Total internal reflection (TIR) is a cornerstone of geometric optics, but it tells an incomplete story. While the incident light is perfectly reflected back into the denser medium, the electromagnetic field does not abruptly vanish at the interface. This simulator visualizes the resulting evanescent wave, a non-propagating field that penetrates into the rarer medium with an intensity that decays exponentially. The core physics is derived from applying Maxwell's equations and the boundary conditions at the interface between two dielectric media with refractive indices n₁ and n₂ (n₁ > n₂). Snell's law, sin θ_t = (n₁/n₂) sin θ_i, predicts that for angles of incidence (θ_i) greater than the critical angle θ_c = arcsin(n₂/n₁), the sine of the transmission angle exceeds 1. The mathematically consistent solution leads to a complex wavevector component perpendicular to the interface, giving rise to an electric field of the form E(z) = E₀ exp(-z/d_p), where z is the distance into the rarer medium. The simulator's primary output is this penetration depth, d_p = λ / (4π √(n₁² sin²θ_i - n₂²)), which scales with the incident wavelength λ and depends sharply on θ_i. The model simplifies reality by assuming ideal, lossless dielectric materials and a perfectly smooth, planar interface. It also treats the light as a monochromatic, plane wave. By interacting with the controls for n₁, n₂, λ, and θ_i, students learn that the evanescent field is a real physical phenomenon, not a mathematical artifact. They can observe how d_p diverges as θ_i approaches θ_c from above, becomes on the order of λ for angles just above θ_c, and shrinks to a small fraction of λ for grazing incidence. This foundational understanding is critical for applications like fiber optic communication, fluorescence microscopy (TIRF), and optical sensors.
Who it's for: Undergraduate physics and engineering students studying wave optics, electromagnetism, or photonics, particularly when covering the Fresnel equations and the full implications of total internal reflection.
Key terms
Total Internal Reflection (TIR)
Evanescent Wave
Penetration Depth
Critical Angle
Refractive Index
Snell's Law
Exponential Decay
Wave Vector
How it works
Plane-wave textbook estimate; real beams have Goos–Hänchen shifts and finite beam widths.
Frequently asked questions
Does the evanescent wave transport energy across the interface?
No, under the ideal conditions modeled here (perfect dielectrics, infinite plane wave), the time-averaged energy flow normal to the interface is zero. The evanescent field stores reactive energy near the surface. However, if a third medium (like a prism or a fluorescent molecule) is brought close to the interface, it can couple to this field, allowing energy transfer in a process called frustrated total internal reflection (FTIR).
Why is the penetration depth important in real-world technology?
The precise, exponential decay of the evanescent field makes it an exquisite probe of surfaces. In Total Internal Reflection Fluorescence (TIRF) microscopy, it selectively excites fluorescent molecules within ~100 nm of a cell's membrane, providing exceptional background rejection. In optical fiber sensors, changes in the evanescent field due to external substances alter the guided light, enabling detection of chemicals or biological agents.
What is a key limitation of this simplified model?
This model assumes an infinite plane wave and a perfectly smooth interface. In reality, laser beams are finite, which means some light can 'tunnel' across a small gap even during TIR (the Goos-Hänchen shift). Furthermore, if the rarer medium is absorptive (has a complex refractive index), the evanescent wave can transfer energy and heat the medium, a principle used in attenuated total reflection (ATR) spectroscopy.
How does the wavelength of light affect the evanescent wave?
The penetration depth d_p is directly proportional to the incident wavelength λ. For a given angle and refractive indices, red light (longer λ) will penetrate farther into the rarer medium than blue light (shorter λ). This scaling is explicit in the equation d_p = λ / (4π √(n₁² sin²θ_i - n₂²)).