Abstract

Observations of the self-deflection effect for an asymmetrical continuous-wave laser beam in sodium vapor are described, and comparisons based on theoretical calculations are made. A self-bending angle as large as eight diffraction widths was recorded, and strong attenuation of the on-axis radiation due to self-bending was measured. At ~200°C the self-deflection angle increased linearly with beam power, and we determined that (n2)max ≃ −10−7 cm2/W for intensities below 220 W/cm2. While numerical calculations, based on an inhomogeneously broadened two-level system, predict strong saturation of the self-bending effect, we observed only moderate saturation.

© 1989 Optical Society of America

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  1. A. E. Kaplan, Pis’ma Zh. Eksp. Teor. Fiz. 9, 58 (1969) [JETP Lett. 9, 33 (1969)].
  2. M. S. Brodin and A. M. Kamuz, Pis’ma Zh. Eksp. Teor. Fiz. 9, 577 (1969) [JETP Lett. 9, 351 (1969)].
  3. A. E. Kaplan, Opt. Lett. 6, 360 (1981).
    [Crossref] [PubMed]
  4. J. A. Hermann, Opt. Commun. 62, 367 (1987); Opt. Quantum Electron. 19, 169 (1987).
    [Crossref]
  5. G. A. Swartzlander and A. E. Kaplan, J. Opt. Soc. Am. B 5, 765 (1988).
    [Crossref]
  6. A. Javan and P. L. Kelley, IEEE J. Quantum Electron. QE-2, 470 (1966).
    [Crossref]
  7. D. H. Close, Phys. Rev. 153, 360 (1967).
    [Crossref]
  8. J. E. Bjorkholm, AT&T Bell Laboratories, Room 4B-423, Holmdel, New Jersey 07733 (personal correspondence).
  9. I. Golub, Y. Beaudoin, and S. L. Chin, Opt. Lett. 13, 488 (1988); J. Opt. Soc. Am. B 5, 2490 (1988).
    [Crossref] [PubMed]
  10. I. C. Khoo, R. R. Michael, T. H. Liu, G. Finn, and A. E. Kaplan, Proc. Soc. Photo-Opt. Instrum. Eng. 613, 43 (1986); I. C. Khoo, G. M. Finn, R. R. Michael, and T. H. Liu, Opt. Lett. 11, 227 (1986).
    [Crossref]
  11. J. E. Bjorkholm and A. Ashkin, Phys. Rev. Lett. 32, 129 (1974).
    [Crossref]
  12. J. E. Bjorkholm, P. W. Smith, W. J. Tomlinson, and A. E. Kaplan, Opt. Lett. 6, 345 (1981).
    [Crossref] [PubMed]
  13. G. A. Swartzlander, H. Yin, and A. E. Kaplan, Opt. Lett. 13, 1011 (1988).
    [Crossref] [PubMed]
  14. J. H. Marburger, “Self-focusing: theory,” in Progress in Quantum Electronics, J. H. Sanders and S. Stenholm, eds. (Pergamon, London, 1975), Vol. 4, p. 60.
    [Crossref]
  15. See, for example, G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), p. 820.
  16. P. L. Kelley, Phys. Rev. Lett. 15, 1005 (1965).
    [Crossref]
  17. V. E. Zakharov and A. B. Shabat, Zh. Eksp. Teor. Fiz. 61, 118 (1971) [Sov. Phys. JETP 34, 62 (1972)].
  18. R. H. Enns, S. S. Rangnekar, and A. E. Kaplan, Phys. Rev. A 35, 466 (1987).
    [Crossref] [PubMed]
  19. R. H. Hardin and R. D. Tappert, SIAM Rev. Chronicle 15, 423 (1973); T. R. Taha and M. J. Ablowitz, J. Comput. Phys. 55, 203 (1984).
    [Crossref]
  20. Note that 1/τN represents the decay rate into either of the ground states. However, the decay into |2〉 is 5/3 times more likely than the decay into |1〉 because of the degeneracy of these two ground states: g1= 3 and g2= 5. The decay time constants into the two states are given by τ1= τN(g1+ g2)/g1= 43.5 nsec and τ2= τN(g1+ g2)/g2= 26.1 nsec. We have ignored the effects; however, if τ1or τ2is used in place of τN, the saturation intensity will be somewhat lower.
  21. H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, Phys. Rev. Lett. 36, 1135 (1976).
    [Crossref]
  22. A. C. Tam and W. Happer, Phys. Rev. Lett. 38, 278 (1977).
    [Crossref]

