Abstract

The z-scan technique has become a standard method to measure 2-photon absorption (2PA) properties of materials used for 2-photon applications. Here we present a completely automated, easily tunable z-scan setup for spectral scanning. An algorithm collecting the required laser beam parameters allows to reliably determine the optimal working window of newly synthesized 2PA photoinitiators (PI) used for two-photon polymerization (2PP) in a wide spectral range. A complete spectrum (3 measurements per wavelength in 10 nm steps) can be obtained within an hour. Matching the wavelength used for 2PP to the maximum 2PA significantly increased the 2PP performance of the system.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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References

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2019 (1)

A. Dobos, W. Steiger, D. Theiner, P. Gruber, M. Lunzer, J. Van Hoorick, S. Van Vlierberghe, and A. Ovsianikov, “Screening of two-photon activated photodynamic therapy sensitizers using a 3D osteosarcoma model,” Analyst 144(9), 3056–3063 (2019).
[Crossref]

2018 (1)

M. Tromayer, A. Dobos, P. Gruber, A. Ajami, R. Dedic, A. Ovsianikov, and R. Liska, “A biocompatible diazosulfonate initiator for direct encapsulation of human stem cells via two-photon polymerization,” Polymer Chem. 22, 2018 (2018).
[Crossref]

2017 (1)

A. Ajami, W. Husinsky, M. Tromayer, P. Gruber, R. Liska, and A. Ovsianikov, “Measurement of degenerate two-photon absorption spectra of a series of developed two-photon initiators using a dispersive white light continuum Z-scan,” Appl. Phys. Lett. 111(7), 071901 (2017).
[Crossref]

2013 (2)

Z. Li, N. Pucher, K. Cicha, J. Torgersen, S. C. Ligon, A. Ajami, W. Husinsky, A. Rosspeintner, E. Vauthey, S. Naumov, T. Scherzer, J. Stampfl, and R. Liska, “A Straightforward Synthesis and Structure–Activity Relationship of Highly Efficient Initiators for Two-Photon Polymerization,” Macromolecules 46(2), 352–361 (2013).
[Crossref]

Z. Li, J. Torgersen, A. Ajami, S. Mühleder, X. Qin, W. Husinsky, W. Holnthoner, A. Ovsianikov, J. Stampfl, and R. Liska, “Initiation efficiency and cytotoxicity of novel water-soluble two-photon photoinitiators for direct 3D microfabrication of hydrogels,” RSC Adv. 3(36), 15939–15946 (2013).
[Crossref]

2012 (1)

G. G. Dubinina, R. S. Price, K. A. Abboud, G. Wicks, P. Wnuk, Y. Stepanenko, M. Drobizhev, A. Rebane, and K. S. Schanze, “Phenylene Vinylene Platinum(II) Acetylides with Prodigious Two-Photon Absorption,” J. Am. Chem. Soc. 134(47), 19346–19349 (2012).
[Crossref]

2011 (1)

M. Drobizhev, N. S. Makarov, S. E. Tillo, T. E. Hughes, and A. Rebane, “Two-photon absorption properties of fluorescent proteins,” Nat. Methods 8(5), 393–399 (2011).
[Crossref]

2010 (1)

J. A. Martín H, M. Santos, and J. de Lope, “Orthogonal variant moments features in image analysis,” Inf. Sci. 180(6), 846–860 (2010).
[Crossref]

2009 (2)

S. J. Jhaveri, J. D. McMullen, R. Sijbesma, L.-S. Tan, W. Zipfel, and C. K. Ober, “Direct Three-Dimensional Microfabrication of Hydrogels via Two-Photon Lithography in Aqueous Solution,” Chem. Mater. 21(10), 2003–2006 (2009).
[Crossref]

M. Pawlicki, H. A. Collins, R. G. Denning, and H. L. Anderson, “Two-Photon Absorption and the Design of Two-Photon Dyes,” Angew. Chem., Int. Ed. 48(18), 3244–3266 (2009).
[Crossref]

2008 (3)

K. Ogawa and Y. Kobuke, “Recent advances in two-photon photodynamic therapy,” Anti-Cancer Agents Med. Chem. 8(3), 269–279 (2008).
[Crossref]

C. R. Mendonca, D. S. Correa, T. Baldacchini, P. Tayalia, and E. Mazur, “Two-photon absorption spectrum of the photoinitiator Lucirin TPO-L,” Appl. Phys. A 90(4), 633–636 (2008).
[Crossref]

N. S. Makarov, M. Drobizhev, and A. Rebane, “Two-photon absorption standards in the 550–1600 nm excitation wavelength range,” Opt. Express 16(6), 4029–4047 (2008).
[Crossref]

2007 (1)

D. S. Corrêa, L. De Boni, L. Misoguti, I. Cohanoschi, F. E. Hernandez, and C. R. Mendonça, “Z-scan theoretical analysis for three-, four- and five-photon absorption,” Opt. Commun. 277(2), 440–445 (2007).
[Crossref]

2006 (2)

C. Jacinto, D. N. Messias, A. A. Andrade, S. M. Lima, M. L. Baesso, and T. Catunda, “Thermal lens and Z-scan measurements: Thermal and optical properties of laser glasses – A review,” J. Non-Cryst. Solids 352(32-35), 3582–3597 (2006).
[Crossref]

D. S. Corrêa, S. L. Oliveira, L. Misoguti, S. C. Zilio, R. F. Aroca, C. J. L. Constantino, and C. R. Mendonça, “Investigation of the Two-Photon Absorption Cross-Section in Perylene Tetracarboxylic Derivatives: Nonlinear Spectra and Molecular Structure,” J. Phys. Chem. A 110(20), 6433–6438 (2006).
[Crossref]

2005 (1)

2004 (4)

M. Balu, J. Hales, D. J. Hagan, and E. W. V. Stryland, “White-light continuum Z-scan technique for nonlinear materials characterization,” Opt. Express 12(16), 3820–3826 (2004).
[Crossref]

L. D. Boni, A. A. Andrade, L. Misoguti, C. R. Mendonça, and S. C. Zilio, “Z-scan measurements using femtosecond continuum generation,” Opt. Express 12(17), 3921–3927 (2004).
[Crossref]

