EUV–soft X-ray spectrometer - monochromator, 

and wavelength-tunable focusing mirror - all in one design

 Published in The Nature Light Journal

A Twist of Fate in Optical Research: From a Budget Quip to Cutting-Edge Innovation

Imagine you're shopping for a top-notch extreme ultraviolet (EUV) beam line grating, but the sales rep cheekily suggests you need at least €50,000 to play in the big leagues, or else, try your luck at Thorlabs or MKS. Sounds discouraging? Well, not for a team of audacious scientists from TU Wien, UCSD San Diego and CAS Beijing.! This seemingly derogatory remark didn't dampen our spirits, says Dimitar Popmintchev and Tenio Popmintchev – researchers from TU Wien and UCSD San Diego. Instead, it fueled a groundbreaking discovery, now celebrated in the Nature Light Journal.

 

Monte-Carlo ray tracing of a-b Optimized spatial beam profile and chirp of a single high-order harmonic. c-d Unoptimized spatial beam profile and chirp of a single high-order harmonic. 

In their remarkable paper, the researchers present a groundbreaking EUV–soft X-ray spectrometer - monochromator, and wavelength-tunable focusing mirror - all integrated into one design. This isn't your ordinary spectrometer; it's a marvel of optical engineering that shatters efficiency records by design, boasting a whopping 60% throughput efficiency and showing promise to overcome 80%. The secret sauce? – Neural network optimization. And it achieves this feat without relying on the crutch of variable line spacing gratings.

 

TP and DP with the set up at TU Wien. Former lab of the 2023 Nobel Laureate in physics Ferenc Krausz.

The team didn't stop there. They harnessed the power of conical diffraction geometry, meticulously optimizing the optical system in a multidimensional parameter space. The result? Optimal imaging performance across a broad spectrum, with the bonus of preserving circular and elliptical polarization states in the first to third diffraction orders. But wait, there's more! The spectrometer is a timekeeper's dream, limiting pulse broadening to a mere 10 fs tail-to-tail and a standard deviation of 2 fs. This precision allows for ultrafast spectroscopic and pump-probe studies with femtosecond accuracy. What's even more exciting is its transformation capabilities. With a single grating, it morphs into a monochromator, extending its prowess to the soft X-ray region with minimal photon loss.

 

Experimental setup showing the torroidal mirror with flat grating. Simple azimuthal rotation of the torroidal mirror result in near perfect compensation of the spatio-temporal aberrations.

This is a game-changer in ultrafast spectroscopy, first and foremost, setting the stage for advanced coherent diffractive imaging of intricate nano- and bio-systems. Now, for the first time dynamic coherent diffraction imaging is feasible in the X-ray water-window, where water is more transparent that the bio- markers, promising unprecedented spatiotemporal precision at the femto-nanometer scale. Second, the design is as well a continuously wavelength-tunable X-ray mirror - a replacement for the inefficient, at best, multilayered EUV mirrors up to the water window oxygen edge and potentially the near keV magnetic L-edges with efficiency between 60%-90%.

One of the design's crowning achievements is its ability to counteract spatial aberrations in the harmonic beam caused by the grating. A simple azimuthal rotation of the toroidal mirror relative to the grating does the trick, ensuring optimal imaging performance over a wide spectrum.Their findings indicate minimal temporal dispersion, promising precision in ultrafast spectroscopic and pump-probe experiments. This level of performance is not just for show; it's a powerful tool for studying micro-scale dynamics across various scientific fields.

In summary, what started as a snide remark about budget constraints sparked a journey leading to an optical masterpiece. This spectrometer is not just another scientific instrument; it's a testament to perseverance, ingenuity, and a bit of humor in the face of adversity. The paper, available at Nature Light Journal, is not just a read; it's an inspiration for anyone who's ever been told they can't afford to dream big.

For more information, please visit the original papers Ref [1].

 Cite as:

[1]. Highly efficient and aberration-free off-plane grating spectrometer and monochromator for EUV—soft X-ray applications. Light Sci Appl 13, 12 (2024). DOI: https://doi.org/10.1038/s41377-023-01342-9 

 

Closed Form Infinite Dispersion Orders in the Lah-Laguerre Optics

 Lah-Laguerre Optics

 Published in arXiv:2011, 30 October 2020
Published in Optics Express 30, 22, 20 October 2022

 

Operating with ultrashort, broadband laser pulses presents a challenge to maintain the pulse phase intact so that the pulse duration or shape does not change and the peak power does not decrease substantially.   

Ultrashort pulse stemming from an anti-resonant fiber

The description of uniform systems allows for the exact implementation of the accumulated phase, i.e., pulse propagation in waveguides or optical fibers. However, balancing the pulse phase becomes an optimization problem when several systems are interconnected. It is no longer practical due to computational speed to match phases ‘shapes’ but rather individual perturbative orders of dispersion. The cited papers have lay the foundation of analytical and more robust description for the chromatic dispersion phenomena that will aid the design of novel optical systems and materials.

The perturbative description of the chromatic dispersion involves several hypergeometrical transforms, the most famous amongst, are the Lah, and Laguerre transforms. The Lah, Laguerre, and several other unnamed forward and inverse transforms form the core of the Lah-Laguerre dispersive optics. In Lah-Laguerre optics, such algorithms allow for incredibly faster computations when solving complicated optimization problems involving phase balancing of different optical systems, at the extereme single cycle waform synthesis.

Using the Lah-Laguerre approach gives the mathematical foundation to evaluate, optimize, and design systems, to output a pulse desired pulse duration balanced to an anticipated chromatic dispersion order. However, due to the experimental uncertainty in measurements of the refractive index or due to the simplified level of description of an optical system, these relations can leave some ambiguity in the estimated chromatic dispersion orders. Fortunately, the inverse transforms relate the Taylor coefficients of the refractive index or optical path to the phase or wavevector. Thus, to put in perspective, a single point phase measurement, provides information for the refractive index or optical path in an extended vicinity of measured frequency. Consequently, this formalism can also facilitate more precise interferometric measurements of the refractive index and aid the design of novel optical materials, nanostructures, and optical systems based on desired dispersion. Furthermore, from a practical point of view, the evaluation speed of the simple hypergeometric series can be competitive even against algorithms such as the fast Fourier transform (FFT). Numerically, the highest order that can be evaluated is solely limited by the computer architecture’s ability to allocate the smallest /largest/ floating-point number.

When pulses have substantial bandwidth it is essential to consider also the higher orders of dispersion. A few example of evaluated chromatic dispersion orders follow below.

The first 10 material dispersion orders of $CaF_2$ material are shown below. More data plots for the material dispersion orders of various materials can be found in Ref.[2].

Material dispersion orders for $CaF_2$

 

The first 10 chromatic dispersion orders of conventional genuine laser pulse compressors are illustrated below.

 

Chromatic dispersion orders in A) grating compressor B) Prism-pair compressor

 

 

For more information, please visit the original papers Ref [1], Ref [2], Ref [3].

Mathematical description. 

