Introduction

Multiphoton absorption is a nonlinear process in which two or more photons of indistinguishable or different frequencies are simultaneously absorbed by a material due its exposure to high intensity laser light [1]. We can have a single-photon process or many distinctive single-photon processes at the same time. In both cases, multiphoton nonlinear make reference to the modification in transmittance of the material as a function of light intensity. Due to the many different effects it produces, multiphoton absorption is of particular interest for various scientific and technological applications such as laser-induced damage [2, 3], fluorescence microscopy [4], optical power limiting [5, 6], fluorescence up-conversion [7, 8], Up converting lasing [9], luminescence imaging [10] and multiphoton spectroscopy [11].

There are numerous experimental techniques for characterizing second-order nonlinear optical processes in materials such as nonlinear absorption. Among them, the open Z-scan method furnishes the simplest and most sensitive experimental technique. This technique initially designed for measurements of two-photon absorption remains valid for those of multiphoton absorption. The basic open Z-scan technique has been well described by Mansoor Cheikh-Bahae et al. [12,13,14], and a brief description of the technique is given here. Along the axis of a focused Gaussian beam assimilated to the Z direction of the laboratory reference frame, a thin sample is translated in the region around the laser beam’s focal point. In this procedure, the transmitted energy in the far field is measured for each position \( z \) of the sample. The term “open aperture” means that the detector collects the entire transmitted light from the sample. The adjustment of the obtained open Z-scan traces by a suitable theoretical expression preliminarily elaborated makes it possible in principle the extraction of material’s nonlinear absorption coefficients.

The establishment of the theoretical expression of the optical intensity transmitted through a nonlinear material is described by means of first-order but nonlinear ordinary differential equation [1]. In the framework of slowly varying envelopes approximation, this equation admits an analytical solution for a single nonlinear absorption process of arbitrary order [15]. However, when several nonlinear processes are active, the corresponding differential equation does not accept an exact analytical solution [16]. Interestingly, different approximate formulas taking into account linear absorption and two simultaneous and successive (in order) nonlinear absorption processes have been given elsewhere [17,18,19,20]. The theoretical treatment of mixing three or more simultaneous nonlinear processes has not been reported to the knowledge of the authors. Note that, our motivations are not purely theoretical, because these phenomena have been observed in experimental situations. For example, Albert et al. [21] have published an excellent review on high-order optical nonlinearities in plasmonic nanocomposites such as metal nanoparticles embedded in dielectric matrix.

In this paper, we investigate theoretically the multiphoton absorption in thin nonlinear materials exhibiting simultaneously linear and \( \left( {N - n} \right) \)-photon absorption processes. An approximate analytical closed-form expression is derived for the normalized transmittance using the weak nonlinearities approximation. Thus, material’s absorption coefficients and incident elliptic Gaussian beam parameters can be extracted by fitting the open Z-scan trace with the established expression. The contribution of each multiphoton process to the total absorption may also be identified and separated accurately.

Transmitted optical intensity

Let us consider a nonlinear material in which prevail simultaneously \( \left( {N - n} \right) \) nonlinear absorption processes, due to its exposure to a Gaussian laser beam. The optical intensity \( I \) evolution inside the material is governed by the following equation [1]:

$$ \frac{{{\text{d}}I}}{{{\text{d}}z^{\prime } }} = - \alpha_{0} I - \alpha_{n} I^{n} - \alpha_{n + 1} I^{n + 1} - \cdots - \alpha_{N} I^{N} = - \alpha_{0} I - \mathop \sum \limits_{k = n \ge 2}^{N} \alpha_{k} I^{k} $$
(1)

where \( \alpha_{0} \) is linear absorption coefficient, \( \alpha_{k} \) the \( k \)-photon absorption coefficient, \( z^{\prime } \) the propagation depth inside the sample, \( n \) and \( N \) are integers such as \( N \) is the highest order of nonlinearity and \( n \) the lowest one \( \left( {2 \le n \le k \le N} \right) \). This first-order nonlinear ordinary differential equation containing polynomial functions of \( I \) is known as polynomial differential equation which contains Abel and Bernoulli equations as particular cases [22]. In general, mathematics does not provide an explicit solution to Eq. (1) except in rare cases. Therefore, numerical methods are often used to solve this equation (for a given \( n \)). The aim of this work is to give another alternative for solving Eq. (1). In our approach, we propose an approximate analytical but closed-form solution of Eq. (1). To do this, we proceed as follows: we derive a solution in the case of three simultaneous and consecutive (in order) nonlinear absorption processes, then, we generalize the result established to any finite number of processes.

