Abstract

For light fields, the coherence in longitudinal direction is governed by both the frequency spectra and angular spectra they possess. In this work, we develop and report a theoretical formulation to demonstrate the effect of the angular spectra of electromagnetic light fields in quantifying their longitudinal spatial coherence. The experimental results obtained by measuring the electromagnetic longitudinal spatial coherence and degree of cross-polarization of uniformly polarized light fields for different angular spectra validate the theoretical findings.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
  2. V. Ryabukho, D. Lyakin, and V. Lychagov, “Longitudinal purely spatial coherence of a light field,” Opt. Spectrosc. 100, 724–733 (2006).
    [Crossref]
  3. V. Ryabukho, D. Lyakin, and V. Lychagov, “Longitudinal coherence length of an optical field,” Opt. Spectrosc. 107, 282–287 (2009).
    [Crossref]
  4. D. Lyakin, P. Ryabukho, and V. Ryabukho, “Mutual spatiotemporal coherence of optical fields in an amplitude-splitting interferometer,” Opt. Spectrosc. 122, 329–337 (2017).
    [Crossref]
  5. I. Abdulhalim, “Competence between spatial and temporal coherence in full field optical coherence tomography and interference microscopy,” J. Opt. A: Pure Appl. Opt. 8, 952–958 (2006).
    [Crossref]
  6. D. V. Lyakin and V. P. Ryabukho, “Longitudinal correlation properties of an optical field with broad angular and frequency spectra and their manifestation in interference microscopy,” Quantum Electron. 43, 949–957 (2013).
    [Crossref]
  7. D. Lyakin, N. Y. Mysina, and V. Ryabukho, “Coherence volume of an optical wave field with broad frequency and angular spectra,” Opt. Spectrosc. 124, 349–359 (2018).
    [Crossref]
  8. V. Ryabukho, L. Maksimova, N. Y. Mysina, D. Lyakin, and P. Ryabukho, “Instantaneous speckle structures in a partially coherent optical wave field with broad frequency and angular spectra,” Opt. Spectrosc. 126, 124–134 (2019).
    [Crossref]
  9. J. Rosen and A. Yariv, “Longitudinal partial coherence of optical radiation,” Opt. Commun. 117, 8–12 (1995).
    [Crossref]
  10. L. M. Soroko, Holography and Coherent Optics (Plenum, 1980).
    [Crossref]
  11. V. Ryabukho, D. Lyakin, and M. Lobachev, “Influence of longitudinal spatial coherence on the signal of a scanning interferometer,” Opt. Lett. 29, 667–669 (2004).
    [Crossref] [PubMed]
  12. H. Ko, “Coherence theory of radio-astronomical measurements,” IEEE Transactions on Antennas Propag. 15, 10–20 (1967).
    [Crossref]
  13. J. Rosen and M. Takeda, “Longitudinal spatial coherence applied for surface profilometry,” Appl. Opt. 39, 4107–4111 (2000).
    [Crossref]
  14. Z. Duan, Y. Miyamoto, and M. Takeda, “Dispersion-free optical coherence depth sensing with a spatial frequency comb generated by an angular spectrum modulator,” Opt. Express 14, 12109–12121 (2006).
    [Crossref] [PubMed]
  15. A. Ahmad, V. Srivastava, V. Dubey, and D. Mehta, “Ultra-short longitudinal spatial coherence length of laser light with the combined effect of spatial, angular, and temporal diversity,” Appl. Phys. Lett. 106, 093701 (2015).
    [Crossref]
  16. I. Abdulhalim, “Spatial and temporal coherence effects in interference microscopy and full-field optical coherence tomography,” Annalen der Physik 524, 787–804 (2012).
    [Crossref]
  17. I. Zeylikovich, “Short coherence length produced by a spatial incoherent source applied for the Linnik-type interferometer,” Appl. Opt. 47, 2171–2177 (2008).
    [Crossref] [PubMed]
  18. J. M. Schmitt, “Optical coherence tomography (oct): a review,” IEEE J. selected topics quantum electronics 5, 1205–1215 (1999).
    [Crossref]
  19. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).
    [Crossref]
  20. 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]
  21. T. Hassinen, J. Tervo, T. Setälä, and A. T. Friberg, “Hanbury Brown-Twiss effect with electromagnetic waves,” Opt. Express 19, 15188–15195 (2011).
    [Crossref] [PubMed]
  22. A. Al-Qasimi, M. Lahiri, D. Kuebel, D. F. James, and E. Wolf, “The influence of the degree of cross-polarization on the Hanbury Brown-Twiss effect,” Opt. Express 18, 17124–17129 (2010).
    [Crossref] [PubMed]
  23. S. Volkov, D. F. James, T. Shirai, and E. Wolf, “Intensity fluctuations and the degree of cross-polarization in stochastic electromagnetic beams,” J. Opt. A: Pure Appl. Opt. 10, 055001 (2008).
    [Crossref]
  24. L.-P. Leppänen, A. T. Friberg, and T. Setälä, “Temporal electromagnetic degree of coherence and Stokes-parameter modulations in Michelson interferometer,” Appl. Phys. B 122, 32 (2016).
    [Crossref]
  25. B. Kanseri and H. C. Kandpal, “Experimental determination of two-point Stokes parameters for a partially coherent broadband light beam,” Opt. Commun. 283, 4558–4562 (2010).
    [Crossref]
  26. L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Measurement of the degree of temporal coherence of unpolarized light beams,” Photonics Res. 5, 156–161 (2017).
    [Crossref]
  27. B. Kanseri, Optical Coherence and Polarization: An Experimental Outlook (Lambert Academic, 2013).

