Abstract

Optical waveguide bends are indispensable to integrated optical systems, and many methods to mitigate bend loss have thus been proposed. Transformation optics (TO) causes light to travel around a bend as if it was propagating in a straight waveguide, eliminating the bend loss. Many reported TO waveguide bends have utilized solid materials, but there are fundamental difficulties for real applications because of their complex fabrication, lack of reconfiguration, and the so-called effective medium condition. Here, we develop a method to overcome these problems using the convection–diffusion of liquids. It enables real-time tunable transformation optical waveguide bends using natural liquid diffusion while still exhibiting the major merits of quasi-conformal mapping. We have experimentally demonstrated bending in visible light by 90 and 180° while preserving the intensity profile at a reasonably high level of fidelity. This work bridges fluid dynamics and optics and has the potential for application in on-chip biological, chemical, and biomedical measurements, as well as detectors and tunable optical systems.

© 2017 Optical Society of America

Full Article  |  PDF Article
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References

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2016 (4)

W. Wu, X. Zhu, Y. Zuo, L. Liang, S. Zhang, X. Zhang, and Y. Yang, “Precise sorting of gold nanoparticles in a flowing system,” ACS Photon. 3, 2497–2504 (2016).
[Crossref]

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[Crossref]

L. Li, X. Q. Zhu, L. Liang, Y. F. Zuo, Y. S. Xu, Y. Yang, Y. J. Yuan, and Q. Q. Huang, “Switchable 3D optofluidic Y-branch waveguides tuned by Dean flows,” Sci. Rep. 6, 38338 (2016).
[Crossref]

2015 (2)

M. Ren, H. Cai, L. K. Chin, K. Radhakrishnan, Y. Gu, G.-Q. Lo, D. L. Kwong, and A. Q. Liu, “Coupled-ring reflector in an external cavity tunable laser,” Optica 2, 940–943 (2015).
[Crossref]

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[Crossref]

2014 (2)

2013 (2)

W. Song and D. Psaltis, “Electrically tunable optofluidic light switch for reconfigurable solar lighting,” Lab Chip 13, 2708–2713 (2013).
[Crossref]

C. Sheng, H. Liu, Y. Wang, S. N. Zhu, and D. A. Genov, “Trapping light by mimicking gravitational lensing,” Nat. Photonics 7, 902–906 (2013).
[Crossref]

2012 (5)

Y. Yang, A. Q. Liu, L. K. Chin, X. M. Zhang, D. P. Tsai, C. L. Lin, C. Lu, G. P. Wang, and N. I. Zheludev, “Optofluidic waveguide as a transformation optics device for lightwave bending and manipulation,” Nat. Commun. 3, 651 (2012).
[Crossref]

L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. 3, 1217 (2012).
[Crossref]

P. Fei, Z. Chen, Y. Men, A. Li, Y. Shen, and Y. Y. Huang, “A compact optofluidic cytometer with integrated liquid-core/PDMS-cladding waveguides,” Lab Chip 12, 3700–3706 (2012).
[Crossref]

Y. Yang, L. K. Chin, J. M. Tsai, D. P. Tsai, N. I. Zheludev, and A. Q. Liu, “Transformation optofluidics for large-angle light bending and tuning,” Lab Chip 12, 3785–3790 (2012).
[Crossref]

Y. Sun and X. Fan, “Distinguishing DNA by analog-to-digital-like conversion by using optofluidic lasers,” Angew. Chem. Int. Ed. 51, 1236–1239 (2012).
[Crossref]

2011 (3)

Y. Yang, A. Q. Liu, L. Lei, L. K. Chin, C. D. Ohl, Q. J. Wang, and H. S. Yoon, “A tunable 3D optofluidic waveguide dye laser via two centrifugal Dean flow streams,” Lab Chip 11, 3182–3187 (2011).
[Crossref]

K. Yao and X. Jiang, “Designing feasible optical devices via conformal mapping,” J. Opt. Soc. Am. B 28, 1037–1042 (2011).
[Crossref]

J. W. Allen, H. Steyskal, and D. R. Smith, “Impedance and complex power of radiating elements under electromagnetic source transformation,” Microw. Opt. Technol. Lett. 53, 1524–1527 (2011).
[Crossref]

2010 (1)

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9, 129–132 (2010).
[Crossref]

2009 (4)

S. Tretyakov, P. Alitalo, O. Luukkonen, and C. Simovski, “Broadband electromagnetic cloaking of long cylindrical objects,” Phys. Rev. Lett. 103, 103905 (2009).
[Crossref]

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[Crossref]

N. I. Landy and W. J. Padilla, “Guiding light with conformal transformations,” Opt. Express 17, 14872–14879 (2009).
[Crossref]

Z. L. Mei and T. J. Cui, “Experimental realization of a broadband bend structure using gradient index metamaterials,” Opt. Express 17, 18354–18363 (2009).
[Crossref]

2008 (6)

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101, 203901 (2008).
[Crossref]

M. Rahm, S. A. Cummer, D. Schuring, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformation,” Phys. Rev. Lett. 100, 063903 (2008).
[Crossref]

D. A. Roberts, M. Rahm, J. B. Pendry, and D. R. Smith, “Transformation-optical design of sharp waveguide bends and corners,” Appl. Phys. Lett. 93, 251111 (2008).
[Crossref]

M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express 16, 11555–11567 (2008).
[Crossref]

D.-H. Kwon and D. H. Werner, “Transformation optical designs for wave collimators, flat lenses and right-angle bends,” New J. Phys. 10, 115023 (2008).
[Crossref]

Y. C. Seow, A. Q. Liu, L. K. Chin, X. C. Li, H. J. Huang, T. H. Cheng, and X. Q. Zhou, “Different curvatures of tunable liquid microlens via the control of laminar flow rate,” Appl. Phys. Lett. 93, 084101 (2008).
[Crossref]