K. J. Schafer, J. M. Hales, M. Balu, K. D. Belfield, E. W. Van Stryland, and D. J. Hagan, “Two-photon absorption cross-sections of common photoinitiators,” J. Photochem. Photobiol., A 162(2-3), 497–502 (2004).
[Crossref]

R. A. Ganeev, A. I. Ryasnyansky, N. Ishizawa, M. Baba, M. Suzuki, M. Turu, S. Sakakibara, and H. Kuroda, “Two- and three-photon absorption in CS2,” Opt. Commun. 231(1-6), 431–436 (2004).
[Crossref]

2003 (2)

2002 (3)

S. Venugopal Rao, N. K. M. Naga Srinivas, and D. Narayana Rao, “Nonlinear absorption and excited state dynamics in Rhodamine B studied using Z-scan and degenerate four wave mixing techniques,” Chem. Phys. Lett. 361(5-6), 439–445 (2002).
[Crossref]

S. M. Mian, S. B. McGee, and N. Melikechi, “Experimental and theoretical investigation of thermal lensing effects in mode-locked femtosecond Z-scan experiments,” Opt. Commun. 207(1-6), 339–345 (2002).
[Crossref]

R. de Nalda, R. del Coso, J. Requejo-Isidro, J. Olivares, A. Suarez-Garcia, J. Solis, and C. N. Afonso, “Limits to the determination of the nonlinear refractive index by the Z-scan method,” J. Opt. Soc. Am. B 19(2), 289 (2002).
[Crossref]

2000 (1)

A. I. Van Den Bulcke, B. Bogdanov, N. De Rooze, E. H. Schacht, M. Cornelissen, and H. Berghmans, “Structural and Rheological Properties of Methacrylamide Modified Gelatin Hydrogels,” Biomacromolecules 1(1), 31–38 (2000).
[Crossref]

1999 (2)

M. Falconieri and G. Salvetti, “Simultaneous measurement of pure-optical and thermo-optical nonlinearities induced by high-repetition-rate, femtosecond laser pulses: application to CS 2,” Appl. Phys. B: Lasers Opt. 69(2), 133–136 (1999).
[Crossref]

M. Falconieri, “Thermo-optical effects in Z -scan measurements using high-repetition-rate lasers,” J. Opt. A: Pure Appl. Opt. 1(6), 662–667 (1999).
[Crossref]

1995 (1)

J. Wang and M. Fiebig, “Bestimmung der Temperaturleitfahigkeit von Toluol und Methanol rnittels laserinduzierter thermischer Gitter,” Heat Mass Transfer 31(1-2), 83–87 (1995)..
[Crossref]

1990 (1)

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[Crossref]

1983 (1)

1962 (1)

M.-K. Hu, “Visual pattern recognition by moment invariants,” IEEE Trans. Inf. Theory 8(2), 179–187 (1962).
[Crossref]

Abboud, K. A.

G. G. Dubinina, R. S. Price, K. A. Abboud, G. Wicks, P. Wnuk, Y. Stepanenko, M. Drobizhev, A. Rebane, and K. S. Schanze, “Phenylene Vinylene Platinum(II) Acetylides with Prodigious Two-Photon Absorption,” J. Am. Chem. Soc. 134(47), 19346–19349 (2012).
[Crossref]

Afonso, C. N.

Ajami, A.

M. Tromayer, A. Dobos, P. Gruber, A. Ajami, R. Dedic, A. Ovsianikov, and R. Liska, “A biocompatible diazosulfonate initiator for direct encapsulation of human stem cells via two-photon polymerization,” Polymer Chem. 22, 2018 (2018).
[Crossref]

A. Ajami, W. Husinsky, M. Tromayer, P. Gruber, R. Liska, and A. Ovsianikov, “Measurement of degenerate two-photon absorption spectra of a series of developed two-photon initiators using a dispersive white light continuum Z-scan,” Appl. Phys. Lett. 111(7), 071901 (2017).
[Crossref]

Z. Li, N. Pucher, K. Cicha, J. Torgersen, S. C. Ligon, A. Ajami, W. Husinsky, A. Rosspeintner, E. Vauthey, S. Naumov, T. Scherzer, J. Stampfl, and R. Liska, “A Straightforward Synthesis and Structure–Activity Relationship of Highly Efficient Initiators for Two-Photon Polymerization,” Macromolecules 46(2), 352–361 (2013).
[Crossref]

Z. Li, J. Torgersen, A. Ajami, S. Mühleder, X. Qin, W. Husinsky, W. Holnthoner, A. Ovsianikov, J. Stampfl, and R. Liska, “Initiation efficiency and cytotoxicity of novel water-soluble two-photon photoinitiators for direct 3D microfabrication of hydrogels,” RSC Adv. 3(36), 15939–15946 (2013).
[Crossref]

S. C. Ligon, M. Tromayer, Z. Li, J. Torgersen, A. Ajami, A. Rosspeintner, S. Naumov, T. Scherzer, J. Stampfl, and R. Liska, “New Developments in Initiators for Two-Photon Polymerization,” 6 Conference: Rad Tech UV/EB 2014At: Rosemont, IL.

Anderson, H. L.

M. Pawlicki, H. A. Collins, R. G. Denning, and H. L. Anderson, “Two-Photon Absorption and the Design of Two-Photon Dyes,” Angew. Chem., Int. Ed. 48(18), 3244–3266 (2009).
[Crossref]

Andrade, A. A.

C. Jacinto, D. N. Messias, A. A. Andrade, S. M. Lima, M. L. Baesso, and T. Catunda, “Thermal lens and Z-scan measurements: Thermal and optical properties of laser glasses – A review,” J. Non-Cryst. Solids 352(32-35), 3582–3597 (2006).
[Crossref]

L. D. Boni, A. A. Andrade, L. Misoguti, C. R. Mendonça, and S. C. Zilio, “Z-scan measurements using femtosecond continuum generation,” Opt. Express 12(17), 3921–3927 (2004).
[Crossref]

Aroca, R. F.

