* [13] Palash Dey, Neeldhara Misra, and Y. Narahari.  Frugal bribery in voting.  Theor. Comput. Sci., 676:15–32, 2017.
  * [14] Palash Dey, Neeldhara Misra, Swaprava Nath, and Garima Shakya.  A parameterized perspective on protecting elections.  Theor. Comput. Sci., 874:15–31, 2021.  doi:10.1016/j.tcs.2021.05.006.
  * [15] Britta Dorn and Dominikus Krüger.  On the hardness of bribery variants in voting with cp-nets.  Ann. Math. Artif. Intell., 77(3-4):251–279, 2016.  URL: http://dx.doi.org/10.1007/s10472-015-9469-3, doi:10.1007/s10472-015-9469-3.
  * [16] Britta Dorn and Ildikó Schlotter.  Multivariate complexity analysis of swap bribery.  Algorithmica, 64(1):126–151, 2012.
  * [17] Jack Edmonds and Richard M Karp.  Theoretical improvements in algorithmic efficiency for network flow problems.  Journal of the ACM (JACM), 19(2):248–264, 1972.
  * [18] Edith Elkind and Piotr Faliszewski.  Approximation algorithms for campaign management.  In International Workshop on Internet and Network Economics, pages 473–482. Springer, 2010.
  * [19] Edith Elkind, Piotr Faliszewski, and Arkadii Slinko.  Swap bribery.  In Proc. 2nd International Symposium on Algorithmic Game Theory (SAGT 2009), pages 299–310. Springer, 2009.
  * [20] Edith Elkind, Piotr Faliszewski, and Arkadii Slinko.  Swap bribery.  In Marios Mavronicolas and Vicky G. Papadopoulou, editors, Algorithmic Game Theory, pages 299–310, Berlin, Heidelberg, 2009. Springer Berlin Heidelberg.
  * [21] Gabor Erdelyi, Edith Hemaspaandra, and Lane A Hemaspaandra.  Bribery and voter control under voting-rule uncertainty.  In Proceedings of the 2014 international conference on Autonomous agents and multi-agent systems, pages 61–68, 2014.
  * [22] Piotr Faliszewski.  Nonuniform bribery.  In Proceedings of the 7th International Joint Conference on Autonomous Agents and Multiagent Systems - Volume 3, AAMAS ’08, page 1569–1572. International Foundation for Autonomous Agents and Multiagent Systems, 2008.
  * [23] Piotr Faliszewski, Rica Gonen, Martin Kouteckỳ, and Nimrod Talmon.  Opinion diffusion and campaigning on society graphs.  In Proc. 27th International Joint Conference on Artificial Intelligence, IJCAI, pages 219–225, 2018.
  * [24] Piotr Faliszewski, Edith Hemaspaandra, and Lane A Hemaspaandra.  The complexity of bribery in elections.  In AAAI, volume 6, pages 641–646, 2006.
  * [25] Piotr Faliszewski, Edith Hemaspaandra, and Lane A Hemaspaandra.  How hard is bribery in elections?  Journal of artificial intelligence research, 35:485–532, 2009.
  * [26] Piotr Faliszewski, Edith Hemaspaandra, Lane A Hemaspaandra, and Jörg Rothe.  Llull and copeland voting computationally resist bribery and constructive control.  Journal of Artificial Intelligence Research, 35:275–341, 2009.
  * [27] Piotr Faliszewski, Edith Hemaspaandra, and Henning Schnoor.  Copeland voting: Ties matter.  In Proceedings of the 7th international joint conference on Autonomous agents and multiagent systems-Volume 2, pages 983–990. Citeseer, 2008\.
  * [28] Piotr Faliszewski, Yannick Reisch, Jörg Rothe, and Lena Schend.  Complexity of manipulation, bribery, and campaign management in bucklin and fallback voting.  In Proc. 13th International Conference on Autonomous Agents and Multiagent Systems (AAMAS), pages 1357–1358, 2014.
  * [29] Piotr Faliszewski, Yannick Reisch, Jörg Rothe, and Lena Schend.  Complexity of manipulation, bribery, and campaign management in bucklin and fallback voting.  Autonomous Agents and Multi-Agent Systems, 29(6):1091–1124, 2015\.
  * [30] Piotr Faliszewski and Jörg Rothe.  Control and bribery in voting.  In Felix Brandt, Vincent Conitzer, Ulle Endriss, Jérôme Lang, and Ariel D. Procaccia, editors, Handbook of Computational Social Choice, pages 146–168. Cambridge University Press, 2016.  doi:10.1017/CBO9781107446984.008.
  * [31] Delbert R Fulkerson.  An out-of-kilter method for minimal-cost flow problems.  Journal of the Society for Industrial and Applied Mathematics, 9(1):18–27, 1961.
  * [32] Michael R. Garey and David S. Johnson.  Computers and Intractability; A Guide to the Theory of NP-Completeness.  W. H. Freeman & Co., USA, 1990.
  * [33] Noam Hazon and Edith Elkind.  Complexity of safe strategic voting.  In International Symposium on Algorithmic Game Theory, pages 210–221. Springer, 2010.
  * [34] Egor Ianovski, Lan Yu, Edith Elkind, and Mark C Wilson.  The complexity of safe manipulation under scoring rules.  In Twenty-Second International Joint Conference on Artificial Intelligence, 2011.
  * [35] Benjamin N Jackson, Patrick S Schnable, and Srinivas Aluru.  Consensus genetic maps as median orders from inconsistent sources.  IEEE/ACM Transactions on computational biology and bioinformatics, 5(2):161–171, 2008.
  * [36] Andrzej Kaczmarczyk and Piotr Faliszewski.  Algorithms for destructive shift bribery.  In Proc. 15th International Conference on Autonomous Agents & Multiagent Systems (AAMAS), pages 305–313, 2016.
  * [37] Orgad Keller, Avinatan Hassidim, and Noam Hazon.  Approximating bribery in scoring rules.  In Proc. 32nd International Conference on Artificial Intelligence (AAAI), pages 1121–1129, 2018.
  * [38] Dušan Knop, Martin Koutecký, and Matthias Mnich.  A unifying framework for manipulation problems.  In Proc. 17th International Conference on Autonomous Agents and MultiAgent Systems (AAMAS), pages 256–264, 2018.
  * [39] Luke Mathieson and Stefan Szeider.  Editing graphs to satisfy degree constraints: A parameterized approach.  Journal of Computer and System Sciences, 78(1):179–191, 2012.  JCSS Knowledge Representation and Reasoning.  URL: https://www.sciencedirect.com/science/article/pii/S0022000011000067, doi:https://doi.org/10.1016/j.jcss.2011.02.001.
  * [40] Nicholas Mattei, Maria Silvia Pini, Francesca Rossi, and Kristen Brent Venable.  Bribery in voting over combinatorial domains is easy.  In ISAIM, 2012.
  * [41] Cynthia Maushagen, Marc Neveling, Jörg Rothe, and Ann-Kathrin Selker.  Complexity of shift bribery in iterative elections.  In Proc. 17th International Conference on Autonomous Agents and MultiAgent Systems (AAMAS), pages 1567–1575, 2018.
  * [42] Svetlana Obraztsova and Edith Elkind.  Optimal manipulation of voting rules.  In Proc. 26th AAAI Conference on Artificial Intelligence (AAAI), pages 2141–2147, 2012.
  * [43] Svetlana Obraztsova and Edith Elkind.  Optimal manipulation of voting rules.  In Proc. 11th International Conference on Autonomous Agents and Multiagent Systems (AAMAS), pages 619–626, 2012.
  * [44] David M. Pennock, Eric Horvitz, and C. Lee Giles.  Social choice theory and recommender systems: Analysis of the axiomatic foundations of collaborative filtering.  In Proc. 17th National Conference on Artificial Intelligence and 12th Conference on on Innovative Applications of Artificial Intelligence, July 30 - August 3, 2000, Austin, Texas, USA., pages 729–734, 2000.  URL: http://www.aaai.org/Library/AAAI/2000/aaai00-112.php.
  * [45] Maria Silvia Pini, Francesca Rossi, and Kristen Brent Venable.  Bribery in voting with soft constraints.  In Proceedings of the AAAI Conference on Artificial Intelligence, volume 27, 2013.
  * [46] Ildikó Schlotter, Piotr Faliszewski, and Edith Elkind.  Campaign management under approval-driven voting rules.  In Proceedings of the AAAI Conference on Artificial Intelligence, volume 25, 2011.
  * [47] Ildikó Schlotter, Piotr Faliszewski, and Edith Elkind.  Campaign management under approval-driven voting rules.  Algorithmica, 77(1):84–115, 2017.
  * [48] Arkadii Slinko and Shaun White.  Nondictatorial social choice rules are safely manipulable.  In Proceedings of the Second International Workshop on Computational Social Choice (COMSOC-2008), pages 403–414, 2008.
  * [49] Arkadii Slinko and Shaun White.  Is it ever safe to vote strategically?  Social Choice and Welfare, 43(2):403–427, 2014.
  * [50] Lirong Xia.  Computing the margin of victory for various voting rules.  In Proceedings of the 13th ACM Conference on Electronic Commerce, pages 982–999, 2012.


DRAFT 45 4 0.85 font=normalsize

#  Elastic bounds for anisotropic layers
\- PREPRINT -

Paolo Vannucci
LMV - Laboratoire de Mathématiques de Versailles UMR8100
Université Paris-Saclay - UVSQ
<EMAIL_ADDRESS>
###### Abstract

The complete set of bounds for the technical constants of an elastic layer,
plate or laminate is given. The bounds are valid in general, also for
completely anisotropic bodies. They are obtained transforming the polar bounds
previously found. These bounds complete the knowledge of classical elasticity
at least in the two-dimensional case and are useful in several situations,
e.g., for determining the correct feasibility domain in design problems or as
necessary conditions for accepting the results of laboratory tests on
anisotropic layers.

Key words:  anisotropy, planar elasticity, elastic bounds, polar formalism

##  1 Introduction

The existence and determination of bounds for the moduli of a material is a
well known topic in the theory of elasticity. As well known, such bounds are
the necessary result of the physical condition imposing the positiveness of
the work done by the applied forces on an elastic body for deforming this one.

This problem can be solved using either purely mathematical, cf. [1], or more
directly mechanically inspired approaches, cf. [2]. For isotropic materials,
this leads to the well known bounds for either the Lamé’s parameters

$3\lambda+2\mu>0,\ \ \mu>0,$ (1)

or for the more commonly used technical parameters, i.e. the Young’s modulus
and the Poisson’s ratio,

$E>0,\ \ -1<\nu<\frac{1}{2}.$ (2)

For anisotropic materials, a, basically mathematic, approach, based upon a
somewhat forgotten theorem, cf. [1], page 340, allows to give a general form
to the bounds of the components of the stiffness (or alternatively of the
compliance) matrix $[C]$,

$\\{\sigma\\}=[C]\\{\varepsilon\\},$ (3)

that traduces, through the Kelvin’s notation, [3, 4], the components of the
fourth-order elastic tensor $\mathbb{E}$ into the components of a $6\times 6$
symmetric matrix, according to the rule

$[C]=\footnotesize\left[\begin{array}[]{llllll}C_{11}=\mathbb{E}_{1111}&C_{12}=\mathbb{E}_{1122}&C_{13}=\mathbb{E}_{1133}&C_{14}=\sqrt{2}\mathbb{E}_{1123}&C_{15}=\sqrt{2}\mathbb{E}_{1131}&C_{16}=\sqrt{2}\mathbb{E}_{1112}\\\
&C_{22}=\mathbb{E}_{2222}&C_{23}=\mathbb{E}_{2233}&C_{24}=\sqrt{2}\mathbb{E}_{2223}&C_{25}=\sqrt{2}\mathbb{E}_{2231}&C_{26}=\sqrt{2}\mathbb{E}_{2212}\\\
&&C_{33}=\mathbb{E}_{3333}&C_{34}=\sqrt{2}\mathbb{E}_{3323}&C_{35}=\sqrt{2}\mathbb{E}_{3331}&C_{36}=\sqrt{2}\mathbb{E}_{3312}\\\
&&&C_{44}=2\mathbb{E}_{2323}&C_{45}=2\mathbb{E}_{2331}&C_{46}=2\mathbb{E}_{2312}\\\
&&&&C_{55}=2\mathbb{E}_{3131}&C_{56}=2\mathbb{E}_{3112}\\\
&&&&&C_{66}=2\mathbb{E}_{1212}\end{array}\right].$ (4)

By this theorem, the positiveness of $[C]$ is get imposing that the six
leading principal minors of $[C]$ are all positive. This approach has the
advantage of clearly fixing the number, six, of bounds to be written in the
most general case; in the presence of some material symmetry, these bounds are
less than six and can be written in an explicit form, cf. [5].

