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自1836年Talbot[1]报道了周期性物体的衍射自成像效应以来, 对Talbot效应的研究和应用逐步深入. 尤其是对分数Talbot效应的理论解释[2], 进一步拓宽了Talbot效应的应用范围, 更加引起研究人员对Talbot效应的重视. 目前, Talbot效应和分数Talbot效应已在光学图形处理[3,4]、光学精密测量[5]、光信息存储[6]、原子光学[7-9]、阵列照明[10,11]等领域得到了广泛应用. 随着半导体激光技术的不断发展, 由半导体激光器组成的周期性阵列其光场同样具有Talbot效应, 值得人们研究. 根据分数Talbot效应[12], 人们开展了半导体激光阵列Talbot外腔锁相的研究[13,14]. 通过激光阵列Talbot外腔锁相技术, 可以提高激光阵列互注入耦合效率[15], 实现激光低阶模式选择, 有效改善大阵列半导体激光器的远场光束质量, 获得近衍射极限的相干激光输出[16]. 由于锁相后的激光阵列具有高简并度和相干性, 亦可将其应用于量子测量技术, 实现以高度相干的激光操纵玻色-爱因斯坦凝聚体(BEC)形成原子密度光栅[17,18]. 因此, 激光阵列的Talbot效应在未来高功率、高光束质量半导体激光器以及量子测量领域蕴藏着广阔的实用前景.
一维激光器阵列的Talbot效应是半导体激光阵列Talbot外腔锁相研究的一个重要方向[19,20], 然而在实际应用中, 一维激光阵列难以达到无限单元周期性排布, 使得一维激光阵列无法产生理想的Talbot效应, 即存在边缘效应. 与一维激光阵列相比, 环形激光阵列在极坐标系下具有角向无限周期性单元结构, 可以近似为无限单元的一维周期性阵列结构[21], 满足Talbot条件.
通常采用笛卡尔坐标系下的分数傅里叶变换计周期性物体的Talbot效应[22,23]. 然而这种方法难以计算环形激光阵列的Talbot效应, 本文在极坐标系下将环形激光阵列的Gyrator变换和菲涅耳衍射相结合, 推导了环形激光阵列的Talbot效应, 获得了其Talbot自成像条件. 通过FDTD Solutions软件模拟得到环形激光阵列在Talbot距离及分数Talbot距离处的自成像情况, 通过与一维激光阵列分数Talbot效应的光场、相位分布情况对比分析, 在分数Talbot距离处均得到了相一致的结果, 并且有效消除了边缘效应, 从而将Talbot效应从一维周期性阵列推广到了环形周期性阵列, 扩展了Talbot效应在环形激光阵列的应用, 为环形激光阵列锁相研究提供了Talbot外腔方法.
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环形激光阵列结构如图1所示, 红色圆表示激光发光单元, r0为环形结构的半径, θ0为相邻发光单元间的圆心角.
图 1 环形激光阵列
Figure 1. Ring laser array.
在极坐标下, 环形激光阵列发光单元的分布方式可以用梳状函数和狄拉克函数表示:
$ \gamma \left( {r,\theta } \right)=\delta \left( {r - {r_0}} \right)\frac{1}{{{\theta _0}}}{\text{comb}}\left( {\frac{\theta }{{{\theta _0}}}} \right), $ 其中r0是环形结构的半径, θ0是相邻发光单元间的圆心角. 激光阵列每一个发光单元为高斯光束, 则发光单元在初始位置的光场可以表示为
$ {E_0}\left( r \right) = {E_0}\exp \left( { - \dfrac{{{r^2}}}{{{\omega ^2_0}}}} \right), $ 其中,
$ E_{0}=|E(0)| $ , ω0是高斯光束的束腰半径. 采用Gyrator变换对(1)式和(2)式形成的初始光场分布做频域分析. 极坐标系下Gyrator变换的积分核函数可写为$\begin{split} &{K_{\text{G}}}\left( {r,\theta ,\rho ,\varphi } \right) = {1}/{{\left| {\sin \alpha } \right|}}\\ &\times\exp \left[ {{\text{i\pi }}\frac{{\left( {{r^2}\sin 2\theta + {\rho ^2}\sin 2\varphi } \right)\cos \alpha - 2r\rho \sin \left( {\theta + \varphi } \right)}}{{\sin \alpha }}} \right],\end{split} $ 其中
$ \alpha $ 为Gyrator变换的阶数. 由此可得极坐标系下的Gyrator变换表达式为[23]$ {G_0}\left( {\rho ,\varphi } \right) = \iint {{E_0}\left( r \right)} \cdot {K_{\text{G}}}\left( {r,\theta ,\rho ,\varphi } \right){\text{d}}r{\text{d}}\theta , $ 由于极坐标系下初始光场分布函数是可分离变量的, 并且
