Abstract

We describe noncollinear third-harmonic Maker fringes for the determination of third-order optical nonlinearities. This method builds on the same principles as conventional Maker fringes experiments, but thanks to the limited interaction length of two intersecting beams, it allows to eliminate the requirement of taking into account the contribution of air to the third-harmonic signal. Thus, third-order susceptibilities can be accurately determined without the need for a vacuum chamber. We review the theoretical underpinnings of the method, and discuss the dependence of the coherence length for harmonic generation on the intersection angle between the beams and the sample orientation, and the effect this has on the Maker fringes pattern.

© 2013 Optical Society of America

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References

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  1. P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, Phys. Rev. Lett. 8, 21 (1962).
    [CrossRef]
  2. J. Jerphagnon and S. K. Kurtz, J. Appl. Phys. 41, 1667 (1969).
    [CrossRef]
  3. J. P. Hermann, Opt. Commun. 9, 74 (1973).
    [CrossRef]
  4. G. R. Meredith, Phys. Rev. B 24, 5522 (1981).
    [CrossRef]
  5. F. Kajzar and J. Messier, Phys. Rev. A 32, 2352 (1985).
    [CrossRef]
  6. F. Krausz and E. Wintner, Appl. Phys. B 49, 479 (1989).
    [CrossRef]
  7. C. L. Callender, C. A. Carere, J. Albert, L.-L. Zhou, and D. J. Worsfold, J. Opt. Soc. Am. B 9, 518 (1992).
    [CrossRef]
  8. D. Faccio, V. Pruneri, and P. Kazansky, Opt. Lett. 25, 1376 (2000).
    [CrossRef]
  9. N. N. Bloembergen and P. S. Pershan, Phys. Rev. 128, 606 (1962).
    [CrossRef]
  10. J. J. Wynne and N. Bloembergen, Phys. Rev. 188, 1211 (1969).
    [CrossRef]
  11. W. N. Herman and L. M. Hayden, J. Opt. Soc. Am. B 12, 416 (1995).

2000 (1)

1995 (1)

1992 (1)

1989 (1)

F. Krausz and E. Wintner, Appl. Phys. B 49, 479 (1989).
[CrossRef]

1985 (1)

F. Kajzar and J. Messier, Phys. Rev. A 32, 2352 (1985).
[CrossRef]

1981 (1)

G. R. Meredith, Phys. Rev. B 24, 5522 (1981).
[CrossRef]

1973 (1)

J. P. Hermann, Opt. Commun. 9, 74 (1973).
[CrossRef]

1969 (2)

J. Jerphagnon and S. K. Kurtz, J. Appl. Phys. 41, 1667 (1969).
[CrossRef]

J. J. Wynne and N. Bloembergen, Phys. Rev. 188, 1211 (1969).
[CrossRef]

1962 (2)

N. N. Bloembergen and P. S. Pershan, Phys. Rev. 128, 606 (1962).
[CrossRef]

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, Phys. Rev. Lett. 8, 21 (1962).
[CrossRef]

Albert, J.

Bloembergen, N.

J. J. Wynne and N. Bloembergen, Phys. Rev. 188, 1211 (1969).
[CrossRef]

Bloembergen, N. N.

N. N. Bloembergen and P. S. Pershan, Phys. Rev. 128, 606 (1962).
[CrossRef]

Callender, C. L.

Carere, C. A.

Faccio, D.

Hayden, L. M.

Herman, W. N.

Hermann, J. P.

J. P. Hermann, Opt. Commun. 9, 74 (1973).
[CrossRef]

Jerphagnon, J.

J. Jerphagnon and S. K. Kurtz, J. Appl. Phys. 41, 1667 (1969).
[CrossRef]

Kajzar, F.

F. Kajzar and J. Messier, Phys. Rev. A 32, 2352 (1985).
[CrossRef]

Kazansky, P.

Krausz, F.

F. Krausz and E. Wintner, Appl. Phys. B 49, 479 (1989).
[CrossRef]

Kurtz, S. K.

J. Jerphagnon and S. K. Kurtz, J. Appl. Phys. 41, 1667 (1969).
[CrossRef]

Maker, P. D.

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, Phys. Rev. Lett. 8, 21 (1962).
[CrossRef]

Meredith, G. R.

G. R. Meredith, Phys. Rev. B 24, 5522 (1981).
[CrossRef]

Messier, J.

F. Kajzar and J. Messier, Phys. Rev. A 32, 2352 (1985).
[CrossRef]

Nisenoff, M.

