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zhouhao
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Member warned to use the homework template for posts in the homework sections of PF.
I am learning principle of optics written by Born&Wolf and confused with the method to derive Snell's laws and law of reflection.
In the textbook,##v_1## is the speed of light in mediator 1 and ##v_2## is the one in mediator 2.
##{\vec s}## is the direction of light and ##{\vec r}## is the position.
As shown in Fig.1,
Incident light,Reflected light and Transmitted light are respectively recognized as plane waves and their behaivor at ##{\vec r}## corresponding to##F(t-\frac{{\vec r}\cdot{\vec s}^{(i)}}{v_1})##,##F(t-\frac{{\vec r}\cdot{\vec s}^{(r)}}{v_1})## ,##F(t-\frac{{\vec r}\cdot{\vec s}^{(t)}}{v_2})##.
If ##{\vec r}## at interface, we have ##t-\frac{{\vec r}\cdot{\vec s}^{(i)}}{v_1}=t-\frac{{\vec r}\cdot{\vec s}^{(r)}}{v_1}=t-\frac{{\vec r}\cdot{\vec s}^{(t)}}{v_2}##.Then we could get ##\frac{s_x^{(i)}}{v_1}=\frac{s_x^{(r)}}{v_1}=\frac{s_x^{(t)}}{v_2}## (Snell's law and Law of reflection included in).
My problem is :
1,Plane wave spreading over all space as shown in Fig.2,it is not like a stright ray(Does this means a light with width combined with many parallel rays behaves like plane wave?);
2,Where is the ##F(t)## with ##{\vec r}=0##,I think if it is originating from somewhere of mediator 1,Reflection rays and Transmitting rays should be expressed as ##F(t-\frac{{{\vec r}_A}\cdot{\vec s}^{(i)}}{v_1}-\frac{({\vec r}-{{\vec r}_A})\cdot{\vec s}^{(r)}}{v_1})##,##F(t-\frac{{{\vec r}_A}\cdot{\vec s}^{(i)}}{v_1}-\frac{({\vec r}-{{\vec r}_A})\cdot{\vec s}^{(t)}}{v_2})##,where A is the point of reflection and transmittion happening.I couldn't conduct the laws from my thoughts.So how does the authors think?
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