December 1, 2013 / Vol. 38, No. 23 / OPTICS LETTERS
5165
Influence of self-focusing of ultrashort laser pulses on optical third-harmonic generation at interfaces E. C. Barbano, S. C. Zílio, and L. Misoguti* Instituto de Física de São Carlos—USP Caixa Postal 369, 13560-590 São Carlos, SP, Brazil *Corresponding author: [email protected] Received September 25, 2013; revised October 30, 2013; accepted October 31, 2013; posted November 1, 2013 (Doc. ID 198338); published November 27, 2013 We report on the third-harmonic generation of femtosecond laser pulses at interfaces. We measured slabs of different types of optical glasses and demonstrated that the asymmetric intensity profile observed for a tightly focused beam can be explained by self-focusing effects. © 2013 Optical Society of America OCIS codes: (160.4330) Nonlinear optical materials; (190.7110) Ultrafast nonlinear optics; (240.4350) Nonlinear optics at surfaces. http://dx.doi.org/10.1364/OL.38.005165
Ultrashort laser pulses are important for the investigation of optical nonlinearities due to their high intensity, broad spectrum, and time-scale properties. Even at low energies, ultrashort pulses are capable of easily inducing third-order nonlinear effects in isotropic materials [1], which have been widely explored for the material’s characterization and in applications such as third-harmonic generation (THG) microscopy [2]. Several studies on the intensity of the third harmonic (TH) generated at interfaces pointed out the need for considering the Gouy phase shift [3–5] and the nonlinearities of the two materials present at the interface. We recently studied the THG of femtosecond pulses in different glass slabs [6,7]. For a tightly focused laser beam, no THG is observed when the beam is focused inside the bulk as expected from the Gouy phase shift. However, THG occurs when the focal plane is near the interfaces because of the discontinuity in the nonlinearity. In experiments where the sample is scanned along the beam propagation direction, the TH generated at the entrance and exit interfaces are unequal [7]. A possible explanation for this asymmetry is that the Fresnel reflection produces a constructive interference at the exit interface [8,9], enhancing the THG there. However, such effect is difficult to be quantified because the Fresnel interference occurs only inside a thin layer (∼λ∕4) smaller than the beam Rayleigh range where the THG occurs. Therefore, it seems worthwhile to investigate more carefully the contributions of linear and nonlinear refractive indices at the interfaces of different materials. The present work reports on THG measurements at the interfaces of several optical glasses carried out with the Z-scan technique [10] at the tightly focused laser beam condition. By carefully controlling the sample position and laser power, we could follow the evolution of the THG intensity at both interfaces of different materials. We found that the TH generated at the entrance interface follows the expected cubic intensity dependence, which is characteristic of a third-order process, but the THG at the exit presents a slope higher than three, indicating that another nonlinear effect is also present. We propose that self-focusing strongly contributes to the THG asymmetry because the positive nonlinear refraction decreases the effective beam waist as the pulse propagates, resulting in laser irradiance greater at the exit 0146-9592/13/235165-04$15.00/0
interface. This unbalances the TH generated at the entrance and exit interfaces. Other authors also observed such asymmetry [11,12], but none of them considered self-focusing of light for its explanation. To fully understand the results presented later, we consider the usual THG cubic dependence on the fundamental beam irradiance [13], which depends on the beam waist radius: 2 ω 3 3 I 3ω sample ∝ χ sample I w ;
(1)
ω where I 3ω sample and I w are the irradiances of the TH and fundamental beam, respectively. Here, we consider only the sample’s nonlinear susceptibility, χ 3 sample , since that of air is negligible. Considering that the beam waist radius changes with the laser power explains why the slope of the THG signal does not follow the cubic power law. In this simple model, we consider that the beam waist for a given laser power, P, is proportional to the sample’s nonlinearity because of the self-focusing process. In the case of a positive nonlinearity, we assume that the beam radius at the exit surface is reduced to
woutput w0 1 − BP;
(2)
where w0 is the radius without self-focusing, BP represents the amount of beam decrease, and B is an empirical constant proportional to the sample’s nonlinearity, n2 . Since the laser irradiance is inversely proportional to the square of the beam radius, we propose the following empirical equation:
I 3ω sample
P A 1 − BP2
3
;
(3)
where A is a constant that depends on the particular sample. The quantity BP changes the slope of the logarithmic THG intensity versus the input power curve, while A adjusts its offset. The self-focusing of light occurs when a laser beam with a Gaussian transverse profile propagates through a medium whose refractive index is given by n n0 n2 I; © 2013 Optical Society of America
(4)
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Fig. 1. Schematic self-focusing action represented by the dashed line for beam focused at the (a) exit and (b) entrance interfaces.
