1.School of Physics and Electronics, Central South University, Changsha 410083, China 2.College of Advanced Interdisciplinary Studies, National University of Defense Technology, Changsha 410073, China
Fund Project:Project supported by the Equipment Pre-research Field Fund, China (Grant No. 6140415020311), the Hunan Provincial Key Laboratory of High Energy Laser Technology Fund, China (Grant No. GNJGJS04), and the Hunan Engineering Research Center of Optoelectronic Inertial Technology, China (Grant No. HN-NUDT1908)
Received Date:06 June 2019
Accepted Date:08 July 2019
Available Online:01 November 2019
Published Online:05 November 2019
Abstract:The vortex beam is a ring-shaped beam whose center intensity or axial intensity is zero in the propagation direction and whose phase has a spiral rising or falling gradient distribution, which is also called a dark hollow beam. Vortex beams have important applications in free-space optical communication, optical micromanipulation, quantum information processing, optical measurement, super-resolution imaging, laser processing, and material processing. In recent years, with the in-depth research on vortex beams, the application requirements for high-power vortex beams also increase. High-power and high-quality vortex beam can be obtained by coherent combining technology. However, the spiral spectrum characteristics of the vortex beam generated by coherent combining technology need further exploring. In this paper, based on the theory of spectral analysis, we derive the position and magnitude of the spiral phase spectral component of the coherent synthetic vortex beam. The numerical results verify the correctness of the theoretical derivation. Based on the above spectral analysis theory, the mode purity of the target synthesis topology charge can be used as the evaluation function to evaluate quality and optimize the parameters for the coherent synthetic vortex beam, and then to quantitatively guide the coherent synthesis of the vortex beam. The results show that with the increase of the number of sub-beams and the radius of the beam waist of the source plane, the reduction of the radius of the bundle ring and the mode purity of the target synthesis topology charge can be improved, and then we can obtain the high-quality vortex beam. This is consistent with the conclusion obtained by using traditional evaluation functions such as power in the bucket. The spiral spectrum analysis of the coherent synthetic vortex beam not only makes up for the lack of evaluation of the spiral phase synthesis effect by the traditional evaluation function, but also has certain reference significance for understanding the nature of the coherent synthesis technique. Keywords:vortex beam/ coherent combining/ spiral spectrum analysis/ evaluation function
对环形排列的高斯光束阵列加载离散涡旋相位, 相干合成n阶BG涡旋光束. 源平面高斯光束阵列及相位分布如图1(a)和图1(b)所示. 具有离散螺旋相位的高斯光束阵列在自由空间的传播方程为[24] 图 1M = 12, n = 2, R = 1.2 mm, w0 = 0.24 mm时的高斯光束阵列 (a)源平面空间分布; (b)源平面相位分布; (c)传输 2 m后合成涡旋光束强度分布; (d)传输2 m后合成涡旋光束相位分布; (e)标准2阶BG涡旋光束强度分布; (f)标准2阶BG涡旋光束相位分布 Figure1. Gaussian beam array with M = 12, n = 2, R = 1.2 mm, w0 = 0.24 mm: (a) Source plane spatial distribution; (b) source plane phase distribution; (c) light field distribution of synthetic vortex beam after 2 m transmission; (d) phase distribution of synthetic vortex beam after 2 m transmission; (e) light field distribution of standard 2nd order BG vortex beam; (f) phase distribution of standard 2nd order BG vortex beam.
以$M = 8$, $n = 1$为例, 计算了w0 = 0.2 mm, R = 2.1 mm, λ = 632.8 nm时, z = 10 m处相干合成涡旋光束的强度和相位分布如图2(a)和图2(b)所示. 根据(8)式计算得到非0谱主成分振幅系数分别是${a_{ - 15}}$, ${a_{ - 7}}$, ${a_1}$, ${a_9}$和${a_{17}}$, 根据(3)式, 将主成分线性叠加后得到z = 10 m处重建光场的强度分布和相位分布如图2(c)和图2(d)所示. 对比图2(a)与图2(c), 图2(b)与图2(d)发现, 螺旋谐波叠加光场与原相干合成涡旋光束的强度分布和相位分布几乎完全一致, 这充分验证了谱分析过程的正确性. 图 2z = 10 m处相干合成涡旋光束的(a)强度分布和(b)光束相位分布; 螺旋谐波重建的(c)强度分布和(d)相位分布 Figure2. Target plane at z = 10 m: (a) Light field distribution of coherent synthetic vortex beam; (b) phase distribution of coherent synthetic vortex beam; (c) light field distribution of spiral harmonic reconstruction light field; (d) phase distribution of spiral harmonic reconstruction light field.
