Abstract

A previously derived condition for the complete destructive interference of partially coherent light emerging from a trio of pinholes in an opaque screen is generalized to the case when the coherence properties of the field are asymmetric. It is shown by example that the interference condition is necessary, but not sufficient, and that the existence of complete destructive interference also depends on the intensity of light emerging from the pinholes and the system geometry; more general conditions for such interference are derived. The phase of the wave field exhibits both phase singularities and correlation singularities, and a number of nonintuitive situations in which complete destructive interference occurs are described and explained.

© 2012 Optical Society of America

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References

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  1. M. S. Soskin and M. V. Vasnetsov, “Singular optics,” in Vol. 42 of Progress in Optics, E. Wolf, ed. (Elsevier, 2001).
  2. M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” in Vol. 53 of Progress in Optics, E. Wolf, ed. (Elsevier, 2009).
  3. G. Gbur, Mathematical Methods for Optical Physics and Engineering (Cambridge University, 2011).
  4. G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117–125 (2003).
    [CrossRef]
  5. H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, “Phase singularities of the coherence functions in Young’s interference pattern,” Opt. Lett. 28, 968–970 (2003).
    [CrossRef]
  6. I. D. Maleev, D. M. Palacios, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation vortices in partially coherent light: theory,” J. Opt. Soc. Am. B 21, 1895–1900 (2004).
    [CrossRef]
  7. G. Gbur and T. D. Visser, “Phase singularities and coherence vortices in linear optical systems,” Opt. Commun. 259, 428–435(2006).
    [CrossRef]
  8. D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
    [CrossRef]
  9. W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96, 073902 (2006).
    [CrossRef]
  10. G. Gbur, T. D. Visser, and E. Wolf, “’Hidden’ singularities in partially coherent wavefields,” Pure Appl. Opt. 6, S239–S242 (2004).
    [CrossRef]
  11. G. V. Bogatyryova, C. V. Fel’de, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, and E. Wolf, “Partially coherent vortex beams with a separable phase,” Opt. Lett. 28, 878–880 (2003).
    [CrossRef]
  12. G. Gbur, T. D. Visser, and E. Wolf, “Complete destructive interference of partially coherent fields,” Opt. Commun. 239, 15–23 (2004).
    [CrossRef]
  13. D. Ambrosini, F. Gori, and D. Paoletti, “Destructive interference from three partially coherent point sources,” Opt. Commun. 254, 30–39 (2005).
    [CrossRef]
  14. L. Basano and P. Ottonello, “Complete destructive interference of partially coherent sources of acoustic waves,” Phys. Rev. Lett. 94, 173901 (2005).
    [CrossRef]
  15. E. Wolf, “Significance and measurability of the phase of a spatially coherent optical field,” Opt. Lett. 28, 5–6 (2003).
    [CrossRef]
  16. C. H. Gan and G. Gbur, “Phase and coherence singularities generated by the interference of partially coherent fields,” Opt. Commun. 280, 249–255 (2007).
    [CrossRef]
  17. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
  18. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

2007 (1)

C. H. Gan and G. Gbur, “Phase and coherence singularities generated by the interference of partially coherent fields,” Opt. Commun. 280, 249–255 (2007).
[CrossRef]

2006 (2)

G. Gbur and T. D. Visser, “Phase singularities and coherence vortices in linear optical systems,” Opt. Commun. 259, 428–435(2006).
[CrossRef]

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96, 073902 (2006).
[CrossRef]

2005 (2)

D. Ambrosini, F. Gori, and D. Paoletti, “Destructive interference from three partially coherent point sources,” Opt. Commun. 254, 30–39 (2005).
[CrossRef]

L. Basano and P. Ottonello, “Complete destructive interference of partially coherent sources of acoustic waves,” Phys. Rev. Lett. 94, 173901 (2005).
[CrossRef]

2004 (4)

G. Gbur, T. D. Visser, and E. Wolf, “’Hidden’ singularities in partially coherent wavefields,” Pure Appl. Opt. 6, S239–S242 (2004).
[CrossRef]

I. D. Maleev, D. M. Palacios, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation vortices in partially coherent light: theory,” J. Opt. Soc. Am. B 21, 1895–1900 (2004).
[CrossRef]

G. Gbur, T. D. Visser, and E. Wolf, “Complete destructive interference of partially coherent fields,” Opt. Commun. 239, 15–23 (2004).
[CrossRef]

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[CrossRef]

2003 (4)

Ambrosini, D.

