August 15, 2014 / Vol. 39, No. 16 / OPTICS LETTERS
4847
Optical limiting properties of a nonlinear multilayer Fabry–Perot resonator containing niobium pentoxide as nonlinear medium Anton A. Ryzhov,1,2,* Inna M. Belousova,1,2 Yanzhi Wang,3 Hongji Qi,3 and Jun Wang3 1
Institute for Laser Physics, S. I. Vavilov State Optical Insitute, Saint Petersburg 199034, Russia 2
3
Laser Optics Department, University ITMO, Saint Petersburg 197101, Russia
Key Laboratory of Materials for High Power Laser, Institute of Optics and Fine Mechanics, Shanghai 201800, China *Corresponding author: [email protected] Received May 20, 2014; revised July 15, 2014; accepted July 15, 2014; posted July 15, 2014 (Doc. ID 212404); published August 13, 2014 The optical limiting effect was numerically simulated and experimentally observed for a 25-layer thin-film Fabry– Perot microresonator by 7 ns laser pulses at 532 nm. The sample, made by vacuum evaporation and consisting of alternating Nb2 O5 and SiO2 layers, has an ultranarrow line of transparency at near 532 nm within a wide spectral band of reflection. By adjusting simulated results in accordance with experimental dependencies of transmittance, reflectance, and absorbance on incident light intensity, the coefficient of optical nonlinearity of Nb2 O5 was estimated at 6 1i · 10−12 cm2 ∕W. © 2014 Optical Society of America OCIS codes: (230.1150) All-optical devices; (230.4170) Multilayers; (250.6715) Switching; (310.6845) Thin film devices and applications. http://dx.doi.org/10.1364/OL.39.004847
In a nonlinear Fabry–Perot resonator indices of reflection and absorption of a medium put between the mirrors depend on light intensity. Such resonators have been well studied since the second half of the 1970s. Theoretically predicted effects of optical bistability, differential gain, and limiting have been experimentally observed for resonators containing Na vapor [1], nonlinear liquids (liquid crystals, nitrobenzene, CS2 ) [2], solid plates of GaAs [3], and InSb [4]. A detailed review of these works have been presented in [5]. However, those resonators were laboratory devices consisting of separate mirrors and a nonlinear medium between them. Multilayer thin-film structures seem to be more applicable for creation of low-threshold nonlinear optical devices because of their small thickness (comparable with a light wavelength) and wholeness. Linear multilayer coatings have been well studied and widely used in optical systems for years. A variety of works have dealt with nonlinear properties of both single and multilayer films [6–13]; however, multilayer optical coatings are still not in use as nonlinear devices. In general, application of a Fabry–Perot resonator for creation of a low-threshold nonlinear optical device is promising for two reasons. Firstly, the resonator is a narrowband optical filter, for which spectral position of the line of transparency depends on the optical distance between the mirrors not on the medium resonant properties. With it a small change in the refractive index of a medium contained between the mirrors results in a significant spectral shift of the line. A small change in the absorption index results in a significant transmittance (T) change at the peak of the line (however, in this case a decrease in T results more from an increase in reflectance (R) than from an increase in absorbance (A) of the whole structure). Secondly, light intensity at the resonant wavelength increases many-fold in the space between the mirrors by interference, which leads to a corresponding decrease in the nonlinear threshold. 0146-9592/14/164847-04$15.00/0
We study the potential of multilayer Fabry–Perot resonators in the capacity of quick-response onewavelength optical limiters. Such limiters can be useful in a variety of laser systems; for example, in a laser rangefinder to protect the detector from intensive reflected (backscattered) radiation. Currently, there is a lack of experimental works realizing the suggested approach, whereas interest in optical power limiting is very active. Husaini et al. [14] have experimentally observed for a nonresonant multilayer structure a decrease in the limiting threshold. The decrease results from an interference several-fold increase of optical-field intensity inside the layers with metal nanoparticles. In this case, sharp spectral characteristics of the structure do not contribute to the nonlinear T characteristic, since the operating wavelength is much above the upper bandgap edge, whereas application of a Fabry–Perot resonator enables it to increase field intensity within its middle layer by several orders of magnitude. In this Letter, we present the results of an experimental and numerical study of nonlinear characteristics of a multilayer resonator at 532 nm. The resonator is made as an optical coating composed of 25 alternating layers of Nb2 O5 and SiO2 [Fig. 1]. The middle Nb2 O5 layer is the thickest and forms the resonator cavity. The formula of the structure is HL6 H 2 LH6 ;
(1)
where H and L indicate layers with high and low refractive indices, respectively. Both are of λ0 ∕4 optical thickness, with λ0 532 nm being the nominal resonant wavelength. The substrate is a 30 mm diameter, 3 mm thickness-fused silica plate. The root-mean-squared roughness of the substrate, measured by an optical profiler, is about 0.5 nm. A set of the thin-film resonators was fabricated at the Shanghai Institute of Optics and Fine Mechanics CAS by © 2014 Optical Society of America
4848
OPTICS LETTERS / Vol. 39, No. 16 / August 15, 2014 Nb2O5 SiO2 Six pairs of λ/4 layers
Nb2O5
SiO2
Middle λ/2 layer
Six pairs of λ/4 layers
Fig. 3. Transmittance dependence on the angle of incidence at low-level radiation at 532 nm.
