## Sources of possible artefacts in the contrast evaluation for the backscattering polarimetric images of different targets in turbid medium

Optics Express, Vol. 17, Issue 26, pp. 23851-23860 (2009)

http://dx.doi.org/10.1364/OE.17.023851

Acrobat PDF (649 KB)

### Abstract

It is known that polarization-sensitive backscattering images of different objects in turbid media may show better contrasts than usual intensity images. Polarimetric image contrast depends on both target and background polarization properties and typically involves averaging over groups of pixels, corresponding to given areas of the image. By means of numerical modelling we show that the experimental arrangement, namely, the shape of turbid medium container, the optical properties of the container walls, the relative positioning of the absorbing, scattering and reflecting targets with respect to each other and to the container walls, as well as the choice of the image areas for the contrast calculations, can strongly affect the final results for both linearly and circularly polarized light.

© 2009 OSA

## 1. Introduction

1. G. D. Lewis, D. L. Jordan, and P. J. Roberts, “Backscattering target detection in a turbid medium by polarization discrimination,” Appl. Opt. **38**(18), 3937–3944 (1999). [CrossRef] [PubMed]

9. F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B **40**(13), 9342–9345 (1989). [CrossRef]

3. R. Nothdurft and G. Yao, “Expression of target optical properties in subsurface polarization-gated imaging,” Opt. Express **13**(11), 4185–4195 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-11-4185. [CrossRef] [PubMed]

4. R. E. Nothdurft and G. Yao, “Effects of turbid media optical properties on object visibility in subsurface polarization imaging,” Appl. Opt. **45**(22), 5532–5541 (2006). [CrossRef] [PubMed]

1. G. D. Lewis, D. L. Jordan, and P. J. Roberts, “Backscattering target detection in a turbid medium by polarization discrimination,” Appl. Opt. **38**(18), 3937–3944 (1999). [CrossRef] [PubMed]

10. G. Yao, “Differential optical polarization imaging in turbid media with different embedded objects,” Opt. Commun. **241**(4-6), 255–261 (2004). [CrossRef]

11. P. W. Zhai, G. W. Kattawar, and P. Yang, “Mueller matrix imaging of targets under an air-sea interface,” Appl. Opt. **48**(2), 250–260 (2009). [CrossRef] [PubMed]

## 2. Methods

### 2.1 Mueller-Stokes Formalism

**S**defined, for any set of orthogonal axes (x, y), as:where

*I*

_{x},

*I*

_{y},

*I*

_{+45°},

*I*

_{-45°}are the intensities which would be measured after passing through linear polarizers oriented along the

*x*,

*y*, + 45° and –45° respectively in the plane perpendicular to the direction of wave propagation, while

*I*

_{L}and

*I*

_{R}would be the intensities transmitted by left and right circular polarizers, and

*E*

_{x},

*E*

_{y}are two orthogonal components of complex Jones vector of the electric field [12]. The Stokes vector is thus defined in terms of directly measurable intensities, which is not the case for the electric field amplitudes involved in the Jones formalism.

### 2.2 Numerical Technique

**r,**direction

**Ω**and time

*t*,

**r**the probability of the of light propagating in direction

**Ω**,

**Q**

_{ext}is the contribution of the external photon sources, μ

_{T}

**=**Nσ

_{T}and μ

_{sca}

**=**Nσ

_{sca}are the total and scattering attenuation coefficients, σ

_{T}and σ

_{sca}are the total and scattering cross-sections of single particle respectively, N is a number density of the particles. The integro-differential Eq. (4) can be transformed by the method of characteristics [14] to the integral inhomogeneous Fredholm equation of the second kind:where

*K*is the scattering integral and

**Q**' is the contribution of the external photon sources. In general no analytical solution is available for Eq. (4a), so we applied the numerical Monte Carlo technique to evaluate

*K*. This statistical approach allowed to solve the problem in realistic (sometimes quite complex) experimental configurations, with straightforward physical interpretation of the results.

