Development of a non-uniform discrete Fourier transform based high speed spectral domain optical coherence tomography system

Kai Wang; Zhihua Ding; Tong Wu; Chuan Wang; Jie Meng; Minghui Chen; Lei Xu

doi:10.1364/OE.17.012121

1. Introduction

Optical coherence tomography is a non-invasive, non-contact imaging modality that uses coherent gating to obtain high-resolution cross-sectional images of tissue microstructure [1]. Compared with conventional time domain OCT which is based on a scanning optical delay line, Fourier domain OCT offers significant advantages in imaging speed and detection sensitivity [2–4]. Fourier domain OCT enables reconstructing the depth profile (A-scan) from the Fourier transform of the acquired spectral data and can be performed in two ways. One is spectral domain OCT (SD-OCT) with a spectrometer and a line-scan CCD, the other is swept source OCT (SS-OCT) employing a rapidly tunable laser source and a point detector. Compared with swept source OCT, spectral domain OCT has the advantage of higher resolution and increased phase stability for functional imaging [5].

Despite well-achieved advances in SD-OCT, limitation in imaging range due to finite resolution of spectrometer remains to be a major drawback. According to sampling principle, theoretical maximum imaging depth is determined by the resolution of the spectrometer. However, practical imaging depth is also limited by the depth dependent sensitivity fall-off, the attenuation of the signal due to washout of interferogram fringe visibility with increasing imaging depth. Several approaches have been proposed to overcome this problem. An inter-pixel shift technique is adopted to double the spectral sampling rate and enhance the signal to noise ratio. But this method based on mechanical movement of the detector hinders the imaging speed to some extent [6]. An optical frequency comb is proposed to reduce the depth dependent sensitivity fall-off, but requires additional tunable fiber Fabry-Perot interferometer [7]. In fact, the sensitivity fall-off of SD-OCT is mainly due to the quality of the spectrometer and could be improved through custom design [8].

Another degrading factor that affects axial resolution and depth dependent sensitivity fall-off is the image reconstruction methods. In order to apply DFT method for correct reconstruction of depth information, spectra data should be linearly sampled in wavenumber (k). Unfortunately, spectra in spectrometer are evenly spread versus wavelength (λ) other than wavenumber, hence recalibration of spectral data such as zero-padding and linear interpolation or cubic spline interpolation prior to DFT is required [9,10]. However, due to the inverse relationship between λ and k, the short wavelength part of the spectrum is more sparsely sampled (in k) than the long wavelength part. This means that if the re-sampled spectrum is interpolated for uniform sampling in k, high frequencies of the spectral fringes are aliased and irretrievably lost. As a result, the computed magnitude at larger depths is underestimated leading to decreased sensitivity. A method using filter bank to overcome the sampling issue is proposed [11], but a complex filter bank must be designed in advance. A linear-in-wavenumber spectrometer is also suggested where depth information can be reconstructed using DFT method without interpolation, but additional prism in the spectrometer complicates the system design [12].

In this paper, we demonstrate a high speed SD-OCT system based on a custom-built spectrometer designed to decrease the depth dependent sensitivity fall-off. After precise spectral calibration of the spectrometer, NDFT of the acquired spectral data is implemented for image reconstruction.

2. Principle and simulation

SD-OCT system is based on a spectral interferometer, where light from reference and sample arms are combined and the resulting interference pattern is spectrally separated, detected and reconstructed into a depth profile. Suppose a multi-layer reflection sample placed in the sample arm of the interferometer, the detected interference signal at the spectrometer can be expressed as

\begin{array}{c} I (k) = S (k) [R_{r} + \sum_{n} R_{n} + 2 \sqrt{R_{r}} \sum_{n} \sqrt{R_{n}} \cos (2 k (z_{_{r}} - z_{n})) + \\ 2 \sum_{n} \sum_{m \neq n} \sqrt{R_{n} R_{m}} \cos (2 k (z_{n} - z_{m}))] \end{array}

