1. Introduction

Measuring weak anisotropies has long been a major topic in optics and sometimes a challenge both in fundamental research and for applications. Focus was long made on optical activity because it is the key to analyze chiral substances such as sugars and drugs in molecular chemistry [1] or to detect parity violation of optical rotation induced by the neutral weak-current interaction in atomic physics [2].

Polarimeters dedicated to weak optical activity measurements are based on null methods [3,4] in order to guarantee the best sensitivity. It usually consists in employing two crossed linear polarizers and a Faraday rotator [5–8] or an electro-optics crystal [9] that makes the input linear polarization state oscillate a few degrees on either side of the crossed position at a frequency f. When the sample is not optically active, the detector receives a signal at frequency 2f, which can be used as a self-reference. If an optically active sample is inserted into the path, another component at frequency f appears whose amplitude is proportional to optical rotation. The ratio between the two components is then independent from fluctuations of the light source and variations of the sample transmission while measuring weak optical rotation. Nevertheless, the modulation frequency is very limited since the magnetic intensity of Faraday coil is reduced when the impedance is raised at high frequency.

Other null-polarimeters based on interferometric schemes exist, although these are not always self-referenced. For instance, a Fabry-Perot cavity [10,11] can be used to accumulate optical rotation as light propagates back and forth between the cavity mirrors thus increasing sensitivity. Mach-Zehnder-based polarimeters were also developed, either for homodyne detection, for which the phase or the amplitude of a reference field is modulated [12], or for heterodyne detection, for which its optical frequency is shifted [13–15]. Common-path spectral interferometric schemes [16,17] were also proposed where spectrum is modulated according to optical activity.

For all the null techniques cited above, measurement speed is limited as none works at a high modulation frequency and they do not make it possible to image linear birefringence with the best sensitivity since they were developed for spatially-unresolved sensitive measurements of optical activity only. Yet linear birefringence is a key polarimetric feature in biological specimen because it is a sensitive indicator of regular arrangements of lipids or proteins [18,19] in mammals, such as collagen and elastin fibers in connective tissues, microtubules and actin filaments in cytoskeleton, or a sensitive indicator of structural (cellulose fibers) and storage (semi-crystalline starch) polysaccharide building blocks in plants.

Our laboratory has recently developed an original method based on the idea of spectrally encoded light polarization [20–22] able to measure the whole polarimetric features of a specimen at 100 kHz [23,24] and that was successfully implemented in a laser scanning microscope [25] to image biological tissues [26]. However, like every Jones/Mueller matrix measurement device [27–30], our technique is operated off null and the price to pay is a drastic loss in sensitivity.

The purpose of the paper is to shed new light on null-polarimetry through an original configuration of spectrally encoded light polarization detection method that we name swept-wavelength null polarimeter. It is able to measure either circular or linear retardance of a sample with high sensitivity and is built to be self-referenced. The method is based on passive polarization optics and a fast wavelength swept laser source. This allows for high speed measurement and a compact design, facilitating its further integration in imaging devices and fast remote sensors. In Section 2, two setups dedicated to either circular or linear retardance will be described theoretically as well as their associated measurement noise. In Section 3, experiments demonstrating high speed circular and linear retardance measurements will be carried out in order to assess the resolution of anisotropy measurements. The device is able to measure linear and circular retardance respectively with 55 × 10⁻⁹ rad/Hz^1/2 (3.1 µdeg/Hz^1/2) and 45 × 10⁻⁹ rad/Hz^1/2 (2.6 µdeg/Hz^1/2) resolution (for a 18 mW optical power at the sample position) and a minimum acquisition time of 10 µs, paving the way to ultrafast sensitive anisotropy measurements.

2. Theoretical model of the polarimeter

The basic principle of the swept-wavelength null polarimeter is schematized in Fig. 1. The optical source is a wavelength swept laser source (SS) that sweeps optical frequency ν(t) = ξt+ξ₀ across a broadband spectral range. The polarimeter is made of two crossed linear polarizers P₁ and P₂ oriented along the vertical (V) and horizontal (H) direction between which are set two thick linear birefringent plates B₁ and B₂ whose thicknesses ae and be are integer multiples of thickness e.

 

Fig. 1. Swept-wavelength null polarimeter for circular and linear retardance measurement. The time-dependent photoelectric signal delivered by the detector D corresponds to optical power versus optical frequency swept in time. P₁ and P₂: linear polarizers. B₁ and B₂: thick linear birefringent plates. (a) For circular retardance (CR) measurements, no further optical component needs to be added. (b) For linear retardance (LR) measurements, two achromatic quarter-wave plates (QW₁ and QW₂) are added in order to measure independently LR and its azimuth α.

