In spectral domain optical coherence tomography depth resolved information is encoded in the cross spectral density function measured with a spectrometer located in the detection arm of an interferometer [8, 9]. A drawback of this method is the presence of coherent noise terms and a strong DC term. Another disadvantage of SD-OCT is that since the detected spectral density is a real function and therefore its Fourier transform is Hermitian, the reconstructed image is symmetrical about zero path difference. As a consequence one cannot distinguish between negative and positive optical path differences with respect to the reference mirror. These effects are of minor importance for measuring thin objects (e. g. retina), because the reference arm can be shifted to a position where the mirror images and DC terms do not affect the image of the measured object. In [10] it is shown, that it is sufficient to subtract the reference arm spectrum (or averaged reference arm spectra) from the measured signal to achieve shot noise limited and autocorrelation free images from the retina. However, if one wants to measure objects with larger depth extension where the whole measurement range is needed (e. g. anterior segment of the eye) the DC and mirror terms heavily decrease the image quality and make it difficult to interpret the collected data. Furthermore, the sensitivity decrease of SD-OCT with distance from the zero position [9] would make a differentiation between positive and negative distances highly desirable because this would allow to place the object within the most sensible measurement range near the reference mirror position.

Fercher et al. [11] first pointed out that this problem can be solved by measuring the phase of the spectral interferometric signal, thus providing access to the complex scattered field. These authors introduced a method based on five measurements at each sampling location to get rid of the DC and mirror terms and to double the measurement range. Wojtkowski et al. [12] were the first to apply this technique to ocular imaging in vitro. However since a phase stability over five measurements is required, this method is not suitable for in vivo measurements. Targowski et al. [13] presented images of the human eye in vivo based on the use of three A-scans per measurement location. However their acquisition speed was 3.6 seconds per image (one image consists of 326 A-lines), a measurement time which makes it difficult to measure patients. Recently Leitgeb et al. [14] introduced a method based on only two measurements which shares most of the advantages of the 5 frame technique. The instrument reported in that work used a piezo mounted reference mirror for phase shifting. This posed severe limits to the measurement speed, preventing in vivo imaging. Furthermore, the original method has the disadvantage that structure terms exactly symmetrical to the reference mirror position are attenuated or suppressed.

The first two terms on the right hand side of equation (1) are DC terms, representing the intensity of light directly back reflected or –scattered at the reference mirror and sample interfaces. The sum of the third term represents autocorrelation terms which are caused by mutual interference of waves scattered within the object. This term can be regarded as a coherent noise that, together with the DC terms, corrupt the true object structure in a reconstructed image. The sum of the last term contains the actual object structure.

The reconstructed object structure is symmetric about the position of the reference mirror (zero position) which can be seen by the fact that the field correlation function Γ occurs twice in the sum, with positive and negative sign of (τ_r –τ_n). The reason for this is that the spectral density is a real function, therefore, its Fourier transform will be Hermitian [15]. The consequence is that the reconstructed image is not only corrupted by the DC and autocorrelation terms, but it will also contain a mirror image of the object, with symmetry about the reference mirror position. If an overlap of image and mirror image is to be avoided, only half of the available image depth can be used. This poses the following limitations [9]: (i) the imaging depth of SD-OCT is limited by the spectrometer resolution, (ii) a tradeoff between imaging depth and depth resolution exists, and (iii) there is a decay of sensitivity with image depth. Typical imaging depths of SD-OCT systems are 1 - 3 mm, too low for imaging deeper structures as, e.g., the anterior chamber of the human eye.

A solution to this problem is to measure the spectral phase and obtain the complex scattered field whose inverse Fourier transform equals the true object structure [11]. To measure the spectral phase, typically 3 to 5 consecutive measurements of spectral intensity are performed, taken with phase shifts in increments of 90° [11-13]. This phase shift can be accomplished by slight shifts of the reference mirror. The disadvantage of this method is that several measurements are necessary which poses considerable demands on the phase stability of the instrument within the time interval needed to perform the 3 to 5 measurements.

