Magnification

Last updated April 02, 2024

Stepwise magnification by 6% per frame into a 39-megapixel image. In the final frame, at about 170x, an image of a bystander is seen reflected in the man's cornea.

Magnification is the process of enlarging the apparent size, not physical size, of something. This enlargement is quantified by a size ratio called optical magnification. When this number is less than one, it refers to a reduction in size, sometimes called de-magnification.

Examples of magnification

Some optical instruments provide visual aid by magnifying small or distant subjects.

A magnifying glass, which uses a positive (convex) lens to make things look bigger by allowing the user to hold them closer to their eye.
A telescope, which uses its large objective lens or primary mirror to create an image of a distant object and then allows the user to examine the image closely with a smaller eyepiece lens, thus making the object look larger.
A microscope, which makes a small object appear as a much larger image at a comfortable distance for viewing. A microscope is similar in layout to a telescope except that the object being viewed is close to the objective, which is usually much smaller than the eyepiece.
A slide projector, which projects a large image of a small slide on a screen. A photographic enlarger is similar.
A zoom lens, a system of camera lens elements for which the focal length and angle of view can be varied.

Size ratio (optical magnification)

Optical magnification is the ratio between the apparent size of an object (or its size in an image) and its true size, and thus it is a dimensionless number. Optical magnification is sometimes referred to as "power" (for example "10× power"), although this can lead to confusion with optical power.

Linear or transverse magnification

For real images, such as images projected on a screen, size means a linear dimension (measured, for example, in millimeters or inches).

Angular magnification

For optical instruments with an eyepiece, the linear dimension of the image seen in the eyepiece (virtual image at infinite distance) cannot be given, thus size means the angle subtended by the object at the focal point (angular size). Strictly speaking, one should take the tangent of that angle (in practice, this makes a difference only if the angle is larger than a few degrees). Thus, angular magnification is given by:

M_{A}={\frac {\tan \varepsilon }{\tan \varepsilon _{0}}}\approx {\frac {\varepsilon }{\varepsilon _{0}}}

where ${\textstyle \varepsilon _{0}}$ is the angle subtended by the object at the front focal point of the objective and ${\textstyle \varepsilon }$ is the angle subtended by the image at the rear focal point of the eyepiece.

For example, the mean angular size of the Moon's disk as viewed from Earth's surface is about 0.52°. Thus, through binoculars with 10× magnification, the Moon appears to subtend an angle of about 5.2°.

By convention, for magnifying glasses and optical microscopes, where the size of the object is a linear dimension and the apparent size is an angle, the magnification is the ratio between the apparent (angular) size as seen in the eyepiece and the angular size of the object when placed at the conventional closest distance of distinct vision: 25 cm from the eye.

By instrument

Single lens

The linear magnification of a thin lens is

M={f \over f-d_{\mathrm {o} }}

where ${\textstyle f}$ is the focal length and ${\textstyle d_{\mathrm {o} }}$ is the distance from the lens to the object. For real images, ${\textstyle M}$ is negative and the image is inverted. For virtual images, ${\textstyle M}$ is positive and the image is upright.

With ${\textstyle d_{\mathrm {i} }}$ being the distance from the lens to the image, ${\textstyle h_{\mathrm {i} }}$ the height of the image and ${\textstyle h_{\mathrm {o} }}$ the height of the object, the magnification can also be written as:

M=-{d_{\mathrm {i} } \over d_{\mathrm {o} }}={h_{\mathrm {i} } \over h_{\mathrm {o} }}

Note again that a negative magnification implies an inverted image.