Nb2O5 SiO2 substrate
Fig. 1.
Structure of the studied multilayer resonator.
ion-assisted ion-beam sputtering. The coating rate was 0.2 nm∕s. The refractive indices of deposited Nb2 O5 and SiO2 layers at 532 nm are 2.31 and 1.48, respectively. These values were determined from the T and R spectra of corresponding single layers. In Fig. 2 the solid line represents the T spectrum of one of the samples in a small vicinity of the transparency line, measured by a spectrophotometer at normal incidence. The figure shows that the real peak of T takes place at 532.4 nm, which slightly exceeds the nominal λ0 532 nm. It does not create a barrier for experimental study at 532 nm because it is possible to shift the resonant wavelength a little lower by giving a tilt to the resonator. The reason is that the optical length difference in the middle layer is proportional to the cosine of the angle of refraction. Figure 3 represents both the experimental and calculated dependencies of T on the angle of incidence at lowlevel 532 nm laser radiation. The measurement was taken by using a pulsed Nd:YAG laser at s-polarization. The maximum of T (T max ) occurs at 4° deviation from the normal incidence position. According to our calculations, taking into account an inaccuracy of the middle-layer thickness, at normal incidence to such a resonator the T max is to be at 532.4 nm, which corresponds to Fig. 2. Figure 3 also shows that T max for the laser beam is about 75%. The spectrophotometric measurement has a lower value. We rely on the value given by the laser
beam and attribute this mismatch to lack of collinear beam configuration in the spectrophotometer. In the absence of absorption, T max of an ideal multilayer resonator is equal to T of the interface between air and the substrate, i.e., 0.96 at the substrate refractive index ns 1.5. In principle, a decrease in T max can be caused by two factors: some extinction in the layers and a difference between reflectances of the mirrors surrounding the middle layer. In the latter case, only an increase in reflectance of the whole structure will occur. By the experimental setup [Fig. 4] we determined 75% T and 15% R at the peak, which indicates a presence of both factors. According to the literary data [15,16], Nb2 O5 films have a significant index of absorption in the visible region. Calculating the T and R spectral characteristics, we selected the imaginary part of the refractive index to model absorption. The refractive index of Nb2 O5 was finally set as nH 2.31 2 · 10−4 i, which at 532 nm is equivalent to the extinction coefficient α 4.8 mm−1 . At the same time, to model a difference between R of the mirrors and the shift of the peak spectral position (from nominal 532 nm to real 532.4 nm), some deviations (under 15%) of layer thicknesses from the nominal values were introduced. Finally, calculation parameters of the resonator were set in such a way that at low intensities the calculated results correspond to the experimental ones as far as possible [Figs. 2 and 3]. Figure 5 represents both the experimental and calculated nonlinear dependences of T, R, and A at three different angles of incidence. A pulsed single-mode Nd:YAG laser was used as a light source (up to 6 mJ pulse energy, 7…10 ns pulse length, and 2 mm beam diameter at 0.86 energy level). The incident pulse energy was regulated by attenuator F1 [Fig. 4].