2019 (1)

V. Ryabukho, L. Maksimova, N. Y. Mysina, D. Lyakin, and P. Ryabukho, “Instantaneous speckle structures in a partially coherent optical wave field with broad frequency and angular spectra,” Opt. Spectrosc. 126, 124–134 (2019).
[Crossref]

2018 (1)

D. Lyakin, N. Y. Mysina, and V. Ryabukho, “Coherence volume of an optical wave field with broad frequency and angular spectra,” Opt. Spectrosc. 124, 349–359 (2018).
[Crossref]

2017 (2)

D. Lyakin, P. Ryabukho, and V. Ryabukho, “Mutual spatiotemporal coherence of optical fields in an amplitude-splitting interferometer,” Opt. Spectrosc. 122, 329–337 (2017).
[Crossref]

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Measurement of the degree of temporal coherence of unpolarized light beams,” Photonics Res. 5, 156–161 (2017).
[Crossref]

2016 (1)

L.-P. Leppänen, A. T. Friberg, and T. Setälä, “Temporal electromagnetic degree of coherence and Stokes-parameter modulations in Michelson interferometer,” Appl. Phys. B 122, 32 (2016).
[Crossref]

2015 (1)

A. Ahmad, V. Srivastava, V. Dubey, and D. Mehta, “Ultra-short longitudinal spatial coherence length of laser light with the combined effect of spatial, angular, and temporal diversity,” Appl. Phys. Lett. 106, 093701 (2015).
[Crossref]

2013 (1)

D. V. Lyakin and V. P. Ryabukho, “Longitudinal correlation properties of an optical field with broad angular and frequency spectra and their manifestation in interference microscopy,” Quantum Electron. 43, 949–957 (2013).
[Crossref]

2012 (1)

I. Abdulhalim, “Spatial and temporal coherence effects in interference microscopy and full-field optical coherence tomography,” Annalen der Physik 524, 787–804 (2012).
[Crossref]

2011 (1)

2010 (2)

A. Al-Qasimi, M. Lahiri, D. Kuebel, D. F. James, and E. Wolf, “The influence of the degree of cross-polarization on the Hanbury Brown-Twiss effect,” Opt. Express 18, 17124–17129 (2010).
[Crossref] [PubMed]

B. Kanseri and H. C. Kandpal, “Experimental determination of two-point Stokes parameters for a partially coherent broadband light beam,” Opt. Commun. 283, 4558–4562 (2010).
[Crossref]

2009 (1)

V. Ryabukho, D. Lyakin, and V. Lychagov, “Longitudinal coherence length of an optical field,” Opt. Spectrosc. 107, 282–287 (2009).
[Crossref]

2008 (2)

I. Zeylikovich, “Short coherence length produced by a spatial incoherent source applied for the Linnik-type interferometer,” Appl. Opt. 47, 2171–2177 (2008).
[Crossref] [PubMed]

