## Longitudinal polarization periodicity of unpolarized light passing through a double wedge depolarizer |

Optics Express, Vol. 20, Issue 25, pp. 27348-27360 (2012)

http://dx.doi.org/10.1364/OE.20.027348

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### Abstract

The polarization characteristics of unpolarized light passing through a double wedge depolarizer are studied. It is found that the degree of polarization of the radiation propagating after the depolarizer is uniform across transverse planes after the depolarizer, but it changes from one plane to another in a periodic way giving, at different distances, unpolarized, partially polarized, or even perfectly polarized light. An experiment is performed to confirm this result. Measured values of the Stokes parameters and of the degree of polarization are in complete agreement with the theoretical predictions.

© 2012 OSA

## 1. Introduction

1. J. P. McGuire and R. A. Chipman, “Analysis of spatial pseudodepolarizers in imaging systems,” Opt. Eng. **29**, 1478–1484 (1990). [CrossRef]

7. J. C. G. de Sande, G. Piquero, and C. Teijeiro, “Polarization changes at Lyot depolarizer output for different types of input beams,” J. Opt. Soc. Am. A **29**, 278–284 (2012). [CrossRef]

*pseudo-depolarizer*is the double wedge depolarizer (DWD), which consists of a pair of uniaxial crystal wedges, with suitably oriented optic axes, placed in contact to form a plate. When a totally and uniformly polarized light impinges on it, this kind of elements produces a periodic variation of the state of polarization across a plane parallel to the output face of the device. The DOP at any point is equal to one but when the Stokes parameters are integrated over a large area compared to the period of the state of polarization, it becomes nearly zero.

1. J. P. McGuire and R. A. Chipman, “Analysis of spatial pseudodepolarizers in imaging systems,” Opt. Eng. **29**, 1478–1484 (1990). [CrossRef]

6. C. Vena, C. Versace, G. Strangi, and R. Bartolino, “Light depolarization by non-uniform polarization distribution over a beam cross section,” J. Opt. A: Pure Appl. Opt. **11**, 125704–10 (2009). [CrossRef]

*z*-positions.

8. F. Gori, M. Santarsiero, S. Vicalvi, and R. Borghi, “Beam coherence-polarization matrix,” Pure and Appl. Opt. **7**, 941–951 (1998). [CrossRef]

16. R. Martínez-Herrero and P. M. Mejías, “On the propagation of random electromagnetic fields with position-independent stochastic behavior,” Opt. Commun. **283**, 4467–4469 (2010). [CrossRef]

## 2. Theory

### 2.1. Preliminaries

*z*axis is orthogonal to the faces of the device (with origin at its exit face) and the thickness of the wedges varies along the

*x*axis. The first wedge has its optic axis along the

*y*direction and thickness (at

*x*= 0) equal to

*d*

_{1}. The second wedge has thickness

*d*

_{2}(at

*x*= 0) and its optic axis is in the

*xy*plane and forms an angle of 45° with respect to the first one.

17. H. Lotem and U. Taor, “Low-loss bireflectant (double reflection) polarization prism,” Appl. Opt. **25**, 1271–1273 (1985). [CrossRef]

19. V. Kuznetsov, D. Faleiev, E. Savin, and V. Lebedev, “Crystal-based device for combining light beams,” Opt. Lett. **34**, 2856–2857 (2009). [CrossRef] [PubMed]

*z*axis, so that they can be characterized through their Jones vector, defined as [20] where

*E*and

_{x}*E*are the components of the electric field along the

_{y}*x*and

*y*axes, at the typical point

**r**. In the most general case, the quantities appearing in the Jones vector are stochastic variables and the polarization characteristics are described through the correlation functions among all the transverse field components. Such correlation functions are collected into the polarization matrix, i.e., with the dagger representing hermitian conjugation and 〈·〉 the ensemble average. The total intensity of the field is defined as the trace of

*P̂*, while the local degree of polarization (DOP) is evaluated as Since the matrix

*P̂*is hermitian and semipositive defined, the DOP only assumes values in the interval [0, 1],

*p*= 1 corresponding to a perfectly polarized field and

*p*= 0 to a completely unpolarized one.