However, and rather surprisingly, for three-dimensional anisotropic bodies the
determination of the bounds for the technical moduli is still an open problem,
only partial results are known in the literature. In particular, cf. [6] or
[7], the following conditions are normally given in the literature:

$\begin{array}[]{c}\forall i,j\in\\{1,2,3\\},\ E_{i}>0,\ \ G_{ij}>0,\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\dfrac{1-2\nu_{12}}{E_{1}}+\dfrac{1-2\nu_{23}}{E_{2}}+\dfrac{1-2\nu_{31}}{E_{3}}>0,\\\
\end{array}$ (5)

Another, rougher, bound is given by Lekhnitskii, [6],

$\nu_{12}+\nu_{23}+\nu_{31}<\frac{3}{2}.$ (6)

These are the only bounds valid for any elastic body, regardless from its
syngony, i.e. they are valid also for triclinic materials. Some other bounds
are known but uniquely for bodies that are at least orthotropic and with the
moduli measured in a reference frame whose axes correspond with some
equivalent directions for the material:

$\begin{array}[]{c}1-\nu_{ij}\nu_{ji}>0\ \ \forall i,j\in\\{1,2,3\\},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
1-\nu_{12}\nu_{21}-\nu_{23}\nu_{32}-\nu_{31}\nu_{13}-2\nu_{32}\nu_{21}\nu_{13}>0,\end{array}$
(7)

conditions that can be transformed, respectively, to

$\begin{array}[]{c}|\nu_{ij}|<\sqrt{\dfrac{E_{i}}{E_{j}}},\vskip 6.0pt plus
2.0pt minus 2.0pt\\\
\nu_{32}\nu_{21}\nu_{13}<\dfrac{1}{2}\left(1-\nu_{32}^{2}\dfrac{E_{2}}{E_{3}}-\nu_{21}^{2}\dfrac{E_{1}}{E_{2}}-\nu_{13}^{2}\dfrac{E_{3}}{E_{1}}\right)<\dfrac{1}{2}.\end{array}$
(8)

All these bounds on the technical constants are found imposing the
positiveness of the strain energy for peculiar stress fields, e.g. pure
extension or shear.

Unlike in the general three-dimensional case, in planar elasticity the problem
has been completely solved in a particular set of elastic moduli, the so-
called polar parameters. However, the corresponding of the polar bounds for
the technical constants have never been given. This is the topic of this
paper, organized as follows: in the next Section, the polar bounds are
recalled, then the bounds for the technical constants in planar elasticity are
obtained and finally some particular cases discussed.

##  2 Polar bounds

The polar formalism was introduced in 1979 by G. Verchery, [8]; a complete
account of the method can be found in [9, 5], here only some elements of this
method, necessary to the developments, are recalled.

By the polar method, the Cartesian components at a direction $\theta$ of a
matrix, e.g. $[C]$, representing, in the Kelvin notation, a plane elastic
tensor, e.g. $\mathbb{E}$, are expressed as

${\begin{array}[]{l}{C_{11}{(}\theta{)}={T}_{0}{+}{2}{T}_{1}{+}{R}_{0}\cos{4}\left({{{\varPhi}}_{0}{-}\theta}\right){+}{4}{R}_{1}\cos{2}\left({{{\varPhi}}_{1}{-}\theta}\right)},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
{C_{12}{(}\theta{)}={-}{T}_{0}{+}{2}{T}_{1}{-}{R}_{0}\cos{4}\left({{{\varPhi}}_{0}{-}\theta}\right)},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ C_{16}(\theta)=\sqrt{2}\left[R_{0}\sin
4\left(\varPhi_{0}-\theta\right)+2R_{1}\sin
2\left(\varPhi_{1}-\theta\right)\right],\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
{C_{22}{(}\theta{)}={T}_{0}{+}{2}{T}_{1}{+}{R}_{0}\cos{4}\left({{{\varPhi}}_{0}{-}\theta}\right){-}{4}{R}_{1}\cos{2}\left({{{\varPhi}}_{1}{-}\theta}\right)},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
{C_{26}{(}\theta{)}=\sqrt{2}\left[{-}{R}_{0}\sin{4}\left({{{\varPhi}}_{0}{-}\theta}\right){+}{2}{R}_{1}\sin{2}\left({{{\varPhi}}_{1}{-}\theta}\right)\right]},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
{C_{66}{(}\theta{)}=2\left[{T}_{0}{-}{R}_{0}\cos{4}\left({{{\varPhi}}_{0}{-}\theta}\right)\right]}.\end{array}}$
(9)

The moduli $T_{0},T_{1},R_{0},R_{1}$ as well as the difference of the angles
$\varPhi_{0}-\varPhi_{1}$ are tensor invariants. The choice of one of the two
polar angles fixes the frame; usually $\varPhi_{1}=0$.

The converse of the previous equations are

$\begin{array}[]{l}T_{0}=\dfrac{1}{8}(C_{11}-2C_{12}+2C_{66}+C_{22}),\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
T_{1}=\dfrac{1}{8}(C_{11}+2C_{12}+C_{22}),\vskip 6.0pt plus 2.0pt minus
2.0pt\\\
R_{0}=\dfrac{1}{8}\sqrt{(C_{11}-2C_{12}-2C_{66}+C_{22})^{2}+8(C_{16}-C_{26})^{2}},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
R_{1}=\dfrac{1}{8}\sqrt{(C_{11}-C_{22})^{2}+2(C_{16}+C_{26})^{2}},\vskip 6.0pt
plus 2.0pt minus 2.0pt\\\ \tan
4\varPhi_{0}=\dfrac{2\sqrt{2}(C_{16}-C_{26})}{C_{11}-2C_{12}-2C_{66}+C_{22}},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ \tan
2\varPhi_{1}=\dfrac{2\sqrt{2}(C_{16}+C_{26})}{C_{11}-C_{22}}.\end{array}$ (10)

The elastic symmetries are determined by the following conditions on the
invariants:

  * •

ordinary orthotropy: $\varPhi_{0}-\varPhi_{1}=K\dfrac{\pi}{4},\
K\in\\{0,1\\}$;

  * •

$R_{0}$-orthotropy: $R_{0}=0$, [10];

  * •

square symmetry: ${R_{1}=0}$;

  * •

isotropy: $R_{0}=R_{1}=0$.

So we see that $T_{0}$ and $T_{1}$ are the isotropy invariants, while
$R_{0},R_{1}$ and $\varPhi_{0}-\varPhi_{1}$ are the anisotropy invariants. The
above relations are valid for any matrix of the elastic type, hence for the
compliance matrix $[S]=[C]^{-1}$ too; we will indicate by
$t_{0},t_{1},r_{0},r_{1},\varphi_{0}$ and $\varphi_{1}$ the polar parameters
of $[S]$.

For the components of the vector, say $\\{L\\}$, representing in the Kelvin
formalism a second-rank symmetric tensor L, the polar formalism gives

$\begin{split}&L_{1}(\theta)=T+R\cos 2(\varPhi-\theta),\vskip 6.0pt plus 2.0pt
minus 2.0pt\\\ &L_{2}(\theta)=T-R\cos 2(\varPhi-\theta),\vskip 6.0pt plus
2.0pt minus 2.0pt\\\ &L_{6}(\theta)=\sqrt{2}R\sin
2(\varPhi-\theta),\end{split}$ (11)

with $T,R$ two invariants, representing respectively the isotropic and the
anisotropic phases of L; $\varPhi$ is an angle determined by the choice of the
frame.

The polar bounds can be found in the following way, [5]: the strain energy
density per unit volume,

$V_{\varepsilon}=\frac{1}{2}\\{\sigma\\}^{\top}\\{\varepsilon\\}=\frac{1}{2}\\{\varepsilon\\}^{\top}[C]\\{\varepsilon\\},$
(12)

can be written using the polar components of $[C]$ and $\\{\varepsilon\\}$
through eqs. (9) and (11):

$V_{\varepsilon}=4T_{1}\ t^{2}+8R_{1}\cos 2(\varPhi_{1}-\varphi)r\
t+2[T_{0}+R_{0}\cos 4(\varPhi_{0}-\varphi)]r^{2},$ (13)

where $t,r$ and $\varphi$ are the polar parameters of $\\{\varepsilon\\}$.
This quantity can be rewritten as

$V_{\varepsilon}=\\{r,t\\}\left[\begin{array}[]{cc}2[T_{0}+R_{0}\cos
4(\varPhi_{0}-\varphi)]&4R_{1}\cos 2(\varPhi_{1}-\varphi)\\\ 4R_{1}\cos
2(\varPhi_{1}-\varphi)&4T_{1}\end{array}\right]\left\\{\begin{array}[]{c}r\\\
t\end{array}\right\\}.$ (14)

The positivity of $V_{\varepsilon}\ \forall\\{r,t\\}$, stating the physical
condition of a positive work done by the applied forces, is ensured if and
only if the matrix in the previous equation is positive definite; by the
already mentioned theorem on the leading principal minors, [1], this happens
if and only if the following two conditions are satisfied:

$\left\\{\begin{array}[]{l}T_{0}+R_{0}\cos 4(\varPhi_{0}-\varphi)\vskip 6.0pt
plus 2.0pt minus 2.0pt\\\ T_{1}[T_{0}+R_{0}\cos
4(\varPhi_{0}-\varphi)]-2R_{1}^{2}\cos^{2}2(\varPhi_{1}-\varphi)>0\end{array}\
\ \ \ \forall\varphi.\right.$ (15)

These two conditions can be transformed to three other inequalities and it can
be proved that one of them is redundant (the complete, technical, proof is
omitted here, the reader is addressed to [11] or to [5]). In the end, one gets
the bounds

$\begin{array}[]{l}T_{0}-R_{0}>0,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
T_{1}\left(T_{0}^{2}-R_{0}^{2}\right)-2R_{1}^{2}[T_{0}-R_{0}\cos
4(\varPhi_{0}-\varPhi_{1})]>0.\end{array}$ (16)

To remark that, being moduli of complex numbers, $R_{0}\geq 0,R_{1}\geq 0$,
which necessarily implies, through eqs. (15) and (16)${}_{1},T_{0}>0,T_{1}>0$.

The above bounds are general, i.e. valid for any type of layer, and are
written in terms of polar invariant, so they are intrinsic bounds, i.e. frame
independent. They can be applied as well to the polar parameters of $[C]$ or
of $[S]$:

$\begin{array}[]{l}t_{0}-r_{0}>0,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
t_{1}\left(t_{0}^{2}-r_{0}^{2}\right)-2r_{1}^{2}[t_{0}-r_{0}\cos
4(\varphi_{0}-\varphi_{1})]>0.\end{array}$ (17)

These last bounds can be obtained following the same procedure but starting
from the stress energy

$V_{\sigma}=\frac{1}{2}\\{\sigma\\}^{\top}[S]\\{\sigma\\}.$ (18)

Of course, the same bounds can be written also for any other elastic-type
planar tensor, like, for instance, the extension and bending stiffness, and
compliance, tensors of laminates, see e.g. [7, 12, 5].

