## Polarization dependence of Z-scan measurement: theory and experiment

Optics Express, Vol. 17, Issue 8, pp. 6397-6406 (2009)

http://dx.doi.org/10.1364/OE.17.006397

Acrobat PDF (278 KB)

### Abstract

Here we report on an extension of common Z-scan method to arbitrary polarized incidence light for measurements of anisotropic third-order nonlinear susceptibility in isotropic medium. The normalized transmittance formulas of closed-aperture Z-scan are obtained for linearly, elliptically and circularly polarized incidence beam. The theoretical analysis is examined experimentally by studying third-order nonlinear susceptibility of CS_{2} liquid. Results show that the elliptically polarized light Z-scan method can be used to measure simultaneously the two third-order nonlinear susceptibility components *χ _{xyyx}
*

^{(3)}and

*χ*

_{xxyy}^{(3)}. Furthermore, the elliptically polarized light Z-scan measurements of large nonlinear phase shift are also analyzed theoretically and experimentally.

© 2009 Optical Society of America

## 1. Introduction

1. I. Fuks-Janczarek, B. Sahraoui, I. V. Kityk, and J. Berdowski, “Electronic and nuclear contributions to the third-order optical susceptibility,” Opt. Commun. **236**, 159 (2004). [CrossRef]

2. 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]

3. 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]

4. M. Lefkir and G. Rivoire, “Influence of transverse effects on measurement of third-order nonlinear susceptibility by self-induced polarization state changes,” J. Opt. Soc. Am. B **14**, 2856–2864 (1997). [CrossRef]

5. M. Sheik-Bahae, A. A. Said, T. H. Wei, 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]

5. M. Sheik-Bahae, A. A. Said, T. H. Wei, 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]

5. M. Sheik-Bahae, A. A. Said, T. H. Wei, 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]

8. J. A. Hermann and R. G. McDuff, “Analysis of spatial scanning with thick optically nonlinear media,” J. Opt. Soc. Am. B **10**, 2056–2064 (1993). [CrossRef]

9. J. G. Tian, W. P. Zang, C. Z. Zhang, and G. Y. Zhang, “Analysis of beam propagation in thick nonlinear media,” Appl. Opt. **34**, 4331–4336 (1995). [CrossRef] [PubMed]

10. T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. Van Stryland, “Eclipsing Z- Scan Measurement of λ/104 Wavefront Distortion,” Opt. Lett. **19**, 317–319 (1994). [CrossRef] [PubMed]

11. J. Wang, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E. W. Van Stryland, “Time-Resolved Z-Scan Measurements of Optical Nonlinearities,” J. Opt. Soc. Am. B **11**, 1009–1017 (1994). [CrossRef]

12. L. Demenicis, A. S. L. Gomes, D. V. Petrov, C.B. de Araújo, C. P. de Melo, C. G. dos Santos, and R. Souto-Maior, “Saturation effects in the nonlinear-optical susceptibility of poly(3-hexadecylthiophene),” J. Opt. Soc. Am. B **14**, 609 (1997). [CrossRef]

13. M. Sheik-Bahae, J. Wang, J.R. DeSalvo, D. J. Hagan, and E. W. Van Stryland, “Measurement of Nondegenerate Nonlinearities using a 2-Color Z-Scan,” Opt. Lett. **17**, 258–260 (1992). [CrossRef] [PubMed]

14. R. Bridges, G. Fischer, and R. Boyd, “Z-scan measurement technique for non-Gaussian beams and arbitrary sample thicknesses,” Opt. Lett. **20**, 1821–1823 (1995). [CrossRef] [PubMed]

15. W. Zhao and P. Palffy-Muhoray, “Z-scan measurements of ?3 using top-hat beams,” Appl. Phys. Lett. **65,**673–675 (1994). [CrossRef]

16. R. DeSalvo, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E. W. Van Stryland, “Z-scan measurements of the anisotropy of nonlinear refraction and absorption in crystals,” Opt. Lett. **18**, 194 (1993). [CrossRef] [PubMed]

18. J. Liang, H. Zhao, and X. Zhou, “Polarization-dependence effects of refractive index change associated with photoisomerization investigated with Z-scan technique,” J. Appl. Phys. **101**, 013106 (2007). [CrossRef]

