Abstract

We present self-induced absorption, a pump/probe-like modulation technique that uses a single monochromatic laser acting simultaneously as both a pump and a probe. The technique is applicable to any system where the phenomenon that us being excited simultaneously induces additional absorption in the beam through a secondary process, leading to a non-linear power component in the beam transmission. The technique is demonstrated on a silicon wafer, where the non-linear transmission is due to free-carrier absorption, and provides information about the recombination lifetime of the semiconductor. Reducing a two-beam technique to a single laser beam simplifies the alignment challenges of traditional dual-beam modulated pump/probe measurements, which require overlap of separate pump and probe lasers on the sample under study.

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

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References

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  22. R. Soref and B. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987).
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  23. 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]
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  27. J. F. Reintjes and J. C. McGroddy, “Indirect two-photon transitions in Si at 1.06 μm,” Phys. Rev. Lett. 30(19), 901–903 (1973).
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    [Crossref]
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  30. P. R. Kidambi, R. Blume, J. Kling, J. B. Wagner, C. Baehtz, R. S. Weatherup, R. Schloegl, B. C. Bayer, and S. Hofmann, “In Situ Observations during Chemical Vapor Deposition of Hexagonal Boron Nitride on Polycrystalline Copper,” Chem. Mater. 26(22), 6380–6392 (2014).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  35. V. Hopfe, D. W. Sheel, C. I. M. A. Spee, R. Tell, P. Martin, A. Beil, M. Pemble, R. Weiss, U. Vogt, and W. Graehlert, “In-situ monitoring for CVD processes,” Thin Solid Films 442(1-2), 60–65 (2003).
    [Crossref]
  36. A. G. Aberle, “Surface passivation of crystalline silicon solar cells: a review,” Prog. Photovolt. Res. Appl. 8(5), 473–487 (2000).
    [Crossref]
  37. J. Schmidt, M. Kerr, and A. Cuevas, “Surface passivation of silicon solar cells using plasma-enhanced chemical-vapour-deposited SiN films and thin thermal SiO 2 /plasma SiN stacks,” Semicond. Sci. Technol. 16(3), 164–170 (2001).
    [Crossref]
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    [Crossref]

2019 (1)

K. M. W. Boyd and R. N. Kleiman, “Quasi-Steady-State Free Carrier Absorption Measurements of Effective Carrier Lifetime in Silicon,” IEEE J. Photovolt. 9(1), 64–71 (2019).
[Crossref]

2017 (1)

W. I. Ndebeka, P. H. Neethling, E. G. Rohwer, C. M. Steenkamp, J. Bergmann, and H. Stafast, “Interband and free charge carrier absorption in silicon at 800 nm: experiments and model calculations,” Appl. Phys. B 123(10), 253 (2017).
[Crossref]

2016 (2)

M. M. Jadidi, R. J. Suess, C. Tan, X. Cai, K. Watanabe, T. Taniguchi, A. B. Sushkov, M. Mittendorff, J. Hone, H. D. Drew, M. S. Fuhrer, and T. E. Murphy, “Tunable Ultrafast Thermal Relaxation in Graphene Measured by Continuous-Wave Photomixing,” Phys. Rev. Lett. 117(25), 257401 (2016).
[Crossref] [PubMed]

M. L. Taheri, E. A. Stach, I. Arslan, P. A. Crozier, B. C. Kabius, T. LaGrange, A. M. Minor, S. Takeda, M. Tanase, J. B. Wagner, and R. Sharma, “Current status and future directions for in situ transmission electron microscopy,” Ultramicroscopy 170, 86–95 (2016).
[Crossref] [PubMed]

2015 (2)

Z.-J. Wang, G. Weinberg, Q. Zhang, T. Lunkenbein, A. Klein-Hoffmann, M. Kurnatowska, M. Plodinec, Q. Li, L. Chi, R. Schloegl, and M.-G. Willinger, “Direct Observation of Graphene Growth and Associated Copper Substrate Dynamics by in Situ Scanning Electron Microscopy,” ACS Nano 9(2), 1506–1519 (2015).
[Crossref] [PubMed]

B. Bein, H.-C. Hsing, S. J. Callori, J. Sinsheimer, P. V. Chinta, R. L. Headrick, and M. Dawber, “In situ X-ray diffraction and the evolution of polarization during the growth of ferroelectric superlattices,” Nat. Commun. 6(1), 10136 (2015).
[Crossref] [PubMed]

2014 (1)

P. R. Kidambi, R. Blume, J. Kling, J. B. Wagner, C. Baehtz, R. S. Weatherup, R. Schloegl, B. C. Bayer, and S. Hofmann, “In Situ Observations during Chemical Vapor Deposition of Hexagonal Boron Nitride on Polycrystalline Copper,” Chem. Mater. 26(22), 6380–6392 (2014).
[Crossref] [PubMed]

2011 (1)

J. García-Martín, J. Gómez-Gil, and E. Vázquez-Sánchez, “Non-Destructive Techniques Based on Eddy Current Testing,” Sensors (Basel) 11(3), 2525–2565 (2011).
[Crossref] [PubMed]

2010 (1)

F. Vetrone, R. Naccache, A. Zamarrón, A. Juarranz de la Fuente, F. Sanz-Rodríguez, L. Martinez Maestro, E. Martín Rodriguez, D. Jaque, J. García Solé, and J. A. Capobianco, “Temperature Sensing Using Fluorescent Nanothermometers,” ACS Nano 4(6), 3254–3258 (2010).
[Crossref] [PubMed]

2007 (1)

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

2006 (1)

E. Langereis, S. B. S. Heil, M. C. M. van de Sanden, and W. M. M. Kessels, “In situ spectroscopic ellipsometry study on the growth of ultrathin TiN films by plasma-assisted atomic layer deposition,” J. Appl. Phys. 100(2), 023534 (2006).
[Crossref]

2004 (1)

J. Isenberg and W. Warta, “Free carrier absorption in heavily doped silicon layers,” Appl. Phys. Lett. 84(13), 2265–2267 (2004).
[Crossref]

2003 (2)