1988 (3)

1987 (2)

R. H. Enns, S. S. Rangnekar, and A. E. Kaplan, Phys. Rev. A 35, 466 (1987).
[Crossref] [PubMed]

J. A. Hermann, Opt. Commun. 62, 367 (1987); Opt. Quantum Electron. 19, 169 (1987).
[Crossref]

1986 (1)

I. C. Khoo, R. R. Michael, T. H. Liu, G. Finn, and A. E. Kaplan, Proc. Soc. Photo-Opt. Instrum. Eng. 613, 43 (1986); I. C. Khoo, G. M. Finn, R. R. Michael, and T. H. Liu, Opt. Lett. 11, 227 (1986).
[Crossref]

1981 (2)

1977 (1)

A. C. Tam and W. Happer, Phys. Rev. Lett. 38, 278 (1977).
[Crossref]

1976 (1)

H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, Phys. Rev. Lett. 36, 1135 (1976).
[Crossref]

1974 (1)

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

1973 (1)

R. H. Hardin and R. D. Tappert, SIAM Rev. Chronicle 15, 423 (1973); T. R. Taha and M. J. Ablowitz, J. Comput. Phys. 55, 203 (1984).
[Crossref]

1971 (1)

V. E. Zakharov and A. B. Shabat, Zh. Eksp. Teor. Fiz. 61, 118 (1971) [Sov. Phys. JETP 34, 62 (1972)].

1969 (2)

A. E. Kaplan, Pis’ma Zh. Eksp. Teor. Fiz. 9, 58 (1969) [JETP Lett. 9, 33 (1969)].

M. S. Brodin and A. M. Kamuz, Pis’ma Zh. Eksp. Teor. Fiz. 9, 577 (1969) [JETP Lett. 9, 351 (1969)].

1967 (1)

D. H. Close, Phys. Rev. 153, 360 (1967).
[Crossref]

1966 (1)

A. Javan and P. L. Kelley, IEEE J. Quantum Electron. QE-2, 470 (1966).
[Crossref]

1965 (1)

P. L. Kelley, Phys. Rev. Lett. 15, 1005 (1965).
[Crossref]

Ashkin, A.

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

Beaudoin, Y.

Bjorkholm, J. E.

J. E. Bjorkholm, P. W. Smith, W. J. Tomlinson, and A. E. Kaplan, Opt. Lett. 6, 345 (1981).
[Crossref] [PubMed]

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

J. E. Bjorkholm, AT&T Bell Laboratories, Room 4B-423, Holmdel, New Jersey 07733 (personal correspondence).

Brodin, M. S.

M. S. Brodin and A. M. Kamuz, Pis’ma Zh. Eksp. Teor. Fiz. 9, 577 (1969) [JETP Lett. 9, 351 (1969)].

Chin, S. L.

Close, D. H.

D. H. Close, Phys. Rev. 153, 360 (1967).
[Crossref]

Enns, R. H.

R. H. Enns, S. S. Rangnekar, and A. E. Kaplan, Phys. Rev. A 35, 466 (1987).
[Crossref] [PubMed]

Finn, G.

I. C. Khoo, R. R. Michael, T. H. Liu, G. Finn, and A. E. Kaplan, Proc. Soc. Photo-Opt. Instrum. Eng. 613, 43 (1986); I. C. Khoo, G. M. Finn, R. R. Michael, and T. H. Liu, Opt. Lett. 11, 227 (1986).
[Crossref]

Gibbs, H. M.

H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, Phys. Rev. Lett. 36, 1135 (1976).
[Crossref]

Golub, I.

Happer, W.

A. C. Tam and W. Happer, Phys. Rev. Lett. 38, 278 (1977).
[Crossref]

Hardin, R. H.

R. H. Hardin and R. D. Tappert, SIAM Rev. Chronicle 15, 423 (1973); T. R. Taha and M. J. Ablowitz, J. Comput. Phys. 55, 203 (1984).
[Crossref]

Hermann, J. A.

J. A. Hermann, Opt. Commun. 62, 367 (1987); Opt. Quantum Electron. 19, 169 (1987).
[Crossref]

Javan, A.

A. Javan and P. L. Kelley, IEEE J. Quantum Electron. QE-2, 470 (1966).
[Crossref]

Kamuz, A. M.

