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Generation of high-quality circular Airy beams in laser resonator

Generation of high-quality circular Airy beams in laser resonator

Zhu Yi-Fan, Geng Tao
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• Abstract

A scheme for forming high-quality circular Airy beams inside the laser resonator is presented theoretically. The desired circular Airy beam can be generated when the common reflective mirror is replaced by a designed diffractive optical element. The mode generated in the proposed cavity can be stimulated by using the so-called eigenvector method. The calculated results show that the parameters of the beams can be controlled by changing the phase distribution of the diffractive optical element. The loss of the generated mode is very low, which is close to that of the fundamental Gaussian mode. The purity of the generated mode is very high, which is much better than that from the phase-only encoding method in Fourier space. The phase distribution of the diffractive optical element needs designing for a fixed resonator length. In practice, the real resonator length may not be equal to the designed resonator length. Thus, the influence of the alignment error of the resonator length is discussed in detail. The results show that the diffraction loss of the proposed system is still very small even when the error reaches up to 2 mm. Meanwhile, the purity of the generated mode decreases little. Then, the influence of etching depth errors and the decenter of the reflective mirrors are discussed in detail. Here we assume that the fluctuations are randomly distributed. The value of the maximum fluctuation is used to represent the etching depth error degree. The results show that the diffraction loss of the proposed system is more sensitive to production error, and the purity of the generated mode is more sensitive to alignment error. Thus, we estimate that the maximum etching depth error should be less than six percent of the wavelength, and the vertical distance between the centers of the two reflective mirrors should be less than 7 μm if one wants to obtain high-quality CAB with high efficiency. The requirements for precision are acceptable for existing microfabrication and operation technologies.

References

 [1] Siviloglou G A, Broky J, Dogariu A, Christodoulides D N 2007 Phys. Rev. Lett. 99 213901 [2] Minovich A E, Klein A E, Neshev D N, Pertsch T, Kivshar Y S, Christodoulides D N 2014 Laser. Photon. Rev. 8 221 [3] Qian J, Liu B Y, Sun H X, Yuan S Q, Yu X Z 2017 Chin. Phys. B 26 114304 [4] 崔省伟, 陈子阳, 胡克磊, 蒲继雄 2013 物理学报 62 094205 Cui S W, Chen Z Y, Hu K L, Pu J X 2013 Acta Phys. Sin. 62 094205 [5] 张泽, 刘京郊, 张鹏, 倪培根, Prakash J, 胡洋, 姜东升, Christodoulides D N, 陈志刚 2013 物理学报 62 034209 Zhang Z, Liu J J, Zhang P, Ni P G, Prakash J, Hu Y, Jiang D S, Christodoulides D N, Chen Z G 2013 Acta Phys. Sin. 62 034209 [6] Guo Y H, Huang Y J, Li X, Pu M B, Gao P, Jin J J, Ma X L, Luo X G 2019 Adv. Opt. Mater. 7 1900503 [7] Li Z, Cheng H, Liu Z, Chen S, Tian J 2016 Adv. Opt. Mater. 4 1230 [8] Fan Q, Zhu W, Liang Y, Huo P, Zhang C, Agrawal A, Huang K, Luo X, Lu Y, Qiu C, Lezec H J, Xu T 2019 Nano Lett. 19 1158 [9] Efremidis N K, Christodoulides D N 2010 Opt. Lett. 35 4045 [10] Papazoglou D G, Efremidis N K, Christodoulides D N, Tzortzakis S 2011 Opt. Lett. 36 1842 [11] Liu K, Koulouklidis A D, Papazoglou D G, Tzortzakis S, Zhang X C 2016 Optica 3 605 [12] Manousidaki M, Papazoglou D G, Farsari M, Tzortzakis S 2016 Optica 3 525 [13] Manousidaki M, Fedorov V Y, Papazoglou D G, Farsari M, Tzortzakis S 2018 Opt. Lett. 4 3 [14] Zhang P, Prakash J, Zhang Z, Mills M S, Efremidis N K, Christodoulides D N, Chen Z 2011 Opt. Lett. 36 2883 [15] Davis J A, Cottrell D M, Zinn J M 2013 Appl. Opt. 52 1888 [16] Davis J A, Cottrell D M, Sand D 2012 Opt. Express 20 13302 [17] 刘正楠, 耿滔, 邓攀 2019 中国激光 46 0209001 Liu Z N, Geng T, Deng P 2019 Chin. J. Lasers 46 0209001 [18] Fox A G, Li T 1961 Bell System Techical Journal 40 453 [19] Chao J, Li B, Cheng Y, Wang Y 2007 Opt. Laser Technol. 39 490 [20] Cheng Y Y, Wang Y Q, Hu J, Li J R 2004 Opt. Commun. 234 1 [21] Bélanger P A, Paré C 1991 Opt. Lett. 16 1057 [22] Leger J R, Chen D, Wang Z 1994 Opt. Lett. 19 108 [23] Jiang Y, Zhu X, Yu W, Shao H, Zheng W, Lu X 2015 Opt. Express 23 29834 [24] Li N, Jiang Y, Huang K, Lu X 2014 Opt. Express 22 22847 [25] Zhou W, Bovik A C, Sheikh H R, Simoncelli E P 2004 IEEE Trans. Image Process 13 600

