Abstract

We present a method that combines the Z-scan technique with nonlinear ellipse rotation (NER) to measure third-order nonlinear susceptibility components. The experimental details are demonstrated, and a comprehensive theoretical analysis is given. The validity of this method is verified by the measurements of the nonlinear susceptibility tensor of a well-characterized liquid, CS2.

© 2007 Optical Society of America

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References

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  1. R. L. Sutherland, ed., Handbook of Nonlinear Optics (Marcel Dekker, New York, 1996).
  2. S. R. Friberg and P. W. Smith, "Nonlinear optical glasses for ultrafast optical switches," IEEE J. Quantum Electron. QE-23, 2089-2094 (1987).
    [CrossRef]
  3. G. Boudebs, M. Chis, and J. P. Bourdin, "Third-order susceptibility measurements by nonlinear image processing," J. Opt. Soc. Am. B 13, 1450-1456 (1996).
    [CrossRef]
  4. P. D. Maker, R. W. Terhune, and C. M. Savage, "Intensity-dependent changes in the refractive index of liquids," Phys. Rev. Lett. 12, 507-509 (1964).
    [CrossRef]
  5. P. D. Maker and R. W. Terhune, "Study of optical effects due to an induced polarization third order in the electric field strength," Phys. Rev. 137, A801-A818 (1965).
    [CrossRef]
  6. M. Sheik-Bahae, A. A. Said, T. H. Vei, D. J. Hagan, and E. W. Van Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
    [CrossRef]
  7. W. Zhao and P. Palffy-Muhoray, "Z-scan technique using top-hat beams," Appl. Phys. Lett. 63, 1613-1615 (1993).
    [CrossRef]
  8. T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. Van Stryland, "Eclipsing Z-scan measurement of Lambda/104 wave-front distortion," Opt. Lett. 19, 317-319 (1994).
    [CrossRef] [PubMed]
  9. R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic Press, San Diego, 2003).
  10. A. J. Van Wonderen, "Influence of transverse effects on self-induced polarization changes in an isotropic Kerr medium," J. Opt. Soc. Am. B 14, 1118-1130 (1997).
    [CrossRef]
  11. M. Lefkir and G. Rivoire, "Influence of transverse effects on measurements of third-order nonlinear susceptibility by self-induced polarization state changes," J. Opt. Soc. Am. B 14, 2856-2864 (1997).
    [CrossRef]
  12. M. Sheik-Bahae and M. P. Hasselbeck, "Third-order optical nonlinearities," in OSA Handbook of Optics, (McGraw-Hill 2001), Vol. IV, Chap. 17.

1997 (2)

1996 (1)

1994 (1)

1993 (1)

W. Zhao and P. Palffy-Muhoray, "Z-scan technique using top-hat beams," Appl. Phys. Lett. 63, 1613-1615 (1993).
[CrossRef]

1990 (1)

M. Sheik-Bahae, A. A. Said, T. H. Vei, D. J. Hagan, and E. W. Van Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

1987 (1)

S. R. Friberg and P. W. Smith, "Nonlinear optical glasses for ultrafast optical switches," IEEE J. Quantum Electron. QE-23, 2089-2094 (1987).
[CrossRef]

1965 (1)

P. D. Maker and R. W. Terhune, "Study of optical effects due to an induced polarization third order in the electric field strength," Phys. Rev. 137, A801-A818 (1965).
[CrossRef]

1964 (1)

P. D. Maker, R. W. Terhune, and C. M. Savage, "Intensity-dependent changes in the refractive index of liquids," Phys. Rev. Lett. 12, 507-509 (1964).
[CrossRef]

Boudebs, G.

Bourdin, J. P.

Chis, M.

Friberg, S. R.

S. R. Friberg and P. W. Smith, "Nonlinear optical glasses for ultrafast optical switches," IEEE J. Quantum Electron. QE-23, 2089-2094 (1987).
[CrossRef]

Hagan, D. J.

T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. Van Stryland, "Eclipsing Z-scan measurement of Lambda/104 wave-front distortion," Opt. Lett. 19, 317-319 (1994).
[CrossRef] [PubMed]

M. Sheik-Bahae, A. A. Said, T. H. Vei, D. J. Hagan, and E. W. Van Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Lefkir, M.

