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Abstract
Recently the acquisition of the time-resolved reflection matrix was demonstrated based on spectral domain optical coherence tomography. In principle, the matrix describes the linear dependence of the OCT signal received from different depths on the field which is incident to the scattering sample. Knowledge of the matrix, hence, enables beam shaping to selectively enhance the received signal, for example to increase the penetration depth when imaging turbid media. We investigate the impact of image artefacts on the approach. Phase conjugation is shown to enhance the OCT signal, but not autocorrelation and mirror artefacts. Imaging applications are demonstrated indicating the potential for future in-vivo studies on biotissues.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical coherence tomography (OCT) enables non-invasive label-free three-dimensional imaging with microscopic resolution. The technique was introduced almost 30 years ago and has been widely employed for medical diagnostics since, for example for retinal imaging. OCT utilizes coherence gating and interferometric detection to determine the time-of-flight distribution of light which is backscattered from a sample. In case the detected light is assumed to be scattered only once, the observed time-of-flight distribution accurately reflects the position of backscattering sample structures and enables non-invasive sub-surface imaging. In practical systems multiple scattered light is detected as well. In this case, the time-of-flight or the corresponding optical path length is statistically distributed and does not reflect the actual penetration depth in the sample. A continuous OCT signal results which does not allow to investigate the sample morphology [1]. The imaging capability with strongly scattering media such as biological tissue, thus, is limited by the ability to suppress contributions from multiple scattered light compared to single-scattered signal contributions. Typical approaches to extend the penetration depth include the choice of weakly scattering light sources, broadband sources which result in an increased temporal resolution or the use of small detectors to enhance the effect of confocal gating.
Scattering of light is a deterministic process if the sample is considered to be static, though, and can be harnessed for imaging. Recently, Badon et al. and Kang et al. demonstrated OCT imaging based on the time-gated reflection matrix [2–4]. In principle, the matrix describes the full-field OCT signal, i.e. the complex-valued field which is backscattered from the sample at a given time-of-flight, which is resulting from sample illumination with different basis modes. Correlations of the reflection matrix allow to suppress signal contributions from multiple scattered light and, thus, enable depth-enhanced imaging [2–4].
In another approach a number of groups demonstrated iterative wavefront shaping to enhance the OCT signal when imaging scattering samples [5–9]. Wavefront shaping approaches were initially demonstrated to enable focusing of light which is transmitted through strongly scattering media [10]. In principle, the technique tries to shape the beam which is incident to the sample in a way that constructive interference is created from scattered light at a single target position behind the medium [10], resulting in a focal spot to be created on top of a diffusive speckle background [9,11]. The spot size is equal to the observed speckle size and, thus, depends on the sample‘s scattering properties rather than on the imaging optics [12]. The peak intensity at the focus relates to the number of independent segments of the shaped wavefront [10,13,14]. A significant intensity enhancement compared to diffusively scattered light can be achieved even with a low-resolution wavefront and with strongly scattering samples [14]. The technique, hence, has to be differentiated from adaptive optics which tries to completely cancel aberrations present in weakly scattering media for diffraction limited focusing and imaging.
Similar to transmission experiments, wavefront shaping approaches can be used to focus backscattered light, as well. Applied to OCT imaging, the technique can be understood to optimize the beam which is incident to the sample such that backscattered light interferes constructively at the detector, resulting in the amplitude of the observed signal to be increased [5–9]. The approach was successfully demonstrated to extend the penetration depth when imaging scattering media such as biological tissue [7]. In contrast to conventional approaches for depth-enhanced OCT imaging, iterative wavefront shaping requires scattering at the sample rather than trying to suppress the corresponding signal contributions. The approach is validated by noting that the technique can enhance signal contributions from weakly scattered or predominantly forward scattered light whose time-of-flight still reflects the sample morphology to some extend [1,15]. These signal contributions become more dominant if low numerical aperture objectives are used or if strongly forward scattering samples are imaged [1,15,16], which is the case for most scanning SD-OCT systems and for most biological tissues [17].
Wavefront shaping techniques require the scattering samples to remain static. When imaging in-vivo biological tissue the scattered field decorrelates at time scales as short as several milliseconds due to tissue movement and blood flow [18,19] and the achieved signal enhancement is lost. Algorithms to optimize the incident wavefront and to capture the OCT signal, thus, need to be faster than that. The speed of wavefront optimization is proportional to the number of degrees of freedom $N$ of the incident wavefront and to the number of acquisitions required to find an optimal value for each degree of freedom. Jang et al. demonstrated an iterative wavefront shaping algorithm for OCT signal enhancement which required 25 acquisitions for each degree of freedom of the wavefront [6]. As a result, the optimization for a single A-scan required 15 seconds, even though the group implemented an efficient data acquisition and utilized a fast spatial light modulator (SLM) for beam shaping. Fiolka et al. demonstrated wavefront shaping with a parallelized algorithm which was introduced by Cui in 2011 [5,20]. The algorithm requires, in principle, two acquisitions for each degree of freedom of the incident wavefront [20].
Choi et al. recently demonstrated the acquisition of the time-resolved reflection matrix with a spectral domain OCT (SD-OCT) system [21]. The matrix describes the linear dependence of time-of-flight resolved backscattered light on the field which is incident to the sample. Optical phase conjugation based on the matrix was demonstrated to optimize the incident beam such that the resulting OCT signal is selectively enhanced [21]. The approach is analogue to iterative wavefront shaping and requires just a single signal acquisition for each degree of freedom of the incident beam to find an optimized wavefront which enhances the OCT signal. The technique, thus, has the potential for applications in high-speed imaging. The impact of image artefacts which are present with SD-OCT systems was not yet discussed, however.
