第 2 章 光輻射的特性
在第 1 章所介紹的四個雷射特性，都是很高的光子簡併態所導致
的。
從波動的觀點得到的光波的模態數與從光子的觀點得到的光子的模態數所得的
結果是一樣的。
相干性的光子描述
光子的基本性质
光波模式和光子狀態、相格
光子的相干性分成時間相干空間相干
時間相干不考慮空間，或簡單來說無論在什麼地方，相互之間的相對相位保持固
定的時間可以維持多久
空間相干不考慮空間，或簡單來說無論在什麼地方，相互之間的相對相位保持固
定的時間可以維持多久。
介紹了光的相干性之後，接著要說明基本的電磁輻射理論。當然，如果這
個電磁輻射是雷射，則電磁輻射理論也就可以用來描述雷射的過程。又因為電磁
輻射是由物質所產生或被物質所吸收，所以所謂的雷射理論。其實就是光與物質
交互作用的理論。如下表所示，我們依據光的古典模型與量子模型以及物質的古
典模型與量子模型，列出了四個電磁輻射理論或雷射理論。然而本章中，在尚未
進入雷射之前，我們先討論前三個電磁輻射理論，包含。
Temporal Coherence and Spatial Coherence-1
Laser Theory
Treatments
Fully Quantum
Semiclassical
Classical
Approach
Rate-Equation
(Quantum
Approach
Approach
Approach
(Lamb Theory)
Interactions
Electrodynamics)
Classical
Schroedinger’s
Schroedinger’s
Oscillators
Equation
Equation
Maxwell’s
Maxwell’s
Quantized Maxwell’s
Photon (Quantized
Equations
Equations
Equations
Fields)
Matters
Energy Levels
Fields
2.1 光的同調性(Coherence)
Consider a wave propagating through space. Coherence is a measure of the
correlation that exists between the phases of the wave measured at different
points. The coherence of a wave depends on the characteristics of its source.
Let us look at a simple example. Imagine two corks bobbing up and down on a wavy
water surface. Suppose the source of the water waves is a single stick moved
harmonically in and out of the water, breaking the otherwise smooth water
surface. There exists a perfect correlation between the motions of the two
corks. They may not bop up and down exactly in phase, one may go up while the
other one goes down, but the phase difference between the positions of the two corks
is constant in time. We say that the source is perfectly coherent. A harmonically
oscillating point source produces a perfectly coherent wave.
When we describe the coherence of light waves, we distinguish two types of
Temporal Coherence and Spatial Coherence-2
coherence.
2.1.1 Temporal Coherence
Temporal coherence is a measure of the correlation between the phases of a light
wave at different points along the direction of propagation. Temporal coherence tells
us how monochromatic a source is.
Assume our source emits waves with wavelength    . Waves with wavelength
 and    , which at some point in space constructively interfere, will
2
destructively interfere after some optical path length lc 
; lc is called the
2
coherence length.
[The phase of a wave propagating into the x-direction is given by
  kx  t . Look at the wave pattern in space at some time t . At some distance
l the phase difference between two waves with wave vectors k1 and k2 which are
   k1  k2  l .
in phase at x  0
  1 , the light is no
longer considered coherent. Interference and diffraction patterns severely loose
contrast.
We therefore have
2 
 2
1   k1  k2  lc  


     
Temporal Coherence and Spatial Coherence-3
       lc  lc  1
     
2
2
lc 
2
]
2
The wave pattern travels through space with speed c.
l
The coherence time tc is tc  c .
c








