Abstract

We have characterized the transverse spatial dependence of the real and the imaginary parts of the complex nonlinear refractive index of a semiconductor-doped glass filter that exhibits absorptive bistability. Using the Z-scan technique combined with an interferometric measurement of the integrated optical thickness, we are able to fit the observed experimental data, assuming a quadratically varying transverse temperature profile in the sample. The transverse variations in the nonlinear refractive index do not scale directly with the size of the incident beam but exhibit marked asymmetries that depend on whether the incident beam is converging or diverging.

© 1999 Optical Society of America

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References

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  1. H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, New York, 1985).
  2. B. S. Wherrett, A. C. Walker, and F. A. P. Tooley, “Nonlinear refraction for cw optical bistability,” in Optical Nonlinearities and Instabilities in Semiconductors, H. Haug, ed. (Academic, San Diego, Calif., 1988), pp. 262–272.
  3. S. W. Koch, “Optical instabilities in semiconductors: theory,” in Optical Nonlinearities and Instabilities in Semiconductors, H. Haug, ed. (Academic, San Diego, Calif., 1988), pp. 273–282.
  4. D. A. B. Miller, “Optical bistability and differential gain resulting from absorption increasing with excitation,” J. Opt. Soc. Am. B 1, 857–864 (1984).
    [CrossRef]
  5. M. Kretschmar, F. Henneberger, H. Rossmann, and I. Haddad, “Dynamical effects in increasing absorption optical bistability of thermal origin,” Phys. Status Solidi B 143, K71–K76 (1987).
    [CrossRef]
  6. W. D. St. John, J. P. Wicksted, and G. Cantwell, “Transverse structures in resonatorless absorptive switching in bulk ZnSe,” J. Appl. Phys. 73, 3013–3017 (1993).
    [CrossRef]
  7. A. K. Kar and B. S. Wherrett, “Thermal dispersive optical bistability and absorptive bistability in bulk ZnSe,” J. Opt. Soc. Am. B 3, 345–350 (1986).
    [CrossRef]
  8. M. R. Taghizadeh, I. Janossy, and S. D. Smith, “Optical bistability in bulk ZnSe due to increasing absorption and self-focussing,” Appl. Phys. Lett. 46, 331–333 (1985).
    [CrossRef]
  9. M. Sheik-Bahae, A. A. Said, and E. W. Van Stryland, “High-sensitivity single beam n2 measurements,” Opt. Lett. 14, 955–957 (1989).
    [CrossRef] [PubMed]
  10. 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]
  11. J. Hollandt, J. Gutowski, and I. Broser, “A detailed study concerning the main aspects influencing thermally induced optical bistability of CdS,” Phys. Status Solidi B 159, 205–211 (1990).
    [CrossRef]
  12. M. Lambsdorff, C. Dornfeld, and C. Klingshirn, “Optical bistability in semiconductors induced by thermal effects,” Z. Phys. B 64, 409–416 (1986).
    [CrossRef]
  13. L. A. Lugiato, “Theory of optical bistability,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1984), Vol. XXI, pp. 115–122.
  14. B. S. Wherret, F. A. P. Tooley, and S. D. Smith, “Absorption switching and bistability in InSb,” Opt. Commun. 52, 301–306 (1984).
    [CrossRef]
  15. J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
    [CrossRef]
  16. J. W. Fleming, “Optical glasses,” in CRC Handbook of Laser Science and Technology, Optical Materials: Part 2, M. J. Weber, ed. (CRC Press, Boca Raton, Fla., 1987), Vol. IV, pp. 69–83.
  17. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).
  18. H. M. Gibbs, G. R. Olbright, N. Peyghambarian, H. E. Schmidt, S. W. Koch, and H. Haug, “Kinks: longitudinal excitation discontinuities in increasing absorption optical bistability,” Phys. Rev. A 32, 692–694 (1985).
    [CrossRef] [PubMed]
  19. A. E. Siegman, “Quasi-fast Hankel transform,” Opt. Lett. 1, 13–15 (1977).
    [CrossRef]

