超快光学 非线性光学

Sharp edges require higher frequencies in the Fourier series.
We hear this as distortion.
Energy
Nonlinear effects in atoms
Another way to look at nonlinear optics is that the potential of the electron is an atom is not a simple quadratic potential.
If the motion isn’t sinusoidal, the medium will also emit the additional frequencies!
Maxwell's Equations in a Medium
The polarization, P , contains the effect of the medium:
Nonlinear optics isn’t something you see everyday.
Sending infrared light into a crystal yielded this display of green light (second-harmonic generation, SHG):
2 = 2nd harmonic!
Harmonic generation is one of many exotic effects that can arise!
Sum- and difference-frequency generation
Suppose there are two different-color beams present:
For weak fields, motion is harmonic, and linear optics prevails. For strong fields (i.e., lasers), anharmonic motion occurs, and higher harmonics occur, both in the motion and the light emission.
Now, suppose the irradiance is high enough that many molecules are excited to the higher-energy state. This state can then act as the lower level for additional excitation. This yields vibrations at all frequencies corresponding to all energy differences between populated states.
1. Intensities of daily life are too weak. 2. Normal light sources are incoherent. 3. The occasional crystal we see has the wrong symmetry (for SHG). 4. Phase-matching is required, and it doesn't usually happen on its own.
Complicated nonlinear-optical effects can occur.
sig
Nonlinear-optical processes are often referred to as:
N-wave-mixing processes
where N is the number of photons involved (including the emitted one). This is a sixwave-mixing process.
We can also imagine this process in terms of the molecular energy levels, using arrows for the photon energies:
Why do nonlinear-optical effects occur? (continued)
Molecules Excited by Laser Light
Laser light causes molecules to vibrate in unison. We say that laser light polarizes the medium.
Accelerating charges emit light. Polarized matter emits light at the frequency at which it is oscillating.
P 0 (1)E (2)E 2 (3)E 3 ...
What are the effects of such nonlinear terms? Consider the second-order term:
SinceE (t) E exp(it) E* exp(it), E (t)2 E2 exp(2it) 2 E 2 E*2 exp(2it)
Nonlinear Optics
Why do nonlinear-optical effects occur? Maxwell's equations in a medium Nonlinear-optical media Second-harmonic generation Sum- and difference-frequency generation Higher-order nonlinear optics The Slowly Varying Envelope Approximation Phase-matching and Conservation laws for photons
Why do nonlinear-optical effects occur?
Recall that, in normal linear optics, a light wave acts on a molecule, which vibrates and then emits its own light wave, which interferes with the original light wave.
Maxwell's Equations in a Nonlinear Medium
Nonlinear optics is what happens when the polarization is the result of higher-order (nonlinear!) terms in the field:
2E1E2 expi(1 2 )t 2E1*E2* expi(1 2 )t Sum-freq gen
2E1E2* expi(1 2 )t 2E1*E2 expi(1 2 )t Diff-freq gen
2 E1 2 2 E2 2
dc rectification
Note also that, when i is negative inside the exp, the E in front has a *.
Nonlinear optics and anharmonic oscillators
A nucleus (in a molecule) also does not have a simple quadratic potential. So its vibrational motion is also nonlinear:
0E
In this simple (and most common) case, the wave equation becomes:
2E 1 2E z2 c02 t2
1 c02
2E t 2
Using the fact that:
00 1/ c02
Simplifying:
2E 1 2E 0
E 0 B 0
E B t
B
1 E
c02 t
0
P t
These equations reduce to the (scalar) wave equation:
2E 1 2E z2 c02 t 2
0
2P t 2
Inhomogeneous Wave Equation
Sine waves of all frequencies are solutions to the wave equation; it’s the polarization that tells which frequencies will occur.
z2
c02 t2
This equation has the solution: E (z, t) E(0, t) cos(t k z)
where
= c k and c = c0 /n and n = (1+)1/2
The induced polarization only changes the refractive index. Dull. If only the polarization contained other frequencies…
Potential due to nucleus
The potential gets very flat out at infinity, so the electron’s motion can easily go nonlinear! Nucleus Position
So an electron’s motion will also depart from a sine wave.
Nonlinear optics allows us to change the color of a light beam, to change its shape in space and time, and to create ultrashort laser pulses.
Why don't we see nonlinear optical effects in our daily life?
The polarization is the driving term for the solution to this equation.
Solving the wave equation in the presence of linear induced polarization
For low irradiances, the polarization is proportional to the incident field:
Nonlinear optics is analogous to nonlinear electronics, which we can observe easily.
Sending a high-volume sine-wave (pure frequency) signal into a cheap amplifier or cheap speakers yields a truncated output signal, more of a square wave than a sine. This square wave has higher frequencies.
E(t) E1 exp(i1t) E1* exp(i1t) E2 exp(i2t) E2* exp(i2t)
So:
E(t)2 E12 exp(2i1t) E1*2 exp(2i1t)
2nd-harmonic gen
E22 exp(2i2t) E2*2 exp(2i2t)
2nd-harmonic gen