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Vector transmission matrix for the

polarization behavior of light

propagation in highly scattering media

Santosh Tripathi,1Richard Paxman,2Thomas Bifano,3and Kimani C.

Toussaint, Jr.4,*

1Department of Electrical and Computer Engineering, University of Illinois at

Urbana-Champaign, 1406, W. Green St., Urbana, Illinois 61801, USA

2Paxman Consulting, 9228 Sunset Lake Dr., Saline, Michagan 48176, USA

3Boston University Photonics Center, 8 Saint Marys St., Boston, Massachusetts 02215, USA

4Department of Mechanical Science and Engineering, University of Illinois at

Urbana-Champaign, 1206, W. Green St., Urbana, Illinois 61801, USA

*ktoussai@illinois.edu

Abstract:

to be useful in controlling the propagation of light in highly scattering

media. In this paper, we present the vector transmission matrix (VTM)

which, unlike the TM, captures both the intensity and polarization trans-

mission property of the scattering medium. We present an experimental

technique for measuring the absolute values of the VTM elements which is

in contrast to existing techniques whereby the TM elements are measured

to within a scaling factor. The usefulness of the VTM is illustrated by

showing that it can be used to both predict and control the magnitude of

the complex polarization ratio of the light focused through the scattering

medium. To the best of our knowledge, this is the first study to show the

possibility of controlling the polarization of the light transmitted through

highly scattering media.

Recently the optical transmission matrix (TM) has been shown

© 2012 Optical Society of America

OCIS codes: (260.5430) Polarization; (120.5410) Polarimetry; (120.0120) Instrumentation,

measurement, and metrology.

References and links

1. I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett.

32, 2309–2311 (2007).

2. S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission

matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett.

104, 100601 (2010).

3. S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,”

Nat. Commun. 1, 81 (2010).

4. I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4,

320–322 (2010).

5. S. M. Popoff, A. Aubry, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Exploiting the time-reversal operator

for adaptive optics, selective focusing, and scattering pattern analysis,” Phys. Rev. Lett. 107, 263901 (2012).

6. D. B. Conkey, A. M. Caravaca-Aguirre, and R. Piestun, “High-speed scattering medium characterization with

application to focusing light through turbid media,” Opt. Express 20, 1733–1740 (2012).

7. Y. Choi, T. Yang, C. Fang-Yen, P. Kang, K. Lee, R. Dasari, M. Feld, and W. Choi, “Overcoming the diffraction

limit using multiple light scattering in a highly disordered medium,” Phys. Rev. Lett. 107, 023902 (2011).

8. A. Dolginov, Y. Gnedin, and N. Silant’ev, “Photon polarization and frequency change in multiple scattering,” J.

Quantum Spectrosc. Ra. 10, 707–754 (1970).

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9. T. Kohlgraf-Owens and A. Dogariu, “Finding the field transfer matrix of scattering media,” Opt. Express 16,

13225–13232 (2008).

10. T. Kohlgraf-Owens and A. Dogariu, “Transmission matrices of random media: Means for spectral polarimetric

measurements,” Opt. Lett. 35, 2236–2238 (2010).

11. C. Prada and M. Fink, “Eigenmodes of the time reversal operator: A solution to selective focusing in multiple-

target media,” Wave Motion 20, 151–163 (1994).

12. C. Prada and J.-L. Thomas, “Experimental subwavelength localization of scatterers by decomposition of the time

reversal operator interpreted as a covariance matrix,” J. Acoust. Soc. Am. 114, 235–243 (2003).

13. J. Pasquesi, S. Schlachter, M. Boppart, E. Chaney, S. Kaufman, and S. Boppart, “In vivo detection of exercise-

induced ultrastructural changes in genetically-altered murine skeletal muscle using polarization-sensitive optical

coherence tomography,” Opt. Express 14, 1547–1556 (2006).

14. R. Rao, M. Mehta, and K. Toussaint Jr., “Fourier transform-second-harmonic generation imaging of biological

tissues,” Opt. Express 17, 14534–14542 (2009).

15. C. Krafft, B. Dietzek, and J. Popp, “Raman and cars microspectroscopy of cells and tissues,” Analyst 134, 1046–

1057 (2009).

16. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley and Sons, 2007).

17. R. C. Gonzalez and R. E. Woods, Digital Image Processing (Prentice Hall, 2007).

18. B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the stokes polarization parameters,” Am.

J. Phys. 75, 163–168 (2007).

19. R. H. Byrd, J. Nocedal, and R. A. Waltz, “Knitro: An integrated package for nonlinear optimization,” in Large

Scale Nonlinear Optimization, G. D. Pillo and F. Giannessi, eds. (Springer Science+Business Media, 2006).

1.Introduction

The ability to control the transmission of light through a highly scattering medium (HSM) has a

multitude of potential applications and has recently garnered special attention [1–7]. In 2007 it

was shown that light can be focused through an HSM by iteratively adjusting the phase profile

of the field incident on it [1]. The study was later systematized in 2010 by the introduction of

the concept of the transmission matrix (TM), each element of which relates the transmission

of an optical field from an incident to an observation point [2]. Since then the ability to fo-

cus light through disordered media has been put to use in a variety of interesting applications

such as ’sub-diffraction’ imaging [4] and image transmission through an HSM [3]. A recent

trend towards using high-speed spatial light modulators for the focusing problem is bound to

further enable a large number of new and exciting applications and developments [5,6]. How-

ever, although the effect of multiple scattering on the polarization of the scattered light has

been investigated for some time [8–10], the control of the polarization state for light trans-

mitted through an HSM has heretofore not been explored. With optical fields, in many cases,

the intensity is the only parameter of concern. However, there are many important applications

where polarization plays a crucial role and warrants consideration. For example, being able to

control the polarization of the light transmitted thorough the HSM could extend the capabilities

of polarization sensitive deep-tissue imaging techniques such as polarization-sensitive optical

coherence tomography [13] as well as various coherent nonlinear microscopy techniques (eg.,

SHG [14] and CARS [15]). It could also be useful in imaging the dipole orientation of the

molecules immobilized in a matrix for single molecule studies, and for the curing of polariza-

tion sensitive polymers in 3D fabrication.

In this paper, we generalize the concept of the TM used in the experiments controlling the

light propagation through an HSM [2,6] to take into account the polarization of optical fields

resulting in the vector transmission matrix (VTM). Further, we outline a method for measuring

the absolute values of the VTM elements. It is in contrast to the TM measurements reported

in the literature where the TMs have been measured upto a scaling factor [5,6]. Moreover, we

show experimentally that the VTMs can be used to optimize the focus of light through an HSM

as well as to predict and control the magnitude of the complex polarization ratio [16] i.e. ratio

of the amplitudes along each polarization basis state of the focused light.

The paper is organized as follows. In Section 2, we present the concept of the VTM and

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present both the theory and an experimental setup for measuring it. In Section 3, we outline a

method for optimizing the polarization components of the focused optical field using the VTM.

Section 4 presents the experimental results and discussion which is followed by the conclusions

in the Section 5.

2.Measurement of the vector transmission matrix

It is well known that the effect of multiple scattering is to scramble the polarization of the input

field [8]. However, since any polarization state can be represented as a linear combination of

two orthogonal basis polarization states [16], to capture the polarization changing behavior of

an HSM, it is sufficient to measure components of the output field along each of the basis polar-

ization states as a function of the basis states at the input. Although any set of basis states can

be chosen, in this paper, we take the canonical x and y polarization directions, corresponding to

the horizontal and vertical direction in the laboratory reference frame as basis states. Assuming

that there are 2M input degrees of freedom and N observation points, the VTM can be defined

to relate the input and output field as

txx

1,1··· txx

...

txx

1,M

...

N,Mtxy

txy

1,1··· txy

...

N,1··· txy

tyy

1,1

...

N,Mtyy

1,M

...

......

N,1··· txx

tyx

1,1

...

tyx

N,M

··· tyx

...

1,M

...

··· tyy

...

1,M

...

