## Imaging the polarization of a light field |

Optics Express, Vol. 21, Issue 4, pp. 4106-4115 (2013)

http://dx.doi.org/10.1364/OE.21.004106

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### Abstract

We describe and analyze a method by which an optical polarization state is mapped to an image sensor. When placed in a Bayesian framework, the analysis allows *a priori* information about the polarization state to be introduced into the measurement. We show that when such a measurement is applied to a single photon, it eliminates exactly one fully polarized state, offering an important insight about the information gained from a single photon polarization measurement.

© 2013 OSA

## 1. Introduction

1. C.A. Skinner, “The polarimeter and its practical applications,” J. Franklin Inst. **196**, 721–750 (1923). [CrossRef]

3. R. Azzam, “Arrangement of four photodetectors for measuring the state of polarization of light,” Opt. Lett. **10**, 309–311 (1985). [CrossRef] [PubMed]

4. H. Luo, K. Oka, E. DeHoog, M. Kudenov, J. Schwegerling, and E. L. Dereniak, “Compact and miniature snapshot imaging polarimeter,” Appl. Opt. **47**, 4413–4417 (2008). [CrossRef] [PubMed]

5. J. Guo and D. Brady, “Fabrication of thin-film micropolarizer arrays for visible imaging polarimetry,” Appl. Opt. **39**, 1486–1492 (2000). [CrossRef]

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

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

*et al.*[8

8. W. Sparks, T. Germer, J. MacKenty, and F. Snik, “Compact and robust method for full Stokes spectropolarimetry,” Appl. Opt. **51**, 5495–5511 (2012). [CrossRef] [PubMed]

*a priori*constraints.

## 2. Stokes parameters and the Poincaré sphere

## 3. Propagation of light through stress-engineered optical elements

9. A. K. Spilman and T. G. Brown, “Stress-induced focal splitting,” Opt. Express **15**, 8411–8421 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-13-8411. [CrossRef] [PubMed]

10. A. K. Spilman and T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt.26, 61–66 (2007), http://ao.osa.org/abstract.cfm?id=119864. [CrossRef]

*μ*m smaller than the outer diameter of the optical flat. Material is removed from the metal housing to create three contact regions at 120°. The high thermal expansion coefficient of the metal ring allows the insertion of the glass window at about 300°C. After cooling, the SEO window acquires a stress distribution of trigonal symmetry, which near the window’s center follows a power law model in which the retardance increases linearly with radius and the orientation of the fast axis precesses with the azimuth. The Jones matrix for a general retarder can be modified as follows to describe the center region of a SEO window: in which (

*r*,

*φ*) is the window polar coordinate, 𝕀 is the 2 × 2 identity matrix, is the pseudo-rotation matrix, and

*c*is the stress-coefficient of the window. This approximate result is only valid in the central part of the window, as can be seen from the insets in Fig. 1. For this reason, a circular aperture of radius

*R*is used to block the outer regions of the window. When the incident uniform beam propagates past this SEO followed by a left-circular analyzer, its resulting transverse irradiance distribution provides a signature that can be unambiguously linked to the polarization state of the incident beam. We will refer to this irradiance distribution by the optical convention of point spread function (PSF). In principle, one can work with PSFs corresponding to any propagation distance away from the SEO. However, it is particularly convenient to consider either the irradiance immediately after the SEO and analyzer, or at a Fourier-conjugate plane, i.e. at the back-focal plane of a lens. While the theoretical treatment that follows is valid for any of these cases, our experiments will use the focused approach for reasons discussed in the concluding remarks.

*μ*m) controlled by ICCapture™, which allows the control of the gamma parameter of the camera and ensures that no pixels are saturated. The SEO window is kept in a fixed position during the course of the experiment, since any change in its orientation would change the orientation of the images.

