70 kHz full 4x4 Mueller polarimeter and simultaneous fiber calibration for endoscopic applications

Sylvain Rivet; Adrian Bradu; Adrian Podoleanu

doi:10.1364/OE.23.023768

1. Introduction

In the fast-developing field of optical imaging for medical diagnosis, the polarization of light has been used in different formats to provide information on the investigated tissue [1–4]. Propagation of a polarized light through scattering tissue imprints polarization changes specific to the medium traversed. The most exhaustive polarimetric technique used to track such changes is the Mueller polarimetry which can track variations in the diattenuation, retardance and depolarization. Mueller polarimetry analyzes the polarization state of backscattered or transmitted light as a function of the polarization state of the incident light and delivers 4x4 real matrices that depend on the polarization properties of the medium. For example retardance has provided relevant information in biological samples such as uterine cervix, burnt skin or retinal nerve [5–7]. Moreover recent studies [8], still preliminary and carried out in free space, show that depolarization imaging can provide different and supplementary information with respect to usual intensity imaging. Indeed, depolarisation property (for linearly or circularly polarized incident light) is linked to the scattering regime that is modified by the presence of tumoral tissues. However these techniques are still limited in scope, at least regarding in vivo investigations because accessing deeper organs requires to incorporate a fiber lead which compromises or complicates the polarization state of the light used. Indeed fiber-based systems using single-mode fibers induce disturbances in the measured polarization due to external factors, such as temperature and mechanical stress.

In 2015 Manhas et al. [9] reported for the first time a full 4x4 Mueller polarimeter for endoscopic applications, i.e. the measurement of diattenuation, retardance and depolarization. Their device is based on: 1) tunable ferroelectric liquid crystals to measure a Mueller matrix within 70 ms, 2) a miniaturized switchable mirror, used sequentially to measure light coming either from the fiber only or light that travelled through the fiber and the medium. Such a set-up can perform polarimetric measurement of the fiber to be subsequently used to compensate the distortion of the polarimetric signal and retrieve the polarimetric response of the sample. Nevertheless this innovative set-up has not been able to overcome two issues: 1) the slow speed of Mueller matrix acquisition that is not compatible with the demands of modern fast laser-scanning imaging systems, 2) the non-instantaneous polarimetric measurement of the fiber and the medium through the fiber, which is a major problem for endoscopy if the fiber is handled during measurement.

This paper aims to address both issues: 1) a set-up is presented, based on spectral coding of polarization (or channeled spectrum polarimetry) performed by passive elements such as birefringent plates. The speed of a Mueller matrix acquisition is then related to the speed of running the spectrometer camera, i.e. 70 kHz (14 μs) in this paper. 2) Advantageously, the polarimetric response of the fiber is measured at the same time as the polarimetric response of the medium through the fiber. This is achieved by separating the two signals thanks to coherence gating principle in an interferometric device illuminated by a broadband source.

This study focuses on the above technical achievements which must be validated before exploring and imaging the specific features of scattering biological samples.

The paper is organized as follows: first the experimental set-up and the procedure to retrieve the sample Mueller matrix are described. Then the systematic errors associated to this device are evaluated. Finally polarimetric measurements of different non-scattering media while deliberately introducing fiber disturbances are presented in order to demonstrate the insensitivity of the polarimetric measurement to fiber disturbances.

2. Experimental set-up and theory

The instant Mueller polarimeter is described in Fig. 1. It is composed of a Super Luminescent Diode (SLD), a passive polarization Coding block (C) made of a linear Polarizer (P1) and two Retarders (R1 and R2), a Non Polarizing Beam Splitter cube (NPBS), two single mode fibers, a 50% Partial Reflector (PaR), a passive polarization Decoding block (D) made of two retarders (R3, R4) and a linear polarizer (P2), and a spectrometer. Light is provided by a broadband superluminescent diode (Superlum S840-B-1-20) centered at 840 nm with 50 nm spectrum width determining a 14 μm-coherence length. The SLD is used in low power mode and the optical power is inferior to 1 mW. The retarders are YVO4 plates (Hg Optronics) with the optical axis set parallel to their surfaces. Their thicknesses are e = 0.4 mm for the Coding block and 5e = 2 mm for the Decoding block. The choice of the thicknesses is crucial for data reduction and conditioning of the system, as shown in [10,11]. The polarizers P1 and P2 used in the Coding and Decoding blocks and the polarizers RLP used for the calibration have a contrast superior to 100,000:1 (Codixx colorPol IR 1100 BC4 CW02). The cube beamsplitter (Thorlabs@ BS011) is a non-polarizing cube. All the above optical elements have a 700 nm −1100 nm antireflection coating. A custom-built spectrometer employs a linear CCD array camera working at 70 kHz with 2048 pixels (Basler spL2048-70km). Typically, the spectrum of the source covers 1100 pixels of the camera. The reference and the probing arms contain 1m-single mode fiber patch cables with a 3 mm diameter protective jacket (Thorlabs P1-780A-FC-1).

Fig. 1 Block diagram of the Mueller polarimeter operating in reflection. SLD: super luminescent diode, C: passive polarization state coding block, D: passive polarization state decoding block, NPBS: non-polarizing beam splitter cube, RM: reference mirror, PaR: 50% partial reflector, PrM: probe mirror. Cube faces are numbered from ① to ④. In the paper, the scattering sample will be replaced by a specific medium, such as a linear birefringence or a linear diattenuator, with the mirror PrM to reflect light to the probe fiber. For the Coding and Decoding blocks, P1, P2: linear polarizers are crossed, R1, R2, R3, R4: YVO4 retarder oriented respectively at 45°, 0°, 0° and 45° according to the orientation of P1.

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2.1. Overall strategy to retrieve the polarimetric properties of the object under test

In order to separate simultaneously the polarization information of the probe fiber and that of the object under test, the principle of coherence gating is used by interfering light reflected from a partial reflector (PaR) placed above the sample with light reflected by the reference mirror (RM).

Light from PaR is only sensitive to the probe fiber, while light reflected by the probe mirror (PrM) behind the sample, is sensitive to both the probe fiber and the sample. Due to the short coherence length of the source, the probe mirror PrM is set at a distance so that light from PrM does not interfere with light from the reference arm RM.

To measure only the polarimetric properties of the sample, knowledge of the two Mueller matrices is necessary, associated to the probe fiber, MPaR, and associated to the combination of probe fiber and sample, MPrM. Then the insensitivity to the fiber is obtained by calculating the product MPaR⁻¹ MPrM.

In section 2.2 we will obtain an expression for the channeled spectrum at the interferometer output, depending on polarimetric properties of both fiber and sample. Then sections 2.3 and 2.4 will provide the transformation matrices in order to retrieve Mueller matrices from the channeled spectrum. Lastly section 2.5 will deal with the calibration steps in order to isolate the matrices MPaR and MPrM and will demonstrate that the product MPaR⁻¹ MPrM is only sensitive to polarimetric properties of the sample.

2.2. Mathematical expression of the channeled spectrum

The channeled spectrum I(ν) can be expressed as the sum of two contributions I = IDC + Ainterference, where the interference term Ainterference is written as follows:

A_{interference (ν, t)} = E_{PaR (ν, t)}^{*} E_{ref (ν)} E x p [i L \frac{2 π}{c} ν] + E_{PaR (ν, t)} E_{ref (ν)}^{*} E x p [- i L \frac{2 π}{c} ν],

for which EPaR and Eref correspond respectively to the electromagnetic field reflected by the partial reflector PaR and by the reference mirror RM. L is the Optical Path Difference (OPD) between the reference arm length and the path length counted to and back from PaR.