*with linearly polarized light was larger than those with circularly and elliptically polarized lights. They tried to explain it by decomposing the polarized light into two perpendicular linear polarized components, but they did not quantitatively analyze the effect of the RIC due to the appearance of perturbation of polarization ellipse and did not give the relationship between normalized transmittance and nonlinear phase shift as in Ref. 5*

_{Tp-v}**26**, 760–769 (1990). [CrossRef]

*χ*

_{xxyy}^{(3)}and

*χ*

_{xyyx}^{(3)}. For molecular orientation and nonresonant electronic nonlinearities, RIC is greatly dependent on polarization state due to the anisotropy of third-order nonlinear susceptibility [19]. By contraries, for thermal induced nonlinearity, excited state nonlinearity and electrostriction, RIC is non-dependent on polarization state. Therefore, in order to obtain more information of third-order nonlinear susceptibility tensor and understand the nonlinear mechanisms from RIC, elliptically and circularly polarized lights should be used in Z-scan measurements. We have demonstrated a method that combines the Z-scan technique with nonlinear ellipse rotation to measure third-order nonlinear susceptibility components [20

20. Z. B. Liu, X. Q. Yan, J. G. Tian, W. Y. Zhou, and W. P. Zang. “Nonlinear ellipse rotation modified Z -scan measurements of third-order nonlinear susceptibility tensor,” Opt.Exp. **15**, 13351 (2007). [CrossRef]

21. Z. B. Liu, X. Q. Yan, W. Y. Zhou, and J. G. Tian, “Evolutions of polarization and nonlinearities in an isotropic nonlinear medium,” Opt. Express **16**, 8144 (2008). [CrossRef] [PubMed]

_{2}for linearly, elliptically and circularly polarized incidence beams were measured and analyzed theoretically. The experimental results agree well with theoretical analyses. Further, the elliptically polarized light Z-scan was used to study intensity dependence of normalized transmittance.

## 2. Theory

*φ*

_{1}( -π/2≤

*φ*

_{1}</2) which is the angle between the linear polarization direction and the slow axis of the λ/4 plate, it can be converted into a polarized beam (

*φ*

_{1}=-π/2, 0 for linearly polarized beam,

*φ*

_{1}=±π/4 for circularly polarized beam, and others for elliptically polarized beam).

_{00}Gaussian beam of waist radius

*w*

_{0}traveling in the +

*z*direction, we can write

**as**

*E**w*

_{z}^{2}=

*w*

^{2}

_{0}(1 +

*z*

^{2}/

*z*

^{2}

_{0}) is the beam radius,

*R*(

*z*) =

*z*(1 +

*z*

_{0}

^{2}/

*z*

^{2}) is the radius of curvature of the wave front at

*Z*,

*z*

_{0}=

*κw*

^{2}

^{0}/2 is the diffraction length of the beam, and

*κ*= 2

*π*/

*λ*is the wave vector.

*E*→

_{0}(

*t*) denotes the radiation electric field vector at the focus and contains the temporal envelope of the laser pulse. The exp[-

*iϕ*(

*z*,

*t*)] term contains all the radically uniform phase variations.

*x*-axis, after passing the λ/4 plate, the electric field can be written as

*δ*

_{1}is the phase retardation,

*x*̂ and

*y*̂ are unit vectors.

*δ*

_{1}=π/2 for a λ/4 plate. The electric field vector of such a beam can always be decomposed into a linear combination of left- and right-hand circular components as [19]

*σ*̂

_{+}= (

*x*̂+

*iy*̂)√2 and

*σ*̂

_{-}= (

*x*̂-

*iy*̂)√2 are the circular-polarization unit vectors,

*E*

_{+}=(

*E*-

_{x}*iE*)/√2,

_{y}*E*

_{-}=(

*E*+

_{x}*iE*)/√2. The other two parameters can be expressed as

_{y}*e*= ∥

*E*

_{+}∣ -∣

*E*

_{-}∥/(∣

*E*

_{+}∣ + ∣

*E*

_{-}∣) , i.e.

