Abstract

Electromagnetic waves propagating in air are visually observed with phase evolution in real time by live electro-optic imaging technique. We show how geometrical and crystallographic arrangements of an electro-optic sensor plate enable the realization of the real-time visual observation of traveling 100-GHz electromagnetic waves. For this purpose, a generation technique for a 100-GHz optical local oscillator signal at 780 nm was newly developed, whose optical wavelength is suitable for the ultra-parallel RF electric field data acquisition by a Si-CMOS image sensor.

© 2010 OSA

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References

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  1. G. S. Settles, Schlieren and shadowgraph techniques: Visualizing phenomena in transparent media (Springer-Verlag, 2001).
  2. T. Kubota and Y. Awatsuji, “Observation of light propagation by holography with a picosecond pulsed laser,” Opt. Lett. 27(10), 815–817 (2002).
    [CrossRef]
  3. R. Muthupillai, D. J. Lomas, P. J. Rossman, J. F. Greenleaf, A. Manduca, and R. L. Ehman, “Magnetic resonance elastography by direct visualization of propagating acoustic strain waves,” Science 269(5232), 1854–1857 (1995).
    [CrossRef] [PubMed]
  4. A. Taflove, and S. C. Hagness, Computational Electrodynamics; the finite-difference time-domain method, 3rd ed. (Artech House, 2005).
    [PubMed]
  5. S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett. 90(10), 107402 (2003).
    [CrossRef] [PubMed]
  6. 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(5801), 977–980 (2006).
    [CrossRef] [PubMed]
  7. J. A. Valdmanis, G. Mourou, and C. W. Gabel, “Picosecond electro-optic sampling system,” Appl. Phys. Lett. 41(3), 211–212 (1982).
    [CrossRef]
  8. K. Yang, G. David, S. V. Robertson, J. F. Whitaker, and L. P. B. Katehi, “Electrooptic Mapping of Near-Field Distributions in Integrated Microwave Circuits,” IEEE Trans. Microw. Theory Tech. 46(12), 2338–2343 (1998).
    [CrossRef]
  9. K. Sasagawa and M. Tsuchiya, “Real-time monitoring system of RF near-field distribution images on the basis of 64-channel parallel electro-optic data acquisition,” IEICE Electron. Express 2(24), 600–606 (2005).
    [CrossRef]
  10. K. Sasagawa, A. Kanno, T. Kawanishi, and M. Tsuchiya, “Live electro-optic imaging system based on ultra-parallel photonic heterodyne for microwave near-fields,” IEEE Trans. Microw. Theory Tech. 55(12), 2782–2791 (2007).
    [CrossRef]
  11. K. Sasagawa, A. Kanno, and M. Tsuchiya, “Instantaneous visualization of K-band electric near-fields by live electrooptic imaging system based on double sideband suppressed carrier modulation,” J. Lightwave Technol. 26(15), 2782–2788 (2008).
    [CrossRef]
  12. M. Tsuchiya, K. Sasagawa, and T. Shiozawa, “Real-time observations and analyses of RF wave propagations by live electrooptic imaging camera,” in Proceedings of 39th European Microwave Conf. (Rome, Italy, 2009) pp. 787–790.
  13. K. Sasagawa, A. Kanno, and M. Tsuchiya, “Real-time digital signal processing for live electro-optic imaging,” Opt. Express 17(18), 15641–15651 (2009).
    [CrossRef] [PubMed]
  14. Live electrooptic imaging camera Web site: http://lei-camera.nict.go.jp/
  15. A. Kanno, K. Sasagawa, and M. Tsuchiya, “W-band live electro-optic imaging system,” in Proceedings of 38th European Microwave Conf. (Amsterdam, The Netherland, 2008) pp. 369–372.
  16. A. Kanno, K. Sasagawa, and M. Tsuchiya, “Phase-resolved visualization of 100 GHz traveling electromagnetic waves by an EO imaging method,” in Proceedings of IEEE conference on Laser and Electro-Optics Society 2008 Annual Meeting (Institute of Electrical and Electronics Engineers, New York, 2008), pp. 218–219.
  17. M. Tsuchiya, A. Kanno, K. Sasagawa, and T. Shiozawa, “Image and/or movie analyses of 100-GHz traveling waves on the basis of real-time observation with a live electrooptic imaging camera,” IEEE Trans. Microw. Theory Tech. 57(12), 3373–3379 (2009).
    [CrossRef]
  18. S. Namba, “Electro-optical effect of zincblende,” J. Opt. Soc. Am. 51(1), 76–79 (1961).
    [CrossRef]
  19. J. Macario, P. Yao, R. Shireen, C. A. Schuetz, S. Shi, and D. W. Prather, “Development of Electro-Optic Phase Modulator for 94 GHz Imaging System,” J. Lightwave Technol. 27(24), 5698–5703 (2009).
    [CrossRef]
  20. K. Sasagawa, A. Kanno, and M. Tsuchiya, “W-band photonic signal generation with carrier and unnecessary sidebands suppressed by second harmonic generation,” in Proceedings of IEEE conference on Laser and Electro-Optics Society 2008 Annual Meeting (Institute of Electrical and Electronics Engineers, New York, 2008), pp. 348–349.
  21. A. Kanno and his colleagues are preparing a paper to describe the optical two-tone signal generation method based on second harmonic generation of modulated light signals.

