Sinusoidal B-M method based spectral domain optical coherence tomography for the elimination of complex-conjugate artifact

doi:10.1364/OE.17.016820

« Show journal navigation

Sinusoidal B-M method based spectral domain optical coherence tomography for the elimination of complex-conjugate artifact

Kai Wang, Zhihua Ding, Yan Zeng, Jie Meng, and Minghui Chen »View Author Affiliations

Optics Express, Vol. 17, Issue 19, pp. 16820-16833 (2009)
http://dx.doi.org/10.1364/OE.17.016820

View Full Text Article

Acrobat PDF (1348 KB)

Journal Search
Article Lookup

Article Tools

Citations

Alert me when this article is cited
Export Citation/Save

Article Navigation

▸ Article Outline
– Abstract
1. Introduction
2. Principle
3. Experiments and results
4. Conclusion
– References and links

Article Tools
Full Text
Article Info
References
Cited By
Figures
Metrics
Download PDF

Viewing Equations in the
HTML Article

Optimal Viewing Recommendations

Full Text
Article Info
References (20)
Cited By
Figures (11)
Metrics

Abstract

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.

1. Introduction

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 [1

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and et, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991). [CrossRef] [PubMed]

]. 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

J. F. de Boer, B. Cense, B. H. Park, M. C. Pierce, G. J. Tearney, and B. E. Bouma, “Improved signal-to-noise ratio in spectral-domain compared with time-domain optical coherence tomography,” Opt. Lett. 28(21), 2067–2069 (2003). [CrossRef] [PubMed]

R. Leitgeb, C. Hitzenberger, and A. Fercher, “Performance of fourier domain vs. time domain optical coherence tomography,” Opt. Express 11(8), 889–894 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-8-889. [CrossRef] [PubMed]

]. However, this Fourier-transform of real-valued spectral data introduces complex conjugate ambiguity and hence the mirror image [4

M. Wojtkowski, A. Kowalczyk, R. Leitgeb, and A. F. Fercher, “Full range complex spectral optical coherence tomography technique in eye imaging,” Opt. Lett. 27(16), 1415–1417 (2002). [CrossRef]

]. 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

R. A. Leitgeb, C. K. Hitzenberger, A. F. Fercher, and T. Bajraszewski, “Phase-shifting algorithm to achieve high-speed long-depth-range probing by frequency-domain optical coherence tomography,” Opt. Lett. 28(22), 2201–2203 (2003). [CrossRef] [PubMed]

J. T. Oh and B.-M. Kim, “Artifact removal in complex frequency domain optical coherence tomography with an iterative least-squares phase-shifting algorithm,” Appl. Opt. 45(17), 4157–4164 (2006). [CrossRef] [PubMed]

], 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 [7

Y. K. Tao, M. Zhao, and J. A. Izatt, “High-speed complex conjugate resolved retinal spectral domain optical coherence tomography using sinusoidal phase modulation,” Opt. Lett. 32(20), 2918–2920 (2007). [CrossRef] [PubMed]

], 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 [8

M. Sarunic, M. A. Choma, C. Yang, and J. A. Izatt, “Instantaneous complex conjugate resolved spectral domain and swept-source OCT using 3x3 fiber couplers,” Opt. Express 13(3), 957–967 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-3-957. [CrossRef] [PubMed]

] or through polarization-based optical demodulation [9

B. J. Vakoc, S. H. Yun, G. J. Tearney, and B. E. Bouma, “Elimination of depth degeneracy in optical frequency-domain imaging through polarization-based optical demodulation,” Opt. Lett. 31(3), 362–364 (2006). [CrossRef] [PubMed]

]. 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 [10

A. Bachmann, R. Leitgeb, and T. Lasser, “Heterodyne Fourier domain optical coherence tomography for full range probing with high axial resolution,” Opt. Express 14(4), 1487–1496 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-4-1487. [CrossRef] [PubMed]

] or electro-optic phase modulator [11

J. Zhang, J. S. Nelson, and Z. Chen, “Removal of a mirror image and enhancement of the signal-to-noise ratio in Fourier-domain optical coherence tomography using an electro-optic phase modulator,” Opt. Lett. 30(2), 147–149 (2005). [CrossRef] [PubMed]

