Quantitative phase imaging (QPI) offers insight into thestructure and dynamics of transparent samples such asbiological cells and their components [1,2]. However,most QPI schemes are transmission-based and largelydo not provide optical sectioning. In contrast, opticalcoherence tomography (OCT) offers excellent depthresolution and can provide phase sensitivity, but theuse of a scanning focused spot in OCT can be limitingto phase sensitivity. To address this concern, commonpath spectral domain OCT measurements have beenshown to provide excellent phase sensitivity [3,4]. Appli-cations have included point measurements on the organof Corti, where scanning the incident beam is acknowl-edged as a source of reduced phase stability [5]. A recenthybrid scheme has included a dual beam method to ex-tend depth of focus to enable common path measure-ments but also saw degraded phase noise [6]. As analternative, swept source phase sensitive OCT can alsoprovide good sensitivity [7] and high imaging speed[8], but challenges in achieving phase stability with aswept source result in reduced sensitivity and spectralsweep range. Parallel OCT schemes provide a path toimproved stability with no scanned beam, enabling highstability across a B-scan [9,10]. However, these works didnot fully exploit the depth resolution of the approach anddid not appreciate that the spatial phase informationcould be used to resolve the complex conjugate (CC)artifact that appears upon Fourier transforming spectraldata [11].In this Letter, we present a phase-sensitive OCT
approach based on using an off-axis reference beam toproduce a spatially varying interference pattern, whichis detected using a parallel spectral domain (PSD)OCT scheme. The off-axis method is often used in digitalholography to enable separation of the interference termfrom the noninterferometric intensities of the sample andreference fields. Although this method typically results insome loss of spatial resolution, the ability to isolate theinterference term using a single data acquisition makes ita powerful, widely used holographic technique. In spite
of its advantages, the off-axis approach has not beenwidely pursued in OCT, as it is not effective for scanningspot schemes due to the need to spatially resolve thefringe.
The experimental scheme is based on a parallelspectral domain OCT system [12], where the interferencepattern across a field of view is detected using the multi-ple channels of an imaging spectrometer (Fig. 1). Thedetected signal can be written as
where Iref , Isam is the intensity of the reference, samplefield at the spatial location x, and wavelength λ. The
Fig. 1. Schematic of PSD OCT system. The properties oflenses L1-L4 are given in the figure. The tilted mirror in thereference arm causes the reference beam to cross the spec-trometer slit at an angle relative to the sample field, producinga spatially varying fringe across the slit.
phase between the two fields appears as a cosine termthat depends both on spatial location and wavelength.When the reference field is made to cross the detectorat an angle θ in the x-direction (parallel to the spectrom-eter entrance slit), the phase term becomes ϕ �x; λ� �kx sin θ where k is the wavevector (2π∕λ).To illustrate this approach, Fig. 2 presents typical data
from an off-axis PSD-OCT measurement where both thesample and reference fields are mirror reflections. Dataare presented for a B-scan comprising 400 spatial chan-nels, spanning 2.5 mm, acquired with a single 18 ms ac-quisition. The spectral data here span 475–725 nm,representing a portion of the spectrum from the FianiumSC-450 light source that is received by the spectrometer.This spectral bandwidth and shape enable a theoreticaldepth resolution of 0.63 μm. Experimentally, the process-ing method described below yields a single reflector peakwidth of 0.78 μm, with the difference due to dispersionnot corrected by the referencing method (see below).The effect of the off-axis reference beam is apparentupon taking a closer look at a narrower spatial and spec-tral range (Fig. 2B). In this plot, a tilted fringe is seen withperiodic components in the wavelength direction, asexpected for SD OCT, but also in the spatial directiondue to the use of the off axis method. Integration ofthis signal across the spatial dimension would result insignificant fringe washout and prevent detection of theinterference signal.This signal can be processed to produce a quantitative
phase image using the following procedure. For eachwavelength component, a Fourier transform is takenwith respect to the spatial dimension. The resulting plot(Fig. 3) shows three components: the noninterferometricintensity (“DC” or autocorrelation) terms, an interfero-metric term at a nonzero spatial frequency, and its CC.The spatial frequency offset of the interferometic termscan be viewed as a spatial carrier frequency that enablesseparation of the different components. Note that the
k-dependence of the offset causes the spatial carrieroffset to vary with wavelength.
These data are processed in Fourier space by firstwindowing the desired signal, effectively eliminatingthe CC and DC terms. The remaining term is recenteredto zero spatial frequency and an inverse Fourier trans-form is performed. The resulting data shown in Figs. 4Aand 4C are thus demodulated and complex valued, pos-sessing both amplitude and phase values. Note that thephase does not vary smoothly due to the systematic shiftof the spatial carrier frequency with wavelength. How-ever, the high phase stability of the system allows thisphase to be accurately measured and corrected usinga single reference B-scan of a pure reflector or feature-less area of the sample. The complex demodulated datafrom each sample B-scan is divided by the reference data,effectively removing both the spatial carrier discontinu-ities in phase as well as dispersion arising from theoptical system (Figs. 4B and 4D).
The demodulated and referenced data can now beprocessed to produce a complex valued, depth-resolved
Fig. 2. Typical interferogram for off axis PSD-OCT approach.(A) Signal across broad spatial and spectral ranges. (B) Insetfrom (A), illustrating the tilted fringe.
