Abstract

A model is presented, which allows to predict the (in)homogeneity of large areas covered with Laser-induced Periodic Surface Structures (LIPSS), based on the laser processing parameters (peak laser fluence and geometrical pulse-to-pulse overlap) and experimentally determined material properties. As such, the model allows to establish optimal processing conditions, given the material properties of the substrate to be processed. The model is experimentally validated over a large range of geometrical pulse-to-pulse overlap values and fluence levels on silicon using a picosecond laser source.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref] [PubMed]
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2018 (3)

J.-M. Romano, A. Garcia-giron, P. Penchev, and S. Dimov, “Triangular laser-induced submicron textures for functionalising stainless steel surfaces,” Appl. Surf. Sci. 440, 162–169 (2018).
[Crossref]

A. H. Lutey, L. Gemini, L. Romoli, G. Lazzini, F. Fuso, M. Faucon, and R. Kling, “Towards laser-textured antibacterial surfaces,” Sci. Reports 8, 1–10 (2018).

C. Kunz, F. A. Müller, and S. Gräf, “Multifunctional hierarchical surface structures by femtosecond laser processing,” Materials 11, 19–26 (2018).
[Crossref]

2017 (4)

C. Yiannakou, C. Simitzi, A. Manousaki, C. Fotakis, A. Ranella, and E. Stratakis, “Cell patterning via laser micro/nano structured silicon surfaces,” Biofabrication 9, 025024 (2017).
[Crossref] [PubMed]

S. V. Kirner, U. Hermens, A. Mimidis, E. Skoulas, C. Florian, F. Hischen, C. Plamadeala, W. Baumgartner, K. Winands, H. Mescheder, J. Krüger, J. Solis, J. Siegel, E. Stratakis, and J. Bonse, “Mimicking bug-like surface structures and their fluid transport produced by ultrashort laser pulse irradiation of steel,” Appl. Phys. A: Mater. Sci. Process. 123, 1–13 (2017).
[Crossref]

I. Gnilitskyi, T. J. Y. Derrien, Y. Levy, N. M. Bulgakova, T. Mocek, and L. Orazi, “High-speed manufacturing of highly regular femtosecond laser-induced periodic surface structures: physical origin of regularity,” Sci. Reports 7, 1–11 (2017).

J. Bonse, S. Höhm, S. V. Kirner, A. Rosenfeld, and J. Krüger, “Laser-induced periodic surface structures — a scientific evergreen,” IEEE J. Sel. Top. Quantum Electron. 23, 9000615 (2017).
[Crossref]

2016 (4)

P. Gregorčič, M. Sedlaček, B. Podgornik, and J. Reif, “Formation of laser-induced periodic surface structures (LIPSS) on tool steel by multiple picosecond laser pulses of different polarizations,” Appl. Surf. Sci. 387, 698–706 (2016).
[Crossref]

D. Puerto, M. Garcia-Lechuga, J. Hernandez-Rueda, A. Garcia-Leis, S. Sanchez-Cortes, J. Solis, and J. Siegel, “Femtosecond laser-controlled self-assembly of amorphous-crystalline nanogratings in silicon,” Nanotechnology 27, 1–8 (2016).
[Crossref]

S. He, J. J. Nivas, A. Vecchione, M. Hu, and S. Amoruso, “On the generation of grooves on crystalline silicon irradiated by femtosecond laser pulses,” Opt. Express 24, 3238 (2016).
[Crossref] [PubMed]

F. A. Müller, C. Kunz, and S. Gräf, “Bio-inspired functional surfaces based on laser-induced periodic surface structures,” Materials 9, 1–29 (2016).
[Crossref]

2015 (3)

I. Paradisanos, C. Fotakis, S. H. Anastasiadis, and E. Stratakis, “Gradient induced liquid motion on laser structured black Si surfaces,” Appl. Phys. Lett. 107, 111603 (2015).
[Crossref]

E. Rebollar, M. Castillejo, and T. A. Ezquerra, “Laser induced periodic surface structures on polymer films: from fundamentals to applications,” Eur. Polym. J. 73, 162–174 (2015).
[Crossref]

A. Rodríguez-Rodríguez, E. Rebollar, M. Soccio, T. A. Ezquerra, D. R. Rueda, J. V. Garcia-Ramos, M. Castillejo, and M. C. Garcia-Gutierrez, “Laser-induced periodic surface structures on conjugated polymers: Poly(3-hexylthiophene),” Macromolecules 48, 4024–4031 (2015).
[Crossref]

2014 (5)

J. Long, P. Fan, M. Zhong, H. Zhang, Y. Xie, and C. Lin, “Superhydrophobic and colorful copper surfaces fabricated by picosecond laser induced periodic nanostructures,” Appl. Surf. Sci. 311, 461–467 (2014).
[Crossref]

J. Lehr and A. M. Kietzig, “Production of homogenous micro-structures by femtosecond laser micro-machining,” Opt. Lasers Eng. 57, 121–129 (2014).
[Crossref]

A. A. Ionin, S. I. Kudryashov, S. V. Makarov, A. A. Rudenko, L. V. Seleznev, D. V. Sinitsyn, E. V. Golosov, Y. R. Kolobov, and A. E. Ligachev, “”Heterogeneous” versus “homogeneous” nucleation and growth of microcones on titanium surface under UV femtosecond-laser irradiation,” Appl. Phys. A: Mater. Sci. Process. 116, 1133–1139 (2014).
[Crossref]

R. Buividas, M. Mikutis, and S. Juodkazis, “Surface and bulk structuring of materials by ripples with long and short laser pulses: recent advances,” Prog. Quantum Electron. 38, 119–156 (2014).
[Crossref]

A. Ruiz de la Cruz, R. Lahoz, J. Siegel, G. F. de la Fuente, and J. Solis, “High speed inscription of uniform, large-area laser-induced periodic surface structures in Cr films using a high repetition rate fs laser,” Opt. Lett. 39, 2491 (2014).
[Crossref] [PubMed]

2013 (4)

J. Bonse, G. Mann, J. Krüger, M. Marcinkowski, and M. Eberstein, “Femtosecond laser-induced removal of silicon nitride layers from doped and textured silicon wafers used in photovoltaics,” Thin Solid Films 542, 420–425 (2013).
[Crossref]

E. Rebollar, J. R. Vázquez de Aldana, I. Martín-Fabiani, M. Hernández, D. R. Rueda, T. A. Ezquerra, C. Domingo, P. Moreno, and M. Castillejo, “Assessment of femtosecond laser induced periodic surface structures on polymer films,” Phys. Chem. Chem. Phys. 15, 11287 (2013).
[Crossref] [PubMed]

J. Eichstädt, G. R. B. E. Römer, and A. J. Huis in’t Veld, “Determination of irradiation parameters for laser-induced periodic surface structures,” Appl. Surf. Sci. 264, 79–87 (2013).
[Crossref]

T. J. Y. Derrien, T. E. Itina, R. Torres, T. Sarnet, and M. Sentis, “Possible surface plasmon polariton excitation under femtosecond laser irradiation of silicon,” J. Appl. Phys. 114, 1–11 (2013).
[Crossref]

2012 (1)

A. A. Ionin, S. I. Kudryashov, S. V. Makarov, L. V. Seleznev, D. V. Sinitsyn, E. V. Golosov, O. A. Golosova, Y. R. Kolobov, and A. E. Ligachev, “Femtosecond laser color marking of metal and semiconductor surfaces,” Appl. Phys. A: Mater. Sci. Process. 107, 301–305 (2012).
[Crossref]

2011 (1)

J. Eichstädt, G. R. Römer, and A. J. Huis in’t Veld, “Towards friction control using laser-induced periodic Surface Structures,” Phys. Procedia 12, 7–15 (2011).
[Crossref]

2010 (2)

B. Dusser, Z. Sagan, H. Soder, N. Faure, J. Colombier, M. Jourlin, and E. Audouard, “Controlled nanostructrures formation by ultra fast laser pulses for color marking,” Opt. Express 18, 2913–2924 (2010).
[Crossref] [PubMed]

J. Bonse and J. Krüger, “Pulse number dependence of laser-induced periodic surface structures for femtosecond laser irradiation of silicon,” J. Appl. Phys. 108, 034903 (2010).
[Crossref]

2009 (1)

J. Bonse, A. Rosenfeld, and J. Krüger, “On the role of surface plasmon polaritons in the formation of laser-induced periodic surface structures upon irradiation of silicon by femtosecond-laser pulses,” J. Appl. Phys. 106, 104910 (2009).
[Crossref]

2008 (2)

M. Halbwax, T. Sarnet, P. Delaporte, M. Sentis, H. Etienne, F. Torregrosa, V. Vervisch, I. Perichaud, and S. Martinuzzi, “Micro and nano-structuration of silicon by femtosecond laser: application to silicon photovoltaic cells fabrication,” Thin Solid Films 516, 6791–6795 (2008).
[Crossref]

M. Groenendijk, “Fabrication of super hydrophobic surfaces by fs laser pulses,” Laser Tech. J. 5, 44–47 (2008).
[Crossref]

2005 (2)

J. Bonse, M. Munz, and H. Sturm, “Structure formation on the surface of indium phosphide irradiated by femtosecond laser pulses,” J. Appl. Phys. 97, 013538 (2005).
[Crossref]

J. Wang and C. Guo, “Ultrafast dynamics of femtosecond laser-induced periodic surface pattern formation on metals,” Appl. Phys. Lett. 87, 1–3 (2005).
[Crossref]

2003 (1)

N. Yasumaru, K. Miyazaki, and J. Kiuchi, “Femtosecond-laser-induced nanostructure formed on hard thin films of TiN and DLC,” Appl. Phys. A: Mater. Sci. Process. 76, 983–985 (2003).
[Crossref]

2002 (1)

A. Sabbah and D. Riffe, “Femtosecond pump-probe reflectivity study of silicon carrier dynamics,” Phys. Rev. B 66, 1–11 (2002).
[Crossref]

1999 (1)

S. Baudach, J. Bonse, and W. Kautek, “Ablation experiments on polyimide with femtosecond laser pulses,” Appl. Phys. A: Mater. Sci. Process. 69, 395–398 (1999).
[Crossref]

1995 (1)

M. A. Green and M. J. Keevers, “Optical Properties of Intrinsic Silicon at 300 K,” Prog. Photovolt. Res. Appl. 3, 1891995).

