Geometric Optics, Vision and Optical Instruments

75 Microscopes

Learning Objectives

  • Describe how a compound microscope forms a magnified image using multiple lenses.
  • Explain the roles of the objective lens and the eyepiece in image formation.
  • Calculate the overall magnification of a compound microscope.
  • Recognize the importance of microscopes in biology, medicine, and healthcare.

The human eye can distinguish an impressive range of objects, from distant mountains to the fine print on a page. However, there is a limit to the amount of detail it can resolve. Cells, bacteria, blood components, and many other biological structures are far too small to be observed without optical instruments. The invention of the microscope revolutionized science by allowing us to visualize structures that were previously invisible, laying the foundation for modern biology and medicine.

A microscope is an optical instrument designed to produce highly magnified images of small objects. Unlike a simple magnifying glass, which uses a single lens, most microscopes contain several lenses that work together. The image produced by one lens serves as the object for the next lens, allowing the overall magnification to become much greater than could be achieved with a single lens. The same principles of ray tracing and the thin lens equation developed earlier in this chapter can be applied to each optical element in sequence.

A researcher looks through the eyepiece of a laboratory microscope while adjusting the focus controls.
Figure 75.1: Modern compound microscopes use multiple lenses to produce highly magnified images of microscopic specimens. Fine-focus controls allow precise adjustment to obtain a sharp image. (Credit: U.S. Navy photo by Tom Watanabe)

The Compound Microscope

The first compound microscopes were developed in Europe during the early seventeenth century by eyeglass makers. Although modern microscopes contain many carefully designed optical elements, the basic principle remains the same: a compound microscope uses two primary lens systems to magnify an object.

The lens closest to the specimen is called the objective lens. Depending on the microscope, objectives typically provide magnifications between 5× and 100×. Modern microscopes usually contain several objectives mounted on a rotating nosepiece, allowing the user to change magnification quickly. Most objectives are parfocal, meaning that only minor refocusing is needed when switching from one objective to another.

The second optical system is the eyepiece, also called the ocular. The eyepiece further magnifies the image created by the objective, producing a final image that can be viewed comfortably by the observer. Together, the objective and eyepiece provide the microscope's total magnification.

Compound microscopes are indispensable tools throughout healthcare and biomedical science. They are routinely used to examine blood cells, identify bacteria, diagnose infectious diseases, evaluate tissue biopsies, analyze microorganisms, and study the structure of cells and organs.

Ray diagram showing image formation in a compound microscope. The objective lens produces a real, inverted, magnified intermediate image, which serves as the object for the eyepiece. The eyepiece then forms a larger virtual image for the observer.
Figure 75.2: Image formation in a compound microscope. The objective lens first produces a real, inverted, magnified intermediate image. This intermediate image becomes the object for the eyepiece, which acts as a magnifier and produces a larger virtual image that is viewed by the observer.

How a Compound Microscope Forms an Image

Image formation in a compound microscope occurs in two stages. First, the specimen is placed just beyond the focal point of the objective lens. Under these conditions, the objective forms a real, inverted, magnified image. This intermediate image serves as the object for the eyepiece.

The eyepiece is positioned so that the intermediate image lies within its focal length. Acting like a magnifying glass, the eyepiece produces a second, much larger virtual image that is observed by the eye. The final image remains inverted relative to the original specimen but appears much larger and is located at a comfortable viewing distance, allowing the eye to remain relaxed.

Because each lens contributes its own magnification, the total magnification of the microscope is simply the product of the magnifications produced by the objective and the eyepiece:

[latex]m=m_{\mathrm{o}}m_{\mathrm{e}}[/latex]

where:

  • [latex]m[/latex] is the overall magnification,
  • [latex]m_{\mathrm{o}}[/latex] is the magnification produced by the objective lens, and
  • [latex]m_{\mathrm{e}}[/latex] is the magnification produced by the eyepiece.

This relationship applies not only to microscopes but also to any optical instrument containing multiple thin lenses or mirrors. Each optical element forms an image that becomes the object for the next element in the system.

Key Concept: Overall Magnification

For a microscope or any other multiple-element optical instrument, the overall magnification equals the product of the magnifications produced by each optical element.

