Abstract

We report a new optical technique to map two-dimensional temperature distributions in liquid solutions based on the thermal motion of fluorescent molecules. We simultaneously capture the fluorescence images of different numerical apertures (NAs) to resolve the temperature-dependent orientations of emission dipoles. In this work, we use two numerical apertures (2NA) prove the concept. This 2NA technique is robust against the intensity variations caused by photobleaching, unsteady illumination and nonuniform molecule distribution. Moreover, as the measured intensity of directional emission is insensitive to polarization changes, this method can be applied to polarizing materials, such as metal surfaces. Under this configuration, the 2NA technique offers another advantage of naturally filtering out the emission background that falls out of collection cones. We foresee the 2NA technique to open a new detection scheme of fluorescence thermometry.

© 2016 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Temperature mapping near plasmonic nanostructures using fluorescence polarization anisotropy

G. Baffou, M.P. Kreuzer, F. Kulzer, and R. Quidant
Opt. Express 17(5) 3291-3298 (2009)

References

  • View by:
  • |
  • |
  • |

  1. G. Kucsko, P. C. Maurer, N. Y. Yao, M. Kubo, H. J. Noh, P. K. Lo, H. Park, and M. D. Lukin, “Nanometre-scale thermometry in a living cell,” Nature 500, 54–57 (2013).
    [Crossref]
  2. S. Kalytchuk, O. Zhovtiuk, S. V. Kershaw, R. Zboril, and A. L. Rogach, “Temperature-dependent exciton and trap-related photoluminescence of CdTe quantum dots embedded in a NaCl matrix: implication in thermometry,” Small 12, 466–476 (2016).
    [Crossref]
  3. J. S. Donner, S. A. Thompson, M. P. Kreuzer, G. Baffou, and R. Quidant, “Mapping intracellular temperature using green fluorescent protein,” Nano Lett. 12, 2107–2111 (2012).
    [Crossref]
  4. K. Okabe, N. Inada, C. Gota, Y. Harada, T. Funatsu, and S. Uchiyama, “Intracellular temperature mapping with a fluorescent polymeric thermometer and fluorescence lifetime imaging microscopy,” Nat. Commun. 3, 705 (2012).
    [Crossref]
  5. P. Löw, B. Kim, N. Takama, and C. Bergaud, “High-spatial-resolution surface-temperature mapping using fluorescent thermometry,” Small 4, 908–914 (2008).
    [Crossref]
  6. G. Baffou, M. P. Kreuzer, F. Kulzer, and R. Quidant, “Temperature mapping near plasmonic nanostructures using fluorescence polarization anisotropy,” Opt. Express 17, 3291–3298 (2009).
    [Crossref]
  7. Y. Yue and X. Wang, “Nanoscale thermal probing,” Nano Rev. 3, 1–11 (2012).
    [Crossref]
  8. J. Wu, T. Y. Kwok, X. Li, W. Cao, Y. Wang, J. Huang, Y. Hong, D. Zhang, and W. Wen, “Mapping three-dimensional temperature in microfluidic chip,” Sci. Rep. 3, 3321 (2013).
  9. K. S. Elvira, X. C. i Solvas, R. C. R. Wootton, and A. J. deMello, “The past, present and potential for microfluidic reactor technology in chemical synthesis,” Nat. Chem. 5, 905–915 (2013).
    [Crossref]
  10. T. Razzaq and C. O. Kappe, “Continuous flow organic synthesis under high-temperature/pressure conditions,” Chem. 5, 1274–1289 (2010).
  11. A. Kruusing, “Underwater and water-assisted laser processing: Part 2–Etching, cutting and rarely used methods,” Opt. Lasers Eng. 41, 329–352 (2004).
    [Crossref]
  12. H. Wallrabe and A. Periasamy, “Imaging protein molecules using FRET and FLIM microscopy,” Current Opinion Biotechnol. 16, 19–27 (2005).
    [Crossref]
  13. W. Srituravanich, L. Pan, Y. Wang, C. Sun, D. B. Bogy, and X. Zhang, “Flying plasmonic lens in the near field for high-speed nanolithography,” Nat. Nanotechnol. 3, 733–737 (2008).
    [Crossref]
  14. S. W. Hell and M. Kroug, “Ground-state-depletion fluorscence microscopy: A concept for breaking the diffraction resolution limit,” Appl. Phys. B 60, 495–497 (1995).
    [Crossref]
  15. W. Y. Shi, Z. Y. Sun, M. Wei, D. G. Evans, and X. Duan, “Tunable photoluminescence properties of fluorescein in a layered double hydroxide matrix by changing the interlayer microenvironment,” J. Phys. Chem. C 114, 21070–21076 (2010).
    [Crossref]
  16. H. F. Arata, P. Löw, K. Ishizuka, C. Bergaud, B. Kim, H. Noji, and H. Fujita, “Temperature distribution measurement on microfabricated thermodevice for single biomolecular observation using fluorescent dye,” Sensors Actuators B-Chem. 117, 339–345 (2006).
    [Crossref]
  17. J. M. Yáñez-Limón, R. Mayen-Mondragón, O. Martínez-Flores, R. Flores-Farias, F. Ruíz, and C. Araujo-Andrade, “Thermal diffusivity studies in edible commercial oils using thermal lens spectroscopy,” Superficies y vacío 18, 31–37 (2005).
  18. E. H. Hellen and D. Axelrod, “Fluorescence emission at dielectric and metal-film interfaces,” J. Opt. Soc. Am. B 4, 337–350 (1987).
    [Crossref]
  19. K. Vasilev, W. Knoll, and M. Kreiter, “Fluorescence intensities of chromophores in front of a thin metal film,” J. Chem. Phys. 120, 3439–3445 (2004).
    [Crossref]
  20. A. Kawski, “Fluorescence anisotropy - theory and applications of rotational depolarization,” Critical Rev. Analytical Chem. 23, 459–529 (1993).
    [Crossref]
  21. N. S. Cheng, “Formula for the viscosity of a glycerol-water mixture,” Industrial Eng. Chem. Res. 47, 3285–3288 (2008).
    [Crossref]
  22. Association, Physical Properties of Glycerol and its Solutions (Glycerine Producers’ Association, 1963).
  23. J. R. Unruh, G. Gokulrangan, G. S. Wilson, and C. K. Johnson, “Fluorescence properties of fluorescein, tetramethyl-rhodamine and texas red linked to a DNA aptamer,” Photochem. Photobiol. 81, 682–690 (2005).
    [Crossref]
  24. A. E. P. Bastos, S. Scolari, M. Stockl, and R. F. M. de Almeida, “Applications of fluorescence lifetime spectroscopy and imaging to lipid domains in vivo,” Imaging Spectroscopic Anal. Living Cells 504, 57–81 (2012).
    [Crossref]
  25. A. Penzkofer and J. Wiedmann, “Orientation of transition dipole-moments of rhodamine-6G determined by excited-state absorption,” Opt. Commun. 35, 81–86 (1980).
    [Crossref]
  26. A. Ali, S. Khan, and F. Nabi, “Volumetric, viscometric and refractive index behaviour of amino acids in aqueous glycerol at different temperatures,” J. Serbian Chem. Soc. 72, 495–512 (2007).
    [Crossref]