根据(9)式可计算得到M不同时, 目标合成拓扑荷n为1的相干合成涡旋光束螺旋谱分布及大小, 如图3(a)—(c)所示. 图 3 相干合成涡旋光束螺旋谱分布及大小(其中n = 1, z = 10 m, w0 = 0.2 mm, R = 2.1 mm) (a) M = 8; (b) M = 12; (c) M = 16 Figure3. Coherent synthetic vortex beam spiral spectrum distribution and size (n = 1, z = 10 m, w0 = 0.2 mm, R = 2.1 mm): (a) M = 8; (b) M = 12; (c) M = 16.
根据(7)和(9)式, 计算了n = 1, w0 = 0.2 mm, R = 2.1 mm, λ = 632.8 nm时, M = 8和M = 16两种情况z = 10 m处相干合成BG涡旋光束的强度分布、相位分布以及螺旋谱分布, 分别如图4(a)、图4(c)、图4(e)和图4(b)、图4(d)、图4(f)所示. M = 8和M = 16时拓扑荷n = 1的螺旋谐波相对功率分别为31.97%和60.61%. 可见, 其他参数相同时, M越大合成涡旋光束越接近标准拓扑荷为1的BG涡旋光束. 进一步对比发现: M = 16时合成涡旋光束环外旁瓣明显较少, 主环能量更高, 光束质量更好; 同时合成相位分布中环上交叉条纹更少, 相位分布更加光滑. 图 4 相干合成BG涡旋光束(n = 1, z = 10 m, w0 = 0.2 mm, R = 2.1 mm) (a) M = 8时强度分布; (b) M = 16时强度分布; (c) M = 8时相位分布; (d) M = 16时相位分布; (e) M = 8时螺旋谱分布; (f) M = 16时螺旋谱分布 Figure4. Coherently synthesized BG vortex beam (n = 1, z = 10 m, w0 = 0.2 mm, R = 2.1 mm): (a) M = 8, light intensity distribution; (b) M =16, light intensity distribution; (c) M = 8, phase distribution; (d) M = 16, phase distribution; (e) M = 8, spiral distribution; (f) M = 16, spiral distribution.
当w0 = 0.2 mm, R = 2.1 mm, z = 10 m 时, 不同目标拓扑荷相干合成BG涡旋光束中心谱纯度Pl随子光束数量M的变化趋势如图5所示. 可见, 随着M不断增大, 非目标螺旋谱分量逐渐减小至0, 目标合成拓扑荷的螺旋谐波纯度不断增加, 趋近100%, 合成光束趋近标准BG涡旋光束. 图 5 不同阶合成涡旋光束拓扑荷模式纯度Pl随子光束数量M的变化趋势(w0 = 0.2 mm, R = 2.1 mm, z = 10 m) Figure5. Variation trend of the spectral purity Pl of the different order synthetic vortex beams with the number of sub-beams M (w0 = 0.2 mm, R = 2.1 mm, z = 10 m).
