Abstract

We developed a method for obtaining three-dimensional images of biological tissues using axial-lateral parallel time domain optical coherence tomography (OCT) with an ultrahigh-speed complementary metal oxide semiconductor (CMOS) camera. The camera obtains a depth-resolved interference image using diffracted light as the reference beam and a linear illumination beam without axial and vertical scans. We can obtain the OCT images (512×512 pixels) at 1,500 frames per second by calculating two sequential images. A sample volume of 5.8×2.8×2.0 (x×y×z) mm3 (corresponding to 512×250×512 pixels) was imaged at six volumes per second in a horizontal mechanical scan. The experimental sensitivity was approximately 76 dB after 2×2-pixel binning. The system was successfully used to image the human finger in vivo.

©2006 Optical Society of America

1. Introduction

Optical coherence tomography (OCT) is a promising biomedical application that permits cross-sectional imaging of biological tissues with high spatial resolution (~10 µm) to depths of a few millimeters [1]. Time domain (TD)-OCT has been used to obtain cross-sectional images by measuring the interference signal as a function of time during axial scanning in the reference arm at each position of a probe beam that scanned laterally in the sample arm. Using the rapid scanning optical delay line at a repetition rate of 4 kHz, 500 lateral points per frame at 8 frames per second (fps) can be achieved [2]. The speed of image acquisition is limited by the necessity for mechanical scanning in two directions (axial and lateral).

Recently, Fourier domain (FD) techniques using spectrometers [3, 4] or frequency-swept lasers [5] have become an active research area because FD-OCT offers higher sensitivity and imaging speed compared to TD-OCT [6]. Spectral domain (SD)-OCT detects the spectrally resolved interference signal with a spectrometer that consists of a high-efficiency diffraction grating and a high-speed line camera at each lateral position. An OCT image consisting of 500 lateral points can be attained at 58.6 fps using the 29.3-kHz line rate of the camera [4].

A parallel detection SD-OCT technique to obtain cross-sectional images without axial and lateral scanning from a single captured image has been demonstrated [7]. More recently, this technique has been developed with linear illumination and a 2-D charge-coupled device (CCD) camera to measure 3-D shapes [8], in vivo real-time imaging of human eye structures [9] and in vivo 3-D dermatological investigation [10]. To obtain OCT images, each A-line must be mapped and interpolated from wavelength- to k-space and subsequently Fourier transformed. It is well known that FD-OCT contains a mirror image and suffers from a reduction in the signal-to-noise ratio (SNR) with increasing depth range. To avoid a mirror image, SD-OCT requires several phase-shifted spectral interference signals [11, 12].

An alternative 2-D detection OCT approach involves parallel TD-OCT techniques that are categorized as transverse (en face) [1321] and longitudinal [2224] cross-sectional imaging methods. One en face imaging technique has been demonstrated using a parallel detection scheme with 2-D smart pixel silicon detector arrays [1315]. Each pixel in the detector array has associated electronics to extract the heterodyne signal generated by the scanning reference mirror. Currently, an array of 58×58 pixels has been achieved, and this technique has also been used to demonstrate 3-D OCT imaging of a sample volume of 0.21×0.21×0.08 mm3 (corresponding to 58×58×58 pixels) at 25 Hz. The other form of en face OCT imaging is full-field (FF) OCT using a CCD camera as the detector array [1621]. As the camera output contains both interference and background (dc) signals, the parallel TD-OCT-based CCD camera requires several phase-shifted interference images to produce an OCT image. In FFOCT, 3-D OCT imaging can be achieved using single axial scanning. For faster axial scanning, the heterodyne signal cannot be detected because of the slow response of the CCD camera. To detect high-frequency signals, FF-OCT using parallel heterodyne detection is necessary, which requires a constant Doppler frequency to synchronize with the sampling function (a rectangle with a duty of 50%) [19].

The longitudinal cross-sectional parallel TD-OCT uses diffracted light as the reference beam and a linear illumination beam [22]. The continuous delay is achieved using a reflective grating in the Littrow configuration of the reference arm. A CCD camera detects the depth-resolved interference image of a sample during an exposure. In principle, OCT images do not involve the problem of mirror images and there is no decrease in the SNR with increasing depth range compared to standard FD-OCT. Using a three-step phase-shifting method, we demonstrated in vivo OCT imaging at 10 fps using an indium gallium arsenide (InGaAs) digital camera operated at 30 fps [24]. We could obtain OCT images of human fingers with a sensitivity of 90 dB for a 2.45×4.80 mm2 (axial×lateral) measurement range. Since the samples must be immobile during the acquisition of three interference images, it was necessary to use a high-speed camera system for parallel TD-OCT.

To our knowledge, this paper is the first demonstration of in vivo OCT imaging of the human finger at over 1,000 fps using axial-lateral parallel TD-OCT with an ultrahigh-speed complementary metal oxide semiconductor (CMOS) camera. The OCT images (512×512 pixels) can be obtained at 1,500 fps by calculating two sequential images. This speed of image acquisition corresponds to an axial scan rate of 768 kHz. Using the mechanical scan of the probe beam, we can image a sample volume of 5.8×2.8×2.0 (x×y×z) mm3 (corresponding to 512×250×512 pixels) at six volumes per second. The experimental sensitivity was approximately 76 dB after 2×2-pixel binning. We describe the sensitivity and spatial resolution of the system and present in vivo OCT images of human fingers.

