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

Thin lenses - 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 $r_2$:
$\frac{r_1+r_2}{r_1-r_2}=\frac{2\left(n^2-1\right)}{n+2}\frac{i+o}{i-o}$

Thin lenses - Spherical aberrations

Light refraction

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

Thin lenses - Light refraction

Refractive indices

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.

Thin lenses - Chromatic (or color) aberrations

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.

Thin lenses - Other lenses

Tech Support

Please submit a message and we will come back to you on short notice.

Precision Point sheet download

Please fill in your details to receive the requested Precision Point sheet.

We use cookies to ensure to give you the best experience on our website. If you continue to use this site we will assume that you are okay with it.