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部分相干光束的2阶统计特性可用交叉谱密度函数来表示,多高斯-谢尔模光束在源平面上的交叉谱密度函数(cross-spectral density, CSD)可以表示为[11]:
$ W\left(\boldsymbol{r}_1, \boldsymbol{r}_2\right)=\sqrt{S\left(\boldsymbol{r}_1\right) S\left(\boldsymbol{r}_2\right)} \eta\left(\boldsymbol{r}_1, \boldsymbol{r}_2\right) $
(1) 式中, S(r)表征强度轮廓分布函数,r1≡(x1, y1),r2≡(x2, y2)是源平面上的任意两点的空间位置坐标,η(r1, r2)表征光束相干结构,多高斯-谢尔模关联结构可表示为:
$ \eta\left(\boldsymbol{r}_1, \boldsymbol{r}_2\right)=\frac{1}{C_0} \sum\limits_{m=1}^M\left(\begin{array}{l} M \\ m \end{array}\right) \frac{(-1)^{m-1}}{m} \exp \left[-\frac{\left(\boldsymbol{r}_2-\boldsymbol{r}_1\right)^2}{2 m \delta^2}\right] $
(2) 式中, M表征多高斯关联函数的模数,为归一化参数,表示二项式系数,σ为光束束腰宽度,δ为光束的相干长度。当M=1时,(1)式转化为高斯-谢尔模光束的交叉谱密度函数表达式。
BORGHI等人证明了光束是否能携带扭曲相位只与相干结构有关,而与光束的振幅无关[20]。因此,为了便于分析,可将(1)式携带扭曲相位,并忽略掉强度可表示为:
$ \bar{W}\left(\boldsymbol{r}_1, \boldsymbol{r}_2\right)=\eta\left(\left|\boldsymbol{r}_1-\boldsymbol{r}_2\right|\right) \exp \left(-\mathrm{i} \mu \boldsymbol{r}_1 \times \boldsymbol{r}_2\right) $
(3) 式中, μ表示扭曲因子。根据部分相干光束模式叠加理论,将(3)式用Mercer展开表示为[39]:
$ \bar{W}\left(\boldsymbol{r}_1, \boldsymbol{r}_2\right)=\sum\limits_{j=0, 1 / 2, 1, \cdots} \sum\limits_{l=-j}^j \varLambda_{j+l} \varPhi_{j, l}{ }^*\left(\boldsymbol{r}_1\right) \varPhi_{j, l}\left(\boldsymbol{r}_2\right) $
(4) 其特征函数Φj, l为正交,Φj, l*表示特征函数的复共轭,特征值Λj+l为非负,当部分相干光束携带扭曲相位后,其特征函数形式可为拉盖尔-高斯型[39]:
$ \begin{gathered} \varPhi_{j, l}=\sqrt{\frac{\mu}{\pi}}\left[\frac{(j-|l|) !}{(j+|l|) !}\right]^{1 / 2}(r \sqrt{\mu})^{2|l|} \times \\ \quad \exp (\mathrm{i} 2 l \theta) \mathrm{L}_{j-|l|}^{2|l|}\left(\mu r^2\right) \exp \left(-\frac{\mu r^2}{2}\right) \end{gathered} $
(5) 式中, j=0, 1/2, 1, …; l=-j, -j+1, …, j; Lj-|l|2|l|为拉盖尔多项式,l代表拓扑荷数,j-|l|为径向指数,2|l|为角向指数,两指数均为非负整数,为方便接下来的证明,(4)式中的空间坐标改写为极坐标形式r≡(r, θ); exp(i2lθ)代表每个模式均带有涡旋相位。从Mercer展开形式可看出,每个模式均携带有涡旋相位,揭示了携带轨道角动量的扭曲光束与涡旋光束之间的内在联系。接下来只需验证W(r1, r2)模式展开后所有特征值是否都是非负实数,且特征值满足Λs=Λj+l,特征值序列{Λs}s=0∞可由相干结构函数积分得到[20]:
$ \varLambda_s=\int \eta(|\boldsymbol{r}|) \mathrm{L}_s\left(\mu r^2\right) \exp \left(-\frac{\mu r^2}{2}\right) \mathrm{d}^2 \boldsymbol{r} $
(6) 式中, r=r2-r1,Ls表示角向指数为0、径向指数为s的拉盖尔多项式,将(2)式代入(6)式计算得到:
$ \begin{gathered} \varLambda_s=\frac{1}{C_0} \sum\limits_{m=1}^M\left(\begin{array}{l} M \\ m \end{array}\right) \frac{(-1)^{m-1}}{m} \times \\ \int_0^{2 \pi} \mathrm{d} \theta \int_0^{\infty} \exp \left[-\frac{r^2}{2 m \delta^2}-\frac{\mu r^2}{2}\right] \mathrm{L}_s\left(\mu r^2\right) r \mathrm{~d} r \end{gathered} $
(7) 最终积分得到特征值序列表达式为:
$ \varLambda_s=\frac{1}{C_0} \sum\limits_{m=1}^M\left(\begin{array}{l} M \\ m \end{array}\right)(-1)^{m-1} 2 \pi \delta^2 \frac{\left(1-\mu m \delta^2\right)^s}{\left(1+\mu m \delta^2\right)^{s+1}} $
(8) 对(8)式分析后可知,当μ≤1/mδ2时,该特征值为非负实数,此时椭圆多高斯-谢尔模光束可以携带扭曲相位。为便于之后的分析,将强度轮廓函数写为椭圆高斯函数,就得到了一类新型椭圆扭曲多高斯-谢尔模(twisted multi-Gaussian-Schell model, TMGSM)光束,其交叉谱密度函数可以表示为:
$ \begin{gathered} W\left(\boldsymbol{r}_1, \boldsymbol{r}_2\right)=\frac{1}{C_0} \sum\limits_{m=1}^M\left(\begin{array}{l} M \\ m \end{array}\right) \frac{(-1)^{m-1}}{m} \times \\ \exp \left(-\frac{r_{1 x}^2+r_{2 x}^2}{4 \sigma_x^2}-\frac{r_{1 y}{ }^2+r_{2 y}{ }^2}{4 \sigma_y^2}\right) \times \\ \exp \left[-\frac{\left(\boldsymbol{r}_1-\boldsymbol{r}_2\right)^2}{2 m \delta^2}\right] \exp \left(-\mathrm{i} \mu \boldsymbol{r}_1 \times \boldsymbol{r}_2\right) \end{gathered} $
(9) 式中, r1x,r1y,r2x,r2y分别为r1,r2在x,y方向上的分量; σx,σy表示x,y方向上的束腰宽度。在傍轴近似下利用Collins公式,椭圆TMGSM光束在像散ABCD光学系统中的传输公式可表示为:
$ \begin{gathered} W\left(\boldsymbol{\rho}_1, \boldsymbol{\rho}_2, z\right)=\frac{1}{\lambda^2 B^2} \iint W\left(\boldsymbol{r}_1, \boldsymbol{r}_2\right) \times \\ \exp \left[-\frac{\mathrm{i} k}{2 B}\left(A \boldsymbol{r}_1^2-2 \boldsymbol{r}_1 \boldsymbol{\rho}_1+D \boldsymbol{\rho}_1^2\right)\right] \times \\ \exp \left[\frac{\mathrm{i} k}{2 B}\left(A \boldsymbol{r}_2^2-2 \boldsymbol{r}_2 \boldsymbol{\rho}_2+D \boldsymbol{\rho}_2^2\right)\right] \mathrm{d}^2 \boldsymbol{r}_1 \mathrm{~d}^2 \boldsymbol{r}_2 \end{gathered} $
(10) 式中, ρ1=(ρ1x, ρ1y)和ρ2=(ρ2x, ρ2y)是输出平面上任意两个位置向量; A, B, C, D为光学系统传递矩阵的元素; λ为光束波长,k=2π/λ是波数。将(9)式代入(10)式,经过积分,就得到了输出平面的交叉谱密度函数解析式:
$ \begin{gathered} W\left(\boldsymbol{\rho}_1, \boldsymbol{\rho}_2, z\right)=\sum\limits_{m=1}^M\left(\begin{array}{l} M \\ m \end{array}\right) \frac{(-1)^{m-1}}{m} \times \\ \frac{k^2}{4 B^2 C_0 \sqrt{N_1 N_3\left(N_4-\varOmega_2\right)\left(N_5-\varOmega_1\right)}} \exp \left[\frac{\mathrm{i} D k}{2 B}\left(\boldsymbol{\rho}_2^2-\boldsymbol{\rho}_1^2\right)\right] \times \\ \exp \left[-\frac{u_1^2}{4 N_1}-\frac{v_1^2}{4 N_3}+\frac{\left(\mathrm{i} v_2-\varPi_2\right)^2}{4\left(N_4-\varOmega_2\right)}+\frac{\left(\mathrm{i} u_2-\varPi_3\right)^2}{4\left(N_5-\varOmega_1\right)}\right] \end{gathered} $