2. Experimental setup

A schematic of our 3-D parallel TD-OCT imaging system is shown in Fig. 1. A non-polarizing cube beam splitter (size: 20 mm) divided the collimated output beam of a superluminescent diode (Qphotonics, SLD QSDM-830-9; center wavelength: λ0=831 nm, -3 dB spectral width: Δλ=26 nm) into signal and reference beams. The theoretical axial resolution was 11.7 µm in air. A cylindrical lens (f=50 mm) was inserted in the signal arm to illuminate the sample with a linear beam. This beam can be moved horizontally by a galvano scanner. The reflective grating was installed in the Littrow configuration, such that first-order diffracted light propagates backward along the incoming path at the reference beam. The Littrow angle, θ, is determined by

2sinθ=λp,

where λ/p is the wavelength-to-period ratio. We used a blazed diffraction grating 12.7×12.7 mm in size, a 1.2-µm period, θ=20.2° first-order Littrow angle, and 70% diffraction efficiency for 830 nm. The imaging depth range, ΔL, generated by the diffraction grating is given by

ΔL=dtanθ,

where d is the beam diameter. The backscattered light from samples and the diffracted light from the grating were imaged onto a CMOS camera (Photron; 512×512 pixels, 17×17 µm pixel size, 10-bit resolution, maximum rate 6,000 fps) using an achromatic imaging lens (f=100 mm, 25.4 mm diameter). The exposure time was set to 1/frame-rate. The diffraction grating was placed on a piezoelectric transducer (PZT) to shift the phase difference between the signal and reference beams. The observed X-Z interference image, I, is described as

I(x,z)=Is+Ir+2[IrIi(Rs(x,z)γ(z))]12cos(ϕsϕr),

where Is, Ir, and Ii are the intensities of the sample beam, reference beam, and incident beam, respectively; Rs is the distribution of the sample reflectance; γ(z) is the amplitude of the modulation, which is determined by the degree of coherence of the light source; ϕs and ϕr are the respective phases of the sample and reference beams; and ⊕ denotes the convolution operator.

 

Fig. 1. Schematic of 3-D axial-lateral parallel time-domain optical coherence tomography. The dashed line is the imaging ray.

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To characterize our system, we applied a three-step phase-shifting method to obtain OCT images of non-scattering media, such as test targets and plane mirrors. We acquired three interference images at phase differences of 0, π/2, and π and then calculated an OCT image as follows:

S=[(I0Iπ2)2+(Iπ2Iπ)2]4=4[IrIi(Rs(x,z)γ(z))].

For in vivo imaging of biological samples, the accuracy of the phase differences between the captured frames was not promising due to sample motion. Consequently, we calculated the squared value of the difference of two sequential images to eliminate the noninterference term for OCT images. The availability of two-image calculations has been demonstrated for in vivo imaging of the anterior eye of the rat using FF-OCT technique [21]. If we obtained two sequential interference images that have π phase difference, the calculated OCT image is described as

S=(I0Iπ)24=4[IrIi(Rs(x,s)γ(z))]cos2(ϕsϕr).

Since the calculation using two captured images only eliminates the noninterference components, the phase terms are still present in the OCT images. The image average procedure is effective to reduce the phase noise [21, 22]. However the temporal resolution is decreased. Therefore we performed pixel binning to reduce the influence of the residual phase terms.

3. Results and discussions

3.1. Lateral resolution

In our system, the vertical and horizontal resolutions depend on the numerical aperture (NA) of the imaging lens and the beam waist of the linear illumination beam, respectively. The cylindrical lens, which is Plano-convex singlet, causes the aberrations. Currently, achromatic cylindrical lenses are not available commercially. Hence we investigated the beam waist experimentally using the commercial available cylindrical lens. For horizontal resolution, we measured the beam waist by moving a beam profiler (Coherent, Beam Master) axially in 100-µm steps, as shown in Fig. 2. The theoretical beam radius, ω, is given as

ω2(z)=ω20[1+(λ0zπω20)2],

where the beam diameter at the waist is 2ω0=(4λ0/π)(f/d). For an expected incident beam diameter of d=8.5 mm, the theoretical curves were close to the experimental results, as shown in Fig. 2. However, the measured beam diameter of 41.8 µm at the waist was much larger than the calculated value of 6.2 µm because of aberrations in the cylindrical lens. The measured depth of focus was about 600 µm, which was eight times larger than the calculated 72 µm.

 

Fig. 2. Measured beam radius around the waist of linear illumination. Solid curves: theoretical values.

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To compare the horizontal and vertical resolution, we measured the intensity profiles of the signal image reflected from a test target (USAF1951) by scanning the galvano mirror. The XY image of a test target was 512×250 (x×y) pixels, as shown in Fig. 3(a), from which we confirmed that the measurement area was 5.8×2.8 (x×y) mm2. Figure 3(b) shows the vertical and horizontal line profiles of Group 4. The Group 4 element 6 (28.5 line pairs/mm) of the test target corresponded to a test bar of about 35 µm. The image contrast of the horizontal profile was lower than that of the vertical profile, which was due to aberrations in the cylindrical lens. For the vertical direction, we were able to resolve the Group 5 element 2 (36 line pairs/mm) of the test target, which corresponded to a test bar of about 28 µm. Considering the beam diameter and the distance between the test pattern and the imaging lens, the theoretical vertical resolution was about 20 µm for an estimated NA of 0.025 in the achromatic imaging lens. Therefore, the experimental value was close to the theoretical one in the vertical direction.

 

Fig. 3. (a) XY image of a test target. The measurement area was 5.8×2.8 (x×y) mm2 (b) Vertical and horizontal line profiles of Group 4.

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3.2. Sensitivity and axial resolution

We measured the sensitivity of our OCT system using the three-step phase-shifting method at different frame rates, as shown in Fig. 4(a). Here, the sample used was a gold mirror with an attenuation of -50 dB. We performed 2×2-pixel binning. The total illumination power used was about 9.3 mW, which corresponded to an optical power of 36.4 µW per A-line (9.3 mW divided by the 256 pixels of the camera). The reference beam power was adjusted by the neutral density (ND) filter until the pixel values were close to the saturation level of the camera at each frame rate. We confirmed that the axial measurement range was ΔL=2.0 mm by moving the test target axially. The experimental sensitivities of 1,000, 3,000, and 6,000 fps were approximately 80, 76, and 72 dB, respectively. The 72 dB sensitivity is the same level as that of high-speed FF-OCT with a CMOS camera operated at 500 fps [21]. However, because our OCT system was less sensitive than FD-OCT instruments [46], the measurable depth of biological tissues was limited to a few hundred microns.