The probe fiber represents the fiber of the endoscope that is subject to twists and movement. The variable t in (1) displays the time dependence of the polarization properties of the probe fiber that are modified by handling.

By using the Jones formalism, fields EPaR and Eref are related to the input field Ein before the coding block thanks to

E_{ref} = J_{D} J_{ref} J_{C} E_{in},

E_{PaR (t)} = J_{D} J_{PaR (t)} J_{C} E_{in},

where JC and JD are the Jones matrices of the Coding and Decoding blocks respectively, Jref that of {the cube from sides ① to ② + the reference fiber + the cube from ② to ④} (Fig. 1). The matrix JPaR is related to the path {the cube from sides ① to ③ + the probe fiber + the cube from ③ to ④}. As it is always possible to calculate Mueller matrices from Jones matrices (Appendix A), let Mref and MPaR(t) be the Mueller matrices corresponding to the Jones matrices Jref and JPaR(t).

Therefore Ainterference depends on the polarimetric properties of the reference fiber (through Mref) and the probe fiber (through MPaR).

Moreover the DC component IDC from the channeled spectrum I(ν) can be written as follows:

I_{DC (ν, t)} = I_{ref (ν)} + I_{PaR (ν, t)} + I_{PrM (ν, t)},

with Iref, IPaR, IPrM corresponding respectively to the intensity of light reflected on the reference mirror RM, on the partial reflector PaR and on the probe mirror PrM (transmitted through the partial reflector PaR).

By using the Mueller-Stokes formalism, intensities Iref, IPaR, IPrM are related respectively to the first element of the Stokes vectors Sref, SPaR, SPrM written as follows

S_{ref} = M_{D} M_{ref} M_{C} S_{in},

S_{P a R (t)} = M_{D} M_{P a R (t)} M_{C} S_{i n},

S_{PrM (t)} = M_{D} M_{PrM (t)} M_{C} S_{in},

where Sin is the Stokes vector corresponds to the input field Ein and the Mueller matrices MD, MC, Mref correspond to the Jones matrices JD, JC and Jref. Therefore IDC depends on the polarimetric properties of the reference fiber (through Mref), the probe fiber (through MPaR) and on the combination of {probe fiber + medium} through MPrM.

Due to the Coding and Decoding blocks, the Fourier transform of the channeled spectrum I(ν) then creates two structures: 1) 13 peaks (indexed from 0 to 12) in real and imaginary parts that are related to the Fourier transform of IDC. 2) 25 complex peaks (indexed from −12 to + 12) that are related to the Fourier transform of Ainterference and whose positions depend on the value L, adjusted in order to separate the two peak structures (Fig. 2). The amplitudes and the relative phases of these 25 peaks do not vary according to L.

Fig. 2 Simulation of the Fourier transform of the channeled spectrum I(ν) according to the OPD L measured between the reference arm length and the path length measured from the Partial Reflector (PaR). By adjusting L, it is possible to measure simultaneously the Fourier transform of IDC, FT{IDC}, and of Ainterference, FT{Ainterference} (Visualization 1).

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In previous work [10], we have demonstrated how to retrieve Mueller matrices from a transformation matrix based on the real and imaginary parts of the Fourier transformation of the electrical signal proportional to the channeled spectrum. This calculation is detailed in section 2.3. A similar strategy is developed in this paper to deal with modulations observed as a result of interferences. The new transformation matrix associated to the interferences components is detailed in section 2.4.

2.3. Mueller Matrix from DC components

The signal, IDC, given by the spectrometer is periodic, and it can be expressed as

I_{D C (ν, t)} = \sum_{m = 0}^{12} [a_{m (t)} \cos (m \frac{2 π Δ n e}{c} ν) + b_{m (t)} \sin (m \frac{2 π Δ n e}{c} ν)],

where ∆n is the birefringence of the material used in the plates of the Coding/Decoding blocks, e is the thickness of the retarder plate in the Coding block, and a_m and b_m are linear combinations of Mueller components m_ij, . Because of the particular set of thicknesses (e,e,5e,5e) of the four retarder plates, a Fourier transformation of the signal leads to 12 complex peaks (12 peaks in real part and 12 peaks in imaginary part) whose magnitudes are expressed as linear combinations of Mueller components, plus a peak at zero frequency, that can be used to determine the 16 Mueller components (Table 1).

Table 1. Magnitude of real and imaginary peaks according to m_ij coefficients. The time dependence in coefficients is omitted to simplify the notation.

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To retrieve the m_ij coefficients, a 25-dimension vector VDC, whose components are the magnitude of the peaks, is defined and the m_ij are put in a 16-dimension vector Xm. From Table 1, the $25 \times 16$ dimension transformation matrix P can be found that satisfies the equation VDC = P Xm and the measurement of VDC provides the m_ij coefficients by the relationship Xm = (P^T P)⁻¹ P^T VDC.

In order to simplify the expressions in the next sections, we call FDC, the function that permits to retrieve the Mueller matrix from the Fourier transform of the spectrum FT{IDC}, and FDC⁻¹, the inverse function as follows

F_{DC} [F T {I_{D C}}] = α \bar{M},

F_{DC}^{- 1} [α \bar{M}] = F T {I_{D C}}

where

\bar{M}

is the normalized Mueller matrix by m₀₀, and α is a coefficient depending on the transmittance of non-polarized light by the material of the sample (equal to 1/2 for a pure linear polarizer), the light intensity and the numerical calculation of the discrete Fourier transform of the signal.

2.4. Mueller Matrix from interferences

The signal, Ainterference, can be expressed as

A_{interference (ν, t)} = \sum_{m = - 12}^{12} c_{m (t)} E x p [i \frac{2 π}{c} ν (L + m \frac{Δ n e}{c})] + C C,

where CC stands for Complex Conjugate. A Fourier transformation of the signal in (11) leads to 25 peaks whose complex values are expressed as r_j and f_j coefficients of the Jones matrices Jref and JPaR (Table 2).

Table 2. Complex values of the peaks according to r_j coefficients of $J_{r e f} = [\begin{matrix} r 1 & r 2 \\ r 3 & r 4 \end{matrix}]$ and f_j coefficients of $J_{P a R} = [\begin{matrix} f 1 & f 2 \\ f 3 & f 4 \end{matrix}]$ depending on the probe fiber. The time dependence in f_j coefficients is omitted to simplify the expressions.

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Because the r_j coefficients are determined previously through a calibration step, only the f_i coefficients now have to be retrieved.

To retrieve the f_j coefficients, a 25-dimension vector Vinterf, whose components are the complex values of the peaks, is defined and the f_j are put in a 4-dimension vector Xf. From Table 2, the $25 \times 4$ dimension transformation matrix Q_r depending on the coefficients r_j can be found for which Vinterf = Q_r Xf and the measurement of Vinterf provide the f_j coefficients by the relationship: Xf = (Q_r^T Q_r)⁻¹ Q_r^T Vinterf.

Thus Xf allows to obtain the Jones matrix JPaR, whence the Mueller-Jones matrix MPaR associated to JPaR (Appendix A). By nature, MPaR has no depolarization.

