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Ultrafast nonlinear refraction measurements of infrared transmitting materials in the mid-wave infrared

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Abstract

We utilize the conventional Z-scan technique to provide absolute measurements of third-order nonlinear refraction coefficients (n2) in the mid-wave infrared at 2 µm and 3.9 µm of common optical materials that have transparency windows spanning this regime. We study a variety of narrow band gap and wide band gap semiconductors, fluoride crystals (BaF2, CaF2, LiF, and MgF2) and optical glasses, and a series of chalcogenide glasses. The n2 is found to span on the order of ∼10−15 to ∼10−12 cm2/W for the semiconductors, ∼10−16 cm2/W for the fluoride crystals and glasses, and ∼10−14 to ∼10−13 cm2/W for the chalcogenides. The experimental results are compared to previous measurements of n2 conducted in the visible and near-infrared along with empirical and theoretical formulations.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

The Kerr nonlinearity arising from the third-order nonlinear refraction coefficient (NLR, of coefficient $\def\upmu{\unicode[Times]{x00B5}}{n_2}$) is important for a wide variety of applications such as all-optical signal processing and all-optical communications [1,2]. For such applications, identifying material systems with large ${n_2}$ coefficients with relatively low loss is paramount in realizing efficiency. In the wavelengths of interest for signal processing and communications (primarily between $1.26 \,\upmu \textrm{m}$ and $1.62 \,\upmu \textrm{m}$), there have been a plethora of studies aimed at studying the nonlinear behavior of many different types of materials that possess a large Kerr nonlinearity [35]. Furthermore, there is an abundancy of commercially available laser sources in this regime. In recent years, there has been much work geared towards developing capable sources at longer wavelength ranges in the mid-wave infrared (MWIR) spanning from $2 \,\upmu \textrm{m}$ to $8\upmu \textrm{m}$ [610] which has the benefit of extending the application space for large ${n_2}$ materials. Although much work has been done to experimentally determine ${n_2}$ in the visible and near-infrared wavelength regimes, there have been relatively few efforts for ${n_2}$ determination in the MWIR to identify such materials.

In this work, we use the conventional Z-scan technique [11] to characterize ${n_2}$ for a range of standard wide band gap and narrow band gap semiconductors, fluoride crystals and optical glasses, and a series of chalcogenide glasses at the MWIR wavelengths of $2\upmu \textrm{m}$ and $3.9\upmu \textrm{m}$. The ${n_2}$ values are compared to previous experimental results in the visible and near-infrared regime as well as theoretical and empirical formulas.

2. Experimental setup

NLR characterization is performed with two different laser sources. For characterization at $2 \,\upmu\textrm{m}$, we used pulses from a Ti:Sapphire amplified laser system (Spectra-Physics, Solstice) producing 800 nm, 3.5 mJ, ∼100 fs (FWHM) pulses at a 1 kHz repetition rate to pump an optical parametric generator/amplifier (OPG/A, Light Conversion, TOPAS-Prime). This OPA generates wavelengths ranging from 1120 nm to 1640 nm corresponding to the signal pulses (vertically polarized) and from 1640 nm to 2800 nm corresponding to the idler pulses (horizontally polarized). For characterization at 3.9 $\upmu\textrm{m}$, we used pulses from a different Ti:Sapphire amplified laser system (Coherent, Hidra-F-100) producing 800 nm, 100 mJ, ∼100 fs (FWHM) pulses at a 10 Hz repetition rate to pump another OPA (Light Conversion, TOPAS-HE-Prime). We pump this OPA with ∼45 mJ to generate wavelengths ranging from 1150 nm to 1650 nm and from 1650 nm to 2650 nm corresponding to the signal and idler pulses, respectively. The 3.9 $\upmu\textrm{m}$ pulses are then generated through difference frequency generation (DFG, horizontally polarized) using a 2 mm thick Type II KTiOAsO4 (KTA) crystal.

Both $2 \,\upmu\textrm{m}$ and 3.9 $\upmu\textrm{m}$ pulses were spatially filtered using an all-reflective focusing geometry as shown in Fig. 1. A CaF2 wedge was used as a pickoff to monitor the energy on a reference detector for the $2 \,\upmu\textrm{m}$ pulses while a ZnSe wedge was used as a pickoff for the 3.9 $\upmu\textrm{m}$ pulses. The pulses were subsequently focused into the sample via a CaF2 lens and a ZnSe lens for the $2 \,\upmu\textrm{m}$ and 3.9 $\upmu\textrm{m}$ pulses, respectively. The detectors used were PbS and PbSe for the $2 \,\upmu\textrm{m}$ and 3.9 $\upmu\textrm{m}$ pulses, respectively.

 figure: Fig. 1.

Fig. 1. Schematic of Z-scan experiment. “CM” stands for concave mirror, “SF” for spatial filter, “50/50” for a 50/50 beamsplitter, “Ir” for iris diaphragm used as the aperture for the closed aperture (CA) detector, “OA” for open-aperture detector, and “Ref” for the reference detector.

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To determine the pulse energy at $2 \,\upmu\textrm{m}$, the 1 kHz rep rate pulses were chopped at a frequency of 26 Hz and the power was directly measured in front of the sample with a broadband THz radiometer with linearity ranging from 0.1 $\upmu\textrm{W}$ to 20 mW. The pulse energy at 3.9 $\upmu\textrm{m}$ was determined by calibrating the reading from the same radiometer with a calibrated pyroelectric detector near the output of the DFG where the energy is relatively high (∼0.5 mJ). The radiometer was then placed directly in front of the sample and a reading was taken to obtain the actual energy. Attenuation for the $2 \,\upmu\textrm{m}$ pulses was achieved with calibrated 1 mm thick glass neutral density (ND) filters and calibrated 1 mm thick Ge ND filters for the 3.9 $\upmu\textrm{m}$ pulses. For both the $2 \,\upmu\textrm{m}$ and 3.9 $\upmu\textrm{m}$ pulses, the minimum beam waist $({{w_0}\textrm{, HW 1}/{\textrm{e}^\textrm{2}}} )$ was measured by one-dimensional knife-edge scans to be 31 $\upmu\textrm{m}$ and 51 $\upmu\textrm{m}$, respectively. This corresponds to a Rayleigh range, ${z_0}$ ($= \pi w_0^2/{\lambda _0}$ where ${\lambda _0}$ is the vacuum wavelength), of 1.5 mm and 2.1 mm for the $2 \,\upmu\textrm{m}$ and 3.9 $\upmu\textrm{m}$ pulses, respectively. The pulse width of 240 fs (FWHM) at 3.9 $\upmu\textrm{m}$ was measured by second-order intensity autocorrelation in a Type I AgGaS2 (AGS) crystal and 90 fs (FWHM) at $2 \,\upmu\textrm{m}$ using a Type I BaB2O4 (BBO) crystal.

3. Results and discussion

NLR and nonlinear absorption (NLA) are both responsible for optical “self-action” meaning that laser light of sufficient irradiance can alter the refractive index, n, and absorption, $\alpha $, of a material. The irradiance-dependent coefficients, n and $\alpha $, are typically expressed up to the third order polarization as follows: $n(I )= {n_0} + {n_2}I$ and $\alpha (I )= {\alpha _0} + {\alpha _2}I$ where the subscript "0" denotes the linear coefficient, I is the irradiance, and ${\alpha _2}$ is the third order NLA (two-photon absorption, 2PA) coefficient. To perform Z-scans, a sample is scanned along the axis of the focus of a beam and the transmittance is recorded as a function of the sample position $T(Z )$ as shown in Fig. 1. Placing a partially closed aperture (CA) in the far-field allows for sensitivity to ${n_2}$ whereas in the case of a fully open aperture (OA), the Z-scan is only sensitive to ${\alpha _2}$, i.e. in the frame of the thin sample approximation limit. Thus, the CA Z-scan is sensitive to both ${n_2}$ and ${\alpha _2}$ whereas the OA Z-scan is sensitive to only ${\alpha _2}$. When analyzing the CA Z-scan data, the ${\alpha _2}$ value from the OA Z-scan may be used, allowing a one parameter fit to obtain ${n_2}$. In certain instances, the OA Z-scan can be divided from the CA Z-scan to obtain a signal devoid of NLA. This is typically referred to as CA/OA which is usually valid when $4\pi {n_2}/({{\lambda_0}{\alpha_2}} )> 1$. The wedge (in Fig. 1) is used as a pick-off to the reference detector in order to reduce the noise in the Z-scan signal as was done in [12]. Note that for the analysis presented herein for materials that exhibit both ${n_2}$ and ${\alpha _2}$ at the wavelength and irradiance levels measured, the complex electric field after the sample (${E_e}$, i.e. Eq. (26) in [11]) is propagated to the aperture using a Huygens-Fresnel integral to obtain the electric field at the aperture, ${E_a}$. For those materials exhibiting only ${n_2}$, five terms of Eq. (9) in [11] are used to obtain ${E_a}$. Eq. (11) from the same reference is subsequently used to obtain the normalized $T(Z )$ in all cases. Note that for all materials studied, there was no observed linear dependence on ${n_2}$ and/or ${\alpha _2}$ with irradiance.

