Three-dimensional Transfer Function Research Articles

Three-Dimensional Transfer Functions of Interference Microscopes

Three-dimensional transfer functions (3D TFs) are generally assumed to fully describe the transfer behavior of optical topography measuring instruments such as coherence scanning interferometers in the spatial frequency domain. Therefore, 3D TFs are supposed to be independent of the surface under investigation resulting in a clear separation of surface properties and transfer characteristics. In this paper, we show that the 3D TF of an interference microscope differs depending on whether the object is specularly reflecting or consists of point scatterers. In addition to the 3D TF of a point scatterer, we will derive an analytical expression for the 3D TF corresponding to specular surfaces and demonstrate this as being most relevant in practical applications of coherence scanning interferometry (CSI). We additionally study the effects of temporal coherence and disclose that in conventional CSI temporal coherence effects dominate. However, narrowband light sources are advantageous if high spatial frequency components of weak phase objects are to be resolved, whereas, for low-frequency phase objects of higher amplitude, the temporal coherence is less affecting. Finally, we present an approach that explains the different transfer characteristics of coherence peak and phase detection in CSI signal analysis.

Analytical transfer function for the nonlinear response of a resonant medium in the spatio-temporal Fourier-transform domain

The nonlinear response of a resonant medium has many applications. To model and find the response of such a medium requires solving the Schrodinger equation (SE), which is a computationally extensive task. In this paper, we develop an analytical model to find the response of a resonant medium due to an applied field by employing the spatio-temporal Fourier-transform (STFT)–domain-based transfer function. A key feature of this approach is the use of the resonant excitation approximation (REA), which amounts to assuming that a group of atoms (or other quantum systems) within a volume element in the STFT domain are excited by only the corresponding volume element in the STFT domain of the field. We first derive the one-dimensional transfer function using an inhomogeneously broadened atomic medium under the REA. Then, we develop the three-dimensional transfer function and show that the analytical model agrees closely with the results obtained via an explicit simulation of the atomic response. As a practical example of the analytical model, we show that it can be used to model a spatio-temporal-correlator-based automatic event recognition system at a speed that is many orders of magnitude faster than solving the SE.

Derivation of various transfer functions of ideal or aberrated imaging systems from the three-dimensional transfer function.

The three-dimensional frequency transfer function for optical imaging systems was introduced by Frieden in the 1960s. The analysis of this function and its partly back-transformed functions (two-dimensional and one-dimensional optical transfer functions) in the case of an ideal or aberrated imaging system has received relatively little attention in the literature. Regarding ideal imaging systems with an incoherently illuminated object volume, we present analytic expressions for the classical two-dimensional x-y-transfer function in a defocused plane, for the axial z-transfer function in the presence of defocusing and for the x-z-transfer function in the presence of a lateral shift δy with respect to the imaged pattern in the x-z-plane. For an aberrated imaging system we use the common expansion of the aberrated pupil function with the aid of Zernike polynomials. It is shown that the line integral appearing in Frieden's three-dimensional transfer function can be evaluated for aberrated systems using a relationship established first by Cormack between the line integral of a Zernike polynomial over a full chord of the unit disk and a Chebyshev polynomial of the second kind. Some new developments in the theory of Zernike polynomials from the last decade allow us to present explicit expressions for the line integral in the case of a weakly aberrated imaging system. We outline a similar, but more complicated, analytic scheme for the case of severely aberrated systems.

How round is the human head?

Binaural microphone arrays are becoming more popular for hearing aids due to their potential to improve speech understanding in noise for hearing impaired listeners. However, such algorithms are often developed using three-dimensional head-related transfer function measurements which are expensive and often limited to a manikin head such as KEMAR. As a result, it is highly desired to use a parametric model for binaural microphone array design on a human head. Human heads have been often modeled using a rigid sphere when diffraction of sound needs to be considered. Although the spherical model may be a reasonable model for first order binaural microphone arrays, recent study has shown that it may not be accurate enough for designing high order binaural microphone arrays for hearing aids on a KEMAR (Merks et al., 2014). In this study, main sources of these errors are further investigated based on numerical simulations as well as three-dimensional measurement data on KEMAR. The implications for further improvement will be discussed.

