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

Figure 3 Illustration of analysis formulation of data region shown below an intensity contour of a defocused readout beam. The square grid shows the coordinate system of the data cells and may be displaced by arbitrary fractions of a data cell from the beam coordinate center. [Adapted from Ref. 11, Fig. 3.]

this subdivision, below a typical defocused Gaussian intensity profile. The coupling coefficient is then expressed as the sum

XX

where ImA is the integral of the intensity over the mth square cell, and Zm is a random variable having values 1 or expðj Þ with probability 1/2. For a fine enough grid, the first (nonrandom) intensity sum over all grid cells is approximately equal to the integral of

twice, in the forward and backward directions. For our purposes, the phase of is unimportant, so we evaluate only the magnitude of 2, which is given by

To give an example of the statistical methodology, we show how the mean of Eq. (7) is found. Steps in derivation of the mean of the second term are straightforward. Define

where we have used the independence of different cells for l 6¼m. The third term of Eq. (7) is rewritten as

Using another integral approximation to the sum, the last summation term in Eq. (11) is A=w2, where w is the Gaussian mode waist at that layer. This approximates the inverse of the number of bit cells, N, in the mode area at a given data layer. Neglecting this last term, assumed to be small for a fine enough grid subdivi-

This result agrees with the expectation that as the interface phase shift or reflectivity vanishes, there is no beam distortion. Similarly, at the limit of zero fill factor, no differential phase exists, causing no distortion. To find the standard deviation of j j2, we have to evaluate j j4. This is done in a manner similar to that of finding the mean, except that products of as many as four sums of random variables occur. Their statistical evaluation requires that these products be separated into terms having sums over the same cells or independent cells. After much algebraic manipulation similar in method to the above derivation, we find that the variance of the coupling coefficient to lowest order in 1=N is

mean and variance apply to a single layer through which the reading beam propagates back and forth. These terms are the same for all layers except for the value of N, which will vary as the beam propagates through the multilayer medium and the Gaussian mode size changes at each layer. For multiple layers on top of the desired data layer, the total amplitude coupling coefficient is the product of the individual layer coefficients. The individual coefficients are statistically independent of each other because the data pit distributions in different layers are uncorrelated. Therefore, the mean of the total coupling coefficient is the product of the individual layer means:

If the ratio of standard deviation to mean for each layer is small, the ratio of the total standard deviation to total mean is approximately

or

If we give the distance of layer l from the reference 0 layers as Ll ¼ l L, then the number of pits in the defocused Gaussian waist is

where L is the layer separation and zR is the Rayleigh range (focal depth) of the Gaussian beam, defined by

with mode waist w0 (the 1=e field radius) at the data layer. A typical value for w0 is 2 m.

Using these results and the relations among the pit radius, cell area, and fill factor, we can find a bound for an infinite sum in Eq. (21),

Equations (16) and (24) give the statistical parameters for the received amplitude coupling coefficient of the beam reflected from a pit-free region (which we will call a 1), because this case will cause an incident Gaussian beam to be reflected on itself. By using the coupling coefficient, we are not concerned with the details of the field scattered out of the receiver mode (a formulation that implicitly neglects multiple scattering back into the mode). If the data at the desired layer were a 0, by design of the pit and focal waist the ideal coupling would be zero. An analytical formation of the exact, nonzero result would have to find the amount of actual reflected field that is scattered back into the receiver mode through scattering processes in both directions of the beam. We have not found a simple analytical technique for this calculation, which is performed only with the numerical method described below. Only the 1’s case will provide an analytical comparison with those numerical results.

14.3.2 Numerical Formulation

The numerical analysis of distortion caused by interfering data layers uses a standard fast Fourier transform (FFT) beam propagation method [21]. The basis of this method is an approximate paraxial simplification of Maxwell’s equations using a slowly varying envelope approximation. Because we are treating each data layer as a thin phase mask separated by a thick homogeneous region from the next mask, at first glance it would appear that the entire propagation between phase masks could be done in one step in the Fourier domain, because there are no variations in the medium to alter the plane wave decomposition of the beam. In principle this would be correct if boundary condition effects were negligible. The use of the unwindowed FFT imposes implicit constraints that the solutions are periodic over the interval of computation. As long as the fields are negligible at the boundaries this would present

no problem. For practical reasons of memory limitations and computation time, however, we have to limit the computation grid to be not much larger than the largest beam waist. To avoid the effects of aliasing (implicit periodic boundary conditions), thin absorbing regions are imposed at the grid boundaries to remove spurious reflected beams. Accurate use of these absorbing layers requires inverse transformation to the spatial domain at several smaller increments through the homogeneous medium between phase masks. Therefore the final technique is the so-called split-step FFT method over several small propagation increments per layers, transforming back and forth from spatial to spatial frequency (wave vector) domains). At each data plane, the spatial domain field is multiplied by the complex exponential phase mask function representing the data pattern.

