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11 反反 反反反 反反反 反反 、、、

11 反射、折射、干涉、繞射. Sections 反射 (reflection) 與折射 (refraction) 干涉 (interference) 繞射 (diffraction)

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Page 1: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

11 反射、折射、干涉、繞射

Page 2: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

Sections

1.反射 (reflection) 與折射 (refraction)

2.干涉 (interference)3.繞射 (diffraction)

Page 3: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

11-1 Reflection and Refraction ( 反射與折射 )

n)(refractio sinsin

n)(reflectio

1122

1

1

nn

Page 4: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

˙ The Index of Refraction

˙ The Stealth Aircraft F-117A

折射率

Page 5: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

Chromatic Dispersion – prisms and gratings

Page 7: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

What produces the blue-green of a Morpho’s wing? 

How do colorshifting inks shift colors?

11-2 Interference – ( 干涉 )

Page 8: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

Huygens’ principle

All points on a wavefront serve as point sources of spherical secondary wavelets.After a time t, the new position of the wavefront will be that of a surface tangent to these secondary wavelets.

Fig. 35-2

Page 9: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

Law of Refraction from Huygens’ principle

Page 10: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

1035-Fig. 35-3

Page 11: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

1135-

Index of Refraction:c

nv

1 2 1 1

1 2 2 2ec hg

vt t

v v v

11

22

sin (for triangle hce)

sin (for triangle hcg)

hc

hc

1 1 1

2 2 2

sin

sin

v

v

1 21 2

and c c

n nv v

1 1 2

2 2 1

sin

sin

c n n

c n n

Law of Refraction: 1 1 2 2sin sinn n

Page 12: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

Phase Difference, Wavelength and Index of Refraction

Page 13: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

13

Wavelength and Index of Refraction

35-

nn n

v v

c c n

nn

v c n cf f

n

The frequency of light in a medium is the same as it is in vacuum

Page 14: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

14

Phase Difference

35-

Fig. 35-4

Since wavelengths in n1 and n2 are different, the two beams may no longer be in phase

11 1

1 1

Number of wavelengths in : n

L L Lnn N

n

22 2

2 2

Number of wavelengths in : n

L L Lnn N

n

2 22 1 2 1 2 1Assuming :

Ln Ln Ln n N N n n

2 1 1/2 wavelength destructive interferenceN N

Page 15: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

Ex.11-1 35-1

wavelength 550.0 nmn2=1.600 andL = 2.600 m

Page 16: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

Young’s Experiment

Page 17: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

17

Coherence

35-

Two sources to produce an interference that is stable over time, if their light has a phase relationship that does not change with time: E(t)=E0cos(wt+f)

Coherent sources: Phase f must be well defined and constant. When waves from coherent sources meet, stable interference can occur - laser light (produced by cooperative behavior of atoms)

Incoherent sources: f jitters randomly in time, no stable interference occurs - sunlight

Page 18: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

18

Fig. 35-13

Intensity and phase

35-

0 0

10 0 2

sin sin ?

2 cos 2 cos

E t E t E t

E E E

2 2 2 10 24 cosE E

22 21 1

02 220 0

4cos 4 cosI E

I II E

phase path length

difference difference

2phase path length2

difference difference

2sind

Eq. 35-22

Eq. 35-23

Page 19: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

E1

E2

19

Intensity in Double-Slit Interference

35-

1 0 2 0sin and sinE E t E E t

2 10 24 cosI I

2sin

d

1 1 12 2 2minima when: sin for 0,1,2, (minima)m d m m

12

2maxima when: for 0,1,2, 2 sin

sin for 0,1,2, (maxima)

dm m m

d m m

Page 20: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

20

Intensity in Double-Slit Interference

35-

Fig. 35-12

avg 02I I

Page 21: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

Ex.11-2 35-2

wavelength 600 nmn2=1.5 andm = 1 → m = 0

Page 22: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

Interference form Thin Films

Page 23: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

23

Reflection Phase Shifts

35-

Fig. 35-16

n1 n2

n1 > n2

n1 n2

n1 < n2

Reflection Reflection Phase ShiftOff lower index 0Off higher index 0.5 wavelength

Page 24: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

24

Phase Difference in Thin-Film Interference

35-

Fig. 35-17

Three effects can contribute to the phase difference between r1 and r2.

