› Modulation Transfer Function (MTF) Simulator Back
Optics

Modulation Transfer Function (MTF) Simulator

Calculate how faithfully a lens transmits the contrast of a scene. Adjust the spatial frequency, aperture, wavelength and sensor pixel pitch to see the diffraction-limited modulation transfer function (MTF), the diffraction cutoff frequency and the contrast transferred at the sensor Nyquist frequency update in real time, and evaluate lens sharpness objectively.

Parameters
Evaluation spatial frequency f
cyc/mm
Fineness of detail at which the MTF is evaluated. Higher = finer bars
Aperture (f-number) N
F-number. The more you stop down (larger N), the lower the cutoff
Wavelength λ
nm
Wavelength of light. The middle of the visible band (green) is about 550 nm
Sensor pixel pitch p
µm
Spacing of adjacent pixels. Sets the Nyquist frequency
Results
—
Diffraction cutoff frequency (cyc/mm)
—
Normalised frequency s
—
MTF (contrast transferred at this frequency, %)
—
Sensor Nyquist frequency (cyc/mm)
—
MTF at the Nyquist frequency (%)
—
Resolution verdict
—
Test target — visualising the contrast loss through the lens

The top band is an ideal sine-wave test target whose spatial frequency rises from left to right; the bottom band is the same target imaged by the lens. Contrast fades out toward the high-frequency (right) side, going to flat grey past the cutoff. The MTF curve and the sweeping evaluation-frequency marker are overlaid below.

MTF curve — contrast transferred vs spatial frequency
Diffraction cutoff frequency vs f-number
Theory & Key Formulas

$$\text{MTF}(s)=\frac{2}{\pi}\left(\arccos s-s\sqrt{1-s^2}\right),\qquad s=\frac{f}{f_{cutoff}},\quad f_{cutoff}=\frac{1}{\lambda\,N}$$

Diffraction-limited MTF for a circular aperture (incoherent imaging). s: normalised spatial frequency (f: evaluation frequency, f_cutoff: diffraction cutoff frequency). N is the f-number and λ is the wavelength; the lens transmits zero contrast above the cutoff f_cutoff (MTF = 0 for s > 1).

$$f_{Nyquist}=\frac{1}{2\,p}$$

Sensor Nyquist frequency. p is the pixel pitch; this is the finest spatial frequency the pixel grid can sample. The lens MTF at this frequency is the yardstick for how well lens and sensor are matched.

What is the Modulation Transfer Function (MTF)?

🙋
I keep seeing an "MTF" graph in lens datasheets — what does it actually show? Isn't it just resolution?
🎓
Good question. Roughly speaking, it is a graph of "how faithfully the lens transmits contrast". A broad area of light and dark — say the edge between a white wall and a black door — comes through with nearly its full contrast. But as the pattern of the scene gets finer and finer, the lens can only render it with weaker and weaker contrast, until beyond a certain fineness the lines blur together into uniform grey. The MTF plots that transmitted fraction of contrast against how fine the detail is — the spatial frequency.
🙋
I see. So how is that different from a number like "resolves 100 line-pairs/mm"?
🎓
That's the key point. A "how many lines can it resolve" number is essentially one pass/fail data point — it says nothing about how much contrast was left just before that. The MTF tells you how crisply detail is resolved across every frequency. Two lenses that both "resolve" 50 lp/mm can be totally different: one at 80% MTF, crisp and clear; the other at 20%, barely visible. That is why the MTF is called the most rigorous and objective measure of sharpness.
🙋
But a really good lens should image fine patterns crisply, right? Build a perfect lens and the MTF could stay at 100%.
🎓
Here is the interesting part — the answer is no. Even a flawless lens with zero aberrations whatsoever is limited by diffraction, because light is a wave. Light passing through a round aperture spreads out and cannot form a true point in the image plane; it forms a small disc instead. As a result, contrast finer than the diffraction cutoff frequency f_cutoff = 1/(λ·N) simply cannot be transmitted at all. That is the diffraction limit — a wall of physics no amount of money can break through.
🙋
So when I stop the aperture down — raise the f-number — how does that cutoff change? People stop down for depth of field all the time.
🎓
There's a famous trade-off there. Look at f_cutoff = 1/(λ·N): the larger N is — that is, the more you stop down — the lower the cutoff. Depth of field gets deeper, but fine-detail contrast keeps falling. This is the diffraction limit every photographer eventually meets. Most lenses show their highest MTF around f/5.6 to f/8, and by f/16 or f/22 the in-focus zone widens but the whole image softens. Drag the slider on the cutoff vs f-number chart below and you'll see that drop.
🙋
You can also enter the sensor pixel pitch. This is about the lens — why does the sensor come into it?
🎓
Because a good system comes down to matching the lens and the sensor. The finest frequency the pixel grid can sample is the "Nyquist frequency", f_Nyquist = 1/(2p). If the lens still has a useful MTF at that Nyquist frequency, it can exploit every pixel. If the MTF is essentially zero there, adding more pixels is wasted. Improving only one side is pointless — the sharpness you actually capture is decided by the balance of both.

