1. Lightwave Fundamentals
Chapter 3
Lightwave Fundamentals
College of Communications Engineering

2. Contents Electromagnetic waves
Dispersion, pulse distortion, information rate
Polarization
Resonant cavities
Reflection at a plane boundary
Critical-angle reflection

3. 3.1 Electromagnetic Waves
Wave Properties
velocity
power
polarization
interference
refraction

4. Wave traveling in the z direction On the figure, t1 < t2 < t3
Electromagnetic Waves
Wave traveling in the z direction
On the figure, t1 < t2 < t3 t1 t3 t2
Electric Field
Position (z)
光波由高速振荡的电场和磁场组成，振荡频率高达10^14Hz。这些场以波的形态高速传播。
图中给出了3个时间点的电场分别，反映了波的前进。
在任何固定的位置上，场的振幅以光的频率变化，每隔一个时间周期重复一次。
Electric field:
(3.1)
At t1 < t2 < t3 , peak amplitude E0 is fixed, Ф = wt - kz

5. Electromagnetic Waves
This is a solution of the wave equation:
Propagation factor:
(3.2)
Frequency: f = v/l
Radian frequency: w = 2pf (rad/s)
Wave peak amplitude: E0
Wave phase: f = wt-kz

6. Electromagnetic Waves
Relationships for the Propagation Factor k
k = w/v = w/(c/n) = wn/c
In free space, n = 1, so that k = ko = w/c
In general, then k = kon
(3.6)
lo: wavelength in free space
l: wavelength in the medium
f is fixed, f = f0
Then,
(3.7)

7. Power Power in a resistor: Pr = V2/R
Power is proportional to the voltage (V) squared.
In an optical beam, define Intensity: I = E2
Since P ∝ E2, P ∝ I.
回忆电路系统中的功率
电磁波由电场E和磁场H组成，光强只和E有关
光波的功率P和电场E^2成正比，因此光波的功率P也和光强成正比。

8. Power :材料的磁导率 : 材料的介电常数 Irradiance S = Power Density (watts/m2)
For a plane wave, the irradiance and intensity are given by:
:材料的磁导率
: 材料的介电常数
Now
We conclude that the intensity is proportional to the irradiance I ∝ S.
辐射强度,光强与辐射强度成正比
(µ/ε)^1/2 波阻抗

9. Electromagnetic Waves
Recall the plane wave given by
(3.1)
This expression represents a wave traveling with zero loss.
If loss occurs, the field is represented by
(3.8)
衰减系数
指数－αZ 应该是无量纲的,Z是距离单位，则 α应该是距离单位的倒数。
 is the attenuation coefficient for E.
The frequency and phase do not vary with loss, only the amplitude of the wave Eoe-az changes with loss.

10. Electromagnetic Waves
P∝E2, for a path length L in a lossy medium, the power diminishes by a factor of:
2 is the attenuation coefficient for P
The corresponding P (or I) reduction in dB is:
This will be a negative number for propagation through a lossy medium.
Define: g (dB/km) in terms of the attenuation coefficient a.
Proof：对数的换低公式
指数－2αL应该是无量纲的，
g = a (proof)
If L is in unit of km, then a is in units of km-1.

11. Electromagnetic Waves
Wave Traveling in a Lossy Medium t1 t2
Electric Field
Distance (z)

12. 3.2 Dispersion, Pulse Distortion, Information Rate
When we write E = Eosin (wt – kz), we imply a single frequency source.
Frequency
Radio oscillators approximate single f pretty well.
Optical sources do not produce single f.
单一频率的光
对于无线电来说我们可以获得很好的近似单一频率的电磁波.即无线电的单频性很好。
对于现在的光源来说产生的光的频谱都有一个范围.很难用单一频率的光去近似。

13. Dispersion, Pulse Distortion, Information Rate
Example： Emission Spectrum of an Optical Source 1
0.5 f
Frequency
Normalized
Power f f1 f2
f = source bandwidth (range of frequencies emitted by the source).
f is the central frequency.

