32. Interference & Diffraction 干涉(擾)和繞(衍)射 Coherence & Interference 同調性和干涉 Double-Slit Interference 雙縫干涉 Multiple-Slit Interference & Diffraction Gratings 多縫干涉和繞射光柵 Interferometry 干涉儀 Huygen’s Principle & Diffraction 惠更斯原理和繞射 The Diffraction Limit 繞射限制

How do optical principles ultimately govern the amount of information that can be stored on a DVD? Ans: Dim of bit >  答： 位元的尺寸 > 

32.1. Coherence & Interference 同調性和干涉 Waves are coherent  they maintain a constant phase relation. ( also same freq & wavelength ) 波動同調  他們保持固定的相位關係。(頻率和波長都一樣) Steady interference patterns occur only when waves are coherent. 穩定的干涉型樣祗有在波動同調時才會出現。 Coherence length = typical length of wave train. 同調長度 = 波列的典型長度  not const. : incoherent 不固定：非同調 Waves from different sources cannot be coherent. 來自不同源頭的波不能同調。 Interference occurs only for waves from same source but travelling on different paths. 祗有來自同一源頭，但行走不同路線的波才能干涉  = const. : coherent  固定：同調

Destructive & Constructive Interference 相長干涉和相消干涉 crest meets trough 峯碰上谷 Destructive interference : 相消干涉 ： crest meets crest 峯碰上峯 Constructive interference : 相長干涉 ： Caveat 但書 :

Application : More CD Music 應用：光碟音樂再探 保護膜 金屬資訊層 Beams reflected from pits out of phase with those from flats. 由坑洞處反射的光束與由平滑處反射的異相。 Reflected beams less intense wherever pit is present. 有坑洞的地方反射光束比較弱。 坑洞 透明塑料 雷射光

32.2. Double-Slit Interference 雙縫干涉 Interference fringes 干涉條紋 Bright fringes 亮紋 :  Dark fringes 暗紋 : L >> d <<  :

Example 32.1. Laser Light 雷射光 Two slits 0.075 mm apart are located 1.5 m from a screen. 兩相距 0.075 mm 的細縫離銀幕 1.5 m 。 Laser light shining through the slits produces an interference pattern whose 3rd order bright fringe is 3.8 cm from the screen center. 照着細縫的雷射光產生一個干涉條紋，其第三階的亮紋離銀幕中心 3.8 cm。 Find the light’s wavelength. 求光的波長。 red light from He-Ne laser 氦-氖雷射的紅光

Intensity in the Interference Pattern 干涉條紋的強度 Neglect difference in amplitude 忽略振幅的差異 : small  小 max. at 最大值在

32.3. Multiple-Slit Interference & Diffraction Gratings 多縫干涉和繞射光柵 For a N-slit system N-縫系統 : Condition for constructive interference 相長干涉的條件： Since 因 Condition for destructive interference is 相消干涉的條件： 主峯 次峯 ( m = integer but not multiples of N ) m = 整數但不是 N 的整倍數

Diffraction Gratings 繞射光柵 二階光譜 一階光譜 Diffraction gratings 繞射光柵: several cm’s across with thousands of slits 幾公分內有幾千個縫 Made by photoreducing images of // lines, 制作： 縮小 // 線的照片， or by ruling with diamond stylus on Al-plated glass. 或以鑽石針在鍍鋁玻璃上劃出。 螢幕 光柵 Two types 兩款 : Transmission gratings 穿透式光柵. Reflection gratings 反射式光柵. 入口細縫 光源 Maxima condition 峯值的條件 position of max depends on  for m  0. m  0時，峯值的位置與  有關。 ( m = order of dispersion) ( m = 散射的位階 )  Spectrometer: sharp lines if N large. 光譜儀： N 大時譜線較窄。

Example 32.2. Grating Spectrometer 光柵光譜儀 Light from glowing hydrogen contains discrete spectral lines called H and H, with wavelengths of 656.3 nm & 486.1 nm, respectively. 發亮的氫氣的光包含一些個別的譜線，其中波長 656.3 nm 和 486.1 nm 的叫 H 和 H 。 Find the 1st order angular separation between these wavelengths in a spectrometer that uses grating with 6000 slits per cm. 求這兩個波長在一個每公分 6000 縫的光柵光譜儀中的一階角度分離量。 maxima condition 峯值條件

Resolving Power 鑑別率 Consider an N-slit grating. 考慮一 N-縫光柵 barely resolved 剛可分辨 Consider an N-slit grating. 考慮一 N-縫光柵 The mth order maximum of a spectral line with wavelength  is at angle max given by 波長為  的 m-階峯值譜線的繞射角度 max 為 強度 The closest minimum is at angle min given by 離它最近的谷值的角度 min 為 角位置 For a spectral line with wavelength   , we have 從另一波長為    的譜線可得 These two lines can be resolved only if 這兩條譜線若可分辨，則 or 或 = resolving power of the grating 光柵的鑑別率

