L12 · Propagation, focus and measurement

Gaussian Beam & Beam Quality

Can a lens make a poor beam diffraction limited? No. An ideal lens can make the beam smaller or less divergent, but shrinking one enlarges the other: the product of waist size and divergence stays fixed, and that product is the beam quality. To measure it, record the beam radius at several distances, locate the waist and work out the quality factor.

Send a beam through free space, a focusing lens or a beam expander and follow its horizontal and vertical widths. Move a profiler along the beam, record how the radius changes, and recover the 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 beam quality at 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}

Moving the profiler keeps the existing records. Changing the source or a lens starts a new acquisition. A beam expander trades spot size for divergence and leaves the beam quality unchanged.

The beam travels in air. The model uses scalar paraxial propagation and ideal thin lenses with unlimited aperture. Axial position zero is the source's horizontal waist. The horizontal and vertical axes are fixed principal axes, and the model does not couple them. The radius is twice the root-mean-square width; for a Gaussian this is the radius where intensity falls to one over e squared. Sources with beam quality one are coherent Gaussian fields and have a physical complex beam parameter. Sources with a higher quality factor use a Gaussian Schell moment model: the intensity keeps the same Gaussian shape, but the angular spread is larger. They are not drawn as a coherent Gaussian with an arbitrary quality factor. Moving the vertical source waist along the axis models simple astigmatism. The covariances propagate through the same ray matrices as the apparatus, and ideal optics keep the quality of each axis fixed. The profiler is a movable sampling plane that does not disturb the beam; it is not an opaque stop. Plots are normalized to their peak. Records hold ideal profile radii, not noisy experimental images. Only planes after the last enabled lens enter the fit. The automatic 13-plane scan adapts to the output caustic and may run past the edge of the apparatus view. Three distinct planes are enough for this noiseless quadratic fit, but not for an ISO-compliant measurement. Finite apertures, aberrations, partial-coherence reconstruction, general rotated astigmatism and detector noise are left out. An output waist inferred upstream of the last lens is virtual: it is extrapolated back from the segment after that lens.

Start with the essentials

Focus question: Can a lens reduce divergence without improving the beam quality factor?
One-sentence intuition: Ideal optics can trade beam size for divergence, but the product of the two, taken from second moments, stays fixed and sets the beam quality; radii measured at several 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 complex beam parameter describes a Gaussian field whose beam quality is one. Its real part is zero at the waist.

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}

The same ray matrices describe the apparatus and carry the beam's second moments through it. A lens with a positive focal length converges the beam; a negative one diverges it.

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}

The beam is centered on the axis here. An ideal optical matrix has determinant one, so it keeps the covariance determinant and the quality along each principal axis unchanged.

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 the divergence is a half-angle. The superscript is part of the quality factor's name, so 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 the squared radius to a quadratic in position, using only the recorded profiles. You need at least three distinct planes to fix all the coefficients; a real measurement needs more planes and an estimate of the noise.

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 spaced by the sum of their focal lengths enlarge a nearly collimated beam and divide its divergence by the same factor. A Gaussian beam of finite size departs slightly from this ray-optics magnification.

Common difficulties

A small spot does not mean high quality

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

Better mental modelBeam quality depends on both the waist radius and the far-field divergence. The focal spot size also depends on the source width, the wavelength and the lens power.

A moment model is not a coherent field

Typical misconceptionYou can put any quality factor into the Gaussian beam parameter and still have the same pure field.

Better mental modelHere, sources with a higher quality factor use a Gaussian Schell moment model. Their intensity can still be Gaussian, but their coherence and angular spread are different.

A teaching scan is not a certified measurement

Typical misconceptionThirteen ideal samples make an ISO-compliant experimental measurement.

Better mental modelThis noiseless scan only demonstrates the fit. A real measurement needs prescribed sampling, calibration, background correction and an uncertainty estimate.

Run the experiment

01
Watch free propagation
Choose Free propagation and move the profiler to several planes. Record three distinct planes.
What to observe: The spot grows as it moves away from the source waist, and the fitted beam quality comes out as one.
02
Focusing does not improve quality
Compare the Focus a Gaussian and Same size, poorer beam quality presets, and scan 13 planes for each.
What to observe: Two sources with the same radius can have different angular spreads. The lens reshapes the caustic, but each quality factor stays the same.
03
Separate the astigmatic axes
Choose Two different waist planes, move the profiler and scan.
What to observe: The horizontal and vertical waists sit at different positions. A spot that looks round at one plane does not mean the two axes share a waist.
04
Trade divergence for size
Choose the Threefold Keplerian expander, compare the radii and divergences, and switch the second lens on and off.
What to observe: The two lenses roughly triple the radius of a nearly collimated beam and cut its divergence to about a third. The beam quality does not change.

Gaussian Beam & Beam Quality

A lens changes focus, not the phase-space area

Core mathematical model

Coherent Gaussian field

Free propagation and thin lenses

Second moments and conserved quality

Radius, divergence and caustic

Recover quality from recorded planes

Keplerian beam expander

Common difficulties

A small spot does not mean high quality

A moment model is not a coherent field

A teaching scan is not a certified measurement

Run the experiment

Watch free propagation

Focusing does not improve quality

Separate the astigmatic axes

Trade divergence for size

Related experiments and articles

Beam shape and beam quality