Absorption can be treated in a similar fashion by multiplying the transmitted power with an additional factor. The reflection of every particle is determined from the collected reflected intensity from every particle, which is , with the power incident on the particle and the fraction of the scattered intensity captured by the collection lens [ 27 ]. Hence, the detected field from the particle is In the model, every particle has a detected field given by the amplitude of 51 and a delay determined by its position.
A simulation of the OCT intensity in the single scattering approximation is shown in Figure 9 where it is averaged over 50 independent realizations of the random particle positions.
The OCT signal in Figure 9 a is in agreement with the single scattering model as the OCT signal decays according to the single scattering model. The height of the numerical OCT signal is with the factor from the discrete inverse Fourier transform. The factor 2 originates from the speckle statistics of the random phasor sum [ 25 ]. The number of particles phasors in a single depth bin is , with the width of a single depth bin. Figure 9 b shows the OCT interference spectrum for the simulated sample and Figure 9 c shows the OCT intensity distribution, evaluated at the point indicated in Figure 9 a , for realizations of the OCT signal.
As expected the distribution of the OCT intensity is an exponential function with a contrast ratio of 1. To demonstrate the use of the numerical OCT model, a two-layered piece of tissue is simulated; see Figure In this article I presented a theoretical and numerical analysis of the most important signal processing steps in Fourier-domain OCT.
This OCT analysis is based on a comparison of the signals in both the - and -domains. Linear spectral sampling and detection is theoretically described and numerically simulated. Good agreement is observed between the analytical model and the numerical simulations. In almost all cases the OCT interference spectrum is nonlinearly sampled. The resulting deterioration of the axial resolution can be removed using nonlinear fast Fourier transforms or, as is most common, linearization of the -domain signal using numerical interpolation [ 30 ].
For nonlinear -domain sampling, the Nyquist depth limit is dependent on the wavenumber as the spectral sampling rate varies over the spectrum. This partial aliasing effect [ 31 ] results in an OCT signal drop at large depths. In general, the effect of nonlinear sampling on the -integration as presented for the case of pixel integration and spectral resolution is not addressed. In most OCT signal analyses the roll-off is described by a -invariant convolution over the wavenumbers, which corresponds to a multiplication of the OCT signal in the -domain.
For small amounts of spectral nonlinearity this is, in general, seen as sufficient to characterize the OCT signal. The quantification of the full effect can be implemented with the presented numerical OCT model by applying a -variant convolution.
Higher order dispersions are easily implemented in the numerical simulations by providing the full material dispersion as described by the Sellmeier equation. In contrast to the usual OCT SNR analysis, which is based on detected power, a -domain representation of the signal to noise is presented for an ideal square source spectrum.
Using the numerical model it is demonstrated how the shot noise of the light detected in the -domain is transformed through the inverse Fourier-transformation to the intensity and amplitude of the complex -domain OCT signal. In this analysis it is shown that for an SNR based on the OCT amplitude, the fundamental shot noise limit is a factor higher than for an intensity based OCT signal analysis [ 19 ]. In case of an intensity based OCT signal, the experimentally determined SNR can be obtained close to the theoretical limit [ 33 ].
For a more realistic Gaussian shaped spectrum the OCT SNR is generally expected to be lower due to the less efficient distribution of optical noise over the detector elements. From the -domain signal and noise description a rigorous derivation of the OCT phase stability is made. In this derivation no use was made of the approximation that the signal is much larger than the noise [ 34 ] or that the noise is orthogonal to the signal [ 35 ].
The derived result is similar to that obtained by Park et al. However, it differs from the result of Choma et al. The numerical model is developed and applied to simulate the OCT signal of a semi-infinite turbid medium. For a semi-infinite turbid medium the simulation matches the single scattering model. The origin of the exponential decay of the OCT signal is well reproduced by modeling the light in the sample after multiple transmission events.
The OCT intensity has an exponential distribution [ 37 ], whereas the amplitude has a Rayleigh distribution, similar to what has been shown by [ 38 ]. The OCT signal intensity from the numerical model incorporates a factor 2 originating from the speckle distribution, which needs to be included in the analytical single scattering OCT model of The one-dimensional OCT model accurately describes OCT measurements of low scattering media for the attenuation [ 20 , 27 ] and the speckle statistics [ 39 ].