In dispersive systems, the phase velocity of a wave depends on its frequency. Effectively a single wavelength monochromatic light will accumulate phase and will experience a time delay compared to the same wave traveling in a vacuum. Such idealized waves have infinite pulse duration – continuous waves. In contrast, when considering many monochromatic waves that interfere constructively over a small spatial extent, create a wavepacket that will experience more dramatic changes. In such cases, we can define a group delay and group velocity dispersion that is a result of collective behavior of the wavelets. The propagation of pulses in matter leads to changes in the phase and pulse duration. In linear light-matter interactions, the spectrum of the wavepacket does not change. The phenomenon is known as chromatic dispersion. The chromatic dispersion has worked both in favor and against some of the most significant innovations of the past decades.

The dispersion relation for the phase is:

$\begin{array}{c}\varphi\hspace{-0.5mm}\left(\omega|\lambda \right) = k \hspace{-0.1mm}\mathrm{(}\omega\mathrm{)}z = \frac{\omega}{c}n \mathrm{(}\omega\mathrm{)}z = \frac{2\pi}{\lambda}n \mathrm{(}\lambda\mathrm{)}z = \frac{\omega}{c}{\it OP} \mathrm{(}\omega\mathrm{)} = \frac{2\pi}{\lambda}{\it OP} \mathrm{(}\lambda\mathrm{)} = \omega\tau \mathrm{(}\omega\mathrm{)} = \frac{2\pi}{\tau_{0}}\tau \mathrm{(}\omega\mathrm{)} \tag{1}\label{myeq1} \end{array}$
where ${\it OP}\mathrm{(}\omega \mathrm{|}\lambda \mathrm{)}$ is the optical path, $n\mathrm{(}\omega \mathrm{|}\lambda \mathrm{)}$ is the refractive index of the medium, $\tau \mathrm{(}\omega \mathrm{|}\lambda \mathrm{)}$ is the corresponding temporal interval, and ${\tau }_0\mathrm{\ }$ is the single-cycle pulse duration for a wavelet with a wavelength $\lambda$. In such a picture, the spectral content of the signal does not change but rather rearranges temporally over a certain band around an average frequency ${\omega }_0$. Historically, the dispersion orders have been defined in the frequency space through the Taylor expansion of the phase $\varphi  \mathrm{(}\omega \mathrm{|}\lambda \mathrm{)}$, or the wavevector $k\hspace{-0.1mm} \mathrm{(}\omega \mathrm{|}\lambda \mathrm{)}$ around the pulse intensity-averaged frequency or the carrier frequency at the pulse peak

The dispersion orders are defined by the Taylor expansion of the phase or the wavevector.

$\begin{array}{c}\varphi \mathrm{(}\omega\mathrm{)} = \varphi\hspace{-1.5mm}\left.\  \right|_{\omega_{0}} + \left. \ \frac{\partial\varphi}{\partial\omega} \right|_{\omega_{0}} \hspace{-1.0mm}\left(\omega - \omega_{0} \right) + \frac{1}{2}\hspace{-1.3mm}\left. \ \frac{\partial^{\hspace{0.3pt}{2}}\varphi}{\partial\omega^{2}} \right|_{\omega_{0}} \hspace{-1.0mm}\left(\omega - \omega_{0} \right)^{2}\  + \ldots + \frac{1}{p!}\hspace{0.0mm}\left. \ \frac{\partial^{\hspace{0.1pt}{p}}\varphi}{\partial\omega^{p}} \right|_{\omega_{0}} \hspace{-1.0mm}\left(\omega - \omega_{0} \right)^{p} + \ldots = \end{array}$
$\begin{array}{c}= \varphi \hspace{-1.6mm} \left. \  \right|_{\omega_{0}} + \tau_{g} \hspace{-1.9mm} \left. \  \right|_{\omega_{0}}\hspace{0.0mm}\left( \omega - \omega_{0} \right) + \frac{1}{2}{\it GDD}\hspace{0.0mm}\left( \omega - \omega_{0} \right)^{2} + \ldots + \frac{1}{p!}{\it POD}\hspace{0.0mm}\left( \omega - \omega_{0} \right)^{p} + \mathrm{\it{R_p}} \tag{2}\label{myeq2} \end{array}$
Such a representation is convenient as it requires knowledge only of a small number of spectral derivatives at a single point. The first derivative ${\left.\frac{\partial \varphi }{\partial \omega }\right|}_{{\omega }_0} \hspace{-0.9mm}=  {\tau }_g$ corresponds to the group delay (${\it GD}$), which results in a temporal lag of the pulse envelope. The presence of higher orders causes distortions in the temporal pulse shape. The second term ${\left.\frac{{\partial }^{\hspace{0.3pt}{2}}\varphi }{\partial {\omega }^{\mathrm{2}}}\right|}_{{\omega }_0} \hspace{-0.9mm}= {\left.\frac{\partial }{\partial \omega }{\tau }_g \hspace{-0.2mm}  \mathrm{(}\omega \mathrm{)}\right|}_{{\omega }_0}$corresponds to the group delay dispersion (${\it GDD}$), and in general,  ${\left.\frac{{\partial }^{\hspace{0.1pt}{p}}\varphi }{\partial {\omega }^p}\right|}_{{\omega }_0} = {\left.\frac{{\partial }^{p\mathrm{-}\mathrm{1}}}{\partial {\omega }^{p\mathrm{-}\mathrm{1}}}{\tau }_g \hspace{-0.2mm}  \mathrm{(}\omega \mathrm{)}\right|}_{{\omega }_0}$ is the $p^{t\hspace{-0.3mm}h}$ order dispersion (${\it POD}$). Finally, $R_{p} = \mathop{max}\limits_{\omega \leq \xi \leq \omega_{0}}\frac{\partial^{p + 1}\varphi(\xi)}{\partial\xi^{p + 1}}\frac{{\ \left( \omega - \omega_{0} \right)}^{p + 1}}{(p + 1)!}$ is the Lagrange error after the first ${\it{p}}$ terms.

 In a similar fashion, the wavevector $k \hspace{-0.1mm}\mathrm{(}\omega \mathrm{|}\lambda \mathrm{)}$ can be expanded in a Taylor series :

