Mueller Matrix Imaging Microscope Using Nanostructured Thin Film Mirrors

Alexander Ruder Author

04/05/2021 Added

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The operation and calibration of a Mueller matrix imaging microscope is demonstrated using nanostructured thin film mirrors for polarization state generation and analysis.

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[00:00:00.030]Hello.
[00:00:00.420]My name is Alex Ruder and I'm a PhD student in the department of electrical and
[00:00:03.990]computer engineering at UNL.
[00:00:05.670]My advisor is professor Mathias Schubert and this presentation is about a Mueller
[00:00:09.570]matrix imaging microscope that we've developed that uses nanostructured thin film
[00:00:13.530]mirrors.
[00:00:14.760]I'm first going to go over some background information on Stokes vectors and
[00:00:18.150]Mueller matrices, and various Mueller matrix instrument principles.
[00:00:21.840]And then I'm going to provide an overview of the instrument construction it's
[00:00:25.350]optical model and the calibration method.
[00:00:27.990]Last we're going to take a look at some Mueller matrix images that were recorded
[00:00:31.320]with the instrument of different anisotropic samples.
[00:00:35.310]So the Stokes.
[00:00:35.880]Vector is a four element column vector that describes the polarization
[00:00:39.180]properties of a beam of light.
[00:00:41.400]And the Mueller matrix is a four by four matrix that fully describes how an
[00:00:45.210]optical element at a given wavelength and orientation will convert the
[00:00:48.930]polarization of an incident beam of light. By multiplying a known stoke
[00:00:53.580]vector with a known Mueller matrix
[00:00:55.170]the resulting modified Stokes vector can be determined.
[00:00:58.620]The Mueller matrix of a sample for a given orientation.
[00:01:02.190]can be measured by varying the input polarization state,
[00:01:05.250]and then observing the resulting polarization properties after the sample.
[00:01:09.960]But because most detectors simply measure intensity and not polarization the
[00:01:13.620]polarization properties after the sample also have to be varied in order to recast
[00:01:18.450]the change in polarization into a change in intensity,
[00:01:22.620]the group of optical components before a sample that generate the instant
[00:01:27.270]polarization state are known as the polarization state generator or PSG.
[00:01:32.460]The group of components after the sample are known as the polarization state
[00:01:36.180]analyzer or detector, PSA, or PSD,
[00:01:39.900]the general optical path of a mule matrix polarimeter begins with the source,
[00:01:44.100]the PSG, the sample,
[00:01:45.630]and finally the PSA because the the detector only records intensity
[00:01:50.190]values. The first row of the PSA,
[00:01:52.440]Mueller matrix only needs to be considered similarly carrying out the
[00:01:56.100]multiplication between the source Stokes vector and PSG Mueller matrix results
[00:02:00.570]in a column vector,
[00:02:02.010]the PSG and PSA vectors are noted here as G and D vectors.
[00:02:06.720]The Mueller matrix can then be rearranged as a vector and the effect of
[00:02:11.370]the different configurations of G and D are rearranged as a matrix known as the
[00:02:16.170]instrument matrix,
[00:02:17.670]carrying out the vector matrix multiplication of the Mueller vector and
[00:02:22.620]instrument matrix results in a vector of intensity.
[00:02:26.220]This gives us a system of linear equations and a minimum of 16,
[00:02:30.810]well chosen configurations of G and D are needed to solve the system.
[00:02:35.370]Typically more are used, which results in an overdetermined system.
[00:02:38.790]We can then solve
[00:02:40.890]this system using ordinary least squares.
[00:02:43.890]And in the case of a spatially varying Mueller matrix polarimeter this system is
[00:02:48.390]solved for each pixel. The optical path of the instrument begins at the source,
[00:02:52.710]which is a fiber collimated lens that is fed with a fiber
[00:02:55.500]coupled green led. The collimated beam is reflected from the first anisotropic
[00:03:00.160]mirror and passes through a field stop to limit the beam diameter.
[00:03:03.490]The beam then passes through the sample and is collected by an objective lens.
[00:03:08.260]The beam exits the objective lens and is reflected from the second
[00:03:11.590]anisotropic mirror,
[00:03:12.790]which is then collected by an achromatic tube lens that forms an image on the
[00:03:16.990]sensor of a monochrome camera.
[00:03:19.000]The mirrors are made using electron beam of operation. An optically thick
[00:03:22.600]titanium base layer was deposited onto thick quartz
[00:03:25.510]substrates. Glancing angle deposition was used to grow a thin
[00:03:30.580]titanium slanted calmer than film on top of this optically
[00:03:33.730]thick base layer. These are images of a similar titanium,
[00:03:37.120]SCTF grown on a Silicon substrate,
[00:03:40.420]the slanted columnar structure of the film results in strong optical anisotropy,
[00:03:45.340]especially as a function of the, as azimuthal angle of the mirror.
[00:03:48.970]Both anisotropic mirrors are attached to stepper motors that rotate
[00:03:52.480]continuously at different speeds. In this case,
[00:03:55.330]the first mirror completes one revolution during the measurement
[00:03:57.790]and the second mirror completes five revolutions. An encoder attached to the
[00:04:01.780]second rotating anisotropic mirror is used to generate a clock signal that is