D2 D3 wedge
F2 flat mirror
Nd:YAG 532 nm
D1 Fig. 2. Transmittance spectrum of the sample in the vicinity of the transparency line at low-level radiation.
F1
sample
Fig. 4. Experimental setup for synchronous measurements of T and R nonlinear dependencies. D1, D2, D3—pulse energy detectors; F1, F2—calibrated sets of filters.
August 15, 2014 / Vol. 39, No. 16 / OPTICS LETTERS
Fig. 5. Calculated and experimental nonlinear characteristics of transmittance (upper plot, black lines), absorbance (upper plot, gray lines), and reflectance (lower plot) at three different angles of incidence.
The light propagation time through the structure is several orders less than the pulse duration. The Nb2 O5 nonlinear response time was also supposed to be much less, so that the nonlinear dependences of T, R, and A were initially calculated as functions of instantaneous incident light intensity. To compare these calculated characteristics with experimental results, the former were reduced to the dependencies on pulse energy by area and time integration (taking spatial and time pulse distributions). In the calculations, SiO2 was assumed to be a linear material and Nb2 O5 a third-order nonlinear material: n n0 n2 I;
(2)
where n0 is the linear part of the Nb2 O5 refractive index, I is the light intensity (W∕cm2 ), and n2 is the nonlinear coefficient (cm2 ∕W). In [17] we described our fielddistribution computing method, which takes into account the dependence of n on I. Although for the current work we developed the method significantly, having switched over to a “right-to-left” computation (i.e., to a calculation of output–input characteristics instead of input–output), for the sake of brevity we do not describe that here. By analogy with the imaginary part of n0 , n2 was fitted so that the calculated characteristics are in accordance
4849
with the experimental results in the best way. The nonlinear coefficient was assumed to be n2 6 1i · 10−12 cm2 ∕W. Such a high n2 is an interesting result by itself. The nonzero imaginary part defines nonlinear absorption. Fitting n2 , we gave preference to a correspondence of 4° characteristics, which were finally well enough. By that we can also see a good correspondence of 2° characteristics, but at 5° the experimental points noticeably deviate from the calculated curve. According to the calculations, at 5°, as E increases, T increases, too, but then decreases. R behaves oppositely, which is easy to understand: Initially the optical thickness of the middle layer is a bit less than λ0 ∕2, and the real part of n increases with increasing E, which means the optical thickness approaches λ0 ∕2 from below for a while, then moves away. At the same time, the imaginary part of n also increases, partly leveling the increase in T at the first part. At the n2 value given above, the calculated T characteristic shows the increase from 45% to 55%, but we did not observe such an increase in the experiment. It could be explained by stronger absorption, but using a larger value of the imaginary part of n in the calculations leads to significant divergences at 2° and 4°. This fact will be investigated further. The maximal pulse energy used in the experiment was about 2 mJ. Such pulses did not cause any irreversible change of the sample. At larger energy values damages of the sample were observed. In conclusion, the presented results obviously demonstrate the optical limiting effect, accompanied with the increase in reflectance, which results from the spectral shift of the transparency line. Although the value of n2 for an Nb2 O5 thin film is not proved by another experiment, the good correspondence between the experimental and calculated nonlinear characteristics indicates that the calculation model is adequate. As opposed to “thick” Fabry–Perot resonators, in which the distance between the mirrors is several orders longer than an optical wavelength, in thin-film resonators that distance is of the same order as the wavelength. As a result, a thin-film resonator gives a relatively wide transparency line, which is usually the sole one within the bandgap of the mirrors. At the same time, transparency-line narrowing leads to a decrease in the nonlinear threshold. At a given resonator interval, the higher the R of the mirrors, the narrower the transparency line. R of the mirrors can be raised by increasing the number of layers. Although optical coatings have been widely used for a long time, making a high-quality filter with a transparency line under 1 nm of width still presents technological difficulties, mainly based on inadequate precision of film-thickness controlling. This work was supported by the Russian Foundation for Basic Research (No. 14-02-00851), the Government of Russian Federation (No. 074-U01), the National Natural Science Foundation of China (No. 61178007), and Science and Technology Commission of Shanghai Municipality (Nano Project No. 11nm0502400, Shanghai Pujiang Program 12PJ1409400). J. W. thanks the financial supports from the National 10000-Talent Program, CAS 100-Talent Program.