$\begin{array}{c}k\hspace{-0.1mm}  \mathrm{(}\omega \mathrm{)}=k{\left.{}\right|}_{{\omega }_0}+{\left.\frac{\partial k}{\partial \omega }\right|}_{{\omega }_0}\hspace{-0.85mm}\left(\omega -{\omega }_0\right)+\frac{1}{2}{\left.\frac{{\partial }^2k}{\partial {\omega }^2}\right|}_{{\omega }_0}\hspace{-0.75mm}{\left(\omega -{\omega }_0\right)}^2\ +\dots +\frac{1}{p!}{\left.\frac{{\partial }^{\hspace{0.1pt}{p}}k}{\partial {\omega }^p}\right|}_{{\omega }_0}\hspace{-0.3mm}{\left(\omega -{\omega }_0\right)}^p+\dots=\end{array}$
$\begin{array}{c} =k_0+v^{-1}_{gr}\hspace{-0.5mm}\left(\omega -{\omega }_0\right)+\frac{1}{2}{\it GDD}{\left(\omega -{\omega }_0\right)}^2+\dots +\frac{1}{p!}{\it POD}{\left(\omega -{\omega }_0\right)}^p+\mathrm{\it{R_p}} \tag{3}\label{myeq3} \end{array}$
Again, the lowest term ${\left.\frac{\partial k}{\partial \omega }\right|}_{{\omega }_0} \hspace{-0.9mm}=\hspace{-0.3mm} \frac{\mathrm{1}}{v_g} \hspace{-0.3mm}=\hspace{-0.3mm} \frac{{\tau }_g}{z}$ represents the inverse group velocity, whereas the second term ${\left.\frac{{\partial }^{\hspace{0.3pt}{2}}k}{\partial {\omega }^{\mathrm{2}}}\right|}_{{\omega }_0} \hspace{-1.0mm}= {\left.\frac{\partial }{\partial \omega }v^{\mathrm{-}\mathrm{1}}_{gr}\right|}_{{\omega }_0}$ represents the ${\it GDD}$. In general, ${\left.\frac{{\partial }^{\hspace{0.1pt}{p}}k}{\partial {\omega }^p}\right|}_{{\omega }_0} \hspace{-0.9mm}= {\left.\frac{{\partial }^{p\mathrm{-}\mathrm{1}}}{\partial {\omega }^{p\mathrm{-}\mathrm{1}}}v^{\mathrm{-}\mathrm{1}}_{gr}\right|}_{{\omega }_0}$ is the $p^{t\hspace{-0.3mm}h}$ order dispersion (${\it POD}$).

The chromatic dispersion orders can be easily evaluated in the frequency domain by obtaining the successive derivatives of the wavevector $k\hspace{-0.1mm}\mathrm{(}\omega \mathrm{)}$ or the phase $\varphi \mathrm{(}\omega \mathrm{)}$. In general: 

$\begin{array}{c}\frac{{\partial }^{\hspace{0.1pt}{p}}}{\partial {\omega }^p}k  \mathrm{(}\omega \mathrm{)}=\frac{1}{c}\left(p\frac{{\partial }^{p-1}}{\partial {\omega }^{p-1}}n  \mathrm{(}\omega \mathrm{)}+\omega \frac{{\partial }^{\hspace{0.1pt}{p}}}{\partial {\omega }^p}n  \mathrm{(}\omega \mathrm{)}\right) \tag{4}\label{myeq4} \end{array}$   
$\begin{array}{c} \frac{{\partial }^{\hspace{0.1pt}{p}}}{\partial {\omega }^p}\varphi    \mathrm{(}\omega \mathrm{)} = \frac{1}{c}\left(p\frac{{\partial }^{p-1}}{\partial {\omega }^{p-1}}{\it OP}   \mathrm{(}\omega \mathrm{)}+\omega \frac{{\partial }^{\hspace{0.1pt}{p}}}{\partial {\omega }^p}{\it OP}   \mathrm{(}\omega \mathrm{)}\right)  \tag{5}\label{myeq5} \end{array}$

The derivatives of any differentiable function $f\mathrm{(}\omega \mathrm{|}\lambda \mathrm{)}$ in the wavelength or the frequency space is specified through a Lah transform as:

$\begin{array}{c} \text{}\hspace{2pt}   \frac{{\partial }^{\hspace{0.1pt}{p}}}{\partial {\omega }^p}f   \mathrm{(}\omega \mathrm{)}={}{\left(\mathrm{-}\mathrm{1}\right)}^p{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^p\sum\limits^p_{m = {0}}{\mathcal{A}\hspace{0.0mm}\mathrm{(}p,m\mathrm{)}\hspace{0.3mm}{\lambda }^m\frac{{\partial }^{\hspace{0.3pt}{m}}}{\partial {\lambda }^m}f  \mathrm{(}\lambda \mathrm{)}} \tag{6}\label{myeq6} \end{array}   \hspace{-1.5em}$
$\begin{array}{c} \text{}\hspace{2pt}   \frac{{\partial }^{\hspace{0.1pt}{p}}}{\partial {\lambda }^p}f  \mathrm{(}\lambda \mathrm{)}={}{\left(\mathrm{-}\mathrm{1}\right)}^p{\left(\frac{\omega }{\mathrm{2}\pi c}\right)}^p\sum\limits^p_{m = {0}}{\mathcal{A}\hspace{0.0mm}\mathrm{(}p,m\mathrm{)}\hspace{0.3mm}{\omega }^m\frac{{\partial }^{\hspace{0.3pt}{m}}}{\partial {\omega }^m}f   \mathrm{(}\omega \mathrm{)}}\tag{7}\label{myeq7} \end{array}   \hspace{-1.5em}$   

The matrix elements of the transformation are the Lah coefficients: $\mathcal{A}\mathrm{(}p,m\mathrm{)} = \frac{p\mathrm{!}}{\left(p\mathrm{-}m\right)\mathrm{!}m\mathrm{!}}\frac{\mathrm{(}p\mathrm{-}\mathrm{1)!}}{\mathrm{(}m\mathrm{-}\mathrm{1)!}}$

Written for the ${\it GDD}$ the above expression states that a constant with wavelength ${\it GDD}$, will have zero higher orders. From a practical point of view, when the ${\it GDD}$ data is experimentally or numerically accessible in the wavelength space, the dispersion orders can be expressed as:

$\begin{array}{c} \frac{{\partial }^{p+2}}{\partial {\omega }^{p+2}} \varphi    \mathrm{(}\omega \mathrm{)}=\frac{{\partial }^{\hspace{0.1pt}{p}}}{\partial {\omega }^p} {\it GDD}={\left(-1\right)}^p{\left(\frac{\lambda }{2\pi c}\right)}^p\sum\limits^p_{m=0}{\mathcal{A}\hspace{0.0mm}\mathrm{(}p,m\mathrm{)}\ {\lambda }^m\frac{{\partial }^{\hspace{0.3pt}{m}}}{\partial {\lambda }^m}{\it GDD}  \mathrm{(}\lambda \mathrm{)}}\tag{8}\label{myeq8} \end{array}$

Substituting equation (6) and (7) expressed for the refractive index $n$ or optical path $OP$ into equation (4) and (5) results in closed-form expressions for the dispersion orders. In general the $p^{th}$  order dispersion (${\it POD}$) is a Laguerre type transform of negative order two:

$\begin{array}{c}\text{}\hspace{2pt}{\it POD}\mathrm{(}n \mathrm{)}=\frac{{\partial }^{\hspace{0.1pt}{p}}}{\partial {\omega }^p} k  \mathrm{(}\omega \mathrm{)}={\left(-1\right)}^p\frac{1}{c}{\left(\frac{\lambda }{2\pi c}\right)}^{p-1}\sum\limits^p_{m=0}{\mathcal{B}\hspace{0.0mm}\mathrm{(}p,m\mathrm{)}\ {\lambda }^m\frac{{\partial }^{\hspace{0.3pt}{m}}}{\partial {\lambda }^m}n \mathrm{(}\lambda \mathrm{)}} \tag{9}\label{myeq9} \end{array}$
$\begin{array}{c}\text{}\hspace{2pt}{\it POD}\mathrm{(}{\it OP} \mathrm{)}=\frac{{\partial }^{\hspace{0.1pt}{p}}}{\partial {\omega }^p} \varphi  \mathrm{(}\omega \mathrm{)}={\left(-1\right)}^p\frac{1}{c}{\left(\frac{\lambda }{2\pi c}\right)}^{p-1}\sum\limits^p_{m=0}{\mathcal{B}\hspace{0.0mm}\mathrm{(}p,m\mathrm{)}\ {\lambda }^m\frac{{\partial }^{\hspace{0.3pt}{m}}}{\partial {\lambda }^m}{\it OP} \mathrm{(}\lambda \mathrm{)}} \tag{10}\label{myeq10} \end{array}$

