Abstract

We propose a novel digital holographic microscopy (DHM) by integrating surface plasmon holographic microscopy (SPHM) with reflection DHM based on the angular and polarization multiplexing techniques. Taking advantages of the high sensitivity of surface plasmon resonance (SPR) and the high reflectivity of gold film, the tiny variations of specimen’s refractive index (RI) can be measured by using SPHM, and meanwhile, the thickness changes of the specimen can be determined by means of reflection DHM. We experimentally monitor the volatilization process of an alcohol-water mixture droplet to verify the validity of the integrated DHM. The proposed microscopy is very promising in the objective-coupling SPR microscopy for multi-information measurements of diverse specimens with low-contrast RI distributions (biomolecules, nanofluids, etc.) in a dynamic and nondestructive way.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]

2018 (2)

H. Lu, X. Gan, D. Mao, B. Jia, and J. Zhao, “Flexibly tunable high-quality-factor induced transparency in plasmonic systems,” Sci. Rep. 8(1), 1558 (2018).
[Crossref] [PubMed]

J. Zhang, S. Dai, J. Zhong, T. Xi, C. Ma, Y. Li, J. Di, and J. Zhao, “Wavelength-multiplexing surface plasmon holographic microscopy,” Opt. Express 26(10), 13549–13560 (2018).
[Crossref] [PubMed]

2017 (3)

2016 (3)

2012 (3)

2011 (2)

Y. H. Huang, H. P. Ho, S. Y. Wu, S. K. Kong, W. W. Wong, and P. Shum, “Phase sensitive SPR sensor for wide dynamic range detection,” Opt. Lett. 36(20), 4092–4094 (2011).
[Crossref] [PubMed]

A. R. Halpern, Y. Chen, R. M. Corn, and D. Kim, “Surface plasmon resonance phase imaging measurements of patterned monolayers and DNA adsorption onto microarrays,” Anal. Chem. 83(7), 2801–2806 (2011).
[Crossref] [PubMed]

2010 (3)

C. Hu, J. Zhong, and J. Weng, “Digital holographic microscopy by use of surface plasmon resonance for imaging of cell membranes,” J. Biomed. Opt. 15(5), 056015 (2010).
[Crossref] [PubMed]

Y. Yanase, T. Hiragun, S. Kaneko, H. J. Gould, M. W. Greaves, and M. Hide, “Detection of refractive index changes in individual living cells by means of surface plasmon resonance imaging,” Biosens. Bioelectron. 26(2), 674–681 (2010).
[Crossref] [PubMed]

I. Kim and K. D. Kihm, “Measuring near-field nanoparticle concentration profiles by correlating surface plasmon resonance reflectance with effective refractive index of nanofluids,” Opt. Lett. 35(3), 393–395 (2010).
[Crossref] [PubMed]

2009 (3)

2008 (2)

J. Y. Lee, T. K. Chou, and H. C. Shih, “Polarization-interferometric surface-plasmon-resonance imaging system,” Opt. Lett. 33(5), 434–436 (2008).
[Crossref] [PubMed]

J. Homola, “Surface plasmon resonance sensors for detection of chemical and biological species,” Chem. Rev. 108(2), 462–493 (2008).
[Crossref] [PubMed]

2007 (2)

K. S. Phillips and Q. Cheng, “Recent advances in surface plasmon resonance based techniques for bioanalysis,” Anal. Bioanal. Chem. 387(5), 1831–1840 (2007).
[Crossref] [PubMed]

W. Yuan, H. P. Ho, C. L. Wong, S. K. Kong, and C. Lin, “Surface plasmon resonance biosensor incorporated in a Michelson interferometer with enhanced sensitivity,” IEEE Sens. J. 7(1), 70–73 (2007).
[Crossref]

2006 (1)

2005 (1)

2004 (1)

G. Steiner, “Surface plasmon resonance imaging,” Anal. Bioanal. Chem. 379(3), 328–331 (2004).
[Crossref] [PubMed]

Ash, W. M.

Chen, C.

L. Ji, X. Sun, G. He, Y. Liu, X. Wang, Y. Yi, C. Chen, F. Wang, and D. Zhang, “Surface plasmon resonance refractive index sensor based on ultraviolet bleached polymer waveguide,” Sens. Actuators B Chem. 244, 373–379 (2017).
[Crossref]

Chen, Y.