The linear transmission spectra of the materials under investigation are shown in Fig. 2 for the (a) semiconductors, (b) glasses and fluoride crystals, and (c) chalcogenide glasses. The spectra from 300 nm to $2 \,\upmu \textrm{m}$ were obtained with a UV-Vis-NIR spectrophotometer (Agilent, Cary 7000) whereas the spectra from $2 \,\upmu \textrm{m}$ to 20 $\upmu \textrm{m}$ were obtained with a FTIR spectrometer (Bruker, Vertex 70). The commercially available semiconductors studied were 3 mm thick ZnS/Cleartran (Meller Optics, SCD3349-02A), 2 mm thick ZnSe (ISP Optics, ZC-W-25-2), 0.35 mm thick undoped [100] GaAs (Wafer Technology, Ltd., Ingot #WV 24326/Un), 0.35 mm thick undoped n-type [100] InP (Wafer Technology, Ltd., Ingot #R7/265/Un), 1 mm thick Si (Andover Corp, IRWS100-25), 1 mm thick Ge (Andover Corp, IRWS200-25), and 0.5 mm thick undoped p-type GaSb (Wafer Technology, Ltd., Ingot #MS/G3/312A/Un). The glass and fluoride crystal samples were 1 mm thick fused silica (Thorlabs, WG41010), 1 mm thick BK7 (Thorlabs, WG11010), 3 mm thick BaF2 (Eksma Optics, 540-7251), 3 mm thick CaF2 (Eksma Optics, 530-6253), 3 mm thick MgF2 (Eksma Optics, 520-5253), and 3 mm thick LiF (Eksma Optics, 510-5253). The chalcogenide samples were all purchased from Amorphous Materials, Inc.: 2 mm thick AMTIR-1 (Ge33As12Se55), 2 mm thick AMTIR-2 (AsSe), 2.5 mm thick AMTIR-4 (AsSe), 2 mm thick AMTIR-5 (AsSe), and 2 mm thick AMTIR-6 (As2S3).

 figure: Fig. 2.

Fig. 2. Linear transmission from 300 nm to 20 $\upmu \textrm{m}$ for the measured (a) semiconductors, (b) glasses and fluoride crystals, and (c) chalcogenide glasses. The legend in (a) is as follows: ZnS/Cleartran (black), ZnSe (red), GaAs (green), InP (blue), Si (orange), Ge (magenta), and GaSb (dark yellow). Fresnel reflections are included in the spectra.

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3.1 Semiconductors

The measured ${n_2}$ values can qualitatively and quantitatively be compared to a theoretical model based on a nonlinear Kramers-Krönig transformation [13]. The formulism incorporates contributions from the AC Stark effect, quadratic Stark effect, Raman, and 2PA among which is based off a well-established two-band parabolic model [14]. Thus for all theoretical calculations presented in this section, the parameters $K = 3100, K^{\prime} = 1.5 \times {10^{ - 8}}, \textrm{and} \,{E_p} = 21.4 \textrm{eV}$ are used. These parameters correspond to nearly material-independent constants for most direct-gap semiconductors and are explicitly defined in [13].

Figure 3(a) shows CA Z-scans of a 3 mm thick ZnS (Cleartran) window measured at 3.9 $\upmu \textrm{m}$ where the open symbols represent the raw Z-scan signal normalized to unity and the solid lines represent the fit for each scan. Each Z-scan trace is fit separately and the average fit from the scans is taken as the value of ${n_2}$. For 3.9 $\upmu \textrm{m}$, the value is found to be ${n_2} = ({5.0 \pm 1.3} )\times {10^{ - 15}}\textrm{c}{\textrm{m}^2}/\textrm{W}$. Similar analysis for the Z-scans taken at $2 \,\upmu \textrm{m}$ yield, ${n_2} = ({5.5 \pm 1.1} )\times {10^{ - 15}}\textrm{c}{\textrm{m}^2}/\textrm{W}$.

 figure: Fig. 3.

Fig. 3. (a) CA Z-scans of ZnS/Cleartran at 3.9 $\upmu \textrm{m}$. The energies of 46 nJ, 94 nJ, 190 nJ, and 382 nJ correspond to peak irradiances of 3.9, 8.0, 16, and 32 GW/cm2, respectively. (b) Theoretical prediction (solid black line) of the ${n_2}$ dispersion of ZnS plotted with experimental data. The experimental data is taken from Ref. [15] for the black circles, Ref. [16] for the green top-pointing triangles, Ref. [13] for the cyan bottom-pointing triangle, Ref. [17] for the blue diamonds, and Ref. [18] for the dark blue left-pointing triangle. The theoretical curve is calculated from Ref. [13]. The red squares are measurements from this work.

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Figure 3(b) shows various experimental ${n_2}$ measurements of ZnS along with the theoretical prediction outlined in [13]. The measured values from this work (solid red squares) are in good agreement with the theoretical model using a band gap of 3.68 eV [19] and the dispersion relation given in [20]. Additionally, this model is shown to have reasonable quantitative and qualitative agreement with previous experimental values in the visible through the MWIR regime confirming the validity of the model for ZnS at least to 4 $\upmu \textrm{m}$.

Figure 4 shows the theoretical ${n_2}$ dispersion curves (solid lines) of ZnSe, GaAs, InP, and GaSb along with experimental values from this work at $2 \,\upmu \textrm{m}$ and 3.9 $\upmu \textrm{m}$ (solid squares). The theoretical curves are calculated using band gap energies of 2.71 eV, 1.42 eV, 1.35 eV, and 0.72 eV for ZnSe, GaAs, InP, and GaSb, respectively [19]. Each experimental data point (solid squares) in the figure represents an average fit of scans performed at multiple energies. Note that the ${n_2}$ value of $({1.2 \pm 0.3} )\times {10^{ - 14}}\textrm{c}{\textrm{m}^2}/\textrm{W}$ for ZnSe at 3.9 $\upmu \textrm{m}$ is a previously published result [24]. There was no observable NLA in the Z-scans of ZnSe, GaAs, and InP at $2 \,\upmu \textrm{m}$ and 3.9 $\upmu \textrm{m}$; therefore, only the ${n_2}$ was used as a fitting parameter for the CA Z-scans.

 figure: Fig. 4.

Fig. 4. Log-linear plot of the theoretical ${n_2}$ dispersion of ZnSe (solid black line), GaAs (solid red line), InP (solid blue line), and GaSb (solid green line) calculated from Ref. [13]. Experimental results obtained via Z-scans at $2 \,\upmu \textrm{m}$ and 3.9 $\upmu \textrm{m}$ from this work are shown as solid squares for each respective sample. The open red circles are experimental data from Ref. [21] and open red triangles from Ref. [22] of GaAs and open black circles from Ref. [23] of ZnSe. The data point at 3.9 $\upmu \textrm{m}$ of ZnSe is a previously published result from Ref. [24]. The inset shows a linear plot of the ${n_2}$ dispersion of GaSb along with the experimental data point at $2 \,\upmu \textrm{m}$ with the horizontal axis expanded having the same horizontal and vertical axes titles as the main figure. The dashed line in the inset represents ${n_2} = 0$.

