Abstract

Silicon based nonlinear photonics has been extensively researched at telecom wavelengths in recent years. However, studies of Kerr nonlinearity in silicon at mid-infrared wavelengths still remain limited. Here, we report the wavelength dependency of third-order nonlinearity in the spectral range from 1.6 µm to 6 µm, as well as multi-photon absorption coefficients in the same range. The third-order nonlinear coefficient n2 was measured with a peak value of 1.65 × 10−13 cm2/W at a wavelength of 2.1 µm followed by the decay of nonlinear refractive index n2 up to 2.6 µm. Our latest measurements extend the wavelength towards 6 µm, which show a sharp decrement of n2 beyond 2.1 µm and steadily retains above 3 µm. In addition, the analysis of three-photon absorption and four-photon absorption processes are simultaneously performed over the wavelength range from 2.3 µm to 4.4 µm. Furthermore, the effect of multi-photon absorption on nonlinear figure of merit in silicon is discussed in detail.

© 2013 Optical Society of America

1. Introduction

The interest in the development of high Kerr nonlinearity with weak multi-photon absorption (MPA) processes has increased over the past decade, because of their importance for ultra-fast all-optical signal processing. Near-infrared (near-IR) and short wave infrared (SWIR) nonlinear optical properties of silicon, namely, the third-order susceptibility χ(3), have been widely studied in the past [1,2]. However, in the mid-infrared (mid-IR), its nonlinearity has not been well characterized. The multi-photon absorption places a fundamental limitation on the usefulness of any high χ(3) material in all-optical switching schemes based on an intensity-dependent refractive index [3], especially strong two-photon absorption (TPA) in silicon. At telecom wavelengths, relatively high peak intensity is normally required to trigger the third-order Kerr nonlinearity, but also leads to a strong TPA [3]. Therefore, the combination of high Kerr nonlinearity and relatively weaker MPA beyond the TPA region makes silicon highly promising for nonlinear applications at mid-IR wavelengths.

Silicon photonics has been identified as a promising material platform for applications in the mid-IR spectral region, especially for building nonlinear optical devices, due to the relatively lower linear and nonlinear absorption within this spectral region [4]. The third-order nonlinear coefficients (n2) and multi-photon absorption processes have been previously reported up to 2.75 µm [5], which shows a decrement of n2 towards longer wavelength from the peak value at 1.9 µm [6,7]. Here, we extend the measurements further to 6 µm in the mid-IR range.

2. Z-scan experiments

We explore the measurements of degenerated MPA coefficients and nonlinear refractive index n2 in silicon, in this broad mid-IR range, using the Z-scan technique [5,7,8]. Z-scan has been widely adopted as a simple single beam technique to obtain MPA coefficients and n2 with the position-dependent transmittance variation by scanning a sample through the focal region of a Gaussian beam [9]. We utilized 150 fs pulses from an optical parametric amplifier source pumped by a 0.8 µm Ti:sapphire regenerative amplifier (Coherent, OperA) at 1 kHz. Combined with a difference frequency generator (DFG), wavelength tunability from 0.24 µm to 10 µm can be achieved. The pulses were focused at normal incidence onto the silicon using a CaF2 lens with 7.5 cm focal length, which minimize the optical loss induced by conventional near-IR optics. The sample is a 675 µm thick p-doped single crystal silicon (~1 × 1015 cm−3 doping concentration), cut with the normal axis along the [100] crystalline direction, such that the incident light is polarized along the [010] direction. The transmitted optical signal is collected by another CaF2 lens with 7.5 cm focal length, and focused onto PbSe detectors (Thorlabs, PDA20H, 1.5-4.8 µm) in reference and signal arms for lock-in detection. Measurements with wavelengths beyond 4 µm are operated by a pyroelectric power probe (Laser Probe, RkP-575, 0.2-20 µm). There will be a 10-15% error tolerance when using the pyroelectric power probe due to thermal effects.

The MPA coefficients and n2 were extracted by analyzing open- and closed-aperture measurements, respectively [8]. When the detection is performed with the presence of aperture, the detected signal has a peak-valley trace, depending on the strength of nonlinear refraction at the focal point. In comparison, in the case of open-aperture detection, the detected signal shows a valley-only trace, which indicates the occurrence of nonlinear MPA, depending on the wavelengths. To emphasize, the determination of beam waist (leads to the calculation of peak intensity) could result approximately 15% error during the experiments depending on the wavelengths, which is considered in addition to the 10-15% error introduced from the pyroelectric power probe by adding errors bars in Fig. 1.

 

Fig. 1 Variation of the MPA coefficients in silicon at mid-IR wavelengths (2.6-4.4 µm).

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The normalized transmittance can be obtained by integrating the intensity I. The total time-integrated transmission for the open aperture is theoretically fitted with Eqs. (1), (2) and (3) obtained from the derivations for MPA coefficients by Sutherland et al. [7,10],

TOA(nPA)=1{1+(n1)αnL[I00/(1+(z/z0)2)]n1}1n1
TOA(3PA)=1{1+2α3Leff'[I00/(1+(z/z0)2)]2}12
TOA(4PA)=1{1+3α4Leff''[I00/(1+(z/z0)2)]3}13

where I00 is the peak intensity, z is the sample position, z0=πω02/λ is the Rayleigh length; ω0 is the beam waist at the focal point (z = 0), λ is the laser wavelength; effective path lengths in the sample for three-photon absorption (3PA) and four-photon absorption (4PA) are given as Leff = [1-exp(−2α0L)]/2α0 and Leff = [1-exp(−3α0L)]/3α0 respectively. To clarify, the α0 is the linear absorption coefficient with wavelength dispersion, which is negligibly small at mid-IR wavelengths. Therefore, the effective path length at α0 0+ is approximately identical to the L (thickness of the sample). In this case, the beam waist ω0 ranged from 38 to 123 µm for different wavelengths. As shown, α3 and α4 present nonlinear absorption coefficients for 3PA and 4PA.