14. Dispersion, Pulse Distortion, Information Rate
Alternatively, we can plot the wavelength emission spectrum as follows: 1
0.5 
Wavelength
Normalized
Power  2 1
Δλ的界定：光功率降到最大值的一半时所对应的波长。
 = linewidth or spectral width

15. Dispersion, Pulse Distortion, Information Rate
Example: If  = 0.82 m,  = 30 nm
so we have 3.7% bandwidth.
The conversion between wavelength and frequency is:
(3.9)
近似等式

16. Dispersion, Pulse Distortion, Information Rate
Proof:
Define the mean wavelength as:
Then,
The mean frequency is: f = c/l
近似式
Now, we have

17. Dispersion, Pulse Distortion, Information Rate
Spectral Widths for Typical Light Sources (table 3.1)
Source Spectral Width  (nm)
LED
Laser Diode
Nd:YAG-Laser (固态钇铝石榴石) 0.1
He-Ne Laser
固态钇铝石榴石

18. Dispersion, Pulse Distortion, Information Rate
If  = 0, (f = 0), the source is perfectly coherent. It is monochromatic（单色）.
Laser diodes are more coherent than LEDs, but are not perfectly coherent.
We will see how source bandwidth limits the information capacity of fiber transmission lines.
光源相干性就非常好. 它是单色的. 激光二极管比 LED相干性好, 但也不是完全相干的单色光. 后面我们将分析光源带宽是怎样影响光通信线路中的信息容量的.

19. 3.2.1 Material Dispersion and Pulse Distortion
Recall that v = c/n.
For glass, n varies with wavelength. Thus, waves of different wavelengths (frequencies) travel at different speeds.
Dispersion:
Wavelength dependent propagation velocity.
Material Dispersion: Dispersion caused by the material.
Waveguide Dispersion: Dispersion caused by the structure of the waveguide.
对于玻璃来说, 不同的波长具有不同折射率n. 那么, 不同波长 (frequencies)的光将具有不同传输速度.
速度随波长变化的特性称为色散. （狭义）和频率f有关的脉冲展宽
广义：所有导致脉冲展宽的因素。
模间色散（多模色散）与f无关，偏振模色散

20. Material Dispersion and Pulse Distortion
Consider a pulse of light emitted by a source which contains a range of wavelengths (say 1, 2, 3).
Input Power
Output Power t 1 2 3
T + 
Arrives last
Arrives first T t
Fastest
wavelength 1 t 2
展宽了 t 3 t
Slowest wavelength

21. Because of dispersion, the components of the input pulse at 1, 2, and 3 travel at different speeds and thus arrive at the receiver at different times. The previous slide displayed how this phenomenon spreads pulses as they travel along a dispersive medium. The output is widened by an amount we label as .
由于色散的存在, 输入光脉冲三个成分 1, 2, and 3 以不同速度在光纤中传输 因此到达接收器时将产生时间差. 前面的幻灯片已经显示了色散介质中脉冲展宽的现象. 我们用参数 定量表征脉冲在色散介质中被展宽了多少.
传输距离越长，脉冲展宽越严重.
光源谱宽越宽，脉冲展宽越严重.

22. Dispersion also distorts an analog signal waveform.
Input Power
Output Power
Pac,in
Pac,out 1 1 2 2 t t
数字系统，色散导致脉冲展宽。
模拟系统，色散导致衰减。
波长1和2波长差很小，波形相似。在输出端，相位变化，导致干涉，造成输出的幅度减小。
Slower wavelength
Pac,out < Pac,in
Information is contained in the amplitude variation.