It has been experimentally established that very high intensities are required to excite multiphoton absorption. For nonresonant transitions, this results in the following relation [23]:

$$ \frac{{\sigma_{n + 1} }}{{\sigma_{n} }}\sim 10^{ - 34} $$
(2)

where \( \sigma_{n + 1} \) is the \( \left( {n + 1} \right) \)-photon absorption cross section and \( \sigma_{n} \) the \( n \)-photon absorption cross section, which are given by the formulas [15]:

$$ \sigma_{n} = \frac{{\alpha_{n} \left( {h\nu } \right)^{n - 1} }}{{N_{0} }} $$
(3)
$$ \sigma_{n + 1} = \frac{{\alpha_{n + 1} \left( {h\nu } \right)^{n} }}{{N_{0} }} $$
(4)

where \( \left( {h\nu } \right) \) is the photon energy and \( N_{0} \) is the number of absorbing entities (atoms, molecules) per unit volume. If we divide Eq. (4) by Eq. (3), we find:

$$ \frac{{\sigma_{n + 1} }}{{\sigma_{n} }} = \frac{{\alpha_{n + 1} }}{{\alpha_{n} }}\left( {h\nu } \right)\sim 10^{ - 34} $$
(5)

In the visible region of the electromagnetic spectrum, the average energy of the corresponding photon is \( h\nu \sim 10^{ - 19} \,{\text{J}} \). Thus, we have:

$$ \frac{{\alpha_{n + 1} }}{{\alpha_{n} }}\sim \frac{{10^{ - 34} }}{{10^{ - 19} }} = 10^{ - 15} $$
(6)

By dividing the relation (6) by the pump beam intensity \( I \), one obtains:

$$ \frac{{\alpha_{n + 1} }}{{\alpha_{n} }}I\sim \frac{{10^{ - 15} }}{I} $$
(7)

For \( I\sim 1\;{\text{GW}}/{\text{cm}}^{2} \), which is a much lower than the intensity required to excite the two-photon absorption process (\( \sim 100 \;{\text{GW}}/{\text{cm}}^{2} \)), we have:

$$ \frac{{\alpha_{n + 1} }}{{\alpha_{n} }}I\sim 10^{ - 24} $$
(8)

Explicitly, the recurrence (8) yields the equations:

$$ \frac{{\alpha_{n + 1} }}{{\alpha_{n} }}I\sim 10^{ - 24} ;\, \frac{{\alpha_{n} }}{{\alpha_{n - 1} }}I\sim 10^{ - 24} ; \ldots ; \frac{{\alpha_{3} }}{{\alpha_{2} }}I\sim 10^{ - 24} ; \;\frac{{\alpha_{2} }}{{\alpha_{0} }}I\sim 10^{ - 24} $$
(9)

The product of these equations is found to be:

$$ \frac{{\alpha_{n + 1} }}{{\alpha_{0} }}I^{n} \sim 10^{ - 24n} \ll 1 $$
(10)

We can see that \( \left( {\alpha_{n + 1} I^{n} /\alpha_{0} } \right) \) is a very small quantity and it is even smaller than the order \( n \) is high, which means that very high intensities are required to excite multiphoton absorption. Therefore, all the terms of second and higher order in \( \left( {\alpha_{n + 1} I^{n} /\alpha_{0} } \right) \), in the Taylor series expansion, can be neglected giving the following approximations:

$$ \left( {1 \pm \frac{{\alpha_{k} }}{{\alpha_{0} }}I^{k} } \right)^{k} \sim \left\{ {\begin{array}{*{20}l} {1: {\text{for}}\,{\text{zeroth - order}}} \hfill & {(11 - {\text{a}})} \hfill \\ {1 \pm \frac{{\alpha_{k} }}{{k\alpha_{0} }}I^{k} : {\text{for}}\,{\text{the}}\,{\text{first - order}}} \hfill & {(11 - {\text{b}})} \hfill \\ \end{array} } \right. $$

here \( k \) is a real number. It should be noted that Eq. (10) means also that we are in the weak nonlinearities regime. Indeed, if we divide Eq. (10) by \( I \) we find:

$$ \frac{{\alpha_{n + 1} }}{{\alpha_{0} }}I^{n} = \frac{{\alpha_{n + 1} I^{n + 1} }}{{\alpha_{0} I}}\sim 10^{ - 24n} \ll 1 $$
(12)

which means that the multiphoton absorption is negligible. In reality, this assumption is contained implicitly through the Taylor series expansion in relation (1).

