A high speed spectral domain optical coherence tomography based on the spatial sinusoidal phase modulation for the elimination of complex-conjugate artifact is presented, where sinusoidal phase modulation of reference arm (M scan) and transverse scanning of sample arm (B scan) are performed simultaneously (sinusoidal B-M method). Herein, the linear phase modulation of the reference arm in conventional linear B-M method is modified to sinusoidal phase modulation. The proposed sinusoidal B-M method relaxes the requirements on the phase-shifting mechanical system and avoids sensitivity fall-off along the transverse direction in contrast to the linear B-M method. A criterion for the relation between transverse over-sampling factor and modulation frequency for optimal complex conjugate rejection is deduced and verified by experiments. Under this criterion, the complex spectral interferogram is reconstructed by harmonic analysis and digital synchronous demodulation. Double imaging depth range on fresh shrimp at A-scan rate of 10 kHz with complex conjugate rejection ratio up to 45dB is achieved.
©2009 Optical Society of America
Optical coherence tomography (OCT) is a non-invasive, non-contact imaging modality that uses coherent gating to obtain high-resolution cross-sectional images of tissue microstructure . Compared with conventional time domain OCT which is based on a scanning optical delay line, spectral domain OCT (SD-OCT) splits frequency components of interference pattern by a grating and collects all of these components simultaneously by a line-scan CCD. Due to the Fourier relation (Wiener-Khintchine theorem between the auto correlation and the spectral power density of the light source), one dimensional in-depth reflectivity profile (A-scan) can be determined by a Fourier-transform of the acquired spectral interferogram [2,3]. However, this Fourier-transform of real-valued spectral data introduces complex conjugate ambiguity and hence the mirror image . In practice, the measured sample has to be positioned at one side of the zero optical path difference (OPD) position to avoid the overlapping mirror images. Therefore, only one half of the depth range is available for OCT imaging.
To restore the full depth range for OCT imaging, many approaches have been proposed to reconstruct a complex spectral interferogram to resolve complex conjugate ambiguity. Phase-shifting method based on multiple spectral interferograms with different phase shifts was used to reconstruct the complex interferogram [5,6], but errors in phase shifts due to departure from nominal phase step as well as polychromatic phase errors result in unsuppressed artifacts. Moreover, a modified phase-shifting method based on the integrating-bucket of sinusoidal phase modulation of the reference arm is also proposed , however a complex quadrature projection algorithm is required to compensate for the chromaticity and any phase variations between integrating bucket acquisitions. Simultaneous detection of the real and imaginary parts of the complex interferogram was proposed either through 3 by 3 fiber coupler  or through polarization-based optical demodulation . However, wavelength-independent splitting ratio and phase shift in the fiber coupler and no birefringence existing in the measured sample should be assumed. Heterodyne detection approach based on acousto-optic frequency shifter  or electro-optic phase modulator  is effective in mirror image suppression, but increases the complexity and the cost of the system. Harmonic lock-in detection based on sinusoidal phase-modulation is proposed to obtain the quadrature components [12,13]. However, the speed for spectral interferogram obtained by lock-in amplifier and scanning monochromator is very slow. Recently, a dispersion-encoded approach based on iterative algorithm is proposed to obtain the full-range image but with more computing time .
Linear B-M method for full depth range OCT imaging is previously proposed and implemented in SD-OCT, where linear phase modulation of the reference arm (M scan) and transverse scanning of sample arm (B scan) are performed simultaneously . The linear B-M method can be recognized as an extension of the conventional phase-shifting method, but with more robust data processing and enhanced suppression of artifacts from polychromatic phase errors. Phase-shift on reference arm required for successive A-scans in the linear B-M method is π/2. This phase-shifting introduces a linear carrier to the detected interference signal versus transversal position. Complex spectral interferograms can be obtained by band-pass filtering or Hilbert transformation along transversal direction. Methods implemented to introduce linear spatial carrier include reference mirror mounted on the PZT driven by saw-tooth or triangular waveforms , piezoelectric fiber stretcher (PFS)  in the reference arm, beam offset at the scanning mirror in the sample arm . However, the phase-shift of π/2 introduced between successive A-scans in the linear B-M method results in accumulated OPD for case with large transverse scanning range. The OPD between reference arm and sample arm caused by linear phase modulation results in sensitivity fall-off along transverse direction, contradicting our pursued target of increasing the imaging depth range. Moreover, linear phase modulation places high demand on the phase-shifting actuator, typically a PZT, which is mainly limited by its slow linear response. An alternative to linear phase modulation is sinusoidal phase modulation, in which the actuator is driven by a sinusoidal waveform. High frequency sinusoidal phase modulation by sine waveform is easier to realize than linear phase modulation by ramp or saw tooth waveforms.