S. Volkov, D. F. James, T. Shirai, and E. Wolf, “Intensity fluctuations and the degree of cross-polarization in stochastic electromagnetic beams,” J. Opt. A: Pure Appl. Opt. 10, 055001 (2008).
[Crossref]

2006 (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, 2669–2671 (2006).
[Crossref]

Z. Duan, Y. Miyamoto, and M. Takeda, “Dispersion-free optical coherence depth sensing with a spatial frequency comb generated by an angular spectrum modulator,” Opt. Express 14, 12109–12121 (2006).
[Crossref] [PubMed]

V. Ryabukho, D. Lyakin, and V. Lychagov, “Longitudinal purely spatial coherence of a light field,” Opt. Spectrosc. 100, 724–733 (2006).
[Crossref]

I. Abdulhalim, “Competence between spatial and temporal coherence in full field optical coherence tomography and interference microscopy,” J. Opt. A: Pure Appl. Opt. 8, 952–958 (2006).
[Crossref]

2004 (1)

2000 (1)

1999 (1)

J. M. Schmitt, “Optical coherence tomography (oct): a review,” IEEE J. selected topics quantum electronics 5, 1205–1215 (1999).
[Crossref]

1995 (1)

J. Rosen and A. Yariv, “Longitudinal partial coherence of optical radiation,” Opt. Commun. 117, 8–12 (1995).
[Crossref]

1967 (1)

H. Ko, “Coherence theory of radio-astronomical measurements,” IEEE Transactions on Antennas Propag. 15, 10–20 (1967).
[Crossref]

Abdulhalim, I.

I. Abdulhalim, “Spatial and temporal coherence effects in interference microscopy and full-field optical coherence tomography,” Annalen der Physik 524, 787–804 (2012).
[Crossref]

I. Abdulhalim, “Competence between spatial and temporal coherence in full field optical coherence tomography and interference microscopy,” J. Opt. A: Pure Appl. Opt. 8, 952–958 (2006).
[Crossref]

Ahmad, A.

A. Ahmad, V. Srivastava, V. Dubey, and D. Mehta, “Ultra-short longitudinal spatial coherence length of laser light with the combined effect of spatial, angular, and temporal diversity,” Appl. Phys. Lett. 106, 093701 (2015).
[Crossref]

Al-Qasimi, A.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).
[Crossref]

Duan, Z.

Dubey, V.

A. Ahmad, V. Srivastava, V. Dubey, and D. Mehta, “Ultra-short longitudinal spatial coherence length of laser light with the combined effect of spatial, angular, and temporal diversity,” Appl. Phys. Lett. 106, 093701 (2015).
[Crossref]

Friberg, A. T.

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Measurement of the degree of temporal coherence of unpolarized light beams,” Photonics Res. 5, 156–161 (2017).
[Crossref]

L.-P. Leppänen, A. T. Friberg, and T. Setälä, “Temporal electromagnetic degree of coherence and Stokes-parameter modulations in Michelson interferometer,” Appl. Phys. B 122, 32 (2016).
[Crossref]

T. Hassinen, J. Tervo, T. Setälä, and A. T. Friberg, “Hanbury Brown-Twiss effect with electromagnetic waves,” Opt. Express 19, 15188–15195 (2011).
[Crossref] [PubMed]

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]

Hassinen, T.

James, D. F.

A. Al-Qasimi, M. Lahiri, D. Kuebel, D. F. James, and E. Wolf, “The influence of the degree of cross-polarization on the Hanbury Brown-Twiss effect,” Opt. Express 18, 17124–17129 (2010).
[Crossref] [PubMed]

S. Volkov, D. F. James, T. Shirai, and E. Wolf, “Intensity fluctuations and the degree of cross-polarization in stochastic electromagnetic beams,” J. Opt. A: Pure Appl. Opt. 10, 055001 (2008).
[Crossref]

Kandpal, H. C.

B. Kanseri and H. C. Kandpal, “Experimental determination of two-point Stokes parameters for a partially coherent broadband light beam,” Opt. Commun. 283, 4558–4562 (2010).
[Crossref]

Kanseri, B.

