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# 1-1 均勻介質中的光波

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(1)

## 第一章

(2)

### 目錄

 1-1 均勻介質中的光波

 1-2 折射率

 1-3 群速度和群折射率

 1-4 磁場、輻照度和波印亭向量

 1-5 斯涅耳定理和內部全反射 (TIR)

 1-6 菲涅耳方程式

 1-7 多重干涉與光學共振器

(3)

(4)

(1)

0

0

## ω − + φ

x

(5)

x

y

### varying electric and magnetic fields which are perpendicular to each other and the direction of propagation, z.

?1999 S.O. Kasap, Optoelectronics (Prentice Hall)

(6)

z Ex = Eosin(ωt　z)

Ex

z

Propagation E

B

k

E and B have constant phase in this xy plane; a wavefront

E

A plane EM wave travelling along z, has the same Ex (or By) at any point in a

given xy plane. All electric field vectors in a given xy plane are therefore in phase.

(7)

### 

，其中 Re 代表實部，因此我們 需擷取計算結束時之任何複數結果的實部。所 以，我們可將寫成

(2) 或

(3)

)]

[exp(

cos

=Re j

0

0

x

x

c

(8)

(4)

0

0

(9)

(5)

(10)

## k

(11)

(6)

r

0

0 2

2 0 2 0

2 2

2 2

2

r

### ∂ ε ε µ

(12)

k

Wave fronts

r

E k Wave fronts

(constant phase surfaces)

z λ

λ

λ

Wave fronts

P

O P

A perfect spherical wave

A perfect plane wave A divergent beam

(a) (b) (c)

(13)

### 

離波源為 r 之任何點的電場可表示為

(7)

(14)

### 

divergence)。腰部越大，發散角越窄，兩者的 關係為

(8)

0

### θ = λ

(15)

y

x Wave fronts

z Beam axis

r Intensity

(a)

(b)

(c) 2wo

θ O

Gaussian

2w

(a) Wavefronts of a Gaussian light beam. (b) Light intensity across beam cross section. (c) Light irradiance (intensity) vs. radial distance r from beam axis (z).

?1999 S.O. Kasap, Optoelectronics (Prentice Hall)

(16)

(17)

(1)

r

0 0

r

(18)

(2)

r

=

=

(19)
(20)
(21)

(22)

(3)

g

(23)

max

### E

max

Wave packet

Two slightly different wavelength waves travelling in the same direction result in a wave packet that has an amplitude variation which travels at the group velocity.

?1999 S.O. Kasap, Optoelectronics (Prentice Hall)

(24)

(4)

(25)

(5) 此式可寫成

(6) 其中

(7)

g

g g

=

g

(26)

g

### n

500 700 900 1100 1300 1500 1700 1900 1.44

1.45 1.46 1.47 1.48 1.49

(27)

(28)

(1)

y y

x

(29)

### A plane EM wave travelling along k crosses an area A at right angles to the direction of propagation. In time ∆t, the energy in the cylindrical volume Av∆t (shown dashed) flows through A .

?1999 S.O. Kasap, Optoelectronics (Prentice Hall)

(30)

(2)

2

0

r

x

2 0) 2 / 1

(

By

2 0 2

0 2

1 2

1

y x

rE B

=

(31)

(3) 給出

(4)

y x r x

r x

r

0 2

0 2

0

2

(32)

(5)

=

2

0

r ×

(33)

(6)

(7)

0

x

2 0

2 0

1

r

= 平均 =

=

r =

2

2 0 3

2 0

0

(34)
(35)
(36)

### (TIR)

(37)

37 n2

z y

O

θi

n1

Ai

λ λ

θr θi

Incident Light Bi Ar

Br

θt θt

λt Refracted Light

Reflected Light kt

At

Bt

B A

B A

A′′

θr ki

kr

A light wave travelling in a medium with a greater refractive index (n1 > n2) suffers reflection and refraction at the boundary.

