Research ArticleAPPLIED PHYSICS

Optomechanical measurement of photon spin angular momentum and optical torque in integrated photonic devices

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Science Advances  09 Sep 2016:
Vol. 2, no. 9, e1600485
DOI: 10.1126/sciadv.1600485
  • Fig. 1 Polarization conversion and optical torque in a wave plate and a birefringent waveguide.

    (A) After passing through a half–wave plate, right-handed circularly (RHC) polarized light is converted to left-handed circularly (LHC) polarized light and transfers an angular momentum of 2ħ per photon to the wave plate and rotates it. Beth’s experiment measured photon angular momentum by detecting the rotation of the wave plate suspended as a torsional pendulum. (B) The polarization state of a hybrid mode in a rectangular waveguide evolves continuously from circular to elliptical to linear as it propagates along the waveguide. The horizontal (vertical) plane plots the Ex (Ey) component of the TE (TM) mode. (C) Linear torque density distribution τ(z) and total torque T(z), which is the integration of τ(z), vary sinusoidally along the waveguide. (D) Distribution of the z component of the torque density τz(x,y) on the waveguide cross section applied by the optical mode, including contributions from dipole and electrostrictive forces. The waveguide geometry is assumed to be 400 nm wide and 340 nm high, and the phase φ is fixed to be 0.

  • Fig. 2 Optomechanical scheme to measure photon angular momentum and optical torque in a waveguide.

    (A) Optical microscope image of the complete integrated photonic device including the suspended waveguide-nanobeam structure (in the red box), the TE and TM mode grating couplers (GC), and an interferometer with a TE-to-TM mode converter (inside the white box) for phase stabilization. Scale bar, 100 μm. (B) Optical transmission spectrum measured from the waveguide coupled to one of the nanocavities on the nanobeam. The resonance mode having a loaded (intrinsic) quality factor of 4.2 × 103 (2.3 × 104) is optimized to detect the rotation of the nanobeam. (C) Schematic of the measurement system. FPC, fiber polarization controller; EOM, electro-optic phase modulator; BS, beam splitter; PBS, polarization beam splitter; PD, photodetector; VOA, variable optical attenuator; ϕ(T), fiber thermo-optic phase shifter. Inset: Scanning electron microscope image of the waveguide and the nanobeam with photonic crystal cavities. Scale bar, 5 μm.

  • Fig. 3 Mechanical characteristics of the torsional optomechanical device.

    (A) Zoom-in scanning electron microscope image showing the suspended waveguide-nanobeam structure. (B) Simulated fundamental torsional mode of the waveguide-nanobeam structure. Also plotted on the yz plane are the normalized angular mode profile θn(z) (in purple) and its representative overlap integrand with torque distribution, Embedded Image (in red), the integration of which along the waveguide yields effective torque Te for a representative waveguide of 10.5 μm long. (C) Broadband thermomechanical noise PSD measured by the probe nanocavity resonance, showing four prominent mechanical modes. They are, with increasing frequency, fundamental out-of-plane torsional, fundamental out-of-plane flapping, fundamental in-plane torsional, and fundamental in-plane flapping modes. (D) Zoom-in view of the fundamental torsional mode resonance at 358.7 kHz, showing a quality factor of 12,000. The PSD unit has been calibrated and converted to rotational angle. (E) The fundamental torsional resonance frequency of devices with various waveguide lengths l. The measurement results show lower frequency than simulation using the typical bulk value of silicon’s elasticity matrix (red line), indicating that the elasticity matrix of the silicon layer in the SOI is effectively lower (blue line).

  • Fig. 4 Measurement of angular momentum and optical torque applied on the waveguide by the light of arbitrary polarization state.