B. Kanseri and H. C. Kandpal, “Experimental determination of two-point Stokes parameters for a partially coherent broadband light beam,” Opt. Commun. 283, 4558–4562 (2010).
[Crossref]

B. Kanseri, Optical Coherence and Polarization: An Experimental Outlook (Lambert Academic, 2013).

Ko, H.

H. Ko, “Coherence theory of radio-astronomical measurements,” IEEE Transactions on Antennas Propag. 15, 10–20 (1967).
[Crossref]

Kuebel, D.

Lahiri, M.

Leppänen, L.-P.

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Measurement of the degree of temporal coherence of unpolarized light beams,” Photonics Res. 5, 156–161 (2017).
[Crossref]

L.-P. Leppänen, A. T. Friberg, and T. Setälä, “Temporal electromagnetic degree of coherence and Stokes-parameter modulations in Michelson interferometer,” Appl. Phys. B 122, 32 (2016).
[Crossref]

Lobachev, M.

Lyakin, D.

V. Ryabukho, L. Maksimova, N. Y. Mysina, D. Lyakin, and P. Ryabukho, “Instantaneous speckle structures in a partially coherent optical wave field with broad frequency and angular spectra,” Opt. Spectrosc. 126, 124–134 (2019).
[Crossref]

D. Lyakin, N. Y. Mysina, and V. Ryabukho, “Coherence volume of an optical wave field with broad frequency and angular spectra,” Opt. Spectrosc. 124, 349–359 (2018).
[Crossref]

D. Lyakin, P. Ryabukho, and V. Ryabukho, “Mutual spatiotemporal coherence of optical fields in an amplitude-splitting interferometer,” Opt. Spectrosc. 122, 329–337 (2017).
[Crossref]

V. Ryabukho, D. Lyakin, and V. Lychagov, “Longitudinal coherence length of an optical field,” Opt. Spectrosc. 107, 282–287 (2009).
[Crossref]

V. Ryabukho, D. Lyakin, and V. Lychagov, “Longitudinal purely spatial coherence of a light field,” Opt. Spectrosc. 100, 724–733 (2006).
[Crossref]

V. Ryabukho, D. Lyakin, and M. Lobachev, “Influence of longitudinal spatial coherence on the signal of a scanning interferometer,” Opt. Lett. 29, 667–669 (2004).
[Crossref] [PubMed]

Lyakin, D. V.

D. V. Lyakin and V. P. Ryabukho, “Longitudinal correlation properties of an optical field with broad angular and frequency spectra and their manifestation in interference microscopy,” Quantum Electron. 43, 949–957 (2013).
[Crossref]

Lychagov, V.

V. Ryabukho, D. Lyakin, and V. Lychagov, “Longitudinal coherence length of an optical field,” Opt. Spectrosc. 107, 282–287 (2009).
[Crossref]

V. Ryabukho, D. Lyakin, and V. Lychagov, “Longitudinal purely spatial coherence of a light field,” Opt. Spectrosc. 100, 724–733 (2006).
[Crossref]

Maksimova, L.

V. Ryabukho, L. Maksimova, N. Y. Mysina, D. Lyakin, and P. Ryabukho, “Instantaneous speckle structures in a partially coherent optical wave field with broad frequency and angular spectra,” Opt. Spectrosc. 126, 124–134 (2019).
[Crossref]

Mehta, D.

A. Ahmad, V. Srivastava, V. Dubey, and D. Mehta, “Ultra-short longitudinal spatial coherence length of laser light with the combined effect of spatial, angular, and temporal diversity,” Appl. Phys. Lett. 106, 093701 (2015).
[Crossref]

Miyamoto, Y.

Mysina, N. Y.

V. Ryabukho, L. Maksimova, N. Y. Mysina, D. Lyakin, and P. Ryabukho, “Instantaneous speckle structures in a partially coherent optical wave field with broad frequency and angular spectra,” Opt. Spectrosc. 126, 124–134 (2019).
[Crossref]

D. Lyakin, N. Y. Mysina, and V. Ryabukho, “Coherence volume of an optical wave field with broad frequency and angular spectra,” Opt. Spectrosc. 124, 349–359 (2018).
[Crossref]

Rosen, J.

J. Rosen and M. Takeda, “Longitudinal spatial coherence applied for surface profilometry,” Appl. Opt. 39, 4107–4111 (2000).
[Crossref]

J. Rosen and A. Yariv, “Longitudinal partial coherence of optical radiation,” Opt. Commun. 117, 8–12 (1995).
[Crossref]

Ryabukho, P.