23. DISPERSION Refractive Index Variation for SiO2 Inflection Point n 1.45n
First Derivative
Second Derivative o
dn2/dl2
n’’ o n’
dn/dl
Inflection point for SiO2 glass occurs near wavelength:

24. Find the amount of pulse spread due to material dispersion.
Let  = time of travel of a pulse over path length L. 1 2 
(t/L)2
(t/L)1
t/L 
t/L 
With No Dispersion
With Dispersion Present
The source linewidth is taken to be (with 2 > 1):
 = 2 - 1

25. The pulse spread per unit length is then:
(3.10)
where 1 is the fastest and 2 is the slowest wavelength.
D(t/L)/Dl = d(t/L)/dl （slope of the curve）
Pulse spread per unit length: D(t/L) = [d(t/L)/dl] Dl (3.12)
Actual spread would be: 1 2 
(t/L)2
(t/L)1
t/L 
脉冲每传输单位距离的展宽

26. D(t/L) = [d(t/L)/dl] Dl = (t/L)’ Dl (3.12)
Two distinct terms determine the pulse spread
1. the slope of the t/L curve
2. the linewidth of the source.
The linewidth will be available from manufacturer's data or must be measured.
Further analysis shows that:
由材料色散导致脉冲展宽中，决定脉冲展宽多少的有两个量. 一是 t/L 曲线的斜率，二是光源的线宽. 一般而言光源线宽可以从光源出厂参数之中获得，也可以通过测量得到. 斜率将在以后内容中再做详细介绍.
斜率：曲线对波长λ求导
(3.13)
The prime and double prime denote first and second derivatives.

27. The pulse travel time is thus:
Proof:
Pulses travel at a speed called the group velocity u. The group velocity is given by:
The pulse travel time is thus:
数字脉冲：包络的速度是群速度u＝dw/dβ
光载波的速度：是相速度
This is the pulse travel time per unit of path length.

28. ( is the free space value)
If n  (), then (/L)’ = 0 and there is no dispersion and no pulse spread.

29. Define material dispersion M :
Combining (3.12) and (3.13):
(3.14)
M (ps/nm/km) is in picoseconds of pulse spread per nanometer of source spectral width and per kilometer of fiber length.
1.3
1.55
0.82
110
-20 M
(ps/(nm.km))
(m)
SiO2
M 单位是ps/nm/km， 含义是当光源线宽1nm，传输长度1km时脉冲展宽多少ps.
M>0，正常色散，M<0,反常色散。

30. 1. For M > 0 (wavelengths < 1.3 mm)
Wavelength 2 arrives before wavelength 1.
Energy at 2 travels faster than energy at 1. (2 > 1)
2. For M < 0 (wavelengths > 1.3 mm)
So that  1 travels faster than wavelength 2.
3. At  1.3 m, M = 0 , and there is no material dispersive pulse spreading.

31. Example: Consider an LED at  = 0. 82 m, L = 10 km, and  = 20 nm
Example: Consider an LED at  = 0.82 m, L = 10 km, and  = 20 nm. Find (/L).
From the graph, at 0.82 mm, M =110 ps/(nm·km).
Change the wavelength to  = 1.5 m,  = 50 nm.
At 1.5 mm, M = -15 ps/nm·km. Then

32. Example:  = 0.82 m,  = 1 nm. M = 110 ps/(nm·km)

33. Between 1200 nm and 1600 nm(near the inflection point), M is given by
Mo = ps/(nm2•km) and o is the zero dispersion
wavelength ( 1300 nm).
Conclusion:
The longer the path the greater the pulse spread.
The greater the source spectral width, the greater the pulse spread.