M. M. Frank, Y. J. Chabal, and G. D. Wilk, “Nucleation and interface formation mechanisms in atomic layer deposition of gate oxides,” Appl. Phys. Lett. 82(26), 4758–4760 (2003).
[Crossref]

V. Hopfe, D. W. Sheel, C. I. M. A. Spee, R. Tell, P. Martin, A. Beil, M. Pemble, R. Weiss, U. Vogt, and W. Graehlert, “In-situ monitoring for CVD processes,” Thin Solid Films 442(1-2), 60–65 (2003).
[Crossref]

2002 (2)

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, “Time-resolved atomic inner-shell spectroscopy,” Nature 419(6909), 803–807 (2002).
[Crossref] [PubMed]

G. Citarella, S. von Aichberger, and M. Kunst, “Microwave photoconductivity techniques for the characterization of semiconductors,” Mater. Sci. Eng. B 91-92, 224–228 (2002).
[Crossref]

2001 (1)

J. Schmidt, M. Kerr, and A. Cuevas, “Surface passivation of silicon solar cells using plasma-enhanced chemical-vapour-deposited SiN films and thin thermal SiO 2 /plasma SiN stacks,” Semicond. Sci. Technol. 16(3), 164–170 (2001).
[Crossref]

2000 (2)

A. G. Aberle, “Surface passivation of crystalline silicon solar cells: a review,” Prog. Photovolt. Res. Appl. 8(5), 473–487 (2000).
[Crossref]

H. P. Ho, S. Y. Wu, and P. K. Chu, “A simplified surface photoreflectance measurement system,” Meas. Sci. Technol. 11(9), 1348–1351 (2000).
[Crossref]

1998 (3)

S. Ghosh and B. M. Arora, “Photoreflectance spectroscopy with white light pump beam,” Rev. Sci. Instrum. 69(3), 1261–1266 (1998).
[Crossref]

J. Linnros, “Carrier lifetime measurements using free carrier absorption transients. I. Principle and injection dependence,” J. Appl. Phys. 84(1), 275–283 (1998).
[Crossref]

J. W. Klaus, A. W. Ott, A. C. Dillon, and S. M. George, “Atomic layer controlled growth of Si3N4 films using sequential surface reactions,” Surf. Sci. 418(1), L14–L19 (1998).
[Crossref]

1996 (1)

R. A. Sinton and A. Cuevas, “Contactless determination of current–voltage characteristics and minority-carrier lifetimes in semiconductors from quasi-steady-state photoconductance data,” Appl. Phys. Lett. 69(17), 2510–2512 (1996).
[Crossref]

1994 (3)

M. B. Suddendorf and M. G. Somekh, “Single beam photoreflectance microscopy system with electronic feedback,” Electron. Lett. 30(5), 398–399 (1994).
[Crossref]

S. W. Glunz, A. B. Sproul, W. Warta, and W. Wettling, “Injection‐level‐dependent recombination velocities at the Si‐SiO 2 interface for various dopant concentrations,” J. Appl. Phys. 75(3), 1611–1615 (1994).
[Crossref]

J. R. Goldman and J. A. Prybyla, “Ultrafast Dynamics of Laser-Excited Electron Distributions in Silicon,” Phys. Rev. Lett. 72(9), 1364–1367 (1994).
[Crossref] [PubMed]

1992 (2)

F. Sanii, F. P. Giles, R. J. Schwartz, and J. L. Gray, “Contactless nondestructive measurement of bulk and surface recombination using frequency-modulated free carrier absorption,” Solid-State Electron. 35(3), 311–317 (1992).
[Crossref]

A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9(3), 405–414 (1992).
[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]

1987 (2)

K. L. Luke and L. Cheng, “Analysis of the interaction of a laser pulse with a silicon wafer: Determination of bulk lifetime and surface recombination velocity,” J. Appl. Phys. 61(6), 2282–2293 (1987).
[Crossref]

R. Soref and B. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987).
[Crossref]

1985 (1)

N. F. Scherer, J. L. Knee, D. D. Smith, and A. H. Zewail, “Femtosecond photofragment spectroscopy: The reaction ICN→ CN+ I,” J. Phys. Chem. 89(24), 5141–5143 (1985).
[Crossref]

1979 (1)

K. G. Svantesson, “Determination of the interband and the free carrier absorption constants in silicon at high-level photoinjection,” J. Phys. Appl. Phys. 12(3), 425–436 (1979).
[Crossref]

1978 (1)

D. K. Schroder, R. N. Thomas, and J. C. Swartz, “Free carrier absorption in silicon,” IEEE J. Solid-State Circuits 13(1), 180–187 (1978).
[Crossref]

1973 (1)

J. F. Reintjes and J. C. McGroddy, “Indirect two-photon transitions in Si at 1.06 μm,” Phys. Rev. Lett. 30(19), 901–903 (1973).
[Crossref]

Aberle, A. G.

A. G. Aberle, “Surface passivation of crystalline silicon solar cells: a review,” Prog. Photovolt. Res. Appl. 8(5), 473–487 (2000).
[Crossref]

Arora, B. M.

S. Ghosh and B. M. Arora, “Photoreflectance spectroscopy with white light pump beam,” Rev. Sci. Instrum. 69(3), 1261–1266 (1998).
[Crossref]

Arslan, I.

M. L. Taheri, E. A. Stach, I. Arslan, P. A. Crozier, B. C. Kabius, T. LaGrange, A. M. Minor, S. Takeda, M. Tanase, J. B. Wagner, and R. Sharma, “Current status and future directions for in situ transmission electron microscopy,” Ultramicroscopy 170, 86–95 (2016).
[Crossref] [PubMed]

Baehtz, C.

P. R. Kidambi, R. Blume, J. Kling, J. B. Wagner, C. Baehtz, R. S. Weatherup, R. Schloegl, B. C. Bayer, and S. Hofmann, “In Situ Observations during Chemical Vapor Deposition of Hexagonal Boron Nitride on Polycrystalline Copper,” Chem. Mater. 26(22), 6380–6392 (2014).
[Crossref] [PubMed]

Bayer, B. C.

