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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 12, Iss. 7 — Apr. 5, 2004
  • pp: 1377–1382
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Nanosecond Z-scan measurements of the nonlinear refractive index of fused silica

Thomas Olivier, Franck Billard, and Hassan Akhouayri  »View Author Affiliations


Optics Express, Vol. 12, Issue 7, pp. 1377-1382 (2004)
http://dx.doi.org/10.1364/OPEX.12.001377


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Abstract

We present the results of Z-scan studies carried out on fused silica at 1064nm and 532nm with two different nanosecond pulse durations. Such measurements in silica and in the nanosecond regime are possible thanks to a high sensitivity setting up of the Z-scan method and in-situ characterizations of the spatio-temporal parameters of the beam. Besides, with the use of a newly adapted numerical simulation only the calibration errors of the measurement devices are significant. In these conditions, we found a higher value of the nonlinear refractive index than in the femtosecond regime and we show that these values depend on pulse duration, which indicates the contribution of nanosecond mechanisms like electrostriction.

© 2004 Optical Society of America

When a high irradiance laser beam is propagated through any transparent medium, photo-induced refractive index variations may lead to self-focusing of the beam. These photo-induced refractive index variations are commonly described by the simple relationship [1

1. P. N. Butcher and D. Cotter, The elements of nonlinear optics (Cambridge University Press, 1990). [CrossRef]

]:

n=n0+γI
(1)

where n 0 is the linear refractive index, I is the intensity of the incident light, expressed inW/m 2 and γ is defined as the nonlinear refractive index, expressed in m 2/W. In the case of multiple mechanisms leading to a refractive index variation, the coefficient γ is a global or an effective parameter that contains all the different contributions. It can be easily demonstrated that the self-focusing process is all the more important that the total power of the beam is high [2

2. R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964). [CrossRef]

]. Self-focusing compensates diffraction and thus may lead to a catastrophic collapse of the beam for a total power greater than a critical power Pc, which is inversely proportional to the nonlinear refractive index γ. Depending on the total power of the beam and its spatial repartition, this self-focusing process may distort the beam by enhancing hot spots inside the beam (small-scale self-focusing) or focus the whole beam (large-scale self-focusing). Even if the power is inferior to the critical power, these two types of self-focusing lead to a local enhancement of the electric field. Therefore, this mechanism plays an important role in the laser damage process. In the nanosecond regime, the laser damage process is very complicated, because fast electronic, as well as opto-mechanical and thermo-optical processes may be involved [3

3. A. Feldman, D. Horowitz, and R. M. Waxler, “Mechanisms for self-focusing in optical glasses,” IEEE J. Quantum Electron. QE-9, 1054–1061 (November 1973). [CrossRef]

]. In order to study the impact of self-focusing on laser damage, it is crucial to have a reliable measurement of the nonlinear refractive index.

For this purpose, we developed a metrology bench based on the Z-scan method [4

4. 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, 4, 760–769 (April 1990). [CrossRef]

]. This method, based on beam distorsion measurements, is particularly suitable for the study of the self-focusing process. Moreover, it is quite sensitive and can be used with nanosecond pulses. In that regime, it then takes into account all kind of mechanisms that may lead to a photo-induced change of the refractive index. In the case of fused silica, the nonlinear refractive index is quite small, roughly 3×10-20 m 2/W [5

5. N. L. Boling, A. J. Glass, and A. Owyoung, “Empirical relationships for predicting nonlinear refractive index changes in optical solids,” IEEE J. Quantum Electron. QE-14, 601–608, 1978. [CrossRef]

], which is about two order of magnitude below the nonlinear refractive index of most of the materials usually studied with the Z-scan method. Moreover, in the nanosecond regime, the maximum intensity that can be reached is limited by the damage fluence of the sample, which is roughly 100J/cm 2 in the case of fused silica at 1064nm [6

6. L. Gallais, J.Y. Natoli, and C. Amra, “Statistical study of single and multiple pulse laser-induced damage in glasses,” Opt. Express 10, 1465–1474 (2002), http://www.opticsexpress.org [CrossRef] [PubMed]

