(0) ,

We define further the equilibrium effective population ≡ → → and the Rabi frequency by the scalar product Ω = μba . E (t )/ ħ , which 415

Results in Physics 11 (2018) 414–421

J.L. Paz et al. M

ξ (t ) , we take its ensemble average over the distributions of states with molecular frequency between ξ and ξ + dξ . Thus, solving the stochastic equations and taking an ensemble average instead of a time average, we have by-passed the problem of solving average involving the multiplicative noise term. At this point it is worth mentioning that this expansion (coherences) is valid as long at the density matrix elements are wide-sense stationary stochastic functions, that is, functions with definite mean values and stationary correlations functions. In our model, both conditions are satisfied for each case of the reduced density matrix elements. With the solution of the coherences 〈ρba (ωk ) 〉ξ at different frequencies, it is possible to evaluate the nonlinear macroscopic polarization components, which result in the tensorial approximation [15] of the form: → P (ωk ) = N

M

∑ ak C2 〈Φ(Bk) 〉ξ ,

χ coupling (ω1) = 2

(13.a)

k=0

M

χ coupling (ω2) =

∑ ak [C2 〈Φ(Bk) 〉ξ

+ C3 〈ΦC(k ) 〉ξ |E (ω2)|2 ],

(13.b)

k=0

0

χ coupling (ω3) =

1 coupling χ (ω1) 2

(13.c)

perturbative conditions, is valid 〈ΦC(k ) 〉ξ = 〈Φ(Bk + 1) 〉ξ = we have defined 〈Φ(Am)〉 = Lξ−1 |Lξ |−2m . The coefficients

In our where pressions, are given by:

〈Φ(Ak + 2) 〉ξ , in all ex-

C1 = i |μba |2 NρD(0) / ħ; C3/ C2 = C2/ C1 = d = −4 |μba |2 (T1/ T2)/ ħ2; ak

∞

(8)

= (−1) k (4S1/ T22) k .

where p (ξ ) is the probability density of events ξ around its average value ω0 , which is taken as:

The M-index in all the previous sums, is the required order to assure the convergence. Using the convolution theorem in the real and imaginary parts of 〈Φ(0) A 〉 expressed as: (0) (0) 〈Φ(0) A 〉ξ = Re 〈Φ A 〉ξ + i Im 〈Φ A 〉ξ ,

(9)

(14.a)

we obtain:

For this case, an OUP is considered, for which the noise has an intensity γ and an exponential correlation function with a decaying rate τ , this is: 〈ξ (t ) ξ (s )〉 = γτ exp(−τ |t −s|) , and where the product γτ is the variance of the distribution. Solving the nonlinear polarizations at different frequencies, we obtain:

Re 〈Φ(0) A 〉ξ =

1 Re Qξ γτ

1 Im Qξ , γτ

and Im 〈Φ(0) A 〉ξ =

(T2−1−iΔ1)(2γτ )−1/2 ,

exp(u2) ,

where Qξ = (1−erf(u)) with u = erf(u) meaning error function evaluated at u, given by:

P (ωm) = χ SV (ωm) E (ωm) + χeff (ωm) E (ωm) + χ coupling (ωm) E (ωa) E (ωb) E ∗ (−ωc ),

(12.c)

Finally, the electric susceptibilities associated to the coupling mechanisms, are given by:

μba 〉ϑ ≡ N 〈〈ρba (ωj , ξ ) 〉g (ω )→ μab 〉ϑ , ∫−∞ dω0 g (ω0) 〈〈ρba (ωj) 〉ξ →

(ξ −ω0 )2 ⎤ 1 exp ⎡− . ⎢ 2γτ ⎥ 2πγτ ⎣ ⎦

+ C3 〈ΦC(k ) 〉ξ |E (ω2)|2 ]|E (ω1)|2 .

k=0

where N is the chemical concentrations of solute molecules, and the external bracket denotes an average over spatial molecular orientations (over the Gaussian distribution of molecular frequencies g (ω0) ). The ensemble average values in the customary, are defined as:

p (ξ ) =

(12.b)

M

∑ ak [C2 〈Φ(Bk) 〉ξ

χ coh (ω3) =

∞

∫−∞ dξXξ p (ξ ),

|E (ω1)|2 ,

k=0

(7)

〈X 〉ξ =

∑ ak C2 〈Φ(Bk) 〉ξ

χ coh (ω2) =

erf(u) =

(10)

where the relationship ωa + ωb = ωc + ωm , holds for values:

