Spectral shaping for non-Gaussian source spectra in optical coherence tomography.

We present a digital spectral shaping technique to reduce the sidelobes (ringing) of the axial point-spread function in optical coherence tomography for non-Gaussian-shaped source spectra. The spectra of two su- perluminescent diodes were combined to generate a spectrum with significant modulation. Images of onion cells demonstrate the improved image quality in a turbid biological sample. A quantitative analysis of the accompanying penalty in signal-to-noise ratio is given.

In optical coherence tomography 1,2 (OCT) the temporal coherence or, equivalently, the source spectral bandwidth governs the axial point-spread function. To increase axial resolution, several methods have been used to generate light with low temporal and high spatial coherence. 3 -8 In virtually all these methods special care must be taken to generate spectral shapes that resemble a Gaussian prof ile. Non-Gaussian spectral shapes will result in the presence of sidelobes in the coherence envelope that will generate spurious structures in OCT images. Many sources that can be used for OCT (e.g., erbium-doped fiber amplif iers and self-phase-modulated spectral broadening of short pulses in microstructure fibers 9 ) do not have Gaussian spectra and therefore require some form of spectral filtering 3,5 or shaping of the source spectrum into a smoother form. 8 In this Letter we demonstrate that these problems can be addressed by a signal processing algorithm without significant penalty to the signal-to-noise ratio.
Digital signal processing was previously used to increase resolution. Kulkarni et al. 10 demonstrated this in a two-dimensional image by an iterative deconvolution with the coherence envelope's point-spread function in the space domain, but this method is computationally intensive. Bashkansky et al. 11 proposed spectral shaping to improve the measured crosscorrelation function. Using a virtually Gaussian source spectrum, they demonstrated an increase in resolution of a single depth prof ile of two glass slides pressed together by deconvolution with the measured power spectrum in the Fourier domain. However, this method for increasing spatial resolution increases the noise f loor significantly.
In this Letter we focus on the reduction of spurious structures in OCT images that have been caused by sidelobes. Previously, sidelobe reduction in a single depth prof ile of a glass slide was demonstrated. 12 Here we present, for the f irst time to our knowledge, the spectral shaping technique applied to imaging of biological structure by use of a source spectrum with significant modulation and analyze the penalty in signal-to-noise ratio that is incurred.
In an OCT system the coherence function is proportional to the Fourier transform of source spectrum S͑k͒: where k 2p͞l is the free-space wave number and Dz z 2 z 0 is the path-length difference between reference and sample waves. The spectral shaping technique shapes the source spectrum by Fourier transforming the interferometric signal in relation (1) and then applying a correction to each Fourier component such that the spectrum becomes Gaussian. After an inverse transform, the ideal coherence function for a Gaussian source spectrum is obtained.
The experimental setup is similar to that reported in Ref. 13, except that in the current setup the design wavelength was 830 nm and no phase modulator was employed. Light from two orthogonally polarized superluminescent diodes (SLDs; SLD-371 and SLD-381, Superlum, Moscow, Russia) was coupled into a single-mode f iber after power efficient combination by a polarizing beam splitter. The individual spectra were in orthogonal states and the source power after combination was 0.9 mW for SLD 1 and 0.3 mW for SLD 2. Source light was then split equally into the sample and reference arms by a f iber 2 3 2 coupler. Axial scanning over 2 mm in depth was done with a rapid scanning optical delay line, 14  lines per second and a 400-kHz center frequency of the interference fringes. In the detection arm, light was split into orthogonal polarization states before being detected. The polarization controller in the detection arm was aligned such that half of the light from each individual source ref lected from the reference arm was sent to each detector, ensuring that each detector sensed the same combination of individual spectra. The polarization controller in the sample arm was aligned to achieve the same objective for sample-arm light. The detector signals were digitized with a dual-channel 12-bit, 5-MHz analog-to-digital conversion board. The sample arm's focal diameter was 25 mm. The sources exhibited a noise peak at 250 kHz, close to the carrier frequency. The fringes detected in the two orthogonal polarization channels were approximately 90 ± out of phase owing to fiber birefringence. We exploited this phase difference to implement balanced detection by subtracting the orthogonal polarization channels, which significantly reduced the 250-kHz noise peak without reducing the signal, resulting in virtually shot-noise-limited detection. However, significant sample birefringence would pose a problem in this nonideal balanced detection scheme.
