Abstract

We consider the interaction of a fully amplitude-modulated, narrow-band laser with a strongly inhomogeneously broadened two-level medium and show that harmonics of the fluorescence of susceptibility, unlike the time-averaged fluorescence, retain the characteristic multiphoton resonance structure that is found in the absence of inhomogeneous broadening. This behavior correlates with the existence of double branches of the corresponding Bloch–Siegert resonance curves. For strong fields, the depth of modulation and the phase shift of the lowest fluorescence harmonic may deviate significantly from the results for weak, broadband excitation.

© 1989 Optical Society of America

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  1. F. Duschinsky, Z. Phys. 81, 7 (1933).
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  2. L. Armstrong, S. Feneuille, J. Phys. B 8, 546 (1975); W. A. McClean, S. Swain, J. Phys. B. 9, 2011 (1976); L. M. Davis, Phys. Rev. Lett. 60, 1258 (1988).
    [CrossRef] [PubMed]
  3. S. Feneuille, M.-G. Schweighofer, G. Oliver, J. Phys. B 9, 2003 (1976).
    [CrossRef]
  4. G. S. Agarwal, N. Nayak, J. Opt. Soc. Am. B 1, 164 (1984).
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  5. P. Thomann, J. Phys. B 9, 2411 (1976); J. Phys. B 13, 1111 (1980); G. S. Agarwal, N. Nayak, J. Phys. B 19, 3385 (1986); W. M. Ruyten, Phys. Rev. A 39, 442 (1989).
    [CrossRef] [PubMed]
  6. R. Saxena, G. S. Agarwal, J. Phys. B 12, 1939 (1979).
    [CrossRef]
  7. L. W. Hillman, J. Krasinski, R. W. Boyd, C. R. Stroud, Phys. Rev. Lett. 52, 1605 (1984); L. W. Hillman, J. Krasinski, K. Koch, C. R. Stroud, J. Opt. Soc. Am. B 2, 211 (1985).
    [CrossRef]
  8. S. Chakmakjian, K. Koch, C. R. Stroud, J. Opt. Soc. Am. B 5, 2015 (1988).
    [CrossRef]
  9. W. M. Ruyten, in Optical Society of America Annual Meeting Technical Digest (Optical Society of America, Washington, D.C., 1988), p. 63.
  10. M. T. Gruneisen, K. R. MacDonald, R. W. Boyd, J. Opt. Soc. Am. B 5, 123 (1988); M. T. Gruneisen, K. R. MacDonald, R. W. Boyd, D. J. Harter, in Optical Society of America Annual Meeting Technical Digest (Optical Society of America, Washington, D.C., 1988), p. 123.
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  12. S. Stenholm, J. Phys. B 5, 878, 890 (1972); F. Ahmad, R. K. Bullough, J. Phys. B 7, L275 (1974).
    [CrossRef]
  13. E. Kyrölä, S. Stenholm, Opt. Comm. 22, 123 (1977); Opt. Comm. 30, 37 (1979).
    [CrossRef]
  14. For Δ = 0, the multiphoton resonance condition can be written as Ω = 2g/x0k ≅ g/k (where x0k is the kth zero of the zeroth-order Bessel function); i.e., Ω is roughly a submultiple of g. Hence the term subharmonic resonances.
  15. E. Arimondo, G. Moruzzi, J. Phys. B 6, 2382 (1973).
    [CrossRef]
  16. W. M. Ruyten, J. W. L. Lewis, J. Opt. Soc. Am. B 5, 2368 (1988).
    [CrossRef]

1988

1987

1984

G. S. Agarwal, N. Nayak, J. Opt. Soc. Am. B 1, 164 (1984).
[CrossRef]

L. W. Hillman, J. Krasinski, R. W. Boyd, C. R. Stroud, Phys. Rev. Lett. 52, 1605 (1984); L. W. Hillman, J. Krasinski, K. Koch, C. R. Stroud, J. Opt. Soc. Am. B 2, 211 (1985).
[CrossRef]

1979

R. Saxena, G. S. Agarwal, J. Phys. B 12, 1939 (1979).
[CrossRef]

1977

E. Kyrölä, S. Stenholm, Opt. Comm. 22, 123 (1977); Opt. Comm. 30, 37 (1979).
[CrossRef]

1976

S. Feneuille, M.-G. Schweighofer, G. Oliver, J. Phys. B 9, 2003 (1976).
[CrossRef]

P. Thomann, J. Phys. B 9, 2411 (1976); J. Phys. B 13, 1111 (1980); G. S. Agarwal, N. Nayak, J. Phys. B 19, 3385 (1986); W. M. Ruyten, Phys. Rev. A 39, 442 (1989).
[CrossRef] [PubMed]

