Abstract

A numerical study of coherent laser pulse evolution in CO2 in the framework of the Maxwell–Schrödinger equations up to the intensity 1013 W/cm2 is presented. The model includes various nonresonant vibrational–rotational transitions usually neglected in low-intensity simulations. Level degeneracy is taken into account, and light-polarization effects are discussed. The limiting intensities up to which various models work are established.

© 1990 Optical Society of America

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References

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  1. J. Allen, J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).
  2. G. L. Lamb, Rev. Mod. Phys. 43, 99 (1971).
    [CrossRef]
  3. G. L. Lamb, Elements of Soliton Theory (Wiley, New York, 1980).
  4. M. E. Crenshaw, C. D. Cantrell, Phys. Rev. A 37, 3338 (1988);Opt. Lett. 13, 386 (1988).
    [CrossRef] [PubMed]
  5. P. B. Corkum, IEEE J. Quantum Electron. QE-21, 216 (1985);in Laser Acceleration of Particles, C. Joshi, T. Katsouleas, eds., AIP Conf. Proc.130, 493 (1985).
    [CrossRef]
  6. J. I. Steinfeld, Molecules and Radiation (Harper & Row, New York, 1974), p. 271.
  7. A. O. Markano, V. T. Platonenko, Sov. J. Quantum Electron. 10, 433, (1980).
    [CrossRef]
  8. V. T. Platonenko, V. D. Taranukhin, Sov. J. Quantum Electron. 13, 1459, (1983).
    [CrossRef]
  9. We define the Rabi frequency as a product of the electric field intensity and the transition dipole moment p12, averaged over the magnetic quantum number M, corresponding to the P(20) transition equal to 0.015 D.10–12 See the discussion at the end of Section 5 and formula (18) defining p12.
  10. A. M. Robinson, Can. J. Phys. 50, 2471 (1972).
    [CrossRef]
  11. I. B. Burak, L. A. Gamss, J. Chem Phys. 65, 5385 (1977).
    [CrossRef]
  12. A. Yariv, Quantum Electronics (Wiley, New York, 1975), p. 550.
  13. G. H. Herzberg, Molecular Spectra and Molecular Structure (Van Nostrand Reinhold, New York, 1945), Vol. 2, pp.14, 21.
  14. S. Chelkowski, A. D. Bandrauk, in Atomic and Molecular Processes with Intense Laser Pulses, A. D. Bandrauk, ed.,Vol. 171 of NATO ASI Series B (Plenum, New York, 1988), pp.57–66.
    [CrossRef]
  15. S. Chelkowski, A. D. Bandrauk, J. Chem. Phys. 89, 3618 (1988).
    [CrossRef]
  16. D. K. Campbell, M. Peyrard, P. Sodano, Physica 19D, 165 (1986).
  17. R. K. Brimacombe, J. Reid, IEEE J. Quantum Electron. QE-19, 1674 (1983).
    [CrossRef]
  18. E. V. Condon, G. H. Shortley, The Theory of Atomic Spectra (Cambridge U. Press, London, 1959), p. 53.
  19. W. F. Ames, Numerical Methods for Partial Differential Equations (Nelson, New York, 1969), p. 180.
  20. F. A. Hopf, C. K. Rhodes, A. Szoke, Phys. Rev. B 1, 2833 (1970).
    [CrossRef]

1988 (2)

M. E. Crenshaw, C. D. Cantrell, Phys. Rev. A 37, 3338 (1988);Opt. Lett. 13, 386 (1988).
[CrossRef] [PubMed]

S. Chelkowski, A. D. Bandrauk, J. Chem. Phys. 89, 3618 (1988).
[CrossRef]

1986 (1)

D. K. Campbell, M. Peyrard, P. Sodano, Physica 19D, 165 (1986).

1985 (1)

P. B. Corkum, IEEE J. Quantum Electron. QE-21, 216 (1985);in Laser Acceleration of Particles, C. Joshi, T. Katsouleas, eds., AIP Conf. Proc.130, 493 (1985).
[CrossRef]

1983 (2)

V. T. Platonenko, V. D. Taranukhin, Sov. J. Quantum Electron. 13, 1459, (1983).
[CrossRef]

R. K. Brimacombe, J. Reid, IEEE J. Quantum Electron. QE-19, 1674 (1983).
[CrossRef]

1980 (1)

A. O. Markano, V. T. Platonenko, Sov. J. Quantum Electron. 10, 433, (1980).
[CrossRef]

1977 (1)

I. B. Burak, L. A. Gamss, J. Chem Phys. 65, 5385 (1977).
[CrossRef]

1972 (1)

A. M. Robinson, Can. J. Phys. 50, 2471 (1972).
[CrossRef]

1971 (1)

G. L. Lamb, Rev. Mod. Phys. 43, 99 (1971).
[CrossRef]

1970 (1)

F. A. Hopf, C. K. Rhodes, A. Szoke, Phys. Rev. B 1, 2833 (1970).
[CrossRef]

Allen, J.