$\alpha = {{\text{π }}}/{2}$ 时, Gyrator变换得到的频域分布可以用零阶贝塞尔恒等式表示为$ {G_0}\left( {\rho ,\varphi } \right) = \int_0^\infty {r{E_0}\left( r \right)} {{\text{J}}_0}\left( {2{\text{π }}r\rho } \right){\text{d}}r. $ 其中J0为零阶贝塞尔函数. 采用菲涅耳传递函数计算环形激光阵列在自由空间传输后的光场分布, 此时观察平面上得到的场分布频谱可以表示为
$ \begin{split} & G\left( {\rho ,\varphi } \right) = {G_0}\left( {\rho ,\varphi } \right)H\left( {\rho ,\varphi } \right) \\ =\;& \exp \left( {{\text{j}}kz} \right)\exp \big[ { - {\text{j\pi }}\lambda z{{\left( {\rho \varphi } \right)}^2}} \big] \\ & \times \int_0^\infty {r{g_0}\left( r \right)} {{\text{J}}_0}\left( {2{\text{π }}r\rho } \right){\text{d}}r, \end{split} $ 其中exp(jkz)是定常数项, λ是激光波长. 在频域
$\left(\rho, \varphi\right)=\left(\dfrac{1}{r_{0}}, \dfrac{n}{\theta_{0}}\right)$ 处对比(5)式和(6)式可以得到, 若要在传输距离$ z $ 处得到自再现像, 则其中二次相位因子必须满足条件:$ \tau =\exp \bigg[ { - {\text{j\pi }}\lambda z{{\left( {\frac{n}{{{r_0}{\theta _0}}}} \right)}^2}} \bigg]=1, $ 此时, 可以得到环形激光阵列的Talbot距离ZT为
$ {Z_{\text{T}}} = \frac{{2{{\left( {{r_0}{\theta _0}} \right)}^2}}}{\lambda }, $ 上述结果表明, 环形激光阵列的Talbot距离与相邻发光单元间圆心角和半径的平方成正比, 与激光波长成反比.
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根据第2节计算的环形阵列Talbot效应, 可以进一步分析得到其分数Talbot效应. 并使用FDTD Solutions有限元分析对基模高斯光束环形阵列在分数Talbot距离处的光场分布和相位分布进行模拟, 模拟结构示意如图2所示, 图中红色实心圆为发光单元, 虚线平面代表相应的观察面.
图 2 仿真模型示意图
Figure 2. Schematic diagram of simulation model.
当
$z = \dfrac{1}{4}{Z_{\text{T}}}$ 时, 二次相位因子$\tau = \exp \Big( { - {\text{j\pi }}\dfrac{{{n^2}}}{2}} \Big)$ . 当n为奇数时,$\tau = {{ - j}}$ ; 当n为偶数时,$ \tau = 1 $ . 此时z平面光场在一个周期内为两种具有不同相位的Talbot子像的叠加, Talbot子像元数为发光单元个数的2倍.当距离
$z = \dfrac{1}{2}{Z_{\text{T}}}$ 时, 二次相位因子$ \tau = - 1 $ , 表明z平面上的光场复振幅分布周期与初始光场相同, 但有半个周期的空间偏移.当
$z = \dfrac{3}{4}{Z_{\text{T}}}$ 时,$\tau = \exp \Big( { - {\text{j\pi }}\dfrac{{{\text{3}}{n^2}}}{2}} \Big)$ . 当n为奇数时,$ \tau = {\text{j}} $ ; 当n为偶数时,$ \tau = 1 $ . 此时z平面光场在一个周期内为两种具有不同相位的Talbot子像的叠加, Talbot子像元数为发光单元的2倍.图3所示为环形激光阵列分数Talbot效应模拟结果, 其光场分布均为归一化之后的结果. 如图3(a), (f)所示, 由28个发光单元组成的激光阵列具有相同的光强和初始相位. 如图3(b), (g)所示, 光束传输到1/4ZT平面时, 子像元数量为发光单元2倍, 子像元光强相同, 相邻子像元相位差为
$ {\text{π }}/2 $ . 如图3(c), (h)所示, 在1/2ZT平面, Talbot子像元数目与发光单元相同, 但是子像的空间位置沿角向方向产生了半个周期的偏移, 子像元光强与发光单元相同, 相邻子像元相位相同. 如图3(d), (i)所示, 在3/4ZT平面呈现数量为发光单元2倍的Talbot子像元, 子像元光强相同, 相邻子像元相位差为$ - {\text{π }}/2 $ . 如图3(e), (j)所示, 在ZT平面, 沿角向产生了空间分布和相位分布与发光单元相同的自再现像.