Maker, P. D.

P. D. Maker and R. W. Terhune, "Study of optical effects due to an induced polarization third order in the electric field strength," Phys. Rev. 137, A801-A818 (1965).
[CrossRef]

P. D. Maker, R. W. Terhune, and C. M. Savage, "Intensity-dependent changes in the refractive index of liquids," Phys. Rev. Lett. 12, 507-509 (1964).
[CrossRef]

Palffy-Muhoray, P.

W. Zhao and P. Palffy-Muhoray, "Z-scan technique using top-hat beams," Appl. Phys. Lett. 63, 1613-1615 (1993).
[CrossRef]

Rivoire, G.

Said, A. A.

M. Sheik-Bahae, A. A. Said, T. H. Vei, D. J. Hagan, and E. W. Van Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Savage, C. M.

P. D. Maker, R. W. Terhune, and C. M. Savage, "Intensity-dependent changes in the refractive index of liquids," Phys. Rev. Lett. 12, 507-509 (1964).
[CrossRef]

Sheik-Bahae, M.

T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. Van Stryland, "Eclipsing Z-scan measurement of Lambda/104 wave-front distortion," Opt. Lett. 19, 317-319 (1994).
[CrossRef] [PubMed]

M. Sheik-Bahae, A. A. Said, T. H. Vei, D. J. Hagan, and E. W. Van Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Smith, P. W.

S. R. Friberg and P. W. Smith, "Nonlinear optical glasses for ultrafast optical switches," IEEE J. Quantum Electron. QE-23, 2089-2094 (1987).
[CrossRef]

Terhune, R. W.

P. D. Maker and R. W. Terhune, "Study of optical effects due to an induced polarization third order in the electric field strength," Phys. Rev. 137, A801-A818 (1965).
[CrossRef]

P. D. Maker, R. W. Terhune, and C. M. Savage, "Intensity-dependent changes in the refractive index of liquids," Phys. Rev. Lett. 12, 507-509 (1964).
[CrossRef]

Van Stryland, E. W.

T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. Van Stryland, "Eclipsing Z-scan measurement of Lambda/104 wave-front distortion," Opt. Lett. 19, 317-319 (1994).
[CrossRef] [PubMed]

M. Sheik-Bahae, A. A. Said, T. H. Vei, D. J. Hagan, and E. W. Van Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Van Wonderen, A. J.

Vei, T. H.

M. Sheik-Bahae, A. A. Said, T. H. Vei, D. J. Hagan, and E. W. Van Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Xia, T.

Zhao, W.

W. Zhao and P. Palffy-Muhoray, "Z-scan technique using top-hat beams," Appl. Phys. Lett. 63, 1613-1615 (1993).
[CrossRef]

Appl. Phys. Lett. (1)

W. Zhao and P. Palffy-Muhoray, "Z-scan technique using top-hat beams," Appl. Phys. Lett. 63, 1613-1615 (1993).
[CrossRef]

IEEE J. Quantum Electron. (2)

S. R. Friberg and P. W. Smith, "Nonlinear optical glasses for ultrafast optical switches," IEEE J. Quantum Electron. QE-23, 2089-2094 (1987).
[CrossRef]

M. Sheik-Bahae, A. A. Said, T. H. Vei, D. J. Hagan, and E. W. Van Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Lett. (1)

Phys. Rev. (1)

P. D. Maker and R. W. Terhune, "Study of optical effects due to an induced polarization third order in the electric field strength," Phys. Rev. 137, A801-A818 (1965).
[CrossRef]

Phys. Rev. Lett. (1)

P. D. Maker, R. W. Terhune, and C. M. Savage, "Intensity-dependent changes in the refractive index of liquids," Phys. Rev. Lett. 12, 507-509 (1964).
[CrossRef]

Other (3)

R. L. Sutherland, ed., Handbook of Nonlinear Optics (Marcel Dekker, New York, 1996).

R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic Press, San Diego, 2003).

