Abstract

We measure the electromagnetic degree of temporal coherence and the associated coherence time for quasi-monochromatic unpolarized light beams emitted by an LED, a filtered halogen lamp, and a multimode He–Ne laser. The method is based on observing at the output of a Michelson interferometer the visibilities (contrasts) of the intensity and polarization-state modulations expressed in terms of the Stokes parameters. The results are in good agreement with those deduced directly from the source spectra. The measurements are repeated after passing the beams through a linear polarizer so as to elucidate the role of polarization in electromagnetic coherence. While the polarizer varies the equal-time degree of coherence consistently with the theoretical predictions and alters the inner structure of the coherence matrix, the coherence time remains almost unchanged when the light varies from unpolarized to polarized. The results are important in the areas of applications dealing with physical optics and electromagnetic interference.

© 2017 Chinese Laser Press

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References

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  1. A. A. Michelson, “The relative motion of the earth and the luminiferous ether,” Am. J. Sci. s3-22, 120–129 (1881).
    [Crossref]
  2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  3. M. Fox, Quantum Optics: An Introduction (Oxford University, 2006).
  4. T. Setälä, J. Tervo, and A. T. Friberg, “Stokes parameters and polarization contrasts in Young’s interference experiment,” Opt. Lett. 31, 2208–2210 (2006).
    [Crossref]
  5. 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, 2669–2671 (2006).
    [Crossref]
  6. L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Interferometric interpretation for the degree of polarization of classical optical beams,” New J. Phys. 16, 113059 (2014).
    [Crossref]
  7. J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett. 29, 536–538 (2004).
    [Crossref]
  8. O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30, 198–200 (2005).
    [Crossref]
  9. A. T. Friberg and T. Setälä, “Electromagnetic theory of optical coherence (invited),” J. Opt. Soc. Am. A 33, 2431–2442 (2016).
    [Crossref]
  10. L.-P. Leppänen, A. T. Friberg, and T. Setälä, “Temporal electromagnetic degree of coherence and Stokes-parameter modulations in Michelson’s interferometer,” Appl. Phys. B 122, 32 (2016).
    [Crossref]
  11. A. Shevchenko, M. Roussey, A. T. Friberg, and T. Setälä, “Ultrashort coherence times in partially polarized stationary optical beams measured by two-photon absorption,” Opt. Express 23, 31274–31285 (2015).
    [Crossref]
  12. E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884–888 (1954).
    [Crossref]
  13. J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137–1143 (2003).
    [Crossref]
  14. T. Setälä, J. Tervo, and A. T. Friberg, “Theorems on complete electromagnetic coherence in the space-time domain,” Opt. Commun. 238, 229–236 (2004).
    [Crossref]
  15. R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009).
  16. J. Tervo, T. Setälä, A. Roueff, P. Réfrégier, and A. T. Friberg, “Two-point Stokes parameters: interpretation and properties,” Opt. Lett. 34, 3074–3076 (2009).
    [Crossref]

2016 (2)

L.-P. Leppänen, A. T. Friberg, and T. Setälä, “Temporal electromagnetic degree of coherence and Stokes-parameter modulations in Michelson’s interferometer,” Appl. Phys. B 122, 32 (2016).
[Crossref]

A. T. Friberg and T. Setälä, “Electromagnetic theory of optical coherence (invited),” J. Opt. Soc. Am. A 33, 2431–2442 (2016).
[Crossref]

2015 (1)

2014 (1)

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Interferometric interpretation for the degree of polarization of classical optical beams,” New J. Phys. 16, 113059 (2014).
[Crossref]

2009 (1)

2006 (2)

2005 (1)

2004 (2)

J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett. 29, 536–538 (2004).
[Crossref]

T. Setälä, J. Tervo, and A. T. Friberg, “Theorems on complete electromagnetic coherence in the space-time domain,” Opt. Commun. 238, 229–236 (2004).
[Crossref]

2003 (1)

1954 (1)

E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884–888 (1954).
[Crossref]

1881 (1)

A. A. Michelson, “The relative motion of the earth and the luminiferous ether,” Am. J. Sci. s3-22, 120–129 (1881).
[Crossref]

Dogariu, A.

Ellis, J.

Fox, M.

M. Fox, Quantum Optics: An Introduction (Oxford University, 2006).

Friberg, A. T.