1993

W. D. St. John, J. P. Wicksted, and G. Cantwell, “Transverse structures in resonatorless absorptive switching in bulk ZnSe,” J. Appl. Phys. 73, 3013–3017 (1993).
[CrossRef]

1990

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. Hollandt, J. Gutowski, and I. Broser, “A detailed study concerning the main aspects influencing thermally induced optical bistability of CdS,” Phys. Status Solidi B 159, 205–211 (1990).
[CrossRef]

1989

1987

M. Kretschmar, F. Henneberger, H. Rossmann, and I. Haddad, “Dynamical effects in increasing absorption optical bistability of thermal origin,” Phys. Status Solidi B 143, K71–K76 (1987).
[CrossRef]

1986

A. K. Kar and B. S. Wherrett, “Thermal dispersive optical bistability and absorptive bistability in bulk ZnSe,” J. Opt. Soc. Am. B 3, 345–350 (1986).
[CrossRef]

M. Lambsdorff, C. Dornfeld, and C. Klingshirn, “Optical bistability in semiconductors induced by thermal effects,” Z. Phys. B 64, 409–416 (1986).
[CrossRef]

1985

M. R. Taghizadeh, I. Janossy, and S. D. Smith, “Optical bistability in bulk ZnSe due to increasing absorption and self-focussing,” Appl. Phys. Lett. 46, 331–333 (1985).
[CrossRef]

H. M. Gibbs, G. R. Olbright, N. Peyghambarian, H. E. Schmidt, S. W. Koch, and H. Haug, “Kinks: longitudinal excitation discontinuities in increasing absorption optical bistability,” Phys. Rev. A 32, 692–694 (1985).
[CrossRef] [PubMed]

1984

D. A. B. Miller, “Optical bistability and differential gain resulting from absorption increasing with excitation,” J. Opt. Soc. Am. B 1, 857–864 (1984).
[CrossRef]

B. S. Wherret, F. A. P. Tooley, and S. D. Smith, “Absorption switching and bistability in InSb,” Opt. Commun. 52, 301–306 (1984).
[CrossRef]

1977

1965

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Broser, I.

J. Hollandt, J. Gutowski, and I. Broser, “A detailed study concerning the main aspects influencing thermally induced optical bistability of CdS,” Phys. Status Solidi B 159, 205–211 (1990).
[CrossRef]

Cantwell, G.

W. D. St. John, J. P. Wicksted, and G. Cantwell, “Transverse structures in resonatorless absorptive switching in bulk ZnSe,” J. Appl. Phys. 73, 3013–3017 (1993).
[CrossRef]

Dornfeld, C.

M. Lambsdorff, C. Dornfeld, and C. Klingshirn, “Optical bistability in semiconductors induced by thermal effects,” Z. Phys. B 64, 409–416 (1986).
[CrossRef]

Gibbs, H. M.

H. M. Gibbs, G. R. Olbright, N. Peyghambarian, H. E. Schmidt, S. W. Koch, and H. Haug, “Kinks: longitudinal excitation discontinuities in increasing absorption optical bistability,” Phys. Rev. A 32, 692–694 (1985).
[CrossRef] [PubMed]

Gordon, J. P.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Gutowski, J.

J. Hollandt, J. Gutowski, and I. Broser, “A detailed study concerning the main aspects influencing thermally induced optical bistability of CdS,” Phys. Status Solidi B 159, 205–211 (1990).
[CrossRef]

Haddad, I.

M. Kretschmar, F. Henneberger, H. Rossmann, and I. Haddad, “Dynamical effects in increasing absorption optical bistability of thermal origin,” Phys. Status Solidi B 143, K71–K76 (1987).
[CrossRef]

Hagan, D. J.

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]

Haug, H.