34. 3.2.2 Solitons
A soliton is a pulse that travel without spreading. The refractive index of glass depends upon the pulse intensity.
This fiber nonlinearity is used to counter the effects of dispersion. The leading edge of the pulse can be slowed down, and the trailing edge speeded up to reduce spreading. Thus, the pulse must be properly shaped.
The nonlinearity is such that solitons are only produced at wavelengths longer than the zero-dispersion wavelength in glass fibers. Compensation to overcome pulse broadening is only possible in the longer wavelength region range 1300 to 1600 nm.
色散带来的脉冲展宽限制了码速。
非线性和反常色散相互作用。
一束光脉冲包含许多不同的频率成分， 不同的频率， 在介质中的传播速度不同。 因此， 光脉冲在光纤中将发生色散， 使得脉宽展宽。 但当具有高强度的极窄单色光脉冲入射到光纤中时， 将产生克尔效应， 即介质的折射率将随着光强发生变化， 由此导致在光脉冲中产生自相位调制(SPM)， 即脉冲前沿产生的相位变化引起频率降低，脉冲后沿产生的相位变化引起频率升高， 于是脉冲前沿比脉冲后沿传播得慢， 从而使脉冲宽度变窄。 当脉冲具有适当的幅度时， 以上两种作用可以恰好抵消， 则脉冲可以保持波形稳定不变地在光纤中传输， 即形成了光孤子。
孤子实际上是在特定条件下从非线性波动方程得到的一个稳定的、 能量有限的不弥散解， 它在传输过程中始终保持其波形和速度不变， 这是由光纤中的色散和非线性效应相互作用而引起的。 基于光孤子的形成机理， 它可以用于光脉冲在光纤中的长距离无畸变传输， 从而构成光孤子通信系统。

35. 单脉冲传输800km时孤子脉冲波形的演变

36. 基态和高阶孤子沿光纤传输时的变化特点

37. Solitons overcome the bandwidth limitations of the fiber, but not the attenuation. Optical amplifiers are needed along the transmission path to maintain the pulse energy above the minimum required for soliton production.
Fiber
Amplifier
孤立子是由非线性场所激发的、能量不弥散的、形态上稳定的准粒子。
虽然孤子可以克服光纤的带宽限制, 但不能克服衰减. 因此就需要光放大器给孤子补充能量， 使光能量不低于产生孤子的要求.

38. 3.2.3 Information Rate
Consider sinusoidal modulation of the light source with modulation frequency f. Modulation period T = 1/f.
Power

39. Information Rate Information Rate Blue: l1 Red: l2 consder T/2 Dt Time
交流分量＝0，只有直流分量
T/2 Dt
Time
This spread reduces the total power variation to zero. Modulation is canceled.

40. Information Rate
The limit on the allowable pulse spread will be taken to be:
(2)

41. Information Rate From (2) we have the requirement that
1/T < 1/(2Dt) (3)
so that the modulation frequency has the limits:
(4)
The maximum modulation frequency is then:
This modulation frequency turns out to be the 3-dB bandwidth. The signal is actually reduced by half (3-dB) at this modulation frequency.
3-dB optical bandwidth:
(5)

42. Information Rate
The total signal loss has two parts and can be expressed by the equation:
(6)
La = Loss due to absorption and scattering (fixed loss).
Lf = Modulation (message) frequency dependent loss.
The modulation frequency dependent loss is given by:
(7)

43. Information Rate Example: Suppose f = f3-dB. Compute the loss.
The equation predicts no modulation frequency loss for modulation frequencies well below the 3-dB frequency.

44. Information Rate Example: Suppose f = 0.1 f3-dB. Compute the loss.
Maximum frequency length product is calculated from (5) as follows:
(5)
(8)
(3.16)

45. Information Rate Find the frequency at which Lf = 1.5 dB.
Use (7) for Lf
(7)
Solving for the frequency at which the loss is 1.5 dB, we obtain
1.5dB光带宽是一个重要参数，它是对应于接收机电功率衰减一半时的频率。也就是说，1.5dB光带宽等于3dB电带宽

46. Information Rate Now consider the photodetection circuit:RL P
Photodetector
Optical Power i
P = incident optic power
i = P detector output current
 = detector responsivity (A/w)
The electrical power in the load resistor RL is:
i和P光成正比

47. Information Rate
Consider two optical power levels P1 and P2 and their corresponding electrical power levels Pe1 and Pe2.
光功率下降1.5dB,电功率下降3dB

