Abstract

We present a method for measuring the transverse electric field profile of a beam of light which allows for direct phase retrieval. The measured values correspond, within a normalization constant, to the real and imaginary parts of the electric field in a plane normal to the direction of propagation. This technique represents a self-referencing method for probing the wavefront characteristics of light.

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References

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  1. J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474(7350), 188–191 (2011).
    [CrossRef] [PubMed]
  2. J. Primot, G. Rousset, and J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 7(9), 1598–1608 (1990).
    [CrossRef]
  3. R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature 353(6340), 144–146 (1991).
    [CrossRef]
  4. A. Zhang, C. H. Rao, Y. D. Zhang, and W. H. Jiang, “Novel detecting methods of Shack-Hartmann wavefront sensor at low light levels,” J. Phys.: Conf. Ser. 48, 190–195 (2006).
    [CrossRef]
  5. G. Sirat and D. Psaltis, “Conoscopic holography,” Opt. Lett. 10(1), 4–6 (1985).
    [CrossRef] [PubMed]
  6. K. Buse and M. Luennemann, “3D imaging: wave front sensing utilizing a birefringent crystal,” Phys. Rev. Lett. 85(16), 3385–3387 (2000).
    [CrossRef] [PubMed]
  7. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–246 (1972).
  8. F. Zernike, “How I Discovered Phase Contrast,” in Nobel Lectures, Physics (Elsevier, 1964), pp. 239–246.
  9. S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral interferometry,” Opt. Lett. 30(15), 1953–1955 (2005).
    [CrossRef] [PubMed]
  10. R. Juanola-Parramon, N. Gonzalez, and G. Molina-Terriza, “Characterization of optical beams with spiral phase interferometry,” Opt. Express 16(7), 4471–4478 (2008).
    [CrossRef] [PubMed]
  11. G. L. Abbas, V. W. S. Chan, and T. K. Yee, “Local-oscillator excess-noise suppression for homodyne and heterodyne detection,” Opt. Lett. 8(8), 419–421 (1983).
    [CrossRef] [PubMed]
  12. J. Goodman, Introduction to Fourier Optics (Roberts & Co Publishers, 2005).
  13. K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
    [CrossRef] [PubMed]

2011 (1)

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474(7350), 188–191 (2011).
[CrossRef] [PubMed]

2008 (1)

2006 (1)

A. Zhang, C. H. Rao, Y. D. Zhang, and W. H. Jiang, “Novel detecting methods of Shack-Hartmann wavefront sensor at low light levels,” J. Phys.: Conf. Ser. 48, 190–195 (2006).
[CrossRef]

2005 (1)

2000 (1)

K. Buse and M. Luennemann, “3D imaging: wave front sensing utilizing a birefringent crystal,” Phys. Rev. Lett. 85(16), 3385–3387 (2000).
[CrossRef] [PubMed]

1996 (1)

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[CrossRef] [PubMed]

1991 (1)

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature 353(6340), 144–146 (1991).
[CrossRef]

1990 (1)

1985 (1)

1983 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–246 (1972).

Abbas, G. L.

Ameer, G. A.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature 353(6340), 144–146 (1991).
[CrossRef]

Bamber, C.

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474(7350), 188–191 (2011).
[CrossRef] [PubMed]

Barnea, Z.

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[CrossRef] [PubMed]

Bernet, S.

Boeke, B. R.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature 353(6340), 144–146 (1991).
[CrossRef]

Browne, S. L.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature 353(6340), 144–146 (1991).
[CrossRef]

Buse, K.

K. Buse and M. Luennemann, “3D imaging: wave front sensing utilizing a birefringent crystal,” Phys. Rev. Lett. 85(16), 3385–3387 (2000).
[CrossRef] [PubMed]

Chan, V. W. S.

Cookson, D. J.

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[CrossRef] [PubMed]

Fontanella, J. C.

Fried, D. L.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature 353(6340), 144–146 (1991).
[CrossRef]

Fugate, R. Q.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature 353(6340), 144–146 (1991).
[CrossRef]

Fürhapter, S.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–246 (1972).

Gonzalez, N.

Gureyev, T. E.

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[CrossRef] [PubMed]

Jesacher, A.