2 π

∞

∑ n=0

(14.b) and where

2n + 1

(−1)n ⎛ 1/ T2−iΔ1 ⎞ ⎜ ⎟ n! (2n + 1) ⎝ 2γτ ⎠

. (15)

〈Φ(Am) 〉ξ ,

m = 2, with a = b = 1, c = 3

we use the recurrence relation For any term m of the function 〈Φ(Am) 〉ξ = Om̂ Re 〈Φ(Am − 1) 〉ξ , where the operator Om̂ is given by: T Om̂ = 2m2 (T2−∂T2−1−i∂Δ1) . Finally, the term 〈A(0) 〉ξ is given by:

m = 3, with a = b = 1, c = 2

〈A(0) 〉ξ =

m = 1, with a = 2, b = 3, c = 1

In Eq. (10), χ SV (ωm) represents the solvent electric susceptibility at frequency ωm . We have defined also χeff (ωm) as the effective susceptibility associated to two-photon processes, which can be written as χeff (ωm) = χ incoh (ωm) + χ coh (ωm) , where χ incoh (ωm) and χ coh (ωm) represent respectively the incoherent and coherent components of the susceptibilities. The incoherent part takes into account the reduction occurring in the relative population due to saturative effects of the pump and probe fields. The coherent components, in turn, are due to population oscillations at the detuning frequency Δ12 . The component χ coup (ωm) is the effective scalar complex susceptibility at frequency ωm due to the coupling processes. At the condition where the maximum of the population pulsation effects occur ω1 ≈ ω2 [16], which implies that Lξ , m ≡ Lξ . It is possible demonstrate that in organic dyes is valid the ∼ condition: 4S1/ T22 |Lξ |2 = Y ≪ 1, and therefore T1 J2 ≪ 1. The Eq. (5) can ∼ ∼ be expressed as ρDdc ∼ ρD(0) (1 + Yξ )−1, and taking Yξ as an expansion parameter, we express the susceptibilities of the following manner:

∑ ak C1 〈Φ(Ak) 〉ξ ,

for values of v = 1,2,3. In this solution we have considered the slowly varying envelope approximation [14]. Here, α v (ω v , z ) is the nonlinear absorption coefficient, defined as:

j = 1, 2, 3, (11)

k=0

α v (ω v , z ) = 2πω v Im χeff (ω v )/ ηv (ω v z ) c,

M

χ coh (ω1)

=

∑ k=0

ak C2 〈Φ(Bk ) 〉ξ

|E (ω2

(16)

Having calculated the expressions for the different nonlinear optical susceptibilities (Eq. (10)), where we explicitly included the stochastic effects of the solvent on the active molecular system, given by the function 〈Φ(Am) 〉ξ , we now study the fields propagation, nonlinear optical properties (absorption and refractive index), and its scattering processes in the condensed medium. Eq. (10) defines the components of the induce polarizations at the different optical frequencies. Solving Maxwell’s equations for these components it is possible to determine the propagation of the different electromagnetic fields along the optical path z: ∼ dE (ω v ) ∼ ∼∗ ∼∗ (z )(E (ω2) δ v,3 + E (ω3) δ v,2) + α v (ω v , z ) E (ω v ) = [Ψ (1,1) v dz ∼∗ (z ) E (ω1) δ v,1] exp(iΔk z z ) + Ψ (2,3) (17) v

M

χ incoh (ωj ) =

−1 Δ1T2−1 ⎞ 1 ⎛ Δ1T2 ⎟⎞ ⎤. exp[(T2−2−Δ12)/2γτ ] ⎡ cos ⎜⎛ ⎟−i sin ⎜ ⎢ ⎥ γτ ⎝ γτ ⎠ ⎦ ⎣ ⎝ γτ ⎠

Ψ (vm, n) (z )

)|2 ,

(18)

where is defined as the coupling factor between the m and n beams, given by:

(12.a) 416

Results in Physics 11 (2018) 414–421

J.L. Paz et al.

