1. Introduction

The measurement of magneto-optical rotation of the polarization plane of light dates back to the 19^th century. In 1846, Faraday [1] discovered the rotation of the polarization plane of light as it propagated through glass placed in a longitudinal magnetic field, thereby discovering the prominent magneto-optical effect (also known as the Faraday effect). Studies of magneto-optical effects have achieved great success and a wide range of applications [2–6]. A significant driving force is the advent of highly reliable, inexpensive, and easily tunable diode lasers, which offer high-quality light sources for research in this field. Moreover, technology and products in this domain continue to evolve [7]. The spin-exchange-relaxation-free (SERF) all-optical magnetometer, based on the measurement of the optical rotation angles of the polarization plane of linearly polarized light, has reached an ultra-high sensitivity of fT/Hz^1/2 [8–16]. Its performance is comparable to that of superconducting quantum interferometers and is expected to replace them in the future. Among the most prominent methods for polarimetry measurements are the beam splitter method [12–16] and polarization modulation techniques [9–11,17,18]. The measurement sensitivity of the Faraday modulation method has reached the order of 10⁻⁸rad or even smaller, but the configuration is relatively complex [1]. A balanced polarimeter based on the beam splitter method can achieve similar shot-noise-limited sensitivity. The beam splitter method has advantages such as fast response, simple structure, practicality, and easy integration. Improving the sensitivity of the beam splitter method is the key to optimizing the practical performance of optical atomic magnetometers.

In 1962, Gozzini et al. [19] used the “balanced polarimeter” beam splitter method for atomic spin detection, which has become one of the most commonly used detection techniques for magneto-optical rotation in optical atomic magnetometers [20–23]. Recently, researchers have primarily concentrated on combining multiple technologies to enhance the measurement accuracy and long-term stability of measurement systems while also aiming to achieve ultrafast measurement capabilities. Polymer optical fibers are also used as sensors to measure rotation angles [24], and the sensitivity of recyclable polymer optical fibers has been improved [25], which may promote the application of fiber optic sensors in the field of photonics. Xing Li et al. [26] proposed the use of a liquid crystal phase retarder combined with a beam splitter differential method to improve measurement accuracy. Huang Jiong et al. [27] proposed an error compression method for optical rotation measurement, which can improve the long-term measurement performance by approximately 3.4 times. Spiliotis et al. [28] proposed a SPH production and detection method for fast magnetometry, it can measure on the ns timescale and with sensitivities of about 10 mG/Hz^1/2 [29].

To enhance the sensitivity of an all-optical atomic magnetometer and minimize the negative impact caused by the absorption of probe light by atoms during optical rotation measurement, Auzinsh M. et al. [30] suggested tuning the wavelength of the probe laser away from the atomic resonance to reduce absorption effects and increasing the intensity of the probe beam to improve measurement sensitivity. However, the magneto-optical rotation effect is related to wavelength detuning, which demands higher sensitivity in optical rotation measurement. The typical “balanced polarimeter” utilizes a polarizing beam splitter to decompose the light beam into orthogonal components, as referenced in various studies [12–16,20–23]. Traditionally, the photon shot noise in each beam is considered independent. Despite new advancements in data processing and technical methods enhancing optical rotation angle measurement performance, the fundamental constraints of photon shot noise remain unaddressed.

In this paper, we introduce a novel measurement method that employs a quasi-Wollaston prism as the beam splitter, where the photon shot noise in each beam is correlated due to quantum entanglement. Moreover, the reference beam provided by this design facilitates the suppression of noise due to intensity fluctuations in the probe beam through synchronous measurement. Compared to the standard balanced polarimeter, our approach has reduced the measurement uncertainty by a factor of 2^1/2. Firstly, we designed the quasi-Wollaston prism, which can separate linearly polarized light beams into three outgoing beams. By synchronously detecting the intensities of these beams as the differential signal and the reference signal, the mathematical model for optical rotation was built. We elucidate that the photon shot noise of each beam is correlated under quantum entanglement. Then, a theoretical analysis of the performance of this measurement method was conducted. Experimental results are consistent with theoretical analysis, indicating that the measurement method proposed in this paper is correct, feasible, and practical. Highly sensitive measurements of the optical rotation angles of the light polarization plane were realized. The method proposed in this paper is applicable to ultra-sensitive measurements of small optical rotation angles, which are induced by magneto-optical effects, as well as to the study of the dynamics of optically pumped atomic states when subjected to electromagnetic fields.

2. Theory

2.1 Principle of optical rotation angle detection using a classic beam splitter

A classic configuration scheme of an all-optical atomic magnetometry system is shown in Fig. 1, with the dashed box indicating the part that uses a beam splitter for detecting the light polarization plane. The pump laser beam L_pump is converted into circularly polarized light after passing through the polarizer P₂ and the λ/4 waveplate. It is then irradiated onto alkali-metal-vapor contained in the cell, which pumps the alkali atoms into a spin-polarized state with angular momentum P_z along the $\hat{\bf z}$ axis. The magnetic field, B, oriented along the ŷ direction, causes the spins to rotate from the pump direction to the probe direction. The resulting projection, P_x, along the probe beam direction is detected as a rotation of the polarization plane of the linearly polarized probe beam.

