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Abstract
The optical angular memory effect (AME) is a basic feature of turbid media and defines the correlation of speckles when the incident light is tilted. AME based imaging through solid scattering media such as ground glass and biomedical tissue has been recently developed. However, in the case of liquid media such as turbid water or blood, the speckle pattern exhibits dynamic time-varying characteristics, which introduces several challenges. The AME of the thick volume dynamic media is particularly different from the layer scatterers. In practice, there are more parameters, e.g., scattering particle size, shape, density, or even the illuminating beam aperture that can influence the AME range. Experimental demonstration of AME phenomenon in liquid dynamic media and confirm the distinctions will contribution to complete the AME theory. In this paper, a dual-polarization speckle detection setup was developed to characterize the AME of dynamic turbid media, where two orthogonal polarized beams were employed for simultaneous detection by a single CCD. The AME of turbid water, milk and blood were measured. The influence of thickness, concentration, particle size and shape, and beam diameter were analyzed. The AME increasement of upon the decrease of beam diameter was tested and verified. The results demonstrate the feasibility of this method for investigating the AME phenomenon and provide guidance for AME based imaging through scattering media.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The wavefront of incident light is disturbed when it encounters rough surfaces or media with inhomogeneous refractive index distributions, such as turbid water, biological tissue and walls [1,2]. Optical imaging through turbid media is a major challenge in many applications such as biomedical imaging, observation of chemical reactions in turbid media, and imaging around corners, as light undergoes multiple scattering when transmitting through such media. Traditionally, it was believed that the scattered photons were detrimental to imaging, whereas only ballistic photons [3] that experienced none or few scatters were selected out by gating methods [4–6] for image reconstruction. For imaging through dynamic turbid water, the multi-frame time-gated holography [7] is proposed. However, the number of ballistic photons decrease rapidly as scattering events accumulate [2], thus limiting the imaging quality.
Interestingly, the scattered photons or the generated speckle patterns show correlations [8], such as the correlation between the transmission and backscattered speckles [9] and the angular memory effect (AME) [10,11] of the transmitted speckles. The AME describes the tilts of the incident light in a certain angle, which result in speckle patterns shift accordingly and maintain its structure. Based on AME, techniques such as speckle correlation [12–17] and wavefront shaping [18–22] were recently developed for high quality imaging through turbid media by the scattered photons. On the other hand, the imaging field of view (FOV) is restricted by the AME range, which is theoretically inversely proportional to the thickness of the turbid media [10]. The AME plays a key role in these scattering imaging techniques. In speckle correlation, imaging background and error increase once the illuminated object size exceeds the AME [16] or the reference point source for cross correlation [14,15] is placed outside the AME range. The AME determines the process for combining the partial images for large FOV imaging and also influences the number of fingerprints that should be used [23]. In wavefront shaping [24], the FOV could also be enlarged by specially designed multi-pupil correction, with one region corresponding to one AME range. The known AME of the specialized scattering media would be highly beneficial and can provide guidelines for imaging [19,25] or an AME-based filter [15] to increase imaging quality.
Studies on AME were carried out [16,26–31], for the better known and usage. Judkewitz et al. [26] showed that the shift memory effect that could jointly work with AME to enlarge the imaging FOV [27]. Schott et al. [28] found that biological tissue with a large anisotropy factor had more than one order of magnitude AME more than the value predicted by theory. Liu et al. [29] provided a more detailed picture of AME formation by considering the scattering times and anisotropy factor. They showed that the AME was a weighted sum of different scattering components, and light that was less scattered had a larger range of AME. Thus, the imaging FOV could be enlarged by selecting the less scattered light [30] or the light within the high-transmission channel [31]. Recently, reports have investigated focusing and imaging through dynamic media [32,33]. However, the AME of dynamic media has not been characterized yet, which has impeded the progress of these techniques for high quality imaging through dynamic liquid media. This is mainly because at least two images of speckle patterns are needed to calculate their correlation and measure the AME of the media [28]. Since the speckle patterns of liquid media such as turbid water and blood change dynamically, acquiring two images may take longer time than the decorrelation rate of the media. Thus, traditional methods have been inadequate for measuring the AME of dynamic media.
