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.