J. Allen, J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).

Ames, W. F.

W. F. Ames, Numerical Methods for Partial Differential Equations (Nelson, New York, 1969), p. 180.

Bandrauk, A. D.

S. Chelkowski, A. D. Bandrauk, J. Chem. Phys. 89, 3618 (1988).
[CrossRef]

S. Chelkowski, A. D. Bandrauk, in Atomic and Molecular Processes with Intense Laser Pulses, A. D. Bandrauk, ed.,Vol. 171 of NATO ASI Series B (Plenum, New York, 1988), pp.57–66.
[CrossRef]

Brimacombe, R. K.

R. K. Brimacombe, J. Reid, IEEE J. Quantum Electron. QE-19, 1674 (1983).
[CrossRef]

Burak, I. B.

I. B. Burak, L. A. Gamss, J. Chem Phys. 65, 5385 (1977).
[CrossRef]

Campbell, D. K.

D. K. Campbell, M. Peyrard, P. Sodano, Physica 19D, 165 (1986).

Cantrell, C. D.

M. E. Crenshaw, C. D. Cantrell, Phys. Rev. A 37, 3338 (1988);Opt. Lett. 13, 386 (1988).
[CrossRef] [PubMed]

Chelkowski, S.

S. Chelkowski, A. D. Bandrauk, J. Chem. Phys. 89, 3618 (1988).
[CrossRef]

S. Chelkowski, A. D. Bandrauk, in Atomic and Molecular Processes with Intense Laser Pulses, A. D. Bandrauk, ed.,Vol. 171 of NATO ASI Series B (Plenum, New York, 1988), pp.57–66.
[CrossRef]

Condon, E. V.

E. V. Condon, G. H. Shortley, The Theory of Atomic Spectra (Cambridge U. Press, London, 1959), p. 53.

Corkum, P. B.

P. B. Corkum, IEEE J. Quantum Electron. QE-21, 216 (1985);in Laser Acceleration of Particles, C. Joshi, T. Katsouleas, eds., AIP Conf. Proc.130, 493 (1985).
[CrossRef]

Crenshaw, M. E.

M. E. Crenshaw, C. D. Cantrell, Phys. Rev. A 37, 3338 (1988);Opt. Lett. 13, 386 (1988).
[CrossRef] [PubMed]

Eberly, J. H.

J. Allen, J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).

Gamss, L. A.

I. B. Burak, L. A. Gamss, J. Chem Phys. 65, 5385 (1977).
[CrossRef]

Herzberg, G. H.

G. H. Herzberg, Molecular Spectra and Molecular Structure (Van Nostrand Reinhold, New York, 1945), Vol. 2, pp.14, 21.

Hopf, F. A.

F. A. Hopf, C. K. Rhodes, A. Szoke, Phys. Rev. B 1, 2833 (1970).
[CrossRef]

Lamb, G. L.

G. L. Lamb, Rev. Mod. Phys. 43, 99 (1971).
[CrossRef]

G. L. Lamb, Elements of Soliton Theory (Wiley, New York, 1980).

Markano, A. O.

A. O. Markano, V. T. Platonenko, Sov. J. Quantum Electron. 10, 433, (1980).
[CrossRef]

Peyrard, M.

D. K. Campbell, M. Peyrard, P. Sodano, Physica 19D, 165 (1986).

Platonenko, V. T.

V. T. Platonenko, V. D. Taranukhin, Sov. J. Quantum Electron. 13, 1459, (1983).
[CrossRef]

A. O. Markano, V. T. Platonenko, Sov. J. Quantum Electron. 10, 433, (1980).
[CrossRef]

Reid, J.

R. K. Brimacombe, J. Reid, IEEE J. Quantum Electron. QE-19, 1674 (1983).
[CrossRef]

Rhodes, C. K.

F. A. Hopf, C. K. Rhodes, A. Szoke, Phys. Rev. B 1, 2833 (1970).
[CrossRef]

Robinson, A. M.

A. M. Robinson, Can. J. Phys. 50, 2471 (1972).
[CrossRef]

Shortley, G. H.

