Abstract

Following the recent experiments of Trebino and Rahn [Opt. Lett. 12, 912 (1987)], I show on general grounds the existence of the fractional Raman resonances in a variety of nonlinear mixing processes in atomic vapors, molecules, and solids. I present theoretical models, both classical and quantum, for such resonances.

© 1988 Optical Society of America

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References

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  1. C. G. B. Garett, IEEE J. Quantum Electron. QE-4, 70 (1968).
    [Crossref]
  2. A. S. Barker, R. Loudon, Rev. Mod. Phys. 44, 18 (1972).
    [Crossref]
  3. G. S. Agarwal, S. S. Jha, Phys. Rev. B 26, 4013 (1982).
    [Crossref]
  4. G. S. Agarwal, S. S. Jha, J. Phys. B 12, 2655 (1979).
    [Crossref]
  5. S. Reynaud, C. Cohen-Tannoudji, J. Phys. B 10, 345 (1977).S. Reynaud, C. Cohen-Tannoudji, J. Phys. B 10, 2311 (1977).
    [Crossref]
  6. R. Trebino, L. A. Rahn [Opt. Lett. 12, 912 (1987)] have seen subharmonic resonances in six-wave mixing geometry in sodium-seeded flame. I am grateful to the authors for showing me their preliminary experimental data before publication.
    [Crossref] [PubMed]
  7. G. S. Agarwal, N. Nayak, Phys. Rev. A 33, 391 (1986).
    [Crossref] [PubMed]
  8. L. Hillman, J. Krasinski, R. W. Boyd, C. R. Stroud, Phys. Rev. Lett. 52, 1605 (1984); S. C. Mehendale, R. G. Harrison, Phys. Rev. A 34, 1613 (1986).
    [Crossref] [PubMed]
  9. R. Trebino [Phys. Rev. A (to be published)] has developed diagrammatic methods for the calculation of the signals characterized by higher-order nonlinearities.
  10. M. Sanjay Kumar, G. S. Agarwal, Phys. Rev. A 33, 1817 (1986).
    [Crossref]
  11. R. Saxena, G. S. Agarwal, J. Phys. B 13, 453 (1980).
    [Crossref]
  12. The explicit form of the renormalized R(3) can be obtained from Ref. 11 by a suitable change of the dipole matrix elements, etc.

1987 (1)

1986 (2)

G. S. Agarwal, N. Nayak, Phys. Rev. A 33, 391 (1986).
[Crossref] [PubMed]

M. Sanjay Kumar, G. S. Agarwal, Phys. Rev. A 33, 1817 (1986).
[Crossref]

1984 (1)

L. Hillman, J. Krasinski, R. W. Boyd, C. R. Stroud, Phys. Rev. Lett. 52, 1605 (1984); S. C. Mehendale, R. G. Harrison, Phys. Rev. A 34, 1613 (1986).
[Crossref] [PubMed]

1982 (1)

G. S. Agarwal, S. S. Jha, Phys. Rev. B 26, 4013 (1982).
[Crossref]

1980 (1)

R. Saxena, G. S. Agarwal, J. Phys. B 13, 453 (1980).
[Crossref]

1979 (1)

G. S. Agarwal, S. S. Jha, J. Phys. B 12, 2655 (1979).
[Crossref]

1977 (1)

S. Reynaud, C. Cohen-Tannoudji, J. Phys. B 10, 345 (1977).S. Reynaud, C. Cohen-Tannoudji, J. Phys. B 10, 2311 (1977).
[Crossref]

1972 (1)

A. S. Barker, R. Loudon, Rev. Mod. Phys. 44, 18 (1972).
[Crossref]

1968 (1)

C. G. B. Garett, IEEE J. Quantum Electron. QE-4, 70 (1968).
[Crossref]

Agarwal, G. S.