## 4. Relation between the PSF and the polarization of the incident beam

*I*

_{R}(

**x**) and

*I*

_{L}(

**x**) are the irradiance profiles corresponding to incident right- and left-circular polarization, respectively, and

**x**= (

*ρ*,

*ϕ*) are the polar coordinates at the CCD. These profiles are ideally axially symmetric. If the CCD were to be placed directly after the SOE and left-circular analyzer,

*I*

_{R,L}would be given by where

*A*is an apodization function (assumed to be constant in what follows). On the other hand, if the beam emerging from the SOE and analyzer is focused by a lens (as it is in our experimental setup),

*I*

_{R,L}are given by where

*f*is the focal distance of the lens, and

*J*is the

_{n}*n*th order Bessel function of the first kind. Figures 2(g)– 2(l) shows the agreement of the PSFs calculated through the substitution of Eq. (5) into Eq. (3) with those measured experimentally.

**u**(

**x**) is defined as Notice that

**u**(

**x**) is a unit vector, which therefore provides a mapping between points

**x**over the CCD to points over the surface of the Poincaré sphere. Notice also from Eq. (6) that, for fully polarized fields, the PSF vanishes at points where

**u**= −

**s**and is maximal at points where

**u**=

**s**.

## 5. Probabilistic estimation of the polarization

**s**is known, the probability density of a photon hitting the point

**x**is a normalized version of the PSF in Eq. (3): where with Φ

_{R,L}representing the total powers for each polarization, and an overline denotes averaging of a function over the

**x**plane with

*w*(

**x**) as a weight factor: Therefore,

**ū**corresponds to the average of the unit vector

**u**(

**x**) mapping

**x**to the Poincaré sphere, weighted by

*w*(

**x**). In the ideal case of a perfectly aligned system and a detector with infinite resolution, the first two components of the constant vector

**ū**vanish due to the dependence of

**u**on

*ϕ*. It is easy to show that the third component can also be made to vanish by choosing the window radius so that Φ

_{R}= Φ

_{L}. In this case, the conditional probability density in Eq. (8) depends linearly on

**s**. In practice, however, it is best not to calculate

*I*

_{R}and

*I*

_{L}but to use measured values [like those in Figs. 2(g)–2(h)] as part of the setup’s calibration, and then to calculate Φ

_{R}, Φ

_{L}, and the averages in Eq. (10) as sums over all pixels rather than as integrals. Due to imperfections in the SOE, the setup’s alignment, and the CCD’s finite extent, pixel orientation and discretization, none of the components of

**ū**will generally vanish exactly. For the experimental data used later in this work,

**ū**= (0.089, −0.026, −0.004). The system’s calibration also uses other known polarizations [e.g., those in Figs. 2(i)–2(l)] to deduce any systematic errors, including the defocus that produces the skewness apparent in the measured point spread functions for the linear polarization states. The details of this procedure are given in [12

12. R. D. Ramkhalawon, A. M. Beckley, and T. G. Brown, “Star test polarimetry using stress-engineered optical elements,” Proc. Spie **8227**, 82270Q–82270Q-8 (2012). [CrossRef]

**s**from the measured locations at the CCD of

*N*photons. By using Eqs. (11) and (13) as well as standard relations for conditional probabilities, the probability density for the polarization given

*N*photon positions is given by Note that the product of

*P*(

**x**

*) in the denominator is independent of*

_{n}**s**and therefore just serves as normalization, and that the distribution

*P*(

**s**) enters as a global factor. The information provided by each detected photon comes from the factors

*P*(

**x**

*|*

_{n}**s**). It is clear from Eq. (8) and the fact that |

**u**| = 1 that, for a given

**x**

*,*

_{n}*P*(

**x**

*|*

_{n}**s**) vanishes exactly for one state of (full) polarization. If the setup is such that

**ū**vanishes, then

*P*(

**x**

*|*

_{n}**s**) is a linear function of

**s**that vanishes at one point over the surface of the Poincaré sphere and is maximal at the antipodal point [see Fig. 3(a)]. If

**ū**does not vanish, the contours of constant

*P*(

**x**

*|*

_{n}**s**) are still planar sections of the Poincaré sphere, but they are generally no longer parallel, so the zero and the maximum are not antipodes in general, as shown in Fig. 3(b). In any case, each detected photon rules out a state of polarization for which

*P*(

**x**

*|*

_{n}**s**) = 0.