Lastly we call Finterf the function that permits to retrieve the Mueller matrix from the Fourier transform of the spectrum FT{ Ainterference }, as follows

F_{interf (J_{ref})} [F T {A_{interference}}] = {\bar{M}}_{P a R},

where

\bar{M} PaR

is the normalized Mueller matrix through m₀₀.

Let us note that since the conditioning of the matrix Q_r depends on the Jones matrix elements r_j of the reference fiber, the propagation of noise could be amplified by calculating its pseudo-inverse (Q_r^T Q_r)⁻¹ Q_r^T. A solution would be to add a linear polarizer in the reference arm before the reference fiber. In this case, whatever the properties of the reference fiber, J_ref would be equal to the Jones matrix of the linear polarizer. Therefore the orientation of the linear polarizer could be chosen in order to optimize the transformation matrix Q_r according to the conditioning.

2.5. Mathematical treatment to measure the polarimetric properties of the medium

It is not possible to measure directly the Mueller matrix $\bar{M} P a R$ associated to the probe fiber and the matrix $\bar{M} PrM$ associated to {probe fiber + medium}. The matrix MPaR will be measured from the interference terms, relying on the calibrated measurement of the reference fiber. The matrix MPrM will be measured from DC terms, after subtracting MPaR and the reference fiber. It will be shown that the product MPaR⁻¹ MPrM is only sensitive to the polarimetric properties of the sample.

2.5.1. Measurement of the normalized matrices $\bar{M} Par$ and $\bar{M} PrM$

For $\bar{M} Par$ , Iref must be measured in a calibration step prior to the start of measurement of the sample. By calculating FDC[FT{Iref}], the Mueller matrix $\bar{M} ref$ is obtained. Then the formalism detailed in Appendix B permits to calculate the Jones matrix Jref associated to $\bar{M} ref$ and we can therefore apply $F_{interf (J_{ref})}$ on Ainterference to obtain $\bar{M} Par$ (see Eq. (12)).

For $\bar{M} PrM$ , we need to measure FT{IPrM} by subtracting FT{Iref} and FT{IPar} from FT{IDC} (see Eq. (4)) and calculate FDC[FT{IPrM}]. FT{IPar} derives from $\bar{M} PaR$ measured above, thanks to the following equation

F T {I_{P a R (t)}} = α_{P a R} F_{DC}^{- 1} [{\bar{M}}_{P a R (t)}],

where α_PaR is a constant depending on several parameters of the set-up (reflectance of the partial reflector, diattenuation of the cube). Thus IPaR must be measured in a previous calibration step to calculate

F D C [F T {I P a R}] = αPaR \bar{M} P a R

and obtain α_PaR.

Figure 3 sums up the different steps to obtain the Mueller matrices $\bar{M} P a R$ and $\bar{M} PrM$ .

Fig. 3 Diagram of steps to obtain the two matrices $\bar{M} P a R$ and $\bar{M} PrM$ . Iref0 and IPar0 are measured before measuring the sample.

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2.5.2. Measurement of the Mueller matrix of the medium $\bar{M} medium$

The matrix $\bar{M} Par$ can be expressed by two Mueller matrices $\bar{M} in$ and $\bar{M} out$ corresponding to the input forward path {the cube from ① to ③ + the probe fiber} and to the output backward path {the probe fiber + the cube from ③ to ④}

{\bar{M}}_{PaR (t)} = {\bar{M}}_{o u t (t)} {\bar{M}}_{i n (t)} .

The measurement of

\bar{M} Par

allows to evaluate polarization effects due to the probe fiber. The matrix

\bar{M} PrM

can be written according to the Mueller matrix of the medium

\bar{M} medium

and to the input and output paths, as follows

{\bar{M}}_{PrM (t)} = {\bar{M}}_{o u t (t)} {\bar{M}}_{m e d i u m} {\bar{M}}_{i n (t)} .

Knowing that we have, for two matrices A and B, the equality

{(A . B)}^{- 1} = B^{- 1} . A^{- 1}

, using (14) and (15) the following product can be written as [12]:

{[{\bar{M}}_{PaR (t)}^{}]}^{- 1} {\bar{M}}_{PrM (t)}^{} = {[{\bar{M}}_{in (t)}^{}]}^{- 1} {\bar{M}}_{medium} {\bar{M}}_{in (t)}

Thus this product is only sensitive to the polarimetric properties of 1) the medium

\bar{M} medium

and 2) the cube in transmission and the probe fiber through

\bar{M} in

. If we consider that the single mode fiber of the probing arm and the non polarizing beamsplitter cube present only retardance effects in transmission, then the matrix

\bar{M} in

is a pure retarder, which means that

\bar{M} in

is a rotation matrix. Thus, the polarization properties of the product

\bar{M} {Par}^{- 1} \bar{M} PrM

are the same as those of

\bar{M} medium

, except for the absolute axis orientation.

In fact an absolute measurement of the Mueller matrix of the medium $\bar{M} medium$ is not possible in an endoscopic configuration. Indeed the measurement of the polarization properties of the hand probe fiber is done after a round-trip in the fiber and, in this case, the absolute axis orientation of $\bar{M} medium$ cannot be known.

Let us note that for a perfect beamsplitter cube (without diattenuation and retardance both in transmission and reflection) Min can be expressed according to Mout and it is not necessary to calculate the inverse of MPar [9], which could reduce the propagation of noise if the matrix MPar is not well-conditioned.

3. Systematic errors

The model presented above does not take into account the systematic errors associated with the elements of this device, i.e. the spectrometer, the Coding and Decoding blocks, the non-polarizing beamsplitter cube (Table 3) and the fiber.

Table 3. Transmission measurements of the non-polarizing beam splitter cube from face ① of the cube to ③ and from ② to ④. Reflection measurements from ① to ② and from ③ to ④. R should be equal to 0 in transmission and π in reflection for a non-retarding medium. Precision is calculated with four different measurements after calibration.

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3.1. Calibration of frequency linearity

The light intensity measured by the spectrometer is a signal that is usually not uniformly sampled according to the optical frequency ν . In order to linearize the spectrometer we have employed a method from the practice of OCT [13]. It consists in measuring the interference term Ainterference without the Coding and Decoding blocks, followed by a procedure to remove the signal chirp due to nonlinearity. This brings the width of the peak in the Fourier Domain to the theoretical value of the coherence length evaluated by using the spectrum width. This procedure leads to higher amplitude and improved resolution peaks obtained by the Fourier transformation of the channeled spectrum I(ν), when using the Coding and Decoding blocks (Fig. 4).

Fig. 4 Experimental amplitude of the Fourier transform of the channeled spectrum I(ν) without frequency calibration (a) and with frequency calibration (b).

