Ординатура / Офтальмология / Английские материалы / Handbook of Optical Coherence Tomography_Bouma, Tearney_2002

In the cross-talk cross-talk double-sum decomposition, the first sum with l ¼ m represents the sum of cross-talk powers from other layers. There is no coherence function reduction of this term, as in the OCT case. The second sum represents power from the coherent addition of the interlayer cross-talk fields and can be greatly reduced by using broadband sources. However, the coherence function appears only linearly, rather than quadratically as in OCT. The previous arguments about making the coupling coefficients real and using undetermined phase factors have been applied. The statistics of this signal are evaluated in the same way as in the OCT case. Note that there is a one-to-one correspondence of the terms within the braces for OCT and DD, but with significant differences between the details of the summation terms in the two cases.

14.4.3 Comparison of OCT and DD Cross-Talk

We now present a quantitative comparison of the evaluation of Eqs. (37) and (40). In the numerical evaluation of the signal distributions for OCT and DD we assume a Gaussian coherence function and use a modified spatial argument instead of a temporal argument, because propagation distance difference is proportional to time delay difference:

The argument z is the distance from the data plane at the beam waist to any crosstalk layer. In this definition, the parameter Le is the 1=e half-width of the scan coherence function. The spatial coherence function used here is the normalized envelope function of the interferometric photocurrent signal produced by scanning a reference mirror with respect to a fixed mirror. Its spatial argument is the scanning mirror displacement, which is half the round-trip difference between the signal and reference paths. By using the mirror displacement argument instead of the round-trip path difference, the depth of reflections from inside the signal medium are shown directly as arguments in the measured autocorrelation function (except for scaling from differences in index of refraction), and the spatial resolution of reflection

sources is determined from Eq. (41). In these normalized numerical calculations, we do not include refractive index scaling, and we set n ¼ 1 in both signal and reference paths. From definition (41), Le can be related to the other standard definition of the coherence length used elsewhere in this volume. If the broadband source has a Gaussian frequency spectrum, the coherence function (with time argument) is the Fourier transform of the spectrum. After converting from time difference to pathlength difference and using wavelength instead of frequency, the full width at halfmaximum coherence length is inversely proportional to the spectral width:

where is the center wavelength and is the spectral full width at half-maximum.

ness to Rayleigh range (L=Lr, where Lr is the same as the previously defined zR), and layer thickness to 1=e scan parameter ðL=LeÞ.

In Fig. 8 we compare the ‘‘0’’ and ‘‘1’’ signal distributions having cross-talk from five layers below and five above the desired data layer, with ratio L=Lr ¼ 1 for different L=Le coherence length ratios. The relative signal distributions are plotted as a function of normalized signal level, with the cross-talk free ‘‘0’’ signal having value 0 and ‘‘1’’ level having value 1. For OCT all cross-talk terms become small for sufficiently small coherence length, and we see significant ‘‘eye-opening’’ for L=Le 1:5 or L=LC 1:8. Since DD always has a residual cross-talk power term independent of coherence length, this particular example shows that no DD ‘‘eye-opening’’ between normalized ‘‘0’’ and ‘‘1’’ levels ever exists for layers so densely packed.

In a second example, shown in Fig. 9, the layer separation has been increased to L=Lr ¼ 2, and the sum is over two interfering layers on either side of the data layer. Now the OCT method has good threshold discrimination for L=Le 1 or L=LC 1:2; that is, the correlation length can be slightly longer if the layer defocusing is increased. The DD method now also has an ‘‘eye-opening’’ at L=Le 1 (L=LC 1:2), but not as large as in the OCT case. What is significant for the DD case is that its performance can be improved by decreasing the coherence length such that Le < L=2.

