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

Lenses - Shift and tilt phenomena - Initial conditions

$f$

= focal length (average)

$n$

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

$r_1,r_2$

= incoming radius, outgoing radius

$o,i$

= 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

Lenses - Shift and tilt phenomena - Lens x = object -x

Lens y = object -y

Lenses - Shift and tilt phenomena - Lens y = object -y

Lens tilt

Lenses - Shift and tilt phenomena - Lens Tilt

Object tilt

Lenses - Shift and tilt phenomena - Object Tilt

Lens y + lens tilt

Lenses - Shift and tilt phenomena - Lens y + lens tilt

Lens x, lens, y, lens tilt, object tilt

Lenses - Shift and tilt phenomena - 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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