Cladding

Core z

y

r φ

Fiber axis

The step index optical fiber. The central region, the core, has greater refractive index than the outer region, the cladding. The fiber has cylindrical symmetry. We use the coordinates r, φ , z to represent any point in the fiber. Cladding is

normally much thicker than shown.

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

圖2.12 步級折射率光纖 ( 示意圖 )。中央區，核心，具有比外圍區，

包層，大的折射率。由於光纖為圓柱對稱，故其內任何一點 P 皆以 r 、 及 z 等座標表示。包層通常比所顯示的為厚。

ϕ

‡ 本質上此為圓柱形的介質波導，其內部核心介 質的折射率為 n1 比外部包層介質的折射率 n2 大。歸一化折射率差(normalized index

difference) 定義為 ^∆

‡ 一個沿著光纖的導引LP模可用一個沿 z 的電場 分佈 的傳播來表示，此電場分佈 ( 或圖 樣 ) 在垂直於纖軸的平面上，因此只和 r 和 相關而與 z 無關。此外，由於存在著兩個邊 界，所以它的特性是由 和 m 兩個整數決定；

因此在一個LP模中的傳播電場分佈是由 給出，並表示為 。故一個 模可用

(1)

) , ( r ϕ

E ϕ

l

) , ( r ϕ E

_lm

LP

lm

LP

_lm

) (

exp )

,

LP E ( r j t z

E = _lm ϕ ω − β _lm

Fiber axis 1

2

3

4

5

Skew ray 1

3 2

4 5 Fiber axis

1

2

3 Meridional ray

1, 3

2

(a) A meridiona ray always

crosses the fibe axis.

(b) A skew ray does not have to cross the fiber axis. It zigzags around the fiber axis.

Illustration of the difference between a meridional ray and a skew ray.

Numbers represent reflections of the ray.

Along the fiber

Ray path projected on to a plane normal to fiber axis

Ray path along the fiber

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

圖2.13 子午光線與斜光線之間的差異的說明。數字代表光線 的反射。

E

r E₀₁

Core

Cladding

The electric field distribution of the fundamental mod in the transverse plane to the fiber axis z. The light intensity is greatest at the center of the fiber. Intensity patterns in LP

₀₁

, LP

₁₁

and LP

₂₁

modes.

(a) The electric field of the fundamental mode

(b) The intensity in the fundamental mode LP

₀₁

(c) The intensity in LP

₁₁

(d) The intensity in LP

₂₁

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

圖2.14 基模在垂直於光纖軸之橫向平面上的電場分佈。最大的 光強度位在光纖的中央。所示為LP₀₁、LP₁₁ 及 LP₂₁ 模的強度圖 樣。

‡ 一般單模光纖具有遠小於較多模光纖的核心半 徑以及較小的 。如果光源波長 足夠小，使 得 V 超過2.405時，單模光纖將變成多模，即 較高的模也將貢獻傳播。讓波長超過而使光纖 變成單模的截止波長 可由下式給出

(4)

∆ λ

λ

c

405 .

2 )

2 (

_{2 1/2}

2 2

1 off

-cut

= a n − n =

V

λ

c

π

0 1 2 3 4 5 6

V b

1

0 0.8

0.6

0.4

0.2

LP

₀₁

LP

₁₁

LP

₂₁

LP

₀₂

2.405

Normalized propagation constant b vs. V-number for a step index fiber for various LP modes.

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

圖2.15 一步級折射率光纖之各種LP模的歸一化傳播常數 b 對 V 數目的曲線圖。

‡ 因LP模的傳播常數隨波導特性以及光源波長而定，所以方便上採

2-4 數值孔徑

Cladding

α < α_max Core

A B

θ < θ_c

A B

θ > θ_c

α > α_max

n

₀

n

₁

n

₂

Lost

Propagates

Maximum acceptance angle α

_max

is that which just gives total internal reflection at the core-cladding interface, i.e.

when α = α

_max

then θ = θ

_c

. Rays with α > α

_max

(e.g. ray B) become refracted and

penetrate the cladding and are eventually lost.

Fiber axis

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

圖2.16 最大可接受角 為剛好在核心-包層的界面上發生內部 全反射，即當 時， 的光線 ( 即光線B) 則變成折射而穿透到包層，最後損失掉。

α

max

α

max

α =

, α α

max

θ

θ =

_c

>

2-5 單模光纖內的色散

τ ^t Spread, τ

0 t λ

Spectrum, λ

λ₁ λ_o λ₂

Intensity Intensity Intensity

Cladding Emitter Core

Very short light pulse

v_g(λ₂) v_g(λ₁) Input

Output

All excitation sources are inherently non-monochromatic and emit within a spectrum, λ, of wavelengths. Waves in the guide with different free space

wavelengths travel at different group velocities due to the wavelength dependence of n₁. The waves arrive at the end of the fiber at different times and hence result in a broadened output pulse.