In this work we provide an analytic framework on how the time-resolved reflection matrix relates to the SD-OCT signal. From this, we demonstrate phase conjugation based on the reflection matrix to selectively enhance only those signal components which reflect the sample morphology but not image artefacts, even though no attempt is made to suppress those artefacts upon acquisition of the matrix. The effect allows to enhance the OCT signal when imaging scattering media and to suppress artefacts at the same time. First imaging applications are presented.
2. Principles
In this section a theoretical framework is developed on how the spectral domain OCT signal depends on the field which is applied to the sample beam. Considering this field to be constructed from a linear superposition of $N$ modes and considering the OCT signal to be discretized at a set of $M$ pixels, the linear dependence can be described by a $M\times N$ matrix, which is the sample‘s time-resolved reflection matrix. The practical acquisition of the reflection matrix is discussed in Sec. 2.2. The application to wavefront optimization at the sample beam to selectively enhance the OCT signal is discussed in Sec. 2.3.
2.1 Definition of the time-resolved reflection matrix
Spectral domain optical coherent tomography is an interferometric technique which is based on the superposition of light reflected from the sample, characterized by it‘s electric field $E_S(\omega )$, with a static reference beam $E_R(\omega )$. The time-of-flight distribution of the backscattered sample beam is taken from the inverse Fourier transform of the resulting power spectral density $|E_R(\omega )+E_S(\omega )|^2$. According to the Wiener Khinchin theorem this signal may be written [22]:
The first two terms in Eq. (1) correspond to the autocorrelation of the reference and sample beam, respectively, and yield DC and autocorrelation artefacts which are well-known for SD-OCT imaging. The third term corresponds to the cross correlation of the two beams which reads $\Gamma _{RS}(\tau ) = \langle E_R(t+\tau )\overline {E_S}(t) \rangle$. $\tau$ describes a relative temporal delay and the angle brackets denote temporal averaging, the bar indicates complex conjugation. The cross correlation $\Gamma _{RS}(\tau )$ yields the time-of-flight distribution of the backscattered sample beam and is termed mutual interference signal. $\Gamma _{RS}$ reflects the sample structure in case signal contributions from multiple scattered light can be neglected and enables imaging. Due to the symmetry of the cross correlation the fourth term in Eq. (1) reads $\Gamma _{SR}(\tau ) = \overline {\Gamma _{RS}}(-\tau )$ and, hence, is found to describe the mirror image of the mutual interference signal.In principle, mutual interference signals and mirror artefacts yield the same information and, hence, either term may be used for sample imaging. Which of the both terms is considered signal and which is assumed to be an image artefact is a matter of definition. In this work we consider $\Gamma _{RS}$ to be the mutual interference signal. Since this definition is essential for the discussion to follow we will provide a short justification: If the distance between the OCT system and the sample is increased an additional temporal delay $t'$ is introduced to the field $E_S$. In this case the cross correlation $\Gamma _{RS}$ reads $\langle E_R(t+\tau ) \overline {E_S}(t+t') \rangle = \langle E_R(t+\tau -t')\overline {E_S}(t) \rangle = \Gamma _{RS}(\tau -t')$. The signal, thus, is shifted to larger temporal delays $\tau$. As a consequence sample features located further away from the OCT system are detected at larger displacements $z = c\tau /2$, i.e. the sample image is retrieved at the correct depth-scale. In contrast, the mirror term $\Gamma _{SR}$ yields the sample‘s image at a reversed scale.
We consider optical propagation in the imaging system and in the scattering sample to be a linear and time invariant process [23]. We further assume a static reference field $E_R(x,y,t)$ at the detector and wavefront control at the sample beam at a single plane in front of the sample only. $E_{in}(x,y,t)$ describes the corresponding field at that plane and can be written from a linear combination of $N$ orthogonal basis modes $e^{in}_n(x,y,t)$ with complex amplitude $A_n$, respectively:
Due to the linearity of propagation, the detected sample beam field reads:
The term $e_n^S$ corresponds to field contributions which are observed at the detector from illumination with the respective basis modes. The explicit function is not relevant for this discussion, but for the sake of completeness we note that $e_n^S$ can be calculated from the Green function $g$ which describes the linear propagation from the plane of the SLM to the sample, scattering at the sample and the subsequent propagation to the detector [23]:Inserting Eq. (3) in the definition of the cross correlation allows to describe the individual SD-OCT signal components (Eq. (1)) in terms of the amplitudes $A_n$ of the sample beam source modes:
The integration is performed over the active detector area $A_{det}$ and the temporal average is taken with respect to the detector integration time.
Obviously, the autocorrelation $\Gamma _{RR}(\tau )$ of the reference beam does not depend on the field at the sample beam. The autocorrelation of the sample beam $\Gamma _{SS}$ is described by Eqs. (6) and (9). The term $\Gamma _{nn'}(\tau )$ corresponds to the cross-correlation of the $n$-th and the $n'$-th source mode at the plane of the detector. The autocorrelation signal detected from the sample beam, thus, does not depend linearly on the field incident to the sample.
