zemax 教程

使用ZEMAX®於設計、優化、公差和分析 Design, optimization, tolerancing and analysis using ZEMAX® 摘要 光學設計軟體ZEMAX®的功能討論可藉由使用ZEMAX去設計和分析一個投影 系統來討論，包括使用透鏡陣列 (lenslet arrays) 來建構聚光鏡 (condenser)。 ABSTRACT A discussion of the capabilities of the optical design program ZEMAX® is followed by a discussion of using ZEMAX to design and analyze a projection system to include the condenser made using lenslet arrays. 簡介 ZEMAX 以非序列性 (non-sequential) 分析工具來結合序列性 (sequential) 描光 程式的傳統功能，且為一套能夠研究所有表面的光學設計和分析的整合性套裝軟 體，並具有研究成像和非成像系統中的雜散光 (stray light) 和鬼影 (ghosting) 的能 力，從簡單的繪圖 (Layout)一直到優化和公差分析皆可達成。 根據過去的經驗，對於光學系統的端對端 (end to end)分析往往是需要兩種不 同的設計和分析工具。一套序列性描光軟體，可用於設計、優化和公差分析，而一 套非序列性或未受限制的 (unconstrained) 描光軟體，可用來分析雜散光、鬼影和一 般的非成像系統分析，包括照明系統。 序列性描光程式這個名詞是與定義一個光學系統為一連串表面的工具有關。所 有的光線打到光學系統之後，會依序的從一個表面到另一個表面穿過這個系統。在 定義的順序上，所有的光線一定會交到所有的表面，否則光路將終止。光線不會跳 過任何中間的表面；光線只能打在每一個已定義的表面一次。若實際光線路徑交到 一個表面上超過一次，如使用在二次描光 (double pass) 中的元件，然後在序列性列 表中，必須再定義超過一次的表面參數。 大部份成像光學系統，如照相機鏡頭、望遠鏡和顯微鏡，可在序列性模式中完 整定義。對於這些系統，序列性描光具有許多優點：非常快、非常彈性和非常普 遍。幾乎任何形狀的光學表面和材質特性皆可建構。在成像系統中，序列性描光最 重要的優點為使用簡單且高精確的方法來做優化和分析。序列性描光的缺點，包括 無法追跡所有可能的光路徑 (即鬼影反射) 和許多無法以序列性方式來描述的光學 系統或元件。 １ 非序列性描光最常用來分析成像系統中的雜散光和鬼影，甚致分析照明和其他 非成像系統。在非序列性描光中，光線入射光學系統後，是自由的沿著實際光學路 徑追跡；一條光線可能打到一個物件 (object) 許多次，而且可能完全未打到其他物 件。此外，非序列性方法可用來分析從光學或機構元件產生的表面散射 (scatter)， 和從場內 (in-field) 和場外 (out-of-field) 的光源所產生的表面反射而形成的鬼影成 像。 INTRODUCTION ZEMAX combines the traditional capabilities of sequential ray tracing programs with non-sequential analysis tools in an integrated package capable of studying all facets of optical design and analysis, from initial layout through optimization and tolerancing with the ability to study stray light and ghosting in imaging as well as non-imaging systems. Historically, two different design and analysis tools were required for end to end analysis of optical systems. A sequential ray tracing program for design, optimization and tolerancing, and a non-sequential, or unconstrained, ray tracing program to analyze stray light, ghosting, and for general non-imaging system analysis, including illumination systems. The term sequential ray tracing program refers to a tool that defines an optical system as a sequential series of surfaces. All rays entering the optical system proceed through the system from surface to surface in a defined, sequential order. All rays must intercept all surfaces in this defined order or the ray path will be terminated. Rays can not skip over any of the intermediate surfaces; rays can only strike each defined surface once. If the actual ray path would intercept a surface more than once, such as a component used in double pass, then the surface parameters must be defined more than once in the sequential listing. Most imaging optical systems, such as camera lenses, telescopes, and microscopes are well defined by the sequential model. Sequential ray tracing has many advantages for these systems: it is very fast, very flexible and very general. Optical surfaces of almost any shape and material properties can be modeled. The most important advantage of sequential ray tracing in imaging systems is that it leads to a straightforward and highly accurate method for optimization and analysis. The disadvantages of sequential ray tracing include the inability to trace