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The CA Z-scans at $2 \,\upmu \textrm{m}$ of GaSb contained both 2PA and NLR. Furthermore, the linear transmittance from Fig. 2(a) along with ${n_0} = 3.78$ at $2 \,\upmu \textrm{m}$ and ${n_0} = 3.72$ at 3.9 $\upmu \textrm{m}$ [25] yields $\alpha _{0,\textrm{GaSb}}^{2\upmu\textrm{m}} = 0.22 \textrm{c}{\textrm{m}^{ - 1}}$ and $\alpha _{0,\textrm{GaSb}}^{3.9\upmu\textrm{m}} = 9.3 \textrm{c}{\textrm{m}^{ - 1}}$, i.e. an increase in absorption at longer wavelengths which was the study of a previous work [26]. Thus, to obtain ${n_2}$, the simultaneously obtained OA Z-scans were fit with ${\alpha _{2,\textrm{GaSb}}} = ({110 \pm 22} ) \textrm{cm}/\textrm{GW}$ using $\alpha _{0,\textrm{GaSb}}^{2\upmu\textrm{m}}$, a factor of ${\sim} 1.7$ larger than the value reported in [27] at 2.05 $\upmu \textrm{m}$. As stated before, there was no observed dependence on ${\alpha _2}$ at $2 \,\upmu \textrm{m}$ with irradiance which indicated that the NLA was, indeed, 2PA even in the presence of relatively large absorption. This allowed a one parameter fit for the CA Z-scans yielding ${n_2} ={-} ({3.5 \pm 0.7} )\times {10^{ - 12}}\textrm{c}{\textrm{m}^\textrm{2}}/\textrm{W}$ at $2 \,\upmu \textrm{m}$ as shown in the inset of Fig. 4. There was no NLA observed for GaSb at 3.9 $\upmu \textrm{m}$. Therefore, using $\alpha _{0,\textrm{GaSb}}^{3.9\upmu\textrm{m}} = 9.3 \textrm{c}{\textrm{m}^{ - 1}}$ yields ${n_2} = ({5.5 \pm 1.4} )\times {10^{ - 12}}\textrm{c}{\textrm{m}^\textrm{2}}/\textrm{W}$ at 3.9 $\upmu \textrm{m}$.

The experimental values agree with the theoretical prediction both quantitatively (within reasonable error) and qualitatively most notably with the magnitude of ${n_2}$ increasing with narrower direct band gap semiconductors. The ${n_2}$ dispersion is shown to be relatively flat from wavelengths corresponding to roughly 0.5 of the band gap to longer wavelengths. Also shown in Fig. 4 are values of ${n_2}$ for GaAs (open red circles) in the MWIR extracted from Ref. [21] and ZnSe (open black circles) from Ref. [23]. These values are in good agreement with the theoretical prediction as well as the measurements from this work.

Table 1 summarizes the ${n_2}$ coefficients of the semiconductors previously depicted along with measurements of Si and Ge. Both Si and Ge exhibit 2PA and NLR at $2 \,\upmu \textrm{m}$, but there is no NLA observed for either at 3.9 $\upmu \textrm{m}$ with the irradiance levels used. Thus, to obtain ${n_2}$ for Si at $2 \,\upmu \textrm{m}$, the OA Z-scans were fit with ${\alpha _{2,\textrm{Si}}} = ({0.85 \pm 0.17} ) \textrm{cm}/\textrm{GW}$ yielding ${n_2} = ({7.7 \pm 1.5} )\times {10^{ - 14}}\textrm{c}{\textrm{m}^\textrm{2}}/\textrm{W}$ when fitting the CA Z-scans. Similarly for Ge, the OA Z-scans were fit with ${\alpha _{2,\textrm{Ge}}} = ({85 \pm 17} ) \textrm{cm}/\textrm{GW}$ (in agreement with the value of $71\textrm{cm}/\textrm{GW}$ reported in [27] at 2.05 $\upmu \textrm{m}$) yielding ${n_2} ={-} ({3.5 \pm 0.7} )\times {10^{ - 13}}\textrm{c}{\textrm{m}^\textrm{2}}/\textrm{W}$ at $2 \,\upmu \textrm{m}$ when fitting the CA Z-scans.

Tables Icon

Table 1. ${n_2}$ coefficients of the studied semiconductors. Note that the value of ZnSe at 3.9 $\upmu\textrm{m}$ is a previously published value from Ref. [24]. The direct and indirect band gap energies are taken from Ref. [19]. The error associated with the measurements at $2 \,\upmu\textrm{m}$ and $3.9 \,\upmu\textrm{m}$ are ${\pm} $ 20% and ${\pm} 25$%, respectively.

The 2PA and NLR coefficients of Si at $2 \,\upmu \textrm{m}$ reported herein are roughly a factor of 2 larger and smaller than the values reported in [28] of ${\alpha _2} \approx 0.4\textrm{cm}/\textrm{GW}$ and ${n_2} \approx 1.1 \times {10^{ - 13}}\textrm{c}{\textrm{m}^\textrm{2}}/\textrm{W}$ at the same wavelength. Additionally, the values of Si reported in [29] are smaller than those reported herein by a factor of ${\sim} 6$ and ${\sim} 4$ for ${\alpha _2}$ and ${n_2}$, respectively, at $2 \,\upmu \textrm{m}$. Gai et al. reported ${n_2}$ values of Si in the MWIR from 3 to 5 $\upmu \textrm{m}$ with an average of $({2.7 \pm 0.5} )\times {10^{ - 14}}\textrm{c}{\textrm{m}^\textrm{2}}/\textrm{W}$ [30] which is 4 times smaller than the value reported in this work at 3.9 $\upmu \textrm{m}$. Finally, we find that NLR coefficients for Si increase slightly with wavelength from 2 to 3.9 $\upmu \textrm{m}$, contrary to these previous reports but observed on a much smaller scale in [31].

Sohn et al. [32] reported the ${n_2}$ dispersion of Ge in the $2 \,\upmu \textrm{m}$ to 5 $\upmu \textrm{m}$ range with a value of $|{{n_2}} |= 4.0 \times {10^{ - 14}}\textrm{c}{\textrm{m}^\textrm{2}}/\textrm{W}$ at $2 \,\upmu \textrm{m}$ and $|{{n_2}} |= 5 \times {10^{ - 14}}\textrm{c}{\textrm{m}^\textrm{2}}/\textrm{W}$ at 4 $\upmu \textrm{m}$. Assuming their value of ${n_2}$ is indeed negative at $2 \,\upmu \textrm{m}$, it is smaller by a factor of ${\sim} 9$. Furthermore, their value at 4 $\upmu \textrm{m}$ is over 50 times smaller than the value reported herein at 3.9 $\upmu \textrm{m}$. Depatie et al. reported a value for Ge of ${\chi ^{(3 )}} = 4 \times {10^{ - 11}}$ esu $({{n_2} = 9.9 \times {{10}^{ - 14}}\textrm{c}{\textrm{m}^\textrm{2}}/\textrm{W}} )$ [33] at 3.8 $\upmu \textrm{m}$ which is over 20 times smaller than the value reported herein at 3.9 $\upmu \textrm{m}$. From Table 1, the ${n_2}$ of Ge and GaSb at 3.9 $\upmu \textrm{m}$ is experimentally determined to be within a factor of ${\sim} 2$ which is in reasonable agreement considering the direct band gap of Ge is 0.14 eV larger than its indirect band gap [34] and should scale similar to the established theory as presented in Fig. 4.

3.2 Fluoride crystals

Figure 5 shows Z-scan measurements of (a) CaF2 and (b) MgF2 at a wavelength of $2 \,\upmu \textrm{m}$. As was done in the analysis of the semiconductors, each energy is separately fit to obtain a value of ${n_2}$ and the average is taken to be the stated value. Thus, the value of ${n_2}$ for CaF2 is found to be $({1.7 \pm 0.34} )\times {10^{ - 16}}\textrm{c}{\textrm{m}^2}/\textrm{W}$ and $({1.1 \pm 0.22} )\times {10^{ - 16}}\textrm{c}{\textrm{m}^2}/\textrm{W}$ for MgF2 at $2 \,\upmu \textrm{m}$.

 figure: Fig. 5.

Fig. 5. CA Z-scans of (a) CaF2 and (b) MgF2 at $2 \,\upmu \textrm{m}$. The input energies of 383 nJ, 749 nJ, and 1.19 $\upmu$J correspond to incident peak irradiances of 260 GW/cm2, 520 GW/cm2, and 820 GW/cm2, respectively. The open symbols represent the data and the solid lines represent fits to the data.