In this paper, we propose a sinusoidal B-M method to eliminate the complex-conjugate artifact in SD-OCT. By sinusoidal phase modulation instead of linear phase modulation, the reference mirror mounted on the PZT vibrates only with small amplitude independent of transverse scanning range and the sensitivity fall-off along the transverse direction in the linear B-M method is thus avoided. In order to obtain optimal complex conjugate rejection, a criterion for the relation between transverse over-sampling factor and modulation frequency is deduced and verified by experiments. And the complex spectral interferogram can be reconstructed by harmonic analysis and digital synchronous demodulation. Under this criterion, the sinusoidal B-M method can achieve double imaging depth range with complex conjugate rejection ratio of 45dB.
2.1. Sinusoidal B-M method
The time sequence for simultaneously B-M scans of the proposed sinusoidal B-M method is illustrated in Fig. 1 , where time sequences of conventional phase-shifting method and linear B-M method are also depicted for comparison and the amplitudes of modulation are plotted in the same scale. In the sinusoidal B-M method, both M-scan and B-scan are executed at the same time. The sinusoidal phase-shifting on the reference arm introduces spatial carrier to the detected interference signal due to the transverse scanning performed simultaneously. The sinusoidal B-M method relaxes the requirements on the phase-shifting actuator as in the phase-shifting method and the linear B-M method. By sinusoidal phase modulation instead of saw-tooth phase modulation as in the conventional phase-shifting method, modulation frequency can be improved and enhanced suppression of artifacts from polychromatic phase errors is feasible. By sinusoidal phase modulation instead of linear phase modulation as in the linear B-M method, the reference mirror mounted on the PZT can vibrate at small amplitude at higher frequency independent of transverse scanning range.
2.2. Criterion for transverse over-sampling factor and modulation frequency
SD-OCT is basically an interferometer, where recombined light from reference and sample arms is spectrally separated, detected in a spectrometer and then reconstructed into a depth reflectivity profile corresponding to a lateral position. The detected interference signal based on the sinusoidal B-M method can be expressed as
In the proposed sinusoidal B-M method, the transverse over-sampling factor and the modulation frequency are of particular interest for the complex reconstruction and needs to be deliberately considered. As to a B-scan image consists of N A-scans covering a transverse range of X, there are N sampling points in the transverse direction of the image and the sampling interval is X/N. So the maximum retrievable spatial frequency according to Nyquist theory is νx = N/2X. For coherent imaging, we assume a characteristic width of the detected spatial frequency spectrum which is determined by the inverse speckle size as νσ = 1/σ . The speckle size can be written as the transverse resolution, where in case of Gaussian optics is expressed as σ = 4λf/πd with λ being the central wavelength, f representing the focal length of the object lens in sample arm, and d denoting the collimated beam diameter at the aperture of the object lens. If the phase change between successive A-scans is ΔΦ, the spatial frequency spectrum of the detected signal is shifted by ± νΦ = ± (ΔΦ/2π)(N/X). If νΦ>>νσ and νΦ<νx, the spatial frequency spectrum can be separated completely, otherwise, the negative part of the spatial spectrum will leak into the positive part leading to a overlapping region and hence unresolved complex ambiguity.