Example 32.3. “Seeing” a Double Star 〝看見〞一組雙星 A double star system appears as a single point to a telescope. 一個雙星系统在望遠鏡中看祗是一個光點。 The H line of its massive stationary star is at  = 656.272 nm; 其較重且靜止的那顆星的 H 線在  = 656.272 nm； that of its orbiting companion is Doppler shifted to  = 656.329 nm while moving away from Earth. 繞着它轉的伴星在朝離開地球的方向走動時，該線卜勒移至  = 656.329 nm。 If a spectrometer has 5000 lines, what order spectrum will resolve these H lines? 如果光譜儀有 500 條線，那一階的光譜可以分辨這兩條 H 線？  m = 3

X-Ray Diffraction X光繞射 X-ray:  ~ 0.1 nm = 1 A Criterion for constructive interference : 相長干涉的準則： Bragg condition 布拉格條件

Other Gratings 其他光柵 Anything with regularly spaced structures can act as a diffraction grating. 任何間隔整齊的結構都可以用來作繞射光柵。 E.g. 例, Rainbow seen on underside of CD or DVD. 光碟或數位多功碟底面常見的彩虹。 Acoustic-optic modulators steers light beams using acoustic waves as gratings. 聲光調製器以聲波為光柵來把光束轉向。 Laser printers & digital copiers also use AOMs for photo-printing. 雷射印表機和數位複印機都用聲光調製器來印照片。

GOT IT 懂嗎 ? 32.2. If you increase the number of slits in a grating while keeping the spacing the same, what happens to 如果你增加細縫數目，但維持間隔距離變，下面各項會怎樣： the positions of the intensity maxima in the interference pattern for a given wavelength; 任一波長的干涉型樣中，峯值條紋的位置； (b) the intensity of the maxima; and 峯值條紋的強度；和 (c) the width of the maxima ? 峯值條紋的寬度。 No change 不變 increases 變大 narrower 變窄

32.4. Interferometry 干涉儀 Interferometry 干涉儀: Split light into beams, 將光分成多條光束， send them on different paths, then 把它們送到不同的路徑上，然後 recombine. 重組。 Usage: Very sensitive technique for measuring displacements, time intervals, & other differences in quantities. 用途： 精密測量，如位移，時段等代表差異的量。

Thin Films 薄膜 Light is partially reflected at both surfaces of a thin film  interference. 光在薄膜的兩面都有部份反射  干涉 180 相差 Upon incident on larger n medium 進入 n 較大的介質時 ： reflected wave is inverted 反射波成反相 (  =  ) Upon incident on smaller n medium: 進入 n 較小的介質時 ： reflected wave suffers no phase-shift 反射波無相差 (  = 0 ) Normal incident; constructive interference: 垂直入射； 相長干涉： 入射光 薄膜 E.g., soap film of varying thickness 例：厚度漸增的肥皂泡膜 無相差

Newton’s ring 牛頓環 Anti-reflection coating 反反射膜 : 3 layers with suitable thickness & 三層厚度適中且

Example 32.4. Soap Film 肥皂膜 A wire loop 20 cm high is dipped in soap solution & then held vertically to produce a film with thickness varying from 0 to 1.0 m. 一高 20 cm 的線圈先浸在肥皂水中，然後豎立以形成一片厚度從 0 增至1.0 m 的薄膜。 The film is illuminated with 650-nm light. 現以 650-nm 的光照着薄膜。 How many bright bands appear? 會出現幾條亮帶？ The film’s refractive index is essentially that of water, 薄膜的折射率與水差不多， n = 1.33. Number of bands = mmax 1 = 3 亮帶數目 i.e., m = 1,2,3

The Michelson Interferometer 邁克森干涉儀 Laser Interferometer Gravitational wave Observatory ( LIGO ) 雷射干涉儀重力波觀察站 Multiple reflection in 4-km arms  effective length of 300 km. 4-km 長的臂內多重反射  有効長度為 300 km

Example 32.5. Sandstorm 沙塵暴 A sandstorm has pitted the aluminum mirrors of a desert solar-energy installation, 一個沙塵暴把沙漠裏一座太陽能設施內的鋁製鏡子都打花， and engineers want to know the depth of the pits. 工程師想知道這些凹洞的深度。 They construct a Michelson interferometer with a sample from one of the pitted mirrors in place of one flat mirror. 他們用其中一個打花了的鏡子去置換一部邁克森干涉儀中的平面鏡。 With 633-nm laser light, the interference pattern in figure results. 在 633-nm的雷射光下，得到下圖中的干涉型樣。 What is the approximate depth of the pit? 凹洞的深度大概是多少？ 這一段相當於徑差為  … Extra path length 額外徑長 = 0.2  …所以這一段相當於 ~0.2 。

