Ultrafast polarization bio-imaging based on coherent detection and time-stretch techniques

Optical polarization imaging has played an important role in many biological and biomedical applications, as it provides a label-free and non-invasive detection scheme to reveal the polarization information of optical rotation, birefringence, and photoelasticity distribution inherent in biological samples. However, the imaging speeds of the previously demonstrated polarization imaging techniques were often limited by the slow frame rates of the arrayed imaging detectors, which usually run at frame rates of several hundred hertz. By combining the optical coherent detection of orthogonal polarizations and the optical timestretch imaging technique, we achieved ultrafast polarization bio-imaging at an extremely fast record line scanning rate up to 100 MHz without averaging. We experimentally demonstrated the superior performance of our method by imaging three slices of different kinds of biological samples with the retrieved Jones matrix and polarization-sensitive information including birefringence and diattenuation. The proposed system in this paper may find potential applications for ultrafast polarization dynamics in living samples or some other advanced biomedical research. © 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement


Introduction
Optical polarization imaging detects the changes in the polarization states of light after passing through or being reflected by polarization-sensitive samples through a combination of a polarizer and an analyzer [1].These changes, which are normally related to optical rotation, birefringence, and photoelastic effect, are difficult to be detected or sensed using conventional light intensity-and phase-based imaging systems [2].Especially in many biological and biomedical applications, such as bulk tissue characterization [3,4], normal and cancerous cells differentiation [5] and in-vivo microcirculation studying [6], optical polarization imaging has becoming a promising detecting and analyzing approach with the advantages of label-free, non-invasion, low damage to samples and sensitive to sub-wavelength structures [7].
To date, several different polarization imaging approaches have been proposed or demonstrated, including conventional single polarized light microscopy [8,9], two polarized lights based Jones matrix microscopy [10] and multiple polarized lights based Mueller matrix imaging system [11].Apart from static biological sample, optical polarization imaging was further extended to observe the dynamic processes, from macroscopic to microscopic like the birefringence change of a muscle tissue [12] to the dynamics of microtubules and molecules [13,14], from plant to animal like the expansion of a plant cell wall [15] to the dynamics of an animal living cell [16].However, in almost all previous polarization imaging techniques the mainly used wide-field light arrayed detectors, such as charge-coupled device (CCD) or complementary metal-oxide semiconductor (CMOS) camera, usually suffer from low frame rates while maintain large pixel counts [17], therefore significantly restricting the imaging speed of the polarization imaging techniques.Since imaging polarization information with an ultrahigh temporal resolution is essential in understanding ultrafast polarization-sensitive dynamics, developing an ultrafast polarization imaging system is highly desirable.
Optical ultrafast time-stretch imaging has been an emerging method that achieves an ultrahigh temporal resolution [18].The essential principle of ultrafast time-stretch imaging includes three steps.Firstly, the spatial information of the sample is encoded into the spectrum of a femtosecond laser pulse through a spatial disperser.Secondly, the modulated laser pulse is stretched into a temporal waveform whose power distribution is proportional to its spectral intensity.Finally, a high-speed optoelectronic detector captures the temporal waveform, which will be processed for image reconstruction.By taking advantage of highspeed optoelectronic components and demodulation techniques in modern optical communication, time-stretch imaging provides a unique way to obtain ultrafast information up to 100 million frames per second.Using time-stretch imaging techniques, several applications have been demonstrated, such as ultrafast tissue and particles imaging, high throughput cell screening, and cellular drug analysis [18][19][20][21][22].However, all these demonstrated time-stretch imaging modalities above focus on either light intensity or phase detection, while a time-stretch imaging modality based on light polarization has not yet been reported.
In this paper, we present a coherent detection and time-stretch based ultrafast polarization imaging system that is capable of delivering spatially dependent Jones matrix of biological samples with a record line scanning rate up to 100 MHz without averaging.Such a capability is enabled by combining an optical coherent detection of orthogonal polarizations and a timestretch technique.Using a phase and polarization diversified homodyne coherent receiver, the imaging system can extract the full field information, including the amplitude, phase, and state of polarization [23], of the light after probing the sample.Thus, all the elements in the Jones matrix of the sample could be directly retrieved, as well as the polarization parameters including diattenuation and birefringence-related phase retardation.

