In order to test the performance of the galvo scanner induced phase shifting technique different measurements were performed. For these measurements the weakly scattering surface of a black-anodized aluminum part was used as a sample. With a power of 1.3 mW onto the sample and a camera integration time of 100 μs, B-scan images consisting of 1000 A-lines were recorded. For a scan angle of 12° the image width was 15 mm. The dynamic range of the images was measured to be 45 dB. By use of the translation stage of the galvo scanner assembly the mirror offset s was set to 24 equidistant positions ranging from s = −1.03 mm to s = 1.12 mm around the x-scanner’s pivot axis. For each step, B-scan images were recorded and hereupon, the phase shift at the surface of the sample object caused by the respective mirror displacement was calculated. The results of these measurements are plotted in for three different positions of the sample surface with respect to the reference mirror. A linear relationship of mirror offset and induced phase shift between adjacent A-scans can be observed.
Measured phase shift Φ as a function of mirror offset s for three different depth positions z. A linear relationship between s and Φ can be observed.
The linearity of the induced path length modulation is demonstrated in . For a mirror offset of 0.56 mm, the cumulated phase shift along one transversal line is plotted versus A-scan number. A linear fit with a slope of 1.633 rad/A-line (which slightly deviates from the desired value Φ = π/2 = 1.571 rad) is shown in red. The offset of −41.553 rad between the measured, unwrapped curve and the fitted curve arises from the back snapping of the galvo scanner. When the scanner is driven by a short negative ramp during the back snapping, complete interference fringe washout – equivalent to total signal loss – occurs. Therefore, for the first ~30 A-scans in , also no phase shift is registered and the offset of 41.553 rad (~30·π/2) arises.
Cumulated (unwrapped) phase shift along a transverse line of a B-scan. The red line is a linear fit with a slope of 1.633 rad/A-scan. R2 = 0.9997 is the coefficient of determination.
The efficiency of complex conjugate suppression as a function of the induced phase shift was investigated using the same data. In , the extinction ratio (ER), i.e. the ratio of the surface signal and its complex conjugate on the other side of the zero path delay, is plotted against mirror offset s. Extinction ratios better than −30 dB can be observed for phase shifts of −π/2 and +π/2, respectively. When going from negative to positive mirror offsets, the resulting phase shifts cause the extinction ratio to flip, i.e. with respect to z the reconstructed full range image will be oriented the opposite way around. The effect can be understood taking into account exp[iπ] = −1. No suppression of the complex conjugate artifact – an extinction ratio of 0 dB – is to be observed for values of Φ = Mπ with M being an integer.
Fig. 6 Extinction ratio of complex conjugate artifact as a function of mirror offset s measured for three different depth positions. For mirror displacements corresponding to phase shifts of −π, 0, and π, the mirror signal is not suppressed, (more ...)
Considering the theoretical description of the complex signal reconstruction shown in section 2, for B(u,k) = B(k)* δ(u) (i.e., assuming a transversally homogeneous object geometry), any phase shift Φ ≠ Mπ should ideally lead to total suppression of complex conjugate signals. Choosing phase shifts in the interval of (Mπ (M+1)π), reconstructed image features will flip from +z to −z and vice versa for even and odd values of M, respectively.
Different physical effects reduce the theoretical performance of our phase shifting technique. First, resulting from the wavelength dependency of equation (5
), a chromatic phase error [13
] will arise when broadband light is being used. The transversal lines of the 2D spectral interferogram will be subject to different phase shifts in the order of
. For the light source used in our experiment, the range of phase shifts is
When s is set such that Φ = π/2, the phase shifts will cover a rather narrow range of ~5° FWHM. However, for larger phase shifts Φ also the chromatic broadening of the covered range of phase shifts is increasing.