Photography

The image recorded by a photographic film or image sensor is always a real image and is usually inverted. When measuring the height of an inverted image using the cartesian sign convention (where the x-axis is the optical axis) the value for $h i$ will be negative, and as a result $M$ will also be negative. However, the traditional sign convention used in photography is "real is positive, virtual is negative".^[1] Therefore, in photography: Object height and distance are always real and positive. When the focal length is positive the image's height, distance and magnification are real and positive. Only if the focal length is negative, the image's height, distance and magnification are virtual and negative. Therefore, the photographic magnification formulae are traditionally presented as^[2]

{\begin{aligned}M&={d_{\mathrm {i} } \over d_{\mathrm {o} }}={h_{\mathrm {i} } \over h_{\mathrm {o} }}\\&={f \over d_{\mathrm {o} }-f}={d_{\mathrm {i} }-f \over f}\end{aligned}}

Magnifying glass

The maximum angular magnification (compared to the naked eye) of a magnifying glass depends on how the glass and the object are held, relative to the eye. If the lens is held at a distance from the object such that its front focal point is on the object being viewed, the relaxed eye (focused to infinity) can view the image with angular magnification

M_{\mathrm {A} }={25\ \mathrm {cm}  \over f}

Here, ${\textstyle f}$ is the focal length of the lens in centimeters. The constant 25 cm is an estimate of the "near point" distance of the eye—the closest distance at which the healthy naked eye can focus. In this case the angular magnification is independent from the distance kept between the eye and the magnifying glass.

If instead the lens is held very close to the eye and the object is placed closer to the lens than its focal point so that the observer focuses on the near point, a larger angular magnification can be obtained, approaching

M_{\mathrm {A} }={25\ \mathrm {cm}  \over f}+1

A different interpretation of the working of the latter case is that the magnifying glass changes the diopter of the eye (making it myopic) so that the object can be placed closer to the eye resulting in a larger angular magnification.

Microscope

The angular magnification of a microscope is given by

M_{\mathrm {A} }=M_{\mathrm {o} }\times M_{\mathrm {e} }

where ${\textstyle M_{\mathrm {o} }}$ is the magnification of the objective and ${\textstyle M_{\mathrm {e} }}$ the magnification of the eyepiece. The magnification of the objective depends on its focal length ${\textstyle f_{\mathrm {o} }}$ and on the distance ${\textstyle d}$ between objective back focal plane and the focal plane of the eyepiece (called the tube length):

M_{\mathrm {o} }={d \over f_{\mathrm {o} }}

The magnification of the eyepiece depends upon its focal length ${\textstyle f_{\mathrm {e} }}$ and is calculated by the same equation as that of a magnifying glass (above).

Note that both astronomical telescopes as well as simple microscopes produce an inverted image, thus the equation for the magnification of a telescope or microscope is often given with a minus sign.^{[ citation needed ]}

Telescope

The angular magnification of an optical telescope is given by

M_{\mathrm {A} }={f_{\mathrm {o} } \over f_{\mathrm {e} }}

in which ${\textstyle f_{\mathrm {o} }}$ is the focal length of the objective lens in a refractor or of the primary mirror in a reflector, and ${\textstyle f_{\mathrm {e} }}$ is the focal length of the eyepiece.

Measurement of telescope magnification

Measuring the actual angular magnification of a telescope is difficult, but it is possible to use the reciprocal relationship between the linear magnification and the angular magnification, since the linear magnification is constant for all objects.

The telescope is focused correctly for viewing objects at the distance for which the angular magnification is to be determined and then the object glass is used as an object the image of which is known as the exit pupil. The diameter of this may be measured using an instrument known as a Ramsden dynameter which consists of a Ramsden eyepiece with micrometer hairs in the back focal plane. This is mounted in front of the telescope eyepiece and used to evaluate the diameter of the exit pupil. This will be much smaller than the object glass diameter, which gives the linear magnification (actually a reduction), the angular magnification can be determined from

M_{\mathrm {A} }={1 \over M}={D_{\mathrm {Objective} } \over {D_{\mathrm {Ramsden} }}}\,.

Maximum usable magnification

With any telescope or microscope, or a lens a maximum magnification exists beyond which the image looks bigger but shows no more detail. It occurs when the finest detail the instrument can resolve is magnified to match the finest detail the eye can see. Magnification beyond this maximum is sometimes called "empty magnification".

