## The polarization properties of a tilted polarizer |

Optics Express, Vol. 21, Issue 22, pp. 27032-27042 (2013)

http://dx.doi.org/10.1364/OE.21.027032

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### Abstract

Polarizers are key components in optical science and technology. Thus, understanding the action of a polarizer beyond oversimplifying approximations is crucial. In this work, we study the interaction of a polarizing interface with an obliquely incident wave experimentally. To this end, a set of Mueller matrices is acquired employing a novel procedure robust against experimental imperfections. We connect our observation to a geometric model, useful to predict the effect of polarizers on complex light fields.

© 2013 OSA

## 1. Introduction

1. Q. Hong, T. Wu, X. Zhu, R. Lu, and S.-T. Wu, “Designs of wide-view and broadband circular polarizers,” Opt. Express **13**, 8318–8331 (2005). [CrossRef] [PubMed]

3. J.-W. Moon, W.-S. Kang, H. yong Han, S. M. Kim, S. H. Lee, Y. gyu Jang, C. H. Lee, and G.-D. Lee, “Wideband and wide-view circular polarizer for a transflective vertical alignment liquid crystal display,” Appl. Opt. **49**, 3875–3882 (2010). [CrossRef] [PubMed]

6. A. Aiello, G. Puentes, D. Voigt, and J. P. Woerdman, “Maximum-likelihood estimation of Mueller matrices,” Opt. Lett. **31**, 817–819 (2006). [CrossRef] [PubMed]

7. D. G. M. Anderson and R. Barakat, “Necessary and sufficient conditions for a Mueller matrix to be derivable from a Jones matrix,” J. Opt. Soc. Am. A **11**, 2305–2319 (1994). [CrossRef]

8. A. Aiello and J. Woerdman, “Physical Bounds to the Entropy-Depolarization Relation in Random Light Scattering,” Phys. Rev. Lett. **94**, 090406 (2005). [CrossRef] [PubMed]

9. R. M. A. Azzam and A. G. Lopez, “Accurate calibration of the four-detector photopolarimeter with imperfect polarizing optical elements,” J. Opt. Soc. Am. A **6**, 1513–1521 (1989). [CrossRef]

12. A. M. Brańczyk, D. H. Mahler, L. A. Rozema, A. Darabi, A. M. Steinberg, and D. F. V. James, “Self-calibrating quantum state tomography,” New J. Phys. **14**, 085003 (2012). [CrossRef]

13. J. Korger, A. Aiello, C. Gabriel, P. Banzer, T. Kolb, C. Marquardt, and G. Leuchs, “Geometric Spin Hall Effect of Light at polarizing interfaces,” Appl. Phys. B **102**, 427–432 (2011). [CrossRef]

## 2. Polarization of a light beam

**=**

*J**E*

_{x}**+**

*x̂**E*

_{y}**of its electric field**

*ŷ***(**

*E***,**

*r**t*) = Re[

**exp(**

*J**i*(

**·**

*k***−**

*r**ωt*))], where

**=**

*k**k*

**is the wave vector. The complex column-vector**

*ẑ***has become known as the**

*J**Jones*vector [15

15. R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. **31**, 488–493 (1941). [CrossRef]

*Stokes*parameters [17]

*I*is the intensity of the light beam transmitted across a linear polarizer oriented at an angle

_{α}*α*with respect to the

**axis and**

*x̂**I*is the right- or left-handed circularly polarized component of the intensity.

_{R,L}*T*and

*M*are called

*Jones*and

*Mueller*matrices, respectively.

## 3. Geometric Polarizer Models

**of wave propagation [4**

*ẑ*4. Y. Fainman and J. Shamir, “Polarization of nonplanar wave fronts,” Appl. Opt. **23**, 3188–3195 (1984). [CrossRef] [PubMed]

*P̂**. FS make use of the transversality of the electric field vector and conclude that the effect of a polarizer reduces to the projection onto an*

_{T}*effective transmitting axis*

*t̂*_{FS}(illustrated in Fig. 1(a)). In their model, the unit vector

*t̂*_{FS}∝

*P̂**− (*

_{T}**·**

*ẑ*

*P̂**)*

_{T}**is found by projecting the polarizer’s transmitting axis**

*ẑ*

*P̂**onto the plane of the electric field perpendicular to the direction of wave propagation*

