Point Spread Function

Point Spread Function

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This article discusses how the Point Spread Function is modelled and interpreted in OpticStudio. The analysis features used are the Spot Diagtam, the FFT PSF, and the Huygens PSF. The merits of each tool will be discussed, as will useful feature settings for the most accurate analysis.
The Point Spread Function (PSF) of an optical system is the irradiance distribution that results from a single point source. (A telescope forming an image of a distant star is a good example.) Although the source may be a point, the image is not. There are two main reasons. First, aberrations in the system will spread the image over a finite area. Second, diffraction effects will also spread the image, even in a system that has no aberrations.
OpticStudio has three basic types of PSF calculations: a geometric (no diffraction) Spot Diagram, the diffraction-based FFT and Huygens PSF. This article will discuss the basic theory and provide some guidance as to the proper use of each type of PSF.
One of the most basic analysis features in OpticStudio is the Spot Diagram. This feature launches many rays from a single source point in object space, traces all the rays through the optical system, and plots the (x, y) coordinates of all the rays relative to some common reference. In this way, the Spot Diagram is a geometric PSF.
The sample optical system used here is a single parabolic F/5 mirror with a focal length of 50 mm. The object is at infinity. This system is a simplified Newtonian telescope, and the included sample file is PSF_Newtonian.ZMX. Here is what the optical system looks like:
The Spot Diagram for two field points, one on-axis and the other at an angle of 2 degrees, is shown below.
Note the Spot Diagram is a collection of points, with each point representing a single ray. There is no interaction or interference between the rays. The Spot Diagram is very effective at showing the effects of the geometric, or ray aberrations of the telescope. The off axis geometric PSF clearly shows the coma and astigmatism of the system. On-axis, however, the Spot Diagram predicts perfect imagery. Is this an accurate representation of the optical system performance? To answer this question for the Spot Diagram results, we need to compare the spot distribution to the diffraction limited response.
A quick way to compare the geometric aberrations to the diffraction limit is to add an Airy Disk reference ellipse to the Spot Diagram. Open the Settings and select Show Airy Disk .
Now the Spot Diagram will indicate the size of the Airy Disk relative to the geometric spot distribution:
On-axis, the spot is much smaller than the Airy Disk, while off axis the spot is much larger than the Airy Disk. This indicates the Spot Diagram is a useful and reasonable indicator of performance off axis only. To compute a more accurate PSF both on and off axis, a consideration of diffraction is required.
Generally speaking, the Spot Diagram is useful if the aberrations are large compared to the diffraction limited performance of the system.
The Fast Fourier Transform (FFT) algorithm has been widely applied to frequency analysis of many electrical and optical systems. Conceptually, the FFT decomposes a spatial distribution into a frequency domain distribution. An excellent discussion of Fourier optics can be found in reference 1 at the end of this article. There is also a summary of diffraction theory in the chapter "Physical Optics Propagation" in the OpticStudio Help System, reference 2. Both of these references describe Fresnel and Fraunhofer diffraction theory.
Most optical imaging systems meet the simplifying assumptions necessary for the Fraunhofer diffraction theory used by the FFT PSF algorithm. The main assumptions are:
The FFT PSF of an optical system is computed as follows. A grid of rays is traced from the source point to the exit pupil. For each ray, the amplitude and optical path difference is used to compute the complex amplitude of a point on the wavefront grid at the exit pupil. The FFT of this grid, appropriately scaled, is then squared to yield the real valued PSF. If the computation is polychromatic, the PSF's are summed incoherently.
To compute an FFT PSF for a sequential system, choose Analyze...PSF...FFT PSF . A sample FFT PSF for the on-axis field point of the Newtonian telescope sample file is shown below. Note, the settings have been modified from the default settings, which is discussed shortly.
Note the familiar Airy Disk shape. This is the expected result for the Newtonian, which is aberration free for the on-axis field point. To generate the picture above, the FFT PSF settings dialog should look like this:
The sampling refers to the grid of rays traced to the entrance pupil. Internally, OpticStudio doubles the size of the grid, filling the region outside of the entrance pupil with zero valued data. Because of this doubling, the output PSF is always on a grid with twice as many points as the sampling grid. If the aberrations are reasonably small, the region of interest is concentrated near the center of the plot. Rather than plot all of these near-zero amplitude points, the display grid may be selected to be smaller than the total grid computed.
There are many ways to display the same underlying PSF data. Try the settings shown below.
Note Display is 128 x 128, Field is 2, Type is Logarithmic, and the Show As is set to False Color. Here is the resulting PSF:
Conceptually, the Huygens PSF is computed by converting each ray on the Spot Diagram into a small plane wave. Recall that a ray models a small piece of a plane wave, and that the ray locally is normal to the wavefront in isotropic media. The plane wave has an amplitude, phase, and direction determined from data associated with the ray from which it is generated. The total irradiance at any point on the image surface may be determined by coherently summing all of the plane waves represented by all of the rays traced. The diffraction based PSF is given directly by this integration over all the rays.
