- #1

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\vec{k_I},\; \vec{k_R},\;\vec{k_T} [/itex] are direction of incidence, reflected and transmitted wave resp.

Unit direction vector of the three waves are:

[tex] \hat k_I =\hat x sin \theta _I \;+\; \hat z cos \theta _I, \;\;\; \hat k_R =\hat x sin \theta _R \;-\; \hat z cos \theta _R, \;\;\; \hat k_T =\hat x sin \theta _T \;+\; \hat z cos \theta _T [/tex]

I denote [itex] E_{0_I} , \; E_{0_R} ,\; E_{0_T} \;[/itex] be incidence, reflected and transmitted E with parallel polarization at z=0 resp. This convention is also used for H field also.

For the normal components of the E ( in z direction):

[tex] \epsilon_1 ( -E_{0_I} sin \theta_I \;+\; E_{0_R} sin \theta_R ) \; \epsilon_2 ( -E_{0_T} sin \theta_T) [/tex] (1)

For tangential components of E and H ( in x direction):

[tex] E_{0_I} cos \theta_I \;+\; E_{0_R} cos \theta_R \;=\; E_{0_T} cos \theta_T [/tex]

[tex] \frac {1}{\mu_1}( H_{0_I} \;+\; H_{0_R} ) \;=\;\frac {1}{ \mu_2 } H_{0_T} \;\Rightarrow\; \frac {1}{\mu_1 v_1}( E_{0_I} \;-\; E_{0_R} ) \;=\; \frac {1}{ \mu_2 } E_{0_T}[/tex] (2)

These will give:

[tex] \tilde E_{0_I} \;-\; \tilde E_{0_R}\;=\; \beta \tilde E_{0_T} \; \hbox { where } \beta \;=\; \frac{\mu_1 v_1}{\mu_2 v_2} [/tex]

[tex] \tilde E_{0_I} \;+\; \tilde E_{0_R} \;=\; \alpha \tilde E_{0_T} \;\hbox { where } \;\alpha \;=\; \frac{ cos \theta_T}{ cos \theta_I} [/tex]

For reflection coef :

[tex] \tilde E_{0_T} \;=\; \frac { \alpha - \beta } { \alpha +\beta} [/tex]

Where the reflected [itex] \tilde E_{0_R} [/itex] is either in phase or 180 deg out of phase depend on whether [itex] \alpha - \beta [/itex] is +ve or -ve respectively.

Here is my problem:

If you look at (1) the [itex]\tilde E_R [/itex] is assumed direction on the plane of indicence while the resulting reflection again defined whether the [itex] \tilde E_R [/itex] is in or out of phase depend on the polarity of [itex] \alpha - \beta [/itex]. How can you define the direction of [itex]\tilde E_R \;[/itex] first and use it to derive the fresnel's equation which can change the direction? I see the same thing on Cheng's and Ulaby's books like this also.