all possible light paths (i.e. ghost reflections) and there are many optical systems or components which can not be described in a sequential fashion. Non-sequential ray tracing is most often used to analyze stray light and ghosting in imaging systems, as well as to analyze illumination and other non-imaging systems. In non-sequential ray tracing rays entering the optical system are free to follow any real optical path; a ray may strike an object multiple times and may entirely miss other objects. Additionally, non-sequential methods can be used to analyze surface scatter from optical or mechanical components as well as visualizing ghost images formed due to surface reflections from in-field and out-of-field sources. ２ ZEMAX 的功能 ZEMAX 可以用於一個完全序列性模式中、一個完全非序性模式中和一個混合 模式中，混合模式對分析具有大部分序列性而卻有一些元件是作用在非序列性方式 的系統，是相當有用的，如導光管 (light pipes) 和屋頂稜鏡 (roof prisms)。 序列性系統需定義視禓角 (field of view)、波長範圍和表面資料。序列性設計 的最重要參數之一，為系統孔徑 (system aperture)。系統孔徑，常指入瞳 (entrance pupil) 或孔徑光欄 (aperture stop)，它限制可從已定義視場入射光學系統的光線。光 學表面可以是折射、反射或繞射。透鏡可以是由均勻或漸變折射率材質所製成。表 面的下彎 (sag) 可以是球面、圓錐面 (conic)、非球面 (aspheric)或藉由多項式或其他 參數函數來定義。也包含了許多繞射光學元件模型。此外，一個使用者自定表面的 功能，允許設計者以撰寫程式的方式來建構任何實際的表面下彎或相位分佈。 一些功能可以用來分析系統，包括數個系統繪圖 (layouts) 類型、匯出 CAD 格 式的表面資訊功能、光點圖 (spot diagrams)、光扇圖 (ray fan) 和光程差圖、繞射調 變轉移函數 (modulation transfer function, MTF) 和點擴散函數 (point spread function, PSF) 圖、包圍圓 (encircled) 和包圍矩形 (ensquared) 的能量資訊、像差計算 ( 塞德 (Seidel) 和策尼克 (Zernike) )、理想或偏斜 (skew) 高斯光束參數計算、極化描光和 波前傳播工具。 ZEMAX FEATURES ZEMAX can be used in a fully sequential mode, a fully non-sequential mode as well as a hybrid mode which is useful for analyzing systems which are largely sequential with some components which behave in a non-sequential fashion, such as light pipes and roof prisms. Sequential systems are defined by a field of view, wavelength range and surface data. One of the most important parameters of the sequential design is the system aperture. The system aperture, often the entrance pupil or the aperture stop, limits the rays that can enter the optical system from the defined field. The optical surfaces can be refractive, reflective or diffractive. Lenses can be made of homogeneous or gradient index materials. The sag of the surface can be spherical, conic, aspheric, or defined by a polynomial or other parametric function. Many surface models for diffractive optical elements are also included. Additionally, a user defined surface capability allows designers to program virtually any arbitrary surface sag or phase profile. Some of the features available to analyze systems include several forms of system layouts, with the ability to export surface information in CAD format, spot diagrams, ray fan and optical path difference plots, diffraction based modulation transfer function (MTF) and point spread function (PSF) plots, encircled and ensquared energy information, aberration calculations (Seidel and Zernike), ideal or skew Gaussian beam parameter calculation, polarization dependent ray tracing and a wavefront propagation tool. ３ 優化 序列性描光軟體的關鍵功能即是可以快速且精確的優化一個光學設計。主要的 優化技巧是以減幅最小均方根 (damped least squares, DLS) 的演算法為基礎，並使 用主動減幅 (active damping)。此外，ZEMAX 包括全域性優化功能，其以結合減褔 最小均方根過程的優化演算法為基礎。優化是以使系統績效函數 (merit function) 的 總值達到最小為基礎。簡單的說，績效函數為一種對一個理想光學系統的數值描 述。重要的是，績效函數代表光學系統的要求性能。