A. R. Halpern, Y. Chen, R. M. Corn, and D. Kim, “Surface plasmon resonance phase imaging measurements of patterned monolayers and DNA adsorption onto microarrays,” Anal. Chem. 83(7), 2801–2806 (2011).
[Crossref] [PubMed]

Cheng, Q.

K. S. Phillips and Q. Cheng, “Recent advances in surface plasmon resonance based techniques for bioanalysis,” Anal. Bioanal. Chem. 387(5), 1831–1840 (2007).
[Crossref] [PubMed]

Cheon, S.

Chou, T. K.

Colomb, T.

Corn, R. M.

A. R. Halpern, Y. Chen, R. M. Corn, and D. Kim, “Surface plasmon resonance phase imaging measurements of patterned monolayers and DNA adsorption onto microarrays,” Anal. Chem. 83(7), 2801–2806 (2011).
[Crossref] [PubMed]

Cuche, E.

Dai, Q.

Dai, S.

Depeursinge, C.

Di, J.

J. Zhang, S. Dai, J. Zhong, T. Xi, C. Ma, Y. Li, J. Di, and J. Zhao, “Wavelength-multiplexing surface plasmon holographic microscopy,” Opt. Express 26(10), 13549–13560 (2018).
[Crossref] [PubMed]

J. Zhang, S. Dai, C. Ma, J. Di, and J. Zhao, “Compact surface plasmon holographic microscopy for near-field film mapping,” Opt. Lett. 42(17), 3462–3465 (2017).
[Crossref] [PubMed]

J. Zhang, S. Dai, C. Ma, J. Di, and J. Zhao, “Common-path digital holographic microscopy for near-field phase imaging based on surface plasmon resonance,” Appl. Opt. 56(11), 3223–3228 (2017).
[Crossref] [PubMed]

C. Ma, J. Di, J. Zhang, Y. Li, T. Xi, E. Li, and J. Zhao, “Simultaneous measurement of refractive index distribution and topography by integrated transmission and reflection digital holographic microscopy,” Appl. Opt. 55(33), 9435–9439 (2016).
[Crossref] [PubMed]

J. Zhang, C. Ma, S. Dai, J. Di, Y. Li, T. Xi, and J. Zhao, “Transmission and total internal reflection integrated digital holographic microscopy,” Opt. Lett. 41(16), 3844–3847 (2016).
[Crossref] [PubMed]

J. Wang, J. Zhao, C. Qin, J. Di, A. Rauf, and H. Jiang, “Digital holographic interferometry based on wavelength and angular multiplexing for measuring the ternary diffusion,” Opt. Lett. 37(7), 1211–1213 (2012).
[Crossref] [PubMed]

W. Sun, J. Zhao, J. Di, Q. Wang, and L. Wang, “Real-time visualization of Karman vortex street in water flow field by using digital holography,” Opt. Express 17(22), 20342–20348 (2009).
[Crossref] [PubMed]

Emery, Y.

Gan, X.

H. Lu, X. Gan, D. Mao, B. Jia, and J. Zhao, “Flexibly tunable high-quality-factor induced transparency in plasmonic systems,” Sci. Rep. 8(1), 1558 (2018).
[Crossref] [PubMed]

Gould, H. J.

Y. Yanase, T. Hiragun, S. Kaneko, H. J. Gould, M. W. Greaves, and M. Hide, “Detection of refractive index changes in individual living cells by means of surface plasmon resonance imaging,” Biosens. Bioelectron. 26(2), 674–681 (2010).
[Crossref] [PubMed]

Granqvist, N.

Greaves, M. W.

Y. Yanase, T. Hiragun, S. Kaneko, H. J. Gould, M. W. Greaves, and M. Hide, “Detection of refractive index changes in individual living cells by means of surface plasmon resonance imaging,” Biosens. Bioelectron. 26(2), 674–681 (2010).
[Crossref] [PubMed]

Grigorenko, A. N.

Halpern, A. R.

A. R. Halpern, Y. Chen, R. M. Corn, and D. Kim, “Surface plasmon resonance phase imaging measurements of patterned monolayers and DNA adsorption onto microarrays,” Anal. Chem. 83(7), 2801–2806 (2011).
[Crossref] [PubMed]

He, G.

