Abstract

Radially polarized beams represent an important member of the family of vector beams, in particular due to the possibility of using them to create strong and tightly focused longitudinal fields, a fundamental property that has been exploited by applications ranging from microscopy to particle acceleration. Since the properties of such a focused beam are intimately related to the Gouy phase shift, proper knowledge of its behavior is crucial. Terahertz microscopic imaging is used to extract the Gouy phase shift of the transverse and longitudinal field components of a tightly focused, radially polarized beam. Since the applied terahertz time-domain approach is capable of mapping the amplitude and phase of an electromagnetic wave in space, we are able to directly trace the evolution of the geometric phase as the wave propagates through the focus. We observe a Gouy phase shift of 2π for the transverse and of π for the longitudinal component. Our experimental procedure is universal and may be applied to determine the geometric phase of other vector beams, such as optical vortices, or even arbitrarily shaped and polarized propagating waves.

© 2016 Optical Society of America

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References

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  57. C. Kübler, R. Huber, S. Tübel, and A. Leitenstorfer, “Ultrabroadband detection of multi-terahertz field transients with GaSe electro-optic sensors: approaching the near infrared,” Appl. Phys. Lett. 85, 3360–3362 (2004).
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2015 (1)

E. A. Nanni, W. R. Huang, K.-H. Hong, K. Ravi, A. Fallahi, G. Moriena, R. J. Dwayne Miller, and F. X. Kartner, “Terahertz-driven linear electron acceleration,” Nat. Commun. 6, 8486 (2015).
[Crossref]

2014 (1)

M. Eisele, T. L. Cocker, M. A. Huber, M. Plankl, L. Viti, D. Ercolani, L. Sorba, M. S. Vitiello, and R. Huber, “Ultrafast multi-terahertz nano-spectroscopy with sub-cycle temporal resolution,” Nat. Photonics 8, 841–845 (2014).
[Crossref]

2013 (4)

S. Waselikowski, C. Fischer, J. Wallauer, and M. Walther, “Optimal plasmonic focusing on a metal disc under radially polarized terahertz illumination,” New J. Phys. 15, 075005 (2013).
[Crossref]

T. C. Petersen, D. M. Paganin, M. Weyland, T. P. Simula, S. A. Eastwood, and M. J. Morgan, “Measurement of the Gouy phase anomaly for electron waves,” Phys. Rev. A 88, 043803 (2013).
[Crossref]

X. Wang, W. Sun, Y. Cui, J. Ye, S. Feng, and Y. Zhang, “Complete presentation of the Gouy phase shift with the THz digital holography,” Opt. Express 21, 2337–2346 (2013).
[Crossref]

X. Pang and T. D. Visser, “Manifestation of the Gouy phase in strongly focused, radially polarized beams,” Opt. Express 21, 8331–8341 (2013).
[Crossref]

2012 (3)

P. Bon, B. Rolly, N. Bonod, J. Wenger, B. Stout, S. Monneret, and H. Rigneault, “Imaging the Gouy phase shift in photonic jets with a wavefront sensor,” Opt. Lett. 37, 3531–3533 (2012).
[Crossref]

M. Yi, K. Lee, J.-D. Song, and J. Ahn, “Terahertz phase microscopy in the sub-wavelength regime,” Appl. Phys. Lett. 100, 161110 (2012).
[Crossref]

S. Winnerl, R. Hubrich, M. Mittendorff, H. Schneider, and M. Helm, “Universal phase relation between longitudinal and transverse fields observed in focused terahertz beams,” New J. Phys. 14, 103049 (2012).
[Crossref]

2011 (3)

A. Adam, “Review of near-field terahertz measurement methods and their applications,” J. Infrared, Millimeter, Terahertz Waves 32, 976–1019 (2011).
[Crossref]

M. Walther and A. Bitzer, “Electromagnetic wave propagation close to microstructures studied by time and phase-resolved THz near-field imaging,” J. Infrared, Millimeter, Terahertz Waves 32, 1020–1030 (2011).
[Crossref]

A. Bitzer, A. Ortner, H. Merbold, T. Feurer, and M. Walther, “Terahertz near-field microscopy of complementary planar metamaterials: Babinet’s principle,” Opt. Express 19, 2537–2545 (2011).
[Crossref]