##  3 The bounds for the technical constants

In order to find the bounds for the technical constants, first, eq. (10) is
written for the components of $[S]$:

$\begin{array}[]{l}t_{0}=\dfrac{1}{8}(S_{11}-2S_{12}+2S_{66}+S_{22}),\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
t_{1}=\dfrac{1}{8}(S_{11}+2S_{12}+S_{22}),\vskip 6.0pt plus 2.0pt minus
2.0pt\\\
r_{0}=\dfrac{1}{8}\sqrt{(S_{11}-2S_{12}-2S_{66}+S_{22})^{2}+8(S_{16}-S_{26})^{2}},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
r_{1}=\dfrac{1}{8}\sqrt{(S_{11}-S_{22})^{2}+2(S_{16}+S_{26})^{2}},\vskip 6.0pt
plus 2.0pt minus 2.0pt\\\ \tan
4\varphi_{0}=\dfrac{2\sqrt{2}(S_{16}-S_{26})}{S_{11}-2S_{12}-2S_{66}+S_{22}},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ \tan
2\varphi_{1}=\dfrac{2\sqrt{2}(S_{16}+S_{26})}{S_{11}-S_{22}}.\end{array}$ (19)

Then, this result is used into eq. (17): the first condition becomes

$\dfrac{1}{8}(S_{11}-2S_{12}+2S_{66}+S_{22})>\dfrac{1}{8}\sqrt{(S_{11}-2S_{12}-2S_{66}+S_{22})^{2}+8(S_{16}-S_{26})^{2}},$
(20)

which gives the two conditions (a third one, stating that the argument of the
square root at the second member must be positive, is redundant)

$\begin{array}[]{l}S_{11}-2S_{12}+2S_{66}+S_{22}>0,\vskip 6.0pt plus 2.0pt
minus 2.0pt\\\ S_{66}(S_{11}-2S_{12}+S_{22})>(S_{16}-S_{26})^{2}.\end{array}$
(21)

To transform eq. (17)2, it is worth to introduce the following polar
invariant, [5], page 145:

$c_{1}=8r_{1}^{2}r_{0}\cos 4(\varphi_{0}-\varphi_{1})$ (22)

which gives

$r_{0}\cos 4(\varphi_{0}-\varphi_{1})=\frac{c_{1}}{8r_{1}^{2}}.$ (23)

The Cartesian expression of $c_{1}$ is known:

$\begin{split}c_{1}=&\frac{1}{64}\left[(S_{11}-S_{22})^{2}-2(S_{16}+S_{26})^{2}\right](S_{11}-2S_{12}-2S_{66}+S_{22})+\\\
&+\frac{1}{8}(S_{11}-S_{22})\left(S_{16}^{2}+S_{26}^{2}\right).\end{split}$
(24)

Finally, eq. (17)2 becomes first

$t_{1}\left(t_{0}^{2}-r_{0}^{2}\right)>2r_{1}^{2}-\frac{c_{1}}{4},$ (25)

then, after some standard passages,

$2S_{12}S_{16}S_{26}+S_{11}S_{22}S_{66}-S_{22}S_{16}^{2}-S_{11}S_{26}^{2}-S_{66}S_{12}^{2}>0.$
(26)

To remark that actually

$2S_{12}S_{16}S_{26}+S_{11}S_{22}S_{66}-S_{22}S_{16}^{2}-S_{11}S_{26}^{2}-S_{66}S_{12}^{2}=\det[S].$
(27)

We get hence the three bounds for the Cartesian components of $[S]$

$\begin{array}[]{l}S_{11}-2S_{12}+2S_{66}+S_{22}>0,\vskip 6.0pt plus 2.0pt
minus 2.0pt\\\ S_{66}(S_{11}-2S_{12}+S_{22})>(S_{16}-S_{26})^{2},\vskip 6.0pt
plus 2.0pt minus 2.0pt\\\
2S_{12}S_{16}S_{26}+S_{11}S_{22}S_{66}-S_{22}S_{16}^{2}-S_{11}S_{26}^{2}-S_{66}S_{12}^{2}>0.\end{array}$
(28)

It is worth noting that this is not the only set of independent bounds that
can be found for the $S_{ij}$s: applying to $[S]$ the already cited theorem on
the leading principal minors, one should get three other bounds:

$\begin{array}[]{l}S_{11}>0,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
S_{11}S_{22}-S_{12}^{2}>0,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
2S_{12}S_{16}S_{26}+S_{11}S_{22}S_{66}-S_{22}S_{16}^{2}-S_{11}S_{26}^{2}-S_{66}S_{12}^{2}>0.\end{array}$
(29)

The main difference between the last two sets of bounds is that conditions
(28) uses exclusively invariant quantities, which is not the case for
conditions (29). Of course, imposing the positivity of the strain energy one
can get similar relations for the $C_{ij}$s.

The passage to the technical constants can be made recalling that, by
definition,

$\begin{array}[]{lll}S_{11}=\dfrac{1}{E_{1}},&S_{12}=-\dfrac{\nu_{12}}{E_{1}},&S_{16}=\dfrac{\eta_{12,1}}{\sqrt{2}E_{1}},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
S_{22}=\dfrac{1}{E_{2}},&S_{66}=\dfrac{1}{2G_{12}},&S_{26}=\dfrac{\eta_{12,2}}{\sqrt{2}E_{2}},\end{array}$
(30)

with $E_{1},E_{2}$ the Young’s moduli in the directions of the two frame axes,
$G_{12}$ the in-plane shear modulus, $\nu_{12}$ the in-plane Poisson’s ratio
and $\eta_{12,1},\eta_{12,2}$ two coefficients of mutual influence of the
second type, [7, 5]. Alternatively, the coefficients of mutual influence of
the first type $\eta_{k,ij}$ can be used, they are linked to the
$\eta_{ij,k}$s by the reciprocity relations

$\frac{\eta_{ij,k}}{E_{k}}=\frac{\eta_{k,ij}}{G_{ij}},\ \ i,j,k\in\\{1,2\\},\
i\neq j.$ (31)

Injecting eq. (30) into eq. (28) gives finally the bounds

$\begin{array}[]{l}\dfrac{1+2\nu_{12}}{E_{1}}+\dfrac{1}{E_{2}}+\dfrac{1}{G_{12}}>0,\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\dfrac{1}{E_{1}^{2}E_{2}^{2}G_{12}}\left\\{E_{1}E_{2}[E_{1}+E_{2}(1+2\nu_{12})]-G_{12}(E_{2}\eta_{12,1}-E_{1}\eta_{12,2})^{2}\right\\}>0,\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\dfrac{1}{E_{1}^{2}E_{2}^{2}G_{12}}\left\\{E_{1}(E_{2}-G_{12}\eta_{12,2}^{2})-E_{2}[E_{2}\nu_{12}^{2}+G_{12}\eta_{12,1}(\eta_{12,1}+2\eta_{12,2}\nu_{12})]\right\\}>0.\end{array}$
(32)

These three bounds are a set of necessary and sufficient conditions for the
elastic energy density is positive for each stress/strain state. They use
invariant quantities, which is undoubtedly an advantage in anisotropic
elasticity.

If, in place of eq. (28) we inject eq. (30) into eq. (29) we will get

$\begin{array}[]{l}\dfrac{1}{E_{1}}>0,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\dfrac{1}{E_{1}E_{2}}-\dfrac{\nu_{12}^{2}}{E_{1}^{2}}>0,\vskip 6.0pt plus
2.0pt minus 2.0pt\\\
\dfrac{1}{E_{1}^{2}E_{2}^{2}G_{12}}\left\\{E_{1}(E_{2}-G_{12}\eta_{12,2}^{2})-E_{2}[E_{2}\nu_{12}^{2}+G_{12}\eta_{12,1}(\eta_{12,1}+2\eta_{12,2}\nu_{12})]\right\\}>0.\end{array}$
(33)

The third condition remains the same, while the two first ones can be
rewritten as

$E_{1}>0,\ \ |\nu_{12}|<\sqrt{\frac{E_{1}}{E_{2}}}.$ (34)

The first one is the well-known condition of positiveness of the Young’s
moduli, while the second one corresponds to the bound (8)1. Also, these two
bounds, or alternatively the fact that $E_{1}>0$ for any direction, imply that
it is $E_{2}>0$ too. Finally, also $G_{12}>0$, which can be proved,
classically, with the mechanical procedure imposing a pure shear stress state
or, using a purely mathematical approach, still using the theorem on leading
principal minors once reordered the vector $\\{\sigma\\}$ representing, in the
Kelvin notation, the stress tensor $\bm{\sigma}$ as
$\\{\sigma\\}=\\{\sigma_{6},\sigma_{1},\sigma_{2}\\}^{\top}$.

Though eq. (32) are a minimal set of necessary and sufficient conditions for
ensuring the positivity of the elastic energy, it is perhaps better, from a
practical point of view, to dispose of more direct bounds, concerning, if
possible, some quantities easy to be measured in laboratory tests. This can be
done transforming eq. (32) and taking into account for the positivity of the
Young’s and shear moduli. Some short passages lead to the set of conditions

$\begin{array}[]{l}E_{1}>0,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
E_{2}>0,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\ G_{12}>0,\vskip 6.0pt plus
2.0pt minus 2.0pt\\\ E_{1}E_{2}+G_{12}[E_{1}+E_{2}(1+2\nu_{12})]>0,\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
E_{1}E_{2}[E_{1}+E_{2}(1+2\nu_{12})]-G_{12}(E_{2}\eta_{12,1}-E_{1}\eta_{12,2})^{2}>0,\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
E_{1}(E_{2}-G_{12}\eta_{12,2}^{2})-E_{2}[E_{2}\nu_{12}^{2}+G_{12}\eta_{12,1}(\eta_{12,1}+2\eta_{12,2}\nu_{12})]>0.\end{array}$
(35)

The above bounds, however, do not use invariant quantities. Though formed by a
redundant number of bounds, this set of conditions is perhaps more interesting
for practical applications, namely for checking the results of laboratory
tests used for characterizing an anisotropic layer or plate. To remark that in
this set of bounds the coefficients of mutual influence enter the problem; in
all the bounds known in the literature, these coefficients were absent.

##  4 Bounds for layers with material symmetries

Let us consider now how bounds (35) change where some kind of material
symmetry is present.

Isotropy. In this case $r_{0}=R_{0}=r_{1}=R_{1}=0$, so the polar bounds reduce
simply to

$t_{0}>0,\ \ t_{1}>0,$ (36)

while for the Cartesian components we have

$S_{11}=S_{22}=t_{0}+2t_{1},\ \ S_{12}=-t_{0}+2t_{1},\ \
S_{66}=2t_{0}=S_{11}-S_{12},\ \ S_{16}=S_{26}=0,$ (37)

which gives

$t_{0}=\frac{S_{11}-S_{12}}{4},\ \ t_{1}=\frac{S_{11}+S_{12}}{4},$ (38)

so the Cartesian bounds are simply

$S_{11}-S_{12}>0,\ \ S_{11}+S_{12}>0.$ (39)

Then, because for isotropy

$E_{1}=E_{2}:=E,\ \ \nu_{12}:=\nu,\ \ G_{12}:=G=\dfrac{E}{2(1+\nu)},\ \
\eta_{12,1}=\eta_{12,2}=0,$ (40)

bounds (35) reduce to

$E>0,\ \ G>0,\ \ 2E^{2}>0,\ \ 2E^{3}(1+\nu)>0,\ \ E^{2}(1-\nu^{2})>0,$ (41)

which of course give the three well known bounds for isotropic planar bodies

$E>0,\ \ -1<\nu<1,$ (42)

or

$\lambda+\mu>0,\ \ \mu>0$ (43)

with the Lamé’s parameters.

Ordinary orthotropy. The polar condition for orthotropy is

$\varphi_{0}-\varphi_{1}=k\frac{\pi}{4},\ \ k\in\\{0,1\\}.$ (44)

The discussion of the mechanical and mathematical differences between the two
types of ordinary orthotropy, $k=0$ or $k=1$, can be found in [9, 5]. However,
for the orthotropic case,

$r_{0}\cos 4(\varphi_{0}-\varphi_{1})=(-1)^{k}r_{0},$ (45)

but eq. (23) does not change, so bounds (35) remain the same: there is no
difference for the two ordinarily orthotropic cases. To remark that bounds
(35) are written in a generic frame, not necessarily in the orthotropic one.
If the axes coincide with the orthotropy directions, then
$\eta_{12,1}=\eta_{12,2}=0$ and bounds (35) become

$\begin{array}[]{l}E_{1}>0,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
E_{2}>0,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\ G_{12}>0,\vskip 6.0pt plus
2.0pt minus 2.0pt\\\ E_{1}+E_{2}(1+2\nu_{12})>0,\vskip 6.0pt plus 2.0pt minus
2.0pt\\\ E_{2}(E_{1}-E_{2}\nu_{12}^{2})>0,\end{array}$ (46)

the condition (35)4 becoming now redundant.