 

Fig. 4. (a) Sensitivities with an attenuation of -50 dB in the sample arm at different frame rates. (b) Measured axial profile and Gaussian curve fitting

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In parallel TD-OCT based on the camera, the sensitivity depends on the reference reflectivity, Rr, the full-well capacity of the camera, ξmax, and the reflectivity of incoherent light, Rinc [17, 20, 24]. When shot noise is assumed to dominate over all other noise and the camera operates at close to saturation, the theoretical minimum detectable reflectivity, R min, is approximately

Rmin=(Rr+2Rinc)22Rrξmax.

High-speed image sensors typically have a full-well capacity of ξmax≈60,000. Since we used 2×2-pixel binning, the actual full-well capacity was equivalent to 4ξmax=240,000. The reference reflectivity was decided by the optical density used of ND filter and the diffraction efficiency in the reference arm. We measured reference reflectivities of Rr=0.5, 1.2, and 2.5% at 1,000, 3,000, and 6,000 fps, respectively, to approach the saturation condition when the measured reflectivity of a plane mirror was Rr=100%. The incoherent component in the signal lights was negligible when a plane mirror was used as the sample. Therefore, these values gave predicted sensitivities of 10log(Rmin) ~-79.8, -76.0, and -72.8 dB at 1,000, 3,000, and 6,000 fps, respectively, which were close to the experimental values. In measuring biological tissues, the influence of incoherent light is not negligible. We measured the reflectivity of incoherent light from human skin to be Rinc=0.1%. Considering Rinc=0.1%, Eq. (7) predicted sensitivities of -76.9, -74.7, and -72.2 dB at 1,000, 3,000, and 6,000 fps, respectively.

The axial profile was rescaled to a linear scale, as shown in Fig. 4(b). The fitted Gaussian curve is also shown in Fig. 4(b). The axial resolution was approximately 11 µm in air, which was close to the theoretical value of 11.7 µm. The influence of the cylindrical lens dispersion was compensated by the ND filter in the reference arm. Since one pixel in the axial direction corresponds to about 3.9 µm (2.0 mm divided by the 512 pixels of the camera) before pixel binning, our setup cannot be applied to ultrahigh-resolution (1–3 µm) OCT directly. If the measurement size of one pixel corresponds to 1 µm, the maximum depth range is 512 µm. Therefore, an ultrahigh resolution OCT could be accomplished by narrowing the axial measurement range.

 

Fig. 5. Signal-to-noise ratio at different depths (a) with a f=50 mm cylindrical lens and (b) without a cylindrical lens.

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Next, we measured the SNR by moving a plane mirror axially in 200-µm steps, as shown in Fig. 5(a). The SNR is the greatest in the center of the axial measurement range and decreases with distance from the center. As a reference, the measured SNR without a cylindrical lens is shown in Fig. 5(b). A decrease in the SNR is not seen in this result. Therefore, the decrease in the SNR was caused by the cylindrical lens. According to Fig. 2, the depth of focus was about 0.6 mm and was shorter than the 2 mm axial measurement range. Using a long focal-length cylindrical lens improved the depth of focus and minimized the resultant decrease in the SNR. However, the horizontal resolution decreased on increasing the beam waist. Since our main purpose was 3-D imaging, a cylindrical lens with a 50-mm focal length was chosen for the in vivo experiments. In the actual measurement of biological tissues, the SNR near the surface of sample would be improved because the backscattered light from surfaces is higher.

3.3. In vivo OCT imaging

First, we obtained OCT images of a healthy human finger in vivo without a horizontal scan. The CMOS camera was used at 3,000 fps and the PZT oscillated sinusoidally at 1.5 kHz to shift the phase between the two sequential interference images. As each OCT image was calculated from two sequentially captured images, the acquisition time of an OCT image was 1/1500 s. In general the fringe contrast in the interference image is reduced and the OCT image is blurred when the samples move during acquisition. The influence of the sample motion would be small in the acquisition time of 1/1500 s. The OCT images are shown using an inverse logarithmic scale. Figure 6(a) shows an OCT image of human skin at a fingertip. The epidermis, dermis, and sweat glands are distinguishable. Figure 6(b) shows an OCT image of a human nail fold region. The nail root was visible beneath the skin. We can see the lunula, which is the whitish, half-moon shape at the base of nail underneath the plate.

 

Fig. 6. OCT images of a human finger in vivo at an acquisition time of 1/1500 s. The measurement range was 5.8×2.0 (x×z) mm2: (a) the skin at the fingertip, (b) the nail fold region.

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Next, the linear probe beam was scanned at 3 Hz using a triangular waveform to measure volume data for a human finger in vivo. Since the CMOS camera was used at 3,000 fps and the horizontal scan was performed at 6 Hz, we obtained 500 interference images per scan, which corresponds to 250 OCT images per scan by calculating two sequential images. Therefore, a 3-D dataset of 512×250×512 pixels (x×y×z) can be obtained at six volumes per second. As the galvano scanner was operated at the same input voltage as in the experiment in Section 3.1, the volume size was 5.8×2.8×2.0 mm3. The sampling interval of the horizontal direction was 11.2 µm (2.8 mm divided by the 250 interference images). Figures 7(a) and 7(b) show measured longitudinal OCT images of a human finger pad at different probe beam positions. These images are also presented on an inverse logarithmic scale. We can distinguish between the epidermis and dermis, and sweat glands appear as a spiral structures in the horny layer. Figures 7(c) and 7(d) show the observed transverse (en face) OCT images from the 3-D dataset. The depths of these images are 424 and 606 µm, respectively. The white arrows indicate the cross-sectional positions of each image. In Fig. 7(c), the black arrows correspond to the sweat glands in Figs. 7(a) and 7(b). Figure 7(e) shows a volume-rendered image produced from the 3-D OCT data. The fingerprint ridges and structures of the epidermis are clearly visible.