where n2 is the nonlinear refractive index proportional to χ 3 sample [14]. For a positive nonlinearity, as is usual for the instantaneous electronic polarization of optical glasses, the refractive index is larger at the center of the laser beam than at the border, resulting in a positive lens whose focal length depends on the laser irradiance and sample nonlinearity [15]. In the THG Z-scan method, the interfaces translate through the laser focal plane. If the beam is tightly focused at the exit interface [Fig. 1(a)], the sample is in the prefocus region, and the nonlinear interaction occurs just in this region. On other hand, when the beam is focused at the entrance interface [Fig. 1(b)], only the post-focus region is important. It is worth mentioning that, in both cases, THG and selffocusing effects occur in a typical Rayleigh range, zR , where the laser irradiance is strong enough to induce such nonlinear effects. As seen in Fig. 1(a), stronger THG is expected due to the reduction of the beam waist at the exit interface. Furthermore, the THG at the entrance interface is not enhanced because the self-focusing changes the beam waist just after the beam propagates by zR , where the irradiance is smaller due to natural beam expansion. In order to verify the validity of the model proposed, we carried out THG Z-scan measurements using a commercial optical parametric amplifier (OPA) (TOPAS, from Light Conversion), synchronously pumped by a chirped-pulse-amplified system (CPA 2001-Clark MXR) with a 775 nm central wavelength and 150 fs pulse duration at 1 kHz repetition rate. The OPA was set to generate pulses at the near IR range (λ 1300 nm) with about 120 fs pulse duration. In this way, the TH is centered on 433 nm. In these wavelengths, the samples used here (silica, BK7, SK11, and F2) present no significant absorption. As the Z-scan method requires, the sample translates through the focus of the laser beam located at z 0. Negative values of z correspond to sample positions between the focusing lens and its focal plane such
that the exit (entrance) interface appears at the negative (positive) z direction. The laser power was carefully adjusted with a calcite polarizer in the range from 0.3 to 0.8 mW (I 0 ∼ 2.4 to 6.4 × 1016 W∕m2 ) and was then focused with a 50 mm focal length lens such that the Rayleigh range was shorter than the sample thickness, allowing us to resolve the two THG signal peaks. To avoid damage to the detector, we used three dichroic mirrors to separate the strong fundamental beam from the TH beam. A lens placed after the sample was used to direct most of the TH light into a large-area silicon PIN detector coupled to a lock-in amplifier. We measured the three optical glasses listed in Table 1 and used fused silica as a reference because it has well-known nonlinearity (n2 2.71 × 10−20 m2 ∕W) [16]. Figure 2 shows that THG peaks generated at the exit are systematically stronger than those generated at the entrance interface for all samples measured, regardless of the power used. All curves exhibit a stronger peak at the exit interface (negative z) and a smaller one at the entrance interface (positive z). Since the samples measured present different linear and nonlinear refractive indices, they can provide good insight on the working principles of THG at interfaces. Silica and F2 have the lowest and the highest linear and nonlinear index of refraction, respectively, while BK7 and SK11 have intermediate values. Z-scan measurements as a function of laser power have similar THG curve shapes, and so we considered only the peak values at the interfaces to establish the power law dependence through the log–log plots of Fig. 3. The solid lines, corresponding to Eq. (3) of our model that takes into account the beam waist decrease, fit very well the experimental data. The slopes of the THG for entrance interfaces (red circles) are exactly three while the slopes for exit interfaces (black squares) are slightly higher than three. As expected, F2 