Abstract

We analyze and test a general approach for efficiently measuring space-variant partially coherent quasi-monochromatic fields using only amplitude masks and free propagation. A phase-space description is presented to analyze approaches of this type and understand their limitations. Three variants of the method are discussed and compared, the first using an aperture mask, the second employing both an obstacle (the exact inverse of the aperture) and a clear mask, and the last combining the previous two. We discuss the advantages and disadvantages of each option.

© 2016 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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2015 (1)

2014 (3)

C. Bamber and J. S. Lundeen, “Observing Dirac’s Classical Phase Space Analog to the Quantum State,” Phys. Rev. Lett. 112, 070405 (2014).
[Crossref]

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. L. Sánchez-Soto, “Wavefront sensing reveals optical coherence,” Nat. Commun. 5, 3275 (2014).
[Crossref] [PubMed]

J. K. Wood, K. A. Sharma, S. Cho, T. G. Brown, and M. A. Alonso, “Using shadows to measure spatial coherence,” Opt. Lett. 39, 4927–4930 (2014).
[Crossref] [PubMed]

2013 (1)

2012 (2)

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
[Crossref]

S. Cho, M. A. Alonso, and T. G. Brown, “Measurement of spatial coherence through diffraction from a transparent mask with a phase discontinuity,” Opt. Lett. 37, 2724–2726 (2012).
[Crossref] [PubMed]

2011 (2)

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

A. I. González and Y. Mejía, “Nonredundant array of apertures to measure the spatial coherence in two dimensions with only one interferogram,” J. Opt. Soc. Am. A 28, 1107–1113 (2011).
[Crossref]

2003 (1)

2000 (1)

D. L. Marks, R. A. Stack, and D. J. Brady, “Astigmatic coherence sensor for digital imaging,” Appt. Opt. 25, 1726–1728 (2000).

1997 (1)

J. Tu and S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997).
[Crossref]

1995 (1)

1993 (3)

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[Crossref] [PubMed]

M. Beck, M. G. Raymer, I. A. Walmsley, and V. Wong, “Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses,” Opt. Lett. 18, 2041–2043 (1993).
[Crossref] [PubMed]

F. Gori, M. Santarsiero, and G. Guattari, “Coherence and the spatial distribution of intensity,” J. Opt. Soc. Am. A 10, 673–679 (1993).
[Crossref]

1987 (1)

J. Bertrand and P. Bertrand, “A tomographic approach to Wigner’s function,” Found. Phys. 17, 397–405 (1987).
[Crossref]

1983 (1)

1980 (1)

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, “The Wigner function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[Crossref]

1974 (1)

1932 (1)

E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[Crossref]

Alieva, T.

Alonso, M. A.

Bamber, C.

C. Bamber and J. S. Lundeen, “Observing Dirac’s Classical Phase Space Analog to the Quantum State,” Phys. Rev. Lett. 112, 070405 (2014).
[Crossref]

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

Bartelt, H. O.

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, “The Wigner function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[Crossref]

Beck, M.

Bertrand, J.

J. Bertrand and P. Bertrand, “A tomographic approach to Wigner’s function,” Found. Phys. 17, 397–405 (1987).
[Crossref]

Bertrand, P.

J. Bertrand and P. Bertrand, “A tomographic approach to Wigner’s function,” Found. Phys. 17, 397–405 (1987).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge, 1999), Ch. 10.
[Crossref]

Brady, D. J.

D. L. Marks, R. A. Stack, and D. J. Brady, “Astigmatic coherence sensor for digital imaging,” Appt. Opt. 25, 1726–1728 (2000).

Brenner, K.-H.

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, “The Wigner function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[Crossref]

Brown, T. G.

Bulabois, J.

Cámara, A.

Cho, S.

Clarke, L.

Courjon, D.

Deschamps, J.

Divitt, S.

Dragoman, D.

Faridani, A.

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[Crossref] [PubMed]

Fleischer, J. W.

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
[Crossref]

González, A. I.

Gori, F.

Guattari, G.

Hradil, Z.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. L. Sánchez-Soto, “Wavefront sensing reveals optical coherence,” Nat. Commun. 5, 3275 (2014).
[Crossref] [PubMed]

Lohmann, A. W.

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, “The Wigner function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[Crossref]

Lundeen, J. S.

C. Bamber and J. S. Lundeen, “Observing Dirac’s Classical Phase Space Analog to the Quantum State,” Phys. Rev. Lett. 112, 070405 (2014).
[Crossref]

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

Marks, D. L.

D. L. Marks, R. A. Stack, and D. J. Brady, “Astigmatic coherence sensor for digital imaging,” Appt. Opt. 25, 1726–1728 (2000).

Mayer, A.

McAlister, D. F.

Mejía, Y.

Motka, L.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. L. Sánchez-Soto, “Wavefront sensing reveals optical coherence,” Nat. Commun. 5, 3275 (2014).
[Crossref] [PubMed]

Novotny, L.

Papoulis, A.

Patel, A.