While most diffraction analyses in OpticStudio assume scalar diffraction theory applies (F/# not too small), the Huygens method can account for the vectoral nature of the electric field if the Use Polarization switch is enabled. All Huygens based analyses consider the full polarization vector and polarization phase aberrations. These computations work by computing data for the Ex, Ey, and Ez components of the polarized electric field separately, then incoherently summing the results. The polarization phase aberrations induced in each orthogonal component of the electric field are considered as any other phase aberration.
Virtually all imaging systems meet the simplifying assumptions necessary for computing the Huygens PSF. However, accurate calculation for the Huygens PSF requires adequate sampling.
The Huygens PSF is not based upon the FFT. The net result is that the Huygens PSF is generally slower than the FFT PSF, but more accurate for those cases where the FFT PSF assumptions do not apply. The most common cases where the FFT PSF assumptions are questionable, and thus the Huygens PSF should be used are when:
The Huygens PSF of an optical system is computed as follows. A grid of rays is traced from the source point to the image surface. For each ray, the amplitude, coordinates, direction cosines, and optical path difference is used to compute the complex amplitude of the plane wave incident at every point on the image space grid. A coherent sum for all rays is performed at every point in the image space grid. The intensity of each point in the image space grid is the square of the resulting complex amplitude sum. If the computation is polychromatic, the PSF's are summed incoherently.
To compute a Huygens PSF for a Sequential system, choose Analyze...PSF...Huygens PSF from the drop-down menu. The Huygens PSF may also be computed for Non-Sequential Component (NSC) systems and this will be discussed shortly. Note the FFT PSF cannot be computed for NSC systems.
The key user-definable parameters for the Huygens PSF are the Pupil Sampling, Image Sampling, and Image Delta. These parameters can be set on the Huygens PSF settings dialog. Open the Settings and change Pupil Sampling , Image Sampling , and Image Delta as shown below.
The Image Delta is the image point spacing in micrometers. The total size of the region where the PSF is computed is the product of the Image Delta and the Image Sampling. Here is the Huygens PSF for the same Newtonian telescope on-axis:
In the Settings , by specifying Field: 2 , we see that the PSF appears as follows off-axis.
The larger the number of rays and image points, the greater the resolution and accuracy of the resulting PSF, at the expense of a longer computation time.
One way to visualize this integration process is to observe the effects of coherent summing of one ray at a time. This can be accomplished with a coherent detector in the Non-Sequential Components feature of OpticStudio. From the supplied article sample files, open HPSF_Integration.ZMX.
This file consists of an elliptical source, a singlet lens, and a Detector Rectangle object. The source generates random rays over a circular region. All the rays exit parallel to the local Z-axis. This source models a collimated source or distant point source. Note the number of Layout Rays is set to 20, while the number of Analysis Rays is set to 1. This latter setting will allow one ray at a time to be traced as will be discussed shortly. The lens is a simple singlet, placed to bring the collimated rays to a good focus on the detector. The detector is defined to be an absorber, with 120 x 120 pixels.
Note that the Detector Rectable Parameter 11, the PSF Wave # is set to 1.
This special mode allows the detector to perform the coherent Huygens PSF integration. Every ray that strikes the detector is converted into a local plane wave that illuminates every pixel on the detector, and the coherent amplitude of the plane wave at every pixel is added to the coherent amplitude already detected. This allows one ray at a time to be traced, if desired, so the effects of summing individual rays can be seen.
To see this integration, open the sample file HPSF_Integration.ZMX , and then choose Analyze...Detector Viewer . In the settings of the Detector Viewer, enable Auto Update . To trace rays for analysis run the Ray Trace by going to Analyze...Ray Trace , and selecting Clear & Trace . Because the source object defines only 1 Analysis Ray, a single random ray is traced and the detector is updated. Click on Trace only (so the detector is not cleared) to trace a second ray. The two rays now traced will coherently interfere like two plane waves making an angle with respect to each other, resulting in a fringe pattern on the detector. Because the rays are random, the fringe pattern will be different every time, and so will not look exactly like the picture below.
Each time Trace is pressed in the Ray Trace dialog box, another ray is added to the sum shown in the Detector Viewer. After 10 rays have been traced the diffraction PSF begins to emerge.
After about 40 rays, features such as the Airy ring can be seen forming.
It takes a few hundred rays before the PSF reasonably converges to the final PSF result.
The only reason to trace one ray at a time is to visualize what the integration is doing. To trace many rays at once, navigate to the Non-Sequential Component Editor (NSCE), and change the number of Analysis Rays on the Source Ellipse object from 1 to 500.
Now reopen the Ray Trace Control and press Clear & Trace. All 500 rays will be traced at once, and the resulting PSF will be displayed in the Detector Viewer window.
Even though the rays are randomly selected the PSF converges to the correct Airy pattern (note that this particular lens is diffraction limited).
1. Goodman, Joseph W., Introduction to Fourier Optics, McGraw Hill 2. OpticStudio Help System Zemax LLC, Kirkland, Washington, United States

