## Influence of the photoinduced focal length of a thin nonlinear material in the Z-scan technique

Optics Express, Vol. 15, Issue 5, pp. 2517-2529 (2007)

http://dx.doi.org/10.1364/OE.15.002517

Acrobat PDF (238 KB)

### Abstract

In this paper the response purely refractive of a thin nonlinear material, in the z-scan technique experiment, is modeled as a lens with a focal length that is a function of some integer power of the incident beam radius. We demonstrate that different functional dependences of the photoinduced lens of a thin nonlinear material give typical z-scan curves with special features. The analysis is based on the propagation of Gaussian beams in the approximation of thin lens and small distortion for the nonlinear sample. We obtain that the position of the peak and valley, the transmittance near the focus and the transmittance far from the Rayleigh range depend on the functional dependence of the focal length. Special values of the power reproduce the results obtained for some materials under cw excitation.

© 2007 Optical Society of America

## 1. Introduction

*z*a characteristic z-scan curve can be obtained. The magnitude and sign of the nonlinearity can be evaluated from the difference between the maximum and minimum transmittance and the shape of the curve, respectively.

1. M. Sheik Bahae, A. A. Said, and E. W. Van Stryland, “High sensitivity single beam n_{2} measurements,” Opt. Lett. **14**, 955–957 (1989). [CrossRef]

2. M. Sheik-Bahae, A. A: Said, T. Wei, D. Hagan, and E. W. Van Stryland, Sensitive measurement of Optical Nonlinearities using a single Beam,” IEEE J. Quantum Electron. **26**, 760–769 (1990). [CrossRef]

3. T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. Van Stryland, “Eclipsing Z-scan measurement of *λ*/10 wavefront distortion,” Opt. Lett. **19**, 317–319 (1994). [CrossRef] [PubMed]

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

5. H. Ma, A. S. L. Gomes, and C. B. de Araujo, “Measurement of nondegenerate optical nonlinearity using a two-color single beam method,” Appl. Phys. Lett. **59**, 2666 (1991). [CrossRef]

6. D. V. Petrov, A. S. L. Gomes, and C. B. de Araujo, “Reflection Z-scan technique for measurements of optical properties surfaces,” Appl. Phys. Lett. **65**, 1067 (1994). [CrossRef]

*et al*., [7

7. P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. Mckay, and R. G. McDuff, “Single-beam z-scan: measurement techniques and analysis,” J. Non Opt. Phys. Mat. **6**, 251–293 (1997). [CrossRef]

8. L. C. Oliveira and S. C. Zilio, “Single beam time-resolved Z-scan measurements of slow absorbers,” Appl. Phys. Lett. **65**, 2121–2123 (1994). [CrossRef]

9. S. J. Sheldon, L. V Knight, and J. M. Thorne, “Laser-induced thermal lens effect: a new theoretical model,” Appl. Opt. **21**, 1663–1669 (1982). [CrossRef] [PubMed]

10. L. Pálfalvi and J. Hebling “Z-scan study of the thermo-optical effect,” Appl. Phys. B **78**, 775–780 (2004) [CrossRef]

11. B. Gu, X. C. Peng, T. Jia, J. P. Ding, J. L. He, and H. T. Wang, “Determinations of third- and fifth-order nonlinearities by the use of the top-hat-beam Z scan: theory and experiment,” J. Opt. Soc. Am. B **22**, 446–452 (2005). [CrossRef]

*m*of the incident beam radius. Under this assumption, a simple model based on the propagation of a Gaussian beam, in the small phase distortion approximation, through a thin sample is analyzed. Obtaining the normalized transmittance, at the far field, which features depend on the focal length of the photoinduced lens in the nonlinear media. The weak lens limit case is initially analyzed in order to obtain analytic formulas for the peak valley position and transmittance difference as functions of

*m*.

## 2. Model

*z*=0 we know the beam waist

*w*

_{0}of the Gaussian beam used to implement the z-scan technique; the thin nonlinear sample can be modeled as a thin lens of focal length

*F*located at a distance z, further that the photodetector, with a small aperture, is located at a distance

*L*(see Fig. 1), and that the different elements of the optical set up does not change the Gaussian distribution, then we can describe the propagation of the beam using the ABCD law.

*L*≫

*z*, where

_{0}*z*is the Rayleigh distance given by

_{0}*z*=

_{0}*πw*/

^{2}_{0}*λ*, with

*λ*the wavelength of the beam. Then it is possible to obtain the normalized transmittance of the z-scan experiment as [12]:

*F*has been assumed. In the following we going to obtain the form of

*F*for a Kerr media of thickness

*d*, with refractive index

*n*and

_{0}*n*are the linear and nonlinear refractive index, respectively. Considering that this sample is illuminated by a Gaussian beam, with intensity

_{2}*P*is the total power and

*w(z)*is the beam radius,

*r*is the radial coordinate.