對於既定的設計，可以適當的 選用好幾個預設的績效函數。對於成像系統，績效函數可用來特別地針對減低光學 像差，藉由限制光線在成像面上的延伸，或使從理想球面的系統波前偏差減至最 小。許多其他的優化參數也用來修改標準績效函數或建立一個使用者自定的績效函 數。 當執行優化時，ZEMAX 對任何使用者建構的系統或表面參數，決定最理想的 值。幾乎任何參數，包括曲率、厚度、玻璃特性、非球面係數和視場或波長資料， 皆可設為變數。可以對可接受的參數值範圍內下限制，以確保可以輕易的建構一個 合理的系統。 OPTIMIZATION The key feature of a sequential ray tracing program is the ability to rapidly and accurately optimize an optical design. The primary optimization technique is based on a damped least squares algorithm using active damping. Additionally, ZEMAX includes global optimization capabilities, which combines a genetically based optimization algorithm with the damped least squares process. Optimization is based on minimizing the total value of the system merit function. Simply stated, the merit function is a mathematical description of an ideal optical system. It is important that the merit function represent the desired performance for the optical system. There are several default merit functions available which can be selected as appropriate for a given design. For imaging systems, merit functions are available which specifically aim at reducing the optical aberrations by limiting the spread of rays on the image surface or to minimize the departure of the system wavefront from an ideal sphere. Many other optimization parameters are also available to modify a standard merit function or to create a user defined merit function. During the optimization process, ZEMAX determines the optimal values for any of the user specified system or surface parameters. Almost any parameter including radius, thickness, glass properties, aspheric coefficients, as well as field or wavelength data can be made variable. Limitations can be placed on the range of acceptable parameter values to insure a reasonable system which can be easily built. 公差分析 在完成光學系統的設計之後，執行公差分析是重要的。公差分析為一種統計的 過程，用來有系統的引入缺陷到光學設計中，以決定任何系統參數的誤差對整體而 ４ 言，如何影響系統性能。公差分析是必須的，因為沒光學元件是光滑的，或者當設 計好時，可以精確的組裝系統。公差分析可用來決定每個系統參數的可接受值範 圍。這個資訊之後可以用來決定任何系統的可能性、以公差範圍內來製造、在指定 的性能水平之上工作。ZEMAX 包括一個廣泛的、完整的公差分析演算法，允許設 計者自由完成任何光學設計的公差。可以決定出相對於任何性能尺寸的公差，且可 以包括任何補償因子的影響，甚致機構的部分或光學的部分將被用來組裝，或系統 的使用。 TOLERANCING After the design of the optical system is completed, it is important to perform a tolerance analysis. Tolerancing is a statistical process which is used to systematically introduce defects into the optical design to determine how an error in any of the system parameters impacts the performance of the system as a whole. Tolerancing is necessary because no optical component will be polished or system assembled exactly as designed. The tolerance analysis is used to determine a range of acceptable values for each system parameter. This information can then be use to determine the likelihood of any system, manufactured within the tolerances, working at or above a specified performance level. ZEMAX includes a comprehensive, integrated tolerancing algorithm which allows the designer complete freedom in the tolerance of any optical design. Tolerances can be determined relative to any performance criteria and can include the effects of any compensator, either mechanical or optical which will be used in the assembly or use of the system. 波前傳播 幾何光線追跡為一般用來描述光的傳播通過一個光學系統的方法。如同先前所 描述的，光線追跡對於分析許多光學系統來說，為一種非常精確的方法，然而這個 模型的實用性有一些限制。