1988 (1)

1982 (1)

H. M. Van Driel, J. E. Sipe, and J. F. Young, “Laser-induced periodic surface structure on solids: a universal phenomenon,” Phys. Rev. Lett. 49, 1955–1958 (1982).
[Crossref]

1981 (1)

P. A. Temple and M. J. Soileau, “Polarization charge model for laser-induced ripple patterns in dielectric materials,” IEEE J. Quantum Electron. 17, 2067–2072 (1981).
[Crossref]

1965 (1)

M. Birnbaum, “Semiconductor surface damage produced by ruby lasers,” J. Appl. Phys. 36, 3688–3689 (1965).
[Crossref]

Amoruso, S.

Anastasiadis, S. H.

I. Paradisanos, C. Fotakis, S. H. Anastasiadis, and E. Stratakis, “Gradient induced liquid motion on laser structured black Si surfaces,” Appl. Phys. Lett. 107, 111603 (2015).
[Crossref]

Audouard, E.

Baudach, S.

S. Baudach, J. Bonse, and W. Kautek, “Ablation experiments on polyimide with femtosecond laser pulses,” Appl. Phys. A: Mater. Sci. Process. 69, 395–398 (1999).
[Crossref]

Baumgartner, W.

S. V. Kirner, U. Hermens, A. Mimidis, E. Skoulas, C. Florian, F. Hischen, C. Plamadeala, W. Baumgartner, K. Winands, H. Mescheder, J. Krüger, J. Solis, J. Siegel, E. Stratakis, and J. Bonse, “Mimicking bug-like surface structures and their fluid transport produced by ultrashort laser pulse irradiation of steel,” Appl. Phys. A: Mater. Sci. Process. 123, 1–13 (2017).
[Crossref]

Becker, F. M.

Birnbaum, M.

M. Birnbaum, “Semiconductor surface damage produced by ruby lasers,” J. Appl. Phys. 36, 3688–3689 (1965).
[Crossref]

Bonse, J.

J. Bonse, S. Höhm, S. V. Kirner, A. Rosenfeld, and J. Krüger, “Laser-induced periodic surface structures — a scientific evergreen,” IEEE J. Sel. Top. Quantum Electron. 23, 9000615 (2017).
[Crossref]

S. V. Kirner, U. Hermens, A. Mimidis, E. Skoulas, C. Florian, F. Hischen, C. Plamadeala, W. Baumgartner, K. Winands, H. Mescheder, J. Krüger, J. Solis, J. Siegel, E. Stratakis, and J. Bonse, “Mimicking bug-like surface structures and their fluid transport produced by ultrashort laser pulse irradiation of steel,” Appl. Phys. A: Mater. Sci. Process. 123, 1–13 (2017).
[Crossref]

J. Bonse, G. Mann, J. Krüger, M. Marcinkowski, and M. Eberstein, “Femtosecond laser-induced removal of silicon nitride layers from doped and textured silicon wafers used in photovoltaics,” Thin Solid Films 542, 420–425 (2013).
[Crossref]

J. Bonse and J. Krüger, “Pulse number dependence of laser-induced periodic surface structures for femtosecond laser irradiation of silicon,” J. Appl. Phys. 108, 034903 (2010).
[Crossref]

J. Bonse, A. Rosenfeld, and J. Krüger, “On the role of surface plasmon polaritons in the formation of laser-induced periodic surface structures upon irradiation of silicon by femtosecond-laser pulses,” J. Appl. Phys. 106, 104910 (2009).
[Crossref]

J. Bonse, M. Munz, and H. Sturm, “Structure formation on the surface of indium phosphide irradiated by femtosecond laser pulses,” J. Appl. Phys. 97, 013538 (2005).
[Crossref]

S. Baudach, J. Bonse, and W. Kautek, “Ablation experiments on polyimide with femtosecond laser pulses,” Appl. Phys. A: Mater. Sci. Process. 69, 395–398 (1999).
[Crossref]

Buividas, R.

R. Buividas, M. Mikutis, and S. Juodkazis, “Surface and bulk structuring of materials by ripples with long and short laser pulses: recent advances,” Prog. Quantum Electron. 38, 119–156 (2014).
[Crossref]

Bulgakova, N. M.

I. Gnilitskyi, T. J. Y. Derrien, Y. Levy, N. M. Bulgakova, T. Mocek, and L. Orazi, “High-speed manufacturing of highly regular femtosecond laser-induced periodic surface structures: physical origin of regularity,” Sci. Reports 7, 1–11 (2017).

Castillejo, M.

A. Rodríguez-Rodríguez, E. Rebollar, M. Soccio, T. A. Ezquerra, D. R. Rueda, J. V. Garcia-Ramos, M. Castillejo, and M. C. Garcia-Gutierrez, “Laser-induced periodic surface structures on conjugated polymers: Poly(3-hexylthiophene),” Macromolecules 48, 4024–4031 (2015).
[Crossref]

E. Rebollar, M. Castillejo, and T. A. Ezquerra, “Laser induced periodic surface structures on polymer films: from fundamentals to applications,” Eur. Polym. J. 73, 162–174 (2015).
[Crossref]

E. Rebollar, J. R. Vázquez de Aldana, I. Martín-Fabiani, M. Hernández, D. R. Rueda, T. A. Ezquerra, C. Domingo, P. Moreno, and M. Castillejo, “Assessment of femtosecond laser induced periodic surface structures on polymer films,” Phys. Chem. Chem. Phys. 15, 11287 (2013).
[Crossref] [PubMed]

Colombier, J.

de la Fuente, G. F.

Delaporte, P.

M. Halbwax, T. Sarnet, P. Delaporte, M. Sentis, H. Etienne, F. Torregrosa, V. Vervisch, I. Perichaud, and S. Martinuzzi, “Micro and nano-structuration of silicon by femtosecond laser: application to silicon photovoltaic cells fabrication,” Thin Solid Films 516, 6791–6795 (2008).
[Crossref]

Derrien, T. J. Y.

I. Gnilitskyi, T. J. Y. Derrien, Y. Levy, N. M. Bulgakova, T. Mocek, and L. Orazi, “High-speed manufacturing of highly regular femtosecond laser-induced periodic surface structures: physical origin of regularity,” Sci. Reports 7, 1–11 (2017).

T. J. Y. Derrien, T. E. Itina, R. Torres, T. Sarnet, and M. Sentis, “Possible surface plasmon polariton excitation under femtosecond laser irradiation of silicon,” J. Appl. Phys. 114, 1–11 (2013).
[Crossref]

Dimov, S.

J.-M. Romano, A. Garcia-giron, P. Penchev, and S. Dimov, “Triangular laser-induced submicron textures for functionalising stainless steel surfaces,” Appl. Surf. Sci. 440, 162–169 (2018).
[Crossref]

Domingo, C.

E. Rebollar, J. R. Vázquez de Aldana, I. Martín-Fabiani, M. Hernández, D. R. Rueda, T. A. Ezquerra, C. Domingo, P. Moreno, and M. Castillejo, “Assessment of femtosecond laser induced periodic surface structures on polymer films,” Phys. Chem. Chem. Phys. 15, 11287 (2013).
[Crossref] [PubMed]

Dusser, B.

Eberstein, M.

J. Bonse, G. Mann, J. Krüger, M. Marcinkowski, and M. Eberstein, “Femtosecond laser-induced removal of silicon nitride layers from doped and textured silicon wafers used in photovoltaics,” Thin Solid Films 542, 420–425 (2013).
[Crossref]

Eichstädt, J.

J. Eichstädt, G. R. B. E. Römer, and A. J. Huis in’t Veld, “Determination of irradiation parameters for laser-induced periodic surface structures,” Appl. Surf. Sci. 264, 79–87 (2013).
[Crossref]

J. Eichstädt, G. R. Römer, and A. J. Huis in’t Veld, “Towards friction control using laser-induced periodic Surface Structures,” Phys. Procedia 12, 7–15 (2011).
[Crossref]

Eichstädt, Justus

Justus Eichstädt, The Spatial Emergence of Laser-Induced Periodic Surface Structures under Lateral Displacement Irradiation Conditions, 2 (University of Twente, 2012).

Etienne, H.

M. Halbwax, T. Sarnet, P. Delaporte, M. Sentis, H. Etienne, F. Torregrosa, V. Vervisch, I. Perichaud, and S. Martinuzzi, “Micro and nano-structuration of silicon by femtosecond laser: application to silicon photovoltaic cells fabrication,” Thin Solid Films 516, 6791–6795 (2008).
[Crossref]

Ezquerra, T. A.