 

Fig. 1. Schematic diagram of a typical all-optical atomic magnetometer detection apparatus.

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The beam splitter polarimetry method typically employs a prism made of crystals as the splitting element. In balanced polarization measurements, aligning the polarization direction of the incident beam at a 45° angle to the polarization analyzer simplifies mathematical processing and enables approximation for small angular deviations. Consequently, both photodetectors receive equal optical power, and any alteration of the polarization plane results in a linear variation in the intensity difference between the detectors, provided that the shift in the polarization plane remains within a few milliradians. The polarizer P₁ is oriented at an angle of approximately 45° with respect to axis of the Wollaston prism (WP). After passing through P₁ and transmitting through the cell, the linearly polarized probe beam, L_probe, with an intensity I₀, incidents onto the splitter prism WP. The prism then splits the probe beam into two linearly polarized beams with orthogonal polarization directions. These beams are received and converted into electrical signals by photodetectors D₁ and D₂. The intensities of the beams are I₁ = I₀cos²(45°+θ) and I₂ = I₀sin²(45°+θ), where 45°+θ is the angle between the light polarization plane and the axis of the beam-splitter cube. Here, θ represents the optical rotation angle. When the outputs are accurately balanced at 45°, θ can be determined as follows (for small rotation angles):

The beam splitter-balanced polarimeter method is insensitive to fluctuations in the intensity of the incident light and to induced ellipticity. If N_ph photons pass through the atomic ensemble with negligible absorption, then the quantum uncertainty in the measurement of θ, limited by photon shot noise, is given by:

The variational sensitivity (short-term resolution) in the measurement of optical rotation angle often presents a practical limitation for atomic magnetometers. The polarization noise is extremely low, in the nanoradian range, even with a moderate laser power of a few milliwatts. However, at low frequencies, technical noise from various sources, which often scales inversely with frequency-such as 1/f noise- exceeds the photon shot noise.

A significant drawback of the classic balanced beam-splitting method is the requirement to know the total intensity of the probe light to normalize the differential signals. Measuring the difference and sum of two signals simultaneously introduces additional technical noise. However, this noise can be circumvented if the intensity of the incident beam is measured in real-time or detected proportionally. The quasi-Wollaston beam-splitting method described below offers a reference beam whose intensity is proportional to that of the incident beam and remains independent of its polarization state.

2.2 Principle of optical rotation measurement using a quasi-Wollaston prism beam splitter

The designed quasi-Wollaston prism (QWP) beam splitter is depicted in Fig. 2(a). Specifically, the QWP shown in Fig. 2(a) consists of an entrance prism and an exit prism, both crafted from uniaxial crystals. The crystal axis of the entrance prism aligns with the main cross-section and the entrance surface of the prism, as illustrated in Fig. 2(b). In contrast, the crystal axis of the exit prism does not align perpendicularly with the main cross-section of the prism. The angle between the crystal axis and the normal of the main cross-section, or the direction of the prism's edge, is denoted as α. Despite this, the crystal axis still lies on the exit surface of the prism, as demonstrated in Fig. 2(c). Linearly polarized incident light passes through the QWP and splits into three outgoing beams, which are subsequently captured by photodetectors and converted into signals.

 

Fig. 2. Schematic diagram of the beam splitter based on a quasi-Wollaston prism (QWP). (a) The linearly polarized probe beam is split into three outgoing beams by the QWP; (b) Structure of the entrance prism; (c) Structure of the exit prism, and (d) The propagation and evolution of the polarization eigenmodes passing through the QWP.

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Linearly polarized light, carrying optical rotation and incident onto the entrance surface of the QWP, is oriented at an angle of 45°+θ relative to the crystal axis. This light is split into two distinct beams with orthogonal linear polarization, denoted as E_o and E_e. Here, we have:

For the E_o component, refraction occurs as it propagates to the interface between the entrance prism and the exit prism. Because the crystal axis of the exit prism is not parallel to the interface, the E_o component undergoes birefringence at the interface. Consequently, it is split into two orthogonal linearly polarized beams, denoted as E_oo and E_oe, within the exit prism, as illustrated in Fig. 2(d). Here, we get:

The propagation direction of E_oo is the same as that of the incident beam, with polarization perpendicular to the crystal axis of the exit prism. Conversely, the propagation direction of E_oe deviates from the incident beam, with polarization parallel to the crystal axis of the exit prism. E_oo and E_oe travel independently backward within the exit prism and further diverge in their propagation directions after experiencing refraction at the exit surface.

For the E_e component within the entrance prism, through analysis similar to that for E_o, it is divided into two orthogonal linearly polarized beams, denoted as E_eo and E_ee, in the exit prism. The propagation direction of E_eo deviates from the incident beam, with polarization perpendicular to the crystal axis of the exit prism. In contrast, the propagation direction of E_ee aligns with the incident beam, with polarization parallel to the crystal axis of the exit prism. E_eo and E_ee travel independently backward within the exit prism and further diverge in their propagation directions after experiencing refraction at the exit surface. Here, we have:

After passing through a quasi-Wollaston prism, linearly polarized light is separated into two orthogonally polarized beams that deviate from the direction of the incident light, and one beam that continues in the same direction as the incident light. The deviation angles between the two orthogonally polarized beams and the incident beam can be calculated based on the structural angle of the prisms and the birefringence indices of the uniaxial crystal material. The beam with an unchanged direction can be considered as the light leakage of the incident probe beam, and its intensity is proportional to the intensity of the incident light. It is also associated with the crystal axis bias angle α of the exit prism. This beam is insensitive to the polarization direction of the incident light. Under balanced polarimeter measurement, this beam can be directly measured as a reference for signal normalization and can also be used as feedback for laser intensity stabilization.

