Abstract

We have measured the spectra of two-photon absorption coefficients for CdSe and GaSe based on a z-scan technique. By using extremely high peak intensities from subpicosecond laser pulses, we observed novel behavior of effective two-photon absorption for CdSe. We compared our results for both crystals with theory as well as with the previous values measured at three and two wavelengths, respectively. For both CdSe and GaSe crystals our measured values are lower than all those reported previously.

© 2001 Optical Society of America

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  1. D. Kovsh, S. Yang, D. J. Hagan, E. W. Van Stryland, “Nonlinear optical beam propagation for optical limiting,” Appl. Opt. 38, 5168–5180 (1999).
    [Crossref]
  2. I. B. Zotova, Y. J. Ding, “Optical parametric oscillators in the presence of strong two-photon absorption for extended applications of nonlinear optical materials,” Opt. Commun. (to be published).
  3. J. M. Ralston, R. K. Chang, “Optical limiting in semiconductors,” Appl. Phys. Lett. 15, 164–166 (1969).
    [Crossref]
  4. F. Bryukner, V. S. Dneprovskii, V. U. Khattatov, “Two-photon absorption in cadmium selenide,” Sov. J. Quantum Electron. 4, 749–751 (1974).
    [Crossref]
  5. J. H. Bechtel, W. L. Smith, “Two-photon absorption in semiconductors with pisosecond laser pulses,” Phys. Rev. B 13, 3515–3522 (1976).
    [Crossref]
  6. D. C. Hanna, A. J. Turner, “Nonlinear absorption measurements in Proustite (Ag3AsS3) and CdSe,” Opt. Quantum Electron. 8, 213–217 (1976).
    [Crossref]
  7. E. W. Van Stryland, M. A. Woodall, W. E. Williams, M. J. Soileau, “Pulse-width and focal-volume dependence of laser-induced breakdown,” in Laser Induced Damage in Optical Materials, Natl. Bur. Stand. (U.S.) Spec. Pub. 638, 589–613 (1981).
  8. M. Bass, E. W. Van Stryland, A. F. Stewart, “Laser calorimetric measurement of two-photon absorption,” Appl. Phys. Lett. 34, 142–144 (1979).
    [Crossref]
  9. V. S. Dneprovskii, S. M. Ok, “Role of absorption by nonequilibrium carriers in determination of two-photon absorption coefficient of CdSe and GaAs crystals,” Sov. J. Quantum Electron. 3, 559–562 (1976).
  10. G.-M. Schucan, R. G. Ispasoiu, A. M. Fox, J. F. Ryan, “Ultrafast two-photon nonlinearities in CdSe near 1.5 µm studied by interferometric autocorrelation,” IEEE J. Quantum Electron. 34, 1374–1379 (1998).
    [Crossref]
  11. A. Penzkofer, M. Schäffner, X. Bao, “Two-photon absorption and resonant non-phase-matched second-harmonic generation,” Opt. Quantum Electron. 22, 351–367 (1990).
    [Crossref]
  12. K. L. Vodopyanov, “Parametric generation of tunable infrared radiation in ZnGeP2 and GaSe pumped at 3 µm,” J. Opt. Soc. Am. B 10, 1723–1729 (1993).
    [Crossref]
  13. N. C. Fernelius, “Properties of gallium selenide single crystal,” Prog. Cryst. Growth Charact. Mater. 28, 275–353 (1994).
    [Crossref]
  14. F. Adduci, I. M. Catalano, A. Cingolani, A. Minafra, “Direct and indirect two-photon processes in layered semiconductors,” Phys. Rev. B 15, 926–931 (1977).
    [Crossref]
  15. K. L. Vodopyanov, S. B. Mirov, V. G. Voevodin, P. G. Schunemann, “Two-photon absorption in GaSe and CdGeAs2,” Opt. Commun. 155, 47–50 (1998).
    [Crossref]
  16. C. R. Pidgeon, B. S. Wherrett, A. M. Johnston, J. Dempsey, A. Miller, “Two-photon absorption in zinc-blende semiconductors,” Phys. Rev. Lett. 42, 1785–1788 (1979).
    [Crossref]
  17. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, E. W. Van Stryland, “Sensitive measurement of optical nonlinearities,” IEEE J. Quantum Electron. 26, 760–769 (1990).
    [Crossref]