D. S. Corrêa, S. L. Oliveira, L. Misoguti, S. C. Zilio, R. F. Aroca, C. J. L. Constantino, and C. R. Mendonça, “Investigation of the Two-Photon Absorption Cross-Section in Perylene Tetracarboxylic Derivatives: Nonlinear Spectra and Molecular Structure,” J. Phys. Chem. A 110(20), 6433–6438 (2006).
[Crossref]

Baba, M.

R. A. Ganeev, A. I. Ryasnyansky, N. Ishizawa, M. Baba, M. Suzuki, M. Turu, S. Sakakibara, and H. Kuroda, “Two- and three-photon absorption in CS2,” Opt. Commun. 231(1-6), 431–436 (2004).
[Crossref]

Baesso, M. L.

C. Jacinto, D. N. Messias, A. A. Andrade, S. M. Lima, M. L. Baesso, and T. Catunda, “Thermal lens and Z-scan measurements: Thermal and optical properties of laser glasses – A review,” J. Non-Cryst. Solids 352(32-35), 3582–3597 (2006).
[Crossref]

Baldacchini, T.

C. R. Mendonca, D. S. Correa, T. Baldacchini, P. Tayalia, and E. Mazur, “Two-photon absorption spectrum of the photoinitiator Lucirin TPO-L,” Appl. Phys. A 90(4), 633–636 (2008).
[Crossref]

Balu, M.

K. J. Schafer, J. M. Hales, M. Balu, K. D. Belfield, E. W. Van Stryland, and D. J. Hagan, “Two-photon absorption cross-sections of common photoinitiators,” J. Photochem. Photobiol., A 162(2-3), 497–502 (2004).
[Crossref]

M. Balu, J. Hales, D. J. Hagan, and E. W. V. Stryland, “White-light continuum Z-scan technique for nonlinear materials characterization,” Opt. Express 12(16), 3820–3826 (2004).
[Crossref]

Belfield, K. D.

K. J. Schafer, J. M. Hales, M. Balu, K. D. Belfield, E. W. Van Stryland, and D. J. Hagan, “Two-photon absorption cross-sections of common photoinitiators,” J. Photochem. Photobiol., A 162(2-3), 497–502 (2004).
[Crossref]

Berghmans, H.

A. I. Van Den Bulcke, B. Bogdanov, N. De Rooze, E. H. Schacht, M. Cornelissen, and H. Berghmans, “Structural and Rheological Properties of Methacrylamide Modified Gelatin Hydrogels,” Biomacromolecules 1(1), 31–38 (2000).
[Crossref]

Bogdanov, B.

A. I. Van Den Bulcke, B. Bogdanov, N. De Rooze, E. H. Schacht, M. Cornelissen, and H. Berghmans, “Structural and Rheological Properties of Methacrylamide Modified Gelatin Hydrogels,” Biomacromolecules 1(1), 31–38 (2000).
[Crossref]

Boni, L. D.

Catunda, T.

C. Jacinto, D. N. Messias, A. A. Andrade, S. M. Lima, M. L. Baesso, and T. Catunda, “Thermal lens and Z-scan measurements: Thermal and optical properties of laser glasses – A review,” J. Non-Cryst. Solids 352(32-35), 3582–3597 (2006).
[Crossref]

Cicha, K.

Z. Li, N. Pucher, K. Cicha, J. Torgersen, S. C. Ligon, A. Ajami, W. Husinsky, A. Rosspeintner, E. Vauthey, S. Naumov, T. Scherzer, J. Stampfl, and R. Liska, “A Straightforward Synthesis and Structure–Activity Relationship of Highly Efficient Initiators for Two-Photon Polymerization,” Macromolecules 46(2), 352–361 (2013).
[Crossref]

Cohanoschi, I.

D. S. Corrêa, L. De Boni, L. Misoguti, I. Cohanoschi, F. E. Hernandez, and C. R. Mendonça, “Z-scan theoretical analysis for three-, four- and five-photon absorption,” Opt. Commun. 277(2), 440–445 (2007).
[Crossref]

Collins, H. A.

M. Pawlicki, H. A. Collins, R. G. Denning, and H. L. Anderson, “Two-Photon Absorption and the Design of Two-Photon Dyes,” Angew. Chem., Int. Ed. 48(18), 3244–3266 (2009).
[Crossref]

Constantino, C. J. L.

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Razzari, L.

Rebane, A.

G. G. Dubinina, R. S. Price, K. A. Abboud, G. Wicks, P. Wnuk, Y. Stepanenko, M. Drobizhev, A. Rebane, and K. S. Schanze, “Phenylene Vinylene Platinum(II) Acetylides with Prodigious Two-Photon Absorption,” J. Am. Chem. Soc. 134(47), 19346–19349 (2012).
[Crossref]

M. Drobizhev, N. S. Makarov, S. E. Tillo, T. E. Hughes, and A. Rebane, “Two-photon absorption properties of fluorescent proteins,” Nat. Methods 8(5), 393–399 (2011).
[Crossref]

N. S. Makarov, M. Drobizhev, and A. Rebane, “Two-photon absorption standards in the 550–1600 nm excitation wavelength range,” Opt. Express 16(6), 4029–4047 (2008).
[Crossref]

Requejo-Isidro, J.

Righini, M.

Rosspeintner, A.

Z. Li, N. Pucher, K. Cicha, J. Torgersen, S. C. Ligon, A. Ajami, W. Husinsky, A. Rosspeintner, E. Vauthey, S. Naumov, T. Scherzer, J. Stampfl, and R. Liska, “A Straightforward Synthesis and Structure–Activity Relationship of Highly Efficient Initiators for Two-Photon Polymerization,” Macromolecules 46(2), 352–361 (2013).
[Crossref]

S. C. Ligon, M. Tromayer, Z. Li, J. Torgersen, A. Ajami, A. Rosspeintner, S. Naumov, T. Scherzer, J. Stampfl, and R. Liska, “New Developments in Initiators for Two-Photon Polymerization,” 6 Conference: Rad Tech UV/EB 2014At: Rosemont, IL.

Ryasnyansky, A. I.

R. A. Ganeev, A. I. Ryasnyansky, N. Ishizawa, M. Baba, M. Suzuki, M. Turu, S. Sakakibara, and H. Kuroda, “Two- and three-photon absorption in CS2,” Opt. Commun. 231(1-6), 431–436 (2004).
[Crossref]

Said, A. A.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[Crossref]

Sakakibara, S.

