Begin with the fundamental TEM₀₀ field
A laser resonator does not support arbitrary field shapes. After every round trip, the light must reproduce the same transverse pattern. The allowed patterns are called transverse modes.
In the common Cartesian notation they are labelled TEMmn. TEM means transverse electromagnetic; the two indices count dark nodal lines in the horizontal and vertical directions. The lowest-order pattern is therefore TEM₀₀: one smooth central spot, without a dark line through it. This is the ideal Gaussian beam described by the calculator.
Under the paraxial approximation, its radially symmetric complex field is
\[ E(r,z)=E_0\,\frac{\omega_0}{\omega(z)} \exp\!\left[-\frac{r^2}{\omega^2(z)}\right] \exp\!\left[-i\left(kz+\frac{kr^2}{2R(z)}-\psi(z)\right)\right] \]This is called a Gaussian beam because its transverse amplitude contains the Gaussian factor \(\exp[-r^2/\omega^2(z)]\). The remaining exponential has magnitude one and therefore changes only the phase, not the intensity.
The coordinates are simple: \(z\) runs along the beam and \(r\) measures distance from its axis. The waist is placed at \(z=0\), where \(\omega_0\) is the minimum beam radius and \(E_0\) is the on-axis field amplitude. After propagation, that radius becomes \(\omega(z)\).
The phase contains three further quantities: the wave number \(k=2\pi/\lambda\), the wavefront curvature \(R(z)\), and the Gouy phase \(\psi(z)\). Their forms will follow from propagation below; they are not extra independent inputs.
From field amplitude to intensity
A detector measures time-averaged intensity, which is proportional to the squared magnitude of the complex field: \(I\propto |E|^2\). Taking the magnitude squared removes the phase exponential and doubles the exponent of the Gaussian amplitude:
\[ I(r,z)=I_0\left[\frac{\omega_0}{\omega(z)}\right]^2 \exp\!\left[-\frac{2r^2}{\omega^2(z)}\right] \]To express the result using total optical power \(P\), require that integrating the intensity over the transverse plane returns that power:
\[ P=\int_0^\infty I(r,z)\,2\pi r\,dr \quad\Longrightarrow\quad I(r,z)=\frac{2P}{\pi\omega^2(z)}\exp\!\left[-\frac{2r^2}{\omega^2(z)}\right] \]Now the meaning of the beam radius follows directly. At \(r=\omega(z)\), the intensity has fallen to \(e^{-2}\approx13.5\%\) of its on-axis value. Thus \(\omega\) is the 1/e² intensity radius; the corresponding beam diameter is \(2\omega\). At the waist, \(I_0=2P/(\pi\omega_0^2)\).
How diffraction sets the beam geometry
The functions \(\omega(z)\), \(R(z)\), and \(\psi(z)\) are not independent additions to the field. Substituting the Gaussian form into the paraxial wave equation fixes all three. The beam radius must evolve as
\[ \omega(z)=\omega_0\sqrt{1+\left(\frac{z}{z_R}\right)^2}, \qquad z_R=\frac{\pi\omega_0^2}{\lambda} \]The Rayleigh range \(z_R\) is therefore the natural propagation length of the beam. It grows with the square of the waist: doubling \(\omega_0\) makes the near-collimated region four times longer.
| Quantity | Symbol | Relation |
|---|---|---|
| Rayleigh range | \(z_R\) | \(z_R = \pi \omega_0^2 / \lambda\) |
| Divergence (half-angle) | \(\theta\) | \(\theta = \lambda / (\pi \omega_0)\) |
| Beam radius vs. z | \(\omega(z)\) | \(\omega(z) = \omega_0 \sqrt{1 + (z/z_R)^2}\) |
| Wavefront curvature | \(R(z)\) | \(R(z) = z\left(1 + (z_R/z)^2\right)\) |
| Beam parameter product | \(\omega_0\,\theta\) | \(\omega_0\,\theta = \lambda / \pi\) |
| Gouy phase | \(\psi(z)\) | \(\psi(z)=\tan^{-1}(z/z_R)\) |
Two regimes now follow from the same radius equation (4). Within ±zᵣ of the waist (the near field) the beam is approximately collimated and the wavefront is nearly flat. Far beyond it (the far field) the radius grows linearly, \(\omega(z) \approx \theta z\), and the wavefront looks spherical, \(R \approx z\). The reciprocity \(\omega_0 \theta = \lambda/\pi\) is the diffraction limit: a smaller waist always diverges faster.
The phase functions are fixed at the same time:
\[ R(z)=z\left[1+\left(\frac{z_R}{z}\right)^2\right], \qquad \psi(z)=\tan^{-1}\!\left(\frac{z}{z_R}\right) \]The wavefront is flat at the waist because \(R(0)\to\infty\). Across the focus, \(\psi\) changes by \(\pi\): this is the Gouy phase shift. In the far field, the radius equation reduces to \(\omega(z)\approx\theta|z|\), giving \(\theta=\lambda/(\pi\omega_0)\).
Why the complex q parameter is useful
Beam size and wavefront curvature always evolve together, so it is useful to store them in one complex number:
\[ \frac{1}{q(z)} = \frac{1}{R(z)} - i\,\frac{\lambda}{\pi \omega^2(z)} \qquad\Longleftrightarrow\qquad q(z)=z+i z_R \]The second form holds when \(z\) is measured from the waist. Free propagation is now simply addition of a real distance, while lenses change the curvature through a matrix transform.
