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Lenses and Geometrical Optics

The action of a simple lens, similar to many of those used in the microscope, is governed by the principles of refraction and reflection and can be understood with the aid of a few simple rules about the geometry involved in tracing light rays through the lens. The basic concepts explored in this discussion, which are derived from the science of Geometrical Optics, will lead to an understanding of the magnification process, the properties of real and virtual images, and lens aberrations or defects.

Introduction to Lenses and Geometrical Optics - The term lens is the common name given to a component of glass or transparent plastic material, usually circular in diameter, which has two primary surfaces that are ground and polished in a specific manner designed to produce either a convergence or divergence of light passing through the material. The optical microscope forms an image of a specimen placed on the stage by passing light from the illuminator through a series of glass lenses and focusing this light either into the eyepieces, on the film plane in a traditional camera system, or onto the surface of a digital image sensor.

Common Aberrations in Lens Systems - Microscopes and other optical instruments are commonly plagued by lens errors that distort the image by a variety of mechanisms associated with defects (commonly referred to as aberrations) resulting from the spherical geometry of lens surfaces. There are three primary sources of non-ideal lens action (errors) that are observed in the microscope. Of the three major classes of lens errors, two are associated with the orientation of wavefronts and focal planes with respect to the microscope optical axis. These include on-axis lens errors such as chromatic and spherical aberration, and the major off-axis errors manifested as coma, astigmatism, and field curvature. A third class of aberrations, commonly seen in stereomicroscopes that have zoom lens systems, is geometrical distortion, which includes both barrel distortion and pincushion distortion.

Lens Interactive Java Tutorials

Simple Bi-Convex Thin Lenses - A simple thin lens has two focal planes that are defined by the geometry of the lens and the relationship between the lens and the focused image. Light rays passing through the lens will intersect and are physically combined at the focal plane, while extensions of the rays passing through the lens will intersect with the rays emerging from the lens at the principal plane. The focal length of a lens is defined as the distance between the principal plane and the focal plane, and every lens has a set of these planes on each side (front and rear). This interactive tutorial explores how changes to focal length and object size affect the size and position of the image formed by a simple thin lens.

Simple Magnification - A typical magnifying glass consists of a single thin bi-convex lens that produces a modest magnification in the range of 1.5x to 30x, with the most common being about 2-4x for reading or studying rocks, stamps, coins, insects, and leaves. Magnifying glasses produce a virtual image that is magnified and upright. This interactive tutorial demonstrates how a simple, thin bi-convex magnifying lens works to produce a magnified virtual image on the retina.

Magnification with a Bi-Convex Lens - Single lenses capable of forming images (like the bi-convex lens) are useful in tools designed for simple magnification applications, such as magnifying glasses, eyeglasses, single-lens cameras, loupes, viewfinders, and contact lenses. This interactive tutorial explores how a simple bi-convex lens can be used to magnify an image.

Image Formation with Converging Lenses - Positive, or converging, thin lenses unite incident light rays that are parallel to the optical axis and focus them at the focal plane to form a real image. This interactive tutorial utilizes ray traces to explore how images are formed by the three primary types of converging lenses, and the relationship between the object and the image formed by the lens as a function of distance between the object and the focal points.

Image Formation with Diverging Lenses - Negative lenses diverge parallel incident light rays and form a virtual image by extending traces of the light rays passing through the lens to a focal point behind the lens. In general, these lenses have at least one concave surface and are thinner in the center than at the edges. This interactive tutorial utilizes ray traces to explore how images are formed by the three primary types of diverging lenses, and the relationship between the object and the image formed by the lens as a function of distance between the object and the focal points.

Geometrical Construction of Ray Diagrams - A popular method of representing a train of propagating light waves involves the application of geometrical optics to determine the size and location of images formed by a lens or multi-lens system. This tutorial explores how two representative light rays can establish the parameters of an imaging scenario.

Perfect Lens Characteristics - The simplest imaging element in an optical microscope is a perfect lens, which is an ideally corrected glass element that is free of aberration and focuses light onto a single point. This tutorial explores how light waves propagate through and are focused by a perfect lens.

Perfect Two-Lens System Characteristics - During investigations of a point source of light that does not lie in the focal plane of a lens, it is often convenient to represent a perfect lens as a system composed of two individual lens elements. This tutorial explores off-axis oblique light rays passing through such a system.

Radius and Refractive Index Effects on Lens Action - The action of a simple bi-convex thin lens is governed by the principles of refraction (which is a function of lens curvature radius and refractive index), and can be understood with the aid of a few simple rules about the geometry involved in tracing light rays through the lens. This interactive tutorial explores how variations in the refractive index and radius of a bi-convex lens affect the relationship between the object and the image produced by the lens.

Optical Aberration Interactive Java Tutorials

Astigmatism - Astigmatism aberrations are similar to comatic aberrations, however these artifacts are not as sensitive to aperture size and depend more strongly on the oblique angle of the light beam. The aberration is manifested by the off-axis image of a specimen point appearing as a line or ellipse instead of a point. Depending on the angle of the off-axis rays entering the lens, the line image may be oriented in either of two different directions, tangentially (meridionally) or sagittally (equatorially). The intensity ratio of the unit image will diminish, with definition, detail, and contrast being lost as the distance from the center is increased.

Chromatic Aberration - Chromatic aberrations are wavelength-dependent artifacts that occur because the refractive index of every optical glass formulation varies with wavelength. When white light passes through a simple or complex lens system, the component wavelengths are refracted according to their frequency. In most glasses, the refractive index is greater for shorter (blue) wavelengths and changes at a more rapid rate as the wavelength is decreased.

Comatic Aberration - Comatic aberrations are similar to spherical aberrations, but they are mainly encountered with off-axis light fluxes and are most severe when the microscope is out of alignment. When these aberrations occur, the image of a point is focused at sequentially differing heights producing a series of asymmetrical spot shapes of increasing size that result in a comet-like (hence, the term coma) shape to the Airy pattern.

Curvature of Field - Modern microscopes deal with field curvature by correcting this aberration using specially designed objectives. These specially-corrected objectives have been named plan or plano (for flat-field) and are the most common type of objective in use today, providing ocular fields ranging between 18 and 26 millimeters, which exhibit sharp detail from center to edge.

Geometrical Distortion - Distortion is an aberration commonly seen in stereoscopic microscopy, which is manifested by changes in the shape of an image rather than the sharpness or color spectrum. The two most prevalent types of distortion, positive and negative (often termed pincushion and barrel, respectively), can often be present in very sharp images that are otherwise corrected for spherical, chromatic, comatic, and astigmatic aberrations. In this case, the true geometry of an object is no longer maintained in the image.

Spherical Aberration - The most serious of the monochromatic defects that occurs with microscope objectives, spherical aberration, causes the specimen image to appear hazy or blurred and slightly out of focus. The effect of spherical aberration manifests itself in two ways: the center remains more in focus than the edges of the image and the intensity of the edges falls relative to that of the center. This defect appears in both on-axis and off-axis image points.

Contributing Authors

Mortimer Abramowitz - Olympus America, Inc., Two Corporate Center Drive., Melville, New York, 11747.

Kenneth R. Spring - Scientific Consultant, Lusby, Maryland, 20657.

Matthew J. Parry-Hill and Michael W. Davidson - National High Magnetic Field Laboratory, 1800 East Paul Dirac Dr., The Florida State University, Tallahassee, Florida, 32310.


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