Abstract

We show that nonlinear optical signals generated by non-phase-matched interactions are strongly suppressed when the interaction volume is finite and localized deep inside the bulk of a homogeneous material, as opposed to the case where the interaction volume extends across a boundary of the material. The suppression in the bulk originates from destructive interference between the signals generated in the two regions where the interaction is gradually turned on and off and depends on the ratio of the coherence length to the characteristic length of the interaction volume.

© 2005 Optical Society of America

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References

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  1. R. W. Boyd, Nonlinear Optics (Academic, San Diego, 1992).
  2. P. Günter, ed., Nonlinear Optical Effects and Materials (Springer, Berlin, 2000).
  3. T. F. Heinz, “Second-order nonlinear optical effects at surfaces and interfaces,” in Nonlinear Surface Electromagnetic Phenomena, H.–E. Ponath and G. I. Stegeman, eds. (Elsevier, Amsterdam, 1991), pp. 353-416.
  4. J. A. Giordmaine, “Nonlinear optical properties of liquids,” Phys. Rev. 138, A1599-A1606 (1965).
    [CrossRef]
  5. P. M. Rentzepis, J. A. Giordmaine, and K. W. Wecht, “Coherent optical mixing in optically active liquids,” Phys. Rev. Lett. 16, 792-794 (1966).
    [CrossRef]
  6. A. P. Shkurinov, A. V. Dubrovskii, and N. I. Koroteev, “Second harmonic generation in an optically active liquid: Experimental observation of a fourth-order optical nonlinearity due to molecular chirality,” Phys. Rev. Lett. 70, 1085-1088 (1993).
    [CrossRef] [PubMed]
  7. P. Fischer, D. S. Wiersma, R. Righini, B. Champagne, and A. D. Buckingham, “Three-wave mixing in chiral liquids,” Phys. Rev. Lett. 85, 4253-4256 (2000).
    [CrossRef] [PubMed]
  8. M. A. Belkin, T. A. Kulakov, K.-H. Ernst, L. Yan, and Y. R. Shen, “Sum-frequency vibrational spectroscopy on chiral liquids: A novel technique to probe molecular chirality,” Phys. Rev. Lett. 85, 4474-4477 (2000).
    [CrossRef] [PubMed]
  9. M. A. Belkin, S. H. Han, X. Wei, and Y. R. Shen, “Sum-frequency generation in chiral liquids near electronic resonance,” Phys. Rev. Lett. 87, 113001 (2001).
    [CrossRef] [PubMed]
  10. G. Berkovic, Y. R. Shen, G. Marowsky, and R. Steinhoff, “Interference between second-harmonic generation from a substrate and from an adsorbate layer,” J. Opt. Soc. Am. B 6, 205-208 (1989).
    [CrossRef]
  11. S. Cattaneo and M. Kauranen, “Determination of second-order susceptibility components of thin films by two-beam second-harmonic generation,” Opt. Lett. 28, 1445-1447 (2003).
    [CrossRef] [PubMed]
  12. S. Cattaneo and M. Kauranen, “Polarization-based identification of bulk contributions in surface nonlinear optics,” Phys. Rev. B 72, 033412 (2005).
    [CrossRef]
  13. S. Cattaneo, “Two-beam surface second-harmonic generation,” Ph.D. thesis, Tampere University of Technology, Tampere, Finland, 2004.
  14. P. Figliozzi, L. Sun, Y. Jiang, N. Matlis, B. Mattern, M. C. Downer, S. P. Withrow, C. W. White, W. L. Mochán, and B. S. Mendoza, “Single-beam and enhanced two-beam second-harmonic generation from silicon nanocrystals by use of spatially inhomogeneous femtosecond pulses,” Phys. Rev. Lett. 94, 047401 (2005).
    [CrossRef] [PubMed]
  15. L. Sun, P. Figliozzi, Y. Q. An, and M. C. Downer, “Nonresonant quadrupolar SHG in isotropic solids using two orthogonally polarized laser beams,” Opt. Lett., in press.
    [PubMed]
  16. J. E. Sipe, “New green-function formalism for surface optics,” J. Opt. Soc. Am. B 4, 481-489 (1987).
    [CrossRef]
  17. P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B 33, 8254–8263 (1986).
    [CrossRef]
  18. J. E. Sipe, V. Mizrahi, and G. I. Stegeman, “Fundamental difficulty in the use of second-harmonic generation as a strictly surface probe,” Phys. Rev. B 35, 9091–9094 (1987).
    [CrossRef]
  19. P. Guyot-Sionnest and Y. R. Shen, “Bulk contribution in surface second-harmonic generation,” Phys. Rev. B 38, 7985-7989 (1988).
    [CrossRef]
  20. Y. R. Shen, “Surface contribution versus bulk contribution in surface nonlinear optical spectroscopy,” Appl. Phys. B 68, 295-300 (1999).
    [CrossRef]

Appl. Phys. B (1)

Y. R. Shen, “Surface contribution versus bulk contribution in surface nonlinear optical spectroscopy,” Appl. Phys. B 68, 295-300 (1999).
[CrossRef]

J. Opt. Soc. Am. B (2)

Nonlinear Surface Electromagnetic Phenom (1)

T. F. Heinz, “Second-order nonlinear optical effects at surfaces and interfaces,” in Nonlinear Surface Electromagnetic Phenomena, H.–E. Ponath and G. I. Stegeman, eds. (Elsevier, Amsterdam, 1991), pp. 353-416.