Principle and experimental setup
With the capability of determining complete information of the electric field, the Jones matrix imaging is able to provide polarization-sensitive information [24].To obtain the Jones matrix of the targeted biological sample, two independent incident polarization states are required in the imaging system [25].In our system, we chose two linear polarization states that are orthogonal to each other, which are referred to as x-and y-polarizations in the rest of the paper.The setup for our ultrafast polarization imaging system is shown in Fig. 1(a), which incorporates the polarization-division multiplexing and coherent detection technique.The optical coherent receiver utilized here consists of two homodyne coherent receivers in a polarization diversity configuration [20] as shown in Fig. 1(b).An optical local oscillator (LO) is required as an absolute phase and polarization reference to restore the phase and polarization information of the incoming optical signal.A polarization beam splitter (PBS) is first used to separate the optical LO and signal into two linear orthogonal polarization components, which are then detected by two homodyne coherent receivers.Based on a 90° optical hybrid and balanced photodetectors, homodyne receiver could output in-phase (I) and quadrature (Q) component of the interference term of the optical LO and signal.I and Q component can then be digitized and used to restore complex full information on the optical complex amp ultrafast quan only a single operating proc Fig. repetition und 1550 nm group um-doped m, which wo linear tration of controller n become axis of the 50 power the pulses by using lled along arm were , we got ust along the pulses These PM al and LO mple and couple the output signal beam out for measurement.In the signal beam, a fiber collimator was used to couple light from the fiber to free space.A quarter wave plate (QWP) was inserted and adjusted to guarantee that successive pulses maintained linear orthogonal polarizations.Then, the pulses in the signal beam were spatially dispersed into a one-dimensional (1-D) spectral rainbow by a volume holographic transmission diffraction grating (Wasatch Photonics, spatial frequency = 600 lines/mm).Therefore, the spatial information of the sample was encoded into the spectrum of the pulse and the average light power illuminating on sample is about 10mW.After interacting with the sample twice (reflection mode with a mirror, Fig. 1(a)), the signal beam that contains the sample information mixed with the LO and detected by an optical digital coherent receiver.A PM variable optical delay line (VODL) in the LO arm was used to fine tune the relative time delay which determines the beat frequency between signal and LO pulses.A real-time oscilloscope (LeCroy LabMaster 10Zi) with a sampling rate of 80 GS/s was used to digitize and acquire the I and Q data of the orthogonal polarization signal from the receiver.I and Q data was sent to a computer to reconstruct the Jones matrix of the sample.The polarizing effects of the sample, including retardance and diattenuation, can be fully characterized and extracted from the retrieved Jones matrix.Our system is a 1D line scan imaging system, this 1D line scan achieved by the spatial dispersion and temporal dispersion of the chirped laser pulse.Thus, the frame rate of this system depends on the pulse repetition of the laser source, i.e. 100MHz.However, it's worth mentioning that the whole exhausting recording time (~200ms) for one line scanning is much longer than the line scanning period (10ns), which is mainly hindered by the data transfer time between computer and the real time oscilloscope.This has no effect on the demonstration of the proposed imaging system.The 2D image acquirement needs the vertical scanning.Therefore, the imaging field of view (FOV) and resolution along the horizontal and vertical direction depend on different factors.The horizontal imaging FOV is up to the length of the 1D dispersed illumination light spot, which mainly depends on the filtered wavelength bandwidth, the spatial frequency of diffraction grating and the NA of the objective lens [27], while the vertical imaging FOV depends on the scanning speed.In addition, the horizontal spatial resolution, which depends on the spatial frequency of diffraction grating, the GVD of the dispersion compensating fiber, and the digitized bandwidth of the coherent receiver [27], is 3μm, nearly the same with the previous system [26].The vertical spatial resolution which only depends on the NA of the objective lens is higher than the horizontal resolution.