48. Information Rate Examples:
A loss of 3 dB in optical power yields a loss of 6 dB in the corresponding electrical power.
A loss of 1.5 dB in optical power yields a loss of 3 dB in the corresponding electrical power.
光功率损耗3dB对应响应电功率损耗6dB.
光功率损耗1.5dB对应电功率损耗3dB

49. Information Rate
We found that the modulation frequency at which the optical loss is 1.5 dB was:
(3.18)
Electrical 3-dB bandwidth length product is:
1.5dB光带宽是一个重要的参数，对应接收机的电功率衰减一半时的频率。
1.5dB光带宽等于3dB电带宽
(3.19)

50. Information Rate Consider a Return-to-Zero (RZ) digital signal. Power tp t
T T T T T T T
Each bit is allotted a time T.
tp = T/2 pulse width
R = 1/T data rate, b/s
每个bit占用时间间隔T,每个脉冲占用半个时隙
在一个码元周期归0，占空比50％，因为归0，高频成分多

51. Information Rate Spectrum of the RZ Signal Power Spectral Density
Frequency
Power
Spectral
Density
(Watts/Hz)
Most of the signal power lies below 1/T Hz, so the required transmission bandwidth by a system is:
绝大部分的信号功率在1/THz以下，所以RZ信号在带宽为1/THz的系统中能充分传输。

52. Information Rate
If the system passes this band of frequencies the pulses will be recognizable. To be conservative, use the 3-dB electrical bandwidth.
The RZ rate length product is then:
如果系统以这个速率传输信号，那么信号可以被接收并被恢复. 保守一点, 我们使用 3-dB 电带宽
(3.20)

53. Information Rate
We obtain the same result by allowing a pulse spread of 70% of the initial pulse duration.
假设最大允许的脉冲展宽等于脉冲宽度的70％
As on the preceding slide.

54. Information Rate Consider the Non-Return-to-Zero (NRZ) digital signal.tp
T T T T T T T
Frequency
Power
Spectral
Density
(Watts/Hz)
Spectrum of the (NRZ) Signal
高频成分少。
Required transmission bandwidth:

55. Information Rate The allowed data rate is:
Use the electrical 3-dB bandwidth:
NRZ rate length product is:
(3.21)
Comparing the results for the RZ and NRZ data rates:

56. BANDWIDTH DATA RATE SUMMARY
Information Rate
BANDWIDTH DATA RATE SUMMARY

57. 3.3 Polarization Linearly polarized:
An electric field points in just one direction, it always points along a single line.
a. Linearly polarized in x direction and traveling in the z direction.
b. linearly polarized in y direction and traveling in the z direction. y z x E v y z x E v
偏振引起偏振模色散，在高速传输时，很严重。
线偏振：对于一个电磁波若电场矢量仅指向一个方向则称为线偏振，因为该指向总是在同一直线上。
下图所示为一个沿z轴方向传播的线偏振波，电场矢量指向x方向.
(a)
(b)

58. Polarization
1. The two orthogonal linear polarizations are the plane wave modes of an unbounded media.
2. They can exist simultaneously.
3. The actual polarization is determined by the polarization of the light source and by other polarization sensitive components in the optical system.
两种正交线偏振波都是在无边界介质中平面波的模式. 他们可以同时存在. 实际光的偏振方向由光源的偏振和光通信系统中其它偏振敏感元件共同决定 .

59. Polarization
If the direction of electric field E varies randomly (as shown) the wave is unpolarized.
Most fibers depolarize the input light. Only special fibers maintain the light polarization. x y E
如果电场矢量的方向发生随机变化，那么它就是非偏振光.
在光纤中传播的光绝大多数都是非偏振光. 只有一些特殊光纤保持偏振光的偏振状态.