Opt. Lett. (2)

S. Cattaneo and M. Kauranen, “Determination of second-order susceptibility components of thin films by two-beam second-harmonic generation,” Opt. Lett. 28, 1445-1447 (2003).
[CrossRef] [PubMed]

L. Sun, P. Figliozzi, Y. Q. An, and M. C. Downer, “Nonresonant quadrupolar SHG in isotropic solids using two orthogonally polarized laser beams,” Opt. Lett., in press.
[PubMed]

Phys. Rev. (1)

J. A. Giordmaine, “Nonlinear optical properties of liquids,” Phys. Rev. 138, A1599-A1606 (1965).
[CrossRef]

Phys. Rev. B (4)

S. Cattaneo and M. Kauranen, “Polarization-based identification of bulk contributions in surface nonlinear optics,” Phys. Rev. B 72, 033412 (2005).
[CrossRef]

P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B 33, 8254–8263 (1986).
[CrossRef]

J. E. Sipe, V. Mizrahi, and G. I. Stegeman, “Fundamental difficulty in the use of second-harmonic generation as a strictly surface probe,” Phys. Rev. B 35, 9091–9094 (1987).
[CrossRef]

P. Guyot-Sionnest and Y. R. Shen, “Bulk contribution in surface second-harmonic generation,” Phys. Rev. B 38, 7985-7989 (1988).
[CrossRef]

Phys. Rev. Lett. (6)

P. Figliozzi, L. Sun, Y. Jiang, N. Matlis, B. Mattern, M. C. Downer, S. P. Withrow, C. W. White, W. L. Mochán, and B. S. Mendoza, “Single-beam and enhanced two-beam second-harmonic generation from silicon nanocrystals by use of spatially inhomogeneous femtosecond pulses,” Phys. Rev. Lett. 94, 047401 (2005).
[CrossRef] [PubMed]

P. M. Rentzepis, J. A. Giordmaine, and K. W. Wecht, “Coherent optical mixing in optically active liquids,” Phys. Rev. Lett. 16, 792-794 (1966).
[CrossRef]

A. P. Shkurinov, A. V. Dubrovskii, and N. I. Koroteev, “Second harmonic generation in an optically active liquid: Experimental observation of a fourth-order optical nonlinearity due to molecular chirality,” Phys. Rev. Lett. 70, 1085-1088 (1993).
[CrossRef] [PubMed]

P. Fischer, D. S. Wiersma, R. Righini, B. Champagne, and A. D. Buckingham, “Three-wave mixing in chiral liquids,” Phys. Rev. Lett. 85, 4253-4256 (2000).
[CrossRef] [PubMed]

M. A. Belkin, T. A. Kulakov, K.-H. Ernst, L. Yan, and Y. R. Shen, “Sum-frequency vibrational spectroscopy on chiral liquids: A novel technique to probe molecular chirality,” Phys. Rev. Lett. 85, 4474-4477 (2000).
[CrossRef] [PubMed]

M. A. Belkin, S. H. Han, X. Wei, and Y. R. Shen, “Sum-frequency generation in chiral liquids near electronic resonance,” Phys. Rev. Lett. 87, 113001 (2001).
[CrossRef] [PubMed]

Other (3)

S. Cattaneo, “Two-beam surface second-harmonic generation,” Ph.D. thesis, Tampere University of Technology, Tampere, Finland, 2004.

R. W. Boyd, Nonlinear Optics (Academic, San Diego, 1992).

P. Günter, ed., Nonlinear Optical Effects and Materials (Springer, Berlin, 2000).

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Figures (4)

Fig. 1.
Fig. 1.

Schematic diagram of noncollinear three-wave mixing. Two plane-wave-like input beams of finite transverse size intersect in a nonlinear medium occupying the region 0 ≤ zd. The function g(z) describes the variation in the strength of the source polarization due to the overlap of the input beams. g(z) is expected to be a smooth, real function that grows from zero to the maximum and then decreases back to zero over the interval a < z < b.

Fig. 2.
Fig. 2.

Second-harmonic (SHG) signal obtained by translating the overlap of the input beams across a 12 mm thick glass plate (BK7). The signal is strongly suppressed when the beams overlap deep in the bulk of the material. The surfaces appear to be closer than the physical thickness of the sample because of refraction.

Fig. 3.
Fig. 3.

Second-harmonic (SHG) signal obtained by translating the overlap of the input beams across a 16.4 mm thick nonlinear crystal (DKDP). The strongest signal is obtained near the crystal surfaces, whereas practically no signal is detected from the bulk of the crystal.

Fig. 4.
Fig. 4.

Nonlinear optical signal arising from a non-phase-matched interaction acting over a finite volume. Phase mismatch leads to infinitesimal phase differences between successive polarization sheets. As the interaction is gradually turned on, the resulting amplitude (A) describes an expanding spiral centered at the origin. The spiral contracts back to the origin when the interaction is gradually turned off. The field amplitudes (arrows) resulting from the two halves of the interaction volume are seen to interfere destructively producing essentially no net signal.

Equations (5)

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0 d e i Δ kz dz = 1 i Δ k ( e i Δ kd 1 ) .
0 d g ( z ) e i Δ kz dz = 1 i Δ k g ( z ) e i Δ kz 0 d 1 i Δ k 0 d g ' ( z ) e i Δ kz dz ,
1 i Δ k 1 L 0 d e i Δ kz dz = 1 i Δ k l c iπL [ e i Δ kd 1 ] .
1 i Δ k g ( 0 ) ( e i Δ kd 1 ) ,
1 i Δ k g ( 0 ) and 1 i Δ k g ( d ) e i Δ kd ,

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