## 6. Discretization of the detector

**x**

*. Therefore, each position*

_{i}**x**

*must be assigned to the pixel at the closest*

_{n}**x**

*, so Eq. (14) becomes where*

_{i}**Ĩ**= (

*Ĩ*

_{1},

*Ĩ*

_{2},...) with

*Ĩ*being the measured number of photons falling in the pixel at

_{i}**x**

*, and Equation (15) gives the probability density in the Poincaré space that the field is in a state of polarization*

_{i}**s**given a detected signal

**Ĩ**at the CCD. As mentioned earlier, the prefactor

*P*(

**s**) in this expression helps preselect the expected states of polarization, and provides the correct Jacobian for the conditional probability density corresponding to the chosen parametrization of

**s**. In the experimental results that follow, we assume that

*P*(

**s**) is constant. The factor exp(−

*q*

_{0}) is also constant, and serves simply as normalization. The main part of Eq. (15) is the factor exp[

*q*(

**s**)]. When the number of detected photons

*N*is large, the distribution given by this factor becomes sharply peaked and, in the absence of noise and experimental errors, this peak is essentially at the true polarization

**s**

_{0}of the incident field. This can be shown by taking the gradient in

**s**and using the fact that, in the limit of small pixel area

*a*and large

*N*,

*Ĩ*≈

_{i}*NaP*(

**x**

*|*

_{i}**s**

_{0}): where in the last step we used ∑

*(*

_{i}P**x**

*|*

_{i}**s**) ≈ 1/

*a*.

*N*, exp[

*q*(

**s**)] is approximately a (generally anisotropic) Gaussian distribution whose maximum is at

**s**

_{0}: where is the Hessian matrix of derivatives of

*q*evaluated at

**s**

_{0}. By using

*Ĩ*≈

_{i}*NaP*(

**x**

*|*

_{i}**s**

_{0}), the components of this matrix can be found to be given approximately by where, as defined in Eq. (10), an overline denotes averaging over the CCD plane with weight factor

*w*(

**x**). The orientation axes and standard-deviation widths of the Gaussian are given, respectively, by the eigenvectors and the inverse of the square root of minus the eigenvalues of .

## 7. Optimal polarimeter

**s**

_{0}(i.e. on the degree of polarization) and not on its direction. For the derivation that follows, we consider the limit of small pixel size, so

**x**is regarded as a continuous variable. The fact that

*R*is chosen such that

**ū**vanishes is not sufficient to ensure optimal performance; the factors 1 +

**u**(

**x**) ·

**s**that make up the probability must also have equal weight. From Eq. (8) we see that (assuming

**ū**vanishes) the weight of this linear function due to an infinitesimal area over the CCD is

*wρ*d

*ρ*d

*ϕ*. Uniformity over the sphere means that this weight should be proportional to sin

*θ*d

*θ*d

*ϕ*, where

*θ*= arccos(

*u*). The condition for the polarimeter to be optimal then states which, after substituting the expressions for

_{z}*w*and

*u*from Eqs. (9) and (7), results in the following condition for

_{z}*I*

_{R,L}:

*I*

_{R}+

*I*

_{L}is constant. The solution to Eq. (21) is then simply where

*I*

_{0}is a constant. That is, the birefringence map should be limited to 0 ≤

*ρ*≤

*R*. For the optimal polarimeter, the Hessian given by Eq. (19) can be calculated analytically.