H. M. Gibbs, G. R. Olbright, N. Peyghambarian, H. E. Schmidt, S. W. Koch, and H. Haug, “Kinks: longitudinal excitation discontinuities in increasing absorption optical bistability,” Phys. Rev. A 32, 692–694 (1985).
[CrossRef] [PubMed]

Henneberger, F.

M. Kretschmar, F. Henneberger, H. Rossmann, and I. Haddad, “Dynamical effects in increasing absorption optical bistability of thermal origin,” Phys. Status Solidi B 143, K71–K76 (1987).
[CrossRef]

Hollandt, J.

J. Hollandt, J. Gutowski, and I. Broser, “A detailed study concerning the main aspects influencing thermally induced optical bistability of CdS,” Phys. Status Solidi B 159, 205–211 (1990).
[CrossRef]

Janossy, I.

M. R. Taghizadeh, I. Janossy, and S. D. Smith, “Optical bistability in bulk ZnSe due to increasing absorption and self-focussing,” Appl. Phys. Lett. 46, 331–333 (1985).
[CrossRef]

Kar, A. K.

Klingshirn, C.

M. Lambsdorff, C. Dornfeld, and C. Klingshirn, “Optical bistability in semiconductors induced by thermal effects,” Z. Phys. B 64, 409–416 (1986).
[CrossRef]

Koch, S. W.

H. M. Gibbs, G. R. Olbright, N. Peyghambarian, H. E. Schmidt, S. W. Koch, and H. Haug, “Kinks: longitudinal excitation discontinuities in increasing absorption optical bistability,” Phys. Rev. A 32, 692–694 (1985).
[CrossRef] [PubMed]

Kretschmar, M.

M. Kretschmar, F. Henneberger, H. Rossmann, and I. Haddad, “Dynamical effects in increasing absorption optical bistability of thermal origin,” Phys. Status Solidi B 143, K71–K76 (1987).
[CrossRef]

Lambsdorff, M.

M. Lambsdorff, C. Dornfeld, and C. Klingshirn, “Optical bistability in semiconductors induced by thermal effects,” Z. Phys. B 64, 409–416 (1986).
[CrossRef]

Leite, R. C. C.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Miller, D. A. B.

Moore, R. S.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Olbright, G. R.

H. M. Gibbs, G. R. Olbright, N. Peyghambarian, H. E. Schmidt, S. W. Koch, and H. Haug, “Kinks: longitudinal excitation discontinuities in increasing absorption optical bistability,” Phys. Rev. A 32, 692–694 (1985).
[CrossRef] [PubMed]

Peyghambarian, N.

H. M. Gibbs, G. R. Olbright, N. Peyghambarian, H. E. Schmidt, S. W. Koch, and H. Haug, “Kinks: longitudinal excitation discontinuities in increasing absorption optical bistability,” Phys. Rev. A 32, 692–694 (1985).
[CrossRef] [PubMed]

Porto, S. P. S.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Rossmann, H.

M. Kretschmar, F. Henneberger, H. Rossmann, and I. Haddad, “Dynamical effects in increasing absorption optical bistability of thermal origin,” Phys. Status Solidi B 143, K71–K76 (1987).
[CrossRef]

Said, A. A.

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]

M. Sheik-Bahae, A. A. Said, and E. W. Van Stryland, “High-sensitivity single beam n2 measurements,” Opt. Lett. 14, 955–957 (1989).
[CrossRef] [PubMed]

Schmidt, H. E.

H. M. Gibbs, G. R. Olbright, N. Peyghambarian, H. E. Schmidt, S. W. Koch, and H. Haug, “Kinks: longitudinal excitation discontinuities in increasing absorption optical bistability,” Phys. Rev. A 32, 692–694 (1985).
[CrossRef] [PubMed]

Sheik-Bahae, M.

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]

M. Sheik-Bahae, A. A. Said, and E. W. Van Stryland, “High-sensitivity single beam n2 measurements,” Opt. Lett. 14, 955–957 (1989).
[CrossRef] [PubMed]

Siegman, A. E.