N,1··· tyx

N,1··· tyy

N,M

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

represents the i polarization component at the mth input point with i = x,y and

m = 1 to M. Similarly, Ei(O)

n

represents the i polarization component at the nth observation

point with n = 1 to N. The VTM is of size 2N ×2M and its element tij

of the j polarization component of the field in the mth input state to i polarization component

of the field on the nth observation point. The TMs reported in the literature [2,6] are a subset

of the VTM shown in Eq. (1). For example, with a y polarized input field incident on an HSM

and the analyzer customarily used in the TM measurements [2,6] at the output oriented along

the x direction, the elements of the VTM shown in the box in Eq. (1) constitute the scalar TM

measured.

Figure 1 shows the experimental setup used for VTM measurements. In the figure, a 45◦-

polarized continuous wave input laser beam is incident upon a non-polarizing beam splitter

BS1 thereby creating a reflected and a transmitted beam. The latter then enters a polarizing

beam splitter PBS which decomposes the beam into horizontally and vertically polarized com-

ponents leading to beams B1 and B2, respectively. B1 and B2 are then respectively steered

by mirrors M1 and M2 onto a second beam splitter BS2 which then directs both beams onto

a nematic liquid crystal spatial light modulator NLC-SLM. As shown in the figure, B1 and

B2 are spatially separate on the NLC-SLM which provides independent phase modulation for

each of the beams. This spatial separation of the beams is achieved by walking B2 vertically

by tilting PBS and M2 appropriately. Since the NLC-SLM used in our experiments provides

phase modulation only for a vertically polarized optical field, B1 is converted to a vertically

polarized beam by passing it through a half-wave plate HWP. The surface of the NLC-SLM

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

Ex(I)

1

...

Ex(I)

M

Ey(I)

1

...

Ey(I)

M

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

=

Ex(O)

1

...

Ex(O)

N

Ey(O)

1

...

Ey(O)

N

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

,

(1)

where Ei(I)

m

n,mgives the contribution

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Fig. 1. Experimental setup to measure the vector transmission matrix. Refer to text for

details.

is then imaged by lens pair L1 and L2 onto the back focal plane of an optical objective Obj1

which focuses the light onto the HSM sample S. The forward scattered optical field is then col-

lected by another objective Obj2, passed through an analyzer P, and subsequently recorded by

a CCD camera. This optical field recorded on the CCD is our control signal. The quarter-wave

plate QWP shown in the setup is flipped out of the beam path while collecting the VTM ele-

ments and flipped into the beam path during polarization measurements. It is worth noting that

the component of the scattered light recorded on the CCD that comes from the beam reflected

off BS1, steered by M3 and retro-reflected by mirror M4 acts as the reference during VTM

measurements. To avoid ambiguity, we define the total reference and total control signals as the

reference and control signals that would be recorded by the CCD in the absence of any optical

component between the Obj2 and CCD.

In our experiments the laser source was an 808-nm diode laser, IQ2C(808-150), acquired

fromPowerTechnology,Inc.Althoughnotshowninthefigure,thebeamcomingoutofthelaser

module was expanded, spatially filtered and passed through a half-wave plate-polarizer combi-

nation prior to being incident on BS1. The beam splitters BS1, PBS and BS2 with respective

part numbers BS013, PBS252 and BS014 were acquired from Thorlabs, Inc. The NLC-SLM

is a 512×512 XY series NLC-SLM acquired from Boulder Nonlinear Systems. The HWP,

QWP and P were acquired respectively from Karl Lambrecht Corporation, CVI Melles Griot

and Thorlabs, Inc. with respective part numbers MWPQA2-12-V800, QWPO-800-06-4-R10

and LPNIR050-MP. The QWP and P were mounted separately on two PRM1Z8E motorized

rotation mount and controller acquired from Thorlabs, Inc. and were controlled through a cus-

tom computer code written in MATLAB and LabVIEW. The camera used in our experiments

was a Hamamatsu ORCA-285G. The Obj1 was an infinity corrected Spencer 10× objective

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with an NA, working distance and parfocal distance of 0.25, 9.1 mm, and 34 mm, respectively.

Similarly, Obj2 was an infinity corrected 45× objective from Reichert with an NA, working

distance and parfocal distance of 0.66, 0.7 mm and 34 mm, respectively. We studied two types

of HSM samples. They were prepared by depositing a ZnO and ethanol mixture, and white nail

polish on microscope slides.