In the linear B-M method where saw-tooth waveform shown in Fig. 2(a) is implemented for phase modulation, the introduced phase term is ϕ(t) = 2πfmt, where fm is the modulation frequency. Corresponding phase shift ΔΦ between successive A-scans is 2πfmT, here T represents the time interval between successive A-scans, i.e. the integration time of line-scan CCD in SD-OCT system. The resulting spatial carrier is νΦ = fmT(N/X), then the condition for complete separation of spatial spectrum of sample as shown in Fig. 2(b) isEqs. (2) and (3) can be combined to beEq. (4) in the linear B-M method is deduced in previously published papers [17,18]. On the other hand, in the sinusoidal B-M method where sinusoidal waveform shown in Fig. 3(a) is implemented for phase modulation, the introduced phase term is ϕ(t) = am(ω)sin2πfmt, where am(ω) is the phase-modulation amplitude, fm is the modulation frequency. Then Eq. (1) can be expressed with respect to time t and expanded as:Eq. (5), the sinusoidal phase modulation can be viewed as the linear summation of different harmonics terms weighted by Bessel functions. Corresponding spatial frequency spectrum along the transverse direction of the sample is illustrated in Fig. 3(b). Since the spatial frequency interval between adjacent harmonics in sinusoidal B-M method is νΦ, it’s evident that frequency separation between harmonics can be guaranteed if separation between negative and positive first-order harmonic pairs is assured, i.e. νΦ>>νσ. Thus, similar to the criterion described by Eq. (4) in the linear B-M method, the modulation frequency in sinusoidal B-M method must obey the following criterion:Eq. (6), which means that the transverse step must be small enough for effective artifact removal.
2.3. Digital synchronous demodulation
Once the frequency of the sinusoidal phase modulation is within the range described by inequality of Eq. (6), the first and second harmonics in Eq. (5) can be extracted by digital synchronous demodulation. To extract the first harmonic term, the spectral interference signal at each spectral point is multiplied by sin2πfmt:Eq. (7), it’s evident that the first harmonic term can be considered as the low-frequency term with respect to t. Then the resulting spectral interference signal at each spectral point is integrated over the interval of nT to obtain the first harmonic term, here T is the period of modulation signal which equals to 1/fm, n is the number of the modulation period. Larger n is helpful in increasing the signal-to-noise ratio by averaging the signal over multiple modulation periods. Similarly, the second and third harmonic terms can be extracted by multiplying the spectral interference signal at each spectral point by sin(2 × 2πfmt) and sin(3 × 2πfmt), respectively. Hence, the extracted harmonic terms are:Eq. (11):
3. Experiments and results
A Superluminescent diode (SLD 371-HP, Superlum Diodes Ltd) with a central wavelength of 835 nm and FWHM bandwidth of 45 nm, corresponding to a coherence length of 6.8 μm, is used to illuminate the system. A polarization independent fiber isolator is placed immediately after the light source to avoid light reflection back to the light source. Then the light is split into the sample and reference arm respectively by a 50/50 fiber coupler. The reference mirror in mounted on a PZT stage (Physik Instrumente) driven by sinusoidal signal. A galvanometer-mounted (6215H, Cambridge Technology) mirror is used for transverse scanning. The light illuminates the sample through a focusing lens with a focal length of 75mm. Light returning from the reference and sample arms are recombined in the fiber coupler and the output interference spectra is detected by a custom-built spectrometer consisting a 60 mm focal length achromatic collimating lens (OZ optics), a 1200lines/mm transmission grating (Wasatch Photonics), and a 150 mm focal length achromatic lens (Edmund optics). The dispersed spectra are focused onto a line-scan CCD camera (ATMEL AVIIVA SM2) with a maximum data transfer rate of 60 MHz, consisting of 2048 pixels, with each pixel at 14 μm by 14 μm in size and 12-bit in digital depth. The spectrometer is designed to measure a wavelength range of about 138 nm centered at 835 nm with spectral resolution of 0.0674 nm, yielding an axial imaging range of 2.56 mm in air. A variable neutral density filter is inserted in the reference arm for light attenuation in order that the maximum intensity on the CCD pixels reaches about 90% of its saturation value. The remaining 10% of the dynamic range (~400 levels) is available to capture the modulation of the spectrum. The polarization controllers are used to maximize the interference fringe visibility. The spectral data are transferred to a computer via a high-speed frame grabber board (PCIe-1430, National Instruments) for data processing. From the sinusoidal phase modulated spectral interferograms, reconstruction of the complex spectral interferograms for doubling of imaging depth range can be realized by an algorithm based on the harmonics analysis and digital synchronous demodulation. The block diagram of the processing procedure in the sinusoidal B-M method is illustrated in Fig. 5 .