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Table 2 lists the ${n_2}$ values of the measured fluoride crystals as well as fused silica and BK7 at 800 nm and $2 \,\upmu \textrm{m}$. Accurate values of ${n_2}$ could not be determined at 3.9 $\upmu \textrm{m}$ due to insufficient signal-to-noise. Also listed in Table 2 are calculations of ${n_2}$ based upon an empirical formula given in [35]:

$${n_2}({\textrm{esu}} )= \frac{{68({{n_d} - 1} ){{({n_d^2 + 2} )}^2}}}{{{\nu _d}{{\left( {1.517 + \frac{{({n_d^2 + 2} )({{n_d} + 1} )}}{{6{n_d}}}{\nu_d}} \right)}^{1/2}}}}{10^{ - 13}}$$
where ${n_d}$ is the index of refraction at the Fraunhofer d-line (587.56 nm) and ${\nu _d}({ = ({{n_d} - 1} )/({{n_F} - {n_c}} )} )$ is the Abbé number where the subscripts F and c are the wavelengths at 486.13 nm and 656.27 nm, respectively. To convert to mks units, the following relation is used [36]: ${n_2}({\textrm{mks}} )= {n_2}({\textrm{esu}} )({40\pi } )/({c{n_0}} )$ where c is the speed of light.

Tables Icon

Table 2. ${n_2}$ values of the studied fluoride crystals and glasses at 800 nm and $2 \,\upmu\textrm{m}$ along with previously published literature values at 532 nm and 1.06 $\upmu\textrm{m}$ of the fluoride crystals and glasses. The error associated with the measurements at 800 nm and $2 \,\upmu\textrm{m}$ are ${\pm} $ 15% and ${\pm} 20$%, respectively. The literature values are as follows: aRef. [11], bRef. [37], cRef. [36], dRef. [38], eRef. [39], fRef. [40], gRef. [41], hRef. [18], iRef. [42], jRef. [43], kRef. [44], lRef. [45], and mRef. [46]. The (+), (#), and (*) denote measurements at 525 nm, 810 nm, and 1.03 $\upmu\textrm{m}$, respectively.

The values calculated from Eq. (1) and listed in Table 2 utilize the dispersion relations given in [47] for BaF2, CaF2, and MgF2, Ref. [48] for LiF, Ref. [49] for fused silica, and Ref. [50] for BK7. There have been a number of ${n_2}$ measurements of fluoride crystals and glasses over the past decades, but all have been relegated to the visible and near-infrared wavelength regime [11,18,3646]. These values are summarized in Table 2.

There is reasonable agreement between the ${n_2}$ values of the fluoride crystals spanning from measurements made in the visible and near-infrared regime to the MWIR measurements presented in this work. This proves that there is minimal NLR dispersion and, thus, ${n_2}$ is expected to be relatively constant from the visible to the MWIR for fluoride crystals. Additionally, the ${n_2}$ value of fused silica at $2 \,\upmu \textrm{m}$ is in good agreement with the established literature precedent of $2.7 \times {10^{ - 16}}\textrm{c}{\textrm{m}^2}/\textrm{W}$ [43] and the ${n_2}$ value of BK7 at $2 \,\upmu\textrm{m}$ is in agreement with previous measurements of $3.6 \times {10^{ - 16}}\textrm{c}{\textrm{m}^2}/\textrm{W}$ at 810 nm and 1.06 $\upmu \textrm{m}$ [44,46]. This, again, proves the relatively flat ${n_2}$ dispersion from the near-infrared to the MWIR for the glasses similar to the fluoride crystals. Since ${n_2}$ for glasses has been shown to be relatively flat up to the edge of its transparency window (see Fig. 2(b)), this suggests that ${n_2}$ for the fluoride crystals should be relatively flat up to the edge of their respective transparency windows. That is, their respective ${n_2}$ values should remain approximately constant at least out to ${\sim} 5 \,\upmu \textrm{m}$ for LiF, ${\sim} 6 \,\upmu \textrm{m}$ for MgF2, ${\sim} 7.5 \,\upmu \textrm{m}$ for CaF2, and ${\sim} 9.5 \,\upmu \textrm{m}$ for BaF2.

3.3 Chalcogenide glasses

Figure 6 shows Z-scan measurements of (a) AMTIR-1 and (b) AMTIR-5 at a wavelength of 3.9 $\upmu \textrm{m}$. As was done in the analysis presented in Sections 3.1 and 3.2, each energy is fit to obtain a value of ${n_2}$ and the average is taken to be the value. Note, again, that there was no observed dependence on ${n_2}$ with irradiance. Thus, the value of ${n_2}$ for AMTIR-1 is found to be $({7.5 \pm 1.9} )\times {10^{ - 14}}\textrm{c}{\textrm{m}^2}/\textrm{W}$ and $({1.6 \pm 0.4} )\times {10^{ - 13}}\textrm{c}{\textrm{m}^2}/\textrm{W}$ for AMTIR-5 at 3.9 $\upmu \textrm{m}$.

 figure: Fig. 6.

Fig. 6. CA Z-scans of (a) AMTIR-1 and (b) AMTIR-5 at 3.9 $\upmu \textrm{m}$. The input energies of 10 nJ, 20 nJ, and 40 nJ correspond to incident peak irradiances of 0.98 GW/cm2, 2.0 GW/cm2, and 3.9 GW/cm2, respectively. The open symbols represent the data and the solid lines represent fits to the data.

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There have been multiple reports aiming to characterize the ${n_2}$ values of various types of chalcogenide glasses in the past few years in the near-infrared and MWIR [4,5153]. For instance, Wang et al. reported ${n_2}$ values at 1.55 $\upmu \textrm{m}$ ranging from $2.0 \times {10^{ - 14}}\textrm{c}{\textrm{m}^2}/\textrm{W}$ to $1.5 \times {10^{ - 13}}\textrm{c}{\textrm{m}^2}/\textrm{W}$ for chalcogenide architectures of the type Ge-As-Se, Ge-Sb-Se, Ge-As-S, Ge-As-Se, and Ge-As-S-Se [4] whereas Lin et al. reported a range of $1.95 \times {10^{ - 14}}\textrm{c}{\textrm{m}^2}/\textrm{W}$ to $1.0 \times {10^{ - 13}}\textrm{c}{\textrm{m}^2}/\textrm{W}$ at 4 $\upmu \textrm{m}$ for Sn-Sb-Se and Ge-Sb-Se architectures [51].

Table 3 summarizes the ${n_2}$ values at $2 \,\upmu \textrm{m}$ and 3.9 $\upmu \textrm{m}$ for the chalcogenide glasses of this study. Of note, AMTIR-2 and AMTIR-5 have very similar ${n_2}$ values at both $2 \,\upmu \textrm{m}$ and 3.9 $\upmu \textrm{m}$ in which their linear transmittance is nearly identical (see Fig. 2(c)). From Table 3, it is observed that for all chalcogenide glasses studied, the ${n_2}$ increases with wavelength. Similar to the fluoride crystals, there have been empirical relations derived to predict the ${n_2}$ of chalcogenide glasses. One such relation is [54]:

$${n_2}({\textrm{c}{\textrm{m}^2}/\textrm{W}} )= \frac{{3.4{{({n_0^2 + 2} )}^3}({n_0^2 - 1} ){d^2}}}{{n_0^2E_s^2}}{10^{ - 16}} $$
where d (in Angstroms) is the mean cation-anion bond length and ${E_s}$ is the Sellmeier gap (in eV). Specifically for As2S3, the parameters have been experimentally determined and found to be ${E_s} = 5.3 \textrm{eV}$ and $d = 0.228 \textrm{nm}$ [54,55]. Substituting those values in Eq. (2) and using the value of ${n_0} = 2.43$ at $2 \,\upmu \textrm{m}$ from [56], we obtain ${n_2} = 2.5 \times {10^{ - 14}}\textrm{c}{\textrm{m}^2}/\textrm{W}$ which is in excellent agreement with the experimental value obtained at $2 \,\upmu \textrm{m}$ (see Table 3). Although the calculated ${n_2}$ value matches the experimental value at $2 \,\upmu \textrm{m}$, the calculated ${n_2}$ value at 3.9 $\upmu \textrm{m}$ decreases to $2.4 \times {10^{ - 14}}\textrm{c}{\textrm{m}^2}/\textrm{W}$ when using ${n_0} = 2.41$, which does not match the experimentally determined value of $({8.3 \pm 2} )\times {10^{ - 14}}\textrm{c}{\textrm{m}^2}/\textrm{W}$. In [57], a nonlinear dispersion relation having a similar form to that of [13] was introduced to predict ${n_2}$ for Se-based chalcogenide glasses at longer wavelengths. But just as Eq. (2), the formalism inherently describes a decreasing value of ${n_2}$ as ${\lambda _0} \to \infty $, counter to what has been experimentally measured in this work, i.e. the ${n_2}$ value is larger at 3.9 $\upmu \textrm{m}$ than at $2 \,\upmu \textrm{m}$. Furthermore, a Miller’s rule type relation also fails when describing the ${n_2}$ of these glasses since the rule predicts ${n_2} \propto {({n_0^2 - 1} )^4}/n_0^2$ [58]. Thus, a more inclusive model is needed to predict the nonlinearities of chalcogenide glasses in the MWIR.