_{T}**.**

*ẑ***. For any orientation of the polarizer, the resulting Jones matrix**

*J*

*P̂**parallel to each other. Analogously to the transmitting case, we interpret the projection of this unit vector*

_{A}

*P̂**as an*

_{A}*effective absorbing axis*If this interpretation holds true, the light field after transmission across an absorbing polarizer becomes As above, the corresponding Jones matrix

*T*= 1 −

_{A}

*ââ*^{T}=

*t̂t̂*^{T}is a projector, where

**=**

*t̂***×**

*ẑ***can be interpreted as the**

*â**effective transmitting axis*as illustrated in Fig. 1(b). While Eq. (5) is structurally equivalent to Fainman and Shamir’s construction, our model coincides with their approach only for normal incidence. Generally, our absorbing model

*T*= 1 −

_{A}

*ââ*^{T}differs from the FS case

19. P. Yeh, “Generalized model for wire grid polarizers,” Proc. SPIE **0307**, 13–21 (1982). [CrossRef]

## 4. Mueller Matrix measurement

*M*describing the device-under-test, and the state of polarization

**. Applied to any Stokes vector, this yields the transmitted intensity.**

*S**M*is unambiguously determined by 16 equations like Eq. (7). However, the measured intensities

*E*denotes experimental values, can be noisy. Thus, acquiring more than 16 values helps to reduce both statistical and systematic errors significantly. To this end, instead of solving a linear system of equations, we pick the Mueller matrix

*M*

^{LS}from the set of all possible Mueller matrices, such that becomes minimal.

*M*is physically acceptable [6

6. A. Aiello, G. Puentes, D. Voigt, and J. P. Woerdman, “Maximum-likelihood estimation of Mueller matrices,” Opt. Lett. **31**, 817–819 (2006). [CrossRef] [PubMed]

8. A. Aiello and J. Woerdman, “Physical Bounds to the Entropy-Depolarization Relation in Random Light Scattering,” Phys. Rev. Lett. **94**, 090406 (2005). [CrossRef] [PubMed]

*H*with non-negative eigenvalues [6

6. A. Aiello, G. Puentes, D. Voigt, and J. P. Woerdman, “Maximum-likelihood estimation of Mueller matrices,” Opt. Lett. **31**, 817–819 (2006). [CrossRef] [PubMed]

*H*=

*H*

^{†}can be expressed using a set of 16 real numbers {

*h*

_{1},...,

*h*

_{16}}. Therefore, these 16 parameters span the vector space of physical Mueller matrices and the set

*M*

^{LS}for the actual Mueller matrix.

**are not known precisely, we can find these parameter employing a procedure similar to the one described above. Interestingly, this requires no a priori information beyond the knowledge that the polarization states**

*S***are prepared using polarizers and birefringent retarders. To this end, the device-under-test is removed from the beam path. Using the same procedure as for the actual measurement, a set of intensities**

*S**M*

^{LS}in Eq. (8) with the identity matrix (Mueller matrix of empty space). Additionally, we express the abstract Stokes vectors

*S**represent horizontally or vertically polarized states, respectively. In our experiment, those are the states transmitted or reflected by a polarizing beam splitter [Fig. 3(a)]. This yields: In particular,*

_{H,V}*ρ*

^{in}and its fast axis oriented at an angle

**axis. Analogously,**

*x̂**α*=

_{i}*α*

_{i}_{+1}−

*α*of both wave plates, where, for example, Δ

_{i}*α*= Δ

_{i}*α*= 22.5°. Thus, our measurement setup is completely described by four parameters,

*ρ*

^{in},

*ρ*

^{out}, which are to be found with this calibration procedure. The set of parameters which minimizes Eq. (10) yields the states of polarization

## 5. Experiment

*ρ*

^{in}=

*π*/2 + 0.008rad,

*ρ*

^{out}=

*π*/2 + 0.019rad). Nevertheless, knowledge of these parameters is crucial to perform a highly accurate Mueller matrix reconstruction.

*ϕ*of the polarizer’s absorbing axis, each for a large number of tilting angles 0° ≤

*θ*≤ 82°. We apply the least-squares method described above to find the Mueller matrices describing our polarizer.

