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Abstract
Seeking a detailed investigation of polarization sensitive optical coherence tomography, electromagnetic numerical simulation of optical coherence tomography is carried out assuming deep diffraction gratings, some of which are in the resonance domain, as a sample to be measured. This is an approach from diffractive optics, and its simplification of the problem makes analysis more focused on its fundamental nature. The combination of the two disciplines provides previously unnoticed issues such as detection of a structure that does not exist.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical coherence tomography (OCT) has developed quickly since its invention in 1991 [1] and has established its status as an indispensable clinical tool in particular in ophthalmology [2] and cardiology [3]. In its conventional use, its signal intensity that corresponds to back-scattering light intensity from a tissue, was solely used to investigate the tissue. The imaging function of OCT can be extended by using other information of the back-scattering light. For example, Doppler OCT measures the flow velocity in the tissue by analyzing the phase of the OCT signal [4,5]. OCT angiography visualizes the flow by analyzing the temporal alteration of OCT signal.
Among such functional extensions of OCT, polarization sensitive OCT (PS-OCT) [6] attracts the researchers with its potential capability to analyze the sub-resolution structure of the tissue. In addition to the conventional structural OCT image, PS-OCT provides polarization sensitive (phase retardation) tomography. The phase retardation is due to form birefringence having its origin in the tissue microstructure with around the size of wavelength [7]. In addition, another quantity measured by PS-OCT, degree-of-polarization-uniformity (DOPU), is known to be sensitive to melanin molecules [8]. And hence, the PS-OCT image may reflects the microstructure of the tissue.
However, the relationship between the microstructure and the PS-OCT signal, which is essentially the set of two OCT signals with two orthogonal polarization states, is not well investigated. This significantly prevents the interpretation of PS-OCT signal in term of the tissue microstructure. An example is cumulative phase retardation and local phase retardation imaging. Phase retardation imaging has been widely used to investigate fibrous and collagenous structures. Although the quantification of retinal nerve fiber layer was successful [9,10] owing to the relatively simple micro-structural property of the retinal nerve fiber layer, its application to the collagenous structure [11–13] is mainly observational and is less quantitative owing to the complex and rather random structure of the collagen. Another example is an interpretation of DOPU. It has been widely used for visualization of retinal melanin distribution [8,14–18]. However, the physical mechanism of DOPU’s sensitivity to melanin is not well understood.
The relationship between the OCT signal and the tissue microstructure may be clarified by numerical simulation of OCT. Perhaps, the most frequently employed approach in OCT is Monte-Carlo method, because it is suitable to investigate stochastic scattering phenomena, which occur in biological samples in OCT measurement. Actually it had been already implemented in1990’s [19], has been developed since, and nowadays becomes a standard simulator of the OCT. Monte Carlo simulation is also applicable for polarization sensitive scattering [20]. However, such scattering based analysis is not appropriate for the analysis of the form birefringence, because here the birefringence is not occurred by the scattering but through wave propagation in microstructure of the tissue. Although other types of Monte Carlo methods were used to analyze PS-OCT [21,22], it only analyzes the effect of stochastic noise to the PS-OCT signal, and not analyzes the interaction between the tissue microstructure and polarization property of the back scattered light. In summary, Monte-Carlo method is not a perfect one to analyze the OCT imaging in the perspective of wave optics or electromagnetic domain.
Electromagnetic numerical simulation would be more promising, because it better treats wave propagation phenomena involving polarization. However, there are very few reports on electromagnetic simulation of the OCT except for some work by Munro, et al. with the finite-difference time-domain (FDTD) method [23] and with the pseudospectral time-domain (PSTD) method [24]. The latter demonstrated even three-dimensional numerical simulation assuming a practical OCT system. Alternatively, in more recent years, there are other electromagnetic approaches in frequency domain based on Mie scattering [25] and a Born series [26] both by Brenner, et al. There are reports on fields analysis having its origin in microscopy [27–29]. While they assume rather stochastic nature of tissue samples, our approach is rigorously electromagnetic and free from Born approximation. This issue is also mentioned in [23].
