2tan1

‡

顯示當 時， ，但其相位變化為

(6)

‡

對於 分量，其相位變化 為

(7)

r

⊥

θ

_i

> θ

_c

| r

_⊥

| = 1

i

i

n

θ φ θ

cos

]

49

‡

當 時，介質2中的電場，為一接近於界面 且沿 z 方向行進的波，此波稱為消散波

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

(8)

‡

式中 為入射波沿 z 軸之波向量，而 為電場透射到介質2中的衰減係數

(attenuation coefficient)，

(9)

c

i

θ

θ >

) (

exp )

, ,

(

²

,

y z t e j t k z

E

_{t ⊥}

=

^{−α y}

ω −

_iz

i i

iz

k

k = sin θ α

2

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

_p r_//

r_⊥

Incidence angle,

θ

_i External reflection

強度、反射比及透射比

‡

對於一個光波在相對介電係數為 的介質中 以速度 v 行進，其光強度 以電場振幅 定義 為

(10)

ε

r

I E

₀

2 0

2

0

1 v ε

_r

ε E

I =

‡

反射比 (reflectance)R是量測反射光對入射光 的強度比，而分別為平行及垂直於入射面的電 場分量定義。反射比 和 定義為

(11)

R

⊥

R

_//

2 2

, 0

2 ,

0

| |

|

|

|

|

⊥

⊥

⊥

⊥

= = r

R

i r

E

E

₂

2 //

//

, 0

2 //

, 0

//

| |

|

|

|

| r

R = =

i r

E

E

‡

反射比由下式簡單地給出

(12)

2

2 1

2 1

// ⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛

+

= −

=

= _⊥

n n

n R n

R

R

‡

對垂直之入射而言，入射和透射光束為垂直，

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

₁

n

₂

n

₃

A B

1-7 多重干涉與光學共振器

‡

由於鏡面上的電場 ( 假設為金屬鍍膜 ) 必須為 零，因此只能用半波長， ，的整數 m 個 去合適此腔的長度 L ，

(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。

‡

只要光頻率 v 和波長 的關係為 ，則這 些模所對應的頻率 為腔的共振頻率

(resonant frequencies)，

(2)

λ υ = c / λ

υ

m

) 2 /(

2 m ; c L

L

m c

_{f f}

m

⎟ = =

⎠

⎜ ⎞

⎝

= ⎛ υ υ

υ

‡

當我們知道腔內的場，我們就可計算強 度 。此外，我們可用反射比

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

+

= −

‡

這些位在滿足 之 的峰值，直接導 致原直觀所推導的式。對於這些共振的，上式 給出

(4)

π m L

k

_m

= k = k

_m

π m L

k

_m =

= − ;

) 1

( ²

0

max

R

I I

‡

法布里-比洛標準具的頻譜寬度 (spectral

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]

頻譜寬度又叫條紋寬度，而為條紋的級

數。

δυ

m

6 .

> 0 R

R R

= −

= ; 1

2

π

1

δυ υ F

F

f m

‡

如果入射的光強度為 ，則有 分數的光 進到腔內以建立式中的 ，並有分數

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

‡

當我們以一個純的正弦波來表示一個電磁波 時，例如用

)

(1)

sin(

_{0 0}

0

t k z

E

E

_x

= 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.

‡

此 範圍為該波列的頻譜寬度 (spectral

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

‡

考慮一個任意的方向 ，並考慮在 y 之任意點 光源所發射的波 (Y) 對在 y = 0 之點光源所發 射的波 (A) 的相位，如果 k 為波向量，

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

)

在文檔中 1-1 均勻介質中的光波 (頁 48-90)