2006 (4)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[Crossref]

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[Crossref]

D. Psaltis, S. R. Quake, and C. Yang, “Developing optofluidic technology through the fusion of microfluidics and optics,” Nature 442, 381–386 (2006).
[Crossref]

2005 (1)

B. T. Mayers, D. V. Vezenov, V. Vullev, and G. M. Whitesides, “Arrays and cascades of fluorescent liquid—liquid waveguides: broadband light sources for spectroscopy in microchannels,” Anal. Chem. 77, 1310–1316 (2005).
[Crossref]

1987 (1)

D. A. Walker, “A fluorescence technique for measurement of concentration in mixing liquids,” J. Phys. E 20, 217–224 (1987).
[Crossref]

Alitalo, P.

S. Tretyakov, P. Alitalo, O. Luukkonen, and C. Simovski, “Broadband electromagnetic cloaking of long cylindrical objects,” Phys. Rev. Lett. 103, 103905 (2009).
[Crossref]

Allen, J. W.

J. W. Allen, H. Steyskal, and D. R. Smith, “Impedance and complex power of radiating elements under electromagnetic source transformation,” Microw. Opt. Technol. Lett. 53, 1524–1527 (2011).
[Crossref]

Bai, G.

Bartal, G.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[Crossref]

Cai, H.

Chai, Y.

Chen, Z.

P. Fei, Z. Chen, Y. Men, A. Li, Y. Shen, and Y. Y. Huang, “A compact optofluidic cytometer with integrated liquid-core/PDMS-cladding waveguides,” Lab Chip 12, 3700–3706 (2012).
[Crossref]

Cheng, T. H.

Y. C. Seow, A. Q. Liu, L. K. Chin, X. C. Li, H. J. Huang, T. H. Cheng, and X. Q. Zhou, “Different curvatures of tunable liquid microlens via the control of laminar flow rate,” Appl. Phys. Lett. 93, 084101 (2008).
[Crossref]

Chin, L. K.

M. Ren, H. Cai, L. K. Chin, K. Radhakrishnan, Y. Gu, G.-Q. Lo, D. L. Kwong, and A. Q. Liu, “Coupled-ring reflector in an external cavity tunable laser,” Optica 2, 940–943 (2015).
[Crossref]

Y. Yang, A. Q. Liu, L. K. Chin, X. M. Zhang, D. P. Tsai, C. L. Lin, C. Lu, G. P. Wang, and N. I. Zheludev, “Optofluidic waveguide as a transformation optics device for lightwave bending and manipulation,” Nat. Commun. 3, 651 (2012).
[Crossref]

Y. Yang, L. K. Chin, J. M. Tsai, D. P. Tsai, N. I. Zheludev, and A. Q. Liu, “Transformation optofluidics for large-angle light bending and tuning,” Lab Chip 12, 3785–3790 (2012).
[Crossref]

Y. Yang, A. Q. Liu, L. Lei, L. K. Chin, C. D. Ohl, Q. J. Wang, and H. S. Yoon, “A tunable 3D optofluidic waveguide dye laser via two centrifugal Dean flow streams,” Lab Chip 11, 3182–3187 (2011).
[Crossref]

Y. C. Seow, A. Q. Liu, L. K. Chin, X. C. Li, H. J. Huang, T. H. Cheng, and X. Q. Zhou, “Different curvatures of tunable liquid microlens via the control of laminar flow rate,” Appl. Phys. Lett. 93, 084101 (2008).
[Crossref]

Cui, T. J.

Cummer, S. A.

M. Rahm, S. A. Cummer, D. Schuring, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformation,” Phys. Rev. Lett. 100, 063903 (2008).
[Crossref]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[Crossref]

Cussler, E. L.

E. L. Cussler, Diffusion: Mass Transfer in Fluid Systems (Cambridge University, 2009).

Fan, X.

Y. Sun and X. Fan, “Distinguishing DNA by analog-to-digital-like conversion by using optofluidic lasers,” Angew. Chem. Int. Ed. 51, 1236–1239 (2012).
[Crossref]

Fei, P.

P. Fei, Z. Chen, Y. Men, A. Li, Y. Shen, and Y. Y. Huang, “A compact optofluidic cytometer with integrated liquid-core/PDMS-cladding waveguides,” Lab Chip 12, 3700–3706 (2012).
[Crossref]

Fung, K. H.

Gabrielli, L. H.

L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. 3, 1217 (2012).
[Crossref]

Genov, D. A.

C. Sheng, H. Liu, Y. Wang, S. N. Zhu, and D. A. Genov, “Trapping light by mimicking gravitational lensing,” Nat. Photonics 7, 902–906 (2013).
[Crossref]

Gu, Y.

Huang, H. J.

Y. C. Seow, A. Q. Liu, L. K. Chin, X. C. Li, H. J. Huang, T. H. Cheng, and X. Q. Zhou, “Different curvatures of tunable liquid microlens via the control of laminar flow rate,” Appl. Phys. Lett. 93, 084101 (2008).
[Crossref]

Huang, Q. Q.

L. Li, X. Q. Zhu, L. Liang, Y. F. Zuo, Y. S. Xu, Y. Yang, Y. J. Yuan, and Q. Q. Huang, “Switchable 3D optofluidic Y-branch waveguides tuned by Dean flows,” Sci. Rep. 6, 38338 (2016).
[Crossref]

Huang, Y. Y.

P. Fei, Z. Chen, Y. Men, A. Li, Y. Shen, and Y. Y. Huang, “A compact optofluidic cytometer with integrated liquid-core/PDMS-cladding waveguides,” Lab Chip 12, 3700–3706 (2012).
[Crossref]

Hui, L.

