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In this video, I derive the first-order and third-order optics formulas by approximating the refraction formula for light using first and third-order Taylor polynomials, respectively. Refraction refers to the change in the direction of light as it passes between different mediums due to a change in speed. Light naturally travels in a way that minimizes time, and this principle helps us derive formulas for how light refracts when entering a material. The distances light travels before and after passing through an optical lens can be calculated using the Law of Cosines, which leads to a complex equation. However, we simplify this equation using Taylor polynomial approximations.
In the first part of the video, I derive the first-order, or Gaussian optics, formula by approximating the cosine function in the distance equations with its first term, which is simply 1.
In the second part, I derive the third-order Gaussian optics formula by using the third-degree Taylor polynomial for cosine, which includes the first two non-zero terms. I also apply a given hint to approximate the inverse of the distances using the first two terms of their respective binomial series. Additionally, I use a second hint to approximate the angle with its sine, which introduces a term involving the height. After working through the algebra, I arrive at the third-order optics formula!
Timestamps:
Exercise 2: Derive 1st Order and 3rd Order Optics Equations for refraction: 0:00
Solution to (a): Approximating cosine with First degree Taylor Polynomial: 0:59
Equation 3: Gaussian optics or 1st order formula: 8:26
Solution to (b): Approximating cosine with Third degree Taylor Polynomial: 9:19
Modifying lengths in form of Binomial series: 17:46
1st Hint: Approximate lengths using first 2 terms of Binomial series: 23:20
Plugging in our approximations into Equation 1: 28:06
2nd Hint: Approximate ɸ with sin(ɸ) to include the height term: 43:48
Equation 4: 3rd order optics formula: 46:37
Exercise 2: Deriving the 1st Order and 3rd Order Optics Formulas