]. Therefore, for a measurement of the nonlinear refractive index near damage threshold, the maximum intensity available is limited to only few GW/cm 2. In these conditions, it would be necessary to have an important beam diameter to reach a high total power and then induce an important self-focusing inside the sample. However, the interpretation of the measured Z-scan curves (normalized transmittance curves) would be difficult as the transmitted power on the photodiodes may not follow linearly the instantaneous output power of the laser. For an accurate metrology purpose, the Z-scan method should be used in its linearity regime, that is to say at a power far below the critical power. Therefore, we chose to work with a small beam radius at the waist position, roughly equal to 15µm at 1/e 2 in intensity. The power is then about 10 times below the critical power, but the on-axis fluence and peak intensity are maximum (roughly 100 to 200J/cm 2 and 5 to 10GW/cm 2 at 1064nm).

Fig. 1. Experimental setup. IRF: infrared filter, λ/2: half-wave plate, P: Glan-Thompson polarizer, 2ω: KTP crystal and half-wave plate, A1, A2: apertures, W1, W2, W3: wedges, RL, SL: reference and signal lenses, RP, SP: reference and signal photodiodes, RM: removable mirror for beam spatial characterization, TS: translation stage.

A calibrated pyrometer is used to measure the total energy of the pulses after the main lens. A removable mirror can be set on the main arm, allowing repetitive characterizations of the beam spatial profile at any position. This is done thanks to a CCD camera with an objective lens. The camera has been calibrated and the resolution of the camera is 0.7µm per pixel. This allows us to perform accurate spatial characterizations of the beam. Figure 2 represents a snapshot of the beam at the waist position and the evolution of the on-axis intensity along the optical axis. As it can be seen, the shape of the beam at the waist position is close to a gaussian shape, but as soon as we look at the on-axis intensity evolution, we find out that the depth of focus is greater in the case of a trimmed Airy beam, which is at the origin of the sensitivity enhancement. The measurements are compared to a simulation of the evolution of the beam profile, using a propagation algorithm based on the Fresnel integral. It must be pointed out that the measurements are in good agreement with the simulations. The simulations have been carried out without any fitting on the spatial parameters of the setup, but by using distances and aperture radii that have been measured on the setup.

Fig. 2. (Left) 2D-beam profile at the waist position at 1064nm. (Right) evolution of the on-axis intensity at 1064nm. The different curves represent simulations (solid lines), measurement (diamonds) and for comparison the case of a Gaussian beam having the same effective area Ae at the waist position (dashed lines).

In order to study now the temporal profile of the laser pulses, we used a 25GHz InGaAs photodiode. It is connected to an oscilloscope with a 2GHz bandwidth via a high frequency wiring. This allows us to perform in-situ temporal characterizations of the instantaneous output power of the laser. As it is shown on Fig. 3, the temporal profile p(t) is complex because the laser is multimode longitudinally. As we are able to measure every temporal profiles of the pulses, this complex behavior will be taken into account in the estimation of the temporal parameters of the beam. Indeed, when the power of the beam is far from the critical power, we checked numerically that the instantaneous normalized transmittance can be decomposed into the following form, as soon as the refractive index change is typically inferior to 10-5:

T(z,t)=1+Δn(t)F(z)
(2)

T(z)=1+γE2πτeAe×F(z)
(3)

where Ae is the effective area, which is evaluated from the 2D-profile as the summation of all the pixels of the image divided by the maximum pixel value and multiplied by the equivalent area of one pixel, that is to say (0.7µm)2 in the case of Fig. 2. E stands for the total energy of the beam inside the sample, and the temporal parameter τe is then defined by:

12πτe=p2(t)dt(p(t)dt)2
(4)

We chose this definition (with the introduction of √2π) because in the case of a gaussian profile having a half-width at 1/e equal to τ, the right-hand term of relation 4 is exactly equal to 1(2πτ). This definition thus simplify comparisons with other works. In our case, this temporal parameter τe will be evaluated exactly as it is defined by using the temporal profile characterization setup that has been described earlier.

Fig. 3. Example of the normalized temporal profile p(t) (normalized output power), measured with a fast photodiode at 1064nm.

