US RE42641 E1 Abstract A method is described for determining depth-resolved backscatter characteristics of scatterers within a sample, comprising the steps of: acquiring a plurality of sets of cross-correlation interferogram data using an interferometer having a sample arm with the sample in the sample arm, wherein the sample includes a distribution of scatterers therein, and wherein the acquiring step includes the step of altering the distribution of scatterers within the sample with respect to the sample arm for substantially each acquisition; and averaging, in the Fourier domain, the cross-correlation interferogram data, thereby revealing backscattering characteristics of the scatterers within the sample.
Claims(60) 1. A method for determining depth-resolved backscatter characteristics of scatterers within a sample, comprising the steps of:
acquiring a plurality of sets of cross-correlation interferogramn data using an interferometer having a sample arm with the sample in the sample arm, wherein the sample includes a distribution of scatterers therein, and wherein the acquiring step includes the step of altering the distribution of scatterers within the sample with respect to the sample arm for substantially each acquisition; and
averaging, in the Fourier domain, the cross-correlation interferogram data, thereby revealing backscattering characteristics of the scatterers within the sample.
2. The method of
calculating a transfer function for each set of cross-correlation interferogram data acquired; and
squaring the magnitude of each transfer function; and
averaging the squared magnitudes.
3. The method of
acquiring auto-correlation interferogram data for the interferometer;
generating, from the auto-correlation interferogram data, an auto-power spectrum;
generating, from the set of cross-correlation interferogram data, a cross-power spectrum; and
obtaining a ratio of the cross-power spectrum to the auto-power spectrum.
4. The method of
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13. A method for determining depth-resolved backscatter characteristics of scatterers within a sample, comprising the steps of:
acquiring auto-correlation data from a low-coherence source interferometer, the low-coherence source interferometer including a sample arm;
acquiring multiple cross-correlation data from the low-coherence source interferometer, wherein the low-coherence source interferometer includes a sample in its sample arm;
obtaining an auto-power spectrum for a windowed portion of the auto-correlation data;
obtaining a cross-power spectrum for a windowed portion of each cross-correlation data;
obtaining a transfer function for each cross-correlation data by taking a ratio of the windowed cross-power spectrum to the auto-power spectrum;
squaring each transfer function; and
averaging the magnitude of the squared transfer functions.
14. An optical coherence tomography system comprising:
an interferometer including an optical radiation source and a sample arm, the interferometer generating a plurality of cross-correlation data outputs for a sample in the sample arm; and
a data processing system, operatively coupled to an output of the interferometer, averaging the cross-correlation data outputs, in the Fourier domain, to reveal backscattering characteristics of scatterers within the sample.
15. The optical coherence tomography system of
16. A method for obtaining optical spectroscopic information from cross-correlation data obtained using low coherence interferometry, comprising:
analyzing cross-correlation data to extract spectral information about a sample, said analyzing comprising performing a time-frequency analysis of the cross-correlation data, and directing an intense pump laser to the sample. 17. The method of claim 16, said analyzing comprising taking the Fourier transform of the cross-correlation data.
18. The method of claim 17, further comprising obtaining several sets of cross-correlation data, and said taking the Fourier transform comprising taking the Fourier transform of several of said sets, and said analyzing comprising averaging the Fourier transform results.
19. The method of claim 17, further comprising calculating a transfer function for the cross-correlation data using the Fourier transform of auto-correlation data.
20. The method of claim 16, further comprising demodulating the cross-correlation data prior to performing a time-frequency analysis of the cross-correlation data.
21. The method of claim 20, said demodulating comprising using coherent demodulation method.
22. The method of claim 16, further comprising using an interferometer to acquire cross-correlation data, wherein the interferometer includes a reference arm and a sample arm, and controlling the depth over which cross-correlation data is acquired.
23. The method of claim 22, wherein said controlling includes the step of limiting a scan length of the reference arm to an area of interest in the sample.
24. The method of claim 16, further comprising using an interferometer to acquire cross-correlation data, and windowing the cross-correlation data to an area of interest in the sample.
25. The method of claim 16, further comprising using an interferometer to acquire cross-correlation data, wherein the interferometer includes a reference arm and the method further comprises the step of monitoring reference arm path length.
26. The method of claim 25, wherein the acquiring includes the step of compensating for velocity fluctuations detected during the monitoring step.
27. The method of claim 16, said directing comprising directing laser energy to the sample such that revealed backscattering characteristics will contain features corresponding to inelastic backscattering characteristics of the scatterers within the sample.
28. The method of claim 16, further comprising directing electromagnetic energy to the sample such that revealed backscattering characteristics will contain features corresponding to inelastic backscattering characteristics of the scatterers within the sample.
29. The method of claim 16, further comprising the step of directing a pump laser to the sample to alter backscattering characteristics of the scatterers within the sample.
30. The method of claim 16, further comprising the step of directing a pump laser to the sample, whereby revealed backscattering characteristics will contain features corresponding to inelastic backscattering characteristics of the scatterers within the sample.
31. The method of claim 16, further comprising the step of directing a pump laser to the sample to alter the spectral characteristics of the sample.
32. The method of claim 16, further comprising altering the spectral characteristics of the sample.
33. The method of claim 32, wherein said altering comprises directing laser energy to the sample.
34. The method of claim 32, wherein said altering comprises effecting stimulated emission.
35. The method of claim 34, further comprising adding external dyes or contrast agents to the sample.
36. The method of claim 32, wherein said altering comprises effecting stimulated Raman scattering.
37. The method of claim 32, further comprising adding external dyes or contrast agents to the sample.
38. The method of claim 32, wherein said altering comprises at least one of stimulated emission, stimulated Raman scattering, coherent anti-Stokes Raman scattering, stimulated Brillouin scattering, stimulated Rayleigh scattering, stimulated Rayleigh-wing scattering, and four-wave mixing.
39. A method for determining depth-resolved backscatter characteristics of scatterers within a sample, comprising the steps of:
acquiring a plurality of sets of cross-correlation interferogram data using an interferometer having a sample arm with the sample in the sample arm, wherein the sample includes a distribution of scatterers therein; and averaging, in the Fourier domain, the cross-correlation interferogram data, thereby revealing backscattering characteristics of the scatterers within the sample. 40. The method of claim 39, further comprising the step of physically altering the distribution of scatterers within the sample.
41. The method of claim 39, further comprising the step of repositioning the sample arm.
42. The method of claim 39, further comprising the step of comparing the backscattering characteristics with control data to diagnose abnormalities or disease within the sample.
