A.1.1

Nonlinear Optics

Nonlinear optics is the area of photonics that studies the phenomena that occur as a consequence of the modification of the optical properties of a material by the presence of light.²⁸ This nonlinear behavior in photonic materials becomes measurable as a result of strong light-matter interaction processes, whose phenomenology may be explained with a simple model of a harmonic oscillator, as the one depicted in Fig. 1.

Figure 1.

Conceptual sketch of (a) a harmonic oscillator based on a spring, where represents an external force that is applied to the mass m, and (b) the analogous description of the dynamics of bound electrons within an atom as a harmonic oscillator with restoring forces provided by atomic electric fields, and under the influence of incident light wave with amplitude .

Let’s suppose that in the mass-spring system from Fig. 1(a) , and then the mass m gets slightly disturbed. The mass m would begin to describe a simple harmonic motion. Assuming that the direction of oscillation is the horizontal axis x, the motion will be described by:

Next, a external force is applied to m. Hooke’s law, , with k the elastic constant of the string and x the strain, states that the spring exerts a force on the mass that is proportional to the elongation of the spring, as long as the strain provoked by the external force on the string remains in the linear zone of the Young modulus of the string material, Fig. 2. If the force is strong enough to deform the string through the nonlinear zone of the Young modulus, Hooke’s law cannot be longer applied to model the motion dynamics, and the restorative force must be described as a polynomial series expansion, this is

Figure 2.

Conceptual graph of the Young modulus of a string. In the linear zone Hooke’s law can be applied to the system. In the nonlinear zone a polynomial series expansion must be used to model the restorative force.

Given this new restoring force formulation, the motion of the string can no longer be described by Eq. (1). Using Eq. (2), the new equation of motion for the spring-mass system under strong external perturbations is:

which happens to be an anharmonic oscillator, where a, b, ⋯ are related to the new frequencies at which the string oscillates and are closely related with the strength of the external force and the string material. The generation of new frequencies of oscillation will depend on the strength of the external force; and for a stronger force, the generation of new frequencies gets encouraged. The distribution of new frequencies that are generated depends mostly on the spring, as there are springs that are more likely to oscillate at certain frequencies than others under these extreme conditions.

Taking the dynamics of this mechanical system as reference, the interaction between an electric field and atoms could be understood as an analogous phenomenon (see Fig. 1(b)). In this case, the mass of the nucleus of each atom is much bigger than the mass of its electrons, it can be assumed that the nucleus stays still and the electrons get accelerated, driven by the incident optical field. When there is no external field, i.e. , the atom will behave like a harmonic oscillator, the electrons will move due the attractive and repulsive forces with the nucleus and other electrons, but they are confined in the vicinity of the atom. Now, when the electric field is applied, if it is strong enough, the oscillation of the electron will follow similar dynamics as those experienced by the mechanical oscillator in Fig. 1(a). So, it would enter in a nonlinear regime, where the motion is now modeled by an anharmonic oscillator. So, the corresponding classical equation of motion is given by:

where e is the charge of the electron and m_e is its mass. Given the existence of an external electric field, the system will oscillate at different frequencies, when the field is strong enough. These frequencies are closely related with the terms ax² + bx³ + ⋯, which are nonlinear terms. In this case, the distribution of new frequencies that can be generated depends on the atom, certain atoms can oscillate at certain frequencies and others cannot. This analogy shows that the light-matter interaction for sufficiently intense optical fields involves the contribution of nonlinear terms that are related with the generation of new frequencies, giving sense to the name of the subject that studies these optical phenomena: “nonlinear optics”. When the the field is not sufficiently intense, the generation of new frequencies is negligible, and the ongoing optical phenomena would belong to the “linear optics” field. Although this analogy provides a first approach to the study of nonlinear optics, for a consistent comprehensive description, more sophisticated models are required, harnessing electrodynamics and quantum mechanics principles.²⁸

In the next section an introduction to the electromagnetic formalism used to explain nonlinear optics is given, elaborating on the nonlinear behavior of the polarization vector for susceptible materials. Additionally, nonlinear phenomena of most interest are presented, and the nonlinear parameters used to quantify the strength of some nonlinearities in photonic materials are discussed.