2016 (1)

S. Kalytchuk, O. Zhovtiuk, S. V. Kershaw, R. Zboril, and A. L. Rogach, “Temperature-dependent exciton and trap-related photoluminescence of CdTe quantum dots embedded in a NaCl matrix: implication in thermometry,” Small 12, 466–476 (2016).
[Crossref]

2013 (3)

J. Wu, T. Y. Kwok, X. Li, W. Cao, Y. Wang, J. Huang, Y. Hong, D. Zhang, and W. Wen, “Mapping three-dimensional temperature in microfluidic chip,” Sci. Rep. 3, 3321 (2013).

K. S. Elvira, X. C. i Solvas, R. C. R. Wootton, and A. J. deMello, “The past, present and potential for microfluidic reactor technology in chemical synthesis,” Nat. Chem. 5, 905–915 (2013).
[Crossref]

G. Kucsko, P. C. Maurer, N. Y. Yao, M. Kubo, H. J. Noh, P. K. Lo, H. Park, and M. D. Lukin, “Nanometre-scale thermometry in a living cell,” Nature 500, 54–57 (2013).
[Crossref]

2012 (4)

Y. Yue and X. Wang, “Nanoscale thermal probing,” Nano Rev. 3, 1–11 (2012).
[Crossref]

J. S. Donner, S. A. Thompson, M. P. Kreuzer, G. Baffou, and R. Quidant, “Mapping intracellular temperature using green fluorescent protein,” Nano Lett. 12, 2107–2111 (2012).
[Crossref]

K. Okabe, N. Inada, C. Gota, Y. Harada, T. Funatsu, and S. Uchiyama, “Intracellular temperature mapping with a fluorescent polymeric thermometer and fluorescence lifetime imaging microscopy,” Nat. Commun. 3, 705 (2012).
[Crossref]

A. E. P. Bastos, S. Scolari, M. Stockl, and R. F. M. de Almeida, “Applications of fluorescence lifetime spectroscopy and imaging to lipid domains in vivo,” Imaging Spectroscopic Anal. Living Cells 504, 57–81 (2012).
[Crossref]

2010 (2)

T. Razzaq and C. O. Kappe, “Continuous flow organic synthesis under high-temperature/pressure conditions,” Chem. 5, 1274–1289 (2010).

W. Y. Shi, Z. Y. Sun, M. Wei, D. G. Evans, and X. Duan, “Tunable photoluminescence properties of fluorescein in a layered double hydroxide matrix by changing the interlayer microenvironment,” J. Phys. Chem. C 114, 21070–21076 (2010).
[Crossref]

2009 (1)

2008 (3)

P. Löw, B. Kim, N. Takama, and C. Bergaud, “High-spatial-resolution surface-temperature mapping using fluorescent thermometry,” Small 4, 908–914 (2008).
[Crossref]

N. S. Cheng, “Formula for the viscosity of a glycerol-water mixture,” Industrial Eng. Chem. Res. 47, 3285–3288 (2008).
[Crossref]

W. Srituravanich, L. Pan, Y. Wang, C. Sun, D. B. Bogy, and X. Zhang, “Flying plasmonic lens in the near field for high-speed nanolithography,” Nat. Nanotechnol. 3, 733–737 (2008).
[Crossref]

2007 (1)

A. Ali, S. Khan, and F. Nabi, “Volumetric, viscometric and refractive index behaviour of amino acids in aqueous glycerol at different temperatures,” J. Serbian Chem. Soc. 72, 495–512 (2007).
[Crossref]

2006 (1)

H. F. Arata, P. Löw, K. Ishizuka, C. Bergaud, B. Kim, H. Noji, and H. Fujita, “Temperature distribution measurement on microfabricated thermodevice for single biomolecular observation using fluorescent dye,” Sensors Actuators B-Chem. 117, 339–345 (2006).
[Crossref]

2005 (3)

J. M. Yáñez-Limón, R. Mayen-Mondragón, O. Martínez-Flores, R. Flores-Farias, F. Ruíz, and C. Araujo-Andrade, “Thermal diffusivity studies in edible commercial oils using thermal lens spectroscopy,” Superficies y vacío 18, 31–37 (2005).

H. Wallrabe and A. Periasamy, “Imaging protein molecules using FRET and FLIM microscopy,” Current Opinion Biotechnol. 16, 19–27 (2005).
[Crossref]

J. R. Unruh, G. Gokulrangan, G. S. Wilson, and C. K. Johnson, “Fluorescence properties of fluorescein, tetramethyl-rhodamine and texas red linked to a DNA aptamer,” Photochem. Photobiol. 81, 682–690 (2005).
[Crossref]

2004 (2)

K. Vasilev, W. Knoll, and M. Kreiter, “Fluorescence intensities of chromophores in front of a thin metal film,” J. Chem. Phys. 120, 3439–3445 (2004).
[Crossref]

A. Kruusing, “Underwater and water-assisted laser processing: Part 2–Etching, cutting and rarely used methods,” Opt. Lasers Eng. 41, 329–352 (2004).
[Crossref]

1995 (1)

S. W. Hell and M. Kroug, “Ground-state-depletion fluorscence microscopy: A concept for breaking the diffraction resolution limit,” Appl. Phys. B 60, 495–497 (1995).
[Crossref]

1993 (1)

A. Kawski, “Fluorescence anisotropy - theory and applications of rotational depolarization,” Critical Rev. Analytical Chem. 23, 459–529 (1993).
[Crossref]

1987 (1)

1980 (1)

A. Penzkofer and J. Wiedmann, “Orientation of transition dipole-moments of rhodamine-6G determined by excited-state absorption,” Opt. Commun. 35, 81–86 (1980).
[Crossref]

Ali, A.