Download Full Size | PDF

The plate B₁ has its slow axis making a small angle β relative to the direction of P₁, and thus transforming the input linearly polarized field into an elliptic polarization state whose two orthogonal components are separated in phase by the phase shift aΔϕ where Δϕ is given by:

Let us write p_⊥=<H|P|V > and p_//=<H|P|H > the elements of the Jones matrix P describing the polarization transformation of the optical elements in the dashed box of Fig. 1, and respectively given between two crossed or parallel polarizers. The element p_⊥ is related to the sample anisotropy knowing that, if there is no anisotropy phenomenon, the Jones matrix P is the identity matrix leading to p_⊥=<H|V>=0. For a sample whose transmission is T_samp, a channeled spectrum is detected that can be described in time by I(t)=T_samp |<H|P|E_inc>|², which leads to:

In order to better identify the role of the angle β in the swept-wavelength null polarimeter, Eq. (3) can be simplified in the case of a weakly anisotropic sample (p_//≈1) and a near null configuration (sin β ≈ β) according to:

The channeled spectrum can be interpreted as a result of interferences between a main electric field probing the anisotropic sample with a power equal to I₀ and two reference fields separated in phase by (a + b)Δϕ and bΔϕ with a weak power equal to β²I₀. If β=0, the sensitivity of the device is limited by the parabolic dependence on the weak anisotropy to be measured and the fluctuations of the source power. If we work near the null configuration (for β≠0), the channeled spectrum is made of interference terms whose amplitudes depend on the product β×p_⊥, which permits to detect linearly the weak anisotropy and amplify it at the same time. On the other hand, the interference between the two reference fields (whose amplitude depends on β²) does not depend on the anisotropy to be measured and can be used as a self-reference. The two terms of interest are then measured independently by Fourier analysis.

Let us resume the theoretical model of the polarimeter without any approximation, enabling anisotropy phenomena to be measured whatever the anisotropy amplitude and the angle β. The channeled spectrum from Eq. (3) is a summation of modulations in time whose amplitudes depend on the transmission of the sample, the anisotropy phenomena, the source intensity I₀ and the angle β of the plate B₁ as follows:

According to the type of retardance (linear or circular) to be measured, the swept-wavelength null polarimeter has to be developed in two different configurations.

2.1. Circular retardance

Circular retardance (CR) originates from the difference in refraction index between left (LCP) and right (RCP) circularly polarized light. Note that LCP and RCP are defined from the point of view of the receiver [31]. A circular retardance phenomenon acts as an optical rotation by an angle θ=CR/2 with CR = 2π(n_RCP-n_LCP)Lν₀/c, where L is the thickness of the sample, n_RCP and n_LCP are the refractive indices of RCP and LCP, and ν₀ is the central optical frequency of the SS.

Figure 1(a) describes a configuration able to measure CR. This is the setup based on two linear polarizers and two linear birefringent plates which is described in the previous paragraph. The anisotropic phenomenon P is then equal to the Jones matrix P_CR of CR, leading to p_//=cos(CR/2) and p_⊥=sin(CR/2). Thus, the ratio γ in Eq. (8) becomes for CR:

The angle θ, and subsequently CR, can thus be retrieved from the measurement of the absolute value of γ_CR.

2.2 Linear retardance

Linear retardance (LR) originates from the difference in refraction index between two orthogonal linearly polarized light whose orientation is named azimuth α. The setup of Fig. 1(a) is unable to measure LR independently from its azimuth α because the main electric field probing the anisotropic sample and the two reference fields separated in phase are linearly polarized. It is common practice to probe linear anisotropy with a circular polarized field by setting an achromatic quarter-wave plate (QW₁) in front of the sample at 45° relative to the polarizer P₁. In that case, the setup is no longer a near null device since light power after the analyzer is not null without the sample and for β=0. In order to keep the null power feature, a second quarter-wave plate (QW₂) has to be set after the sample at -45° relative to the polarizer P₁.

Figure 1(b) describes a setup suitable for the measurement of linear retardance. The Jones matrix P corresponds to the product of the Jones matrices for the two quarter-wave plates J_QW and the linear retardance sample P_LR, whose expression is given by:

A linear retardance phenomenon is characterized by LR = 2π(n_slow-n_fast)Lν₀/c, and the azimuth α of the slow axis. For LR, p_//=cos(LR/2) and p_⊥=sin(LR/2) e ^–j2α, and the ratio γ becomes:

The measurement of the modulus of γ_LR provides LR while its argument is proportional to azimuth α.

2.3 Noise analysis

The detected photocurrent can be expressed as i(t) = ρI(t), where ρ is the detector responsivity (A/W) and intensity I is expressed in optical power. The detector photocurrent consists of both signal i_s and noise i_n that are digitized and sampled into a finite number of data points N_S over T_acq that is inferior or equal to the duty cycle T_rep of the SS. Noise can be quantified by an equivalent power I_n for which the standard deviation (SD) of the power ΔI_n within the detection bandwidth B for a monochromatic source whose power is constant in time, as:

Let $\tilde{F}$ denote the Fourier transform of the digitized current i(t_m) with t_m= m T_acq/N_s given by:

Because the detector responsivity depends on wavelength and each time interval of the signal corresponds to different optical frequencies, photocurrent i(t) could be different from optical intensity I(t). In order to simplify, we will assume that i(t) is proportional to I(t) by i(t) = ρ_eff I(t) with ρ_eff the effective detector responsivity associated to the broadband source. Thus measuring peak values ${\tilde{F}_a}$ and ${\tilde{F}_{a + b}}$ at frequencies af₀ and (a + b)f₀ provides the complex-value amplitudes F_a and F_b from respectively ${\tilde{F}_a} = {N_s}\,{\rho _{eff}}\,{F_a}$ and ${\tilde{F}_{a + b}} = {N_s}\,{\rho _{eff}}\,{F_{a + b}}$.

