Modulation Transfer Function
While in-clinic contrast sensitivity tests give us an idea of the IOL performance in real world, optical bench testing of IOLs is preferable in many ways because it eliminates certain variables like pupil size, lens centration and alignment, light adjustments, accommodation, etc. Ideally, with any multifocal IOL the patient should get crisp vision at its designated focal points, minimal disturbing optical effects or dysphotopsia, and a large depth of focus. The ability of a multifocal IOL to give its patient a great distance and near vision, a large depth of focus and minimal photic phenomenon are interrelated. Thus a multifocal IOL providing a greater depth of focus, may come with some amount of drop in optical quality, or vice-e-versa.
While defocus curves generated by real world implantation of the EDOF or multifocal IOL gives us an idea of the lens ability to provide a range of functional vision, the Modulation Transfer Function (MTF) gives an understanding of the optical quality provided by the lens in a laboratory set up.
By definition the MTF is a measure of the ability of lens to transfer contrast at a particular resolution from the object to the image. It is therefore the ability of the lens to transfer the details of the object to the image. Thus it is a ratio of image contrast to object contrast.
The Modulation Transfer Function (MTF) is defined as the modulation (Mi) of the image divided by the modulation (Mo) of the object :
Ideally in terms of percentage the MTF of a lens should be 100 % or the value of the MTF ratio should be 1. However, most lenses, including the human eye, are not a perfect optical system. As a result, when rays of light from an object pass through them, they undergo certain degree of degradation. Therefore, what researchers are interested in. is to understand how far is the degradation of image noticed when compared to the object resolution.
The Resolution is the ability of the lens to distinguish minute details. With humans, our ability to distinguish two sperate objects as distinctly separate objects, is met only when the rays of light from the two objects subtend at an angle of 1 min of arc at the nodal point of eye ( refer to the article Decoding Snellen Chart in 4 mins https://www.quickguide.org/post/decoding-snellen-s-chart-in-4-mins ). In terms of MTF and in a laboratory set up, this is described as the ability to distinguish the highest number of lines per millimeter. Thought MTF is the measure of the contrast that the optics of the lens system offer, yet resolution and contrast are closely linked. You cannot have great MTF values with a lens system that offers great contrast but poor resolution (refer to my article 'Understanding the Point Source Function and Strehl Ratio of a lens system' https://www.quickguide.org/post/points-spread-function ).
One line pair consists of one white and one black line. The more the lines per millimeter (lp/mm), the more degradation in quality of image transferred from the object. The more lines per millimeter, the more the edges of the lines get blurred. As the blur increases the white and black lines or bars overlap. As the white and black lines or bars blur together they leave no pure white or pure black bars. The difference between the brightest and darkest regions diminishes and contrast is lost.
The number of lines or bars per millimeter determines spatial frequency, that is, the spacing of lines per unit interval. This interval could be expressed in lines per millimeter (lp/mm) or degrees per cycle of vison (cpd), where each degree of vision has 60 minutes. High spatial frequency refers to higher lines per millimeter or higher cycles per degree of vision. High spatial frequency is therefore accompanied by higher degradation of the quality of image transferred by the lens.
Therefore, the modulation transfer function plot describes the modulation of the lens as the object increases or decreases in complexity. The X axis represents the number of lines per millimeter (spatial frequency) while the Y axis represents the modulation score. As spatial frequency or lines per millimeter increases the MTF value decreases as a thumb rule for any lens.
Fig 1 : On the left the object as lines per mm; on the right the image formed as delivered by the lens.
The Contrast or modulation ratio is the difference between the brightest part and the darkest part of an image. This difference is quantified as a ratio and hence the term contrast ratio. In the spatial frequency described above and in Figure 1, we have alternate white and dark lines. We can measure the amount of light coming from each white and dark lines. The contrast/modulation ratio is arrived by simply averaging the difference in maximum and minimum light intensities. Thus contrast/ modulation ratio is described by the equation :
Where I is the intensity; Imax is the maximum intensity; and Imin is the minimum intensity of the grating.
Fig 2 The Imax is the maximum intensity that we get from the white lines and Imin is the minimum intensity that we get from the dark or black lines. The original output refers to the object contrast which is theoretically 1 in the equation. The output signal refers to the image contrast.
Or in other words,
In Figure3, you can see a very high contrast or modulation transferred by the lens as the difference in light transmitted through white and dark bars are very high giving a very high contrast or modulation ratio.
In Fig 4 you can notice a very poor contrast or modulation as the difference between light transmitted by the white and black bars or lines are minimum.
For measurement of the MTF a standard set up is created. The light source or laser is usually monochromatic light of around 550 nm approximately as it represents peak human photopic sensitivity. A pin hole is placed to minimize diffraction at the light source. The collimating lens is used to keep the laser light collimated or parallel with minimal spreading or scattering of light. The aperture is set to around 3.0 mm to mimic average diurnal human pupil. An artificial eye is created with IOL placed in it. Simulating this environment helps in understanding IOL performance inside the eye as it mimics clinical conditions. A diffraction limited microscope is used to magnify the image at the focal point. The camera is used to pick up the image and the MTF is analyzed.
