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The modulation transfer function (MTF) is an optical bench measurement used by engineers to evaluate the performance of a lens or a lens system. In its most basic sense, the MTF is a method for describing the contrast sensitivity of a lens system. For the human eye, this could be thought of as its “visual performance.”
Modulation transfer is the ability of a lens system to transfer an object’s contrast to its image. Modulation is therefore defined as the ratio of image contrast to object contrast. Ideally, it would be one or 100%. Modulation transfer plots describe the modulation of a lens system as the object’s complexity increases. Therefore, the Y-axis represents modulation, and the X-axis represents spatial frequency, measured in line pairs per millimeter. As you would expect, as the spatial frequency increases, the modulation of any lens system decreases. Outside of optical engineering, most are unfamiliar with the importance of MTF because there are no easy ways to standardize it.
The most common way that MTF is explained is as an analogy to sound. Just as in optical imaging, audio recordings do not perfectly duplicate the original. A sound consists of many individual frequencies, or pure tones, simultaneously reaching the ear. Two parameters characterize a pure tone: the frequency, or tone, and the loudness, or volume. A sine wave typically represents a pure tone. The horizontal distance between peaks determines the frequency, and the vertical distance from peak to valley determines the volume. Of course, most sounds are a mixture of many hundreds of different frequencies, each with its own volume. This would be a complex sound. In much the same way, an optical image is composed of many spatial frequencies and varying amounts of contrast.
When a sound is recorded, each component frequency gets recorded, but the process invariably changes the volume of each frequency. The playback is a combination of tones that constitute the original sound, but typically at varying volumes. This change in volume for a specific tone causes the recorded sound to differ from the original. Likewise, when an image is projected or recorded, the contrast typically differs from the original object by small or large amounts.
If a single pure tone is recorded, the frequency of the recording matches the frequency of the original, but the recorded volume usually changes. The ratio of the recorded volume to the original volume would be the measured response of an audio system to that particular frequency. If an audio system has different responses to different frequencies, the recorded sound will not match the original. However, if the audio system has the same response to all frequencies, the recorded sound will duplicate the original sound. A graph of response vs. frequency, known as a frequency response curve, is typically constructed to show the fidelity, or frequency response, of any audio system. High-end audiophiles spend much of their time (and most of their money) chasing this.
Optical images of any kind can be analyzed similarly. The tricky part here is that the optical analogy of a pure tone is a sine wave grating, or SWG. The horizontal peak-to-peak distance determines the frequency of the sine wave grating. The sine wave grating contrast is indicated by the difference in brightness between the brightest and darkest points, and is analogous to the volume of a tone. One difference between sound and optics is that SWGs also have an orientation, which can be vertical, horizontal, or oblique.
Unlike a sine wave grating, which gradually changes from dark to light, another pattern, known as Ronchi rulings, changes abruptly and is instead based on a square wave rather than a sine wave. Snellen figures (the standard visual acuity eye chart) are essentially Ronchi rulings. This is why Snellen acuity is a poor measure of visual performance. Contrast sensitivity testing, using charts that are essentially SWGs, provides a more comprehensive evaluation of visual performance; however, this test is more complicated to administer and is poorly understood by those outside of optics and ophthalmology, such as insurance companies.
For Ronchi rulings, there are basically dark bars and light bars, and we can measure the amount of light coming from each. The maximum amount of light will come from the light bars and the minimum from the dark bars. If the light in a lens system is measured in terms of transmittance (T), we can define modulation according to the following equation:
- Modulation = Mc = (Tmax – Tmin ) / (Tmax + Tmin)
Where Tmax is the maximum transmittance of the grating and Tmin is the minimum transmittance. When we examine the ratio of transmission between the light and dark bars, we are measuring contrast. We can look at a sine wave grating in the same manner.
Now, let’s assume that you have a sine wave grating of a specific frequency (u) and modulation (contrast), and its image is passed through a lens. The modulation of the image can now be measured. The modulation transfer function at a specific frequency, MTF(u), is defined as the modulation, Mi, of the image divided by the modulation of the object, Mo, and is described by the following:
- MTF(u) = Mi / Mc
The magnitude of MTF (u) versus u is typically what is plotted and what you will see on MTF graphs.
Now back to the audio analogy. Just as a typical sound is a mixture of many pure frequencies, optical images are also mixtures of many SWGs. The image of a single SWG has the same frequency and orientation as the original SWG, but the contrast is continuously decreased. The ratio of the image SWG contrast and the object SWG contrast is the transfer factor. The transfer factor is always between 0 and 1, and different frequencies have different transfer factors. The graph of transfer factor vs. frequency is the modulation transfer function and is analogous to the frequency response curve of an audio system.
The MTF of an ideal optical system (one with no loss of contrast or detail) would be a horizontal line. Of course, this is impossible to achieve. At some point, the MTF becomes 0, which is known as the cutoff frequency. A SWG with a frequency exceeding the cutoff will image as uniform gray, with no variation in contrast. In other words, SWGs with frequencies above the cutoff do not appear in the image. SWGs with frequencies below the cutoff appear in the image, but at reduced contrast when compared to the original. The cutoff frequency roughly corresponds to the resolution. The MTF provides a more comprehensive evaluation of optical performance than resolution, but it is more challenging to measure and interpret.
The human eye can be thought of in the same way as any optical system, with two basic components: the cornea and the lens. The cornea is an optical structure with positive spherical aberration, and the lens is an optical structure with negative spherical aberration. The lens, with its negative spherical aberration, significantly reduces the effect of the cornea’s positive spherical aberration.
Intraocular lenses used in ophthalmic surgery are generally spherical, are made of either silicone or plastic, and have a single index of refraction. They are also not generally aspheric, and as such, produce positive spherical aberration because pencils of light traveling through the visual axis (known as paraxial rays) are bent less than those that travel through an area away from the visual axis (known as marginal rays).
The greater the power of a spherical intraocular lens, the more spherical aberration it will produce. And the more the spherical aberration, the more the MTF is degraded, almost like unwanted noise in a poor-quality sound recording. With an intraocular lens of a non-physiologic power, such as +35.00 diopters, there is an increase in spherical aberration on the order of the square of each doubling of diopteric strength. This means that there is roughly four times as much spherical aberration for a +35.00 diopter intraocular lens as there would be at an intraocular lens implant power of +17.50 diopters.
As stated above, the difference between the bending of paraxial rays and the marginal rays is the amount of either positive or negative spherical aberration. The human cornea exhibits naturally occurring positive spherical aberration, while the human lens displays naturally occurring negative spherical aberration. The net result is that these two structures together produce an optical system with a very low amount of positive spherical aberration. But suppose an older style spherical intraocular lens is implanted (which has positive spherical aberration). In that case, this increases the total amount of spherical aberration and degrades the image quality at larger pupil sizes.
Overall, improving the modulation transfer function of the human eye represents an effort to achieve the optimal visual experience. The newest generation of intraocular lenses aims to achieve this based on the aforementioned scientific principles.