Since signal analysis is carried out in Fourier domain, noise has to be written in Fourier domain as well. This has been studied in details for swept source optical coherence tomography (SS-OCT) [32–34]. For sampled data of noise current mutually uncorrelated, it has been demonstrated that the SD of noise in Fourier domain is theoretically given by $\Delta {\tilde{F}_n} = \rho \,\sqrt {{N_s}/2} \,\Delta {I_n}$. In practice noise is impacted by window functions used for Fourier transform and detector features such as internal photo-electronic signal gain and excess noise factors for avalanche photodiodes (APD) [35]. We have therefore decided to describe noise by an effective SD power I_n,eff according to:

In order to determine the theoretical resolution of the setup, we focus on CR phenomenon by assuming that polarizers have infinite extinction ratio, the angle β is weak and larger than the optical rotation θ to be detected. Thus, the ratio γ between the peak values at af₀ and (a + b)f₀ is equal to $|\mathrm{\gamma } |= {\mathrm{\theta } / {\mathrm{\beta } = }}|{{{{{\tilde{F}}_{a + b}}} / {{{\tilde{F}}_a}}}} |$ (Eq. (9)), and for uncorrelated peaks, the squared SD of θ is related to $\Delta {\tilde{F}_{n,\,eff}}$ by:

The resolution Δθ of the measurement depends on the features of the detector but also on the intensity I₀ at the sample position and the angle β that have to be taken special care of when minimizing the resolution. Without excess intensity noise, Eq. (16) leads to:

Resolution Δθ is given for a measurement over the duty cycle T_rep of the SS and can be improved if the measurement is averaged over time. Then it is more convenient to define resolution Δθ per unit bandwidth as follows:

Equation (19) shows that Δθ could be infinitely small if the source intensity I₀ is infinitely large. In fact, the finite extinction ratio of the polarizers needs to be taken into account. Non-ideal linear polarizers can be described by the following Jones matrix:

3. Experimental results

The setup designed for optical anisotropy measurement is shown in Fig. 1, the swept-source (SS) was a laser from Axsun Tech. Inc. that sweeps wavelength over a spectral range of 100 nm around 1050 nm at a rate of 100 kHz, i.e. a duty cycle of T_rep= 10 µs. Optical power was around 18 mW at the sample. The plates B₁ and B₂ were YVO₄ crystals whose thicknesses were respectively 2 mm and 0.4 mm, which corresponds to a = 5, b = 1 and e = 0.4 mm. The plate B₁ was mounted on a motorized rotation stage (Newport SR50PP, minimum increment: 0.004°, accuracy: ±0.0020°, uni-directional repeatability: ±0.0035°, bi-directional repeatability: ±0.0016°) to adjust precisely the angle β. The polarizers P₁ and P₂ were Glan-Taylor polarizers with extinction ratio of 10⁵ (=1/ɛ²). The electrical signal delivered by an avalanched photodiode (APD module C12703SPL, Hamamatsu) was digitized by a data acquisition board (DAQ, ATS9350 digitizer, AlazarTech) that was triggered by a Mach-Zehnder interferometer clock signal (k-clock) from SS. The electrical signal was then sampled into a finite number of data points N_S= 1328 over T_acq= 4.4 µs, whose time intervals were equivalent to a constant optical frequency interval, avoiding re-sampling of the signal. Although the power saturation of the detector was given at 8.8 µW, optical power at the detector position did not exceed 6 µW (I_sat= 6 µW) in order to work in the linear range of the detector.

3.1. Circular retardance

The device dedicated to circular anisotropy is shown in Fig. 1(a). In order to easily characterize the setup, the rotation θ_motor of the analyzer P₂ controlled by a motorized rotation stage (Newport URS100PP, minimum increment: 0.0002°, accuracy: ±0.008°, uni-directional repeatability: ±0.00035°, bi-directional repeatability: ±0.0044) was used to simulate the rotation of the polarization plane of linearly polarized light induced by an optically active sample. This makes it simple and accurate to obtain a change in the optical rotation θ for a CR anisotropy. In order to illustrate the principle of the method, several channeled spectra were recorded for different θ_motor in Figs. 2(a1-a3). Angle β equal to 0.46° was chosen close to β_sat in order to nearly saturate the detector while working in its linear range. When θ_motor increased, modulations appeared that were measured by Fourier transforming channeled spectra in Fig. 2(a4). As expected peak ${\tilde{F}_6}$ (corresponding to ${\tilde{F}_{a + b}}$) as well as peak ${\tilde{F}_1}$ (${\tilde{F}_b}$) were sensitive to optical rotation since ${\tilde{F}_6} \approx {\tilde{F}_1} \propto \mathrm{\theta }\,\mathrm{\beta }$ for small optical rotations, while peak ${\tilde{F}_5}$ (${\tilde{F}_a}$) was only sensitive to the angle β knowing that ${\tilde{F}_5} \propto {\mathrm{\beta }^2}$. Notice that peak ${\tilde{F}_4}$ (${\tilde{F}_{a - b}}$) did not vary significantly because ${\tilde{F}_4} \propto \mathrm{\theta }\,{\mathrm{\beta }^3}$. The gray filling curve represents the Fourier transform of the optical spectrum (without any modulation) and indicates that laser spectrum displays slight modulations too.

