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Through analysis of near-field beam profiles, we propose a method using Shannon entropy to assess the development of small-scale self-focusing during laser propagation and amplification in high-power laser systems. In this method, the entropy curve that corresponds to increasing B integral displays an evident turning point at which small-scale self-focusing starts to rapidly develop. In contrast to classical methods using contrast, modulation, or power spectral density, the proposed method provides the B integral criterion more clearly and objectively. This approach is an optimization method that can be utilized in the design and operation of high-power laser systems.
© 2016 Optical Society of America
1. Introduction
Localized, extremely high-intensity modulation induced by small-scale self-focusing (SSFF) leads to serious optics damage in high-power laser systems for inertial confinement fusion; this damage limits the energy output and threatens the safe operation of the system [1–5]. Assessing SSFF development and judging the occurrence of SSFF is therefore necessary for ensuring a “safe” high-power laser system design. The two main influence factors on SSFF of output beam in high-power laser systems are the noise sources and the nonlinear effect that accumulates during laser propagation and amplification. Break-up integral (B integral) represents the nonlinear phase shift and depends on the intensity, medium properties and structures of system [1,6,7]. For a particular high-power laser system the noise sources are relatively unchanged. Hence, B integral corresponding to that SSFF begins to rapidly develop is usually treated as criterion of a high-power laser system [7–10]. And it is needed to find out a method to provide the B integral criterion for such a system. Several classical methods can be applied to identify the B integral criterion in high-power laser systems. These methods include analyzing changes in near-field contrast, modulation [11], and power spectral density (PSD) [12] with increasing B integral. However, these methods do not provide a clear criterion but yield a curve that varies with B integral. Criterion, such as in National Ignition Facility (NIF) [3], is based on human experiential judgment in some degree. Therefore, a method for assessment of SSFF development and establishment of a corresponding B integral criterion must be proposed.
Classical methods reflect SSFF development by analyzing changes in near-field images in terms of contrast, modulation, or PSD curves as a function of increasing B integral. In image processing, Shannon entropy [13] is an important index that reveals image information. This index is widely used in medical and biological imaging, image quality assessment, image recognition, image reconstruction, color rendering, and semiconductor lasers [14–19]. Nevertheless, few studies have used Shannon entropy to represent SSFF development in high-power laser systems.
In this paper, we investigated SSFF development by using Shannon entropy. With increasing B integral, the curve of Shannon entropy displays an evident turning point at which SSFF begins to rapidly develop. Compared with classical analyses using contrast, modulation, and PSD, the proposed method provides an alternative, intuitive, and objective means to obtain the B integral criterion for the design and operation of high-power laser systems.
2. Theoretical analysis
Shannon entropy is a characterization index that depends on the probability distribution of statistical quantities. For the probability distribution , Shannon entropy is defined as:
This equation holds the following assumptions: if and only if all but one are zero, the entropy has a minimum value of zero; the entropy has a maximum value of for a given N, when all are equal (i.e., 1/N); and any change toward equalization of the probabilities increases the entropy [13]. Shannon entropy can be perceived as related to changes in probability distribution, especially the probability distribution equalization.High-frequency modulation in high-power laser systems exhibits extremely fast nonlinear growth under the upgrade of operating fluence; this process is known as SSFF. With the development of SSFF, the beam quality will correspondingly degrade. And the fluence distribution of beam will also change. As illustrated in Figs. 1(a) and 1(b), the entire output near-field is smooth and almost in the low-fluence state when the system is operated under low fluence; moreover, the SSFF effect is minimal. With increasing operating fluence, SSFF develops, and the output near-field consists of relatively low- and high-fluence parts. Finally, SSFF intensifies when the operating fluence is extremely high. In this case, many parts of the output near-field are maintained in a relatively high-fluence state. We performed statistical analysis on fluence for each output near-field. With SSFF development, the percentage of relatively high fluence increases gradually and that of relatively low fluence decreases. Consequently, the fluence probability distribution moves towards equalization and then reverts back to its relatively unequal state [Fig. 1(c)]. Then, we could calculate the Shannon entropy from the probability distribution of fluence. Given the properties of Shannon entropy, the Shannon entropy calculated from the fluence probability distribution will increase first and then decrease with SSFF development. Hence, a maximum point in the entropy curve will be formed as SSFF starts to develop rapidly. And the B integral criterion of a high-power laser system can be identified.
Fig. 1 (a) Different output near-field profiles, (b) corresponding fluence distribution along the horizontal direction, and (c) schematic of changes in fluence probability distribution with varying B integrals.
3. Numerical simulation and experiment
3.1 Numerical simulation
We adopt the SG99 code [21] to simulate a prototype of the SG-III facility (TIL facility). The spatial distribution and time duration of the input pulse are 8 order super-Gaussian and 0.3 ns, respectively. The spatial resolution in the simulations is 1024 × 1024. The cavity spatial filter diameter is 2.8 cm, and the focal length of the spatial filter lens is 900 cm. Furthermore, 21 beam output from the booster amplifier and the corresponding values between the booster amplifier are archived by changing the injected energy from 0.01 J to 1 J.
