## Band limited data reconstruction in modulated polarimeters |

Optics Express, Vol. 19, Issue 16, pp. 14976-14989 (2011)

http://dx.doi.org/10.1364/OE.19.014976

Acrobat PDF (929 KB)

### Abstract

Data processing for sequential in time polarimeters based on the Data Reduction Matrix technique yield polarization artifacts in the presence of time varying signals. To overcome these artifacts, polarimeters are designed to operate at higher and higher speeds. In this paper we describe a band limited reconstruction algorithm that allows the measurement and processing of temporally varying Stokes parameters without artifacts. An example polarimeter consisting of a rotating retarder and polarizer is considered, and conventional processing methods are compared to a band limited reconstruction algorithm for the example polarimeter. We demonstrate that a significant reduction in error is possible using these methods.

© 2011 OSA

## 1. Introduction

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

3. J. S. Tyo, M. P. Rowe, E. N. Pugh, and N. Engheta, “Target detection in optically scattering media by polarization-difference imaging,” Appl. Opt. **35**, 1855–1870 (1996). [CrossRef] [PubMed]

3. J. S. Tyo, M. P. Rowe, E. N. Pugh, and N. Engheta, “Target detection in optically scattering media by polarization-difference imaging,” Appl. Opt. **35**, 1855–1870 (1996). [CrossRef] [PubMed]

7. V. Thilak, D. G. Voelz, and C. D. Creusere, “Image segmentation from multi-look passive polarimetric imagery,” in “*Proc. SPIE 6682*,” , J. A. Shaw and J. S. Tyo, eds. (SPIE, Bellingham, WA, 2007), p. 668206. [CrossRef]

8. D. J. Diner, A. Davis, B. Hancock, G. Gutt, R. A. Chipman, and B. Cairns, “Dual-photoeleastic-modulator-based polarimetric imaging concept for aerosol remote sensing,” Appl. Opt. **46**, 8428–8445 (2007). [CrossRef] [PubMed]

10. D. W. Tyler, A. M. Phenis, A. B. Tietjen, M. Virgen, J. D. Mudge, J. S. Stryjewski, and J. A. Dank, “First high-resolution passive polarimetric images of boosting rocket exhaust plumes,” in “*Proc. SPIE vol. 7461: Polarization Science and Remote Sensing IV*,” , J. A. Shaw and J. S. Tyo, eds. (SPIE, Bellingham, WA, 2009), p. 74610J.

*I*,

_{x}*I*,

_{y}*I*

_{45}, and

*I*

_{135}are the fluxes observed through ideal linear polarizers at the indicated orientations, and

*I*and

_{L}*I*are fluxes through ideal left- and right-circular polarizers, respectively. Optical detectors cannot measure

_{R}*s*

_{1},

*s*

_{2}, and

*s*

_{3}directly. Instead, optical polarimeters modify the detected flux of the optical field in a polarization-dependent manner, and then reconstruct the Stokes parameters through an inversion process [1

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

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

12. R. M. A. Azzam, I. M. Elminyawi, and A. M. El-Saba, “General analysis and optimization of the four-detector photopolarimeter,” J. Opt. Soc. Am. A **5**, 681–689 (1988). [CrossRef]

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

16. K. Oka and T. Kaneko, “Compact complete imaging polarimeter using birefringent wedge prisms,” Opt. Express **11**, 1510–1519 (2003). [CrossRef] [PubMed]

17. K. Oka and T. Kato, “Spectroscopic polarimetry with a channeled spectrum,” Opt. Lett. **24**, 1475–1477 (1999). [CrossRef]

19. L. Gendre, A. Foulonneau, and L. Bigué, “High-speed imaging acquisition of stokes linearly polarized components using a single ferroelectric liquid crystal modulator,” in “*Proc. SPIE vol. 7461: Polarization Science and Remote Sensing IV*,” , J. A. Shaw and J. S. Tyo, eds. (SPIE, Bellingham, WA, 2009), p. 74610G.

*wavefront division polarimeters*and

*modulated polarimeters*. Wavefront division polarimeters split the light into multiple channels and make the constituent polarization measurements with independent hardware in each channel. Modulated polarimeters introduce a polarimetric modulation in time, space, wavelength, or some combination thereof, and measure the modulated signal using a single detector or detector array. The polarization signal is then determined by demodulating the information carried in the polarization-dependent side bands.

