Abstract

The unique superiority of transformation optics devices designed from coordinate transformation is their capability of recovering both ray trajectory and optical path length in light manipulation. However, very few experiments have been done so far to verify this dual-recovery property from viewpoints of both ray trajectory and optical path length simultaneously. The experimental difficulties arise from the fact that most previous optical transformation optics devices only work at the nano-scale; the lack of intercomparison between data from both optical path length and ray trajectory measurement in these experiments obscured the fact that the ray path was subject to a subwavelength lateral shift that was otherwise not easily perceivable and, instead, was pointed out theoretically [B. Zhang et al. Phys. Rev. Lett. 104, 233903, 2010]. Here, we use a simple macroscopic transformation optics device of phase-preserved optical elevator, which is a typical birefringent optical phenomenon that can virtually lift an optical image by a macroscopic distance, to demonstrate decisively the unique optical path length preservation property of transformation optics. The recovery of ray trajectory is first determined with no lateral shift in the reflected ray. The phase preservation is then verified with incoherent white-light interferometry without ambiguity and phase unwrapping.

© 2013 OSA

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References

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    [CrossRef] [PubMed]
  2. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006).
    [CrossRef] [PubMed]
  3. L. S. Dolin, “On a possibility of comparing three-dimensional electromagnetic systems with inhomogeneous filling,” Izv. Vyss. Ucebn. Zaved. Radiofiz.4, 964–967 (1961).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  18. T. Ergin, J. Fischer, and M. Wegener, “Optical phase cloaking of 700 nm light waves in the far field by a three-dimensional carpet cloak,” Phys. Rev. Lett.107(17), 173901 (2011).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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2012 (1)

J. Wang, Y. Xu, H. Chen, and B. Zhang, “Ultraviolet dielectric hyperlens with layered graphene and boron nitride,” J. Mater. Chem.22(31), 15863–15868 (2012).
[CrossRef]

2011 (4)

U. Leonhardt, “To invisibility and beyond,” Nature471(7338), 292–293 (2011).
[CrossRef] [PubMed]

B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett.106(3), 033901 (2011).
[CrossRef] [PubMed]

X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nat Commun2, 176 (2011).
[CrossRef] [PubMed]

T. Ergin, J. Fischer, and M. Wegener, “Optical phase cloaking of 700 nm light waves in the far field by a three-dimensional carpet cloak,” Phys. Rev. Lett.107(17), 173901 (2011).
[CrossRef] [PubMed]

2010 (2)

B. Zhang, T. Chan, and B. I. Wu, “Lateral shift makes a ground-plane cloak detectable,” Phys. Rev. Lett.104(23), 233903 (2010).
[CrossRef] [PubMed]

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science328(5976), 337–339 (2010).
[CrossRef] [PubMed]

2009 (3)

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater.8(7), 568–571 (2009).
[CrossRef] [PubMed]

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics3(8), 461–463 (2009).
[CrossRef]

U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science323(5910), 110–112 (2009).
[CrossRef] [PubMed]

2008 (1)

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett.101(20), 203901 (2008).
[CrossRef] [PubMed]

2007 (2)

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315(5819), 1686 (2007).
[CrossRef] [PubMed]

Y. Luo, L. J. Arauz, J. E. Castillo, J. K. Barton, and R. K. Kostuk, “Parallel optical coherence tomography system,” Appl. Opt.46(34), 8291–8297 (2007).
[CrossRef] [PubMed]

2006 (4)

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express14(18), 8247–8256 (2006).
[CrossRef] [PubMed]

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations,” Phys. Rev. B74(7), 075103 (2006).
[CrossRef]

U. Leonhardt, “Optical conformal mapping,” Science312(5781), 1777–1780 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

2000 (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

1999 (1)

J. M. Schmitt, “Optical coherence tomography (OCT): a review,” IEEE J. Sel. Top. Quantum Electron.5(4), 1205–1215 (1999).
[CrossRef]

1961 (1)

L. S. Dolin, “On a possibility of comparing three-dimensional electromagnetic systems with inhomogeneous filling,” Izv. Vyss. Ucebn. Zaved. Radiofiz.4, 964–967 (1961).

