Introduction to waves
Waves have crests and troughs.The crest of a wave is sometimes called a wave front.
The shape of a wave is determined
by its wave front.
In physics, a wave front is the locus (a line, or, in a wave propagating in 3 dimensions, a surface) of points having the same phase.
1
rr
E(x,t)=E0cos(kx-ωt+ϕ0)
Waves have cycles, frequency, and amplitude, just like
oscillations.
The frequency of a wave tells how often each point oscillates.
The wavelength of a wave is the length of one complete cycle.
The amplitude of a wave is the maximum movement from equilibrium.
2
X
rr
E(x,t)=E0cos(kx-ωt+ϕ0)
空间参量与时间参量
空间周期
λ
时间周期T=
λ
v
空间频率f=
1
λ
1
时间频率υ=
T
2π
传播数k=2πf=
λ
2π
时间角频率ω=2πυ=
T
ω=kv
•简谐波相位的多种表达形式
⎛zt⎫⎛z⎫φ=2π -⎪+φ0=ω -t⎪+φ0
⎝λT⎭⎝v⎭⎛z⎫
=2π -vt⎪+φ0=2π(fz-υt)+φ0
⎝λ⎭
3
rrrrr
E(r,t)=E0cos(k⋅r-ωt+ϕ0)
f(z,t)=Acos⎡kz-vtω=kv⎤()⎣⎦
Transverse Wave:
• Note how the wave pattern definitely moves
4
to the right.
• However any particular point (look at the blue one) just moves transversely (i.e., up and down) to the direction of the wave.
Wave Velocity (Wave Phase Velocity): • The wave velocity is defined as the wavelength divided by the time it takes a wavelength (green) to pass by a fixed point
(blue).
5
The phase velocityof a wave is the rate at which the phase of the wave propagates in space. This is the velocity at which the phase of any single frequency component of the wave will propagate. We can pick one particular phase of the wave (for example the crest) and it would appear to travel at the phase velocity.
vphase
●
2π/τλ===k2π/λτ
ω
0.0,
r1n
..r1
Snapshots of a wave with angular frequency ωare shown at 3 times:
Which of the following expressions describes this wave?
=0
(a) y = sin(kx-ωt)
(b) y = sin(kx+ωt) (c) y =cos(kx+ωt)
π
=2ω
In what direction is this wave traveling?
t=
x
πω
(a) +x direction (b) 6
0.0,
r1n
..r1
=0
π
=2ω
πt=ω
x
•We claim this wave moves in the -x direction.•The orange dot marks a point of constant phase.
•
It clearly moves in the -x direction as time increases!!
⏹ Wave-fronts are helpful for drawing pictures of interfering waves.
⏹ A wave's wave-fronts sweep along at the speed of light.
⏹ A plane wave has flat wave-fronts throughout all space. It also has infinite energy. It doesn’t exist in reality.
7
空间参量与时间参量
空间周期
λ
时间周期T=
λ
v
空间频率f=
1
λ
1
时间频率υ=
T
2π
传播数k=2πf=
λ
2π
时间角频率ω=2πυ=
T
ω=kv
rrrrr
E(r,t)=E0cos(k⋅r-ωt+ϕ0) urr
E(z,t)=E0cos⎡kz-vtω=kv⎤()⎣⎦
8
复数表示的优点
•将波函数中与空间坐标有关的因子和时间坐标有关的因子分离•简化运算运算规则
•线性运算：（加、减、乘常数、除常数、微分、积分），将复数形式代入进行运算，最后结果取实部
rrrrr
E(r,t)=E0cos(k⋅r-ωt+ϕ0)
rrrrrrrrrk⋅r-ωt+ϕ0(r)⎤i⎡k⋅r+ϕ0(r)⎤-iωt°r±i⎡±⎦=E⎣⎦E(r,t)=E0e⎣0ee
In Rectangular Coordinate System
•乘除：先分别取实部，再乘除。
9
In Spherical Coordinate System
常用的微分运算表达式为
A
ψ(r,t)=cos⎡k(r-vt)⎤⎣⎦r
10
Aik(r-vt)
ψ(r,t)=e
r
A spherical waveis a constant-frequency wave whose wavefronts(surfaces of constant phase) are
parallel concentric spheres of constant amplitude normal to the phase velocity vector.
When the distance from the source is very large, a spherical wave can be locally
approximated as a plane wave.