2010 (2)

A. Bitzer, A. Ortner, and M. Walther, “Terahertz near-field microscopy with subwavelength spatial resolution based on photoconductive antennas,” Appl. Opt. 49, E1–E6 (2010).
[Crossref]

I. da Paz, M. Nemes, S. Padua, C. Monken, and J. P. de Faria, “Indirect evidence for the Gouy phase for matter waves,” Phys. Lett. A 374, 1660–1662 (2010).
[Crossref]

2009 (5)

2008 (2)

H. P. Urbach and S. F. Pereira, “Field in focus with a maximum longitudinal electric component,” Phys. Rev. Lett. 100, 123904 (2008).
[Crossref]

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[Crossref]

2007 (8)

H. Chen, Q. Zhan, Y. Zhang, and Y.-P. Li, “The Gouy phase shift of the highly focused radially polarized beam,” Phys. Lett. A 371, 259–261 (2007).
[Crossref]

H. Kandpal, S. Raman, and R. Mehrotra, “Observation of Gouy phase anomaly with an interferometer,” Opt. Lasers Eng. 45, 249–251 (2007).
[Crossref]

J. Deibel, M. Escarra, N. Berndsen, K. Wang, and D. Mittleman, “Finite-element method simulations of guided wave phenomena at terahertz frequencies,” Proc. IEEE 95, 1624–1640 (2007).
[Crossref]

G. Chang, C. J. Divin, C.-H. Liu, S. L. Williamson, A. Galvanauskas, and T. B. Norris, “Generation of radially polarized terahertz pulses via velocity-mismatched optical rectification,” Opt. Lett. 32, 433–435 (2007).
[Crossref]

B. Hao and J. Leger, “Experimental measurement of longitudinal component in the vicinity of focused radially polarized beam,” Opt. Express 15, 3550–3556 (2007).
[Crossref]

W. Zhu, A. Agrawal, and A. Nahata, “Direct measurement of the Gouy phase shift for surface plasmon-polaritons,” Opt. Express 15, 9995–10001 (2007).
[Crossref]

M. A. Seo, A. J. L. Adam, J. H. Kang, J. W. Lee, S. C. Jeoung, Q. H. Park, P. C. M. Planken, and D. S. Kim, “Fourier-transform terahertz near-field imaging of one-dimensional slit arrays: mapping of electric-field-, magnetic-field-, and Poynting vectors,” Opt. Express 15, 11781–11789 (2007).
[Crossref]

Z. Wang, Z. Zeng, R. Li, and Z. Xu, “Measurement of Gouy phase shift by use of supercontinuum spectral interference,” Chin. Opt. Lett. 5, S183–S185 (2007).

2005 (5)

2004 (4)

G. Miyaji, N. Miyanaga, K. Tsubakimoto, K. Sueda, and K. Ohbayashi, “Intense longitudinal electric fields generated from transverse electromagnetic waves,” Appl. Phys. Lett. 84, 3855–3857 (2004).
[Crossref]

C. Kübler, R. Huber, S. Tübel, and A. Leitenstorfer, “Ultrabroadband detection of multi-terahertz field transients with GaSe electro-optic sensors: approaching the near infrared,” Appl. Phys. Lett. 85, 3360–3362 (2004).
[Crossref]

F. Lindner, G. G. Paulus, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius, and F. Krausz, “Gouy phase shift for few-cycle laser pulses,” Phys. Rev. Lett. 92, 113001 (2004).
[Crossref]

J. H. Chow, G. de Vine, M. B. Gray, and D. E. McClelland, “Measurement of Gouy phase evolution by use of spatial mode interference,” Opt. Lett. 29, 2339–2341 (2004).
[Crossref]

2003 (3)

A. Whiting, A. Abouraddy, B. Saleh, M. Teich, and J. Fourkas, “Polarization-assisted transverse and axial optical superresolution,” Opt. Express 11, 1714–1723 (2003).
[Crossref]

N. C. R. Holme, B. C. Daly, M. T. Myaing, and T. B. Norris, “Gouy phase shift of single-cycle picosecond acoustic pulses,” Appl. Phys. Lett. 83, 392–394 (2003).
[Crossref]

