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According to the generalized Huygens-Fresnel principle, we derive the analytical formula for the complex degree of coherence of the echo light field under the von Karman atmospheric turbulence spectrum condition. Based on split-step beam propagation method of the turbulent phase screen and the target surface model, the fold pass propagation simulation of the laser in the turbulent atmosphere is realized. The dynamic speckle characteristics on the image plane are consistent with the experimental phenomenon. Firstly, the simulation results of the complex degree of coherence and phase structure function of the mirrored reflection echo light field are compared with the theoretical values, which verifies the correctness of the algorithm. Based on this, the complex degree of coherence of the echo light field reflected by the optical rough surface is calculated and analyzed. The results show that on a double-path turbulent flow path of 1.1 km, in other words, it transmits 2.2 km in unfolded mode, the spatial coherence of the echo light field is very sensitive to the root mean square value of height. When the root-mean-square value of height is close to the wavelength, the coherence is seriously degraded. When the correlation length of the target surface is much larger than the atmospheric coherence length, the coherence length of the echo light field is relatively close to the set spatial coherence length. When the correlation length of the target surface is close to the atmospheric coherence length, the influence of the rough surface of the target on the beam coherence cannot be ignored. When the correlation length of the target surface is much smaller than the atmospheric coherence length, the target surface characteristics have a dominant influence on the echo coherence, the spatial coherence of the light field is seriously degraded, and the echo is close to incoherent light. Considering the smooth target reflection surface, the greater the strength of turbulence, the faster the complex coherence decreases with space. The atmospheric coherence diameter
${r_0}$ can be calculated further according to the complex degree of coherence. For the Pearson correlation coefficient the simulation value and theoretical value are both 0.998, which indicates that the atmospheric coherence diameter calculated by the complex degree of coherence has a high correlation with the theoretical value. This research provides a theoretical basis for the coherent detection scheme of echoes from rough surfaces in the turbulent atmosphere. The simulation algorithm extracts the target surface features by analyzing the variation of the complex coherence of laser echo signals in the turbulent atmosphere with the spatial distance, and also provides a method of using the known target surface to obtain path turbulence information.-
Keywords:
- turbulent atmosphere /
- laser echo light field /
- complex degree of coherence /
- phase structure function
[1] John W S 1978 Laser Beam Propagation in the Atmosphere (Berlin: Springer) pp129–170
[2] Tatarskii V I, Ishimaru A, Zavorotny V U 1993 Wave Propagation in Random Media (scintillation) (Bellingham: SPIE) p343
[3] Jing X, Hou Z H, Wu Y, Qin L A, He F, Tan F F 2013 Opt. Lett. 38 3445
Google Scholar
[4] Vorontsov M A, Kolosov V 2005 J. Opt. Soc. Am. A 22 126
Google Scholar
[5] Mahon R, Moore C I, Ferraro M, Rabinovich W S, Suite M R 2016 Appl. Opt. 55 5172
Google Scholar
[6] Mahon R, Moore C I, Ferraro M, Rabinovich W S, Suite M R 2012 Appl. Opt. 51 6147
Google Scholar
[7] Mahon R, Ferraro M S, Goetz P G, Moore C I, Murphy J, Rabinovich W S 2015 Appl. Opt. 54 F96