presents the uppermost slope because it has the highest nonlinear refractive index. The fitting of the experimental data with Eq. (3) allows finding the sample nonlinearity using the value of silica as a reference. By comparing the relative B values of each fitting, one can estimate n2 . Table 1 shows the results achieved by this procedure (THG slope). As we mentioned earlier, the beam radius should change as a function of sample nonlinearity and input power [15]. This can be corroborated by noticing that the THG peaks at the exit interfaces are systematically narrower than those generated at the entrance interfaces. Figure 4 presents the normalized THG peaks obtained for silica at a laser power of 0.8 mW, where the
Table 1. Parameters of Optical Glasses Used in Measurementsa n0 ne n0 Sample (433 nm) (546.1 nm) (1300 nm) Silica BK7 SK11 F2
1.46693 1.52697 1.57564 1.64271
1.46008 1.51872 1.56605 1.62408
1.44692 1.50370 1.54965 1.59813
Thickness (mm) 1.225 0.925 1.155 1.165
n2 (theory) from Eqs. (6) n2 (Experimental) n2 (Experimental) and (7) ×2.53× THG Slope ×2.71× THG Width ×2.71× 10−20 m2 ∕W 10−20 m2 ∕W 10−20 m2 ∕W 1 1.3 1.5 3.8
1 1.7 1.8 3.8
1 1.6 1.7 2.8
Theoretical and experimental nonlinearities (n2 ) of the samples. The experimental values were obtained from ratios of B values using silica as a reference. a
December 1, 2013 / Vol. 38, No. 23 / OPTICS LETTERS
Silica BK7 SK11 F2
0.8
Normalized THG [arb. units]
THG Intensity [arb. units]
1.0
0.6 0.4 0.2 0.0 -1.0
-0.5
0.0 z (mm)
0.5
different widths can be easily seen. As predicted by the Gaussian beam theory, the waist radius is related to the Rayleigh range and thus, by measuring zR of the TH, we can estimate the fundamental beam waist radius by taking into account that we are dealing with a thirdorder process. Using a Gaussian function to fit the peaks in all sample interfaces, it is possible to follow how the widths change with the laser power. Basically, the widths (FWHM) at the entrance increase as a function of the laser power, while those at the exit decrease. Such behavior can be explained by an equation similar to Eq. (2): W W 0 1 BP;
(5)
where W and W 0 are the THG peak widths with and without self-focusing, respectively. The positive (negative) signal is relative to the entrance (exit) interface. Table 1 shows the results achieved by this procedure (THG width). Exit Entrance
0.8 0.6 0.4 0.2
0.0
0.2
0.4
Fig. 4. Normalized TH peaks generated at the entrance and exit interfaces for silica obtained with 0.8 mW pump power. The points are the experimental results, and the lines are fittings with Gaussian functions.
Despite the noise in the measured widths, Fig. 5 shows a trend of broadening (entrance) and shortening (exit) of the THG peaks. The effect is more evident in F2 because of its high nonlinearity. Broadening and shortening of the THG peaks are explained by the self-focusing effect as the TH is generated near the focal plane. When the nonlinear sample is placed after the focus position (entrance interface), the selffocusing effect tends to recollimate the beam, and zR increases. On other hand, for a sample positioned before the focal plane (exit interface), zR decreases. The broadening of zR at the entrance interface does not affect the fundamental laser irradiance, as most of the TH is generated near that interface, where the self-focusing did not take place yet because of the lack of beam propagation. These explain why the THG at the entrance interfaces follows the cubic power law. In the case of exit interfaces, most of the THG occurs in the vicinity of the output face where the beam waist compression already
(4.0)
0.32 (3)
Silica
BK7
0.01
(5.3)
(4.1)
1
(3)
(3)
0.1
SK11
0.01 0.4
0.6
Exit Entrance
0.28
0.1
THG Peak Width (mm)
THG Signal [arb. units]
-0.2
(3.7)
(3)
0.2
-0.4
Sample Displacement (mm)
Fig. 2. THG Z-scan signal for silica, BK7, SK11, and F2 obtained at a laser power of ∼0.44 mW. An offset of 0.1 between consecutive curves was introduced for better visualization.
