# Gaussian beam interference

• davidbenari
In summary, the phase shift of a gaussian beam and a plane wave is ##\pi##. This means that if you measure interference before and after you'll see the minimums and maximums of intensity invert.

#### davidbenari

The superposition of a gaussian beam and a plane wave generates a pattern of rings whose phase shift before and after the focal plane (of the gaussian beam) is ##\pi##. This means that if you measure interference before and after you'll see the minimums and maximums of intensity invert.

We can create the superposition via

##(e^{-ikz}+\frac{w_0}{w(z)}e^{-r^2/w(z)^2}e^{-i(kz+\frac{kr^2}{2R(z)}-\psi(z))} ) * (e^{ikz}+\frac{w_0}{w(z)}e^{-r^2/w(z)^2}e^{i(kz+\frac{kr^2}{2R(z)}-\psi(z))} )##

The obtained expression is

##1+\frac{w_0^2}{w(z)^2}\exp(-2r^2/w(z)^2) + 2 \frac{w_0}{w(z)} e^{-r^2/w(z)^2} \cos (\frac{kr^2}{2R(z)}-\psi(z))##

Remember ##R(z)=z[1+(z_R/z)^2]## and ##\psi(z)=\textrm{arctan}(z/z_R)##

MY PROBLEM IS:

This equation doesn't predict the pi phase shift!

Consider the argument of the cosine function.

##R(z^+)= R##

At ##z^+##

The argument is

##\frac{kr^2}{2R}-\pi/2##

at ##z=-z^+##

The argument is

##\frac{-kr^2}{2R}+\pi/2##

Now we know ##cos(\theta)=cos(-\theta)## therefore there is no pi phase shift.

But I know as a matter of fact that there should be a pi phase shift( I've observed it!).

But I don't understand what's happening here mathematically.

Any help will be very much appreciated.

By ##\pi## phase change, they must mean the Gouy phase ##\psi(z)##. As you go from ##-\infty## to ##\infty##, this term undergoes a change of ##\pi##.

Yeah, but the interference term isn't changing by ##\pi## which would explain the inversion of fringes. How else can one explain this inversion of fringes ? The sources I've seen just say "it's due to the gouy phase as can be seen from the interference term".

Is there any way to write a coordinate free gaussian beam so as to make ##k## negative some times? This would fix a lot of things!

## What is a Gaussian beam interference?

Gaussian beam interference refers to the phenomenon in which two or more Gaussian beams of light overlap and interfere with each other, resulting in a complex interference pattern. This is a common occurrence in optics and is used in various applications such as interferometry and holography.

## How is Gaussian beam interference different from other types of interference?

Gaussian beam interference is different from other types of interference, such as Young's double-slit interference, because it involves the interference of light beams with a Gaussian intensity distribution. This results in a distinct interference pattern with a central peak and concentric rings of varying intensity.

## What factors affect Gaussian beam interference?

The intensity and spatial profiles of the individual beams, the angle of incidence, and the distance between the beams are some of the factors that can affect Gaussian beam interference. Any changes in these factors can alter the interference pattern and its characteristics.

## How is Gaussian beam interference used in real-world applications?

Gaussian beam interference has various applications in fields such as optics, photonics, and telecommunications. It is used in interferometers to measure tiny displacements, in holography to create 3D images, and in laser systems to shape and manipulate the intensity distribution of the beam.

## What are some challenges in working with Gaussian beam interference?

One of the main challenges in working with Gaussian beam interference is achieving a stable and precise alignment of the beams. This is crucial for obtaining accurate interference patterns. Additionally, the intensity and profile of the beams can change due to factors such as thermal effects, which can also affect the interference pattern.