Google Scholar
[8] Mei H P, Ye H, Kang L, Qian X M, Huang H H, Huang Y B, Zhu W Y, Wu X Q, Rao R Z 2019 OSA Continuum 2 1938
Google Scholar
[9] 张骏昕, 梅海平, 沈刘晶, 黄印博, 罗福, 吴小龑 2021 中国激光 48 0105001
Google Scholar
Zhang J X, Mei H P, Shen L J, Huang Y B, Luo F, Wu X Y 2021 Chin. J. Lasers 48 0105001
Google Scholar
[10] 沈刘晶, 梅海平, 任益充, 张骏昕 2021 中国激光 48 119
Google Scholar
Shen L J, Mei H P, Ren Y C, Zhang J X 2021 Chin. J. Las. 48 119
Google Scholar
[11] Bashkansky M, Lucke R L, Funk E, Rickard L J, Reintjes J 2002 Opt. Lett. 27 1983
Google Scholar
[12] 党文佳, 曾晓东, 冯喆珺 2013 62 024204
Google Scholar
Dang W J, Zeng X D, Feng Z J 2013 Acta Phys. Sin. 62 024204
Google Scholar
[13] Liu Y T, Zeng X D, Cao C Q, Feng Z J, Lai Z, Fan Z J, Wang T, Cui S B, Ding J H 2019 Opt. Commun. 440 171
Google Scholar
[14] Liu Y T, Zeng X D, Cao C Q, Feng Z J, Lai Z 2019 Opt. Lett. 44 5896
Google Scholar
[15] Andrews L C, Miller W B, Ricklin J C 1993 Appl. Opt. 32 5918
Google Scholar
[16] Andrews L C, Young C Y, Miller W B 1996 J. Opt. Soc. Am. A 13 851
Google Scholar
[17] Andrews L C, Phillips R L, Miller W B 1997 Appl. Opt. 36 698
Google Scholar
[18] Korotkova O, Andrews L C 2002 Laser Radar Technology and Applications VII Orlando, FL, United States, July 29, 2002 p4723
[19] Andrews L C, Phillips R L 2005 Laser Beam Propagation Through Random Media (2nd Ed.) (Bellingham: SPIE) pp179–218
[20] Martin J M, Flatte S M 1988 Appl. Opt. 27 2111
Google Scholar
[21] Martin J M, Flatte S M 1990 J. Opt. Soc. Am. A 7 838
Google Scholar
[22] Rubio J A, Belmonte A, Comeron A 1999 Opt. Eng. 38 1462
Google Scholar
[23] Frehlich R 2000 Appl. Opt. 37 393
[24] Welch G, Phillips R 1990 J. Opt. Soc. Am. A 7 578
Google Scholar
[25] Nelson D H, Walters D L, MacKerrow E P, Schmitt M J, Quick C R, Porch W M, Petrin R R 2000 Appl. Opt. 39 1857
Google Scholar
[26] Belmonte A 2000 Appl. Opt. 39 5426
Google Scholar
[27] Belmonte A 2005 Opt. Express 13 9598
Google Scholar
[28] Li Y Q, Wang L G, Wu Z S 2019 Optik 179 244
Google Scholar
[29] Gbur G, Wolf E 2002 J. Opt. Soc. Am. A 19 1592
[30] Schmidt, Jason D 2010 Numerical Simulation of Optical Wave Propagation with Examples in MATLAB (Bellingha: SPIE) pp157–163
[31] McGlamery B L 1976 Proc. SPIE 0074 225
Google Scholar
[32] Sedmak G 2004 Appl. Opt. 43 4527
Google Scholar
[33] Harding C M, Johnston R A, Lane R G 1999 Appl. Opt. 38 2161
Google Scholar
[34] Assemat F, Wilson R W, Gendron E 2006 Opt. Express 14 988
Google Scholar
[35] Kuga Y, Phu P 1996 J. Electromagn. 10 451
Google Scholar
[36] Herraez M A, Burton D R, Lalor M J, Gdeisat M A 2002 Appl. Opt. 41 7437
Google Scholar
[37] 饶瑞中, 王海燕 2012 光学涡旋在湍流大气中的传播 (上海: 上海交通大学出版社) 第216页
Rao R Z, Wang H Y 2012 Propagation of Optical Vortices in Turbulent Atmosphere (Shanghai: Shanghai Jiaotong University Press) p216 (in Chinese)
[38] Tao Z W, Ren Y C, Abdukirim A, Liu S W, Rao R Z 2021 J. Opt. Soc. Am. A 38 1120
Google Scholar
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图 1 激光经过湍流大气双程传输示意图(展开模式)
Figure 1. Schematic of double-passage retroreflected laser propagation through atmospheric turbulence (unfolded).
图 2
${\delta _{\text{h}}} = {10^{ - 2}}\lambda $ 时, 不同表面相关长度的粗糙面 (a)$ {L_x} = 0.1{\text{ m}} $ ; (b)$ {L_x} = 0.2{\text{ m}} $ ; (c)$ {L_x} = 0.3{\text{ m}} $ ; (d)$ {L_x} = 0.4{\text{ m}} $ Figure 2. Gaussian rough surface with different characteristic parameters for
${\delta _{\text{h}}} = {10^{ - 2}}\lambda $ : (a)$ {L_x} = 0.1{\text{ m}} $ ; (b)$ {L_x} = 0.2{\text{ m}} $ ; (c)$ {L_x} = 0.3{\text{ m}} $ ; (d)$ {L_x} = 0.4{\text{ m}} $ .
图 3 基于无限长相位屏和随机粗糙面的回波光场仿真模型(展开模型)
Figure 3. Simulation model of echo light field based on infinitely long phase screen and random rough surface (unfolded).
