By detecting the material modulation for the polarization of incident light, the composition and fine structure of the material can be revealed by using specialized optical techniques and computational methods [1,2]. Compared to intensity-based methods, polarization technologies can obtain multi-dimensional characteristics and improve the sensitivity and specificity of material detection. Currently, polarization measurement is attracting increasing attention across various research fields, including material characterization [3], biomedical studies [4], and clinical applications [5].

Interferometric imaging, including optical coherence tomography (OCT) and holographic microscopy, can extract three-dimensional (3D) structures from interference signals. Integrating interferometric imaging with polarization measurement has become an effective approach for obtaining 3D polarization information. Among these methods, polarization-sensitive OCT (PS-OCT) [6–8], as a representative interferometric imaging method with polarization measurement function, can offer high-resolution, non-invasive, and non-destructive imaging [9,10]. By introducing polarization modules, PS-OCT provides a series of polarization parameters related to birefringence and dichroism, demonstrating great potential in ophthalmology [11], cardiology [12], dermatology [13], and more.

With the growing demand for diagnostic applications, polarization measurement methods of interferometric imaging have continually evolved. At present, there are three kinds of methods. (1) Single-input methods [14–17] use the single polarized probe light and can obtain polarization information in a single acquisition. However, this method limits the variety of polarization parameters and may introduce phase jump errors [17]. (2) Multi-input methods [18–20] use multiple polarized probe lights and offer accurate and comprehensive polarization parameters. However, multiple signals are acquired in a time sequence, decreasing the acquisition speed. (3) Single-detector-based methods [21–23] simplify the setup. However, they rely on the time-sequenced or the space-division method, limiting the acquisition speed or the axial imaging range.

As a representative multi-input method, polarization modulation methods control the polarization state of the probe light by an electro-optic modulator [19,24], obtaining accurate and rich polarization information while reducing the acquisition speed. Modulating polarization in the spectral domain rather than the time domain offers the potential to overcome the limitation of acquisition speed. Currently, this idea is primarily used in polarimetry [25–27] represented by polarization spectral intensity modulation (PSIM). These methods employ wavelength-sensitive birefringent materials and the polarizer to modulate polarization as an intensity function of wavelength, followed by demodulating the spectrum to extract polarization information.

However, integrating PSIM into interferometric imaging has challenges. PSIM causes the polarization to change continuously with the wavelength, which makes it difficult to meet the achromatic polarization state required for current interferometric polarization modulation methods. Besides, PSIM introduces an intensity modulation on the spectrum, which could interfere with depth-resolved imaging. To address these challenges, specific polarization calculation methods and spectral-polarization-modulation (SPM) strategies should be developed to mitigate intensity modulation and ensure compatibility with interferometric imaging.

In this study, we proposed a rapid and simple interferometric polarization measurement method and integrated it into an OCT system to establish an advancing interferometric imaging system called SPM-OCT. This method can extract birefringent and dichroic parameters from the polarization-modulated signal without reducing the acquisition rate or the axial imaging range. The standard polarization elements, a skinned mouse leg, and gold nanorods (GNRs) were imaged by SPM-OCT and a home-made PS-OCT, respectively. A comparison of their imaging results proved that the SPM-OCT system can obtain the birefringence and dichroic parameters precisely and avoid the phase jump errors in the polarization image.

Besides, label-free interferometric imaging methods face challenges in distinguishing molecular signals from the background or the incoherent processes. To achieve molecular imaging, specific probes are typically required [28]. GNRs, as multifunctional probes, offer great potential for interferometric molecular imaging in oncology [29], ophthalmology [30], and respiratory medicine [31]. Intensity-based OCT molecular imaging requires a probe with different scattering intensities than the tissue [32,33]. Although studies have shown that GNRs can enhance the intensity of OCT [34], it is still difficult for intensity-based OCT to distinguish GNRs from biological tissues with high contrast. By comparing the polarization parameters of biological tissues and GNRs, it was found that the dichroic parameters obtained from our proposed method can distinguish GNRs from biological tissues.

In this study, we proposed a polarization parameter calculation based on polarization modulation in interferometry. The schematic diagram of the interferometric system is shown in Fig. 1. Assume that the linear polarization direction of the light source is modulated in the $\eta$ domain. (The $\eta$ domain can be the time domain or wavenumber domain, etc.) The Jones vector of this light source electric field $E_l$ can be expressed as $$E_l=\left[\begin{array}{c}{E}_{lH}\\ E_{lV}\end{array}\right]=E_0·\left[\begin{array}{c}\cos ({\omega }_0·\eta )\\ \sin ({\omega }_0·\eta )\end{array}\right],$$(1)where $E_0$ is the electric field scalar; ${\omega }_0$ is the modulation frequency of the linear polarization direction modulated in the $\eta$ domain.

The light output from the light source is divided into the reference arm and the sample arm through the beam splitter, respectively. The reference light returns directly to the beam splitter without changing the polarization state, whose electric field ($E_r$) can be expressed as $E_r=E_l$. In the sample arm, light illuminates the sample and is reflected into the beam splitter. The Jones matrix of the sample is defined as $J_s$. Therefore, the electric field of the sample light can be expressed as $$E_s=J_s·J_s·E_l·e^{-ik\mathrm{\Delta }z},$$(2)where $k$ is the wavenumber of the sample light; $\mathrm{\Delta }z$ is the optical length difference between the reference and sample arms.

