We report on a novel method combining achromatic complex FDOCT signal reconstruction with a common path and dual beam configuration. The complex signal reconstruction allows resolving the complex ambiguity of the Fourier transform and to enhance the achievable depth range by a factor of two. The dual beam configuration shares the property of high phase stability with common path FDOCT. This is of importance for a proper complex signal reconstruction and is in particular useful in combination with handheld probes such as in endoscopy and catheter applications. The advantage of the presented approach is the flexibility to choose arbitrarily positioned interfaces in the sample arm as reference together with the possibility to compensate for dispersion. The method and first experimental results are presented and its properties concerning SNR and dynamic range are discussed.
© 2007 Optical Society of America
Fourier domain optical coherence tomography (FDOCT) has nowadays reached large acceptance in the biomedical imaging community due to the sensitivity advantage together with the possibility of high resolution imaging at high acquisition speed [1-7]. Recent realizations based on swept source technology achieve unprecedented scan speeds of several 100kHz with high phase accuracy [8-10]. Still, drawbacks of FDOCT are the depth dependent sensitivity as well as the complex ambiguity of the FDOCT signal leading to disturbing mirror structures as well as maximum depth ranging restrictions. A potential candidate to remove those artifacts is heterodyne FDOCT, both for the spectrometer-based  as well as for the swept source modality [12-14].
Nevertheless, in particular spectrometer-based FDOCT needs high phase stability between successive spectra. Any phase noise due to sample motion or mechanical beam scanning will cause signal degradation as well as insufficient suppression of mirror terms. This will be especially critical for in-vivo measurements. Another source of phase instabilities are fiber-based setups in case of employing handheld scanners where moving the sample arm fiber introduces unwanted phase changes.
A solution to above problems is a common path configuration where sample and reference beam travel through the same fiber to the sample or most generally to an applicator. For the true common path concept a prominent sample arm reflection serves as reference signal in which case reference and sample field exhibit maximum relative phase stability. Particularly phase contrast schemes profit of the enhanced phase stability enabling highly sensitive optical path length variations [15-19]. The other common path variant is to have a separate reference arm by placing the interferometer into the hand piece or applicator, as was demonstrated by Tumlinson et al. with an endoscope configuration .
The concept of a common path with a prominent sample reflection as a reference captivates by its simplicity due to the fact that it does not need an extra interferometer. As already mentioned a prominent reflection (R1) situated close to the sample structure (R2) plays the role of the reference arm (see Fig. 1), resulting in a relative delay of 2Δz. Such a configuration presents extremely high phase stability; values down to 18pm for spectrometer-based  and 39pm for swept-source based  OCT systems have already been reported. However, not much flexibility is offered to the user since the reference reflector must always be close to the sample structure. Also beam scanning might be problematic if the probe scans not telecentrically in order to guarantee a stable reference reflection intensity. Usually a glass window may serve as a reference interface. Nevertheless, the thickness of the glass plate will reduce the achievable depth range apart from the possibility of ghost terms due to the reflections on both glass interfaces. Using the interface that is closer to the sample as reference might improve the situation but the drawback will still be a changing reference reflectivity and thus a changing OCT signal if the sample touches the interface. The only sensible application to profit from the extraordinary phase stability of such configuration seems to be coherent phase microscopy [15-17, 19].
The motivation of this work is to introduce a dual beam FDOCT variant that profits from the high phase stability of a common path configuration if used in conjunction with handheld applicators, without sacrificing measurement depth range, and keeping the flexibility for beam scanning as well as the possibility of dispersion balancing. In the following we present the method with a detailed discussion of signal-to-noise and dynamic range issues. Finally we demonstrate the feasibility to perform in-vivo measurements employing spectrometer-based heterodyne FDOCT.