光線模型的限制是因為光線追跡而產生，每條光線是獨 立的，即，一條光線傳播的路徑是可以完全決定的，而不受其他光線的影響。光線 之間不會發生干涉 (interference)。若光線與一個限制孔徑或遮擋物的表面相交，光 線不是被擋住就是通過，但在其他方面，光線路徑是不受影響的，光線不會發生繞 射現像。ZEMAX 包含了數個光線為基礎的繞射計算，包括 PSF 和 MTF，為包括 單階的 Fraunhofer 計算，波前從近場 (出瞳) 傳播到遠場 (成像面)。這些計算只可以 在成像系統中執行，且在非常接近成像的表面。 當以光線為基礎的方法不適用時，ZEMAX 中的物理光學傳播 (Physical Optics Propagation, POP) 工具可用來分析系統，包括表面非常接近焦點、表面遠離焦點但 接近繞射孔徑 (diffracting apertures) 和平行光的傳播，或行經長距離後的近似平面 波前。 使用 POP，任何波前，包括高斯和混合的高階模態光束、帽蓋形 (top hat) 分佈 或任意的使用者自定波前，皆可以傳播通過任何 ZEMAX 的設計檔案。幾乎支援 ５ 所有表面型態。從任何場點來的波前可插入光學系統中的任何位置 (不會僅能在已 定義的物面位罝)。波前傳播通過每個表面，且相位強度的資訊可以在每個表面做 計算。POP 功能是非常有用的用於空間濾波 (spatial filters)、分析光束成形的光學 或任何其他與干涉和繞射有關的光學系統，如繞射菲涅耳區域平板 (Fresnel Zone Plates)。 WAVEFRONT PROPAGATION Geometrical ray tracing is the method generally used to describe the propagation of light through an optical system. As described previously, ray tracing is a very accurate method for analyzing many optical systems, however there are some limitations to the applicability of this model. The limitations to the ray model arise because in ray tracing, each ray is an independent entity, that is, the path a ray follows is entirely deterministic, it is not affected by other rays. No interference occurs between rays. If rays intersect a surface with a limiting aperture or obscuration, ray is either blocked or it passes through, but the ray path is otherwise unaffected, the rays do not diffract. ZEMAX includes several ray based diffraction calculations, including PSF and MTF, which include a single step Fraunhofer calculation which propagates the wavefront from the near field (exit pupil) to the far field (image). These calculations can only be performed for imaging systems, at surfaces very near the image. The physical optics propagation (POP) tool in ZEMAX is used to analyze systems where the ray based methods are not appropriate, including surfaces very near a focus, surfaces far from focus but near diffracting apertures and for the propagation of collimated, or nearly collimated wavefronts propagating long distances. With POP, any wavefront, including Gaussian and mixed higher mode beams, top hat distribution or any arbitrary user specified wavefront, can be propagated through any ZEMAX design file. Almost all surface types are supported. The wavefront can be inserted at any location in the optical system (not just at the defined object location), from any field point. The wavefront is the propagated through the remaining surfaces and the intensity of phase information can be calculated each surface. POP calculations are very useful for modeling spatial filters, analyzing beam shaping optics or any other optical system which depends on interference or diffraction, such as diffractive Fresnel Zone Plates. 極化光線追跡 光可以光線來類比，並以位置、方向、相位和振幅來表示之。然而，光也與電 場有關。電場的方向與傳播方向互相垂直，且隨著時間的改變，其方向和大小亦會 有所變化。當光線傳播通過一光學系統時，其極化狀態一般並無需有守恆的關係存 在。 當極化光線傳播通過一個光學系統時，極化光線追跡可追蹤光線極化的狀態。 介質間的界面，包括空氣、玻璃和金屬，會導致衰減 (diattenuation) 和延緩 ６ (retardance) 的變化產生，和改變極化橢圓 (polarization ellipse) 的形狀。而入射角和 波長的關係則會減少表面的傳輸。這些在傳輸相位上的變化，稱為極化像差 (polarization aberrations)。這些像差會導致 MTF 和 Strehl ratio 值變小，此外亦會使 得系統性能降低。實際上，這些像差並沒有不同於任何其他成像的像差。ZEMAX 可以在任何類型的光學表面上，完整建構出任何薄膜干涉的鍍膜層。極化光線追跡 的計算，包括薄膜鍍膜層、體積吸收 (volumetric absorption) 和波長與入射角的影 響，能更精確的預測真實系統的性能。 