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3.2. Thickness errors of YVO4 retarder plates and response of the detection

As was shown in previous work [10,11], the accuracy of the thickness of the retarder plates in the Coding and Decoding plates and the thermal expansion of the retarder plates result in a variation between imaginary and real parts of Fourier peaks. The calibration method applied in [14] used two reference linear polarizers (RLP) placed in front of faces ① and ④ of the cube (not shown in Fig. 1). Thus the polarimetric response of the system depends on the orientation of the two RLPs whatever the polarimetric features of the cube beamsplitter, the fibers and the sample. It results in a specific set of equations that can be easily inverted with an adequate choice of the relative orientation of the RLPs. By comparing the amplitude and the phase of the theoretical peaks with those of the experimental peaks [15], we can measure and numerically compensate: 1) the response of the spectrometer that corresponds to the roll-off in the peak amplitudes in Fourier domain due to the finite spectral width of each wavelength channel (Fig. 5 and 2) the thickness errors that are written in the form of phase errors ϕ₂, ϕ₃, and ϕ₄ corresponding respectively to the second, third, and fourth wave plate, the first wave plate setting the reference thickness. The matrix P, used for the measurement of Mueller matrices from DC components, must be modified due to the new relationships between the complex peaks and the Mueller coefficients in the Fourier domain. These relationships can be found in [15].

Fig. 5 Experimental response (arbitrary units) of the detection by comparing the amplitude of the experimental peaks to the theoretical peaks. This curve corresponds to the roll-off due to the spectrometer.

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In the same way, the roll-off and the thickness of Yvo4 plates must be measured and compensated for interference terms. To do so, the matrix Q_r, used for the measurement of Mueller matrices from interference components, must be modified because the interference term Ainterference becomes

A_{interference (ν, t)} = \sum_{m = - 12}^{12} d_{m (t)} E x p [i \frac{2 π}{c} ν (L + m (\frac{Δ n e}{c} + ϕ_{w})) + i φ_{m}] + C C,

where φ_m is an additional phase, which depends on ϕ_p (p = 2,3,4) and d_m is a function that depends on the Jones coefficients, f_j. The expressions of φ_m and d_m are given in Table 4 of Appendix C. Moreover as a sinusoidal signal can be interpreted as a sine, or cosine function according to its position in the analysis window, an additional phase, ϕ_w, has to be considered and measured.

Table 4. Complex values of the peaks according to r_j coefficients of $J_{r e f} = [\begin{matrix} r 1 & r 2 \\ r 3 & r 4 \end{matrix}]$ and f_j coefficients of $J_{P a R} = [\begin{matrix} f 1 & f 2 \\ f 3 & f 4 \end{matrix}]$ depending on the probe fiber and phase errors of the YVO4 retarder plates.

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3.3. Nonpolarizing beamsplitter cube

In Eq. (16), it was shown that the polarization properties of the product $\bar{M} {Par}^{- 1} \bar{M} PrM$ are the same as those of the $\bar{M} medium$ if the matrix $\bar{M} in$ corresponding to the input forward path {the cube from ① to ③ + the probe fiber} is considered as a pure retarder.

In order to verify this assumption, the full polarimetric response of the beamsplitter cube has been measured (Table 3) before inserting it into the set-up. Matrix decomposition using Lu and Chipman decomposition [16] was used to obtain the polarimetric parameters R (retardance), D (diattenuation), knowing that D = 1 for a perfect polarizer and D = 0 for a non-polarizing medium, and P_d (depolarization index) of the cube beamsplitter for which P_d = 0 for a perfectly depolarizing medium and P_d = 1 for a non-depolarizing medium.

These results show that the cube in transmission can be considered as a perfect non polarizing cube because the diattenuation of the transmission is close to zero, which validates the simplification of the matrix $\bar{M} in$ as a pure retarder.

Thus, the properties of the product $\bar{M} PaR {(t)}^{- 1} \bar{M} PrM (t)$ are the same as those of the Mmedium, except for the absolute axis orientation and this result is true whatever the diattenuation of the cube in reflection and whatever the retardance of the cube in reflection and transmission.

3.4. Optical fiber

In this paper, the reference fiber remains static during the measurement and we have assumed that its polarimetric properties are unchanged (unlike the handled probe fiber). In order to verify this assumption, the Mueller matrix corresponding to the reference fiber has been measured for one minute in Fig. 6. The variation of the retardance R = 1.212 ± 0.003 rad (mean ± 2 standard deviation) is inferior to 1% and corresponds to noise. Mean values and standard deviations are calculated from measured values in time. All polarimetric values in the paper have been calculated in the same way.

Fig. 6 Experimental retardance of the reference fiber in time.

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However, a depolarization effect equal to Pd = 0.941 ± 0.004 has been measured, certainly coming from the dispersion of the fiber. Indeed when the snapshot Mueller polarimeter is used to measure Mueller matrices, we assume that all the optical elements (except for the retarder plates of the Coding and Decoding blocks) are achromatic, i.e. their polarimetric properties are the same within the 50 nm-spectral bandwidth of the source. Modeling the fiber as a pure retarder is therefore a simplification and can decrease the accuracy of the measurement of the medium.

4. Results and discussion

During endoscopy, the fiber link is handled, which induces birefringence. The polarimetric measurements of different media while handling the probe fiber are presented here in order to prove the tolerance of the set-up to disturbances due to the fiber birefringence.

In order to study the ability of this system to compensate disturbances, polarimetric measurements have been carried out in time and recorded every second according to the following protocol: the fiber is untouched, then it is laterally shaken fast by hand (between 3 and 5 periods per second with an amplitude of 5 to 10 cm), and finally the fiber is left untouched.

The media used as samples are 1) “ vacuum”, 2) a linear retarder and 3) a linear polarizer. Due to the reflection configuration without scattering, the backward path followed by light is exactly the same as the forward path, which means that it is only possible to measure linear retarders and linear diattenuators. In this preliminary paper, scattering samples are not studied. The accuracy (mean value) and the uncertainties (twice the standard deviation) of the polarimetric values are calculated in the specific protocol presented here. The relevance of these values for biological samples would depend on the conditions of in vivo explorations (amplitude of the movement of the fiber, type of deformation such as bending or twist) and specific objectives of the measurement (absolute or relative measurement).

4.1. “Vacuum” sample

In Fig. 1, the medium is removed to create a “vacuum” sample. In order to assess the ability of the device to provide measurements unaffected by the probe fiber, the retardance of the “vacuum” was measured in time while handling the fiber. Figure 7(b) shows the evolution of the experimental retardance of the fiber during handling. This is measured from $\bar{M} PaR (t)$ . The blue line in Fig. 7(a) shows the retardance of the medium acquired using the polarimetric properties of the probe fiber $\bar{M} PaR (t 0)$ obtained in a previous measurement. The blue line retardance is then measured from the product $\bar{M} PaR {(t 0)}^{- 1} \bar{M} PrM (t)$ . The red line in Fig. 7(a) shows the retardance of the medium acquired using the polarimetric properties of the fiber $\bar{M} PaR (t)$ measured simultaneously. This red line retardance is then measured from the product $\bar{M} PaR {(t)}^{- 1} \bar{M} PrM (t)$ . For an instantaneous correction of the fiber effect (red line), the “vacuum” retardance is equal to 0.04 ± 0.04 rad, comparing to 0.1 ± 0.1 rad for a correction before the handling of the fiber.

Fig. 7 Experimental retardance of the “vacuum” sample (a) and the probe fiber (b) in time. Red line: retardance of the medium by using the matrix MPaR(t) measured from the interference components simultaneously. Blue line: retardance of the medium by using the matrix MPaR(t0) measured once previously. Diattenuation value of the “vacuum” equals to 0.012 ± 0.004, depolarization index equal to 0.937 ± 0.006 for $\bar{M} PaR {(t)}^{- 1} \bar{M} PrM (t)$ and $\bar{M} PaR {(t 0)}^{- 1} \bar{M} PrM (t)$ .