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

圖2.17 所有的激發光源本質上皆為非單色光並發射出一頻譜波 長 。由於折射率 n₁ 為波長的函數，故波導中不同自由波長的波 將以不同的群速度行進。波將以不同的時間到達光纖的末端，因而 導致一個加寬的光脈衝。

λ

∆

‡ 色散通常以單位長度的擴展表示並由下列給出

‡ 在 (1) 式中，由於有限的輸入頻譜， 因而 成為群延遲時間上的擴展。若 為基模的傳播 常數，則由定義

(3)

L τ /

∆ β

01

ω τ β

d d

₀₁

1 =

=

g

g

v

0

1.2 1.3 1.4 1.5 1.6

1.1 -30

20 30

10

-20 -10

λ (µm)

D^m

D^m + D^w

D_w λ₀

Dispersion coefficient (ps km^-1 nm^-1)

Material dispersion coefficient (D_m) for the core material (taken as SiO₂), waveguide dispersion coefficient (D_w) (a = 4.2 µm) and the total or chromatic dispersion coefficient D_ch (= D^m + D_w) as a

function of free space wavelength, λ.

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

圖2.18 核心材料 的材料色散係數 、波導色散係數 以及總或色彩色散係數 為自由空間波長的函 數。

)

(SiO₂ (D_m) (D_m

+

= _m

ch D

D ( D_w)

)

波導色散

‡ 如圖2.17，我們若使用一個具有頻譜 之很短 的光脈衝當作輸入，則因波導色散，輸出光脈 衝單位長度的加寬或色散， ，可由下列求 得

(4)

λ

∆ L

τ /

∆

τ _{= ∆} λ

∆ | D

_w

|

L

‡ 它是隨波導特性 ( 不瑣碎的方式 ) 而定，且在 的範圍，可由下列近似給出

(5)

2.4 5

.

1 < V <

2 2 2

2

2 ) 2

(

984 .

1

cn a

D

_w

N

π

≈

g

色彩色散或總色散

‡ 在一階近似時，此兩個色散效應可簡單地相加 而使單位長度的總色散變成

λ (6)

τ _{= + ∆}

∆ | D

_m

D

_w

|

L

外形與偏振色散效應

‡ 如果 隨波長改變，則光源中不同的波長將有 不同的群速度並經不同的群延遲而導致脈衝的 加寬。外形色散是色彩色散的一部分，因其隨 輸入的頻譜 變化

(7)

∆

λ

∆

τ _{= ∆} λ

∆ | D _p |

L

Core

z

n1 x // x

n1 y // y

Ey

Ex

Ex

Ey

E

∆τ = Pulse spread

Input light pulse

Output light pulse

t

t

∆τ Intensity

Suppose that the core refractive index has different values along two orthogonal directions corresponding to electric field oscillation direction (polarizations). We can take x and y axes along these directions. An input light will travel along the fiber with E_x and E_y polarizations having different group velocities and hence arrive at the output at different times

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

圖2.19 假設核心折射率在相應於電場振盪方向 ( 偏振 ) 的兩個正 交方向上的值不相同；我們讓 x 和 y 軸沿此二方向，則一沿著光纖 行進的輸入光其 Ex 和 Ey 偏振具有不同的群速度因而以不同的時 間到達輸出端。

20

-10 -20

-30 10

1.1 1.2 1.3 1.4 1.5 1.6 1.7 0

30

λ (µm)

D_m

D_w

D_ch = D_m + D_w λ₁

Dispersion coefficient (ps km ^-1 nm^-1)

λ₂

n

r

Thin layer of cladding with a depressed index

Dispersion flattened fiber example. The material dispersion coefficient (D

_m

) for the core material and waveguide dispersion coefficient (D

_w

) for the doubly clad fiber result in a flattened small chromatic dispersion between λ

₁

and λ

₂

_.

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

圖2.20 色散平坦化光纖的例子。核心材料的材料色散係數

以及雙包層光纖的波導色散係數 導致 和 之間小而平坦的 色彩色散。

) (D_m

λ

1

λ

₂

)

( D

_w

2-6 位元率、色散、電以及光的

帶寬

0 t

Emitter

Very short light pulses

Input Output

Fiber

Photodetector Digital signal

Information Information

0 t

~2 τ

_1/2

T

t

Output Intensity Input Intensity

² τ

1/2

An optical fiber link for transmitting digital information and the effect of dispersion in the fiber on the output pulses.