2.1 Dual beam
A dual beam configuration is an extension of a common path setup presented in the previous paragraph. Instead of a single light beam travelling the common path to the reference (R1) and the sample (R2) as illustrated in Fig. 1, two beams delayed by an optical path length 2ΔzILS enter the common path and travel together to the reference and sample (see Fig. 2(a)). In this case, again, both reference and sample light share the same path and exhibit therefore high relative phase stability. This concept has been adapted for time domain OCT in particular for precise eye length measurements in order to remove artefacts due to axial proband motion [22, 23].
In the simplest case, a single reflecting sample surface and one reference reflector cause four light fields with relative respective delays. Depending on the optical distance Δz between reference R1 and sample R2 and the introduced delay Δz ILS within the interferometric light source (ILS) (see Fig. 2(a)), a perfect match between the two fields can be achieved, as illustrated in Fig. 2(a) for Δz ILS=Δz0. As a matter of fact, all light fields present within the unambiguous depth range of the Fourier domain system (spectrometer-based or swept source) are coherently summing up and contribute to the detected interference signal. This clearly has a strong adverse effect on the achievable system dynamic range. However, the potential of the dual beam configuration lies in the possibility to choose an arbitrarily distant interface in the common path as reference by matching the respective delay Δz ILS of the interferometric light source as illustrated in Fig. 2(b).
In the most general way, the intensity of the total optical field impinging on the camera array in case of a single reflecting sample surface can be written as:
with being the detected reference and sample light fields respectively, with I(r,s)R,S=IR,Sρ2r,s being the light intensity contributions at the detector and ρr being the amplitude reflectivity of R1 in Fig. 2 and ρs the sample amplitude reflectivity (R2 in Fig. 2), both accounting also for coupling losses, additional losses on optical elements and the diffraction grating efficiency. The upper indexes (r) or (s) indicate whether the contribution is coming from the reference beam IR or the sample beam IS of the ILS respectively. k stands for the wave number and zR,S are the integral optical path lengths travelled by the respective light fields. The shading in Eq. (1) visualizes the different contributions to the signal generation: the green shaded elements correspond to the four DC terms; the yellow elements are the complex conjugates to the ones on the bottom left side of the DC terms; the red shaded elements are zero if the reference surface is placed far away from the sample surface (see Fig. 2(b)) such that the coherence function becomes zero and no interference will occur anymore; for the same reason the blue shaded elements would vanish as well due to the matched delay ΔzILS≈Δz between the two fields ER and ES.
Hence one is finally left with the DC components as well as the actual sample-reference cross-correlation term in the lower left corner of Eq. (1) together with its complex conjugate. The delay Δz ILS can be used to adjust the position of the sample structure within the unambiguous depth range.
In case of the reference interface being close to the actual sample structure one encounters all terms given in Eq. (1). One could introduce a dispersion unbalance between the reference (R1) and the sample (R2) field, and place double the dispersion into the reference arm of the ILS. Different undesired cross correlation terms would then be attenuated since they experienced double or quadruple dispersion whereas the actual structure terms were dispersion corrected.
Nevertheless one still suffers from the complex conjugate mirror terms that lead to a reduced maximum system depth range and might obstruct the structure reconstruction.
2.2 Heterodyne dual beam
The concept of heterodyne spectrometer-based FDOCT was already discussed by Bachmann et al. : slight detuning of two acousto-optic frequency shifters in the reference and sample arm of the interferometer causes an achromatic beating signal of frequency Ω=|ωR-ωS| detected by the sensor. By quadrature detecting this timely varying signal the full complex signal can be reconstructed and the unambiguous depth range is doubled. For this purpose the detector is locked to four times the beating frequency, resulting in π/2 phase shifted copies of the time dependent interference signal components. The frequency-shifted light fields can be written as:
with ω0 being the light frequency and ω R,S the frequency shift induced by the acousto-optic frequency shifters. The resulting signal detected by the line scan camera therefore becomes, for the case where reference and sample are well separated (see Fig. 2(b)):
with Ψ containing all time-independent phase terms. Beside the additional DC terms I(s)R(k) and I(r)S(k), this signal is equal to a standard heterodyne FDOCT configuration and has the same properties with respect to the suppression of mirror terms. Dual beam heterodyne FDOCT therefore allows for displacing the actual sample structure along the full doubled depth range by adjusting the distance Δz ILS.