43. The method of claim 42, further comprising the steps of incorporating a sample probe of the interferometer into an endoscope or surgical instrument, and scanning the endoscope or surgical instrument along a portion of a patient's gastrointestinal tract tissue to diagnose abnormalities or disease within the patient's gastrointestinal tract tissue, wherein the control data includes data corresponding to backscattering characteristics of relatively normal gastrointestinal tract tissue.
44. The method of claim 39, wherein the acquiring cross-correlation interferogram data step or the averaging step includes the step of controlling the depth over which cross-correlation interferogram data is averaged.
45. The method of claim 44, wherein the interferometer includes a reference arm and the controlling step includes the step of limiting a scan length of the reference arm to an area of interest in the sample.
46. The method of claim 44, wherein the controlling step includes the step of windowing the cross-correlation interferogram data to an area of interest in the sample.
47. The method of claim 39, wherein the interferometer includes a reference arm and the method further comprises the step of monitoring reference arm path length, wherein the acquisition step includes the step of compensating for velocity fluctuations detected during the monitoring step.
48. The method of claim 39, further comprising the step of directing an intense pump laser to the sample, whereby the revealed backscattering characteristics will contain features corresponding to inelastic backscattering characteristics of the scatterers within the sample.
49. A method of rapidly determining cross-power spectra from cross-correlation data obtained using low coherence interferometry, comprising the steps of
passing the cross-correlation data through a bank of narrow bandpass filters, and using the output from the narrow bandpass filters as a representation or spectral estimation of cross-power spectrum. 50. The method of claim 49, said passing comprising passing demodulated cross-correlation data, and selecting the center frequency of the bank of narrow bandpass filters according to the demodulation frequency.
51. A method for obtaining information concerning a characteristic associated with a sample from cross-correlation data obtained using low coherence interferometry, comprising:
effecting spectral alterations in the sample from which the cross-correlation data is obtained, and analyzing the cross-correlation data to extract information pertaining to the characteristic associated with the sample. 52. The method of claim 51, further comprising using at least one of dye and contrast agent to enhance said effecting.
53. The method of claim 51, said effecting comprising at least one of using stimulated emission, using stimulated Raman scattering, using coherent anti-Stokes Raman scattering, using stimulated Brillouin scattering, using stimulated Rayleigh scattering, using stimulated Rayleigh-wing scattering, and using four-wave mixing.
54. The method of claim 51, said effecting comprising directing laser energy to the sample.
55. The method of claim 54, said directing of laser energy comprising using a beam splitter to direct both low coherence interferometer light and laser energy to the sample.
56. The method of claim 54, said directing of laser energy comprising using a wavelength division multiplexer to direct both low coherence interferometer light and laser energy to the sample.
57. The method of claim 54, said effecting comprising directing a time varying incident electromagnetic energy input to the sample, and further comprising detecting light from the sample synchronously with the modulation of the incident electromagnetic energy.
58. The method of claim 54, said effecting comprising directing pulsed incident electromagnetic energy input to the sample, and further comprising detecting light from the sample using gated integration technique.
59. The method of claim 54, said effecting comprising directing pulsed incident electromagnetic energy input to the sample, and further comprising, detecting, in a timed relation to the pulsed incident electromagnetic energy, light from the sample.
60. The method of claim 59, said detecting in a timed relation comprising using gated integration technique.
Description This application claims priority under 35 U.S.C. 119 from Provisional Application Ser. No. 60/048,237, filed Jun. 2, 1997, the entire disclosure of which is incorporated herein by reference. Optical imaging of a biological specimen has always been a formidable and challenging task because the complex microscopic structure of tissues causes strong scattering of the incident radiation. Strong scattering in tissue at optical wavelengths is due to particulate scattering from cellular organelles and other microscopic particles, as well as to refractive index variations arising within and between cell and tissue layers. For over a century, conclusive diagnosis of many diseases of cellular origin (such as cancer) has been performed by the process of excisional biopsy, comprising the identification, removal, histological preparation and optical microscopic examination of suspect tissue samples. Many developments have taken place to aid the pathologist in interpretation of histological microstructure, primarily the development of a wide variety of histochemical stains specific to the biochemistry of tissue microstructures. This technique provides sufficient resolution to visualize individual cells within the framework of the surrounding gross tissue structure. In the last several years, a revolution has been stimulated in the field of ultra-high resolution microscopy for biomedical applications. Ultra-high resolution microscopy allows visualization of sub-cellular and sub-nuclear structures. This has resulted in the invention of tools for high resolution optical imaging, including near field scanning microscopy, standing wave fluorescence microscopy, and digital deconvolution microscopy. These technologies are primarily designed for imaging features at or near the surface of materials. Inhomogeneties of the refractive indices inside a biological specimen leading to multiple scattering limit the probing depth of these techniques. Thus, considerable effort is required to cut and preserve the samples in order to prepare the specimen to the requirements of the microscope. In medical applications, this means that suspect tissue sites identified using minimally invasive diagnostic technologies such as endoscopy must still be acquired and processed via routine histological examination. This step introduces significant delay and expense. The invention of confocal microscopy and its advanced development in the past few years have provided the researcher the capability to study biological specimens including living organisms without the need for tissue resection and histological processing. However, the presence of multiple scattering in samples limits confocal microscopy to specimens which are thin and mostly transparent. There is a need, therefore, for new optical methods capable of in vivo imaging deeper inside highly scattering tissues and other biological specimens. Optical coherence tomography (“OCT”) is a technology that allows for noninvasive, cross-sectional optical imaging in biological media with high spatial resolution and high sensitivity. OCT is an extension of low coherence or white-light interferometry, in which a low temporal coherence light source is utilized to obtain precise localization of reflections internal to a probed structure along an optic axis (i.e., as a function of depth into the sample). In OCT, this technique is extended to enable scanning of the probe beam in a direction perpendicular to the optical axis, building up a two-dimensional reflectivity data