#### 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

= incoming radius, outgoing radius

= 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

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

#### Refractive indices

Material | n [-] |
---|---|

Vacuum | 1 |

Air | 1.0003 |

Water | 1.33 |

Ethanol | 1.36 |

Flint glass | 1.62 |

Crown glass | 1.52 |

Fused silica | 1.46 |

PMMA | 1.49 |

Diamond | 2.42 |

#### 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

Denomination | λ [nm] |
---|---|

γ-ray | <0.01 |

X-ray | 0.01 - 10 |

UV-ray (far - near) | 10 - 390 |

Indigo | 390 - 450 |

Blue (cyan = 490) | 450 - 490 |

Green | 490 - 580 |

Yellow | 580 - 600 |

Orange | 600 - 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.