图 4 接收平面光斑及像焦平面上的散斑分布, 前两列目标反射面为平面镜
$ {r_0} = 0.1{\text{ m}} $ , 接收平面光斑 (a)和焦平面光斑 (b);$ {r_0} = 0.04{\text{ m}} $ , 接收平面光斑(c)和焦平面光斑(d); 后两列目标反射面为高斯粗糙面$ {r_0} = 0.1{\text{ m}} $ , 接收平面光斑(e)和焦平面光斑(f);$ {r_0} = 0.04{\text{ m}} $ , 接收平面光斑(g)和像平面光斑(h)Figure 4. Light intensity at the receiver plane and the speckle intensity on the image plane, the first two columns of target reflecting surface are plane mirrors:
$ {r_0} = 0.1{\text{ m}} $ , the light intensity at the receiver plane (a) and the light intensity at focal plane (b);$ {r_0} = 0.04{\text{ m}} $ , the light intensity at the receiver plane (c) and the light intensity at focal plane (d); the second two columns of target reflection surfaces are Gaussian rough surfaces:$ {r_0} = 0.1{\text{ m}} $ , the light intensity at the receiver plane (e) and the light intensity at focal plane (f);$ {r_0} = 0.04{\text{ m}} $ , the light intensity at the receiver plane (g) and the light intensity at focal plane (h).
图 5 不同大气条件下, 接收平面处的回波光强和相位(目标反射面为理想平面镜) (a) 自由空间的接收光斑; (b)
${r_0}$ = 0.1 m, 经过湍流的回波光斑;${r_0}$ = 0.1 m, 回波光场的(c)包裹相位和(d)解包裹相位Figure 5. The echo light intensity and phase at the pupil plane under different atmospheric conditions (the target reflecting surface are plane mirrors): (a) Receiving light intensity in free space; (b) receiving light intensity in turbulence atmosphere at
${r_0}$ = 0.1 m; wrapped phase (c) and unwrapped phase (d) of the echo light intensity at${r_0}$ = 0.1 m.
图 6 不同大气条件下, 入瞳处的回波光强和相位分布(目标反射面为理想平面镜) (a) 自由空间接收光斑; (b)
${r_0}$ = 0.04 m, 经湍流的回波光斑;${r_0}$ = 0.04 m, 回波光场的(c) 包裹相位和(d) 解包裹相位Figure 6. The echo light intensity and phase at the pupil plane under different atmospheric conditions (the target reflecting surface are plane mirrors): (a) Receiving light intensity in free space; (b) receiving light intensity in turbulence atmosphere at
${r_0}$ = 0.04 m; wrapped phase (c) and unwrapped phase (d) of the echo light field at${r_0}$ = 0.04 m.
图 7
${r_0}$ 不同时, 回波光场的PSF随间距的变化 (a)${r_0}$ = 0.10 m; (b)${r_0}$ = 0.04 mFigure 7. PSF of the echo light field at different
${r_0}$ : (a)${r_0}$ = 0.10 m; (b)${r_0}$ = 0.04 m.
图 8
${r_0}$ 不同时, 回波光场的DOC随间距的变化 (a)${r_0}$ = 0.10 m; (b)${r_0}$ = 0.04 mFigure 8. DOC of the echo light field at different
${r_0}$ : (a)${r_0}$ = 0.10 m; (b)${r_0}$ = 0.04 m.
图 9 不同
$ {\delta _{\text{h}}} $ 对应的粗糙反射面、回波接收光强分布及DOC ($ {L_x} = 0.0532{\text{ m}} $ ) (a1)${\delta _{\text{h}}} = {10^{ - 2}}\lambda $ , (a2)${\delta _{\text{h}}} = {10^{ - 1}}\lambda $ , (a3)${\delta _{\text{h}}} = $ $ {10^0}\lambda $ ; (b1)—(b3) 对应的回波接收光斑; (c1)—(c3) 对应的回波接收光场的DOCFigure 9. (a1)–(a3) Rough reflection surface, echo receiving light intensity and DOC of the receiving light field corresponding to different
${\delta _{\text{h}}}$ ($ {L_x} = 0.0532{\text{ m}} $ ): (a1)${\delta _{\text{h}}} = {10^{ - 2}}\lambda $ , (a2)${\delta _{\text{h}}} = {10^{ - 1}}\lambda $ , (a3)${\delta _{\text{h}}} = {10^0}\lambda $ ; (b1)–(b3) the echo receiving light intensity; (c1)–(c3) the DOC of the echo light field.