M. S. Brodin and A. M. Kamuz, Pis’ma Zh. Eksp. Teor. Fiz. 9, 577 (1969) [JETP Lett. 9, 351 (1969)].

Kaplan, A. E.

G. A. Swartzlander and A. E. Kaplan, J. Opt. Soc. Am. B 5, 765 (1988).
[Crossref]

G. A. Swartzlander, H. Yin, and A. E. Kaplan, Opt. Lett. 13, 1011 (1988).
[Crossref] [PubMed]

R. H. Enns, S. S. Rangnekar, and A. E. Kaplan, Phys. Rev. A 35, 466 (1987).
[Crossref] [PubMed]

I. C. Khoo, R. R. Michael, T. H. Liu, G. Finn, and A. E. Kaplan, Proc. Soc. Photo-Opt. Instrum. Eng. 613, 43 (1986); I. C. Khoo, G. M. Finn, R. R. Michael, and T. H. Liu, Opt. Lett. 11, 227 (1986).
[Crossref]

J. E. Bjorkholm, P. W. Smith, W. J. Tomlinson, and A. E. Kaplan, Opt. Lett. 6, 345 (1981).
[Crossref] [PubMed]

A. E. Kaplan, Opt. Lett. 6, 360 (1981).
[Crossref] [PubMed]

A. E. Kaplan, Pis’ma Zh. Eksp. Teor. Fiz. 9, 58 (1969) [JETP Lett. 9, 33 (1969)].

Kelley, P. L.

A. Javan and P. L. Kelley, IEEE J. Quantum Electron. QE-2, 470 (1966).
[Crossref]

P. L. Kelley, Phys. Rev. Lett. 15, 1005 (1965).
[Crossref]

Khoo, I. C.

I. C. Khoo, R. R. Michael, T. H. Liu, G. Finn, and A. E. Kaplan, Proc. Soc. Photo-Opt. Instrum. Eng. 613, 43 (1986); I. C. Khoo, G. M. Finn, R. R. Michael, and T. H. Liu, Opt. Lett. 11, 227 (1986).
[Crossref]

Korn, G. A.

See, for example, G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), p. 820.

Korn, T. M.

See, for example, G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), p. 820.

Liu, T. H.

I. C. Khoo, R. R. Michael, T. H. Liu, G. Finn, and A. E. Kaplan, Proc. Soc. Photo-Opt. Instrum. Eng. 613, 43 (1986); I. C. Khoo, G. M. Finn, R. R. Michael, and T. H. Liu, Opt. Lett. 11, 227 (1986).
[Crossref]

Marburger, J. H.

J. H. Marburger, “Self-focusing: theory,” in Progress in Quantum Electronics, J. H. Sanders and S. Stenholm, eds. (Pergamon, London, 1975), Vol. 4, p. 60.
[Crossref]

McCall, S. L.

H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, Phys. Rev. Lett. 36, 1135 (1976).
[Crossref]

Michael, R. R.

I. C. Khoo, R. R. Michael, T. H. Liu, G. Finn, and A. E. Kaplan, Proc. Soc. Photo-Opt. Instrum. Eng. 613, 43 (1986); I. C. Khoo, G. M. Finn, R. R. Michael, and T. H. Liu, Opt. Lett. 11, 227 (1986).
[Crossref]

Rangnekar, S. S.

R. H. Enns, S. S. Rangnekar, and A. E. Kaplan, Phys. Rev. A 35, 466 (1987).
[Crossref] [PubMed]

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, Zh. Eksp. Teor. Fiz. 61, 118 (1971) [Sov. Phys. JETP 34, 62 (1972)].

Smith, P. W.

Swartzlander, G. A.

Tam, A. C.

A. C. Tam and W. Happer, Phys. Rev. Lett. 38, 278 (1977).
[Crossref]

Tappert, R. D.

R. H. Hardin and R. D. Tappert, SIAM Rev. Chronicle 15, 423 (1973); T. R. Taha and M. J. Ablowitz, J. Comput. Phys. 55, 203 (1984).
[Crossref]

Tomlinson, W. J.

Venkatesan, T. N. C.

H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, Phys. Rev. Lett. 36, 1135 (1976).
[Crossref]

Yin, H.

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, Zh. Eksp. Teor. Fiz. 61, 118 (1971) [Sov. Phys. JETP 34, 62 (1972)].