A.1.2

Nonlinear Polarization

In linear optics, when an electric field interacts with a material, an induced polarization arises, which depends linearly on the electric field intensity, and is given by:

where χ⁽¹⁾ is known as the linear electric susceptibility and is related to the (linear) refractive index by . As it was shown before, if the strength of the field is large enough, the nonlinearities of the interaction between the field and the material becomes important. The analytical relation between the electric field and the resulting polarization must be generalized, which is often done by expressing the polarization in terms of a power series of the field:

where χ⁽²⁾, χ⁽³⁾, ⋯ are called the second, third, … order nonlinear optical susceptibilities. For the sake of simplicity, let’s assume that the fields and are scalar quantities, which is a reductionist assumption as the nonlinear susceptibilities depend greatly on the polarization of light, but this holds for specific conditions (e.g., linearly polarized light, oriented along one of the material’s principal axes) and helps to understand the phenomenon. The amount of significant terms required for the polarization vector expansion in power series depends on the strength of the field and the material itself; as they are only considered when they account for measurable phenomena. Nonlinear effects manifestations become significant when the strength of the field is comparable with the atomic electric fields, which have the order of magnitude of ,²⁸ where e is the charge of the electron, ϵ₀ is the free space permittivity, and a₀ is the Bohr radius. So, the nonlinear phenomena associated to the second order susceptibility χ⁽²⁾ will be relevant when

The same holds for the third order susceptibility χ⁽³⁾:

and so on. Since experimental observations are almost always direct measurements of irradiance, an approximated threshold for the intensity required to excite measurable nonlinearities may be formulated as:

Knowing this, the most commonly used materials in nonlinear photonics exhibit phenomena associated to the second-order susceptibility, and third-order susceptibility, as their corresponding nonlinear effects are usually more easily excited than nonlinearities of higher order. And P⁽²⁾(t) = ϵ₀χ⁽²⁾E²(t) is referred as the second-order nonlinear polarization, which gives account of the nonlinear phenomena associated to χ⁽²⁾, and P⁽³⁾(t) = ϵ₀χ⁽³⁾E³(t) is referred as the third-order nonlinear polarization, which gives account of the nonlinear phenomena associated to χ⁽³⁾. In order to detect significant nonlinear phenomena, an intense electromagnetic field must propagate through a material medium with relatively high nonlinear susceptibilities.

Then, to complete the presentation of the formalism, Eq. (6) is formulated as

where P^L(t) is the linear polarization in Eq. (5) and P^NL(t) contains all the nonlinears terms. Next, Eq. (9) can be replaced in the wave equation for propagation in a material, so that²⁸

where c₀ is the speed of light in free space and μ₀ is the magnetic permeability in free space. Finally, Eq. (10) leads to the wave equation in nonlinear optical media

where P^NL = ϵ₀ [χ⁽²⁾E²(t) + χ⁽³⁾E³(t)], and only low-order nonlinear effects are considered. According to the form of P^NL different phenomena can occur, and by solving Eq. (11) it is found that new frequencies can be generated, and other effects as self-focusing and defocusing, self-phase modulation, nonlinear absorption, among others can be found as response to external very intense electric fields. In general, P^NL(t) is a vector and is called nonlinear polarization vector, but in this case the scalar approximation is held.

A.1.3

Nonlinear Phenomena

The different nonlinear phenomena that are achievable for sufficiently intense electric field depends mainly on the material that is used as a media for the nonlinear propagation described in Eq. (11). This dependence is given by the nonlinear susceptibility χ, which in general is a tensor with multiple entries. Due symmetries in the material crystalline structure some entries nullifies and therefore some phenomena are encouraged or not. For more information see references 28 and 29. The nonlinear phenomena can be classified into groups according to their relation with the nonlinear susceptibility. Thus, there are second-order nonlinear phenomena and third-order nonlinear phenomena. Usually materials that present important second-order susceptibility do not have an important third-order susceptibility, and vice versa.

