# Extending the Effective Ranging Depth of Spectral Domain Optical Coherence Tomography by Spatial Frequency Domain Multiplexing

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{0}

^{2}/4nδλ and is fundamentally limited by the center wavelength λ

_{0}of the light source, the spectral resolution δλ, and the refractive index n. The spectral resolution is δλ = Δλ/N when the spectrometer is pixel-limited. In this equation, Δλ represents the spectral bandwidth of the light source and N is the data number of the digitized axial spectral interference signal that covers the bandwidth. In fact, there is a sensitivity roll-off issue for FDOCT, which poses a further limitation to the practical imaging depth range. For the case of SDOCT, the sensitivity roll-off effect is due to the finite pixel number of the line scanning camera. Due to this effect, the visibility of the spectral interference fringe is maximized when the optical path difference (OPD) is zero and decays as the path length difference increases. Thus, the effective ranging depth is defined as the imaging depth at which the sensitivity has decayed −6 dB from the peak value [2]. For FDOCT, the complex conjugate ambiguity further limits the imaging depth to one half of the full-range imaging space. Another limitation on the imaging depth of the FDOCT is posed by the finite depth of focus of the objective lens.

## 2. Methods

#### 2.1. Spatial Frequency Domain Multiplexing Spectral Domain Optical Coherence Tomography (SDOCT) System

_{A}and Z

_{B}in Figure 1. Thus, two sets of the spectral interferogram corresponding to the depths Z

_{A}and Z

_{B}can be acquired. In the sample arm, the light beam just incidents upon the pivot of the GS0 for sample scanning. The light beams in the reference arms incident upon the GS1 and GS2 with the different offset distances s

_{1}and s

_{2}away from the pivots, respectively, as shown in Figure 1.

_{A}and Z

_{B}will be shifted to the different center spatial frequencies away from the zero spatial frequency.

#### 2.2. Demodulation Algorithm for Spatial Frequency Domain Multiplexing SDOCT

_{A}and Z

_{B}, respectively. ν

_{c1}and ν

_{c2}are the center spatial carrier frequencies, which are proportional to the pivot-offset distances. For reconstruction of the complex valued spectral interferogram, two band-pass filters with center frequencies of ν

_{c1}and ν

_{c2}are applied to the spatial frequency spectrum (step (ii) in Figure 2). We then obtain the separated spatial frequency content ${C}_{A}(\nu +{\nu}_{c1},k)$ and ${C}_{B}(\nu +{\nu}_{c2},k)$.

_{A}and Z

_{B}can be calculated by inversely Fourier transforming the complex-valued linear-in-k spectral interferogram ${\tilde{I}}_{A}\left(x,k\right)$ and ${\tilde{I}}_{B}\left(x,k\right)$ along the wavenumber direction. Finally, an extended range OCT image is formed by concatenating the image regions that correspond to the effective ranging depths of the two full-range OCT images.

#### 2.3. The Center Spatial Frequency and Parameters Design Criterion

_{1}and s

_{2}. The phase shift between successive axial scans (A-scan) induced by the galvo scanner can be expressed as [9]

_{i}are all responsible to the image reconstruction quality. Thus, to avoid crosstalk and fully utilize the proposed method for extended range imaging, the center spatial carrier frequencies and the system parameters must be chosen carefully.

## 3. Experiments and Results

^{2}diameter focal spot on the sample with a confocal parameter of 3.7 mm. A 50/50 fiber coupler splits the light beam into the sample arm and the reference arms. In the sample arm, the light beam is deflected by a 2D galvo scanner and focused via a 75 mm achromatic lens L0 to the sample. Within the reference arm, a wide bandwidth non-polarizing beam splitter (NPBS, Daheng Optics Inc., Beijing, China) is used to divide the reference optical power into the two separate reference arms. The fiber collimator on a translation stage in the reference arm is used to adjust the overall delay of the two channels. The single-pass OPD between Reference Arms I and II is set to 3 mm. The GS1 and GS2 are the X and Y mirror of a pair of large beam 2D GS system (GVS012, Thorlabs Inc., Newton, NJ, USA), respectively. The focus lengths of achromatic lens L1and L2 are both 30 mm. The plane mirror M1 and M2 are the wide bandwidth dielectric coated mirrors covering the wavelength range of 800 to 900 nm. The spectral interference signal is detected by a custom-built spectrometer, which consists of a collimator (f = 60 mm, OZ optics Inc., Ottawa, ON, Canada), a transmissive diffraction grating (1800 lines/mm), a camera lens (f =105 mm, Nikon Inc., Tokyo, Japan), and a line-scan CMOS camera (sp2048-70km, Basler AG Inc., Ahrensburg, German). The spectral resolution of the spectrometer is 0.065 nm, allowing for a theoretical effective imaging depth range of 3.5 mm.