POLARIZATION RAY TRACING Light, as represented by rays is described by a location, direction, phase and amplitude. However, there is also an electric field associated with the light. The electric field is oriented perpendicularly to the direction of propagation and may be time varying in orientation and magnitude. The state of the polarization is generally not conserved as the rays propagate though the optical system Polarization ray tracing tracks the state of polarization of a ray as it propagates through an optical system. Interfaces between media, including air, glass and metals, introduce changes to the diattenuation and retardance, changing the shape of the polarization ellipse. Surface transmission is also reduced as a function of angle of incidence and wavelength. These variations in transmitted phase are referred to as polarization aberrations. These aberrations result in a reduction of the MTF and Strehl ratio, and otherwise degrade system performance. Physically, these aberrations are no different than any other imaging aberration. ZEMAX can fully model any thin film interference coatings placed on any type of optical surface. Polarization ray tracing calculations include the effects of thin film coatings, volumetric absorption, and wavelength and incident angle to more accurately predict true system performance. 非序列性分析 使用非序列性分析，光學元件必須以真實物件來表示，而不是以表面來表示。 並允許以光線追跡的方法來決定物件被光線打到的順序。這不僅包括物件的順序， 也包括物件上的特殊表面或小刻面 (facet) 的順序。當考量控制雜散光、分析成像 系統中的鬼影、設計照明和其他非成像光學系統時，這樣的功能是重要的。 非序列性分析不會被已定義的系統孔徑所限制。任何光分佈種類的光源可以放 置在光學空間中的任何位置。從任何光源發射出的光線可以任何實際有意義的方向 傳播。打到光學或機構元件上的一條特定光線的順序，可藉由光線的位置、目前傳 播的方向和其他元件的位置來決定。被光打到的物件將是沿光線傳播方向上最接近 光源的物件。此外，在非序列性空間中，在每個光線表面交點上，任何光線可以分 裂 (split) 成任何數目的子光線。每條子光線將與父段中的一些能量有關。這樣一 來，便允許追跡所有的折射、反射、散射和繞射能量路徑。建構諸如菲涅耳反射的 影響來分析成像系統的鬼影，和在粗糙的或光滑的機構或光學元件上建構散射面來 ７ 研究雜散光影響皆是必須的。每一條子光線的能量可以藉由極化光線追跡來決定， 這對研究鬼影的相對強度和建構諸如干涉儀 (interferometers) 的系統，包括整形平 板 (shearing plates) ，是很重要的。對於散射特性，使用者可自由的指定任何表面 粗糙的種類，然後分裂子光線的散射分佈模型。 檢測面裝置是用在非序列性分析中。檢測面為奇特的表面，不是平的就是彎曲 的，是用來量測入射在檢測面位置上的能量。可以賦予檢測面表面性質來模仿真實 檢測器的效果，包括用於 Narcissus 分析的反饋 (self-reflection)。可量測包括非同調 或同調發光、同調相位和發光強度。並提供幅射度 (Radiometric) 或光度 (photometric) 資訊。 複雜的篩選功能可用於從指定的光源而來所截取的能量資訊， 包括僅觀看鬼影、折射、反射、散射或繞射能量，並在所指定的物件上做限制。在 追蹤雜散光和鬼影問題上，這些篩選功能 (filters) 是很重要的。所篩選的光線資訊 也可以用來產生以通過篩選功能為基礎的光源光線資料。這個功能可以用於反向光 線追跡的分析。 以上所述為 ZEMAX 的概述。有許多其他的功能還沒有提到。我們現在將應 用 ZEMAX 於特定問題：一個投影系統的設計和分析，包括聚光鏡 (condenser) 和 投影機的設計。 NONSEQUENTIAL ANALYSIS With non-sequential analysis, optical components must be described as true objects, rather than as a collection of surfaces. This allows the ray trace to determine the sequence in which objects are struck by the ray. This includes not only the order of the objects, but also the particular surface or facet on the object. This capability is important when considering controlling stray light, analyzing ghosts in imaging systems, as well as for designing illumination and other non-imaging optical systems. Non-sequential analysis is not constrained by a defined system aperture. Sources of any type of light distribution can be position anywhere in the optical space. Rays launched from any of the sources can propagate in any physically meaningful direction. The order in which a particular ray strikes the optical or mechanical components is determined by the position of