Abstract

Most approaches to high-capacity 3D optical data storage (ODS) require confinement of the writing action to a specified depth in the writing medium. This is achieved by a nonlinear photoresponse, usually two-photon absorption, which requires a pulsed long-wavelength source. Fluorescence photobleaching of a dye/polymer composite can be used at a short wavelength to store data at the diffraction limit in a layered storage medium. In this work, the writing response of a bleachable dye/polymer system illuminated with single pulses of various duration obtained from a modulated 405 nm wavelength CW laser was studied. A transition from a linear to nonlinear writing mechanism was observed near the microsecond time scale. Concentration-dependent measurements indicate that a photothermal mechanism accounts for the nonlinear response in the short pulse, higher power regime. This nonlinear response may be useful for realizing terabyte scale ODS in multilayered polymer media.

© 2014 Optical Society of America

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2013 (1)

M. D. Mottaghi and C. Dwyer, “Thousand-fold increase in optical storage density by polychromatic address multiplexing on self-assembled DNA nanostructures,” Adv. Mater. 25, 3593–3598 (2013).
[Crossref]

2012 (1)

C. Ryan, C. W. Christenson, B. Valle, A. Saini, J. Lott, J. Johnson, D. A. Schiraldi, C. Weder, E. Baer, K. D. Singer, and J. Shan, “Roll-to-roll fabrication of multilayer films for high capacity optical data storage,” Adv. Mater. 24, 5222–5226 (2012).
[Crossref]

2011 (2)

Z. Lu, Y. Liu, W. Hu, X. W. D. Lou, and C. M. Li, “Rewritable multicolor fluorescent patterns for multistate memory devices with high data storage capacity,” Chem. Commun. 47, 9609–9611 (2011).
[Crossref]

F.-K. Bruder, R. Hagen, T. Rölle, M.-S. Weiser, and T. Fäcke, “From the surface to volume: concepts for the next generation of optical-holographic data-storage materials,” Angew. Chem. Int. Ed. 50, 4552–4573 (2011).
[Crossref]

2009 (3)

A. S. Dvornikov, E. P. Walker, and P. M. Rentzepis, “Two-photon three-dimensional optical storage memory,” J. Phys. Chem. A 113, 13633–13644 (2009).
[Crossref]

C. O. Yanez, C. D. Andrade, S. Yao, G. Luchita, M. V. Bondar, and K. D. Belfield, “Photosensitive polymeric materials for two-photon 3D WORM optical data storage systems,” ACS Appl. Mater. Interfaces 1, 2219–2229 (2009).
[Crossref]

V. Ostroverkhov, B. L. Lawrence, X. Shi, E. P. Boden, and C. Erben, “Micro-holographic storage and threshold holographic recording materials,” Jpn. J. Appl. Phys. 48, 03A035 (2009).
[Crossref]

2007 (1)

2006 (1)

G. Donnert, C. Eggeling, and S. W. Hell, “Major signal increase in fluorescence microscopy through dark state relaxation,” Nat. Mater. 4, 81–86 (2006).
[Crossref]

2005 (1)

C. Eggeling, A. Volkmer, and C. A. M. Seidel, “Molecular photobleaching kinetics of rhodamine 6G by one- and two-photon induced confocal fluorescence microscopy,” Chem. Phys. Chem. 6, 791–804 (2005).
[Crossref]

2004 (2)

R. Zondervan, F. Kulzer, M. A. Kol’chenko, and M. Orrit, “Photobleaching of rhodamine 6G in poly(vinyl alcohol) at the ensemble and single-molecule levels,” J. Phys. Chem. A 108, 1657–1665 (2004).
[Crossref]

N. Shida, T. Higuchi, Y. Hosoda, H. Miyoshi, A. Nakano, and K. Tsuchiya, “Multilayer optical read-only-memory disk applicable to Blu-ray disc standard using a photopolymer sheet with a recording capacity of 100  GB,” Jpn. J. Appl. Phys. 43, 4983–4986 (2004).
[Crossref]

1999 (1)

M. Talhavini and T. Atvars, “Photostability of xanthene molecules trapped in poly (vinyl alcohol) (PVA) matrices,” J. Photochem. Photobiol. A 120, 141–149 (1999).
[Crossref]

1998 (1)