The DC and auto-correlation terms due to internal interferences between sample structure fields can be further eliminated using a differential complex signal reconstruction according to :
with being the complex reconstructed interference signal of two adjacent spectra recorded at an arbitrary time instance t0.
2.3 Sensitivity and dynamic range
Sensitivity and dynamic range (DR) are important issues in spectrometer-based FDOCT. In practice, the DR depends on the reference light power being set close to the saturation level of the detector in order to achieve maximum sensitivity. It is evident that the dual beam configuration will present smaller sensitivity than standard FDOCT due to the presence of a second strong DC signal I(r)S(k) not serving as reference signal for coherent amplification but reducing CCD dynamics. We would therefore like to comment more in detail on DR and sensitivity of the dual beam configuration as compared to the standard configuration in spectrometer-based FDOCT.
In §2.1 we defined the beam intensities in the ILS (see Fig. 2(a)) to be IR and IS respectively. The corresponding amount of generated photoelectrons [2, 5] is then NR,S(k)=I(r,s)R,S (k) β (k) Apixel with as the photon conversion factor with the reduced Planck constant ħ, the vacuum light speed c, τ the integration time of the camera, η(k) the detector quantum efficiency, ħkc the photon energy in vacuum, and Apixel the size of a detector pixel. We further express the total spectrally integrated number of photoelectrons as function of the spectral peak value as Ntot=αN(k0), with k0 being the center wave number where the detected spectrum is assumed to have its maximum. For a Gaussian spectrum with the spectral FWHM being imaged onto N/m pixels, i.e. Δk(n)FWHM=N/m, we have , where N is the total number of detector pixels, and m defines the ratio of N to the FWHM. In case of a rectangular spectrum αrect=N. According to Eq. (3) the signal term can be written as:
with Nref(k)≡β(k)IR (k)ρ2r Apixel and Nsample (k)≡β(k)IS (k)ρ2s Apixel. An additional assumption we make is that the presence of a reference surface in the sample arm does not influence the ratio of sample to reference reflectivity ρs/ρr significantly, which means that the transmittance of the reference surface is high. With the approximation ρ2r≫ρ2s, we consider only those fields for the DC term that are reflected at the reference interface R1:
where we define a load factor γ as the ratio between DC level and the pixel saturation level Nsat. This definition will be useful for our dynamic range discussion since the maximum sample signal will clearly depend on the remaining pixel capacity. We would further like to find the optimum ratio ξ between the ILS intensities IR and IS. With the definition IS≡ξIR and Eq. (6), the number of photoelectrons corresponding to the sample signal becomes:
The signal-to-noise ratio (SNR) can be defined as SNR=〈SOCT2〉/σ̂ 2, with 〈∙〉 being the time average, being the signal peak at the position z0=Δz-ΔzILS after Fourier Transform (FT) and the noise variance after FT. Following  the squared OCT signal reads 〈SOCT2〉=(αNAC(k0)/(2N))2. The noise variances before and after FT are related via 2=σ2/N. For shot-noise limited detection it can be expressed by the pixel-averaged total DC signal with Eq. (6) as 2≈(1/N) (αγNsat/N). Together with Eq. (7), the SNR in this case becomes:
We observe firstly that the SNR increases linearly with the load factor γ. Secondly, the SNR expression reaches a maximum for ξ=1, or IR=IS. In words, the two interferometer arms of the ILS should have the same intensity in order to achieve a maximum SNR in dual beam interferometry. This is an important conclusion which will facilitate the following comparison of dual beam to standard FDOCT.