### 2.2. Totally polarized input light

*z*axis with uniform and linear polarization along the

*x*and the

*y*direction, respectively. In Fig. 1(b) the notations used in the following are introduced. In particular, the wave-vector direction of a typical plane wave propagating in each medium is shown, together with the transmission coefficients at each interface. Note that, due to the used geometry, the wave vectors of all waves lye in the

*xz*plane. The superscript (

*ξ*=

*x,y*), when it occurs, always refers to the original polarization of the wave impinging on the device. The first subindex of the transmission coefficients refers to the polarization direction of the field in the incidence medium:

*x*or

*y*, if the incidence medium is air,

*o*or

*e*(from ordinary and extraordinary, respectively) if the incidence medium is the crystal. The same convention holds for the second subindex, but referred to the transmission medium. The angle that each wave vector inside the second wedge forms with the

*z*axis will be referred to as

*β*, where

_{στ}*σ*=

*o*,

*e*denotes the character (ordinary or extraordinary) of the wave in the first wedge and

*τ*=

*o*,

*e*denotes the character of the wave in the second wedge. Finally,

*γ*denotes the refraction angle of the corresponding

_{στ}*στ*wave at the exit surface. Due to the geometry of the DWD, all the waves propagate forming large angles with respect to the optic axes of the crystals. As a consequence, the effects of optical activity can be neglected [21

21. S. C. McClain, L. W. Hillman, and R. A. Chipman, “Polarization ray tracing in anisotropic optically active media. II. Theory and physics,” J. Opt. Soc. Am. A **10**, 2383–2393 (1993). [CrossRef]

*x*direction. Its Jones vector is of the form where

*A*is the amplitude of the field across the input face of the DWD and

*in*stands for input.

*z*-component, the Jones vector of the output field,

*z*> 0 can be evaluated taking the effects of all interfaces and propagation distances into account. After some calculations, it turns out to be (see Appendix) with and

*d*=

*d*

_{1}+

*d*

_{2}is the total thickness of the device,

*k*is the vacuum wave number,

*k*is the ordinary wave number and

_{o}*k*is the wave number corresponding to the

_{oe}*oe*extraordinary wave. This wave number, as well as the angles

*β*and

_{oe}*γ*, can be calculated by repeatedly applying the Snell law, together with the relation obtained from the index ellipse for a wave travelling at an angle

_{oe}*φ*with respect to the optic axis of the second wedge [22].

*oo*) propagates along the

*z*axis, while the other one (

*oe*) propagates along a direction that forms the angle

*γ*with respect to the

_{oe}*z*axis. Since the transmission coefficients are real quantities, the polarizations of the two waves are linear, but directed along different directions. The sum of such two waves produces a periodic variation of the polarization state across the transverse plane, with period

*L*= 2

_{o}*π*/(

*k*|sin

*γ*|), in a similar way to that generated by a polarization grating [23

_{oe}23. F. Gori, “Measuring Stokes parameters by means of a polarization grating,” Opt. Lett. **24**, 584–586 (1999). [CrossRef]

24. G. Piquero, R. Borghi, and M. Santarsiero, “Gaussian Schell-model beams propagating through polarization gratings,” J. Opt. Soc. Am. A **18**, 1399–1405 (2001). [CrossRef]

26. V. Arrizón, E. Tepichin, M. Ortíz-Gutiérrez, and A.W. Lohmann, “Fresnel diffraction at l/4 of the Talbot distance of an anisotropic grating,” Opt. Commun. **127**, 171–175 (1996). [CrossRef]

28. Z. Bomzon, A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Polarization Talbot self-imaging with computer-generated, space-variant subwavelength dielectric gratings,” Appl. Opt. **41**, 5218–5222 (2002). [CrossRef] [PubMed]

