在進入『矩陣光學』之前，先作個簡單的總結。相信讀者已經清楚明白維基百科

幾何光學

高斯光學

幾何光學中研究和討論光學系統理想成像性質的分支稱為高斯光學 ，或稱近軸光學。它通常只討論對某一軸線(即光軸)具有旋轉對稱性的光學系統。如果從物點發出的所有光線經光學系統以後都交於同一點，則稱此點是物點的完善像。

光學影像

如果物點在垂軸平面上移動時，其完善像點也在垂軸平面上作線性移動，則此光學系統成像是理想的。可以證明，非常靠近光軸的細小物體，其每個物點都以很細的、很靠近光軸的單色光束被光學系統成像時，像是完善的。這表明，任何實際的光學系統(包括單個球面、單個透鏡)的近軸區都具有理想成像的性質。

為便於一般地了解光學系統的成像性質和規律，在研究近軸區成像規律的基礎上建立起被稱為理想光學系統的光學模型。這個模型完全撇開具體的光學系統結構，僅以幾對基本點的位置以及一對基本量的大小來表征。

根據基本點的性質能方便地導出成像公式，從而可以了解任意位置的物體被此模型成像時，像的位置、大小、正倒和虛實等各種成像特性和規律。反過來也可以根據成像要求求得相應的光學模型。任何具體的光學系統都能與一個等效模型相對應，對於不同的系統，模型的差別僅在於基本點位置和焦距大小有所不同而已。

高斯光學的理論是進行光學系統的整體分析和計算有關光學參量的必要基礎。

利用光學系統的近軸區可以獲得完善成像，但沒有什麼實用價值。因為近軸區只有很小的孔徑(即成像光束的孔徑角)和很小的視場(即成像範圍)，而光學系統的功能，包括對物體細節的分辨能力、對光能量的傳遞能力以及傳遞光學資訊的多少等，正好是被這兩個因素所決定的。要使光學系統有良好的功能，其孔徑和視場要遠比近軸區所限定的為大。

當光學系統的孔徑和視場超出近軸區時，成像質量會逐漸下降。這是因為自然點發出的光束中，遠離近軸區的那些光線在系統中的傳播光路偏離理想途徑，而 不再相交於高斯像點(即理想像點)之故。這時，一點的像不再是一個點，而是一個模糊的彌散斑；物平面的像不再是一個平面，而是一個曲面，而且像相對於物還 失去了相似性。所有這些成像缺陷，稱為像差。

有關光學系統的一些要求

一個光學系統須滿足一系列要求，包括：放大率、物像共軛距、轉像和光軸轉折等 高斯光學要求；孔徑和視場等性能要求，以及校正像差和成像質量等方面的要求。這些要求都需要在設計時予以考慮和滿足。因此，光學系統設計工作應包括：對光 學系統進行整體安排，並計算和確定系統或系統的各個組成部分的有關高斯光學參量和性能參量；選取或確定系統或系統各組成部分的結構形式並計算其初始結構參 量；校正和平衡像差；評價像質。

像差與光學系統結構參量(如透鏡厚度、透鏡表面曲率半徑等)之間的關係極其複雜，不可能以具體的函數.式表達出來，因而無法採用聯立方程式之類的辦法直接由像差要求計算出系統的精確結構參量。現在能做到的是求得滿足初級像差要求的解。

初級像差是實際像差的近似表示，僅在孔徑和視場較小時能反映實際的像差情況，因此，按初級像差要求求得的解只是初始的結構參量，需對其進行修改才能 達到像差的進一步校正和平衡，在這一過程中，傳統的做法是根據追跡光線得到的像差數據及其在系統各面上的分布情況，進行分析、判斷，找出對像差影響大的參 量，加以修改，然後再追跡光線求出新的像差數據加以訐價。如此反覆修改 ，直到把應該考慮的各種像差都校正和平衡到符合要求為止。這是一個極其繁複和費時很 多的過程。

 

詞條之內容。可以知道『光線追跡』

Ray tracing (physics)

In physics, ray tracing is a method for calculating the path of waves or particles through a system with regions of varying propagation velocity, absorption characteristics, and reflecting surfaces. Under these circumstances, wavefronts may bend, change direction, or reflect off surfaces, complicating analysis. Ray tracing solves the problem by repeatedly advancing idealized narrow beams called rays through the medium by discrete amounts. Simple problems can be analyzed by propagating a few rays using simple mathematics. More detailed analyses can be performed by using a computer to propagate many rays.