The peak intensity I00 at the sample is determined as I00 = laser_power / [(πω02) × pulse_repetitions × pulse_width] × 40%, due to Fresnel reflection and back scattering at the one-side polished sample surface (40% of the measured incident intensity). Fresnel reflection is normally stated as 30% if both sides are well polished. In our case, single-side polished sample is used. Therefore, in addition to Fresnel reflection, backscattering caused by unpolished side gives rise to additional loss which is ~10% from transmission measurements.

By fitting the experimental data with these curves, MPA coefficients can be estimated. Peak intensity of 57 GW/cm2 is applied, the MPA coefficients are extracted from the open aperture Z-scan as shown in Fig. 1. When going from longer to shorter wavelength, we observed a steep increment of MPA coefficients towards the transition wavelength between 3PA and 4PA (the Si bandgap energy Eg = 1.1 eV are approximately three times of photon energy at 3.3 µm, four times at 4.4 µm and five times at 5.5 µm, correspondently). In Fig. 1, beyond 2.2 µm where 3PA dominates, the 3PA obeys the two-band theory of MPA [11,12], increasing dramatically with the red-shift of wavelengths and rolling over before reaching the 3PA-4PA transition wavelength. Within the 4PA dominant region, the 4PA follows the similar rule as 3PA, apart from a significant drop in nonlinear absorption coefficients. By further extending the wavelength beyond 4.4 µm, 5PA was not observed at an even higher peak intensity of 100GW/cm2, where the first experimental 5PA was only recently reported by Q. Zheng et al [13].

Overall, the nonlinear absorption still shows a dramatic drop in magnitude with wavelength dispersion. Additionally, the MPA induced free-carrier absorption (FCA) can induce an error for calculation of MPA coefficients. Especially at longer wavelengths, the FCA coefficient is proportional to λ2, thus, the error will increase gradually once approaching the 4PA region, but the total contribution to the MPA is believed to be comparably small due to low doping concentration [14].

The n2 can be obtained by analyzing closed-aperture Z-scan measurements [14], where the transmitted light through silicon sample is filtered by an aperture, with a transmittance of 40%. Figure 2 shows selected Z-scan results for Si and SiO2 reference samples: Fig. 2(a) and (b) show the normalized closed- (40%) and open-aperture transmittance for 675µm-thick Si at 2.6 µm. Figure 2(c) shows normalized transmittance versus z-coordinate with the presence of a 40% transmitted aperture for a 1mm-thick SiO2 at 1.55 µm. For the proof of accurately analytic fit, the n2 of SiO2 is measured to be 2.98 × 10−16 cm2/W, which is almost identical to two reference values of 2.2 × 10−16 cm2/W [15] and 2.96 × 10−16 cm2/W [16] found in the literature. A closed-aperture Z-scan example at a longer wavelength of 3.8 µm is also fitted in Fig. 2(d), with a n2 value of 5.8 × 10−14 cm2/W. The closed-aperture data, TCA, are fitted to Eq. (4) [17],

TCA=1+4Δφ0(z/z0)[1+(z/z0)2][9+(z/z0)2]
where Δφ0 is the phase change of the laser beam due to nonlinear refraction. Subsequently, the n2 value can be calculated through Eq. (5):

 

Fig. 2 a) Closed and b) open aperture Z-scan curve of a 675µm-thick Si at 2.6 µm; c) Closed aperture Z-scan curve of a 1mm-thick SiO2 substrate at 1.55 µm; d)Closed aperture Z-scan curve of a 675µm-thick Si at 3.8 µm.

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n2=Δφ0λ2πI00Leff

In Fig. 2(a), at a peak intensity of 57 GW/cm2, the n2 value at 2.6 µm is calculated to be 2.5 × 10−14 cm2/W with the presence of 3PA. The 3PA coefficient at this wavelength is determined as 2 × 10−21 cm3/W2 as shown in Fig. 2(b). Furthermore, the dispersion of n2 from 1.6 to 6 µm is measured and shown in Fig. 3. In these measurements, any errors induced from the determination of intensity could translate to errors in the accuracy of the n2 calculation, including the pulse energy fluctuation, background noise, thermal effects in power detection and beam waist determination. The displayed error bars in Fig. 3 indicate the considerations of all distributing uncertainty factors listed above, which is estimated by taking a few independent measurements under the identical conditions.

 

Fig. 3 Variation of n2 in silicon at mid-IR wavelengths from 1.6 to 6 µm: crosses (data from ref. 7) and diamonds (data measured by SUTD-NUS).

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A sharp increment of n2 from telecom wavelength towards mid-IR range is observed with a peak at 2.1 µm, following by a dip at longer wavelength, which matches with the reference data in the literature (red crosses in Fig. 3) [7]. In order to further explore the dispersion of n2 at mid-IR wavelengths, the Z-scan measurements are extended to longer wavelength beyond 2PA region up to 6 µm. According to Kramers-Kronig (KK) relations [18], two-photon resonance leads to the peak of n2 at the edge of transition wavelength (2.2 µm), which is in line with measured results in Fig. 3. The value of n2 appears to decay rapidly after entering 3PA region, and saturates at a relatively constant value of 5.2 × 10−14 cm2/GW. The variation of n2 appears to be consistent with KK transform [19,20] as shown in Fig. 3. These results are a strong indication of consistent strength of Kerr nonlinearity at mid-IR wavelength.