**M**is the Mueller matrix of the sample under study and the value of

**D**is

15. B. Kaplan, G. Ledanois, and B. Drévillon, “Mueller matrix of dense polystyrene latex sphere suspensions: measurements and monte carlo simulation,” Appl. Opt. **40**(16), 2769–2777 (2001). [CrossRef] [PubMed]

### 2.2 Modelled set-up

*g*= 0.917 and a scattering coefficient μ

_{s}= 0.734 cm

^{−1}for 1 µm diameter polystyrene particle in water. The concentration of particles was fixed at 3.6 ·10

^{7}particles/cm

^{3}to ensure the transport mean free path (mfp')

*l*

_{s}' = 1/(μ

_{s}(1-

*g*)) = 16.4 cm be equal to the depth of container. Due to the large value of anisotropy factor the enhancement of forward light scattering with small fraction of light to be scattered back was expected within the container volume. Consequently, one would not expect that the lateral dimensions of the container, which were smaller than transport mean free path

*l*

_{s}' but larger than mean free path

*l*

_{s}= 1/μ

_{s}= 1.36 cm, would have any significant influence on the backscattering polarimetric image of the sample.

_{p}= 1.59 and n

_{water}= 1.33 respectively, with vanishing imaginary part (no absorption). The distance from the backscattered light detector to the surface of the sample was fixed at 70 cm.

^{2}) targets were immersed into the suspension of polystyrene particles in water (see Fig. 1(b), top view). Similar target configuration was used for the polarimetric imaging of the combined multiple targets in [4

4. R. E. Nothdurft and G. Yao, “Effects of turbid media optical properties on object visibility in subsurface polarization imaging,” Appl. Opt. **45**(22), 5532–5541 (2006). [CrossRef] [PubMed]

_{R}= 1.21 + i·6.93), the Mueller matrix of the dielectric-metal interface was calculated according to [16

16. R. Kalibjian, “Stokes polarization vector and Mueller matrix for a corner-cube reflector,” Opt. Commun. **240**(1-3), 39–68 (2004). [CrossRef]

^{2}in order to reduce the statistical noise of calculations.

3. R. Nothdurft and G. Yao, “Expression of target optical properties in subsurface polarization-gated imaging,” Opt. Express **13**(11), 4185–4195 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-11-4185. [CrossRef] [PubMed]

5. S. G. Demos and R. R. Alfano, “Optical polarization imaging,” Appl. Opt. **36**(1), 150–155 (1997). [CrossRef] [PubMed]

*I*

_{co}and

*I*

_{cr}represent co-polarized and cross-polarized light intensities with respect to the incident beam. For an incident linearly-horizontal polarized beam

*I*

_{co}is the intensity of backscattered light at horizontal detection,

*I*

_{cr}is the intensity at vertical detection. When an incident beam is right-circularly polarized,

*I*

_{co}is the intensity of backscattered light at clockwise detection and

*I*

_{cr}is the intensity at counter-clockwise detection. As follows from Eq. (1), (3) and (8) the OSC values for linearly and circularly polarized light can be expressed in terms of Mueller matrix coefficients: if we define a linearly-horizontal polarized beam to be parallel to the x axis of the laboratory coordinate system. The corresponding OSC

_{L}and OSC

_{C}values were calculated for each pixel (i, j) of the images taken at different target immersion depths.

_{L}

^{tar}and OSC

_{C}

^{tar}values was performed over the pixels (i, j) from T - central quarter of the area of each target. The values of OSC

_{L}

^{b}and OSC

_{C}

^{b}for the scattering background were calculated using the pixels (i, j) from B - 0.5 cm width stripe, surrounding target area (see Fig. 1b).

## 3. Results and discussion

### 3.1 Mutual impact of multiple targets

_{11}, M

_{22}*, M

_{33}* and M

_{44}* are presented in Fig. 2 . All calculated off-diagonal elements of Mueller matrix were about three orders of magnitude less than diagonal elements, i. e. they were statistically equal to zero. This means that the sample behaved as a pure depolarizer. The values of M

_{22}* were almost everywhere equal to the absolute values of M

_{33}*, as expected for an isotropic medium [17].