When the thin nonlinear material exhibit linear absorption \( \left( {\alpha_{0} } \right) \), \( n \)-photon absorption \( \left( {\alpha_{n} } \right) \),\( \left( {n + 1} \right) \)-photon absorption \( \left( {\alpha_{n + 1} } \right) \), and \( \left( {n + 2} \right) \)-photon absorption \( \left( {\alpha_{n + 2} } \right) \), Eq. (1) is reduced to:

$$ \frac{{{\text{d}}I}}{{{\text{d}}z^{\prime } }} = - \alpha_{0} I - \alpha_{n} I^{n} - \alpha_{n + 1} I^{n + 1} - \alpha_{n + 2} I^{n + 2} $$
(13)

To solve this equation, we are now going to use a series of substitutions similar to that used for solving the Bernoulli equation (general form \( y = x^{1 - p} \)) with the zeroth-order approximation (11-a):

$$ \begin{aligned} & I = \left( {u - \frac{{\alpha_{n} }}{{\alpha_{0} }}} \right)^{{\frac{1}{1 - n}}} \Rightarrow \\ & \frac{1}{1 - n}\frac{{{\text{d}}u}}{{{\text{d}}z^{\prime } }} = - \alpha_{0} u - \alpha_{n + 1} u^{{\frac{1}{1 - n}}} \left( {1 - \frac{{\alpha_{n} }}{{\alpha_{0} u}}} \right)^{{\frac{1}{1 - n}}} - \alpha_{n + 2} u^{{\frac{2}{1 - n}}} \left( {1 - \frac{{\alpha_{n} }}{{\alpha_{0} u}}} \right)^{{\frac{2}{1 - n}}} \\ & \approx - \alpha_{0} u - \alpha_{n + 1} u^{{\frac{1}{1 - n}}} - \alpha_{n + 2} u^{{\frac{2}{1 - n}}} \\ \end{aligned} $$
(14)
$$ u = \left( {s - \frac{{\alpha_{n + 1} }}{{\alpha_{0} }}} \right)^{{ - \frac{1 - n}{n}}} \Rightarrow - \frac{1}{n}\frac{{{\text{d}}s}}{{{\text{d}}z^{\prime } }} = - \alpha_{0} s - \alpha_{n + 2} s^{{ - \frac{1}{n}}} \left( {1 - \frac{{\alpha_{n + 1} }}{{\alpha_{0} s}}} \right)^{{ - \frac{1}{n}}} \approx - \alpha_{0} s - \alpha_{n + 2} s^{{ - \frac{1}{n}}} $$
(15)
$$ s = m^{{\frac{n}{n + 1}}} \Rightarrow \frac{{{\text{d}}m}}{{{\text{d}}z^{\prime } }} = \left( {n + 1} \right)\alpha_{0} m + \left( {n + 1} \right)\alpha_{n + 2} $$
(16)

As we’ll see, this lead to a first-order linear differential equation that we can easily solve for \( m \), and once we have this, we can also get the solution to the original differential equation by going back the chain of substitutions:

$$ m = Ae^{{\alpha_{0} \left( {n + 1} \right)z'}} - \frac{{\alpha_{n + 2} }}{{\alpha_{0} }} $$
(17)

In terms of \( I \):

$$ I = \left[ {\left[ {\left[ {Ae^{{\alpha_{0} \left( {n + 1} \right)z^{\prime}}} - \frac{{\alpha_{n + 2} }}{{\alpha_{0} }}} \right]^{{\frac{n}{n + 1}}} - \frac{{\alpha_{n + 1} }}{{\alpha_{0} }}} \right]^{{\frac{n - 1}{n}}} - \frac{{\alpha_{n} }}{{\alpha_{0} }}} \right]^{{\frac{1}{1 - n}}} $$
(18)

Applying the initial condition \( I\left( {z^{\prime } = 0} \right) = I_{\text{inc}} \), where \( I_{\text{inc}} \) is the incident optical intensity on the sample, and solving for the constant of integration \( A \) gives:

$$ A = \left[ {\left( {I_{\text{inc}}^{1 - n} + \frac{{\alpha_{n} }}{{\alpha_{0} }}} \right)^{{\frac{n}{n - 1}}} + \frac{{\alpha_{n + 1} }}{{\alpha_{0} }}} \right]^{{\frac{n + 1}{n}}} + \frac{{\alpha_{n + 2} }}{{\alpha_{0} }} $$
(19)

