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双焦光学系统要求在变焦前后像面保持不变。其实现方法有两种:一是增加一个补偿组补偿焦面移动,该方法会增加机械结构的复杂性;二是在透镜移动前后遵循物像交换原则[13-14],保证物像共轭距不变,该方法机械结构简单。文中拟采用第二种设计方式实现。
采用物像交换原则设计双焦光学系统,按前固定组、变倍组和后固定组进行光学划分。其中:前固定组用于会聚无穷远物体的光线;变倍组用于改变焦距,达到变倍的目的;后固定组用于校正像差。
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依据高斯提出的理想光学系统几何光学理论,研究近轴区成像规律建立理想光学系统成像模型,计算双焦光学系统光学结构的初始解。
图1给出了轴向双焦光学系统结构,当光学系统位于长焦时,
${l_{2l}}$ 、${l'}\!\!_{2l}$ 为变倍组的物距和像距,d23为后固定组和变倍组的间隔。当光学系统位于短焦时,${l_{2s}}$ 、${l'}\!\!_{2s}$ 为变倍组的物距和像距,d12为变倍组和前固定组间隔。AA′为变倍组的共轭距。(1)变倍组相关参数计算
令变倍组焦距为
$f_2'$ ,系统的放大倍率为$\beta $ ,则短焦和长焦时变倍组的垂轴放大倍率分别为:$${\beta _{2s}} = - {\beta ^{ - 1/2}}$$ (1) $${\beta _{2l}} = - {\beta ^{1/2}}$$ (2) 由
$$\frac{1}{{l_{2s}'}} - \frac{1}{{{l_{2s}}}} = \frac{1}{{f_2'}}$$ (3) 可得短焦时变倍组的物距以及像距:
$${l_{2s}} = \left(\frac{1}{{{\beta _{2s}}}} - 1\right)f_2'$$ (4) $${l}_{2s}' = (1 - {\beta _{2s}})f_2'$$ (5) 由物像交换原则可得长焦时变倍组的物距和像距:
$${l_{2l}} = - {l'}_{2s} = - (1 - {\beta _{2s}})f_2'$$ (6) $${l}_{2l}' = - {l_{2s}} = - \left(\frac{1}{{{\beta _{2s}}}} - 1\right)f_2'$$ (7) 当光学系统从短焦到长焦时,变倍组移动的距离为:
$$\Delta = \left(\frac{1}{{{\beta _{2s}}}} - {\beta _{2s}}\right)f_2' = \left(\frac{1}{{{\beta _{2s}}}} - \frac{1}{{{\beta _{2l}}}}\right)f_2'$$ (8) (2)前固定组焦距的计算
在对目标进行搜索和识别时,成像光学系统的物距可以认为是无穷远,则进入前固定组的光线是平行光入射且其光线的出射交点会聚于前固定组的焦平面上,该焦平面即为变倍组的物面。由此可以求出前固定组的焦距:
$$f_1' = {d_{12}} + {l_{2s}} = {d_{12}} + \left(\frac{1}{{{\beta _{2s}}}} - 1\right)f_2'$$ (9) (3)后固定组焦距的计算
当光学系统位于短焦时,令系统的总光焦度为
$\phi $ ,后固定组的光焦度为${\phi _3}$ ,前固定组和变倍组的等效组总光焦度为${\phi _{12}}$ ,等效组像方主平面与后固定组间隔为${d_{123}}$ ,则可以求出后固定组的焦距$f_3'$ :$$\phi {\rm{ = }}{\phi _{12}}{\rm{ + }}{\phi _3}{\rm{ - }}{d_{123}}{\phi _{12}}{\phi _3}$$ (10) $$f_3'{\rm{ = }}\frac{1}{{{\phi _3}}}$$ (11) -
由于进入前固定组的光线为平行光,可得:
$${y_2} = \frac{{f_l'{l_{2l}}}}{{2Ff_1'}}$$ (12) $${y_3} = \frac{{{y_2}({\rm{ - }}l_{2l}' + {d_{23}})}}{{{\rm{ - }}l_{2l}'}}$$ (13) 式中:
$f_l'$ 为光学系统位于长焦时的焦距。为了研究的方便性,针对一个具体问题进行设计分析,根据表1给出的具体设计指标要求,展开论述。
表 1 光学系统设计指标
Table 1. Design index of optical system
Parameter Value Wavelength 486-656 nm Field of view 8.6°/2.9° Focal length 40 (WFOV) mm 120 (NFOV) mm Spot diagram <0.5 pixel@0-0.707 fields
<0.7 pixel@0.707-1 fieldsMTF 100 lp/mm>0.3 Temperature −20-45 ℃ CCD pixel 5 μm 在双焦光学系统设计时,首先要确定变倍组焦距。由公式(8)可以看出变倍组的焦距决定了其移动范围,若焦距过大,则移动范围过大;若焦距过小,则其承担的光焦度过大,会引入较大像差导致公差敏感。其次要确定前固定组焦距,由公式(9)可以看出,前固定组焦距和
