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Propagation characteristics of focused astigmatic Gaussian beams in Kerr nonlinear media

Hu Jing Wang Huan Ji Xiao-Ling

Hu Jing, Wang Huan, Ji Xiao-Ling. Propagation characteristics of focused astigmatic Gaussian beams in Kerr nonlinear media. Acta Phys. Sin., 2021, 70(7): 074205. doi: 10.7498/aps.70.20201661
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Propagation characteristics of focused astigmatic Gaussian beams in Kerr nonlinear media

Hu Jing, Wang Huan, Ji Xiao-Ling
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  • When a powerful laser beam propagates in a Kerr nonlinear medium, the Kerr effect on the beam propagation characteristics is very significant. The astigmatic laser beams are often encountered in practice. Until now, much work has been carried out on the propagation characteristics of astigmatic laser beams in linear media, but a few researches have been reported about the propagation of astigmatic laser beams through nonlinear media. To the best of our knowledge, the propagation or the transformation of astigmatic laser beams through an optical system in a Kerr nonlinear medium has not been investigated. In this paper, the propagation characteristics of focused astigmatic Gaussian beams in a nonlinear Kerr medium are studied. The Kerr effect on the beam astigmatism and the focal shift of focused astigmatic Gaussian beams are investigated in detail, and the self-focusing focal length and focus control of focused astigmatic Gaussian beams in the Kerr nonlinear medium are also studied. For the beam spreading case, the analytical formula for each of the beam width, the beam waist position, and the focal shift of focused astigmatic Gaussian beams in the Kerr nonlinear medium is derived. It is shown that in the self-focusing medium, as the beam power increases (i.e. the self-focusing effect becomes stronger), the beam astigmatism becomes stronger, but the focal shift decreases. However, in a self-defocusing medium, as the beam power increases (i.e. the self-defocusing effect becomes stronger), the beam astigmatism becomes weaker, but the focal shift increases. On the other hand, for the beam self-focusing case, the analytical formula of the self-focusing focal length of focused astigmatic Gaussian beams in the Kerr nonlinear medium is derived. It is found that the number of foci can be controlled by applying beam astigmatism. The results obtained in this paper are of theoretical and practical significance.
      PACS:
      42.65.Hw(Phase conjugation; photorefractive and Kerr effects)
      42.65.-k(Nonlinear optics)
      42.60.Jf(Beam characteristics: profile, intensity, and power; spatial pattern formation)
      42.30.Lr(Modulation and optical transfer functions)
      Corresponding author: Ji Xiao-Ling, jiXL100@163.com
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 61775152)

    像散光束在实际应用中常会遇到, 例如, 当激光垂直入射非旋转对称光学系统或倾斜入射到光学元件表面时, 光束将会发生像散[1-3]. 像散会严重影响激光传输特性. 吴逢铁教授课题组研究了像散对轴棱锥衍射特性的影响与修正[4], 并采用理论和实验方法研究了像散Bessel光束自重建特性[5]. 林强教授课题组推导出部分相干扭曲各向异性高斯谢尔模型光束(具有复杂像散特性的部分相干光)传输的张量ABCD定律[6], 并用实验研究了复杂像散椭圆光束的轨道角动量[7]. 王绍民教授课题组[8]采用张量ABCD定律研究了复杂像散高斯激光束的对称变换问题. Tari等[9]提出采用对称透镜校正像散高斯光束的像散和椭圆度. 冯国英教授课题组[10]采用理论和实验方法研究了像散椭圆高斯光束的M2因子矩阵. 蔡阳健教授课题组研究了部分相干扭曲各向异性高斯谢尔模型光束在大气湍流中的传输特性[11]以及在色散和吸收介质中的传输特性[12]. 张逸新教授课题组[13]研究了大气湍流中像散对高斯涡旋激光束传输和成像的影响. 然而, 目前的研究大都仅局限于像散光束在线性介质的传输特性[4-13].

    当高功率激光通过Kerr非线性介质传输时, Kerr 效应将会严重影响激光传输特性和光束质量[14-16]. 最近, 我们课题组解析求解了非线性薛定谔方程, 并推导出了部分相干激光在非线性Kerr介质中传输的ABCD定律[17]; 研究发现部分相干脉冲光比完全相干脉冲光在避免材料发生光学损伤方面更具优势[18]; 推导出了部分相干光通过Kerr介质中光束系统传输变换公式, 并发现Kerr效应可以调控光斑位置和尺寸[19], 文献[19]已被遴选为“Spotlight on Optics”论文.