the ray, its current direction of propagation, and the position of the other components. The struck object will be that object which is closest to the source along the direction of the ray being propagated. Additionally, within the non-sequential space, any ray can be split into any number of child rays at each ray-surface intersection. Each child ray will be associated with some of the energy in the parent segment. This allows tracking of all refracted, reflected, scattered and diffracted energy paths. This is necessary to model effects such as Fresnel reflection to analyze ghosting in imaging systems as well as modeling scatter from rough or polished mechanical or optical components to study stray light effects. The energy in each child ray can be determined by polarization ray tracing, this is important for studying the relative intensity of ghosts as well as for modeling systems such as interferometers, including shearing plates. For scattering, the ８ user is free to specify any type of surface roughness, and then split the child rays model the scatter distribution. Detector devices are used for non-sequential analysis. Detectors are pixilated surfaces, either flat or curved which are used to measure the energy incident at the detector location. Detectors can be provided with surface properties to mimic the effects of the actual detector, including self-reflection to be used for Narcissus analysis. Measurements include incoherent or coherent irradiance, coherent phase and radiant intensity. Radiometric or photometric information can be provided. Complex filtering can be applied to the captured energy information to include looking at only ghosted, refracted, reflected, scattered or diffracted energy, to limit these based on specific objects, from specific sources. These filters are important in tracking stray light and ghosting problems. The filtered ray information can also be used to generate source ray data based on rays passing the filter. This can be used for reverse ray tracing analysis. The preceding is an overview of ZEMAX. There are many other features which have not been considered. We will now apply ZEMAX to a specific problem: design and analysis of a projection system, to include design of both the condenser and the projector. 投影系統的設計和分析 我們將看到使用 ZEMAX 來設計和分析由一聚光鏡配件所組成的投影系統， 其能在投影鏡頭的輸入端提供均勻的照明。一個 8 mm 長的燈絲光源所發出的光在 經過聚光鏡後，將在底片閘 (film gate) 上形成一均勻的能量分佈。投影機會產生一 個 640 x 480 mm 的影像到距輸出端 2000 mm 遠的螢幕上。兩個次要配件中的每一 個次要配件將可序列性的設計 (優化)，然後將這個系統組合起來，所以能計算出整 個系統的照明和成像性質。 DESIGN AND ANALYSIS OF A PROJECTION SYSTEM We will look at using ZEMAX to design and analyze a projection systems consisting of a condenser assembly to provide uniform illumination at the input of a projection lens. The condenser will provide a uniform distribution of energy at the film gate from an 8 mm long filament source. The projector will provide a 640 x 480 mm image to a screen 2000 mm from the output. Each of the two subassemblies will be designed (optimized) sequentially and then the systems will be combined so the illumination and imaging properties of the full system can be evaluated. 聚光鏡的設計 聚光鏡必須收集從光源發出來的光，並在底片閘 (film gate) 上形成一均勻分佈 的光。此處考慮的聚光鏡將由一準直鏡配件所組成，用來使沿著兩個透鏡陣列中的 第一個透鏡陣列傳播的初始光源能量平行。第一透鏡陣列，為場透鏡陣列 (field array)，用於收