2 Introduction to Holographic Data Storage Systems
2.2 Channel Model
2.2.1 Complete Channel Model
A schematic diagram of a 4-fL (focal length) holographic data storage system is shown in Figure 2.2. Each of the two Fourier lenses in this 4-fL architecture performs a Fourier-transform operation, so that the light from Spatial Light Modulator (SLM) is imaged onto the Charged Coupled Detector (CCD). The storage of the Fourier holograms instead of image holograms serves a similar purpose as interleaving, in which the information is distributed in a different transformed domain so as to decrease the possibility of burst errors. And the placement of the aperture in the back of the first Fourier lens is to reduce the size of effective recording area on the holographic medium so that storage density could be increased. As the aperture windows the Fourier transform of the signal beam, it also acts as a low-pass filter whose bandwidth
[6]
Figure 2.2 Schematic diagram of a holographic data storage system in the 4-fL architecture
is determined by the aperture width, thus introducing the inter-pixel interference at the same time. One more thing to note about system is that, here we have assumed the system is pixel matched, which means that each of the SLM pixels is imaged onto one CCD pixel. Pixel-matched imaging is favorable in real implementation since it helps the realization of high data rates.
Based on the 4-fL architecture, a channel model could be developed, as is illustrated in Figure 2.3. The important elements in this model will be introduced respectively as follows.
Figure 2.3 Block diagram of a complete channel model
- dij: The input binary data sequence, dij, takes on values in the set {1, 1/ε}. While a pixel ONE takes the value 1, a ZERO pixel takes the value 1/ε, in which ε is the amplitude contrast ratio, i.e., the ratio of amplitudes of the bit 1 and the bit 0. This non-ideal effect is referred to as “the limited contrast ratio of the input SLM”.
- p(x,y): The SLM’s pixel shape function can be formulated as
Where α represents the SLM’s linear fill factor (namely the square root of the area fill factor for a square pixel). The fill factor is the ratio of the area of the active part of a pixel to the total area of a pixel. The symbol Δ represents the pixel width, which is assumed to be same for both SLM and CCD, and
∏ ( )
x is the unitrectangular function. Now, the output from the SLM could be expressed as:
( )
( , ) kl , (2.2)
k l
s x y
=∑∑ d p x k
− Δ − Δy l
Where
k and l refer to the pixel location along the x and y direction.
- hA(x,y): As mentioned earlier, the aperture leads to a low-pass behavior, whose frequency response is represented as HA(fx,fy) in Figure 2.3. The effect of the Fourier transform, HA(fx,fy) , and the inverse Fourier transform combined together is equal to convolving the SLM output with an impulse response, hA(x,y). The width of hA(x,y) is inversely proportional to the frequency plane aperture width.
For a square aperture of width D, the impulse response of the aperture is described
as
f represents the lens’s focal length. And in this way, the counterpart in frequency
Ldomain, HA(fx,fy), is an ideal low-pass filter with a cut-off frequency equal to
2 L
The impulse response hA(x,y), or commonly referred to as the point-spread function (PSF), may be the most critical ingredient in the channel model, since it compactly characterizes the extent of inter-pixel interference. Now, the signal through the PSF can be expressed as
( )
Where ⊗ represents a 2-D convolution, and h(x,y), which integrate the effects of the SLM pixel shape function and PSF, is referred to as the pixel-spread function (PxSF). From (2.1) and (2.5), it could be observed that the extent of IPI also depends on the SLM fill factor. A high SLM fill factor would broaden the PxSF, while a low one tends to increase the PxSF roll-off.
-
∫∫
| |i 2: The CCD is inherently a square-law device and tends to detect the intensity of incident light. It transforms the signal from the continuous domain to the discrete domain by integrating it spatially. Thus, the output from the CCDdetector combined with the noises can be described as below:
Where
β
represents the CCD linear fill factor. Now we know that the CCD fillfactor also has some connection with the extent of IPI. As a high CCD fill factor normally implies higher signal levels, it also contributes to channel nonlinearity and results in more IPI. And no
(i,j) and n
e(i,j) represents the term of optical noise
and electrical noise respectively. Optical noise results from optical scatter, laser speckle, etc., and is generally modeled as a circularly symmetric Gaussian random process with zero mean and variance No. Electrical noise arises from the electronics in the CCD array, and is normally modeled as a additive white Gaussian noise with zero mean and variance Ne.In Summary, the complete channel model is formulated as:
( )
There are still a few imperfections in a holographic data storage channel that are not considered, such as the inter-page interference, lens aberration, misalignment (including magnification, tilt, rotation, and translation differences) between SLM and CCD, and so on. However, these effects are either beyond the scope of our research or
could be conveniently integrated with the above architecture, so they are not specifically mentioned earlier. In addition, these effects are regarded as having minor influences when compared to the inter-pixel interference and noises.
For the simulations, pages of size 512×512 pixels are applied, with the parameters:
λ ,
f , and
LΔ
, set as 515 nm, 89 mm, and 18 μm, respectively. These parameters areselected as the same as in a paper on the IBM DEMON system [8].
To facilitate adjustment on operating conditions and make the simulation independent from λ ,
f ,
LΔ
, and D, the normalized aperture width, w, is defined asN
w D
=
D
, in which Nf
LD
=λ
Δ is the Nyquist aperture width. To make the description of the extent of IPI even more intuitive, the normalized half-blur width, W, of the point-spread function (PSF) is introduced and defined as
D
N 1W
=D
= . Thus, W = 1w
corresponds to a system with Nyquist aperture width, which also implies a system with critical sampling so that the first zero of PSF coincides with the center of the nearest pixel. While a larger W corresponds to severer IPI, a W larger than 1 suggests operation beyond the classical resolution limit.
For simplicity, the amplitude contrast ratio (ε), the SLM fill factor (α ), and CCD fill factor (
β
) are all set to 1 in the simulation. In fact, their influences can somehowbe depicted through the adjustment of W. For the setting of the noises, it is observed in [6] that the system under an electrical-noise-dominated condition and that under an
optical-noise-dominated condition yield extremely similar simulation results, so, again, for the reason of compactness, we choose to consider only the case when electrical noise dominates. We use signal-to-noise ratio (SNRe) to characterize the extent of electrical noise, and it is defined as SNRe=10 log10(0.52/N0).
Finally, we define two channel conditions that apply the complete channel model in our simulations hereafter: CH-1, with W = 1, represents the case of medium IPI, and CH-2, with W = 1.25, represents the case of severe IPI.