P. R. Kidambi, R. Blume, J. Kling, J. B. Wagner, C. Baehtz, R. S. Weatherup, R. Schloegl, B. C. Bayer, and S. Hofmann, “In Situ Observations during Chemical Vapor Deposition of Hexagonal Boron Nitride on Polycrystalline Copper,” Chem. Mater. 26(22), 6380–6392 (2014).
[Crossref] [PubMed]

Beil, A.

V. Hopfe, D. W. Sheel, C. I. M. A. Spee, R. Tell, P. Martin, A. Beil, M. Pemble, R. Weiss, U. Vogt, and W. Graehlert, “In-situ monitoring for CVD processes,” Thin Solid Films 442(1-2), 60–65 (2003).
[Crossref]

Bein, B.

B. Bein, H.-C. Hsing, S. J. Callori, J. Sinsheimer, P. V. Chinta, R. L. Headrick, and M. Dawber, “In situ X-ray diffraction and the evolution of polarization during the growth of ferroelectric superlattices,” Nat. Commun. 6(1), 10136 (2015).
[Crossref] [PubMed]

Bennett, B.

R. Soref and B. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987).
[Crossref]

Bergmann, J.

W. I. Ndebeka, P. H. Neethling, E. G. Rohwer, C. M. Steenkamp, J. Bergmann, and H. Stafast, “Interband and free charge carrier absorption in silicon at 800 nm: experiments and model calculations,” Appl. Phys. B 123(10), 253 (2017).
[Crossref]

Blume, R.

P. R. Kidambi, R. Blume, J. Kling, J. B. Wagner, C. Baehtz, R. S. Weatherup, R. Schloegl, B. C. Bayer, and S. Hofmann, “In Situ Observations during Chemical Vapor Deposition of Hexagonal Boron Nitride on Polycrystalline Copper,” Chem. Mater. 26(22), 6380–6392 (2014).
[Crossref] [PubMed]

Boyd, K. M. W.

K. M. W. Boyd and R. N. Kleiman, “Quasi-Steady-State Free Carrier Absorption Measurements of Effective Carrier Lifetime in Silicon,” IEEE J. Photovolt. 9(1), 64–71 (2019).
[Crossref]

Bristow, A. D.

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

Cai, X.

M. M. Jadidi, R. J. Suess, C. Tan, X. Cai, K. Watanabe, T. Taniguchi, A. B. Sushkov, M. Mittendorff, J. Hone, H. D. Drew, M. S. Fuhrer, and T. E. Murphy, “Tunable Ultrafast Thermal Relaxation in Graphene Measured by Continuous-Wave Photomixing,” Phys. Rev. Lett. 117(25), 257401 (2016).
[Crossref] [PubMed]

Callori, S. J.

B. Bein, H.-C. Hsing, S. J. Callori, J. Sinsheimer, P. V. Chinta, R. L. Headrick, and M. Dawber, “In situ X-ray diffraction and the evolution of polarization during the growth of ferroelectric superlattices,” Nat. Commun. 6(1), 10136 (2015).
[Crossref] [PubMed]

Capobianco, J. A.

F. Vetrone, R. Naccache, A. Zamarrón, A. Juarranz de la Fuente, F. Sanz-Rodríguez, L. Martinez Maestro, E. Martín Rodriguez, D. Jaque, J. García Solé, and J. A. Capobianco, “Temperature Sensing Using Fluorescent Nanothermometers,” ACS Nano 4(6), 3254–3258 (2010).
[Crossref] [PubMed]

Chabal, Y. J.

M. M. Frank, Y. J. Chabal, and G. D. Wilk, “Nucleation and interface formation mechanisms in atomic layer deposition of gate oxides,” Appl. Phys. Lett. 82(26), 4758–4760 (2003).
[Crossref]

Cheng, L.

K. L. Luke and L. Cheng, “Analysis of the interaction of a laser pulse with a silicon wafer: Determination of bulk lifetime and surface recombination velocity,” J. Appl. Phys. 61(6), 2282–2293 (1987).
[Crossref]

Chi, L.

Z.-J. Wang, G. Weinberg, Q. Zhang, T. Lunkenbein, A. Klein-Hoffmann, M. Kurnatowska, M. Plodinec, Q. Li, L. Chi, R. Schloegl, and M.-G. Willinger, “Direct Observation of Graphene Growth and Associated Copper Substrate Dynamics by in Situ Scanning Electron Microscopy,” ACS Nano 9(2), 1506–1519 (2015).
[Crossref] [PubMed]

Chinta, P. V.

B. Bein, H.-C. Hsing, S. J. Callori, J. Sinsheimer, P. V. Chinta, R. L. Headrick, and M. Dawber, “In situ X-ray diffraction and the evolution of polarization during the growth of ferroelectric superlattices,” Nat. Commun. 6(1), 10136 (2015).
[Crossref] [PubMed]

Chu, P. K.

H. P. Ho, S. Y. Wu, and P. K. Chu, “A simplified surface photoreflectance measurement system,” Meas. Sci. Technol. 11(9), 1348–1351 (2000).
[Crossref]

Citarella, G.

G. Citarella, S. von Aichberger, and M. Kunst, “Microwave photoconductivity techniques for the characterization of semiconductors,” Mater. Sci. Eng. B 91-92, 224–228 (2002).
[Crossref]

Crozier, P. A.

M. L. Taheri, E. A. Stach, I. Arslan, P. A. Crozier, B. C. Kabius, T. LaGrange, A. M. Minor, S. Takeda, M. Tanase, J. B. Wagner, and R. Sharma, “Current status and future directions for in situ transmission electron microscopy,” Ultramicroscopy 170, 86–95 (2016).
[Crossref] [PubMed]

Cuevas, A.

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[Crossref]

Scrinzi, A.

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, “Time-resolved atomic inner-shell spectroscopy,” Nature 419(6909), 803–807 (2002).
[Crossref] [PubMed]

Seibert, M. M.

