2.1.

Stokes Vector

Polarization is the oscillation orientation of the electrical vector of a propagating light wave. This is generally an elliptical state that includes linear and circular components. Light can also be depolarized, in which the oscillations are randomly oriented. The polarization state of light can be quantified through a Stokes vector $\mathit{S}$ comprising four Stokes parameters as per Eq. (1):

Each of the Stokes parameters corresponds to the difference between two orthogonal states of polarization, using a Cartesian coordinate frame of reference. $S_0$ is the total intensity of polarized light and can be calculated as sum of any two orthogonal components. $S_1$ is the difference between horizontally and vertically linear polarized components, $S_2$ is the difference between linear polarized components $+45\mathrm{deg}$ and $-45\mathrm{deg}$ from the horizontal, and $S_3$ is the difference in intensity between right- and left-hand circular polarized components. The Stokes parameters can be determined from six intensity measurements rather than from the sixteen measurements required form the Mueller matrix formalism. In addition, due to the relationships between the Stokes parameters, it is possible to measure a Stokes vector using only four intensity measurements. These factors make the Stokes vector a convenient object to measure. From the Stokes parameters, one can calculate the shape and orientation of the polarization ellipse and depolarization metrics.

If propagating through a turbid medium, polarized light will experience modifications on its initial state of polarization (SOP). Depolarization is the strongest of these modifications, which provides the best signal to noise ratio, as depolarization is a result of scattering, a phenomenon present in large quantities in all turbid media. Polarization is lost quickly; just several scattering events are enough to destroy the initial polarization. It is expected that the average path length of polarization preserving photons is of the order of the transport path length, which is about a few millimeters in tissue.¹¹

From the Stokes vector, we can calculate depolarization metric called the degree of polarization (DOP) as in Eq. (2),¹²

The DOP represents the fraction of backscattered light that maintains the initial SOP. It ranges from 0 for fully depolarized to 1 for fully polarized light and could be used for tissue characterization.⁸ Values of DOP that carry information lie between 0 and 1, as the backscattered signal should be partially polarized.

While indicative of tissue optical properties, DOP measurements are also affected by the wavelength of the illuminating light source. In the same tissue sample, red light penetrates deeper and undergoes mostly volumetric scattering. In contrast, blue light has a shallower penetration and carries mostly surface information. The difference in illuminated tissue volume affects DOP due to the difference in cellular composition of those volumes.

2.2.

Speckle

While noncoherent light can be used for polarimetry, we have opted to use partially coherent light to take advantage of the modification of wave phase-encoded in speckle. When coherent or partially coherent light is scattered it generates speckle, a stochastic interference pattern in the form of dark and bright spots. Speckle patterns are composed of numerous speckles that can each have an individual size and shape, intensity, and SOP.¹³ Each speckle is the result of the interference of stochastically distributed light of similar SOPs. The SOP and corresponding Stokes vector may vary from speckle to speckle and in certain scattering conditions, the polarization states among the speckles become non-uniform, forming a phenomenon called polarization speckle.^14,15 The composition of polarization speckle is highly sensitive to the phase of light, and thus speckle patterns generated from illuminating tissue can contain additional valuable information on tissue morphology. For example, speckle has been found useful in measuring skin surface roughness, with applications in skin cancer detection.⁹

2.3.

Polarization Speckle Design Considerations

We have two primary design considerations in measuring polarization speckle. The first consideration is ensuring the development of an informative polarization speckle field. Polarization and coherence are mutually connected,¹⁶ and light depolarization occurs at comparable propagating distances to light decorrelation (the decay of polarization matching the decay of coherence). As mentioned previously, only DOP measurements between 0 and 1, convey tissue information. Therefore, emerging light is preferably partially polarized and correspondingly partially coherent. A partially coherent speckle pattern is produced when the average photon path length difference in tissue is on the order of the temporal coherence length of the illuminating source.¹⁷ If the path length difference is too large, then the returning light will fail to be coherent and a pattern will not appear. To ensure partially polarized backscattered light, we require a laser with a coherence length on the order of millimeters to match the expected depolarization length.¹¹ This is one of the main features that influenced the choice of laser in the design of this probe, as expanded upon in Sec. 3.2.

The second consideration is the validity of spatial averaging of polarization within a speckle field. Since each speckle will maintain its own Stokes vector, and the SOPs would be distributed stochastically over the detection area, then spatially averaging a measurement across an increasing number of speckles may force measured DOP toward 0.¹⁴ We can adjust the detection geometry to ensure an appropriate number of speckles appear on the detector area. Essentially, the number of speckles on the detector must be large enough to ensure accurate mean intensity measurements, but not too large such that the stokes parameters are artificially reduced due to spatial averaging. In this article, we present two studies for this design consideration: calculating the proper relationship between detector size and the distance between object and detector, and ensuring the detection of a uniform field. The first of these studies is listed in Sec. 3.3. The probe’s response to field uniformity is detailed in Sec. 4.2.

