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Dispersion of the nonlinear refractive index in sapphire Arkady Major, Fumiyo Yoshino, Irkalis Nikolakakos, J. Stewart Aitchison, and Peter W. E. Smith Department of Electrical and Computer Engineering, University of Toronto, 10 King’s College Road, Toronto M5S 3G4, Canada Received October 23, 2003 The nonlinear refractive index, n2 , of sapphire was experimentally measured in the 550 – 1550-nm wavelength range by use of a picosecond Z-scan technique. It was found that in this spectral region the value of n2 decreases monotonically from ⬃3.3 3 10216 to ⬃2.8 3 10216 cm2 兾W. An empirical expression for the wavelength dependence of the nonlinear refractive index in the 270 – 1550-nm range was obtained. © 2004 Optical Society of America OCIS codes: 190.4720, 190.4400, 140.3380.
Determination of the dispersion of the nonlinear refractive index in solids is important not only from a fundamental point of view but also for practical applications. For example, the performance of nonlinear all-optical switching and signal processing elements can be improved and the generation of ultrashort pulses from a mode-locked laser optimized with respect to wavelength. It is also well known that the key issue in the generation of ultrashort laser pulses is proper compensation for the combined effects of chromatic dispersion and self-phase modulation that occur in the cavity components and in the laser crystal itself.1 – 4 In the few-optical-cycle pulse regime in which the pulse spectrum spreads over a large bandwidth 共.100 nm兲 these phenomena become extremely important.2 – 4 It is known that both the refractive index, n, and the nonlinear refractive index, n2 , of the crystal exhibit dispersion; however, for most materials, only the linear index has been well characterized experimentally. The dispersion of the nonlinear refractive index is usually ignored by assumption of a constant value for n2 . Taking into account the frequency dependence of n2 , one would determine the nonlinear phase change wNL 共v兲 acquired by the broadband pulse through self-phase modulation (and subsequently the dispersive contribution of self-phase modulation to the group-velocity dispersion) as a convolution of pulse power spectrum I 共v兲 and the spectrum of n2 共v兲. Hence, detailed knowledge of the dispersion of n2 will increase understanding of the pulse generation process and add additional degrees of freedom to control the process. In this Letter we report on broadband characterization of the nonlinear refractive index of sapphire, which was proved to be one of the best laser crystal hosts, for example, Ti:sapphire laser crystals,2,3 for generation of ultrashort pulses, as well as a common reference material in nonlinear optics experiments. Using a broadly tunable picosecond optical parametric amplifier for the Z-scan technique,5 we made systematic measurements of the nonlinear refractive index of sapphire in the 550 –1550-nm range with the electric field vector parallel to the c axis (p polarization, k⬜c axis), as it would be in a laser. We found that in this wavelength region the value of the nonlinear refractive index decreases monotonically 0146-9592/04/060602-03$15.00/0
from ⬃3.3 3 10216 to ⬃2.8 3 10216 cm2 兾W. Using these results and the previously reported values, we can well approximate the dispersion of n2 in sapphire in the 270– 1550-nm range by a double exponential decay. Commonly accepted values of the nonlinear refractive index of sapphire6 as well as theoretical studies based on the Kramers– Krönig model7 show excellent agreement with the data. To date the experimental data on the nonlinear refractive index of sapphire have been scarce and were limited to several common wavelengths, as shown in Fig. 1 for data from Refs. 6 and 8– 13 as well as in this Letter. The figure reveals several surprising facts: There is actually only one measurement of the nonlinear refractive index near the gain peak of Ti:sapphire crystal; the usually accepted value of n2 for Ti:sapphire crystal6 is determined at 1064 nm; there are no data on the nonlinear refractive index at wavelengths above 1064 nm. The dashed curve in Fig. 1 shows the theoretical scaling of n2 calculated by assumption of a simple two-parabolic-band model.7 The nonlinear index of refraction is determined from a Kramers– Krönig relation and is given by p ∂ µ 4 3 105 pK Ep hv ¯ , n2 共v兲 苷 G2 (1) cn0 2 Eg 4 Eg where c is the speed of light (in meters per second), n0 is the linear refractive index, Eg is the bandgap
Fig. 1. Data from the literature: KK, Kramers –Krönig relation; s, p polarizations. © 2004 Optical Society of America
March 15, 2004 / Vol. 29, No. 6 / OPTICS LETTERS
energy (⬃7.3 eV for sapphire), Ep is a Kane momentum parameter and has a value of ⬃21 eV, K is a materialdependent constant equal to 0.86 3 1028 when Eg and Ep are in electron volts, and n2 is in square centimeters per watt. The function G2 共 hv兾E ¯ g 兲 depends on the band-structure model and is a function only of the ratio of the photon energy to the energy gap of the material [or the ratio of cutoff wavelength lg (艐170 nm) to the wavelength of interest l]. It should be noted, however, that this theoretical model provides only a general description of the dispersion in the n2 values for wide-bandgap materials and should be viewed as a guideline only. To characterize the nonlinear refractive index of sapphire we used a standard implementation of the single-beam, Z-scan technique.5 The laser source consisted of a chirped-pulse amplif ied, picosecond Ti:sapphire laser that pumped a broadly tunable optical parametric amplif ier at a repetition rate of 1 kHz. The details of the experimental setup used can