A. Rodríguez-Rodríguez, E. Rebollar, M. Soccio, T. A. Ezquerra, D. R. Rueda, J. V. Garcia-Ramos, M. Castillejo, and M. C. Garcia-Gutierrez, “Laser-induced periodic surface structures on conjugated polymers: Poly(3-hexylthiophene),” Macromolecules 48, 4024–4031 (2015).
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E. Rebollar, M. Castillejo, and T. A. Ezquerra, “Laser induced periodic surface structures on polymer films: from fundamentals to applications,” Eur. Polym. J. 73, 162–174 (2015).
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E. Rebollar, J. R. Vázquez de Aldana, I. Martín-Fabiani, M. Hernández, D. R. Rueda, T. A. Ezquerra, C. Domingo, P. Moreno, and M. Castillejo, “Assessment of femtosecond laser induced periodic surface structures on polymer films,” Phys. Chem. Chem. Phys. 15, 11287 (2013).
[Crossref] [PubMed]

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J. Heitz, B. Reisinger, M. Fahrner, C. Romanin, J. Siegel, and V. Svorcik, “Laser-induced periodic surface structures (LIPSS) on polymer surfaces,” Int. Conf. on Transparent Opt. Networks pp. 1–4 (2012).

Fan, P.

J. Long, P. Fan, M. Zhong, H. Zhang, Y. Xie, and C. Lin, “Superhydrophobic and colorful copper surfaces fabricated by picosecond laser induced periodic nanostructures,” Appl. Surf. Sci. 311, 461–467 (2014).
[Crossref]

Faucon, M.

A. H. Lutey, L. Gemini, L. Romoli, G. Lazzini, F. Fuso, M. Faucon, and R. Kling, “Towards laser-textured antibacterial surfaces,” Sci. Reports 8, 1–10 (2018).

Faure, N.

Florian, C.

S. V. Kirner, U. Hermens, A. Mimidis, E. Skoulas, C. Florian, F. Hischen, C. Plamadeala, W. Baumgartner, K. Winands, H. Mescheder, J. Krüger, J. Solis, J. Siegel, E. Stratakis, and J. Bonse, “Mimicking bug-like surface structures and their fluid transport produced by ultrashort laser pulse irradiation of steel,” Appl. Phys. A: Mater. Sci. Process. 123, 1–13 (2017).
[Crossref]

Fotakis, C.

C. Yiannakou, C. Simitzi, A. Manousaki, C. Fotakis, A. Ranella, and E. Stratakis, “Cell patterning via laser micro/nano structured silicon surfaces,” Biofabrication 9, 025024 (2017).
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I. Paradisanos, C. Fotakis, S. H. Anastasiadis, and E. Stratakis, “Gradient induced liquid motion on laser structured black Si surfaces,” Appl. Phys. Lett. 107, 111603 (2015).
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A. H. Lutey, L. Gemini, L. Romoli, G. Lazzini, F. Fuso, M. Faucon, and R. Kling, “Towards laser-textured antibacterial surfaces,” Sci. Reports 8, 1–10 (2018).

Garcia-giron, A.

J.-M. Romano, A. Garcia-giron, P. Penchev, and S. Dimov, “Triangular laser-induced submicron textures for functionalising stainless steel surfaces,” Appl. Surf. Sci. 440, 162–169 (2018).
[Crossref]

Garcia-Gutierrez, M. C.

A. Rodríguez-Rodríguez, E. Rebollar, M. Soccio, T. A. Ezquerra, D. R. Rueda, J. V. Garcia-Ramos, M. Castillejo, and M. C. Garcia-Gutierrez, “Laser-induced periodic surface structures on conjugated polymers: Poly(3-hexylthiophene),” Macromolecules 48, 4024–4031 (2015).
[Crossref]

Garcia-Lechuga, M.

D. Puerto, M. Garcia-Lechuga, J. Hernandez-Rueda, A. Garcia-Leis, S. Sanchez-Cortes, J. Solis, and J. Siegel, “Femtosecond laser-controlled self-assembly of amorphous-crystalline nanogratings in silicon,” Nanotechnology 27, 1–8 (2016).
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Garcia-Leis, A.

D. Puerto, M. Garcia-Lechuga, J. Hernandez-Rueda, A. Garcia-Leis, S. Sanchez-Cortes, J. Solis, and J. Siegel, “Femtosecond laser-controlled self-assembly of amorphous-crystalline nanogratings in silicon,” Nanotechnology 27, 1–8 (2016).
[Crossref]

Garcia-Ramos, J. V.

A. Rodríguez-Rodríguez, E. Rebollar, M. Soccio, T. A. Ezquerra, D. R. Rueda, J. V. Garcia-Ramos, M. Castillejo, and M. C. Garcia-Gutierrez, “Laser-induced periodic surface structures on conjugated polymers: Poly(3-hexylthiophene),” Macromolecules 48, 4024–4031 (2015).
[Crossref]

Gemini, L.

A. H. Lutey, L. Gemini, L. Romoli, G. Lazzini, F. Fuso, M. Faucon, and R. Kling, “Towards laser-textured antibacterial surfaces,” Sci. Reports 8, 1–10 (2018).

Gnilitskyi, I.

I. Gnilitskyi, T. J. Y. Derrien, Y. Levy, N. M. Bulgakova, T. Mocek, and L. Orazi, “High-speed manufacturing of highly regular femtosecond laser-induced periodic surface structures: physical origin of regularity,” Sci. Reports 7, 1–11 (2017).

Golosov, E. V.

A. A. Ionin, S. I. Kudryashov, S. V. Makarov, A. A. Rudenko, L. V. Seleznev, D. V. Sinitsyn, E. V. Golosov, Y. R. Kolobov, and A. E. Ligachev, “”Heterogeneous” versus “homogeneous” nucleation and growth of microcones on titanium surface under UV femtosecond-laser irradiation,” Appl. Phys. A: Mater. Sci. Process. 116, 1133–1139 (2014).
[Crossref]

A. A. Ionin, S. I. Kudryashov, S. V. Makarov, L. V. Seleznev, D. V. Sinitsyn, E. V. Golosov, O. A. Golosova, Y. R. Kolobov, and A. E. Ligachev, “Femtosecond laser color marking of metal and semiconductor surfaces,” Appl. Phys. A: Mater. Sci. Process. 107, 301–305 (2012).
[Crossref]

Golosova, O. A.

A. A. Ionin, S. I. Kudryashov, S. V. Makarov, L. V. Seleznev, D. V. Sinitsyn, E. V. Golosov, O. A. Golosova, Y. R. Kolobov, and A. E. Ligachev, “Femtosecond laser color marking of metal and semiconductor surfaces,” Appl. Phys. A: Mater. Sci. Process. 107, 301–305 (2012).
[Crossref]

Gräf, S.

C. Kunz, F. A. Müller, and S. Gräf, “Multifunctional hierarchical surface structures by femtosecond laser processing,” Materials 11, 19–26 (2018).
[Crossref]

F. A. Müller, C. Kunz, and S. Gräf, “Bio-inspired functional surfaces based on laser-induced periodic surface structures,” Materials 9, 1–29 (2016).
[Crossref]

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M. A. Green and M. J. Keevers, “Optical Properties of Intrinsic Silicon at 300 K,” Prog. Photovolt. Res. Appl. 3, 1891995).

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P. Gregorčič, M. Sedlaček, B. Podgornik, and J. Reif, “Formation of laser-induced periodic surface structures (LIPSS) on tool steel by multiple picosecond laser pulses of different polarizations,” Appl. Surf. Sci. 387, 698–706 (2016).
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M. Groenendijk, “Fabrication of super hydrophobic surfaces by fs laser pulses,” Laser Tech. J. 5, 44–47 (2008).
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J. Wang and C. Guo, “Ultrafast dynamics of femtosecond laser-induced periodic surface pattern formation on metals,” Appl. Phys. Lett. 87, 1–3 (2005).
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M. Halbwax, T. Sarnet, P. Delaporte, M. Sentis, H. Etienne, F. Torregrosa, V. Vervisch, I. Perichaud, and S. Martinuzzi, “Micro and nano-structuration of silicon by femtosecond laser: application to silicon photovoltaic cells fabrication,” Thin Solid Films 516, 6791–6795 (2008).
[Crossref]

He, S.

Heitz, J.

J. Heitz, B. Reisinger, M. Fahrner, C. Romanin, J. Siegel, and V. Svorcik, “Laser-induced periodic surface structures (LIPSS) on polymer surfaces,” Int. Conf. on Transparent Opt. Networks pp. 1–4 (2012).

Hermens, U.

S. V. Kirner, U. Hermens, A. Mimidis, E. Skoulas, C. Florian, F. Hischen, C. Plamadeala, W. Baumgartner, K. Winands, H. Mescheder, J. Krüger, J. Solis, J. Siegel, E. Stratakis, and J. Bonse, “Mimicking bug-like surface structures and their fluid transport produced by ultrashort laser pulse irradiation of steel,” Appl. Phys. A: Mater. Sci. Process. 123, 1–13 (2017).
[Crossref]

Hernández, M.

E. Rebollar, J. R. Vázquez de Aldana, I. Martín-Fabiani, M. Hernández, D. R. Rueda, T. A. Ezquerra, C. Domingo, P. Moreno, and M. Castillejo, “Assessment of femtosecond laser induced periodic surface structures on polymer films,” Phys. Chem. Chem. Phys. 15, 11287 (2013).
[Crossref] [PubMed]

Hernandez-Rueda, J.

D. Puerto, M. Garcia-Lechuga, J. Hernandez-Rueda, A. Garcia-Leis, S. Sanchez-Cortes, J. Solis, and J. Siegel, “Femtosecond laser-controlled self-assembly of amorphous-crystalline nanogratings in silicon,” Nanotechnology 27, 1–8 (2016).
[Crossref]

Hischen, F.

S. V. Kirner, U. Hermens, A. Mimidis, E. Skoulas, C. Florian, F. Hischen, C. Plamadeala, W. Baumgartner, K. Winands, H. Mescheder, J. Krüger, J. Solis, J. Siegel, E. Stratakis, and J. Bonse, “Mimicking bug-like surface structures and their fluid transport produced by ultrashort laser pulse irradiation of steel,” Appl. Phys. A: Mater. Sci. Process. 123, 1–13 (2017).
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J. Bonse, S. Höhm, S. V. Kirner, A. Rosenfeld, and J. Krüger, “Laser-induced periodic surface structures — a scientific evergreen,” IEEE J. Sel. Top. Quantum Electron. 23, 9000615 (2017).
[Crossref]

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Huis in’t Veld, A. J.