 

Fig. 7. 3-D OCT images of a human finger pad in vivo: (a, b) longitudinal OCT images in the X-Z plane, (c, d) transverse OCT images. The white arrows indicate the cross-sectional positions. (e) Volume-rendered image. The volume size was 5.8×2.8×2.0 (x×y×z) mm3.

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4. Conclusion

We demonstrated ultrahigh-speed OCT imaging at 1,500 fps using an axial-lateral parallel TD-OCT with a CMOS camera. Moreover, 3-D imaging of 5.8×2.8×2.0 mm3 (512×250×512 [x×y×z] pixels) was performed at 6 Hz using a single mechanical scan. The increase in the volume rate was made possible by increasing the scan rate of the probe beam. Since the experimental sensitivity was approximately 76 dB after 2×2-pixel binning, this sensitivity proved sufficient to image typical biological tissues at penetration depths of a few hundred microns. The advantages of this technique are that it is artifact-free and has roughly the same SNR for increasing depth range compared to standard FD-OCT. We found that this SNR depended on the depth of focus of a cylindrical lens. Although the method of acquiring two sequential images to obtain one OCT image may be a disadvantage when compared to a parallel SD-OCT that uses a single frame, this drawback can be overcome by using a high-speed camera. Therefore, we believe that our axial-lateral parallel TD-OCT system with a high-speed camera has the potential for high-speed imaging in scattering samples and can have diverse applications.

Acknowledgments

This study was supported by Industrial Technology Research Grant Program in ’05 from New Energy and Industrial Technology Development Organization (NEDO) of Japan.

References and Links

1. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991). [CrossRef]   [PubMed]  

2. A. M. Rollins, M. D. Kulkarni, S. Yazdanfar, R. Ungarunyawee, and J. A. Izatt, “In vivo video rate optical coherence tomography,” Opt. Express 3, 219–229 (1998). [CrossRef]   [PubMed]  

3. G. Häusler and M. W. Lindner, “Coherence radar” and “spectral radar”-new tools for dermatological diagnosis,” J. Biomed. Opt. 3, 21–31 (1998). [CrossRef]  

4. N. Nassif, B. Cense, B. H. Park, S. H. Yun, T. C. Chen, B. E. Bouma, G. J. Tearney, and J. F. de Boer, “In vivo human retinal imaging by ultrahigh-speed spectral domain optical coherence tomography,” Opt. Lett. 29, 480–482 (2004). [CrossRef]   [PubMed]  

5. S. H. Yun, G. J. Tearney, J. F. de Boer, N. Iftimia, and B. E. Bouma, “High-speed optical frequency-domain imaging,” Opt. Express 11, 2953–2963 (2003). [CrossRef]   [PubMed]  

6. R. A. Leitgeb, C. K. Hitzenberger, and A. F. Fercher, “Performance of fourier domain vs. time domain optical coherence tomography,” Opt. Express 11, 889–894 (2003). [CrossRef]   [PubMed]  

7. A. Zuluaga and R. Richards-Kortum, “Spatially resolved spectral interferometry for determination of subsurface structure,” Opt. Lett. 24, 519–521 (1999). [CrossRef]  

8. T. Endo, Y. Yasuno, S. Makita, M. Itoh, and T. Yatagai, “Profilometry with line-field Fourier-domain interferometry,” Opt. Express 13, 695–701 (2005). [CrossRef]   [PubMed]  

9. B. Grajciar, M. Pircher, A. Fercher, and R. Leitgeb, “Parallel Fourier domain optical coherence tomography for in vivo measurement of the human eye,” Opt. Express 13, 1131–1137 (2005). [CrossRef]   [PubMed]  

10. Y. Yasuno, T. Endo, S. Makita, G. Aoki, M. Itoh, and T. Yatagai, “Three-dimensional line-field Fourier domain optical coherence tomography for in vivo dermatological investigation,” J. Biomed. Opt. 11, 014014–014020 (2006). [CrossRef]   [PubMed]  

11. M. Wojtkowski, A. Kowalczyk, R. Leitgeb, and A. F. Fercher, “Full range complex spectral optical coherence tomography technique in eye imaging,” Opt. Lett. 27, 1415–1417 (2002). [CrossRef]  

12. R. A. Leitgeb, C. K. Hitzenberger, A. F. Fercher, and T. Bajraszewski, “Phase-shifting algorithm to achieve high-speed long-depth-range probing by frequency-domain optical coherence tomography,” Opt. Lett. 28, 2201–2203 (2003). [CrossRef]   [PubMed]  

13. S. Bourquin, P. Seitz, and R. P. Salathé, “Optical coherence topography based on a two-dimensional smart detector array,” Opt. Lett. 26, 512–514 (2001). [CrossRef]  

14. M. Ducros, M. Laubsher, B. Karamater, S. Bourquin, T. Lasser, and R. P. Salathé, “Parallel optical coherence tomography in scattering samples using a two-dimensional smart-pixel detector array,” Opt. Commun. 202, 29–35 (2002). [CrossRef]  

15. M. Laubscher, M. Ducros, B. Karamata, T. Lasser, and R. Salathé, “Video-rate three-dimensional optical coherence tomography,” Opt. Express 10, 429–435 (2002). [PubMed]  

