
1.IntroductionOptical coherence tomography (OCT) uses the short temporal coherence properties of broadband light to extract structural information from heterogeneous samples such as tissue. Two approaches are recognized to generate 2D and 3D OCT images: singlepoint raster scanning^{1, 2, 3, 4} and parallel/widefield imaging.^{5, 6, 7, 8} Speckle noise is a major obstacle in measuring specific structural properties (e.g., birefringence^{9, 10, 11}) of tissue specimens and observing specific features that approach the diffraction limit.^{12} Although speckle in OCT images is still being investigated, a common view is that the effect originates in distorted wavefronts of light returning from tissue that result in random appearing constructive and destructive features in the interferogram. Figure 1 shows OCT images of in vivo primate retina and porcine mesenteric artery that present highcontrast speckle in regions of strong backscatter in two different biological tissue samples. The appearance of speckle in the retinal nerve fiber layer (RNFL) of Fig. 1(a) does not vary with depth, but rather present relatively constant over the depth. The media and adventitia area of Fig. 1(b) also display a gradual distribution of the speckle feature, indicating that the speckle is not dependent on depth because multiple backscattering and random phase variation of the returning light dominate the statistical properties of the speckle. Speckle reduction methods have been addressed by several investigators and include postsignal processing,^{13, 14, 15, 16, 17, 18} angular compounding,^{19, 20, 21, 22} and use of spatially incoherent illumination.^{23} Postsignal processing techniques have been reported, including a zeroamplitude procedure,^{18} deconvolution, and rotating kernel transformation. The zeroamplitude procedure (ZAP), which operates in the complex number domain, shows speckle reduction in OCT images but blurs boundaries between tissue structures. Other complexnumberdomain processing methods applied to OCT include iterative point deconvolution (i.e., CLEAN in Ref. 16), and constrained iterative deconvolution.^{17} Deconvolution techniques require some prior knowledge of the point spread function of the imaging optics,^{24} as well as the optical properties of the imaged sample. The computation time of the rotating kernel transformation (RKT) technique, applied by Rogowsk and Brezinski^{14} increases substantially with kernel size. Angular compounding methods are based on acquiring/averaging multiple beams each with a different incidence/reflecting angle from the sample but have the disadvantages of either timerestricted implementation or high complexity. Use of spatially incoherent illumination with a thermal light source has neither time constraints nor high complexity, but suffers from reduced SNR due to the low photon number per spatial mode.^{23} Since the average number of photons within a coherence volume for thermal light sources is of the order of unity, these sources are nondegenerate.^{25} Use of quasihomogeneous (partially spatially coherent) illumination has long been recognized to improve lateral resolution by decreasing speckle in recorded images.^{25, 26, 27} Although partially spatially coherent illumination can be easily implemented using a multimode optical fiber,^{28} its application in OCT imaging has not been reported, possibly due to problems associated with modal dispersion and alignment constraints. We present a mathematical analysis and an experimental result showing a simple and rapid method to reduce speckle in OCT imaging using a partially spatially coherent light source generated by a multimode fiber. Although the coherence length of this source is reduced, the photon number per spatial mode is much larger than unity. The paper is organized as follows. First, we derive an interferometric imaging equation for an OCT system utilizing a GaussianSchell model light source. A numerical simulation and experimental setup are described in Sec. 2. Experimental results are presented in Sec. 3 and discussion in Sec. 4. Summary and conclusions follow in Sec. 5. 