Volumetric Mode Sorter based on Phase Holography
Proposed project for UCLA Research in Industrial Projects for Students (RIPS) 2011
Industrial Sponsor:
HRL Laboratories, LLC
Thaddeus Ladd
Objective: Design computer generated holograms for sorting optical orbital angular momentum
modes.
Overview: Free-space optical communication may employ multiple degrees of freedom of light
to communicate information. Some, such as the phase or longitudinal mode, are time-decoded,
but others are measured by spatial sorting at an appropriate filter and subsequent detection by
spatially separated detectors. A simple example is to use multiple wavelengths, which are
routinely sorted using a prism, a ruled grating, or a holographic grating. Another example is
polarization, which may be spatially sorted using a polarizing beam splitter. A degree of
freedom of recent interest is the transverse spatial profile of a free-space beam, which may be
characterized into a discrete set of transverse modes. In particular, Laguerre-Gauss modes are
convenient due to their well-defined orbital angular momentum (OAM). Unlike wavelength or
polarization, methods to sort different OAM modes are substantially more complex; some
proposals suggest integrating thousands of interferometers, prisms, and waveplates. A more
realistic possibility is the use of a computer generated hologram (CGH); that is, a surface or
volume of arbitrary refractive index designed to affect light strictly by interference.
HRL Laboratories, LLC, is developing in-house methods for direct-writing of volume
CGHs, and seeks new algorithms for designing mode-sorting CGHs that sort more modes with
more efficiency than the current state-of-the art. New CGH designs in conjunction with
advances in direct-writing of CGHs may have high impact in reducing the bandwidth and power
consumption in free-space optical communication links.
State of the Art: Existing work to sort OAM modes using surface CGHs have been
performed [1], but the efficiency and scalability of these initial efforts are uncertain. Theoretical
considerations [2] indicate that small CGHs can accomplish the task if the CGH employs
sufficient variation over a reasonably sized volume, rather than a single surface. Recent work
generating multiple images from a single volume CGH [3] provide seed concepts for developing
mode-sorting CGHs via systematic searching. More recently, an OAM mode sorter was
designed using a conformal mapping achieved by a pair of spatial light modulators and a
lens [4]. This solution provides a strong existence proof for good OAM-mode-sorting elements,
although improvements in efficiency and in the number of distinguishable modes are desired.
Core Methods/Background: The design of a volumetric mode sorter is an inverse scattering
problem. That is, we know the input and desired output modes of our diffractive volume
scatterer; from these, we hope to deduce the variation of refractive index in the volume. In this
regard, the problem has a similar goal to computational tomography (CT) used in medical x-ray
and ultrasound. The challenge in the present problem is that a successful mode-sorter is likely to
use interference effects not accounted for in the low-order approximations used by x-ray CT, for
example, nor is a mode-sorter likely to have the high periodicity that enables higher order
techniques from crystallography. Solutions to high-order inverse scattering problems frequently
require computer searching, guided by intuition drawn from low-order methods. A summary of
both low-order and high-order scattering formalisms, especially the various orders of the Born
approximation, are introduced in classical optics texts, such as Born & Wolf, Ch. 13 [5]; a more
detailed and recent textbook in this area is by Chew [6]. A recent review more strongly focused
on the design of volumetric CGHs, with particular emphasis on the methods behind the result in
Ref. 3, is found in Ref. 7.
The methods employed to design a mode-sorter require only classical wave optics.
Quantum effects are not important, except that a desired goal for a mode-sorter should be
consistent with single photons. In practice, this means only that the waveform intensities
resulting from different OAM beams scattering from the device have small spatial overlap, so
that a single click from an array of single-photon counters determines the OAM state
unambiguously.
Tasks:
1. Develop search algorithm for design of mode-sorting CGH.
2. Devise robust metrics for characterizing the performance of a mode-sorting CGH.
3. Test mode-sorting CGH's from Task 1 using metrics from Task 2.
References:
[1] R. Celechovsky and Z. Bouchal, Optical implementation of the vortex information channel,
New J. Phys. 9, 328–328 (2007).
[2] D. A. B. Miller, Fundamental limit for optical components, J. Opt. Soc. Am. B, 24, A1–A18 (2007).
[3] T. D. Gerke and R. Piestun, Aperiodic volume optics, Nature Photon. 4, 188 (2010).
[4] G. Berkhout, M. Lavery, J. Courtial, M. Beijersbergen, and M. Padgett, Efficient Sorting of
Orbital Angular Momentum States of Light, Phys. Rev. Lett. 105, (2010).
[5] M. Born and E. Wolf, Principles of Optics, 7th ed., Cambridge University Press (1999).
[6] W. C. Chew, Waves and fields in inhomogeneous media, New York: Van Nostrand Reinhold
(1990).
[7] R. Piestun and J. Shamir, Synthesis of Three-Dimensional Light Fields and Applications,
Proc. IEEE 90, 222 (2002).
Desired Skills/Coursework/Machine Languages: It is helpful for the student to have taken
some form of introduction to diffraction or wave scattering theory, either via a traditional
E&M or optics course, a PDE/Fourier Analysis course with substantial work on wave
equations, or engineering courses on imaging methods.
Computer skills are a must. Both scientific computing/numerical analysis background
and traditional computer science (i.e. optimization/parallelization of searching/sorting
routines) are required, either via formal coursework or prior experience. The preferred
languages for the sponsor are C++ and/or Matlab.