= Gaussian beam
{c}
{title2=$I(r,z)=2P e^{-2r^2/w(z)^2}/(\pi w(z)^2)$}
{wiki}
A fundamental <Gaussian beam> is an ideal beam of <light> whose transverse <wave amplitude> has a <Gaussian function> profile. Here consider a circularly symmetric beam in <vacuum>, with propagation along $z$, two transverse coordinates $x,y$, and <radius> $r=\sqrt{x^2+y^2}$ from the <optical axis>. With the <beam waist> at $z=0$, total <power> $P>0$, and <vacuum> <wavelength> $\lambda$, its time-averaged <light intensity>, meaning <irradiance> in $\mathrm{W\,m^{-2}}$, is
$$
I(r,z)=\frac{2P}{\pi w(z)^2}\exp\!\left(-\frac{2r^2}{w(z)^2}\right),
\qquad w(z)=w_0\sqrt{1+\left(\frac{z}{z_R}\right)^2},
\qquad z_R=\frac{\pi w_0^2}{\lambda}.
$$
Here $w_0$ is the minimum beam <radius>, and $z_R$ is the <Rayleigh range>. Far from the <beam waist>, $w(z)\sim\theta|z|$ with <beam divergence> half-angle $\theta=\lambda/(\pi w_0)$. These are the fundamental circular <Gaussian beam> laws within the scalar <paraxial approximation>, appropriate when $w_0\gg\lambda$ and the <beam divergence> is small. The underlying envelope solves the <parabolic wave equation> exactly; this does not make it an exact vector solution of <Maxwell equations>. https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/8e54d1625e9e71eb5b4e4998e6967e4e_chapter2.pdf[Franz Kärtner, MIT 6.974 notes, 2006, §2.4, equations 2.219 and 2.227–2.238]{ref}.
The <radius> convention matters. Writing $U$ for the complex <wave amplitude> envelope, the same $w(z)$ is the <wave amplitude> $1/e$ <radius> and the <light intensity> $1/e^2$ <radius>:
$$
\frac{|U(r,z)|}{|U(0,z)|}=e^{-r^2/w(z)^2},
\qquad \frac{I(r,z)}{I(0,z)}=e^{-2r^2/w(z)^2}.
$$
At $r=w(z)$ these ratios are $e^{-1}\approx0.367879$ and $e^{-2}\approx0.135335$, respectively. The <light intensity> $1/e$ <radius> is instead $w(z)/\sqrt2$. None of these radii is a sharp edge: the ideal <Gaussian function> has nonzero tails at every finite <radius>. Kogelnik and Li explicitly use the <wave amplitude> $1/e$ <radius> in their fundamental-mode treatment. https://dielslab.unm.edu/sites/default/files/Kogelnik66_0.pdf[Herwig Kogelnik and Tingye Li, Laser Beams and Resonators, Applied Optics 5, 1550–1567, 1966, §3.2]{ref}.
For a numerical example, take <vacuum> <wavelength> $\lambda=632.8\,\mathrm{nm}$ and <beam waist> <radius> $w_0=0.500\,\mathrm{mm}$. Substitution gives <Rayleigh range> $z_R\approx1.24114754\,\mathrm m$ and far-field <beam divergence> half-angle $\theta\approx0.402852992\,\mathrm{mrad}$. At $z=z_R$, the beam <radius> is $w=\sqrt2w_0\approx0.707107\,\mathrm{mm}$ and the on-axis <irradiance> is half its waist value. At the same <wavelength>, halving the <beam waist> <radius> to $0.250\,\mathrm{mm}$ quarters the <Rayleigh range> to $0.310286885\,\mathrm m$ and doubles the <beam divergence> half-angle to $0.805705984\,\mathrm{mrad}$. At fixed <power>, that smaller <beam waist> also quadruples the peak <irradiance>.
