1-1 均勻介質中的光波

第一章

光的波性質

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

1-1 均勻介質中的光波

平面電磁波

‡

)

cos(

0

ω − + φ

= E t kz

E

E

z

Direction of Propagation

B

z x

y

k

An electromagnetic wave is a travelling wave which has time

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

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

圖1.1 一個電磁波為一個具有時變電場及磁場的行進波，且二者相 互垂直並垂直傳播方向。

z E_x = E_osin(ωt　z)

E_x

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 E_x (or B_y) at any point in a

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

‡

為 ，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

φ

φ

)]

( exp )

exp(

[ )

,

( z t E

j

j t kz

E

= ^Re φ ω −

)]

( exp [

) ,

( z t E j t kz

E

= Re

ω −

‡

)

cos(

) ,

( r t = E

ω t − k ⋅ r + φ

E

‡

(5)

‡

δ t

δ z t

z δ δ /

ω ₌ υλ

=

= dt k v dz

) 2

( ω = πυ

y

z

k

Direction of propagation

r O

θ

E(r,t)

r′

馬克斯威波方程與發散波

‡

(6)

ε

µ

ε

2 0 2 0

2 2

2 2

2

t E z

E y

E x

E

r

∂

= ∂

∂ + ∂

∂ + ∂

∂

∂ ε ε µ

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)

‡

(7)

)

cos( t kr r

E = A ω −

‡

divergence)。腰部越大，發散角越窄，兩者的 關係為

(8)

θ

2

) 2

( 2 4

w

π

θ = λ

y

x Wave fronts

z Beam axis

r Intensity

(a)

(b)

(c) 2w_o

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)

圖1.5 (a) 高斯光束的波前，(b) 光束橫切面的光強度，(c) 光輻 照度 ( 強度 ) 與光束軸之徑向距離間的關係。

1-2 折射率

‡

(1)

ε

0 0

1 µ ε ε

=

v

‡

(2)

r

n

c

ε

v

1-3 群速度和群折射率

‡

dk

d ω

g

= v

δ k

δω ω

ω + δω ω ? δω δ k

E

E

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)

圖1.6 兩波長稍微不同且在同方向行進的波形成一個具有振幅 變化並以群速度行進的波包。

‡

⎥⎦

⎢⎣ ⎤

⎥ ⎡

⎦

⎢ ⎤

⎣

= ⎡

= λ

π

ω λ ²

) ( n k c

v

‡

(5) 此式可寫成

(6) 其中

(7)

λ λ ω

d n dn

c dk

d

−

=

= )

g

( 介質

v

g g

(

v N

=

c

λ λ d n dn

N

= −

N

n

500 700 900 1100 1300 1500 1700 1900 1.44

1.45 1.46 1.47 1.48 1.49

Wavelength (nm)

1-4 磁場、輻照度和波印亭向量

‡

(1)

[1]

y y

x

B

n B c

E = v =

z

Propagation direction

E

B

k

Area A v ∆t

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)

圖1.8 一平面電磁波沿著k行進並通過一垂直於傳播方向的面積A。在 時間，圓柱體積 Av∆t ( 虛線所示 ) 內的能量流過A。

‡

給出。同樣地，空間中任一區域的磁場 B_y 具 有能量密度 。故 E_x 和 B_y的能量密度相 同，

(2)

2

)

2 / 1

( ε ε

E

2 0) 2 / 1

(

µ

2 0 2

0 2

1 2

1

y x

rE B

ε µ

ε

‡

因此，在時間內 將有 之電磁波體積通 過，此體積內的能量因而被接收。如果S為流 過單位面積的電磁波功率，則

(3) 給出

(4)

∆ t

∆

t v

t A v

∆ t

之能量 單位時間單位體積流過

= S

y x r x

r x

r

E E B

t A

E t

S A ε ε ε ε ε ε

0 2

0 2

0

)

)(

(

v v v

=

∆ =

= ∆

‡

B

E

S

v

ε

ε

‡

(6)

‡

(7)

)

0

sin( t E

E

= ω

2 0

2 0

1

E

S v ε ε

I

n

=

c /

v ε

n

2 0 3

2 0

0

( 1 . 33 10 )