Square-symmetry. This special type of orthotropy is determined by the polar
condition $r_{1}=R_{1}=0$. The polar bounds (17) become

$\begin{array}[]{l}t_{0}-r_{0}>0,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
t_{1}(t_{0}^{2}-r_{0}^{2})>0,\end{array}$ (47)

which, being $r_{0}>0$, reduce to

$t_{0}-r_{0}>0,\ \ t_{1}>0.$ (48)

For the Cartesian components we have, for any direction,

$S_{11}=S_{22},\ \ S_{26}=-S_{16},$ (49)

so conditions (28) become

$\begin{array}[]{l}S_{11}-S_{12}+S_{66}>0,\vskip 6.0pt plus 2.0pt minus
2.0pt\\\ S_{66}(S_{11}-S_{12})>2S_{16}^{2},\vskip 6.0pt plus 2.0pt minus
2.0pt\\\ S_{11}+S_{12}>0.\end{array}$ (50)

For the technical constants it is

$E_{1}=E_{2},\ \ \eta_{12,2}=-\eta_{12,1}\frac{E_{2}}{E_{1}},$ (51)

so bounds (35) become

$\begin{array}[]{l}E_{1}>0,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
G_{12}>0,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
E_{1}+2G_{12}(1+\nu_{12})>0,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
E_{1}(1+\nu_{12})-2G_{12}\eta_{12,1}^{2}>0,\vskip 6.0pt plus 2.0pt minus
2.0pt\\\ 1-\nu_{12}>0.\end{array}$ (52)

When written in one of the two symmetry frames, shifted of $\pi/4$, because
$\eta_{12,1}=0$ we get

$\begin{array}[]{l}E_{1}>0,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
G_{12}>0,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
E_{1}+2G_{12}(1+\nu_{12})>0,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
-1<\nu_{12}<1.\end{array}$ (53)

$R_{0}$-orthotropy. This special orthotropy is characterized by $R_{0}=0$,
[10]. However, unlike the case of square symmetry, this does not imply that
$r_{0}=0$, but that

$r_{0}=\frac{r_{1}^{2}}{t_{1}},\ \ k=0,$ (54)

so that bounds (17) become

$\begin{array}[]{l}t_{0}-\dfrac{r_{1}^{2}}{t_{1}}>0,\vskip 6.0pt plus 2.0pt
minus 2.0pt\\\
t_{1}\left(t_{0}+\dfrac{r_{1}^{2}}{t_{1}}\right)-2r_{1}^{2}>0.\end{array}$
(55)

Rather surprisingly, the bounds on the $S_{ij}$s, eq. (28) reduce to only two,
the third one being a product of the first two ones (this can be checked using
eqs. (19)1,2,4 into the previous equations):

$\begin{array}[]{l}S_{11}+2S_{12}+S_{22}>0,\vskip 6.0pt plus 2.0pt minus
2.0pt\\\
2(S_{11}S_{22}-S_{12}^{2})+S_{66}(S_{11}+2S_{12}+S_{22})-(S_{16}+S_{26})^{2}>0.\end{array}$
(56)

Injecting eq. (30) into these conditions, the bounds for the technical
constants are obtained as well:

$\begin{array}[]{l}E_{1}>0,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
E_{2}>0,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\ G_{12}>0,\vskip 6.0pt plus
2.0pt minus 2.0pt\\\ E_{1}+E_{2}(1-2\nu_{12})>0,\vskip 6.0pt plus 2.0pt minus
2.0pt\\\
E_{1}^{2}(E_{2}-G_{12}\eta_{12,2}^{2})-E_{2}^{2}G_{12}(\eta_{12,1}^{2}+4\nu_{12}^{2})+\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
+E_{1}E_{2}[E_{2}(1-2\nu_{12})+2G_{12}(2-\eta_{12,1}\eta_{12,2})]>0.\end{array}$
(57)

$r_{0}$-orthotropy. This is the compliance dual of the previous case and it
characterizes some special materials like paper,[13]. In such a case, the
polar bounds (17) reduce to

$\begin{array}[]{l}t_{0}>0,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
t_{0}t_{1}-2r_{1}^{2}>0.\end{array}$ (58)

Concerning the components of $[S]$,

${\begin{array}[]{l}{S_{11}{(}\theta{)}={t}_{0}{+}{2}{t}_{1}{+}{4}{r}_{1}\cos{2}\left({{{\varphi}}_{1}-\theta}\right)},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
{S_{12}{(}\theta{)}={-}{t}_{0}{+}{2}{t}_{1}},\vskip 6.0pt plus 2.0pt minus
2.0pt\\\ S_{16}(\theta)=S_{26}{(}\theta{)}=2\sqrt{2}r_{1}\sin
2\left(\varphi_{1}-\theta\right),\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
{S_{22}{(}\theta{)}={t}_{0}{+}{2}{t}_{1}{-}{4}{r}_{1}\cos{2}\left({{{\varphi}}_{1}-\theta}\right)},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ {S_{66}{(}\theta{)}=2{t}_{0}}.\end{array}}$
(59)

It is worth noting that $S_{12}$ and $S_{66}$ are isotropic and that from the
previous equations it is immediate to get

$t_{0}=\frac{1}{2}S_{66},\ \
t_{1}=\frac{1}{2}\left(S_{12}+\frac{S_{66}}{2}\right),\ \
2r_{1}^{2}=\frac{1}{32}\left[(S_{11}-S_{22})^{2}+8S_{16}^{2}\right],$ (60)

so bounds (58) become

$S_{66}>0,\ \ \ 4S_{66}(S_{66}+2S_{12})-(S_{11}-S_{22})^{2}-8S_{16}^{2}>0$
(61)

which for the technical parameters gives

$E_{1}>0,\ \ E_{2}>0,\ \ G_{12}>0,\ \
E_{1}E_{2}^{2}(E_{1}-4\nu_{12}G_{12})-(E_{2}-E_{1})^{2}G_{12}^{2}-4\eta_{12,1}^{2}E_{2}^{2}G_{12}^{2}>0.$
(62)

A much simpler form of these bounds can be obtained for a frame that coincides
with the orthotropy axes ($\varphi_{1}-\theta=0$); in such a case, from eq.
(59) we get $S_{16}=0$ and

$t_{0}=\frac{1}{4G_{12}},\ \
t_{1}=\frac{1}{2}\left(\frac{1}{4G_{12}}-\frac{\nu_{12}}{E_{1}}\right),\ \
r_{1}=\frac{1}{4}\left(\frac{1+\nu_{12}}{E_{1}}-\frac{1}{2G_{12}}\right),$
(63)

so that bounds (62) become

$E_{2}>0,\ \ G_{12}>0,\ \ E_{1}>G_{12}(1+\nu_{12})^{2}.$ (64)

##  5 Conclusion

The results found above fill the gap for what concerns the elastic bounds of
the technical constants of an anisotropic layer or plate. What is important to
point out is that they are written in a generic frame, so involving, and this
is the first time, also the coefficients of mutual influence of the second
type (through reciprocity relations, the use of the coefficients of the first
type is straightforward). The interest of these bounds is first theoretical,
then practical: the determination of the elastic parameters of a layer or of a
plate is particularly difficult when the body is anisotropic. Disposing of
bounds, serving as a necessary control for the acceptance of the experimental
measures, is rather important in practice. The same can be said for correctly
defining the feasibility domain in some design problem, e.g. concerning
laminated plates made of composite, anisotropic layers.

## References

  * [1] Hohn FE.  _Elementary matrix algebra_.  New York, NY: MacMillan, 1958.
  * [2] Love AEH.  _A treatise on the mathematical theory of elasticity_.  New York, NY: Dover, 1944.
  * [3] Thomson W - Lord Kelvin.  Elements of a mathematical theory of elasticity.  _Philosophical Transations of the Royal Society_ 1856; 146: 481–498.
  * [4] Thomson W - Lord Kelvin.  Mathematical theory of elasticity.  _Encyclopedia Britannica_ 1878; 7: 819–825.
  * [5] Vannucci P.  _Anisotropic elasticity_.  Berlin, Germany: Springer, 2018.
  * [6] Lekhnitskii SG.  _Theory of elasticity of an anisotropic elastic body_.  San Francisco, CA: English translation (1963) by P. Fern, Holden-Day, 1950.
  * [7] Jones RM.  _Mechanics of composite materials. Second Edition_.  Philadelphia, PA: Taylor & Francis, 1999.
  * [8] Verchery G.  Les invariants des tenseurs d’ordre 4 du type de l’élasticité.  In _Proc. of Colloque Euromech 115 (Villard-de-Lans, 1979): Comportement mécanique des matériaux anisotropes_. Paris: Editions du CNRS, 1982, pp. 93–104.
  * [9] Vannucci P.  Plane anisotropy by the polar method.  _Meccanica_ 2005; 40: 437–454.
  * [10] Vannucci P.  A special planar orthotropic material.  _Journal of Elasticity_ 2002; 67: 81–96.
  * [11] Vannucci P and Desmorat B.  Analytical bounds for damage induced planar anisotropy.  _International Journal of Solids and Structures_ 2015; 60-61: 96–106.
  * [12] Gay D.  _Composite Materials Design and Applications - Third Edition_.  Boca Raton, FL: CRC Press, 2014.
  * [13] Vannucci P.  On special orthotropy of paper.  _Journal of Elasticity_ 2010; 99: 75–83.


# Factor 30 pulse compression by hybrid multi-pass multi-plate spectral
broadening

Marcus Seidel  Prannay Balla  Deutsches Elektronen-Synchrotron DESY,
Notkestrasse 85, 22607 Hamburg, Germany.  Helmholtz-Institute Jena,
Fröbelstieg 3,07743 Jena, Germany.  GSI Helmholtzzentrum für
Schwerionenforschnung GmbH, Planckstrasse 1, 64291 Darmstadt, Germany.  Chen
Li  Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, 22607 Hamburg,
Germany.  Gunnar Arisholm  FFI (Norwegian Defence Research Establishment), P.
O. Box 25, NO-2027 Kjeller, Norway.  Lutz Winkelmann  Deutsches Elektronen-
Synchrotron DESY, Notkestrasse 85, 22607 Hamburg, Germany.  Ingmar Hartl
Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, 22607 Hamburg,
Germany.  Christoph M. Heyl  Deutsches Elektronen-Synchrotron DESY,
Notkestrasse 85, 22607 Hamburg, Germany.  Helmholtz-Institute Jena,
Fröbelstieg 3,07743 Jena, Germany.  GSI Helmholtzzentrum für
Schwerionenforschnung GmbH, Planckstrasse 1, 64291 Darmstadt, Germany.

###### Abstract

As Ultrafast laser technology advances towards ever higher peak and average
powers, generating sub-50 fs pulses from laser architectures that exhibit best
power-scaling capabilities remains a major challenge. Here, we present a very
compact and highly robust method to compress 1.24 ps pulses to 39 fs by means
of only a single spectral broadening stage which neither requires vacuum parts
nor custom-made optics. Our approach is based on the hybridization of the
multi-plate continuum and the multi-pass cell spectral broadening techniques.
Their combination leads to significantly higher spectral broadening factors in
bulk material than what has been reported from either method alone. Moreover,
our approach efficiently suppresses adverse features of single-pass bulk
spectral broadening. We use a burst mode Yb:YAG laser emitting pulses with 80
MW peak power that are enhanced to more than 1 GW after post-compression. With
only 0.19 % rms pulse-to-pulse energy fluctuations, the technique exhibits
excellent stability. Furthermore, we have measured state-of-the-art spectral-
spatial homogeneity and good beam quality of M${}^{2}=1.2$ up to a spectral
broadening factor of 30. Due to the method’s simplicity, compactness and
scalability, it is highly attractive for turning a high-power picosecond laser
into an ultrafast light source that generates pulses of only a few tens of
femtoseconds duration.

