Magnifying definition

The size of the airy disk is primarily useful in the context of pixel size. The following interactive tool shows a single airy disk compared to pixel size for several camera models:

Note how most of the lines in the fabric are still resolved at f/11, but have slightly lower small-scale contrast or acutance (particularly where the fabric lines are very close). This is because the airy disks are only partially overlapping, similar to the effect on adjacent rows of alternating black and white airy disks (as shown on the right). By f/22, almost all fine lines have been smoothed out because the airy disks are larger than this detail.

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

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]

Note: CF = "crop factor" (commonly referred to as the focal length multiplier); assumes square pixels, 4:3 aspect ratio for compact digital and 3:2 for SLR. *Calculator assumes that your camera sensor uses the typical bayer array.

where f {\textstyle f} is the focal length, d o {\textstyle d_{\mathrm {o} }} is the distance from the lens to the object, and x 0 = d 0 − f {\textstyle x_{0}=d_{0}-f} as the distance of the object with respect to the front focal point. A sign convention is used such that d 0 {\textstyle d_{0}} and d i {\displaystyle d_{i}} (the image distance from the lens) are positive for real object and image, respectively, and negative for virtual object and images, respectively. f {\textstyle f} of a converging lens is positive while for a diverging lens it is negative.

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

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Here, f {\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.

Camera Type Digital SLR with CF of 1.6X Digital SLR with CF of 1.5X Digital SLR with CF of 1.3X Digital SLR with 4/3" sensor 35 mm (full frame) Digital compact with 1/3" sensor Digital compact with 1/2.3" sensor Digital compact with 1/2" sensor Digital compact with 1/1.8" sensor Digital compact with 2/3" sensor Digital compact with a 1" sensor APS 6x4.5 cm 6x6 cm 6x7 cm 5x4 inch 10x8 inch

M = − d i d o = h i h o {\displaystyle M=-{d_{\mathrm {i} } \over d_{\mathrm {o} }}={h_{\mathrm {i} } \over h_{\mathrm {o} }}}

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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.

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

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°.

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

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

Even when a camera system is near or just past its diffraction limit, other factors such as focus accuracy, motion blur and imperfect lenses are likely to be more significant. Diffraction therefore limits total sharpness only when using a sturdy tripod, mirror lock-up and a very high quality lens.

Some diffraction is often ok if you are willing to sacrifice sharpness at the focal plane in exchange for sharpness outside the depth of field. Alternatively, very small apertures may be required to achieve sufficiently long exposures, such as to induce motion blur with flowing water. In other words, diffraction is just something to be aware of when choosing your exposure settings, similar to how one would balance other trade-offs such as noise (ISO) vs shutter speed.

Light rays passing through a small aperture will begin to diverge and interfere with one another. This becomes more significant as the size of the aperture decreases relative to the wavelength of light passing through, but occurs to some extent for any aperture or concentrated light source.

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

The image magnification along the optical axis direction M L {\displaystyle M_{L}} , called longitudinal magnification, can also be defined. The Newtonian lens equation is stated as f 2 = x 0 x i {\displaystyle f^{2}=x_{0}x_{i}} , where x 0 = d 0 − f {\textstyle x_{0}=d_{0}-f} and x i = d i − f {\textstyle x_{i}=d_{i}-f} as on-axis distances of an object and the image with respect to respective focal points, respectively. M L {\displaystyle M_{L}} is defined as

For additional reading on this topic, also see the addendum: Digital Camera Diffraction, Part 2: Resolution, Color & Micro-Contrast

In practice, the diffraction limit doesn't necessarily bring about an abrupt change; there is actually a gradual transition between when diffraction is and is not visible. Furthermore, this limit is only a best-case scenario when using an otherwise perfect lens; real-world results may vary.

Are smaller pixels somehow worse? Not necessarily. Just because the diffraction limit has been reached (with large pixels) does not necessarily mean an image is any worse than if smaller pixels had been used (and the limit was surpassed); both scenarios still have the same total resolution (even though the smaller pixels produce a larger file). However, the camera with the smaller pixels will render the photo with fewer artifacts (such as color moiré and aliasing). Smaller pixels also give more creative flexibility, since these can yield a higher resolution if using a larger aperture is possible (such as when the depth of field can be shallow). On the other hand, when other factors such as noise and dynamic range are considered, the "small vs. large" pixels debate becomes more complicated...