The phase mask for each layer is generated by using random numbers to determine the presence or absence of a circular data pit in each grid cell. Also, the coordinate center of each intermediate layer’s frame of reference with respect to the beam center is offset by a random amount, between 1=2 of a cell width in each direction. Both of these randomization steps are used to simulate each layer of a multilayer CD-ROM. Once generated, each random data mask is retained for use in the reverse propagation computation. Each layer has a 19 19 array of data cells in a 256 256 element computation grid. The value of used in the analytical and numerical calculations is 0.503. The calculations are performed with as many as eight data layers through which the beam propagates. At the layer where the beam comes to a focus, we examine the two cases of reflection from a ‘‘1’’ (no phase alteration) or a ‘‘0’’ ( reflected phase shift region centered with respect to the focal point). For both the analytical and numerical cases the coupling coefficients are normalized with respect to the reflectivity from the desired data layer. Also, to isolate the effects of propagation distortion, the Fresnel reflectivity and power loss at each intermediate layer are neglected. For large numbers of layers, there would be such a practical requirement for small reflectivity to avoid excessive beam attenuation. We will discuss this point below in more detail.

A single measurement sample is found by calculating the forward and backward beam propagation through the out-of-focus data layers, with a midpoint reflection from the centered, in-focus data layer (from a ‘‘0’’ or ‘‘1’’), and then numerically calculating the beam overlap with the receiver mode. The histograms and statistics for the overall coupling coefficients are found from Monte Carlo repetition of this process with different randomly generated data layers. Calculations and data display were performed using Fortran code on a Macintosh 68040 processor computer. Two-dimensional patterns of the field intensity at various depths were saved for example members of the ensemble, to allow visualization of different degrees of beam distortion. Illustrations of such a comparison for different reflectivities are shown in Fig. 4 (see color plate), for propagation back and forth through eight data layers, with a layer thickness equal to the Gaussian beam Rayleigh range. At the top of Fig. 4 are examples of ‘‘1’’ and ‘‘0’’ beam intensity profiles for a Fresnel reflectivity of R ¼ 0:001. Intensities are shown on a logarithmic false-color scale, with each gradation of color corresponding to a 2 dB increment. The intensities at the edges of the frames approach zero because of the absorbing boundary conditions. In each frame pair, the left-hand part shows the distorted beam and the righthand part the undistorted (receiver) beam. At the bottom of the figure is a similar comparison (using the same randomly generated data layers) for the case of

Figure 4 Images of beam intensities after propagation back and forth through eight data interfaces with separation equal to a Rayleigh length. Interface reflectivity R and data at focus are indicated in each pair of profiles (Bit ¼ 1 means no pit; Bit ¼ 0 means pit in focal plane). The left-hand image of each pair is the undistorted intensity with no data in the interfering layers. The images are shown in logarithmic false color scales, with each gradation equal to 2 dB. (See color plate.) [From Ref. 11, Fig. 4.]

R ¼ 0:04. As shown previously [Eq. (3)], the transmitted phase shift through a pit

region is proportional to the square root of R, and the difference in propagation p

quality for R ¼ 0:033 and 0:2ðR ¼ 0:001 and 0.04) is dramatic. Thus, it is crucial for good beam quality and successful readout from multilayer media that the reflectivity per layer interface be very low. However, low reflectivity can stress the receiver signal-to-noise ratio as discussed later.

14.3.3 Comparison of Analytical and Numerical Formulations

Graphs of the amplitude coupling coefficient for the case of eight data layers are shown in Fig. 5. The graphs illustrated correspond to the beam profile examples of Fig. 4, except that many propagation cases were evaluated to generate the graphs. The vertical bars mark the positions of the mean values of the amplitude coupling coefficient. Note the gaps and scale changes in different positions of the horizontal axes. The data have been normalized to convert them to probability densities, such that the integral of each of the four curves (zeros and ones for two reflectivity values) is unity (the integrals may not appear equal because of the axes’ scale changes). The numerical values of the means and standard deviations shown in the figure were found from the ensemble of coupling coefficient results. The analytic comparisons for the ‘‘1’’ data were derived using Eqs. (12), (14), (19), and (22). Given the approx-

Figure 5 Graphs of amplitude-coupling coefficients after propagating back and forth through eight data layers, each separated by a Rayleigh length. (a) R ¼ 0:001; (b) R ¼ 0:04. Note the breaks and scale changes in the bottom axes and the difference in scales between (a) and (b). [From Ref. 10, Fig. 2.]

imations involved in these formulas, the agreement is surprisingly good. This agreement gives confidence in the validity of using the analytical results in cases where the numerical modeling would be impractical, as for larger numbers of data layers.

