Abstract

An experimental setup to measure the three-dimensional phase-intensity distribution of an infrared laser beam in the focal region has been presented. It is based on the knife-edge method to perform a tomographic reconstruction and on a transport of intensity equation-based numerical method to obtain the propagating wavefront. This experimental approach allows us to characterize a focalized laser beam when the use of image or interferometer arrangements is not possible. Thus, we have recovered intensity and phase of an aberrated beam dominated by astigmatism. The phase evolution is fully consistent with that of the beam intensity along the optical axis. Moreover, this method is based on an expansion on both the irradiance and the phase information in a series of Zernike polynomials. We have described guidelines to choose a proper set of these polynomials depending on the experimental conditions and showed that, by abiding these criteria, numerical errors can be reduced.

© 2012 OSA

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2012

2011

2010

R. L. Olmon, P. M. Krenz, B. A. Lail, L. V. Saraf, G. D. Boreman, and M. B. Raschke, “Determination of electric-field, magnetic-field, and electric-current distributions of infrared optical antennas: A near-field optical vector network analyzer,” Phys. Rev. Lett.105, 167403 (2010).
[CrossRef]

2009

G. Volpe, S. Cherukulappurath, R. J. Parramon, G. Molina-Terriza, and R. Quidant, “Controlling the Optical Near Field of Nanoantennas with Spatial Phase-Shaped Beams,” Nano Lett.93608–3611 (2009).
[CrossRef] [PubMed]

2008

E. Cubukcu, N. Yu, E. J. Smythe, L. Diehl, K. B. Crozier, and F. Capasso, “Plasmonic Laser Antennas and Related Devices,” J. Sel. T. Q. Elec.14, 1448–1461 (2008).
[CrossRef]

J. M. Rico-García, L. M. Sánchez-Brea, and J. Alda, “Application of tomographic techniques to the spatial-response mapping of antenna-coupled detectors in the visible,” Appl. Opt.47, 768–775 (2008).
[CrossRef] [PubMed]

2006

S. Barbero and L. N. Thibos, “Error analysis and correction in wavefront reconstruction from the transport-of-intensity equation,” Opt. Eng.45, 094001 (2006).
[CrossRef]

G. M. Dai, “Zernike aberration coefficients transformed to and from Fourier series coefficients for wavefront representation,” Opt. Lett.31, 501–503 (2006).
[CrossRef] [PubMed]

S. V. Pinhasi, R. Alimi, S. Eliezer, and L. Perelmutter “Fast optical computerized topography,” Phys. Lett. A374, 2798–2800 (2006).
[CrossRef]

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. H. Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A23, 1179–1200(2006).
[CrossRef]

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nature Phys.2, 101–104 (2006).
[CrossRef]

2005

J. Alda, J. M. Rico-García, J. M. López-Alonso, and G. D. Boreman, “Optical antennas for nanophotonics applications,” Nanotechnology16, S230, (2005).
[CrossRef]

J. M. López-Alonso, B. Monacelli, J. Alda, and G. D. Boreman, “Uncertainty analysis in the measurement of the spatial responsivity of infrared antennas,” Appl. Opt.44, 4557–4568 (2005).
[CrossRef] [PubMed]

2003

2001

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B72, 109–113 (2001).
[CrossRef]

1999

1998

1997

1996

1990

1983

1982

1977

1976

1972

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik35, 237–246 (1972).

Acosta, E.

Alda, J.

Alimi, R.

S. V. Pinhasi, R. Alimi, S. Eliezer, and L. Perelmutter “Fast optical computerized topography,” Phys. Lett. A374, 2798–2800 (2006).
[CrossRef]

Alonso, J.

Banzer, P.

Barbero, S.

S. Barbero and L. N. Thibos, “Error analysis and correction in wavefront reconstruction from the transport-of-intensity equation,” Opt. Eng.45, 094001 (2006).
[CrossRef]

Barty, A.

Beetz, T.

Bernabéu, E.

Boreman, G. D.

Born, M.

M. Born and E. Wolf, Principles of Optics, (Pergamon Press, 1980).

Braat, J. M.

J. M. Braat, S. Van Haver, A. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point spread functions,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2008), 349–468.
[CrossRef]

Brener, I.

Cai, Z.

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nature Phys.2, 101–104 (2006).
[CrossRef]

Capasso, F.

E. Cubukcu, N. Yu, E. J. Smythe, L. Diehl, K. B. Crozier, and F. Capasso, “Plasmonic Laser Antennas and Related Devices,” J. Sel. T. Q. Elec.14, 1448–1461 (2008).
[CrossRef]

Carney, P. S.

