Total Internal Reflection Simulator: Critical Angle & Numerical Aperture of Optical Fibers

simulator beginner ~8 min
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NA = 0.242 — acceptance half-angle 14.0 deg

A fiber with core index 1.48 and cladding index 1.46 yields a numerical aperture of 0.242 and an acceptance half-angle of 14.0 degrees — typical of standard graded-index multimode fiber.

Formula

θ_c = arcsin(n_clad / n_core)
NA = sqrt(n_core² - n_clad²)
θ_acceptance = arcsin(NA / n_outside)

Light Trapped in Glass

When light travels from a denser medium (glass core) to a less dense medium (glass cladding), Snell's law dictates that rays hitting the interface at shallow angles refract outward. Below a specific critical angle, the refracted ray would need to bend past 90 degrees, which is physically impossible — the light is instead perfectly reflected back into the core. This total internal reflection is the foundation of every optical fiber ever drawn.

Numerical Aperture

The numerical aperture quantifies how much light a fiber can capture. It depends only on the refractive index difference between core and cladding: NA = sqrt(n1^2 - n2^2). A higher NA means a wider acceptance cone and easier coupling, but also more guided modes and potentially more dispersion. Fiber designers balance NA against bandwidth requirements.

Critical Angle and Acceptance Cone

The critical angle at the core-cladding boundary sets the maximum bounce angle for guided rays. Translating this through Snell's law at the fiber entrance face gives the acceptance half-angle — the cone within which launched light will be guided. Rays outside this cone refract through the cladding and are lost within centimetres.

Single-Mode vs Multimode

When the core is small enough (typically under 9 um at 1550 nm), only the fundamental mode propagates, eliminating modal dispersion entirely. Larger cores support hundreds of modes, each travelling at a slightly different group velocity. This simulation visualises ray paths inside the fiber, showing how changing the indices and core size affects acceptance, guiding, and the transition from multimode to single-mode operation.

FAQ

What is total internal reflection in fiber optics?

Total internal reflection (TIR) occurs when light in a denser medium (fiber core) strikes the core-cladding interface at an angle less than the critical angle, causing 100% reflection back into the core. This is the fundamental mechanism that confines light within an optical fiber over many kilometres.

How is numerical aperture calculated?

NA = sqrt(n1^2 - n2^2) where n1 is the core refractive index and n2 is the cladding index. A larger NA means the fiber accepts light from a wider cone. Typical single-mode fibers have NA around 0.12, while multimode fibers range from 0.2 to 0.5.

What is the critical angle of an optical fiber?

The critical angle theta_c = arcsin(n2/n1) is the maximum angle (measured from the interface normal) at which TIR still occurs. Rays hitting the interface at angles greater than theta_c refract into the cladding and are lost.

What is the difference between single-mode and multimode fiber?

Single-mode fiber has a core diameter of about 8-9 um, supporting only one spatial mode and enabling long-distance telecom. Multimode fiber (50 or 62.5 um core) supports hundreds of modes, is easier to couple light into, but suffers modal dispersion that limits bandwidth-distance product.

Sources

Embed

<iframe src="https://homo-deus.com/lab/fiber-optics/total-internal-reflection/embed" width="100%" height="400" frameborder="0"></iframe>
View source on GitHub