In Cylindrical Coordinate System
常用的微分表达式为
Harmonic cylindrical wave
° ψ
(
ρ,t)=
A
A
e
ik(ρ-vt)
ψ(
ρ,t)=
cos⎡⎣k(ρ-vt)⎤⎦
A perfect cylindrical wave
The length of wave train of a single photon
•单色均匀平面光波是一种理想模型，严格的单色平面
光波在时间和空间上都无限扩展，实际上是不存在的。
•实际上，普通光源，原子发出的光波是由一段段有限长的称为波列的光波组成的；每段波列，其振幅在持续时间内保持不变或缓慢变化，前后各段之间没有固定的相位关系，甚至光矢量的振动方向也不同。•时间上有限、空间上有限的实际光波，根据傅立叶变换，可以表示为不同频率、不同传播方向的单色波的叠加。
E(z,t)=∑E0lcos(klz-ωlt)
l
N
E(t)=⎰+∞
E(ν)exp(-i2πνt)d-1
-∞
ν=F
[E(ν)]
⎧eE(t)=⎨
⎩
-βt
e,t≥0
0,t
-i2πν0t
E(ν)=⎰e
-∞
+∞
-βt
e
-i2πν0t
e
i2πνt
dt=
i2πν-ν0+iβ
E(ν=E(ν)E(ν)=
2
*
1
4πν-ν0+β
2
2
2
(t-t0)2
ψ (r,t)=A2τ2
cos⎡⎣k(z-vt)⎤⎦
0e
-
Light source
Cd Lamp Kr Lamp Hg Lamp Ne Lamp Incandescent
Lamp He-Ne Laser
(CW) He-Ne Laser (Single Mode)
λ0(nm) Δλ(nm)
643.8 605.8 546.1 632.8 550 632.8 632.8
1.3*10-3 5.5*10-3
5
2.0*10-3
Τ L0
good 1.1ns 0.32m good 0.22ns 0.067m poor 0.2ps 60μm good 0.67ns 0.2m Very
300 ~3fs ~1μm
poor good
~1.3*10-3 ~1ns 0.2~0.3m 1.0*10-9
Very
1.3ms
good
4*105 m
The group velocityof a wave is the velocity with which the variations in the shape of the wave's amplitude (modulationor envelopeof the wave) propagate through space.
vgroup
dω=dk
21
22
Introduction to waves
Waves have crests and troughs.The crest of a wave is sometimes called a wave front.
The shape of a wave is determined
by its wave front.
In physics, a wave front is the locus (a line, or, in a wave propagating in 3 dimensions, a surface) of points having the same phase.
1
rr
E(x,t)=E0cos(kx-ωt+ϕ0)
Waves have cycles, frequency, and amplitude, just like
oscillations.
The frequency of a wave tells how often each point oscillates.
The wavelength of a wave is the length of one complete cycle.
The amplitude of a wave is the maximum movement from equilibrium.
2
X
rr
E(x,t)=E0cos(kx-ωt+ϕ0)
空间参量与时间参量
空间周期
λ
时间周期T=
λ
v
空间频率f=
1
λ
1
时间频率υ=
T
2π
传播数k=2πf=
λ
2π
时间角频率ω=2πυ=
T
ω=kv
•简谐波相位的多种表达形式
⎛zt⎫⎛z⎫φ=2π -⎪+φ0=ω -t⎪+φ0
⎝λT⎭⎝v⎭⎛z⎫
=2π -vt⎪+φ0=2π(fz-υt)+φ0
⎝λ⎭
3
rrrrr
E(r,t)=E0cos(k⋅r-ωt+ϕ0)
f(z,t)=Acos⎡kz-vtω=kv⎤()⎣⎦
Transverse Wave:
• Note how the wave pattern definitely moves
4
to the right.
• However any particular point (look at the blue one) just moves transversely (i.e., up and down) to the direction of the wave.
Wave Velocity (Wave Phase Velocity): • The wave velocity is defined as the wavelength divided by the time it takes a wavelength (green) to pass by a fixed point
(blue).
5
The phase velocityof a wave is the rate at which the phase of the wave propagates in space. This is the velocity at which the phase of any single frequency component of the wave will propagate. We can pick one particular phase of the wave (for example the crest) and it would appear to travel at the phase velocity.
vphase
●
2π/τλ===k2π/λτ
ω
0.0,
r1n
..r1
Snapshots of a wave with angular frequency ωare shown at 3 times:
Which of the following expressions describes this wave?