Tables Icon

Table 3. ${n_2}$ coefficients of the selected chalcogenide glasses. The error associated with the measurements at $2 \,\upmu\textrm{m}$ and $3.9 \,\upmu\textrm{m}$ are ${\pm} $ 20% and ${\pm} 25$%, respectively.

4. Conclusions

We have presented ${n_2}$ measurements of wide band gap and narrow band gap semiconductors, chalcogenide glasses, and optical glasses and fluoride crystals in the mid-wave infrared at $2 \,\upmu \textrm{m}$ and 3.9 $\upmu \textrm{m}$ using the conventional Z-scan technique. The ${n_2}$ measurements of the direct gap semiconductors were in agreement qualitatively and quantitatively with theoretical predictions based upon a Kramers-Krönig transformation. The ${n_2}$ value of Si was observed to slightly increase from $2 \,\upmu \textrm{m}$ to 3.9 $\upmu \textrm{m}$. The ${n_2}$ dispersion of fluoride crystals was found to be relatively flat from the visible to $2 \,\upmu \textrm{m}$, indicating that the ${n_2}$ should also be relatively flat extending to the edge of their respective transparency window in the MWIR. The ${n_2}$ values of the chalcogenide glasses were also observed to increase with wavelength.

Disclosures

The authors declare no conflicts of interest.

References

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References

  • View by:

  1. A. E. Willner, S. Khaleghi, M. R. Chitgarha, and O. F. Yilmaz, “All-Optical Signal Processing,” J. Lightwave Technol. 32(4), 660–680 (2014).
    [Crossref]
  2. E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, “Roadmap of optical communications,” J. Opt. 18(6), 063002 (2016).
    [Crossref]
  3. M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett. 82(18), 2954–2956 (2003).
    [Crossref]
  4. T. Wang, X. Gai, W. Wei, R. Wang, Z. Yang, X. Shen, S. Madden, and B. Luther-Davies, “Systematic z-scan measurements of the third order nonlinearity of chalcogenide glasses,” Opt. Mater. Express 4(5), 1011–1022 (2014).
    [Crossref]
  5. J. Toulouse, “Optical Nonlinearities in Fibers: Review, Recent Examples, and Systems Applications,” J. Lightwave Technol. 23(11), 3625–3641 (2005).
    [Crossref]
  6. R. I. Woodward, M. R. Majewski, N. Macadam, G. Hu, T. Albrow-Owen, T. Hasan, and S. D. Jackson, “Q-switched Dy:ZBLAN fiber lasers beyond 3 μm: comparison of pulse generation using acousto-optic modulation and inkjet-printed black phosphorus,” Opt. Express 27(10), 15032–15045 (2019).
    [Crossref]
  7. M. Seidel, X. Xiao, S. A. Hussain, G. Arisholm, A. Hartung, K. T. Zawilski, P. G. Schunemann, F. Habel, M. Trubetskov, V. Pervak, O. Pronin, and F. Krausz, “Multi-watt, multi-octave, mid-infrared femtosecond source,” Sci. Adv. 4(4), eaaq1526 (2018).
    [Crossref]
  8. E. A. Anashkina, A. V. Andrianov, V. V. Dorofeev, and A. V. Kim, “Toward a mid-infrared femtosecond laser system with suspended-core tungstate–tellurite glass fibers,” Appl. Opt. 55(17), 4522–4530 (2016).
    [Crossref]
  9. E. Sorokin, N. Tolstik, I. T. E. D. H. G. Sorokina, and P. Moulton, “Mid-Infrared Femtosecond Lasers,” in Advanced Solid-State Lasers Congress, OSA Technical Digest (online) (Optical Society of America, 2013), AF1A.3.
  10. J. S. Sanghera, L. B. Shaw, and I. D. Aggarwal, “Chalcogenide Glass-Fiber-Based Mid-IR Sources and Applications,” IEEE J. Sel. Top. Quantum Electron. 15(1), 114–119 (2009).
    [Crossref]
  11. M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
    [Crossref]
  12. H. Ma, A. S. L. Gomes, and C. B. d. Araujo, “Measurements of nondegenerate optical nonlinearity using a two-color single beam method,” Appl. Phys. Lett. 59(21), 2666–2668 (1991).
    [Crossref]
  13. M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, “Dispersion of bound electron nonlinear refraction in solids,” IEEE J. Quantum Electron. 27(6), 1296–1309 (1991).
    [Crossref]
  14. E. W. Van Stryland, M. Woodall, H. Vanherzeele, and M. Soileau, “Energy band-gap dependence of two-photon absorption,” Opt. Lett. 10(10), 490–492 (1985).
    [Crossref]
  15. S. R. Flom, G. Beadie, S. S. Bayya, B. Shaw, and J. M. Auxier, “Ultrafast Z-scan measurements of nonlinear optical constants of window materials at 772, 1030, and 1550 nm,” Appl. Opt. 54(31), F123–F128 (2015).
    [Crossref]
  16. T. D. Krauss and F. W. Wise, “Femtosecond measurement of nonlinear absorption and refraction in CdS, ZnSe, and ZnS,” Appl. Phys. Lett. 65(14), 1739–1741 (1994).
    [Crossref]
  17. J. He, Y. Qu, H. Li, J. Mi, and W. Ji, “Three-photon absorption in ZnO and ZnS crystals,” Opt. Express 13(23), 9235–9247 (2005).
    [Crossref]
  18. R. Adair, L. L. Chase, and S. A. Payne, “Nonlinear refractive index of optical crystals,” Phys. Rev. B 39(5), 3337–3350 (1989).
    [Crossref]
  19. S. M. Sze, Physics of Semiconductor Devices (John Wiley & Sons, 1981).
  20. M. Debenham, “Refractive indices of zinc sulfide in the 0.405–13-µm wavelength range,” Appl. Opt. 23(14), 2238–2239 (1984).
    [Crossref]
  21. W. C. Hurlbut, Y.-S. Lee, K. L. Vodopyanov, P. S. Kuo, and M. M. Fejer, “Multiphoton absorption and nonlinear refraction of GaAs in the mid-infrared,” Opt. Lett. 32(6), 668–670 (2007).
    [Crossref]
  22. J. M. Hales, S.-H. Chi, T. Allen, S. Benis, N. Munera, J. W. Perry, D. McMorrow, D. J. Hagan, and E. W. Van Stryland, “Third-Order Nonlinear Optical Coefficients of Si and GaAs in the Near-Infrared Spectral Region,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2018), JTu2A.59.
  23. M. R. Ferdinandus, J. Gengler, M. Tripepi, and C. Liebig, “Measurements of Optical Nonlinearities at Mid-IR Wavelengths Using a Modified Z-Scan Technique,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (Optical Society of America, 2019), SF2G.4.
  24. K. Werner, M. G. Hastings, A. Schweinsberg, B. L. Wilmer, D. Austin, C. M. Wolfe, M. Kolesik, T. R. Ensley, L. Vanderhoef, A. Valenzuela, and E. Chowdhury, “Ultrafast mid-infrared high harmonic and supercontinuum generation with n2 characterization in zinc selenide,” Opt. Express 27(3), 2867–2885 (2019).
    [Crossref]
  25. S. Adachi, “Optical dispersion relations for GaP, GaAs, GaSb, InP, InAs, InSb, AlxGa1−xAs, and In1−xGaxAsyP1−y,” J. Appl. Phys. 66(12), 6030–6040 (1989).
    [Crossref]
  26. A. Chandola, R. Pino, and P. S. Dutta, “Below bandgap optical absorption in tellurium-doped GaSb,” Semicond. Sci. Technol. 20(8), 886–893 (2005).
    [Crossref]
  27. T. J. Wagner, M. J. Bohn, J. R. A. Coutu, L. P. Gonzalez, J. M. Murray, K. L. Schepler, and S. Guha, “Measurement and modeling of infrared nonlinear absorption coefficients and laser-induced damage thresholds in Ge and GaSb,” J. Opt. Soc. Am. B 27(10), 2122–2131 (2010).
    [Crossref]
  28. A. D. Bristow, N. Rotenberg, and H. M. v. Driel, “Two-photon absorption and Kerr coefficients of silicon for 850–2200 nm,” Appl. Phys. Lett. 90(19), 191104 (2007).
    [Crossref]
  29. Q. Lin, J. Zhang, G. Piredda, R. W. Boyd, P. M. Fauchet, and G. P. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett. 91(2), 021111 (2007).
    [Crossref]
  30. X. Gai, Y. Yu, B. Kuyken, P. Ma, S. J. Madden, J. Van Campenhout, P. Verheyen, G. Roelkens, R. Baets, and B. Luther-Davies, “Nonlinear absorption and refraction in crystalline silicon in the mid-infrared,” Laser Photonics Rev. 7(6), 1054–1064 (2013).
    [Crossref]
  31. T. Wang, N. Venkatram, J. Gosciniak, Y. Cui, G. Qian, W. Ji, and D. T. H. Tan, “Multi-photon absorption and third-order nonlinearity in silicon at mid-infrared wavelengths,” Opt. Express 21(26), 32192–32198 (2013).
    [Crossref]
  32. B.-U. Sohn, C. Monmeyran, L. C. Kimerling, A. M. Agarwal, and D. T. H. Tan, “Kerr nonlinearity and multi-photon absorption in germanium at mid-infrared wavelengths,” Appl. Phys. Lett. 111(9), 091902 (2017).
    [Crossref]
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2019 (3)