Jiang, W. H.

A. Zhang, C. H. Rao, Y. D. Zhang, and W. H. Jiang, “Novel detecting methods of Shack-Hartmann wavefront sensor at low light levels,” J. Phys.: Conf. Ser. 48, 190–195 (2006).
[CrossRef]

Juanola-Parramon, R.

Luennemann, M.

K. Buse and M. Luennemann, “3D imaging: wave front sensing utilizing a birefringent crystal,” Phys. Rev. Lett. 85(16), 3385–3387 (2000).
[CrossRef] [PubMed]

Lundeen, J. S.

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474(7350), 188–191 (2011).
[CrossRef] [PubMed]

Molina-Terriza, G.

Nugent, K. A.

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[CrossRef] [PubMed]

Paganin, D.

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[CrossRef] [PubMed]

Patel, A.

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474(7350), 188–191 (2011).
[CrossRef] [PubMed]

Primot, J.

Psaltis, D.

Rao, C. H.

A. Zhang, C. H. Rao, Y. D. Zhang, and W. H. Jiang, “Novel detecting methods of Shack-Hartmann wavefront sensor at low light levels,” J. Phys.: Conf. Ser. 48, 190–195 (2006).
[CrossRef]

Ritsch-Marte, M.

Roberts, P. H.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature 353(6340), 144–146 (1991).
[CrossRef]

Rousset, G.

Ruane, R. E.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature 353(6340), 144–146 (1991).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–246 (1972).

Sirat, G.

Stewart, C.

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474(7350), 188–191 (2011).
[CrossRef] [PubMed]

Sutherland, B.

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474(7350), 188–191 (2011).
[CrossRef] [PubMed]

Tyler, G. A.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature 353(6340), 144–146 (1991).
[CrossRef]

Wopat, L. M.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature 353(6340), 144–146 (1991).
[CrossRef]

Yee, T. K.

Zhang, A.

A. Zhang, C. H. Rao, Y. D. Zhang, and W. H. Jiang, “Novel detecting methods of Shack-Hartmann wavefront sensor at low light levels,” J. Phys.: Conf. Ser. 48, 190–195 (2006).
[CrossRef]

Zhang, Y. D.

A. Zhang, C. H. Rao, Y. D. Zhang, and W. H. Jiang, “Novel detecting methods of Shack-Hartmann wavefront sensor at low light levels,” J. Phys.: Conf. Ser. 48, 190–195 (2006).
[CrossRef]

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

J. Phys.: Conf. Ser. (1)

A. Zhang, C. H. Rao, Y. D. Zhang, and W. H. Jiang, “Novel detecting methods of Shack-Hartmann wavefront sensor at low light levels,” J. Phys.: Conf. Ser. 48, 190–195 (2006).
[CrossRef]

Nature (2)

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474(7350), 188–191 (2011).
[CrossRef] [PubMed]

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature 353(6340), 144–146 (1991).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Optik (Stuttg.) (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–246 (1972).

Phys. Rev. Lett. (2)

K. Buse and M. Luennemann, “3D imaging: wave front sensing utilizing a birefringent crystal,” Phys. Rev. Lett. 85(16), 3385–3387 (2000).
[CrossRef] [PubMed]

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[CrossRef] [PubMed]

Other (2)

F. Zernike, “How I Discovered Phase Contrast,” in Nobel Lectures, Physics (Elsevier, 1964), pp. 239–246.

J. Goodman, Introduction to Fourier Optics (Roberts & Co Publishers, 2005).

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Figures (5)

Fig. 1
Fig. 1

Experimental setup. The output from a diode laser is injected into a single-mode fibre (SM) and exits at the focus of a collimating lens. The polarization of the beam is set to 45° with a linear polarizer (Pol) immediately after the fibre. A reverse bullseye (apodizing) filter (RB) could be inserted after the polarizer to modify the beam profile. The intention is to measure the real and imaginary parts of the transverse electric field of the beam along the x dimension at the λ/2 plate (λ/2 sliver). The lens was masked off with a rectangular aperture that transmitted a narrow strip of light across most of the beamwidth in x. This λ/2 plate can be scanned across the beam to map out the real and imaginary field profiles. A Fourier Transform (FT) lens takes the Fourier transform of the beam. Only a small portion is transmitted through a slit at the focus. The distance between the half-wave plate and the FT lens is 1m, the focal length of the FT lens (f). Likewise the pinhole is 1m from the FT lens. By measuring the 0° and 90° polarization components of the beam after a polarizing beamsplitter (PBS) with two photodiodes (PDx & PDy) and taking the difference over the sum, a measure proportional to the real part of the field is determined. If a quarter-wave (λ/4) plate is inserted before the PBS, the left-handed and right-handed components of circular polarization can be measured, and the abovementioned ratio is proportional to the imaginary part of the field.