2

1 c nE nE

S = = ×

=

ε

I

1-5 斯涅耳定理和內部全反射

(TIR)

37 n₂

z y

O

θ_i

n₁

A_i

λ λ

θ_r θ_i

Incident Light B_i A_r

B_r

θ_t θ_t

λ_t Refracted Light

Reflected Light k_t

A_t

B_t

B′ A

B A′

A′′

θ_r k_i

k_r

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)

圖1.9 一光波在折射率 較大的介質中行進時於邊界 發生反射及折射。

)

( n

> n

‡

及 ，所以

(1) 或

(2)

B

B B

A ′ = ′ / sin θ A

A B

A ′ = ′ / sin θ

t i

t B t

A

θ

θ

2

1

v

v

′ =

1 2 2

1

sin sin

n n

t

i = =

v v θ

θ

‡

(3)

θ

θ

sin

n n

c

=

θ

1-6 菲涅耳方程式

41

n2

θ_i θ_i ^{n1 > n2}

Incident light

θ_t

Transmitted (refracted) light

Reflected light

k_t

θ_i>θ_c θ_c

TIR θ_c

Evanescent wave

k_i k_r

(a) (b) (c)

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

> θ_c and total internal reflection (TIR).

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

圖1.10 在較密介質中行進的光波入射到一較疏的介質中。隨入射角 對的 關係，由折射率比決定 ，此波可能透射 ( 折射 ) 或反射。

(a) ； (b) ；(c) 且內部全反射 (TIR)。

θc θ_c

θ

θ < θ =θ θ >θ

k_i

n₂

n₁ > n₂ θ_t=90

Evanescent wave

Reflected wave Incident

wave

θ_i θ_r

E_r,//

E_r,⊥

E_i,⊥ E_i,//

E_t,⊥

(b) θ_{i >} θ_c 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 θ_i θ_r

Incident wave

θ_t

Transmitted wave

E_i,//

E_i,⊥

E_r,//

E_t,//

E_t,⊥ E_r,⊥

Reflected wave

k_t

k_r

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) θ_i < θ_c then some of the wave is transmitted into the less dense medium. Some of the wave is reflected.

E_i,⊥

‡

(1a)

‡

(1b)

1 2 / n

n

n

E

2 / 1 2

2

2 / 1 2

2

, 0

, 0

] sin

[ cos

] sin

[ cos

i i

i i

i r

n n E

E

θ θ

θ θ

− +

−

= −

=

⊥

⊥ ⊥

r

2 / 1 2

2 ,

0 , 0

] sin

[ cos

cos 2

i i

i i

t

E n E

θ θ

θ

−

= +

=

⊥

⊥ ⊥

t

‡

‡

(3)

i i

i i

i r

n n

n n

E E

θ θ

θ θ

cos ]

sin [

cos ]

sin [

2 2

/ 1 2 2

2 2

/ 1 2 2

//

, 0

//

, 0

//

− +

−

= −

= r

2 / 1 2 2

2 //

, 0

//

, 0

//

cos [ sin ]

cos 2

i i

i i

t

n n

n E

E

θ θ

θ

−

= +

= t

E

r

t

=

+

‡

(4)

) 0 ( θ

=

2 1

2 1

//

n n

n n

+

= −

= r

r

‡

(5)

//

= 0 r

θ

tan n

=

θ

47

Internal reflection: (a) Magnitude of the reflection coefficients r

and r

vs. angle of incidence θ

for n

= 1.44 and n

= 1.00. The critical angle is 44? (b) The corresponding phase changes φ

and φ

vs. incidence angle.

φ

φ

(b)

60 120 180

Incidence angle,

θ

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

|

r

|

r

θ_c

θ_p

Incidence angle,

θ

(a)

Magnitude of reflection coefficients Phase changes in degrees

0 10 20 30 40 50 60 70 80 90 θ_c

θ_p

TIR

0

−60

−120

−180

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

圖1.12 內反射：(a) 與 時之反射係數 及 的大小對入射角 的關係，臨界角為44^。；(b) 對應的相位改變 及 對入射角的關係。

44 .