Frequently Asked Questions

The MTF (Modulation Transfer Function) describes how faithfully an optical system reproduces contrast, expressed as a function of how fine the detail is — the spatial frequency. Broad areas of light and dark come through at almost full contrast, but as the pattern gets finer the contrast weakens, and beyond a certain frequency the lines blur into uniform grey. The MTF plots that transmitted fraction of contrast against spatial frequency (cycles/mm). It is the single most rigorous, objective measure of sharpness because it tells you not just whether detail is resolved but how crisply — far more informative than a resolution number alone.
Because light is a wave, even a flawless lens with zero aberrations is limited by diffraction. Light passing through a circular aperture spreads out and cannot form a true point in the image plane — it forms a small disc (the Airy disc) instead. As a result, the lens transmits exactly zero contrast for detail finer than the diffraction cutoff frequency f_cutoff = 1/(λ·N), where λ is the wavelength and N is the f-number. This diffraction limit is a hard physical wall that sets the ceiling for lens design and aperture choice.
The more you stop down, the lower the diffraction cutoff frequency f_cutoff = 1/(λ·N) becomes, and the less fine-detail contrast the lens can transmit. Stopping down for depth of field softens fine detail in return — this is the diffraction limit every photographer eventually meets. Balancing aberrations against diffraction, most lenses show their highest MTF around f/5.6 to f/8; by f/16 or f/22 the depth of field grows but overall sharpness actually drops. The tool's cutoff-frequency vs f-number chart shows this fall-off.
The Nyquist frequency f_Nyquist = 1/(2·pixel pitch) is the finest spatial frequency the pixel grid can sample. If the lens does not still deliver useful contrast (MTF) at that frequency, you cannot exploit the sensor's resolution. Conversely, if the lens MTF stays high well beyond Nyquist, the high-frequency content that cannot be sampled shows up as moiré and false colour (aliasing). To match lens and sensor well, you size the pixel pitch and aperture with the MTF at the Nyquist frequency in mind.

Real-World Applications

Evaluating cameras and interchangeable lenses: The MTF curves printed in lens-maker datasheets are the most trustworthy objective data for choosing a product. The MTF at low spatial frequency (10 lp/mm) reflects overall image "punch" and contrast, while the MTF at high spatial frequency (30-50 lp/mm) reflects fine-detail resolving power. Computing the diffraction-limited MTF, as this tool does, gives a starting point for estimating how far a lens sits from the design ideal and which aperture is sharpest.

Industrial cameras and machine vision: In visual inspection and dimensional measurement, the required spatial frequency is worked back from the smallest defect to detect, and you check that the lens still has a sufficient MTF there. Sweeping the illumination wavelength, aperture and sensor pixel pitch to find a lens-and-aperture combination whose diffraction cutoff sits well above the inspection frequency, with adequate MTF at the Nyquist frequency, is exactly the calculation this tool performs.