K. E. Schmidt, J. Schulz, M. M. Seibert, R. L. Shoeman, R. Sierra, H. Soltau, D. Starodub, F. Stellato, S. Stern, and L. Strüder, “Time-resolved protein nanocrystallography using an X-ray free-electron laser,” (2011).

Sharma, R.

M. L. Taheri, E. A. Stach, I. Arslan, P. A. Crozier, B. C. Kabius, T. LaGrange, A. M. Minor, S. Takeda, M. Tanase, J. B. Wagner, and R. Sharma, “Current status and future directions for in situ transmission electron microscopy,” Ultramicroscopy 170, 86–95 (2016).
[Crossref] [PubMed]

Sheel, D. W.

V. Hopfe, D. W. Sheel, C. I. M. A. Spee, R. Tell, P. Martin, A. Beil, M. Pemble, R. Weiss, U. Vogt, and W. Graehlert, “In-situ monitoring for CVD processes,” Thin Solid Films 442(1-2), 60–65 (2003).
[Crossref]

Sheik-Bahae, M.

A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9(3), 405–414 (1992).
[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]

Shoeman, R. L.

K. E. Schmidt, J. Schulz, M. M. Seibert, R. L. Shoeman, R. Sierra, H. Soltau, D. Starodub, F. Stellato, S. Stern, and L. Strüder, “Time-resolved protein nanocrystallography using an X-ray free-electron laser,” (2011).

Sierra, R.

K. E. Schmidt, J. Schulz, M. M. Seibert, R. L. Shoeman, R. Sierra, H. Soltau, D. Starodub, F. Stellato, S. Stern, and L. Strüder, “Time-resolved protein nanocrystallography using an X-ray free-electron laser,” (2011).

Sinsheimer, J.

B. Bein, H.-C. Hsing, S. J. Callori, J. Sinsheimer, P. V. Chinta, R. L. Headrick, and M. Dawber, “In situ X-ray diffraction and the evolution of polarization during the growth of ferroelectric superlattices,” Nat. Commun. 6(1), 10136 (2015).
[Crossref] [PubMed]

Sinton, R. A.

R. A. Sinton and A. Cuevas, “Contactless determination of current–voltage characteristics and minority-carrier lifetimes in semiconductors from quasi-steady-state photoconductance data,” Appl. Phys. Lett. 69(17), 2510–2512 (1996).
[Crossref]

Smith, D. D.

N. F. Scherer, J. L. Knee, D. D. Smith, and A. H. Zewail, “Femtosecond photofragment spectroscopy: The reaction ICN→ CN+ I,” J. Phys. Chem. 89(24), 5141–5143 (1985).
[Crossref]

Soltau, H.

K. E. Schmidt, J. Schulz, M. M. Seibert, R. L. Shoeman, R. Sierra, H. Soltau, D. Starodub, F. Stellato, S. Stern, and L. Strüder, “Time-resolved protein nanocrystallography using an X-ray free-electron laser,” (2011).

Somekh, M. G.

M. B. Suddendorf and M. G. Somekh, “Single beam photoreflectance microscopy system with electronic feedback,” Electron. Lett. 30(5), 398–399 (1994).
[Crossref]

Soref, R.

R. Soref and B. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987).
[Crossref]

Spee, C. I. M. A.

V. Hopfe, D. W. Sheel, C. I. M. A. Spee, R. Tell, P. Martin, A. Beil, M. Pemble, R. Weiss, U. Vogt, and W. Graehlert, “In-situ monitoring for CVD processes,” Thin Solid Films 442(1-2), 60–65 (2003).
[Crossref]

Sproul, A. B.

S. W. Glunz, A. B. Sproul, W. Warta, and W. Wettling, “Injection‐level‐dependent recombination velocities at the Si‐SiO 2 interface for various dopant concentrations,” J. Appl. Phys. 75(3), 1611–1615 (1994).
[Crossref]

Stach, E. A.

M. L. Taheri, E. A. Stach, I. Arslan, P. A. Crozier, B. C. Kabius, T. LaGrange, A. M. Minor, S. Takeda, M. Tanase, J. B. Wagner, and R. Sharma, “Current status and future directions for in situ transmission electron microscopy,” Ultramicroscopy 170, 86–95 (2016).
[Crossref] [PubMed]

Stafast, H.

W. I. Ndebeka, P. H. Neethling, E. G. Rohwer, C. M. Steenkamp, J. Bergmann, and H. Stafast, “Interband and free charge carrier absorption in silicon at 800 nm: experiments and model calculations,” Appl. Phys. B 123(10), 253 (2017).
[Crossref]

Starodub, D.

K. E. Schmidt, J. Schulz, M. M. Seibert, R. L. Shoeman, R. Sierra, H. Soltau, D. Starodub, F. Stellato, S. Stern, and L. Strüder, “Time-resolved protein nanocrystallography using an X-ray free-electron laser,” (2011).

Steenkamp, C. M.

W. I. Ndebeka, P. H. Neethling, E. G. Rohwer, C. M. Steenkamp, J. Bergmann, and H. Stafast, “Interband and free charge carrier absorption in silicon at 800 nm: experiments and model calculations,” Appl. Phys. B 123(10), 253 (2017).
[Crossref]

Stellato, F.

K. E. Schmidt, J. Schulz, M. M. Seibert, R. L. Shoeman, R. Sierra, H. Soltau, D. Starodub, F. Stellato, S. Stern, and L. Strüder, “Time-resolved protein nanocrystallography using an X-ray free-electron laser,” (2011).

Stern, S.

K. E. Schmidt, J. Schulz, M. M. Seibert, R. L. Shoeman, R. Sierra, H. Soltau, D. Starodub, F. Stellato, S. Stern, and L. Strüder, “Time-resolved protein nanocrystallography using an X-ray free-electron laser,” (2011).

Strüder, L.

K. E. Schmidt, J. Schulz, M. M. Seibert, R. L. Shoeman, R. Sierra, H. Soltau, D. Starodub, F. Stellato, S. Stern, and L. Strüder, “Time-resolved protein nanocrystallography using an X-ray free-electron laser,” (2011).

Suddendorf, M. B.