3. Calculation of nonlinear figure of merit and discussion

The manipulations and usefulness of Kerr nonlinearity are always limited by MPA at high peak intensities and subsequently generated free carrier absorption (FCA) [2]. Especially in the 2PA region, the strong nonlinear absorption induced at high optical intensities strongly presents. For instance, as the application of optical switching, a low nonlinear absorption and higher n2 are ideal. Therefore, here, we introduce the nonlinear figure of merit (FOM) to be n2/λIα3PA and n2/λI2α4PA [5], corresponding to 3PA and 4PA dominant regions, with peak pump intensities ranging from 35 to 57GW/cm2.

According to Boggess et al. [21], the FCA coefficient σFCA at 1064nm is 5 × 10−18 cm2, due to σFCA is proportional to λ2, thus, the corresponding σFCA at 2600nm can be estimated to be 5 × 10−17. By our calculation, the absorption change due to high-peak-intensity induced FCA (ΔαFCA) is determined as 0.07 cm−1, where the absorption change due to 3PA Δα3PA is estimated as 0.84 cm−1. Therefore, we can conclude that the high-peak-intensity induced FCA counts less than 10% error in our experiments. The doping induced FCA is expressed as NoσFCA (N0 = doping concentration), which has been calculated with a value of 0.03 cm−1. As discussed, the FCA absorption caused by high peak intensity is calculated to be 0.07 cm−1, which gives a total FCA value of 0.1 cm−1 by adding up peak-intensity-induced FCA and doping inducing FCA. To compare, the absorption change due to 3PA is 0.84 cm−1. The total FCA induced (0.1 cm−1) remains approximately 10% of total nonlinear absorption. Therefore, the FCA originated from doping has limited impact on determining the MPA coefficients.

In Fig. 4, we use normalized FOM data at the 2PA region from F. Gholami et al. [5] to bridge up with our FOM results beyond 2.2 µm. It can be clearly observed in Fig. 4 that the FOM is lower at short wave-IR region, and increases sharply with increasing wavelengths, due to decaying MPA in the mid-IR region as shown in Fig. 1. Especially when it moves from near-IR towards the transition wavelength for 2PA to 3PA at 2.2 µm, the FOM rises as the effect of 2PA diminishes. In the region of 3PA, in contrast to that for 2PA, 3PA coefficient exhibits an increment for the red-shift of the wavelengths, resulting in a drop in FOM.

 

Fig. 4 Nonlinear figure of merit (FOM) with presence of MPA: crosses (data from ref. 7) and stars (data measured by SUTD-NUS).

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As discussed above, TPA is forbidden beyond 2.2 μm and hence, three-photon absorption determines FOM by n2/λIα3PA, that indicates that FOM decreases if n2 decreases, α3 increases or wavelength increases. The data in Fig. 3 shows that n2 decreases at λ > 2.2um. The theory used in ref. 10 shows that α3 should increase with wavelength in this spectral range, which is consistent with our data in Fig. 1. All these factors lead to the deceasing trend at λ > 2.2 um.

Furthermore, the value of n2 is observed to decay beyond 2PA region, thus, the calculated FOM rolls over until it reaches the 3PA-4PA transition wavelength at 3.3 µm. Due to the dramatic decrement of MPA in 4PA dominant region, the nonlinear FOM exhibits a steady rise beyond 3PA region, which is almost identical to the magnitudes in the 2PA dominant short wave-IR region [19].

4. Conclusion

In summary, we report the measurements of degenerated MPA and Kerr nonlinear coefficients covering the near-IR and mid-IR spectral region up to 6 µm. The results indicate a good strength of Kerr nonlinearity and significant enhancement in nonlinear FOM at longer mid-IR wavelengths (3-6 µm), which appears to be a promising candidate for on-chip all-optical signal processing.

Acknowledgments

Funding from the SUTD-ZJU Collaborative Research Grant, SUTD-MIT International Design Centre and the NUSNNI NanoCore of National University of Singapore is gratefully acknowledged.

References and links

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]  

2. 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]  

3. V. Mizrahi, K. W. Delong, G. I. Stegeman, M. A. Saifi, and M. J. Andrejco, “Two-photon absorption as a limitation to all-optical switching,” Opt. Lett. 14(20), 1140–1142 (1989). [CrossRef]   [PubMed]  

4. S. Pearl, N. Rotenberg, and H. M. van Driel, “Three photon absorption in silicon for 2300-3300 nm,” Appl. Phys. Lett. 93(13), 131102 (2008). [CrossRef]  

5. F. Gholami, S. Zlatanovic, A. Simic, L. Liu, D. Borlaug, N. Alic, M. Nezhad, Y. Fainman, and S. Radic, “Third-order nonlinearity in silicon beyond 2350 nm,” Appl. Phys. Lett. 99(8), 081102 (2011). [CrossRef]  

6. N. Hon, R. Soref, and B. Jalali, “The third-order nonlinear optical coefficients of Si, Ge, and Si1-xGex in the midwave and longwave infrared,” Appl. Phys. Lett. 110, 011301 (2011).