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

Raymer, M. G.

Rehacek, J.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. L. Sánchez-Soto, “Wavefront sensing reveals optical coherence,” Nat. Commun. 5, 3275 (2014).
[Crossref] [PubMed]

Rodrigo, J. A.

Sánchez-Soto, L. L.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. L. Sánchez-Soto, “Wavefront sensing reveals optical coherence,” Nat. Commun. 5, 3275 (2014).
[Crossref] [PubMed]

Santarsiero, M.

Sharma, K. A.

Situ, G.

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
[Crossref]

Smithey, D. T.

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[Crossref] [PubMed]

Stack, R. A.

D. L. Marks, R. A. Stack, and D. J. Brady, “Astigmatic coherence sensor for digital imaging,” Appt. Opt. 25, 1726–1728 (2000).

Stewart, C.

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

Stoklasa, B.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. L. Sánchez-Soto, “Wavefront sensing reveals optical coherence,” Nat. Commun. 5, 3275 (2014).
[Crossref] [PubMed]

Sutherland, B.

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

Tamura, S.

J. Tu and S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997).
[Crossref]

Tu, J.

J. Tu and S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997).
[Crossref]

Waller, L.

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
[Crossref]

Walmsley, I. A.

Wigner, E. P.

E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge, 1999), Ch. 10.
[Crossref]

Wong, V.

Wood, J. K.

Appt. Opt. (1)

D. L. Marks, R. A. Stack, and D. J. Brady, “Astigmatic coherence sensor for digital imaging,” Appt. Opt. 25, 1726–1728 (2000).

Found. Phys. (1)

J. Bertrand and P. Bertrand, “A tomographic approach to Wigner’s function,” Found. Phys. 17, 397–405 (1987).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

Nat. Commun. (1)

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. L. Sánchez-Soto, “Wavefront sensing reveals optical coherence,” Nat. Commun. 5, 3275 (2014).
[Crossref] [PubMed]

Nat. Photonics (1)

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
[Crossref]

Nature (1)

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

Opt. Commun. (1)

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, “The Wigner function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[Crossref]

Opt. Express (1)

Opt. Lett. (4)

Optica (1)

Phys. Rev. (1)

E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[Crossref]

Phys. Rev. E (1)

J. Tu and S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997).
[Crossref]

Phys. Rev. Lett. (2)

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[Crossref] [PubMed]

C. Bamber and J. S. Lundeen, “Observing Dirac’s Classical Phase Space Analog to the Quantum State,” Phys. Rev. Lett. 112, 070405 (2014).
[Crossref]

Other (1)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge, 1999), Ch. 10.
[Crossref]

Supplementary Material (2)

NameDescription
» Visualization 1: MOV (3800 KB)      regions where the A function vanishes for varying values of z and w
» Visualization 2: MOV (4421 KB)      change in coherence function as obstacle position changes

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Figures (8)

Fig. 1
Fig. 1 Plots of (a) ��̃(1), (b) ��̃(2), and (c) ��̃(3).
Fig. 2
Fig. 2 Regions in ambiguity phase space that are zeros of ��̃(1) (blue areas and lines), ��̃(2) (yellow lines), and ��̃(3) (dotted red lines). Note that for |x′| > w the zero lines of ��̃(2) and ��̃(3) coincide. The background green-level distribution represents J ¯ ˜ ( p x / z ; x ). The inset shows how the inclination of this distribution depends on z.
Fig. 3
Fig. 3 Regions where the functions ��̃(−x′/z, x′) vanish, for λ = 532nm, z = 470mm, and for w = (a) .27mm and (b) .76mm. In both figures, blue areas are the zeros of ��̃(1), yellow line are the zeros of ��̃(2), and dotted red lines are the zeros of ��̃(3). Visualization 1 shows the effect of the variation of w and z on the location of these zeros.
Fig. 4
Fig. 4 Diagram of the experimental setup. Just after a relayed focus, the dotted line indicates the location of an obstruction (shown in the top image) inserted to create an inhomogeneous field. The bottom images indicate three examples of data taken with their respective masks displayed on the SLM.
Fig. 5
Fig. 5 Estimations of found using (a,d,g,j,m,p) an aperture, (b,e,h,k,n,q) the difference of an open mask and and obstacle, and (c,f,i,l,o,r) an aperture plus an open mask minus and obstacle. The three rows correspond to three test fields with decreasing coherence widths. The plots in (a–i) used w = 0.27 mm (so that 2w2/λz = 0.583) while those in (j–r) used w = 0.76 mm (so that 2w2/λz = 4.62). In all cases, the plot ranges are |x′, y′| ≤ 1.7 mm, and the color schemes for amplitude (normalized to peak value) and phase are as indicated by the palette on the right.
Fig. 6
Fig. 6 Measured coherence distributions and the results of using Hermite-Gaussian fitting. Parts (a) and (c) correspond to parts (c) and (f) of Fig. 5, while parts (b) and (d) show the corresponding results when the Hermite-Gaussian fitting procedure is used. In all cases, the range of separations in x′ and y′ is from −1.7 mm to 1.7 mm.
Fig. 7
Fig. 7 Changes in coherence as a function of point separation x′ within a range of −0.714 mm to 0.714 mm, for nine locations of the pivot point x0 (the center of the aperture/obstacle) over the test plane with a spacing of 1.425mm. Visualization 2 shows an animation with a finer sampling in x0 with spacing of 0.45 mm.
Fig. 8
Fig. 8 (a) The basic estimate for [also shown in Fig. 7(e)] and (b) a corrected estimate that includes the corrections in the second term of Eq. (19).

