Download Multiplexing in high-density optical data storage using the orbital

Multiplexing in high-density optical data storage using the orbital
angular momentum of light beams
Robbert Jan Voogd, Joseph J.M. Braat*
Optics Research Group, Department of Imaging Science and Technology,
Faculty of Applied Sciences, Delft University of Technology
Lorentzweg 1, 2628 CJ Delft, The Netherlands
(Paper presented at the meeting of the Danish Optical Society, November 2005)
ABSTRACT
Multiple information can be stored in one storage location by imparting fine structure to the optical effects (or ‘pits’) on
a disc. We have chosen a fine structure such that the optical beam reflected from the disc shows orbital angular
momentum. Because of the electromagnetic conservation law that applies to angular momentum it should be possible to
detect the amount of angular momentum both in the optical far-field or in the image plane of the optical disc (nearfield). The orbital angular momentum is created by a sub-structure in each optical ‘pit’ that approaches a continuous
helical phase structure by means of discrete phase levels in a quadrant arrangement. The reflected light beam shows an
optical vortex and the intensity distribution in the focal region close to the image plane gives rise to a rotation that
depends on the helical sense of the orbital angular momentum of the beam. By reducing the phase excursion from its
basic 2π-value to a lower one, the vortex can be made to move from the center of the beam to an off-axis position, thus
showing a clear departure from circular symmetry in the intensity pattern; the angular position of the off-axis intensity
minimum is directly related to the orientation of the substructure on the optical disc. In this way, by angle-sensitive
detection, we can discriminate between a certain number of angular orientations of the substructure in an optical ‘pit’
and use a combination of screw sense and angular position for multiplexing purposes. By exploiting the two possible
screw senses and e.g. eight angular positions, we obtain in total 16=24 possible choices, leading to 4 stored bits per pit.
A possible further increase of capacity will be strongly determined by the signal to noise ratio of the optical signal
derived from the disc.
Keywords: Orbital angular momentum, optical vortices, multiplexing, optical data storage
1. INTRODUCTION
The standard approach for increasing the capacity of an optical disc is to reduce the value of the quantity λ/NA, the
diffraction unit, where λ is the wavelength, and NA the numerical aperture of the scanning objective. The optical signal
that is derived from the disc is sliced with respect to a ‘decision level’ that determines if the instantaneous value of the
detector signal is a logical ‘one’ or ‘zero’. The possibility of an increase in capacity has been demonstrated by allowing
more than one decision level, e.g. in two- or three-level recording schemes. However, the increase in capacity has been
rather marginal, several tens of percentages instead of factors of two or more. An increase in spatial density has also
been realized by a reduction of fabrication and system tolerances; in this way, the capacity increase going from the CDto the DVD-system has been a factor of 7 instead of the factor of 3 that would have been expected from the scaling in
the quantity λ/NA. The proposed Blu ray system is the most recent high-capacity system that has exploited to a practical
maximum the λ/NA range by switching to 400 nm wavelength and the highest possible numerical aperture (0.85). A
still higher capacity could be obtained by switching to ‘immersion’ read-out and recording using a so-called Solid
Immersion Lens (SIL)1. In this option, the transmission of the focused beam from the SIL-lens to the recording layer
relies on tunneling through a minute air gap whose thickness should not exceed the value of typically 30 nm.
*
Author for correspondence. E-mail: [email protected]
Experiments have shown that this is practically possible by using an active control system for the gap distance;
however, the functioning of such a system in a mass-consumer playing unit with removable discs is still questionable.
A problem encountered with higher-capacity systems is the data retrieval time. The track spacing is reduced
according to the lower value of λ/NA and, at a fixed maximum rotational speed, this leads to a proportionally longer
data retrieval time. In this way, with the practical limit of 100 Hz for the rotational speed, we find a total disc read-out
time of typically 2 minutes for a CD-disc but at least 10 minutes for the Blu ray disc. With respect to recording and
erasure, the numbers are even more conservative. Using the multiplexing option, the data rate is automatically
increased by the multiplexing factor and, for a multiplexed disk at equal spatial density, the gain is fully obtained from
the in-track direction while the track spacing is preferably kept unaltered. Because the λ/NA-value of a multiplexed disc
system is generally unchanged, the specifications and tolerances of the system are not tightened as it is the case in
systems with a reduced value of the diffraction unit λ/NA.
