   Chapter 11.11, Problem 34E

Chapter
Section
Textbook Problem

# (a) Derive Equation 3 for Gaussian optics from Equation 1 by approximating cos ϕ in Equation 2 by its first-degree Taylor polynomial.(b) Show that if cos ϕ is replaced by its third-degree Taylor polynomial in Equation 2, then Equation 1 becomes Equation 4 for third-order optics. [Hint: Use the first two terms in the binomial series for l o − 1 and l i − 1 . Also, use ϕ ≈ sin ϕ ]

To determine

(a)

To derive:

The equation for Gaussian optics.

Explanation

1) Concept:

Tnx=fa+f'ax-a++fnxn! is nth Degree Taylor polynomial of f(x)

2) Calculation:

Equation 1  for Gaussian optics is

n1l0+n2li=1Rn2sili-n1s0l0

Where,(equation 2),

l0=R2+s0+R2-2Rs0+Rcos

li=R2+si-R2+2Rsi-Rcos

Approximate cos  by its first degree Taylor polynomial.

The first degree Taylor polynomial  cos is T1= 1

Therefore,

cos1 for small values of

Therefore l0, l1 becomes

l0=R2+s0+R2-2Rs0+R

l0=R2+s02+2s0R+R2-2R

To determine

(b)

To show:

If cos  is replaced by its second degree Taylor polynomial, then the equation becomes equation 4 for Third order Gaussian optics.

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