V. Ryabukho, L. Maksimova, N. Y. Mysina, D. Lyakin, and P. Ryabukho, “Instantaneous speckle structures in a partially coherent optical wave field with broad frequency and angular spectra,” Opt. Spectrosc. 126, 124–134 (2019).
[Crossref]

D. Lyakin, P. Ryabukho, and V. Ryabukho, “Mutual spatiotemporal coherence of optical fields in an amplitude-splitting interferometer,” Opt. Spectrosc. 122, 329–337 (2017).
[Crossref]

Ryabukho, V.

V. Ryabukho, L. Maksimova, N. Y. Mysina, D. Lyakin, and P. Ryabukho, “Instantaneous speckle structures in a partially coherent optical wave field with broad frequency and angular spectra,” Opt. Spectrosc. 126, 124–134 (2019).
[Crossref]

D. Lyakin, N. Y. Mysina, and V. Ryabukho, “Coherence volume of an optical wave field with broad frequency and angular spectra,” Opt. Spectrosc. 124, 349–359 (2018).
[Crossref]

D. Lyakin, P. Ryabukho, and V. Ryabukho, “Mutual spatiotemporal coherence of optical fields in an amplitude-splitting interferometer,” Opt. Spectrosc. 122, 329–337 (2017).
[Crossref]

V. Ryabukho, D. Lyakin, and V. Lychagov, “Longitudinal coherence length of an optical field,” Opt. Spectrosc. 107, 282–287 (2009).
[Crossref]

V. Ryabukho, D. Lyakin, and V. Lychagov, “Longitudinal purely spatial coherence of a light field,” Opt. Spectrosc. 100, 724–733 (2006).
[Crossref]

V. Ryabukho, D. Lyakin, and M. Lobachev, “Influence of longitudinal spatial coherence on the signal of a scanning interferometer,” Opt. Lett. 29, 667–669 (2004).
[Crossref] [PubMed]

Ryabukho, V. P.

D. V. Lyakin and V. P. Ryabukho, “Longitudinal correlation properties of an optical field with broad angular and frequency spectra and their manifestation in interference microscopy,” Quantum Electron. 43, 949–957 (2013).
[Crossref]

Saastamoinen, K.

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Measurement of the degree of temporal coherence of unpolarized light beams,” Photonics Res. 5, 156–161 (2017).
[Crossref]

Schmitt, J. M.

J. M. Schmitt, “Optical coherence tomography (oct): a review,” IEEE J. selected topics quantum electronics 5, 1205–1215 (1999).
[Crossref]

Setälä, T.

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Measurement of the degree of temporal coherence of unpolarized light beams,” Photonics Res. 5, 156–161 (2017).
[Crossref]

L.-P. Leppänen, A. T. Friberg, and T. Setälä, “Temporal electromagnetic degree of coherence and Stokes-parameter modulations in Michelson interferometer,” Appl. Phys. B 122, 32 (2016).
[Crossref]

T. Hassinen, J. Tervo, T. Setälä, and A. T. Friberg, “Hanbury Brown-Twiss effect with electromagnetic waves,” Opt. Express 19, 15188–15195 (2011).
[Crossref] [PubMed]

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]

Shirai, T.

S. Volkov, D. F. James, T. Shirai, and E. Wolf, “Intensity fluctuations and the degree of cross-polarization in stochastic electromagnetic beams,” J. Opt. A: Pure Appl. Opt. 10, 055001 (2008).
[Crossref]

Soroko, L. M.

L. M. Soroko, Holography and Coherent Optics (Plenum, 1980).
[Crossref]

Srivastava, V.

A. Ahmad, V. Srivastava, V. Dubey, and D. Mehta, “Ultra-short longitudinal spatial coherence length of laser light with the combined effect of spatial, angular, and temporal diversity,” Appl. Phys. Lett. 106, 093701 (2015).
[Crossref]

Takeda, M.

Tervo, J.

Volkov, S.

S. Volkov, D. F. James, T. Shirai, and E. Wolf, “Intensity fluctuations and the degree of cross-polarization in stochastic electromagnetic beams,” J. Opt. A: Pure Appl. Opt. 10, 055001 (2008).
[Crossref]

Wolf, E.

