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Mathematically, we may represent the object plane field as:
:<math> O(x_o,y_o) = \
i.e., as a sum over weighted impulse functions, although this is also really just stating the sifting property of 2D delta functions (discussed further below). Rewriting the object transmittance function in the form above allows us to calculate the image plane field as the superposition of the images of each of the individual impulse functions, i.e., as a superposition over weighted point spread functions in the image plane using the ''same'' weighting function as in the object plane, i.e., <math>O(x_o,y_o)</math>. Mathematically, the image is expressed as:
:<math>I(x_i,y_i) = \
in which <math display="inline">\mbox{PSF}(x_i/M-u,y_i/M-v)</math> is the image of the impulse function <math> \delta(x_o-u,y_o-v)</math>.
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[[Image:SquarePost.svg|Square Post Function|right|thumb|220px]]
We imagine the object plane as being decomposed into square areas such as this, with each having its own associated square post function. If the height, ''h'', of the post is maintained at 1/w<sup>2</sup>, then as the side dimension ''w'' tends to zero, the height, ''h'', tends to infinity in such a way that the volume (integral) remains constant at 1. This gives the 2D impulse the sifting property (which is implied in the equation above), which says that when the 2D impulse function, δ(''x'' − ''u'',''y'' − ''v''), is integrated against any other continuous function, {{nowrap|''f''(''u'',''v'')}}, it "sifts out" the value of ''f'' at the ___location of the impulse,
The concept of a perfect point source object is central to the idea of PSF. However, there is no such thing in nature as a perfect mathematical point source radiator; the concept is completely non-physical and is rather a mathematical construct used to model and understand optical imaging systems. The utility of the point source concept comes from the fact that a point source in the 2D object plane can only radiate a perfect uniform-amplitude, spherical wave — a wave having perfectly spherical, outward travelling phase fronts with uniform intensity everywhere on the spheres (see [[Huygens–Fresnel principle]]). Such a source of uniform spherical waves is shown in the figure below. We also note that a perfect point source radiator will not only radiate a uniform spectrum of propagating plane waves, but a uniform spectrum of exponentially decaying ([[Evanescent wave|evanescent]]) waves as well, and it is these which are responsible for resolution finer than one wavelength (see [[Fourier optics]]). This follows from the following [[Fourier transform]] expression for a 2D impulse function,
:<math>\delta (x,y) \propto \
[[Image:PSF.svg|Truncation of Spherical Wave by Lens|right|thumb|400px]]
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The figure above illustrates the truncation of the incident spherical wave by the lens. In order to measure the point spread function — or impulse response function — of the lens, a perfect point source that radiates a perfect spherical wave in all directions of space is not needed. This is because the lens has only a finite (angular) bandwidth, or finite intercept angle. Therefore, any angular bandwidth contained in the source, which extends past the edge angle of the lens (i.e., lies outside the bandwidth of the system), is essentially wasted source bandwidth because the lens can't intercept it in order to process it. As a result, a perfect point source is not required in order to measure a perfect point spread function. All we need is a light source which has at least as much angular bandwidth as the lens being tested (and of course, is uniform over that angular sector). In other words, we only require a point source which is produced by a convergent (uniform) spherical wave whose half angle is greater than the edge angle of the lens.
Due to intrinsic limited resolution of the imaging systems, measured PSFs are not free of uncertainty.<ref>{{Cite journal|last1=Ahi|first1=Kiarash|last2=Shahbazmohamadi|first2=Sina|last3=Asadizanjani|first3=Navid|date=July 2017 |title=Quality control and authentication of packaged integrated circuits using enhanced-spatial-resolution terahertz time-___domain spectroscopy and imaging|url=https://www.researchgate.net/publication/318712771|journal=Optics and Lasers in Engineering|volume=104|pages=274–284|doi=10.1016/j.optlaseng.2017.07.007|bibcode=2018OptLE.104..274A}}</ref> In imaging, it is desired to suppress the side-lobes of the imaging beam by [[apodization]] techniques. In the case of transmission imaging systems with Gaussian beam distribution, the PSF is modeled by the following equation:<ref>{{Cite journal|last=Ahi|first=K.|date=November 2017|title=Mathematical Modeling of THz Point Spread Function and Simulation of THz Imaging Systems|journal=IEEE Transactions on Terahertz Science and Technology|volume=7|issue=6|pages=747–754|doi=10.1109/tthz.2017.2750690|issn=2156-342X|bibcode=2017ITTST...7..747A|s2cid=11781848}}</ref>
:<math>\mathrm{PSF}(f, z) = I_r(0,z,f)\exp\left[-z\alpha(f)-\dfrac{2\rho^2}{0.36{\frac{cka}{\text{NA}f}}\sqrt{{1+\left ( \frac{2\ln 2}{c\pi}\left ( \frac{\text{NA}}{0.56k} \right )^2 fz\right )}^2}}\right],</math>
where ''k-factor'' depends on the truncation ratio and level of the irradiance, ''NA'' is numerical aperture, ''c'' is the speed of light, ''f'' is the photon frequency of the imaging beam, ''I<sub>r</sub>'' is the intensity of reference beam, ''a'' is an adjustment factor and <math>\rho</math> is the radial position from the center of the beam on the corresponding ''z-plane''.
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| journal = Journal of Microscopy
| title = The point-spread function of a confocal microscope: its measurement and use in deconvolution of 3-D data
| author1=P. J. Shaw |author2=D. J. Rawlins
|
| issue = 2
| pages = 151–165
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=== Ophthalmology ===
Point spread functions have recently become a useful diagnostic tool in clinical [[ophthalmology]]. Patients are measured with a [[Shack–Hartmann wavefront sensor|Shack-Hartmann]] [[wavefront sensor]], and special software calculates the PSF for that patient's eye. This method allows a physician to simulate potential treatments on a patient, and estimate how those treatments would alter the patient's PSF. Additionally, once measured the PSF can be minimized using an adaptive optics system. This, in conjunction with a [[Charge-coupled device|CCD]] camera and an adaptive optics system, can be used to visualize anatomical structures not otherwise visible ''in vivo'', such as cone photoreceptors.<ref>{{Cite journal|last1=Roorda|first1=Austin|last2=Romero-Borja|first2=Fernando|last3=Iii|first3=William J. Donnelly|last4=Queener|first4=Hope|last5=Hebert|first5=Thomas J.|last6=Campbell|first6=Melanie C. W. |author6-link=Melanie Campbell|date=2002-05-06|title=Adaptive optics scanning laser ophthalmoscopy|journal=Optics Express|language=EN|volume=10|issue=9|pages=405–412|doi=10.1364/OE.10.000405|issn=1094-4087| bibcode=2002OExpr..10..405R |pmid=19436374|s2cid=21971504|doi-access=free}}</ref>
==See also==
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|s2cid=5318333
}}
* {{cite journal
|title=Multi-scale Optics for Enhanced Light Collection from a Point Source
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