2.Theory and MethodsFigure 2 shows a twobeam imaging Michelson interferometer. Broadband and partially spatially coherent light is emitted from a source and illuminates the sample and a reference mirror, which are imaged onto a CCD camera by lens L3. Spatial coordinates at the sample, reference, and CCD camera are denoted by $(\xi ,\zeta )$ , $({\xi}^{\prime},{\zeta}^{\prime})$ , and $(x,z)$ respectively, where $\xi $ , ${\xi}^{\prime}$ , and $x$ are in a plane perpendicular to light propagation, while $\zeta $ , ${\zeta}^{\prime}$ , and $z$ are along a line parallel to light propagation. The amplitude of light ${A}_{s}$ at the CCD camera plane produced from the sample reflection can be written as a coherent integral^{29} of amplitudes reflected from the sample modulated by pointspread function $\left[h(\bullet )\right]$ and the local reflectivity $\left[r\left(\xi \right)\right]$ , Eq. 1$${A}_{s}(x,\nu )={\int}_{\infty}^{\infty}{A}_{i}(\xi ,\nu )r\left(\xi \right)\mathrm{exp}(2\pi j\nu \tau )h(x\xi ,\nu )\mathrm{d}\xi ,$$Eq. 2$${A}_{r}(x,\nu )={\int}_{\infty}^{\infty}{A}_{i}({\xi}^{\prime},\nu ){r}_{\mathrm{mir}}h(x{\xi}^{\prime},\nu )\mathrm{d}{\xi}^{\prime},$$Eq. 3$${I}_{d}(x,\nu ,\tau )=\u27e8{\mid {A}_{s}(x,\nu ,\tau )+{A}_{r}(x,\nu )\mid}^{2}\u27e9={I}_{s}(x,\nu )+{I}_{r}(x,\nu )+2\phantom{\rule{0.2em}{0ex}}\mathrm{Re}\u27e8{A}_{s}(x,\nu ,\tau ){A}_{r}^{*}(x,\nu )\u27e9,$$4.The complex fringe signal optical frequency $\left(\nu \right)$ is given by, Eq. 5$${I}_{f}(x,\nu ,\tau )=\u27e8{A}_{s}(x,\nu ,\tau ){A}_{r}^{*}(x,\nu )\u27e9={r}_{\mathrm{mir}}{\int}_{\infty}^{\infty}{\int}_{\infty}^{\infty}\u27e8{A}_{i}(\xi ,\nu ){A}_{i}^{*}({\xi}^{\prime},\nu )\u27e9{r}^{*}\left(\xi \right)\mathrm{exp}(j2\pi \nu \tau )h(x\xi ,\nu ){h}^{*}(x{\xi}^{\prime},\nu )\mathrm{d}\xi \phantom{\rule{0.2em}{0ex}}\mathrm{d}{\xi}^{\prime}.$$By summing over optical frequencies $\left(\nu \right)$ the expression for the complex fringe signal ${I}_{f}(x,\tau )$ becomes Eq. 6$${I}_{f}(x,\tau )={\int}_{\infty}^{\infty}\u27e8{A}_{s}(x,\nu ,\tau ){A}_{r}^{*}(x,\nu )\u27e9\mathrm{d}\nu .$$Eq. 7$${I}_{f}(x,\tau )={r}_{\mathrm{mir}}{\int}_{\infty}^{\infty}{\int}_{\infty}^{\infty}{\int}_{\infty}^{\infty}W(\xi ,{\xi}^{\prime},\nu )r\left(\xi \right)\mathrm{exp}(j2\pi \nu \tau )h(x\xi ){h}^{*}(x{\xi}^{\prime})\mathrm{d}\xi \phantom{\rule{0.2em}{0ex}}\mathrm{d}{\xi}^{\prime}\phantom{\rule{0.2em}{0ex}}\mathrm{d}\nu ,$$Eq. 8$$W({\xi}_{1},{\xi}_{2},\omega )=\sum _{n}{\alpha}_{n}\left(\omega \right){\phi}_{n}^{*}({\xi}_{1},\omega ){\phi}_{n}({\xi}_{2},\omega ),$$Eq. 9$$W(\xi ,{\xi}^{\prime},\nu )={\left[S(\xi ,\nu )\right]}^{1\u22152}{\left[S({\xi}^{\prime},\nu )\right]}^{1\u22152}g(\xi {\xi}^{\prime},\nu ),$$Eq. 10$$S(\xi ,\nu )={A}^{2}\left(\nu \right)\mathrm{exp}[{\xi}^{2}\u22152{\sigma}_{s}^{2}\left(\nu \right)],$$Eq. 11$$g(\xi {\xi}^{\prime},\nu )=\mathrm{exp}[{(\xi {\xi}^{\prime})}^{2}\u22152{\sigma}_{g}^{2}\left(\nu \right)],$$If the power spectral density ${A}^{2}\left(\nu \right)$ is given as Eq. 13$${A}^{2}\left(\nu \right)=\mathrm{exp}[\frac{{(\nu {\nu}_{0})}^{2}}{2\Delta {\nu}^{2}}],$$Eq. 14$${I}_{f}(x,\tau )=\sqrt{\pi}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(i\pi \tau {\nu}_{0})\mathrm{exp}(\frac{{\pi}^{2}{\tau}^{2}}{2\Delta {\nu}^{2}})\int \int \mathrm{sinc}\left[\frac{D}{cz}{\nu}_{0}(x\xi )\right]\mathrm{sinc}\left[\frac{D}{cz}{\nu}_{0}(x{\xi}^{\prime})\right]r\left(\xi \right)[\frac{1}{2\Delta {\nu}^{2}}+A(\xi ,{\xi}^{\prime})]\mathrm{exp}[A(\xi ,{\xi}^{\prime}){\pi}^{2}{\tau}^{2}]\mathrm{d}\xi \phantom{\rule{0.2em}{0ex}}\mathrm{d}{\xi}^{\prime},$$
$$A(\xi ,{\xi}^{\prime})=\frac{{\sigma}_{g}{\left({\nu}_{0}\right)}^{2}({\xi}^{2}+{\xi}^{2})+2{\sigma}_{s}{\left({\nu}_{0}\right)}^{2}{(\xi {\xi}^{\prime})}^{2}}{4{\sigma}_{s}{\left({\nu}_{0}\right)}^{2}{\sigma}_{g}{\left({\nu}_{0}\right)}^{2}{\nu}_{0}^{2}}.$$
We note that the complex fringe signal
${I}_{f}(x,\tau )$
in Eq. 14 contains the OCT complex fringe signal with carrier frequency
$\left[\mathrm{exp}(i\pi \tau {\nu}_{0})\right]$