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

8. N. Venkatram, R. Sathyavathi, and D. N. Rao, “Size dependent multiphoton absorption and refraction of CdSe nanoparticles,” Opt. Express 15(19), 12258–12263 (2007). [CrossRef]   [PubMed]  

9. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Special 30th anniversary feature: sensitive measurement of optical nonlinearities using a single beam,” IEEE LEOS Newsletter 21, 17–35 (2007).

10. R. L. Sutherland, D. G. Mclean, and S. Kirkpatrick, Handbook of Nonlinear Optics, 2nd ed. (Marcel Dekker, 2003) p. 528.

11. 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]   [PubMed]  

12. B. S. Wherrett, “Scaling rules for multiphoton interband absorption in semiconductors,” J. Opt. Soc. Am. B 1(1), 67–72 (1984). [CrossRef]  

13. Q. Zheng, H. Zhu, S. Chen, C. Tang, E. Ma, and X. Chen, “Frequency-upconverted stimulated emission by simultaneous five-photon absorption,” Nat. Photonics 7(3), 234–239 (2013). [CrossRef]  

14. D. K. Schroder, R. N. Thomas, and J. C. Swartz, “Free carrier absorption in silicon,” IEEE Trans. Electron. Dev. 25(2), 254–261 (1978). [CrossRef]  

15. J. Wang, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E. W. Van Stryland, “Time-resolved Z-scan measurements of optical nonlinearities,” J. Opt. Soc. Am. B. 11(6), 1009–1017 (1994). [CrossRef]  

16. A. Boskovic, S. V. Chernikov, J. R. Taylor, L. Gruner-Nielsen, and O. A. Levring, “Direct continuous-wave measurement of n2 in various types of telecommunication fiber at 1.55 µm,” Opt. Lett. 21(24), 1966–1968 (1996). [CrossRef]   [PubMed]  

17. T. Kato, Y. Suetsugu, M. Takagi, E. Sasaoka, and M. Nishimura, “Measurement of the nonlinear refractive index in optical fiber by the cross-phase-modulation method with depolarized pump light,” Opt. Lett. 20(9), 988–990 (1995). [CrossRef]   [PubMed]  

18. 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]  

19. M. Sheik-Bahae, D. J. Hagan, and E. W. Van Stryland, “Dispersion and band-gap scaling of the electronic Kerr effect in solids associated with two-photon absorption,” Phys. Rev. Lett. 65(1), 96–99 (1990). [CrossRef]   [PubMed]  

20. C. Bree, A. Demircan, and G. Steinmeyer, “Kramers-Kronig relations and high-order nonlinear susceptibilities,” Phys. Rev. A 85(3), 033806 (2012). [CrossRef]  

21. T. F. Boggess, K. Bohnert, K. Mansour, S. C. Moss, I. W. Boyd, and A. L. Smirl, “Simultaneous measurement of the two-photon coefficient and free-carrier cross section above the bandgap of crystalline silicon,” IEEE J. Quantum Electron. 22(2), 360–368 (1986). [CrossRef]  

References

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  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]
  2. 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]
  3. V. Mizrahi, K. W. Delong, G. I. Stegeman, M. A. Saifi, and M. J. Andrejco, “Two-photon absorption as a limitation to all-optical switching,” Opt. Lett. 14(20), 1140–1142 (1989).
    [Crossref] [PubMed]
  4. S. Pearl, N. Rotenberg, and H. M. van Driel, “Three photon absorption in silicon for 2300-3300 nm,” Appl. Phys. Lett. 93(13), 131102 (2008).
    [Crossref]
  5. F. Gholami, S. Zlatanovic, A. Simic, L. Liu, D. Borlaug, N. Alic, M. Nezhad, Y. Fainman, and S. Radic, “Third-order nonlinearity in silicon beyond 2350 nm,” Appl. Phys. Lett. 99(8), 081102 (2011).
    [Crossref]
  6. N. Hon, R. Soref, and B. Jalali, “The third-order nonlinear optical coefficients of Si, Ge, and Si1-xGex in the midwave and longwave infrared,” Appl. Phys. Lett. 110, 011301 (2011).
  7. A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850-2200 nm,” Appl. Phys. Lett. 90(19), 191104 (2007).
    [Crossref]
  8. N. Venkatram, R. Sathyavathi, and D. N. Rao, “Size dependent multiphoton absorption and refraction of CdSe nanoparticles,” Opt. Express 15(19), 12258–12263 (2007).
    [Crossref] [PubMed]
  9. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Special 30th anniversary feature: sensitive measurement of optical nonlinearities using a single beam,” IEEE LEOS Newsletter 21, 17–35 (2007).
  10. R. L. Sutherland, D. G. Mclean, and S. Kirkpatrick, Handbook of Nonlinear Optics, 2nd ed. (Marcel Dekker, 2003) p. 528.
  11. 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] [PubMed]
  12. B. S. Wherrett, “Scaling rules for multiphoton interband absorption in semiconductors,” J. Opt. Soc. Am. B 1(1), 67–72 (1984).
    [Crossref]
  13. Q. Zheng, H. Zhu, S. Chen, C. Tang, E. Ma, and X. Chen, “Frequency-upconverted stimulated emission by simultaneous five-photon absorption,” Nat. Photonics 7(3), 234–239 (2013).
    [Crossref]
  14. D. K. Schroder, R. N. Thomas, and J. C. Swartz, “Free carrier absorption in silicon,” IEEE Trans. Electron. Dev. 25(2), 254–261 (1978).
    [Crossref]
  15. J. Wang, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E. W. Van Stryland, “Time-resolved Z-scan measurements of optical nonlinearities,” J. Opt. Soc. Am. B. 11(6), 1009–1017 (1994).
    [Crossref]
  16. A. Boskovic, S. V. Chernikov, J. R. Taylor, L. Gruner-Nielsen, and O. A. Levring, “Direct continuous-wave measurement of n2 in various types of telecommunication fiber at 1.55 µm,” Opt. Lett. 21(24), 1966–1968 (1996).
    [Crossref] [PubMed]
  17. T. Kato, Y. Suetsugu, M. Takagi, E. Sasaoka, and M. Nishimura, “Measurement of the nonlinear refractive index in optical fiber by the cross-phase-modulation method with depolarized pump light,” Opt. Lett. 20(9), 988–990 (1995).
    [Crossref] [PubMed]
  18. 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]
  19. M. Sheik-Bahae, D. J. Hagan, and E. W. Van Stryland, “Dispersion and band-gap scaling of the electronic Kerr effect in solids associated with two-photon absorption,” Phys. Rev. Lett. 65(1), 96–99 (1990).
    [Crossref] [PubMed]
  20. C. Bree, A. Demircan, and G. Steinmeyer, “Kramers-Kronig relations and high-order nonlinear susceptibilities,” Phys. Rev. A 85(3), 033806 (2012).
    [Crossref]
  21. T. F. Boggess, K. Bohnert, K. Mansour, S. C. Moss, I. W. Boyd, and A. L. Smirl, “Simultaneous measurement of the two-photon coefficient and free-carrier cross section above the bandgap of crystalline silicon,” IEEE J. Quantum Electron. 22(2), 360–368 (1986).
    [Crossref]