16. M. Lebec, L. Blanchot, H. Saint-jalmes, E. Beaurepaire, and A. C. Boccara, “Full-field optical coherence microscopy,” Opt. Lett. 23, 244–246 (1998). [CrossRef]  

17. A. Dubois, L. Vabre, A. C. Boccara, and E. Beaurepaire, “High-resolution full-field optical coherence tomography with a Linnik microscope,” Appl. Opt. 41, 805–812 (2002). [CrossRef]   [PubMed]  

18. L. Vabre, A. Dubois, and A. C. Boccara, “Thermal-light full-field optical coherence tomography,” Opt. Lett. 27, 530–532 (2002). [CrossRef]  

19. M. Akiba, K. P. Chan, and N. Tanno, “Full-field optical coherence tomography by two-dimensional heterodyne detection with a pair of CCD camera,” Opt. Lett. 28, 816–818 (2003). [CrossRef]   [PubMed]  

20. A. Dubois, K. Grieve, G. Moneron, R. Lecaque, L. Vabre, and C. Boccara, “Ultrahigh-resolution full-field optical coherence tomography,” Appl. Opt. 43, 2874–2883 (2004). [CrossRef]   [PubMed]  

21. K. Grieve, A. Dubois, M. Simonutti, M. Paques, J. Sahel, J. Le Gargasson, and C. Boccara, “In vivo anterior segment imaging in the rat eye with high speed white light full-field optical coherence tomography,” Opt. Express 13, 6286–6295 (2005). [CrossRef]   [PubMed]  

22. I. Zeylikovich, A. Gilerson, and R. R. Alfano, “Nonmechanical grating-generated scanning coherence microscopy,” Opt. Lett. 23, 1797–1799 (1998). [CrossRef]  

23. I. Zeylikovich and R. R. Alfano, “Performing selected optical measurements with optical coherence domain reflectometry,” US patent: 6,437,867, (2002).

24. Y. Watanabe, K. Yamada, and M. Sato, “In vivo nonmechanical scanning grating-generated optical coherence tomography using an InGaAs digital camera,” Opt. Commun. 261, 376–380 (2006). [CrossRef]  

References

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  1. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
    [Crossref] [PubMed]
  2. A. M. Rollins, M. D. Kulkarni, S. Yazdanfar, R. Ungarunyawee, and J. A. Izatt, “In vivo video rate optical coherence tomography,” Opt. Express 3, 219–229 (1998).
    [Crossref] [PubMed]
  3. G. Häusler and M. W. Lindner, “Coherence radar” and “spectral radar”-new tools for dermatological diagnosis,” J. Biomed. Opt. 3, 21–31 (1998).
    [Crossref]
  4. N. Nassif, B. Cense, B. H. Park, S. H. Yun, T. C. Chen, B. E. Bouma, G. J. Tearney, and J. F. de Boer, “In vivo human retinal imaging by ultrahigh-speed spectral domain optical coherence tomography,” Opt. Lett. 29, 480–482 (2004).
    [Crossref] [PubMed]
  5. S. H. Yun, G. J. Tearney, J. F. de Boer, N. Iftimia, and B. E. Bouma, “High-speed optical frequency-domain imaging,” Opt. Express 11, 2953–2963 (2003).
    [Crossref] [PubMed]
  6. R. A. Leitgeb, C. K. Hitzenberger, and A. F. Fercher, “Performance of fourier domain vs. time domain optical coherence tomography,” Opt. Express 11, 889–894 (2003).
    [Crossref] [PubMed]
  7. A. Zuluaga and R. Richards-Kortum, “Spatially resolved spectral interferometry for determination of subsurface structure,” Opt. Lett. 24, 519–521 (1999).
    [Crossref]
  8. T. Endo, Y. Yasuno, S. Makita, M. Itoh, and T. Yatagai, “Profilometry with line-field Fourier-domain interferometry,” Opt. Express 13, 695–701 (2005).
    [Crossref] [PubMed]
  9. B. Grajciar, M. Pircher, A. Fercher, and R. Leitgeb, “Parallel Fourier domain optical coherence tomography for in vivo measurement of the human eye,” Opt. Express 13, 1131–1137 (2005).
    [Crossref] [PubMed]
  10. Y. Yasuno, T. Endo, S. Makita, G. Aoki, M. Itoh, and T. Yatagai, “Three-dimensional line-field Fourier domain optical coherence tomography for in vivo dermatological investigation,” J. Biomed. Opt. 11, 014014–014020 (2006).
    [Crossref] [PubMed]
  11. M. Wojtkowski, A. Kowalczyk, R. Leitgeb, and A. F. Fercher, “Full range complex spectral optical coherence tomography technique in eye imaging,” Opt. Lett. 27, 1415–1417 (2002).
    [Crossref]
  12. R. A. Leitgeb, C. K. Hitzenberger, A. F. Fercher, and T. Bajraszewski, “Phase-shifting algorithm to achieve high-speed long-depth-range probing by frequency-domain optical coherence tomography,” Opt. Lett. 28, 2201–2203 (2003).
    [Crossref] [PubMed]
  13. S. Bourquin, P. Seitz, and R. P. Salathé, “Optical coherence topography based on a two-dimensional smart detector array,” Opt. Lett. 26, 512–514 (2001).
    [Crossref]
  14. M. Ducros, M. Laubsher, B. Karamater, S. Bourquin, T. Lasser, and R. P. Salathé, “Parallel optical coherence tomography in scattering samples using a two-dimensional smart-pixel detector array,” Opt. Commun. 202, 29–35 (2002).
    [Crossref]
  15. M. Laubscher, M. Ducros, B. Karamata, T. Lasser, and R. Salathé, “Video-rate three-dimensional optical coherence tomography,” Opt. Express 10, 429–435 (2002).
    [PubMed]
  16. M. Lebec, L. Blanchot, H. Saint-jalmes, E. Beaurepaire, and A. C. Boccara, “Full-field optical coherence microscopy,” Opt. Lett. 23, 244–246 (1998).
    [Crossref]
  17. A. Dubois, L. Vabre, A. C. Boccara, and E. Beaurepaire, “High-resolution full-field optical coherence tomography with a Linnik microscope,” Appl. Opt. 41, 805–812 (2002).
    [Crossref] [PubMed]
  18. L. Vabre, A. Dubois, and A. C. Boccara, “Thermal-light full-field optical coherence tomography,” Opt. Lett. 27, 530–532 (2002).
    [Crossref]
  19. M. Akiba, K. P. Chan, and N. Tanno, “Full-field optical coherence tomography by two-dimensional heterodyne detection with a pair of CCD camera,” Opt. Lett. 28, 816–818 (2003).
    [Crossref] [PubMed]
  20. A. Dubois, K. Grieve, G. Moneron, R. Lecaque, L. Vabre, and C. Boccara, “Ultrahigh-resolution full-field optical coherence tomography,” Appl. Opt. 43, 2874–2883 (2004).
    [Crossref] [PubMed]
  21. K. Grieve, A. Dubois, M. Simonutti, M. Paques, J. Sahel, J. Le Gargasson, and C. Boccara, “In vivo anterior segment imaging in the rat eye with high speed white light full-field optical coherence tomography,” Opt. Express 13, 6286–6295 (2005).
    [Crossref] [PubMed]
  22. I. Zeylikovich, A. Gilerson, and R. R. Alfano, “Nonmechanical grating-generated scanning coherence microscopy,” Opt. Lett. 23, 1797–1799 (1998).
    [Crossref]
  23. I. Zeylikovich and R. R. Alfano, “Performing selected optical measurements with optical coherence domain reflectometry,” US patent: 6,437,867, (2002).
  24. Y. Watanabe, K. Yamada, and M. Sato, “In vivo nonmechanical scanning grating-generated optical coherence tomography using an InGaAs digital camera,” Opt. Commun. 261, 376–380 (2006).
    [Crossref]

