Abstract

The dependence of the Rayleigh-type optical mixing signal intensity on the incident pump and probe frequencies for different values of T1/T2 is shown in a three-dimensional plot for a homogeneously broadened line. The symmetry properties of the signal in this space and of the nonlinear process that generates the line are related. These properties are used to show the dependence of the measured spectra on the way in which the frequencies of the incident fields are varied. Furthermore, this plot in frequency space yields information that may be used for the determination of the relaxation times T1 and T2.

© 1990 Optical Society of America

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  1. T. Yajima, H. Souma, Y. Ishida, Phys. Rev. A 17, 324 (1978).
    [CrossRef]
  2. H. Souma, T. Yajima, Y. Taira, J. Phys. Soc. Jpn. 48, 2040 (1980).
    [CrossRef]
  3. H. Souma, E. J. Heilweil, R. M. Hochstrasser, J. Chem. Phys. 76, 5693 (1982).
    [CrossRef]
  4. Y. Masumoto, S. Shionoya, H. Okamoto, Opt. Commun. 53, 385 (1985).
    [CrossRef]
  5. F. García Golding, A. Marcano O., Phys. Rev. A 32, 1526 (1985).
    [CrossRef]
  6. J. L. Paz, H. J. Franco, I. Reif, A. Marcano O., F. García Golding, Phys. Rev. A 37, 9 (1988).
    [CrossRef]
  7. T. Yajima, H. Souma, Phys. Rev. A 17, 309 (1978).
    [CrossRef]
  8. S. Haroche, F. Hartman, Phys. Rev. A 6, 1280 (1972).
    [CrossRef]
  9. B. R. Mollow, Phys. Rev. A 5, 2217 (1972).
    [CrossRef]
  10. R. W. Boyd, M. G. Raymer, P. Narum, D. J. Harter, Phys. Rev. A 24, 411 (1981).
    [CrossRef]
  11. G. S. Agarwal, N. Nayak, J. Opt. Soc. Am. B 1, 264 (1984);Phys. Rev. A 33, 391 (1986).
  12. G. I. Toptygina, E. E. Fradkin, Zh. Eksp. Teor. Fiz. 182, 429 (1982) [Sov. Phys. JETP 55, 246 (1982)].
  13. R. W. Boyd, M. Sargent, J. Opt. Soc. Am. B 5, 99 (1988).
    [CrossRef]
  14. M. T. Gruneisen, K. R. MacDonald, R. W. Boyd, J. Opt. Soc. Am. B 5, 123 (1988).
    [CrossRef]
  15. F. Y. Wu, S. Ezequiel, M. Ducloy, B. R. Mollow, Phys. Rev. Lett. 38, 1077 (1977).
    [CrossRef]
  16. G. Grynberg, E. Le Bihan, M. Pinard, J. Phys(Paris) 47, 1321 (1986).
  17. G. Khitrova, J. K. Valley, H. M. Gibbs, Phys. Rev. Lett. 60, 1126 (1988).
    [CrossRef] [PubMed]

1988 (4)

J. L. Paz, H. J. Franco, I. Reif, A. Marcano O., F. García Golding, Phys. Rev. A 37, 9 (1988).
[CrossRef]

R. W. Boyd, M. Sargent, J. Opt. Soc. Am. B 5, 99 (1988).
[CrossRef]

M. T. Gruneisen, K. R. MacDonald, R. W. Boyd, J. Opt. Soc. Am. B 5, 123 (1988).
[CrossRef]

G. Khitrova, J. K. Valley, H. M. Gibbs, Phys. Rev. Lett. 60, 1126 (1988).
[CrossRef] [PubMed]

1986 (1)

G. Grynberg, E. Le Bihan, M. Pinard, J. Phys(Paris) 47, 1321 (1986).

1985 (2)

Y. Masumoto, S. Shionoya, H. Okamoto, Opt. Commun. 53, 385 (1985).
[CrossRef]

F. García Golding, A. Marcano O., Phys. Rev. A 32, 1526 (1985).
[CrossRef]

1984 (1)