G. S. Agarwal, N. Nayak, Phys. Rev. A 33, 391 (1986).
[Crossref] [PubMed]

M. Sanjay Kumar, G. S. Agarwal, Phys. Rev. A 33, 1817 (1986).
[Crossref]

G. S. Agarwal, S. S. Jha, Phys. Rev. B 26, 4013 (1982).
[Crossref]

R. Saxena, G. S. Agarwal, J. Phys. B 13, 453 (1980).
[Crossref]

G. S. Agarwal, S. S. Jha, J. Phys. B 12, 2655 (1979).
[Crossref]

Barker, A. S.

A. S. Barker, R. Loudon, Rev. Mod. Phys. 44, 18 (1972).
[Crossref]

Boyd, R. W.

L. Hillman, J. Krasinski, R. W. Boyd, C. R. Stroud, Phys. Rev. Lett. 52, 1605 (1984); S. C. Mehendale, R. G. Harrison, Phys. Rev. A 34, 1613 (1986).
[Crossref] [PubMed]

Cohen-Tannoudji, C.

S. Reynaud, C. Cohen-Tannoudji, J. Phys. B 10, 345 (1977).S. Reynaud, C. Cohen-Tannoudji, J. Phys. B 10, 2311 (1977).
[Crossref]

Garett, C. G. B.

C. G. B. Garett, IEEE J. Quantum Electron. QE-4, 70 (1968).
[Crossref]

Hillman, L.

L. Hillman, J. Krasinski, R. W. Boyd, C. R. Stroud, Phys. Rev. Lett. 52, 1605 (1984); S. C. Mehendale, R. G. Harrison, Phys. Rev. A 34, 1613 (1986).
[Crossref] [PubMed]

Jha, S. S.

G. S. Agarwal, S. S. Jha, Phys. Rev. B 26, 4013 (1982).
[Crossref]

G. S. Agarwal, S. S. Jha, J. Phys. B 12, 2655 (1979).
[Crossref]

Krasinski, J.

L. Hillman, J. Krasinski, R. W. Boyd, C. R. Stroud, Phys. Rev. Lett. 52, 1605 (1984); S. C. Mehendale, R. G. Harrison, Phys. Rev. A 34, 1613 (1986).
[Crossref] [PubMed]

Loudon, R.

A. S. Barker, R. Loudon, Rev. Mod. Phys. 44, 18 (1972).
[Crossref]

Nayak, N.

G. S. Agarwal, N. Nayak, Phys. Rev. A 33, 391 (1986).
[Crossref] [PubMed]

Rahn, L. A.

Reynaud, S.

S. Reynaud, C. Cohen-Tannoudji, J. Phys. B 10, 345 (1977).S. Reynaud, C. Cohen-Tannoudji, J. Phys. B 10, 2311 (1977).
[Crossref]

Sanjay Kumar, M.

M. Sanjay Kumar, G. S. Agarwal, Phys. Rev. A 33, 1817 (1986).
[Crossref]

Saxena, R.

R. Saxena, G. S. Agarwal, J. Phys. B 13, 453 (1980).
[Crossref]

Stroud, C. R.

L. Hillman, J. Krasinski, R. W. Boyd, C. R. Stroud, Phys. Rev. Lett. 52, 1605 (1984); S. C. Mehendale, R. G. Harrison, Phys. Rev. A 34, 1613 (1986).
[Crossref] [PubMed]

Trebino, R.

IEEE J. Quantum Electron. (1)

C. G. B. Garett, IEEE J. Quantum Electron. QE-4, 70 (1968).
[Crossref]

J. Phys. B (3)

G. S. Agarwal, S. S. Jha, J. Phys. B 12, 2655 (1979).
[Crossref]

S. Reynaud, C. Cohen-Tannoudji, J. Phys. B 10, 345 (1977).S. Reynaud, C. Cohen-Tannoudji, J. Phys. B 10, 2311 (1977).
[Crossref]

R. Saxena, G. S. Agarwal, J. Phys. B 13, 453 (1980).
[Crossref]

Opt. Lett. (1)

Phys. Rev. A (2)

G. S. Agarwal, N. Nayak, Phys. Rev. A 33, 391 (1986).
[Crossref] [PubMed]

M. Sanjay Kumar, G. S. Agarwal, Phys. Rev. A 33, 1817 (1986).
[Crossref]

Phys. Rev. B (1)