    迄今为止, 像散椭圆光束在非线性介质中传输特性的研究较少. 郭旗教授课题组[20]研究了椭圆高斯光束在强非局域非线性介质中的传输特性. Goncharenko等[21]研究了旋转椭圆高斯光束在梯度折射率非线性介质中的传输. 此外, 在局域Kerr非线性介质中, Cornolti等[22]的研究表明椭圆高斯光的自陷功率随光束椭圆度的增大而增大, Singh等[23]研究了椭圆高斯光的自聚焦和自相位调制动力学问题. 然而, 像散激光束通过含光学系统的Kerr介质传输变换的研究还未涉及. 本文研究了Kerr非线性介质中聚焦像散高斯光束的传输特性, 主要包括: Kerr效应对光束像散特性和焦移特性的影响, 以及聚焦像散高斯光束的自聚焦焦距和光束焦点调控.

    激光在Kerr介质中的传输特性由Kerr效应和衍射效应共同决定. 高斯激光诱导介质折射率变化引起的非线性作用与衍射作用之比$\eta = $$ {{\left( {{n_2}{I_0}{k^2}w_1^2} \right)}/ {{n_0}}}$[17], 其中光强${I_0} = {{2 P}/ {\left( {{\text{π}}w_1^2} \right)}}$, k为波数, w1为入射光束宽, n0n2分别为介质的线性和非线性折射率, n2 > 0(即η > 0)与n2 < 0 (即η < 0)分别对应Kerr自聚焦介质与Kerr自散焦介质. η < 1时激光在传输的过程中逐渐发散, η > 1时激光逐渐会聚. η < 1对应于激光在自散焦介质中传输, 或者在自聚焦介质中传输, 但满足光束功率P < Pcr, Pcr为光束自聚焦临界功率; η > 1对应于激光在自聚焦介质中传输, 且满足P > Pcr. 本文第2—第4节研究η < 1情况(在自聚焦介质中满足P < n0π/(2n2k 2)时, 能够保持光束类高斯轮廓分布), 第5节讨论η > 1情况.

    高斯光束以束腰入射, 通过Kerr介质中像散薄透镜传输(如图1所示). 高斯光束通过像散薄透镜后, 其光场分布为可表示为

    图 1  Kerr非线性介质中聚焦像散高斯光束传输的示意图
    Fig. 1.  Schematic diagram of a focused astigmatic Gaussian beam propagating in Kerr nonlinear media.

    $\begin{split} &{E_0}\left( {x,y,0} \right) = \sqrt {{I_0}} \exp \left[ { - \left( {\frac{{{x^2} + {y^2}}}{{w_1^2}}} \right)} \right]\\ &\times\exp \left[ {{\rm{i}}k\left( {\frac{{{x^2}+{y^2}}}{{2f}}} \right)} \right]\exp \left[ { - {\rm{i}}k{C_6}\left( {{x^2} - {y^2}} \right)} \right],\end{split}$

    (1)

    其中, f为透镜焦距, C6为像散系数, 波数k = 2π/λ, λ为波长. 对于简单像散, xy方向的光束复参数q可表示为[17]

    $\frac{1}{{{q_j}}} = \frac{1}{{{R_j}}} - {\rm{i}}{M^2}\frac{\lambda }{{{\rm{\pi }}w_j^2}},~~(j = x,y),$

    (2)

    其中Rjwj分别为光束曲率半径和束宽, M 2 = (1-η)1/2为高斯光束在Kerr介质中的M 2因子[17]. 需要说明的是, 本文研究的是完全相干光, 文献[17]研究的是部分相干光, 本文(2)式是文献[17]中相应公式在空间相干宽度趋于∞时的特例.

    根据(2)式, (1)式可简化为

    ${E_0}\left( {x,y,0} \right) = \sqrt {{I_0}} \exp \left[ { - \frac{{{\rm{i}}k}}{2}\left( {\frac{{{x^2}}}{{{q_{1x}}}} + \frac{{{y^2}}}{{{q_{1y}}}}} \right)} \right],$

    (3)

    其中${q_{1 j}}$z = 0处的光束复参数, 1/R1x = 2C6 –1/f和1/R1y = –2C6 – 1/f分别为xy方向光束曲率半径.