=0
(a) y = sin(kx-ωt)
(b) y = sin(kx+ωt) (c) y =cos(kx+ωt)
π
=2ω
In what direction is this wave traveling?
t=
x
πω
(a) +x direction (b) 6
0.0,
r1n
..r1
=0
π
=2ω
πt=ω
x
•We claim this wave moves in the -x direction.•The orange dot marks a point of constant phase.
•
It clearly moves in the -x direction as time increases!!
⏹ Wave-fronts are helpful for drawing pictures of interfering waves.
⏹ A wave's wave-fronts sweep along at the speed of light.
⏹ A plane wave has flat wave-fronts throughout all space. It also has infinite energy. It doesn’t exist in reality.
7
空间参量与时间参量
空间周期
λ
时间周期T=
λ
v
空间频率f=
1
λ
1
时间频率υ=
T
2π
传播数k=2πf=
λ
2π
时间角频率ω=2πυ=
T
ω=kv
rrrrr
E(r,t)=E0cos(k⋅r-ωt+ϕ0) urr
E(z,t)=E0cos⎡kz-vtω=kv⎤()⎣⎦
8
复数表示的优点
•将波函数中与空间坐标有关的因子和时间坐标有关的因子分离•简化运算运算规则
•线性运算：（加、减、乘常数、除常数、微分、积分），将复数形式代入进行运算，最后结果取实部
rrrrr
E(r,t)=E0cos(k⋅r-ωt+ϕ0)
rrrrrrrrrk⋅r-ωt+ϕ0(r)⎤i⎡k⋅r+ϕ0(r)⎤-iωt°r±i⎡±⎦=E⎣⎦E(r,t)=E0e⎣0ee
In Rectangular Coordinate System
•乘除：先分别取实部，再乘除。
9
In Spherical Coordinate System
常用的微分运算表达式为
A
ψ(r,t)=cos⎡k(r-vt)⎤⎣⎦r
10
Aik(r-vt)
ψ(r,t)=e
r
A spherical waveis a constant-frequency wave whose wavefronts(surfaces of constant phase) are
parallel concentric spheres of constant amplitude normal to the phase velocity vector.
When the distance from the source is very large, a spherical wave can be locally
approximated as a plane wave.
In Cylindrical Coordinate System
常用的微分表达式为
Harmonic cylindrical wave
° ψ
(
ρ,t)=
A
A
e
ik(ρ-vt)
ψ(
ρ,t)=
cos⎡⎣k(ρ-vt)⎤⎦
A perfect cylindrical wave
The length of wave train of a single photon
•单色均匀平面光波是一种理想模型，严格的单色平面
光波在时间和空间上都无限扩展，实际上是不存在的。
•实际上，普通光源，原子发出的光波是由一段段有限长的称为波列的光波组成的；每段波列，其振幅在持续时间内保持不变或缓慢变化，前后各段之间没有固定的相位关系，甚至光矢量的振动方向也不同。•时间上有限、空间上有限的实际光波，根据傅立叶变换，可以表示为不同频率、不同传播方向的单色波的叠加。
E(z,t)=∑E0lcos(klz-ωlt)
l
N
E(t)=⎰+∞
E(ν)exp(-i2πνt)d-1
-∞
ν=F
[E(ν)]
⎧eE(t)=⎨
⎩
-βt
e,t≥0
0,t
-i2πν0t
E(ν)=⎰e
-∞
+∞
-βt
e
-i2πν0t
e
i2πνt
dt=
i2πν-ν0+iβ
E(ν=E(ν)E(ν)=
2
*
1
4πν-ν0+β
2
2
2
(t-t0)2
ψ (r,t)=A2τ2
cos⎡⎣k(z-vt)⎤⎦
0e
-
Light source
Cd Lamp Kr Lamp Hg Lamp Ne Lamp Incandescent
Lamp He-Ne Laser
(CW) He-Ne Laser (Single Mode)
λ0(nm) Δλ(nm)
643.8 605.8 546.1 632.8 550 632.8 632.8
1.3*10-3 5.5*10-3
5
2.0*10-3
Τ L0
good 1.1ns 0.32m good 0.22ns 0.067m poor 0.2ps 60μm good 0.67ns 0.2m Very
300 ~3fs ~1μm
poor good
~1.3*10-3 ~1ns 0.2~0.3m 1.0*10-9
Very
1.3ms
good
4*105 m
The group velocityof a wave is the velocity with which the variations in the shape of the wave's amplitude (modulationor envelopeof the wave) propagate through space.
vgroup
dω=dk
21
22