Besides, numerical simulations of the propagation of the trimmed Airy beam inside the nonlinear sample have been carried out. Indeed, in the case of a trimmed Airy beam combined with a sample thickness greater than the depth of focus, no analytical model is available. Hence, it is crucial to provide an accurate description of the propagation of any beam shape inside the nonlinear sample, in order to provide a reliable value of the nonlinear refractive index. We developed a general simulation tool based on the split-step fast Fourier transform method [9

9. M. D. Feit and J. A. Fleck, “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B 5, 3, 633–640 (March 1988). [CrossRef]

] and we adapted this method to the case of the circular symmetry by using Fast Hankel Transforms [10

10. A. J. S. Hamilton, “Uncorrelated modes of the nonlinear power spectrum,” Mon. Not. R. Astron. Soc. 312, 257, (2000). [CrossRef]

]. This simulation tool allows us to take into account any beam shape (with circular symmetry), any sample thickness and any type and strength of the refractive index variation.

In these conditions, accurate measurements have been carried out on different types of fused silica (herasil and suprasil). An example of the measurements is represented on Fig. 4 with the corresponding simulations. The experimental curves are really close to the simulation curves and this, without any fit on the spatial parameters of the beam. The only parameter that is fitted to obtain the correspondence between experimental and simulation curves is the value of the nonlinear refractive index. This good correlation has been made possible because of our accurate characterization of the beam and the use of our adapted simulations that makes no approximations on the beam shape.

As the real spatio-temporal parameters of the beam are taken into account in our simulations and interpretations, the remaining absolute error on the nonlinear refractive index estimation is the summation of the calibration errors of the characterization tools, which is evaluated to be 12% at 1064nm and 16% at 532nm. Seven samples made of different fused silica types and with different thicknesses have been characterized by this method. No change greater than 5% has been noticed from one sample to the other. However, these samples have been tested with two different pulse durations (7ns and 20ns) and it has been noticed that the nonlinear refractive index decreases with the pulse duration.

Fig. 4. Normalized transmittance curves obtained on a 5mm-thick sample of fused silica at 1064nm (left) and at 532nm (right). The dots represent the experimental results and the solid lines represent the simulations.

We thus developed a very sensitive metrology setup based on the Z-scan method that can measure nonlinear refractive indexes as small as the one of fused silica, and this in the nanosecond regime and close to the damage threshold of the sample. Besides, accurate characterizations of the real spatio-temporal parameters of the beam and the use of a versatile simulation tool allow us to provide a very reliable value of the nonlinear refractive index. In these conditions, the dependence of the nonlinear refractive index of fused silica on the pulse duration have been noticed, as could be expected.

Acknowledgments

We would like to thank CEA/CESTA, Bordeaux, France, for financial support.

References and links

1.

P. N. Butcher and D. Cotter, The elements of nonlinear optics (Cambridge University Press, 1990). [CrossRef]

2.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964). [CrossRef]

3.

A. Feldman, D. Horowitz, and R. M. Waxler, “Mechanisms for self-focusing in optical glasses,” IEEE J. Quantum Electron. QE-9, 1054–1061 (November 1973). [CrossRef]

4.

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, 4, 760–769 (April 1990). [CrossRef]

5.

N. L. Boling, A. J. Glass, and A. Owyoung, “Empirical relationships for predicting nonlinear refractive index changes in optical solids,” IEEE J. Quantum Electron. QE-14, 601–608, 1978. [CrossRef]

6.

L. Gallais, J.Y. Natoli, and C. Amra, “Statistical study of single and multiple pulse laser-induced damage in glasses,” Opt. Express 10, 1465–1474 (2002), http://www.opticsexpress.org [CrossRef] [PubMed]

7.

T. Olivier, F. Billard, and H. Akhouayri, “Z-scan theoretical and experimental studies for accurate measurements of nonlinear refractive index and absorption of optical glasses near damage threshold,” presented at the 35th Laser Damage Symposium, Boulder, United-States, Sept 2003.

8.

B. K. Rhee, J. S. Byun, and E. W. Van Stryland, “Z-scan using circularly symmetric beams,” J. Opt. Soc. Am. B 13, 2720–2723 (December 1996). [CrossRef]

9.