    依据Kerr介质中的ABCD定律[17], 像散高斯光传输至z处的复参数${q_{2 j}}$可表示为

    $\frac{1}{{{q_{2j}}}} = \frac{{{1 / {{q_{1j}}}}}}{{1 + {z/ {{q_{1j}}}}}}.$

    (4)

    将(2)式代入(4)式中, 分离实部与虚部后可得像散高斯光束在Kerr介质中传输的曲率半径R2j与束宽w2j分别为

    $ {R_{2j}}=\frac{{{{(1 - {z / f} \pm 2{C_6}z)}^2} + {{({z / Z})}^2}}}{{{{-(1-{z / f})} / f}\pm2{C_6}(1-2{z / f})+z({1 / {{Z^2}}}+4C_6^2)}},$

    (5)

    $w_{2j}^2 = w_1^2\left[{\left(1 - \frac{z}{f} \pm 2{C_6}z\right)^2} + {\left(\frac{z}{Z}\right)^2}\right],$

    (6)

    其中$Z = {{{\text{π}}w_1^2} / {{M^2}\lambda }}$为光束在Kerr非线性介质中的瑞利长度[17], j = x时取“+”号, j = y时取“–”号. (5)式和(6)式表明, 一般情况下, xy方向的光束曲率半径和束宽均不相等, 光斑呈椭圆形状. 但是, 当z = f时, 有w2y = w2x, 即为圆高斯光束. 值得指出的是, 这一结论在线性介质中也是成立的. 另外, 当C6 = 0时, (5)式和(6)式简化为无像散时的结果.

    值得指出的是, 虽然Kerr效应和衍射效应均与光束束宽有关, 但η参数描述Kerr效应与衍射效应作用之比, 并且可得到$ \eta= 2n_2k^2P/(n_0\pi ) $与束宽无关, 故$ M^2 = (1-\eta)^{1/2} $也与束宽无关. 另一方面, 我们已证明, 在线性介质中, 像散高斯光束的M 2 = 1[24], 即像散不改变高斯光束的M 2因子. 在非线性Kerr介质中, 只要光束保持类高斯轮廓分布, 像散光束在xy方向的M 2因子的表达式应相同, 因此分别引入xy方向的ABCD定律来研究Kerr非线性介质中聚焦像散高斯光束的传输特性是合理的[25], 即(4)式成立.

    令(5)式中R2j → ∞, 可得到通过透镜后像散高斯光束(聚焦像散高斯光束)在Kerr介质中的束腰位置为

    ${s_{2j}} = \frac{{{1 / f} \mp 2{C_6}}}{{{{\left( {{1 / f} \mp 2{C_6}} \right)}^2} + {{\left( {{1 / Z}} \right)}^2}}},$

    (7)

    式中, j = x时取“–”号, j = y时取“+”号. 由于像散作用, 一般情况下有s2xs2y, 即xy方向光束束腰位置不重合. 但是, 当$ C_6 = (1/f^2 + 1/Z^2)^{1/2}/2 $时, 有s2x = s2y = f/2 (即xy方向光束束腰位置重合), 该结论在线性介质中也成立.

    除特别说明, 本文数值计算采用参数如下: f = 0.2 m, w1 = 0.4 mm. 自聚焦介质中: n0 = 2.4, n2 = 2 × 10–13 cm2/W(As2S3玻璃), λ = 1.06 µm; 自散焦介质中: n0 = 1.56, n2 = –1.5 × 10–13 cm2/W(合成含有3, 4-二烷氧基噻吩的可溶性多恶唑), λ = 0.532 µm.

    由(7)式可知, s2y > 0, 且s2y随着C6的增大而减小. 其物理原因是: 由于像散作用, y方向光束相当于被焦距为1/(2C6)的凸透镜会聚. Kerr介质中x方向束腰位置s2xC6的变化如图2所示, 由图可知, s2xC6的变化关于C6 = 1/(2f )对称, 且存在一个最大值和一个最小值. 令(7)式中∂s2x/∂C6 = 0可得: 当$ C_{6,\min }= 1/(2f) - 1/(2Z) $$ C_{6,\max} = 1/(2f ) + 1/(2Z) $时, s2x分别有最大值和最小值, 即

    图 2  束腰位置s2x随像散系数C6的变化
    Fig. 2.  Beam waist position s2x versus the astigmatic coefficient C6.

    ${s_{2x,\max }} = \frac{Z}{2},\;{s_{2x,\min }} = - \frac{Z}{2}.$

    (8)

    利用像散系数C6可调节束腰位置, 例如当$ C_6 < 1/(2f) $时, 光束会聚, s2x > 0; 当$ C_6 > 1/(2f) $时, 光束发散, s2x < 0 (即束腰位置位于透镜左方). 此外, 容易得到: ΔC6 = C6,max C6,min = 1/Z, Δs2x = s2x, maxs2x, min = Z, 它们仅与光束瑞利长度有关. 自聚焦介质中(η > 0), η越大(自聚焦作用越强), 则C6,mins2x,max越大, 而C6,maxs2x,min越小, 那么ΔC6越小, Δs2x越大. 自散焦介质中(η < 0), |η|越大(自散焦作用越强), 则C6,mins2x,max越小, 而C6,maxs2x,min越大, 那么ΔC6越大, Δs2x越小. 其物理原因是: 自聚焦作用越强, 则Z越大; 自散焦作用越强, 则Z越小.

