Abstract

The nonlinear signal in a degenerate four-wave mixing experiment as a result of diffraction of a spatially distributed third-order nonlinearity is studied theoretically. The approach has been applied for the case of monochromatic and ultrashort pulses. As a consequence, the degenerate four-wave mixing signal is found to be independent of light-coherence parameters and cannot be explained as a product of diffraction on a dynamic grating. In this model only the fast component of the signal (the coherent artifact) is related directly to third-order nonlinear processes. The main features (the origins, temporal shapes, dependencies of pump intensity, and modulation effects) of the coherent artifact and a slow-decay component are discussed theoretically and experimentally.

© 2000 Optical Society of America

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References

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  1. R. A. Fisher, ed., Optical Phase Conjugation (Academic, New York, 1983).
  2. P. G. R. Smith and P. Ewart, “Spectral line shape of resonant four-wave mixing induced by broad-bandwidth lasers,” Phys. Rev. A 54, 2347–2355 (1996).
    [CrossRef] [PubMed]
  3. M. Kaczmarek, D. R. Meacher, and P. Ewart, “Time dependence of degenerate four-wave mixing with broad bandwidth pulsed lasers,” J. Mod. Opt. 37, 1561–1571 (1990).
    [CrossRef]
  4. J. G. Fujimoto and T. K. Yee, “Diagramatic density matrix theory of transient four-wave mixing and the measurements of transient phenomena,” IEEE J. Quantum Electron. QE-22, 1215–1228 (1986).
    [CrossRef]
  5. L. Banyai, E. Reitsamer, D. B. Tran Thoai, and H. Hang, “Quantum kinetics of femtosecond four-wave mixing in semiconductors,” J. Opt. Soc. Am. B 13, 1278–1283 (1996).
    [CrossRef]
  6. J. Oberie, G. Jonusauskas, E. Abraham and C. Rulliere, “Third-order optical nonlinearities of excited states in diphenyl-polyene derivatives: a subpicosecond study,” Opt. Commun. 124, 616–627 (1996).
    [CrossRef]
  7. K. S. Wong, S. G. Han, and Z. V. Vardeny, “Studies of resonant and preresonant femtosecond degenerate four wave mixing in unoriented conducting polymers,” J. Appl. Phys. 70, 1896–1898 (1991).
    [CrossRef]
  8. G. M. Carter, “Excited-state dynamics and temporally resolved nonresonant nonlinear optical processes in polydiacetylenes,” J. Opt. Soc. Am. B 4, 1018–1024 (1987).
    [CrossRef]
  9. Y. M. Cheung and S. K. Gayen, “Optical nonlinearities of tea studied by Z-scan and four-wave mixing techniques,” J. Opt. Soc. Am. B 11, 636–643 (1994).
    [CrossRef]
  10. Y. Pang and P. N. Prasad, “Photoinduced processes and resonant third-order nonlinearity in poly (3-dodecylthiophene) studies by femtosecond time resolved degenerate four wave mixing,” J. Chem. Phys. 93, 2201–2204 (1990).
    [CrossRef]
  11. K. S. Wong and Z. V. Vardeny, “Measurements of χ(3)(ω, ω, −ω, ω) in conducting polymers at λ=620 nm,” Synth. Met. 49, 13–20 (1992).
    [CrossRef]
  12. L. Yang, R. Dorsinville, P. P. Ho, W. K. Zou, N. L. Yang, and R. R. Alfano, “Intensity dependence of picosecond nonlinear response time of polydiacetylene,” Appl. Phys. Lett. 53, 2008–2010 (1988).
    [CrossRef]
  13. M. K. Casstevens, M. Samoc, J. Pfleger, and P. N. Prasad, “Dynamics of third-order nonlinear optical processes in Langmuir–Blodgett and evaporated films of phthalocyanines,” J. Chem. Phys. 92, 2019–2024 (1990).
    [CrossRef]
  14. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984).
  15. S. Rozouvan and W. Schrepp, “Degenerate four-wave mixing measurements on thin polymer films,” Appl. Spectrosc. (to be published).
  16. S. Rozouvan, “Commutative spatial and time symmetry of degenerate four-wave mixing measurements,” J. Opt. Soc. Am. B 16, 768–773 (1999).
    [CrossRef]
  17. E. Hecht, Optik (Addison-Wesley, London, 1989).
  18. R. J. Bell, Introductory Fourier Transform Spectroscopy (Academic, New York, 1972).