When the diameter of the airy disk's central peak becomes large relative to the pixel size in the camera (or maximum tolerable circle of confusion), it begins to have a visual impact on the image. Once two airy disks become any closer than half their width, they are also no longer resolvable (Rayleigh criterion).

Technical Note: Independence of Focal Length Since the physical size of an aperture is larger for telephoto lenses (f/4 has a 50 mm diameter at 200 mm, but only a 25 mm diameter at 100 mm), why doesn't the airy disk become smaller? This is because longer focal lengths also cause light to travel farther before hitting the camera sensor -- thus increasing the distance over which the airy disk can continue to diverge. The competing effects of larger aperture and longer focal length therefore cancel, leaving only the f-number as being important (which describes focal length relative to aperture size).

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

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]

For an ideal circular aperture, the 2-D diffraction pattern is called an "airy disk," after its discoverer George Airy. The width of the airy disk is used to define the theoretical maximum resolution for an optical system (defined as the diameter of the first dark circle).

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

This should not lead you to think that "larger apertures are better," even though very small apertures create a soft image; most lenses are also quite soft when used wide open (at the largest aperture available). Camera systems typically have an optimal aperture in between the largest and smallest settings; with most lenses, optimal sharpness is often close to the diffraction limit, but with some lenses this may even occur prior to the diffraction limit. These calculations only show when diffraction becomes significant, not necessarily the location of optimum sharpness (see camera lens quality: MTF, resolution & contrast for more on this).

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The longitudinal magnification is always negative, means that, the object and the image move toward the same direction along the optical axis. The longitudinal magnification varies much faster than the transverse magnification, so the 3-dimensional image is distorted.

Diffraction thus sets a fundamental resolution limit that is independent of the number of megapixels, or the size of the film format. It depends only on the f-number of your lens, and on the wavelength of light being imaged. One can think of it as the smallest theoretical "pixel" of detail in photography. Furthermore, the onset of diffraction is gradual; prior to limiting resolution, it can still reduce small-scale contrast by causing airy disks to partially overlap.

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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 hi 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]

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.

With any telescope, microscope or 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".

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.

M A = tan ⁡ ε tan ⁡ ε 0 ≈ ε ε 0 {\displaystyle M_{A}={\frac {\tan \varepsilon }{\tan \varepsilon _{0}}}\approx {\frac {\varepsilon }{\varepsilon _{0}}}}

Since the size of the airy disk also depends on the wavelength of light, each of the three primary colors will reach its diffraction limit at a different aperture. The calculation above assumes light in the middle of the visible spectrum (~550 nm). Typical digital SLR cameras can capture light with a wavelength of anywhere from 450 to 680 nm, so at best the airy disk would have a diameter of 80% the size shown above (for pure blue light).

As two examples, the Canon EOS 20D begins to show diffraction at around f/11, whereas the Canon PowerShot G6 begins to show its effects at only about f/5.6. On the other hand, the Canon G6 does not require apertures as small as the 20D in order to achieve the same depth of field (due to its much smaller sensor size).

Camera Canon EOS 1Ds Canon EOS 1Ds Mk II Canon EOS 1Ds Mk III, 5D Mk II Canon EOS 1D Canon EOS 1D Mk II Canon EOS 1D Mk III Canon EOS 1D Mk IV Canon EOS 1D X Canon EOS 5D Canon EOS 5D Mk III Canon EOS 7D,60D,550D,600D,650D,1D C Canon EOS 50D, 500D Canon EOS 40D, 400D, 1000D Canon EOS 30D, 20D, 350D Canon EOS 1100D Canon PowerShot G1 X Canon PowerShot G11, G12, S95 Canon PowerShot G9, S100 Canon PowerShot G6 Nikon D3, D3S / D700 Nikon D40, D50, D70 Nikon D4 Nikon D60, D80, D3000 Nikon D3X Nikon D2X, D90, D300, D5000 Nikon D800 Nikon D5100, D7000 Sony SLT-A65, SLT-A77, NEX-7 Sony DSC-RX100