[latex]m=m_{\mathrm{o}}m_{\mathrm{e}}[/latex]

Increasing the magnification of either the objective or the eyepiece increases the overall magnification of the instrument.

Example: Calculating the Magnification of a Compound Microscope

A specimen is placed 6.20 mm from the objective lens of a compound microscope. The objective has a focal length of 6.00 mm, the eyepiece has a focal length of 50.0 mm, and the distance between the two lenses is 23.0 cm. Calculate the overall magnification of the microscope.

Strategy

The microscope contains two lenses that work in sequence. First, use the thin lens equation to determine the image formed by the objective lens and calculate its magnification. This intermediate image then becomes the object for the eyepiece, which forms the final image. The overall magnification is the product of the magnifications of the two lenses.

Solution

Step 1: Calculate the magnification of the objective lens.

The magnification produced by the objective is

[latex]m_{\mathrm{o}}=-\frac{d_{\mathrm{i}}}{d_{\mathrm{o}}}[/latex]

The object distance is

[latex]d_{\mathrm{o}}=6.20\ \text{mm}[/latex]

To find the image distance, use the thin lens equation:

[latex]\frac{1}{d_{\mathrm{i}}}=\frac{1}{f_{\mathrm{o}}}-\frac{1}{d_{\mathrm{o}}}[/latex]

Substituting the known values gives

[latex]\frac{1}{d_{\mathrm{i}}}=\frac{1}{6.00\ \text{mm}}-\frac{1}{6.20\ \text{mm}}=0.00538\ \text{mm}^{-1}[/latex]

Therefore,

[latex]d_{\mathrm{i}}=186\ \text{mm}[/latex]

The objective magnification is

[latex]m_{\mathrm{o}}=-\frac{186}{6.20}=-30.0[/latex]

Step 2: Calculate the magnification of the eyepiece.

The intermediate image formed by the objective acts as the object for the eyepiece. Since the lenses are separated by 230 mm, the object distance for the eyepiece is

[latex]d_{\mathrm{o}}'=230\ \text{mm}-186\ \text{mm}=44.0\ \text{mm}[/latex]

Use the thin lens equation again:

[latex]\frac{1}{d_{\mathrm{i}}'}=\frac{1}{f_{\mathrm{e}}}-\frac{1}{d_{\mathrm{o}}'}[/latex]
[latex]\frac{1}{d_{\mathrm{i}}'}=\frac{1}{50.0\ \text{mm}}-\frac{1}{44.0\ \text{mm}}=-0.00273\ \text{mm}^{-1}[/latex]

Thus,

[latex]d_{\mathrm{i}}'=-367\ \text{mm}[/latex]

The eyepiece magnification is

[latex]m_{\mathrm{e}}=-\frac{d_{\mathrm{i}}'}{d_{\mathrm{o}}'}=-\frac{-367}{44.0}=8.34[/latex]

Step 3: Calculate the overall magnification.

[latex]m=m_{\mathrm{o}}m_{\mathrm{e}}[/latex]
[latex]m=(-30.0)(8.34)\approx-250[/latex]

The microscope therefore produces an overall magnification of approximately

[latex]\boxed{m\approx-250}[/latex]

Discussion

Both the objective lens and the eyepiece contribute significantly to the final magnification. The negative sign indicates that the final image is inverted relative to the original specimen. The eyepiece produces a virtual image located approximately 367 mm to the left of the eyepiece, allowing the observer to view the specimen comfortably.

This example illustrates the general method used to analyze any multi-element optical system. Each lens is treated independently, and the image produced by one lens becomes the object for the next. Modern microscopes may contain many additional lenses to reduce optical aberrations, but the same sequence of image formation applies.

Clinical Connection

Although modern laboratory microscopes often display the objective magnification (such as 40×) and eyepiece magnification (typically 10×), giving a total magnification of 400×, magnification alone does not determine image quality. The ability to distinguish two nearby structures depends primarily on the microscope's resolution, which is determined by factors such as the wavelength of light and the numerical aperture of the objective lens. This distinction is especially important in clinical laboratories when identifying bacteria, blood cells, parasites, and tissue structures.