M. B. Suddendorf and M. G. Somekh, “Single beam photoreflectance microscopy system with electronic feedback,” Electron. Lett. 30(5), 398–399 (1994).
[Crossref]

Suess, R. J.

M. M. Jadidi, R. J. Suess, C. Tan, X. Cai, K. Watanabe, T. Taniguchi, A. B. Sushkov, M. Mittendorff, J. Hone, H. D. Drew, M. S. Fuhrer, and T. E. Murphy, “Tunable Ultrafast Thermal Relaxation in Graphene Measured by Continuous-Wave Photomixing,” Phys. Rev. Lett. 117(25), 257401 (2016).
[Crossref] [PubMed]

Sushkov, A. B.

M. M. Jadidi, R. J. Suess, C. Tan, X. Cai, K. Watanabe, T. Taniguchi, A. B. Sushkov, M. Mittendorff, J. Hone, H. D. Drew, M. S. Fuhrer, and T. E. Murphy, “Tunable Ultrafast Thermal Relaxation in Graphene Measured by Continuous-Wave Photomixing,” Phys. Rev. Lett. 117(25), 257401 (2016).
[Crossref] [PubMed]

Svantesson, K. G.

K. G. Svantesson, “Determination of the interband and the free carrier absorption constants in silicon at high-level photoinjection,” J. Phys. Appl. Phys. 12(3), 425–436 (1979).
[Crossref]

Swartz, J. C.

D. K. Schroder, R. N. Thomas, and J. C. Swartz, “Free carrier absorption in silicon,” IEEE J. Solid-State Circuits 13(1), 180–187 (1978).
[Crossref]

Taheri, M. L.

M. L. Taheri, E. A. Stach, I. Arslan, P. A. Crozier, B. C. Kabius, T. LaGrange, A. M. Minor, S. Takeda, M. Tanase, J. B. Wagner, and R. Sharma, “Current status and future directions for in situ transmission electron microscopy,” Ultramicroscopy 170, 86–95 (2016).
[Crossref] [PubMed]

Takeda, S.

M. L. Taheri, E. A. Stach, I. Arslan, P. A. Crozier, B. C. Kabius, T. LaGrange, A. M. Minor, S. Takeda, M. Tanase, J. B. Wagner, and R. Sharma, “Current status and future directions for in situ transmission electron microscopy,” Ultramicroscopy 170, 86–95 (2016).
[Crossref] [PubMed]

Tan, C.

M. M. Jadidi, R. J. Suess, C. Tan, X. Cai, K. Watanabe, T. Taniguchi, A. B. Sushkov, M. Mittendorff, J. Hone, H. D. Drew, M. S. Fuhrer, and T. E. Murphy, “Tunable Ultrafast Thermal Relaxation in Graphene Measured by Continuous-Wave Photomixing,” Phys. Rev. Lett. 117(25), 257401 (2016).
[Crossref] [PubMed]

Tanase, M.

M. L. Taheri, E. A. Stach, I. Arslan, P. A. Crozier, B. C. Kabius, T. LaGrange, A. M. Minor, S. Takeda, M. Tanase, J. B. Wagner, and R. Sharma, “Current status and future directions for in situ transmission electron microscopy,” Ultramicroscopy 170, 86–95 (2016).
[Crossref] [PubMed]

Taniguchi, T.

M. M. Jadidi, R. J. Suess, C. Tan, X. Cai, K. Watanabe, T. Taniguchi, A. B. Sushkov, M. Mittendorff, J. Hone, H. D. Drew, M. S. Fuhrer, and T. E. Murphy, “Tunable Ultrafast Thermal Relaxation in Graphene Measured by Continuous-Wave Photomixing,” Phys. Rev. Lett. 117(25), 257401 (2016).
[Crossref] [PubMed]

Tell, R.

V. Hopfe, D. W. Sheel, C. I. M. A. Spee, R. Tell, P. Martin, A. Beil, M. Pemble, R. Weiss, U. Vogt, and W. Graehlert, “In-situ monitoring for CVD processes,” Thin Solid Films 442(1-2), 60–65 (2003).
[Crossref]

Thomas, R. N.

D. K. Schroder, R. N. Thomas, and J. C. Swartz, “Free carrier absorption in silicon,” IEEE J. Solid-State Circuits 13(1), 180–187 (1978).
[Crossref]

Uiberacker, M.

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, “Time-resolved atomic inner-shell spectroscopy,” Nature 419(6909), 803–807 (2002).
[Crossref] [PubMed]

van de Sanden, M. C. M.

E. Langereis, S. B. S. Heil, M. C. M. van de Sanden, and W. M. M. Kessels, “In situ spectroscopic ellipsometry study on the growth of ultrathin TiN films by plasma-assisted atomic layer deposition,” J. Appl. Phys. 100(2), 023534 (2006).
[Crossref]

van Driel, H. M.

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

Van Stryland, E. W.

A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9(3), 405–414 (1992).
[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]

Vázquez-Sánchez, E.

J. García-Martín, J. Gómez-Gil, and E. Vázquez-Sánchez, “Non-Destructive Techniques Based on Eddy Current Testing,” Sensors (Basel) 11(3), 2525–2565 (2011).
[Crossref] [PubMed]

Vetrone, F.

F. Vetrone, R. Naccache, A. Zamarrón, A. Juarranz de la Fuente, F. Sanz-Rodríguez, L. Martinez Maestro, E. Martín Rodriguez, D. Jaque, J. García Solé, and J. A. Capobianco, “Temperature Sensing Using Fluorescent Nanothermometers,” ACS Nano 4(6), 3254–3258 (2010).
[Crossref] [PubMed]

Vogt, U.

V. Hopfe, D. W. Sheel, C. I. M. A. Spee, R. Tell, P. Martin, A. Beil, M. Pemble, R. Weiss, U. Vogt, and W. Graehlert, “In-situ monitoring for CVD processes,” Thin Solid Films 442(1-2), 60–65 (2003).
[Crossref]

von Aichberger, S.