The analytical forms for the OCT and DD interlayer cross-talk [Eqs. (37) and (40)] show great similarity. The first two terms are identical, but the last two terms differ in their dependence of the coherence function. In the limit of very broad bandwidth sources, the coherence function approaches a delta function. In this limit, all the cross-talk terms of the OCT system vanish, and perfect bit decisions can be made. For the DD case, the residual randomly fluctuating powers detected from the interfering layers remain. If the layers are closely spaced with respect to a Rayleigh range, then the resulting cross-talk will be significant and cause bit errors. If the layers are more widely spaced (yielding poorer volume data density), then the geometric defocusing factors (included in the reflective coupling coefficients) will minimize the detected power from interfering (but out-of-focus) data layers, allowing reliable bit decisions. In general, shorter coherence lengths will allow DD systems to have significantly higher volume densities (closer layer spacing) up to a point where the layer spacing becomes comparable to a Rayleigh range (one or more interfering

layers are within the focus). At that point, OCT systems can be used to achieve even higher volume density. OCT systems have the additional advantage of greater system receiver sensitivity, discussed below.

14.5SYSTEM SENSITIVITY ISSUES

Signals from different layers will have some intrinsic differences in strength because of attenuation and distortion. In this section, we discuss these differences and their relation to system performance. We will compare parameters for an OCT system, a low power broadband DD system, and a more conventional higher power DD system. Typical parameters that will be used for the example calculations are a

The first case we consider is an OCT system with a broadband source having 1 mW total power equally divided between signal and reference beams. A heterodyne detection sensitivity of 100 photons/bit can be achieved for an uncorrected bit error rate (BER) of 10 9. At the above data rate, this sensitivity corresponds to a mini-

mum detectable power of Pmin ¼ 100fDh ¼ 1:45 10 10 W. To allow for practical implementation losses and the diversion of some signal power for tracking consid-

erations, Pmin is derated by a factor of 10, giving a minimum detectable power of 1:45 10 9 W. This means that the smallest allowable reflectivity from a data layer is bounded by the ratio of the minimum detectable power to the input signal power,

The optical power in the diffraction-limited mode of the signal beam will be attenuated by scattering and reflection as it passes through data layers. To limit signal strength variation with data layer depth, we impose a constraint that the nominal read signal not drop below 90% of its initial value from the first or top data layer. This ensures a nearly constant receiver eye-opening as different data layers are addressed, and reduces any need for automatic gain control. From the previous propagation analysis, we found that scattering introduces a loss factor of 1.85 times the Fresnel reflectivity. If we include this in a total effective reflectivity, then Reff ¼ 2:85R. The normalized read-field amplitude (appropriate for heterodyne detection, and not the intensity) after going back and forth through N data layers (traversing each of N layers twice) is (1 Reff ÞN , which we require to be greater than

0.9. In the limit of small R, this is equivalent to NReff lnð0:9Þ ¼ 0:105. Using the smallest allowable value of R from the detection limit, Rmin;eff ¼ 2:85 2:9 10 6 ¼ 8:3 10 6, which gives a maximum value for N ¼ 1:3 104. This bound establishes

a limit on the number of layers in an OCT system solely from signal sensitivity and attenuation factors. Material uniformity problems may require the refractive index difference to be larger than the limit from detection sensitivity. Also, other practical engineering problems such as excessive total thickness and focusing aberration provide a much smaller limit.

408 Chinn and Swanson

As an example of a practical aberration problem, we have examined focal depth issues for a typical aspheric focusing lens of the type used in standard CD systems, with a numerical aperture of 0.55, at a wavelength of 780 nm, optical disk thickness of 1.2 mm, and polycarbonate disk refractive index of 1.57. The phase function at the lens pupil is calculated from the optical path difference as a function of ray angle from the focal spot at the design depth in the medium, 1.2 mm. This phase function is assumed to be flattened by the aspheric lens as it collimates the beam originating from the focus at the data layer. Conversely, the same phase function is imposed on the incident flat-phase collimated beam, to focus it at the required depth. Then with the surface of the medium translated with respect to the lens, a second phase function at the lens is recalculated from the rays originating at a new focal spot at a different depth in the medium. Because the focal depth is no longer at the optimal point in the medium for which the lens was designed, there is some residual spherical aberration, which is given by the difference in the two phase functions less an unimportant radially constant factor. The amount of the mode returning into the collimated beam is calculated by finding the overlap of the phase distortion, exp½j ðrÞ&, across the beam profile,