When applied to problems of electromagnetic radiation, ray tracing often relies on approximate solutions to Maxwell’s equations that are valid as long as the light waves propagate through and around objects whose dimensions are much greater than the light’s wavelength. Ray theory does not describe phenomena such as interference and diffraction, which require wave theory (involving the phase of the wave).

Technique

Ray tracing works by assuming that the particle or wave can be modeled as a large number of very narrow beams (rays), and that there exists some distance, possibly very small, over which such a ray is locally straight. The ray tracer will advance the ray over this distance, and then use a local derivative of the medium to calculate the ray’s new direction. From this location, a new ray is sent out and the process is repeated until a complete path is generated. If the simulation includes solid objects, the ray may be tested for intersection with them at each step, making adjustments to the ray’s direction if a collision is found. Other properties of the ray may be altered as the simulation advances as well, such as intensity, wavelength, or polarization. The process is repeated with as many rays as are necessary to understand the behavior of the system.

Ray tracing of a beam of light passing through a medium with changing refractive index. The ray is advanced by a small amount, and then the direction is re-calculated.

Optical design

Ray tracing may be used in the design of lenses and optical systems, such as in cameras, microscopes, telescopes, and binoculars, and its application in this field dates back to the 1900s. Geometric ray tracing is used to describe the propagation of light rays through a lens system or optical instrument, allowing the image-forming properties of the system to be modeled. The following effects can be integrated into a ray tracer in a straightforward fashion:

• Dispersion leads to chromatic aberration
• Polarization
  • Crystal optics
  • Fresnel equations
• Laser light effects
• Thin film interference (optical coating, soap bubble) can be used to calculate the reflectivity of a surface.

For the application of lens design, two special cases of wave interference are important to account for. In a focal point, rays from a point light source meet again and may constructively or destructively interfere with each other. Within a very small region near this point, incoming light may be approximated by plane waves which inherit their direction from the rays. The optical path length from the light source is used to compute the phase. The derivative of the position of the ray in the focal region on the source position is used to obtain the width of the ray, and from that the amplitude of the plane wave. The result is the point spread function, whose Fourier transform is the optical transfer function. From this, the Strehl ratio can also be calculated.

The other special case to consider is that of the interference of wavefronts, which, as stated before, are approximated as planes. When the rays come close together or even cross, however, the wavefront approximation breaks down. Interference of spherical waves is usually not combined with ray tracing, thus diffraction at an aperture cannot be calculated.

These techniques are used to optimize the design of the instrument by minimizing aberrations, for photography, and for longer wavelength applications such as designing microwave or even radio systems, and for shorter wavelengths, such as ultraviolet and X-ray optics.

Before the advent of the computer, ray tracing calculations were performed by hand using trigonometry and logarithmic tables. The optical formulas of many classic photographic lenses were optimized by roomfuls of people, each of whom handled a small part of the large calculation. Now they are worked out in optical design software. A simple version of ray tracing known as ray transfer matrix analysis is often used in the design of optical resonators used in lasers. The basic principles of the most frequently used algorithm could be found in Spencer and Murty’s fundamental paper: “General ray tracing Procedure”.^[1]

 

對『光學系統』設計的重要性。

為著方便學習，接續若干篇章，將介紹功能強大好用的『Ipython』 互動 Shell 。

 

IPython provides a rich architecture for interactive computing with:

• A powerful interactive shell.
• A kernel for Jupyter.
• Support for interactive data visualization and use of GUI toolkits.
• Flexible, embeddable interpreters to load into your own projects.
• Easy to use, high performance tools for parallel computing.

To get started with IPython in the Jupyter Notebook, see our official example collection. Our notebook gallery is an excellent way to see the many things you can do with IPython while learning about a variety of topics, from basic programming to advanced statistics or quantum mechanics.

To learn more about IPython, you can watch our videos and screencasts, download our talks and presentations, or read our extensive documentation. IPython is open source (BSD license), and is used by a range of other projects; add your project to that list if it uses IPython as a library, and please don’t forget to cite the project.

IPython supports Python 2.7 and 3.3 or newer. Our older 1.x series supports Python 2.6 and 3.2.

 

，權充利用『IPython』 notebook 筆記本操作『SymPy』、『NumPy』、『SciPy』、…… 入門之導引。希望讀者能充分掌握善用工具也！！