Question

4. Two lenses /¡ and f2 are separated by a distance d. a) Evaluate the ray transfer matrix that describes the optical system. b) What is the effective focal length of the optical system when d-fi +2? c) When d-fi+f2, what is the angular magnification of the system? d) Under what circumstances is the image of an object at infinity inverted? e) What optical instrument is this?

Answer 1

The ray transfer matrix for a thin lens is

1 1

and for a blank space is

1

PART A:

01 1 10 01

$\Rightarrow M=\begin{bmatrix} 1 & d\\ -\frac{1}{f_2}& -\frac{d}{f_2} +1\end{bmatrix}\begin{bmatrix} 1 & 0\\ -\frac{1}{f_1}& 1 \end{bmatrix}$

fi + 1 f2

$\Rightarrow M=\begin{bmatrix} 1-\frac{d}{f_1} & L\\ -\frac{1}{f_2}-\frac{1}{f_1}+\frac{d}{f_1f_2} &1 -\frac{d}{f_2} \end{bmatrix}$

PART B:

The equivalent focal length of the system is

$-\frac{1}{f_{eq}}=-\frac{1}{f_2}-\frac{1}{f_1}+\frac{d}{f_1f_2}$

$\Rightarrow \frac{1}{f_{eq}}=\frac{1}{f_2}+\frac{1}{f_1}-\frac{d}{f_1f_2}$

When

$d=f_1+f_2$

$\frac{1}{f_{eq}}=\frac{1}{f_2}+\frac{1}{f_1}-\frac{f_1+f_2}{f_1f_2}$

$\frac{1}{f_{eq}}=\frac{f_1+f_2}{f_1f_2}-\frac{f_1+f_2}{f_1f_2} = 0$

$f_{eq}=\frac{1}{0}=\infty$

So, the equivalent focal length of the system is infinity.

PART C:

The angular magnification is

$D=1-\frac{d}{f_2}$

When

$d=f_1+f_2$

$D=1-\frac{f_1+f_2}{f_2}=\frac{f_2-f_1-f_2}{f_2}=-\frac{f_1}{f_2}$

So, the angular magnification is

D=-f f2

PART D:

When the object is placed at the focus of a convex lens, the image formed is at infinity and inverted.