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Collimated light is light whose propagation directions occupy a narrow angular range. In geometrical optics, ideal collimation is represented by parallel light rays. A collimator approximates this by placing a small source at the focal position of a lens. A reticle projected by an autocollimator provides a practical example: the outgoing pattern travels as an approximately collimated beam before reflection from a test mirror.^[ref]

A finite beam still undergoes diffraction. For an ideal Gaussian beam in free space, let $w_0$ be its beam waist radius, measured where the intensity has fallen to $1/e^2$ of its axial value, and let $\lambda$ be its wavelength. Its far-field beam divergence half-angle and Rayleigh range areThe radius at distance $z$ from the beam waist is $w(z)=w_0\sqrt{1+(z/z_R{)}^2}$. Thus small beam divergence can make a beam nearly constant in width over a useful distance without making it perfectly parallel everywhere.^[ref]

For an original example, choose $\lambda =500\mathrm{nm}$ and $w_0=1\mathrm{mm}$. Then $\theta \simeq 0.159\mathrm{mrad}$ and $z_R\simeq 6.28\mathrm{m}$. At $z=z_R$ the radius is $\sqrt{2}\mathrm{mm}$, so even this small-divergence beam eventually widens. Collimation describes propagation directions; optical coherence describes phase correlations and is a separate property.

The encircled radiant energy of a Gaussian beam is the radiant energy inside a centred circular receiver at a specified transverse plane. For fixed Gaussian beam radius $w$ and total laser pulse energy $U$,The radiant exposure of a Gaussian beam contains the radial integration and aperture-capture plot. Its $w$ is the irradiance $e^{-2}$ radius, as specified in the primary beam account.^[ref]

An original annular example distinguishes energy near an axis from energy between two radii. Subtracting two encircled values givesChoose $U=5\mu \mathrm{J}$, $w=0.50\mathrm{mm}$, $R_1=0.25\mathrm{mm}$ and $R_2=0.50\mathrm{mm}$. The annulus receives $5(e^{-1/2}-e^{-2})\approx 2.356\mu \mathrm{J}$, or 47.12 percent of the full beam energy. A bright central point does not imply that most energy lies in an arbitrarily small disk around it.

Indeed, the radial radiant-energy density of a Gaussian beam is $dQ/dR=2\pi RH(R)$, where $H$ is local radiant exposure. For this profile it is largest at $R=w/2$, even though $H$ is largest at zero radius. Increasing a centred aperture and comparing integrated detector readings can test the cumulative prediction. Detector linearity, aperture alignment and the finite collection boundary must be controlled; a displaced aperture is not described by the centred formula.

Francesco D'Angelo is an optical specialist associated with Edmund Optics, whose own laser-optics staff directory identifies him as a Senior Solutions Engineer.^[ref] He is the named author of the technical note Propagation of a Gaussian Beam Through a Thin Lens by Francesco D'Angelo, which explains Gaussian beam propagation and transformation by a thin lens.

The note relates beam waist, beam divergence and Rayleigh range, then uses the complex beam parameter to calculate how a thin lens changes the beam. This connects an optical component specification to the actual position and size of a focused beam, instead of treating every incident beam as a perfectly parallel bundle of rays.^[ref]

An illustrative consequence of the note's ideal Gaussian beam relation $w_0\theta =\lambda /\pi$ is easy to check. At a fixed wavelength $\lambda =1\mu \mathrm{m}$, a beam waist radius of $0.5\mathrm{mm}$ gives a far-field beam divergence half-angle of approximately $0.637\mathrm{mrad}$. Halving the beam waist radius doubles that half-angle. Focusing to a smaller spot and obtaining lower beam divergence cannot be chosen independently within this model.^[ref]

The employer's laser-optics staff directory includes a labeled portrait.

