Approximating Optical Refraction using Taylor Polynomials

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2024-09-04 00:58

In this video, I demonstrate how Taylor polynomials are used in optical physics to simplify the complex refraction equation. The refraction path lengths of light outside and inside a refractive medium, like a glass lens, can be derived using the Law of Cosines. However, the resulting equation is often cumbersome to work with. In first-order optics, we approximate these lengths using the first-order Taylor polynomial for cosine, where cos(x) is approximated as 1. For larger angles, a more accurate approximation is needed, so we use the third-degree Taylor polynomial instead, a method known as third-order optics.

Timestamps:

Applications to Optical Physics: 0:00
Equation 1 derived from Fermat's principle that light travels to minimize time taken: 1:12
Index of refraction is ratio of speed of light in a vacuum over the speed of light in the medium: 2:40
Recap on the Law of Cosines: 2:54
Derivation of Equation 2: Lengths used in Equation 1: 4:33
Gauss' Linear approximation for Equation 2 using the first degree Taylor polynomial: 12:22
Equation 3 uses the Linear Approximation of Equation 2 into Equation 1: 15:24
- Gaussian or first-order optics: 16:35
More accurate approximation uses a Taylor Polynomial of degree 2: 16:47
- Equation 4: Updating Equation 1 with the more accurate approximation into Equation 2: 17:42
- Third-order optics: 18:53