Smith, S. D.

M. R. Taghizadeh, I. Janossy, and S. D. Smith, “Optical bistability in bulk ZnSe due to increasing absorption and self-focussing,” Appl. Phys. Lett. 46, 331–333 (1985).
[CrossRef]

B. S. Wherret, F. A. P. Tooley, and S. D. Smith, “Absorption switching and bistability in InSb,” Opt. Commun. 52, 301–306 (1984).
[CrossRef]

St. John, W. D.

W. D. St. John, J. P. Wicksted, and G. Cantwell, “Transverse structures in resonatorless absorptive switching in bulk ZnSe,” J. Appl. Phys. 73, 3013–3017 (1993).
[CrossRef]

Taghizadeh, M. R.

M. R. Taghizadeh, I. Janossy, and S. D. Smith, “Optical bistability in bulk ZnSe due to increasing absorption and self-focussing,” Appl. Phys. Lett. 46, 331–333 (1985).
[CrossRef]

Tooley, F. A. P.

B. S. Wherret, F. A. P. Tooley, and S. D. Smith, “Absorption switching and bistability in InSb,” Opt. Commun. 52, 301–306 (1984).
[CrossRef]

Van Stryland, E. W.

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]

M. Sheik-Bahae, A. A. Said, and E. W. Van Stryland, “High-sensitivity single beam n2 measurements,” Opt. Lett. 14, 955–957 (1989).
[CrossRef] [PubMed]

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]

Wherret, B. S.

B. S. Wherret, F. A. P. Tooley, and S. D. Smith, “Absorption switching and bistability in InSb,” Opt. Commun. 52, 301–306 (1984).
[CrossRef]

Wherrett, B. S.

Whinnery, J. R.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Wicksted, J. P.

W. D. St. John, J. P. Wicksted, and G. Cantwell, “Transverse structures in resonatorless absorptive switching in bulk ZnSe,” J. Appl. Phys. 73, 3013–3017 (1993).
[CrossRef]

Appl. Phys. Lett.

M. R. Taghizadeh, I. Janossy, and S. D. Smith, “Optical bistability in bulk ZnSe due to increasing absorption and self-focussing,” Appl. Phys. Lett. 46, 331–333 (1985).
[CrossRef]

IEEE J. Quantum Electron.

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. Appl. Phys.

W. D. St. John, J. P. Wicksted, and G. Cantwell, “Transverse structures in resonatorless absorptive switching in bulk ZnSe,” J. Appl. Phys. 73, 3013–3017 (1993).
[CrossRef]

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

B. S. Wherret, F. A. P. Tooley, and S. D. Smith, “Absorption switching and bistability in InSb,” Opt. Commun. 52, 301–306 (1984).
[CrossRef]

Opt. Lett.

Phys. Rev. A

H. M. Gibbs, G. R. Olbright, N. Peyghambarian, H. E. Schmidt, S. W. Koch, and H. Haug, “Kinks: longitudinal excitation discontinuities in increasing absorption optical bistability,” Phys. Rev. A 32, 692–694 (1985).
[CrossRef] [PubMed]

Phys. Status Solidi B

M. Kretschmar, F. Henneberger, H. Rossmann, and I. Haddad, “Dynamical effects in increasing absorption optical bistability of thermal origin,” Phys. Status Solidi B 143, K71–K76 (1987).
[CrossRef]

J. Hollandt, J. Gutowski, and I. Broser, “A detailed study concerning the main aspects influencing thermally induced optical bistability of CdS,” Phys. Status Solidi B 159, 205–211 (1990).
[CrossRef]

Z. Phys. B

M. Lambsdorff, C. Dornfeld, and C. Klingshirn, “Optical bistability in semiconductors induced by thermal effects,” Z. Phys. B 64, 409–416 (1986).
[CrossRef]

Other

L. A. Lugiato, “Theory of optical bistability,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1984), Vol. XXI, pp. 115–122.

H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, New York, 1985).