set, used to create a cross-sectional, gray-scale or false-color image of internal tissue backscatter. Many studies have suggested the use of elastic backscatter or reflectance as a noninvasive diagnostic tool for early detection of several human diseases, including cancer. The use of backscattered light is based on the fact that many tissue pathologies are accompanied by architectural changes at the cellular and sub-cellular level, for example the increase in the nuclear to cytoplasmic volume ratio accompanying neoplastic conversion. In the near infra-red (NIR) zone, the elastic scattering properties of the tissue are most strongly affected by changes in tissue features whose dimensions are on the same order as the NIR wavelength. Preliminary success in diagnosing cancer in the bladder, skin, and gastrointestinal tissues has been reported with techniques based on elastic backscatter spectroscopy. However, currently implemented spectroscopic systems do not incorporate depth resolution and thus cannot provide information on the degree of infiltration or cancer staging. Although elastic backscatter spectra can be collected with confocal techniques, the turbidity of biological samples in combination with the point spread function of confocal microscopes limit the penetration depth for acquiring spatially selective spectra to no greater than a few hundred micrometers in most tissues. Many tissue samples have features of interest located at a depth more than that can be probed by confocal techniques, but less than that of other sub-surface imaging modalities such as ultrasound. Accordingly, there is a need for a spectroscopy system that is capable of obtaining depth-resolved elastic backscatter spectra from a sample. Inelastic scattering processes including fluorescence and Raman spectroscopy have also been exploited for noninvasive disease diagnosis. Unlike elastic scattering events, in which the incident and scattered radiation are at the same frequency, in inelastic scattering events all or part of the incident optical energy is temporarily absorbed by the atoms and/or molecules of the subject tissue, before being remitted at a different (usually lower) optical frequency. Thus, inelastic scattering processes serve as intimate probes of tissue biochemistry. Several studies have reported on laser-induced fluorescence spectroscopy as a potential early cancer diagnostic in the skin, breast, respiratory, gastrointestinal, and urogential tracts. Additional studies have reported on the more biochemically specific Raman spectroscopy for characterization of atherosclerotic lesions in the coronary arteries, as well as for early cancer detection in the gastrointestinal tract and cervix. In all studies of fluorescence, Raman, and other inelastic scattering spectroscopies in human tissues to date, means have not been available to resolve the depth of the scattered signal with micrometer-scale resolution. There is thus a need for a spectroscopy system capable of obtaining depth-resolved inelastic backscatter spectra as well as elastic backscatter spectra from a sample, such as could be obtained by extending Optical Coherence Tomography to detect inelastically scattered light. The depth resolved elastic and inelastic backscattering spectroscopic information could aid in the detection of the shapes and sizes of lesions in an affected organ and could thus assist in accurate staging of diseases such as cancer. The inelastic scattering spectroscopies based on spontaneous fluorescence and spontaneous Raman scattering which have been used in medical diagnostic applications to date are not suitable for combination with Optical Coherence Tomography because they are incoherent scattering processes, and thus the scattered light would not be detected with OCT. However, coherent inelastic scattering processes do exist, in particular the process of stimulated emission is the coherent analog of spontaneous emission, and stimulated Raman scattering is the coherence analog of spontaneous Raman scattering. Other stimulated coherent scattering processes also occur which may find future application in medical diagnostics, for example coherent anti-Stokes Raman scattering (CARS) and four-wave mixing (FWM). All of these coherent inelastic scattering processes require the presence of pump energy which is converted into signal energy in a coherent gain process. Thus, by virtue of their coherence, stimulated coherent gain processes are suitable for combination with Optical Coherence Tomography to allow for depth resolution of the location of the inelastic scattering events. Therefore, there exists a need for a system which allows for this combination. The present invention provides a technique for depth-resolved coherent backscatter spectroscopy. This technique is an extension of OCT technology. U.S. patent application Ser. No. 09/040,128, filed Mar. 17, 1998, the disclosure of which is incorporated herein by reference, describes an improved OCT system that utilizes a transfer function model, where the impulse response is interpreted as a description of actual locations of reflecting and scattering sites within the tissue. Estimation of the impulse response provides the true axial complex reflectivity profile of the sample with the equivalent of femtosecond ranging resolution. An interferogram obtained having the sample replaced with a mirror, is the auto-correlation function of the source optical waveform. The interferogram obtained with the tissue in the sample is the measured output of the system and is known as the cross-correlation function. By deconvolving an impulse response profile from the output interferometric signal, a more accurate description of the tissue sample is obtained. This model assists in calculating the spectral characteristics of optical elements over the bandwidth of the source by analysis of the interference resulting from internal tissue reflections. This model may also be extended to spectrally analyze light backscattered from particles in turbid media. Because individual scatterers in a turbid specimen may be considered as being essentially randomly distributed in space, the ensemble average of transfer functions obtained from cross-correlation data windowed to a specific region within the sample reveals the backscattering characteristics of the scatterers localized to that region. From this model, it is determined that the squared magnitude of the frequency domain transfer function correlates with the backscatter spectrum of scatterers. Accordingly, the present invention provides a system or method for determining depth resolved backscatter characteristics of scatterers within a sample which includes the means for, or step of averaging (in the Fourier domain) the interferogram data obtained over a region of the sample. In one embodiment of the present invention, the system or method includes the means for, or steps of: (a) acquiring auto-correlation interferogram data from the Low-Coherence interferometer; (b) acquiring multiple sets of cross-correlation interferogram data from the Low-Coherence interferometer having the sample under analysis in the sample arm, where the distribution of scatterers within the sample has been altered for each acquisition (e.g., by squeezing or stretching the tissue sample) or where the sample arm is repositioned slightly for each acquisition; (c) obtaining an auto-power spectrum by performing a Fourier transform on auto-correlation data; (d) obtaining a cross-power spectrum for the windowed portion of each cross-correlation data by performing a Fourier transform on the windowed portion of each cross-correlation data set; (e) obtaining a transfer function from the ratio of each cross-power spectrum to the auto-power spectrum; (f) squaring each transfer function obtained in step (e) and (g) averaging the magnitude of the