3.1.

Polarizing Filters

Mathematically, it is possible to determine the Stokes vector of light using any four intensity measurements taken from unique polarization states on the surface of the Poincaré sphere, a geometrical model of polarization. In practice, this appears in division of focal plane polarimetry methods that measure polarization with four separate detectors in space, with devices such as pixel polarization cameras. Studies in this area have noted that systematic error decreases if the measurement points around the Poincaré sphere are equidistant in space.¹⁸ However, our probe is designed with the simplest configuration to implement; that of four identical film circular polarizing filters in four independent rotation angles. This scheme takes advantage of the assumption that the sum of intensities measured from two orthogonal polarizing analyzer orientations (any two perpendicular linear polarizations, or left- and right-hand circular polarizations) are the same regardless of which two orientations they are. The full details of this scheme can be found in our previous work.¹⁰ This is a resource-efficient scheme first described by $c$¹⁹ that requires only one type of polarization film, a circular polarizer comprising a linear polarizer and quarter-wave plate, as opposed to four unique elliptical filters.

To set the laser diode’s initial polarization to match the device detectors’ axes, a polarizing filter is placed before the laser. This is a wire grid linear polarizing filter (ThorLabs, WPL12-VIS). When the device is set for initial circular polarization, this is followed by a film circular polarizing filter. To note, this is the same film used as analyzers for the detector; an intentional choice for resource efficiency.

3.2.

Laser Diode

We selected a laser diode similar in wavelength to ones previously used for speckle work (660 nm, 120 mW, Thorlabs, HL6545MG).⁷ This laser diode was tested through Michaelson interferometry to determine if the coherence length of our selected laser diode is on the order of the path length that the light would travel through tissue. The temporal coherence length was found to be $\sim 3000\mu \mathrm{m}$, which is on the order of the photon path length for red light in skin, and therefore an appropriate length for speckle generation.

According to the American National Standard for Safe Use of Lasers,²⁰ the maximum permissible exposure (MPE) for skin exposure to a laser beam for durations between ${10}^{-7}$ and 10 $s$ is described by the equation

3.3.

Device Geometry

Several aspects of the probe’s geometry address polarization speckle. First, we must ensure that the speckle field is statistically uniform at the device’s detection plane, which requires a detection plane in the far diffraction zone. For speckle, it was shown that, instead of the common formula, the distance $z$ between the detector and the object for which the far zone occurs obeys the condition in Eq. (4):²¹

The second aspect is how to limit the amount of speckles $N$ averaged over the detection area while ensuring a sufficient amount are detected to reduce statistical error. In free-space geometry the number of speckles on a detector can be evaluated as in Eq. (5),²²

Fig. 1

Plotting speckle area versus device height. The horizontal line indicates $13{\mathrm{mm}}^2$, the size of our photodiodes.

For final device height, we are given a functional range between 2.5 and 30 cm. This range also works well with other restrictions that device height would generate. As we are observing backscattered light, the intensity of the signal decreases with the inverse-square of the distance. Secondly, a smaller height would make the device more portable, and a probe with a length greater than 20 cm would be unwieldy.

A series of tests were done to determine the effect of the probe’s height on its measurements. These tests were carried out validate our theoretical model of polarization speckle development and test for the appropriate distance ($Z$) for the device. DOP measurements using both linear (${\mathrm{DOP}}_L$) and circular (${\mathrm{DOP}}_C$) initial polarizations were taken using the probe at heights of 5, 10, 15, and 20 cm over different RMS roughnesses of silicone skin phantoms (heights chosen to sample from a practical range). The properties of these phantoms have been previous reported,¹⁰ along with their recipe.²⁴ The matrix of the phantoms was made from Smooth-On brand MoldMax 10T silicone and was pigmented to resemble human skin tones with a combination of Silc-pig silicone pigments (Smooth-On, Inc.). This silicone begins as a viscous resin, and is mixed with a curing agent at room temperature to hold its form. The silicone was molded with a metal roughness standard (Microshurf #334 comparator, Rubert+Co Ltd., Cheadle). The phantom roughnesses were verified by a WYKO NT2000 optical profilometer (Veeco, Tucson, Arizona). The testing scheme is shown in Fig. 2 and the results are in the following Fig. 3. For each measurement in Fig. 3, 10 measurements were taken and averaged from points on the corresponding roughness area of the phantom. Error bars indicate the standard deviation of measurements.