2018 (1)

M. Seidel, X. Xiao, S. A. Hussain, G. Arisholm, A. Hartung, K. T. Zawilski, P. G. Schunemann, F. Habel, M. Trubetskov, V. Pervak, O. Pronin, and F. Krausz, “Multi-watt, multi-octave, mid-infrared femtosecond source,” Sci. Adv. 4(4), eaaq1526 (2018).
[Crossref]

2017 (2)

B.-U. Sohn, C. Monmeyran, L. C. Kimerling, A. M. Agarwal, and D. T. H. Tan, “Kerr nonlinearity and multi-photon absorption in germanium at mid-infrared wavelengths,” Appl. Phys. Lett. 111(9), 091902 (2017).
[Crossref]

R. Lin, F. Chen, X. Zhang, Y. Huang, B. Song, S. Dai, X. Zhang, and W. Ji, “Mid-infrared optical properties of chalcogenide glasses within tin-antimony-selenium ternary system,” Opt. Express 25(21), 25674–25688 (2017).
[Crossref]

2016 (3)

B. Qiao, F. Chen, Y. Huang, P. Zhang, S. Dai, and Q. Nie, “Investigation of mid-infrared optical nonlinearity of Ge20SnxSe80−x ternary chalcogenide glasses,” Mater. Lett. 162, 17–19 (2016).
[Crossref]

E. A. Anashkina, A. V. Andrianov, V. V. Dorofeev, and A. V. Kim, “Toward a mid-infrared femtosecond laser system with suspended-core tungstate–tellurite glass fibers,” Appl. Opt. 55(17), 4522–4530 (2016).
[Crossref]

E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, “Roadmap of optical communications,” J. Opt. 18(6), 063002 (2016).
[Crossref]

2015 (3)

2014 (3)

2013 (2)

X. Gai, Y. Yu, B. Kuyken, P. Ma, S. J. Madden, J. Van Campenhout, P. Verheyen, G. Roelkens, R. Baets, and B. Luther-Davies, “Nonlinear absorption and refraction in crystalline silicon in the mid-infrared,” Laser Photonics Rev. 7(6), 1054–1064 (2013).
[Crossref]

T. Wang, N. Venkatram, J. Gosciniak, Y. Cui, G. Qian, W. Ji, and D. T. H. Tan, “Multi-photon absorption and third-order nonlinearity in silicon at mid-infrared wavelengths,” Opt. Express 21(26), 32192–32198 (2013).
[Crossref]

2012 (1)

2010 (1)

2009 (1)

J. S. Sanghera, L. B. Shaw, and I. D. Aggarwal, “Chalcogenide Glass-Fiber-Based Mid-IR Sources and Applications,” IEEE J. Sel. Top. Quantum Electron. 15(1), 114–119 (2009).
[Crossref]

2007 (3)

W. C. Hurlbut, Y.-S. Lee, K. L. Vodopyanov, P. S. Kuo, and M. M. Fejer, “Multiphoton absorption and nonlinear refraction of GaAs in the mid-infrared,” Opt. Lett. 32(6), 668–670 (2007).
[Crossref]

A. D. Bristow, N. Rotenberg, and H. M. v. Driel, “Two-photon absorption and Kerr coefficients of silicon for 850–2200 nm,” Appl. Phys. Lett. 90(19), 191104 (2007).
[Crossref]

Q. Lin, J. Zhang, G. Piredda, R. W. Boyd, P. M. Fauchet, and G. P. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett. 91(2), 021111 (2007).
[Crossref]

2005 (3)

2003 (1)

M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett. 82(18), 2954–2956 (2003).
[Crossref]

2000 (1)

1998 (1)

1996 (1)

R. DeSalvo, A. A. Said, D. J. Hagan, E. W. Van Stryland, and M. Sheik-Bahae, “Infrared to ultraviolet measurements of two-photon absorption and n/sub 2/in wide bandgap solids,” IEEE J. Quantum Electron. 32(8), 1324–1333 (1996).
[Crossref]

1994 (1)

T. D. Krauss and F. W. Wise, “Femtosecond measurement of nonlinear absorption and refraction in CdS, ZnSe, and ZnS,” Appl. Phys. Lett. 65(14), 1739–1741 (1994).
[Crossref]

1993 (1)

1992 (1)

R. Adair, L. L. Chase, and S. A. Payne, “Dispersion of the nonlinear refractive index of optical crystals,” Opt. Mater. 1(3), 185–194 (1992).
[Crossref]

1991 (3)

M. E. Lines, “OXIDE GLASSES FOR FAST PHOTONIC SWITCHING - A COMPARATIVE-STUDY,” J. Appl. Phys. 69(10), 6876–6884 (1991).
[Crossref]

H. Ma, A. S. L. Gomes, and C. B. d. Araujo, “Measurements of nondegenerate optical nonlinearity using a two-color single beam method,” Appl. Phys. Lett. 59(21), 2666–2668 (1991).
[Crossref]

M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, “Dispersion of bound electron nonlinear refraction in solids,” IEEE J. Quantum Electron. 27(6), 1296–1309 (1991).
[Crossref]

1990 (1)

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[Crossref]

1989 (2)

R. Adair, L. L. Chase, and S. A. Payne, “Nonlinear refractive index of optical crystals,” Phys. Rev. B 39(5), 3337–3350 (1989).
[Crossref]

S. Adachi, “Optical dispersion relations for GaP, GaAs, GaSb, InP, InAs, InSb, AlxGa1−xAs, and In1−xGaxAsyP1−y,” J. Appl. Phys. 66(12), 6030–6040 (1989).
[Crossref]

1987 (1)

1985 (1)

1984 (2)

M. Debenham, “Refractive indices of zinc sulfide in the 0.405–13-µm wavelength range,” Appl. Opt. 23(14), 2238–2239 (1984).
[Crossref]

M. E. Lines, “Scattering losses in optic fiber materials. I. A new parametrization,” J. Appl. Phys. 55(11), 4052–4057 (1984).
[Crossref]

1980 (2)

H. H. Li, “Refractive index of alkaline earth halides and its wavelength and temperature derivatives,” J. Phys. Chem. Ref. Data 9(1), 161–290 (1980).
[Crossref]

D. Depatie and D. Haueisen, “Multiline phase conjugation at 4 µm in germanium,” Opt. Lett. 5(6), 252–254 (1980).
[Crossref]

1978 (1)

N. L. Boling, A. J. Glass, and A. Owyoung, “Empirical relationships for predicting non-linear refractive-index changes in optical solids,” IEEE J. Quantum Electron. 14(8), 601–608 (1978).
[Crossref]

1977 (1)

D. Milam, M. J. Weber, and A. J. Glass, “Nonlinear refractive index of fluoride crystals,” Appl. Phys. Lett. 31(12), 822–825 (1977).
[Crossref]

1976 (2)

H. H. Li, “Refractive index of alkali halides and its wavelength and temperature derivatives,” J. Phys. Chem. Ref. Data 5(2), 329–528 (1976).
[Crossref]

D. Milam and M. J. Weber, “Measurement of nonlinear refractive-index coefficients using time-resolved interferometry: Application to optical materials for high-power neodymium lasers,” J. Appl. Phys. 47(6), 2497–2501 (1976).
[Crossref]

1974 (1)

M. Levenson, “Feasibility of measuring the nonlinear index of refraction by third-order frequency mixing,” IEEE J. Quantum Electron. 10(2), 110–115 (1974).
[Crossref]

1965 (1)

Abouraddy, A. F.

