Abstract

We present a very simple method for measuring the spatial coherence of quasi-monochromatic fields through the comparison of two measurements of the radiant intensity with and without a small obscuration at the test plane. From these measurements one can measure simultaneously the field’s coherence at all pairs of points whose centroid is the centroid of the obstacle. This method can be implemented without the need of any refractive or diffractive focusing elements.

© 2014 Optical Society of America

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References

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  21. S. Cho, M. A. Alonso, and T. G. Brown, Opt. Lett. 37, 2724 (2012).
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  22. S. Cho, “Studies on partially coherent fields and coherence measurement methods,” Doctorate thesis (University of Rochester, 2013).

2014

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. L. Sánchez-Soto, Nat. Commun. 5, 3275 (2014).
[CrossRef]

2013

2012

S. Cho, M. A. Alonso, and T. G. Brown, Opt. Lett. 37, 2724 (2012).
[CrossRef]

L. Waller, G. Situ, and J. W. Fleischer, Nat. Photonics 6, 474 (2012).
[CrossRef]

2011

2008

2007

G. A. Swartzlander and R. I. Hernandez-Aranda, Phys. Rev. Lett. 99, 163901 (2007).
[CrossRef]

C. Rydberg and J. Bengtsson, Opt. Express 15, 13613 (2007).
[CrossRef]

2006

2000

C. C. Cheng, M. G. Raymer, and H. Heier, J. Mod. Opt. 47, 1237 (2000).
[CrossRef]

D. L. Marks, R. A. Stack, and D. J. Brady, Appl. Opt. 25, 1726 (2000).

1999

1998

1996

1994

M. G. Raymer, M. Beck, and D. F. McAlister, Phys. Rev. Lett. 72, 1137 (1994).
[CrossRef]

1992

K. A. Nugent, Phys. Rev. Lett. 68, 2261 (1992).
[CrossRef]

1982

1980

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, Opt. Commun 32, 32 (1980).
[CrossRef]

1938

F. Zernike, Physica 5, 785 (1938).
[CrossRef]

Alieva, T.

Alonso, M. A.

Baltes, K. M.

Barbastathis, G.

Bartelt, H. O.

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, Opt. Commun 32, 32 (1980).
[CrossRef]

Beck, M.

M. G. Raymer, M. Beck, and D. F. McAlister, Phys. Rev. Lett. 72, 1137 (1994).
[CrossRef]

Bengtsson, J.

Borghi, R.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), Chap. 10.

Brady, D. J.

Brenner, K.-H.

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, Opt. Commun 32, 32 (1980).
[CrossRef]

Brown, D. P.

Brown, T. G.

Cámara, A.

Cheng, C. C.

C. C. Cheng, M. G. Raymer, and H. Heier, J. Mod. Opt. 47, 1237 (2000).
[CrossRef]

Cho, S.

S. Cho, M. A. Alonso, and T. G. Brown, Opt. Lett. 37, 2724 (2012).
[CrossRef]

S. Cho, “Studies on partially coherent fields and coherence measurement methods,” Doctorate thesis (University of Rochester, 2013).

Fleischer, J. W.

L. Waller, G. Situ, and J. W. Fleischer, Nat. Photonics 6, 474 (2012).
[CrossRef]

González, A. I.

Heier, H.

C. C. Cheng, M. G. Raymer, and H. Heier, J. Mod. Opt. 47, 1237 (2000).
[CrossRef]

Hernandez-Aranda, R. I.

G. A. Swartzlander and R. I. Hernandez-Aranda, Phys. Rev. Lett. 99, 163901 (2007).
[CrossRef]

Hradil, Z.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. L. Sánchez-Soto, Nat. Commun. 5, 3275 (2014).
[CrossRef]

Iaconis, C.

Jauch, H. P.

Konforti, N.

Lohmann, A. W.

D. Mendlovic, G. Shabtay, A. W. Lohmann, and N. Konforti, Opt. Lett. 23, 1084 (1998).
[CrossRef]

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, Opt. Commun 32, 32 (1980).
[CrossRef]

Marks, D. L.

McAlister, D. F.

M. G. Raymer, M. Beck, and D. F. McAlister, Phys. Rev. Lett. 72, 1137 (1994).
[CrossRef]

Mejía, Y.

Mendlovic, D.

Motka, L.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. L. Sánchez-Soto, Nat. Commun. 5, 3275 (2014).
[CrossRef]

Nugent, K. A.

K. A. Nugent, Phys. Rev. Lett. 68, 2261 (1992).
[CrossRef]

Petruccelli, J. C.

Raymer, M. G.

C. C. Cheng, M. G. Raymer, and H. Heier, J. Mod. Opt. 47, 1237 (2000).
[CrossRef]

M. G. Raymer, M. Beck, and D. F. McAlister, Phys. Rev. Lett. 72, 1137 (1994).
[CrossRef]

Rehacek, J.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. L. Sánchez-Soto, Nat. Commun. 5, 3275 (2014).
[CrossRef]

Rodrigo, J. A.

Rydberg, C.

Sánchez-Soto, L. L.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. L. Sánchez-Soto, Nat. Commun. 5, 3275 (2014).
[CrossRef]

Santarsiero, M.

Shabtay, G.

Situ, G.

L. Waller, G. Situ, and J. W. Fleischer, Nat. Photonics 6, 474 (2012).
[CrossRef]

Stack, R. A.

