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Nanophotonic waveguides are a promising platform to trap cold atoms using red-and blue-detuned evanescent-field optical dipole forces. The asymmetric structure of integrated waveguides leads to a large birefringence that is not encountered in cylindrically symmetric optical nanofibers. We have studied both theoretically and experimentally the modal properties and suitability of silicon nitride rib waveguides for cold-atom trapping. The dependence of the modal effective index on the rib width is explored experimentally by measuring beat lengths between propagating modes. These measurements are made using a novel spatial Fourier analysis technique based on conventional far-field imaging of elastic scattering from the waveguide. We find that the beat length between the lowest order TE00 and TM00 modes is approximately 5µm, in excellent agreement with numerical calculations. We propose to take advantage of this birefringence and mode structure to create novel, one-dimensional periodic trapping potentials for atoms within the evanescent field of the waveguide.
© 2016 Optical Society of America
1. Introduction
Atoms confined in optical fields with mode areas comparable to the atomic cross section can exhibit enhanced atom-light interactions [1, 2]. Recent experiments demonstrating this enhancement are a critical step towards realizing applications in quantum networks [3] and provide a rich space for exploring atom-photon interactions and collective atomic effects [4]. Optical platforms exhibiting small mode areas that can efficiently couple to atoms include hollow optical fibers [5–7], photonic crystal cavities [8, 9], and optical nanofibers (ONFs) [2, 10–18], which exhibit efficient coupling between the atom and propagating modes in a 1D system. ONFs are appealing in part because they have almost unity transmission [18], have analytic mode solutions, and atoms can be loaded into an ONF-based optical dipole trap from a standard magneto-optical trap. Challenges remain with ONFs, however, including fragility, reliable fabrication, optical power handling, and limitations on geometries. These limitations have led to recent proposals to use integrated rib waveguides [19–23] for improved scalability and robustness.
The fundamental confinement principle both in integrated rib waveguides and in ONFs is the generation of an optical dipole trapping potential from a superposition of red-detuned and blue-detuned evanescent fields. However, some key differences in the optical mode properties of these waveguides that are relevant for atom trapping and for atom-photon coupling have not been thoroughly explored to date. While ideal cylindrical waveguides have no preferred polarization axis, rectangular nanophotonic waveguides are strongly birefringent, with the lowest order quasi-transverse electric (TE) and quasi-transverse magentic (TM) modes usually having markedly distinct propagation constants. Because atom-photon interactions depend heavily on optical polarization, knowledge of waveguide birefringence and modal field component is of significant importance.
In this work, we investigate birefringence in integrated silicon nitride (Si3N4) rib waveguides suitable for trapping cold atoms and for achieving strong coupling between atomic ensembles and optical modes. We measure beat lengths between the TE and TM optical modes using direct imaging of scattered light. Because of the high waveguide birefringence, the modal beat lengths are only a few microns, which is substantially less than the extent of a typical trapped atom ensemble. Our measurements of beat length agree well with numerical calculations, and the calculations also show the dependence of the propagation characteristics on waveguide geometry. We show that, unlike cylindrically-symmetric ONFs, this birefringence can produce a spin-dependent potential with a ~5 µm periodicity when superpositions of the TE and TM modes are used. Confinement of atoms in such a 1-dimensional periodic potential with period greater than the optical wavelength is one possible avenue towards individually addressable quantum registers for quantum information applications [24].
2. Waveguide geometry and modes
We have created integrated waveguide optical dipole traps, by using a 175 nm thick low-pressure chemical vapor deposition (LPCVD) SiN layer deposited onto a 5 µm thick thermal silicon dioxide layer [25, 26]. The waveguides are patterned using fixed-beam-moving-stage electron-beam lithography and etched to a depth of 100 nm using a reactive-ion plasma etch. Compared to fully-etched ridge waveguides, rib waveguides have a lower propagation loss due to reduced modal overlap with and scattering from the sidewalls, and they enable a single-mode cutoff at a larger width. Waveguides with nominal widths varying from 0.75 µm to 2.0 µm were fabricated. Fig. 1(b) shows a false-color scanning-electron microscope image of a nominally 0.75 µm wide waveguide fabricated in this manner. Fig. 1(c) and Fig. 1(d) show the intensity of the lowest order TE and TM modes supported by the waveguide, respectively, calculated using a finite-element mode solver at a wavelength of 780 nm (Electromagnetic Waves, Comsol Multiphysics). The TE (TM) modes are predominantly polarized with an in-plane, x (out-of-plane, y) electric field.