2006 (2)

Y. Watanabe, K. Yamada, and M. Sato, “In vivo nonmechanical scanning grating-generated optical coherence tomography using an InGaAs digital camera,” Opt. Commun. 261, 376–380 (2006).
[Crossref]

Y. Yasuno, T. Endo, S. Makita, G. Aoki, M. Itoh, and T. Yatagai, “Three-dimensional line-field Fourier domain optical coherence tomography for in vivo dermatological investigation,” J. Biomed. Opt. 11, 014014–014020 (2006).
[Crossref] [PubMed]

2005 (3)

2004 (2)

2003 (4)

2002 (5)

2001 (1)

1999 (1)

1998 (4)

1991 (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Akiba, M.

Alfano, R. R.

I. Zeylikovich, A. Gilerson, and R. R. Alfano, “Nonmechanical grating-generated scanning coherence microscopy,” Opt. Lett. 23, 1797–1799 (1998).
[Crossref]

I. Zeylikovich and R. R. Alfano, “Performing selected optical measurements with optical coherence domain reflectometry,” US patent: 6,437,867, (2002).

Aoki, G.

Y. Yasuno, T. Endo, S. Makita, G. Aoki, M. Itoh, and T. Yatagai, “Three-dimensional line-field Fourier domain optical coherence tomography for in vivo dermatological investigation,” J. Biomed. Opt. 11, 014014–014020 (2006).
[Crossref] [PubMed]

Bajraszewski, T.

Beaurepaire, E.

Blanchot, L.

Boccara, A. C.

Boccara, C.

Bouma, B. E.

Bourquin, S.

M. Ducros, M. Laubsher, B. Karamater, S. Bourquin, T. Lasser, and R. P. Salathé, “Parallel optical coherence tomography in scattering samples using a two-dimensional smart-pixel detector array,” Opt. Commun. 202, 29–35 (2002).
[Crossref]

S. Bourquin, P. Seitz, and R. P. Salathé, “Optical coherence topography based on a two-dimensional smart detector array,” Opt. Lett. 26, 512–514 (2001).
[Crossref]

Cense, B.

Chan, K. P.

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Chen, T. C.

de Boer, J. F.

Dubois, A.

Ducros, M.

M. Ducros, M. Laubsher, B. Karamater, S. Bourquin, T. Lasser, and R. P. Salathé, “Parallel optical coherence tomography in scattering samples using a two-dimensional smart-pixel detector array,” Opt. Commun. 202, 29–35 (2002).
[Crossref]

M. Laubscher, M. Ducros, B. Karamata, T. Lasser, and R. Salathé, “Video-rate three-dimensional optical coherence tomography,” Opt. Express 10, 429–435 (2002).
[PubMed]

Endo, T.

Y. Yasuno, T. Endo, S. Makita, G. Aoki, M. Itoh, and T. Yatagai, “Three-dimensional line-field Fourier domain optical coherence tomography for in vivo dermatological investigation,” J. Biomed. Opt. 11, 014014–014020 (2006).
[Crossref] [PubMed]

T. Endo, Y. Yasuno, S. Makita, M. Itoh, and T. Yatagai, “Profilometry with line-field Fourier-domain interferometry,” Opt. Express 13, 695–701 (2005).
[Crossref] [PubMed]

Fercher, A.

Fercher, A. F.

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Fujimoto, J. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Gilerson, A.

Grajciar, B.

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Grieve, K.

Häusler, G.

G. Häusler and M. W. Lindner, “Coherence radar” and “spectral radar”-new tools for dermatological diagnosis,” J. Biomed. Opt. 3, 21–31 (1998).
[Crossref]

Hee, M. R.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Hitzenberger, C. K.

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Iftimia, N.

Itoh, M.

