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Abstract
The forked grating coupler (FGC) is an optical vortex interface for silicon photonics. Using the structure of a Bragg grating coupler with a calculated forked hologram, the FGC couples optical vortex modes into confined waveguide modes of a photonic integrated circuit. Design methodologies are given, as well as measured performance data from fabricated devices. Data are analyzed with a variety of metrics. The effectiveness of design features are evaluated. Advanced FGC designs are demonstrated with focused forked gratings, allowing feed length to be reduced, and with apodization improving vortex mode fidelity. Some configurations achieve over 25 dB multiplexing crosstalk isolation.
© 2017 Optical Society of America
1. Introduction
Grating couplers are a well-studied [1–4] and favored technology for photonic integrated circuit (PIC) interfaces to or from optical fiber, yet there appears to be no basic grating coupler structure developed in hardware that can directly and efficiently couple an optical vortex (OV) beam to or from a photonic chip.
Optical vortex beams carry orbital angular momentum (OAM) [5] and have a variety of potential applications, including particle manipulation [6], micro-fabrication [7], and optical [8] and quantum [9] communications. Of particular interest here is the use of OAM modes in optical fiber multiplexing [10].
Many methods for generating and manipulating vortex beams and other high-order spatial modes have been developed. Vortex beams can be emitted directly by a laser [11], or a vortex can be introduced into a beam by means of bulk optics such as spiral phase plates [12], or spatial light modulators [13]. Most relevant to the grating coupler application is that it is possible to create OV beams by means of a diffractive element such as a forked grating [14,15].
Although OAM generation with bulk optics is useful for research, future commercial spatial-mode multiplexing applications will demand micro-scale, low-profile interface solutions built within photonic ICs. This miniaturization requirement for OAM beam interfaces suggests the use of metamaterials [16–19]. These techniques require different scattering geometry, reflective or transmissive. Specific to the application of a grating coupler, approaches that couple a vortex beam into or out of a waveguide are of key interest, particularly through the use of a diffracting grating on the surface of one or more waveguides [20]. In contrast, techniques to make a vortex beam propagating parallel to the slab are also valuable for chip-perimeter edge-coupling applications [21]. Many of these techniques are theoretical or simulation studies, but waveguide coupled, vortex-capable integrated optics devices have been fabricated [22–24], and more hardware results are on the horizon.
In this work we have developed, fabricated, and characterized the forked grating coupler (FGC) sketched in Fig. 1. The FGC integrates a forked hologram structure with a grating coupler attempting to retain the advantages of both structures, including convenience of placement on the die, low profile, simple fabrication, reasonable bandwidth, small size commensurate with optical fiber mode-field diameters, and CMOS fabrication process compatibility.
Fig. 1 The forked grating coupler (a) with 1) 12 um × 12 um forked grating, 2) adiabatic taper, 3) buried oxide SiO2, 4) 500 nm × 220 nm silicon waveguide feed, 5) silicon carrier wafer. Incident optical vortex (OV) shown with related coordinate systems. (b) close up of grating implemented with uniform width grooves.
An important design parameter for the FGC is the size of the holographic grating. Because the FGC is envisioned to be a vortex-fiber interface, we have studied 12 um × 12 um gratings exclusively. Their size is commensurate with the size of typical high-index ring “vortex” fiber (ring diameter ≈ 10 um) [25] supporting vortex modes with light of 1550 nm wavelength. A 12 um grating creates OAM beams with a waist diameter that can optimally couple directly to the mode field diameter of such fibers. Nevertheless, the size of a forked grating can be varied to optimize performance for other applications, as suggested in [20].
2. Theory
The basic phase match design methodology based on Bragg theory and two-dimensional effective index analysis that works well for the conventional, non-vortex, uniform grating couplers [2,3] must be extended to handle the variable grooves in a so-called “forked” pattern found in the FGC. The conventional grating coupler apodization design methodology is also extended to handle a short focal length feed taper.
2.1. Phase match
In order to calculate the forked grating pattern from a phase match condition, consider an incident OV beam propagating through cladding toward the origin of the grating. The central beam axis is entirely in the y, z plane. The beam hits a substrate that begins at z = 0 in the x, y plane. A beam oriented coordinate system x′, y′, z′ is aligned with the beam. The azimuthal angle about the OV beam, φ is measured relative to the x′ axis. The incidence angle, θ is measured between the z and z′ axis. The angle θ also represents a rotation transformation between beam and substrate coordinates. The rotation is about the x = x′ axis. The axes arrangement is shown in Fig. 1(a).