3.1 Calculation of the scaling coefficient β
From Eq. (11), it can be seen that the scaling coefficient β is essential for reconstruction of the complex spectral interferograms. Actually, β is dependent on the peak-to-peak voltage amplitude (Vpp) of the sinusoidal waveform applied to PZT and the wavelength distribution on CCD. In order to determine β, firstly the wavelength distribution on CCD in the spectrometer is firstly determined by a commercial available mercury argon lamp which is illustrated in Fig. 6(a) . Herein, seven characteristic spectral lines from the lamp are dispersed and then detected by the CCD in the spectrometer. The indexed pixel numbers of all characteristic spectral lines are recorded and a third-order polynomial fitting is performed to them. It can be seen that wavelength versus CCD pixel number is not linear due to residual aberrations and perhaps misalignment of the spectrometer. Then spectral interferograms are obtained with a mirror used as the sample when the integration time of CCD is set to be 100 µs (corresponding A-scan rate is 10 KHz), the initial phase and the frequency of sinusoidal signal is 0 and 1250 Hz, respectively. With a stationary sample arm, a 2048 (pixel) × 1024 (point) 2D spectral interferogram is acquired for different Vpp of the sinusoidal signal, where the two dimensions of the matrix can be specified as wavelength axis and time axis, respectively. Then the 2D spectral interferograms are digitally 1D Fourier transformed along the time axis to retrieve the harmonic terms. After harmonics analysis of the acquired 2D spectral interferogram with different Vpp, we find that when the Vpp is set to be 1.0 V, the power spectrum of the time dependent spectral interferogram is mainly concentrated in the first- and second-order harmonic terms which are illustrated in Fig. 6(b), herein the wavelength is 849.7 nm. Since the first- and second-order harmonic terms are used for complex spectral interferogram reconstruction, higher complex conjugate rejection ratio can be realized due to the better utilization of total power under this condition. Thus, the Vpp of the sinusoidal signal applied to PZT is set to be 1.0 V for following imaging on biological sample.
Then the ratio of the amplitude of the first- and third-order harmonic terms at different wavelength can be calculated from Eq. (10) which is illustrated in Fig. 7(a) . In order to remove the unwanted phase noise during the sinusoidal phase modulation, the calculated ratio versus wavelength is low-pass filtered to obtain the smoothed curve. Finally, an iterative procedure is performed upon the smoothed curve (ratio(ω)) within wavelength range from 805nm to 885nm to determine the optimum am(ω) by minimizing the least square differences ΔU, i.e.,Fig. 7(b), herein the am 0 is about 2.19. It should be mentioned that the fitting curve and corresponding am(ω) are extended to the wavelength range from about 770nm to 910nm covering the whole pixel range on the CCD array. Such extension makes no difference because of the negligible contribution from outside wavelength range from 805nm to 885nm.
The calculated am(ω) under the least square fitting and the corresponding scaling coefficient β according to Eq. (9) are illustrated in Fig. 8(a) and 8(b), respectively. Since the phase modulation is achieved by dithering the reference mirror, the modulation amplitude am(ω) is related to the displacement of the PZT by , here d is the displacement amplitude of the PZT movement, c is the light velocity. Thus, from the calculated modulation amplitude am 0 at central wavelength, d can be calculated by. The displacement amplitude d is calculated to be about 145.7nm irrespective of transverse scanning range, which is much smaller than the linear B-M method where the PZT is typical driven by a sawtooth wave with amplitude of 25 µm . Thus, with the sinusoidal B-M method, benefits of low requirement on PZT and the avoidance of sensitivity fall-off along transverse direction in linear B-M method are achieved. However, the disadvantage of sinusoidal linear B-M method in contrast to linear B-M method is the complicated procedure for complex reconstruction and less efficiency due to several convolved copies of the real and imaginary frequencies. Finally, with the scaling coefficient β, the complex spectral interferograms can be constructed by the real and image parts of the complex spectral interferogram, i. e. the second and first harmonic terms.
3.2 Determination of the transverse over-sampling factor and modulation frequency
For optimal complex conjugate rejection, the introduced spatial carrier frequency must be high enough to separate the positive and negative parts of the spatial frequency spectrum of the sample, as well as the spatial frequency spectrum located at adjacent harmonic terms, as shown in Fig. 3(b). In order to verify the criterion described by the inequality of Eq. (6), the spatial frequency spectrum along the transverse direction of the sample with various modulation frequency of the sinusoidal waveform and various transverse over-sampling factor are obtained. There are two data sets from the experiments on a fresh shrimp, namely, a fixed modulation frequency of the sinusoidal waveform with changing transverse over-sampling factor and a fixed transverse over-sampling factor with changing modulation frequency of the sinusoidal waveform. Herein, the integration time of CCD is set to be 100 µs and the focal length of the focusing lens is 75 mm for both cases.