The returned sample light and the reference light interfere with each other at the beam splitter. The interference terms $A_H$ and $A_V$ in the horizontally ($H$) and vertically ($V$) polarized channels are $$A_H(k,\eta )=E_{rH}^{*}{E}_{sH}+E_{rH}{E}_{sH}^{*},$$(3)$$A_V(k,\eta )=E_{rV}^{*}{E}_{sV}+E_{rV}{E}_{sV}^{*}.$$(4)

The interference signal ($L$) obtained by the detector is $$L(k,\eta )=A_H(k,\eta )+A_V(k,\eta ).$$(5)

On the one hand, assume that the sample at the optical length difference $\mathrm{\Delta }z$ is birefringent and the Jones matrix of the birefringent sample $J_b$ can be expressed as $$J_b({\delta }_b,{\theta }_b)=\left[\begin{array}{cc}{\cos }^2({\theta }_b)+{\sin }^2({\theta }_b)e^{-i{\delta }_b}& \cos ({\theta }_b)\sin ({\theta }_b)(1-e^{-i{\delta }_b})\\ \cos ({\theta }_b)\sin ({\theta }_b)(1-e^{-i{\delta }_b})& {\cos }^2({\theta }_b)e^{-i{\delta }_b}+{\sin }^2({\theta }_b)\end{array}\right],$$(6)where ${\delta }_b$ is the retardation of the birefringent sample; ${\theta }_b$ is the optic axis of the birefringent sample.

Based on Eqs. (1)–(6), the interference signal of the birefringent sample is proportional to the following formula: $$L(k,\eta )\propto \cos (k·\mathrm{\Delta }z){\cos }^2({\theta }_b-\frac{{\omega }_0}{2}·\eta )+\cos (k·\mathrm{\Delta }z+2{\delta }_b){\sin }^2({\theta }_b-\frac{{\omega }_0}{2}·\eta ).$$(7)

By performing an inverse Fourier transform on the interference signal in the $k$ domain to obtain the modulation signal in the $z$ domain and squaring it, the square amplitude of the modulation signal of birefringence in the $z$ domain ($T_b$) can be obtained as $$T_b(\mathrm{\Delta }z,\eta )={|{\mathcal{F}}_k^{-1}(L(k,\eta ))|}^2\propto 1+{\cos }^2({\delta }_b)+{\sin }^2({\delta }_b)\cos (4{\theta }_b-2{\omega }_0·\eta ).$$(8)

Then, the $T_b(\mathrm{\Delta }z,\eta )$ is inversely Fourier transformed from the $\eta$ domain to the $\omega$ domain as follows: $$M_b(\mathrm{\Delta }z,\omega )={\mathcal{F}}_{\eta }^{-1}(T_b(\mathrm{\Delta }z,\eta )).$$(9)

Based on Eqs. (8) and (9), the amplitudes of $M_b$ at $\omega =0$ and $\omega =2{\omega }_0$ are proportional to the following formula: $$|M_b(\mathrm{\Delta }z,0)|\propto 1+{\cos }^2({\delta }_b),$$(10)$$|M_b(\mathrm{\Delta }z,2{\omega }_0)|\propto {\sin }^2({\delta }_b).$$(11)

Besides, the phase of $M_b$ at $\omega =2{\omega }_0$ is as follows: $$\text{Angle}(M_b(\mathrm{\Delta }z,2{\omega }_0))=4{\theta }_b.$$(12)

The retardation ${\delta }_b$ and the optic axis ${\theta }_b$ of the birefringent sample can be calculated as follows: $${\delta }_b=\arcsin \left(\sqrt{\frac{2|M_b(\mathrm{\Delta }z,2{\omega }_0)|}{(|M_b(\mathrm{\Delta }z,0)|+|M_b(\mathrm{\Delta }z,2{\omega }_0)|)}}\right),$$(13)$${\theta }_b=\frac{\text{Angle}(M_b(\mathrm{\Delta }z,2{\omega }_0))}{4}.$$(14)

On the other hand, assume that the sample is dichroic and the Jones matrix of the sample $J_d$ can be expressed as $$J_d(D_H,D_V,{\theta }_d)=\left[\begin{array}{cc}{D}_H{\cos }^2({\theta }_d)+D_V{\sin }^2({\theta }_d)& (D_H-D_V)\cos ({\theta }_d)\sin ({\theta }_d)\\ (D_H-D_V)\cos ({\theta }_d)\sin ({\theta }_d)& D_H{\sin }^2({\theta }_d)+D_V{\cos }^2({\theta }_d)\end{array}\right],$$(15)where $D_H$ and $D_V$ are attenuation ratios in horizontal and vertical directions of the dichroic sample, respectively; ${\theta }_d$ is the transmission axis of the dichroic sample.