Microscopy and semiconductor lithography: In microscope objectives and the projection lenses of exposure tools, the diffraction limit directly governs resolution. Lithography prints ever-finer patterns by shortening the wavelength λ (deep- and extreme-ultraviolet) and raising the numerical aperture to lift the cutoff frequency — the MTF concept is itself the roadmap of miniaturisation. The relationship that a shorter wavelength raises the cutoff can be felt directly with the wavelength slider here.

Astronomical telescopes and observation instruments: Under ideal conditions with no atmospheric turbulence, a telescope's resolution is set by the diffraction limit of its aperture. A larger aperture (effectively a smaller f-number) raises the cutoff frequency and lets finer celestial structure be separated. The MTF is also used to estimate the limits of sharpening (deconvolution) in image processing, underpinning the principle that high-frequency content the sensor never captured cannot be recovered in post.

Common Misconceptions and Pitfalls

The most common assumption is that "higher resolution (more resolved lines) means a better MTF". The resolved-line figure is a single value — how many lines can still be told apart just before contrast falls to zero — and says nothing about how much contrast remained before that point. A lens whose MTF stays gently high and a lens that crawls along just above the resolution limit can post the very same "resolved-line" number yet look completely different in perceived sharpness. In practice, compare the MTF value at the spatial frequency you actually use (say 30 lp/mm) rather than the resolution-limit number.

Next is the belief that "the more you stop down, the sharper it gets". The diffraction-limited MTF this tool computes assumes zero aberrations, so the MTF degrades monotonically as you stop down and f_cutoff drops. A real lens adds aberrations on top: it is soft near wide open from aberrations, soft when stopped down too far from diffraction, and sharpest somewhere in between (often f/5.6 to f/8). Stopping down to f/16 or f/22 "for more depth of field" widens the in-focus zone but reliably lowers overall sharpness. Deep DOF and high MTF are a trade-off.

Finally, the misconception that "the MTF is determined by the lens alone". The MTF of the image a user finally sees is the product of the MTFs of the lens, the sensor (pixel pitch and low-pass filter) and the processing pipeline (sharpening). Even if the lens MTF is high at Nyquist, a coarse pixel pitch cannot sample it and moiré appears; processing sharpness lifts apparent contrast but cannot recover information the lens never transmitted (above the cutoff). The MTF is a whole-system quantity — do not judge final image quality from the lens figures alone.

How to Use

  1. Enter spatial frequency (cyc/mm) in freqNum or drag freqRange slider to set the object detail scale you want to evaluate.
  2. Set lens f-number (fnumNum) and wavelength (wlNum, typically 550 nm for green light) to define diffraction-limited performance.
  3. Input pixel pitch (ppNum in micrometers) to calculate normalized frequency relative to sensor Nyquist limit and read MTF percentage output.
  4. Compare MTF at Nyquist against diffraction cutoff frequency—values below 5% indicate aliasing risk on the sensor.

Worked Example

A 50 mm lens at f/2.8 imaging at 550 nm wavelength yields diffraction cutoff of approximately 0.774 cyc/mm. With a 4.0 micrometer pixel pitch (Nyquist = 125 cyc/mm), evaluate a scene detail at 30 cyc/mm: normalized frequency s = 30/0.774 = 38.8, producing MTF ≈ 18%. At sensor Nyquist (125 cyc/mm), MTF drops to 2%, triggering an aliasing warning for fine textures like fabric weave.

Practical Notes

  1. Smaller f-numbers (f/1.4) push diffraction cutoff higher, preserving contrast in high-frequency detail; stopped down to f/8, cutoff halves, softening texture rendition.
  2. Shorter wavelengths (405 nm UV) improve MTF at identical f-number and frequency versus 650 nm red; use for inspection optics.
  3. Pixel pitch trade-off: 2.4 micrometer pixels raise Nyquist to 208 cyc/mm but introduce photon noise; 6.0 micrometer sensors reduce Nyquist to 83 cyc/mm but gain SNR in surveillance.
  4. MTF below 10% at operational frequencies indicates unsharp masking or deconvolution post-processing is needed to recover edge definition.