C. Eggeling, J. Widengren, and R. Rigler, “Photobleaching of fluorescent dyes under conditions used for single-molecule detection: evidence of two-step photolysis,” Anal. Chem. 702651–2659 (1998).
[Crossref]

1997 (1)

P. Bojarski, “Concentration quenching and depolarization of rhodamine 6G in the presence of fluorescent dimers in polyvinyl alcohol films,” Chem. Phys. Lett. 278, 225–232 (1997).
[Crossref]

1996 (1)

L. Song, C. A. Varma, J. W. Verhoeven, and H. J. Tanke, “Influence of the triplet excited state on the photobleaching kinetics of fluorescein in microscopy,” Biophys. J. 70, 2959–2968 (1996).
[Crossref]

1994 (1)

H. Fukumura and H. Masuhara, “The mechanism of dopant-induced laser ablation. Possibility of cyclic multiphotonic absorption in excited states,” Chem. Phys. Lett. 221, 373–378 (1994).
[Crossref]

1981 (1)

1980 (1)

P. Hammond, “Comparison of experimental and theoretical excited-state spectra for rhodamine 6G,” IEEE J. Quantum Electron. 16, 1157–1160 (1980).
[Crossref]

1975 (1)

J. Bechtel, “Heating of solid targets with laser pulses,” J. Appl. Phys. 461585–1593 (1975).

Andrade, C. D.

C. O. Yanez, C. D. Andrade, S. Yao, G. Luchita, M. V. Bondar, and K. D. Belfield, “Photosensitive polymeric materials for two-photon 3D WORM optical data storage systems,” ACS Appl. Mater. Interfaces 1, 2219–2229 (2009).
[Crossref]

Atvars, T.

M. Talhavini and T. Atvars, “Photostability of xanthene molecules trapped in poly (vinyl alcohol) (PVA) matrices,” J. Photochem. Photobiol. A 120, 141–149 (1999).
[Crossref]

Baer, E.

C. Ryan, C. W. Christenson, B. Valle, A. Saini, J. Lott, J. Johnson, D. A. Schiraldi, C. Weder, E. Baer, K. D. Singer, and J. Shan, “Roll-to-roll fabrication of multilayer films for high capacity optical data storage,” Adv. Mater. 24, 5222–5226 (2012).
[Crossref]

Bechtel, J.

J. Bechtel, “Heating of solid targets with laser pulses,” J. Appl. Phys. 461585–1593 (1975).

Belfield, K. D.

C. O. Yanez, C. D. Andrade, S. Yao, G. Luchita, M. V. Bondar, and K. D. Belfield, “Photosensitive polymeric materials for two-photon 3D WORM optical data storage systems,” ACS Appl. Mater. Interfaces 1, 2219–2229 (2009).
[Crossref]

Boden, E. P.

V. Ostroverkhov, B. L. Lawrence, X. Shi, E. P. Boden, and C. Erben, “Micro-holographic storage and threshold holographic recording materials,” Jpn. J. Appl. Phys. 48, 03A035 (2009).
[Crossref]

Bojarski, P.

P. Bojarski, “Concentration quenching and depolarization of rhodamine 6G in the presence of fluorescent dimers in polyvinyl alcohol films,” Chem. Phys. Lett. 278, 225–232 (1997).
[Crossref]

Bondar, M. V.

C. O. Yanez, C. D. Andrade, S. Yao, G. Luchita, M. V. Bondar, and K. D. Belfield, “Photosensitive polymeric materials for two-photon 3D WORM optical data storage systems,” ACS Appl. Mater. Interfaces 1, 2219–2229 (2009).
[Crossref]

Bruder, F.-K.

F.-K. Bruder, R. Hagen, T. Rölle, M.-S. Weiser, and T. Fäcke, “From the surface to volume: concepts for the next generation of optical-holographic data-storage materials,” Angew. Chem. Int. Ed. 50, 4552–4573 (2011).
[Crossref]

Chon, J. W. M.

Christenson, C. W.

C. Ryan, C. W. Christenson, B. Valle, A. Saini, J. Lott, J. Johnson, D. A. Schiraldi, C. Weder, E. Baer, K. D. Singer, and J. Shan, “Roll-to-roll fabrication of multilayer films for high capacity optical data storage,” Adv. Mater. 24, 5222–5226 (2012).
[Crossref]

Donnert, G.