 The inverse transforms relate the Taylor coefficients of the refractive index or the optical path to the wavevector or the phase.
$\begin{array}{c}\text{}\hspace{2pt}{\lambda }^p\frac{{\partial }^{\hspace{0.1pt}{p}}}{\partial {\lambda }^p}n \mathrm{(}\lambda \mathrm{)}={\left(-1\right)}^p\frac{c}{\omega }\sum\limits^p_{m=0}{\mathcal{B}\hspace{0.0mm}\mathrm{(}p,m\mathrm{)}\hspace{0.5mm}{\omega }^m\frac{{\partial }^{\hspace{0.3pt}{m}}}{\partial {\omega }^m} k  \mathrm{(}\omega \mathrm{)} }\tag{11}\label{myeq11} \end{array}$
$\begin{array}{c}\text{}\hspace{2pt}{\lambda }^p\frac{{\partial }^{\hspace{0.1pt}{p}}}{\partial {\lambda }^p}{\it OP} \mathrm{(}\lambda \mathrm{)}={\left(-1\right)}^p\frac{c}{\omega }\sum\limits^p_{m=0}{\mathcal{B}\hspace{0.0mm}\mathrm{(}p,m\mathrm{)}\hspace{0.5mm}{\omega }^m\frac{{\partial }^{\hspace{0.3pt}{m}}}{\partial {\omega }^m} \varphi   \mathrm{(}\omega \mathrm{)}\ }\tag{12}\label{myeq12} \end{array}$

The matrix elements of the transforms are the unsigned Laguerre coefficients of order negative two:  $\mathcal{B}\hspace{0.0mm}\mathrm{(}p,m\mathrm{)} = \frac{p\mathrm{!}}{\left(p\mathrm{-}m\right)\mathrm{!}m\mathrm{!}}\frac{\mathrm{(}p\mathrm{-}\mathrm{2)!}}{\mathrm{(}m\mathrm{-}\mathrm{2)!}}$

The polynomial sums form sequential polynomials $G^{\left(\alpha \right)}_p\mathrm{(}x\mathrm{)}$ . The corresponding generating function can be expressed as:

$\begin{array}{c} G^{  \left(\alpha \right)}_p\mathrm{(}x\mathrm{)}=x^{-\alpha }\frac{d^p}{dx^p}\left(x^{p+\alpha }f \mathrm{(}x\mathrm{)}\right)=\sum\limits^p_{m=0}{\mathcal{C}\mathrm{(}p+\alpha ,p-m\mathrm{)}\frac{p!}{m!}x^m}f^{\left(m\right)} \mathrm{(}x\mathrm{)}\tag{13}\label{myeq13} \end{array}$

The first four chromatic dispersion orders are well-known in the literature. Using the above-mentioned Lah-Laguerre optical formalism, the first ten dispersion orders, written for the wavevector, can be explicitly written in closed-form expressions as:

$\begin{array}{l}\hspace{-55pt}\text{C I}.\hspace{2pt}\boldsymbol{{\it GD}} = \frac{\partial }{\partial \omega }k\hspace{-0.3mm}  \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(n  \mathrm{(}\omega \mathrm{)}+\omega \frac{\partial n  \mathrm{(}\omega \mathrm{)}}{\partial \omega }\right) = {-}\frac{\mathrm{1}}{c}G^{\left(\mathrm{-}\mathrm{2}\right)}_{\mathrm{1}} \mathrm{(}\lambda \mathrm{)}=\frac{\mathrm{1}}{c}\left(n \mathrm{(}\lambda \mathrm{)}-\lambda \frac{\partial n \mathrm{(}\lambda \mathrm{)}}{\partial\lambda }\right) = v^{\mathrm{-}\mathrm{1}}_{gr}\tag{14}\label{myeq14} \end{array} \hspace{-0.5em}$
The group refractive index $n_g$is defined in terms of the group velocity $v_{gr}$: $n_g\enspace   = \enspace  cv^{\mathrm{-}\mathrm{1}}_{gr}$.
$\begin{array}{l}\hspace{-115pt}\text{C II}.\hspace{2pt}\boldsymbol{{\it GDD}} = \frac{{\partial }^{\hspace{0.3pt}{2}}}{\partial {\omega }^{\mathrm{2}}}k\hspace{-0.3mm}  \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(\mathrm{2}\frac{\partial n\mathrm{(}\omega \mathrm{)}}{\partial \omega }+\omega \frac{{\partial }^{\hspace{0.3pt}{2}}n\mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{2}}}\right) = \frac{\mathrm{1}}{c}\left(\frac{\lambda }{\mathrm{2}\pi c}\right)G^{\left(\mathrm{-}\mathrm{2}\right)}_{\mathrm{2}} \mathrm{(}\lambda \mathrm{)} = \\= \frac{\mathrm{1}}{c}\left(\frac{\lambda }{\mathrm{2}\pi c}\right)\left({\lambda }^{\mathrm{2}}\frac{{\partial }^{\hspace{0.3pt}{2}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}\right)\tag{15}\label{myeq15} \end{array} \hspace{-0.5em}$
$\begin{array}{l}\hspace{-99pt}\text{C III}.\hspace{2pt}\boldsymbol{{\it TOD}} = \frac{{\partial }^{\hspace{0.3pt}{3}}}{\partial {\omega }^{\mathrm{3}}}k\hspace{-0.3mm}  \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(\mathrm{3}\frac{{\partial }^{\hspace{0.3pt}{2}}n\mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{2}}}+\omega \frac{{\partial }^{\hspace{0.3pt}{3}}n\mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{3}}}\right) = {-}\frac{\mathrm{1}}{c}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{2}}G^{\left(\mathrm{-}\mathrm{2}\right)}_{\mathrm{3}} \mathrm{(}\lambda \mathrm{)} = \\  = {-} \frac{\mathrm{1}}{c}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{2}}\Bigl(\mathrm{3}{\lambda }^{\mathrm{2}}\frac{{\partial }^{\hspace{0.3pt}{2}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}} +{\lambda }^{\mathrm{3}}\frac{{\partial }^{\hspace{0.3pt}{3}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}\Bigr)\tag{16}\label{myeq16} \end{array} \hspace{-0.5em}$
$\begin{array}{l}\hspace{-102pt}\text{C IV}.\hspace{2pt}\boldsymbol{{\it FOD}} = \frac{{\partial }^{\hspace{0.3pt}{4}}}{\partial {\omega }^{\mathrm{4}}}k\hspace{-0.3mm}  \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(\mathrm{4}\frac{{\partial }^{\hspace{0.3pt}{3}}n\mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{3}}}+\omega \frac{{\partial }^{\hspace{0.3pt}{4}}n\mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{4}}}\right) = \frac{\mathrm{1}}{c}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{3}}G^{\left(\mathrm{-}\mathrm{2}\right)}_{\mathrm{4}} \mathrm{(}\lambda \mathrm{)} = \\  = \frac{\mathrm{1}}{c}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{3}}\Bigl(\mathrm{12}{\lambda }^{\mathrm{2}}\frac{{\partial }^{\hspace{0.3pt}{2}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}    +\mathrm{8}{\lambda }^{\mathrm{3}}\frac{{\partial }^{\hspace{0.3pt}{3}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+{\lambda }^{\mathrm{4}}\frac{{\partial }^{\hspace{0.3pt}{4}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}\Bigr)\tag{17}\label{myeq17} \end{array} \hspace{-0.5em}$
$\begin{array}{l}\hspace{-67pt}\text{C V}.\hspace{2pt}\boldsymbol{{\it FiOD}} = \frac{{\partial }^{\hspace{0.3pt}{5}}}{\partial {\omega }^{\mathrm{5}}}k\hspace{-0.3mm}  \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(\mathrm{5}\frac{{\partial }^{\hspace{0.3pt}{4}}n  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{4}}}+\omega \frac{{\partial }^{\hspace{0.3pt}{5}}n  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{5}}}\right)={-}\frac{\mathrm{1}}{c}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{4}}G^{\left(\mathrm{-}\mathrm{2}\right)}_{\mathrm{5}} \mathrm{(}\lambda \mathrm{)} = \\  = {-}\frac{\mathrm{1}}{c}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{4}} \Bigl(\mathrm{60}{\lambda }^{\mathrm{2}}\frac{{\partial }^{\hspace{0.3pt}{2}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+\mathrm{60}{\lambda }^{\mathrm{3}}\frac{{\partial }^{\hspace{0.3pt}{3}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{15}{\lambda }^{\mathrm{4}}\frac{{\partial }^{\hspace{0.3pt}{4}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+{\lambda }^{\mathrm{5}}\frac{{\partial }^{\hspace{0.3pt}{5}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}\Bigr)\tag{18}\label{myeq18} \end{array} \hspace{-0.5em}$
$\begin{array}{l}\hspace{-35pt}\text{C VI}.\hspace{2pt}\boldsymbol{{\it SiOD}} = \frac{{\partial }^{\hspace{0.3pt}{6}}}{\partial {\omega }^{\mathrm{6}}}k\hspace{-0.3mm}  \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(\mathrm{6}\frac{{\partial }^{\hspace{0.3pt}{5}}n  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{5}}}+\omega \frac{{\partial }^{\hspace{0.3pt}{6}}n  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{6}}}\right) = \frac{\mathrm{1}}{c}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{5}}G^{\left(\mathrm{-}\mathrm{2}\right)}_{\mathrm{6}} \mathrm{(}\lambda \mathrm{)} = \\  = \frac{\mathrm{1}}{c}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{5}}\Bigl(\mathrm{360}{\lambda }^{\mathrm{2}}\frac{{\partial }^{\hspace{0.3pt}{2}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}} +\mathrm{480}{\lambda }^{\mathrm{3}}\frac{{\partial }^{\hspace{0.3pt}{3}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{180}{\lambda }^{\mathrm{4}}\frac{{\partial }^{\hspace{0.3pt}{4}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\mathrm{24}{\lambda }^{\mathrm{5}}\frac{{\partial }^{\hspace{0.3pt}{5}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}+{\lambda }^{\mathrm{6}}\frac{{\partial }^{\hspace{0.3pt}{6}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}}\Bigr)\tag{19}\label{myeq19} \end{array} \hspace{-0.5em}$