G. S. Agarwal, N. Nayak, J. Opt. Soc. Am. B 1, 264 (1984);Phys. Rev. A 33, 391 (1986).

1982 (2)

G. I. Toptygina, E. E. Fradkin, Zh. Eksp. Teor. Fiz. 182, 429 (1982) [Sov. Phys. JETP 55, 246 (1982)].

H. Souma, E. J. Heilweil, R. M. Hochstrasser, J. Chem. Phys. 76, 5693 (1982).
[CrossRef]

1981 (1)

R. W. Boyd, M. G. Raymer, P. Narum, D. J. Harter, Phys. Rev. A 24, 411 (1981).
[CrossRef]

1980 (1)

H. Souma, T. Yajima, Y. Taira, J. Phys. Soc. Jpn. 48, 2040 (1980).
[CrossRef]

1978 (2)

T. Yajima, H. Souma, Y. Ishida, Phys. Rev. A 17, 324 (1978).
[CrossRef]

T. Yajima, H. Souma, Phys. Rev. A 17, 309 (1978).
[CrossRef]

1977 (1)

F. Y. Wu, S. Ezequiel, M. Ducloy, B. R. Mollow, Phys. Rev. Lett. 38, 1077 (1977).
[CrossRef]

1972 (2)

S. Haroche, F. Hartman, Phys. Rev. A 6, 1280 (1972).
[CrossRef]

B. R. Mollow, Phys. Rev. A 5, 2217 (1972).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal, N. Nayak, J. Opt. Soc. Am. B 1, 264 (1984);Phys. Rev. A 33, 391 (1986).

Boyd, R. W.

Ducloy, M.

F. Y. Wu, S. Ezequiel, M. Ducloy, B. R. Mollow, Phys. Rev. Lett. 38, 1077 (1977).
[CrossRef]

Ezequiel, S.

F. Y. Wu, S. Ezequiel, M. Ducloy, B. R. Mollow, Phys. Rev. Lett. 38, 1077 (1977).
[CrossRef]

Fradkin, E. E.

G. I. Toptygina, E. E. Fradkin, Zh. Eksp. Teor. Fiz. 182, 429 (1982) [Sov. Phys. JETP 55, 246 (1982)].

Franco, H. J.

J. L. Paz, H. J. Franco, I. Reif, A. Marcano O., F. García Golding, Phys. Rev. A 37, 9 (1988).
[CrossRef]

Gibbs, H. M.

G. Khitrova, J. K. Valley, H. M. Gibbs, Phys. Rev. Lett. 60, 1126 (1988).
[CrossRef] [PubMed]

Golding, F. García

J. L. Paz, H. J. Franco, I. Reif, A. Marcano O., F. García Golding, Phys. Rev. A 37, 9 (1988).
[CrossRef]

F. García Golding, A. Marcano O., Phys. Rev. A 32, 1526 (1985).
[CrossRef]

Gruneisen, M. T.

Grynberg, G.

G. Grynberg, E. Le Bihan, M. Pinard, J. Phys(Paris) 47, 1321 (1986).

Haroche, S.

S. Haroche, F. Hartman, Phys. Rev. A 6, 1280 (1972).
[CrossRef]

Harter, D. J.

R. W. Boyd, M. G. Raymer, P. Narum, D. J. Harter, Phys. Rev. A 24, 411 (1981).
[CrossRef]

Hartman, F.

S. Haroche, F. Hartman, Phys. Rev. A 6, 1280 (1972).
[CrossRef]

Heilweil, E. J.

H. Souma, E. J. Heilweil, R. M. Hochstrasser, J. Chem. Phys. 76, 5693 (1982).
[CrossRef]

Hochstrasser, R. M.

H. Souma, E. J. Heilweil, R. M. Hochstrasser, J. Chem. Phys. 76, 5693 (1982).
[CrossRef]

Ishida, Y.

T. Yajima, H. Souma, Y. Ishida, Phys. Rev. A 17, 324 (1978).
[CrossRef]

Khitrova, G.