E. V. Condon, G. H. Shortley, The Theory of Atomic Spectra (Cambridge U. Press, London, 1959), p. 53.

Sodano, P.

D. K. Campbell, M. Peyrard, P. Sodano, Physica 19D, 165 (1986).

Steinfeld, J. I.

J. I. Steinfeld, Molecules and Radiation (Harper & Row, New York, 1974), p. 271.

Szoke, A.

F. A. Hopf, C. K. Rhodes, A. Szoke, Phys. Rev. B 1, 2833 (1970).
[CrossRef]

Taranukhin, V. D.

V. T. Platonenko, V. D. Taranukhin, Sov. J. Quantum Electron. 13, 1459, (1983).
[CrossRef]

Yariv, A.

A. Yariv, Quantum Electronics (Wiley, New York, 1975), p. 550.

Can. J. Phys. (1)

A. M. Robinson, Can. J. Phys. 50, 2471 (1972).
[CrossRef]

IEEE J. Quantum Electron. (2)

R. K. Brimacombe, J. Reid, IEEE J. Quantum Electron. QE-19, 1674 (1983).
[CrossRef]

P. B. Corkum, IEEE J. Quantum Electron. QE-21, 216 (1985);in Laser Acceleration of Particles, C. Joshi, T. Katsouleas, eds., AIP Conf. Proc.130, 493 (1985).
[CrossRef]

J. Chem Phys. (1)

I. B. Burak, L. A. Gamss, J. Chem Phys. 65, 5385 (1977).
[CrossRef]

J. Chem. Phys. (1)

S. Chelkowski, A. D. Bandrauk, J. Chem. Phys. 89, 3618 (1988).
[CrossRef]

Phys. Rev. A (1)

M. E. Crenshaw, C. D. Cantrell, Phys. Rev. A 37, 3338 (1988);Opt. Lett. 13, 386 (1988).
[CrossRef] [PubMed]

Phys. Rev. B (1)

F. A. Hopf, C. K. Rhodes, A. Szoke, Phys. Rev. B 1, 2833 (1970).
[CrossRef]

Physica (1)

D. K. Campbell, M. Peyrard, P. Sodano, Physica 19D, 165 (1986).

Rev. Mod. Phys. (1)

G. L. Lamb, Rev. Mod. Phys. 43, 99 (1971).
[CrossRef]

Sov. J. Quantum Electron. (2)

A. O. Markano, V. T. Platonenko, Sov. J. Quantum Electron. 10, 433, (1980).
[CrossRef]

V. T. Platonenko, V. D. Taranukhin, Sov. J. Quantum Electron. 13, 1459, (1983).
[CrossRef]

Other (9)

We define the Rabi frequency as a product of the electric field intensity and the transition dipole moment p12, averaged over the magnetic quantum number M, corresponding to the P(20) transition equal to 0.015 D.10–12 See the discussion at the end of Section 5 and formula (18) defining p12.

J. I. Steinfeld, Molecules and Radiation (Harper & Row, New York, 1974), p. 271.

G. L. Lamb, Elements of Soliton Theory (Wiley, New York, 1980).

J. Allen, J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).

E. V. Condon, G. H. Shortley, The Theory of Atomic Spectra (Cambridge U. Press, London, 1959), p. 53.

W. F. Ames, Numerical Methods for Partial Differential Equations (Nelson, New York, 1969), p. 180.

A. Yariv, Quantum Electronics (Wiley, New York, 1975), p. 550.

G. H. Herzberg, Molecular Spectra and Molecular Structure (Van Nostrand Reinhold, New York, 1945), Vol. 2, pp.14, 21.

S. Chelkowski, A. D. Bandrauk, in Atomic and Molecular Processes with Intense Laser Pulses, A. D. Bandrauk, ed.,Vol. 171 of NATO ASI Series B (Plenum, New York, 1988), pp.57–66.
[CrossRef]

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Figures (15)

Fig. 1
Fig. 1

Schematic diagram of the CO2 energy levels used in the model calculations. The circled numbers 1–5 indicate levels used in the calculations of transition amplitudes presented in Fig. 5 below. The values of the transition frequencies are given in inverse centimeters; ω1/2π = 2320 cm−1.

Fig. 2
Fig. 2

Schematic diagram of energy levels used in the model calculations presented in Fig. 3.