In this paper, we adopt electromagnetic numerical simulation designed for the analysis of diffractive optics (DO). DO is an optical element based technology fully utilizing wave nature of light and boasting three major advantages: generating arbitrary wavefronts, hybridizing multiple functions and smallness. Because of this, size of interest in recent DO research is mainly comparable to or smaller than wavelength of light. On the other hand, axial and lateral resolutions of the OCT are typically an order of a few to 20 µm [30]. Although the two disciplines, the OCT and DO has nothing in common as if living in mutually exclusive worlds, the polarization sensitive electromagnetic simulation of OCT will provide previously unnoticed aspects for both disciplines as we show here. In order to make the problem simplest, we concentrate on the polarization sensitive OCT signal of a depth-scan (A-scan), and our assumed samples are one-dimensional periodic structures normally illuminated by a plane wave in classical mount. Electromagnetic wave propagation is simulated with the finite-difference time-domain (FDTD) method. We understand this is the most suitable one among many simulators for electromagnetic wave propagation, because nature of phenomena in the OCT is in time-domain. In comparison to the work by Munro [23,24] and Brenner [26], the present study is mainly focused on polarization dependent property of A-scan profile, and not two dimensional image formation. In one hand, it limits the direct applicability of the study to the analysis of general 2D imaging properties of OCT. But on the other hand, it highlights very detailed characteristic of OCT A-scan and enables its physical interpretation such as in Section 3.4.
2. Analysis model and method
2.1 OCT model
We follow a technique of so-called spectral domain OCT (SD-OCT) [31,32] as shown in Fig. 1: an incident light wave from a broadband source is divided into two paths to a reference mirror and a sample, then reflected waves are coupled constructing an interfered time signal $f(t)$ of electric field, which enters a spectrometer consisting of a diffraction grating and a line sensor.
By assuming the spectrometer has an perfect spectral resolution, it gives a Fourier transform of $f(t)$
at a detector surface of the line sensor, and its output signal is a power spectrum $|\tilde {F}(\nu )|^{2}$. Here, tilde symbol $\, \tilde {} \,$ denotes a complex variable. This is the end of experimental part of the SD-OCT and subsequent signal processing of inverse Fourier transform of the power spectrum is performed by a computer, and finally obtained time function is a desired OCT signal. This $I(t)$ is usually presented as a function of depth as $I(2z/c)$, where $z$ is a depth in the air (or equivalently vacuum) and $c$ is the light speed in vacuum. Here the refractive index of the air is 1.0. Note that the inverse Fourier transform in Eq. (2) yields four terms. The first two are the self-interference signals of the object beam and the reference beam, which appear at around $t$ = 0. In addition, two cross-interference signals appear; one is the OCT signal and the other is its complex conjugate signal, as a so-called mirror image or coherence artifact [31]. By applying a proper path-length difference between the reference and the object arms, the former and the latter appeare in positive and negative in-air depth positions. Similarly, we can safely omit the negative frequency components in our simulation.In our numerical simulation, we obtain $f(t)$ by using FDTD method as described in details in Section 2.3, and then follow the standard SD-OCT processing through Eqs. (1) to (3).
2.2 Grating sample model
The sample of our analysis is a rectangular nondispersive dielectric grating free-standing in air by assuming its refractive index of 1.0. This is normally illuminated by a plane wave of Fourier transform limited temporal Gaussian pulse. Representation of this diffraction problem with a framework of the FDTD method is Fig. 2. As the problem is periodic in the $x$ (lateral) direction both in terms of structure and electromagnetic fields, only one period needs to be computed with periodic boundary condition in the $x$ direction. This periodicity results in our assumption that the lateral resolution of OCT is sufficiently larger than the grating period. For the $z$ (axial) direction, perfectly matched layers (PMLs) are placed at the both ends. The PML is the most frequently employed boundary condition in the FDTD method: the analyzed space is surrounded by artificial absorbing medium [33].
Refractive indices are chosen $n_{1}=1.0$ and $n_{2}=1.2$ to simulate low index contrast situation assuming biomedical applications. Note that this choice is a sort of normalization of a problem as often employed in numerical simulation. In the present study, $h=20$ µm is fixed to guarantee sufficient separation between reflections at the two grating boundaries in the $z$ direction. Our samples are ten gratings whose periods range from $d$ = 0.4 to 9.6 µm.
2.3 Analysis by the FDTD method
As to practical implementation of the FDTD method, unit cell sizes are 25 or 37.5 nm in the $x$ direction depending on grating period and 25 nm in the $z$ direction. Total length of computation area in the $z$ direction is 23.75 µm.
Incident pulsed wave is generated at a source plane with total field/scattered field scheme [34,35] and reflected waves are evaluated at an observation plane. Central wavelength $\lambda$ and full width at half maximum (FWHM) waveband of the pulse $\Delta \lambda$ in air are 1.0 µm and 100 nm, respectively, thus its FWHM pulse width is 14.8 fs. These light source specifications indicate the in-air axial resolution of the OCT is 4.4 µm.