Some second-order nonlinear phenomena are:

Some third-order nonlinear phenomena are:

This work is focused on the Two-Photon Absorption (TPA) and the Nonlinear Kerr Effect, since most of third-order nonlinear phenomena rely on those effects. To explain the change of the refractive index due a high intensity field, the nonlinear polarization for a beam interacting with a media with third-order susceptibility is described as:²⁸

As it has been shown, the polarization, and the electric field are time dependent. In nonlinear optics, it is pertinent to analyze the frequencies involved in the light-matter interaction. Therefore, it is useful to take the Fourier transform of the polarization vector to analyze the frequency dependency of the interaction. So, the total polarization will be given by the Fourier transform of Eq. (9):

where χ_eff = χ⁽¹⁾ + 3χ⁽³⁾|E(ω)|² is the effective permittivity. This is the permittivity of the material taking into account the third-order susceptibility. Under the linear approximation, the refractive index can be written in terms of the permittivity⁴³ as

so, the refractive index is generalized for nonlinear studies by taking into account the third order-susceptibility, which gives rise to the expression

Using Eq. (14), the linear refractive index can be defined as , and Eq. (15) takes the form

As the quotient is a very small number, the binomial approximation (1 + x)ⁿ ≈ 1 + nx for x << 1 may be applied. So, the refractive index definition is written as

The electric field can be written as:

therefore, the irradiance:

Replacing Eq. (19) into Eq. (17), the refractive index becomes dependent from irradiance by taking into account the third-order susceptibility. This is:

But the refractive index has both real and imaginary parts:⁴³

where Re{n} is commonly known as refractive index and α is related to the losses by absorption, called the absorption coefficient. So, Eq. (20) yields:

where the third-order susceptibility is in general a complex value quantity, and the complex-valued n₀ was decomposed into its real and imaginary parts in a similar way as it was done for n in Eq. (21). So, equating the real and imaginary parts of Eq. (22), it is obtained for the refractive index and the absorption coefficient:

where

is called the nonlinear refractive index, and

is called the two-photon absorption coefficient. Now it become evident, by inspecting Eqs. (23), that in general the refractive index is irradiance-dependent, but this is only relevant when the medium of propagation has significant third-order susceptibility.

A.2.1

Modeling approach and slowly varying envelope approximation

To describe the optical fields involved in a Z-scan system, a wave optics model is employed. In the following derivations, light is assumed to preserve its linear polarization state; so anisotropic effects within the sample will not be considered. And the external medium will be assumed to be air. The laser light could either carry a constant power over time, or be pulsed to attain greater instantaneous power levels. And in this work, optical nonlinearities are assumed to appear instantaneously, right after the material is exposed to the intense optical excitation. This approximation is certainly accurate in the cases in which pulsed light is used, with very short pulse duration with respect to the time response of material’s thermal nonlinear optical effects. Before the sample, the focused laser light is modeled as a Gaussian beam, following the analytical formulation given by the

Eq. (25),^{21, 43}

where z and r are axial and radial coordinates, k₀ is the vacuum wavenumber of the beam, E₀(t) is the temporally-modulated electric field’s complex amplitude at focus, w₀ is the radius of the beam at focus, exp [−iφ(z, t)] is a conventional phase propagation term, and

After the sample, the propagation and power calculations must take into account the radially-varying phase-shift and power attenuation introduced by the sample; which produce deviations from the Gaussian formulation. Fourier optics propagation methods are very useful for these far-field calculations, especially for numerical implementations.

The propagation of the optical field within the sample is, in general, influenced by the linear and nonlinear absorption, linear and nonlinear refraction, and natural diffraction processes. If the sample is thin (i.e., with thickness L shorter than beam’s Rayleigh length z₀²¹), the diffraction and refraction phenomena may be neglected for the study of the evolution of the beam within the sample. This approach is known as slowly varying envelope approximation (SVEA), and entails that the beam transverse profile preserves its size (see Fig. 4). Thereby, the nonlinear phase-shift that results from the propagation throughout the nonlinear medium is only taken into account for the far field propagation, between the sample and the detector. It is important to notice that this approximation does not imply that the irradiance distribution of the beam is not tracked within the sample: it gets attenuated as consequence of absorption processes, and its decay rate distribution is determinant in the description of the considered nonlinear effects.

Figure 4.

Slowly varying envelope approximation, for thin media, facilitates the analytical description of the nonlinear phenomena under study. It assumes that the beam is neither converging, nor diverging within the sample.

