第四章
受激發射元件 - 雷射
目錄
4-1 受激發射及光子放大
4-2 受激發射速率及愛因斯坦係數
4-3 光纖放大器
4-4 氣體雷射:氦 - 氖雷射
4-5 氣體雷射之輸出光譜
4-6 雷射共振條件
4-7 雷射二極體之原理
4-8 異質結構雷射二極體
目錄(續)
4-9 雷射二極體之基本特性
4-10 穩態半導體之速率方程式
4-11 光纖通信之光發射器
4-12 單頻固態雷射
4-13 量子井元件
4-14 豎直空腔面發射雷射 (VCSELs)
4-15雷射放大器
4-1 受激發射及光子放大
E
1E
2h υ
(a) Absorption
h υ
(b) Spontaneous emission
h υ
(c) Stimulated emission
In h υ
Out h υ
E
2E
2E
1E
1Absorption, spontaneous (random photon) emission and stimulated emission.
?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
E1 hυ13
E2 Metastable state
E1 E3
E2
hυ32
E1
E3
E2
E1 E3
E2
hυ2 hυ21
Coherent photons OUT
(a) (b) (c) (d)
E3
The principle of the LASER. (a) Atoms in the ground state are pumped up to the energy level E3 by incoming photons of energy hυ13 = E3–E1. (b) Atoms at E3 rapidly decay to the metastable state at energy level E2 by emitting photons or emitting lattice vibrations; hυ32 = E3–E2. (c) As the states at E2 are long-lived, they quickly become populated and there is a population inversion between E2 and E1. (d) A random photon (from a spontaneous decay) of energy hυ21 = E2–E1 can initiate stimulated
emission. Photons from this stimulated emission can themselves further stimulate emissions leading to an avalanche of stimulated emissions and coherent photons being emitted.
?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
IN
4-2 受激發射速率及愛因斯坦係
數
考慮如圖4.1的介質,E 1 能階每單位體積含有個 N1 原 子,且 E 2 能階每單位體積含有 N 2 個原子,吸收光子 後使原子由 E 1 能階躍昇至 E 2 能階的數目正比於N1及 每單位體積具有 能量之光子;兩者同時考 慮,則由 E 1 躍昇至 E 2 的速率與輻射時的能量密度有 關,因此,向上躍昇的速率可寫為
(1)
1
2
E
E h υ = −
)
1 (
12
12 B N ρ h υ
R =
這裡必須強調在第 (5) 式的普朗克定律僅滿足於熱平衡的情況,
我們就是基於這樣的情況來決定愛因斯坦係數。在雷射操作期 間,當然 P(hv) 並不如算式 (5) 所述,事實上是較等式 (5) 所述 要來得大,由等式 (1) 至等式 (5),我們可得到
4-3 光纖放大器
Energy of the Er
3+ ion in the glass fiber
E 1
0
1.54 eV 1.27 eV
0.80 eV E 2
E 3 E ′ 3
1550 nm 1550 nm
In
Out
980 nm
Non-radiative decay
Pump
Energy diagram for the Er3+ ion in the glass fiber medium and light amplification by stimulated emission from E2 to E1. Dashed arrows indicate radiationless
transitions (energy emission by lattice vibrations)
Signal in
SpliceSignal out Er
3+-doped
fiber (10 - 20 m)
Wavelength-selectivecoupler
Pump laser diode
Splice
λ = 1550 nm λ = 1550 nm
λ = 980 nm
Termination Optical
isolator
Optical isolator
A simplified schematic illustration of an EDFA (optical amplifier). The erbium-ion doped fiber is pumped by feeding the light from a laser pump diode, through a coupler, into the erbium ion doped fiber.
?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
4-4 氣體雷射:氦 - 氖雷射
Current regulated HV power supply
Flat mirror (Reflectivity = 0.999) Concave mirror (Reflectivity = 0.985)
He-Ne gas mixture
Laser beam Very thin tube
A schematic illustration of the He-Ne laser
?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
(1s
2) (1s
12s
1)
0 20.61 eV
He
(2p
6)
Ground states
(2p
55s
1)
Ne
(2p
53p
1)
(2p
53s
1)
Collisions
Lasing emission 632.8 nm
~600 nm
Collisions with the walls
Fast spontaneous decay 20.66 eV
Electron impact
The principle of operation of the He-Ne laser. He-Ne laser energy levels
(for 632.8 nm emission).