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Consequently these curves show that the measurement of the material polarimetric properties is less affected thanks to simultaneous correction of the fiber effect. Nevertheless, the correction is not perfect, which means that the Mueller matrices measured from DC components and interferometric components are not rigorously the same.

Lastly depolarization has been measured with a depolarization index equal to 0.937 ± 0.006 for $\bar{M} PaR {(t)}^{- 1} \bar{M} PrM (t)$ and $\bar{M} PaR {(t 0)}^{- 1} \bar{M} PrM (t)$ . This is due to the matrix $\bar{M} PrM (t)$ that contains the depolarization effect of the probe fiber.

4.2. Double-pass quarter wave plate

In this section, the studied medium is a zero-order quarter wave plate at 830 nm (Thorlabs). Figure 8(b) shows the evolution of the experimental retardance of the fiber during handling.

Fig. 8 Experimental retardance of a double-pass quarter wave plate (a) and the sample fiber (b) in time. Red line: retardance of the material by using the matrix MPaR(t) measured from interference components simultaneously. Blue line: retardance of the medium by using the matrix MPaR(t0) measured prior to handling. Diattenuation value of the material equals to 0.071 ± 0.002, depolarization index Pd equals to 0.876 ± 0.007 for $\bar{M} PaR {(t)}^{- 1} \bar{M} PrM (t)$ and $\bar{M} PaR {(t 0)}^{- 1} \bar{M} PrM (t)$ . Pd is inferior to one obtained for “vacuum” due to the chromaticity of the quarter wave plate.

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We have measured, in time, the retardance evolution of the quarter wave plate in a double-pass configuration by using $\bar{M} PaR (t 0)$ and $\bar{M} PaR (t)$ as in 4.1. For an instantaneous correction of the fiber effect (red line in Fig. 8(a)), the birefringent medium retardance is equal to 3.05 ± 0.02 rad, comparing to 3.05 ± 0.04 rad for a correction before the handling of the fiber (blue line in Fig. 8(a)). There again, the influence of the probe fiber on the measurement of the material polarimetric properties is reduced by measuring the fiber polarimetric properties during handling. Moreover, the variation of the retardance in both cases is less significant than in 4.1. This result could be explained by the fact that the birefringent medium has a strong retardance, which diminishes fiber effects on the polarimetric measurement of the medium. Lastly, the mean value of the retardance should be π for a double-pass through an ideal zero-order quarter wave plate. This is not the case, because the quarter wave plate is for 830 nm and the spectrum source is centered at 840 nm. Due to this, the mean value of the retardance should be equal to 3.1 rad, which leads to a relative bias inferior to 2% by comparing it to the measured retardance (3.05 rad).

Moreover, the depolarization index Pd is measured and is equal to 0.876 ± 0.007, which is inferior to the Pd value obtained for the “vacuum” sample. This effect is due to the chromaticity of the wave plate and the broadband spectrum of the source that increase depolarization effects.

4.3. Linear polarizer

Let us now evaluate the linear polarizer (Codixx colorPol IR 1100). Figure 9(b) shows the evolution of the experimental diattenuation of the fiber during handling.

Fig. 9 Experimental diattenuation of a linear polarizer (a) and of the probe fiber (b) in time. Red line: diattenuation of the medium by using the matrix MPaR(t) measured from interference simultaneously. Blue dots: diattenuation of the material by using the matrix MPaR(t0) measured prior to handling. Depolarization index of the material is equal to 0.95 ± 0.07 for $\bar{M} PaR {(t)}^{- 1} \bar{M} PrM (t)$ and $\bar{M} PaR {(t 0)}^{- 1} \bar{M} PrM (t)$ .

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Here the curves (red line and blue dots in Fig. 9(a)) are exactly the same. In order to understand this result, we can calculate the product $\bar{M} PaR {(t 0)}^{- 1} \bar{M} PrM (t)$ from (14) and (15) as follows

{[{\bar{M}}_{PaR (t 0)}^{}]}^{- 1} {\bar{M}}_{PrM (t)}^{} = {[{\bar{M}}_{in (t0)}^{}]}^{- 1} {[{\bar{M}}_{out (t0)}^{}]}^{- 1} [{\bar{M}}_{out (t)}^{}] {\bar{M}}_{medium} {\bar{M}}_{in (t)} .

For a diattenuation of the cube in reflection,

\bar{M} out

can be written as a product between a diattenuation matrix of the cube and a pure retarder due to the cube and the probe fiber, as follows

{\bar{M}}_{o u t}^{} = {\bar{M}}_{D}^{(c u b e)} {\bar{M}}_{R}^{(o u t)} .

Hence (18) becomes:

{[{\bar{M}}_{PaR (t 0)}^{}]}^{- 1} {\bar{M}}_{PrM (t)}^{} = {[{\bar{M}}_{i n (t0)}^{}]}^{- 1} {[{\bar{M}}_{R (t0)}^{(o u t)}]}^{- 1} [{\bar{M}}_{R (t)}^{(o u t)}] {\bar{M}}_{medium} {\bar{M}}_{i n (t)}^{} .

As we have retarder matrices, i.e. rotation matrices, before and after

\bar{M} medium

, the diattenuation value of the product

\bar{M} PaR {(t 0)}^{- 1} \bar{M} PrM (t)

is equal to the diattenuation value of

\bar{M} medium

, which is the value of the product

\bar{M} PaR {(t)}^{- 1} \bar{M} PrM (t)

.

Diattenuation is equal to 0.99 ± 0.03 presenting a more important variation during the handling of the fiber than in the other cases ( ± 0.004 for “vacuum”, ± 0.002 for quarter wave plate). The increased influence of the handled fiber comes from the fact that IPrM is half that obtained for the “vacuum” or the quarter wave plate experiments. In this case, it is more difficult to eliminate the contribution of the fiber to the DC components when the matrix $\bar{M} PrM$ is extracted (see Part 2.5.1). The reflectance of the partial reflector should then be chosen so that IPrM is not too inferior to IPaR.

This problem illustrates the trade-off concerning the choice of light quantity coming from the reference arm and the partial mirror. Indeed, the intensity measured by the detector is the sum of three contributions: the sample, the reference arm and the calibration mirror. The dynamic range of the detector must then be shared between these contributions. In order not to decrease the sensitivity for the actual sample measurement, the intensity from the sample must not be too weak according to the two other contributions. A variable pinhole is then placed in the reference arm in order to control Iref and adjust Iref = IPaR, which maximizes the interference contrast. Moreover the reflectance of the partial mirror should be chosen so that IPaR is inferior to IPrM.

5. Conclusion

In this paper the measurement of complete Mueller matrices of a material through a single mode fiber, was demonstrated and validated on several samples. A special configuration is used to attenuate the birefringence fluctuation in the fiber lead. Measuring simultaneously the polarimetric response of the fiber and of the material enables to compensate the disturbances of the fiber. The polarimetric measurement is achieved at a high repetition rate, allowed by a fast reading time of the camera. This polarimeter could be easily developed with a swept source and a photodiode to gain in simplicity and rapidity [14].