G. Khitrova, J. K. Valley, H. M. Gibbs, Phys. Rev. Lett. 60, 1126 (1988).
[CrossRef] [PubMed]

Le Bihan, E.

G. Grynberg, E. Le Bihan, M. Pinard, J. Phys(Paris) 47, 1321 (1986).

MacDonald, K. R.

Marcano O., A.

J. L. Paz, H. J. Franco, I. Reif, A. Marcano O., F. García Golding, Phys. Rev. A 37, 9 (1988).
[CrossRef]

F. García Golding, A. Marcano O., Phys. Rev. A 32, 1526 (1985).
[CrossRef]

Masumoto, Y.

Y. Masumoto, S. Shionoya, H. Okamoto, Opt. Commun. 53, 385 (1985).
[CrossRef]

Mollow, B. R.

F. Y. Wu, S. Ezequiel, M. Ducloy, B. R. Mollow, Phys. Rev. Lett. 38, 1077 (1977).
[CrossRef]

B. R. Mollow, Phys. Rev. A 5, 2217 (1972).
[CrossRef]

Narum, P.

R. W. Boyd, M. G. Raymer, P. Narum, D. J. Harter, Phys. Rev. A 24, 411 (1981).
[CrossRef]

Nayak, N.

G. S. Agarwal, N. Nayak, J. Opt. Soc. Am. B 1, 264 (1984);Phys. Rev. A 33, 391 (1986).

Okamoto, H.

Y. Masumoto, S. Shionoya, H. Okamoto, Opt. Commun. 53, 385 (1985).
[CrossRef]

Paz, J. L.

J. L. Paz, H. J. Franco, I. Reif, A. Marcano O., F. García Golding, Phys. Rev. A 37, 9 (1988).
[CrossRef]

Pinard, M.

G. Grynberg, E. Le Bihan, M. Pinard, J. Phys(Paris) 47, 1321 (1986).

Raymer, M. G.

R. W. Boyd, M. G. Raymer, P. Narum, D. J. Harter, Phys. Rev. A 24, 411 (1981).
[CrossRef]

Reif, I.

J. L. Paz, H. J. Franco, I. Reif, A. Marcano O., F. García Golding, Phys. Rev. A 37, 9 (1988).
[CrossRef]

Sargent, M.

Shionoya, S.

Y. Masumoto, S. Shionoya, H. Okamoto, Opt. Commun. 53, 385 (1985).
[CrossRef]

Souma, H.

H. Souma, E. J. Heilweil, R. M. Hochstrasser, J. Chem. Phys. 76, 5693 (1982).
[CrossRef]

H. Souma, T. Yajima, Y. Taira, J. Phys. Soc. Jpn. 48, 2040 (1980).
[CrossRef]

T. Yajima, H. Souma, Y. Ishida, Phys. Rev. A 17, 324 (1978).
[CrossRef]

T. Yajima, H. Souma, Phys. Rev. A 17, 309 (1978).
[CrossRef]

Taira, Y.

H. Souma, T. Yajima, Y. Taira, J. Phys. Soc. Jpn. 48, 2040 (1980).
[CrossRef]

Toptygina, G. I.

G. I. Toptygina, E. E. Fradkin, Zh. Eksp. Teor. Fiz. 182, 429 (1982) [Sov. Phys. JETP 55, 246 (1982)].

Valley, J. K.

G. Khitrova, J. K. Valley, H. M. Gibbs, Phys. Rev. Lett. 60, 1126 (1988).
[CrossRef] [PubMed]

Wu, F. Y.

F. Y. Wu, S. Ezequiel, M. Ducloy, B. R. Mollow, Phys. Rev. Lett. 38, 1077 (1977).
[CrossRef]

Yajima, T.

