Focus-extension by Depth-encoded Synthetic Aperture in Optical Coherence Tomography References and Links

We present a novel method to extend the depth-of-focus of Optical Coherence Tomography (OCT). OCT is an interferometric imaging technique that provides depth-resolved scattering information. The axial resolution in OCT is provided by the coherence gate and is invariant over the full image depth. The lateral resolution is determined by the beam parameters such as wavelength and numerical aperture. The Rayleigh range determines the depth range over which the lateral resolution can be maintained. The lateral resolution is often sacrificed to maintain relatively long Rayleigh range. In this study, we propose to use a depth-encoded synthetic aperture detection scheme to extend the depth range over which a sharp focus can be maintained beyond the Rayleigh range. An annular phase plate is inserted into the light path in the sample arm, which gives rise to three separate images in a single B-scan, corresponding to three different optical path length encoded apertures. These three images are coherently summed after phase-manipulation to reconstruct a new image with a lateral resolution that is maintained over a five times larger depth range. " Micrometer-scale resolution imaging of the anterior eye in vivo with optical coherence tomography, " Arch. Optical Doppler tomographic imaging of fluid flow velocity in highly scattering media, " Opt.resolution imaging of the human esophagus and stomach in vivo using optical coherence tomography, " Gastrointest.speed fiber based polarization-sensitive optical coherence tomography of in vivo human skin, " Opt.stabilized optical frequency domain imaging at 1-µm for the measurement of blood flow in the human choroid, " Opt. Three-dimensional coronary artery microscopy by intracoronary optical frequency domain imaging, " JACC Cardiovasc. An optical coherence microscope with enhanced resolving power in thick tissue, " Opt. Dynamic focus control in high-speed optical coherence tomography based on a microelectromechanical mirror, " Opt. Commun. In vivo endoscopic multi-beam optical coherence tomography, " Phys. High-resolution optical coherence tomography over a large depth range with an axicon lens, " Opt. Extended focus depth for Fourier domain optical coherence microscopy, " Opt. Binary-phase spatial filter for real-time swept-source optical coherence microscopy, " Opt. Non-iterative numerical method for laterally superresolving Fourier domain optical coherence tomography, " Opt. Improved lateral resolution in optical coherence tomography by digital focusing using two-dimensional numerical diffraction method, " Opt. Optical phantoms of varying geometry based on thin building blocks with controlled optical properties, " J. speed miniature motorized endoscopic probe for optical frequency domain imaging, " Opt. High-precision, …


Introduction
Optical coherence tomography (OCT) is an interferometric technique that provides crosssectional images of biological tissue [1].It has become an essential tool for clinical diagnosis and disease monitoring in ophthalmology and it shows great potential in other clinical areas such as dermatology, cardiology and gastroenterology [2][3][4][5][6][7].The axial resolution of OCT imaging at moderate numerical aperture is determined primarily by the light source as illustrated by the equation: ).The depth of focus is generally defined as twice the Rayleigh range.Current broadband laser sources permit axial resolutions below 10 μm over several millimeters of image depth.However, it is impossible to achieve ~10 μm transverse resolution over a depth range of several millimeters using Gaussian beams and normal objectives.The relatively short focal-depth of imaging optics limits the application of high lateral resolution OCT to thin slices.
To date, various methods have been proposed to address the limited depth-of-focus of OCT imaging.Examples are dynamic focus [9,10], multi-focus [11] and focal-depth extension methods [12][13][14][15][16][17].Dynamic focusing synchronizes the focal plane position with coherence gate scanning, which is more suitable for time-domain OCT [9,10].The multifocus approach requires parallel acquisition at different focal depths [11].In comparison, depth-of-focus extension has been more successful.Till now, the methods proposed for focusdepth extension can be categorized into three groups: (1) Bessel beam illumination with an axicon lens [12,13], (2) phase apodization [14] and (3) digital refocusing [15][16][17].The axicon and phase apodization approaches are inefficient in illumination and collection from the extended focus, leading to signal losses of tens of decibels [13,14].Interferometric synthetic aperture methods (ISAM) have been developed for digital refocusing by solving the inverse scattering problem [15].It is computationally expensive and requires phase stable acquisition of consecutive depth profiles.Other digital refocusing methods based on deconvolution or scalar diffraction models also require phase-stable OCT measurements [16,17].
In this study, we propose a novel method to extend the focus-depth, which is compatible with endoscopy and computationally inexpensive.It is analogous to synthetic aperture radar.The idea is to use an annular phase plate to separate the light into different optical apertures that are encoded by a depth offset in an OCT A-scan.This leads to the acquisition of multiple OCT images in parallel in a single B-scan.Manipulation of the phase of these images and coherent summation allows the construction of a new image with a significantly extended depth-of-focus.An intuitive description of the principle of this novel depth-encoded synthetic aperture method is illustrated in Fig. 1.In section 2.1 a more thorough mathematical description is given.To explain the method, the wavefront of the scattered field is analyzed.An annular phase plate is inserted into the sample arm at the back focal plane of the imaging lens.The annular phase plate consists of a 3-mm thick polycarbonate plate with a circular hole at the center.Approximately half of the scattered light travels through the central hole (central beam), and half of the light travels through the polycarbonate plate (edge beam).When the object is in-focus, the collected light goes through the phase plate with a planar wavefront.The edge beam is delayed with respect to the central beam due to the longer optical path length, given by ( ) ).The newly constructed image has a comparable lateral resolution as the image acquired without the phase plate.When the object is defocused (Fig. 1(c)) (in this example above the focal plane), the edge beam undergoes the same delay due to the phase plate as in the in-focus case.In addition, a small extra delay ( z