Based on Eqs. (1)–(5) and (15), the interference signal of a dichroic sample is proportional to the following formula: $$L(k,\eta )\propto \cos (k·\mathrm{\Delta }z)((D_H^2-D_V^2)\cos (2{\theta }_d-{\omega }_0\mathrm{\Delta }\eta )+(D_H^2+D_V^2)).$$(16)

By performing an inverse Fourier transform on the interference signal in the $k$ domain to obtain the modulation signal in the $z$ domain, the amplitude of the modulation signal of dichroism in the $z$ domain ($T_d$) can be expressed as follows: $$T_d(\mathrm{\Delta }z,\eta )=|{\mathcal{F}}_k^{-1}(L(k,\eta ))|\propto (D_H^2-D_V^2)\cos (2{\theta }_d-{\omega }_0·\eta )+(D_H^2+D_V^2).$$(17)

Then, the $T_d(\mathrm{\Delta }z,\eta )$ is inversely Fourier transformed in the $\eta$ domain to obtain the modulation signal in the $\omega$ domain, $$M_d(\mathrm{\Delta }z,\omega )={\mathcal{F}}_{\eta }^{-1}(T_d(\mathrm{\Delta }z,\eta )).$$(18)

Based on Eqs. (17) and (18), the amplitudes of $M_d$ at $\omega =0$ and $\omega ={\omega }_0$ are proportional to the following formula: $$|M_d(\mathrm{\Delta }z,0)|\propto D_H^2+D_V^2,$$(19)$$|M_d(\mathrm{\Delta }z,{\omega }_0)|\propto D_H^2-D_V^2.$$(20)

Besides, the phase of $M_d$ at $\omega ={\omega }_0$ is as follows: $$\text{Angle}(M_d(\mathrm{\Delta }z,{\omega }_0))=2{\theta }_d.$$(21)

We defined the diattenuation ratio ${\delta }_d$ to describe the magnitude of the dichroism. Based on Eqs. (17) and (18), the diattenuation ratio ${\delta }_d$ and the transmission axis ${\theta }_d$ of the dichroic sample can be calculated as $${\delta }_d=\frac{|D_H^2-D_V^2|}{|D_H^2+D_V^2|}=\frac{|M_d(\mathrm{\Delta }z,{\omega }_0)|}{|M_d(\mathrm{\Delta }z,0)|},$$(22)$${\theta }_d=\frac{\text{Angle}(M_d(\mathrm{\Delta }z,{\omega }_0))}{2}.$$(23)

It can be seen that when the modulation frequency of the source is ${\omega }_0$, the birefringence and dichroic parameters can be calculated by demodulating the modulation signal at the $2{\omega }_0$ and ${\omega }_0$ frequencies, respectively. That is to say, the frequency of the modulation signal corresponding to the birefringence information is twice the frequency of the modulation signal corresponding to the dichroism information.

In this study, we proposed an SPM method that can obtain a complete polarization modulation signal in the spectral domain by a single acquisition. The SPM module is shown in Fig. 2. It includes the vertical linear polarizer (LP), the multi-order wave plate (MP) with the fast axis direction being $\pi /4$ rad to the LP direction, and the vertical achromatic quarter wave plate (AP).

The light passing the polarizer is in a vertical linear polarization state. Its Jones vector ($E_{\mathrm{in}}$) can be expressed as $$E_{\mathrm{in}}=E_0·\left[\begin{array}{c}1\\ 0\end{array}\right],$$(24)where $E_0$ is the electric field scalar.

The wave plate can be considered as a birefringent device. Based on Eq. (6), the Jones matrix of the MP with the fast axis direction being $\pi /4$ rad to the vertical linear polarizer direction can be expressed as $$J_{\mathrm{MP}}=J_b({\delta }_{\mathrm{MP}},\pi /4)=\left[\begin{array}{cc}1/2+(1/2)·e^{-i{\delta }_{\mathrm{MP}}}& (1/2)·(1-e^{-i{\delta }_{\mathrm{MP}}})\\ (1/2)·(1-e^{-i{\delta }_{\mathrm{MP}}})& (1/2)·e^{-i{\delta }_{\mathrm{MP}}}+1/2\end{array}\right],$$(25)where ${\delta }_{\mathrm{MP}}=\mathrm{\Delta }n·d·\chi$ is the retardation of the MP; $\mathrm{\Delta }n$ is the birefringence of the MP; $d$ is the thickness of the MP; $\chi$ is the wavenumber of the light. It should be noted that to clearly explain the principle of the proposed method, we use $k$ and $\chi$ to represent the wavenumber in OCT imaging and the SPM method, respectively.

Besides, the Jones matrix of the AP with the fast axis direction being 0 rad to the vertical linear polarizer direction can be expressed as $$J_{\mathrm{AP}}=J_b(\pi /2,0)=\left[\begin{array}{cc}1& 0\\ 0& -i\end{array}\right].$$(26)

Therefore, the Jones vector of the output light from the SPM module ($E_{\mathrm{out}}$) can be expressed as $$E_{\mathrm{out}}=J_{\mathrm{AP}}·J_{\mathrm{MP}}·E_{\mathrm{in}}\propto \left[\begin{array}{c}\cos ({\delta }_{\mathrm{MP}}(\chi )/2)\\ \sin ({\delta }_{\mathrm{MP}}(\chi )/2)\end{array}\right]=\left[\begin{array}{c}\cos ({\gamma }_m·\chi )\\ \sin ({\gamma }_m·\chi )\end{array}\right],$$(27)where ${\gamma }_m=\mathrm{\Delta }n·d/2$ represents half the product of the birefringence and thickness of the MP.

