Magnificationformula for lens

The perceived magnification of an object, thanks to the use of powerful telephoto lenses, comes from the reduced projection of the object onto relatively small sensors. If that projected image can change the size, let's say by a factor of two, we say that the lens has a 2× zoom.

To explain how the MTF is computed, first we need to define the Point Spread Function (PSF), the transfer function (TF), and their relationship with the MTF. The Point Spread Function (PSF) of an imaging system is the resulting irradiance distribution when the object is a point source. For coherent light, the PSF is related to the Fourier transform of the pupil function’s complex amplitude. For incoherent light, the PSF is related to the Fourier transform of the pupil function’s intensity. Therefore, the coherent and incoherent PSF relate as follows:

Figure 6 shows the MTF chart from a lens that would meet these requirements for all fields. It is important to know the sensor characteristics during the design process since in many cases the Nyquist frequency of the sensor is only a small fraction of the diffraction cutoff frequency fc.

Holds a Ph.D. degree in mathematics obtained at Jagiellonian University. An active researcher in the field of quantum information and a lecturer with 8+ years of teaching experience. Passionate about everything connected with maths in particular and science in general. Fond of coding, she has a keen eye for detail and perfectionistic leanings. A bookworm, an avid language learner, and a sluggish hiker. She is hopelessly addicted to coffee & chocolate – the darker and bitterer, the better. See full profile

A lens is a device made of a material with a different refraction index to air (there can even be electromagnetic lenses that act on electric currents). This and its shape allows it to bend rays of light as they come into contact with it.

Maybe you expected the magnification to be a bigger number, something like 10×10\times10× or 20×20\times20×, like the values you see on binoculars or telescopes (we made an entire calculator for that, check out our telescope magnification calculator).

🔎 The word "focus" comes from Latin for "fireplace". This is because the Romans believed that their ancestral gods were located in the fireplace, or hearth, and so would direct (or focus) their worship towards it.

Their lenses are usually manufactured with a focal length of 25 cm25\ \text{cm}25 cm. If you use the lens to look at an object closer to it than that distance, you create a virtual image of the object.

Figure 2: On-axis MTF plot of an optical system compared to the diffraction-limited MTF plot. The output image depicts excellent contrast at low frequencies and poor contrast at high frequencies, mimicking the MTF plot.

Therefore, the MTF can be computed by calculating the complex autocorrelation of the pupil function, or by computing the Fourier transform of the incoherent PSF. Using the definition of the MTF, the OTF can be written in complex form as:

It is sometimes useful to measure through-focus MTF, which shows the MTF for selected frequencies as a function of defocus position. Another informative MTF chart is MTF vs field points, which shows the MTF for selected frequencies as a function of field points. Examples of these plots are shown in Figure 4.

The Modulation Transfer Function (MTF) is a performance metric that measures the degradation of contrast in the image compared to contrast in the object. As contrast degrades, distinguishing small features in the image becomes increasingly harder for an observer. There are several factors that cause contrast degradation, such as diffraction effects, optical aberrations, and vignetting. Therefore, the MTF can be used to evaluate and compare optical systems.

Lens formula

The zoom describes how much the lens's focal length can change by (there are such things as zoom lenses). A typical 18/5518/5518/55 lens will have its zoom defined by:

The incoherent transfer function of a linear, shift-invariant system is given by the Fourier transform of the incoherent point spread function:

The magnification of a lens is an absolute measure of how much the height of a real image differs from the object's height. Remember, that in a camera, the real image forms on the sensor (or on the film, if you're old school).

In photography, the magnification of a lens is the ratio between the height of the image projected onto the sensor or film of the camera and the height of the real image you are taking a picture of.

How tocalculatemagnificationof a drawing

Another useful consideration is that for small aberrations, the degradation of the MTF occurs near half the diffraction cutoff frequency. This is because the MTF at zero frequency is always unity and the MTF at the diffraction cutoff frequency is zero. If we measure the aberrated MTF as a ratio of the aberration-free MTF, the result is a function that sags down in the middle.

The values of hhh and ggg are hidden in the further magnification properties section of our calculator, so if you need to know either of these, just click the button!