2013 (1)

Q. Zheng, H. Zhu, S. Chen, C. Tang, E. Ma, and X. Chen, “Frequency-upconverted stimulated emission by simultaneous five-photon absorption,” Nat. Photonics 7(3), 234–239 (2013).
[Crossref]

2012 (1)

C. Bree, A. Demircan, and G. Steinmeyer, “Kramers-Kronig relations and high-order nonlinear susceptibilities,” Phys. Rev. A 85(3), 033806 (2012).
[Crossref]

2011 (2)

F. Gholami, S. Zlatanovic, A. Simic, L. Liu, D. Borlaug, N. Alic, M. Nezhad, Y. Fainman, and S. Radic, “Third-order nonlinearity in silicon beyond 2350 nm,” Appl. Phys. Lett. 99(8), 081102 (2011).
[Crossref]

N. Hon, R. Soref, and B. Jalali, “The third-order nonlinear optical coefficients of Si, Ge, and Si1-xGex in the midwave and longwave infrared,” Appl. Phys. Lett. 110, 011301 (2011).

2008 (1)

S. Pearl, N. Rotenberg, and H. M. van Driel, “Three photon absorption in silicon for 2300-3300 nm,” Appl. Phys. Lett. 93(13), 131102 (2008).
[Crossref]

2007 (5)

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]

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

N. Venkatram, R. Sathyavathi, and D. N. Rao, “Size dependent multiphoton absorption and refraction of CdSe nanoparticles,” Opt. Express 15(19), 12258–12263 (2007).
[Crossref] [PubMed]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Special 30th anniversary feature: sensitive measurement of optical nonlinearities using a single beam,” IEEE LEOS Newsletter 21, 17–35 (2007).

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] [PubMed]

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]

1996 (1)

1995 (1)

1994 (1)

J. Wang, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E. W. Van Stryland, “Time-resolved Z-scan measurements of optical nonlinearities,” J. Opt. Soc. Am. B. 11(6), 1009–1017 (1994).
[Crossref]

1990 (2)

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]

M. Sheik-Bahae, D. J. Hagan, and E. W. Van Stryland, “Dispersion and band-gap scaling of the electronic Kerr effect in solids associated with two-photon absorption,” Phys. Rev. Lett. 65(1), 96–99 (1990).
[Crossref] [PubMed]

1989 (1)

1986 (1)

T. F. Boggess, K. Bohnert, K. Mansour, S. C. Moss, I. W. Boyd, and A. L. Smirl, “Simultaneous measurement of the two-photon coefficient and free-carrier cross section above the bandgap of crystalline silicon,” IEEE J. Quantum Electron. 22(2), 360–368 (1986).
[Crossref]

1984 (1)

1978 (1)

D. K. Schroder, R. N. Thomas, and J. C. Swartz, “Free carrier absorption in silicon,” IEEE Trans. Electron. Dev. 25(2), 254–261 (1978).
[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]

Alic, N.

F. Gholami, S. Zlatanovic, A. Simic, L. Liu, D. Borlaug, N. Alic, M. Nezhad, Y. Fainman, and S. Radic, “Third-order nonlinearity in silicon beyond 2350 nm,” Appl. Phys. Lett. 99(8), 081102 (2011).
[Crossref]

Andrejco, M. J.

Boggess, T. F.

T. F. Boggess, K. Bohnert, K. Mansour, S. C. Moss, I. W. Boyd, and A. L. Smirl, “Simultaneous measurement of the two-photon coefficient and free-carrier cross section above the bandgap of crystalline silicon,” IEEE J. Quantum Electron. 22(2), 360–368 (1986).
[Crossref]

Bohnert, K.

T. F. Boggess, K. Bohnert, K. Mansour, S. C. Moss, I. W. Boyd, and A. L. Smirl, “Simultaneous measurement of the two-photon coefficient and free-carrier cross section above the bandgap of crystalline silicon,” IEEE J. Quantum Electron. 22(2), 360–368 (1986).
[Crossref]

Borlaug, D.