*α*is typically small. In such cases, the following approximations hold [21

21. S. C. McClain, L. W. Hillman, and R. A. Chipman, “Polarization ray tracing in anisotropic optically active media. II. Theory and physics,” J. Opt. Soc. Am. A **10**, 2383–2393 (1993). [CrossRef]

29. S. C. McClain, L. W. Hillman, and R. A. Chipman, “Polarization ray tracing in anisotropic optically active media. I. Algorithms,” J. Opt. Soc. Am. A **10**, 2371–2382 (1993). [CrossRef]

30. Z. Zhang and H. J. Caufield, “Reflection and refraction by interfaces of uniaxial crystals,” Opt. & Laser Technol. **28**, 549–553 (1996). [CrossRef]

*oo*plane wave, which propagates along the

*z*axis, is linearly polarized at −45°, while the other one (

*oe*) is linearly polarized at +45°. Therefore, the polarization state across a transverse plane periodically varies along the

*x*axis, from linear to circular and vice versa, with a period depending on the angle between the two propagation directions. This is at the basis of the use of such a device as a “depolarizer”: although the local degree of polarization must be unitary at every point, it vanishes when the average polarization is considered over spatial regions having size much larger than the transverse period of polarization state.

*y*direction. In this case the Jones vector is given by

*z*> 0 the Jones vector turns out to be with and Here,

*k*is the wave number for an extraordinary wave propagating perpedicularly to the optic axis, whereas

_{e}*k*is the wave number corresponding to an extraordinary wave that travels inside the right wedge in the direction given by

_{ee}*β*(see Fig. 1(b)).

_{ee}*ee*and

*eo*) with different linear polarization states, propagating along different directions, and the same considerations hold as the ones made after Eq. (6).

*α*[21

21. S. C. McClain, L. W. Hillman, and R. A. Chipman, “Polarization ray tracing in anisotropic optically active media. II. Theory and physics,” J. Opt. Soc. Am. A **10**, 2383–2393 (1993). [CrossRef]

29. S. C. McClain, L. W. Hillman, and R. A. Chipman, “Polarization ray tracing in anisotropic optically active media. I. Algorithms,” J. Opt. Soc. Am. A **10**, 2371–2382 (1993). [CrossRef]

30. Z. Zhang and H. J. Caufield, “Reflection and refraction by interfaces of uniaxial crystals,” Opt. & Laser Technol. **28**, 549–553 (1996). [CrossRef]

*T*≃

_{y}*T*, the approximated expression of the output field reads

_{x}*γ*−

_{eo}*γ*≃ −

_{ee}*γ*, that is, approximately equal to that formed by the two waves of the previous case, but in the opposite sense. This means that the polarization pattern of the output field has the same structure and period of the one obtained when the incident wave is polarized along

_{oe}*x*.

### 2.3. Unpolarized input light

*I*

_{0}the intensity of the input wave and set the intensity of each component wave to

*I*

_{0}/2, the polarization matrix across the entrance surface of the device, evaluated from Eq. (2) with

*P̂*as the sum of the polarization matrices pertinent to each component. The corresponding degree of polarization, from Eq. (3), turns out to be zero.

_{in}*x*and

*y*components of the incident field. Since such contributions form two different periodic polarization patterns at any

*z*plane with approximately the same transverse period, they give rise to a periodic structure both in

*x*and

*z*directions. This behavior can be observed in Fig. 2(a) and (b) where

*s*

_{1}(

*x*,

*z*) and

*s*

_{3}(

*x*,

*z*) Stokes parameters are represented. Both

*s*

_{1}(

*x*,

*z*) and

*s*

_{3}(

*x*,

*z*) show the same behavior with a quarter-period delay in the

*x*direction.