R. A. Ganeev, A. I. Ryasnyansky, N. Ishizawa, M. Baba, M. Suzuki, M. Turu, S. Sakakibara, and H. Kuroda, “Two- and three-photon absorption in CS2,” Opt. Commun. 231(1-6), 431–436 (2004).
[Crossref]

Salvetti, G.

M. Falconieri and G. Salvetti, “Simultaneous measurement of pure-optical and thermo-optical nonlinearities induced by high-repetition-rate, femtosecond laser pulses: application to CS 2,” Appl. Phys. B: Lasers Opt. 69(2), 133–136 (1999).
[Crossref]

Santos, M.

J. A. Martín H, M. Santos, and J. de Lope, “Orthogonal variant moments features in image analysis,” Inf. Sci. 180(6), 846–860 (2010).
[Crossref]

Schacht, E. H.

A. I. Van Den Bulcke, B. Bogdanov, N. De Rooze, E. H. Schacht, M. Cornelissen, and H. Berghmans, “Structural and Rheological Properties of Methacrylamide Modified Gelatin Hydrogels,” Biomacromolecules 1(1), 31–38 (2000).
[Crossref]

Schafer, K. J.

K. J. Schafer, J. M. Hales, M. Balu, K. D. Belfield, E. W. Van Stryland, and D. J. Hagan, “Two-photon absorption cross-sections of common photoinitiators,” J. Photochem. Photobiol., A 162(2-3), 497–502 (2004).
[Crossref]

Schanze, K. S.

G. G. Dubinina, R. S. Price, K. A. Abboud, G. Wicks, P. Wnuk, Y. Stepanenko, M. Drobizhev, A. Rebane, and K. S. Schanze, “Phenylene Vinylene Platinum(II) Acetylides with Prodigious Two-Photon Absorption,” J. Am. Chem. Soc. 134(47), 19346–19349 (2012).
[Crossref]

Scherzer, T.

Z. Li, N. Pucher, K. Cicha, J. Torgersen, S. C. Ligon, A. Ajami, W. Husinsky, A. Rosspeintner, E. Vauthey, S. Naumov, T. Scherzer, J. Stampfl, and R. Liska, “A Straightforward Synthesis and Structure–Activity Relationship of Highly Efficient Initiators for Two-Photon Polymerization,” Macromolecules 46(2), 352–361 (2013).
[Crossref]

S. C. Ligon, M. Tromayer, Z. Li, J. Torgersen, A. Ajami, A. Rosspeintner, S. Naumov, T. Scherzer, J. Stampfl, and R. Liska, “New Developments in Initiators for Two-Photon Polymerization,” 6 Conference: Rad Tech UV/EB 2014At: Rosemont, IL.

Sheik-Bahae, M.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[Crossref]

E. W. Van Stryland and M. Sheik-Bahae, “Z-scan measurements of optical nonlinearities,” Charact. Tech. Tabul. Org. Nonlinear Mater.655–692 (1998).

Siegman, A. E.

A. E. Siegman, “Defining, measuring, and optimizing laser beam quality,” in Laser Resonators and Coherent Optics: Modeling, Technology, and Applications (International Society for Optics and Photonics, 1993), Vol. 1868, pp. 2–13.

Sijbesma, R.

S. J. Jhaveri, J. D. McMullen, R. Sijbesma, L.-S. Tan, W. Zipfel, and C. K. Ober, “Direct Three-Dimensional Microfabrication of Hydrogels via Two-Photon Lithography in Aqueous Solution,” Chem. Mater. 21(10), 2003–2006 (2009).
[Crossref]

Solis, J.

Stampfl, J.

Z. Li, N. Pucher, K. Cicha, J. Torgersen, S. C. Ligon, A. Ajami, W. Husinsky, A. Rosspeintner, E. Vauthey, S. Naumov, T. Scherzer, J. Stampfl, and R. Liska, “A Straightforward Synthesis and Structure–Activity Relationship of Highly Efficient Initiators for Two-Photon Polymerization,” Macromolecules 46(2), 352–361 (2013).
[Crossref]

Z. Li, J. Torgersen, A. Ajami, S. Mühleder, X. Qin, W. Husinsky, W. Holnthoner, A. Ovsianikov, J. Stampfl, and R. Liska, “Initiation efficiency and cytotoxicity of novel water-soluble two-photon photoinitiators for direct 3D microfabrication of hydrogels,” RSC Adv. 3(36), 15939–15946 (2013).
[Crossref]

S. C. Ligon, M. Tromayer, Z. Li, J. Torgersen, A. Ajami, A. Rosspeintner, S. Naumov, T. Scherzer, J. Stampfl, and R. Liska, “New Developments in Initiators for Two-Photon Polymerization,” 6 Conference: Rad Tech UV/EB 2014At: Rosemont, IL.

J. Stampfl, R. Liska, and A. Ovsianikov, Multiphoton Lithography: Techniques, Materials, and Applications (John Wiley & Sons, 2016).

Stegeman, G.

F. Yoshino, S. Polyakov, M. Liu, and G. Stegeman, “Observation of Three-Photon Enhanced Four-Photon Absorption,” Phys. Rev. Lett. 91(6), 063902 (2003).
[Crossref]

Steiger, W.

A. Dobos, W. Steiger, D. Theiner, P. Gruber, M. Lunzer, J. Van Hoorick, S. Van Vlierberghe, and A. Ovsianikov, “Screening of two-photon activated photodynamic therapy sensitizers using a 3D osteosarcoma model,” Analyst 144(9), 3056–3063 (2019).
[Crossref]

Stepanenko, Y.

G. G. Dubinina, R. S. Price, K. A. Abboud, G. Wicks, P. Wnuk, Y. Stepanenko, M. Drobizhev, A. Rebane, and K. S. Schanze, “Phenylene Vinylene Platinum(II) Acetylides with Prodigious Two-Photon Absorption,” J. Am. Chem. Soc. 134(47), 19346–19349 (2012).
[Crossref]

Stryland, E. W. V.

Suarez-Garcia, A.

Suzuki, M.