60. 3.4 Resonant Cavities A
1、让任意一个光波从左边的镜子传向右边的镜子，如A图所示。绿波在右边的镜子处发生发射，因此这个波经历了一次180度相移。从A图我们可以看出，这个波在其相位上发生了中断，在这里应该是不可能的，也就是说，这个谐振器不支持这个波。 B
2、在图B中，在右边的镜子处，这个波也发生了一个180度相移，然后继续传播，在左边的镜子处，同样经历了一个180度相移，然后继续传播。因此，图B所示的波有着一个稳定的模式，我们称之为驻波

61. Resonant Cavities (3.22) Design:
Standing-wave pattern in a cavity (m = 4)
Design:
The cavity must be an integral number of half wavelengths long to support a wave.
The wavelength is that in the medium filling the cavity.
腔长必须是支持波的半波长整数倍. 波长应该是填充该腔介质中的波长.
在右边的镜子处，这个波也发生了一个180度相移，然后继续传播，在左边的镜子处，同样经历了一个180度相移，然后继续传播。因此，图中所示的波有着一个稳定的模式，我们称之为驻波

62. Resonant Cavities The resonant wavelengths are: (3.23)
The corresponding resonant frequencies are:

63. Resonant Cavities Cavity Resonant frequencies Frequency
This picture shows the longitudinal modes of the cavity.
The resonant frequency spacing is:
谐振腔支持的纵模. 下面我们将讨论纵模间隔，即可以谐振的频率之间的间隔.
(3.25)

64. Resonant Cavities
The free space wavelength spacing corresponding to fc is c calculated from:
(3.26)
This equation refers to the free space wavelengths.

65. Resonant Cavities Example: Consider an AlGaAs laser cavity.
L = 0.3 mm = 300 m; n = 3.6; o = 0.82 m.
Find the cavity resonant wavelength spacing c.

66. Resonant Cavities
Example: Suppose the AlGaAs, LD has a spectral width of 2 nm. Determine the number of longitudinal modes in the output.
2 nm
0.82 m
Gain (AlGaAs) 
Cavity Resonances
c
0.82 m

67. Resonant Cavities Laser Output 2 nm c  0.82 m
The laser emits 6 longitudinal modes.
A laser emitting only one longitudinal mode is a single-mode laser.

68. 3.5 Reflection at a Plane Boundary
Consider normal incidence of light at a boundary.
This is referred to as Fresnel Reflection. n1 n2
Incident Wave
Transmitted Wave
Reflected Wave
Boundary
Reflection Coefficient:

69. Reflection at a Plane Boundary
Reflection Coefficient:

70. Reflection at a Plane Boundary
Define Reflectance R (反射比) as:
反射比
This result is valid for normal incidence.

71. Reflection at a Plane Boundary
For air-to-glass, compute the transmitted power.
4% power reflected. 96% power transmitted.
In dB, the transmitted power is:
10 lg (0.96) = dB
Typically we round this off to 0.2 dB (omitting the minus sign). This is called the Fresnel loss.
菲涅尔损耗

72. Reflection at a Plane Boundary Perpendicular Polarization (s)
Consider arbitrary incidence:  Et
n n2 Er  t r i  Ei
Perpendicular Polarization (s)
垂直偏振

73. Reflection at a Plane Boundary Parallel Polarization (p)
Consider arbitrary incidence: Et
n n2 Er t r i Ei
平行偏振
Parallel Polarization (p)
平行偏振

74. Reflection at a Plane Boundary
Plane of Incidence(入射平面)
Defined by the normal to the boundary and the ray direction of the incident beam. x z
定义边界法线和入射波传播的方向构成的平面为入射平面.
Incident
Boundary
The xz plane is the plane of incidence in this example.

75. Reflection at a Plane Boundary
Fresnel’s Law of Reflection
For parallel polarization, the reflection coefficient:
(3.29)
For perpendicular polarization, the reflection coefficient:
(3.30)
Note that  may be complex.