${d_{12}}$ 有关,${d_{12}}$ 的选取主要考虑以下两个因素:一是系统的总长,二是主光线进入变倍组的入射高度和入射角度。由公式(9)、(12)可得,${d_{12}}$ 越大,${y_2}$ 越小,引入像差越少。最后确定后固定组的焦距,要考虑筒长以及与变倍组之间的距离${d_{23}}$ ,在不与变倍组相碰撞的前提下,由公式(13)可以看出,${d_{23}}$ 越小,${y_3}$ 越小,引入像差越小。此次设计共采用七片透镜,前固定组和变倍组各两片透镜,后固定组三片透镜。根据上述选择分析,结合物像交换原则和高斯光学求解方法,得出各组焦距和间隔如下:
$f_1' = 115 \;{\rm{mm}}$ ,$f_2' = {\rm{ - 35}}{\rm{mm}}$ ,$f_3' = 40\;{\rm{mm}}$ ,${d_{12}}$ =20 mm,${d_{23}}$ =10 mm。 -
首先将上述得到的焦距等数据输入ZEMAX中,将各组的间隔以及焦距大小设为变量,在评价函数中加入操作数EFFL控制焦距;再加入操作数ZTHI让长焦和短焦时总长相等,且前固定组和后固定组间隔相等,来保证焦距改变时变化的透镜组只有变倍组。另外,为保证像面稳定,需要在多重结构中将短焦时光学系统的后截距设为变量,长焦时光学系统的后截距设为拾取。进行优化即可得到图2所示的近轴光学系统光路图。优化后系统评价函数由0.000000038变为0,各组焦距和间隔分别为
$f_1' = 115.925\;{\rm{mm}}$ ,$f_2' = $ $ {\rm{ - 35}}{\rm{.111}}{\rm{mm}}$ ,$f_3'{\rm{ = }}39.688\;{\rm{mm}}$ ,${d_{12}}{\rm{ = }}20\;{\rm{mm}}$ ,${d_{23}}{\rm{ = }}10.008\;{\rm{mm}}$ 。 -
近轴替代的核心思想是通过使用标准透镜来替代无像差的近轴面,进而再将标准透镜带入的像差优化到设计要求范围。
替换原则:第一步替换后固定组。采用透镜逐个插入法进行替代,在透镜插入系统时,要求其对系统的光焦度贡献为0,这一步的目的是使插入前后不改变任何光线轨迹。为了在不改变光学系统的情况下完成替代,需要逐步增加近轴面的焦距值,使近轴面的焦距接近无穷大,而插入的标准透镜组的焦距值接近近轴面最初的焦距值,然后删掉近轴面。在改变焦距的同时需要添加基本操作数控制球差、彗差等初级像差,并且需要控制透镜的厚度以及折射率,最后再将模型玻璃替换成真实玻璃。第一步的优化至关重要,若替换完后其像差过大,需要进一步优化,将RMS半径优化到6个像元以内,以便于后续替换过程能更好地平衡像差。第一步优化后光路图如图3所示,透镜参数如表2所示。
表 2 第一步优化后镜片参数
Table 2. Lens parameters after the first step optimization
Radius/mm Thickness/mm Glass 75.183 4.46 Zk13 −23.174 1.063 −20.456 1.977 Sf6 −43.443 11.686 914.672 2.267 Sk11 −39.321 50.766 第二步替换掉变倍组。方法与第一步类似,逐步插入透镜,将近轴面的焦距逐步优化到无穷大,使加入的透镜焦距接近最开始近轴面的焦距,然后删掉近轴面,再次进行优化。第二步优化后光路图如图4所示,透镜参数如表3所示。
表 3 第二步优化后透镜参数
Table 3. Lens parameters after the second step optimization
Radius/mm Thickness/mm Glass −53.012 1.999 P-sf68 −33.681 0.998 −35.382 1.999 N-lak33b 69.948 56.318/1.357 128.881 3.276 Zk13 −34.662 1.524 −26.716 1.999 Sf6 −60.543 0.999 117.307 8.127 Sk11 −44.078 64.357 第三步替换掉前固定组。与第一步的步骤一样,应用相同的过程来替换近轴面,即得到的初始结构如图5所示。表4是第三步优化后透镜参数,表5是每一步优化后的像差大小。
表 4 第三步优化后透镜参数
Table 4. Lens parameters after the third step optimization
Surface Radius/mm Thickness/mm Glass 1 57.870 10 Zk12 2 −77.106 2 Kzfs8 3 193.419 13.083/59.282 4 −42.419 4.997 P-sf68 5 −28.545 1 6 −27.789 2 N-lak33b 7 77.385 47.197/0.997 8 78.631 6.895 Zk13 9 −26.74 1.612 10 −24.316 2 Sf6 11 −62.578 1 12 60.429 2.496 Sk11 13 −80.595 55.744 表 5 优化过程中像差大小
Table 5. The size of aberration during optimization