4-5 氣體雷射之輸出光譜
當我們考慮氣體分子在雷射管內之馬克斯威運動分 佈,我們發現在相對於頻率頻譜之強度分佈圖中其兩 邊之半強度點 ( 半最大值之全寬FWHM) 之線寬
可表示為
2 /
υ 1
∆
(a) Optical gain vs. wavelength characteristics (called the optical gain curve) of the lasing medium. (b) Allowed modes and their wavelengths due to stationary EM waves within the optical cavity. (c) The output spectrum (relative intensity vs. wavelength) is determined by satisfying (a) and (b) simultaneously, assuming no cavity losses.
(c)
Relative intensity
δλm
Optical Gain
λ
Allowed Oscillations (Cavity Modes)
δλm (b)
L
Stationary EM oscillations Mirror Mirror
λ λ
Doppler broadening
m(λ/2) = L (a)
λο
?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
由雷射兩端鏡面之反射可在共振腔 L 內形成相反方向 之行進波,這些相反方向之建設性干涉行進波可形成 駐波,這也就是固定電磁共振。部份共振能量會經由 99% 反射率之反射鏡彈出,這恰好與附屬天線之 LC 電路由其共振場輻射能量之原理相似。只有特定波長 之駐波可維持在光共振腔內,正如在音樂儀器中只存 在特定聲波,任何在共振腔內之駐波必須滿足其半波 長 之整數倍恰好等於共振腔長度 L
(4) 2
λ /
L m ⎟ =
⎠
⎜ ⎞
⎝
⎛
2
λ
Optical gain between FWHM points
δλm
(a) 5 modes
(b) 4 modes
Number of laser modes depends on how the cavity modes intersect the optical gain curve.
In this case we are looking at modes within the linewidth
∆λ1/2. λ
Cavity modes
?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
圖4.9 共振腔模數與光增益曲線相交程度決定雷射之模數。在 這樣的情形中我們可看出在
∆ λ
1/2 線寬中之模4-6 雷射共振條件
P+
δ
PLaser medium
x
δ
x P(a) A laser medium with an optical gain (b) The optical gain curve of the medium. The dashed line is the approximate derivation in the text.
hυ
E2
E1
υ Optical Gain
υ
ο∆ υ
(a) (b)
g(
υ )
g(υ
o)
?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
圖4.10 (a) 具有光增益的雷射介質 (b) 介質的光增益曲線。虛線是
增益係數 g 主要用以描述在 E 1 至 E 2 之受激發射超過
光子吸收的情況下,共振腔內單位長度所增加之雷射
發射強度。我們知道受激發射與吸收速率之差值 ( 參
閱4.2節中之算式 (1) 與等式 (2)) 即等於同調性光子濃
度改變之淨速率,也就是
考慮如圖4.11所示在共振腔內某點具內部光能量 P i 之 電磁波並朝向標示為 1 之鏡面方向前進,當行進至標 示為 1 之鏡面時電磁波會被反射並朝標示為 2 之鏡面 方向行進;同樣地,行進至標示為 2 之鏡面時,電磁 波會被反射,設其回至出發點之最後光功率為 P f 。在 穩定情況下,振盪的情況並未建立且能量亦不會衰
滅,這表示 P f 必須等於 P i ,因此在整個來回的過程中 不會有光功率損耗,這表示淨往返光學增益 (net
round - trip optical gain) G op 必須為 1
L P
iP
fR
1R
2Steady state EM oscillations
Optical cavity resonator
Reflecting surface
Reflecting surface
Cavity axis x 2 1
E
fE
i?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
圖4.11 光學共振腔
Simplified description of a laser oscillator. (N
2− N
1) and coherent output power (P o ) vs. pump rate under continuous wave steady state operation.