## 8. Theoretical comparison with a standard polarimeter

*P*

_{0}is a normalization constant,

**Ĩ**

^{St}is a vector whose six components represent the number of photons detected by each of the detectors, and

**s**which vanish exactly at one point over the surface of the Poincaré sphere. However, in this case, these linear functions are always aligned with the three Cartesian coordinates, so only the six reference full states of polarization can be “ruled out” by the detection of a photon. In the large

*N*limit, the probability density and the components of the Hessian of

*q*

^{St}(

**s**) = ln[

*P*

^{St}(

**s**|

**Ĩ**

^{St})] evaluated at

**s**=

**s**

_{0}can be shown to be given approximately by where

*s*

_{0n}are the components of the polarization

**s**

_{0}of the incident field, and

*δ*is the Kronecker delta. Given the diagonal nature of

_{m,n}^{St}, the orientation axes of the Gaussian distribution for exp[

*q*

^{St}(

**s**)] is always that of the axes

*s*

_{1},

*s*

_{2}, and

*s*

_{3}. Both for the standard and the new polarimeter, the Hessian is proportional to the number of detected photons

*N*, so the standard-deviation widths scale as

*N*

^{−1/2}.

*s*1–

*s*3 plane for the SEO-based polarimeter according to Eq. (19), for the optimal polarimeter described earlier, and for the theoretical model of a standard polarimeter, according Eqs. (25), all for

*N*= 2500 (recall that these spreads scale as

*N*

^{−1/2}). The black ellipses represent the standard-deviation cross-section of the Gaussian distributions over this plane for several values of

**s**

_{0}, while the radius of the underlying green circles represents the standard deviation cross-section in the direction of

*s*

_{2}(normal to the plane). Notice that, when the degree of polarization is small (i.e. for

**s**

_{0}near the center of the Poincaré sphere), both SEO and standard polarimeters present very similar standard deviations, which are nearly isotropic and roughly of size

*I*

_{R}and

*I*

_{L}, which resulted in

**ū**not vanishing exactly.] However, for a highly polarized field, the uncertainty in the results of the standard polarimeter (particularly in the radial direction) depend significantly on whether the polarization is nearly one of the six ones used as references, while for the proposed system the behavior is significantly more uniform, resembling that of the optimal case shown in Figure 4(b).

## 9. Experimental results

*q*(

**s**), are compared in Table 1 to results given by a commercial polarimeter. Because the most general measurement of polarization is measurement within a volume, we provide a root sum square (RSS) difference with the commercial polarimeter and the equivalent volumetric error (the fraction of volume of the full Poincaré sphere represented by that uncertainty). Since the degree of polarization (DoP) is important in its own right, we separately report the DoP and the difference between our measurement and that of the reference.

## 10. Concluding remarks

*et al*[8

8. W. Sparks, T. Germer, J. MacKenty, and F. Snik, “Compact and robust method for full Stokes spectropolarimetry,” Appl. Opt. **51**, 5495–5511 (2012). [CrossRef] [PubMed]

11. A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express **18**, 10777–10785 (2010). [CrossRef] [PubMed]

13. D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A. **64**, 052312–27 (2001). [CrossRef]

*a priori*information, we can explore the implications of polarization measurements at very low light levels. For a very small number of independent photons, the formalism suggests that each photon represents an individual stochastic event (absorption at a particular pixel) described by a spatial probability density function. Figure 3(c) illustrates the character of this distribution for the case of a sequence of two photons which are, in this case, generated by a numerically weighted PSF. While it is clear that each photon location yields some result (in a maximum likelihood sense) of the state of the field, it is more striking that the location of each event eliminates precisely one fully polarized state from the possible states that comprise the light field. For quantum measurements, the method projects the polarization state of the photon onto a continuous variable space; since the method is inherently probabilistic, it may provide an important tool for exploring novel quantum states using correlative imaging. Replacing our analyzer with a polarization beam splitter and measuring coincidence between two sensors would open up new avenues for the direct imaging of quantum states.