\Image[/gaussian-beam-waist-and-divergence.png]
{title=<Gaussian beam> radii at 632.8 nm: a smaller <beam waist> gives a shorter <Rayleigh range> and greater <beam divergence>}
The contours in the left panel are $r=\pm w(z)$ in a plane through the <optical axis>; they mark an <light intensity> level, not an outer boundary. The right panel compares the exact model radii with the far-field asymptotes $w=\theta z$. The horizontal and vertical axes use different length units, so the drawn slopes are not physical <angles>.
The normalization expresses conserved <power> during free propagation. In transverse <polar coordinates>, an <annulus> has <area> $2\pi r\,dr$, so
$$
\int_{\mathbb R^2}I\,dx\,dy
=2\pi\int_0^\infty I(r,z)r\,dr=P,
\qquad \frac{P(r<a)}{P}=1-e^{-2a^2/w(z)^2}.
$$
A proof outline is the substitution $s=2r^2/w(z)^2$ in this <integral>. Spreading reduces peak <irradiance> as $w^{-2}$ while increasing transverse <area> as $w^2$, leaving <power> unchanged. In particular, the <disk> of <radius> $w(z)$ contains $1-e^{-2}\approx0.8646647168$, or $86.4665\%$, of the total <power>; the local <light intensity> at its rim is only $13.5335\%$ of the on-axis <light intensity>. These are different quantities.
\Image[/gaussian-beam-power-profile.png]
{title=<Gaussian beam> <radius> conventions and the fraction of <power> inside a centered <disk>}
The left panel compares normalized <wave amplitude> magnitude and <irradiance>; the right panel integrates the transverse <area> to show enclosed <power>. For an ideal absorbing circular <aperture> of <radius> $a$ centered on the <optical axis>, the same fraction gives the <power> passing the <aperture> at that plane. An <aperture> of <radius> $w$ clips about $13.5335\%$ of the incident <power>; <radius> $2w$ clips only $e^{-8}\approx0.000335463$, or $0.0335463\%$. After clipping, extra <diffraction> means the transmitted field no longer obeys the unmodified ideal <Gaussian beam> law. Every finite <aperture> cuts some ideal tail. Kogelnik and Li distinguish their large-<aperture> beam analysis from their finite-<aperture> <diffraction> analysis. https://dielslab.unm.edu/sites/default/files/Kogelnik66_0.pdf[Kogelnik and Li, 1966, §§3–4]{ref}.
The useful tradeoff is that a small bright spot cannot remain narrow indefinitely. An <optical lens> can create a smaller <beam waist>, but <diffraction> then makes the beam spread faster. The particularly simple relation $w_0\theta=\lambda/\pi$ ties localization directly to <beam divergence>. A proof outline for the propagation law starts with $U(x,y,0)=U_0e^{-(x^2+y^2)/w_0^2}$. The <parabolic wave equation> separates into two transverse factors, and a <Gaussian integral> in each coordinate gives
$$
U(x,y,z)=\frac{U_0}{q(z)}\exp\!\left(-\frac{x^2+y^2}{w_0^2q(z)}\right),
\qquad q(z)=1+i\frac{z}{z_R},\qquad k=\frac{2\pi}{\lambda}.
$$
Here $q$ is dimensionless and $k$ is the <wavenumber>; the carrier convention is $e^{i(kz-\omega t)}$, with <angular frequency> $\omega$. Taking the squared <complex modulus> gives the stated $w(z)$ and its $w^{-2}$ <light intensity> scaling. Each transverse factor contributes $q^{-1/2}$, explaining the $q^{-1}$ prefactor here and the different prefactor in the preserved <Gaussian beam with one transverse coordinate>.
Historically, <Gaussian beam> theory developed alongside early <laser> resonator theory and lens-guided transmission. In their 1966 review, Kogelnik and Li credit Boyd and Gordon, and Goubau and Schwering, with early beam treatments; they summarize earlier work rather than claim the whole construction as a new discovery. Their paper brings beam propagation, focusing and finite-<aperture> <diffraction> into a common practical account. https://dielslab.unm.edu/sites/default/files/Kogelnik66_0.pdf[Kogelnik and Li, 1966, introduction and §3]{ref}.
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