A. Ali, S. Khan, and F. Nabi, “Volumetric, viscometric and refractive index behaviour of amino acids in aqueous glycerol at different temperatures,” J. Serbian Chem. Soc. 72, 495–512 (2007).
[Crossref]

Arata, H. F.

H. F. Arata, P. Löw, K. Ishizuka, C. Bergaud, B. Kim, H. Noji, and H. Fujita, “Temperature distribution measurement on microfabricated thermodevice for single biomolecular observation using fluorescent dye,” Sensors Actuators B-Chem. 117, 339–345 (2006).
[Crossref]

Araujo-Andrade, C.

J. M. Yáñez-Limón, R. Mayen-Mondragón, O. Martínez-Flores, R. Flores-Farias, F. Ruíz, and C. Araujo-Andrade, “Thermal diffusivity studies in edible commercial oils using thermal lens spectroscopy,” Superficies y vacío 18, 31–37 (2005).

Axelrod, D.

Baffou, G.

J. S. Donner, S. A. Thompson, M. P. Kreuzer, G. Baffou, and R. Quidant, “Mapping intracellular temperature using green fluorescent protein,” Nano Lett. 12, 2107–2111 (2012).
[Crossref]

G. Baffou, M. P. Kreuzer, F. Kulzer, and R. Quidant, “Temperature mapping near plasmonic nanostructures using fluorescence polarization anisotropy,” Opt. Express 17, 3291–3298 (2009).
[Crossref]

Bastos, A. E. P.

A. E. P. Bastos, S. Scolari, M. Stockl, and R. F. M. de Almeida, “Applications of fluorescence lifetime spectroscopy and imaging to lipid domains in vivo,” Imaging Spectroscopic Anal. Living Cells 504, 57–81 (2012).
[Crossref]

Bergaud, C.

P. Löw, B. Kim, N. Takama, and C. Bergaud, “High-spatial-resolution surface-temperature mapping using fluorescent thermometry,” Small 4, 908–914 (2008).
[Crossref]

H. F. Arata, P. Löw, K. Ishizuka, C. Bergaud, B. Kim, H. Noji, and H. Fujita, “Temperature distribution measurement on microfabricated thermodevice for single biomolecular observation using fluorescent dye,” Sensors Actuators B-Chem. 117, 339–345 (2006).
[Crossref]

Bogy, D. B.

W. Srituravanich, L. Pan, Y. Wang, C. Sun, D. B. Bogy, and X. Zhang, “Flying plasmonic lens in the near field for high-speed nanolithography,” Nat. Nanotechnol. 3, 733–737 (2008).
[Crossref]

Cao, W.

J. Wu, T. Y. Kwok, X. Li, W. Cao, Y. Wang, J. Huang, Y. Hong, D. Zhang, and W. Wen, “Mapping three-dimensional temperature in microfluidic chip,” Sci. Rep. 3, 3321 (2013).

Cheng, N. S.

N. S. Cheng, “Formula for the viscosity of a glycerol-water mixture,” Industrial Eng. Chem. Res. 47, 3285–3288 (2008).
[Crossref]

de Almeida, R. F. M.

A. E. P. Bastos, S. Scolari, M. Stockl, and R. F. M. de Almeida, “Applications of fluorescence lifetime spectroscopy and imaging to lipid domains in vivo,” Imaging Spectroscopic Anal. Living Cells 504, 57–81 (2012).
[Crossref]

deMello, A. J.

K. S. Elvira, X. C. i Solvas, R. C. R. Wootton, and A. J. deMello, “The past, present and potential for microfluidic reactor technology in chemical synthesis,” Nat. Chem. 5, 905–915 (2013).
[Crossref]

Donner, J. S.

J. S. Donner, S. A. Thompson, M. P. Kreuzer, G. Baffou, and R. Quidant, “Mapping intracellular temperature using green fluorescent protein,” Nano Lett. 12, 2107–2111 (2012).
[Crossref]

Duan, X.

W. Y. Shi, Z. Y. Sun, M. Wei, D. G. Evans, and X. Duan, “Tunable photoluminescence properties of fluorescein in a layered double hydroxide matrix by changing the interlayer microenvironment,” J. Phys. Chem. C 114, 21070–21076 (2010).
[Crossref]

Elvira, K. S.

K. S. Elvira, X. C. i Solvas, R. C. R. Wootton, and A. J. deMello, “The past, present and potential for microfluidic reactor technology in chemical synthesis,” Nat. Chem. 5, 905–915 (2013).
[Crossref]

Evans, D. G.

W. Y. Shi, Z. Y. Sun, M. Wei, D. G. Evans, and X. Duan, “Tunable photoluminescence properties of fluorescein in a layered double hydroxide matrix by changing the interlayer microenvironment,” J. Phys. Chem. C 114, 21070–21076 (2010).
[Crossref]

Flores-Farias, R.

J. M. Yáñez-Limón, R. Mayen-Mondragón, O. Martínez-Flores, R. Flores-Farias, F. Ruíz, and C. Araujo-Andrade, “Thermal diffusivity studies in edible commercial oils using thermal lens spectroscopy,” Superficies y vacío 18, 31–37 (2005).

Fujita, H.

H. F. Arata, P. Löw, K. Ishizuka, C. Bergaud, B. Kim, H. Noji, and H. Fujita, “Temperature distribution measurement on microfabricated thermodevice for single biomolecular observation using fluorescent dye,” Sensors Actuators B-Chem. 117, 339–345 (2006).
[Crossref]

Funatsu, T.

K. Okabe, N. Inada, C. Gota, Y. Harada, T. Funatsu, and S. Uchiyama, “Intracellular temperature mapping with a fluorescent polymeric thermometer and fluorescence lifetime imaging microscopy,” Nat. Commun. 3, 705 (2012).
[Crossref]

Gokulrangan, G.

J. R. Unruh, G. Gokulrangan, G. S. Wilson, and C. K. Johnson, “Fluorescence properties of fluorescein, tetramethyl-rhodamine and texas red linked to a DNA aptamer,” Photochem. Photobiol. 81, 682–690 (2005).
[Crossref]

Gota, C.