Fig. 1 (a): The geometry of our evanescent SiN rib waveguides. (b): A false-color SEM of a waveguide facet. (c): The intensity of the quasi-TE00 mode, which is predominantly polarized with an in-plane (lateral) electric field.(d): The intensity of the quasi-TM00 mode, which is predominantly polarized with an out-of-plane (vertical) electric field.
The number of modes allowed in a waveguide depends on the wavelength, height, width, and refractive indices of the core and cladding. Each guided mode has an associated propagation constant , where k is the free-space wavevector 2π/λ. For our sample, the thickness and indices of the layers are fixed, but the rib width can be varied. The calculated dependence of this effective index on rib width at a wavelength of 780 nm is shown in Fig. 2. For rib widths less than approximately 0.7 µm, the waveguide supports only the fundamental TE mode, TE00, and the fundamental TM mode, TM00. As the width is increased, additional modes are supported, though not necessarily populated. A number of modal anti-crossings are shown, indicating strong coupling between TE and TM modes of opposite parity.
Fig. 2 The dependence of neff on the rib width for each allowed waveguide mode at a wavelength of 780nm. Outlined regions depict widths for which modes are strongly coupled.
A useful method to characterize the propagating modes in a waveguide is by measuring the beat length, zb, due to interference between two modes. It is defined by
In an ideal ONF, the lowest order (HE11) modes are degenerate and there is no birefringence. This degeneracy is strongly lifted in rib waveguides due to the asymmetric geometry, leading to high birefringence (as shown in Fig. 2) and beat lengths that are only a few microns.3. Waveguide dipole potential
In waveguides with sub-wavelength dimensions, there is a significant evanescent field that interacts with the atoms and can be used for confinement. The total potential is the sum of the optical dipole potential and the van der Waals potential, UOD + UvdW. The optical dipole potential is given by [27]
where the first term in the square brackets is the scalar light shift and the second term is the vector light shift (the tensor light shift is insignificant here).Nanoscale waveguides and ONFs can produce significant vector light shifts that vary transversally and longitudinally when the trapping light is beating between two eigenmodes. The ellipticity, ϵ, can be expressed as
where e is the normalized polarization vector of the optical field such that the vector light shift characterizes the ellipticity of the optical field projected onto the quantization axis of the trap. Also, I(x, y) is the evanescent intensity, ΓD1 = 2π × 5.75 MHz and ΓD2 = 2π × 6.07 MHz are the linewidths of their respective transitions, ΔDi is the detuning from that transition, gF = [F(F + 1) + S(S + 1) âĂŞI(I + 1)]/[F(F + 1)] is the Lande g-factor, and F is the total angular momentum operator. The dipole potential must overcome the van der Waals surface term, which for sub-micron distances is approximated by [19] where y is the distance from the top of the waveguide.The fundamental trapping mechanism of an integrated optical dipole trap is analogous to that of ONFs and will only briefly be described here [2]. A stable trapping potential is formed in the evanescent fields of two laser colors propagating through the waveguides, as shown in Fig. 3. A red-detuned laser attracts atoms to the waveguide, while a blue-detuned laser, which has shorter range, repels them from the waveguide surface and overcomes UvdW. The resulting potential has a trap minimum near 100 nm from the surface for optical powers on the order of 10 mW [28]. An appealing feature of rib waveguides compared to ONFs is that trapping can be accomplished with negligible vector light shifts when both the blue-detuned and the red-detuned fields propagate in the TE00 mode.
Fig. 3 Finite-element calculation of the optical dipole potential of a 1 µm wide SiN waveguide, using the TE00 modes at a blue-detuned wavelength of 760 nm (10.5 mW) and a red-detuned wavelength of 1000 nm (25 mW). (a): Two-dimensional transverse plane of the optical dipole potential. The blue island above the waveguide core shows a region of potential minimum suitable for trapping, approximately 125 nm above the waveguide surface. (b): The potential profile along an out-of-plane (y) path from the waveguide surface to 500 nm above, including the van der Waals potential, at the position indicated by the dashed line in (a).