P. S. Carney, B. Deutsch, A. A. Govyadinov, and R. Hillenbrand, “Phase in nanooptics,” ACS Nano6, 8 – 12 (2012).
[CrossRef] [PubMed]

Chapman, H. N.

Chen, W.

Chen, X.

Cherukulappurath, S.

G. Volpe, S. Cherukulappurath, R. J. Parramon, G. Molina-Terriza, and R. Quidant, “Controlling the Optical Near Field of Nanoantennas with Spatial Phase-Shaped Beams,” Nano Lett.93608–3611 (2009).
[CrossRef] [PubMed]

Codreanu, L.

Crozier, K. B.

E. Cubukcu, N. Yu, E. J. Smythe, L. Diehl, K. B. Crozier, and F. Capasso, “Plasmonic Laser Antennas and Related Devices,” J. Sel. T. Q. Elec.14, 1448–1461 (2008).
[CrossRef]

Cubukcu, E.

E. Cubukcu, N. Yu, E. J. Smythe, L. Diehl, K. B. Crozier, and F. Capasso, “Plasmonic Laser Antennas and Related Devices,” J. Sel. T. Q. Elec.14, 1448–1461 (2008).
[CrossRef]

Cui, C.

Dai, G. M.

Deutsch, B.

P. S. Carney, B. Deutsch, A. A. Govyadinov, and R. Hillenbrand, “Phase in nanooptics,” ACS Nano6, 8 – 12 (2012).
[CrossRef] [PubMed]

Diehl, L.

E. Cubukcu, N. Yu, E. J. Smythe, L. Diehl, K. B. Crozier, and F. Capasso, “Plasmonic Laser Antennas and Related Devices,” J. Sel. T. Q. Elec.14, 1448–1461 (2008).
[CrossRef]

Dirksen, P.

J. M. Braat, S. Van Haver, A. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point spread functions,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2008), 349–468.
[CrossRef]

Dorn, R.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B72, 109–113 (2001).
[CrossRef]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B72, 109–113 (2001).
[CrossRef]

Eddins, S. L.

R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing Using MATLAB, (Gatesmark Publishing2008).

Eliezer, S.

S. V. Pinhasi, R. Alimi, S. Eliezer, and L. Perelmutter “Fast optical computerized topography,” Phys. Lett. A374, 2798–2800 (2006).
[CrossRef]

Fienup, J. R.

Firester, A. H.

Fumeaux, C.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik35, 237–246 (1972).

Ginn, J. C.

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B72, 109–113 (2001).
[CrossRef]

Gonzalez, R. C.

R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing Using MATLAB, (Gatesmark Publishing2008).

Govyadinov, A. A.

P. S. Carney, B. Deutsch, A. A. Govyadinov, and R. Hillenbrand, “Phase in nanooptics,” ACS Nano6, 8 – 12 (2012).
[CrossRef] [PubMed]

Gureyev, T. E.

Halas, N. J.

M. W. Knight, H. Sobhani, P. Nordlander, and N. J. Halas, “Photodetection with active optical antenna,” Science332, 702–704, (2011).
[CrossRef] [PubMed]

He, H.

Heller, M. E.

Herrmann, W.

Hillenbrand, R.

P. S. Carney, B. Deutsch, A. A. Govyadinov, and R. Hillenbrand, “Phase in nanooptics,” ACS Nano6, 8 – 12 (2012).
[CrossRef] [PubMed]

Howells, M. R.

Huber, C.

Jacobsen, C.

Janssen, A. E. M.

J. M. Braat, S. Van Haver, A. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point spread functions,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2008), 349–468.
[CrossRef]

Kinzel, E. C.

Kneubhl, F. K.

Knight, M. W.

M. W. Knight, H. Sobhani, P. Nordlander, and N. J. Halas, “Photodetection with active optical antenna,” Science332, 702–704, (2011).
[CrossRef] [PubMed]

Krenz, P. M.

R. L. Olmon, P. M. Krenz, B. A. Lail, L. V. Saraf, G. D. Boreman, and M. B. Raschke, “Determination of electric-field, magnetic-field, and electric-current distributions of infrared optical antennas: A near-field optical vector network analyzer,” Phys. Rev. Lett.105, 167403 (2010).
[CrossRef]

Lail, B. A.

E. C. Kinzel, J. C. Ginn, R. L. Olmon, D. J. Shelton, B. A. Lail, I. Brener, M. B. Sinclair, M. B. Raschke, and G. D. Boreman, “Phase resolved near-field mode imaging for the design of frequency-selective surfaces,” Opt. Express20, 11986–11993 (2012).
[CrossRef] [PubMed]

R. L. Olmon, P. M. Krenz, B. A. Lail, L. V. Saraf, G. D. Boreman, and M. B. Raschke, “Determination of electric-field, magnetic-field, and electric-current distributions of infrared optical antennas: A near-field optical vector network analyzer,” Phys. Rev. Lett.105, 167403 (2010).
[CrossRef]

Leuchs, G.