1975

L. Armstrong, S. Feneuille, J. Phys. B 8, 546 (1975); W. A. McClean, S. Swain, J. Phys. B. 9, 2011 (1976); L. M. Davis, Phys. Rev. Lett. 60, 1258 (1988).
[CrossRef] [PubMed]

1973

E. Arimondo, G. Moruzzi, J. Phys. B 6, 2382 (1973).
[CrossRef]

1972

S. Stenholm, J. Phys. B 5, 878, 890 (1972); F. Ahmad, R. K. Bullough, J. Phys. B 7, L275 (1974).
[CrossRef]

1933

F. Duschinsky, Z. Phys. 81, 7 (1933).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal, N. Nayak, J. Opt. Soc. Am. B 1, 164 (1984).
[CrossRef]

R. Saxena, G. S. Agarwal, J. Phys. B 12, 1939 (1979).
[CrossRef]

Arimondo, E.

E. Arimondo, G. Moruzzi, J. Phys. B 6, 2382 (1973).
[CrossRef]

Armstrong, L.

L. Armstrong, S. Feneuille, J. Phys. B 8, 546 (1975); W. A. McClean, S. Swain, J. Phys. B. 9, 2011 (1976); L. M. Davis, Phys. Rev. Lett. 60, 1258 (1988).
[CrossRef] [PubMed]

Boyd, R. W.

Chakmakjian, S.

Duschinsky, F.

F. Duschinsky, Z. Phys. 81, 7 (1933).
[CrossRef]

Feneuille, S.

S. Feneuille, M.-G. Schweighofer, G. Oliver, J. Phys. B 9, 2003 (1976).
[CrossRef]

L. Armstrong, S. Feneuille, J. Phys. B 8, 546 (1975); W. A. McClean, S. Swain, J. Phys. B. 9, 2011 (1976); L. M. Davis, Phys. Rev. Lett. 60, 1258 (1988).
[CrossRef] [PubMed]

Gruneisen, M. T.

Hillman, L. W.

L. W. Hillman, J. Krasinski, R. W. Boyd, C. R. Stroud, Phys. Rev. Lett. 52, 1605 (1984); L. W. Hillman, J. Krasinski, K. Koch, C. R. Stroud, J. Opt. Soc. Am. B 2, 211 (1985).
[CrossRef]

Keefer, D.

Koch, K.

Krasinski, J.

L. W. Hillman, J. Krasinski, R. W. Boyd, C. R. Stroud, Phys. Rev. Lett. 52, 1605 (1984); L. W. Hillman, J. Krasinski, K. Koch, C. R. Stroud, J. Opt. Soc. Am. B 2, 211 (1985).
[CrossRef]

Kyrölä, E.

E. Kyrölä, S. Stenholm, Opt. Comm. 22, 123 (1977); Opt. Comm. 30, 37 (1979).
[CrossRef]

Lewis, J. W. L.

MacDonald, K. R.

Moruzzi, G.

E. Arimondo, G. Moruzzi, J. Phys. B 6, 2382 (1973).
[CrossRef]

Nayak, N.

Oliver, G.

S. Feneuille, M.-G. Schweighofer, G. Oliver, J. Phys. B 9, 2003 (1976).
[CrossRef]

Ruyten, W. M.

W. M. Ruyten, J. W. L. Lewis, J. Opt. Soc. Am. B 5, 2368 (1988).
[CrossRef]

W. M. Ruyten, in Optical Society of America Annual Meeting Technical Digest (Optical Society of America, Washington, D.C., 1988), p. 63.

Saxena, R.

R. Saxena, G. S. Agarwal, J. Phys. B 12, 1939 (1979).
[CrossRef]

Schweighofer, M.-G.

S. Feneuille, M.-G. Schweighofer, G. Oliver, J. Phys. B 9, 2003 (1976).
[CrossRef]

Stenholm, S.

E. Kyrölä, S. Stenholm, Opt. Comm. 22, 123 (1977); Opt. Comm. 30, 37 (1979).
[CrossRef]

S. Stenholm, J. Phys. B 5, 878, 890 (1972); F. Ahmad, R. K. Bullough, J. Phys. B 7, L275 (1974).
[CrossRef]

Stroud, C. R.

S. Chakmakjian, K. Koch, C. R. Stroud, J. Opt. Soc. Am. B 5, 2015 (1988).
[CrossRef]

L. W. Hillman, J. Krasinski, R. W. Boyd, C. R. Stroud, Phys. Rev. Lett. 52, 1605 (1984); L. W. Hillman, J. Krasinski, K. Koch, C. R. Stroud, J. Opt. Soc. Am. B 2, 211 (1985).
[CrossRef]

Thomann, P.

