# Optical Coherence Imaging

Optical techniques play a significant role in medicine as they promise safe and low-cost solutions to many problems. Optical interferometry has been one of the preferred tools employed by scientists to help gain insights into various fundamentals of basic science and nature for more than three centuries. While the early experiments on interference by Boyle, Hooke, and Newton date back to the 17th century, the understanding of the physics behind this phenomenon continuously evolved over the next two centuries until physicists concurred on the wave/particle dual nature of light. Over the last one and a half decades, the field of medicine has benefitted extensively from the rapid progress in the instrumentation and technology of low coherence imaging techniques, broadly classified under the terms low-coherence interferometry (LCI) or optical coherence tomography (OCT). OCT is analogous to ultrasound, but instead of using sound waves, it uses low-coherence (broadband) light. The roots of the OCT technique can be traced back to the early developments of optical low-coherence reflectometry (OLCR) in the telecommunication industry during the late 1980s. The technique was aimed at finding faults or reflection sites in miniature optical waveguides and optical fibers. OCT exploits the short temporal coherence of a broadband light source, and enables high-resolution, non-invasive, *in vivo* imaging of microscopic structures in scattering tissues up to depths of approximately 2 mm, depending on the tissue type. For wavelengths ranging from 800 to 1300 nm, attenuation of light is due largely to scattering, rather than absorption. Simply stated, by measuring the time it takes the reflected light to return to the detector, an image can be created. It is possible to measure this time through low coherence interferometry using a Michelson type interferometer. This, in turn, allows us to deduce the precise location of a reflection.

One of the major breakthroughs in the OCT technology has been the theoretical and experimental demonstration of the superior sensitivity performance of Fourier-domain OCT (FD-OCT) systems over time-domain OCT (TD-OCT) systems. Over the last several years, the continuous improvements and maturation of Fourier-domain methods resulted in the development of robust and portable FD-OCT systems, and have provided a significant advantage to clinical OCT imaging as rapid image acquisition rates enabled scanning of larger tissue volumes while reducing artifacts due to patient motion.

## Time-domain and Spectral-domain OCT

The goal of OCT is to obtain 3D images of biological tissues. Axial sectioning in OCT is achieved through 'coherence gating, i.e., when the sample and reference beams of the interferometer can interfere coherently. Therefore, the axial resolution of OCT is related to the coherence length of the source. If one has to obtain measurements in 3D, one must measure the signals as a function of three independent dimensions. In time-domain OCT, the signal is detected as a function of *x, y,* and *t.* In Spectral-domain OCT, the signal is detected as a function of *x, y, *and λ. In the former case, the data in *t *can be converted to* z* using the speed-of-light in the medium. In the latter case, Fourier transform is used. A standard Fourier-domain OCT system can be described by the generalized schematic shown in Figure 1. Although the operating mechanisms of time-domain OCT (TD-OCT) and Fourier-domain OCT (FD-OCT) systems differ as different broadband sources and detection and signal processing schemes are employed, the basic principle is the same and can be explained with the help of the same schematic block diagram (Figure 1a). Light from the broadband source is split into two arms of the Michelson interferometer using a beam splitter or a 50/50 fiber optic coupler. The light incident on the tissue undergoes partial backscattering due to the presence of discrete as well as a continuum of reflection sites at different depths within the tissue. At the output of the interferometer, the backscattered light from the tissue is then recombined with the light from the reference arm and the interference signal recorded at the detection end is then used to extract axial structural information of the tissue, which can be represented as an A-scan (Figure 1b-c). The beam incident on the tissue is scanned laterally and a series of A-scans are collected and then used to obtain the cross-sectional image of the tissue as shown in Figure 2.