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OPTICS LETTERS / Vol. 39, No. 16 / August 15, 2014
References 1. H. Gibbs, S. McCall, and T. Venkatesan, Phys. Rev. Lett. 36, 1135 (1976). 2. T. Bischofberger and Y. R. Shen, Appl. Phys. Lett. 32, 156 (1978). 3. H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann, Appl. Phys. Lett. 35, 451 (1979). 4. D. Miller, S. D. Smith, and A. Johnston, Appl. Phys. Lett. 35, 658 (1979). 5. E. Abraham and S. D. Smith, Meas. Sci. Technol. 15, 33 (1982). 6. E. Lidorikis, Q. Li, and C. Soukoulis, Phys. Rev. B 54, 10249 (1996). 7. E. Lidorikis, K. Busch, Q. Li, C. Chan, and C. Soukoulis, Phys. Rev. B 56, 15090 (1997).
8. R. Wang, J. Dong, and D. Y. Xing, Phys. Rev. E 55, 6301 (1997). 9. P. Tran, Opt. Lett. 21, 1138 (1996). 10. P. Hou, Y. Chen, J. Shi, M. Shen, X. Chen, and Q. Wang, Europhys. Lett. 81, 64003 (2008). 11. Q. Li, C. Chan, K. Ho, and C. Soukoulis, Phys. Rev. B 53, 15577 (1996). 12. X. Liu, J. W. Haus, and M. S. Shahriar, Opt. Express 17, 2696 (2009). 13. J. He and M. Cada, Appl. Phys. Lett. 61, 2150 (1992). 14. S. Husaini, H. Teng, and V. M. Menon, Appl. Phys. Lett. 101, 111103 (2012). 15. T. Hashimoto and T. Yoko, Appl. Opt. 34, 2941 (1995). 16. C.-C. Lee, C.-L. Tien, and J.-C. Hsu, Appl. Opt. 41, 2043 (2002). 17. I. M. Belousova and A. A. Ryzhov, Opt. Spectrosc. 112, 902 (2012).
4847
Optical limiting properties of a nonlinear multilayer Fabry–Perot resonator containing niobium pentoxide as nonlinear medium Anton A. Ryzhov,1,2,* Inna M. Belousova,1,2 Yanzhi Wang,3 Hongji Qi,3 and Jun Wang3 1
Institute for Laser Physics, S. I. Vavilov State Optical Insitute, Saint Petersburg 199034, Russia 2
3
Laser Optics Department, University ITMO, Saint Petersburg 197101, Russia
Key Laboratory of Materials for High Power Laser, Institute of Optics and Fine Mechanics, Shanghai 201800, China *Corresponding author: [email protected] Received May 20, 2014; revised July 15, 2014; accepted July 15, 2014; posted July 15, 2014 (Doc. ID 212404); published August 13, 2014 The optical limiting effect was numerically simulated and experimentally observed for a 25-layer thin-film Fabry– Perot microresonator by 7 ns laser pulses at 532 nm. The sample, made by vacuum evaporation and consisting of alternating Nb2 O5 and SiO2 layers, has an ultranarrow line of transparency at near 532 nm within a wide spectral band of reflection. By adjusting simulated results in accordance with experimental dependencies of transmittance, reflectance, and absorbance on incident light intensity, the coefficient of optical nonlinearity of Nb2 O5 was estimated at 6 1i · 10−12 cm2 ∕W. © 2014 Optical Society of America OCIS codes: (230.1150) All-optical devices; (230.4170) Multilayers; (250.6715) Switching; (310.6845) Thin film devices and applications. http://dx.doi.org/10.1364/OL.39.004847
In a nonlinear Fabry–Perot resonator indices of reflection and absorption of a medium put between the mirrors depend on light intensity. Such resonators have been well studied since the second half of the 1970s. Theoretically predicted effects of optical bistability, differential gain, and limiting have been experimentally observed for resonators containing Na vapor [1], nonlinear liquids (liquid crystals, nitrobenzene, CS2 ) [2], solid plates of GaAs [3], and InSb [4]. A detailed review of these works have been presented in [5]. However, those resonators were laboratory devices consisting of separate mirrors and a nonlinear medium between them. Multilayer thin-film structures seem to be more applicable for creation