Fig. 3
Fig. 3

Pulse shapes, expressed as electric field × transition dipole moment, ∊p12 (cm−1) for the level scheme of Fig. 2. Transition dipole moments: p12 = p23 = 0.015 D, n0 = 5.85 × 1017 molecules/ cm3, input peak intensity I = 2.35 × 1013 W/cm2. The detuning Δ/ (2π) is (a) 98 and (b) 32 cm−1. The incoming light is resonant with the P(20) lasing transition from the 001 to the 100 vibrational value.

Fig. 4
Fig. 4

Pulse shapes, expressed as electric field × transition dipole moment, ∊p12 (cm−1), for the two-level model with the same values of p12, n0, and I as in Fig. 3.

Fig. 5
Fig. 5

Transition probabilities |c3|2, |c5|2, and |c1|2 as functions of time for π/2 pulses at intensities (a) I = 2.35 × 1013 W/cm2 and (b) I = 9.3 × 1013 W/cm2. Initially the system was in level 2. Values of transition dipole moments used: P12 = 0.015 D, P24 = 10 × p12, P 23 = 2 p 24, p 35 = 3 p 23 (the prefactors for the last p’s correspond to harmonic oscillator transition moments).

Fig. 6
Fig. 6

Schematic diagram of energy sublevels of two vibrational levels of CO2, 001 and 100, included in the calculations of Section 5. Before the pulse arrival the molecule was assumed to be in the state characterized by J0 distributed according to Boltzmann’s law.19

Fig. 7
Fig. 7

Pulse shapes expressed as electric field × transition dipole moment for all transitions ΔJ = ±1, including level degeneracy, at z = 0, z = 2z1, z = 4z1 and z = 6z1, where z1 = [5.58/n0(19] × 1018 cm. The input (at z = 0) I0 and output (at z = 6z1) I pulse intensities are 1010 and 6.8 × 1011 W/cm2, respectively. p12 = 0.015 D.

Fig. 8
Fig. 8

Pulse shapes at z = 6z1 and z = 20z1, the input pulse being the last pulse from Fig. 7, obtained in the following different models (linear polarization): (a) a model including level degeneracy and P(J) and R(J) transitions, based on Eqs. (11), (16), and (17); (b) the model with nondegenerate levels, only P(J) transitions included, a model similar to the model of Ref. 6; (c) a model with nondegenerate levels, P(J) and R(J) transitions included.

Fig. 9
Fig. 9

Pulse shapes in the low-intensity regime at z = 0, (a) z = 4z1 and (b) z = 8z1. The input pulse intensity at z = 0 is 4 × 103 W/cm2.

Fig. 10
Fig. 10

Pulse shapes in the nonlinear regime for linear light polarization at (a) z = 0, z = 2z1, and z = 4z1 and at (b) z = 20z1. The input (at z = 0) and output (at z = 20z1) pulse intensities are 1010 and 7.4 × 1012 W/cm2, respectively. The input pulse area defined by Eq. (8) is 0.066 × π.

Fig. 11
Fig. 11

Pulse shapes in the nonlinear regime for circular light polarization at the same distances and for the same input intensity as in Fig. 10.

Fig. 12
Fig. 12

Pulse shapes in the nonlinear regime for (a) linear light polarization and (b) circular polarization at z = 0 and z = 8z1. The input (at z = 0) and output (at z = 8z1) pulse intensities are 2.2 × 1012 and 1.3 × 1013 W/cm2, respectively. The input pulse area is 2.8 × π.

Fig. 13
Fig. 13

Pulse shapes in the nonlinear regime for (a) linear light polarization and (b) circular polarization at z = 0 and z = 4z1. The input (at z = 0) pulse intensity is 1010. The input pulse area is 1.26 × π.

Fig. 14
Fig. 14

Pulse shapes for the medium consisting of two degenerate-level systems [only P(20) transitions included] at z = 0, z = 20z1, and z = 40z1 for (a) linearly and (b) circularly polarized light.

Fig. 15
Fig. 15

Polarizibility function F(σ) for light propagation in degenerate two-level systems [Eqs. (23) and (24)]. σ is the pulse area defined by Eq. (22) Triangles, linear light polarization; crosses, circular polarization.