The magnitude of this complex coupling factor is squared to allow for the twopass coupling of the beam into the diffraction-limited mode, as it couples to the focal spot on incidence and to the collimated beam on reflection. In the calculation, the focus displacement inside the disk was treated as the independent variable, and the lens-to-disk separation was adjusted slightly from the paraxial focal condition to minimize the rms phase deviation in the aperture (in a manner similar to using the ‘‘circle of least confusion’’ for aberrated foci). By doing this, a slight amount of defocus partially compensates for the spherical aberration. In Fig. 10 is plotted

the two-pass field coupling j lensj2 as a function of interior layer offset from the design optimum. Also shown is the residual displacement of the disk to minimize

the phase error. The formula above expresses the coupling for any mode shape (e.g., Gaussian). We have used a uniform beam intensity across the lens aperture, which gives a worst-case estimate, because high-angle phase distortion is weighted more heavily. For simple, fixed high numerical aperture lenses, the aberration-induced focal degradation may be the most stringent limit on the available depth of data storage. In more sophisticated systems, a multielement microzoom lens or compensating plate could be used to compensate for this aberration and greatly extend the available depth of storage [22]. In the above example, assuming a usable focal range of 0.2 mm, a layer separation of 20 m would allow 10 layers with no optics compensation.

Next, we examine two cases of DD systems, the first with 1 mW power (using the same broadband source considered above) and a second with 10 mW power, such as a semiconductor laser. The laser will have much longer coherence length, leading to worse cross-talk problems. For a thermally limited DD receiver using a pin photodiode, a detector sensitivity is approximately 10,000 photons/bit. Quantum-limited direct detection can be achieved with low-noise optical amplifiers, but their expense probably precludes their use in the type of systems we envisage,

Figure 10 Aberration-induced OCT signal reduction (coupling) and optimized focal shift (right-hand axis) as a function of focal layer offset.

even if low excess noise can be achieved. At the previous bit rate, this sensitivity is

N data layers is (1 Reff Þ , which is also required to be greater than 0.9. Note the factor of 2 in the exponent, resulting from intensity, rather than amplitude, attenua-

These limits on allowable numbers of layers are two and one order(s) of magnitude less than that of the OCT system. All of the previous limit relations are summarized in Table 1.

In general, we note that an OCT system is about 100 times as sensitive as a DD system. In principle, this advantage can be used to achieve 100 times as many layers for the same sensitivity or 100 times the sensitivity for the same number of layers. This sensitivity advantage may also enable the use of lower power sources, particularly in the blue spectral region where only limited power and lifetime have been demonstrated. More important, it is imperative to have low reflectivity per layer to achieve high volume density. Otherwise, the beam quality and point spread

function will degrade from scattering, and interlayer reflective cross-talk will be high. OCT system sensitivity advantages can be used to accommodate these low distortion and cross-talk requirements for low reflectivity interfaces. Finally, as noted above, other practical considerations such as medium thickness and variable aberration at different focal depth may limit the number of layers. Here, too, the OCT method has the advantage of being able to use more closely spaced layers to maximize their number within a fixed depth range in systems where aberration compensation is not used.

14.6EXPERIMENTAL RESULTS

14.6.1 Experimental Setup

As a first demonstration and verification of OCT and DD readout, we measured small-field images of a standard, single-layer CD-ROM and prototype multilayer disks. A diagram of the experimental apparatus is shown in Fig. 11. The broadband incoherent source is a fiber-coupled superluminescent LED emitting near 830 nm, whose spectrum is shown in Fig. 12a. The spectrum was measured using a commercial optical spectrum analyzer at its maximum resolution, 0.1 nm. Data from several adjacent spectral windows were acquired, stored, and concatenated to provide a large-bandwidth, high resolution spectrum required for accurate Fourier analysis. The coherence function of the source was measured with the apparatus described below by scanning the reference mirror position. The measured LED coherence function is shown in Fig. 12b, with an FWHM scan coherence length of 9:4 m. There is good agreement between the experimentally measured coherence function and that obtained from the Fourier transform of the measured frequency spectrum. Because the measured scan distance is half the change in reference displacement, the usual coherence length is 2 9:4 m ¼ 18:8 m. Using the definition of OCT resolution as half the usual coherence length, we see that our measured FWHM scan