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We propose a ring laser array structure and study the Talbot effect. By comparing with the one-dimensional laser array, the near-field distribution of the identical intensity of the ring laser array without edge effect is obtained, which can be more conducive to improving the capability of the external cavity phase locking of lasers. In this paper, Talbot effect and self imaging condition of ring laser array in polar coordinates are calculated by using Gyrator canonical transformation. The sub-image distribution at the fractional Talbot distance is further analyzed. The optical profile and phase distribution of the ring laser array at the fractional Talbot distance are simulated, which are mutually verified with the theoretical calculation results. At a quarter of the Talbot distance, the number of sub-images is twice that of the emitters. The light intensities of the sub-images are identical, and thedifference in phase between adjacent sub-images is
${\pi }/{2}$ . At half of the Talbot distance, the number of sub-images is the same as that of the emitters, while the spatial position of sub-image is shifted by half a cycle along the angular direction. Moreover, the sub-images with twice the number of the emitters are present in three quarters of the Talbot distance. The light intensities of the sub-images are identical and the difference in phase between adjacent sub-images is$- {\pi }/{2}$ . Further, the Talbot images with the same spatial and phase distribution as the emitters are generated along the angular direction at the Talbot distance. The optical profile and phase distribution of one dimensional laser array at the fractional Talbot distance are also simulated by FDTD Solutions for comparison. It is found that the edge effect of one-dimensional laser array leads to the uneven distribution of near-field light intensity, in which the intensity of light spot on the edge of array is significantly lower than that in the center of array. While the Talbot sub-images of ring laser array with identical light intensity are obtained. Therefore, We consider that the ring laser array can effectively eliminate the edge effect. The results are helpful in studying the external cavity phase locking of ring laser arrays and its applications in the field of high brightness coherent laser and quantum measurement.-
Keywords:
- ring laser array /
- fractional Talbot effect /
- self imaging /
- gyrator transformation /
- coherent array
[1] Talbot H F 1836 The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 9 1
[2] Berry M V, Klein S 1996 Int. J. Opt. 43 2139
[3] 吴玲玲, 吴国俊, 仓玉萍, 陈良益 2010 光子学报 39 1723
Google Scholar
Wu L L, Wu G J, Cang Y P, Chen L Y 2010 Acta Photon. Sin. 39 1723
Google Scholar
[4] Javidi B, Lancis J, Tajahuerce E, Andres P, Martinez-Leon L, Araiza-Esquivel M A 2011 Appl. Opt. 50 96
Google Scholar
[5] Chai S J, Fekete J, McDowall P, Coop S, Andersen M F 2018 Phys. Rev. A 97 033616
Google Scholar
[6] Qiu T, Xia L, Ma H, Zheng C, Chen L 2016 Opt. Commun. 35 20
[7] Chen H X, Xiong D Z, Wang P J 2010 Chin. Opt. Lett. 8 348
Google Scholar
[8] Chapman M S, Ekstrom C R, Hammond T D, Schmiedmayer J, Tannian B E, Wehinger S, Pritchard D E 1995 Phys. Rev. A 51 14
Google Scholar