*z*plane (diversely to what happens with

*s*

_{1}(

*x*,

*z*) and

*s*

_{3}(

*x*,

*z*)) and ii) the DOP at the exit of the ”depolarizer” varies from zero to one with the propagation distance, in a periodic way. This means that, for a totally unpolarized input light, the field is perfectly polarized at some transverse planes at the exit of the device.

*δ*and

_{o}*δ*appearing in Eq. (23). In fact, retaining only the terms up to the second order in

_{e}*α*, the phases in Eqs. (8) and (15) can be written as where

*n*and

_{e}*n*are the extraordinary and ordinary refractive index of the crystal. Using such an approximation, the Stokes vector becomes and the DOP of the light after the DWD turns out to be i.e., it is independent of the lateral variable

_{o}*x*and varies from zero to one in a periodic way as a function of the propagation distance

*z*.

*m*= 0, 1, 2...), the DOP reaches its maximum value, equal to unity. There, totally polarized light field is obtained. Note that at such distances,

*s*

_{1}(

*x*,

*Z*) and

_{m}*s*

_{2}(

*x*,

*Z*) are sinusoidal functions of variable

_{m}*x*with maximum amplitude. On the other hand, for distances

*s*

_{1}(

*x*,

*z*) =

_{m}*s*

_{2}(

*x*,

*z*) = 0 and the DOP vanishes, so that the field is completely unpolarized across such planes. It must be noted that consecutive

_{m}*z*-planes where DOP is maximum and minimum are separated by the distance

*Z*−

_{m}*z*=

_{m}*z*/4.

_{T}## 3. Experiment

*λ*= 632.8 nm), linearly polarized along the vertical (

*y*) direction, were used. At the output of one of these lasers a half-wave plate (HWP) rotated at 45° was placed to obtain a linear polarization along the horizontal (

*x*) axis. A neutral density filter (F) was used to adjust the output power of one of the lasers. The two beams were combined by using a polarizing beam splitter (PBS) and expanded by means of a telescope (a 20× microscope objective MO and a collimating lens L with 200 mm focal length). The resulting wave (unpolarized and approximately plane) was sent onto a DWD. The latter (Thorlabs, DPU-25-A) consisted of two quartz wedges (

*n*= 1.5426,

_{o}*n*= 1.5517) [31

_{e}31. G. Ghosh, “Dispersion-equation coefficients for the refractive index and birefringence of calcite and quartz crystals,” Opt. Commun. **163**, 95–102 (1999). [CrossRef]

*α*= 2°. A more precise value of

*α*was measured by analyzing the propagated pattern in the far zone, giving

*α*= 2.17°.

*p*and the Stokes parameters of the output field were measured by means of a polarimeter (Thorlabs PAX5710VIS-T-TXP) at different

_{out}*x*positions, at several distances from the exit surface of the DWD. The detection surface of the polarimeter was circular with diameter of 300

*μ*m.

*p*was practically constant (within 5%) across the transverse direction for all

_{out}*z*planes (see Fig. 4). The measured DOP dependence on the propagation distance is represented in Fig. 5 (dots), together with the theoretical curve (solid line), calculated from Eq. (27). It can be observed that, at the exit of the DWD, the measured DOP is nearly zero, but it grows as the light propagates and reaches a maximum value, near unity, for a propagation distance around 2.75 m. Around this distance, the light is nearly totally polarized. Then, the DOP decreases to nearly zero when the propagation distance is around 5.5 m. This behavior is periodically repeated with increasing propagation distance. A complete agreement between calculated curve and experimental points is obtained.