Alekseyev, L. V.

Arauz, L. J.

Barbastathis, G.

B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett.106(3), 033901 (2011).
[CrossRef] [PubMed]

Bartal, G.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater.8(7), 568–571 (2009).
[CrossRef] [PubMed]

Barton, J. K.

Brenner, P.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science328(5976), 337–339 (2010).
[CrossRef] [PubMed]

Cardenas, J.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics3(8), 461–463 (2009).
[CrossRef]

Castillo, J. E.

Chan, T.

B. Zhang, T. Chan, and B. I. Wu, “Lateral shift makes a ground-plane cloak detectable,” Phys. Rev. Lett.104(23), 233903 (2010).
[CrossRef] [PubMed]

Chen, H.

J. Wang, Y. Xu, H. Chen, and B. Zhang, “Ultraviolet dielectric hyperlens with layered graphene and boron nitride,” J. Mater. Chem.22(31), 15863–15868 (2012).
[CrossRef]

Chen, X.

X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nat Commun2, 176 (2011).
[CrossRef] [PubMed]

Dolin, L. S.

L. S. Dolin, “On a possibility of comparing three-dimensional electromagnetic systems with inhomogeneous filling,” Izv. Vyss. Ucebn. Zaved. Radiofiz.4, 964–967 (1961).

Engheta, N.

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations,” Phys. Rev. B74(7), 075103 (2006).
[CrossRef]

Ergin, T.

T. Ergin, J. Fischer, and M. Wegener, “Optical phase cloaking of 700 nm light waves in the far field by a three-dimensional carpet cloak,” Phys. Rev. Lett.107(17), 173901 (2011).
[CrossRef] [PubMed]

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science328(5976), 337–339 (2010).
[CrossRef] [PubMed]

Fischer, J.

T. Ergin, J. Fischer, and M. Wegener, “Optical phase cloaking of 700 nm light waves in the far field by a three-dimensional carpet cloak,” Phys. Rev. Lett.107(17), 173901 (2011).
[CrossRef] [PubMed]

Gabrielli, L. H.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics3(8), 461–463 (2009).
[CrossRef]

Jacob, Z.

Jiang, K.

X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nat Commun2, 176 (2011).
[CrossRef] [PubMed]

Kostuk, R. K.

Lee, H.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315(5819), 1686 (2007).
[CrossRef] [PubMed]

Leonhardt, U.

U. Leonhardt, “To invisibility and beyond,” Nature471(7338), 292–293 (2011).
[CrossRef] [PubMed]

U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science323(5910), 110–112 (2009).
[CrossRef] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science312(5781), 1777–1780 (2006).
[CrossRef] [PubMed]

Li, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater.8(7), 568–571 (2009).
[CrossRef] [PubMed]

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett.101(20), 203901 (2008).
[CrossRef] [PubMed]

Lipson, M.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics3(8), 461–463 (2009).
[CrossRef]

Liu, X.

B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett.106(3), 033901 (2011).
[CrossRef] [PubMed]

Liu, Z.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315(5819), 1686 (2007).
[CrossRef] [PubMed]

Luo, Y.

X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nat Commun2, 176 (2011).
[CrossRef] [PubMed]

B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett.106(3), 033901 (2011).
[CrossRef] [PubMed]

Y. Luo, L. J. Arauz, J. E. Castillo, J. K. Barton, and R. K. Kostuk, “Parallel optical coherence tomography system,” Appl. Opt.46(34), 8291–8297 (2007).
[CrossRef] [PubMed]

Narimanov, E.

Pendry, J. B.

X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nat Commun2, 176 (2011).
[CrossRef] [PubMed]

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science328(5976), 337–339 (2010).
[CrossRef] [PubMed]

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett.101(20), 203901 (2008).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

Poitras, C. B.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics3(8), 461–463 (2009).
[CrossRef]

Salandrino, A.