It can be seen that the Jones vector of the output light is the same as the expression of the polarization-modulated light source required for interferometric imaging with polarization modulation [Eq. (1)]. To avoid potential dimensional misunderstandings, we used $\gamma$ to specifically refer to the polarization modulation frequency in the $\chi$ domain. Therefore, by applying the SPM module, a rapid SPM-OCT system can be established, and the polarization parameters can be obtained from the spectral polarization modulation signal.

The schematic diagram of the SPM-OCT system is shown in Fig. 3. The source light is emitted from a super luminescent diode (SLD) (BLM2-D, Superlum) with a center wavelength of 840 nm and a 3 dB bandwidth of 100 nm. Then, the light passes through a three-paddle polarization controller (PC) and is coupled to the SPM module. The SPM module includes a vertical linear polarizer (LP), two multi-order wave plates (MPs) (WPMH05M-830, Thorlabs) with the fast axis direction being 45 deg to the LP direction, and a vertical achromatic quarter wave plate (AP) (AQWP05M-980, Thorlabs). The purpose of using two MPs is to generate a suitable polarization modulation frequency that is easier to distinguish.

The output light of the SPM module is separated into the sample and reference arms by a 50:50 non-polarizing beam splitter (BS). The reference light passes a dispersion compensator (DC) and a neutral density filter (ND) and is reflected by a reference mirror (RM). In the sample arm, after backscattering from the sample through the galvanometer scanning mirror (GV) and the scanning lens (SL), the sample light returns to the BS and interferes with the reference light. Then, the interference light is acquired by the commercial spectrometer (SP) (Cobra-S 800, Wasatch Photonics). The optical fiber used to connect the spectrometer to the collimator is a non-polarization-maintaining, low-birefringence optical fiber. Our system is performed with an axial resolution of $\sim 3.4\mathrm{\mu m}$ in air and a transverse resolution of $\sim 8\mathrm{\mu m}$ in the focal plane. The spectrometer operates at a 25 kHz line rate. Factors limiting the resolution include the central wavelength and bandwidth of the light source as well as the numerical aperture and aberrations of the objective.

Besides, to verify the imaging capability of the SPM-OCT system, we also used the imaging result from a home-made single-input PS-OCT system as a reference. The accuracy of this PS-OCT system has been verified [35], which has been applied in many biological studies [36–38].

The flowchart of the SPM-OCT imaging algorithm is shown in Fig. 4(a). The specific steps are as follows.

Step 1: Acquire the spectral signal $S(k)$ obtained by the SPM-OCT system.

Step 2.1: Perform a one-dimensional (1D) inverse Fourier transform (${\mathcal{F}}^{-1}$) on the interference signal in the $k$ domain to obtain the spatial signal $S(z)$ in the $z$ domain as follows: $$S(z)={\mathcal{F}}_k^{-1}(S(k)).$$(28)

Step 2.2: The intensity information $I(z)$ of SPM-OCT is calculated as follows: $$I(z)={|S(z)|}^2.$$(29)

Step 3.1: Split the spectral signal $S(k)$ to obtain the split-spectral interference signal $L(k,\chi )$. The schematic diagram of the split-spectrum method is shown in Fig. 4(b) The interference signal is divided into multiple narrow-band interference signals by multiplying the interference signal with narrow-bandwidth Gaussian windows.

The premise for our proposed method is to distinguish the signal of the modulation frequency corresponding to the birefringence parameter (${\gamma }_b=2{\gamma }_m$) and the signal of the modulation frequency corresponding to the dichroism parameter (${\gamma }_d={\gamma }_m$) in the $\gamma$ domain. The number and bandwidth of windows should be set to reduce the overlap between the signal of ${\gamma }_b$ and ${\gamma }_d$. The window interval is equal to the quotient of the spectral bandwidth of the light source and the number of windows. According to the actual measurement results, we set the following parameters. The number of windows is 64. The bandwidth of each window is about $850\mathrm{rad}·{\mathrm{cm}}^{-1}$. The window interval is about $150\mathrm{rad}·{\mathrm{cm}}^{-1}$.