?1999 S.O. Kasap, Optoelectronics (Prentice Hall)

1

2

(38)

(1) 或

(2)

i

t

t i

sin

sin

2

1

=

=

1 2 2

1

sin sin

t

i = =

(39)

(3)

t

c 1

2

c

(40)

### 1-6 菲涅耳方程式

(41)

41

n2

θi θi n1 > n2

Incident light

θt

Transmitted (refracted) light

Reflected light

kt

θi>θc θc

TIR θc

Evanescent wave

ki kr

(a) (b) (c)

Light wave travelling in a more dense medium strikes a less dense medium. Depending on the incidence angle with respect to θc, which is determined by the ratio of the refractive indices, the wave may be transmitted (refracted) or reflected. (a) θi < θc (b) θi = θc (c) θi

> θc and total internal reflection (TIR).

?1999 S.O. Kasap, Optoelectronics (Prentice Hall)

(a) ； (b) ；(c) 且內部全反射 (TIR)。

θc θc

θ

θ < θ =θ θ >θ

(42)

ki

n2

n1 > n2 θt=90

Evanescent wave

Reflected wave Incident

wave

θi θr

Er,//

Er,⊥

Ei, Ei,//

Et,⊥

(b) θi > θc then the incident wave suffers total internal reflection.

However, there is an evanescent wave at the surface of the medium.

z y

x into paper θi θr

Incident wave

θt

Transmitted wave

Ei,//

Ei,⊥

Er,//

Et,//

Et, Er,⊥

Reflected wave

kt

kr

Light wave travelling in a more dense medium strikes a less dense medium. The plane of incidence is the plane of the paper and is perpendicular to the flat interface between the

(a) θi < θc then some of the wave is transmitted into the less dense medium. Some of the wave is reflected.

Ei,⊥

(43)

(1a)

(1b)

1 2 / n

=

2 / 1 2

2

2 / 1 2

2

, 0

, 0

i i

i i

i r

2 / 1 2

2 ,

0 , 0

i i

i i

t

(44)

(3)

i i

i i

i r

2 2

/ 1 2 2

2 2

/ 1 2 2

//

, 0

//

, 0

//

2 / 1 2 2

2 //

, 0

//

, 0

//

i i

i i

t

//

//

//

(45)

### 

例如，將垂直入射角 的情況放進菲涅耳 方程式，我們發現

(4)

i

2 1

2 1

//

(46)

(5)

//

p

2 p

(47)

47

//

i

1

2

//

//

(b)

60 120 180

Incidence angle,

### θ

i

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 10 20 30 40 50 60 70 80 90

|

// |

|

|

θc

θp

Incidence angle,

### θ

i

(a)

Magnitude of reflection coefficients Phase changes in degrees

0 10 20 30 40 50 60 70 80 90 θc

θp

TIR

0

−60

−120

−180

?1999 S.O. Kasap, Optoelectronics (Prentice Hall)

1

2

//

i

//

(48)

(6)

(7)

i

c

i

i

2 1 2

2

//

//

i n

1 [sin ]

tan 1

2 1 2 2

//

=

+

(49)

49

### 

(evanescent wave) 且沿著 z 前進，其電場隨 著進入介質2而減少，即

(8)

### 

(attenuation coefficient)，

(9)

c

i

2

,

t

α y

iz

i i

iz

2

2 1 2

2

2 1 2

2 2 sin 1

⎟⎟

⎜⎜

= i

(50)

//

### vs. angle of incidence θ for n = 1.00 and n = 1.44.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0 10 20 30 40 50 60 70 80 90

p r//

r

Incidence angle,

### θ

i External reflection

(51)
(52)
(53)

(10)

r

0

2 0

0

r

(54)

(11)

//

2 2

, 0

2 ,

0

i r

2

2 //

//

, 0

2 //

, 0

//

i r

(55)

(12)

2

2 1

2 1

// ⎟⎟

⎜⎜

+

=

=

=

(56)

(13) 或

(14)