1
Exit Entrance
1.0
0.0
1.0
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F2 0.2
0.4
0.6
0.24 0.20 0.16 0.32 0.28 0.24 0.20
0.8
Input Laser Power (mW)
Fig. 3. THG peak values at the interfaces as a function of incident laser power for silica, BK7, SK11, and F2. The symbols are the experimental data, while the solid lines are results obtained with the model proposed. The numbers in parenthesis indicate the slopes.
BK7
Silica
0.16 0.2
SK11 0.4
F2 0.6
0.2
0.4
0.6
0.8
Input Laser Power (mW)
Fig. 5. FWHM of THG peaks measured as a function of the pump laser power for silica, BK7, SK11, and F2 glasses. The widths are obtained by fitting all peaks with a Gaussian function. The solid lines are obtained from Eq. (5).
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took place due to self-focusing and propagation along the Rayleigh distance. Due to this reduction of the beam waist radius, the laser irradiance increases and leads to a THG law with an exponent greater than cubic. In order to compare the measured nonlinearity, we can use the empirical expression given in [17] and [18], where n2 can be theoretically derived from the linear index of refraction, ne (at λ 546.1 nm) and the Abbe number, ve : n2 10−13 esu
68ne − 1n2e 22
h i1∕2 ; 2 e 1 ve 1.517 ne 2n v e 6ne
(6)
which can be converted to SI units according to 2 m 4.19 × 10−7 n2 esu: n2 W ne
(7)
Such theoretical values of n2 , although obtained for the visible range, indicate the right trends of the nonlinearities in these four samples. Considering that there is a typical wide dispersion on the value of the nonlinearity [14,16], it is possible to say that this new method used here leads to good values of the nonlinearities, and it allows making further comparison with other values obtained by other methods. Table 1 summarizes the results obtained by our proposed method to understand the THG at interfaces. The slightly higher nonlinearities obtained with the slope may indicate a small contribution of the Fresnel reflection interference [5]. In conclusion, we have studied the THG at interfaces of different optical glasses using fs laser pulses. The TH intensities asymmetry could be explained by the selffocusing effect, which increases the laser irradiance at the output interface due to the reduction of the beam waist. This study is important for fundamental understanding of the THG process in transparent materials at the tightly focused condition. Also, for the application’s
point of view, it is important for studying the material’s third-order nonlinearity [1] and its use in the THG scanning laser microscopy [2,4]. This work was supported by the FAPESP (grant 2011/ 22787-7) and CNPq from Brazil.