Step 3.2: Calculate the modulation signals of birefringence ($M_b(z,\gamma )$) and dichroism ($M_d(z,\gamma )$) as follows: $$M_b(z,\gamma )={\mathcal{F}}_{\chi }^{-1}({|{\mathcal{F}}_k^{-1}(L(k,\chi ))|}^2),$$(30)$$M_d(z,\gamma )={\mathcal{F}}_{\chi }^{-1}(|{\mathcal{F}}_k^{-1}(L(k,\chi ))|).$$(31)

Step 3.3: Calculate the retardation ${\delta }_b$, the optic axis ${\theta }_b$, the diattenuation ratio ${\delta }_d$, and the transmission axis ${\theta }_d$ of the sample as follows: $${\delta }_b=\arcsin \left(\sqrt{\frac{2|M_b(z,2{\gamma }_m)|}{(|M_b(z,0)|+|M_b(z,2{\gamma }_m)|)}}\right),$$(32)$${\theta }_b=\frac{\text{Angle}(M_b(z,2{\gamma }_m))}{4},$$(33)$${\delta }_d=\frac{|D_H^2-D_V^2|}{|D_H^2+D_V^2|}=\frac{|M_d(z,{\gamma }_m)|}{|M_d(z,0)|},$$(34)$${\theta }_d=\frac{\text{Angle}(M_d(z,{\gamma }_m))}{2},$$(35)where ${\gamma }_m=\mathrm{\Delta }n·d/2$ is half of the total retardation of two MPs. Here, the range of ${\delta }_b$ is $[0,\pi /2]$; the range of ${\theta }_b$ is $[0,\pi /2]$; the range of ${\delta }_d$ is [0,1]; the range of ${\theta }_d$ is $[0,\pi ]$. The polarization parameters calculated by this method are accumulative parameters.

The reason why the SPM-OCT system does not have phase jumps can be deduced theoretically. For a typical single-input PS-OCT, the interference signals of two polarization channels ($P_H$ and $P_V$) are both required to calculate the optic axis [39]. The product ($P$) of $P_V$ and the conjugate of $P_H$ is proportional to the following equation: $$P=P_V{P}_H^{*}\propto \sin ({\delta }_b^{\prime })\cos ({\delta }_b^{\prime })\exp (i\varphi ),$$(36)where $\varphi =\pi -2{\theta }_b^{\prime }$. ${\delta }_b^{\prime }$ and ${\theta }_b^{\prime }$ are the retardation and optic axis measured by the single-input PS-OCT, respectively.

The optic axis is calculated from the phase of $P$ as follows: $${\theta }_b^{\prime }=\pi /2-\text{Angle}(P)/2.$$(37)

It can be seen from Eq. (36) that the calculated phase ($\text{Angle}(P)$) is affected by the sign of the ${\theta }_b^{\prime }$ term coefficient ($\sin ({\delta }_b^{\prime })\cos ({\delta }_b^{\prime })$) [39]. If the ${\delta }_b^{\prime }$ is located in the first or third quadrant, the coefficient of the ${\theta }_b^{\prime }$ term is positive, and a correct ${\theta }_b^{\prime }$ is calculated. But if ${\delta }_b^{\prime }$ is in the second or fourth quadrant, the coefficient of the ${\theta }_b^{\prime }$ term is negative, which can result in $+\pi /2$ or $-\pi /2$ phase wrapping to the calculated ${\theta }_b^{\prime }$. This is the reason why phase jump errors occur in single-input PS-OCT.

Compared with the typical single-input PS-OCT, the SPM-OCT calculates the optic axis based on the Eq. (8). The coefficient of the ${\theta }_b$ term is ${\sin }^2({\delta }_b)$, which is always positive. Therefore, there is no phase jump in the optic axis image calculated by the SPM-OCT method.

The contrast-to-noise ratio (CNR) is used to evaluate the imaging contrast between the biological tissue and GNR. The CNR is defined as follows [40]: $$\mathrm{CNR}=\frac{N_{\mathrm{GNR}}-N_{\text{Tissue}}}{\sqrt{{\sigma }_{\text{Tissue}}^2}},$$(38)where $N_{\mathrm{GNR}}$ and $N_{\text{Tissue}}$ are the mean values within the GNR and biological tissue regions, respectively, and ${\sigma }_{\text{Tissue}}^2$ is the variance of the values within the biological tissue region.

The BALB/c nude mouse was used for the imaging experiment. After the mouse was sacrificed by anesthetic overdose, the skin of its leg was removed, and the skinned leg was imaged by PS-OCT and SPM-OCT, respectively.

To make a GNR phantom, the GNR was dispersed in the agar solution, then the agar solution was solidified to form a GNR phantom, and the phantom was imaged using PS-OCT and SPM-OCT, respectively. All experimental protocols using animals were approved by the Institutional Animal Care Committee of Nankai University.

To evaluate the accuracy of polarization measurements of the SPM-OCT system, a standard quarter-wave plate (QWP) (AQWP05M-980, Thorlabs) and a linear polarizer (LP) (LPVIS050-MP2, Thorlabs) were scanned by the SPM-OCT system, respectively. The QWP was used to verify the accuracy of birefringence parameters, including retardation (${\delta }_b$) and the optic axis (${\theta }_b$). The LP was used to verify the accuracy of dichroic parameters, including the diattenuation ratio (${\delta }_d$) and transmission axis (${\theta }_d$).

The modulation signals obtained in the $\chi$ domain under different orientations of QWP and LP are shown in Figs. 5(a1) and 5(a2), respectively. It can be seen that the frequencies of the modulation signals generated by QWP and LP are different, and their phases are related to the orientations of these devices. To compare the modulation signal of QWP and LP, the signals at a specific orientation in Figs. 5(a1) and 5(a2) are selected, respectively. Then, these two signals are inverse Fourier transformed from the $\chi$ domain to the $\gamma$ domain, as shown in Fig. 5(b). It can be seen that the frequency (${\gamma }_b$) of the modulation signal generated by the birefringent sample (QWP) is twice the frequency (${\gamma }_d$) of the modulation signal generated by the dichroic sample (LP), which is consistent with our theoretical derivation of SPM-OCT.