The fluence statistics of each near-field is obtained, and the intensity Shannon entropy is calculated. In the statistic, based on Eq. (1), i is the fluence value of the beam and pi is the occurrence frequency of that fluence value equals to i. And the value of N is the gray level of the diagnostic device. In numerical simulation, the fluence value may not be integer, we treat the nearby value as the same. The intensity Shannon entropy values corresponding to is depicted as red curve in Fig. 2(a). As predicted, the intensity entropy increases first and then decreases with increasing , which has a turning point at.
Fig. 2 (a) Intensity Shannon entropy, and (b) contrast and modulation curves corresponding to different ΔB values.
For comparison, we also calculate the contrast and modulation of output near-field corresponding to different values [Fig. 2(b)]. Contrast is defined as ratio of variance to average value, which could describe the discrete degree of fluence value. Modulation is the ratio of peak value to average value. So the better beam quality corresponds to smaller contrast and modulation value. While the nonlinear growth is not serious, the beam is relatively smooth, and the contrast and modulation will keep a relative low value. Once the nonlinear growth is serous, the beam quality will degrades sharply, and the contrast and modulation will correspondingly increase. From Fig. 2(b), we could see that the contrast and modulation increases with increasing . In particular, both parameters sharply increase at from 1.5 to 2, reflecting the dramatic decrease in beam quality. However, the curves do not exhibit an evident turning point. By contrast, the proposed near-field intensity Shannon entropy curve exhibits obvious turning points at which SSFF begins to develop rapidly. The turning point is also in the range that contrast and modulation increase obviously.
3.2 Experiment
The high-fluence experiment is conducted in TIL facility. The duration of laser pulse injection is 0.3 ns, the output beam aperture is 29 cm, the cavity spatial filter diameter is 2.3 mm, and the focal length of the spatial filter lens is 900 cm. A 12-bit CCD is used to obtain beam output beams after booster amplifier of the TIL facility. As shown in Fig. 3, output near-field is relatively smooth when operated under low fluence. However, the output near-field becomes flat and exhibits beam splitting when operated under high fluence.
Fig. 3 Output near-field of the TIL facility and the fluence distribution along the vertical direction.
In our experiment, the 12bit CCD decides that the N = 4096. The intensity Shannon entropy of near-field is calculated and shown in Fig. 4(a). The red square points denote the intensity Shannon entropy. This parameter increases first and then decreases, with a turning point at around . The tendency of the Shannon entropy curve is exactly the same with the theoretical analysis and numerical simulation. The turning point is slightly different from the numerical simulation. It clould be caused by the magnitude of the input power and phase of the noise. We also calculate the contrast and modulation of beam near-field [Fig. 4(b)]. Contrast and modulation increases with increasing , particularly rising sharply at from 1.75 to 2.25. B-integral is a traditional criterion which is directly connected with the nonlinear physics process, which is generally represented by modulation and contrast of laser near-field. While the intensity Shannon entropy represents the laser intensity distribution which is a direct result of SSFF and it has the advantage of giving an obvious turning point where the SSFF starts to grow rapidly. So the intensity Shannon entropy has the potential applications as a new criterion in high power-laser systems design and operation.
Fig. 4 (a) Intensity Shannon entropy, and (b) contrast and modulation of the output near-field corresponding to different ΔB values in the TIL facility.
4. Conclusion
By theoretical analysis, numerical simulation, and experiment, we report that output near-field intensity Shannon entropy can reflect the development of SSFF in high-power laser systems. The intensity Shannon entropy curve with respect to increasing B integral exhibit obvious turning point. The turning point corresponds to the point at which SSFF begins to develop sharply. The Shannon entropy method could provide the B integral criterion value of a given high-power laser system, which is significant for the design and operation of such system. Unlike classical methods that analyze contrast, modulation, or PSD curves, the method detects the SSFF occurrence and provides the B integral criterion more clearly. The proposed approach also reduces the dependency on expert experience. At present, the results of the study demonstrate that Shannon entropy could be used to characterize laser beams.
Acknowledgments
This work was supported by the SG-III laser facility upgrade project. We also thank the SG-III prototype facility teams for their assistance in the experiments.