As shown in Fig. 2(a), After passing through the quasi-Wollaston prism, the linearly polarized light beam is split into three beams, which are captured by the photodetectors D₁, D₂, and D₃, and converted into signals I_1QWP, I_2QWP, and I_REF, respectively. Considering the incident probe light intensity is I_0QWP, it can be demonstrated that:

From Eqs. (8) and (9), it is evident that the differential signal between the outputs of the two balanced photodetectors, D₁ and D₂, when multiplied by tan²(α) and normalized by twice the output of photodetector D₃, represents a measurement of the amplitude of the optical rotation angle. Furthermore, by analyzing Eqs. (3) through (8), we can deduce that the intensity of each beam is related to that of the incident beam. Consequently, the photon shot noise in each beam is not independent but exists in a state of quantum entanglement.

This measurement method is insensitive to fluctuations in the incident light intensity and benefits from the synchronous detection of the differential signal and the reference signal. The shot noise for each detector is correlated due to quantum entanglement, leading to a reduction in measurement uncertainty. Although the signal from each detector is reduced under the same incident light power compared to the typical beam splitter method, this can be compensated for by increasing the probe beam intensity by a factor of 1/cos²(α). However, as will be seen below, this increase is unnecessary because the reference intensity is less one-tenth of the total power.

In the case of θ = 0, the rations I_1QWP/I_0QWP and I_2QWP/I_0QWP are both equal to cos²(α)/2, while I_REF/I_0QWP equals sin²(α). The proportions of light power received by each detector as a function of α are shown in Fig. 4(a). Here the ratio I_1QWP/I_0QWP and I_2QWP/I_0QWP coincide, as indicated by the red line, and the blue line represents the variation of I_REF/I_0QWP with α. As α increase, I_1QWP/I_0QWP and I_2QWP/I_0QWP decrease, whereas I_REF/I_0QWP increases. The value of α, along with the power of the incident probe beam, determines the output amplitudes of the photodetectors and ultimately the uncertainty in the measurement of θ. Assuming N_ph photons pass through the atomic ensemble during measurement time τ with negligible absorption, the uncertainty in the measurement of θ, limited by photon shot noise as derived from Eq. (9), is given by:

 

Fig. 3. The schematic diagram of the experimental system setup utilizes a quasi-Wollaston prism as a beam splitter.

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Fig. 4. (a) Light intensity ratios of the three outgoing beams from the quasi-Wollaston prism as a function of the bias angle α; (b) Uncertainty increases, compared to the classical balanced polarimetry method, as a function of optical rotation angles.

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Equation (10) is derived from the uncertainty principle, which dictates that the photon numbers in each light beam fluctuate according to a separate Poisson distribution. However, as mentioned above, this does not align with our design. The explanation is as follows: The quasi-Wollaston prism cube functions as a physical beam splitter for light intensity. Due to quantum entanglement, the photon shot noise impacting the photodetectors is correlated; it is impossible to determine through which beam the noise photons will pass, but the numbers received by the detectors are proportional to the beam splitting ratios. Assuming N_ph photons pass through the atomic ensemble during measurement time τ with negligible absorption, the total shot noise uncertainty is given by the square root of N_ph. Consequently, the photon shot noise uncertainty for each detector, due to quantum entanglement, is

The uncertainty of θ, derived from a detailed analysis using Eq. (9), is given by:

3. Experimental system design and its performance analysis

3.1 Experimental system design

In this section, we present the design parameters of the QWP (quasi-Wollaston prism) and theoretically analyze the system's performance for the measurement method that utilizes QWP. Our approach is applicable to both unmodulated and modulated intensity probe beams. Figure 3 illustrates the schematic of the experimental setup used in our study. Typically, Faraday modulation techniques reduce noise arising from intensity fluctuations in the probe beam and from other sources. The amplitude of first-harmonic of the signal is a measure of the polarization rotation caused by the sample and is also proportional to the modulation amplitude. The essence of Faraday modulation is to introduce an amplitude modulation to the signals. Photo-elastic modulators (PEMs) can also be utilized for polarimetry, offering the advantage of higher modulation frequencies. In our setup, we employ a photo-elastic modulator to modulate the intensity of the incident probe light without altering the plane of polarization, while synchronously detecting the signals. As depicted in Fig. 3, the photo-elastic modulator is positioned between two Glan-laser polarizers. Each polarizer is oriented such that its transmission axis is at a 45° angle to the axis of the modulator. The Glan-laser polarizer on the right is used to produce linearly polarized probe light, while the Glan-laser polarizer on the left is used to control the overall light intensity. The PEM operates in DC mode when powered off and switches to light intensity modulation mode when powered on. To test and calibrate system performance, we utilize a rotation stage equipped with a half-waveplate to induce optical rotation.