K. Okabe, N. Inada, C. Gota, Y. Harada, T. Funatsu, and S. Uchiyama, “Intracellular temperature mapping with a fluorescent polymeric thermometer and fluorescence lifetime imaging microscopy,” Nat. Commun. 3, 705 (2012).
[Crossref]

Harada, Y.

K. Okabe, N. Inada, C. Gota, Y. Harada, T. Funatsu, and S. Uchiyama, “Intracellular temperature mapping with a fluorescent polymeric thermometer and fluorescence lifetime imaging microscopy,” Nat. Commun. 3, 705 (2012).
[Crossref]

Hell, S. W.

S. W. Hell and M. Kroug, “Ground-state-depletion fluorscence microscopy: A concept for breaking the diffraction resolution limit,” Appl. Phys. B 60, 495–497 (1995).
[Crossref]

Hellen, E. H.

Hong, Y.

J. Wu, T. Y. Kwok, X. Li, W. Cao, Y. Wang, J. Huang, Y. Hong, D. Zhang, and W. Wen, “Mapping three-dimensional temperature in microfluidic chip,” Sci. Rep. 3, 3321 (2013).

Huang, J.

J. Wu, T. Y. Kwok, X. Li, W. Cao, Y. Wang, J. Huang, Y. Hong, D. Zhang, and W. Wen, “Mapping three-dimensional temperature in microfluidic chip,” Sci. Rep. 3, 3321 (2013).

Inada, N.

K. Okabe, N. Inada, C. Gota, Y. Harada, T. Funatsu, and S. Uchiyama, “Intracellular temperature mapping with a fluorescent polymeric thermometer and fluorescence lifetime imaging microscopy,” Nat. Commun. 3, 705 (2012).
[Crossref]

Ishizuka, K.

H. F. Arata, P. Löw, K. Ishizuka, C. Bergaud, B. Kim, H. Noji, and H. Fujita, “Temperature distribution measurement on microfabricated thermodevice for single biomolecular observation using fluorescent dye,” Sensors Actuators B-Chem. 117, 339–345 (2006).
[Crossref]

Johnson, C. K.

J. R. Unruh, G. Gokulrangan, G. S. Wilson, and C. K. Johnson, “Fluorescence properties of fluorescein, tetramethyl-rhodamine and texas red linked to a DNA aptamer,” Photochem. Photobiol. 81, 682–690 (2005).
[Crossref]

Kalytchuk, S.

S. Kalytchuk, O. Zhovtiuk, S. V. Kershaw, R. Zboril, and A. L. Rogach, “Temperature-dependent exciton and trap-related photoluminescence of CdTe quantum dots embedded in a NaCl matrix: implication in thermometry,” Small 12, 466–476 (2016).
[Crossref]

Kappe, C. O.

T. Razzaq and C. O. Kappe, “Continuous flow organic synthesis under high-temperature/pressure conditions,” Chem. 5, 1274–1289 (2010).

Kawski, A.

A. Kawski, “Fluorescence anisotropy - theory and applications of rotational depolarization,” Critical Rev. Analytical Chem. 23, 459–529 (1993).
[Crossref]

Kershaw, S. V.

S. Kalytchuk, O. Zhovtiuk, S. V. Kershaw, R. Zboril, and A. L. Rogach, “Temperature-dependent exciton and trap-related photoluminescence of CdTe quantum dots embedded in a NaCl matrix: implication in thermometry,” Small 12, 466–476 (2016).
[Crossref]

Khan, S.

A. Ali, S. Khan, and F. Nabi, “Volumetric, viscometric and refractive index behaviour of amino acids in aqueous glycerol at different temperatures,” J. Serbian Chem. Soc. 72, 495–512 (2007).
[Crossref]

Kim, B.

P. Löw, B. Kim, N. Takama, and C. Bergaud, “High-spatial-resolution surface-temperature mapping using fluorescent thermometry,” Small 4, 908–914 (2008).
[Crossref]

H. F. Arata, P. Löw, K. Ishizuka, C. Bergaud, B. Kim, H. Noji, and H. Fujita, “Temperature distribution measurement on microfabricated thermodevice for single biomolecular observation using fluorescent dye,” Sensors Actuators B-Chem. 117, 339–345 (2006).
[Crossref]

Knoll, W.

K. Vasilev, W. Knoll, and M. Kreiter, “Fluorescence intensities of chromophores in front of a thin metal film,” J. Chem. Phys. 120, 3439–3445 (2004).
[Crossref]

Kreiter, M.

K. Vasilev, W. Knoll, and M. Kreiter, “Fluorescence intensities of chromophores in front of a thin metal film,” J. Chem. Phys. 120, 3439–3445 (2004).
[Crossref]

Kreuzer, M. P.

J. S. Donner, S. A. Thompson, M. P. Kreuzer, G. Baffou, and R. Quidant, “Mapping intracellular temperature using green fluorescent protein,” Nano Lett. 12, 2107–2111 (2012).
[Crossref]

G. Baffou, M. P. Kreuzer, F. Kulzer, and R. Quidant, “Temperature mapping near plasmonic nanostructures using fluorescence polarization anisotropy,” Opt. Express 17, 3291–3298 (2009).
[Crossref]

Kroug, M.

S. W. Hell and M. Kroug, “Ground-state-depletion fluorscence microscopy: A concept for breaking the diffraction resolution limit,” Appl. Phys. B 60, 495–497 (1995).
[Crossref]

Kruusing, A.

A. Kruusing, “Underwater and water-assisted laser processing: Part 2–Etching, cutting and rarely used methods,” Opt. Lasers Eng. 41, 329–352 (2004).
[Crossref]

Kubo, M.

G. Kucsko, P. C. Maurer, N. Y. Yao, M. Kubo, H. J. Noh, P. K. Lo, H. Park, and M. D. Lukin, “Nanometre-scale thermometry in a living cell,” Nature 500, 54–57 (2013).
[Crossref]

Kucsko, G.

G. Kucsko, P. C. Maurer, N. Y. Yao, M. Kubo, H. J. Noh, P. K. Lo, H. Park, and M. D. Lukin, “Nanometre-scale thermometry in a living cell,” Nature 500, 54–57 (2013).
[Crossref]

Kulzer, F.

Kwok, T. Y.

J. Wu, T. Y. Kwok, X. Li, W. Cao, Y. Wang, J. Huang, Y. Hong, D. Zhang, and W. Wen, “Mapping three-dimensional temperature in microfluidic chip,” Sci. Rep. 3, 3321 (2013).