References 1. G. C. Bjorklund, IEEE J. Quantum Electron. QE-11, 287 (1975). 2. Y. Barad, H. Eisenberg, H. Horowitz, and Y. Silberberg, Appl. Phys. Lett. 70, 922 (1997). 3. L. G. Gouy, C. R. Acad. Sci. Paris 110, 1251 (1890). 4. V. Shcheslavskiy, G. I. Petrov, S. Saltiel, and V. V. Yakovlev, J. Struc. Biol. 147, 42 (2004). 5. K. D. Moll, D. Homoelle, A. L. Gaeta, and R. W. Boyd, Phys. Rev. Lett. 88, 153901 (2002). 6. I. A. Heisler, L. Misoguti, S. C. Zilio, E. V. Rodriguez, and C. B. de Araujo, Appl. Phys. Lett. 92, 091109 (2008). 7. E. C. Barbano, J. P. Siqueira, C. R. Mendonça, L. Misoguti, and S. C. Zílio, Proc. SPIE 7917, 79170T (2011). 8. M. D. Crisp, N. L. Boling, and G. Dubé, Appl. Phys. Lett. 21, 364 (1972). 9. W. Koechner, Solid-State Laser Engineering, 4th ed. (Springer-Verlag, 1996). 10. M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. 26, 760 (1990). 11. D. Stoker, M. F. Becker, and J. W. Keto, Phys. Rev. A 71, 061802(R) (2005). 12. J. M. Schins, T. Schrama, J. Squier, G. J. Brakenhoff, and M. Muller, J. Opt. Soc. Am. B 19, 1627 (2002). 13. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008). 14. I. Rau, F. Kajzar, J. Luc, B. Sahraoui, and G. Boudebs, J. Opt. Soc. Am. B 25, 1738 (2008). 15. V. Magni, G. Cerullo, and S. De Silvestri, Opt. Commun. 96, 348 (1993). 16. D. Milam, Appl. Opt. 37, 546 (1998). 17. S. Hazarika and S. Rai, Opt. Mater. 30, 462 (2007). 18. N. L. Boling, A. J. Glass, and A. Owyoung, IEEE J. Quantum Electron. QE-14, 601 (1978).
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Influence of self-focusing of ultrashort laser pulses on optical third-harmonic generation at interfaces E. C. Barbano, S. C. Zílio, and L. Misoguti* Instituto de Física de São Carlos—USP Caixa Postal 369, 13560-590 São Carlos, SP, Brazil *Corresponding author: [email protected] Received September 25, 2013; revised October 30, 2013; accepted October 31, 2013; posted November 1, 2013 (Doc. ID 198338); published November 27, 2013 We report on the third-harmonic generation of femtosecond laser pulses at interfaces. We measured slabs of different types of optical glasses and demonstrated that the asymmetric intensity profile observed for a tightly focused beam can be explained by self-focusing effects. © 2013 Optical Society of America OCIS codes: (160.4330) Nonlinear optical materials; (190.7110) Ultrafast nonlinear optics; (240.4350) Nonlinear optics at surfaces. http://dx.doi.org/10.1364/OL.38.005165
Ultrashort laser pulses are important for the investigation of optical nonlinearities due to their high intensity, broad spectrum, and time-scale properties. Even at low energies, ultrashort pulses are capable of easily inducing third-order nonlinear effects in isotropic materials [1], which have been widely explored for the material’s characterization and in applications such as third-harmonic generation (THG) microscopy [2]. Several studies on the intensity of the third harmonic (TH) generated at interfaces pointed out the need for considering the Gouy phase shift [3–5] and the nonlinearities of the two materials present at the interface. We recently studied the THG of femtosecond pulses in different glass slabs [6,7]. For a tightly focused laser beam, no THG is observed when the beam is focused inside the bulk as expected from the Gouy phase shift. However, THG occurs when the focal plane is near the interfaces because of the discontinuity in the nonlinearity. In experiments where the sample is scanned along the beam propagation direction, the TH generated at the entrance and exit interfaces are unequal [7]. A possible explanation for this asymmetry is that the Fresnel reflection produces a constructive interference at the exit interface [8,9], enhancing the THG there. However, such effect is difficult to be quantified because the Fresnel interference occurs only inside a thin layer (∼λ∕4) smaller than the beam Rayleigh range where the THG occurs. Therefore, it seems worthwhile to investigate more carefully the contributions of linear and nonlinear refractive indices at the interfaces of different materials. The present work reports on THG measurements at the interfaces of several optical glasses carried out with the Z-scan technique [10] at the tightly focused laser beam condition. By carefully controlling the sample position and laser power, we could follow the evolution of the THG intensity at both interfaces of different materials. We found that the TH generated at the entrance interface follows the expected cubic intensity dependence, which is characteristic of a third-order process, but the THG at the exit presents a slope higher than three, indicating that another nonlinear effect is also present. We propose that self-focusing strongly contributes to the THG asymmetry because the positive nonlinear refraction decreases the effective beam waist as the pulse propagates, resulting in laser irradiance greater at the exit 0146-9592/13/235165-04$15.00/0