(11) 其中,使用了一系列参数替换:
$ \begin{gathered} N_1=\frac{1}{4 \sigma_x^2}+\frac{1}{2 m \delta^2}+\frac{\mathrm{i} A k}{2 B} ; N_2=\frac{1}{4 \sigma_x^2}+\frac{1}{2 m \delta^2}-\frac{\mathrm{i} A k}{2 B} ; \\ N_3=\frac{1}{4 \sigma_y^2}+\frac{1}{2 m \delta^2}+\frac{\mathrm{i} A k}{2 B}; \end{gathered} $
$ \begin{gathered} N_4=\frac{1}{4 \sigma_y^2}+\frac{1}{2 m \delta^2}-\frac{\mathrm{i} A k}{2 B} ; N_5=N_2-\frac{\eta_1^2}{4\left(N_4-\varOmega_2\right)} ; \\ u_1=\frac{k \rho_{1 x}}{B} ; u_2=\frac{k \rho_{2 x}}{B} ; v_1=\frac{k \rho_{1 y}}{B} ; v_2=\frac{k \rho_{2 y}}{B} ; \end{gathered} $
$ \begin{gathered} \varOmega_1=\frac{1}{4 N_1 m^2 \delta^4}-\frac{\mu^2}{4 N_3} ; \varOmega_2=\frac{1}{4 N_3 m^2 \delta^4}-\frac{\mu^2}{4 N_1} ; \\ \varPi_1=\frac{i u_1}{2 N_1 m \delta^2}-\frac{v_1 \mu}{2 N_3} ; \end{gathered} $
$ \begin{gathered} \varPi_2=\frac{\mathrm{i} v_1}{2 N_3 m \delta^2}+\frac{u_1 \mu}{2 N_1} ; \varPi_3=\varPi_1+\frac{\left(\varPi_2-\mathrm{i} v_2\right) \eta_1}{2\left(N_4-\varOmega_2\right)} \\ \eta_1=\frac{\mathrm{i} \mu}{2 N_3 m \delta^2}-\frac{\mathrm{i} \mu}{2 N_1 m \delta^2} \end{gathered} $
(12) 接下来,本文中将研究椭圆TMGSM光束在梯度折射率光纤中的传输特性。梯度折射率光纤的折射率分布可由下式表示:
$ n^2=\left\{\begin{array}{l} n_0{ }^2\left(1-\beta^2 \rho^2\right), \left(\rho^2 \leqslant R_0{ }^2\right) \\ n_0{ }^2\left(1-\beta^2 R_0{ }^2\right), \left(\rho^2>R_0{ }^2\right) \end{array}\right. $
(13) 其中,
$ \beta=\frac{1}{R_0}\left[1-\frac{n_1^2}{n_0^2}\right]^{\frac{1}{2}} $
(14) 式中,R0为纤芯半径,ρ为空间任意点距纤芯中心的距离,β为梯度折射率系数,n0为光纤中心折射率,n1为光纤包层折射率,梯度折射率光纤的ABCD传输矩阵可表示为:
$ \left[\begin{array}{ll} A & B \\ C & D \end{array}\right]=\left[\begin{array}{cc} \cos (\beta z) & \frac{\sin (\beta z)}{n_0 \beta} \\ -n_0 \beta \sin (\beta z) & \cos (\beta z) \end{array}\right] $
(15) 在这里用一种特定的梯度折射率光纤,纤芯为掺杂锗的二氧化硅(掺杂质量分数为0.079的氧化锗),包层为二氧化硅制成,折射率由塞米尔方程计算:
$ n^2(\omega)=1+\sum\limits_{p=1}^3 \frac{B_p \omega_p{ }^2}{\omega_p^2-\omega^2} $