Li, X.

J. Wu, T. Y. Kwok, X. Li, W. Cao, Y. Wang, J. Huang, Y. Hong, D. Zhang, and W. Wen, “Mapping three-dimensional temperature in microfluidic chip,” Sci. Rep. 3, 3321 (2013).

Lo, P. K.

G. Kucsko, P. C. Maurer, N. Y. Yao, M. Kubo, H. J. Noh, P. K. Lo, H. Park, and M. D. Lukin, “Nanometre-scale thermometry in a living cell,” Nature 500, 54–57 (2013).
[Crossref]

Löw, P.

P. Löw, B. Kim, N. Takama, and C. Bergaud, “High-spatial-resolution surface-temperature mapping using fluorescent thermometry,” Small 4, 908–914 (2008).
[Crossref]

H. F. Arata, P. Löw, K. Ishizuka, C. Bergaud, B. Kim, H. Noji, and H. Fujita, “Temperature distribution measurement on microfabricated thermodevice for single biomolecular observation using fluorescent dye,” Sensors Actuators B-Chem. 117, 339–345 (2006).
[Crossref]

Lukin, M. D.

G. Kucsko, P. C. Maurer, N. Y. Yao, M. Kubo, H. J. Noh, P. K. Lo, H. Park, and M. D. Lukin, “Nanometre-scale thermometry in a living cell,” Nature 500, 54–57 (2013).
[Crossref]

Martínez-Flores, O.

J. M. Yáñez-Limón, R. Mayen-Mondragón, O. Martínez-Flores, R. Flores-Farias, F. Ruíz, and C. Araujo-Andrade, “Thermal diffusivity studies in edible commercial oils using thermal lens spectroscopy,” Superficies y vacío 18, 31–37 (2005).

Maurer, P. C.

G. Kucsko, P. C. Maurer, N. Y. Yao, M. Kubo, H. J. Noh, P. K. Lo, H. Park, and M. D. Lukin, “Nanometre-scale thermometry in a living cell,” Nature 500, 54–57 (2013).
[Crossref]

Mayen-Mondragón, R.

J. M. Yáñez-Limón, R. Mayen-Mondragón, O. Martínez-Flores, R. Flores-Farias, F. Ruíz, and C. Araujo-Andrade, “Thermal diffusivity studies in edible commercial oils using thermal lens spectroscopy,” Superficies y vacío 18, 31–37 (2005).

Nabi, F.

A. Ali, S. Khan, and F. Nabi, “Volumetric, viscometric and refractive index behaviour of amino acids in aqueous glycerol at different temperatures,” J. Serbian Chem. Soc. 72, 495–512 (2007).
[Crossref]

Noh, H. J.

G. Kucsko, P. C. Maurer, N. Y. Yao, M. Kubo, H. J. Noh, P. K. Lo, H. Park, and M. D. Lukin, “Nanometre-scale thermometry in a living cell,” Nature 500, 54–57 (2013).
[Crossref]

Noji, H.

H. F. Arata, P. Löw, K. Ishizuka, C. Bergaud, B. Kim, H. Noji, and H. Fujita, “Temperature distribution measurement on microfabricated thermodevice for single biomolecular observation using fluorescent dye,” Sensors Actuators B-Chem. 117, 339–345 (2006).
[Crossref]

Okabe, K.

K. Okabe, N. Inada, C. Gota, Y. Harada, T. Funatsu, and S. Uchiyama, “Intracellular temperature mapping with a fluorescent polymeric thermometer and fluorescence lifetime imaging microscopy,” Nat. Commun. 3, 705 (2012).
[Crossref]

Pan, L.

W. Srituravanich, L. Pan, Y. Wang, C. Sun, D. B. Bogy, and X. Zhang, “Flying plasmonic lens in the near field for high-speed nanolithography,” Nat. Nanotechnol. 3, 733–737 (2008).
[Crossref]

Park, H.

G. Kucsko, P. C. Maurer, N. Y. Yao, M. Kubo, H. J. Noh, P. K. Lo, H. Park, and M. D. Lukin, “Nanometre-scale thermometry in a living cell,” Nature 500, 54–57 (2013).
[Crossref]

Penzkofer, A.

A. Penzkofer and J. Wiedmann, “Orientation of transition dipole-moments of rhodamine-6G determined by excited-state absorption,” Opt. Commun. 35, 81–86 (1980).
[Crossref]

Periasamy, A.

H. Wallrabe and A. Periasamy, “Imaging protein molecules using FRET and FLIM microscopy,” Current Opinion Biotechnol. 16, 19–27 (2005).
[Crossref]

Quidant, R.

J. S. Donner, S. A. Thompson, M. P. Kreuzer, G. Baffou, and R. Quidant, “Mapping intracellular temperature using green fluorescent protein,” Nano Lett. 12, 2107–2111 (2012).
[Crossref]

G. Baffou, M. P. Kreuzer, F. Kulzer, and R. Quidant, “Temperature mapping near plasmonic nanostructures using fluorescence polarization anisotropy,” Opt. Express 17, 3291–3298 (2009).
[Crossref]

Razzaq, T.

T. Razzaq and C. O. Kappe, “Continuous flow organic synthesis under high-temperature/pressure conditions,” Chem. 5, 1274–1289 (2010).

Rogach, A. L.

S. Kalytchuk, O. Zhovtiuk, S. V. Kershaw, R. Zboril, and A. L. Rogach, “Temperature-dependent exciton and trap-related photoluminescence of CdTe quantum dots embedded in a NaCl matrix: implication in thermometry,” Small 12, 466–476 (2016).
[Crossref]

Ruíz, F.

J. M. Yáñez-Limón, R. Mayen-Mondragón, O. Martínez-Flores, R. Flores-Farias, F. Ruíz, and C. Araujo-Andrade, “Thermal diffusivity studies in edible commercial oils using thermal lens spectroscopy,” Superficies y vacío 18, 31–37 (2005).

Scolari, S.

A. E. P. Bastos, S. Scolari, M. Stockl, and R. F. M. de Almeida, “Applications of fluorescence lifetime spectroscopy and imaging to lipid domains in vivo,” Imaging Spectroscopic Anal. Living Cells 504, 57–81 (2012).
[Crossref]

Shi, W. Y.