L.-P. Leppänen, A. T. Friberg, and T. Setälä, “Temporal electromagnetic degree of coherence and Stokes-parameter modulations in Michelson’s interferometer,” Appl. Phys. B 122, 32 (2016).
[Crossref]

A. T. Friberg and T. Setälä, “Electromagnetic theory of optical coherence (invited),” J. Opt. Soc. Am. A 33, 2431–2442 (2016).
[Crossref]

A. Shevchenko, M. Roussey, A. T. Friberg, and T. Setälä, “Ultrashort coherence times in partially polarized stationary optical beams measured by two-photon absorption,” Opt. Express 23, 31274–31285 (2015).
[Crossref]

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Interferometric interpretation for the degree of polarization of classical optical beams,” New J. Phys. 16, 113059 (2014).
[Crossref]

J. Tervo, T. Setälä, A. Roueff, P. Réfrégier, and A. T. Friberg, “Two-point Stokes parameters: interpretation and properties,” Opt. Lett. 34, 3074–3076 (2009).
[Crossref]

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, 2669–2671 (2006).
[Crossref]

T. Setälä, J. Tervo, and A. T. Friberg, “Stokes parameters and polarization contrasts in Young’s interference experiment,” Opt. Lett. 31, 2208–2210 (2006).
[Crossref]

T. Setälä, J. Tervo, and A. T. Friberg, “Theorems on complete electromagnetic coherence in the space-time domain,” Opt. Commun. 238, 229–236 (2004).
[Crossref]

J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137–1143 (2003).
[Crossref]

Korotkova, O.

Leppänen, L.-P.

L.-P. Leppänen, A. T. Friberg, and T. Setälä, “Temporal electromagnetic degree of coherence and Stokes-parameter modulations in Michelson’s interferometer,” Appl. Phys. B 122, 32 (2016).
[Crossref]

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Interferometric interpretation for the degree of polarization of classical optical beams,” New J. Phys. 16, 113059 (2014).
[Crossref]

Mandel, L.

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

Martínez-Herrero, R.

R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009).

Mejías, P. M.

R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009).

Michelson, A. A.

A. A. Michelson, “The relative motion of the earth and the luminiferous ether,” Am. J. Sci. s3-22, 120–129 (1881).
[Crossref]

Piquero, G.

R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009).

Réfrégier, P.

Roueff, A.

Roussey, M.

Saastamoinen, K.

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Interferometric interpretation for the degree of polarization of classical optical beams,” New J. Phys. 16, 113059 (2014).
[Crossref]

Setälä, T.

L.-P. Leppänen, A. T. Friberg, and T. Setälä, “Temporal electromagnetic degree of coherence and Stokes-parameter modulations in Michelson’s interferometer,” Appl. Phys. B 122, 32 (2016).
[Crossref]

A. T. Friberg and T. Setälä, “Electromagnetic theory of optical coherence (invited),” J. Opt. Soc. Am. A 33, 2431–2442 (2016).
[Crossref]

A. Shevchenko, M. Roussey, A. T. Friberg, and T. Setälä, “Ultrashort coherence times in partially polarized stationary optical beams measured by two-photon absorption,” Opt. Express 23, 31274–31285 (2015).
[Crossref]

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Interferometric interpretation for the degree of polarization of classical optical beams,” New J. Phys. 16, 113059 (2014).
[Crossref]

J. Tervo, T. Setälä, A. Roueff, P. Réfrégier, and A. T. Friberg, “Two-point Stokes parameters: interpretation and properties,” Opt. Lett. 34, 3074–3076 (2009).
[Crossref]

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, 2669–2671 (2006).
[Crossref]

T. Setälä, J. Tervo, and A. T. Friberg, “Stokes parameters and polarization contrasts in Young’s interference experiment,” Opt. Lett. 31, 2208–2210 (2006).
[Crossref]

T. Setälä, J. Tervo, and A. T. Friberg, “Theorems on complete electromagnetic coherence in the space-time domain,” Opt. Commun. 238, 229–236 (2004).
[Crossref]

J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137–1143 (2003).
[Crossref]

Shevchenko, A.

Tervo, J.

Wolf, E.

O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30, 198–200 (2005).
[Crossref]

E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884–888 (1954).
[Crossref]

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

Am. J. Sci. (1)

A. A. Michelson, “The relative motion of the earth and the luminiferous ether,” Am. J. Sci. s3-22, 120–129 (1881).
[Crossref]

Appl. Phys. B (1)

L.-P. Leppänen, A. T. Friberg, and T. Setälä, “Temporal electromagnetic degree of coherence and Stokes-parameter modulations in Michelson’s interferometer,” Appl. Phys. B 122, 32 (2016).
[Crossref]

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

New J. Phys. (1)

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Interferometric interpretation for the degree of polarization of classical optical beams,” New J. Phys. 16, 113059 (2014).
[Crossref]

Nuovo Cimento (1)

E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884–888 (1954).
[Crossref]

Opt. Commun. (1)

T. Setälä, J. Tervo, and A. T. Friberg, “Theorems on complete electromagnetic coherence in the space-time domain,” Opt. Commun. 238, 229–236 (2004).
[Crossref]

Opt. Express (2)

Opt. Lett. (5)

Other (3)

R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009).