1999 (1)

1996 (3)

L. Banyai, E. Reitsamer, D. B. Tran Thoai, and H. Hang, “Quantum kinetics of femtosecond four-wave mixing in semiconductors,” J. Opt. Soc. Am. B 13, 1278–1283 (1996).
[CrossRef]

P. G. R. Smith and P. Ewart, “Spectral line shape of resonant four-wave mixing induced by broad-bandwidth lasers,” Phys. Rev. A 54, 2347–2355 (1996).
[CrossRef] [PubMed]

J. Oberie, G. Jonusauskas, E. Abraham and C. Rulliere, “Third-order optical nonlinearities of excited states in diphenyl-polyene derivatives: a subpicosecond study,” Opt. Commun. 124, 616–627 (1996).
[CrossRef]

1994 (1)

1992 (1)

K. S. Wong and Z. V. Vardeny, “Measurements of χ(3)(ω, ω, −ω, ω) in conducting polymers at λ=620 nm,” Synth. Met. 49, 13–20 (1992).
[CrossRef]

1991 (1)

K. S. Wong, S. G. Han, and Z. V. Vardeny, “Studies of resonant and preresonant femtosecond degenerate four wave mixing in unoriented conducting polymers,” J. Appl. Phys. 70, 1896–1898 (1991).
[CrossRef]

1990 (3)

Y. Pang and P. N. Prasad, “Photoinduced processes and resonant third-order nonlinearity in poly (3-dodecylthiophene) studies by femtosecond time resolved degenerate four wave mixing,” J. Chem. Phys. 93, 2201–2204 (1990).
[CrossRef]

M. Kaczmarek, D. R. Meacher, and P. Ewart, “Time dependence of degenerate four-wave mixing with broad bandwidth pulsed lasers,” J. Mod. Opt. 37, 1561–1571 (1990).
[CrossRef]

M. K. Casstevens, M. Samoc, J. Pfleger, and P. N. Prasad, “Dynamics of third-order nonlinear optical processes in Langmuir–Blodgett and evaporated films of phthalocyanines,” J. Chem. Phys. 92, 2019–2024 (1990).
[CrossRef]

1988 (1)

L. Yang, R. Dorsinville, P. P. Ho, W. K. Zou, N. L. Yang, and R. R. Alfano, “Intensity dependence of picosecond nonlinear response time of polydiacetylene,” Appl. Phys. Lett. 53, 2008–2010 (1988).
[CrossRef]

1987 (1)

1986 (1)

J. G. Fujimoto and T. K. Yee, “Diagramatic density matrix theory of transient four-wave mixing and the measurements of transient phenomena,” IEEE J. Quantum Electron. QE-22, 1215–1228 (1986).
[CrossRef]

Abraham, E.

J. Oberie, G. Jonusauskas, E. Abraham and C. Rulliere, “Third-order optical nonlinearities of excited states in diphenyl-polyene derivatives: a subpicosecond study,” Opt. Commun. 124, 616–627 (1996).
[CrossRef]

Alfano, R. R.

L. Yang, R. Dorsinville, P. P. Ho, W. K. Zou, N. L. Yang, and R. R. Alfano, “Intensity dependence of picosecond nonlinear response time of polydiacetylene,” Appl. Phys. Lett. 53, 2008–2010 (1988).
[CrossRef]

Banyai, L.

Carter, G. M.

Casstevens, M. K.

M. K. Casstevens, M. Samoc, J. Pfleger, and P. N. Prasad, “Dynamics of third-order nonlinear optical processes in Langmuir–Blodgett and evaporated films of phthalocyanines,” J. Chem. Phys. 92, 2019–2024 (1990).
[CrossRef]

Cheung, Y. M.

Dorsinville, R.