IEEE J. Quantum Electron. (1)

A. Javan and P. L. Kelley, IEEE J. Quantum Electron. QE-2, 470 (1966).
[Crossref]

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

Opt. Commun. (1)

J. A. Hermann, Opt. Commun. 62, 367 (1987); Opt. Quantum Electron. 19, 169 (1987).
[Crossref]

Opt. Lett. (4)

Phys. Rev. (1)

D. H. Close, Phys. Rev. 153, 360 (1967).
[Crossref]

Phys. Rev. A (1)

R. H. Enns, S. S. Rangnekar, and A. E. Kaplan, Phys. Rev. A 35, 466 (1987).
[Crossref] [PubMed]

Phys. Rev. Lett. (4)

H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, Phys. Rev. Lett. 36, 1135 (1976).
[Crossref]

A. C. Tam and W. Happer, Phys. Rev. Lett. 38, 278 (1977).
[Crossref]

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

P. L. Kelley, Phys. Rev. Lett. 15, 1005 (1965).
[Crossref]

Pis’ma Zh. Eksp. Teor. Fiz. (2)

A. E. Kaplan, Pis’ma Zh. Eksp. Teor. Fiz. 9, 58 (1969) [JETP Lett. 9, 33 (1969)].

M. S. Brodin and A. M. Kamuz, Pis’ma Zh. Eksp. Teor. Fiz. 9, 577 (1969) [JETP Lett. 9, 351 (1969)].

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

I. C. Khoo, R. R. Michael, T. H. Liu, G. Finn, and A. E. Kaplan, Proc. Soc. Photo-Opt. Instrum. Eng. 613, 43 (1986); I. C. Khoo, G. M. Finn, R. R. Michael, and T. H. Liu, Opt. Lett. 11, 227 (1986).
[Crossref]

SIAM Rev. Chronicle (1)

R. H. Hardin and R. D. Tappert, SIAM Rev. Chronicle 15, 423 (1973); T. R. Taha and M. J. Ablowitz, J. Comput. Phys. 55, 203 (1984).
[Crossref]

Zh. Eksp. Teor. Fiz. (1)

V. E. Zakharov and A. B. Shabat, Zh. Eksp. Teor. Fiz. 61, 118 (1971) [Sov. Phys. JETP 34, 62 (1972)].

Other (4)

J. H. Marburger, “Self-focusing: theory,” in Progress in Quantum Electronics, J. H. Sanders and S. Stenholm, eds. (Pergamon, London, 1975), Vol. 4, p. 60.
[Crossref]

See, for example, G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), p. 820.

J. E. Bjorkholm, AT&T Bell Laboratories, Room 4B-423, Holmdel, New Jersey 07733 (personal correspondence).

Note that 1/τN represents the decay rate into either of the ground states. However, the decay into |2〉 is 5/3 times more likely than the decay into |1〉 because of the degeneracy of these two ground states: g1= 3 and g2= 5. The decay time constants into the two states are given by τ1= τN(g1+ g2)/g1= 43.5 nsec and τ2= τN(g1+ g2)/g2= 26.1 nsec. We have ignored the effects; however, if τ1or τ2is used in place of τN, the saturation intensity will be somewhat lower.

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Figures (10)

Fig. 1
Fig. 1

Right-triangular intensity profile, of size w0 and peak intensity I0, is incident upon a thin nonlinear-refractive medium (n = n0 + n2I) of length L. A nonlinear prism is induced in the medium, causing the transmitted rays to be deflected by an angle θNL.

Fig. 2
Fig. 2

Schematic diagram of the experiment. See the text for a complete description. The inset shows a razor blade inserted into the beam to form a semi-Gaussian beam profile, which is then imaged into the sodium cell of length L. The beam is self-deflected away from the razor image because n2 < 0.

Fig. 3
Fig. 3

Far-field intensity profiles obtained at 202°C are shown by the solid curves for both linear (a) and nonlinear (b) propagation. Numerical calculations based on a slab-beam/thin-film analysis are shown as dotted curves. (c) The self-deflection angle is plotted against the input beam power (circles with error bars). The linear relationship indicates that Δn = n2I, where n2 = −1.4 × 10−7 cm2/W. The centroid of the intensity distribution of each of the measured profiles is marked with a diamond.