Equations (7) and (10) yield the mutual interference signal. The term $\Gamma _{Rn}(\tau )$ describes the cross correlation of the reference beam and the $n$-th source mode which is backscattered from the sample. Eq. (8) describes the mirror signal. We note that the the mirror signal $\Gamma _{SR}$ depends linearly on the amplitudes $A_n$ of the source modes. In contrast, the mutual interference signal depends linearly on the complex conjugated amplitudes. This behavior is not surprising since mutual interference and mirror signals are complex conjugated with respect to each other due to the hermitian symmetry of the SD-OCT signal. The linear dependence of the mutual interference signal on the complex-conjugated sample beam field is also evident from the definition of the cross-correlation $\Gamma _{RS}$.
In practical SD-OCT systems the signal is digitally detected at a discrete frequency raster and the time-domain signal is calculated from the inverse discrete Fourier transform (IDFT) instead as from the continuous transform. Hence, the time domain signal $I(\tau )$ is evaluated at a set of $M$ discrete points $\tau _m = m \Delta _\tau + \tau _0$ (or $z_m = m \Delta _z + z_0$) which are determined by the spectral sampling. Thus, the linear dependence of mirror signal components on the field incident to the sample can be quantified by the $M \times N$ matrix $R_{mn}^{mirror}$ (compare Eq. (8)):
We are interested in a linear description of the mutual interference signal, however. This is achieved by considering the complex-conjugated signal (compare Eq. (7)):
From these definitions we find $R_{mn}^{mirror} = R_{(-m)n}$. Eq. (13) yields our definition of the time-resolved reflection matrix $R_{mn}$ which describes the linear dependence of the mutual interference signal on the field at the sample beam. The complex conjugation of the signal has no impact on optical phase conjugation which enhances the amplitude $|\Gamma _{RS}|$ of the signal and which is discussed in Sec. 2.3.2.2 Matrix acquisition
The reflection matrix is acquired experimentally by sequentially applying each source mode individually to the sample [21,24]. In case the $n'$-th mode is applied with unity amplitude $A_{n'} =1$, the detected mutual interference signal reads $\Gamma _{RS}(\tau _m) =\Gamma _{Rn'}(\tau _m)$ (Eq. (7)). The $n'$-th column of the reflection matrix, thus, is found from the complex conjugate of the OCT signal according to our definition (Eq. (14)). The full matrix is acquired by iterating all modes and, hence, requires $N$ sequential signal acquisitions.
In practical SD-OCT systems additional artefact signals are present. The impact is discussed in Sec. 3. At this point we consider the acquired reflection matrix to describe the mutual interference signal only. We note that phase shifting approaches can be used to actively suppress image artefacts during the acquisition of the reflection matrix. Taking, for example, a conventional 5-step algorithm [25], the approach is easily verified from Eq. (1) and Eqs. (5) to (8) to cancel all signal components except $\Gamma _{RS}$.
2.3 Phase conjugation
Once the reflection matrix $R_{mn}$ is determined, the SD-OCT signal can selectively be enhanced by using a conventional phase conjugation approach [21,23,24]. We consider signal enhancement at a single target pixel $m_t$. Ideally, the OCT signal at this pixel probes the field which is backscattered from a distinct depth and lateral position in the sample. An optimal wavefront which is applied to the sample and which enhances the signal received from the target is found by calculating the amplitudes $A_n^{opt}$ of source modes (compare Eq. (2)) from the conjugated reflection matrix and superimposing all modes with their respective amplitude applied [21,23,24]:
The effect on the mutual interference signal which is detected in this case is found by inserting into Eq. (13): At the target pixel $m_t$ all contributions from the individual modes to the OCT signal (summands in Eq. (16)) match in phase and a large signal amplitude $|\Gamma _{RS}(\tau _{m_t})|$ results for the superimposed signal. This effect is well-known from previous transmission and reflection matrix experiments [21,23,24].The physical effect of phase conjugation on the SD-OCT signal is understood by considering our definition of the reflection matrix (Eqs. (10) and (14)). The phase of the $m_t$-th matrix column probes the (temporally averaged) phase difference between the static reference beam and the individual source modes which are backscattered from the sample and detected at the time-of-flight which corresponds to the $m_t$-th pixel of the OCT signal. Neglecting the effects of multiple scattering, the sample beam can be assumed to be backscattered from a single target depth. The phase conjugation approach actively manipulates the complex amplitude of the source modes such that the individual contributions to the complex OCT signal are aligned to the real axis and, hence, match in phase. If the superimposed wavefront is applied to the sample, the individual signal contributions are summed up and, hence, and a large amplitude results for the final signal.
We emphasize the fact that the reference beam remains static throughout the process. This implies that the phase conjugation algorithm actually aligns the contributions from individual source modes to the field $E_S$ of the sample beam in phase at the plane of the detector. As a consequence, phase conjugation enhances the field amplitude of light which is backscattered from the target depth to the detector. An increase of the detected OCT signal results as well.
3. Phase conjugation in the presence of image artefacts
In conventional transmission or reflection matrix approaches interferometric techniques such as phase shifting interferometry are required to determine the phase of the scattered field and the complex-valued reflection or transmission matrix. The approach is necessitated by real-world detectors being able to capture the intensity of scattered light only. In contrast, spectral domain OCT is already an interferometric imaging technique itself and does not require additional approaches to determine the phase of the backscattered field. Indeed, the phase of the backscattered wave is encoded in the offset of spectral interference fringes and can be acquired from the complex-valued inverse Fourier transform of the raw spectral data (compare Eqs. (7) and (10)). The experimentally determined spectral raw data is real-valued, on the other hand. As a consequence, the SD-OCT signal is hermitian symmetric and image artefacts result (Eq. (1)).
