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

• z_R – Rayleigh range ¹, the distance along the axis of propagation from the beam wait such that `w(z) = sqrt(2)w`.  Since the beam half-width is a radial distance, the Rayleigh range marks where the cross-sectional area of the beam (defined by the beam half-width) has doubled.
• Θ - the full-angular width ², another measure of how quickly the beam spreads.  Bigger `Theta` means faster spread.
• w - The beam half-width ³, where the radial distance `r` from the axis of propagation is such that `E(r) = E_0/e`
• w₀ - The beam waist is the smallest beam half-width for a given beam (where the beam is "skinniest").

Gaussian Beams

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.

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.

We can rework `w(z)` to solve for `w_0`:
`w_0 = +sqrt(w^2 - (lambda*z)/pi)`

Rayleigh Range

`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.

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.

Full-Angular Width

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.

`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.

1. ^ Hecht, Eugene. "Modern Optics: Lasers and Other Topics." Optics. Reading, MA: Addison-Wesley, 2002. 618. Print.
2. ^ Hecht, Eugene. "Modern Optics: Lasers and Other Topics." Optics. Reading, MA: Addison-Wesley, 2002. 618. Print.
3. ^ Hecht, Eugene. "Modern Optics: Lasers and Other Topics." Optics. Reading, MA: Addison-Wesley, 2002. 618. Print.
4. ^ Hecht, Eugene. "Modern Optics: Lasers and Other Topics." Optics. Reading, MA: Addison-Wesley, 2002. 618. Print.
5. ^ Hecht, Eugene. "Modern Optics: Lasers and Other Topics." Optics. Reading, MA: Addison-Wesley, 2002. 618. Print.
6. ^ Hecht, Eugene. "Modern Optics: Lasers and Other Topics." Optics. Reading, MA: Addison-Wesley, 2002. 618. Print.