##  1 Introduction

Figure 1: Overview of single stage compression factors achieved by bulk-based
MPCs and the multi-plate continuum technique. The light gray and blue shaded
areas enclose the typical operation regimes of the respective compression
techniques. References to the experiments with the highest reported single-
stage compression factors are provided.

Femtosecond light pulses are used for a variety of applications including
time-resolved fundamental science, material processing and medical
applications [1]. Present ultrafast laser developments reach for increasing
peak- and average powers [2] in order to advance applications in these fields,
for instance by laser-particle acceleration [3], nonlinear attosecond science
[4] and raw-event detection utilized in coincidence spectroscopy [5]. Today’s
most common high-power ultrafast light sources rely on active Yb-ions. A major
downside of these lasers is the inability to directly deliver sub-100 fs
pulses. Therefore, a multitude of applications requiring shorter pulses cannot
readily be addressed. To overcome this limitation, spectral broadening and
subsequent pulse compression have become a thriving research topic [6].
Different spectral broadening methods methods are commonly employed. Due to
their excellent power-handling capabilities, bulk nonlinear media and multi-
pass cells (MPCs) employing bulk- or gas-based nonlinear spectral broadening
present a vital alternative to fibers or hollow-core capillaries. This was
facilitated by the inventions of MPC spectral broadening [7] and multi-plate
continuum generation [8]. In both approaches, the introduced quasi-waveguide
results in much better spatial homogeneities of the broadband spectra than in
bare bulk experiments which were earlier proposed as a simple method for
extending optical pulse bandwidths [9].

Here, we present the combination of both techniques in a single, compact
spectral broadening stage. We demonstrate unprecedented high pulse compression
factors of more than 30 (see Fig. 1), corresponding to a reduction of the FWHM
pulse duration from 1240 fs to 39 fs. Despite strong self-phase modulation, we
efficiently suppress spatial spectrum variations that result from single-pass
bulk spectral broadening in the critical self-focusing regime [10]. For
comparison, the highest pulse compression factors reported from single multi-
plate continuum stages are 8.3 [11] and 9.6 [12]. Large compression factors
have been achieved with gas-filled MPCs at multi-mJ pulse energy levels [13,
14, 15], but only a few single bulk MPC stages were reported to be able to
achieve more than 10-fold compression. Those methods, however, either relied
on custom-made dispersive mirrors [16] or multi-ps long input pulses being
insensitive to dispersion [17, 18]. In contrast, the implementation of our
method requires only off-the-shelf optics and enables strong spectral
broadening by high B-integrals per pass. Therefore, our compression scheme is
a compact alternative to more complex multi-stage setups [12, 19, 20, 21, 22,
23] and presents a highly attractive method to generate sub-50 fs pulses from
a high-power picosecond laser.

##  2 Materials and Methods

Figure 2: Spectral broadening setup. Waveplate WP1 and thin-film polarizers
TFP1 and TFP2 are used for polarization cleaning of the Yb:YAG amplifier
output. Mode-matching is adjusted with the lenses L1 and L2 as well as the
translation stage with mirrors M2 and M3 on it. The mirror M4 is lower than
the other mirrors, such that the ingoing beam hits M4 on the top and the
outgoing beam is passing it. The focal length of both MPC mirrors M6 and M7 is
100 mm. Pick-off mirror M5 has only a 4 mm vertical aperture to not clip the
beam despite of the 32 roundtrips. The six 1 mm thin Kerr media are centered
in the about 385 mm long MPC, mounted in lens tubes and spaced by 16 mm. The
lens L3 is used for collimation, the waveplate WP2 for readjusting
p-polarization to match the specification of the chirped mirrors CM1 and CM2.
A wedge is used for sending the beam to all diagnostics except from a FROG-
device which is behind the compressor.

###  2.1 Experimental Design

We compressed the output of a multi-stage Yb:YAG amplifier operating in 10 Hz
burst mode being matched to the pulse trains of the free-electron laser FLASH
at DESY [24, 25]. The system delivered pulses centered at 1030 nm wavelength
with 112 $\mu$J energy, near Fourier-transform limited 1240 fs pulse duration
and a peak power of about 80 MW. Frequency resolved optical gating (FROG)
measurements of the pulses are provided in supplement S4. The maximal in-burst
average power with minimized pulse-to-pulse energy fluctuations was 112 W. The
corresponding 1 MHz pulse repetition rate could be maintained over the burst
duration of 800 $\mu$s. To avoid saturation of our measurement devices, we
used an acousto-optic modulator to adjust the number of pulses in the bursts
and their amplitude [25].

Whereas a high-pressure gas-filled multi-pass cell could have been used at our
peak power level [26], we decided to spectrally broaden the pulses in a bulk-
based MPC. This obviated the need for a complex overpressure system. The setup
is described in the caption of Fig. 2. The employed multilyaer MPC mirrors
were standard quarter-wave stacks with 200 mm radius of curvature and 50.8 mm
diameter. All Kerr media used were stock items. In this way, the setup was
kept as simple as possible. We used two focusing lenses and a translation
stage for adjusting mode-matching while monitoring the beam size in the
targeted focal plane inside the MPC.

We measured pulse durations by second-harmonic FROG with a commercial device
(Mesa Photonics). In the diagnostics section of the setup, we used a compact
grating spectrometer or an optical spectrum analyzer (OSA) for spectrum
measurements and a Spiricon M200 M2-meter for beam quality evaluation. An
ANDOR Kymera 193 Czerny-Turner-type spectrograph with a 10 $\mu$m horizontal
entrance slit and a grating in the $2f$-plane with 500 lines/mm was used for
measuring spectral-spatial homogeneity. We placed a FLIR grasshopper CCD
camera in the $4f$ plane that recorded the dispersed spectrum along the
horizontal axis and a line-out of the beam along the vertical axis. The
wavelength axis was calibrated by comparison to a spectrum recorded with the
compact grating spectrometer. The spectral response of the spectrograph was
not determined. The beam profiles were measured with the same camera, which
had 3.69$\,\mu$m $\times$ 3.69$\,\mu$m pixel size.

###  2.2 Simulations

We support our experimental data by nonlinear beam propagation simulations. We
employed the SISYFOS code [27] which was previously used for bulk spectral
broadening simulations [10]. The Herriott-cell shown in Fig. 2 was modeled
under the following approximations: the beam was always on the optical axis of
the system, a fundamental Gaussian beam ($M^{2}=1$) was propagated,
reflectivity and dispersion of the MPC mirrors as well as transmission of the
anti-reflective (AR) coatings were taken from theoretical data provided by the
suppliers. We included the Kerr nonlinearity of air but we neglected the non-
instantaneous Raman contributions to the nonlinearities of fused silica and
air. Self-steeping was implicitly considered by solving the wave equation in
the frequency domain.

Test simulations with both transverse dimensions and a single transverse
dimension, exploiting cylindrical symmetry, led to the same outcome.
Therefore, we used the computationally less expensive reduced spatial
dimensionality with 512 radial grid points. The time and frequency grid,
respectively, consisted of 2048 points to properly model the narrow initial
and the broad final spectrum.

The Kerr media’s nonlinear refractive index, $n^{\text{sim}}_{2}=2.5\cdot
10^{-16}$cm2/W, used in simulations was deduced from a comparison between
simulation and an MPC experiment with factor ten spectral broadening. It is in
good agreement with literature values [28]. We define the spectral broadening
factor as the ratio between initial and broadened spectrum transform-limited
FWHM pulse durations.

##  3 Results

###  3.1 Hybrid multi-plate, multi-pass approach

Figure 3: (a) Comparison of simulated peak irradiances inside the MPC shown in
Fig. 2(a) for 100 $\mu$J, 1250 fs Gaussian input pulses in case of linear
(blue line) and nonlinear (red line) mode-matching. The crosses indicate the
locations of the silica plates. Nonlinear mode-matching significantly reduces
the maximal peak irradiance. Residual peak irradiance oscillations are caused
by the imprecision of our analytical model. Best mode-matching is accomplished
for 75 $\mu$J input pulses according to (c). (b) Output spectra after 32
roundtrips corresponding to irradiance evolution from (a) and comparison to
the input spectrum. (c) Simulated dependence of maximal peak irradiance in the
Kerr media (I${}_{\text{max}}$) on input pulse energy (E${}_{\text{p}}$) for a
nonlinear mode-matching point of about 75 $\mu$J where
I${}_{\text{max}}$/E${}_{\text{p}}$ reaches its minimum. The monotonic
increase of I${}_{\text{max}}$ obviates the need for modifying the input beam
during energy ramp-up in a nonlinear mode-matching setting.

We conducted initial experiments with single silica plates of 3 mm to 10 mm
thickness inside an approximately 350 mm long MPC. The thick Kerr media
exhibited two drawbacks: First, in addition to spectral broadening, we
observed an accumulation of spectral power near the fundamental wavelength of
1030 nm with increasing input pulse energy, indicating a power-dependent mode-
mismatch. We even damaged the Kerr medium after extending the MPC to 380 mm
length. Second, we measured the spectral phase of pulses compressed by
approximately a factor of ten and retrieved a characteristic phase kink near
the fundamental wavelength which is known from bulk spectral broadening [10].
We attribute these observations to nonlinear beam reshaping inside the Kerr
medium. This is supported by a set of free beam nonlinear propagation
simulations that yield a spectral broadening factor of 1.3. The simulations
are described in supplement S1. We found that an intense beam propagating
through a short medium undergoes significantly less deformation compared to a
less intense beam propagating through a longer medium. This indicates that
limiting the B-integral per pass, as proposed in the original MPC spectral
broadening patent [29], may not be necessary to suppress nonlinear space-time
coupling. The B-integral per Kerr medium in multi-plate continuum generation
was clearly higher than in bulk MPC experiments. Nonetheless, good spatial-
spectral homogeneity was reported [12, 30]. Consequently, we used this
approach and distributed the nonlinear phase accumulation among a sequence of
multiple thin plates inside the Herriott-cell.

Furthermore, Kerr lensing plays a crucial role in multi-plate continuum
generation because it establishes a quasi-waveguide [10, 30]. We also included
the self-focusing effect in the mode-matching calculations (see supplement S2)
of our hybrid multi-plate, multi-pass scheme. We refer to this as nonlinear
mode-matching in bulk MPCs. Herriott pattern, cell length and beam curvature
at the mirrors correspond to the linear mode-matching calculations but
including the Kerr effect yields smaller spot sizes at the mirrors and larger
ones in the cell center (see Fig. S1).

Fig. 3(a) shows the simulated peak irradiance inside the MPC illustrated in
Fig. 2. The difference between linear and nonlinear mode-matching is striking.
Whereas for linear mode-matching the peak irradiance in the Kerr media reaches
up to 2 TW/cm2, which would cause coating damage in an experiment, the peak
irradiance stays below 500 GW/cm2 for nonlinear mode-matching, preventing
self-focusing induced damage. Consequently, high peak intensities in the
nonlinear media can be obtained over all passes by accounting for Kerr
lensing. This, in turn, enables higher spectral broadening factors from a
single stage. A recent theory paper has also pointed out the importance of
nonlinear mode-matching in gas-filled MPCs that operate in the sub-critical
focusing regime [31]. The irradiance enhancement predicted for the bulk MPC
studied here is, however, more than 10 times higher than in the numerical
example of ref. [31].

Our simulations indicate that the shape of the nonlinearly broadened spectra
indicate the quality of mode-matching (see Fig. 3(b)). With nonlinear mode-
matching, the outermost spectral lobes exhibit the highest spectral power. By
contrast, excessive spectral power is concentrated near the fundamental
wavelength if only the linear cavity is mode-matched. This is in agreement
with our experimental observations. We note that a distinct peak around 1030
nm is also caused by unbroadened input pulse pedestals or pre/post pulses.
This peak is, however, independent of power launched into the MPC.

###  3.2 Spectral broadening and pulse compression experiments

In order to verify the occurrence of self-focusing and to determine an
experimental nonlinear refractive index, we conducted Z-scan-type measurements
inside the MPC (see supplement S3). We used the result to calculate the beam
parameters for non-linear mode-matching and aligned the MPC correspondingly
(see supplement S2 for details). We did not adjust any optics while increasing
the input power but expected the laser beam to be matched to the Herriott-cell
at full input energy. This is possible because the maximal peak irradiances in
the Kerr media monotonically increase with input pulse energy according to our
simulations shown in Fig. 3(c).