1

= 1

n n

= 1 . 00 r

r

θ

φ

φ

‡

(6)

‡

(7)

r

θ

> θ

| r

| = 1

i

i

n

θ φ θ

cos

] [sin

2 tan 1

2 1 2

2

−

⎟ =

⎠

⎜ ⎞

⎝

⎛

φ

E

i n

θ π θ

φ

tan 1

2 1 2 2

//

= −

⎟⎞

⎜⎛ +

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

θ

θ >

) (

exp )

, ,

(

,

y z t e j t k z

E

=

ω −

i i

iz

k

k = sin θ α

2 1 2

2

2 1 2

2 2 sin 1

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ ⎟⎟ −

⎠

⎜⎜ ⎞

⎝

= ⎛ _i

n n

n θ

λ

α π

The reflection coefficients r

and r

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

θ

r_⊥

Incidence angle,

θ

強度、反射比及透射比

‡

(10)

ε

I E

2 0

2

1 v ε

ε E

I =

‡

(11)

R

R

2 2

, 0

2 ,

0

| |

|

|

|

|

⊥

⊥

⊥

⊥

= = r

R

i r

E

E

2 //

//

, 0

2 //

, 0

//

| |

|

|

|

| r

R = =

i r

E

E

‡

(12)

2

2 1

2 1

// ⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛

+

= −

=

= _⊥

n n

n R n

R

R

‡

且透射比定義為

(13) 或

(14)

2 1

2 2

, 0 1

2 , 0

2

| |

|

|

|

|

⊥

⊥

⊥ ⊥

⎟⎟

⎠

⎜⎜ ⎞

⎝

= ⎛

= t

T n

n E

n

E n

i 2 t

//

1 2 2

//

, 0 1

2 //

, 0 2

//

| |

|

|

|

| t

T ⎟⎟

⎠

⎜⎜ ⎞

⎝

= ⎛

= n

n E

n

E n

i t

2 1 //

4 n n

= +

=

= T

T

T

d

Semiconductor of photovoltaic device Antireflection

coating Surface

Illustration of how an antireflection coating reduces the

n

n

n

A B

1-7 多重干涉與光學共振器

‡

(1)

2 λ /

"

"

3 , 2 ,

2 ⎟ = = 1

⎠

⎜ ⎞

⎝

⎛ L m

m λ

A

B L

M₁ M₂ m = 1

m = 2

m = 8

Relative intensity

υ δυ_m

υ_m υ_{m + 1}

υ_{m - 1}

(a) (b) (c)

R ~ 0.4 R ~ 0.8

1 υ_f

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)

圖1.16 法布里-比洛光學腔以及其特性的圖例。(a) 反射波干涉，(b) 只有特定波長的電磁駐波，模態，可容許在腔內，(c) 各種模態的強 度對頻率的關係；R為鏡面反射比，R越小意指由腔的損失越高。

‡

(resonant frequencies)，

(2)

λ υ = c / λ

υ

) 2 /(

2 m ; c L

L

m c

m

⎟ = =

⎠

⎜ ⎞

⎝

= ⎛ υ υ

υ

‡

進一步簡化此式子。經過代數處理後的結果為 (3)

2 cavity cavity = | E |

I R = r ²

) ( sin 4

) 1

( ^{2 2}

0 cavity

R kL R

I I

+

= −

‡

(4)

π m L

k

= k = k

π m L

k

= − ;

) 1

( ²

0

max

R

I I

‡

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)

[1]

數。

δυ

6 .

> 0 R

R R

= −

= ; 1

2

π

δυ υ F

F

f m

‡

的 離開腔成為透射強度 。因此，

(6)

incident

I

) 1

( − R

cavity

I

) 1

( − R

cavity

I

cavity

I

) ( sin 4

) 1

(

) 1

(

2 2

2 incident

d transmitte

R kL R

I R

I − +

= −

L λ λ_m

λ_{m - 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)

圖1.17

1-8 古斯-亨琴移動與光學隧道作

用

θ_i

n2

n1 > n2

Incident light

Reflected light θ_r

∆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)

圖1.18 內部全反射的反射光束顯示在界面上已被橫向位移一個量

∆ z

θ_i

n2

n1 > n²

Incident light

Reflected light

θ_r

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 Incident

light Reflected

light

θ_i > θ_c TIR

(a)

Glass prism

θ_i > θ_c 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)

圖1.20 (a) 在玻璃稜鏡長的面上的入射光歷經TIR，稜鏡偏折光線；

(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。

1-9 時間與空間的同調性

‡

)

sin(

0

t k z

E

E

= ω −

P Time

Q Field

υ Amplitude

υ_ο

−∞ ∞

Time

(a)

υ Amplitude

υ_ο

∆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 υ_o. (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.