**Figure 1. (a) Schematic diagram of a generalized OCT system. (b) Black: Measured interferogram, Red: Background subtracted signal for a single A-scan. (c) Fourier-transform of the background-subtracted signals before (red) and after (blue) dispersion compensation. **

**Figure 2. OCT image reconstruction. Cross-sectional OCT images are typically obtained by assembling a series of A-scans (axial depth-profiles) as the incident beam is scanned in a lateral/transverse direction.**

**Theoretical formulation of signal extraction in OCT**

If we assume a wavelength-independent splitting ratio for the coupler or beam-splitter, then the broadband source light output propagating into each arm of the interferometer can be written as:

Here, *s(k)* is the electric field amplitude spectrum, *k* is the wavenumber, and *z* is the propagation distance. The reference arm typically has a variable attenuator or neutral density filter to adjust the reference light power level. The attenuated electric field reflected by the reference mirror is given as

where R_{R }= |r_{R}|^{2} is the attenuated power reflectivity of the reference arm. The light incident on the tissue will undergo backreflection and backscattering from multiple sites due the presence of scattering particles and refractive index variations within the tissue. The backscattered photons returning from the sample arm can be described as the convolution of the incident field and the backscattering function and can be written as:

where r_{S}(z) is the depth-dependent amplitude reflectivity function of the sample. The sample and reference electric fields are recombined at the beamsplitter and are given by:

The incident light is then converted into photo-current by optical detectors, which are square law intensity detection devices. The generated photo-current is proportional to the time average of the incident electric field multiplied by its complex conjugate and is given by:

where ρ is the detector responsivity (Ampere/Watt) and S(k) = |s(k)|^{2} is the normalized spectral power density of the source. The first two terms on the right hand side of the equation represent the DC component of the current and self-interference. The final term in this equation accounts for the interference between the reference and sample electric fields and is used to extract the axial depth profile or structural information in OCT. When simplified, the AC component of the photocurrent can be written as follows:

The goal of various OCT signal processing techniques is to extract the depth-dependent reflectivity function (R_{S} (Δz)) of the sample in order to obtain its axial structural profile information. In a SD-OCT system, the light output from the interferometer is directed to a spectrometer. Various spectral slices of the combined broadband output from the reference and sample arms are spatially encoded using a collimator, diffraction grating, and a linear detector array. Resampling of the data obtained from the linear detector array is performed in order to correct for the nonlinear spatial mapping of wavenumbers. After resampling and subtraction of the DC background, the depth profile structural information can be obtained by performing the inverse Fourier transform operation.

In swept-source OCT, the broad bandwidth optical source is replaced by a rapid-scanning laser source. By rapidly sweeping the source wavelength over a broad wavelength range, and collecting all the scattering information at each wavelength and at each position, the composition of the collected signal is equivalent to the spectral-domain OCT technique. Collected spectral data is then inverse Fourier transformed to recover the spatial depth-dependent information. Swept-source OCT systems are advantageous for their extremely fast scan rates, on the order of 50,000 to several MHz axial scans per second. In FD-OCT systems, the interference signal is distributed and integrated over many spectral slices and is inverse Fourier transformed to obtain the depth-dependent reflectivity profile of the sample. However, in TD-OCT systems, the interference signal of the broadband fields is integrated over time as the reference path delay is modulated in a periodic manner with constant speed to obtain the same information.

### Axial resolution and axial range of spectral-domain OCT

While the transverse resolution is controlled by the objective lens at the sample arm, the axial resolution, *δ**z*, and the axial range, *z _{R}*, captured in a single volume is dependent on the spectral bandwidth of the camera, Δ

*λ*, and the number of spectral samples,

*M*.

Assuming that the light source used covers the bandwidth of the detection completely,*δz* can be estimated as the coherence length of a source with the spectral bandwidth of the camera. Given that the spectral sampling of the camera is the vector **λ***={λ _{1},λ_{2},. . .,λ_{M}}*, where

*λ*

_{M}− λ_{1}= Δ*λ*,

Since OCM is reliant on backscattering, a sample that is axially displaced by a distance *d* displaces the beam by a distance of *2d*. Therefore, the axial range can be estimated as half of the coherence length of a source with the bandwidth equivalent to adjacent spectral samples.