G. Citarella, S. von Aichberger, and M. Kunst, “Microwave photoconductivity techniques for the characterization of semiconductors,” Mater. Sci. Eng. B 91-92, 224–228 (2002).
[Crossref]

Wagner, J. B.

M. L. Taheri, E. A. Stach, I. Arslan, P. A. Crozier, B. C. Kabius, T. LaGrange, A. M. Minor, S. Takeda, M. Tanase, J. B. Wagner, and R. Sharma, “Current status and future directions for in situ transmission electron microscopy,” Ultramicroscopy 170, 86–95 (2016).
[Crossref] [PubMed]

P. R. Kidambi, R. Blume, J. Kling, J. B. Wagner, C. Baehtz, R. S. Weatherup, R. Schloegl, B. C. Bayer, and S. Hofmann, “In Situ Observations during Chemical Vapor Deposition of Hexagonal Boron Nitride on Polycrystalline Copper,” Chem. Mater. 26(22), 6380–6392 (2014).
[Crossref] [PubMed]

Wang, J.

Wang, Z.-J.

Z.-J. Wang, G. Weinberg, Q. Zhang, T. Lunkenbein, A. Klein-Hoffmann, M. Kurnatowska, M. Plodinec, Q. Li, L. Chi, R. Schloegl, and M.-G. Willinger, “Direct Observation of Graphene Growth and Associated Copper Substrate Dynamics by in Situ Scanning Electron Microscopy,” ACS Nano 9(2), 1506–1519 (2015).
[Crossref] [PubMed]

Warta, W.

J. Isenberg and W. Warta, “Free carrier absorption in heavily doped silicon layers,” Appl. Phys. Lett. 84(13), 2265–2267 (2004).
[Crossref]

S. W. Glunz, A. B. Sproul, W. Warta, and W. Wettling, “Injection‐level‐dependent recombination velocities at the Si‐SiO 2 interface for various dopant concentrations,” J. Appl. Phys. 75(3), 1611–1615 (1994).
[Crossref]

Watanabe, K.

M. M. Jadidi, R. J. Suess, C. Tan, X. Cai, K. Watanabe, T. Taniguchi, A. B. Sushkov, M. Mittendorff, J. Hone, H. D. Drew, M. S. Fuhrer, and T. E. Murphy, “Tunable Ultrafast Thermal Relaxation in Graphene Measured by Continuous-Wave Photomixing,” Phys. Rev. Lett. 117(25), 257401 (2016).
[Crossref] [PubMed]

Weatherup, R. S.

P. R. Kidambi, R. Blume, J. Kling, J. B. Wagner, C. Baehtz, R. S. Weatherup, R. Schloegl, B. C. Bayer, and S. Hofmann, “In Situ Observations during Chemical Vapor Deposition of Hexagonal Boron Nitride on Polycrystalline Copper,” Chem. Mater. 26(22), 6380–6392 (2014).
[Crossref] [PubMed]

Wei, T. H.

Wei, T.-H.

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]

Weinberg, G.

Z.-J. Wang, G. Weinberg, Q. Zhang, T. Lunkenbein, A. Klein-Hoffmann, M. Kurnatowska, M. Plodinec, Q. Li, L. Chi, R. Schloegl, and M.-G. Willinger, “Direct Observation of Graphene Growth and Associated Copper Substrate Dynamics by in Situ Scanning Electron Microscopy,” ACS Nano 9(2), 1506–1519 (2015).
[Crossref] [PubMed]

Weiss, R.

V. Hopfe, D. W. Sheel, C. I. M. A. Spee, R. Tell, P. Martin, A. Beil, M. Pemble, R. Weiss, U. Vogt, and W. Graehlert, “In-situ monitoring for CVD processes,” Thin Solid Films 442(1-2), 60–65 (2003).
[Crossref]

Westerwalbesloh, T.

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, “Time-resolved atomic inner-shell spectroscopy,” Nature 419(6909), 803–807 (2002).
[Crossref] [PubMed]

Wettling, W.

S. W. Glunz, A. B. Sproul, W. Warta, and W. Wettling, “Injection‐level‐dependent recombination velocities at the Si‐SiO 2 interface for various dopant concentrations,” J. Appl. Phys. 75(3), 1611–1615 (1994).
[Crossref]

Wilk, G. D.

M. M. Frank, Y. J. Chabal, and G. D. Wilk, “Nucleation and interface formation mechanisms in atomic layer deposition of gate oxides,” Appl. Phys. Lett. 82(26), 4758–4760 (2003).
[Crossref]

Willinger, M.-G.

Z.-J. Wang, G. Weinberg, Q. Zhang, T. Lunkenbein, A. Klein-Hoffmann, M. Kurnatowska, M. Plodinec, Q. Li, L. Chi, R. Schloegl, and M.-G. Willinger, “Direct Observation of Graphene Growth and Associated Copper Substrate Dynamics by in Situ Scanning Electron Microscopy,” ACS Nano 9(2), 1506–1519 (2015).
[Crossref] [PubMed]

Wu, S. Y.

H. P. Ho, S. Y. Wu, and P. K. Chu, “A simplified surface photoreflectance measurement system,” Meas. Sci. Technol. 11(9), 1348–1351 (2000).
[Crossref]

Yakovlev, V.

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, “Time-resolved atomic inner-shell spectroscopy,” Nature 419(6909), 803–807 (2002).
[Crossref] [PubMed]

Young, J.

Zamarrón, A.

F. Vetrone, R. Naccache, A. Zamarrón, A. Juarranz de la Fuente, F. Sanz-Rodríguez, L. Martinez Maestro, E. Martín Rodriguez, D. Jaque, J. García Solé, and J. A. Capobianco, “Temperature Sensing Using Fluorescent Nanothermometers,” ACS Nano 4(6), 3254–3258 (2010).
[Crossref] [PubMed]

Zewail, A. H.

N. F. Scherer, J. L. Knee, D. D. Smith, and A. H. Zewail, “Femtosecond photofragment spectroscopy: The reaction ICN→ CN+ I,” J. Phys. Chem. 89(24), 5141–5143 (1985).
[Crossref]

Zhang, Q.