Here, S(k) is the source power spectral density, R_r and R_n are reflectivity coefficients of the reference arm and the n^th reflector within the sample; z_r and z_n are optical paths of the reference arm and the n^th reflector relative to a common reference, for example the recombining coupler. The first two terms in the brackets of Eq. (1) are the non-interferometric contributions. The third term donates the cross-interferometric signal, and the fourth term represents the mutual interference of reflectors within the sample. Performing DFT of Eq. (1) yields

\begin{array}{c} F T^{- 1} [I (k)] = Γ (z) \otimes {R_{r} δ (0) + \sum_{n} R_{n} δ (0) + 2 \sqrt{R_{r}} \sum_{n} \sqrt{R_{n}} δ (z \pm z_{r n}) \\ + 2 \sum_{n} \sum_{m \neq n} \sqrt{R_{n} R_{m}} δ (z \pm z_{n m})} \end{array}

Where

Γ (z)

represents the envelope of the coherence function of the source and z_rn = z_r-z_n, z_nm = z_n-z_m. The third term in the brackets of Eq. (2) contains depth information and the other terms represent the DC background and the autocorrelation noises. It can be seen that DFT of Eq. (1) into Eq. (2) corresponds variable pairs of depth z and wavenumber k. For coherence-function-limited depth resolution, the spectra should be evenly sampled in wavenumber. Unfortunately, the dispersed spectrum in spectrometer is evenly spread in wavelength other than wavenumber, thus additional re-sampling and interpolation prior to DFT are necessary. Otherwise, degradation of resolution is un-avoided during image reconstruction. Alternatively, if we take the spectra data as an irregular sampling and implement NDFT of the spectral data directly without re-sampling and interpolation, then the depth information can be correctly reconstructed without degradation as

A (z_{m}) = \sum_{i = 0}^{N - 1} I (k_{i}) e^{- j \frac{2 π}{Δ K} k_{i} m}, m = 0, 1, ... N - 1

Here, z_m denotes the depth coordinate, ΔK is the wavenumber range, I(k_i) is the sampled signal corresponding to wavenumber k_i at CCD pixel indexed by i. Equation (3) can be written in matrix form as

A = D I

Where

A = [\begin{array}{l} A (z_{0}) \\ A (z_{1}) \\ ⋮ \\ A (z_{N - 1}) \end{array}]

I = [\begin{array}{l} I [k_{0}] \\ I [k_{1}] \\ ⋮ \\ I [k_{N - 1}] \end{array}]

D = [\begin{array}{c} 1 & 1 & \dots & 1 \\ p_{0}^{- 1} & p_{1}^{- 1} & \dots & p_{N - 1}^{- 1} \\ p_{0}^{- 2} & p_{1}^{- 2} & \dots & p_{N - 1}^{- 2} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ p_{0}^{- (N - 1)} & p_{1}^{- (N - 1)} & \dots & p_{N - 1}^{- (N - 1)} \end{array}]

In Eq. (7), p_n is expressed by

p_{n} = \exp (j \frac{2 π}{Δ K} k_{i}), i = 0, 1, \dots N - 1

The matrix D is called Vander monde matrix fully determined by wavelength distribution on the CCD plane. Once wavelength distribution on CCD plane of the spectrometer is calibrated, Vander monde matrix D can be pre-calculated and stored. Depth information is thus retrieved by direct matrix multiplication of Vander monde matrix D and the acquired spectral data according to Eq. (4).

Considering an ideal case of linear sampling in wavenumber space, the defined depth dependent sensitivity fall-off factor using DFT method for reconstruction is given by [8,13]

R_{D F T} (z) = \frac{\sin (p R z)}{p R z} \exp (- \frac{a^{2} R^{2} z^{2}}{4 \ln 2})

Where a is the diffraction spot-size of spectral components on CCD plane which is not constant but varied as a function of CCD pixel position, it can be represented by an effective constant spot-size [7]. p denotes the CCD pixel size, and R is the reciprocal linear dispersion. For minimized depth dependent sensitivity fall-off, these parameters determined by spectrometer should be deliberately considered.