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

M. Fox, Quantum Optics: An Introduction (Oxford University, 2006).

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Figures (7)

Fig. 1.
Fig. 1. Illustration of Michelson’s interferometer. An incident beam is split into arms 1 and 2 by a BS. The fields reflect back from the mirrors (M) and propagate to the output via the BS again. The length of arm 2 can be adjusted by translating the mirror, yielding to a time delay τ between the fields from the two arms.
Fig. 2.
Fig. 2. Illustration of the used measurement setup. Collimating optics render the light from the source beam-like. The beam is split into arms 1 and 2 with a nonpolarizing BS. The mirror (M) in arm 2 is translated by a piezo element, yielding a controllable time difference τ between the fields from the two arms, and their interference is observed with a CMOS camera. To transform the variations of the polarization Stokes parameters into intensity modulation, suitable quarter-wave plates Q(θ), where θ is the angle the fast axis of the wave plate created with the x axis, are placed into the two arms.
Fig. 3.
Fig. 3. Measured spectra of the considered light sources: an LED with the center wavelength of 633.8 nm (solid red line), filtered light from a halogen lamp (dashed blue line) with the bandwidth of 10 nm and center wavelength of 634.5 nm, and a He–Ne laser of the wavelength 632.8 nm (yellow vertical line).
Fig. 4.
Fig. 4. Intensity, S0, of the LED source as a function of τ at the output of Michelson’s interferometer. Inset shows a magnified section over a short τ range demonstrating the oscillatory behavior of S0(τ).
Fig. 5.
Fig. 5. Illustration of the temporal coherence properties of an LED source. (a) Visibilities of the Stokes-parameter modulations for unpolarized LED light. (b) Temporal electromagnetic degree of coherence obtained from the visibilities (blue curve, maximum value γ(0)=0.68, coherence time τc=24  fs) and by Fourier transforming the spectrum (red crosses). Panels (c) and (d) are as in (a) and (b), respectively, but for polarized LED light. In (d) the maximum value is γ(0)=0.9 and τc=22  fs.
Fig. 6.
Fig. 6. Temporal coherence properties of filtered light from a halogen lamp. (a) Visibilities of the Stokes-parameter modulations for unpolarized light. (b) Temporal electromagnetic degree of coherence obtained from the visibilities (blue curve, maximum value γ(0)=0.66, coherence time τc=53  fs) and by Fourier transforming the spectrum of the light (red crosses). Panels (c) and (d) are as in (a) and (b), respectively, but for polarized lamp light. In (d) the maximum value is γ(0)=0.94 and τc=57  fs.
Fig. 7.
Fig. 7. Temporal coherence properties of a He–Ne laser source. (a) Visibilities of the Stokes-parameter modulations when the beam is unpolarized. (b) Temporal electromagnetic degree of coherence (maximum value γ(0)=0.66, coherence time τc=0.71  ns). Panels (c) and (d) are as in (a) and (b), respectively, but for a polarized laser beam. In (d) the maximum value is γ(0)=0.88 and τc=0.63  ns.

Equations (18)

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γ2(τ)=tr[Γ(τ)Γ(τ)]tr2[Γ(0)],
S0(τ)=Γxx(τ)+Γyy(τ),
S1(τ)=Γxx(τ)Γyy(τ),
S2(τ)=Γxy(τ)+Γyx(τ),
S3(τ)=i[Γyx(τ)Γxy(τ)],
Γ(τ)=12n=03Sn(τ)σn,
σ0=[1001],σ1=[1001],σ2=[0110],σ3=[0ii0].
S0=Jxx+Jyy,
S1=JxxJyy,
S2=Jxy+Jyx,
S3=i(JyxJxy),
γ2(τ)=12n=03|γn(τ)|2,
Sn(τ)=12Sn(i)+12S0(i)|γn(i)(τ)|cos[αn(τ)+Δϕω0τ],
Vn(τ)=max[Sn(τ)]min[Sn(τ)]max[S0(τ)]+min[S0(τ)]=|γn(i)(τ)|,
γ(i)(τ)=[12n=03Vn2(τ)]1/2.
n=13Vn2(τ)=V02(τ)=|γs(τ)|2,
τc2=τ2γ2(τ)dτγ2(τ)dτ,
γ2(0)=12(P2+1),

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