Fig. 2
Fig. 2

Real (green dots) and imaginary (blue triangles) measurements of the transverse electric field. The intensity has been calculated as the quadrature sum of the real and imaginary parts. The error bars are statistical and equal to one standard deviation based on the variation in multiple runs.

Fig. 3
Fig. 3

Phase calculation based on real and imaginary measurements. The error bars are statistical at one standard deviation based on the variation in multiple runs.

Fig. 4
Fig. 4

Measurement with apodizing filter installed to modify beam shape. Comparison of the intensity measured by our technique (data points with error bars) with an intensity measurement made with a power meter (solid line). The error bars are statistical and equal to one standard deviation based on the variation in multiple runs.

Fig. 5
Fig. 5

Real (green dots) and imaginary (blue triangles) measurements of the transverse electric field with a glass plate halfway across the beam. The inset is the phase calculated over the central region of the measurement, showing a clear discontinuity. The intensity is calculated from the measured data.

Equations (15)

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E ( r ,t )=( ε x , ε y , ε z ) E 0 ( x,y,z )exp( i2πνt ),
TEF P ( x,y )= E 0 ( x,y,0 ).
1 2 ( cosθ+sinθ ) and 1 2 ( cosθsinθ ).
E x y ( x,y,0 )= 1 2 ( ( 1w( x,y ) )+w(x,y)( cosθ±sinθ ) ) E 0 ( x,y,0 ),
E x y ( x,y, z f )= i λf exp( i 2πf λ ) 1 2 ( ( 1w( x',y' ) )+w(x',y')( cosθ±sinθ ) ) × E 0 ( x , y ,0 )exp( i 2π λf ( x x+ y y ) )d x d y .
E x y ( 0,0, z f )= i λf exp( i 2πf λ ) 1 2 ( ( 1w( x',y' ) )+w(x',y')( cosθ±sinθ ) ) . × E 0 ( x , y ,0 )d x d y
I x y = E x y * ( 0,0, z f ) E x y ( 0,0, z f ) = 1 2 ( λf ) 2 [ | E allspace | 2 + ( cosθ1±sinθ ) 2 | E waveplate | 2 +( cosθ1±sinθ )Re( E allspace E waveplate ) ]
E allspace = allspace E 0 ( x , y ,0 )d x d y E waveplate = areaofwaveplate E 0 ( x , y ,0 )d x d y .
I x I y I x + I y = 2sinθ( Re( E allspace E waveplate )+2( cosθ1 ) | E waveplate | 2 ) | E allspace | 2 +2( cosθ1 )Re( E allspace E waveplate )2( cosθ1 ) | E waveplate | 2 .
I x I y I x + I y 2sinθRe( E allspace E waveplate ) | E allspace | 2 .
I x I y I x + I y 2sinθ( Re( E φ waveplate ) ) | E allspace | .
E R L ( 0,0, z f )= i λf exp( i 2πf λ )( ( 1±i )( E allspace +( cosθ1 ) E waveplate )+( 1i ) E waveplate )
I R L = E R L * ( 0,0, z f ) E R L ( 0,0, z f ) = 1 ( λf ) 2 [ | E allspace | 2 +( ( cosθ1 ) 2 + sin 2 θ ) | E waveplate | 2 +2( cosθ1 )Re( E allspace E waveplate ) ±2sinθ( Re( E allspace )Im( E waveplate )Im( E allspace )Re( E wavplate ) ) ]
I R I L I R + I L 2sinθ( Im( E ϕ waveplate ) ) | E allspace |
Phase=arctan( Re( E φwaveplate ) Im( E φwaveplate ) )

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