with envelope
$\left[\mathrm{exp}({\pi}^{2}{\tau}^{2}\u22152\Delta {\nu}^{2})\right]$
, but also a multiplicative term, which is the integral over spatial coordinates
$[\xi ,{\xi}^{\prime}]$
. The integral over
$\xi $
and
${\zeta}^{\prime}$
accumulates in regions where the point spread functions are nonzero and is proportional to
$r\left(\xi \right)$
. Because the phase of
$r\left(\xi \right)$
can fluctuate with sample position
$\left(\xi \right)$
, the multiplicative term introduces speckle.Because the exact form of the complex reflectivity of the sample $\left[r\left(\xi \right)\right]$ is not generally known, we evaluate the statistics of ${I}_{f}(x,\tau )$ . We assume that the local complex reflectivity $\left[r\left(\xi \right)\right]$ in a sample has a fixed amplitude distribution $\left[r\left(\xi \right)\right]$ with randomly varying but uniformly distributed phase^{19} $\left({\varphi}_{k}\right)$ . The complex fringe signal at position $x$ in Eq. 14 can be viewed as a sum of all field amplitudes reflected back within a coherence patch. Each field amplitude reflected from the sample contributes as a complexvalued phasor at position $x$ in the observation plane. The complex fringe signal with speckle can be analyzed as a random phasor sum: Eq. 15$${I}_{f}=a\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\left(i\theta \right)=\frac{1}{N}\sum _{k=1}^{N}{\alpha}_{k}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\left(i{\varphi}_{k}\right).$$16.2.1.Numerical SimulationBecause speckle in OCT arises from coherent interference of backscattered light from the sample, we investigate speckle reduction by considering a GaussianSchell model source with different spatial coherence lengths $\left({\sigma}_{g}\right)$ used in conjunction with a 1D rough surface. The complex fringe signal [Eq. 14] is computed along a line using software written by the author. We assume pupil of lens L3 has a diameter of $1\phantom{\rule{0.3em}{0ex}}\mathrm{cm}$ and is placed $10\phantom{\rule{0.3em}{0ex}}\mathrm{cm}$ away from the sample or reference mirror giving a numerical aperture $\left(\mathrm{NA}\right)=0.1$ . The width of the point spread function (PSF) is calculated to be $1.2{\lambda}_{0}\u2215\mathrm{NA}=8.5\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$ assuming ${\lambda}_{0}=850\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ . Grid spacing for $\xi $ and ${\xi}^{\prime}$ is $0.34\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$ or 25 times smaller than the PSF width. Beam spot diameter $\left({\sigma}_{s}\right)$ at the sample and the reference mirror is set at ${\sigma}_{s}=100\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$ and several spatial coherent lengths $\left({\sigma}_{g}\right)$ of the GaussianSchell model source are chosen to observe the speckle reduction effect. The sample reflectivity is set at unity between 40 and $60\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$ and zero elsewhere. Phases vary from $\pi $ to $\pi $ with uniformly distributed random probability. The image plane is positioned at $10\phantom{\rule{0.3em}{0ex}}\mathrm{cm}$ from lens L3 and complex fringe signal $\left[{I}_{f}\left(x\right)\right]$ space is computed over $[x=0,100\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}]$ at discrete points separated by $1\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$ . 2.2.Experimental SetupA depthresolving OCT system based on a twobeam Michelson interferometer was constructed to investigate speckle reduction using a partially spatial coherent source. Partially spatial coherent light was generated from a coherent light source by using a long segment of highNA multimode optical fiber. For coherent illumination, we used a single mode fiber with a cuffoff wavelength ${\lambda}_{c}=800\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ . Both single and multimode optical fibers are placed in the source path of a Michelson interferometer to investigate effect of source coherence length on speckle reduction. First, we used a $100\text{}\mathrm{m}$ long, 0.48NA multimode glass optical fiber with a $200\text{}\mu \mathrm{m}$ core diameter, producing about 63,000 coherent spatial modes.