Jiang, X.

Jim, K. L.

Johnson, S. G.

L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. 3, 1217 (2012).
[Crossref]

Justice, B. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[Crossref]

Kundtz, N.

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9, 129–132 (2010).
[Crossref]

Kwon, D.-H.

D.-H. Kwon and D. H. Werner, “Transformation optical designs for wave collimators, flat lenses and right-angle bends,” New J. Phys. 10, 115023 (2008).
[Crossref]

Kwong, D. L.

Landy, N. I.

Lei, L.

Y. Yang, A. Q. Liu, L. Lei, L. K. Chin, C. D. Ohl, Q. J. Wang, and H. S. Yoon, “A tunable 3D optofluidic waveguide dye laser via two centrifugal Dean flow streams,” Lab Chip 11, 3182–3187 (2011).
[Crossref]

Leonhardt, U.

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[Crossref]

Li, A.

P. Fei, Z. Chen, Y. Men, A. Li, Y. Shen, and Y. Y. Huang, “A compact optofluidic cytometer with integrated liquid-core/PDMS-cladding waveguides,” Lab Chip 12, 3700–3706 (2012).
[Crossref]

Li, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[Crossref]

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101, 203901 (2008).
[Crossref]

Li, L.

L. Li, X. Q. Zhu, L. Liang, Y. F. Zuo, Y. S. Xu, Y. Yang, Y. J. Yuan, and Q. Q. Huang, “Switchable 3D optofluidic Y-branch waveguides tuned by Dean flows,” Sci. Rep. 6, 38338 (2016).
[Crossref]

Li, X. C.

Y. C. Seow, A. Q. Liu, L. K. Chin, X. C. Li, H. J. Huang, T. H. Cheng, and X. Q. Zhou, “Different curvatures of tunable liquid microlens via the control of laminar flow rate,” Appl. Phys. Lett. 93, 084101 (2008).
[Crossref]

Liang, L.

W. Wu, X. Zhu, Y. Zuo, L. Liang, S. Zhang, X. Zhang, and Y. Yang, “Precise sorting of gold nanoparticles in a flowing system,” ACS Photon. 3, 2497–2504 (2016).
[Crossref]

L. Li, X. Q. Zhu, L. Liang, Y. F. Zuo, Y. S. Xu, Y. Yang, Y. J. Yuan, and Q. Q. Huang, “Switchable 3D optofluidic Y-branch waveguides tuned by Dean flows,” Sci. Rep. 6, 38338 (2016).
[Crossref]

Y. Shi, X. Q. Zhu, L. Liang, and Y. Yang, “Tunable focusing properties using optofluidic Fresnel zone plates,” Lab Chip 16, 4554–4559 (2016).
[Crossref]

L. Liang, Y. F. Zuo, W. Wu, X. Q. Zhu, and Y. Yang, “Optofluidic restricted imaging, spectroscopy and counting of nanoparticles by evanescent wave using immiscible liquids,” Lab Chip 16, 3007–3014 (2016).
[Crossref]

Y. Shi, L. Liang, X. Q. Zhu, X. M. Zhang, and Y. Yang, “Tunable self-imaging effect using hybrid optofluidic waveguides,” Lab Chip 15, 4398–4403 (2015).
[Crossref]

Lide, D. R.

D. R. Lide, Handbook of Chemistry and Physics, 87th ed. (CRC Press, 2007), Chap. 8, p. 57.

Lin, C. L.

Y. Yang, A. Q. Liu, L. K. Chin, X. M. Zhang, D. P. Tsai, C. L. Lin, C. Lu, G. P. Wang, and N. I. Zheludev, “Optofluidic waveguide as a transformation optics device for lightwave bending and manipulation,” Nat. Commun. 3, 651 (2012).
[Crossref]

Lipson, M.

L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. 3, 1217 (2012).
[Crossref]

Liu, A. Q.

M. Ren, H. Cai, L. K. Chin, K. Radhakrishnan, Y. Gu, G.-Q. Lo, D. L. Kwong, and A. Q. Liu, “Coupled-ring reflector in an external cavity tunable laser,” Optica 2, 940–943 (2015).
[Crossref]

Y. Yang, L. K. Chin, J. M. Tsai, D. P. Tsai, N. I. Zheludev, and A. Q. Liu, “Transformation optofluidics for large-angle light bending and tuning,” Lab Chip 12, 3785–3790 (2012).
[Crossref]

Y. Yang, A. Q. Liu, L. K. Chin, X. M. Zhang, D. P. Tsai, C. L. Lin, C. Lu, G. P. Wang, and N. I. Zheludev, “Optofluidic waveguide as a transformation optics device for lightwave bending and manipulation,” Nat. Commun. 3, 651 (2012).
[Crossref]

Y. Yang, A. Q. Liu, L. Lei, L. K. Chin, C. D. Ohl, Q. J. Wang, and H. S. Yoon, “A tunable 3D optofluidic waveguide dye laser via two centrifugal Dean flow streams,” Lab Chip 11, 3182–3187 (2011).
[Crossref]

Y. C. Seow, A. Q. Liu, L. K. Chin, X. C. Li, H. J. Huang, T. H. Cheng, and X. Q. Zhou, “Different curvatures of tunable liquid microlens via the control of laminar flow rate,” Appl. Phys. Lett. 93, 084101 (2008).
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L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. 3, 1217 (2012).
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Liu, H.

C. Sheng, H. Liu, Y. Wang, S. N. Zhu, and D. A. Genov, “Trapping light by mimicking gravitational lensing,” Nat. Photonics 7, 902–906 (2013).
[Crossref]

Lo, G.-Q.