L. Ji, X. Sun, G. He, Y. Liu, X. Wang, Y. Yi, C. Chen, F. Wang, and D. Zhang, “Surface plasmon resonance refractive index sensor based on ultraviolet bleached polymer waveguide,” Sens. Actuators B Chem. 244, 373–379 (2017).
[Crossref]

Hide, M.

Y. Yanase, T. Hiragun, S. Kaneko, H. J. Gould, M. W. Greaves, and M. Hide, “Detection of refractive index changes in individual living cells by means of surface plasmon resonance imaging,” Biosens. Bioelectron. 26(2), 674–681 (2010).
[Crossref] [PubMed]

Hiragun, T.

Y. Yanase, T. Hiragun, S. Kaneko, H. J. Gould, M. W. Greaves, and M. Hide, “Detection of refractive index changes in individual living cells by means of surface plasmon resonance imaging,” Biosens. Bioelectron. 26(2), 674–681 (2010).
[Crossref] [PubMed]

Ho, H. P.

Y. H. Huang, H. P. Ho, S. Y. Wu, S. K. Kong, W. W. Wong, and P. Shum, “Phase sensitive SPR sensor for wide dynamic range detection,” Opt. Lett. 36(20), 4092–4094 (2011).
[Crossref] [PubMed]

W. Yuan, H. P. Ho, C. L. Wong, S. K. Kong, and C. Lin, “Surface plasmon resonance biosensor incorporated in a Michelson interferometer with enhanced sensitivity,” IEEE Sens. J. 7(1), 70–73 (2007).
[Crossref]

Homola, J.

J. Homola, “Surface plasmon resonance sensors for detection of chemical and biological species,” Chem. Rev. 108(2), 462–493 (2008).
[Crossref] [PubMed]

Hong, B. H.

Hu, C.

C. Hu, J. Zhong, and J. Weng, “Digital holographic microscopy by use of surface plasmon resonance for imaging of cell membranes,” J. Biomed. Opt. 15(5), 056015 (2010).
[Crossref] [PubMed]

Huang, Y. H.

Ji, L.

L. Ji, X. Sun, G. He, Y. Liu, X. Wang, Y. Yi, C. Chen, F. Wang, and D. Zhang, “Surface plasmon resonance refractive index sensor based on ultraviolet bleached polymer waveguide,” Sens. Actuators B Chem. 244, 373–379 (2017).
[Crossref]

Jia, B.

H. Lu, X. Gan, D. Mao, B. Jia, and J. Zhao, “Flexibly tunable high-quality-factor induced transparency in plasmonic systems,” Sci. Rep. 8(1), 1558 (2018).
[Crossref] [PubMed]

Jiang, H.

Jussila, H.

Kabashin, A. V.

Kaneko, S.

Y. Yanase, T. Hiragun, S. Kaneko, H. J. Gould, M. W. Greaves, and M. Hide, “Detection of refractive index changes in individual living cells by means of surface plasmon resonance imaging,” Biosens. Bioelectron. 26(2), 674–681 (2010).
[Crossref] [PubMed]

Kihm, K. D.

Kim, D.

A. R. Halpern, Y. Chen, R. M. Corn, and D. Kim, “Surface plasmon resonance phase imaging measurements of patterned monolayers and DNA adsorption onto microarrays,” Anal. Chem. 83(7), 2801–2806 (2011).
[Crossref] [PubMed]

Kim, H.

Kim, I.

Kim, M. K.

Kong, S. K.

Y. H. Huang, H. P. Ho, S. Y. Wu, S. K. Kong, W. W. Wong, and P. Shum, “Phase sensitive SPR sensor for wide dynamic range detection,” Opt. Lett. 36(20), 4092–4094 (2011).
[Crossref] [PubMed]

W. Yuan, H. P. Ho, C. L. Wong, S. K. Kong, and C. Lin, “Surface plasmon resonance biosensor incorporated in a Michelson interferometer with enhanced sensitivity,” IEEE Sens. J. 7(1), 70–73 (2007).
[Crossref]

Krzewina, L.

Lee, B. J.

Lee, J. S.

Lee, J. Y.

Li, E.

Li, S.

Li, Y.

Lin, C.

W. Yuan, H. P. Ho, C. L. Wong, S. K. Kong, and C. Lin, “Surface plasmon resonance biosensor incorporated in a Michelson interferometer with enhanced sensitivity,” IEEE Sens. J. 7(1), 70–73 (2007).
[Crossref]

Liu, Y.