Principle of depth-encoded synthetic aperture OCT
occurs due to the curvature of the wavefront.This small extra delay is equivalent to the delay as described in our earlier publication on self interference fluorescence microscopy (SIFM) [18].An image with improved focus can be constructed by correcting the edge beam wavefront for both the terms    and z  (Fig. 1(d)).

Theory
Mathematically, the detected OCT signal   I k in spectral domain OCT and optical frequency domain imaging (OFDI) for a single scattering object can be expressed by the following equation: where k is the wave number, α is the square root of the scattering object reflectivity at depth z.
  r I k and   s I k on the right-hand side of Eq. ( 1) are the wavelength-dependent intensity reflected from the reference arm and sample arm, respectively, and are also called DC terms.The third term is the interference between the reference arm and sample arm that contains the depth information.The DC terms are from now on omitted since they carry no depth information and only the interference term is retained.
When a phase plate is inserted into the sample arm, there are three different paths for the sample arm light to reach the detector via a scattering object in the sample: the first light path passes through the center of the phase plate both on the way to the scattering object and back.The second path passes through the center on the way to the scattering object and travels back through the edge, or, equivalently, passes through the edge on the way to the scattering object and travels back through the center.Finally, the third light path passes through the edge of the phase plate both on the way to the scattering object and back.
These three paths create three different depth encoded images, where each image corresponds to the light detected through a distinct circular (or annular) aperture.The interference of the sample arm light with a local oscillator field (the reference field) provides both the amplitude and phase of these three images.By adding these three images coherently, and manipulating the phase in a way similar to synthetic aperture radar or phased array ultrasonics, a single image can be constructed for which the lateral resolution is preserved over a much larger range than the Rayleigh range of the original Gaussian beam.
The interference term for a single object with the phase plate in the beam path is thus given by: where  is the single pass optical path length difference between the central beam and edge beam, C.C. indicates the complex conjugate, and z  is the small extra optical path length difference between the central beam and edge beam resulting from the defocus-induced wavefront curvature, which becomes zero when the object is in focus.
To facilitate the image processing of summing coherently the three terms in Eq. ( 2) into a single image, the k value is written as 0 k k k    for the terms that contain products with z  and z  .As an example, the second term on the right hand side of Eq. ( 2) (the middle image) can be rewritten as: The wavelength-dependent phase term ( k z   ) can be neglected since z  is smaller than a wavelength, and the optical bandwidth or the tuning range of the laser source is in general smaller than ± 5% of the central wavelength.Then, Eq. ( 3) can be written as follows with the constant phase (k-independent) terms separated out: The third term on the right hand side of Eq. ( 2) (the bottom image) is now given by: compared to Eq. ( 1) for the OCT signal detected without phase plate, an extra oscillation phase (  ) and a constant phase  are added in Eqs. ( 4) and ( 5), resulting from the phase plate and defocus effect.The phase term ( 0 k z  ) is negative when the focal plane is below the imaging target and positive when the focal plane is above the imaging target.We hypothesize that the focus can be recovered by correcting the defocus-induced 0 k z  in the detected OCT signal.