76. Reflection at a Plane Boundary
Plots of p and s for n1 = 1 (air), n2 = 1.48 (glass)
Parallel (rp)
Perpendicular (rs) rs rp
Angle of incidence (qi)

77. Reflection at a Plane Boundary
From equation (3.29) for parallel polarization, we can get total transmission (no reflection) if
The angle satisfying this equation is the Brewster angle B. The solution is:
Compute B for air-to-glass and glass-to-air:
For n1 = 1, n2 = 1.5
For n1 = 1.5, n2 = 1
For perpendicular polarization there is no Brewster angle. No  i s.t. Equ.3.30 = 0.

78. Reflection at a Plane Boundary
Antireflection Coatings
We have just seen that we can transmit a beam from one material to another without reflection under Brewster angle conditions. We can also transmit with no (or very little) reflection by placing a coating between the two materials. n1 n2 n3
l/4
The thickness of the coating is a quarter wavelength. The reflectance R for this configuration is:
防反射涂层

79. Reflection at a Plane Boundary
Clearly, the reflectance becomes zero if:
A coating that reduces the reflectance is called an antireflection (AR) coating .消反射涂覆
Example: Compute the reflectance when a quarter wavelength of magnesium fluoride (氟化镁n = 1.38) is coated onto a piece of glass (n = 1.5).
Solution:
The reflectance is:
当四分之一波长厚的氟化镁(n = 1.38)涂在一块玻璃(n = 1.5) 上反射比为多少
4% → 1.4%

80. 3.6 Critical Angle Reflection
Fresnel’s Law of Reflection
For parallel polarization:
(3.29)
For perpendicular polarization:
(3.30)
菲涅尔反射定律

81. Critical Angle Reflection
From equations (3.29) and (3.30), we find that
The incident angle satisfying this equation is the angle whose sine is given by:
(3.34)
Call the solution c, the critical angle.
全反射临界角
c exists only if n1 > n2. That is, travel from a high index to a low index material. This result is valid for both polarizations.

82. Critical Angle ReflectionIf
then
is purely imaginary.
Under this condition, equations (3.29) and (3.30) can be written in the form:
纯虚数, j 为虚数单位
where A, B, C, and D are real and j is the imaginary term

83. Critical Angle Reflection
Then:
We conclude that there is complete reflection (called critical angle reflection) for all rays which satisfy the condition:
发生全反射

84. Critical Angle Reflection
Consider waves undergoing critical angle reflections:
n n2
In region n1 we have a standing wave due to the
interference of the incident and reflected waves.

85. Critical Angle Reflection
In region n2 the electric field is not zero. The
boundary conditions require the electric field to be continuous at the boundary. The field in n2 termed as evanescent is a decaying exponentially carrying no power.
where the attenuation coefficient is given by
在折射率为 n2 区域电场其实不为零. 边界条件要求电场在边界处连续. 在折射率为 n2 的区域仍有电场，电场随着导边界距离的增加按指数衰减.

86. Critical Angle Reflection
Consider a wave where
Evanescent Wave E z
Standing Wave
n n2
Envelope
e-z
这里衰减因子和前面提到的衰减系数不同，衰减系数是指功率的实际损耗，
这里衰减因子并不具有这样的含义，仅仅指电磁波回到入射区之前，场在第二种介质中要传播多远。
The decaying wave carries no power in the z-direction.
衰减波在第二种介质中并没有功率损失.这里衰减因子和前面提到的衰减系数不同，衰减系数是指功率的实际损耗，
这里衰减因子并不具有这样的含义，仅仅指电磁波回到入射区之前，场在第二种介质中要传播多远。
At the critical angle,
In this case, there is no decay. The wave penetrates deeply into the second medium.

87. Critical Angle Reflection
As i increases,  increases and the decay becomes
greater.
 = 0,  = c
|E|
e-z
i > c z
As i increases from qc towards 90o, a increases and the evanescent field penetrates less and less into the second medium. qi

88. LIGHTWAVE FUNDAMENTALS
Pulse spread
3-dB bandwidths
Rate-length products
Reflectance
Critical angle reflections