The phase of the inward traveling OV beam will be χφ + kz′, where χ is the topological charge of the beam and k is the cladding wavenumber. The incident OV beam induces a guided waveguide mode moving in the x, y plane. Because the waveguide tends to be many wavelengths wide under the grating, all the modes here will have an approximately identical effective propagation phase parameter, βeff, very close to that of the infinite slab mode, thereby greatly simplifying the analysis. In general, βeff is a slowly varying function of x and y, as the presence of the grating alters the propagation velocity in the guide depending on the pitch, duty cycle, and depth of the grating at that location.
The phase of the incident beam and the guided mode are assumed to match at z = 0. A conventional equation expressing this phase match condition [15] can be crafted assuming we want the captured waveguide mode to propagate purely in the y direction. This focus-at-infinity geometry requires a long adiabatic taper to funnel the slab mode from the 12 um width of the grating down to a suitably narrow (500 nm) single-mode waveguide. To avoid this excessively long feed taper, the phase match condition is altered such that the slab mode propagates cylindrically to a focal point (x, y) = (0, −R) some relatively short distance, R, from the center of the grating. The cylindrical coordinate, r, ϕ, origin is placed at this focal point. If we assume slowly varying structures per the Wentzel–Kramers–Brillouin (WKB) approximation [26], the phase of this focused mode will be , where βeff has a gradual r, ϕ dependence.
By equating the phase of the focused mode with the phase of the inward traveling OV beam, and eliminating the beam axis coordinates with a rotation transformation, we can formulate the focused phase match criteria for the FGC with this transcendental equation
where k is the cladding wavenumber, and with , ϕ = arctan(x/(y + R)) as polar coordinates from an origin at the feed focus. The integer m specifies a particular grating groove. Solving Eq. (1) for a range of m over a sector in the r, ϕ plane yields the grating groove design. For the non-vortex case, topological charge χ = 0, Eq. (1) yields focused grating lines that are uniformly spaced ellipses [3]. For the vortex χ ≠ 0 case, it yields an elliptically distorted forked grating pattern.The slowly varying phase parameter βeff(r, ϕ) was estimated through an iterative process. First, a wide range of grating profiles at the fixed etch depth are simulated with finite difference time domain (FDTD) methods. From these an estimate of βeff is interpolated, parameterized by groove pitch, p, and duty cycle d; we denote the resulting estimate Beff(p, d). Next, the initial pitch and duty-cycle fields, p(r, ϕ) and d(r, ϕ), are calculated for the grating. With these known, we have βeff(r, ϕ) ≈ Beff(p(r, ϕ), d(r, ϕ)) The duty-cycle field is fixed by the apodization synthesis. The pitch field, on the other hand, is unknown until after the phase match equation is solved to give the grating groove locations. The first time through, a constant pitch p0 = p(r, ϕ) is estimated. On subsequent iterations, the pitch field can be calculated from the previous solution of Eq. (1).
2.2. Apodization design
Equation (1) considers phase only, but to achieve good vortex mode fidelity, aperture amplitude control (apodization) is also critical. The natural transverse taper of the TE0 mode already provides a reasonable amplitude match [1] to the transverse component of a vortex mode. For this reason, herein we have only considered longitudinal (y direction) apodization.
The amplitude of the diffracted radiation emitted at any point in the grating face represents a kind of waveguide propagation loss. This diffracted exitance is expressed by a local attenuation parameter, α, that applies to the propagating wave in the waveguide at that location. α is determined by the local grating geometry, primarily the groove width.
For a desired transversely uniform (not dependent on x) amplitude apodization function, A(y), that is non zero for a grating region extending from y = 0 to y = L, it is possible to calculate α(y) for an infinity-focused grating using the well-known general method of analyzing “leaky wave” structures [27]. Propagation losses other than grating diffraction are assumed negligible and grating structures must vary slowly (WKB approximation). Under those assumptions, several authors [1,4] have designed conventional, infinity-focused, apodized grating couplers using the equation
where , and 0 < η < 1 is an efficiency factor chosen by design, representing the fraction of incident power to be scattered by the entire grating.In the case of a near focused grating coupler, the power density in the cylindrical wave through the short feed naturally declines with a 1/r dependence. For a desired uniform (not dependent on ϕ) apodization function, A(r), it can be shown that the expression for α is
where , and r0 and rL are the radial distances at the start and end of the grating respectively.Because of the negative S(rL)/η term in the numerator, the value of α(r) given by Eq. (3) is not necessarily positive. Therefore, unlike the infinity-focused case covered by Eq. (2), in the near focused case not all apodization functions are potentially realizable. The 1/r spreading loss is a negative bias that is difficult to overcome for certain profiles, a fact that seems under-appreciated in the focused grating coupler literature.