The model is simple to use and can easily be adapted for testing OCT attenuation quantification [ 40 ] or tissue segmentation algorithms [ 41 ].
Although it assumes light from the sample arm to be incident perpendicular to the sample, the effect of focusing and back-coupling efficiency [ 22 ] can be easily implemented by adding a depth dependent confocal back-coupling function. The numerical OCT model for a turbid medium does incorporate the effect of dependent scattering in its dependence on [ 42 ], however, multiple scattering effects are not incorporated. More elaborate analytical models [ 43 ] or Monte Carlo simulations [ 44 ] can be used to study the OCT signal in these cases [ 45 ].
The OCT model can be extended to incorporate time dependent scattering processes such as present in the case of Doppler OCT and speckle dynamics [ 46 ]. In conclusion, I presented an overview of analytical expressions for the Fourier-domain OCT signal after sampling, in dispersive media, with noise, and for a scattering medium. A numerical model is developed to simulate the OCT signal. Good agreement is observed between analytical and numerical results.
Inserting the polynomial expansion of 22 up to quadratic order in the variable , is obtained. The first term of the Taylor expansion is The second term of the Taylor expansion is The third term is Hence, the total phase is as used in A cosine sampled at points and period points is given by. The inverse DFT of this cosine is The peak value for positive occurs when the denominator of the second term is zero, that is, when.
With a similar approach it can be demonstrated that. From the definition of the discrete Fourier transform 41 , the correlation between and is calculated as follows: which, for yields zero. Hence and are uncorrelated. For a zero mean Gaussian random variable with variance , the variance of and equivalently is calculated as Using the identities and the variance of is Following a similar derivation it can be demonstrated that the imaginary part has an identical variance.
The phase of the OCT signal in the complex plane is with the real and imaginary parts of the signal with noise. Defining the argument of the arctan as the variation of the angle is Consider the variable to be a random variable with real and imaginary parts having mean and variance. Then, using standard error propagation [ 47 ], is The variance of the phase is derived as When the real and imaginary variances are equal, is which is equal to The author thanks M.
Trull, J. This is an open access article distributed under the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Article of the Year Award: Outstanding research contributions of , as selected by our Chief Editors.
Read the winning articles. Journal overview. Special Issues. Academic Editor: Rainer Leitgeb. Received 28 Sep Revised 01 Dec Accepted 19 Jan Published 22 Mar Abstract In this work the theory of the optical coherence tomography OCT signal after sampling, in dispersive media, with noise, and for a turbid medium is presented.
Introduction Optical coherence tomography OCT is an optical imaging technique that is rapidly progressing into various application fields. Figure 1. Schematic representation of the Fourier-domain OCT system. The length of the reference and sample arm is indicated with and the unidirectional distance from the zero-delay point with. Table 1. Figure 2. The effect of sampling and detection on the OCT signal. The green curve indicates the OCT signal roll-off of Figure 3.
The lines are a guide to the eye. Table 2. Figure 4. Also indicated is the displacement in air green dashed line. Also indicated is the dispersion-free FWHM peak width green line. Figure 5. Figure 6. Schematic of the construction of the OCT phase in the -domain. The OCT signal is represented by the long arrow. A single noise realization is represented by the short arrow and is a vector from the Gaussian distributed noise around the origin. The OCT signal including the noise is constructed from the vectorial addition of the two contributions and leads to the variation of the angle.
Figure 7. Simulation of the phase variance of the OCT signal circles as a function of SNR compared to the theoretical predication of 47 solid line. The inset shows the histogram of the phase distribution of the OCT signal at. Figure 8. Schematic overview of the conversion of the three-dimensional distribution of particles a to a one-dimensional distribution in depth b.
Figure 9. Simulation of the intensity OCT signal. The dashed red line indicates the single scattering OCT model, and the arrow indicates the location where the distribution of the OCT signal is determined. Figure Simulation of the OCT signal intensity for a two-layer turbid sample.
