Abstract

Studying the coherence of an optical field is typically compartmentalized with respect to its different physical degrees of freedom (DoFs)—spatial, temporal, and polarization. Although this traditional approach succeeds when the DoFs are uncoupled, it fails at capturing key features of the field’s coherence if the DOFs are indeed correlated—a situation that arises often. By viewing coherence as a “resource” that can be shared among the DoFs, it becomes possible to convert the entropy associated with the fluctuations in one DoF to another DoF that is initially fluctuation-free. Here, we verify experimentally that coherence can indeed be reversibly exchanged—without loss of energy—between polarization and the spatial DoF of a partially coherent field. Starting from a linearly polarized spatially incoherent field—one that produces no spatial interference fringes—we obtain a spatially coherent field that is unpolarized. By reallocating the entropy to polarization, the field becomes invariant with regard to the action of a polarization scrambler, thus suggesting a strategy for avoiding the deleterious effects of a randomizing system on a DoF of the optical field.

© 2017 Optical Society of America

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Corrections

30 August 2017: A typographical correction was made to the author affiliations.


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References

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2017 (1)

2016 (3)

D. Guzman-Silva, R. Brüning, F. Zimmermann, C. Vetter, M. Grafe, M. Heinrich, S. Nolte, M. Duparre, A. Aiello, M. Ornigotti, and A. Szameit, “Demonstration of local teleportation using classical entanglement,” Laser Photon. Rev. 10, 317–321 (2016).
[Crossref]

H. E. Kondakci and A. F. Abouraddy, “Diffraction-free pulsed optical beams via space-time correlations,” Opt. Express 24, 28659–28668 (2016).
[Crossref]

K. J. Parker and M. A. Alonso, “Longitudinal ISO-phase condition and needle pulses,” Opt. Express 24, 28669–28677 (2016).
[Crossref]

2015 (4)

A. Aiello, F. Toppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

S. Berg-Johansen, F. Töppel, B. Stiller, P. Banzer, M. Ornigotti, E. Giacobino, G. Leuchs, A. Aiello, and C. Marquardt, “Classically entangled optical beams for high-speed kinematic sensing,” Optica 2, 864–868 (2015).
[Crossref]

S. M. Hashemi Rafsanjani, M. Mirhosseini, O. S. Magana-Loaiza, and R. W. Boyd, “State transfer based on classical nonseparability,” Phys. Rev. A 92, 023827 (2015).
[Crossref]

K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and B. E. A. Saleh, “Optical coherency matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref]

2014 (2)

A. F. Abouraddy, K. H. Kagalwala, and B. E. A. Saleh, “Two-point optical coherency matrix tomography,” Opt. Lett. 39, 2411–2414 (2014).
[Crossref]

F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
[Crossref]

2013 (2)

V. Venkataraman, K. Saha, and A. L. Gaeta, “Phase modulation at the few-photon level for weak-nonlinearity-based quantum computing,” Nat. Photonics 7, 138–141 (2013).
[Crossref]

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
[Crossref]

2011 (2)

2010 (1)

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref]

2009 (1)

2007 (3)

2006 (1)

2004 (3)

2003 (2)

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
[Crossref]

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

2002 (1)

A. F. Abouraddy, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Quantum entanglement and the two-photon Stokes parameters,” Opt. Commun. 201, 93–98 (2002).
[Crossref]

2001 (1)

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

2000 (1)

1998 (1)

1987 (1)

1938 (1)

F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica 5, 785–795 (1938).
[Crossref]

Abouraddy, A. F.

A. F. Abouraddy, “What is the maximum attainable visibility by a partially coherent electromagnetic field in Young’s double-slit interference?” Opt. Express 25, 18320–18331 (2017).
[Crossref]

H. E. Kondakci and A. F. Abouraddy, “Diffraction-free pulsed optical beams via space-time correlations,” Opt. Express 24, 28659–28668 (2016).
[Crossref]

K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and B. E. A. Saleh, “Optical coherency matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref]

A. F. Abouraddy, K. H. Kagalwala, and B. E. A. Saleh, “Two-point optical coherency matrix tomography,” Opt. Lett. 39, 2411–2414 (2014).
[Crossref]

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
[Crossref]

A. F. Abouraddy, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Quantum entanglement and the two-photon Stokes parameters,” Opt. Commun. 201, 93–98 (2002).
[Crossref]

Aiello, A.