squared transfer functions to reveal backscattering characteristics of scatterers resident within that window. Based upon the model described above, any form of coherent spectroscopy can be performed in a depth-resolved manner. Accordingly, the present invention also provides a system and method for performing stimulated-emission spectroscopic optical coherence tomography (SE-SPOCT). Such a system or method includes the means for, or steps of directing an intense pump laser at the appropriate frequency to induce depth- and frequency-dependent gain in the sample volume interrogated. The depth resolved spectrum obtained according to the steps discussed above will thus contain features corresponding to the frequency-dependent round-trip gain experienced by the OCT source radiation (inelastic backscattering characteristics of the scatterers resident within the window). In a detailed embodiment of SE-SPOCT, the pump laser is cycled on and off and gated detection is performed to separate the elastic backscattering characteristics from the inelastic backscattering characteristics. Additionally, the pump laser may be modulated at a certain frequency and gated detection is performed to separate the elastic backscattering characteristics from the inelastic backscattering characteristics. Similar to SE-SPOCT, the present invention also provides a system or method for performing stimulated Raman scattering spectroscopic OCT (SRS-SPOCT). In this embodiment, the system or method includes the means for, or the step of directing a high intensity pump light into the sample interaction region. The depth resolved spectrum obtained according to the steps discussed above will thus contain localized peaks in the depth-resolved backscatter spectrum that provide substantial vibrational/rotational spectral information of the scatterers within the sample. Accordingly, it is an object of the present invention to provide a system and method for acquiring depth-resolved backscatter spectra of a sample, utilizing OCT. It is a further object of the present invention to provide a system and method for performing stimulated-emission spectroscopic OCT. And it is a further objective of the present invention to provide a system or method for performing stimulated Raman scattering spectroscopic OCT. These and other objects and advantages of the present invention will be apparent from the following description, the attached drawings and the appended claims. I. Michelson Interferometer As shown in Those of ordinary skill in the art will recognize that, although it is preferred to scan the sample probe In developing the present invention, a unique transfer function model has been developed for OCT interaction with the sample, where the impulse response is interpreted as a description of the actual locations of the reflecting and scattering sites within the sample. Based upon this model, the transfer function of the system can be calculated from the source auto-power spectrum and the cross-power spectrum of the electric fields in the reference and sample arms. The estimation of the impulse response from the transfer function provides the true axial complex reflectivity profile of the sample with the equivalent of femtosecond temporal resolution. Extending this model, it is determined that the squared magnitude of the frequency domain transfer function correlates with the backscatter spectrum of scatterers within the sample. In particular, because individual scatterers in a turbid specimen are randomly distributed in space, the ensemble average of transfer functions obtained from cross-correlation data windowed to a specific region within the sample reveals the backscattering characteristics of the scatterers localized to that region. II. Model for Low Coherence Interferometry in Thick Scattering Media The present invention is based on a systems theory model which treats the interaction of the low coherence interferometer with the specimen as a linear shift invariant (LSI) system. For the scan lengths of a few mm, typical of OCT imaging, it is assumed that group velocity dispersion is negligible and the group and phase velocities are equal. In the NIR region absorption causes negligible attenuation as compared to the attenuation due to multiple scattering within tissues. This model does not take into account the attenuation of light due to multiple scattering as well as absorption. An optical wave with an electric field with space and time dependence expressed in scalar form as 2{tilde over (e)} The fields returning from the reference and sample arms again get separated 50/50 into the arms consisting of the source and detector. Therefore the interference of {tilde over (e)} Note that {tilde over (R)} We represent the optical field interaction with the LSI sample as a transfer function {tilde over (H)}(k) (where k is wavenumber) whose inverse Fourier transform is the impulse response {tilde over (h)}(z):
where represents convolution. Note that shift invariance allows omission of the terms vt in this expression. The convolution theorem leads to {tilde over (E)} _{s}(k)={tilde over (E)}_{i}(k){tilde over (H)}*(k) (c)
where {tilde over (E)} _{s}(k), {tilde over (E)}_{i}(k), and {tilde over (H)}(k) are the Fourier transforms of {tilde over (e)}_{s}(z), {tilde over (e)}_{i}(z), and {tilde over (h)}(z), respectively. {tilde over (H)}(k) is the system transfer function. The LSI assumption provides:
{tilde over (e)} _{s}(vt−2l_{s})={tilde over (e)}_{i}(vt−2l_{s}){tilde over (h)}*(−(vt−2l_{s})) (d)
Inserting Eq. (d) in Eq. (a) leads to {tilde over (R)} _{is}(Δl)={tilde over (R)}_{ii}(Δl){tilde over (h)}(Δl), {tilde over (S)}(k)={tilde over (S)}_{ii}(k){tilde over (H)}(k) (e)
{tilde over (H)}(k)={tilde over (S)} _{is}(k){tilde over (S)}_{ii}(k) (f)
Note that according to the Wiener-Khinchin theorem, Fourier transforming the autocorrelation and cross-correlation functions gives us {tilde over (S)} This analysis will be useful when raw interferograms are measured. It is convenient to measure the complex envelopes of the interferometric data by demodulating {tilde over (R)} R Therefore let's analyze the system using complex envelopes of the electric fields and the impulse response. We represent the optical field complex envelope interaction with the LSI sample as a transfer function H(k) (where k is wavenumber) whose inverse Fourier transform is the impulse response h(z):
_{s}(k)=√2E_{i}(k)H*(k) (1a)
where represents the convolution operation. E _{s}(k) and E_{i}(k) are Fourier transforms of e_{s}(z) and e_{i}(z), respectively. Note that {tilde over (h)}(vt−z)=h(vt−z)exp[j(k_{0}(vt−z)] and {tilde over (H)}(k)=H(k+k_{0}). Also the LSI assumption leads to
√2e _{s}(vt−l_{s})=√2e_{i}(vt−l_{s})h(−(vt−l_{s})). (1b)
The fields returning from the reference and sample arms again get separated 50/50 into the arms consisting of the source and detector. Therefore the interference of {tilde over (e)} _{i }(vt−2l_{r}) and {tilde over (e)}_{s }(vt−2l_{s}) is incident on the detector. The reference arm length can be varied by various means. One such method is mechanically scanning the reference mirror causing sweeping of the reference arm length l_{r}. Since the detector response time (e.g., nanosecond to microsecond for typical optical receivers) is much longer than the optical wave period (˜10^{−15 }second) the complex envelope of the photocurrent generated by a square law detector is proportional to
i _{D}˜<[e_{i}(vt−2l_{r})+e_{s}(vt−2l_{s})][e_{i}(vt−2l_{s})+e_{s}vt−2l_{s})]*>, =<[e_{i}(vt)+e_{s}(vt+D_{l})][e_{i}(vt)+e_{s}(vt+D_{l})]*>, (2)
where <> denotes integrating over the detector response time which is long compared to the electric field period, and 2(l _{s}−l_{r})=D_{l }is the round trip optical path length difference. When the reference arm length is scanned at a constant velocity, after filtering out the dc components, the time varying components of Eq. 2 reduce to