Lu, C.

Y. Yang, A. Q. Liu, L. K. Chin, X. M. Zhang, D. P. Tsai, C. L. Lin, C. Lu, G. P. Wang, and N. I. Zheludev, “Optofluidic waveguide as a transformation optics device for lightwave bending and manipulation,” Nat. Commun. 3, 651 (2012).
[Crossref]

Luukkonen, O.

S. Tretyakov, P. Alitalo, O. Luukkonen, and C. Simovski, “Broadband electromagnetic cloaking of long cylindrical objects,” Phys. Rev. Lett. 103, 103905 (2009).
[Crossref]

Mayers, B. T.

B. T. Mayers, D. V. Vezenov, V. Vullev, and G. M. Whitesides, “Arrays and cascades of fluorescent liquid—liquid waveguides: broadband light sources for spectroscopy in microchannels,” Anal. Chem. 77, 1310–1316 (2005).
[Crossref]

Mei, Z. L.

Men, Y.

P. Fei, Z. Chen, Y. Men, A. Li, Y. Shen, and Y. Y. Huang, “A compact optofluidic cytometer with integrated liquid-core/PDMS-cladding waveguides,” Lab Chip 12, 3700–3706 (2012).
[Crossref]

Mock, J. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[Crossref]

Ohl, C. D.

Y. Yang, A. Q. Liu, L. Lei, L. K. Chin, C. D. Ohl, Q. J. Wang, and H. S. Yoon, “A tunable 3D optofluidic waveguide dye laser via two centrifugal Dean flow streams,” Lab Chip 11, 3182–3187 (2011).
[Crossref]

Padilla, W. J.

Pendry, J. B.

M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express 16, 11555–11567 (2008).
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M. Rahm, S. A. Cummer, D. Schuring, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformation,” Phys. Rev. Lett. 100, 063903 (2008).
[Crossref]

D. A. Roberts, M. Rahm, J. B. Pendry, and D. R. Smith, “Transformation-optical design of sharp waveguide bends and corners,” Appl. Phys. Lett. 93, 251111 (2008).
[Crossref]

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101, 203901 (2008).
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D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[Crossref]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref]

Psaltis, D.

W. Song and D. Psaltis, “Electrically tunable optofluidic light switch for reconfigurable solar lighting,” Lab Chip 13, 2708–2713 (2013).
[Crossref]

D. Psaltis, S. R. Quake, and C. Yang, “Developing optofluidic technology through the fusion of microfluidics and optics,” Nature 442, 381–386 (2006).
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D. Psaltis, S. R. Quake, and C. Yang, “Developing optofluidic technology through the fusion of microfluidics and optics,” Nature 442, 381–386 (2006).
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Rahm, M.

D. A. Roberts, M. Rahm, J. B. Pendry, and D. R. Smith, “Transformation-optical design of sharp waveguide bends and corners,” Appl. Phys. Lett. 93, 251111 (2008).
[Crossref]

M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express 16, 11555–11567 (2008).
[Crossref]

M. Rahm, S. A. Cummer, D. Schuring, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformation,” Phys. Rev. Lett. 100, 063903 (2008).
[Crossref]

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Roberts, D. A.

M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express 16, 11555–11567 (2008).
[Crossref]

D. A. Roberts, M. Rahm, J. B. Pendry, and D. R. Smith, “Transformation-optical design of sharp waveguide bends and corners,” Appl. Phys. Lett. 93, 251111 (2008).
[Crossref]

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J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[Crossref]

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M. Rahm, S. A. Cummer, D. Schuring, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformation,” Phys. Rev. Lett. 100, 063903 (2008).
[Crossref]

Seow, Y. C.

Y. C. Seow, A. Q. Liu, L. K. Chin, X. C. Li, H. J. Huang, T. H. Cheng, and X. Q. Zhou, “Different curvatures of tunable liquid microlens via the control of laminar flow rate,” Appl. Phys. Lett. 93, 084101 (2008).
[Crossref]

Shen, Y.

P. Fei, Z. Chen, Y. Men, A. Li, Y. Shen, and Y. Y. Huang, “A compact optofluidic cytometer with integrated liquid-core/PDMS-cladding waveguides,” Lab Chip 12, 3700–3706 (2012).
[Crossref]

Sheng, C.

C. Sheng, H. Liu, Y. Wang, S. N. Zhu, and D. A. Genov, “Trapping light by mimicking gravitational lensing,” Nat. Photonics 7, 902–906 (2013).
[Crossref]

Shi, Y.

Y. Shi, X. Q. Zhu, L. Liang, and Y. Yang, “Tunable focusing properties using optofluidic Fresnel zone plates,” Lab Chip 16, 4554–4559 (2016).
[Crossref]

Y. Shi, L. Liang, X. Q. Zhu, X. M. Zhang, and Y. Yang, “Tunable self-imaging effect using hybrid optofluidic waveguides,” Lab Chip 15, 4398–4403 (2015).
[Crossref]

Simovski, C.

S. Tretyakov, P. Alitalo, O. Luukkonen, and C. Simovski, “Broadband electromagnetic cloaking of long cylindrical objects,” Phys. Rev. Lett. 103, 103905 (2009).
[Crossref]

Smith, D. R.

J. W. Allen, H. Steyskal, and D. R. Smith, “Impedance and complex power of radiating elements under electromagnetic source transformation,” Microw. Opt. Technol. Lett. 53, 1524–1527 (2011).
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N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9, 129–132 (2010).
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M. Rahm, S. A. Cummer, D. Schuring, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformation,” Phys. Rev. Lett. 100, 063903 (2008).
[Crossref]

D. A. Roberts, M. Rahm, J. B. Pendry, and D. R. Smith, “Transformation-optical design of sharp waveguide bends and corners,” Appl. Phys. Lett. 93, 251111 (2008).
[Crossref]

M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express 16, 11555–11567 (2008).
[Crossref]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[Crossref]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref]

Song, W.