In this paper, a dual-polarization speckle detection setup is proposed for recording two orthogonal polarized speckle patterns simultaneously on different parts of the same CCD sensor. During exposure (typically 0.2 ms in experiment), the two beams experience the same scattering and distortion, thus retaining the correlation of the two speckle patterns. By tuning the incident angle of the two beams, a series of speckles pairs are obtained and used for calculating the angular dependent correlation profile. The AME is defined by the full width at e-1 of the maximum of the angular correlation profile. First, the AME of turbid water is measured by the proposed experimental setup, where the influence of thickness, concentration, particle size and beam diameter are analyzed. Then, the AME of other dynamic scattering media such as milk and blood is measured. From the results of the influence of the beam diameter, it is found that the AME can be increased by decreasing the beam diameter.
2. Methods
The AME theory was proposed by Feng et al. [10], and the angular speckle correlation is given by:
where, C is the correlation between the reference speckle and the speckle when incident beam is tilted by an angle of θ, k is the wave vector and d is the thickness of media. Following Eq. (1), the AME range was chosen as the full width at e-1 of the maximum correlation, Δθ≈ 3.63k-1d-1, which is inversely proportional to d. The AME theory has been demonstrated in relatively static scatterers such as ground glass [11], layers of powder [16] and biological tissues [28]. The AME theory is the basic fundamental to characterize the speckle correlation. However, for scatterers with large anisotropy factor (g) and transport mean free path (TMFP, l*), the thickness assumption $L \gg {l^\ast }$ for Eq. (1) would be easily broken [28]. Thus, the AME range for these samples is higher than that predicated by Δθ from Eq. (1). For liquid dynamic samples with volumetric scattering such as turbid water and blood, the problem is more complex as the AME is influenced by parameters such as thickness, concentration, scattering particle size and shape. To measure the AME of specialized dynamic scattering media, the proposed dual-polarized speckle detection setup as shown in Fig. 1.The laser beam (RGB Lasersystems, NOVAPRO, 685 nm) was split into two orthogonal polarizations. The vertical polarized beam (Beam 1) was orthogonally incident onto the scattering media and produced the reference speckle. To ensure an appropriate sampling rate, the speckles that were 30 mm behind the scattering media were selected by an iris with a 1 mm diameter (Iris 2) and imaged onto the left part of CCD2 (The Imaging Source, DMK 33UP5000, 60 fps @ 2560 × 2048 pixels, pixel size 4.8 × 4.8 µm2) with 5× objective (Nikon, 5×, NA 0.15) and a lens with a focal length of 200 mm (Lens 5). The focal plane of the objective was just coincident with Iris 2. The horizontal polarized beam (Beam 2) was incident onto scattering media with a controllable angle by tilting the mirror M1. Its corresponding speckles were imaged by the right part of CCD2, as shown in the insert of Fig. 1. M1 was imaged onto the surface of the scattering media by Lens 1 and Lens 2 to avoid the lateral shift of the beam when tilting M1. The image plane of Lens 2 was coincident with Iris 1. The focal lengths of Lens 1 (f = 50 mm) and Lens 2 (f = 200 mm) were chosen such that a 4× magnification was achieved, which in turn increased the corresponding angular resolution. Lens 3 and Lens 4 were chosen to be identical to Lens 1 and Lens 2, respectively, in order to maintain the similarity of Beam 1 and Beam 2. Iris 1 was placed at the surface of the scatterers and was used to control the incident beam diameter, which was 6 mm in the experiment. The speckles of Beam 1 and Beam 2 were recorded by CCD2 simultaneously. This was done such that the dynamic change of the scatter did not influence the correlations of the two speckles. CCD1 (the same model as CCD2) was used to measure the angle between the two beams by recording their interference fringes. The series angle correlation profile was recorded by tuning the incident angle of M1, and the AME was calculated from the correlation profile.
In the experiment, the turbid water is made of silicon dioxide microspheres (SiO2, with 2, 5, 10 and 20 µm diameter) mixed with distilled water. The dimensionless concentration of turbid water is calculated by R = msolid/(mwater + msolid), where msolid is the mass of SiO2 powder and mwater is the mass of water. This formula was adapted from Ref. [33] with a minor modification.