∼ ∼ Ψ (vm, n) (z ) = 2πiω v χ coup (ω v ) E (ωm) E (ωn )/ ηv (ω v , z ) c,

considerations, the functions ϒ¯1(z ) and ϒ¯2 (z ) can be taken again approximately constant and equal to their values at the origin, this is: ϒ r(2) (z ) = ϒ r(2) (0) with r = 1, 2. Then, the normalized intensity of the FWM signal can be expressed in this approximation as:

(19)

and where ηv (ω v z ) is the nonlinear refraction index, defined as:

ηv (ω v z ) = [η02 + 4π Re χeff (ω v )]1/2 ,

(20)

d ln |Ψ3|2 ⎞ ⎤ ∼ ∼ 2 z f (w )/|w | , dS ⎠ ⎥ ⎦

η0 is the refractive index of the solvent: where η0 = [1 + 4π Re χ SV (ω v )]1/2 . In Eq. (17) the z component of the propagation vector mismatch is defined as Δk z :

(I3(2) (z )/ I2 (0)) = |Ψ 3(0) |2 exp ⎡−⎛2α3 + α1(S0) S0 ⎢ ⎝ ⎣

ω Δk z ≃ [2η1−(η2 + η3)cos φ], c

∼) = e (w + w ) z−e (w − w ) z−e−(w − w ) z + e−(w + w ) z f (w where and ∼ = Re[w ∼] + i Im[w ∼]. w ∼ = (ϒ¯ (2) )2−4ϒ¯ (2) . ∼= 1 w ∼ = 1 Re[w ∼ ] + i Im[w ∼ ] and w Also w 0 0 0 0 1 2 2 2 Evaluating the two roots in this expression, we can obtain the analytic signal intensity in the second approximation as:

⎜

(25) ∼ ∼∗

(21) → → where φ is the angle between k1 and k2 . Decoupling the individual equations in Eq. (17), we can obtain for the FWM signal the following expression: ∼ ∼ d 2E3 (z ) dE (z ) ∼ + ϒ¯1(z ) 3 + ϒ¯2 (z ) E3 (z ) = 0 2 (22) dz dz

∼ ∼∗

∼ ∼∗

∼] z ) + sin2 (Im[w ∼] z ) d ln |Ψ3|2 ⎞ ⎤ ⎡ sinh2 (Re[w ⎤ z −⎛2α3 + α1(S0) S0 ∼]2 + Im[w ∼]2 ⎥ ⎥ ⎢ dS Re[ w ⎝ ⎠ ⎦⎣ ⎦ (26) . ⎜

ln Ψ 3(1,1) ⎞

d Δk ⎞ −iΔk−iz ⎛ dz ⎠ ⎝ ⎠

∼ ∼∗

I3(2) (z ) = 4 |Ψ 3(0) |2 exp ⎡ ⎢ I2 (0) ⎣

The coefficients of this wave equation, are given by:

d ϒ¯1(z ) = α2 + α3−⎜⎛ dz ⎝

⎟

⎟

⎟

(23.a) (1,1)

d ln Ψ 3 dα ϒ¯2 (z ) = α2 α3−Ψ 3(1,1) Ψ 2(1,1) ∗ + ⎛ 3 ⎞−α3 ⎛⎜ dz dz ⎝ ⎠ ⎝

Third approximation

⎞−iΔkα −izα ⎛ d Δk ⎞ ⎟ 3 3 ⎝ dz ⎠ ⎠ (23.b)

We take the derivative of the saturation parameter more accurately and incorporate it into the spatial evolution of the FWM signal field. A simple expression allowing this, and also including the two other approximations above as particular limiting cases, is given by:

Note that in Eq. (22) we have considered Ψ1(2,3) (z ) ≈ 0 , since this factor determines the photonic coupling of two low intensity beams (probe and generated FWM signal). In Eq. (17) we can see that for the pump ∼ ∼ beam dE1/ dz = −α1 (ω1, z ) E1, which can be transformed in terms of the saturation parameter to dS1/ dz = −2α1 (ω1, z ) S1. The z dependence in Eqs. (22) comes from the corresponding dependence on both the pumpintensity along the optical axis and its spatial derivative.

dS1 4S = −2φS0 exp ⎡−2φ + 0 ⎤, dz b ⎦ ⎣ and b = 1 + where φ = variation of S1, the coefficients ϒ¯1(3) (z ) and

Δ12 T22 . With ϒ¯ 2(3) (z ) take

this form of the the form:

d Δk ϒ¯1(3) = 2α3−iΔk + 2φS0 exp(−2φz + 4S0/ b) ⎡iz ⎣ dS i ⎛ d Im[Ψ3] d Re[Ψ3] ⎞ + −Im[Ψ3] Re[Ψ3] dS dS ⎠ |Ψ3|2 ⎝

Optical propagation studies The manner of propagation of the FWM signal along the optical path z depends on how the intensity of the pump field changes along the axis z via the functions ϒ¯1 and ϒ¯2 . In the following, we explore three cases about of the propagation of the signal beam.