L. Yang, R. Dorsinville, P. P. Ho, W. K. Zou, N. L. Yang, and R. R. Alfano, “Intensity dependence of picosecond nonlinear response time of polydiacetylene,” Appl. Phys. Lett. 53, 2008–2010 (1988).
[CrossRef]

Ewart, P.

P. G. R. Smith and P. Ewart, “Spectral line shape of resonant four-wave mixing induced by broad-bandwidth lasers,” Phys. Rev. A 54, 2347–2355 (1996).
[CrossRef] [PubMed]

M. Kaczmarek, D. R. Meacher, and P. Ewart, “Time dependence of degenerate four-wave mixing with broad bandwidth pulsed lasers,” J. Mod. Opt. 37, 1561–1571 (1990).
[CrossRef]

Fujimoto, J. G.

J. G. Fujimoto and T. K. Yee, “Diagramatic density matrix theory of transient four-wave mixing and the measurements of transient phenomena,” IEEE J. Quantum Electron. QE-22, 1215–1228 (1986).
[CrossRef]

Gayen, S. K.

Han, S. G.

K. S. Wong, S. G. Han, and Z. V. Vardeny, “Studies of resonant and preresonant femtosecond degenerate four wave mixing in unoriented conducting polymers,” J. Appl. Phys. 70, 1896–1898 (1991).
[CrossRef]

Hang, H.

Ho, P. P.

L. Yang, R. Dorsinville, P. P. Ho, W. K. Zou, N. L. Yang, and R. R. Alfano, “Intensity dependence of picosecond nonlinear response time of polydiacetylene,” Appl. Phys. Lett. 53, 2008–2010 (1988).
[CrossRef]

Jonusauskas, G.

J. Oberie, G. Jonusauskas, E. Abraham and C. Rulliere, “Third-order optical nonlinearities of excited states in diphenyl-polyene derivatives: a subpicosecond study,” Opt. Commun. 124, 616–627 (1996).
[CrossRef]

Kaczmarek, M.

M. Kaczmarek, D. R. Meacher, and P. Ewart, “Time dependence of degenerate four-wave mixing with broad bandwidth pulsed lasers,” J. Mod. Opt. 37, 1561–1571 (1990).
[CrossRef]

Meacher, D. R.

M. Kaczmarek, D. R. Meacher, and P. Ewart, “Time dependence of degenerate four-wave mixing with broad bandwidth pulsed lasers,” J. Mod. Opt. 37, 1561–1571 (1990).
[CrossRef]

Oberie, J.

J. Oberie, G. Jonusauskas, E. Abraham and C. Rulliere, “Third-order optical nonlinearities of excited states in diphenyl-polyene derivatives: a subpicosecond study,” Opt. Commun. 124, 616–627 (1996).
[CrossRef]

Pang, Y.

Y. Pang and P. N. Prasad, “Photoinduced processes and resonant third-order nonlinearity in poly (3-dodecylthiophene) studies by femtosecond time resolved degenerate four wave mixing,” J. Chem. Phys. 93, 2201–2204 (1990).
[CrossRef]

Pfleger, J.

M. K. Casstevens, M. Samoc, J. Pfleger, and P. N. Prasad, “Dynamics of third-order nonlinear optical processes in Langmuir–Blodgett and evaporated films of phthalocyanines,” J. Chem. Phys. 92, 2019–2024 (1990).
[CrossRef]

Prasad, P. N.

M. K. Casstevens, M. Samoc, J. Pfleger, and P. N. Prasad, “Dynamics of third-order nonlinear optical processes in Langmuir–Blodgett and evaporated films of phthalocyanines,” J. Chem. Phys. 92, 2019–2024 (1990).
[CrossRef]

Y. Pang and P. N. Prasad, “Photoinduced processes and resonant third-order nonlinearity in poly (3-dodecylthiophene) studies by femtosecond time resolved degenerate four wave mixing,” J. Chem. Phys. 93, 2201–2204 (1990).
[CrossRef]

Reitsamer, E.

Rozouvan, S.

Rulliere, C.

J. Oberie, G. Jonusauskas, E. Abraham and C. Rulliere, “Third-order optical nonlinearities of excited states in diphenyl-polyene derivatives: a subpicosecond study,” Opt. Commun. 124, 616–627 (1996).
[CrossRef]

Samoc, M.

