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Understanding the Point Spread Function and Strehl Ratio of a lens system

The image of an object produced on the retina by the Intra Ocular Lens (IOL) will not be as perfect as the object in itself. There will be some degradation of image due to diffraction effect of pupil (spreading of light when light passes through a circular aperture). Smaller the aperture, that is smaller the pupil size, larger is the diffraction. The amount of surface roughness of the IOL, or the inhomogeneities within the IOL matrix, will lead to scattering of light, which will further degrade the image quality. Aberrations present in the IOL, like spherical aberration, will also impact image quality.


Optical scientists are interested to understand the optical quality of the lens, and the clarity and resolution of the image of an object that the lens can offer. Therefore optical bench test systems in a laboratory are often set up to test the optical quality of an IOL. A key part of this testing is the Point Spread Function (PSF). PSF is the ability of the lens to capture an object point source of light, as a point source of light on the image plane. That is, in an ideal lens, the image of the point source of light (the object) will be captured exactly as a point source of light. But we do not live in an ideal world. Thus, in reality, the image of the point source of light is spread out. How much of the light is spread out, is measured by PSF. Thus, point spread function or PSF is a measurement of blur of a point source of light in the image plane and it defines what the point source of light will in reality look like.


Fig 1 showing a blurred image of a point source of light.quickguide.org


Figure 3-A bright central intensity followed by concentric circles of bright and dark rings.

If the point source of light is below the resolution limit of a lens, it would create a spot that is larger than the object itself and will create a blurry edge (see Figure 1). If this image is studied further through contrast enhancement methods, then we will see a bright intensity center, surrounded by dark and bright concentric rings (Figure 3). If we draw a line trying to capture the bright and dark intensity points of this image, we will get a line that looks similar to a gaussian beam profile, that is one high peak at center, followed by small hills and depressions in the foot of the central peak representing the concentric rings Fig 2(b). The best intensity or best focused spot at the center is called Airy disc. The area beyond the central intensity or brightest maxima is called Airy pattern that captures the concentric bright and dark regions. Thus the light intensity of the Airy disc goes to the primary diffraction order, while the concentric rings of Airy pattern goes to other diffractive orders.






Figure 2b showing the central airy disc with a line diagram super imposed to show the maximum intensity or PSF value. Beyond the Airy disc is the Airy pattern that is represented by concentric circles of white and black shades arising due to diffraction:quickguide.org

In Figure 2a, a point source of light is captured by the lens on the image plane. Figure 2(b) is the point spread function of this image, with a bright central intensity spot, followed by concentric bright and dark rings. If we analyze this image more, we can obtain a line that has a central peak followed by wavy pattern. The central intensity diameter is the Airy disc, followed by Airy pattern.


Resolution

The importance of PSF is that it provides an understanding of the resolution that a lens or any optical system can offer. The resolution of an optical system is its ability to distinguish between two objects that are in close proximity from one another. The human eye can distinguish between two object if rays of light subtend at an angle of 1 min of arc (1/60 of a degree) at the nodal point of eye. The nodal point of eye is about 17 mm in front of the retina and should fall within the human crystalline lens (refer to the article Decoding Snellen Chart in 4 mins ! ). Similarly, the point spread function helps us in understanding the resolution limit of the optical system.


Rayleigh criterion


Figure 4. the distance from the center of the PSF to the first destructive interference band is where another objects center of PSF should not overlap

When two objects are in close proximity to each other they create a diffraction pattern such that each of the two objects will create an Airy pattern and a Airy disc area. In order to be able to see the two objects separate and distinct, they must not be closer together than the distance from the center of the PSF to the first destructive interference band of the destructive pattern (figure 4 see the red arrow). In other words, the limit of resolution is reached when the center of the diffraction pattern (place of highest intensity of light) is over the first minima or trough of the other diffractive pattern (fig 5).






Figure 5


To express Rayleigh criterion in terms of PSF, consider the following illustration in Figure 6.