+

d Im[Ψ3]2 ⎞ ⎤ 1 ⎛ d Re[Ψ3]2 + 2 dS dS 2 |Ψ3| ⎝ ⎠⎥ ⎦ ⎜

⎟

(28.a)

ϒ¯ 2(3) = α32−|Ψ3|2 −iΔkα3 + 2φS0 exp(−2φz + 4S0

First approximation

iα d Im[Ψ3] d Re[Ψ3] ⎞ −Im[Ψ3] / b) ⎡ 32 ⎛Re[Ψ3] ⎢ |Ψ3| ⎝ dS dS ⎠ ⎣

In this case, we consider dS1/ dz = 0 , that take S1 (z ) and ϒ¯1, ϒ¯2 to be constant and equal to their values at the origin. With this approximation, Eq. (22) can be solved analytically for the normalized intensity of the FWM signal, giving:

(I3(1) (z )/ I2 (0)) = |Ψ 3(0) |2 exp[−2α3 z ][sinh2 (wz )/ w 2],

(27)

2 dc 2πω1 μab ρD T2/ ħη1 c

+

α3 ⎛ d Re[Ψ3]2 d Im[Ψ3]2 ⎞ d Δk ⎞ ⎛ dα3 ⎞ ⎤ + + iα3 z ⎛ − dS dS 2 |Ψ3|2 ⎝ ⎝ dS ⎠ ⎝ dS ⎠ ⎥ ⎠ ⎦ ⎜

⎟

(28.b)

(24)

An analytical solution for these more complex forms for the parameters ϒ¯1(3) (z ) and ϒ¯ 2(3) (z ) is no longer feasible, and consequently, the solution of Eq. (22) has to be solved numerically. In what follows, we can consider this level of approximation (the numerical one) in the propagation of the FWM signal as an exact solution of the problem, and compare its solutiońs behavior with that of the two first limiting cases to evaluate how good an approximation those analytical cases can be to the exact solution.

The boundary condition E3 (0) = 0 and E2 (0) = E2(0) have been used. In ϒ¯1(1) (z ) = 2α3−iΔk z ; this case then, we obtain: (1,1) (1) (1,1) 2 2 2 w = |Ψ 3 | −(Δk z )2 /4 , ϒ¯ 2 (z ) = α3 −iα3 Δk z−|Ψ 3 | , and (0) 2 (1,1) (1,1) ∗ |Ψ 3 | = Ψ 2 (0)Ψ 3 (0) . Notice that we also approximated to their α2 (ω2 , 0) ≈ α3 (ω3 , 0) initial values the parameters and Ψ 2(1,1) (0) ≈ Ψ 3(1,1) (0) . Second approximation

Results and discussion Since the approximation dS1/ dz = 0 is rather restrictive and not generally applicable, we can consider a second approximation level in which the spatial derivative of the saturation parameter can vary along the optical axis. At a first level we take a linear variation dS1/ dz = −2α1(S0) S0 for the pump propagation. However, even in this case ϒ¯1(z ) and ϒ¯2 (z ) might remain practically constant along the optical path, when either the optical length is small enough or the z dependence of the functions ϒ¯1(z ) and ϒ¯2 (z ) is weak. With these

In this section, we study the propagation of the FWM signal along the optical path for a two-level system in interaction with the fields and immersed in a thermal bath. In these calculations, Eqs. (10)–(13) have been truncated at the M value at which the relative difference between two-consecutive order terms was smaller than 1%. Also in our study, parameters corresponding to a typical molecular system, the organic dye Green Malachite, were used to generate sensible realistic values for 417

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Fig. 1. Signal intensity I3 of FWM signal vs. optical path z (cm) for S1 = 0.05 (in z = 0) and N = 0.01 mM (a), 0.1 mM (b), 1.0 mM (c) and 10 mM (d), in the three cases of propagation: first (analytical) approximation (p), second (analytical) approximation (q), and third (numerical) approximation (r).

