Abstract

A new approach is proposed for estimating the degree of coherence of optical waves. The possibility of transformation of the spatial polarization distribution in the measured spatial intensity distribution for determining the degree of correlation of superposing waves, linearly polarized in the plane of incidence, is shown.

© 2009 OSA

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References

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  1. M. Born, and E. Wolf, Principles of Optics, (Pergamon, Oxford, 1980).
  2. O. V. Angelsky, N. N. Dominikov, P. P. Maksimyak, and T. Tudor, “Experimental revealing of polarization waves,” Appl. Opt. 38(14), 3112–3117 (1999).
    [CrossRef]
  3. T. Setälä, J. Tervo, and A. T. Friberg, “Stokes parameters and polarization contrasts in Young’s interference experiment,” Opt. Lett. 31(14), 2208–2210 (2006).
    [CrossRef] [PubMed]
  4. T. Setälä, J. Tervo, and A. T. Friberg, “Contrasts of Stokes parameters in Young’s interference experiment and electromagnetic degree of coherence,” Opt. Lett. 31(18), 2669–2671 (2006).
    [CrossRef] [PubMed]
  5. O. V. Angelsky, S. B. Yermolenko, C. Yu. Zenkova, and A. O. Angelskaya, “Polarization manifestations of correlation (intrinsic coherence) of optical fields,” Appl. Opt. 47(29), 5492–5499 (2008).
    [PubMed]
  6. O. V. Angelsky, C. Yu. Zenkova, M. P. Gorsky, and N. V. Gorodyns’ka, “Feasibility of estimating the degree of coherence of waves at the near field,” Appl. Opt. 48(15), 2784–2788 (2009).
    [CrossRef] [PubMed]
  7. N. Huse, A. Schönle, and S. W. Hell, “Z-polarized confocal microscopy,” J. Biomed. Opt. 6(4), 480–484 (2001).
    [CrossRef]
  8. A. Apostol and A. Dogariu, “Non-Gaussian statistics of optical near-fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(2 Pt 2), 025602 (2005).
    [CrossRef] [PubMed]
  9. T. Tudor, “Waves. Amplitude waves. Intensity waves,” Journal of Optics-Paris (Nouv. Rev. Opt.) 22, 291–296 (1991).
  10. T. Tudor, “Polarization waves as observable phenomena,” J. Opt. Soc. Am. A 14(8), 2013–2020 (1997).
    [CrossRef]

2009 (1)

2008 (1)

2006 (2)

2005 (1)

A. Apostol and A. Dogariu, “Non-Gaussian statistics of optical near-fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(2 Pt 2), 025602 (2005).
[CrossRef] [PubMed]

2001 (1)

N. Huse, A. Schönle, and S. W. Hell, “Z-polarized confocal microscopy,” J. Biomed. Opt. 6(4), 480–484 (2001).
[CrossRef]

1999 (1)

1997 (1)

1991 (1)

T. Tudor, “Waves. Amplitude waves. Intensity waves,” Journal of Optics-Paris (Nouv. Rev. Opt.) 22, 291–296 (1991).

Angelskaya, A. O.

Angelsky, O. V.

Apostol, A.

A. Apostol and A. Dogariu, “Non-Gaussian statistics of optical near-fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(2 Pt 2), 025602 (2005).
[CrossRef] [PubMed]

Dogariu, A.

A. Apostol and A. Dogariu, “Non-Gaussian statistics of optical near-fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(2 Pt 2), 025602 (2005).
[CrossRef] [PubMed]

Dominikov, N. N.

Friberg, A. T.

Gorodyns’ka, N. V.

Gorsky, M. P.

Hell, S. W.

N. Huse, A. Schönle, and S. W. Hell, “Z-polarized confocal microscopy,” J. Biomed. Opt. 6(4), 480–484 (2001).
[CrossRef]

Huse, N.

N. Huse, A. Schönle, and S. W. Hell, “Z-polarized confocal microscopy,” J. Biomed. Opt. 6(4), 480–484 (2001).
[CrossRef]

Maksimyak, P. P.

Schönle, A.

N. Huse, A. Schönle, and S. W. Hell, “Z-polarized confocal microscopy,” J. Biomed. Opt. 6(4), 480–484 (2001).
[CrossRef]

Setälä, T.

Tervo, J.

Tudor, T.

Yermolenko, S. B.

Zenkova, C. Yu.

Appl. Opt. (3)

J. Biomed. Opt. (1)

N. Huse, A. Schönle, and S. W. Hell, “Z-polarized confocal microscopy,” J. Biomed. Opt. 6(4), 480–484 (2001).
[CrossRef]

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

Journal of Optics-Paris (Nouv. Rev. Opt.) (1)

T. Tudor, “Waves. Amplitude waves. Intensity waves,” Journal of Optics-Paris (Nouv. Rev. Opt.) 22, 291–296 (1991).

Opt. Lett. (2)

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

A. Apostol and A. Dogariu, “Non-Gaussian statistics of optical near-fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(2 Pt 2), 025602 (2005).
[CrossRef] [PubMed]

Other (1)

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

Supplementary Material (6)

» Media 1: MOV (1813 KB)     
» Media 2: MOV (1720 KB)     
» Media 3: MOV (1759 KB)     
» Media 4: MOV (1777 KB)     
» Media 5: MOV (1806 KB)     
» Media 6: MOV (1758 KB)     

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Figures (10)