  c , we have
. We can write
lc 
2
2
tc 
1

If we know the wavelength or frequency spread of a light source, we can calculate lc
and tc . We cannot observe interference patterns produced by division of amplitude,
such as thin-film interference if the optical path difference greatly exceeds lc .
c
Temporal coherence is the measure of the average correlation between the value of
a wave at every pair of times separated by delay τ. In other words, it characterizes
Temporal Coherence and Spatial Coherence-4
how well a wave can interfere with itself at a different time. The delay over which the
phase or amplitude wanders by a significant amount (and hence the correlation
decreases by significant amount) is defined as the coherence time  c . At τ=0 the
degree of coherence is perfect whereas it drops significantly by delay  c . The
coherence length lc is defined as the distance the wave travels in time  c .
The amplitude of a single frequency wave as a function of time t (red) and a copy
of the same wave delayed by τ(green). The coherence time of the wave is infinite
since it is perfectly correlated with itself for all delays τ.
The amplitude of a wave whose phase drifts significantly in time τc as a function of
time t (red) and a copy of the same wave delayed by 2τc (green). At any particular
time t the wave can interfere perfectly with its delayed copy. But, since half the time
Temporal Coherence and Spatial Coherence-5
the red and green waves are in phase and half the time out of phase, when averaged
over t any interference disappears at this delay.
Waves of different frequencies (i.e. colors) interfere to form a pulse if they are
coherent.
Spectrally incoherent light interferes to form continuous light with a randomly
varying phase and amplitude.
我 們 可 以 用
Michelson
Interference
來 說 明
Temporal Coherence and Spatial Coherence-6
Temporal
coherence
(Monochromaticity).
首先，我們知道相同頻率  但相位  不同的兩道光分別可以表示成
 E1  E10eit 1 

i t  2 
 E2  E20e
這兩道光產生的干射光譜強度 I 為
*
I   E1  E2   E1  E2 
i  
 i  
  E10    E20   E10 E20  e  1 2   e  1 2  
2
2
2
2
 2 I 0  2 I 0 cos  , as E10  E20  I 0
其中   1  2 。
假設有一個非單色光源是由 1 和 2 兩個強度相同的單色光所組成的，我
們將看看這束光在通過干射儀之後的干射條紋的變化與 1 2 的關係為何？
相同頻率  但相位  不同的兩道光分別可以表示成
 E1  E10eit 1 

i t  2 
 E2  E20e
這兩道光產生的干射光譜強度 I 為
Temporal Coherence and Spatial Coherence-7
*
I   E1  E2   E1  E2 
i  
 i  
  E10    E20   E10 E20  e  1 2   e  1 2  
2
2
2
2
 2 I 0  2 I 0 cos  , as E10  E20  I 0
其中   1  2
每個波長的單色光相干疊加之後的光強度為
I    I 0 1  cos  
其中相位差   k L ； L 為 Michelson Interferometer 移動臂所移動的距離。
又k 
2

 I  k L   I  L   I 0 1  cos  k L  
所以，波長為 1 和 2 所產生的干射光譜強度分別為
2

 I1  L   I10 1  cos  k1L   , where k1  

1

 I 2  L   I 20 1  cos  k2 L   , where k2  2



2
若兩道光的強度相等，即 I 0  I10  I 20 ，則光的總強度會等於兩束光的非相干
疊加，即
I  L   I1  L   I 2  L 
 I 0  2  cos  k1L   cos  k2 L  


 k

 2 I 0 1  cos 
L  cos  k L  
 2



其中 k 
k1  k2
, k  k1  k2 。
2
因為 k  k ，所以干射光的強度 I  L  將會被 k 所調變。(因為 k 相對小於
k ，所以必須要有比較長的 L 才會觀察到 k 的變化；反之，只要有比較短的 L
就可觀察到 k 的變化。)
Temporal Coherence and Spatial Coherence-8
I  L  1  0 ，
若
對應於
k

