31. Images & Optical Instruments 影像和光學器材

31. Images & Optical Instruments 影像和光學器材 Images with Mirrors 鏡子成像 Images with Lens 鏡片(透鏡)成像 Refraction in Lenses: The Details 鏡片內的折射：詳情 Optical Instruments 光學器材

How does laser surgery provide permanent vision correction? 雷射手術如何達成永久的視力改善？ Ans: Laser light reshapes cornea to adjust the focal point. 答：雷射光把角膜的形狀改變，以修正其焦點。

Light ray = line (or curve)  wave front. 光線= 垂直於波前的直(或曲) 線。 Geometrical (or ray) optics 幾何(或線) 光學 : Light ray in homegeneous medium = straight line. 均勻介質中的光線 = 直線 Valid when 適用於 L >> . Real image : image location is the point of convergence of actual light rays. 實像：像的位置是真實光線的聚合點。 Can be shown on screen. 可以在螢幕上顯示。 Virtual image : some or all of the light rays that converge to form the image are virtual (straight line extension of the actual rays). 虛像：聚合成像的光線中有一些或全部是虛構的 (真實光線的延伸) 。 Can’t be shown on screen. 不可以在螢幕上顯示。

31.1. Images with Mirrors 鏡子成像 物像 Virtual image 虛像鏡 Short-cut 捷徑: OP = PO Front-to-back reversal. 前後倒置 Right-to-left-handed coord. 右變左手座標

GOT IT 懂嗎 ? 31.1. You stand in front of a plane mirror whose top is at the same height as the top of your head. 你站在一塊上邊跟你的頭頂一樣高的平面鏡子前面。 Approximately how far down must the mirror extend for you to see your full image? 這鏡子大概要往下延伸多少才能讓你看到你的全身？ Ans. Half your height. 答：你身高的一半

Curved Mirrors 彎曲的鏡子 Spherical aberration 球面像差 equal angles 此二角相等 conic sections 圓錐曲線: f = e d Normal 法線 f 焦點鏡軸準線 d 切線 Parabola 拋物線 : e = 1 Parabolic mirror 拋物面鏡 Hubble telescope 哈伯望遠鏡: Spherical mirror (upper) 球面鏡 (上) Parabolic mirror (lower) 拋物面鏡 (下) Spherical aberration is small for paraxial rays. 近軸光線的球面像差較小。

Tactics 31.1: Paraxial Ray Tracing with Spherical Mirrors策略31.1 ：球面鏡的近軸光線追蹤 C F • Ray // axis reflected through focus. // 鏡軸的光線反射後通過焦點。 • Ray through focus reflected // axis. 通過焦點的光線反射後 // 鏡軸。 • Ray striking through mirror mid-point reflects symmetrically. • 打到鏡的中點的光線對稱地反射。 • Ray through mirror center reflects upon itself. • 通過鏡心的光線朝原路反射。

Concave Spherical Mirror 球面凹鏡 實，倒，縮小像虛，正，放大像實，倒，放大像 Bear and image are both in front of the mirror. 熊和像都在鏡前。

Convex Mirrors 凸鏡 Convex mirror.凸鏡 Image always virtual, upright, and reduced in size. 成像一定是虛，正，且縮小。 Wide-angle view. 廣角觀

The Mirror Equation 球面鏡公式 Shaded triangles are similar: 兩有色的三角形相似： h < 0 Mirror Equation 球面鏡公式 R = radius of mirror 鏡子的半徑

Table 31.1. Image Formation with Mirrors: Sign Conventions

Example 31.1. Hubble Space Telescope 哈伯太空望遠鏡 During assembly, a technician stood 3.85 m in front of the concave mirror of the HST. 在組裝時，一個技師站在哈伯太空望遠鏡的凹鏡前 3.85 m 處。 Let the focal length of the telescope be 5.52 m. Find 設望遠鏡的焦距為 5.52 m 。求技師成像的 • (a) the location, and (a) 位置，和 • the magnification (b) 放大率。 • of the technician’s image. A technician standing in front of the Hubble Space Telescope mirror. 一個技師站在哈伯太空望遠鏡前。 ( Virtual image; behind mirror ) 虛像；在鏡後  ( Upright; enlarged 正立；放大 )