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref]

2002 (1)

T. Feurer, N. S. Stoyanov, D. W. Ward, and K. A. Nelson, “Direct visualization of the Gouy phase by focusing phonon polaritons,” Phys. Rev. Lett. 88, 257402 (2002).
[Crossref]

2001 (1)

2000 (3)

R. W. McGowan, R. A. Cheville, and D. Grischkowsky, “Direct observation of the Gouy phase shift in THz impulse ranging,” Appl. Phys. Lett. 76, 670–672 (2000).
[Crossref]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[Crossref]

J. Arlt and M. J. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical bottle beam,” Opt. Lett. 25, 191–193 (2000).
[Crossref]

1999 (3)

S. Hunsche, S. Feng, H. G. Winful, A. Leitenstorfer, M. C. Nuss, and E. P. Ippen, “Spatiotemporal focusing of single-cycle light pulses,” J. Opt. Soc. Am. A 16, 2025–2028 (1999).
[Crossref]

H.-C. Kim and Y. H. Lee, “Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation,” Opt. Commun. 169, 9–16 (1999).
[Crossref]

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, “Direct observation of the Gouy phase shift with single-cycle terahertz pulses,” Phys. Rev. Lett. 83, 3410–3413 (1999).
[Crossref]

1996 (2)

P. Hariharan and P. A. Robinson, “The Gouy phase shift as a geometrical quantum effect,” J. Mod. Opt. 43, 219–221 (1996).

D. G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett. 21, 9–11 (1996).
[Crossref]

1995 (1)

1994 (1)

1993 (1)

R. Simon and N. Mukunda, “Bargmann invariant and the geometry of the Guoy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
[Crossref]

1984 (1)

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. A 392, 45–57 (1984).
[Crossref]

1980 (1)

1890 (1)

L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” C. R. Acad. Sci. 110, 1251 (1890).

Abouraddy, A.

Adam, A.

A. Adam, “Review of near-field terahertz measurement methods and their applications,” J. Infrared, Millimeter, Terahertz Waves 32, 976–1019 (2011).
[Crossref]

Adam, A. J. L.

Agrawal, A.

Ahn, J.

M. Yi, K. Lee, J.-D. Song, and J. Ahn, “Terahertz phase microscopy in the sub-wavelength regime,” Appl. Phys. Lett. 100, 161110 (2012).
[Crossref]

Arlt, J.

Baltuška, A.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius, and F. Krausz, “Gouy phase shift for few-cycle laser pulses,” Phys. Rev. Lett. 92, 113001 (2004).
[Crossref]

Baumann, S. M.

Berndsen, N.

J. Deibel, M. Escarra, N. Berndsen, K. Wang, and D. Mittleman, “Finite-element method simulations of guided wave phenomena at terahertz frequencies,” Proc. IEEE 95, 1624–1640 (2007).
[Crossref]

Berry, M. V.

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. A 392, 45–57 (1984).
[Crossref]

Bitzer, A.

Bon, P.

Bonod, N.

Boyd, R. W.

Chang, G.

Chen, H.

H. Chen, Q. Zhan, Y. Zhang, and Y.-P. Li, “The Gouy phase shift of the highly focused radially polarized beam,” Phys. Lett. A 371, 259–261 (2007).
[Crossref]

Cheville, R. A.

R. W. McGowan, R. A. Cheville, and D. Grischkowsky, “Direct observation of the Gouy phase shift in THz impulse ranging,” Appl. Phys. Lett. 76, 670–672 (2000).
[Crossref]

Chong, C. T.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[Crossref]

Chow, J. H.

Cocker, T. L.

M. Eisele, T. L. Cocker, M. A. Huber, M. Plankl, L. Viti, D. Ercolani, L. Sorba, M. S. Vitiello, and R. Huber, “Ultrafast multi-terahertz nano-spectroscopy with sub-cycle temporal resolution,” Nat. Photonics 8, 841–845 (2014).
[Crossref]

Cui, Y.

da Paz, I.

I. da Paz, M. Nemes, S. Padua, C. Monken, and J. P. de Faria, “Indirect evidence for the Gouy phase for matter waves,” Phys. Lett. A 374, 1660–1662 (2010).
[Crossref]

Daly, B. C.