L. Ji, X. Sun, G. He, Y. Liu, X. Wang, Y. Yi, C. Chen, F. Wang, and D. Zhang, “Surface plasmon resonance refractive index sensor based on ultraviolet bleached polymer waveguide,” Sens. Actuators B Chem. 244, 373–379 (2017).
[Crossref]

Lu, H.

H. Lu, X. Gan, D. Mao, B. Jia, and J. Zhao, “Flexibly tunable high-quality-factor induced transparency in plasmonic systems,” Sci. Rep. 8(1), 1558 (2018).
[Crossref] [PubMed]

Ma, C.

Magistretti, P. J.

Mao, D.

H. Lu, X. Gan, D. Mao, B. Jia, and J. Zhao, “Flexibly tunable high-quality-factor induced transparency in plasmonic systems,” Sci. Rep. 8(1), 1558 (2018).
[Crossref] [PubMed]

Marquet, P.

Park, J. S.

Patskovsky, S.

Phillips, K. S.

K. S. Phillips and Q. Cheng, “Recent advances in surface plasmon resonance based techniques for bioanalysis,” Anal. Bioanal. Chem. 387(5), 1831–1840 (2007).
[Crossref] [PubMed]

Qin, C.

Rappaz, B.

Rauf, A.

Shih, H. C.

Shum, P.

Steiner, G.

G. Steiner, “Surface plasmon resonance imaging,” Anal. Bioanal. Chem. 379(3), 328–331 (2004).
[Crossref] [PubMed]

Sun, W.

Sun, X.

L. Ji, X. Sun, G. He, Y. Liu, X. Wang, Y. Yi, C. Chen, F. Wang, and D. Zhang, “Surface plasmon resonance refractive index sensor based on ultraviolet bleached polymer waveguide,” Sens. Actuators B Chem. 244, 373–379 (2017).
[Crossref]

Sun, Z.

Wang, F.

L. Ji, X. Sun, G. He, Y. Liu, X. Wang, Y. Yi, C. Chen, F. Wang, and D. Zhang, “Surface plasmon resonance refractive index sensor based on ultraviolet bleached polymer waveguide,” Sens. Actuators B Chem. 244, 373–379 (2017).
[Crossref]

Wang, G. P.

Wang, J.

Wang, L.

Wang, Q.

Wang, X.

L. Ji, X. Sun, G. He, Y. Liu, X. Wang, Y. Yi, C. Chen, F. Wang, and D. Zhang, “Surface plasmon resonance refractive index sensor based on ultraviolet bleached polymer waveguide,” Sens. Actuators B Chem. 244, 373–379 (2017).
[Crossref]

Weng, J.

C. Hu, J. Zhong, and J. Weng, “Digital holographic microscopy by use of surface plasmon resonance for imaging of cell membranes,” J. Biomed. Opt. 15(5), 056015 (2010).
[Crossref] [PubMed]

Wong, C. L.

W. Yuan, H. P. Ho, C. L. Wong, S. K. Kong, and C. Lin, “Surface plasmon resonance biosensor incorporated in a Michelson interferometer with enhanced sensitivity,” IEEE Sens. J. 7(1), 70–73 (2007).
[Crossref]

Wong, W. W.

Wu, S. Y.

Xi, T.

Yanase, Y.

Y. Yanase, T. Hiragun, S. Kaneko, H. J. Gould, M. W. Greaves, and M. Hide, “Detection of refractive index changes in individual living cells by means of surface plasmon resonance imaging,” Biosens. Bioelectron. 26(2), 674–681 (2010).
[Crossref] [PubMed]

Yang, H.

Yi, Y.

L. Ji, X. Sun, G. He, Y. Liu, X. Wang, Y. Yi, C. Chen, F. Wang, and D. Zhang, “Surface plasmon resonance refractive index sensor based on ultraviolet bleached polymer waveguide,” Sens. Actuators B Chem. 244, 373–379 (2017).
[Crossref]

Yuan, W.

W. Yuan, H. P. Ho, C. L. Wong, S. K. Kong, and C. Lin, “Surface plasmon resonance biosensor incorporated in a Michelson interferometer with enhanced sensitivity,” IEEE Sens. J. 7(1), 70–73 (2007).
[Crossref]

Zhang, D.