R. A. Ganeev, A. I. Ryasnyansky, N. Ishizawa, M. Baba, M. Suzuki, M. Turu, S. Sakakibara, and H. Kuroda, “Two- and three-photon absorption in CS2,” Opt. Commun. 231(1-6), 431–436 (2004).
[Crossref]

Tan, L.-S.

S. J. Jhaveri, J. D. McMullen, R. Sijbesma, L.-S. Tan, W. Zipfel, and C. K. Ober, “Direct Three-Dimensional Microfabrication of Hydrogels via Two-Photon Lithography in Aqueous Solution,” Chem. Mater. 21(10), 2003–2006 (2009).
[Crossref]

Tayalia, P.

C. R. Mendonca, D. S. Correa, T. Baldacchini, P. Tayalia, and E. Mazur, “Two-photon absorption spectrum of the photoinitiator Lucirin TPO-L,” Appl. Phys. A 90(4), 633–636 (2008).
[Crossref]

Theiner, D.

A. Dobos, W. Steiger, D. Theiner, P. Gruber, M. Lunzer, J. Van Hoorick, S. Van Vlierberghe, and A. Ovsianikov, “Screening of two-photon activated photodynamic therapy sensitizers using a 3D osteosarcoma model,” Analyst 144(9), 3056–3063 (2019).
[Crossref]

Tillo, S. E.

M. Drobizhev, N. S. Makarov, S. E. Tillo, T. E. Hughes, and A. Rebane, “Two-photon absorption properties of fluorescent proteins,” Nat. Methods 8(5), 393–399 (2011).
[Crossref]

Torgersen, J.

Z. Li, J. Torgersen, A. Ajami, S. Mühleder, X. Qin, W. Husinsky, W. Holnthoner, A. Ovsianikov, J. Stampfl, and R. Liska, “Initiation efficiency and cytotoxicity of novel water-soluble two-photon photoinitiators for direct 3D microfabrication of hydrogels,” RSC Adv. 3(36), 15939–15946 (2013).
[Crossref]

Z. Li, N. Pucher, K. Cicha, J. Torgersen, S. C. Ligon, A. Ajami, W. Husinsky, A. Rosspeintner, E. Vauthey, S. Naumov, T. Scherzer, J. Stampfl, and R. Liska, “A Straightforward Synthesis and Structure–Activity Relationship of Highly Efficient Initiators for Two-Photon Polymerization,” Macromolecules 46(2), 352–361 (2013).
[Crossref]

S. C. Ligon, M. Tromayer, Z. Li, J. Torgersen, A. Ajami, A. Rosspeintner, S. Naumov, T. Scherzer, J. Stampfl, and R. Liska, “New Developments in Initiators for Two-Photon Polymerization,” 6 Conference: Rad Tech UV/EB 2014At: Rosemont, IL.

Tromayer, M.

M. Tromayer, A. Dobos, P. Gruber, A. Ajami, R. Dedic, A. Ovsianikov, and R. Liska, “A biocompatible diazosulfonate initiator for direct encapsulation of human stem cells via two-photon polymerization,” Polymer Chem. 22, 2018 (2018).
[Crossref]

A. Ajami, W. Husinsky, M. Tromayer, P. Gruber, R. Liska, and A. Ovsianikov, “Measurement of degenerate two-photon absorption spectra of a series of developed two-photon initiators using a dispersive white light continuum Z-scan,” Appl. Phys. Lett. 111(7), 071901 (2017).
[Crossref]

S. C. Ligon, M. Tromayer, Z. Li, J. Torgersen, A. Ajami, A. Rosspeintner, S. Naumov, T. Scherzer, J. Stampfl, and R. Liska, “New Developments in Initiators for Two-Photon Polymerization,” 6 Conference: Rad Tech UV/EB 2014At: Rosemont, IL.

Turu, M.

R. A. Ganeev, A. I. Ryasnyansky, N. Ishizawa, M. Baba, M. Suzuki, M. Turu, S. Sakakibara, and H. Kuroda, “Two- and three-photon absorption in CS2,” Opt. Commun. 231(1-6), 431–436 (2004).
[Crossref]

Van Den Bulcke, A. I.

A. I. Van Den Bulcke, B. Bogdanov, N. De Rooze, E. H. Schacht, M. Cornelissen, and H. Berghmans, “Structural and Rheological Properties of Methacrylamide Modified Gelatin Hydrogels,” Biomacromolecules 1(1), 31–38 (2000).
[Crossref]

Van Hoorick, J.

A. Dobos, W. Steiger, D. Theiner, P. Gruber, M. Lunzer, J. Van Hoorick, S. Van Vlierberghe, and A. Ovsianikov, “Screening of two-photon activated photodynamic therapy sensitizers using a 3D osteosarcoma model,” Analyst 144(9), 3056–3063 (2019).
[Crossref]

Van Stryland, E. W.

K. J. Schafer, J. M. Hales, M. Balu, K. D. Belfield, E. W. Van Stryland, and D. J. Hagan, “Two-photon absorption cross-sections of common photoinitiators,” J. Photochem. Photobiol., A 162(2-3), 497–502 (2004).
[Crossref]

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[Crossref]

E. W. Van Stryland and M. Sheik-Bahae, “Z-scan measurements of optical nonlinearities,” Charact. Tech. Tabul. Org. Nonlinear Mater.655–692 (1998).

Van Vlierberghe, S.

A. Dobos, W. Steiger, D. Theiner, P. Gruber, M. Lunzer, J. Van Hoorick, S. Van Vlierberghe, and A. Ovsianikov, “Screening of two-photon activated photodynamic therapy sensitizers using a 3D osteosarcoma model,” Analyst 144(9), 3056–3063 (2019).
[Crossref]

Vauthey, E.

Z. Li, N. Pucher, K. Cicha, J. Torgersen, S. C. Ligon, A. Ajami, W. Husinsky, A. Rosspeintner, E. Vauthey, S. Naumov, T. Scherzer, J. Stampfl, and R. Liska, “A Straightforward Synthesis and Structure–Activity Relationship of Highly Efficient Initiators for Two-Photon Polymerization,” Macromolecules 46(2), 352–361 (2013).
[Crossref]

Venugopal Rao, S.

