## Nonlinear ellipse rotation modified Z-scan measurements of third-order nonlinear susceptibility tensor

Optics Express, Vol. 15, Issue 20, pp. 13351-13359 (2007)

http://dx.doi.org/10.1364/OE.15.013351

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### 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, CS_{2}.

© 2007 Optical Society of America

## 1. Introduction

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]

*χ*

^{(3)}

*, and its history goes back to the classic work of Maker*

_{xyyx}*et al*. [4

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]

*χ*

^{(3)}in a single experimental setup for isotropic materials. It is not necessary for the beams employed in the experiment to be perfect TEM

_{00}Gaussian modes as long as they are well-characterized. The time dependence of the nonlinearity can be studied readily. However, the disadvantages of this technique include the fact that only the modulus of

*χ*

^{(3)}(

*i.e*., |χ

^{(3)}|) can generally be measured. A far more complicated experimental apparatus is needed; in general, the technique must be supplemented with another measurement to extract the real and imaginary parts of

*χ*

^{(3)}. Hence, different measurements are usually required to unravel the underlying physical mechanism by varying parameters such as irradiance and polarization state, or even the use of different measurement methods [1].

*χ*

^{(3)}

*,*

_{xxyy}*χ*

^{(3)}

*, and*

_{xyxy}*χ*

^{(3)}

*[9], which have not been taken into account in Z-scan measurements as yet. NER and other nonlinear polarization dynamics are owing to the existence of*

_{xyyx}*χ*

^{(3)}

*. Moreover, polarization-dependent NLR can be observed in the isotropic medium based on the effect of*

_{xyyx}*χ*

^{(3)}

*[9]. Therefore, it is expected that different tensor components of*

_{xyyx}*χ*

^{(3)}can be obtained by Z-scan measurements using linearly, circularly, and elliptically polarized light. In this work, the polarization dependence of NLR in CS

_{2}is studied. This molecule has been studied thoroughly in NER and presents large molecular reorientation nonlinearities in the subnanosecond regime. We combine the Z-scan technique and NER to carry out a sensitive and simple measurement of

*χ*

^{(3)}

*. The theoretical analysis is made and the obtained transmittance formulae allow direct estimation of the component*

_{xyyx}*χ*

^{(3)}

*.*

_{xyyx}## 2. NER modified Z-scan technique

_{00}beam for absolute measurements is required. Sample distortions or wedges, or a tilting of the sample during translation, can cause the beam to walk off the far-field aperture. This produces unwanted fluctuations in the detected signal. Even if these are kept under control, beam jitter will produce the same effect. The technique cannot be used to measure off-diagonal elements of the susceptibility tensor except when a second nondegenerate frequency beam is employed.

_{+}and E

_{-}, respectively. Without NLA or scattering, when the elliptically polarized beam propagates through the nonlinear medium, the orientation of the polarization ellipse will rotate an angle

*θ*as a function of input intensity under a given sample length

*d*. The beam going through the nonlinear medium passes through the second quarter-wave plate oriented crosswise to the first and then a polarizer oriented for extinction of the beam in the absence of the nonlinear sample. NER can determine only one component

*χ*

^{(3)}

*, but not the total third-order nonlinear susceptibility*

_{xyyx}*χ*

^{(3)}.

*θ*of the polarization ellipse depends upon the intensity of the input beam, the tight-focus geometry in Z-scan can be also used in NER. It is possible to combine the advantages of Z-scan and NER to measure the component

*χ*

^{(3)}

*simply and sensitively. Figure 1(a) gives the configuration of the NER modified Z-scan method, which is the same as that of an open-aperture Z-scan except that two paralleled polarizers and two crossed quarter-wave plates are used. When the sample is far away from focus, the beam irradiance is low and NLR is negligible; the polarization state remains unchanged for incident and transmitted beams. Hence, all of the transmitted irradiance through the sample is collected into detector D2, and the transmittance [D2/D1, in Fig. 1(a)] remains relatively constant. As the sample is brought closer into focus, the beam irradiance increases, thus leading to the rotation of the polarization ellipse. The rotation of the polarization ellipse permits only part of the transmitted irradiance to pass the second polarizer, and a decrease in the measured transmittance occurs. Such Z-scan traces with NER are expected to be symmetric with respect to the focus (*