Figure 3 shows in an intuitive way the signal contributions on camera pixel level at spectral position k0 with equal load factor γ (detected signal when sample light is blocked) where we assume the cosine in Eq. (3) to be 1. Since maximum SNR is achieved for both arms of the ILS at equal intensity (ξ=1) we can write IR=IS=I/2. The dotted region indicates the light intensity reflected by the reference surface R1 which does not contribute to coherent amplification – but still contributes to shot noise and burdens the sample with additional light power. Hence, the effective reference signal for dual beam is only half that of the standard configuration with equal noise floor which results in a decreased SNR. According to Fig. 3 the SNR can be expressed as:
However, the maximum SNR is the same for both configurations as it is limited by the saturation value of the camera pixel. This implies the relation for the maximum sample reflectivity assuming equal reference signal:
The sensitivity ∑ on the other hand is defined as the inverse of the smallest detectable sample reflectivity i.e. for SNR≡1. From Eq. (9) and with the same load factor γ for both configurations, we can write:
which is equivalent to a -6dB disadvantage in sensitivity for dual beam as compared to standard FDOCT. Together with Eq. (10) we can deduce the following relation:
i.e. the ratio between maximum and minimum sample reflectivity remains the same.
This relation leads us directly to the implications to DR which is defined as the ratio between the maximum to the minimum SNR. For a given reference intensity and load factor γ, the maximum SNR is achieved for the maximum sample reflectivity . Since the minimum SNR depends on the minimum sample reflectivity and considering Eq. (12), the DR will remain the same for dual beam and standard FDOCT:
One could be tempted to increase SNR by increasing the load factor γ (cf. Eq. (8)). However, changing γ from e.g. 0.7 to 0.8 increases the SNR by less than +1dB while decreasing the DR already by -4dB (cf. Eq. (13)). The situation becomes even worse for larger load factors.
A Mach-Zehnder like interferometer setup as shown in Fig. 4 was built. The spectrometer consists of a collimator with a focal length of 80mm, a transmission diffraction grating (1200lines/mm), an objective (CL) with a focal length of 135mm and a line scan camera (ATMEL AVIIVA M2, 2048pixel, 12bit) driven at 17.4kHz line rate. The light source (LS) is a Ti:Sapphire laser with center wavelength at 800nm and a bandwidth (FWHM) of 130nm. The effectively by the spectrometer detected bandwidth (FWHM) is 90nm due to spectral transmittance losses along the total system, i.e. coupling losses. The maximum depth range (after complex signal reconstruction) is 4mm and the axial resolution in air is 4µm. The signal drop-off along the depth range is approximately -7dB/mm with a sensitivity close to the zero delay of about 95dB with 2x1.1mW light power incident on the sample and a load factor γ of 0.8. Using Eq. (8) the theoretical sensitivity is calculated to be ∑dual≈101dB with ξ=1, α=800, γ=0.8, Nsat≈1.2·105 and ρr2≈1.4·10-3. The reference arm length can be adjusted by means of a translation stage (TS). Beam splitting and recombination is realized by a fiber coupler (FC) and a 50:50 beam splitter (BS) respectively.
The peculiarity of the proposed system is the light source module comprising an interferometer with two acousto-optic frequency shifters (AOFS) (AA Opto-Electronic SA with optical packaging by Cube Optics AG, ωR=2π·100MHz, ωS=2π·100MHz+4.35kHz). Since our acousto-optic elements are based on a birefringent crystal (tellurium dioxide (TeO2)) light has to enter these devices in a controlled, linear polarization state. In addition, in order to maximize interference contrast, the light field states at the common path input have to be oriented accordingly, employing polarization control paddles (PC) (see Fig. 4). The sample is finally illuminated by two frequency shifted copies of the original light field. The dispersion compensation (Disp) in the reference arm of the ILS pre-compensates for the additional dispersion induced by the wedge plate and the lens f2 of the hand piece.
The hand piece consists of a scanning unit based on a single mirror tip/tilt scanner (X/Y scan) . It is placed in the back focal plane of lens f2, allowing for two-dimensional transverse scanning of the sample. The glass wedge with a deviation angle of 2° (ϑ≈3.1°) is used in order to create a single well defined reference reflex at the front surface. Such a configuration can be seen as auto-collimation and the reference signal intensity is adjusted by slightly tilting the glass wedge. The theoretical beam width on the sample is 26.5µm (1/e2-intensity) with a Rayleigh range of 1.3mm and is defined by the ratio of the focal lengths (f1=15mm, f2=75mm) used in the handheld probe and the mode field diameter of the one meter single mode (SM) fiber. With a transverse scanning speed across the sample of 40mm/s the resulting transverse over-sampling is approximately 12x.
