Gaussian Beams and the ABCD Matrix

Beam propagation, focusing, and the ABCD matrix method — from first principles to interactive optical design.

Key Takeaways

A Gaussian beam is fully defined by its wavelength and waist size. The Rayleigh range $z_{R} = π w_{0}^{2} / λ$ sets the scale for everything.
The ABCD matrix transforms the complex beam parameter $q$ through any paraxial optical system — one matrix multiplication replaces full diffraction integrals.
You can never focus a Gaussian beam to a point: the minimum spot size is set by the diffraction limit $w_{0} = λ f / (π w_{in})$ .

If you point a laser at a wall across the room, you see a bright spot. Point it at a wall across a football field, and the spot is larger. Point it at the Moon (NASA has been doing this since 1969, bouncing laser pulses off retroreflectors the Apollo astronauts left behind) and the spot is about 2 km across when it arrives. The beam spreads, and it spreads in a very specific way that follows from Maxwell's equations applied to the paraxial wave approximation.

A Gaussian beam is the simplest solution to that equation. It describes how real laser beams propagate, how they focus when you put a lens in front of them, how they expand when you chain two lenses together, and why you can never, no matter how clever your optics, focus a beam to a point. Once you understand the geometry of a Gaussian beam, a remarkable thing happens: the same matrix method that ray optics uses for tracing rays through lenses also propagates Gaussian beams. You get the full behavior of a diffracting, focusing, expanding laser beam from a 2×2 matrix multiplication. That matrix is called the ABCD matrix, and it is the workhorse of laser optical design.

This tutorial walks through the physics step by step. By the end you should be able to look at any optical system, write down its ABCD matrix, and predict exactly how a laser beam transforms through it.

The Gaussian Beam

A Gaussian beam has three defining properties. Its intensity profile transverse to the propagation direction is a Gaussian function. Its wavefronts are curved in a specific way that changes smoothly as the beam propagates. And it has a single location along its length, called the waist, where the beam is smallest and the wavefronts are flat.

At any plane z along the propagation direction, the intensity profile is:

I (r, z) = I_{0} (z) \cdot \exp ⁣ (- \frac{2 r^{2}}{w (z)^{2}})

The parameter $w (z)$ is called the beam radius at position z. It is the radius at which the intensity has dropped to $1 / e^{2} \approx 13.5 %$ of the peak. The beam radius evolves with distance as:

w (z) = w_{0} \sqrt{1 + {(\frac{z}{z_{R}})}^{2}}

where $w_{0}$ is the beam radius at the waist ( $z = 0$ ) and $z_{R}$ is the Rayleigh range, the distance over which the beam radius grows from $w_{0}$ to $\sqrt{2} \cdot w_{0}$ . The Rayleigh range is:

z_{R} = \frac{π w_{0}^{2}}{λ}

This is the single most important equation in Gaussian beam optics. It tells you how tightly you can focus a beam (small $w_{0}$ means short $z_{R}$ ) and how quickly it diverges (short $z_{R}$ means fast divergence). It is the depth of focus of a laser beam.

Let me make that concrete. Take a helium-neon laser with $λ = 632.8$ nm and a waist radius of 500 μm (a typical collimated beam from a commercial HeNe). The Rayleigh range is:

z_{R} = \frac{π \cdot (500 \times 10^{- 6})^{2}}{632.8 \times 10^{- 9}} = 1.241 m

So the beam stays roughly the same size for about a meter, then starts to spread noticeably. If I instead focus that HeNe beam down to a 10 μm waist using a lens, the Rayleigh range becomes:

z_{R} = \frac{π \cdot (10 \times 10^{- 6})^{2}}{632.8 \times 10^{- 9}} = 0.496 mm

Half a millimeter. You have to position your sample with micron precision to stay in focus. This is why tightly focused lasers are demanding to align, and why there is always a tradeoff: tight focus versus depth of field.

Beam Width Evolution

w₀ = 500.0 μm

λ = 632.8 nm

Rayleigh range z_R1.241 m

Divergence θ402.85 μrad

Divergence

Far from the waist ( $z ≫ z_{R}$ ), the beam radius grows linearly with distance:

w (z) \to \frac{λ}{π w_{0}} \cdot z

The half-angle of that asymptotic cone is the far-field divergence angle:

θ = \frac{λ}{π w_{0}}

This is the second fundamental equation. Notice what it says: the divergence is inversely proportional to the waist size. If you make the beam tighter at the waist, it diverges faster in the far field. You cannot beat this tradeoff by clever optical design, because it is baked into Maxwell's equations. The product of waist and divergence is a constant:

w_{0} \cdot θ = \frac{λ}{π}

This quantity is called the beam parameter product, or BPP. For a perfect Gaussian beam at 632.8 nm, BPP = 0.201 mm·mrad. This is the diffraction limit.

