Abstract

We have extended the conventional Z-scan theory by employing an aberration-free approximation of a Gaussian beam through a nonlinear medium and derived a simple analytical formula for Z-scan transmittance, including the effects of both nonlinear absorption and nonlinear refraction, which could be applicable to the sample with large nonlinear phase shifts. We verified the extended Z-scan theory in an amorphous chalcogenide As2S3 thin film by measuring the Z-scan transmittance with both open and closed apertures. The nonlinear refractive index γ=7.6×10-5 cm2/W and the nonlinear absorption coefficient β=1.6 cm/W of As2S3 were measured at subbandgap 633-nm illumination.

© 1999 Optical Society of America

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References

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  1. N. F. Mott and E. A. Davis, Electronic Processes in Non-Crystalline Materials (Clarendon, Oxford, 1979).
  2. F. Yonezawa, ed., Fundamental Physics of Amorphous Semiconductors (Springer-Verlag, Berlin, 1981).
  3. D. W. Hall, M. A. Newhouse, N. F. Borrelli, W. H. Dumbaugh, and D. L. Weidman, “Nonlinear optical susceptibilities of high-index glasses,” Appl. Phys. Lett. 54, 1293–1295 (1989).
    [CrossRef]
  4. C. H. Kwak, J. T. Kim, and S. S. Lee, “Scalar and vector holographic gratings recorded in a photoanisotropic amorphous As2S3 thin film,” Opt. Lett. 13, 437–439 (1988).
    [CrossRef] [PubMed]
  5. C. H. Kwak, S. Y. Park, H. M. Kim, E.-H. Lee, and C. M. Kim, “Dammann gratings for multispot array generation by using photoinduced anisotropic materials,” Opt. Commun. 88, 249–257 (1992).
    [CrossRef]
  6. C. W. Slinger, A. Zakery, P. J. S. Ewen, and A. E. Owen, “Photodoped chalcogenides as potential infrared holographic media,” Appl. Opt. 31, 2490–2498 (1992).
    [CrossRef] [PubMed]
  7. C. H. Kwak, J. T. Kim, and S. S. Lee, “Nonlinear optical image processing in photoanisotropic amorphous As2S3 thin film,” Appl. Opt. 28, 737–739 (1989).
    [CrossRef] [PubMed]
  8. M. Asobe, K. Suzuki, T. Kanamori, and K. Kubodera, “Nonlinear refractive index measurement in chalcogenide-glass fibers by self-phase modulation,” Appl. Phys. Lett. 60, 1153–1154 (1992).
    [CrossRef]
  9. H. Nishihara, M. Haruna, and T. Suhara, Optical Integrated Circuits (McGraw-Hill, New York, 1989).
  10. F. Michelotti, E. Fazio, F. Senesi, M. Bertolotti, V. Chumash, and A. Andriesh, “Nonlinearity and photostructural changes in glassy As2S3 thin films,” Opt. Commun. 101, 74–78 (1993).
    [CrossRef]
  11. R. Rangel-Rojo, T. Kosa, E. Hajto, P. J. S. Ewen, A. E. Owen, A. K. Kar, and B. S. Wherrett, “Near-infrared optical nonlinearities in amorphous chalcogenides,” Opt. Commun. 109, 145–150 (1994).
    [CrossRef]
  12. H. Nasu, Y. Ibara, and K. Kubodera, “Optical third-harmonic generation from some high-index glasses,” J. Non-Cryst. Solids 110, 229–234 (1989).
    [CrossRef]
  13. 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, 760–769 (1990).
    [CrossRef]
  14. M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Solieau, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
    [CrossRef]
  15. C. H. Kwak and S. S. Lee, “Density matrix treatment of photodarkening kinetics in amorphous chalcogenide As2S3 thin films,” Appl. Opt. 27, 2858–2862 (1988).
    [CrossRef] [PubMed]
  16. A. L. Dawar, P. K. Shishodia, G. Chauhan, J. C. Joshi, C. Jagadish, and P. C. Mathur, “Effect of UV exposure on optical properties of amorphous As2S3 thin films,” Appl. Opt. 29, 1971–1973 (1990).
    [CrossRef] [PubMed]

1994 (1)