Using successively the first-order approximation (11-b), the expression of \( A \) becomes:

$$ \begin{aligned} & A = \left[ {\frac{1}{{I_{\text{inc}}^{n} }}\left( {1 + \frac{{\alpha_{n} }}{{\alpha_{0} }}I_{\text{inc}}^{n - 1} } \right)^{{\frac{n}{n - 1}}} + \frac{{\alpha_{n + 1} }}{{\alpha_{0} }}} \right]^{{\frac{n + 1}{n}}} + \frac{{\alpha_{n + 2} }}{{\alpha_{0} }} \\ & \approx \frac{1}{{I_{\text{inc}}^{n + 1} }}\left( {1 + \frac{{n\alpha_{n} }}{{\left( {n - 1} \right)\alpha_{0} }}I_{\text{inc}}^{n - 1} + \frac{{\alpha_{n + 1} }}{{\alpha_{0} }}I_{\text{inc}}^{n} } \right)^{{\frac{n + 1}{n}}} + \frac{{\alpha_{n + 2} }}{{\alpha_{0} }} \\ & \approx \frac{1}{{I_{\text{inc}}^{n + 1} }}\left( {1 + \frac{{\left( {n + 1} \right)\alpha_{n} }}{{\left( {n - 1} \right)\alpha_{0} }}I_{\text{inc}}^{n - 1} + \frac{{\left( {n + 1} \right)\alpha_{n + 1} }}{{n\alpha_{0} }}I_{\text{inc}}^{n} + \frac{{\alpha_{n + 2} }}{{\alpha_{0} }}I_{\text{inc}}^{n + 1} } \right) \\ \end{aligned} $$
(20)

Substituting Eq. (20) into Eq. (18) and using successively the first-order approximation (11-b), we obtain:

$$ \begin{aligned} & I = \left[ {\left[ {\frac{{e^{{n\alpha_{0} z^{\prime } }} }}{{I_{\text{inc}}^{n} }}\left[ {1 + \frac{{\left( {n + 1} \right)\alpha_{n} }}{{\left( {n - 1} \right)\alpha_{0} }}I_{\text{inc}}^{n - 1} + \frac{{\left( {n + 1} \right)\alpha_{n + 1} }}{{n\alpha_{0} }}I_{\text{inc}}^{n} + \frac{{\alpha_{n + 2} }}{{\alpha_{0} }}\left( {1 - e^{{ - \left( {n + 1} \right)\alpha_{0} z^{\prime } }} } \right)I_{\text{inc}}^{n + 1} } \right]^{{\frac{n}{n + 1}}} - \frac{{\alpha_{n + 1} }}{{\alpha_{0} }}} \right]^{{\frac{n - 1}{n}}} - \frac{{\alpha_{n} }}{{\alpha_{0} }}} \right]^{{\frac{1}{1 - n}}} \\ & \approx \left[ {\frac{{e^{{\left( {n - 1} \right)\alpha_{0} z^{\prime } }} }}{{I_{\text{inc}}^{n - 1} }}\left[ {1 + \frac{{n\alpha_{n} }}{{\left( {n - 1} \right)\alpha_{0} }}I_{\text{inc}}^{n - 1} + \frac{{\alpha_{n + 1} }}{{\alpha_{0} }}\left( {1 - e^{{ - n\alpha_{0} z^{\prime } }} } \right)I_{\text{inc}}^{n} + \frac{{n\alpha_{n + 2} }}{{\left( {n + 1} \right)\alpha_{0} }}\left( {1 - e^{{ - \left( {n + 1} \right)\alpha_{0} z^{\prime } }} } \right)I_{\text{inc}}^{n + 1} } \right]^{{\frac{n - 1}{n}}} - \frac{{\alpha_{n} }}{{\alpha_{0} }}} \right]^{{\frac{1}{1 - n}}} \\ & \approx \left[ {\frac{{e^{{\left( {n - 1} \right)\alpha_{0} z^{\prime } }} }}{{I_{\text{inc}}^{n - 1} }}\left[ {1 + \frac{{\alpha_{n} }}{{\alpha_{0} }}I_{\text{inc}}^{n - 1} \left( {1 - e^{{ - \left( {n - 1} \right)\alpha_{0} z^{\prime } }} } \right) + \frac{{\left( {n - 1} \right)\alpha_{n + 1} }}{{n\alpha_{0} }}\left( {1 - e^{{ - n\alpha_{0} z^{\prime } }} } \right)I_{\text{inc}}^{n} + \frac{{\left( {n - 1} \right)\alpha_{n + 2} }}{{\left( {n + 1} \right)\alpha_{0} }}\left( {1 - e^{{ - \left( {n + 1} \right)\alpha_{0} z^{\prime } }} } \right)I_{\text{inc}}^{n + 1} } \right]} \right]^{{\frac{1}{1 - n}}} \\ & \approx I_{\text{inc}} e^{{ - \alpha_{0} z^{\prime } }} \left[ {1 - \alpha_{n} \left( {\frac{{1 - e^{{ - \left( {n - 1} \right)\alpha_{0} z^{\prime } }} }}{{\left( {n - 1} \right)\alpha_{0} }}} \right)I_{\text{inc}}^{n - 1} - \alpha_{n + 1} \left( {\frac{{1 - e^{{ - n\alpha_{0} z^{\prime } }} }}{{n\alpha_{0} }}} \right)I_{\text{inc}}^{n} - \alpha_{n + 2} \left( {\frac{{1 - e^{{ - \left( {n + 1} \right)\alpha_{0} z^{\prime } }} }}{{\left( {n + 1} \right)\alpha_{0} }}} \right)I_{\text{inc}}^{n + 1} } \right] \\ \end{aligned} $$
(21)