G. Donnert, C. Eggeling, and S. W. Hell, “Major signal increase in fluorescence microscopy through dark state relaxation,” Nat. Mater. 4, 81–86 (2006).
[Crossref]

Dvornikov, A. S.

A. S. Dvornikov, E. P. Walker, and P. M. Rentzepis, “Two-photon three-dimensional optical storage memory,” J. Phys. Chem. A 113, 13633–13644 (2009).
[Crossref]

Dwyer, C.

M. D. Mottaghi and C. Dwyer, “Thousand-fold increase in optical storage density by polychromatic address multiplexing on self-assembled DNA nanostructures,” Adv. Mater. 25, 3593–3598 (2013).
[Crossref]

Eggeling, C.

G. Donnert, C. Eggeling, and S. W. Hell, “Major signal increase in fluorescence microscopy through dark state relaxation,” Nat. Mater. 4, 81–86 (2006).
[Crossref]

C. Eggeling, A. Volkmer, and C. A. M. Seidel, “Molecular photobleaching kinetics of rhodamine 6G by one- and two-photon induced confocal fluorescence microscopy,” Chem. Phys. Chem. 6, 791–804 (2005).
[Crossref]

C. Eggeling, J. Widengren, and R. Rigler, “Photobleaching of fluorescent dyes under conditions used for single-molecule detection: evidence of two-step photolysis,” Anal. Chem. 702651–2659 (1998).
[Crossref]

Erben, C.

V. Ostroverkhov, B. L. Lawrence, X. Shi, E. P. Boden, and C. Erben, “Micro-holographic storage and threshold holographic recording materials,” Jpn. J. Appl. Phys. 48, 03A035 (2009).
[Crossref]

Evans, R. A.

Fäcke, T.

F.-K. Bruder, R. Hagen, T. Rölle, M.-S. Weiser, and T. Fäcke, “From the surface to volume: concepts for the next generation of optical-holographic data-storage materials,” Angew. Chem. Int. Ed. 50, 4552–4573 (2011).
[Crossref]

Fukumura, H.

H. Fukumura and H. Masuhara, “The mechanism of dopant-induced laser ablation. Possibility of cyclic multiphotonic absorption in excited states,” Chem. Phys. Lett. 221, 373–378 (1994).
[Crossref]

Gu, M.

Hagen, R.

F.-K. Bruder, R. Hagen, T. Rölle, M.-S. Weiser, and T. Fäcke, “From the surface to volume: concepts for the next generation of optical-holographic data-storage materials,” Angew. Chem. Int. Ed. 50, 4552–4573 (2011).
[Crossref]

Hammond, P.

P. Hammond, “Comparison of experimental and theoretical excited-state spectra for rhodamine 6G,” IEEE J. Quantum Electron. 16, 1157–1160 (1980).
[Crossref]

Hell, S. W.

G. Donnert, C. Eggeling, and S. W. Hell, “Major signal increase in fluorescence microscopy through dark state relaxation,” Nat. Mater. 4, 81–86 (2006).
[Crossref]

Higuchi, T.

N. Shida, T. Higuchi, Y. Hosoda, H. Miyoshi, A. Nakano, and K. Tsuchiya, “Multilayer optical read-only-memory disk applicable to Blu-ray disc standard using a photopolymer sheet with a recording capacity of 100  GB,” Jpn. J. Appl. Phys. 43, 4983–4986 (2004).
[Crossref]

Hosoda, Y.

N. Shida, T. Higuchi, Y. Hosoda, H. Miyoshi, A. Nakano, and K. Tsuchiya, “Multilayer optical read-only-memory disk applicable to Blu-ray disc standard using a photopolymer sheet with a recording capacity of 100  GB,” Jpn. J. Appl. Phys. 43, 4983–4986 (2004).
[Crossref]

Hu, W.

Z. Lu, Y. Liu, W. Hu, X. W. D. Lou, and C. M. Li, “Rewritable multicolor fluorescent patterns for multistate memory devices with high data storage capacity,” Chem. Commun. 47, 9609–9611 (2011).
[Crossref]

Johnson, J.

C. Ryan, C. W. Christenson, B. Valle, A. Saini, J. Lott, J. Johnson, D. A. Schiraldi, C. Weder, E. Baer, K. D. Singer, and J. Shan, “Roll-to-roll fabrication of multilayer films for high capacity optical data storage,” Adv. Mater. 24, 5222–5226 (2012).
[Crossref]

Kol’chenko, M. A.