*M*is less then 10

_{ab}^{−3}. Furthermore, our data indicates that the results are also accurate. The sample, we have studied is a linear polarizer. For normal incidence (

*θ*= 0°), the transmittance across such a polarizer does not depend on the helicity of the incident beam and the transmitted beam is linearly polarized. The corresponding Mueller matrix elements |

*M*

_{03}| < 0.01 and |

*M*

_{30}| < 0.01 clearly vanish for all relevant measurements. Thus, we estimate systematic errors to be below 10

^{−2}.

*θ*. This behaviour cannot be described by a perfect projector as in the geometric models discussed earlier. Thus, we propose to generalize the projection rule, Eq. (6), to include two transmission coefficients

*τ*and

_{a}*τ*for states of polarization parallel and perpendicular to the effective absorbing axis:

_{t}*θ*and

*ϕ*. In Fig. 5, we demonstrate that this model accurately agrees with our observation for different configurations.

22. A. Beer, “Bestimmung der Absorption des rothen Lichts in farbigen Flüssigkeiten,” Ann. Phys. **162**, 78–88 (1852). [CrossRef]

*τ*|

_{a}^{2}> 0 of Eq. (12b) accounts for the transmittance for crossed polarization, i.e. the fact that even if the electric field is polarized parallel to the effective absorbing axis, the absorption is not 100%. The phase of the complex parameter

*τ*indicates that this field component is scattered with a phase determined by the orientation of the nano-particles relative to the incoming wave.

_{a}*θ*< 45°, the observation agrees with the prediction of the geometric absorbing model

*T*. Close to grazing incidence

_{A}*θ*→ 90°, the latter deviates, which we can understand in a physical picture. The particles embedded in our polarizer are cigar-shaped [23

23. S. Polizzi, A. Armigliato, P. Riello, N. F. Borrelli, and G. Fagherazzi, “Redrawn phase-separated borosilicate glasses: A TEM investigation,” Microsc. Microanal. M. **8**, 157–165 (1997). [CrossRef]

24. S. Polizzi, P. Riello, G. Fagherazzi, and N. Borrelli, “The microstructure of borosilicate glasses containing elongated and oriented phase-separated crystalline particles,” J. Non-Cryst. Solids **232–234**, 147–154 (1998). [CrossRef]

*P̂**. By design, the wavelength is close to the resonance of the particles’ long axes. At normal incidence, the scattering and absorption is strong for states of polarization parallel to the long axis and negligible in the orthogonal case.*

_{A}

*P̂**takes part in the interaction. Thus, the effect of a single particle decreases proportionally to cos(*

_{A}*θ*) as the coupling becomes less efficient.

*τ*(

_{a}*θ*) and

*τ*(

_{t}*θ*), which can be directly measured.

## 6. Conclusion

25. R. C. Thompson, J. R. Bottiger, and E. S. Fry, “Measurement of polarized light interactions via the Mueller matrix,” Appl. Opt. **19**, 1323–1332 (1980). [CrossRef] [PubMed]

26. E. Compain, S. Poirier, and B. Drevillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and mueller-matrix ellipsometers,” Appl. Opt. **38**, 3490–3502 (1999). [CrossRef]

## Acknowledgments

## References and links

1. | Q. Hong, T. Wu, X. Zhu, R. Lu, and S.-T. Wu, “Designs of wide-view and broadband circular polarizers,” Opt. Express |

2. | Q. Hong, T. X. Wu, R. Lu, and S.-T. Wu, “Wide-view circular polarizer consisting of a linear polarizer and two biaxial films,” Opt. Express |

3. | J.-W. Moon, W.-S. Kang, H. yong Han, S. M. Kim, S. H. Lee, Y. gyu Jang, C. H. Lee, and G.-D. Lee, “Wideband and wide-view circular polarizer for a transflective vertical alignment liquid crystal display,” Appl. Opt. |

4. | Y. Fainman and J. Shamir, “Polarization of nonplanar wave fronts,” Appl. Opt. |

5. | A. Aiello, C. Marquardt, and G. Leuchs, “Nonparaxial polarizers,” Opt. Lett. |

