1. Introduction

The scientific community has recently renewed its interest in laser wavelengths significantly beyond the well-established visible-to-near-infrared region around 1 µm. This resurgence is fueled, on one hand, by the favorable wavelength scaling of certain parameters crucial for plasma physics and high-energy physics, such as the ponderomotive potential $\propto \lambda ^2$ and the critical plasma density $\propto 1/\lambda ^2$. On the other hand, recent advancements in powerful ultrafast lasers operating at mid-wave- and long-wave infrared (MWIR, LWIR) wavelengths have also played a significant role [1].

Currently, amplifying picosecond pulses in high-pressure $\textrm{CO}_{2}$ laser amplifiers [2] remains the sole method to achieve multi-terawatt peak power levels in the LWIR spectrum. Recent advancements in this field have included transitioning to mixed-isotope gas mixtures, adopting the chirped-pulse amplification scheme, and implementing a solid-state, optical-parametric-amplifier-based front-end seed system, culminating in the achievement of 5 TW peak power in 2 ps pulses at 9.2 µm [3]. This operational regime introduces challenges as the transparent optics in LWIR laser systems start exhibiting nonlinear (intensity-dependent) variations in optical properties, such as refractive index and absorption. These effects can potentially degrade laser system performance; for instance, nonlinear refraction-related self-focusing can lead to optical damage, and nonlinear absorption can induce premature saturation-like behavior in laser amplifiers. Conversely, when harnessed effectively, nonlinear interactions can enable the design of innovative laser sub-systems, such as post-compressors that utilize the self-phase modulation effect to broaden the pulse’s spectrum following amplification [4]. Given these dynamics, acquiring reliable data on the nonlinear response of materials under intense LWIR laser pulses has become crucial.

In this work, we report on characterizing the nonlinear optical responses of various infrared materials that are transparent at $\textrm{CO}_{2}$ laser wavelengths. It is important to note that our focus has been on materials suitable for use in high-energy laser systems, meaning those that can be fabricated in sufficiently large sizes for use in high-energy laser beam optics. Nonlinear crystals typically used for wavelength-conversion applications are outside the scope of this study. Our primary goal is to provide empirical data that is directly applicable to laser system design, rather than delving into the theoretical interpretation of the observed dependencies, which could be explored in future research.

In the sections that follow, we elucidate our measurement methodology for evaluating nonlinear refraction and absorption in materials, leveraging the "single-shot" output from a multi-terawatt LWIR laser at the Accelerator Test Facility of Brookhaven National Laboratory (ATF). We further describe the material samples used in our research, presenting literature data on the linear refraction properties of these materials. We supplement this information with our findings on the linear absorption characteristics of two materials where such absorption proves significant. Lastly, we present the measurement results of nonlinear optical properties for the eighteen materials under examination — nonlinear refraction was quantified in seventeen of these materials and nonlinear absorption in eleven. We then discuss their significance in broadening our understanding of material responses to ultra-intense optical pulses within the LWIR range.

2. Methods

2.1 Experiment

The nonlinear refractive index, denoted as $n_2$, is defined as the proportionality coefficient in the first-approximation expression for the intensity-dependent index of refraction: $n = n_0 + n_2 I$, where $I$ represents the intensity of the optical field and $n_0$ is the refractive index in the zero-intensity limit. The effective value of $n_2$ measured in an experiment can be influenced by the temporal characteristics of the laser pulse, due to the finite characteristic times of processes contributing to intensity-dependent refraction [5]. Given the applied nature of this research, which focuses on measuring nonlinear optical properties pertinent to the engineering of high-peak-power LWIR lasers, we did not attempt to deduce the intrinsic $n_2$ in the zero pulse duration limit. Instead, we centered our efforts on the precise measurement of its effective value specific to the pulse format of a state-of-the-art multi-TW LWIR laser. While this approach to some extent simplifies the task, a primary limitation is the inability to use the de-facto standard Z-scan method for characterizing nonlinear refraction [6], given the single-shot operation regime of our laser. To address this, we developed a method for single-shot measurement of nonlinear refraction, described in detail in [4] and briefly summarized here. Additionally, we introduced a modification to the experimental setup to facilitate single-shot characterization of the nonlinear absorption of sample materials. Figure 1 provides simplified schematics of the setups used for studying both nonlinear refraction and absorption, shown in parts (a) and (b), respectively.