Airy Diameter: 21.3 µm Camera Canon EOS 1Ds Canon EOS 1Ds Mk II Canon EOS 1Ds Mk III, 5D Mk II Canon EOS 1D Canon EOS 1D Mk II Canon EOS 1D Mk III Canon EOS 1D Mk IV Canon EOS 1D X Canon EOS 5D Canon EOS 5D Mk III Canon EOS 7D,60D,550D,600D,650D,1D C Canon EOS 50D, 500D Canon EOS 40D, 400D, 1000D Canon EOS 30D, 20D, 350D Canon EOS 1100D Canon PowerShot G1 X Canon PowerShot G11, G12, S95 Canon PowerShot G9, S100 Canon PowerShot G6 Nikon D3, D3S / D700 Nikon D40, D50, D70 Nikon D4 Nikon D60, D80, D3000 Nikon D3X Nikon D2X, D90, D300, D5000 Nikon D800 Nikon D5100, D7000 Sony SLT-A65, SLT-A77, NEX-7 Sony DSC-RX100 Pixel Diameter: 6.9 µm

Although the above diagrams help give a feel for the concept of diffraction, only real-world photography can show its visual impact. The following series of images were taken on the Canon EOS 20D, which typically exhibits softening from diffraction beyond about f/11. Move your mouse over each f-number to see how these impact fine detail:

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M A = 1 M = D O b j e c t i v e D R a m s d e n . {\displaystyle M_{\mathrm {A} }={1 \over M}={D_{\mathrm {Objective} } \over {D_{\mathrm {Ramsden} }}}\,.}

Typically, magnification is related to scaling up visuals or images to be able to see more detail, increasing resolution, using microscope, printing techniques, or digital processing. In all cases, the magnification of the image does not change the perspective of the image.

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Another complication is that sensors utilizing a Bayer array allocate twice the fraction of pixels to green as red or blue light, and then interpolate these colors to produce the final full color image. This means that as the diffraction limit is approached, the first signs will be a loss of resolution in green and pixel-level luminosity. Blue light requires the smallest apertures (highest f-stop) in order to reduce its resolution due to diffraction.

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.

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.

This calculator shows a camera as being diffraction limited when the diameter of the airy disk exceeds what is typically resolvable in an 8x10 inch print viewed from one foot. Click "show advanced" to change the criteria for reaching this limit. The "set circle of confusion based on pixels" checkbox indicates when diffraction is likely to become visible on a computer at 100% scale. For a further explanation of each input setting, also see the depth of field calculator.

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:

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

The form below calculates the size of the airy disk and assesses whether the camera has become diffraction limited. Click on "show advanced" to define a custom circle of confusion (CoC), or to see the influence of pixel size.

Note: above airy disk will appear narrower than its specified diameter (since this is defined by where it reaches its first minimum instead of by the visible inner bright region).

For real images, M {\textstyle M} is negative and the image is inverted. For virtual images, M {\textstyle M} is positive and the image is upright.

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.

M = d i d o = h i h o = f d o − f = d i − f f {\displaystyle {\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}}}

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Since the divergent rays now travel different distances, some move out of phase and begin to interfere with each other — adding in some places and partially or completely canceling out in others. This interference produces a diffraction pattern with peak intensities where the amplitude of the light waves add, and less light where they subtract. If one were to measure the intensity of light reaching each position on a line, the measurements would appear as bands similar to those shown below.

Diffraction is an optical effect which limits the total resolution of your photography — no matter how many megapixels your camera may have. It happens because light begins to disperse or "diffract" when passing through a small opening (such as your camera's aperture). This effect is normally negligible, since smaller apertures often improve sharpness by minimizing lens aberrations. However, for sufficiently small apertures, this strategy becomes counterproductive — at which point your camera is said to have become diffraction limited. Knowing this limit can help maximize detail, and avoid an unnecessarily long exposure or high ISO speed.

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.

As a result of the sensor's anti-aliasing filter (and the Rayleigh criterion above), an airy disk can have a diameter of about 2-3 pixels before diffraction limits resolution (assuming an otherwise perfect lens). However, diffraction will likely have a visual impact prior to reaching this diameter.

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