Field Aberration First step Second step Third step WFOV S1 0.039841 0.028905 0.009558 S2 −0.001334 −0.002104 0.000991 S3 0.003687 0.001254 −0.001397 S4 0.004753 0.000664 0.001780 S5 0.000835 0.006641 0.008469 CL −0.002214 −0.001471 −0.001162 CT −0.001464 −0.001599 −0.000406 Maximum RMS radius/μm 25.203 11.011 5.852 Minimum RMS radius/μm 21.081 10.054 3.684 NFOV S1 0.039844 −0.026449 0.000056 S2 −0.001333 0.001632 −0.000895 S3 0.003687 −0.000991 0.000350 S4 0.004753 0.000664 0.001780 S5 0.000835 0.000643 0.000416 CL −0.002215 0.003065 −0.001473 CT −0.001464 −0.000774 0.000831 Maximum RMS radius/μm 25.368 11.711 8.691 Minimum RMS radius/μm 20.564 8.984 4.626
Solving initial structure of bifocal system according to theory of paraxial optics
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摘要: 为了快速得到双焦光学系统初始结构,依据近轴光学理论设计一款双焦、双视场光学系统。通过高斯光学理论及物像交换原则求解光学系统近轴光学元件的初始位置,分组将标准透镜插入到求解位置上,通过逐步增大近轴元件的焦距优化镜组及镜片间隔,使插入的透镜组焦距趋向该近轴元件焦距的理论计算值。以此类推完成每个镜组的优化设计。通过该方法设计了焦距为40/120 mm、视场为8.6°/2.9°的光学系统,所有镜片均为球面。在奈奎斯特频率100 lp/mm处,120 mm焦距时调制传递函数为0.58,接近衍射极限;40 mm焦距时调制传递函数为0.52。设计结果表明该方法适用于双视场光学系统,可快速得到光学系统初始结构,简化了设计难度。Abstract: In order to get the initial structure of the bifocal optical system quickly, a bifocal and dual-field optical system was designed according to the theory of paraxial optics. The initial position of the optical elements near the axis of the optical system was solved by Gauss optics theory and the principle of object and image exchange. The standard lens was inserted into the solution position by grouping. The lens spacing was optimized by gradually increasing the focal length of the elements near the axis, so that the focal length of the inserted lens set approached the theoretical calculated value of the focal length of the element near the axis. Then the method was used to completes the optimal design of each lens set. An optical system with a focal length of 40/120 mm and a field of view of 8.6°/2.9° was designed by this method. All the lenses were spherical. At Nyquist frequency of 100 lp/mm, the modulation transfer function at 120 mm focal length was 0.55, close to the diffraction limit. The modulation transfer function at a focal length of 40 mm was 0.4. The design results show that this method is suitable for dual-field optical system, and the initial structure of optical system can be obtained quickly, which greatly reduces the difficulty of design.
