## Nonlinear refractive index of porcine cornea studied by z-scan and self-focusing during femtosecond laser processing

Optics Express, Vol. 18, Issue 4, pp. 3700-3707 (2010)

http://dx.doi.org/10.1364/OE.18.003700

Acrobat PDF (285 KB)

### Abstract

We have investigated the nonlinear refractive index of ex-vivo pig cornea by a combined approach using the standard z-scan technique on extracted corneas or corneal slices, as well as studying the deviations caused by self-focusing during femtosecond laser processing of the pig eyes. The experiments yield consistently an upper limit of 1.2 MW for the critical power of self-focusing in porcine cornea, and a value of 2·10^{−19} m^{2}/W for its nonlinear refractive index. We also demonstrate that due to this nonlinear refraction the cutting depth of typical fs-laser surgery processing in cornea may depend considerably, albeit in a well controllable way, on the laser parameters.

© 2010 OSA

## 1. Introduction

2. H. K. Soong and J. B. Malta, “Femtosecond lasers in ophthalmology,” Am. J. Ophthalmol. **147**(2), 189–197, e2 (2009). [CrossRef]

3. T. Juhasz, F. H. Loesel, R. M. Kurtz, C. Horvath, J. F. Bille, and G. Mourou, ““Corneal Refractive Surgery with Femtosecond Lasers,” IEEE J. Sel. Top. Quantum Electron. **5**(4), 902–910 (1999). [CrossRef]

4. H. Lubatschowski, G. Maatz, A. Heisterkamp, U. Hetzel, W. Drommer, H. Welling, and W. Ertmer, “Application of ultrashort laser pulses for intrastromal refractive surgery,” Graefes Arch. Clin. Exp. Ophthalmol. **238**(1), 33–39 (2000). [CrossRef] [PubMed]

*P*), the intensity dependent refractive index

_{c}*n*=

*n*⋅

_{0}+ n_{2}*I*(

*n*: linear;

_{0}*n*: nonlinear refractive index of material;

_{2}*I*: time averaged intensity of laser) causes a different focal position as the one predicted by linear Gaussian beam optics. Depending on whether

*n*is positive or negative, the focus moves backward or forward along the direction of the beam. If the

_{2}*n*of cornea turns out to be large enough to cause focal shifts of several µm, nonlinear refraction would be an important issue for the precision of fs-laser-based eye surgery. As the nonlinearity of cornea as a biological material may vary appreciably from eye to eye, knowledge of the corneal nonlinear refractive index appears to be crucial for precise cutting, in particular when, e.g., lenticle cutting is intended.

_{2}*P*and

_{c}*n*of cornea have been published to date, although a lot of work has been dedicated to investigations concerning the processing of cornea with fs-lasers. For instance, theoretical models using water as a model substance for cornea were developed for plasma formation [6

_{2}6. J. Noack and A. Vogel, “Laser-induced plasma formation in water at nanosecond to femtosecond time scales: calculation of thresholds, absorption coefficients, and energy density,” IEEE J. Quantum Electron. **35**(8), 1156–1167 (1999). [CrossRef]

7. C. L. Arnold, A. Heisterkamp, W. Ertmer, and H. Lubatschowski, “Streak formation as side effect of optical breakdown during processing the bulk of transparent Kerr media with ultra-short laser pulses,” Appl. Phys. B **80**(2), 247–253 (2005). [CrossRef]

8. C. L. Arnold, A. Heisterkamp, W. Ertmer, and H. Lubatschowski, “Computational model for nonlinear plasma formation in high NA micromachining of transparent materials and biological cells,” Opt. Express **15**(16), 10303–10317 (2007). [CrossRef] [PubMed]

9. A. Heisterkamp, T. Ripken, T. Mamom, W. Drommer, H. Welling, W. Ertmer, and H. Lubatschowski, “Nonlinear side effects of fs pulses inside corneal tissue during photodisruption,” Appl. Phys. B **74**(4-5), 419–425 (2002). [CrossRef]