33.1.2.源平面束腰半径 -->
3.1.2.源平面束腰半径
根据(7)和(9)式, 计算了n = 1, M = 12, R = 2.1 mm, λ = 632.8 nm时w0 = 0.15 mm和w0 = 0.3 mm两种情况z = 10 m处相干合成BG涡旋光束的强度分布、相位分布以及螺旋谱分布, 分别如图6(a)、图6(c)、图6(e)和图6(b)、图6(d)、图6(f)所示. w0 = 0.15 mm和w0 = 0.3 mm时拓扑荷n = 1的螺旋谐波相对功率分别为35.12%和67.75%. 可见, 其他参数相同时, w0越大合成涡旋光束越接近标准拓扑荷为1的BG涡旋光束. 进一步对比发现: w0 = 0.3 mm时合成涡旋光束环外旁瓣明显较少, 主环能量更高, 光束质量更好; 同时合成相位分布中, 环上交叉条纹更少, 相位分布更加光滑. 图 6 相干合成BG涡旋光束(n = 1, z = 10 m, M = 12, R = 2.1 mm) (a) w0 = 0.15 mm时强度分布; (b) w0 = 0.3 mm时强度分布; (c) w0 = 0.15 mm时相位分布; (d) w0 = 0.3 mm时相位分布; (e) w0 = 0.15 mm时螺旋谱分布; (f) w0 = 0.3 mm时螺旋谱分布 Figure6. Coherently synthesized BG vortex beam (n = 1, z = 10 m, M = 12, R = 2.1 mm): (a) w0 = 0.15 mm, light intensity distribution; (b) w0 = 0.3 mm, light intensity distribution; (c) w0 = 0.15 mm, phase distribution; (d) w0 = 0.3 mm, phase distribution; (e) w0 = 0.15 mm, spiral distribution; (f) w0 = 0.3 mm, spiral distribution.
当M = 12, R = 2.1 mm, z = 10 m时, 不同目标拓扑荷相干合成BG涡旋光束中心谱纯度Pl随子光束束腰半径w0的变化趋势如图7所示. 可见, 随着w0不断增大, 非目标拓扑荷螺旋谱分量逐渐减小至0, 目标合成拓扑荷的螺旋谐波纯度不断增加, 趋近100%, 合成光束趋近标准BG涡旋光束. 图 7 不同阶合成涡旋光束拓扑荷模式纯度Pl随子光束束腰半径w0的变化(M = 12, R = 2.1 mm, z = 10 m) Figure7. Variation trend of the spectral purity Pl of the different order synthetic vortex beams with sub beam waist radius w0 (M = 12, R = 2.1 mm, z = 10 m).
33.1.3.组束环半径 -->
3.1.3.组束环半径
根据(7)和(9)式, 计算了n = 1, M = 12, w0 = 0.2 mm, λ = 632.8 nm时, R = 1 mm和R = 2.2 mm两种情况z = 10 m处相干合成BG涡旋光束的强度分布、相位分布以及螺旋谱分布, 分别如图8(a)、图8(c)、图8(e)和图8(b)、图8(d)、图8(f)所示. R = 1 mm和R = 2.2 mm时拓扑荷n = 1的螺旋谐波相对功率分别为88.92%和43.62%. 可见, 其他参数相同时, R越小合成涡旋光束越接近标准拓扑荷为1的BG涡旋光束. 进一步对比发现: R = 1 mm时合成涡旋光束环外旁瓣明显较少, 主环能量更高, 光束质量更好; 同时合成相位分布中环上交叉条纹更少, 相位分布更加光滑. 图 8 相干合成BG涡旋光束(n = 1, z = 10 m, M = 12, w0 = 0.2 mm) (a) R = 1 mm时强度分布; (b) R = 2.2 mm时强度分布; (c) R = 1 mm时相位分布; (d) R = 2.2 mm时相位分布; (e) R = 1 mm时螺旋谱分布; (f) R = 2.2 mm时螺旋谱分布 Figure8. Coherently synthesized BG vortex beam (n = 1, z = 10 m, M = 12, w0 = 0.2 mm): (a) R = 1 mm, light intensity distribution; (b) R = 2.2 mm, light intensity distribution; (c) R = 1 mm, phase distribution; (d) R = 2.2 mm, phase distribution; (e) R = 1 mm, spiral distribution; (f) R = 2.2 mm, spiral distribution.
当M = 12, w0 = 0.2 mm, z = 10 m时, 不同目标拓扑荷相干合成BG涡旋光束中心谱纯度Pl随组束环半径R的变化趋势如图9所示. 由图9可见, 随着R不断增大, 非目标拓扑荷螺旋谱分量逐渐增大, 目标合成拓扑荷的螺旋谐波纯度不断减小, 趋近0, 更难相干合成标准BG涡旋光束. 图 9 不同阶合成涡旋光束拓扑荷模式纯度Pl随组束环半径R的变化(M = 12, w0 = 0.2 mm, z = 10 m) Figure9. Variation trend of the spectral purity Pl of the different order synthetic vortex beams with beam ring radius R (M = 12, w0 = 0.2 mm, z = 10 m).