fluidistic
Nov30-10, 12:24 PM
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 \vec E _1 (r_1 ,t)=\vec E_{01} (r_1)} e^{i(kr_1 -\omega t + \varepsilon _1)} and \vec E _2 (r_2 ,t)=\vec E_{02} (r_1)} e^{i(kr_2 -\omega t + \varepsilon _2)}.
First questions: Hecht's was always meticulous writing \vec k \cdot \vec x for plane waves, now he dropped the vector notation? I don't understand why. Ok k and r are parallels in this case so \vec k \cdot \vec r =kr, 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 r_1 and r_2 are the radii of the spherical wavefronts overlapping at P; they specify the distances from the sources to P. In this case \delta = k(r_1-r_2)+(\varepsilon _1 - \varepsilon _2)."
In case you wonder, P is just a considered point over a screen far away from the sources. \delta is the phase difference according to Hecht.
I do not understand why \delta is worth what it's worth. I realize that the difference in optical path of the waves emitted by both sources is \frac{(r_1-r_2)}{n} where n is the refractive index of the medium. How do you reach \delta form it?
So he's talking about spherical waves emitted by 2 sources. He says that the waves can be written under the form \vec E _1 (r_1 ,t)=\vec E_{01} (r_1)} e^{i(kr_1 -\omega t + \varepsilon _1)} and \vec E _2 (r_2 ,t)=\vec E_{02} (r_1)} e^{i(kr_2 -\omega t + \varepsilon _2)}.
First questions: Hecht's was always meticulous writing \vec k \cdot \vec x for plane waves, now he dropped the vector notation? I don't understand why. Ok k and r are parallels in this case so \vec k \cdot \vec r =kr, 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 r_1 and r_2 are the radii of the spherical wavefronts overlapping at P; they specify the distances from the sources to P. In this case \delta = k(r_1-r_2)+(\varepsilon _1 - \varepsilon _2)."
In case you wonder, P is just a considered point over a screen far away from the sources. \delta is the phase difference according to Hecht.
I do not understand why \delta is worth what it's worth. I realize that the difference in optical path of the waves emitted by both sources is \frac{(r_1-r_2)}{n} where n is the refractive index of the medium. How do you reach \delta form it?