D. Guzman-Silva, R. Brüning, F. Zimmermann, C. Vetter, M. Grafe, M. Heinrich, S. Nolte, M. Duparre, A. Aiello, M. Ornigotti, and A. Szameit, “Demonstration of local teleportation using classical entanglement,” Laser Photon. Rev. 10, 317–321 (2016).
[Crossref]

S. Berg-Johansen, F. Töppel, B. Stiller, P. Banzer, M. Ornigotti, E. Giacobino, G. Leuchs, A. Aiello, and C. Marquardt, “Classically entangled optical beams for high-speed kinematic sensing,” Optica 2, 864–868 (2015).
[Crossref]

A. Aiello, F. Toppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
[Crossref]

Alonso, M. A.

Al-Qasimi, A.

Banzer, P.

Berg-Johansen, S.

Borghi, R.

Born, M.

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

Boyd, R. W.

S. M. Hashemi Rafsanjani, M. Mirhosseini, O. S. Magana-Loaiza, and R. W. Boyd, “State transfer based on classical nonseparability,” Phys. Rev. A 92, 023827 (2015).
[Crossref]

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light (Wiley, 1998).

C. Brosseau and A. Dogariu, “Symmetry properties and polarization descriptors for an arbitrary electromagnetic wavefield,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2006), Vol. 49, Chap. 4, pp. 315–380.

Brüning, R.

D. Guzman-Silva, R. Brüning, F. Zimmermann, C. Vetter, M. Grafe, M. Heinrich, S. Nolte, M. Duparre, A. Aiello, M. Ornigotti, and A. Szameit, “Demonstration of local teleportation using classical entanglement,” Laser Photon. Rev. 10, 317–321 (2016).
[Crossref]

Di Giuseppe, G.

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
[Crossref]

Dogariu, A.

Duparre, M.

D. Guzman-Silva, R. Brüning, F. Zimmermann, C. Vetter, M. Grafe, M. Heinrich, S. Nolte, M. Duparre, A. Aiello, M. Ornigotti, and A. Szameit, “Demonstration of local teleportation using classical entanglement,” Laser Photon. Rev. 10, 317–321 (2016).
[Crossref]

Eberly, J. H.

Ellis, J.

Friberg, A. T.

Gaeta, A. L.

V. Venkataraman, K. Saha, and A. L. Gaeta, “Phase modulation at the few-photon level for weak-nonlinearity-based quantum computing,” Nat. Photonics 7, 138–141 (2013).
[Crossref]

Gamel, O.

Giacobino, E.

A. Aiello, F. Toppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

S. Berg-Johansen, F. Töppel, B. Stiller, P. Banzer, M. Ornigotti, E. Giacobino, G. Leuchs, A. Aiello, and C. Marquardt, “Classically entangled optical beams for high-speed kinematic sensing,” Optica 2, 864–868 (2015).
[Crossref]

F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
[Crossref]

Gil, J. J.

Gori, F.

Grafe, M.

D. Guzman-Silva, R. Brüning, F. Zimmermann, C. Vetter, M. Grafe, M. Heinrich, S. Nolte, M. Duparre, A. Aiello, M. Ornigotti, and A. Szameit, “Demonstration of local teleportation using classical entanglement,” Laser Photon. Rev. 10, 317–321 (2016).
[Crossref]

Guzman-Silva, D.

D. Guzman-Silva, R. Brüning, F. Zimmermann, C. Vetter, M. Grafe, M. Heinrich, S. Nolte, M. Duparre, A. Aiello, M. Ornigotti, and A. Szameit, “Demonstration of local teleportation using classical entanglement,” Laser Photon. Rev. 10, 317–321 (2016).
[Crossref]

Hashemi Rafsanjani, S. M.

S. M. Hashemi Rafsanjani, M. Mirhosseini, O. S. Magana-Loaiza, and R. W. Boyd, “State transfer based on classical nonseparability,” Phys. Rev. A 92, 023827 (2015).
[Crossref]

Heinrich, M.