Y. Yasuno, T. Endo, S. Makita, G. Aoki, M. Itoh, and T. Yatagai, “Three-dimensional line-field Fourier domain optical coherence tomography for in vivo dermatological investigation,” J. Biomed. Opt. 11, 014014–014020 (2006).
[Crossref] [PubMed]

T. Endo, Y. Yasuno, S. Makita, M. Itoh, and T. Yatagai, “Profilometry with line-field Fourier-domain interferometry,” Opt. Express 13, 695–701 (2005).
[Crossref] [PubMed]

Izatt, J. A.

Karamata, B.

Karamater, B.

M. Ducros, M. Laubsher, B. Karamater, S. Bourquin, T. Lasser, and R. P. Salathé, “Parallel optical coherence tomography in scattering samples using a two-dimensional smart-pixel detector array,” Opt. Commun. 202, 29–35 (2002).
[Crossref]

Kowalczyk, A.

Kulkarni, M. D.

Lasser, T.

M. Ducros, M. Laubsher, B. Karamater, S. Bourquin, T. Lasser, and R. P. Salathé, “Parallel optical coherence tomography in scattering samples using a two-dimensional smart-pixel detector array,” Opt. Commun. 202, 29–35 (2002).
[Crossref]

M. Laubscher, M. Ducros, B. Karamata, T. Lasser, and R. Salathé, “Video-rate three-dimensional optical coherence tomography,” Opt. Express 10, 429–435 (2002).
[PubMed]

Laubscher, M.

Laubsher, M.

M. Ducros, M. Laubsher, B. Karamater, S. Bourquin, T. Lasser, and R. P. Salathé, “Parallel optical coherence tomography in scattering samples using a two-dimensional smart-pixel detector array,” Opt. Commun. 202, 29–35 (2002).
[Crossref]

Le Gargasson, J.

Lebec, M.

Lecaque, R.

Leitgeb, R.

Leitgeb, R. A.

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Lindner, M. W.

G. Häusler and M. W. Lindner, “Coherence radar” and “spectral radar”-new tools for dermatological diagnosis,” J. Biomed. Opt. 3, 21–31 (1998).
[Crossref]

Makita, S.

Y. Yasuno, T. Endo, S. Makita, G. Aoki, M. Itoh, and T. Yatagai, “Three-dimensional line-field Fourier domain optical coherence tomography for in vivo dermatological investigation,” J. Biomed. Opt. 11, 014014–014020 (2006).
[Crossref] [PubMed]

T. Endo, Y. Yasuno, S. Makita, M. Itoh, and T. Yatagai, “Profilometry with line-field Fourier-domain interferometry,” Opt. Express 13, 695–701 (2005).
[Crossref] [PubMed]

Moneron, G.

Nassif, N.

Paques, M.

Park, B. H.

Pircher, M.

Puliafito, C. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Richards-Kortum, R.

Rollins, A. M.

Sahel, J.

Saint-jalmes, H.

Salathé, R.

Salathé, R. P.

M. Ducros, M. Laubsher, B. Karamater, S. Bourquin, T. Lasser, and R. P. Salathé, “Parallel optical coherence tomography in scattering samples using a two-dimensional smart-pixel detector array,” Opt. Commun. 202, 29–35 (2002).
[Crossref]

S. Bourquin, P. Seitz, and R. P. Salathé, “Optical coherence topography based on a two-dimensional smart detector array,” Opt. Lett. 26, 512–514 (2001).
[Crossref]

Sato, M.

Y. Watanabe, K. Yamada, and M. Sato, “In vivo nonmechanical scanning grating-generated optical coherence tomography using an InGaAs digital camera,” Opt. Commun. 261, 376–380 (2006).
[Crossref]

Schuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Seitz, P.

Simonutti, M.

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Swanson, E. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Tanno, N.

Tearney, G. J.

Ungarunyawee, R.

Vabre, L.

Watanabe, Y.

Y. Watanabe, K. Yamada, and M. Sato, “In vivo nonmechanical scanning grating-generated optical coherence tomography using an InGaAs digital camera,” Opt. Commun. 261, 376–380 (2006).
[Crossref]

Wojtkowski, M.

Yamada, K.

Y. Watanabe, K. Yamada, and M. Sato, “In vivo nonmechanical scanning grating-generated optical coherence tomography using an InGaAs digital camera,” Opt. Commun. 261, 376–380 (2006).
[Crossref]

Yasuno, Y.

Y. Yasuno, T. Endo, S. Makita, G. Aoki, M. Itoh, and T. Yatagai, “Three-dimensional line-field Fourier domain optical coherence tomography for in vivo dermatological investigation,” J. Biomed. Opt. 11, 014014–014020 (2006).
[Crossref] [PubMed]

T. Endo, Y. Yasuno, S. Makita, M. Itoh, and T. Yatagai, “Profilometry with line-field Fourier-domain interferometry,” Opt. Express 13, 695–701 (2005).
[Crossref] [PubMed]

Yatagai, T.

Y. Yasuno, T. Endo, S. Makita, G. Aoki, M. Itoh, and T. Yatagai, “Three-dimensional line-field Fourier domain optical coherence tomography for in vivo dermatological investigation,” J. Biomed. Opt. 11, 014014–014020 (2006).
[Crossref] [PubMed]

T. Endo, Y. Yasuno, S. Makita, M. Itoh, and T. Yatagai, “Profilometry with line-field Fourier-domain interferometry,” Opt. Express 13, 695–701 (2005).
[Crossref] [PubMed]

Yazdanfar, S.

Yun, S. H.

Zeylikovich, I.

I. Zeylikovich, A. Gilerson, and R. R. Alfano, “Nonmechanical grating-generated scanning coherence microscopy,” Opt. Lett. 23, 1797–1799 (1998).
[Crossref]

I. Zeylikovich and R. R. Alfano, “Performing selected optical measurements with optical coherence domain reflectometry,” US patent: 6,437,867, (2002).