 

Fig. 1. Experimental setup configurations for studying nonlinear refraction (a) and nonlinear absorption (b). The two configurations differ by the addition of an $\sim 1:1$ imaging lens in the setup used for studying nonlinear absorption. Additionally, samples used for measuring nonlinear refraction are typically considerably thicker than those used for observing nonlinear absorption.

Download Full Size | PDF

The central technical aspect of these experiments lies in producing a high-quality, highly reproducible, azimuthally symmetric Gaussian-like beam and analyzing the effects of passing it through a material sample to characterize the intensity-dependent interaction. Variation of the intensity-related phase shift across the beam, manifesting as self-focusing, enables the evaluation of the nonlinear refractive index, while variations in transmittance across the beam provide information on nonlinear absorption. It is imperative that the beam’s intensity be sufficiently high to achieve a regime of measurable nonlinear interaction.

To generate the required intense, high-quality probe beam, we employed a lossy but robust scheme. Here, we directed the high-peak-power pulse, which can reach up to a nominal 5 TW (with an energy of 10 J and pulse duration of 2 ps) [3], through a hard-edge Teflon aperture with a 12.88 mm diameter. We then allowed the beam to propagate over a distance of 14 m before it interacted with the material sample. This approach transformed the original top-hat-like beam profile, which had an energy-dependent size and significant intensity variations, into a near-ideal, theoretically predictable beam profile defined by long-range diffraction and a negligible effect from the original beam’s intensity distribution. In most cases, the effect of nonlinear refraction in air was negligible; however, for high-energy shots, its contribution was accurately characterized and incorporated into the propagation model [7]. This inclusion was crucial for precisely predicting both the spatial and temporal pulse profiles at the sample. An example of a measured beam profile at the hard-edge aperture is depicted in Fig. 2(a), while the beam profile at the sample location is shown in Fig. 2(b).

 

Fig. 2. (a) Typical high-energy beam profile at the Teflon aperture (the size of the aperture is outlined by the dashed line). (b) Beam profile at the sample. (c) Example of a self-focused beam used for extracting information on the nonlinear refractive index. (d) Example of the beam after passing through a sample during the measurement of nonlinear absorption. All profiles are individually normalized.

Download Full Size | PDF

After the beam passes through the sample, it is attenuated by reflecting approximately 3 % of its energy off the surface of a $\textrm{BaF}_{2}$ wedge window. The most of the energy is transmitted through the window and is measured using a pyroelectric joulemeter (Gentec-EO QE95). In the setup configuration designed for analyzing nonlinear refraction through self-focusing, as depicted in Fig. 1(a), the beam is allowed to propagate 1.6 m after the sample before reaching the pyroelectric-array beam profiler (Ophir Spiricon Pyrocam IV). This ensures that self-focusing is manifested as a reduction in the beam size. An example of a measured self-focused beam is shown in Fig. 2(c). For the setup configuration that characterizes nonlinear absorption, a lens with a focal length of $F={0.4}\,\textrm{m}$ is positioned between the sample and the beam profiler to achieve approximately a 1:1 imaging of the sample’s back surface. This provides a beam profile immediately after the beam exits the sample. An example of such a profile, exhibiting pronounced absorption in the intense central region of the beam, can be seen in Fig. 2(d). Comparing the joulemeter readings with the integrated signal from the beam profiler enables absolute intensity calibration. It also allows for the use of the beam profiler in measuring energies that fall below the joulemeter’s sensitivity threshold.

