Generalized directional coupling for high-precision manipulation of the optical phase for classical and quantum light

Summary form only given. The manipulation of the phase of classical and quantum light is a major key to quantum photonics [1]. Integrated optics has several advantages over bulk optical systems, most notably in robustness, handling and size. Writing three-dimensional waveguides circuits into transparent materials using an ultrafast laser is thereby a new emerging technology [2; 3] with possible applications in quantum communication [4], and quantum computing [5]. In our work, we experimentally demonstrate the high-precision manipulation of the optical phase of classical and quantum light, using a generalized directional coupler.The behavior of light in directional coupler consisting of two waveguides can be described by the Hamiltonian H=T+V1+V2, where T is the kinetic energy and Vi is the potential energy of waveguide i = 1 and i = 2. The wavefunction of the system is constructed as a superposition of the single waveguide modes Ψ=a|1+b|2, and the Schrödinger equation of the coupler reads as HΨ=ksmΨ, where ksm denotes the propagation constant of the supermode. The propagation constants of the supermodes can be calculated straightforwardly as ksm1,2=k0±κ(1∓ σ)/(1-σ2) , where k0 is the propagation constant of the individual waveguides, κ=2|V2|1=1|V1|2 represents the coupling constant, and σ=1|=2|1is the direct overlap of the individual waveguide modes. For σ → 0 the well known distribution of ksm1,2 is preserved. As ksm = (ksm1+ksm2)/2>k0, in a generalized directional coupler a phase delay Δφ is introduced that readswith lC = π/κ as the coupling length. The phase delay depends only on σ, which itself is a function of the spacing between the waveguides. As the spacing can be controlled in a very accurate manner, this approach allows the highly precise manipulation of the optical phase along a directional coupler. In order- to measure this effect, we set up a tuned Mach-Zehnder (MZ) interferometer with identical lengths of the interferometer arms. A third waveguide was applied to one of the arms, forming an additional directional coupler in which the light completely couples forth and back with no intensity losses in the relevant interferometer arm (Fig. 1(a)). However, as the change in the propagation constant in this arm results in a phase delay, at the output of the MZ the ratio of intensities (for classical light) and the count rates (for single photons) changes. To obtain an all-reflecting coupler (ARC) for different coupling constants, the respective length of a coupler has to be adjusted and the associated phase delays are shown in Fig. 1(b). Due to decreasing σ for larger distances between the waveguides, the accumulated phase difference shrinks, even if the length of the coupler increases. Normalizing the phase difference by length allows the direct calculation of the change of the propagation constant caused by the ARC (Fig. 1(c)).In conclusion, we demonstrated a highly precise method for the manipulation of the optical phase in integrated waveguide structures by employing generalized directional coupling. The propagation constants of the supermodes in the structures are controlled by the spacing between the waveguides in the ARC. Our approach holds for classical as well as quantum light, and has the potential to find its use in a multitude of integrated-optical devices.