N. C. R. Holme, B. C. Daly, M. T. Myaing, and T. B. Norris, “Gouy phase shift of single-cycle picosecond acoustic pulses,” Appl. Phys. Lett. 83, 392–394 (2003).
[Crossref]

de Faria, J. P.

I. da Paz, M. Nemes, S. Padua, C. Monken, and J. P. de Faria, “Indirect evidence for the Gouy phase for matter waves,” Phys. Lett. A 374, 1660–1662 (2010).
[Crossref]

de Vine, G.

Deibel, J.

J. Deibel, M. Escarra, N. Berndsen, K. Wang, and D. Mittleman, “Finite-element method simulations of guided wave phenomena at terahertz frequencies,” Proc. IEEE 95, 1624–1640 (2007).
[Crossref]

J. Deibel, M. Escarra, and D. Mittleman, “Photoconductive terahertz antenna with radial symmetry,” Electron. Lett. 41, 226–228 (2005).
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G. Miyaji, N. Miyanaga, K. Tsubakimoto, K. Sueda, and K. Ohbayashi, “Intense longitudinal electric fields generated from transverse electromagnetic waves,” Appl. Phys. Lett. 84, 3855–3857 (2004).
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Supplementary Material (1)

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Figures (10)

Fig. 1.
Fig. 1.

Schematic illustration of the terahertz near-field microscope.

Fig. 2.
Fig. 2.

(a) Sketch of the radial emitter and (b) photograph of the radial emitter. The blue arrows indicate the polarization of the emitted field.

Fig. 3.
Fig. 3.

(a) Measured field map of a radially polarized terahertz pulse plotted as it passes through the focus. Color code has been normalized to the maximum field strengths. (b) Corresponding field transients at the position of highest field amplitudes. (c) Real part of the electric field amplitude plotted at a frequency of 1.35 THz. (d) Frequency dependence of the field amplitude of the pulses in (b).

Fig. 4.
Fig. 4.

Illustration of the procedure used to project the two-dimensional field maps onto only one dimension.

Fig. 5.
Fig. 5.

Transverse component of the radially polarized beam: (a) projected and integrated field data and fit and (b) comparison of fit with and without Gouy phase term.

Fig. 6.
Fig. 6.

Transverse component of the radially polarized beam: (a) Gouy phase shift at 1.35 THz and (b) compilation of several other frequencies.

Fig. 7.
Fig. 7.

Field maps in the frequency domain of the longitudinal component of the radially polarized beam at (a) 0.8 THz and (b) 1.35 THz.

Fig. 8.
Fig. 8.

Longitudinal component of the radially polarized beam: (a) projected and integrated field data and fit and (b) comparison of fit with and without Gouy phase term.

Fig. 9.
Fig. 9.

Longitudinal component of the radially polarized beam: (a) Gouy phase shift at 1.35 THz and (b) compilation of several other frequencies.

Fig. 10.
Fig. 10.

Illustration of the phase shift of the transverse and longitudinal electric field components of a radially polarized beam.

Equations (9)

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E rad ( ρ , z , t ) = E 0 w 0 w 2 ( z ) ρ ^ l 0 ( ρ , z ) · exp [ i ( k z 2 arctan z z 0 ) ] exp [ i ω t ] ,
l 0 ( ρ , z ) = ρ exp [ i k ρ 2 2 R ( z ) ] exp [ ρ 2 w 2 ( z ) ] .
ϕ Gouy ( z ) = 2 arctan z z 0 ,
E ( ρ ) = E 0 | ρ | exp ( ρ 2 w 0 2 ) ,
M ( z is ) = z is R ( z is ) = z 0 2 z is .
ρ i 2 + ( z j M ( z is ) ) 2 = R ( z is ) 2 ,
z is 3 z is ( ρ i 2 + z j 2 2 z 0 2 ) 2 z j z 0 2 = 0 .
E ( z ) = A sin ( 2 π λ z + ζ + n arctan ( z z 0 ) ) ,
E ( z ) = A sin ( 2 π λ z + ϕ Gouy ( z ) ) ,

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