3.2 Parameters design of the quasi-Wollaston prism

The light intensity ratios of the three outgoing beams are related to the crystal axis bias angle α, as depicted in Fig. 4(a) and according to Eq. (8). When θ equals 0, both the rations I_1QWP/I_0QWP and I_2QWP/I_0QWP are equal to cos²(α)/2, while the ratio I_REF/I_0QWP equals to sin²(α). The ratios I_1QWP/I_0QWP and I_2QWP/I_0QWP coincide, as indicated by the red line, and the blue line represents the variation of I_REF/I_0QWP with α. The two curves intersect at 35.264 degrees, when cos²(α)/2 equals sin²(α). The intersection points get displaced when θ deviates from 0, as shown in Fig. 4(a). In our design, with α set to approximately 17.5°, the ratios I_1QWP/I_0QWP and I_2QWP/I_0QWP are both roughly 0.455, while I_REF/I_0QWP is about 0.09. This balance ensures that all three photodetectors receive sufficient light power without significantly increasing measurement uncertainty. As Fig. 4(b) illustrates, the measurement uncertainty is independent of α and remains consistent at small optical rotation angles. To achieve uncertainty beyond the conventional quantum limit in measurement, it is entirely possible at a small optical rotation angle. As depicted in Fig. 4(b), within a significant range close to 0, the measurement uncertainty of the optical rotation angle is smaller than the conventional quantum limit. Notably, within a range of a few milliradians, the measurement uncertainty is almost 0.707 times that of conventional quantum noise limited. Therefore, our validation experiment will also be conducted within this range.

The prism features a 15 mm × 15 mm square entry pupil, which is more than adequate for the 3.25 mm waist diameter of the collimated probe beam. The essence of the prism’s aperture design is to ensure that there’s no beam obstruction of the beam on both the entry and exit surfaces of the prism. Apertures larger than 15 mm are also feasible, whereas those smaller than 10 mm are not recommended, as the waist of Gaussian beams does not encapsulate all energy. Beam separation at the prism’s exit surface can lead to beam obstruction or edge diffraction effects, thus potentially impairing the performance of photodetectors. The structural angles of the two right-angle prisms that comprise the quasi-Wollaston prism cube are both 45°, which is typically in factory. In our experiments, we used two quasi-Wollaston prisms made of calcite and α-BBO, respectively. The beam separation angle directed toward D₁ and D₂ is approximately 19.42° for the quasi-Wollaston prism made of calcite. For the quasi-Wollaston prism made of α-BBO, the beam separation angle is approximately 13.95°, with both beams being asymmetric to the direction of the incident probe beam.

3.3 Performance analysis utilizes an unmodulated probe beam

Under the condition of no modulation of the incident probe beam (DC mode), and for small optical rotation angles, the measurement uncertainty limited by photon shot noise is of the same order as that of the balanced polarimetry method when 0 < θ << α < π/4. The uncertainty can be improved by increasing the probe beam power. The I_REF beam can be used for monitoring light intensity and can also serve as feedback for light intensity stabilization. The uncertainty limited by photon shot noise for 2 mW incident light intensity at 770 nm (which is used in a K magnetometer) is

For the intensity-unmodulated probe beam, the PEM is powered off, so no retardation is applied to it. Signals from the differential channel and the reference channel are converted into output voltages, V_diff and V_c, respectively. We introduce the coefficients η₁ and η₃, which are associated with quantum efficiency [31] and the photoelectric conversion gain factors of the photodetectors, and then we get:

A pair of balanced-amplified photodetectors is used in the measurement. To prevent saturation of the balanced amplifier, the power difference between the optical input ports should be below the saturation power level. The maximum saturation power is approximately 30 µW for a typical balanced amplifier. Under the situation mentioned above, the measurement range is determined by the condition I_2QWP-I_1QWP< 30 µW, from which we obtain the maximum measurement range θ_max< 8.25 mrad.

One advantage of using unmodulated light mode is that the system becomes simpler and the PEM module can be entirely eliminated. Additionally, there's no need to use a lock-in amplifier to gather photoelectric signals. High-precision digital voltage or current acquisition cards are appropriate, which significantly reduces the system construction cost.

3.4 Performance analysis utilizes intensity-modulated probe beam

For the optical setup depicted in Fig. 3, the equations for the intensities of the light beams reaching the detectors can be derived as a function of time when in modulated mode. The general expression for the intensity function, I’, is given by:

Typically, the second harmonic is utilized for synchronous detection of the linearly polarized flux. The differential signal, which is the difference between the outputs of the two photodetectors D₁ and D₂, is produced by a high-speed transimpedance amplifier, specifically (I_2QWP-I_1QWP). The reference signal is the obtained from the output of the amplified photodetector D₃. The differential signal, I_diff, is:

We denote the modulated differential signal as S and the reference signal as S_c, which are defined as follows (for small rotation angles):

An important specification for a balanced amplifier is the common-mode rejection ratio (CMRR). Equal optical path lengths are critical for common-mode noise suppression, especially at high frequencies. Any difference in path length will introduce a phase difference between the two signals, which will decrease the noise reduction capability of the balanced detectors. In our experimental setup, the difference in optical path lengths for the balanced amplified photodetectors is less than 0.5 mm for both quasi-Wollaston prisms (QWPs) made of calcite and α-BBO. This suggests a CMRR better than 60 dB (modulation frequency 0.1 MHz, PDB210A, Thorlabs).