2 1

2 2

, 0 1

2 , 0

2

i 2 t

//

1 2 2

//

, 0 1

2 //

, 0 2

//

i t

2 1 //

(57)
(58)
(59)
(60)
(61)
(62)
(63)
(64)

1

2

3

(65)
(66)
(67)

(68)

(1)

### m λ

(69)

A

B L

M1 M2 m = 1

m = 2

m = 8

Relative intensity

υ δυm

υm υm + 1

υm - 1

(a) (b) (c)

R ~ 0.4 R ~ 0.8

1 υf

Schematic illustration of the Fabry-Perot optical cavity and its properties. (a) Reflected waves interfere. (b) Only standing EM waves, modes, of certain wavelengths are allowed in the cavity. (c) Intensity vs. frequency for various modes. R is mirror reflectance and lower R means higher loss from the cavity.

?1999 S.O. Kasap, Optoelectronics (Prentice Hall)

(70)

### 

(resonant frequencies)，

(2)

m

f f

m

(71)

### 

2 cavity cavity = | E |

I R = r 2

) ( sin 4

) 1

( 2 2

0 cavity

R kL R

I I

+

=

(72)

(4)

m

m

m =

= ;

) 1

( 2

0

max

(73)

### 

width)[1] 所定義之個別模強度在最大值一 半時的全寬度 (FWHM)。當 時，它的值 可直接由下列計算出

(5)

數。

m

2

1

f m

(74)

(6)

incident

I

cavity

cavity

cavity

### I

) ( sin 4

) 1

(

) 1

(

2 2

2 incident

d transmitte

R kL R

I R

I +

=

(75)

L λ λm

λm - 1 Fabry-Perot etalon

Partially reflecting plates

Output light Input light

Transmitted light

### Transmitted light through a Fabry-Perot optical cavity.

?1999 S.O. Kasap, Optoelectronics (Prentice Hall)

(76)
(77)
(78)

### 用

(79)

θi

n2

n1 > n2

Incident light

Reflected light θr

∆z

Virtual reflecting plane

Penetration depth, δ

z y

The reflected light beam in total internal reflection appears to have been laterally shifted by an amount ∆z at the interface.

### A B

?1999 S.O. Kasap, Optoelectronics (Prentice Hall)

(80)

θi

n2

n1 > n2

Incident light

Reflected light

θr

### When medium B is thin (thickness d is small), the field penetrates to the BC interface and gives rise to an attenuated wave in medium C.

z y

d n1

A B C

(81)

81 Incident

light Reflected

light

θi > θc TIR

(a)

Glass prism

θi > θc FTIR

(b) n1

n1

n2 n

1

B = Low refractive index transparent film ( n

2)

C A A

Reflected

Transmitted

(a) A light incident at the long face of a glass prism suffers TIR; the prism deflects the light.

(b) Two prisms separated by a thin low refractive index film forming a beam-splitter cube.

The incident beam is split into two beams by FTIR.

Incident light

?1999 S.O. Kasap, Optoelectronics (Prentice Hall)

(b) 兩稜鏡以低折射率的薄膜分開以形成一個光束分光器立方體，入射 光束以FTIR分成兩道光束。

(82)

(83)

(1)

0 0

0

x

### = ω −

(84)

P Time

Q Field

υ Amplitude

υο

−∞

Time

(a)

υ Amplitude

υο

υ = 1/∆t Time

(b)

P Q

l = c∆t

Space

∆t

(c)

υ Amplitude

(a) A sine wave is perfectly coherent and contains a well-defined frequency υo. (b) A finite wave train lasts for a duration ∆t and has a length l. Its frequency spectrum extends over

υ = 1/∆t. It has a coherence time ∆t and a coherence length l. (c) White light exhibits practically no coherence.