In figure 6a the two point objects (red and green dots) have a large separation. Therefore the diffractive pattern created, that is the Airy Disc and the Airy pattern of each point source of object are separated from each other, and thus the two objects will be seen as two separate objects.

In figure 6b, the two points are overlapping, such that the diffractive pattern are also over one another. Therefore two objects will not be seen as two separate objects, and the observer will see it as one.

In figure 6c, the limit of resolution is explained. Herein, the two objects are very close to each other, such that the center of the diffractive pattern or point of highest light intensity of the red dot is on the first minima of the green dot. In this case the Rayleigh resolution criteria has been reached.




While the PSF gives us an understanding of the resolution of the imaging system (in this case our intra ocular lens), the modulation transfer function or MTF provides us information on contrast that a lens can provide. As indicated in my article Modulation Transfer Function (https://www.quickguide.org/post/modulation-transfer-function-mtf ) , 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.


Resolution and contrast, are inseparable. A good optical system, including our object of interest, the IOL, should provide both. If the resolution is high but the contrast is poor, or vice versa, the optical clarity is lost. Consider the below image, figure 7 as an illustration. In figure 7, the y-axis denotes the resolution while the x-axis is the contrast. When either contrast or resolution is low, the image is not clear, especially at high spatial frequency (when the number of black and white bars are more).

So while the PSF of a lens gives us an idea of the resolution it provides, the MTF of the lens will provide us the contrast or modulation, that is so important in low light conditions.


Figure 7: source : journal of human sport and exercise; Basilo Pueo, Unversity of Alicante

Diffraction limited


In the MTF graph of a lens, the diffraction limited MTF and the real MTF of the lens is shown. The diffraction limited MTF is the maximum possible MTF for a perfect aberration free optical system. The MTF cannot cross the Diffraction limit as the lens MTF will be limited by diffraction of the aperture.

In a perfect optical system, one which has no aberrations, the image formed will still be not as perfect as the object. That is the optical system, whether a microscope, camera or an intra ocular lens (the pupil works like an aperture in the eye) will have a limit to its resolution due to diffraction. This diffraction or spreading of light happens when light passes through the aperture of the optical system. If the optical system has achieved this resolution limit, it is called diffraction limited. In other words, the aberration free optical system or lens cannot achieve any further improvement in its image quality. This diffraction limit of the optical system is inversely proportional to the aperture diameter. That is more the aperture size, less the diffraction, and therefore the diffraction limit of resolution is higher. If diffraction limit is higher, then the maximum intensity of light in the point spread function will be higher and the resolution of the image will be more.


Strehl Ratio

So we understand that it is impossible for an optical system to provide resolution beyond the diffraction limit of the aperture. Accepting the diffraction limit, what then can be the highest resolution offered by optical system? This will be the PSF when all aberrations in the optical system is corrected, like spherical aberration, lens tilt, etc. If such a lens system is developed then its PSF will be very close to diffraction limited resolution. The strehl ratio is the ratio of the peak height of PSF of the lens in question to that of an aberration free lens. In other words, it is the resolution or peak intensity of light in the Airy disc achieved with the lens compared to the resolution or peak intensity that can be achieved with a completely aberration free lens. You could also think of it as the ratio of peak light intensity in the Airy disc to that of an ideal lens. Strehl ratio ranges from 0 to 1. In optical systems like the microscope, a Strehl ratio of .80 is considered to be very high. That in such case, the drop in intensity of light is less than 20% to that of a diffraction limited and aberration free optical system.



Figure 8: Above left image 256 points of laser beams projected on the entrance pupil to reach retina. Bottom left image, shows the RSD in reality. In the middle images the vertical and horizontal PSF (XY direction) are shown which are derived from RSD. In the right are the values of various lower and higher order aberrations derived from the RSD" quickguide.org

As a diagnostic device the iTrace gives the PSF and Strehl ratio of the patient. The iTrace calculates these by analyzing the retinal spot diagram (RSD). The iTrace projects a set of points on the entrance pupil of eye and a retinal spot diagram is created from the wavefront that leaves the entrance pupil of eye after reaching the retina. The retinal spot diagram or RSD contains all information related to patient's aberration in eye (Fig 8). Smaller the RSD, higher is the resolution of eye, that is less the aberrations. From the RSD the PSF is calculated, which gives an idea of how the patient sees a point source of light. The Modular Transfer Function (MTF) maps is provided by the iTrace through fourier transformation of the retinal spot diagram (RSD) images to MTF.