of low-threshold nonlinear optical devices because of their small thickness (comparable with a light wavelength) and wholeness. Linear multilayer coatings have been well studied and widely used in optical systems for years. A variety of works have dealt with nonlinear properties of both single and multilayer films [6–13]; however, multilayer optical coatings are still not in use as nonlinear devices. In general, application of a Fabry–Perot resonator for creation of a low-threshold nonlinear optical device is promising for two reasons. Firstly, the resonator is a narrowband optical filter, for which spectral position of the line of transparency depends on the optical distance between the mirrors not on the medium resonant properties. With it a small change in the refractive index of a medium contained between the mirrors results in a significant spectral shift of the line. A small change in the absorption index results in a significant transmittance (T) change at the peak of the line (however, in this case a decrease in T results more from an increase in reflectance (R) than from an increase in absorbance (A) of the whole structure). Secondly, light intensity at the resonant wavelength increases many-fold in the space between the mirrors by interference, which leads to a corresponding decrease in the nonlinear threshold. 0146-9592/14/164847-04$15.00/0
We study the potential of multilayer Fabry–Perot resonators in the capacity of quick-response onewavelength optical limiters. Such limiters can be useful in a variety of laser systems; for example, in a laser rangefinder to protect the detector from intensive reflected (backscattered) radiation. Currently, there is a lack of experimental works realizing the suggested approach, whereas interest in optical power limiting is very active. Husaini et al. [14] have experimentally observed for a nonresonant multilayer structure a decrease in the limiting threshold. The decrease results from an interference several-fold increase of optical-field intensity inside the layers with metal nanoparticles. In this case, sharp spectral characteristics of the structure do not contribute to the nonlinear T characteristic, since the operating wavelength is much above the upper bandgap edge, whereas application of a Fabry–Perot resonator enables it to increase field intensity within its middle layer by several orders of magnitude. In this Letter, we present the results of an experimental and numerical study of nonlinear characteristics of a multilayer resonator at 532 nm. The resonator is made as an optical coating composed of 25 alternating layers of Nb2 O5 and SiO2 [Fig. 1]. The middle Nb2 O5 layer is the thickest and forms the resonator cavity. The formula of the structure is HL6 H 2 LH6 ;
(1)
where H and L indicate layers with high and low refractive indices, respectively. Both are of λ0 ∕4 optical thickness, with λ0 532 nm being the nominal resonant wavelength. The substrate is a 30 mm diameter, 3 mm thickness-fused silica plate. The root-mean-squared roughness of the substrate, measured by an optical profiler, is about 0.5 nm. A set of the thin-film resonators was fabricated at the Shanghai Institute of Optics and Fine Mechanics CAS by © 2014 Optical Society of America
4848
OPTICS LETTERS / Vol. 39, No. 16 / August 15, 2014 Nb2O5 SiO2 Six pairs of λ/4 layers
Nb2O5
SiO2
Middle λ/2 layer
Six pairs of λ/4 layers
Fig. 3. Transmittance dependence on the angle of incidence at low-level radiation at 532 nm.
Nb2O5 SiO2 substrate
Fig. 1.
Structure of the studied multilayer resonator.