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

圖2.22 一光纖連接以傳送數位訊息以及光纖中的色散對輸出脈衝 的效應。

‡ 脈衝可傳送的最大位元率，或簡稱位元率 B ， 大約為

(1) )

2 /(

1 ∆ τ _{1 / 2}

2 / 1

5 . 0

τ

≈ ∆

B

t Output optical power

∆ τ

1/2

T = 4 σ

1

0.5

0.61 2 σ

A Gaussian output light pulse and some tolerable intersymbol

interference between two consecutive output light pulses (y-axis in relative units). At time t = σ from the pulse center, the relative magnitude is e ^-1/2 = 0.607 and full width root mean square (rms) spread is ∆ τ _rms = 2 σ .

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

圖2.23 高斯輸出脈衝及兩連續輸出脈衝 ( y 軸為相對單位 ) 間之可 容忍的碼間干擾。距脈衝中心為 處的相對大小

為 ，而全寬度均方根(ms) 延展為

a

t = 607

.

2

0

/

1

=

e

−

∆ τ

_rms

= 2 σ

‡ 位元率 B 若以來表示 則需兩連續輸出光脈 衝之波峰間的分隔為 4 ，如圖2.23所示。因 此，

(2)

σ σ

σ 25 .

≈ 0

B

‡ 如果 D _ch 為色彩色散係數，則輸出光脈衝的均 方根色散為 ，而乘積 BL ，稱為乘積

值，並由下列給出

(3)

σ

λ

LD

ch

σ

λ

σ | |

25 . 0 25

. 0

D

ch

BL ≈ L =

‡ 體色散若以一個均方根色散 表示則可由各個 均方根色散求得

(4)

σ

2

intramodal 2

intermodal

2 σ σ

σ = +

0 t

P_i = Input light power

Emitter

Optical Input

Optical Output Fiber

Photodetector Sinusoidal signal

Sinusoidal electrical signal t

0 t f

1 kHz 1 MHz 1 GHz

P_o / P_i

f_op 0.1

0.05 f = Modulation frequency

An optical fiber link for transmitting analog signals and the effect of dispersion in the fiber on the bandwidth, f

_op

.

P_o = Output light power

Electrical signal (photocurrent)

f_el 0.7071

1 kHz 1 MHz 1 GHz f

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

圖2.24 用來傳送類比訊號的光纖連接以及光纖中的色散對帶寬 的效應。

‡ 如果光纖的色散特性為高斯的，則 σ (5)

19 . 75 0

.

0 ≈

≈ B

f

_op

2-7 斜射率 (GRIN) 光纖

n

₁

n

₂

2 1 3 O

n

n

₁

2 1 3

n

n

₂

O O' O''

n

₂

(a) Multimode step index fiber. Ray paths are different so that rays arrive at different times.

(b) Graded index fiber.

Ray paths are different but so are the velocities along the paths so that all the rays arrive at the same time.

2 3

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

圖2.25 (a) 多模步級折射率光纖，光線路徑不同故光線以不同 的時間到達；

nb

n_c

O O'

Ray 1 A

B'

B

θ_A θ_B

θ_{B' Ray 2}

M θ_{B' c/n}

b

c/n_a 21

B'' n_a

a b

c We can visualize a graded index fiber by imagining a stratified

medium with the layers of refractive indices n_a > n_b > n_c ... Consider two close rays 1 and 2 launched from O at the same time but with slightly different launching angles. Ray 1 just suffers total internal reflection.

Ray 2 becomes refracted at B and reflected at B'.

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

圖2.26 我們可藉由想像一個層狀的介質，其層折射率

為 ，而瞭解斜射率光纖。考慮兩個靠近的光線 1及2，以相同的時間但不同的發射角由O發射。光線1正好遭 到內部全反射，而光線2先在 B 折射而後在 反射。

c

…

b

a

n n

n > >

B′

n decreases step by step from one layer to next upper layer; very thin layers.

Continuous decrease in n gives a ray path changing continuously.

TIR TIR

(a) A ray in thinly stratifed medium becomes refracted as it passes from one layer to the next upper layer with lower n and eventually its angle satisfies TIR (b) In a medium where n decreases continuously the path of the ray bends

continuously.

(a) (b)

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

圖2.27 (a) 薄層狀介質中的光線當它由一層通過下一個折

射率 n 較低的上層時變成折射直至最後它的角度滿足TIR；

在文檔中 介質波導與光纖 (頁 31-81)