For practical case of nonlinear sampling in wavenumber space, the depth dependent sensitivity fall-off factor R_{Interpolation} using DFT with interpolation method can be thought as the ideal sensitivity fall-off factor given by Eq. (9) convolved with an interpolation dependent degeneration function P_{Interpolation}, i.e.,

R_{I n t e r p o l a t i o n} (z) = R_{D F T} (z) \otimes P_{I n t e r p o l a t i o n} (δ k, z)

Where, δk is the uniform interval of interpolation in wavenumber space. On the other hand, the depth dependent sensitivity fall-off factor R_NDFT using NDFT without interpolation is given by

R_{N D F T} (z) = R_{D F T} (z) \otimes P_{N D F T} (z)

Where P_NDFT is the point spread function (PSF) of the non-uniform sampling determined by

P_{N D F T} (z) = \frac{1}{Δ K} | \sum_{i = 0}^{N - 1} e^{- j \frac{2 π}{Δ K} k_{i} z} |

Similarly to the Vander monde matrix D, P_NDFT is also fully determined by the wavelength distribution. As P_NDFT expressed by Eq. (12) approaches a delta function [14], R_NDFT becomes more like R_DFT as compared with R_{Interpolation}.

Assuming linear wavelength distribution on CCD plane, simulations on sensitivity fall-off factor versus depth z are performed with following typical parameters, i.e., N = 2048, p = 14 μm, central wavelength λ₀ of the Gaussian spectrum distribution is 835 nm, wavelength spacing δλ between adjacent CCD pixels is 0.067 nm, $δ k = δ λ / λ_{0}^{2}$ , R = 1/δk, and a is 30 μm. The calculated depth dependent sensitivity fall-offs based on different reconstruction methods are illustrated in Fig. 1 , where the interpolation adopted is cubic spline, and simulation based on DFT without interpolation is also given for comparison. The comparison results demonstrate that for a practical SD-OCT system, image reconstruction based on NDFT method improves the sensitivity fall-off especially at large depth in contrast to DFT with and without interpolation.

Fig. 1 Numerical simulations on depth dependent sensitivity fall-off based on different reconstuction methods

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3. System and experiment

A schematic of the high-speed SD-OCT system is illustrated in Fig. 2 . 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. After passing through a polarization independent isolator, the light is split into the sample and reference arm respectively by a 50/50 fiber coupler. A galvanometer-mounted (6215H, Cambridge Technology) mirror is used to scan the probe light transversely on the sample. The light illuminates the sample through an imaging lens with a focal length of 40 mm. 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. 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 [15,16]. 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.

Fig. 2 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, NDF is the neutral density filter.

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The optical layout of the spectrometer and its Zemax simulation are illustrated in Fig. 3 . 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. The grating is positioned under Littrow condition, and the dispersed light is focused onto the line-scan CCD by a focusing lens (TECHSPEC^® Near IR doublets, Edmund optics) special designed for wavelength range from 750 nm to 1100 nm. The spot size of focused light is designed to be smaller than the CCD pixel width for optimum spectral resolution and minimum sensetivity fall-off. For calibration purpose, a mercury argon lamp is used as a source for the spectrometer. Seven characteristic spectral lines from the lamp are dispersed and then detected by the CCD in the spectrometer. We record the indexed pixel numbers of all characteristic spectral lines and perform a third-order polynomial fitting to them. As shown in Fig. 4 , wavelength versus CCD pixel number is not linear due to residual aberrations and perhaps misalignment of the spectrometer.

Fig. 3 Optical layout of the costom-built spectrometer (a) and its Zemax simulation of ray tracing trajectories of two edge wavelenghs corresponding to 3dB bandwidth of the source (b).