Collected reflection data at the observation plane is numerically Fourier transformed along $x$ direction and only the zeroth order (DC) component is extracted. This operation corresponds to a common assumption that OCT captures only the pure backward scattering light for the imaging, and it is somehow rational by considering relatively low effective NA of OCT (see also Section 4.1). Reference wave from a mirror and object wave from a sample are obtained in separate computation, and then superposed to construct a time function $f(t)$.
During this process, the object wave is numerically delayed by 7,000 time steps from the reference wave. This numerical delay is to emulate an in-air path-length difference of 109 µm between the reference and the sample arms. Although the distance of 109 µm is shorter than the most of practical OCT configuration, it does not harm the generality of our analysis. It is because the simulated OCT signal, i.e. cross-interference between the reference and object beams, does not overlap the self-interference signals of object and reference beams. This is to follow actual experimental setup necessary to distinguish a signal term of $I(t)$.
An example of this operation for $d=0.4$ µm in s polarization is portrayed in Fig. 3, where electric field vector is oscillating in the $x$ direction in Fig. 2. Time function $f(t)$ consists of three peaks: the first, second and third ones correspond to reflection from a reference mirror, an front and back surfaces of a sample, respectively. Here, the total number of time steps in computation is $2^{15}$, a single time step being equivalent to a return distance 15.6 nm in air. Finally obtained OCT signal is $I(t)$, in which data area between the third and forth peaks are of our interest, i.e. the OCT signal, which carry the information on the sample.
Fig. 3. Example of $f(t)$ and $I(t)$ for $d=0.4$ µm in s polarization. The vertical scale is linear.
3. Results and interpretation
3.1 Result summary
Obtained OCT signals $I(2z/c)$ for $0.4\le d \le 9.6$ µm are summarized in Fig. 4. What is presented here is an extraction of a signal part of $I(t)$ in Fig. 3 for each period $d$. The first and second peaks correspond to reflection from the entrance and exit planes of grating layers, respectively. Solid and broken curves denote s- and p-polarization states, respectively.
Fig. 4. Period dependence of OCT signal $I(2z/c)$. Solid and broken curves denotes s- and p-polarization states, respectively. The vertical scale is linear.
3.2 Subwavelength gratings
When a grating period $d<1$ µm, it is called subwavelength grating and is known to exhibit strong polarization dependence of diffraction efficiency [36]. When normally illuminated, it generates only zeroth diffraction order both in reflection and transmission in our analysis model. Then, difference in shapes of the OCT signals between two polarization states is observed both in first and second peaks. This shows contrast to other values of $d>1$, where the height of the first peak is the same for the both polarization states.
3.3 Peak height
In some cases such as $d=1.2, 1.6$, and 3.2 µm, the first peaks are lower than the second ones. It is not surprising in our present model, because our samples are purely periodic gratings, which generate higher diffraction order waves, and only zeroth reflection order is considered for analysis, i.e. higher diffraction orders carrying significant energy are filtered out. Also, this effect is different between polarization states, in particular in the gratings in resonance-domain, in which transverse feature sizes are comparable to the wavelength of light [37].
3.4 Peak distortion
Among the cases of $d>1$ µm in Fig. 4, the most noticeable is s-polarization curve of $d=1.2$ µm in which the tail of the second peak is strongly distorted as if there exists an associated small peak. This distortion can be explained as follows.
What occurs in general around the grating layer is schematically illustrated in Fig. 5, where directions of propagating and evanescent waves are shown with dark gray and white arrows, respectively. In the present case of a Gaussian pulse $\Delta \lambda /\lambda =0.1$ normally incident on a grating $d/\lambda =1.2$, only zeroth and first diffraction orders exist both in reflection and transmission for wavelength components between 0.6 and 1.2 µm, and only zeroth order over 1.2 µm. On the other hand, evanescent waves upon diffraction are coupled with waveguide modes running along a grating vector, i.e. the grating layer behaves as if a waveguide, because its refractive index is effectively higher than surrounding media, in this particular case vacuum or air. This phenomenon is known as a resonant waveguide grating (RWG) [38]. As a result, the already coupled waveguide modes running inside the grating layer are re-coupled with propagating waves, indicated as dark gray arrows in Fig. 5. This means some fraction of incident optical energy is getting out of the waveguide and reflected and transmitted with certain degree of time delay.