[9] Zhang Z Y, Liu X, Zhang D, Sheng J T, Zhang Y Q, Zhang Y P, Xiao M 2018 Phys. Rev. A 97 013603
Google Scholar
[10] Wang H, Zhang Z, Lu C, Wang Q 2003 Opt. Commun. 222 69
Google Scholar
[11] Wang H S 2009 International Conference on Information Engineering and Computer Science Wuhan, China, December 19–20, 2009 p1
[12] 华建文 1997 中国激光 000 163
Google Scholar
Hua J W 1997 Chin. J. Lasers 000 163
Google Scholar
[13] Shichijo T, Miyamoto T 2019 Jpn. J. Appl. Phys. 58 SJJC01.1
[14] Sanders S, Waarts R G, Nam D W, Welch D F, Flood K M 1993 Proceedings of LEOS San Jose, CA, USA, November 15–18, 1993 p590
[15] Kandidov V P, Kondrat'ev A V 1997 Quantum Electron. 27 234
Google Scholar
[16] Liu L R 1989 Opt. Lett. 14 1312
Google Scholar
[17] Li C, Zhou T W, Zhai Y Y, Yue X G, Xiang J G, Yang S F, Xiong W, Chen X Z 2017 Phys. Rev. A 95 033821
Google Scholar
[18] Li C, Zhou T W, Zhai Y Y, Yue X G, Xiang J G, Yang S F, Xiong W, Chen X Z 2017 Chin. Phys. Lett. 34 084207
Google Scholar
[19] 陆丹 2004 硕士学位论文(成都: 四川大学)
Lu D 2004 M. S. Dissertation (Chengdu: Sichuan University) (in Chinese)
[20] 严地勇 2003 硕士学位论文 (成都: 四川大学)
Yan D Y 2003 M. S. Dissertation (Chengdu: Sichuan University) (in Chinese)
[21] 杨哲宁, 乐阳阳, 洪煦昊, 赵瑞智, 陆蓉儿, 冯霞, 许亚光, 袁旭东, 张超, 秦亦强, 朱永元 2020 物理学报 69 034201
Google Scholar
Yang Z N, Yue Y Y, Hong X H, Zhao R Z, Lu R E, Feng X, Xu Y G, Yuan X D, Zhang C, Qin Y Q, Zhu Y Y 2020 Acta Phys. Sin. 69 034201
Google Scholar
[22] 孙琼阁, 马金鹏, 杨瑀, 李辰, 刘正君, 刘树田 2014 光学学报 34 0711004
Google Scholar
Sun Q G, Ma J P, Yang Y, Li C, Liu Z J, Liu S T 2014 Acta Opt. Sin. 34 0711004
Google Scholar
[23] 马金鹏 2009 硕士学位论文 (哈尔滨: 哈尔滨工业大学)
Ma J P 2009 M. S. Dissertation (Harbin: Harbin Institute of Technology) (in Chinese)
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图 1 环形激光阵列
Figure 1. Ring laser array.
图 2 仿真模型示意图
Figure 2. Schematic diagram of simulation model.
图 3 环形激光阵列Talbot效应 (a)初始光斑空间分布; (b) 1/4 Talbot距离处光斑空间分布; (c) 1/2 Talbot距离处光斑空间分布; (d) 3/4 Talbot距离处光斑空间分布; (e) Talbot距离处光斑空间分布; (f)初始光源相位分布; (g) 1/4 Talbot距离处相位分布; (h) 1/2 Talbot距离处相位分布; (i) 3/4 Talbot距离处相位分布; (j) Talbot距离处相位分布
Figure 3. The Talbot effect of ring laser array: (a) The initial optical profile; (b) optical profile at 1/4 Talbot distance; (c) optical profile at 1/2 Talbot distance; (d) optical profile at 3/4 Talbot distance; (e) optical profile at Talbot distance; (f) phase distribution of the initial light source; (g) phase distribution at 1/4 Talbot distance; (h) phase distribution at 1/2 Talbot distance; (i) phase distribution at 3/4 Talbot distance; (j) phase distribution at Talbot distance.
图 4 一维阵列Talbot效应 (a)初始光场分布; (b)1/4 Talbot距离处光斑空间分布; (c) 1/2 Talbot距离处光斑空间分布; (d) 3/4 Talbot距离处光斑空间分布; (e) Talbot距离处光斑空间分布; (f)初始光源相位分布; (g) 1/4 Talbot距离处相位分布; (h) 1/2Talbot距离处相位分布; (i) 3/4 Talbot距离处相位分布; (j)Talbot距离处相位分布
Figure 4. The Talbot effect of one-dimensional array: (a) The initial optical profile; (b) optical profile at 1/4 Talbot distance; (c) optical profile at 1/2 Talbot distance; (d) optical profile at 3/4 Talbot distance; (e) optical profile at Talbot distance; (f) phase distribution of the initial light source; (g) phase distribution at 1/4 Talbot distance; (h) phase distribution at 1/2 Talbot distance; (i) phase distribution at 3/4 Talbot distance; (j) phase distribution at Talbot distance.
