## Signal-to-noise analysis of Stokes parameters in division of focal plane polarimeters |

Optics Express, Vol. 18, Issue 25, pp. 25815-25824 (2010)

http://dx.doi.org/10.1364/OE.18.025815

Acrobat PDF (1235 KB)

### Abstract

An analysis of the temporal noise in the Stokes parameters computed by division of focal plane polarimeters is presented. Theoretical estimations of the Stokes parameter signal-to-noise ratios for CCD polarization imaging sensors with both 4-polarizer and 2-polarizer micropolarization filter arrays are derived. The theoretical derivation is verified with measurements from an integrated polarization imaging sensor composed of a CCD imaging array and aluminum nanowire polarization filters. The measured data obtained from the CCD polarimeters matches the theoretical derivations of the temporal noise model of the Stokes parameters.

© 2010 OSA

## 1. Introduction

1. J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. **45**(22), 5453–5469 (2006). [CrossRef] [PubMed]

2. D. H. Goldstein, “Mueller matrix dual-rotating retarder polarimeter,” Appl. Opt. **31**(31), 6676–6683 (1992). [CrossRef] [PubMed]

3. K. J. Voss and Y. Liu, “Polarized radiance distribution measurements of skylight. I. System description and characterization,” Appl. Opt. **36**(24), 6083–6094 (1997). [CrossRef] [PubMed]

4. C. J. Zappa, M. L. Banner, H. Schultz, A. Corrada-Emmanuel, L. B. Wolff, and J. Yalcin, “Retrieval of short ocean wave slope using polarimetric imaging,” Meas. Sci. Technol. **19**(5), 055503 (2008). [CrossRef]

6. J. S. Tyo, “Hybrid division of aperture/division of a focal-plane polarimeter for real-time polarization imagery without an instantaneous field-of-view error,” Opt. Lett. **31**(20), 2984–2986 (2006). [CrossRef] [PubMed]

7. J. L. Pezzaniti and D. B. Chenault, “A division of aperture MWIR imaging polarimeter,” Proc. SPIE **5888**, 58880V (2005). [CrossRef]

8. V. Gruev, R. Perkins, and T. York, “CCD polarization imaging sensor with aluminum nanowire optical filters,” Opt. Express **18**(18), 19087–19094 (2010). [CrossRef] [PubMed]

13. X. Zhao, A. Bermak, F. Boussaid, and V. G. Chigrinov, “Liquid-crystal micropolarimeter array for full Stokes polarization imaging in visible spectrum,” Opt. Express **18**(17), 17776–17787 (2010). [CrossRef] [PubMed]

8. V. Gruev, R. Perkins, and T. York, “CCD polarization imaging sensor with aluminum nanowire optical filters,” Opt. Express **18**(18), 19087–19094 (2010). [CrossRef] [PubMed]

13. X. Zhao, A. Bermak, F. Boussaid, and V. G. Chigrinov, “Liquid-crystal micropolarimeter array for full Stokes polarization imaging in visible spectrum,” Opt. Express **18**(17), 17776–17787 (2010). [CrossRef] [PubMed]

*et al.*where birefringent materials and thin film polarization filters were monolithically integrated with a custom CMOS imaging chip [10

10. A. Andreou and Z. Kalayjian, “Polarization imaging: principles and integrated polarimeters,” IEEE Sens. J. **2**(6), 566–576 (2002). [CrossRef]

10. A. Andreou and Z. Kalayjian, “Polarization imaging: principles and integrated polarimeters,” IEEE Sens. J. **2**(6), 566–576 (2002). [CrossRef]

11. M. Momeni and A. H. Titus, “An analog VLSI chip emulating polarization vision of Octopus retina,” IEEE Trans. Neural Netw. **17**(1), 222–232 (2006). [CrossRef] [PubMed]

13. X. Zhao, A. Bermak, F. Boussaid, and V. G. Chigrinov, “Liquid-crystal micropolarimeter array for full Stokes polarization imaging in visible spectrum,” Opt. Express **18**(17), 17776–17787 (2010). [CrossRef] [PubMed]