A challenging part of the measurement process was ensuring that nonlinear absorption did not affect the measurement of nonlinear refraction, and vice versa, that the measurement of nonlinear absorption was not affected by the microfilamentation inside the sample that can occur due to nonlinear refraction when the nonlinear phase shift accumulated in the beam passing through a sample (B-integral) exceeds $\sim \pi$ radians. To decouple these two nonlinear effects, we utilized different sample geometries for the two measurements: In the experimental configuration shown in Fig. 1(a), the sample was sufficiently thick to allow achieving quantifiable self-focusing (at B-integral 1–3 rad) at intensities below the onset of nonlinear absorption. In the configuration of Fig. 1(b) for measuring nonlinear absorption, the thinnest possible samples were used. For some materials, several trial-and-error attempts were needed to determine the optimal sample geometry. In the subsequent section, we list only the samples used in the final measurements, the results of which are reported here.

As a final remark on the experiment details, we note that for materials with weak optical nonlinearity, where the highest pulse energy was required, additional attenuation of the beam before the beam profiler was arranged by reflecting the beam from up to three BK7 glass blanks that, at 9.2 µm, have $\sim {40}{\% }$ reflection.

2.2 Data analysis

To analyze the measurement data, we first convert the 2-dimensional measured beam profile into a one-dimensional curve representing fluence as a function of radial distance from the beam’s center. This is achieved through an azimuthal projection. In this method, we ’average out’ the values around concentric circles centered on the beam’s centroid and assign each average to a radial coordinate corresponding to the radius of that circle. Then, depending on whether we are analyzing nonlinear refraction or nonlinear absorption data, we use the following procedures.

Nonlinear refraction

Nonlinear absorption

Accurate information regarding the temporal structure of the laser pulse is crucial for interpreting the results of measurements related to nonlinear optical properties. Figure 3 presents the pulse profile utilized in our analysis. This temporal profile was predicted based on theoretical model calculations for the multi-TW LWIR laser at ATF. It was further validated experimentally through autocorrelation measurements [3].

 

Fig. 3. Temporal pulse structure as predicted theoretically and confirmed experimentally [3], employed in the data analysis process.

Download Full Size | PDF

Data analysis relies on linear optical properties, such as the linear refractive index and its wavelength dispersion, as well as linear absorption. These properties are either sourced from the literature or measured in preparatory experiments. Since this information is also pertinent for any high-power laser design utilizing the studied materials, we include it in the subsequent section.

3. Studied materials and their linear optical properties

3.1 Halides of alkali and alkaline earth metals

Halides of alkali and alkaline earth metals typically form crystalline structures with a high ionic character, resulting in a wide transparency range that extends from the visible to the MWIR and LWIR regions. These materials tend to have low refractive indices and high damage thresholds, which are crucial properties for high-power laser applications. Furthermore, their optical homogeneity and the ability to be grown as large single crystals also contribute to their widespread use in the field of optics.

In particular, KCl and NaCl, despite their relative high susceptibility to moisture and softness, are often used in optics for high-power $\textrm{CO}_{2}$ lasers. $\textrm{BaF}_{2}$ is a more attractive material due to its significantly higher hardness and much lower solubility in water; however, it has strong absorption at 10.6 µm (the most efficient line of conventional $\textrm{CO}_{2}$ lasers), which significantly limits its use in these lasers. Other readily available materials from this group that may be of interest for LWIR laser applications include NaF, KBr, and CsI.

We obtained at least two samples of each of the materials listed in Table 1. The smaller samples were intended for studies on nonlinear absorption, while the larger samples were designated for measuring nonlinear refraction. Additionally, we used the larger samples to measure linear absorption at $\textrm{CO}_{2}$ laser wavelengths, results of which are presented later in this section. Two identical samples of $\textrm{BaF}_{2}$ were ordered from different suppliers located on different continents to test for possible inconsistencies; no measurable differences in optical properties of these samples were observed.

Table 1. Samples of halides of alkali and alkaline earth metals.

View Table | View all tables in this article

The following dispersion formulae from the literature were used in our data analysis. When multiple formulae are available in the literature for a given material, we chose the one that a) has a claimed validity range covering the LWIR region, and b) is most frequently cited by other authors. Hence, these formulae can be recommended for use in the design of LWIR laser optics. Wavelength $\lambda$ is expressed in micrometers, and the validity range of each formula is also provided.