One advantage of intensity modulation of the incident probe light is that it can suppress noise arising from fluctuations and other causes, such as 1/f noise. Another advantage is that the optical rotation angle, θ, is obtained through the second harmonic amplitudes of the signals from both channels, which are proportional to the incident light intensity. After calibration, the constant k can be obtained with high precision. The retardation amplitude of the PEM is not a strict limitation for the measurement, as it affects the second harmonic amplitudes of the signals from the two channels synchronously.

4. Experimental results and discussion

4.1 Experiment setup and direct evidence

In our study, the experiment setup, as shown in Fig. 3, includes a photo-elastic modulator (PEM200, Hinds instrument, Inc.) situated between two Glan-laser polarizers (GL15-B, Thorlabs), each aligned at a 45° angle relative to the modulator axis. We employ a tunable diode laser (DL pro, TOPTICA) as the light source, which is fiber-coupled and collimated to produce a probe beam with an output waist diameter of 3.25 mm. A rotation stage equipped with a stepper motor and a half-waveplate (WPH05M-780, Thorlabs) is utilized to induce optical rotation. The half-waveplate's initial principal axis is aligned with the incident light's polarization direction, which is balanced with the QWP-based polarimeter. The optical rotation angle θ is controlled by rotating the half-waveplate. A quasi-Wollaston prism, in conjunction with photodetectors, functions as a polarimeter. Balanced amplified photodetectors (PDB210A, Thorlabs), comprising two well-matched photodiodes (D₁ and D₂) and an ultra-low noise, high-speed transimpedance amplifier, produce an output voltage (RF output) that is proportional to the difference in photocurrents from the two photodiodes. An additional amplified photodetector (D₃) (PDAPC2, Thorlabs), whose output is independent of θ, serves as the reference signal. For signal processing and recording, we use a lock-in amplifier (Zurich Instrument, HF2LI), with both reference and differential signals input synchronously. Practically, a focal lens is employed to focus the split beams onto the photodetectors. In experiment using a QWP made of α-BBO, the lens is omitted, whereas it is included when using a QWP made of calcite. The QWP, focal lens, balanced amplified photodetectors, and the amplified photodetector are all housed in a box to shield against stray light, as depicted in Fig. 5(a) (computers, laser controller, and rotation stage controller are not shown). Cross-sectional views of this assembly are presented in Fig. 5(b) and Fig. 5(c). The photo-elastic modulator operates at a modulation frequency of 50 kHz, resulting in an intensity modulation frequency of 100 kHz for the probe beam. The modulator's retardation amplitude is approximately π, enabling the intensity modulation depth to reach 100%. The rotation stage, driven by a stepper motor, has a resolution of 0.001° and features closed-loop control via a laser-engraved circular grating ruler with a resolution of 0.0005°.

 

Fig. 5. Picture of the experimental system setup utilizing a quasi-Wollaston prism as a beam splitter. (a) Picture of the experimental system; (b) Perspective view of the QWP-based polarimeter made of calcite; (c) Perspective view of the QWP-based polarimeter made of α-BBO.

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We present direct evidence that the quasi-Wollaston prism splits an incident beam into three separate outgoing beams. Through this prism, an object can produce three images, as demonstrated in Fig. 6. The central image exhibits the lowest clarity, with its intensity falling below 9% of the total power, resulting in a diminished contrast ratio.

 

Fig. 6. Evidences that the quasi-Wollaston prism separates an incident beam into three outgoing beams. (a) Image showing the performance of a QWP made of calcite; (b) Image showing the performance of a QWP made of α-BBO.

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4.2 Experimental results using unmodulated probe beam intensity

For the unmodulated probe beam intensity, the PEM was powered off, thus no retardation was applied to the photo-elastic modulator. Signals from the differential channel and the reference channel were converted to output voltages V_diff and V_c, respectively.

In this experiment, data recording was carried out incrementally. The rotation stage rotated stepwise from -0.032° to +0.032° (-558.505 µrad to +558.505 µrad), with each step corresponding to a rotation angle of 0.002° (34.907 µrad). This effectively changed the optical rotation angles θ from -0.064° to +0.064° (-1117.011 µrad to +1117.011 µrad), with each adjustment being 0.004° (69.813 µrad). At each position, both the reference and differential signals were simultaneously input into the lock-in amplifier. The amplitudes of these signals (V_diff and V_c) were demodulated, sampled, and recorded by the lock-in amplifier. Every time the rotation stage rotated to a specific position, it collected demodulated values of the differential and reference signals for a duration of 2 s. The signs of V_diff were judged from the original signals displayed on the scope. For an unmodulated beam intensity, the demodulation value was 2^1/2 times the absolute value of the input amplitude. The demodulation frequency for the lock-in amplifier was set to 0 Hz for both channels. The sampling rate for the demodulation value is 3598 Hz, and the data was segmented into100 parts per second, meaning that there were 36 demodulation data samples within a measurement time of 0.01 s. These 36 samples were used for averaging values and calculating uncertainty. Due to rotation stage's resolution of 0.001°, its zero point did not coincide with the zero point of the polarimeter. We analyzed the measurement uncertainty and sensitivity of the optical rotation angles. Additionally, we examined the stability and changes in uncertainty of both the reference and differential signals. The background noise of the detector connected to the lock-in amplifier was measured when the laser light was turned off. An analysis was also conducted on the sources of noise in the measurement of optical rotation angles and their impact on the measurement results.