What about real lasers, which are never perfect Gaussians? Every real beam has a beam quality factor $M^{2} \geq 1$ that tells you how many times worse than a diffraction-limited Gaussian it is. A high-quality HeNe has $M^{2} \approx 1.0$ (effectively perfect). A typical single-mode diode laser has $M^{2} \approx 1.1$ to 1.3. A multimode diode bar might have $M^{2}$ in the tens or higher. All the physics that follows still applies to real beams. You just replace $λ$ with $λ_{eff} = M^{2} \cdot λ$ in the Rayleigh range and divergence formulas:

z_{R} = \frac{π w_{0}^{2}}{M^{2} λ}, θ = \frac{M^{2} λ}{π w_{0}}, BPP = M^{2} \cdot \frac{λ}{π}

Mental model: HeNe is effectively perfect, a good single-mode diode is close, a multimode bar is terrible.

For our 500 μm HeNe, the divergence is:

θ = \frac{632.8 \times 10^{- 9}}{π \cdot 500 \times 10^{- 6}} = 403 μ rad

After 10 meters of propagation, the beam radius has grown to about 4 mm, eight times its waist value.

Wavefront Curvature

The intensity profile is Gaussian, but the phase structure matters too. A Gaussian beam has curved wavefronts, and the radius of curvature $R (z)$ of those wavefronts evolves as:

R (z) = z (1 + {(\frac{z_{R}}{z})}^{⁣ 2})

At the waist ( $z = 0$ ), $R$ is infinite: the wavefronts are flat. As you move away, $R$ decreases until it reaches its minimum at $z = z_{R}$ , where $R = 2 z_{R}$ . Further out ( $z ≫ z_{R}$ ), $R \approx z$ , which is the geometric limit of a spherical wave emanating from the waist location. Three special cases to remember:

At the waist: $R = \infty$ (flat wavefronts)
At $z = z_{R}$ : $R = 2 z_{R}$ (minimum radius of curvature)
Far field: $R \approx z$ (looks like a spherical wave from the waist)

So a Gaussian beam, even far from its waist, is not quite a spherical wave. It has a Gaussian amplitude profile superimposed on a spherical wavefront, and the "center" of that spherical wave is the waist location, not the lens that created the beam. This is why a Gaussian beam focused by a lens does not, in general, form its waist at the lens's geometric focal point. Ray optics misses this subtlety, and ABCD matrices correct it.

The Complex Beam Parameter q

Carrying around both $w (z)$ and $R (z)$ separately is bookkeeping we would rather skip. The trick that makes Gaussian beam optics tractable is to combine them into a single complex number:

\frac{1}{q (z)} = \frac{1}{R (z)} - i \frac{λ}{π w^{2} (z)}

The real part of $1 / q$ carries the wavefront curvature, and the imaginary part carries the beam radius. Equivalently:

q (z) = z + i z_{R}

where z is measured from the waist. This is a beautifully compact notation. A single complex number captures everything about the beam at a given plane: its spot size, its wavefront curvature, and its distance from the waist.

At the waist, $q = i z_{R}$ (pure imaginary). Far from the waist, $q \approx z$ (almost real). The evolution of q through free space is the simplest rule in all of Gaussian beam optics: as the beam propagates a distance d in vacuum or a uniform medium, q just shifts by d:

q (z + d) = q (z) + d

This looks trivial, but it contains all the diffraction physics. The imaginary part ( $z_{R}$ ) stays constant, the real part grows, and when you extract $w (z)$ and $R (z)$ from q, you get the correct diffracting beam.

The Complex Beam Parameter q

z = 0.0 mm

w₀ = 500 μm, λ = 632.8 nm, z_R = 1241.1 mm

w(z)500.000 μm

R(z)∞ (flat)

Re(q)0.00 mm

Im(q)1241.15 mm

ABCD Matrices

Now for the connection to ray optics. In the paraxial approximation, a light ray at a given plane is described by two numbers: its height y above the optical axis and its angle θ relative to the axis. A 2-vector $(y, θ)$ represents the ray. When the ray passes through an optical element or propagates a distance, the new $(y^{'}, θ^{'})$ is related to the old one by a 2×2 matrix:

(\begin{array}{c} y^{'} \\ θ^{'} \end{array}) = (\begin{array}{cc} A & B \\ C & D \end{array}) (\begin{array}{c} y \\ θ \end{array})

The primitive matrices for the most common elements are:

Element	ABCD matrix
Free space, distance d	[[1, d], [0, 1]]
Thin lens, focal length f	[[1, 0], [−1/f, 1]]
Curved mirror, radius R	[[1, 0], [−2/R, 1]]
Flat refractive interface, n₁ → n₂	[[1, 0], [0, n₁/n₂]]

Thick lenses, curved interfaces, and windows are just compositions of these primitives. The flat-interface matrix comes from the same refraction geometry explained in our Fresnel Equations tutorial. The Gaussian Beam Propagator handles all of these automatically, so you rarely need to build the compound matrices by hand.

For a cascade of elements, you multiply the matrices in reverse order (the rightmost matrix acts on the ray first):

M_{total} = M_{N} \dots M_{2} \cdot M_{1}

That is the standard result from geometrical optics. What is remarkable, and not at all obvious, is that the exact same matrices propagate the complex q-parameter of a Gaussian beam. If a beam has parameter $q_{1}$ before an optical element with matrix $[\begin{matrix} A & B \\ C & D \end{matrix}]$ , then after the element:

q_{2} = \frac{A q_{1} + B}{C q_{1} + D}

This is the ABCD law for Gaussian beams. Kogelnik derived it in 1965. It means you can take all the matrices you learned for ray tracing and reuse them for full diffraction calculations. Every lens, mirror, and free-space gap you put in the path of a laser beam transforms q through a rational function whose coefficients are the entries of the system matrix.

Let me walk through a worked example to show how this actually gets used.

Worked Example: Focusing a Collimated Beam

Take a green laser with $w_{0} = 1$ mm and $λ = 532$ nm. Its Rayleigh range is:

z_{R} = \frac{π \cdot (10^{- 3})^{2}}{532 \times 10^{- 9}} = 5.905 m

A very long Rayleigh range. The beam is essentially collimated over any lab-scale distance. Now put a lens of focal length $f = 100$ mm right at the waist and ask: where is the new waist, and how small is it?

Step 1. Write down the inputs. At the lens, $q_{1} = i \cdot 5.905$ m (pure imaginary because we are at the input waist). The lens matrix has $- 1 / f = - 10 m^{- 1}$ :

M_{lens} = (\begin{array}{cc} 1 & 0 \\ - 10 m^{- 1} & 1 \end{array})

Step 2. Apply the ABCD law. With $A = 1$ , $B = 0$ , $C = - 10 m^{- 1}$ , $D = 1$ :

q_{2} = \frac{A q_{1} + B}{C q_{1} + D} = \frac{q_{1}}{1 - 10 q_{1}}

Substitute $q_{1} = i \cdot 5.905$ m:

q_{2} = \frac{i \cdot 5.905}{1 - 10 \cdot i \cdot 5.905} = \frac{i \cdot 5.905}{1 - 59.05 i}

Step 3. Simplify. Rationalize the denominator (multiply by the complex conjugate) and work out the arithmetic. The result, in meters, is:

q_{2} \approx - 0.1 + 0.00169 i

Step 4. Interpret. This is where the physics is. In the q-parameter, the real part is the signed distance from the nearest waist, and the imaginary part is the Rayleigh range at that waist.

$Re (q_{2}) = - 0.1$ m → the new waist is located 100 mm downstream of the lens. That is the back focal plane, exactly where ray optics would predict a collimated beam to focus.
$Im (q_{2}) = 0.00169$ m → the new Rayleigh range is 1.69 mm. Much shorter than the 5.9 m input, because the focused waist is much smaller.
From $z_{R} = π w_{0}^{2} / λ$ , we can back out the new waist radius: $w_{0, new} = \sqrt{λ \cdot z_{R} / π} = \sqrt{532 \times 10^{- 9} \cdot 1.69 \times 10^{- 3} / π}$ = 16.93 μm.

Cross-check against the textbook shortcut for focusing a collimated Gaussian beam:

w_{0, focus} \approx \frac{λ f}{π w_{0, in}} = \frac{532 \times 10^{- 9} \cdot 0.1}{π \cdot 10^{- 3}} = 16.93 μ m

Same answer. The ABCD formalism reproduces the result you would get from the specialized formula, but it does so for any sequence of optical elements, not just a single lens.

And $z_{R} = 1.69$ mm is the entire depth of focus of the focused beam. If you want to image something here, you had better position the sample to within a few hundred microns.