Example 31.2. Jurassic Park 侏儸紀公園 Convex side-view mirror: Objects in mirror are closer than they seem. 車旁的後視凸鏡：鏡中的物體比看似的近。 If the curvature radius of the mirror is 12 m and the T. Rex is 9.0 m from the mirror, 如果鏡的曲度半徑是12 m，且暴龍離鏡子 9.0 m ， by what factor does the dinosaur appear reduced in size? 暴龍看來縮小了幾倍？ (image is upright & smaller) 成像正立且較小

31.2. Images with Lens 鏡片(透鏡)成像 Convex lens 凸透鏡 Concave lens 凹透鏡 Thin lens: Light rays bend just once going through the lens. 薄透鏡：光線通過透鏡時祇彎曲一次。

Tactics 31.2: Ray Tracing with Thin Lens策略31.2 ：薄透鏡的光線追蹤 passes thru undeflected 原方向出來 entering // axis // 鏡軸進入 passes thru focus 通過焦點出來 entering thru center 朝中點進入

Lens Images by Ray Tracing 追蹤光線為透鏡成像 倒立，縮小，實像倒立，放大，實像正立，放大，虛像虛像

Getting Quantitative: The Lens Equation來點量化：透鏡方程式收斂的 (f > 0) Shaded triangles similar: 兩個有色的三角形相似：發散的 (f < 0) lens equation 透鏡方程式

Table 31.2. Image Formation with Lens: Sign Convention

GOT IT 懂嗎 ?. 31.3. You look through a lens at this page and see the words enlarged and right-side up. 你用一個透鏡來看這一頁，文字都放大而且正立。 Is the image you observe real or virtual? 你看到的像是實或虛？ Is the lens concave or convex? 透鏡是凹或凸？

Example 31.3. Fine Print 小字印刷 You ‘re using a magnifying glass (converging lens) with a 30-cm focal length to read a telephone book. 你用一個焦距為 30-cm 的放大鏡 (收斂透鏡 )來看電話簿。 How far from the page should you hold the lens in order to see the print enlarged three times? 如果要把字體放大三倍，透鏡應該離頁面多遠？ Image is upright enlarged so it must be virtual. 成像正立且放大，所以一定是虛的。 Hence 因此, h > 0, s < 0.

31.3. Refraction in Lenses: The Details鏡片內的折射：詳情 Refraction at a Curved Surface in the Paraxial Approximation 近軸假定下，彎曲表面的折射 Snell’s law: 史耐爾定律： paraxial approx. 近軸假定  Green line is tangent to lens surface at point A. 綠線與鏡面相切於 A 點。 It merges with segment AB in the paraxial approx. 在近軸假定下，它與 AB 弦融合。 T(BOA): T(BOC): T(BAC): T(IBA): T(BCI): Snell’s law: 史耐爾定律

Example 31.4. Cylindrical Aquarium 圓筒形魚缸 An aquarium consists of a thin-walled plastic tube 70 cm in diameter. 一個魚缸由一條直徑 70 cm 的薄壁塑料管組成。 For a cat looking directly into the aquarium, 對一頭直視魚缸的貓來說， what is the apparent distance to a fish 15 cm from the aquarium wall? 一條離魚缸壁 15 cm 的魚，看似有多遠？ Plastic wall thin  negligible. 塑料壁薄  可忽略 nwater = 1.333 Top view 下視圖

Lenses, Thick & Thin 透鏡，厚和薄的 O1 I1 = O2  I2 LHS 左邊 (O1 I1 ) : RHS 右邊 (O2 I2 ) : Thin lens薄透鏡 (t 0) : Lensmaker’s formula 磨鏡人的公式

Common Lens Types 常見的透鏡形式 平-凸双凸新月凸面平-凹双凹新月凹面

Example 31.5. Plano-Convex Lens 平-凸透鏡 Find an expression for the focal length of the plano-convex lens, given refractive index n and radius R for the curved surface. 已知折射率 n 和曲面半徑 R，求平-凸透鏡的焦距。平-凸 Object on left hand side: 物體在透鏡左方： Object on right hand side: 物體在透鏡右方： same result 結果相同