13. H. Kogelnik and T. Li, “Laser beams and Resonators,” Appl. Opt. **5**, 1550–1567 (1966). [CrossRef] [PubMed]

*f*= -

*l*/

*C*. Then considering a thin sample, i.e. when

*d*tends to 0, the focal length of the Kerr media is given by

14. J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples” J. Appl. Phys. **36**, 3–8 (1965). [CrossRef]

*P*is the absorbed power, (∂

_{abs}*n*/∂

*T*) is the change of refractive index with the temperature. In this case the focal length depends on the second power of the beam radius.

_{T}*a*is a constant with the adequate units, it can have parameters of the material, and

_{m}*m*is an integer number.

*m*, however, some materials can exhibit a nonlinear response than can be probably modeled as the sum of more than one dependence of

*w(z)*on

*m*[15

15. P. A. Márquez Aguilar, J. J. Sánchez Mondragón, S. Stepanov, and G. Bloch, “Z-scan experiments with cubic photorefractive crystal Bi_{12}Ti_{20},” Opt. Commun. **118**, 165–174 (1995). [CrossRef]

## 3. Weak lens approximation

*F*≥

*z*, approximation that we call weak lens, obtaining the following expression:

_{0}, z*F*=

_{0m}*a*is the shortest focal length of the photoinduced lens. From this expression it is possible to calculate the position of the peak and valley of the z-scan curve, giving the following relation

_{m}w^{m}_{0}*k*=

*2π*/

*λ*.

*m*to be larger than 1 in order to obtain real positions and transmittances, however in Eq. (13) the value

*m*=

*1*produces also a nonconstant transmittance with

*z*that no presents a peak or valley. Note that

*Δz*

_{p-ν}and

*ΔT*

_{p-ν}depend in a complicated way on the value of

*m*, besides

*ΔT*

_{p-ν}is proportional to the inverse of

*w*

_{0}^{m-2}.

*m*. The parameters were adjusted to have the same value of

*ΔT*

_{p-ν}in order to see the main features of the curves for different values of

*m*. As it can see the curves follow the typical shape of a z-scan curve, except for

*m*=1: a prefocal minimum and a post focal maximum, for a positive nonlinearity (the opposite for the negative one), located in a symmetric position with respect to

*z*=0 and similar amplitude with respect to 1. However, the curves present differences in: the peak and valley position, the slope of the linear part (near the waist of the beam) and the decay or growing of the transmittance in the wings (far from the waist of the beam).

- a) when |
*z*|≪*z*, in this case the transmittance takes on the following form:_{0}that represents a linear behavior with slope*2*/*F*, that means that depends inversely on_{0m}*w*._{0}^{m} - b) When |
*z*|≪*z*, in this case the normalized transmittance takes the following form:_{0}

*z*

^{m-1}. Then depending on the value of

*m*, the normalized transmittance reaches the peak or valley in a faster or slower way.

*m*will determine the main features of the z-curve because it define the separation between the peak and valley, the dependence of

*ΔT*

_{p-ν}with the beam waist and the dependence of the normalized transmittance in the wings with

*z*. Next we present in detail some special values of

*m*. Note that any real value can be used, however we are going to restrict to only integers.

### 3.1 Special case m=4.

*m*we obtain the following relations:

*F*=

_{04}*a*.

_{4}w^{4}_{0}*Δz*

_{p-ν}agrees very well with that reported in the small distortion approximation of a sample with a Kerr nonlinearity [16

16. M. Sheik-Bahae, A. A. Said, D. Hagan, M. J. Soileau, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. **38**, 1228–1235 (1991). [CrossRef]

*z*. It is important to note that

^{3}*ΔT*

_{p-ν}∝

*w*

_{0}^{-2}, for this value of

*m*. Then changing the beam waist and using the same sample and incident power an inverse quadratic change of

*ΔT*

_{p-ν}must be obtained, see Fig. 3.

### 3.2 Special case m=2

*F02*=

*a*.