Fig. 2

Diagram of device height testing.

Fig. 3

(a) DOP versus device height for initial linear and (b) circular polarizations. Measurements shown on three different RMS surface roughnesses (9, 18, and $34\mu \mathrm{m}$). Error bars indicate standard deviation of 10 measurements.

Three tendencies can be observed. First, the DOP for circularly polarized light is consistently lower than the DOP for linearly polarized light, which implies that the skin phantoms are in the Rayleigh regime of depolarization. Second, DOP is smaller for rougher surfaces, demonstrating that roughness introduces additional depolarization.^25,26

The third observation is that for both modes of input light and all roughnesses, the DOP increases from 5 to 10 cm but saturates beyond 10 cm. The decrease in DOP from 10 to 5 cm is explainable as in decreasing $Z$ we approach a Fresnel zone of diffraction.⁸ This would cause the path difference between central and peripheral rays to become larger and exceed the temporal coherence length, leading to polarization and light coherence decay. After 10 cm, since the speckle area increases while the detector area remains constant, the number of speckles on the detector is decreasing, but the average polarization and their metrics such as DOP could stay the same if the speckle field is statistically stationary. The complete extent of DOP versus roughness versus device height is still subject to further study, but recognizing the additional preference for a lower height to increase signal strength, we chose a final height of 10 cm.

3.4.

Completed Prototype Design

Figure 4 shows a model and photo of the probe head, and a view of its internal structure to illustrate the major functional blocks. At this prototype stage of development, the probe head does not contain purely electronic components. These are kept in a connected casing that is also connected to a computer. The probe’s measurements are recorded on this computer through a LabVIEW UI. In a more developed prototype, the electronic and computation hardware could be minimized in both component cost and size. The radial distance at which the photodiodes are placed is determined by two design constraints; the first being the device’s handheld size, and the second being the acceptance angle (15 deg) of the polarizing filters used. As shown in Fig. 2, the photodiodes are 6 mm from the laser diode at the center, such that an angle within the film’s acceptance ($\theta =4\mathrm{deg}<15\mathrm{deg}$) is observed between the center of the photodiode and the incident site. The outer dimensions of the probe head are 12.5 cm in length and 5 cm in diameter.

Fig. 4

(a) Model of the Stokes polarimetry probe. (b) Photo of the probe held in hand. (c) Expanded view of probe components. Includes (1) targeting cone, (2) and (5) casing, (3) components to hold polarizing film, and (4) laser diode and photodiodes.

The tested prototype is primarily composed of fitted and pressed 3D-printed components. The inside of the targeting cone shown in Fig. 4 (1) is coated with optically dampening velvet to eliminate ambient reflections. To aim the probe, a sticker is applied centered on the target site within which the targeting cone fits. These stickers are a standardized and common landmarking tool in dermatology, to assist in labeling sites of interest and observing their scale in photographs. With the press of a button (not shown in the figure) the laser diode (4) is fired, passing through an input polarizer (3), and striking the target. The backscattered light is observed equally by the photodiodes (4) through their respective polarizing filters (3), which are all held in the same plane to allow for uniform speckle field distribution.

4.1.

Polarized Light Generation—Circular Polarizing Film

The circular polarizing film (Bolder Vision Optik) was specified at a center wavelength of 660 nm. This film is a laminated combination of linear polarizer and quarter-wave plate. The first test we performed is the analysis of this film in conjunction with the laser diode as a circular PSG. While the film was made to tight specifications, a laser diode is not perfectly monochromatic, which likely reduces the film’s effectiveness by some margin. For this test, laser light was passed through this circular polarizing film. A wire-grid linear polarizer was placed in the optical path as an analyzer and was rotated as transmission was measured.

By observing the fluctuation in intensity between these two angles of the linear polarizer, we can determine the ratio of the two arms of the beam’s polarization ellipse, which informs us of the probe’s initial polarization state. For perfectly circular polarized light, there should be no fluctuation in intensity. In the presence of ellipticity, the measurement will be different at the angles perpendicular and parallel to the ellipse’s major axis, as shown in Fig. 5. We measured an intensity ratio difference of 15:14. Computing the ellipticity angle from the ratio results in $\chi =43\mathrm{deg}$. When being used as part of a circular PSG, the resulting normalized $S_3=\sin (2\chi )=0.998$, with a corresponding linear component of $S_1=\cos (2\chi )=0.068$. It is worth noting that the linear component is represented by both $S_1$ and $S_2$, but in this experiment the azimuthal angle of the ellipse was arbitrary, so we model the entire linear component using $S_1$. The resulting normalized polarizance vector of the circular polarizing film is ${[1,0.068,0,0.998]}^T$, which makes the light slightly elliptical.