$\begin{array}{l}\hspace{-50pt}\text{C VII}.\hspace{2pt}\boldsymbol{{\it SeOD}} = \frac{{\partial }^{\hspace{0.3pt}{7}}}{\partial {\omega }^{\mathrm{7}}}k\hspace{-0.3mm}  \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(\mathrm{7}\frac{{\partial }^{\hspace{0.3pt}{6}}n  \mathrm{(}\omega \mathrm{)}}{{\partial \omega }^{\mathrm{6}}}+\omega \frac{{\partial }^{\hspace{0.3pt}{7}}n  \mathrm{(}\omega \mathrm{)}}{{\partial \omega }^{\mathrm{7}}}\right)  = {-}\frac{\mathrm{1}}{c}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{6}}G^{\left(\mathrm{-}\mathrm{2}\right)}_{\mathrm{7}} \mathrm{(}\lambda \mathrm{)}  = \\   =\hspace{-0.7mm}  {-}\frac{\mathrm{1}}{c}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{6}}\hspace{-0.5mm} \Bigl(\mathrm{2520}{\lambda }^{\mathrm{2}}\frac{{\partial }^{\hspace{0.3pt}{2}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+\mathrm{4200}{\lambda }^{\mathrm{3}}\frac{{\partial }^{\hspace{0.3pt}{3}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{2100}{\lambda }^{\mathrm{4}}\frac{{\partial }^{\hspace{0.3pt}{4}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\mathrm{420}{\lambda }^{\mathrm{5}}\frac{{\partial }^{\hspace{0.3pt}{5}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}+\\   +\mathrm{35}{\lambda }^{\mathrm{6}}\frac{{\partial }^{\hspace{0.3pt}{6}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}}+{\lambda }^{\mathrm{7}}\frac{{\partial }^{\hspace{0.3pt}{7}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{7}}}\Bigr)\tag{20}\label{myeq20} \end{array} \hspace{-1.5em}$
$\begin{array}{l}\hspace{-39pt}\text{C VIII}.\hspace{2pt}\boldsymbol{{\it EOD}} = \frac{{\partial }^{\hspace{0.3pt}{8}}}{\partial {\omega }^{\mathrm{8}}}k\hspace{-0.3mm}  \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(\mathrm{8}\frac{{\partial }^{\hspace{0.3pt}{7}}n  \mathrm{(}\omega \mathrm{)}}{{\partial \omega }^{\mathrm{7}}}+\omega \frac{{\partial }^{\hspace{0.3pt}{8}}n  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{8}}}\right) = \frac{\mathrm{1}}{c}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{7}}G^{\left(\mathrm{-}\mathrm{2}\right)}_{\mathrm{8}} \mathrm{(}\lambda \mathrm{)}=\\  =  \frac{\mathrm{1}}{c}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{7}}\Bigl(\mathrm{20160}{\lambda }^{\mathrm{2}}\frac{{\partial }^{\hspace{0.3pt}{2}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}} +\mathrm{40320}{\lambda }^{\mathrm{3}}\frac{{\partial }^{\hspace{0.3pt}{3}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{25200}{\lambda }^{\mathrm{4}}\frac{{\partial }^{\hspace{0.3pt}{4}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\mathrm{6720}{\lambda }^{\mathrm{5}}\frac{{\partial }^{\hspace{0.3pt}{5}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}+\\ +\mathrm{840}{\lambda }^{\mathrm{6}}\frac{{\partial }^{\hspace{0.3pt}{6}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}} +\mathrm{48}{\lambda }^{\mathrm{7}}\frac{{\partial }^{\hspace{0.3pt}{7}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{7}}}+{\lambda }^{\mathrm{8}}\frac{{\partial }^{\hspace{0.3pt}{8}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{8}}}\Bigr)\tag{21}\label{myeq21} \end{array} \hspace{-1.5em}$
$\begin{array}{l}\hspace{-27pt}\text{C IX}.\hspace{2pt}\boldsymbol{{\it NOD}} = \frac{{\partial }^{\hspace{0.3pt}{9}}}{\partial {\omega }^{\mathrm{9}}}k\hspace{-0.3mm}  \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(\mathrm{9}\frac{{\partial }^{\hspace{0.3pt}{8}}n  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{8}}}+\omega \frac{{\partial }^{\hspace{0.3pt}{9}}n  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{9}}}\right) = {-}\frac{\mathrm{1}}{c}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{8}}G^{\left(\mathrm{-}\mathrm{2}\right)}_{\mathrm{9}} \mathrm{(}\lambda \mathrm{)}=\\  = {-}\frac{\mathrm{1}}{c}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{8}}\Bigl(\mathrm{181440}{\lambda }^{\mathrm{2}}\frac{{\partial }^{\hspace{0.3pt}{2}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+\mathrm{423360}{\lambda }^{\mathrm{3}}\frac{{\partial }^{\hspace{0.3pt}{3}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{317520}{\lambda }^{\mathrm{4}}\frac{{\partial }^{\hspace{0.3pt}{4}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\mathrm{105840}{\lambda }^{\mathrm{5}}\frac{{\partial }^{\hspace{0.3pt}{5}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}+\\  +\mathrm{17640}{\lambda }^{\mathrm{6}}\frac{{\partial }^{\hspace{0.3pt}{6}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}}+\mathrm{1512}{\lambda }^{\mathrm{7}}\frac{{\partial }^{\hspace{0.3pt}{7}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{7}}}+\mathrm{63}{\lambda }^{\mathrm{8}}\frac{{\partial }^{\hspace{0.3pt}{8}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{8}}}+{\lambda }^{\mathrm{9}}\frac{{\partial }^{\hspace{0.3pt}{9}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{9}}}\Bigr)\tag{22}\label{myeq22} \end{array} \hspace{-1.5em}$
 $\begin{array}{l}\hspace{-30pt}\text{C X}.\hspace{2pt}\boldsymbol{{\it TeOD}} = \frac{{\partial }^{\hspace{0.3pt}{10}}}{\partial {\omega }^{\mathrm{10}}}k\hspace{-0.3mm}  \mathrm{(}\omega \mathrm{)} = \frac{\mathrm{1}}{c}\left(\mathrm{10}\frac{{\partial }^{\hspace{0.3pt}{9}}n  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{9}}}+\omega \frac{{\partial }^{\hspace{0.3pt}{10}}n  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{10}}}\right) = \frac{\mathrm{1}}{c}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{9}}G^{\left(\mathrm{-}\mathrm{2}\right)}_{\mathrm{10}} \mathrm{(}\lambda \mathrm{)}=\\  = \frac{\mathrm{1}}{c}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{9}}\Bigl(\mathrm{1814400}{\lambda }^{\mathrm{2}}\frac{{\partial }^{\hspace{0.3pt}{2}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+\mathrm{4838400}{\lambda }^{\mathrm{3}}\frac{{\partial }^{\hspace{0.3pt}{3}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{4233600}{\lambda }^{\mathrm{4}}\frac{{\partial }^{\hspace{0.3pt}{4}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\\  +{1693440}{\lambda }^{\mathrm{5}}\frac{{\partial }^{\hspace{0.3pt}{5}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}+\mathrm{352800}{\lambda }^{\mathrm{6}}\frac{{\partial }^{\hspace{0.3pt}{6}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}}+\mathrm{40320}{\lambda }^{\mathrm{7}}\frac{{\partial }^{\hspace{0.3pt}{7}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{7}}}+\mathrm{2520}{\lambda }^{\mathrm{8}}\frac{{\partial }^{\hspace{0.3pt}{8}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{8}}}+\mathrm{80}{\lambda }^{\mathrm{9}}\frac{{\partial }^{\hspace{0.3pt}{9}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{9}}}+\\  +{\lambda }^{\mathrm{10}}\frac{{\partial }^{\hspace{0.3pt}{10}}n \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{10}}}\Bigr)\tag{23}\label{myeq23} \end{array} \hspace{-1.5em}$   

