Abstract

By use of an expansion in Gaussian functions we solve the problem of the propagation of a laser beam that has been passed through a thin slab of nonlinearly refracting material. We compare the theoretical results for beam profile with experimental measurements for the semiconductor InSb and confirm that self-defocusing has been observed by D. A. B. Miller, M. H. Mozolowski, A. Miller, and S. D. Smith [Opt. Commun. 27, 133 (1978)]. The deduced value for the nonlinear refractive index at ~5 K is −6 × 10−5 cm2 W−1 at 1886 cm−1. This implies a third-order optical nonlinearity χ(3) ~ 10−2 esu, much larger than any previously reported for a solid.

© 1979 Optical Society of America

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References

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  1. D. A. B. Miller, M. H. Mozolowski, A. Miller, S. D. Smith, Opt. Commun. 27, 133 (1978).
    [Crossref]
  2. H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
    [Crossref] [PubMed]
  3. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
  4. D. A. B. Miller, S. D. Smith, Appl. Opt. 17, 3804 (1978).
    [Crossref] [PubMed]
  5. D. G. Seiler, L. K. Hanes, Opt. Commun. 28, 326 (1979).
    [Crossref]
  6. D. A. B. Miller, Ph.D. thesis (Heriot-Watt University, Edinburgh, 1979).
  7. P. Lavallard, R. Bichard, C. Benoît à la Guillaume, Phys. Rev. 16, 2804 (1977).
    [Crossref]
  8. A. V. Nurmikko, Opt. Commun. 16, 365 (1976).
    [Crossref]
  9. D. A. B. Miller, S. D. Smith, B. S. Wherrett, Phys. Rev. Lett. (submitted for publication); see also G. D. Holah, J. Dempsey, D. A. B. Miller, B. S. Wherrett, A. Miller, in Proceedings of the 14th International Conference on the Physics of Semiconductors, Edinburgh, 1978, Institute of Physics Conference Series No. 43 (Institute of Physics, London, 1979), p. 505.
  10. Compare J. E. Bjorkholm, A. Ashkin, Phys. Rev. Let. 32, 129 (1974).
    [Crossref]
  11. D. A. B. Miller, S. D. Smith, A. Johnston, Appl. Phys. Lett. (submitted for publication).

1979 (1)

D. G. Seiler, L. K. Hanes, Opt. Commun. 28, 326 (1979).
[Crossref]

1978 (2)

D. A. B. Miller, M. H. Mozolowski, A. Miller, S. D. Smith, Opt. Commun. 27, 133 (1978).
[Crossref]

D. A. B. Miller, S. D. Smith, Appl. Opt. 17, 3804 (1978).
[Crossref] [PubMed]

1977 (1)

P. Lavallard, R. Bichard, C. Benoît à la Guillaume, Phys. Rev. 16, 2804 (1977).
[Crossref]

1976 (1)

A. V. Nurmikko, Opt. Commun. 16, 365 (1976).
[Crossref]

1974 (1)

Compare J. E. Bjorkholm, A. Ashkin, Phys. Rev. Let. 32, 129 (1974).
[Crossref]

1966 (1)

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Ashkin, A.

Compare J. E. Bjorkholm, A. Ashkin, Phys. Rev. Let. 32, 129 (1974).
[Crossref]

Benoît à la Guillaume, C.

P. Lavallard, R. Bichard, C. Benoît à la Guillaume, Phys. Rev. 16, 2804 (1977).
[Crossref]

Bichard, R.

P. Lavallard, R. Bichard, C. Benoît à la Guillaume, Phys. Rev. 16, 2804 (1977).
[Crossref]

Bjorkholm, J. E.

Compare J. E. Bjorkholm, A. Ashkin, Phys. Rev. Let. 32, 129 (1974).
[Crossref]

Hanes, L. K.

D. G. Seiler, L. K. Hanes, Opt. Commun. 28, 326 (1979).
[Crossref]

Johnston, A.

D. A. B. Miller, S. D. Smith, A. Johnston, Appl. Phys. Lett. (submitted for publication).

Kogelnik, H.

Lavallard, P.

P. Lavallard, R. Bichard, C. Benoît à la Guillaume, Phys. Rev. 16, 2804 (1977).
[Crossref]

Li, T.

Miller, A.

D. A. B. Miller, M. H. Mozolowski, A. Miller, S. D. Smith, Opt. Commun. 27, 133 (1978).
[Crossref]

Miller, D. A. B.

D. A. B. Miller, M. H. Mozolowski, A. Miller, S. D. Smith, Opt. Commun. 27, 133 (1978).
[Crossref]

D. A. B. Miller, S. D. Smith, Appl. Opt. 17, 3804 (1978).
[Crossref] [PubMed]

D. A. B. Miller, S. D. Smith, A. Johnston, Appl. Phys. Lett. (submitted for publication).

D. A. B. Miller, Ph.D. thesis (Heriot-Watt University, Edinburgh, 1979).

D. A. B. Miller, S. D. Smith, B. S. Wherrett, Phys. Rev. Lett. (submitted for publication); see also G. D. Holah, J. Dempsey, D. A. B. Miller, B. S. Wherrett, A. Miller, in Proceedings of the 14th International Conference on the Physics of Semiconductors, Edinburgh, 1978, Institute of Physics Conference Series No. 43 (Institute of Physics, London, 1979), p. 505.