As an example of this effect, Fig. 6 shows electric fields of the pulse after propagating several different distances inside the grating in the depth direction, where the grating depth is assumed infinite. It should be noted that these plots are not OCT signals, but the electric fields within the grating. Each plot represents a time function of the electric field after propagating certain length in the grating as indicated in the caption of Fig. 6. It is seen that as a pulse propagates, coupling and re-coupling continue and thus the tail of the pulse is extended. This is the main reason for the distorted second OCT signal peak. A degree of the distortion including time delay depends on various conditions, perhaps mainly an effective index of the grating layer. Although there is a report on designing an RWG [39], it is not so easy to quantify the distortion shape. It should be mentioned that this phenomenon seems more noticeable when the relative grating period $d/\lambda$ approaches to unity, i.e. the center wavelength of the object beam.
Fig. 6. Propagation distance dependence of an electric field of a pulse within a grating layer which is assumed infinitely deep. Each value at the right end denotes a distance the pulse propagates.
In order to look into this issue in a little more detail, gratings with $d=1.1, 1.2$, and 1.3 µm are compared for the same depth of $h=20$ µm under s-polarization illumination. Note that this shows a series of processes to obtain OCT signals such as in Fig. 4 and the cases of $d=1.1$ and 1.3 are employed for comparison and not included in Fig. 4. Second peaks of reflected electric fields from samples, i.e. a part of time signals $f(t)$, in Fig. 7 clearly show that a pulse tail is more distorted and extended for a shorter grating period $d$. Note that Fig. 7 does not show OCT signals, but electric fields of the object beam. Accordingly, the spectral interference signal, i.e. $|\tilde {F}(\nu )|^{2}$, exhibits more significant contribution of peripheral frequency components to the distorted pulse shape for a shorter period $d$ as found in Fig. 8.
Fig. 8. Power spectra $|\tilde {F}(\nu )|^{2}$ for $d=1.1$, 1.2, and 1.3 µm. The vertical scale is linear.
Finally, second peaks of desired OCT signals are given in Fig. 9, where a curve for $d=1.3$ µm is, unlike ones for $d=1.1$ and 1.2 µm, not distorted, but slightly stretched in rightward direction.
Fig. 9. Second peak shapes of OCT signal $I(2z/c)$ for $d=1.1$, 1.2, and 1.3 µm. The vertical scale is linear.
3.5 Peak shift
Another issue noticeable in Fig. 4 is shifting of the second peak positions in increasing grating period $d$. On the other hand, peak position of the first peaks can be regarded practically unchanged, its ranges being 0.30 and mere 0.047 µm in round-trip distance for s- and p-polarization states, respectively. Considering this fact, in DO point of view, the most likely cause would be change of effective index, because the value is dependent on grating period. This is summarized in Fig. 10, where the second peak positions and corresponding effective indices of grating layers are plotted for both s- and p- polarization states. An effective index is defined as
where $k$ and $\gamma$ are a wavenumber in vacuum and $z$ component of a wavenumber in the grating layer, respectively. The values of $n_\textrm {eff}$ are obtained with Fourier modal method (FMM), which is the most widely used for electromagnetic grating analysis in DO [40,41]. Alternatively, $n_\textrm {eff}$ can be obtained as a solution of a well-known eigenvalue equation [36]. When an effective index of the medium increases, corresponding vacuum distance increases. Figure 10 indicates that this effect significantly contributes to the shift of second peak, but there should be something extra to fully explain the phenomenon.4. Discussion
4.1 Limitation of numerical aperture
In an actual OCT system, a sample with periodic structure such as a grating is illuminated by a focused beam and thus electromagnetic field scattered by the sample grating is not theoretically periodic. In addition, owing to finite acceptance numerical apertures of a focusing lens and an optical fiber, not only normally reflected wave but also some extra angular components contribute to final OCT signal.
In this sense, our approach could seem oversimplified. We would like to emphasize, however, it can remove some practical limitations and simplify the problem, resulting in concentration on physical phenomena. For example, assuming a practical model of focused-beam illumination, electromagnetic numerical simulation requires tough boundary condition in the $x$ direction, because even slight unwanted numerical reflection affects the obtained OCT signal significantly in particular in resonant problems like ours. On the other hand, our approach does not have to worry about it at all. Therefore, our observed data exhibits very fundamental nature of OCT signal processing, which is free from practical detail of experimental setup.
Even though our assumption of normal incidence is for fundamental study, it is still reasonable in many practical OCT study, because the numerical aperture (NA) of OCT is normally low. For example, a beam diameter of 2.4 mm and an objective focal length of 60 mm result in an effective NA of 0.02. (The values are for the prototype of an anterior eye segment OCT [42].) And hence, the present analysis is still reasonable for many cases.