F. Gholami, S. Zlatanovic, A. Simic, L. Liu, D. Borlaug, N. Alic, M. Nezhad, Y. Fainman, and S. Radic, “Third-order nonlinearity in silicon beyond 2350 nm,” Appl. Phys. Lett. 99(8), 081102 (2011).
[Crossref]

Boskovic, A.

Boyd, I. W.

T. F. Boggess, K. Bohnert, K. Mansour, S. C. Moss, I. W. Boyd, and A. L. Smirl, “Simultaneous measurement of the two-photon coefficient and free-carrier cross section above the bandgap of crystalline silicon,” IEEE J. Quantum Electron. 22(2), 360–368 (1986).
[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]

Bree, C.

C. Bree, A. Demircan, and G. Steinmeyer, “Kramers-Kronig relations and high-order nonlinear susceptibilities,” Phys. Rev. A 85(3), 033806 (2012).
[Crossref]

Bristow, A. D.

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

Chen, S.

Q. Zheng, H. Zhu, S. Chen, C. Tang, E. Ma, and X. Chen, “Frequency-upconverted stimulated emission by simultaneous five-photon absorption,” Nat. Photonics 7(3), 234–239 (2013).
[Crossref]

Chen, X.

Q. Zheng, H. Zhu, S. Chen, C. Tang, E. Ma, and X. Chen, “Frequency-upconverted stimulated emission by simultaneous five-photon absorption,” Nat. Photonics 7(3), 234–239 (2013).
[Crossref]

Chernikov, S. V.

Delong, K. W.

Demircan, A.

C. Bree, A. Demircan, and G. Steinmeyer, “Kramers-Kronig relations and high-order nonlinear susceptibilities,” Phys. Rev. A 85(3), 033806 (2012).
[Crossref]

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]

Fainman, Y.

F. Gholami, S. Zlatanovic, A. Simic, L. Liu, D. Borlaug, N. Alic, M. Nezhad, Y. Fainman, and S. Radic, “Third-order nonlinearity in silicon beyond 2350 nm,” Appl. Phys. Lett. 99(8), 081102 (2011).
[Crossref]

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.

Garcia, H.

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

Gholami, F.

F. Gholami, S. Zlatanovic, A. Simic, L. Liu, D. Borlaug, N. Alic, M. Nezhad, Y. Fainman, and S. Radic, “Third-order nonlinearity in silicon beyond 2350 nm,” Appl. Phys. Lett. 99(8), 081102 (2011).
[Crossref]

Gruner-Nielsen, L.

Hagan, D. J.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Special 30th anniversary feature: sensitive measurement of optical nonlinearities using a single beam,” IEEE LEOS Newsletter 21, 17–35 (2007).

J. Wang, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E. W. Van Stryland, “Time-resolved Z-scan measurements of optical nonlinearities,” J. Opt. Soc. Am. B. 11(6), 1009–1017 (1994).
[Crossref]

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]

M. Sheik-Bahae, D. J. Hagan, and E. W. Van Stryland, “Dispersion and band-gap scaling of the electronic Kerr effect in solids associated with two-photon absorption,” Phys. Rev. Lett. 65(1), 96–99 (1990).
[Crossref] [PubMed]

Hon, N.

N. Hon, R. Soref, and B. Jalali, “The third-order nonlinear optical coefficients of Si, Ge, and Si1-xGex in the midwave and longwave infrared,” Appl. Phys. Lett. 110, 011301 (2011).

Hurlbut, W. C.

Jalali, B.

N. Hon, R. Soref, and B. Jalali, “The third-order nonlinear optical coefficients of Si, Ge, and Si1-xGex in the midwave and longwave infrared,” Appl. Phys. Lett. 110, 011301 (2011).

Kato, T.

Kuo, P. S.

Lee, Y. S.

Levring, O. A.

Lin, Q.

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]

Liu, L.

F. Gholami, S. Zlatanovic, A. Simic, L. Liu, D. Borlaug, N. Alic, M. Nezhad, Y. Fainman, and S. Radic, “Third-order nonlinearity in silicon beyond 2350 nm,” Appl. Phys. Lett. 99(8), 081102 (2011).
[Crossref]

Ma, E.

Q. Zheng, H. Zhu, S. Chen, C. Tang, E. Ma, and X. Chen, “Frequency-upconverted stimulated emission by simultaneous five-photon absorption,” Nat. Photonics 7(3), 234–239 (2013).
[Crossref]

Mansour, K.

T. F. Boggess, K. Bohnert, K. Mansour, S. C. Moss, I. W. Boyd, and A. L. Smirl, “Simultaneous measurement of the two-photon coefficient and free-carrier cross section above the bandgap of crystalline silicon,” IEEE J. Quantum Electron. 22(2), 360–368 (1986).
[Crossref]

Mizrahi, V.

Moss, S. C.

T. F. Boggess, K. Bohnert, K. Mansour, S. C. Moss, I. W. Boyd, and A. L. Smirl, “Simultaneous measurement of the two-photon coefficient and free-carrier cross section above the bandgap of crystalline silicon,” IEEE J. Quantum Electron. 22(2), 360–368 (1986).
[Crossref]

Nezhad, M.

F. Gholami, S. Zlatanovic, A. Simic, L. Liu, D. Borlaug, N. Alic, M. Nezhad, Y. Fainman, and S. Radic, “Third-order nonlinearity in silicon beyond 2350 nm,” Appl. Phys. Lett. 99(8), 081102 (2011).
[Crossref]

Nishimura, M.

Pearl, S.