$\textrm{BaF}_{2}$ [9].

Due to the relatively large thickness of the samples used for measuring the nonlinear refractive index, and owing to the absence of consistent numerical data in the literature on their linear absorption in the LWIR range, we conducted preliminary measurements. These involved examining their transmittance at four wavelengths using a continuous-wave $\textrm{CO}_{2}$ laser (Access Lasers L4G-FC). The results were corrected for Fresnel reflection on the sample surfaces and are presented in Fig. 4. For each measurement, the signal from the powermeter (Gentec-EO UP17P) was integrated over a one-minute period to achieve a signal-to-noise ratio greater than 1000. However, due to the slow drift in laser power and potential deviations in surface losses from theoretical Fresnel reflection, we conservatively estimated the absolute measurement accuracy as ranging between 0.001 and 0.003 cm⁻¹, with the higher values corresponding to materials with higher indices of refraction.

 

Fig. 4. Measured linear internal absorption of compounds of alkali and alkaline earth metals, with fits of $\textrm{BaF}_{2}$ and NaF data using an empirical formula.

Download Full Size | PDF

As depicted in Fig. 4, no measurable absorption was detected in KCl, NaCl, KBr, and CsI. For $\textrm{BaF}_{2}$ and NaF, we fitted the experimental absorption data with a simple exponential formula, convenient for practical use in modeling laser beam interactions in the wavelength range of 9–11 µm. Wavelength $\lambda$ is measured in µm, and the linear absorption coefficient $\alpha _0$ in cm⁻¹.

It is worth noting that Eq. (7) is a refined version of the expression we suggested in [4]. The relatively low linear absorption of $\textrm{BaF}_{2}$ at 9.2 µm makes it an attractive option for the optics of ultrafast high-peak-power lasers operating around this wavelength, especially considering its low Fresnel reflection losses ($<{3}{\% }$ per surface at normal incidence).

3.2 Halides and chalcogenides of transition and post-transition metals

Halides and chalcogenides of transition and post-transition metals display a broad spectrum of transparency in the infrared region, coupled with relatively high refractive indices compared to the previously discussed group of materials. This makes them indispensable in the field of infrared optics and imaging systems. These compounds possess a narrower bandgap than the halides of alkali and alkaline earth metals, leading to the transparency range starting at generally longer wavelengths. The pronounced Fresnel reflection in these materials necessitates the use of antireflection coatings or a Brewster orientation.

ZnSe stands out as the predominant material in this category for LWIR applications. Traditional growth methods enable the fabrication of large boules, making it ideal for crafting high-average-power IR optics. These methods yield a grain polycrystalline structure. ZnS, CdTe, and KRS-5 (a mixed TlBr-TlI crystal with a 2:3 ratio of Br to I atoms) are other renowned materials in infrared applications. The final selection among these often hinges on the desired transparency range, as well as the mechanical and chemical resistance requirements. All these compounds exhibit high transparency in the wavelength range of the $\textrm{CO}_{2}$ laser. While AgBr and AgCl are less frequently utilized in optical applications, their transparency spanning the vast visible-to-LWIR spectrum and their availability in larger sizes (owing to their use in particle detectors) merited their inclusion in our study.

Table 2 enumerates the samples employed in our nonlinear refraction (thicker samples) and nonlinear absorption (thinner samples) measurements. It’s noteworthy that procuring a CdTe sample of an appropriate thickness for a reliable measurement of $n_2$ proved challenging, and as such, these measurements have been omitted from the results in the ensuing section. After listing the materials, we provide the dispersion formulae recommended for designing laser optics.

Table 2. Samples of halides and chalcogenides of transition and post-transition metals

View Table | View all tables in this article

AgBr

To the best of our knowledge, a dispersion formula for AgBr valid in the LWIR range has not been reported previously. To address this gap, we combined data on the refractive indices of AgBr from two publications, [11] and [12], and applied the Sellmeier equation to fit it. This effort resulted in Eq. (9) for the linear index of refraction of AgBr, valid over a wide wavelength range, as shown in Fig. 5.

