Exercise 2: Deriving the 1st Order and 3rd Order Optics Formulas

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

Full video, notes, and playlists:

- Full video and playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0F76sIU8xm09oqBTq1mlry3
- HIVE notes: https://peakd.com/hive-128780/@mes/infinite-sequences-and-series-applications-of-taylor-polynomials
- Infinite Sequences and Series: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz .

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