] 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

A. B. Vakhtin, K. A. Peterson, and D. J. Kane, “Resolving the complex conjugate ambiguity in Fourier-domain OCT by harmonic lock-in detection of the spectral interferogram,” Opt. Lett. 31(9), 1271–1273 (2006). [CrossRef] [PubMed]

,13

A. B. Vakhtin, K. A. Peterson, and D. J. Kane, “Demonstration of complex-conjugate-resolved harmonic Fourier-domain optical coherence tomography imaging of biological samples,” Appl. Opt. 46(18), 3870–3877 (2007). [CrossRef] [PubMed]

]. 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 [14

B. Hofer, B. Považay, B. Hermann, A. Unterhuber, G. Matz, and W. Drexler, “Dispersion encoded full range frequency domain optical coherence tomography,” Opt. Express 17(1), 7–24 (2009), http://www.opticsinfobase.org/abstract.cfm?uri=oe-17-1-7. [CrossRef] [PubMed]

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 [15

Y. Yasuno, S. Makita, T. Endo, G. Aoki, M. Itoh, and T. Yatagai, “Simultaneous B-M-mode scanning method for real-time full-range Fourier domain optical coherence tomography,” Appl. Opt. 45(8), 1861–1865 (2006). [CrossRef] [PubMed]

]. 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 [16

R. K. Wang, “In vivo full range complex Fourier domain optical coherence tomography,” Appl. Phys. Lett. 90, 054103–1-054103–3 (2007).

], piezoelectric fiber stretcher (PFS) [17

S. Vergnole, G. Lamouche, and M. L. Dufour, “Artifact removal in Fourier-domain optical coherence tomography with a piezoelectric fiber stretcher,” Opt. Lett. 33(7), 732–734 (2008). [CrossRef] [PubMed]

] in the reference arm, beam offset at the scanning mirror in the sample arm [18

R. A. Leitgeb, R. Michaely, T. Lasser, and S. C. Sekhar, “Complex ambiguity-free Fourier domain optical coherence tomography through transverse scanning,” Opt. Lett. 32(23), 3453–3455 (2007). [CrossRef] [PubMed]

]. 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. Principle

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.

Fig. 1 Time sequences for B-M scans in (a) conventional phase-shifting method, (b) linear B-M method, and (c) sinusoidal B-M method.

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

I_{signal} (ω, x) = I_{R} (ω) + I_{S} (ω, x) + 2 {[I_{R} (ω) I_{S} (ω, x)]}^{\frac{1}{2}} \cos [Δ φ_{S} (ω, x) + φ (x)],

(1)

where ω is the optical frequency, x is the transverse position on sample and related to time t through the transverse scanning, I_R(ω) and I_s(ω,x) are the light intensities coming from the reference and sample arms of the interferometer respectively, Δϕ_s(ω,x) is the transverse position dependent phase delay of the sample with respect to the reference arm, ϕ(x) is the transverse position dependent phase introduced by phase modulation in the reference arm during transverse sample scanning.

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 ν_x = 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 ν_σ = 1/σ [18

]. 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 ± ν_Φ = ± (ΔΦ/2π)(N/X). If ν_Φ>>ν_σ and ν_Φ<ν_x , 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) = 2πf_mt, where f_m is the modulation frequency. Corresponding phase shift ΔΦ between successive A-scans is 2πf_mT, 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 ν_Φ = f_mT(N/X), then the condition for complete separation of spatial spectrum of sample as shown in Fig. 2(b) is

f_{m} T (N / X) > > 1 / σ \Rightarrow f_{m} T > > (X / N) / σ,

(2)

f_{m} T (N / X) < N / 2 X \Rightarrow f_{m} T < \frac{1}{2} .

(3)

We define the ratio between the transverse resolution σ and the transverse step size (X/N) as transverse over-sampling factor, i.e., ρ = σ/(X/N). With the relationship of

f_{A - s c a n} = 1 / T

, Eqs. (2) and (3) can be combined to be

\frac{1}{ρ} f_{A - s c a n} < < f_{m} < \frac{1}{2} f_{A - s c a n} .