A. Al-Qasimi, M. Lahiri, D. Kuebel, D. F. James, and E. Wolf, “The influence of the degree of cross-polarization on the Hanbury Brown-Twiss effect,” Opt. Express 18, 17124–17129 (2010).
[Crossref] [PubMed]

S. Volkov, D. F. James, T. Shirai, and E. Wolf, “Intensity fluctuations and the degree of cross-polarization in stochastic electromagnetic beams,” J. Opt. A: Pure Appl. Opt. 10, 055001 (2008).
[Crossref]

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).
[Crossref]

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

Yariv, A.

J. Rosen and A. Yariv, “Longitudinal partial coherence of optical radiation,” Opt. Commun. 117, 8–12 (1995).
[Crossref]

Zeylikovich, I.

Annalen der Physik (1)

I. Abdulhalim, “Spatial and temporal coherence effects in interference microscopy and full-field optical coherence tomography,” Annalen der Physik 524, 787–804 (2012).
[Crossref]

Appl. Opt. (2)

Appl. Phys. B (1)

L.-P. Leppänen, A. T. Friberg, and T. Setälä, “Temporal electromagnetic degree of coherence and Stokes-parameter modulations in Michelson interferometer,” Appl. Phys. B 122, 32 (2016).
[Crossref]

Appl. Phys. Lett. (1)

A. Ahmad, V. Srivastava, V. Dubey, and D. Mehta, “Ultra-short longitudinal spatial coherence length of laser light with the combined effect of spatial, angular, and temporal diversity,” Appl. Phys. Lett. 106, 093701 (2015).
[Crossref]

IEEE J. selected topics quantum electronics (1)

J. M. Schmitt, “Optical coherence tomography (oct): a review,” IEEE J. selected topics quantum electronics 5, 1205–1215 (1999).
[Crossref]

IEEE Transactions on Antennas Propag. (1)

H. Ko, “Coherence theory of radio-astronomical measurements,” IEEE Transactions on Antennas Propag. 15, 10–20 (1967).
[Crossref]

J. Opt. A: Pure Appl. Opt. (2)

I. Abdulhalim, “Competence between spatial and temporal coherence in full field optical coherence tomography and interference microscopy,” J. Opt. A: Pure Appl. Opt. 8, 952–958 (2006).
[Crossref]

S. Volkov, D. F. James, T. Shirai, and E. Wolf, “Intensity fluctuations and the degree of cross-polarization in stochastic electromagnetic beams,” J. Opt. A: Pure Appl. Opt. 10, 055001 (2008).
[Crossref]

Opt. Commun. (2)

J. Rosen and A. Yariv, “Longitudinal partial coherence of optical radiation,” Opt. Commun. 117, 8–12 (1995).
[Crossref]

B. Kanseri and H. C. Kandpal, “Experimental determination of two-point Stokes parameters for a partially coherent broadband light beam,” Opt. Commun. 283, 4558–4562 (2010).
[Crossref]

Opt. Express (3)

Opt. Lett. (2)

Opt. Spectrosc. (5)

D. Lyakin, N. Y. Mysina, and V. Ryabukho, “Coherence volume of an optical wave field with broad frequency and angular spectra,” Opt. Spectrosc. 124, 349–359 (2018).
[Crossref]

V. Ryabukho, L. Maksimova, N. Y. Mysina, D. Lyakin, and P. Ryabukho, “Instantaneous speckle structures in a partially coherent optical wave field with broad frequency and angular spectra,” Opt. Spectrosc. 126, 124–134 (2019).
[Crossref]

V. Ryabukho, D. Lyakin, and V. Lychagov, “Longitudinal purely spatial coherence of a light field,” Opt. Spectrosc. 100, 724–733 (2006).
[Crossref]

V. Ryabukho, D. Lyakin, and V. Lychagov, “Longitudinal coherence length of an optical field,” Opt. Spectrosc. 107, 282–287 (2009).
[Crossref]

D. Lyakin, P. Ryabukho, and V. Ryabukho, “Mutual spatiotemporal coherence of optical fields in an amplitude-splitting interferometer,” Opt. Spectrosc. 122, 329–337 (2017).
[Crossref]

Photonics Res. (1)