To measure the VTM we divide the SLM into 256 segments and use half of the segments to

control the x polarized component B1 and the other half to control the y polarized component

B2 resulting in M = 128. Since the contribution of each control segment at the input to an

observation point at the output is weak, using the canonical basis results in a low signal-to-

noise ratio; therefore we start by measuring the VTM elements using a Hadamard basis at the

input [2, 6]. As in [2], we also use four-point phase shifting interferometry. To measure the

VTM elements of the form tyj

element of the Hadamard basis, the analyzer transmission axis is oriented along y, whereas to

measure the VTM elements of the form txj

The VTM elements with Hadamard basis at the input are converted to VTM elements with the

canonical elements at the input using a standard Hadamard-to-canonical transformation [17].

These transformed VTM elements are of the form

?

where IRi

nth observation point, respectively, and ?φ is the phase of the reference signal subtracted from

the phase of the control signal. A VTM element should relate the input at the mth input degree

of freedom to the output at the nth observation point and as such should be independent of the

reference signal; however, as can be seen from Eq. (2), the measured VTM elements are scaled

by a factor

IRi

components of the total reference signal as reference in the phase shifting measurements. Since

the x and y components of the total reference signal can have a phase difference between them,

the measured phase values for Txj

elements together this phase offset needs to be corrected for.

In our experiments we solve both of these problems by measuring the Stokes vector

?

by orientating the polarizer along x direction and measuring the value of intensity on the CCD

as a function of the orientation of the QWP [18]. During these measurements a beam block is

placed in front of the PBS in Fig. 1. From the Stokes vector we calculate the x and y components

of the total reference signal as [16]

n,Hm, where the subscript Hmin the VTM represents the mth basis

n,Hmthe analyzer transmission axis is oriented along x.

Tij

n,m=

IRi

n,mISi

n,mei?φ,

(2)

n,mand ISi

n,mare the i polarization components of the reference and control signals at the

?

n,mwhich depends on the reference signal. Further, Txj

n,mand Tyj

n,muse the x and y

n,mand Tyj

n,mhave an offset between them. To use all the VTM

SR,0

n,m

SR,1

n,m

SR,2

n,m

SR,3

n,m

??

for the total reference signal. The Stokes vector is measured

IRx

n,m=1

2

?SR,0

?SR,0

n,m+SR,1

n,m

?

?,

(3)

and

IRy

n,m=1

2

n,m−SR,1

n,m

(4)

respectively. Similarly, the phase angle between the x and y components of the reference signal

ϕ is calculated as [16]

ϕ =1

2

?

?SR,2

n,m+iSR,3

n,m

?.

(5)

Using these values the absolute values of the VTM elements are calculated as

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Fig.2.Asimplifiedblockdiagramoutliningtheprocedurefollowedincalculatingthephase

image for polarization optimization. A given phase profile is changed iteratively until the

polarization predicted by the VTM corresponding to the phase profile is similar to the

desired polarization. At each iteration, corrections to the phase image are calculated based

on both the nature of the change in polarization error and the phase image from previous

iterations.

txj

n,m=

Txj

n,m

?

IRx

n,m

,

(6)

and

tyj

n,m=

Tyj

n,m

?

IRy

n,m

eiϕ,

(7)

where we take the phase of the x component of the reference signal as the datum for all of

the phases involved in the experiments. As has been shown previously [2,6], controlling only

intensity through the HSM does not require determination of the absolute value of the TM

elements. However, for control of the polarization state, it is necessary to be able to tune the

relative weights ofthe polarization components along basis statesboth in termsof the amplitude

and phase, thus knowing the VTM elements to within a scaling factor only is of limited use.

In addition, even for intensity only control, knowing the absolute values of the TM or VTM

elements can facilitate quantitative control on the intensity, e.g., delivering a specified amount

of light to a particular optical mode.