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[1] Talbot H F 1836 The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 9 1
[2] Berry M V, Klein S 1996 Int. J. Opt. 43 2139
[3] 吴玲玲, 吴国俊, 仓玉萍, 陈良益 2010 光子学报 39 1723
Google Scholar
Wu L L, Wu G J, Cang Y P, Chen L Y 2010 Acta Photon. Sin. 39 1723
Google Scholar
[4] Javidi B, Lancis J, Tajahuerce E, Andres P, Martinez-Leon L, Araiza-Esquivel M A 2011 Appl. Opt. 50 96
Google Scholar
[5] Chai S J, Fekete J, McDowall P, Coop S, Andersen M F 2018 Phys. Rev. A 97 033616
Google Scholar
[6] Qiu T, Xia L, Ma H, Zheng C, Chen L 2016 Opt. Commun. 35 20
[7] Chen H X, Xiong D Z, Wang P J 2010 Chin. Opt. Lett. 8 348
Google Scholar
[8] Chapman M S, Ekstrom C R, Hammond T D, Schmiedmayer J, Tannian B E, Wehinger S, Pritchard D E 1995 Phys. Rev. A 51 14
Google Scholar
[9] Zhang Z Y, Liu X, Zhang D, Sheng J T, Zhang Y Q, Zhang Y P, Xiao M 2018 Phys. Rev. A 97 013603
Google Scholar
[10] Wang H, Zhang Z, Lu C, Wang Q 2003 Opt. Commun. 222 69
Google Scholar
[11] Wang H S 2009 International Conference on Information Engineering and Computer Science Wuhan, China, December 19–20, 2009 p1
[12] 华建文 1997 中国激光 000 163
Google Scholar
Hua J W 1997 Chin. J. Lasers 000 163
Google Scholar
[13] Shichijo T, Miyamoto T 2019 Jpn. J. Appl. Phys. 58 SJJC01.1
[14] Sanders S, Waarts R G, Nam D W, Welch D F, Flood K M 1993 Proceedings of LEOS San Jose, CA, USA, November 15–18, 1993 p590
[15] Kandidov V P, Kondrat'ev A V 1997 Quantum Electron. 27 234
Google Scholar
[16] Liu L R 1989 Opt. Lett. 14 1312
Google Scholar
[17] Li C, Zhou T W, Zhai Y Y, Yue X G, Xiang J G, Yang S F, Xiong W, Chen X Z 2017 Phys. Rev. A 95 033821
Google Scholar
[18] Li C, Zhou T W, Zhai Y Y, Yue X G, Xiang J G, Yang S F, Xiong W, Chen X Z 2017 Chin. Phys. Lett. 34 084207
Google Scholar
[19] 陆丹 2004 硕士学位论文(成都: 四川大学)
Lu D 2004 M. S. Dissertation (Chengdu: Sichuan University) (in Chinese)
[20] 严地勇 2003 硕士学位论文 (成都: 四川大学)
Yan D Y 2003 M. S. Dissertation (Chengdu: Sichuan University) (in Chinese)
[21] 杨哲宁, 乐阳阳, 洪煦昊, 赵瑞智, 陆蓉儿, 冯霞, 许亚光, 袁旭东, 张超, 秦亦强, 朱永元 2020 物理学报 69 034201
Google Scholar
Yang Z N, Yue Y Y, Hong X H, Zhao R Z, Lu R E, Feng X, Xu Y G, Yuan X D, Zhang C, Qin Y Q, Zhu Y Y 2020 Acta Phys. Sin. 69 034201
Google Scholar
[22] 孙琼阁, 马金鹏, 杨瑀, 李辰, 刘正君, 刘树田 2014 光学学报 34 0711004
Google Scholar
Sun Q G, Ma J P, Yang Y, Li C, Liu Z J, Liu S T 2014 Acta Opt. Sin. 34 0711004
Google Scholar
[23] 马金鹏 2009 硕士学位论文 (哈尔滨: 哈尔滨工业大学)
Ma J P 2009 M. S. Dissertation (Harbin: Harbin Institute of Technology) (in Chinese)
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