(85)

### 

width) 而隨時間同調長度 而定，並由下列 給出

(2)

### ∆ υ 1

(86)

c

(a)

Time

(b)

A

B

∆t

Interference No interference No interference

Space

c P

Q Source

Spatially coherent source

An incoherent beam

(c)

(a) Two waves can only interfere over the time interval ∆t. (b) Spatial coherence involves

(87)

### 10-1 繞射原理

(88)

Light intensity pattern

Incident light wave

Diffracted beam

Circular aperture

A light beam incident on a small circular aperture becomes diffracted and its light intensity pattern after passing through the aperture is a diffraction pattern with circular bright rings (called Airy rings). If the screen is far away from the aperture, this would be a

(89)

Incident plane wave

New wavefront A secondary wave source

(a) (b)

Another new

wavefront (diffracted)

θ z

(a) Huygens-Fresnel principles states that each point in the aperture becomes a source of secondary waves (spherical waves). The spherical wavefronts are separated by λ. The new wavefront is the envelope of the all these spherical wavefronts. (b) Another possible

wavefront occurs at an angle θ to the z-direction which is a diffracted wave.

?1999 S.O. Kasap, Optoelectronics (Prentice Hall)

89

(90)

(1)

(91)

(2)

) (

==

= y a

y

(

) 0

exp( sin

)

(92)

(3)

θ

2 sin 1

ka

j

### =

(93)

θ

A

ysinθ y

Y

θ

θ = 0 δy

δy z

Screen Incident

light wave

θ

R = Large

θ

c b

Light intensity a

y

y

z

(a) (b)

(a) The aperture is divided into N number of point sources each occupying δy with

amplitude ∝ δy. (b) The intensity distribution in the received light at the screen far away from the aperture: the diffraction pattern

Incident light wave

?1999 S.O. Kasap, Optoelectronics (Prentice Hall)

；(b) 遠離孔徑之屏幕上所接受的光的強度分佈：繞射圖 案。

(94)

(4)

2

sin

2

; 1 ) ( sinc )

0 ( 2 sin

1

2 sin sin 1

)

( 2

2

ka ka

ka a

C

=

=

= I

I

(95)

(5)

(96)

(97)

(6)

### sin =

(98)
(99)
(100)

d

z y

Incident light wave

Diffraction grating

One possible diffracted beam

θ a

Intensity

y

m = 0 m = 1

m = -1 m = 2

m = -2

Zero-order First-order

First-order Second-order Second-order Single slit

diffraction envelope

dsinθ

(a) (b )

(101)

### 

若此量為波長的整數倍，則所有來自成對之狹 縫的波將互為建設性干涉，

(7)

(102)

(8)

i

m

m

i

### λ

(103)

103

Incident light wave

m = 0 m = -1 m = 1

Zero-order First-orde First-order

(a) Transmission grating (b) Reflection grating Incident

light wave

Zero-order First-order

First-order

(a) Ruled periodic parallel scratches on a glass serve as a transmission grating. (b) A reflection grating. An incident light beam results in various "diffracted" beams. The

zero-order diffracted beam is the normal reflected beam with an angle of reflection equal to the angle of incidence.

?1999 S.O. Kasap, Optoelectronics (Prentice Hall)

(104)

(105)

### R

?1999 S.O. Kasap, Optoelectronics (Prentice Hall)

(106)
(107)

1

2

### n

3

B1

A1 A2 A3 A0

C1 B2

B3

B4 B5

C2 C3 B6

Thin film coating of refractive index n2 on a semiconductor device

?1999 S.O. Kasap, Optoelectronics (Prentice Hall)

(108)

L

Fabry-Perot etalon

Output light Input light

θ

θ

θ

θ

monochromatic

source Screen

Screen

k

FP etalon

Fabry-Perot optical resonator and the Fabry-Perot interferometer (schematic)

(109)

n1

n3

n2

Air

Glass substrate

Thin layer

(a)

Glass substrate Thin layer

Prism Laser light

(b)

(a) Light propagation along an optical guide. (b) Coupling of laser light into a thin layer - optical guide - using a prism. The light propagates along the thin layer.

?1999 S.O. Kasap, Optoelectronics (Prentice Hall)

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