_{11}upper row, Fig. 2) show that the visibility of the absorbing target drops faster with immersion depth than that of the reflecting and scattering targets. With the chosen scattering coefficient μ

_{s}and combined target configuration the absorbing target becomes almost invisible on the unpolarized intensity images when the immersion depth

*d*exceeds critical value of 0.24·

*l*

_{s}'. The same tendency is observed in the polarimetric images (M

_{22}*, M

_{33}* and M

_{44}* rows in Fig. 2) but the absorbing target image fades even faster (critical value

*d*= 0.18·

*l*

_{s}').

_{22}* and M

_{33}* rows in Fig. 2) compared with circular polarized image (M

_{44}* row, Fig. 2).

_{L}and OSC

_{C}calculated with Eqs. (9)-(11) for the reflecting, absorbing, scattering targets and surrounding turbid background are presented in Fig. 3 versus target immersion depth expressed in a transport mean free path

*l*

_{s}' .

_{L}value is higher then OSC

_{C}value at any target immersion depth. Despite the multiple scattering of the light within the container volume (mean free path

*l*

_{s}= 1.36 cm), due to the high anisotropy factor

*g*the scattered light direction is slowly randomized. Consequently the frame of reference for the linear polarized eigenvectors remains almost unchanged, thus preserving the OSC

_{L}values after short path backscattering sequences. At the same time the flip of helicity of the backscattered circularly polarized light affects the OSC

_{C}values significantly. Both linear and circular OSC values of the reflecting and scattering targets show opposite trends with increasing depth.

_{L}and OSC

_{C}of absorbing target show non-monotonous variation with the immersion depth increase (see Fig. 3(c), solid symbols). To understand this phenomenon we repeated the calculations with the same parameters but without reflecting and scattering targets.

_{L}and OSC

_{C}decrease monotonously with the immersion depth (see Fig. 3(c), open symbols). For example, at

*d*= 0.18·

*l*

_{s}' the value of OSC

_{L}of the single absorbing target increases and the value of OSC

_{C}decreases (see Fig. 3(c), open symbols) because we removed the shadowing of the absorbing target by the backscattering light cone of scattering target (see Fig. 2, M

_{22}* and M

_{44}* rows).

*d*> 0.3·

*l*

_{s}' both OSC

_{L}and OSC

_{C}values of single absorbing target were equal to the corresponding values for scattering background (see Fig. 3(c), (d), open symbols), hence, according to Eq. (12) single absorbing target in turbid medium was invisible at depths larger than 0.3·

*l*

_{s}'.

_{C}of scattering background (see Fig. 2, (M

_{44}* row) and Fig. 3(d) (solid symbols)). The calculated value of scattering background OSC

_{L}remained almost constant, because the broadening of light cone backscattering from the reflecting target was accompanied by the broadening of light cone backscattering from the scattering target and their respective OSC

_{L}values have an opposite trend (see Fig. 2, (M

_{22}*row) and Fig. 3(a),(b),(d)).

_{22}*, M

_{33}* (and OSC

_{L}value consequently) of the scattering background are higher for the pixels in the corners of the container compared to the central part. The values of OSC

_{L,C}

^{b}are used for image contrast evaluation, so the choice of pixels for the spatial averaging can affect the calculated image contrast.

### 3.2 Impact of container walls on the contrast evaluation

_{L}and OSC

_{C}of the scattering background medium we performed simulations of the sample described above but without any target immersed, varying the material of the wall. First we assumed that the walls were absorbing and afterwards we modelled a container with plastic walls of 0.5 mm thickness and n

_{wall}= 1.58. We performed spatial averaging of simulated Mueller matrix coefficients over the pixels of whole top surface of the sample. The results are presented in Tab. 1. The simulated optical properties of the container wall and the cross-section of container in (x, y) plane are given in the first and second columns respectively.

_{22}* exceeded the corresponding absolute value of M

_{33}* by 50% (see Tab. (1)).