(4)

And a similar inequality of Eq. (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) = a_m(ω)sin2πf_mt, where a_m(ω) is the phase-modulation amplitude, f_m is the modulation frequency. Then Eq. (1) can be expressed with respect to time t and expanded as:

\begin{array}{l} I_{signal} (ω, t) = I_{R} (ω) + I_{S} (ω, t) \\ + 2 {[I_{R} (ω) I_{S} (ω, t)]}^{\frac{1}{2}} \times {J_{0} [a_{m} (ω)] \\ - 2 J_{1} [a_{m} (ω)] \sin (2 π f_{m} t) \sin (Δ ϕ_{S} (ω, t)) \\ + 2 J_{2} [a_{m} (ω)] \cos (2 \times 2 π f_{m} t) \cos (Δ ϕ_{S} (ω, t)) \\ - 2 J_{3} [a_{m} (ω)] \sin (3 \times 2 π f_{m} t) \sin (Δ ϕ_{S} (ω, t)) \\ + 2 J_{4} [a_{m} (ω)] \cos (4 \times 2 π f_{m} t) \cos (Δ ϕ_{S} (ω, t)) - \dots} . \end{array}

(5)

From 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 ν_Φ , 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. ν_Φ>>ν_σ . 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:

\frac{1}{ρ} f_{A - s c a n} < < f_{m} < \frac{1}{n} f_{A - s c a n},

(6)

here n is the order of the harmonics term. Thus, with a given A-scan rate, the over-sampling factor ρ must be large enough to satisfy Eq. (6), which means that the transverse step must be small enough for effective artifact removal.

Fig. 2 (a) Saw-tooth waveform for phase modulation in the reference arm and (b) spatial frequency spectrum from the detected signal versus transverse scanning in the linear B-M method.

Fig. 3 (a) Sinusoidal waveform for phase modulation in the reference arm and (b) spatial frequency spectrum from detected interference signal versus transverse scanning in the sinusoidal B-M method.

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 sin2πf_mt:

\begin{array}{l} I_{signal} (ω, t) \sin (2 π f_{m} t) = I_{R} (ω) \sin (2 π f_{m} t) + I_{S} (ω, t) \sin (2 π f_{m} t) \\ + 2 {[I_{R} (ω) I_{S} (ω, t)]}^{\frac{1}{2}} \times {J_{0} [a_{m} (ω)] \sin (2 π f_{m} t) \\ - J_{1} [a_{m} (ω)] \sin (Δ ϕ_{S} (ω, t)) (1 - 2 \cos (2 \times 2 π f_{m} t)) \\ + J_{2} [a_{m} (ω)] \cos (Δ ϕ_{S} (ω, t)) (\sin (3 \times 2 π f_{m} t) - \sin (2 π f_{m} t)) \\ - J_{3} [a_{m} (ω)] \sin (Δ ϕ_{S} (ω, t)) (\cos (2 \times 2 π f_{m} t) - \cos (4 \times 2 π f_{m} t)) + \dots} . \end{array}

(7)

From 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/f_m , 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 × 2πf_mt) and sin(3 × 2πf_mt), respectively. Hence, the extracted harmonic terms are:

H_{1} [ω, Δ ϕ_{S} (ω, t)] = - 4 J_{1} [a_{m} (ω)] {[I_{R} (ω) I_{S} (ω, t)]}^{\frac{1}{2}} \sin (Δ ϕ_{S} (ω, t)),

(8-1)

H_{2} [ω, Δ ϕ_{S} (ω, t)] = 4 J_{2} [a_{m} (ω)] {[I_{R} (ω) I_{S} (ω, t)]}^{\frac{1}{2}} \cos (Δ ϕ_{S} (ω, t)),

(8-2)

H_{3} [ω, Δ ϕ_{S} (ω, t)] = - 4 J_{3} [a_{m} (ω)] {[I_{R} (ω) I_{S} (ω, t)]}^{\frac{1}{2}} \sin (Δ ϕ_{S} (ω, t)) .

