THIN LENSES – PRACTICAL IMPLEMENTATION

Engineering Fundamentals

INTRODUCTION

This sheet focusses 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.

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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

= focal length (average)

nn_m

= refractive index of lens, medium

r_1, r_2

= incoming radius, outgoing radius

o,i

= 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

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.

Thin lenses - Spherical aberrations

Minimize this effect via a good choice for r_1 and r_2:
\frac{r_1+r_2}{r_1-r_2}=\frac{2(n^2-1)}{n+2}\frac{i+o}{i-o}

Light refraction
Thin lenses - Light refraction
Refraction 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).

Thin lenses - Chromatic (or color) aberrations

Minimize this effect by minimizing the spectrum of wavelengths of the light source.

Wavelengths
Denomination λ [nm]
γ-ray1e-5- 0.1
X-ray0.001 - 100
UV-ray (far - near)200 - 300
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

_{Visible\ light:\ 390\ \sim\ 770\ nm}

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

This page uses QuickLaTeX to display formulas.