By calculating the modulation signals obtained by measuring QWP and LP, the birefringence and dichroic parameters are obtained. A comparison of measured and theoretical values of birefringence parameters, including retardation (${\delta }_b$) and the optic axis (${\theta }_b$), is shown in Fig. 5(c1). As the optic axis changes, the measured retardation (blue squares) is close to the theoretical value (blue dashed line). The measured mean value and standard deviation are 87.6° and 0.2°, respectively, which are quite close to the theoretical value of 90° with a small error. The measured optic axis (pink triangles) increases almost linearly with the increase of QWP orientation (pink dashed line), indicating that the optic axis obtained by SPM-OCT can characterize the optic axis of the birefringent sample.

As described in our previous study [35], the standard deviation of the retardation measured by PS-OCT is 4.31°. Besides, the optic axis values measured by PS-OCT slightly deviate from the ground truth. Comparing the validation results of SPM-OCT and PS-OCT, it can be seen that SPM-OCT has a lower retardation error than PS-OCT. The factors of SPM-OCT measurement errors include the retardation error of the achromatic quarter wave plate (AP) and the linear relationship between the wave numbers and the retardation of multi-order wave plates (MPs).

Besides, a comparison of measured and theoretical values of dichroic parameters, including the diattenuation ratio (${\delta }_d$) and transmission axis (${\theta }_d$), is shown in Fig. 5(c2). The diattenuation ratio of LP is extremely high, so its theoretical value can be regarded as 1. The measured diattenuation ratio (blue squares) slightly fluctuates around the theoretical value (blue dashed line) with the change in the orientation of LP. The measured mean value and standard deviation are 0.99 and 0.06, respectively, close to the theoretical value. The measured transmission axis (pink triangles) increases almost linearly with the change of LP orientation (pink dashed line), proving that the transmission axis obtained by SPM-OCT can accurately characterize the dichroic direction.

Figures 5(d1)–5(d4) are the retardation, optic axis, diattenuation ratio, and transmission axis B-scan images of the measured polarization elements at several angles. Angles shown in Figs. 5(d1) and 5(d2) correspond to the fast axis of the QWP. Angles shown in Figs. 5(d3) and 5(d4) correspond to the transmission axis of the LP. The QWP is constructed by aligning the fast axis of a birefringent plate with the slow axis of another birefringent plate. The LP consists of ellipsoid nanoparticles embedded in glass. The sample light enters the element and focuses on its lower surface [the horizontal line shown in Fig. 5(d)], whose polarization information is used to verify the polarization measurement capability of the system. The other interfaces of the polarization element are far from the focal plane, which are almost invisible due to their weak intensity. The different depth positions of the QWP and LP are because we adjusted the optical path length of the reference arm to accommodate the different thicknesses of these elements.

In summary, the consistency between the polarization measurement results and theoretical values demonstrates the high precision and reliability of the SPM-OCT system in polarization measurement.

The skinned mouse leg includes low-birefringence (lipid) and high-birefringence (muscle) tissues, making it an ideal biological tissue for evaluating the polarization measurement capability of the imaging system. To analyze the polarization measurement capability of SPM-OCT in biological tissues, SPM-OCT and PS-OCT were used to image the skinned mouse leg. Figures 6(a)–6(c) are the intensity, intensity-retardation, and intensity-optic axis B-scan images of the skinned mouse leg, respectively. Labels 1 and 2 represent the images obtained by PS-OCT and SPM-OCT, respectively. Among them, intensity-retardation and intensity-optic axis images are two-parameter merged images, which use intensity information to determine its brightness and polarization information to determine its hue. The optic axis ranges of SPM-OCT and PS-OCT are $\text{[}0,\pi /2]$ and $\text{[}0,\pi ]$, respectively, so their color bars are different. To make the color bar expression range of SPM-OCT and PS-OCT consistent, we extended the color bar of SPM-OCT so that it can express the range of $\text{[}0,\pi ]$.

Comparing Figs. 6(a1) and 6(a2), it can be seen that the intensity images obtained by PS-OCT and SPM-OCT are similar, and the high resolution of these systems ensures the clear presentation of lipid texture (circled by the yellow dotted line). As shown in Figs. 6(b1) and 6(b2), the retardation images obtained by PS-OCT and SPM-OCT are also similar. Among them, the lipid tissue shows low retardation values due to its low birefringence, while the muscle tissue with a high birefringence shows the retardation fluctuating with depth [38].

