# Thin lenses: Shift and tilt phenomena

## Engineering Fundamentals

#### Introduction

Mounting thin lenses to their mechanical interfaces comprises imperfections to the initially designed optical paths due to manufacturing and assembly tolerances.

#### Quantification

This sheet provides qualitative information about the phenomena because derivation of the shifts and tilts of the image as a result of the lens and/or object movement is extensive and non-transparent.

#### Initial conditions $f$

= focal length (average)

$n$

= refractive index of lens (medium air/vacuum: $n_m=1$)

$r_1,r_2$

$o,i$

= object distance, image distance

Focus distance $f$, object $o$ and image $i$ distance: $\frac{1}{f}=\frac{1}{i}+\frac{1}{o}$
Magnification: $m\ =\frac{i}{o}$

#### Lens x = object -x #### Lens y = object -y #### Lens tilt #### Object tilt #### Lens y + lens tilt #### Lens x, lens, y, lens tilt, object tilt #### Lens y = object -y

Due to manufacturing also the incoming and outgoing radii of the lens can differ. Consequently, the focus distance is affected:
$\frac{1}{f}=\left(n-1\right)\left(\frac{1}{r_1}-\frac{1}{r_2}\right)$

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