1999 (1)

1998 (2)

G.-M. Schucan, R. G. Ispasoiu, A. M. Fox, J. F. Ryan, “Ultrafast two-photon nonlinearities in CdSe near 1.5 µm studied by interferometric autocorrelation,” IEEE J. Quantum Electron. 34, 1374–1379 (1998).
[Crossref]

K. L. Vodopyanov, S. B. Mirov, V. G. Voevodin, P. G. Schunemann, “Two-photon absorption in GaSe and CdGeAs2,” Opt. Commun. 155, 47–50 (1998).
[Crossref]

1994 (1)

N. C. Fernelius, “Properties of gallium selenide single crystal,” Prog. Cryst. Growth Charact. Mater. 28, 275–353 (1994).
[Crossref]

1993 (1)

1990 (2)

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

A. Penzkofer, M. Schäffner, X. Bao, “Two-photon absorption and resonant non-phase-matched second-harmonic generation,” Opt. Quantum Electron. 22, 351–367 (1990).
[Crossref]

1981 (1)

E. W. Van Stryland, M. A. Woodall, W. E. Williams, M. J. Soileau, “Pulse-width and focal-volume dependence of laser-induced breakdown,” in Laser Induced Damage in Optical Materials, Natl. Bur. Stand. (U.S.) Spec. Pub. 638, 589–613 (1981).

1979 (2)

M. Bass, E. W. Van Stryland, A. F. Stewart, “Laser calorimetric measurement of two-photon absorption,” Appl. Phys. Lett. 34, 142–144 (1979).
[Crossref]

C. R. Pidgeon, B. S. Wherrett, A. M. Johnston, J. Dempsey, A. Miller, “Two-photon absorption in zinc-blende semiconductors,” Phys. Rev. Lett. 42, 1785–1788 (1979).
[Crossref]

1977 (1)

F. Adduci, I. M. Catalano, A. Cingolani, A. Minafra, “Direct and indirect two-photon processes in layered semiconductors,” Phys. Rev. B 15, 926–931 (1977).
[Crossref]

1976 (3)

V. S. Dneprovskii, S. M. Ok, “Role of absorption by nonequilibrium carriers in determination of two-photon absorption coefficient of CdSe and GaAs crystals,” Sov. J. Quantum Electron. 3, 559–562 (1976).

J. H. Bechtel, W. L. Smith, “Two-photon absorption in semiconductors with pisosecond laser pulses,” Phys. Rev. B 13, 3515–3522 (1976).
[Crossref]

D. C. Hanna, A. J. Turner, “Nonlinear absorption measurements in Proustite (Ag3AsS3) and CdSe,” Opt. Quantum Electron. 8, 213–217 (1976).
[Crossref]

1974 (1)

F. Bryukner, V. S. Dneprovskii, V. U. Khattatov, “Two-photon absorption in cadmium selenide,” Sov. J. Quantum Electron. 4, 749–751 (1974).
[Crossref]

1969 (1)

J. M. Ralston, R. K. Chang, “Optical limiting in semiconductors,” Appl. Phys. Lett. 15, 164–166 (1969).
[Crossref]

Adduci, F.

F. Adduci, I. M. Catalano, A. Cingolani, A. Minafra, “Direct and indirect two-photon processes in layered semiconductors,” Phys. Rev. B 15, 926–931 (1977).
[Crossref]

Bao, X.

A. Penzkofer, M. Schäffner, X. Bao, “Two-photon absorption and resonant non-phase-matched second-harmonic generation,” Opt. Quantum Electron. 22, 351–367 (1990).
[Crossref]

Bass, M.

M. Bass, E. W. Van Stryland, A. F. Stewart, “Laser calorimetric measurement of two-photon absorption,” Appl. Phys. Lett. 34, 142–144 (1979).
[Crossref]

Bechtel, J. H.