In order to properly reconstruct the complex signal as described in §2.2, special attention has to be paid to the synchronization of the camera with the resulting beating frequency (cf. inlet A of Fig. 4). The complex differential reconstruction needs two pairs of complex reconstructed spectra Ĩ(k) (thus in total four by 90° retarded acquisitions) which is realized by externally triggering the camera frame grabber (see (b) in inlet A of Fig. 4). Frequency shifters and trigger signal generators are linked and synchronized via a common 10MHz time base. The exposure time τ (see (c) in inlet A of Fig. 4) is 45µs. The complex spectra Ĩ(k) (see §2.2) are finally reconstructed using two successively recorded spectra as indicated by (d) in inlet A of Fig. 4.
With the extension shown in inlet B of Fig. 4 we had the flexibility to compare the phase stability of the following three configurations:
- Common path: By blocking the dual beam arm and placing a thin glass plate instead of the mirror in inlet B.
- Dual beam: By blocking the external reference arm (inlet B). For phase stability measurement a mirror was used as sample without X/Y scanner.
- Standard: Cross-correlation between mirror of inlet B and mirror at sample position of the dual beam arm (top right corner in Fig. 4).
Dual beam and standard FDOCT could be measured simultaneously by adjusting the two respective reference signals R1 (for dual beam) and mirror of inlet B (for standard) to γ≈0.4 each.
4. Results and discussion
In order to demonstrate the advantage of dual beam versus standard FDOCT in terms of phase stability, the previously described three configurations were used (Fig. 4). For each configuration the SNR of the signal peak was adjusted to approximately 26.5dB. The phase fluctuations at the signal peak position were measured while the system was unperturbed and perturbed respectively. The perturbation consisted in bending and moving the sample arm fiber guiding during measurement. The resulting standard deviations of the phase fluctuations σ Δϕ are shown in Table 1. The measured values are close to shot noise limited phase stability defined by the relation . For the perturbed case of the standard FDOCT configuration no clear value could be measured since the phase fluctuations are strongly varying (see Fig. 5).
The phase signals were extracted after FFT at the mean signal peak positions. By touching and bending the SM fiber, the signal of the standard setup is heavily perturbed, even resulting in up to 100µm signal peak shift in depth. This displacement is caused by a change in optical path length due to a stress-induced change in refractive index. Both signal peaks were again adjusted to approximately the same SNR≈26.5dB. The strong fluctuations of the standard signal peak intensity are mainly due to fringe washout and stress-induced polarization state changes in the perturbed fiber, resulting in reduced interference fringe contrast. These measurements proof clearly the advantage of dual beam FDOCT over standard FDOCT for employing fiberized handheld applicators.
In the following we demonstrate the feasibility of the introduced dual beam FDOCT principle to perform in-vivo imaging of human skin on the finger tip of a male subject. For this task we employed a fiberized handheld probe with a single mirror tip/tilt scanner. The reference reflex was realized by placing a wedge glass plate into the collimated beam before the scanner, generating a stable reference light intensity. Attention had to be paid to the positioning of the beam with respect to the scanner pivot since slight misalignment introduces unwanted phase shifts during scanning. The total distance Δz between reference and sample was 200mm which had to be pre-compensated by adjusting Δz ILS within the ILS. The recorded tomograms consist of 1100 depth scans each, covering a transverse range of 2.5mm. The measurements had been performed by first adjusting the focal plane to the zero delay using a mirror and then placing the sample structure across this position.