W. Song and D. Psaltis, “Electrically tunable optofluidic light switch for reconfigurable solar lighting,” Lab Chip 13, 2708–2713 (2013).
[Crossref]

Starr, A. F.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[Crossref]

Steyskal, H.

J. W. Allen, H. Steyskal, and D. R. Smith, “Impedance and complex power of radiating elements under electromagnetic source transformation,” Microw. Opt. Technol. Lett. 53, 1524–1527 (2011).
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Sun, Y.

Y. Sun and X. Fan, “Distinguishing DNA by analog-to-digital-like conversion by using optofluidic lasers,” Angew. Chem. Int. Ed. 51, 1236–1239 (2012).
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Tang, C. Y.

Tang, C.-Y.

Tao, L.

Tretyakov, S.

S. Tretyakov, P. Alitalo, O. Luukkonen, and C. Simovski, “Broadband electromagnetic cloaking of long cylindrical objects,” Phys. Rev. Lett. 103, 103905 (2009).
[Crossref]

Tsai, D. P.

Y. Yang, A. Q. Liu, L. K. Chin, X. M. Zhang, D. P. Tsai, C. L. Lin, C. Lu, G. P. Wang, and N. I. Zheludev, “Optofluidic waveguide as a transformation optics device for lightwave bending and manipulation,” Nat. Commun. 3, 651 (2012).
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Y. Yang, L. K. Chin, J. M. Tsai, D. P. Tsai, N. I. Zheludev, and A. Q. Liu, “Transformation optofluidics for large-angle light bending and tuning,” Lab Chip 12, 3785–3790 (2012).
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Tsai, J. M.

Y. Yang, L. K. Chin, J. M. Tsai, D. P. Tsai, N. I. Zheludev, and A. Q. Liu, “Transformation optofluidics for large-angle light bending and tuning,” Lab Chip 12, 3785–3790 (2012).
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Tsang, Y. H.

Valentine, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
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B. T. Mayers, D. V. Vezenov, V. Vullev, and G. M. Whitesides, “Arrays and cascades of fluorescent liquid—liquid waveguides: broadband light sources for spectroscopy in microchannels,” Anal. Chem. 77, 1310–1316 (2005).
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Vullev, V.

B. T. Mayers, D. V. Vezenov, V. Vullev, and G. M. Whitesides, “Arrays and cascades of fluorescent liquid—liquid waveguides: broadband light sources for spectroscopy in microchannels,” Anal. Chem. 77, 1310–1316 (2005).
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D. A. Walker, “A fluorescence technique for measurement of concentration in mixing liquids,” J. Phys. E 20, 217–224 (1987).
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Y. Yang, A. Q. Liu, L. K. Chin, X. M. Zhang, D. P. Tsai, C. L. Lin, C. Lu, G. P. Wang, and N. I. Zheludev, “Optofluidic waveguide as a transformation optics device for lightwave bending and manipulation,” Nat. Commun. 3, 651 (2012).
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Wang, Q. J.

Y. Yang, A. Q. Liu, L. Lei, L. K. Chin, C. D. Ohl, Q. J. Wang, and H. S. Yoon, “A tunable 3D optofluidic waveguide dye laser via two centrifugal Dean flow streams,” Lab Chip 11, 3182–3187 (2011).
[Crossref]

Wang, Y.

C. Sheng, H. Liu, Y. Wang, S. N. Zhu, and D. A. Genov, “Trapping light by mimicking gravitational lensing,” Nat. Photonics 7, 902–906 (2013).
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Werner, D. H.

D.-H. Kwon and D. H. Werner, “Transformation optical designs for wave collimators, flat lenses and right-angle bends,” New J. Phys. 10, 115023 (2008).
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B. T. Mayers, D. V. Vezenov, V. Vullev, and G. M. Whitesides, “Arrays and cascades of fluorescent liquid—liquid waveguides: broadband light sources for spectroscopy in microchannels,” Anal. Chem. 77, 1310–1316 (2005).
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W. Wu, X. Zhu, Y. Zuo, L. Liang, S. Zhang, X. Zhang, and Y. Yang, “Precise sorting of gold nanoparticles in a flowing system,” ACS Photon. 3, 2497–2504 (2016).
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L. Liang, Y. F. Zuo, W. Wu, X. Q. Zhu, and Y. Yang, “Optofluidic restricted imaging, spectroscopy and counting of nanoparticles by evanescent wave using immiscible liquids,” Lab Chip 16, 3007–3014 (2016).
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Xu, D.

Xu, Y. S.

L. Li, X. Q. Zhu, L. Liang, Y. F. Zuo, Y. S. Xu, Y. Yang, Y. J. Yuan, and Q. Q. Huang, “Switchable 3D optofluidic Y-branch waveguides tuned by Dean flows,” Sci. Rep. 6, 38338 (2016).
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Yang, C.

D. Psaltis, S. R. Quake, and C. Yang, “Developing optofluidic technology through the fusion of microfluidics and optics,” Nature 442, 381–386 (2006).
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Yang, Y.

