Surprising applications of Newton's hyperbolism transform of curves in Fourier-transform spectroscopy

TL;DR

This work addresses the link between Bloch-phase-space representations and spectral line shapes by introducing the Newton hyperbolism transform, a geometric method that maps circles and ellipses in phase space to Lorentzian lines in FT spectroscopy. It generalizes the transform to a parametric family with $P_p=(\zeta\eta_E \frac{x}{|y|^p},y)$, enabling a continuous transition from ellipses to Lorentzians and revealing how the phase $\varphi$ governs absorption vs dispersion modes. A key contribution is the identification of finite-support, truncated parabolic lines at $p\approx0.5$ for absorption-mode spectra, accompanied by a new window function $W(t)$ that yields parabolic line shapes in the frequency domain; dispersion-mode lines yield sharp semi-ellipses with minimal tails. The authors validate the approach via simulations and an NMR apodization demonstration on methyl acetate, discuss robustness to imperfect decay compensation and noise, and propose potential benefits for multidimensional spectroscopy and automated peak analysis. Overall, the work offers a simple, geometry-driven framework that connects phase-space pictures with observable spectra and suggests practical avenues for novel spectral representations and processing.

Abstract

The Fourier transform (FT) represents a key tool in modern spectroscopy which drastically reduces measurement times and helps to improve the signal-to-noise ratio in spectra. Fourier transforming exponentially decaying time domain signals gives Lorentzian line shapes which can be manipulated by apodization methods. The underlying transitions of spectral lines can be visualized by a Bloch vector or equivalent phase-space representations. Here, we study and generalize a surprisingly elegant geometric transform, the hyperbolism of curves originally found by Isaac Newton, which allows to transform ellipses into Lorentzian lines, and vice versa. With this, we show that the Bloch picture and especially corresponding phase-space representations are directly geometrically related to the Lorentzian line shape. We also introduce a novel continuous parametrization of Newton's transform which results in further interesting line shapes. In particular, we find that truncated parabolic lines with finite support can be obtained by the half transform and introduce a new apodization approach to replicate this line shape in experimental spectra. We discuss concrete applications in nuclear magnetic resonance spectroscopy.

Figures (20)

• Figure 1: In magnetic resonance spectroscopy, being given an initial transverse magnetization vector $\vec{M}(0)$ that evolves with a frequency $\omega_0$ and transverse relaxation time $T_2$, phase-sensitive spectral lines can be obtained (A) by calculating the time evolution $\vec{M}(t)$ of $\vec{M}(0)$ to obtain the free induction decay (FID) which can be Fourier transformed. This gives a complex spectrum of which only the real part is commonly observed. (B) alternatively, it is straightforward to analyze the initial transverse magnetization components of $\vec{M}(0)$ which can be mapped to pure absorption or dispersion mode Lorentzians that, when combined directly, give the spectral line shape. (C) circles corresponding to the intersection of the phase-space representation DROPSBEADS in polar form LeinerWQST of $\vec{M}(0)$ can be directly transformed to corresponding spectral lines by the so-called hyperbolism or Newton transform (NT) of curves as will be shown in the following.
• Figure 2: Schematic representation of the so-called hyperbolism or Newton transform of curves illustrated by (cut and half-inverted) rubber sheet geometry before and after applying the required deformation. In case (a), a red circle drawn centered in the upper half-plane gives a pure absorption mode Lorentzian ($\varphi = 0^\circ$), whereas the curve obtained in case (b) corresponds to an equal mixture of absorption and dispersion Lorentzian modes ($\varphi = 45^\circ$). The checkerboard reveals the applied stretchings. The right half-plane in the original picture has grey instead of white squares to illustrate the included lower half-plane mirroring operation.
• Figure 3: Any single-transition operator $I^{k,l}$ can be represented as a Bloch vector $\vec{M}^{k,l}$ or an equivalent phase-space representation given by a spherical function (red and green shape).
• Figure 4: Full Newton hyperbolism transform protocol for three examples of transverse Bloch vectors corresponding to an uncoupled spin or to multiplet components (single-transitions) of coupled spins. The shown Bloch vectors correspond to phase angles $\varphi = 0^\circ$, $90^\circ$ and $237^\circ$. The transform procedure is illustrated using the previously introduced checkerboard image (right column) to clarify the required geometric operations. (a) the Bloch vector and an equivalent spherical function represented by a circle encompassing the vector are drawn into a coordinate system with axes corresponding to the Cartesian components $M_x$ and $M_y$ of the examined single-transition. (b) the coordinates are transformed according to the chosen detector setup by a reflection across the indicated dashed diagonal. Note that for the transformation from step (a) to (b), we exemplarily assume the case where a real-part time-domain signal is detected along the x-axis, and the imaginary part is detected along the y-axis. The lower half-plane is then (c) reflected through the y-axis giving a cartoon like spectrum composed of circular arcs. (d) the latter is then transformed into Lorentzian lines as introduced in eq. \ref{['eq:S4']}. (e) finally, physically relevant parameters such as the damping constant and the amplitude (corresponding to the length of the projected Bloch vector), can be introduced by associated scaling in both coordinate dimensions which gives a physically accurate spectral line with correct line shapes as specified in Eq. \ref{['eq:S6']}. Here, $r = 0.5$ and $k = 0.4$ Hz. Each step is schematically visualized by a corresponding checkerboard. Green arrows illustrate the applied transformations in steps (a) to (c).
• Figure 5: Continuous parametric transform of circles into associated Lorentzians with a transform parameter $p \in [0;1]$ incremented in steps of $\Delta p = \frac{1}{8}$. Read from left to right, the phase angles $\varphi$ defining the position of the circle correspond to a pure absorption mode ($\varphi = 0^\circ$), a pure dispersion mode ($\varphi = 90^\circ$) and an arbitrary case ($\varphi = 237^\circ$). The presented figures show the resulting line shapes for the denoted values of $p$. Grey dotted lines mark the coordinate axes.