 

Fig. 2. (a) and (b) Circular retardance is simulated by rotating the analyzer (P₂) of the polarimeter by an angle θ_motor. Figures (a1), (a2) and (a3) correspond to the channeled spectra measured during the swept of the SS for three different angles of P₂. (a4) Fourier Transform of the channeled spectra obtained for three different angles of P₂. (b1) and (b2) Average of 10⁴ successive optical rotation measurements versus the rotation of P₂ with two different increments. (c) Circular retardance induced by a Faraday rotator powered by an alternative current. (c1) Average of 10⁴ successive optical rotation measurements versus time for a 0.5 Hz alternative current. (c2) Optical rotation measurement versus time for a 5 kHz alternative current without averaging. Optical rotation measurements have been carried out by subtracting the ratio γ with/without the rotation of P₂ or with/without the Faraday rotator (background correction).

Download Full Size | PDF

From the measurement of the ratio $\mathrm{\gamma } = {\tilde{F}_6}/{\tilde{F}_5}$ in Eq. (9), optical rotation can be calculated. In practice, peak ${\tilde{F}_6}$ was preferred to peak ${\tilde{F}_1}$ due to the proximity of the high peak DC that adds noise to peak ${\tilde{F}_1}$. The average of 10⁴ successive optical rotation measurements θ_mean, i.e. an average over 0.1 s, was plotted versus the rotation θ_motor of the analyzer P₂ with an increment of 0.01 deg (π/2 × 10⁻⁴ rad) and 0.001 deg (π/2 × 10⁻⁵ rad) respectively in Fig. 2(b1) and Fig. 2(b2). Experimental curves being in good agreement with theory, they can be used to calibrate the setup. However, the increment of 0.001 deg reaches the limit of the rotation stage. In order to illustrate the high sensitivity of our device, optical rotation induced by a Faraday rotator whose magnetic field oscillates weakly in time was measured. For a 0.5 Hz oscillation, the variation of optical rotation was slow enough to average data and ensure a good resolution of oscillations by increasing SNR as Fig. 2(c1) shows. On the other hand, Fig. 2(c2) illustrates another interest of our device that is to follow faster variations of polarimetric anisotropies since the repetition rate of the optical rotation measurement is equal to 100 kHz (sweep repetition rate of the SS). In this example, the Faraday rotator was modulated at 5 kHz, so it was not possible to average measurements leading to a lowering of the SNR. The performance of the polarimeter in terms of resolution will be detailed in Section 3.3.

Our swept-wavelength null polarimeter is tolerant regarding thickness errors and dispersion of plates B₁ and B₂. Indeed measurements are made at the maximum of Fourier peaks independently from peak positions (related to the absolute waveplate thicknesses) and peak widths (related to the waveplate retardation dispersion) as long as peaks are separable. As far as axis alignment errors of the polarizer and the plates B₁ and B₂ are concerned, the Fourier transformation of the channeled spectrum is displayed in real-time during the alignment of each plate in order to cancel peaks, with the resolution of the null-method itself. However a residual amplitude can be present at the peak positions if the linear polarizers are slightly elliptical, or in the case of modulations in the laser spectrum itself. This residual amplitude (see the gray filling curve and the red line in Fig. 2(a4)) is then removed by a background subtraction.

3.2. Linear retardance

The setup designed for linear anisotropy is shown in Fig. 1(b). This is the same setup as in Fig. 1(a) except for the addition of two true zero order quarter-wave plates at 1050 nm made of MgF₂ material of thickness of 22.65 µm and cemented on a glass substrate. The choice of true zero-order quarter wave plates rather than achromatic quarter wave plates was motivated by the higher quality of extinction with the formers, while still exhibiting slow spectral variations within the bandwidth of the SS. As easy as it is to generate weak optical rotations by rotating the analyzer or using magneto-active materials in order to study the sensitivity of the polarimeter, it is equally difficult to benefit from weak and homogeneous birefringent materials. For this purpose, two identical wave plates made of quartz were joined in such a way that the fast axis of the former was aligned with the slow axis of the latter as sketched in Fig. 3(a). Although these plates are considered similar, a small thickness difference was still present due to the limited accuracy of the manufacturing process leading to a residual retardance estimated to π/150 rad. From the amplitude and the phase of the ratio $\mathrm{\gamma } = {\tilde{F}_6}/{\tilde{F}_5}$, the small linear retardance LR and azimuth α of this custombuilt waveplate were calculated by means of Eq. (11). Figures 3(a1) and 3(a2) respectively display LR and α versus the rotation α_manual of the custombuilt waveplate around the propagation axis. Azimuth measurement was in good agreement with manual rotation of the custombuilt waveplate. LR varied slightly according to α_manual because the laser beam did not probe exactly the same area of the custombuilt waveplate during its rotation and the inhomogeneous thicknesses of the wave plates were then revealed.