Because the longitudinal electric field component, Ez, is significant in nanophotonic systems, the field ellipticity, and thus the vector light shift, depends on the specific modes that are excited. Ez can be comparable in strength with the transverse component, especially for TM modes, and is π/2 out of phase with the transverse polarization components, leading to nonzero transverse ellipticity when a single TE or TM mode is excited. Ez vanishes on the waveguide’s axis of symmetry for the TE00 mode, but is maximized on that axis for the TM00 mode. Thus, for either pure TE or TM modes, there is no longitudinal component of ellipticity, and for the TE00 mode the transverse ellipticity is zero along the central waveguide axis, as shown in Fig. 4.
Fig. 4 The arrows indicate the transverse ellipticity (ϵ) of the TE00 mode at 760 nm, and the surface plot is the product of the magnitude of the transverse ellipticity and the intensity. Since the longitudinal ellipticity in this case is exactly zero, the ellipticity amplitude at x=0 vanishes.
Although the trap fields can be formed using pure transverse modes of the waveguide, most atom-photon interactions depend crucially on control, or at least knowledge, of the local optical polarization, and thus the waveguide birefringence cannot be neglected. In a cylindrically-symmetric waveguide such as an ONF formed from standard optical fiber, the static medium birefringence can be neglected, with beat length scales greater than several centimeters. For the waveguides considered here, the measured beat lengths are substantially smaller than the extent of a typical trapped atom sample and therefore must be considered.
The high degree of birefringence affects the atom-light interactions for on-resonant, near-resonant, and far-off-resonant beams, creating a unique environment for atomic physics experiments. In a simple picture that neglects Ez at the trap location, a beam propagating along z with equal contributions of TE00 and TM00 at the trap height oscillates between σ+ and σ− polarization (with quantization axis along z) over half the beat period. Despite the complicated orientation of the ellipticity in subwavelength waveguides, which have a non-zero Ez, a field that is launched with approximately equal vertical (y) and in-plane (x) linear polarization will have locations at which the dominant ellipticity component is longitudinal. This oscillation of the longitudinal component of the ellipticity is shown in Fig. 5 for a wavelength of 760 nm with equal amounts of TE00 and TM00 modes present.
Fig. 5 (a): Finite-element calculation of the longitudinal (z) component of the ellipticity (iϵ × ϵ*) in the case of simultaneous excitation of the TE00 and TM00 modes at 760 nm. This component of the ellipticity oscillates at the beat length for these two modes. All spatial dimensions are in microns.
For resonant and near-resonant beams, the birefringence can be used to prepare spatially-modulated spin states by optical pumping. By launching the superposition of TE00 and TM00 modes near resonance in a longitudinal magnetic field, the alternating circular polarization will produce a periodic array of ensembles with opposite spin. Additionally, when probing the atoms with off-resonant light, the optical polarization can be further rotated by the trapped atoms [29], and the atoms effectively introduce mode coupling through their birefringence. In ONFs, phase delays as large as 1 mrad per atom have been measured [30]. This Faraday effect has been a useful nondestructive way to measure the number of trapped atoms with low uncertainty [31]. Faraday rotation and related dispersive, nondestructive, spin-dependent measurement techniques may be employed to implement spin squeezing of one or more atomic ensembles trapped adjacent to a waveguide [32].