M. K. Casstevens, M. Samoc, J. Pfleger, and P. N. Prasad, “Dynamics of third-order nonlinear optical processes in Langmuir–Blodgett and evaporated films of phthalocyanines,” J. Chem. Phys. 92, 2019–2024 (1990).
[CrossRef]

Smith, P. G. R.

P. G. R. Smith and P. Ewart, “Spectral line shape of resonant four-wave mixing induced by broad-bandwidth lasers,” Phys. Rev. A 54, 2347–2355 (1996).
[CrossRef] [PubMed]

Tran Thoai, D. B.

Vardeny, Z. V.

K. S. Wong and Z. V. Vardeny, “Measurements of χ(3)(ω, ω, −ω, ω) in conducting polymers at λ=620 nm,” Synth. Met. 49, 13–20 (1992).
[CrossRef]

K. S. Wong, S. G. Han, and Z. V. Vardeny, “Studies of resonant and preresonant femtosecond degenerate four wave mixing in unoriented conducting polymers,” J. Appl. Phys. 70, 1896–1898 (1991).
[CrossRef]

Wong, K. S.

K. S. Wong and Z. V. Vardeny, “Measurements of χ(3)(ω, ω, −ω, ω) in conducting polymers at λ=620 nm,” Synth. Met. 49, 13–20 (1992).
[CrossRef]

K. S. Wong, S. G. Han, and Z. V. Vardeny, “Studies of resonant and preresonant femtosecond degenerate four wave mixing in unoriented conducting polymers,” J. Appl. Phys. 70, 1896–1898 (1991).
[CrossRef]

Yang, L.

L. Yang, R. Dorsinville, P. P. Ho, W. K. Zou, N. L. Yang, and R. R. Alfano, “Intensity dependence of picosecond nonlinear response time of polydiacetylene,” Appl. Phys. Lett. 53, 2008–2010 (1988).
[CrossRef]

Yang, N. L.

L. Yang, R. Dorsinville, P. P. Ho, W. K. Zou, N. L. Yang, and R. R. Alfano, “Intensity dependence of picosecond nonlinear response time of polydiacetylene,” Appl. Phys. Lett. 53, 2008–2010 (1988).
[CrossRef]

Yee, T. K.

J. G. Fujimoto and T. K. Yee, “Diagramatic density matrix theory of transient four-wave mixing and the measurements of transient phenomena,” IEEE J. Quantum Electron. QE-22, 1215–1228 (1986).
[CrossRef]

Zou, W. K.

L. Yang, R. Dorsinville, P. P. Ho, W. K. Zou, N. L. Yang, and R. R. Alfano, “Intensity dependence of picosecond nonlinear response time of polydiacetylene,” Appl. Phys. Lett. 53, 2008–2010 (1988).
[CrossRef]

Appl. Phys. Lett. (1)

L. Yang, R. Dorsinville, P. P. Ho, W. K. Zou, N. L. Yang, and R. R. Alfano, “Intensity dependence of picosecond nonlinear response time of polydiacetylene,” Appl. Phys. Lett. 53, 2008–2010 (1988).
[CrossRef]

IEEE J. Quantum Electron. (1)

J. G. Fujimoto and T. K. Yee, “Diagramatic density matrix theory of transient four-wave mixing and the measurements of transient phenomena,” IEEE J. Quantum Electron. QE-22, 1215–1228 (1986).
[CrossRef]

J. Appl. Phys. (1)

K. S. Wong, S. G. Han, and Z. V. Vardeny, “Studies of resonant and preresonant femtosecond degenerate four wave mixing in unoriented conducting polymers,” J. Appl. Phys. 70, 1896–1898 (1991).
[CrossRef]

J. Chem. Phys. (2)

Y. Pang and P. N. Prasad, “Photoinduced processes and resonant third-order nonlinearity in poly (3-dodecylthiophene) studies by femtosecond time resolved degenerate four wave mixing,” J. Chem. Phys. 93, 2201–2204 (1990).
[CrossRef]