Firstly, the transverse step size is varied by changing of the transverse scanning range at fixed number of A-scans (2000), resulting transverse over-sampling factor of 4, 18 and 32, respectively. The corresponding results of the transverse Fourier transform with respect to time t are illustrated in Figs. 9(a) -9(c). It can be seen that with a fixed modulation frequency, larger transverse over-sampling factor is better for separation of the transverse spatial spectrum. Secondly, the transverse over-sampling factor is fixed at 16, and the modulation frequency of the sinusoidal waveform is set to be 500Hz, 800Hz and 1250Hz, respectively. The results of the transverse Fourier transform with respect to time t are illustrated in Figs. 9(d)-9(f). It is clear that with higher modulation frequency, the transverse spatial spectrum can be well separated without overlap, which is helpful for complex conjugate rejection.
3.3 Imaging on biological sample
In order to evaluate the performance of the system implemented with the proposed sinusoidal B-M method for in vivo imaging, the dorsal part of a fresh shrimp is used as the biological sample. The integration time of the CCD is set to be 100µs (corresponding A-scan rate is 10 KHz), and the Vpp and the frequency of sinusoidal signal is 1.0 V and 1250 Hz, respectively, which means that exactly eight spectral interferograms are collected during one phase modulation period. In order to meet the condition described by the inequality of Eq. (6) and obtain the maximum complex conjugate rejection ratio, the transverse over-sampling factor is firstly chosen to be 32. Thus in the digital synchronous demodulation procedure, integration is performed over 32 data points i.e. 4 phase modulation periods. With the extracted first and second harmonic terms and the pre-calculated scaling coefficient β demonstrated in Fig. 8(b), the complex spectral interferograms can be constructed whose inverse Fourier transform is free from complex conjugate ambiguous. Figure 10 shows the cross-sectional images of the fresh shrimp and corresponding A-scan signals (indicated by arrows) obtained by conventional SD-OCT system and full range complex SD-OCT system using the proposed method with the transverse over-sampling factor of 32, respectively. The image size is about 2mm × 4.5mm formed by 4800 A-scans. And the resulting typical complex conjugate rejection ratio is about 45 dB. With the transverse over-sampling factor of 8 comparable to that adopted in the linear B-M method [17,18], the dorsal part of the fresh shrimp is also imaged with the same reconstruction procedure except that the digital synchronous demodulation is performed only within one modulation period. The results are demonstrated in Fig. 11 with the image size of 2mm × 4.5mm formed by 1200 A-scans. A typical complex conjugate rejection ratio of about 41dB is achieved. From the imaging results, it can be seen that with the conventional reconstruction method, the structure of the sample is deteriorated by the overlapping mirror image. By use of the proposed method, the structure of the sample is correctly reconstructed and the mirror image, DC and autocorrelation terms are almost completely suppressed. With the over-sampling factor of 8 comparable to the that adopted in the linear B-M method, complex conjugate free image is obtained with satisfactory quality, while with increased over-sampling factor, the complex conjugate rejection ratio is increased from 41dB to 45dB due to the improved SNR of the digital synchronous demodulation.
We have developed a full range complex spectral domain optical coherence tomography system based on the spatial sinusoidal phase modulation. From the detected interference spectra under sinusoidal B-M scans, recovery of the complex spectra and expanding the accessible imaging depth range by a factor of 2 is realized by harmonics analysis and digital synchronous demodulation. A criterion for transverse over-sampling factor and modulation frequency of the sinusoidal waveform is deduced and confirmed for optimal complex conjugate rejection. The feasibility of the system implemented with the proposed sinusoidal B-M method is assured by in vivo imaging of dorsal part of the fresh shrimp at 10 KHz A-scan rate with different transverse over-sampling factor. The complex conjugate, DC, and autocorrelation artifacts are rejected to the background noise level with complex conjugate rejection ratio up to 45 dB.
This work was supported by National High Technology Research and Development Program of China (2006AA02Z4E0) and Natural Science Foundation of China (60878057).
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