^{32} Second, we used a $30\text{}\mathrm{m}$ long single mode optical fiber (Rifocs Corp.) similar to that used in a conventional OCT setup. In both cases, a modelocked Ti:sapphire laser with an output centered at $850\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ and spectral width $\Delta \lambda \sim 50\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ is coupled into the fibers and used as source light. When coupled into a single mode fiber, light gives a ${\mathrm{TEM}}_{00}$ spatial mode. In the first setup (Fig. 3 ), light is coupled into a multimode fiber to generate many spatial modes. The $100\text{}\mathrm{m}$ long multimode fiber is mounted on a spool to reduce any variations in mechanical stress that might introduce variable mode coupling. Following the multimode fiber, lens L2 directs light into both sample and reference paths through a broadband beamsplitter (BS). In the second setup using a singlemode fiber, the multimode fiber is removed from the source path of the interferometer. Reflected light from both sample and reference paths are imaged onto a CCD camera (FASTCAM Super $10\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ ) using a focusing lens (L3). A voice coil with a retroreflector is used in the reference path to provide variable delay $\left(\tau \right)$ between sample and reference paths. The voice coil is driven by a function generator creating a sinusoidal wave at $0.1\phantom{\rule{0.3em}{0ex}}\mathrm{Hz}$ and $10\phantom{\rule{0.3em}{0ex}}{\mathrm{V}}_{\mathrm{p}\text{}\mathrm{p}}$ . Scan range is $1.6\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ , yielding a fringe carrier frequency ${f}_{c}=752\phantom{\rule{0.3em}{0ex}}\mathrm{Hz}$ . Frame rate of the CCD camera is set at $3000\phantom{\rule{0.3em}{0ex}}\text{frames}\u2215\mathrm{s}$ sampling the fringe at four points per cycle or twice the Nyquist limit. Integration time of the camera is $50\phantom{\rule{0.3em}{0ex}}\mu \mathrm{s}$ . Output images from the CCD camera are recorded in the digital video (DV) format to avoid any loss in spatial resolution. Incoherent demodulation is used to determine the envelope of the complex fringe signal [Eq. 15] from recorded 3D data cubes. Fringe amplitude is determined at each pixel in time (Fig. 4 ). Other than incoherent demodulation no additional image enhancement is used. A millfinished aluminum plate is used as a scattering surface. The rough surface of the aluminum plate produces speckles caused by distorted wavefronts in reflected light. The aluminum plate was tilted slightly to produce a uniform phase gradient and produce linear fringes in the CCD. For comparison, the same physical location covering a $2\text{}\mathrm{mm}$ beam diameter on the aluminum plate is imaged with both single and multimode fiber systems. To maximize fringe visibility and to avoid saturation of CCD photocells, light intensity incident on the CCD camera was adjusted by using a neutral density filter in the detection path of the interferometer. 3.ResultsEffect of speckle reduction in OCT imaging was simulated using a GaussianSchell model source and Eq. 14 for ${I}_{f}(x,\tau )$ . For this simulation, the illuminating beam spot $\left[{\sigma}_{s}\right]$ was fixed at $100\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$ . Simulation results shown in Fig. 5 and Table 1 are determined for a depthresolving OCT system with several different spatial coherent lengths $\left({\sigma}_{g}\right)$ corresponding to a GaussianSchell model source. In Fig. 5, the dotted lines represent the envelope of the complex fringe signal without speckle, while solid lines are with speckle. The specklefree case was simulated by removing the random phase term in Eq. 15. Coherent illumination $({\sigma}_{g}={\sigma}_{s}=100\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m})$ produces the greatest speckle [Fig. 5(a)]. For incoherent illumination, spatial coherence