L  0 
2
2
則

L 
2
2
2
1

2
1 2
1 12
1 12


2 2  1 2 

c 1
c 1

2  1   2 2 
當  或  越小，越趨向單色光，則 L 越大；換言之，從 L 可測量出  或
 的值。
2.1.2 Spatial Coherence
Spatial coherence is a measure of the correlation between the phases of a light wave at
different points transverse to the direction of propagation. Spatial coherence tells us
how uniform the phase of the wave front is.
 , two
A distance L
slits separated by a distance greater than d c 
interference pattern. We call
 d c2
4
L
will no longer produce an
2
the coherence area of the source.
[At time t look at a source of width  a perpendicular distance L from a
screen. Look at two points (P1 and P2) on the screen separated by a distance d .
Temporal Coherence and Spatial Coherence-9
Light waves emitted from the two edges of the source have a some definite phase
difference right in the center between the to points at some time t . A ray traveling
 to point P2 must travel a distance
from the left edge of
d sin 
farther then a
2
ray traveling to the center. The path of a ray traveling from the right edge of  to
point P2 travel is approximately
d sin 
shorter then the path to the center. The
2
path difference for the two rays therefore is d sin  , which introduces a phase
  
2 d sin 

. For the distance from P1 to P2 we therefore get a
phase difference   2  
4 d sin 

. Wavelets emitted from the two edges of
the source are that are in phase at P1 at time t are are out of phase by
P2 at the same time t . We have sin  

2L
 
2 d 
L
When   1 , the light is no longer considered coherent.
Temporal Coherence and Spatial Coherence-10
4 d sin 

at
  1 --> d c 
L
.]
2
雷射的空間相干性(橫向相干性)和方向性(指向性)：空間相干性是指光源在同
一時刻在不同空間，各點發出的光波相位關聯程度。其實光束的空間相干性和它
的方向性是同義的。
 

x
Diffraction Limited：  d  

D
我們先以示意圖來說明
Assuming Ac is a coherence area, then
A plane wave with an infinite coherence length.
Temporal Coherence and Spatial Coherence-11
A wave with a varying profile (wavefront) and infinite coherence length.
The wave with finite coherence length, as mentioned above, is passed through a
pinhole. The emerging wave has infinite coherence area. The coherence length (or
coherence time) are unchanged by the pinhole.
我們可以再用雙狹縫干射(Young’s double-slit interferometer)來說明如下：
Temporal Coherence and Spatial Coherence-12
如上圖所示的 Young’s Interference，因為光源不可能是真正的點光源，所以實際
上由光源到屏幕上的雙狹縫干涉是由在 Δx 範圍內的光源所造成的，如果想要在
屏幕上形成清楚的亮暗條紋，最好的情況是 So’所發出的光和由 So 發出的光是
“相同”的，因為 So 是在光源的中間的位置，所以如果 So’和 So 是“相同”的，則
可用”單狹縫繞射“的分析方式來理解，得知整個光源 Δx 內都是“相同”的。但是
當 So’和 So 之間有差異時，原來的亮暗之間開始重疊，最差的情況就是最暗和
最亮發生重合←(由亮暗紋要成灰灰一片沒有條紋)
假設如上圖所示，原來由 So’，So 至 P 點造成亮紋，即
( So ' AP  So ' BP)  ( So AP  So BP)  0 →亮紋
如果要得到暗紋，最簡單的想法就是通過 A、B 兩個狹縫的光都是黑的，也就是
So’和 So 兩個“光源”的光程差最小為
即
則

。
2
x

 sin  
2
2
x
x
 sin  

2
2
(
x Lx
)
2 R
Temporal Coherence and Spatial Coherence-13
得 x 


其中 Δx 的物理意義為，在 Δx 範圍內所發的光，如果是 Coherence，則必須在 Δθ
的角度內的狹縫才能產生干涉效應。也就是說如果 Spatial Coherence 愈高，即
Δx 愈大，發散角 Δθ 就必須愈小。或只可以說，因為我們發現雷射光的發散角很
小即方向性很高，所以雷射光的 Spatial Coherence 很高，當然反過來說因為雷射
的 Spatial Coherence 造成方向性很好，所以只能再一個小角度範圍內作干涉實
驗，一般的光源，因為 Spatial Coherence 差，所以必須要先通過一個小孔，把光
束過濾後，再作干涉實驗。而實際上，我們可以看到通過小孔的光束發散角很大，
這當然不完全是因為繞射的原因。
Temporal Coherence and Spatial Coherence-14