interface. This unbalances the TH generated at the entrance and exit interfaces. Other authors also observed such asymmetry [11,12], but none of them considered self-focusing of light for its explanation. To fully understand the results presented later, we consider the usual THG cubic dependence on the fundamental beam irradiance [13], which depends on the beam waist radius: 2 ω 3 3 I 3ω sample ∝ χ sample I w ;
(1)
ω where I 3ω sample and I w are the irradiances of the TH and fundamental beam, respectively. Here, we consider only the sample’s nonlinear susceptibility, χ 3 sample , since that of air is negligible. Considering that the beam waist radius changes with the laser power explains why the slope of the THG signal does not follow the cubic power law. In this simple model, we consider that the beam waist for a given laser power, P, is proportional to the sample’s nonlinearity because of the self-focusing process. In the case of a positive nonlinearity, we assume that the beam radius at the exit surface is reduced to
woutput w0 1 − BP;
(2)
where w0 is the radius without self-focusing, BP represents the amount of beam decrease, and B is an empirical constant proportional to the sample’s nonlinearity, n2 . Since the laser irradiance is inversely proportional to the square of the beam radius, we propose the following empirical equation:
I 3ω sample
P A 1 − BP2
3
;
(3)
where A is a constant that depends on the particular sample. The quantity BP changes the slope of the logarithmic THG intensity versus the input power curve, while A adjusts its offset. The self-focusing of light occurs when a laser beam with a Gaussian transverse profile propagates through a medium whose refractive index is given by n n0 n2 I; © 2013 Optical Society of America
(4)
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OPTICS LETTERS / Vol. 38, No. 23 / December 1, 2013
Fig. 1. Schematic self-focusing action represented by the dashed line for beam focused at the (a) exit and (b) entrance interfaces.
where n2 is the nonlinear refractive index proportional to χ 3 sample [14]. For a positive nonlinearity, as is usual for the instantaneous electronic polarization of optical glasses, the refractive index is larger at the center of the laser beam than at the border, resulting in a positive lens whose focal length depends on the laser irradiance and sample nonlinearity [15]. In the THG Z-scan method, the interfaces translate through the laser focal plane. If the beam is tightly focused at the exit interface [Fig. 1(a)], the sample is in the prefocus region, and the nonlinear interaction occurs just in this region. On other hand, when the beam is focused at the entrance interface [Fig. 1(b)], only the post-focus region is important. It is worth mentioning that, in both cases, THG and selffocusing effects occur in a typical Rayleigh range, zR , where the laser irradiance is strong enough to induce such nonlinear effects. As seen in Fig. 1(a), stronger THG is expected due to the reduction of the beam waist at the exit interface. Furthermore, the THG at the entrance interface is not enhanced because the self-focusing changes the beam waist just after the beam propagates by zR , where the irradiance is smaller due to natural beam expansion. In order to verify the validity of the model proposed, we carried out THG Z-scan measurements using a commercial optical parametric amplifier (OPA) (TOPAS, from Light Conversion), synchronously pumped by a chirped-pulse-amplified system (CPA 2001-Clark MXR) with a 775 nm central wavelength and 150 fs pulse duration at 1 kHz repetition rate. The OPA was set to generate pulses at the near IR range (λ 1300 nm) with about 120 fs pulse duration. In this way, the TH is centered on 433 nm. In these wavelengths, the samples used here (silica, BK7, SK11, and F2) present no significant absorption. As the Z-scan method requires, the sample translates through the focus of the laser beam located at z 0. Negative values of z correspond to sample positions between the focusing lens and its focal plane such