In this way, the irradiance profile evolution within the sample is described by the Eq. (27), and the accumulated phase-shift can be calculated using Eq. (28):²¹

To avoid confusion, ℓ is used to denote the coordinate along the propagation axis for calculations within the sample, and z is used to denote position along the propagation axis at which the sample is placed (coordinate of the sample’s input facet). ℓ = 0 at sample’s input facet, and z = 0 at beam’s focus.

A.2.2

Nonlinear phase-shift: for negligible nonlinear absorption

In general, the attenuation coefficient α in Eq. (27) can have an irradiance-independent part and an irradiance-dependent one (23b). If the nonlinear absorption processes have a very low probability to occur, the power attenuation in the material is essentially linear, and α ≈ α₀. In this case, the solution to Eq. (27) becomes

This expression for the irradiance evolution allows to calculate the accumulated irradiance-dependent phase-shift variation. Thus, using Eqs. (23a), (28), and (29), for a sample with thickness L, it is found that

It is possible to notice that the last factor in Eq. (30b) has length dimension, and it can be regarded as an effective length for the nonlinear phenomenon. This means that the nonlinear phase-shift variation that is accumulated through the propagation in the lossy medium with thickness L, could be equally calculated as the one that would be accumulated after the constant-irradiance propagation through a non-lossy medium with thickness L_eff; being

Then, the nonlinear phase-shift is calculated for media without nonlinear absorption as follows²¹

where I(ℓ = 0) = I(z, r, t) is the irradiance distribution of the Gaussian beam, derived from the Eq. (25); with r and z evaluated at sample’s input face plane, and I₀(t) evaluated at beam’s focus.

A.2.3

Nonlinear phase-shift: considering nonlinear absorption

If the nonlinear absorption is appreciable, the irradiance profile evolution is described by the differential equation

whose solution, by separation of variables, requires the calculation of the antiderivative

It may be obtained by partial fractions decomposition, as follows:

Thus, a expression for the irradiance dependence on ℓ is derived:

If the irradiance profile at the input facet is known, the parameter K^(D) is calculated as

and Eq. (36) takes the form²¹

All parameters K^(·) are arbitrary constants.

Last expression describes the evolution of the irradiance, taking into account the nonlinear absorption effects, and allows to calculate the corresponding irradiance-dependent phase-shift variation. Using Eqs. (23a), (28) and (38) it is found that

if the denominator’s expression is expanded and the substitution is introduced, one obtains

which can be easily solved by introducing the second substitution , as it yields

By replacing u and 𝜐, the following expression results for the nonlinear phase-shift variation profile:²¹

A.2.4

Formulation in terms of measurable quantities

Within the wave optics approach, beam’s irradiance and phase profiles are the physical entities directly affected by the nonlinearities of the material. However, to connect this theoretical framework with the observations carried out in the laboratory, it is necessary to derive a description of the nonlinearities manifestation in terms of average optical power measurements.

As it was explained at the start of this section, the nonlinear absorption may be directly associated with the deviations from the linear regime that appear in open-aperture measurements. Therefore, it is convenient to find the analytical expression for the optical power that would be measured after the sample, given the irradiance profile previously obtained (Eq. (38)). For this aim, the irradiance profile is integrated across the sample’s output facet plane. This power value is expected to be same for any detection plane after that sample, since the propagation in air is assumed to be lossless. Considering that I(ℓ = 0) = I(z, r, t) is the irradiance profile of the Gaussian beam, with r and z evaluated at sample’s input facet plane, and I₀(t) evaluated at beam’s focus, the integration process is summarized as follows:

with and Ω = β_TPA L_eff to ease the notation. Then, with u = −2r²/w²(z) and 𝜐 = Λ + Ωe^u, the integration by substitution yields

To express Eq. (43f) in terms of measurable values of input power, a similar integration procedure is carried out for the input irradiance profile; i.e., Gaussian beam’s irradiance I(ℓ = 0) = I(z, r, t). In this case, the substitution u = −2r²/w²(z) and the definition of w²(z) from Eqs. (26) are employed.

