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光学系统设计参数见表1。
表 1 光学设计参数
Table 1. Parameter of optical design
System parameter Requirements Spectral band/μm 0.4-5.0 Focal length/mm 400 F number 3 Field of view/(°) 2ω=8 Real entrance pupil Aperture stop front;placed outside the
optical system structure所设计的中继光学系统的工作谱段较宽,反射式光学系统能通过所有谱段的光波,不受色差和二级光谱色差的影响,适用于宽光谱、大波段的光学系统。反射式光学系统主要分为同轴反射式和离轴反射式两种,同轴反射光学系统普遍存在中心遮拦现象,该现象使进入光学系统的能量减少,从而影响成像质量。离轴反射系统则避免了这种现象,其中离轴三反光学系统在设计时拥有较多的自由度,能够满足更高的设计要求,从而提供更好的成像质量。离轴三反结构的确定是以同轴三反为基础,同轴三反结构图如图1所示。
其中,d1为主镜与次镜间的距离,d2为次镜与三镜间的距离,F为物方焦点,
$ {l}_{f} $ 为物方焦点与主镜的距离。根据几何光学、三级像差等理论,可得同轴情况下三反系统结构参数与遮拦比的关系为:
$$ {R}_{1}=\frac{2\left[{\alpha }_{1}{\left(1-{\alpha }_{2}\right)}^{2}+{\alpha }_{2}{\left(1-{\alpha }_{1}\right)}^{2}\right]}{1-{\alpha }_{1}}{f}' $$ (1) $$ {R}_{2}=\frac{2{\alpha }_{1}\left[{\alpha }_{1}{\left(1-{\alpha }_{2}\right)}^{2}+{\alpha }_{2}{\left(1-{\alpha }_{1}\right)}^{2}\right]}{1+{\alpha }_{1}\left({\alpha }_{2}-2\right)}{f}' $$ (2) $$ {R}_{3}=\frac{2{\alpha }_{1}\left[{\alpha }_{1}{\left(1-{\alpha }_{2}\right)}^{2}+{\alpha }_{2}{\left(1-{\alpha }_{1}\right)}^{2}\right]}{1+{\alpha }_{1}\left({\alpha }_{1}+{\alpha }_{2}-3\right)}{f}' $$ (3) $$ {d}_{1}=\left[{\alpha }_{1}{\left(1-{\alpha }_{2}\right)}^{2}+{\alpha }_{2}{\left(1-{\alpha }_{1}\right)}^{2}\right]{f}' $$ (4) $$ {d}_{2}=-\left[{\alpha }_{1}{\left(1-{\alpha }_{2}\right)}^{2}+{\alpha }_{2}{\left(1-{\alpha }_{1}\right)}^{2}\right]{f}' $$ (5) $$ {d}_{3}={\alpha }_{1}{\alpha }_{2}{f}' $$ (6) $$ {l}_{f}=\dfrac{{R}_{1}}{2}×\dfrac{2{d}_{1}\left({R}_{3}+2{d}_{2}-{R}_{2}\right)+{R}_{2}\left({R}_{3}+2{d}_{2}\right)}{2{d}_{1}\left({R}_{3}+2{d}_{2}-{R}_{2}\right)+{R}_{2}\left({R}_{3}+2{d}_{2}\right)-{R}_{1}\left({R}_{3}+2{d}_{2}-{R}_{2}\right)}= \dfrac{{\alpha }_{1}-\left(1+{\alpha }_{1}{\alpha }_{2}\right)\left[{\alpha }_{1}{\left(1-{\alpha }_{2}\right)}^{2}+{\alpha }_{2}{\left(1-{\alpha }_{1}\right)}^{2}\right]}{{\alpha }_{2}{\alpha }_{1}^{2}}{f}' $$ (7) 式中:R1、R2、R3分别为主镜、次镜及三镜的曲率半径;α1、α2为次镜对主镜与三镜对次镜的遮拦比;
$ {f}' $ 为总焦距[14]。光学系统一般通过孔径光阑与光学系统的物方焦平面重合或接近实现像方远心或准远心,像方远心光路的出瞳位置位于无穷远处或较远处,出射光各视场主光线接近平行,易实现瞳孔匹配、光路衔接、分光等[15]。且光学系统具有实入瞳的需求,光学系统孔径光阑位置要位于主镜前,此时物方焦点为实焦点,即
$ {l}_{f} $ 小于零。根据三级像差理论得到光学系统中的球差、彗差、像散的系数表达式为:
$$ \begin{split} {{S}}_{{\text{Ⅰ}}}=&\frac{1}{4}\left[\left({e}_{1}^{2}-1\right)-{e}_{2}^{2}{\alpha }_{1}{\left(1+{\alpha }_{1}\right)}^{3}+{e}_{3}^{2}{\alpha }_{1}{\left(1+{\alpha }_{1}\right)}^{4}-\right.\\ &\left.{\alpha }_{1}^{2}\left(1+{\alpha }_{1}\right){\left(1-{\alpha }_{1}\right)}^{2}\right] \end{split} $$ (8) $$ \begin{split} {{S}}_{{\text{Ⅱ}}}=&\frac{1}{4}\left[{e}_{2}^{2}{(1-\alpha }_{1}{)\left(1+{\alpha }_{1}\right)}^{3}+{e}_{3}^{2}{(\alpha }_{1}^{2}-2){\left(1+{\alpha }_{1}\right)}^{3}-\right.\\ &\left.{\alpha }_{1}^{5}{+2{\alpha }_{1}^{4}+\alpha }_{1}^{3}-3{\alpha }_{1}^{2}-1\right] \end{split} $$ (9) $$ \begin{split} {{S}}_{{\text{Ⅲ}}}=&\frac{1}{4{\alpha }_{1}}\left[{e}_{2}^{2}{\left(1-{\alpha }_{1}\right)}^{5}+{e}_{3}^{2}{{(\alpha }_{1}^{2}-2)}^{2}{\left(1+{\alpha }_{1}\right)}^{2}-{\alpha }_{1}^{6}+3{\alpha }_{1}^{5}- \right.\\ &\left.6\alpha _{1}^{3}+2{\alpha }_{1}^{2}{+\alpha }_{1}+1\right] \end{split} $$ (10) 式中:e12、e22
、e32 为主镜、次镜及三镜的二次非球面系数。为了使结构更合理,对三级像差进行校正,令SⅠ、SⅡ、SⅢ均为零,求解出各镜的二次非球面系数[16]。根据上述分析及参考文献[17],取遮拦比为α1 =0.5,α2=1,已知焦距为400 mm,根据公式(1)~(7)计算可得,R1=−400 mm、R2=−200 mm、R3=−400 mm、d1=−100 mm、d2=100 mm、d3=−200 mm、 $ {l}_{f} $ =−200 mm。又根据公式(8)~(10)求得二次非球面系数e12=1.13、e22 =−0.38、e32 =−0.26。 在离轴三反光学系统中,孔径光阑通常位于主镜或次镜上。孔径光阑位于主镜上,光学系统的结构较紧凑,可在中间像面加入消杂散光光阑,有利于小视场的光学系统设计;孔径光阑位于次镜上,主镜与三镜关于次镜对称,光学系统可以较好地校正轴外视场的像差,适用于大视场光学系统的设计[14]。文中光学系统的孔径光阑位于主镜前满足实入瞳需求,且中继系统要与前面的光学系统或光学器件衔接,其必须位于整体光学系统的结构之外,位置示意图如图2所示。根据上述计算的初始结构参数,且孔径光阑要与光学系统的物方焦平面接近,因此孔径光阑距主镜的距离等于
$ {l}_{f} $ ,将光阑前置量设为200 mm,此时孔径光阑也位于结构外,同轴三反初始结构如图3所示。在确保光学系统的孔径光阑位于光学系统结构外的条件下,对上述同轴三反初始结构进行光阑或视场离轴,镜面倾斜,达到避开镜面相互之间的遮拦,使其既满足实入瞳、像方远心需求,又具有良好的成像质量。
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在同轴三反结构的基础上进行光阑离轴后,系统实现无遮拦,离轴量为220 mm,此时由于孔径光阑具有偏心量,导致其位于结构内。在优化过程中,以各镜的曲率半径、二次非球面系数、偏心量、倾斜角度及镜间距作为优化变量,但优化后不能同时满足孔径光阑位于结构外和像质良好的要求,并且随着光阑前置量的增加,光学系统愈不对称,光学系统的垂轴像差也愈大,像差校正难度提升,通过逐步提升各镜非球面系数对像质的影响也已经很微弱,此时,通过优化现有变量已经无法使光学系统在孔径光阑位于结构外与像质良好达到平衡,需要引入新的优化变量。
自由曲面是一类非旋转对称面型,具有较多的自由度,能够有效地校正像差,提高光学系统的成像质量。近年来,随着加工技术的提升,越来越多的研究者将其运用于成像系统中,从而提升光学系统性能,基于自由曲面的成像系统设计关键之一就是选择合适的描述方法[18]。
自由曲面的描述方法一般有参数描述法和多项式描述,参数描述法主要包括贝塞尔曲面、B样条曲面、非均匀有理B样条等,但存在精度不高、加工检测难等问题。多项式描述法一般有泽尼克多项式、Gauss多项式、XY多项式等,在离轴三反光学系统中的自由曲面大都采用泽尼克多项式和XY多项式描述。泽尼克多项式具有在圆域内正交、各多项式的系数之间相互独立、系数间影响较小等特点,表达式为:
$$ {\textit{z}}=\frac{c{r}^{2}}{1+\sqrt{1-(1+k){c}^{2}{r}^{2}}}+{\sum }_{i=1}^{n}{A}_{i}{Z}_{i}\left(\rho ,\psi \right) $$ (11) 式中:z为矢高;r为径向值;c为顶点曲率半径;k为二次曲面系数;等号后第一项为标准二次曲面表达式;Ai为泽尼克子项系数;Zi为泽尼克多项式第i项[19]。
XY多项式描述的曲面具有能够校正非对称像差、精度高、易于加工等特点,广泛应用于成像光学系统中。XY多项式的表达式为:
$$ \begin{array}{l} {\textit{z}}\left( {x,y} \right) = \dfrac{{c\left( {{x^2} + {y^2}} \right)}}{{1 + \sqrt {1 - \left( {k + 1} \right){c^2}\left( {{x^2} + {y^2}} \right)} }} + \displaystyle\sum\nolimits_{j = 2}^J {{a_j}{x^m}{y^n}} \\ j = \dfrac{{{{\left( {m + n} \right)}^2} + m + 3n}}{2} + 1 \end{array} $$ (12) 式中:c为顶点曲率半径;k为圆锥常数;aj为xmyn项的系数,j为xmyn的总项数,m、n均为非负整数且之和大于1。通常自由曲面XY多项式的阶数最多为八次,高阶数会使优化效率降低,由于自由曲面的变量个数较多,模型较为复杂,且非旋转对称面型导致传统的像差理论不再适用,在展开后续优化过程中较为较困难,目前一般采用循序渐进的优化模式,逐步将非球面替换成自由曲面,在此过程中逐步加入像差约束和更为严格的结构控制条件,根据光学系统的像差情况选取表达式中合适的系数进行优化[20]。该模式使每一步优化具有较好的起点,提升了光线追迹速度,缩短了优化时间,也避免了面型产生突变[21]。