*s*

_{1}(

*x*,

*z*),

*s*

_{2}(

*x*,

*z*) and

*s*

_{3}(

*x*,

*z*) Stokes parameters (normalized to the total intensity

*I*

_{0}) at transverse planes located at different

*z*distances behind the DWD (red circles:

*s*

_{1}; green down triangles:

*s*

_{2}; blue up triangles:

*s*

_{3}). At

*z*= 0.01 m, the all three Stokes parameters are approximately zero, representing unpolarized light. For

*z*= 0.50 m, the Stokes parameters

*s*

_{1}(

*x*,

*z*) and

*s*

_{3}(

*x*,

*z*) follow sinusoidal dependences

*vs*the transverse variable

*x*(with 0.28 maximum amplitude and a quarter-period delay), while

*s*

_{2}(

*x*,

*z*) ≈ 0. Similar results are observed for

*z*= 1.50 m and

*z*= 2.75 m but with larger maximum amplitude (0.72 and 0.97, respectively) than in the previous case. Theoretical curves obtained by means of Eq. (26) are also represented for these distances. A very good agreement with experimental data is observed.

## 4. Conclusions

*pseudo-depolarizers*and their effect is to produce a periodic transverse variation of the polarization state of the field at their output, as the one produced by a polarization grating. As a consequence, the DOP of the output field, evaluated averaging the Stokes parameters over a sufficiently large area, turns out to be nearly zero.

## Appendix

*z*components are considered, so that the field of a plane wave, linearly polarized along

*x*, normally incident on the DWD can be written as where

*A*is its amplitude and

*x*is the

_{i}*x*-coordinate across the entrance surface of the device.

*x,z*> 0) at the exit of the DWD is obtained by evaluating the optical path length along the line sketched in Fig. 1(b), which represents a flux line of the wave vector inside the crystal. The amplitude of the field is derived using the transmission coefficients pertinent to every interface along the optical path.

*d*

_{1}+

*x*tan

_{i}*α*) inside the first wedge as an ordinary wave, the field becomes

*oo*) and an extraordinary wave (

*oe*). The corresponding Snell’s laws are

*oo*wave propagates along the

*z*axis up to the exit surface of the device, where the field turns out to be and after the exit surface it becomes

*oe*field across the inner side of the exit DWD face is [29

29. S. C. McClain, L. W. Hillman, and R. A. Chipman, “Polarization ray tracing in anisotropic optically active media. I. Algorithms,” J. Opt. Soc. Am. A **10**, 2371–2382 (1993). [CrossRef]

30. Z. Zhang and H. J. Caufield, “Reflection and refraction by interfaces of uniaxial crystals,” Opt. & Laser Technol. **28**, 549–553 (1996). [CrossRef]

*x*=

_{oe}*d*

_{2}tan

*β*+

_{oe}*x*(1 − tan

_{i}*α*tan

*β*) and (

_{oe}*e*,

_{oe,x}*e*,

_{oe,y}*e*)

_{oe,z}*(with the superscript*

^{T}*T*denoting transpose) is a unitary vector perpendicular to the

*oe*ray vector that lies in the plane formed by the second crystal optic axis and the wave vector of the

*oe*wave. The angle

*β*is obtained from Eqs. (9) and (34), taking into account that

_{oe}*oe*field propagating beyond the DWD turns out to be

*γ*is obtained on applying the Snell’s law at the exit surface. On replacing the

_{oe}*x*value given in Eq. (31), the latter equation becomes

_{i}*z*component of the above field is neglected, the superposition of the

*oo*and

*oe*waves in Eqs. (36) and (39) gives the field in Eq. (6).

*y*, it is found that

*eo*wave, and

*ee*wave. Again, on assuming as negligible the

*z*components of such fields, their superposition gives rise to the field expressed in Eq. (13).

## Acknowledgments

## References and links

1. | J. P. McGuire and R. A. Chipman, “Analysis of spatial pseudodepolarizers in imaging systems,” Opt. Eng. |

2. | S. C. McClain, R. A. Chipman, and L. W. Hillman“Aberrations of a horizontal/vertical depolarizer,” Appl. Opt. |

3. | M. El Sherif, M.S. Khalil, S. Khodeir, and N. Nagib, “Simple depolarizers for spectrophotometric measurements of anisotropic samples,” Opt. & Laser Technol. |

4. | G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Computer-generated infrared depolarizer using space-variant subwavelength dielectric gratings,” Opt. Lett. |