[00:04:05.830]fed the camera. This serves as a frame acquisition trigger pulse,
[00:04:09.700]and by triggering frame acquisitions this way,
[00:04:12.760]the different angular positions of each rotating mirror are precisely known for
[00:04:16.900]each frame of
[00:04:17.420]the measurement. In the current setup 128 frames are recorded for a single
[00:04:22.420]measurement.
[00:04:22.960]The time varying Stokes parameters for each mirror are modeled as a fourth order
[00:04:27.010]Fourier series where each vector element has unique Fourier coefficients.
[00:04:32.230]The coefficients are treated as images,
[00:04:34.210]which results in a per-pixel model of the instrument,
[00:04:37.660]carrying out the vector matrix vector multiplication gives a stack of
[00:04:42.430]intensity images where, in this case, consists of 128 images.
[00:04:46.870]The Fourier coefficient images are initially unknown and must be determined using a
[00:04:51.340]calibration process. Several especially homogenous samples with different,
[00:04:55.930]mostly ideal, Mueller matrices are measured. In this case,
[00:04:59.470]we took measurements of air as well as a polarizer and a wave plate with the
[00:05:04.090]polarizer and wave plate measured at different rotation angles. Regression
[00:05:09.070]analysis is then used to find the best fitting Fourier coefficient images that
[00:05:13.150]result in the closest match generated intensity to the
[00:05:18.000]measured calibration image stacks.
[00:05:20.910]The resulting coefficient images look like for both PSG,
[00:05:24.120]which is the bottom four rows. And the PSD,
[00:05:27.660]which is the upper four rows. On the left side
[00:05:31.290]You can see that, at first,
[00:05:32.700]it doesn't look like there's significant spatial variation between each of the
[00:05:36.510]elements when they're all set to the same scale.
[00:05:39.030]But if we allow each frame to rescale (on the right side),
[00:05:42.450]but keep the zero value color mapped to white, you can see that there is spatial
[00:05:46.890]variation across a significant number of the elements here in the coefficient
[00:05:51.240]images.
[00:05:51.630]Now the instrument is ready to accurately measure Mueller matrices of
[00:05:55.800]unknown samples. As a sanity check,
[00:05:58.580]the recorded intensity measurement for the straight-through air measurement is
[00:06:02.330]determined. Because air doesn't appreciably alter the polarization
[00:06:06.560]properties of light over short distances,
[00:06:08.600]the Mueller matrix should ideally be an identity matrix,
[00:06:11.840]which is close to what we see here. Across the diagonal
[00:06:15.080]the values are very close to one while there is optical activity and the off
[00:06:19.340]diagonal elements in the image,
[00:06:21.230]the values of each of these frames have been scaled up by twenty-five times.
[00:06:24.680]So the resulting measurement is actually quite close to what it should be.
[00:06:28.700]Moving on to other samples. this is a patterned titanium slanted
[00:06:31.640]columnar thin-film deposited onto a glass slide,
[00:06:35.180]the same type of structure that the rotating anisotropic mirrors are made out
[00:06:39.620]of. Um, you can see that there's significant optical activity in the one-two
[00:06:43.490]and two-one elements of the Mueller matrix.
[00:06:46.490]And if we take the same region in the sample and rotate it in the plane by
[00:06:51.320]45 degrees, then repeat the measurement.
[00:06:54.230]We can see that the optical activity propagates from the one-two
[00:06:58.640]and two-one Mueller matrix elements into the one-three and three-one
[00:07:03.500]matrix elements. Last,
[00:07:05.690]we have a birefringent resolution target from Thorlabs. Observing this
[00:07:10.490]without polarizers, it just looks like a piece of glass.
[00:07:13.400]But when this resolution target is placed between two crossed polarizers,
[00:07:17.210]the pattern,
[00:07:18.260]on it can be seen and rotating the sample between the cross polarizers causes
[00:07:22.910]the pattern to invert as it's rotated.
[00:07:26.060]So the sample acts as a half wave plate with a homogenous phase shift over the
[00:07:30.320]sample plane,
[00:07:31.220]but different fast axis orientations in the background and pattern regions.
[00:07:35.960]This is reflected in these Mueller matrix images.
[00:07:39.350]We can see element on one has almost no optical activity,
[00:07:42.950]which indicates that the sample isn't strongly absorbing or scattering the light
[00:07:47.780]and the near lack of activity in the first row in and column also indicates
[00:07:52.220]that the sample isn't highly polarizing. So in conclusion,
[00:07:55.700]we were able to use nanostructure thin film mirrors as polarization,
[00:07:59.690]state generator and analyzer and Mueller matrix imaging microscope,
[00:08:04.190]A spatially varying Fourier series approach was used to calibrate the instrument
[00:08:08.690]and model the optical properties of each of the mirrors,
[00:08:12.870]especially varying samples were imaged with good results.
[00:08:15.890]And in the instrument shows a promising optical simplification for Wheeler
[00:08:20.750]matrix imaging.
[00:08:21.920]Thank you for listening to my presentation and thank you to all of my
[00:08:24.980]collaborators that helped make this work possible.

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Mueller Matrix Imaging Microscope Using Nanostructured Thin Film Mirrors

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