R. Zondervan, F. Kulzer, M. A. Kol’chenko, and M. Orrit, “Photobleaching of rhodamine 6G in poly(vinyl alcohol) at the ensemble and single-molecule levels,” J. Phys. Chem. A 108, 1657–1665 (2004).
[Crossref]

Kulzer, F.

R. Zondervan, F. Kulzer, M. A. Kol’chenko, and M. Orrit, “Photobleaching of rhodamine 6G in poly(vinyl alcohol) at the ensemble and single-molecule levels,” J. Phys. Chem. A 108, 1657–1665 (2004).
[Crossref]

Lawrence, B. L.

V. Ostroverkhov, B. L. Lawrence, X. Shi, E. P. Boden, and C. Erben, “Micro-holographic storage and threshold holographic recording materials,” Jpn. J. Appl. Phys. 48, 03A035 (2009).
[Crossref]

Li, C. M.

Z. Lu, Y. Liu, W. Hu, X. W. D. Lou, and C. M. Li, “Rewritable multicolor fluorescent patterns for multistate memory devices with high data storage capacity,” Chem. Commun. 47, 9609–9611 (2011).
[Crossref]

Li, X.

Liu, Y.

Z. Lu, Y. Liu, W. Hu, X. W. D. Lou, and C. M. Li, “Rewritable multicolor fluorescent patterns for multistate memory devices with high data storage capacity,” Chem. Commun. 47, 9609–9611 (2011).
[Crossref]

Lott, J.

C. Ryan, C. W. Christenson, B. Valle, A. Saini, J. Lott, J. Johnson, D. A. Schiraldi, C. Weder, E. Baer, K. D. Singer, and J. Shan, “Roll-to-roll fabrication of multilayer films for high capacity optical data storage,” Adv. Mater. 24, 5222–5226 (2012).
[Crossref]

Lou, X. W. D.

Z. Lu, Y. Liu, W. Hu, X. W. D. Lou, and C. M. Li, “Rewritable multicolor fluorescent patterns for multistate memory devices with high data storage capacity,” Chem. Commun. 47, 9609–9611 (2011).
[Crossref]

Lu, Z.

Z. Lu, Y. Liu, W. Hu, X. W. D. Lou, and C. M. Li, “Rewritable multicolor fluorescent patterns for multistate memory devices with high data storage capacity,” Chem. Commun. 47, 9609–9611 (2011).
[Crossref]

Luchita, G.

C. O. Yanez, C. D. Andrade, S. Yao, G. Luchita, M. V. Bondar, and K. D. Belfield, “Photosensitive polymeric materials for two-photon 3D WORM optical data storage systems,” ACS Appl. Mater. Interfaces 1, 2219–2229 (2009).
[Crossref]

Masuhara, H.

H. Fukumura and H. Masuhara, “The mechanism of dopant-induced laser ablation. Possibility of cyclic multiphotonic absorption in excited states,” Chem. Phys. Lett. 221, 373–378 (1994).
[Crossref]

McLeod, R. R.

R. R. McLeod, Encyclopedia of Computer Science and Engineering, B. W. Wah, ed. (Wiley, 2008), pp. 2069–2082.

Miyoshi, H.

N. Shida, T. Higuchi, Y. Hosoda, H. Miyoshi, A. Nakano, and K. Tsuchiya, “Multilayer optical read-only-memory disk applicable to Blu-ray disc standard using a photopolymer sheet with a recording capacity of 100  GB,” Jpn. J. Appl. Phys. 43, 4983–4986 (2004).
[Crossref]

Mottaghi, M. D.

M. D. Mottaghi and C. Dwyer, “Thousand-fold increase in optical storage density by polychromatic address multiplexing on self-assembled DNA nanostructures,” Adv. Mater. 25, 3593–3598 (2013).
[Crossref]

Nakano, A.

N. Shida, T. Higuchi, Y. Hosoda, H. Miyoshi, A. Nakano, and K. Tsuchiya, “Multilayer optical read-only-memory disk applicable to Blu-ray disc standard using a photopolymer sheet with a recording capacity of 100  GB,” Jpn. J. Appl. Phys. 43, 4983–4986 (2004).
[Crossref]

Orrit, M.