The recorded signal was reconstructed following the differential complex scheme from §2.2 (see Fig. 6(c)). The dynamic range within the tomogram is about 40dB with a system sensitivity of 95dB. One observes that the DC term is strongly suppressed as compared to its original amplitude (directly after Fast Fourier Transform (FFT)). If we define the DC suppression ratio as DCsuppress≡DC 2×2/DCFFT, with DC2×2 being the DC signal value in Fig. 6(c) and DCFFT the one in Fig. 6(a) we have DCsuppress=-47dB. The fact that the DC term is not fully suppressed is explained by the presence of slight intensity fluctuations throughout the tomogram. These fluctuations were measured to be in the kHz-range with a standard deviation of 0.33%.
In Fig. 6 we compare the differential complex reconstruction technique (Eq. (4)) (Fig. 6(c)) to the standard complex reconstruction based on two adjacent lines Ĩ(k,t0) with background correction (Fig. 6(d)). The background for the tomogram is obtained by averaging of all transversally recorded spectra. The brightness of the tomograms was adjusted by first normalising the intensity to that of a common bright structure (sweat gland) and then setting the minimum of the intensity scale bar to the calculated noise floor. The maximum scale bar value is given by the highest intensity in the tomogram. This results in a linear gray scale spanning over a DR of 28.5dB for standard complex reconstruction and 31dB for the differential complex reconstruction. As expected, the SNR for the differential complex method is better by approximately +3dB.as compared to the standard complex reconstruction. It can also be observed that DC suppression works slightly better for the differential complex approach (Fig. 6(c) and (d))
The tomogram in Fig. 6(a) shows the measured data with standard reconstruction employing straight forward FFT reconstruction. One can clearly see that the structure had been measured across the zero delay due to the presence of mirror structures. Figure 6(b) finally shows a standard reconstruction as in Fig. 6(a) but with background subtraction in post-processing. Again, a slight DC term remains together with sample structure obstructing mirror terms.
Investigating the mirror term suppression within different 2D tomograms for bright scattering structures, the suppression ratio can be measured to be better than -15dB. Higher over-sampling would increase the suppression ratio as one remains tighter within the speckle pattern .
Figure 7(a) shows a 3D data set of a human finger tip, consisting of 66 2D tomograms and reconstructed using the differential complex scheme. The total recording time was 4.5s. By performing edge detection on each individual 2D tomogram, the user has access e.g. to a thickness map of the epidermis as illustrated in Fig. 7(b). The red frame in Fig. 7(a) indicates the position of the 2D tomograms presented in Fig. 6 within the 3D data cube. The rudimentary DC peak at the zero-delay, visible in Fig. 6(c), was removed from Fig. 7(a) by first setting it to zero and afterwards interpolating the intensities in post processing.
The demonstrated principle can easily be adapted for endoscopic OCT as well as for common path ophthalmic imaging. In particular the phase stability can be enhanced by placing the reference to one of the scanning prism interfaces in an endoscope, or by using actually a sample reflection such as at the cornea front surface as reference . In the latter case one could achieve complete axial proband motion suppression which is especially interesting for functional imaging extensions such as Doppler FDOCT [27-30]. Still, using dual beam FDOCT in conjunction with illumination power limited applications such as in ophthalmology one would have a -6dB sensitivity disadvantage which cannot be compensated by simply increasing illumination power.
Finally, one should mention that the principle of dual-beam heterodyne FDOCT can equally be used for swept source FDOCT. The latter would have the advantage of larger dynamic range, as well as the high A-scan rates of modern swept-sources.
For the first time dual beam FDOCT was presented allowing for phase sensitive measurements even through long probing fibers and employing handheld probes. A detailed theoretical analysis of the sensitivity and the dynamic range capabilities of a dual beam configuration was presented showing a -6dB disadvantage in sensitivity with an equal dynamic range as compared to standard FDOCT. The concept was tested on a dual beam heterodyne FDOCT setup employing a handheld probe with a single mirror tip/tilt scanner and performing in-vivo measurements on human skin.
The authors acknowledge technical support from Cube Optics AG and thank the Swiss National Fond (project 205321-109704/1) and the Swiss Academy of Engineering Sciences (SATW project TK 01/06) for financial support.
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