P. Marchenko, S. Orlov, C. Huber, P. Banzer, S. Quabis, U. Peschel, and G. Leuchs, “Interaction of highly focused vector beams with a metal knife-edge,” Opt. Express19, 7244–7249 (2011).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B72, 109–113 (2001).
[CrossRef]

López-Alonso, J. M.

J. Alda, J. M. Rico-García, J. M. López-Alonso, and G. D. Boreman, “Optical antennas for nanophotonics applications,” Nanotechnology16, S230, (2005).
[CrossRef]

J. M. López-Alonso, B. Monacelli, J. Alda, and G. D. Boreman, “Uncertainty analysis in the measurement of the spatial responsivity of infrared antennas,” Appl. Opt.44, 4557–4568 (2005).
[CrossRef] [PubMed]

Marchenko, P.

Marchesini, S.

Miao, J.

Molina-Terriza, G.

G. Volpe, S. Cherukulappurath, R. J. Parramon, G. Molina-Terriza, and R. Quidant, “Controlling the Optical Near Field of Nanoantennas with Spatial Phase-Shaped Beams,” Nano Lett.93608–3611 (2009).
[CrossRef] [PubMed]

Monacelli, B.

Noll, R. J.

Nordlander, P.

M. W. Knight, H. Sobhani, P. Nordlander, and N. J. Halas, “Photodetection with active optical antenna,” Science332, 702–704, (2011).
[CrossRef] [PubMed]

Novotny, L.

L. Novotny and N. Van Hulst, “Antennas for light,” Nature Phot.5, 83–90 (2011).
[CrossRef]

Noy, A.

Nugent, K. A.

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nature Phys.2, 101–104 (2006).
[CrossRef]

T. E. Gureyev and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A13(8), 1670–1682 (1996).
[CrossRef]

Olmon, R. L.

E. C. Kinzel, J. C. Ginn, R. L. Olmon, D. J. Shelton, B. A. Lail, I. Brener, M. B. Sinclair, M. B. Raschke, and G. D. Boreman, “Phase resolved near-field mode imaging for the design of frequency-selective surfaces,” Opt. Express20, 11986–11993 (2012).
[CrossRef] [PubMed]

R. L. Olmon, P. M. Krenz, B. A. Lail, L. V. Saraf, G. D. Boreman, and M. B. Raschke, “Determination of electric-field, magnetic-field, and electric-current distributions of infrared optical antennas: A near-field optical vector network analyzer,” Phys. Rev. Lett.105, 167403 (2010).
[CrossRef]

Orlov, S.

Parramon, R. J.

G. Volpe, S. Cherukulappurath, R. J. Parramon, G. Molina-Terriza, and R. Quidant, “Controlling the Optical Near Field of Nanoantennas with Spatial Phase-Shaped Beams,” Nano Lett.93608–3611 (2009).
[CrossRef] [PubMed]

Paterson, D.

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nature Phys.2, 101–104 (2006).
[CrossRef]

Peele, A. G.

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nature Phys.2, 101–104 (2006).
[CrossRef]

Perelmutter, L.

S. V. Pinhasi, R. Alimi, S. Eliezer, and L. Perelmutter “Fast optical computerized topography,” Phys. Lett. A374, 2798–2800 (2006).
[CrossRef]

Peschel, U.

Pinhasi, S. V.

S. V. Pinhasi, R. Alimi, S. Eliezer, and L. Perelmutter “Fast optical computerized topography,” Phys. Lett. A374, 2798–2800 (2006).
[CrossRef]

Quabis, S.

P. Marchenko, S. Orlov, C. Huber, P. Banzer, S. Quabis, U. Peschel, and G. Leuchs, “Interaction of highly focused vector beams with a metal knife-edge,” Opt. Express19, 7244–7249 (2011).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B72, 109–113 (2001).
[CrossRef]

Quidant, R.

G. Volpe, S. Cherukulappurath, R. J. Parramon, G. Molina-Terriza, and R. Quidant, “Controlling the Optical Near Field of Nanoantennas with Spatial Phase-Shaped Beams,” Nano Lett.93608–3611 (2009).
[CrossRef] [PubMed]

Quiney, H. M.

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nature Phys.2, 101–104 (2006).
[CrossRef]

Raschke, M. B.