S. Venugopal Rao, N. K. M. Naga Srinivas, and D. Narayana Rao, “Nonlinear absorption and excited state dynamics in Rhodamine B studied using Z-scan and degenerate four wave mixing techniques,” Chem. Phys. Lett. 361(5-6), 439–445 (2002).
[Crossref]

Wang, J.

J. Wang and M. Fiebig, “Bestimmung der Temperaturleitfahigkeit von Toluol und Methanol rnittels laserinduzierter thermischer Gitter,” Heat Mass Transfer 31(1-2), 83–87 (1995)..
[Crossref]

Wei, T.-H.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[Crossref]

Wicks, G.

G. G. Dubinina, R. S. Price, K. A. Abboud, G. Wicks, P. Wnuk, Y. Stepanenko, M. Drobizhev, A. Rebane, and K. S. Schanze, “Phenylene Vinylene Platinum(II) Acetylides with Prodigious Two-Photon Absorption,” J. Am. Chem. Soc. 134(47), 19346–19349 (2012).
[Crossref]

Wnuk, P.

G. G. Dubinina, R. S. Price, K. A. Abboud, G. Wicks, P. Wnuk, Y. Stepanenko, M. Drobizhev, A. Rebane, and K. S. Schanze, “Phenylene Vinylene Platinum(II) Acetylides with Prodigious Two-Photon Absorption,” J. Am. Chem. Soc. 134(47), 19346–19349 (2012).
[Crossref]

Yoshino, A.

Yoshino, F.

F. Yoshino, S. Polyakov, M. Liu, and G. Stegeman, “Observation of Three-Photon Enhanced Four-Photon Absorption,” Phys. Rev. Lett. 91(6), 063902 (2003).
[Crossref]

Zilio, S. C.

D. S. Corrêa, S. L. Oliveira, L. Misoguti, S. C. Zilio, R. F. Aroca, C. J. L. Constantino, and C. R. Mendonça, “Investigation of the Two-Photon Absorption Cross-Section in Perylene Tetracarboxylic Derivatives: Nonlinear Spectra and Molecular Structure,” J. Phys. Chem. A 110(20), 6433–6438 (2006).
[Crossref]

L. D. Boni, A. A. Andrade, L. Misoguti, C. R. Mendonça, and S. C. Zilio, “Z-scan measurements using femtosecond continuum generation,” Opt. Express 12(17), 3921–3927 (2004).
[Crossref]

Zipfel, W.

S. J. Jhaveri, J. D. McMullen, R. Sijbesma, L.-S. Tan, W. Zipfel, and C. K. Ober, “Direct Three-Dimensional Microfabrication of Hydrogels via Two-Photon Lithography in Aqueous Solution,” Chem. Mater. 21(10), 2003–2006 (2009).
[Crossref]

Analyst (1)

A. Dobos, W. Steiger, D. Theiner, P. Gruber, M. Lunzer, J. Van Hoorick, S. Van Vlierberghe, and A. Ovsianikov, “Screening of two-photon activated photodynamic therapy sensitizers using a 3D osteosarcoma model,” Analyst 144(9), 3056–3063 (2019).
[Crossref]

Angew. Chem., Int. Ed. (1)

M. Pawlicki, H. A. Collins, R. G. Denning, and H. L. Anderson, “Two-Photon Absorption and the Design of Two-Photon Dyes,” Angew. Chem., Int. Ed. 48(18), 3244–3266 (2009).
[Crossref]

Anti-Cancer Agents Med. Chem. (1)

K. Ogawa and Y. Kobuke, “Recent advances in two-photon photodynamic therapy,” Anti-Cancer Agents Med. Chem. 8(3), 269–279 (2008).
[Crossref]

Appl. Opt. (1)

Appl. Phys. A (1)

C. R. Mendonca, D. S. Correa, T. Baldacchini, P. Tayalia, and E. Mazur, “Two-photon absorption spectrum of the photoinitiator Lucirin TPO-L,” Appl. Phys. A 90(4), 633–636 (2008).
[Crossref]

Appl. Phys. B: Lasers Opt. (1)

M. Falconieri and G. Salvetti, “Simultaneous measurement of pure-optical and thermo-optical nonlinearities induced by high-repetition-rate, femtosecond laser pulses: application to CS 2,” Appl. Phys. B: Lasers Opt. 69(2), 133–136 (1999).
[Crossref]

Appl. Phys. Lett. (1)

A. Ajami, W. Husinsky, M. Tromayer, P. Gruber, R. Liska, and A. Ovsianikov, “Measurement of degenerate two-photon absorption spectra of a series of developed two-photon initiators using a dispersive white light continuum Z-scan,” Appl. Phys. Lett. 111(7), 071901 (2017).
[Crossref]

Biomacromolecules (1)

A. I. Van Den Bulcke, B. Bogdanov, N. De Rooze, E. H. Schacht, M. Cornelissen, and H. Berghmans, “Structural and Rheological Properties of Methacrylamide Modified Gelatin Hydrogels,” Biomacromolecules 1(1), 31–38 (2000).
[Crossref]

Chem. Mater. (1)

S. J. Jhaveri, J. D. McMullen, R. Sijbesma, L.-S. Tan, W. Zipfel, and C. K. Ober, “Direct Three-Dimensional Microfabrication of Hydrogels via Two-Photon Lithography in Aqueous Solution,” Chem. Mater. 21(10), 2003–2006 (2009).
[Crossref]

Chem. Phys. Lett. (1)

S. Venugopal Rao, N. K. M. Naga Srinivas, and D. Narayana Rao, “Nonlinear absorption and excited state dynamics in Rhodamine B studied using Z-scan and degenerate four wave mixing techniques,” Chem. Phys. Lett. 361(5-6), 439–445 (2002).
[Crossref]

Heat Mass Transfer (1)

J. Wang and M. Fiebig, “Bestimmung der Temperaturleitfahigkeit von Toluol und Methanol rnittels laserinduzierter thermischer Gitter,” Heat Mass Transfer 31(1-2), 83–87 (1995)..
[Crossref]

IEEE J. Quantum Electron. (1)

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[Crossref]

IEEE Trans. Inf. Theory (1)

M.-K. Hu, “Visual pattern recognition by moment invariants,” IEEE Trans. Inf. Theory 8(2), 179–187 (1962).
[Crossref]