*et al.*[15

15. J. S. Tyo, C. F. LaCasse, and B. M. Ratliff, “Total elimination of sampling errors in polarization imagery obtained with integrated microgrid polarimeters,” Opt. Lett. **34**, 3187–3189 (2009). [CrossRef] [PubMed]

21. B. M. Ratliff, C. F. Lacasse, and J. S. Tyo, “Quantifying ifov error and compensating its effects in dofp polarimeters,” Opt. Express **17**, 9112 – 9125 (2009). [CrossRef] [PubMed]

*et al.*. [15

15. J. S. Tyo, C. F. LaCasse, and B. M. Ratliff, “Total elimination of sampling errors in polarization imagery obtained with integrated microgrid polarimeters,” Opt. Lett. **34**, 3187–3189 (2009). [CrossRef] [PubMed]

15. J. S. Tyo, C. F. LaCasse, and B. M. Ratliff, “Total elimination of sampling errors in polarization imagery obtained with integrated microgrid polarimeters,” Opt. Lett. **34**, 3187–3189 (2009). [CrossRef] [PubMed]

## 2. Polarimetric Data Reduction Matrix method for DoT Polarimeters

**W**from affecting the measurement.

*f*

_{0}(in units of rotations per second). The analyzer vector as a function of time is and Eq. (3) becomes Equation (8) indicates that the

*s*

_{0}information is contained in the base band, (unmodulated DC) term, the

*s*

_{1}information is contained in the in-phase (cosine) side band at 2

*f*

_{0}, and the

*s*

_{2}information is contained in the quadrature (sine) side band at 2

*f*

_{0}. The polarization information could be demodulated using frequency-domain methods, which is often done for systems using photo elastic modulators operating at very high rates (∼10s of kHz) [4

4. M. P. Silverman and W. Strange, “Object delineation within turbid media by backscattering of phase modulated light,” Opt. Commun. **144**, 7–11 (1997). [CrossRef]

8. D. J. Diner, A. Davis, B. Hancock, G. Gutt, R. A. Chipman, and B. Cairns, “Dual-photoeleastic-modulator-based polarimetric imaging concept for aerosol remote sensing,” Appl. Opt. **46**, 8428–8445 (2007). [CrossRef] [PubMed]

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

25. J. S. Tyo, “Design of optimal polarimeters: maximization of SNR and minimization of systematic errors,” Appl. Opt. **41**, 619–630 (2002). [CrossRef] [PubMed]

26. F. Goudail and A. Beniere, “Estimation precision of the linear degree of polarization and of the angle of polarization in the presence of different types of noises,” Appl. Opt. **49**, 683–693 (2010). [CrossRef] [PubMed]

## 3. A Linear Systems Formalism for Data Reduction

**S**as approximately constant in computing the PDRM with only a few exceptions [8

8. D. J. Diner, A. Davis, B. Hancock, G. Gutt, R. A. Chipman, and B. Cairns, “Dual-photoeleastic-modulator-based polarimetric imaging concept for aerosol remote sensing,” Appl. Opt. **46**, 8428–8445 (2007). [CrossRef] [PubMed]

**34**, 3187–3189 (2009). [CrossRef] [PubMed]

16. K. Oka and T. Kaneko, “Compact complete imaging polarimeter using birefringent wedge prisms,” Opt. Express **11**, 1510–1519 (2003). [CrossRef] [PubMed]

17. K. Oka and T. Kato, “Spectroscopic polarimetry with a channeled spectrum,” Opt. Lett. **24**, 1475–1477 (1999). [CrossRef]

*I*(

*t*) in Eq. (2) yields where the tilde indicates Fourier transform and * is the convolution operator. Equation (9) is a deconvolution problem that can be inverted by careful design of the analyzer Stokes parameters

**A**(

*t*). For the example in Eq. (8) side bands in frequency space carry the polarization information as

*W*(as seen in Fig. 1). The conditions for no overlap to occur in the frequency domain for the rotating analyzer polarimeter are and where

_{B}*f*is the temporal sampling frequency of the polarimeter. Other types of DoT polarimeters with different specific modulation schemes have their own band limit requirements but a criteria similar to Eq. (11) can be derived for other modulation schemes by applying a no side lobe overlap condition in the Fourier domain.

