# Lenses

### Introduction

Lenses are transparent optical components that use refraction to focus or collimate electromagnetic radiation.

### Properties of lenses

Lenses are characterized by their focal length. The focal length is given by the lens maker's formula:

1/f(lambda) = [n(lambda) - 1][1/R_{1} - 1/R_{2}]

where f is the focal length, n is refractive index, and R_{1} and R_{2} are the radii of curvature of the two surfaces of the lens.
*Illustration of lens properties:*

Focus spot size of light from infinity:

spot diameter = lambda * f / pi * D

where lambda is wavelength, f is focal length, and D is diameter of the incoming beam.

### Images and ray tracing

The position of an image can be determined by tracing three lines in a diagram:
- parallel to the optical axis and through the focal point
- through the center of the lens
- through the focal point to the lens and then parallel to the optical axis

*Finding an image by ray tracing*

The position of an image can be found from the ray trace or from:

1/f = 1/x_{o} + 1/x_{i}

where f is the lens focal length, x_{o} is distance of the object from the lens, and x_{i} is the distance of the image from the lens.

Magnification, M, is the ratio of the image size to the object size, and is equal to:

M = x_{i} / x_{o}

where x_{i} and x_{o} are the distances of the image and object from the lens, respectively.

### Light collection

The light collection efficiency is the solid angle that an optic makes with an object. The f-number describes this angle:

f-number: f/# = l/d

where l is distance and d is diameter of lens.
The solid angle that a lens collects is approximately:

OMEGA = pi d^{2} / 4 l^{2}

The fraction of light that an optic collects is this solid angle divided by the total 4 pi steradians:

fraction collected = OMEGA /4pi = pi d^{2} / 16 pi l^{2} = 1 / 16 (f/#)^{2}.

Lenses (and curved mirrors) do not focus light perfectly. Chromatic and spherical aberations occur on-axis and coma and astigmitism occur off-axis.
#### Chromatic aberration

Chromatic aberration occurs due to the variation of refractive index with wavelength for a lens material (there is no chromatic aberation in curved mirrors). This wavelength dependence results in slightly different focal lengths for different wavelengths of light. Compound lenses, called achromats, can reduce or eliminate chromatic aberation because the components are chosen such that the variation in refractive index as a function of wavelength cancels out.
#### Spherical aberration

Spherical aberation results because the actual focal point of a light ray depends on its distance from the optic axis.
#### Coma

Coma is caused by the distortion of a wavefront as it encounters an optic asymmetrically. The result for collimated incoming light is a circle instead of a point image. The light rays farther from the optic axis have more severe aberation and the resulting image looks like a comet-shaped series of circles.
#### Astigmatism

The projection of an optic off-axis looks squashed in one direction. The squashed direction focuses light to a greater extent than the normal dimension. The result is two line images.
#### Minimizing aberrations

- work on or near the optic axis
- use compound lenses (achromats, doublets, triplets) which can be designed to reduce chromatic aberation, spherical aberation, and coma
- use computer optimized aspheric lenses

### Related Topics

### Further Information

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Copyright © 1996 by Brian M. Tissue

updated 2/21/96