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结合目前已经研发出的曲面传感器[2]以及笔者课题组的前期设计[5],选用像素尺寸大小为1.25 μm的曲面传感接收器,像面尺寸为5.2 mm
$ \times $ 3.9 mm,全视角为100°,奎奈斯特频率为400 lp/mm,表1为系统设计参数。表 1 光学设计参数
Table 1. Optical design prescription
Parameter Value Waveband/nm 486-656 (Visible light) Relative aperture 1/1.8 Full field of view/(°) 100 Focal length/mm 2.7 Total length/mm $ \leqslant $2.7 -
由同心透镜的特性可知,同心反射式透镜结构主要需要对轴向球差和色差进行校正。文中利用参考文献[8]中的光路计算方法,结合像差理论,通过求解同心反射式透镜的轴向球差的极小值计算系统的初始结构。
参考文献[8]中的同心透镜光路计算方法,同心反射式透镜的光路如图2所示。
设物点O为光轴上一点,各表面入射光线与水平方向的夹角分别为
${\theta _a}(a = 0,\cdots,{\rm{5}})$ ,符号逆时针为正,顺时针为负;${\varphi _b}(b = 1,2,3,4)$ 分别为光线在各个面的入射角;${\varphi _c}'(c = 1,2,3,4)$ 分别为光线在各个面的折射角;${K_0}$ 与$ {K}_{1} $ 分别定义为曲率中心C到O点和曲率中心C到I点的距离;${r_d}(d = 1,\cdots,6)$ 分别为曲率中心C到出射光线和入射光线的直线距离,即$C{M_1}$ 的距离为${r_1}$ ,$C{M_{\rm{2}}}$ 的距离为${r_2}$ ,……;${\delta _e}(e = 1,\cdots,4)$ 分别为光线在折射面上偏折的角度;${M_f}(f = 1,\cdots,6)$ 分别为曲率中心C点到各个光线的垂直交点;${P_g}(g = 1,\cdots,4)$ 分别为入射光线与各个折射面的交点。设入射光线依次经过表面1、2、3、4、5,$ {R}_{1} $ 、$ {R}_{2} $ 分别为两个球面光学表面的曲率半径,$ {n}_{1} $ 、$ {n}_{2} $ 分别为透镜Ⅰ、Ⅱ的折射率,则系统的球差${L_s}$ 可利用公式(1)进行计算:$${L_s}{\rm{ = }}\frac{1}{{{{({K_1})}_n}}} - \frac{1}{{{{({K_{\rm{1}}})}_p}}}{\rm{ }}$$ (1) 式中:
${({K}_{1})}_{n}、{({K}_{\rm{1}})}_{p}$ 分别为实际光路计算和近轴光路计算结果。由图2可知:
$${\delta _1} = {\theta _0} - {\theta _1} = {\varphi _1} - {\varphi _1}'$$ (2) $${\delta _2} = {\theta _1} - {\theta _2} = {\varphi _2} - {\varphi _2}'$$ (3) $${\delta _3} = {\theta _{\rm{4}}} - {\theta _{\rm{3}}} = {\varphi _3}' - {\varphi _3}$$ (4) $${\delta _4} = ({\theta _5} + \pi ) + (\pi - {\theta _4}) = {\varphi _4}' - {\varphi _4}$$ (5) 由于表面3为反射面,
${\theta _3} = \pi - {\theta _2}$ ,可得:$$ \begin{split} {\theta _5} =& - {\theta _0} + {\varphi _1} - {\varphi _1}' + {\varphi _2} - {\varphi _2}' - {\varphi _3} {\rm{ }} +{\varphi _3}' -\\ & {\varphi _4} + {\varphi _4}' - \pi \\ \end{split} $$ (6) 又由于
$\alpha = {\theta _5} + \pi $ ,$$ \begin{split} \alpha =& - {\theta _0} + {\varphi _1} - {\varphi _1}' + {\varphi _2} - {\varphi _2}' - {\varphi _3} {\rm{ }} + {\varphi _3}' - \\ & {\varphi _4} + {\varphi _4}' \\ \end{split} $$ (7) 在△
$ CI{M_6}$ 中,根据正弦定理:$${K_1} = \frac{{{r_{\rm{6}}}}}{{\sin \alpha }}$$ (8) 在△