Stoklasa, B.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. L. Sánchez-Soto, Nat. Commun. 5, 3275 (2014).
[CrossRef]

Swartzlander, G. A.

G. A. Swartzlander and R. I. Hernandez-Aranda, Phys. Rev. Lett. 99, 163901 (2007).
[CrossRef]

Tian, L.

Waller, L.

L. Waller, G. Situ, and J. W. Fleischer, Nat. Photonics 6, 474 (2012).
[CrossRef]

Walmsley, I. A.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), Chap. 10.

Zernike, F.

F. Zernike, Physica 5, 785 (1938).
[CrossRef]

Appl. Opt.

J. Mod. Opt.

C. C. Cheng, M. G. Raymer, and H. Heier, J. Mod. Opt. 47, 1237 (2000).
[CrossRef]

J. Opt. Soc. Am. A

Nat. Commun.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. L. Sánchez-Soto, Nat. Commun. 5, 3275 (2014).
[CrossRef]

Nat. Photonics

L. Waller, G. Situ, and J. W. Fleischer, Nat. Photonics 6, 474 (2012).
[CrossRef]

Opt. Commun

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, Opt. Commun 32, 32 (1980).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

G. A. Swartzlander and R. I. Hernandez-Aranda, Phys. Rev. Lett. 99, 163901 (2007).
[CrossRef]

K. A. Nugent, Phys. Rev. Lett. 68, 2261 (1992).
[CrossRef]

M. G. Raymer, M. Beck, and D. F. McAlister, Phys. Rev. Lett. 72, 1137 (1994).
[CrossRef]

Physica

F. Zernike, Physica 5, 785 (1938).
[CrossRef]

Other

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), Chap. 10.

S. Cho, “Studies on partially coherent fields and coherence measurement methods,” Doctorate thesis (University of Rochester, 2013).

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Figures (3)

Fig. 1.
Fig. 1.

(a) Autocorrelation A0 for window functions with rectangular shape A(x)=rect(x/w) (black) and Gaussian shape A(x)=exp(πx2/w2) (gray). (b), (c) Function A¯0(x) for (b) a binary phase mask A(x)=sgn(x) and (c) a transparent mask with a localized obscuration of rectangular shape a(x)=rect(x/w) (black) or Gaussian shape a(x)=exp(πx2/w2) (gray).

Fig. 2.
Fig. 2.

(a) Diagram of the optical set-up. (b)–(g) Real parts of J¯0(0,x)/J¯0(0,0) obtained by using square obscurations of side (b) 1.855 mm and (d), (f), and (g) 3.81 mm, and circular obscurations of diameter (d) 1.855 mm and (e) 3.81 mm. In (b)–(e) the obscuration was at the center of the SLM, while in (f)/(g) it was shifted to the left/right by 2.93 mm. The insets in (b)–(g) show the measured intensities without and with the obscuration.

Fig. 3.
Fig. 3.

Plots of |J¯0(0,y¯,x,y)/J¯0(0,y¯,0,0)|, for (a) x=0 and (b) y=0. The curves in the insets show three slices corresponding to y¯=3.71 (black), 0 (gray), and 3.71 mm (pale gray), respectively.

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

I(p)=k2πJ(x1;x2)exp[ik(x2x1)p]dx1dx2=k2πJ¯(x¯;x)exp(ikxp)dxdx¯,
IA(p;x0)=k2πA*(τx2)A(τ+x2)×J¯(x0+τ;x)exp(ikxp)dxdτ,
J¯(x0+τ;x)=n=0τnn!nJ¯x¯n(x0;x).
IA=k2πn=0nJ¯x¯n(x0;x)An(x)exp(ikxp)dx,
An(x)=τnn!A*(τx2)A(τ+x2)dτ.
J¯(x0;x)A0(x)IA(p;x0)exp(ikxp)dp.
Δ(p;x0)=I(p)IA(p;x0)=k2πn=0nJ¯x¯n(x0;x)A¯n(x)exp(ikxp)dx,
A¯n(x)=τnn![1A*(τx2)A(τ+x2)]dτ.
n=0nJ¯x¯n(x0;x)A¯n(x)=Δ(p;x0)exp(ikxp)dp.
A¯n(x)=1n!|x|/2|x|/2τndτ=[1+(1)n]|x|n+1(n+1)!.
J¯(x0;x)A¯0(x)Δ(p;x0)exp(ikxp)dp.
J¯(x¯;x)J¯0(x¯;x)=Δ(p;x¯)exp(ikxp)dpA¯0(x),
A¯0(x)=2a(x)dxa(τx2)a(τ+x2)dτ,
J¯N(x¯;x)=J¯0(x¯;x)n=1NnJ¯N1x¯n(x¯;x)A¯n(x)A¯0(x).
J¯2(x¯;x)=J¯0(x¯;x)2J¯0x¯2(x¯;x)A¯2(x)A¯0(x).
Δ(p;x0)=I(p)IA(p;x0)=k24π2n,m=0n+mJ¯x¯ny¯m(x0;x)A¯n,m(x)×exp(ikx·p)dxdy,
A¯n,m(x)=τnηmn!m![a(τx2)+a(τ+x2)a(τx2)a(τ+x2)]dτdη,
J¯0(x¯;x)=Δ(p;x¯)exp(ikx·p)dpdqA¯0,0(x),
J¯N(x¯;x)=J¯0(x¯;x)n=1Nn=0nnJ¯N1x¯ny¯nn(x¯;x)A¯n,nn(x)A¯0,0(x).

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