It is important to note that the optic axis images obtained by single-input PS-OCT may display erroneous information known as the phase jump. For example, the direction of the anisotropic structure of muscle fibers determines the value of the optic axis. Due to the directional continuity of the muscle fibers, the optic axis of the normal muscle usually does not have drastic changes within the imaging range of OCT. However, in the optic axis image obtained by single-input PS-OCT, a part of the muscle will show erroneous numerical mutations. As shown in Figs. 6(b1) and 6(c1), the region corresponding to the retardation changes from 0 to $\pi /2$ along the depth [from blue to red displayed in Fig. 6(b)] shows correct optic axis values. However, the region where retardation changes from $\pi /2$ to 0 along the depth appears with wrong optic axis values, called the phase jump. [The phase jump area is circled by a white dotted line in Fig. 6(c1)]. In comparison, the optic axis images obtained by SPM-OCT suppressed the error phenomenon of the phase jump. Comparing Figs. 6(c1) and 6(c2), the optic axis values of the whole muscle obtained by SPM-OCT show correct optic axis values, and there is no phase jump area.

Due to its three-dimensional imaging capability, OCT can not only provide B-scan images that display the tomographic structure but also provide en face images that are similar to microscope images. Figures 7(a)–7(c) are the intensity, intensity-retardation, and intensity-optic axis en face images of the skinned mouse leg, respectively. Labels 1 and 2 represent the images obtained by PS-OCT and SPM-OCT, respectively. All en face images are single axial slices. As shown in Figs. 7(a) and 7(b), the intensity and retardation of the sample presented by SPM-OCT are similar to those presented by PS-OCT, proving high imaging resolution and accurate retardation measurement of the SPM-OCT. In terms of the presentation of the optic axis image, the phase jump problem will have a great impact on the en face image of single-input PS-OCT. As shown in Fig. 7(c1), the phase jump area showed obvious erroneous values, resulting in serious misinterpretation of the tissue state. In comparison, as shown in Fig. 7(c2), the optic axis image of SPM-OCT shows accurate values without the phase jump problem.

Then, we enlarged the fiber tissue shown in the en face image to compare the imaging quality of PS-OCT and SPM-OCT. The images of labels 3 and 4 are enlarged images of the fiber tissue selected by the dotted lines in the images of labels 1 and 2, respectively. It can be seen that both PS-OCT and SPM-OCT can display the intensity and polarization information of fiber bundles (marked by black dotted lines). The intensity and retardation information obtained by SPM-OCT is the same as the correct information obtained by PS-OCT. The optic axis image obtained by SPM-OCT avoids the phase jump phenomenon.

The comparison results in this section proved that SPM-OCT not only can obtain various polarization parameters compared to the available PS-OCT but also can avoid the phase jump errors in the optic axis image, which greatly reduces the misreading of polarization information.

In this section, the SPM-OCT was used to image the skinned mouse leg and GNR phantom, respectively, and explored the feasibility of distinguishing biological tissues and GNRs based on SPM-OCT. Figures 8(a)–8(d) are the en face images of the retardation, optic axis, diattenuation ratio, and transmission axis, respectively. Labels 1 and 2 represent the images of the skinned mouse leg and GNR phantom, respectively.

The skinned mouse leg image mainly shows two types of tissues: muscle and lipid. As shown in Fig. 8(a1), muscles could show high retardation values due to their high birefringence, while low birefringent lipids show low retardation values. The optic axis image [Fig. 8(b1)] reflects the direction of the birefringent sample. The diattenuation ratio image [Fig. 8(c1)] shows the magnitude of the dichroism. It can be seen that the diattenuation ratio value of muscle and lipid is small, which is consistent with the current understanding that biological tissues have low dichroism [6]. The transmission axis image [Fig. 8(d1)] shows the transmitting direction of the dichroism. It can be seen that although the dichroism of biological tissues is small, Fig. 8(d1) can still show the different transmission axes of biological tissue.

The images of the GNR phantom show several GNRs dispersed in the solid agar. As shown in Fig. 8(a2), GNRs with birefringence show a certain retardation. The optic axis image [Fig. 8(b2)] shows the birefringent direction of the GNR. The diattenuation ratio image [Fig. 8(c2)] shows that the GNR has higher dichroism than biological tissue. The transmission axis image shows the dichroic direction of the GNR. It can be seen that there are differences in the optic axis and the transmission axis. This is because the GNR is not completely dispersed into individual particles, and the optic axis and the transmission axis reflect different combined effects of multiple GNRs. Comparing the polarization images of biological tissues and GNRs, the most obvious difference is the huge difference in the diattenuation ratio values between them. This difference is expected to be a key parameter for distinguishing biological tissues and GNRs.

To evaluate the effect of SPM-OCT in distinguishing biological tissues and GNRs, we quantified and compared the retardation and diattenuation of muscle, lipid, and GNR. The quantified areas of muscle and lipid are selected by dotted lines in Figs. 8(a1) and 8(c1). The box plots of the quantification results are shown in Fig. 9. The distribution of the cross-coupling coefficients is shown in the boxplots (Fig. 9). The 25%–75% indicates the range from the first quartile (Q1) to the third quartile (Q3), which is represented by a square box. The difference between the Q3 and the Q1 is called the interquartile range (IQR). The 1.5IQR rule is a statistical method used for identifying outliers in a dataset. The 1.5IQR range indicates the range from $\mathrm{Q}1-1.5\times \mathrm{IQR}$ to $\mathrm{Q}3+1.5\times \mathrm{IRQ}$. Data points outside the 1.5IQR region are considered as outliers. The non-outlier region indicates the range of actual data points within the 1.5IQR range.