图 10
${\delta _{\text{h}}} = {10^{ - 1}}\lambda $ , 不同$ {L_x} $ 对应的粗糙反射面、回波接收光强分布及DOC (a1)$ {L_x} = {10^6}\lambda = 0.532{\text{ m}} $ ; (a2)${L_x} = 5 \times {10^5}\lambda = $ $ 0.266{\text{ m}}$ ; (a3)${L_x} = 2 \times {10^5}\lambda = 0.1064{\text{ m}}$ ; (a4)${L_x} = {10^5}\lambda = 0.0532{\text{ m}}$ ; (a5)${L_x} = 5 \times {10^4}\lambda = 0.0266{\text{ m}}$ ; (b1)—(b5) 对应的回波接收光斑; (c1)—(c5) 对应的回波接收光场的DOCFigure 10. Rough reflection surface, echo receiving light intensity and DOC of receiving light field corresponding to different
${L_x}$ with${\delta _{\text{h}}} = {10^{ - 1}}\lambda $ : (a1)$ {L_x} = {10^6}\lambda = 0.532{\text{ m}} $ ; (a2)${L_x} = 5 \times {10^5}\lambda = 0.266{\text{ m}}$ ; (a3)${L_x} = 2 \times {10^5}\lambda = 0.1064{\text{ m}}$ ; (a4)${L_x} = {10^5}\lambda = $ $ 0.0532{\text{ m}}$ ; (a5)${L_x} = 5 \times {10^4}\lambda = 0.0266{\text{ m}}$ ; (b1)–(b5) corresponding echo receiving light intensity; (c1)–(c5) corresponding DOC of the echo light field.
图 11 不同湍流强度条件下的回波光场以及对应的DOC随间距的变化 (a)
$ {\delta }_{\text{h}}={10}^{-1}\lambda \text{ }, {L}_{x}={10}^{6}\lambda $ 时的目标反射面; (b1)—(b6) r0 = 0.02, 0.04, 0.06, 0.08, 0.10, 0.12 m时的接收光斑 ; (c)${r_0}$ 不同时, 回波光场的DOCFigure 11. The echo light field and corresponding DOC under different turbulence intensity conditions change with the spacing: (a) The target reflection surface under the condition of
${\delta _{\text{h}}} = {10^{ - 1}}\lambda {\text{ }}, {\text{ }}{L_x} = {10^6}\lambda $ ; (b1)–(b6) received spot at r0 = 0.02, 0.04, 0.06, 0.08, 0.10, 0.12 m; (c) DOC of the echo light field under different${r_0}$ .表 1 回波光场仿真参数选取
Table 1. Echo wave light field simulation parameter selection.
参数类型 参数名称 数值$/{\text{m}}$ 无限长相位屏参数 大气相干长度 0.02—0.12 相位屏尺寸 1 外尺度 50 传输光束参数 波长 $0.532 \times {10^{ - 6}}$ 传输路径$2 L$ 2200 束腰半径 0.2 接收屏 接收屏尺寸 1 
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表 2 仿真获得的
$r_0'$ 与设定参数${r_0}$ 的对比Table 2. The setting parameters
$r_0'$ obtained by simulation compared with${r_0}$ .设定参数${r_0}$/m 仿真值$r_0'$/m 0.120 0.123 0.100 0.107 0.080 0.083 0.060 0.058 0.040 0.042 0.020 0.018
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[1] John W S 1978 Laser Beam Propagation in the Atmosphere (Berlin: Springer) pp129–170
[2] Tatarskii V I, Ishimaru A, Zavorotny V U 1993 Wave Propagation in Random Media (scintillation) (Bellingham: SPIE) p343
[3] Jing X, Hou Z H, Wu Y, Qin L A, He F, Tan F F 2013 Opt. Lett. 38 3445
Google Scholar
[4] Vorontsov M A, Kolosov V 2005 J. Opt. Soc. Am. A 22 126
Google Scholar
[5] Mahon R, Moore C I, Ferraro M, Rabinovich W S, Suite M R 2016 Appl. Opt. 55 5172
Google Scholar
[6] Mahon R, Moore C I, Ferraro M, Rabinovich W S, Suite M R 2012 Appl. Opt. 51 6147
Google Scholar
[7] Mahon R, Ferraro M S, Goetz P G, Moore C I, Murphy J, Rabinovich W S 2015 Appl. Opt. 54 F96
Google Scholar
[8] Mei H P, Ye H, Kang L, Qian X M, Huang H H, Huang Y B, Zhu W Y, Wu X Q, Rao R Z 2019 OSA Continuum 2 1938