Pump rate
Threshold pump rate
(N
2− N
1)
thN
2− N
1Threshold population inversion
P
o= Lasing output power
(N
2− N
1) and P
o 所以我們可推導出一般之模態條件
L (11)
m m ⎟ =
⎠
⎜ ⎞
⎝
⎛ 2 n
λ
Wave fronts Spherical
mirror
Optical cavity
TEM00 TEM10
TEM01 TEM11
TEM00 TEM10
TEM01 TEM11 (b)
(c) (d)
Laser Modes (a) An off-axis transverse mode is able to self-replicate after one round trip. (b) Wavefronts in a self-replicating wave (c) Four low order transverse cavity modes and their fields. (d) Intensity patterns in the modes of (c).
(a)
?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
光強度如減弱,其減弱程度可以用 因子來表 示,其中 為吸收係數,且這個因子主要用以表示 x 方向隨距離之衰減程度。同樣的,我們以 來表 示功率的增加,其中 g 為每單位長度之光增益我們稱 之為介質之光增益係數,且定義為每單位長度(距離)光 功率改變之比例。沿 x 方向上任一點之光功率p與同調 光子濃度 N ph 及其能量 hv 成正比。這些同調之光子以 c/n 速度傳送,其中 n 為相對折射係數,因此極短時 間 在雷射管內之傳送距離 所以 g 可寫為
(5)
) exp( − α x
) exp(−
α x
α
t n c
x δ
δ = ( / )
δ t
N
P N ph ph
δ δ δ
δ δ
δ n
g = = =
4-7 雷射二極體之原理
p
+n
+EF n
(a) Eg Ev
Ec
Ev
H o l es i n V B
El ectro ns i n C B
Junction
El ectro ns
Ec
p
+Eg
V
n
+(b)
EF n
eV EF p
The energy band diagram of a degenerately doped p-n with no bias. (b) Band diagram with a sufficiently large forward bias to cause population inversion and hence stimulated emission.
In v ers i o n reg i o n
EF p
Ec
Ec eVo
?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
hυ Eg
Optical gain
EF n − EF p
Optical absorption 0
Energy
Ec
Ev
CB
VB
(a) The density of states and energy distribution of electrons and holes in the conduction and valence bands respectively at T ≈ 0 in the SCL
under forward bias such that E
Fn− E
Fp> E
g. Holes in the VB are empty states. (b) Gain vs. photon energy.
Density of states Electrons in CB
Holes in VB
= Empty states EF n
EF p eV
At T > 0 At T = 0
(a) (b)
?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
L Electrode Current
GaAs
n + GaAs p +
Cleaved surface mirror
Electrode Active region
(stimulated emission region)
A schematic illustration of a GaAs homojunction laser diode. The cleaved surfaces act as reflecting mirrors.
L
Typical output optical power vs. diode current (I) characteristics and the corresponding output spectrum of a laser diode.
λ Laser
λ Laser
Optical P ower
Optical P ower
0 I
λ
Optical P ower
LED
It h Spontaneous
emission
Stimulated emission Optical Power
?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
4-8 異質結構雷射二極體
Refractive index
Photon density
Active region
∆n ~ 5%
2 eV
Holes in VB Electrons in CB
AlGaAs AlGaAs
1.4 eV
Ec
Ev Ec
Ev (a)
(b)
p
n p
∆Ec
(a) A double
heterostructure diode has two junctions which are between two different bandgap semiconductors (GaAs and AlGaAs).
2 eV
(b) Simplified energy band diagram under a large forward bias.
Lasing recombination takes place in the p- GaAs layer, the active layer
(~0.1 µm)
(c) Higher bandgap materials have a lower refractive index
(d) AlGaAs layers provide lateral optical confinement.
(c)
(d)
?1999 S.O. Kasap, Optoelectronics (Prentice Hall) GaAs
圖4.18 (a) 兩種不同能隙半導體 (GaAs與AlGaAs) 所形成兩個接面 的雙異質結構二極體式意圖 (b) 在大的順向偏壓下簡化之能帶圖,
Oxide insulator
Stripe electrode
Substrate
Electrode
Active region where J > Jt h. (Emission region)
p-GaAs (Contacting layer)
n-GaAs (Substrate) p-GaAs (Active layer)
Current paths
L W
Cleaved reflecting surface Elliptical
laser beam p-AlxGa
1-xAs (Confining layer) n-AlxGa
1-xAs (Confining layer) 2 1 3
Cleaved reflecting surface
Substrate
Oxide insulation
n-AlGaAs p + -AlGaAs (Contacting layer)
n-GaAs (Substrate) p-GaAs (Active layer)
n-AlGaAs (Confining layer) p-AlGaAs (Confining layer)
Schematic illustration of the cross sectional structure of a buried heterostructure laser diode.