*n*

_{0}is the linear refractive index,

*A*= 6

*χ*

_{xxyy}^{(3)}and

*B*= 6

*χ*

_{xyyx}^{(3)}. Substituting Eq. (4) into Eq. (5), we can rewrite Eq. (5) as follows:

*δ*

_{n±}is different for two components, the left- and right-hand circular components propagate with different phase velocities, thus, the polarization direction will rotate as the elliptically polarized beam propagates through the nonlinear medium. This is called as nonlinear ellipse rotation [19,3

3. 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]

*z*′ is the propagation depth in the sample. If nonlinear absorption of the medium is negligible, the phase shifts at the exit surface for the two circular components can be obtained by substituting Eq. (6) into Eq. (7) and solving the equations:

*ϕ*

_{+,0}(

*t*) and Δ

*ϕ*

_{-,0}(

*t*) are on-axis phase shift at the focus of left- and right-hand circular components, respectively. They are defined as

*L*=[1-exp(-

_{eff}*αL*)]/

*α*: is the effective length,

*L*is the sample length,

*α*is the linear absorption coefficient and Δ

*n*

_{±0}(

*t*) = 2

*π*[

*A*+ (1±sin

*δ*

_{1}sin2

*φ*

_{1})

*B*/2]∣

*E*

_{0}(

*t*)∣

^{2}/

*n*

_{0}are the on-axis RICs of the two circular components at the focal plane.

**26**, 760–769 (1990). [CrossRef]

*i*Δ

*ϕ*

_{±}(

*z*,

*r*,

*t*)] in Eq. (11) can be expanded as

*d*as the propagation distance in free space from the medium to the aperture plane, and

*g*=1 +

*d*/

*R*(

*z*), the other parameters in Eq. (13) are expressed as

*r*

_{a}is the radius of the aperture,

*S*= 1- exp(-2

*r*

^{2}

_{a}/

*w*

^{2}

_{a}) is the aperture linear transmittance,

*w*

_{a}is the beam radius at the aperture in the linear regime. Under the far-field condition of

*z*

_{0}≪

*d*, the normalized transmittance of Eq. (19) can be calculated as:

*Y*

_{a}=

*r*

_{a}/

*Dω*

_{0}is a dimensionless aperture radius,

*x*=

*z*/

*z*

_{0}is the dimensionless sample position, and

*D*=

*d*/

*z*

_{0}is the dimensionless distance from the sample to the aperture plane. From Eq. (21), it is seen that there is no nonlinear phase shifts coupling terms of left- and right- hand lights, which means the two circular components independently propagate and reach the detector. The transmittance at the detector plane is the weighted average of the two components. By using L’Hôpital’s rule, the normalized transmittance for closed-aperture (CA) case can be obtained from Eq. (20) as

*T*can be written as

_{CA}*ϕ*=

_{eff}*kπ*[2

*A*+ (1-sin

^{2}

*δ*

_{1}sin

^{2}2

*φ*

_{1})

*B*]∣

*E*

_{0}(

*t*)∣

^{2}

*L*/

_{eff}*n*

_{0}is effective nonlinear phase shift at the focus on the Z axis. If the first quarter-wave plate is taken away or the angle

*φ*

_{1}is set at 0 deg, above formula will return to the normal Z-scan transmittance formula in Ref. 5

**26**, 760–769 (1990). [CrossRef]

*A*and

*B*values from one experiment, one possible way is to abandon first-order approximation by taking more terms in Eqs. (26) and (20) as normalized transmittance fitting formula because the coefficients of high-order terms are non-correlation. According to Chen et al’s works [22

22. S. Q. Chen, Z. B. Liu, W. P. Zang, J. G. Tian, W. Y. Zhou, F. Song, and C. P. Zhang, “Study on Z-scan characteristics for a large nonlinear phase shift,” J. Opt. Soc. Am. B **22**, 1911 (1997). [CrossRef]

**26**, 760–769 (1990). [CrossRef]

*cw*situation) results can be extended to transient effects induced by pulsed radiation. For a Gaussian pulse, the average refractive index is

*n*

_{±,0}is the peak-on-axis index change at the focus.

