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消色差複合透鏡之模擬與分析

Modeling and analysis of the hybrid achromatic lens. 消色差複合透鏡之模擬與分析. Department of Electrical Engineering Southern Taiwan University of Technology Tainan. 林正峰，林世堃，杜彥錫. CONTENTS. Introduction of the hybrid lens Profile of the diffractive surface Hybrid lens and aberrations

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消色差複合透鏡之模擬與分析

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  1. Modeling and analysis of the hybrid achromatic lens 消色差複合透鏡之模擬與分析 Department of Electrical Engineering Southern Taiwan University of Technology Tainan 林正峰，林世堃，杜彥錫

  2. CONTENTS • Introduction of the hybrid lens • Profile of the diffractive surface • Hybrid lens and aberrations • Thickness comparison and ray fan plot • Fabrication errors • Simulation of diffraction efficiency • PSF and MTF of the hybrid lens • Summary

  3. Introduction of the hybrid lens • Advantages： • Correct the chromatic aberration and other aberrations • Reduce weight, volume, and cost + = Refractive lens Diffractive lens Hybrid lens d-line(587nm)、C-line(656nm) and F-line(486nm) nd-vd

  4. Phase shift function r -2π -4π -6π -8π f :焦距 r: 半徑 由於 如果將修改成modulo 2π的相位結構，我們便可得到blazed-type Fresnel 透鏡的相位函數 m為第幾個zone的意思，rm為第m個zone的內半徑。

  5. Profile of the diffractive surface Topt為繞射面的最大深度，即 ， 為設計波長 0th zone 1st zone T(r) TOPT m-th zone r r2 r1 rm rmax 繞射面的表面輪廓(profile)T(r)可表示為：

  6. Design procedures of the hybrid lens Simulate the SPDT Define Purpose、Environment and Fabrication technology Simulate diffraction efficiencies of diffractive surfaces by different types of diamond turning tools and different tools radii. Find achromatic coefficient(A)and spherical aberration coefficient(G) Use the commercial optical system design software to optimize Compare and analyze Produce diffractive profile

  7. Hybrid lens and aberrations Spherical aberration of refractive surface: y: radius(aperture/2)Φref:optical power C:conjugate parameter n: refractive index B:bending parameter Marginal ray y h f Chief ray B=1 convex to left and plane to right

  8. Hybrid lens and aberration • Assume the object is at infinity, i.e., C=-1. In addition, the refractive • index for the diffractive surface is infinite, so we can find the relationship: • Using the equation below, we can find spherical aberration coefficient (G): • By the commercial optical system design software–OSLO, we can optimize the coefficients A and G.

  9. Design cases Doublet (SF2/BAK1) Hybrid lens(Glass) Hybrid lens(PMMA) Eyepiece (BK7/SF61) Eyepiece (Hybrid lens) (Glass/PMMA) Usp 2,829,560 case1 • EFL : 99.76mm, aperture/2 : 25mm, F/#:2 • Principal wavelength: 0.589mm, other wavelengths: 0.486mm and 0.656mm, view angle: 3o case2 • EFL : 100mm, aperture/2 : 64mm, F/#:3.3 • Principal wavelength: 0.589mm, other wavelengths: 0.486mm and 0.656mm, view angle: 25.17o

  10. Original Design Hybrid Lens (BK7) Hybrid Lens (PMMA) Thickness comparison Doublet 24.81mm 20.68mm 10mm Eyepiece 64.81mm 15mm 20mm Eyepiece Doublet

  11. Ray fan plot of the doublet (a)doublet (b) hybrid lens BK7 (c) hybrid lens PMMA • 縱軸實際光束與參考光束(chief ray )在像面上的位移。 • 橫軸為歸一化(normalize)後的入孔(entrance pupil)座標 • 左圖為子午光束擷取曲線(meridional ray intercept curve)，右圖為徑向光束擷取曲線(sagittal ray intercept curve) • 上中下圖分別為全視場 、0.7視場及0視場

  12. Ray fan plot of the eyepiece (c)hybrid lens PMMA (a) Doublet (b) hybrid lens BK7

  13. Fabrication errors (round tool ) Round tool △d x △x • To model effects of fabrication errors, we assume the hybrid lens is fabricated by single-point diamond turning technique. Diffractive surface • The △x is fabrication error in x direction ，△d is fabrication error of depth. The diffractive surface obtained by using a 50mm round tool