    本文采用像散参数β = w2y/w2x表征光束的像散程度. 根据(6)式可得Kerr介质中光束像散参数β

    $\beta = {\left[ {\frac{{{Z^2}{{\left( {f - z - 2f{C_6}z} \right)}^2} + {{(zf)}^2}}}{{{Z^2}{{\left( {f - z + 2f{C_6}z} \right)}^2} + {{(zf)}^2}}}} \right]^{{1 / 2}}}.$

    (9)

    β越远离数值1, 光斑呈现椭圆状越明显, 即光束像散越厉害. 由(9)式可知, 当z = f时, 有β = 1, 即在透镜焦平面处光束为圆高斯分布, 该结论与Kerr效应及像散系数C6均无关.

    Kerr介质中像散参数β随相对传输距离z/f的变化如图3所示. 由图可知, β存在一个最小值和一个最大值. 令(9)式中β/∂z = 0可得: 当${z_{\min }} = $$ {\left[ {{1 / f} + {{\left( {{1 / {{Z^2}}} + 4 C_6^2} \right)}^{{1 / 2}}}} \right]^{ - 1}}$${z_{\max }} =\Big[ {1 / f} - ( {1 / {{Z^2}}} + $$ 4 C_6^2 )^{{1 / 2}} \Big]^{ - 1}$时, β分别有最小值和最大值, 即

    图 3  像散参数β随相对传输距离z/f的变化 (a)不同像散系数; (b)不同Kerr效应强度
    Fig. 3.  Astigmatic parameter β versus the relative propagation distance z/f: (a) Different values of the astigmatic coefficient; (b) different strength of the Kerr effect.

    ${\beta _{\min }} = {\left[ {\frac{{{Z^2}{{\left( {\sqrt {{1 / {{Z^2} + 4C_6^2}}} - 2{C_6}} \right)}^2} + 1}}{{{Z^2}{{\left( {\sqrt {{1 / {{Z^2} + 4C_6^2}}} + 2{C_6}} \right)}^2} + 1}}} \right]^{{1 / 2}}},\tag{10a}$

    ${\beta _{\max }} = {\left[ {\frac{{{Z^2}{{\left( {\sqrt {{1 / {{Z^2} + 4C_6^2}}} + 2{C_6}} \right)}^2} + 1}}{{{Z^2}{{\left( {\sqrt {{1 / {{Z^2} + 4C_6^2}}} - 2{C_6}} \right)}^2} + 1}}} \right]^{{1 / 2}}}.\tag{10b}$

    由(10)式可知, βminβmax仅与瑞利长度和像散系数有关, 而与透镜焦距无关. 并且, 当z → ∞时, 像散参数β将趋于一个稳定值, 根据(9)式可得该值为

    $\mathop {\lim }\limits_{z \to \infty } \beta = \sqrt {1 + \frac{{8{C_6}}}{{f\left[ {{{(2{C_6} - {1 / f})}^2} + {{{1 / Z}}^2}} \right]}}} .$

    (11)

    图3(a)表明, Kerr介质中, C6越大, β越远离数值1, 即光束像散越厉害. 图3(b)表明, 自聚焦介质中(η > 0), η越大(自聚焦作用越强), β的值越远离数值1, 即光束像散越厉害; 自散焦介质中(η < 0), |η|越大(自散焦作用越强), β的值越接近数值1, 即光束像散越弱.

    聚焦像散高斯光束在Kerr介质中的场分布表示为

    $\begin{split} E(x,y,z) =\;& \sqrt {{I_0}} {\left[ {\frac{{w_1^2}}{{{w_{2x}}{w_{2y}}}}} \right]^{{1 / 2}}}\\ &\times\exp \left[ { - \frac{{{\rm{i}}k}}{2}\left( {\frac{{{x^2}}}{{{q_{2x}}}} + \frac{{{y^2}}}{{{q_{2y}}}}} \right)} \right].\end{split}$

    (12)

    根据(12)式, 可得光强分布为

    $\begin{split} I(x,y,z) = \;&E(x,y,z){E^ * }(x,y,z) \\ =\;& \frac{{{I_0}w_1^2}}{{{w_{2x}}{w_{2y}}}}\exp \left[ { - 2\left( {\frac{{{x^2}}}{{w_{2x}^2}} + \frac{{{y^2}}}{{w_{2y}^2}}} \right)} \right].\end{split}$

    (13)