 

Fig. 3. (a) Custombuilt waveplate made of two wave plates in order to obtain an equivalent waveplate whose linear retardance LR is weak. (a1) Linear retardance of sample (a) versus its rotation, (a2) Azimuth versus the rotation of sample (a). (b) Glass plate deformed by piezoelectric transducer powered by an alternative current. (b1) Average of 10⁴ successive linear retardance measurements versus time. Linear retardance measurements have been carried out by subtracting the ratio γ with and without the waveplate (background correction). Only the alternative variation is shown to easily see the sensitivity of the polarimeter.

Download Full Size | PDF

In order to go further and visualize the sensitivity of the polarimeter, a glass plate was subjected to strain between two metallic plates, one fixed and one attached to a piezoelectric transducer as shown in Fig. 3(b). When the piezoelectric transductor was supplied with alternative voltage, LR was photoelastically induced and varied in the same way. Figure 3(b1) displays the average of 10⁴ successive linear retardance measurements named LR_mean versus time, demonstrating the high sensitivity of the polarimeter according to linear retardance measurements too. Demonstration of high speed measurement of weak linear retardance could not be done in the same way as in Section 3.1 for circular retardance because of the limited cut-off frequency of the piezoelectric transducer used for this experiment.

Similarly to the waveplates B₁ and B₂, the axis alignment of quarter-wave plates QW₁ and QW₂ is adjusted by minimizing the amplitude ${\tilde{F}_6}$ with the resolution of our null-method. However a residual linear birefringence may appear if the thicknesses of the two quarter-wave plates are not identical, which leads to a non-zero value of ${\tilde{F}_6}$. Background subtraction is then an efficient method to remove the residual birefringence without affecting the retardance measurement [36] as long as the sample retardance is weak.

3.3 Noise analysis

The resolution of polarization measurements depends on the features of the detector but also on the choice of the angle β and the optical power at the sample as detailed in Section 2.3. From the standard deviation of the amplitude of the peak ${\tilde{F}_6}$ (${\tilde{F}_{a + b}}$), the standard deviation $\Delta I_{n,\,eff}^{}$ of the equivalent power was measured versus the root mean square I_RMS of the optical power at the detector position. Figure 4(a) shows that $\Delta I_{n,\,eff}^2$ can be fitted by a linear curve (with n_receiver= 1.79 pW/Hz^1/2 and n_shot= 2.26 × 10⁻⁶ pW/Hz in relation to Eq. (14)) up to I_RMS= 3 µW for which the detector is closed to saturation. This linear behavior indicates that excess intensity noise can be neglected, validating the assumption done in Eq. (17).

 

Fig. 4. (a) Standard deviation of the equivalent power measured versus the root mean square I_RMS of the optical power at the detector position. (b) Standard deviation of circular retardance (CR) according to the optical power I₀ at the sample position and the maximum power I_max driven by the angle β of the plate B₁: β=0.46 deg for I_max= I_sat, β=0.38 deg for I_max= 0.7 I_sat, β=0.29 deg for I_max= 0.4 I_sat. (c) Allan deviation log-log plot of linear (Δ_ALR) and circular (Δ_ACR) retardance versus averaging time T_average. For T_average= T_rep= 10⁻⁵ s, there is any average.

Download Full Size | PDF

In Section 2.3, it is demonstrated that resolution is optimal when optical power I₀ probing the sample is as large as possible and the angle β is chosen to nearly saturate the detector. In order to verify experimentally this result, standard deviation of circular retardance was measured according to I₀ and β from Eq. (9) with the setup described in Fig. 1(a). For each value of I₀ controlled by the rotation of a linear polarizer set in front of the polarimeter, 3 values of β were chosen so that the maximum intensity I_max of the channeled spectrum reaches respectively the saturation of the detector, 0.7 times the saturation and 0.4 times the saturation. Figure 4(b) plots the standard deviation of circular retardance for each value of I₀ and β. Whatever I₀, ΔCR is minimum when β is adjusted to reach the limit of the detector saturation. Moreover, when β=β_sat, the blue curve in Fig. 4(b) confirms the theoretical approach since the larger I₀, the smaller ΔCR. Thus the best ΔCR at 10 µs acquisition time is equal to 1.2 × 10⁻⁵ rad. This result is valid for linear retardance too as shown in Fig. 4(c) at 10 µs acquisition time, i.e. for T_average equal to 10⁻⁵ s. Standard deviation of LR has been measured from Eq. (11) with the setup described in Fig. 1(b).

If the purpose of the polarimeter is not to measure fast variations of the polarimetric anisotropies in time but to improve resolution, averaging measurements can be carried out. To check the stability over time of the setup and determine the highest number of measurements that can be averaged, the Allan deviation [37,38], also known as two-sample standard deviation was performed. The Allan deviation is defined so that it has the same value as the standard deviation for white noise, i.e. Gaussian noise of uniform spectral power density. However examination of a log-log plot of Allan deviation versus sampling period allows different noise types to be distinguished by the slope of the plot in particular time regions, which is not the case for standard deviation. For instance a region of slope −0.5 corresponds to Gaussian noise while a region of slope 0.5 describes random-walk noise. In general a 1/f ⁿ noise corresponds to a region of slope (n−1)/2. Allan deviations for circular and linear retardance measurements were calculated respectively with the setup described in Fig. 1(a) and Fig. 1(b), according to averaging time. By increasing the averaging time, the Allan deviation Δ_ACR and Δ_ALR shown in Fig. 4(c) indicates that resolution drops with a white-noise dependence to the value of 1.4 × 10⁻⁸ rad at 10 s for CR, and the value of 7 × 10⁻⁸ rad at 0.65 s for LR, which confirms our assumption of white noise in Section 2.3, leading to a CR sensitivity of 45 × 10⁻⁹ rad/Hz^1/2 (2.6 µdeg/Hz^1/2) and a LR sensitivity of 55 × 10⁻⁹ rad/Hz^1/2 (3.1 µdeg/Hz^1/2). However, the behaviors of ΔCR and ΔLR changed respectively from 10 s and 0.65 s, with a slope value close to 1, driven by mechanical or thermal drifts. The theoretical estimation of the measurement resolution including the contribution of the relevant noise sources (Eq. (21)) gives a value of ΔCR = 2Δθ=42 × 10⁻⁹ rad/ Hz^1/2 that is the same order of magnitude as experimental resolution.