length was set much smaller $({\sigma}_{g}=0.01\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m})$ than diameter $\left({\sigma}_{s}\right)$ of the illuminating beam [Fig. 5(d)]. Two additional spatial coherence lengths $({\sigma}_{g}=10,1\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m})$ in Figs. 5(b) and 5(c), respectively, were investigated to demonstrate speckle reduction with decreasing coherence length. To assess speckle reduction quantitatively, speckle rootmeansquare (rms) values were calculated by subtracting computed complex fringe signal with speckle from the specklefree case. Speckle rms variation with spatial coherence length $\left({\sigma}_{g}\right)$ is shown in Table 1. Entries in Table 1 were computed by averaging 30 simulations for each spatial coherence length $\left({\sigma}_{g}\right)$ . Note the logarithmic scale to display ${\sigma}_{g}$ . Table 1Speckle rms variation according to spatial coherence length (σg) .
To observe difference in sample illumination through single or multimode fibers, light from the reference path was blocked and the aluminum plate was placed in the sample path. Figures 7(a) and 7(b) show intensity images of the sample when illuminated by singlemode and multimode fibers, respectively. Due to a small mode field diameter $\left(5\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}\right)$ at the singlemode fiber tip, Airy rings in the illuminated area are observed in Fig. 7(a), whereas relatively flat illumination is evident in the multimode case [Fig. 7(b)]. After restoring the reference path, an interference pattern was measured by the CCD camera corresponding to a $1.6\text{}\mathrm{mm}$ pathlength delay between sample and reference paths. During measurement of the interference pattern, no significant motion artifacts in the fringes, caused by environmental changes, were observed. Measured interference patterns were demodulated using an incoherent demodulation algorithm ( ). Figures 8(a) and 8(b) show OCT crosssectional images of the sample at a fixed height, with the normalized complex fringe signal represented along the vertical axis, recorded with singlemode and multimode fiber systems, respectively. When using the multimode fiber for illumination, speckle is significantly reduced [Fig. 8(b)] compared to that recorded using the singlemode fiber [Fig. 8(a)]. Demodulated interference patterns at a fixed depth recorded with single and multimode fiber systems are shown in (a) and (b) respectively. For comparison, both images were recorded at the same position on the aluminum plate, so surface microirregularities of the surface are identical. The image in (a) was recorded after rotating the aluminum plate $20\phantom{\rule{0.3em}{0ex}}\text{degrees}$ counterclockwise. The image recorded using the multimode fiber, (b), shows continuous linear fringes corresponding to tilt of the aluminum plate, whereas the image recorded using single mode fiber for illumination, (a), displays disrupted and fragmented bright grey bands over the surface due to speckle. 4.DiscussionLight emitted from a GaussianSchell model source considered here has a variable degree of spatial coherence from incoherent to fully coherent. When partially spatial coherent light is launched into the interferometer, the beam may be regarded as an assembly of mutually uncorrelated finitesized phase cells. In OCT, light input into the interferometer is divided into two beams, which illuminate the sample and the reference mirror. While mirror reflectivity in the reference path is considered constant, the sample has a complex reflectivity function and backscatters incident light. When light reflected from sample and reference paths recombine at the detection plane, cells of the beams overlap and partially interfere. Interference can occur only for light originating from the same phase cell in sample and reference paths. The speckled appearance of the images in is the effect of interference between light backscattered from different particles within a coherence area. With a decrease in the coherence