11. M. P. Poudel, “Study of self-focusing effect induced by femtosecond photodisruption on model substances,” Opt. Lett. **34**(3), 337–339 (2009). [CrossRef] [PubMed]

*P*and the

_{c}*n*of pig cornea are studied using a z-scan experiment [12] based on a 300 fs-laser at 1030 nm wavelength. (ii) Using the same laser source, intrastromal cutting is performed with repetition rates up to 350 kHz; discussing, in terms of self-focusing theory, the differences of achieved and intended cutting depths obtained at various laser parameters, we arrive at an independent determination of

_{2}*n*of porcine cornea. The consistency of the values derived from the different methods allows us to discuss the implications of the nonlinear refractive index for fs corneal surgery.

_{2}## 2. Experimental setup

## 3. Results and discussions

13. M. Falconieri, “Thermo-optical effects in Z-scan measurements using high-repetition-rate lasers,” J. Opt. A, Pure Appl. Opt. **1**(6), 662–667 (1999). [CrossRef]

14. J. Kampmeier, B. Radt, R. Birngruber, and R. Brinkmann, “Thermal and biomechanical parameters of porcine cornea,” Cornea **19**(3), 355–363 (2000). [CrossRef] [PubMed]

15. S. Venkatesh, S. Guthrie, F. R. Cruickshank, R. T. Bailey, W. S. Foulds, and W. R. Lee, “Thermal lens measurements in the cornea,” Br. J. Ophthalmol. **69**(2), 92–95 (1985). [CrossRef] [PubMed]

_{0}| ≤ z

_{R}) allowed a reliable evaluation of an n

_{2}value, while the other had to be discarded because of the above given reasons. Several scans of comparable quality have been analyzed by the method described in [16

16. M. Yin, H. P. Li, S. H. Tang, and W. Ji, “Determination of nonlinear absorption and refraction by single Z-scan method,” Appl. Phys. B **70**(4), 587–591 (2000). [CrossRef]

*n*

_{2}= 2·10

^{−19}m

^{2}/W. Besides the complete data evaluation, we also used the setup for z-scan experiments with varying intensity to estimate directly the critical power for self-focusing. The peak laser power above which the z-scans gave the first indication of nonlinear refractivity in the cornea, was experimentally found to be

*P*≈1.2 MW. Since the flaps are very thin and the transmitted light can only accumulate a nonlinear phase shift over a short path length, this value can only be interpreted as an upper limit for the critical power. With λ being the laser wavelength, the relation between P

_{c}_{c}and n

_{2}is given by [17

17. W. Liu and S. L. Chin, “Direct measurement of the critical power of femtosecond Ti:sapphire laser pulse in air,” Opt. Express **13**(15), 5750–5755 (2005). [CrossRef] [PubMed]

*n*≥ 1·10

_{2}^{−19}m

^{2}/W. This is nicely compatible with the above given results from the complete fitting of the z-scans, and proves independently that the nonlinear refractive index of cornea clearly exceeds that of water [18

18. E. T. J. Nibbering, M. A. Franco, B. S. Prade, G. Grillon, C. Le Blanc, and A. Mysyrowicz, “Measurement of the nonlinear refractive index of transparent materials by spectral analysis after nonlinear propagation,” Opt. Commun. **119**(5-6), 479–484 (1995). [CrossRef]

19. P. P. Ho and R. R. Alfano, “Optical Kerr effects in liquids,” Phys. Rev. A **20**(5), 2170–2187 (1979). [CrossRef]

^{2}and 1.8 J/cm

^{2}[4

4. H. Lubatschowski, G. Maatz, A. Heisterkamp, U. Hetzel, W. Drommer, H. Welling, and W. Ertmer, “Application of ultrashort laser pulses for intrastromal refractive surgery,” Graefes Arch. Clin. Exp. Ophthalmol. **238**(1), 33–39 (2000). [CrossRef] [PubMed]

20. H. Sun, M. Han, M. H. Niemz, and J. F. Bille, “Femtosecond laser corneal ablation threshold: dependence on tissue depth and laser pulse width,” Lasers Surg. Med. **39**(8), 654–658 (2007). [CrossRef] [PubMed]