6. | A. Aiello, G. Puentes, D. Voigt, and J. P. Woerdman, “Maximum-likelihood estimation of Mueller matrices,” Opt. Lett. |

7. | D. G. M. Anderson and R. Barakat, “Necessary and sufficient conditions for a Mueller matrix to be derivable from a Jones matrix,” J. Opt. Soc. Am. A |

8. | A. Aiello and J. Woerdman, “Physical Bounds to the Entropy-Depolarization Relation in Random Light Scattering,” Phys. Rev. Lett. |

9. | R. M. A. Azzam and A. G. Lopez, “Accurate calibration of the four-detector photopolarimeter with imperfect polarizing optical elements,” J. Opt. Soc. Am. A |

10. | D. H. Goldstein, “Mueller matrix dual-rotating retarder polarimeter,” Appl. Opt. |

11. | B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys. |

12. | A. M. Brańczyk, D. H. Mahler, L. A. Rozema, A. Darabi, A. M. Steinberg, and D. F. V. James, “Self-calibrating quantum state tomography,” New J. Phys. |

13. | J. Korger, A. Aiello, C. Gabriel, P. Banzer, T. Kolb, C. Marquardt, and G. Leuchs, “Geometric Spin Hall Effect of Light at polarizing interfaces,” Appl. Phys. B |

14. | J. Korger, A. Aiello, V. Chille, P. Banzer, C. Wittmann, N. Lindlein, C. Marquardt, and G. Leuchs, “Observation of the geometric spin Hall effect of light,” arXiv:1303.6974 (2013). |

15. | R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. |

16. | J. N. Damask, |

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

18. | K. Kim, L. Mandel, and E. Wolf, “Relationship between Jones and Mueller matrices for random media,” J. Opt. Soc. Am. A |

19. | P. Yeh, “Generalized model for wire grid polarizers,” Proc. SPIE |

20. | S.-Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A |

21. | J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta |

22. | A. Beer, “Bestimmung der Absorption des rothen Lichts in farbigen Flüssigkeiten,” Ann. Phys. |

23. | S. Polizzi, A. Armigliato, P. Riello, N. F. Borrelli, and G. Fagherazzi, “Redrawn phase-separated borosilicate glasses: A TEM investigation,” Microsc. Microanal. M. |

24. | S. Polizzi, P. Riello, G. Fagherazzi, and N. Borrelli, “The microstructure of borosilicate glasses containing elongated and oriented phase-separated crystalline particles,” J. Non-Cryst. Solids |

25. | R. C. Thompson, J. R. Bottiger, and E. S. Fry, “Measurement of polarized light interactions via the Mueller matrix,” Appl. Opt. |

26. | E. Compain, S. Poirier, and B. Drevillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and mueller-matrix ellipsometers,” Appl. Opt. |

**OCIS Codes**

(120.5410) Instrumentation, measurement, and metrology : Polarimetry

(260.2130) Physical optics : Ellipsometry and polarimetry

(260.5430) Physical optics : Polarization

**ToC Category:**

Physical Optics

**History**

Original Manuscript: August 20, 2013

Revised Manuscript: September 27, 2013

Manuscript Accepted: October 10, 2013

Published: October 31, 2013

**Citation**

Jan Korger, Tobias Kolb, Peter Banzer, Andrea Aiello, Christoffer Wittmann, Christoph Marquardt, and Gerd Leuchs, "The polarization properties of a tilted polarizer," Opt. Express **21**, 27032-27042 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-22-27032