_{s}## 4. Unifying the PDRM and Linear Systems Methods for General Modulated Polarimeters

**S**(

*x*,

*y*,

*t*,

*λ*). The signal is measured with a polarimeter that contains a single detector with an integration window

*d*(

*x*,

*y*,

*t*,

*λ*), a system impulse response function

*h*(

*x*,

*y*,

*t*,

*λ*), and a polarimetric modulation described by

**A**(

*x*,

*y*,

*t*,

*λ*). The modulated flux is The functions in Eq. (12) are not necessarily linear, shift-invariant (LSI) functions. When the system is not LSI, the convolution integrals are replaced by overlap integrals of the more general form rather than the more familiar convolution form for LSI systems where in both cases the input is

*f*(

*x*) and the impulse response is

*h*(

*x*). Furthermore, the impulse response

*h*(

*x*,

*y*,

*t*,

*λ*) is assumed to be scalar in Eq. (12). In reality, the system is described by a polarimetric impulse response matrix that relates how the optical system alters the polarization state between object and image plane before it is ever sampled [29

29. J. P. McGuire and R. A. Chipman, “Diffraction image formation in optical systems with polarization aberrations. I: formulation and example,” J. Opt. Soc. Am. A **7**, 1614–1626 (1990). [CrossRef]

**I**. The first part of the pseudo inverse in Eq. (6) is For example, a rotating retarder polarimeter might make

*N*= 16 measurements as the retarder is rotated from 0° to 360°, or a DoFP polarimeter might be decomposed into 4-element (2 × 2) super pixels. In the linear systems formalism, a similar formation of

**Z**is where

*A*,

_{i}*A*are as defined in Eq. (2). The

_{j}**W**

^{T}**W**part of the pseudo inverse is the integral of the product of the modulation functions contained in

**A**, so the inversion of this quantity describes how to separate the Stokes parameters.

**Z**and

**Z**

^{−1}are diagonal matrices for modulation schemes where the modulators

*A*are orthogonal over the integral. The quantity

_{j}**Z**

^{−1}will be referred to as the

*modulator inner product inversion matrix*.

**A**,

**Z**

^{−1}will be constant with respect to the initial phase of the modulation. However, in cases where the polarimeter varies in time or in space, as would be the case when there is drift in the absolute angular position of a freely running rotating retarder or in the absolute phase of oscillation of the PEM-based systems [8

**46**, 8428–8445 (2007). [CrossRef] [PubMed]

**Z**will not be constant and needs to be computed separately at each reconstruction point. The more general

**Z**in the time domain can be calculated by where

*w*(

*t*) is the reconstruction window used for the estimation of the Stokes signal. If

*w*is a rectangular window with a length corresponding to integer periods of all modulation frequencies in the system, Eq. (17) reduces to Eq. (16); in general arbitrary window shapes and modulation schemes need the weighted inner product matrix elements calculated for all space time and/or wavelength.

**W**

*term in Eq. (15), examine how it operates on the modulated flux in the standard PDRM formalism, The matrix*

^{T}**W**

*plays two roles. First,*

^{T}**W**

*acts as the homodyne in the demodulation process, since the matrix*

^{T}**W**includes the modulation strategy of

**A**(

*x*,

*y*,

*t*,

*λ*). The homodyne process remodulates the modulated signal. When the modulation is made up of a superposition of sinusoids, multiplication by a homodyne creates a superposition of signals, with one component of the signal unmodulated (centered at base band) and other components of the superposition are copies of the signal modulated at higher frequencies. These copies can then be filtered out with a low pass filter, leaving only the component of the signal that is unmodulated. Multiplying the input Stokes parameters by

**W**and then by

**W**

*is equivalent to mixing with a carrier frequency once to move the base band signal up to the side bands and a second time to create a copy at base band along with spurious copies at higher frequencies. In communications theory, the next step is to low pass filters to eliminate the spurious high frequency copies. The low pass filter is implicitly included in the matrix multiplication*

^{T}**W**

^{T}**W**, and this is the second role of

**W**

*. However, in the matrix multiplication the low pass filter has a rectangular footprint (in time, space, and/or wavelength). This rectangular footprint has a sinc-function frequency response that allows leakage of spurious high frequency signals into the reconstruction as shown below.*

^{T}**A**is a4 × 1 vector with each element a scalar function of space time and wavelength, not the 4 ×

*N*matrix that

**W**conventionally implies. The quantity

*w*(

*x*,

*y*,

*t*,

*λ*) is the windowing function in Eq. (17); this ensures that the reconstruction algorithm properly unfixes the modulators over the particular window that has been chosen for the estimation. Separating

*w*from the inversion process allows for the optimization of the low pass filter operation and for the control over which portions of the frequency domain are included in the estimation. The advantage of this will become clear in the following section. In the special case when the

*Z*matrix is constant over the modulation, the final estimation can be written as

## 5. Discussion

*W*and hence will have errors.