2009 (3)

2008 (1)

2007 (1)

K. Sasagawa, A. Kanno, T. Kawanishi, and M. Tsuchiya, “Live electro-optic imaging system based on ultra-parallel photonic heterodyne for microwave near-fields,” IEEE Trans. Microw. Theory Tech. 55(12), 2782–2791 (2007).
[CrossRef]

2006 (1)

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(5801), 977–980 (2006).
[CrossRef] [PubMed]

2005 (1)

K. Sasagawa and M. Tsuchiya, “Real-time monitoring system of RF near-field distribution images on the basis of 64-channel parallel electro-optic data acquisition,” IEICE Electron. Express 2(24), 600–606 (2005).
[CrossRef]

2003 (1)

S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett. 90(10), 107402 (2003).
[CrossRef] [PubMed]

2002 (1)

1998 (1)

K. Yang, G. David, S. V. Robertson, J. F. Whitaker, and L. P. B. Katehi, “Electrooptic Mapping of Near-Field Distributions in Integrated Microwave Circuits,” IEEE Trans. Microw. Theory Tech. 46(12), 2338–2343 (1998).
[CrossRef]

1995 (1)

R. Muthupillai, D. J. Lomas, P. J. Rossman, J. F. Greenleaf, A. Manduca, and R. L. Ehman, “Magnetic resonance elastography by direct visualization of propagating acoustic strain waves,” Science 269(5232), 1854–1857 (1995).
[CrossRef] [PubMed]

1982 (1)

J. A. Valdmanis, G. Mourou, and C. W. Gabel, “Picosecond electro-optic sampling system,” Appl. Phys. Lett. 41(3), 211–212 (1982).
[CrossRef]

1961 (1)

Awatsuji, Y.

Cummer, S. A.

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(5801), 977–980 (2006).
[CrossRef] [PubMed]

David, G.

K. Yang, G. David, S. V. Robertson, J. F. Whitaker, and L. P. B. Katehi, “Electrooptic Mapping of Near-Field Distributions in Integrated Microwave Circuits,” IEEE Trans. Microw. Theory Tech. 46(12), 2338–2343 (1998).
[CrossRef]

Economou, E. N.

S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett. 90(10), 107402 (2003).
[CrossRef] [PubMed]

Ehman, R. L.

R. Muthupillai, D. J. Lomas, P. J. Rossman, J. F. Greenleaf, A. Manduca, and R. L. Ehman, “Magnetic resonance elastography by direct visualization of propagating acoustic strain waves,” Science 269(5232), 1854–1857 (1995).
[CrossRef] [PubMed]

Foteinopoulou, S.

S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett. 90(10), 107402 (2003).
[CrossRef] [PubMed]

Gabel, C. W.

J. A. Valdmanis, G. Mourou, and C. W. Gabel, “Picosecond electro-optic sampling system,” Appl. Phys. Lett. 41(3), 211–212 (1982).
[CrossRef]

Greenleaf, J. F.

R. Muthupillai, D. J. Lomas, P. J. Rossman, J. F. Greenleaf, A. Manduca, and R. L. Ehman, “Magnetic resonance elastography by direct visualization of propagating acoustic strain waves,” Science 269(5232), 1854–1857 (1995).
[CrossRef] [PubMed]

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(5801), 977–980 (2006).
[CrossRef] [PubMed]

Kanno, A.