4.2.1 Results in DC mode

For different optical rotation angles, the ratios of the demodulated values (after compensating for background offset) of the differential signals to the reference signals, as well as the corresponding fitted characterization line, are presented in Fig. 7(a). The relationship between the optical rotation angles and the ratios can be expressed as: θ = -51.446 + 783.546 × (V_diff/V_c), where the unit of θ is microradians (µrad). The linearity, R², is 0.99975, indicating that the measurement of the optical rotation angles exhibits excellent linearity. Figure 7(b) displays the maximum standard deviations of the ratios. Based on k_DC = 783.546(±2.202) and the maximum standard deviation of V_diff/V_c, which is 6.0 × 10⁻⁶, the measurement uncertainty of the optical rotation angles is 4.70(1) nrad for each measurement duration of 0.01 seconds. Consequently, the sensitivity (short-term resolution) of the measurement is 4.70(1) nrad × (0.01 s)^1/2 = 4.70(1) × 10⁻¹⁰rad/Hz^1/2.

 

Fig. 7. Experimental results for the QWP made of α-BBO in DC mode. (a) The ratios of the demodulated amplitudes of differential signals to reference signals as a function of optical rotation angles; (b) The uncertainties associated with these ratios plotted against optical rotation angles; (c) The ratios of the demodulated amplitudes of differential signals to reference signals as a function of optical rotation angles in a repetitive measurement; (d) The uncertainties associated with these ratios plotted against optical rotation angles in a repetitive measurement.

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The experimental results of the system’s repeatability are shown in Fig. 7(c) and 7(d). For different optical rotation angles, the ratios of the demodulated values of the differential signals to the reference signals, as well as the corresponding fitted characterization line, are presented in Fig. 7(c). The relationship between the optical rotation angles and the ratios can be expressed as: θ = -27.097 + 741.395 × (V_diff/V_c). The linearity, R², is 0.99979, indicating that the repetitive measurement exhibits excellent linearity. Figure 7(d) shows the maximum standard deviations of the ratios. Based on k_DC = 741.395(±1.926) and the maximum standard deviation of V_diff/V_c, which is 6.169 × 10⁻⁶, the measurement uncertainty of the optical rotation angles is 4.57(1) nrad for each measurement duration of 0.01 seconds. It is worth noting that the uncertainty and sensitivity of the measurement results are still superior to conventional quantum noise limits at each optical rotation angle position.

The change in the coefficient (k_DC) in the repeated experimental results is attributed to the use of the rotation stage input value with errors as the standard true value to calibrate the response curve. This calibration process in line with the closed-loop control accuracy of only 0.0005 degrees of the rotation stage that we utilized. To achieve more stable and accurate linear fitting coefficients, additional high-precision auxiliary equipment is required to synchronously measure the angle of the rotation stage and input it as the true value of the angle.

Due to the closed-loop error of the rotation stage being 0.0005 degrees, the expected change in the fitting coefficient k_DC is approximately ±2.2% (± 2^1/2× 0.0005/0.032), as confirmed in the repeatability experiment. Consequently, the standard deviation of the method in sequential optical rotation angle cycles will be close to 0.0005 degrees. This limitation is determined by experimental conditions, particularly the accuracy of the rotation stage that establishes the standard angle.

When the optical rotation angle is set to positions of 0, 279.253, 488.692, 767.945, 977.384 µrad, the data for the reference signal sampled over 2 seconds is shown in Fig. 8(a). Each sample point represents the average of 36 subsamples taken within 0.01 seconds, and the corresponding uncertainty for the sample points is presented in Fig. 8(b). Similarly, the data for the differential signal over 2 seconds is depicted in Fig. 8(c), with each sample point being the average of 36 subsamples within 0.01 seconds; the corresponding uncertainty is illustrated in Fig. 8(d). The background noise is also sampled, and the data for it over 2 seconds is shown in Fig. 8(e), with the uncertainty for each sample point displayed in Fig. 8(f). The ratio of the differential signal to the reference signal, after compensating for background offset, is shown in Fig. 8(g), and the corresponding uncertainty for each sample point is indicated in Fig. 8(h). From Fig. 8(h), it is evident that for a sampling time of 0.01 seconds, the measurement uncertainty of the ratio remains within a certain range. Throughout the experiment, the maximum measurement uncertainty for a single 0.01-second measurement was 6.0 × 10⁻⁶, as previously mentioned, corresponding to an optical rotation angle measurement uncertainty of 4.7 nrad.

