*s.greg.jones@narelab.com; phone 1 222 555-1234; fax 1 222 555-876; narelab.com
Optimization of holographic data storage system based on Seidel
aberrations reduction
Ren-Chung Liu*
a, Shiuan-Huei Lin
b, Ken-Yuh Hsu
a aDepartment of Photonics, National Chiao-Tung University, 1001 University Road, Hsinchu, Taiwan
300;
bDepartment of Electro-physics, National Chiao-Tung University, 1001 University Road,
Hsinchu, Taiwan 300
ABSTRACT
In this research, we investigate the influence of Seidel aberrations on the point spread function and the probability density function of holographic data storage systems, and thus the storage capacity and bit error rate of storage system can be obtained. The aberrations tolerances of storage systems with different numerical aperture are obtained. Optimization on BER and SC of holographic data storage systems by reducing Seidel aberrations will be demonstrated numerically.
Keywords: hologram, Seidel aberration, data storage
1. INTRODUCTION
Volume holographic data storage is regarded as a potential technology for next generation optical storage in the era of cloud computing. It has attracted intense research effort from both academia and industries [1]. Storage capacity (SC) and bit-error-rate (BER) are two parameters that are often used to evaluate the performance of holographic data storage systems. Bit errors can be induced by many factors such as material scattering, device imperfections, system misalignment, intra-page cross-talk between pixels of the same page, inter-page cross-talk between different pages, and optical aberrations, etc. When bit error occurs, it will require more error correction codes and thus the storage capacity will be degraded.
In this talk, we consider the influence of optical aberrations on BER and SC of holographic data storage systems. We describe the influence of aberrations on the point spread function and the probability density function (PDF), and thus the SC and BER can be developed. Finally, computer simulation will be presented.
2. SEIDEL ABERRATION AND MODEL ANALYSIS OF ABERRATIVE HDSS
2.1 Seidel aberration
The most popular model to describe the aberration was proposed by W. Hamilton. The aberrations are composed by Seidel aberrations and higher order aberrations. In general, the image quality is most affected by Seidel aberrations. Hence, we investigate the influence of Seidel aberrations on volume holographic storage. The generalized imaging system is shown as Fig.1.
object plane image plane
ξ η u ν (ρ,θ) Figure 1.
where η is the height of the object, (ρ,θ) is the position of the pupil in polar-coordinate. The Seidel aberrations can be classified to five types, called spherical, coma, astigmatism, field curvature, and distortion aberrations. These aberrations can be represented in Seidel polynomials:
ij kjk iW
W , , cos ………..……….(1)
where iWjk is the wavefront aberration coefficient represented by wavelengths. It means the phase errors of wavefront
when the light passes the interface of lens. These five kinds of Seidel aberration are listed in Tab.1 in polar and Cartesian coordinate.
aberration type polar coodinate Cartesian coordinate
spherical 4 40 0W
4 2 2 40 0W x y coma
1
3 1
31 1W cos W
x y
y 2 2 31 1
astigmatism
2
2 2
22 2W cos 2 2 22 2W
y field curvature 2 2 20 2W
2 2 2 20 2W
x y distortion
3
1 1
11 3W cos W y 3 11 3
Tab.1 Seidel polynomials represented in polar and Cartesian coordinates.
2.2 Equivalent model of aberrative HDSS for point-spread-function
All the imaging system with complex optical elements can be simplified to be an equivalent model composed with input(object) plane, entrance pupil, optical system, exit pupil, and output(image) plane, consequently. The equivalent model is shown as Fig.2
Fig.2 Equivalent model of aberrative HDSS for point-spread-function
For a linear system, the relationship between the input amplitude O(ξ,η) and output amplitude I(u,v) in coherent light case can be written as eq.(2)：
O
h
u
v
d
d
v
u
I
,
,
c,
;
,
(2) where h(u,v;ξ,η) is the impulse response(point spread function,PSF) of the system. For an imaging system, the impulse response is the Fourier transform of the exit pupil P(x,y)：
P
x
y
i
x
y
dxdy
h
c
,
,
exp
2
(3)In an aberrative imaging system, the pupil function P’(x,y) is modified to be the original pupil function P(x,y) multiplied with the aberration term exp[j2πW(x,y)/λ] [2]：
x y P x y j W
x y P' , , exp 2 ,
(4)object plane entrance pupil exit pupil image plane optical system ξ η u ν x y
Input plane (SLM) HMS 1 2 3 4
0
6w,.A (.w,_), 6wA (.w,4). II M (,w, x Output plane (CCD) Input plane (SLM) pupil Ï Output plane (CCD) 01011... 1i
01011... tNILLINJrg-.
Gq.uW.sU n.
u Wnhire lAccin.
__L_ _L_ __LJ_J_where W(x,y) is the effective length error induced by aberrations. Generally there are several lenses in a HDSS. The total aberrations can be regarded as the summation of the Seidel aberrations coefficient of each interface. Fig.3 shows that multiple surfaces of HDSS can be simplified to the generalized model.