Numerical Aperture and Resolution

Most laboratory microscopes can achieve magnifications of up to approximately 1500×, with a theoretical optical resolution of about 0.2 μm. However, magnification alone does not determine image quality. A blurry image viewed at higher magnification still lacks detail. The ability to distinguish two closely spaced objects depends primarily on the microscope's resolution, which is strongly influenced by the design of the objective lens.

Modern microscope objectives contain multiple carefully designed lenses that minimize optical aberrations while maximizing image quality. Three important characteristics describe an objective lens:

  • Magnification
  • Numerical aperture (NA)
  • Working distance

Among these, the numerical aperture (NA) is often the most important because it determines how efficiently the objective gathers light from the specimen.

The numerical aperture is defined as

[latex]\mathrm{NA}=n\sin\alpha[/latex]

where

  • [latex]n[/latex] is the refractive index of the medium between the specimen and the objective, and
  • [latex]\alpha[/latex] is one-half of the acceptance angle of the objective.

A larger acceptance angle allows the objective to collect more light from the specimen. As the numerical aperture increases, the microscope captures more information from fine details, resulting in improved resolution. For example, a 0.75 NA objective can resolve much finer structures than a 0.10 NA objective, even if both have similar magnification.

(a) The numerical aperture of a microscope objective is determined by the maximum cone of light that can enter the lens. (b) A camera lens collects light over a similar acceptance angle, with the aperture diameter controlling the amount of light entering the system.
Figure 75.3: (a) The numerical aperture (NA) measures the light-gathering ability of a microscope objective. (b) The same optical principles determine how much light enters a camera lens through its aperture.

Clinical Connection

In clinical laboratories, increasing magnification does not necessarily improve diagnosis. For example, identifying bacteria in a Gram stain or distinguishing subtle features of blood cells requires excellent resolution, which depends much more on numerical aperture than on magnification. This is why laboratory microscopes often use high-NA objectives together with immersion oil.

Working Distance

Although numerical aperture determines image quality, it does not indicate how close the objective must be to the specimen. This is described by the working distance, which is the distance between the front surface of the objective lens and the specimen when the image is in focus.

As magnification and numerical aperture increase, the working distance generally becomes smaller. High-power objectives must therefore be positioned extremely close to the specimen. This allows them to collect light over a larger angle but also increases the risk of damaging the coverslip, specimen, or objective lens if the microscope is focused carelessly.

It is important to distinguish the working distance from the focal length. Because microscope objectives contain many optical elements, the focal length is measured within the lens assembly, whereas the working distance is measured from the front of the objective to the specimen.

F-Number and Light Collection

Another quantity used in optics is the f-number, written as [latex]f/\#[/latex]. It describes how much light reaches the image plane in cameras and other optical instruments.

[latex]\frac{f}{\#}=\frac{f}{D}\approx\frac{1}{2\,\mathrm{NA}}[/latex]

where [latex]f[/latex] is the focal length of the lens and [latex]D[/latex] is the diameter of the aperture.

A smaller f-number corresponds to a larger aperture, allowing more light to enter the optical system. This improves image brightness and is especially useful in low-light photography. Larger f-numbers reduce the amount of light entering the lens but increase the depth of field, making more of the scene appear sharply focused.

The concept of numerical aperture also applies to optical fibers, which must efficiently collect and guide light.

Light entering an optical fiber within its acceptance angle is guided by total internal reflection. The maximum acceptance angle determines the fiber's numerical aperture.
Figure 75.4: The numerical aperture of an optical fiber determines the maximum angle at which light can enter the fiber and still be transmitted efficiently.

Immersion Objectives

Can the numerical aperture be greater than 1? Surprisingly, the answer is yes. This is possible by placing a transparent liquid between the objective lens and the microscope slide. The liquid minimizes refraction at the glass-air interface and allows the objective to collect light over a larger range of angles.

Common immersion media include water, glycerol, and immersion oil. Because immersion oil has a refractive index very close to that of glass, it provides the greatest improvement in light collection and image resolution.

Comparison of light rays entering a microscope objective through air, water, and immersion oil. Immersion liquids allow more rays to enter the objective, increasing numerical aperture and resolution.
Figure 75.5: Immersion media reduce refraction between the specimen and the objective lens. Oil immersion collects the largest cone of light, producing the highest numerical aperture and the greatest resolution.