For far-off-resonant propagating modes, the modal birefringence can lead to unique 1D potential energy surfaces not possible in ONFs. Because the vector light shift depends on the ellipticity of the trapping beam, the potential can be modulated on a length scale that is tunable with the wavelength. As seen in Eq. 2, the vector light shift can be on the same order as the scalar light shift for mF levels of opposite sign. Such a waveguide with a spatially oscillating longitudinal ellipticity would produce islands of potential wells, also spaced by the beat length. Specifically, using 7.5 mW of blue-detuned power at 760 nm and 25 mW of red-detuned power at 1000 nm with Rb87 optically pumped to the F = 1, mF = 1 state with quantization axis along the waveguide direction, Eq. 2 can be used with our electromagnetic numerical solver to illustrate these trap islands. The result of such a calculation is shown in Fig. 6. The blue islands represent traps of depth approximately 0.7 mK at a height of 100 nm above the waveguide surface (comparable to ONF traps [2]) and, after initial coupling artifacts attenuate, are indeed spaced at the TE00-TM00 beat length. A cross-section of the potential is shown in Fig. 6(b), with the central blue area indicating the location of the potential wells above the waveguide. Fig. 6(c) shows the potential energy for a linear path extending in the y direction from the waveguide surface, at the location indicated by the dashed line in Fig. 6(b). An array of atoms trapped in the long-period one-dimensional lattice formed by mode beats could be employed as qubits that are individually addressable by transversely propagating, diffraction-limited laser beams. Compared to the period of a lattice formed by a standing wave from counterpropagating light, which is fixed at λ/2neff, the longitudinal trap period of our mode-beat lattice can be controlled by modifying the waveguide dimensions. For example, a thicker SiN core or a narrower rib etch would both lead to decreased birefriengence and thus an increased beat length. Thus, not only does a mode-beat lattice lead to a longitudinal trap that is easier to address optically due to its larger size, but it also can be modified along the same waveguide by lithographic patterning.
Fig. 6 (a): Finite-element calculation of the optical dipole potential for F = 1, mF = 1 atoms 140 nm above the top of a 1 µm wide SiN waveguide with simultaneous excitation of copropagating TE00 and TM00 modes. The blue islands show regions of potential wells suitable for trapping, spaced at the TE00-TM00 beat length of the blue-detuned field. The red-detuned field is launched with an electric field polarized in-plane, and the blue-detuned field is launched with equal components of in-plane and out-of-plane polarization. All spatial dimensions are in microns. (b): The dipole potential on a cross-section in the xy (transverse) plane at the location indicated in (a). (c): The potential profile along an out-of-plane (y) path from the waveguide surface to 500 nm above, including the van der Waals potential, at the position indicated by the line in (a) and (b).
4. Experimental waveguide characterization
In optical fibers, a number of techniques can be used to measure beat lengths [33], but are typically unable to resolve the short beat lengths of nanophotonic waveguides. For integrated waveguides, we experimentally verify the modal propagation by measuring the beat lengths between mode pairs using far-field imaging of polarization-dependent scattered light in the waveguide. Rayleigh scattering (RS) has been used in optical fibers [34–36] and nanofibers [37] to measure polarization states and beat lengths between modes. Because the scattered light has dipolar contributions, its amplitude and polarization are dependent on the local polarization of light propagating in the waveguide. When two orthogonal modes copropagate, the waveguide birefringence causes the polarization of the propagating field, and hence the scattered intensity pattern, to oscillate with beat length zb. These oscillations are observed by far-field imaging by suitable choice of viewing direction. For our waveguides, we use the geometry shown in Fig. 7, in which images are acquired at a 45-degree angle out-of-plane to increase contrast in the scattered light [36] from TE-TM beats. Viewing from a direction normal to the waveguide, as is typically done out of convenience, should produce minimal contrast because the strengths of the TE and TM scattering do not change and their intensity distributions are similar. Fig. 7 shows a typical scattering image for a nominally 1 µm wide waveguide with light launched with a polarization of 45° to the in-plane (x) and out-of-plane (y) directions. Such a polarization would populate both TE and TM waveguide modes.
Fig. 7 45-degree waveguide polarization-dependent far-field imaging for mode-beating measurements. Shown on the right is a typical scattering image from light at 780 nm launched into a 1 µm wide waveguide (not to scale).