14. V. Gruev, A. Ortu, N. Lazarus, J. Van der Spiegel, and N. Engheta, “Fabrication of a dual-tier thin film micropolarization array,” Opt. Express **15**(8), 4994–5007 (2007). [CrossRef] [PubMed]

8. V. Gruev, R. Perkins, and T. York, “CCD polarization imaging sensor with aluminum nanowire optical filters,” Opt. Express **18**(18), 19087–19094 (2010). [CrossRef] [PubMed]

9. V. Gruev, J. Van der Spiegel, and N. Engheta, “Dual-tier thin film polymer polarization imaging sensor,” Opt. Express **18**(18), 19292–19303 (2010). [CrossRef] [PubMed]

**18**(17), 17776–17787 (2010). [CrossRef] [PubMed]

**18**(18), 19087–19094 (2010). [CrossRef] [PubMed]

*S*

_{0}, is a measure of the total intensity of the light incident on the sensor. The second and third Stokes parameters,

*S*

_{1}and

*S*

_{2}, provide information about the linear polarization state of the incident light. The fourth Stokes parameter,

*S*

_{3}, provides information about the circular polarization state of the incident light and cannot be determined using only linear polarization filters, thus making it not measureable by the type of sensor described in this paper.

15. J. S. Tyo, “Optimum linear combination strategy for an N-channel polarization-sensitive imaging or vision system,” J. Opt. Soc. Am. A **15**(2), 359–366 (1998). [CrossRef]

16. F. Goudail and A. Bénière, “Estimation precision of the degree of linear polarization and of the angle of polarization in the presence of different sources of noise,” Appl. Opt. **49**(4), 683–693 (2010). [CrossRef] [PubMed]

*I*

_{0}is the intensity of the 0° filtered light wave,

*I*

_{90}is the intensity of the 90° filtered light wave, and so on. An imaging sensor capable of characterizing partially polarized light based on Eqs. (1) through (3) must employ four linear polarization filters offset by 45° together with an array of imaging elements.

*I*

_{tot}is the unfiltered intensity value. An imaging sensor capable of characterizing partially polarized light based on Eqs. (4) through (6) must employ two linear polarization filters offset by 45° together with an array of imaging elements and record the light intensity of the incoming light wave without any polarization filters.

## 2. Pixilated polarization imaging sensor architecture

^{-}/ADU and a total readout noise of 18 e

^{-}.

## 3. Theoretical signal-to-noise analysis for the polarization imaging sensor

^{-}to the noise at operating temperature (40°C). Therefore, for simplicity, dark current will be ignored in this analysis.

*I*is the mean signal, in units of electrons, generated by the incident light on a pixel.

_{photo}*I*, the standard deviation of that pixel's signal is given by Eq. (10):Due to the differences in the Stokes parameter calculations and in light intensities received by the pixels under the differently oriented polarization filters, the two pixilated filter array architectures result in different signal-to-noise ratios for the calculated Stokes parameters. Next, we derive the theoretical noise model for the two pixilated filter array imaging architectures.

_{i}### 3.1 Signal-to-noise analysis for a sensor with a 4-polarizer filter array

*S*

_{0}

*= I*

_{0}

*+ I*

_{90}

*= I*

_{45}

*+ I*

_{135}, the uncertainties in the calculated Stokes parameters all reduce to the same formula, expressed in terms of the mean calculated value of

*S*and are presented by Eq. (17):Finally, the signal-to-noise ratios (SNR) for the mean calculated Stokes parameters may be expressed via Eq. (18) through Eq. (20): Note that because the values of

_{0}*S*and

_{1}*S*can be negative, their absolute values are taken in the numerator of their respective signal-to-noise calculations.