M. D. Feit and J. A. Fleck, “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B 5, 3, 633–640 (March 1988). [CrossRef]

10.

A. J. S. Hamilton, “Uncorrelated modes of the nonlinear power spectrum,” Mon. Not. R. Astron. Soc. 312, 257, (2000). [CrossRef]

11.

D. Milam and M. J. Weber, “Measurement of nonlinear refractive-index coefficients using time-resolved interferometry: Application to optical materials for high-power neodynium lasers,” J. Appl. Phys. 47, 6, 2497–2501 (June 1976). [CrossRef]

12.

W. T. White III, W. L Smith, and D. Milam, “Direct measurement of the nonlinear refractive-index coefficient gamma at 355 nm in fused silica and BK-10 glass,” Opt. Lett. 9, 11, 10–12 (January 1984). [CrossRef]

OCIS Codes
(120.3940) Instrumentation, measurement, and metrology : Metrology
(160.6030) Materials : Silica
(190.0190) Nonlinear optics : Nonlinear optics
(190.5940) Nonlinear optics : Self-action effects
(260.5950) Physical optics : Self-focusing
(320.4240) Ultrafast optics : Nanosecond phenomena

ToC Category:
Research Papers

History
Original Manuscript: February 11, 2004
Revised Manuscript: March 26, 2004
Published: April 5, 2004

Citation
Thomas Olivier, Franck Billard, and Hassan Akhouayri, "Nanosecond Z-scan measurements of the nonlinear refractive index of fused silica," Opt. Express 12, 1377-1382 (2004)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-7-1377


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References

  1. P. N. Butcher and D. Cotter, The elements of nonlinear optics (Cambridge University Press, 1990). [CrossRef]
  2. R. Y. Chiao, E. Garmire and C. H. Townes, �??Self-trapping of optical beams,�?? Phys. Rev. Lett. 13, 479-482 (1964). [CrossRef]
  3. A. Feldman, D. Horowitz and R. M.Waxler, �??Mechanisms for self-focusing in optical glasses,�?? IEEE J. Quantum Electron. QE-9, 1054-1061 (November 1973). [CrossRef]
  4. 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, 4, 760-769 (April 1990). [CrossRef]
  5. N. L. Boling, A. J. Glass and A. Owyoung, �??Empirical relationships for predicting nonlinear refractive index changes in optical solids,�?? IEEE J. Quantum Electron. QE-14, 601-608, 1978. [CrossRef]
  6. L. Gallais, J.Y. Natoli and C. Amra, �??Statistical study of single and multiple pulse laser-induced damage in glasses,�?? Opt. Express 10, 1465-1474 (2002), <a href="http://www.opticsexpress.org">http://www.opticsexpress.org</a> [CrossRef] [PubMed]
  7. T. Olivier, F. Billard and H. Akhouayri, �??Z-scan theoretical and experimental studies for accurate measurements of nonlinear refractive index and absorption of optical glasses near damage threshold,�?? presented at the 35th Laser Damage Symposium, Boulder, United-States, Sept 2003.
  8. B. K. Rhee, J. S. Byun and E. W. Van Stryland, �??Z-scan using circularly symmetric beams,�?? J. Opt. Soc. Am. B 13, 2720-2723 (December 1996). [CrossRef]
  9. M. D. Feit and J. A. Fleck, �??Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,�?? J. Opt. Soc. Am. B 5, 3, 633-640 (March 1988). [CrossRef]
  10. A. J. S. Hamilton, �??Uncorrelated modes of the nonlinear power spectrum,�?? Mon. Not. R. Astron. Soc. 312, 257, (2000). [CrossRef]
  11. D. Milam and M. J. Weber, �??Measurement of nonlinear refractive-index coefficients using time-resolved interferometry: Application to optical materials for high-power neodynium lasers,�?? J. Appl. Phys. 47, 6, 2497-2501 (June 1976). [CrossRef]
  12. W. T. White III, W. L Smith and D. Milam, �??Direct measurement of the nonlinear refractive-index coefficient gamma at 355 nm in fused silica and BK-10 glass,�?? Opt. Lett. 9, 11, 10-12 (January 1984). [CrossRef]

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