G. S. Agarwal, S. S. Jha, Phys. Rev. B 26, 4013 (1982).
[Crossref]

Phys. Rev. Lett. (1)

L. Hillman, J. Krasinski, R. W. Boyd, C. R. Stroud, Phys. Rev. Lett. 52, 1605 (1984); S. C. Mehendale, R. G. Harrison, Phys. Rev. A 34, 1613 (1986).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

A. S. Barker, R. Loudon, Rev. Mod. Phys. 44, 18 (1972).
[Crossref]

Other (2)

R. Trebino [Phys. Rev. A (to be published)] has developed diagrammatic methods for the calculation of the signals characterized by higher-order nonlinearities.

The explicit form of the renormalized R(3) can be obtained from Ref. 11 by a suitable change of the dipole matrix elements, etc.

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Equations (16)

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ω l ω s = ω R / n ,
V = C i j k x i x j Q k ,
x ( ω ) = 1 e α e ( ω ) · ( ω ) , Q ( ω ) 1 z α ion ( ω ) · ( ω ) .
p ( 3 ) ( 2 ω l ω s ) = 2 e 2 α e ( 2 ω l ω s ) · C · α e ( ω l ) * ( ω t ) Q ( ω l ω s ) ,
Q ( ω l ω s ) = 1 z 2 e 2 α ion ( ω l ω s ) · C · α e ( ω l ) × α e * ( ω s ) · * ( ω l ) ( ω s )
p ( 2 ω l ω s ) = χ ( 5 ) ( ω l , ω l , ω s , ω s ω s ) 2 ( ω l ) × * 2 ( ω s ) ( ω s ) ,
e x ( 2 ω l ω s ) = 2 e α e ( 2 ω l ω s ) · C · x ( ω s ) Q ( 2 ω l 2 ω s ) ,
Q ( 2 ω l 2 ω s ) = 1 z 2 α ion ( 2 ω l 2 ω s ) C · x ( 2 ω l ω s ) x * ( ω s ) ,
p ( 2 ω l ω s ) = 2 e 2 z 2 α e ( 2 ω l ω s ) · C · x ( ω s ) [ α ion ( 2 ω l 2 ω s ) C · x * ( ω s ) p ( 3 ) ( 2 ω l ω s ) ] ,
ω l ω s = ω R / 2
e x ( 3 ω l 2 ω s ) = 2 e α e ( 3 ω l 2 ω s ) C · x ( ω l ) Q ( 2 ω l 2 ω s ) ,
e x ( 3 ω l ω s ) = 2 e α e ( 3 ω l 2 ω s ) C · x ( ω s ) Q ( 3 ω l 3 ω s ) ,
Q ( 3 ω l 3 ω s ) = 1 z α ion ( 3 ω l 3 ω s ) C · x * ( ω s ) x ( 3 ω l 2 ω s )
H = [ l exp ( i ω l t ) + s exp ( i ω s t ) ] ( d 13 | 1 3 | + d 12 | 1 2 | ) + H . c . ,
p ( 2 ω l ω s ) = p ( 3 ) ( 2 ω l ω s ) × { 1 + i | d 12 · s | 2 ћ 2 [ Γ 23 + i ( ω 23 2 ω l + 2 ω s ) ] ( ω 13 ω l ) + } .
p μ ( 2 ω l ω s ) = α β R μ α β ( 2 ) ( δ , δ ) α ( ω l ) β ( ω l ) , R μ α β ( 2 ) ( δ , δ ) = N 2 i j k ρ ι i ( 0 ) ( d i k d k j d j i { [ μ α β ( Λ j i δ ) ( Λ k i 2 δ ) + β α μ ( Λ i j 2 δ ) ( Λ i k δ ) α μ β ( Λ j k 2 δ ) ( Λ j i δ ) β μ α ( Λ j k 2 δ ) ( Λ i k δ ) ] + d i k α d k i β d j j μ ( 1 Λ i k δ + 1 Λ k i δ ) × [ C k j ( 2 δ ) C i j ( 2 δ ) ] } + ( α β ) ) , δ = ω l ω s .

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