L12 · Propagation, focus and measurement

Gaussian Beam & Beam Quality

Propagate two beam axes through free space, a focusing lens or a beam expander. Move a profiler, record a caustic, and recover beam quality from the measured second-moment radii.

Recorded distinct planes $\text{—}$

Horizontal radius at profiler $\text{—}$

Vertical radius at profiler $\text{—}$

Conserved horizontal beam quality $\text{—}$

Conserved vertical beam quality $\text{—}$

Horizontal quality fitted from records $\text{—}$

Vertical quality fitted from records $\text{—}$

Output horizontal waist position $\text{—}$

Output vertical waist position $\text{—}$

Output horizontal waist radius $\text{—}$

Output vertical waist radius $\text{—}$

Output horizontal caustic range $\text{—}$

Output vertical caustic range $\text{—}$

Horizontal far-field half-angle $\text{—}$

Vertical far-field half-angle $\text{—}$

Field-parameter applicability $\text{—}$

Horizontal Gaussian q real part at profiler $\text{—}$

Horizontal Gaussian q imaginary part $\text{—}$

L12 · Beam workbench

Measure quality across several planes

Enable first lensApply the first lens matrix at its physical plane.Enable second lensPlace a positive lens downstream of the first lens position.Use a negative first lensThe focal-length slider sets its magnitude.Vacuum wavelength

\lambda

1064\,\mathrm{nm}

Source horizontal waist radius

w_{0x}

0.4\,\mathrm{mm}

Source vertical waist radius

w_{0y}

0.4\,\mathrm{mm}

Source horizontal beam quality

M_x^2

1

Source vertical beam quality

M_y^2

1

Vertical source waist displacement

z_{0y}

0\,\mathrm{cm}

First lens axial position

z_1

40\,\mathrm{cm}

First lens focal-length magnitude

|f_1|

20\,\mathrm{cm}

Second lens separation

d

60\,\mathrm{cm}

Second lens focal length

f_2

40\,\mathrm{cm}

Movable profiler axial position

z_D

70\,\mathrm{cm}

Move the profiler without clearing records. Changing source or lenses starts a new acquisition. A beam expander trades spot size against divergence and preserves quality.

Air, scalar paraxial propagation and ideal thin lenses of infinite aperture. The source horizontal waist defines axial zero; horizontal and vertical axes are fixed principal axes without cross-axis coupling. Radius is twice the root-mean-square width, equal to the intensity one-over-e-squared radius for a Gaussian. Unit-quality sources are coherent Gaussian fields and admit a physical complex beam parameter. Higher-quality-factor sources use a Gaussian Schell moment model with the same Gaussian intensity shape and greater angular variance; they are not rendered as a coherent Gaussian with an arbitrary quality factor. The vertical source waist can be displaced to model simple astigmatism. Covariances propagate through the same ray matrices as the apparatus; ideal optics conserve each axis quality. The profiler is a movable, non-perturbing sampling plane, not an opaque stop; plots use peak normalization. Records contain ideal profile radii, not noisy experimental images. Only planes after the final enabled lens enter the fit. The automatic 13-plane scan adapts to the output caustic and may extend beyond the apparatus view. Three distinct planes suffice for this noiseless quadratic fit, not for ISO-compliant measurement. Finite apertures, aberration, partial-coherence reconstruction, general rotated astigmatism and detector noise are omitted. Inferred output waists upstream of the final lens are virtual extrapolations of its downstream segment.

Start with the essentials

Focus question: Can a lens reduce divergence without improving the beam quality factor?
One-sentence intuition: Size and divergence trade against each other under ideal optics. Their invariant second-moment product sets quality, while multiple sampled planes locate the waist and recover that product.

Core mathematical model

Coherent Gaussian field

\begin{aligned}q&=z-z_0+iz_R\\z_R&=\pi w_0^2/\lambda\\w^2(z)&=w_0^2\left[1+\left(\frac{z-z_0}{z_R}\right)^2\right]\end{aligned}

This physical complex parameter applies to unit-quality Gaussian fields. At the waist its real part vanishes.