area, speckle reduction is expected, as indicated in Table 1, but as Fig. 5(d) illustrates speckle still exists despite a very small spatial coherence length $({\sigma}_{g}=0.01\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m})$ corresponding to incoherent illumination. An interesting point when looking at the complex fringe signal [Eq. 14] as a random phasor sum is that speckle increases with the second moment of the complex fringe signal, which follows the second moment $\left(\overline{{\alpha}^{2}}\right)$ of amplitude $\left({\alpha}_{k}\right)$ , not the random phase distribution $\left({\varphi}_{k}\right)$ . The result of spatial coherence length dependence on the second moment of the complex fringe signal is shown at Fig. 6. The second moment $\overline{{\alpha}^{2}}$ of the intensity fluctuation increases as the degree of coherence increases and approaches unity for a spatially coherent beam. The second moment never reaches zero even at the smallest spatial coherence length. Producing incoherent sources has an inherent tradeoff in view of radiation efficiency and the degeneracy parameter.^{25} The radiant emittance of a partially spatial coherent source is expressed^{30} by where the radiation efficiency $\left({C}_{g}\right)$ is a proportionality factor between radiant emittance $\left[{E}_{\nu}\left(\rho \right)\right]$ and spectral density $\left[S(\rho ,\nu )\right]$ , and $\rho $ is a position vector of a radiation source point. The radiation efficiency increases monotonically with ${\sigma}_{g}$ , from zero for a completely incoherent source to unity for a completely coherent source.^{30} The degeneracy parameter indicates average number of identical photons (same polarization sate) that are contained in a coherence volume. For example, use of spatially incoherent illumination with a thermal light source suffers from a low degeneracy number.^{23} The advantage of using a multimode fiber to produce partially spatially coherent light is high number of photons per coherence volume. The source and multimode fiber used in experiments reported here provided $0.3\times {10}^{9}$ photons per coherence volume substantially higher than that provided by a thermal source. Spatial coherence length at the tip of the multimode fiber was calculated^{33, 34} to be $22\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$ by
$${\sigma}_{g}={\left\{\frac{4{\sigma}_{0}^{2}}{{(\pi D\mathrm{NA}\u22152\lambda )}^{2}1}\right\}}^{1\u22152},$$
where
$D$
is a core diameter of multimode fiber.Another advantage of this method to modify coherence properties of the light source is relative simplicity over other methods including use of a ground glass plate or liquid crystals.^{30} In Eq. 1 we assumed amplitude of light returning from the sample is the product of the incident amplitude $\left[{A}_{i}(\xi ,\nu )\right]$ and the local reflectivity $\left[r\left(\xi \right)\right]$ , which is reasonable for the scattering object studied. In tissue, however, the amplitude of light returning from the sample will be a complex function of ${A}_{i}(\xi ,\nu )$ due to multiple photon paths in the tissue and spatial variation of the refractive index of the tissue. Speckle reduction using partially spatial coherent sources can be also understood as an incoherent mode summation. When broadband light is coupled into a multimode fiber, a number of orthogonal spatial modes are generated which are spatially and temporally coherent. Each spatial mode propagates through the multimode fiber at a distinct group velocity producing modal dispersion. After propagating a sufficient length along the fiber, spatial modes become temporally decorrelated due to differences in group velocity. Detected interference is an incoherent sum of interferograms formed between spatial modes. Because each mode forms a statistically independent speckle field, mode summation has the effect of reducing speckle by averaging. The appearance of unbroken grooves in the multimode fiber case [Fig. 9(b) ] is consistent with the speckle reduction effect resulting from superposition