## 3. Experimental result and discussions

*w*

_{0}of 18 μm was incident to a 1 mm quartz cell containing CS

_{2}. The on-axis peak intensity

*I*

_{0}was 5.23 GW/cm

^{2}. The extensively studied isotropic nonlinear medium, CS

_{2}, was chosen as the sample. CS

_{2}is transparent and has very small linear absorption coefficients

*α*

_{0}(<10

^{-3}cm

^{-1}) in the visible region [23

23. Z. B. Liu, Y. L. Liu, B. Zhang, W. Y. Zhou, J. G. Tian, W. P. Zang, and C. P. Zhang, “Nonlinear absorption and optical limiting properties of carbon disulfide in a short-wavelength region,” J. Opt. Soc. Am. B **24**, 1101 (2007). [CrossRef]

^{-11}cm/W, so we can ignore the nonlinear absorption in our experimental conditions. The physical mechanism leading to nonlinear refraction of CS

_{2}in subnansecond regime is molecular orientation, and the ratio of

*B*to

*A*should be 6 [19]. In order to make the reference light be synchronous with the signal light, we placed the beam splitter between the λ/4 plate and the lens. Polarization states could be changed by altering the λ/4 plate. These experiments were carried out at room temperature.

*T*of Z-scan is dependent on the angle

*φ*

_{1}, i.e., the ellipticity

*e*. Figure 2 gives the CA Z-scan curves of linear, circular, and elliptical polarizations of CS

_{2}. The angle

*φ*

_{1}was set 78 degree (i.e.

*e*=0.2126) to create an elliptically polarized light. It can be seen that the NLR is largest for a linearly polarized beam and smallest for a circularly polarized beam, which is consistent with Liang’s experimental result [18

18. J. Liang, H. Zhao, and X. Zhou, “Polarization-dependence effects of refractive index change associated with photoisomerization investigated with Z-scan technique,” J. Appl. Phys. **101**, 013106 (2007). [CrossRef]

*m*+

*n*=3) to experimental results, the theoretical fits are in good agreement with experimental results.

*n*=

*π*(2

*A*+

*B*)∣

*E*∣

^{2}/

*n*

_{0}[19], we can not distinguish the

*B*and

*A*values (only 2

*A*+

*B*is obtained) from linearly polarized light Z-scan. The (

*A*+

*B*/2) from the fit of linear polarization component was 9.232×10

^{-12}esu. For circular polarization case, from Eq. (5) the RIC equals to 2

*πA*∣

*E*∣

^{2}/

*n*

_{0}for both left- and right-hand lights, thus, only one value

*A*can be obtained from the fit. The

*A*from the fit in Fig. 2 is 2.241×10

^{-12}esu (i.e.

*χ*

_{xxyy}^{(3)}= 3.735×10

^{-13}esu ). Comparing the two cases, we could obtain

*B*= 1.398×10

^{-11}esu (i.e.

*χ*

_{xyyx}^{(3)}= 2.330×10

^{-12}esu). The ratio

*B*to

*A*is 6.240, which agrees very well with theoretical value for molecular orientation. The ratio of

*B*to

*A*is 6 for molecular orientation, 1 for nonresonant electronic response, and 0 for electrostriction [19].

*B*and

*A*values could be determined from Eq. (10). The fitting values of

*B*and

*A*for elliptically polarized case is 1.313×10

^{-11}and 2.182×10

^{-12}esu respectively (i.e.

*χ*

_{xyyx}^{(3)}=2.189×10

^{-12}esu, and

*χ*

_{xxyy}^{(3)}, =3.637×10

^{-13}esu). The ratio

*B*to

*A*was 6.019. The

*A*and

*B*values from these fits agree well with theoretical simulation [24

24. K. Kiyohara, K. Kamada, and K. Ohta, “Orientational and collision-induced contribution to third-order nonlinear optical response of liquid CS_{2},” J. Chem. Phys. **112**, 6338, (2000). [CrossRef]

3. 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]

26. R. Volle, V. Boucher, K. D. Dorkenoo, R. Chevalier, and X. N. Phu, “Local polarization state observation and third-order nonlinear susceptibility measurements by self-induced polarization state changes method,” Opt. Commun. **182**, 443 (2000). [CrossRef]

*χ*

^{(3)}= 2

*χ*

_{xxyy}^{(3)}+

*χ*

_{xyyx}^{(3)}) obtained from any polarized light experiment is identical. Indeed the polarization state of the light just changes the contribution of

*B*to nonlinear refractive index as shown in Eq. (6) since both polarization state and nonlinear refractive index are determined by angle

*φ*

_{1}.