W. Y. Shi, Z. Y. Sun, M. Wei, D. G. Evans, and X. Duan, “Tunable photoluminescence properties of fluorescein in a layered double hydroxide matrix by changing the interlayer microenvironment,” J. Phys. Chem. C 114, 21070–21076 (2010).
[Crossref]

Solvas, X. C. i

K. S. Elvira, X. C. i Solvas, R. C. R. Wootton, and A. J. deMello, “The past, present and potential for microfluidic reactor technology in chemical synthesis,” Nat. Chem. 5, 905–915 (2013).
[Crossref]

Srituravanich, W.

W. Srituravanich, L. Pan, Y. Wang, C. Sun, D. B. Bogy, and X. Zhang, “Flying plasmonic lens in the near field for high-speed nanolithography,” Nat. Nanotechnol. 3, 733–737 (2008).
[Crossref]

Stockl, M.

A. E. P. Bastos, S. Scolari, M. Stockl, and R. F. M. de Almeida, “Applications of fluorescence lifetime spectroscopy and imaging to lipid domains in vivo,” Imaging Spectroscopic Anal. Living Cells 504, 57–81 (2012).
[Crossref]

Sun, C.

W. Srituravanich, L. Pan, Y. Wang, C. Sun, D. B. Bogy, and X. Zhang, “Flying plasmonic lens in the near field for high-speed nanolithography,” Nat. Nanotechnol. 3, 733–737 (2008).
[Crossref]

Sun, Z. Y.

W. Y. Shi, Z. Y. Sun, M. Wei, D. G. Evans, and X. Duan, “Tunable photoluminescence properties of fluorescein in a layered double hydroxide matrix by changing the interlayer microenvironment,” J. Phys. Chem. C 114, 21070–21076 (2010).
[Crossref]

Takama, N.

P. Löw, B. Kim, N. Takama, and C. Bergaud, “High-spatial-resolution surface-temperature mapping using fluorescent thermometry,” Small 4, 908–914 (2008).
[Crossref]

Thompson, S. A.

J. S. Donner, S. A. Thompson, M. P. Kreuzer, G. Baffou, and R. Quidant, “Mapping intracellular temperature using green fluorescent protein,” Nano Lett. 12, 2107–2111 (2012).
[Crossref]

Uchiyama, S.

K. Okabe, N. Inada, C. Gota, Y. Harada, T. Funatsu, and S. Uchiyama, “Intracellular temperature mapping with a fluorescent polymeric thermometer and fluorescence lifetime imaging microscopy,” Nat. Commun. 3, 705 (2012).
[Crossref]

Unruh, J. R.

J. R. Unruh, G. Gokulrangan, G. S. Wilson, and C. K. Johnson, “Fluorescence properties of fluorescein, tetramethyl-rhodamine and texas red linked to a DNA aptamer,” Photochem. Photobiol. 81, 682–690 (2005).
[Crossref]

Vasilev, K.

K. Vasilev, W. Knoll, and M. Kreiter, “Fluorescence intensities of chromophores in front of a thin metal film,” J. Chem. Phys. 120, 3439–3445 (2004).
[Crossref]

Wallrabe, H.

H. Wallrabe and A. Periasamy, “Imaging protein molecules using FRET and FLIM microscopy,” Current Opinion Biotechnol. 16, 19–27 (2005).
[Crossref]

Wang, X.

Y. Yue and X. Wang, “Nanoscale thermal probing,” Nano Rev. 3, 1–11 (2012).
[Crossref]

Wang, Y.

J. Wu, T. Y. Kwok, X. Li, W. Cao, Y. Wang, J. Huang, Y. Hong, D. Zhang, and W. Wen, “Mapping three-dimensional temperature in microfluidic chip,” Sci. Rep. 3, 3321 (2013).

W. Srituravanich, L. Pan, Y. Wang, C. Sun, D. B. Bogy, and X. Zhang, “Flying plasmonic lens in the near field for high-speed nanolithography,” Nat. Nanotechnol. 3, 733–737 (2008).
[Crossref]

Wei, M.

W. Y. Shi, Z. Y. Sun, M. Wei, D. G. Evans, and X. Duan, “Tunable photoluminescence properties of fluorescein in a layered double hydroxide matrix by changing the interlayer microenvironment,” J. Phys. Chem. C 114, 21070–21076 (2010).
[Crossref]

Wen, W.

J. Wu, T. Y. Kwok, X. Li, W. Cao, Y. Wang, J. Huang, Y. Hong, D. Zhang, and W. Wen, “Mapping three-dimensional temperature in microfluidic chip,” Sci. Rep. 3, 3321 (2013).

Wiedmann, J.

A. Penzkofer and J. Wiedmann, “Orientation of transition dipole-moments of rhodamine-6G determined by excited-state absorption,” Opt. Commun. 35, 81–86 (1980).
[Crossref]

Wilson, G. S.

J. R. Unruh, G. Gokulrangan, G. S. Wilson, and C. K. Johnson, “Fluorescence properties of fluorescein, tetramethyl-rhodamine and texas red linked to a DNA aptamer,” Photochem. Photobiol. 81, 682–690 (2005).
[Crossref]

Wootton, R. C. R.

K. S. Elvira, X. C. i Solvas, R. C. R. Wootton, and A. J. deMello, “The past, present and potential for microfluidic reactor technology in chemical synthesis,” Nat. Chem. 5, 905–915 (2013).
[Crossref]

Wu, J.

J. Wu, T. Y. Kwok, X. Li, W. Cao, Y. Wang, J. Huang, Y. Hong, D. Zhang, and W. Wen, “Mapping three-dimensional temperature in microfluidic chip,” Sci. Rep. 3, 3321 (2013).

Yáñez-Limón, J. M.

J. M. Yáñez-Limón, R. Mayen-Mondragón, O. Martínez-Flores, R. Flores-Farias, F. Ruíz, and C. Araujo-Andrade, “Thermal diffusivity studies in edible commercial oils using thermal lens spectroscopy,” Superficies y vacío 18, 31–37 (2005).

Yao, N. Y.

G. Kucsko, P. C. Maurer, N. Y. Yao, M. Kubo, H. J. Noh, P. K. Lo, H. Park, and M. D. Lukin, “Nanometre-scale thermometry in a living cell,” Nature 500, 54–57 (2013).
[Crossref]

Yue, Y.

Y. Yue and X. Wang, “Nanoscale thermal probing,” Nano Rev. 3, 1–11 (2012).
[Crossref]

Zboril, R.