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• 图 1  (a) CAB初始面的光强分布; (b) CAB初始面的相位分布; (c) CAB的侧面光强分布

Figure 1.  (a) Intensity distributions of the CAB at the initial plane; (b) phase distributions of the CAB at the initial plane; (c) intensity distributions of the CAB during propagation in the r-z plane.

图 2  谐振腔示意图

Figure 2.  Schematic of the laser resonator configuration for CAB generation.

图 3  不同参数条件下, 使用Fox-Li方法计算获得的腔内光场模式分布　(a) r0 = 1 mm, w = 0.2 mm和a = 0.15; (b) r0 = 1.1 mm, w = 0.22 mm和a = 0.17; (c) r0 = 1.2 mm, w = 0.25 mm和a = 0.2

Figure 3.  Calculation results of the intensity distributions of the modes by using Fox-Li method with different parameters: (a) r0 = 1 mm, w = 0.2 mm and a = 0.15; (b) r0 = 1.1 mm, w = 0.22 mm and a = 0.17; (c) r0 = 1.2 mm, w = 0.25 mm and a = 0.2.

图 4  理想CAB和使用不同方法产生的光束的径向光强分布　(a) r0 = 1 mm, w = 0.2 mm和a = 0.15; (b) r0 = 1.1 mm, w = 0.22 mm和a = 0.17; (c) r0 = 1.2 mm, w = 0.25 mm和a = 0.2

Figure 4.  Radial intensity distributions of the ideal CAB and the beams produced by different methods: (a) r0 = 1 mm, w = 0.2 mm and a = 0.15; (b) r0 = 1.1 mm, w = 0.22 mm and a = 0.17; (c) r0 = 1.2 mm, w = 0.25 mm and a = 0.2.

图 5  理想CAB和使用不同方法产生的光束的光轴光强分布　(a) r0 = 1 mm, w = 0.2 mm和a = 0.15; (b) r0 = 1.1 mm, w = 0.22 mm和a = 0.17; (c) r0 = 1.2 mm, w = 0.25 mm和a = 0.2

Figure 5.  On-axis intensity contrast of the ideal CAB and the beams produced by different methods: (a) r0 = 1 mm, w = 0.2 mm and a = 0.15; (b) r0 = 1.1 mm, w = 0.22 mm and a = 0.17; (c) r0 = 1.2 mm, w = 0.25 mm and a = 0.2.

图 6  光束参数为${r_0} = 1\;{\rm{mm}}$, $w = 0.2\;{\rm{mm}}$$a = 0.15$时, 系统对准误差对产生光束质量的影响　(a)基模的$\left| \gamma \right|$以及S与腔长误差${\delta _{\rm{l}}}$的关系; (b)基模的$\left| \gamma \right|$以及S与同轴度误差${\delta _{\rm{d}}}$的关系

Figure 6.  The influence of the alignment errors on formation of the fundamental mode with ${r_0} = 1\;{\rm{mm}}$, $w = 0.2\;{\rm{mm}}$ and $a = 0.15$: (a) $\left| \gamma \right|$ and S of the fundamental mode as a function of $\delta_{\rm l}$; (b) $\left| \gamma \right|$ and S of the fundamental mode as a function of $\delta _{\rm d}$

图 7  基模的$\left| \gamma \right|$以及S${\delta _{\rm{h}}}$的关系

Figure 7.  $\left| \gamma \right|$ and S of the fundamental mode as a function of ${\delta _{\rm{h}}}$.