_{xyyx}*z*= 0); they have minimum transmittance at focus. The coefficient of

*χ*

^{(3)}

*can be calculated easily from such transmittance.*

_{xyyx}## 3. Theory

*z*-axis. The electric field can be written as

*E*=

_{lin}*E*(

_{0}*r*,

*z*)exp[

*i*(

*kz*-

*ωt*)], where

*k*=

*nω*/

*c*is the wave vector,

*ω*is the frequency of light, and

*n*is the linear refractive index of the medium. We assume a stationary regime, and thus the amplitude

*E*

_{0}(

*r*,

*z*) does not depend on the time

*t*. When the linearly polarized beam passes the λ/4 plate with angle α [with -

*π*/2≤

*α*≤

*π*/2 being the angle between the linear polarization direction and the λ/4 plate slow axis—see Fig. 1(b)], it can be converted into an elliptically polarized beam. The electric field

*E*of such an elliptically polarized beam can always be decomposed into a linear combination of the

*x*- and

*y*-direction components (

*E*and

_{x}*E*), or left- and right-hand circular components (

_{y}*E*

_{+}and

*E*

_{−}) with the unitary transformation

*E*= (

_{+}*E*-

_{x}*iE*)√2 and

_{y}*E*

_{−}=(

*E*+

_{x}*iE*)/√2. If we define the λ/4 plate slow axis as the

_{y}*x*-axis, the electric field of

*x*- and

*y*-direction components can be written as

*π*/2 for the λ/4 plate.

*χ*

^{(3)}

*(ω=ω+ω−ω) , the condition of intrinsic permutation symmetry requires that*

_{ijkl}*χ*

^{(3)}

*be equal to*

_{xxyy}*χ*

^{(3)}

*. Hence, there are only two independent elements of the susceptibility tensor describing the NLR of an isotropic medium. Following the notation of Maker*

_{xyxy}*et al*. [4

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]

**P**

^{NL}can be written as

*A*= 6

*χ*

^{(3)}

*and*

_{xxyy}*B*= 6

*χ*

^{(3)}

*. Hence, the total refractive indexes of two circular components is given by [9]*

_{xyyx}*δn*

_{±}is different for two circular components and depends upon ellipticity

*e*, where

*e*=||

*E*

_{+}|-|

*E*

_{−}||/(|

*E*

_{+}| + |

*E*

_{−}|) . First, we consider two specific cases. One case is for a circularly polarized beam with

*e*=1; this means that only one of two circular components is present. Thus, the change of the refractive index can be given by

*δn*=(2

*π*/

*n*

_{0})

*A*|

*E*|

^{2}, which clearly depends on A but not B. The NLR coefficient

*n*

_{2}of the total beam should reach a minimum in this case. Another case is for a linearly polarized beam. Since a linearly polarized beam is a combination of equal amounts of left-and right-hand circular components (i.e. |

*E*|

^{2}= 2|

*E*

_{+}|

^{2}=2|

*E*

_{−}|

^{2}), the change of the refractive index can be given by

*δn*= (2

*π*/

*n*

_{0})(

*A*+

*B*/2)|

*E*|

^{2}, and

*n*

_{2}reaches a maximum. For the case of arbitrary ellipticity,

*n*

_{2}can be written as

*q*= (1-

*e*)/(1+

*e*), and

*n*

_{2cir}and

*n*

_{2lin}are the NLR coefficients in the case of circularly and linearly polarized beams, respectively.

*δn*

_{±}and cause the rotation of the polarization ellipse of the transmitted wave [see Fig. 1(c)]. The angle of rotation can be written as

*Q*= -(2

*πω*/

*cn*)

*Bd*sinα cosα and

*d*is the sample length. In new

*x*́-

*y*́ coordinates taken along the major and minor axes of the ellipse, we can write the electric field