L. Ji, X. Sun, G. He, Y. Liu, X. Wang, Y. Yi, C. Chen, F. Wang, and D. Zhang, “Surface plasmon resonance refractive index sensor based on ultraviolet bleached polymer waveguide,” Sens. Actuators B Chem. 244, 373–379 (2017).
[Crossref]

Zhang, J.

Zhao, J.

H. Lu, X. Gan, D. Mao, B. Jia, and J. Zhao, “Flexibly tunable high-quality-factor induced transparency in plasmonic systems,” Sci. Rep. 8(1), 1558 (2018).
[Crossref] [PubMed]

J. Zhang, S. Dai, J. Zhong, T. Xi, C. Ma, Y. Li, J. Di, and J. Zhao, “Wavelength-multiplexing surface plasmon holographic microscopy,” Opt. Express 26(10), 13549–13560 (2018).
[Crossref] [PubMed]

J. Zhang, S. Dai, C. Ma, J. Di, and J. Zhao, “Compact surface plasmon holographic microscopy for near-field film mapping,” Opt. Lett. 42(17), 3462–3465 (2017).
[Crossref] [PubMed]

J. Zhang, S. Dai, C. Ma, J. Di, and J. Zhao, “Common-path digital holographic microscopy for near-field phase imaging based on surface plasmon resonance,” Appl. Opt. 56(11), 3223–3228 (2017).
[Crossref] [PubMed]

C. Ma, J. Di, J. Zhang, Y. Li, T. Xi, E. Li, and J. Zhao, “Simultaneous measurement of refractive index distribution and topography by integrated transmission and reflection digital holographic microscopy,” Appl. Opt. 55(33), 9435–9439 (2016).
[Crossref] [PubMed]

J. Zhang, C. Ma, S. Dai, J. Di, Y. Li, T. Xi, and J. Zhao, “Transmission and total internal reflection integrated digital holographic microscopy,” Opt. Lett. 41(16), 3844–3847 (2016).
[Crossref] [PubMed]

J. Wang, J. Zhao, C. Qin, J. Di, A. Rauf, and H. Jiang, “Digital holographic interferometry based on wavelength and angular multiplexing for measuring the ternary diffusion,” Opt. Lett. 37(7), 1211–1213 (2012).
[Crossref] [PubMed]

W. Sun, J. Zhao, J. Di, Q. Wang, and L. Wang, “Real-time visualization of Karman vortex street in water flow field by using digital holography,” Opt. Express 17(22), 20342–20348 (2009).
[Crossref] [PubMed]

Zhong, J.

Anal. Bioanal. Chem. (2)

K. S. Phillips and Q. Cheng, “Recent advances in surface plasmon resonance based techniques for bioanalysis,” Anal. Bioanal. Chem. 387(5), 1831–1840 (2007).
[Crossref] [PubMed]

G. Steiner, “Surface plasmon resonance imaging,” Anal. Bioanal. Chem. 379(3), 328–331 (2004).
[Crossref] [PubMed]

Anal. Chem. (1)

A. R. Halpern, Y. Chen, R. M. Corn, and D. Kim, “Surface plasmon resonance phase imaging measurements of patterned monolayers and DNA adsorption onto microarrays,” Anal. Chem. 83(7), 2801–2806 (2011).
[Crossref] [PubMed]

Appl. Opt. (3)

Biomed. Opt. Express (1)

Biosens. Bioelectron. (1)

Y. Yanase, T. Hiragun, S. Kaneko, H. J. Gould, M. W. Greaves, and M. Hide, “Detection of refractive index changes in individual living cells by means of surface plasmon resonance imaging,” Biosens. Bioelectron. 26(2), 674–681 (2010).
[Crossref] [PubMed]

Chem. Rev. (1)

J. Homola, “Surface plasmon resonance sensors for detection of chemical and biological species,” Chem. Rev. 108(2), 462–493 (2008).
[Crossref] [PubMed]

IEEE Sens. J. (1)

W. Yuan, H. P. Ho, C. L. Wong, S. K. Kong, and C. Lin, “Surface plasmon resonance biosensor incorporated in a Michelson interferometer with enhanced sensitivity,” IEEE Sens. J. 7(1), 70–73 (2007).
[Crossref]