L. Li, X. Q. Zhu, L. Liang, Y. F. Zuo, Y. S. Xu, Y. Yang, Y. J. Yuan, and Q. Q. Huang, “Switchable 3D optofluidic Y-branch waveguides tuned by Dean flows,” Sci. Rep. 6, 38338 (2016).
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Y. Shi, X. Q. Zhu, L. Liang, and Y. Yang, “Tunable focusing properties using optofluidic Fresnel zone plates,” Lab Chip 16, 4554–4559 (2016).
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L. Liang, Y. F. Zuo, W. Wu, X. Q. Zhu, and Y. Yang, “Optofluidic restricted imaging, spectroscopy and counting of nanoparticles by evanescent wave using immiscible liquids,” Lab Chip 16, 3007–3014 (2016).
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W. Wu, X. Zhu, Y. Zuo, L. Liang, S. Zhang, X. Zhang, and Y. Yang, “Precise sorting of gold nanoparticles in a flowing system,” ACS Photon. 3, 2497–2504 (2016).
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Y. Shi, L. Liang, X. Q. Zhu, X. M. Zhang, and Y. Yang, “Tunable self-imaging effect using hybrid optofluidic waveguides,” Lab Chip 15, 4398–4403 (2015).
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Y. Yang, L. K. Chin, J. M. Tsai, D. P. Tsai, N. I. Zheludev, and A. Q. Liu, “Transformation optofluidics for large-angle light bending and tuning,” Lab Chip 12, 3785–3790 (2012).
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Y. Yang, A. Q. Liu, L. K. Chin, X. M. Zhang, D. P. Tsai, C. L. Lin, C. Lu, G. P. Wang, and N. I. Zheludev, “Optofluidic waveguide as a transformation optics device for lightwave bending and manipulation,” Nat. Commun. 3, 651 (2012).
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Y. Yang, A. Q. Liu, L. Lei, L. K. Chin, C. D. Ohl, Q. J. Wang, and H. S. Yoon, “A tunable 3D optofluidic waveguide dye laser via two centrifugal Dean flow streams,” Lab Chip 11, 3182–3187 (2011).
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Yao, J.

Yao, K.

Yoon, H. S.

Y. Yang, A. Q. Liu, L. Lei, L. K. Chin, C. D. Ohl, Q. J. Wang, and H. S. Yoon, “A tunable 3D optofluidic waveguide dye laser via two centrifugal Dean flow streams,” Lab Chip 11, 3182–3187 (2011).
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Yuan, Y. J.

L. Li, X. Q. Zhu, L. Liang, Y. F. Zuo, Y. S. Xu, Y. Yang, Y. J. Yuan, and Q. Q. Huang, “Switchable 3D optofluidic Y-branch waveguides tuned by Dean flows,” Sci. Rep. 6, 38338 (2016).
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J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
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Zhang, S.

W. Wu, X. Zhu, Y. Zuo, L. Liang, S. Zhang, X. Zhang, and Y. Yang, “Precise sorting of gold nanoparticles in a flowing system,” ACS Photon. 3, 2497–2504 (2016).
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Zhang, X.

W. Wu, X. Zhu, Y. Zuo, L. Liang, S. Zhang, X. Zhang, and Y. Yang, “Precise sorting of gold nanoparticles in a flowing system,” ACS Photon. 3, 2497–2504 (2016).
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C.-Y. Tang, G. Bai, K. L. Jim, X. Zhang, K. H. Fung, Y. Chai, Y. H. Tsang, J. Yao, and D. Xu, “Lensed water-core Teflon-amorphous fluoroplastics optical fiber,” J. Lightwave Technol. 32, 1538–1542 (2014).
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C. Y. Tang, X. Zhang, Y. Chai, L. Hui, L. Tao, and Y. H. Tsang, “Controllable parabolic lensed liquid-core optical fiber by using electrostatic force,” Opt. Express 22, 20948–20953 (2014).
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J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[Crossref]

Zhang, X. M.

Y. Shi, L. Liang, X. Q. Zhu, X. M. Zhang, and Y. Yang, “Tunable self-imaging effect using hybrid optofluidic waveguides,” Lab Chip 15, 4398–4403 (2015).
[Crossref]

Y. Yang, A. Q. Liu, L. K. Chin, X. M. Zhang, D. P. Tsai, C. L. Lin, C. Lu, G. P. Wang, and N. I. Zheludev, “Optofluidic waveguide as a transformation optics device for lightwave bending and manipulation,” Nat. Commun. 3, 651 (2012).
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Zheludev, N. I.

Y. Yang, A. Q. Liu, L. K. Chin, X. M. Zhang, D. P. Tsai, C. L. Lin, C. Lu, G. P. Wang, and N. I. Zheludev, “Optofluidic waveguide as a transformation optics device for lightwave bending and manipulation,” Nat. Commun. 3, 651 (2012).
[Crossref]

Y. Yang, L. K. Chin, J. M. Tsai, D. P. Tsai, N. I. Zheludev, and A. Q. Liu, “Transformation optofluidics for large-angle light bending and tuning,” Lab Chip 12, 3785–3790 (2012).
[Crossref]

Zhou, X. Q.

Y. C. Seow, A. Q. Liu, L. K. Chin, X. C. Li, H. J. Huang, T. H. Cheng, and X. Q. Zhou, “Different curvatures of tunable liquid microlens via the control of laminar flow rate,” Appl. Phys. Lett. 93, 084101 (2008).
[Crossref]

Zhu, S. N.

C. Sheng, H. Liu, Y. Wang, S. N. Zhu, and D. A. Genov, “Trapping light by mimicking gravitational lensing,” Nat. Photonics 7, 902–906 (2013).
[Crossref]

Zhu, X.

W. Wu, X. Zhu, Y. Zuo, L. Liang, S. Zhang, X. Zhang, and Y. Yang, “Precise sorting of gold nanoparticles in a flowing system,” ACS Photon. 3, 2497–2504 (2016).
[Crossref]

Zhu, X. Q.