The transmitted optical intensity at the exit plane of the sample is obtained by substituting \( z^{\prime} \) by the length \( L \) of the sample:

$$ I_{\text{out}} \left( {x,y,z,t} \right) = I_{\text{inc}} e^{{ - \alpha_{0} L}} \left[ {1 - \left( {\alpha_{n + 2} L_{\text{eff}}^{{\left( {n + 1} \right)}} I_{\text{inc}}^{n + 1} + \alpha_{n + 1} L_{\text{eff}}^{\left( n \right)} I_{\text{inc}}^{n} + \alpha_{n} L_{\text{eff}}^{{\left( {n - 1} \right)}} I_{\text{inc}}^{n - 1} } \right)} \right] $$
(22)

where \( L_{\text{eff}}^{{\left( {n - 1} \right)}} \), \( L_{\text{eff}}^{\left( n \right)} \) and \( L_{\text{eff}}^{{\left( {n - 1} \right)}} \) are the effective lengths of order \( \left( {n - 1} \right), n \) and \( \left( {n + 1} \right) \), respectively given by the formulas:

$$ L_{\text{eff}}^{{\left( {n + 1} \right)}} = \frac{{1 - e^{{ - \left( {n + 1} \right)\alpha_{0} L}} }}{{\alpha_{0} \left( {n + 1} \right)}} (23 - {\text{a}}) $$
$$ L_{\text{eff}}^{\left( n \right)} = \frac{{1 - e^{{ - n\alpha_{0} L}} }}{{n\alpha_{0} }}(23 - {\text{b}}) $$
$$ L_{\text{eff}}^{{\left( {n - 1} \right)}} = \frac{{1 - e^{{ - \left( {n - 1} \right)\alpha_{0} L}} }}{{\alpha_{0} \left( {n - 1} \right)}} (23 - {\text{c}}) $$

Following the same procedure, the result obtained in Eq. (22) can be easily generalized to the case of \( \left( {N - n} \right) \) successive nonlinear absorption processes:

$$ I_{\text{out}} \left( {x,y,z,t} \right) = I_{\text{inc}} e^{{ - \alpha_{0} z}} \left( {1 - \mathop \sum \limits_{k = n}^{k = N} \alpha_{k} L_{\text{eff}}^{{\left( {k - 1} \right)}} I_{\text{inc}}^{k - 1} } \right) $$
(24)

which represents the approximate solution of Eq. (1).

Transmitted optical power

Let us consider that the incident beam is of elliptic spatial profile. In Cartesian coordinate, the form of the electric field pattern of an elliptic Gaussian beam propagating along the \( + z \) direction can be written as [24, 25]:

$$ \begin{aligned} & E\left( {x,y,z,t} \right) = E_{0} \left( t \right)\left( {\frac{{W_{ox} W_{oy} }}{{W_{x} \left( z \right)W_{y} \left( z \right)}}} \right)^{{\frac{1}{2}}} \exp \left( { - \frac{{x^{2} }}{{W_{x}^{2} \left( z \right)}} - \frac{{y^{2} }}{{W_{y}^{2} \left( z \right)}}} \right) \\ & \exp \left( { - \frac{{ikx^{2} }}{{2R_{x} \left( z \right)}} - \frac{{iky^{2} }}{{2R_{y} \left( z \right)}}} \right) \exp \left( { - ikz + i\theta \left( z \right)} \right) \\ \end{aligned} $$
(25)

