KAHIBARO
Table of Contents
Seeing More Than the Eye Alone
Optical instruments are devices that use mirrors, lenses, or both to improve human vision. They can make distant objects appear closer, make tiny objects appear larger, or project images in a useful way. The basic ideas behind these instruments come from reflection, refraction, image formation, and magnification, but here the focus is on how these ideas are combined in real devices.
The human eye itself is an optical system, so many instruments are designed to work together with it. In most cases, an instrument either changes the angle under which the eye sees an object, or forms an intermediate image that the eye can observe more easily.
An optical instrument does not usually change the actual size of the object. It changes the way the object appears to the observer, often by increasing its apparent size or making it easier to focus on.
Angular Size and Apparent Enlargement
When we look at an object, what matters for perception is not only its actual size, but the angle it subtends at the eye. A distant mountain may be huge, but it appears small because the angle is small. A tiny insect close to the eye may appear large because the angle is larger.
Optical instruments often work by increasing this viewing angle. If an object of height $h$ is viewed from distance $d$, the small viewing angle is approximately
$$
\theta \approx \frac{h}{d}
$$
If an instrument allows the eye to see the object under a larger angle $\theta'$, then the object appears magnified.
This leads to the idea of angular magnification,
$$
M = \frac{\theta'}{\theta}
$$
where $\theta$ is the angle seen by the unaided eye, and $\theta'$ is the angle seen through the instrument.
For many optical instruments, the most useful definition of magnification is angular magnification,
$$
M = \frac{\theta'}{\theta}
$$
because the instrument changes how large the object appears to the eye.
The Simple Magnifier
The simplest optical instrument is the magnifying glass, also called a simple magnifier. It is a converging lens used to view a small nearby object. Without a lens, the eye has a limit to how close it can focus. This nearest comfortable viewing distance is called the near point. For a typical relaxed adult eye, it is often taken as about $25 \, \text{cm}$.
A magnifying glass allows the object to be placed closer to the eye than would otherwise be possible, while still producing a virtual image that the eye can focus on. The image appears larger because it subtends a larger angle.
If the final image is formed at the near point, the angular magnification of a simple magnifier is approximately
$$
M = 1 + \frac{D}{f}
$$
where $D$ is the near-point distance, usually taken as $25 \, \text{cm}$, and $f$ is the focal length of the lens.
If the final image is formed at infinity, which is more comfortable for the eye, then
$$
M = \frac{D}{f}
$$
A shorter focal length gives greater magnification.
For a simple magnifier,
$$
M = \frac{D}{f}
$$
for an image at infinity, and
$$
M = 1 + \frac{D}{f}
$$
for an image at the near point.
The Compound Microscope
A microscope is used to view very small objects. It uses two converging lenses. The lens close to the object is called the objective, and the lens near the eye is called the eyepiece.
The objective first forms a real, inverted, enlarged intermediate image of the small object. Then the eyepiece acts as a magnifier and produces a larger virtual image of that intermediate image.
So the microscope magnifies in two stages. First, the objective provides linear magnification. Second, the eyepiece provides angular magnification.
The total magnification is approximately the product of these two effects:
$$
M \approx m_o M_e
$$
where $m_o$ is the magnification of the objective and $M_e$ is the angular magnification of the eyepiece.
For a common ideal form of compound microscope, the total magnification is often written as
$$
M \approx \frac{L}{f_o}\frac{D}{f_e}
$$
where $L$ is the tube length, $f_o$ is the focal length of the objective, $f_e$ is the focal length of the eyepiece, and $D$ is the near-point distance.
A microscope usually uses a very short focal length objective to obtain a large first magnification.
For an ideal compound microscope,
$$
M \approx \frac{L}{f_o}\frac{D}{f_e}
$$
A short focal length objective is the main reason high magnification is possible.
The Astronomical Telescope
A telescope is used to observe very distant objects. Because distant objects send nearly parallel rays into the instrument, the objective lens or mirror forms a real image near its focal plane. This image is then viewed by an eyepiece.
In a refracting astronomical telescope, both the objective and the eyepiece are converging lenses. The objective has a long focal length and large diameter so that it can collect more light and form a clear image. The eyepiece then magnifies that image.
For normal adjustment, the final image is at infinity, which is comfortable for the eye. The angular magnification is
$$
M = -\frac{f_o}{f_e}
$$
where $f_o$ is the focal length of the objective and $f_e$ is the focal length of the eyepiece. The negative sign means the image is inverted.
A large objective focal length increases magnification, while a short eyepiece focal length also increases magnification.
For an astronomical telescope in normal adjustment,
$$
M = -\frac{f_o}{f_e}
$$
The negative sign indicates that the final image is inverted.