Choi et al. demonstrated the direct acquisition of the time-resolved reflection matrix from the SD-OCT signal without further artefact suppression [21]. The reflection matrix can be determined from the acquisition of $N$ raw spectra in this case. This number exactly matches the number of degrees of freedom of the beam incident to the sample and yields at least a three-fold increase in acquisition speed compared to any phase shifting algorithm. In principle, the reflection matrix can not be acquired faster than that. The impact of artefact signals which are present in the SD-OCT signal in this case was not discussed by the group, however.
In this section we investigate the impact of artefacts on optical phase conjugation with the depth-resolved reflection matrix. In most practical applications the individual components of the OCT signal are separated with respect to the axial scan axis. DC- and autocorrelation artefacts, for example, are detected close to $z=0$. The position of mutual interference and mirror signal components depends on the length of the reference arm, on the other hand. Typically the reference arm is aligned in a way that mutual interference signals do not overlap with other signal components.
To enhance the OCT signal at the target pixel $m_t$, the optimized wavefront for phase conjugation is calculated from the $m_t$-th row of the reflection matrix (Eq. (15)). This row corresponds to the signal which is detected from the target depth only. Hence, to investigate the impact of image artefacts on the acquisition of the reflection matrix and on subsequent phase conjugation we may assume the OCT signal at the target to be dominated by just one signal component. Based in this assumption we demonstrate phase conjugation to enhance the mutual interference signal only.
3.1 Target signal dominated by mutual interference signals
We consider the signal at the target $m_t$ to be dominated by mutual interference signals. In this case the $m_t$-th row of the experimentally acquired reflection matrix $R^{obs}_{mn}$ matches the same row of the matrix $R_{mn}$ which describes mutual interference signals only (Eq. (13)). Phase conjugation with the matrix $R^{obs}_{mn}$, hence, has the same effect as phase conjugation with the ideal matrix $R_{mn}$ and selectively enhances the mutual interference signal at the target depth (Sec. 2.3). Due to the hermitian symmetry of the conventional SD-OCT signal, the signal at the mirror position of the target is enhanced simultaneously.
Furthermore, we expect an additional selective enhancement of autocorrelation artefacts. In Sec. 2.3 we gave reasons that phase conjugation with the reflection matrix actually enhances the amplitude of light which is backscattered from the target depth to the detector. The increase of the mutual interference signal amplitude results from the increased sample beam amplitude and from interference with the static reference beam. Autocorrelation artefacts result from mutual interference with light which is backscattered from another strongly reflecting sample layer, for example the sample front surface. The field which is backscattered from this layer may take a random phase and amplitude distribution in the case of beam shaping for signal enhancement at another depth. The interference with light which backscattered from the target depth is enhanced nonetheless, since more light is scattered from this depth to the detector.
Typically, autocorrelation signals are dominated by a few strongly reflecting layers present in the sample. The autocorrelation signal, thus, takes the form of the mutual interference signal which is shifted by the absolute position of the dominant reflecting layer. In the case of optical phase conjugation we expect a simultaneous enhancement of the autocorrelation signal which corresponds to the shifted position of the target depth.
3.2 Target signal dominated by mirror signals
Mirror artefacts depend linearly on the field of the sample beam without complex-conjugation of the OCT signal and, hence, may be written in matrix notation as well (Eq. (11)). We consider the OCT signal at the target $m_t$ to be dominated by mirror artefacts. When illuminating the sample with an isolated mode, which is the case during the experimental acquisition of the reflection matrix, the resulting signal reads $\Gamma _{SR}(\tau _{m_t})=R_{m_tn'}^{mirror}$ at the target. Due to our definition of the time-resolved reflection matrix we save the complex-conjugate of this signal to the corresponding elements of the experimentally determined reflection matrix, i.e. $R_{m_tn}^{obs} = \overline {R^{mirror}_{m_tn}}$.
The phase conjugation algorithm calculates the wavefront incident to the sample from the complex conjugate of $R_{mn}^{obs}$ (Eq. (15)). Inserting into Eq. (11) reveals the effect on the signal amplitude in case the signal at the target is dominated by mirror artefacts:
In general, this corresponds to a sum of random phasors and, opposed to phase conjugation with the mutual interference signal, the individual summands do not match in phase at the target $m=m_t$. Hence, no significant enhancement of the signal amplitude results.We note that it is a matter of definition which signal components are considered mutual interference and which are considered mirror signals. Both components faithfully reflect the sample morphology and enable imaging. With the phase conjugation algorithm presented in this work only mutual interference signals according to our definition are enhanced while no similar effect on mirror signals is expected. The algorithm can easily be changed to yield the opposite behavior, i.e. to enhance what we define to be mirror signals but not the mutual interference signals. This is readily achieved by taking the experimentally observed reflection matrix $R_{mn}^{obs}$ directly from the complex OCT signal instead as from the complex conjugated one. The matrix yields a linear description of mirror artefacts (Eq. (11)) but not of mutual interference signals (Eq. (14)) in this case and phase conjugation enhances mirror artefacts only. This behavior is easily verified analogue to the previous discussion.