K. Sasagawa, A. Kanno, and M. Tsuchiya, “Real-time digital signal processing for live electro-optic imaging,” Opt. Express 17(18), 15641–15651 (2009).
[CrossRef] [PubMed]

M. Tsuchiya, A. Kanno, K. Sasagawa, and T. Shiozawa, “Image and/or movie analyses of 100-GHz traveling waves on the basis of real-time observation with a live electrooptic imaging camera,” IEEE Trans. Microw. Theory Tech. 57(12), 3373–3379 (2009).
[CrossRef]

K. Sasagawa, A. Kanno, and M. Tsuchiya, “Instantaneous visualization of K-band electric near-fields by live electrooptic imaging system based on double sideband suppressed carrier modulation,” J. Lightwave Technol. 26(15), 2782–2788 (2008).
[CrossRef]

K. Sasagawa, A. Kanno, T. Kawanishi, and M. Tsuchiya, “Live electro-optic imaging system based on ultra-parallel photonic heterodyne for microwave near-fields,” IEEE Trans. Microw. Theory Tech. 55(12), 2782–2791 (2007).
[CrossRef]

Katehi, L. P. B.

K. Yang, G. David, S. V. Robertson, J. F. Whitaker, and L. P. B. Katehi, “Electrooptic Mapping of Near-Field Distributions in Integrated Microwave Circuits,” IEEE Trans. Microw. Theory Tech. 46(12), 2338–2343 (1998).
[CrossRef]

Kawanishi, T.

K. Sasagawa, A. Kanno, T. Kawanishi, and M. Tsuchiya, “Live electro-optic imaging system based on ultra-parallel photonic heterodyne for microwave near-fields,” IEEE Trans. Microw. Theory Tech. 55(12), 2782–2791 (2007).
[CrossRef]

Kubota, T.

Lomas, D. J.

R. Muthupillai, D. J. Lomas, P. J. Rossman, J. F. Greenleaf, A. Manduca, and R. L. Ehman, “Magnetic resonance elastography by direct visualization of propagating acoustic strain waves,” Science 269(5232), 1854–1857 (1995).
[CrossRef] [PubMed]

Macario, J.

Manduca, A.

R. Muthupillai, D. J. Lomas, P. J. Rossman, J. F. Greenleaf, A. Manduca, and R. L. Ehman, “Magnetic resonance elastography by direct visualization of propagating acoustic strain waves,” Science 269(5232), 1854–1857 (1995).
[CrossRef] [PubMed]

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(5801), 977–980 (2006).
[CrossRef] [PubMed]

Mourou, G.

J. A. Valdmanis, G. Mourou, and C. W. Gabel, “Picosecond electro-optic sampling system,” Appl. Phys. Lett. 41(3), 211–212 (1982).
[CrossRef]

Muthupillai, R.

R. Muthupillai, D. J. Lomas, P. J. Rossman, J. F. Greenleaf, A. Manduca, and R. L. Ehman, “Magnetic resonance elastography by direct visualization of propagating acoustic strain waves,” Science 269(5232), 1854–1857 (1995).
[CrossRef] [PubMed]

Namba, S.

Pendry, J. B.

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(5801), 977–980 (2006).
[CrossRef] [PubMed]

Prather, D. W.

Robertson, S. V.

K. Yang, G. David, S. V. Robertson, J. F. Whitaker, and L. P. B. Katehi, “Electrooptic Mapping of Near-Field Distributions in Integrated Microwave Circuits,” IEEE Trans. Microw. Theory Tech. 46(12), 2338–2343 (1998).
[CrossRef]

Rossman, P. J.

R. Muthupillai, D. J. Lomas, P. J. Rossman, J. F. Greenleaf, A. Manduca, and R. L. Ehman, “Magnetic resonance elastography by direct visualization of propagating acoustic strain waves,” Science 269(5232), 1854–1857 (1995).
[CrossRef] [PubMed]

Sasagawa, K.

K. Sasagawa, A. Kanno, and M. Tsuchiya, “Real-time digital signal processing for live electro-optic imaging,” Opt. Express 17(18), 15641–15651 (2009).
[CrossRef] [PubMed]

M. Tsuchiya, A. Kanno, K. Sasagawa, and T. Shiozawa, “Image and/or movie analyses of 100-GHz traveling waves on the basis of real-time observation with a live electrooptic imaging camera,” IEEE Trans. Microw. Theory Tech. 57(12), 3373–3379 (2009).
[CrossRef]

K. Sasagawa, A. Kanno, and M. Tsuchiya, “Instantaneous visualization of K-band electric near-fields by live electrooptic imaging system based on double sideband suppressed carrier modulation,” J. Lightwave Technol. 26(15), 2782–2788 (2008).
[CrossRef]

K. Sasagawa, A. Kanno, T. Kawanishi, and M. Tsuchiya, “Live electro-optic imaging system based on ultra-parallel photonic heterodyne for microwave near-fields,” IEEE Trans. Microw. Theory Tech. 55(12), 2782–2791 (2007).
[CrossRef]

K. Sasagawa and M. Tsuchiya, “Real-time monitoring system of RF near-field distribution images on the basis of 64-channel parallel electro-optic data acquisition,” IEICE Electron. Express 2(24), 600–606 (2005).
[CrossRef]

Schuetz, C. A.

