ELEC 8501: The Fourier Transform and Its Applications

Jianbing Xu Sep.21, 2011

Fourier Transform in full range, Fourier-domainOptical Coherence Tomography

Optical Coherence Tomography (OCT) is an emerging imaging modality which can providemicrometer-scale cross-sectional images of tissue microstructures [1]. Compared to ConventionalTime-Domain OCT, Fourier Domain OCT [2] attracts more attention due to its potential for highersensitivity, elimination of depth scanning and higher acquisition speed. However, a major limitationof this technique is that the Fourier transform of the real-valued spectral signal from the balanceddetector is Hermitian symmetric [3], which will results in a ”mirror image” of the sample structurewith respect to the zero phase delay. This will lead to ambiguity in interpreting the reconstructedOCT images, therefore limits the usable ranging depth and requires the sample to be placed entirelywithin the positive or negative space. Full range OCT is a concept which is trying to overcome thisdrawback and thus can make full use of the positive and negative space simultaneously. To achievefull range OCT, complex conjugate spectral interferogram need to be obtained out of the real-valued spectral fringes acquired by detectors. Different methods have been proposed to achieve thecomplex conjugate spectrum to eliminate the “mirror image”. Phase modulation has been provento be an effective way to eliminate mirror images. It modulates the phase of reference arm, andtherefore extra phase shifts can be introduced. By choosing an appropriate carrier frequency, onecan separate the negative and positive terms of the Fourier transformed interference signal fromeach other [4].

In Fourier Domain OCT, the coherence fringe spectrum including phase shifts can be expressedas follows:

𝐼(𝑘, 𝑡) =∑𝑧

𝐼(𝑥,𝑧)(𝑘)𝑒𝑥𝑝{𝑖[𝜑𝑠𝑟(𝑥, 𝑧) + 𝜑𝑠ℎ𝑖𝑓𝑡𝑖𝑛𝑔(𝑡) + 𝜑0]}. (1)

Where k is the wavenumber of the signal, x and z represent the lateral and axial position atthe sample, x and t are related by the equation, 𝑥 = 𝑣𝑥 ⋅𝑡 , where 𝑣𝑥 is a scanning speed of the beamtowards the sample. 𝐼(𝑥,𝑧)(𝑘) is an interference intensity from all points along an A-scan, 𝜑𝑠𝑟(𝑥, 𝑧) isa phase term indicating the phase different between the sample and reference arm during scanning,𝜑𝑠ℎ𝑖𝑓𝑡𝑖𝑛𝑔(𝑡)includes phase shifts both caused by proposed phase modulation and involuntary samplemotion, and 𝜑0 denotes the initial phase. The image sensor can only detect the real part of theintensity value of Eq. (1)

𝐼(𝑘, 𝑡) =∑𝑧

𝐼(𝑥,𝑧)(𝑘) cos{𝑖[𝜑𝑠𝑟(𝑥, 𝑧) + 𝜑𝑠ℎ𝑖𝑓𝑡𝑖𝑛𝑔(𝑡) + 𝜑0]}. (2)

Here we use 𝜑𝑡𝑜𝑡𝑎𝑙 to replace 𝜑𝑠𝑟(𝑥, 𝑧) + 𝜑𝑠ℎ𝑖𝑓𝑡𝑖𝑛𝑔(𝑡) + 𝜑0 for simplicity. To reconstruct thecomplex representation of the signal, one can incorporate Fourier transformation and bandpassfiltering. The Fourier transform of the interference fringes I(𝑘, 𝑡) along time (sample lateral struc-ture)into the frequency space I(k,w) creates two symmetric complex conjugate terms, which wasshown as follows:

𝐼(𝑘,𝑤) =∑𝑧

𝐼(𝑥,𝑧)(𝑘) ⋅ 𝜋 ⋅ [𝛿(𝑤 + 𝜑𝑡𝑜𝑡𝑎𝑙) + 𝛿(𝑤 − 𝜑𝑡𝑜𝑡𝑎𝑙)]. (3)

The two terms of Equation (3) describes complex conjugate artifact images around zero phasedelay. To eliminate one of the complex conjugate terms, a bandpass filter scaled by a factor of 2,

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Eq. (4), is applied to Eq. (3), Here, scaling by two is done intended to preserves the signal energyduring the filtering process.

H(𝑤) =

{2, ∣𝑤 − 𝑤𝑠ℎ𝑖𝑓𝑡𝑖𝑛𝑔∣ < 𝐵

2 .

0, otherwise.(4)

Where 𝑤𝑠ℎ𝑖𝑓𝑡𝑖𝑛𝑔 is an angular frequency by phase shifts, and B is the bandwidth of the filter.The multiplication of Eqs. (3)and (4) will result in

𝐼(𝑘,𝑤) =∑𝑧

𝐼(𝑥,𝑧)(𝑘) ⋅ 𝜋 ⋅ [𝛿(𝑤 + 𝜑𝑡𝑜𝑡𝑎𝑙)]. (5)

Which represents only the positive side of the complex conjugate values. If the inverse Fouriertransform of the filtered output is taken, it will yield a complex representation of the detectedsignal.

𝐼(𝑘, 𝑡) =∑𝑧

𝐼(𝑥,𝑧)(𝑘)𝑒𝑥𝑝{𝑖[𝜑𝑠𝑟(𝑥, 𝑧) + 𝜑𝑠ℎ𝑖𝑓𝑡𝑖𝑛𝑔(𝑡) + 𝜑0]}. (6)

Finally, the inverse Fourier transform along k of Eq. (6)into the z space will be able to generatethe complex conjugate-free image.

The following figures (1) shows an example of how the full range OCT can eliminate “mirrorterms” and thus double imaging depth [4].

Fig. 1 Images of rabbit cornea obtained with FDOCT systems (a) without and (b) with imple-mentation of full range OCT.

References

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

[2] R. Huber, M. Wojtkowski, and J. G. Fujimoto, ”Fourier Domain Mode Locking (FDML): Anew laser operating regime and applications for optical coherence tomography,” Optics Express14(8), 3225-3237 (2006).

[3] D. Y. Kim, J. S. Werner, and R. J. Zawadzki, ”Comparison of phase-shifting techniques forin vivo full-range, high-speed Fourier-domain optical coherence tomography (Journal Paper),” JBiomed Opt 15, 056011 (2010).

[4] J. Zhang, J. S. Nelson, and Z. Chen, ”Removal of a mirror image and enhancement of thesignal-to-noise ratio in Fourier-domain optical coherence tomography using an electro-optic phasemodulator,” Opt Lett 30, 147-149 (2005).

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