3.3 Target signal dominated by DC and autocorrelation signals
In case the OCT signal at the target position $m_t$ is dominated by autocorrelation artefacts, the corresponding row of the measured reflection matrix reads (Eqs. (5) and (6)):
The signal is composed from the reference beam autocorrelation $\Gamma _{RR}$ and the term $\Gamma _{nn}$ which is the autocorrelation of the $n$-th mode after reflection at the sample (Eq. (9)). The reference beam autocorrelation is considered static and does not depend on which mode is applied to the sample beam. $\Gamma _{RR}$, hence, is constant for all columns of $R_{m_tn}^{obs}$. In contrast, the dependence of the sample beam autocorrelation $\Gamma _{SS}$ on the incident beam in principle is fully described by the term $\Gamma _{nn'}$ (Eq. (6)). The dependence is not linear, however, and the cross-terms $\Gamma _{n,n'\neq n}$ which yield the mutual interference of different modes are not accessed experimentally since only isolated modes are applied during the acquisition of the reflection matrix. The phase conjugation algorithm relies on the reflection matrix to yield a linear description of the signal which is to be enhanced, on the other hand. This condition is not met if the OCT signal is dominated by autocorrelation and DC artefacts. Hence, no significant signal enhancement is expected.
3.4 Overlapping signal components
Our analytic discussion on the effect of phase conjugation is based on the assumption that the OCT signal at the target depth is dominated by only one signal component. In practice, this condition is not always met. If the length of the reference arm is chosen to be too short, for example, mutual interference signals may overlap with autocorrelation or with mirror artefacts. In general, mutual interference signals and image artefacts which are detected at the same time-of-flight result from reflections at different depths in the sample, though. The signal components, hence, can be considered to be uncorrelated.
We do expect the phase conjugation algorithm to work to some extend even if experimental noise is detected during the acquisition of the reflection matrix. In fact, for the theoretical discussion given in Sec. 2.3 we considered the phase conjugation algorithm to align signal contributions which are resulting from the individual modes exactly in phase. The amplitude of the superimposed OCT signal is enhanced even in case all complex-valued contributions are only roughly aligned to one direction, though. Previous works utilized the approach for wavefront shaping with binary amplitude-only spatial light modulators [26,27]. Phase errors can result from experimental noise and a lower signal enhancement results compared to ideal phase conjugation. The impact of uncorrelated image artefacts overlapping with the mutual interference signal is expected to be similar to the effect of noise, i.e. we expect a limited enhancement of mutual interference signals even in case image artefacts are detected at the same position.
4. Experimental validation
We demonstrate the acquisition of the depth resolved reflection matrix and subsequent phase conjugation experimentally with the SD-OCT system which was reported previously by our group [11]. A schematic of the setup is illustrated in Fig. 1(a). The system is based on a double-interferometer design with imbalanced arm lengths. The optical path length of the effective reference beam (blue) and the effective sample beam (red) is matched and, hence, mutual interference from these beams is detected with the OCT system. A liquid crystal phase-only spatial light modulator allows to manipulate both beams independently. The shaped beams do not pass the SLM again prior to detection at the spectrograph. The experimental design, thus, aligns well to the assumptions which were made for the theoretical discussion and enables the acquisition of the reflection matrix. Details are given in our previous work [11]. The approach is demonstrated with a sample consisting of a stack of multiple layers of pergamin paper (Whatman 2122, GE Healthcare, UK) with a thickness of 30 µm each. The sample‘s extinction coefficient was determined to be $30\pm 10$ mm−1 by measuring the intensity of unscattered light transmitted through the sample.
Fig. 1. (a) Experimental design. The reference (blue) and sample beam (red) are manipulated independently with a single phase-only SLM. Abbreviations: SLM spatial light modulator, spec spectrograph. (b) Signal acquired with a random and with a flat phase pattern (panel (c)) applied to the sample beam, respectively. A five step phase shifting algorithm is used to suppress image artefacts. Phase shifting is conducted by applying a uniform phase offset to the reference beam. (d) Phase pattern found by the phase conjugation algorithm to enhance the OCT signal at a depth of 0.73 mm and (e) at 0.90 mm. The signals which are acquired once the respective phase patterns are applied to the sample beam are illustrated in panel (b).
We start with the reflection matrix which is acquired from the mutual interference SD-OCT signal alone. To this end we utilize the SLM to apply a set of 1024 different Hadamard modes to the sample beam. For each mode we apply a set of five different phase shifts to the reference beam and acquire the complex-valued SD-OCT signal from a conventional five step phase shifting algorithm [25]. The conjugated signal is then saved to the column of the reflection matrix which corresponds to the tested mode, respectively (Sec. 2.2). Fig. 1 illustrates the effect of subsequent optical phase conjugation on the acquired OCT signal. The algorithm finds random-like phase patterns (Fig. 1 panels (d) and (e)) which result, after application to the sample beam, in a significant and localized increase in signal intensity which is detected from the target depth.
For a quantitative comparison we note that the OCT signal which results from sample illumination with a flat wavefront is subject to speckle. Hence, instead we use the average signal amplitude which results from sample illumination with a random wavefront for comparison. This data can be taken directly from the reflection matrix without further acquisition by averaging the amplitudes $|R_{mn}|$ of the respective matrix rows. The approach corresponds to the signal one would receive from a speckle-compounding algorithm [28]. In addition we illustrate the 95-th percentile of the signal amplitude. This number gives a good estimate for the maximal amplitude which is observed with random wavefront sample illumination. As an effect of optical phase conjugation we observe a local signal enhancement which significantly exceeds the amplitude expected from illumination with a random wavefront. Compared to the average signal amplitude at the target we observe an enhancement of 14.99 dB (31.56-fold linear enhancement) and 11.42 dB (13.88-fold linear enhancement) for the two scans illustrated on Fig. 1, respectively.