A large part of this paper deals with the calibration process that ensures an effective extraction of the polarimetric information of the medium. The birefringence of the cube used to divert light to the object and back and its diattenuation in reflection have been taken into account. Some residual effects of the fiber on the sample measurement could be due to the depolarization of the fiber, which has not been dealt with in this paper. This effect could be decreased by using a narrower bandwidth and appropriate thicknesses for the coding and decoding plates. Particular attention must also be paid to the balance choice of the reflectance of the partial reflector (PaR) according to the light coming from the material.

This paper presents a theoretical frame and calibration steps for a procedure to deliver full Mueller matrix measurements through a single mode fiber, connecting the core of an imaging instrument to its probe head, remotely inserted within the patient. These preliminary results are obtained with non-scattering media. The following steps will be to image scattering samples and refine the process in relation to a specific medium, so that polarimetric contrasts are effectively enhanced to suit medical needs.

6 Appendix A: Mueller (Jones) matrix related to Jones matrix [17]

Let J be a Jones matrix. The Mueller matrix M associated to J is given by the following expression:

M = A (J \otimes J_{}^{*}) A^{- 1},

where

\otimes

stands for the Kronecker product and A is the matrix:

A = [\begin{matrix} 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & - 1 \\ 0 & 1 & 1 & 0 \\ 0 & i & - i & 0 \end{matrix}] .

The Mueller matrix M is called a Mueller-Jones matrix because this Mueller matrix has no depolarization.

7 Appendix B: Jones matrix related to Mueller-Jones [18]

Let M be a Mueller matrix and m_i_,_j its 16 elements. We can define a matrix N written as follows:

N = \sum_{i = 0}^{3} \sum_{j = 0}^{3} m_{i, j} σ_{i} \otimes σ_{j},

where

σ_{i}

are the Pauli matrices and

\otimes

is the Kronecker product.

The N matrix has 4 eigenvalues λ_i and 4 eigenvectors W_i written according to the elements of Jones matrices J⁽ⁱ⁾.

W_{i} \propto [\begin{matrix} J_{11}^{(i)} \\ J_{12}^{(i)} \\ J_{21}^{(i)} \\ J_{22}^{(i)} \end{matrix}] .

If M is a Mueller-Jones matrix, i.e. without depolarization, N has only one non-zero eigenvalue and the elements of the corresponding eigenvector are the elements of the Jones matrix.

If M is not a Mueller-Jones matrix, due to noise or depolarization, the only eigenvector to be considered should be that corresponding to the highest eigenvalue.

8 Appendix C: Complex values of interference peaks

Acknowledgments

The authors thank the laboratory LSOL and in particular G. Leroux for technical assistance in the building of the coding and decoding blocks. S. Rivet acknowledges the European Research Council under a Marie-Curie Intra-European Fellowship for Career Development, No. 625509. A. Bradu and A. Gh. Podoleanu acknowledge the support of the European Research Council (ERC) (http://erc.europa.eu) COGATIMABIO 249889. A. Podoleanu is also supported by the NIHR Biomedical Research Centre at Moorfields Eye Hospital NHS Foundation Trust and the UCL Institute of Ophthalmology.

References and links

1. R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med. 7(11), 1245–1248 (2001). [CrossRef] [PubMed]

2. S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7(3), 329–340 (2002). [CrossRef] [PubMed]

3. N. Ghosh and I. A. Vitkin, “Tissue polarimetry: concepts, challenges, applications, and outlook,” J. Biomed. Opt. 16(11), 110801 (2011). [CrossRef] [PubMed]

4. M. Dubreuil, P. Babilotte, L. Martin, D. Sevrain, S. Rivet, Y. Le Grand, G. Le Brun, B. Turlin, and B. Le Jeune, “Mueller matrix polarimetry for improved liver fibrosis diagnosis,” Opt. Lett. 37(6), 1061–1063 (2012). [CrossRef] [PubMed]

5. A. Pierangelo, A. Nazac, A. Benali, P. Validire, H. Cohen, T. Novikova, B. H. Ibrahim, S. Manhas, C. Fallet, M.-R. Antonelli, and A. D. Martino, “Polarimetric imaging of uterine cervix: a case study,” Opt. Express 21(12), 14120–14130 (2013). [CrossRef] [PubMed]

6. B. H. Park, C. Saxer, S. M. Srinivas, J. S. Nelson, and J. F. de Boer, “In vivo burn depth determination by high-speed fiber-based polarization sensitive optical coherence tomography,” J. Biomed. Opt. 6(4), 474–479 (2001). [CrossRef] [PubMed]

7. B. Cense, T. C. Chen, B. H. Park, M. C. Pierce, and J. F. de Boer, “Thickness and Birefringence of Healthy Retinal Nerve Fiber Layer Tissue Measured with Polarization-Sensitive Optical Coherence Tomography,” Invest. Ophthalmol. Vis. Sci. 45(8), 2606–2612 (2004). [CrossRef] [PubMed]

8. A. Pierangelo, A. Benali, M.-R. Antonelli, T. Novikova, P. Validire, B. Gayet, and A. De Martino, “Ex-vivo characterization of human colon cancer by Mueller polarimetric imaging,” Opt. Express 19(2), 1582–1593 (2011). [CrossRef] [PubMed]

9. S. Manhas, J. Vizet, S. Deby, J.-C. Vanel, P. Boito, M. Verdier, A. De Martino, and D. Pagnoux, “Demonstration of full 4×4 Mueller polarimetry through an optical fiber for endoscopic applications,” Opt. Express 23(3), 3047–3054 (2015). [CrossRef] [PubMed]

10. M. Dubreuil, S. Rivet, B. Le Jeune, and J. Cariou, “Snapshot Mueller matrix polarimeter by wavelength polarization coding,” Opt. Express 15(21), 13660–13668 (2007). [CrossRef] [PubMed]

11. P. Lemaillet, S. Rivet, and B. Le Jeune, “Optimization of a snapshot Mueller matrix polarimeter,” Opt. Lett. 33(2), 144–146 (2008). [CrossRef] [PubMed]

12. S. Makita, M. Yamanari, and Y. Yasuno, “Generalized Jones matrix optical coherence tomography: performance and local birefringence imaging,” Opt. Express 18(2), 854–876 (2010). [CrossRef] [PubMed]

13. Y. Yasuno, V. D. Madjarova, S. Makita, M. Akiba, A. Morosawa, C. Chong, T. Sakai, K.-P. Chan, M. Itoh, and T. Yatagai, “Three-dimensional and high-speed swept-source optical coherence tomography for in vivo investigation of human anterior eye segments,” Opt. Express 13(26), 10652–10664 (2005). [CrossRef] [PubMed]

14. A. Le Gratiet, S. Rivet, M. Dubreuil, and Y. Le Grand, “100 kHz Mueller polarimeter in reflection configuration,” Opt. Lett. 40(4), 645–648 (2015). [CrossRef] [PubMed]

15. M. Dubreuil, S. Rivet, B. Le Jeune, and J. Cariou, “Systematic errors specific to a snapshot Mueller matrix polarimeter,” Appl. Opt. 48(6), 1135–1142 (2009). [CrossRef] [PubMed]