$C{P_4}{M_6}$ 中,有$\sin ({\varphi _4}') = {r_6}/{R_1}$ 。同理,$\sin ({\varphi _4}) = $ $ {r_5}/{R_1}$ 。由折射定律可知,${r_{\rm{6}}} = {r_{\rm{5}}}{n_1}/{n_0}$ ,由此可得:$$ \begin{split} {r_{\rm{6}}} =& {r_{\rm{5}}}\dfrac{{{n_1}}}{{{n_0}}} = {r_{\rm{4}}}\dfrac{{{n_2}}}{{{n_1}}} \cdot \dfrac{{{n_1}}}{{{n_0}}} = {r_2}\dfrac{{{n_1}}}{{{n_2}}} \cdot \dfrac{{{n_2}}}{{{n_1}}} \cdot \dfrac{{{n_1}}}{{{n_0}}} {\rm{ }} = \\ & {r_1}\dfrac{{{n_0}}}{{{n_1}}} \cdot \dfrac{{{n_1}}}{{{n_2}}} \cdot \dfrac{{{n_2}}}{{{n_1}}} \cdot \dfrac{{{n_1}}}{{{n_0}}} = {r_1} \\ \end{split} $$ (9) 由公式(7)~(9)可得:
$$ \begin{split} \frac{1}{{{K_1}}} =& \dfrac{1}{{{r_1}}}\sin ( - {\theta _0} + {\varphi _1} - \varphi {'_1} + {\varphi _2} - \varphi {'_2} {\rm{ }} -\\ & {\varphi _3} + \varphi {'_3} - {\varphi _4} + \varphi {'_4}) \\ \end{split} $$ (10) 根据正弦定理和折射定律,可将公式(10)中除
${\theta _0}$ 外的角度值均由${r_1}$ 和系统参数(${R_1},{R_2},n$ )表示,此处省略。最终计算得到:$$ \begin{split} \dfrac{1}{{{K_{\rm{1}}}}} =& \dfrac{1}{{{r_1}}}\arcsin \Bigg[2\Bigg({\arcsin }\dfrac{{{r_1}}}{{{R_1}}} - {\arcsin }\dfrac{{{n_0}{r_1}}}{{{n_1}{R_1}}} {\rm{ }} +\\ & {\arcsin }\dfrac{{{n_0}{r_{\rm{1}}}}}{{{n_1}{R_2}}} - {\arcsin }\dfrac{{{n_0}{r_{\rm{1}}}}}{{{n_2}{R_2}}}\Bigg) - {\theta _0}\Bigg] \\ \end{split} $$ (11) 当入射光线平行于光轴入射时,
${r_1} = h$ ,$h$ 为光线相对于光轴的入射高度,${\theta _{\rm{0}}} = 0$ 。此时公式表示为:$$ \begin{split} \dfrac{1}{({K}_{\rm{1}}{)}_{{n}}}=&\dfrac{1}{h}\mathrm{arsin}\Bigg[2({\mathrm{arcsin}}\dfrac{{h}_{}}{{R}_{1}}-{\mathrm{arcsin}}\dfrac{{h}_{}}{{n}_{1}{R}_{1}}\rm{ }+\\ &{\mathrm{arcsin}}\dfrac{h}{{n}_{1}{R}_{2}}-{\mathrm{arcsin}}\dfrac{h}{{n}_{2}{R}_{2}})\Bigg] \end{split} $$ (12) 当入射光线为近轴光线时,
${\theta _{}} \approx \sin {\theta _{}}$ ,${\theta _{\rm{0}}} = {r_1}/{K_0}$ ,在近轴光线中${K_0}$ 也可视为无穷大。此时公式表示为:$$ \frac{1}{({K}_{\rm{1}}{)}_{p}}=\frac{2}{{R}_{1}}-\frac{2}{{n}_{1}{R}_{1}}+\frac{2}{{n}_{1}{R}_{2}}-\frac{2}{{n}_{2}{R}_{2}}$$ (13) 结合公式(1)、(12)和(13)可以得到同心反射式结构的球差计算公式为:
$$ \begin{split} {L_S} =& \dfrac{1}{h}\arcsin \Bigg[2\Bigg({\arcsin }\dfrac{{{h_{}}}}{{{R_1}}} - {\arcsin }\dfrac{{{h_{}}}}{{{n_1}{R_1}}} + {\arcsin }\dfrac{h}{{{n_1}{R_2}}} -\\ & {\arcsin }\dfrac{h}{{{n_2}{R_2}}}\Bigg)\Bigg] - \Bigg(\dfrac{2}{{{R_1}}} - \dfrac{2}{{{n_1}{R_1}}} + \dfrac{2}{{{n_1}{R_2}}} - \dfrac{2}{{{n_2}{R_2}}}\Bigg) \\ \end{split} $$ (14) 式中:
${R_1}$ 和${R_2}$ 分别为同心透镜表面的曲率半径;${n_1}$ 和${n_2}$ 为同心透镜的折射率;h为入射光的高度,即系统入瞳大小的一半。将上述五个参数作为变量,根据设计要求加入限制条件,当系统球差
${L_s}$ 为极小值时求解系统初始结构。限制条件主要考虑以下几方面:(1)由表1可知,在确定了系统F数和焦距之后,可以求出系统入瞳大小,即确定了h值大小;
(2)根据常规手机镜头光学系统总长大小,可以确定
${R_1}$ <2.7 mm;(3)为了避免透镜组交界处发生全反射现象,尽量使
${n_1}$ >${n_2}$ 。手机镜头材料一般选用塑料材质,由于材料种类有限,可依据经验先确定两片透镜的材料,在初始结构确定后,选择折射率相近的材料进行替换,然后通过光学设计软件优化进一步减小系统色差。利用以上限制条件,经过编制程序计算,求出一组光学系统的初始结构如表2所示。
表 2 初始结构参数
Table 2. Initial structural prescription
Surface type Radius/mm Thickness/mm Glass OBJ $\infty $ $\infty $ 1 1.885 1.047 1.66,20.4 2 0.832 0.830 1.59,30.9 STO $\infty $ −0.830 MIRROR 4 0.832 −1.047 1.66,20.4 5 1.885 −0.5 IMA 2.385 -
利用光学设计软件进行优化,进一步提升了系统的成像质量,得到了符合设计要求的手机镜头结构[9-10]。优化后手机镜头为两片式结构,系统F数为1.8,焦距为2.7 mm,系统总长为2.7 mm,全视场角为100°,材料选用日本塑料OKP-A2和具有热塑性的树脂PS,设计结构如图3所示。
图4为手机镜头的MTF曲线图,在200 lp/mm处,0.7视场的MTF值分别大于0.62,全视场的MTF值均大于0.55;在400 lp/mm处,0.7视场的MTF值大于0.34,全视场的MTF值大于0.23。
图5为系统的点列图,由于同心透镜存在渐晕,会引起艾里斑在渐晕处增大,导致不同视场的像差不相同。手机镜头允许的最大弥散斑半径要最大不超过1.5/NR[11],NR为CMOS图像传感器的分辨率,该设计选取的曲面传感接收器像元尺寸为1.25 μm,可以计算出像面允许的最大弥散斑半径为3.75 μm。从图5中可以看出全视场RMS半径均小于1.2 μm,满足手机镜头成像要求。
图6为手机镜头的相对照度图,手机镜头相对照度越低,在光线微弱处拍出的照片就越不清晰[12]。从图6中可以看出,全视场相对照度均大于0.64,满足手机镜头成像要求。
为了获得手机镜头更好的成像质量,需要在探测器前加入红外滤光片。随着光学镀膜技术的发展,目前很多光学系统已经采用镀红外截止膜的方式实现红外滤光[13],可应用于文中设计的手机镜头;也可以在探测器前加入球形红外滤光片[5],结构如图7所示。
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公差分析是光学系统设计的一个重要环节,如果公差要求过严,将对生产制造提出更高要求,增加制造成本,不利于手机镜头批量化生产。利用光学设计软件进行公差分析,表3为系统的公差设定值。TFRN为各表面曲率半径的公差,用光圈表示;TSDX、TSDY分别为两个方向的偏心;TTHI为元件的绝对位置和镜片厚度的公差;TSTX、TSTY为光学元件的倾斜公差;TEDX、TEDY为元件的偏心公差[14]。
表 3 光学系统参数公差
Table 3. Tolerance of the optical system
Item Value TFRN/fringes ±1.5 TSDX, TSDY/mm ±0.013 TTHI/mm ±0.013 TSTX, TSTY/(°) ±0.02 TEDX, TEDY/mm ±0.013 TETX, TETY/(°) ±0.02 将MTF值作为公差敏感度,运用蒙特卡洛法对公差进行分析,设定500组随机误差数,对截止频率处的MTF值进行分析[15],分析结果表明90%以上的样本优于0.20。通过以上分析说明,设计的手机镜头公差要求宽松,采用直接装配的方式即可满足精度要求,适用于手机镜头的批量化生产。
Design of miniaturization concentric reflective mobile phone lens