R. Zondervan, F. Kulzer, M. A. Kol’chenko, and M. Orrit, “Photobleaching of rhodamine 6G in poly(vinyl alcohol) at the ensemble and single-molecule levels,” J. Phys. Chem. A 108, 1657–1665 (2004).
[Crossref]

Ostroverkhov, V.

V. Ostroverkhov, B. L. Lawrence, X. Shi, E. P. Boden, and C. Erben, “Micro-holographic storage and threshold holographic recording materials,” Jpn. J. Appl. Phys. 48, 03A035 (2009).
[Crossref]

Rentzepis, P. M.

A. S. Dvornikov, E. P. Walker, and P. M. Rentzepis, “Two-photon three-dimensional optical storage memory,” J. Phys. Chem. A 113, 13633–13644 (2009).
[Crossref]

Rigler, R.

C. Eggeling, J. Widengren, and R. Rigler, “Photobleaching of fluorescent dyes under conditions used for single-molecule detection: evidence of two-step photolysis,” Anal. Chem. 702651–2659 (1998).
[Crossref]

Rölle, T.

F.-K. Bruder, R. Hagen, T. Rölle, M.-S. Weiser, and T. Fäcke, “From the surface to volume: concepts for the next generation of optical-holographic data-storage materials,” Angew. Chem. Int. Ed. 50, 4552–4573 (2011).
[Crossref]

Ryan, C.

C. Ryan, C. W. Christenson, B. Valle, A. Saini, J. Lott, J. Johnson, D. A. Schiraldi, C. Weder, E. Baer, K. D. Singer, and J. Shan, “Roll-to-roll fabrication of multilayer films for high capacity optical data storage,” Adv. Mater. 24, 5222–5226 (2012).
[Crossref]

Saini, A.

C. Ryan, C. W. Christenson, B. Valle, A. Saini, J. Lott, J. Johnson, D. A. Schiraldi, C. Weder, E. Baer, K. D. Singer, and J. Shan, “Roll-to-roll fabrication of multilayer films for high capacity optical data storage,” Adv. Mater. 24, 5222–5226 (2012).
[Crossref]

Schiraldi, D. A.

C. Ryan, C. W. Christenson, B. Valle, A. Saini, J. Lott, J. Johnson, D. A. Schiraldi, C. Weder, E. Baer, K. D. Singer, and J. Shan, “Roll-to-roll fabrication of multilayer films for high capacity optical data storage,” Adv. Mater. 24, 5222–5226 (2012).
[Crossref]

Schneider, M. B.

Seidel, C. A. M.

C. Eggeling, A. Volkmer, and C. A. M. Seidel, “Molecular photobleaching kinetics of rhodamine 6G by one- and two-photon induced confocal fluorescence microscopy,” Chem. Phys. Chem. 6, 791–804 (2005).
[Crossref]

Shan, J.

C. Ryan, C. W. Christenson, B. Valle, A. Saini, J. Lott, J. Johnson, D. A. Schiraldi, C. Weder, E. Baer, K. D. Singer, and J. Shan, “Roll-to-roll fabrication of multilayer films for high capacity optical data storage,” Adv. Mater. 24, 5222–5226 (2012).
[Crossref]

Shi, X.

V. Ostroverkhov, B. L. Lawrence, X. Shi, E. P. Boden, and C. Erben, “Micro-holographic storage and threshold holographic recording materials,” Jpn. J. Appl. Phys. 48, 03A035 (2009).
[Crossref]

Shida, N.

N. Shida, T. Higuchi, Y. Hosoda, H. Miyoshi, A. Nakano, and K. Tsuchiya, “Multilayer optical read-only-memory disk applicable to Blu-ray disc standard using a photopolymer sheet with a recording capacity of 100  GB,” Jpn. J. Appl. Phys. 43, 4983–4986 (2004).
[Crossref]

Singer, K. D.

C. Ryan, C. W. Christenson, B. Valle, A. Saini, J. Lott, J. Johnson, D. A. Schiraldi, C. Weder, E. Baer, K. D. Singer, and J. Shan, “Roll-to-roll fabrication of multilayer films for high capacity optical data storage,” Adv. Mater. 24, 5222–5226 (2012).
[Crossref]

Song, L.