Figure 9 : Above left PSF of lens, below left PSF of cornea, above right PSF of the total eye:quickguide.org

In Figure 9 the PSF of the lens, cornea and the total eye of a patient is displayed. Note the PSF value is provided of each of the images.











Fig 10: VSOTF display in iTrace.

Visulal Strehl Optical Focus Transfer Function (VSOTF) in iTrace.


The VSOTF display in the iTrace is a feature that helps you to understand the depth of focus in association with the image quality. In Figure 10 you can see the peak of the depth of focus curve denoted in red line looking like a Gaussian curve display (blue arrow). The peak resolution (VSOTF, also called by company as showing contrast) is near about 80% and at half of that height represents the depth of focus (yellow arrow). So the DOF at 50% of this peak is around .74 diopters.


With more and more interest in intermediate vision, a good lens should be able to give depth of focus without compromising the image quality. The height of the red cone line gives us an understanding of the image quality while at 50 percent of the peak is the depth of focus.


A VSOTF value of 60 to 80 percent is generally regarded as very good. This display could be utilized after an enhanced monofocal/EDOF or a presbyopia correcting IOL implantation. The information in this display will tell the clinician if the depth of focus achieved is coming at a significant cost to the contrast of the patient. So patients with an EDOF IOL getting a depth of focus of more than 2 diopters of either plus or minus side of defocus with a high VSOTF value of more than 60% should be very happy with their lens.


Clinical correlation of of iTrace objective VOSTF with subjective depth of focus of patients

At the 41st congress of ESCRS in Vienna (2023), Mayank Nanavaty and team presented their findings on correlation of objective measurements of iTrace VOSTF with subjective defocus curve done on Eyhance & RayOne implanted patients. They presented the data from 43 eyes implanted with Eyhance and 42 eyes implanted with Rayone IOLs. A 25% to 30% threshold of VSOTF was found to closely correlate with defocus curve of patients obtained subjectively. That is, while the company claims 50% of VOSTF threshold as the depth of focus value, clinically Nanavaty and his team found this threshold value of 25-30 percent.


Nanavaty in his paper presented in the ESCRS 2023 named OBJECTIVE QUANTIFICATION OF THE DEPTH OF FOCUS IN PSEUDOPHAKIC EYES, concluded the following:

  • The clinically derived depth of focus can be comparable to 25% or 30% threshold of VSOTF values (Nanavaty threshold) on the iTrace aberrometer.

  • The VSOTF in the iTrace can be useful to replace the cumbersome method of deriving depth of focus from the defocus curves.

  • This can have further use in clinical studies related to modern enhanced monofocals, EDOF, and trifocals.






(to be continued...)





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References:

1. Principles and Clinical Applications of Ray-Tracing aberrometry (Part I)

Alfredo Castillo Gómez and co-authors; JOURNAL OF EMMETROPIA - VOL 3, APRIL-JUNE

2. Effect of Spherical Aberration on the Optical Quality after Implantation of Two Different Aspherical Intraocular Lenses- Michael Lasta, and co-authors ; https://doi.org/10.1155%2F2017%2F8039719

3. TELESCOPES: RESOLUTION, SEEING AND PLATE SCALE REFERENCE: Chapter 6, Birney, Gonzalez & Oesper; http://srmastro.uvacreate.virginia.edu/astr313/lectures/telescopes/telescopes_res.html

4. Objective Quantification Of Depth Of Focus In Pseudophakic Eyes, Mayank Nanavaty

Session: Evaluation testing, 41st Congress of the ESCRS (2023)





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