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摘要: 基于现代社会对手机镜头高像素、小型化的要求,基于同心透镜原理设计了同心反射式手机镜头。通过光路计算,求解了系统的球差表达式,进一步获得了系统的初始结构。利用光学设计软件设计了光学系统,镜头采用像元大小为1.25 μm的曲面传感器,光学系统的F数为1.8,焦距为2.7 mm,最大全视场角为100°,系统总长为2.7 mm。设计结果表明,在空间截止频率400 lp/mm处,0.7视场的调制传递函数均大于0.34,全视场的调制传递函数均大于0.23,各视场的弥散斑半径均小于艾里斑。在全视场内,相对照度高于0.64,该设计满足手机镜头成像要求。Abstract: Based on the requirements of modern society for high pixels and miniaturization of mobile phone lenses, based on the principle of concentric lenses, a concentric reflective mobile phone lens was designed. Through optical path calculation, the spherical aberration expression of the system was solved, and the initial structure of the system was further obtained. The optical system was designed using optical design software, the lens adopted a curved sensor with a pixel size of 1.25 μm, the F number of the optical system was 1.8, the focal length was 2.7 mm, the maximum full field of view was 100°, and the total system length was 2.7 mm. The results show that at the spatial cut-off frequency of 400 lp/mm, the modulation transfer functions of the 0.7 field of view are all greater than 0.34, and the modulation transfer functions of the full field of view are all greater than 0.23. The radius of the diffuse spot in each field of view is smaller than the Airy disk. In the full field of view, the relative illuminance is higher than 0.64. This design meets the imaging requirements of mobile phone lenses.
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Key words:
- optical design /
- mobile phone lens /
- concentric structure /
- reflective type /
- curved surface sensors
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表 1 光学设计参数
Table 1. Optical design prescription
Parameter Value Waveband/nm 486-656 (Visible light) Relative aperture 1/1.8 Full field of view/(°) 100 Focal length/mm 2.7 Total length/mm $ \leqslant $ 2.7表 2 初始结构参数
Table 2. Initial structural prescription
Surface type Radius/mm Thickness/mm Glass OBJ $\infty $ $\infty $ 1 1.885 1.047 1.66,20.4 2 0.832 0.830 1.59,30.9 STO $\infty $ −0.830 MIRROR 4 0.832 −1.047 1.66,20.4 5 1.885 −0.5 IMA 2.385 表 3 光学系统参数公差
Table 3. Tolerance of the optical system
Item Value TFRN/fringes ±1.5 TSDX, TSDY/mm ±0.013 TTHI/mm ±0.013 TSTX, TSTY/(°) ±0.02 TEDX, TEDY/mm ±0.013 TETX, TETY/(°) ±0.02 -
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