Fig. 1
Fig. 1

Scheme of superposition of plane waves linearly polarized at the incidence plane and converging at angle of 90°. Magnitudes |η(1,2)|=1 , |η(1,3)|=1 and |η(2,3)|=1 determine the correlation degree between the corresponding waves; E(1) , Vφ=2ijtr[W(Q1,Q1,0]  tr[W(Q2,Q2,0)]φij(1)(r)+φij(2)(r)+φij(3)(r)|ηij(1,2)|+ 2ijtr[W(Q1,Q1,0]  tr[W(Q3,Q3,0)]φij(1)(r)+φij(2)(r)+φij(3)(r)|ηij(1,3)|cos[φ1]+ and E(3) are amplitudes of the 2ijtr[W(Q2,Q2,0]  tr[W(Q3,Q3,0)]φij(1)(r)+φij(2)(r)+φij(3)(r)|ηij(2,3)|cos[φ2], superimposed waves.

Fig. 2
Fig. 2

Polarization distribution in the plane of superposition of two plane, linearly polarized waves of equal intensities along axis OX shown by red. Red fragment at the left illustrates homogeneous intensity distribution.. The dependence of the field intensity shown in relative units from coordinate (right upper part of a figure) shows the mechanism for obtaining homogeneous intensity in case of a half-period shift of the interference distributions of equal visibility for the Ex and Ez components. The resulting visibility V=0. Negative magnitudes of x-coordinate are explained by the fact that the introduced coordinate «0» corresponds to the initial point of convergence of waves, where the resulting polarization is linear having a phase difference is zero. Media 1.

Fig. 3
Fig. 3

Video illustrating changes of modulation depth of the resulting interference distribution for change in time of a phase of the reference wave synchronized with changes of the resulting state of polarization along the axis OX (red). Red and black correspond to the distribution caused by interference of Ex components and Ez components, respectively. The VMD M=max[Vφ]min[Vφ] is determined by the difference of maximal and minimal magnitudes of visibility, which in this case equals unity (i.e. max[Vφ]=1 and min[Vφ]=0 ). Media 2.

Fig. 4
Fig. 4

Video illustrating that spatial polarization takes place at the incidence plane. Symbols are the same as in previous case. Media 3.

Fig. 5
Fig. 5

Optical arrangement for holographic experiment: Bs1 and Bs2, beam splitters; M1, M2, and M3, mirrors; P1, P2, and P3, polarizers; PR, prism; IL, immersion liquid; H, hologram.

Fig. 7
Fig. 7

Video illustrating polarization modulation when superimposing orthogonally linearly polarized waves converging at an angle of 90° are not strictly coherent. Brown shows the lines corresponding to the contribution from incoherent component of a beam 1 (Fig. 6), green shows the lines corresponding to the coherent “residual” component of beam 1, blue shows the resulting polarization distribution. Other symbols are the same as in the previous media files. Media 4.

Fig. 6
Fig. 6

Scheme of superposition of plane waves linearly polarized at the incidence plane and converging at an angle of 90°. Magnitudes |η(1,2)|=0.5 , |η(1,3)|=0.5 and |η(2,3)|=1 determine the correlation degree between the corresponding waves; E(1) , E(2) and E(3) are amplitudes of the superimposed waves.

Fig. 8
Fig. 8

Video illustrating changes of the modulation depth of the resulting interference distribution for temporal changing the phase of the reference wave synchronized with changes of the states of polarization along axis OX. Modulo of the correlation degree of superposing waves 1 and 2 |η(1,2)|=0.5 . The VMD is 0.5. Media 5.

Fig. 9
Fig. 9

Illustration of the fact that the field is left polarized at the incidence plane. Media 6.

Fig. 10
Fig. 10

а) Scheme for interference of three plane waves linearly polarized in the plane of incidence, when the convergence angle differs from 90°. b) Intensity distribution in the incidence plane: interference of Ez-components of interacting waves; the resulting distribution. с) Polarization modulation at the incidence plane.

Equations (5)

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E(r,t)=E(Q1,t)exp(ikR1)R1+E(Q2,t)exp(ikR2)R2+E(Q3,t)exp(ikR3)R3
I(r)=ij{φij(1)(r)+φij(2)(r)+φij(3)(r)+2tr[W(Q1,Q1,0)]  tr[W(Q2,Q2,0)]|ηij(1,2)|cos[αij(1,2)]cos[δ1]++2tr[W(Q1,Q1,0)]  tr[W(Q3,Q3,0)]|ηij(1,3)|cos[αij(1,3)]cos[δ2]++2tr[W(Q2,Q2,0)]  tr[W(Q3,Q3,0)]|ηij(2,3)|cos[αij(2,3)]cos[δ3]}
Vφ=2ijtr[W(Q1,Q1,0]  tr[W(Q2,Q2,0)]φij(1)(r)+φij(2)(r)+φij(3)(r)|ηij(1,2)|+2ijtr[W(Q1,Q1,0]  tr[W(Q3,Q3,0)]φij(1)(r)+φij(2)(r)+φij(3)(r)|ηij(1,3)|cos[φ1]+2ijtr[W(Q2,Q2,0]  tr[W(Q3,Q3,0)]φij(1)(r)+φij(2)(r)+φij(3)(r)|ηij(2,3)|cos[φ2],
M=max[Vφ]min[Vφ]=2mijtr[W(Qm,Qm,0]  tr[W(Q3,Q3,0)]φij(m)(r)+φij(3)(r)|ηij(m,3)| for   m = 1, 2; i, j = x, z
|η(1,2)|=Vcos[Δθ],

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