Optics: Converging waves

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Hi. A spherical wave ##e^{i(kr-\omega t)}## diverging from a single point ##(x=0,y=0,z=-z_0)## can be approximated as a parabolic wave in the paraxial case around the z-axis. I.e., ##k r = k \sqrt{x^2+y^2+z^2} \simeq k (z +\frac{x^2+y^2}{2z})##.

OK, then let's say a lens is placed such that its optical axis coincides with the ##z## axis and its focus points are at ##-z_0## and ##z_0##. In this case, the outgoing parabolic wave from ##-z_0## will be focused into the point ##z_0##. My question is, how is this to be modeled mathematically? Intuitively I would guess that ##k r \simeq k ( z - \frac{x^2+y^2}{2z})##, but what is ##kr## equal to in the accompanying case of a converging spherical wave? Something ala ##e^{i(kr + \omega t)} e^{i \phi}##, where ##\phi## is some phase factor?

I would appreciate it if you guys could help me in clearing this stuff up :)

Thanks
 
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blue_leaf77
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In this case, the outgoing parabolic wave from −z0-z_0 will be focused into the point z0
If this is converging lens, then the rays will be focused at infinity, that is the outgoing rays are collimated.
 
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If this is converging lens, then the rays will be focused at infinity, that is the outgoing rays are collimated.
crap, yeah you're right. I was thinking in terms of rays from the object plane being focused into the image plane, but I mixed it up. sorry.

But anyway, do you know the mathematical form of waves converging to a single point?
 
  • #4
blue_leaf77
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converging spherical wave must be the inverse of the diverging one, the sign of the wavevector k must change (better change the sign of k rather than the sign of ##\omega t## as you did above, this will mean time reversal).
 
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