ELEC 8501: The Fourier Transform and Its Applications Jianbing Xu Sep.21, 2011 Fourier Transform in full range, Fourier-domain Optical Coherence Tomography Optical Coherence Tomography (OCT) is an emerging imaging modality which can provide micrometer-scale cross-sectional images of tissue microstructures [1]. Compared to Conventional Time-Domain OCT, Fourier Domain OCT [2] attracts more attention due to its potential for higher sensitivity, elimination of depth scanning and higher acquisition speed. However, a major limitation of this technique is that the Fourier transform of the real-valued spectral signal from the balanced detector is Hermitian symmetric [3], which will results in a ”mirror image” of the sample structure with respect to the zero phase delay. This will lead to ambiguity in interpreting the reconstructed OCT images, therefore limits the usable ranging depth and requires the sample to be placed entirely within the positive or negative space. Full range OCT is a concept which is trying to overcome this drawback and thus can make full use of the positive and negative space simultaneously. To achieve full 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 the complex conjugate spectrum to eliminate the “mirror image”. Phase modulation has been proven to be an effective way to eliminate mirror images. It modulates the phase of reference arm, and therefore extra phase shifts can be introduced. By choosing an appropriate carrier frequency, one can separate the negative and positive terms of the Fourier transformed interference signal from each other [4]. In Fourier Domain OCT, the coherence fringe spectrum including phase shifts can be expressed as follows: (,)= ∑ (,) (){[ (, )+ ℎ ()+ 0 ]}. (1) Where k is the wavenumber of the signal, x and z represent the lateral and axial position at the sample, x and t are related by the equation, = ⋅ , where is a scanning speed of the beam towards the sample. (,) () is an interference intensity from all points along an A-scan, (, ) is a phase term indicating the phase different between the sample and reference arm during scanning, ℎ ()includes phase shifts both caused by proposed phase modulation and involuntary sample motion, and 0 denotes the initial phase. The image sensor can only detect the real part of the intensity value of Eq. (1) (,)= ∑ (,) () cos{[ (, )+ ℎ ()+ 0 ]}. (2) Here we use to replace (, )+ ℎ ()+ 0 for simplicity. To reconstruct the complex representation of the signal, one can incorporate Fourier transformation and bandpass filtering. The Fourier transform of the interference fringes I(,) along time (sample lateral struc- ture)into the frequency space ¯ I(k,w) creates two symmetric complex conjugate terms, which was shown as follows: ¯ (,)= ∑ (,) () ⋅ ⋅ [( + )+ ( − )]. (3) The two terms of Equation (3) describes complex conjugate artifact images around zero phase delay. To eliminate one of the complex conjugate terms, a bandpass filter scaled by a factor of 2, 1