where \( E_{0} \left( t \right) \) is the on-axis electric field amplitude at the focus, i.e., \( E_{0} = E\left( {0,0} \right) \) and containing the temporal envelope of the electric field,\( k \) is the wave number in free space, \( W_{x,y} \left( z \right) \) are the beam widths or beam waists, \( W_{0x,y} \) are the waists radius, \( R_{x,y} \left( z \right) \) are the wavefront radius of curvature for each dimension \( \left( {x,y} \right) \) and \( \theta \left( z \right) \) is the on-axis phase shift of the beam. The time factor \( \exp \left( { - i\omega t} \right) \) has been omitted in Eq. (1), where \( \omega \) is the angular frequency of the elliptical Gaussian beam.

According to Eq. (25), the intensity of the incident beam can be expressed as follows:

$$ I_{\text{inc}} \left( {x,y,z,t} \right) = \left| {E\left( {x,y,z,t} \right)} \right|^{2} = I_{0} h\left( t \right)\left( {\frac{{W_{0x} W_{0y} }}{{W_{x} W_{y} }}} \right){ \exp }\left( { - \frac{{2x^{2} }}{{W_{x}^{2} }} - \frac{{2y^{2} }}{{W_{y}^{2} }}} \right) $$
(26)

where \( I_{0} = I\left( {0,0,0} \right) \) is the on-axis intensity of the incident beam at the focus, i.e.,\( I_{0} = I\left( {0,0} \right), h\left( t \right) \) is function of time describing the temporal profile of the incident beam pulse.

The second step of our model is the calculation of the transmitted optical power. The latter can be determined from the following relation:

$$ P_{\text{out}} \left( {z,t} \right) = \mathop \int \limits_{ - \infty }^{ + \infty } \mathop \int \limits_{ - \infty }^{ + \infty } I_{\text{out}} \left( {x,y,z,t} \right){\text{d}}x{\text{d}}y $$
(27)

Substituting Eq. (26) into Eq. (24), the expression (27) becomes:

$$ P_{\text{out}} \left( {z,t} \right) = I_{0} h\left( t \right)\left( {\frac{{W_{0x} W_{0y} }}{{W_{x} W_{y} }}} \right)\exp \left( { - \alpha_{0} L} \right)\mathop \smallint \limits_{ - \infty }^{ + \infty } \mathop \smallint \limits_{ - \infty }^{ + \infty } \left[ {1 - \mathop \sum \limits_{k = n}^{k = N} C_{k} \left( {\exp \left( { - \frac{{2x^{2} }}{{W_{x}^{2} }} - \frac{{2y^{2} }}{{W_{y}^{2} }}} \right)} \right)^{k - 1} } \right]\exp \left( { - \frac{{2x^{2} }}{{W_{x}^{2} }} - \frac{{2y^{2} }}{{W_{y}^{2} }}} \right){\text{d}}x{\text{d}}y $$
(28)

where we have introduced the quantity \( C_{k} \) defined as follows:

$$ C_{k} = \alpha_{k} L_{\text{eff}}^{{\left( {k - 1} \right)}} \left( {I_{0} h\left( t \right)\frac{{W_{0x} W_{0y} }}{{W_{x} W_{y} }}} \right)^{k - 1} $$
(29)

To calculate the integral in Eq. (28), we first performing the following substitutions:

$$ \left\{ {\begin{array}{*{20}c} {x = \frac{{r^{2} }}{{W_{y} }}\cos \theta } & {\left( {30 - {\text{a}}} \right)} \\ {y = \frac{{r^{2} }}{{W_{x} }}\sin \theta } & {\left( {30 - {\text{b}}} \right)} \\ {{\text{d}}x{\text{d}}y = \frac{{2r^{3} }}{{W_{x} W_{y} }}{\text{d}}r{\text{d}}\theta } & {\left( {30 - {\text{c}}} \right)} \\ \end{array} } \right. $$

where \( \left( {r,\theta } \right) \) are the polar coordinates. After a first integration over \( \theta \), one obtain:

$$ P_{T} \left( {z,t} \right) = 4\pi I_{0} h\left( t \right)\frac{{W_{0x} W_{0y} }}{{\left( {W_{x} W_{y} } \right)^{2} }}e^{{ - \alpha_{0} L}} \mathop \smallint \limits_{0}^{ + \infty } \left[ {1 - \mathop \sum \limits_{k = n}^{k = N} C_{k} \left( {\exp \left( { - \frac{{2r^{4} }}{{\left( {W_{x} W_{y} } \right)^{2} }}} \right)} \right)^{k - 1} } \right]\exp \left[ { - \frac{{2r^{4} }}{{\left( {W_{x} W_{y} } \right)^{2} }}} \right]r^{3} {\text{d}}r $$
(31)