 

Fig. 8. Data for optical rotation set to position of 0, 279.253, 488.692, 767.945, and 977.384 microradians. (a) The reference signals; (b) Uncertainty of the reference signals; (c) Differential signals; (d) Uncertainty of the differential signals; (e) Background offset of the differential signal and reference signals; (f) Uncertainty of the background noise from the differential and reference channels; (g) Ratios of the differential signal to the reference signal; (h) Uncertainty of the ratios.

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4.2.2 Analysis of the impact of noise sources in DC mode

Although background offset compensation can yield the true value of the signal, the impact of background offset and noise on the measurement results depends on the measurement method and the data processing technique employed. In our experiment, the reference signal is independent of the optical rotation angle throughout the entire testing process and depends solely on the intensity of the incident probe beam and the structure of the prism. According to Eq. (13), its compensation merely constitutes a minor correction to the system parameter (k_DC) and does not alter the dependency of the optical rotation angle on the ratio. The actual collected reference signal's noise includes photon shot noise, background noise (detector noise, electronic circuit noise) and air current-induced light intensity disturbance. According to the principle of composite uncertainty, the uncertainty of the true value of the reference signal will increase due to the data processing method, which in turn increases the uncertainty of the ratio V_diff/V_c, adversely affecting the measurement uncertainty. Ideally, the uncertainty of the true value of the reference signal should be less than the standard deviation of the actual collected reference signal. When the background offset of the reference signal channel is relatively small, this compensation is not necessary.

Similarly, for differential signals, background offset compensation can reveal the true value of the signal; however, this adjustment only corrects the signal's zero-crossing point–that is, the translation of the fitting curve along the horizontal axis–and does not change the relationship between the optical rotation angle and the ratio. The background noise of the differential channel is merely one-tenth of the measurement signal noise and does not significantly affect the uncertainty of differential signals. Therefore, when obtaining the system parameter k_DC according to Eq. (13), although our calculation accounts for the background shift, compensating for the background offset of the measurement signal is not necessary, and background compensation cannot significantly improve the uncertainty of the measurement results. This is because the background noise of the two channels is on par with the noise of the reference signal, which is only one-tenth of the noise in the differential signal. As observed from Fig. 8(b), (d), (f), and (h), the uncertainty of the measurement results primarily stems from the noise of the differential signal. The noise in differential signals is mainly constrained by the common mode rejection ratio of the amplified balanced photodetectors. Fluctuations in the probe beam intensity will induce strong noise in the differential signal, which is more pronounced in the light intensity modulation mode described below.

The error sources affecting the calibration of k_DC also include the closed-loop control accuracy of the rotation stage. Owing to the use of multiple pairs of measurement data for linear fitting, the positioning accuracy errors tend to each other out due to averaging. At each optical rotation angle position, the mechanical vibrations and air currents induce changes in the optical rotation angle during the measurement process, which are additional sources of noise in differential signals. The noise caused by vacuum quantum fluctuations was ignored in the experiment.

All noise is detrimental to measurements, as it hinders our ability to accurate assess a value and limits the smallest change we can detect. Small changes in the optical rotation angle may be masked by noise. Excessive noise can cause the measurement results to deviate from the true value. Any measures that can reduce noise in measurements are beneficial.

4.3 Experiment results using modulated probe beam intensity

For the modulated probe beam intensity, a modulation frequency of 50 kHz was applied to the PEM, with A₀ ≈ π, allowing the intensity modulation depth of the probe beam to reach 100%. Signals from the differential channel and the reference channel were converted into output voltages, S and S_c, respectively. In this experiment, data recording was carried out incrementally. The rotation stage rotated stepwise from -0.07° to +0.07° (-1221.73 µrad to +1221.73 µrad), effectively changed the optical rotation angles θ from -0.14° to +0.14° (-2443.46 µrad to +2443.461 µrad). At each position, both the reference and differential signals were simultaneously input into the lock-in amplifier. Every time the rotation stage rotated to a specific position, it collected demodulated values of the differential and reference signals for a duration of 2 seconds. The signs of S were judged from the original signals displayed on the oscilloscope. For modulated beam intensity, the demodulation value was 1/2^1/2 times the absolute value of the AC component amplitudes of the input voltages when the lock-in amplifier's demodulation frequency was locked to that of the reference channel. The sampling rate for the demodulation value was 3598 Hz, and the data was segmented into100 parts per second, meaning that there were 36 demodulation data samples within a measurement time of 0.01 seconds. These 36 samples were used for averaging values and calculating uncertainty. We analyzed the measurement uncertainty and sensitivity of the optical rotation angles.

4.3.1 Results in probe beam intensity modulation mode

For different optical rotation angles, the ratios of the demodulated values of the differential signals to the reference signals are shown in Fig. 9(a). Only 31 data points, ranging from 349.066 µrad to 2443.416 µrad, are used for line fitting, and the corresponding fitted characterization line is displayed in Fig. 9(a). The relationship between the optical rotation angles and the ratios can be expressed as: θ = -245.902 + 2224.214×|S/S_c|, where the unit of θ is µrad. The linearity, R² = 0.99969, indicates that the measurement of the optical rotation angles has good linearity within the fitted line range. The standard deviations of the ratios are shown in Fig. 9(b). According to k_AC = 2224.214(±7.29) and the maximum standard deviation of |S/S_c| which is 2.2 × 10⁻⁵, the measurement uncertainty of the optical rotation angle is 48.9 nrad for each measurement duration time of 0.01 seconds. Consequently, the sensitivity (short-term resolution) of the measurement is 48.9(2) nrad × (0.01 s)^1/2 = 4.89(2) × 10⁻⁹rad/Hz^1/2.

