Оптикалық жүйедегі нүктелік таралу функциясы

Response in an optical imaging system

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 point object, that is considered as a spatial impulse. In functional terms, it is the spatial domain version (i. e., the inverse Fourier transform) of the optical transfer function (OTF) of an imaging system. It is a useful concept in Fourier optics, astronomical imaging, medical imaging, electron microscopy and other imaging techniques such as 3D microscopy (like in confocal laser scanning microscopy) and fluorescence microscopy. The degree of spreading (blurring) in the image of a point object for an imaging system is a measure of the quality of the imaging system. In non coherent imaging systems, such as fluorescent microscopes, telescopes or optical microscopes, the image formation process is linear in the image intensity and described by a linear system theory. This means that when two objects A and B are imaged simultaneously by a non coherent imaging system, the resulting image is equal to the sum of the independently imaged objects. In other words: the imaging of A is unaffected by the imaging of B and vice versa, owing to the non interacting property of photons. In space invariant systems, i. e. those in which the PSF is the same everywhere in the imaging space, the image of a complex object is then the convolution of that object and the PSF. The PSF can be derived from diffraction integrals.

By virtue of the linearity property of optical non coherent imaging systems, i. e.,

Image(Object1 + Object2) = Image(Object1) + Image(Object2)

the image of an object in a microscope or telescope as a non coherent imaging system can be computed by expressing the object plane field as a weighted sum of 2D impulse functions, and then expressing the image plane field as a weighted sum of the images of these impulse functions. This is known as the superposition principle, valid for linear systems. The images of the individual object plane impulse functions are called point spread functions (PSF), reflecting the fact that a mathematical point of light in the object plane is spread out to form a finite area in the image plane. (In some branches of mathematics and physics, these might be referred to as Green's functions or impulse response functions. PSFs are considered impulse response functions for imaging systems. When the object is divided into discrete point objects of varying intensity, the image is computed as a sum of the PSF of each point. As the PSF is typically determined entirely by the imaging system (that is, microscope or telescope), the entire image can be described by knowing the optical properties of the system. This imaging process is usually formulated by a convolution equation. In microscope image processing and astronomy, knowing the PSF of the measuring device is very important for restoring the (original) object with deconvolution. For the case of laser beams, the PSF can be mathematically modeled using the concepts of Gaussian beams. For instance, deconvolution of the mathematically modeled PSF and the image, improves visibility of features and removes imaging noise. 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:

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, Ir is the intensity of reference beam, a is an adjustment factor and is the radial position from the center of the beam on the corresponding z plane.

The diffraction theory of point spread functions was first studied by Airy in the nineteenth century. He developed an expression for the point spread function amplitude and intensity of a perfect instrument, free of aberrations (the so called Airy disc). The theory of aberrated point spread functions close to the optimum focal plane was studied by Zernike and Nijboer in the 1930–40s. A central role in their analysis is played by Zernike's circle polynomials that allow an efficient representation of the aberrations of any optical system with rotational symmetry. Recent analytic results have made it possible to extend Nijboer and Zernike's approach for point spread function evaluation to a large volume around the optimum focal point. This extended Nijboer Zernike (ENZ) theory allows studying the imperfect imaging of three dimensional objects in confocal microscopy or astronomy under non ideal imaging conditions. The ENZ theory has also been applied to the characterization of optical instruments with respect to their aberration by measuring the through focus intensity distribution and solving an appropriate inverse problem.

In microscopy, experimental determination of PSF requires sub resolution (point like) radiating sources. Quantum dots and fluorescent beads are usually considered for this purpose. Theoretical models as described above, on the other hand, allow the detailed calculation of the PSF for various imaging conditions. The most compact diffraction limited shape of the PSF is usually preferred. However, by using appropriate optical elements (e. g., a spatial light modulator) the shape of the PSF can be engineered towards different applications.

In observational astronomy, the experimental determination of a PSF is often very straightforward due to the ample supply of point sources (stars or quasars). The form and source of the PSF may vary widely depending on the instrument and the context in which it is used. For radio telescopes and diffraction limited space telescopes, the dominant terms in the PSF may be inferred from the configuration of the aperture in the Fourier domain. In practice, there may be multiple terms contributed by the various components in a complex optical system. A complete description of the PSF will also include diffusion of light (or photo electrons) in the detector, as well as tracking errors in the spacecraft or telescope. For ground based optical telescopes, atmospheric turbulence (known as astronomical seeing) dominates the contribution to the PSF. In high resolution ground based imaging, the PSF is often found to vary with position in the image (an effect called anisoplanatism). In ground based adaptive optics systems, the PSF is a combination of the aperture of the system with residual uncorrected atmospheric terms.

The PSF is also a fundamental limit to the conventional focused imaging of a hole, with the minimum printed size being in the range of 0.6 0.7 wavelength/NA, with NA being the numerical aperture of the imaging system. For example, in the case of an EUV system with wavelength of 13.5 nm and NA=0.33, the minimum individual hole size that can be imaged is in the range of 25 29 nm. A phase shift mask has 180 degree phase edges which allow finer resolution.