To determine the system MTF, a sinusoidal pattern with perfect contrast is used as the object. Image contrast is defined as:

The MTF is a function of a 2D spatial-frequency vector, but it is usually presented as a 1D plot. In this type of representation, it is useful to compare the response to that of a perfect system. In an aberration-free system, the contrast is only degraded by diffraction effects. The dotted line in Figure 2 shows the MTF for the aberration-free system, which is labeled as “Diffraction Limit”.

In the case of violent kangaroos, it may be better to go for the second option: that's why camera manufacturers sell extension tubes, short rings to mount between the lens and the body, which end up increasing hhh by some precious millimeters.

Experimentally, there are different ways to measure the frequency response of an optical system. Two examples are the slanted-edge target and the three-bar target. These targets are easy to use since they just need to be imaged. The three-bar target is a common method to measure resolution, it consists of three parallel bars of certain width and separation that corresponds to a specific spatial frequency. To measure system resolution, multiple three-bar patterns at specific frequencies are arranged in a single chart, and the resulting image is inspected to determine the smallest features that can be resolved. A popular target with these characteristics is the USAF 1951 chart shown in Figure 7. The measured resolution from a three-bar target might differ from the predicted CTF, since the CTF assumes an infinitely long pattern of bars. In a pattern with a finite number of bars, there can be end effects that reduce the contrast of the bars at the ends of the pattern. For this reason, the measured contrast might be less than the predicted by the CTF.

For example, assume a digital sensor with a pixel size of 7.4 μm x 7.4 μm. According to the Nyquist theorem, the highest frequency that can be resolved is 1 cycle/(2 x 7.4 um) = 0.0676 cycles/μm ≈ 68 cycles/mm.  Based on the sensor characteristics, a typical performance requirement for the lens could be:

The optical designer doesn’t necessarily aim to achieve the diffraction limit performance. The desired MTF curve is based on design requirements. The lens specifications are usually of the form of an MTF value for specific frequencies. The MTF specification may come, for instance, from the sensor pixel size and the Nyquist theorem, which describes the maximum spatial frequency that the sensor can resolve.

where is the phase. The MTF can only take positive values, but the OTF can be negative when there is phase reversal. For the image of the bar pattern, this would result in contrast reversal, meaning that white features become dark and dark features become light. In an MTF plot, the behavior is that the MTF curve reaches zero and then “bounces”. It is important to be aware of this behavior. After the bounce, the MTF technically increases even though it reached zero at a lower frequency. Another possible impact is that if the MTF is used in the error function during optimization, phase reversal may produce local minima.

Figure 1 shows the response of the system to a single frequency, however, the MTF measures the response for continuous frequencies up to the cutoff frequency fc. The cutoff frequency, which is the highest frequency the system can resolve, is given by:

How to compute magnificationin physics

Since that beast would be too dangerous to photograph at a close distance, we suggest you use a 500 mm500\ \text{mm}500 mm telephoto lens. We advise you to keep your distance, let's say 150 meters (but remember that a kangaroo can reach a maximum speed of 70 m/s70\ \text{m/s}70 m/s). Insert these values into our magnification of a lens calculator, which will return:

A camera is nothing but lenses and a sensor. At least in theory! To understand how it works, we need to explore the world of optics.

Other common system specifications are Strehl ratio and RMS wavefront error. For small aberrations, specifying the Strehl ratio is equivalent to specifying the RMS wavefront error. The Strehl ratio is defined as the ratio of the peak intensity of the measured PSF to the peak intensity of the perfect PSF. The MTF is related to the inverse Fourier transform of the incoherent PSF. As a result, it is possible to express the Strehl ratio as the integral under the entire MTF curve, including frequencies beyond the Nyquist frequency (which are irrelevant). For this reason, in situations where Nyquist is well below the diffraction cutoff, it makes more sense to specify MTF at frequencies between zero and the Nyquist frequency, rather than specifying the Strehl ratio or RMS wavefront error.

When you are snapping a picture, you don't usually know the values of hhh and ggg, but you know the focal length for sure, and you likely know the distance between you and your subject. These two quantities are enough for you to calculate the magnification of your lens!

The magnification of a lens with focal length 55 mm at a distance of 100 m is m = 0.0005506. To calculate it, follow the steps:

So far, we have described sine-wave targets that represent a single frequency. The response of the system to a square-wave pattern is called Contrast Transfer Function (CTF). The CTF may be a more appropriate performance metric for systems that image objects with square-wave features, such as barcode readers. Computationally, the CTF can be approximated by summing a series of sine wave response values.