of many interferograms each representing mutually incoherent spatial modes. As samplereference path length delay increases, the delay may compensate the mode separation introduced by the multimode fiber, and the decorrelated modes become correlated again. To avoid this intermodal interference, the multimode fiber should be sufficiently long so that the modal separation is much greater than maximum samplereference delay time in scanning. To verify no intermodal interference was present in our system, signal to noise ratios for single mode and multimode cases were measured and compared across the full scanning range $\left(1.6\phantom{\rule{0.3em}{0ex}}\mathrm{mm}\right)$ of the voice coil stage. A mirror was placed in the sample path with a photoreceiver and a $100\text{}\mu \mathrm{m}$ pinhole in the detection path replaced the CCD camera. We express noise sources in terms of the photocurrent variance ${\sigma}_{i}^{2}$ . Primary noise sources include receiver $\left({\sigma}_{\mathrm{re}}^{2}\right)$ , and shot noise $\left({\sigma}_{\mathrm{sh}}^{2}\right)$ . Intensity noise was omitted because of the small contribution from a mode locked laser.^{35} Receiver noise was calculated from the manufacturer’s specification. Shot noise is given by, where ${I}_{\mathrm{dc}}$ represents the mean detector photocurrent, and $B$ is the electronic detection bandwidth, and $q$ is the charge on an electron.Total photocurrent variance is given by Measured and calculated shot noises are compared in Table 2 . The shot noise is calculated from the mean detector photocurrent ${I}_{\mathrm{dc}}$ . Measured shot noise is comparable with calculated values.Table 2Noise comparison between multimode and singlemode source fiber setups.
We define SNR as peaktopeak current of the coherence function divided by the total photocurrent standard variance $\left({\sigma}_{i}\right)$ . SNRs were 80 and $79\phantom{\rule{0.3em}{0ex}}\mathrm{dB}$ over $1.6\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ of reference and sample path delays for singlemode and multimode fiber cases, respectively. The Small difference in SNR between coherent and partially spatial coherent cases indicates degradation of SNR due to the multimode source fiber was insignificant and supports the hypothesis that mode separation between adjacent modes is longer than the scanning range $\left(1.6\phantom{\rule{0.3em}{0ex}}\mathrm{mm}\right)$ after $100\phantom{\rule{0.3em}{0ex}}\mathrm{m}$ of travel through the multimode fiber. Mode crosscoupling in the fiber was below the measurement limit over the entire scan $\left(1.6\phantom{\rule{0.3em}{0ex}}\mathrm{mm}\right)$ . 5.ConclusionWe proposed and demonstrated a speckle reduction method using a partially spatial coherent source in a Michelson interferometer in a parallel detecting OCT system. Speckle in interferograms recorded using a partially spatial coherent source is substantially reduced compared to the fully coherent case. No degradation in SNR is observed, and no measurable mode crosscoupling is observed in the multimode fiber. A partially spatial coherent source is also preferable for en face imaging not only for reducing speckle but also for eliminating Airy rings in the image caused by the small field diameter of most singlemode fibers. Our simulations model a partially spatially coherent source with a GaussianSchell model source to indicate speckle reduction in OCT interferograms. As the physical viewpoint, our experiments explain that superposition of coherent modes has the effect of reducing speckle by averaging because each mode forms a statistically independent speckle field. The results of both simulations and experiments indicate that broadband light sources with reduced spatial coherence that provide a large number of photons per coherence volume may be effectively utilized to reduce speckle in OCT interferograms. AcknowledgmentsThis research has been supported by National Institutes of Health (NIH) (R01 EB002021) and the Texas ATP. ReferencesD. Huang,
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