1. Differences in reflection conditions

2. Difference in path length traveled.

3. Differences in the media in which the waves travel. One must use the wavelength in each medium ( l / n), to calculate the phase.

2

0

Page 25: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

25

Equations for Thin-Film Interference

35-

2

odd number odd number2 wavelength = (in-phase waves)

2 2 nL

½ wavelength phase difference to difference in reflection of r1 and r2

22 integer wavelength = integer (out-of-phase waves)nL

22

n n

1

22

2 for 0,1,2, (maxima-- bright film in air)L m mn

2

2 for 0,1,2, (minima-- dark film in air)L m mn

Page 26: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

26

Color Shifting by Paper Currencies,paints and Morpho Butterflies

35-

weak mirror

looking directly down : red or red-yellow tilting : green

better mirror

soap film

Page 27: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

大藍魔爾蝴蝶

Page 28: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

雙狹縫干涉之強度

Page 29: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

Ex.11-3 35-3 water film

thickness 320 nmn2=1.33

m = 0, 1700 nm, infraredm = 1, 567 nm, yellow-greenm = 2, 340 nm, ultraviolet

Page 30: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

Ex.11-4 35-4 anti-reflection coating

Page 31: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

Ex.11-5 35-5 thin air wedge

Page 32: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

32

Fig. 35-23

Michelson Interferometer

35-

1 22 2 (interferometer)L d d

1

2 (slab of material of thickness

placed in front of ) mL L

L M

Page 33: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

33

Determining Material thickness L

35-

2= (number of wavelengths

in same thickness of air)

a

LN

2 2= = (number of wavelengths

in slab of material)

mm

L LnN

2 2 2- = = n-1 (difference in wavelengths

for paths with and without

thin slab)

m a

Ln L LN N

Page 34: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

34

Diffraction Pattern from a single narrow slit.

11-3 Diffraction and the Wave Theory of Light

36-

Centralmaximum

Side or secondarymaxima

Light

Fresnel Bright Spot.

Brightspot

Light These patterns cannot be explained using geometrical optics (Ch. 34)!

Page 35: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

The Fresnel Bright Spot (1819)

Newton

corpuscle Poisson Fresnel wave

Page 36: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

Diffraction by a single slit

Page 37: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

單狹縫繞射之強度

Page 38: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

Double-slit diffraction (with interference)

Single-slit diffraction

雙狹縫與單狹縫

Page 39: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

3936-

Diffraction by a Single Slit: Locating the first minimum

sin sin2 2

aa

(first minimum)

Page 40: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

4036-

Diffraction by a Single Slit: Locating the Minima

(second minimum)sin sin 24 2

aa

(minima-dark fringes)sin , for 1,2,3a m m

Page 41: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

Ex.11-6 36-1 thin air wedge

Page 42: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

42Fig. 36-7

Intensity in Single-Slit Diffraction, Qualitatively

36-

phase path length2

difference difference

2sinx

N=18 q = 0 q small 1st min.1st side

max.