4. Discussions

The interest of the swept-wavelength null polarimeter lies in its sensitivity due to the important relative weight of the polarimetric signal in comparison to the dynamic range of the detector. Let us consider a linear birefringent sample with retardance LR. When the bias angle β is adjusted so that the intensity at the detector position is close to the saturation of the detector (I_sat ≈ I₀(2β)²), the amplitude of the polarimetric signal is then I₀ LR β/2, leading to a relative weight of the polarimetric signal equal to LR/(8β) in comparison to the dynamic range of the detector. The smaller the angle β, the higher the relative weight of the polarimetric signal. Notice that the intensity I₀ at the sample plane has to be maximum to ensure the best sensitivity (see Fig.4b). A resolution of 3.1 µdeg/Hz^1/2 has been measured with this new polarimeter.

In comparison, many polarimetric devices like Mueller matrix polarimeters operate off null and thus cannot reach this sensitivity. Indeed the intensity I₀ has to be adjusted not to saturate the detector (I_sat ≈ I₀) and the amplitude of the polarimetric signal is proportional to I₀. Thus the relative weight of the polarimetric signal is equal to LR in comparison to the dynamic range of the detector. To demonstrate the loss of sensitivity in off-null polarimeters, the angle β has been adjusted at 45° in order to operate off-null like in Mueller polarimeters. In this case, without sample, the intensity I₀ should be drastically reduced so as not to saturate the detector. The amplitude of the peak ${\tilde{F}_6}$ (corresponding to ${\tilde{F}_{a + b}}$) is theoretically equal to I₀ LR/8 and has been measured 10³ times at 10 µs acquisition time, which has given a resolution on LR equal to 253 µdeg/Hz^1/2. This result is consistent with our Mueller polarimeter based on spectrally encoded light polarization [25] with which we share the same detector and laser source.

However Mueller matrix polarimeters are able to provide all the polarimetric signature of the sample (linear retardance/diattenuation with its azimuth, circular retardance/diattenuation and depolarization) and can be corrected from systematic errors [39] by using known polarimetric elements as measurement standards. The ability to sort polarimetric properties is very efficient for thick and scattering samples [40]. By contrast, the swept-wavelength null polarimeter was designed to measure only circular or linear retardance depending on the choice of the configuration (Fig. 1(a) or Fig. 1(b)). Due to its high sensitivity, the device can be used to measure weak polarimetric effects such as optical activity in solutions or birefringence in cells or histological sections of tissues. Let us focus on the configuration able to measure linear retardance (Fig. 1(b)). One should note that in the case of a pure linear dichroic sample, this configuration is sensitive to linear diattenuation and its azimuth since the ratio $\mathrm{\gamma } = {\tilde{F}_6}/{\tilde{F}_5}$ is equal to

Lastly, the assessment of the accuracy of our technique is difficult because it would require measurement standards known with a precision of 3.1 µdeg/Hz^1/2. In practice no commercial component reaches this level of precision. In order to limit systematic errors, the swept-wavelength null polarimeter is built with pure birefringent crystals avoiding polarimetric inhomogeneity like liquid crystals for instance. The study of accuracy in our polarimeter will be the subject of another paper which will deal with both hardware and numerical solutions to limit systematic errors.

5. Conclusion

Our null-polarimeter has been shown to measure accurately linear and circular retardance respectively with 55 × 10⁻⁹ rad/Hz^1/2 (3.1 µdeg/Hz^1/2) and 45 × 10⁻⁹ rad/Hz^1/2 (2.6 µdeg/Hz^1/2) resolutions (for a 18 mW optical power at the sample position) and with a minimum acquisition time of 10 µs, paving the way for ultrafast and ultrasensitive polarimetric measurements.