In order to evaluate the integral in Eq. (31), we are going to make the following substitutions:

$$ \left\{ {\begin{array}{*{20}l} {q = \exp \left[ { - \frac{{2r^{4} }}{{\left( {W_{x} W_{y} } \right)^{2} }}} \right]} \hfill & {\left( {32 - {\text{a}}} \right)} \hfill \\ {{\text{d}}q = - \frac{{8r^{3} }}{{\left( {W_{x} W_{y} } \right)^{2} }}\exp \left[ { - \frac{{2r^{4} }}{{\left( {W_{x} W_{y} } \right)^{2} }}} \right]{\text{d}}r} \hfill & {\left( {32 - {\text{b}}} \right)} \hfill \\ \end{array} } \right. $$

which gives:

$$ P_{T} \left( {z,t} \right) = P_{\text{inc}} \left( t \right)e^{{ - \alpha_{0} L}} \mathop \smallint \limits_{0}^{1} \left( {1 - \mathop \sum \limits_{k = n}^{k = N} C_{k} q^{k - 1} } \right){\text{d}}q $$
(33)

where \( P_{\text{inc}} \left( t \right) \) is the incident optical power on the sample given by the relationship:

$$ P_{\text{inc}} \left( t \right) = \frac{\pi }{2}I_{0} h\left( t \right)W_{0x} W_{0y} $$
(34)

The evaluation of the integral in Eq. (33) gives the following expression for the transmitted power:

$$ P_{T} \left( {z,t} \right) = P_{\text{inc}} \left( t \right)e^{{ - \alpha_{0} L}} \left[ {1 - \mathop \sum \limits_{k = n}^{k = N} \frac{{C_{k} }}{k}} \right] $$
(35)

Normalized energy transmittance

The last step of our model is the calculation of the open Z-scan normalized transmittance. This quantity is defined as:

$$ T\left( z \right) = \frac{{\mathop \smallint \nolimits_{ - \infty }^{ + \infty } P_{T} \left( {z,t} \right){\text{d}}t}}{{\mathop \smallint \nolimits_{ - \infty }^{ + \infty } P_{inc} \left( {z,t} \right){\text{d}}t}} $$
(36)

Substituting Eqs. (34) and (35) into Eq. (36) and using the following expressions of \( W_{x} \) and \( W_{y} \):

$$ W_{x,y} = W_{0x,y} \left[ {1 + \left( {\frac{{z - z_{0x,y} }}{{z_{{R_{x,y} }} }}} \right)^{2} } \right]^{{\frac{1}{2}}} $$
(37)

the expression of the normalized transmittance as a function of z and the characteristics of the elliptic Gaussian beam can be expressed as:

$$ T\left( z \right) = e^{{ - \alpha_{0} L}} \left[ {1 - \mathop \sum \limits_{k = n}^{k = N} \frac{{\alpha_{k} F_{k} }}{{\left[ {\left[ {1 + \left( {\frac{{z - z_{0x} }}{{z_{{R_{x} }} }}} \right)^{2} } \right]\left[ {1 + \left( {\frac{{z - z_{0y} }}{{z_{{R_{y} }} }}} \right)^{2} } \right]} \right]^{{\frac{k - 1}{2}}} }}} \right] $$
(38)

where we have introduced the coefficients:

$$ F_{k} = \frac{{L_{\text{eff}}^{{\left( {k - 1} \right)}} I_{0}^{k - 1} \mathop \smallint \nolimits_{ - \infty }^{ + \infty } \left( {h\left( t \right)} \right)^{k} }}{{k\mathop \smallint \nolimits_{ - \infty }^{ + \infty } h\left( t \right){\text{d}}t}} $$
(39)