Equations (22)

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I A ( x 0 ; x ) = 1 λ 2 z 2 J ( x 1 ; x 2 ) A * ( x 1 x 0 ) A ( x 2 x 0 ) exp [ i k ( x 2 x ) 2 ( x 1 x ) 2 2 z ] d 2 x 1 d 2 x 2 ,
I A ( x 0 ; x ) = 1 λ 2 z 2 J ¯ ( x 0 + τ ; x ) A * ( τ x 2 ) A ( τ + x 2 ) exp [ i k ( x 0 + τ x ) x z ] d 2 τ d 2 x .
Δ ( x 0 ; x ) = 1 λ 2 z 2 J ¯ ( x 0 + τ ; x ) 𝒜 ( τ , x ) exp [ i k ( x 0 + τ x ) x z ] d 2 τ d 2 x ,
Δ ˜ ˜ ( p ; x ) = 1 λ 4 Δ ( x 0 ; x ) exp [ i k ( x x z x 0 p ) ] d 2 x d 2 x 0 ,
J ¯ ˜ ( p ; x ) = 1 λ 2 J ¯ ( x ; x ) exp ( i k x p ) d 2 x ,
𝒜 ˜ ( p ; x ) = 1 λ 2 𝒜 ( τ ; x ) exp ( i k τ p ) d 2 τ ,
Δ ˜ ˜ ( p ; x ) = J ¯ ˜ ( p x z ; x ) 𝒜 ˜ ( p ; x ) ,
𝒜 ˜ ( 1 ) ( p ; x ) = 1 λ 2 a * ( τ x 2 ) a ( τ + x 2 ) exp ( i k τ p ) d 2 τ .
𝒜 ˜ ( 2 ) ( p ; x ) = a ˜ ( p ) exp ( i k x p 2 ) + a ˜ * ( p ) exp ( i k x p 2 ) 𝒜 ˜ ( 1 ) ( p ; x ) ,
𝒜 ˜ ( 2 ) ( p ; x ) = 2 a ˜ ( p ) cos ( i k x p 2 ) 𝒜 ˜ ( 1 ) ( p ; x ) .
𝒜 ˜ ( 3 ) ( p ; x ) = a ˜ ( p ) exp ( i k x p 2 ) + a ˜ * ( p ) exp ( i k x p 2 ) = 2 a ˜ ( p ) cos ( i k x p 2 ) ,
𝒜 ˜ ( 1 ) ( p ; x ) = sin [ k ( w | x | ) p / 2 ] π p Θ ( w | x | ) ,
𝒜 ˜ ( 3 ) ( p ; x ) = 2 sin ( k w p / 2 ) π p cos ( k x p / 2 ) ,
𝒜 ˜ ( 2 ) ( p ; x ) = 𝒜 ˜ ( 3 ) ( p ; x ) 𝒜 ˜ ( 1 ) ( p ; x ) ,
𝒜 ˜ ( 1 ) ( p ; x ) = sin [ k ( w | x | p x / 2 ) ] π p x sin [ k ( w | y | p y / 2 ) ] π p y Θ ( w | x | ) Θ ( w | y | ) ,
𝒜 ˜ ( 3 ) ( p ; x ) = 2 sin ( k w p x / 2 ) π p x sin ( k w p y / 2 ) π p y cos ( k x p / 2 ) ,
𝒜 ˜ ( 2 ) ( p ; x ) = 𝒜 ˜ ( 3 ) ( p ; x ) 𝒜 ˜ ( 1 ) ( p ; x ) .
J ¯ ( x ¯ , x ) = J ¯ ˜ ( p , x ) exp ( i k x ¯ p ) d 2 p = Δ ˜ ˜ ( p + x / z ; x ) 𝒜 ˜ ( p x / z ; x ) exp ( i k x ¯ p ) d 2 p .
J ¯ ( x 0 , x ) = [ 1 + f 1 x 0 + ] Δ ˜ ( x 0 , x ) 𝒜 ˜ ( x / z ; x ) ,
f 1 ( x , z ) = p 𝒜 ˜ ( x / z ; x ) i k 𝒜 ˜ ( x / z ; x ) ,
Δ ˜ ( x 0 , x ) = 1 λ 2 Δ ( x 0 + τ , x ) exp [ i k τ x z ] d 2 x .
J ¯ ( x 0 , x ) m , n = 0 N a m n HG m ( x W ) HG n ( x W ) ,

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