(8-3)

It can be seen that H₁ and H₂ represent the imaginary and real parts of the complex spectral interferogram, respectively. In order to construct the complex interferogram, an additional scaling coefficient β must be applied to equalize the amplitude of H₁ and that of H₂ :

β = \frac{J_{1} [a_{m} (ω)]}{J_{2} [a_{m} (ω)]} .

(9)

The scaling coefficient β depends on the modulation amplitude a_m(ω). a_m(ω) is wavelength dependent, and can be extracted from

\frac{J_{1} [a_{m} (ω)]}{J_{3} [a_{m} (ω)]} = H_{1} / H_{3} .

(10)

Once the modulation amplitude is determined, the scaling coefficient is obtained. Then the complex interferogram can be constructed. Its Fourier transform is free from complex-conjugate ambiguity and removal of the DC and autocorrelation terms, as shown in Eq. (11):

f (z) = F^{- 1} {β H_{2} [ω, Δ ϕ_{S} (ω)] - i H_{1} [ω, Δ ϕ_{S} (ω)]} .

(11)

3. Experiments and results

The proposed sinusoidal B-M method for elimination of the complex-conjugate artifact is implemented in the developed SD-OCT system [19

K. Wang, Z. Ding, T. Wu, C. Wang, J. Meng, M. Chen, and L. Xu, “Development of a non-uniform discrete Fourier transform based high speed spectral domain optical coherence tomography system,” Opt. Express 17(14), 12121–12131 (2009), http://www.opticsinfobase.org/abstract.cfm?uri=oe-17-14-12121. [CrossRef] [PubMed]

,20

Y. Zhang, X. Li, L. Wei, K. Wang, Z. Ding, and G. Shi, “Time-domain interpolation for Fourier-domain optical coherence tomography,” Opt. Lett. 34(12), 1849–1851 (2009). [CrossRef] [PubMed]

] shown in Fig. 4 .

Fig. 4 Schematic diagram of the established SD-OCT system, where OI is the optical isolator, FC is the 3dB fiber coupler, PC is the polarization controller, DG is the diffraction grating, and NDF is the neutral density filter.

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 .

Fig. 5 Block diagram of the processing procedure in the sinusoidal B-M method

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 (V_pp ) 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 V_pp 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 V_pp , we find that when the V_pp 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 V_pp of the sinusoidal signal applied to PZT is set to be 1.0 V for following imaging on biological sample.

Fig. 6 (a) The calibration curve of wavelength distribution on the CCD array. (b) Fourier transform of interference signal at 849.7nm when the V_pp of sinusoidal signal is set to be 1.0V.

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 a_m (ω) by minimizing the least square differences ΔU, i.e.,

Δ U = \sum_{ω} {[r a t i o (ω) - (J_{1} [a_{m}^{0} ω / ω_{0}] / J_{3} [a_{m}^{0} ω / ω_{0}])]}^{2}

(12)

Here,

a_{m} (ω) = a_{m}^{0} ω / ω_{0} = a_{m}^{0} λ_{0} / λ

and a_m ⁰ is the modulation amplitude at central wavelength, λ ₀ is the central wavelength, and ω ₀ is the central optical frequency. The choice of fitting range from 805nm to 885nm is due to the spectral distribution of the source implemented in the system, outside above range the spectrum intensity is very weak and can be disregarded. The smoothed ratio of the amplitude of the first- and third-order harmonic terms at different wavelength and its corresponding least square fitting are illustrated in Fig. 7(b), herein the a_m ⁰ is about 2.19. It should be mentioned that the fitting curve and corresponding a_m (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.

Fig. 7 (a) Calculated ratio of the amplitude of the first- and third-order harmonic term (H₁/ H₃) with respect to wavelength and (b) smoothed curve after low-pass filtering and corresponding least square fitting with a_m ⁰ of 2.19.