16. S. Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13(5), 1106–1113 (1996). [CrossRef]

17. R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42(5), 293–297 (1982). [CrossRef]

18. S. R. Cloude, “Group theory and polarization algebra,” Optik (Stuttg.) 75(1), 26–36 (1986).

Peak	a_m	b_m
0	$16 m_{00} + 8 m_{02} - 8 m_{20} - 4 m_{22}$	0
1	$8 m_{01} - 4 m_{21}$	0
2	$- 4 m_{02} + 2 m_{22}$	$- 4 m_{03} + 2 m_{23}$
3	$2 m_{12}$	$- 2 m_{13}$
4	$- 4 m_{11}$	0
5	$- 8 m_{10} - 4 m_{12}$	0
6	$- 4 m_{11}$	0
7	$2 m_{12}$	$2 m_{13}$
8	$- m_{22} + m_{33}$	$m_{23} + m_{32}$
9	$2 m_{21}$	$- 2 m_{31}$
10	$4 m_{20} + 2 m_{22}$	$- 4 m_{30} - 2 m_{32}$
11	$2 m_{21}$	$- 2 m_{31}$
12	$- m_{22} - m_{33}$	$- m_{23} + m_{32}$

Peak	c_m
−12	$- r_{1} f_{4}^{*}$
−11	$r_{1} f_{3}^{} - r_{2} f_{4}^{}$
−10	$f_{3}^{} (2 r_{1} + r_{2}) + f_{4}^{} (2 r_{2} + r_{1})$
−9	$r_{1} f_{3}^{} - r_{2} f_{4}^{}$
−8	$- r_{2} f_{3}^{*}$
−7	$r_{1} f_{2}^{} - r_{3} f_{4}^{}$
−6	$- r_{1} f_{1}^{} + r_{2} f_{2}^{} + r_{3} f_{3}^{} - r_{4} f_{4}^{}$
−5	$- f_{1}^{} (2 r_{1} + r_{2}) - f_{2}^{} (2 r_{2} + r_{1}) + f_{3}^{} (2 r_{3} + r_{4}) + f_{4}^{} (2 r_{4} + r_{3})$
−4	$- r_{1} f_{1}^{} + r_{2} f_{2}^{} + r_{3} f_{3}^{} - r_{4} f_{4}^{}$
−3	$r_{2} f_{1}^{} - r_{4} f_{3}^{}$
−2	$f_{2}^{} (- 2 r_{1} + r_{3}) + f_{4}^{} (- 2 r_{3} + r_{1})$
−1	$f_{1}^{} (2 r_{1} - r_{3}) + f_{2}^{} (- 2 r_{2} + r_{4}) + f_{3}^{} (2 r_{3} - r_{1}) + f_{4}^{} (- 2 r_{4} + r_{2})$
0	$\begin{array}{l} f_{1}^{} (4 r_{1} + 2 r_{2} - 2 r_{3} - r_{4}) + f_{2}^{} (4 r_{2} + 2 r_{1} - 2 r_{4} - r_{3}) \\ + f_{3}^{} (4 r_{3} + 2 r_{4} - 2 r_{1} - r_{2}) + f_{4}^{} (4 r_{4} + 2 r_{3} - 2 r_{2} - r_{1}) \end{array}$
1	$f_{1}^{} (2 r_{1} - r_{3}) + f_{2}^{} (- 2 r_{2} + r_{4}) + f_{3}^{} (2 r_{3} - r_{1}) + f_{4}^{} (- 2 r_{4} + r_{2})$
2	$f_{1}^{} (- 2 r_{2} + r_{4}) + f_{3}^{} (2 r_{4} + r_{2})$
3	$r_{1} f_{2}^{} - r_{3} f_{4}^{}$
4	$- r_{1} f_{1}^{} + r_{2} f_{2}^{} + r_{3} f_{3}^{} - r_{4} f_{4}^{}$
5	$- f_{1}^{} (2 r_{1} + r_{2}) - f_{2}^{} (2 r_{2} + r_{1}) + f_{3}^{} (2 r_{3} + r_{4}) + f_{4}^{} (2 r_{4} + r_{3})$
6	$- r_{1} f_{1}^{} + r_{2} f_{2}^{} + r_{3} f_{3}^{} - r_{4} f_{4}^{}$
7	$r_{2} f_{1}^{} - r_{4} f_{3}^{}$
8	$- r_{3} f_{2}^{*}$
9	$r_{3} f_{1}^{} - r_{4} f_{2}^{}$
10	$f_{1}^{} (2 r_{3} + r_{4}) + f_{2}^{} (2 r_{4} + r_{3})$
11	$r_{3} f_{1}^{} - r_{4} f_{2}^{}$
12	$- r_{4} f_{1}^{*}$

	NPBS	R (rad)	D	Pd
transmission	from ① to ③	0.17 ± 0.02	0.03 ± 0.01	1.002 ± 0.009
transmission	from ② to ④	0.19 ± 0.02	0.04 ± 0.01	0.99 ± 0.01
reflection	from ① to ②	2.84 ± 0.04	0.06 ± 0.02	1.01 ± 0.02
reflection	from ③ to ④	2.41 ± 0.02	0.06 ± 0.02	1.00 ± 0.02