B. S. Wherrett, A. C. Walker, and F. A. P. Tooley, “Nonlinear refraction for cw optical bistability,” in Optical Nonlinearities and Instabilities in Semiconductors, H. Haug, ed. (Academic, San Diego, Calif., 1988), pp. 262–272.

S. W. Koch, “Optical instabilities in semiconductors: theory,” in Optical Nonlinearities and Instabilities in Semiconductors, H. Haug, ed. (Academic, San Diego, Calif., 1988), pp. 273–282.

J. W. Fleming, “Optical glasses,” in CRC Handbook of Laser Science and Technology, Optical Materials: Part 2, M. J. Weber, ed. (CRC Press, Boca Raton, Fla., 1987), Vol. IV, pp. 69–83.

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).

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Figures (10)

Fig. 1
Fig. 1

Optical bistability curve obtained with our experimental setup.

Fig. 2
Fig. 2

Graphic solution of a heat-flow equation that corresponds to Eqs. (2) and (4). Dotted curve, A(ΔT), the variation of the sample absorption with temperature. For the range of incident intensities that give rise to the family of straight lines between the curves labeled I2 and I3 there are multiple solutions.

Fig. 3
Fig. 3

Experimental curve of absorption as a function of temperature with several experimental points corresponding to the upper and lower branches (high- and low-transmission states) of the bistability region. A(T)/F={1-exp[-α(T)d]} represents the fraction of the incident light absorbed by the sample. These values were taken at an ambient temperature of 20 °C.

Fig. 4
Fig. 4

(a) Experimental setup that we employed. DIN, DT, and DR are photodiodes. DIN and DT monitor the incident and the transmitted powers, respectively; DR monitors the interference of the incident laser beam caused by reflections from the first and second uncoated surfaces of the doped glass filter. NDF, neutral-density filter; BS, beam splitter. (b) Schematic of the setup for the Z-scan experiment.

Fig. 5
Fig. 5

Incident power, transmitted power, and interference fringes as a function of time observed during a bistable cycle.

Fig. 6
Fig. 6

Ratio of transmission for a full aperture and a closed aperture as a function of the sample position relative to the focus.

Fig. 7
Fig. 7

(a) Transverse curvature of the real part of the nonlinear refractive index [nd]2 as a function of the sample position relative to the focus. (b) Transverse curvature of the nonlinear absorption coefficient α2 normalized by the wave vector k as a function of the sample position relative to the focus.

Fig. 8
Fig. 8

Expected transmission ratio of a Z-scan experiment when the imaginary part of the transverse refractive index is set to zero. These curves were obtained for an incident power of 95.2 mW.

Fig. 9
Fig. 9

Parameter S as a function of the sample positions relative to the beam focus. This parameter represents the transverse area of the beam exiting the sample when transverse nonlinearities are taken into account compared with the beam area that would have resulted if the complex refractive index were independent of the radial coordinate (i.e. when [nd]2 and α2 are zero).

Fig. 10
Fig. 10

Value of the real part of the transverse refractive index [nd]2 multiplied by the beam size w(z) at the entrance of the sample as a function of the sample’s position relative to the beam focus.

Equations (13)

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

Tt=A(T)ICd+D2T,
A(T)=F{1-exp[-α(T)d]},
D2T-(ΔT)τ,
A(ΔT)=CdI(r=0)τ(ΔT).
Pin [1-exp(-αd)]=2πrdiffusionΔTκ,
T(r)T0-½T2r2,
[nd(r)]=[nd]0-½[nd]2r2,
α(r)=α0-½α2r2,
1q˜(z)1R(z)-i λπw2(z),
ABCDcos(γ˜d)(n0γ˜)-1 sin(γ˜d)-n0γ˜ sin(γ˜d)cos(γ˜d).
n0γ˜2d=[nd]2-iα2 λ0d2π.
T=1-exp2a2πλIm1q˜a.
T(z)=1-0.63(L2/ZR)Im(1/q˜a),

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