3. Results and discussion
3.1 AME range of turbid water
From Eq. (1), the AME range is inversely proportional to the thickness of the media. However, the influence of other factors such as concentration, particle size and shape of the scatters is unknown. To investigate the influence of these parameters, the AME of various turbid water samples was measured, as shown in Fig. 2. A series of speckle pairs of the two beams were obtained by tuning the incident angle of Beam 2, as shown in the inset of Fig. 1. The angular dependent correlation profile was obtained by calculating the correlation of each speckle pair, as shown in Fig. 2(a). Then, the correlation profiles were fitted with Eq. (1) and the AMEs were measured as the full width at e-1 of the maximum correlation.
Fig. 2. AMEs of turbid water (concentration of 0.0244) at different thicknesses. (a) Angular correlation profiles of turbid water under different thicknesses; (b) The calculated AME (red mark) from (a) and AME theory (green dashed line).
To validate the working of the setup, the angular correlation profile and corresponding AMEs were measured for turbid water with concentration of 0.0244 (1 g SiO2 of 5 µm diameter mixed with 40 g water) as a function of d, as shown in Fig. 2. The AMEs were calculated from the correlation profiles and fitted with inverse relationship to d, as shown in Fig. 2(b). Since the measured AME from the experiment is found to be inversely proportional to the thickness of the scattering media, it can be concluded that the system is reliable. However, the AMEs are about one order of magnitude larger than the theoretical predications obtained by Eq. (1). This phenomenon is same as the increased AME in biological tissue given by Schott et al [28], the high anisotropy factor and strong forward scattering is the mainly reason for the incensement. The anisotropy factor of the turbid water of 5 µm diameter SiO2 is 0.966 calculated by Mie theory.
The concentration, particle size and particle shape will also influence AME significantly as they determine the scattering angle and scattering times. To investigate the influence of concentration, the AMEs of 1 mm thick turbid water (5 µm diameter SiO2) at different concentrations were measured, as shown in Fig. 3(a)-(b). Similar to the case of thickness, the relationship between AME and concentration could also be fitted by an inverse function 1/R, as shown in Fig. 3(b). To investigate the influence of particle size and shape, the AMEs of 2 mm-thick turbid water samples with 2-20 µm diameters ball shaped and 5 µm irregular shaped SiO2 were measured when the concentration was 0.0361. The correlation profile and corresponding AME are shown in Fig. 3(c)-(d). It can be observed that the turbid water composed of smaller particles exhibits a much smaller AME range, while the shape of particles does not have any major influence on the AME.
Fig. 3. AMEs of turbid water as a function of different concentrations and particle sizes. (a) Angular correlation profile of turbid water under different concentrations; (b) The calculated AME from (a) and fitted by 1/R; (c) Angular correlation profile of turbid water with different particle sizes; (d) The calculated AME from (c).
In order to verify the efficacy of the method for realistic samples encountered in biological applications, the AMEs of milk and pig blood were measured, as shown in Fig. 4(a). The calculated AME of 0.5 mm thick milk, 0.2 mm and 0.5 mm thick blood were determined to be 0.316, 0.415 and 0.115 degrees, respectively. In addition to the properties of scattering media, properties of the light source, such as beam diameter can also influence the AME. Iris 1 in the experimental setup was used to control the beam diameter in order to ascertain its influence on AME. When the beam diameter was decreased from 6 mm to 2 mm, the AME range increased. Both pig blood and turbid water were used for demonstration, as shown in Fig. 4(b). Compared with 6 mm beam diameter, the AME of 0.2 mm-thick blood at 2 mm diameter was increased to 0.453 degrees, exhibiting an increasement of about 9% (calculated by (0.453-0.415)/0.415). The AME of 2 mm-thick turbid water samples (10 µm SiO2) with a concentration of 0.0244 was increased from 0.237 to 0.288 degrees, thereby showing an increasement of about 21%.
Fig. 4. Correlation profiles of milk and pig blood. (a) The correlation profiles of milk and pig blood at different thickness; (b) Increase in the correlation of turbid water and blood by decreasing the aperture from 6 mm to 2 mm.