The calculated a_m (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 a_m (ω) is related to the displacement of the PZT by

a_{m} (ω) = 2 ω d / c = 4 π d / λ

, here d is the displacement amplitude of the PZT movement, c is the light velocity. Thus, from the calculated modulation amplitude a_m ⁰ at central wavelength, d can be calculated by

d = a_{m}^{0} λ_{0} / 4 π

. 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 [16

R. K. Wang, “In vivo full range complex Fourier domain optical coherence tomography,” Appl. Phys. Lett. 90, 054103–1-054103–3 (2007).

]. 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.

Fig. 8 (a) Calculated wavelength dependent phase-modulation amplitude a_m (ω) under the least square fitting and (b) corresponding calculated scaling coefficient β with respect to wavelength.

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.

Fig. 9 Transverse Fourier transformation spectra with respect to time t under different conditions: (a)-(c) with a fixed modulation frequency of the sinusoidal waveform at 1250Hz and different transverse over-sampling factor corresponding to 4, 18 and 32, respectively. (d)-(f) with a fixed transverse over-sampling factor of 16 and different modulation frequency of the sinusoidal waveform corresponding to 500 Hz, 800Hz, 1250Hz, respectively.

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 V_pp 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.

Fig. 10 (a) Picture of the fresh shrimp under imaging; Tomogram of dorsal part of the fresh shrimp denoted by the line segment in (a) with (b) standard and (c) complex reconstruction with the transverse over-sampling factor of 32; and the corresponding A-scans (d) and (e) indicated by the arrows.

Fig. 11 (a) Picture of the fresh shrimp under imaging; Tomogram of dorsal part of the fresh shrimp denoted by the line segment in (a) with (b) standard and (c) complex reconstruction with the transverse over-sampling factor of 8; and the corresponding A-scans (d) and (e) indicated by the arrows.

4. Conclusion

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.

Acknowledgements

This work was supported by National High Technology Research and Development Program of China (2006AA02Z4E0) and Natural Science Foundation of China (60878057).