  14. Fabrication error ( half-radiused tool ) Diffractive surface half-radiused tool x Errors of depth Diffractive surface obtained by using a 1mm half-radiused tool

  15. Diffraction efficiency (case 1-BK7) Doublet, (Round tool) Doublet, (half-radiused tool)

  16. PSF&MTF (hybrid lens) PSF PSF A2： Doublet, BK7,半R刀 籃色：理想 綠色：2μm 紅色：50μm 淡籃色：0.5mm 粉紅色：1mm A1：Doublet, BK7,全R刀 籃色：理想 綠色：2μm 紅色： 12μm 淡籃色：32μm 粉紅色：47μm MTF MTF 設計編號：A1

  17. MTF (different wavelengths) PSF MTF Ideal (No fabrication errors) d-line(587nm) C-line(656nm) F-line(486nm) Round tool- (47um) PSF MTF

  18. Summary • Comparing with conventional achromats, the hybrid lens not only can correct chromatic aberration and spherical aberration, but also can reduce weight and volume. • Diffraction efficiency of the diffractive surface will decrease due to fabrication errors, which depend on types of tools used and tool radii. • PSF and MTF will deteriorate when tool size increases. • Under wavelengths other than the design wavelength, diffraction efficiency of the diffractive surface will decrease.

  19. Reference [1] D.A.Buralli, G.M.Morris,” Design of a wide filed diffractive landscape lens” Appl. Opt Vol 8(18), pp3950-3959 ,(1989) [2] W. T. Welfford ,“Aberration of optical system” Hilger,Bristol,1986,pp226-239 [3] W. A. Kleinhans, “Aberration of curred zone plates and Fresnel lens” Appl. Opt Vol 16(6) ,PP.1701-1704 (1977) [4] L.N.Hazra,C. A.Delisle, “Higher order kinoform lenses :diffraction efficiency and aberrational properties” Optical Engineering, Vol 36(5), pp1500-1507(1997) [5] C.G.Blough, “A new approach to chromatic aberrations unilizing ahybrid surface” OSA Diffractive Optics PP272-274(1998) [6] T. Stone, N.George, “Hybrid diffractive-refractive lenses and achromats” ,Appl. Opt Vol 27(14) ,PP. 2960-2971 (1988) [7] M. J. Riedl ,“Diamond-turned diffractive optical elements for the infrared”,SPIE Vol 2540 pp257-269 (1995) [8] 趙麗萍,“折衍混合單透鏡代替雙腔合望遠鏡的設計研究”,光學學報Vol.18 (2), pp223-227,(1998) [9] G.J.Swanson,”Binary optics technology:the theory and design of multi-level diffravtive optical elements,”U.S Department of Commerce. National Technical information Service ,Aug(1989) [10] J.W.Goodman,” fourier optics “,(1996) [11] M.Yamagata,Y. Tanaka,T.Sasano ,”Efficiency Simulation for Diamond-Turned Diffractive Lenses”,Jpn.J.Appl.Phys,Vol.37,pp3695-3700,(1998) [12] J.A.Cox, B.Fritz, T.Werner,” Process error limitations on optics performance “, SPIE vol.1551,(1991) [13] D.A.Buralli,G.M.Morris,J.R.Roger, “Optical performance of holographic kinoforms”, applied optics,vol .28,no.5,(1989)

  20. 阿貝數與折射率 • 無色光學玻璃： 按色散分二類，色散小是冕類(K)，色散大是火石類(F) • 冕類: 氟冕(FK)、輕冕(QK)、磷冕(PK)、重磷冕(ZPK)、冕(K)、重冕(ZK)、鋇冕(BaK)、鑭冕(LaK)、鈦冕(TiK)、特冕(TK) • 火石類: 輕火石(QF)、火石(F)、重火石(ZF)、鋇火石(BaF)、重鋇火石(ZBaF)、鑭火石(LaF)、重鑭火石(ZLaF)、鈦火石(TiF)、冕火石(KF)、特種火石(TF)

  21. Phase shit function Plane wave r λ Z f A(P)以及f(P)分別是此電磁波在P點的振幅及相位，而v是此電磁波的頻率。 電場u(P,t)亦可以複數表示，即 其中 U(P)=A(P)exp[-jf(P)]為u(P,t)的相位(phase)或複數振幅(complex amplitude)。