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Measurement of the degree of temporal coherence of unpolarized light beams,” Photonics Res. 5, 156–161 (2017).
[Crossref]

Quantum Electron. (1)

D. V. Lyakin and V. P. Ryabukho, “Longitudinal correlation properties of an optical field with broad angular and frequency spectra and their manifestation in interference microscopy,” Quantum Electron. 43, 949–957 (2013).
[Crossref]

Other (4)

L. M. Soroko, Holography and Coherent Optics (Plenum, 1980).
[Crossref]

B. Kanseri, Optical Coherence and Polarization: An Experimental Outlook (Lambert Academic, 2013).

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

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 Schematic diagram representing the geometrical angular spectrum (2θ) of a source (σ). Symbols are described in the text.
Fig. 2
Fig. 2 Plot of EMDOC vs longitudinal shift (path difference between two arms of the Michelson interferometer) for the source having (a) uniform intensity distribution [Eq. (7)], and (b) Gaussian distribution [Eq. (9)]. Colour codes for curves having angular spectrum θ (in radian): 0 (temporal) black, 0.02 red, 0.04 mustard, 0.06 blue, 0.1 purple and 0.2 green.
Fig. 3
Fig. 3 (a) Scheme of the experimental set-up for investigating electromagnetic longitudinal spatial coherence. (b) Degree of polarization of the source measured for different values of the angular spectrum showing constant behaviour. Notations are described in the text.
Fig. 4
Fig. 4 Plot of EMDOC (blue, left) and DOCP (orange, right) versus path difference for angular spectra (θ) ranging from of 0.016rad to 0.13rad, where blue ticks are for experimental EMDOC points and red line is its fitting with Eq. (9). Orange ticks correspond to experimental DOCP points and green line represents its fitting with straight line.
Fig. 5
Fig. 5 (a) Plot of EMDOC (left) and DOCP (right) versus path difference for purely temporal coherence (without MO), where orange ticks correspond to experimental data and green line is its straight line fit. Blue ticks denote experimentally measured EMDOC fitted using Eq. (9) with red line. (b) Plot of longitudinal coherence length (Lc) versus angular spectra of the source. Black ticks represent experimental data and red curve is its theoretical fit.

Equations (10)

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E i ( z m , t , α ) = A i ( t , α ) r m exp ( ι k r m ) ; for ( i = x , y ) ,
W i j ( z 1 , z 2 , θ ) = z 2 E i * ( z 1 , t , α ) E j ( z 2 , t , α ) d ζ d η ;
W i j ( z 1 , z 2 , θ ) = 2 π z 2 0 θ E i * ( z 1 , t , α ) E j ( z 2 , t , α ) α d α .
W i j ( z m , z n , θ ) = π z 2 θ 2 2 exp ( ι k Δ z 4 ) sinc ( k Δ z θ 2 4 ) W i j ( z m , z n , 0 ) ; ( for m , n = 1 , 2 ) ,
γ e ( z 1 , z 2 , θ ) = tr [ W ( z 1 , z 2 , θ ) . W ( z 2 , z 1 , θ ) ] tr W ( z 1 , z 1 , θ ) tr W ( z 2 , z 2 , θ ) .
γ e ( z 1 , z 2 , θ ) = 1 2 n = 0 3 | V j ( z 1 , z 2 , θ ) | 2 .
γ e ( z 1 , z 2 , θ ) = ( 2 k Δ z θ 2 ) 2 ( 1 cos k Δ z θ 2 2 ) γ e ( z 1 , z 2 , 0 ) ,
W i j ( z 1 , z 2 , θ ) = π z 2 θ 2 2 ( 1 + ι k Δ z θ 2 4 ) ( 1 exp ( 2 ( 1 + ι k Δ z θ 2 4 ) ) ) W i j ( z 1 , z 2 , 0 ) .
γ e ( z 1 , z 2 , θ ) = ( 1 + e 4 2 e 2 cos k Δ z θ 2 2 ) ( 1 e 2 ) 2 [ 1 + ( k Δ z θ 2 4 ) 2 ] γ e ( z 1 , z 2 , 0 ) .
γ e 2 ( z 1 , z 2 , θ ) = 1 2 [ 1 + P ( z 1 , z 2 , θ ) ] V 0 ( z 1 , z 2 , θ ) .

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