Recently, an alternative method has been presented that requires only two successive measurements with a 90° phase shift between the two measurements [14]. The thus recorded spectral intensities can be regarded as quadrature components from which a complex spectral intensity can be generated: where I(ν, Δ = 90°) represents the spectral intensity recorded after a phase shift of Δ = 90°. An inverse Fourier transform of this signal now yields a first complex time domain signal : where and are the DC autocorrelation terms of eq. (1). Similarly, the inverse Fourier transform of the complex conjugate spectral intensity signal yields a second complex time domain signal : We can now calculate the modulus of these signals to obtain real-valued time domain signals: These signals still contain the DC and autocorrelation terms, however, structure terms and their mirror terms are now separated. Signal S₁ contains the direct structure term, signal S₂ the mirror term. If we calculate the difference: we can eliminate the DC and autocorrelation terms and obtain a signal with positive intensities corresponding to the real structure and negative intensities corresponding to the mirror image. We finally multiply the signal with the Heaviside step function Φ[ΔS(τ)] to obtain only the positive values that correspond to the real structure:

While the method described so far has the advantage of effectively eliminating DC, autocorrelation, and mirror terms with only two measurements, it also has a drawback: it suppresses real structure terms that are exactly symmetric about the reference mirror position. This can be seen, if we assume two signals located at τ₁ and τ₂ with τ_r - τ₁ = −(τ_r - τ₂). In this case ΔS(τ) will be zero in case of equal reflectivities of the two symmetric backscatterers (in case of different reflectivities, a reduced signal corresponding to the difference in reflectivities will be observed).

To overcome this problem, we use additional information inherently present in the acquired data to determine the position of symmetric real structure terms (as opposed to mirror terms). For this purpose we calculate the inverse Fourier transforms of the two acquired real-valued spectral intensities and obtain their modulus. We will show that the difference between these signals indicates the position of symmetric real structure terms. (In the following discussion, we will omit the DC and autocorrelation terms since they are not affected by the 90° phase shift and it is therefore evident that, after subtraction of the two Fourier transformed signals, the corresponding signal peaks will be removed). The two modulus signals (without DC and AC) M₁ and M₂ are: and the absolute value of their difference: contains information on the location of symmetric sample structure terms. Two cases have to be considered:

To ensure that the reference arm power is near the saturation limit of the linescan camera (thus providing maximum sensitivity), a variable neutral density filter was added to the reference arm. After the beams are recombined at the NPBS they are coupled into a single mode polarization maintaining fiber (PMF) and guided to a spectrometer. With a half wave plate in front of the input fiber coupler the polarization state is aligned parallel to one of the birefringent axes of the PMF to excite only one polarization mode in the PMF, thus propagating a single plane parallel polarization state to the spectrometer. The light emitted from the fiber tip was collimated by a fiber coupler with a focal length of 50 mm. In front of the spectrometer we added another half wave plate to adjust the linear output of the PMF to the efficiency maximum of the diffraction grating.

The spectrometer consists of a reflection grating (1200 lines/mm), a camera lens with a focal length of 200 mm, and a 2048 element line scan CCD camera with a pixel size of 14 × 14 μm² (Atmel Aviiva M2 CL 2014). The maximum line rate of the camera is 29 kHz and via camera link and a high speed frame grabber card (PCI 1428 National Instruments) data could be transferred continuously to a personal computer. The resolution of the camera is 12 bit per pixel. Since a spectrometer collects data as a function of wavelength, but the Fourier transform relationship is between time (distance) and frequency (wavenumber), the wavelength dependent data are re-sampled by linear interpolation to be equidistant in frequency space [10].

We used an object lens L1 with a focal length of 80 mm and a beam diameter of 1.5 mm, providing a transversal resolution of 55 μm and a depth of focus of 5.7 mm. This large depth of focus was necessary to image the whole anterior chamber with approximately constant transverse resolution, independent of imaging depth, because in SD-OCT dynamic focusing is not possible.

