# B Definition of transmission & reflection in TE/TM

#### Dimani4

Hi,
I have a question in definition of reflection/transmission coefficients in TE/TM modes.
Let's see TE polarization case.

The reflection coefficient for the magnetic field is defined as:

However the transmission coefficient for the magnetic field is defined as:

Now, let's see the TM mode.

Here, transmission coefficient for electric field is defined as:

And the reflection coefficient is defined as;

My question is: why the reflection coefficient is defined the only by the parallel component of the magnetic (Hz for TE) and electric field (Ez for TM) but the transmission coefficient is defined by the whole component (Et/Ei or Ht/Hi of the electric/magnetic field?

Thank you.

The pictures are taken from the attached pdf file (EM13).

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If I understood your question properly, the subscript $t$ which you think may stand for "whole" or "total" stands for "transmitted" rather than "total".

#### Dimani4

If I understood your question properly, the subscript $t$ which you think may stand for "whole" or "total" stands for "transmitted" rather than "total".
Yes, the subscript here means transmitted but also a total. Check it out the pdf file I've attached (for instance page 180 for TM).

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Yes, the subscript here means transmitted but also a total. Check it out the pdf file I've attached (for instance page 180 for TM).
The transmitted vector has the same polarization as the incident vector. There isn't any discrepancy here. I have been through this material quite a number of times, (the first time in an optics course in approximately 1976), and I have never observed anything anomalous about it.

#### Dimani4

The transmitted vector has the same polarization as the incident vector. There isn't any discrepancy here. I have been through this material quite a number of times, (the first time in an optics course in approximately 1976), and I have never observed anything anomalous about it.
you can't say it. take a look for example the polarization for H in the TE case or polarization of E in the TM case. Their polarization is dependent for the angles of incidence (θi) and transmission (θt). My confusion or "misunderstanding" is why the definition for transmission and reflection coefficient is different.

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you can't say it. take a look for example the polarization for H in the TE case or polarization of E in the TM case. Their polarization is dependent for the angles of incidence (θi) and transmission (θt). My confusion or "misunderstanding" is why the definition for transmission and reflection coefficient is different.
Please try to explain the difficulty you are having in more detail. I don't see any problem with the equations.

#### Dimani4

Please try to explain the difficulty you are having in more detail. I don't see any problem with the equations.
Let's take TE case.
The reflection coefficient for magnetic field is defined as :

Transmission coefficient is:

My question is why reflection coefficient is not Γ(H)=Hr/Hi but Hzr/Hzi or vise a versa why transmission coefficient is not τ(H)=Hzt/Hzi but Ht/Hi ? I see that the definitions here are not consistent.

Thank you.

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Just giving a quick answer, (I may study this in more detail, but I think I may have picked out the point of confusion), the z-direction is the direction perpendicular to the interface for these optics problems, while y is the transverse direction. In one of the boundary conditions of Maxwell's equations, $\nabla \times E=-\frac{\partial{B}}{\partial{t}}$, the y-component is involved in the derivation, while for the equation $\nabla \cdot E=0$, only the z-component is needed. I'll look at this in more detail if this doesn't answer your question, but I'm thinking it might suffice.

#### Dimani4

Just giving a quick answer, (I may study this in more detail, but I think I may have picked out the point of confusion), the z-direction is the direction perpendicular to the interface for these optics problems, while y is the transverse direction. In one of the boundary conditions of Maxwell's equations, $\nabla \times E=-\frac{\partial{B}}{\partial{t}}$, the y-component is involved in the derivation, while for the equation $\nabla \cdot E=0$, only the z-component is needed. I'll look at this in more detail if this doesn't answer your question, but I'm thinking it might suffice.
Thank you for your quick response but here I guess z is transverse to the incidence plane (z actually is in this plane), y is perpendicular to this plane and as x is also perpendicular as you can see here;

Please explain it in more details because I never paid attention on this point till now... :)

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Thank you for your quick response but here I guess z is transverse to the incidence plane and x is perpendicular as you can see here;
View attachment 220214

Please explain it in more details because I never paid attention on this point till now... :)
I looked at the "link"=yes, they have an odd choice of $z$. My guess is that most everything else they did was correct, and you can read through their derivation and see if it makes sense. Alternatively, try reading another source on the same topic. It would take a while for me to go through this author's write-up in detail, but this topic, in general, has never presented any major problems of any kind. If it is problematic, it is likely that is is due to this particular author rather than the concept itself.

#### Dimani4

I looked at the "link"=yes, they have an odd choice of $z$. My guess is that most everything else they did was correct, and you can read through their derivation and see if it makes sense. Alternatively, try reading another source on the same topic. It would take a while for me to go through this author's write-up in detail, but this topic, in general, has never presented any major problems of any kind. If it is problematic, it is likely that is is due to this particular author rather than the concept itself.
yes. the derivations of the formulas are right but they didn't explain why these definitions (transmission and reflection coefficients) are not consistent. the coefficients are also right but I still dont understand (they say "we can define". you can't say "we can define" because this coefficient you can measure and the definition of the reflection coefficient is the reflected field/incidence field and nothing more.)
Anyway, they didnt explain why for reflection you take only the transverse field but for transmission you take the whole field. do you have any idea?

Actually if you do Hr/Hi you have the same coefficient as they get but without minus sign i.e. Eyr/Eyi but it really odd because you have the same reflection coefficient for electric and magnetic field (TE case page 173).
Thank you.

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The boundary conditions are generally of the form $E_i +E_r=E_t$. etc. Suggest you try reading through a derivation of normal incidence before doing the more complex case. I haven't studied this author's derivation sufficiently to really comment on it, but if it is written up properly, there really should be no ambiguity. $\\$ The Kirchhoff Fresnel relations are written up in many textbooks including J.D. Jackson's Classical Electrodynamics, where I studied them in detail as a graduate student. Thousands upon thousands of students have read through these derivations, and I believe they are free of errors. $\\$ Since it is in the same material, and the ratio of electric field to magnetic field is the same for the transverse electromagnetic wave, (other than a possible minus sign), you will get the same reflection coefficient for electric field as you do for the magnetic field.

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#### Dimani4

The boundary conditions are generally of the form $E_i +E_r=E_t$. etc. Suggest you try reading through a derivation of normal incidence before doing the more complex case. I haven't studied this author's derivation sufficiently to really comment on it, but if it is written up properly, there really should be no ambiguity. $\\$ The Kirchhoff Fresnel relations are written up in many textbooks including J.D. Jackson's Classical Electrodynamics, where I studied them in detail as a graduate student. Thousands upon thousands of students have read through these derivations, and I believe they are free of errors. $\\$ Since it is in the same material, and the ratio of electric field to magnetic field is the same for the transverse electromagnetic wave, (other than a possible minus sign), you will get the same reflection coefficient for electric field as you do for the magnetic field.
yes, I understand the tangential boundary conditions for electric field as for the magnetic field. $H_i +H_r=H_t$ but still the question of reflection and transmission coefficient definition still remains. This pdf file presents very good detailed analysis for the TE and TM case but this point is not so clear, like the author doesn't want to focus on this specific point. I'm not saying that this is an error I say I have some misunderstanding.

"Definition of transmission & reflection in TE/TM"

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