R. Rangel-Rojo, T. Kosa, E. Hajto, P. J. S. Ewen, A. E. Owen, A. K. Kar, and B. S. Wherrett, “Near-infrared optical nonlinearities in amorphous chalcogenides,” Opt. Commun. 109, 145–150 (1994).
[CrossRef]

1993 (1)

F. Michelotti, E. Fazio, F. Senesi, M. Bertolotti, V. Chumash, and A. Andriesh, “Nonlinearity and photostructural changes in glassy As2S3 thin films,” Opt. Commun. 101, 74–78 (1993).
[CrossRef]

1992 (3)

M. Asobe, K. Suzuki, T. Kanamori, and K. Kubodera, “Nonlinear refractive index measurement in chalcogenide-glass fibers by self-phase modulation,” Appl. Phys. Lett. 60, 1153–1154 (1992).
[CrossRef]

C. H. Kwak, S. Y. Park, H. M. Kim, E.-H. Lee, and C. M. Kim, “Dammann gratings for multispot array generation by using photoinduced anisotropic materials,” Opt. Commun. 88, 249–257 (1992).
[CrossRef]

C. W. Slinger, A. Zakery, P. J. S. Ewen, and A. E. Owen, “Photodoped chalcogenides as potential infrared holographic media,” Appl. Opt. 31, 2490–2498 (1992).
[CrossRef] [PubMed]

1991 (1)

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Solieau, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[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, 760–769 (1990).
[CrossRef]

A. L. Dawar, P. K. Shishodia, G. Chauhan, J. C. Joshi, C. Jagadish, and P. C. Mathur, “Effect of UV exposure on optical properties of amorphous As2S3 thin films,” Appl. Opt. 29, 1971–1973 (1990).
[CrossRef] [PubMed]

1989 (3)

C. H. Kwak, J. T. Kim, and S. S. Lee, “Nonlinear optical image processing in photoanisotropic amorphous As2S3 thin film,” Appl. Opt. 28, 737–739 (1989).
[CrossRef] [PubMed]

D. W. Hall, M. A. Newhouse, N. F. Borrelli, W. H. Dumbaugh, and D. L. Weidman, “Nonlinear optical susceptibilities of high-index glasses,” Appl. Phys. Lett. 54, 1293–1295 (1989).
[CrossRef]

H. Nasu, Y. Ibara, and K. Kubodera, “Optical third-harmonic generation from some high-index glasses,” J. Non-Cryst. Solids 110, 229–234 (1989).
[CrossRef]

1988 (2)

Andriesh, A.

F. Michelotti, E. Fazio, F. Senesi, M. Bertolotti, V. Chumash, and A. Andriesh, “Nonlinearity and photostructural changes in glassy As2S3 thin films,” Opt. Commun. 101, 74–78 (1993).
[CrossRef]

Asobe, M.

M. Asobe, K. Suzuki, T. Kanamori, and K. Kubodera, “Nonlinear refractive index measurement in chalcogenide-glass fibers by self-phase modulation,” Appl. Phys. Lett. 60, 1153–1154 (1992).
[CrossRef]

Bertolotti, M.

F. Michelotti, E. Fazio, F. Senesi, M. Bertolotti, V. Chumash, and A. Andriesh, “Nonlinearity and photostructural changes in glassy As2S3 thin films,” Opt. Commun. 101, 74–78 (1993).
[CrossRef]

Borrelli, N. F.

D. W. Hall, M. A. Newhouse, N. F. Borrelli, W. H. Dumbaugh, and D. L. Weidman, “Nonlinear optical susceptibilities of high-index glasses,” Appl. Phys. Lett. 54, 1293–1295 (1989).
[CrossRef]

Chauhan, G.

Chumash, V.

F. Michelotti, E. Fazio, F. Senesi, M. Bertolotti, V. Chumash, and A. Andriesh, “Nonlinearity and photostructural changes in glassy As2S3 thin films,” Opt. Commun. 101, 74–78 (1993).
[CrossRef]

Dawar, A. L.

Dumbaugh, W. H.

D. W. Hall, M. A. Newhouse, N. F. Borrelli, W. H. Dumbaugh, and D. L. Weidman, “Nonlinear optical susceptibilities of high-index glasses,” Appl. Phys. Lett. 54, 1293–1295 (1989).
[CrossRef]

Ewen, P. J. S.