Within the realizability limits imposed by Eq. (3), the design process for α is incorporated into the iterative procedure described at the end of section 2.1. As with βeff, empirical estimates for αeff are extracted from the results of a parameterized simulation of grating groove geometries, yielding αeff(p, d). Based on these estimates, grating duty-cycle is modulated versus r, ϕ in the grating to achieve the α required to set the diffracted wave amplitude at that location.
Beyond the constraint presented by 1/r loss, given the minimum feature size limitations of fabrication, achievable α can possibly be further constrained by the range of realizable duty cycle.
3. Methods
A gallery of four different variant FGC designs were synthesized using the above theory. The designs are tuned for an incident beam angle θ=25° and 1550 nm wavelength light. This off-normal beam angle is chosen to avoid second-order back reflection for all pitches in the forked grating. Polarization of the beam is assumed linear, parallel to the grating, and TE within the feed waveguide. As shown in Table 1, the prototypes spanned a selection of two different vortex charges (χ) and two different apodization functions.
Table 1. Design variants of fabricated prototype FGC devices
Apodization exponential devices have no applied apodization. They use constant, 225 nm groove width, naturally creating an exponentially decaying grating illumination, with the early part of the grating radiating the most. In contrast, flat apodization attempts to implement an A(r) = 1 constant amplitude profile. The exponential and flat apodizations can use a very short 48 um feed focal length yet still be realizable with with Eq. (3). Other functions with low tails (e.g. Hamming) would require longer feed lengths.
The efficiency values (η) given in Table 1 are the design efficiencies used for calculating the α profile with Eq. (3). As such, they represent a theoretical upper bound for overall device efficiency. They do not include other loss factors: substrate wave loss, back reflection, mode mismatch, etc.
3.1. Device fabrication
The designs were implemented by Applied Nanotools Inc. starting with an SOI wafer with 220 nm thick silicon and 2 um buried oxide. Fabrication used electron-beam lithography and a two-etch process by an inductively coupled plasma reactive-ion etching system. Gratings are shallow etched down nominally 70 nm and waveguides outlines are etched fully down 220 nm to the buried oxide. Grating dimensions are approximately 12 um × 12 um for all devices. After etching, a 2 um protective top coat of SiO2 was applied.
The nominal 70 nm partial etch yielded an average depth of 76.5 nm at profilimeter test pads. Using a scanning electron microscope (SEM) at a cleaved edge, etch depth was measured at 77 nm for eight locations. Feature size error was typically +2 nm (slight dilation). Inter-layer overlay accuracy averaged 30 nm. Sidewall slope was ≈ 80 deg. Typical straight waveguide losses for this process were measured at ≈ 1.5 dB/cm for both TE and TM polarization. SEM images of typical devices are shown in Fig. 2.
Fig. 2 SEM photos of typical FGC devices. The χ = 1 device (a) with flat apodization has: 1) continuous grating groove etched down 70 nm, 2) Si waveguide taper, and 3) buried oxide exposed after waveguide outlining etch. Also, shown are a (b) non-apodized (exponential) χ = 1, (c) flat apodized χ = 2, and (d) non-apodized χ = 2 devices.
3.2. Measurement technique
The device characterization setup in Fig. 3(a) uses a Mach-Zehnder interferometer to produce interference images between the spherically diverging vortex-beam and a collimated reference beam. The source of light is a 1549.12 nm CW laser coupled to a polarization maintaining (PM) fiber, which is split into reference and test fibers using a 90/10% power divider. Using an XYZ stage, the test fiber is aligned with the input coupler on the chip. Figure 3(b) sketches the device under test (DUT) circuit on the chip. The input coupler is a conventional grating coupler tuned for an incident beam angle of −35°. This connects to a 500 nm × 220 nm silicon waveguide that, after being split in a y-branch to a power reference output, extends a short distance to the tapered feed of the FGC device under test. The FGC then radiates a vortex beam +25° from normal.
Fig. 3 (a) Experimental setup for characterizing the vortex mode fidelity of prototype FGC devices under test (DUT). (b) Simplified plan view of DUT.