P. Thomann, J. Phys. B 9, 2411 (1976); J. Phys. B 13, 1111 (1980); G. S. Agarwal, N. Nayak, J. Phys. B 19, 3385 (1986); W. M. Ruyten, Phys. Rev. A 39, 442 (1989).
[CrossRef] [PubMed]

Appl. Opt.

J. Opt. Soc. Am. B

J. Phys. B

L. Armstrong, S. Feneuille, J. Phys. B 8, 546 (1975); W. A. McClean, S. Swain, J. Phys. B. 9, 2011 (1976); L. M. Davis, Phys. Rev. Lett. 60, 1258 (1988).
[CrossRef] [PubMed]

S. Feneuille, M.-G. Schweighofer, G. Oliver, J. Phys. B 9, 2003 (1976).
[CrossRef]

P. Thomann, J. Phys. B 9, 2411 (1976); J. Phys. B 13, 1111 (1980); G. S. Agarwal, N. Nayak, J. Phys. B 19, 3385 (1986); W. M. Ruyten, Phys. Rev. A 39, 442 (1989).
[CrossRef] [PubMed]

R. Saxena, G. S. Agarwal, J. Phys. B 12, 1939 (1979).
[CrossRef]

S. Stenholm, J. Phys. B 5, 878, 890 (1972); F. Ahmad, R. K. Bullough, J. Phys. B 7, L275 (1974).
[CrossRef]

E. Arimondo, G. Moruzzi, J. Phys. B 6, 2382 (1973).
[CrossRef]

Opt. Comm.

E. Kyrölä, S. Stenholm, Opt. Comm. 22, 123 (1977); Opt. Comm. 30, 37 (1979).
[CrossRef]

Phys. Rev. Lett.

L. W. Hillman, J. Krasinski, R. W. Boyd, C. R. Stroud, Phys. Rev. Lett. 52, 1605 (1984); L. W. Hillman, J. Krasinski, K. Koch, C. R. Stroud, J. Opt. Soc. Am. B 2, 211 (1985).
[CrossRef]

Z. Phys.

F. Duschinsky, Z. Phys. 81, 7 (1933).
[CrossRef]

Other

For Δ = 0, the multiphoton resonance condition can be written as Ω = 2g/x0k ≅ g/k (where x0k is the kth zero of the zeroth-order Bessel function); i.e., Ω is roughly a submultiple of g. Hence the term subharmonic resonances.

W. M. Ruyten, in Optical Society of America Annual Meeting Technical Digest (Optical Society of America, Washington, D.C., 1988), p. 63.

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Figures (4)

Fig. 1
Fig. 1

Dc fluorescence r0 (upper left) and in-phase component r2 cos φ2 of the fluorescence modulated at 2Ω, for ΩT1 = ΩT2/2 = 10. The projections of the maxima of r0 and the zeros of r2 cos φ2 (intersections with the shaded plane) are shown in the limit ΩT1, ΩT2 → ∞. Note the doubly branched resonance structure for the component at 2Ω.

Fig. 2
Fig. 2

Bloch–Siegert resonance curves for harmonic components of the fluorescence at 2Ω (above) and the susceptibility at Ω (below). Solid curves coincide with the Bloch–Siegert curves of the dc component. Circles, squares, and triangles mark zeros of the zeroth-, first-, and second-order Bessel functions, respectively. N denotes the quantum number of the resonances.

Fig. 3
Fig. 3

Inhomogeneously broadened values of the dc and 2Ω components of (a) the fluorescence and (b) the imaginary part of the susceptibility at Ω as a function of the interaction strength for ΩT1 = ΩT2/2 = 2. The dashed line represents the constant field approximation of relation (3). The resonance structure remains where the corresponding Bloch–Siegert curves are doubly branched.

Fig. 4
Fig. 4

Inhomogeneously broadened values of (a) the depth of modulation 2/0 and (b) the phase φ 2 ¯ of the fluorescence modulated at 2Ω for ΩT1 = 2 and various ratios T1/T2. For 2g/Ω → 0, the IHB results are equal to those for weak, broadband excitation (circles), independent of T2.

Equations (3)

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ρ 22 ( t ) = r 0 + n = 1 r 2 n cos ( 2 n Ω t - φ 2 n )
( 1 + δ n , 0 ) r 2 n = δ n , 0 - J 0 ( 2 g / Ω ) J 2 n ( 2 g / Ω ) .
r 0 ¯ ( π g 2 T 1 / Ω ) ( 1 + 2 g 2 T 1 T 2 ) - 1 / 2

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