ion-assisted ion-beam sputtering. The coating rate was 0.2 nm∕s. The refractive indices of deposited Nb2 O5 and SiO2 layers at 532 nm are 2.31 and 1.48, respectively. These values were determined from the T and R spectra of corresponding single layers. In Fig. 2 the solid line represents the T spectrum of one of the samples in a small vicinity of the transparency line, measured by a spectrophotometer at normal incidence. The figure shows that the real peak of T takes place at 532.4 nm, which slightly exceeds the nominal λ0 532 nm. It does not create a barrier for experimental study at 532 nm because it is possible to shift the resonant wavelength a little lower by giving a tilt to the resonator. The reason is that the optical length difference in the middle layer is proportional to the cosine of the angle of refraction. Figure 3 represents both the experimental and calculated dependencies of T on the angle of incidence at lowlevel 532 nm laser radiation. The measurement was taken by using a pulsed Nd:YAG laser at s-polarization. The maximum of T (T max ) occurs at 4° deviation from the normal incidence position. According to our calculations, taking into account an inaccuracy of the middle-layer thickness, at normal incidence to such a resonator the T max is to be at 532.4 nm, which corresponds to Fig. 2. Figure 3 also shows that T max for the laser beam is about 75%. The spectrophotometric measurement has a lower value. We rely on the value given by the laser
beam and attribute this mismatch to lack of collinear beam configuration in the spectrophotometer. In the absence of absorption, T max of an ideal multilayer resonator is equal to T of the interface between air and the substrate, i.e., 0.96 at the substrate refractive index ns 1.5. In principle, a decrease in T max can be caused by two factors: some extinction in the layers and a difference between reflectances of the mirrors surrounding the middle layer. In the latter case, only an increase in reflectance of the whole structure will occur. By the experimental setup [Fig. 4] we determined 75% T and 15% R at the peak, which indicates a presence of both factors. According to the literary data [15,16], Nb2 O5 films have a significant index of absorption in the visible region. Calculating the T and R spectral characteristics, we selected the imaginary part of the refractive index to model absorption. The refractive index of Nb2 O5 was finally set as nH 2.31 2 · 10−4 i, which at 532 nm is equivalent to the extinction coefficient α 4.8 mm−1 . At the same time, to model a difference between R of the mirrors and the shift of the peak spectral position (from nominal 532 nm to real 532.4 nm), some deviations (under 15%) of layer thicknesses from the nominal values were introduced. Finally, calculation parameters of the resonator were set in such a way that at low intensities the calculated results correspond to the experimental ones as far as possible [Figs. 2 and 3]. Figure 5 represents both the experimental and calculated nonlinear dependences of T, R, and A at three different angles of incidence. A pulsed single-mode Nd:YAG laser was used as a light source (up to 6 mJ pulse energy, 7…10 ns pulse length, and 2 mm beam diameter at 0.86 energy level). The incident pulse energy was regulated by attenuator F1 [Fig. 4].
D2 D3 wedge
F2 flat mirror
Nd:YAG 532 nm
D1 Fig. 2. Transmittance spectrum of the sample in the vicinity of the transparency line at low-level radiation.
F1
sample
Fig. 4. Experimental setup for synchronous measurements of T and R nonlinear dependencies. D1, D2, D3—pulse energy detectors; F1, F2—calibrated sets of filters.
August 15, 2014 / Vol. 39, No. 16 / OPTICS LETTERS
Fig. 5. Calculated and experimental nonlinear characteristics of transmittance (upper plot, black lines), absorbance (upper plot, gray lines), and reflectance (lower plot) at three different angles of incidence.
The light propagation time through the structure is several orders less than the pulse duration. The Nb2 O5 nonlinear response time was also supposed to be much less, so that the nonlinear dependences of T, R, and A were initially calculated as functions of instantaneous incident light intensity. To compare these calculated characteristics with experimental results, the former were reduced to the dependencies on pulse energy by area and time integration (taking spatial and time pulse distributions). In the calculations, SiO2 was assumed to be a linear material and Nb2 O5 a third-order nonlinear material: n n0 n2 I;
(2)
where n0 is the linear part of the Nb2 O5 refractive index, I is the light intensity (W∕cm2 ), and n2 is the nonlinear coefficient (cm2 ∕W). In [17] we described our fielddistribution computing method, which takes into account the dependence of n on I. Although for the current work we developed the method significantly, having switched over to a “right-to-left” computation (i.e., to a calculation of output–input characteristics instead of input–output), for the sake of brevity we do not describe that here. By analogy with the imaginary part of n0 , n2 was fitted so that the calculated characteristics are in accordance