Y. Shi, X. Q. Zhu, L. Liang, and Y. Yang, “Tunable focusing properties using optofluidic Fresnel zone plates,” Lab Chip 16, 4554–4559 (2016).
[Crossref]

L. Li, X. Q. Zhu, L. Liang, Y. F. Zuo, Y. S. Xu, Y. Yang, Y. J. Yuan, and Q. Q. Huang, “Switchable 3D optofluidic Y-branch waveguides tuned by Dean flows,” Sci. Rep. 6, 38338 (2016).
[Crossref]

L. Liang, Y. F. Zuo, W. Wu, X. Q. Zhu, and Y. Yang, “Optofluidic restricted imaging, spectroscopy and counting of nanoparticles by evanescent wave using immiscible liquids,” Lab Chip 16, 3007–3014 (2016).
[Crossref]

Y. Shi, L. Liang, X. Q. Zhu, X. M. Zhang, and Y. Yang, “Tunable self-imaging effect using hybrid optofluidic waveguides,” Lab Chip 15, 4398–4403 (2015).
[Crossref]

Zuo, Y.

W. Wu, X. Zhu, Y. Zuo, L. Liang, S. Zhang, X. Zhang, and Y. Yang, “Precise sorting of gold nanoparticles in a flowing system,” ACS Photon. 3, 2497–2504 (2016).
[Crossref]

Zuo, Y. F.

L. Liang, Y. F. Zuo, W. Wu, X. Q. Zhu, and Y. Yang, “Optofluidic restricted imaging, spectroscopy and counting of nanoparticles by evanescent wave using immiscible liquids,” Lab Chip 16, 3007–3014 (2016).
[Crossref]

L. Li, X. Q. Zhu, L. Liang, Y. F. Zuo, Y. S. Xu, Y. Yang, Y. J. Yuan, and Q. Q. Huang, “Switchable 3D optofluidic Y-branch waveguides tuned by Dean flows,” Sci. Rep. 6, 38338 (2016).
[Crossref]

ACS Photon. (1)

W. Wu, X. Zhu, Y. Zuo, L. Liang, S. Zhang, X. Zhang, and Y. Yang, “Precise sorting of gold nanoparticles in a flowing system,” ACS Photon. 3, 2497–2504 (2016).
[Crossref]

Anal. Chem. (1)

B. T. Mayers, D. V. Vezenov, V. Vullev, and G. M. Whitesides, “Arrays and cascades of fluorescent liquid—liquid waveguides: broadband light sources for spectroscopy in microchannels,” Anal. Chem. 77, 1310–1316 (2005).
[Crossref]

Angew. Chem. Int. Ed. (1)

Y. Sun and X. Fan, “Distinguishing DNA by analog-to-digital-like conversion by using optofluidic lasers,” Angew. Chem. Int. Ed. 51, 1236–1239 (2012).
[Crossref]

Appl. Phys. Lett. (2)

Y. C. Seow, A. Q. Liu, L. K. Chin, X. C. Li, H. J. Huang, T. H. Cheng, and X. Q. Zhou, “Different curvatures of tunable liquid microlens via the control of laminar flow rate,” Appl. Phys. Lett. 93, 084101 (2008).
[Crossref]

D. A. Roberts, M. Rahm, J. B. Pendry, and D. R. Smith, “Transformation-optical design of sharp waveguide bends and corners,” Appl. Phys. Lett. 93, 251111 (2008).
[Crossref]

J. Lightwave Technol. (1)

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

J. Phys. E (1)

D. A. Walker, “A fluorescence technique for measurement of concentration in mixing liquids,” J. Phys. E 20, 217–224 (1987).
[Crossref]

Lab Chip (7)

Y. Shi, X. Q. Zhu, L. Liang, and Y. Yang, “Tunable focusing properties using optofluidic Fresnel zone plates,” Lab Chip 16, 4554–4559 (2016).
[Crossref]

W. Song and D. Psaltis, “Electrically tunable optofluidic light switch for reconfigurable solar lighting,” Lab Chip 13, 2708–2713 (2013).
[Crossref]

L. Liang, Y. F. Zuo, W. Wu, X. Q. Zhu, and Y. Yang, “Optofluidic restricted imaging, spectroscopy and counting of nanoparticles by evanescent wave using immiscible liquids,” Lab Chip 16, 3007–3014 (2016).
[Crossref]

P. Fei, Z. Chen, Y. Men, A. Li, Y. Shen, and Y. Y. Huang, “A compact optofluidic cytometer with integrated liquid-core/PDMS-cladding waveguides,” Lab Chip 12, 3700–3706 (2012).
[Crossref]

Y. Yang, A. Q. Liu, L. Lei, L. K. Chin, C. D. Ohl, Q. J. Wang, and H. S. Yoon, “A tunable 3D optofluidic waveguide dye laser via two centrifugal Dean flow streams,” Lab Chip 11, 3182–3187 (2011).
[Crossref]

Y. Shi, L. Liang, X. Q. Zhu, X. M. Zhang, and Y. Yang, “Tunable self-imaging effect using hybrid optofluidic waveguides,” Lab Chip 15, 4398–4403 (2015).
[Crossref]

Y. Yang, L. K. Chin, J. M. Tsai, D. P. Tsai, N. I. Zheludev, and A. Q. Liu, “Transformation optofluidics for large-angle light bending and tuning,” Lab Chip 12, 3785–3790 (2012).
[Crossref]

Microw. Opt. Technol. Lett. (1)

J. W. Allen, H. Steyskal, and D. R. Smith, “Impedance and complex power of radiating elements under electromagnetic source transformation,” Microw. Opt. Technol. Lett. 53, 1524–1527 (2011).
[Crossref]

Nat. Commun. (2)

Y. Yang, A. Q. Liu, L. K. Chin, X. M. Zhang, D. P. Tsai, C. L. Lin, C. Lu, G. P. Wang, and N. I. Zheludev, “Optofluidic waveguide as a transformation optics device for lightwave bending and manipulation,” Nat. Commun. 3, 651 (2012).
[Crossref]