In Figure 1 the combined spectrum of SLD1 and SLD2 is shown. We calculated the spectrum from the average of the square root of the power spectrum of 500 A-lines (10,000 samples per A-line, zero padded to 16,384) by Fourier transforming the interferometric responses to a single surface. The dashed curve is a Gaussian f it with a center wavelength of 833 nm and a FWHM of 49 nm to the spectrum, determined by the zeroth, first, and second moments of the spectral density. The ratio of this ideal Gaussian source spectrum and the measured spectral density def ines a spectral correction curve, SC͑l͒. We obtained the spectrally filtered response of each depth profile by multiplying the Fourier transform of each individual depth profile by SC͑l͒ and performing the inverse transform. The coherence function envelope was obtained by digital quadrature demodulation with a spatial resolution of 1.6 mm. Figure 2 shows the coherence envelope before and after spectral shaping for a single surface (glass slide) and the sibelobe reduction at specific locations. The increase in the noise f loor far from the coherence envelope was 8.5 dB, and the FWHM of the coherence envelope was 6.0 6.5 mm.
To demonstrate that spectral shaping is not only feasible for (highly) ref lective single surfaces but actually improves images of biological structures, we imaged a section of an onion's skin. A total of six images were acquired at the same location. It took 1 s to acquire each image, scanning a width of 1 mm, and each image was processed both without and with spectral shaping. To obtain the SC͑l͒ for each individual image we determined the source spectral density from the image data by calculating the square root of the power spectrum for each A-line in the image and averaging over all A-lines (500). We used the zeroth moment of this spectral density and the first and second moments of the glass slide's spectral density to determine the ideal Gaussian source spectrum. As can be seen from Fig. 1, the spectral density as determined from one image is a good approximation of the glass slide's spectrum. The image spectrum was scaled vertically such that the ideal Gaussians for the spectra of the image and of the glass slide exactly overlapped. Figure 3 shows side by side the uncorrected and the spectrally shaped images after the six individual images have been summed to reduce background noise. Both images were gray-scale coded over a dynamic range of 40 dB (from maximum signal in the image to the noise level) and cropped to 600 mm 3 480 mm (width 3 depth). The image was magnified in the depth direction by a factor of 2 to increase the visibility of the sidelobes. Processing data acquired in 1 s required 2 s on a 1.3 GHz Pentium IV processor.
Throughout the image, spurious structures were visible about strongly scattering objects that correspond to the positions of the sibelobes. The spectrally shaped image showed a significant reduction of these spurious structures. Figure 4 shows single depth prof iles averaged over four A-lines before and after spectral correction and the sidelobe reduction achieved in the area that lies between the arrows in Fig. 3. The increase in the noise f loor determined by averaging of the signal in the air above the sample surface was only 0.9 dB.
Increase in the noise f loor can be estimated as follows: The intensity ref lected from the sample arm is proportional to the square of the interferometric intensity, I ͑z͒ in relation (1). Curve SC͑l͒ gives a multiplication factor for each Fourier component f ͑l͒,    which consists of a signal f s ͑l͒ and a noise f n ͑l͒ contribution. From Parseval's theorem, the average noise level ͗I n 2 ͑z͒͘ is equal to ͗j f n ͑l͒j 2 ͘, where angle brackets denote averaging. Thus the increase in the noise level is given by R SC 2 ͑l͒ j f n ͑l͒j 2 dl͞ R j f n ͑l͒j 2 dl, which, assuming a white-noise spectrum, reduces to 10 log͗SC 2 ͑l͒͘ in decibels. For the glass slide we calculated a value of 8.65 dB, which corresponds well to a measured value of 8.5 dB, and for the image we calculated a value of 0.97 dB, which corresponds well to a measured value of 0.9 dB. The 8.5-dB increase in the noise level for the glass slide is due mainly to the large value of SC͑l͒ from 855 to 867 nm (Fig. 1).
The peak values of the coherence envelopes are proportional to R j f s ͑l͒jdl. Equal peak heights before and after spectral shaping are maintained if R j f s ͑l͒jdl R SC͑l͒ jf s ͑l͒jdl. This condition can be approximately satisf ied by use of the zeroth moment of the spectral density j f ͑l͒j as the zeroth moment for the ideal Gaussian. Doing so results in virtually equal values of the coherence peaks for uncorrected and spectrally filtered depth prof iles, as can observed from Figs. 2 and 4. The optimal choice of the f irst and second moments for the ideal Gaussian in a particular measurement can be determined by choice of the maximum allowable increase in the noise f loor.
Absorption-and wavelength-dependent scattering will alter the ref lected spectrum and thus the coherence function in all OCT systems. To some extent, spectral shaping is able to correct for these effects on the coherence function, because the average ref lected spectrum is used as a source reference. However, the method is not able to correct for changes in the ref lected spectrum as a function of depth. In conclusion, the spectral shaping procedure enables non-Gaussian source spectra to be used in OCT, which simplifies the development of low-cost broadband sources.