Editing Point spread function (section)

==Theory==
The point spread function may be independent of position in the object plane, in which case it is called ''shift invariant''. In addition, if there is no distortion in the system, the image plane coordinates are linearly related to the object plane coordinates via the [[magnification]] ''M'' as:

:<math>(x_i, y_i) = (M x_o, M y_o)</math>.

If the imaging system produces an inverted image, we may simply regard the image plane coordinate axes as being reversed from the object plane axes. With these two assumptions, i.e., that the PSF is shift-invariant ''and'' that there is no distortion, calculating the image plane convolution integral is a straightforward process.

Mathematically, we may represent the object plane field as:

:<math> O(x_o,y_o) = \iint O(u,v) ~ \delta(x_o-u,y_o-v) ~ du\, dv</math>

i.e., as a sum over weighted impulse functions, although this is also really just stating the shifting 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) = \iint O(u,v) ~ \mathrm{PSF}(x_i/M-u , y_i/M-v) \, du\, dv</math>

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>.

The 2D impulse function may be regarded as the limit (as side dimension ''w'' tends to zero) of the "square post" function, shown in the figure below.

[[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''&nbsp;&minus;&nbsp;''u'',''y''&nbsp;&minus;&nbsp;''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, i.e., at the point {{nowrap|(''x'',''y'')}}.

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  \iint e^{j(k_x x + k_y y)} \, d k_x\, d k_y</math>

[[Image:PSF.svg|Truncation of Spherical Wave by Lens|right|thumb|400px]]

The quadratic [[lens (optics)|lens]] intercepts a ''portion'' of this spherical wave, and refocuses it onto a blurred point in the image plane. For a single [[lens (optics)|lens]], an on-axis point source in the object plane produces an [[Airy disc]] PSF in the image plane. It can be shown (see [[Fourier optics]], [[Huygens–Fresnel principle]], [[Fraunhofer diffraction]]) that the field radiated by a planar object (or, by reciprocity, the field converging onto a planar image) is related to its corresponding source (or image) plane distribution via a [[Fourier transform]] (FT) relation. In addition, a uniform function over a circular area (in one FT domain) corresponds to {{nowrap|''J''<sub>1</sub>(''x'')/''x''}} in the other FT domain, where {{nowrap|''J''<sub>1</sub>(''x'')}} is the first-order [[Bessel function]] of the first kind. That is, a uniformly-illuminated circular aperture that passes a converging uniform spherical wave yields an [[Airy disk]] image at the focal plane. A graph of a sample [[Airy disk]] is shown in the adjoining figure.

[[Image:Airy-3d.svg|[[Airy disk]]|right|thumb|300px]]

Therefore, the converging (''partial'') spherical wave shown in the figure above produces an [[Airy disc]] in the image plane. The argument of the function {{nowrap|''J''<sub>1</sub>(''x'')/''x''}} is important, because this determines the ''scaling'' of the Airy disc (in other words, how big the disc is in the image plane). If Θ<sub>max</sub> is the maximum angle that the converging waves make with the lens axis, ''r'' is radial distance in the image plane, and [[wavenumber]] ''k''&nbsp;=&nbsp;2π/λ where λ&nbsp;=&nbsp;wavelength, then the argument of the function is: {{nowrap|kr tan(Θ<sub>max</sub>)}}. If Θ<sub>max</sub> is small (only a small portion of the converging spherical wave is available to form the image), then radial distance, r, has to be very large before the total argument of the function moves away from the central spot. In other words, if Θ<sub>max</sub> is small, the Airy disc is large (which is just another statement of Heisenberg's [[uncertainty principle]] for Fourier Transform pairs, namely that small extent in one domain corresponds to wide extent in the other domain, and the two are related via the ''[[space-bandwidth product]]''). By virtue of this, high [[magnification]] systems, which typically have small values of Θ<sub>max</sub> (by the [[Abbe sine condition]]), can have more blur in the image, owing to the broader PSF. The size of the PSF is proportional to the [[magnification]], so that the blur is no worse in a relative sense, but it is definitely worse in an absolute sense.

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''.