_{B}### 5.1. Rotating Retarder Polarimeter

*δ*rotating at a frequency

*f*

_{0}followed by an fixed ideal linear analyzer with the orientation 0° such that analyzer as function of time is

*d*(

_{n}*t*) =

*δ*(

*t*−

*nt*

_{0}) at a point

*h*(

*x*,

*y*) =

*δ*(

*x*,

*y*). With the periodic sampling there are only samples at the sample number

*A*

_{2}and

*A*

_{3}. In Eq. (22)

*Z*

_{12}=

*Z*

_{21}≠ 0 because the modulators

*A*

_{0}and

*A*

_{1}are not orthogonal; both have DC components.

### 5.2. Example: Band Limited Polarization Scene

*W*≤ 2

_{B}*f*

_{0}and

**A**.

**A**(

**t**)

*I*(

*t*) using each of the four modulation functions in Eq. (21). The shaded region in Fig. 5 and all other following frequency domain plots indicates the band limit criteria for the modeled polarimeter. The first and second modulators place the

*s*

_{0}and

*s*

_{1}information in base band. The third modulator places the

*s*

_{2}information at base band, and the fourth modulator places the

*s*

_{3}information at base band.

**Z**

^{−1}separates

*s*

_{0}and

*s*

_{1}and equalizes the amplitude of all channels, yielding the result in Fig. 6.

*w*does not reject the contributions from these copies, self-error (error in reconstruction resulting from the desired signal) and cross-error (error caused by channel cross talk) will occur. The artifacts caused by these copies are different from aliasing since there is no overlap in the frequency domain of these channels; aliasing will be discussed in section 5.3

*x*) = 1 for |

*x*| < 1 and 0 otherwise.

*s*

_{0}information showing up as high frequency error in

*s*

_{0}signal) and cross error (e.g. base band

*s*

_{0}information showing up as high frequency error in the

*s*

_{1}signal), as well as base band signal attenuation.

**46**, 8428–8445 (2007). [CrossRef] [PubMed]

### 5.3. Example: High Frequency Aliasing

*W*) as in Eq. (11) the previous section,

_{B}*I*) is shown in Fig. 9(a), while the Fourier transform split into components are shown in Fig. 9(b). Figure 9 corresponds Fig. 4 (Media 1) from the previous example, and is another singal frame from the animation showing how the measured flux behaves as signal bandwidth increases.

## 6. Conclusions

*N*-element window in calculating the inversion of the measurement matrix

*W*. Since the excitation in the conventional PDRM is assumed to be constant, equal weighting only guarantees proper reconstruction for constant signals. The band limited approach presented in this manuscript improves upon the conventional polarimetric data reduction matrix (time limited reconstruction) method for properly band limited time dependent Stokes parameters. When Eq. (11) is satisfied, error-free polarimetric reconstruction is possible for noise-free excitations. Equation (20) shows how to implement a band limited reconstruction algorithm for general modulated polarimeters. This result relaxes the requirement for high modulation frequencies to reduce error in DoT polarimeters [18, 19

19. L. Gendre, A. Foulonneau, and L. Bigué, “High-speed imaging acquisition of stokes linearly polarized components using a single ferroelectric liquid crystal modulator,” in “*Proc. SPIE vol. 7461: Polarization Science and Remote Sensing IV*,” , J. A. Shaw and J. S. Tyo, eds. (SPIE, Bellingham, WA, 2009), p. 74610G.

## References and links

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

2. | L. J. Cheng, M. Hamilton, C. Mahoney, and G. Reyes, “Analysis of AOTF hyperspectral imaging,” in “Proceedings of SPIE Vol. 2231, Algorithms for Multispectral and Hyperspectral Imagery,” , A. Iverson, ed. (SPIE, Bellingham, WA, 1994), pp. 158–166. |

3. | J. S. Tyo, M. P. Rowe, E. N. Pugh, and N. Engheta, “Target detection in optically scattering media by polarization-difference imaging,” Appl. Opt. |

4. | M. P. Silverman and W. Strange, “Object delineation within turbid media by backscattering of phase modulated light,” Opt. Commun. |

5. | D. B. Chenault and J. L. Pezzaniti, “Polarization imaging through scattering media,”(SPIE, Bellingham, WA, 2000), pp. 124 – 133. |