Electrode
?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
圖4.20 埋入式異質結構雷射二極體之橫截面結構示意圖。
4-9 雷射二極體之基本特性
Height, H
Width W Length, L
The laser cavity definitions and the output laser beam characteristics.
Fabry-Perot cavity Dielectric mirror
Diffraction limited laser beam
?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
778 780 782
Po = 1 mW Po = 5 mW Relative optical power
λ (nm) Po = 3 mW
Output spectra of lasing emission from an index guided LD.
At sufficiently high diode currents corresponding to high optical power, the operation becomes single mode. (Note:
Relative power scale applies to each spectrum individually and not between spectra)
?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
0 20 40 60 80 0
2 4 6 8 10
P o (mW)
I (mA)
0 °C 25 °C
50 °C
Output optical power vs. diode current as three different temperatures. The threshold current shifts to higher temperatures.
?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
圖4.23 在不同溫度下輸出光功率與二極電流之關係
λo (nm)
Mode hopping
20 30 40 50
Case temperature (° C) Single mode
776 778 780 782 784 786 788
20 30 40 50
Case temperature (° C) Single mode
20 30 40 50
Multimode
Case temperature (° C)
Peak wavelength vs. case temperature characteristics. (a) Mode hops in the output spectrum of a single mode LD. (b) Restricted mode hops and none over the temperature range of interest (20 - 40 °C). (c) Output spectrum from a multimode LD.
(a) (b) (c)
?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
4-10 穩態半導體之速率方程式
等式的第二項為受激發射速率,其大小取決於導帶可 利用之電子濃度及主動區之光子濃度 N ph 。 N ph 為光 共振腔所激發之同調輻射光子濃度,也就是共振腔的 一個模態。當電流增加時,更多的泵浦作用伴隨產
生, N ph 亦隨之增加 ( 由光共振腔所協助產生 ) 之後,
激發性發射多於自發性發射 ( 如圖4.17),輸出光功率 P o 正比於 N ph 。
考慮共振腔中同調輻射之光子濃度,在穩態下:
共振腔同調輻射光子之損耗速率激發性發射速率即
在半導體雷射科學中,閥值電子濃度 n th 及閥值電流 I th 通常在受激發射多於自發性發射且整體損耗機制為
之情況。這僅發生於注入 n 達到 n th 時,其中 n th 為閥值濃度。由等式 (2),也就是當
(3) τ
phph
th C
n τ
= 1
Simplified and idealized description of a semiconductor laser diode based on rate equations. Injected electron concentration n and coherent radiation output power P o vs. diode current I.
I
I
thn
thn
n
Threshold population inversion
P
oP
o= Lasing output power ∝ N
ph4-11 光纖通信之光發射器
Typical optical power output vs. forward current for a LED and a laser diode.
Current
0Light power
Laser diode
LED
100 mA 50 mA
5 mW 10 mW
4-12 單頻固態雷射
Corrugated
dielectric structure Distributed Bragg reflector
(a) (b)
A B
Λ
q( λ
B/2n) = Λ
Active layer
(a) Distributed Bragg reflection (DBR) laser principle. (b) Partially reflected waves at the corrugations can only constitute a reflected wave when the wavelength
satisfies the Bragg condition. Reflected waves A and B interfere constructive when q( λ
B/2n) = Λ.
?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
例如:兩個不同的反射波,其光程差為 ,其中 為 波浪週期,這可由只有在介質中之波長為 的整數倍 方向可形成建設性干涉,這樣的波長我們稱為布拉格 波長 ,此外同相位干涉之條件可表示為
Λ 2 Λ 2
λ
BΛ
Λ
Active layer Corrugated grating Guiding layer
(a)
(a) Distributed feedback (DFB) laser structure. (b) Ideal lasing emission output. (c) Typical output spectrum from a DFB laser.
