What happens to the electric field's phase upon reflection?

[Gabriel Maia]

I have a problem with the phase of an electric field as it is reflected by and transmitted through a dielectric interface.

At the boundary between the two media, all waves must exist simultaneously and the tangential component must be equal on both sides of the interface, right? Therefore for all time t and for all points on the interface we have that

\hat{n} \times \vec{E}_{i} + \hat{n} \times \vec{E}_{r} = \hat{n} \times \vec{E}_{t}

where \hat{n} is the unitary vector orthogonal to the interface plane and \vec{E} is the electric field, meaning the indices i, r and t, 'incoming', 'reflected', and 'transmitted'.

The electric fields have the form

\vec{E}_{\lambda} = \vec{E}_{0,\lambda}\exp\left[i\left( \vec{k}_{\lambda}\cdot \vec{r} - \omega_{\lambda}\,t \right)\right]

I'm reading a text where the author says that \omega_{i}=\omega_{r}=\omega_{t} and that \vec{k}_{i}\cdot \vec{r}=\vec{k}_{r}\cdot \vec{r}=\vec{k}_{t}\cdot \vec{r}

and to prove so he starts by assuming that the amplitudes of \vec{E}_{0,\lambda} are equal.

My question is... are they? Because if they are I can understand how he derives this conclusion without problems, but these amplitudes encompass the Fresnel coefficients, don't they? So they are not in general equal, which means the frequencies are not equal as well.

So, are the frequencies and the dot product \vec{k}_{\lambda}\cdot \vec{r} equal or not? If they are, how can I prove this keeping in mind the Fresnel coefficients? I believe I just need to understand better the assumption that the amplitudes are equal. Thank you very much.

 

[rude man]

I have a problem with the phase of an electric field as it is reflected by and transmitted through a dielectric interface.

At the boundary between the two media, all waves must exist simultaneously and the tangential component must be equal on both sides of the interface, right? Therefore for all time t and for all points on the interface we have that

\hat{n} \times \vec{E}_{i} + \hat{n} \times \vec{E}_{r} = \hat{n} \times \vec{E}_{t}

doesn't sound right but for the moment I won't question it

where \hat{n} is the unitary vector orthogonal to the interface plane and \vec{E} is the electric field, meaning the indices i, r and t, 'incoming', 'reflected', and 'transmitted'.

The electric fields have the form

\vec{E}_{\lambda} = \vec{E}_{0,\lambda}\exp\left[i\left( \vec{k}_{\lambda}\cdot \vec{r} - \omega_{\lambda}\,t \right)\right]

This is not the form of the reflected wave

I'm reading a text where the author says that \omega_{i}=\omega_{r}=\omega_{t} and that \vec{k}_{i}\cdot \vec{r}=\vec{k}_{r}\cdot \vec{r}=\vec{k}_{t}\cdot \vec{r}

This sounds much more resonable to me: Incident = reflected (with proper phase relationship) + transmitted.

and to prove so he starts by assuming that the amplitudes of \vec{E}_{0,\lambda} are equal.

My question is... are they? Because if they are I can understand how he derives this conclusion without problems, but these amplitudes encompass the Fresnel coefficients, don't they? So they are not in general equal, which means the frequencies are not equal as well.

Frequency is the one thing always preserved in any linear medium. Lambda can and does change, but not frequency.

So, are the frequencies and the dot product \vec{k}_{\lambda}\cdot \vec{r} equal or not? If they are, how can I prove this keeping in mind the Fresnel coefficients? I believe I just need to understand better the assumption that the amplitudes are equal. Thank you very much.

Don't know how much help I've given, hope some.

 

[NFuller]

I have a problem with the phase of an electric field as it is reflected by and transmitted through a dielectric interface.

At the boundary between the two media, all waves must exist simultaneously and the tangential component must be equal on both sides of the interface, right? Therefore for all time t and for all points on the interface we have that

\hat{n} \times \vec{E}_{i} + \hat{n} \times \vec{E}_{r} = \hat{n} \times \vec{E}_{t}

where \hat{n} is the unitary vector orthogonal to the interface plane and \vec{E} is the electric field, meaning the indices i, r and t, 'incoming', 'reflected', and 'transmitted'.

The electric fields have the form

\vec{E}_{\lambda} = \vec{E}_{0,\lambda}\exp\left[i\left( \vec{k}_{\lambda}\cdot \vec{r} - \omega_{\lambda}\,t \right)\right]

I'm reading a text where the author says that \omega_{i}=\omega_{r}=\omega_{t} and that \vec{k}_{i}\cdot \vec{r}=\vec{k}_{r}\cdot \vec{r}=\vec{k}_{t}\cdot \vec{r}

I don't see any major issues up to this point.

and to prove so he starts by assuming that the amplitudes of ⃗E0,λE→0,λ\vec{E}_{0,\lambda} are equal.

This is either wrong or just very bad notation. If the author is saying that the magnitude of the electric field of the incident, reflected, and transmitted waves are equal, then that statement is wrong.