i _{D}˜R_{is}(Δl)=<e_{i}(vt)e_{s}(+Δl)> (3)
which is just the cross-correlation between the complex envelopes of the fields returning from the reference and sample aims. i _{D }is the complex envelope of the corresponding current at the photoreceiver output. We can also obtain the autocorrelation function of the source field R_{ii}(Dl) by performing the same operation with a mirror in the sample arm, in which case h(z)=d(z) and e_{s}(z)=e_{i}(z)R_{ii}(Δl)=<e_{i}(vt)e_{i}*(vt)+Δl)>. For an optical source with a well characterized spectrum, the form of the autocorrelation function R_{ii}(Dl) is calculated explicitly by computing the inverse Fourier transform of the power spectral density. For a superluminescent diode source approximated by a Gaussian power spectrum, we obtain
According to the Wiener-Khinchin theorem, Fourier transforming the autocorrelation and cross-correlation functions gives us S _{ii}(k) and S_{is}(k). S_{ii}(k) and S_{is}(k) are the auto-power and cross-power spectral densities, respectively. We form an estimate of the transfer function H(k) in an arbitrary turbid sample which may contain many closely spaced reflections by Fourier transforming both sides of Eq. 5:
S _{is}(k)=S_{ii}(k)H(k), H(k)=S_{is}(k)/S_{ii}(k), h(z)H(k) (6)
where a Fourier transform pair is related by . In practice, the correlation functions defined in Eq. a1, a2 and 3 are hard to measure. The measured (or estimated) correlation functions are influenced by the properties of the optical elements, the measurement electronics, and data acquisition systems, and various noise sources. Therefore what we measure are “estimates” of {tilde over (R)} Similarly the measured (or estimated) power spectra are influenced by the properties of the optical elements, the measurement electronics, and data acquisition systems, and various noise sources. Therefore what we measure are “estimates” of {tilde over (S)} While we describe the specific case of a device which uses infra-red light source, the spectroscopy procedure is applicable to any interferometric device illuminated by any electromagnetic radiation source. In Eqs. b, c, 1a, 1b, and 5 we describe the light-specimen interaction as a linear shift invariant system. We describe the deconvolution methods based on Eqs. e,f, and 6. It should be apparent to a person skilled in the art that the interaction described by Eqs. b, c, e,f, 1a, 1b, 5, and 6 can be exploited by many other methods in space/time domain as well as frequency domain including iterative deconvolution methods, etc. This model also forms the basis of “blind” deconvolution methods which do not use a priori information about the auto-correlation function but assume that it convolves with the impulse response. The true transfer function H(k) is rarely estimated or measured. In most practical cases, what we get is an “estimate” of the transfer function which is different than the true transfer function. One can obtain this estimate in various ways. One such method is taking the ratio of cross-power spectrum and the auto-power spectrum and taking the complex conjugate of the ratio. Next we interpret the impulse response h(z) and the transfer function H(k). All information regarding the spatially varying and frequency dependent complex reflectivity of the sample is contained in the space domain function h(z). The interpretation for deconvolution is an approximation of that for spectroscopy. In the case of deconvolution, we model tissue as a body having several layers of materials possessing different refractive indices. The impulse response can be interpreted as a description of the actual locations and reflectivities of reflecting sites within the sample arising from index of refraction inhomogeneities. The impulse response takes a form of a series of spikes (i.e., delta functions) which are located at the reflection sites while dealing with discrete data. These spikes have he In order to describe the impulse response quantitatively, it is convenient to deal with discrete representations of the impulse response and crosscorrelation function. It will also be helpful to do so since we measure quantized values of the discrete representations of cross-correlation functions using a computer. We assume that M samples of R R _{ii}(n) and h(n) are discrete representations of the autocorrelation function and the target impulse response, respectively. Now, as discussed earlier, h(n) assumes a form of a series of spikes (i.e., discrete delta functions) which are located at the reflection sites. In a tissue sample, one usually does not find a reflection site at every sample. Therefore the probability of occurrence of an interface at a sample is much less than one. A random sequence of zeros and ones is known as a Bernoulli sequence. If the adjacent elements in such a sequence are completely unrelated, then such a sequence is known as a white sequence (since the power spectrum of such a sequence is white). We could possibly represent the impulse response by a Bernoulli event sequence b(n). The sequence provides a one every time an interface occurs and a zero in the absence of a reflector at the sample point. However, the amplitude of the reflectivity is not constant and fluctuates randomly due to refractive index inhomogeneities and hence we need to multiply this sequence by a Gaussian random number generator g(n), g(n) is also a white sequence. Therefore
h(n)=b(n)g(n). (8) This essentially means that when at every point in b(n) a one occurs (i.e., a reflection site occurs), we turn on our Gaussian random number generator and replace the one by the output of the generator. The value of this random number represents the reflectivity at that point. If the reflectivities are complex, one can generate a complex Gaussian random number. We are assuming that all reflections occur at the interfaces and the effects of point scatterers do not interfere with the process of impulse response estimation. In order to perform spatially resolved spectroscopy, a more complicated interpretation of the transfer function is required. For a monochromatic wave with an amplitude of one and zero initial phase incident on a reflector, the reflected field {tilde over (e)} Let us examine the complex envelope h(n) quantitatively. For a homogeneous medium, we can write
where b(n) is a Bernoulli sequence as discussed earlier and c(n) is the inverse Fourier transform of C(k). C(k) denotes spectrally dependent electric field backscattering coefficient of the scatterers. We call c(n) as the specific impulse response of a single scatterer. Thus the frequency dependent backscattering cross-section of particles is represented by |C(k)| ^{2}. The Fourier transform of b(n) is given by B(k) and E{|B(k)|^{2}} represents the expectation value (i.e., the statistical average) of |B(k)|^{2 }and is equivalent to the power spectral density of the Bernoulli sequence. For a white sequence, this can be assumed to be equal to a constant (in this case can be assumed to be 1 for simplicity) for all wavenumbers. Fourier transforming both sides of Eq. 10 provides
{tilde over (H)}(k)=B(k)C(k). (11) Taking magnitude square of both sides gives, |{tilde over (H)}(k)| ^{2}=|B(k)|^{2}|C(k)|^{2}. (12)
Obtaining ensemble averages on both sides yields, E{|{tilde over (H)}(k)| ^{2}}=E{|B(k)|^{2}|C(k)|^{2}}. (13)
Since C(k) is not a randomly varying function, we get E{|H(k)| ^{2}}=E{|B(k)|^{2}}|C(k)|^{2}=1+|C(k)|^{2}=C(k)|^{2}. (14)
Thus averaging several measurements of |H(k)| Now as indicated in Eq.(c), {tilde over (H)}(k) is the quantity that actually interacts with the electric field. Using
|{tilde over (C)}(k)| In an inhomogeneous medium (such as a tissue specimen), a mixture of various particles within a few source coherence lengths can be described as