M. K. Casstevens, M. Samoc, J. Pfleger, and P. N. Prasad, “Dynamics of third-order nonlinear optical processes in Langmuir–Blodgett and evaporated films of phthalocyanines,” J. Chem. Phys. 92, 2019–2024 (1990).
[CrossRef]

J. Mod. Opt. (1)

M. Kaczmarek, D. R. Meacher, and P. Ewart, “Time dependence of degenerate four-wave mixing with broad bandwidth pulsed lasers,” J. Mod. Opt. 37, 1561–1571 (1990).
[CrossRef]

J. Opt. Soc. Am. B (4)

Opt. Commun. (1)

J. Oberie, G. Jonusauskas, E. Abraham and C. Rulliere, “Third-order optical nonlinearities of excited states in diphenyl-polyene derivatives: a subpicosecond study,” Opt. Commun. 124, 616–627 (1996).
[CrossRef]

Phys. Rev. A (1)

P. G. R. Smith and P. Ewart, “Spectral line shape of resonant four-wave mixing induced by broad-bandwidth lasers,” Phys. Rev. A 54, 2347–2355 (1996).
[CrossRef] [PubMed]

Synth. Met. (1)

K. S. Wong and Z. V. Vardeny, “Measurements of χ(3)(ω, ω, −ω, ω) in conducting polymers at λ=620 nm,” Synth. Met. 49, 13–20 (1992).
[CrossRef]

Other (5)

R. A. Fisher, ed., Optical Phase Conjugation (Academic, New York, 1983).

E. Hecht, Optik (Addison-Wesley, London, 1989).

R. J. Bell, Introductory Fourier Transform Spectroscopy (Academic, New York, 1972).

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984).

S. Rozouvan and W. Schrepp, “Degenerate four-wave mixing measurements on thin polymer films,” Appl. Spectrosc. (to be published).

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Figures (4)

Fig. 1
Fig. 1

Experimental setup for DFWM measurements: 1, Ar+ (24-W cw, 514 nm); 2, Ti:sapphire (1-W, 700–950-nm, 90 fs) lasers; 3, Ti:sapphire oscillator (Coherent REGA 900); 4, near-infrared optical parametrical amplifier (Coherent OPA 9800); 5, λ/4 and λ/2 plates; 6a, 6b, chopper controller and chopper; 7, lens; 8, sample; 9, aperture; 10, Si photodiode; 11, low noise preamplifier; 12, lock-in amplifier; 13, PC with IEEE 488 interface card; 14, step-motor controller; 15, delay stage.

Fig. 2
Fig. 2

Normalized spectra of OPA 9800 lasing lines.

Fig. 3
Fig. 3

DFWM signal as a function of the input power density. Shapes of the DFWM signal corresponded to low-power and high-power points (relevant points are marked by arrows).

Fig. 4
Fig. 4

DFWM signal from CS2, λ=1.25 mkm.

Equations (29)