D. Ambrosini, F. Gori, and D. Paoletti, “Destructive interference from three partially coherent point sources,” Opt. Commun. 254, 30–39 (2005).
[CrossRef]

Basano, L.

L. Basano and P. Ottonello, “Complete destructive interference of partially coherent sources of acoustic waves,” Phys. Rev. Lett. 94, 173901 (2005).
[CrossRef]

Bogatyryova, G. V.

Dennis, M. R.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” in Vol. 53 of Progress in Optics, E. Wolf, ed. (Elsevier, 2009).

Duan, Z.

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96, 073902 (2006).
[CrossRef]

Fel’de, C. V.

Gan, C. H.

C. H. Gan and G. Gbur, “Phase and coherence singularities generated by the interference of partially coherent fields,” Opt. Commun. 280, 249–255 (2007).
[CrossRef]

Gbur, G.

C. H. Gan and G. Gbur, “Phase and coherence singularities generated by the interference of partially coherent fields,” Opt. Commun. 280, 249–255 (2007).
[CrossRef]

G. Gbur and T. D. Visser, “Phase singularities and coherence vortices in linear optical systems,” Opt. Commun. 259, 428–435(2006).
[CrossRef]

G. Gbur, T. D. Visser, and E. Wolf, “’Hidden’ singularities in partially coherent wavefields,” Pure Appl. Opt. 6, S239–S242 (2004).
[CrossRef]

G. Gbur, T. D. Visser, and E. Wolf, “Complete destructive interference of partially coherent fields,” Opt. Commun. 239, 15–23 (2004).
[CrossRef]

H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, “Phase singularities of the coherence functions in Young’s interference pattern,” Opt. Lett. 28, 968–970 (2003).
[CrossRef]

G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117–125 (2003).
[CrossRef]

G. Gbur, Mathematical Methods for Optical Physics and Engineering (Cambridge University, 2011).

Gori, F.

D. Ambrosini, F. Gori, and D. Paoletti, “Destructive interference from three partially coherent point sources,” Opt. Commun. 254, 30–39 (2005).
[CrossRef]

Hanson, S. G.

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96, 073902 (2006).
[CrossRef]

Maleev, I. D.

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[CrossRef]

I. D. Maleev, D. M. Palacios, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation vortices in partially coherent light: theory,” J. Opt. Soc. Am. B 21, 1895–1900 (2004).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Marathay, A. S.

I. D. Maleev, D. M. Palacios, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation vortices in partially coherent light: theory,” J. Opt. Soc. Am. B 21, 1895–1900 (2004).
[CrossRef]

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[CrossRef]

Miyamoto, Y.

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96, 073902 (2006).
[CrossRef]

O’Holleran, K.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” in Vol. 53 of Progress in Optics, E. Wolf, ed. (Elsevier, 2009).

Ottonello, P.

L. Basano and P. Ottonello, “Complete destructive interference of partially coherent sources of acoustic waves,” Phys. Rev. Lett. 94, 173901 (2005).
[CrossRef]

Padgett, M. J.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” in Vol. 53 of Progress in Optics, E. Wolf, ed. (Elsevier, 2009).

Palacios, D. M.

I. D. Maleev, D. M. Palacios, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation vortices in partially coherent light: theory,” J. Opt. Soc. Am. B 21, 1895–1900 (2004).
[CrossRef]

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[CrossRef]

Paoletti, D.

D. Ambrosini, F. Gori, and D. Paoletti, “Destructive interference from three partially coherent point sources,” Opt. Commun. 254, 30–39 (2005).
[CrossRef]

Polyanskii, P. V.

Ponomarenko, S. A.

Schouten, H. F.