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表 1 光学系统设计指标
Table 1. Design index of optical system
Parameter Value Wavelength 486-656 nm Field of view 8.6°/2.9° Focal length 40 (WFOV) mm 120 (NFOV) mm Spot diagram <0.5 pixel@0-0.707 fields
<0.7 pixel@0.707-1 fieldsMTF 100 lp/mm>0.3 Temperature −20-45 ℃ CCD pixel 5 μm 表 2 第一步优化后镜片参数
Table 2. Lens parameters after the first step optimization
Radius/mm Thickness/mm Glass 75.183 4.46 Zk13 −23.174 1.063 −20.456 1.977 Sf6 −43.443 11.686 914.672 2.267 Sk11 −39.321 50.766 表 3 第二步优化后透镜参数
Table 3. Lens parameters after the second step optimization
Radius/mm Thickness/mm Glass −53.012 1.999 P-sf68 −33.681 0.998 −35.382 1.999 N-lak33b 69.948 56.318/1.357 128.881 3.276 Zk13 −34.662 1.524 −26.716 1.999 Sf6 −60.543 0.999 117.307 8.127 Sk11 −44.078 64.357 表 4 第三步优化后透镜参数
Table 4. Lens parameters after the third step optimization
Surface Radius/mm Thickness/mm Glass 1 57.870 10 Zk12 2 −77.106 2 Kzfs8 3 193.419 13.083/59.282 4 −42.419 4.997 P-sf68 5 −28.545 1 6 −27.789 2 N-lak33b 7 77.385 47.197/0.997 8 78.631 6.895 Zk13 9 −26.74 1.612 10 −24.316 2 Sf6 11 −62.578 1 12 60.429 2.496 Sk11 13 −80.595 55.744 表 5 优化过程中像差大小
Table 5. The size of aberration during optimization
Field Aberration First step Second step Third step WFOV S1 0.039841 0.028905 0.009558 S2 −0.001334 −0.002104 0.000991 S3 0.003687 0.001254 −0.001397 S4 0.004753 0.000664 0.001780 S5 0.000835 0.006641 0.008469 CL −0.002214 −0.001471 −0.001162 CT −0.001464 −0.001599 −0.000406 Maximum RMS radius/μm 25.203 11.011 5.852 Minimum RMS radius/μm 21.081 10.054 3.684 NFOV S1 0.039844 −0.026449 0.000056 S2 −0.001333 0.001632 −0.000895 S3 0.003687 −0.000991 0.000350 S4 0.004753 0.000664 0.001780 S5 0.000835 0.000643 0.000416 CL −0.002215 0.003065 −0.001473 CT −0.001464 −0.000774 0.000831 Maximum RMS radius/μm 25.368 11.711 8.691 Minimum RMS radius/μm 20.564 8.984 4.626 -
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