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fluidistic
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I'm reading through Hecht's book on Optics and I fail to understand something. I think it's the third edition, page 380, chapter 9 (Interference).
So he's talking about spherical waves emitted by 2 sources. He says that the waves can be written under the form [tex]\vec E _1 (r_1 ,t)=\vec E_{01} (r_1)} e^{i(kr_1 -\omega t + \varepsilon _1)}[/tex] and [tex]\vec E _2 (r_2 ,t)=\vec E_{02} (r_1)} e^{i(kr_2 -\omega t + \varepsilon _2)}[/tex].
First questions: Hecht's was always meticulous writing [tex]\vec k \cdot \vec x[/tex] for plane waves, now he dropped the vector notation? I don't understand why. Ok k and r are parallels in this case so [tex]\vec k \cdot \vec r =kr[/tex], but he never justified it, I find it very strange. I'm likely missing something. Any help to understand here will be very welcome.
Then he went to say "The terms [tex]r_1[/tex] and [tex]r_2[/tex] are the radii of the spherical wavefronts overlapping at P; they specify the distances from the sources to P. In this case [tex]\delta = k(r_1-r_2)+(\varepsilon _1 - \varepsilon _2)[/tex]."
In case you wonder, P is just a considered point over a screen far away from the sources. [tex]\delta[/tex] is the phase difference according to Hecht.
I do not understand why [tex]\delta[/tex] is worth what it's worth. I realize that the difference in optical path of the waves emitted by both sources is [tex]\frac{(r_1-r_2)}{n}[/tex] where n is the refractive index of the medium. How do you reach [tex]\delta[/tex] form it?
So he's talking about spherical waves emitted by 2 sources. He says that the waves can be written under the form [tex]\vec E _1 (r_1 ,t)=\vec E_{01} (r_1)} e^{i(kr_1 -\omega t + \varepsilon _1)}[/tex] and [tex]\vec E _2 (r_2 ,t)=\vec E_{02} (r_1)} e^{i(kr_2 -\omega t + \varepsilon _2)}[/tex].
First questions: Hecht's was always meticulous writing [tex]\vec k \cdot \vec x[/tex] for plane waves, now he dropped the vector notation? I don't understand why. Ok k and r are parallels in this case so [tex]\vec k \cdot \vec r =kr[/tex], but he never justified it, I find it very strange. I'm likely missing something. Any help to understand here will be very welcome.
Then he went to say "The terms [tex]r_1[/tex] and [tex]r_2[/tex] are the radii of the spherical wavefronts overlapping at P; they specify the distances from the sources to P. In this case [tex]\delta = k(r_1-r_2)+(\varepsilon _1 - \varepsilon _2)[/tex]."
In case you wonder, P is just a considered point over a screen far away from the sources. [tex]\delta[/tex] is the phase difference according to Hecht.
I do not understand why [tex]\delta[/tex] is worth what it's worth. I realize that the difference in optical path of the waves emitted by both sources is [tex]\frac{(r_1-r_2)}{n}[/tex] where n is the refractive index of the medium. How do you reach [tex]\delta[/tex] form it?