that the exit (entrance) interface appears at the negative (positive) z direction. The laser power was carefully adjusted with a calcite polarizer in the range from 0.3 to 0.8 mW (I 0 ∼ 2.4 to 6.4 × 1016 W∕m2 ) and was then focused with a 50 mm focal length lens such that the Rayleigh range was shorter than the sample thickness, allowing us to resolve the two THG signal peaks. To avoid damage to the detector, we used three dichroic mirrors to separate the strong fundamental beam from the TH beam. A lens placed after the sample was used to direct most of the TH light into a large-area silicon PIN detector coupled to a lock-in amplifier. We measured the three optical glasses listed in Table 1 and used fused silica as a reference because it has well-known nonlinearity (n2 2.71 × 10−20 m2 ∕W) [16]. Figure 2 shows that THG peaks generated at the exit are systematically stronger than those generated at the entrance interface for all samples measured, regardless of the power used. All curves exhibit a stronger peak at the exit interface (negative z) and a smaller one at the entrance interface (positive z). Since the samples measured present different linear and nonlinear refractive indices, they can provide good insight on the working principles of THG at interfaces. Silica and F2 have the lowest and the highest linear and nonlinear index of refraction, respectively, while BK7 and SK11 have intermediate values. Z-scan measurements as a function of laser power have similar THG curve shapes, and so we considered only the peak values at the interfaces to establish the power law dependence through the log–log plots of Fig. 3. The solid lines, corresponding to Eq. (3) of our model that takes into account the beam waist decrease, fit very well the experimental data. The slopes of the THG for entrance interfaces (red circles) are exactly three while the slopes for exit interfaces (black squares) are slightly higher than three. As expected, F2 presents the uppermost slope because it has the highest nonlinear refractive index. The fitting of the experimental data with Eq. (3) allows finding the sample nonlinearity using the value of silica as a reference. By comparing the relative B values of each fitting, one can estimate n2 . Table 1 shows the results achieved by this procedure (THG slope). As we mentioned earlier, the beam radius should change as a function of sample nonlinearity and input power [15]. This can be corroborated by noticing that the THG peaks at the exit interfaces are systematically narrower than those generated at the entrance interfaces. Figure 4 presents the normalized THG peaks obtained for silica at a laser power of 0.8 mW, where the
Table 1. Parameters of Optical Glasses Used in Measurementsa n0 ne n0 Sample (433 nm) (546.1 nm) (1300 nm) Silica BK7 SK11 F2
1.46693 1.52697 1.57564 1.64271
1.46008 1.51872 1.56605 1.62408
1.44692 1.50370 1.54965 1.59813
Thickness (mm) 1.225 0.925 1.155 1.165
n2 (theory) from Eqs. (6) n2 (Experimental) n2 (Experimental) and (7) ×2.53× THG Slope ×2.71× THG Width ×2.71× 10−20 m2 ∕W 10−20 m2 ∕W 10−20 m2 ∕W 1 1.3 1.5 3.8
1 1.7 1.8 3.8
1 1.6 1.7 2.8
Theoretical and experimental nonlinearities (n2 ) of the samples. The experimental values were obtained from ratios of B values using silica as a reference. a
December 1, 2013 / Vol. 38, No. 23 / OPTICS LETTERS
Silica BK7 SK11 F2
0.8
Normalized THG [arb. units]
THG Intensity [arb. units]
1.0
0.6 0.4 0.2 0.0 -1.0
-0.5
0.0 z (mm)
0.5
different widths can be easily seen. As predicted by the Gaussian beam theory, the waist radius is related to the Rayleigh range and thus, by measuring zR of the TH, we can estimate the fundamental beam waist radius by taking into account that we are dealing with a thirdorder process. Using a Gaussian function to fit the peaks in all sample interfaces, it is possible to follow how the widths change with the laser power. Basically, the widths (FWHM) at the entrance increase as a function of the laser power, while those at the exit decrease. Such behavior can be explained by an equation similar to Eq. (2): W W 0 1 BP;
(5)
where W and W 0 are the THG peak widths with and without self-focusing, respectively. The positive (negative) signal is relative to the entrance (exit) interface. Table 1 shows the results achieved by this procedure (THG width). Exit Entrance
0.8 0.6 0.4 0.2
0.0
0.2
0.4
Fig. 4. Normalized TH peaks generated at the entrance and exit interfaces for silica obtained with 0.8 mW pump power. The points are the experimental results, and the lines are fittings with Gaussian functions.
