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对于受光阑限制的色散会聚透镜系统,其传输矩阵为:
$ \boldsymbol{M}=\left[\begin{array}{ll} A & B \\ C & D \end{array}\right]=\left[\begin{array}{cc} 1-2 / f(\lambda) & z \\ -1 / f(\lambda) & 1 \end{array}\right] $
(1) 式中,A, B, C, D分别为传输矩阵的矩阵元,z为研究的考察面到初始入射面的距离,f(λ)为与波长λ相关的色散透镜焦距,且f(λ)=(n0-1)f0/[n(λ)-1],n0为中心波长λ0对应的折射率,f0为λ0对应的焦距,n(λ)为对应波长λ的折射率。考虑色散透镜的材料为熔融石英,则其折射率n(λ)由参考文献[19]中给出。当宽带TEM22模HG光束通过这一系统后,对于光束的每一频率分量,考察面z处的场分布为:
$ \begin{array}{*{20}{c}} {{E_{22}}(x, y, z, \omega ) = \frac{{{\rm{i}}k}}{{2{\rm{ \mathit{ π} }}B}}\exp ( - {\rm{i}}kz) \times }\\ {\int_{ - a}^a {\int_{ - a}^a {{E_{22}}} } \left( {{x_0}, {y_0}, 0, \omega } \right)\exp \left\{ { - \frac{{{\rm{i}}k}}{{2B}}\left[ {A\left( {x_0{}^2 + y_0{}^2} \right) - } \right.} \right.}\\ {\left. {\left. {2\left( {{x_0}x + {y_0}y} \right) + D\left( {{x^2} + {y^2}} \right)} \right]} \right\}{\rm{d}}{x_0}{\rm{d}}{y_0}} \end{array} $
(2) 式中,E22(x0, y0, 0, ω)是初始入射面的场分布,a是正方形硬边光阑中心孔的半宽,k=2π/λ, 为波数,x和y为考察面坐标,ω为频率。
为简单起见,考虑初始入射面z=0处的空间分量E22(x0, y0, 0)和频谱分量f(ω)可分离,即E22(x0, y0, 0, ω)=E22(x0, y0, 0)f(ω),且:
$ \begin{array}{c} E_{22}\left(x_{0}, y_{0}, 0\right)=\mathrm{H}_{2}\left(\frac{\sqrt{2} x_{0}}{w_{0}}\right) \mathrm{H}_{2}\left(\frac{\sqrt{2} y_{0}}{w_{0}}\right) \times \\ \exp \left[-\frac{\mathrm{i} k}{2 q_{0}}\left(x_{0}{}^{2}+y_{0}{}^{2}\right)\right] \end{array} $
(3) 式中,H2()为厄米多项式,w0为对应的基模高斯光束束腰宽度,并且1/q0=-iλ/(πw02),q0为z=0处的q参量。假设f(ω)为典型的高斯型,即:
$ f(\omega)=\frac{a_{6}}{\sqrt{2} \omega_{0} \gamma} \exp \left[-\frac{a_{\mathrm{G}}{}^{2}\left(\omega-\omega_{0}\right)^{2}}{\omega_{0}{}^{2} \gamma^{2}}\right] $
(4) 式中,γ=Δω/ω0为相对带宽,Δω为带宽,ω0为中心频率,aG=(2ln2)1/2。
光束通过透镜后,由傅里叶逆变换,可以得到任一点处在时间域的场分布为:
$ \begin{array}{c} {E_{22}}\left( {x, {\rm{ }}y, {\rm{ }}z, {\rm{ }}t} \right) = \\ \frac{1}{{{\rm{2 \mathit{ π} }}}}\int_{ - \infty }^\infty {{E_{22}}} (x, y, z, \omega )\exp ({\rm{i}}\omega t){\rm{d}}\omega \end{array} $
(5) 式中,时间t=z/c,c是光速。为推导和计算的方便,仅考虑TEM22模HG光束的1维情况,2维情况下的结果是类似的。因此,积分得到TEM22模HG光束1维场分布为:
$ \begin{array}{*{20}{c}} {{E_{22}}(x, z) = \frac{S}{{{T^3}}}\left\{ { - 4\left( {\sqrt {\alpha T} - \frac{{Qx}}{{{w_0}}}} \right) \times } \right.}\\ {\exp \left[ { - {{\left( {\sqrt {\alpha T} + \frac{{Qx}}{{{w_0}}}} \right)}^2}} \right] - 4\left( {\sqrt {\alpha T} + \frac{{Qx}}{{{w_0}}}} \right) \times }\\ {\exp \left[ { - {{\left( {\sqrt {\alpha T} \frac{{Qx}}{{{w_0}}}} \right)}^2}} \right] + }\\ {\left. {\sqrt {\rm{ \mathit{ π} }} {W_x}\left[ {2 - T + 4{Q^2}{{\left( {\frac{x}{{{w_0}}}} \right)}^2}} \right]} \right\}} \end{array} $