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In this article, we’ll talk you through the Point Spread Function.
Suppose that you have a tiny fluorescent object, such as a 10 nm-diameter fluorescent bead or even a single fluorescent molecule, and you try to observe it under a fluorescence microscope.
Provided that the object is bright enough, even though it is well below the resolution limit of your microscope you can still see the object, but it will appear larger than it really is.
Diffraction of light, which determines the microscope’s resolution limit, blurs out any point-like object to a certain minimal size and shape called the Point Spread Function (PSF).
The PSF, then, is the three-dimensional image of a point-like object under the microscope.
The PSF is usually taller than it is wide (like an American football standing on its tip) because optical microscopes have a worse resolution in the depth direction than in the lateral direction.
The PSF varies depending on the wavelength of the light you are viewing: shorter wavelengths of light (such as blue light, 450 nm) result in a smaller PSF, while longer wavelengths (such as red light, 650 nm) result in a larger PSF and, therefore, worse resolution.
Also, the Numerical Aperture (NA) of the objective lens that you use affects the size and shape of the PSF.
A high-NA objective gives you a nice small PSF and, therefore, better resolution, as demonstrated in Figure 1 (an interactive version of this figure is available over at ZEISS Online Tutorials ).
Surprisingly, the magnification of the objective lens does not affect the PSF—only the NA and wavelength matter.
You can use beads to measure PSFs for the objective lenses on your microscope to determine the resolution of each lens and also to see what condition each lens is in: the PSF of a damaged objective lens is often large and possibly skewed in one direction or another.

Figure 1: Interactive tutorial—The resolving power of an objective determines the size of the Airy diffraction pattern formed, and the radius of the central disk is determined by the combined NAs of the objective and condenser. Left image: image produced by smaller NA. Right image: increased resolution of image by increasing NA. You can access the interactive version of this tutorial over at ZEISS Online Campus.
Sometimes, a real specimen does indeed have single point-like fluorescent objects nicely separated from each other.
For example, cancer researchers studying the complicated process of DNA repair often irradiate cells and look to see what proteins localize on punctate sites of double-strand breaks.
These nuclear foci are small enough that you are actually observing the microscope’s PSF when you image them.
In many cases, however, your specimen is a complicated arrangement of closely spaced fluorophores, and the PSF is not apparent in your images.
Nevertheless, the PSF is hard at work, blurring out every fluorescent structure in your specimen as if tracing out the fine details with a fat paintbrush.

Figure 2: BSC-1 African Green Monkey Kidney Cells, 63x, stained with DAPI, Alexa 488 (Tubulin), Alexa 568 (TOMM20). ZEISS Axio Imager.Z2, Axiocam 506 mono, ApoTome.2 . Deconvolved image (ZEN software) on the right. Sample courtesy of Michael W. Davidson, The Florida State University.
Now, if we take the trouble to measure the PSF for a particular objective lens on our microscope, could we use what we know about the shape and size of the PSF to somehow undo its blurring influence in our specimen?
The answer is, “Yes!”. Mathematically, the blurring of the PSF with the actual arrangement of fluorophores in the specimen is called a convolution of the specimen with the PSF:
It’s no accident that the symbol for convolution ( ∗ \ast ∗ ) looks like a multiplication since the two operators are related.
We don’t know what the actual specimen looks like, but we record its image in the microscope, and we can also record the PSF.
An operation called deconvolution reverses the effect of the PSF on the specimen, much like the division operator reverses the effect of multiplication.
Practically speaking, we can never fully know what the specimen looks like, but by using iterative deconvolution algorithms we can remove some of the PSF’s blurring influence, particularly in the depth direction where the blur is worst.
Figure 2 shows the results of deconvolution.
Hopefully, this article has provided you with a good grasp of what the Point Spread Function is, how it affects your image, and what you can do to minimize these effects.
If you’re interested in delving deeper, head over to ZEISS Online Campus where you can find a chapter on The Point Spread Function , which goes into more detail about the maths underlying the PSF.
Originally published November 26, 2014. Reviewed and republished May 2021.
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The point spread function (PSF) describes the response of a focused optical imaging system to a point source or point object. A more general term for the PSF is the system's impulse response; the PSF is the impulse response or impulse response function (IRF) of a focused optical imaging system.The PSF in many contexts can be thought of as the extended blob in an image that represents a single ...
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