R. Rangel-Rojo, T. Kosa, E. Hajto, P. J. S. Ewen, A. E. Owen, A. K. Kar, and B. S. Wherrett, “Near-infrared optical nonlinearities in amorphous chalcogenides,” Opt. Commun. 109, 145–150 (1994).
[CrossRef]

C. W. Slinger, A. Zakery, P. J. S. Ewen, and A. E. Owen, “Photodoped chalcogenides as potential infrared holographic media,” Appl. Opt. 31, 2490–2498 (1992).
[CrossRef] [PubMed]

Fazio, E.

F. Michelotti, E. Fazio, F. Senesi, M. Bertolotti, V. Chumash, and A. Andriesh, “Nonlinearity and photostructural changes in glassy As2S3 thin films,” Opt. Commun. 101, 74–78 (1993).
[CrossRef]

Hagan, D. J.

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Solieau, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[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, 760–769 (1990).
[CrossRef]

Hajto, E.

R. Rangel-Rojo, T. Kosa, E. Hajto, P. J. S. Ewen, A. E. Owen, A. K. Kar, and B. S. Wherrett, “Near-infrared optical nonlinearities in amorphous chalcogenides,” Opt. Commun. 109, 145–150 (1994).
[CrossRef]

Hall, D. W.

D. W. Hall, M. A. Newhouse, N. F. Borrelli, W. H. Dumbaugh, and D. L. Weidman, “Nonlinear optical susceptibilities of high-index glasses,” Appl. Phys. Lett. 54, 1293–1295 (1989).
[CrossRef]

Ibara, Y.

H. Nasu, Y. Ibara, and K. Kubodera, “Optical third-harmonic generation from some high-index glasses,” J. Non-Cryst. Solids 110, 229–234 (1989).
[CrossRef]

Jagadish, C.

Joshi, J. C.

Kanamori, T.

M. Asobe, K. Suzuki, T. Kanamori, and K. Kubodera, “Nonlinear refractive index measurement in chalcogenide-glass fibers by self-phase modulation,” Appl. Phys. Lett. 60, 1153–1154 (1992).
[CrossRef]

Kar, A. K.

R. Rangel-Rojo, T. Kosa, E. Hajto, P. J. S. Ewen, A. E. Owen, A. K. Kar, and B. S. Wherrett, “Near-infrared optical nonlinearities in amorphous chalcogenides,” Opt. Commun. 109, 145–150 (1994).
[CrossRef]

Kim, C. M.

C. H. Kwak, S. Y. Park, H. M. Kim, E.-H. Lee, and C. M. Kim, “Dammann gratings for multispot array generation by using photoinduced anisotropic materials,” Opt. Commun. 88, 249–257 (1992).
[CrossRef]

Kim, H. M.

C. H. Kwak, S. Y. Park, H. M. Kim, E.-H. Lee, and C. M. Kim, “Dammann gratings for multispot array generation by using photoinduced anisotropic materials,” Opt. Commun. 88, 249–257 (1992).
[CrossRef]

Kim, J. T.

Kosa, T.

R. Rangel-Rojo, T. Kosa, E. Hajto, P. J. S. Ewen, A. E. Owen, A. K. Kar, and B. S. Wherrett, “Near-infrared optical nonlinearities in amorphous chalcogenides,” Opt. Commun. 109, 145–150 (1994).
[CrossRef]

Kubodera, K.

M. Asobe, K. Suzuki, T. Kanamori, and K. Kubodera, “Nonlinear refractive index measurement in chalcogenide-glass fibers by self-phase modulation,” Appl. Phys. Lett. 60, 1153–1154 (1992).
[CrossRef]

H. Nasu, Y. Ibara, and K. Kubodera, “Optical third-harmonic generation from some high-index glasses,” J. Non-Cryst. Solids 110, 229–234 (1989).
[CrossRef]

Kwak, C. H.

Lee, E.-H.

C. H. Kwak, S. Y. Park, H. M. Kim, E.-H. Lee, and C. M. Kim, “Dammann gratings for multispot array generation by using photoinduced anisotropic materials,” Opt. Commun. 88, 249–257 (1992).
[CrossRef]

Lee, S. S.

Mathur, P. C.

Michelotti, F.

