## 1.

## Introduction

Fourier domain optical coherence tomography (FDOCT) techniques have reshaped the OCT landscape due to demonstrated sensitivities that are two to three orders of magnitude greater than their time domain counterparts.^{1, 2, 3} The shift to FDOCT, however, is not without costs. FDOCT techniques are limited in imaging depth and suffer from two important sources of artifacts. The first, called “complex conjugate ambiguity,” arises because the Fourier transform of the real-valued spectral interferometric signal is Hermitian symmetric. This results in sample reflectors at a positive displacement,
$+\Delta x$
, with respect to the reference reflector, being superimposed on those at a negative displacement,
$-\Delta x$
. The second source of artifact, termed “dc artifact,” originates from the noninterferometric light and autocorrelation from sample reflectors, which transform to
$\Delta x=0$
, and thereby obscure reflectors positioned at zero pathlength difference. These artifacts can be removed by retrieval of the complex interferometric signal. Additionally, retrieval of the complex interferometric signal effectively doubles the imaging depth by removal of the complex conjugate ambiguity.

Previous techniques for acquiring the complex FD interferometric signal have relied on collecting the in-phase and quadrature (
$\pi \u22152$
-shifted) components generated by phase shifting interferometry^{4} or
$3\times 3$
interferometry.^{5, 6} These techniques are constrained to homodyne insofar as both the reference and sample arm optical fields have the same phase velocity. Homodyne detection is required for spectral domain OCT (SDOCT) systems that employ spectrometers coupled to charge accumulation detectors such as charge-coupled devices (CCDs) and photodiode arrays. In swept-source OCT (SSOCT), however, a current-generating photodiode is employed, which enables the spectral interferometric signal to be encoded with a characteristic heterodyne beat frequency. Phase-shifting interferometry requires acquisition of multiple images and therefore is not instantaneous.
$3\times 3$
interferometry is an instantaneous technique for acquisition of the complex interferometric signal. However,
$N\times N$
fiber couplers are wavelength dependent, making it difficult to accurately reconstruct the full complex interferometric signal.

Two techniques for complex conjugate artifact removal using heterodyne SSOCT have previously been reported. The first described a system utilizing electro-optic phase modulators to introduce a heterodyne beat frequency in the reference arm.^{7} A drawback to this technique is that electro-optic modulators are polarization dependent and highly dispersive, thus requiring methods for controlling the polarization and compensation of the group-velocity dispersion. The second technique utilized acousto-optic modulators for heterodyne SSOCT.^{8} This system entails an interferometric topology in which the frequency-shifting components are placed between the sample and the receiver. Frequency-shifting elements, such as electro- and acousto-optic modulators, are typically lossy. Placing these elements after the sample prevents the ability to compensate power losses due to light exposure limitations on *in vivo* samples. Additionally, both published techniques required oversampling of the instrinsic signal bandwidth due to the heterodyne modulation, and the latter technique required a computationally intensive algorithm for resampling the acquired data into frequency space.

In this paper, we present a heterodyne SSOCT system that enables efficient detection of the complex interferometric signal, thus having the potential for increased signal-to-noise ratio (SNR). Additionally, we use wavenumber triggering,^{9, 10} thereby eliminating the necessity to resample the acquired data in software. We show that heterodyne detection in SSOCT enables complete resolution of complex conjugate ambiguity and the removal of noninterferometric signals. We also show that frequency shifting the reference arm optical field upshifts the cross-interferometric signal to a user-tunable frequency that corresponds to an electronic pathlength mismatch between the interferometer arms. This electronic pathlength mismatching recenters the A-scan at an offset that can be far from dc, which effectively resolves the complex conjugate ambiguity problem. While it is also possible, in principle, to upshift the cross-interferometric signal by placing the sample focus at a large pathlength mismatch, this physical pathlength mismatching leads to significant signal attenuation due to amplitude falloff secondary to finite source linewidths.^{4, 5} We show that frequency shifting provides a method for complex conjugate ambiguity resolution that circumvents signal falloff that occurs by placing samples at a large pathlength mismatch. Through electronic demodulation, we gain access to the in-phase and quadrature components of the interferometric signal and enables wavenumber triggering, which eliminates the necessity for oversampling.

## 2.

## Theoretical Analysis

## 2.1.

### Imaging Depth

The imaging depth achieved using FDOCT systems is limited by two mechanisms, the spectral sampling interval (which limits the maximum depth observable) and the system spectral resolution (which leads to a falloff of SNR with depth). The maximum imaging depth in FDOCT systems is described as^{2}

^{11}For SSOCT systems, the spectral sampling interval is limited by the sampling rate of the temporally sweeping source frequency. Over $4.0\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ maximum imaging depths have been achieved for $1.3\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$ SSOCT systems.

^{12}

The second parameter that limits the imaging depth of FDOCT systems is falloff. The sensitivity of FDOCT systems degrades as a function of imaging depth due to fringe washout. The
$-3\phantom{\rule{0.3em}{0ex}}\mathrm{dB}$
falloff depth, derived from the analysis reported by Yun ^{11} is given by

## 2

$$\Delta {z}_{-3\mathrm{dB}}=\frac{2\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}2}{{\delta}_{r}k},$$^{11}In comparison, the $-3\phantom{\rule{0.3em}{0ex}}\mathrm{dB}$ imaging depth for $1.3\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$ SSOCT is $3.7\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ , more than double the distance achieved using SDOCT. Imaging depth capabilities over $4\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ can be valuable for applications such as endoscopy, small animal imaging, and human anterior segment imaging.

## 2.2.

### Heterodyne SSOCT

In FDOCT, the photocurrent signal generated by $n$ reflectors is given by

## 3

$$i\left(k\right)\propto S\left(k\right)\{{R}_{R}+\sum _{n}{R}_{n}+2\sqrt{{R}_{R}}\sum _{n}\sqrt{{R}_{n}}\phantom{\rule{0.2em}{0ex}}\mathrm{cos}\left[2k({z}_{R}-{z}_{n})\right]+2\sum _{n}\sum _{m\ne n}{\left({R}_{n}{R}_{m}\right)}^{1\u22152}\phantom{\rule{0.2em}{0ex}}\mathrm{cos}\left[2k({z}_{n}-{z}_{m})\right]\}.$$In SSOCT systems, the wavenumber is parameterized in time $t$ by the relationship $k={k}_{0}+t(\mathrm{d}k\u2215\mathrm{d}t)$ , where ${k}_{0}$ is the starting wavenumber, and $\mathrm{d}k\u2215\mathrm{d}t$ is the nonlinear sweep velocity. This sweeping leads to the conversion of pathlength differences in the auto- and cross-terms to an electronic frequency in $i\left(t\right)$ , the time-varying photocurrent. The cross-frequencies have instantaneous values of ${\omega}_{n}=(\mathrm{d}k\u2215\mathrm{d}t)({z}_{R}-{z}_{n})$ , while the autofrequencies have instantaneous values of ${\omega}_{\mathrm{nm}}=(\mathrm{d}k\u2215\mathrm{d}t)({z}_{n}-{z}_{m})$ . If the reference arm optical field is shifted by some beat frequency ${\omega}_{D}$ , then the time-varying photocurrent becomes

## 4

$$i\left(t\right)\propto S\left(t\right)\{{R}_{R}+\sum _{n}{R}_{n}+2\sqrt{{R}_{R}}\sum _{n}\sqrt{{R}_{n}}\phantom{\rule{0.2em}{0ex}}\mathrm{cos}[({\omega}_{n}+{\omega}_{D})t+{\varphi}_{n}]+2\sum _{n}\sum _{m\ne n}{\left({R}_{n}{R}_{m}\right)}^{1\u22152}\phantom{\rule{0.2em}{0ex}}\mathrm{cos}({\omega}_{nm}t+{\varphi}_{nm})\},$$The cross-interferometric signal can be recovered by bandpass filtering around ${\omega}_{D}$ with a noise equivalent bandwidth of $\mathrm{NEB}=2{z}_{\mathrm{max}}\Delta k\u2215\Delta t$ . If demodulation is performed the bandpassed signal is electronically mixed with orthogonal local oscillators with frequency ${\omega}_{D}$ . In this case, the in-phase (real) and quadrature (imaginary) parts of the complex interferometric signal can be recovered:

## 5.

## 3.

## Experimental Setup

We constructed the heterodyne SSOCT setup shown in Fig. 1 using a fiber-based swept laser source (Micron Optics, Inc., ${\lambda}_{0}=1310\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ , $\Delta \lambda =100\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ , 250 Hz sweep rate), acousto-optic modulators (ACM-1002AA6 IntraAction, Corp.), and a New Focus balanced photoreceiver. The acousto-optic modulators (AOs) had a common center frequency of $100\phantom{\rule{0.3em}{0ex}}\mathrm{MHz}$ , one having a user-adjustable offset $\left({\omega}_{D}\right)$ from that frequency. The diffraction efficiency of the AOs was measured to be 60%, and using the optical setup depicted in the figure, the maximum diffracted optical bandwidth recoupled into the circulator was $64\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ . To trigger each line acquisition, a $250\phantom{\rule{0.3em}{0ex}}\mathrm{Hz}$ clock was provided by the swept-laser source. Spectral interferogram samples evenly spaced in wavenumber were clocked into the data acquisition system by using a fiber Fabry-Perot interferometer (FFPI, Micron Optics, Inc.). Ten percent of the swept-laser source output was delivered to the FFPI and the output was detected using a New Focus 125 MHz photoreceiver. The detected signal was passed through a $2.1\phantom{\rule{0.3em}{0ex}}\mathrm{MHz}$ low-pass filter and electronic gain to produce an electronic comb, where each $5\phantom{\rule{0.3em}{0ex}}\mathrm{V}$ peak was evenly spaced in wavenumber. The wavenumber spacing was $0.1\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ , given by the free spectral range of the FFPI.

The heterodyne interferometric signal collected using the balanced photoreceiver was high-pass filtered $\left(500\phantom{\rule{0.3em}{0ex}}\mathrm{kHz}\right)$ to remove the spectral and autocorrelation artifacts, then demodulated (RF Micro Devices RF2713) by mixing with a local oscillator (LO) of frequency ${\omega}_{D}$ . The in-phase and quadrature components were low-pass filtered ( $500\phantom{\rule{0.3em}{0ex}}\mathrm{kHz}$ cutoff frequency) and digitized in dual analog-to-digital (A/D) channels (National Instruments PCI 6115) using the clocking signal from the wavenumber trigger. The output power of the swept-laser source was $500\phantom{\rule{0.3em}{0ex}}\mu \mathrm{W}$ . There was approximately $-6\phantom{\rule{0.3em}{0ex}}\mathrm{dB}$ source power attenuation in the system prior to the sample ( $-3\phantom{\rule{0.3em}{0ex}}\mathrm{dB}$ through the AO and $-2.9\phantom{\rule{0.3em}{0ex}}\mathrm{dB}$ insertion loss into the fiber of the optical circulator), resulting in $60\phantom{\rule{0.3em}{0ex}}\mu \mathrm{W}$ illumination on the sample (cornea).

The SNR of the system, using a $-60\phantom{\rule{0.3em}{0ex}}\mathrm{dB}$ calibrated reflector near zero pathlength difference, was measured to be $99\phantom{\rule{0.3em}{0ex}}\mathrm{dB}$ . The predicted SNR, assuming same signal power, was $112\phantom{\rule{0.3em}{0ex}}\mathrm{dB}$ . The heterodyne SSOCT setup described here, places the AO in front of the sample. This design enables compensation of optical power loss from the AO and, therefore, the SNR of the system could be increased by use of a higher power laser source.

## 4.

## Results

To illustrate the behavior of the cross-interferometric term in Eq. 3 with the reference arm frequency shifted with respect to the sample arm, we show in Fig. 2 the fringe patterns of typical interferograms at two pathlength differences, centered around zero, for homodyne [Figs. 2(a) and 2(b)] and heterodyne [Figs. 2(c) and 2(d)] SSOCT setups. The homodyne setup was measured by setting ${\omega}_{D}$ , the frequency difference between the two AOs, to zero. The fringe frequency for the cross-correlation term, when ${\omega}_{D}=0$ , was identical for equivalent positive [Fig. 2(a)] and negative [Fig. 2(b)] displacements, and it was thus ambiguous whether the pathlength difference was positive or negative. However, when ${\omega}_{D}=20\phantom{\rule{0.3em}{0ex}}\mathrm{kHz}$ , the fringe frequency for the positive displacement [Fig. 2(c)] was higher than for a negative displacement [Fig. 2(d)], as expected from Eq. 4, and therefore the positive and negative locations were resolved.