‡

width) 而隨時間同調長度 而定，並由下列 給出

(2)

υ

∆

∆ t

∆ t

=

∆ υ 1

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

10-1 繞射原理

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

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)

圖1.24 (a) 惠更斯-菲涅耳原理敘述孔徑上每一點皆成為一個次級子 波 ( 球面波 ) 源，這些球面波前均以 分開，新的波前為所有這些 球面波的波封；(b) 另外的可能波前發生在與方向 z 夾 角的方位 上，其為繞射波。

λ θ

89

‡

則波 Y 對 A 為 異相。因此從之點光源 所發射的波具有一個場 為

(1)

λ π /

= 2 k

) sin exp(

)

( δ θ

δ E ∝ y − jky δ E

θ sin ky

θ

‡

(2)

) (

θ E

∫ =⁼ −

= ^{y a}

y

y jky

C

E

θ

δ

θ

‡

θ

θ θ

θ

2 sin 1

2 sin sin 1

) (

2 sin 1

ka

ka a

Ce E

ka

j

⎟

⎠

⎜ ⎞

⎝

⎛

=

−

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)

圖1.25 (a) 孔徑分成個數目的點源，每個佔據 且其大

小 ；(b) 遠離孔徑之屏幕上所接受的光的強度分佈：繞射圖 案。

δ y δ y

∝

‡

(4)

|

) (

| E θ I

θ β

β θ

θ

θ

2

; 1 ) ( sinc )

0 ( 2 sin

1

2 sin sin 1

)

( ²

2

ka ka

ka a

C

=

=

⎥⎥

⎥⎥

⎦

⎤

⎢⎢

⎢⎢

⎣

⎡ ⎟

⎠

⎜ ⎞

⎝

′ ⎛

= I

I

‡

"

, 2 ,

1

;

sin = m = ± ±

a

m λ

θ

The rectangular aperture of dimensions a × b on the left gives the diffraction pattern on the right.

a

b

‡

(6)

θ )

( a

D = λ

D

θ 1 . 22 λ

sin =

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 )

繞射光柵

‡

"

,

2 ,

1 ,

0

;

sin = m m = ± ±

d θ λ

‡

(8)

θ

θ

"

, 2 ,

1 , 0

; )

sin

(sin − = m m = ± ±

d θ

θ

λ

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)

圖1.29 (a) 玻璃上之週期性平行刻痕當做一透射光柵；(b) 反射光 柵。一入射光束形成各種“繞射”光束，零級繞射光束為垂直的反射光 束，具有等於入射角的反射角。

First order

γ

Normal to grating plane Normal to

face

d

γ

Two confocal spherical mirrors reflect waves to and from each other. F is the focal point and R is the radius. The optical cavity contains a Gaussian beam

Wave front

Spherical mirror Optical cavity

Spherical mirror

A B

L

R

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

2 θ R F

圖1.31 兩個共焦的球面鏡彼此來回相互反射光波。F 為焦點而 R 為半徑；此光學腔含有一個高斯光束。

n

n

n

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)

圖1.32 折射率的薄膜鍍在一半導體元件上。

L

Fabry-Perot etalon

Output light Input light

θ

θ

θ

θ

Broad Lens

monochromatic

source Screen

Screen

k

FP etalon

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

n1

n3

n2

Air

Glass substrate

Thin layer

(a)

Glass substrate Thin layer

Prism Laser light

d = Adjustable coupling gap

(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)

圖1.34 (a) 沿光學波導的光傳播；(b) 雷射光利用一個稜鏡耦合 到一個薄層-光學波導，光沿此薄層傳播。