4849
with the experimental results in the best way. The nonlinear coefficient was assumed to be n2 6 1i · 10−12 cm2 ∕W. Such a high n2 is an interesting result by itself. The nonzero imaginary part defines nonlinear absorption. Fitting n2 , we gave preference to a correspondence of 4° characteristics, which were finally well enough. By that we can also see a good correspondence of 2° characteristics, but at 5° the experimental points noticeably deviate from the calculated curve. According to the calculations, at 5°, as E increases, T increases, too, but then decreases. R behaves oppositely, which is easy to understand: Initially the optical thickness of the middle layer is a bit less than λ0 ∕2, and the real part of n increases with increasing E, which means the optical thickness approaches λ0 ∕2 from below for a while, then moves away. At the same time, the imaginary part of n also increases, partly leveling the increase in T at the first part. At the n2 value given above, the calculated T characteristic shows the increase from 45% to 55%, but we did not observe such an increase in the experiment. It could be explained by stronger absorption, but using a larger value of the imaginary part of n in the calculations leads to significant divergences at 2° and 4°. This fact will be investigated further. The maximal pulse energy used in the experiment was about 2 mJ. Such pulses did not cause any irreversible change of the sample. At larger energy values damages of the sample were observed. In conclusion, the presented results obviously demonstrate the optical limiting effect, accompanied with the increase in reflectance, which results from the spectral shift of the transparency line. Although the value of n2 for an Nb2 O5 thin film is not proved by another experiment, the good correspondence between the experimental and calculated nonlinear characteristics indicates that the calculation model is adequate. As opposed to “thick” Fabry–Perot resonators, in which the distance between the mirrors is several orders longer than an optical wavelength, in thin-film resonators that distance is of the same order as the wavelength. As a result, a thin-film resonator gives a relatively wide transparency line, which is usually the sole one within the bandgap of the mirrors. At the same time, transparency-line narrowing leads to a decrease in the nonlinear threshold. At a given resonator interval, the higher the R of the mirrors, the narrower the transparency line. R of the mirrors can be raised by increasing the number of layers. Although optical coatings have been widely used for a long time, making a high-quality filter with a transparency line under 1 nm of width still presents technological difficulties, mainly based on inadequate precision of film-thickness controlling. This work was supported by the Russian Foundation for Basic Research (No. 14-02-00851), the Government of Russian Federation (No. 074-U01), the National Natural Science Foundation of China (No. 61178007), and Science and Technology Commission of Shanghai Municipality (Nano Project No. 11nm0502400, Shanghai Pujiang Program 12PJ1409400). J. W. thanks the financial supports from the National 10000-Talent Program, CAS 100-Talent Program.
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OPTICS LETTERS / Vol. 39, No. 16 / August 15, 2014
References 1. H. Gibbs, S. McCall, and T. Venkatesan, Phys. Rev. Lett. 36, 1135 (1976). 2. T. Bischofberger and Y. R. Shen, Appl. Phys. Lett. 32, 156 (1978). 3. H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann, Appl. Phys. Lett. 35, 451 (1979). 4. D. Miller, S. D. Smith, and A. Johnston, Appl. Phys. Lett. 35, 658 (1979). 5. E. Abraham and S. D. Smith, Meas. Sci. Technol. 15, 33 (1982). 6. E. Lidorikis, Q. Li, and C. Soukoulis, Phys. Rev. B 54, 10249 (1996). 7. E. Lidorikis, K. Busch, Q. Li, C. Chan, and C. Soukoulis, Phys. Rev. B 56, 15090 (1997).
8. R. Wang, J. Dong, and D. Y. Xing, Phys. Rev. E 55, 6301 (1997). 9. P. Tran, Opt. Lett. 21, 1138 (1996). 10. P. Hou, Y. Chen, J. Shi, M. Shen, X. Chen, and Q. Wang, Europhys. Lett. 81, 64003 (2008). 11. Q. Li, C. Chan, K. Ho, and C. Soukoulis, Phys. Rev. B 53, 15577 (1996). 12. X. Liu, J. W. Haus, and M. S. Shahriar, Opt. Express 17, 2696 (2009). 13. J. He and M. Cada, Appl. Phys. Lett. 61, 2150 (1992). 14. S. Husaini, H. Teng, and V. M. Menon, Appl. Phys. Lett. 101, 111103 (2012). 15. T. Hashimoto and T. Yoko, Appl. Opt. 34, 2941 (1995). 16. C.-C. Lee, C.-L. Tien, and J.-C. Hsu, Appl. Opt. 41, 2043 (2002). 17. I. M. Belousova and A. A. Ryzhov, Opt. Spectrosc. 112, 902 (2012).