Figure 3 demonstrates that the signal falloff remained centered at zero pathlength difference, even as the electronic frequency was shifted. Figure 3(a) shows falloff centered at zero pathlength for homodyne SSOCT. By shifting the zero pathlength difference frequency to ${\omega}_{D}=1\phantom{\rule{0.3em}{0ex}}\mathrm{MHz}$ , the image contents were moved away from dc and bandpass filtered to remove the spectral artifacts. After filtering and demodulation, the falloff was recentered around zero frequency and the $-3.5\phantom{\rule{0.3em}{0ex}}\mathrm{dB}$ signal attenuation out to a $4.0\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ scan depth was preserved.

In Fig. 4 , complete removal of complex conjugate ambiguity and dc artifacts is demonstrated using heterodyne SSOCT. In the A-scan acquired using homodyne SSOCT [Fig. 4(a)], the complex conjugate ambiguity artifact is clearly shown. At zero pathlength difference a strong dc peak from the noninterferometric spectral artifact is also present. By shifting the frequency of the reference arm by ${\omega}_{D}=1\phantom{\rule{0.3em}{0ex}}\mathrm{MHz}$ , the complex conjugate ambiguity was completely resolved [Fig. 4(b)]. Also, the spectral artifact located at dc was removed. There is, however, still present a small dc artifact that originated from the demodulation board. The output signals from the demodulation board contained a small $(<0.1\phantom{\rule{0.3em}{0ex}}\mathrm{V})$ dc offset. As a result, this dc offset appeared as a small peak located at zero pathlength difference in the A-scans. The dynamic range of the system was limited to $25\phantom{\rule{0.3em}{0ex}}\mathrm{dB}$ by the input requirements of the demodulation board $(<80\phantom{\rule{0.3em}{0ex}}\mathrm{mV})$ . As a result, we were able to demonstrate $25\phantom{\rule{0.3em}{0ex}}\mathrm{dB}$ extinction of the complex conjugate ambiguity artifact. In theory, however, the capability of the system to remove this artifact is not limited.

*In vivo* images acquired using homodyne and heterodyne SSOCT techniques are shown in Figs. 5(a) and 5(b)
, respectively. These images are of the anterior segment of a human eye and were constructed using 500 lines/image acquired at a rate of
$250\phantom{\rule{0.3em}{0ex}}\mathrm{Hz}$
. In the homodyne SSOCT image, the iris, cornea, and sclera were obscured by the mirror image arising from the complex conjugate ambiguity artifact. Using heterodyne SSOCT, however, the artifact was completely removed and the cornea, iris, and sclera were no longer obscured. It is evident that at least
$6\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$
of imaging depth is necessary to visualize the entire anterior segment.

## 5.

## Conclusions

We demonstrated a heterodyne SSOCT technique using AOs for complex conjugate ambiguity and dc artifact removal. We also presented an efficient heterodyne SSOCT design that minimizes power losses after the sample, enabling an increased SNR. This technique enables the zero pathlength difference to be shifted away from electronic dc with no penalty due to finite laser linewidth falloff. Combined with coherent demodulation, this technique provides access to the entire complex interferometric signal without oversampling, as the method also enables data acquisition using wavenumber triggering, thus eliminating computationally intensive resampling algorithms. Although we had limited available laser source power and dynamic range, we presented a technique for heterodyne SSOCT with wavenumber triggering using electronic demodulation. We also demonstrated *in vivo* human anterior segment images where more than
$6\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$
imaging depth is required.

## Acknowledgments

This work was supported by National Institute of Health, R21 EB000243. We greatly appreciate the collaboration with K. Hsu at Micron Optics, Inc.

## References

*In vivo*dynamic human retinal blood flow imaging using ultra-high-speed spectral domain optical Doppler tomography,” Opt. Express, 11 (25), 3490 –3497 (2003). 1094-4087 Google Scholar