J. Eichstädt, G. R. B. E. Römer, and A. J. Huis in’t Veld, “Determination of irradiation parameters for laser-induced periodic surface structures,” Appl. Surf. Sci. 264, 79–87 (2013).
[Crossref]

J. Eichstädt, G. R. Römer, and A. J. Huis in’t Veld, “Towards friction control using laser-induced periodic Surface Structures,” Phys. Procedia 12, 7–15 (2011).
[Crossref]

Ionin, A. A.

A. A. Ionin, S. I. Kudryashov, S. V. Makarov, A. A. Rudenko, L. V. Seleznev, D. V. Sinitsyn, E. V. Golosov, Y. R. Kolobov, and A. E. Ligachev, “”Heterogeneous” versus “homogeneous” nucleation and growth of microcones on titanium surface under UV femtosecond-laser irradiation,” Appl. Phys. A: Mater. Sci. Process. 116, 1133–1139 (2014).
[Crossref]

A. A. Ionin, S. I. Kudryashov, S. V. Makarov, L. V. Seleznev, D. V. Sinitsyn, E. V. Golosov, O. A. Golosova, Y. R. Kolobov, and A. E. Ligachev, “Femtosecond laser color marking of metal and semiconductor surfaces,” Appl. Phys. A: Mater. Sci. Process. 107, 301–305 (2012).
[Crossref]

Itina, T. E.

T. J. Y. Derrien, T. E. Itina, R. Torres, T. Sarnet, and M. Sentis, “Possible surface plasmon polariton excitation under femtosecond laser irradiation of silicon,” J. Appl. Phys. 114, 1–11 (2013).
[Crossref]

Jee, Y.

Jourlin, M.

Juodkazis, S.

R. Buividas, M. Mikutis, and S. Juodkazis, “Surface and bulk structuring of materials by ripples with long and short laser pulses: recent advances,” Prog. Quantum Electron. 38, 119–156 (2014).
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Kautek, W.

S. Baudach, J. Bonse, and W. Kautek, “Ablation experiments on polyimide with femtosecond laser pulses,” Appl. Phys. A: Mater. Sci. Process. 69, 395–398 (1999).
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M. A. Green and M. J. Keevers, “Optical Properties of Intrinsic Silicon at 300 K,” Prog. Photovolt. Res. Appl. 3, 1891995).

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J. Lehr and A. M. Kietzig, “Production of homogenous micro-structures by femtosecond laser micro-machining,” Opt. Lasers Eng. 57, 121–129 (2014).
[Crossref]

Kirner, S. V.

J. Bonse, S. Höhm, S. V. Kirner, A. Rosenfeld, and J. Krüger, “Laser-induced periodic surface structures — a scientific evergreen,” IEEE J. Sel. Top. Quantum Electron. 23, 9000615 (2017).
[Crossref]

S. V. Kirner, U. Hermens, A. Mimidis, E. Skoulas, C. Florian, F. Hischen, C. Plamadeala, W. Baumgartner, K. Winands, H. Mescheder, J. Krüger, J. Solis, J. Siegel, E. Stratakis, and J. Bonse, “Mimicking bug-like surface structures and their fluid transport produced by ultrashort laser pulse irradiation of steel,” Appl. Phys. A: Mater. Sci. Process. 123, 1–13 (2017).
[Crossref]

Kiuchi, J.

N. Yasumaru, K. Miyazaki, and J. Kiuchi, “Femtosecond-laser-induced nanostructure formed on hard thin films of TiN and DLC,” Appl. Phys. A: Mater. Sci. Process. 76, 983–985 (2003).
[Crossref]

Kling, R.

A. H. Lutey, L. Gemini, L. Romoli, G. Lazzini, F. Fuso, M. Faucon, and R. Kling, “Towards laser-textured antibacterial surfaces,” Sci. Reports 8, 1–10 (2018).

Kolobov, Y. R.

A. A. Ionin, S. I. Kudryashov, S. V. Makarov, A. A. Rudenko, L. V. Seleznev, D. V. Sinitsyn, E. V. Golosov, Y. R. Kolobov, and A. E. Ligachev, “”Heterogeneous” versus “homogeneous” nucleation and growth of microcones on titanium surface under UV femtosecond-laser irradiation,” Appl. Phys. A: Mater. Sci. Process. 116, 1133–1139 (2014).
[Crossref]

A. A. Ionin, S. I. Kudryashov, S. V. Makarov, L. V. Seleznev, D. V. Sinitsyn, E. V. Golosov, O. A. Golosova, Y. R. Kolobov, and A. E. Ligachev, “Femtosecond laser color marking of metal and semiconductor surfaces,” Appl. Phys. A: Mater. Sci. Process. 107, 301–305 (2012).
[Crossref]

Krazer, A.

A. Krazer, Lehrbuch der Thetafunktionen (B.G. Teubner, 1903).

Krüger, J.

J. Bonse, S. Höhm, S. V. Kirner, A. Rosenfeld, and J. Krüger, “Laser-induced periodic surface structures — a scientific evergreen,” IEEE J. Sel. Top. Quantum Electron. 23, 9000615 (2017).
[Crossref]

S. V. Kirner, U. Hermens, A. Mimidis, E. Skoulas, C. Florian, F. Hischen, C. Plamadeala, W. Baumgartner, K. Winands, H. Mescheder, J. Krüger, J. Solis, J. Siegel, E. Stratakis, and J. Bonse, “Mimicking bug-like surface structures and their fluid transport produced by ultrashort laser pulse irradiation of steel,” Appl. Phys. A: Mater. Sci. Process. 123, 1–13 (2017).
[Crossref]

J. Bonse, G. Mann, J. Krüger, M. Marcinkowski, and M. Eberstein, “Femtosecond laser-induced removal of silicon nitride layers from doped and textured silicon wafers used in photovoltaics,” Thin Solid Films 542, 420–425 (2013).
[Crossref]

J. Bonse and J. Krüger, “Pulse number dependence of laser-induced periodic surface structures for femtosecond laser irradiation of silicon,” J. Appl. Phys. 108, 034903 (2010).
[Crossref]

J. Bonse, A. Rosenfeld, and J. Krüger, “On the role of surface plasmon polaritons in the formation of laser-induced periodic surface structures upon irradiation of silicon by femtosecond-laser pulses,” J. Appl. Phys. 106, 104910 (2009).
[Crossref]

Kudryashov, S. I.

A. A. Ionin, S. I. Kudryashov, S. V. Makarov, A. A. Rudenko, L. V. Seleznev, D. V. Sinitsyn, E. V. Golosov, Y. R. Kolobov, and A. E. Ligachev, “”Heterogeneous” versus “homogeneous” nucleation and growth of microcones on titanium surface under UV femtosecond-laser irradiation,” Appl. Phys. A: Mater. Sci. Process. 116, 1133–1139 (2014).
[Crossref]

A. A. Ionin, S. I. Kudryashov, S. V. Makarov, L. V. Seleznev, D. V. Sinitsyn, E. V. Golosov, O. A. Golosova, Y. R. Kolobov, and A. E. Ligachev, “Femtosecond laser color marking of metal and semiconductor surfaces,” Appl. Phys. A: Mater. Sci. Process. 107, 301–305 (2012).
[Crossref]

Kunz, C.

C. Kunz, F. A. Müller, and S. Gräf, “Multifunctional hierarchical surface structures by femtosecond laser processing,” Materials 11, 19–26 (2018).
[Crossref]

F. A. Müller, C. Kunz, and S. Gräf, “Bio-inspired functional surfaces based on laser-induced periodic surface structures,” Materials 9, 1–29 (2016).
[Crossref]

Lahoz, R.

Lazzini, G.

A. H. Lutey, L. Gemini, L. Romoli, G. Lazzini, F. Fuso, M. Faucon, and R. Kling, “Towards laser-textured antibacterial surfaces,” Sci. Reports 8, 1–10 (2018).

Lehr, J.

J. Lehr and A. M. Kietzig, “Production of homogenous micro-structures by femtosecond laser micro-machining,” Opt. Lasers Eng. 57, 121–129 (2014).
[Crossref]

Levy, Y.

I. Gnilitskyi, T. J. Y. Derrien, Y. Levy, N. M. Bulgakova, T. Mocek, and L. Orazi, “High-speed manufacturing of highly regular femtosecond laser-induced periodic surface structures: physical origin of regularity,” Sci. Reports 7, 1–11 (2017).

Ligachev, A. E.

A. A. Ionin, S. I. Kudryashov, S. V. Makarov, A. A. Rudenko, L. V. Seleznev, D. V. Sinitsyn, E. V. Golosov, Y. R. Kolobov, and A. E. Ligachev, “”Heterogeneous” versus “homogeneous” nucleation and growth of microcones on titanium surface under UV femtosecond-laser irradiation,” Appl. Phys. A: Mater. Sci. Process. 116, 1133–1139 (2014).
[Crossref]

A. A. Ionin, S. I. Kudryashov, S. V. Makarov, L. V. Seleznev, D. V. Sinitsyn, E. V. Golosov, O. A. Golosova, Y. R. Kolobov, and A. E. Ligachev, “Femtosecond laser color marking of metal and semiconductor surfaces,” Appl. Phys. A: Mater. Sci. Process. 107, 301–305 (2012).
[Crossref]

Lin, C.

J. Long, P. Fan, M. Zhong, H. Zhang, Y. Xie, and C. Lin, “Superhydrophobic and colorful copper surfaces fabricated by picosecond laser induced periodic nanostructures,” Appl. Surf. Sci. 311, 461–467 (2014).
[Crossref]

Long, J.