Page 43: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

43

Intensity and path length difference

36-

Fig. 36-9

12sin

2

E

R mE

R

121

2

sinmEE

22

2

sin m

m m

I EI I

I E

2sina

Page 44: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

44

Here we will show that the intensity at the screen due to a single slit is:

36-Fig. 36-8

Intensity in Single-Slit Diffraction, Quantitatively

2

sin (36-5)mI I

1where sin (36-6)

2

a

, for 1, 2,3m m In Eq. 36-5, minima occur when:

sin , for 1,2,3

or sin , for 1,2,3

(minima-dark fringes)

am m

a m m

If we put this into Eq. 36-6 we find:

Page 45: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

Ex.11-7 36-2

1, 1, 2,3,

2m m

Page 46: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

46

Diffraction by a Circular Aperture

36-

Distant point source, e,g., star

lens

Image is not a point, as expected from geometrical optics! Diffraction is responsible for this image pattern

dq

Light

q

a

Light

q

a

sin 1.22 (1st min.- circ. aperture)d

sin 1.22 (1st min.- single slit)a

Page 47: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

47

Rayleigh’s Criterion: two point sources are barely resolvable if their angular separation qR results in the central maximum of the diffraction pattern of one source’s image is centered on the first minimum of the diffraction pattern of the other source’s image.

Resolvability

36-

Fig. 36-11

small1sin 1.22 1.22 (Rayleigh's criterion)

R

R d d

Page 48: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

Ex.11-8 36-3

D = 2.0 mmd = 1.5 mm

Page 49: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

11-4.9 Diffraction – ( 繞射 )

Why do the colors in a pointillism painting change with viewing distance?

Page 50: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

Ex.11-9 36-4

d = 32 mmf = 24 cmλ= 550 nm

Page 51: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

51

Diffraction by a Double Slit

36-

Two vanishingly narrow slits a<<l

Single slit a~l

Two Single slits a~l

2

2 sincos (double slit)mI I

sind

sina

Page 52: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

Ex.11-10 36-5

d = 32 mmf = 24 cmλ= 550 nm

2

sin

sin for 0,1,2,

a

d m m

Page 53: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

53

Diffraction Gratings

36-

Fig. 36-18 Fig. 36-19

sin for 0,1,2 (maxima-lines)d m m

Fig. 36-20

Page 54: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

54

Width of Lines

36-

Fig. 36-22

hwsin , sin hw hwNd

hw (half width of central line)Nd

hw (half width of line at )cosNd

Fig. 36-21

Page 55: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

55

Separates different wavelengths (colors) of light into distinct diffraction

lines

Grating Spectroscope

36-

Fig. 36-23

Fig. 36-24

Page 56: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

Compact Disc

Page 57: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

57

Optically Variable Graphics

36-

Fig. 36-27

Page 58: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

58

Gratings: Dispersion

36-

sind m

Differential of first equation (what change in angle does a change in wavelength produce?)

Angular position of maxima

cosd d md

For small angles

cosd m and d d

cos

m

d

(dispersion defined)D

(dispersion of a grating) (36-30)cos

mD

d

Page 59: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

59

Gratings: Resolving Power

36-

hw cosNd

Substituting for q in calculation on previous slide

Rayleigh's criterion for half-width to resolve two lines

hw

mN

R Nm

avg (resolving power defined)R

(resolving power of a grating) (36-32)R Nm

Page 60: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

60

Dispersion and Resolving Power Compared

36-

Page 61: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

61

X-rays are electromagnetic radiation with wavelength ~1 Å = 10-10 m (visible light ~5.5x10-7 m)

X-Ray Diffraction

36-

Fig. 36-29

X-ray generation

X-ray wavelengths to short to be resolved by a standard optical grating

1 1 1 0.1 nmsin sin 0.0019

3000 nm

m

d

Page 62: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

62

d ~ 0.1 nm

→ three-dimensional diffraction grating

Diffraction of x-rays by crystal

36-

Fig. 36-30

2 sin for 0,1, 2 (Bragg's law)d m m

Page 63: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

6336-

Fig. 36-31

X-Ray Diffraction, cont’d

2 050 045 or 0.2236

20

ad a d a

Page 64: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)

Structural Coloring by Diffraction

Page 65: 11 反射、折射、干涉、繞射. Sections  反射 (reflection) 與折射 (refraction)  干涉 (interference)  繞射 (diffraction)