The characterization of the nonlinear optical properties of materials is an active field of research because of its many potential applications such as in optical limiter and in all-optical photonics devices [1]. The operation of optical limiters is usually based on the phenomenon of nonlinear absorption, especially those multiphoton ones, for instance, two-photon absorption (2PA) and three-photon absorption (3PA) [2]. Furthermore, many materials exhibit multiphoton absorption process under femtosecond laser pulses [2–7] and have been widely investigated. So it is of great importance to fully characterize the multiphoton absorption properties. In particular, great effort has been devoted to the determination of multiphoton absorption coefficients from the configuration of the Z-scan traces [5–8], intensity dependence of Z-scan technique [4, 5] and optical limiting performance [2]. However, no satisfactory theoretical analysis is available to identify and separate the contributions of 2PA and 3PA to the whole nonlinear absorption.

In this Letter, we present a theoretical study on optical beam propagation in a thin nonlinear medium with simultaneous 2PA and 3PA, a situation that exists, for example, in polydiacetylenes at 1320 nm with 100 fs pulses [4]. The analytical expression for the Z-scan traces, for the first time to our knowledge, are obtained and could straightforwardly determine the 2PA and 3PA coefficients simultaneously. Furthermore, we compare the analytic solutions with numerical simulations in detail.

Assuming a Gaussian beam traveling along the +z direction (the origin of z coordinate is located at the focusing plane of the tightly focused Gaussian beam), the optical intensity can be written as

where ω ²(x)=${\omega }_0^2$ (1+x ²) is the beam radius, ω ₀ is the waist radius of the Gaussian beam, x = z/z ₀ is the relative position of the sample and z ₀=${k\omega }_0^2$/2 is the diffraction length of the beam, k = 2π/λ, λ is the laser wavelength, and I ₀ is the on-axis optical intensity of the Gaussian beam at the focus.

Now we consider a Gaussian laser beam traveling in the +z direction within a nonlinear sample exhibiting linear absorption (α ₀) and (n+1)-photon absorption (β_n , where n ≥ 1 is a integer). Under the thin sample approximation, that is L << z ₀, where L is the geometrical length of the sample (this condition is easy satisfied in the practically experimental configurations), the light beam propagation through the sample is governed by the following equation [1]

Here z′ is the propagation length inside the sample. The intensity transmitted through the sample is given by

The normalized transmittance of the open aperture Z-scan for only (n+1)-photon absorption can be written as

where ψ_n =Ψ_n /(1+x ²), Ψ_n = (nβ_n$I_0^n$ $L_{\mathit{\text{eff}}}^{\left(n\right)}$ ^1/n is the (n+1)-photon absorption on-axis peak phase shift, $L_{\mathit{\text{eff}}}^{\left(n\right)}$ = [1-exp(-n α ₀ L)]/n α ₀ is the effective sample length taking into account of (n+1)- photon absorption, and ₂ F ₁[] is the hypergeometric function [9].

It is note that for the pure 2PA case (n = 1) Eq. (4) is degenerated into the well-known existing form in the literature [8]

For the pure 3PA process (n = 2), the normalized transmittance is

One may distinguish whether the material possesses pure two- or three-photon absorption process or not from the above expressions (5) and (6) [5, 6]. However, in most case the material exhibits simultaneous 2PA and 3PA, a situation that exists, for example, in polydiacetylenes [4] and chalcogenide glasses [10]. A very paramount problem is how to identify and separate the contributions of them.

When a Gaussian beam propagates in a thin nonlinear sample with simultaneous 2PA and 3PA, the optical intensity I inside the sample satisfies the equation as follows

In general, it is impossible to get an analytic solution for Eq. (7). Although many approximate solutions have been provided [2, 10, 11], it is difficult to obtain the expression of the normalized Z-scan transmittance. Here we pay our attention to the Z-scan trace instead of the optical intensity transmitted through the sample, because the Z-scan experiment only gives the normalized transmittance and then extracts the information of the nonlinear properties of materials. This may be a more practicable approach and an effective avenue. For this purpose, we present a theoretical treatment, which introduces a so-called coupling function f(x, Ψ ₁, Ψ ₂). The normalized transmittance of the Z-scan for the nonlinear medium with simultaneous 2P A and 3P A can be constructed as follows

where T ₂(x, Ψ ₁) and T ₃(x, Ψ ₂) are still given by Eqs. (5) and (6), respectively. It must be noted that the condition f(x, Ψ ₁, Ψ ₂)=1 should be satisfied when the contribution of 2PA and/or 3PA is absent. Based on a numerical fitting, the following expression was found

This empirical formula has only an utmost error of 1% from the extract numerical results for Ψ ₁ ≤ π and Ψ ₂ ≤ π. We can easily check that when β₁= 0 or β₂= 0, Eq. (8) degenerates into Eq. (6) or Eq. (5), respectively.