J. H. Bechtel, W. L. Smith, “Two-photon absorption in semiconductors with pisosecond laser pulses,” Phys. Rev. B 13, 3515–3522 (1976).
[Crossref]

Bryukner, F.

F. Bryukner, V. S. Dneprovskii, V. U. Khattatov, “Two-photon absorption in cadmium selenide,” Sov. J. Quantum Electron. 4, 749–751 (1974).
[Crossref]

Catalano, I. M.

F. Adduci, I. M. Catalano, A. Cingolani, A. Minafra, “Direct and indirect two-photon processes in layered semiconductors,” Phys. Rev. B 15, 926–931 (1977).
[Crossref]

Chang, R. K.

J. M. Ralston, R. K. Chang, “Optical limiting in semiconductors,” Appl. Phys. Lett. 15, 164–166 (1969).
[Crossref]

Cingolani, A.

F. Adduci, I. M. Catalano, A. Cingolani, A. Minafra, “Direct and indirect two-photon processes in layered semiconductors,” Phys. Rev. B 15, 926–931 (1977).
[Crossref]

Dempsey, J.

C. R. Pidgeon, B. S. Wherrett, A. M. Johnston, J. Dempsey, A. Miller, “Two-photon absorption in zinc-blende semiconductors,” Phys. Rev. Lett. 42, 1785–1788 (1979).
[Crossref]

Ding, Y. J.

I. B. Zotova, Y. J. Ding, “Optical parametric oscillators in the presence of strong two-photon absorption for extended applications of nonlinear optical materials,” Opt. Commun. (to be published).

Dneprovskii, V. S.

V. S. Dneprovskii, S. M. Ok, “Role of absorption by nonequilibrium carriers in determination of two-photon absorption coefficient of CdSe and GaAs crystals,” Sov. J. Quantum Electron. 3, 559–562 (1976).

F. Bryukner, V. S. Dneprovskii, V. U. Khattatov, “Two-photon absorption in cadmium selenide,” Sov. J. Quantum Electron. 4, 749–751 (1974).
[Crossref]

Fernelius, N. C.

N. C. Fernelius, “Properties of gallium selenide single crystal,” Prog. Cryst. Growth Charact. Mater. 28, 275–353 (1994).
[Crossref]

Fox, A. M.

G.-M. Schucan, R. G. Ispasoiu, A. M. Fox, J. F. Ryan, “Ultrafast two-photon nonlinearities in CdSe near 1.5 µm studied by interferometric autocorrelation,” IEEE J. Quantum Electron. 34, 1374–1379 (1998).
[Crossref]

Hagan, D. J.

D. Kovsh, S. Yang, D. J. Hagan, E. W. Van Stryland, “Nonlinear optical beam propagation for optical limiting,” Appl. Opt. 38, 5168–5180 (1999).
[Crossref]

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

Hanna, D. C.

D. C. Hanna, A. J. Turner, “Nonlinear absorption measurements in Proustite (Ag3AsS3) and CdSe,” Opt. Quantum Electron. 8, 213–217 (1976).
[Crossref]

Ispasoiu, R. G.

G.-M. Schucan, R. G. Ispasoiu, A. M. Fox, J. F. Ryan, “Ultrafast two-photon nonlinearities in CdSe near 1.5 µm studied by interferometric autocorrelation,” IEEE J. Quantum Electron. 34, 1374–1379 (1998).
[Crossref]

Johnston, A. M.

C. R. Pidgeon, B. S. Wherrett, A. M. Johnston, J. Dempsey, A. Miller, “Two-photon absorption in zinc-blende semiconductors,” Phys. Rev. Lett. 42, 1785–1788 (1979).
[Crossref]

Khattatov, V. U.

F. Bryukner, V. S. Dneprovskii, V. U. Khattatov, “Two-photon absorption in cadmium selenide,” Sov. J. Quantum Electron. 4, 749–751 (1974).
[Crossref]

Kovsh, D.

Miller, A.