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Pi(3)(r,t)=χijkl(3)(E1j+E2j+E3j)×(E1k+E2k+E3k)(E1l+E2l+E3l),
E4(K, t)iλ0z 12π3V P(3)(r, t)exp(iKr)d3r,
E4(k, t)iλ0z 12π3Vχ(3)[E1 sin(iω0t+ik1r)+E2 sin(iω0t+ik2r)+E3 sin(iω0t+ik3r)]3 exp(iKr)d3r.
E4,i(K, t)iλ0z 12π3Vjklt1,t2,t3=13χijkl(3) sin(iω0t+ikt1r)sin(iω0t+ikt2r)sin(iω0t+ikt3r)Et1,jEt2,kEt3,l exp(iKr)d3r=-i4λ0z 12π3Vjklt1,t2,t3=13χijkl(3)[sin(3iω0t+ir(kt1+kt2+kt3)+sin(iω0t+ir(-kt1-kt2+kt3)+sin(iω0t+ir(-kt1+kt2-kt3)+sin(iω0t+ir(kt1-kt2-kt3)]Et1,j×Et2,kEt3,l exp(ikr)d3r.
δ(x-x0)=1/2π- exp[-iy(x-x0)]dy,
E4,i(K, t)-i4λ0z 12π2 exp(iω0t)δ[|(k1+k2-k3-K)|]χijkl(3)Et1,jEt2,kEt3,lVd3r.
P˜i(3)(r, t)χijkl(3)E1jE2kE3l×exp[iω0t+i(k1+k2-k3)r],
E(k, t)=d=1NA(d) exp(iwdt+ikdr+φd).
E4(K, t)iλ0z 12π3Vχ(3)d=1N[E1(d) sin(iωdt+ik1dr+φd)+E2(d) sin(iωdt+ik2dr+φd)+E3(d) sin(iωdt+ik3dr+φd)]3×exp(iKr)d3r.
E4i(K,t)iλ0z 12π3V d=1Nt=13j,k,l=13 χijkl(3)E1j(d)E2k(d)E3i(d)×[sin(iω1dt+ik1dr+φ1d)sin(iω2dt+ik2dr+φ2d)sin(iω3dt+ik3dr+φ3d)]×exp(ik0dr)d3r=iλ0z 12π3V d1,d2,d3=1N t1,t2,t3=13χijkl(3)Et1,j(d1)Et2,k(d2)Et3,l(d3)×sin(iωt1,d1t+ikt1,d1r+φt1,d1)sin(iωt2,d2t+ikt2,d2r+φt2,d2)sin(iωt3,d3t+ikt3,d3r+φt3,d3)×exp(iKr)d3r
=-i4λ0z 12π3V d1,d2,d3=1Nt1,t2,t3=13χijkl(3)Et1,j(d1)Et2,k(d2)Et3,l(d3)×{sin[i(ωt1,d1+ωt2,d2+ωt3,d3)t+i(kt1,d1+kt2,d2+kt3,d3)r+φt1,d1+φt2,d2+φt3,d3]+sin[i(-ωt1,d1-ωt2,d2+ωt3,d3)t+i(-kt1,d1-kt2,d2-kt3,d3)r-φt1,d1-φt2,d2+φt3,d3]+sin[i(ωt1,d1-ωt2,d2-ωt3,d3)t+i(kt1,d1-kt2,d2-kt3,d3)r+φt1,d1-φt2,d2-φt3,d3]+sin[i(-ωt1,d1+ωt2,d2-ωt3,d3)t+i(-kt1,d1+kt2,d2-kt3,d3)r-φt1,d1+φt2,d2-φt3,d3]}×exp(iKr)d3r.
ωt1,d1+ωt2,d2-ωt3,d3=ωx,
ωt1,d1/c0aωt1,d1/c+-ωt2,d2/c0aωt2,d2/c-0-ωt3,d3/caωt3,d3/c 1a2+1
=0ωx/caωx/c 1a2+1.
E4i(k, t)i4λ0z 12π2d=1N exp(iωdt+φd)×δ(|k1,d+k2,d+k3,d-K|)×χijkl(3)Et1,j(d)Et2,k(d)Et3,l(d)Vd3r.
E4(t)χ(3)-E1(t1+t)dt1×-E2(t2)E3(t1+t2)dt2=χ(3)[E1(E2E3)].
exp(-At2)[exp(-At2)exp(-At2)]exp(-At2/3),
τDFWM=τpulse3.
I=I1+I2+2I1I2 Re[γ11(τ)],
γ11(τ)=E(t)E*(t+τ)|E|2
a(t)=F-1(A(ω))=- A(ω)exp(iωt)dω,
b(t)=F-1(B(ω))=- B(ω)exp(iωt)dω,
c(t)=F-1(C(ω))=- C(ω)exp(iωt)cω.
F- a*(t1)b(t1+t2)dt2
=1/(2π)- - a*(t1)b(t1+t2)dt1×exp(-iωt2)dt2=1/(2π)- a*(t1)×- b(t1+t2)exp(-iωt2)dt2dt1=- a*(t1)exp(-iωt1)B(ω)dt1=2πA(ω)B(ω),
F-1(A(ω)B(ω))=2π- a*(t1)b(t1+t2)dt1.
F-1(A(ω)B(ω)C(ω))
=1/2π- a(t+t1)dt×C(ω)B(ω)exp(-iωt1)dω*=(2π)2- a(t+t1)dt- b(t2)c*(t1+t2)dt2
=(2π)2a(t)(b(t)c(t)).

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