_{2}### 3.2 Signal-to-noise analysis for a sensor with a 2-polarizer filter array

*I*

_{tot},

*I*

_{0}and

*I*

_{45}in terms of

*S*

_{0},

*S*

_{1}and

*S*

_{2}using Eq. (4) through Eq. (6) gives the estimated uncertainties in the calculated Stokes parameters using the 2-polarizer array sensor in terms of the mean calculated values for the parameters. The uncertainties for the three Stokes parameters are presented by Eq. (27) through Eq. (29): Finally, the signal-to-noise ratios of the calculated Stokes parameters are given by Eq. (30) through Eq. (32): Comparing Eq. (18) through Eq. (20) with Eq. (30) through Eq. (32) respectively shows that the signal-to-noise ratio for

*S*

_{0}is always slightly better for the 2-polarizer architecture, though the two converge as

*S*

_{0}gets larger. This is due to the fact that the dominant noise source for high light intensities is the shot noise and the read noise term contributions are insignificant compared to the shot noise contributions. The signal-to-noise ratios for

*S*

_{1}and

*S*

_{2}are functions of

*S*

_{1}and

*S*

_{2}respectively for the 2-polarizer architecture but independent of

*S*

_{1}and

*S*

_{2}for the 4-polarizer architecture. Furthermore, even when the signal-to-noise ratios for

*S*

_{1}and

*S*

_{2}for the 2-polarizer architecture are at their maximum (i.e.

*S*

_{1}or

*S*

_{2}equal to -

*S*

_{0}) the signal-to-noise for the 4-polarizer architecture is larger. Thus, the signal-to-noise for

*S*

_{1}and

*S*

_{2}for the 4-polarizer architecture are always better than those for the 2-polarizer architecture. Another observation can be made that the 4-polarizer architecture allows for uniform sampling of the incoming light wave on the Poincare sphere. In other words, the 4 linear polarization filters are equally spaced i.e. equally offset by 45° and allow for an optimal SNR of the Stokes parameters [20

20. D. S. Sabatke, M. R. Descour, E. L. Dereniak, W. C. Sweatt, S. A. Kemme, and G. S. Phipps, “Optimization of retardance for a complete Stokes polarimeter,” Opt. Lett. **25**(11), 802–804 (2000). [CrossRef]

22. Z Z. Wang, J. S. Tyo, and M. M. Hayat, “Generalized signal-to-noise ratio for spectral sensors with correlated bands,” J. Opt. Soc. Am. A **25**(10), 2528–2534 (2008). [CrossRef]

_{0}parameter is described by Eq. (34):For the 2-polarizer filter array, a neighborhood of 2 by 2 pixels contains two intensity pixels. Hence, the first Stokes parameter for this sensor which is described by Eq. (4), can also be computed as presented by Eq. (35):In Eq. (35),

*I*and

_{tot,1}*I*are the intensity values of both pixels without polarization filters in a neighborhood of 2 by 2 pixels. The SNR for the S

_{tot,2}_{0}parameter in this sensor is described by Eq. (36). Equations (34) and (36) allow for a comparison of the SNR of the first Stokes parameter when computed on a neighborhood of 2 by 2 pixels. The SNR of the 2-polarizer filter array is better when compared to 4-polarizer filter array as previously concluded.

## 4. Experimental methods and results

*S*

_{1}and

*S*

_{2}could be varied while maintaining a constant

*S*

_{0}.

*S*

_{0},

*S*

_{1}and

*S*

_{2}for both array architectures were calculated for each set of 1000 images. All measurements were taken with the sensor at an operating temperature of 40°C.

*S*

_{0}parameter in a 2 by 2 neighborhood of pixels for both filter array architectures are presented in Fig. 2 . Both data sets closely follow their respective theoretical curves, with average deviations from the modeled curve of 2.46% and 1.83% for the 4-polarizer and 2-polarizer array architectures respectively. As predicted by the theoretical model, the signal-to-noise for the 2-polarizer filter array is slightly greater than that of the 4-polarizer filter array at low values of

*S*

_{0}and converges with that of the 4-polarizer array at higher values of

*S*

_{0}.