Gravity wave propagation in the realistic atmosphere based on a three-dimensional transfer function model

Abstract. In order to study the filter effect of the background winds on the propagation of gravity waves, a three-dimensional transfer function model is developed on the basis of the complex dispersion relation of internal gravity waves in a stratified dissipative atmosphere with background winds. Our model has successfully represented the main results of the ray tracing method, e.g. the trend of the gravity waves to travel in the anti-windward direction. Furthermore, some interesting characteristics are manifest as follows: (1) The method provides the distribution characteristic of whole wave fields which propagate in the way of the distorted concentric circles at the same altitude under the control of the winds. (2) Through analyzing the frequency and wave number response curve of the transfer function, we find that the gravity waves in a wave band of about 15–30 min periods and of about 200–400 km horizontal wave lengths are most likely to propagate to the 300-km ionospheric height. Furthermore, there is an obvious frequency deviation for gravity waves propagating with winds in the frequency domain. The maximum power of the transfer function with background winds is smaller than that without background winds. (3) The atmospheric winds may act as a directional filter that will permit gravity wave packets propagating against the winds to reach the ionospheric height with minimum energy loss.

Study of Internal Gravity Waves Propagation on the Basis of the Three-Dimensional Transfer Function Model

在考虑背景风场及大气耗散的条件下,建立了3维内重力波传输函数数值模式.分析了300 km高度3维传输函数在频率波数域的特性,并以近地面单位脉冲点源为激发源,得到了内重力波在3维空间中的时空分布.讨论了不同时空尺度地面方波源激发的内重力波在电离层高度的能量分布特征.结果表明,(1)对内重力波而言,背景大气相当于一个带通滤波器,只有波动周期和波长分别在15～30 min和200～400 km之间的重力波扰动最容易上传到300km高度;(2)在背景风场的作用下水平面上以同心圆扩散的波阵面以及垂直方向上成漏斗状的波阵面发生了变形,并且逆风方向比顺风方向更有利于声重力波由对流层向电离层高度传播;(3)300km高度对时间尺度和空间尺度分别在20～30 min和150～250 km之间的地面方波源响应的总能量最强.

Calculation of vectorial three-dimensional transfer functions in large-angle focusing systems.

The optical transfer function (OTF) is used in describing imaging systems in the Fourier domain. So far the calculation of the OTF of a large-aperture imaging system has been difficult because the vectorial nature of light breaks the cylindrical symmetry of the pupil function. We derive a simple line integral solution for calculating the vectorial three-dimensional OTF. We further extend this approach to imaging through a planar interface of two media with mismatched refractive indices. In general, our formalism allows for calculation of the Fourier transform of any product of two arbitrary vector components of the electromagnetic field. Arbitrary phase or amplitude modifications of the pupil function can be taken into account.

Three-dimensional transfer functions of coherent anti-Stokes Raman scattering microscopy.

The three-dimensional coherent transfer function of confocal coherent anti-Stokes Raman scattering microscopy was derived theoretically. The three-dimensional optical transfer function was also derived under the weak-contrast assumption. The effect of a pinhole in front of the detector on the optical transfer function was estimated, and it was found that the cutoff frequency of the optical transfer function is independent of the pinhole. Micrometer-order spatial resolution along the optical axis was also experimentally demonstrated.

The three-dimensional transfer function and phase space mappings

Abstract For paraxial wave-fields in two dimensions, it is known that the defocused OTF is given by a section through the ambiguity function. This is no longer true for high aperture fields. It is shown that there is a simple relationship, even for high aperture fields, between the two-dimensional (one transverse and one longitudinal) generalized OTF and the spectral correlation function. The connection with the Wigner distribution function is also discussed. The generalization to three-dimensional wavefields is also considered.

Readout of three-dimensional optical memories

The method of recording data in a volume memory determines the spatial frequency content. We use a threedimensional transfer function approach to show that this has serious implications for the choice of readout system. In particular, one cannot use a ref lection confocal system to read a photopolymer-based memory. A solution to this problem is proposed.

Closed-form impulse responses of discrete-domain multidimensional filters

It is known that useful two-dimensional (2-D) and three-dimensional (3-D) discrete-domain recursive transfer functions may be designed by applying the RID bilinear transformation to the continuous-domain transfer functions of prototype MD inductance-resistance networks. Closed-form expressions are derived for the impulse responses of these 2- and 3-D discrete-domain filters. >

Three-dimensional transfer functions in 4Pi confocal microscopes

We investigate three-dimensional transfer functions in order to understand image formation in the recently proposed 4Pi confocal microscope [ J. Opt. Soc. Am. A9, 2159 ( 1992);Opt. Commun.93, 277 ( 1992)]. Both nonfluorescence and fluorescence imaging methods are considered, and the effects of apodization functions are taken into account. It is shown that for a 4Pi confocal system of finite-sized aperture the cutoff spatial frequency in the axial direction is larger than that in the transverse direction; this behavior is opposite that of conventional confocal systems, explaining the observation of high axial resolution in 4Pi confocal systems. In addition, the three-dimensional optical transfer function for a conventional confocal fluorescence microscope of high aperture is presented.