Written for the phase $\varphi$, the first ten dispersion orders can be expressed as a function of wavelength using the Lah transforms as:

$\begin{array}{l}\hspace{-127pt}\text{B I}.\hspace{2pt} \frac{\partial \varphi\mathrm{(}\omega \mathrm{)}}{\partial \omega }={-}\left(\frac{\lambda }{\mathrm{2}\pi c}\right)G^{\left(\mathrm{-}\mathrm{1}\right)}_{\mathrm{1}} \mathrm{(}\lambda \mathrm{)} = {-}\left(\frac{\mathrm{2}\pi c}{{\omega }^{\mathrm{2}}}\right)\frac{\partial \varphi  \mathrm{(}\omega \mathrm{)}}{\partial \lambda } = {-}\left(\frac{{\lambda }^{\mathrm{2}}}{\mathrm{2}\pi c}\right)\frac{\partial \varphi \mathrm{(}\lambda \mathrm{)}}{\partial \lambda }\tag{24}\label{myeq24}\end{array}$
$\begin{array}{l}\hspace{-73pt}\text{B II}.\hspace{2pt}\frac{{\partial }^{\hspace{0.3pt}{2}}\varphi  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{2}}} = \frac{\partial }{\partial \omega }\left(\frac{\partial \varphi  \mathrm{(}\omega \mathrm{)}}{\partial \omega }\right) = {\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{2}}G^{\left(\mathrm{-}\mathrm{1}\right)}_{\mathrm{2}} \mathrm{(}\lambda \mathrm{)} = {\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{2}}\left(\mathrm{2}\lambda \frac{\partial \varphi \mathrm{(}\lambda \mathrm{)}}{\partial \lambda }+{\lambda }^{\mathrm{2}}\frac{{\partial }^{\hspace{0.3pt}{2}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}\right)\tag{25}\label{myeq25} \end{array}$
$\begin{array}{l}\hspace{-60pt}\text{B III}.\hspace{2pt}\frac{{\partial }^{\hspace{0.3pt}{3}}\varphi  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{3}}}={-}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{3}}G^{\left(\mathrm{-}\mathrm{1}\right)}_{\mathrm{3}} \mathrm{(}\lambda \mathrm{)} = {-}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{3}}\left(\mathrm{6}\lambda \frac{\partial \varphi \mathrm{(}\lambda \mathrm{)}}{\partial \lambda }+\mathrm{6}{\lambda }^{\mathrm{2}}\frac{{\partial }^{\hspace{0.3pt}{2}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+{\lambda }^{\mathrm{3}}\frac{{\partial }^{\hspace{0.3pt}{3}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}\right)\tag{26}\label{myeq26} \end{array}$
$\begin{array}{l}\hspace{-57pt}\text{B IV}.\hspace{2pt}\frac{{\partial }^{\hspace{0.3pt}{4}}\varphi  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{4}}}={}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{4}}G^{\left(\mathrm{-}\mathrm{1}\right)}_{\mathrm{4}} \mathrm{(}\lambda \mathrm{)} = {\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{4}}\Bigl(\mathrm{24}\lambda \frac{\partial \varphi \mathrm{(}\lambda \mathrm{)}}{\partial \lambda }+\mathrm{36}{\lambda }^{\mathrm{2}}\frac{{\partial }^{\hspace{0.3pt}{2}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+\mathrm{12}{\lambda }^{\mathrm{3}}\frac{{\partial }^{\hspace{0.3pt}{3}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\\ +{\lambda }^{\mathrm{4}}\frac{{\partial }^{\hspace{0.3pt}{4}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}\Bigr) \tag{27}\label{myeq27} \end{array}$
$\begin{array}{l}\hspace{-35pt}\text{B V}.\hspace{2pt}\frac{{\partial }^{\mathrm{5}}\varphi  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{5}}}={-}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{5}}G^{\left(\mathrm{-}\mathrm{1}\right)}_{\mathrm{5}} \mathrm{(}\lambda \mathrm{)} = {-}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{5}}\Bigl(\mathrm{120}\lambda \frac{\partial \varphi \mathrm{(}\lambda \mathrm{)}}{\partial \lambda }+\mathrm{240}{\lambda }^{\mathrm{2}}\frac{{\partial }^{\hspace{0.3pt}{2}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}    +\mathrm{120}{\lambda }^{\mathrm{3}}\frac{{\partial }^{\hspace{0.3pt}{3}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\\+\mathrm{20}{\lambda }^{\mathrm{4}}\frac{{\partial }^{\hspace{0.3pt}{4}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+{\lambda }^{\mathrm{5}}\frac{{\partial }^{\hspace{0.3pt}{5}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}\Bigr) \tag{28}\label{myeq28} \end{array}\hspace{-.5em}$
$\begin{array}{l}\hspace{-37pt}\text{B VI}.\hspace{2pt}\frac{{\partial }^{\hspace{0.3pt}{6}}\varphi  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{6}}}={}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{6}}G^{\left(\mathrm{-}\mathrm{1}\right)}_{\mathrm{6}} \mathrm{(}\lambda \mathrm{)} = {\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{6}}\Bigl(\mathrm{720}\lambda \frac{\partial \varphi \mathrm{(}\lambda \mathrm{)}}{\partial \lambda }+\mathrm{1800}{\lambda }^{\mathrm{2}}\frac{{\partial }^{\hspace{0.3pt}{2}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+\mathrm{1200}{\lambda }^{\mathrm{3}}\frac{{\partial }^{\hspace{0.3pt}{3}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\\  +\mathrm{300}{\lambda }^{\mathrm{4}}\frac{{\partial }^{\hspace{0.3pt}{4}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\mathrm{30}{\lambda }^{\mathrm{5}}\frac{{\partial }^{\hspace{0.3pt}{5}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}\mathrm{\ +}{\lambda }^{\mathrm{6}}\frac{{\partial }^{\hspace{0.3pt}{6}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}}\Bigr)\tag{29}\label{myeq29} \end{array} \hspace{-.5em}$
$\begin{array}{l}\hspace{-45pt}\text{B VII}.\hspace{2pt} \frac{{\partial }^{\hspace{0.3pt}{7}}\varphi  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{7}}}={-}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{7}}G^{\left(\mathrm{-}\mathrm{1}\right)}_{\mathrm{7}} \mathrm{(}\lambda \mathrm{)} = {-}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{7}} \Bigl(\mathrm{5040}\lambda \frac{\partial \varphi \mathrm{(}\lambda \mathrm{)}}{\partial \lambda }+\mathrm{15120}{\lambda }^{\mathrm{2}}\frac{{\partial }^{\hspace{0.3pt}{2}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+\\  +\mathrm{12600}{\lambda }^{\mathrm{3}}\frac{{\partial }^{\hspace{0.3pt}{3}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{4200}{\lambda }^{\mathrm{4}}\frac{{\partial }^{\hspace{0.3pt}{4}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\mathrm{630}{\lambda }^{\mathrm{5}}\frac{{\partial }^{\hspace{0.3pt}{5}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}+\mathrm{42}{\lambda }^{\mathrm{6}}\frac{{\partial }^{\hspace{0.3pt}{6}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}}+{\lambda }^{\mathrm{7}}\frac{{\partial }^{\hspace{0.3pt}{7}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{7}}} \Bigr)\tag{30}\label{myeq30} \end{array} \hspace{-.0em}$
$\begin{array}{l}\hspace{-27pt}\text{B VIII}.\hspace{2pt}\frac{{\partial }^{\hspace{0.3pt}{8}}\varphi  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{8}}}={}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{8}}G^{\left(\mathrm{-}\mathrm{1}\right)}_{\mathrm{8}} \mathrm{(}\lambda \mathrm{)} = {\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{8}}\Bigl(\mathrm{40320}\lambda \frac{\partial \varphi \mathrm{(}\lambda \mathrm{)}}{\partial \lambda }+\mathrm{141120}{\lambda }^{\mathrm{2}}\frac{{\partial }^{\hspace{0.3pt}{2}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+\\  +\mathrm{141120}{\lambda }^{\mathrm{3}}\frac{{\partial }^{\hspace{0.3pt}{3}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{58800}{\lambda }^{\mathrm{4}}\frac{{\partial }^{\hspace{0.3pt}{4}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\mathrm{11760}{\lambda }^{\mathrm{5}}\frac{{\partial }^{\hspace{0.3pt}{5}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}+\mathrm{1176}{\lambda }^{\mathrm{6}}\frac{{\partial }^{\hspace{0.3pt}{6}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}}+\mathrm{56}{\lambda }^{\mathrm{7}}\frac{{\partial }^{\hspace{0.3pt}{7}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{7}}}+\\  +{\lambda }^{\mathrm{8}}\frac{\partial^{\hspace{0.3pt}{8}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial{\lambda }^{\mathrm{8}}}\Bigr)\tag{31}\label{myeq31} \end{array} \hspace{-0.5em}$
$\begin{array}{l}\hspace{-45pt}\text{B IX}.\hspace{2pt}\frac{{\partial }^{\hspace{0.3pt}{9}}\varphi  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{9}}}={-}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{9}}G^{\left(\mathrm{-}\mathrm{1}\right)}_{\mathrm{9}} \mathrm{(}\lambda \mathrm{)} = {-}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{9}}\Bigl(\mathrm{362880}\lambda \frac{\partial \varphi \mathrm{(}\lambda \mathrm{)}}{\partial \lambda }+\mathrm{1451520}{\lambda }^{\mathrm{2}}\frac{{\partial }^{\hspace{0.3pt}{2}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+\\  +\mathrm{1693440}{\lambda }^{\mathrm{3}}\frac{{\partial }^{\hspace{0.3pt}{3}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{846720}{\lambda }^{\mathrm{4}}\frac{{\partial }^{\hspace{0.3pt}{4}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\mathrm{211680}{\lambda }^{\mathrm{5}}\frac{{\partial }^{\hspace{0.3pt}{5}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}+\mathrm{28224}{\lambda }^{\mathrm{6}}\frac{{\partial }^{\hspace{0.3pt}{6}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}}+\\+\mathrm{2016}{\lambda }^{\mathrm{7}}\frac{{\partial }^{\hspace{0.3pt}{7}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{7}}}+\mathrm{72}{\lambda }^{\mathrm{8}}\frac{{\partial }^{\hspace{0.3pt}{8}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{8}}}+{\lambda }^{\mathrm{9}}\frac{\partial ^{\mathrm{9}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{9}}}\Bigr)\tag{32}\label{myeq32} \end{array} \hspace{-0.5em}$
$\begin{array}{l}\hspace{-32pt}\text{B X}.\hspace{2pt}\frac{{\partial }^{\hspace{0.3pt}{10}}\varphi  \mathrm{(}\omega \mathrm{)}}{\partial {\omega }^{\mathrm{10}}}={}{\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{10}}G^{\left(\mathrm{-}\mathrm{1}\right)}_{\mathrm{10}} \mathrm{(}\lambda \mathrm{)} = {\left(\frac{\lambda }{\mathrm{2}\pi c}\right)}^{\mathrm{10}}\Bigl(\mathrm{3628800}\lambda \frac{\partial \varphi \mathrm{(}\lambda \mathrm{)}}{\partial \lambda }+\mathrm{16329600}{\lambda }^{\mathrm{2}}\frac{{\partial }^{\hspace{0.3pt}{2}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{2}}}+\\+\mathrm{21772800}{\lambda }^{\mathrm{3}}\frac{{\partial }^{\hspace{0.3pt}{3}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{3}}}+\mathrm{12700800}{\lambda }^{\mathrm{4}}\frac{{\partial }^{\hspace{0.3pt}{4}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{4}}}+\mathrm{3810240}{\lambda }^{\mathrm{5}}\frac{{\partial }^{\hspace{0.3pt}{5}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{5}}}+\mathrm{635040}{\lambda }^{\mathrm{6}}\frac{{\partial }^{\hspace{0.3pt}{6}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{6}}}+\\+\mathrm{60480}{\lambda }^{\mathrm{7}}\frac{{\partial }^{\hspace{0.3pt}{7}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{7}}}  +\mathrm{3240}{\lambda }^{\mathrm{8}}\frac{{\partial }^{\hspace{0.3pt}{8}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{8}}}+\mathrm{90}{\lambda }^{\mathrm{9}}\frac{{\partial }^{\hspace{0.3pt}{9}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{9}}}+{\lambda }^{\mathrm{10}}\frac{{\partial }^{\hspace{0.3pt}{10}}\varphi \mathrm{(}\lambda \mathrm{)}}{\partial {\lambda }^{\mathrm{10}}}\Bigr)\tag{33}\label{myeq33} \end{array} \hspace{-0.5em}$ 