Jones matrix analysis
Having described the system setup and the operating procedures, we provide a detailed analysis for reconstructing the Jones matrix of the sample.The Jones vector (E IN ) of the input signal beam is transformed into the output Jones vector (E OUT ) by where J IN and J OUT represent the Jones matrices of the input and output optical bers at the signal beam respectively, J ST is the round-trip Jones matrix of the sample, J QI and J QB are the Jones matrices of the QWP for incident and back reflected lights, respectively, J GI and J GB are the Jones matrices of the grating for incident and back reflected light, respectively.According to the Jones reversibility theorem [28], we know where the symbol T stands for taking the transpose of a matrix.Here, the input SMF fiber length of OC is about 2 meters, and we have tapped it onto the optical table in the experiment, thus the diattenuation of the input SMF fiber can be ignored, then the Jones matrix of the input fiber, J IN , can be modeled as linear retarder [29].The Jones matrix of the QWP, J QI , is also a linear retarder with adjustable angle.Therefore, we can use the QWP to compensate for the elliptical polarization transformation induced by the input fiber and make the incident pulses before the grating maintaining linear.With a specific adjusted angle of the optic axis of the QWP, the J IN and J QI can have a relationship as follows: The utilized diffraction grating has a diffraction efficiency (η) of 90% around 1550 nm.While the phase retardation induced by the grating of linear orthogonal polarized lights may be different, we can still express J GI as ,0;0, , where φ g is the phase retardation difference between the linear orthogonal polarized lights induced by the grating.With equations above, we can rewrite the Eqs.(1) into, ' ' where E IN' represents the electric fields of input pulses after the grating, with its polarization linear orthogonal.The electric fields of the incident pulses with linear orthogonal polarization states are denoted as E INx and E INy .Although pulses with linear orthogonal polarizations are temporally separated, we incorporate them into a single matrix for mathematical convenience, as follows: where φ is the relative phase difference between the two linear orthogonal polarized pulses after the grating, it is induced by the grating and the different travelling length in process 2 and 3. A digital coherent receiver was used to measure the interferometric signals.For each polarized pulse, the coherent receiver output I iX , Q iX , I iY , and Q iY (i = x, y) from four channels, where X and Y are referred to as two orthogonal polarization components of the coherent receiver.Here, I iX and Q iX correspond to the in-phase and quadrature part of the interference term of the X component in the coherent receiver of the i-polarized pulse (i = x, y), while I iY and Q iY correspond to the in-phase and quadrature part of the interference term of the Y component in the coherent receiver.After getting the output data of orthogonal polarized pulses, we can form an S matrix based on the digital coherent measurement [30].
Here, R represents the responsivity of the photodiode.E LOX and E LOY are the X and Y electric elds components of the LO pulse in the coherent receiver.The linear orthogonal polarized pulses in the LO arm transmit along the fast and slow axis of the PM fiber, and when it reaches the coherent receiver, the linear orthogonal polarized pulses split equally into the X and Y channels of the detector.Thus, we can write the Jones matrix of digital coherent as Here, 1 ϕ is the relative phase difference between the linear orthogonal polarizations of LO at the coherent receiver, P is the average power of the LO light.With Eq. ( 7), we can rewrite the Eq. ( 6) as Similar to the fiber-optic components based polarization-sensitive optical coherence tomography (PS-OCT) system [31], a reference measurement is needed to obtain the round trip Jones matrix of the sample J ST .When nothing is placed at the sample position, we assume, where 2 ϕ represents the phase delay caused by the free space propagation.Actually, the reference measurement was achieved by a blank line scan imaging, then a whole image could be reconstructed base on it.The blank reference signal can be expressed as (10) By multiplying the inverse matrix of S ref with S, we get, We note that some terms can be canceled out, and we can finally get an explicit expression of the round trip Jones matrix of the sample We note here that 2 e iϕ is a constant phase term, which has no influence on retrieving the information of the sample.To solve for the unknown phase φ, we used the fact that the round trip Jones matrix of the sample J ST is a transposed symmetric matrix [32].Having obtained the round trip Jones matrix of the sample, we can also transform it into an equivalent Mueller matrix [30], J J J J J J J J J J J J J J J J J J J J J J J J J J J J J J J J (13) where the ⊗ represents the Kronecker tensor product and U is the 4 × 4 Jones-Mueller transformation matrix, Unlike a Jones matrix, the Mueller matrix can be applied to all kinds of optical system and it shows a more explicit intensity transformation property over the Jones matrix under some situations.For example, M 11 element represents a complete intensity transformation property of the sample without the influence of polarization property.Having obtained the Jones matrix of the sample, we can use two different methods, matrix diagonalization [31] or polar decomposition [33], to retrieve the polarization parameters, such as diattenuation and phase retardation.It is noted that matrix diagonalization and polar decomposition are equivalent in fact.Based on matrix polar decomposition, we can express retardance (R) and diattenuation (D) with only the Jones matrix and the polarization parameters [33] as, ( where the tr and det are the trace and determinant of the matrix, respectively, the † denotes the conjugate transpose of the matrix.