S. Kalytchuk, O. Zhovtiuk, S. V. Kershaw, R. Zboril, and A. L. Rogach, “Temperature-dependent exciton and trap-related photoluminescence of CdTe quantum dots embedded in a NaCl matrix: implication in thermometry,” Small 12, 466–476 (2016).
[Crossref]

Zhang, D.

J. Wu, T. Y. Kwok, X. Li, W. Cao, Y. Wang, J. Huang, Y. Hong, D. Zhang, and W. Wen, “Mapping three-dimensional temperature in microfluidic chip,” Sci. Rep. 3, 3321 (2013).

Zhang, X.

W. Srituravanich, L. Pan, Y. Wang, C. Sun, D. B. Bogy, and X. Zhang, “Flying plasmonic lens in the near field for high-speed nanolithography,” Nat. Nanotechnol. 3, 733–737 (2008).
[Crossref]

Zhovtiuk, O.

S. Kalytchuk, O. Zhovtiuk, S. V. Kershaw, R. Zboril, and A. L. Rogach, “Temperature-dependent exciton and trap-related photoluminescence of CdTe quantum dots embedded in a NaCl matrix: implication in thermometry,” Small 12, 466–476 (2016).
[Crossref]

Appl. Phys. B (1)

S. W. Hell and M. Kroug, “Ground-state-depletion fluorscence microscopy: A concept for breaking the diffraction resolution limit,” Appl. Phys. B 60, 495–497 (1995).
[Crossref]

Chem. (1)

T. Razzaq and C. O. Kappe, “Continuous flow organic synthesis under high-temperature/pressure conditions,” Chem. 5, 1274–1289 (2010).

Critical Rev. Analytical Chem. (1)

A. Kawski, “Fluorescence anisotropy - theory and applications of rotational depolarization,” Critical Rev. Analytical Chem. 23, 459–529 (1993).
[Crossref]

Current Opinion Biotechnol. (1)

H. Wallrabe and A. Periasamy, “Imaging protein molecules using FRET and FLIM microscopy,” Current Opinion Biotechnol. 16, 19–27 (2005).
[Crossref]

Imaging Spectroscopic Anal. Living Cells (1)

A. E. P. Bastos, S. Scolari, M. Stockl, and R. F. M. de Almeida, “Applications of fluorescence lifetime spectroscopy and imaging to lipid domains in vivo,” Imaging Spectroscopic Anal. Living Cells 504, 57–81 (2012).
[Crossref]

Industrial Eng. Chem. Res. (1)

N. S. Cheng, “Formula for the viscosity of a glycerol-water mixture,” Industrial Eng. Chem. Res. 47, 3285–3288 (2008).
[Crossref]

J. Chem. Phys. (1)

K. Vasilev, W. Knoll, and M. Kreiter, “Fluorescence intensities of chromophores in front of a thin metal film,” J. Chem. Phys. 120, 3439–3445 (2004).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Phys. Chem. C (1)

W. Y. Shi, Z. Y. Sun, M. Wei, D. G. Evans, and X. Duan, “Tunable photoluminescence properties of fluorescein in a layered double hydroxide matrix by changing the interlayer microenvironment,” J. Phys. Chem. C 114, 21070–21076 (2010).
[Crossref]

J. Serbian Chem. Soc. (1)

A. Ali, S. Khan, and F. Nabi, “Volumetric, viscometric and refractive index behaviour of amino acids in aqueous glycerol at different temperatures,” J. Serbian Chem. Soc. 72, 495–512 (2007).
[Crossref]

Nano Lett. (1)

J. S. Donner, S. A. Thompson, M. P. Kreuzer, G. Baffou, and R. Quidant, “Mapping intracellular temperature using green fluorescent protein,” Nano Lett. 12, 2107–2111 (2012).
[Crossref]

Nano Rev. (1)

Y. Yue and X. Wang, “Nanoscale thermal probing,” Nano Rev. 3, 1–11 (2012).
[Crossref]

Nat. Chem. (1)

K. S. Elvira, X. C. i Solvas, R. C. R. Wootton, and A. J. deMello, “The past, present and potential for microfluidic reactor technology in chemical synthesis,” Nat. Chem. 5, 905–915 (2013).
[Crossref]

Nat. Commun. (1)

K. Okabe, N. Inada, C. Gota, Y. Harada, T. Funatsu, and S. Uchiyama, “Intracellular temperature mapping with a fluorescent polymeric thermometer and fluorescence lifetime imaging microscopy,” Nat. Commun. 3, 705 (2012).
[Crossref]

Nat. Nanotechnol. (1)

W. Srituravanich, L. Pan, Y. Wang, C. Sun, D. B. Bogy, and X. Zhang, “Flying plasmonic lens in the near field for high-speed nanolithography,” Nat. Nanotechnol. 3, 733–737 (2008).
[Crossref]

Nature (1)

G. Kucsko, P. C. Maurer, N. Y. Yao, M. Kubo, H. J. Noh, P. K. Lo, H. Park, and M. D. Lukin, “Nanometre-scale thermometry in a living cell,” Nature 500, 54–57 (2013).
[Crossref]

Opt. Commun. (1)

A. Penzkofer and J. Wiedmann, “Orientation of transition dipole-moments of rhodamine-6G determined by excited-state absorption,” Opt. Commun. 35, 81–86 (1980).
[Crossref]

Opt. Express (1)

Opt. Lasers Eng. (1)

A. Kruusing, “Underwater and water-assisted laser processing: Part 2–Etching, cutting and rarely used methods,” Opt. Lasers Eng. 41, 329–352 (2004).
[Crossref]

Photochem. Photobiol. (1)

J. R. Unruh, G. Gokulrangan, G. S. Wilson, and C. K. Johnson, “Fluorescence properties of fluorescein, tetramethyl-rhodamine and texas red linked to a DNA aptamer,” Photochem. Photobiol. 81, 682–690 (2005).
[Crossref]

Sci. Rep. (1)

J. Wu, T. Y. Kwok, X. Li, W. Cao, Y. Wang, J. Huang, Y. Hong, D. Zhang, and W. Wen, “Mapping three-dimensional temperature in microfluidic chip,” Sci. Rep. 3, 3321 (2013).