in dependence of the penetration depth (Z optical length) in the medium for a saturation parameter of S1 = 0.1 (in z = 0) and the same concentrations as before. Again for cases 1 and 3 the intensity peaks at about the same position, but interestingly, for the lower concentrations the approximation provided by the simplified case 1 (that is, when the saturation parameter does not change along the propagation path), lies within the accepted tolerance for an approximation. In this case then, when the pump intensity also remains constant, such an approximation for the propagation of the FWM could be used in replacement of the exact calculation. But then again, this holds only near the summit of the intensity distribution. For high chemical concentrations again the calculation based on this analytical approximation fall far from the region of accepted tolerance. Also, it can be observed that for lower concentrations, the shape and symmetry of the solutions for the first analytical case approaches to that of the exact solution. Here, the maximum shifts well off from the numerical case’s one, towards lower z. It is also interesting to notice that the second analytical approximation curves are always asymmetric with respect to the center of the numerical solution, and its maximum falls always a little bit shorter in z than that of the exact case. We can specify that at very low concentration values, the two analytical cases coincide quite well with the numerical in both intensity and position of the maximum intensity. Since the study system has a very low number of molecules in the optical path, there is no opacity or change in pumping intensity. Results for a larger value of the saturation parameter (S1 = 0.4 (in z = 0)) are shown in Fig. 3, and for the same concentrations. In this Figure, we have represented the intensity I3 of the Four-wave mixing

the calculations. For this well-known dye, the transition dipole moment is μba = 2.81 × 10−18 erg1/2 cm3/2 and the Bohr frequency is ω0 = 3.06 × 1015 s−1. Fig. 1 shows the propagation of the corresponding intensities I3 of the FWM signal along the optical axis z (in cm), for values of saturation parameter S1 = 0.05 (in z = 0) and different values of chemical concentration (in 10−3 Munits) for the three different approximations described above. These are noted like this: first approximation (p), second approximation (q) and third approximation (r). In all figures, a dashed curve is included to limit the range for the admitted tolerance of ± 10% around the exact numerical calculations (r). In these pictures for S1 = 0.05 (in z = 0) it is observed that for cases 2 and 3 (p, r) there is a maximum at about the same position along the optical axis, independently of the dye concentration (z ≈ 0.02 cm) . When evaluating the quality of the approximation that each of the two analytical cases provide compared with the exact numerical case, it is clear that in both cases, and all concentrations included in Fig. 1, it is not satisfactory since all curves fall far out of the accepted tolerance region. Also, notice that the symmetry corresponding to the second analytical solution is different to that of the exact numerical case for all concentrations. We note that in all the cases studied, that employing a constant pumping intensity along the optical path, reproduces the numerical result in setting the z position of the maximum. In this case the actual intensity of the Four-wave mixing signal, is underestimated when compared to the exact case, taken as the numerical case. Under these conditions of saturation and chemical concentrations, it is not convenient to use the second approximation level, since it shifts the maximum towards lower optical length values, compared to the exact case. In Fig. 2, the signal intensity (I3 of the FWM signal) is again plotted 418

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Fig. 2. Signal intensity I3 of the FWM signal vs. optical path (Z in cm) for S1 = 0.1 (in z = 0) and N = 0.01 mM (a), 0.1 mM (b), 1.0 mM (c) and 10 mM (d), in the three cases of propagation: first (analytical) approximation (p), second (analytical) approximation (q), and third (numerical) approximation (r).

In Fig. 4, we have represented the intensity I3 of the Four-wave mixing signal as function of the optical path z (in cm). Fig. 4 shows results obtained when variation of the rate between relaxation times for the FWM signal intensity distribution are allowed upon penetration in the medium for the three approximations here considered. Independently of such rate, between the limits here considered, the shape, symmetry, and position of maxima remain similar to those already shown. Interestingly, when T1 < T2 , the first analytical approximation overestimates the intensity of the FWM signal respect to that of the numerical case which differs to what happens when the time rate is inverted and the pump beam intensity remains constant. In Table 1 the validity for the three approximations presented here within the tolerance limits set in this work, are summarized. The tolerance limits in this table are ± 10%in intensity and ± 0.6 cm in the position along the optical axis z. Three different saturation parameters and four referential concentrations are included. The same type of summary in Table 2 for three different rates between the relaxation times, is presented.

signal as function of the optical path z (in cm). For this saturation parameter the behavior observed before becomes a trend. Now, for concentrations 0.01 and 0.1 mM, the first analytical approximation provides profiles more similar to the exact ones than the other cases considered there. Additionally, independently of the chemical concentration, both the first and the numerical approximation keep their maxima in the same positions as before (z ≈ 0.02 cm) . For larger concentrations again, the approximations differ considerably within the allowed levels of tolerance with a behavior similar to that discussed for the previous referential saturation parameter. In particular the maxima corresponding to the second analytical approximation lie below z = 0.01 cm . Another point worth making is that at higher levels of chemical concentration for the dye, the intensity of the FWM signal is reduced very strongly in relation to lower values, in factors as important as 80 to 800, when saturation goes up 4 times, from 0.1 to 0.4. This is consisting with the fact that increasing the concentration of the dye makes the solution more opaque thus reducing strongly and nonlinearly the intensity of the FWM signal. As this occurs, the intensity of the pump beam becomes nearly constant and the processes of recombination with the probe beam necessary to generate the FWM signal become inexistent. Here, we notice that for high pump intensity and high chemical concentrations to consider that along the optical trajectory the pump intensity is not modified, is an error. The result in this first approximation overestimates the result when compared with the exact case.