*E*as

*x*

*y*

*x*

*x*̂cos

*θ*-

*y*̂sin

*θ*and

*y*

*x*̂sin

*θ*+

*y*̂cos

*θ*. Therefore, by transforming the electric field vector from

*x*́-

*y*́ coordinates to

*x*-

*y*coordinates,

**E**can be written as

*φ*(-

*π*/2≤

*φ*≤

*π*/2) relative to the

*x*-axis, the output electric field

**E**

_{out}through the analyzer can be written as

_{00}Gaussian beam of beam waist radius

*w*

_{0}traveling in the +

*z*direction, we can write the input

*E*as

*E*

_{00}is the on-axis electric field at focus,

*w*

_{z}^{2}=

*w*

_{0}

^{2}(1 +

*z*

^{2}/

*z*

_{0}

^{2}) is the beam radius,

*R*=

_{z}*z*(1 +

*z*

_{0}

^{2}/

*z*

^{2}) is the radius of curvature of the wavefront at

*z*, and

*z*

_{0}=

*kw*

_{0}

^{2}/2 is the diffraction length of the beam. For the spatial Gaussian beam, the influence of transverse effects on self-induced polarization changes must be considered [10

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]

**E**

_{out}up to infinity:

*z*as follows:

*ε*

_{0}is the permittivity of vacuum and

*P*=

*E*

_{00}

^{2}

*w*

^{2}

_{0}/

*w*

_{z}^{2}. In the case of linear propagation (i.e.,

*θ*= 0), the transmitted power can be written as

*i.e.*,

*α*=

*φ*and Δ =

*π*/2. The normalized transmittance

*T*(

*z*) can be written as

*T*(

*z*) can be written as

*R*= 1/(sin

^{2}

*α*sin

^{2}

*φ*+ cos

^{2}

*α*cos

^{2}

*φ*).

*θ*(

*t*)〉:

## 4. Experimental results and discussions

_{2}This molecule has been studied extensively by many experimental methods and now is widely used as a reference sample. As the nonlinearities of CS

_{2}mainly rely on the molecular reorientation effect in the subnanosecond regime, the high ratio of

*B*to

*A*attains 6 [9] and the NLR coefficient

*n*

_{2}=3.4×10

^{-18}m

^{2}/W for linearly polarized light [6

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]

*w*

_{0}of 22 μm was incident to a 1 mm cell containing CS

_{2}. The on-axis peak intensity

*I*

_{0}was 5.93GW/cm

^{2}. To keep the intensities of input beams fixed for different polarization states, linear, elliptical, and circular polarizations were realized by altering only the angle between the first polarizer and the λ/4 plate with

*α*= 0,

*α*= -22.5°, and

*α*= -45°,

*i.e.*, the ellipticity

*e*= 0,

*e*= 0.4142 , and

*e*= 1, respectively. The negative sign of

*α*implies that the beam is a left-handed elliptical polarization after passing through the λ/4 plate.

*n*

_{2}, fitted by using the theoretical model of Ref[6

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]

^{-18}, 1.6×10

^{-18}, and 0.78×10

^{-18}m

^{2}/W for linear, circular, and elliptical polarization, respectively. From Eq. (3), one can get the ratio of

*n*

_{2lin}to

*n*

_{2cir}to be 1 +

*B*/2

*A*. For molecular reorientation nonlinearities,

*n*

_{2lin}/

*n*

_{2cir}=4 owing to

*B*/

*A*= 6. It is obvious that

*n*

_{2lin}/

*n*

_{2cir}= 3.8 obtained experimentally agrees well with the theoretical one. Note that the change of NLR for different polarization states depends upon

*B*but not

*A*. It is identical to NER in which the rotation of the elliptical axis is caused only by

*B*.

*A*= 2.16×10

^{-20}m

^{2}/V

^{2}, since the changes of refractive index and absorption only depend on

*A*in the case of circular polarization. And then,

*B*can be determined to be 12.8×10

^{-20}m

^{2}/V

^{2}by the Z-scan experimental results of linear polarization or elliptical polarization, which is 5.9 times as large as that of

*A*. Therefore,

*B/A*is closer to that of theoretical analysis.

*B*directly, we used the experimental setup combining Z-scan and NER as shown in Fig. 1(a), and two paralleled polarizers and two crossed quarter-wave plates were additionally used. The circular symbols in Fig. 3 represent the experimental results of the Z-scan. Using Eq. (14) to fit the experimental data, we obtain coefficient

*B*to be 12.6×10

^{-20}m

^{2}/V

^{2}, which is identical to the results of closed aperture Z-scan measurements within errors. If the second λ/4 plate is removed and other experimental conditions are kept unchanged, the transmittance change of the polarization ellipse through the analyzer due to the rotation of axis position can be observed directly. The valley of the Z-scan curve has a larger magnitude than that with the second λ/4 plate, indicating that the removal of a λ/4 plate can enhance the sensitivity of Z-scan measurements.

