Prove that phase-conjugate waves satisfy Maxwell's equations

[Haorong Wu]

Homework Statement

Let $\mathbf E =\mathbf E_0 ( \mathbf r) e^{-i \omega t}$, $\mathbf H =\mathbf H_0 ( \mathbf r) e^{-i \omega t}$ be some solutions of the Maxwell equations. If the medium is not absorbing, then prove that the phase-conjugate waves $$\mathbf E_{PC} =\mathbf E_0^* ( \mathbf r) e^{-i \omega t}, \mathbf H_{PC} =\mathbf H_0^* ( \mathbf r) e^{-i \omega t} $$ also satisfy the Maxwell equation.

Relevant Equations

Maxwell equations

This is the second part of a problem. In the first part of the problem, I have proven that $\mathbf E^* =\mathbf E_0^* ( \mathbf r) e^{i \omega t}$ satisfies the Maxwell equations.

Then, in this part of the problem, I tried to first prove that $\mathbf E^{'} =\mathbf E_0 ( \mathbf r) e^{i \omega t}$ satisfies the Maxwell equations, since by conjugation, the waves in the question would satisfy the Maxwell equations.

Now, $\mathbf E^{'} =\mathbf E_0 ( \mathbf r) e^{i \omega t}=\mathbf E_0 ( \mathbf r) e^{-i \omega t}e^{2i \omega t}=\mathbf E e^{2i \omega t}$. Similarly, $\mathbf B^{'} =\mathbf B e^{2i \omega t}$.

Substitute them into $\nabla \times \mathbf E + \frac {\partial \mathbf B} {\partial t}=0$ yields $$(\nabla \times \mathbf E)e^{2i \omega t}+ \frac {\partial \mathbf B} {\partial t} e^{2i \omega t} + \mathbf B \cdot 2i \omega e^{2i \omega t} =0$$

Since $\mathbf E =\mathbf E_0 ( \mathbf r) e^{-i \omega t}$, $\mathbf H =\mathbf H_0 ( \mathbf r) e^{-i \omega t}$ are some solutions of the Maxwell equations, then $(\nabla \times \mathbf E)+ \frac {\partial \mathbf B} {\partial t} =0$, leaving $$ \mathbf B \cdot 2i \omega =0$$.

I do not know where I did wrong.

 

[Abhishek11235]

$\mathbf B$ is only function of $\mathbf r$

 

[Haorong Wu]

$\mathbf B$ is only function of $\mathbf r$

I am sorry I do not follow you. $\mathbf B$ is proportional to $\mathbf H$. Since $\mathbf H =\mathbf H_0 ( \mathbf r) e^{-i \omega t}$, $\mathbf B$ should depend on $t$, as well.

 

[vanhees71]

In this case all fields are assumed to be time-dependent by a factor $\exp(-\mathrm{i} \omega t)$.

 

[Haorong Wu]

In this case all fields are assumed to be time-dependent by a factor $\exp(-\mathrm{i} \omega t)$.

Thanks!. But I am sorry I am not familiar with the expression that "be time-dependent by a factor". Do you mean the fields need the extra phase-term?

 

[vanhees71]

The idea is to consider fields with harmonic time dependence, i.e., for any field $f$ and ansatz is made of the form
$$f(t,\vec{x})=f_0(\omega,\vec{x}) \exp(-\mathrm{i} \omega t).$$

 

[kent davidge]

The idea is to consider fields with harmonic time dependence, i.e., for any field $f$ and ansatz is made of the form
$$f(t,\vec{x})=f_0(\omega,\vec{x}) \exp(-\mathrm{i} \omega t).$$

Does this type of field represent a monochromatic wave?

 

[vanhees71]

I don't like "monochromatic", but yes, it's often expressed in this way. One should however be aware of the fact that color is a physiological notion, which is much more complicated than the frequency of a plane-wave mode of the electromagnetic field!

 

[Haorong Wu]

The idea is to consider fields with harmonic time dependence, i.e., for any field $f$ and ansatz is made of the form
$$f(t,\vec{x})=f_0(\omega,\vec{x}) \exp(-\mathrm{i} \omega t).$$

I solved the problem in other way. I noted that the phase-conjugate wave travels in the opposite direction, or equivalently, travels with a time-inversal. So the time derivatives in the Maxwell equations become $\partial _{-t}$. Then the problem is solved.

I am not sure whether this method is connected to your statement.

Thanks, @vanhees71 .