Adachi, S.

S. Adachi, “Optical dispersion relations for GaP, GaAs, GaSb, InP, InAs, InSb, AlxGa1−xAs, and In1−xGaxAsyP1−y,” J. Appl. Phys. 66(12), 6030–6040 (1989).
[Crossref]

Adair, R.

R. Adair, L. L. Chase, and S. A. Payne, “Dispersion of the nonlinear refractive index of optical crystals,” Opt. Mater. 1(3), 185–194 (1992).
[Crossref]

R. Adair, L. L. Chase, and S. A. Payne, “Nonlinear refractive index of optical crystals,” Phys. Rev. B 39(5), 3337–3350 (1989).
[Crossref]

R. Adair, L. L. Chase, and S. A. Payne, “Nonlinear refractive-index measurements of glasses using three-wave frequency mixing,” J. Opt. Soc. Am. B 4(6), 875–881 (1987).
[Crossref]

Agarwal, A. M.

B.-U. Sohn, C. Monmeyran, L. C. Kimerling, A. M. Agarwal, and D. T. H. Tan, “Kerr nonlinearity and multi-photon absorption in germanium at mid-infrared wavelengths,” Appl. Phys. Lett. 111(9), 091902 (2017).
[Crossref]

Aggarwal, I. D.

J. S. Sanghera, L. B. Shaw, and I. D. Aggarwal, “Chalcogenide Glass-Fiber-Based Mid-IR Sources and Applications,” IEEE J. Sel. Top. Quantum Electron. 15(1), 114–119 (2009).
[Crossref]

G. Lenz, J. Zimmermann, T. Katsufuji, M. E. Lines, H. Y. Hwang, S. Spalter, R. E. Slusher, S. W. Cheong, J. S. Sanghera, and I. D. Aggarwal, “Large Kerr effect in bulk Se-based chalcogenide glasses,” Opt. Lett. 25(4), 254–256 (2000).
[Crossref]

Agrawal, G. P.

Q. Lin, J. Zhang, G. Piredda, R. W. Boyd, P. M. Fauchet, and G. P. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett. 91(2), 021111 (2007).
[Crossref]

Agrell, E.

E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, “Roadmap of optical communications,” J. Opt. 18(6), 063002 (2016).
[Crossref]

Albrow-Owen, T.

Allen, T.

J. M. Hales, S.-H. Chi, T. Allen, S. Benis, N. Munera, J. W. Perry, D. McMorrow, D. J. Hagan, and E. W. Van Stryland, “Third-Order Nonlinear Optical Coefficients of Si and GaAs in the Near-Infrared Spectral Region,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2018), JTu2A.59.

Anashkina, E. A.

Andrianov, A. V.

Arisholm, G.

M. Seidel, X. Xiao, S. A. Hussain, G. Arisholm, A. Hartung, K. T. Zawilski, P. G. Schunemann, F. Habel, M. Trubetskov, V. Pervak, O. Pronin, and F. Krausz, “Multi-watt, multi-octave, mid-infrared femtosecond source,” Sci. Adv. 4(4), eaaq1526 (2018).
[Crossref]

Austin, D.

Auxier, J. M.

Baets, R.

X. Gai, Y. Yu, B. Kuyken, P. Ma, S. J. Madden, J. Van Campenhout, P. Verheyen, G. Roelkens, R. Baets, and B. Luther-Davies, “Nonlinear absorption and refraction in crystalline silicon in the mid-infrared,” Laser Photonics Rev. 7(6), 1054–1064 (2013).
[Crossref]

Bayya, S. S.

Beadie, G.

Benis, S.

J. M. Hales, S.-H. Chi, T. Allen, S. Benis, N. Munera, J. W. Perry, D. McMorrow, D. J. Hagan, and E. W. Van Stryland, “Third-Order Nonlinear Optical Coefficients of Si and GaAs in the Near-Infrared Spectral Region,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2018), JTu2A.59.

Bohn, M. J.

Boling, N. L.

N. L. Boling, A. J. Glass, and A. Owyoung, “Empirical relationships for predicting non-linear refractive-index changes in optical solids,” IEEE J. Quantum Electron. 14(8), 601–608 (1978).
[Crossref]

Boucaud, P.

T. K. P. Luong, V. L. Thanh, A. Ghrib, M. E. Kurdi, and P. Boucaud, “Making germanium, an indirect band gap semiconductor, suitable for light-emitting devices,” Adv. Nat. Sci.: Nanosci. Nanotechnol. 6(1), 015013 (2015).
[Crossref]

Bowers, J. E.

E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, “Roadmap of optical communications,” J. Opt. 18(6), 063002 (2016).
[Crossref]

Boyd, R. W.

Q. Lin, J. Zhang, G. Piredda, R. W. Boyd, P. M. Fauchet, and G. P. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett. 91(2), 021111 (2007).
[Crossref]

R. W. Boyd, Nonlinear Optics, 3rd Edition (Academic Press, Inc., 2008), p. 640.

Brandt-Pearce, M.

E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, “Roadmap of optical communications,” J. Opt. 18(6), 063002 (2016).
[Crossref]

Bristow, A. D.

A. D. Bristow, N. Rotenberg, and H. M. v. Driel, “Two-photon absorption and Kerr coefficients of silicon for 850–2200 nm,” Appl. Phys. Lett. 90(19), 191104 (2007).
[Crossref]

Chandola, A.

A. Chandola, R. Pino, and P. S. Dutta, “Below bandgap optical absorption in tellurium-doped GaSb,” Semicond. Sci. Technol. 20(8), 886–893 (2005).
[Crossref]

Chase, L. L.

R. Adair, L. L. Chase, and S. A. Payne, “Dispersion of the nonlinear refractive index of optical crystals,” Opt. Mater. 1(3), 185–194 (1992).
[Crossref]

R. Adair, L. L. Chase, and S. A. Payne, “Nonlinear refractive index of optical crystals,” Phys. Rev. B 39(5), 3337–3350 (1989).
[Crossref]

R. Adair, L. L. Chase, and S. A. Payne, “Nonlinear refractive-index measurements of glasses using three-wave frequency mixing,” J. Opt. Soc. Am. B 4(6), 875–881 (1987).
[Crossref]

Chen, F.

R. Lin, F. Chen, X. Zhang, Y. Huang, B. Song, S. Dai, X. Zhang, and W. Ji, “Mid-infrared optical properties of chalcogenide glasses within tin-antimony-selenium ternary system,” Opt. Express 25(21), 25674–25688 (2017).
[Crossref]

B. Qiao, F. Chen, Y. Huang, P. Zhang, S. Dai, and Q. Nie, “Investigation of mid-infrared optical nonlinearity of Ge20SnxSe80−x ternary chalcogenide glasses,” Mater. Lett. 162, 17–19 (2016).
[Crossref]

Cheong, S. W.

Chi, S.-H.

J. M. Hales, S.-H. Chi, T. Allen, S. Benis, N. Munera, J. W. Perry, D. McMorrow, D. J. Hagan, and E. W. Van Stryland, “Third-Order Nonlinear Optical Coefficients of Si and GaAs in the Near-Infrared Spectral Region,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2018), JTu2A.59.

Chitgarha, M. R.

Chowdhury, E.

Chraplyvy, A. R.

E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, “Roadmap of optical communications,” J. Opt. 18(6), 063002 (2016).
[Crossref]

Coutu, J. R. A.