*S*

_{1}and

*S*

_{2}parameters in a 2 by 2 neighborhood of pixels for both filter array architectures at 3 different incident intensities are presented in Fig. 3 and Fig. 4 respectively. For both the

*S*

_{1}and

*S*

_{2}signal-to-noise ratios, the data follows the theoretical curves very closely for all three

*S*

_{0}values. For the

*S*

_{1}signal-to-noise, the average deviations from the theoretical model for the lowest, middle and highest intensity values are 2.04%, 2.86% and 2.98% for the 4-polarizer architecture and 1.93%, 2.16% and 2.06% for the 2-polarizer architecture. For the

*S*

_{2}signal-to-noise, the average deviations from the theoretical model for the lowest, middle and highest intensity values are 1.89%, 2.63% and 2.60% for the 4-polarizer architecture and 2.06%, 2.10% and 2.09% for the 2-polarizer architecture.

*S*

_{1}and

*S*

_{2}values that range from -

*S*

_{0}to +

*S*

_{0}, however likely due to the non-ideal response of the micropolarizers this full range is not achieved, particularly at low incident intensities. While this complication does not change the theoretical model for the sensor noise, this problem does affect the sensor accuracy. A calibration routine would be required to increase the accuracy of measurements taken with the sensor; however implementation of such a routine is beyond the scope of this paper. A similar problem with division of focal plane polarimeters in the long wave infrared spectrum (LWIR) was observed by Bowers et al. [23

23. D. L. Bowers, J. K. Boger, L. D. Wellems, S. E. Ortega, M. P. Fetrow, J. E. Hubbs, W. T. Black, B. M. Ratliff, and J. S. Tyo, “Unpolarized calibration and nonuniformity correction for long-wave infrared microgrid imaging polarimeters,” Opt. Eng. **47**(4), 046403 (2008). [CrossRef]

24. J. S. Tyo and H. Wei, “Optimizing imaging polarimeters constructed with imperfect optics,” Appl. Opt. **45**(22), 5497–5503 (2006). [CrossRef] [PubMed]

## 5. Conclusion

*S*

_{0}for the 2-polarizer filter array at low light intensities and better signal-to-noise ratios for

*S*

_{1}and

*S*

_{2}for the 4-polarizer filter array. Thus, the choice of filter array architecture for a polarization imaging sensor is dependent on the relative importance of the measurements of

*S*

_{0}versus

*S*

_{1}and

*S*

_{2}for a particular application.

## Acknowledgment

## References and links

1. | J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. |

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

3. | K. J. Voss and Y. Liu, “Polarized radiance distribution measurements of skylight. I. System description and characterization,” Appl. Opt. |

4. | C. J. Zappa, M. L. Banner, H. Schultz, A. Corrada-Emmanuel, L. B. Wolff, and J. Yalcin, “Retrieval of short ocean wave slope using polarimetric imaging,” Meas. Sci. Technol. |

5. | R. M. A. Azzam, “Division-of-amplitude photopolarimeter (DOAP) for the simultaneous measurement of all four Stokes parameters of light,” Opt. Acta (Lond.) |

6. | J. S. Tyo, “Hybrid division of aperture/division of a focal-plane polarimeter for real-time polarization imagery without an instantaneous field-of-view error,” Opt. Lett. |

7. | J. L. Pezzaniti and D. B. Chenault, “A division of aperture MWIR imaging polarimeter,” Proc. SPIE |

8. | V. Gruev, R. Perkins, and T. York, “CCD polarization imaging sensor with aluminum nanowire optical filters,” Opt. Express |

9. | V. Gruev, J. Van der Spiegel, and N. Engheta, “Dual-tier thin film polymer polarization imaging sensor,” Opt. Express |

10. | A. Andreou and Z. Kalayjian, “Polarization imaging: principles and integrated polarimeters,” IEEE Sens. J. |

11. | M. Momeni and A. H. Titus, “An analog VLSI chip emulating polarization vision of Octopus retina,” IEEE Trans. Neural Netw. |