Figure 2(a) illustrates the signal in case the target depth is scanned through the sample. Each row in the illustration corresponds to another point-optimized A-scan. The sample‘s mutual interference signal is evident in the right hemisphere (first and fourth quadrant). Additional image artefacts are suppressed due to the five step phase shifting algorithm. The diagonal elements correspond to the signal which is received at the respective target depths for the phase conjugation algorithm. We observe these elements to be significantly enhanced at the first quadrant of the illustration, i.e. the algorithm takes only at those depths where a mutual interference signal is detected effect and does not produce an artificial signal at those positions where no sample is located.
Fig. 2. (a) SD-OCT signal acquired with a five step phase shifting algorithm for artefact suppression and optimized for signal enhancement at different target depths. Quadrants are labeled by roman numbers. (b) Intensity at the diagonal elements of panel (a) only. (c,d) Same as (a) and (b), but taken from the conventional SD-OCT signal without artefact suppression.
Figure 2(b) presents the intensity at the diagonal elements only. Indeed, at the position at which a OCT signal is detected from the sample a significant signal enhancement results. The slope of the OCT signal decreases at a depth of about 1 mm. This indicates that the signal at this depth actually is dominated by multiple scattered light from sample layers which are actually shallower than the supposed target depth [1,16]. The signal, thus, does not necessarily reflect the sample morphology. We observe the phase conjugation algorithm to enhance this signal, as well. We further observe additional ghost images at depths of −1 mm and 1.8 mm which are also enhanced by phase conjugation.
We repeated the experiment at the same sample but without suppressing image artefacts during the acquisition of the reflection matrix and during subsequent phase conjugation (Fig. 2(c)). The reflection matrix, hence, is acquired directly from the complex-valued SD-OCT signal. In contrast to the previous data, additional strong autocorrelation artefacts close to $z=0$ as well as mirror artefacts in the left hemisphere (second and third quadrant) are evident.
The phase conjugation algorithm behaves differently depending on which kind of signal is observed at the target, i.e. at the diagonal elements. At label (1) the target depth corresponds to autocorrelation artefacts. Hence, at the target itself (at the diagonal) and at other positions in the scan no signal enhancement is observed. In contrast, at label (2) mutual interference signal components are targeted (compare Fig. 2(a)). In this case a signal enhancement at the target position results. Additionally, we observe a signal enhancement which appears as a line parallel to the diagonal elements at the position of the ghost image (1.8 mm). Furthermore we observe a local enhancement of autocorrelation artefacts. This signal corresponds to interference of light which is reflected from the target depth with light reflected at the sample front surface. The signal, thus, appears as a line parallel to the diagonal elements, as well. Due to the hermitian symmetry, the corresponding mirror signals which are evident in the second quadrant are enhanced, too. At label (3) the ghost image is targeted. A signal enhancement similar to the case when the mutual interference signal is targeted results. The lower half of Fig. 2(c) (third and fourth quadrant) corresponds to the case autocorrelation and mirror artefacts only are targeted. No significant enhancement of the OCT signal is evident.
Figure 2(d) reflects the in-target signal amplitude, i.e. the diagonal elements of Fig. 2(c). Obviously, we observe a significant signal enhancement only at those depths where a mutual interference signal (label (2), compare Fig. 2(b)) or a ghost image is observed (label (3)). At other depths, the in-target signal amplitude resulting from phase conjugation is comparable to the level which is achieved with sample illumination with a random wavefront.
Figure 3 illustrates the enhancement of the optimized OCT signal (Fig. 2(d)) compared to the average amplitude which is observed with a random wavefront. We note that the amplitude of the DC signal effectively probes the average power of the detected raw spectral data. This allows us to determine the detected sample beam power from the data illustrated in Fig. 2(c) and to investigate the effect of phase conjugation on it. The result is drawn to Fig. 3 which demonstrates that the power detected from the sample beam is increased in case the OCT signal is enhanced by the phase conjugation algorithm. This supports our claim that phase conjugation based on the OCT signal indeed enhances the amplitude of light which is backscattered from the sample to the detector. The OCT signal probes the amplitude of light which is backscattered from the target depth only. Hence, some quantitative differences compared to the observed enhancement of total sample beam power, which contains contributions from depths other than the target depth as well, are observed.
Fig. 3. OCT signal and sample beam power enhancement. The enhancement of the OCT signal is taken from Fig. 2(d). The sample beam power is recovered from the amplitude of the DC signal after subtraction of the reference beam power. The dashed line marks unity enhancement.
5. Imaging applications
In a first application we utilize phase conjugation to directly enhance the OCT signal similar to previous works [6–8,29]. In contrast to these approaches which are based on iterative wavefront shaping, imaging with the time-resolved reflection matrix and subsequent phase conjugation is demonstrated, for the first time.
For each A-scan we acquire the time-resolved reflection matrix from the conventional SD-OCT signal without phase shifting. Analogue to the data presented in Fig. 2(d), a fully optimized depth-scan is subsequently captured by using the phase conjugation algorithm for point-wise signal enhancement at different depths and stitching the scan from the in-target optimized signal only. The effect of phase conjugation on the OCT signal quickly is lost in case the sample is laterally scanned. We observed this effect in our previous works which discussed OCT signal enhancement based on iterative wavefront shaping [9,11]. Hence, to optimize a full cross-sectional B-scan or volume scan the procedure in general has to be repeated at each lateral scan position again, i.e. for each scan position a new reflection matrix is captured.