Lens Aberrations 透鏡像差 Spherical aberration 球面像差 Stopped down  better focus 閉鎖光圈  改善聚焦 Chromatic aberration 色差 : cause 成因 : n = n( ). Minimized by using composite lens. 可用複式透鏡減低之 Astigmatism : cause : different R in different direction. 散光: 成因 : 不同方向同 R。

31.4. Optical Instruments 光學器材 The Eye 眼睛 Corrective power 矯正力 P = 1 / f. [P] = diopter 折(屈)光度 = m1 Myopic (nearsighted) 近視 Corrective: Divergent lens. 矯正：發散透鏡 Hyperopic (farsighted) 遠視 Corrective: Convergent lens. 矯正：收斂透鏡

Application: Laser Vision Correction應用：雷射視力矯正 LASIK procedure 雷射屈光角膜塑形術

Conceptual Example31.6. Contact Lens Mix-Up 搞亂了隱形眼鏡 You and your roommate have gotten your boxes of disposable contact lenses mixed up. 你和你的室友把你們兩盒丟棄式隱形眼鏡搞混了。 One box is marked “1.75 diopter”, the other “+2.5 diopter”. 一盒寫着 “1.75 屈光度” ，另一盒 “+2.5 屈光度 ” 。 You are farsighted and your roommate is nearsighted. 你是遠視而你的室友是近視。 Which lenses are yours? 那些鏡片是你的？ Converging lens correct far-sightedness. 聚焦鏡片糾正遠視  f > 0  P = 1 / f > 0

Making the Connection 連起來 What’s the focal length of your contact? 你的隱形眼鏡的焦距為何?

Example 31.6. Lost Your Glasses! 眼鏡丟了! You’ve lost your reading glasses; 你的老花眼鏡丟了， without them, your eyes can’t focus closer than 70 cm. 沒有它，你的眼睛不能聚焦比70 cm近的東西。 Nonprescription reading glasses come in 0.25-diopter increments. 零售的老花眼鏡的度數以 0.25-diopter 為單位。 Which glasses should you buy so you can focus at the standard 25-cm near point? 你應該買那種眼鏡才能聚焦到標準的 25-cm 近點？ Aim: make object at 25 cm appears to be at 70 cm. 目標：使 25 cm 處的東西好像在 70 cm 。 i.e., s = 25 cm , s = 70 cm. Ans: buy glasses with P = 2.5 diopters 答：買 P = 2.5 屈光度的眼鏡

Cameras 相機 Works like the eye. 像眼睛一樣

Magnifiers & Microscopes 放大鏡和顯微鏡 Closest distance eye can focus is 25 cm (near point). 眼睛能聚焦的最短距離是 25 cm (近點) 。 Angular magnification 角放大率 : Simple magnifier 簡單放大鏡

from eyepiece 離目鏡 Compound microscope 複式顯微鏡 For from objective 離物鏡  For the eyepiece: 目鏡的角放大率 Compound telescope 複式顯微鏡 Overall magnification: 整體放大率：

Telescopes 望遠鏡 Refracting telescope 折射式望遠鏡 Object inverted 像是倒的 Extra diverging eyepiece or set of reflecting prisms needed to get upright image. 必需外加發散式目鏡或一組反射棱鏡才能得正立像。

Reflecting Telescope 反射式望遠鏡 • Advantages 優點 : • No chromatic aberration 無色差 . • Mirror (adjustable) can be much larger (~10m) than lens (~1m). • (可調)鏡子 (~10m) 可以做成比透鏡 (~1m) 大得多。 Only light gathering power is important for astronomical telescopes. 天文望遠鏡祇重視取光能力。 Cassegrain design 卡塞格林式 (large telescopes) 大型望遠鏡光譜儀或其他儀器 Newtonian design 牛頓式 (small telescopes 小型望遠鏡) Giant Magellan Telescope巨大麥哲倫望遠鏡 (2016)

31. Images & Optical Instruments 影像和光學器材