12. | T. Tokuda, S. Sato, H. Yamada, K. Sasagawa, and J. Ohta, “Polarisation-analysing CMOS photosensor with monolithically embedded wire grid polarizer,” Electron. Lett. |

13. | X. Zhao, A. Bermak, F. Boussaid, and V. G. Chigrinov, “Liquid-crystal micropolarimeter array for full Stokes polarization imaging in visible spectrum,” Opt. Express |

14. | V. Gruev, A. Ortu, N. Lazarus, J. Van der Spiegel, and N. Engheta, “Fabrication of a dual-tier thin film micropolarization array,” Opt. Express |

15. | J. S. Tyo, “Optimum linear combination strategy for an N-channel polarization-sensitive imaging or vision system,” J. Opt. Soc. Am. A |

16. | F. Goudail and A. Bénière, “Estimation precision of the degree of linear polarization and of the angle of polarization in the presence of different sources of noise,” Appl. Opt. |

17. | V. Gruev, J. Van der Spiegel, and N. Engheta, “Nano-wire Dual Layer Polarization Filter,” Proc. IEEE ISCAS,561–564 (2009). |

18. | B. Razavi, Design of Analog CMOS Integrated Circuits (McGraw-Hill, New York, NY, 2001). |

19. | R. Philip, Bevington and D. Keith Robinson, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1992). |

20. | D. S. Sabatke, M. R. Descour, E. L. Dereniak, W. C. Sweatt, S. A. Kemme, and G. S. Phipps, “Optimization of retardance for a complete Stokes polarimeter,” Opt. Lett. |

21. | J. S. Tyo, “Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error,” Appl. Opt. |

22. | Z Z. Wang, J. S. Tyo, and M. M. Hayat, “Generalized signal-to-noise ratio for spectral sensors with correlated bands,” J. Opt. Soc. Am. A |

23. | D. L. Bowers, J. K. Boger, L. D. Wellems, S. E. Ortega, M. P. Fetrow, J. E. Hubbs, W. T. Black, B. M. Ratliff, and J. S. Tyo, “Unpolarized calibration and nonuniformity correction for long-wave infrared microgrid imaging polarimeters,” Opt. Eng. |

24. | J. S. Tyo and H. Wei, “Optimizing imaging polarimeters constructed with imperfect optics,” Appl. Opt. |

**OCIS Codes**

(110.4280) Imaging systems : Noise in imaging systems

(120.5410) Instrumentation, measurement, and metrology : Polarimetry

(230.5440) Optical devices : Polarization-selective devices

(110.5405) Imaging systems : Polarimetric imaging

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: September 16, 2010

Revised Manuscript: October 28, 2010

Manuscript Accepted: November 4, 2010

Published: November 24, 2010

**Citation**

Robert Perkins and Viktor Gruev, "Signal-to-noise analysis of Stokes parameters in division of focal plane polarimeters," Opt. Express **18**, 25815-25824 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-25-25815