SD-OCT has the advantage that no moving parts are necessary to obtain a depth scan. However, the achievable depth range Δz_max is given by the Nyquist limit of spectral resolution and is fixed by the spectrometer settings; i.e.: it depends on the wavelength resolution δλ of the spectrometer and the center wavelength λ₀ of the light source [9, 10]. In normal SD-OCT systems Δz_max = λ₀²/(4δλ). In a full range complex SD-OCT system, this range is doubled. The measured depth range of our system is 2.95 mm with standard SD-OCT and 5.9 mm with complex SD-OCT, which is in good agreement with the design range of 6 mm. The sensitivity (ratio between the signal of a perfectly reflecting mirror and mean noise) of our system was 98 dB with an integration time of 100 μs and a power of 1.1 mW onto the cornea. This provided corneal images with a signal-to-noise ratio (SNR, defined as the ratio between the strongest signal obtained from the cornea and the mean noise) of 39 dB.

Figure 2c shows signal ΔS = S₁ – S₂. The DC term, the mirror image, and the fixed pattern noise terms are removed. The signal ΔS⁺ displaying the object structure is obtained after multiplication of ΔS⁺ by the Heaviside step function that eliminates the negative peak (fig. 2d). From fig. 2c one can easily imagine that symmetric structure terms (i.e., positive and negative peaks at the same abscissa position) would cancel each other.

To test the capability of our method to reconstruct the symmetric structure terms and thus be used to image scattering tissue with high speed in vivo, we imaged the human anterior chamber of a healthy volunteer. With a total of 1600 spectra per image (800 complex spectra equal to 800 full depth A-lines) the total recording time for one image was 160 ms. Figure 3 shows the results of these measurements.

After initial data processing steps comprising removement of the DC signal and fixed pattern noise (including a signal caused probably by internal reflection in the reference arm), and re-sampling the data to be equidistant in frequency space, fig. 3a was calculated by inverse Fourier transforming the real valued spectra I(ν). The overlapping mirror images are clearly visible. Furthermore residual DC and fixed pattern noise terms (straight vertical lines) can be seen. Hence it is difficult to recognize morphological details. In a next step the original two-frame technique [14] was applied, yielding signal ΔS⁺(τ), cf. fig. 3b. The mirror image is removed and the DC and fixed pattern noise terms are reduced. However, some shadow-like artifacts can be observed at positions where backscattering sites in the sample are symmetric with respect to the zero position (cf. areas where the real image of the cornea overlaps with its mirror image in fig. 3a; due to the statistical distribution of backscattering sites and intensities, there is no complete extinction of the corneal signals in these areas).

The modified two-frame technique presented in this paper solves a problem of the original version of the algorithm, the suppression of symmetric structure terms. Although the original version provides images that can be used to derive the shape of the different structures of the anterior chamber and to determine the boundaries of, e.g., iris and cornea with great accuracy (cf. fig. 3b), the shadows observed at object structures symmetric to the zero position degrade the image quality and prevent a quantitative determination of signal intensities. These image artifacts are likely to be more critical in strongly scattering tissue. The new, enhanced algorithm solves this problem. The quantity ΔM(τ) is used to determine the locations of symmetric structure terms. It should be pointed out that the measured value of ΔM(τ) cannot be used directly as a measure of the signal magnitude of these symmetric terms. The reason is that the measured value of ΔM deviates from the value provided by eq. (8) by a factor that depends on the initial phase offset of the cosine terms of which I(ν) consists (the phase offset was assumed to be zero in the derivation of eq.(8)). This factor varies between 1 (zero phase offset) and 0 (45° phase offset). Therefore, ΔM(τ) is just used to locate the positions of the symmetric structure terms; their values are obtained by gating signal S₁(τ) with the positions where ΔM(τ) exceeds a threshold larger than noise. Only in rare cases, where the initial phase offset of a cosine term in the real-valued intensity signal I(ν) will be close to 45°, the method might fail (because then ΔM will be zero). However, given the random distribution of backscattering sites in scattering tissue, only a very small fraction of data points will be missed for this reason, and in a logarithmic intensity plot, as usual in OCT, this effect will probably be negligible.

Although the phase shift provided by the electro optic phase modulator has the required rectangular waveform, there are additional sources of phase instabilities that can degrade the results. Before we discuss the different sources of phase errors, we analyze their possible consequences.