Mozolowski, M. H.

D. A. B. Miller, M. H. Mozolowski, A. Miller, S. D. Smith, Opt. Commun. 27, 133 (1978).
[Crossref]

Nurmikko, A. V.

A. V. Nurmikko, Opt. Commun. 16, 365 (1976).
[Crossref]

Seiler, D. G.

D. G. Seiler, L. K. Hanes, Opt. Commun. 28, 326 (1979).
[Crossref]

Smith, S. D.

D. A. B. Miller, M. H. Mozolowski, A. Miller, S. D. Smith, Opt. Commun. 27, 133 (1978).
[Crossref]

D. A. B. Miller, S. D. Smith, Appl. Opt. 17, 3804 (1978).
[Crossref] [PubMed]

D. A. B. Miller, S. D. Smith, A. Johnston, Appl. Phys. Lett. (submitted for publication).

D. A. B. Miller, S. D. Smith, B. S. Wherrett, Phys. Rev. Lett. (submitted for publication); see also G. D. Holah, J. Dempsey, D. A. B. Miller, B. S. Wherrett, A. Miller, in Proceedings of the 14th International Conference on the Physics of Semiconductors, Edinburgh, 1978, Institute of Physics Conference Series No. 43 (Institute of Physics, London, 1979), p. 505.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Wherrett, B. S.

D. A. B. Miller, S. D. Smith, B. S. Wherrett, Phys. Rev. Lett. (submitted for publication); see also G. D. Holah, J. Dempsey, D. A. B. Miller, B. S. Wherrett, A. Miller, in Proceedings of the 14th International Conference on the Physics of Semiconductors, Edinburgh, 1978, Institute of Physics Conference Series No. 43 (Institute of Physics, London, 1979), p. 505.

Appl. Opt. (2)

Opt. Commun. (3)

D. G. Seiler, L. K. Hanes, Opt. Commun. 28, 326 (1979).
[Crossref]

D. A. B. Miller, M. H. Mozolowski, A. Miller, S. D. Smith, Opt. Commun. 27, 133 (1978).
[Crossref]

A. V. Nurmikko, Opt. Commun. 16, 365 (1976).
[Crossref]

Phys. Rev. (1)

P. Lavallard, R. Bichard, C. Benoît à la Guillaume, Phys. Rev. 16, 2804 (1977).
[Crossref]

Phys. Rev. Let. (1)

Compare J. E. Bjorkholm, A. Ashkin, Phys. Rev. Let. 32, 129 (1974).
[Crossref]

Other (4)

D. A. B. Miller, S. D. Smith, A. Johnston, Appl. Phys. Lett. (submitted for publication).

D. A. B. Miller, S. D. Smith, B. S. Wherrett, Phys. Rev. Lett. (submitted for publication); see also G. D. Holah, J. Dempsey, D. A. B. Miller, B. S. Wherrett, A. Miller, in Proceedings of the 14th International Conference on the Physics of Semiconductors, Edinburgh, 1978, Institute of Physics Conference Series No. 43 (Institute of Physics, London, 1979), p. 505.

D. A. B. Miller, Ph.D. thesis (Heriot-Watt University, Edinburgh, 1979).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

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Figures (2)

Fig. 1
Fig. 1

Experimental transmission and far-fieldbeam width (defined as the diameter at half peak intensity) against peak intensity. Laser line, 1882 cm−1; spot diameter, 208 μm; InSb sample 7.5 mm long, antireflection coated on both ends, NDNA ≃ 3.8 × 10−14 cm−3 (n-type); temperature ≃5 K.

Fig. 2
Fig. 2

Experimental (solid lines) and theoretical (broken lines) intensity profiles (a) in the near field at 7 cm from the sample and (b) in the far field at 189 cm. Data for 130-mW beam of 1.67-mm spot diameter on laser line at 1886 cm−1; InSb sample and temperature as for Fig. l. Theoretical profiles are shown for the self-defocusing condition (x = 3.5, dashed lines) in the near and far field and for the self-focusing condition (x = −3.5, dot–dashed line) in the near field to emphasize that the experimental results originate from a defocusing. In the far field, the focusing and defocusing results are practically identical. The theoretical and experimental plots are normalized to give the same power levels.

Equations (7)

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n ( r ) = n 1 + n 2 I ( r ) ,
4 n 2 L 2 w 0 2 n 1 I p 1 ,
E ( r , 0 ) = E ( 0 , 0 ) exp [ ( - r 2 w 0 2 ) + i x exp ( - 2 r 2 w 0 2 ) ] .
x = - ω c n 2 I p ( 1 - e - α L α ) ,
E ( r , 0 ) = E ( 0 , 0 ) 0 ( i x ) m m ! exp ( - r 2 w m 2 ) ,
E ( r , z ) = E ( 0 , 0 ) 0 ( i x ) m m ! ( 1 + z 2 d m 2 ) - 1 / 2 × exp [ - r 2 w m 2 ( z ) - i k r 2 2 R m ( z ) - i P m ( z ) ] ,
I ( 0 , z ) I ( 0 , 0 ) π z 2 4 d 0 2 x - 1 erf ( - i x ) 2 ,

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