L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. 3, 1217 (2012).
[Crossref]

Nat. Mater. (2)

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[Crossref]

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9, 129–132 (2010).
[Crossref]

Nat. Photonics (1)

C. Sheng, H. Liu, Y. Wang, S. N. Zhu, and D. A. Genov, “Trapping light by mimicking gravitational lensing,” Nat. Photonics 7, 902–906 (2013).
[Crossref]

Nature (1)

D. Psaltis, S. R. Quake, and C. Yang, “Developing optofluidic technology through the fusion of microfluidics and optics,” Nature 442, 381–386 (2006).
[Crossref]

New J. Phys. (1)

D.-H. Kwon and D. H. Werner, “Transformation optical designs for wave collimators, flat lenses and right-angle bends,” New J. Phys. 10, 115023 (2008).
[Crossref]

Opt. Express (4)

Optica (1)

Phys. Rev. Lett. (3)

M. Rahm, S. A. Cummer, D. Schuring, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformation,” Phys. Rev. Lett. 100, 063903 (2008).
[Crossref]

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101, 203901 (2008).
[Crossref]

S. Tretyakov, P. Alitalo, O. Luukkonen, and C. Simovski, “Broadband electromagnetic cloaking of long cylindrical objects,” Phys. Rev. Lett. 103, 103905 (2009).
[Crossref]

Sci. Rep. (1)

L. Li, X. Q. Zhu, L. Liang, Y. F. Zuo, Y. S. Xu, Y. Yang, Y. J. Yuan, and Q. Q. Huang, “Switchable 3D optofluidic Y-branch waveguides tuned by Dean flows,” Sci. Rep. 6, 38338 (2016).
[Crossref]

Science (3)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[Crossref]

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[Crossref]

Other (2)

D. R. Lide, Handbook of Chemistry and Physics, 87th ed. (CRC Press, 2007), Chap. 8, p. 57.

E. L. Cussler, Diffusion: Mass Transfer in Fluid Systems (Cambridge University, 2009).

Supplementary Material (2)

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» Supplement 1       Supplemental materials
» Visualization 1       Video

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Figures (7)

Fig. 1.
Fig. 1. Main concept of liquid waveguide bends. (a) Counterflow convection–diffusion process. The dotted black lines indicate the flowing direction. The concentration at a point contains different sources from different input positions. (b) Calculated RI profile (color map) as a result of the diffusion process in (a). The black arrows represent the light path. (c) Two-layer structure of the optofluidic device. The channel in the lower layer is the liquid transformation waveguide bends. (d) Top view of the fabricated chip. The inset shows the cross section at the middle of the bend, in which the red and yellow arrows represent the flowing directions of the glycol flow and water flow, respectively.
Fig. 2.
Fig. 2. Analysis of RI profile. (a) RI profile when the glycol flow rate is 16.84  nl/min. (b) RI profile at the middle of the bend. The dotted black line represents the RI profile required by an ideal TO bend. The solid lines denote the RI profiles of the liquid bend when the glycol flow rate is 16.84  nl/min (red), 50  nl/min (blue), 100  nl/min (green), and 200  nl/min (black). The radii of 27.5, 0, and 27.5 μm represent r1 (inner boundary), r2 (middle radius), and r3 (outer boundary), respectively. (c) RI profile at three different radii when the glycol flow rate is 16.84  nl/min. The dotted black lines represent the RI profile for ideal TO bends. The solid lines are the RI profiles of the liquid bend at r1, r2, and r3.
Fig. 3.
Fig. 3. Simulated and experimental RI profiles when the glycol flow rate is [(a), (d)] 449.14, [(b), (e)] 112.28, and [(c), (f)] 16.84  nl/min. Panels (g), (h), and (i) show RI profiles along the solid black lines in (d), (e), and (f), respectively.
Fig. 4.
Fig. 4. Liquid bends of 90° at a glycol flow rate of 16.84  nl/min. Panels (a) and (b) show simulated and measured light propagation, respectively. Panels (c) and (d) show intensity profiles along the observation lines in (a) and (b), respectively. Line 1 (4) and 3 (6) represent the input and output, respectively.
Fig. 5.
Fig. 5. (a) Simulated and (b) experimental light propagation in 180° liquid bends. Panels (c) and (d) show intensity profiles along the observation lines in (a) and (b), respectively.
Fig. 6.
Fig. 6. Light beam profiles of the liquid bends. (a) Input and (b) output light of 90° bends, and (c) output light of 180° bends. Panels (d) and (g), (e) and (h), and (f) and (i) show light intensity profiles of (a), (b), and (c), respectively.
Fig. 7.
Fig. 7. Cross section of light in liquid bends using a ray-tracing module: (a) 90, (b) 180, and (c) 270° bends. θ=0 represents the input light. θ=π/2, π, and 3π/2 represent the output light.

Equations (10)

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Ct=D2CUC+R,
2(nc2)=0.
C(r,θ)=|C1(r,θ)+C2(r,θ)|r+|C1(r,θ)+C2(r,θ)|θ,
UrCr=D2  Cr2,UθrCθ=D2  Cθ2,
14·2πr1·a·v1=Q1,14·2πr2·a·v2=Q2.
v¯=v1+v22=1πa(Q1r1+Q2r2),
14·2πr1·dupper·v1=Q1,14·2πr2·dupper·v2=Q2.
v1t1=s,v2t2=s.
t¯=t1+t22=sπdupper4(r1Q1+r2Q2),
H=v¯t¯=sdupper4a(2+r2Q1r1Q2+r1Q2r2Q1),

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