    令(13)式中x = y = 0, 可得Kerr介质中光束的轴上光强为

    $I\left( {0,0,z} \right) = \frac{{{I_0}w_1^2}}{{{w_{2x}}{w_{2y}}}}.$

    (14)

    轴上光强最大值的位置${z_{I - \max }}$由方程∂I (0, 0, z)/∂z = 0确定. 对(14)式求偏导I(0, 0, z), 即

    $\begin{split} &\frac{{\partial I\left( {0,0,z} \right)}}{{\partial z}} \\ &= \frac{{ - {I_0}w_1^2\left( {{w_{2y}}{{\partial {w_{2x}}} / {\partial z + {w_{2x}}{{\partial {w_{2y}}} / {\partial z}}}}} \right)}}{{{{\left( {{w_{2y}}{w_{2x}}} \right)}^2}}} = 0,\end{split}$

    (15)

    可得: 在${z_{I - \max }}= U + V - l/3$ 处有最大光强, 其中$U = [-u/2 + (u^2/4 + v^3/27)^{1/2}]^{1/3}$, $V = [-u/2- $$ (u^2/4 + v^3/27)^{1/2}]^{1/3}$, $u = m-l^2/3,~v = 2(l/3)^3- $$ ml/3+n, ~n=-1/(f\xi)$, $m= ( {3 / {{f^2}}}+ {1 / {{Z^2}}}- 4 C_6^2 ) / \xi$, $l = {{3( {4 C_6^2-{1 / {{f^2}}}-{1 / {{Z^2}}}} )} / {( {f\xi } )}}$, $\xi = [(2C_6+ 1/f)^2 + $$ 1/Z^2][(2C_6 - 1/f)^2 + 1/Z^2]$.

    像散高斯光的光斑面积可表示为

    $S(z) = {\rm{\pi }}{w_{2x}}{w_{2y}}.$

    (16)

    本文中, 把像散高斯光的光斑面积最小位置处定义为束腰位置${z_{S - \min }}$, 即束腰位置${z_{S - \min }}$由方程∂S(z)/∂z = 0确定. 将(6)式代入(16)式, 并对S(z)求偏导:

    $\frac{{\partial S\left( z \right)}}{{\partial z}} = {\rm{\pi }}\left( {{w_{2y}}{{\partial {w_{2x}}} / {\partial z + {w_{2x}}{{\partial {w_{2y}}} / {\partial z}}}}} \right) = 0.$

    (17)

    对比(15)式与(17)式可知: ${z_{I - \max }} = {z_{S - \min }}$, 即聚焦像散高斯光束束腰位置与轴上最大光强位置重合, 该结论与Kerr效应无关(见图4). 特别地, 当C6 = (1/f 2 + 1/Z 2)1/2/2时, 有s2x = s2y = f/2, 则${z_{S - \min }}$ = f/2.

    图 4  (a)轴上光强I(0, 0, z)和(b)光斑面积S(z)随相对传输距离z/f的变化, C6 = 0.2 m–1
    Fig. 4.  (a) Axial intensity and (b) the area of beam spot versus the relative propagation distance z/f.

    把聚焦像散高斯光束的束腰位置认为是实际焦面, 那么几何焦面与实际焦平面的间距Δ定义为焦移, 即

    $\Delta = f - {z_{S - \min }}.$

    (18)

    图5为Kerr介质中聚焦像散高斯光束的焦移Δ随像散系数C6的变化. 由图5可知, C6越大, 则Δ越大. 自聚焦介质中, η越大(自聚焦作用越强), 则Δ越小; 自散焦介质中, |η|越大(自散焦作用越强), 则Δ越大.

    图 5  焦移Δ随像散系数C6的变化 (a)自聚焦介质; (b)自散焦介质
    Fig. 5.  Focal shift Δ versus the astigmatic coefficient C6: (a) Self-focusing media; (b) self-defocusing media.

    当高功率激光通过自聚焦介质传输, 且P > Pcr (即η > 1)时, 激光在传输过程中逐渐会聚, 即发生自聚焦现象. 理想情况下, 当束宽趋于零时, 激光束会聚于一点, 其传输距离称为自聚焦焦距zf[26-28]. 实际上, 激光在接近焦点时已不再是类高斯轮廓分布了, 因此自聚焦焦距zf仅是一种理想情形, 但它仍然具有一定的参考意义, 文献[26-28]均采用该法研究了自聚焦焦距.