J. Biomed. Opt. (1)

C. Hu, J. Zhong, and J. Weng, “Digital holographic microscopy by use of surface plasmon resonance for imaging of cell membranes,” J. Biomed. Opt. 15(5), 056015 (2010).
[Crossref] [PubMed]

Opt. Express (3)

Opt. Lett. (9)

J. Zhang, S. Dai, C. Ma, J. Di, and J. Zhao, “Compact surface plasmon holographic microscopy for near-field film mapping,” Opt. Lett. 42(17), 3462–3465 (2017).
[Crossref] [PubMed]

J. Zhang, C. Ma, S. Dai, J. Di, Y. Li, T. Xi, and J. Zhao, “Transmission and total internal reflection integrated digital holographic microscopy,” Opt. Lett. 41(16), 3844–3847 (2016).
[Crossref] [PubMed]

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Optica (1)

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H. Lu, X. Gan, D. Mao, B. Jia, and J. Zhao, “Flexibly tunable high-quality-factor induced transparency in plasmonic systems,” Sci. Rep. 8(1), 1558 (2018).
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Sens. Actuators B Chem. (1)

L. Ji, X. Sun, G. He, Y. Liu, X. Wang, Y. Yi, C. Chen, F. Wang, and D. Zhang, “Surface plasmon resonance refractive index sensor based on ultraviolet bleached polymer waveguide,” Sens. Actuators B Chem. 244, 373–379 (2017).
[Crossref]

Other (1)

M. K. Kim, Digital Holographic Microscopy (Springer, 2011), Chap. 11.

Supplementary Material (1)

NameDescription
» Visualization 1       The dynamic changes of the 3D thickness distributions of an alcohol-water mixture droplet during the volatilization process.

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Figures (8)

Fig. 1
Fig. 1 Three-layer Kretschmann configuration. SPW: surface plasmon wave.
Fig. 2
Fig. 2 (a) Theoretical curve of Δφ versus n3. The thickness and dielectric constant of the gold layer are 48.8 nm and –11.740 + 1.2611i, respectively; the wavelength and incident angle of the light wave are 632.8nm and 72.82°, respectively; the RI of the prism is 1.5151. (b) 5th polynominal fitting curve of n3 versus Δφ.
Fig. 3
Fig. 3 Experimental setup of the integrated DHM. FC: fiber coupler; BE1, 2: beam expanders; L1-3: lenses; HP1, 2: half-wave plates; PBS1, 2: polarized beam splitters; M1-5: mirrors; BS1-4: beam splitters; LWDMO1, 2: long working distance microscope objectives; Op and Rp: object and reference beams in SPHM; Os and Rs: object and reference beams in reflection DHM.
Fig. 4
Fig. 4 (a) Composite background hologram. (b) Corresponding spatial spectra.
Fig. 5
Fig. 5 Measurement results of an alcohol-water mixture droplet with the volume ratio of 2:3 at the initial time. Relative reflection phase distribution (a) Δϕ1(x, y) and (d) Δϕ2(x, y) of the specimen. (b) Undistorted phase image of (a). (c) One dimensional RI distribution of the specimen along the black line in (b). (e) 3D thickness distribution of the specimen. (f) Thickness profile of the specimen along the black line in (d).
Fig. 6
Fig. 6 Measurement results of monitoring the volatilization process of an alcohol-water mixture droplet with the volume ratio of 1:2. (a)-(d) Δϕ1, RI, Δϕ2 and maximum thickness variations with time of the specimen, respectively.
Fig. 7
Fig. 7 Measurement results of the 3D thickness distributions of the alcohol-water mixture droplet at (a) 0s, (b) 9s and (c) 42s, respectively. (d) Corresponding thickness profiles of the specimen (Visualization 1).
Fig. 8
Fig. 8 Sensitivities of the specimen RI measurement based on TIR and SPR.

Equations (4)

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r=r( ε 1 , ε 2 , ε 3 ,d,λ,θ),
n 3 (x,y)= p 1 [Δφ(x,y)] 5 + p 2 [Δφ(x,y)] 4 + p 3 [Δφ(x,y)] 3 + p 4 [Δφ(x,y)] 2 + p 5 Δφ(x,y)+ p 6 ,
Δ ϕ 2 (x,y)= 4π[ n 3 (x,y) n 0 (x,y)]h(x,y) λ ,
h(x,y)= Δ ϕ 2 (x,y)λ 4π[ n 3 (x,y)1] .

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