Free propagation and thin lenses

\begin{aligned}P(d)&=\begin{pmatrix}1&d\\0&1\end{pmatrix}\\F(f)&=\begin{pmatrix}1&0\\-1/f&1\end{pmatrix}\\q^{\prime}&=(Aq+B)/(Cq+D)\end{aligned}

Use the same matrices for the apparatus and beam moments. A positive focal length converges; a negative one diverges.

Second moments and conserved quality

\begin{aligned}\Sigma&=\begin{pmatrix}\langle x^2\rangle&\langle x\theta\rangle\\\langle x\theta\rangle&\langle\theta^2\rangle\end{pmatrix}\\\Sigma^{\prime}&=T\Sigma T^{\mathsf T}\\M_x^2&=\frac{4\pi}{\lambda}\sqrt{\det\Sigma}\end{aligned}

Centroids are zero here. A unit-determinant ideal optical matrix preserves the covariance determinant and each principal-axis quality.

Radius, divergence and caustic

\begin{aligned}w_\sigma&=2\sqrt{\langle x^2\rangle}\\w_\sigma^2(z)&=w_{\sigma0}^2+\Theta^2(z-z_0)^2\\M^2&=\pi w_{\sigma0}\Theta/\lambda\end{aligned}

The radius is twice the root-mean-square width and divergence is a half-angle. The superscript belongs to the quality-factor name; do not square the slider value again.

Recover quality from recorded planes

\begin{aligned}w_\sigma^2(z)&=a+bz+cz^2\\z_0&=-b/(2c)\\w_{\sigma0}^2&=a-b^2/(4c)\\M^2&=\frac{\pi}{\lambda}\sqrt{ac-b^2/4}\end{aligned}

Fit radius squared to a quadratic using only recorded profile values. Fewer than three distinct planes cannot determine all coefficients; real measurements need more sampling and noise assessment.

Keplerian beam expander

\begin{aligned}d&=f_1+f_2\\|\mathcal M|&=f_2/f_1\\w_{\rm out}&\simeq|\mathcal M|w_{\rm in}\\\Theta_{\rm out}&\simeq\Theta_{\rm in}/|\mathcal M|\end{aligned}

Two positive lenses separated by their summed focal lengths exchange size and divergence for a nearly collimated input. A finite Gaussian wavefront gives a small departure from the ray-optics magnification.

Common difficulties

Small spot is not high quality

Typical misconceptionThe smallest focal spot must have the best beam quality.

Better mental modelQuality uses both waist radius and far-field divergence; source width, wavelength and lens power also affect spot size.

Generalized moments are not a coherent field

Typical misconceptionAn arbitrary quality factor can be inserted into physical Gaussian q while retaining the same pure field.

Better mental modelHigher-quality-factor sources here are Gaussian Schell moment models. Their intensity can be Gaussian while coherence and angular spread differ.

A teaching scan is not a certification

Typical misconceptionThirteen ideal samples prove ISO-compliant experimental measurement.

Better mental modelThis noiseless acquisition demonstrates the fit. Real work requires prescribed sampling, calibration, background correction and uncertainty assessment.

Run the experiment

01
Learn free propagation
Choose free propagation and move the profiler through several planes. Record three distinct planes.
What to observe: The spot grows away from its source waist; the fitted quality is one.
02
Focus without repairing quality
Compare focus and poorer-quality presets; scan 13 planes for each.
What to observe: The same source radius can hide different angular variance. The lens changes the caustic but preserves each quality factor.
03
Separate astigmatic axes
Choose two waist planes, move the profiler and scan.
What to observe: Horizontal and vertical waists occur at different positions; one round-looking spot need not mean a common waist.
04
Trade divergence for size
Choose the threefold expander, compare radii and divergence, and toggle the second lens.
What to observe: The two-lens train approximately triples a nearly collimated beam radius and divides its divergence by three; its quality remains unchanged.