Spectral and temperature dependence of two-photon and free-carrier absorption in InSb

The nonlinear absorption spectrum of InSb was measured using a combination of tunable $\sim 160\text{fs}$, $\sim 10\text{ps}$, and $\sim 150\text{ns}$ IR sources along with a cryostat for controlling the sample temperature to vary the band gap energy from 0.17 to 0.23 eV. The measured nonlinear optical properties in InSb are consistent with those predicted by the models which include two- and three-photon absorption (2PA and 3PA), multiphoton generated free-carrier absorption (FCA) and various recombination mechanisms. Temperature-dependent Z-scan and nonlinear transmission measurements yield information on the temperature and spectral dependence of 2PA, FCA, and carrier recombination processes of Shockley-Read-Hall, and Auger mechanisms. We find good agreement between the measured and the modeled nonlinear properties is possible only when the recently predicted temperature dependence of the FCA is considered. The wavelength- and temperature-dependent 2PA and 3PA coefficients in InSb were experimentally obtained. The inferred values of the 2PA and 3PA are consistent with the scaling rules of a simple two-parabolic band model. We further determine recombination rates from nonlinear transmittance of nanosecond pulses of ${\text{CO}}_2$ laser.

Figure 1

(Color online) FCA spectral theory (solid lines) from Ref. 19 and data (open circles) from linear absorption measurements in a 0.65-mm-thick InSb sample at (a) 300 K vs wavelength and (b) $10.6\mu \text{m}$ vs temperature.

Figure 2

(Color online) Open-aperture Z-scan data at several input pulse energies using 300 K a 0.54-mm-thick InSb sample at (a) $9\mu \text{m}$ fit with ${\tau }_{\text{FWHM}}=370\text{fs}$, $w_0=45\mu \text{m}$, ${\alpha }_2=2.9\text{cm}/\text{MW}$ and (b) $11.5\mu \text{m}$ fit with ${\tau }_{\text{FWHM}}=500\text{fs}$, $w_0=63\mu \text{m}$, ${\alpha }_2=2.0\text{cm}/\text{MW}$.

Figure 3

(Color online) Linear transmission data in a 0.65-mm-thick InSb sample obtained at (a) $9.6\mu \text{m}$ and (b) $10.6\mu \text{m}$, using $\sim 10\text{ps}$ FWHM pulsewidths, compared to transmission using theory from Ref. 19 for FCA and the same theoretical absorption reduced by 25%.

Figure 4

(Color online) Open-aperture Z-scan data using 300 K InSb at (a) $9.6\mu \text{m}$ fit with ${\tau }_{\text{FWHM}}=10\text{ps}$, $w_0=140\mu \text{m}$, ${\alpha }_2=3.4\text{cm}/\text{MW}$, ${\sigma }_{\text{FCA}}=7.3\times {10}^{-16}{\text{cm}}^2$, (b) $10.6\mu \text{m}$ fit with ${\tau }_{\text{FWHM}}=10\text{ps}$, $w_0=140\mu \text{m}$, ${\alpha }_2=3.5\text{cm}/\text{MW}$, ${\sigma }_{\text{FCA}}=6.8\times {10}^{-16}{\text{cm}}^2$, and 82 K InSb (c) $9.6\mu \text{m}$ fit with ${\tau }_{\text{FWHM}}=10\text{ps}$, $w_0=140\mu \text{m}$, ${\alpha }_2=1.0\text{cm}/\text{MW}$, ${\sigma }_{\text{FCA}}=3.0\times {10}^{-15}{\text{cm}}^2$, (d) $12\mu \text{m}$ fit with ${\tau }_{\text{FWHM}}=10\text{ps}$, $w_0=140\mu \text{m}$, ${\alpha }_3=0.025{\text{cm}}^3/{\text{MW}}^2$, ${\sigma }_{\text{FCA}}=3.4\times {10}^{-15}{\text{cm}}^2$.

Figure 5

(Color online) Degenerate pump-probe data in InSb at $10\mu \text{m}$ with cross-polarized beams; both the (a) short and (b) long delay curves are fit with the same parameters: ${\tau }_{\text{FWHM}}=10\text{ps}$, ${\sigma }_{\text{FCA}}=6.8\times {10}^{-16}{\text{cm}}^{-2}$, $w_0=500\mu \text{m}$, $L=450\mu \text{m}$, ${\alpha }_{2,\text{S}}=1.4\text{cm}/\text{MW}$, ${\alpha }_{2,\text{C}}=1.9\text{cm}/\text{MW}$, $C=6.5\times {10}^{-26}{\text{cm}}^6/\text{s}$. Linearly polarized beams are used in these experiments.

Figure 6

(Color online) Comparison of 2PA values obtained from fitting femtosecond and picosecond Z-scan data with Wherrett’s theory, Ref. 6, scaled by an empirical factor and more recent theory, Ref. 19 at (a) 80 K and (b) 300 K.

Figure 7

(Color online) Picosecond Z-scan results for 2PA vs. temperature and vs photon energy normalized to the band gap vs temperature (a) and energy (b) at $9.6\mu \text{m}$ and vs temperature (c) and energy (d) at $10.6\mu \text{m}$ compared to theory.

Figure 8

(Color online) (a) 2PA coefficient scaled (by isolating the $E_g^{-3}$ in Eq. (1) from picosecond Z-scan experiments and nanosecond results at 300 K and 80 K compared to Wherrett’s scaling theory from Ref. 6 where the solid line is generated using $K_2=3.1{\text{eV}}^{5/2}\text{cm}/\text{MW}$, the dashed line a value of $K_2=1.9{\text{eV}}^{5/2}\text{cm}/\text{MW}$, and the dotted line a value of $K_2=1.3{\text{eV}}^{5/2}\text{cm}/\text{MW}$. (b) Same as for (a) except that here the data is scaled from Eq. (1) to isolate the $F_2$ spectral dependence.

Figure 9

(Color online) Nonlinear transmission data up to the damage threshold at $\lambda =10.6\mu \text{m}$ of InSb (a) 300 K fit with $w_0=180\mu \text{m}$, ${\sigma }_{\text{FCA}}=6.75\times {10}^{-16}{\text{cm}}^2$, $\tau =2\mu \text{s}$, $B=0.7\times {10}^{-10}{\text{cm}}^3/\text{s}$, ${\alpha }_2=9.5\text{cm}/\text{MW}$, $C=7\times {10}^{-26}{\text{cm}}^6/\text{s}$ and (b) 80 K fit with $w_0=180\mu \text{m}$, ${\sigma }_{\text{FCA}}=32.4\times {10}^{-16}{\text{cm}}^2$, $\tau =2\mu \text{s}$, $B=3.3\times {10}^{-10}{\text{cm}}^3/\text{s}$, ${\alpha }_2=0.5\text{cm}/\text{MW}$, $C=30\times {10}^{-26}{\text{cm}}^6/\text{s}$. The drop in transmission near 1 mJ indicates the onset of damage. Inset shows the input pulse from the nanosecond ${\text{CO}}_2$ laser which is included in the modeling.

Figure 10

(Color online) Comparison of values obtained from fitting nanosecond data for 2PA (a) with a constant FCA cross section and (b) with a temperature-dependent FCA cross section.

Figure 11

(Color online) 2PA picosecond data vs temperature controlled band gap at (a) $9.6\mu \text{m}$ and (b) $10.6\mu \text{m}$ with and without exciton enhancement.

Figure 12

(Color online) Picosecond and nanosecond 2PA data at $10.6\mu \text{m}$ compared to theory with and without exciton enhancement.