This is a pretty neat way to glow up. Light in the fiber optic cable zooms through the core by constantly bouncing off of the edge of the molded plastic, a principle called "total internal reflection" (which also kinda sounds like a very expensive meditation island retreat). When the light pipe doesn't have total reflection, you can get a nifty lighting effect from the light illuminating the plastic tubing.

Beamdivergencecalculator

`z_R`, the Rayleigh range, is the distance (`z_R`) from the beam waist where the circle of radius `w(z_R)` doubles in area compared to the area of the circle defined by the beam waist.  In other words, `z_R equiv z : w(z) = sqrt(2) w_0`.  See Picture 2.  A smaller `z_R` means the beam spreads faster.  A larger `z_R` means the beam spreads slower.

Gaussian beamsoftware

Lasercalculator

Using the definition of `z_R` and the equation for `w(z)`, we see5 that `z_R = (pi w_0^2)/lambda`.   As we can see from the equation for `z_R`, beams with a smaller waist or larger wavelength spread faster.  Beams with a larger waist or smaller wavelength spread slower.

The TEM00 mode in a resonator has a Gaussian profile; the beam extends to infinity, but it quickly approaches 0 as it does so.  As such, a common way to describe the "size" of the beam is its beam half-width (w).  The beam half-width is the radial distance from the central axis where the electric field drops to `1/e` of its value at the central axis.  I.e. `E(w) = E_0/e`.   More generally, the electric field is shown by `E(r) = E_0e^(-r^2/w^2)`, where r is the radial distance from the central axis.  Since the intensity of a beam is proportional to the square of the Electric Field- `I(r) = I_0 e^(-r^2/w^2)` - the Intensity is even more clumped up around the central axis, with `I(w) = I_0 /e^2 approx (.14) I_0`.  This means that the majority of the beam's power is transferred where `r le w`, which supports the use of `w` to define the size of a Gaussian beam.

The Gaussian Beam Calculator calculates the beam half-width, Rayleigh range, and full-angular width of a Gaussian Beam (TEM00).

Gaussian beam

If you buy less than a full reel, you'll get a single strip, but it will be a cut piece from a reel which may or may not have a connector on it. If the piece comes from the end of the reel, the connector may be on the output end of the strip!

For `z`>>`z_R`, `w(z)` is roughly linear.  This means we can approximate the angle `Theta` between the two edges of the beam very far from the waist.  See Picture 3.  With a quick limit and the small angle approximation, we conclude6 that `Theta = (2 lambda)/(pi w_0)`.  Larger `Theta` means the beam spreads faster.

Adafruit NeoPixel Digital RGB LED strips come to us in 4 or 5 meter reels with a 2 or 3-pin JST SM connector on each end and separated power/ground wires as shown in the picture below. If you order a full 4 or 5 meters, you get the full reel with both connectors installed.

All you need is a way to get a ton of photons into your light pipe, and that's what this is! It's Fiber Optic Light Source in a cool white color. It's a high powered LED cluster, you need to power it with 9-12V and 100mA of current, the metal body acts as a heat sink. Then, installation is a breeze. Just loosen the clear knob on the LED emitter and slide one of our fiber optic tubes.

By applying the definition of `w` to the solutions to a TEM00 in a resonator, we reach the equation4 `w = w_0 sqrt(1 + (lambda * z)/(pi * w_0^2))`, where `z=0` at the "waist" of the beam, (`w_0`).  See Picture 1.  The equation `w(z)` traces out a hyperbola, so it's more curved in the middle and approaches a straight line farther away.

`Theta` is very much related to `z_R`, and they tell us essentially the same thing about the spread of Gaussian Beams .  Higher wavelength beams spread faster.  Beams that were initially focused to a smaller waist spread faster.