*z*for an externally focused Gaussian beam entering the medium is given by:where

_{sf}*k*is the wave vector,

*w*the beam radius at the entrance into the nonlinear medium,

*P*the peak power,

*z*the position of the focus without self-focusing and

_{0}*w*is the beam waist of the focused laser beam. As in our setup the laser light is travelling through a liquid layer first, we have to introduce the possible effects of the physiological saline on radius and curvature of the entrance beam. Although the nonlinear refractive index of water is smaller than that of cornea, it still suffices to induce an additional shift of the geometrical focus backward to the laser direction. Taking the nonlinear refractive index of the physiological saline as 5·10

_{0}^{−20}m

^{2}/W as in [19

19. P. P. Ho and R. R. Alfano, “Optical Kerr effects in liquids,” Phys. Rev. A **20**(5), 2170–2187 (1979). [CrossRef]

*w*is the beam radius at the location of the Kerr medium,

_{Kerr}*l*is the Kerr medium length (in our case the liquid thickness),

*P*is the power of the beam and

*a = 1.723*for a circular beam. The Kerr lens diminishes the beam radius at the entrance into the cornea increasing the convergence of the beam. By presence of the Kerr medium alone, the geometrical focus will be displaced by a distance

*z*, decreasing the thickness of the produced flaps. This effect will become stronger with increasing laser power. A schematic of the beam propagation is shown in the inset of Fig. 1.

_{Kerr}*w*and the position of the new focus

_{Kerr}*z*were first calculated using Eq. (3), then the beam radius in front of the cornea

_{Kerr}*w*was deduced from the lens transformation of Gaussian beams and it was used in Eq. (2) to calculate

*z*. The values calculated by this procedure using

_{sf}*n*

_{2}= 2·10

^{−19}m

^{2}/W for cornea are in good agreement with the experimentally measured values, as can bee seen in Fig. 3 and 4 (solid lines representing the theoretical values). The remaining, very small deviation of theoretical and experimental values may be due to imperfections of the laser beam, or due to the fact that Eq. (2) is an approximate solution for the self-focusing position.

## 4. Conclusions

*n*

_{2}= 2·10

^{−19}m

^{2}/W). One can see that this value is four times larger than the one of water, showing that the cornea has a stronger third order nonlinear response. Since corneal tissue typically contains more than 70% water, it is an obvious conclusion that the main contribution to its nonlinear optical behaviour comes from the collagen fibrils. Turning around the argument, our investigation demonstrates that modelling optical nonlinearity of cornea with the nonlinear parameters of water is not a very good approximation.

## Acknowledgments

## References and links

1. | H. Lubatschowski, “Overview of Commercially Available Femtosecond Lasers in Refractive Surgery,” J. Refract. Surg. |

2. | H. K. Soong and J. B. Malta, “Femtosecond lasers in ophthalmology,” Am. J. Ophthalmol. |

3. | T. Juhasz, F. H. Loesel, R. M. Kurtz, C. Horvath, J. F. Bille, and G. Mourou, ““Corneal Refractive Surgery with Femtosecond Lasers,” IEEE J. Sel. Top. Quantum Electron. |

4. | H. Lubatschowski, G. Maatz, A. Heisterkamp, U. Hetzel, W. Drommer, H. Welling, and W. Ertmer, “Application of ultrashort laser pulses for intrastromal refractive surgery,” Graefes Arch. Clin. Exp. Ophthalmol. |

5. | R. W. Boyd, |

6. | J. Noack and A. Vogel, “Laser-induced plasma formation in water at nanosecond to femtosecond time scales: calculation of thresholds, absorption coefficients, and energy density,” IEEE J. Quantum Electron. |

7. | C. L. Arnold, A. Heisterkamp, W. Ertmer, and H. Lubatschowski, “Streak formation as side effect of optical breakdown during processing the bulk of transparent Kerr media with ultra-short laser pulses,” Appl. Phys. B |