 

Fig. 1. Analytical Z-scan traces (solid lines) compared with the numerical simulations (discrete symbols) for different intensities at I ₀= 15 GW/cm² (squares), 20 GW/cm² (circles) and 25 GW/cm² (triangles), respectively.

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Now let us compare our analytic results with numerical simulations. Taking below parameters for the nonlinear material [4] as L= 0.01 cm, α ₀= 1 cm^-1, β₁= 5 cm/GW and β₂= 0.1 cm³/GW², we shows the intensity-dependent Z-scan traces obtained by our analytic solution (solid lines) and numerical simulations (discrete symbols) in Fig. 1 for I ₀= 15, 20 and 25 GW/cm², respectively. One can see that our analytic solution agrees well with the numerical simulations.

Figures 2, 3 and 4 show the analytical solutions (solid lines) and numerical simulations (circles) of the normalized transmittance of the valley T_V when one parameter of the nonlinear medium changes. Here the parameters used in our investigation, if they are not the changing one, are L= 0.1 cm, α ₀= 1 cm^-1, β₂= 8 cm/GW, β₂= 2 cm³/GW² and I ₀= 3 GW/cm², which are the typical parameter values [10] for chalcogenide glasses. It is also verified that our analytical solution agrees very well with the numerical simulation, although the analytical solution in Fig. 4 exhibits a relatively large deviation from the numerical simulation. However, the relative errors of T_V are found to be at most 1% within the range 0 ≤ β ₂ ≤ 5 cm³/GW² and the other parameters are L= 0.1 cm, α ₀= 1 cm^-1, α₁= 8 cm/GW and I ₀= 3 GW/cm².

 

Fig. 2. The normalized transmittance at the valley T_V as a function of the linear absorption coefficient α ₀ obtained by numerical simulation (circles) and our analytic solution (solid lines), respectively. The parameters are inside the figures.

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Fig. 3. The normalized transmittance at the valley T_V as a function of the 2PA coefficient β₁ obtained by numerical simulation (circles) and our analytic solution (solid lines), respectively. The parameters are inside the figures.

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To identify whether the material possesses simultaneous 2PA and 3PA or not, our analytic expression (8) associated with the empirical formula (9) could directly distinguish and give out the contributions of 2PA and 3PA by fitting the Z-scan experimental data. Furthermore, it is of great importance to perform the Z-scan experiment at various intensities of radiation and then to give the dependence of T_V on I ₀, as depicted by circles in Fig. 5. The theoretical fit T_V vs I ₀ using Eqs. (8) and (9) when x= 0 allows us to estimate the coefficients related to 2PA and 3PA simultaneously, with the known values of L= 0.1 cm and α ₀= 1 cm^-1. The fitting results are obtained β₁ = 8.012 cm/GW and β₂= 1.956 cm³/GW², with the relative errors of 0.15% and 2.2% with respect to the theoretical values β₁ = 8 cm/GW and β₂= 2 cm³/GW², respectively.

 

Fig. 4. The normalized transmittance at the valley T_V as a function of 3PA coefficient β₂ obtained by numerical simulation (circles) and our analytic solution (solid lines), respectively. The parameters are inside the figures.

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Fig. 5. The normalized transmittance at the valley T_V as a function of the on-axis peak intensity I ₀ obtained by numerical simulation (circles) and our analytic solution (solid lines), respectively. The parameters are inside the figures.

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In summary, the Z-scan formula for thin sample with simultaneous 2PA and 3PA is obtained, to the best of our knowledge, by introducing a coupling function. The result agrees very well with exact numerical simulation. It allows us to easily identify and directly estimate the 2PA and 3PA coefficients of the thin medium with simultaneous 2PA and 3PA.