在上述光学系统中引入自由曲面,首先分别在各镜上尝试引入自由曲面,优化后发现主镜、三镜为自由曲面时对光阑前置带来的像差校正及像质的影响较大,因此,将主镜及三镜的面型均设为XY多项式表达的自由曲面。优化时,为避免面型产生突变,采用循序渐进的优化模式,并且由于离轴三反光学系统各镜的偏心及倾斜均在YOZ平面,X项只选取偶次项作为优化变量,优化后光阑的前置量为240 mm,但光学系统畸变残余量较大,将次镜用XY多项式表示的自由曲面替换,并逐步提升各镜自由曲面阶数,迭代优化后,光学系统的孔径光阑位于结构外且像质良好,满足设计要求。在后续成像系统中,系统的成像方案将采用像面前分光模式在0.4~1.6 μm、1.6~5.0 μm谱段分别成像,从而获取全波段图像信息。
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设计出的宽谱段实入瞳远心中继光学系统结构图如图4所示,结构参数见表2,光学系统的光阑前置量为240 mm,出瞳距为1049.32 mm,为实入瞳准远心光学系统,各镜均为XY多项式描述的自由曲面,参数见表3,利用MTF、畸变曲线、像差曲线及点列图对系统的设计结果进行了分析评价,如图5~图8所示,在用MTF进行评价时,选取了不同的空间频率(50、40、30、20、10 lp/mm)。
表 2 离轴三反光学系统结构参数
Table 2. Parameters of off-axis three-mirror optical system
Radius/mm Thickness/mm Decenter/mm Tilt/(°) First mirror −458.7815 −107.5917 58.71 0.19 Second mirror −167.2861 134.6101 8.81 10.09 Third mirror −355.8577 −107.8693 −256.96 −28.66 表 3 各镜的自由曲面参数
Table 3. Parameters of free-form surface of each mirror
Parameters First mirror Second mirror Third mirror x2 −7.4433e-005 0.0003 1.8133e-005 y2 3.2701e-005 0.0004 1.8497e-005 x2y 3.6196e-007 1.5541e-006 −3.4655e-007 y3 1.8164e-007 1.2668e-006 6.6196e-008 x4 1.1032e-009 1.2601e-008 7.4445e-010 x2y2 1.5319e-009 1.6041e-008 −2.6963e-009 y4 7.6358e-010 9.3326e-009 2.0005e-009 x2y3 3.1651e-013 3.5955e-011 1.7453e-011 x4y −2.0905e-012 −1.9364e-010 1.8144e-011 y5 −2.8706e-013 −9.5697e-012 −8.4075e-012 x6 2.4865e-015 2.0033e-013 6.4256e-014 x4y2 −3.6701e-015 −2.0303e-013 −5.8383e-014 x2y4 6.3076e-016 4.5629e-013 −5.2467e-014 y6 −3.5416e-016 6.6398e-014 1.6164e-014 由MTF图可以看出,在空间频率50 lp/mm处,MTF最小为0.540,在空间频率40 lp/mm处,MTF最小为0.631,在空间频率30 lp/mm处,MTF最小为0.724,在空间频率20 lp/mm处,MTF最小为0.820,在空间频率10 lp/mm处,MTF最小为0.911,且均接近于衍射极限;由光学系统的畸变曲线可以看出,最大畸变小于1%;从点列图及像差曲线可以看出,各视场各谱段点列图均方根都远小于艾里斑半径,像面的弥散斑直径均在像元尺寸之内,能量分布较集中,该中继光学系统具有良好的成像质量。
Design of telecentric relay optical system with broadband and real entrance pupil
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摘要: 红外光谱成像系统、光场成像系统、光学显微系统、偏振干涉成像系统、复眼成像系统、环带式全景光学系统、多尺度成像系统及头戴式增强显示系统等光学系统中,通常需要中继光学系统来实现光路衔接、配曈、偏转等。研究了现有中继光学系统结构,介绍了光阑前置即具有实入曈的像方远心离轴三反光学系统的设计方法及自由曲面的描述方法,完成了满足设计参数的具有实入瞳的远心中继光学系统仿真设计,系统各镜采用XY多项式描述的自由曲面离轴三反光学系统结构。CODEV仿真设计结果表明,在工作谱段0.4~5.0 μm、焦距400 mm、F数3、视场角2ω=8°下,系统MTF(Modulation Transfer Function)接近于衍射极限,畸变小于1%,成像质量良好。