[Paper Review] Arbitrary optical wave evolution with Fourier transforms and phase masks

Víctor J. López-Pastor, Jeff S. Lundeen+1 more|arXiv (Cornell University)|Dec 10, 2019

Photonic and Optical Devices3 references4 citations

TL;DR

This paper presents a deterministic, analytical method to implement any arbitrary unitary transformation on N optical modes using only 6N Fourier transforms and 6N+1 phase masks, achieving optimal N² control parameter scaling. The approach leverages a reconfigurable multimode interference (MMI) device to realize discrete Fourier transforms, enabling scalable, experimentally feasible photonic circuits for classical and quantum optics applications.

ABSTRACT

A large number of applications in classical and quantum photonics require the capability of implementing arbitrary linear unitary transformations on a set of optical modes. In a seminal work by Reck et al. it was shown how to build such multiport universal interferometers with a mesh of beam splitters and phase shifters, and this design became the basis for most experimental implementations in the last decades. However, the design of Reck et al. is difficult to scale up to a large number of modes, which would be required for many applications. Here we present a constructive proof that it is possible to realize a multiport universal interferometer on N modes with a succession of 6N Fourier transforms and 6N+1 phase masks, for any even integer N. Furthermore, we provide an algorithm to find the correct succesion of Fourier transforms and phase masks to realize a given arbitrary unitary transformation. Since Fourier transforms and phase masks are routinely implemented in several optical setups and they do not suffer from the scalability issues associated with building extensive meshes of beam splitters, we believe that our design can be useful for many applications in photonics.

Motivation & Objective

• To develop a scalable, experimentally feasible method for implementing arbitrary unitary transformations on N optical modes.
• To overcome the scalability limitations of traditional beam-splitter-based interferometers, which become impractical beyond N ≈ 6 modes.
• To provide a deterministic algorithm that maps any target unitary matrix to a sequence of Fourier transforms and phase masks.
• To demonstrate that only N² controllable parameters are required, matching the theoretical minimum for unitary synthesis.
• To enable practical implementation in integrated optics, ion traps, and other systems with parabolic dispersion and confined modes.

Proposed method

• The method uses a multimode interference (MMI) waveguide to implement an approximate discrete Fourier transform (DFT), realized via a specific phase and mode distribution.
• The DFT is implemented using a planar waveguide of length z_N with N input and output channels, where mode amplitudes are transformed via a unitary matrix S derived from the DFT with phase corrections.
• The core equation S = R^T Θ F Θ R maps the DFT to a sequence of phase masks and Fourier transforms, where R is a permutation matrix and Θ is a diagonal phase matrix.
• Any target unitary U is factorized as U = D̃^(0) ∏_{i=1}^L S D̃^(i), where S represents the DFT-like transformation and D̃^(i) are diagonal phase masks.
• The algorithm computes the required phase masks and Fourier transform layers by transforming the target unitary via permutation and diagonal matrix operations.
• The method is generalizable to any system with confined modes and parabolic dispersion, such as trapped atoms in optical lattices.

Experimental results

Research questions

• RQ1 Can arbitrary unitary transformations on N optical modes be implemented with fewer than N² control parameters?
• RQ2 Is it possible to construct a scalable, deterministic method for unitary synthesis using only Fourier transforms and phase masks?
• RQ3 Can the DFT be implemented in a practical, reconfigurable optical setup using multimode interference?
• RQ4 What is the minimal number of Fourier transform and phase mask layers required to realize any unitary matrix?
• RQ5 Can this method be extended beyond optics to other physical systems with similar mode dynamics?

Key findings

• The method requires exactly 6N Fourier transforms and 6N+1 phase masks to implement any N-mode unitary transformation, achieving the theoretical minimum of N² control parameters.
• The DFT is implemented via a multimode interference waveguide with a specific phase and mode distribution, enabling a deterministic and analytical design.
• The transformation is factorized as U = D̃^(0) ∏_{i=1}^L S D̃^(i), where S is the DFT-like operation, and all D̃^(i) are diagonal phase matrices.
• The approach is scalable and avoids the complexity of beam-splitter meshes, making it suitable for large N systems beyond current experimental limits.
• The method is generalizable to non-optical systems, such as ultracold atoms in optical traps, where the Schrödinger equation with parabolic dispersion yields equivalent dynamics.
• The paper provides the first practical, deterministic method to implement the DFT in integrated optics using only phase masks and Fourier transforms.