The radial radiant-energy density of a Gaussian beam gives energy per increment of radius, rather than per unit surface area. For local radiant exposure $H(R)$ and encircled radiant energy of a Gaussian beam $Q(R)$,Its units are $\mathrm{J}{\mathrm{m}}^{-1}$, while $H$ has units $\mathrm{J}{\mathrm{m}}^{-2}$. The primary Gaussian profile uses $w$ as the irradiance $e^{-2}$ radius.^[ref]

For an original narrow-annulus comparison, take $U=1\mu \mathrm{J}$ and $w=0.80\mathrm{mm}$. The central disk from zero to $0.020\mathrm{mm}$ receives about $1.249\mathrm{nJ}$. The annulus from $0.390$ to $0.410\mathrm{mm}$ receives about $30.32\mathrm{nJ}$. Both span the same radial width, but the outer annulus has much more area. Equal radial width is not equal receiving area.

The shell density is maximal at $R=w/2$ and vanishes at $R=0$, despite the local exposure being largest on axis. Dividing $g$ by $U$ gives a probability density function over nonnegative radius: it is a Rayleigh distribution with scale $w/2$. This is an energy-weighted mathematical representation, not an additional photon-counting assumption. Differences between successive centred-aperture energy measurements estimate shell energies; dividing by their radial separation approximates $g$ when that separation is sufficiently small.

The radiant exposure of a Gaussian beam is its time-integrated transverse irradiance. At a fixed plane normal to a circular Gaussian beam, suppose the Gaussian beam radius $w$ is unchanged during a pulse and denotes the radius where irradiance is $e^{-2}$ of the on-axis value. With total laser pulse energy $U$,The primary beam account specifies this radius convention and the corresponding irradiance normalization; the radiant exposure parent derives its time integral.^[ref]

An original aperture example asks how much of that energy a finite receiver collects. Integrating over a centred disk of radius $R$ gives the encircled radiant energy of a Gaussian beam:For stipulated $U=2\mu \mathrm{J}$, $w=0.80\mathrm{mm}$ and $R=0.40\mathrm{mm}$, the receiver collects only $39.35$percent, or $0.7869\mu \mathrm{J}$. Its spatially averaged radiant exposure is about $1.566\mathrm{J}{\mathrm{m}}^{-2}$, compared with on-axis $H(0)\approx 1.989\mathrm{J}{\mathrm{m}}^{-2}$. Dividing the total $U$ by this receiver's area would count energy that missed it.

To collect a chosen fraction $0<f<1$ under this ideal model, the centred aperture needs $R=w\sqrt{-\log (1-f)/2}$. For95percent, $R\approx 1.224w$. A possible experimental check measures integrated received energy through several known apertures and compares their ratios. Aperture centring, detector linearity and any profile change during the pulse affect that comparison. A time-varying radius must remain inside the time integral rather than being replaced by an unspecified average radius.

Wavefront curvature describes the change in the normal direction across a surface of equal wave phase. A plane wavefront has zero curvature; a spherical wavefront has curvature magnitude $1/|R|$ for radius $R$. A general wavefront can have different principal curvatures in different transverse directions. The sign convention must be stated when distinguishing converging and diverging waves.^[ref]

In a paraxial transverse plane, choose the convention in which a diverging spherical wave has $R>0$. Its quadratic phase variation is approximatelyThe corresponding local ray angle is approximately $r/R$. For illustrative $R=0.50\mathrm{m}$ and $r=2\mathrm{mm}$, the angle is $0.004$ radians and the optical path difference is $r^2/(2R)=4\mu \mathrm{m}$. At wavelength $500\mathrm{nm}$, that is eight phase cycles. A modest geometrical departure can therefore produce a substantial optical phase difference.

For an ideal Gaussian beam with its waist at $z=0$,The reciprocal form remains finite at the waist: its value is zero, so the wavefront there is locally planar even though the beam radius is minimal. At $z=z_R$, $R=2z_R$. Beam size and curvature are distinct beam parameters.^[ref]

A thin lens changes the curvature according to $1/R^{\prime }=1/R-1/f$ in this convention, while the beam radius at the lens is unchanged in the thin-element approximation.^[ref] Thus a lens can alter convergence without an immediate jump in the beam width.

Experimentally, curvature can be inferred from a measured phase profile or local wavefront slopes. Phase unwrapping and the chosen reference matter because an intensity image in one plane does not itself provide the quadratic phase. A curved wavefront is not necessarily aberrated: the appropriate ideal reference can itself be spherical or Gaussian-beam curvature.