Abstract: The relay optical system was widely used in optical systems such as infrared spectral imaging system, light field imaging system, optical microscopy system, polarization interference imaging system, compound eye imaging system, ring-belt panoramic optical system, multi-scale imaging system and head-mounted enhanced display system, etc, which can link up, match pupil and deflect optical systems. The structures of the existing relay optical system were studied. The design method of telecentric off-axis three-mirror optical system with front aperture and the description method of free-form surface were introduced. According to the design parameters, the telecentric relay optical system with broadband and real entrance pupil was completed. The system was an off-axis three-mirror optical system structure, and each mirror was a free-form surface described by XY polynomial. The results of CODEV software simulation show that the MTF of the system is close to the diffraction limit, the distortion is less than 1%, and the imaging quality is good with the working spectral range of 0.4-5.0 μm, f'=400 mm, F/3 and 2ω=8°.
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Key words:
- optical design /
- relay system /
- telecentric /
- off-axis three-mirror /
- free-form surface
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表 1 光学设计参数
Table 1. Parameter of optical design
System parameter Requirements Spectral band/μm 0.4-5.0 Focal length/mm 400 F number 3 Field of view/(°) 2ω=8 Real entrance pupil Aperture stop front;placed outside the
optical system structure表 2 离轴三反光学系统结构参数
Table 2. Parameters of off-axis three-mirror optical system
Radius/mm Thickness/mm Decenter/mm Tilt/(°) First mirror −458.7815 −107.5917 58.71 0.19 Second mirror −167.2861 134.6101 8.81 10.09 Third mirror −355.8577 −107.8693 −256.96 −28.66 表 3 各镜的自由曲面参数
Table 3. Parameters of free-form surface of each mirror
Parameters First mirror Second mirror Third mirror x2 −7.4433e-005 0.0003 1.8133e-005 y2 3.2701e-005 0.0004 1.8497e-005 x2y 3.6196e-007 1.5541e-006 −3.4655e-007 y3 1.8164e-007 1.2668e-006 6.6196e-008 x4 1.1032e-009 1.2601e-008 7.4445e-010 x2y2 1.5319e-009 1.6041e-008 −2.6963e-009 y4 7.6358e-010 9.3326e-009 2.0005e-009 x2y3 3.1651e-013 3.5955e-011 1.7453e-011 x4y −2.0905e-012 −1.9364e-010 1.8144e-011 y5 −2.8706e-013 −9.5697e-012 −8.4075e-012 x6 2.4865e-015 2.0033e-013 6.4256e-014 x4y2 −3.6701e-015 −2.0303e-013 −5.8383e-014 x2y4 6.3076e-016 4.5629e-013 −5.2467e-014 y6 −3.5416e-016 6.6398e-014 1.6164e-014 -
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