The intensity scale is logarithmic. References D. Huang, E. Swanson, C. Lin et al. Laan, R. Bremmer, M. Aalders, and K. Dubey, D. Mehta, A. Anand, and C. Fercher, W. Drexler, C. Hitzenberger, and T. Tomlins and R. Izatt, M. Choma, and A. Zetie, S. Adams, and R. Oppenheim and A. Huber, B. Biedermann, W. Wieser, C. Eigenwillig, and T. Van Der Horst, J. Bijster, and J. Fercher, C. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T.
Hillman and D. Leitgeb, C. An interference effect fast modulations in intensity are seen at the detector only if the time travelled by light in the reference and sample arms is nearly equal. Thus the presence of interference serves as a relative measure of distance travelled by light. Optical coherence tomography leverages this concept by replacing the mirror in the sample arm with the sample to be imaged Figure 2.
The reference arm is then scanned in a controlled manner and the resulting light intensity is recorded on the detector. The rapid modulation interference pattern occurs when the mirror is nearly equidistant to one of the reflecting structures in the sample, and can be processed to register the presence of that structure. The distance between two mirror locations where interference occurs corresponds to the optical distance between two reflecting structures of the sample in the path of the beam.
Even though the light beam passes through different structures in the sample, the low-coherence interferometry described above helps to distinguish the amount of reflection from each unique structure in the path of the beam. In doing the so, the material scattering, and hence structure, can be measured as a function of depth. Exploring the transverse or x-y localization of the sample structure is simpler. The broadband light source beam that is used in OCT is focused to a small spot on the order of a few micrometers and scanned over the sample.
By mapping the sample in x and y with a scanning arm while collecting depth information using interferometry, a complete 3D picture of the sample can be constructed. Instead of recording intensity at different locations of the reference mirror, the intensity is recorded as a function of the wavelengths or frequencies of the light. The intensity modulations when measured as a function of frequency are called spectral interference. The rate of variation of intensity over different frequencies is indicative of the location of the different reflecting layers in the samples.
It can be shown that a Fourier transform of spectral interference data provides information equivalent to that which would be obtained by moving the reference mirror Figure 3. There are two common methods of measuring spectral interference in OCT: spectral-domain and swept-source.
In spectral-domain OCT SD-OCT , a broadband light source delivers many wavelengths to the sample, and all are measured simultaneously using a spectrometer as the detector. In swept-source OCT SS-OCT , the light source is swept through a range of wavelengths and the temporal output of the detector is converted to spectral interference. Fourier-domain OCT allows for much faster imaging than scanning of the sample arm mirror in the interferometer, as all the back reflections from the sample are being measured simultaneously.
This speed increment introduced by Fourier-domain OCT has opened a whole new arena of applications for the technology. Live video, in-vivo OCT imaging can be easily obtained using commercial systems, allowing it to be used for process monitoring and guided surgery. The axial and transverse resolution of an OCT system are independent. The axial depth resolution is related to the bandwidth, or the coherence length, of the source. It should be noted that this is the spectrum measured at the detector and may differ from the spectrum of the source, due to the response of optical components and the detector itself.
The above equation holds only for Gaussian spectra. For a spectrum of arbitrary shape, the axial spread function should be estimated to understand the achievable resolution and any artifacts like side lobes. Plots of the axial resolution equation for three different central wavelengths is found in figure 5, showing how axial resolution is affected by bandwidth of the source for different common operating bands in the near infrared.
The imaging depth of OCT is primarily limited by the depth of penetration of the light source in the sample. Additionally, in Fourier-domain OCT, the depth is limited by the finite number of pixels and optical resolution of the spectrometer. Optical coherence tomography of the anterior segment of the eye.
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Support Center Support Center. External link. Please review our privacy policy. Intensity information acquired in time domain; no complex conjugate image. No moving reference mirror required; higher sensitivity than TD-OCT; high scanning speed and axial resolution have been attained. No moving reference mirror required; Higher sensitivity than TD-OCT; very high scanning speeds can be attained; minimal signal drop-off with depth.
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