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Fig. 4 The calibration curve of wavelength distribution on the CCD array

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In order to study the performance of the custom-built spectrometer and the affecting factors on depth dependent sensitivity fall-off of the developed SD-OCT system, a mirror is used as the sample for system characterization. 13 depth positions of the mirror are selected for depth dependent measurement. Corresponding to every depth position of the mirror, the CCD is positioned at the focus plane and defocus plane of the focusing lens in the spectrometer for comparison. In order to reduce the fluctuation of the noise floor, 100 consecutive interference spectra are acquired at an acquisition speed of 35μs per A-line, and then the depth dependent PSFs of the system are obtained by averaging over the 100 PSFs after data processing. Measured depth dependent sensitivity fall-off and Zemax simulation of spot diagrams for typical spectral components on CCD plane corresponding to CCD located at focused position and defocused position in the spectrometer are shown in Fig. 5 . The defocused length in Figs. 5(b) and 5(d) is set to be 500 μm adjusted via a precise Five-Axis aligner (9082M, Newfocus). Zemax simulation of spot diagrams corresponding to 5 spectral components including 812.5 nm, 823 nm, 835 nm, 847 nm and 857.5 nm are given for CCD position at focused plane shown in Fig. 5(c) and defocused plane shown in Fig. 5(d), respectively. In case of CCD positioned at the focused plane of the focusing lens in the spectrometer shown in Fig. 5(c), the diameters of the central three circles and the marginal two circles are 14 μm and 30 μm, respectively. It is evident that spot sizes of spectral components within 3dB bandwidth of the source are comparable to the size of CCD pixel. Correspondingly, measured sensitivity fall-off shown in Fig. 5(a) demonstrates depth dependent sensitivity fall-off by 16.1 dB over 2 mm. In case of CCD positioned at the defocused plane shown in Fig. 5(d), the diameters of the central three circles and the marginal two circles are 40 μm and 70 μm, respectively. It is evident that spot sizes of spectral components within 3dB bandwidth of the source are much larger than the size of the CCD pixel. Correspondingly, measured sensitivity fall-off shown in Fig. 5(b) demonstrates depth dependent sensitivity fall-off by 35.2 dB over the same depth range. This increased sensitivity fall-off shown in Fig. 5(b) in comparison with Fig. 5(a) is due to mismatching of spectral spot with pixel size, leading to spectral “cross talk” and hence washout of the interferogram fringe visibility at large imaging depth. Therefore, diffraction spot size of spectral components on CCD plane should not be larger than pixel size of the CCD array and improved sensitivity fall-off can be achieved by more complex optical design of the focusing lens and deliberately alignment and positioning of CCD in the spectrometer.

Fig. 5 Measured depth dependent sensitivity fall-off (a and b) and Zemax simulation of spot diagrams for typical spectral components on CCD plane (c and d) corresponding to CCD located at focused position and defocused position in the spectrometer, respectively.

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Depth dependent PSFs shown in Fig. 5 is reconstructed through conventional interpolation combined with DFT. Actually reconstruction method plays an important role in affecting depth dependent sensitivity fall-off of the system. In order to confirm the improved performance of NDFT method for imaging reconstruction, different reconstruction approaches are adopted and compared in terms of axial resolution and depth dependent sensitivity fall-off.

Figure 6(a) illustrates a typical spectral interferogram after subtraction of DC term measured with a mirror as the sample at the depth of about 450 μm. Three approaches including NDFT without interpolation, DFT with interpolation and DFT without interpolation are implemented for depth profile reconstruction. The reconstructed PSFs are shown in Fig. 6(b). It can be seen that the axial resolution reconstructed by NDFT method is about 7.5 μm, comparable to that by DFT with interpolation method and higher than that by DFT without interpolation method. Peak value of the reconstructed PSF by NDFT method is the highest among the three reconstruction approaches.

Fig. 6 (a) Recorded spectral interferogram of a mirror as the sample at the depth of 450 μm and corresponding (b) PSFs reconstructed using NDFT method (blue line), DFT method with (red) and without interpolation (black).