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.
为了对比环形周期阵列与一维周期阵列的Talbot效应, 同样模拟了一维基模高斯光束阵列的分数Talbot效应, 模型中一维基模高斯光束阵列的发光单元与上述环形阵列发光单元保持相同参数, 模拟结果如图4所示. 将图4与图3相比, 可以明显地观察到一维激光阵列与环形激光阵列的一致性和差异. 二者在1/4ZT平面、1/2ZT平面、3/4ZT平面以及ZT平面的Talbot子像元数目和空间位置以及相邻子像元的相位差具有一致性. 然而, 从图4可以看出, 一维激光阵列的Talbot效应存在明显的边缘衰减效应, 其Talbot子像元的光强在阵列边缘处逐渐降低, 并且传输距离越远, 衰减效果越明显. 这是由于一维激光阵列无法构成无限单元的周期分布, 阵列边缘处不能实现理想的Talbot效应. 然而环形激光阵列在极坐标系下沿角向可以视为无限延伸的周期分布, 因此不存在边缘衰减效应, Talbot子像元始终保持等光强分布.
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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本文将笛卡尔坐标系下的Gyrator变换应用到极坐标系中, 并结合菲涅耳衍射, 通过理论计算和模拟仿真研究了环形激光阵列结构的Talbot效应. 计算了环形激光阵列的Talbot成像条件并分析了其分数 Talbot效应, 计算结果表明, 在偶数倍1/2 Talbot距离处, 子像和原始发光单元一致; 在奇数倍1/2 Talbot距离处子像位置沿角向旋转1/2周期, 子像元数量无变化; 在奇数倍1/4 Talbot距离处, 子像元数目增加至发光单元的2倍. 使用FDTD Solutions软件对环形激光阵列和一维激光阵列的分数Talbot效应进行仿真模拟和比较分析表明, 基于极坐标系下Gyrator变换方法得到的环形激光阵列的Talbot效应与一维激光阵列Talbot效应变化规律具有一致性, 但消除了一维激光阵列的边缘衰减效应, 可以获得等光强分布的Talbot子像. 本文认为利用环形激光阵列的Talbot效应, 在图像光学加密、无掩膜光刻蚀、阵列照明等领域有潜在的应用价值. 并且基于环形激光结构Talbot效应的外腔锁相设计可使激光输出模式得到选择, 在互注入锁相后输出得到相干光, 在高亮度相干激光领域和量子测量领域得到应用.
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本文对环形激光阵列的Talbot效应进行了研究, 利用Gyrator正则变换推导了极坐标下环形激光阵列的Talbot效应和自成像条件, 并进一步分析了其在分数Talbot距离处的成像规律. 通过FDTD Solutions软件对环形激光阵列在分数Talbot距离处的空间分布、相位分布进行模拟, 得到与理论计算相一致的结果. 通过与一维激光阵列分数Talbot效应的空间分布、相位分布情况进行对比分析, 环形激光阵列可有效消除一维激光阵列的Talbot边缘效应, 获得等光强分布的Talbot自再现像, 扩展了Talbot效应在环形激光相干阵列锁相的应用.
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关键词:
- 环形激光阵列 /
- 分数Talbot效应 /
- 自成像 /
- Gyrator变换 /
- 相干阵列
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
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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
Fig. 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
Fig. 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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