•  [1] Siviloglou G A, Broky J, Dogariu A, Christodoulides D N 2007 Phys. Rev. Lett. 99 213901 [2] Minovich A E, Klein A E, Neshev D N, Pertsch T, Kivshar Y S, Christodoulides D N 2014 Laser. Photon. Rev. 8 221 [3] Qian J, Liu B Y, Sun H X, Yuan S Q, Yu X Z 2017 Chin. Phys. B 26 114304 [4] 崔省伟, 陈子阳, 胡克磊, 蒲继雄 2013 物理学报 62 094205 Cui S W, Chen Z Y, Hu K L, Pu J X 2013 Acta Phys. Sin. 62 094205 [5] 张泽, 刘京郊, 张鹏, 倪培根, Prakash J, 胡洋, 姜东升, Christodoulides D N, 陈志刚 2013 物理学报 62 034209 Zhang Z, Liu J J, Zhang P, Ni P G, Prakash J, Hu Y, Jiang D S, Christodoulides D N, Chen Z G 2013 Acta Phys. Sin. 62 034209 [6] Guo Y H, Huang Y J, Li X, Pu M B, Gao P, Jin J J, Ma X L, Luo X G 2019 Adv. Opt. Mater. 7 1900503 [7] Li Z, Cheng H, Liu Z, Chen S, Tian J 2016 Adv. Opt. Mater. 4 1230 [8] Fan Q, Zhu W, Liang Y, Huo P, Zhang C, Agrawal A, Huang K, Luo X, Lu Y, Qiu C, Lezec H J, Xu T 2019 Nano Lett. 19 1158 [9] Efremidis N K, Christodoulides D N 2010 Opt. Lett. 35 4045 [10] Papazoglou D G, Efremidis N K, Christodoulides D N, Tzortzakis S 2011 Opt. Lett. 36 1842 [11] Liu K, Koulouklidis A D, Papazoglou D G, Tzortzakis S, Zhang X C 2016 Optica 3 605 [12] Manousidaki M, Papazoglou D G, Farsari M, Tzortzakis S 2016 Optica 3 525 [13] Manousidaki M, Fedorov V Y, Papazoglou D G, Farsari M, Tzortzakis S 2018 Opt. Lett. 4 3 [14] Zhang P, Prakash J, Zhang Z, Mills M S, Efremidis N K, Christodoulides D N, Chen Z 2011 Opt. Lett. 36 2883 [15] Davis J A, Cottrell D M, Zinn J M 2013 Appl. Opt. 52 1888 [16] Davis J A, Cottrell D M, Sand D 2012 Opt. Express 20 13302 [17] 刘正楠, 耿滔, 邓攀 2019 中国激光 46 0209001 Liu Z N, Geng T, Deng P 2019 Chin. J. Lasers 46 0209001 [18] Fox A G, Li T 1961 Bell System Techical Journal 40 453 [19] Chao J, Li B, Cheng Y, Wang Y 2007 Opt. Laser Technol. 39 490 [20] Cheng Y Y, Wang Y Q, Hu J, Li J R 2004 Opt. Commun. 234 1 [21] Bélanger P A, Paré C 1991 Opt. Lett. 16 1057 [22] Leger J R, Chen D, Wang Z 1994 Opt. Lett. 19 108 [23] Jiang Y, Zhu X, Yu W, Shao H, Zheng W, Lu X 2015 Opt. Express 23 29834 [24] Li N, Jiang Y, Huang K, Lu X 2014 Opt. Express 22 22847 [25] Zhou W, Bovik A C, Sheikh H R, Simoncelli E P 2004 IEEE Trans. Image Process 13 600
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• Received Date:  16 July 2019
• Accepted Date:  16 September 2019
• Available Online:  05 December 2019
• Published Online:  01 January 2020

Generation of high-quality circular Airy beams in laser resonator

Corresponding author: Geng Tao, Tao_Geng@hotmail.com
• Engineering Research Center of Optical Instruments and Systems, Ministry of Education, Shanghai Key Laboratory of Modern Optics and Systems, School of Optical-Electrical and ComputerEngineering, University of Shanghai for Science and Technology, Shanghai 200093, China

Abstract: A scheme for forming high-quality circular Airy beams inside the laser resonator is presented theoretically. The desired circular Airy beam can be generated when the common reflective mirror is replaced by a designed diffractive optical element. The mode generated in the proposed cavity can be stimulated by using the so-called eigenvector method. The calculated results show that the parameters of the beams can be controlled by changing the phase distribution of the diffractive optical element. The loss of the generated mode is very low, which is close to that of the fundamental Gaussian mode. The purity of the generated mode is very high, which is much better than that from the phase-only encoding method in Fourier space. The phase distribution of the diffractive optical element needs designing for a fixed resonator length. In practice, the real resonator length may not be equal to the designed resonator length. Thus, the influence of the alignment error of the resonator length is discussed in detail. The results show that the diffraction loss of the proposed system is still very small even when the error reaches up to 2 mm. Meanwhile, the purity of the generated mode decreases little. Then, the influence of etching depth errors and the decenter of the reflective mirrors are discussed in detail. Here we assume that the fluctuations are randomly distributed. The value of the maximum fluctuation is used to represent the etching depth error degree. The results show that the diffraction loss of the proposed system is more sensitive to production error, and the purity of the generated mode is more sensitive to alignment error. Thus, we estimate that the maximum etching depth error should be less than six percent of the wavelength, and the vertical distance between the centers of the two reflective mirrors should be less than 7 μm if one wants to obtain high-quality CAB with high efficiency. The requirements for precision are acceptable for existing microfabrication and operation technologies.

Reference (25)

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