J. Long, P. Fan, M. Zhong, H. Zhang, Y. Xie, and C. Lin, “Superhydrophobic and colorful copper surfaces fabricated by picosecond laser induced periodic nanostructures,” Appl. Surf. Sci. 311, 461–467 (2014).
[Crossref]

Lutey, A. H.

A. H. Lutey, L. Gemini, L. Romoli, G. Lazzini, F. Fuso, M. Faucon, and R. Kling, “Towards laser-textured antibacterial surfaces,” Sci. Reports 8, 1–10 (2018).

Makarov, S. V.

A. A. Ionin, S. I. Kudryashov, S. V. Makarov, A. A. Rudenko, L. V. Seleznev, D. V. Sinitsyn, E. V. Golosov, Y. R. Kolobov, and A. E. Ligachev, “”Heterogeneous” versus “homogeneous” nucleation and growth of microcones on titanium surface under UV femtosecond-laser irradiation,” Appl. Phys. A: Mater. Sci. Process. 116, 1133–1139 (2014).
[Crossref]

A. A. Ionin, S. I. Kudryashov, S. V. Makarov, L. V. Seleznev, D. V. Sinitsyn, E. V. Golosov, O. A. Golosova, Y. R. Kolobov, and A. E. Ligachev, “Femtosecond laser color marking of metal and semiconductor surfaces,” Appl. Phys. A: Mater. Sci. Process. 107, 301–305 (2012).
[Crossref]

Mann, G.

J. Bonse, G. Mann, J. Krüger, M. Marcinkowski, and M. Eberstein, “Femtosecond laser-induced removal of silicon nitride layers from doped and textured silicon wafers used in photovoltaics,” Thin Solid Films 542, 420–425 (2013).
[Crossref]

Manousaki, A.

C. Yiannakou, C. Simitzi, A. Manousaki, C. Fotakis, A. Ranella, and E. Stratakis, “Cell patterning via laser micro/nano structured silicon surfaces,” Biofabrication 9, 025024 (2017).
[Crossref] [PubMed]

Marcinkowski, M.

J. Bonse, G. Mann, J. Krüger, M. Marcinkowski, and M. Eberstein, “Femtosecond laser-induced removal of silicon nitride layers from doped and textured silicon wafers used in photovoltaics,” Thin Solid Films 542, 420–425 (2013).
[Crossref]

Martín-Fabiani, I.

E. Rebollar, J. R. Vázquez de Aldana, I. Martín-Fabiani, M. Hernández, D. R. Rueda, T. A. Ezquerra, C. Domingo, P. Moreno, and M. Castillejo, “Assessment of femtosecond laser induced periodic surface structures on polymer films,” Phys. Chem. Chem. Phys. 15, 11287 (2013).
[Crossref] [PubMed]

Martinuzzi, S.

M. Halbwax, T. Sarnet, P. Delaporte, M. Sentis, H. Etienne, F. Torregrosa, V. Vervisch, I. Perichaud, and S. Martinuzzi, “Micro and nano-structuration of silicon by femtosecond laser: application to silicon photovoltaic cells fabrication,” Thin Solid Films 516, 6791–6795 (2008).
[Crossref]

Mescheder, H.

S. V. Kirner, U. Hermens, A. Mimidis, E. Skoulas, C. Florian, F. Hischen, C. Plamadeala, W. Baumgartner, K. Winands, H. Mescheder, J. Krüger, J. Solis, J. Siegel, E. Stratakis, and J. Bonse, “Mimicking bug-like surface structures and their fluid transport produced by ultrashort laser pulse irradiation of steel,” Appl. Phys. A: Mater. Sci. Process. 123, 1–13 (2017).
[Crossref]

Mikutis, M.

R. Buividas, M. Mikutis, and S. Juodkazis, “Surface and bulk structuring of materials by ripples with long and short laser pulses: recent advances,” Prog. Quantum Electron. 38, 119–156 (2014).
[Crossref]

Mimidis, A.

S. V. Kirner, U. Hermens, A. Mimidis, E. Skoulas, C. Florian, F. Hischen, C. Plamadeala, W. Baumgartner, K. Winands, H. Mescheder, J. Krüger, J. Solis, J. Siegel, E. Stratakis, and J. Bonse, “Mimicking bug-like surface structures and their fluid transport produced by ultrashort laser pulse irradiation of steel,” Appl. Phys. A: Mater. Sci. Process. 123, 1–13 (2017).
[Crossref]

Miyazaki, K.

N. Yasumaru, K. Miyazaki, and J. Kiuchi, “Femtosecond-laser-induced nanostructure formed on hard thin films of TiN and DLC,” Appl. Phys. A: Mater. Sci. Process. 76, 983–985 (2003).
[Crossref]

Mocek, T.

I. Gnilitskyi, T. J. Y. Derrien, Y. Levy, N. M. Bulgakova, T. Mocek, and L. Orazi, “High-speed manufacturing of highly regular femtosecond laser-induced periodic surface structures: physical origin of regularity,” Sci. Reports 7, 1–11 (2017).

Moreno, P.

E. Rebollar, J. R. Vázquez de Aldana, I. Martín-Fabiani, M. Hernández, D. R. Rueda, T. A. Ezquerra, C. Domingo, P. Moreno, and M. Castillejo, “Assessment of femtosecond laser induced periodic surface structures on polymer films,” Phys. Chem. Chem. Phys. 15, 11287 (2013).
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C. Kunz, F. A. Müller, and S. Gräf, “Multifunctional hierarchical surface structures by femtosecond laser processing,” Materials 11, 19–26 (2018).
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F. A. Müller, C. Kunz, and S. Gräf, “Bio-inspired functional surfaces based on laser-induced periodic surface structures,” Materials 9, 1–29 (2016).
[Crossref]

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J. Bonse, M. Munz, and H. Sturm, “Structure formation on the surface of indium phosphide irradiated by femtosecond laser pulses,” J. Appl. Phys. 97, 013538 (2005).
[Crossref]

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Orazi, L.

I. Gnilitskyi, T. J. Y. Derrien, Y. Levy, N. M. Bulgakova, T. Mocek, and L. Orazi, “High-speed manufacturing of highly regular femtosecond laser-induced periodic surface structures: physical origin of regularity,” Sci. Reports 7, 1–11 (2017).

Paradisanos, I.

I. Paradisanos, C. Fotakis, S. H. Anastasiadis, and E. Stratakis, “Gradient induced liquid motion on laser structured black Si surfaces,” Appl. Phys. Lett. 107, 111603 (2015).
[Crossref]

Penchev, P.

J.-M. Romano, A. Garcia-giron, P. Penchev, and S. Dimov, “Triangular laser-induced submicron textures for functionalising stainless steel surfaces,” Appl. Surf. Sci. 440, 162–169 (2018).
[Crossref]

Perichaud, I.

M. Halbwax, T. Sarnet, P. Delaporte, M. Sentis, H. Etienne, F. Torregrosa, V. Vervisch, I. Perichaud, and S. Martinuzzi, “Micro and nano-structuration of silicon by femtosecond laser: application to silicon photovoltaic cells fabrication,” Thin Solid Films 516, 6791–6795 (2008).
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Plamadeala, C.

S. V. Kirner, U. Hermens, A. Mimidis, E. Skoulas, C. Florian, F. Hischen, C. Plamadeala, W. Baumgartner, K. Winands, H. Mescheder, J. Krüger, J. Solis, J. Siegel, E. Stratakis, and J. Bonse, “Mimicking bug-like surface structures and their fluid transport produced by ultrashort laser pulse irradiation of steel,” Appl. Phys. A: Mater. Sci. Process. 123, 1–13 (2017).
[Crossref]

Podgornik, B.

P. Gregorčič, M. Sedlaček, B. Podgornik, and J. Reif, “Formation of laser-induced periodic surface structures (LIPSS) on tool steel by multiple picosecond laser pulses of different polarizations,” Appl. Surf. Sci. 387, 698–706 (2016).
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Puerto, D.

D. Puerto, M. Garcia-Lechuga, J. Hernandez-Rueda, A. Garcia-Leis, S. Sanchez-Cortes, J. Solis, and J. Siegel, “Femtosecond laser-controlled self-assembly of amorphous-crystalline nanogratings in silicon,” Nanotechnology 27, 1–8 (2016).
[Crossref]

Ranella, A.

C. Yiannakou, C. Simitzi, A. Manousaki, C. Fotakis, A. Ranella, and E. Stratakis, “Cell patterning via laser micro/nano structured silicon surfaces,” Biofabrication 9, 025024 (2017).
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W. S. Rasband, ImageJ 1.50i (U.S. National Institutes of Health, Bethesda, Maryland, USA, 2016).

Rebollar, E.

E. Rebollar, M. Castillejo, and T. A. Ezquerra, “Laser induced periodic surface structures on polymer films: from fundamentals to applications,” Eur. Polym. J. 73, 162–174 (2015).
[Crossref]

A. Rodríguez-Rodríguez, E. Rebollar, M. Soccio, T. A. Ezquerra, D. R. Rueda, J. V. Garcia-Ramos, M. Castillejo, and M. C. Garcia-Gutierrez, “Laser-induced periodic surface structures on conjugated polymers: Poly(3-hexylthiophene),” Macromolecules 48, 4024–4031 (2015).
[Crossref]

E. Rebollar, J. R. Vázquez de Aldana, I. Martín-Fabiani, M. Hernández, D. R. Rueda, T. A. Ezquerra, C. Domingo, P. Moreno, and M. Castillejo, “Assessment of femtosecond laser induced periodic surface structures on polymer films,” Phys. Chem. Chem. Phys. 15, 11287 (2013).
[Crossref] [PubMed]

Reif, J.