## Acknowledgments

## References and links

1. | C.A. Skinner, “The polarimeter and its practical applications,” J. Franklin Inst. |

2. | E. Collett, |

3. | R. Azzam, “Arrangement of four photodetectors for measuring the state of polarization of light,” Opt. Lett. |

4. | H. Luo, K. Oka, E. DeHoog, M. Kudenov, J. Schwegerling, and E. L. Dereniak, “Compact and miniature snapshot imaging polarimeter,” Appl. Opt. |

5. | J. Guo and D. Brady, “Fabrication of thin-film micropolarizer arrays for visible imaging polarimetry,” Appl. Opt. |

6. | J. Tyo, D. Goldstein, D. Chenault, and J. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. |

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

8. | W. Sparks, T. Germer, J. MacKenty, and F. Snik, “Compact and robust method for full Stokes spectropolarimetry,” Appl. Opt. |

9. | A. K. Spilman and T. G. Brown, “Stress-induced focal splitting,” Opt. Express |

10. | A. K. Spilman and T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt.26, 61–66 (2007), http://ao.osa.org/abstract.cfm?id=119864. [CrossRef] |

11. | A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express |

12. | R. D. Ramkhalawon, A. M. Beckley, and T. G. Brown, “Star test polarimetry using stress-engineered optical elements,” Proc. Spie |

13. | D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A. |

**OCIS Codes**

(260.1440) Physical optics : Birefringence

(260.5430) Physical optics : Polarization

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: December 6, 2012

Revised Manuscript: January 18, 2013

Manuscript Accepted: January 19, 2013

Published: February 11, 2013

**Citation**

Roshita D. Ramkhalawon, Thomas G. Brown, and Miguel A. Alonso, "Imaging the polarization of a light field," Opt. Express **21**, 4106-4115 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-4-4106

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### References

- C.A. Skinner, “The polarimeter and its practical applications,” J. Franklin Inst.196, 721–750 (1923). [CrossRef]
- E. Collett, Polarized Light: Fundamentals and Applications (CRC Press, 1992).
- R. Azzam, “Arrangement of four photodetectors for measuring the state of polarization of light,” Opt. Lett.10, 309–311 (1985). [CrossRef] [PubMed]
- H. Luo, K. Oka, E. DeHoog, M. Kudenov, J. Schwegerling, and E. L. Dereniak, “Compact and miniature snapshot imaging polarimeter,” Appl. Opt.47, 4413–4417 (2008). [CrossRef] [PubMed]
- J. Guo and D. Brady, “Fabrication of thin-film micropolarizer arrays for visible imaging polarimetry,” Appl. Opt.39, 1486–1492 (2000). [CrossRef]
- J. Tyo, D. Goldstein, D. Chenault, and J. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt.45, 5453–5469 (2006). [CrossRef] [PubMed]
- F. Gori, “Measuring Stokes parameters by means of a polarization grating,” Opt. Lett.24,584–586 (1999). [CrossRef]
- W. Sparks, T. Germer, J. MacKenty, and F. Snik, “Compact and robust method for full Stokes spectropolarimetry,” Appl. Opt.51, 5495–5511 (2012). [CrossRef] [PubMed]
- A. K. Spilman and T. G. Brown, “Stress-induced focal splitting,” Opt. Express15, 8411–8421 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-13-8411 . [CrossRef] [PubMed]
- A. K. Spilman and T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt.26, 61–66 (2007), http://ao.osa.org/abstract.cfm?id=119864 . [CrossRef]
- A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express18, 10777–10785 (2010). [CrossRef] [PubMed]
- R. D. Ramkhalawon, A. M. Beckley, and T. G. Brown, “Star test polarimetry using stress-engineered optical elements,” Proc. Spie8227, 82270Q–82270Q-8 (2012). [CrossRef]
- D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A.64, 052312–27 (2001). [CrossRef]

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