Soskin, M. S.

Swartzlander, G. A.

I. D. Maleev, D. M. Palacios, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation vortices in partially coherent light: theory,” J. Opt. Soc. Am. B 21, 1895–1900 (2004).
[CrossRef]

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[CrossRef]

Takeda, M.

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96, 073902 (2006).
[CrossRef]

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” in Vol. 42 of Progress in Optics, E. Wolf, ed. (Elsevier, 2001).

Visser, T. D.

G. Gbur and T. D. Visser, “Phase singularities and coherence vortices in linear optical systems,” Opt. Commun. 259, 428–435(2006).
[CrossRef]

G. Gbur, T. D. Visser, and E. Wolf, “Complete destructive interference of partially coherent fields,” Opt. Commun. 239, 15–23 (2004).
[CrossRef]

G. Gbur, T. D. Visser, and E. Wolf, “’Hidden’ singularities in partially coherent wavefields,” Pure Appl. Opt. 6, S239–S242 (2004).
[CrossRef]

H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, “Phase singularities of the coherence functions in Young’s interference pattern,” Opt. Lett. 28, 968–970 (2003).
[CrossRef]

G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117–125 (2003).
[CrossRef]

Wang, W.

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96, 073902 (2006).
[CrossRef]

Wolf, E.

G. Gbur, T. D. Visser, and E. Wolf, “’Hidden’ singularities in partially coherent wavefields,” Pure Appl. Opt. 6, S239–S242 (2004).
[CrossRef]

G. Gbur, T. D. Visser, and E. Wolf, “Complete destructive interference of partially coherent fields,” Opt. Commun. 239, 15–23 (2004).
[CrossRef]

G. V. Bogatyryova, C. V. Fel’de, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, and E. Wolf, “Partially coherent vortex beams with a separable phase,” Opt. Lett. 28, 878–880 (2003).
[CrossRef]

H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, “Phase singularities of the coherence functions in Young’s interference pattern,” Opt. Lett. 28, 968–970 (2003).
[CrossRef]

E. Wolf, “Significance and measurability of the phase of a spatially coherent optical field,” Opt. Lett. 28, 5–6 (2003).
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

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

Opt. Commun. (5)

G. Gbur and T. D. Visser, “Phase singularities and coherence vortices in linear optical systems,” Opt. Commun. 259, 428–435(2006).
[CrossRef]

G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117–125 (2003).
[CrossRef]

G. Gbur, T. D. Visser, and E. Wolf, “Complete destructive interference of partially coherent fields,” Opt. Commun. 239, 15–23 (2004).
[CrossRef]

D. Ambrosini, F. Gori, and D. Paoletti, “Destructive interference from three partially coherent point sources,” Opt. Commun. 254, 30–39 (2005).
[CrossRef]

C. H. Gan and G. Gbur, “Phase and coherence singularities generated by the interference of partially coherent fields,” Opt. Commun. 280, 249–255 (2007).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. Lett. (3)

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[CrossRef]

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96, 073902 (2006).
[CrossRef]

L. Basano and P. Ottonello, “Complete destructive interference of partially coherent sources of acoustic waves,” Phys. Rev. Lett. 94, 173901 (2005).
[CrossRef]

Pure Appl. Opt. (1)

G. Gbur, T. D. Visser, and E. Wolf, “’Hidden’ singularities in partially coherent wavefields,” Pure Appl. Opt. 6, S239–S242 (2004).
[CrossRef]

Other (5)

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” in Vol. 42 of Progress in Optics, E. Wolf, ed. (Elsevier, 2001).

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” in Vol. 53 of Progress in Optics, E. Wolf, ed. (Elsevier, 2009).

G. Gbur, Mathematical Methods for Optical Physics and Engineering (Cambridge University, 2011).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

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Figures (10)

Fig. 1.
Fig. 1.

(a) Intensity pattern of an LG01 mode in the waist plane. (b) Phase contour of the corresponding wave field in the waist plane. Here the width of the beam is w0=1mm.

Fig. 2.
Fig. 2.