Google Scholar
[9] 张骏昕, 梅海平, 沈刘晶, 黄印博, 罗福, 吴小龑 2021 中国激光 48 0105001
Google Scholar
Zhang J X, Mei H P, Shen L J, Huang Y B, Luo F, Wu X Y 2021 Chin. J. Lasers 48 0105001
Google Scholar
[10] 沈刘晶, 梅海平, 任益充, 张骏昕 2021 中国激光 48 119
Google Scholar
Shen L J, Mei H P, Ren Y C, Zhang J X 2021 Chin. J. Las. 48 119
Google Scholar
[11] Bashkansky M, Lucke R L, Funk E, Rickard L J, Reintjes J 2002 Opt. Lett. 27 1983
Google Scholar
[12] 党文佳, 曾晓东, 冯喆珺 2013 62 024204
Google Scholar
Dang W J, Zeng X D, Feng Z J 2013 Acta Phys. Sin. 62 024204
Google Scholar
[13] Liu Y T, Zeng X D, Cao C Q, Feng Z J, Lai Z, Fan Z J, Wang T, Cui S B, Ding J H 2019 Opt. Commun. 440 171
Google Scholar
[14] Liu Y T, Zeng X D, Cao C Q, Feng Z J, Lai Z 2019 Opt. Lett. 44 5896
Google Scholar
[15] Andrews L C, Miller W B, Ricklin J C 1993 Appl. Opt. 32 5918
Google Scholar
[16] Andrews L C, Young C Y, Miller W B 1996 J. Opt. Soc. Am. A 13 851
Google Scholar
[17] Andrews L C, Phillips R L, Miller W B 1997 Appl. Opt. 36 698
Google Scholar
[18] Korotkova O, Andrews L C 2002 Laser Radar Technology and Applications VII Orlando, FL, United States, July 29, 2002 p4723
[19] Andrews L C, Phillips R L 2005 Laser Beam Propagation Through Random Media (2nd Ed.) (Bellingham: SPIE) pp179–218
[20] Martin J M, Flatte S M 1988 Appl. Opt. 27 2111
Google Scholar
[21] Martin J M, Flatte S M 1990 J. Opt. Soc. Am. A 7 838
Google Scholar
[22] Rubio J A, Belmonte A, Comeron A 1999 Opt. Eng. 38 1462
Google Scholar
[23] Frehlich R 2000 Appl. Opt. 37 393
[24] Welch G, Phillips R 1990 J. Opt. Soc. Am. A 7 578
Google Scholar
[25] Nelson D H, Walters D L, MacKerrow E P, Schmitt M J, Quick C R, Porch W M, Petrin R R 2000 Appl. Opt. 39 1857
Google Scholar
[26] Belmonte A 2000 Appl. Opt. 39 5426
Google Scholar
[27] Belmonte A 2005 Opt. Express 13 9598
Google Scholar
[28] Li Y Q, Wang L G, Wu Z S 2019 Optik 179 244
Google Scholar
[29] Gbur G, Wolf E 2002 J. Opt. Soc. Am. A 19 1592
[30] Schmidt, Jason D 2010 Numerical Simulation of Optical Wave Propagation with Examples in MATLAB (Bellingha: SPIE) pp157–163
[31] McGlamery B L 1976 Proc. SPIE 0074 225
Google Scholar
[32] Sedmak G 2004 Appl. Opt. 43 4527
Google Scholar
[33] Harding C M, Johnston R A, Lane R G 1999 Appl. Opt. 38 2161
Google Scholar
[34] Assemat F, Wilson R W, Gendron E 2006 Opt. Express 14 988
Google Scholar
[35] Kuga Y, Phu P 1996 J. Electromagn. 10 451
Google Scholar
[36] Herraez M A, Burton D R, Lalor M J, Gdeisat M A 2002 Appl. Opt. 41 7437
Google Scholar
[37] 饶瑞中, 王海燕 2012 光学涡旋在湍流大气中的传播 (上海: 上海交通大学出版社) 第216页
Rao R Z, Wang H Y 2012 Propagation of Optical Vortices in Turbulent Atmosphere (Shanghai: Shanghai Jiaotong University Press) p216 (in Chinese)
[38] Tao Z W, Ren Y C, Abdukirim A, Liu S W, Rao R Z 2021 J. Opt. Soc. Am. A 38 1120
Google Scholar
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