Inf. Sci. (1)

J. A. Martín H, M. Santos, and J. de Lope, “Orthogonal variant moments features in image analysis,” Inf. Sci. 180(6), 846–860 (2010).
[Crossref]

J. Am. Chem. Soc. (1)

G. G. Dubinina, R. S. Price, K. A. Abboud, G. Wicks, P. Wnuk, Y. Stepanenko, M. Drobizhev, A. Rebane, and K. S. Schanze, “Phenylene Vinylene Platinum(II) Acetylides with Prodigious Two-Photon Absorption,” J. Am. Chem. Soc. 134(47), 19346–19349 (2012).
[Crossref]

J. Non-Cryst. Solids (1)

C. Jacinto, D. N. Messias, A. A. Andrade, S. M. Lima, M. L. Baesso, and T. Catunda, “Thermal lens and Z-scan measurements: Thermal and optical properties of laser glasses – A review,” J. Non-Cryst. Solids 352(32-35), 3582–3597 (2006).
[Crossref]

J. Opt. A: Pure Appl. Opt. (1)

M. Falconieri, “Thermo-optical effects in Z -scan measurements using high-repetition-rate lasers,” J. Opt. A: Pure Appl. Opt. 1(6), 662–667 (1999).
[Crossref]

J. Opt. Soc. Am. B (2)

J. Photochem. Photobiol., A (1)

K. J. Schafer, J. M. Hales, M. Balu, K. D. Belfield, E. W. Van Stryland, and D. J. Hagan, “Two-photon absorption cross-sections of common photoinitiators,” J. Photochem. Photobiol., A 162(2-3), 497–502 (2004).
[Crossref]

J. Phys. Chem. A (1)

D. S. Corrêa, S. L. Oliveira, L. Misoguti, S. C. Zilio, R. F. Aroca, C. J. L. Constantino, and C. R. Mendonça, “Investigation of the Two-Photon Absorption Cross-Section in Perylene Tetracarboxylic Derivatives: Nonlinear Spectra and Molecular Structure,” J. Phys. Chem. A 110(20), 6433–6438 (2006).
[Crossref]

Macromolecules (1)

Z. Li, N. Pucher, K. Cicha, J. Torgersen, S. C. Ligon, A. Ajami, W. Husinsky, A. Rosspeintner, E. Vauthey, S. Naumov, T. Scherzer, J. Stampfl, and R. Liska, “A Straightforward Synthesis and Structure–Activity Relationship of Highly Efficient Initiators for Two-Photon Polymerization,” Macromolecules 46(2), 352–361 (2013).
[Crossref]

Nat. Methods (1)

M. Drobizhev, N. S. Makarov, S. E. Tillo, T. E. Hughes, and A. Rebane, “Two-photon absorption properties of fluorescent proteins,” Nat. Methods 8(5), 393–399 (2011).
[Crossref]

Opt. Commun. (3)

S. M. Mian, S. B. McGee, and N. Melikechi, “Experimental and theoretical investigation of thermal lensing effects in mode-locked femtosecond Z-scan experiments,” Opt. Commun. 207(1-6), 339–345 (2002).
[Crossref]

R. A. Ganeev, A. I. Ryasnyansky, N. Ishizawa, M. Baba, M. Suzuki, M. Turu, S. Sakakibara, and H. Kuroda, “Two- and three-photon absorption in CS2,” Opt. Commun. 231(1-6), 431–436 (2004).
[Crossref]

D. S. Corrêa, L. De Boni, L. Misoguti, I. Cohanoschi, F. E. Hernandez, and C. R. Mendonça, “Z-scan theoretical analysis for three-, four- and five-photon absorption,” Opt. Commun. 277(2), 440–445 (2007).
[Crossref]

Opt. Express (4)

Phys. Rev. Lett. (1)

F. Yoshino, S. Polyakov, M. Liu, and G. Stegeman, “Observation of Three-Photon Enhanced Four-Photon Absorption,” Phys. Rev. Lett. 91(6), 063902 (2003).
[Crossref]

Polymer Chem. (1)

M. Tromayer, A. Dobos, P. Gruber, A. Ajami, R. Dedic, A. Ovsianikov, and R. Liska, “A biocompatible diazosulfonate initiator for direct encapsulation of human stem cells via two-photon polymerization,” Polymer Chem. 22, 2018 (2018).
[Crossref]

RSC Adv. (1)

Z. Li, J. Torgersen, A. Ajami, S. Mühleder, X. Qin, W. Husinsky, W. Holnthoner, A. Ovsianikov, J. Stampfl, and R. Liska, “Initiation efficiency and cytotoxicity of novel water-soluble two-photon photoinitiators for direct 3D microfabrication of hydrogels,” RSC Adv. 3(36), 15939–15946 (2013).
[Crossref]

Other (6)

J. J. Moré, “The Levenberg-Marquardt algorithm: Implementation and theory,” in Numerical Analysis, G. A. Watson, ed. (Springer Berlin Heidelberg, 1978), Vol. 630, pp. 105–116.

S. C. Ligon, M. Tromayer, Z. Li, J. Torgersen, A. Ajami, A. Rosspeintner, S. Naumov, T. Scherzer, J. Stampfl, and R. Liska, “New Developments in Initiators for Two-Photon Polymerization,” 6 Conference: Rad Tech UV/EB 2014At: Rosemont, IL.

ISO 11146-1:2005, “Lasers and laser-related equipment – Test methods for laser beam widths, divergence angles and beam propagation ratios,” http://www.iso.org/cms/render/live/en/sites/isoorg/contents/data/standard/03/36/33625.html .

A. E. Siegman, “Defining, measuring, and optimizing laser beam quality,” in Laser Resonators and Coherent Optics: Modeling, Technology, and Applications (International Society for Optics and Photonics, 1993), Vol. 1868, pp. 2–13.

E. W. Van Stryland and M. Sheik-Bahae, “Z-scan measurements of optical nonlinearities,” Charact. Tech. Tabul. Org. Nonlinear Mater.655–692 (1998).

J. Stampfl, R. Liska, and A. Ovsianikov, Multiphoton Lithography: Techniques, Materials, and Applications (John Wiley & Sons, 2016).