D. Guzman-Silva, R. Brüning, F. Zimmermann, C. Vetter, M. Grafe, M. Heinrich, S. Nolte, M. Duparre, A. Aiello, M. Ornigotti, and A. Szameit, “Demonstration of local teleportation using classical entanglement,” Laser Photon. Rev. 10, 317–321 (2016).
[Crossref]

James, D.

James, D. F. V.

O. Gamel and D. F. V. James, “Causality and complete positivity of classical polarization maps,” Opt. Lett. 36, 2821–2823 (2011).
[Crossref]

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

Kagalwala, K. H.

K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and B. E. A. Saleh, “Optical coherency matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref]

A. F. Abouraddy, K. H. Kagalwala, and B. E. A. Saleh, “Two-point optical coherency matrix tomography,” Opt. Lett. 39, 2411–2414 (2014).
[Crossref]

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
[Crossref]

Kim, K.

Kondakci, H. E.

H. E. Kondakci and A. F. Abouraddy, “Diffraction-free pulsed optical beams via space-time correlations,” Opt. Express 24, 28659–28668 (2016).
[Crossref]

K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and B. E. A. Saleh, “Optical coherency matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref]

Korotkova, O.

Kwiat, P. G.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

Leuchs, G.

A. Aiello, F. Toppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

S. Berg-Johansen, F. Töppel, B. Stiller, P. Banzer, M. Ornigotti, E. Giacobino, G. Leuchs, A. Aiello, and C. Marquardt, “Classically entangled optical beams for high-speed kinematic sensing,” Optica 2, 864–868 (2015).
[Crossref]

F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
[Crossref]

Magana-Loaiza, O. S.

S. M. Hashemi Rafsanjani, M. Mirhosseini, O. S. Magana-Loaiza, and R. W. Boyd, “State transfer based on classical nonseparability,” Phys. Rev. A 92, 023827 (2015).
[Crossref]

Mandel, L.

Marquardt, C.

A. Aiello, F. Toppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

S. Berg-Johansen, F. Töppel, B. Stiller, P. Banzer, M. Ornigotti, E. Giacobino, G. Leuchs, A. Aiello, and C. Marquardt, “Classically entangled optical beams for high-speed kinematic sensing,” Optica 2, 864–868 (2015).
[Crossref]

F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
[Crossref]

Martínez-Herrero, R.

Mejías, P. M.

Mirhosseini, M.

S. M. Hashemi Rafsanjani, M. Mirhosseini, O. S. Magana-Loaiza, and R. W. Boyd, “State transfer based on classical nonseparability,” Phys. Rev. A 92, 023827 (2015).
[Crossref]

Mujat, M.

Mukunda, N.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref]

Munro, W. J.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

Nolte, S.

D. Guzman-Silva, R. Brüning, F. Zimmermann, C. Vetter, M. Grafe, M. Heinrich, S. Nolte, M. Duparre, A. Aiello, M. Ornigotti, and A. Szameit, “Demonstration of local teleportation using classical entanglement,” Laser Photon. Rev. 10, 317–321 (2016).
[Crossref]

Ornigotti, M.

D. Guzman-Silva, R. Brüning, F. Zimmermann, C. Vetter, M. Grafe, M. Heinrich, S. Nolte, M. Duparre, A. Aiello, M. Ornigotti, and A. Szameit, “Demonstration of local teleportation using classical entanglement,” Laser Photon. Rev. 10, 317–321 (2016).
[Crossref]

S. Berg-Johansen, F. Töppel, B. Stiller, P. Banzer, M. Ornigotti, E. Giacobino, G. Leuchs, A. Aiello, and C. Marquardt, “Classically entangled optical beams for high-speed kinematic sensing,” Optica 2, 864–868 (2015).
[Crossref]

Parker, K. J.

Peres, A.

A. Peres, Quantum Theory: Concepts and Methods (Kluwer Academic, 1995).

Qian, X.-F.

Saha, K.

V. Venkataraman, K. Saha, and A. L. Gaeta, “Phase modulation at the few-photon level for weak-nonlinearity-based quantum computing,” Nat. Photonics 7, 138–141 (2013).
[Crossref]

Saleh, B. E. A.