Fig. 3. Fourier transform of interferogram in Fig. 1, illustratingseparation of interferometric term from autocorrelation termsas a function of spatial frequency (f x) and wavelength (log scalemagnitude). Spatial carrier frequency varies with wavelengthand the complex conjugate (CC) of this term is visible at thenegative spatial frequency.
Fig. 4. Demodulated data showing both amplitude (A) andphase (C), where discontinuities due to wavelength shift of spa-tial carrier frequency are seen. (B) Amplitude and (D) phasemap after referencing to pure reflector.
A-scan for each transverse position. Figure 5A shows theB-scan obtained from the data shown in Figs. 1–3, dem-onstrating the elimination of the CC artifact and suppres-sion of the autocorrelation intensities. The CC iseffectively eliminated, presenting an amplitude that is51.7 dB lower than the primary signal. The autocorrela-tion intensity is not completely eliminated due to thesignal processing allowing some of this term to pass, pro-ducing a suppression of 29.7 dB. The phase data are seento vary parabolically across the field of view due to theimaging optics (Fig. 5B). However, this trend is consis-tent and stable and thus can be subtracted. Upon localfitting of this shape and subtracting the fitted curve,the resulting phase variation is Δϕ � 16.1 mrad, corre-sponding to an optical path length, OPL � Δϕ∕�2k� �0.77 nm across the field of view (Fig. 5C), with the factorof two arising from the double pass in the reflectiongeometry. This value is identical to the temporal stabilityseen for a given point, also found to be 16.1 mrad.A significant trade-off exists with employing the off-
axis approach. Because of the use of spatial bandwidthto separate the signals, there is a loss of spatial resolutionin one direction. Figure 6 presents an image of a phaseonly resolution chart demonstrating the difference inspatial resolution. The highest expected resolution of13.3 μm for the vertical direction is based on the sizeof the CCD pixels in the spectrometer (20 μm) dividedby the optical magnification (3×) times a factor of 2
for the Nyquist criterion. However, the spatial filter usedto process the off-axis hologram necessarily reduces thisby a factor of 4× to 53.2 μm. Indeed the smallest well re-solved bar in the resolution target is 49.6 μm per pair.Note that there is no loss of resolution in the horizontaldirection. The target is translated 10 μm between B-scansand a resolution of 20 μm is expected. The best resolutionin this direction is 19.68 μm per line pair.
As a final demonstration of this approach, we present athree-dimensional volumetric image of the resolution tar-get atop a coverglass (Fig. 7). In this image we can see theflat phase of the coverglass (bottom right) and the detailsof the resolution target (bottom left). While the phasestability in the horizontal direction here remains stable,we see a variation due to the physical translation of the
Fig. 5. (A) Processed data presented as a B-scan, inset showslogarithmic scale of one A-scan, at location of white dotted line,to illustrate CC suppression. (B) Phase varies parabolically.(C) After subtracting trend, phase standard deviation is16.1 mrad, corresponding to an OPL sensitivity of 0.77 nm.
Fig. 6. Imaging example showing difference in resolution be-tween vertical and horizontal directions due to processing ofoff-axis interferogram.
Fig. 7. Volumetric rendering of three-dimensional phase sam-ple. Since the third dimension is scanned by physical translatingthe sample, there is a significant loss of phase stability in thisdirection.
sample. Thus the inherent stability of the approach(16.1 mrad) is reduced to between 28 and 50 mrad acrossthe field of view, after polynomial subtraction. These var-iations correspond to 1.2–2.4 nm in optical path lengthvariation, a significant loss of stability compared to asingle B-scan. The origin of this loss of stability is clearlythe need to scan the sample using a translation stage.However, this only becomes a limitation when construct-ing a volumetric image.It is worth noting that some variants of our approach
have been previously reported in the literature [13,14]. Aholoscopy approach used a swept source with an off axisreference [13], demonstrating the principle of CC sup-pression but not quantifying it. Further, this work didnot explore phase imaging applications, and did notevaluate the phase stability of their method, which isexpected to be more limited for the swept source. Theapproach by Witte et al. [14] uses an off-axis referencebeam to mitigate the effects of short coherence length,similar to a previous approach by Rinehart et al. [15].However, neither of these off-axis approaches obtaineddepth-resolved measurements but rather used lowcoherence light to implement depth gating as a meansto reject out of focus light.In summary, we have presented a modality for phase
imaging within OCT that uses an off axis reference fieldto separate signal components by spatial frequency. Byusing a parallel frequency domain OCT system, we areable to detect a spatial fringe along the direction ofthe spectrometer slit. The main advantage of this ap-proach is that no scanning is needed along this spatialdirection and thus the spatial stability of the measure-ments is seen to vary less than 0.8 nm across a B-scan.In addition, this approach offers the significant benefit ofcompletely eliminating the CC artifact, potentially dou-bling the effective depth range of a given detectionscheme. Although the signals near zero path delay (DCartifact) are not completely eliminated, they are effec-tively localized and thus do not confuse the signal tothe same degree as the CC artifact. Further measures,such as subtracting the intensity of the reference andsignal fields prior to processing, could further reducethe effect of the zero path delay artifact. The approach
that we have described here offers significant benefits,but these are not without cost. The spatial resolutionalong the direction of the spatial fringe is reduced bya factor of 4. This arrangement offers advantages whenusing low coherence interferometry for phase imaging,and this trade-off is likely acceptable for many practicalapplications.
This work was supported by the National ScienceFoundation (CBET-1039562), the Research School+,the European Space Agency, and the RWTÜV Stiftung.
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