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, Phys. Rev. Lett. 8, 21 (1962).
[CrossRef]

Pershan, P. S.

N. N. Bloembergen and P. S. Pershan, Phys. Rev. 128, 606 (1962).
[CrossRef]

Pruneri, V.

Savage, C. M.

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, Phys. Rev. Lett. 8, 21 (1962).
[CrossRef]

Terhune, R. W.

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, Phys. Rev. Lett. 8, 21 (1962).
[CrossRef]

Wintner, E.

F. Krausz and E. Wintner, Appl. Phys. B 49, 479 (1989).
[CrossRef]

Worsfold, D. J.

Wynne, J. J.

J. J. Wynne and N. Bloembergen, Phys. Rev. 188, 1211 (1969).
[CrossRef]

Zhou, L.-L.

Appl. Phys. B (1)

F. Krausz and E. Wintner, Appl. Phys. B 49, 479 (1989).
[CrossRef]

J. Appl. Phys. (1)

J. Jerphagnon and S. K. Kurtz, J. Appl. Phys. 41, 1667 (1969).
[CrossRef]

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

Opt. Commun. (1)

J. P. Hermann, Opt. Commun. 9, 74 (1973).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. (2)

N. N. Bloembergen and P. S. Pershan, Phys. Rev. 128, 606 (1962).
[CrossRef]

J. J. Wynne and N. Bloembergen, Phys. Rev. 188, 1211 (1969).
[CrossRef]

Phys. Rev. A (1)

F. Kajzar and J. Messier, Phys. Rev. A 32, 2352 (1985).
[CrossRef]

Phys. Rev. B (1)

G. R. Meredith, Phys. Rev. B 24, 5522 (1981).
[CrossRef]

Phys. Rev. Lett. (1)

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, Phys. Rev. Lett. 8, 21 (1962).
[CrossRef]

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Figures (2)

Fig. 1.
Fig. 1.

Wavevector diagram depicting refraction at the sample interface and the wavevectors of interacting and generated waves, i.e., the two incident waves (k1 and k2), the nonlinear polarization (kp), the radiated signal inside the sample (kt), and the “reflected” signal wave outside the sample (kr). In this diagram, all wavevectors are drawn to scale assuming θ=50° and a refractive index of the fundamental and the third-harmonic wave arbitrarily chosen to be 1.5 and 1.6, respectively. The relationships between kp, kt, and kr are the same as for the case in which the nonlinear polarization is generated by only one beam, or by any other nonlinear optical process. The inset box contains a diagram of the experimental beam arrangement corresponding to this configuration, with the two incident beams and the two generated THG beams, of which one is detected.

Fig. 2.
Fig. 2.

Phase mismatch factors for THG generation and noncollinear THG Maker fringes for different crossing angles between the beams. Curves are labeled with the corresponding value of α in panel (a) and are drawn with the same pattern and the same sequence in panels (b) and (c). All curves are for fused silica and for a wavelength of 2.1 μm, with n(ω)=1.4364 and n(ωp)=1.4553. (a) Coherence length π/Δk, equal to λ/[6(n(ωp)n(ω)] at normal incidence (zero rotation angle) and α=0. (b) Factor determining the amplitude of the generated wave in Eq. (8), equal to [n2(ωp)n2(ω)]1 at normal incidence and α=0. (c) Resulting noncollinear THG Maker fringes from Eq. (11) and a sample thickness of 1 mm. (d) Experimental result in a 1 mm fused silica plate and theoretical fit, obtained by a slight adjustment of the plate thickness used for the calculations.

Equations (12)

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E(ω)(r,t)=E(a)2eiωt+ik1·r+E(b)2eiωt+ik2·r+c.c.,
Pi(NL)(kp)=34ε0j,k,l=13χijkl(3)Ej(a)Ek(b)El(b),
k1a=k{sin(θα),cos(θα)},
k2a=k{sin(θ+α),cos(θ+α)},
ki={ki,ya,[n(ω)k]2[ki,ya]2},
kp=2k1+k2,
Et(r)=E0+ωp2/c2kt2kp2P(NL)ε0ei(kpkt)·r.
Et(d)=P(NL)ε0kr2kt2kp2[eiΔkdkp,zkr,zkt,zkr,z],
kr={kp,y,[kr]2[kp,y]2},
kt={kp,y,[kt]2[kp,y]2}.
P=[(t1as)2t2ast3sa]2|Et(d)|2,
L=w0sinα,

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