Z.-J. Wang, G. Weinberg, Q. Zhang, T. Lunkenbein, A. Klein-Hoffmann, M. Kurnatowska, M. Plodinec, Q. Li, L. Chi, R. Schloegl, and M.-G. Willinger, “Direct Observation of Graphene Growth and Associated Copper Substrate Dynamics by in Situ Scanning Electron Microscopy,” ACS Nano 9(2), 1506–1519 (2015).
[Crossref] [PubMed]

ACS Nano (2)

F. Vetrone, R. Naccache, A. Zamarrón, A. Juarranz de la Fuente, F. Sanz-Rodríguez, L. Martinez Maestro, E. Martín Rodriguez, D. Jaque, J. García Solé, and J. A. Capobianco, “Temperature Sensing Using Fluorescent Nanothermometers,” ACS Nano 4(6), 3254–3258 (2010).
[Crossref] [PubMed]

Z.-J. Wang, G. Weinberg, Q. Zhang, T. Lunkenbein, A. Klein-Hoffmann, M. Kurnatowska, M. Plodinec, Q. Li, L. Chi, R. Schloegl, and M.-G. Willinger, “Direct Observation of Graphene Growth and Associated Copper Substrate Dynamics by in Situ Scanning Electron Microscopy,” ACS Nano 9(2), 1506–1519 (2015).
[Crossref] [PubMed]

Appl. Phys. B (1)

W. I. Ndebeka, P. H. Neethling, E. G. Rohwer, C. M. Steenkamp, J. Bergmann, and H. Stafast, “Interband and free charge carrier absorption in silicon at 800 nm: experiments and model calculations,” Appl. Phys. B 123(10), 253 (2017).
[Crossref]

Appl. Phys. Lett. (4)

R. A. Sinton and A. Cuevas, “Contactless determination of current–voltage characteristics and minority-carrier lifetimes in semiconductors from quasi-steady-state photoconductance data,” Appl. Phys. Lett. 69(17), 2510–2512 (1996).
[Crossref]

M. M. Frank, Y. J. Chabal, and G. D. Wilk, “Nucleation and interface formation mechanisms in atomic layer deposition of gate oxides,” Appl. Phys. Lett. 82(26), 4758–4760 (2003).
[Crossref]

J. Isenberg and W. Warta, “Free carrier absorption in heavily doped silicon layers,” Appl. Phys. Lett. 84(13), 2265–2267 (2004).
[Crossref]

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

Chem. Mater. (1)

P. R. Kidambi, R. Blume, J. Kling, J. B. Wagner, C. Baehtz, R. S. Weatherup, R. Schloegl, B. C. Bayer, and S. Hofmann, “In Situ Observations during Chemical Vapor Deposition of Hexagonal Boron Nitride on Polycrystalline Copper,” Chem. Mater. 26(22), 6380–6392 (2014).
[Crossref] [PubMed]

Electron. Lett. (1)

M. B. Suddendorf and M. G. Somekh, “Single beam photoreflectance microscopy system with electronic feedback,” Electron. Lett. 30(5), 398–399 (1994).
[Crossref]

IEEE J. Photovolt. (1)

K. M. W. Boyd and R. N. Kleiman, “Quasi-Steady-State Free Carrier Absorption Measurements of Effective Carrier Lifetime in Silicon,” IEEE J. Photovolt. 9(1), 64–71 (2019).
[Crossref]

IEEE J. Quantum Electron. (2)

R. Soref and B. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987).
[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]

IEEE J. Solid-State Circuits (1)

D. K. Schroder, R. N. Thomas, and J. C. Swartz, “Free carrier absorption in silicon,” IEEE J. Solid-State Circuits 13(1), 180–187 (1978).
[Crossref]

J. Appl. Phys. (4)

S. W. Glunz, A. B. Sproul, W. Warta, and W. Wettling, “Injection‐level‐dependent recombination velocities at the Si‐SiO 2 interface for various dopant concentrations,” J. Appl. Phys. 75(3), 1611–1615 (1994).
[Crossref]

J. Linnros, “Carrier lifetime measurements using free carrier absorption transients. I. Principle and injection dependence,” J. Appl. Phys. 84(1), 275–283 (1998).
[Crossref]

E. Langereis, S. B. S. Heil, M. C. M. van de Sanden, and W. M. M. Kessels, “In situ spectroscopic ellipsometry study on the growth of ultrathin TiN films by plasma-assisted atomic layer deposition,” J. Appl. Phys. 100(2), 023534 (2006).
[Crossref]

K. L. Luke and L. Cheng, “Analysis of the interaction of a laser pulse with a silicon wafer: Determination of bulk lifetime and surface recombination velocity,” J. Appl. Phys. 61(6), 2282–2293 (1987).
[Crossref]

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

J. Phys. Appl. Phys. (1)

K. G. Svantesson, “Determination of the interband and the free carrier absorption constants in silicon at high-level photoinjection,” J. Phys. Appl. Phys. 12(3), 425–436 (1979).
[Crossref]

J. Phys. Chem. (1)

N. F. Scherer, J. L. Knee, D. D. Smith, and A. H. Zewail, “Femtosecond photofragment spectroscopy: The reaction ICN→ CN+ I,” J. Phys. Chem. 89(24), 5141–5143 (1985).
[Crossref]

Mater. Sci. Eng. B (1)

G. Citarella, S. von Aichberger, and M. Kunst, “Microwave photoconductivity techniques for the characterization of semiconductors,” Mater. Sci. Eng. B 91-92, 224–228 (2002).
[Crossref]

Meas. Sci. Technol. (1)

H. P. Ho, S. Y. Wu, and P. K. Chu, “A simplified surface photoreflectance measurement system,” Meas. Sci. Technol. 11(9), 1348–1351 (2000).
[Crossref]

Nat. Commun. (1)

B. Bein, H.-C. Hsing, S. J. Callori, J. Sinsheimer, P. V. Chinta, R. L. Headrick, and M. Dawber, “In situ X-ray diffraction and the evolution of polarization during the growth of ferroelectric superlattices,” Nat. Commun. 6(1), 10136 (2015).
[Crossref] [PubMed]