Our derivations in section 2 were based on a phase shift Δ = 90° between alternate A-lines. Only in this case, the separation of structure and mirror terms in signals S₁(τ) and S₂(τ) will be perfect. If Δ deviates from 90°, signal S₁(τ) will contain residual mirror terms, and signal S₂(τ) will contain residual structure terms. However, these residual terms in S₁ and S₂ have no influence on the suppression of the mirror image in the actually displayed signal ΔS⁺(τ) (or F(τ)) (as long as the residual mirror term is not stronger than the real structure term). This can best be seen in an example where a large deviation from the ideal value of Δ = 90° is introduced. Figure 4 shows such an example, where Δ 40° (sample: attenuated mirror). From fig. 4a, which shows signals S₁ and S₂, it can be seen that the residual mirror term has ~ half of the amplitude of the real structure term. After calculating ΔS (fig. 4b), the real structure term is reduced in height (by a factor of ~ 2), and a corresponding negative mirror term is observed. In a final step, ΔS⁺ is calculated (fig. 4c). The negative mirror term is eliminated by the step function, the reduced structure term remains. This example shows that phase errors have no influence on the effectiveness of the algorithm to eliminate mirror terms. Instead, the effect of phase errors is to reduce the signal magnitude of real structure terms (because the residual mirror terms are subtracted). The consequence is a reduction of SNR. Simulations have shown that a phase error of 30° causes a SNR drop by 3 dB. This indicates that our algorithm is rather immune to small phase errors. If, on the other hand, very large phase errors occur (> 90°), real image and mirror image are reversed (cf. figure 5).

We now discuss possible sources of phase errors and the impact they might have on the results. One source of phase error can be interferometric jitter. However, since we use a very short integration time of 100 μs, this can easily be avoided. A second possible error source is polychromatic phase error. The phase shift introduced by the electro optic modulator depends on the wavelength. If a perfect 90° phase shift is introduced for the center wavelength λ₀, the shift will deviate from 90° for the other wavelengths of the broadband source. With λ₀ = 821 nm and δλ = 25 nm, as used in our setup, the polychromatic phase error within the FWHM bandwidth will not exceed 1.4°. This can be neglected in the light of the previous discussion.

Another source of error is a phase change due to the galvanometer scanner movement between the two A-scans necessary to compute the complex signal. This effect has to be divided into two different sources of phase error. The first phase change arises from the sample path length variation with varying scanner angle. The sample path length shows a quadratic dependence on the angle of the scanner [20]. An additional phase change occurs, if the beam incident upon the scanning mirror does not fall on its axis of rotation [21]. This path length change depends on the off pivot point distance and on the angle of the scanner. The corresponding phase instabilities can be quite large. The example shown in fig. 4 was taken from an experiment where the incident beam position on the scanner was not exactly the pivot point of the scanner.

To avoid SNR degradation, the incident beam onto the scanners has to be carefully adjusted, ensuring pivot point illumination. It is clear that the effect of scanner movement strongly depends on the scanner velocity and the integration time. Therefore measurements might be difficult to realize, where a fast scanner velocity is necessary (e. g. 3D measurements). Further quantitative investigations are necessary to look into this problem in more detail. One solution would be to drive the scanner with a stepwise waveform. However, up to now no scanners with an appropriate response time (due to the very short integration time used in modern SD-OCT systems) are available.

A further source of phase errors are involuntary movements of the subject. In case of large movements, signals similar to those in figure 5 can be observed. The flip of the image approximately at its middle position occurred due to an involuntary movement of the subject which caused a phase error larger than 90°. In this case, signals S₁ and S₂ are switched, which, after calculation of ΔS⁺, results in a flip of the image. This motion induced artifact must not be confused with the so called phase washout. A detailed description of this phenomenon can be found in [22].

The previous discussion has shown that some phase instabilities can be a problem with our technique. A possible solution to this problem will be to further decrease the integration time for one A-scan. Our camera allows integration times down to 35 μs. This should be possible without decreasing the signal to noise ratio, if the light power is increased. In our application, the light is focused on the cornea and divergent on the retina (divergence angle 19 mrad). This corresponds to viewing of an extended light source, for which the ANSI standards [23] allow an illuminating power of 7.6 mW for illumination times > 15 s in the case of the parameters (λ, α) used in our setup.