_{1}(n)+b_{2}(n)c_{2}(n)+b_{3}(n)c_{3}(n) (15)
where b _{i}(n) {i is a natural number} is a Bernoulli sequence describing positions of ith type of scatterers having the specific impulse response c_{i}(n). Let b_{i}(n) be white processes. Suppose these processes are statistically independent of each other. Fourier domain representation of Eq. 15 is
{tilde over (H)}(k)=B _{1}(k)C_{1}(k)+B_{2}(k)C_{2}(k)+B_{3}(k)C_{3}(k)+ (16)
where B _{i}(k) and C_{i}(k) are Fourier transforms of b_{i}(n) and c_{i}(n), respectively. Taking magnitude squares on both sides yields,
The lower row on the right hand side of Eq. 18 can be written as Again as indicated in Eq.(c), {tilde over (H)}(k) is the quantity that actually interacts with the electric field. Using {tilde over (H)}(k)=H(k+k
In summary, for a monochromatic wave incident on a reflector, {tilde over (H)}(k) is the reflection coefficient (i.e., backscattering coefficient) at a wavenumber k. {tilde over (H)}(k) is real for many particles. |{tilde over (H)}(k)| The impulse response h(z) essentially is a convolution of two functions, viz., a function b(z) which describes locations of scatterers and a function c(z) which is the inverse Fourier transform of C(k). C(k) denotes spectrally dependent electric field backscattering coefficient of the scatterers. The wavenumber dependent backscattering cross-section of the particles is represented by |C(k)| ^{2 }for a white sequence is assumed to be equal to a constant (1 for simplicity), averaging estimates of |H(k)|^{2 }(obtained from different locations in a homogeneous section of the specimen) provides an estimate of |C(k)|^{2 }which is the elastic backscatter spectrum of scatterers residing within that homogeneous region. The short source coherence length allows to control the depth over which the spectral information needs to be measured. This is achieved by selecting the reference arm scan to an area of interest in the sample, or alternatively by windowing the area of interest from one or more full length reference arm scans. While computing the spectra of scatterers located deep inside a specimen, one should note that C(k) has the information regarding the backscatter spectrum of scatterers as well as round-trip spectral filtering due to the intervening medium. Thus the interferometric signal contains depth resolved information about both the spatial distribution of scattering centers within a tissue sample, as well as about the spectral characteristics of the individual scatterers.
The system design gets simplified if we coherently demodulate the interferometric data. C(k) would be estimated using such a system. However, |{tilde over (C)}(k)| The above model also allows one to measure actual spectrum of the light in the sample arm. The spectrum of light in the sample arm is defined as
_{ss}(k)={tilde over (E)}_{s}(k){tilde over (E)}*_{s}(k)
Using {tilde over (E)} _{s}(k)={tilde over (E)}_{i}(k){tilde over (H)}*(k) we get
_{ss}(k), i.e., compute E{{tilde over (E)}_{s}(k){tilde over (E)}*_{s}(k)}. We denote E{{tilde over (E)}_{s}(k){tilde over (E)}*_{s}(k)} by {tilde over (S)}_{ss}(k), i.e., {tilde over (S)}_{ss}(k)=E{ _{ss}(k)}
Thus S Following from the above model, the present invention provides a system and method for determining depth resolved backscatter characteristics of scatterers within a sample by averaging (in the Fourier domain) the interferogram data obtained over a region of the sample. In particular, the backscatter characteristics are obtained according to the steps as illustrated in As shown in As will be appreciated by those of ordinary skill in the art, there are several ways to obtain the auto-power spectrum for an OCT system, all of which are within the scope of the present invention. For example, the auto-correlation data can be measured using a strong reflector which is a part of the specimen itself; the auto-correlation function can be modeled using the information about the radiation source; and the auto-correlation function can be also calculated using the knowledge of the source power spectral density. For instance, the inverse Fourier transform of the measured source power spectrum would provide an estimate of the auto-correlation function. Additionally, since the auto-power spectrum is nothing but the source power spectrum, the auto-power spectrum can be obtained using the knowledge of the source. For instance, the source power spectrum measured using any spectrometer or a spectrum analyzer would provide an estimate of the auto-power spectrum. As indicated in steps The number of sets of cross-correlation data to obtain depends upon the desired estimation accuracy of E{|{tilde over (H)}(k)| If, in step As indicated in step As indicated in step As indicated in step As shown in The fiber It will be apparent to one of ordinary skill in the art that there are many known methods and/or mechanisms for injecting the above reference arm delay, other than a translating reference mirror. All of these methods, of course, are within the scope of the present invention. Alternative reference arm optical delay strategies include those which modulate the length of the reference arm optical fiber by using a piezo-electric fiber stretcher, methods based on varying the path length of the reference arm light by passing the light through rapidly rotating cubes or other rotating optical elements, and methods based on Fourier-domain pulse-shaping technology which modulate the group delay of the reference arm light by using an angularly scanning mirror to impose a frequency-dependent phase on the reference arm light after having been spectral dispersed. This latter technique, which is the first to have been shown capable of modulating the reference arm light fast enough to acquire OCT images at video rate, depends upon the fact that the inverse Fourier transform of a phase ramp in the frequency domain is equal to a group delay in the time domain. This latter delay line is also highly dispersive, in that it can impose different phase and group delays upon the reference arm light. For such a dispersive delay line, the OCT interferogram fringe spacing depends upon the reference arm phase delay, while the position of the interferogram envelope at any time depends upon the reference arm group delay. All types of delay lines can be described as imposing a Doppler shift frequency The analog interferogram data signal The cross-power spectrum data is then sent to a processing algorithm Each estimate of the transfer function H(k) Once all of the transfer function estimates {tilde over (H)}′(k) Note that the operations described herein have been performed and tested in software using packages such as LabVIEW and MATLAB. It is also within the scope of the invention that these operations be performed by using hardware DSP devices and circuitry. For example, the Fourier transform algorithm The backscatter spectrum C(k) It will be apparent to those of ordinary skill in the art that a demodulation step/device may be incorporated to the system of A preliminary demonstration of depth resolved spectroscopy using OCT is provided in IV. Depth Resolved Backscatter Fourier Transform Spectroscopy Application In many diseases structural changes occur at the cellular and sub-cellular level in the affected organ. Examples of such changes include enlargement and changes in shapes of nuclei in colonic adenoma (which is a precursor to colonic cancer) and an increased population of inflammatory cells in ulcerative colitis. These scatterers in the tissue have dimensions comparable to the near infra-red wavelengths of the light source (e.g., 1250 nm to 1350 nm for the SLD in our laboratory). Therefore the scattering process is best described by the phenomenon of “Mie scattering” for which the backscattering cross-sections of these scattering sites are highly frequency dependent. In a typical