3. Phase image calculation for polarization control

The VTM of an HSM captures the medium’s polarization changing behavior, and thus it is

possible to predict the output polarization given the phase profile at the input. This ability

can be used to calculate the phase image to be displayed on the NLC-SLM that can result in

the desired polarization components at the output. To calculate the required phase profile, one

can do a direct matrix inversion of Eq. (1). However, in general, it results in a required input

field which can be realized only through both the phase and amplitude modulation of the input

field-a capability which is not present in our current setup (we use a phase only SLM in the

setup). In a similar situation, in intensity only studies [2], the results of the inversion have been

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Fig. 3. Speckle field before and after optimization. An enhancement of 41× was observed

with 256 control segments.

normalized. However, that cannot be done for polarization control as the type of normalization

performed in [2] would change the relative strength of the orthogonal polarization components

resulting in a polarization state different from the desired one. Thus, we resort to an iterative

optimization process to calculate the phase image. A conceptual block diagram describing the

basic steps in the calculation of the phase image is shown in Fig. 2. The process starts with

a given phase image. Using the VTM the output polarization for the given phase image is

predicted. The error between the desired and the predicted polarization is evaluated and is used

to control the correction factor to be applied to the phase image. The nature of the change in

errors as a function of phase images obtained from previous iterations determines the nature of

the correction to be applied to the phase image.

Although Fig. 2 illustrates the concept, the actual steps followed in the optimization process

are more complicated. Rather than optimizing for the polarization itself, we optimize for the

intensity under the constraint that the predicted polarization be close to the desired polarization

within a given tolerance limit. In our calculations, we characterize the polarization by the com-

plex polarization ratio defined by two parameters: φ and R. These parameters are defined as the

difference between the phase and the ratio of the magnitudes of the y and x polarization com-

ponents, respectively [16]. The optimization problem to be solved is depicted in Eq. (8) and is

solved by using the interior point method available in the KNITRO [19] optimization package

accessed from MATLAB. As in any interior or barrier method, KNITRO breaks the optimiza-

tion problem into a set of barrier sub-problems controlled by a barrier parameter. The algorithm

then repeats the process of solving the sub-problem and reducing the barrier parameter until the

optimization problem is solved [19].

⎧

⎪

We note that the iterative optimization followed in our case is a purely computational process

as we utilize the information about the HSM contained in the VTM only. This approach is in

contrast to the iterative optimization process followed, for example in [1] for intensity control,

where one has to physically measure the intensity corresponding to each phase profile.

Maximize

? Ei(I)

m

I

?

? Ei(I)

m

?

subject to

⎪

⎪

⎪

⎩

⎨

?????R2−

???∑Ey(O)

+?∑Ex(O)

n

???

2

???∑Ex(O)

n

???

2

?????< εr

?

φ −?∑Ey(O)

nn

?

mod 2π < εp

.

(8)

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Fig. 4. Plot of the observed versus predicted polarization state to illustrate the value of the

vector transmission matrix in predicting the output polarization state.

4. Results and discussion

To elucidate the usefulness of the VTM in predicting the polarization of the light focused

through an HSM we carried out a series of experiments. The VTMs were at first measured

and then used to design phase images that would focus light through the HSM. Initially, the

optimization of the intensity of the focused light was carried out without considering the po-

larization information contained in the VTMs. Figure 3 shows a representative speckle pattern

before and after optimization. For 256 control segments on the NLC-SLM, we were able to

achieve an intensity improvement of more than 40× which is consistent with what has been

previously reported in the literature using a similar number of control segments [2]; however,

for basic polarization studies an improvement by a factor of 10× was found sufficient. Once

the light was focused, the predicted output polarization of the phase image that resulted in the

focused light was then calculated by using the VTM and compared with the actual polarization

observed. The reference signal is blocked during the assessment of the observed polarization

state.

A scatter plot of the observed versus predicted ratio of the intensities of the y and x polariza-

tion components have been plotted in Fig. 4. The diagonal line represents the ideal case of equal

values of the predicted and the observed ratios and is shown for reference. Any deviation of the

observed points from the diagonal line represents the error in the observed values. From the re-

sults we see that the observed ratios follow the predicted ratios to a large extent. The values are

specially close for ratios between 0.1 and 10. For these ratios, the average relative error is 66%.