Sensors Actuators B-Chem. (1)

H. F. Arata, P. Löw, K. Ishizuka, C. Bergaud, B. Kim, H. Noji, and H. Fujita, “Temperature distribution measurement on microfabricated thermodevice for single biomolecular observation using fluorescent dye,” Sensors Actuators B-Chem. 117, 339–345 (2006).
[Crossref]

Small (2)

S. Kalytchuk, O. Zhovtiuk, S. V. Kershaw, R. Zboril, and A. L. Rogach, “Temperature-dependent exciton and trap-related photoluminescence of CdTe quantum dots embedded in a NaCl matrix: implication in thermometry,” Small 12, 466–476 (2016).
[Crossref]

P. Löw, B. Kim, N. Takama, and C. Bergaud, “High-spatial-resolution surface-temperature mapping using fluorescent thermometry,” Small 4, 908–914 (2008).
[Crossref]

Superficies y vacío (1)

J. M. Yáñez-Limón, R. Mayen-Mondragón, O. Martínez-Flores, R. Flores-Farias, F. Ruíz, and C. Araujo-Andrade, “Thermal diffusivity studies in edible commercial oils using thermal lens spectroscopy,” Superficies y vacío 18, 31–37 (2005).

Other (1)

Association, Physical Properties of Glycerol and its Solutions (Glycerine Producers’ Association, 1963).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1 Schematic of working principle and experimental setup of 2NA technique. (a) Excitation, thermal rotation and emission of fluorescent molecules in solution. Molecules in the solution are excited using dark field illumination and rotate before emitting fluorescence under a temperature gradient generated by a heater. CS: coverslip; HT: heater; SL: solution. (b) Effect of thermal rotation on emission: at low/large temperature, rotation angle of the molecule is small/large before emission. τF is the lifetime of transition dipole moment. (c) 2NA setup: experimental configuration for simultaneous collection of fluorescence using two different numerical apertures (NAs). One beam path is limited by a limiting aperture at the conjugate plane of pupil plane to have a small imaging NA. Temperature map is calculated based the ratio between fluorescence intensities collected by two CCDs. OBJ: objective; M: mirror; TL: tube lens; BS: beamsplitter; LA: limiting aperture; CCD: charge-coupled device (camera); PP: pupil plane; CPP: conjugate pupil plane.
Fig. 2
Fig. 2 Optical characterization of fluorescein in glycerol. (a) Experimental configuration to study emission spectra of fluorescein in glycerol. Polarization of incident light is controlled by a half-wave plate and the emission is collected using a spectrometer. Inset: chemical structure of fluorescein. (b) Excitation and emission spectra of fluorescein. (c) Spectrum peak intensity as a function of rotation angle of the half-wave plate. When the detection angle is at 90 °, the peak intensity has a sinusoidal oscillation. While the peak intensity is a constant when the detection angle is at an angle smaller than 20 °.
Fig. 3
Fig. 3 Characterization of fabricated heater. (a) Scanning electron microscope (SEM) image of aluminum heater on glass substrate. Scale bar: 20 μm. (b) Enlarged view of the edge of the heater. Scale bar: 1 μm. (c) Height profile across the edge of the heater measured using atomic force microscopy (AFM). (d) Schematic of cross-sectional view of the simulation model (blocks are not to scale). The dotted line indicates the line of symmetry. (e) Temperature profiles along a line right underneath the heater at different time instants. (f) Temperature distribution inside solution at time instant 35 ms.
Fig. 4
Fig. 4 Comparison between ratiometric measurement and intensity measurement. (a) Time sequences of heating, illumination and CCD exposure. Each measurement cycle contains five heating pulses, five illumination pulses and one CCD exposure. Insets: enlarged view of time sequence of heating (upper panel) and illumination (middle panel) within one CCD exposure. (b) Simultaneously measured intensities from CCD1 (left axis) and CCD2 (right axis) under conditions indicated in panel a. (c) Calculated ratio between simultaneously collected fluorescence intensities shown in panel b. Inset: A fluorescence image. The solid rectangular highlights the area on the heater (25 in width)that is used to extract the information of intensity and ratio.
Fig. 5
Fig. 5 Map of ratio change and the corresponding map of temperature. (a) 2D map of intensity of two CCDs. Two dotted lines divide each panel into three regions: glass (I), edge (II) and metal (III). Scale bar: 3 μm. (b) Calibration curves of ratio change as a function of temperature for glass and metal surfaces. (c) 2D experimental map of ratio change and temperature converted from panel a based on the calibration curves in panel b. (d) Temperature profiles on glass along the direction perpendicular to the boundary between glass and metal film. Length is measured from left to right. Trend lines are plotted to aid visualization.
Fig. 6
Fig. 6 Theoretical study of 2NA method. (a) Orientation annotation: observation plane (θ, ϕ); emission dipole plane (θ′, ϕ′); rotational cone of emission dipole (θ″). z0 is the height of dipole μ measured from the bottom surface of the heated coverslip. (b) The normalized total emitted power of a dipole as a function of height z0 in fixed-amplitude model at emission wavelength equaling to 518 nm. The normalization value is the same asymptotic value as z0 → ∞ for dipoles oriented both parallel (horizontal dipole) and perpendicularly (vertical dipole) to the interface. (c) The ratio change as a function of temperature for a dipole in free space. Vertical excitation means excitation field E is along interface normal while tilted excitation corresponds to the case where the angle between E and interface normal is π/6.

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

μ = μ cos θ z ^ + μ sin θ cos ( ϕ ϕ ) p ^ + μ sin θ ( ϕ ϕ ) s ^
I = n 2 Z 0 [ cos 2 θ | E μ z | 2 + 1 2 sin 2 θ ( | E μ p | 2 + | E μ s | 2 ) ]
F = [ cos 2 θ η z + 1 2 sin 2 θ ( η p + η s ) ] exp ( t τ F )
P t = D r [ 1 sin θ θ ( sin θ P θ ) ]
6 D r = k B T V ν ( T ) 1 τ R
cos 2 θ = cos 2 θ e cos 2 θ + 1 2 sin 2 θ e sin 2 θ
F ( z 0 , t ) = ( 3 cos 2 θ e 1 ) 2 ( 2 η z η p η s ) 3 exp [ ( 1 τ R + 1 τ F ) t ] + ( η p + η s + η z ) 3 exp ( t τ F )
F ¯ ( z 0 ) = [ ( 3 cos 2 θ e 1 ) 2 ( 2 η z η p η s ) 3 1 1 + τ F / τ R + ( η p + η s + η z ) 3 ] τ F
1 Δ r = A 1 Δ g + B

Metrics