histopathological assessment, changes in cellular structure are examined. These include changes in cellular as well as nuclear sizes and shapes and clustering patterns of the cells or nuclei. Variations in morphology affect the elastic scattering properties of cells. This variation in backscattering properties of the tissue microstructure could be exploited to diagnose various diseases. Depth resolved elastic backscattering spectroscopic information could aid in detecting the shapes and sizes of the lesions in an affected organ and thus assist in accurate staging of diseases such as the cancer. Extracellular architecture as well as the sizes and shapes of cells and cellular components exhibit variations in different tissue types as well as in different layers within the same tissue. The frequency dependence of the elastic backscattering properties of biological materials is closely related to these morphological changes. Therefore this spectral information can be used in detecting shapes, sizes or refractive indices of various particles at different depths in the tissue specimen. This can be achieved since the backscattered spectrum is a function of shape, size and the refractive index of the particles and the refractive index of the surrounding medium. This spectral information may also provide contrast mechanisms based on differential backscattering spectroscopic properties of the sites localized within different depth regions of the specimen. Range gated spectra thus obtained can be used to develop various contrast enhancement mechanisms. For instance, an OCT image obtained with complex envelope information can be displayed as a gray scale image using amplitude information. It is possible to compute spectra resolved at different depths in regular intervals using complex envelope data. This spectral information can be “color” coded (e.g., one can encode the peaks or widths of the spectra in various colors). Thus the gray scale image can be supplemented with this “color” information resulting in a colored OCT image. Better differentiation between the layers in OCT images could be achieved by using such a color information. This spectral information could also be used to complement the diagnostic capability of OCT. Three dimensionally resolved spectroscopy could assist in studying the extent of infiltration of a disease such as cancer, database of spectral signatures of various layers of normal and abnormal tissue samples. The pathological states at different layers could be determined by comparing the spectra acquired at these layers with those in the database. Two dimensional lateral scanning of the tissue samples may provide the degree of invasion of diseases accurately. V. Methods to Obtain Spectra at a Given Depth As described in section III, the cross-correlation function {tilde over (R)}′ Here we elaborate on a method to obtain segments of {tilde over (R)}′ Next, at a predetermined point past the starting depth, another depth window of {tilde over (R)} This operation can be summarized by the following equation (known as windowed Fourier transform (WFT) equation): Thus N samples of {tilde over (S)} If {tilde over (R)}′ Here we elaborate on a method to obtain segments of {tilde over (R)}′ Next, at a predetermined point past the starting depth, another depth window of R This operation can be summarized by the following equation: Thus the spatial resolution of the spectral estimate is given by the window size (NDz), the larger the window—the lower the spatial resolution. But spectral estimation precision k Several types of analysis windows may be used in the circuit/algorithm, including rectangular, Bartlett (triangular), Hamming, Hanning, Blackman windows. It is well known to those skilled in the art that the choice of window may affect the power spectrum estimation accuracy. In our experiments we used a simple rectangular window. An alternative implementation of the window is to pad the N-point analysis window with zeros on either side, increasing its length in order to enhance the frequency precision. The size (length) of the window is indicated as NDz which must be shorter than the entire A-scan length L. While a user may choose the window length as short as he wishes, it may be apparent to those of ordinary skill in the art that due to the interference from the reflectors located within a coherence length, the axial resolution is limited by the coherence length. Therefore it is desirable to choose the window length longer than or equal to the coherence length. The above steps for converting the interferogram data into power spectrum data may be performed by a bank of narrow-band band-pass filters (NBPF), where each NBPF passes a particular wavenumber along the power spectrum wavenumber scale. The outputs of each NBPF may be input directly into the transfer function calculation. This method eliminates the need for the windowed Fourier transform circuit/algorithm and, may also eliminate the need for the coherent demodulation circuit/algorithm. Thus, this method provides faster and cheaper signal processing. These NBPF may also be implemented by a bank of demodulators and a corresponding bank of low-pass filters, where each demodulator demodulates the data at a particular wavenumber along the power spectrum frequency scale and each corresponding low-pass filter filters the output of the demodulator to only pass a narrow band of wavenumbers as desired by users. These NBPF can be applied directly to {tilde over (R)}′ Thus in this alternate embodiment of the present invention, the necessity of a windowed Fourier transform step to produce the power spectra may {tilde over (S)} As shown in Alternatively, as shown in If f Finally, as shown in It will be apparent to one of ordinary skill in the art that the array of complex NBPFs may be replaced by two arrays of NBPFs, one array for the in-phase data, and one array for the quadrature data; The frequency f The banks of NBPF, discussed above, may also be replaced by a bank of demodulators and low-pass filters, where the demodulation frequency is same as the center frequency of the corresponding BPF and each corresponding low-pass filter has a bandwidth as desired by users. It should be obvious to one skilled in the art that the approach of using a bank of NBPF's or demodulators to obtain spatially-localized frequency information is quite general, and is not necessarily limited to the case of using all NBPF's symmetrically distributed around the reference arm Doppler shift frequency, or around baseband or at evenly spaced frequencies, or even all with the same pass bandwidth. The user may have the option to select any frequency to monitor, with any bandwidth desired. VI. Method to Measure Actual Spectrum of the Light in the Sample Arm Following from the model described above in Section II, the present invention also provides a system/method for determining or measuring the actual spectrum {tilde over (S)} As shown in It is advantageous, in the above data processing scheme, that the auto-correlation and cross-correlation functions be measured with the sub-micron accuracy. Therefore, to enhance the accuracy of the low coherence interferogram acquisition, a long coherence length calibration interferometer As shown in The illumination source The analog signal As will be apparent to those of ordinary skill in the art, there are many way to use the calibration interferometer VIII. Stimulated Coherent Spectroscopic Optical Coherence Tomogaphy (SC-SPOCT) The present invention also provides a system and method for performing three-dimensionally resolved coherent spectroscopy by taking advantage of the depth-resolving capability of OCT. In principle, any coherent scattering process may be detected by OCT, as long as the scattered light remains within the bandwidth of the OCT source light spectrum. In