For more information, please visit the original papers Ref [1], Ref [2], Ref [3].

 Cite as:

[1]. D. Popmintchev, et al., "Analytical Lah-Laguerre optical formalism for perturbative chromatic dispersion ", Optics Express 30, 22,  pp. 40779-40808, 20 October 2022 DOI: https://doi.org/10.1364/OE.457139, DOI: 10.1364/OE.457139.

[2]. D. Popmintchev, et al., “Dispersion orders of various materials,” figshare (2022), https://doi.org/10.6084/m9.figshare.19236792. 

[3]. D. Popmintchev, et al., "Theory of the Chromatic Dispersion, Revisited", arXiv:2011, 30  October 2020, https://doi.org/10.48550/arXiv.2011.00066, DOI: 10.48550/arXiv.2011.00066.

 

Attosecond - picometer resolution science is .. a hand away.

 

12 Octave Coherent High Harmonics’ X-ray Supercontinua Enable Ultra-Precise Measurements on a Tabletop.

Published in Phys. Rev. Lett. 120, 093002 March 2018

      

Coherent X-rays are an ideal probe of the ultra-fast and ultra-small atomic and molecular world, due to their nature of having extremely short wavelength, short pulse duration, unique ability to penetrate thick and opaque objects combined with elemental specific fingerprints from the atomic absorption edges.

The large synchrotron and free electron laser facilities are the forefront of science, however tabletop sources are becoming more and more capable of performing complimentary invivo studies in the biological and material sciences. The other alternative source, the high harmonics, originate from a highly nonlinear process of energy upshift, when a high power laser is focused into a gas with intensities close to the binding atomic potentials. In this process thousands of photons can add to create a single photon with X-ray energy.

      

While Ultrafast spectroscopies in the visible and infrared (IR) energy regions can probe non-equilibrium dynamics on the ultrafast time scale, they cannot determine the chemical nature of the excited state dynamics with elemental selectivity. Alternatively, coherent X-ray spectroscopies have an important additional advantage of being able to uncover oxidation state, magnetic state, or charge localization to specific elements, giving information about the electronic structures, (i.e., valence, bands and charge), as well as orbital and spin ordering phenomena.

Today, atomic structure can be determined routinely using incoherent X-ray crystallography techniques. Unlike X-ray crystallography, which requires samples to be crystalline, XAFS spectroscopy can extract dynamic local structure information from various phases (i.e crystals, gases, low concentration solutions, disordered solids etc.), with some prior knowledge. Making it a viable computational imaging technique at the attosecond - picometer scale on up.

The near- and extended- X-ray absorption fine structure (XAFS) is a universal response of matter interacting with X-ray light near an absorption edge. This quantum phenomenon can be understood in a simple three-step model, where an incident X-ray is absorbed by an atom, which leads to the ejection of a photoelectron from a core-shell. Then the photoelectron is scattered from neighboring non-excited atoms, and the quantum interferences of the outgoing and incoming scattered electron waves lead to an energy-dependent variation of the X-ray absorption probability. With some prior knowledge, these techniques can provide full spatio-temporal and chemical imaging information (4+1D) of the local atomic structure of materials, including 1) the types of atoms that are present, coordination numbers (number of neighboring atoms at particular distances), inter-atomic distances, as well as disorder (using EXAFS), and 2) the oxidation state, and coordination chemistry (i.e., symmetries, isomerism, etc.) (using NEXAFS).

Using coherent high harmonic supercontinua laser-like beams, high quality extended edge coherent absorption spectroscopy of materials have been observed in the near-keV region for the first time, to obtain structural and chemical information. Published in Phys. Rev. Lett. 2018, DOI: 10.1103/PhysRevLett.120.093002

In the near future, because high harmonic X-ray bursts emerge as an isolated femtosecond-to-attosecond pulses, it will be possible to perform extensive spatio-temporal and chemical imaging of the local atomic structure of materials.

 The attosecond -picometer resolution science is .. a hand away.

cite as:

[1]. Popmintchev, Dimitar, et. al., Phys. Rev. Lett. 120, 093002, 1 March 2018 DOI: Phys. Rev. Lett. 2018, DOI: 10.1103/PhysRevLett.120.093002

[2]. Popmintchev, Dimitar, "Quantum and Extreme Nonlinear Optics Design of Coherent Ultrafast X-ray Light and Applications" ISBN 9781369490015 2016 https://www.amazon.com/dp/B01N9G9JVZ

[3]. Popmintchev, Tenio, et. al., Science  Vol. 336, Issue 6086, pp. 1287-1291 08 Jun 2012  DOI: Science, DOI: 10.1126/science.1218497