To provide a more precise description of the turbid media, the optical depth (OD), which defined the exponential relationship between the remaining ballistic photons after passing through the media and the input photons, are measured as listed in Table 1. OD is obtained by the same experimental measurement as shown in Ref. [3]. As OD is the product of the attenuation coefficient (µ) and the thickness d, then the attenuation coefficients could be calculated by dividing OD with thickness. The corresponding µ are also shown in Table 1. The attenuation due to absorption is negligible for all the turbid water, pig blood (at 685 nm) [34] and milk [35], thus the scattering coefficient (µs) is approximate to the attenuation coefficient, µs≈µ. The difference between the measured attenuation coefficients and the reported value for blood [34] and milk [35] may due to the different light wavelengths and samples used. The scattering coefficient of turbid water can be calculated by Mie theory and Ref. [36], which are 1.8 and 3.9 mm-1 for 20 µm and 10 µm diameter SiO2 at concentration 0.0244 respectively. The average scattering coefficients measured by experiment are 2.0 mm-1 and 5.2 mm-1, which are slightly higher than the theoretical value. This difference may due to the errors in such as the particle size and the concentration.
Table 1. The measured optical depth
For analyzing the distinction between the measured AME and the theoretical value calculated by Eq. (1), the TMFP of turbid water is calculated. Take turbid water of 20 µm and 10 µm diameter SiO2 at concentration 0.0244 for example. The mean free path, which is inversed to the scattering coefficient, are 0.5 mm and 0.19 mm (l = 1/µs). The anisotropy factor g of 20 µm and 10 µm diameter SiO2 calculated by Mie theory are 0.954 and 0.955. The TMFP (l*=l/(1-g)) are 10.9 and 4.2 mm respectively. As it is analyzed in Ref. [28], for biological samples with large anisotropy factor, the TMFP is longer than the thickness and violate the assumption $L \gg {l^\ast }$, which could be the reason for the measured AME range larger than predicted by Eq. (1). As the TMFP of turbid water here also violate the assumption, this could also be the reason for the AME higher than Eq. (1) prediction.
The images of the SiO2 particles are shown in Fig. 5 to describe the turbid media. The decorrelation time [21] τ, which is defined as the time when the speckle correlation coefficient decreases to e-1, was used to describe the time-scale of the speckles change of the dynamic media. The speckle decorrelation profiles of two kinds of turbid water (4 mm thick, 0.0244 concentration of 5 µm SiO2; 1 mm thick, 0.0244 concentration of 20 µm SiO2), 0.5 mm and 0.2 mm thick blood, and 0.5 mm thick milk were measured as shown in Fig. 6(a). The decorrelation times of the samples are 0.54, 1.48, 12.2, 7.1 and 0.59 ms respectively. For the best of AME measurement, the speckle patterns should be stable or relative stable during the exposure time of the camera. In current experiments, the typical exposure time is 0.2 ms, which is faster than the decorrelation of the samples and sufficient for the AME measurement. Moreover, the influence of the exposure time on the accuracy of AME measurement was further investigated and the 0.5 mm thick blood (τ=7.1 ms) was used for demonstration. The angular correlation profiles are measured at exposure time of 1, 8, and 31 ms respectively. The results are shown in Fig. 6(b) and indicate that the measured AME are close to the ground truth even when the exposure time not much longer than the decorrelation time of dynamic media. This may because during the exposure, the acquired speckles could be taken as a sum up of several uncorrelated speckles of the dynamic media at different independent status. If the exposure time is short, the independent statuses are less and the crosstalk between each other will not influence the measured AME much. Nevertheless, if the exposure time is much longer than the decorrelation time (more than 4 times as in the experiment demonstration), the crosstalk will be ambitious and show impact on the measured angular correlation and the final AME, as it is shown by the green dashed line in Fig. 6(b). The speckles acquired at different exposure time (t) are also shown in Fig. 6(b) for proven. The sum-up speckles gotten at longer exposure time possess low contrast with evident background.
Fig. 5. Images of the SiO2 particles. (a)-(c) Images of 2, 5, 20 µm diameter ball shaped SiO2; (d) Image of 5 µm irregular shaped SiO2; Scale Bar: 5 µm.
Fig. 6. Speckle decorrelation time and the measured correlation profile at different exposure time. (a) The speckle decorrelation time of the dynamic media; (b) The angular correlation profile of 0.5 mm thick blood measured at different exposure time.