(6) 式中,${T = 1 - \frac{{{\rm{i \mathsf{ π} }}{F_{\rm{w}}}[z - f(\lambda )]/f(\lambda )}}{{z/f(\lambda )}}}$,${Q = \frac{{{\rm{i \mathsf{ π} }}{F_{\rm{w}}}}}{{{T^{1/2}}z/f(\lambda )}}}$,${S = \frac{{{\rm{i}}{F_{\rm{w}}}}}{{z/f(\lambda )}}\exp \left\{ {\left[ {{Q^2} - \frac{{{\rm{i \mathsf{ π} }}{F_{\rm{w}}}}}{{z/f(\lambda )}}} \right]{{\left( {\frac{x}{{{w_0}}}} \right)}^2}} \right\}}$,${\left. {{W_x} = {\mathop{\rm erf}\nolimits} (\sqrt {\alpha T} - Qx/{w_0}} \right) + {\mathop{\rm erf}\nolimits} \left( {\sqrt {\alpha T} + Qx/{w_0}} \right)}$。另有α=(a/w0)2和Fw=w02/分别是截断参量和光束相关的菲涅耳数,erf()是误差函数。最后得到TEM22模HG光束通过色散透镜的1维场分布:
$ {E_{22}}(x, z, t) = \frac{1}{{\sqrt {{\rm{2 \mathit{ π} }}} }}\int_{ - \infty }^\infty {{E_{22}}} (x, z, \omega )\exp ({\rm{i}}\omega t){\rm{d}}\omega $
(7) 光强分布则为I22(x, z, t)=E22(x, z, t)2。
色散会聚透镜系统中的微米焦开关
Micron focal switch in dispersion focused lens system
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摘要: 为了研究宽带TEM22模厄米-高斯光束通过受光阑限制色散会聚透镜系统的微米焦开关现象, 采用数值计算方法对TEM22模厄米-高斯光束的光强分布进行了分析, 取得了传输轴上光强分布的数据。结果表明, 带宽变化引起轴上TEM22模厄米-高斯光束两个光强极大的相对大小改变, 结果导致光强主极大位置跃变; 当相对带宽γ=0.231且菲涅耳数Fw=100时, 光强主极大位置跃变距离为2.5μm, 出现微米焦开关现象; 带宽和菲涅耳数是影响光束微米焦开关的重要因素; 菲涅耳数较大时, 较窄的带宽就可以诱导微米焦开关, 反之则需要较宽的带宽。此研究结果有助于光通信技术中微纳光学器件的设计和制作。Abstract: In order to know micron focal switch of polychromatic TEM22 mode Hermite-Gaussian (HG) beams passing through an apertured dispersion lens system, numerical calculation examples were used to study the beam's intensity distributions. The data of the axial intensity distributions were achieved. It is found that variation of the bandwidth results in change of two maximal intensities of TEM22 mode (HG) beams and thus position of the principle maximal intensity shifts rapidly. When relative bandwidth γ is 0.231 and Fresnel number Fw is 100, the position varies 2.5μm and the phenomenon of micron focal switch presents. The bandwidth and Fresnel number are important factors that induce the micron focal switch. A narrow bandwidth is enough to induce the focal switch of TEM22 mode (HG) beams in the system with large Fresnel number whereas a broad bandwidth is required in the system with small Fresnel number. Results in this paper help the design and manufacture of micro and nano optical devices in optical communication technique.
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Key words:
- laser physics /
- micron focal switch /
- bandwidth /
- Fresnel number
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