In this paper we will briefly introduce the multiplexing method in optical data storage based on the introduction
of orbital angular momentum in the reflected beam2. We will describe the first experiments with specially designed
structures that have been fabricated with the aid of an Ion Beam Pattern Generator and the measured intensity
distribution in the focal region of the imaged multiplex structures.
2. BASIC THEORY
The angular momentum of a (focused) beam of light comprises an intrinsic part due to the photon spin (units ± h or
± h / 2π per photon) and a part that can be associated with the geometrical structure of a light beam and we briefly recall
the basic theory of this type of beams.2,3,4 Beams carrying net spin momentum have to be non plane-polarized and the
rotation sense of the generally elliptically polarized light determines the sign of the spin angular momentum.
Geometrically introduced orbital angular momentum is encountered when a light beam has passed through e.g. a
discontinuous phase plate whose transmission function is given, in cylindrical coordinates, by5
t (r , ϕ , z ) = exp(i lϕ ) exp (ikz )
(1)
implying a phase increase over one full turn of l 2π where l is known as the topological charge, and k = 2π/λ. We
assume a wave propagating in the z direction as exp[-i(ω t - k z)]. We have dropped the term exp(-iω t) since in the
laboratory frame, the dynamics occur as the beam propagates through space not time.6
If l is an integer, the orbital angular momentum has been increased by an amount of l h . Of course, l does not have to
be limited to integer values; in that case, the expectation value for the angular momentum simply takes on the analog
value of l h . For the total beam, the angular momentum per unit area and per second is given by the quantity l I / 2πν
with I being the intensity of the beam and ν the frequency of the light. The wavefront with an l = 1 topological charge is
thus a helical function and is usually known as an optical vortex. Optical vortices may be formed using phase objects
with a spiral phase depth. However, as Levenson et. al.7 have pointed out, it is not necessary to have a continuous phase
structure to induce a topological charge. In Ref. 6 the authors used a quadrant structure with a full φ = 2π phase depth to
form a vortex phase mask for optical lithography to print sub-wavelength features. A stepwise approximation to a spiral
with a quadrant staircase structure (QSS) is sufficient (see Fig. 1). Kim et. al.8 have also shown that non-spiral phase
plates are sufficient to induce a helical wavefront in a gaussian beam. We shall use this approach to numerically model
QSSs with dimension typically of ~0.6λ/NA, to simulate the typical size of a structure on a recording surface. The
change in depth d (for the largest step) from the reference phase plane obeys, in reflection, the expression:
3 φλ
d= .
4 4πn0
(2)
where φ = lϕ (0 ≤ ϕ < 2π) is the phase of the light and n0 is the refractive index of the incident medium, usually air, so
that n0 ≈ 1. The factor ¾ arises from the fact that the step with the largest depth is the third of the four quadrants. The
optical system is modeled using the Kirchhoff approximation and also the vectorial diffraction approach within
DIFFRACT software9, in which a given amplitude distribution is Fourier-decomposed into plane waves and
subsequently, the waves are superimposed after propagation in either the Fresnel or Fraunhofer regimes.10 We shall
show that the vortex that exists within the background field gives rise to a rotation in the focal region of the emergent
beam, as a direct result of the transfer of orbital angular momentum within the beam. It is this phenomenon that we
shall exploit to propose a new method for multiplexing information from a single diffraction spot.
Fig. 1: A schematic of the simulated optical system. The incident Gaussian beam is focused onto a quadrant phase structure shown.
The dimensions are of the order of a diffraction unit, and the transmissivity is unity. The region around the structure has a
transmissivity of zero. The output beam is focused using L2 and we investigate the images of the vortex 25-45 wavelengths around
the nominal focal point at z0 (one focal depth amounts to approximately 8 λ in the simulated system).
We have modeled isolated QSSs in an optical configuration equivalent to the one shown in Fig. 1. Although the
configuration shows a transmissive mask, it is optically equivalent to a reflective phase object that would be
encountered in reality. A plane polarized lowest order Gaussian light beam is focused down onto a QSS phase mask
placed at the focal plane of two identical lenses L1 (of a given NA, typically 0.5, and focal length 10000λ), and the
emergent beam is focused by the lens L2 (f = 20000λ, NA = 0.25) and propagated in the z-direction to the focal region
where the images are recorded typically ±mλ each side of the focal plane at z0 (see Fig. 1), where m is an integer
typically 25-45 (note that the focal depth in this detection region approximately equals 8λ) . The amplitude transmission
through the four quadrants is set at unity. In our typical simulation the laser spot size was larger that the QSS dimension
of ~0.6λ/NA, which gave rise to weak amplitude, specific to the wavefront helicity, at the focal region due to the large
light intensity from outside of the quadrants. For this reason, we added an additional mask around the quadrants with t
= 0. It has been shown numerically that the rotation of optical vortices can be detected.11 Also, Arlt12 has shown
experimentally, that the characteristic rotation around the beam axis can be used to determine the handedness of the
vortex. We propose to utilize this phenomenon of rotation within the Rayleigh range resulting in the variation of the
intensity distribution to propose a new method for multiplexing information on an optical disc.
