Question on phasors at boundary of oblique incidence.

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A plane wave travels in ##\hat k_I=\hat x \sin\theta_I+\hat z \cos \theta_I## direction hitting a boundary formed by xy plane ( z=0). The incidence wave is in the plane of incident formed by xz plane where y=0.

We let ##\tilde E_I(\vec k_I)= \hat x E_{I_x}+\hat y E_{I_y}+\hat z E_{I_z} =E_{0I}e^{-j\vec k_I\cdot \vec r}##. This means ##\tilde E_{0I}## has to have x, y and z components ##\Rightarrow\;\tilde E_{0I}=\hat x E_{0I_x}+\hat y E_{0I_y}+\hat z E_{0I_z}##

But at ##z=0##, ##E_{0I_z}## has to be zero!
If we let ##E_{0I_z}=0##, then it won't work for the vector where z is not 0! how do I resolve this? Only way I can think of is ##E_{0I_z}## is a function of z and it's zero at z=0. Am I right?

thanks
 
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