Finally, if the used laser source is pulsed, it is important to take into consideration that the measurements made by a photodetector correspond to average optical power, and previously derived expression are valid for instantaneous power values. Thus, a relation must be established between the instantaneous peak power and the detected average power. To simplify the analytical model, the temporal envelope of the pulse is assumed to have a rectangular shape. This means that for a laser source with pulse duration τ and repetition rate f_rep = 1/T, the instantaneous optical power is given by

since the average power and peak power are related by their dependence with the energy of the pulse ɛ_pulse:

Then, using Eqs. (26), (44c) and (45b), the outgoing optical power expression from Eq. (43f) can be written in terms of average power values, as follows:²¹

Eq. (46) constitutes a ready-to-use expression to analyze experimental open-aperture input/output power measurements.

To obtain the corresponding theoretical description for the closed-aperture power measurements, it is necessary to calculate the electric field profile at the aperture plane, taking into account the details of the propagation between the sample and the aperture. In this work, this step is carried out by numerical means, using Fourier optics propagation methods. Thereby, the analytical formulation that is needed for this numerical implementation is the one describing the electric field’s complex amplitude profile at the sample’s output facet plane. The norm of this complex amplitude can be obtained from the expression for the output irradiance profile (Eq. (38)), which multiplied by the phase factor that accounts for the nonlinear phase-shift variation (Eq. (42b)) yields²¹

In the previous expression, E(ℓ = 0) and I(ℓ = 0) correspond to the Gaussian beam’s electric field and irradiance profiles, E(z, r, t) and I(z, r, t), evaluated at sample’s input facet plane; which may be obtained from Eq. (25).

To illustrate the functional form of the output signals predicted by the model, transmission curves for open-aperture and closed-aperture setup configurations are presented in Fig. 5. Different input power levels were considered, to illustrate the typical evolution of the nonlinear material’s response as incident power increases. Fig. 5(a) shows the optical power transmission predicted by Eq. (46) for open-aperture measurements; and Fig. 5(b) presents the scaled version of these open-aperture curves, taking as power reference the (maximum) transmission obtained in the linear range of the Z-scan. Likewise, Fig.5(c) shows the power transmission for the closed-aperture case, obtained by numerically calculating the propagation of the electric field profile described by Eq. (47b). The numerical implementation used for this calculation is detailed in section B. Fig. 5(d) presents the normalized version of these closed-aperture curves, after the nonlinear absorption effects subtraction, and taking as power reference the transmission obtained in the linear range of the closed-aperture Z-scan.

Figure 5.

Typical power transmission curves theoretically predicted by the previously derived model, for different input power levels. Curves of the same color correspond to the same input power. Sample position is measured in arbitrary length units.

A.3.1

Experimental Setup

Fig. 6(a) and diagram 6(b) illustrate the optical bench setup used in the Z-Scan experiment in free-space. Table 2 details the optical elements that the experimental setup comprises.

Figure 6.

Table 2.

Experimental Equipment

A.3.2

Sample Preparation and Mounting

Before beginning the Z-Scan process one must take proper care while mounting the sample. Be sure to wear gloves to avoid damaging the sample. One can mount a sample onto an iris that is secured in the open position with double sided tape, taking care not to obscure the beam path with the tape.

Figure 7.

Mounting the sample.

The sample can then be mounted onto the translation stage by carefully screwing the iris mount into the post holder that is mounted onto the translation stage. Ensure the sample is perpendicular to the beam path in the experimental setup.

A.3.3

Laser Activation

With the sample mounted, it is time to turn on the laser. Begin by turning on the computer connected to the laser at the optical bench. Locate the ‘ELMO Software’.

Once the ‘Elmo software’ is running, one will see the startup menu in Fig. 9(a). Select the ‘Subsystems’ tab as seen in Fig. 9(b) and press the ‘Elmo is On’ and ‘Elma is On’ buttons. Now you will see the text to the right of these buttons turns from orange to green and the text in the buttons changes as seen in Fig. 9(c).

Figure 8.

Elmo Control V3 Software

Figure 9.

Double check the Elmo laser system to see the lights have turned green. Finally pull up the laser safety shutter to allow the laser pulses to exit the laser system and follow the beam path. Make sure to check the beam path with a photo-phosphorescent viewing card to ensure the beam is aligned properly.

A.3.4

Pre-Measurement Procedures

Set the aperture size for the adjustable iris to ensure that for all of our scans the aperture size is the same. Begin by taking the power measurement with the Thorlabs power-meter immediately before the lens that lies before the sample stage. This power reading should be the same for all the aperture size calibrations, in our lab we use 100 mW.