H. Souma, T. Yajima, Y. Taira, J. Phys. Soc. Jpn. 48, 2040 (1980).
[CrossRef]

T. Yajima, H. Souma, Y. Ishida, Phys. Rev. A 17, 324 (1978).
[CrossRef]

T. Yajima, H. Souma, Phys. Rev. A 17, 309 (1978).
[CrossRef]

J. Chem. Phys. (1)

H. Souma, E. J. Heilweil, R. M. Hochstrasser, J. Chem. Phys. 76, 5693 (1982).
[CrossRef]

J. Opt. Soc. Am. B (3)

J. Phys(Paris) (1)

G. Grynberg, E. Le Bihan, M. Pinard, J. Phys(Paris) 47, 1321 (1986).

J. Phys. Soc. Jpn. (1)

H. Souma, T. Yajima, Y. Taira, J. Phys. Soc. Jpn. 48, 2040 (1980).
[CrossRef]

Opt. Commun. (1)

Y. Masumoto, S. Shionoya, H. Okamoto, Opt. Commun. 53, 385 (1985).
[CrossRef]

Phys. Rev. A (7)

F. García Golding, A. Marcano O., Phys. Rev. A 32, 1526 (1985).
[CrossRef]

J. L. Paz, H. J. Franco, I. Reif, A. Marcano O., F. García Golding, Phys. Rev. A 37, 9 (1988).
[CrossRef]

T. Yajima, H. Souma, Phys. Rev. A 17, 309 (1978).
[CrossRef]

S. Haroche, F. Hartman, Phys. Rev. A 6, 1280 (1972).
[CrossRef]

B. R. Mollow, Phys. Rev. A 5, 2217 (1972).
[CrossRef]

R. W. Boyd, M. G. Raymer, P. Narum, D. J. Harter, Phys. Rev. A 24, 411 (1981).
[CrossRef]

T. Yajima, H. Souma, Y. Ishida, Phys. Rev. A 17, 324 (1978).
[CrossRef]

Phys. Rev. Lett. (2)

G. Khitrova, J. K. Valley, H. M. Gibbs, Phys. Rev. Lett. 60, 1126 (1988).
[CrossRef] [PubMed]

F. Y. Wu, S. Ezequiel, M. Ducloy, B. R. Mollow, Phys. Rev. Lett. 38, 1077 (1977).
[CrossRef]

Zh. Eksp. Teor. Fiz. (1)

G. I. Toptygina, E. E. Fradkin, Zh. Eksp. Teor. Fiz. 182, 429 (1982) [Sov. Phys. JETP 55, 246 (1982)].

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Figures (9)

Fig. 1
Fig. 1

Three-dimensional plot of the RTOM signal intensity as a function of Δ1 = ω1ω0 and Δ2 = ω2ω0 for the case of T1 = T2.

Fig. 2
Fig. 2

Three-dimensional plot of the RTOM signal intensity as a function of Δ1 = ω1ω0 and Δ2 = ω2ω0 for the case of T1 = 10T2.

Fig. 3
Fig. 3

Contour plot of the RTOM signal intensity for T1 = T2.

Fig. 4
Fig. 4

Contour plot of the RTOM signal intensity for T1 = 10T2.

Fig. 5
Fig. 5

Cuts of the three-dimensional plot of Fig. 2 along the diagonals Δ1 = Δ2 (curve a) and Δ1 = −Δ2 (curve b).

Fig. 6
Fig. 6

RTOM signal symmetry properties in the frequency space. The points A, B, C, and D have the following coordinates: A ( Δ 1 0 , Δ 2 0 ), B ( Δ 1 0 , 2 Δ 1 0 Δ 2 0 ), C ( Δ 1 0 , 2 Δ 1 0 + Δ 2 0 ), D ( Δ 1 0 , Δ 2 0 ). The values of frequencies ω1 and ω2 at these points are shown at the bottom. For clarity, the magnitudes of the frequency detuning are exaggerated.

Fig. 7
Fig. 7

Cuts of the three-dimensional plot of Fig. 2 along the horizontal Δ1 = 0 (curve a) and the vertical Δ2 = 0 (curve b). For the horizontal spectrum j = 2, and for the vertical spectrum j = 1.

Fig. 8
Fig. 8

Cuts of the three-dimensional plot of Fig. 1 along the horizontal Δ1 = 1/2T2 (curve a) and the vertical Δ2 = 1/2T2 (curve b). For the horizontal spectrum j = 2, and for the vertical spectrum j = 1.