For a good quality telescope operating in good atmospheric conditions, the maximum usable magnification is limited by diffraction. In practice it is considered to be 2× the aperture in millimetres or 50× the aperture in inches; so, a 60 mm diameter telescope has a maximum usable magnification of 120×.^{[ citation needed ]}

With an optical microscope having a high numerical aperture and using oil immersion, the best possible resolution is 200 nm corresponding to a magnification of around 1200×. Without oil immersion, the maximum usable magnification is around 800×. For details, see limitations of optical microscopes.

Small, cheap telescopes and microscopes are sometimes supplied with the eyepieces that give magnification far higher than is usable.

The maximum relative to the minimum magnification of an optical system is known as zoom ratio.

"Magnification" of displayed images

Magnification figures on pictures displayed in print or online can be misleading. Editors of journals and magazines routinely resize images to fit the page, making any magnification number provided in the figure legend incorrect. Images displayed on a computer screen change size based on the size of the screen. A scale bar (or micron bar) is a bar of stated length superimposed on a picture. When the picture is resized the bar will be resized in proportion. If a picture has a scale bar, the actual magnification can easily be calculated. Where the scale (magnification) of an image is important or relevant, including a scale bar is preferable to stating magnification.

Related Research Articles

In optics, aberration is a property of optical systems, such as lenses, that causes light to be spread out over some region of space rather than focused to a point. Aberrations cause the image formed by a lens to be blurred or distorted, with the nature of the distortion depending on the type of aberration. Aberration can be defined as a departure of the performance of an optical system from the predictions of paraxial optics. In an imaging system, it occurs when light from one point of an object does not converge into a single point after transmission through the system. Aberrations occur because the simple paraxial theory is not a completely accurate model of the effect of an optical system on light, rather than due to flaws in the optical elements.

$<span class="mw-page-title-main">Lens</span> Optical device which transmits and refracts light$

A lens is a transmissive optical device that focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a compound lens consists of several simple lenses (elements), usually arranged along a common axis. Lenses are made from materials such as glass or plastic and are ground, polished, or molded to the required shape. A lens can focus light to form an image, unlike a prism, which refracts light without focusing. Devices that similarly focus or disperse waves and radiation other than visible light are also called "lenses", such as microwave lenses, electron lenses, acoustic lenses, or explosive lenses.

<span class="mw-page-title-main">Optics</span> Branch of physics that studies light

Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviolet, and infrared light. Light is a type of electromagnetic radiation, and other forms of electromagnetic radiation such as X-rays, microwaves, and radio waves exhibit similar properties.

In optics, the numerical aperture (NA) of an optical system is a dimensionless number that characterizes the range of angles over which the system can accept or emit light. By incorporating index of refraction in its definition, NA has the property that it is constant for a beam as it goes from one material to another, provided there is no refractive power at the interface. The exact definition of the term varies slightly between different areas of optics. Numerical aperture is commonly used in microscopy to describe the acceptance cone of an objective, and in fiber optics, in which it describes the range of angles within which light that is incident on the fiber will be transmitted along it.

The focal length of an optical system is a measure of how strongly the system converges or diverges light; it is the inverse of the system's optical power. A positive focal length indicates that a system converges light, while a negative focal length indicates that the system diverges light. A system with a shorter focal length bends the rays more sharply, bringing them to a focus in a shorter distance or diverging them more quickly. For the special case of a thin lens in air, a positive focal length is the distance over which initially collimated (parallel) rays are brought to a focus, or alternatively a negative focal length indicates how far in front of the lens a point source must be located to form a collimated beam. For more general optical systems, the focal length has no intuitive meaning; it is simply the inverse of the system's optical power.

The optical microscope, also referred to as a light microscope, is a type of microscope that commonly uses visible light and a system of lenses to generate magnified images of small objects. Optical microscopes are the oldest design of microscope and were possibly invented in their present compound form in the 17th century. Basic optical microscopes can be very simple, although many complex designs aim to improve resolution and sample contrast.

In photography, angle of view (AOV) describes the angular extent of a given scene that is imaged by a camera. It is used interchangeably with the more general term field of view.