^{−1}. According to the values of the sample beam diameter d, the center wavelength, the focus length of the sample objective and Equation (7), the theoretical BW of the spatial frequency content is calculated to be 22.5 mm

^{−1}. In order to suppress the crosstalk of the two spatial frequency contents of the two depths signal, the difference of the two center spatial carrier frequencies must be greater than 67.5 mm

^{−1}according to Equation (8). Considering that the optimal conjugate suppression ratio occurs when $\delta \phi $ is $\pi /2$ [9], each of the two center spatial carrier frequencies needs to be close to the central region of the positive spatial frequency range. Thus, the two center spatial carrier frequencies ν

_{c1}and ν

_{c2}are set to 56.5 and 123.5 mm

^{−1}, respectively. According to Equation (4), the phase shift $\delta {\phi}_{1}$ and $\delta {\phi}_{2}$ between successive axial scans are calculated to be $0.3\pi $ and $0.7\pi $. The angular velocity of the galvo scanners is set to 0.93 rad/s. The exposure time of the CMOS camera is set to 18 μs. According to Equation (3) and the above parameters, the pivot-offset distances on the GS1 and GS2 mirrors are calculated to be 1.8 and 3.9 mm, respectively. The pivot-offset distance is adjusted via the diagonal translation of the scanners to keep the light beam at the center of the focusing optics.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Schematic of the spatial frequency domain multiplexing spectral domain optical coherence tomography (SDOCT) system. SLD: super luminescent diode; FC: fiber coupler; NPBS: non-polarizing beam splitter; M: reference mirror; GS: galvanometer scanner; L: lens; SMP: sample; COL: collimator; PC: polarization controller; DG: diffraction grating; COMP: computer; CMOS: complementary metal oxide semiconductor.

**Figure 2.**Data processing flowchart of the spatial frequency domain multiplexing SDOCT system. FT: Fourier Transform.

**Figure 3.**Sketch of the two reference arms in the spatial frequency domain multiplexing SDOCT system. s

_{1}and s

_{2}: the pivot-offset distances; L1 and L2: the focal lens in the reference arms; M1 and M2: the plane mirrors; f: focal length of the focal lens.

**Figure 4.**Schematic of the choice of the center spatial carrier frequencies. Crosstalk between the spatial spectra occurs in (

**a**), and the two spatial spectra can be well separated through parameters design as shown in (

**b**).

**Figure 5.**The measured sensitivity curves corresponding to the Reference Arm I and Reference Arm II.

**Figure 6.**(

**a**) The photograph of the artificial phantom of a step comprised of two blocks of sellotape. The red line is the lateral scan position on the phantom; (

**b**,

**c**) the cross-sectional optical coherence tomography (OCT) image of the phantom acquired with the conventional single-reference-arm approach; (

**d**) the extended range OCT image acquired with the proposed two-reference-arm spatial frequency domain multiplexing approach. The blurry horizontal white line is the residual direct current (DC) term.

**Figure 7.**Cross-sectionalOCT image of the swine adipose tissue, (

**a**,

**b**) acquired with the conventional single-reference-arm approach. (

**c**) The extended range OCT image is obtained with the two-reference–arm spatial frequency domain multiplexing approach. The blurry horizontal white line is the residual DC terms. (

**d**) A-scans at the position marked by the red arrows in (

**a**,

**b**) demonstrate the advantage of the ranging depth extension.

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**MDPI and ACS Style**

Wu, T.; Wang, Q.; Liu, Y.; Wang, J.; He, C.; Gu, X.
Extending the Effective Ranging Depth of Spectral Domain Optical Coherence Tomography by Spatial Frequency Domain Multiplexing. *Appl. Sci.* **2016**, *6*, 360.
https://doi.org/10.3390/app6110360

**AMA Style**

Wu T, Wang Q, Liu Y, Wang J, He C, Gu X.
Extending the Effective Ranging Depth of Spectral Domain Optical Coherence Tomography by Spatial Frequency Domain Multiplexing. *Applied Sciences*. 2016; 6(11):360.
https://doi.org/10.3390/app6110360

**Chicago/Turabian Style**

Wu, Tong, Qingqing Wang, Youwen Liu, Jiming Wang, Chongjun He, and Xiaorong Gu.
2016. "Extending the Effective Ranging Depth of Spectral Domain Optical Coherence Tomography by Spatial Frequency Domain Multiplexing" *Applied Sciences* 6, no. 11: 360.
https://doi.org/10.3390/app6110360