be found elsewhere.14 The Z-scan measurements were performed on a 1-mm-thick sapphire crystal cut such that the measurements with Ekc axis (p polarization, k⬜c axis) were possible. Approximately 1-ps-long pulses with energies in the range of approximately 0.15 3.0 mJ were used. Two slightly different focusing arrangements were chosen for different sets of wavelengths. In most measurements the focused spot size diameter was approximately 16 mm for the 550 – 900 nm and approximately 54 mm for the 1250–1550-nm wavelength regions, which resulted in the maximum on-axis peak intensities in an approximate range of 150 300 GW兾cm2 . A sample of the experimental Z-scan trace is presented in Fig. 2. The vertical axis of Fig. 2 plots normalized transmission, where T0 is the initial or linear transmission, and the horizontal axis plots distance normalized to the Rayleigh range Z0 of the beam. The theoretical fit (shown as a solid curve) was calculated according to the procedure outlined in Ref. 15; a Gaussian pulse shape and beam prof ile were assumed. The estimated uncertainty of our measurements is 625 30%, depending on wavelength, and is common to all Z-scan measurements. The results of the Z-scan measurements in the 550– 1550-nm wavelength region are presented in Fig. 3. The data agree reasonably well with the previously reported values as well as displaying a trend similar to that predicted by the Kramers –Krönig transformation–based model. In this wavelength range the nonlinear refractive index decreases monotonically from ⬃3.3 3 10216 to ⬃2.8 3 10216 cm2 兾W and can be well approximated by a linear regression. There is, however, a noticeable disagreement with the theory for wavelengths below 500 nm in the region where the nonlinear refraction is accompanied by the onset of the two-photon absorption. One of the reasons for this deviation is uncertainty in the value of the bandgap energy of the sapphire, which was determined from the absorption edge measurements.7 To get a simple, compact, and practical empirical expression for the wavelength dependence of n2 in the 270 –1550-nm range [function G2 in
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Eq. (1) has 23 terms] we fitted the data with a double exponential decay function given by n2 3 10216 苷 n2 0 1 N1 exp关2共l 2 l0 兲兾l1 兴 1 N2 exp关2共l 2 l0 兲兾l2 兴 ,
(2)
where l is in nanometers and n2 is in square centimeters per watt. The values of the other f it parameters are as follows: n2 0 , 2.5 cm2 兾W; l0 , 266.0 nm; N1 , 2.3 cm2 兾W; l1 , 46.6 nm; N2 , 1.0 cm2 兾W; l2 , 1086.3 nm. Equation (2) is plotted as a solid curve in Fig. 3. At this point it should be noted, however, that Eq. (2) has no underlying physical theory that leads to this expression and therefore should be viewed as the best fit to the available experimental data only. It is apparent from Fig. 3 that the exponential form of the data fit is dictated mainly by the two points measured by DeSalvo et al. at 355 and 266 nm.6 Our reasons for including these two data points in the wavelength window of the fit stem from the facts that they do not contradict the general trend predicted by the theoretical model and that our own data are in excellent agreement with the other two data points at 532 and 1064 nm reported in the same paper.6 In conclusion, a systematic characterization of the nonlinear refractive index of sapphire crystal in the 550– 1550-nm range was performed. It was found that in this wavelength region the value of the
Fig. 2. Sample Z-scan trace.
Fig. 3. range.
Results of the experiments in the 550– 1550-nm
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nonlinear refractive index decreases monotonically from ⬃3.3 3 10216 to ⬃2.8 3 10216 cm2 兾W and can be well approximated by a linear dependence. Using these results and the previously reported values, we obtained a simple empirical expression (the best fit to the data) for the dispersion of n2 in the 270 – 1550-nm range. We believe that this report has added one more step in characterizing the fundamental properties of solid-state materials and will serve for present and future applications. The authors acknowledge the assistance of V. Sukhovatkin with the experiments and helpful discussions with D. J. F. Cooper and I. V. Mel’nikov. A. Major’s e-mail address is
[email protected]. References 1. J. A. Valdmanis, R. L. Fork, and J. P. Gordon, Opt. Lett. 10, 131 (1985). 2. U. Morgner, F. X. Kärtner, S. H. Cho, Y. Chen, H. A. Haus, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, and T. Tschudi, Opt. Lett. 24, 411 (1999). 3. D. H. Sutter, G. Steinmeyer, L. Gallman, N. Matuschek, F. Morier-Genoud, U. Keller, V. Scheuer, G. Angelow, and T. Tschudi, Opt. Lett. 24, 631 (1999). 4. P. C. Wagenblast, U. Morgner, F. Grawert, T. R. Schibli, F. X. Kärtner, V. Scheuer, G. Angelow, and M. J. Lederer, Opt. Lett. 27, 1726 (2002).
5. M. Sheik-Bahae, A. A. Said, and E. W. Van Stryland, Opt. Lett. 14, 955 (1989). 6. R. DeSalvo, A. A. Said, D. J. Hagan, E. W. Van Stryland, and M. Sheik-Bahae, IEEE J. Quantum Electron. 32, 1324 (1996). 7. M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. 27, 1296 (1991). 8. R. Adair, L. L. Chase, and S. A. Payne, Phys. Rev. B 39, 3337 (1989). 9. M. J. Weber, D. Milam, and W. L. Smith, Opt. Eng. 17, 463 (1978). 10. M. D. Levenson, IEEE J. Quantum Electron. 10, 110 (1974). 11. D. W. Fradin, IEEE J. Quantum Electron. 9, 954 (1973). 12. C. R. Ginliano and M. H. Marburger, Phys. Rev. Lett. 27, 905 (1971). 13. S. Smolorz and F. Wise, Opt. Lett. 23, 1381 (1998). 14. A. Major, I. Nikolakakos, J. S. Aitchison, A. I. Ferguson, N. Langford, and P. W. E. Smith, Appl. Phys. B 77, 433 (2003). 15. M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. 26, 760 (1990).