In the interesting and practical case of an incident circular Gaussian beam, we have:

$$ \begin{array}{*{20}c} {z_{0x} = z_{oy} = z_{0} } & {\left( {40 - {\text{a}}} \right)} \\ \end{array} $$
$$ \begin{array}{*{20}c} {z_{{R_{x} }} = z_{{R_{y} }} = z_{R} } & {\left( {40 - {\text{b}}} \right)} \\ \end{array} $$
$$ \begin{array}{*{20}c} {h\left( t \right) = e^{{ - t^{2} }} } & {\left( {40 - {\text{c}}} \right)} \\ \end{array} $$
$$ \begin{array}{*{20}c} {\mathop \smallint \limits_{ - \infty }^{ + \infty } h\left( t \right){\text{d}}t = \pi } & {\left( {40 - {\text{d}}} \right)} \\ \end{array} $$
$$ \begin{array}{*{20}c} {\mathop \smallint \limits_{ - \infty }^{ + \infty } \left( {h\left( t \right)} \right)^{k} = \frac{\pi }{{\sqrt[{}]{k}}}} & {\left( {40 - {\text{e}}} \right)} \\ \end{array} $$
$$ \begin{array}{*{20}c} {F_{k} = \frac{{L_{\text{eff}}^{{\left( {k - 1} \right)}} I_{0}^{k - 1} }}{{k\sqrt[{}]{k}}}} & {\left( {40 - {\text{f}}} \right)} \\ \end{array} $$

and the expression of normalized transmission (38) is reduced to:

$$ T\left( z \right) = e^{{ - \alpha_{0} L}} \left[ {1 - \mathop \sum \limits_{k = n}^{k = N} \frac{{\alpha_{k} F_{k} }}{{\left[ {\left( {1 + \left( {\frac{{z - z_{0} }}{{z_{R} }}} \right)^{2} } \right)} \right]^{k - 1} }}} \right] $$
(41)

Equations (38) and (40) give the normalized transmission for any number of simultaneous and successive (in order) nonlinear absorption processes. Thus, they can be used to adjust the experimental traces obtained by the open Z-scan technique in order to determine the nonlinear absorption coefficients \( \alpha_{k} \left( {k \ge 2} \right) \) and the parameters \( z_{0} \) and \( z_{R} \) of the incident elliptic Gaussian beam.

It is important to note that the expression (41) allows identifying and separating the contributions of the different absorption processes. Indeed, if we neglected linear absorption \( \left( {\alpha_{0} = 0} \right) \), the normalized transmittance can be expressed as:

$$ T\left( z \right) = 1 - \mathop \sum \limits_{k = n}^{k = N} T_{kPA} $$
(42)

where \( T_{kPA} \) is the k-photon normalized transmittances, defined as:

$$ T_{kPA} = \frac{{\alpha_{k} F_{k} }}{{\left[ {\left( {1 + \left( {\frac{{z - z_{0} }}{{z_{R} }}} \right)^{2} } \right)} \right]^{k - 1} }} $$
(43)

It is also possible to compare the contributions of two successive (in order) absorption processes by performing the following ratio:

$$ \frac{{T_{\left( k \right)PA} \left( z \right)}}{{T_{{\left( {k - 1} \right)PA}} \left( z \right)}} = \left( {\frac{{k^{\prime } \sqrt {k^{\prime } } }}{k\sqrt k }} \right)\left( {\frac{{\alpha_{k} }}{{\alpha_{{k^{\prime}}} }}} \right)\frac{{I_{0}^{{k - k^{\prime } }} }}{{\left[ {1 + \left( {\frac{{z - z_{0} }}{{z_{R} }}} \right)^{2} } \right]^{{k - k^{\prime } }} }} $$
(44)

where \( \mathop {\lim }\limits_{{\alpha_{0} \to 0}} \left( {L_{\text{eff}}^{\left( k \right)} /L_{\text{eff}}^{{\left( {k^{\prime } } \right)}} } \right) = 1 \). For a given position \( z \) and knowing the values of the parameters \( \alpha_{k + 1} , \alpha_{k} \) and \( z_{R} \) from the Z-scan traces fitting, one can estimate the ratio \( \left( {T_{\left( k \right)PA} /T_{{k^{\prime}PA}} \left( z \right)} \right) \).

Conclusions

On the basis of the weak nonlinearities approximation, we have developed an open Z-scan theory for thin nonlinear optical material with concurrence of \( \left( {N - n} \right) \) nonlinear absorption processes. The proposed normalized transmittance approximate analytical but closed formula alloying the measure of nonlinear absorption coefficients and some characteristics of the incident elliptic Gaussian beam. Finally, it should be stressed that the Z-scan theory presented in this paper is proved to be useful for identifying and separating the contribution of each absorption process.