Note that when the predicted ratios are too large or small, the observed ratios do not follow the

predicted values as faithfully. We expect that this behavior stems from the interplay between

the non-ideal experimental conditions, e.g., diffraction at the NLC-SLM, decorrelation of the

system, error in the measurement of the characterstics of the reference signal and the stringent

requirment on the accuracy of the model of the system when the ratios are too large or small.

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Fig. 5. Plot comparing the experimentally obtained (closed markers) and targeted (open

markers) ratios of the intensities of the polarization components y and x. The dashed curve

shows the trajectory of all possible ratios. Each ratio can represent any point on a unique

circle on the Poincar´ e sphere; the inset shows the circle corresponding to the targeted po-

larization ratio of 1 outlined on the sphere.

Using the procedure outlined in Section 3, we calculated the phase images that would result

in the ratios of 0.05, 0.5, 1, 2, and 20 for the intensities of the y and x polarization components.

These phase images were then displayed on the NLC-SLM to modify the phase profile of the

light incident on the HSM. The strength of the x and the y components of the focused light

was then measured through the Stokes vector measurement. Due to the fact that our system

had a speckle decorrelation time of ∼1 hr, the experiments were repeated with different sets

of VTMs and different observation points. This is equivalent to repeating the experiment for

different samples. For each desired ratio, the intensities of the measured x and y polarization

components were normalized by the total intensity then plotted on the xy coordinate plane along

with the standard deviations along the x and y directions. The results are shown in Fig. 5. The

targeted values and the trajectory of all possible ratios is also shown for reference. Comparing

the targeted and observed values we see that the observed values again follow the targeted ones.

Here we note that in Fig. 5 the trajectory of all possible ratios maps an arc of a unit circle

in the first quadrant of the xy coordinate plane. Since the ratios of the y and x values change

nonlinearly along the circumference of a circle, the plot has a nonlinear scale for the ratios.

Note that due to the nonlinear nature of the scale, the large error bars for ratios near one do not

signify large relative error compared to that for very small or very large ratios. Moreover, in our

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experiments the available input degrees-of-freedom was found to be insufficient to control both

the relative phase and intensities of the polarization components which limited us to exploring

the control of the relative intensities only. Since the relative phases were neglected, each point

shown in Fig. 5 could represent any point on a unique circle on the Poincar´ e sphere whereby

the locus of a polarization with fixed intensity ratio is a circle normal to the x-axis. The inset

in Fig. 5 shows a Poincar´ e sphere with the dashed circle corresponding to an intensity ratio of

1 outlined on it. We are working on modifying the experimental setup to allow for amplitude

modulation at the input which we expect to facilitate finer control over the input field and help

us explore the complete state of polarization control of the light focused thorough the highly

scattering medium .

5. Conclusion

We presented a vector transmission matrix for the study of the polarization behavior of light

propagation in highly scattering media. For M input field control segments and N observation

points, the size of the vector transmission matrix is 2N×2M whereas the size for a scalar trans-

mission matrix with the same number of input field control segments and observation points is

N ×M. We also discussed a method for experimentally measuring the absolute values of the

elements of the vector transmission matrix. The polarization behavior of the highly scattering

media encoded in these elements was shown to be useful in predicting the magnitude of the

complex polarization ratio of the light focused through such media. It was further shown that

it is possible to calculate a phase profile for the light incident on the highly scattering medium

that would deliver the desired magnitude of the complex polarization ratio at the targeted ob-

servation point. We are working on refining our experimental technique to extend our control to

the phase of the complex polarization ratio which is necessary to exercise the complete control

over the sate of polarization of the transmitted light. The ability to predict and control the po-

larization of the light transmitted through the highly scattering media has potential to be useful

in a wide range of imaging, metrology, and fabrication problems.

Acknowledgments

We acknowledge financial support from the National Academies Keck Futures Initiative (NAS

NAKFI IS10). S.T. and K.C.T. also acknowledge partial support from an NSF CAREER award

(NSF DBI 09-54155).

#168000 - $15.00 USD

(C) 2012 OSA

Received 4 May 2012; revised 15 Jun 2012; accepted 16 Jun 2012; published 29 Jun 2012

2 July 2012 / Vol. 20, No. 14 / OPTICS EXPRESS 16076