    根据文献[17]中的(4)式, 可得到聚焦像散高斯光束(P > Pcr)在自聚焦介质传输时束宽传输方程为

    $w_j^2 = w_{1j}^2\left[ {{{\left( {1 + \frac{z}{{{R_{1j}}}}} \right)}^2} + \frac{{4{z^2}\left( {1 - \eta } \right)}}{{{k^2}w_{1j}^4}}} \right], \;\;{(j = x,y}) ,$

    (19)

    其中, w1x = w1y = w1, 1/R1x = 2C6–1/f, 1/R1y = –2C6–1/f. 令(19)式等于零, 可得到聚焦像散高斯光束在自聚焦介质中传输的自聚焦焦距为:

    $\frac{1}{{{z_{f1}}}} = \frac{1}{f} + 2{C_6} + \frac{{2\sqrt {\eta - 1} }}{{kw_1^2}},\tag{20a}$

    $\frac{1}{{{z_{f2}}}} = \frac{1}{f} - 2{C_6} - \frac{{2\sqrt {\eta - 1} }}{{kw_1^2}}.\tag{20b}$

    实际上, 当z = zf1时, 有wy = 0; 当z = zf2时, 有wx = 0.

    值得指出的是, 自聚焦焦距需满足zf > 0. 由(20a)式可知: f > zf1 > 0, 即该焦点始终存在. 令(20b)式中${1 / f} - 2 C_6' = {{2{{\left( {\eta - 1} \right)}^{{1 / 2}}}} / {\left( {kw_1^2} \right)}}$, 可得

    $C_6' = \frac{1}{{2f}} - \frac{{\sqrt {\eta - 1} }}{{kw_1^2}}.$

    (21)

    ${C_6} < C_6'$时, zf2 > 0; 当${C_6} > C_6'$时, zf2 < 0 (舍去).

    图6为自聚焦焦距zf随像散系数C6的变化. 由图6可知, 当${C_6} < C_6'$时, 光束有两个焦点; 当${C_6} > C_6'$时, 光束仅有一个焦点. 因此, 通过调控像散系数C6可以控制光束焦点的个数.

    图 6  自聚焦焦距zf随像散系数C6的变化
    Fig. 6.  Self-focusing focal length zf versus the astigmatic coefficient C6.

    本文研究了Kerr非线性介质中聚焦像散高斯光束的传输特性. 在光束扩展情况下(η < 1), 推导出了聚焦像散高斯光束在Kerr非线性介质中传输的束宽、束腰位置和焦移的解析公式, 研究了Kerr效应对光束像散特性和焦移特性的影响. 研究表明: 束腰位置s2x随像散系数C6的变化关于C6 = 1/(2f)轴对称, 且存在一个最大值和一个最小值, f为透镜焦距. 自聚焦介质中, η越大(自聚焦作用越强), 光束像散越厉害, 但焦移越小; 自散焦介质中, |η|越大(自散焦作用越强), 光束像散越弱, 但焦移越大. 在光束自聚焦情况下(η > 1), 研究了聚焦像散高斯光束的自聚焦焦距, 推导出了自聚焦焦距的解析公式. 研究表明: 通过调控像散系数C6可以控制光束焦点的个数(一个或两个焦点). 本文研究结果具有理论和实际应用意义.

    [1]

    Hanna D C 1969 IEEE J. Quantum Electron. 5 483Google Scholar

    [2]

    Zhao B, Li Z 1998 Appl. Opt. 37 2563Google Scholar

    [3]

    Thaning A, Jaroszewicz Z, Friberg A T 2003 Appl. Opt. 42 9Google Scholar

    [4]

    江新光, 吴逢铁 2008 物理学报 57 4202Google Scholar

    Jiang X G, Wu F T 2008 Acta Phys. Sin. 57 4202Google Scholar

    [5]

    杨艳飞, 陈婧, 吴逢铁, 胡润, 张惠忠, 胡汉青 2018 物理学报 67 224201Google Scholar

    Yang Y F, Chen J, Wu F T, Hu R, Zhang H Z, Hu H Q 2018 Acta Phys. Sin. 67 224201Google Scholar

    [6]

    Lin Q, Cai Y J 2002 Opt. Lett. 27 216Google Scholar

    [7]

    董一鸣, 徐云飞, 张璋, 林强 2006 物理学报 55 5755Google Scholar

    Dong Y M, Xu Y F, Zhang Z, Lin Q 2006 Acta Phys. Sin. 55 5755Google Scholar

    [8]

    Zhao D M, Lin Q, Wang S M 1994 Opt. Quantum Electron. 26 903Google Scholar

    [9]

    Tari T, Richter P 1992 Opt. Quantum Electron. 24 S865Google Scholar

    [10]

    刘晓丽, 冯国英, 李玮, 唐淳, 周寿桓 2013 物理学报 62 194202Google Scholar

    Liu X L, Feng G Y, Li W, Tang C, Zhou S H 2013 Acta Phys. Sin. 62 194202Google Scholar

    [11]