particular, we disclose a method for depth-resolving stimulated coherent scattering processes using a system based upon the SPOCT concept. Stimulated coherent scattering processes involve a transfer of energy from photons in an intense pump beam and from excited states of an atom or molecule to a typically weaker probe beam. The probe photon frequency is typically Stokes shifted (i.e., is at a lower frequency) from the pump photon frequency. Examples of stimulated coherent scattering processes include stimulated emission, stimulated Raman scattering, coherent anti-Stokes Raman scattering (in which case the probe photon frequency is higher than the pump photon frequency) stimulated Brillouin scattering, stimulated Rayleigh scattering, stimulated Rayleigh-wing scattering, four-wave mixing, and others which are well known to those practiced in the art. In typical stimulated scattering experiments, the pump photons are provided by an intense laser pump source, and the probe photons are either provided by a weak probe beam or are provided by incoherent scattering processes or noise. The operation of a laser, for example, is based on stimulated emission of radiation at the laser oscillation frequency which builds up from optical noise present in the laser cavity due to a spontanous (i.e., incoherent) emission background. In stimulated Raman scattering experiments, the probe beam may either be supplied as a weak source at a frequency corresponding to the Raman transition to be interrogated in the sample, or it may also be allowed to build up from incoherent spontaneous Raman scattering noise. In either case, the intense pump radiation sets up a condition in which the probe experiences frequency-dependent coherent gain in the medium. The frequency dependence of the gain is determined by the medium's specific atomic and molecular composition; thus probing the frequency dependence of the stimulated gain may serve as a sensitive probe of tissue biochemistry with applications in medical diagnostics. In typical experiments of this type, the probe radiation is monochromatic and the gain experienced in traversing an excited medium is measured using either direct, gated, or synchronous detection techniques as the frequency of the probe radiation is scanned. The primary idea of stimulated coherent spectroscopic OCT is to take advantage of the broad spectral content of OCT probe light in combination with a separate pump light beam to perform stimulated coherent spectroscopy over the whole source spectrum at once, while simultaneously using the short coherence length of the OCT probe light to depth resolve the resulting stimulated scattering spectrum. Referring again to For stimulated scattering processes which do not require phase matching between the pump and probe beams, the pump beam may be directed into the sample at any angle with respect to the probe beam. A convenient design is thus to combine the pump and probe beams coaxially, as illustrated in The concept and process of SC-SPOCT is illustrated schematically in As illustrated in In general, the spectrum of light returning from the sample {tilde over (S)}
The actual stimulated coherent gain experienced in tissues in most cases will be very small, perhaps altering the sample arm power spectrum by as little as 1 part in 10 Methods for using gated or synchronous detection to achieve this objective are illustrated in Gated detection of stimulated coherent scattering is illustrated schematically in Synchronous detection of stimulated coherent scattered light is illustrated in IX. Stimulated-Emission Spectroscopic Optical Coherence Tomography (SE-SPOCT) Stimulated emission spectroscopic optical coherence tomography (SE-SPOCT) is a specific implementation of stimulated coherent spectroscopic OCT which obtains a depth-resolved stimulated emission spectrum of a sample. SE-SPOCT may be used to image uptake of high quantum efficiency laser dyes or infrared-emitting exogenous fluorescent probes into biological specimens. Examples of the latter include probes for measurement of intra-cellular pH (e.g. carbocyanine, with an excitation maximum at 780 nm and an emission maximum at 795 nm), and nucleic acids (e.g. IR-132, with its excitation maximum at 805 nm and its emission maximum at 835 nm). A suitable pump laser X. Raman Scattering Spectroscopic OCT(SRS-SPOCT) Similar to SE-SPOCT, the present invention also provides a system or method for performing stimulated Raman scattering spectroscopic OCT (SRS-SPOCT). In this embodiment, the system or method includes the means for, or the step of directing a high intensity pump light into the sample interaction region. Resonant gain experienced by the low-coherence probe light will appear as localized peaks in the depth-resolved backscatter spectrum of the sample. The SC-SPOCT concept is well suited to Raman spectroscopy since SRS spectra have sharp spectral features, and can be obtained in any desired wavelength range by selection of the pump laser frequency. Thus substantial vibrational/rotational spectral information can be collected using relatively narrow bandwidth OCT probe sources. In addition, coherent detection of Raman signals avoids incoherent fluorescent noise typical of other systems. Although SRS signals have not previously been observed in turbid media, coherent Raman gain spectroscopy has been demonstrated with low-power (including cw) laser sources using nonlinear interferometry. SRS-SPOCT may be implemented using a modelocked, Q-switched Nd:YAG laser operating at 1060 nm as a pump source and a femtosecond Cr:Forsterite laser with a bandwidth extending from 1250 nm-1350 nm as the SPOCT probe. Assuming SRS gain cross sections typical of organic solvents, modelocked peak pump powers of ˜1 MW focused to the OCT probe beam spot size will be sufficient to generate ˜1% stimulated Raman gain over a 100 μm window depth in samples. The wavenumber shift available for depth-resolved spectral acquisition with this SRS-SPOCT implementation will encompass 1100-2000 cm XI. Conclusion While describing the present invention, we talk about the scanning Michelson interferometer where the reference arm length is mechanically scanned by translating the reference mirror. It is to be understood that the inventions described herein are applicable to any interferometric device which estimates the correlation functions described above. The deconvolution algorithms are also applicable to any device which measures the auto-power spectra and cross-power spectra. Thus, the present invention is applicable to any device capable of measuring any of the above mentioned quantities whether the device operates in free space or is fiber optically integrated. Also, the present invention is equally applicable in situations where a measuring device is coupled to an endoscope or a catheter or any other diagnostic instrument. The transfer function was described above as a function of spatial frequency (i.e., wavenumber k=2p/l,l is wavelength in the medium). It is to be understood that the transfer function can also be estimated using our methods as a function of optical frequency f′. Note that f′ can be related to k by f′=ck/(2p) where c is phase velocity in the medium at that wavenumber. Also, it is obvious that the transfer functions can also be expressed as a function of w=2pf′ or k Although a low temporal coherence source is useful in making the measurements in OCDR and OCT, it is to be understood that a high temporal coherence source can also be used with the spectroscopy methods of the present invention. Having described the invention in detail and by reference to the drawings, it will be apparent that modification and variations are possible without departing from the scope of the invention as defined in the following claims. Patent Citations
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