This technique benefits from the latest advances in Optical Coherent Tomography (OCT) [41] technology in terms of light sources. Indeed, both OCT and the reported new swept-wavelength null-polarimeter are spectral interferograms and all the relevant information is contained in the modulations of channeled spectra (the depth-resolved reflectivity profile for OCT, optical anisotropy phenomena for swept-wavelength null-polarimetry). OCT technologies are continually evolving and several ways of improving our technique could be considered: (1) developing new swept-sources in the visible range [42] to benefit from higher anisotropies at shorter wavelengths, (2) amplifying optical power of swept-sources [43] in order to further increase sensitivity. For instance in the case of an extinction ratio contrast of 10⁵ and a saturation power I_sat equal to 10 µW, the sensitivity is optimal for ${I_0} = \left( {\sqrt 2 - 1} \right){I_{sat}}/{\varepsilon ^2}$, i.e. I₀= 420 mW, leading to a 12 × 10⁻⁹ rad/ Hz^1/2 (0.7 µdeg/Hz^1/2) resolution (with n_receiver= 1.79 pW/Hz^1/2 and n_shot= 2.26 × 10⁻⁶ pW/Hz), (3) increasing the speed of the swept-sources [44] up to several MHz.

The immediate next step will be to implement our swept-wavelength null-polarimeter in an optical scanning microscope in order to image very weak birefringent structures in tissue sections as well as cell cultures currently beyond the scope of polarization microscopy.

Appendix

Table 1. Expressions of the peak amplitudes^a

View Table

Funding

Région Bretagne; Conseil Général du Finistère.

Acknowledgment

The authors thank Mr. Gaël Le Roux for technical assistance in achieving the mechanical items, the Conseil Général du Finistère and the Région Bretagne.

Disclosures

The authors declare that there are no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. K. W. Busch and M. A. Busch, Chiral analysis (Elsevier Science, 2006).

2. B. P. Das, “Atomic parity non-conservation: Present status and future prospects,” Pramana - J Phys 50(6), 515–519 (1998). [CrossRef]  

3. H. Rudolph, “Photoelectric polarimeter attachment,” J. Opt. Soc. Am. 45(1), 50–59 (1955). [CrossRef]  

4. Z. Yang and H. Jia, “Improving the accuracy of optical measurement based on optical null methods by curve-fitting,” Rev. Sci. Instrum. 84(5), 053104 (2013). [CrossRef]  

5. H. Cary, R. C. Hawes, P. B. Hooper, J. J. Duffield, and K. P. George, “A recording spectropolarimeter,” Appl. Opt. 3(3), 329–337 (1964). [CrossRef]  

6. J. Badoz, “Mesures photométriques de faibles biréfringences et de très petits pouvoirs rotatoires,” J. Phys. Rad. 17(S11), 143–149 (1956).

7. B. Rabinovitch, W. F. March, and R. L. Adams, “Noninvasive glucose monitoring of the aqueous humor of the eye: Part I. Measurement of very small optical rotations,” Diabetes Care 5(3), 254–258 (1982). [CrossRef]  

8. A. Arnaud, F. Silveira, E. M. Frins, A. Dubra, C. D. Perciante, and J. A. Ferrari, “Precision synchronous polarimeter with linear response for the measurement of small rotation angles,” Appl. Opt. 39(16), 2601–2604 (2000). [CrossRef]  

9. Z. Weissman and D. Goldberg, “Self-referenced, microdegree, optical rotation polarimeter for biomedical applications: an analysis,” J. Biomed. Opt. 21(7), 071104 (2015). [CrossRef]  

10. Y. Le Grand and A. Le Floch, “Measurement of small optical activities by use of helicoidal waves,” Opt. Lett. 17(5), 360–362 (1992). [CrossRef]  

11. F. Della Valle, A. Ejlli, U. Gastaldi, G. Messineo, E. Milotti, R. Pengo, G. Ruoso, and G. Zavattini, “The PVLAS experiment: measuring vacuum magnetic birefringence and dichroism with a birefringent Fabry-Perot cavity,” Eur. Phys. J. C 76(1), 24 (2016). [CrossRef]  

12. S. Spielman, K. Fesler, C. B. Eom, T. H. Geballe, M. M. Fejer, and A. Kapitulnik, “Test for nonreciprocal circular birefringence in YBa₂Cu₃O₇ thin films as evidence for broken time-reversal symmetry,” Phys. Rev. Lett. 65(1), 123–126 (1990). [CrossRef]  

13. T. Mitsui and K. Sakurai, “Precise measurement of the refractive index and optical rotatory power of a suspension by a delayed optical heterodyne technique,” Appl. Opt. 35(13), 2253–2258 (1996). [CrossRef]  

14. K.-H. Chen, C.-C. Hsu, and D.-C. Su, “Interferometric optical sensor for measuring glucose concentration,” Appl. Opt. 42(28), 5774–5776 (2003). [CrossRef]  

15. D. He, B. Xie, and S. Feng, “Null polarimetry near shot noise limit at 1 Hz,” Rev. Sci. Instrum. 87(4), 043102 (2016). [CrossRef]  

16. F. Preda, A. Perri, J. Réhault, B. Dutta, J. Helbing, G. Cerullo, and D. Polli, “Time-domain measurement of optical activity by an ultrastable common-path interferometer,” Opt. Lett. 43(8), 1882–1885 (2018). [CrossRef]  

17. Y. Nishiyama, S. Ishikawa, and H. Nagatani, “Phase-stable optical activity measurement by common-path spectral interferometry,” Opt. Lett. 45(20), 5868–5871 (2020). [CrossRef]  

18. S. Inoué, Collected Works of Shinya Inoué: microscopes, living cells, and dynamic molecules (World Scientific Publishing, 2008).

