## Finding the field transfer matrix of scattering media

Optics Express, Vol. 16, Issue 17, pp. 13225-13232 (2008)

http://dx.doi.org/10.1364/OE.16.013225

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### Abstract

When illuminated by temporally coherent light, multiply scattering media produce speckle patterns that in many situations are unpolarized on spatial averaging. As a result, the underlying field statistics are assumed to be Gaussian and information about them can be extracted from intensity-intensity correlations. However, such an approach cannot be applied to any scattering medium where the interaction leads to partially developed speckle patterns. We present a general procedure to directly measure the field transfer matrix of a linear medium without regard to the scattering regime. Experimental results demonstrate the ability of our procedure to correctly measure field transfer matrices and use them to recover the polarization state of incident illumination.

© 2008 Optical Society of America

## 1. Introduction

2. I. Freund, “Stokes-vector reconstruction,” Opt. Lett. **15**, 1425–1427 (1990). [CrossRef] [PubMed]

4. S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. **61**, 834–837 (1988). [CrossRef] [PubMed]

5. I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. **61**, 2328–2331 (1988). [CrossRef] [PubMed]

2. I. Freund, “Stokes-vector reconstruction,” Opt. Lett. **15**, 1425–1427 (1990). [CrossRef] [PubMed]

6. I. Freund, “Looking through walls and around corners,” Physica A **168**, 49–65 (1990). [CrossRef]

7. R. H. Brown and R. Q. Twiss, “Interferometry of the intensity fluctuations in light. I. basic theory: the correlation between photons in coherent beams of radiation,” P. Roy. Soc. L. and A. Mat. **242**, 300–324 (1957). [CrossRef]

*assumption*regarding the statistics of the field.

## 2. Theory of field transfer

**r**that illuminates the second scatterer at position

_{1}**r**may be expressed as

_{2}**is a complex tensor that expresses the magnitude and phase of the coupling between incident and scattered field components. It also includes transferring the scattered field to the point**α ¯

**r**. Now, replacing

_{2}**E**

_{inc}with

**E**

_{scat,1}and

**(**α ¯

**r**,

_{2}**r**) with

_{1}**(**α ¯

**r**,

_{3}**r**), we obtain the field at position

_{2}**r**on the third scatterer. This process can be continued until the field reaches the detector at position

_{3}**r**to obtain the contribution of a particular scattering path to the field at the detector:

**r**to

_{i}**r**

_{i+1}as well as information about the scattering event at

**r**. Since we are interested in the transfer of the field through the medium, not in the particular details of how the transfer occurs, the mixing of scattering and propagation information is not important.

_{i}**r**due to all illuminated input points can be written as

**ê**

_{inc}, can be factored out of the sum so long as its polarization is spatially constant, and we can express the resulting output field in terms of a single transfer matrix,

**, which subsumes the intensity profile of the illumination. For a given experimental geometry and illumination source,**α ˜
(r)

**is only a function of detector location; however, if the spatial intensity profile of the illumination is modified,**α ˜
(r)

**will change as well because the intensity profile acts as a weighting function for the contribution of each scattering path. We emphasize that the tilded quantities represent the only measurable parameters of the scattering material because it is not practically possible to separate the contributions of individual paths from the detected intensity. It is also important to note that the illuminated points can have an arbitrary spatial extent and spatial intensity profile on the random scattering medium as long as the scattering paths remain coherent with one another.**α ˜
(r)

**has nine elements with eighteen unknowns: nine coupling magnitudes and nine phases. Because the scattering is not isotropic,**α ˜

**depends on the direction of propagation of the incident light, and it is difficult to measure the full transfer matrix for an arbitrary geometry. However, if the scattering medium is surrounded by an isotropic medium, the electric field of the illumination is confined to a plane and can be decomposed into two orthogonal polarization states with a phase between them. If the scattered fields are allowed to propagate away from the scattering medium before detection, they can also be decomposed into two orthogonal polarization states and a phase term. In this situation, the transfer matrix has only four elements consisting of eight unknowns, which can be determined by illuminating the scattering medium with appropriately polarized light. Moreover, we can choose one of the elements of the transfer matrix to be real since we cannot measure absolute phase at a point and only compare the intensities between points. Additional simplifications can be introduced by realizing that it is not necessary to characterize all seven of the remaining unknowns simultaneously. If a polarizer oriented along the x-axis is placed between the scattering medium and the detector,**α ˜

α ˜

_{21}=

α ˜

_{22}=0, and there are only three unknowns that need to be characterized. The detected intensity at point

**r**is then given by

*θ*is the phase between the x and y components of the incident field, and E

_{x}and E

_{y}are their respective magnitudes. In Eq. (4),

*is the phase introduced by the coupling of E*ϕ ˜

_{y}into a scattered x polarized field, and

α ˜

_{11}and

α ˜

_{12}are the magnitudes of the coupling of the incident x and y polarized fields, respectively, into scattered fields polarized along x. The polarization of the scattered field and the elements of

**measured are determined by the orientation of the final polarizer.**α ˜

## 3. Calibration and field recovery procedure

α ˜

_{11}≠0 and

α ˜

_{12}≈0, while the points shown in green have a transfer matrix of

α ˜

_{11}≈0 and

α ˜

_{12}≠0. Because the output points represented by the blue and green areas couple only one of the two input field components, they measure the x and y components of the unknown field. The points shown in red on the other hand have

α ˜

_{11}≈

α ˜

_{12}and contain the information about the phase of the unknown field because both of the input field components are coupled into the measured intensity.