Fig.3 The generalized model of HDSS with multiple elements
The total aberration coefficient is the summation of the aberration coefficient of each surface, show as eq.(5).
M m m jk i jk i k j i jk iW H W W H W 1 where , cos , , (5)2.3 Probability density function, bit-error-rate, and storage capacity
The intensity of input data bit in a holographic storage system is binary in “0”(dark state) or “1”(bright state). These data can be distorted by system noises during hologram recording and reconstruction steps. As a result, the intensity distribution of the output data will be changed to a gray level function. This distribution function can be described by a probability density function P(I), where I represents the light intensity of the data bit. Fig.4 shows the schematic diagram of the volume holographic data storage system together with typical PDF distributions of input and output data.
It is obvious that when the distributions of output data bit”0” and bit”1” overlap, the error bits will occur. The BER is defined as [3]:
t t I I I I I P I P BER 0 1 1 255 0 0 (6) where π0 and π1 are the ratios of the numbers of bit”0” and bit”1”, and P0(I) and P1(I) are PDF for the data, respectively.In analogue to communication theory a storage capacity (SC) per bit is defined as [3]:
255 0 1 1 0 0 0 2 0 0 0 0 1 1 1 2 1 1 log log I P I P I I P I P I P I P I P I P SC (7)Input data Output data
I
P(I) P(I)
I
I
tIt
Fig.4 Volume holographic data storage system and the PDF distributions
of input and
output data.
P0(I)
10 pages 1000x1000 pixels /page
Set NA of HDSS
Set Seidel aberration coefficient ;W,, in wave
Calculate PSF and the output pages
Calculate PDF and BER
BER> 0.4;
Seidel aberration Tolerance is obtained
Increase NA of HDSS
Increase ;WJk
From Eq. (6) and Eq. (7) we see that if the probability density functions of the output data are obtained, then BER and SC of the system can be calculated. The probability density function can be obtained from the point spread function of the holographic system by taking into account of the influence of Seidel aberrations.
3. SIMULATION
3.1 Aberration tolerance of HDSS with different NA
Next we will simulate the aberration tolerance of HDSS with different numerical aperture(NA). The flow chart of simulation is shown in fig.5. In the beginning, ten data pages randomly distributed with black and white pixels are generated. The resolution of each page is 1000x1000pixels. Then set the initial NA of HDSS to be 0.1. Then set aberration coefficients with incremental quantity 0.1 wavelength. After NA, aberration coefficient are set, PSF of HDSS can be obtained from eq.(4). Output data pages can be calculated from eq.(2). Then PDF of output data can be calculated. According to eq.(6) and eq.(7), BER and SC can be get. If BER is zero, increase aberration coefficient by 0.1 and run the loop again until BER is no longer zero. The aberration tolerance of HDSS with NA 0.1 is obtained. Next increase the numerical aperture, and run the loop again.
Fig.5 Flow chart for finding aberration tolerances of HDSS with different NA
Tab.2 shows simulation results of aberration tolerances of HDSS with different NA. From this table, we can know that for low NA HDSS, generally BER is dominated by the spherical aberration. For high NA HDSS, BER is dominated by distortion, field curvature, and astigmatism, respectively.
Tab.2 Aberration tolerances of HDSS with different NA
NA 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Spherical 0 W 40 0.3 3.4 16 50 120 260 300 Coma 1 W 31 0.4 3.1 9.5 26 44 70 110 Astigmatism 2W22 0.2 0.7 1.5 2.6 4 5.6 7.7 Field Curv. 2W20 0.12 0.36 0.78 1.31 2.1 2.9 3.9 Distortion 3W11 0.3 0.6 0.9 1.2 1.5 1.8 2.1
3.2 Optimal CCD pixel size of HDSS
The information capacity of each page (bits/page) is dependent on the resolution of SLM and CCD. However, the minimum bit size is limited by the PSF of HDSS. Next we simulate the PSF of a HDSS with 0.07NA. The pixel pitch of
SI \I(8.1pa) 240mm I000z1000pisels LI Material 148.5mm
..
93.6mm CCD(6.7pm) 66.7mm Volume holographic storage systemCCD pixels /bit 3x3 4 PDF BER 0.14 7.5x10-5 0 Bits /page 110 kb 62.5 kb 40 kb 27.6 kb
SLM and CCD are 8.1 and 6.7 um, respectively. Two identical lens with focal length 150mm and BK7 material are used to achieve pixel match. The system arrangement is plotted in fig.6.
Fig.6 The system arrangement of HDSS1 with NA0.07
The wavelength is set to be 532nm. The software Zemax was used to obtain the aberration coefficients. Tab.3 shows the aberration coefficients of Seidel aberration of the HDSS1 and aberration tolerance with 0.07 NA.