However, since the axial reconstruction is obtained through Fourier transform which contains both the sample features on one half of the range and its complex conjugate in the other half, the axial range can be more accurately described as

Given that there are *M* pixels axially spanning *z _{R}*, the size of each pixel in

*z*, Δ

*z*, can be calculated as

Figure 3a shows the distribution of Δ*z* and *δz* for a central wavelength of 550 nm and different bandwidths assuming that *2π/λ* is linear. Firstly, Δ*z ≤ δz* which maintains sampling criteria. Secondly, by tuning the bandwidth appropriately, the axial resolution of the OCT setup can be varied based on the applications. Figure 3b shows the distribution of *z _{R}* for different values of Δ

*λ*and

*M*. It is apparent that

*z*is tunable over a wide range of values. For instance, for an OCT setup where

_{R}*M*= 40 and Δ

*λ*=250 nm,

*z*= 12 μm and

_{R }*δz*=1.27 μm, which was used to image thin samples such as densely populated cell cultures. However, for imaging extended biological tissues such as the retina, the desired values for

*z*is closer to 100-200 μm. Therefore, as marked in Fig. 3b, if M =200 and Δλ=100nm, then

_{R}*z*≈ 150 μm and the corresponding

_{R }*δz*is approximately 2 μm.

**Figure 3. (a) Plot of axial resolution (δz) in pink and axial pixel size (Δz) in cyan for different spectral bandwidths (Δλ) for a central wavelength of 550 nm. (b) Map of the one-sided axial range for different Δλ and number of spectral samples. The two indicated points indicate the parameters ideal for cellular imaging and tissue imaging. **

Ideally, to maintain Nyquist sampling, *δz > 2Δz. *Finally, typically in low-coherence interferometry, the coherence length (L_{c}), and therefore, the axial resolution is estimated using the following equation

### Functional extensions of OCT

#### Magnetomotive OCT

Magnetomotive optical coherence tomography (MM-OCT) is a novel extension of OCT for imaging a distribution of magnetic molecular imaging agents in biological specimens. In the MM-OCT system, the magnetic field is generated using an electromagnet and in vivo imaging is performed while the specimen is placed within the gradient of the magnetic field. Usually, a ring-shaped, water-cooled solenoid is used and the sample is placed such that the z-axis of A-scans is along the central axis or open bore of the solenoid ring. The gradient of the magnetic field applies force on the magnetic nanoparticles (MNPs) resulting in their motion. These MNPs are often attached to the tissue matrix and hence the force applied to these nanoparticles is coupled with viscoelastic restoring forces, resulting in nanometer-scale displacements that can be detected by a conventional or phase-sensitive OCT system. The dynamic magnetomotion of the MNPs will be a function of applied magnetic forces and restoring forces from the microscopic tissue matrix, and hence these magnetomotive-based scattering signals can also be used to study the biomechanical properties of the tissues. MM-OCT provides high specificity and sensitivity by exploiting the large difference between the magnetic volume susceptibilities of MNPs and biological tissue.

#### Polarization-Sensitive OCT

Polarization-sensitive optical coherence tomography (PS-OCT) maps depth- and spatially-resolved changes in the polarization state of light induced by anisotropic tissue properties. By exploiting the interaction between the polarization state of light and tissue, additional structural and functional (physiological) information can be extracted. PS-OCT incorporates standard OCT principles, but also tracks the incident and backscattered polarization state of the light. The basic experimental set-up of a PS-OCT system is similar to that of a standard OCT system, however, it has several additional components such as linear polarizers, polarizer controllers, and a polarization beam-splitter with an extra photodetector within the detection arm of the interferometer.

#### Spectroscopic OCT

Spectroscopic OCT (SOCT) maps spatially localized spectral absorption and backscattering information of endogenous molecules as well as exogenous contrast agents in tissues by detecting and processing the interferometric OCT signal. Time-frequency analysis is commonly used to extract the SOCT signal from OCT data. For a TD-OCT system, the SOCT signal can be obtained by applying a short-time Fourier transform (STFT) to a shortened time-delay window that is computationally scanned across the temporal interferometric data. In a FD-OCT system, the short-frequency Fourier transform can be used to obtain the SOCT signal. One of the major limitations of SOCT signal processing arises from the fact that there is an inherent trade-off between spectral and depth (spatial) resolution due to the uncertainty principle. For example, if one wants to obtain better spectral resolution, then the width of the STFT time-delay window would need to be increased, resulting in decreased spatial depth resolution.