Nature (1)

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, “Time-resolved atomic inner-shell spectroscopy,” Nature 419(6909), 803–807 (2002).
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Phys. Rev. Lett. (3)

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Figures (4)

Fig. 1
Fig. 1 Schematic Diagram of Experimental Apparatus.
Fig. 2
Fig. 2 Visual representation of single-beam self-induced absorption signal at different temporal points along a single excitation period. (a) Laser beam passing through wafer from top to bottom, with the AC and DC components represented by red and green shading, respectively. The AC component generates a free-carrier population at the same frequency with a phase lag of ϕFCA = tan–1 ωτ, shown for the peak condition of Eq. (10) when ωτ = 1. The DC component is periodically attenuated by free-carrier absorption. Band-to-band absorption effects are not shown since they have been removed during signal normalization of Eq. (8). (b) Magnitude of the free-carrier population, n, as a function of time. (c) Decomposition of the signal, S, from the transmitted laser beam. The ‘MOD’ component is due to the AC modulation of the laser itself. The ‘FCA’ component is due to periodic free-carrier absorption of the DC component of the laser and is proportional to the free-carrier population and opposite in sign. Equation (9) includes the superposition of these signals.
Fig. 3
Fig. 3 Comparison of single-beam self-induced absorption and dual-beam modulated pump/probe techniques. The single-beam data is given in absolute units whereas the dual-beam data is scaled arbitrarily for comparison. Symbols represent experimental datapoints and solid lines represent the best-fit to the frequency dependence in Eq. (10). The intensity of the pump laser illuminating the silicon wafer is 1.412 W/cm2 for both single and dual-beam data sets.
Fig. 4
Fig. 4 Plot of Ya/K for various incident powers. The plot is linear as expected, and yields a FCA cross section of σFCA = (3.95 ± 0.1) × 10−10 μm2.

Equations (27)

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P( t )= 0 T 2 P 0 π w 2 e 2 r 2 w 2  ( 1+mcosωt ) e αW α FCA W ( 2πrdr ).
P( t )= 0 T 4 P 0 w 2 e 2 r 2 w 2 ( 1+mcosωt ) e αW ( 1 α FCA W ) rdr.
N DC = 2 P 0 λ f a τ hcπ w 2 W ,
N AC = 2 P 0 λ f a τ hcπ w 2 W m 1+ ω 2 τ 2 ,
P( t )=T P 0 ( 1+mcosωt ) e αW ( 1 1 2 σ FCA ( N DC + N AC cos( ωt ϕ FCA ) )W ).
P( t )=T P 0 e αW [ 1 σ FCA W 2 N DC σ FCA W 4 m N AC cos ϕ FCA +( 1 σ FCA W 2 N DC σ FCA W 2m N AC cos ϕ FCA )mcosωt σ FCA W 2 N AC sin ϕ FCA sinωt σ FCA W 2 m N AC cos ϕ FCA cos2ωt σ FCA W 2 m N AC sin ϕ FCA sin2ωt ].
S ω ( t )=ζ( ω )T P 0 e αW [ ( 1 σ FCA W 2 N DC σ FCA W 2m N AC cos ϕ FCA )mcosωt σ FCA W 2 N AC sin ϕ FCA sinωt ],
S ω ( t )=ζ( ω )T P 0 e αW m [ ( 1 σ FCA P 0 λ f a τ hcπ w 2 σ FCA P 0 λ f a τ hcπ w 2 1 1+ ω 2 τ 2 )cosωt σ FCA P 0 λ f a τ hcπ w 2 ωτ 1+ ω 2 τ 2 sinωt ].
s( t )=( 1 σ FCA P 0 λ f a τ hcπ w 2 σ FCA P 0 λ f a τ hcπ w 2 1 1+ ω 2 τ 2 )cosωt σ FCA P 0 λ f a τ hcπ w 2 ωτ 1+ ω 2 τ 2 sinωt.
Y= σ FCA P 0 λ f a τ hcπ w 2 [ ωτ 1+ ω 2 τ 2 ].
Y a = σ FCA K P 0 ,
K= λ f a τ hcπ w 2 .
τ min = hcν I 0 λ f a σ FCA ,
n i ( t )= 8 ϕ 0 α f a e αW 2 W( 1 e αW ) k A k e t τ k ,
τ k = ( τ b 1 + α k 2 D ) 1 ,
A k = sin( α k W 2 ) ( α 2 + α k 2 )( α k W+sin α k W ) × [ αsinh( αW 2 )cos( α k W 2 )+ α k cosh( αW 2 )sin( α k W 2 ) ].
n( t )= 0 t n i ( t t )g( t )d t  .
n ˜ ( t )= 8ϕα f a e αW 2 W( 1 e αW ) k A k τ k ( 1+m 1 1+iω τ k e iωt ).
n ˜ ( r,t )= 16 P 0 λα f a e αW 2 hcπ w 2 W( 1 e αW ) e 2 r 2 w 2 k A k τ k ( 1+m 1 1+iω τ k e iωt ).
n ˜ ( r,t )= 2 P 0 λ f a τ hcπ w 2 W e 2 r 2 w 2 ( 1+m 1 1+iωτ e iωt ),
τ= ( τ b 1 +2S/W ) 1 .
n( r,t )= 2 P 0 λ f a τ hcπ w 2 W e 2 r 2 w 2 ( 1+ m 1+ ω 2 τ 2 ( cosωt+ωτsinωt )  ).
n( r,t )= 2 P 0 λ f a τ hcπ w 2 W e 2 r 2 w 2 ( 1+ m 1+ ω 2 τ 2 cos( ωt ϕ FCA )  ),
ϕ FCA = tan 1 ωτ.
n( r,t )= e 2 r 2 w 2 ( N DC + N AC cos( ωt ϕ FCA ) ),
N DC = 2 P 0 λ f a τ hcπ w 2 W ,
N AC = 2 P 0 λ f a τ hcπ w 2 W m 1+ ω 2 τ 2 .

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