Optical power
λ (nm) 0.1 nm Ideal lasing emission
λ B λ
(b) (c)
?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
圖4.28 (a) 分佈回饋 (DFB) 雷射結構,(b) 理想之雷射輸出,(c) 傳統之DFB雷射之輸出頻譜。
可允許之DFB模態不僅與布拉格波長有關,而且相關 於 做對稱的分佈, 為一允許之DFB雷射光模態 則
(2)
λ
Bλ m
) 1 2 (
2
+
±
= m
L
B B
m
n
λ λ
λ
Active layer
L D
(a)
Cleaved-coupled-cavity (C
3) laser
λ
Cavity Modes
λ
λ
In L In D In both
L and D(b)
?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
圖4.29 劈裂–耦合–共振腔(C3)雷射結構
4-13 量子井元件
A quantum well (QW) device. (a) Schematic illustration of a quantum well (QW) structure in which a thin layer of GaAs is sandwiched between two wider bandgap semiconductors (AlGaAs). (b) The conduction electrons in the GaAs layer are confined (by E
c) in the x-direction to a small length d so that their energy is quantized. (c) The density of states of a two-dimensional QW. The density of states is constant at each quantized energy level.
AlGaAs AlGaAs
GaAs
y z
x
d
Ec
Ev
d
E1 E2 E3
g(E)
Density of states E
QW Bulk
n = 1 Eg2
Eg1
E ² Ecn = 2
Bulk QW
² Ev
(a) (b) (c)
Dy
Dz
?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
E
cE
vE
1E′
1h υ = E
1? E′
1E
In single quantum well (SQW) lasers electrons are injected by the forward current into the thin GaAs layer which serves as the active layer. Population inversion between E1 and E
′
1 is reached even with a small forward current which results in stimulated emissions.Active layer Barrier layer E
cE
vE
A multiple quantum well (MQW) structure.
Electrons are injected by the forward current into active layers which are quantum wells.
?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
4-14 豎直空腔面發射雷射
(VCSELs)
Contact
Surface emission
Dielectric mirror
Contact Substrate λ /4n
1Active layer
λ /4n
2Dielectric mirror
4-15雷射放大器
Simplified schematic illustrations of two types of laser amplifiers
Pump current
Active region
AR = Antireflection coating
AR
Signal in Signal out
(a) Traveling wave amplifier (a) Fabry-Perot amplifier Partial mirror Partial mirror
?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
4-16 全像術
Real cat imag Virtual cat image
(b)
Photographic plate
(a)
Hologram Laser
beam Through beam
Observer
Cat Mirror
Laser beam
Ecat(x,y) Eref(x,y)
Ecat(x,y) Eref(x,y)
A highly simplified illustration of holography. (a) A laser beam is made to interfere with
the diffracted beam from the subject to produce a hologram. (b) Shining the laser beam
through the hologram generates a real and a virtual image.
其中我們利用一般複數大小平方的定義,即複數與共 軛複數的乘積 ( 星號的部份 ),因此
∗ (3)
∗
∗
∗ + + +
= UU U U U U U U
y
x , )
r r r rI (
(1s2) (1s12s1)
0
20.6 eV
He
(2p6)
Ground states
Ne
(2p53p1)
Lasing emissions
Collisions with the walls Electron impact
(1s12s1) (2p54s1)
(2p53s1)
~600 nm
Fast spontaneous decay
(2p55s1)
1152 nm
1118 nm 1523 nm 19.8 eV
632.8 nm
543.5 nm
(2p54p1)
3.39 µm Collisions
Ar atom ground state Ar+ ion
ground state 4p levels
4s
0
15.75 eV
488 nm
514.5 nm
72 nm Energy
Pumping
p
+Eg
V
n
+EFp
EFn e−
h+ A
Ev B Ec
Ev Ec
The energy band diagram of a degenerately doped p-n with with a sufficiently large forward bias to just cause population inversion where A and B overlap.
?1999 S.O. Kasap, Optoelectronics (Prentice Hall)