*any unknown field*from its speckle pattern simply because the medium does not produce a sufficient number of independent combinations of the field.

_{x}, E

_{y}, and

*θ*, are determined by solving the following system of equations for each group of detectors

**r**denotes the location of the point in the detection plane rather than the location of a scatterer.

_{i}## 4. Experimental demonstration

α ˜

_{11}≫

α ˜

_{12}or

α ˜

_{11}≪

α ˜

_{12}, and the smaller of the two can be approximated as 0 so that transfer matrices very close to the Δ axis are moved onto the axis. Also, there seems to be some clustering of the points near

*ϕ*=-1. Even though the transfer matrices are not uniformly distributed on the sphere, our measurements show that the sample produces a sufficiently large number of substantially different mixings of the incident field. This is the only requirement for recovering the state of polarization of the incident field.

α ˜

_{11}and

α ˜

_{12}randomly distributed uniformly between 0 and 1 and

ϕ ˜

_{12}to be randomly distributed uniformly between 0 and 2π. The magnitudes of the coupling matrices were then scaled so that the resulting calculated intensity distribution was similar to the actual data for unit strength electric field inputs. Gaussian white noise with a signal to noise ratio of 34 was then added to the intensity image to simulate the detector noise in a real measurement. Speckle images were generated for both the calibration and test states and processed using the same code as the experimental data. Our simulations indicate that the most significant source of error in the data collection and processing is the noise in the detector itself. For simulated data with no noise, the solid angle covered by the measurement data on the Poincare sphere (white dots in Fig. 4) is approximately 0.013 steradians. When noise comparable to the noise of the detector used in the experiment is added to the simulated data before processing, the spread of the recovered states increases to 0.048 steradians. The remaining error is likely due to mechanical instabilities in the experiment.

## 5. Conclusions

## References and links

1. | J. W. Goodman, |

2. | I. Freund, “Stokes-vector reconstruction,” Opt. Lett. |

3. | P. K. Rastogi, ed., |

4. | S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. |

5. | I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. |

6. | I. Freund, “Looking through walls and around corners,” Physica A |

7. | R. H. Brown and R. Q. Twiss, “Interferometry of the intensity fluctuations in light. I. basic theory: the correlation between photons in coherent beams of radiation,” P. Roy. Soc. L. and A. Mat. |

8. | C. Brosseau, |

9. | L. Mandel and E. Wolf, |

**OCIS Codes**

(030.6140) Coherence and statistical optics : Speckle

(120.5410) Instrumentation, measurement, and metrology : Polarimetry

(290.4210) Scattering : Multiple scattering

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: June 17, 2008

Revised Manuscript: July 30, 2008

Manuscript Accepted: August 11, 2008

Published: August 13, 2008

**Citation**

Thomas Kohlgraf-Owens and Aristide Dogariu, "Finding the field transfer matrix of scattering media," Opt. Express **16**, 13225-13232 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-13225

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### References

- J. W. Goodman, Speckle Phenomena in Optics 1st Ed. (Roberts & Co., Englewood, Co, 2007).
- I. Freund, "Stokes-vector reconstruction," Opt. Lett. 15, 1425-1427 (1990). [CrossRef] [PubMed]
- P. K. Rastogi, ed., Digital Speckle Pattern Interferometry & Related Techniques, 1st Ed. (Wiley, New York, 2001).
- S. Feng, C. Kane, P. A. Lee, and A. D. Stone, "Correlations and fluctuations of coherent wave transmission through disordered media," Phys. Rev. Lett. 61, 834-837 (1988). [CrossRef] [PubMed]
- I. Freund, M. Rosenbluh, and S. Feng, "Memory effects in propagation of optical waves through disordered media," Phys. Rev. Lett. 61, 2328-2331 (1988). [CrossRef] [PubMed]
- I. Freund, "Looking through walls and around corners," Physica A 168, 49-65 (1990). [CrossRef]
- R. H. Brown and R. Q. Twiss, "Interferometry of the intensity fluctuations in light. I. basic theory: the correlation between photons in coherent beams of radiation," P. Roy. Soc. L. A. Mat. 242, 300-324 (1957). [CrossRef]
- C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (John Wiley & Sons, Inc., New York, 1998).
- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, 1st Ed. (Cambridge UP, New York, 1995).

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