Final comments A molecular two-level system immersed in a thermal bath in interaction with electromagnetic fields, has been treated with a perturbation approach. The solute-solvent interaction has been associated with a random Bohr frequency shift in the molecular system akin to a 419

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Fig. 3. Signal intensity I3 of the FWM signal vs. optical path z (in cm) for S1 = 0.4 (in z = 0) and N = 0.01 mM (a), 0.1 mM (b), 1.0 mM (c) and 10 mM (d), in the three cases of propagation: first (analytical) approximation (p), second (analytical) approximation (q), and third (numerical) approximation (r).

comparisons, it becomes clear that the second approximation for the variation along the path of the saturation parameter (the case of linear decrease of S) is mostly far from the exact case, especially in the location of the maxima. On the other hand, the symmetry of the calculated function around that value, provides thus a less useful approach than expected. Interestingly, the case of constant saturation, that is, with no variation along z, can under given condition, fulfill the 10% tolerance restriction indicated for a correct approximation. This case also corresponds to the simplest solution. Both the maxima and the symmetry around them are very close to the exact case for ranges of the indicated above basic parameters. Finally, these studies underline the importance of the longitudinal and transversal relaxation times for the propagation of the FWM signal. In all cases studied here, compromises and competence between the mechanisms leading to saturation, the chemical concentration of the species present, and the time scale of the solute-solvent interaction mediated dissipation. The validity of the ansatz whereby the pump beam is taken to remain constant along the optical axis has been confirmed in these calculations. In this way, propagation and interaction mechanisms can be approached without the need to call for analytically complex or demanding numerical solutions.

widening of the upper energy level. Both the macroscopic polarization behavior and the field amplitude variation in the propagation through the medium have been studied as a function of the probe beam second order amplitude. The apparition of new photonic processes related to couplings between incident and FWM-generated fields, has been observed. These processes reintroduce energy in the fields that are normally absorbed along the optical axis. Three approximation levels for the FWM-signal propagation have been used to describe these processes, two of which produce analytical solutions. The first of these analytical solutions corresponds to a constant saturation parameter along the whole optical path, and the second one to a linear decrease on penetration of such saturation parameter, respectively. Comparison of these two analytical solutions with the all-numerical approach, which we consider as the ‘exact’ one, permit determining conditions, whereby the analytical solutions are equivalent, within a 10% tolerance, to the numerical case. These conditions correspond to ranges of values of the saturation parameter, chemical concentration, and the noise intensity defining parameter, where the first-order consideration of the pump beam amplitude under the two analytical approximations can considered sufficient instead of the full numerical or transcendental solutions. The corresponding comparisons are evaluated by using the corresponding profiles along the z optical axis for the optical properties in these three cases, especially in the neighborhood of the maxima obtained in the exact case. The analysis involves both the absolute value of the optical property and the symmetry around the maxima. In these

Acknowledgements The authors acknowledges the financial support provided by the 420

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Fig. 4. Signal intensity I3 of the FWM signal vs. optical path z (in cm) in the three cases of propagation: first (analytical) approximation (p), second (analytical) approximation (q), and third (numerical) approximation (r), for cases of (a)T1/ T2 = 1, (b) T1/ T2 = 0.1, and (c) T1/ T2 = 10 . Table 1 Validity of the different approaches in propagation studies for different values of solute concentrations and saturation parameters.

S1 = 0.05 (in z = 0) S1 = 0.1 (in z = 0) S1 = 0.4 (in z = 0)

N = 0.01 mM

N = 0.1 mM

N = 0.5 mM

N = 1 mM

N = 10 mM

(r) (p,q,r) (r)

(r) (p,q,r) (r)

(r) (p,r) (r)

(p,r) (r) (r)

(r) (p,r) (r)

Table 2 Validity of the different approaches in propagation studies for different values of relaxation times. T1 = 0.1T2

T1 = T2

T1 = 10T2

(r)

(p,r)

(r)

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