Despite the noise in the measured widths, Fig. 5 shows a trend of broadening (entrance) and shortening (exit) of the THG peaks. The effect is more evident in F2 because of its high nonlinearity. Broadening and shortening of the THG peaks are explained by the self-focusing effect as the TH is generated near the focal plane. When the nonlinear sample is placed after the focus position (entrance interface), the selffocusing effect tends to recollimate the beam, and zR increases. On other hand, for a sample positioned before the focal plane (exit interface), zR decreases. The broadening of zR at the entrance interface does not affect the fundamental laser irradiance, as most of the TH is generated near that interface, where the self-focusing did not take place yet because of the lack of beam propagation. These explain why the THG at the entrance interfaces follows the cubic power law. In the case of exit interfaces, most of the THG occurs in the vicinity of the output face where the beam waist compression already
(4.0)
0.32 (3)
Silica
BK7
0.01
(5.3)
(4.1)
1
(3)
(3)
0.1
SK11
0.01 0.4
0.6
Exit Entrance
0.28
0.1
THG Peak Width (mm)
THG Signal [arb. units]
-0.2
(3.7)
(3)
0.2
-0.4
Sample Displacement (mm)
Fig. 2. THG Z-scan signal for silica, BK7, SK11, and F2 obtained at a laser power of ∼0.44 mW. An offset of 0.1 between consecutive curves was introduced for better visualization.
1
Exit Entrance
1.0
0.0
1.0
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F2 0.2
0.4
0.6
0.24 0.20 0.16 0.32 0.28 0.24 0.20
0.8
Input Laser Power (mW)
Fig. 3. THG peak values at the interfaces as a function of incident laser power for silica, BK7, SK11, and F2. The symbols are the experimental data, while the solid lines are results obtained with the model proposed. The numbers in parenthesis indicate the slopes.
BK7
Silica
0.16 0.2
SK11 0.4
F2 0.6
0.2
0.4
0.6
0.8
Input Laser Power (mW)
Fig. 5. FWHM of THG peaks measured as a function of the pump laser power for silica, BK7, SK11, and F2 glasses. The widths are obtained by fitting all peaks with a Gaussian function. The solid lines are obtained from Eq. (5).
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took place due to self-focusing and propagation along the Rayleigh distance. Due to this reduction of the beam waist radius, the laser irradiance increases and leads to a THG law with an exponent greater than cubic. In order to compare the measured nonlinearity, we can use the empirical expression given in [17] and [18], where n2 can be theoretically derived from the linear index of refraction, ne (at λ 546.1 nm) and the Abbe number, ve : n2 10−13 esu
68ne − 1n2e 22
h i1∕2 ; 2 e 1 ve 1.517 ne 2n v e 6ne
(6)
which can be converted to SI units according to 2 m 4.19 × 10−7 n2 esu: n2 W ne
(7)
Such theoretical values of n2 , although obtained for the visible range, indicate the right trends of the nonlinearities in these four samples. Considering that there is a typical wide dispersion on the value of the nonlinearity [14,16], it is possible to say that this new method used here leads to good values of the nonlinearities, and it allows making further comparison with other values obtained by other methods. Table 1 summarizes the results obtained by our proposed method to understand the THG at interfaces. The slightly higher nonlinearities obtained with the slope may indicate a small contribution of the Fresnel reflection interference [5]. In conclusion, we have studied the THG at interfaces of different optical glasses using fs laser pulses. The TH intensities asymmetry could be explained by the selffocusing effect, which increases the laser irradiance at the output interface due to the reduction of the beam waist. This study is important for fundamental understanding of the THG process in transparent materials at the tightly focused condition. Also, for the application’s
point of view, it is important for studying the material’s third-order nonlinearity [1] and its use in the THG scanning laser microscopy [2,4]. This work was supported by the FAPESP (grant 2011/ 22787-7) and CNPq from Brazil.
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