8. | C. L. Arnold, A. Heisterkamp, W. Ertmer, and H. Lubatschowski, “Computational model for nonlinear plasma formation in high NA micromachining of transparent materials and biological cells,” Opt. Express |

9. | A. Heisterkamp, T. Ripken, T. Mamom, W. Drommer, H. Welling, W. Ertmer, and H. Lubatschowski, “Nonlinear side effects of fs pulses inside corneal tissue during photodisruption,” Appl. Phys. B |

10. | K. Plamann, V. Nuzzo, D. Peyrot, F. Deloison, M. Savoldelli, and J. M. Legeais, “Laser parameters, focusing optics and side effects in femtosecond laser corneal surgery”, Proc. SPIE |

11. | M. P. Poudel, “Study of self-focusing effect induced by femtosecond photodisruption on model substances,” Opt. Lett. |

12. | E. W. Van Stryland, M. Sheik-Bahae, |

13. | M. Falconieri, “Thermo-optical effects in Z-scan measurements using high-repetition-rate lasers,” J. Opt. A, Pure Appl. Opt. |

14. | J. Kampmeier, B. Radt, R. Birngruber, and R. Brinkmann, “Thermal and biomechanical parameters of porcine cornea,” Cornea |

15. | S. Venkatesh, S. Guthrie, F. R. Cruickshank, R. T. Bailey, W. S. Foulds, and W. R. Lee, “Thermal lens measurements in the cornea,” Br. J. Ophthalmol. |

16. | M. Yin, H. P. Li, S. H. Tang, and W. Ji, “Determination of nonlinear absorption and refraction by single Z-scan method,” Appl. Phys. B |

17. | W. Liu and S. L. Chin, “Direct measurement of the critical power of femtosecond Ti:sapphire laser pulse in air,” Opt. Express |

18. | E. T. J. Nibbering, M. A. Franco, B. S. Prade, G. Grillon, C. Le Blanc, and A. Mysyrowicz, “Measurement of the nonlinear refractive index of transparent materials by spectral analysis after nonlinear propagation,” Opt. Commun. |

19. | P. P. Ho and R. R. Alfano, “Optical Kerr effects in liquids,” Phys. Rev. A |

20. | H. Sun, M. Han, M. H. Niemz, and J. F. Bille, “Femtosecond laser corneal ablation threshold: dependence on tissue depth and laser pulse width,” Lasers Surg. Med. |

21. | L. M. Liu, |

**OCIS Codes**

(170.3660) Medical optics and biotechnology : Light propagation in tissues

(190.0190) Nonlinear optics : Nonlinear optics

**ToC Category:**

Medical Optics and Biotechnology

**History**

Original Manuscript: November 17, 2009

Revised Manuscript: January 11, 2010

Manuscript Accepted: January 11, 2010

Published: February 5, 2010

**Virtual Issues**

Vol. 5, Iss. 5 *Virtual Journal for Biomedical Optics*

**Citation**

M. Miclea, U. Skrzypczak, S. Faust, F. Fankhauser, H. Graener, and G. Seifert, "Nonlinear refractive index of porcine cornea studied by z-scan and self-focusing during femtosecond laser processing," Opt. Express **18**, 3700-3707 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-4-3700