To adjust the power of the laser, turn the gear of the Half-Wave Plate’s mount until the power-meter reads the desired value. Finally, move the power meter immediately behind our adjustable aperture and measure the power as we adjust the aperture size to a predetermined amplitude, in our lab we use 18 mW. This aperture size calibration is needed if a post-mounted iris is used as blocking element in closed-aperture measurements. This process can be bypassed if another element was used, with a fixed aperture size, such as a precision pinhole.

Figure 10.

(a) Menlo Systems Elmo Femtosecond Laser with blue lights indicating that the laser is turned off. (b) Elmo Femtosecond Laser with green lights indicating that the laser is turned on.

Figure 11.

(a) Measure power before pre-sample lens. (b) Adjusting the laser power with the half-wave plate mount. (c) Adjusting the iris aperture size.

A.3.5

Z-Scan Procedures

The Z-Scan consists of two types of scans, to be performed sequentially. The closed-aperture (CA) and open-aperture (OA) scans, as mentioned in previous section. To fully characterize the sample material, one must perform several Z-scans, making up a power sweep. A sweep is a series of scans over a range of incident power levels. For efficiency one should begin each set of scans by setting the aperture size as described in the Pre-Measurement Procedures section. After the aperture size is set, locate the power-meter immediately before the sample to adjust the incident power level.

Figure 12.

Setting the laser power before the sample.

Remember that the power can be adjusted by turning the mount of the half-wave plate as seen in Fig. 11(a). Once at the desired power level, it is time to set up the closed-aperture Z-Scan (followed by an open-aperture Z-Scan). Make sure to move the power-meter directly behind the iris aperture and turn off any lights on the optical bench. Connect the power-meter to a laptop and turn on the Thorlabs Optical Power Monitor software.

Once connected to the software window, find the settings menu on the default display page and find the beam parameters section. Under ‘profile’, make sure that ‘Gaussian’ is selected on the drop down menu. Select the ‘Monitoring’ tab on the top left menu bar. Upon selection a ‘Table Configuration’ window will pop up, prompting one to confirm which parameter is being measured. In this case make sure it says ‘Power’ and press the ‘OK’ button on the bottom right of the window before continuing.

The final configuration step for the Z-Scan data collection process will be managed in the ‘Settings’ menu on the right side of the ‘Monitoring’tab. Follow the settings in Fig. 14(b) for the Z-Scan. Make sure to change the ‘File Path’ to an appropriate folder. Before each Z-Scan make sure to change the ‘File Name’ so that the data from a previous Z-Scan is not overwritten. Double-Check to make sure there are 171 Samples and the ‘Number of Samples’ selection dot is marked. Lastly, adjust the ‘Saving Interval’ to a time-frame that is comfortable (3-5 seconds is recommended). Before starting each scan, ensure that the sample’s translation stage is set to the zero marker.

Figure 13.

Figure 14.

Figure 15.

(a) Translation stage set to the zero marker. (b) Translation stage moved ten ticks.

When fully prepared for the Z-Scan to start, press the ‘Start Logging’ button. Upon pressing this button, the system will immediately begin taking measurements. One will have the ‘Saving Interval’ time-frame to move the translation stage in increments of 10 on the adjustment knob (note that each small tick on the know is 0.01 mm, so moving by ten of those ticks for each measurement step is a movement of 0.1 mm or 100 μm for the sample). Remember that this translation stage utilizes gears inside the turning mechanism. Be careful to turn the adjustment knob precisely in units of 10 ticks and to not overshoot the ticks, re-adjusting after overturning the ticks can lead to inaccurate step distance increments in the recorded data. Once the Z-Scan is complete there will be 171 recorded measurements automatically saved into an Excel spreadsheet at the location of the preset file path. After completing a closed-aperture Z-Scan, reset the translation stage to the zero marker and open the adjustable iris aperture.

Figure 16.

Open Aperture Z-Scan Set-up.

The power-meter must now be moved to a position close to the focal point of the post-sample lens for measurement of the open-aperture Z-Scan. The post-sample lens is just used to ensure that all the optical power of the expanded beam is captured by the sensitive area of the photodetector. Follow the same procedures for the closed-aperture Z-Scan for the open-aperture Z-Scan, ensuring that the ‘File Name’ has been modified as to not overwrite the previous scan data. Once the open-aperture Z-Scan is complete, follow the Pre-Measurement procedures to reset the iris aperture size and then adjust the laser power level with the half-wave plate for another desired power level to continue with a Z-Scan power sweep.