 

Fig. 5. Sellmeier fit of AgBr refractive index data from Refs. [11] (Schröter) and [12] (McCarthy).

Download Full Size | PDF

AgCl [13]

3.3 Chalcogenide glasses

Chalcogenide glasses are amorphous compounds primarily composed of chalcogen elements and exhibit transparency in the infrared spectrum. Although they have been known for several decades, their utilization in MWIR and LWIR laser optics remains limited. For this study, we procured samples of three chalcogenide glasses available in sufficiently large sizes, making them suitable for our single-shot method of measuring the nonlinear refractive index. The availability of material in large dimensions is also essential for high-power laser optics applications, which we are keen on investigating. It’s worth noting that glasses of the same composition might be marketed under different names by various manufacturers. In Table 3, the sample list specifies the supplier’s name followed by the name under which the material was purchased. Subsequent to the list, dispersion formulae for the evaluated chalcogenide glasses are provided.

Table 3. Samples of chalcogenide glasses

View Table | View all tables in this article

$\textrm{Ge}_{33}\textrm{As}_{12}\textrm{Se}_{55}$ (AMTIR-1, IRG22, IRG201) [18]

$\textrm{Ge}_{10}\textrm{As}_{40}\textrm{Se}_{50}$ (IRG24, IRG207) [18]

$\textrm{Ge}_{28}\textrm{Sb}_{12}\textrm{Se}_{60}$ (IRG25, IRG205) [18]

It’s worth noting that although the linear absorption in the relatively thin chalcogenide glass samples used in our study is insubstantial, this might not be the case for thicker optics. According to the material datasheets [18], the absorption coefficients at 9.2 µm are 0.0125 cm⁻¹ for SCHOTT IRG22, 0.0079 cm⁻¹ for IRG24, and 0.0208 cm⁻¹ for IRG25 glasses.

3.4 Elemental and III-V compound semiconductors

The final group of materials we studied comprises three prominent semiconductors often used in LWIR optics: Si, Ge, and GaAs. Owing to their narrow bandgap, these materials are opaque in the visible spectrum and are known for their high affinity for multiphoton absorption. Their wide availability in large-sized wafers, a consequence of their significant role in the semiconductor industry, coupled with their potential use in active optical switching elements for LWIR laser systems [19], warrants their inclusion in our study. Details of the samples used are provided in Table 4, followed by the recommended dispersion formulae.

Table 4. Samples of elemental and III-V compound semiconductors

View Table | View all tables in this article

GaAs [20]

4. Results

Results of the studies of nonlinear optical properties of infrared materials performed with a 9.2 µm, 2 ps probe pulse are presented in Fig. 6 and Fig. 7 for the nonlinear refractive index and nonlinear absorption, respectively. The numerical data for $n_2$ and the onset of nonlinear absorption $\Phi _{NLA}$ are summarized in Table 5. In this table, sample materials are arranged according to their linear refractive index $n_0$. For the reader’s convenience, we also provide the group velocity dispersion $\beta _2$ for each material at 9.2 µm, calculated using the following Eq. (21) and dispersion formulae reported in Section 3.

 

Fig. 6. Measured values of nonlinear refractive indices $n_2$ of seventeen materials plotted as a function of linear refractive index $n_0$ at 9.2 µm. The fitting curve is provided for visual convenience only and doesn’t represent a physical model.

Download Full Size | PDF

 

Fig. 7. Transmitted fluence as a function of input fluence, both corrected for Fresnel reflections on the sample’s surfaces. The deviation from the diagonal line indicates the onset of nonlinear absorption; approximate values of the onset are listed in Table 5.

Download Full Size | PDF

Table 5. Optical properties of LWIR materials at 9.2 µm for a 2 ps pulse.

View Table | View all tables in this article

We would like to add the following comments and clarifications regarding the presented results.

5. Discussion

The primary contribution of this work is the significant extension of the wavelength range for measuring the nonlinear optical properties of transparent materials. With a few exceptions, neither the nonlinear refraction nor the nonlinear absorption of the studied materials have been previously examined in the long-wave infrared spectral range.