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### References

- H. Lubatschowski, “Overview of Commercially Available Femtosecond Lasers in Refractive Surgery,” J. Refract. Surg. 24(1), 102–107 (2008).
- H. K. Soong and J. B. Malta, “Femtosecond lasers in ophthalmology,” Am. J. Ophthalmol. 147(2), 189–197, e2 (2009). [CrossRef]
- T. Juhasz, F. H. Loesel, R. M. Kurtz, C. Horvath, J. F. Bille, and G. Mourou, ““Corneal Refractive Surgery with Femtosecond Lasers,” IEEE J. Sel. Top. Quantum Electron. 5(4), 902–910 (1999). [CrossRef]
- H. Lubatschowski, G. Maatz, A. Heisterkamp, U. Hetzel, W. Drommer, H. Welling, and W. Ertmer, “Application of ultrashort laser pulses for intrastromal refractive surgery,” Graefes Arch. Clin. Exp. Ophthalmol. 238(1), 33–39 (2000). [CrossRef] [PubMed]
- R. W. Boyd, Nonlinear Optics (Elsevier Inc. - 2008), Chapter 7.
- J. Noack and A. Vogel, “Laser-induced plasma formation in water at nanosecond to femtosecond time scales: calculation of thresholds, absorption coefficients, and energy density,” IEEE J. Quantum Electron. 35(8), 1156–1167 (1999). [CrossRef]
- C. L. Arnold, A. Heisterkamp, W. Ertmer, and H. Lubatschowski, “Streak formation as side effect of optical breakdown during processing the bulk of transparent Kerr media with ultra-short laser pulses,” Appl. Phys. B 80(2), 247–253 (2005). [CrossRef]
- C. L. Arnold, A. Heisterkamp, W. Ertmer, and H. Lubatschowski, “Computational model for nonlinear plasma formation in high NA micromachining of transparent materials and biological cells,” Opt. Express 15(16), 10303–10317 (2007). [CrossRef] [PubMed]
- A. Heisterkamp, T. Ripken, T. Mamom, W. Drommer, H. Welling, W. Ertmer, and H. Lubatschowski, “Nonlinear side effects of fs pulses inside corneal tissue during photodisruption,” Appl. Phys. B 74(4-5), 419–425 (2002). [CrossRef]
- K. Plamann, V. Nuzzo, D. Peyrot, F. Deloison, M. Savoldelli, and J. M. Legeais, “Laser parameters, focusing optics and side effects in femtosecond laser corneal surgery”, Proc. SPIE 6844, W0–W10 (2008)
- M. P. Poudel, “Study of self-focusing effect induced by femtosecond photodisruption on model substances,” Opt. Lett. 34(3), 337–339 (2009). [CrossRef] [PubMed]
- E. W. Van Stryland, M. Sheik-Bahae, Characterization Techniques and Tabulations for Organic Nonlinear Materials, M. G. Kuzyk and C. W. Dirk eds., (Marcel Dekker, Inc. 1998) 655–692.
- M. Falconieri, “Thermo-optical effects in Z-scan measurements using high-repetition-rate lasers,” J. Opt. A, Pure Appl. Opt. 1(6), 662–667 (1999). [CrossRef]
- J. Kampmeier, B. Radt, R. Birngruber, and R. Brinkmann, “Thermal and biomechanical parameters of porcine cornea,” Cornea 19(3), 355–363 (2000). [CrossRef] [PubMed]
- S. Venkatesh, S. Guthrie, F. R. Cruickshank, R. T. Bailey, W. S. Foulds, and W. R. Lee, “Thermal lens measurements in the cornea,” Br. J. Ophthalmol. 69(2), 92–95 (1985). [CrossRef] [PubMed]
- M. Yin, H. P. Li, S. H. Tang, and W. Ji, “Determination of nonlinear absorption and refraction by single Z-scan method,” Appl. Phys. B 70(4), 587–591 (2000). [CrossRef]
- W. Liu and S. L. Chin, “Direct measurement of the critical power of femtosecond Ti:sapphire laser pulse in air,” Opt. Express 13(15), 5750–5755 (2005). [CrossRef] [PubMed]
- E. T. J. Nibbering, M. A. Franco, B. S. Prade, G. Grillon, C. Le Blanc, and A. Mysyrowicz, “Measurement of the nonlinear refractive index of transparent materials by spectral analysis after nonlinear propagation,” Opt. Commun. 119(5-6), 479–484 (1995). [CrossRef]
- P. P. Ho and R. R. Alfano, “Optical Kerr effects in liquids,” Phys. Rev. A 20(5), 2170–2187 (1979). [CrossRef]
- H. Sun, M. Han, M. H. Niemz, and J. F. Bille, “Femtosecond laser corneal ablation threshold: dependence on tissue depth and laser pulse width,” Lasers Surg. Med. 39(8), 654–658 (2007). [CrossRef] [PubMed]
- L. M. Liu, Photonic devices (Cambridge University Press, 2005), Chapter 9.

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