ISO standard requirements for all IOLs is an MTF value of 0.43 at a spatial frequency of 100 lines/mm using an aperture of 3.0 mm in diameter. The 3.0 mm aperture mimics the average of human pupil in diurnal vision. For non aspheric IOLs, or IOLs of other design (say meniscus) which significantly limits the attainment of this value, in no case should the MTF value at 100 lines/mm be less than 0.28.
In article Decoding Snellen Chart in 4 mins https://www.quickguide.org/post/decoding-snellen-s-chart-in-4-mins ) we had learned that in order to distinguish two points as distinctly separate rays of light must subtend at a minimum angle of 1 min of arc or 1/60th of a degree at the nodal point of eye. The equivalent of this in MTF is 30 cpd or 100 lines per mm. To understand the minimum angle of resolution (MAR) of spatial frequency we apply the following formula :
MAR = 60mins/2 x SF ; where MAR is the minimum angle of resolution and SF is the spatial frequency.
For example to calculate the minimum angle of resolution (MAR) or 30 cpd
MAR = 60/2 x 30 = 1' min of arc.
Remember this is exactly the minimum angle of resolution required for seeing each limb of the letter E of the Snellen Visual Acuity chart. Hence, 30cpd (100 lines/mm) is equivalent to reading 6/6 or 20/20 of the Snellen Chart.
While ISO standard is applicable to MTF of the IOL, another way of understanding the image quality transferred by an IOL is the measurement of Strehl Ratio. The Strehl Ratio (SR), is a measure of the overall imaging quality of the tested design compared with that of the perfect diffraction-limited design. Thus the Strehl Ratio can be measured by dividing the intensity of the real or measured IOL point spread function by the intensity of the diffraction limited ideal point spread function.
Fig 6 explains the MTF values of Alcon Vivity lens when compared to the AcrySof IQ for a 20.0 diopter lens in a model eye with 50 lines/mm at 3 mm aperture. The Y axis shows the MTF plotted while the X axis represents the positive and negative powered lenses dialed causing drop in modulation of targets and subsequently plotted.
With zero defocus, the AcrySof IQ ( SN60WF) shows better MTF values. However, with a defocus between -1.0 and -2.0 diopter lenses (corresponding to a distance of 100 cm and 50 cm respectively from eye), the Vivity ( DFT015 ) shows better MTF values signifying its extended range of vision capability.
Most multifocal IOL companies quote their diffractive efficiency in terms of MTF values at a fixed aperture and at a given wavelength of light. They usually experiment with a longer wavelength wave light ( between 500 nm to 550 nm). MTF scores will not only be determined by the spatial frequency, but also the aperture size and the wavelength of light being experimented with. Longer wavelength of mono chromatic light will have lower MTF values compared to shorter (say blue ) wavelength mono chromatic light (Fig 7). This is because diffraction in itself is limiting on contrast. Since diffraction is more with longer wavelength (diffraction involves some light loss due to destructive interference) of light, MTF is poorer with longer wavelength of light than with shorter wavelength of light.
To explain this more, resolution and contrast are closely related. Resolution is determined by Airy Disc pattern ( explained in the article Understanding the Point Source Function and Strehl Ratio of a lens system https://www.quickguide.org/post/points-spread-function ). Airy disk is calculated by the following equation
Airy Disk = 1.22 x wavelength of light / 2 N.A
where NA stands for numerical aperture. Smaller Airy Disk leads to higher resolution. With a shorter wavelength of light the Airy Disk will be smaller per the equation that would lead to a higher resolution or intensity of light.
A question may arise here that if shorter wavelength of light provides higher MTF values, why do ophthalmic companies experiment with longer wavelength of light, usually beyond 500 nm? Theoretically, though 405 nm wavelength of light (Violet) should give you higher MTF values, yet optical designs with diffraction gratings (steps or rings) are limited by their capabilities to provide resolution or contrast at that wavelength of light. You see, the diffractive steps designed in the multifocal IOL of any company, can diffract light with maximum efficiency with longer wavelength of light while shorter wavelength of light will pass through the steps or rings less diffracted. The shorter the wavelength of light, less it bends by diffraction. This will actually limit the MTF values of the near and intermediate though MTF of distance image will be higher as the distance MTF will be still obtained due to the refraction of light by the base curvature of the lens.
Thus when comparing the optics of different diffractive multifocal IOL, one must consider the line/mm (spatial frequency), the aperture size, and the wavelength of light that is being experimented with.
Consider the following points when comparing MTF of different lenses:
Is the MTF on-axis or off-axis. Manufacturers most often quote MTF for on-axis but are silent about MTF off-axis, that is how the lens performs with small amounts of decentration (in eye we know there are natural decentrations too).
At what wavelength of light, aperture size and spatial frequency is the MTF derived from.
1. EN/ISO 11979-2: Ophthalmic Implants Intraocular Lenses d Part 2: Optical Properties and Test Methods, Geneva, International Organization for Standardization, 1999
3. Imaging quality of intraocular lenses; Rainer Rawer, Wilhelm Stork, Christoph W. Spraul, Christian Lingenfelder; J CATARACT REFRACTIVE SURG - VOL 31, AUGUST 2005
4. MTF Curves and Lens Performance, Section 2.6 of the Imaging Resource Guide. Edmund Optics (https://www.edmundoptics.in/knowledge-center/application-notes/imaging/mtf-curves-and-lens-performance/)