Tab.3 the aberration coefficients of the HDSS1 and aberration tolerance Wmax with 0.07 NA Aberration coefficient iWjk Wmax
Spherical 0W40 8.9 0.3
Coma 1W31 -2.6 0.4
Astigmatism 2W22 1.2 0.2
Field Curvature 2W20 0.5 0.12
Distortion 3W11 -0.6 0.3
The PSF of HDSS1 can be obtained from eq.(1) and eq.(4). Then ten input data pages O(ξ,η) are generated randomly distributed with black and white pixels. The resolution of each page is 1000x1000 pixels. The output data pages I(u,v) captured on CCD can be obtained by using eq.(2). In order to optimize the bit size, 3x3, 4x4, 5x5, and 6x6 pixels are used to present 1 bit respectively. Then PDF was calculated from the output data pages by counting the bit numbers versus intensity level. Tab.4 shows the PDF, bits/pa ge, BER, and SC of HDSS1 with different bit sizes.
Tab.4 PDF, bits/pa ge, BER, and SC of HDSS1 with different bit sizes.
The results shows that one bit represented by 5x5 pixels is good enough to achieve low BER (<10-3, upper bond of BER for error bit correction) and high capacity. Fig.7 is the PSF of HDSS1 in the position (3.35, 3.35)mm of output plane on
PSF original corrected 4x4 pixels /bit (62.5kbits/page) 5x5 pixels /bit (40kbits/page) PDF BER 0.01 6x10-4 PDF BER 7.5x10-5 0 6x6 pixels /bit (27.6kbits/page) PDF BER 0 0 SLM(8.Ip.m) L1 Triplet
n
CCD(6.7pm) 157mm52mm Y V
143mm 1000x1000pixels 7.6 - 12mm Aberration (Corrected) Spherical oW40 Coma 1W31 0.34 -1.53 Astigmatism 2W32 1.96Field Curvature Mao 1.69
Distortion 3W11 -0.17
CCD. The full width of PSF is around 30 μm. It is a good criterion to choose the CCD pixel size similar to the full width of PSF of HDSS.
Fig.7 The PSF of HDSS1 in the position (3.35, 3.35)mm of output plane on CCD.
From Tab.3, we found that spherical aberration is the most serious one compared to the aberration tolerances. Next, an improved system HDSS2 is proposed by reducing the spherical aberration. Fig.8 shows the system arrangement of HDSS2 and corresponding aberration coefficients. The lens models of HDSS2 are Y45-171, Y45-165, Y32-996, and Y45-165, respectively from Edmund Company. The spherical aberration is reduced from 8.9 to 0.34 by changing the imaging system.
Fig.8 The system arrangement of HDSS2 and corresponding aberration coefficients
Tab.5 shows the PDF, bits/page, BER, and SC of HDSS1(original) and HDSS2(corrected) with different bit/pixel sizes. Tab.5 PDF, bits/page, BER, and SC of HDSS1 and HDSS2 with different bit/pixel sizes.
3.3 HDSS Performance improvement by using circular SLM
Because of the circular symmetry of Seidel aberration, the PSFs in output plane are also circularly symmetric. Fig.9 shows the distributions of PSFs of HDSS1 in output plane. The spacing of adjacent sub-photo is 0.84μm. It can be seem that PSFs are circularly symmetric. Hence, bit data in each page are arranged in circular shape to be displayed on circular SLM.
30 μm
6.7 mm 6.7 mm
O O
O
O
O
O
O
s
O O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
e
e
O
O
O
O
e
e
O
e
O
O
e
e
e
O
O
O
O 00
O
00
O 00
E DE
O 00
D DD.
O 00
O 00
O
00
Fig.9 The distributions of PSFs of HDSS1 in output plane
Tab.6 shows the BER and SC with rectangular and circular data page. The BER of circular data pages can achieve lower BER than rectangular data pages.
Tab.6 The BER and SC with rectangular and circular data page.
4. CONCLUSION
A method for evaluating the influence of Seidel aberrations on BER and SC of HDSS is proposed. The
aberration tolerances of HDSS with different NA are figured out by taking into account different kinds of
Seidel aberrations. For low NA HDSS without aberration correction, spherical aberration is the most serious
aberration beyond the tolerance. An improvement design by reducing spherical aberration coefficient from
8.9 to 0.3 wavelengths has been demonstrated for achieving lower BER and higher SC. Due to the circular
symmetry of PSF of HDSS, the performance of HDSS can be further improved by using circular SLM to
display the data pages.
BER SC
SLM rectangular circular
7.5x10-5 1.4x10-5
REFERENCES
[1]
Hans J. Coufal, Demetri Psaltis, Glenn T. Sincerbox, Holographic Data Storage, Springer, New York,
2000.
[2] Joseph W. Goodman, "Introduction to Fourier Optics", third edition, Englewood Colorado, ROBERTS ＆ COMPANY, p.145-147, 2005.
[3]
Neifeld, M. A. and W. C. Chou. "Information theoretic limits to the capacity of volume holographic
optical memory." Applied Optics 36(2): Optical Society of America, 514-517, 1997.
ACKNOWLEDGEMENT
Financial support by National Science Council, Taiwan under contracts NSC101-2221-E-009-112-MY3 and AUT Ministry of Education, Taiwan are gratefully acknowledged.