Schurig, D.

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(5801), 977–980 (2006).
[CrossRef] [PubMed]

Shi, S.

Shiozawa, T.

M. Tsuchiya, A. Kanno, K. Sasagawa, and T. Shiozawa, “Image and/or movie analyses of 100-GHz traveling waves on the basis of real-time observation with a live electrooptic imaging camera,” IEEE Trans. Microw. Theory Tech. 57(12), 3373–3379 (2009).
[CrossRef]

Shireen, R.

Smith, D. R.

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(5801), 977–980 (2006).
[CrossRef] [PubMed]

Soukoulis, C. M.

S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett. 90(10), 107402 (2003).
[CrossRef] [PubMed]

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(5801), 977–980 (2006).
[CrossRef] [PubMed]

Tsuchiya, M.

M. Tsuchiya, A. Kanno, K. Sasagawa, and T. Shiozawa, “Image and/or movie analyses of 100-GHz traveling waves on the basis of real-time observation with a live electrooptic imaging camera,” IEEE Trans. Microw. Theory Tech. 57(12), 3373–3379 (2009).
[CrossRef]

K. Sasagawa, A. Kanno, and M. Tsuchiya, “Real-time digital signal processing for live electro-optic imaging,” Opt. Express 17(18), 15641–15651 (2009).
[CrossRef] [PubMed]

K. Sasagawa, A. Kanno, and M. Tsuchiya, “Instantaneous visualization of K-band electric near-fields by live electrooptic imaging system based on double sideband suppressed carrier modulation,” J. Lightwave Technol. 26(15), 2782–2788 (2008).
[CrossRef]

K. Sasagawa, A. Kanno, T. Kawanishi, and M. Tsuchiya, “Live electro-optic imaging system based on ultra-parallel photonic heterodyne for microwave near-fields,” IEEE Trans. Microw. Theory Tech. 55(12), 2782–2791 (2007).
[CrossRef]

K. Sasagawa and M. Tsuchiya, “Real-time monitoring system of RF near-field distribution images on the basis of 64-channel parallel electro-optic data acquisition,” IEICE Electron. Express 2(24), 600–606 (2005).
[CrossRef]

Valdmanis, J. A.

J. A. Valdmanis, G. Mourou, and C. W. Gabel, “Picosecond electro-optic sampling system,” Appl. Phys. Lett. 41(3), 211–212 (1982).
[CrossRef]

Whitaker, J. F.

K. Yang, G. David, S. V. Robertson, J. F. Whitaker, and L. P. B. Katehi, “Electrooptic Mapping of Near-Field Distributions in Integrated Microwave Circuits,” IEEE Trans. Microw. Theory Tech. 46(12), 2338–2343 (1998).
[CrossRef]

Yang, K.

K. Yang, G. David, S. V. Robertson, J. F. Whitaker, and L. P. B. Katehi, “Electrooptic Mapping of Near-Field Distributions in Integrated Microwave Circuits,” IEEE Trans. Microw. Theory Tech. 46(12), 2338–2343 (1998).
[CrossRef]

Yao, P.

Appl. Phys. Lett. (1)

J. A. Valdmanis, G. Mourou, and C. W. Gabel, “Picosecond electro-optic sampling system,” Appl. Phys. Lett. 41(3), 211–212 (1982).
[CrossRef]

IEEE Trans. Microw. Theory Tech. (3)

K. Yang, G. David, S. V. Robertson, J. F. Whitaker, and L. P. B. Katehi, “Electrooptic Mapping of Near-Field Distributions in Integrated Microwave Circuits,” IEEE Trans. Microw. Theory Tech. 46(12), 2338–2343 (1998).
[CrossRef]

M. Tsuchiya, A. Kanno, K. Sasagawa, and T. Shiozawa, “Image and/or movie analyses of 100-GHz traveling waves on the basis of real-time observation with a live electrooptic imaging camera,” IEEE Trans. Microw. Theory Tech. 57(12), 3373–3379 (2009).
[CrossRef]

K. Sasagawa, A. Kanno, T. Kawanishi, and M. Tsuchiya, “Live electro-optic imaging system based on ultra-parallel photonic heterodyne for microwave near-fields,” IEEE Trans. Microw. Theory Tech. 55(12), 2782–2791 (2007).
[CrossRef]