To the best of our knowledge, no experimental data on the nonlinear refractive index of CsI, AgBr, and chalcogenide glasses IRG24 and IRG25 have been published before this work for any wavelength. For the other thirteen materials for which we report the $n_2$ value at 9.2 µm, it is useful to compare with available literature values. We recently introduced a database of nonlinear refractive indices as part of the refractiveindex.info database [24], which currently comprises a considerable fraction of published $n_2$ data. In Fig. 8 and Fig. 9, we present the newly acquired values alongside the data from the database. The available data is significantly scattered, which can be attributed partly to the different temporal structures of the pulses used by various authors to probe nonlinear refraction and partly to the typically high measurement error bars, which sometimes reach up to 50%. Nonetheless, aggregating multiple datasets on the same plot often illuminates general trends, and the new data point at LWIR provided by this work contributes significantly. In this context, we would like to draw the reader’s attention to the plot corresponding to $\textrm{BaF}_{2}$. The data point at 9.2 µm reinforces the trend of decreasing $n_2$ with wavelength, a trend that was only faintly discernible in previous data. Considering the trends in the remaining plots, we can assert that for none of the studied materials are there significant discrepancies between the newly acquired data for $n_2$ at 9.2 µm and the value range where it might be anticipated.

 

Fig. 8. Comparison of $n_2$ values obtained in this work at 9.2 µm with literature data for different wavelengths included in refractiveindex.info database [24] (continued in Fig. 9).

Download Full Size | PDF

 

Fig. 9. (continued from Fig. 8) Comparison of $n_2$ values obtained in this work at 9.2 µm with literature data for different wavelengths included in refractiveindex.info database. Data from following publications are used: Adair [25]; Bristow [26]; DeSalvo [27]; Ensley [23]; Flom [28]; Jansonas [29]; Hurlbut [30]; Lin [31]; Milam [32]; Patwardhan [33]; Pigeon [34]; Sheik-Bahae [35]; Werner [36].

Download Full Size | PDF

Regarding the data on nonlinear absorption, its direct comparison with previously published data–obtained under different wavelengths and pulse durations–is challenging. This stems from the intricate nature of nonlinear absorption processes. Such processes might entail multi-photon and tunneling effects which are heavily reliant on photon energies. Additionally, they are influenced by the accumulation of highly-absorbing free charge carriers in the conduction zone. This makes the process dependent on both intensity and fluence. We posit that the results presented here lay down an empirical foundation for impending theoretical explorations, which are beyond the ambit of this study. However, it’s worth noting that nonlinear absorption in semiconductors has garnered substantial attention since the inception of lasers [37]. A noteworthy early work, akin in approach to ours, explored nonlinear absorption in germanium using an 80 ns pulse from a $\textrm{CO}_{2}$ laser operating at 10.6 µm [38]. This study measured a substantially lower onset of nonlinear absorption–4 MW/cm²–compared to nearly 2 GW/cm² we report for a 2 ps pulse. This discrepancy underscores the pronounced absorption due to free-carriers accumulated during the protracted pulse in the former study. Additionally, our prior study on the nonlinear absorption of Ge using a 5 ps LWIR pulse, along with a rudimentary model we devised to account for the accumulation of free carriers [39], aligns seamlessly with the current findings.

The datasets acquired in this study–pertaining to the nonlinear refractive index and absorption curves–are arguably among the most comprehensive of their kind. They were acquired under uniform conditions, leveraging identical probe pulses, experimental setups and data analysis methodologies. Consequently, this cohesive dataset stands as a potent resource for validating theoretical models in subsequent research endeavors.

The primary focus of this paper is to present experimental data on the nonlinear optical properties of infrared materials, particularly those applicable in high-peak-power LWIR lasers. For selecting materials for specific optical elements in nonlinear photonics applications, we offer the following guidelines:

6. Conclusion

In this study, we have undertaken a comprehensive investigation into the nonlinear optical properties of transparent materials in the long-wave infrared (LWIR) spectral range. Our primary contributions can be summarized as follows: