## Abstract

Phase sensing implementations of spectral domain optical coherence tomography (SDOCT) have demonstrated the ability to measure nanometer-scale temporal and spatial profiles of samples. However, the phase information suffers from a 2π ambiguity that limits observations of larger sample displacements to lengths less than half the source center wavelength. We introduce a synthetic wavelength phase unwrapping technique in SDOCT that uses spectral windowing and corrects the 2π ambiguity, providing accurate measurements of sample motion with information gained from standard SDOCT processing. We demonstrate this technique by using a common path implementation of SDOCT and correctly measure phase profiles from a phantom phase object and human epithelial cheek cells which produce multiple wrapping artifacts. Using a synthetic wavelength for phase unwrapping could prove useful in Doppler or other phase based implementations of OCT.

© 2009 Optical Society of America

## 1. Introduction

There has been much work in recent years to develop quantitative phase imaging techniques capable of detecting sub-wavelength dynamics in biological samples. Techniques such as phase contrast [1] and differential interference contrast microscopy[2] have been used to image transparent samples qualitatively by converting phase differences to intensity differences, but these methods suffer from a non-linear phase to amplitude conversion and thus do not directly provide quantitative data. Phase shifting interferometry[3], digital holography[4,5], Fourier phase microscopy[6,7], and Hilbert phase microscopy[8,9] are interferometric techniques capable of detecting nanometer scale features in biological specimens. These quantitative methods have applications in the realm of cellular imaging due to their high sensitivity and use of intrinsic contrast agents but lack the ability to obtain depth-resolved measurements from an optically thick sample.

The development of Fourier-domain OCT (FDOCT), particularly spectrometer-based spectral-domain systems with no moving parts (spectral-domain OCT or SDOCT), have greatly enhanced the phase stability of OCT systems due to improvements to the system SNR over time domain systems [10,11,12]. Common path implementations have given rise to a new class of functional, nanometer-scale sensitive quantitative phase microscopies termed spectral domain phase microscopy (SDPM) or spectral domain optical coherence phase microscopy (SDOCPM) [13,14], which have the advantage of being able to obtain depth-resolved information from a given sample with milliradian phase stability.

In SDPM, phase information is obtained from Fourier processing as in standard SDOCT. The phase of a given depth sample represents sub-coherence length changes in the optical path length through the sample as a function of time or a lateral dimension. Phase changes can be caused by changes in the sample refractive index, position of scattering objects, or both. SDPM is capable of producing depth-resolved phase maps measuring the motion of a dynamic sample throughout its volume. A common path geometry allows for cancellation of common mode noise, and such systems have experimental phase sensitivities as low as 53 picometers [13]. Common path systems have been used for lateral profiling of cells [14], and have also been used to study cytoplasmic streaming in Amoeba using Doppler flow [15], cytoskeletal rheology[16], and contractile motion of beating cardiomyocytes[17].

All the aforementioned phase imaging modalities suffer from phase wrapping artifacts. The phase in the OCT signal is linearly related to the sample motion over time. However, the measured phase is limited to a range of -π to +π. Phase wrapping occurs when a change in phase between consecutive measurements is such that the total phase change falls outside this range and thus yields an ambiguous result. SDPM operates in a reflection geometry, so changes in a sample reflector’s position greater than half of the source center wavelength will induce phase wrapping. Software implementations for phase unwrapping have been used in SDPM, but are complex and computationally intensive [14,18]. A technique to resolve the 2π ambiguity in low coherence interferometery used a dispersion imbalance in the sample arm and polarization effects to simultaneously detect two interferograms from different lateral locations on a sample [19]. This method required two spectrometer channels as well as additional polarization optics which add complexity to the optical setup.

The concept of using multiple wavelengths to obtain precise length measurements was first described by Tilford[20]. Use of two or three illuminating wavelengths to perform more robust phase unwrapping was first introduced in phase shifting interferometry[21] and has since been applied in digital holography[22, 23, 24] and phase imaging interference microscopy[25]. The difference in the phase information obtained at two or more wavelengths can be cast in terms of an equivalent phase that would be obtained at a longer synthetic wavelength, Λ, which is a function of each of the imaging wavelengths. This allows wrap-free measurements of changes in optical path length less than Λ/2, which can be significantly larger than any of the imaging wavelengths alone. A similar technique using a combination of low coherence and continuous wave sources was used to detect phase crossings in a Michelson interferometer allowing for phase wrap removal[26]. These techniques require the use of multiple sources, which also complicate the optical setup. A similar technique in broadband interferometric confocal microscopy measured the group optical path delay through a sample by detecting relative phase changes between different wavelengths, allowing for measurement of the group refractive index while avoiding phase ambiguity [27].

This paper presents the theoretical analysis and experimental verification of synthetic wavelength-based phase unwrapping in OCT. Synthetic wavelength phase unwrapping may be applied to OCT data and can correctly resolve sample motions that are larger than λ_{o}/2. Image processing in OCT uses the Fourier transform of a broadband spectrum. By windowing the signal spectrum before applying the Fourier transform, phase information at multiple center wavelengths may be obtained. A similar procedure to the multi-wavelength method previously used in other phase imaging modalities may then be applied for correct phase unwrapping, except that in OCT only a single source is needed due to the large spectral bandwidth used. Such a procedure could be applied to phase based implementations of OCT in both the Fourier or time domain including Doppler, common path, or other phase based implementations.

## 2. SD-OCT dual-wavelength phase unwrapping theory

#### 2.1 SD-OCT interferometric signal and phase wrapping

For the case of a single reflector, the SD-OCT interferometric signal detected by a spectrometer as a function of wavenumber is given by

where *S(k)* is the source spectral density, *R _{R}* and

*R*are the reference and sample reflectivities respectively,

_{s}*n*is the index of refraction, and

*x*+

*Δx*is the distance between the reference and sample reflectors.

*x*accounts for the discrete sampling of the detector in the

*x*-domain while

*Δx*represents subresolution changes in the sample position around

*x*[15]. The first two terms in Eq. (1) represent DC terms while the third term contains the interferometric data of interest in OCT. Taking a Fourier transform of Eq. (1) and ignoring the DC terms yields a depth reflectivity profile of the sample, known as an A-scan, with peaks located at ±2nx. These peaks can be described as

in which *S* is the total source power, *E(2nx)* is the coherence envelope function, and *k _{o}* is the source center wavenumber. The phase of the detected signal at time

*t*can be used to detect subresolution deviations in the sample with respect to the phase at a reference time,

_{j}*t*. These deviations are related to the detected phase as

_{o}in which *Δx* is the subresolution motion of the sample reflector, *λ _{o}* is the center wavelength of the source,

*Δφ*is the phase difference between times

*t*and

_{j}*t*, and

_{o}*m*is an integer number of half wavelengths. The factor of 2 in the denominator accounts for the double pass optical path length due to the reflection geometry of the optical setup used in SDOCT.

A given phase value is not an absolute determination of *Δx*, but can potentially represent *Δx* plus any integer number of half wavelengths. Without *a priori* knowledge of the sample motion or structure, there is no way to know the exact value of *m* in Eq. (3), as any displacement that is a multiple of *λ _{o}*/2 will induce phase wrapping. If

*m*= ±1, simply adding or subtracting 2π to the phase can correctly unwrap the artifact. However, if |

*m*| ≤ 2, it will be impossible to unwrap the phase using this simple method. A larger

*Δx*may be correctly measured without phase wrapping if a larger

*λ*is used. This is the basis for multi-wavelength unwrapping in other phase imaging modalities and will be applied here to OCT.

_{o}#### 2.2 Synthetic wavelength phase unwrapping

For two given wavelengths, *λ _{1}* and

*λ*, a longer synthetic wavelength Λ may be defined as

_{2}The synthetic wavelength has a corresponding synthetic phase, *Δφ _{syn}*, that can be calculated by the difference of the phase measurements made at each of the two single wavelengths then adding 2π to the result whenever

*Δφ*is less than zero [22].

_{syn}*Δφ*is the phase one would obtain had Λ been the actual illumination wavelength used and thus allows for wrap-free phase measurements of a large

_{syn}*Δx*.

*Δφ*may still suffer from phase wrapping if Λ is not chosen to be sufficiently large. It should also be noted that the noise level in the synthetic wavelength phase map suffers from noise magnification due to amplification by Λ, as is discussed in the next section.

_{syn}In OCT, *λ _{1}* and

*λ*can correspond to the center wavelengths of two different subsets of the broadband source spectrum. The broadband spectrum acquired during SD-OCT imaging can be divided by applying Gaussian windows to different portions of the spectrum as shown in Fig. 1. The spectrum is first interpolated to be evenly spaced in wavenumber. The DC components of the signal are isolated by applying a Fourier transform low pass filter. The raw spectrum is then divided by the DC spectrum which leaves the interferometric term from Eq. (1). The Gaussian windows are applied to the remaining interference signal. The resulting signals have the form

_{2}$${i}_{2}\left(k\right)\propto {e}^{\frac{-{\left(k-{k}_{2}\right)}^{2}}{\Delta {k}_{2}^{2}}}\mathrm{cos}\left(2\mathrm{kn}\left(x+\Delta x\right)\right)$$

with *k _{1}* <

*k*being the centers of the two Gaussian windows and

_{2}*Δk*and

_{1}*Δk*as their bandwidths. Though in principle the windows may be of any desirable shape, Gaussian windows offer a convenient spectral shaping tool due to their Fourier transform properties and for their ability to suppress sidelobe artifacts, allowing for separation of closely spaced reflectors[28,29]. A side effect is that spectral reshaping also decreases the SNR in OCT images [28,29], which affects the phase stability as described in the following section. Applying the windows in the described manner allows the resulting spectra to be Gaussian shaped regardless of the form of the original spectrum. Performing a Fourier transform separately to the two windowed spectra yields two sets of phase information, and this process allows the group optical path length to be measured from the difference of the observed phases. In order to determine the exact center wavelengths of the two resulting spectra, a center of mass calculation over each windowed spectrum was performed, which for Gaussian windows simply yields their respective center wavelengths.

_{2}#### 2.3 Synthetic wavelength windowing and noise analysis

In the shot-noise limit, the phase sensitivity, *δφ*, of a reflector in an acquired OCT interferogram is given by [15]

and the SNR is described as

where *ρ(k)* is the detector responsivity, *S(k)* is the spectral density function detected, *Δt* is the integration time, and *e* is the electronic charge [12]. The SNR used here is the SNR of the sample reflector of interest which determines the sensitivity of the phase measurement[13]. The phase sensitivity can be converted to a displacement sensitivity by combining Eq. (6) with Eq. (3) for *m* = 0 giving

Several important observations can be made regarding Eqs. (6), (7), and (8). *δφ* depends upon the integrated power at the detector, and *δx* is affected both by *δφ* and *λ _{o}*. Each of the two spectra produced by windowing has phase noise,

*δφ*and

_{1}*δφ*, respectively. Because

_{2}*Δφ*=

_{syn}*φ*-

_{1}*φ*, the synthetic wavelength phase noise,

_{2}*δφ*depends on a quadrature addition of the noise levels of each spectra. The noise of each of the two spectra is correlated according to the amount of overlap of the two windows, which can be described by their covariance. Thus, the synthetic wavelength has a phase noise given by

_{syn}where *cov(δφ _{1},δφ_{2})* is the covariance of the phase noise between the two spectra. As a first simple model for the covariance, we assume

in which a(k) is a function describing the amount of overlap of the two Gaussian windows described as

with β = (k_{1}Δk_{2}+k_{2}Δk_{1})/(Δk_{1}+Δk_{2}) being the point at which the two windows intersect. If it is assumed that Δk_{1} = Δk_{2}, then as $\underset{{k}_{1}-{k}_{2}}{lim}\to 0,a\left(k\right)\to 1$ corresponding to complete overlap of the two windows. In this case, *δφ _{syn}* = 0 as

*cov(φ*=φ

_{1}^{2},φ_{2}^{2})_{1}

^{2}+φ

_{2}

^{2}. If the two windows are greatly separated,

*a(k)*= 0, indicating that

*δφ*is equal to

_{syn}*δφ*and

_{1}*δφ*added in quadrature.

_{2}Windowing the original spectrum reduces both the bandwidth and SNR. The SNR becomes

where the Gaussian in the integrand is centered at k_{1} and has a bandwidth of *Δk*. The SNR of the windowed spectrum is lower than that of the original spectrum. This effectively degrades *δφ _{syn}*, and the reduced bandwidth results in a loss of axial resolution in each of the images generated from the windowing procedure. It is desirable to use broad windows to preserve the axial resolution. If the synthetic wavelength Λ were to replace

*λ*in Eq. (8) and

_{o}*δφ*replaced

_{syn}*δφ*, the synthetic wavelength displacement sensitivity,

*δφ*, is given as

_{syn}The maximum displacement in the sample that can be detected without phase wrapping is determined as

As $\underset{\mid {\lambda}_{1}-{\lambda}_{2}\mid}{lim}\to 0$, Λ continuously increases. Larger displacements can be resolved by placing the two windows closer together. Doing so also increases the SNR of each windowed spectra by integrating over a larger area of the spectral density function. However, Eq. (13) reveals that *δφ _{syn}* also scales with Λ.

It is important to note the distinction between *δφ _{syn}* and

*δx*in this analysis.

_{syn}*δφ*approaches 0 as the separation between the windows decreases because the synthetic phase is based upon the difference between the measured phases of each window. When the windows are completely overlapped, they possess complete correlation and thus their covariance is equal to the sum of their variances, which from Eq. (9) yields

_{syn}*δφ*= 0. However,

_{syn}*δx*is not zero because Λ approaches ∞ in this case. For situations where window separation is large, Λ becomes small while

_{syn}*δφ*becomes large because the SNR of the windowed spectra degrades with increasing distance from the source center. Thus Λ and

_{syn}*δφ*act to balance each other, keeping

_{syn}*δx*non-zero and finite.

_{syn}These observations indicate that the optimal choice of window placement over the original spectrum is determined by balancing the separation of the windows about the source center such that Λ is sufficiently large to unwrap expected phase jumps and yet still maintains sufficient phase stability to keep *δx _{syn}* low. Figure 2 illustrates the results of this analysis. Experimental data is obtained from the surface of a glass coverslip as described in the Methods section. Equation (13) is plotted theoretically using Eq. (9) and the SNR dependent expressions to determine

*δφ*. The measured SNR for the lower wavelength window decreased from 46.3 dB to 39.4 dB while that of the higher wavelength window decreased from 46.6 dB to 43 dB as the wavelength separation increased. Because the power of the source was less concentrated for wavelengths far from the source center, the strength of the interference fringes was also reduced at these wavelengths. Experimentally,

_{syn}*δφ*is determined by the standard deviation of the

_{syn}*δφ*measurements.

_{syn}Figure 2 shows that *δx _{syn}* increases (resolution degrades) as the two windows are moved very close together or very far apart. It is noted that both the independently measured displacement sensitivity and the SNR-calculated sensitivity follow the same general trend for windows that are separated by more than 20 nm. However, the theoretical model does not correctly predict the experimental trend of

*δx*for highly overlapped windows (|λ

_{syn}_{1}- λ

_{2}| < 20 nm) as

*δφ*does not decrease fast enough to balance the rapid increase in Λ. The covariance model in Eqs. (10) and (11) could be an oversimplification of the process governing the effects of window overlap.

_{syn}Furthermore, it is critical that the coherence length for each of the resulting windows remains the same in order to ensure proper comparison of the phase at each pixel in depth. Because the synthetic wavelength method relies upon pixel to pixel comparisons between the two phase maps generated from windowing, differences in the coherence lengths can cause the location of a scatterer to shift resulting in errors in the synthetic phase measurement.

#### 2.4 Noise reduction

The noise level *δφ _{syn}* can be reduced to that of

*δφ*by using the synthetic wavelength image as a reference for correctly unwrapping the single wavelength image [22,25,30]. After unwrapping using the synthetic wavelength method, the correct

_{o}*Δx*in Eq. (3) is determined but contains a high level of noise. Simply dividing the synthetic wavelength result by an integer number of

*λ*/2 allows for calculation of the appropriate value of

_{o}*m*. This allows the data to be recast in terms of the source center wavelength. The correctly unwrapped image has the same level of noise as is expected in the single wavelength case. However, areas of the image that possess noise levels of

*δx*>

*λ*would cause a miscalculation of

_{o}/4*m*resulting in spikes at these locations throughout the single wavelength corrected image. These spikes are within +/- 1 wrap and can be removed through a more simple unwrapping technique or filtering.

#### 2.5 Algorithm summary

The algorithm is summarized as follows:

- Acquire raw OCT spectrum
- Interpolate data to be evenly spaced in k.
- Perform FFT on data, low pass filter, then perform iFFT to obtain DC spectrum.
- Divide interpolated spectrum by DC spectrum.
- Apply two different Gaussian windows to resulting interference fringes.
- Perform FFT separately to each of the two newly formed spectra.
- Extract phase information from each FFT.
- Subtract one phase dataset from the other and add 2π if the difference is less than zero.

The resulting phase data will be unwrapped according to the synthetic wavelength. Conversion of the phase difference to a physical displacement measurement is obtained by substituting *Λ* and *Δφ _{syn}* =

*φ*into Eq. (3). Reducing the level of noise in the image can then be accomplished by dividing the resulting displacement profile by an integer number of

_{1}-φ_{2}*λ*and then adding the corresponding amount of wraps to the single wavelength data.

_{o}/2## 3. Methods

Our SDPM microscope shown in Fig. 3 consisted of a fiber based Michelson interferometer with the sample arm inserted via a documentation port into an inverted microscope (Zeiss Axiovert 200). A modelocked Titanium-Sapphire laser (Femtolasers, Femtosource, λ_{o} ~ 790 nm Δλ ~ 70 nm) was used as the source with a measured axial resolution of 5.7 μm in air. The custom spectrometer contained a linescan CCD (Atmel, A VIVA, 2048 pixel, 19 kHz). The sample arm used two galvo mirrors to allow raster scanning of the sample. A microscope objective (Zeiss, 40x, 0.6 NA) focused light onto the sample giving a lateral resolution with a calculated diffraction limit of 1.6 μm. This objective was capable of resolving the smallest bars on a USAF test chart with 4.4 μm spacing per line pair (~ 2.2 μm per bar) in good agreement with the expected resolution.

To analyze the effects of windowing on the noise of the synthetic wavelength image, a 150 μm thick glass coverslip was placed in the sample arm and 500 A-scans were acquired at a single point with 150 μs integration time per A-scan. Following the synthetic wavelength windowing procedure described above, dual Gaussian windows with 40 nm band widths were applied to each of the original spectra. The windows were equidistant from the source center. The synthetic phase was calculated by taking a difference of the phase measurements obtained from Fourier transformation of each pair of windows. This procedure was repeated for various values of Λ. The phase stability was measured experimentally by taking the standard deviation of *δφ _{syn}* at the peak pixel location of the top coverslip surface in the A-scan profile.

*δx*was calculated using the measured phase stability in Eq. (13). Theoretically, the phase stability depends on the SNR, which was calculated by taking the intensity at the A-scan peak squared divided by the variance of the noise floor over a region near the coverslip surface peak. The theoretical

_{syn}*δx*was calculated by combining Eqs. (6), (9) and (13). Figure 2 shows the results of this analysis.

_{syn}## 4. Results and discussion

An atomic force microscope (AFM) calibration grating (TZ011, Mikromasch Inc., ~10 μm pitch, ~ 1.5 μm step height) was used to validate the synthetic wavelength technique. The grating was spaced 150μm away from the coverslip surface to avoid phase corruption effects[31]. This caused a loss in signal due to the coverslip’s distance away from the axial focus. Calculating the loss of intensity for backscatterers away from the focal plane[32] revealed a loss in SNR of -84 dB. Using Eq. (7), the theoretical sensitivity of our system for a perfect reflector may be calculated using ρ = 0.9 with 25 mW power in the sample arm at 150 μs integration time, yielding 130 dB sensitivity. This allowed detection of reflectors beyond the depth of focus of our objective.

We applied the synthetic wavelength unwrapping method to data obtained by laterally scanning the sample beam across the surface of the calibration step grating, which consisted of SiO_{2} steps coated with Si_{3}N_{4} on a silicon wafer, to obtain a dataset consisting of 100×50 A-scans covering a 20 × 6 μm area with an integration time of 150 μs. The 1.5 μm step height is roughly twice the center wavelength of the source and yet is less than its coherence length. Thus, the grating is expected to produce multiple wrapping artifacts in the phase data.

The index of refraction for Si_{3}N_{4} has a trivial complex component at 790 nm[33], implying normally incident light undergoes a π phase shift upon reflection. However, the index of refraction of silicon at 790 nm (n = 3.673 and k = 5×10^{-3})[33] could potentially cause a non-π phase shift in reflected light. The phase shift deviation from π for light incident upon a silicon surface from air may be describe as [34]

where n_{o}=1 is the index of air, n_{1}=3.673 is the real component of the index of silicon, and k_{1} = 5×10^{-3} is its imaginary index. The value of p was calculated to be less than 1 mrad, which is much less than the phase difference caused by the calibration grating step height (approximately 400 mrad at the synthetic wavelength). Thus, the difference in materials between the peaks and valleys on the grating was not expected to affect the phase measurement significantly.

Figure 4(a) shows the raw, wrapped phase data obtained from the calibration step grating. Figure 4(b) shows results of the synthetic wavelength unwrapping. Two Gaussian windows were used with 50 nm bandwidths using *Λ*=41 μm. The phase of the peak pixel location in an A-scan corresponding to the peak surface of the grating for each spectrum was used to calculate *Δφ _{syn}* and yielded an SNR of 32.3 dB and 33.0 dB for the lower and higher wavelength windows respectively. The step height was measured by taking the difference of the average peaks and valley of the grating as outlined in Fig. 4(b). The standard deviation over the valley indicated was 116 nm. The dual-wavelength method correctly measures an average step height of 1.51 +/- 0.14 μm. Figure 4(c) shows a cross-sectional profile of the grating comparing the synthetic wavelength result with unwrapping performed by a simple one dimensional 2π addition/subtraction algorithm from Matlab. The Matlab algorithm only measures an average step height of 70 nm due to the method’s inability to correctly detect multiple wrapping. AFM profiling of the grating (Digital Instruments 3100, 0.5Å height resolution) measured an average step height of 1.52 μm. This confirms that using a synthetic wavelength can correctly unwrap a phase profile, even in the presence of multiply wrapped phase. Using values for

*Λ*from 16.5 μm to 94.7 μm (|λ

_{1}-λ

_{2}| = 40 nm to 8 nm, respectively) yielded similar results for the measured step height. Using a smaller

*Λ*resulted in underestimating the calculated step height, potentially due to errors in the phase measurement resulting from windowing the original spectral data far from the source center.

Matlab uses a simple algorithm that unwraps the phase by searching for phase discontinuities along a one dimensional path. If the phase jump between two neighboring pixels in an image is greater than π or less than -π this is interpreted as a wrap, and 2π is subtracted or added to smooth the phase profile. Thus, if more than one wrap is contained within the pixel, it will not be correctly unwrapped. It is also possible that a true absolute phase jump greater than π will be misinterpreted and perceived by Matlab to be a wrap. Another scenario is that a true wrap may not be detected if the resulting phase change between neighboring pixels does not yield an absolute phase difference greater than π. Thus, unwrapping using the phase from two different pixels still results in an ambiguous phase measurement. The synthetic wavelength method only uses the information contained within a single data point to generate an unwrapped profile and can utilize a sufficiently large *Λ* to ensure correct unwrapping. This is possible because of the broadband spectrum of the source and the ability to access its phase information by the Fourier transform. The phase at a single pixel in a given phase image can be measured unambiguously regardless of the phase in the surrounding pixels.

Using the synthetic wavelength phase map as a reference, the original single wavelength image was corrected by dividing the synthetic wavelength image by λ_{o}/2 of the source to obtain an integer amount for *m* in Eq. (3). This result was then added to the single wavelength phase map to produce an intermediate image. The result in Fig. 5(a) shows regions throughout the dataset containing spikes or regions where the phase was incorrectly determined. The image was corrected by subtracting the intermediate image from the synthetic wavelength image and then adding +/-2π to areas of the difference map in excess of |π|. However, some spikes still remained in regions of the image containing high amounts of noise[22]. These areas are within a +/- 2π wrap (or equivalently, +/- λ_{o}/2) from the correct value and can be removed through a simpler software unwrapping method that searches for these sharp steps[30]. A simple unwrapping filter consisting of a 3×3 window was used to compare the phase of the center pixel to an average of the phases of the neighboring pixels. If a phase difference in excess of |π| was detected, then the center pixel was unwrapped by the addition or subtraction of 2π. The result of this step is shown in Fig. 5(b). The noise level for the corrected image, taken as the standard deviation of the measure values in the valley of the grating, was 3 nm. The measured value for the average step height was 1.51 +/- 0.01 μm. Portions of the image in Fig. 5(b) contain noise artifacts that were unable to be removed by the unwrapping filter due to high noise levels in the image which corrupt the single wavelength image after correction. However, the amplified noise level introduced in the synthetic wavelength image was reduced to the single wavelength noise limit while measuring the correct step height in the presence of multiple wrapping artifacts.

We also applied synthetic wavelength unwrapping to phase measurements from human epithelial cheek cells. The cells were placed upon the top surface of a coverslip which had anti-reflective coating to help reduce the effects of phase corruption [31]. The phase profile of a group of cells is shown in Fig. 6. Images consist of 1000×100 lines acquired using a 20× objective (NA=0.5, 1.9 μm spot size) to give a wider field of view for visualizing a group of cells. Initially a wrapped phase profile was obtained in Fig. 6(a) and was then unwrapped using the synthetic wavelength method with *Λ*=20.4μm. As can be seen from the bright field microscope image in Fig. 6(c), single cells as well as a cluster of cells stacked together are present. The cell cluster should be expected to introduce multiple wrapping artifacts due to its thickness whereas the single cells may produce only a single wrap. After applying the synthetic wavelength algorithm, noise reduction to the single wavelength image was performed. Additionally, a 3×3 median filter was used to smooth the image. A clear picture of the cell height above the coverslip surface is obtained in Fig. 6(b). The heights of both single cells and the cell cluster are resolved with the regions of the original image containing either single or multiple phase wraps corrected. The standard deviation over the region indicated by the box in Fig. 6(b) was 31 nm for the synthetic wavelength image and 7 nm for the single wavelength corrected image demonstrating the effects of single wavelength noise reduction.

## 5. Conclusion

We have presented a synthetic wavelength processing method to allow for correct phase unwrapping in phase sensitive implementations of OCT. Using two spectral windows on the detected broadband spectrum allows phase information from different wavelengths to be obtained. The phase information can then be related to that of a longer synthetic wavelength to measure large phase jumps in a given sample that would normally induce phase wrapping. Though not developed here, the two window method may potentially be extrapolated to use a continuous range of wavelengths, similar to [27], to allow even more robust phase unwrapping. This method may prove useful in other phase based implementations of OCT where phase wrapping is problematic, such as in Doppler or polarization sensitive OCT or in applications such as cellular imaging.

## Acknowledgments

The authors would like to acknowledge Justin Migacz for his assistance in performing the AFM measurements. Funding for this project was provided by the Center for Biomolecular and Tissue Engineering at Duke University and NIH grant R21 EB 006338.

## References and links

**1. **O. W. Richards, “Phase Difference Microscopy,” Nature **154**, 672 (1944). [CrossRef]

**2. **H. Gundlach, “Phase contrast and differential interference contrast instrumentation and applications in cell, developmental, and marine biology,” Opt. Eng. **32**, 3223–3228, (1993). [CrossRef]

**3. **K. Creath, “Phase-shifting speckle interferometry,” Appl. Opt. **24**, 3053–3058, (1985). [CrossRef]

**4. **E. Cuche, F. Bevilacqua, and C. Depeursinge, “Digital Holography for quantitative phase-contrast imaging,” Opt. Lett. **24**, 291–293, (1999). [CrossRef]

**5. **C. J. Mann, L. Yu, C. Lo, and M. K. Kim, “High-resolution quantitative phase-contrast microscopy by digital holography,” Opt. Express **13**, 8693–8698, (2005). [CrossRef]

**6. **G. Popescu, L. P. Deflores, J.C. Vaughan, K. Badizadegan, H. Iwai, R. R. Dasari, and M. S. Feld, Fourier phase microscopy for investigation of biological structures and dynamics,” Opt. Lett. **29**, 2503–2505, (2004). [CrossRef]

**7. **N. Lue, W. Choi, G. Popescu, T. Ikeda, R. R. Dasari, K. Badizadegan, and M. S. Feld, “Quantitative phase imaging of live cells using fast Fourier phase microscopy,” Appl. Opt. **46**, 1836–1842, (2007). [CrossRef]

**8. **T. Ikeda, G. Popescu, R. R. Dasari, and M.S. Feld, “Hilbert phase microscopy for investigating fast dynamics in transparent systems,” Opt. Lett. **30**, 1165–1167, (2005). [CrossRef]

**9. **G. Popescu, T. Ikeda, C. A. Best, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Erythrocyte structure and dynamics quantified by Hilbert phase microscopy,” J. Biomed. Opt. **10**, 060503 (2005). [CrossRef]

**10. **R. Leitgeb, C. K. Hitzenberger, and A. F. Fercher, “Performance of Fourier domain vs. time domain optical coherence tomography,” Opt. Express **11**, 889–894, (2003). [CrossRef]

**11. **J.F. de Boer, B. Cense, B. H. Park, M. C. Pierce, G. J. Tearney, and B. E. Bouma, “Improved signal-to-noise ratio in spectral-domain compared with time-domain optical coherence tomography,” Opt. Lett. **28**, 2067–2069, (2003). [CrossRef]

**12. **M. A. Choma, M. V. Sarunic, C. Yang, and J. A. Izatt, “Sensitivity advantage of swept source and Fourier domain optical coherence tomography,” Opt. Express **11**, 2183–2189 (2003). [CrossRef]

**13. **M. A. Choma, A. K. Ellerbee, C. Yang, T. L. Creazzo, and J. A. Izatt, “Spectral-domain phase microscopy,” Opt. Lett. **30**, 1162–1164 (2005). [CrossRef]

**14. **C. Joo, T. Akkin, B. Cense, B. H. Park, and J. F. de Boer, “Spectral-domain optical coherence phase microscopy for quantitative phase-contrast imaging,” Opt. Lett. **30**, 2131–2133, (2005). [CrossRef]

**15. **M. A. Choma, A. K. Ellerbee, S. Yazdanfar, and J. A. Izatt, “Doppler flow imaging of cytoplasm streaming using spectral domain phase microscopy,” J. Biomed. Opt. **11**, 024014 (2006). [CrossRef]

**16. **E. J. McDowell, A. K. Ellerbee, M. A. Choma, B. E. Applegate, and J. A. Izatt, “Spectral domain phase microscopy for local measurements of cytoskeletal rheology in single cells,” J. Biomed. Opt. **04400**, (2007).

**17. **A. K. Ellerbee, T. L. Creazzo, and J. A. Izatt, “Investigating nanoscale cellular dynamics with cross-sectional spectral domain phase microscopy,” Opt. Express **15**, 8115 – 8124, (2007). [CrossRef]

**18. **D. C. Ghilglia and M. D. Pritt, *Two Dimensional Phase Unwrapping: Theory, Algorithms, and Software*. (Wiley, 1998)

**19. **C.K. Hitzenberger, M. Sticker, R. Leitgeb, and A.F. Fercher, “Differential phase measurements in low-coherence interferometry without 2π ambiguity,” Opt. Lett. **26**, 1864–1866, (2001). [CrossRef]

**20. **C. R. Tilford, “Analytical procedure for determining lengths from fractional fringes,” Appl. Opt. **16**, 1857–1860 (1977). [CrossRef]

**21. **Y. Cheng and J. C. Wyant, “Two-wavelength phase shifting interferometry,” Appl. Opt. **23**, 4539–4543 (1984). [CrossRef]

**22. **J. Gass, A. Dakoff, and M. K. Kim, “Phase imaging without 2π ambiguity by multiwavelength digital holography,” Opt. Lett. **28**, 1141–1143, (2003) [CrossRef]

**23. **J. Kuhn, T. Colomb, F. Montfort, F. Charrière, Y. Emery, E. Cuche, P. Marquet, and C. Depeursinge, “Realtime dual-wavelength digital holographic microscopy with a single hologram acquisition,” Opt. Express **15**, 7231–7242, (2007). [CrossRef]

**24. **S. Tamano, M. Otaka, and Y. Hayasaki, “Two-wavelength phase-shifting low-coherence digital holography,” Jpn. J. Appl. Phys. **47**, 8844–8847 (2008). [CrossRef]

**25. **N. Warnasooriya and M.K. Kim, “LED-based multi-wavelength phase imaging interference microscopy,” Opt. Express **15**, 9239–9247, (2007). [CrossRef]

**26. **C. Yang, A. Wax, R.R. Dasari, and M.S. Feld, “2π ambiguity-free optical distance measurement with subnanometer precision with a novel phase-crossing low-coherence interferometer,” Opt. Lett. **27**, 77–79, (2002). [CrossRef]

**27. **D. L. Marks, S.C. Schlachter, A.M. Zysk, and S.A. Boppart, “Group refractive index reconstruction with broadband interferometric confocal microscopy,” J. Opt. Soc. Am. A **25**, 1156–1164 (2008). [CrossRef]

**28. **R. Tripathi, N Nassif, J. S. Nelson, B. H. Park, and J. F. de Boer, “Spectral shaping for non-Gaussian source spectra in optical coherence tomography,” Opt. Lett. **27**, 406–408, (2002). [CrossRef]

**29. **D.L. Marks, P.S. Carney, and S.A. Boppart, “Adaptive spectral apodization for sidelobe reduction in optical coherence tomography images,” J. Biomed. Opt. **9**, 1281–1287 (2004). [CrossRef]

**30. **A. Khmaladze, A. Restrepo-Martínez, M.K. Kim, R. Castañeda, and A. Blandón, “Simultaneous Dual-Wavelength Reflection Digital Holography Applied to the Study of the Porous Coal Samples,” Appl. Opt. **47**, 3203–3210 (2008). [CrossRef]

**31. **A. K. Ellerbee and J.A. Izatt, “Phase retrieval in low-coherence interferometric microscopy,” Opt. Lett. **32**, 388–390, (2007). [CrossRef]

**32. **J.A. Izatt and M.A. Choma, “Theory of Optical Coherence Tomography,” in *Optical Coherence Tomography: Technology and Applications*,
W. Drexler and J.G. Fujimoto, eds. (Springer, 2008). [CrossRef]

**33. **D.R. Lide, ed. *CRC Handbook of Chemistry and Physics*, (CRC Press, 2001–2002).

**34. **T. Doi, K. Toyoda, and Y. Tanimura, “Effects of phase changes on reflection and their wavelength dependence in optical profilometry,” Appl. Opt. **36**, 7157–7161, (1997). [CrossRef]