_{2}w^{2}_{0}*ΔZ*

_{p-ν}increase with respect to the case of

*m*=

*4*, in fact this value coincide with references [18

18. C. Hu and J. R. Whinnery, “New thermo optical measurement method and comparison with other methods,” Appl. Opt. **12**, 72–79 (1973). [CrossRef] [PubMed]

19. F. L. S. Cuppo, A. M. F. Neto, S. L. Gómez, and P. Palffy-Muhoray, “Termal-lens model compared with the
Sheik-Bahae formalism in interpreting Z-scan experiments on lyotropic liquid crystals,” J. Opt. Soc. Am. B **19**, 1342–1348 (2002). [CrossRef]

*z*.

*ΔT*

_{p-ν}does not depend on the beam waist, then the z-scan curves with different lenses, keeping all the other parameters constant, must have the same transmittance difference, see Fig. 4. This fact represent a significant difference with respect to the case

*m*=

*4*.

### 3.3 Special case m=3

*F*

_{03}=

*a*

_{3}

*w*

_{0}

^{3}.

*m*,

*Δz*

_{p-ν}is smaller than that obtained for

*m*=2 and larger than that for

*m*=4. The transmittance in the wings follows a dependence inverse with

*z*. The dependence of

^{2}*ΔT*

_{p-ν}is inverse linear with

*w*, see Fig. 5.

_{0}### 3.4 Special case m = 1

*w*. In Fig. 6 we plot the normalized transmittance for different beam waists. Note that the curves continue being very symmetric.

_{0}## 4. Complete formula

*F*is of the same order or smaller than

_{0m}*z*. In this case is not possible to obtain formulas for the peak-valley position and transmittance difference, as in the weak approximation. In the following analysis we restrict the maximum value of the normalized transmittance to 5 in all the curves, greater values means phase differences on axis greater than 2

_{0}*π*, then light from different radial distances of the beam could interfere losing the beam its Gaussian distribution.

*m*present a sharp peak and a broad valley. The peak moves to the right in the case of a positive photoinduced lens and the opposite for a negative one. The valley moves to the position

*z*=

*0*. As consequence,

*ΔZ*

_{p-ν}grew as

*F*decreased. In general this case was characterized by asymmetric curves.

_{0m}*m*continues determining some features of the z-scan curves, we present typical results obtained for different values of

*m*. For

*m*= 4, and different values of the ratio

*F*/

_{0m}*z*, see Fig.07, we can see that the peak is very sharp compared with the valley. The normalized transmittance in the valley, in some cases, almost reached the value of zero and it was located at

_{0}*z*=0. The normalized transmittances in the wings almost reach the value of one. Note that the same behavior for the z-scan curves was reported in [20

20. C. H. Kwak, Y. L. Lee, and S. G. Kim, “Analysis of asymmetric Z-scan measurement for large optical nonlinearities in an amorphous As_{2}S_{3} thin film,” J. Opt. Soc. Am. B **16**, 600–604, (1999). [CrossRef]

_{2}S

_{3}considering large phase shifts. They also reported a formula (Eq. 11 in Ref. [20

20. C. H. Kwak, Y. L. Lee, and S. G. Kim, “Analysis of asymmetric Z-scan measurement for large optical nonlinearities in an amorphous As_{2}S_{3} thin film,” J. Opt. Soc. Am. B **16**, 600–604, (1999). [CrossRef]

*ΔΦ*; that can be reproduced after some algebraic manipulation of our Eq. 1 to give

_{0}*x*=

*z*/

*z*and the relation between

_{o}*ΔΦ*and our parameters is;

_{0}*m*=2 gave z-scan curves with different features. The peak was not as sharp as in the case of

*m*=

*4*and the valley never reach the value of zero. The normalized transmittance in the wings was different for each curve: positive values of

*z*gave greater differences than the negative ones, Fig. 9. When the waist of the beam was changed the obtained peak-valley transmittance difference was practically the same. This result was the same than that obtained in the weak lens approximation, then this characteristic is maintained for this value of

*m*, Fig. 10.

*m*=

*1*, the curves obtained for different values of the ratio

*F*/

_{0}*z*are shown in Fig. 11. They are asymmetric and small changes in the ratio produce large amplitude curves.

_{0}*ΔT*

_{p-ν}is the parameter that has been used to evaluate the magnitude of the nonlinearity of the tested sample. We obtain that different values of

*m*gave the same

*ΔT*

_{p-ν}when

*F*/

_{0m}*z*> 10, differences are obtained when

_{0}*F*/

_{0m}*z*is smaller, see Fig. 12.

_{0}*ΔZ*

_{p-ν}as function of

*F*/

_{0m}*z*presents a clear dependence with

_{0}*m*, see Fig. 13. Values of

*F*/

_{0m}*z*> 6 gave a magnitude of

_{0}*ΔZ*

_{p-ν}that follows the relation obtained by Eq. 12, weak lens approximation, for each

*m*. While values of

*F*/

_{0n}*z*< 6 gave larger differences, however it is possible to associate a unique value of

_{0}*m*depending on the

*F*/

_{0m}*z*ratio.