K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and B. E. A. Saleh, “Optical coherency matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref]

A. F. Abouraddy, K. H. Kagalwala, and B. E. A. Saleh, “Two-point optical coherency matrix tomography,” Opt. Lett. 39, 2411–2414 (2014).
[Crossref]

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
[Crossref]

A. F. Abouraddy, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Quantum entanglement and the two-photon Stokes parameters,” Opt. Commun. 201, 93–98 (2002).
[Crossref]

Santarsiero, M.

Sergienko, A. V.

A. F. Abouraddy, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Quantum entanglement and the two-photon Stokes parameters,” Opt. Commun. 201, 93–98 (2002).
[Crossref]

Setälä, T.

Simon, B. N.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref]

Simon, R.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref]

Simon, S.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref]

Stiller, B.

Szameit, A.

D. Guzman-Silva, R. Brüning, F. Zimmermann, C. Vetter, M. Grafe, M. Heinrich, S. Nolte, M. Duparre, A. Aiello, M. Ornigotti, and A. Szameit, “Demonstration of local teleportation using classical entanglement,” Laser Photon. Rev. 10, 317–321 (2016).
[Crossref]

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J. Opt. Soc. Am. A (3)

Laser Photon. Rev. (1)

D. Guzman-Silva, R. Brüning, F. Zimmermann, C. Vetter, M. Grafe, M. Heinrich, S. Nolte, M. Duparre, A. Aiello, M. Ornigotti, and A. Szameit, “Demonstration of local teleportation using classical entanglement,” Laser Photon. Rev. 10, 317–321 (2016).
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V. Venkataraman, K. Saha, and A. L. Gaeta, “Phase modulation at the few-photon level for weak-nonlinearity-based quantum computing,” Nat. Photonics 7, 138–141 (2013).
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K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
[Crossref]

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A. Aiello, F. Toppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
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F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
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Opt. Commun. (1)

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Opt. Express (5)

Opt. Lett. (10)

Optica (1)

Phys. Lett. A (1)

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
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S. M. Hashemi Rafsanjani, M. Mirhosseini, O. S. Magana-Loaiza, and R. W. Boyd, “State transfer based on classical nonseparability,” Phys. Rev. A 92, 023827 (2015).
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D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
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C. Brosseau and A. Dogariu, “Symmetry properties and polarization descriptors for an arbitrary electromagnetic wavefield,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2006), Vol. 49, Chap. 4, pp. 315–380.

A. Peres, Quantum Theory: Concepts and Methods (Kluwer Academic, 1995).

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Figures (6)

Fig. 1.
Fig. 1.

Concept of an optical-coherence converter. (a) Starting with a polarized but spatially incoherent field ( S p = 0 and S s = 1 , S = S p + S s = 1 ), coherence is converted from polarization to the spatial DoF, thereby yielding an unpolarized but spatially coherent field ( S p = 1 and S s = 0 ) but without introducing further fluctuations (fixed total entropy S = 1 ). The device thus converts the statistical fluctuations (and the attendant entropy) from one DoF to the other. (b) When a polarized but spatially incoherent field is incident on a double slit, no interference fringes are observed. (c) After converting coherence from polarization to the spatial DoF, high-visibility (but unpolarized) interference fringes appear.

Fig. 2.
Fig. 2.

(a) Schematic depicting the input field preparation (source) and characterization (measurement). The field at points a and b is spatially incoherent but fully polarized (scalar). F, filter; P, polarizer; L, lens; PA, polarization analyzer; CCD, charge-coupled device camera. (b) A coherency converter maps the spatially incoherent but polarized field at a and b to a spatially coherent but unpolarized field at a and b . (c) Schematic of the optical setup for the coherence-converter. A biconvex lens (L: f = 20    cm ) images a and b to a and b , respectively, with 2 × magnification. The delay lines enable matching pairs of paths within the source temporal coherence length. HWP, half-wave plate; PBS, polarizing beam splitter; BS, beam splitter. The planes at which the coherency matrices G 1 , G 2 , and G 3 are reconstructed are marked.

Fig. 3.
Fig. 3.