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### References

- J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45(22), 5453–5469 (2006). [CrossRef] [PubMed]
- D. H. Goldstein, “Mueller matrix dual-rotating retarder polarimeter,” Appl. Opt. 31(31), 6676–6683 (1992). [CrossRef] [PubMed]
- K. J. Voss and Y. Liu, “Polarized radiance distribution measurements of skylight. I. System description and characterization,” Appl. Opt. 36(24), 6083–6094 (1997). [CrossRef] [PubMed]
- C. J. Zappa, M. L. Banner, H. Schultz, A. Corrada-Emmanuel, L. B. Wolff, and J. Yalcin, “Retrieval of short ocean wave slope using polarimetric imaging,” Meas. Sci. Technol. 19(5), 055503 (2008). [CrossRef]
- R. M. A. Azzam, “Division-of-amplitude photopolarimeter (DOAP) for the simultaneous measurement of all four Stokes parameters of light,” Opt. Acta (Lond.) 29, 685–689 (1982).
- J. S. Tyo, “Hybrid division of aperture/division of a focal-plane polarimeter for real-time polarization imagery without an instantaneous field-of-view error,” Opt. Lett. 31(20), 2984–2986 (2006). [CrossRef] [PubMed]
- J. L. Pezzaniti and D. B. Chenault, “A division of aperture MWIR imaging polarimeter,” Proc. SPIE 5888, 58880V (2005). [CrossRef]
- V. Gruev, R. Perkins, and T. York, “CCD polarization imaging sensor with aluminum nanowire optical filters,” Opt. Express 18(18), 19087–19094 (2010). [CrossRef] [PubMed]
- V. Gruev, J. Van der Spiegel, and N. Engheta, “Dual-tier thin film polymer polarization imaging sensor,” Opt. Express 18(18), 19292–19303 (2010). [CrossRef] [PubMed]
- A. Andreou and Z. Kalayjian, “Polarization imaging: principles and integrated polarimeters,” IEEE Sens. J. 2(6), 566–576 (2002). [CrossRef]
- M. Momeni and A. H. Titus, “An analog VLSI chip emulating polarization vision of Octopus retina,” IEEE Trans. Neural Netw. 17(1), 222–232 (2006). [CrossRef] [PubMed]
- T. Tokuda, S. Sato, H. Yamada, K. Sasagawa, and J. Ohta, “Polarisation-analysing CMOS photosensor with monolithically embedded wire grid polarizer,” Electron. Lett. 45(4), 228–230 (2009). [CrossRef]
- X. Zhao, A. Bermak, F. Boussaid, and V. G. Chigrinov, “Liquid-crystal micropolarimeter array for full Stokes polarization imaging in visible spectrum,” Opt. Express 18(17), 17776–17787 (2010). [CrossRef] [PubMed]
- V. Gruev, A. Ortu, N. Lazarus, J. Van der Spiegel, and N. Engheta, “Fabrication of a dual-tier thin film micropolarization array,” Opt. Express 15(8), 4994–5007 (2007). [CrossRef] [PubMed]
- J. S. Tyo, “Optimum linear combination strategy for an N-channel polarization-sensitive imaging or vision system,” J. Opt. Soc. Am. A 15(2), 359–366 (1998). [CrossRef]
- F. Goudail and A. Bénière, “Estimation precision of the degree of linear polarization and of the angle of polarization in the presence of different sources of noise,” Appl. Opt. 49(4), 683–693 (2010). [CrossRef] [PubMed]
- V. Gruev, J. Van der Spiegel, and N. Engheta, “Nano-wire Dual Layer Polarization Filter,” Proc. IEEE ISCAS,561–564 (2009).
- B. Razavi, Design of Analog CMOS Integrated Circuits (McGraw-Hill, New York, NY, 2001).
- R. Philip, Bevington and D. Keith Robinson, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1992).
- D. S. Sabatke, M. R. Descour, E. L. Dereniak, W. C. Sweatt, S. A. Kemme, and G. S. Phipps, “Optimization of retardance for a complete Stokes polarimeter,” Opt. Lett. 25(11), 802–804 (2000). [CrossRef]
- J. S. Tyo, “Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error,” Appl. Opt. 41(4), 619–630 (2002). [CrossRef] [PubMed]
- Z Z. Wang, J. S. Tyo, and M. M. Hayat, “Generalized signal-to-noise ratio for spectral sensors with correlated bands,” J. Opt. Soc. Am. A 25(10), 2528–2534 (2008). [CrossRef]
- D. L. Bowers, J. K. Boger, L. D. Wellems, S. E. Ortega, M. P. Fetrow, J. E. Hubbs, W. T. Black, B. M. Ratliff, and J. S. Tyo, “Unpolarized calibration and nonuniformity correction for long-wave infrared microgrid imaging polarimeters,” Opt. Eng. 47(4), 046403 (2008). [CrossRef]
- J. S. Tyo and H. Wei, “Optimizing imaging polarimeters constructed with imperfect optics,” Appl. Opt. 45(22), 5497–5503 (2006). [CrossRef] [PubMed]

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