_{0}## 5. Conclusion

*w(z)*to some integer power

*m*. Gaussian beam propagation and thin lens approximation were used to obtain an expression for the on axis far field normalized transmittance. The obtained z-scan curves present different features according to the value of

*m*. Approximation to weak lens allowed to obtain analytic formulas for the peak-valley position and transmittance difference. Showing that, the peak-valley position difference is strongly dependent on the value of

*m*. Another parameter that is clearly affected by

*m*is the peak-valley transmittance difference for different beam radius used. Then it is not necessary to suppose what dependence, on the photoinduced lens, will present the sample to characterize, this can be determined if the waist of the beam is known or analyzing the change in transmittance difference for different focusing lenses.

## References and links

1. | M. Sheik Bahae, A. A. Said, and E. W. Van Stryland, “High sensitivity single beam n |

2. | M. Sheik-Bahae, A. A: Said, T. Wei, D. Hagan, and E. W. Van Stryland, Sensitive measurement of Optical Nonlinearities using a single Beam,” IEEE J. Quantum Electron. |

3. | T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. Van Stryland, “Eclipsing Z-scan measurement of |

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

5. | H. Ma, A. S. L. Gomes, and C. B. de Araujo, “Measurement of nondegenerate optical nonlinearity using a two-color single beam method,” Appl. Phys. Lett. |

6. | D. V. Petrov, A. S. L. Gomes, and C. B. de Araujo, “Reflection Z-scan technique for measurements of optical properties surfaces,” Appl. Phys. Lett. |

7. | P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. Mckay, and R. G. McDuff, “Single-beam z-scan: measurement techniques and analysis,” J. Non Opt. Phys. Mat. |

8. | L. C. Oliveira and S. C. Zilio, “Single beam time-resolved Z-scan measurements of slow absorbers,” Appl. Phys. Lett. |

9. | S. J. Sheldon, L. V Knight, and J. M. Thorne, “Laser-induced thermal lens effect: a new theoretical model,” Appl. Opt. |

10. | L. Pálfalvi and J. Hebling “Z-scan study of the thermo-optical effect,” Appl. Phys. B |

11. | B. Gu, X. C. Peng, T. Jia, J. P. Ding, J. L. He, and H. T. Wang, “Determinations of third- and fifth-order nonlinearities by the use of the top-hat-beam Z scan: theory and experiment,” J. Opt. Soc. Am. B |

12. | M. D. Iturbe Castillo, J. J. Sánchez-Mondragón, and S. I. Stepanov, “Peculiarities of Z-scan technique in liquids with nonlinearity (steady regime),” Optik |

13. | H. Kogelnik and T. Li, “Laser beams and Resonators,” Appl. Opt. |

14. | J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples” J. Appl. Phys. |

15. | P. A. Márquez Aguilar, J. J. Sánchez Mondragón, S. Stepanov, and G. Bloch, “Z-scan experiments with cubic photorefractive crystal Bi |

16. | M. Sheik-Bahae, A. A. Said, D. Hagan, M. J. Soileau, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. |

17. | R. Torres Quintero, L. Zambrano-Valencia, R. S. Bermúdez-Cruz, and M. Takur, “Z-scan like results produced by linear optical approximation of a nonlinear material,” Rev. Mex. Fis , |

18. | C. Hu and J. R. Whinnery, “New thermo optical measurement method and comparison with other methods,” Appl. Opt. |

19. | F. L. S. Cuppo, A. M. F. Neto, S. L. Gómez, and P. Palffy-Muhoray, “Termal-lens model compared with the
Sheik-Bahae formalism in interpreting Z-scan experiments on lyotropic liquid crystals,” J. Opt. Soc. Am. B |

20. | C. H. Kwak, Y. L. Lee, and S. G. Kim, “Analysis of asymmetric Z-scan measurement for large optical nonlinearities in an amorphous As |

**OCIS Codes**

(190.4400) Nonlinear optics : Nonlinear optics, materials

(190.5940) Nonlinear optics : Self-action effects

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: December 8, 2006

Manuscript Accepted: February 9, 2007

Published: March 5, 2007

**Citation**

Edmundo Reynoso Lara, Zulema Navarrete Meza, M. David Iturbe Castillo, Carlos G. Treviño Palacios, Erwín Martí Panameño, and M. Luis Arroyo Carrasco, "Influence of the photoinduced focal length of a thin nonlinear material in the Z-scan technique," Opt. Express **15**, 2517-2529 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-5-2517