E. C. Kinzel, J. C. Ginn, R. L. Olmon, D. J. Shelton, B. A. Lail, I. Brener, M. B. Sinclair, M. B. Raschke, and G. D. Boreman, “Phase resolved near-field mode imaging for the design of frequency-selective surfaces,” Opt. Express20, 11986–11993 (2012).
[CrossRef] [PubMed]

R. L. Olmon, P. M. Krenz, B. A. Lail, L. V. Saraf, G. D. Boreman, and M. B. Raschke, “Determination of electric-field, magnetic-field, and electric-current distributions of infrared optical antennas: A near-field optical vector network analyzer,” Phys. Rev. Lett.105, 167403 (2010).
[CrossRef]

Rico-García, J. M.

J. M. Rico-García, L. M. Sánchez-Brea, and J. Alda, “Application of tomographic techniques to the spatial-response mapping of antenna-coupled detectors in the visible,” Appl. Opt.47, 768–775 (2008).
[CrossRef] [PubMed]

J. Alda, J. M. Rico-García, J. M. López-Alonso, and G. D. Boreman, “Optical antennas for nanophotonics applications,” Nanotechnology16, S230, (2005).
[CrossRef]

Riege, S. P. H.

Rios, S.

Roddier, F.

Rosen, R.

Rothuizen, H.

Sánchez-Brea, L. M.

Saraf, L. V.

R. L. Olmon, P. M. Krenz, B. A. Lail, L. V. Saraf, G. D. Boreman, and M. B. Raschke, “Determination of electric-field, magnetic-field, and electric-current distributions of infrared optical antennas: A near-field optical vector network analyzer,” Phys. Rev. Lett.105, 167403 (2010).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik35, 237–246 (1972).

Sayre, D.

Schaefer, J. A.

Shapiro, D.

Shelton, D. J.

Sheng, P.

Sinclair, M. B.

Smythe, E. J.

E. Cubukcu, N. Yu, E. J. Smythe, L. Diehl, K. B. Crozier, and F. Capasso, “Plasmonic Laser Antennas and Related Devices,” J. Sel. T. Q. Elec.14, 1448–1461 (2008).
[CrossRef]

Sobhani, H.

M. W. Knight, H. Sobhani, P. Nordlander, and N. J. Halas, “Photodetection with active optical antenna,” Science332, 702–704, (2011).
[CrossRef] [PubMed]

Soto, M.

Spence, J. C. H.

Teague, M. R.

Thibos, L. N.

S. Barbero and L. N. Thibos, “Error analysis and correction in wavefront reconstruction from the transport-of-intensity equation,” Opt. Eng.45, 094001 (2006).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Experimental setup diagram. The inset shows the working region where the beam path is cut by the knife.

Fig. 2
Fig. 2

Normalized intensity measurement (dashed line), and deconvolved signal computed with the PSF of the detector (dotted line). Difference between both curves is plotted below.

Fig. 3
Fig. 3

(a) Normalized intensity profiles obtained at different angles (arranged in columns), also called sinogram. (b) Beam irradiance map generated from this sinogram via the inverse Radon transform. Astigmatism is clearly seen.

Fig. 4
Fig. 4

Evolution of β as a function of the radial polynomial degree n (evaluated within the range 0 < k < kmax). For the experimental conditions described in this paper kmax = 25.6. The β parameter is above 99% until n = 21.

Fig. 5
Fig. 5

Representation of Zernikes arranged according to their radial and azimuthal parameters ( Z n m). As an example, if experimental conditions are defined with nmax = 4 and |m|max = 2, only the polynomials within the gray area should be employed.

Fig. 6
Fig. 6

(a) Irradiance contour maps reconstructed using the tomographic technique. The planes are separated Δz = 108 μm. (b) Five phase maps can be retrieved from seven irradiance maps.

Fig. 7
Fig. 7

(a) Phase map reconstructed at the focal plane using N′ Zernike coefficients. (b) Difference between the phase map reconstructed with N′ and one with N (magnitude in radians).

Equations (8)

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φ = i N φ i Z i .
φ N = N f M ( N ) 1 δ z I ( N ) ,
M i j = 0 2 π 0 R I P ( ρ / R , θ ) Z i ( ρ / R , θ ) Z j ( ρ / R , θ ) ρ d ρ d θ ,
δ z I = I P + 1 ( x , y , Δ z ) I P 1 ( x , y , Δ z ) .
Z n m ( ρ , θ ) = R n m ( ρ ) Radial part { 2 cos | m | θ ( m > 0 ) 1 ( m = 0 ) 2 sin | m | θ ( m < 0 ) , Angular part
k max = R Δ ρ .
β = 0 k max | 𝔕 n ( k ) | 2 d k 0 | 𝔕 n ( k ) | 2 d k .
2 Δ θ = 2 π | m | max .

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