To resolve the mode beats, a set of Fourier transforms (FFT’s) are carried out along the two or three columns of pixels corresponding to the waveguide location. These beats would be visible as peaks in the Fourier image at spatial frequencies corresponding to the beat lengths allowed by that waveguide width. As in ONFs [37], however, the waveguides exhibit substantial excess surface scattering that does not have the same polarization dependence as the mode beating. This excess scattering can be reduced by subtracting two images taken with orthogonal superpositions of TE and TM [36]: +45° linear polarization (polarizer in Fig. 7 transmit axis oriented +45° to the waveguide axis) and −45° linear polarization (polarizer in Fig. 7 transmit axis oriented −45° to the waveguide axis). The π-phase shift between these mode-beats amplifies the TE-TM beating after image subtraction, but the excess scattering, which is proportional only to the propagating power, remains constant and is subtracted. The Fourier transform of this subtracted image for the 1-µm-wide waveguide is shown in blue in Fig. 8, which clearly shows two beat periods at 5.0±0.1 µm and 15.6±0.8 µm, corresponding to TE00-TM00 beating and TE10-TM00 beating, respectively. The error in these measurements derives from a combination of the spatial resolution and the field of view. In addition, by subtracting the image obtained with a polarization of 0° (polarizer transmit axis oriented perpendicular to the waveguide axis) from that obtained with a polarization of 90° (polarizer transmit axis oriented parallel to the waveguide axis), any beating between TE modes should remain, while unpolarized sidewall scattering, as well as TE-TM beating should be cancelled. The resulting Fourier transform of this differential image is shown in red in Fig. 8. The TE00-TE10 beat at 7.4 µm is clearly resolved, and the TE00-TM00 beat is entirely suppressed. Interestingly, the TE10-TM00 beat in the 90°-0° plot is still visible, likely due to the strong in-plane polarization component present in the TM00 mode.
Fig. 8 Measured beat lengths of a nominally 1 µm wide waveguide at 780 nm using our differential Fourier transform method. (a): The FFT of data obtained with the polarizer oriented −45° subtracted from data obtained with the polarizer oriented at +45°. (b): The FFT of data obtained with the polarizer oriented 0° subtracted from data obtained with the polarizer oriented at +90°.
By carrying out this analysis on a number of waveguide segments of varying widths, we can both measure beat periods and understand whether the beating is primarily from TE-TM modes, or TE-TE modes. TM-TM beats are not expected to be observed due to the increased noise at low spatial frequencies (large beat lengths) using this method. These data are plotted in Fig. 9 and compared to the expected beat lengths as calculated from the effective indices shown in Fig. 2. The agreement between the calculated beat lengths and the measured lengths is excellent, indicating that our model is accurately predicting not only the beat lengths, but the ellipticity, polarization, and evanescent extent of our fields. The curves for some of these intermodal beat lengths end abruptly, indicating the cutoff rib width for 780 nm light. Below a rib width of approximately 0.70 µm, only the TE00 and TM00 modes exist, though the TM00 has strong TE10 character. In Rayleigh scattering, the only way to observe beating behavior between TE modes is to spatially resolve the light propagating in the waveguide, because the integrated power along a given polarization direction is constant [34].
Fig. 9 Measured beat length vs. waveguide width for four different waveguides. The lines are calculated from the effective indices shown in Fig. 2
5. Conclusion
We have measured and calculated birefringence in rib waveguides with parameters suitable for atom trapping. Although the atom trapping mechanism in rib waveguides is similar to that of ONFs, the mode degeneracies of ONFs are strongly lifted in these waveguides, resulting in beat lengths of only a few microns. Such lengths are significantly smaller than a typical trapped atom sample, and require important consideration for resonant atom-photon interactions (e.g. optical pumping through the waveguide) and near-resonant Faraday rotation effects. This birefringence can lead to unique periodic optical potentials with not observed in ONFs. In ONFs, the orthogonal transverse polarizations of the two fundamental modes are used to reduce vector light shifts, but this is not required in the rib waveguides, where both the blue- and red-detuned trapping beams can be in the TE00 mode and provide stable trapping at a minimum in the vector light shift profile [20]. If the TE00 and TM00 modes are simultaneously excited for the blue-detuned mode, the birefringence and periodic vector light shifts can create longitudinal traps along the waveguides for full three-dimensional confinement, with periodicity greater than the optical wavelength in the longitudinal direction.
We have also demonstrated a technique to measure mode-beating in waveguides, and we showed that the beat lengths measured by this technique, approximately 5 µm between the TE00 and TM00 modes, agree very well with our models. The technique is based on Fourier spatial analysis of far-field scattering from the waveguides, and is applicable to any waveguide geometry. The scattering data used for this technique is obtained with the same imaging techniques already employed in typical waveguide setups. Though the waveguides studied here are simple structures that can be easily modeled with cross-section eigenmode solvers, this technique is particularily useful for more complicated structures whose properties vary longitudinally and whose modal content is much less straightforward.
Funding
Office of Naval Research and the Army Research Laboratory.
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