Complete complex conjugate resolved heterodyne swept source optical coherence tomography using a dispersive optical delay line: erratum.

We correct an error in our previous paper [Biomed. Opt. Express 2, 1218 (2011)] which led to an erroneous conclusion that a dispersive optical delay line (DODL) used in a swept source optical coherence tomography (SSOCT) system generated a pure phase modulation allowing for complex conjugate artifact removal in Fourier domain OCT via optical heterodyning. We now understand that an alternate phenomenon known as coherence revival was responsible for the observed phase modulation, while the DODL provided a compact means of generating a large group delay with readily adjustable group velocity dispersion compensation.

In our recent paper [1], we asserted that the use of a dispersive optical delay line (DODL) in a swept source optical coherence tomography (SSOCT) system resulted in the generation of a pure phase modulation that allowed for resolution of the complex conjugate ambiguity via optical heterodyning. This assertion was incorrect, and resulted from an error in our derivation of the axial position shift in a DODL.

Heterodyne SSOCT techniques function by creating an axial position shift between the peak of the fall-off profile and the axial position encoded at electronic DC [2]. The peak of the fall-off profile occurs at the zero pathlength difference (ZPD) position, where the group delays of the reference and sample arms are matched. The axial position shift is thus defined as the difference between the net group delay difference and the net phase delay difference between the arms. Under the assumption of a non-dispersive sample (i.e. equal group and phase delays to the reflection site), this corresponds to the difference between the round-trip group and phase delays of the DODL. In the original paper [1], we erroneously defined the axial position shift (in an inline equation on page 1222) as ∆z = 2πM/k0.

To derive the correct axial position shift of the DODL, we re-iterate the round-trip phase and group delays as correctly derived in the original paper as (asterisked equation numbers correspond to equations from the original paper):

Here, λ is the instantaneous central wavelength of the sweeping laser linewidth, λ0 is the central wavelength of the laser tuning bandwidth, and M is the slope of the wavelength-dependant phase delay of the DODL (in mm/nm). The correct axial position shift is thus the difference between these group and phase delays:

The photocurrent from an SSOCT system in which the reference arm phase delay varies with wavelength was correctly derived in Eq. (7) of the original manuscript [1]. Making use of the correct definition of Δz, this equation can be re-expressed as

where k is the instantaneous central wavenumber, z is the phase pathlength corresponding to the nth sample reflector, z0 is the phase pathlength of the reference reflector at the central wavelength, and M is the slope of the wavelength-dependent reference phase pathlength (in mm/nm). Since the ZPD position occurs where the group delays are matched, we define a new term Δz to denote the group delay difference between the nth sample reflector and the ZPD position:

This expression assumes the sample is non-dispersive, and includes the extra group delay introduced in the reference arm by the DODL in Eq. (20*). Combining Eqs. (22) and (23) and making use of the definition in Eq. (21) yields

From this expression, it is clear that the axial position shift created by the DODL leads only to a constant (wavenumber-independent) phase shift in the photocurrent signal that does not produce any separation between ZPD and DC.

Despite this error, the experimental results we reported in [1] (which we stand by) demonstrate that a phase modulation enabling complex conjugate artifact removal was present when using the DODL which was not observed when using the standard reference arm. We now understand that this phase modulation was created by the laser itself through a mechanism known as coherence revival. We discuss coherence revival-based heterodyne SSOCT in detail in a separate manuscript [3].

Briefly, coherence revival refers to a phenomenon where an interferometer employing an external cavity tunable laser (ECTL) exhibits interference fringes when the two arms are mismatched by the laser’s cavity length [4]. This cavity length is often several orders of magnitude longer than the source coherence length. This phenomenon can be understood as resulting from the interference of sequential optical waveforms emanating from sequential optical cavity round trips within the laser. Thus, an interferometer whose arms are offset by a cavity length is analogous to one in which a virtual copy of the laser cavity has been placed in the shorter arm. If the optical pathlength of the laser cavity varies over the course of the laser sweep, this results in a phase modulation that creates a heterodyne SSOCT signal.

Unlike the DODL, whose pathlength depends on wavelength but is fixed in time, the laser cavity optical pathlength changes over the course of the sweep, but is essentially constant with respect to optical frequency over the integration time of each individual spectral channel. Since the group delay is the derivative of instantaneous phase shift with respect to frequency, the cavity optical pathlength variation does not create an offsetting group delay.

We have demonstrated that at least two commercially available swept source ECTL’s exhibit this coherence revival behavior with inherent phase modulation [3]. However, in both cases, the variation in the cavity optical pathlength was not linear with the instantaneous central wavelength of the laser sweep. As a result, while the SSOCT signals acquired in this configuration were indeed phase modulated, after Fourier transformation, the signals exhibited severe degradation in the axial point spread function (PSF), analogous to the effects of material dispersion. In [1], we believe we inadvertently used the DODL to correct for group velocity dispersion by displacing the diffraction grating from the focal plane of the lens [5]. Thus, even though the DODL was not responsible for creating the phase modulation, it did serve to correct the degradation in the axial PSF of the system. A DODL may indeed be a valuable tool in the coherence revival CCR technique, as it provides a compact means of generating a large delay while also providing direct optical means to correct degradation of the axial PSF.