5. | V. A. Bagan, B. L. Davydov, and I. E. Samartsev, “Characteristics of Cornu depolarisers made from quartz and paratellurite optically active crystals,” Quant. Electron. |

6. | C. Vena, C. Versace, G. Strangi, and R. Bartolino, “Light depolarization by non-uniform polarization distribution over a beam cross section,” J. Opt. A: Pure Appl. Opt. |

7. | J. C. G. de Sande, G. Piquero, and C. Teijeiro, “Polarization changes at Lyot depolarizer output for different types of input beams,” J. Opt. Soc. Am. A |

8. | F. Gori, M. Santarsiero, S. Vicalvi, and R. Borghi, “Beam coherence-polarization matrix,” Pure and Appl. Opt. |

9. | F. Gori, M. Santarsiero, R. Borghi, and G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun. |

10. | G. P. Agrawal and E. Wolf, “Propagation-induced polarization changes in partially coherent optical beams,” J. Opt. Soc. Am. A |

11. | E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A |

12. | F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, “Effects of coherence on the degree of polarization in a Young interference pattern,” Opt. Lett. |

13. | M. Salem and E. Wolf “Coherence-induced polarization changes in light beams,” Opt. Lett. |

14. | T. D. Visser, D. Kuebel, M. Lahiri, T. Shirai, and E. Wolf, “Unpolarized light beams with different coherence properties,” J. Mod. Opt. |

15. | F. Gori, J. Tervo, and J. Turunen, “Correlation matrices of completely unpolarized beams,” Opt. Lett. |

16. | R. Martínez-Herrero and P. M. Mejías, “On the propagation of random electromagnetic fields with position-independent stochastic behavior,” Opt. Commun. |

17. | H. Lotem and U. Taor, “Low-loss bireflectant (double reflection) polarization prism,” Appl. Opt. |

18. | L. V. Alekseeva, I. V. Povkh, V. I. Stroganov, B. I. Kidyarov, and P. G. Pasko, “Four-ray splitting in optical crystals,” J. Opt. Technol. |

19. | V. Kuznetsov, D. Faleiev, E. Savin, and V. Lebedev, “Crystal-based device for combining light beams,” Opt. Lett. |

20. | E. Wolf, |

21. | S. C. McClain, L. W. Hillman, and R. A. Chipman, “Polarization ray tracing in anisotropic optically active media. II. Theory and physics,” J. Opt. Soc. Am. A |

22. | M. Born and E. Wolf, |

23. | F. Gori, “Measuring Stokes parameters by means of a polarization grating,” Opt. Lett. |

24. | G. Piquero, R. Borghi, and M. Santarsiero, “Gaussian Schell-model beams propagating through polarization gratings,” J. Opt. Soc. Am. A |

25. | H. F. Talbot, “Facts relating to optical science,” Phil. Mag. |

26. | V. Arrizón, E. Tepichin, M. Ortíz-Gutiérrez, and A.W. Lohmann, “Fresnel diffraction at l/4 of the Talbot distance of an anisotropic grating,” Opt. Commun. |

27. | J. Tervo and J. Turunen, “Transverse and longitudinal periodicities in fields produced by polarization gratings,” Opt. Commun. |

28. | Z. Bomzon, A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Polarization Talbot self-imaging with computer-generated, space-variant subwavelength dielectric gratings,” Appl. Opt. |

29. | S. C. McClain, L. W. Hillman, and R. A. Chipman, “Polarization ray tracing in anisotropic optically active media. I. Algorithms,” J. Opt. Soc. Am. A |

30. | Z. Zhang and H. J. Caufield, “Reflection and refraction by interfaces of uniaxial crystals,” Opt. & Laser Technol. |

31. | G. Ghosh, “Dispersion-equation coefficients for the refractive index and birefringence of calcite and quartz crystals,” Opt. Commun. |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(260.5430) Physical optics : Polarization