    (A) The quadrature resonance responses (root mean square value) of the device with l = 10.5 μm at its fundamental torsional mode when the phase φ0 is varied and the optical power is kept at 95 μW. The results show that both the magnitude and the sign of the driving optical torque are changed because of φ0. (B) The measured amplitude of modulated torque δ Te over phase modulation δφ (blue symbols) depends sinusoidally (red line) on the phase φ0 as in Eq. 2. The maximum torque Tm(l) is determined from this result. (C) The optical torque Tm(l) measured (symbols) when the fraction of TE mode power PTE is varied while the total power P is kept constant, for two devices with l = 10.5 μm (blue) and 9 μm (red). The results show predicted semicircular dependence (lines). (D) The optical torque Tm(l) measured (symbols) when the total power P is varied, for the same two devices. The results show linear dependence below 100 μW (lines). (E) The optical torque Tm(l) for devices with various waveguide lengths l and, correspondingly, Δkl. Fitting the data with the theoretical model (line) yields the waveguide birefringence Δn and the coefficient of angular momentum transfer per photon η. The error bars reflect the measurements from multiple devices for each length.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/2/9/e1600485/DC1

    Theory of linear optical torque density in silicon waveguide

    Photonic crystal nanocavity design

    Waveguide-nanobeam junction design

    TE-to-TM mode converter design

    Simulation of mechanical modes

    Effective torsional simple harmonic oscillator model

    Calibration of optomechanical measurement transduction factors

    Effective torque

    Resonance response of the effective torsional simple harmonic oscillator

    Comparison of the excitation of the torsional and flapping modes

    fig. S1. Simulated torque density distributions inside a silicon waveguide (bulk contribution) suspended in air.

    fig. S2. Simulated surface torque density distribution along waveguide surfaces.

    fig. S3. Simulated coefficient η for waveguides with various widths but with a fixed height of 340 nm.

    fig. S4. Simulated mode profile (TE field) of the photonic crystal nanocavity.

    fig. S5. Finite-difference time-domain simulation results showing the transmission of TE and TM mode through the junction structure.

    fig. S6. Scanning electron micrograph of the mode convertor.

    fig. S7. Simulated mechanical mode profiles of the suspended silicon waveguide and nanobeam.

    fig. S8. Simulated normalized mode profiles of the suspended silicon waveguide and nanobeam.

    fig. S9. Comparison of the excitation of the out-of-plane torsional and flapping modes under identical experimental conditions.

    table S1. Simulated resonance frequencies of the mechanical modes.

    table S2. Simulated parameters of the effective torsional simple harmonic oscillator.

    References (4143)

  • Supplementary Materials

    This PDF file includes:

    • Theory of linear optical torque density in silicon waveguide
    • Photonic crystal nanocavity design
    • Waveguide-nanobeam junction design
    • TE-to-TM mode converter design
    • Simulation of mechanical modes
    • Effective torsional simple harmonic oscillator model
    • Calibration of optomechanical measurement transduction factors
    • Effective torque
    • Resonance response of the effective torsional simple harmonic oscillator
    • Comparison of the excitation of the torsional and flapping modes
    • fig. S1. Simulated torque density distributions inside a silicon waveguide (bulk contribution) suspended in air.
    • fig. S2. Simulated surface torque density distribution along waveguide surfaces.
    • fig. S3. Simulated coefficient η for waveguides with various widths but with a fixed height of 340 nm.
    • fig. S4. Simulated mode profile (TE field) of the photonic crystal nanocavity.
    • fig. S5. Finite-difference time-domain simulation results showing the transmission of TE and TM mode through the junction structure.
    • fig. S6. Scanning electron micrograph of the mode convertor.
    • fig. S7. Simulated mechanical mode profiles of the suspended silicon waveguide and nanobeam.
    • fig. S8. Simulated normalized mode profiles of the suspended silicon waveguide and nanobeam.
    • fig. S9. Comparison of the excitation of the out-of-plane torsional and flapping modes under identical experimental conditions.
    • table S1. Simulated resonance frequencies of the mechanical modes.
    • table S2. Simulated parameters of the effective torsional simple harmonic oscillator.
    • References (4143)

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