    Cai Y J, He S L 2006 Appl. Phys. Lett. 89 041117Google Scholar

    [12]

    Cai Y J, Lin Q, Ge D 2002 J. Opt. Soc. Am. A 19 2036Google Scholar

    [13]

    赵贵燕, 张逸新, 王建宇, 贾建军 2010 物理学报 59 1378Google Scholar

    Zhao G Y, Zhang Y X, Wang J Y, Jia J J 2010 Acta Phys. Sin. 59 1378Google Scholar

    [14]

    Soljacic M, Segev M, Coskun T, Christodoulides D N, Vishwanath A 2000 Phys. Rev. Lett. 84 467Google Scholar

    [15]

    Mitchell M, Chen Z G, Shih M F, Segev M 1996 Phys. Rev. Lett. 77 490Google Scholar

    [16]

    Sun C, Dylov D V, Fleischer J W 2009 Opt. Lett. 34 3003Google Scholar

    [17]

    Wang H, Ji X L, Zhang H, Li X Q, Deng Y 2019 Opt. Lett. 44 743Google Scholar

    [18]

    Wang H, Ji X L, Deng Y, Li X Q, Yu H 2020 Opt. Lett. 45 710Google Scholar

    [19]

    Hu J, Wang H, Ji X L, Deng Y, Chen L F 2020 J. Opt. Soc. Am. A 37 1282Google Scholar

    [20]

    王形华, 郭旗 2005 物理学报 54 3183Google Scholar

    Wang X H, Guo Q 2005 Acta Phys. Sin. 54 3183Google Scholar

    [21]

    Goncharenko A M, Logvin Y A, Samson A M, Shapovalov P S 1991 Opt. Commun. 81 225Google Scholar

    [22]

    Cornolti F, Lucchesi M, Zambon B 1990 Opt. Commun. 75 129Google Scholar

    [23]

    Singh T, Saini N S, Kaul S S 2000 Pramana-J. Phys. 55 423Google Scholar

    [24]

    季小玲, 吕百达 2000 强激光与粒子束 4 12

    Ji X L, Lü B D 2000 High Power Laser and Particle Beams 4 12

    [25]

    Guo S F, Tian Q 2010 Chin. Phys. B 6 19Google Scholar

    [26]

    Porras M A, Alda J, Bernabeu E 1993 Appl. Opt. 30 32Google Scholar

    [27]

    Yariv A, Yeh P 1978 Opt. Commun. 2 27Google Scholar

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    Miller R I, Roberts T G 1987 Appl. Opt. 21 26Google Scholar

    期刊类型引用(2)

    1. 艾亦章,吕奇霖,李世文,马再如,王方,刘红婕,杜泉. 方形超高斯光束在Kerr介质中的强度演化特性. 强激光与粒子束. 2022(04): 25-31 . 百度学术
    2. 邓凌,李晓庆,穆轶,季小玲. 风控热晕下椭圆激光光束质量的优化研究. 中国激光. 2022(04): 35-45 . 百度学术

    其他类型引用(5)

  • 图 1  Kerr非线性介质中聚焦像散高斯光束传输的示意图

    Figure 1.  Schematic diagram of a focused astigmatic Gaussian beam propagating in Kerr nonlinear media.

    图 2  束腰位置s2x随像散系数C6的变化

    Figure 2.  Beam waist position s2x versus the astigmatic coefficient C6.

    图 3  像散参数β随相对传输距离z/f的变化 (a)不同像散系数; (b)不同Kerr效应强度

    Figure 3.  Astigmatic parameter β versus the relative propagation distance z/f: (a) Different values of the astigmatic coefficient; (b) different strength of the Kerr effect.

    图 4  (a)轴上光强I(0, 0, z)和(b)光斑面积S(z)随相对传输距离z/f的变化, C6 = 0.2 m–1

    Figure 4.  (a) Axial intensity and (b) the area of beam spot versus the relative propagation distance z/f.

    图 5  焦移Δ随像散系数C6的变化 (a)自聚焦介质; (b)自散焦介质

    Figure 5.  Focal shift Δ versus the astigmatic coefficient C6: (a) Self-focusing media; (b) self-defocusing media.

    图 6  自聚焦焦距zf随像散系数C6的变化

    Figure 6.  Self-focusing focal length zf versus the astigmatic coefficient C6.