(a) Four measurements required to reconstruct the spatial coherence matrix G S for a scalar field at a and b . The intensity pattern is recorded with both slits open (left), and two measurements are made: the intensity on the optical axis (red dot) and at the location midway along the first expected fringe location calculated from the slit separation (green dot). No fringes are observed here since the field is spatially incoherent. Next, the intensity on the optical axis is recorded when a (left) and b (right) are blocked (the red dots; see Refs. [7,8] for details). (b) Plot depicting graphically the real parts of the elements of the spatial-polarization coherency matrix G 1 for the source plane as reconstructed from OCmT that utilizes the measurements in (a) when carried out in conjunction with polarization measurements. (c) Plot graphically depicting the elements of the theoretically expected coherency matrix G = 1 2 diag { 1,0 , 1,0 } , corresponding to a scalar H-polarized field that is spatially incoherent [Eq. (3)].

Fig. 4.
Fig. 4.

(a) Schematic for the setup to combine two linearly polarized fields from a and b that are statistically independent or spatially incoherent ( S p = 0 and S s = 1 ) into a and b whereupon the field becomes unpolarized but spatially coherent ( S p = 1 and S s = 0 ), without loss of power or increase in total entropy S = 1 . HWP, half-wave plate rotated to implement the transformation H V ; PBS, polarizing beam splitter. (b) Malus curves for fields at the two input ports of the PBS highlight the linear polarization (one orthogonal to the other) and that for the field at the output port highlights its random polarization. The dashed and continuous lines are the flat and sinusoidal curves associated with unpolarized and V-polarized light, respectively. (c) Graphical depiction of the elements of the full coherency matrix G 2 is obtained experimentally and (d) expected theoretically.

Fig. 5.
Fig. 5.

(a) Schematic for the coherence converter that transforms two linearly polarized, spatially incoherent fields (at a and b ) into two randomly polarized mutually coherent fields (at a and b ). (b) Graphical depiction of the real part of the entries of the experimentally reconstructed G 3 via OCmT. (c) The theoretical expectation for G 3 . (d) Interference patterns obtained by overlapping the fields from a and b after a polarization projection, with high-visibility fringes observed in all cases. The top panels are CCD camera images and the lower panels are obtained by integrating the fringes vertically. The main reasons for the different interference-fringe visibilities observed after the various polarization projections are experimental inaccuracies in aligning the two spatial modes and superposing them at the PBS and unequal slit sizes at a and b . As a result, H and V projections display higher visibility than that for the diagonal and circular projections. (e) Visibility as a function of a relative delay inserted between the fields at a and b before overlapping them at the CCD camera [Fig. 2(c)] for the diagonal polarization projection case in (d).

Fig. 6.
Fig. 6.

Effect of a polarization scrambler on the field when introduced at two different planes. (a) A polarization scrambler is placed at the G 2 plane (after the PBS at a ) has no effect on the visibility of the interference pattern measured at the G 3 plane, as shown in the right panels. (b) A polarization scrambler placed before a destroys the visibility. (a, b) The measurements in the right panels are averaged over the polarization shown in Fig. 5(d).

Equations (7)

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G = ( G a a HH G a a HV G a b HH G a b HV G a a VH G a a VV G a b VH G a b VV G b a HH G b a HV G b b HH G b b HV G b a VH G b a VV G b b VH G b b VV ) ,
G s ( r ) = ( G a a HH + G a a VV G a b HH + G a b VV G b a HH + G b a VV G b b HH + G b b VV ) , G p ( r ) = ( G a a HH + G b b HH G a a HV + G b b HV G a a VH + G a a VH G a a VV + G b b VV ) ,
G 1 = 1 2 ( 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ) = 1 2 ( 1 0 0 1 ) s ( 1 0 0 0 ) p ,
G 2 = 1 2 ( 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 ) = ( 1 0 0 0 ) s 1 2 ( 1 0 0 1 ) p .
G 3 = 1 4 ( 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 ) = 1 2 ( 1 1 1 1 ) s 1 2 ( 1 0 0 1 ) p .
G 2 = j = 1 4 p j { I ^ U ^ p ( j ) } G 2 { I ^ U ^ p ( j ) } ,
G 3 = { Λ ^ b I } G 3 { Λ ^ b I } + j = 1 4 p j { Λ ^ a U ^ p ( j ) } G 3 { Λ ^ a U ^ p ( j ) } ,