A.3.6

Post-measurement Procedures

Having collected the (OA and CA) Z-scan measurements, for different incident power levels, save all data in a single CSV file, and use the data processing program²⁶ to extract the nonlinear parameters. Check section B for more information on the parameter extraction process, and check the software user manual in the software repository²⁶ to correctly launch the program.

B.1.1

Logarithmic function-based fitting

The first method comprises an exact logarithmic fitting, directly applying Eq. (49). Thereby, no data standardization is required. A vs curve is obtained for every z coordinate, by extracting the corresponding measurement of every Z-scan made in the input power sweep. So, every vs curve has as many points as incident power levels were tested. Then, leveraging an optimization fitting function, it is possible to find the corresponding value of F for every z coordinate of the scans, which provides a numerical description of the F(z) function associated to the setup under consideration. Having this information, it only remains to perform a lorentzian fitting of F(z); which yields the values for the lorentzian amplitude-related value A, half-width at half-maximum σ, and center value μ. These values have a direct relationship with the parameters involved in the analytical definition of F(z). So, by comparing Eq. (48) with the canonical form of a lorentzian function, one gets

from which it is immediately obtained that

B.1.2

Taylor expansion-based fitting

The remaining two methods use a series expansion to represent the logarithmic function in Eq. (49). This allows the use of less sophisticated fitting software, and may provide more stable fittings for experimental measurements with a small number of Z-scans (i.e., power sweep performed for few incident power values).

Thus, using the Maclaurin power series for the analytical function ln[1 + x], and considering the first two terms of the expansion, it is possible to find a polynomial expression to approximate the relation between the incident optical power and outgoing optical power . Using this strategy it is found that

Then, one could either find F(z) performing a linear regression of the ratio vs curve for each z coordinate, in a similar way as it was done in section B.1.1, or directly solving Eq. (52) for F(z) and analyzing the resulting lorentzian for every incident power level within the experimental power sweep. These are regarded as the second and third fitting methods. Each of these entails a different way of standardizing data before proceeding with the fitting.

For the second method, the linear regression is performed after having adjusted experimental data according to the expression

which means that measured output data must get divided by the constant e^−α₀L and the corresponding incident power value , and with this standardized data set a new -dependent curve is obtained for every z coordinate of the performed Z-scans. The slopes a(z) of all the calculated linear regressions would store the information required to numerically determine F(z), given that

b(z) is the set of intercepts obtained with the linear regression applied for every z coordinate. Eq. (54) predicts that they are all values expected to be close to 1.

Having the numerical description of the F(z) function associated to the setup under consideration, it is possible to perform a second lorentzian fitting and extract w₀ and β_TPA, following the same procedure used in section B.1.1.

For the third method, data standardization is made according to the expression

In this case, the first fitting to obtain F(z) is not required, since standardizing data as shown in Eq. (55) enables to immediately obtain the numerical description of the F(z). If this standardization is applied to data for every incident power level, it is possible to apply a different lorentzian fitting for every standardized data set: one per incident power level considered in the power sweep. So, following the same procedure used in section B.1.1, every lorentzian fitting would provide different pair of values for w₀ and β_TPA. All of them are averaged in order to improve method’s robustness, and return the final estimations for w₀ and β_TPA.

Figure 18.

Illustrative curves of the open-aperture fitting methods, applied to the measurement data set from Fig. 17. All solid blue and dashed red plots correspond to experimental and fitting curves, respectively. (a) Example of an exact logarithmic fitting, for the vs curve evaluated for z = 0.0085 m. (b) Lorentzian second fitting of the numerical description of F(z), obtained following the first fitting method. (c) Example of a linear regression fitting, for the standardized vs curve evaluated for z = 0.0085 m. (d) Lorentzian second fitting of the numerical description of F(z), obtained following the second fitting method. (e-f) Examples of the lorentzian fittings of the standardized data, according to Eq. (55), for two of the incident power levels considered in the power sweep: 63 mW and 111 mW.