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations,” Phys. Rev. B74(7), 075103 (2006).
[CrossRef]

Schmitt, J. M.

J. M. Schmitt, “Optical coherence tomography (OCT): a review,” IEEE J. Sel. Top. Quantum Electron.5(4), 1205–1215 (1999).
[CrossRef]

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

Smith, D. R.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

Stenger, N.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science328(5976), 337–339 (2010).
[CrossRef] [PubMed]

Sun, C.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315(5819), 1686 (2007).
[CrossRef] [PubMed]

Tyc, T.

U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science323(5910), 110–112 (2009).
[CrossRef] [PubMed]

Valentine, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater.8(7), 568–571 (2009).
[CrossRef] [PubMed]

Wang, J.

J. Wang, Y. Xu, H. Chen, and B. Zhang, “Ultraviolet dielectric hyperlens with layered graphene and boron nitride,” J. Mater. Chem.22(31), 15863–15868 (2012).
[CrossRef]

Wegener, M.

T. Ergin, J. Fischer, and M. Wegener, “Optical phase cloaking of 700 nm light waves in the far field by a three-dimensional carpet cloak,” Phys. Rev. Lett.107(17), 173901 (2011).
[CrossRef] [PubMed]

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science328(5976), 337–339 (2010).
[CrossRef] [PubMed]

Wu, B. I.

B. Zhang, T. Chan, and B. I. Wu, “Lateral shift makes a ground-plane cloak detectable,” Phys. Rev. Lett.104(23), 233903 (2010).
[CrossRef] [PubMed]

Xiong, Y.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315(5819), 1686 (2007).
[CrossRef] [PubMed]

Xu, Y.

J. Wang, Y. Xu, H. Chen, and B. Zhang, “Ultraviolet dielectric hyperlens with layered graphene and boron nitride,” J. Mater. Chem.22(31), 15863–15868 (2012).
[CrossRef]

Zentgraf, T.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater.8(7), 568–571 (2009).
[CrossRef] [PubMed]

Zhang, B.

J. Wang, Y. Xu, H. Chen, and B. Zhang, “Ultraviolet dielectric hyperlens with layered graphene and boron nitride,” J. Mater. Chem.22(31), 15863–15868 (2012).
[CrossRef]

B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett.106(3), 033901 (2011).
[CrossRef] [PubMed]

B. Zhang, T. Chan, and B. I. Wu, “Lateral shift makes a ground-plane cloak detectable,” Phys. Rev. Lett.104(23), 233903 (2010).
[CrossRef] [PubMed]

Zhang, J.

X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nat Commun2, 176 (2011).
[CrossRef] [PubMed]

Zhang, S.

X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nat Commun2, 176 (2011).
[CrossRef] [PubMed]

Zhang, X.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater.8(7), 568–571 (2009).
[CrossRef] [PubMed]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315(5819), 1686 (2007).
[CrossRef] [PubMed]

Appl. Opt. (1)

IEEE J. Sel. Top. Quantum Electron. (1)

J. M. Schmitt, “Optical coherence tomography (OCT): a review,” IEEE J. Sel. Top. Quantum Electron.5(4), 1205–1215 (1999).
[CrossRef]

Izv. Vyss. Ucebn. Zaved. Radiofiz. (1)

L. S. Dolin, “On a possibility of comparing three-dimensional electromagnetic systems with inhomogeneous filling,” Izv. Vyss. Ucebn. Zaved. Radiofiz.4, 964–967 (1961).