Three-dimensional transfer functions for high-aperture systems

The scalar three-dimensional optical transfer function is derived without using the paraxial approximation. The weak-object transfer function for a partially coherent system with equal condenser and objective apertures and the coherent transfer function for a confocal transmission system are of identical form. The coherent transfer function of a confocal reflection system is also derived. Both uniform angular illumination and systems obeying the sine condition are considered. In all cases the transfer functions can be expressed analytically.

Three-dimensional Coherent Transfer Functions for Confocal Imaging with Unequal Annular Lenses

Abstract We report, in this paper, the three-dimensional (3D) coherent functions for a confocal scanning system consisting of two unequal annular lenses. Analytical expressions for the 3D coherent transfer functions are derived for both transmission and reflection cases. Various numerical plots of the 3D coherent transfer function are given for different confocal systems including dark field microscopes. Our method is then generalized to the fibre-optical confocal system in which optical fibres rather than pinholes are employed for illumination and collection. In this case, the 3D coherent transfer function in transmission can also be obtained analytically, whilst that in reflection cannot be expressed by analytical functions, but by a single integral. Finally, images of a grating of periodic square structures are simulated to reveal the significance of the derived 3D coherent transfer functions.

Image formation in two-mode fiber-based confocal microscopes

A description is given of a confocal scanning microscope in which a two-mode optical fiber is used both to launch the light into the microscope and to detect the image signal. There is a discussion of the variety of imaging modes that result from variation of the combination of fiber modes used to launch and to detect the signal. The form of the three-dimensional transfer function is considered in all the cases, and the effect of the finite size of the fiber core is included.

Three-dimensional Transfer Function Analysis of a Confocal Fluorescence Microscope With a Finite-sized Source and Detector

Abstract The influence of a finite source and detector on the three-dimensional imaging of an incident light fluorescent confocal scanning optical microscope is investigated on the basis of its three-dimensional optical-transfer function. It is shown that the optical resolution in such set-ups remains the same as with punctiform pinholes if the source and the detecting pinhole radii are both smaller than the size of the Airy disc.

Dependence of two- and three-dimensional optical transfer functions on pinhole radius in a coherent confocal microscope—comments

The three-dimensional transfer function described by Drazic [ J. Opt. Soc. Am. A9, 725 ( 1992)] for a weakly modulated object in a confocal reflection microscope is shown to be invalid because it neglects both the bilinear terms and the phase difference between different planes.

Confocal Microscopy with Detector Offset

Abstract Three-dimensional image formation in confocal scanning microscopes in which the point detector has been offset laterally is described. Analytical expressions for the three-dimensional transfer functions for both the reflection and transmission cases are presented and the imaging that can occur if the detector is offset is discussed.

Three-dimensional Transfer Function of Coherent Confocal Microscopes with Extended Source and Detector

We consider the effect of the finite size of the source and of the detector on the three-dimensional transfer function of an incident light coherent confocal scanning microscope. Up to now the source has always been modelled as being punctiform. We will show that a finite source alters the 3D resolution and hence the imaging properties of such a microscope much more than a finite detector of the same size. It is also the aim of this paper to determine the greatest size of the source and the detector which still preserves the 3D resolution attained with a confocal microscope.

State-space models of 3-D transfer function matrix

An algorithm for obtaining two different state-space models of all three-dimensional (3-D) transfer function matrices is given. The resulting 3-D models are the 3-D analogue of the Roesser 2-D model and Eising 2-D model. The relation between these two models is also given. The approach is based on the construction of the Hankel matrix from the polynomial coefficients of a negative power series expansion of 3-D, 2-D, and 1-D transfer function matrices, respectively. The proposed algorithm results in a few large system matrices for the 3-D model analogue of the Roesser 2-D model, while there are more system matrices for the 3-D model analogue of the Eising 2-D model but of a smaller order.