# How is momentum conserved in phase mismatch?

• Korak Biswas
In summary, classical EM theory states that shining light of frequency ω on a second order non-linear medium creates a radiation of frequency 2ω, with the amplitude dependent on the momentum difference between the incident and created fields. However, there is a debate on where this momentum mismatch comes from, and it is suggested that the fields transfer momentum to matter. This is explored in depth in the paper https://arxiv.org/abs/0710.0461, which is considered to resolve the debate.

#### Korak Biswas

From classical EM theory, we know that if we shine light of frequency ω on a second order non-linear medium, a radiation of frequency 2ω is created. The amplitude of the radiation of frequency 2ω is dependent on the momentum difference between the incident field and the created field. But I can't understand where this momentum mismatch comes from. The momentum should be conserved always.

The momentum of the field plus the matter is conserved. The momentum of the field alone or the matter alone need not be conserved.

Korak Biswas
Dale said:
The momentum of the field plus the matter is conserved. The momentum of the field alone or the matter alone need not be conserved.
Thanks for your reply. I guessed it. The fields somehow transfer some amount of momentum to matter. But I couldn't explain this using Maxwell's equations. Will you please elaborate?

Korak Biswas said:
This is a longstanding debate in the literature, but my favorite paper on the topic is this one (which I think completely resolves the debate)

https://arxiv.org/abs/0710.0461

Dale said:
This is a longstanding debate in the literature, but my favorite paper on the topic is this one (which I think completely resolves the debate)

https://arxiv.org/abs/0710.0461
Thanks once again. I will go through this and come back to you if necessary.

Dale

## 1. How is momentum conserved in phase mismatch?

When two objects with different momenta collide, the total momentum of the system before and after the collision must be equal. This means that if one object gains momentum, the other must lose an equal amount of momentum to maintain the overall balance. This is known as the conservation of momentum.

## 2. What is phase mismatch in the context of momentum conservation?

Phase mismatch occurs when the phases of two colliding objects do not match up perfectly. This can happen when the two objects have different velocities or directions of motion. In this case, the total momentum of the system is still conserved, but the individual momenta of the objects may change due to the phase mismatch.

## 3. How does phase mismatch affect the conservation of momentum?

In order for momentum to be conserved in a phase mismatch scenario, the change in momentum of one object must be equal and opposite to the change in momentum of the other object. This means that if one object gains momentum, the other must lose an equal amount of momentum, and vice versa.

## 4. Can momentum be conserved if there is a phase mismatch?

Yes, momentum can still be conserved even in the presence of phase mismatch. As long as the total momentum of the system remains constant, the conservation of momentum is still valid.

## 5. Are there any real-life examples of phase mismatch and momentum conservation?

Yes, there are many real-life examples of phase mismatch and momentum conservation. For instance, when a car collides with a stationary object, the total momentum of the system is conserved, but the individual momenta of the car and the object may change due to the difference in their velocities. Another example is a pool game, where the total momentum of the cue ball and the object ball must be equal before and after the collision, even if there is a phase mismatch in their movements.