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### References

- Q. Hong, T. Wu, X. Zhu, R. Lu, and S.-T. Wu, “Designs of wide-view and broadband circular polarizers,” Opt. Express13, 8318–8331 (2005). [CrossRef] [PubMed]
- Q. Hong, T. X. Wu, R. Lu, and S.-T. Wu, “Wide-view circular polarizer consisting of a linear polarizer and two biaxial films,” Opt. Express13, 10777–10783 (2005). [CrossRef] [PubMed]
- J.-W. Moon, W.-S. Kang, H. yong Han, S. M. Kim, S. H. Lee, Y. gyu Jang, C. H. Lee, and G.-D. Lee, “Wideband and wide-view circular polarizer for a transflective vertical alignment liquid crystal display,” Appl. Opt.49, 3875–3882 (2010). [CrossRef] [PubMed]
- Y. Fainman and J. Shamir, “Polarization of nonplanar wave fronts,” Appl. Opt.23, 3188–3195 (1984). [CrossRef] [PubMed]
- A. Aiello, C. Marquardt, and G. Leuchs, “Nonparaxial polarizers,” Opt. Lett.34, 3160–3162 (2009). [CrossRef] [PubMed]
- A. Aiello, G. Puentes, D. Voigt, and J. P. Woerdman, “Maximum-likelihood estimation of Mueller matrices,” Opt. Lett.31, 817–819 (2006). [CrossRef] [PubMed]
- D. G. M. Anderson and R. Barakat, “Necessary and sufficient conditions for a Mueller matrix to be derivable from a Jones matrix,” J. Opt. Soc. Am. A11, 2305–2319 (1994). [CrossRef]
- A. Aiello and J. Woerdman, “Physical Bounds to the Entropy-Depolarization Relation in Random Light Scattering,” Phys. Rev. Lett.94, 090406 (2005). [CrossRef] [PubMed]
- R. M. A. Azzam and A. G. Lopez, “Accurate calibration of the four-detector photopolarimeter with imperfect polarizing optical elements,” J. Opt. Soc. Am. A6, 1513–1521 (1989). [CrossRef]
- D. H. Goldstein, “Mueller matrix dual-rotating retarder polarimeter,” Appl. Opt.31, 6676–6683 (1992). [CrossRef] [PubMed]
- B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys.75, 163 (2007). [CrossRef]
- A. M. Brańczyk, D. H. Mahler, L. A. Rozema, A. Darabi, A. M. Steinberg, and D. F. V. James, “Self-calibrating quantum state tomography,” New J. Phys.14, 085003 (2012). [CrossRef]
- J. Korger, A. Aiello, C. Gabriel, P. Banzer, T. Kolb, C. Marquardt, and G. Leuchs, “Geometric Spin Hall Effect of Light at polarizing interfaces,” Appl. Phys. B102, 427–432 (2011). [CrossRef]
- J. Korger, A. Aiello, V. Chille, P. Banzer, C. Wittmann, N. Lindlein, C. Marquardt, and G. Leuchs, “Observation of the geometric spin Hall effect of light,” arXiv:1303.6974 (2013).
- R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am.31, 488–493 (1941). [CrossRef]
- J. N. Damask, Polarization Optics in Telecommunications (Springer, 2005).
- M. Born and E. Wolf, Principles of Optics (Pergamon Pr., Oxford, 1999), 7th ed.
- K. Kim, L. Mandel, and E. Wolf, “Relationship between Jones and Mueller matrices for random media,” J. Opt. Soc. Am. A4, 433–437 (1987). [CrossRef]
- P. Yeh, “Generalized model for wire grid polarizers,” Proc. SPIE0307, 13–21 (1982). [CrossRef]
- S.-Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A13, 1106–1113 (1996). [CrossRef]
- J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta33, 185–189 (1986). [CrossRef]
- A. Beer, “Bestimmung der Absorption des rothen Lichts in farbigen Flüssigkeiten,” Ann. Phys.162, 78–88 (1852). [CrossRef]
- S. Polizzi, A. Armigliato, P. Riello, N. F. Borrelli, and G. Fagherazzi, “Redrawn phase-separated borosilicate glasses: A TEM investigation,” Microsc. Microanal. M.8, 157–165 (1997). [CrossRef]
- S. Polizzi, P. Riello, G. Fagherazzi, and N. Borrelli, “The microstructure of borosilicate glasses containing elongated and oriented phase-separated crystalline particles,” J. Non-Cryst. Solids232–234, 147–154 (1998). [CrossRef]
- R. C. Thompson, J. R. Bottiger, and E. S. Fry, “Measurement of polarized light interactions via the Mueller matrix,” Appl. Opt.19, 1323–1332 (1980). [CrossRef] [PubMed]
- E. Compain, S. Poirier, and B. Drevillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and mueller-matrix ellipsometers,” Appl. Opt.38, 3490–3502 (1999). [CrossRef]

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