 

Fig. 9. Experimental results of a QWP made of α-BBO under light intensity modulation mode. (a) The ratios of the demodulated amplitudes of differential signals to reference signals plotted against optical rotation angles; (b) The uncertainties of the ratios plotted against optical rotation angles.

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Under light intensity modulation mode, the system should operate within the linear aera until the common mode effect is eliminated. The accuracy is limited by the nonlinearity of using 2θ to approximate sin2θ. Below 2.5 mrad, the largest error from this approximation would be 2 × 10⁻⁸rad, which is small but lager than the measurement resolution.

4.3.2 Impact of common-mode rejection in light intensity modulation mode

As mentioned above, equal optical path lengths are critical for common-mode noise suppression. Even though the difference in optical path lengths for balanced amplified photodetectors is less than 0.5 mm, the difference in wire lengths between the photodetectors and the transimpedance amplifier input ports introduced a phase difference. This decreased the noise reduction capability of the balanced detectors. The consequent effects include distortion of the zero-cross performance in light intensity modulation mode and reduced sensitivity across the entire range. In AC input coupling mode, Fig. 10(a) displays the differential signal obtained from experiments near a zero optical rotation angle, and it can be observed that the shape of the differential signal is disrupted. Under DC input coupling mode, the ideal differential signal waveform is depicted in Fig. 10(b), whereas in AC input coupling mode, the waveform is shifted along the longitudinal axis. In the presence of common mode interference, even if the optical rotation angle is zero, the differential signal in light intensity modulation mode still exhibits AC components. The theoretical simulation is represented by the curve in Fig. 10(c).

 

Fig. 10. The impact of common-mode rejection in light intensity modulation mode. (a) Experimental results; (b) Ideal differential signals with an infinite common-mode rejection ratio in DC input coupling mode; (c) Theoretical simulation prediction of the differential signal with common-mode interference (-10 mm difference in optical path lengths).

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There are several measures available to mitigate common mode interference. The first is to employ a highly transparent glass compensation plate to compensate for the equivalent optical path difference caused by any factor. The second is to alter the installation position of the detectors directly, although this is more feasible for split detectors.

4.4 Analysis of the impact of using a half-wave plate

In our experiment setup, a half-waveplate, which is specific to the wavelength of 780 nm, is used to produce the optical rotation angles. However, the laser wavelength used here is 770 nm, resulting in ellipticity of the incident polarization and nonlinearity in the optical rotation angles. Theoretical analysis shows that the ellipticity is less than 5.1 × 10⁻⁵ for θ < 2.5 mrad. The ellipticity of the probe beam decreases the differential signals, and with only minor changes in the system parameter constant k, the measurement sensitivity and uncertainty remain of the same order.

When the wavelength of the half-wave plate used coincides with that of the light source, the probe beam behind the half-wave plate will be strictly linearly polarized. In this case, the energy of the minor-axis in the elliptical beam due to wavelength mismatch will revert to the linearly polarized light. Consequently, the negative effects caused by wavelength detuning are eliminated, enhancing the differential signal while keeping the reference signal independent of the optical rotation angle. The ratios of differential signals to reference signals will slightly increase. However, since the optical rotation angle is related to wavelength detuning, under the same rotation angle of the rotation stage, the optical rotation angle produced by the rotation of the half-wave plate will be exactly twice the rotation angle of the rotation stage when the wavelength is consistent. Compared to using a half-wave plate with wavelength detuning, the resulting optical rotation angle will experience a slight increase, with a relative amplitude of change of about 4 × 10⁻⁴. As the rotation stage angle is used as the true value to calibrate the system, this will lead to a slight change (decrease) in the system parameter k_DC, with a relative amplitude of about 4 × 10⁻⁴. The differences between the differential signals generated by a half-wave plate with the same wavelength as the light source and the differential signals generated by a half-wave plate with wavelength detuning are shown in Fig. 11.

 

Fig. 11. The differences between the differential signals generated by a detuned half-wave plate and the differential signals generated by a half-wave plate that matches the light source wavelength. The center wavelength of the light source is 770 nm.

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5. Conclusion

In conclusion, we propose a novel method for measuring small optical rotation angles based on beam splitter polarimetry detection using a quasi-Wollaston prism under quantum entanglement. The measurement uncertainty of optical rotation under quantum entanglement surpasses the conventional photon shot noise limitation. We achieved highly sensitive measurement of the optical rotation angles of the light polarization plane. The primary limitation under AC mode is the common mode rejection ratio. The DC mode is recommended for its simpler system configuration.

In future research, our main focus will be on enhancing the common mode rejection ratio to improve the performance of modulation modes within our system. We anticipate suppressing low-frequency noise in signals modulated to high frequencies, thereby achieving improved measurement uncertainty.