A second effect, by which the range of phase shifts applied is broadened, originates from the non-zero diameter of the collimated beam at the galvo mirror surface. When the sample beam hits the galvo scanner mirror, a Gaussian distribution of phase shifts Φ is produced due to the Gaussian intensity distribution in the beam’s cross-section incident at different offsets s
from the pivot axis. For a mirror offset s
with respect to the beam center, which produces a central phase shift Φ0
, the arising distribution of phase shifts G
can be approximated as
) = 2k
is the central phase shift as defined in section 2.2 and ΔΦFWHM
is the full width at half maximum (FWHM
) of the distribution caused by a FWHM
beam diameter BD
. Now, taking into account that any phase shift
will reproduce the image at either depth +z
dependent on the parity of M
, all components shifted by Φ in an interval with odd M
will give rise to a signal on one side of zero path delay z
= 0, whereas all phase shifts in an even-M
’ed interval will contribute to a signal on the other side. Since the extinction ratio ER
was defined as the quotient of the signal fraction on one side of zero path delay and the signal fraction on the other side, one can theoretically estimate ER
by integrating positively and negatively shifted fragments of G
and calculating their ratio as
The first element of the integrands, Θ(sin(±Φ)), is the Heaviside function applied to the sine of Φ and −Φ; NC
(Φ) denotes the noise contribution of the phase shifts. The numerator contains all contributions of G
for even values of M
; the denominator in turn contains the contributions for odd-valued M
The theoretical model for mirror term suppression was employed for an estimation of ER
. For this first approximation of ER
, the influence of the chromatic phase error mentioned above was ignored because it is considerably smaller. By means of a beam profiler (Dataray, USA), the beam’s FWHM
was measured to be ~500 μm. Using this value for BD
and the set parameter values used for the ER
measurements, a theoretical curve ER
(dashed plot in ) was computed with equations (6
) and (7
). NC(Φ) = 0 was assumed. It can be seen in that the measured values are in good agreement with the theoretical prediction; still, slight deviations are observable. The minimum and the maximum of the theoretical curve are closer to s
= 0 than the results of the measurement, which indicates that the measured central phase shifts Φ0
are smaller than the expected ones. An actually slightly smaller total scanning angle Δβmax
which is proportional to Φ0
might be responsible for this deviation.
As a third effect, the width of the distribution of spatial frequencies due to the transverse object structure has to be considered. For a sample without any variations of structure along the B-lines along x-direction (to which our test object can be approximated), the spectrum in spatial frequency domain can be expressed by B(u,k) = B(k)* δ(u) with δ(u) being the Dirac delta function. Of course, for sample structures exhibiting variations along x-direction (as it is usually the case for biological tissue), the spectrum B(u,k) will cover the corresponding range in u-domain. It was mentioned as one condition for total complex signal reconstruction in section 2.1 that the range covered by B(u,k) must not exceed half of the spatial frequency space. Any overlap of the spectral components in the positive and negative ranges will induce complex conjugate artifacts. Hence, for structures with a broad spectrum in spatial frequency domain an inferior ER is to be expected.
Since the method discussed in this paper is a dynamic phase shifting technique (i.e., the phase is shifted not stepwise but continuously), one has to consider fringe washout as a source of signal degradation. The larger mirror offset s
is chosen, the higher will be the speed of the scanning mirror and the stronger will be the fringe washout. Yun et al. have first described this movement-induced fringe washout as a SNR
degrading factor of sin2
being the phase shift induced by axial movement [19
]. For phase shifts of |Φ0
| ≤ π/2, as they are used in this paper, the SNR
penalty caused by fringe washout will be less than 1 dB; for |Φ0
| = π, the theoretical signal loss amounts to almost 4 dB. When even larger phase shifts are produced, signal degradation will considerably reduce image quality. Also, it is crucial for the algorithm to have a certain phase relation between neighboring A-scans; therefore, the distance between adjacent A-scans must not exceed the focal beam diameter.
In order to evaluate the FRC SD-OCT system performance for in vivo imaging, we imaged the anterior chamber of the eye of a healthy human volunteer. The integration time of the spectrometer CCD camera was set to 100 μs per A-scan; the sample was exposed to a light power of 1.3 mW which is well below the safety limits given by the ANSI and IEC standards for the safe use of lasers [20
]. For the total scanning angle of 12° the mirror offset s
was adjusted such that a phase shift of π/2 required for efficient removal of the complex conjugate artifact was generated. The images shown in show results derived from the same spectral data. For , the spectral interferograms were (after fixed pattern noise removal and remapping to k
-space) processed by an inverse FFT of each real-valued spectral interferogram (conventional SD-OCT). The true object structure is obscured by the overlapping mirror image. was calculated from the same data; however, here the additional processing step described in section 2.1 was performed in order to recover the complex valued spectral data. Two remaining transverse line artifacts due to an internal reflection and the complex signals’ DC terms that could not be removed by the algorithm were eliminated by cutting out the respective B-lines and interpolating the intermediate values [22
]. As can be seen in , the complex conjugate mirror image could sufficiently be suppressed.
Fig. 7 B-scan images of human anterior eye segment in vivo. (A) Tomogram obtained by inverse FFT of the real-valued spectral data. (B) Full-range image obtained by reconstruction of the complex-valued spectral data. Image size: 14 (x) × 6.6 (z, optical (more ...)