Fig. 9
Fig. 9

Cuts of the three-dimensional plot of Fig. 2 along the horizontal Δ1 = 1/2T2 (curve a) and the vertical Δ2 = 1/2T2 (curve b). For the vertical spectrum j = 1, and for the horizontal spectrum j = 2.

Tables (2)

Tables Icon

Table 1 Angle (α) between the Horizontal Axes and the Line, Which Contains the Maxima of the Vertical Spectra, As a Function of T1/T2

Tables Icon

Table 2 Summary of the Results

Equations (16)

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d ρ D d t = 2 i ( H ab ρ ba ρ ab H ba ) 1 T 1 [ ρ D ρ D ( 0 ) ] ,
d ρ ba d t = i H ba ρ D ( 1 T 2 + i ω 0 ) ρ ba = d ρ ab * d t ,
H ba = [ μ ba · E 1 exp ( i ω 1 t ) μ ba · E 2 exp ( i ω 2 t ) ] + c . c . ,
ρ ba ( ω 3 ) = ( 2 i 3 ) [ ( μ ba · E 1 ) 2 ( μ ab · E 2 * ) Γ 1 ( Δ ) D 3 ] × ( 1 D 1 + 1 D 2 * ) ρ D ( 0 ) exp ( i ω 3 t ) ,
Γ 1 ( Δ ) = 1 / T 1 i ( Δ ) , Δ = ω 1 ω 2 , D j = 1 / T 2 + i ( ω 0 ω j ) , j = 1 , 2 , 3.
P = N ρ ba μ ab ,
I ( Δ 1 , Δ ) = I max 3 T 1 2 T 2 4 ( 4 T 2 2 + Δ 2 ) ( 1 T 1 2 + Δ 2 ) ( 1 T 2 2 + Δ 1 2 ) [ 1 T 2 2 + ( Δ 1 Δ ) 2 ] [ 1 T 2 2 + ( Δ 1 + Δ ) 2 ] ,
I ( Δ 1 , Δ ) = I ( Δ 1 , Δ ) .
I ( Δ 1 , Δ ) = I ( Δ 1 , Δ ) .
I ( Δ 1 , Δ ) = I ( Δ 1 , Δ ) .
I ( Δ 1 , Δ ) = I ( | Δ 1 | , | Δ | ) .
I ( δ 1 δ 2 ) I max = { ( T 2 2 T 1 ) 2 [ 4 + ( δ 1 δ 2 ) 2 ] [ ( T 2 T 1 ) 2 + ( δ 1 δ 2 ) 2 ] } { 1 / ( 1 + δ 2 2 ) ( 1 + δ 1 2 ) [ 1 + ( 2 δ 1 + δ 2 ) 2 ] } .
x = 1 2 ( Δ 1 Δ 2 ) and y = 1 2 ( Δ 1 + Δ 2 ) .
I ( x , y ) = I max 4 T 1 2 T 2 4 ( 2 T 2 ) 2 + 2 x 2 [ ( 1 T 1 ) 2 + 2 x 2 ] [ ( 1 T 2 ) 2 + 1 2 ( x + y ) 2 ] [ ( 1 T 2 ) 2 + 1 2 ( y x ) 2 ] [ ( 1 T 2 ) 2 + 1 2 ( 3 x + y ) 2 ] .
1 [ ( 2 T 2 ) 2 + ( x 0 + y ) 2 ] [ ( 2 T 2 ) 2 + ( y x 0 ) 2 ] [ ( 2 T 2 ) 2 + ( 3 x 0 + y ) 2 ] .
( 2 T 2 ) 2 + x 2 [ ( 1 2 T 1 ) 2 + x 2 ] [ ( 2 T 2 ) 2 + ( x + y 0 ) 2 ] [ ( 2 T 2 ) 2 + ( y 0 x ) 2 ] [ ( 2 3 T 2 ) 2 + ( x + y 0 3 ) 2 ] .

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