*φ*are shown in Fig. 4(a) for linear output at low input power (

*I*

_{0}< 10

^{6}W/cm

^{2}) and nonlinear output at high input power (

*I*

_{0}= 5.93GW/cm

^{2}). A 19° rotation of the polarization ellipse at nonlinear output relative to that of linear output can be observed, while no obvious change of ellipticity occurs. Figure 4(b) gives the ratio of transmitted power at nonlinear output to that of linear output. The solid line is the theoretical fit with

*B*= 12.6×10

^{-20}m

^{2}/V

^{2}, which is easily obtained by assuming

*z*= 0 in Eq. (15). In other words, the ratio in Fig. 4(b) also shows the valley change of the NER-modified Z-scan without the second λ/4 plate as the analyzer rotates. Therefore, if we change the orientation of the analyzer

*φ*, Z-scan curves will give different profiles as shown in Fig. 5. First,

*φ*=90° means that the analyzer is along the minor axis of the polarization ellipse; hence, the rotation of the polarization ellipse causes the increase of transmitted power through the analyzer, and the Z-scan curve exhibits a peak structure as shown in Fig. 5(a). For

*φ*=80° and 76°, the normalized transmittance first decreases and then increases as the sample moves toward focus [see Figs. 5(b) and 5(c)]. The decrease of normalized transmittance is caused by the relative rotation of the analyzer to the minor axis of the polarization ellipse and is terminated while the analyzer is along the minor axis. Thereafter, the polarization ellipse continues to rotate as the sample moves continually, since the maximum rotation angle of polarization ellipse attains 19° (80° + 19° = 99° and 76° + 19° = 95° are larger than 90°). Therefore, after the analyzer reaches the minor axis, the analyzer will be away from the minor axis as the input intensity further increases, leading to the increase of transmittance. In the case of

*φ*= 55° (55° + 19° < 90°), the Z-scan curve exhibits a valley structure as shown in Fig. 5(d), since the analyzer rotates towards but does not reach the minor axis of the polarization ellipse.

*χ*

^{(3)}

_{xyyx}, the curves of the NER modified Z-scan are similar to those of an open-aperture Z-scan with multiphoton absorption or saturable absorption. Furthermore, the experimental setup of the NER modified Z-scan is also similar to an open-aperture Z-scan and has less strict conditions on the beam profile than a closed-aperture Z-scan. This is because the nonlinear effect measured by an open-aperture Z-scan depends upon the amplitude change but not the phase distortion of the beam.

## 5. Conclusion

## Acknowledgments

## References and links

1. | R. L. Sutherland, ed., |

2. | S. R. Friberg and P. W. Smith, “Nonlinear optical glasses for ultrafast optical switches,” IEEE J. Quantum Electron. |

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

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

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. |

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. |

7. | W. Zhao and P. Palffy-Muhoray, “Z-scan technique using top-hat beams,” Appl. Phys. Lett. |

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. |

9. | R. W. Boyd, |

10. | A. J. Van Wonderen, “Influence of transverse effects on self-induced polarization changes in an isotropic Kerr medium,” J. Opt. Soc. Am. B |

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 |

12. | M. Sheik-Bahae and M. P. Hasselbeck, “Third-order optical nonlinearities,” in |

**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: July 23, 2007

Revised Manuscript: September 4, 2007

Manuscript Accepted: September 4, 2007

Published: September 28, 2007

**Citation**

Zhi-Bo Liu, Xiao-Qing Yan, Jian-Guo Tian, Wen-Yuan Zhou, and Wei-Ping Zang, "Nonlinear ellipse rotation modified Z-scan measurements of third-order nonlinear susceptibility tensor," Opt. Express **15**, 13351-13359 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-20-13351

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### References

- R. L. Sutherland, ed., Handbook of Nonlinear Optics (Marcel Dekker, New York, 1996).
- S. R. Friberg and P. W. Smith, "Nonlinear optical glasses for ultrafast optical switches," IEEE J. Quantum Electron. QE-23, 2089-2094 (1987). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- W. Zhao and P. Palffy-Muhoray, "Z-scan technique using top-hat beams," Appl. Phys. Lett. 63, 1613-1615 (1993). [CrossRef]
- 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]
- R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic Press, San Diego, 2003).
- 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]
- 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]
- 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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