It was observed that the AME could be increased by decreasing the beam diameter. This may be attributed to the increase in the speckle grain size and the lateral correlation length. For example, the speckles behind 0.2 mm thick blood at 6 mm diameter and 2 mm diameter are shown in Fig. 7 for comparison. The AME increase phenomenon could be explained by the two phase mask model [29]. The limited AME range is a consequence of the relative shift between the speckle wavefront and scattering media when the incident angle of light is tilted. The speckles generated by the first phase mask transmit to the surface of the second one and then generate the final output speckle pattern behind it. When incident light is tilted, the speckles at the surface of the second phase mask shift accordingly. The AME depends on the lateral correlation length, which is defined by the light speckle size incident onto the second phase mask and phase gradient of the mask. If the incident beam diameter is decreased, the speckle size i.e., the lateral correlation length could be increased. Accordingly, the correlation angle and AME range could be increased. This feature could be used for increasing the AME and imaging FOV. But on the other hand, the aperture of scatterers or the speckle grain size determines the imaging resolution [13]. Thus, increasing AME and imaging FOV by decreasing the aperture will reduce the resolution, and the trade-off should be considered.
Fig. 7. The comparison of the speckles at different beam diameter. (a) Speckles at 6 mm beam diameter; (b) Speckles at 2 mm beam diameter.
The key concept of the proposed dual speckle detection setup for measuring AME lies in simultaneous separation and detection of speckles from two orthogonal polarized beams by a single CCD. The two beams will be mixed together if the scattering media changes the polarization of light. The crosstalk is measured by blocking the light path of Beam 1, and the results are shown in Fig. 8. For quantitative evaluation, the crosstalk intensity ratio is calculated by: P = Ic/(Io + Ic), where Io is the remained light intensity in the original channel, and Ic is the crosstalk intensity in the blocked channel. The correlation between the crosstalk and the original speckle are also calculated, which are all lower than 0.01. The low intensity and correlation indicate that the crosstalk in the experiment contribute little to the measured AME, and could be neglected. The media with highly density and thickness will be more likely to change the polarization of light. For the polarization sensitive media that fully mixed the two beams together, the setup proposed here may not work. To measure the AME of these polarization sensitive dynamic media, a modified “dual-wavelength separating detection” setup could be designed, where two beams with different wavelengths [37] are used to illuminate the media and the speckles of the two beams could be separately by dichroic mirror and detected by a CCD simultaneously.
Fig. 8. The crosstalk of the two beams. (a) Turbid water of 4 mm thick, 0.0244 concentration of 5 µm SiO2; (b) 0.2 mm thick blood; (c) 0.5 mm thick blood; (d) 0.5 mm thick milk.
The application of the proposed dual-polarization detection setup for measuring AME of liquid dynamic media is demonstrated. AME of turbid water with different thicknesses, concentrations, particle sizes and shapes were measured. According to the Mie theory, the scattering angle increased as the particle size decreased, which resulted in a decrease of AME. Additionally, it was observed that the particle shape did not influence the AME. This was evident from the fact that 5 µm irregular shaped particles took almost the same AME as ball shaped particles. However, this is a conjecture as it is challenging to accurately evaluate the diameter of irregularly shaped particles. The proposed method could be used to characterize the AME of specialized scattering media, which may be challenging to examine from theoretical approaches.
Although the OD of 0.2 mm-thick blood is higher than that of turbid water, the AME range of the blood is higher, as shown in Fig. 4. This indicates that AME-based imaging methods are preferred for thin and strong scattering media, whereas gating methods based on ballistic photons work better for thick and weak scattering media.
4. Conclusion
In summary, a dual-polarization speckle detection system was proposed in this paper. Based on dual-polarization detection, the AMEs of dynamic turbid water, milk and blood were measured and analyzed. The influence of thickness, concentration, particle size and shape, and the beam diameter were also analyzed. Particles with irregular shapes took almost the same AME range with ball shapes. The AME was increased by decreasing the beam diameter, which could be used for enlarging the imaging FOV.
Funding
National Natural Science Foundation of China (62105359, 61905277, 61991452, 62005309); Key Research and Development Projects of Shaanxi Province (2020GY-008, 2021GY-079, 2022SF-87); China Postdoctoral Science Foundation (2020M673522).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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