Peak	φ_m	d_m
−12	$- (ϕ_{2} + ϕ_{3} + ϕ_{4} + π)$	$r_{1} f_{4}^{*}$
−11	$- (ϕ_{3} + ϕ_{4})$	$r_{1} f_{3}^{} - r_{2} f_{4}^{}$
−10	$- (ϕ_{3} + ϕ_{4})$	$2 (r_{1} f_{3}^{} + r_{2} f_{4}^{}) + e^{i ϕ 2} r_{2} f_{3}^{} + e^{- i ϕ 2} r_{1} f_{4}^{}$
−9	$- (ϕ_{3} + ϕ_{4})$	$r_{1} f_{3}^{} - r_{2} f_{4}^{}$
−8	$ϕ_{2} - ϕ_{3} - ϕ_{4} + π$	$r_{2} f_{3}^{*}$
−7	$- (ϕ_{2} + ϕ_{4})$	$r_{1} f_{2}^{} - r_{3} f_{4}^{}$
−6	$- ϕ_{4}$	$- r_{1} f_{1}^{} + r_{2} f_{2}^{} + r_{3} f_{3}^{} - r_{4} f_{4}^{}$
−5	$- ϕ_{4}$	$2 (- r_{1} f_{1}^{} - r_{2} f_{2}^{} + r_{3} f_{3}^{} + r_{4} f_{4}^{}) + e^{i ϕ 2} (r_{4} f_{3}^{} - r_{2} f_{1}^{}) + e^{- i ϕ 2} (r_{3} f_{4}^{} - r_{1} f_{2}^{})$
−4	$- ϕ_{4}$	$- r_{1} f_{1}^{} + r_{2} f_{2}^{} + r_{3} f_{3}^{} - r_{4} f_{4}^{}$
−3	$ϕ_{2} - ϕ_{4}$	$r_{2} f_{1}^{} - r_{4} f_{3}^{}$
−2	$- ϕ_{2}$	$- 2 (r_{1} f_{2}^{} + r_{3} f_{4}^{}) + e^{i (ϕ 3 - ϕ 4)} r_{3} f_{2}^{} + e^{- i (ϕ 3 - ϕ 4)} r_{1} f_{4}^{}$
−1	0	$2 (r_{1} f_{1}^{} - r_{2} f_{2}^{} + r_{3} f_{3}^{} - r_{4} f_{4}^{}) + e^{i (ϕ 3 - ϕ 4)} (r_{4} f_{2}^{} - r_{3} f_{1}^{}) + e^{- i (ϕ 3 - ϕ 4)} (r_{2} f_{4}^{} - r_{1} f_{3}^{})$
0	$π$	$\begin{array}{l} - 4 (r_{1} f_{1}^{} + r_{2} f_{2}^{} + r_{3} f_{3}^{} + r_{4} f_{4}^{}) - 2 e^{i ϕ 2} (r_{2} f_{1}^{} + r_{4} f_{3}^{}) - 2 e^{- i ϕ 2} (r_{1} f_{2}^{} + r_{3} f_{4}^{}) \\ + 2 e^{i (ϕ 3 - ϕ 4)} (r_{3} f_{1}^{} + r_{4} f_{2}^{}) + 2 e^{- i (ϕ 3 - ϕ 4)} (r_{1} f_{3}^{} + r_{2} f_{4}^{}) \\ + e^{i (ϕ 2 + ϕ 3 - ϕ 4)} (r_{4} f_{1}^{} + r_{2} f_{3}^{}) + e^{- i (ϕ 2 + ϕ 3 - ϕ 4)} (r_{1} f_{4}^{} + r_{3} f_{2}^{}) \end{array}$
1	0	$2 (r_{1} f_{1}^{} - r_{2} f_{2}^{} + r_{3} f_{3}^{} - r_{4} f_{4}^{}) + e^{i (ϕ 3 - ϕ 4)} (r_{4} f_{2}^{} - r_{3} f_{1}^{}) + e^{- i (ϕ 3 - ϕ 4)} (r_{2} f_{4}^{} - r_{1} f_{3}^{})$
2	$ϕ_{2}$	$- 2 (r_{2} f_{1}^{} + r_{4} f_{3}^{}) + e^{i (ϕ 3 - ϕ 4)} r_{4} f_{1}^{} + e^{- i (ϕ 3 - ϕ 4)} r_{2} f_{3}^{}$
3	$- (ϕ_{2} - ϕ_{4})$	$r_{1} f_{2}^{} - r_{3} f_{4}^{}$
4	$ϕ_{4}$	$- r_{1} f_{1}^{} + r_{2} f_{2}^{} + r_{3} f_{3}^{} - r_{4} f_{4}^{}$
5	$ϕ_{4}$	$2 (- r_{1} f_{1}^{} - r_{2} f_{2}^{} + r_{3} f_{3}^{} + r_{4} f_{4}^{}) + e^{i ϕ 2} (r_{4} f_{3}^{} - r_{2} f_{1}^{}) + e^{- i ϕ 2} (r_{3} f_{4}^{} - r_{1} f_{2}^{})$
6	$ϕ_{4}$	$- r_{1} f_{1}^{} + r_{2} f_{2}^{} + r_{3} f_{3}^{} - r_{4} f_{4}^{}$
7	$ϕ_{2} + ϕ_{4}$	$r_{2} f_{1}^{} - r_{4} f_{3}^{}$
8	$- (ϕ_{2} - ϕ_{3} - ϕ_{4} + π)$	$r_{3} f_{2}^{*}$
9	$ϕ_{3} + ϕ_{4}$	$r_{3} f_{1}^{} - r_{4} f_{2}^{}$
10	$ϕ_{3} + ϕ_{4}$	$2 (r_{3} f_{1}^{} + r_{4} f_{2}^{}) + e^{i ϕ 2} r_{4} f_{1}^{} + e^{- i ϕ 2} r_{3} f_{2}^{}$
11	$ϕ_{3} + ϕ_{4}$	$r_{3} f_{1}^{} - r_{4} f_{2}^{}$
12	$ϕ_{2} + ϕ_{3} + ϕ_{4} + π$	$r_{4} f_{1}^{*}$

Peak	a_m	b_m
0	$16 m_{00} + 8 m_{02} - 8 m_{20} - 4 m_{22}$	0
1	$8 m_{01} - 4 m_{21}$	0
2	$- 4 m_{02} + 2 m_{22}$	$- 4 m_{03} + 2 m_{23}$
3	$2 m_{12}$	$- 2 m_{13}$
4	$- 4 m_{11}$	0
5	$- 8 m_{10} - 4 m_{12}$	0
6	$- 4 m_{11}$	0
7	$2 m_{12}$	$2 m_{13}$
8	$- m_{22} + m_{33}$	$m_{23} + m_{32}$
9	$2 m_{21}$	$- 2 m_{31}$
10	$4 m_{20} + 2 m_{22}$	$- 4 m_{30} - 2 m_{32}$
11	$2 m_{21}$	$- 2 m_{31}$
12	$- m_{22} - m_{33}$	$- m_{23} + m_{32}$

70 kHz full 4x4 Mueller polarimeter and simultaneous fiber calibration for endoscopic applications

Abstract

1. Introduction

2. Experimental set-up and theory

2.1. Overall strategy to retrieve the polarimetric properties of the object under test

2.2. Mathematical expression of the channeled spectrum

2.3. Mueller Matrix from DC components

2.4. Mueller Matrix from interferences

2.5. Mathematical treatment to measure the polarimetric properties of the medium

2.5.1. Measurement of the normalized matrices $\bar{M} Par$ and $\bar{M} PrM$

2.5.2. Measurement of the Mueller matrix of the medium $\bar{M} medium$

3. Systematic errors

3.1. Calibration of frequency linearity

3.2. Thickness errors of YVO4 retarder plates and response of the detection

3.3. Nonpolarizing beamsplitter cube

3.4. Optical fiber

4. Results and discussion

4.1. “Vacuum” sample

4.2. Double-pass quarter wave plate

4.3. Linear polarizer

5. Conclusion

6 Appendix A: Mueller (Jones) matrix related to Jones matrix [17]

7 Appendix B: Jones matrix related to Mueller-Jones [18]

8 Appendix C: Complex values of interference peaks

Acknowledgments

References and links

Supplementary Material (1)

Cited By

Figures (9)

Tables (4)

Equations (24)

Optics Express

Abstract

1. Introduction

2. Experimental set-up and theory

2.1. Overall strategy to retrieve the polarimetric properties of the object under test

2.2. Mathematical expression of the channeled spectrum

2.3. Mueller Matrix from DC components

2.4. Mueller Matrix from interferences

2.5. Mathematical treatment to measure the polarimetric properties of the medium

2.5.1. Measurement of the normalized matrices M¯Par and M¯PrM

2.5.2. Measurement of the Mueller matrix of the medium M¯medium

3. Systematic errors

3.1. Calibration of frequency linearity

3.2. Thickness errors of YVO4 retarder plates and response of the detection

3.3. Nonpolarizing beamsplitter cube

3.4. Optical fiber

4. Results and discussion

4.1. “Vacuum” sample

4.2. Double-pass quarter wave plate

4.3. Linear polarizer

5. Conclusion

6 Appendix A: Mueller (Jones) matrix related to Jones matrix [17]

7 Appendix B: Jones matrix related to Mueller-Jones [18]

8 Appendix C: Complex values of interference peaks

Acknowledgments

References and links

Supplementary Material (1)

Cited By

Figures (9)

Tables (4)

Equations (24)

Optics Express

2.5.1. Measurement of the normalized matrices $\bar{M} Par$ and $\bar{M} PrM$

2.5.2. Measurement of the Mueller matrix of the medium $\bar{M} medium$