*A*and

*B*values could be simultaneously obtained from single experiment by taking more terms in Eq. (26) during experimental data analysis. Furthermore the ratio of

*B*to

*A*can be used to determine the nonlinear mechanism. However the values of

*A*and

*B*can not be simultaneously obtained from circularly or linearly polarized light Z-scan no matter how many terms are taken for the transmittance fitting formula. Thus, circularly and linearly polarized light have to be used together in Z-scan experiments as in Ref. 20

20. Z. B. Liu, X. Q. Yan, J. G. Tian, W. Y. Zhou, and W. P. Zang. “Nonlinear ellipse rotation modified Z -scan measurements of third-order nonlinear susceptibility tensor,” Opt.Exp. **15**, 13351 (2007). [CrossRef]

*χ*

_{xxyy}^{(3)}and

*χ*

_{xyyx}^{(3)}. It can be easily understood from Eq. (21), the condition for determining

*A*and

*B*is that Δ

*ϕ*

_{+,0}(

*t*) and Δ

*ϕ*

_{-,0}(

*t*) should be simultaneously obtained from fitting and they must be different, only elliptically polarized light can satisfy this requirement. Compared with circularly and linearly polarized lights Z-scans, the advantage of elliptically polarized light Z-scan is that two nonlinear susceptibility components can be simultaneously determined in a single Z-scan measurement.

^{2}). The angle

*φ*

_{1}was set 78 deg for these experiments, and the linear transmittance

*S*is 15%. It is seen that the anti-symmetry of the peak and valley has been destroyed by large intensities. Blue lines are the fits using Eq. (20) (terms are taken to

*m*+

*n*≤30 in order to eliminate the errors from missing terms [22

22. S. Q. Chen, Z. B. Liu, W. P. Zang, J. G. Tian, W. Y. Zhou, F. Song, and C. P. Zhang, “Study on Z-scan characteristics for a large nonlinear phase shift,” J. Opt. Soc. Am. B **22**, 1911 (1997). [CrossRef]

*B*value is identical for all the intensities and the ratio

*B*to

*A*is 6.

*B*=1.372×10

^{-11}esu chosen for this fitting was from best fit result of Fig. 3a. The

*B*value is set to be identical for all the intensities because it is usually assumed to be independent of intensity in the χ

_{xyyx}

^{(3)}measurements [3

**12**, 507–509 (1964). [CrossRef]

*T*) as shown in Fig. 4(a). The experimental data do not match the fitting results for higher intensities. It can be seen that the peak-valley difference for fixed

_{p-v}*B*is not proportional to the light intensity but is reaching its saturation value. According to Eq. (27), the effective nonlinear phase shift at the focus is proportional to the intensity, so the Δ

*T*should also be proportional to the intensity as the green dash line shown, which has been widely used as a criterion to judge the nonlinear type [18

_{p-v}18. J. Liang, H. Zhao, and X. Zhou, “Polarization-dependence effects of refractive index change associated with photoisomerization investigated with Z-scan technique,” J. Appl. Phys. **101**, 013106 (2007). [CrossRef]

*T*and intensity means that the first-order approximation cannot be used for large nonlinear phase shift case. For large nonlinear phase shift case, although Δ

_{p-v}*T*~

_{p-v}*I*

_{0}may present a saturation, the nonlinear phase shift (so does the RIC) still increases with intensity. In order not to wrongly regard large nonlinear phase shift case as saturated nonlinear effect, the proportional relationship between Δ

*T*and intensity under first-order approximation could not simply treated as criteria to determine whether it is a Kerr nonlinear effect or not. In this case, more terms in Eqs. (20) and (26) should be taken to analyze the experimental data and get the nonlinear phase shifts, and then judge the nonlinear type of the medium.

_{p-v}*B*fitting and experimental results can be ignored. Moreover, during our experiments, no obvious scattering was observed. According to the literatures [7,25,27

27. R.A. Ganeev, A.I. Ryasnyansky, M. Suzuki, N. Ishizawa, M. Turu, S. Sakakibara, and H. Kuroda, “Nonlinear refraction in CS_{2},” Appl. Phys. B **78**, 433 (2004). [CrossRef]

^{2}. Since the intensities (<25 GW/cm

^{2}) used in our experiments are much smaller than the critical intensity, the discrepancy between fixed

*B*fitting and experimental results should not mainly arise from nonlinear absorption. No nonlinearity for the quartz cell was observed in our experiments. In addition, the self-defocusing effects of acoustic generation and heat accumulation can also be excluded, because the repetition rate of picosecond pulses in our experiments was too low to bring this type self-defocusing effects [27