Fig 4(a) shows a set of measured broadened spectra that were recorded behind
the MPC while the laser pulse energy was increased. The derived dependence of
the spectral broadening factor on the measured output pulse energy is plotted
in Fig 4(b). The dispersion of the six Kerr plates yielded saturation of self-
phase modulation at high input powers. Sub-40 fs Fourier-transform limits were
reached at about 75 $\mu$J output pulse energy. The highest achieved spectral
broadening factor was 33, corresponding to a reduction of bandwidth-limited
pulse duration from 1.2 ps to 36 fs. We note that the Fourier tranform-limit
was independent of the number of pulses per laser burst, i.e., the same
spectral broadening characteristics were also obtained with 112 W average
power over the maximal 800 $\mu$s burst duration. The measured spectra in Fig.
4(a) exhibit a central spectral peak at both low and high output energies. It
was caused by a pedestal of the input pulses which we measured by FROG (see
supplement S4). Aside from that, the spectral shape agrees well with the
simulated one in Fig. 3(b) up to approximately 50 $\mu$J output energy. For
higher energies, the spectral modulations flatten out at parts of the spectrum
and the symmetry with respect to the fundamental wavelength decreases. We
attribute this to the spurious four-wave mixing which was qualitatively
predicted in simulations (see supplement S5) and was experimentally confirmed
as shown in Fig. 4(c). The generation of spectral side-bands was previously
observed in gas-filled MPC experiments [13, 32] and was studied explicitly in
a simulation paper [33]. First signatures could be detected approximately 50
dB below the maximal power spectral density for 30 $\mu$J output pulse energy.
At maximum power, the amplitude of this short wavelength feature becomes one
order of magnitude stronger. We note that the mixed frequencies lie outside of
the reflection band of our dielectric optics, and thus the measured spectrum
does not represent the magnitude and central frequency of the side-bands
generated inside the MPC. Our simulation also predicts spectral components
around 1125 nm (see Fig. S5(b)) which are outside of our detection range and
could not be observed.

Figure 4: (a) Broadened spectra after 32 roundtrips in a 385 mm long MPC
detected with a response function corrected grating spectrometer. The spectra
are offset by the output energy measured with an energy meter and corrected by
the wedge W1 transmission. (b) Corresponding, spectral broadening factors as a
function of output pulse energy. The transform-limited pulse duration at full
power is more than 30 times shorter than at the input. The transmission is
constantly about 80 %. (c) Spectral side-bands, becoming detectable for more
than 30 $\mu$J output energy. These are excited by four-wave mixing as
qualitatively predicted by simulations. The spectra are normalized to the
maximum of the respective self-phase modulated part near 1030 nm. (d) Spectrum
stability measured over 17 hours with one recording per second. Figure 5: (a)
Retrieved FROG spectrum and spectral phase of the post-compressed pulses. (b)
Retrieved pulse shape compared to the transform limited pulse for 65 $\mu$J
pulse energy.

Despite the large spectral broadening factor, the measured spectrum was highly
stable as Fig. 4(d) demonstrates. From one recording per second in an
overnight measurement ($\approx$17 hours measurement time), we computed a
standard deviation of the Fourier-transform limited pulse duration of only 0.1
fs at 35.7 fs mean value. The pulse-to-pulse energy fluctuations after the MPC
exhibited only a 0.19 % standard deviation relative to the mean output energy.
The measurements were performed with a single pulse per burst. Spectrum and
pulse energy were logged in parallel.

To shorten the spectrally broadened pulses, we set-up a chirped-mirror
compressor inducing a group delay dispersion (GDD) of -200 fs2 per reflection.
For best compression, we changed the number of bounces in steps of four and
slightly varied the input pulse energy to the MPC. By means of second harmonic
FROG, we could at best retrieve a FWHM duration of 39 fs (Fig. 5),
corresponding to a factor 32 pulse duration shortening in comparison to the
1.24 ps input shown in supplement S4. The required 72 bounces off the chirped
mirrors applied a total GDD of -14 400 fs2 to the pulses and resulted in only
78 % transmission of the compressor. Moreover, we measured 5 % depolarization
after waveplate WP2. This is caused by the large nonlinear phase shift in the
MPC which effectively amplifies cross-polarized light. Therefore, the usable
pulse energy accumulated to 65 $\mu$J which yielded a peak power of 1.1 GW. By
contrast to other bulk-MPC or multi-plate experiments [19, 20, 34], we did not
observe a phase kink near the fundamental wavelength (Fig. 5(a)).

Figure 6: (a) Measured spectra across the vertical beam axis for a 36 fs
Fourier transform-limit. (b) Near-field (NF) and far-field (FF) profiles, also
recorded at 36 fs Fourier transform-limit. (c) Experimentally determined
$M^{2}$ parameters for 32 roundtrips (red markers) and 8 roundtrips (blue
markers).

###  3.3 Beam characterization

Fig. 6(a) shows a camera image after the $4f$-spectrograph at full spectral
broadening. The frequency content is nearly uniformly distributed across the
beam axis. We calculated a mean spectral homogeneity along the vertical beam
axis of $\bar{V}_{y}=97.3$ %, following the figure of merit proposed by
Weitenberg et al. [35] (see supplement S6). A homogeneity of
$\bar{V}_{y}=97.3$ % was also measured for a gas-filled MPC with comparable
spectral broadening from 1.3 ps to 39 fs bandwidth limit [13]. This
underscores that the detrimental spatial effects, which are typical for
single-pass spectral broadening in the critical self-focusing regime, have
been largely suppressed by the multi-plate multi-pass approach.

Although the near- and far-field beam profiles for the highest spectral
broadening factor displayed in Fig. 6(b) look close to Gaussian, we determined
a distinct increase of the $M^{2}$ parameters to $1.6\times 1.5$ at full input
power. Fig 6(c) shows that we measured $M^{2}$ values of about 1.2 for all
input energies up to 90 $\mu$J, corresponding to a spectral broadening factor
of about 30. However, for higher input energies, a gradual decrease of beam
quality was recorded. A similar observation was reported in the first MPC
paper by Schulte et al. [7] and attributed to thermal effects. I our work, we
could run experiments at only 10 Hz pulse repetition rate and can thus exclude
thermal effects. In ref. [36], the degradation was interpreted as an onset of
bulk-like broadening. This is, however, in contradiction to our compression
and spectral homogeneity measurements. An experiment with the same input
energy but considerably less roundtrips (blue markers Fig 6(c)) suggests that
the higher $M^{2}$ values are caused by parasitic nonlinear effects in the
MPC. We suspect that the four-wave mixing process (see supplement S4, Fig.
4(c)) between broadened spectrum and unbroadened mode-mismatched pulse
pedestal introduced non-periodic losses in the MPC which led to reshaping of
the beam. Therefore, input pulse shaping could further improve the output beam
quality.

##  4 Discussion

We have presented a hybrid bulk multi-pass, multi-plate method for spectral
broadening of ultrashort laser pulses. This novel approach yielded
significantly higher single-stage compression factors than previously reported
from the individual techniques. In total, the laser beam passed 384 Kerr media
in our setup. On the one hand, this is practically impossible to implement in
a pure multi-plate scheme where less than ten passes are typical. On the other
hand, to our knowledge all bulk-MPC papers roughly followed the initial
proposal that the B-integral per pass through the Herriott-cell should not
exceed $\pi/5$ [29]. The simulations shown in Fig. 3 predict B-integrals per
pass of up to $B\approx 4\pi/5$ resulting from an average peak irradiance
$<I_{p}>=285$ GW/cm2 inside the Kerr media. This was achieved through the
introduction of multiple thin plates and explains why the presented 32-fold
pulse duration reduction has not been accomplished with the bulk-MPC technique
alone. It is only comparable to the best results demonstrated with gas-filled
MPCs which operate in the sub-crititcal self-focusing regime [13, 14, 15]. By
contrast to merely increasing the number of roundtrips in the MPC to attain
large nonlinear phase shifts, the nonlinear mode-matching technique minimizes
peak intensity variations in the Kerr media, and thus enables large spectral
broadening factors by high intensities and not primarily by long propagation
lengths in the nonlinear material. This is essential if dispersion induces
pulse stretching, i.e. for generating ultrashort pulses. Despite the large
single-pass B-integrals and operating approximately 20 times above the
critical power of the nonlinear material, we could vastly suppress undesired
features of bulk spectral broadening, such as, spectral inhomogeneity across
the beam profile and the typical phase kink near the input wavelength. This
was enabled by distributing the Kerr nonlinearity along the MPC instead of
using one thick fused silica window. The demonstrated approach was implemented
with standard quarter-wave stack mirrors and off-the-shelf Kerr media. Neither
a vacuum or high-pressure compatible enclosure, nor dispersive MPC mirrors
were needed. Moreover, the setup was very compact and excellent stability was
demonstrated. This makes the method highly attractive for turning a high-power
picosecond laser into an ultrafast light source generating sub-50 fs pulses.
Whereas we measured constantly good beam quality up to a spectral broadening
factor of 30, the $M^{2}$ value increased from 1.2 to 1.6 upon reducing the
Fourier-transform limit further from 40 fs to 36 fs. We attribute this to
parasitic four-wave mixing in the MPC which, according to our simulations,
could be circumvented by pedestal-free input pulses.

The main obstacle for generating even shorter pulses with our approach is the
bandwidth of the used optics. The MPC mirrors support durations of about 30
fs. Replacing tantalum pentoxide by titania as high index material of the
quarter-wave stack mirrors would enable the generation of about 20 fs pulses.
By using enhanced metallic mirrors, even few-cycle pulses were demonstrated
[14, 37]. Conventional anti-reflection coatings would lead to clearly lower
MPC transmissions if larger bandwidths with the same number of roundtrips are
targeted. However, nanotextured surfaces exhibit very low Fresnel reflections
over a large spectral range [38]. This opens the perspective of few-cycle
pulse generation by the hybrid multi-pass multi-plate approach. Moreover, the
Kerr media can be moved closer to the MPC mirrors if laser peak power
increases [7], and thus the proposed scheme can be employed in the 5 MW to 5
GW range without the need for Herriott-cells exceeding usual optical table
lengths. For certain high peak power applications, bulk-MPCs may present
interesting alternatives to gas-filled MPCs. Gas ionization and mirror damage
thresholds differ by orders of magnitudes. Therefore, gas-filled MPCs are
usually operated near the stability edge. In contrast, bulk Kerr media and
dielectric mirrors exhibit comparable damage thresholds. Consequently, very
compact Herriott-cells that do not compromise spectral broadening are
possible. Bulk spectral broadening is also envisioned for PW class lasers
[39]. Similar to high-power multi-pass amplifier research, where thermal
lenses need to be considered, power-scaling of the bulk-MPC approach would
require to find a compact resonator design that can handle Kerr lensing. A
possible solution is the use of folded bow-tie cavities [40].

In conclusion, the hybrid multi-pass, multi-plate approach has enabled record-
high single-stage pulse compression factors for these bulk spectral broadening
methods. The demonstrated efficiency, beam and pulse quality is comparable to
less compact and more complex multi-stage setups. In contrast to gas-filled
MPCs, no sealed enclosure of the setup is required and our approach can be
applied to a large peak power range. Therefore, it is a very attractive method
for pulse-compression of state-of-the-art laser systems and opens perspectives
for novel high repetition rate extreme light sources operating in the few-
cycle or TW to PW peak power regime.

## Acknowledgments

We thank Caterina Vidoli for proof-reading the manuscript and we acknowledge
DESY (Hamburg, Germany), a member of the Helmholtz Association HGF, for the
provision of experimental facilities.

### Author Contributions

M. Seidel conceived the hybrid spectral broadening scheme, conducted
experiments as well as simulations and led the writing of the manuscript. P.
Balla implemented the initial setup and conducted first experiments. C. Li and
L. Winkelmann aligned the laser used for the experiments. G. Arisholm
developed the SISYFOS code and adapted it to MPC simulations. L. Winkelmann,
I. Hartl and C. M. Heyl initiated the project. I. Hartl and C.M. Heyl
conceived the initial setup and supervised the project. All authors revised
the manuscript and contributed to the writing of it.