Fig. 5

Diagram of polarization ellipse of the probe’s initial elliptically polarized light.

4.2.

Polarized Light Measurement—Error Evaluation

Once the probe’s detection scheme was assembled, we simulated polarized backscattered light to evaluate the error present in the system. These experiments were performed using the framework show in Fig. 6. The probe head (E) was centered within the beam originating from (A) and expanded using a magnifying lens (B). Following this, adding a rotatable linear polarizer (C) allows the beam to be swept through all angles of linear polarization. By placing a fixed quarter-wave plate after the linear polarizer (D), we can also sweep through degrees of elliptical polarization. To note, this process created monochromatic polarized light but not polarization speckle. The intensity readings of the four photodiodes were recorded, and then combined into a Stokes vector, normalized such that the first element $S_0$ is 1.

Fig. 6

Diagram of field uniformity test with depiction of expanded laser light. (A) Laser diode. (B) Magnifying lens. (C) Rotatable linear polarizer. (D) Non-rotating quarter-wave plate. (E) Probe head.

Measuring the Stokes parameters while sweeping through all angles of linear polarization and degrees of elliptical polarization would generate a sinusoidal function. For a perfect polarimeter tracking a linear sweep, the measurements of $S_1$ and $S_2$ would perfectly fit a sine function, and $S_3$ would read a constant 0. For an elliptical sweep, all three Stokes parameters would fit a sinusoidal function. To evaluate the error of these measurements, we fitted each of our measurements to a sine function (or with 0 in the case of linear sweep $S_3$) and calculated the RMS error, assuming that the fitted function approximates the ideal. In each case, the Stokes parameters were normalized by dividing by $S_0$ (which represents intensity), such that $S_0=1$. These fittings are shown in Fig. 7. In addition, we calculated the RMS error of our measured DOPs for each measurement against an ideal DOP of 1. In this testing scheme, the DOP of every state should be close to 1, as this is a direct polarization measurement without scattering or speckle generation. These results are reported in Table 1.

Fig. 7

Linear and elliptical polarization measurements compared against fitted sine curves.

Table 1

Total RMS error analysis.

From these tests, we can also investigate the sources of error for this probe. We measured the variance in measurements when presented with the same depolarized light over time. This allows us to approximate the random error present in our measurements, separate from the systematic error caused by our PSA scheme and ellipticity of our filters. We found that by propagating error from the photodiode measurements up to DOP calculation, the final random error is $\sim 0.02$.

This finding indicates that most of our error is generated by systematic sources, which is also observable in the data from the sweeps. The measured points become more distant from the curve at specific angles in the sweep, and in fact, the Stokes parameters exceed a magnitude of 1 at those angles, which should be an impossible value. These values occur due to the Stokes parameter calculations relying on the measurements close to the extinction angle of the polarizing filters, where photodiode sensitivity is greatly reduced. Errors caused by non-uniformity of the field across the detector face will be amplified at these angles, which can result in calculating Stokes parameters with absolute values >1. It is anticipated that this error could be reduced by selecting a different PSA filter scheme, as expanded upon below.

The errors present in this probe can be considered in the greater context of Mueller matrix polarimeter error analysis. As reported by Tyo,²⁷ the error inherent to an PSG or PSA can be calculated from their characteristic matrix, a matrix formed from their polarizance vectors. To reduce error, the matrix must be well-conditioned, a property that indicates a small change in output when provided with a small change in input. This is observed mathematically through the matrix’s condition number, a value calculated from the ratio of the largest and smallest singular values of the characteristic matrix. This condition number spans the range of 1 to infinity, where 1 indicates perfect conditioning. Our PSA’s characteristic matrix is in Eq. (6):

More detail of this calculation can be found in established work.^27–29 From our PSA, we can calculate a condition number of $\sim 3.2$, which is greater than Tyo’s calculated minimum possible condition number of $\sqrt{3}$. Bruce et al.²⁸ experimentally determined the error present in Mueller matrix polarimeters of various condition numbers. It was estimated that a Mueller matrix polarimeter with a condition number of $\sim 3.2$ would see 10% RMS error (0.1 for a normalized Mueller matrix) if the signal is subject to a noise level of 5%. While we expect that the error of a Stokes vector polarimeter would be less than that of a Mueller matrix polarimeter due to the fewer number of measurements required, this estimate corresponds to the experimental error of our findings. As these estimates match, we consider our results to indicate good performance given the lesser robustness of this probe relative to standard polarimetric devices.