C. R. Pidgeon, B. S. Wherrett, A. M. Johnston, J. Dempsey, A. Miller, “Two-photon absorption in zinc-blende semiconductors,” Phys. Rev. Lett. 42, 1785–1788 (1979).
[Crossref]

Minafra, A.

F. Adduci, I. M. Catalano, A. Cingolani, A. Minafra, “Direct and indirect two-photon processes in layered semiconductors,” Phys. Rev. B 15, 926–931 (1977).
[Crossref]

Mirov, S. B.

K. L. Vodopyanov, S. B. Mirov, V. G. Voevodin, P. G. Schunemann, “Two-photon absorption in GaSe and CdGeAs2,” Opt. Commun. 155, 47–50 (1998).
[Crossref]

Ok, S. M.

V. S. Dneprovskii, S. M. Ok, “Role of absorption by nonequilibrium carriers in determination of two-photon absorption coefficient of CdSe and GaAs crystals,” Sov. J. Quantum Electron. 3, 559–562 (1976).

Penzkofer, A.

A. Penzkofer, M. Schäffner, X. Bao, “Two-photon absorption and resonant non-phase-matched second-harmonic generation,” Opt. Quantum Electron. 22, 351–367 (1990).
[Crossref]

Pidgeon, C. R.

C. R. Pidgeon, B. S. Wherrett, A. M. Johnston, J. Dempsey, A. Miller, “Two-photon absorption in zinc-blende semiconductors,” Phys. Rev. Lett. 42, 1785–1788 (1979).
[Crossref]

Ralston, J. M.

J. M. Ralston, R. K. Chang, “Optical limiting in semiconductors,” Appl. Phys. Lett. 15, 164–166 (1969).
[Crossref]

Ryan, J. F.

G.-M. Schucan, R. G. Ispasoiu, A. M. Fox, J. F. Ryan, “Ultrafast two-photon nonlinearities in CdSe near 1.5 µm studied by interferometric autocorrelation,” IEEE J. Quantum Electron. 34, 1374–1379 (1998).
[Crossref]

Said, A. A.

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

Schäffner, M.

A. Penzkofer, M. Schäffner, X. Bao, “Two-photon absorption and resonant non-phase-matched second-harmonic generation,” Opt. Quantum Electron. 22, 351–367 (1990).
[Crossref]

Schucan, G.-M.

G.-M. Schucan, R. G. Ispasoiu, A. M. Fox, J. F. Ryan, “Ultrafast two-photon nonlinearities in CdSe near 1.5 µm studied by interferometric autocorrelation,” IEEE J. Quantum Electron. 34, 1374–1379 (1998).
[Crossref]

Schunemann, P. G.

K. L. Vodopyanov, S. B. Mirov, V. G. Voevodin, P. G. Schunemann, “Two-photon absorption in GaSe and CdGeAs2,” Opt. Commun. 155, 47–50 (1998).
[Crossref]

Sheik-Bahae, M.

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

Smith, W. L.

J. H. Bechtel, W. L. Smith, “Two-photon absorption in semiconductors with pisosecond laser pulses,” Phys. Rev. B 13, 3515–3522 (1976).
[Crossref]

Soileau, M. J.

E. W. Van Stryland, M. A. Woodall, W. E. Williams, M. J. Soileau, “Pulse-width and focal-volume dependence of laser-induced breakdown,” in Laser Induced Damage in Optical Materials, Natl. Bur. Stand. (U.S.) Spec. Pub. 638, 589–613 (1981).

Stewart, A. F.

M. Bass, E. W. Van Stryland, A. F. Stewart, “Laser calorimetric measurement of two-photon absorption,” Appl. Phys. Lett. 34, 142–144 (1979).
[Crossref]

Turner, A. J.

D. C. Hanna, A. J. Turner, “Nonlinear absorption measurements in Proustite (Ag3AsS3) and CdSe,” Opt. Quantum Electron. 8, 213–217 (1976).
[Crossref]

Van Stryland, E. W.

