# How to find the center of a paraboloidal wave

• semc
In summary, by multiplying the paraboloidal wave and the transmittance through a thin lens, we can obtain the equation U(r)=Aexp(-jnkd)exp[jk(\frac{x^2+y^2}{2f}-z)], which represents a paraboloidal wave centered at z. By multiplying this equation with the lens phase exp[ik(x^2+y^2)/(2f)], we can determine the center of the paraboloidal wave, which will be located at a distance f from the lens.
semc

## Homework Statement

Show that the paraboloidal wave centered at the point z1 is converted by a lens of focal length f into a paraboloidal wave centered about z2 where 1/z1+1/z2 =1/f

## Homework Equations

Equation of paraboloidal wave and transmittance through a thin lens.

## The Attempt at a Solution

I multiplied the paraboloidal wave and the transmittance through the thin lens and I got
U(r)=Aexp(-jnkd)exp[jk($\frac{x^2+y^2}{2f}$-z)]
But how do I continue to show that this paraboloidal wave is centered at f?

It looks like your lens is located at z=0; the parabolic wave centered at z is basically a spherical wave centered at z1 where z1 is faraway from the lens, i.e., z1^2>>x^2+y^2, so that the wavefront at the lens ~ exp[ik(x^2+y^2)/(2z1)]. (Taylor expansion of the phase ikR where R^2=x^2+y^2+z^2) Now multiply by the lens phase exp[ik(x^2+y^2)/(2f)] and figure out where the center is. Double check the sign conventions since I haven't done these for years.

## 1. How do I determine the center of a paraboloidal wave?

To find the center of a paraboloidal wave, you need to locate the point where the curvature of the wave is the greatest. This point is known as the focal point and is located along the axis of symmetry of the paraboloid.

## 2. Can I use mathematical equations to find the center of a paraboloidal wave?

Yes, you can use mathematical equations to find the center of a paraboloidal wave. The equation for a paraboloid is x^2 + y^2 = 4ax, where a is the distance from the origin to the focal point. By solving this equation, you can determine the coordinates of the focal point and thus, the center of the wave.

## 3. Is it possible to find the center of a paraboloidal wave experimentally?

Yes, it is possible to find the center of a paraboloidal wave experimentally. This can be done by measuring the distance between the focal point and various points on the wave's surface. The point with the greatest distance from the focal point will be the center of the wave.

## 4. How does the shape of a paraboloidal wave affect its center?

The shape of a paraboloidal wave does not affect its center. The center will always be located at the focal point, regardless of the size or curvature of the wave.

## 5. Can the center of a paraboloidal wave change over time?

No, the center of a paraboloidal wave will not change over time. As long as the shape and size of the wave remain constant, the center will always be located at the focal point. However, if the wave is distorted or manipulated in some way, the center may shift accordingly.

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