Experimental results
In order to demonstrate the validity of the quantitative polarization information retrieved by the system, we firstly imaged a birefringent resolution target (thorlabs, R2L2S1B) which has two N-BK7 protective glass covers and a liquid crystal polymer layer inside.The fast axis of the liquid crystal polymer layer is designed to aligned parallel to the side of the glass covers.
The target has a retardation of 280 ± 20 nm and a maximum resolution of 18 line pairs per millimeter.Due to the relative low resolution of the target, we used two normal lens with 75mm focal length as the objective lens to achieve a larger FOV.The images obtained by conventional polarizing microscopy and our system are shown below in Fig. 2. All images have the same scale of 400 × 500 µm.As shown in Fig. 2(b), the phase retardation retrieved by our system is a bit more than 1 rad, the standard retardation of 280 nm equals to 1.135 rad when converted to phase retardation around 1550 nm illumination wavelength, which basically matches with our results.As shown in Fig. 2(b)-(h), they are slightly distorted, we thought it would be result from two main causes.Firstly, the resolution between the spectral dispersion direction and the orthogonal scan direction are different.The spectral dispersion direction which corresponding to the vertical direction in the images reconstructed here has a relatively lower resolution than that of the scan direction, so the straight lines paralleled to the scan direction in Fig. 2(b)-(h) distorted more.Secondly, the 10% energy instability of the laser pulses will also result in signal distortion as an amplitude noise.We did not provide the bright field image of the target because of the actual birefringence resolution target is transparent, as shown in Fig. 2(i).Similarly, the M 11 image in Fig. 2(d) also shows a negligible intensity changing map.The Mueller matrix element M 11 just represents the complete amplitude transformation property of the sample.The last carcinoma (B BCC continue for higher thr carcinoma ac imaging syste our system re birefringence around the co the connectiv can acquire t transformation obtained usin comparing to and decompo assessment on     recording time length is finite, which is caused by the utilized oscilloscope.In the proposed system here, the line scan imaging data was firstly stored in the memory of the oscilloscope until it's full, and then transmitted to the computer through an Ethernet cable.Due to the limited memory depth of the oscilloscope, the imaging recording time length is finite.This is a drawback of current "off-line" imaging system that was used for experimental demonstration.Actually, an in-line real time imaging system with infinite imaging time length can be realized with embedded hardware processing techniques.
Additionally, as we know, birefringence and diattenuation represent different polarization properties of samples, birefringence shows the refractive index property, and diattenuation shows the transmittance or absorption property when light polarized parallel and polarized perpendicular to an orientation axis.Although diattenuation is negligible in most biological tissues, we can still make it an extension for polarizing imaging [38,39].In the reference article, diattenuation was used to analysis large or long fibrous protein based biological tissue, such as brain tissue or retinal nerve fiber.
In summary, by combining the optical coherent detection of orthogonal polarizations and time-stretch techniques, we developed a polarization imaging system to image polarizationsensitive information of biological samples with an ultrafast line scanning rate up to 100 MHz.We demonstrated the capability of this imaging system by imaging a standard birefringence resolution target and three slices of different biological samples, including the plant stem cross cut, the spirogyra conjugation, and the skin basal cell carcinoma sample.We anticipate that this ultrafast polarization imaging system will become a feasible approach for imaging ultrafast polarization-sensitive dynamics in future in vivo biological and biomedical applications.

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