6. | Y. Y. Schechner, S. G. Narasimhan, and S. K. Nayar, “Polarization-based vision through haze,” in “ACM SIGGRAPH ASIA 2008 courses,”(ACM, New York, NY, USA, 2008), SIGGRAPH Asia ’08, pp. 71:1–71:15 |

7. | V. Thilak, D. G. Voelz, and C. D. Creusere, “Image segmentation from multi-look passive polarimetric imagery,” in “ |

8. | D. J. Diner, A. Davis, B. Hancock, G. Gutt, R. A. Chipman, and B. Cairns, “Dual-photoeleastic-modulator-based polarimetric imaging concept for aerosol remote sensing,” Appl. Opt. |

9. | K. Sassen, “Polarization in LIDAR,” in “ |

10. | D. W. Tyler, A. M. Phenis, A. B. Tietjen, M. Virgen, J. D. Mudge, J. S. Stryjewski, and J. A. Dank, “First high-resolution passive polarimetric images of boosting rocket exhaust plumes,” in “ |

11. | R. A. Chipman, “Polarimetry,” in “ |

12. | R. M. A. Azzam, I. M. Elminyawi, and A. M. El-Saba, “General analysis and optimization of the four-detector photopolarimeter,” J. Opt. Soc. Am. A |

13. | J. L. Pezzaniti and D. B. Chenault, “A division of aperture MWIR imaging polarimeter,” in “ |

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

15. | J. S. Tyo, C. F. LaCasse, and B. M. Ratliff, “Total elimination of sampling errors in polarization imagery obtained with integrated microgrid polarimeters,” Opt. Lett. |

16. | K. Oka and T. Kaneko, “Compact complete imaging polarimeter using birefringent wedge prisms,” Opt. Express |

17. | K. Oka and T. Kato, “Spectroscopic polarimetry with a channeled spectrum,” Opt. Lett. |

18. | R. M. E. Illing, “High-speed fieldable imaging stokes vector polarimeter,” in “ |

19. | L. Gendre, A. Foulonneau, and L. Bigué, “High-speed imaging acquisition of stokes linearly polarized components using a single ferroelectric liquid crystal modulator,” in “ |

20. | M. H. Smith, J. B. Woodruff, and J. D. Howe, “Beam wander considerations in imaging polarimetry,” in “ |

21. | B. M. Ratliff, C. F. Lacasse, and J. S. Tyo, “Quantifying ifov error and compensating its effects in dofp polarimeters,” Opt. Express |

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

23. | J. S. Tyo and T. S. Turner, “Variable retardance, Fourier transform imaging spectropolarimeters for visible spectrum remote sensing,” Appl. Opt. |

24. | A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part I,” Opt. Eng. |

25. | J. S. Tyo, “Design of optimal polarimeters: maximization of SNR and minimization of systematic errors,” Appl. Opt. |

26. | F. Goudail and A. Beniere, “Estimation precision of the linear degree of polarization and of the angle of polarization in the presence of different types of noises,” Appl. Opt. |

27. | R. A. Chipman, “Polarization analysis of optical systems,” Opt. Eng. |

28. | R. A. Chipman, “Polarimetric impulse response,” in “ |

29. | J. P. McGuire and R. A. Chipman, “Diffraction image formation in optical systems with polarization aberrations. I: formulation and example,” J. Opt. Soc. Am. A |

**OCIS Codes**

(120.2130) Instrumentation, measurement, and metrology : Ellipsometry and polarimetry

(120.5410) Instrumentation, measurement, and metrology : Polarimetry

(110.5405) Imaging systems : Polarimetric imaging

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: May 13, 2011

Revised Manuscript: June 29, 2011

Manuscript Accepted: July 1, 2011

Published: July 20, 2011

**Citation**

Charles F. LaCasse, Russell A. Chipman, and J. Scott Tyo, "Band limited data reconstruction in modulated polarimeters," Opt. Express **19**, 14976-14989 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-16-14976