Angular resolution describes the ability of any image-forming device such as an optical or radio telescope, a microscope, a camera, or an eye, to distinguish small details of an object, thereby making it a major determinant of image resolution. It is used in optics applied to light waves, in antenna theory applied to radio waves, and in acoustics applied to sound waves. The colloquial use of the term "resolution" sometimes causes confusion; when an optical system is said to have a high resolution or high angular resolution, it means that the perceived distance, or actual angular distance, between resolved neighboring objects is small. The value that quantifies this property, θ, which is given by the Rayleigh criterion, is low for a system with a high resolution. The closely related term spatial resolution refers to the precision of a measurement with respect to space, which is directly connected to angular resolution in imaging instruments. The Rayleigh criterion shows that the minimum angular spread that can be resolved by an image forming system is limited by diffraction to the ratio of the wavelength of the waves to the aperture width. For this reason, high resolution imaging systems such as astronomical telescopes, long distance telephoto camera lenses and radio telescopes have large apertures.

An optical telescope is a telescope that gathers and focuses light mainly from the visible part of the electromagnetic spectrum, to create a magnified image for direct visual inspection, to make a photograph, or to collect data through electronic image sensors.

The field of view (FOV) is the angular extent of the observable world that is seen at any given moment. In the case of optical instruments or sensors, it is a solid angle through which a detector is sensitive to electromagnetic radiation. It is further relevant in photography.

<span class="mw-page-title-main">Objective (optics)</span> Lens or mirror in optical instruments

In optical engineering, an objective is an optical element that gathers light from an object being observed and focuses the light rays from it to produce a real image of the object. Objectives can be a single lens or mirror, or combinations of several optical elements. They are used in microscopes, binoculars, telescopes, cameras, slide projectors, CD players and many other optical instruments. Objectives are also called object lenses, object glasses, or objective glasses.

In photography and cinematography, perspective distortion is a warping or transformation of an object and its surrounding area that differs significantly from what the object would look like with a normal focal length, due to the relative scale of nearby and distant features. Perspective distortion is determined by the relative distances at which the image is captured and viewed, and is due to the angle of view of the image being either wider or narrower than the angle of view at which the image is viewed, hence the apparent relative distances differing from what is expected. Related to this concept is axial magnification – the perceived depth of objects at a given magnification.

$<span class="mw-page-title-main">Airy disk</span> Diffraction pattern in optics$

In optics, the Airy disk and Airy pattern are descriptions of the best-focused spot of light that a perfect lens with a circular aperture can make, limited by the diffraction of light. The Airy disk is of importance in physics, optics, and astronomy.

A magnifying glass is a convex lens that is used to produce a magnified image of an object. The lens is usually mounted in a frame with a handle. Beyond its primary function of magnification, this simple yet ingenious tool serves a variety of purposes. It can be employed to focus sunlight, harnessing the Sun's rays to create a concentrated hot spot at the lens's focus, which is often used for starting fires.

An eyepiece, or ocular lens, is a type of lens that is attached to a variety of optical devices such as telescopes and microscopes. It is named because it is usually the lens that is closest to the eye when someone looks through an optical device to observe an object or sample. The objective lens or mirror collects light from an object or sample and brings it to focus creating an image of the object. The eyepiece is placed near the focal point of the objective to magnify this image to the eyes. The amount of magnification depends on the focal length of the eyepiece.

<span class="mw-page-title-main">Point spread function</span> Response in an optical imaging system

The point spread function (PSF) describes the response of a focused optical imaging system to a point source or point object. A more general term for the PSF is the system's impulse response; the PSF is the impulse response or impulse response function (IRF) of a focused optical imaging system. The PSF in many contexts can be thought of as the extended blob in an image that represents a single point object, that is considered as a spatial impulse. In functional terms, it is the spatial domain version of the optical transfer function (OTF) of an imaging system. It is a useful concept in Fourier optics, astronomical imaging, medical imaging, electron microscopy and other imaging techniques such as 3D microscopy and fluorescence microscopy.