(16) 式中,ωp=2πc/λp,Bp,λp均为特定材料下的塞米尔系数,c为光速,ω为λ=632.8nm时对应的角频率。对纯二氧化硅,塞米尔系数B1=0.6961663, B2=0.4079426, B3=0.8974794,λ1=0.0684043μm, λ2=0.1162414μm, λ3=9.896161μm。对掺杂质量分数为0.079锗的二氧化硅,其塞米尔系数B1=0.7136824, B2=0.4254807, B3=0.8964226,λ1=0.0617167μm, λ2=0.1270814μm,λ3=9.896161μm。取光纤纤芯半径为25μm,通过(13)式、(14)式和(16)式可以得到n0=1.46977, n1=1.45702, β=5.25726mm-1,从而确定ABCD传输矩阵的具体参数。部分相干光束的光强和相干度可由交叉光谱密度函数表示为:
$ S(\boldsymbol{\rho})=W(\boldsymbol{\rho}, \boldsymbol{\rho}) $
(17) $ \eta\left(\boldsymbol{\rho}_1, \boldsymbol{\rho}_2\right)=\frac{W\left(\boldsymbol{\rho}_1, \boldsymbol{\rho}_2\right)}{\sqrt{S\left(\boldsymbol{\rho}_1\right)} \sqrt{S\left(\boldsymbol{\rho}_2\right)}} $
(18)
扭曲多高斯光束在梯度折射率光纤中的传输特性
Propagation characteristics of twisted multi-Gaussian beams in gradient index fibers
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摘要: 为了得到一种新型椭圆扭曲多高斯-谢尔模(TMGSM)光束, 采用Mercer模式展开的方法进行了理论分析和验证, 证明了多高斯-谢尔模关联结构可携带扭曲相位, 详细研究了其在梯度折射率光纤中传输时的光强和相干度演化。结果表明, 椭圆TMGSM光束在梯度折射率光纤中传输时, 光强和相干度分布随着传输距离的增加发生周期性旋转, 并在0.5L(L为周期)的整数倍处偏转π/2, 其旋转角速度呈非线性变化且与扭曲因子的大小有关; 增大多高斯模数, 焦平面处光强分布的平顶区域增大, 相干度分布轮廓变小。此研究结果在光纤通信、聚焦成像、光学捕获等方面具有潜在的应用前景。
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关键词:
- 激光光学 /
- 扭曲相位 /
- Mercer模式展开理论 /
- 多高斯关联 /
- 梯度折射率光纤
Abstract: In order to obtain a new elliptically twisted multi-Gaussian-Schell model beam, a Mercer model expansion method was adopted, the theoretical analysis and verification were carried out. It is proved that the twisted phase can be carried in the multi-Gaussian-Shell model correlation structure, and the intensity and coherence evolution of the beam propagating in gradient index fibers was studied. The results show that when elliptically twisted multi-Gaussian-Schell model beams are propagating in the graded index fibers, the distribution of intensity and coherence rotates periodically with the increase of transmission distance, and the deflection is π/2 degrees at the integer multiples of 0.5L (L is a cycle). The rotation angular velocity is nonlinear and related to the size of the twist factor. With the increase of multi-Gaussian modulus, the flat top region of the intensity distribution at the focal plane increases, and the coherence distribution contour becomes smaller. The research results have potential applications in optical fiber communication, focused imaging, optical capture and so on. -
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