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### References

- J. S. Tyo, D. H. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45, 5453–5469 (2006). [CrossRef] [PubMed]
- L. J. Cheng, M. Hamilton, C. Mahoney, and G. Reyes, “Analysis of AOTF hyperspectral imaging,” in “Proceedings of SPIE Vol. 2231, Algorithms for Multispectral and Hyperspectral Imagery,” , A. Iverson, ed. (SPIE, Bellingham, WA, 1994), pp. 158–166.
- J. S. Tyo, M. P. Rowe, E. N. Pugh, and N. Engheta, “Target detection in optically scattering media by polarization-difference imaging,” Appl. Opt. 35, 1855–1870 (1996). [CrossRef] [PubMed]
- M. P. Silverman and W. Strange, “Object delineation within turbid media by backscattering of phase modulated light,” Opt. Commun. 144, 7–11 (1997). [CrossRef]
- D. B. Chenault and J. L. Pezzaniti, “Polarization imaging through scattering media,”(SPIE, Bellingham, WA, 2000), pp. 124 – 133.
- Y. Y. Schechner, S. G. Narasimhan, and S. K. Nayar, “Polarization-based vision through haze,” in “ACM SIGGRAPH ASIA 2008 courses,”(ACM, New York, NY, USA, 2008), SIGGRAPH Asia ’08, pp. 71:1–71:15
- V. Thilak, D. G. Voelz, and C. D. Creusere, “Image segmentation from multi-look passive polarimetric imagery,” in “Proc. SPIE 6682 ,” , J. A. Shaw and J. S. Tyo, eds. (SPIE, Bellingham, WA, 2007), p. 668206. [CrossRef]
- D. J. Diner, A. Davis, B. Hancock, G. Gutt, R. A. Chipman, and B. Cairns, “Dual-photoeleastic-modulator-based polarimetric imaging concept for aerosol remote sensing,” Appl. Opt. 46, 8428–8445 (2007). [CrossRef] [PubMed]
- K. Sassen, “Polarization in LIDAR,” in “LIDAR: Range-resolved optical remote sensing of the atmosphere ,” ,C. Weitkamp, ed. (Springer, 2005), pp. 19–42.
- D. W. Tyler, A. M. Phenis, A. B. Tietjen, M. Virgen, J. D. Mudge, J. S. Stryjewski, and J. A. Dank, “First high-resolution passive polarimetric images of boosting rocket exhaust plumes,” in “Proc. SPIE vol. 7461: Polarization Science and Remote Sensing IV ,” , J. A. Shaw and J. S. Tyo, eds. (SPIE, Bellingham, WA, 2009), p. 74610J.
- R. A. Chipman, “Polarimetry,” in “Handbook of Optics ,” , M. Bass, ed. (McGraw-Hill, 2009), 3rd ed.
- R. M. A. Azzam, I. M. Elminyawi, and A. M. El-Saba, “General analysis and optimization of the four-detector photopolarimeter,” J. Opt. Soc. Am. A 5, 681–689 (1988). [CrossRef]
- J. L. Pezzaniti and D. B. Chenault, “A division of aperture MWIR imaging polarimeter,” in “Proceedigns of SPIE vol. 5888: Polarization Science and Remote Sensing II ,” , J. A. Shaw and J. S. Tyo, eds. (SPIE, Bellingham, WA, 2005), p. 5888OV.
- A. G. Andreou and Z. K. Kalayjian, “Polarization imaging: principles and integrated polarimeters,” IEEE Sens. J. 2, 566–576 (2002). [CrossRef]
- J. S. Tyo, C. F. LaCasse, and B. M. Ratliff, “Total elimination of sampling errors in polarization imagery obtained with integrated microgrid polarimeters,” Opt. Lett. 34, 3187–3189 (2009). [CrossRef] [PubMed]
- K. Oka and T. Kaneko, “Compact complete imaging polarimeter using birefringent wedge prisms,” Opt. Express 11, 1510–1519 (2003). [CrossRef] [PubMed]
- K. Oka and T. Kato, “Spectroscopic polarimetry with a channeled spectrum,” Opt. Lett. 24, 1475–1477 (1999). [CrossRef]
- R. M. E. Illing, “High-speed fieldable imaging stokes vector polarimeter,” in “Proceedigns of SPIE vol. 5888: Polarization Science and Remote Sensing II ,” , J. A. Shaw and J. S. Tyo, eds. (SPIE, Bellingham, WA, 2005), p. 58880X.
- L. Gendre, A. Foulonneau, and L. Bigué, “High-speed imaging acquisition of stokes linearly polarized components using a single ferroelectric liquid crystal modulator,” in “Proc. SPIE vol. 7461: Polarization Science and Remote Sensing IV ,” , J. A. Shaw and J. S. Tyo, eds. (SPIE, Bellingham, WA, 2009), p. 74610G.
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- B. M. Ratliff, C. F. Lacasse, and J. S. Tyo, “Quantifying ifov error and compensating its effects in dofp polarimeters,” Opt. Express 17, 9112 – 9125 (2009). [CrossRef] [PubMed]
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