Gaussian Beam

Choosing the shortest practical Galilean or Keplerian beam expander

I remember that when I first started designing optical systems, I was always unsure how to start choosing the lenses for my 4f systems. Sometimes it is just the available lenses in the lab that set the constraint. But sometimes you want the 4f system to be as short as possible. But where is the limit? Last year, a friend of mine at Caltech showed me a really cool trick that I had not heard before, and I thought it might be useful to share.

The useful trick is to start from the largest internal ray angle that we are willing to tolerate before aberrations become problematic. For fixed incoming and outgoing beam radii, this angle gives us the shortest practical telescope and a direct way to choose suitable focal lengths.

Start with the actual question

We assume that the beam entering the telescope is collimated and has radius \(r\), and that the required output radius is \(R>r\).

Here \(r\) and \(R\) are beam radii. For the Gaussian radius convention used on this site, see Gaussian beam equations: intensity and beam radius.

There are two cases:

  • Galilean: a negative first lens and a positive second lens.
  • Keplerian: a positive first lens and a positive second lens.

The standard magnification and spacing formulas are given in Laser beam expander design.

The new question is not how an expander works in general. It is:

For these fixed radii, what is the shortest lens separation compatible with the largest ray angle we are willing to use?

Choose the largest acceptable ray angle

Let \(\alpha\) be the magnitude of the marginal ray angle inside the telescope. For a small angle in air,

\[ \tan\alpha\approx\alpha\approx\mathrm{NA}_{\rm work}. \]

The subscript “work” is intentional. This is the working marginal-ray angle inside the telescope. It is not the source NA and not the residual divergence of the expanded beam.

A larger working angle makes either layout shorter. We therefore choose the largest value that the lenses and the required optical quality can tolerate, which we call \(\mathrm{NA}_{\rm work,max}\).

Galilean negative lens, then positive lens r R αmax dG,min f₁,G < 0 f₂,G > 0 Keplerian two positive lenses r R real focus αmax dK,min f₁,K > 0 f₂,K > 0
Both layouts are drawn at \(\alpha_{\max}\approx\mathrm{NA}_{\rm work,max}\). The Galilean beam only grows from \(r\) to \(R\). The Keplerian beam first contracts from \(r\) to a real focus and then expands from zero to \(R\).

Galilean telescope

In the Galilean layout, the negative lens makes the marginal ray diverge from radius \(r\) to radius \(R\). The beam therefore grows by \(R-r\) between the lenses:

\[ R-r=d_{\rm G}\tan\alpha. \]

At the chosen angular limit,

\[ \boxed{ f_{1,\rm G}\approx-\frac{r}{\mathrm{NA}_{\rm work,max}}, \qquad f_{2,\rm G}\approx\frac{R}{\mathrm{NA}_{\rm work,max}}, \qquad d_{\rm G,min}\approx\frac{R-r}{\mathrm{NA}_{\rm work,max}}. } \]

The signed afocal spacing is \(d_{\rm G}=f_{1,\rm G}+f_{2,\rm G}\). Since the first focal length is negative, this is the difference between the focal length magnitudes.

Keplerian telescope

In the Keplerian layout, the first positive lens brings the marginal ray from radius \(r\) to a real focus. The ray then diverges from that focus to radius \(R\) at the second positive lens. The total distance is therefore set by \(r+R\):

\[ r+R=d_{\rm K}\tan\alpha. \]

At the same chosen angular limit,

\[ \boxed{ f_{1,\rm K}\approx\frac{r}{\mathrm{NA}_{\rm work,max}}, \qquad f_{2,\rm K}\approx\frac{R}{\mathrm{NA}_{\rm work,max}}, \qquad d_{\rm K,min}\approx\frac{R+r}{\mathrm{NA}_{\rm work,max}}. } \]

Both focal lengths are positive, and the afocal spacing is \(d_{\rm K}=f_{1,\rm K}+f_{2,\rm K}\).

The difference between the two layouts

Layout Lens signs Minimum separation Internal focus
Galilean \(-,+\) \((R-r)/\mathrm{NA}_{\rm work,max}\) No real focus
Keplerian \(+,+\) \((R+r)/\mathrm{NA}_{\rm work,max}\) Real focus

For the same radii and the same maximum working angle,

\[ \boxed{ d_{\rm K,min}-d_{\rm G,min} =\frac{2r}{\mathrm{NA}_{\rm work,max}}. } \]

The Galilean telescope is shorter because its virtual focus lets the beam expand from \(r\) to \(R\). The Keplerian telescope must first shrink the beam from \(r\) to zero and then grow it from zero to \(R\).

Include the source collimator

If the telescope follows a source collimator of focal length \(f_0\), use the collimation relation from Gaussian beam equations: thin lens focusing in reverse:

\[ r\approx f_0\,\mathrm{NA}_{\rm src}. \]

Ignoring the small mechanical gap before the telescope, the two total lengths are

\[ \boxed{ L_{\rm G,min}\approx f_0+\frac{R-r}{\mathrm{NA}_{\rm work,max}}, \qquad L_{\rm K,min}\approx f_0+\frac{R+r}{\mathrm{NA}_{\rm work,max}}. } \]

What sets the angular limit?

The calculation does not predict \(\mathrm{NA}_{\rm work,max}\). It turns a chosen optical quality limit into the shortest possible paraxial layout. The limit must come from the actual lenses and the required wavefront quality.

A very common choice is to keep the working NA around \(0.1\). At \(0.1~\mathrm{rad}\), replacing \(\tan\alpha\) by \(\alpha\) changes the geometry by only about 0.3%, so it is a sensible small angle scale. That fact alone does not guarantee low lens aberration.

In practice, choose the most restrictive limit from:

  • the usable NA and clear aperture of both lenses;
  • the spherical and chromatic aberration allowed at the operating wavelength;
  • lens form, orientation, and beam fill;
  • damage threshold and coating limits;
  • alignment sensitivity and available stock focal lengths.

The distinction between lens clear aperture, illuminated beam diameter, effective f number, and effective NA is covered in F-number and numerical aperture for Gaussian beams.

The formula gives the shortest layout permitted by an aberration limit. It does not replace the real lens aberration calculation used to choose that limit.

Using the result in GaussianBeam

Use the equations above to generate the initial lens choices:

  1. Fix \(r\), \(R\), and \(\mathrm{NA}_{\rm work,max}\).
  2. For a Galilean layout, enter a negative \(f_1\) and a positive \(f_2\).
  3. For a Keplerian layout, enter positive values for both \(f_1\) and \(f_2\).
  4. In either case, constrain the separation to the signed expression \(f_1+f_2\).
  5. Verify the exact Gaussian waist, curvature, and aperture margins in the planner.

Exact Gaussian propagation uses the \(q\)-parameter relation already derived in Gaussian beam equations: the complex q parameter; it is not repeated here. GaussianBeam remains a paraxial Gaussian model, so final aberration validation requires real lens data or higher order ray tracing.

The whole argument in one line

\[ \boxed{ \text{largest acceptable internal angle} \;\Longrightarrow\; f_1,\ f_2,\ d_{\min} } \]

Both telescopes become shorter as the internal ray angle grows. Choose the largest angle that the real optics can tolerate, then calculate the focal lengths from that boundary. For the same radii and angular limit, the Galilean layout is shorter, while the Keplerian layout provides a real internal focus.