S. Pearl, N. Rotenberg, and H. M. van Driel, “Three photon absorption in silicon for 2300-3300 nm,” Appl. Phys. Lett. 93(13), 131102 (2008).
[Crossref]

Piredda, G.

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]

Quochi, F.

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

Radic, S.

F. Gholami, S. Zlatanovic, A. Simic, L. Liu, D. Borlaug, N. Alic, M. Nezhad, Y. Fainman, and S. Radic, “Third-order nonlinearity in silicon beyond 2350 nm,” Appl. Phys. Lett. 99(8), 081102 (2011).
[Crossref]

Rao, D. N.

Rotenberg, N.

S. Pearl, N. Rotenberg, and H. M. van Driel, “Three photon absorption in silicon for 2300-3300 nm,” Appl. Phys. Lett. 93(13), 131102 (2008).
[Crossref]

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

Said, A. A.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Special 30th anniversary feature: sensitive measurement of optical nonlinearities using a single beam,” IEEE LEOS Newsletter 21, 17–35 (2007).

J. Wang, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E. W. Van Stryland, “Time-resolved Z-scan measurements of optical nonlinearities,” J. Opt. Soc. Am. B. 11(6), 1009–1017 (1994).
[Crossref]

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]

Saifi, M. A.

Sasaoka, E.

Sathyavathi, R.

Schroder, D. K.

D. K. Schroder, R. N. Thomas, and J. C. Swartz, “Free carrier absorption in silicon,” IEEE Trans. Electron. Dev. 25(2), 254–261 (1978).
[Crossref]

Sheik-Bahae, M.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Special 30th anniversary feature: sensitive measurement of optical nonlinearities using a single beam,” IEEE LEOS Newsletter 21, 17–35 (2007).

J. Wang, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E. W. Van Stryland, “Time-resolved Z-scan measurements of optical nonlinearities,” J. Opt. Soc. Am. B. 11(6), 1009–1017 (1994).
[Crossref]

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]

M. Sheik-Bahae, D. J. Hagan, and E. W. Van Stryland, “Dispersion and band-gap scaling of the electronic Kerr effect in solids associated with two-photon absorption,” Phys. Rev. Lett. 65(1), 96–99 (1990).
[Crossref] [PubMed]

Simic, A.

F. Gholami, S. Zlatanovic, A. Simic, L. Liu, D. Borlaug, N. Alic, M. Nezhad, Y. Fainman, and S. Radic, “Third-order nonlinearity in silicon beyond 2350 nm,” Appl. Phys. Lett. 99(8), 081102 (2011).
[Crossref]

Smirl, A. L.

T. F. Boggess, K. Bohnert, K. Mansour, S. C. Moss, I. W. Boyd, and A. L. Smirl, “Simultaneous measurement of the two-photon coefficient and free-carrier cross section above the bandgap of crystalline silicon,” IEEE J. Quantum Electron. 22(2), 360–368 (1986).
[Crossref]

Soref, R.

N. Hon, R. Soref, and B. Jalali, “The third-order nonlinear optical coefficients of Si, Ge, and Si1-xGex in the midwave and longwave infrared,” Appl. Phys. Lett. 110, 011301 (2011).

Stegeman, G. I.

Steinmeyer, G.

C. Bree, A. Demircan, and G. Steinmeyer, “Kramers-Kronig relations and high-order nonlinear susceptibilities,” Phys. Rev. A 85(3), 033806 (2012).
[Crossref]

Suetsugu, Y.

Swartz, J. C.

D. K. Schroder, R. N. Thomas, and J. C. Swartz, “Free carrier absorption in silicon,” IEEE Trans. Electron. Dev. 25(2), 254–261 (1978).
[Crossref]

Takagi, M.

Tang, C.

Q. Zheng, H. Zhu, S. Chen, C. Tang, E. Ma, and X. Chen, “Frequency-upconverted stimulated emission by simultaneous five-photon absorption,” Nat. Photonics 7(3), 234–239 (2013).
[Crossref]

Taylor, J. R.

Thomas, R. N.

D. K. Schroder, R. N. Thomas, and J. C. Swartz, “Free carrier absorption in silicon,” IEEE Trans. Electron. Dev. 25(2), 254–261 (1978).
[Crossref]

van Driel, H. M.

S. Pearl, N. Rotenberg, and H. M. van Driel, “Three photon absorption in silicon for 2300-3300 nm,” Appl. Phys. Lett. 93(13), 131102 (2008).
[Crossref]

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

Van Stryland, E. W.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Special 30th anniversary feature: sensitive measurement of optical nonlinearities using a single beam,” IEEE LEOS Newsletter 21, 17–35 (2007).

J. Wang, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E. W. Van Stryland, “Time-resolved Z-scan measurements of optical nonlinearities,” J. Opt. Soc. Am. B. 11(6), 1009–1017 (1994).
[Crossref]

M. Sheik-Bahae, D. J. Hagan, and E. W. Van Stryland, “Dispersion and band-gap scaling of the electronic Kerr effect in solids associated with two-photon absorption,” Phys. Rev. Lett. 65(1), 96–99 (1990).
[Crossref] [PubMed]

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]

Venkatram, N.

Vodopyanov, K. L.

Wang, J.

J. Wang, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E. W. Van Stryland, “Time-resolved Z-scan measurements of optical nonlinearities,” J. Opt. Soc. Am. B. 11(6), 1009–1017 (1994).
[Crossref]

Wei, T. H.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Special 30th anniversary feature: sensitive measurement of optical nonlinearities using a single beam,” IEEE LEOS Newsletter 21, 17–35 (2007).