It can be seen that the median retardation value of the lipid is low, while that of the muscle is high. The retardation of GNR is higher than that of the lipids but lower than that of the muscle. The CNR of GNR relative to lipid and muscle in the retardation image is 3 dB and 0 dB, respectively, which shows that it is difficult to identify GNR in lipid and muscle using the retardation image.

In contrast, the diattenuation ratio of GNR is higher than that of biological tissue. This result confirms that the diattenuation values of most biological tissues are extremely low [41] and that GNRs have high diattenuation [42]. Outliers with high values are associated with the aggregation of multiple GNRs. Suppose a multidimensional polarization vector describes the polarization properties of the GNR. When there are multiple GNRs in the detection spot, the polarization vector detected by the system is the superposition of the polarization vectors of multiple GNRs, resulting in a change in the obtained polarization information.

The CNR of GNR relative to lipid and muscle in the diattenuation image is 25 dB and 20 dB, respectively, which are significantly higher than that of some intensity-based OCT molecular imaging methods [32,33]. This result demonstrated that the dichroic parameters obtained from SPM-OCT can distinguish GNRs from biological tissues.

Integrating polarization measurement with interferometric imaging has shown great potential for generating three-dimensional structural and polarization data simultaneously. However, these integrations, represented by PS-OCT, still face two major challenges. First, there is a trade-off between the acquisition time and the diversity of polarization parameters. Second, the dual-detector configurations increase the complexity and cost of the system, while single-detector-based methods face constraints in acquisition speed and imaging range. So far, we have developed various single-input PS-OCT systems, effectively applied in biological tissue evaluation [35–37] and high-resolution dataset construction [43–45]. Building upon our extensive research, it is imperative to develop a compact and efficient interferometric polarization measurement method.

To address these problems, we conducted the following work in this study. (1) An efficient SPM method was proposed to extract birefringence and dichroism parameters by interferometric imaging, meeting the demand for rapid and multi-parameter polarization measurement. (2) The SPM method was successfully integrated into an OCT system, creating an advanced interferometric imaging system (SPM-OCT) capable of capturing multi-dimensional polarization parameters without sacrificing imaging speed or axial range. The improvement in imaging speed of SPM-OCT is achieved by reducing acquisition times. Compared with the multi-input PS-OCT system using time multiplexing, the acquisition speed of SPM-OCT is twice that of dual-input PS-OCT and three times that of triple-input PS-OCT. As a single-input system, SPM-OCT can obtain the birefringence and dichroism parameters through a single acquisition, achieving a better suppression effect on motion artifacts caused by breathing and heartbeat. (3) The imaging results demonstrated that the SPM-OCT system can provide both birefringence and dichroic parameters while mitigating phase jump errors, differentiating biological tissues from GNRs based on the dichroic parameter.

At present, most of the interferometric polarization modulation methods belong to time-sequenced mechanisms based on electro-optic modulator [18,24]. Since many interferometric imaging methods are wide-spectrum-based methods, we designed an SPM method to generate the polarization modulation signal in the wavenumber domain. The modulation signals can be obtained through a single acquisition, thus reducing the acquisition time. Besides, the system structure only changes the light source of a conventional interferometric imaging system. Therefore, the system design is very beneficial to endow existing systems with polarization detection functions, making it highly valuable for the product upgrading of interferometric imaging systems.

The SPM-OCT combines the OCT and the SPM method, which has the advantages of rapid acquisition, comprehensive polarization parameters, and simplified single-detector configuration. The imaging capability of SPM-OCT was verified by imaging standard polarization elements, biological tissues, and GNR phantoms, proving the high reliability, accuracy, and application value of the SPM-OCT system. It should be noted that structurally anisotropic nanomaterials, including gold nanorods, usually have significantly higher dichroism than biological tissues. The high sensitivity of SPM-OCT to dichroic materials makes it a potential method for achieving OCT molecular imaging.

Future research will proceed in the following directions. (1) Combine the SPM method with various interferometric imaging methods to endow and optimize their polarization detection function. (2) Incorporate the depolarization calculation methods from existing PS-OCT frameworks and design parameters to describe depolarization and local polarization obtained by SPM-OCT. (3) Refine both hardware components and signal processing techniques to further improve the resolution and accuracy of the system. (4) Investigate ways to expand the design and application of the SPM-OCT system in the field of medical diagnostics.

In this study, we developed the SPM method to perform simple and rapid interferometric polarization measurement and constructed the SPM-OCT system to extract birefringence and dichroic parameters without reducing the acquisition rate or the axial imaging range. Through imaging experiments on standard polarization elements, biological tissues, and GNR phantoms, we demonstrated that our method not only offers accurate birefringent and dichroic parameters but also effectively avoids phase jump errors. The ability of SPM-OCT to differentiate GNRs from biological tissues further underscores its potential for clinical and research applications. The efficient polarization measurement mechanism of the SPM method is anticipated to promote the development of interferometric imaging and inspire new research directions in polarization imaging technology.