We demonstrate the approach with a phantom consisting of alternating layers of pergamin paper and cover glass slides (CG15CH, Thorlabs, USA). The sample was aligned with a tilt of ten degree with respect to the optical axis and is scanned perpendicular to the tilt axis. The conventional B-scan taken at the sample is illustrated in Fig. 4(a). The signal is subject to speckle noise and, since the sample is not fully kept to one side of the coherence gate, mutual interference and mirror signals can not directly be told apart. Fig. 4(b) illustrates a scan taken with a compounding algorithm. Similar to the previous illustrations, we directly took the compound signal from the average amplitude of the reflection matrix. With the compounding algorithm the speckle contrast is reduced and sample features at larger penetration depths become visible. Fig. 4(c) illustrates the image which is taken with the phase conjugation approach. To increase the acquisition speed the number of independent modes from which the reflection matrix is acquired was reduced to $N=256$. As a consequence of phase conjugation the amplitude of mutual interference signal components is significantly enhanced. Image artefacts are evident in the scan, nonetheless, and overlap with the mutual interference signal which reflects the sample structure. The amplitude of image artefacts is not enhanced compared to the conventional acquisition (Fig. 4(a)), on the other hand.
Fig. 4. OCT scan taken at a layered scattering phantom. (a) Conventional scan. (b) Scan taken with a speckle compounding algorithm. (c) Enhanced scan taken with phase conjugation. (d) Enhanced scan with background subtraction.
We exploit this feature to suppress image artefacts in post-processing: We estimate the amplitude of the non-enhanced signal components from the 95-th percentile of the signal amplitudes which are saved to the reflection matrix (compare Figs. 1 and 2). The percentile is taken for each row of the matrix, i.e. for each depth, individually. This background signal is subtracted from the optimized scan which is illustrated in Fig. 4 panel (c). The result is illustrated in Fig. 4(d) and is found to indeed remove image artefacts. The amplitude of the background signal is overestimated by our algorithm at positions where strong mirror signals are observed and results in some dark spots which are evident in the final image.
The phase conjugation algorithm does not produce a signal at positions where no reflection from the sample is expected. We do find a significant signal enhancement between the individual layers of the sample, however. At these depths only the clear cover glass slides are located. From their bulk-volume no reflection is expected. This signal, hence, corresponds to multiple scattered light which is reflected at the previous layers and which is enhanced by phase conjugation as well. To investigate whether an actual benefit for depth-enhanced imaging exists we estimate the signal-to-noise ratio (SNR) of the optimized scan from the contrast of single-scattered and multiple scattered signal contributions. Fig. 5(a) illustrates the average amplitude of the six right-most A-scans from Fig. 4(d) as well as the comparable signal captured with the compounding algorithm (Fig. 4(b)). The SNR corresponds to the peak-to-value distance observed in Fig. 5. We estimate the SNR by fitting a third-order polynomial baseline to the scans and plotting the baseline-subtracted data to Fig. 5(b). From this data we find the peak-to-valley distance to be increased by about 4 dB with the phase conjugation algorithm. The resulting effect on the OCT image is evident when observing the signal which is received from the sixth paper layer located at a depth of 1.7 mm. This signal is clearly visible with the scan optimized by phase conjugation.
Fig. 5. (a) Average of the six right-most A-scans in Fig. 4, panels (b) and (d). The dashed line indicates the fitted signal baseline. (b) Baseline-subtracted signals.
Finally, we demonstrate the approach for imaging with biological tissue. We prepared a sample from a food-quality chicken thigh and captured the optimized OCT signal similar to the previous acquisition. The images resulting from the conventional SD-OCT signal, from speckle compounding and from phase conjugation are plotted to Fig. 6 with respect to the same dynamic range of 30 dB. Obviously, the image contrast is enhanced with the phase conjugation algorithm compared to the other approaches. As a consequence, the signal corresponding to the epidermis can clearly be distinguished from that of the dermis which is found below a depth of approximately 1.1 mm. Furthermore, at those strongly reflecting sample layers where a signal enhancement results from phase conjugation we find a significant reduction of speckle noise which is comparable to the effect of the compounding algorithm. This feature can be understood from the phase conjugation algorithm creating constructive interference from the backscattered field for each pixel of the OCT scan individually. The resulting image, hence, can be considered to be stitched from bright speckle only.
Fig. 6. Phase conjugation with a tissue sample cut from a chicken thigh. (a) Conventional OCT signal. (b) Signal taken with a speckle compounding algorithm. (c) Enhanced scan taken with phase conjugation. (d) Enhanced scan with background subtraction. All scans are plotted to the same dynamic range.
6. Discussion
In this work we demonstrated optical phase conjugation with the time-resolved reflection matrix to selectively enhance mutual interference components of the spectral domain OCT signal only, even though additional image artefacts are not suppressed. In principle, optical phase conjugation with the reflection matrix is based on the assumption that the matrix provides a linear description of the signal. With SD-OCT systems this condition is met either for mutual interference or for mirror components, depending on the definition of the reflection matrix. The condition is not met for both signal components at the same time, however, and not for DC and autocorrelation signals. Phase conjugation with the reflection matrix, hence, enhances only those OCT signal components which provide an unambiguous sample image.