Physical optics simulation
To explore the relationship between the phase factor 0 k z  and the defocus distance we performed a physical optics simulation using Zemax software (Radiant Zemax, Redmond, WA, USA).The simulation model (Fig. 2(a)) comprises: (1) a Gaussian source with a 1/e 2 diameter of 9.2 μm in a medium with refractive index 1.44 to simulate the average refractive index of the phantom described in section 3, (2) an 18 mm focal length objective (LSM02, Thorlabs, Newton, NJ, USA) to collect the light, (3) a zone plate to simulate the annular phase plate (optical thickness: 0.68 mm and a hole with a diameter of 1.8 mm), (4) a collimator (focal length: 18.67 mm) to couple the light into the fiber and (5) a collection fiber (mode field diameter: 9.2 μm).This model was used to calculate the total coupling efficiency of the light from the point source to the single mode fiber as a function of wavelength and defocus distance [18].In addition, we also simulated the point spread function (PSF) of the light beam at different defocused positions without the phase plate to determine the exact intensity distribution.To simulate the experimental situation, a 300-μm thick layer with refractive index of 1.44 is defined in the physical optics model as shown in Fig. 3(a); then the irradiance cross-section, also called PSF, on a surface in the middle of the layer is calculated as the layer is moved away from the lens along the optical axis.The calculated PSF is convolved with a Gaussian window which has a 5-μm width (diameter of the spheres used in experiment) to enable a good comparison between experiment and simulation.This yields the single pass PSF.In order to account for the PSF of both the incident and the scattered beam Fig. 3(b) shows the squared irradiance along lateral (x) and axial (z) dimensions.Below the focal plane the lateral intensity distribution shows the expected profile for a focused Gaussian beam, but above the focal plane (between objective and focal plane), the lateral intensity distribution exhibits multiple-maxima.A phantom was created by embedding 5-μm melamine spheres into a silicone polymer [19].Since the depth range of the systems was 5-mm in air, the phantom was made 300 μm thick (optical thickness: 432 μm).This ensured that three images from different optical apertures could be depth-encoded without overlap.In the experiment, the focal plane was initially positioned below the phantom and then the phantom was translated away from the objective, along the optic axis, in steps of 40 μm.B-scans were acquired at each depth.

Results
Fig. 5. Refocusing Process: A single B-scan contains three images of the same object at different depths (i.e., top, middle and bottom); the first step is to correct the depth-separation due to the phase plate for the middle and bottom images (blue text box).Then, the phase of the middle and bottom images are further manipulated (orange text box) and coherently added to the top image to construct a new image with improved lateral resolution.The four colored windows are zoomed-in images of two spheres to better visualize the refocusing process.The image is 1.28 mm wide and its aspect ratio is 3:1.
Figure 5 shows an experimental OCT image of the scattering sample with the annular phase plate and the result of the refocusing process.The phantom was defocused by about 1.5 times the Rayleigh range.The original B-scan on the left side of Fig. 5 contains three depth-encoded images corresponding to three different apertures.These three separated images are coherently added after applying the refocusing algorithm.The refocusing algorithm consists of two steps: (1) The first step is to depth-decode the middle and bottom images to the depth position of the top image to correct the depth-offset due to the phase plate ( ( ) ).This can be done using the Fourier shift-theorem: ( ) ( )exp( ).