References and links

1.	D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and et, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991). [CrossRef] [PubMed]
2.	J. F. de Boer, B. Cense, B. H. Park, M. C. Pierce, G. J. Tearney, and B. E. Bouma, “Improved signal-to-noise ratio in spectral-domain compared with time-domain optical coherence tomography,” Opt. Lett. 28(21), 2067–2069 (2003). [CrossRef] [PubMed]
3.	R. Leitgeb, C. Hitzenberger, and A. Fercher, “Performance of fourier domain vs. time domain optical coherence tomography,” Opt. Express 11(8), 889–894 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-8-889. [CrossRef] [PubMed]
4.	M. Wojtkowski, A. Kowalczyk, R. Leitgeb, and A. F. Fercher, “Full range complex spectral optical coherence tomography technique in eye imaging,” Opt. Lett. 27(16), 1415–1417 (2002). [CrossRef]
5.	R. A. Leitgeb, C. K. Hitzenberger, A. F. Fercher, and T. Bajraszewski, “Phase-shifting algorithm to achieve high-speed long-depth-range probing by frequency-domain optical coherence tomography,” Opt. Lett. 28(22), 2201–2203 (2003). [CrossRef] [PubMed]
6.	J. T. Oh and B.-M. Kim, “Artifact removal in complex frequency domain optical coherence tomography with an iterative least-squares phase-shifting algorithm,” Appl. Opt. 45(17), 4157–4164 (2006). [CrossRef] [PubMed]
7.	Y. K. Tao, M. Zhao, and J. A. Izatt, “High-speed complex conjugate resolved retinal spectral domain optical coherence tomography using sinusoidal phase modulation,” Opt. Lett. 32(20), 2918–2920 (2007). [CrossRef] [PubMed]
8.	M. Sarunic, M. A. Choma, C. Yang, and J. A. Izatt, “Instantaneous complex conjugate resolved spectral domain and swept-source OCT using 3x3 fiber couplers,” Opt. Express 13(3), 957–967 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-3-957. [CrossRef] [PubMed]
9.	B. J. Vakoc, S. H. Yun, G. J. Tearney, and B. E. Bouma, “Elimination of depth degeneracy in optical frequency-domain imaging through polarization-based optical demodulation,” Opt. Lett. 31(3), 362–364 (2006). [CrossRef] [PubMed]
10.	A. Bachmann, R. Leitgeb, and T. Lasser, “Heterodyne Fourier domain optical coherence tomography for full range probing with high axial resolution,” Opt. Express 14(4), 1487–1496 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-4-1487. [CrossRef] [PubMed]
11.	J. Zhang, J. S. Nelson, and Z. Chen, “Removal of a mirror image and enhancement of the signal-to-noise ratio in Fourier-domain optical coherence tomography using an electro-optic phase modulator,” Opt. Lett. 30(2), 147–149 (2005). [CrossRef] [PubMed]
12.	A. B. Vakhtin, K. A. Peterson, and D. J. Kane, “Resolving the complex conjugate ambiguity in Fourier-domain OCT by harmonic lock-in detection of the spectral interferogram,” Opt. Lett. 31(9), 1271–1273 (2006). [CrossRef] [PubMed]
13.	A. B. Vakhtin, K. A. Peterson, and D. J. Kane, “Demonstration of complex-conjugate-resolved harmonic Fourier-domain optical coherence tomography imaging of biological samples,” Appl. Opt. 46(18), 3870–3877 (2007). [CrossRef] [PubMed]
14.	B. Hofer, B. Považay, B. Hermann, A. Unterhuber, G. Matz, and W. Drexler, “Dispersion encoded full range frequency domain optical coherence tomography,” Opt. Express 17(1), 7–24 (2009), http://www.opticsinfobase.org/abstract.cfm?uri=oe-17-1-7. [CrossRef] [PubMed]
15.	Y. Yasuno, S. Makita, T. Endo, G. Aoki, M. Itoh, and T. Yatagai, “Simultaneous B-M-mode scanning method for real-time full-range Fourier domain optical coherence tomography,” Appl. Opt. 45(8), 1861–1865 (2006). [CrossRef] [PubMed]
16.	R. K. Wang, “In vivo full range complex Fourier domain optical coherence tomography,” Appl. Phys. Lett. 90, 054103–1-054103–3 (2007).
17.	S. Vergnole, G. Lamouche, and M. L. Dufour, “Artifact removal in Fourier-domain optical coherence tomography with a piezoelectric fiber stretcher,” Opt. Lett. 33(7), 732–734 (2008). [CrossRef] [PubMed]
18.	R. A. Leitgeb, R. Michaely, T. Lasser, and S. C. Sekhar, “Complex ambiguity-free Fourier domain optical coherence tomography through transverse scanning,” Opt. Lett. 32(23), 3453–3455 (2007). [CrossRef] [PubMed]
19.	K. Wang, Z. Ding, T. Wu, C. Wang, J. Meng, M. Chen, and L. Xu, “Development of a non-uniform discrete Fourier transform based high speed spectral domain optical coherence tomography system,” Opt. Express 17(14), 12121–12131 (2009), http://www.opticsinfobase.org/abstract.cfm?uri=oe-17-14-12121. [CrossRef] [PubMed]
20.	Y. Zhang, X. Li, L. Wei, K. Wang, Z. Ding, and G. Shi, “Time-domain interpolation for Fourier-domain optical coherence tomography,” Opt. Lett. 34(12), 1849–1851 (2009). [CrossRef] [PubMed]

OCIS Codes
(100.2650) Image processing : Fringe analysis
(110.4500) Imaging systems : Optical coherence tomography
(120.5060) Instrumentation, measurement, and metrology : Phase modulation

ToC Category:
Imaging Systems

History
Original Manuscript: July 24, 2009
Revised Manuscript: August 19, 2009
Manuscript Accepted: August 25, 2009
Published: September 4, 2009

Virtual Issues
Vol. 4, Iss. 11 Virtual Journal for Biomedical Optics

Citation
Kai Wang, Zhihua Ding, Yan Zeng, Jie Meng, and Minghui Chen, "Sinusoidal B-M method based spectral domain optical coherence tomography for the elimination of complex-conjugate artifact," Opt. Express 17, 16820-16833 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-19-16820

Sort: Author | Year | Journal | Reset

References

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Click here to see a list of articles that cite this paper