Equations (34)

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P = Ω 2 ( Δ 2 + Ω 2 ) 1 sin 2 [ ( Δ 2 + Ω 2 ) 1 / 2 t / 2 ) ] ,
c 2 ( 2 E / z 2 ) ( 2 E / t 2 ) = 4 π ( 2 P / t 2 ) ,
( c / t ) + c ( c / z ) = 2 ω π i Q .
E x ( t , z ) = ( 1 / 2 ) c ( t , z ) exp ( ikz ω t ) + c . c . = cos [ k z ω t + φ ( t , z ) ] ,
P x ( t , z ) = ( 1 / 2 ) Q ( t , z ) exp ( ikz ω t ) + c . c . ,
E y ( t , z ) = ± i ( 1 / 2 ) c ( t , z ) exp ( ikz ω t ) + c . c . ,
P y ( t , z ) = ± i ( 1 / 2 ) Q ( t , z ) exp ( ikz ω t ) + c . c . ,
P = n 0 ψ ( t , z ) | μ | ψ ( t , z ) .
ψ ( t , z ) = j c j exp ( i E j t / ) ψ j .
i ċ j = E ( t , z ) m p j , m exp ( i ω j , m t ) c m ,
S = ( p 12 / ) ( t , z ) d t
n ( J 0 ) = exp { B [ J 0 ( J 0 + 1 ) 19 × 20 ] / k T } × n 0 ( 19 ) ( 2 J 0 + 1 ) / ( 2 × 19 + 1 ) ,
ψ ( t , z , J 0 , M 0 ) = J odd q J ( t , J 0 , M 0 ) × exp [ i E ( J ) t / + ikz ] J , M 0 + J even q J ( t , J 0 , M 0 ) × exp [ i E ( J ) t / ] J , M 0 .
2 i q ˙ J = c { ρ + ( J , M 0 ) exp [ i Δ + ( J ) t ] q J + 1 + p ( J , M 0 ) exp [ i Δ ( J ) t ] q J 1 }
2 i q ˙ J = c * { p ( J + 1 , M 0 ) exp [ i Δ ( J + 1 ) t ] q J + 1 + p + ( J 1 , M 0 ) exp [ i Δ + ( J 1 ) t ] q J 1 }
Δ + ( J ) = 2 B ( 19 J ) , Δ ( J ) = 2 B ( 20 + J )
p + ( J , M 0 ) = J + 1 , M 0 | μ x | J , M 0 = p + ( J , 0 ) [ 1 M 0 2 / ( J + 1 ) 2 ] 1 / 2 ,
p ( J , M 0 ) = J 1 , M 0 | μ x | J , M 0 = p + ( J 1 , M ) ,
p + ( J , 0 ) = R 12 ( J + 1 ) [ ( 2 J + 1 ) ( 2 J + 3 ) ] 1 / 2 ,
p + ( J , M 0 ) = J + 1 , M 0 1 | μ x i μ y | J , M 0 = p + ( J , 0 ) { [ 1 M 0 / ( J + 1 ) ] × [ 1 ( M 0 1 ) / ( J + 1 ) ] 1 / 2 ,
p ( J , M 0 ) = J 1 , M 0 1 | μ x i μ y | J , M 0 = p + ( J 1 , M 0 + 1 ) ,
p + right ( J , M ) = p left ( J + 1 , M + 1 ) ,
p right ( J , M ) = p left ( J + 1 , M + 1 ) .
P = J 0 = 1 , 3 , 35 [ n ( J 0 ) / ( 2 J 0 + 1 ) ] × M 0 = J 0 J 0 ψ ( t , z , J 0 , M 0 ) | μ | ψ ( t , z , J 0 , M 0 ) .
Q = A J 0 = 1 , 3 , 35 [ n ( J 0 ) / ( 2 J 0 + 1 ) ] × M 0 = J 0 J odd q J { p + ( J , M 0 ) q J + 1 * exp [ i Δ + ( J ) t ] + p ( J , M 0 ) q J 1 * exp [ i Δ ( J ) t ] } ,
c c / z = 2 ω π i Q ( τ , z ) ,
p 12 2 = M = J 0 J 0 p + ( J 0 , M ) 2 / ( 2 J 0 + 1 ) = R 12 2 ( J 0 + 1 ) / [ 3 ( 2 J 0 + 1 ) ] .
Γ = 4 π ω T 2 p 12 2 n ( 19 ) / ( c ) ,
z 1 = z 0 n 0 ( 19 ) / n ( 19 ) ,
z 1 = 5.58 × 10 18 cm 4 / n ( 19 ) = 0.095 psec 1 T 2 / Γ .
c c / z = 2 π ω p 12 n ( 19 ) sin σ ,
σ = ( p 12 / ) τ ( τ , z ) d τ .
c c / z = 2 π ω p 12 n ( 19 ) F ( σ ) ,
F ( σ ) = M p + ( 19 , M ) / p 12 sin [ p + ( 19 , M ) σ / p 12 ] .

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