Hanbury Brown and Twiss measurements in curved space

Journal name:
Nature Photonics
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Published online

When Hanbury Brown and Twiss (HBT) proposed their technique of intensity correlation measurements1, 2, 3 to examine the angular size of stars in the visible range, they challenged the common conception of quantum mechanics and kicked off a discussion that led to the establishment of quantum optics4, 5, 6. In this Letter we revisit this fundamental technique and study its implications in the presence of space curvature. To this end we theoretically and experimentally investigate the evolution of speckle patterns propagating along two-dimensional surfaces of constant positive and negative Gaussian curvature, defying the notion that light always gains spatial coherence during free-space propagation. We also discuss the measurability of the traversed space's curvature utilizing HBT from an inhabitant's point of view. Through their symmetry, surfaces with constant Gaussian curvature act as analogue models for universes possessing non-vanishing cosmological constants.

At a glance


  1. Collection of surfaces with constant Gaussian curvature K.
    Figure 1: Collection of surfaces with constant Gaussian curvature K.

    The Gaussian curvature is defined as the inverse product of the two principal radii of curvature R1,2 and is independent of the embedding in 3D space. Here we make use of the mean radius of curvature . The sum of the interior angles of a triangle on a curved surface depends on both the Gaussian curvature and the enclosed area and in general is different from π. The circumference C of a circle depends on the radius r in a nontrivial way as shown. a, Different variations of a surface with zero Gaussian curvature realized by folding a flat sheet. b, Negatively curved saddle shape. Note that the radii R1,2 are assigned different signs when their respective centres are on opposite sides of the surface. c, Positively curved sphere with radius R1 = R2 = R.

  2. Experimental set-up.
    Figure 2: Experimental set-up.

    a, Hourglass-type figure. The phase modulated Gaussian envelope field distribution at the back side of a linear spatial light modulator is imaged onto the coupling facet of a waveguide spanning along the surface of the glass figure. A prism is utilized to optimize the front coupling. A CCD camera is moved about the figure's axis of symmetry collecting column-wise images along the azimuthal angle. b, Half-sphere. The whole field distribution is rotated by a dove prism and then imaged onto the entrance face of the half-sphere. Inside the half-sphere the light is guided by total internal reflection. c,d, Coordinate systems as used to obtain the raw data depicted in the first column of Fig. 3. Images are taken stepwise along the azimuthal angle φ, and then rectified to vertically appear in the proper distance t along the figures' profile curves (with t = 0 at the bodies' waists). Exemplary images of actual Gaussian beam propagation as seen in the experiment are depicted aside. Scale bars, 1 cm. c, The laser beam is coupled into the waveguide on the bottom right (the aiding prism is indicated by the white outline), winds around the figure's waist and hits the upper facet after slightly more than one turn. d, Coupling to the half-sphere with a microscope objective.

  3. Experimental measurements of the second-order DOC function.
    Figure 3: Experimental measurements of the second-order DOC function.

    a,d,g, Realization of speckle evolution along the hourglass (a) and along the sphere for zS < R (d) and zS > R (g). The beams are coupled into the surfaces on the left and propagate to the right (indicated by red arrows). For the evaluation only a central region is considered (framing white lines). Geodesics running perpendicular to the propagation direction z are represented by white lines. In the vicinity of the beam's centre these lines correspond to the transverse x direction as used in the central subfigure column. Additional light spots are caused by reflections or parasitic beams running through the bulk of the figure. b,e,h, Second-order DOC function g(2)x) depending on the proper shift Δx in the transverse direction for four different propagation distances (corresponding to the coloured squares in c, f and i). Solid lines are measurements, dashed lines represent Gaussian fits. c,f,i, Correlation length ρ as obtained from Gaussian fits to g(2)x, z). Error bars are confidence intervals of the fits. For small correlation lengths, deviations from theory are due to the limited experimental resolution caused by the finite scatterer and CCD pixel size. c, The solid blue line is a fit with equation (4), the solid green line is a comparison with zero Gaussian curvature for identical initial values. f,i, Solid red lines are fits with equation (5).

  4. Simulations of speckle evolution on curved surfaces.
    Figure 4: Simulations of speckle evolution on curved surfaces.

    The field propagates along the figures' equators (t = 0) with R = R0 = 10 mm and propagation constant k = 104 mm−1. The ratio ρ0/w0 is kept constant for all cases. Sample realizations are depicted on the left and the evolution of the transverse correlation length ρ(z) on the right, which in each case was derived from 1,000 phase realizations. a, Negatively curved surface (K < 0) with ρ0 = 0.5 · ρstat. The blue line in the right subfigure is a plot of equation (4). b,c,d, Positively curved surface (K > 0) with (b) short speckle diffraction length (zS < R), ρ0 = 0.1 · ρstat, (c) long speckle diffraction length (zS > R), ρ0 = 10 · ρstat and (d) the static case (zS = R), and ρ0 = ρstat = 14 μm. The red lines are plots of equation (5).


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  1. Institute of Optics, Information and Photonics, Friedrich-Alexander University Erlangen-Nürnberg (FAU), Haberstr. 9a, Erlangen 91058, Germany

    • Vincent H. Schultheiss,
    • Sascha Batz &
    • Ulf Peschel
  2. Institute of Condensed Matter Theory and Solid State Optics, Friedrich-Schiller-University Jena, Max-Wien-Platz 1, Jena 07743, Germany

    • Ulf Peschel


V.H.S. and S.B. derived the theory, V.H.S. and U.P. conceived and designed the experiments, V.H.S. performed the experiments and analysed the data, S.B. provided the MATLAB code for the simulations, V.H.S, S.B. and U.P. co-wrote the paper.

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