_{L}value calculated with Eq. (8) can depend on the orientation of the linear polarizer with respect to the (x, y) axes of the laboratory coordinate system. For example, if we define a linearly-horizontal polarized beam to be at 45° to the x axis of the laboratory coordinate system, the Eq. (9), expressing the OSC

_{L}value in terms of Mueller matrix coefficients, transforms intoThe values of OSC

_{L}

^{b}calculated using either Eq. (9) or Eq. (13) will be different, because M

_{22}*≠M

_{33}*. This means that for the studied sample the orthogonal state contrast of the scattering background medium for linear polarization (OSC

_{L}

^{b}) is not invariant under the linear polarizer plane rotation by 45° around z axis. In the case of completely absorbing walls, there are no photons coming back from the walls to the scattering medium. When the walls of container reflect photons back to the scattering medium, the polarization of these photons contribute to the polarimetric image of the sample carrying back the information about the optical properties of the boundaries and their position with respect to the linear polarizer plane orientation. The square cross-section sample is not invariant under rotation by 45° around z axis. Thus, for given scattering coefficient μ

_{s}the value of OSC

_{L}

^{b}and, consequently, the contrast value (see Eq. (12)) for linearly polarized light will strongly depend on the orientation of linear polarizer plane with respect to the side and diagonal of the square cross-section of the sample.

_{wall}= 1.58. The cross-section of the cylindrical container in (x, y) plane is invariant under any rotation around z axis. Consequently, due to an azimuthal symmetry of the sample the calculated averaged coefficients M

_{22}* and M

_{33}* become equal (see Tab. (1)).

## 4. Conclusions

_{s}and transport mean free path

*l*

_{s}' the walls of the container can actively interfere with the scattered polarized light. The lack of sample azimuthal symmetry in (x, y) plane perpendicular to the direction of wave propagation can affect the OSC image of the sample for linearly polarized light. It can induce significant errors in the differential polarization image contrast evaluation. To avoid these errors we need to measure complete Mueller matrix of the sample or degree of polarization

*ρ*

_{s}of backscattering light (see Eq. (2),(3)).

_{L}value of the scattering background medium for the container with square cross-section depends on the exact orientation of the linear polarizer plane, being maximal for the horizontal linear-polarized light direction parallel to the walls of square cross-section and minimal when this direction is parallel to the diagonals of the square.

_{s}, used in our simulations was quite low. However, we believe that not the exact value of scattering coefficient (or mean free path) is important in the experiments, but rather the ratio of transport mean free path l

_{s}' to the characteristic dimension L of the container with scattering sample. Obviously, the decrease of ratio l

_{s}'/L will reduce the influence of the above mentioned artefacts on polarimetric image contrast. The use of point source illumination directed towards the centre of sample surface will also decrease the impact of container walls on polarimetric image contrast.

## Acknowledgements

## References and links

1. | G. D. Lewis, D. L. Jordan, and P. J. Roberts, “Backscattering target detection in a turbid medium by polarization discrimination,” Appl. Opt. |

2. | G. W. Kattawar and M. J. Raković, “Virtues of mueller matrix imaging for underwater target detection,” Appl. Opt. |

3. | R. Nothdurft and G. Yao, “Expression of target optical properties in subsurface polarization-gated imaging,” Opt. Express |

4. | R. E. Nothdurft and G. Yao, “Effects of turbid media optical properties on object visibility in subsurface polarization imaging,” Appl. Opt. |

5. | S. G. Demos and R. R. Alfano, “Optical polarization imaging,” Appl. Opt. |

6. | S. G. Demos, H. Radousky, and R. Alfano, “Deep subsurface imaging in tissues using spectral and polarization filtering,” Opt. Express |

7. | H. Shao, Y. He, Y. Shao, and H. Ma, “Contrast enhancement subsurface optical imaging with different incident polarization states”, Proc. of SPIE, |

8. | G. C. Giakos, A. Molhokar, A. Orozco, V. Kumar, S. Sumrain, D. Mehta, A. Maniyedath, N. Ojha, and A. Medithe, “Laser imaging through scattering media”, IMTC 04. Proc. of the 21st IEEE, |

9. | F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B |

10. | G. Yao, “Differential optical polarization imaging in turbid media with different embedded objects,” Opt. Commun. |

11. | P. W. Zhai, G. W. Kattawar, and P. Yang, “Mueller matrix imaging of targets under an air-sea interface,” Appl. Opt. |