27. R.A. Ganeev, A.I. Ryasnyansky, M. Suzuki, N. Ishizawa, M. Turu, S. Sakakibara, and H. Kuroda, “Nonlinear refraction in CS_{2},” Appl. Phys. B **78**, 433 (2004). [CrossRef]

*B*fitting and experimental results may be the temperature changes because of absorption. According to references [19,24

24. K. Kiyohara, K. Kamada, and K. Ohta, “Orientational and collision-induced contribution to third-order nonlinear optical response of liquid CS_{2},” J. Chem. Phys. **112**, 6338, (2000). [CrossRef]

*B*value is inverse to temperature. The temperature was higher for larger radiation intensity, so the

*B*value should be smaller for larger radiation intensity. In order to check whether the discrepancy could come from the decreasing of

*B*value, the fits with different

*B*were carried out. In the new fitting (terms in Eq. (20) is taken until

*m*+

*n*=30), the

*B*values are altered to get the best fit for different intensities. The criterion of best fit is that the peak-valley transmittance difference from fit is nearly identical to the experimental Δ

*T*, when the ratio of

_{p-v}*B*to

*A*was fixed at the theoretical value 6. The new fitting curves are shown as red dashed lines in Fig. 3, and the Δ

*T*and

_{p-v}*B*values obtained from the new fits are shown in Fig. 4. Comparing the two fitting results in Fig. 3, we find that the new fits are much better than the former ones (fixed

*B*fits), and the

*B*value varies with intensity. The

*B*value used in new fits decreases with the intensity as shown in Fig.4b, which is consistent with theoretical prediction [19,24

24. K. Kiyohara, K. Kamada, and K. Ohta, “Orientational and collision-induced contribution to third-order nonlinear optical response of liquid CS_{2},” J. Chem. Phys. **112**, 6338, (2000). [CrossRef]

*B*value is just qualitative, from which we can conclude that the

*B*value decreases with intensity. If one wants to quantitatively determine the relationship between

*B*value and intensity, the temperature change of the liquid should be studied fully.

## 4. Conclusion

_{2}medium. From the analysis, we found that the elliptically polarized light Z-scan could be used to simultaneously measure the two third-order nonlinear susceptibility components

*χ*

_{xyyx}

^{(3)}and

*χ*

_{xxyy}

^{(3)}. Furthermore, the ratio of

*B*to

*A*was determined to be about 6, which agrees very well with theoretical value for molecular orientation. The normalized transmittance formulas of elliptically polarized light Z-scan were also examined for large nonlinear phase shifts.

## Acknowledgments

## References and links

1. | I. Fuks-Janczarek, B. Sahraoui, I. V. Kityk, and J. Berdowski, “Electronic and nuclear contributions to the third-order optical susceptibility,” Opt. Commun. |

2. | G. Boudebs, M. Chis, and J. P. Bourdin, “Third-order susceptibility measurements by nonlinear image processing,” J. Opt. Soc. Am. B |

3. | P. D. Maker, R. W. Terhune, and C. M. Savage, “Intensity-dependent changes in the refractive index of liquids,” Phys. Rev. Lett. |

4. | M. Lefkir and G. Rivoire, “Influence of transverse effects on measurement of third-order nonlinear susceptibility by self-induced polarization state changes,” J. Opt. Soc. Am. B |

5. | M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. |

6. | P. B. Chapple, J. Straromlynska, J. A. Hermann, T. J. Mckay, and R. G. McDuff, “Single-beam Z-scan: measurement techniques and analysis,” J. Nonlinear Opt. Phys. Mater. |

7. | E. W. Van Stryland and M. Sheik-Bahae, “Z-scan Measurements of Optical Nonlinearities,” in |

8. | J. A. Hermann and R. G. McDuff, “Analysis of spatial scanning with thick optically nonlinear media,” J. Opt. Soc. Am. B |

9. | J. G. Tian, W. P. Zang, C. Z. Zhang, and G. Y. Zhang, “Analysis of beam propagation in thick nonlinear media,” Appl. Opt. |

10. | T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. Van Stryland, “Eclipsing Z- Scan Measurement of λ/104 Wavefront Distortion,” Opt. Lett. |

11. | J. Wang, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E. W. Van Stryland, “Time-Resolved Z-Scan Measurements of Optical Nonlinearities,” J. Opt. Soc. Am. B |