Healthcare Application

Oil immersion objectives are used routinely in clinical microbiology and hematology laboratories. A 100× oil immersion objective allows healthcare professionals to identify bacteria, examine blood smears for parasites such as Plasmodium (malaria), evaluate blood cell morphology, and observe fine cellular details that would be difficult or impossible to resolve using air objectives.

Field of View and Specimen Illumination

Only a small portion of a specimen is visible through the microscope at one time. This visible region is called the field of view. As magnification increases, the field of view becomes smaller, requiring the user to move either the specimen or the objective to examine different regions.

Modern digital microscopes often automate this process by scanning adjacent fields of view and combining them into a single high-resolution image. This technique is widely used in digital pathology, where an entire tissue section can be examined on a computer screen.

Microscopes also require adequate illumination because microscopic structures reflect or transmit relatively little light. Condensers concentrate light onto the specimen, and different illumination methods emphasize different specimen features.

Examples of microscope illumination methods including transmitted light, mirror illumination, dark-field illumination, and reflected laser illumination.
Figure 75.6: Common illumination methods used in microscopy. Different techniques emphasize different specimen properties and are selected according to the application.

Beyond Optical Microscopes

Although visible-light microscopes remain the most widely used instruments in biology and medicine, other types of microscopes provide much higher resolution. Electron microscopes use beams of electrons instead of visible light, allowing individual atoms to be imaged under appropriate conditions. Because electrons are easily scattered by air molecules, these instruments operate inside high-vacuum chambers.

Modern electron microscopes can achieve magnifications exceeding 50 million×. Their images are collected electronically and displayed on computers rather than viewed directly through an eyepiece.

Electron microscopy has transformed many areas of science and medicine, including virology, cell biology, materials science, and nanotechnology. During the 1990s, Pratibha L. Gai developed the Environmental Transmission Electron Microscope (ETEM), enabling researchers to observe individual atoms during chemical reactions.

Researcher standing beside a transmission electron microscope used for atomic-scale imaging.
Figure 75.7: Transmission electron microscopes use electron beams instead of visible light to produce images with atomic-scale resolution. (Credit: Oak Ridge National Laboratory)
Scanning electron microscope image showing bacteria attached to the surface of a grain of sand.
Figure 75.8: Scanning electron microscopy produces detailed three-dimensional images of biological specimens, such as bacterial communities growing on a grain of sand. (Credit: Anthony D'Onofrio, William H. Fowle, Eric J. Stewart, Kim Lewis, The Lewis Laboratory, Northeastern University)

Take-Home Investigation: Build a Water Lens

Look through an empty transparent plastic bottle and describe the image you observe. Next, partially fill the bottle with water and compare the image. Finally, use the bottle as a lens to form an image of a bright distant object, such as a window or lamp, on a sheet of paper.

Estimate the focal length of the water lens and investigate how the focal length changes as the water level inside the bottle increases or decreases. Explain your observations using the principles of refraction and lens curvature.

Section Summary

  • A compound microscope is a multiple-element optical instrument in which the image produced by one lens becomes the object for the next lens.
  • Image formation in a microscope is analyzed by applying the thin lens equation to each optical element in sequence. The intermediate image formed by the objective serves as the object for the eyepiece.
  • The overall magnification of a compound microscope is the product of the magnifications produced by the objective and the eyepiece:
[latex]m=m_{\mathrm{o}}m_{\mathrm{e}}[/latex]
  • The objective lens produces the first magnified image, while the eyepiece acts as a magnifier that enlarges this intermediate image for comfortable viewing.
  • Magnification and resolution are not the same. Increasing magnification alone does not reveal additional detail unless the microscope also has sufficient resolving power.
  • The numerical aperture (NA) describes the light-gathering ability and resolving power of an objective lens and is given by
[latex]\mathrm{NA}=n\sin\alpha[/latex]
  • Objectives with larger numerical apertures collect more light and produce higher-resolution images, although they generally have shorter working distances.
  • Immersion media such as oil or water increase the numerical aperture by reducing refraction between the specimen and the objective lens, allowing finer details to be resolved.
  • The f-number describes the light-gathering ability of a lens and is related to the numerical aperture by
[latex]\frac{f}{\#}=\frac{f}{D}\approx\frac{1}{2\,\mathrm{NA}}[/latex]
  • Modern microscopes use a variety of illumination methods—including transmitted light, reflected light, dark-field illumination, and laser illumination—to optimize image quality for different specimens and applications.