M. Sheik-Bahae and M. P. Hasselbeck, "Third-order optical nonlinearities," in OSA Handbook of Optics, (McGraw-Hill 2001), Vol. IV, Chap. 17.

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Figures (5)

Fig. 1.
Fig. 1.

(a) Experimental arrangement for NER modified Z-scan. D1 and D2 are the detectors. (b) and (c) are the geometry of incident and the transmitted polarization ellipse, respectively. The slow axis of the first λ/4 plate is taken along the x-axis. α is the angle between the first polarizer direction and the x-axis. θ is the rotating angle of the polarization ellipse induced by nonlinearity. φ is the angle of the analyzer.

Fig. 2.
Fig. 2.

Z-scan curves for linear, circular, and elliptical polarization obtained by traditional closed aperture Z-scan.

Fig. 3.
Fig. 3.

Z-scan curves of NER modified Z-scan with and without the second λ/4 plate. The polarization directions of the polarizer and the analyzer parallel each other (α=φ=-22.5°). The solid lines are the theoretical fittings with B = 12.6×10-20m2/V2 and I 0 = 5.93 GW/cm2.

Fig. 4.
Fig. 4.

(a) Normalized transmitted power through the analyzer at linear output and nonlinear output obtained by rotating the analyzer. (b) The ratio of nonlinear transmitted power to linear transmitted power as a function of rotation angle of the analyzer.

Fig. 5.
Fig. 5.

Z-scan curves of NER modified Z-scan without second λ/4 plate at (a) φ=90°, (b) φ = 80°, (c) φ=76°, and (d) φ=55°. The solid lines are the theoretical fittings with B=12.6×10-20m2/V2 and I 0=5.93 GW/cm2.

Equations (19)

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E x = E 0 cos α exp [ i ( kz ωt ) ] ,
E y = E 0 sin α exp [ i ( kz ωt + Δ ) ] ,
P NL = A ( E E * ) E + 1 2 B ( E E ) E * ,
n ± = n 0 + 2 π n 0 [ A E ± 2 + ( A + B ) E 2 ] .
n 2 = ( 1 q 2 ) n 2 cir + 2 qn 2 lin 1 + q 2 ,
θ = 1 2 ( n + n ) ω c d = πω cn ( E _ 2 E + 2 ) d = QE 0 2 ,
E ( x y ) = E 0 ( cos α x ̂ + i sin α y ̂ ) exp [ i ( kz ωt ) ] ,
E = E 0 [ ( cos α cos θ i sin α sin θ ) x ̂ + ( cos α sin θ + i sin α cos θ ) y ̂ ] exp [ i ( kz ωt ) ] ,
E out = E 0 [ ( cos α cos θ i sin α sin θ ) ( cos Δ + i sin Δ ) cos φ + ( cos α sin θ + i sin α cos θ ) sin φ ] exp [ i ( kz ωt ) ] .
E 0 ( r , z ) = E 00 w 0 w z exp [ r 2 w z 2 ik r 2 2 R z ] ,
P ( z ) = ε 0 n 0 0 + E out 2 r d r .
P non ( z ) = ε 0 n 0 cπw z 2 8 Q { 1 2 cos 2 α [ sin ( 2 QP ) cos 2 φ cos ( 2 QP ) sin 2 φ cos Δ + sin 2 φ cos Δ ] + QP ( 1 + sin 2 α sin 2 φ sin Δ ) }
P lin = ε 0 n 0 c π E 00 2 w 0 2 4 ( sin 2 α sin 2 φ + cos 2 α cos 2 φ + 1 2 sin 2 α sin 2 φ sin Δ ) .
T ( z ) = P non ( z ) P lin .
T ( z ) = 1 + sin 2 2 α 2 + cos 2 2 α 4 Q P sin ( 2 Q P ) .
T ( z ) = R 2 + sin 2 φ + sin 2 ( Q P φ ) 4 Q P R cos 2 α ,
θ ( t ) = θ E 0 ( t ) 2 d t E 0 ( t ) 2 d t .
θ ( t ) = θ 2 .
Re ( χ ( 3 ) ) = 4 ε 0 n 0 2 c 3 n 2 .

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