Cui, Y.

d. Araujo, C. B.

H. Ma, A. S. L. Gomes, and C. B. d. Araujo, “Measurements of nondegenerate optical nonlinearity using a two-color single beam method,” Appl. Phys. Lett. 59(21), 2666–2668 (1991).
[Crossref]

Dai, S.

R. Lin, F. Chen, X. Zhang, Y. Huang, B. Song, S. Dai, X. Zhang, and W. Ji, “Mid-infrared optical properties of chalcogenide glasses within tin-antimony-selenium ternary system,” Opt. Express 25(21), 25674–25688 (2017).
[Crossref]

B. Qiao, F. Chen, Y. Huang, P. Zhang, S. Dai, and Q. Nie, “Investigation of mid-infrared optical nonlinearity of Ge20SnxSe80−x ternary chalcogenide glasses,” Mater. Lett. 162, 17–19 (2016).
[Crossref]

Debenham, M.

Delfyett, P. J.

Depatie, D.

DeSalvo, R.

R. DeSalvo, A. A. Said, D. J. Hagan, E. W. Van Stryland, and M. Sheik-Bahae, “Infrared to ultraviolet measurements of two-photon absorption and n/sub 2/in wide bandgap solids,” IEEE J. Quantum Electron. 32(8), 1324–1333 (1996).
[Crossref]

R. DeSalvo, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E. W. Van Stryland, “Z-scan measurements of the anisotropy of nonlinear refraction and absorption in crystals,” Opt. Lett. 18(3), 194–196 (1993).
[Crossref]

Ding, B.

Ding, P.

Dinu, M.

M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett. 82(18), 2954–2956 (2003).
[Crossref]

Dorofeev, V. V.

Dutta, P. S.

A. Chandola, R. Pino, and P. S. Dutta, “Below bandgap optical absorption in tellurium-doped GaSb,” Semicond. Sci. Technol. 20(8), 886–893 (2005).
[Crossref]

Eggleton, B. J.

E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, “Roadmap of optical communications,” J. Opt. 18(6), 063002 (2016).
[Crossref]

Ensley, T. R.

Fauchet, P. M.

Q. Lin, J. Zhang, G. Piredda, R. W. Boyd, P. M. Fauchet, and G. P. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett. 91(2), 021111 (2007).
[Crossref]

Fejer, M. M.

Ferdinandus, M. R.

M. R. Ferdinandus, J. Gengler, M. Tripepi, and C. Liebig, “Measurements of Optical Nonlinearities at Mid-IR Wavelengths Using a Modified Z-Scan Technique,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (Optical Society of America, 2019), SF2G.4.

Fischer, J. K.

E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, “Roadmap of optical communications,” J. Opt. 18(6), 063002 (2016).
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Flom, S. R.

Gai, X.

T. Wang, X. Gai, W. Wei, R. Wang, Z. Yang, X. Shen, S. Madden, and B. Luther-Davies, “Systematic z-scan measurements of the third order nonlinearity of chalcogenide glasses,” Opt. Mater. Express 4(5), 1011–1022 (2014).
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X. Gai, Y. Yu, B. Kuyken, P. Ma, S. J. Madden, J. Van Campenhout, P. Verheyen, G. Roelkens, R. Baets, and B. Luther-Davies, “Nonlinear absorption and refraction in crystalline silicon in the mid-infrared,” Laser Photonics Rev. 7(6), 1054–1064 (2013).
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Figures (6)

Fig. 1.
Fig. 1. Schematic of Z-scan experiment. “CM” stands for concave mirror, “SF” for spatial filter, “50/50” for a 50/50 beamsplitter, “Ir” for iris diaphragm used as the aperture for the closed aperture (CA) detector, “OA” for open-aperture detector, and “Ref” for the reference detector.
Fig. 2.
Fig. 2. Linear transmission from 300 nm to 20 $\upmu \textrm{m}$ for the measured (a) semiconductors, (b) glasses and fluoride crystals, and (c) chalcogenide glasses. The legend in (a) is as follows: ZnS/Cleartran (black), ZnSe (red), GaAs (green), InP (blue), Si (orange), Ge (magenta), and GaSb (dark yellow). Fresnel reflections are included in the spectra.
Fig. 3.
Fig. 3. (a) CA Z-scans of ZnS/Cleartran at 3.9 $\upmu \textrm{m}$. The energies of 46 nJ, 94 nJ, 190 nJ, and 382 nJ correspond to peak irradiances of 3.9, 8.0, 16, and 32 GW/cm2, respectively. (b) Theoretical prediction (solid black line) of the ${n_2}$ dispersion of ZnS plotted with experimental data. The experimental data is taken from Ref. [15] for the black circles, Ref. [16] for the green top-pointing triangles, Ref. [13] for the cyan bottom-pointing triangle, Ref. [17] for the blue diamonds, and Ref. [18] for the dark blue left-pointing triangle. The theoretical curve is calculated from Ref. [13]. The red squares are measurements from this work.
Fig. 4.
Fig. 4. Log-linear plot of the theoretical ${n_2}$ dispersion of ZnSe (solid black line), GaAs (solid red line), InP (solid blue line), and GaSb (solid green line) calculated from Ref. [13]. Experimental results obtained via Z-scans at $2 \,\upmu \textrm{m}$ and 3.9 $\upmu \textrm{m}$ from this work are shown as solid squares for each respective sample. The open red circles are experimental data from Ref. [21] and open red triangles from Ref. [22] of GaAs and open black circles from Ref. [23] of ZnSe. The data point at 3.9 $\upmu \textrm{m}$ of ZnSe is a previously published result from Ref. [24]. The inset shows a linear plot of the ${n_2}$ dispersion of GaSb along with the experimental data point at $2 \,\upmu \textrm{m}$ with the horizontal axis expanded having the same horizontal and vertical axes titles as the main figure. The dashed line in the inset represents ${n_2} = 0$.
Fig. 5.
Fig. 5. CA Z-scans of (a) CaF2 and (b) MgF2 at $2 \,\upmu \textrm{m}$. The input energies of 383 nJ, 749 nJ, and 1.19 $\upmu$J correspond to incident peak irradiances of 260 GW/cm2, 520 GW/cm2, and 820 GW/cm2, respectively. The open symbols represent the data and the solid lines represent fits to the data.
Fig. 6.
Fig. 6. CA Z-scans of (a) AMTIR-1 and (b) AMTIR-5 at 3.9 $\upmu \textrm{m}$. The input energies of 10 nJ, 20 nJ, and 40 nJ correspond to incident peak irradiances of 0.98 GW/cm2, 2.0 GW/cm2, and 3.9 GW/cm2, respectively. The open symbols represent the data and the solid lines represent fits to the data.

Tables (3)

Tables Icon

Table 1. n 2 coefficients of the studied semiconductors. Note that the value of ZnSe at 3.9 µ m is a previously published value from Ref. [24]. The direct and indirect band gap energies are taken from Ref. [19]. The error associated with the measurements at 2 µ m and 3.9 µ m are ± 20% and ± 25 %, respectively.

Tables Icon

Table 2. n 2 values of the studied fluoride crystals and glasses at 800 nm and 2 µ m along with previously published literature values at 532 nm and 1.06 µ m of the fluoride crystals and glasses. The error associated with the measurements at 800 nm and 2 µ m are ± 15% and ± 20 %, respectively. The literature values are as follows: aRef. [11], bRef. [37], cRef. [36], dRef. [38], eRef. [39], fRef. [40], gRef. [41], hRef. [18], iRef. [42], jRef. [43], kRef. [44], lRef. [45], and mRef. [46]. The (+), (#), and (*) denote measurements at 525 nm, 810 nm, and 1.03 µ m , respectively.

Tables Icon

Table 3. n 2 coefficients of the selected chalcogenide glasses. The error associated with the measurements at 2 µ m and 3.9 µ m are ± 20% and ± 25 %, respectively.

Equations (2)

Equations on this page are rendered with MathJax. Learn more.

n 2 ( esu ) = 68 ( n d 1 ) ( n d 2 + 2 ) 2 ν d ( 1.517 + ( n d 2 + 2 ) ( n d + 1 ) 6 n d ν d ) 1 / 2 10 13
n 2 ( c m 2 / W ) = 3.4 ( n 0 2 + 2 ) 3 ( n 0 2 1 ) d 2 n 0 2 E s 2 10 16

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