Conceptual Questions

  1. Geometric optics describes the behavior of light interacting with objects that are much larger than its wavelength. Why is it still appropriate to use geometric optics to analyze image formation in a microscope?
  2. The final image produced by the compound microscope shown in Figure 75.2 cannot be projected directly onto a screen. Could additional lenses or mirrors be used to project this image? Explain your reasoning.
  3. Why is the objective lens designed to produce a real intermediate image rather than a highly magnified virtual image? Consider how the eyepiece functions in the microscope.
  4. What advantages do oil immersion objectives provide compared with standard air objectives?
  5. The numerical aperture (NA) is an important parameter for both microscope objectives and optical fibers. In what ways is the concept of numerical aperture similar in these two optical systems?

Problem Exercises

  1. A compound microscope has an overall magnification of 800×. The objective lens provides a magnification of 200×.
    1. What is the magnification of the eyepiece?
    2. If the microscope also has objectives with magnifications of 100× and 400×, what total magnifications are possible using the same eyepiece?
  2. A microscope objective has a focal length of 0.150 cm, and the specimen is located 0.155 cm from the objective.
    1. What magnification is produced by the objective?
    2. What is the overall magnification if an 8× eyepiece is used?
  3. A microscope objective has a focal length of 0.500 cm.
    1. Where must an object be placed so that the objective produces a magnification of [latex]-400[/latex]?
    2. Where should a 5.00 cm focal-length eyepiece be positioned to provide an additional fourfold (4.00×) magnification?
  4. You replace one microscope objective with another that has a different numerical aperture.
    1. Determine the acceptance angle for a [latex]1.40\,\mathrm{NA}\,60\times[/latex] oil immersion objective.
    2. Compare this angle with that of a [latex]0.65\,\mathrm{NA}\,40\times[/latex] air objective.
    3. Which objective would you use first to locate an area of interest on a specimen? Explain your reasoning.
  5. An amoeba is located 0.305 cm from the objective lens of a microscope whose objective has a focal length of 0.300 cm. The eyepiece has a focal length of 2.00 cm and is positioned 20.0 cm from the objective (see Figure 75.3).
    1. Where is the intermediate image formed by the objective?
    2. What is the magnification produced by the objective?
    3. Where is the final image formed by the eyepiece?
    4. What magnification is produced by the eyepiece?
    5. What is the overall magnification of the microscope?
  6. A microscope is initially equipped with a [latex]0.10\,\mathrm{NA}\,4\times[/latex] objective. It is then switched to a [latex]0.65\,\mathrm{NA}\,40\times[/latex] objective.
    1. Calculate the acceptance angle for each objective.
    2. Compare the two values and discuss how they affect image brightness and resolution.
    3. Which objective would you use first to locate a specimen on a microscope slide?
  7. Unreasonable Results. A microscope is reported to have a 0.500 cm focal-length objective lens, a 5.00 cm focal-length eyepiece, and an overall magnification of 250,000×. Evaluate whether these values are physically reasonable for a conventional optical microscope. Explain your reasoning.

Glossary

compound microscope
An optical microscope that uses two or more lens systems—typically an objective lens and an eyepiece—to produce a highly magnified image.
objective lens
The lens closest to the specimen. It forms the first real, magnified image and largely determines the microscope's resolution.
eyepiece (ocular)
The lens or combination of lenses closest to the observer's eye. It magnifies the intermediate image formed by the objective lens.
numerical aperture (NA)
A measure of the light-gathering ability and resolving power of an objective lens, defined by

[latex]\mathrm{NA}=n\sin\alpha[/latex]

where [latex]n[/latex] is the refractive index of the medium between the specimen and the objective, and [latex]\alpha[/latex] is half of the acceptance angle of the objective.

definition

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Introductory Physics for the Health and Life Sciences II Copyright © 2012 by OSCRiceUniversity is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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