Focus a Collimated Beam

w₀ = 1.00 mm

λ = 532 nm

f = 100 mm

Output waist w₀'16.93 μm

Waist position z'150.00 mm

Output z_R1.69 mm

Output divergence10.000 mrad

Open in Gaussian Beam Propagator →

A Second Example: The Beam Expander

Laser beams are often too small for the application. A LIDAR system needs a collimated beam with very low divergence over kilometers. You enlarge the beam using a beam expander: two lenses spaced so that their focal points coincide. There are two flavors.

A Galilean beam expander uses one diverging lens (negative f) and one converging lens (positive f) placed $f_{1} + f_{2}$ apart (which, with $f_{1} < 0$ , is smaller than $f_{2}$ ). A Keplerian beam expander uses two converging lenses placed $f_{1} + f_{2}$ apart, with an intermediate focus between them. The magnification in both cases is $∣ f_{2} / f_{1} ∣$ . Galileans have no intermediate focus, so they do not ionize air with high-power lasers. Keplerians have an intermediate focus where you can place a pinhole to spatially filter the beam.

Take a HeNe beam with $w_{0} = 500$ μm, $λ = 632.8$ nm, and expand it 4× using lenses with $f_{1} = - 25$ mm and $f_{2} = 100$ mm separated by 75 mm. You can work out $M_{sys} = M_{L 2} \cdot M_{free} \cdot M_{L 1}$ by hand, or load the Galilean Beam Expander preset in the Gaussian Beam Propagator and click to see the system matrix. Either way, the afocal portion (the two lenses and the space between them) comes out to:

M_{afocal} = (\begin{array}{cc} 4 & 0.075 \\ 0 & 0.25 \end{array})

The A element is 4 and the C element is 0. That is all the physics you need. $A = 4$ is the magnification: the output beam radius is 4 times the input. $C = 0$ means the system is afocal: a collimated input gives a collimated output, because the ABCD law with $C = 0$ preserves the real part of $1 / q$ (the wavefront curvature) under the transformation.

After the expander, the beam has $w_{0} \approx 2$ mm. Its Rayleigh range has grown by a factor of $4^{2} = 16$ (because $z_{R}$ scales with $w_{0}^{2}$ ), to about 19.9 m. And its divergence has shrunk by a factor of 4, to about 101 μrad. The expander trades beam size for low divergence, which is exactly why you use it.

Beam Expander Comparison

M = 4×

w₀ = 0.5 mm

λ = 633 nm

Output waist500.00 μm

Output divergence402.98 μrad

ABCD matrix[4.00, 75.0 mm; 0.0000, 0.25]

Output waist10.98 μm

Output divergence18.354 mrad

ABCD matrix[-4.00, 125.0 mm; 0.0000, -0.25]

Open Galilean preset in Propagator →

Summary

Gaussian beams follow two simple formulas: $w (z) = w_{0} \sqrt{1 + (z / z_{R})^{2}}$ with $z_{R} = π w_{0}^{2} / λ$ , and $θ = λ / (π w_{0})$ . Everything else is bookkeeping. The complex beam parameter $q = z + i z_{R}$ packages the spot size and wavefront curvature into a single number. The ABCD matrices you learned in ray optics propagate q through any paraxial optical system via the rule $q_{2} = (A q_{1} + B) / (C q_{1} + D)$ . You can cascade any number of elements by multiplying matrices.

The beauty of the ABCD formalism is that a single matrix multiplication replaces pages of physical reasoning. Design a beam expander. Propagate through a window. Focus into a sample. Every calculation reduces to the same recipe: write down the matrices, multiply them, apply the ABCD law, read off the answer. You are now fluent in the language that every laser physicist speaks.

Going Further

If you want to build systems with Gaussian beams, try our Gaussian Beam Propagator. It is a visual chain builder where you add elements one at a time, see the beam envelope update in real time, and get exact values for output waist, Rayleigh range, divergence, and beam parameter product. All the examples in this tutorial are preloaded as presets. For glass and mirror coating data to plug into thick-lens or curved-interface calculations, see our Material Database.

For the deeper physics, the canonical references are:

A. E. Siegman, Lasers (University Science Books, 1986). Chapters 16-17 on Gaussian beams and ABCD matrices. The original treatise.
H. Kogelnik and T. Li, "Laser beams and resonators," Applied Optics 5, 1550 (1966). The paper that introduced the ABCD law for Gaussian beams.
B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 3rd ed. 2019). Chapter 3 is the cleanest textbook presentation.

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