### Conflicts of Interest

The authors declare that there is no conflict of interest regarding the
publication of this article.

### Data Availability

Data can be obtained from the corresponding author upon reasonable request.

## Supplementary Materials

##  S1 Simulated single pass enhancements

We used the SISYFOS package [27] to simulate nonlinear propagation of an
intense Gaussian beam through fused silica and compared the pulse peak
irradiance at the beginning and at the end of propagation. We varied
propagation length, i.e. Kerr medium thickness, pulse energy and input beam
size while keeping a fixed spectral broadening factor of about 1.3 (Table S1).
Whereas the irradiance enhancement within the nonlinear medium for 150 $\mu$J
pulse energy and a 1 mm long plate was about 2 %, the enhancement was more
than three times larger for a 3 mm long Kerr medium pumped at lower intensity.
This is caused by self-focusing which leads to a highly nonlinear increase of
on-axis irradiance in extended Kerr media if the critical power is exceeded
[10].

Table S1: Simulation of irradiance enhancement inside a Kerr medium for a fixed broadening factor of 1.3 and Gaussian input pulses with $\Delta t_{p}=750$ fs duration. B-integral | $l$ | $w$ | $E_{p}$ | $\mathcal{E}$
---|---|---|---|---
$0.28\pi$ | 1 mm | 117 $\mu$m | 100 $\mu$J | 3.4 %
$0.27\pi$ | 1 mm | 147 $\mu$m | 150 $\mu$J | 2.1 %
$0.27\pi$ | 3 mm | 147 $\mu$m | 50 $\mu$J | 6.5 %
$l$: Kerr medium length, $w$: $1/e^{2}$ beam radius, $E_{p}$: pulse energy,
$\mathcal{E}$: irradiance enhancement.

##  S2 Nonlinear mode-matching

A Herriott-type cell is usually operated as a $q$-parameter preserving
resonator [41, 42]. It, therefore, must satisfy the following ABCD matrix
relation for Gaussian beams:

$\displaystyle q$
$\displaystyle\stackrel{{\scriptstyle!}}{{=}}\frac{\tilde{M}_{\text{MPC}}(1,1)q+\tilde{M}_{\text{MPC}}(1,2)}{\tilde{M}_{\text{MPC}}(2,1)q+\tilde{M}_{\text{MPC}}(2,2)},$
(1) $\displaystyle\text{where
}\tilde{M}^{\text{lin}}_{\text{MPC}}=\tilde{L}(f)\tilde{P}(d).$ (2)

$\tilde{L}$ denotes the thin lens ABCD matrix, $f$ the MPC mirror focal
length, $\tilde{P}$ is the propagation ABCD matrix and $d$ the cell length
which must be long enough to prevent optics damage. The beam radius of
curvature (RoC) must be $2f$ at both MPC mirror planes for matching the
q-parameter to the eigenmode of the MPC [41, 43]. The beam size for mode-
matching is inferred from the cell length. Eq. (2) holds for a passive cavity
or linear propagation. Kerr lensing, however, modifies the ABCD matrix to:

$\displaystyle\tilde{M}_{\text{MPC}}^{\text{nl}}=\tilde{L}(f)\tilde{P}(d_{MK})\left[\tilde{K}(n,l,\gamma)\tilde{P}(d_{K})\right]^{N-1}\tilde{K}(n,l,\gamma)\tilde{P}(d_{MK}),$
(3)

where $d_{MK}$ is the distance from the MPC mirror to the outer Kerr medium,
$\tilde{K}$ is the Kerr lens ABCD matrix depending on the refractive index
$n$, the plate length $l$, and the parameter $\gamma$, $d_{K}$ the plate
spacing and $N$ the number of Kerr media. Only symmetric cells are considered.
Matching again the RoCs of beam and mirrors at the mirrors leads to smaller
spot sizes on the mirrors due to the additional lenses inside the MPC. We used
the Kerr lens ABCD matrix derived in ref. [44] for a quadratic duct and added
a scaling parameter $\kappa$ to account for the Gaussian beam shape [45]:

$\displaystyle\tilde{K}(l,\gamma)$ $\displaystyle=\begin{pmatrix}\cos(\gamma
l)&\sin(\gamma l)/(n\gamma)\\\ n\gamma\sin(\gamma l)&\cos(\gamma
l)\end{pmatrix},$ (4) $\displaystyle\gamma$
$\displaystyle=\frac{\kappa}{w^{2}}\left(\frac{8n_{2}P_{p}}{\pi
n}\right)^{1/2},$ (5)

where $w$ is the beam radius, $n_{2}$ the nonlinear refractive index and
$P_{p}$ the pulse peak power. After having fixed the nonlinear refractive
index $n_{2}^{\text{sim}}$ (see section 2.2), the $\kappa$ parameter was
derived from a Z-scan-type simulation resembling the setup shown in Fig.
S2(a). The Gaussian beam width at the camera plane was computed for eight
Kerr-lens positions and fitted by (4). The comparison between the best-fit
$\gamma$ from (4) and the analytical $\gamma$ from (5) yielded $\kappa=0.43$.

Figure S1: (a) Inverse radius of curvature ($1/\text{RoC}$) and (b) beam
radius (w) for a mode-matched MPC without nonlinear media (black solid lines)
and with six Kerr lenses (red solid lines). Only the radii of curvature at the
mirrors are identical. Otherwise, beam sizes and curvatures differ
considerably during a single pass. The red dashed lines show inverse beam
curvature and beam radius for a nonlinear mode-matching setting and a weak
beam, i.e. negligible Kerr lensing. This configuration was used to align the
MPC.

Fig. S1 shows an example of single-pass ABCD matrix calculations for a cold
Herriott-cell where linear mode-matching applies and the MPC with six Kerr
media that we experimentally studied. In both cases, the radii of curvature at
the mirrors are $-5$ m after and $+5$ m before reflection. Otherwise, the
q-parameters evolve very differently inside the MPC. The linearly mode-matched
beam is about 50 % larger at the mirrors than the nonlinearly mode-matched
beam. This does not cause damage issues since the damage threshold of the Kerr
media is critical. The focus spot size of the nonlinearly mode-matched MPC is
larger than that of the linearly mode-matched cell. This is different from
nonlinear mode-matching in a gas-filled cell where the beam divergence is
effectively reduced by the gas nonlinearity [31]. By contrast, a weak quasi-
guiding structure is observable in Fig. S1. The slope of the radius of
curvature changes significantly at every Kerr medium resulting in much smaller
beam size variations near the center of the MPC than in the linearly mode-
matched case that exhibits usual Gaussian beam diffraction.

To align the MPC in our experiments, we calculated the beam parameter
$q_{\text{M}}$ on the cavity mirror that fulfilled Eqs. (1) and (3) for the
maximal input energy of 112 $\mu$J. Subsequently, we computed the beam path
for linear propagation (red dashed lines in Fig. S1) since alignment was done
at low input power. By placing a beam profiler into the predicted focal plane
and varying the distance between the lenses L1 and L2 (see Fig. 2), as well as
the position of the translation stage with the mirrors M2 and M3, the measured
beam waist was match to our calculations. Afterward, the silica plates were
inserted and the energy was ramped up to 112 $\mu$J without further
modifications of the mode-matching optics.

Figure S2: (a) The Z-scan experiments were done in the MPC arrangement shown
in Fig. 2 using all optics up to mirror M5. A beam sample BS was inserted and
the beam was blocked (B) in front of the curved mirror M6. The beam sizes were
measured with the camera (CAM) (b) Measured beam diameters in x- and
y-direction for 90 $\mu$J and 100 $\mu$J input pulse energies. The spot sizes
were fitted by applying Eqs. (3)-(5) and optimizing $n_{2}^{\text{exp}}$
(solid lines). The observation of varying beam sizes manifest the impact of
Kerr-lensing on the beam inside the MPC. Figure S3: (a) Measured and (b)
retrieved FROG traces of the input pulses in logarithmic scale. The grid size
was 256 x 256 points and the FROG error 0.2 %. (c) Left axis: Retrieved
spectrum (solid blue line) and spectrum recorded with an optical spectrum
analyzer (dashed light blue line, OSA). Right axis: Retrieved spectral phase
(red line). (d) Retrieved pulse (red line) and Fourier transform-limited pulse
(black line) assuming 112$\,\mu$J pulse energy.

##  S3 Z-scans

Before conducting spectral broadening experiments, we measured an effective
$n_{2}^{\text{exp}}$ by a Z-scan type measurement. This was done because of
the beam shape dependence of the $\gamma$ parameter and the uncertainty in the
determination of the peak power $P_{p}$. We moved a set of three Kerr media
along the MPC shown in Fig. 2 and measured the beam size variations at the MPC
mirror M6 plane (see Fig. S2(a)). By the ABCD matrix calculus from Eqs.
(3)-(5), we determined $n^{\text{exp}}_{2}=2.25\cdot 10^{-16}$cm2/W. The
calculated spot sizes reproduced the measured ones well as Fig. S2(b)
exhibits. The derived nonlinear refractive index is in good agreement with the
used simulation and the literature values [28].

Figure S4: (a) Measured and (b) retrieved FROG traces in logarithmic scale.
The grid size was 1024 x 1024 points and the FROG error 0.4 %. (c) Comparison
of spectra corresponding to the compressed pulse retrieval. We measured with a
compact grating spectrometer (red dashed line) as well as with an optical
spectrum analyzer (gray solid line, OSA). As in Fig. 5, the blue solid line
shows the spectrum derived from the FROG trace. Figure S5: (a) Simulated
Evolution of spectral broadening for Gaussian input pulses (see Fig. 3) in
comparison to (b) input pulses with the shape of Fig. 2(b) and the same peak
power. The latter case shows clear signatures of parasitic four-wave mixing.
The spectra were normalized at each propagation length.

##  S4 FROG measurements

To determine the experimental compression factor and to understand the
generation of the spectral side-bands shown in Fig. 4(d), we measured the
input pulse shape by FROG. The results are shown in Fig. S3. We retrieved a
nearly transform-limited pulse of 1.24 ps duration which, however, exhibits a
pedestal, i.e., a non-Gaussian shape. For the maximum pulse energy of 112
$\mu$J, the pulse peak power was about 80 MW.

Moreover, we provide additional information about the FROG measurement
resulting in the 39 fs pulses displayed in Fig. 5. The measured and retrieved
FROG traces as well as a comparison between measured and retrieved spectra are
shown in Fig. S4. Although the FROG error was 0.4 %, the fundamental spectrum
measured with an optical spectrum analyzer (OSA) only reproduced the wiggles
of the FROG spectrum well, whereas the spectral amplitudes differed. We note
that there are additional side lobes around 990 nm and 1070 nm visible in the
OSA spectrum which are neither present in the retrieved FROG spectrum, nor in
the one measured with a grating spectrometer. These might emerge from
additional nonlinearity accumulated by steering the beam into the OSA. Due to
the burst-mode operation of our laser, relatively high pulse energies are
needed to measure a spectrum with an OSA. On the other hand, our silicon-based
compact grating spectrometer cannot resolve the fast oscillations near the
fundamental wavelength. We note that the combination of the spectra shown in
Fig. S4(c) with the retrieved phase from Fig. 5(a) results only in a pulse
duration variation from 36 fs to 39 fs.

##  S5 Parasitic four-wave mixing

We simulated the spectrum evolution inside the MPC for Gaussian pulses and the
retrieved pulses shown in Fig. S3(d). Whereas the spectral broadening of
Gaussian pulses, visualized in Fig. S5(a), resembles one-dimensional textbook
examples for self-phase-modulation [46], simulations with the experimentally
measured input pulses result in more power near the fundamental wavelength, a
pronounced peak at the long-wave edge of the self-phase modulated spectrum
and, most striking, the generation spectral side-bands. These resemble the
experimental observation shown in Fig. 4(d).