F. Michelotti, E. Fazio, F. Senesi, M. Bertolotti, V. Chumash, and A. Andriesh, “Nonlinearity and photostructural changes in glassy As2S3 thin films,” Opt. Commun. 101, 74–78 (1993).
[CrossRef]

Nasu, H.

H. Nasu, Y. Ibara, and K. Kubodera, “Optical third-harmonic generation from some high-index glasses,” J. Non-Cryst. Solids 110, 229–234 (1989).
[CrossRef]

Newhouse, M. A.

D. W. Hall, M. A. Newhouse, N. F. Borrelli, W. H. Dumbaugh, and D. L. Weidman, “Nonlinear optical susceptibilities of high-index glasses,” Appl. Phys. Lett. 54, 1293–1295 (1989).
[CrossRef]

Owen, A. E.

R. Rangel-Rojo, T. Kosa, E. Hajto, P. J. S. Ewen, A. E. Owen, A. K. Kar, and B. S. Wherrett, “Near-infrared optical nonlinearities in amorphous chalcogenides,” Opt. Commun. 109, 145–150 (1994).
[CrossRef]

C. W. Slinger, A. Zakery, P. J. S. Ewen, and A. E. Owen, “Photodoped chalcogenides as potential infrared holographic media,” Appl. Opt. 31, 2490–2498 (1992).
[CrossRef] [PubMed]

Park, S. Y.

C. H. Kwak, S. Y. Park, H. M. Kim, E.-H. Lee, and C. M. Kim, “Dammann gratings for multispot array generation by using photoinduced anisotropic materials,” Opt. Commun. 88, 249–257 (1992).
[CrossRef]

Rangel-Rojo, R.

R. Rangel-Rojo, T. Kosa, E. Hajto, P. J. S. Ewen, A. E. Owen, A. K. Kar, and B. S. Wherrett, “Near-infrared optical nonlinearities in amorphous chalcogenides,” Opt. Commun. 109, 145–150 (1994).
[CrossRef]

Said, A. A.

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Solieau, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[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, 760–769 (1990).
[CrossRef]

Senesi, F.

F. Michelotti, E. Fazio, F. Senesi, M. Bertolotti, V. Chumash, and A. Andriesh, “Nonlinearity and photostructural changes in glassy As2S3 thin films,” Opt. Commun. 101, 74–78 (1993).
[CrossRef]

Sheik-Bahae, M.

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Solieau, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[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, 760–769 (1990).
[CrossRef]

Shishodia, P. K.

Slinger, C. W.

Solieau, M. J.

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Solieau, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

Suzuki, K.

M. Asobe, K. Suzuki, T. Kanamori, and K. Kubodera, “Nonlinear refractive index measurement in chalcogenide-glass fibers by self-phase modulation,” Appl. Phys. Lett. 60, 1153–1154 (1992).
[CrossRef]

Van Stryland, E. W.

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Solieau, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[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, 760–769 (1990).
[CrossRef]

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, 760–769 (1990).
[CrossRef]

Weidman, D. L.

D. W. Hall, M. A. Newhouse, N. F. Borrelli, W. H. Dumbaugh, and D. L. Weidman, “Nonlinear optical susceptibilities of high-index glasses,” Appl. Phys. Lett. 54, 1293–1295 (1989).
[CrossRef]

Wherrett, B. S.

R. Rangel-Rojo, T. Kosa, E. Hajto, P. J. S. Ewen, A. E. Owen, A. K. Kar, and B. S. Wherrett, “Near-infrared optical nonlinearities in amorphous chalcogenides,” Opt. Commun. 109, 145–150 (1994).
[CrossRef]

Zakery, A.

Appl. Opt. (4)

Appl. Phys. Lett. (2)

M. Asobe, K. Suzuki, T. Kanamori, and K. Kubodera, “Nonlinear refractive index measurement in chalcogenide-glass fibers by self-phase modulation,” Appl. Phys. Lett. 60, 1153–1154 (1992).
[CrossRef]

D. W. Hall, M. A. Newhouse, N. F. Borrelli, W. H. Dumbaugh, and D. L. Weidman, “Nonlinear optical susceptibilities of high-index glasses,” Appl. Phys. Lett. 54, 1293–1295 (1989).
[CrossRef]

IEEE J. Quantum Electron. (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, 760–769 (1990).
[CrossRef]

