# Thin lenses: Practical implementation

## Engineering Fundamentals

#### Introduction

This sheet focuses on the use of spherical lenses in opto-mechanical applications, not on the design of a lens itself. The thin lens equation, chromatic and spherical aberrations, wave lengths and the use of GRIN lenses is discussed.

#### Spherical lenses versus parabolic lenses

Spherical lenses have a focal region (see ‘Spherical aberrations’) whereas parabolic lenses comprise an exact focal point. However, a spherical surface is much more cost-efficient to manufacture (grinding/polishing) and therefore they are (still) often used.

#### Spherical thin lenses

$f$
$n, n_m$
$r_1,r_2$
$o,i$

= focal length (average)
= refractive index of lens, medium
= object distance, image distance

#### Spherical thin lens equations

Focus distance: $\frac{1}{f}=\left(\frac{n-n_m}{n_m}\right)\left(\frac{1}{r_1}-\frac{1}{r_2}\right)=\frac{1}{i}+\frac{1}{o}$
Magnification: $m=-\frac{i}{o}$

#### Lens Shift / Tilt

Through manufacturing tolerances, the lens and/or the object and image can shift or tilt in relation to the nominal position. See Thin lenses: Shift and tilt phenomena Precision Point sheet.

#### Spherical aberrations

Further from the lens axis the refraction of light due to a spherical shaped lens is larger. Therefore, the rays do not focus in the same point causing the image to become blurry. Minimize this effect via a good choice for $r_1$ and $r_2$:
$\frac{r_1+r_2}{r_1-r_2}=\frac{2\left(n^2-1\right)}{n+2}\frac{i+o}{i-o}$

#### Light refraction

$\frac{sin\alpha_1}{sin\alpha_2}=\frac{n}{n_m}$

Materialn [-]
Vacuum1
Air1.0003
Water1.33
Ethanol1.36
Flint glass1.62
Crown glass1.52
Fused silica1.46
PMMA1.49
Diamond2.42

#### Chromatic (or color) aberrations

The refractive index $n$ decreases with increasing wavelength. Therefore, the image becomes ‘fringed’. Therefore, suppliers often specify the focus distance  $f$ per wavelength ($\lambda$). Minimize this effect by minimizing the spectrum of wavelengths of the light source.

#### Wavelengths

Denominationλ [nm]
γ-ray<0.01
X-ray0.01 - 10
UV-ray (far - near)10 - 390
Indigo390 - 450
Blue (cyan = 490)450 - 490
Green490 - 580
Yellow580 - 600
Orange600 - 620
Red (dark red > 700)620 - 770
Fiber optics (infrared)800 - 1e6
Infrared (near far)1800 - 4e4

#### Other lenses

GRIN lenses are treated such that they comprise a GRadient in the INdex of refraction over the lens axis. This gradient may be designed in circular, parabolic or sinusoidal shapes, whereby the possibility arises to place $f$ and $i$ arbitrary.

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