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### References

- M. Sheik Bahae, A. A. Said, and E. W. Van Stryland, "High sensitivity single beam n2 measurements," Opt. Lett. 14, 955-957 (1989). [CrossRef]
- M. Sheik-Bahae, A. A: Said, T. Wei, D. Hagan, E. W. Van Stryland, and E. W. Van Stryland, "Sensitive measurement of Optical Nonlinearities using a single Beam," IEEE J. Quantum Electron. 26, 760-769 (1990). [CrossRef]
- T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. Van Stryland, "Eclipsing Z-scan measurement of λ/10 wave-front distortion," Opt. Lett. 19, 317-319 (1994). [CrossRef] [PubMed]
- W. Zhao and P. Palffy-Muhoray, "Z-scan technique using top-hat beams," Appl. Phys. Lett. 63, 1613-1615 (1993). [CrossRef]
- H. Ma, A. S. L. Gomes, and C. B. de Araujo, "Measurement of nondegenerate optical nonlinearity using a two-color single beam method," Appl. Phys. Lett. 59, 2666 (1991). [CrossRef]
- D. V. Petrov, A. S. L. Gomes, and C. B. de Araujo, "Reflection Z-scan technique for measurements of optical properties surfaces," Appl. Phys. Lett. 65, 1067 (1994). [CrossRef]
- P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. Mckay, and R. G. McDuff, "Single-beam z-scan: measurement techniques and analysis," J. Nonlinear Opt. Phys. Mater. 6, 251-293 (1997). [CrossRef]
- L. C. Oliveira and S. C. Zilio, "Single beam time-resolved Z-scan measurements of slow absorbers," Appl. Phys. Lett. 65, 2121-2123 (1994). [CrossRef]
- S. J. Sheldon, L. V Knight, and J. M. Thorne, "Laser-induced thermal lens effect: a new theoretical model," Appl. Opt. 21, 1663-1669 (1982). [CrossRef] [PubMed]
- L. Pálfalvi and J. Hebling "Z-scan study of the thermo-optical effect," Appl. Phys. B 78, 775-780 (2004) [CrossRef]
- B. Gu, X. C. Peng, T. Jia,J. P. Ding, J. L. He, and H. T. Wang, "Determinations of third- and fifth-order nonlinearities by the use of the top-hat-beam Z scan: theory and experiment," J. Opt. Soc. Am. B 22,446-452 (2005). [CrossRef]
- M. D. Iturbe Castillo, J. J. Sánchez-Mondragón, and S. I. Stepanov, "Peculiarities of Z-scan technique in liquids with nonlinearity (steady regime)," Optik 100, 49-56 (1995).
- H. Kogelnik and T. Li, "Laser beams and Resonators," Appl. Opt. 5,1550-1567 (1966). [CrossRef] [PubMed]
- J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, "Long-transient effects in lasers with inserted liquid samples" J. Appl. Phys. 36, 3-8 (1965). [CrossRef]
- P. A. Márquez Aguilar, J. J. Sánchez Mondragón, S. Stepanov, and G. Bloch, "Z-scan experiments with cubic photorefractive crystal Bi12Ti20," Opt. Commun. 118,165-174 (1995). [CrossRef]
- M. Sheik-Bahae, A. A. Said, D. Hagan, M. J. Soileau, E. W. Van Stryland, "Nonlinear refraction and optical limiting in thick media," Opt. Eng. 38, 1228-1235 (1991). [CrossRef]
- R. Torres Quintero, L. Zambrano-Valencia, R. S. Bermúdez-Cruz, and M. Takur, "Z-scan like results produced by linear optical approximation of a nonlinear material," Rev. Mex. Fis, 46, 586-592 (2000).
- C. Hu and J. R. Whinnery, "New thermo optical measurement method and comparison with other methods," Appl. Opt. 12, 72-79 (1973). [CrossRef] [PubMed]
- F. L. S. Cuppo, A. M. F. Neto, S. L. Gómez and P. Palffy-Muhoray, "Termal-lens model compared with the Sheik-Bahae formalism in interpreting Z-scan experiments on lyotropic liquid crystals," J. Opt. Soc. Am. B 19,1342-1348 (2002). [CrossRef]
- C. H. Kwak, Y. L. Lee and S. G. Kim, "Analysis of asymmetric Z-scan measurement for large optical nonlinearities in an amorphous As2S3 thin film," J. Opt. Soc. Am. B 16,600-604 (1999). [CrossRef]

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