Phase Manipulation
Refocusing Process + the sum of the magnitude of the top image with the shifted middle and bottom images (incoherent magnitude sum) and then squared the magnitude.The optimal frequency-shift factor was determined by maximizing the squared magnitude integrated over the incoherently summed image.The ratio of the optimal frequency-shift factors for the bottom and middle images was 1.9996 ± 0.0017.
(2) The second step is to correct the defocus-induced additional phase change ψ .This phase change is wavenumber-independent and therefore can be corrected simply by applying a constant phase factor to the Fourier-transformed spectrum which undergoes a frequencyshift in the first step.Then, all three complex images are coherently summed to reconstruct a new image.This can be expressed as: ).The images with two spheres as an example in the colored windows in Fig. 5 provide detailed insight into the reconstruction process and how the focus is improved.The phase manipulation leads to constructive interference on the center part of the focus spot and destructive interference on the edge part of the focus spot (i.e., side lobe in middle and bottom images).This combined constructive and deconstructive interference process improves the lateral resolution and also increases the total backscattered intensity of the refocused image.Figure 6 is a direct comparison between three different imaging modes for different positions of the lens focus: 1) the full beam image without phase plate, 2) the truncated Gaussian beam image (top image of the original B-scan), and 3) the refocused image, where the three images are shown as the top, middle and bottom in each subfigure.The subfigures (Figs.6(a)-6(h)) represent the images of the phantom being moved away from the objective at 80-μm step size.Images were acquired with 40-μm step size but not all images are shown due to limited space.Due to the refractive index mismatch, the step size should be scaled by the refractive index of the phantom to represent the actual focal plane shift [9].Thus, the physical phantom movement of 80 μm leads to a 115.2-μm displacement of the focal plane.From Figs. 6(a) to 6(h), it is seen that the spheres in the truncated beam images appeared to have a very slow change in width over the defocus process.This can be explained by the fact that the light beam is truncated from 3.4 mm to 1.8 mm (diameter) by the phase plate which results in an extension of Rayleigh range at the cost of an increased lateral focal spot size.In contrast, the corresponding full beam image without phase plate experienced a much more rapid change in the sphere's width during the same defocus process.For the refocused images, it is obvious that as compared to the full beam image, the refocused image had comparable focus size and maintained the focus over a much larger depth range.
To better evaluate the performance of the refocusing algorithm we selected five spheres and labeled them with numbers 1 to 5 in Fig. 6(d).These were used as examples to quantify the focus performance.Figure 7 shows the spheres' intensity profiles on a linear scale and normalized for better comparison.From top to bottom row profiles are shown as a function of the translation of the phantom away from the objective with 80-μm step size, corresponding to Fig. 6.Overall, the five spheres showed a similar profile change (broad-narrow-broad) and the best focus occurred at the fourth row from the top.For the truncated beam and refocused images, the sphere's profiles were all single-peak shaped.For the full beam image however, spheres 1, 4 and 5 showed a profile with sidelobes in the first two rows.Spheres 2 and 3 showed an irregular shape.The behavior of spheres 1, 4 and 5 is consistent with the physical optics simulation (Fig. 3(b)), showing that the lateral profile at the depth between objective and actual focal plane exhibits multiple-maxima.Spheres 2 and 3 do not show this behavior.We hypothesize that spheres 2 and 3 were not exactly centered in the B-scan cross section and therefore were only partially sampled by the full beam.To further quantify the changes discussed above, all the intensity profiles in Fig. 7 were fitted with function to determine the full width at half maximum (FWHM).In Figs.8(a)-8(e) the FWHM is plotted as a function of the phantom displacement relative to the focal plane.Clearly, for all five spheres, the slowest variation in FWHM during defocus occurred for the truncated Gaussian beam image (for this image the resolution is always poor).The most rapid FWHM change is observed for the full beam image.As for the refocused image, it was focused better than the truncated Gaussian beam image over the whole range.This indicates that our refocusing algorithm not only extends the depth-focus as beam truncation does, but also produces better resolution than beam truncation does.As compared to the full beam image, the refocused image yielded a comparable resolution within a short range around actual focal plane.Moreover, the resolution of the refocused image degraded much more slowly than that of the full beam image.For example, spheres 1 and 5 showed the smallest FWHM in the refocused image and full beam image when the phantom was at around 200 μm.As the phantom position increased from 200 μm, the FWHM increased much more quickly for the full beam image than for the refocused image.To determine the difference in depth-of-focus between the full beam and the refocused beam, the slope of the FWHM as a function of phantom position at the right side of the focus position in Figs.8(a)-8(e) was determined.On average, the FWHM increase in the full beam image was about 5 times faster than the refocused image.Figure 8(f) depicts a comparison between the full beam FWHM measured from the experiment and the FWHM calculated from the physical optics simulation.The experimental results match the simulation very well.To assess the efficiency of the refocusing algorithm in comparison with the truncated beam and the Gaussian beam, the backscattered intensity measured with the OCT system was determined.The energy defined as the backscattered intensity integrated over the area of the five selected spheres was calculated for three imaging modes (i.e., truncated beam, refocused, full beam imaging).In Fig. 9, it is clearly seen that, in truncated beam imaging mode, the five spheres (Figs.9(a)-9(e)) showed a slow continuous change in energy (increase-maximumdecrease).For the refocused imaging and full beam imaging modes, spheres 2 and 3 show a qualitatively different energy curve than spheres 1, 4 and 5, especially in full beam imaging mode.This behavior is attributed to the assumption that spheres 2 and 3 are only partially sampled by the full beam as noted before.Therefore, these two spheres were excluded from the energy comparison between the refocus imaging and full beam imaging.Spheres 1, 4 and 5 (Figs.9(a), 9(d) and 9(e)) experienced a very similar energy change over different phantom positions in both the refocused imaging and full beam imaging.Moreover, the energy difference between those two imaging modes was also very consistent among those three spheres.
The energy efficiency of our refocusing algorithm is evaluated on spheres 1, 4 and 5 at three depths around the optimal focus position (phantom at 200, 240, and 280 μm).The results show that our refocusing algorithm suffered only from a −1.9 to −4.1 dB energy efficiency loss as compared to the full beam imaging.This is close to the expected value (−3 dB) based on the theoretical considerations described above.This indicates that our