J. Non-Cryst. Solids (1)

H. Nasu, Y. Ibara, and K. Kubodera, “Optical third-harmonic generation from some high-index glasses,” J. Non-Cryst. Solids 110, 229–234 (1989).
[CrossRef]

Opt. Commun. (3)

C. H. Kwak, S. Y. Park, H. M. Kim, E.-H. Lee, and C. M. Kim, “Dammann gratings for multispot array generation by using photoinduced anisotropic materials,” Opt. Commun. 88, 249–257 (1992).
[CrossRef]

F. Michelotti, E. Fazio, F. Senesi, M. Bertolotti, V. Chumash, and A. Andriesh, “Nonlinearity and photostructural changes in glassy As2S3 thin films,” Opt. Commun. 101, 74–78 (1993).
[CrossRef]

R. Rangel-Rojo, T. Kosa, E. Hajto, P. J. S. Ewen, A. E. Owen, A. K. Kar, and B. S. Wherrett, “Near-infrared optical nonlinearities in amorphous chalcogenides,” Opt. Commun. 109, 145–150 (1994).
[CrossRef]

Opt. Eng. (1)

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Solieau, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

Opt. Lett. (1)

Other (3)

N. F. Mott and E. A. Davis, Electronic Processes in Non-Crystalline Materials (Clarendon, Oxford, 1979).

F. Yonezawa, ed., Fundamental Physics of Amorphous Semiconductors (Springer-Verlag, Berlin, 1981).

H. Nishihara, M. Haruna, and T. Suhara, Optical Integrated Circuits (McGraw-Hill, New York, 1989).

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Figures (3)

Fig. 1
Fig. 1

Theoretical plots of closed-aperture Z-scan transmittance for purely nonlinear refractions: (a) the conventional small-phase-shift theory of Eq. (12) derived from Sheik-Bahae et al.13 and (b) the large-nonlinear-phase-shift theory of Eq. (11).

Fig. 2
Fig. 2

Normalized open-aperture Z-scan data from an As2S3 thin film at 633-nm wavelength with an intensity of I0=1.5 kW/cm2. Solid curve, a theoretical curve from Eq. (18) with q00=1.1 and β=1.6 cm/W.

Fig. 3
Fig. 3

Closed-aperture (on-axis) Z-scan data from an As2S3 thin film at 633-nm wavelength with an intensity of I0=1.5 kW/cm2. Solid curve, a theoretical curve from Eq. (17) with q00=1.1 and ΔΦ0=6.0.

Equations (21)

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

n=n0+n22|E|2=n0+γI,
E(r, z)=E0w0w(z)exp-r2w2+ikr22Rexp[-iϕ(z)],
nn0-n22E02w02w22r2w2=n0-Δn(r, z).
Δϕ(r, z)=-Δϕ0(z)2r2w2,
Ee(r, z, Δϕ)=E(r, z)exp[iΔϕ(r, z)]=E(0, z)exp-ikr22qe,
1qe=1R-2λπw2Δϕ0-iλπw2.
qa=qe+D.
wa2(z)=w2g2+λDπw22,
Ra(z)=D1-gg2+λDπw22-1,
g=1+1R-2λπw2Δϕ0D.
Ea(r, z, Δϕ0)=E(0, z)wwaexp[iη(z)]exp-ikr22qa
wwaexp[iη(z)]=g+iλDπw2-1.
1qa=1Ra-iλπwa2.
Tphase(z)=|Ea(0, z, Δϕ0)|2|Ea(0, z, Δϕ0=0)|2=11-4x(1+x2)2ΔΦ0+4(1+x2)3ΔΦ02,
Tphase(z)=1+4xΔΦ0(1+x2)(9+x2),
Ie=I(r, z)exp(-αL)1+q(r, z),
Δϕ(r, z)=kγβln[1+q(r, z)],
Ee=I(r, z)exp(-αL)1+βLeffI(r, z)1/2 exp(iΔϕ)=E(r, z)exp(-αL/2)(1+q)(ikγ/β-1/2).
Δϕ(r, z)-kγβq01+q02r2w2,
Tphase+absorption=11-(4x-η)(1+x2)2(1+q0)ΔΦ0+(4+η2)(1+x2)3(1+q0)2ΔΦ02,
Topen(z)=ln[1+q0(z)]q0(z).

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