L. Song, C. A. Varma, J. W. Verhoeven, and H. J. Tanke, “Influence of the triplet excited state on the photobleaching kinetics of fluorescein in microscopy,” Biophys. J. 70, 2959–2968 (1996).
[Crossref]

Talhavini, M.

M. Talhavini and T. Atvars, “Photostability of xanthene molecules trapped in poly (vinyl alcohol) (PVA) matrices,” J. Photochem. Photobiol. A 120, 141–149 (1999).
[Crossref]

Tanke, H. J.

L. Song, C. A. Varma, J. W. Verhoeven, and H. J. Tanke, “Influence of the triplet excited state on the photobleaching kinetics of fluorescein in microscopy,” Biophys. J. 70, 2959–2968 (1996).
[Crossref]

Tsuchiya, K.

N. Shida, T. Higuchi, Y. Hosoda, H. Miyoshi, A. Nakano, and K. Tsuchiya, “Multilayer optical read-only-memory disk applicable to Blu-ray disc standard using a photopolymer sheet with a recording capacity of 100  GB,” Jpn. J. Appl. Phys. 43, 4983–4986 (2004).
[Crossref]

Valle, B.

C. Ryan, C. W. Christenson, B. Valle, A. Saini, J. Lott, J. Johnson, D. A. Schiraldi, C. Weder, E. Baer, K. D. Singer, and J. Shan, “Roll-to-roll fabrication of multilayer films for high capacity optical data storage,” Adv. Mater. 24, 5222–5226 (2012).
[Crossref]

Varma, C. A.

L. Song, C. A. Varma, J. W. Verhoeven, and H. J. Tanke, “Influence of the triplet excited state on the photobleaching kinetics of fluorescein in microscopy,” Biophys. J. 70, 2959–2968 (1996).
[Crossref]

Verhoeven, J. W.

L. Song, C. A. Varma, J. W. Verhoeven, and H. J. Tanke, “Influence of the triplet excited state on the photobleaching kinetics of fluorescein in microscopy,” Biophys. J. 70, 2959–2968 (1996).
[Crossref]

Volkmer, A.

C. Eggeling, A. Volkmer, and C. A. M. Seidel, “Molecular photobleaching kinetics of rhodamine 6G by one- and two-photon induced confocal fluorescence microscopy,” Chem. Phys. Chem. 6, 791–804 (2005).
[Crossref]

Walker, E. P.

A. S. Dvornikov, E. P. Walker, and P. M. Rentzepis, “Two-photon three-dimensional optical storage memory,” J. Phys. Chem. A 113, 13633–13644 (2009).
[Crossref]

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Figures (6)

Fig. 1.
Fig. 1.

(a) Absorption spectra of films with varying dye concentrations, normalized to the peak values, showing no significant dimer formation. (b) FL spectra of same films, normalized only to thickness and beam intensity. There is a large bathochromic shift indicative of FL absorption and re-emission.

Fig. 2.
Fig. 2.

Integrated FL and absorption from spectra in Fig. 1. Absorption increases linearly with concentration but FL saturates, indicating a reduction in FL QY expected from the FL absorption and re-emission. Inset: calculated relative QY versus concentration.

Fig. 3.
Fig. 3.

Bleaching for different exposure times in the 1.0 wt. % sample. The excitation power is altered to keep the fluence constant. This shows that there are bleaching pathways being activated at higher powers that dominate a linear response from S1/T1 states.

Fig. 4.
Fig. 4.

(a) (False color online) confocal FL image of spots written in the 1.0 wt. % sample at 400 ns and 36MW/cm2. Scale bar is 2 μm. (b) Bleaching versus power in the 1.0 wt. % sample at various exposure times demonstrating ability to write for short times and with a nonlinear response.

Fig. 5.
Fig. 5.

FWHM of the bleached spots, obtained from Gaussian fits, versus bleaching. FWHM of the writing beam was measured to be 580 nm, and accounting for diffraction, written spots are no larger than the writing beam.

Fig. 6.
Fig. 6.

Bleaching versus exposure time for multiple concentrations and power. Only high concentrations and large absorbed powers will yield measurable response for sub-microsecond times. Circled points are discussed explicitly in the manuscript. An incident power of 80 mW gives an intensity of 30MW/cm2.

Equations (1)

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ΔT(r=0,t)=(1QY)I0αd2Kln(1+4κtd2),

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