are the central wavelength and the optical bandwidth of the source, respectively[8].The transverse resolution depends on the wavelength  and beam size (d) of the laser light as well as the objective's focal length (f).In the focus, the lateral resolution is given by the focus diameter: single-sided depth range over which the lateral resolution is maintained within a factor of 2 is given by the Rayleigh range (

Fig. 1 .
Fig. 1.Principle of the depth-encoded synthetic aperture method for extending the depth-offocus of optical coherence tomography.The drawing shows the source of the scattered light (orange dot), the lens, the beam diameter as a function of the axial position defined by Gaussian optics (solid green), the reconstructed beam diameter after image processing (dotted green), the wavefronts before and after the phase plate (blue and red lines) in Figs.(a) and (c).Δz and n are the thickness and refractive index of the annular phase plate (PP).(a) The scattering object is located in the focus of the lens.Solid blue and red lines indicate the sections of the wavefront that pass through the inner and outer sections of the annular phase plate, respectively.The outer section is delayed by a term ( 1) z n ∆ − .(b) By application of the Fourier shift theorem, the delayed part of the wavefront is shifted back (dotted red lines) and the original image is reconstructed.(c) When the scattering object is out of focus (here above the focal plane) the wavefront is curved.This leads to a small extra optical path length difference δz.(d) By compensating both the shift ( 1) z n ∆ − induced by the phase plate and the extra term δz a refocused image with improved lateral resolution (indicated by the dotted green lines) can be obtained.See section 2.1 for details.
with z ∆ the thickness of the phase plate, and n the refractive phase plate (Fig.1(a)).In a single OCT B-scan, the central beam image and the edge beam image are encoded to two different depth locations separated by   1 z n   .A new image can be constructed by correcting the delay (or depth-separation two images (Fig. 1(b)

Figure 2 (
b) presents an example of the coupling efficiency as a function of wavelength.It shows constructive and destructive interference over the source spectrum caused by the optical path length difference for the core and edge beams through the phase plate.Because of the defocus-induced phase factor 0 k z  , the phase of this interference pattern shifts when the point source is moved away from the focal plane (Fig.2(b)).As shown in Fig.2(c), the phase shift is linearly dependent on the defocus within a region of approximately 400 μm around the focal plane.This corresponds to a region of approximately 3.5 times the Rayleigh length.The slope of the curve in the linear region is 7.25 mrad/μm.The physical optics model thus predicts a slope of 7.25 mrad/μm for the defocus induced phase shift of the middle image (single pass through the edge of the phase plate).Since for the bottom image the light travels twice through the edge of the phase plate, a slope of 14.5 mrad/μm is expected.The sinusoidal modulation of the coupling efficiency as a function of wavelength (Fig.2(b)), introduced by the phase plate, indicates that the phase plate reduces the OCT signal by 50%.

Fig. 2 .
Fig. 2. (a) Physical optics model including the lenses, the medium, the collection fiber and the phase plate.(b) Coupling efficiency as a function of wavelength for three different positions of the source with respect to the focal plane.(c) Phase shift of the coupling efficiency function in (b) as a function of defocus.The slope of the linear region is 7.25 mrad/μm.

Fig. 3 .
Fig. 3. Light intensity distribution of the focused beam in the sample.(a) The lateral and depth intensity distribution is evaluated at the plane indicated by the dotted line as a function of axial location of the sample with respect to the lens.(b) Color coded intensity distribution as a function over x (lateral) and z (depth) dimensions.The image is normalized to each row for better visibility of the lateral structure.Scale bars are 20 μm in x dimension and 200 μm in z dimension.