12. | L. Demenicis, A. S. L. Gomes, D. V. Petrov, C.B. de Araújo, C. P. de Melo, C. G. dos Santos, and R. Souto-Maior, “Saturation effects in the nonlinear-optical susceptibility of poly(3-hexadecylthiophene),” J. Opt. Soc. Am. B |

13. | M. Sheik-Bahae, J. Wang, J.R. DeSalvo, D. J. Hagan, and E. W. Van Stryland, “Measurement of Nondegenerate Nonlinearities using a 2-Color Z-Scan,” Opt. Lett. |

14. | R. Bridges, G. Fischer, and R. Boyd, “Z-scan measurement technique for non-Gaussian beams and arbitrary sample thicknesses,” Opt. Lett. |

15. | W. Zhao and P. Palffy-Muhoray, “Z-scan measurements of ?3 using top-hat beams,” Appl. Phys. Lett. |

16. | R. DeSalvo, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E. W. Van Stryland, “Z-scan measurements of the anisotropy of nonlinear refraction and absorption in crystals,” Opt. Lett. |

17. | S. J. Wagner, J. Meier, A. S. Helmy, J. S. Aitchison, D. Modotto, M. Sorel, and D. C. Hutchings, “Polarization-Dependent Nonlinear Refraction in GaAs/AlAs Superlattice Waveguides,” in |

18. | J. Liang, H. Zhao, and X. Zhou, “Polarization-dependence effects of refractive index change associated with photoisomerization investigated with Z-scan technique,” J. Appl. Phys. |

19. | R. W. Boyd, |

20. | Z. B. Liu, X. Q. Yan, J. G. Tian, W. Y. Zhou, and W. P. Zang. “Nonlinear ellipse rotation modified Z -scan measurements of third-order nonlinear susceptibility tensor,” Opt.Exp. |

21. | Z. B. Liu, X. Q. Yan, W. Y. Zhou, and J. G. Tian, “Evolutions of polarization and nonlinearities in an isotropic nonlinear medium,” Opt. Express |

22. | S. Q. Chen, Z. B. Liu, W. P. Zang, J. G. Tian, W. Y. Zhou, F. Song, and C. P. Zhang, “Study on Z-scan characteristics for a large nonlinear phase shift,” J. Opt. Soc. Am. B |

23. | Z. B. Liu, Y. L. Liu, B. Zhang, W. Y. Zhou, J. G. Tian, W. P. Zang, and C. P. Zhang, “Nonlinear absorption and optical limiting properties of carbon disulfide in a short-wavelength region,” J. Opt. Soc. Am. B |

24. | K. Kiyohara, K. Kamada, and K. Ohta, “Orientational and collision-induced contribution to third-order nonlinear optical response of liquid CS |

25. | R. L. Sutherland, |

26. | R. Volle, V. Boucher, K. D. Dorkenoo, R. Chevalier, and X. N. Phu, “Local polarization state observation and third-order nonlinear susceptibility measurements by self-induced polarization state changes method,” Opt. Commun. |

27. | R.A. Ganeev, A.I. Ryasnyansky, M. Suzuki, N. Ishizawa, M. Turu, S. Sakakibara, and H. Kuroda, “Nonlinear refraction in CS |

28. | |

29. | B. Gu, J. Chen, Y. Fan, J. Ding, and H. Wang, “Theory of Gaussian beam Z scan with simultaneous third-and fifth-order nonlinear refraction based on a Gaussian decomposition method,” J. Opt. Soc. Am. B |

30. | R. A. Ganeev, M. Baba, M. Morita, A. I Ryasnyansky, M. Suzuki, M. Turu, and H. Kuroda, “Fifth-order optical nonlinearity of pseudoisocyanine solution at 529 nm,” J. Opt. A: Pure Appl. Opt. |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(190.3270) Nonlinear optics : Kerr effect

(190.4710) Nonlinear optics : Optical nonlinearities in organic materials

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: February 4, 2009

Revised Manuscript: March 15, 2009

Manuscript Accepted: March 15, 2009

Published: April 2, 2009

**Citation**

Xiao-Qing Yan, Zhi-Bo Liu, Xiao-Liang Zhang, Wen-Yuan Zhou, and Jian-Guo Tian, "Polarization dependence of Z-scan measurement: theory and experiment," Opt. Express **17**, 6397-6406 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-8-6397

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