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Figures (7)

Fig. 1.
Fig. 1. 2PA compounds used in this study. Rhodamine B is a laser dye, which was chosen as a reference standard for the setup [23]. M2CMK has been used in various 2PP applications [24,25]. The water-soluble PI DAS is used for 2PP structuring of biocompatible hydrogels [11,26].
Fig. 2.
Fig. 2. Beam path from the tunable fs-laser to the sample. A waveplate and polarizing beam splitter attenuate the input laser power. After the chopper and before mirror 2, the beam is expanded by 4x. The expanders consists of two parabolic mirrors. A lens focuses the beam. A motorized stage moves the sample in and out of focus. Two diodes record the measurement- and reference signal while a mechanical chopper allows to adjust the on/off duration of the signal. A flip mirror allows to redirect the beam to an auto correlator to measure the pulse duration.
Fig. 3.
Fig. 3. (a) Focal position for over the laser tuning range. Using a Galilean beam expander caused large shifts in the focal position if the collimation was not manually adjusted for each wavelength. In contrast, a reflective expander collimates the beam over the entire laser spectrum, causing only negligible shifts in focus. (b) Thermal z-scan comparison of Rhodamine B (10 mM in methanol at 800 nm) measured with and without chopper. The signal drop without chopper (straight line) was four times larger than when a chopper was used (squares). The measurement intensity was 37.1 GW cm−2. A notable sharpening of the transmission curve for continuous exposure resulted in 89% of the signal change within $\pm 1.13 \cdot {z_{R}}$, compared to 69% for the chopper signal.
Fig. 4.
Fig. 4. (a) Four different chopper exposure times were selected to study the impact of heat accumulation due to the high laser repetition rates on the extracted cross section. For each exposure time the cross section was measured at 74 and 104 GW cm−2 and the relative change in cross section was calculated. At 450 µs exposure time the change in extracted cross section was more than 20% indicating thermal effects. In contrast at chopper on times of 210 µs and below such behavior was not observed. (b) 2PA-spectrum of rhodamine B. The solid blue squares show data measured using the presented tunable z-scan setup with standard deviation below 5%. Black triangles are reference values measured using an OPA-based system [23], whereas dash-dotted data were measured using a z-scan setup based on white light continuum (WLC). No error bars for WLC are provided in source [12]. The data follow similar trends exhibiting high absorption in the region between 760 and 840 nm and lower absorption for longer wavelengths. Close to 700 nm the values obtained with the tunable system and the referenced amplified setup increase.
Fig. 5.
Fig. 5. (a) 2PA studies of M2CMK (10 mM solution in tetrahydrofuran) at 800 nm. In the pure 2PA case ${q_{0}}$ is expected to follow a linear trend, which was observed for measurement powers up to 18.5 GW cm−2 (solid line). From 22 GW cm−2 on ${q_{0}}$ followed an exponential trend (dashed line). (b) This exponential increase was also visible in the recorded transmission signal, where a notable sharpening of the curve can be seen between 15.0 GW cm−2 and 37.5 GW cm−2. While at 15.0 GW cm−2 65% of the signal drop occurs within a distance of $\pm 1.13 \cdot {z_{R}}$ around the focus, at 37.5 GW cm−2 77% of the signal drop is happening within this section. A Rayleigh length of ${z_{fit}} = 0.5 \cdot {z_{R}}$ was required to describe the signal curve at 37.5 GW cm−2, indicating that the material response was not limited to 2PA processes only.
Fig. 6.
Fig. 6. (a) 2PA-spectra of 10 mM M2CMK in methanol and 10 mM DAS in PBS. M2CMK z-scans results obtained with the tunable system (blue squares) were compared to the referenced WLC z-scan measurements (solid line)) [12]. While higher than the values obtained by WLC z-scan the spectral behavior of ${{\boldsymbol \sigma }_{2}}$ followed a comparable trend with the absorption maximum between 720 and 800 nm and a decrease above 900 nm. The 2PA spectrum of 10 mM DAS in PBS exhibits a maximum absorption of 90 GM at 700 nm (triangles) and was comparable with the WLC z-scan (dash-dotted line) [11]. No error bars for WLC are provided in sources. (b) Matching the wavelength used for 2PP to the absorption peak resulted in a significant reduction of the polymerization threshold for GelMA (95% degree of substitution, dissolved in PBS) with 2 mM DAS as PI.
Fig. 7.
Fig. 7. Measurement algorithm. Illustration of the automated measurement algorithm used in this setup. For a given wavelength an intensity range is selected. For each intensity, the z-scan is carried out m-times. As a first criterion the sharpening of the transmission curve is judged by comparing ${z_{fit}}$ from direct fit and ${z_{R}}$ obtained via beamprofiler. If results get rejected due to this process, the available ${q_{0}}(c)$ are fitted using linear regression. If this behavior is met, the measurement is accepted. If not, the power range is reduced by one increment ($\Delta I$). This procedure is repeated until the criteria are met or the minimum intensity has been reached. To increase measurement efficiency, measurements for ${I_{c}} = {I_{0}} + c \cdot \Delta I$ with $c \in \{ 1,2\ldots ,n\} $ can be used from the previous iteration.

Tables (1)

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Table 1. Uncertainties in the setup parameters can significantly affect the calculated 2PA cross section (${\sigma _{2}}$) by up to 30%. Change in ${\sigma _{2}}$ calculated by taking reference data for a 2PA absorbing compound with ${\sigma _{2}} = 150$ and Eq. (1) for a focused laser beam with 20 µm diameter, 70 fs pulse duration and power of 500 mW.

Equations (6)

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T ( z ) = m = 0 q 0 ( σ 2 , z , 0 ) m ( m + 1 ) 3 / 2
q 0 ( σ 2 , z , 0 ) = β ( σ 2 ) I 0 ( ω 0 , P ) L e f f ( ω ( z ) ω 0 ) 2
I 0 ( ω 0 , P ) = 4 ln ( 2 ) π P π ω 0 2 R τ
σ 2 = σ 2 ( λ ) = h c λ β ( λ ) N A ρ
ω ( z ) = ω 0 1 + ( z z 0 z R ) 2
z R z R , G a u s s = M 2 , z R , G a u s s = π ω 0 2 λ