P. Gregorčič, M. Sedlaček, B. Podgornik, and J. Reif, “Formation of laser-induced periodic surface structures (LIPSS) on tool steel by multiple picosecond laser pulses of different polarizations,” Appl. Surf. Sci. 387, 698–706 (2016).
[Crossref]

Reisinger, B.

J. Heitz, B. Reisinger, M. Fahrner, C. Romanin, J. Siegel, and V. Svorcik, “Laser-induced periodic surface structures (LIPSS) on polymer surfaces,” Int. Conf. on Transparent Opt. Networks pp. 1–4 (2012).

Riffe, D.

A. Sabbah and D. Riffe, “Femtosecond pump-probe reflectivity study of silicon carrier dynamics,” Phys. Rev. B 66, 1–11 (2002).
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Rodríguez-Rodríguez, A.

A. Rodríguez-Rodríguez, E. Rebollar, M. Soccio, T. A. Ezquerra, D. R. Rueda, J. V. Garcia-Ramos, M. Castillejo, and M. C. Garcia-Gutierrez, “Laser-induced periodic surface structures on conjugated polymers: Poly(3-hexylthiophene),” Macromolecules 48, 4024–4031 (2015).
[Crossref]

Romanin, C.

J. Heitz, B. Reisinger, M. Fahrner, C. Romanin, J. Siegel, and V. Svorcik, “Laser-induced periodic surface structures (LIPSS) on polymer surfaces,” Int. Conf. on Transparent Opt. Networks pp. 1–4 (2012).

Romano, J.-M.

J.-M. Romano, A. Garcia-giron, P. Penchev, and S. Dimov, “Triangular laser-induced submicron textures for functionalising stainless steel surfaces,” Appl. Surf. Sci. 440, 162–169 (2018).
[Crossref]

Römer, G. R.

J. Eichstädt, G. R. Römer, and A. J. Huis in’t Veld, “Towards friction control using laser-induced periodic Surface Structures,” Phys. Procedia 12, 7–15 (2011).
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Römer, G. R. B. E.

J. Eichstädt, G. R. B. E. Römer, and A. J. Huis in’t Veld, “Determination of irradiation parameters for laser-induced periodic surface structures,” Appl. Surf. Sci. 264, 79–87 (2013).
[Crossref]

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A. H. Lutey, L. Gemini, L. Romoli, G. Lazzini, F. Fuso, M. Faucon, and R. Kling, “Towards laser-textured antibacterial surfaces,” Sci. Reports 8, 1–10 (2018).

Rosenfeld, A.

J. Bonse, S. Höhm, S. V. Kirner, A. Rosenfeld, and J. Krüger, “Laser-induced periodic surface structures — a scientific evergreen,” IEEE J. Sel. Top. Quantum Electron. 23, 9000615 (2017).
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J. Bonse, A. Rosenfeld, and J. Krüger, “On the role of surface plasmon polaritons in the formation of laser-induced periodic surface structures upon irradiation of silicon by femtosecond-laser pulses,” J. Appl. Phys. 106, 104910 (2009).
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A. A. Ionin, S. I. Kudryashov, S. V. Makarov, A. A. Rudenko, L. V. Seleznev, D. V. Sinitsyn, E. V. Golosov, Y. R. Kolobov, and A. E. Ligachev, “”Heterogeneous” versus “homogeneous” nucleation and growth of microcones on titanium surface under UV femtosecond-laser irradiation,” Appl. Phys. A: Mater. Sci. Process. 116, 1133–1139 (2014).
[Crossref]

Rueda, D. R.

A. Rodríguez-Rodríguez, E. Rebollar, M. Soccio, T. A. Ezquerra, D. R. Rueda, J. V. Garcia-Ramos, M. Castillejo, and M. C. Garcia-Gutierrez, “Laser-induced periodic surface structures on conjugated polymers: Poly(3-hexylthiophene),” Macromolecules 48, 4024–4031 (2015).
[Crossref]

E. Rebollar, J. R. Vázquez de Aldana, I. Martín-Fabiani, M. Hernández, D. R. Rueda, T. A. Ezquerra, C. Domingo, P. Moreno, and M. Castillejo, “Assessment of femtosecond laser induced periodic surface structures on polymer films,” Phys. Chem. Chem. Phys. 15, 11287 (2013).
[Crossref] [PubMed]

Ruiz de la Cruz, A.

Sabbah, A.

A. Sabbah and D. Riffe, “Femtosecond pump-probe reflectivity study of silicon carrier dynamics,” Phys. Rev. B 66, 1–11 (2002).
[Crossref]

Sagan, Z.

Sanchez-Cortes, S.

D. Puerto, M. Garcia-Lechuga, J. Hernandez-Rueda, A. Garcia-Leis, S. Sanchez-Cortes, J. Solis, and J. Siegel, “Femtosecond laser-controlled self-assembly of amorphous-crystalline nanogratings in silicon,” Nanotechnology 27, 1–8 (2016).
[Crossref]

Sarnet, T.

T. J. Y. Derrien, T. E. Itina, R. Torres, T. Sarnet, and M. Sentis, “Possible surface plasmon polariton excitation under femtosecond laser irradiation of silicon,” J. Appl. Phys. 114, 1–11 (2013).
[Crossref]

M. Halbwax, T. Sarnet, P. Delaporte, M. Sentis, H. Etienne, F. Torregrosa, V. Vervisch, I. Perichaud, and S. Martinuzzi, “Micro and nano-structuration of silicon by femtosecond laser: application to silicon photovoltaic cells fabrication,” Thin Solid Films 516, 6791–6795 (2008).
[Crossref]

Sedlacek, M.

P. Gregorčič, M. Sedlaček, B. Podgornik, and J. Reif, “Formation of laser-induced periodic surface structures (LIPSS) on tool steel by multiple picosecond laser pulses of different polarizations,” Appl. Surf. Sci. 387, 698–706 (2016).
[Crossref]

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A. A. Ionin, S. I. Kudryashov, S. V. Makarov, A. A. Rudenko, L. V. Seleznev, D. V. Sinitsyn, E. V. Golosov, Y. R. Kolobov, and A. E. Ligachev, “”Heterogeneous” versus “homogeneous” nucleation and growth of microcones on titanium surface under UV femtosecond-laser irradiation,” Appl. Phys. A: Mater. Sci. Process. 116, 1133–1139 (2014).
[Crossref]

A. A. Ionin, S. I. Kudryashov, S. V. Makarov, L. V. Seleznev, D. V. Sinitsyn, E. V. Golosov, O. A. Golosova, Y. R. Kolobov, and A. E. Ligachev, “Femtosecond laser color marking of metal and semiconductor surfaces,” Appl. Phys. A: Mater. Sci. Process. 107, 301–305 (2012).
[Crossref]

Sentis, M.

T. J. Y. Derrien, T. E. Itina, R. Torres, T. Sarnet, and M. Sentis, “Possible surface plasmon polariton excitation under femtosecond laser irradiation of silicon,” J. Appl. Phys. 114, 1–11 (2013).
[Crossref]

M. Halbwax, T. Sarnet, P. Delaporte, M. Sentis, H. Etienne, F. Torregrosa, V. Vervisch, I. Perichaud, and S. Martinuzzi, “Micro and nano-structuration of silicon by femtosecond laser: application to silicon photovoltaic cells fabrication,” Thin Solid Films 516, 6791–6795 (2008).
[Crossref]

Siegel, J.

S. V. Kirner, U. Hermens, A. Mimidis, E. Skoulas, C. Florian, F. Hischen, C. Plamadeala, W. Baumgartner, K. Winands, H. Mescheder, J. Krüger, J. Solis, J. Siegel, E. Stratakis, and J. Bonse, “Mimicking bug-like surface structures and their fluid transport produced by ultrashort laser pulse irradiation of steel,” Appl. Phys. A: Mater. Sci. Process. 123, 1–13 (2017).
[Crossref]

D. Puerto, M. Garcia-Lechuga, J. Hernandez-Rueda, A. Garcia-Leis, S. Sanchez-Cortes, J. Solis, and J. Siegel, “Femtosecond laser-controlled self-assembly of amorphous-crystalline nanogratings in silicon,” Nanotechnology 27, 1–8 (2016).
[Crossref]

A. Ruiz de la Cruz, R. Lahoz, J. Siegel, G. F. de la Fuente, and J. Solis, “High speed inscription of uniform, large-area laser-induced periodic surface structures in Cr films using a high repetition rate fs laser,” Opt. Lett. 39, 2491 (2014).
[Crossref] [PubMed]

J. Heitz, B. Reisinger, M. Fahrner, C. Romanin, J. Siegel, and V. Svorcik, “Laser-induced periodic surface structures (LIPSS) on polymer surfaces,” Int. Conf. on Transparent Opt. Networks pp. 1–4 (2012).

Simitzi, C.

C. Yiannakou, C. Simitzi, A. Manousaki, C. Fotakis, A. Ranella, and E. Stratakis, “Cell patterning via laser micro/nano structured silicon surfaces,” Biofabrication 9, 025024 (2017).
[Crossref] [PubMed]

Sinitsyn, D. V.

A. A. Ionin, S. I. Kudryashov, S. V. Makarov, A. A. Rudenko, L. V. Seleznev, D. V. Sinitsyn, E. V. Golosov, Y. R. Kolobov, and A. E. Ligachev, “”Heterogeneous” versus “homogeneous” nucleation and growth of microcones on titanium surface under UV femtosecond-laser irradiation,” Appl. Phys. A: Mater. Sci. Process. 116, 1133–1139 (2014).
[Crossref]

A. A. Ionin, S. I. Kudryashov, S. V. Makarov, L. V. Seleznev, D. V. Sinitsyn, E. V. Golosov, O. A. Golosova, Y. R. Kolobov, and A. E. Ligachev, “Femtosecond laser color marking of metal and semiconductor surfaces,” Appl. Phys. A: Mater. Sci. Process. 107, 301–305 (2012).
[Crossref]

Sipe, J. E.