Zuluaga, A.

Appl. Opt. (2)

J. Biomed. Opt. (2)

Y. Yasuno, T. Endo, S. Makita, G. Aoki, M. Itoh, and T. Yatagai, “Three-dimensional line-field Fourier domain optical coherence tomography for in vivo dermatological investigation,” J. Biomed. Opt. 11, 014014–014020 (2006).
[Crossref] [PubMed]

G. Häusler and M. W. Lindner, “Coherence radar” and “spectral radar”-new tools for dermatological diagnosis,” J. Biomed. Opt. 3, 21–31 (1998).
[Crossref]

Opt. Commun. (2)

M. Ducros, M. Laubsher, B. Karamater, S. Bourquin, T. Lasser, and R. P. Salathé, “Parallel optical coherence tomography in scattering samples using a two-dimensional smart-pixel detector array,” Opt. Commun. 202, 29–35 (2002).
[Crossref]

Y. Watanabe, K. Yamada, and M. Sato, “In vivo nonmechanical scanning grating-generated optical coherence tomography using an InGaAs digital camera,” Opt. Commun. 261, 376–380 (2006).
[Crossref]

Opt. Express (7)

Opt. Lett. (9)

A. Zuluaga and R. Richards-Kortum, “Spatially resolved spectral interferometry for determination of subsurface structure,” Opt. Lett. 24, 519–521 (1999).
[Crossref]

N. Nassif, B. Cense, B. H. Park, S. H. Yun, T. C. Chen, B. E. Bouma, G. J. Tearney, and J. F. de Boer, “In vivo human retinal imaging by ultrahigh-speed spectral domain optical coherence tomography,” Opt. Lett. 29, 480–482 (2004).
[Crossref] [PubMed]

M. Lebec, L. Blanchot, H. Saint-jalmes, E. Beaurepaire, and A. C. Boccara, “Full-field optical coherence microscopy,” Opt. Lett. 23, 244–246 (1998).
[Crossref]

L. Vabre, A. Dubois, and A. C. Boccara, “Thermal-light full-field optical coherence tomography,” Opt. Lett. 27, 530–532 (2002).
[Crossref]

M. Akiba, K. P. Chan, and N. Tanno, “Full-field optical coherence tomography by two-dimensional heterodyne detection with a pair of CCD camera,” Opt. Lett. 28, 816–818 (2003).
[Crossref] [PubMed]

M. Wojtkowski, A. Kowalczyk, R. Leitgeb, and A. F. Fercher, “Full range complex spectral optical coherence tomography technique in eye imaging,” Opt. Lett. 27, 1415–1417 (2002).
[Crossref]

R. A. Leitgeb, C. K. Hitzenberger, A. F. Fercher, and T. Bajraszewski, “Phase-shifting algorithm to achieve high-speed long-depth-range probing by frequency-domain optical coherence tomography,” Opt. Lett. 28, 2201–2203 (2003).
[Crossref] [PubMed]

S. Bourquin, P. Seitz, and R. P. Salathé, “Optical coherence topography based on a two-dimensional smart detector array,” Opt. Lett. 26, 512–514 (2001).
[Crossref]

I. Zeylikovich, A. Gilerson, and R. R. Alfano, “Nonmechanical grating-generated scanning coherence microscopy,” Opt. Lett. 23, 1797–1799 (1998).
[Crossref]

Science (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Other (1)

I. Zeylikovich and R. R. Alfano, “Performing selected optical measurements with optical coherence domain reflectometry,” US patent: 6,437,867, (2002).

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Figures (7)

Fig. 1.
Fig. 1. Schematic of 3-D axial-lateral parallel time-domain optical coherence tomography. The dashed line is the imaging ray.
Fig. 2.
Fig. 2. Measured beam radius around the waist of linear illumination. Solid curves: theoretical values.
Fig. 3.
Fig. 3. (a) XY image of a test target. The measurement area was 5.8×2.8 (x×y) mm2 (b) Vertical and horizontal line profiles of Group 4.
Fig. 4.
Fig. 4. (a) Sensitivities with an attenuation of -50 dB in the sample arm at different frame rates. (b) Measured axial profile and Gaussian curve fitting
Fig. 5.
Fig. 5. Signal-to-noise ratio at different depths (a) with a f=50 mm cylindrical lens and (b) without a cylindrical lens.
Fig. 6.
Fig. 6. OCT images of a human finger in vivo at an acquisition time of 1/1500 s. The measurement range was 5.8×2.0 (x×z) mm2: (a) the skin at the fingertip, (b) the nail fold region.
Fig. 7.
Fig. 7. 3-D OCT images of a human finger pad in vivo: (a, b) longitudinal OCT images in the X-Z plane, (c, d) transverse OCT images. The white arrows indicate the cross-sectional positions. (e) Volume-rendered image. The volume size was 5.8×2.8×2.0 (x×y×z) mm3.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

2 sin θ = λ p ,
Δ L = d tan θ ,
I ( x , z ) = I s + I r + 2 [ I r I i ( R s ( x , z ) γ ( z ) ) ] 1 2 cos ( ϕ s ϕ r ) ,
S = [ ( I 0 I π 2 ) 2 + ( I π 2 I π ) 2 ] 4 = 4 [ I r I i ( R s ( x , z ) γ ( z ) ) ] .
S = ( I 0 I π ) 2 4 = 4 [ I r I i ( R s ( x , s ) γ ( z ) ) ] cos 2 ( ϕ s ϕ r ) .
ω 2 ( z ) = ω 2 0 [ 1 + ( λ 0 z π ω 2 0 ) 2 ] ,
R min = ( R r + 2 R inc ) 2 2 R r ξ max .

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