D. Kovsh, S. Yang, D. J. Hagan, E. W. Van Stryland, “Nonlinear optical beam propagation for optical limiting,” Appl. Opt. 38, 5168–5180 (1999).
[Crossref]

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

E. W. Van Stryland, M. A. Woodall, W. E. Williams, M. J. Soileau, “Pulse-width and focal-volume dependence of laser-induced breakdown,” in Laser Induced Damage in Optical Materials, Natl. Bur. Stand. (U.S.) Spec. Pub. 638, 589–613 (1981).

M. Bass, E. W. Van Stryland, A. F. Stewart, “Laser calorimetric measurement of two-photon absorption,” Appl. Phys. Lett. 34, 142–144 (1979).
[Crossref]

Vodopyanov, K. L.

K. L. Vodopyanov, S. B. Mirov, V. G. Voevodin, P. G. Schunemann, “Two-photon absorption in GaSe and CdGeAs2,” Opt. Commun. 155, 47–50 (1998).
[Crossref]

K. L. Vodopyanov, “Parametric generation of tunable infrared radiation in ZnGeP2 and GaSe pumped at 3 µm,” J. Opt. Soc. Am. B 10, 1723–1729 (1993).
[Crossref]

Voevodin, V. G.

K. L. Vodopyanov, S. B. Mirov, V. G. Voevodin, P. G. Schunemann, “Two-photon absorption in GaSe and CdGeAs2,” Opt. Commun. 155, 47–50 (1998).
[Crossref]

Wei, T. H.

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

Wherrett, B. S.

C. R. Pidgeon, B. S. Wherrett, A. M. Johnston, J. Dempsey, A. Miller, “Two-photon absorption in zinc-blende semiconductors,” Phys. Rev. Lett. 42, 1785–1788 (1979).
[Crossref]

Williams, W. E.

E. W. Van Stryland, M. A. Woodall, W. E. Williams, M. J. Soileau, “Pulse-width and focal-volume dependence of laser-induced breakdown,” in Laser Induced Damage in Optical Materials, Natl. Bur. Stand. (U.S.) Spec. Pub. 638, 589–613 (1981).

Woodall, M. A.

E. W. Van Stryland, M. A. Woodall, W. E. Williams, M. J. Soileau, “Pulse-width and focal-volume dependence of laser-induced breakdown,” in Laser Induced Damage in Optical Materials, Natl. Bur. Stand. (U.S.) Spec. Pub. 638, 589–613 (1981).

Yang, S.

Zotova, I. B.

I. B. Zotova, Y. J. Ding, “Optical parametric oscillators in the presence of strong two-photon absorption for extended applications of nonlinear optical materials,” Opt. Commun. (to be published).

Appl. Opt. (1)

Appl. Phys. Lett. (2)

J. M. Ralston, R. K. Chang, “Optical limiting in semiconductors,” Appl. Phys. Lett. 15, 164–166 (1969).
[Crossref]

M. Bass, E. W. Van Stryland, A. F. Stewart, “Laser calorimetric measurement of two-photon absorption,” Appl. Phys. Lett. 34, 142–144 (1979).
[Crossref]

IEEE J. Quantum Electron. (2)

G.-M. Schucan, R. G. Ispasoiu, A. M. Fox, J. F. Ryan, “Ultrafast two-photon nonlinearities in CdSe near 1.5 µm studied by interferometric autocorrelation,” IEEE J. Quantum Electron. 34, 1374–1379 (1998).
[Crossref]

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

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

Laser Induced Damage in Optical Materials (1)

E. W. Van Stryland, M. A. Woodall, W. E. Williams, M. J. Soileau, “Pulse-width and focal-volume dependence of laser-induced breakdown,” in Laser Induced Damage in Optical Materials, Natl. Bur. Stand. (U.S.) Spec. Pub. 638, 589–613 (1981).

