超快光学 第05章 色散
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Etot (t ) 2E0 cos(avet )cos(t )
Adding waves of two different frequencies yields the product of a rapidly varying cosine (ave) and a slowly varying cosine ().
Pulse compression Chirped mirrors源自文库
Dispersion in Optics
The dependence of the refractive index on wavelength has two effects on a pulse, one in space and the other in time. Dispersion disperses a pulse in space (angle):
In phase
It’s usually very difficult to see optical beats because they occur on a time scale that’s too fast to detect. This is why we say that beams of different colors don’t interfere, and we only see the average intensity.
1 2
2
and
1 2
2
1 ave 2 ave
Etot (t ) Re{E0 exp i(avet t ) E0 exp i(avet t )} Re{E0 exp(iavet )[exp(it ) exp(it )]} Re{2 E0 exp(iavet ) cos(t )}
Out of phase
Individual waves
Sum
Etot ( z, t ) 2E0 cos(kave z avet ) Pulse(kz t )
Irradiance
2p/max Pulse separation: 2p/min time
Group Velocity
So the group velocity equals the phase velocity when dn/d = 0, such as in vacuum. But n usually increases with , so dn/d > 0, and: vg < v
The group velocity is less than the phase velocity in non-absorbing regions.
? ( ) E in
H()
? ( ) E out
? ( ) H ( ) E ? ( ) E out in
where H() is the transfer function of the device/medium:
~
Optical device or medium
exp[ ( ) L / 2] for a medium
amplitude
The phase velocity comes from the rapidly varying part: v = ave / kave
What about the other velocity—the velocity of the pulse amplitude? Define the group velocity: vg /k Taking the continuous limit, we define the group velocity as:
Traveling-Wave Beats
In phase Out of phase In phase Out of phase In phase
Individual waves Sum
Envelope
Intensity
z
Seeing Beats
However, a sum of many frequencies will yield a train of well-separated pulses:
taking E0 to be real.
Let Similiarly, So :
ave
Etot ( z, t ) Re{E0 exp i(kave z kz avet t ) E0 exp i (kave z kz avet t )} Re{E0 exp i (kave z avet ) exp i (kz t ) exp[i (kz t )]} Re{2 E0 exp i (kave z avet ) cos(kz t )} 2 E0 cos(kave z avet ) cos(kz t )
c0 l0 dn v g / 1 n n d l0
Spectral Phase and Optical Devices
Recall that the effect of a linear passive optical device (i.e., windows, filters, etc.) on a pulse is to multiply the frequency-domain field by a transfer function:
If n1 n2 ,
Phase and Group Velocities
In vacuum
Unrealistic
Most common case
Possible
Unrealistic
Unrealistic
Calculating the group velocity
vg d /dk
Now, is the same in or out of the medium, but k = k0 n, where k0 is the k-vector in vacuum, and n is what depends on the medium. So it's easier to think of as the independent variable:
Use the chain rule :
dn dn d l0 d d l0 d d l0 2p c0 2p c0 l02 2 2 d (2p c0 / l0 ) 2p c0
Now, l0 2p c0 / , so : Recalling that :
we have : or :
Dispersion and Ultrashort Pulses
Angular dispersion and group-velocity dispersion Phase and group velocities Group-delay dispersion
Negative groupdelay dispersion
Angular dispersion dn/dl
qout(blue) > qout(red)
Dispersion also disperses a pulse in time: Group delay dispersion or Chirp d2n/dl2
vgr(blue) < vgr(red)
Both of these effects play major roles in ultrafast optics.
Calculating group velocity vs. wavelength
We more often think of the refractive index in terms of wavelength, so let's write the group velocity in terms of the vacuum wavelength l0.
vg = (c0 /n) / (1+ dn/d) = v / (1+ dn/d)
Except in regions of anomalous dispersion (near a resonance and which are absorbing), dn/d is positive. So vg < v for most frequencies!
c0 k1 c0 k2 vg k n1k1 n2 k2
where k1 and k2 are the k-vector magnitudes in vacuum.
If n1 n2 n,
c0 k1 k2 c0 vg v phase velocity n k1 k2 n v g v c0 / n
vg d /dk
Group velocity is not equal to phase velocity if the medium is dispersive (i.e., n varies).
Evaluate the group velocity for the two-frequency case:
Light-wave beats (continued): E tot(z,t) = 2E0 cos(kavez–avet) Pulse(kz–t)
carrier wave
This is a rapidly oscillating wave: [cos(kavez–avet)] with a slowly varying amplitude: [2E0 Pulse(kz–t)]
c dn v g 0 / 1 n n d 2p c0 dn l02 c0 v g / 1 nl0 d l0 2p c0 n
H ( ) BH ( ) exp[iH ( )]
When two functions of different frequency interfere, the result is beats.
Etot (t ) Re{E0 exp(i1t ) E0 exp(i2t )}
Let:
taking E0 to be real.
ave
When two waves of different frequency interfere, the result is beats.
In phase Out of phase In phase Out of phase In phase
Individual waves Sum
2p/
Envelope
v g dk / d
1
Using k = n() / c0, calculate: dk /d = (n + dn/d) / c0 vg c0 / (n dn/d) = (c0 /n) / (1 + /n dn/d ) Finally:
dn v g v / 1 n d
Irradiance
time
When two waves of different frequency interfere, they also produce beats.