We further presented, for the first time to our knowledge, the application of phase conjugation for direct OCT image enhancement, similar to previous works based on iterative wavefront optimization [6–8,29]. The speed of the imaging approach, in principle, depends on the number $N$ of independent modes which are incident to the sample and which are tested by the reflection matrix. Since the optimized A-scan is stitched from a set of point-optimized scans, another set of $L$ acquisitions is required whereas $L$ is the number of pixels in the axial field of view. Jang et al. presented an algorithm which requires a set of $(25N+L)$ acquisitions to capture a single optimized A-scan [6]. Due to the implementation of a fast data processing algorithm and due to the use of a high-speed digital micromirror device for wavefront manipulation the acquisition of a full optimized A-scan could be demonstrated within 15.4 s for values of $N=300$ and $L=200$, nonetheless [7]. The acquisition of the reflection matrix presented in this work requires no phase shifting and no iterative optimization approach. Hence, a full A-scan can be captured with a set of $(N+L)$ acquisitions. The algorithm potentially yields a significant increase in acquisition speed compared to the previous approach. For the proof-of-concept experiment presented in this work a slow phase-only spatial light modulator operating at a frame rate of 6 Hz was used, though. As a consequence, the acquisition of a single A-scan for the image presented in Fig. 6 took 4 minutes, respectively ($N=256, L=377$).
We note that both techniques currently are too slow for in-vivo OCT imaging, especially since the effect of wavefront shaping quickly decorrelates and, hence, acquisitions times in the millisecond range are required [18,19]. In principle, the total acquisition time depends on a number of parameters. The number of acquisitions required per degree of freedom of the optimized wavefront, i.e. per incident mode, is determined by the algorithm which is used to capture the reflection matrix. With the presented algorithm the matrix is directly captured from the SD-OCT signal and, hence, only one measurement is necessary [21]. This is the minimal value if the reflection matrix is not to be underdetermined. The time required for a single acquisition depends on the experimental device. So far, in this and in the previous work the acquisition of the time-resolved reflection matrix was demonstrated with liquid-crystal phase only spatial light modulators only [21]. Recently, Conkey et al. demonstrated a digital micromirror device to be used for the high-speed acquisition of the optical transmission matrix [30]. The approach is based on off-axis holography with the binary amplitude-only wavefront shaper and can in principle be implemented with the algorithm presented in this work to enable high-speed acquisition of the time-resolved reflection matrix.
With the image acquisition algorithm presented in this work another set of $L$ point-optimized signals has to be taken once the reflection matrix is determined, where $L$ is the number of pixels in the axial field of view of the OCT system. This number can be reduced by using the phase-conjugation algorithm not to focus to a single target pixel but to an axially extended target. Enhancing the OCT signal at two pixels at the same time can reduce the number $L$ by a factor two, for example. Previous works demonstrated the resulting signal enhancement to be decreased in the case spatially extended targets are chosen for phase conjugation or iterative wavefront shaping [31]. In OCT imaging the sample‘s reflectivity profile in general is an inhomogeneous function of depth, on the other hand. Phase conjugation, thus, is expected to predominantly enhance the signal at strongly reflecting layers which are present in the extended target window. As a consequence, a significant signal enhancement and a high image contrast can result even when working with large target sizes. Further experimental validation is needed on this approach, though. In another approach a singular value decomposition may be used to identify strongly reflecting layers from the depth-resolved reflection matrix [3,32]. With this technique the wavefronts corresponding to the largest eigenvalues of the matrix can be played back to the sample to focus light to these layers and to construct the enhanced A-scan signal [32]. The number of additional acquisitions $L$ corresponds to the number of eigenvalues which are used in this case. Alternatively, an optimized OCT image may be acquired directly from the reflection matrix similar to previous full-field approaches [3] ($L=0$).
Finally, we note that we did observe a significant enhancement of the OCT signal with the phase conjugation algorithm. This finding aligns well with previous experimental results [6–9,11,21,29]. We found signal contributions which result from multiple scattered light to be enhanced as well, on the other hand. These signal contributions obstruct the image which is received from the sample and, hence, limit the contrast and signal-to-noise ratio of the OCT system. The imaging capability of OCT systems with scattering media is based on the rejection of multiple scattered light compared to weakly or single scattered signal contributions. The impact of strongly multiple scattered light on iterative wavefront shaping or phase conjugation with OCT systems has not yet been discussed in the previous works and needs to be subject to further careful analysis.
Recently Jeong et al. demonstrated non-invasive focusing to scattering media by capturing the time-gated (full-field) reflection matrix and playing the eigenmodes which correspond to the largest eigenvalues of the matrix back to the sample [33]. The group demonstrated that this approach predominantly focuses light to strongly reflecting targets embedded to the sample. In another publication the same group demonstrated iterative wavefront shaping with a full-field OCT system to preferably couple light to these eigenmodes and to focus light to a strongly reflecting target, as well [34]. We do expect a similar effect for phase conjugation based on the time-resolved reflection matrix as presented in this work. In this case, phase conjugation preferably enhances OCT signal contributions which result from scattering at strongly reflecting layers compared to contributions from diffusively multiple scattered light. The approach, thus, can indeed be assumed to enhance the signal-to-noise ratio with turbid samples such as biological tissue. Further systematic experimental analysis is required, though, to demonstrate this capability.
7. Conclusion
To summarize, we presented an analytic framework on how the time-resolved reflection matrix relates to the SD-OCT signal. We demonstrated theoretically and experimentally that phase conjugation with the matrix allows to selectively enhance the OCT signal which is detected at a chosen target depth, but not image artefacts. In a first application we demonstrated the approach for direct OCT signal enhancement when imaging scattering media. As an effect of phase conjugation the signal-to-noise ratio is observed to be increased compared to speckle compounding and the speckle contrast is reduced. We are confident in the technique to motivate future devices for depth-enhanced OCT imaging, for example for non-invasive medical diagnostics with strongly scattering samples such as the human skin.
Funding
Niedersächsisches Ministerium für Wissenschaft und Kultur (78904-63-6/16); Deutsche Forschungsgemeinschaft (390833453, EXC 2122).
Disclosures
The authors declare no conflicts of interest.
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