Optics 090 · Waveguides, structured light, and modern optics

Gaussian Beam & Gouy Phase

How do the width and phase of a focused Gaussian beam change as it passes through the focus? The beam narrows to a waist, then widens: one Rayleigh range from the waist it is the square root of two times wider, and far away it spreads at a fixed divergence angle. A lens moves the waist and changes its size. Through the focus the Gouy phase adds an extra half cycle of phase, which has to be counted when matching the beam to a cavity mode.

Focus a Gaussian beam on a 3D bench and watch the lens transform its complex beam parameter. Sample the mode fields, follow the Gouy phase continuously along with the sign of the wavefront curvature, measure beam quality from second moments and freeze detector records.

Sampled second-moment radius

w_D

$0.50$

Continuous fundamental-basis Gouy lag

\Psi_{00}

$50\%$

Ideal target-mode coupling

\eta

$0.00\pi$

Model regime $\text{valid model regime}$

Model checked against analytic results

Gaussian Beam & Gouy Phase

Input second-moment waist radius; waist at origin

w_{D,0}

280\,\mu\mathrm m

Vacuum wavelength in ideal air

\lambda_0

633\,\mathrm{nm}

Specified incoherent modal mixture

M^2

1

Thin-lens power; zero removes lens

1/f

12\,\mathrm{m}^{-1}

Lens position from input waist

z_L

100\,\mathrm{mm}

Detector distance downstream of the lens

d

100\,\mathrm{mm}

Focusing changes more than the spot size: the beam's phase also slips behind that of a plane wave. Move the thin lens and the detector, follow one continuously changing complex beam parameter, and see why two spots of equal size can still fail to match. Raising the beam-quality factor mixes in other modes incoherently, in a specified proportion.

The beam propagates through air in the scalar paraxial approximation, and the thin lens is ideal with an infinite aperture. The beam-quality factor comes from a specified incoherent mix of the fundamental and first-order modes, so the total field has no unique phase. The 3D display uses fixed schematic scales, and the ideal detector window adapts to the beam. Aberrations and clipping are not modeled, and no measured instrument is reproduced.

Physics tutorial

Audit the physics of Gaussian Beam & Gouy Phase

BackgroundA beam carries both an intensity profile and a phase front. One complex parameter records both: $\frac1q=\frac1R-i\frac{\lambda_0}{\pi w_b^2}$ . A lens changes the curvature at once but cannot change the beam radius instantly. Propagation after the lens then forms a new waist.

Why it mattersHow do the width and phase of a focused Gaussian beam change as it passes through the focus?

Start with the essentials

Focus question: How do the width and phase of a focused Gaussian beam change as it passes through the focus?
One-sentence intuition: Mode matching measures how well two fields overlap, not whether the bright spots are the same size: $\eta=\sum_j p_j\left|\int u_{\rm target}^{*}u_j\,d^2r\right|^2$ . A curvature mismatch lowers the coupling even when the intensity radii agree. For incoherent modes, each overlap is squared first and the resulting powers are then added.

Core mathematical model

Coordinates, radii and the complex parameter

q(z)=z-z_w+iz_R,\quad z_R=\frac{\pi w_{b,0}^2}{\lambda_0},\quad w_D=2\sigma_x=\sqrt{M^2}\,w_b

Air has a refractive index of one. Lengths are calculated in meters, while the controls show micrometers, nanometers and millimeters. The optical axis points from the source to the detector, and the input waist sets the absolute origin. The basis radius is where the amplitude falls to one over e of its peak; the second-moment radius equals it only for a pure fundamental mode. This radius is not a diameter or a wavefront radius of curvature.

A thin lens changes phase, not instantaneous intensity

q_+=\frac{q_-}{1-q_-/f},\quad q_d=q_++d,\quad\frac1{R_+}=\frac1{R_-}-\frac1f

The lossless lens with an infinite aperture multiplies the field by a quadratic phase, and the beam width is the same on both faces. The real and imaginary parts of the output parameter give the position of the new waist and its Rayleigh range, including virtual waists. Setting the power to zero removes the lens, but its position still serves as the reference for the detector distance downstream.

A continuous Gouy phase and an explicit mode mixture

E_{mn}\propto H_m(\sqrt2x/w_b)H_n(\sqrt2y/w_b)e^{-r^2/w_b^2}e^{ikr^2/(2R)-i(m+n+1)\Psi_{00}}

The convention uses a positive forward carrier and a negative temporal phase. The forward carrier is removed, and the Gouy lag is measured from the original waist. On axis through the lens the phase is continuous, so the output arctangent must not be reset without its offset. In free space, the fundamental mode builds up a total lag of pi as it passes from far before the waist to far beyond it. This is a phase relative to a plane wave; it does not mean the light is slowed, and it is not a group delay.

Beam quality does not determine a unique field

p_{00}=2-M^2,\quad p_{10}=p_{01}=\frac{M^2-1}{2},\quad I=\sum_jp_j|u_j|^2,\quad1\leq M^2\leq2

This is one particular incoherent mixture, rotationally symmetric, with a common mode parameter. As the mode weights change, the second-moment waist is held at the control value, so the basis waist changes too. Each component has its own Gouy order, but the mixture has no single electric-field phase. Many other beams share the same quality factor; this model does not identify them or imitate an arbitrary source.

Common difficulties

Can one quality number supply one Gouy phase?

Typical misconceptionMultiply the wavelength in a Gaussian field by the quality factor, and you still have a physical coherent field.

Better mental modelNo. Second-moment propagation by itself does not fix the coherence or the modal phase. Here an explicit, normalized mode mixture sets the intensity and the coupling, and each component keeps the physical wavelength and its own phase.

Run the experiment

01
Scene 1: Follow a real focus
Start in free space. Change the input waist and the wavelength, move the detector, and compare the sampled second-moment radius with the caustic. The waist sits at the original coordinate origin.
What to observe: At a waist the wavefront curvature is zero, so the radius of curvature is infinite, while the intensity radius reaches its finite minimum.
02
Scene 2: Remove the lens
Choose the lens preset. Move the lens and adjust its power. In general the output waist is not one focal length behind the lens. Across a real waist the curvature changes sign, while the beam radius stays finite.
What to observe: The target mode is a separate ideal reference, not a reconstruction from detector data. When two modes propagate without loss, moving their common observation plane does not change their full-plane overlap.
03
Scene 3: Incoherent modal mixture
Raise the beam quality factor. Weight moves from the fundamental to an equal, incoherent pair of first-order modes. Watch the spot change, follow the two Gouy branches and see the coupling to the centered mode drop, then freeze the actual fields.
What to observe: In 3D, only the small plane connected to the outgoing beam is a detector. The caustics, phase plots, mode bars and frozen records sit in the separate data row. Brightness and the unequal 3D scales are display choices, not changes in measured power.

Gaussian Beam & Gouy Phase

Audit the physics of Gaussian Beam & Gouy Phase

Core mathematical model

Coordinates, radii and the complex parameter

A thin lens changes phase, not instantaneous intensity

A continuous Gouy phase and an explicit mode mixture

Beam quality does not determine a unique field

Common difficulties

Can one quality number supply one Gouy phase?

Run the experiment

Scene 1: Follow a real focus

Scene 2: Remove the lens

Scene 3: Incoherent modal mixture

Related experiments and articles

Beam shape and beam quality