12. | S. Huard, |

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

14. | G. I. Bell, and S. Glasstone, |

15. | B. Kaplan, G. Ledanois, and B. Drévillon, “Mueller matrix of dense polystyrene latex sphere suspensions: measurements and monte carlo simulation,” Appl. Opt. |

16. | R. Kalibjian, “Stokes polarization vector and Mueller matrix for a corner-cube reflector,” Opt. Commun. |

17. | H. C. Van de Hulst, |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(260.5430) Physical optics : Polarization

(290.1350) Scattering : Backscattering

(110.0113) Imaging systems : Imaging through turbid media

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: September 28, 2009

Revised Manuscript: November 4, 2009

Manuscript Accepted: November 6, 2009

Published: December 14, 2009

**Citation**

Tatiana Novikova, Arnaud Bénière, François Goudail, and Antonello De Martino, "Sources of possible artefacts in the contrast evaluation for the backscattering polarimetric images of different targets in turbid medium," Opt. Express **17**, 23851-23860 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-26-23851

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### References

- G. D. Lewis, D. L. Jordan, and P. J. Roberts, “Backscattering target detection in a turbid medium by polarization discrimination,” Appl. Opt. 38(18), 3937–3944 (1999). [CrossRef] [PubMed]
- G. W. Kattawar and M. J. Raković, “Virtues of mueller matrix imaging for underwater target detection,” Appl. Opt. 38(30), 6431–6438 (1999). [CrossRef] [PubMed]
- R. Nothdurft and G. Yao, “Expression of target optical properties in subsurface polarization-gated imaging,” Opt. Express 13(11), 4185–4195 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-11-4185 . [CrossRef] [PubMed]
- R. E. Nothdurft and G. Yao, “Effects of turbid media optical properties on object visibility in subsurface polarization imaging,” Appl. Opt. 45(22), 5532–5541 (2006). [CrossRef] [PubMed]
- S. G. Demos and R. R. Alfano, “Optical polarization imaging,” Appl. Opt. 36(1), 150–155 (1997). [CrossRef] [PubMed]
- S. G. Demos, H. Radousky, and R. Alfano, “Deep subsurface imaging in tissues using spectral and polarization filtering,” Opt. Express 7(1), 23–28 (2000), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-7-1-23 . [CrossRef] [PubMed]
- H. Shao, Y. He, Y. Shao, and H. Ma, “Contrast enhancement subsurface optical imaging with different incident polarization states”, Proc. of SPIE, 6047, 60470Z(1–6), (2006).
- G. C. Giakos, A. Molhokar, A. Orozco, V. Kumar, S. Sumrain, D. Mehta, A. Maniyedath, N. Ojha, and A. Medithe, “Laser imaging through scattering media”, IMTC 04. Proc. of the 21st IEEE, 1, 433 - 437 (2004).
- F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40(13), 9342–9345 (1989). [CrossRef]
- G. Yao, “Differential optical polarization imaging in turbid media with different embedded objects,” Opt. Commun. 241(4-6), 255–261 (2004). [CrossRef]
- P. W. Zhai, G. W. Kattawar, and P. Yang, “Mueller matrix imaging of targets under an air-sea interface,” Appl. Opt. 48(2), 250–260 (2009). [CrossRef] [PubMed]
- S. Huard, The Polarization of Light (Wiley, New York, 1997).
- R. A. Chipman, “Polarimetry” in Handbook of Optics, 2nd ed. M. Bass ed. (McGraw Hill, New York, 1995), vol. 2, chap 22.
- G. I. Bell, and S. Glasstone, Nuclear Reactor Theory (Van Nostrand Reinhold, New York, 1970).
- B. Kaplan, G. Ledanois, and B. Drévillon, “Mueller matrix of dense polystyrene latex sphere suspensions: measurements and monte carlo simulation,” Appl. Opt. 40(16), 2769–2777 (2001). [CrossRef] [PubMed]
- R. Kalibjian, “Stokes polarization vector and Mueller matrix for a corner-cube reﬂector,” Opt. Commun. 240(1-3), 39–68 (2004). [CrossRef]
- H. C. Van de Hulst, Light scattering by Small Particles (Dover, New York, 1981).

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