32.5. Huygen’s Principle & Diffraction 惠更斯原理和繞射 All points on a wave front acts as point sources of spherically propagating wavelets. 波前上每一點都是送出球面波的波源。 New wave front is the surface tangent to all these wavelets. 新的波前就是與這些球面波全部相切的面。

Diffraction 繞射 Plane wave incident on barrier with hole. 平面波入射有洞的屏障。 Small hole ( d <  ) : significant diffraction 小洞 ( d <  ) : 明顯的繞射 Large hole ( d >> ) : negligible diffraction 大洞 ( d >> ) : 幾無繞射 Geometrical optics are valid for systems with dimensions >>  幾何光學在系統的尺寸 >>  時適用

Single-Slit Diffraction 單縫繞射 Light from y cancelled by that from y+a/2 if : 從 y 來的光若被 y+a/2 來的消掉則： or Light from y cancelled by that from y+a/4 if : 從 y 來的光若被 y+a/4 來的消掉則： or … Destructive interference 相消干涉: a >  m = 0 gives  = 0 is the central maximum 得 為中央峯值。 c.f. maxima of multi-slit diffraction 與多縫繞射的峯值比較

Intensity in Single-Slit Diffraction 單縫繞射的強度 Phase difference from points at ends of slit: 從縫兩端來的相差： is the intensity of central maximum. 為中央峯值的強度 Minima at 谷值在 or

Multiple-Slits & Other Diffraction Systems 多縫和其他繞射系統 Slit width not negligible 縫寬不可忽略 Diffraction patterns from light passing sharp edges 光經過鋒利邊沿時的繞射型樣 Straight edge 直邊 Circular aperture with crosshairs 有十字準線的圓形光圈

32.6. The Diffraction Limit 繞射限制 2 far-away incoherent sources illuminating a slit 兩個在遠處的非同調光源照在一個細縫上 2 single-slit diffraction patterns ( but no interference pattern ). 兩個單縫繞射型樣 (無干涉型樣) Large angular separation: diffraction patterns distinct 角度分離量大：繞射型樣清晰 Small angular separation: diffraction patterns overlap 角度分離量小：繞射型樣重叠 Note: angular separation of the sources = that of the patterns =  注： 光源的角度分離量 = 型樣的 = 

一型樣的中央峯值與另一型樣的第一個谷值重叠。 Fully resolved 完全解析 Barely resolved (Rayleigh criterion ) 剛好解析 ( 瑞雷準則 ) Unresolved 不能解析 Rayleigh criterion 瑞雷準則 : Central max. of one pattern coincides with 1st min. of the other. 一型樣的中央峯值與另一型樣的第一個谷值重叠。

1st min. of single slit diffraction is given by 單縫繞射的第一個谷值在 For optical systems  << a  光學系統 Rayleigh criterion for a slit becomes 單縫的瑞雷準則變成 Rayleigh criterion for a circular aperture : 圓形光圈的瑞雷準則： Diffraction limit : minimum size of resolvable object set by diffraction. 繞射限制：繞射所定的最大解析度 (能分辨的物體最小的尺寸)。 Interferometer: combine data from widely separated telescopes to produce a large effective aperture size. 干涉儀：連接相距甚遠的望遠鏡的資料，可得到一個很大的有効光圈。 Commonly used in radio telescopes. 常用於無線電波望遠鏡。

GOT IT 懂嗎 ? 32.3. You try to resolve details within a cell but they look fuzzy even at the highest power of your microscope. 你想看到細胞內的細節，可是它們在你的顯微鏡最大倍數下還是模模糊糊的。 Which of the following might help 以下那一項會有幫助 : substitute an eyepiece with shorter focal length, 把目鏡換焦距較短的， put a red filter over the white light source used to illuminate the microscope slide; or 把紅色濾片蓋在照明載玻片的白光源上；或 (c) put a blue filter over the white light source? 把藍色濾片蓋在白光源上？

Example 32.6. Asteroid Alert 小行星警訊 An asteroid 20106 km away appears on a collision course with Earth. 一個在 20106 km 處的小行星好像會撞上地球。 What’s the minimum size for the asteroid that could be resolved with the 2.4-m-diameter diffraction-limited Hubble Space Telescope, using 550-nm reflected sunlight? 在 550-nm 的反射日光下，直徑 2.4 m，繞射限制的哈伯太空望遠鏡可以看到的最小小行星有多大？ Minimum size for the asteroid that could be resolved : 能分辨的最小小行星：

Application: CD to DVD to Blu-ray 應用：從光碟到數位通用碟到藍光碟 laser: IR 雷射：紅外線 Content: 74 min audio 存量： 74分鐘音樂 laser: red light 雷射：紅光 Content: 2 hr vidio 存量： 2小時電影 laser: blue light 雷射：藍光 Content: 4.5 hr HD vidio 存量： 4.5小時高質電影