Opt. Commun. (1)

K. L. Vodopyanov, S. B. Mirov, V. G. Voevodin, P. G. Schunemann, “Two-photon absorption in GaSe and CdGeAs2,” Opt. Commun. 155, 47–50 (1998).
[Crossref]

Opt. Quantum Electron. (2)

A. Penzkofer, M. Schäffner, X. Bao, “Two-photon absorption and resonant non-phase-matched second-harmonic generation,” Opt. Quantum Electron. 22, 351–367 (1990).
[Crossref]

D. C. Hanna, A. J. Turner, “Nonlinear absorption measurements in Proustite (Ag3AsS3) and CdSe,” Opt. Quantum Electron. 8, 213–217 (1976).
[Crossref]

Phys. Rev. B (2)

J. H. Bechtel, W. L. Smith, “Two-photon absorption in semiconductors with pisosecond laser pulses,” Phys. Rev. B 13, 3515–3522 (1976).
[Crossref]

F. Adduci, I. M. Catalano, A. Cingolani, A. Minafra, “Direct and indirect two-photon processes in layered semiconductors,” Phys. Rev. B 15, 926–931 (1977).
[Crossref]

Phys. Rev. Lett. (1)

C. R. Pidgeon, B. S. Wherrett, A. M. Johnston, J. Dempsey, A. Miller, “Two-photon absorption in zinc-blende semiconductors,” Phys. Rev. Lett. 42, 1785–1788 (1979).
[Crossref]

Prog. Cryst. Growth Charact. Mater. (1)

N. C. Fernelius, “Properties of gallium selenide single crystal,” Prog. Cryst. Growth Charact. Mater. 28, 275–353 (1994).
[Crossref]

Sov. J. Quantum Electron. (2)

V. S. Dneprovskii, S. M. Ok, “Role of absorption by nonequilibrium carriers in determination of two-photon absorption coefficient of CdSe and GaAs crystals,” Sov. J. Quantum Electron. 3, 559–562 (1976).

F. Bryukner, V. S. Dneprovskii, V. U. Khattatov, “Two-photon absorption in cadmium selenide,” Sov. J. Quantum Electron. 4, 749–751 (1974).
[Crossref]

Other (1)

I. B. Zotova, Y. J. Ding, “Optical parametric oscillators in the presence of strong two-photon absorption for extended applications of nonlinear optical materials,” Opt. Commun. (to be published).

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Figures (6)

Fig. 1
Fig. 1

Experimental setup for the z-scan technique.

Fig. 2
Fig. 2

Dots, typical transmittance by translating a CdSe crystal across the beam waist of a Gaussian beam. Solid curve, best nonlinear least-squares fit by use of Eq. (3).

Fig. 3
Fig. 3

Dots, spectrum of the TPA coefficient by fitting the profile obtained with the z-scan technique (see Fig. 1) with our theory at an average incident power of 20 mW (the peak laser intensity inside CdSe is 3.7 × 1010 W/cm2). Solid curve, theoretical result based on the parabolic band model.

Fig. 4
Fig. 4

Transmittance versus peak laser power at two different wavelengths at the top half of the CdSe bandgap: dots, measurements; solid curve, theory based on a constant value of the TPA coefficient.

Fig. 5
Fig. 5

Transmittance versus peak laser power at two different wavelengths at the lower half of the CdSe bandgap: dots, measurements; solid curve, theory based on a constant value of the TPA coefficient.

Fig. 6
Fig. 6

Dots, spectrum of the TPA coefficient by fitting the profile obtained with the z-scan technique (see, e.g., Fig. 2) with our theory at an average incident power of 500 mW (the peak laser intensity outside GaSe is 3.2 × 108 W/cm2). Solid curve, theoretical result based on the three-band model.

Tables (2)

Tables Icon

Table 1 Measured TPA Coefficients for CdSe

Tables Icon

Table 2 Measured TPA Coefficients for GaSe

Equations (7)

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

dIidz=-βIi2,
Eix, y, z=Ei,0w0wz exp-ikz-ηz-r21w2z+i k2Rz-t2τ2,
T=T021+βT0nP2λtan-1z+l/2z0-tan-1z-l/2z0,
β=4π e4c2Pn2Eg314πε02fωEg,
P=-i/ms|pz|z=c0.441/2/137.03,
fα=2α-12α32152α-13α-12×4α-12α3/2+22α+13/2,
Icr=2ωσhτ1-R0,

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