  22. wavefront aberration 光(ray) 入孔(entrance pupil) y hx x hy h 為歸一化後的瞳孔座標(normalized radius coordinate) h 是經過歸一化(normalized)的光入射高度 為 的瞳孔極座標之相角(polar pupil coordinate) 光軸(optical axis) 物點(object point) Welford提出的波前像差(wavefront aberration) SI～SV分別表示球差(spherical aberration)、慧差(coma)、像散(astigmatism)、場曲(curvature of field )及畸變(distortion)

  23. 焦距的運算

  24. 光學塑膠特性 特　　性 Acryl 系列 Styrene 系列 Polycarbonate 系列 Polyolefin 系列 Polydiethylene glycol bis allylcarbonate (簡稱) PMMA PS PC TPX PEDC(CR-39) 折射率nd 1.492 1.592 1.584 1.466 1.499 阿貝數Vd 58 31 31 61 58 透光率(%） 94 91 92 90 92 飽和吸溼率(%) 2.0 0.1 0.4 0.1 ~ 熱變形溫度(℃) 100 90 130 90 70 成型收縮率(%) 0.3～0.5 0.2～0.4 0.4～0.6 1.5～3 ～14 比重 1.19 1.06 1.20 0.87 1.32 資料來源：光學KOGAKU (Japanese Journal of Optics)、工研院化工所〝光學塑膠之現狀與展望〞

  25. 各種光學塑膠之優缺點 優　　　　點 缺　　　　點 PMMA 極佳透明性與耐候性，成形性良好，光學同向性佳，高阿貝數 吸水率較大，表面硬度低，耐溶劑性差 PS 成形性與機械特性都非常好，吸水率低，高折射率 耐候性差，容易產生雙折射 PC 耐熱性與耐衝擊性極佳，吸水性小 雙折射率大 TPX 比重小，低吸水性，耐化學藥品性佳，近紫外光波長透過範圍最廣 成形收縮率大，鍍膜密著性差 CR-39 耐磨耗性極佳，透光性佳，耐熱性及耐溶劑性極佳 成形收縮率極大，成形時無法完全聚合，殘餘烷基含量達10% 資料來源：化工所〝光學塑膠之現況與展望〞報告、光學KOGAKU(Japanese Journal of Optics)

  26. 玻璃與塑膠之特性比較 玻　　　璃 優　　　點 塑　　　膠 缺　　　點 光學塑膠 材料成本低，重量輕，可以模造(molding)。可大量，快速生產，能承受機械衝擊，裝鏡架可併入模造，設計上很方便 表面硬度低，不耐刮，折射率隨溫度改變，易因壓力產生雙折射，易吸收液體而改變性質，鍍膜不易 折射率 1.5～1.9以上 1.3～1.7 阿貝數 20～65以上 25～55 雙折射 不產生雙折射 會產生雙折射 全光線透過率 85～95 90以上 光譜範圍 370～1500mm以上 400～1100nm 溫度 軟化點溫度:500～700℃ 熱變形溫度：70～130℃ 線性膨脹係數 70～130 X 10-7 玻璃的10倍 比重 2.2～7.3 1.0～1.5 玻璃與塑膠之特性比較 光學塑膠之優缺點 資料來源：化工所〝光學塑膠之現況與展望〞報告、光學KOGAKU(Japanese Journal of Optics)

  27. Transmittance function d(x) d n 0 x : wave在透鏡內所引起的相位延遲 : wave離開透鏡在空氣中所引起的相位延遲 • k為wave number等於2 π /λ，n為透鏡材料的折射率，而繞射面的最大深度為d，在x點上的深度為d(x) (厚度函數) To MTF To MTF/PSF(hybrid lens)

  28. Point spread function Bessel function J0:zero order, first type Bessel function

  29. Modulation Transfer Function 1 MTF 0.5 ：觀察面座標 0 (line/mm)

  30. Test and verify • F/# : 10, F: 50mm aperture : 5mm material : PMMA • Principal wavelength: 0.58756μm • F/# : 10, F: 254 mm, aperture : 25.4mm , material : silica, order:8 • Principal wavelength: 0.6328μm

  31. 可見波長之像差 使用case1 材料為BK7單個doublet 波長為可見光380-790nm時 使用case1 材料為BK7單個hybrid 波長為可見光380-790nm時

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