Gaussian Beam Focusing Simulator — How It Works

This interactive Gaussian beam focusing simulator lets engineers, researchers, and students visualize how a laser beam behaves when passing through a focusing optical element. By adjusting the input beam diameter and focal length in real time, you can explore the fundamental physics that govern laser spot size, beam divergence, and depth of focus, all critical parameters in photonics system design.

The tool models a TEM₀₀ Gaussian beam — the ideal single-mode laser profile used as the reference standard in optical engineering. The spot size is measured at the 1/e² intensity point, which is the industry standard definition for beam waist radius w₀.

The Physics Behind the Simulation

At the core of this simulator is the Gaussian beam waist equation, which defines the minimum achievable spot radius at the focal point of a lens:

Where w₀ is the focused spot radius, λ is the laser wavelength (1064 nm in this simulator, a standard Nd:YAG reference), f is the focal length of the optical element, and w is the input beam radius at the lens. This relationship defines the diffraction limit, the theoretical minimum spot size achievable with a given optical setup, beyond which no lens can focus tighter regardless of quality.

One of the most counter-intuitive results visible in this tool: to achieve a smaller focused spot, you need a larger input beam. A wider beam fills more of the lens aperture, which increases the numerical aperture and allows for tighter diffraction-limited focusing.

Optical Elements Explained

The simulator includes five optical element types commonly used in photonics and laser systems, each with distinct focusing characteristics:

• Bi-Convex Lens: The standard converging element for general-purpose laser focusing. Suitable for most industrial and laboratory applications where aberration correction is not critical.
• Plano-Convex Lens: Optimized for infinite-conjugate focusing (collimated input beam). Reduces spherical aberration compared to a bi-convex for typical focusing setups.
• Achromatic Doublet: A two-element cemented lens corrected for both chromatic and spherical aberration. Preferred for broadband or multi-wavelength laser systems.
• Aspheric Lens: Provides the highest precision focusing with near-zero spherical aberration. Essential for tight-focus, high-NA applications such as fiber coupling and micro-machining.
• Bi-Concave Lens: A diverging element that expands the beam and creates a virtual focus behind the lens. Commonly used as the first stage in beam expander configurations.

Key Engineering Trade-Offs in Laser Focusing

Designing a laser focusing system always involves balancing competing optical parameters. This simulator makes those trade-offs immediately visible:

Focal Length vs. Spot Size vs. Working Distance

Reducing focal length f tightens the focused spot significantly but it also reduces the working distance between the lens and the target, and dramatically narrows the depth of focus. Short focal lengths are used in laser micro-machining and cutting, while longer focal lengths are required in remote sensing, rangefinding, and marking applications where standoff distance matters.

Input Beam Diameter vs. Spot Size vs. Optic Size

Expanding the input beam w before the focusing lens reduces the final spot size proportionally. However, this demands larger, higher-cost optics to avoid clipping the beam edges, a condition that would introduce diffraction rings and degrade beam quality. The optimal beam diameter is typically chosen to fill 1/e² of the clear aperture of the focusing lens.

Divergence Angle After Focus

The output divergence angle θ is equally important: a tightly focused beam diverges rapidly after the waist. This determines how quickly the beam expands beyond the focal point and is a critical parameter in fiber coupling efficiency, material processing depth, and free-space propagation distance.