An interesting result of the evaluation of j jT for the ‘‘1’’ case is its analytical

showing a signal loss due to scattering that resembles a reflectivity loss.

We discuss the difference between OCT and DD signals in more detail below. For purposes of comparing the propagation results with these two methods, recall that DD will sense the received intensity, which will be proportional to the square of the amplitude coupling coefficient. The OCT signal results from the heterodyne mixing of a constant reference field and varying signal field, the latter of which is linearly proportional to the amplitude coupling coefficient. The beam distortion itself is independent of the means of detection, so the only difference in the OCT and DD analyses is in the choice of evaluating either the received field or intensity of the distorted beam. Note that in finding the average intensity of the received beam using the square of Eq. (25), the average power loss effects of data scattering resemble Fresnel losses but may be even larger. For example, if ¼ 0:5, the coefficient of R in Eq. (25) is 1.85. If power loss due to reflectivity alone were considered, the coefficient of R would be 1, corresponding to the usual power attenuation factor 1 R per interface. When both effects are considered, the total effective reflectivity for loss considerations is equivalent to 2:85R.

The effects of scattering of the light in a multilayer optical storage medium can be severe. It is important that the reflectivity per layer (which is proportional to the scattering coefficient) be kept small so that the average received light level is nearly constant as a function of depth and so the variance around these levels does not cause bit errors.

14.4CROSS-TALK ANALYSIS

In this section we discuss the effects of signal interference by reflections from layers other than the desired one. We assume that the Fresnel reflectivity from each data layer interface is small, so multiple back-and-forth reflections can be neglected. This approximation is also justified in light of the larger geometrical defocusing factors for multiply reflected beams. We also assume that the signal received from a spurious reflection can be calculated when using beams that are undistorted by propagation through the data layers. This approximation has been validated by the above propagation calculations if the Fresnel reflectivity and data propagation phase shift are small enough, as they need be for good system performance. Because distortion of the spuriously reflected beam would reduce the received cross-talk signal, we are doing a worst-case analysis. The main factors that are included for both OCT and DD are the geometrical defocusing reduction of signals reflected from out-of-focus layers, the effects of random data on this reflection, and the effects of source coherence on the total cross-talk signal.

These points are illustrated schematically in Fig. 6, which shows ray paths of incident (solid) and reflected (dashed) beams from the signal layer (at the beam focus) and one interfering data layer. The phase front curvatures are also indicated by corresponding thinner lines perpendicular to the beams. The arrow diagrams at the right edge of the figure are phasor representations of the relative time dependence of the reflected signal and cross-talk fields. The reflected cross-talk field is shown as an arc of blurred phase (from the temporal decorrelation) whose mean is tipped (from differences in propagation distances) and reduced (from defocusing and data effects) with respect to the reflected signal phasor (vertical arrow). The mean tipping angle has a randomness associated with unknown layer thickness variations relative to an optical wavelength.

Figure 6 Cross-sectional illustration of beams used in calculation of reflective feedback effects. The heavy solid line is an iso-intensity ray incident and reflected from the signal layer, and the thin solid line is a perpendicular phase front. The heavy dashed line is a ray incident and reflected from one cross-talk layer, and the thin dashed line is its perpendicular phase front. The right-hand arrow diagrams represent temporal phasors for the reflected signal and cross-talk fields, described more fully in the text. [From Ref. 11, Fig. 6.]

Let the amplitude of the field that is incident on the desired data be

where EðtÞ is the slowly varying envelope of a carrier at radian frequency !0. The temporal behavior of EðtÞ will depend on its coherence properties. The field quantities are defined as amplitudes of the receiver spatial mode, whose spatial dependence is not shown. The reflected field detected at the receiver is

In Eq. (27) a constant shift of time arguments with respect to the incident field is unimportant and is omitted. The signal variable di has values 1 or 0, depending on whether a data pit is absent (giving unity normalized signal) or present (giving zero coupling into the receiver mode). The cross-talk summation is over reflections from all layers except the desired data layer, including those above and below it. The quantities l are reflective amplitude coupling coefficients similar to those defined in the propagation analysis. They will be described in more detail below. The temporal arguments of the cross-talk fields are shifted because of their different propagation distances with respect to the signal layer.

The reflective coupling coefficient from layer l is

ð

l ¼ Ef ðrÞEb ðrÞZðrÞ dA ð28Þ