3. MODELLING RESULTS AND EXPERIMENT
We analyzed the intensity distributions of QSSs for phase depths (φ) of 2π, 3π/2, π and π/2 radians at the focal point at
z0. Figure 2 shows that there are significant differences in the spatial intensity variation. We find that a detection system
e.g a quad-detector would yield significantly different voltage values for the various sum/difference combinations for
the four phase depth values shown here. Having established that there is an intensity profile variation as a function of φ,
we proceeded to investigate the predicted field rotation in the focal region. These numerical results are presented next.
Fig. 2: 3D (top) and 2D (bottom) intensity distributions at the focal plane for QSS phase depths of (a) 2π, (b) 3π/2, (c) π and (d) π/2
radians. As expected the 2π phase depth shows clearly a phase singularity giving rise to zero intensity at the center. A detection
system e.g. a quad detector at the focal point would yield differing values of intensity for the various sum/difference combinations.
The vertical scale in the 3D plots is linear and in arbitrary units. The intensity profiles are approximately 6λ in extent at the 1/e2
level.
We shall now present numerical simulation data on the intensity distribution as a function of z from -40λ to +40λ
through the focal plane, for a quadrant staircase structure with a phase depth of 3π/2 radians (Fig. 3).
Fig. 3: Rotation of the image in the focal region for a maximum phase depth of φ = 3π/2. The images are taken at (a) z0 - 45λ, (b) z0 30λ, (c) z0 - 15λ, (d) z0 (focus), (e) z0 + 15λ, (f) z0 + 30λ, (g) z0 + 45λ. The sense of the rotation is again right-handed.
Apart from the handedness of the optical beam, we can also detect the off-axis position of the vortex that has moved in
the vertical direction for the particular geometry of Fig. 3. This enables an azimuth-sensitive detection of the
substructures on the disc and is our basic source for multiplexing. In Fig. 4 we show the experimental structures that
have been machined
with the aid of an ion
beam pattern generator;
their typical size is 1
μm2 per sub-quadrant
and in Figures (5)-(7) we
show the experimental
results.
Fig. 4. The machined quadrant structures for testing the orbital angular momentum multiplexing of data.
φ=3π/2
Fig. 5. Measured (top row) and simulated intensity distributions for a structure with a phase depth of 3π/2
radians at axial positions from 1.2mm in front of the nominal focus to 1.2mm behind the nominal focus in
steps of 0.4mm. The total image size is 134x134 μm and the beam waist is 50 μm (λ=633 nm; one focal depth
is approximately 8 mm).
φ=π
Fig. 6. Same legend as Fig. 5, but now for a maximum phase excursion of π radians.
φ=π
Fig. 7. Same legend as Fig. 6, but now with a rotation of 90 degrees of the quadrant substructure and changed
handedness.
From the experimental data it is evident that the azimuthal dependence of the sub-structures can be clearly visualized in
detector plane. With an assembly of sub-detectors, the azimuth of the detected pattern can be electronically determined
and used for multiplexing of the data. The final limit in the number of useful angular positions is determined by the
signal to noise ration of the detected signal and the cross-talk from other effects, both neighbours in the track direction
and in the adjacent track portions.
4. SUMMARY
By numerical modeling and experiment we have shown that a staircase-like quadrant phase object can be used to impart
orbital angular momentum to a coherent gaussian light beam. The angular momentum quantity, due to its intrinsic
conservation law, is transported to the image region. The beam rotation along the optical axis and the angular position
of the beam vortex can be detected with the aid of a composite detector. We have fabricated quadrant staircase
structures using ion beam milling and experimentally shown that the theoretically expected results can basically be
exploited for data multiplexing in optical storage.
ACKNOWLEGEMENT
Part of this research was carried out in the framework of the European project IST-2000-26479 (SLAM).
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