Etot ( z, t ) Re{E0 exp i(k1 z 1t ) E0 exp i(k2 z 2t )} kave k1 k2 k k and k 1 2 2 2 1 2 and 1 2 2 2
Adding waves of two different frequencies yields the product of a rapidly varying cosine (ave) and a slowly varying cosine ().
Pulse compression Chirped mirrors源自文库
Dispersion in Optics
The dependence of the refractive index on wavelength has two effects on a pulse, one in space and the other in time. Dispersion disperses a pulse in space (angle):
In phase
It’s usually very difficult to see optical beats because they occur on a time scale that’s too fast to detect. This is why we say that beams of different colors don’t interfere, and we only see the average intensity.
1 2
2
and
1 2
2
1 ave 2 ave
Etot (t ) Re{E0 exp i(avet t ) E0 exp i(avet t )} Re{E0 exp(iavet )[exp(it ) exp(it )]} Re{2 E0 exp(iavet ) cos(t )}
Out of phase
Individual waves
Sum
Etot ( z, t ) 2E0 cos(kave z avet ) Pulse(kz t )
Irradiance
2p/max Pulse separation: 2p/min time
Group Velocity
So the group velocity equals the phase velocity when dn/d = 0, such as in vacuum. But n usually increases with , so dn/d > 0, and: vg < v
The group velocity is less than the phase velocity in non-absorbing regions.
? ( ) E in
H()
? ( ) E out
? ( ) H ( ) E ? ( ) E out in
where H() is the transfer function of the device/medium:
~
Optical device or medium
exp[ ( ) L / 2] for a medium
amplitude
The phase velocity comes from the rapidly varying part: v = ave / kave
What about the other velocity—the velocity of the pulse amplitude? Define the group velocity: vg /k Taking the continuous limit, we define the group velocity as:
Traveling-Wave Beats
In phase Out of phase In phase Out of phase In phase
Individual waves Sum
Envelope
Intensity
z
Seeing Beats
However, a sum of many frequencies will yield a train of well-separated pulses:
taking E0 to be real.
Let Similiarly, So :
ave
Etot ( z, t ) Re{E0 exp i(kave z kz avet t ) E0 exp i (kave z kz avet t )} Re{E0 exp i (kave z avet ) exp i (kz t ) exp[i (kz t )]} Re{2 E0 exp i (kave z avet ) cos(kz t )} 2 E0 cos(kave z avet ) cos(kz t )
c0 l0 dn v g / 1 n n d l0
Spectral Phase and Optical Devices
Recall that the effect of a linear passive optical device (i.e., windows, filters, etc.) on a pulse is to multiply the frequency-domain field by a transfer function:
If n1 n2 ,
Phase and Group Velocities
In vacuum
Unrealistic
Most common case
Possible
Unrealistic
Unrealistic
Calculating the group velocity
vg d /dk
Now, is the same in or out of the medium, but k = k0 n, where k0 is the k-vector in vacuum, and n is what depends on the medium. So it's easier to think of as the independent variable:
Use the chain rule :
dn dn d l0 d d l0 d d l0 2p c0 2p c0 l02 2 2 d (2p c0 / l0 ) 2p c0
Now, l0 2p c0 / , so : Recalling that :
we have : or :
Dispersion and Ultrashort Pulses
Angular dispersion and group-velocity dispersion Phase and group velocities Group-delay dispersion
Negative groupdelay dispersion
Angular dispersion dn/dl
qout(blue) > qout(red)
Dispersion also disperses a pulse in time: Group delay dispersion or Chirp d2n/dl2
vgr(blue) < vgr(red)
Both of these effects play major roles in ultrafast optics.
Calculating group velocity vs. wavelength
We more often think of the refractive index in terms of wavelength, so let's write the group velocity in terms of the vacuum wavelength l0.
vg = (c0 /n) / (1+ dn/d) = v / (1+ dn/d)
Except in regions of anomalous dispersion (near a resonance and which are absorbing), dn/d is positive. So vg < v for most frequencies!
c0 k1 c0 k2 vg k n1k1 n2 k2
where k1 and k2 are the k-vector magnitudes in vacuum.
If n1 n2 n,
c0 k1 k2 c0 vg v phase velocity n k1 k2 n v g v c0 / n
vg d /dk
Group velocity is not equal to phase velocity if the medium is dispersive (i.e., n varies).
Evaluate the group velocity for the two-frequency case:
Light-wave beats (continued): E tot(z,t) = 2E0 cos(kavez–avet) Pulse(kz–t)
carrier wave
This is a rapidly oscillating wave: [cos(kavez–avet)] with a slowly varying amplitude: [2E0 Pulse(kz–t)]
c dn v g 0 / 1 n n d 2p c0 dn l02 c0 v g / 1 nl0 d l0 2p c0 n
H ( ) BH ( ) exp[iH ( )]
When two functions of different frequency interfere, the result is beats.
Etot (t ) Re{E0 exp(i1t ) E0 exp(i2t )}
Let:
taking E0 to be real.
ave
When two waves of different frequency interfere, the result is beats.
In phase Out of phase In phase Out of phase In phase
Individual waves Sum
2p/
Envelope
v g dk / d
1
Using k = n() / c0, calculate: dk /d = (n + dn/d) / c0 vg c0 / (n dn/d) = (c0 /n) / (1 + /n dn/d ) Finally:
dn v g v / 1 n d
Irradiance
time
When two waves of different frequency interfere, they also produce beats.
Etot ( z, t ) Re{E0 exp i(k1 z 1t ) E0 exp i(k2 z 2t )} kave k1 k2 k k and k 1 2 2 2 1 2 and 1 2 2 2