H. M. Van Driel, J. E. Sipe, and J. F. Young, “Laser-induced periodic surface structure on solids: a universal phenomenon,” Phys. Rev. Lett. 49, 1955–1958 (1982).
[Crossref]

Skoulas, E.

S. V. Kirner, U. Hermens, A. Mimidis, E. Skoulas, C. Florian, F. Hischen, C. Plamadeala, W. Baumgartner, K. Winands, H. Mescheder, J. Krüger, J. Solis, J. Siegel, E. Stratakis, and J. Bonse, “Mimicking bug-like surface structures and their fluid transport produced by ultrashort laser pulse irradiation of steel,” Appl. Phys. A: Mater. Sci. Process. 123, 1–13 (2017).
[Crossref]

Soccio, M.

A. Rodríguez-Rodríguez, E. Rebollar, M. Soccio, T. A. Ezquerra, D. R. Rueda, J. V. Garcia-Ramos, M. Castillejo, and M. C. Garcia-Gutierrez, “Laser-induced periodic surface structures on conjugated polymers: Poly(3-hexylthiophene),” Macromolecules 48, 4024–4031 (2015).
[Crossref]

Soder, H.

Soileau, M. J.

P. A. Temple and M. J. Soileau, “Polarization charge model for laser-induced ripple patterns in dielectric materials,” IEEE J. Quantum Electron. 17, 2067–2072 (1981).
[Crossref]

Solis, J.

S. V. Kirner, U. Hermens, A. Mimidis, E. Skoulas, C. Florian, F. Hischen, C. Plamadeala, W. Baumgartner, K. Winands, H. Mescheder, J. Krüger, J. Solis, J. Siegel, E. Stratakis, and J. Bonse, “Mimicking bug-like surface structures and their fluid transport produced by ultrashort laser pulse irradiation of steel,” Appl. Phys. A: Mater. Sci. Process. 123, 1–13 (2017).
[Crossref]

D. Puerto, M. Garcia-Lechuga, J. Hernandez-Rueda, A. Garcia-Leis, S. Sanchez-Cortes, J. Solis, and J. Siegel, “Femtosecond laser-controlled self-assembly of amorphous-crystalline nanogratings in silicon,” Nanotechnology 27, 1–8 (2016).
[Crossref]

A. Ruiz de la Cruz, R. Lahoz, J. Siegel, G. F. de la Fuente, and J. Solis, “High speed inscription of uniform, large-area laser-induced periodic surface structures in Cr films using a high repetition rate fs laser,” Opt. Lett. 39, 2491 (2014).
[Crossref] [PubMed]

Stratakis, E.

S. V. Kirner, U. Hermens, A. Mimidis, E. Skoulas, C. Florian, F. Hischen, C. Plamadeala, W. Baumgartner, K. Winands, H. Mescheder, J. Krüger, J. Solis, J. Siegel, E. Stratakis, and J. Bonse, “Mimicking bug-like surface structures and their fluid transport produced by ultrashort laser pulse irradiation of steel,” Appl. Phys. A: Mater. Sci. Process. 123, 1–13 (2017).
[Crossref]

C. Yiannakou, C. Simitzi, A. Manousaki, C. Fotakis, A. Ranella, and E. Stratakis, “Cell patterning via laser micro/nano structured silicon surfaces,” Biofabrication 9, 025024 (2017).
[Crossref] [PubMed]

I. Paradisanos, C. Fotakis, S. H. Anastasiadis, and E. Stratakis, “Gradient induced liquid motion on laser structured black Si surfaces,” Appl. Phys. Lett. 107, 111603 (2015).
[Crossref]

Sturm, H.

J. Bonse, M. Munz, and H. Sturm, “Structure formation on the surface of indium phosphide irradiated by femtosecond laser pulses,” J. Appl. Phys. 97, 013538 (2005).
[Crossref]

Svorcik, V.

J. Heitz, B. Reisinger, M. Fahrner, C. Romanin, J. Siegel, and V. Svorcik, “Laser-induced periodic surface structures (LIPSS) on polymer surfaces,” Int. Conf. on Transparent Opt. Networks pp. 1–4 (2012).

Temple, P. A.

P. A. Temple and M. J. Soileau, “Polarization charge model for laser-induced ripple patterns in dielectric materials,” IEEE J. Quantum Electron. 17, 2067–2072 (1981).
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Torregrosa, F.

M. Halbwax, T. Sarnet, P. Delaporte, M. Sentis, H. Etienne, F. Torregrosa, V. Vervisch, I. Perichaud, and S. Martinuzzi, “Micro and nano-structuration of silicon by femtosecond laser: application to silicon photovoltaic cells fabrication,” Thin Solid Films 516, 6791–6795 (2008).
[Crossref]

Torres, R.

T. J. Y. Derrien, T. E. Itina, R. Torres, T. Sarnet, and M. Sentis, “Possible surface plasmon polariton excitation under femtosecond laser irradiation of silicon,” J. Appl. Phys. 114, 1–11 (2013).
[Crossref]

Van Driel, H. M.

H. M. Van Driel, J. E. Sipe, and J. F. Young, “Laser-induced periodic surface structure on solids: a universal phenomenon,” Phys. Rev. Lett. 49, 1955–1958 (1982).
[Crossref]

Vázquez de Aldana, J. R.

E. Rebollar, J. R. Vázquez de Aldana, I. Martín-Fabiani, M. Hernández, D. R. Rueda, T. A. Ezquerra, C. Domingo, P. Moreno, and M. Castillejo, “Assessment of femtosecond laser induced periodic surface structures on polymer films,” Phys. Chem. Chem. Phys. 15, 11287 (2013).
[Crossref] [PubMed]

Vecchione, A.

Vervisch, V.

M. Halbwax, T. Sarnet, P. Delaporte, M. Sentis, H. Etienne, F. Torregrosa, V. Vervisch, I. Perichaud, and S. Martinuzzi, “Micro and nano-structuration of silicon by femtosecond laser: application to silicon photovoltaic cells fabrication,” Thin Solid Films 516, 6791–6795 (2008).
[Crossref]

Walser, M. R.

Wang, J.

J. Wang and C. Guo, “Ultrafast dynamics of femtosecond laser-induced periodic surface pattern formation on metals,” Appl. Phys. Lett. 87, 1–3 (2005).
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E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University, 1920), 3rd ed.

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S. V. Kirner, U. Hermens, A. Mimidis, E. Skoulas, C. Florian, F. Hischen, C. Plamadeala, W. Baumgartner, K. Winands, H. Mescheder, J. Krüger, J. Solis, J. Siegel, E. Stratakis, and J. Bonse, “Mimicking bug-like surface structures and their fluid transport produced by ultrashort laser pulse irradiation of steel,” Appl. Phys. A: Mater. Sci. Process. 123, 1–13 (2017).
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Wolfram Research, I.

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Xie, Y.

J. Long, P. Fan, M. Zhong, H. Zhang, Y. Xie, and C. Lin, “Superhydrophobic and colorful copper surfaces fabricated by picosecond laser induced periodic nanostructures,” Appl. Surf. Sci. 311, 461–467 (2014).
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Yasumaru, N.

N. Yasumaru, K. Miyazaki, and J. Kiuchi, “Femtosecond-laser-induced nanostructure formed on hard thin films of TiN and DLC,” Appl. Phys. A: Mater. Sci. Process. 76, 983–985 (2003).
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Yiannakou, C.

C. Yiannakou, C. Simitzi, A. Manousaki, C. Fotakis, A. Ranella, and E. Stratakis, “Cell patterning via laser micro/nano structured silicon surfaces,” Biofabrication 9, 025024 (2017).
[Crossref] [PubMed]

Young, J. F.

H. M. Van Driel, J. E. Sipe, and J. F. Young, “Laser-induced periodic surface structure on solids: a universal phenomenon,” Phys. Rev. Lett. 49, 1955–1958 (1982).
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Zhang, H.

J. Long, P. Fan, M. Zhong, H. Zhang, Y. Xie, and C. Lin, “Superhydrophobic and colorful copper surfaces fabricated by picosecond laser induced periodic nanostructures,” Appl. Surf. Sci. 311, 461–467 (2014).
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Zhong, M.