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For comparison of reconstruction method on the depth dependent sensitivity fall-off, these three methods are adopted to reconstruct the depth dependent sensitivity fall-off using the same spectral data obtained in above-mentioned 13 depth dependent measurements. The reconstructed results are demonstrated in Fig. 7 . Based on Zemax simulation of spot diagrams for typical spectral components on CCD plane corresponding to CCD located at focused position, an expected sensitivity fall-off is also shown in Fig. 7. This expected sensitivity fall-off is calculated according to Eq. (9) with an effective constant spot size of 42 µm, which is determined by the variable spot size on CCD plane derived from quadratic fitting to Zemax data [8]. It is clear that the sensitivity fall-off based on NDFT method is more close to the expected one in contrast to other reconstruction methods. The discrepancy between expected sensitivity fall-off and that based on NDFT method might be due to underestimation of optical aberrations in Zemax simulation as well as misalignment of the spectrometer. Nevertheless, NDFT method improves the sensitivity at large depth and gives the best reconstruction outcome among all approaches adopted for reconstruction. This conclusion is accordant with our theoretical simulations demonstrated in Fig. 1.

Fig. 7 Reconstructed depth dependent sensitivity fall-off based on three methods and the theoretical expected sensitivity fall-off.

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In vivo OCT imaging on biological sample is conducted to confirm the performance of the NDFT based SD-OCT system. Figure 8 shows the in vivo B-scan images on human finger from volunteer using reconstruction method based on NDFT, DFT with interpolation and DFT without interpolation, respectively. The image size is about 4.5 mm by 1.8 mm formed by 500 A-lines at A-scan rate of 29 kHz. It can be seen that the image reconstructed by NDFT method presents the highest quality among all three reconstruction approaches. Improved SNR at large depth and hence deep imaging is achieved as evident in Fig. 8(a) in comparison with Figs. 8(b) and 8(c). Reconstruction time with a PC of Intel E2600 CPU and 2G RAM is about 3.87 s and 2.35 s for NDFT method and DFT with interpolation method, respectively. NDFT method is relatively slower than DFT with interpolation method due to the well-established FFT algorithm. Anyway, this is not a big difference in computation time. Faster PC and more elaborated NDFT algorithm should overcome such defect in reconstruction speed.

Fig. 8 In vivo OCT images on human finger from volunteer based on (a) NDFT method, (b) DFT with interpolation method, and (c) DFT without interpolation method. Zero OPD: zero optical-path difference, SC: stratum corneum, SD: sweat gland duct, DEJ: dermis-epidermis junction

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4. Conclusion

Depth dependent sensitivity fall-off is the major drawback of SD-OCT system which limits the imaging depth. There are mainly two reasons accounting for sensitivity fall-off of SD-OCT system, one is the quality of the spectrometer, the other is the image reconstruction method. To obtain minimized depth dependent sensitivity fall-off, deliberately design and optical alignment of the spectrometer in combination with suitable reconstruction method is necessary. We develop a SD-OCT system based on a custom-built spectrometer in combination with NDFT for image reconstruction. Zemax simulations and sensitivity fall-off measurements under two alignment states of the spectrometer confirms that diffraction spot size of spectral components on CCD plane should not be larger than pixel size of the CCD array, otherwise spectral “cross talk” will lead to increased sensitivity fall-off at large imaging depth. The developed system realizes a coherence-function-limited axial resolution of 7.5μm in air at an A-scan rate of 29 kHz. Both theoretical simulations and experiments demonstrate that NDFT based SD-OCT system achieves the minimized depth dependent sensitivity fall-off in contrast to conventional reconstruction methods.

Acknowledgement

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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Development of a non-uniform discrete Fourier transform based high speed spectral domain optical coherence tomography system

Abstract

1. Introduction

2. Principle and simulation

3. System and experiment

4. Conclusion

Acknowledgement

References and links

Cited By

Figures (8)

Equations (12)

Optics Express