In paraxial ray optics, an ABCD matrix maps input ray height \(r_1\) and angle \(\alpha_1\) to the output ray:
\[ \begin{pmatrix}r_2\\\alpha_2\end{pmatrix} =\begin{pmatrix}A&B\\C&D\end{pmatrix} \begin{pmatrix}r_1\\\alpha_1\end{pmatrix} \]The letters \(A\), \(B\), \(C\), and \(D\) are just the four matrix entries—\(D\) is not a distance. \(A\) and \(D\) are dimensionless, \(B\) has units of length, and \(C\) has units of inverse length. The same matrix transforms a Gaussian beam by
\[ q' = \frac{A q + B}{C q + D} \]Chain the matrices of a system and apply (8) once to propagate through all of it. The two building blocks: free space of length d is \(\left(\begin{smallmatrix}1 & d\\ 0 & 1\end{smallmatrix}\right)\); a thin lens of focal length f is \(\left(\begin{smallmatrix}1 & 0\\ -1/f & 1\end{smallmatrix}\right)\). This is exactly what the calculator does internally.
Derive the thin-lens transform
Take an input waist \(\omega_0\) a distance \(s\) before a lens of focal length \(f\). At the waist, \(q=i z_R\). Propagating to the lens adds \(s\), so the incident beam parameter there is
\[ q_L=s+i z_R,\qquad z_R=\frac{\pi\omega_0^2}{\lambda} \]A thin lens has \(A=D=1\), \(B=0\), and \(C=-1/f\). Substituting those entries into the ABCD rule (8) gives
\[ \frac{1}{q_{\mathrm{out}}}=\frac{1}{q_L}-\frac{1}{f} \]Immediately after the lens, describe the new Gaussian beam by its waist distance \(s'\) and Rayleigh range \(z_R'\): \(q_{\mathrm{out}}=-s'+i z_R'\). The minus sign appears because the new waist is still a positive distance ahead of the lens. Separating the real and imaginary parts now produces the quantities plotted by the lens calculator.
The waist magnification is \(m = \omega_0'/\omega_0 = f / \sqrt{(s-f)^2 + z_R^2}\). As \(z_R \to 0\) both expressions collapse to the geometric results (\(1/s + 1/s' = 1/f\), \(m = f/|s-f|\)) — the Gaussian correction is entirely in the \(z_R\) terms.
Waist at the front focal plane (\(s = f\))
Setting \(s = f\) removes the first term and gives a clean, much-used pair:
\[ s'=f,\qquad \omega_0' = \frac{\lambda f}{\pi \omega_0} = f\,\theta \]A waist at the front focal plane images to a waist at the back focal plane, with a size set only by the focal length and the input divergence \(\theta = \lambda/(\pi \omega_0)\). This is the Fourier relationship between the two focal planes, and the basis of fiber collimators (put the fiber tip at the focal plane and the output waist radius is \(f\,\theta\)).
Collimated input (\(z_R \gg f\))
For a beam that is essentially collimated at the lens with radius \(\omega_L\), the focus lands one focal length away with spot radius
\[ \omega_0' \approx \frac{\lambda f}{\pi \omega_L} \]the familiar diffraction-limited focused-spot formula.
Open a worked fiber-to-telescope example →
Where the paraxial model is valid
Everything above is a paraxial result: it assumes small angles, so that \(\sin\theta \approx \tan\theta \approx \theta\) and the field is a slowly varying envelope on the carrier wave. The Gaussian beam is the leading-order mode of the paraxial wave equation, so the model is self-consistent only while the divergence stays small — formally \(\theta = \lambda/(\pi \omega_0) \ll 1\).
Where it breaks down, in practical terms:
- Good to ~1–2% for divergence half-angle \(\theta \lesssim 0.3\ \mathrm{rad}\) (~17°), i.e. numerical aperture \(\mathrm{NA} \lesssim 0.3\), equivalently waist \(\omega_0 \gtrsim \lambda\).
- Degrades as the waist shrinks toward the wavelength (NA → 0.5 and beyond): the real focal spot is larger than the paraxial prediction, and a longitudinal (on-axis) field component appears, so the beam is no longer purely transverse.
- Fails for tight focusing (high-NA objectives, optical tweezers, NA ≳ 0.6): polarization and vector-diffraction effects dominate. Use a vector diffraction model (Richards–Wolf) there, not Gaussian beam formulas.
For a focusing lens, define the effective f-number from the beam that actually illuminates it: \(N=f/D_{\mathrm{beam}}=f/(2\omega_L)\). In air and at small angles, \(\mathrm{NA}_{\mathrm{eff}}\approx1/(2N)\). Read what effective f-number and NA mean for a Gaussian beam →
How M² describes a real beam
Real beams diverge faster than the ideal by the beam quality factor \(M^2\ge1\):
\[ M^2=\frac{\pi\omega_0\theta}{\lambda}, \qquad \theta=\frac{M^2\lambda}{\pi\omega_0} \]An ideal TEM₀₀ beam has \(M^2=1\). A larger value means more divergence and poorer focusability; for Gaussian-like beams, use the propagation equations above with \(\lambda\to M^2\lambda\). Read what M² means and how it is measured →