On the other hand, so called OCT microscopy (OCM) [43,44] has recently attracted wide attention of researchers. Since the NA of OCM, typically 0.06 to more than 0.1, is significantly higher than conventional OCT, we need further modification in the model and method to analyze such high NA OCMs.
4.2 Limitation of sample structure
In the present analysis, only purely periodic structures were analyzed. Although such structure must be unrealistic in particular in biological tissue, this simplified analysis can have at least two utilities.
Firstly, it gives us intuitive model to understand and interpret the OCT and PS-OCT signals. Secondly, such simplified model can be used as a unit block to represent more complicated structures, since any structure can be represented by the superposition of locally periodic structures. Although careful design of analysis frame work is required, the present analysis can be used as a unit module of the analysis of further complicated structures. Note, however, while this concept is generally acceptable in lateral direction [45], each consisting structure must be well separated, i.e. farther than coherence length, in axial direction.
Needless to say, increasing complexity of sample structure in axial direction requires large analysis area sometimes including three-dimensional and eventually results in heavy computational load. For readers’s reference, a typical CPU time to obtain a single A-scan result in Fig. 4 ranges from 17 s for $d=0.4$ µm to 295 s for $d=9.6$ µm with a Fortran workstation with two 2.93 GHz Xeon processor (4 core) and 48 GB RAM.
4.3 Potential PS-OCT analysis
The current study aims at exposing the behavior of OCT signals of p- and s-polarization states as the most fundamental OCT components. On the other hand, several secondary PS-OCT signals can be computed from these fundamental OCT signals, such as cumulative phase retardation [46,47], local phase retardation [21,48,49], degree-of-polarization-uniformity (DOPU) [15] and depolarization index [50]. Because large variety of such secondary PS-OCT signals exist, the analysis of such signal exceeds the scope of the present paper. However, it is an important research subject for the future study.
5. Conclusions
We have attempted to numerically simulate PS-OCT through electromagnetic wave propagation with the FDTD method and subsequent signal processing. Although the present study analyzed only simplified grating structures, our prime motivation is investigating effects of tissue microstructure on OCT signal, and polarization plays an important role. We believe that this is the very reason the PS-OCT is required. For this purpose, we adopted rather unorthodox approach, i.e. diffractive optics, to simulating OCT. As mentioned earlier, DO is based on optical elements and when their feature sizes are comparable to wavelength of light, their optical behavior must be treated electromagnetically. Then, DO has ample experience of handling various sorts of electromagnetic scattering problems, e.g. rectangular, blazed, triangular, rounded and even random ones in surface relief in addition to the one with gradually modulated index profile [51]. Its method and vast amount of knowledge should be truly useful also for OCT.
In applying concept of DO to OCT, a problem to be solved is simplified, i.e. electromagnetic wave propagation in tissues with periodic structures. This enables us to concentrate on fundamental aspect of wave propagation inside the tissue, unnecessary to worry about unwanted numerical error associated with a boundary condition, and accurate analysis becomes possible. Although our simplified problem may be scarcely encountered in biomedical applications, it turns realistic when the OCT is applied to industrial objects [52,53]. In addition, we should not exclude possibility of regular or periodic structures appearing within an illuminated area of a sample even in biomedical applications. Please note that resonant effects should be weakened in non-periodic structures.
The most interesting outcome of our work should be peak shape distortion observed with a grating of $d=1.2$ µm in s polarization. This result indicates that under some conditions the OCT may exhibit small peak artefacts which mimic non-existing structures, whether a sample is biomedical or industrial. This may cause serious consequence in some cases. It is not, however, easy to solve this issue. In addition, it should be emphasized that effective index of the medium plays an important role, e.g. in shift of peak position.
Our analysis is of course a sort of forward problem: there exists a deterministic structure first and wave propagation is analyzed. On the other hand, actual OCT is to solve a inverse problem: an OCT signal is acquired at first, and then a sample structure is guessed from the measured signal.
Then, our important task in future is to bridge over the two problems and this is certainly necessary to improve accuracy of OCT.
Funding
Japan Science and Technology Agency (15K13371, 18H01893, 18J13841).
Acknowledgment
H. Fujibuchi is currently with Kochi Broadcasting Co., Ltd., Kochi, 780-8550, Japan.
Disclosures
HI and HF declare no conflicts of interest. YY discloses following conflicts of interest; Topcon (F), Tomey Corp. (F, P), Nikon (F), Yokogawa (F), Kao (F).
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