Reflecting Telescopes
Not all telescopes use lenses as objectives. Many important telescopes use mirrors. A reflecting telescope uses a concave mirror to collect light from distant objects and bring it to focus. An eyepiece then magnifies the image.
Reflecting telescopes have important advantages. Large lenses are difficult to make and support, and they can produce chromatic aberration because different wavelengths refract differently. Mirrors do not suffer from chromatic aberration in the same way, and they are often easier to build in very large sizes.
A reflecting telescope can collect a great amount of light, which is essential in astronomy. Often, the brightness and resolution of a telescope are more important than simply increasing magnification.
Resolving Power and Light Gathering
Two important properties of optical instruments are magnification and resolving power. Magnification makes an image appear larger, but resolving power determines how much detail can actually be distinguished.
If two points are very close together, an instrument may show them as one blurred spot instead of two distinct points. The ability to separate them is the resolving power.
Larger objective lenses or mirrors generally improve resolution and also collect more light. The light-gathering power is roughly proportional to the area of the objective opening, so for diameter $D$,
$$
\text{Light-gathering power} \propto D^2
$$
This is why large telescopes are so useful. They do not just magnify more, they also allow faint objects to be seen and fine detail to be resolved.
High magnification alone does not guarantee a better image. Good optical performance also requires high resolving power and sufficient light collection.
The Terrestrial Telescope and Binoculars
An astronomical telescope forms an inverted image, which is acceptable for viewing stars and planets. For viewing objects on Earth, an upright image is usually preferred. A terrestrial telescope includes additional optics to make the final image upright.
Binoculars are essentially two small telescopes placed side by side, one for each eye. They usually contain prisms that both fold the light path into a compact shape and correct the image orientation. Using both eyes gives a more natural and comfortable view.
Binoculars are often labeled by two numbers, such as $8 \times 40$. The first number is the magnification, and the second is the diameter of the objective lens in millimeters. Thus, $8 \times 40$ binoculars magnify by a factor of 8 and have objective lenses of diameter $40 \, \text{mm}$.
The Camera as an Optical Instrument
A camera is another important optical instrument. Its goal is not mainly to help the eye directly, but to form a real image on a sensor or film. A converging lens focuses light from the object onto the image plane.
Unlike a magnifier or telescope, the camera produces a real image that can be recorded. To focus on objects at different distances, the distance between the lens and the sensor may be adjusted, or the lens shape may be changed internally.
The image formed by a simple camera lens is usually real and inverted. Camera quality depends on accurate focusing, good light collection, and reduction of optical defects.
The Eye and Vision Correction
The eye is itself an optical instrument. The cornea and lens focus light onto the retina, where the image is detected. A normal eye can adjust focus for objects at different distances by changing the shape of its lens.
Sometimes the eye does not focus properly. In myopia, or nearsightedness, distant objects are focused in front of the retina. A diverging corrective lens is used. In hyperopia, or farsightedness, nearby objects are difficult to focus on, and a converging corrective lens is used.
This shows that optical instruments are not only for science or astronomy. Even ordinary eyeglasses are practical optical devices that modify the formation of images to improve vision.
Comparison of Common Optical Instruments
Different optical instruments are designed for different purposes. Their optical arrangements reflect those purposes.
| Instrument | Main purpose | Main optical parts | Typical image role |
|---|---|---|---|
| Magnifying glass | View small nearby objects | One converging lens | Virtual enlarged image |
| Microscope | View tiny objects in great detail | Objective and eyepiece | Real intermediate image, then virtual enlarged image |
| Telescope | View distant objects | Objective and eyepiece, or mirror and eyepiece | Real image formed by objective, then magnified |
| Binoculars | View distant terrestrial objects with both eyes | Two telescopes plus prisms | Upright magnified view |
| Camera | Record images | Lens and sensor or film | Real image on detector |
| Eyeglasses | Correct vision | Converging or diverging lenses | Adjust focus onto retina |
Main Ideas to Remember
Optical instruments extend the ability of the eye. A magnifier helps with small nearby objects. A microscope gives large magnification for tiny objects. A telescope helps us see distant objects by collecting light and increasing angular size. A camera records real images, and eyeglasses correct defects in vision.
Although the designs differ, the underlying aim is similar. Each instrument controls light so that the image becomes more useful to the observer.
Key formulas for common optical instruments:
$$
M_{\text{magnifier}} = \frac{D}{f} \quad \text{or} \quad 1 + \frac{D}{f}
$$
$$
M_{\text{microscope}} \approx \frac{L}{f_o}\frac{D}{f_e}
$$
$$
M_{\text{telescope}} = -\frac{f_o}{f_e}
$$
These formulas are idealized, but they capture the main behavior of the instruments.