In optics, the exit pupil is a virtual aperture in an optical system. Only rays which pass through this virtual aperture can exit the system. The exit pupil is the image of the aperture stop in the optics that follow it. In a telescope or compound microscope, this image is the image of the objective element(s) as produced by the eyepiece. The size and shape of this disc is crucial to the instrument's performance, because the observer's eye can see light only if it passes through the aperture. The term exit pupil is also sometimes used to refer to the diameter of the virtual aperture. Older literature on optics sometimes refers to the exit pupil as the Ramsden disc, named after English instrument-maker Jesse Ramsden.

The Barlow lens, named after Peter Barlow, is a diverging lens which, used in series with other optics in an optical system, increases the effective focal length of an optical system as perceived by all components that are after it in the system. The practical result is that inserting a Barlow lens magnifies the image. A real Barlow lens is not a single glass element, because that would generate chromatic aberration, and spherical aberration if the lens is not aspheric. More common configurations use three or more elements for achromatic correction or apochromatic correction and higher image quality.

<span class="mw-page-title-main">Curved mirror</span> Mirror with a curved reflecting surface

A curved mirror is a mirror with a curved reflecting surface. The surface may be either convex or concave. Most curved mirrors have surfaces that are shaped like part of a sphere, but other shapes are sometimes used in optical devices. The most common non-spherical type are parabolic reflectors, found in optical devices such as reflecting telescopes that need to image distant objects, since spherical mirror systems, like spherical lenses, suffer from spherical aberration. Distorting mirrors are used for entertainment. They have convex and concave regions that produce deliberately distorted images. They also provide highly magnified or highly diminished (smaller) images when the object is placed at certain distances.

In photography, a long-focus lens is a camera lens which has a focal length that is longer than the diagonal measure of the film or sensor that receives its image. It is used to make distant objects appear magnified with magnification increasing as longer focal length lenses are used. A long-focus lens is one of three basic photographic lens types classified by relative focal length, the other two being a normal lens and a wide-angle lens. As with other types of camera lenses, the focal length is usually expressed in a millimeter value written on the lens, for example: a 500 mm lens. The most common type of long-focus lens is the telephoto lens, which incorporate a special lens group known as a telephoto group to make the physical length of the lens shorter than the focal length.

References

↑ Ray, Sidney F. (2002). Applied Photographic Optics: Lenses and Optical Systems for Photography, Film, Video, Electronic and Digital Imaging. Focal Press. p. 40. ISBN 0-240-51540-4.
↑ Kingslake, Rudolph (1992). Optics in Photography. Bellingham, Washington: SPIE Optical Engineering Press. p. 32. ISBN 0-8194-0763-1. "If a lens is thin, or if we can guess at the position of the principal planes, we can readily construct from [ $1/ d i + 1/ d o = 1/ f$ and $M = d i / d o$ ] the following simple rules that it is well to bear in mind. They refer specifically to the case of a positive lens forming a real image of a real object, all distances and the magnification being assumed to be positive quantities. If virtual images are involved, it is better to return to the original formulas, [previously stated]. The equations are [ $d o = f (1 + 1/M)$ and $d i = f (1 + M)$ ]."

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[1] Ray, Sidney F. (2002). Applied Photographic Optics: Lenses and Optical Systems for Photography, Film, Video, Electronic and Digital Imaging. Focal Press. p. 40. ISBN 0-240-51540-4.

[2] Kingslake, Rudolph (1992). Optics in Photography. Bellingham, Washington: SPIE Optical Engineering Press. p. 32. ISBN 0-8194-0763-1. "If a lens is thin, or if we can guess at the position of the principal planes, we can readily construct from [ $1/ d i + 1/ d o = 1/ f$ and $M = d i / d o$ ] the following simple rules that it is well to bear in mind. They refer specifically to the case of a positive lens forming a real image of a real object, all distances and the magnification being assumed to be positive quantities. If virtual images are involved, it is better to return to the original formulas, [previously stated]. The equations are [ $d o = f (1 + 1/M)$ and $d i = f (1 + M)$ ]."

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