J. Mater. Chem. (1)

J. Wang, Y. Xu, H. Chen, and B. Zhang, “Ultraviolet dielectric hyperlens with layered graphene and boron nitride,” J. Mater. Chem.22(31), 15863–15868 (2012).
[CrossRef]

Nat Commun (1)

X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nat Commun2, 176 (2011).
[CrossRef] [PubMed]

Nat. Mater. (1)

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater.8(7), 568–571 (2009).
[CrossRef] [PubMed]

Nat. Photonics (1)

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics3(8), 461–463 (2009).
[CrossRef]

Nature (1)

U. Leonhardt, “To invisibility and beyond,” Nature471(7338), 292–293 (2011).
[CrossRef] [PubMed]

Opt. Express (1)

Phys. Rev. B (1)

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations,” Phys. Rev. B74(7), 075103 (2006).
[CrossRef]

Phys. Rev. Lett. (5)

T. Ergin, J. Fischer, and M. Wegener, “Optical phase cloaking of 700 nm light waves in the far field by a three-dimensional carpet cloak,” Phys. Rev. Lett.107(17), 173901 (2011).
[CrossRef] [PubMed]

B. Zhang, T. Chan, and B. I. Wu, “Lateral shift makes a ground-plane cloak detectable,” Phys. Rev. Lett.104(23), 233903 (2010).
[CrossRef] [PubMed]

B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett.106(3), 033901 (2011).
[CrossRef] [PubMed]

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett.101(20), 203901 (2008).
[CrossRef] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

Science (5)

U. Leonhardt, “Optical conformal mapping,” Science312(5781), 1777–1780 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315(5819), 1686 (2007).
[CrossRef] [PubMed]

U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science323(5910), 110–112 (2009).
[CrossRef] [PubMed]

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science328(5976), 337–339 (2010).
[CrossRef] [PubMed]

Other (3)

D. Malacara, Optical Shop Testing. (2nd Edition, John Wiley Inc., Wiley-Interscience, 2007).

E. P. Goodwin and J. C. Wyant, Field Guide to Interferometric Optical Testing. (SPIE Press, 2006).

E. J. Post, Formal Structure of Electromagnetics: General Covariance and Electromagnetics (Interscience, 1962).

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Figures (6)

Fig. 1
Fig. 1

(a) A cuboidal phase-preserved optical elevator is made of a single piece of natural birefringent crystal with the geometrical parameters of H = 19.8mm, L = 40mm, and W = 10mm. (b) The phase-preserved elevating design virtually lifts a sheet to a height of h = 2.2mm, enlarges the volumetric space under the sheet, and further provides the camouflage capability of exterior environment.

Fig. 2
Fig. 2

Simulations of phase-preserved optical elevating effects, performed with light propagation using the software COMSOL Multiphysics. The refractive index of the surrounding media is assumed to be 1.66. Light at 561nm with the incident angle of θ passing through the phase-preserved elevator system (a) is identical to its corresponding expansive virtual space without phase-preserved elevators (b).

Fig. 3
Fig. 3

Phase-preserved optical elevating using green, blue, and red laser beams. a, Schematic diagram of the experimental setup. The laser beam goes through a mask pattern “S•G”, and a CCD camera is monitoring the phase-preserved elevating effect above the tank. The surrounding medium in the glass tank is first filled with a transparent liquid of refractive index ~1.66. The images captured on the CCD camera when the light through “G” is reflected by a mirror sheet placed at a height of h~2.2mm. Meanwhile, light through “S•” is reflected from the plain mirror sheet without the phase-preserved elevator (b), and with the phase-preserved elevator above the plain mirror (c).

Fig. 4
Fig. 4

Optical characterization of the phase-preserved optical elevating in free space using a low-coherence interferometer with a white light source and a linear polarizer. The elevating device is rotated with different incident angle for phase measurement.

Fig. 5
Fig. 5

Schematic of the optical paths of TM wave (TM) and TE wave (TE) in the elevator within a surrounding background medium with refractive index nb.

Fig. 6
Fig. 6

Elevated height with respect to different incident angles: theoretical (black line) and experimental (red diamond) results.

Equations (4)

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x=x' y= H Hh y'+ H hH h z=z'
ε=μ= J J T det(J) =( Hh/H 0 0 0 H/Hh 0 0 0 Hh/H )
ε=( (Hh/H) 2 0 0 1 )
OPD= n 1 h cosα n b htanαsinθ=h n 1 2 n b 2 sin 2 θ

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