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]

Wherrett, B. S.

Zhang, J.

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]

Zheng, Q.

Q. Zheng, H. Zhu, S. Chen, C. Tang, E. Ma, and X. Chen, “Frequency-upconverted stimulated emission by simultaneous five-photon absorption,” Nat. Photonics 7(3), 234–239 (2013).
[Crossref]

Zhu, H.

Q. Zheng, H. Zhu, S. Chen, C. Tang, E. Ma, and X. Chen, “Frequency-upconverted stimulated emission by simultaneous five-photon absorption,” Nat. Photonics 7(3), 234–239 (2013).
[Crossref]

Zlatanovic, S.

F. Gholami, S. Zlatanovic, A. Simic, L. Liu, D. Borlaug, N. Alic, M. Nezhad, Y. Fainman, and S. Radic, “Third-order nonlinearity in silicon beyond 2350 nm,” Appl. Phys. Lett. 99(8), 081102 (2011).
[Crossref]

Appl. Phys. Lett. (6)

S. Pearl, N. Rotenberg, and H. M. van Driel, “Three photon absorption in silicon for 2300-3300 nm,” Appl. Phys. Lett. 93(13), 131102 (2008).
[Crossref]

F. Gholami, S. Zlatanovic, A. Simic, L. Liu, D. Borlaug, N. Alic, M. Nezhad, Y. Fainman, and S. Radic, “Third-order nonlinearity in silicon beyond 2350 nm,” Appl. Phys. Lett. 99(8), 081102 (2011).
[Crossref]

N. Hon, R. Soref, and B. Jalali, “The third-order nonlinear optical coefficients of Si, Ge, and Si1-xGex in the midwave and longwave infrared,” Appl. Phys. Lett. 110, 011301 (2011).

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

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

IEEE J. Quantum Electron. (2)

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]

T. F. Boggess, K. Bohnert, K. Mansour, S. C. Moss, I. W. Boyd, and A. L. Smirl, “Simultaneous measurement of the two-photon coefficient and free-carrier cross section above the bandgap of crystalline silicon,” IEEE J. Quantum Electron. 22(2), 360–368 (1986).
[Crossref]

IEEE LEOS Newsletter (1)

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Special 30th anniversary feature: sensitive measurement of optical nonlinearities using a single beam,” IEEE LEOS Newsletter 21, 17–35 (2007).

IEEE Trans. Electron. Dev. (1)

D. K. Schroder, R. N. Thomas, and J. C. Swartz, “Free carrier absorption in silicon,” IEEE Trans. Electron. Dev. 25(2), 254–261 (1978).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Opt. Soc. Am. B. (1)

J. Wang, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E. W. Van Stryland, “Time-resolved Z-scan measurements of optical nonlinearities,” J. Opt. Soc. Am. B. 11(6), 1009–1017 (1994).
[Crossref]

Nat. Photonics (1)

Q. Zheng, H. Zhu, S. Chen, C. Tang, E. Ma, and X. Chen, “Frequency-upconverted stimulated emission by simultaneous five-photon absorption,” Nat. Photonics 7(3), 234–239 (2013).
[Crossref]

Opt. Express (1)

Opt. Lett. (4)

Phys. Rev. A (1)

C. Bree, A. Demircan, and G. Steinmeyer, “Kramers-Kronig relations and high-order nonlinear susceptibilities,” Phys. Rev. A 85(3), 033806 (2012).
[Crossref]

Phys. Rev. Lett. (1)

M. Sheik-Bahae, D. J. Hagan, and E. W. Van Stryland, “Dispersion and band-gap scaling of the electronic Kerr effect in solids associated with two-photon absorption,” Phys. Rev. Lett. 65(1), 96–99 (1990).
[Crossref] [PubMed]

Other (1)

R. L. Sutherland, D. G. Mclean, and S. Kirkpatrick, Handbook of Nonlinear Optics, 2nd ed. (Marcel Dekker, 2003) p. 528.

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Figures (4)

Fig. 1
Fig. 1 Variation of the MPA coefficients in silicon at mid-IR wavelengths (2.6-4.4 µm).
Fig. 2
Fig. 2 a) Closed and b) open aperture Z-scan curve of a 675µm-thick Si at 2.6 µm; c) Closed aperture Z-scan curve of a 1mm-thick SiO2 substrate at 1.55 µm; d)Closed aperture Z-scan curve of a 675µm-thick Si at 3.8 µm.
Fig. 3
Fig. 3 Variation of n2 in silicon at mid-IR wavelengths from 1.6 to 6 µm: crosses (data from ref. 7) and diamonds (data measured by SUTD-NUS).
Fig. 4
Fig. 4 Nonlinear figure of merit (FOM) with presence of MPA: crosses (data from ref. 7) and stars (data measured by SUTD-NUS).

Equations (5)

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

T OA( nPA ) = 1 {1+(n1) α n L[ I 00 / (1+ ( z/ z 0 ) 2 )] n1 } 1 n1
T OA( 3PA ) = 1 {1+2 α 3 L eff ' [ I 00 / (1+ ( z/ z 0 ) 2 )] 2 } 1 2
T OA( 4PA ) = 1 {1+3 α 4 L eff '' [ I 00 / (1+ ( z/ z 0 ) 2 )] 3 } 1 3
T CA =1+ 4Δ φ 0 (z/ z 0 ) [1+ (z/ z 0 ) 2 ][9+ (z/ z 0 ) 2 ]
n 2 = Δ φ 0 λ 2π I 00 L eff

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