J. Long, P. Fan, M. Zhong, H. Zhang, Y. Xie, and C. Lin, “Superhydrophobic and colorful copper surfaces fabricated by picosecond laser induced periodic nanostructures,” Appl. Surf. Sci. 311, 461–467 (2014).
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Appl. Phys. A: Mater. Sci. Process. (5)

A. A. Ionin, S. I. Kudryashov, S. V. Makarov, L. V. Seleznev, D. V. Sinitsyn, E. V. Golosov, O. A. Golosova, Y. R. Kolobov, and A. E. Ligachev, “Femtosecond laser color marking of metal and semiconductor surfaces,” Appl. Phys. A: Mater. Sci. Process. 107, 301–305 (2012).
[Crossref]

N. Yasumaru, K. Miyazaki, and J. Kiuchi, “Femtosecond-laser-induced nanostructure formed on hard thin films of TiN and DLC,” Appl. Phys. A: Mater. Sci. Process. 76, 983–985 (2003).
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S. Baudach, J. Bonse, and W. Kautek, “Ablation experiments on polyimide with femtosecond laser pulses,” Appl. Phys. A: Mater. Sci. Process. 69, 395–398 (1999).
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S. V. Kirner, U. Hermens, A. Mimidis, E. Skoulas, C. Florian, F. Hischen, C. Plamadeala, W. Baumgartner, K. Winands, H. Mescheder, J. Krüger, J. Solis, J. Siegel, E. Stratakis, and J. Bonse, “Mimicking bug-like surface structures and their fluid transport produced by ultrashort laser pulse irradiation of steel,” Appl. Phys. A: Mater. Sci. Process. 123, 1–13 (2017).
[Crossref]

A. A. Ionin, S. I. Kudryashov, S. V. Makarov, A. A. Rudenko, L. V. Seleznev, D. V. Sinitsyn, E. V. Golosov, Y. R. Kolobov, and A. E. Ligachev, “”Heterogeneous” versus “homogeneous” nucleation and growth of microcones on titanium surface under UV femtosecond-laser irradiation,” Appl. Phys. A: Mater. Sci. Process. 116, 1133–1139 (2014).
[Crossref]

Appl. Phys. Lett. (2)

I. Paradisanos, C. Fotakis, S. H. Anastasiadis, and E. Stratakis, “Gradient induced liquid motion on laser structured black Si surfaces,” Appl. Phys. Lett. 107, 111603 (2015).
[Crossref]

J. Wang and C. Guo, “Ultrafast dynamics of femtosecond laser-induced periodic surface pattern formation on metals,” Appl. Phys. Lett. 87, 1–3 (2005).
[Crossref]

Appl. Surf. Sci. (4)

P. Gregorčič, M. Sedlaček, B. Podgornik, and J. Reif, “Formation of laser-induced periodic surface structures (LIPSS) on tool steel by multiple picosecond laser pulses of different polarizations,” Appl. Surf. Sci. 387, 698–706 (2016).
[Crossref]

J.-M. Romano, A. Garcia-giron, P. Penchev, and S. Dimov, “Triangular laser-induced submicron textures for functionalising stainless steel surfaces,” Appl. Surf. Sci. 440, 162–169 (2018).
[Crossref]

J. Eichstädt, G. R. B. E. Römer, and A. J. Huis in’t Veld, “Determination of irradiation parameters for laser-induced periodic surface structures,” Appl. Surf. Sci. 264, 79–87 (2013).
[Crossref]

J. Long, P. Fan, M. Zhong, H. Zhang, Y. Xie, and C. Lin, “Superhydrophobic and colorful copper surfaces fabricated by picosecond laser induced periodic nanostructures,” Appl. Surf. Sci. 311, 461–467 (2014).
[Crossref]

Biofabrication (1)

C. Yiannakou, C. Simitzi, A. Manousaki, C. Fotakis, A. Ranella, and E. Stratakis, “Cell patterning via laser micro/nano structured silicon surfaces,” Biofabrication 9, 025024 (2017).
[Crossref] [PubMed]

Eur. Polym. J. (1)

E. Rebollar, M. Castillejo, and T. A. Ezquerra, “Laser induced periodic surface structures on polymer films: from fundamentals to applications,” Eur. Polym. J. 73, 162–174 (2015).
[Crossref]

IEEE J. Quantum Electron. (1)

P. A. Temple and M. J. Soileau, “Polarization charge model for laser-induced ripple patterns in dielectric materials,” IEEE J. Quantum Electron. 17, 2067–2072 (1981).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

J. Bonse, S. Höhm, S. V. Kirner, A. Rosenfeld, and J. Krüger, “Laser-induced periodic surface structures — a scientific evergreen,” IEEE J. Sel. Top. Quantum Electron. 23, 9000615 (2017).
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 Left: SEM micrograph of LSFL and grooves on the surface of silicon processed with 100 laser pulses on the same location at a wavelength of λ = 1030nm pulse duaration of tp = 6.7ps, peak fluence of F0 = 1.03J/cm2, pulse frequency of f = 1kHz, beam diameter of 32.6μm; Right: lower ( F th low) and upper ( F th up) fluence thresholds for the formation of LSFL are indicated overlaid on a Gaussian fluence profile. The arrow indicates the direction of the E-field of the laser polarization E⃗.
Fig. 2
Fig. 2 Accumulated fluence Fa(x, y), its maximum F a max and its minimum F a min value, as well as upper N eff ξ up 1 F th up ( 1 ) and lower N eff ξ up 1 F th low ( 1 ) fluence thresholds in between which LIPSS can occur.
Fig. 3
Fig. 3 SEM micrographs of Laser-induced modifications processed with NOS =2 and NOS =125 overscans on silicon.
Fig. 4
Fig. 4 Experimentally derived upper and lower accumulated fluence thresholds as a function of the number of overscans on silicon samples at f = 1kHz with λ = 1030nm and tp = 6.7ps.
Fig. 5
Fig. 5 Upper ( N eff F th up) and lower ( N eff F th low) accumulated LIPSS fluence threshold and calculated allowed minimum ( N eff F 0 min) and maximum ( N eff F 0 max) fluence levels. The material dependent parameters F th low ( 1 ) = 1.49 J / cm 2, F th up ( 1 ) = 1.99 J / cm 2, ξlow = 0.74 and ξup = 0.76, corresponding to this graph were found in the previous subsection.
Fig. 6
Fig. 6 SEM micrographs of laser processed areas at an overlap of OL = 0.5 (a–d) and OL = 0.9 (e–h) at various fluence levels. These used peak (F0) and accumulated fluence levels (Fa) are indicated in the images. The accumulated fluence levels are a product of the peak fluence levels and Neff. All micrographs were obtained from 50° tilted samples at the same magnification, except for Fig. 6(a), which shows a magnified top view of LSFL and molten regions originating from overlapping consecutive pulses. The arrows indicate the scanning direction v⃗ and the laser polarization direction E⃗.
Fig. 7
Fig. 7 FFT frequency maps from the same surface structures processed with an overlap OL = 0.9 as in Fig. 6(e) to 6(h) in the same order. The periodicities (frequencies) of the surface structures on the vertical and horizontal axis are normalized to the laser wavelength of λ = 1030nm. The SEM micrographs on which these FFT maps are based, were taken from 0° angle (top view). The rotation from 7a, 7b to 7c and 7d originates from different rotations of the sample when analyzed with the SEM.
Fig. 8
Fig. 8 Segments of Fig. 5, including identified surface structures (data points) in SEM micrographs using 2D FFT maps, all as a function of the overlap (OL) compared to model (curves). For the sake of clarity, the errorbars are not shown. For each studied overlap value, the identified surface morphologies of the processed areas are indicated for the used accumulated fluence. Note that the values of the vertical axes shift for every segment.

Tables (1)

Tables Icon

Table 1 Laser processing parameters which were used to validate the model, as well as identified resulting surface structures. Beam diameter was d = 32.6μm and pulse frequency was f = 1kHz

Equations (23)

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F th ( N OS ) = F th ( 1 ) N OS ξ 1 ,
F a min F th low ( 1 ) N OS ξ low 1 ,
F a max F th up ( 1 ) N OS ξ up 1 ,
F ( x , y ) = F 0 exp ( 8 ( x 2 + y 2 ) d 2 ) ,
Q L = 1 v d f .
F a ( x , y ) = F 0 n = m = exp ( 8 [ ( x n Δ x ) 2 + ( y m Δ y ) 2 ] d 2 ) ,
θ 3 ( h , q ) = n = q n 2 exp ( 2 n i h ) ,
F a ( x , y ) = F 0 π 8 d 2 f 2 v 2 N eff θ 3 [ f π x v , exp ( d 2 f 2 π 2 8 v 2 ) ] θ 3 , x θ 3 [ f π y v , exp ( d 2 f 2 π 2 8 v 2 ) ] θ 3 , y .
E P = F 0 exp ( 8 ( x 2 + y 2 ) d 2 ) d x d y = F 0 π 8 d 2 .
N eff = π 8 d Δ x d Δ y = π 8 ( d f v ) 2 = π 8 ( OL 1 ) 2 ,
F a ( x , y ) = F 0 π 8 ( OL 1 ) 2 N eff θ 3 [ π x d ( OL 1 ) , exp ( π 2 8 ( OL 1 ) 2 ) ] θ 3 , x · θ 3 [ π y d ( OL 1 ) , exp ( π 2 8 ( OL 1 ) 2 ) ] θ 3 , y .
θ 3 ( 0 , q ) = n = n = q n 2 = 1 + 2 n = 1 n = q n 2 = 1 + 2 q + 2 q 4 + 2 q 9 +
max { θ 3 , x } = max { θ 3 , y } [ 1 + 2 exp ( π 2 8 ( 1 OL ) 2 ) ] .
min { θ 3 , x } = min { θ 3 , y } [ 1 2 exp ( π 2 8 ( 1 OL ) 2 ) ] .
F a min F 0 π 8 ( OL 1 ) 2 [ 1 2 exp ( π 2 8 ( 1 OL ) 2 ) ] 2 ,
F a max F 0 π 8 ( OL 1 ) 2 [ 1 + 2 exp ( π 2 8 ( 1 OL ) 2 ) ] 2 .
F a min F th low ( 1 ) N eff ξ low ,
F a max F th up ( 1 ) N eff ξ up .
F th low ( 1 ) N eff ξ low 1 [ 1 2 exp ( π N eff ) ] 2 F 0 min F 0 F th up ( 1 ) N eff ξ up 1 [ 1 + 2 exp ( π N eff ) ] 2 F 0 max .
OL min = 1 π 2 2 [ ln ( 2 + 4 F th up ( 1 ) Δ F + 4 ( F th up ( 1 ) ( F th up ( 1 ) Δ F ) Δ F 2 ) 1 2 ) ] 1 2 ,
F th low ( N OS ) = F 0 ( N OS ) exp ( 2 d th low ( N OS ) 2 d 2 ) ,
F th up ( N OS ) = F 0 ( N OS ) exp ( 2 d th up ( N OS ) 2 d 2 ) ,
N e ( 1 R ) F a line E [ α + ( 1 R ) F a line β 2 2 π τ ] ,