  • [1]

    Hanna D C 1969 IEEE J. Quantum Electron. 5 483Google Scholar

    [2]

    Zhao B, Li Z 1998 Appl. Opt. 37 2563Google Scholar

    [3]

    Thaning A, Jaroszewicz Z, Friberg A T 2003 Appl. Opt. 42 9Google Scholar

    [4]

    江新光, 吴逢铁 2008 物理学报 57 4202Google Scholar

    Jiang X G, Wu F T 2008 Acta Phys. Sin. 57 4202Google Scholar

    [5]

    杨艳飞, 陈婧, 吴逢铁, 胡润, 张惠忠, 胡汉青 2018 物理学报 67 224201Google Scholar

    Yang Y F, Chen J, Wu F T, Hu R, Zhang H Z, Hu H Q 2018 Acta Phys. Sin. 67 224201Google Scholar

    [6]

    Lin Q, Cai Y J 2002 Opt. Lett. 27 216Google Scholar

    [7]

    董一鸣, 徐云飞, 张璋, 林强 2006 物理学报 55 5755Google Scholar

    Dong Y M, Xu Y F, Zhang Z, Lin Q 2006 Acta Phys. Sin. 55 5755Google Scholar

    [8]

    Zhao D M, Lin Q, Wang S M 1994 Opt. Quantum Electron. 26 903Google Scholar

    [9]

    Tari T, Richter P 1992 Opt. Quantum Electron. 24 S865Google Scholar

    [10]

    刘晓丽, 冯国英, 李玮, 唐淳, 周寿桓 2013 物理学报 62 194202Google Scholar

    Liu X L, Feng G Y, Li W, Tang C, Zhou S H 2013 Acta Phys. Sin. 62 194202Google Scholar

    [11]

    Cai Y J, He S L 2006 Appl. Phys. Lett. 89 041117Google Scholar

    [12]

    Cai Y J, Lin Q, Ge D 2002 J. Opt. Soc. Am. A 19 2036Google Scholar

    [13]

    赵贵燕, 张逸新, 王建宇, 贾建军 2010 物理学报 59 1378Google Scholar

    Zhao G Y, Zhang Y X, Wang J Y, Jia J J 2010 Acta Phys. Sin. 59 1378Google Scholar

    [14]

    Soljacic M, Segev M, Coskun T, Christodoulides D N, Vishwanath A 2000 Phys. Rev. Lett. 84 467Google Scholar

    [15]

    Mitchell M, Chen Z G, Shih M F, Segev M 1996 Phys. Rev. Lett. 77 490Google Scholar

    [16]

    Sun C, Dylov D V, Fleischer J W 2009 Opt. Lett. 34 3003Google Scholar

    [17]

    Wang H, Ji X L, Zhang H, Li X Q, Deng Y 2019 Opt. Lett. 44 743Google Scholar

    [18]

    Wang H, Ji X L, Deng Y, Li X Q, Yu H 2020 Opt. Lett. 45 710Google Scholar

    [19]

    Hu J, Wang H, Ji X L, Deng Y, Chen L F 2020 J. Opt. Soc. Am. A 37 1282Google Scholar

    [20]

    王形华, 郭旗 2005 物理学报 54 3183Google Scholar

    Wang X H, Guo Q 2005 Acta Phys. Sin. 54 3183Google Scholar

    [21]

    Goncharenko A M, Logvin Y A, Samson A M, Shapovalov P S 1991 Opt. Commun. 81 225Google Scholar

    [22]

    Cornolti F, Lucchesi M, Zambon B 1990 Opt. Commun. 75 129Google Scholar

    [23]

    Singh T, Saini N S, Kaul S S 2000 Pramana-J. Phys. 55 423Google Scholar

    [24]

    季小玲, 吕百达 2000 强激光与粒子束 4 12

    Ji X L, Lü B D 2000 High Power Laser and Particle Beams 4 12

    [25]

    Guo S F, Tian Q 2010 Chin. Phys. B 6 19Google Scholar

    [26]

    Porras M A, Alda J, Bernabeu E 1993 Appl. Opt. 30 32Google Scholar

    [27]

    Yariv A, Yeh P 1978 Opt. Commun. 2 27Google Scholar

    [28]

    Miller R I, Roberts T G 1987 Appl. Opt. 21 26Google Scholar

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  • 期刊类型引用(2)

    1. 艾亦章,吕奇霖,李世文,马再如,王方,刘红婕,杜泉. 方形超高斯光束在Kerr介质中的强度演化特性. 强激光与粒子束. 2022(04): 25-31 . 百度学术
    2. 邓凌,李晓庆,穆轶,季小玲. 风控热晕下椭圆激光光束质量的优化研究. 中国激光. 2022(04): 35-45 . 百度学术

    其他类型引用(5)

Metrics
  • Abstract views:  6807
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  • Cited By: 7
Publishing process
  • Received Date:  09 October 2020
  • Accepted Date:  16 November 2020
  • Available Online:  25 March 2021
  • Published Online:  05 April 2021

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