##  S6 Homogeneity calculation

We calculated the figure of merit proposed by Weitenberg et al. [35] in the
following way:

$\displaystyle V_{y}(y_{i})$
$\displaystyle=\left[\sum_{j}\left\\{I(\lambda_{j},y_{i})I_{0}(\lambda_{j})\right\\}^{1/2}\right]^{2}\left[\sum_{j}I(\lambda_{j},y_{i})\sum_{k}I_{0}(\lambda_{k})\right]^{-1},$
(6) $\displaystyle\bar{V}_{y}$
$\displaystyle=\sum_{i}V_{y}(y_{i})P(y_{i})\left[\sum_{j}P(y_{j})\right]^{-1},$
(7)

where $\lambda_{j}$ are the camera pixels in x-direction representing
wavelength, $y_{i}$ are the camera pixels in y-direction representing the
vertical beam axis, $I(\lambda_{j},y_{i})$ are the camera counts,
$I_{0}(\lambda_{j})$ the on-axis counts and
$P(y_{i})=\sum_{j}I(\lambda_{j},y_{i})$ is the vertical beam profile. We
summed over the $6\sigma$ width of $P(y_{i})$ and over the full width at 0.2 %
of the maximum of the average spectrum. The level corresponded to our signal
to noise ratio. Due to the isotropy of all optics, we did not explicitly
measure the homogeneity in x-direction.

## References

  * [1] Gerard Mourou  “Nobel Lecture: Extreme light physics and application”  In _Reviews of Modern Physics_ 91.3, 2019, pp. 030501  DOI: 10.1103/RevModPhys.91.030501
  * [2] Hanieh Fattahi et al.  “Third-generation femtosecond technology”  In _Optica_ 1.1, 2014, pp. 45  DOI: 10.1364/OPTICA.1.000045
  * [3] Félicie Albert et al.  “2020 roadmap on plasma accelerators”  In _New Journal of Physics_ 23.3, 2021, pp. 031101  DOI: 10.1088/1367-2630/abcc62
  * [4] I. Orfanos et al.  “Attosecond pulse metrology”  In _APL Photonics_ 4.8, 2019, pp. 080901  DOI: 10.1063/1.5086773
  * [5] Sara Mikaelsson et al.  “A high-repetition rate attosecond light source for time-resolved coincidence spectroscopy”  In _Nanophotonics_ 10.1, 2021, pp. 117–128  DOI: doi:10.1515/nanoph-2020-0424
  * [6] Tamas Nagy, Peter Simon and Laszlo Veisz  “High-energy few-cycle pulses: post-compression techniques”  In _Advances in Physics: X_ 6.1, 2021, pp. 1845795  DOI: 10.1080/23746149.2020.1845795
  * [7] Jan Schulte et al.  “Nonlinear pulse compression in a multi-pass cell”  In _Optics Letters_ 41.19, 2016, pp. 4511  DOI: 10.1364/OL.41.004511
  * [8] Chih-Hsuan Lu et al.  “Generation of intense supercontinuum in condensed media”  In _Optica_ 1.6, 2014, pp. 400  DOI: 10.1364/OPTICA.1.000400
  * [9] Claude Rolland and P.. Corkum  “Compression of high-power optical pulses”  In _Journal of the Optical Society of America B_ 5.3, 1988, pp. 641  DOI: 10.1364/JOSAB.5.000641
  * [10] Marcus Seidel et al.  “All solid-state spectral broadening: an average and peak power scalable method for compression of ultrashort pulses”  In _Optics Express_ 24.9, 2016, pp. 9412  DOI: 10.1364/OE.24.009412
  * [11] Pei-Chi Huang et al.  “Polarization control of isolated high-harmonic pulses”  In _Nature Photonics_ 12.6, 2018, pp. 349–354  DOI: 10.1038/s41566-018-0145-0
  * [12] Chih-Hsuan Lu et al.  “Greater than 50 times compression of 1030 nm Yb:KGW laser pulses to single-cycle duration”  In _Optics Express_ 27.11, 2019, pp. 15638  DOI: 10.1364/OE.27.015638
  * [13] Martin Kaumanns et al.  “Multipass spectral broadening of 18 mJ pulses compressible from 13 ps to 41 fs”  In _Optics Letters_ 43.23, 2018, pp. 5877  DOI: 10.1364/OL.43.005877
  * [14] Prannay Balla et al.  “Postcompression of picosecond pulses into the few-cycle regime”  In _Optics Letters_ 45.9, 2020, pp. 2572  DOI: 10.1364/OL.388665
  * [15] Martin Kaumanns et al.  “Spectral broadening of 112 mJ, 13 ps pulses at 5 kHz in a LG ${}_{\textrm{10}}$ multipass cell with compressibility to 37 fs”  In _Optics Letters_ 46.5, 2021, pp. 929  DOI: 10.1364/OL.416734
  * [16] Sebastian Gröbmeyer et al.  “Self-compression at 1 um wavelength in all-bulk multi-pass geometry”  In _Applied Physics B_ 126.10, 2020, pp. 159  DOI: 10.1007/s00340-020-07506-4
  * [17] Jiajun Song et al.  “Generation of 172 fs pulse from a Nd: YVO4 picosecond laser by using multi-pass-cell technique”  In _Applied Physics B_ 127.4, 2021, pp. 50  DOI: 10.1007/s00340-021-07587-9
  * [18] Jiajun Song et al.  “Generation of 601 fs pulse from an 8 kHz Nd:YVO4 picosecond laser by multi-pass-cell spectral broadening”  In _Chinese Optics Letters_ 19.9, 2021, pp. 093201  DOI: 10.3788/COL202119.093201
  * [19] Kilian Fritsch et al.  “All-solid-state multipass spectral broadening to sub-20 fs”  In _Optics Letters_ 43.19, 2018, pp. 4643  DOI: 10.1364/OL.43.004643
  * [20] Edoardo Vicentini et al.  “Nonlinear pulse compression to 22 fs at 156 uJ by an all-solid-state multipass approach”  In _Optics Express_ 28.4, 2020, pp. 4541  DOI: 10.1364/OE.385583
  * [21] Gaia Barbiero et al.  “Broadband terahertz solid-state emitter driven by Yb:YAG thin-disk oscillator”  In _Journal of Physics B: Atomic, Molecular and Optical Physics_ 53.12, 2020, pp. 125601  DOI: 10.1088/1361-6455/ab8049
  * [22] Chia-Lun Tsai et al.  “Efficient nonlinear compression of a mode-locked thin-disk oscillator to 27 fs at 98 W average power”  In _Optics Letters_ 44.17, 2019, pp. 4115  DOI: 10.1364/OL.44.004115
  * [23] G. Barbiero et al.  “Efficient nonlinear compression of a thin-disk oscillator to 8.5 fs at 55 W average power”  In _arXiv:2108.06244 [physics.optics]_ , 2021  URL: https://arxiv.org/abs/2108.06244
  * [24] C. Li, L. Winkelmann and I. Hartl  “Flexible Pulse-Shape Picosecond Front-End for XFEL Photocathode Lasers”  In _Conference on Lasers and Electro-Optics_ San Jose, California: OSA, 2019, pp. SF3I.4  DOI: 10.1364/CLEO˙SI.2019.SF3I.4
  * [25] Marcus Seidel et al.  “Ultrafast MHz-rate burst-mode pump-probe laser for the FLASH FEL facility based on nonlinear compression of ps-level pulses from an Yb-amplifier chain” arXiv: 2105.05882  In _arXiv:2105.05882 [physics]_ , 2021  URL: http://arxiv.org/abs/2105.05882
  * [26] L. Lavenu et al.  “High-power two-cycle ultrafast source based on hybrid nonlinear compression”  In _Optics Express_ 27.3, 2019, pp. 1958  DOI: 10.1364/OE.27.001958
  * [27] Gunnar Arisholm and Helge Fonnum  “Simulation System For Optical Science (SISYFOS) – tutorial,version 2” 21/01183, FFI-rapport  Norwegian Defence Research Establishment (FFI), 2021  URL: https://www.ffi.no/en/publications-archive/simulation-system-for-optical-science-sisyfos-tutorial-version-2
  * [28] David Milam  “Review and assessment of measured values of the nonlinear refractive-index coefficient of fused silica”  In _Applied Optics_ 37.3, 1998, pp. 546  DOI: 10.1364/AO.37.000546
  * [29] Andreas Vernaleken et al.  “Method and arrangement for the spectral broadening of laser pulses for non-linear pulse compression”, 2015  URL: https://patents.google.com/patent/DE102014007159A1/en
  * [30] Yu-Chen Cheng, Chih-Hsuan Lu, Yuan-Yao Lin and A.. Kung  “Supercontinuum generation in a multi-plate medium”  In _Optics Express_ 24.7, 2016, pp. 7224  DOI: 10.1364/OE.24.007224
  * [31] Marc Hanna et al.  “Nonlinear beam matching to gas-filled multipass cells”  In _OSA Continuum_ 4.2, 2021, pp. 732  DOI: 10.1364/OSAC.418247
  * [32] P. Russbueldt et al.  “Scalable 30 fs laser source with 530 W average power”  In _Optics Letters_ 44.21, 2019, pp. 5222  DOI: 10.1364/OL.44.005222
  * [33] Marc Hanna et al.  “Hybrid pulse propagation model and quasi-phase-matched four-wave mixing in multipass cells”  In _Journal of the Optical Society of America B_ 37.10, 2020, pp. 2982  DOI: 10.1364/JOSAB.395789
  * [34] Chia-Lun Tsai et al.  “Nonlinear Compression of Intense Optical Pulses at 1.55 um by Multiple Plate Continuum Generation”  In _Journal of Lightwave Technology_ 37.19, 2019, pp. 5100–5107  DOI: 10.1109/JLT.2019.2929287
  * [35] Johannes Weitenberg et al.  “Multi-pass-cell-based nonlinear pulse compression to 115 fs at 75 uJ pulse energy and 300 W average power”  In _Optics Express_ 25.17, 2017, pp. 20502  DOI: 10.1364/OE.25.020502
  * [36] Marc Hanna et al.  “Nonlinear temporal compression in multipass cells: theory”  In _Journal of the Optical Society of America B_ 34.7, 2017, pp. 1340  DOI: 10.1364/JOSAB.34.001340
  * [37] Michael Müller et al.  “Multipass cell for high-power few-cycle compression”  In _Optics Letters_ 46.11, 2021, pp. 2678  DOI: 10.1364/OL.425872
  * [38] Douglas S. Hobbs, Bruce D. MacLeod and Ernest Sabatino III  “Continued advancement of laser damage resistant optically functional microstructures”  In _Laser-Induced Damage in Optical Materials: 2012_ 8530 SPIE, 2012, pp. 141–163  URL: https://doi.org/10.1117/12.976909
  * [39] E A Khazanov, S Yu Mironov and G A Mourou  “Nonlinear compression of high-power laser pulses: compression after compressor approach”  In _Physics-Uspekhi_ 62.11, 2019, pp. 1096–1124  DOI: 10.3367/UFNe.2019.05.038564
  * [40] Christoph M. Heyl et al.  “High-energy bow tie multi-pass cells for nonlinear spectral broadening applications”  In _submitted for publication_ , 2021
  * [41] D. Herriott, H. Kogelnik and R. Kompfner  “Off-Axis Paths in Spherical Mirror Interferometers”  In _Applied Optics_ 3.4, 1964, pp. 523  DOI: 10.1364/AO.3.000523
  * [42] Alphan Sennaroglu and James Fujimoto  “Design criteria for Herriott-type multi-pass cavities for ultrashort pulse lasers”  In _Optics Express_ 11.9, 2003, pp. 1106  DOI: 10.1364/OE.11.001106
  * [43] Donald R. Herriott and Harry J. Schulte  “Folded Optical Delay Lines”  In _Applied Optics_ 4.8, 1965, pp. 883  DOI: 10.1364/AO.4.000883
  * [44] Anthony E. Siegman  “Lasers”  Sausalito, California: University Science Books, 1986
  * [45] Li Yan, Yuan-Qun Liu and C.H. Lee  “Pulse temporal and spatial chirping by a bulk Kerr medium in a regenerative amplifier”  In _IEEE Journal of Quantum Electronics_ 30.9, 1994, pp. 2194–2202  DOI: 10.1109/3.309883
  * [46] Govind Agrawal  “Self-Phase Modulation”  In _Nonlinear Fiber Optics_ Elsevier, 2013, pp. 87–128  DOI: 10.1016/B978-0-12-397023-7.00004-8