The magnification of a lens is an absolute value that depends on the focal length of the lens itself, while the zoom is a relative quantity that describes how much you can change the focal length of a lens by, thus changing its magnification.

where Imax and Imin  are maximum and minimum irradiances, respectively. The maximum value the contrast can take is 1, and the minimum value is 0. A cycle of the sinusoidal pattern corresponds to a black and a white fringe. Eash cycle is also referred to as a line pair (lp). Spatial frequency is often measured in line pairs or cycles per unit distance, which is typically mm. Thus, the spatial frequency unit is commonly represented as lp/mm or cycles/mm. For afocal systems, the units are converted to angular frequencies, typically measured in cycles/milliradian.

Magnificationformula for mirror

How tocalculatemagnificationmicroscope

However, when talking cameras, the magnification is usually a really small number. The number followed by a ×\times× is the zoom.

Now consider that the sensor is at most a few centimeters wide, while you can take a picture of the Eiffel Tower, which is 330 m330\ \text{m}330 m tall. Even from afar and with a powerful telephoto lens, you'll always get a magnification that is much smaller than you expect when taking pictures with a camera.

Lenses can focus or "unfocus" light rays. In this tool, we will only consider converging lenses. Their main feature is the ability to focus every ray entering the lens parallel to the optical axis at a specific point, the focus.

Physicist by education, scientist by vocation. He holds a master’s degree in complex systems physics with a specialization in quantum technologies. If he is not reading something, he is outdoors trying to enjoy every bit of nature around him. He uses his memory as an advantage, and everything you will read from him contains at least one “I read about this five years ago” moment! See full profile

Another method to experimentally measure the MTF is the slanted-edge target. This method is different in that frequency information can be obtained from a single target. The edge must be at a specified small angle with respect to the pixel array of the sensor, as shown in Figure 7. When the input is a step function instead of a point source, the irradiance distribution at the image is an edge spread function. The MTF can then be determined from the Fourier transform of the derivative of the edge spread function.

Thanks to the properties of similar triangles, we can compute the magnification of a lens also using the distances between the object/image and the lens:

Our lens magnification calculator will focus on the world of lenses in photography, finally explaining what magnification is, why it is different from zoom, and much more!

For off-axis field points, the MTF result depends on the orientation of the bar pattern. It is common to show the tangential (T) and radial directions (R) for each field point, as shown in Figure 3. In a tangential target, the bars are tangent to the image circle, and in a radial target, the bars are parallel to the radial direction.

If you'd like to know more, try our others optic calculators dedicated to lenses, like the thin lens equation calculator or the lens maker equation calculator.

Since, in most cases (unless you are using a microscope), the lens shrinks the object, the magnification value is less than 1.

How to compute magnificationof a microscope

CODE V is a powerful optical design tool. Its advanced analysis tools include the computation of the MTF by evaluating the autocorrelation of the pupil function, and the computation of the CTF by calculating a series summation of sine wave response values. In CODE V, MTF metrics are used throughout the design process for analyzing, optimizing, and tolerancing optical systems.

First thing – the upward facing arrow on the left of the image is the object we are looking at. The rays of light coming from it hit the lens. The one parallel to the optic axis (the topmost line) gets focused and so converges on the focus. The ray passing through the center of the lens meets the focused ray on the other side of the lens, which creates a flipped image called the real image of the object.

Expand the further magnification properties section to see the variable extension tube. We set it at 0 mm0\ \text{mm}0 mm by default, but change it according to your needs!

Magnificationformula Biology

As you can see, now the rays on the right side of the lens do not converge. We are dealing in terms of virtual images, which originate from the virtual continuations of the rays, creating a non reversed image of the object.

Imagine you are taking a picture of a huge kangaroo, let's say two meters tall and weighing 95 kg95\ \text{kg}95 kg, like the one that terrorized Brisbane a few years ago.

Figure 3: Off-axis MTF plots for the tangential and radial directions. In CODE V, the target bars are relative to the Y axis. Radial MTF corresponds to vertical bars and tangential MTF corresponds to horizontal bars.

🔎 Lenses and their properties have been known by humanity for a long time. However, only in the 13ᵗʰ century did lens-making skills reach a level of refinement that allowed for the construction of glasses, telescopes, and much more!