IEICE Electron. Express (1)

K. Sasagawa and M. Tsuchiya, “Real-time monitoring system of RF near-field distribution images on the basis of 64-channel parallel electro-optic data acquisition,” IEICE Electron. Express 2(24), 600–606 (2005).
[CrossRef]

J. Lightwave Technol. (2)

J. Opt. Soc. Am. (1)

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. Lett. (1)

S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett. 90(10), 107402 (2003).
[CrossRef] [PubMed]

Science (2)

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(5801), 977–980 (2006).
[CrossRef] [PubMed]

R. Muthupillai, D. J. Lomas, P. J. Rossman, J. F. Greenleaf, A. Manduca, and R. L. Ehman, “Magnetic resonance elastography by direct visualization of propagating acoustic strain waves,” Science 269(5232), 1854–1857 (1995).
[CrossRef] [PubMed]

Other (8)

A. Taflove, and S. C. Hagness, Computational Electrodynamics; the finite-difference time-domain method, 3rd ed. (Artech House, 2005).
[PubMed]

G. S. Settles, Schlieren and shadowgraph techniques: Visualizing phenomena in transparent media (Springer-Verlag, 2001).

M. Tsuchiya, K. Sasagawa, and T. Shiozawa, “Real-time observations and analyses of RF wave propagations by live electrooptic imaging camera,” in Proceedings of 39th European Microwave Conf. (Rome, Italy, 2009) pp. 787–790.

Live electrooptic imaging camera Web site: http://lei-camera.nict.go.jp/

A. Kanno, K. Sasagawa, and M. Tsuchiya, “W-band live electro-optic imaging system,” in Proceedings of 38th European Microwave Conf. (Amsterdam, The Netherland, 2008) pp. 369–372.

A. Kanno, K. Sasagawa, and M. Tsuchiya, “Phase-resolved visualization of 100 GHz traveling electromagnetic waves by an EO imaging method,” in Proceedings of IEEE conference on Laser and Electro-Optics Society 2008 Annual Meeting (Institute of Electrical and Electronics Engineers, New York, 2008), pp. 218–219.

K. Sasagawa, A. Kanno, and M. Tsuchiya, “W-band photonic signal generation with carrier and unnecessary sidebands suppressed by second harmonic generation,” in Proceedings of IEEE conference on Laser and Electro-Optics Society 2008 Annual Meeting (Institute of Electrical and Electronics Engineers, New York, 2008), pp. 348–349.

A. Kanno and his colleagues are preparing a paper to describe the optical two-tone signal generation method based on second harmonic generation of modulated light signals.

Supplementary Material (2)

» Media 1: MOV (451 KB)     
» Media 2: MOV (615 KB)     

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Figures (4)

Fig. 1
Fig. 1

Schematic of experimental setup for visualization of aerially traveling W-band electromagnetic waves. AR: anti-reflection coat, CCD: charge-coupled device, DSP: digital signal processor, EO: electro optic, HR: high reflection, LED: light-emitting diode, PBS: polarization beam splitter, PC: personal computer.

Fig. 2
Fig. 2

(a) Side and (b) top views for the geometrical configuration of 1-mm-thick (100) ZnTe EO sensor plate with respect to an opening of WR-10 waveguide flange are schematically drawn. The latter acts as a point source of spherically traveling W-band electromagnetic waves. The broken line in (b) indicates the metal film at the surface of an electromagnetic-wave reflector, which is also shown in a CCD image in Fig. 4(a).

Fig. 3
Fig. 3

(a) Schematic diagram of 100-GHz optical LO generator. Spectra of the modulated optical signal in the 1550-nm band are shown for (b) before and (c) after the optical band elimination filter (BEF). Optical power spectrum after the second harmonic generation is shown in (d). BEF: band elimination filter, EDFA: erbium-doped fiber amplifier, PPLN: periodically-poled lithium niobate.

Fig. 4
Fig. 4

Experimental results of visual observations of W-band electromagnetic waves traveling in air. (a) Top-view photograph of the geometrical configuration. Acquired images of phasor, magnitude and phase for an electromagnetic-wave reflector at (b) 45° and (c) at an obtuse angle. The phasor images are extracted from the corresponding real-time videos (Media 1 and Media 2 for (b) and (c), respectively). The solid lines in those images indicate the surface of the reflector as a guide for eyes.

Equations (1)

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S(i,j)=A(i,j)cos[δft+φ(i,j)]

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