References and links
1. R. W. Boyed, S. G. Lukishova, and Y. Shen, Self-focusing: Past and Present (Springer, 2009),chap. 8.
2. D. Villate, N. Blanchot, and C. Rouyer, “Beam breakup integral measurement on high-power laser chains,” Opt. Lett. 32(5), 524–526 (2007). [CrossRef] [PubMed]
3. Janson Carpenter and Don Correll, eds, “ICF annual report” (1997).
4. C. A. Haynam, P. J. Wegner, J. M. Auerbach, M. W. Bowers, S. N. Dixit, G. V. Erbert, G. M. Heestand, M. A. Henesian, M. R. Hermann, K. S. Jancaitis, K. R. Manes, C. D. Marshall, N. C. Mehta, J. Menapace, E. Moses, J. R. Murray, M. C. Nostrand, C. D. Orth, R. Patterson, R. A. Sacks, M. J. Shaw, M. Spaeth, S. B. Sutton, W. H. Williams, C. C. Widmayer, R. K. White, S. T. Yang, and B. M. Van Wonterghem, “National Ignition Facility laser performance status,” Appl. Opt. 46(16), 3276–3303 (2007). [CrossRef] [PubMed]
5. A. M. Rubenchik, S. K. Turitsyn, and M. P. Fedoruk, “Modulation instability in high power laser amplifiers,” Opt. Express 18(2), 1380–1388 (2010). [CrossRef] [PubMed]
6. V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3(11), 307–310 (1966).
7. M. S. Kochetkova, M. A. Martyanov, A. K. Poteomkin, and E. A. Khazanov, “Experimental observation of the small-scale self-focusing of a beam in the nondestructive regime,” Quantum Electron. 39(10), 923–927 (2009). [CrossRef]
8. W. Williams, J. Trenholme, C. Orth, S. Haney, R. Sacks, J. Auerbach, J. Lawson, M. Henesian, K. Jancaitis, and P. Renard, “ICF annual report” (1996).
9. M. S. Kuzmina, M. A. Martyanov, A. K. Poteomkin, E. A. Khazanov, and A. A. Shaykin, “Theoretical and experimental study of laser radiation propagating in a medium with thermally induced birefringence and cubic nonlinearity,” Opt. Express 19(22), 21977–21988 (2011). [CrossRef] [PubMed]
10. S. Mironov, V. Lozhkarev, G. Luchinin, A. Shaykin, and E. Khazanov, “Suppression of small-scale self-focusing of high-intensity femtosecond radiation,” Appl. Phys. B 113(1), 147–151 (2013). [CrossRef]
11. M. Henesian, P. Renard, J. Auerbach, J. Caird, B. Ehrlich, S. Haney, J. Hunt, J. Lawson, K. Manes, D. Milam, R. Sacks, L. Seppala, I. Smith, R. Speck, C. Thompson, B. VanWonterghem, P. Wegner, T. Weiland, C. Widmayer, W. Williams, and J. B. Trenholme, “The use of propagation of Beamlet and Nova to ensure a “safe” National Ignition Facility laser system design,” Proc. SPIE 3047, 0277 (1997). [CrossRef]
12. Z.-T. Peng, J. Feng, L.-Q. Liu, Q.-H. Zhu, C. Bo, Z. Kun, L. Hua, Q.-Q. Zhang, X.-F. Cheng, D.-B. Jiang, H.-J. Liu, and H.-S. Peng, “Power spectra density estimation of quality of the laser beam passing through an self-focusing media,” Wuli Xuebao 52(1), 87–90 (2003).
13. C. E. Shannon, “A mathematic theory of communication,” Bell Syst. Tech. J. 27(3), 379–423 (1948). [CrossRef]
14. P. Miao, Z. Chao, Y. Zhang, N. Li, and N. V. Thakor, “Entropy analysis reveals a simple linear relation between laser speckle and blood flow,” Opt. Lett. 39(13), 3907–3910 (2014). [CrossRef] [PubMed]
15. G. Liu, S. Yousefi, Z. Zhi, and R. K. Wang, “Automatic estimation of point-spread-function for deconvoluting out-of-focus optical coherence tomographic images using information entropy-based approach,” Opt. Express 19(19), 18135–18148 (2011). [CrossRef] [PubMed]
16. X. Chen, H. Zhao, P. Liu, B. Zhou, and W. Ren, “Automatic salient object detection via maximum entropy estimation,” Opt. Lett. 38(10), 1727–1729 (2013). [CrossRef] [PubMed]
17. K. R. Gosselin and M. W. Renfro, “Reconstruction of three-dimensional chemiluminescence images with a maximum entropy deconvolution algorithm,” Appl. Opt. 51(11), 1671–1680 (2012). [CrossRef] [PubMed]
18. L. L. A. Price, “Entropy, color, and color rendering,” J. Opt. Soc. Am. A 29(12), 2557–2565 (2012). [CrossRef] [PubMed]
19. J. P. Toomey, D. M. Kane, and T. Ackemann, “Complexity in pulsed nonlinear laser systems interrogated by permutation entropy,” Opt. Express 22(15), 17840–17853 (2014). [CrossRef] [PubMed]
20. J. M. Elson and J. M. Bennett, “Calculation of the power spectral density from surface profile data,” Appl. Opt. 34(1), 201–208 (1995). [CrossRef] [PubMed]
21. J. Su, W. Wang, F. Jing, Z. Peng, Q. Zhang, L. Liu, X. Cheng, D. Hu, J. Yuan, X. Wei, and X. Zhang, “The code SG99 for high-power laser propagation and its applications,” Proc. SPIE 5627, 527–531 (2005). [CrossRef]