Illustration of the three-pinhole geometry under consideration. The distance between the pinholes is taken to be d=1mm.

Fig. 3.
Fig. 3.

Relation of the correlations between the three pinholes in terms of μ1 and μ2.

Fig. 4.
Fig. 4.

Illustration of μ2 as a function of μ1.

Fig. 5.
Fig. 5.

(a) Intensity pattern on the observation plane (z=2m), for μ1=1/3. (b) Phase contour of the cross-spectral density on the observation plane when the reference point is at (x1,y1)=(1mm,1mm). (c) Phase contour of the cross-spectral density on the observation plane when the reference point is at (x1,y1)=(1mm,0.8mm). For all cases, μ1=1/3, and μ2=7/9.

Fig. 6.
Fig. 6.

(a) Intensity pattern on the observation plane (z=2m). (b) Phase contour of the cross-spectral density on the observation plane. Here μ1=0, μ2=1, and the reference point is at (x1,y1)=(1mm,1mm).

Fig. 7.
Fig. 7.

(a) Intensity pattern on the observation plane (z=2m). (b) Phase contour of the cross-spectral density on the observation plane. Here μ1=1/2, μ2=0, and the reference point is at (x1,y1)=(1mm,1mm).

Fig. 8.
Fig. 8.

Illustration of the values of μ12 that give complete destructive interference, as a function of μ13 and μ23.

Fig. 9.
Fig. 9.

(a) Intensity pattern and (b) phase contour of the cross-spectral density, with μ13=2/3, μ23=1/4, and μ12=0.5550. The corresponding field amplitudes are A1=0.6550, A2=0.5042, and A3=0.5627. The reference point is at (x1,y1)=(1mm,1mm).

Fig. 10.
Fig. 10.

(a) Intensity pattern and (b) phase contour of the cross-spectral density, with μ12=1/2, μ23=3/2, and μ13=0. The corresponding field amplitudes are A1=0.3536, A2=0.7071, and A3=0.6124. The reference point is at (x1,y1)=(1mm,1mm).

Equations (30)

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Re(μ0)=1N1.
W(r1,r2,ω)U*(r1,ω)U(r2,ω),
W(r1,r2,ω)=I(r1,ω)I(r2,ω)μ(r1,r2,ω),=A(r1,ω)A(r2,ω)μ(r1,r2,ω).
I(r,ω)=W(r,r,ω),
μ(r1,r2,ω)=W(r1,r2,ω)I(r1,ω)I(r2,ω).
Un(P,ω)=ika22πU0(Qn,ω)eikRnRn,
I(P,ω)=n=1NUn*(P,ω)m=1NUm(P,ω).
W0(Q1,Q2,ω)=U0*(Q1,ω)U0(Q2,ω),
(2πka2)2I(P,ω)=xMx,
x=[A0(Q1)eikR1/R1A0(Q2)eikR2/R2A0(Q3)eikR3/R3],
M=[1μ12μ13μ12*1μ23μ13*μ23*1],
1|μ12|2|μ23|2|μ13|2+μ12μ23μ13*+μ12*μ23*μ13=0.
Mx(P)=0.
My=0,
y=[A0(Q1)A0(Q2)A0(Q3)].
13μ02+2μ03=0.
y=A0[111];
Un(Pj,ω)=ika22πU0(Qn,ω)eikRnjRnj,
W(P1,P2,ω)=n=1NUn*(P1,ω)m=1NUm(P2,ω).
W(P1,P2,ω)=(2πka2)2,I(P,ω)=x(P1)Mx(P2),
x(Pj)=[A0(Q1)eikR1j/R1jA0(Q2)eikR2j/R2jA0(Q3)eikR3j/R3j].
1μ12μ2[μ2μ12]+μ12[μ21]=0.
μ2=1,μ2=2μ121.
A1+A2+A3=0.
y=A0[112μ1].
W12,3=(U1+U2)*U3=U1*U3+U2*U3.
μ12,3=W12,3I12I3,
U1*U3=A1A3μ1=1,
U2*U3=A2A3μ1=1,
μ122+μ232=1,

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