Fig. 4 .FC
Fig. 4. Schematic of Optical Frequency Domain Imaging (OFDI) system.PP: Annular Phase Plate; FC: Fiber-optic Coupler; CIR: Fiber-optic Circulator; CL: Collimator; L: Lens; M: Mirror; FBG: Fiber Bragg Grating; PR: Photoreceiver; BD: Balanced DetectorThe in-house-built OFDI system consists of a wavelength-swept laser source, sample and reference arms, galvo mirror scanning system, optical trigger generation and data acquisition card as shown in Fig.4.The light source centered at 1310 nm sweeps from 1258 to 1366 nm at a frequency of 50 KHz (Axsun Technologies Inc., MA, USA).The output power of the laser light is about 29 mW.The light from the source is split by a 90/10 fiber coupler.A small portion (10%) is sent to a fiber bragg grating (FBG, 1269.86 nm, O/E land Inc., LaSalle, QC, CA) and the reflected light is detected by a photoreceiver (model 1811, 125-MHz, New Focus, San Jose, CA, USA) to generate an optical trigger pulse.The large portion (90%) is further split into the sample arm (50%) and the reference arm (50%) by a 50/50 fiber coupler.The back-reflected light from the sample arm is collected by the same coupler and transferred to another 50/50 fiber coupler for balanced-detection with a balanced amplified photodetector (PDB430C, 800~1700 nm, 350 MHz, Thorlabs, Newton, NJ, USA).The light in the reference arm is coupled to the aforementioned 50/50 coupler after a free space delay line to match the reference and sample arm path length.A special optical coherence tomography scanning lens with 18 mm focal length (LSM02, Thorlabs, Newton, NJ, USA) is used for delivering the laser light to the sample and collecting the back-scattered photons.The annular phase plate (PP) is fabricated by drilling a 1.8 mm (diameter) hole in the center of a transparent polycarbonate slide of 3 mm thickness.It is positioned at the back focal plane of the scanning objective which is relayed from the objective by a pair of achromatic lenses with a focal length of 60 mm.Lateral scanning is implemented through a galvo mirror scanning system which is driven by a 16-bit analog voltage output device (PCIe-6259, National Instruments, Austin, TX, USA).Saw-tooth waveforms are generated by PCIe-6259 and sent to the galvo-scanner driver board for raster scanning.The OCT signal detected by the balanced detector is filtered by a low pass filter (BLP-70+, Mini-circuits, Brooklyn, NY, USA) to block any signals at the spectrum before and after the frequency-shift, respectively; and φ ∆ is the frequency-shift factor for depth-decoding the middle and bottom images relative to the top image in the spectral domain.Based on the theoretical description from section 2 we expect that φ ∆ is equal to .Experimentally, we estimated frequency-shift factor ranges for both middle image and bottom image according to the optical thickness of the phase plate.Next, over a dense 2-D mesh of frequency-shift factors for the middle and bottom images, we calculated (n-1)) exp(i2ΔkΔz(n-1))Depth Decode exp(i(k 0 Δz(n-1)+k 0 δz)) exp(i2(k 0 Δz(n-1)+k 0 δz))

Fig. 6 .
Fig. 6. (a)-(h): Intensity images acquired when the phantom is moved away from the objective at a physical step size of 80 μm.Each subfigure includes three imaging modes: (I) full beam image at top, (II) truncated Gaussian beam image at middle, and (III) refocused image at bottom.All the images are presented on a logarithmic scale with the colormap clipped to 75% of the maximum intensity of the image.The image is 1.28 mm wide and its aspect ratio is 3:1.

Fig. 7 .
Fig.7.Lateral intensity profiles of the selected spheres in Fig.6for different phantom positions (defocused depths): (a) to (e) are spheres 1 to 5, respectively.The rows from top to bottom represent the different phantom positions with 80-μm step size, corresponding to the images in Fig.6from top (Fig.6(a)) to bottom (Fig.6(h)).Black: Refocused image; Red: Truncated Gaussian beam image; Blue: Full beam image.
are determined by maximizing the squared absolute value of the coherent sum (which is the backscattered intensity) integrated over the target depth range of the reconstructed or refocused image ( refocused S