Fig. 4
Fig. 4

Experimental results obtained at 212°C are shown by the solid curves for both linear (a) and nonlinear (b) propagation. Results of our numerical calculations are depicted by the dotted curves. (c) Self-deflection angle plotted against the input-beam power (circles with error bars). Rather than increasing linearly with intensity (solid curve), the self-deflection effect apparently saturates, The centroid of the intensity distribution of each of the measured profiles is marked with a diamond.

Fig. 5
Fig. 5

Self-deflection angle as a function of the normalized detuning frequency. The experimental data are represented as circles with error bars. The solid curve is a numerical calculation that is discussed in the text. The diamonds indicate the position of the centroid of the intensity distribution of the profiles.

Fig. 6
Fig. 6

Far-field intensity profiles across a narrow slice of the beam cross section are plotted. The liquid-sodium temperature was 197°C. The peak of the self-deflected beam, shown as the solid curve in (b), is offset from the peak of the linearly propagated beam, shown as the solid curve in (a), by 1.6 mrad. The dashed curve shows the magnitude of the self-deflected profile, measured at a near-resonant frequency, in comparison with the linear profile measured at an off-resonant frequency.

Fig. 7
Fig. 7

A large self-deflection angle, measuring 8.4θD, was achieved at 214°C. The linear (a) and nonlinear (b) profiles have been arbitrarily normalized.

Fig. 8
Fig. 8

The on-axis far-field beam intensity is plotted against the input beam power. The solid curve is a fit to the analytical solution for a Kerr-nonlinear medium, with n2 = −1.7 × 10−7 cm2/W.

Fig. 9
Fig. 9

The normalized profiles of self-deflected beams that were obtained at different laser frequencies are shown, and their relative magnitudes are depicted in the inset. The beam centered at zero (A) was measured far off-resonance, and the others (B, C, and D) were measured at −3.9, −2.9, and −1.9 GHz, respectively, from the resonance.

Fig. 10
Fig. 10

(a) Treating sodium as an inhomogeneously broadened two-level system, we computed values of the intensity-dependent refractive index, Δn(I), finding, in contrast to our experimental results, that strong saturation should occur. The inset shows the functions Re[w] (1) and −Im[w] (2) in their linear states, with their saturated states depicted by 1′ and 2′, respectively. As observed experimentally, our numerical calculations predict that the maximum nonlinear effect should occur at Δ ≃ −1 (and Δ ≃ +1). (b) A comparison of the self-deflected profiles of slab semi-Gaussian beams in saturated (solid curve) and Kerr (dotted curve) nonlinear media shows that a saturated nonlinear response does not produce subpeaks.

Equations (15)

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θ NL / θ D = - n 2 k L I 0 / 2 ,
I ( θ x , 0 ) | 0 w r d r ( 1 - r / w ) 1 / 2 exp ( - i n 2 I 0 k L r / w ) × - π / 2 π / 2 d ϕ exp ( - i k θ x r cos ϕ ) | 2 ,
ξ on - axis - π / 2 + π / 2 d θ 0 r d r I 0 1 / 2 exp ( - r 2 / w 0 2 ) × exp [ - i n 2 k L I 0 exp ( - 2 r 2 / w 0 2 ) ] .
I on - axis [ C 2 ( η NL ) + S 2 ( η NL ) ] w 0 4 / n 2 k L ,             η NL = ( 2 n 2 k L I 0 / π ) 1 / 2 ,
2 i k E / z = 2 E / x 2 + k 2 ( 2 / 0 ) E 2 E ,
- + E ( x , z ) 2 d x = C 1 = inv .
( 2 i k ) - 1 - + ( E E x * - E * E x ) d x = C 2 = inv ,
x c - + x E ( x ) 2 d x / C 1
C 1 d x c / d z = - + x d x ( E * E z + E E z * ) = - + d x ( x / 2 i k ) ( E * E x x - E E x x * ) ,
d x c / d z x ( E * E x - E E x * ) | - + - - ( E * E x - E E x * ) d x .
d x c / d z - ( E * E x - E E x * ) d x = const . ,
x c = z c 2 / c 1 + x c ( z = 0 ) ,
n ( ν , I ) - 1 = - δ n Im [ w ( Δ + i η ) ] ,
E - 1 d E / d z = - 2 π ν δ n ( 1 + I / I s ) - 1 / 2 / c × Re [ w ( Δ + i η ) ] ,
Δ n n 2 I ,             n 2 = ( ln 2 ) 3 / 2 λ 6 N / ( 32 π 6 Δ 3 h c Δ ν D 3 τ N 2 ) .

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