Editing Fresnel equations (section)

=== ''p'' components ===
For the ''p'' polarization, the incident, reflected, and transmitted {{math|'''E'''}} fields are parallel to the red arrows and may therefore be described by their components in the directions of those arrows. Let those components be {{math|''E''<sub>i</sub>, ''E''<sub>r</sub>, ''E''<sub>t</sub>{{hsp}}}} (redefining the symbols for the new context). Let the reflection and transmission coefficients be {{math|''r''<sub>p</sub>}} and {{math|''t''<sub>p</sub>}}. Then, if the incident {{math|'''E'''}} field is taken to have unit amplitude, we have
{{NumBlk|:|<math>\begin{align}
  E_\text{i} &=         e^{i\mathbf{k}_\text{i}\mathbf{\cdot r}}\\
  E_\text{r} &= r_{p\,} e^{i\mathbf{k}_\text{r}\mathbf{\cdot r}}\\
  E_\text{t} &= t_{p\,} e^{i\mathbf{k}_\text{t}\mathbf{\cdot r}}.
\end{align}</math>|{{EquationRef|17}}}}
If the {{math|'''E'''}} fields are in the directions of the red arrows, then, in order for {{math|'''k''', '''E''', '''H'''}} to form a right-handed orthogonal triad, the respective {{math|'''H'''}} fields must be in the {{math|−''z''}}-direction ("into the page") and may therefore be described by their components in that direction. This is consistent with the adopted sign convention, namely that a positive reflection or transmission coefficient is one that preserves the direction of the transverse field {{larger|(}}the {{math|'''H'''}} field in the case of the ''p'' polarization{{larger|)}}. The agreement of the ''other'' field with the red arrows reveals an alternative definition of the sign convention: that a positive reflection or transmission coefficient is one for which the field vector in the plane of incidence points towards the same medium before and after reflection or transmission.<ref>This agrees with Born & Wolf, 1970, p.{{hsp}}38, Fig.{{tsp}}1.10.</ref>

So, for the incident, reflected, and transmitted {{math|'''H'''}} fields, let the respective components in the {{math|−''z''}}-direction be {{math|''H''<sub>i</sub>, ''H''<sub>r</sub>, ''H''<sub>t</sub>}}. Then, since {{math|''H'' {{=}} ''YE''}},
{{NumBlk|:|<math>\begin{align}
  H_\text{i} &=\, Y_1 e^{i\mathbf{k}_\text{i}\mathbf{\cdot r}}\\
  H_\text{r} &=\, Y_1 r_{p\,} e^{i\mathbf{k}_\text{r}\mathbf{\cdot r}}\\
  H_\text{t} &=\, Y_2 t_{p\,} e^{i\mathbf{k}_\text{t}\mathbf{\cdot r}}.
\end{align}</math>|{{EquationRef|18}}}}

At the interface, the tangential components of the {{math|'''E'''}} and {{math|'''H'''}} fields must be continuous; that is,
{{NumBlk|:|<math>\left.\begin{align}
  E_\text{i}\cos\theta_\text{i} - E_\text{r}\cos\theta_\text{i} &= E_\text{t}\cos\theta_\text{t}\\
  H_\text{i} + H_\text{r} &= H_\text{t}
\end{align}~~\right\}~~~\text{at}~~ y=0\,.</math>|{{EquationRef|19}}}}
When we substitute from equations ({{EquationNote|17}}) and ({{EquationNote|18}}) and then from ({{EquationNote|7}}), the exponential factors again cancel out, so that the interface conditions reduce to
{{NumBlk|:|<math>\begin{align}
  \cos\theta_\text{i} - r_\text{p}\cos\theta_\text{i} &=\, t_\text{p}\cos\theta_\text{t}\\
  Y_1 + Y_1 r_\text{p} &=\, Y_2 t_\text{p} \,.
\end{align}</math>|{{EquationRef|20}}}}
Solving for {{math|''r''<sub>p</sub>}} and {{math|''t''<sub>p</sub>}}, we find
{{NumBlk|:|<math>r_\text{p}=\frac{Y_2\cos\theta_\text{i}-Y_1\cos\theta_\text{t}}{Y_2\cos\theta_\text{i}+Y_1\cos\theta_\text{t}}</math>|{{EquationRef|21}}}}
and
{{NumBlk|:|<math>t_\text{p}=\frac{2Y_1\cos\theta_\text{i}}{Y_2\cos\theta_\text{i}+Y_1\cos\theta_\text{t}}\,.</math>|{{EquationRef|22}}}}
At ''normal incidence'' {{math|(''θ''<sub>i</sub> {{=}} ''θ''<sub>t</sub> {{=}} 0)}} indicated by an additional subscript 0, these results become
{{NumBlk|:|<math>r_\text{p0}=\frac{Y_2-Y_1}{Y_2+Y_1}</math>|{{EquationRef|23}}}}
and
{{NumBlk|:|<math>t_\text{p0}=\frac{2Y_1}{Y_2+Y_1}\,.</math>|{{EquationRef|24}}}}
At {{itco|''grazing incidence''}} {{math|(''θ''<sub>i</sub> → 90°)}}, we again have {{math|cos{{tsp}}''θ''<sub>i</sub> → 0}}, hence {{math|''r''<sub>p</sub> → −1}} and {{math|''t''<sub>p</sub> → 0}}.

Comparing ({{EquationNote|23}}) and ({{EquationNote|24}}) with ({{EquationNote|15}}) and ({{EquationNote|16}}), we see that at ''normal'' incidence, under the adopted sign convention, the transmission coefficients for the two polarizations are equal, whereas the reflection coefficients have equal magnitudes but opposite signs. While this clash of signs is a disadvantage of the convention, the attendant advantage is that the signs agree at ''grazing'' incidence.