(240.5440) Optics at surfaces : Polarization-selective devices

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: October 10, 2012

Manuscript Accepted: November 1, 2012

Published: November 20, 2012

**Citation**

Juan Carlos G. de Sande, Massimo Santarsiero, Gemma Piquero, and Franco Gori, "Longitudinal polarization periodicity of unpolarized light passing through a double wedge depolarizer," Opt. Express **20**, 27348-27360 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-25-27348

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### References

- J. P. McGuire and R. A. Chipman, “Analysis of spatial pseudodepolarizers in imaging systems,” Opt. Eng.29, 1478–1484 (1990). [CrossRef]
- S. C. McClain, R. A. Chipman, and L. W. Hillman“Aberrations of a horizontal/vertical depolarizer,” Appl. Opt.31, 2326–2331 (1992). [CrossRef] [PubMed]
- M. El Sherif, M.S. Khalil, S. Khodeir, and N. Nagib, “Simple depolarizers for spectrophotometric measurements of anisotropic samples,” Opt. & Laser Technol.28, 561–563 (1996). [CrossRef]
- G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Computer-generated infrared depolarizer using space-variant subwavelength dielectric gratings,” Opt. Lett.28, 1400–1402 (2003). [CrossRef] [PubMed]
- V. A. Bagan, B. L. Davydov, and I. E. Samartsev, “Characteristics of Cornu depolarisers made from quartz and paratellurite optically active crystals,” Quant. Electron.39, 73–78 (2009). [CrossRef]
- C. Vena, C. Versace, G. Strangi, and R. Bartolino, “Light depolarization by non-uniform polarization distribution over a beam cross section,” J. Opt. A: Pure Appl. Opt.11, 125704–10 (2009). [CrossRef]
- J. C. G. de Sande, G. Piquero, and C. Teijeiro, “Polarization changes at Lyot depolarizer output for different types of input beams,” J. Opt. Soc. Am. A29, 278–284 (2012). [CrossRef]
- F. Gori, M. Santarsiero, S. Vicalvi, and R. Borghi, “Beam coherence-polarization matrix,” Pure and Appl. Opt.7, 941–951 (1998). [CrossRef]
- F. Gori, M. Santarsiero, R. Borghi, and G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun.163, 159–163 (1999). [CrossRef]
- G. P. Agrawal and E. Wolf, “Propagation-induced polarization changes in partially coherent optical beams,” J. Opt. Soc. Am. A17, 2019–2023 (2000). [CrossRef]
- E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A312, 263–267 (2003). [CrossRef]
- F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, “Effects of coherence on the degree of polarization in a Young interference pattern,” Opt. Lett.31, 688–690 (2006). [CrossRef] [PubMed]
- M. Salem and E. Wolf “Coherence-induced polarization changes in light beams,” Opt. Lett.33, 1180–1182 (2008). [CrossRef] [PubMed]
- T. D. Visser, D. Kuebel, M. Lahiri, T. Shirai, and E. Wolf, “Unpolarized light beams with different coherence properties,” J. Mod. Opt.56, 1369–1374 (2009). [CrossRef]
- F. Gori, J. Tervo, and J. Turunen, “Correlation matrices of completely unpolarized beams,” Opt. Lett.34, 1447–1449 (2009). [CrossRef] [PubMed]
- R. Martínez-Herrero and P. M. Mejías, “On the propagation of random electromagnetic fields with position-independent stochastic behavior,” Opt. Commun.283, 4467–4469 (2010). [CrossRef]
- H. Lotem and U. Taor, “Low-loss bireflectant (double reflection) polarization prism,” Appl. Opt.25, 1271–1273 (1985). [CrossRef]
- L. V. Alekseeva, I. V. Povkh, V. I. Stroganov, B. I. Kidyarov, and P. G. Pasko, “Four-ray splitting in optical crystals,” J. Opt. Technol.39, 441–443 (2002). [CrossRef]
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