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特殊关联部分相干光束在源平面的交叉谱密度(cross-spectral density CSD)函数可表示为[21]:
$ W_0\left(\boldsymbol{r}_1, \boldsymbol{r}_2\right)=\tau^*\left(\boldsymbol{r}_1\right) \tau\left(\boldsymbol{r}_2\right) \mu\left(\boldsymbol{r}_2-\boldsymbol{r}_1\right) $
(1) 式中:r1与r2是相对源平面传输过程中的两个随机位置矢量;*表示复共轭;r表示相对源平面的位置矢量,令r=r1=r2, τ(r)具有高斯分布[21]:
$ \tau(\boldsymbol{r})=\exp \left(-\frac{\boldsymbol{r}^2}{2 w_0{ }^2}\right) $
(2) 式中:w0为光束腰宽。光谱相干度为[21]:
$ \begin{gathered} \mu\left(\boldsymbol{r}_2-\boldsymbol{r}_1\right)=\mathrm{L}_n^0\left(\frac{r_{\mathrm{d}}^2}{2 d_0^2}\right) \exp \left(-\frac{r_{\mathrm{d}}^2}{2 d_0^2}\right)- \\ \frac{i \sqrt{2}(n-1 / 2)!}{2 n!} \frac{r_{\mathrm{d}}}{d_0} \times \\ \quad \exp \left(-\frac{r_{\mathrm{d}}^2}{2 d_0^2}\right) \mathrm{L}_{n-1 / 2}^1\left(\frac{r_{\mathrm{d}}^2}{2 d_0^2}\right) \cos \varphi_{\mathrm{d}} \end{gathered} $
(3) 式中:Ln0(·)表示拉盖尔函数;n为阶数;i为虚数单位;rd与φd是rd的幅度与相位,rd= r2-r1;d0为相干长度。
将式(2)、式(3)代入到式(1),得到源平面的交叉谱密度函数为:
$ \begin{gathered} W_0\left(\boldsymbol{r}_1, \boldsymbol{r}_2\right)=\exp \left(-\frac{\boldsymbol{r}_1^2+\boldsymbol{r}_2^2}{2 w_0^2}\right) \times \\ {\left[\mathrm{L}_n^0\left(\frac{r_{\mathrm{d}}^2}{2 d_0^2}\right) \exp \left(-\frac{r_{\mathrm{d}}{ }^2}{2 d_0^2}\right)-\frac{\mathrm{i} \sqrt{2}(n-1 / 2)!}{2 n!} \times\right.} \\ \left.\frac{r_{\mathrm{d}}}{d_0} \exp \left(-\frac{r_{\mathrm{d}}^2}{2 d_0^2}\right) \mathrm{L}_{n-1 / 2}^1\left(\frac{r_{\mathrm{d}}^2}{2 d_0{ }^2}\right) \cos \varphi_{\mathrm{d}}\right] \end{gathered} $
(4) -
在近轴条件下,PCCL光束通过ABCD光学系统传输后的交叉谱密度函数可表示为[23]:
$ \begin{gathered} W\left(\boldsymbol{\rho}_1, \boldsymbol{\rho}_2, z\right)=\left(\frac{k}{2 \pi B}\right)^2 \iint W_0\left(\boldsymbol{r}_1, \boldsymbol{r}_2\right) \times \\ \exp \left\{-\frac{\mathrm{i} k}{2 B}\left[A\left(\boldsymbol{r}_1{ }^2-\boldsymbol{r}_2{ }^2\right)-2\left(\boldsymbol{\rho}_1 \boldsymbol{r}_1-\boldsymbol{\rho}_2 \boldsymbol{r}_2\right)+\right.\right. \\ \left.\left.D\left(\boldsymbol{\rho}_1{ }^2-\boldsymbol{\rho}_2{ }^2\right)\right]\right\} \mathrm{d} \boldsymbol{r}_1 \mathrm{~d} \boldsymbol{r}_2 \end{gathered} $
(5) 式中:k=2π/λ为波数;ρ1和ρ2是焦平面的位置矢量;f为聚焦透镜焦距;z表示光束经焦平面后的传输距离[19]。
$ \left[\begin{array}{ll} A & B \\ C & D \end{array}\right]=\left[\begin{array}{cc} -z / f & f+z \\ -1 / f & 1 \end{array}\right] $
(6) 令ρ1=ρ2=ρ,可得到几何焦平面附近的光强分布为:
$ \begin{gathered} I(\boldsymbol{\rho}, z)=\left(\frac{k}{2 \pi B}\right)^2 \iint W_0\left(\boldsymbol{r}_1, \boldsymbol{r}_2\right) \times \\ \exp \left\{\frac{\mathrm{i} k z}{2 f B}\left(\boldsymbol{r}_1^2-\boldsymbol{r}_2^2\right)+\frac{\mathrm{i} k}{B} \boldsymbol{\rho}\left(\boldsymbol{r}_1-\boldsymbol{r}_2\right)\right\} \mathrm{d} \boldsymbol{r}_1 \mathrm{~d} \boldsymbol{r}_2 \end{gathered} $
(7) 定义一个新函数:
$ F(\boldsymbol{r})=\exp \left[-\left(\frac{1}{2 w_0{ }^2}-\frac{\mathrm{i} k A}{2 B}\right) \boldsymbol{r}^2\right] $
(8) 将式(4)、式(8)代入式(5)得:
$ \begin{aligned} & W(\boldsymbol{\rho}, \boldsymbol{\rho}, z)=\frac{1}{\lambda^2 B^2} \iiint F^*\left(r_1\right) F\left(r_2\right) \times \\ & \exp \left[-\mathrm{i} k\left(\boldsymbol{r}_2-\boldsymbol{r}_1\right) \cdot\left(\boldsymbol{v}_1+\boldsymbol{v}_2\right)\right] p_1\left(\boldsymbol{v}_1\right) \times \\ & \exp \left[-\frac{\mathrm{i} k}{B}\left(\boldsymbol{r}_2-\boldsymbol{r}_1\right) \cdot \boldsymbol{\rho}\right] \mathrm{d}^2 \boldsymbol{r}_1 \mathrm{~d}^2 \boldsymbol{r}_2 \mathrm{~d}^2 \boldsymbol{v}_1 \mathrm{~d}^2 \boldsymbol{v}_2 \end{aligned} $
(9) 式中:p1(v1)是任意参量v1 ≡(vx, vy)值的非负函数,转化为极坐标形式的可分函数[22]:
$ p_1\left(\boldsymbol{v}_1\right)=\frac{k^2 d_0{ }^2\left(k d_0 \boldsymbol{v}_1\right)^{2 n}}{2^n n!\pi} \exp \left(\frac{k^2 d_0{ }^2 \boldsymbol{v}_1{ }^2}{2}\right) \cos ^2\left(\frac{\theta}{2}\right) $
(10) 引入变换:rd= r2-r1, rs=(r2+r1)/2,将F*(r1)与F(r2)写成傅里叶变换的形式:
$ \left\{\begin{array}{l} F^*\left(\boldsymbol{r}_{\mathrm{s}}-\frac{\boldsymbol{r}_{\mathrm{d}}}{2}\right)=\left(\frac{k}{2 \pi}\right)^2 \int \tilde{F}^*(\boldsymbol{u}) \times \\ \exp \left[-\mathrm{i} k \boldsymbol{u} \cdot\left(\boldsymbol{r}_{\mathrm{s}}-\frac{\boldsymbol{r}_{\mathrm{d}}}{2}\right)\right] \mathrm{d}^2 \boldsymbol{u} \\ F\left(\boldsymbol{r}_{\mathrm{s}}+\frac{\boldsymbol{r}_{\mathrm{d}}}{2}\right)=\left(\frac{k}{2 \pi}\right)^2 \int \tilde{F}(\boldsymbol{u}) \times \\ \exp \left[\mathrm{i} k \boldsymbol{u} \cdot\left(\boldsymbol{r}_{\mathrm{s}}+\frac{\boldsymbol{r}_{\mathrm{d}}}{2}\right)\right] \mathrm{d}^2 \boldsymbol{u} \end{array}\right. $
(11) 式中:$\tilde{F}^*(\boldsymbol{u}), \tilde{F}(\boldsymbol{u}) $表示F*(r1)和F(r2)的傅里叶变换,u表示与傅里叶变换相关的矢量。
把式(10)、式(11)代入式(9),得到:
$ \begin{gathered} \exp W(\boldsymbol{\rho}, \boldsymbol{\rho}, z)=\frac{1}{\lambda^2 B^2} \iiint \int \tilde{F}^*\left(\boldsymbol{u}_1\right) \tilde{F}\left(\boldsymbol{u}_2\right) p_1\left(\boldsymbol{v}_1\right) \times\\ \exp \left[-\mathrm{i} k \boldsymbol{u}_1 \cdot\left(\boldsymbol{r}_{\mathrm{s}}-\frac{\boldsymbol{r}_{\mathrm{d}}}{2}\right)\right] \times \\ \exp \left[\mathrm{i} k \boldsymbol{u}_2 \cdot\left(\boldsymbol{r}_{\mathrm{s}}+\frac{\boldsymbol{r}_{\mathrm{d}}}{2}\right)\right] \exp \left(-\frac{\mathrm{i} k}{z} \boldsymbol{r}_{\mathrm{d}} \cdot \boldsymbol{\rho}\right) \times \\ \exp \left[-\mathrm{i} k \boldsymbol{r}_{\mathrm{d}} \cdot\left(\boldsymbol{v}_1+\boldsymbol{v}_2\right)\right] \mathrm{d}^2 \boldsymbol{r}_{\mathrm{s}} \mathrm{~d}^2 \boldsymbol{r}_{\mathrm{d}} \mathrm{~d}^2 \boldsymbol{v}_1 \mathrm{~d}^2 \boldsymbol{v}_2 \mathrm{~d}^2 \boldsymbol{u}_1 \mathrm{~d}^2 \boldsymbol{u}_2 \end{gathered} $
(12) 式中:u1=u2=u=ρ /B。
经过冗长的积分计算,得到PCCL光束在焦平面处的交叉谱密度表达式为:
$ \begin{gathered} W(\boldsymbol{\rho}, \boldsymbol{\rho}, z)=\frac{1}{F(B)} \exp \left[-\frac{1}{w_0{ }^2 F(B)} \rho^2\right]\left\{\left[\frac{P_1}{F(B)}\right]^n \times\right. \\ \mathrm{L}_n^0\left(-\frac{2 B^2 \rho^2}{k^2 d_0{ }^2 w_0{ }^4 P_1 F(B)}\right)+\frac{\sqrt{2}(n-1 / 2)!}{n!} \times \\ \left.\frac{B \rho \cos \varphi_d}{d_0 k w_0{ }^2 P_1}\left[\frac{P_1}{F(B)}\right]^{n+1 / 2} \mathrm{~L}_{n-1 / 2}^1\left[-\frac{2 B^2 \rho^2}{k^2 d_0{ }^2 w_0{ }^4 P_1 F(B)}\right]\right\} \end{gathered} $
(13) 定义:
$ \left\{\begin{array}{l} M=\frac{1}{2 w_0{ }^2}-\frac{\mathrm{i} k A}{2 B} \\ F(B)=A^2+\frac{B^2}{k^2 w_0{ }^2}\left(\frac{1}{w_0{ }^2}+\frac{2}{d_0{ }^2}\right) \\ P_1=A^2+\frac{B^2}{k^2 w_0{ }^4} \end{array}\right. $
(14) -
当粒子半径a0远小于激光波长λ(a0≤λ/20)时,粒子被称为瑞利粒子,根据瑞利散射理论,聚焦光场对瑞利粒子的梯度力Fgrad和散射力Fscat为[19]:
$ $
(15) 式中:ez为光束传输方向上的单位矢量;c为光速;η=np/nm表示相对折射率;nm为捕获微粒折射率;np为介质折射率。
PCCL光束对瑞利粒子作用力的理论研究
Theoretical study on the force of PCCL beam on Rayleigh particles
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摘要: 为了探究部分相干月牙形光束(PCCL)对瑞利粒子的捕获特性,采用广义惠更斯-菲涅耳原理推导了PCCL光束经过ABCD光学系统传输后的交叉谱密度及对瑞利粒子作用力的表达式,对聚焦PCCL光束在焦平面的归一化光强和对瑞利粒子作用力的分布情况进行了软件模拟和理论分析,得到了聚焦PCCL光束的光强和捕获力随光束参量以及焦距变化的规律。结果表明,选取光束阶数n=3、初始腰宽w0=0.1 mm、相干长度d0=6 mm时,经焦距f=12 mm的透镜聚焦后的PCCL光束对高、低折射率粒子均能实现捕获;由于PCCL光束在焦平面处呈现离轴光强分布,峰值光强不在坐标原点,通过调节光束阶数、初始腰宽、相干长度和焦距,可实现对聚焦PCCL光强及瑞利粒子作用力的调控,从而实现聚焦PCCL光束对不同位置处瑞利粒子的灵活捕获。该研究结果为实际聚焦PCCL光束稳定捕获微粒提供了理论依据。Abstract: To investigate the capture characteristics of Rayleigh particles by partially coherent crescent like beam (PCCL). The generalized Huygens Fresnel principle was adopted to derive the expression of the cross spectral density and the force on Rayleigh particles of PCCL beams transmitted through an ABCD optical system. Software simulation and theoretical analysis were conducted on the normalized intensity and force distribution on Rayleigh particles of a focused PCCL beam in the focal plane, and the patterns of the intensity and capture force of the focused PCCL beam changing with the beam parameters and focal length were obtained. The results show that when the beam order n=3, the initial waist width w0=0.1 mm, and the coherence length d0=6 mm, the PCCL beam focused by the lens with a focal length of f=12 mm, both high and low refractive index particles can be captured. Since the PCCL beam presents an off-axis light intensity distribution in the focal plane, and the peak light intensity is not at the coordinate origin, the focused PCCL light intensity and the force of Rayleigh particles can be controlled by adjusting the beam order, initial waist width, coherence length and focal length, so as to realize the flexible capture of Rayleigh particles at different positions by focusing the PCCL beam. The results of this study provide a theoretical basis for the stable capture of particles by focusing PCCL beams.
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