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

20. K. Oka and T. Kato, “Spectroscopic polarimetry with a channeled spectrum,” Opt. Lett. 24(21), 1475–1477 (1999). [CrossRef]  

21. N. Hagen, K. Oka, and E. L. Dereniak, “Snapshot Mueller matrix spectropolarimeter,” Opt. Lett. 32(15), 2100–2102 (2007). [CrossRef]  

22. 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]  

23. M. Dubreuil, S. Rivet, B. Le Jeune, and L. Dupont, “Time-resolved switching analysis of a ferroelectric liquid crystal by snapshot Mueller matrix polarimetry,” Opt. Lett. 35(7), 1019–1021 (2010). [CrossRef]  

24. 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]  

25. A. Le Gratiet, M. Dubreuil, S. Rivet, and Y. Le Grand, “Scanning Mueller polarimetric microscopy,” Opt. Lett. 41(18), 4336–4339 (2016). [CrossRef]  

26. S. Rivet, M. Dubreuil, A. Bradu, and Y. Le Grand, “Fast spectrally encoded Mueller optical scanning microscopy,” Sci Rep 9(1), 3972 (2019). [CrossRef]  

27. N.-J. Jan, J. L. Grimm, H. Tran, K. L. Lathrop, G. Wollstein, R. A. Bilonick, H. Ishikawa, L. Kagemann, J. S. Schuman, and I. A. Sigal, “Polarization microscopy for characterizing fiber orientation of ocular tissues,” Biomed. Opt. Express 6(12), 4705–4718 (2015). [CrossRef]  

28. O. Arteaga, M. Baldrís, J. Antó, A. Canillas, E. Pascual, and E. Bertran, “Mueller matrix microscope with a dual continuous rotating compensator setup and digital demodulation,” Appl. Opt. 53(10), 2236–2244 (2014). [CrossRef]  

29. M.-R. Antonelli, A. Pierangelo, T. Novikova, P. Validire, A. Benali, B. Gayet, and A. De Martino, “Mueller matrix imaging of human colon tissue for cancer diagnosis: how Monte Carlo modelling can help in the interpretation of experimental data,” Opt. Express 18(10), 10200–102008 (2010). [CrossRef]  

30. T. Huang, R. Meng, J. Qi, Y. Liu, X. Wang, Y. Chen, R. Liao, and H. Ma, “Fast Mueller matrix microscope based on dual dofp polarimeters,” Opt. Lett. 46(7), 1676–1679 (2021). [CrossRef]  

31. X. Xie and J. D. Simon, “Picosecond circular dichroism spectroscopy: a Jones matrix analysis,” J. Opt. Soc. Am. B 7(8), 1673–1684 (1990). [CrossRef]  

32. W. Drexler and J. G. Fujimoto, Optical coherence tomography: technology and applications (Springer,2015).

33. S. H. Yun, G. J. Tearney, J. F. de Boer, N. Iftimia, and B. E. Bouma, “High-speed optical frequency-domain imaging,” Opt. Express 11(22), 2953–2963 (2003). [CrossRef]  

34. B. Johnson, W. Atia, D. C. Flanders, M. Kuznetsov, B. D. Goldberg, N. Kemp, and P. Whitney, “SNR of swept SLEDs and swept lasers for OCT,” Opt. Express 24(10), 11174–11186 (2016). [CrossRef]  

35. R. J. McIntyre, “Multiplication Noise in Uniform Avalanche Diodes,” IEEE Trans. Electron Devices ED-13(1), 164–168 (1966). [CrossRef]  

36. M. Shribak and R. Oldenbourg, “Techniques for fast and sensitive measurements of two-dimensional birefringence distributions,” Appl. Opt. 42(16), 3009–3017 (2003). [CrossRef]  

37. D. W. Allan, “Statistics of atomic frequency standards,” in Proceedings of the IEEE54(2), 221–230 (1966).

38. D. W. Allan, “Should classical variance be used as a basic measure in standards metrology?” IEEE Trans. Instrum. Meas. IM-36(2), 646–654 (1987). [CrossRef]  

39. 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]  

40. F. Boulvert, B. Boulbry, G. Le Brun, B. Le Jeune, S. Rivet, and J. Cariou, “Analysis of the depolarizing properties of irradiated pig skin,” J. Opt. A: Pure Appl. Opt. 7(1), 21–28 (2005). [CrossRef]  

41. J. Fujimoto, “Optical coherence tomography for ultrahigh resolution in vivo imaging,” Nat. Biotechnol. 21(11), 1361–1367 (2003). [CrossRef]  

42. S. Karpf and B. Jalali, “Frequency-doubled FDML-MOPA laser in the visible,” Opt. Lett. 44(24), 5913–5916 (2019). [CrossRef]  

43. S. Karpf and B. Jalali, “Fourier-domain mode-locked laser combined with a master-oscillator power amplifier architecture,” Opt. Lett. 44(8), 1952–1955 (2019). [CrossRef]  

44. B. Potsaid, V. Jayaraman, J. G. Fujimoto, J. Jiang, P. J. S. Heim, and A. E. Cable, “MEMS tunable VCSEL light source for ultrahigh speed 60kHz - 1 MHz axial scan rate and long range centimeter class OCT imaging,” Proc. SPIE 8213, 82130M (2012). [CrossRef]