The FGC output beam is imaged on the camera through a microscope objective (FL=1.2 cm). Focus is adjusted such that the lens produces a trans-side vortex beam waist 5 Rayleigh lengths (approx 9 cm) from the imaging camera. Given the power available from the laser, this distance produces a reasonable approximation to a far-field image amplitude and spherical phase (4% curvature difference), while still providing enough image power to remain within the dynamic range of the camera. Effective beam waist magnification in this geometry is 18.5X. The reference beam is emitted by a collimator and then folded through mirrors to combine with the vortex beam in a non-polarizing beam splitter. The curvature radius of the coaxially aligned collimated reference beam (≈ 20 meters) is effectively a plane wave relative to 9 cm curvature radius of the vortex beam.
The intensity profile is directly measured on the camera, but phase is inferred from interferograms. These interference images are forked-shaped patterns warped by the spherical phase component of the diverging beam, by misalignment phase tilt, and by the vortex phase aberrations we seek to measure. A single interferogram is insufficient to determine the phase aberration. There are three unknown variables: at any given pixel, neither the amplitude nor the phase is known for the vortex beam, nor is the amplitude known for the reference beam. If, however, N ≥ 3 interferograms are recorded, each with a unique and known reference phase increment, the system of N equations and three unknowns is, in principle, analytically solvable for each pixel [28]. Unfortunately, in practice it is not easy to introduce precisely known reference-beam phase shifts.
To avoid the difficulty of phase increment calibration, an inter-frame intensity correlation (IIC) algorithm [29] calculates a least-mean-square estimate of the wavefront phase. The IIC algorithm processes the sequence of interference images, such as shown in Fig. 4, each acquired with a different reference phase shift that randomly span 2π radians. These shifts are easily introduced by tiny mechanical perturbations. The robust IIC algorithm requires no knowledge of the exact phase shift introduced. Wavefront phase can be recovered with as few as five phase shifts, but ten are used for good sampling of the full 2π radians, as well as to lower the noise and interference fringe “print through” in the phase image.
Fig. 4 Input for the IIC algorithm [29] is this typical sequence of ten, spiral interferograms. Also shown are test and reference intensity images (on top and bottom far right) acquired from a typical χ = +1 flat apodized FGC.
The raw phase image from the IIC algorithm still contains the spherical phase component of the diverging vortex beam, as well as some phase tilt because of axis misalignment. These incidental phase components are removed before any metrics related to phase (e.g. charge spectrum) are calculated. An approximate spherical phase correction is estimated from the distance of the image relative to the beam waist. An approximate phase tilt correction is estimated from the centroid of the Fourier transform of the complex vortex image. Using these as a starting point, fine corrections are then computed by maximizing the overlap integral between the ideal vortex phase distribution and the approximately corrected phase image.
When the grating coupler is used as an interface to optical fiber, one important measure of performance is the through-loss associated with the interface, typically expressed as the parameter s21. In a multiplexing application, various crosstalk isolation terms are also of interest. When the full, complex optical field of the coupler is known from measurements, such as have been acquired herein, the mode-match loss or crosstalk terms for the application are readily predicted by overlap integrals. Although critically important for each particular application, the dependence on an application-specific mode field distribution muddies the choice of a “best” general metric for evaluating the overall quality of FGC prototypes in the abstract. Moreover, even if an application mode field is picked, it still remains unclear which application parameter to optimize: s21, or one of the crosstalk terms, or perhaps some other figure of merit such as a signal-to-crosstalk ratio.
In this work we have chosen to calculate mode match and crosstalk using an application-generic measurement circle that is located in the measured field based on features within that field. First, the location of singular points in the phase image recovered by IIC are accurately established by convolving a small, ideal vortex kernel with the measured phase, yielding a vortex density field with sharp, easily located maximum at each singularity. Next, a circle is drawn through the point of maximum intensity and about the centroid of the phase singularities. Finally, vortex fidelity metrics are calculated from the measured phase and amplitude about this circle. The ring of peak radial intensity reveals valuable information about modal content and has been used by others [25] for characterizing vortex fidelity; however, our method is novel in that it uses the self calibrating IIC algorithm for complete phasefront recovery, and the measured null centroid is used to define the center rather than an a priori geometrical center.
Final processed amplitude and corrected phase images from typical FGC devices are shown in Fig. 5. The axes are spherical azimuth and elevation divergence angles relative to the nominal beam axis at the device. The azimuth plane is parallel to the longitudinal (y) axis on the DUT.
Fig. 5 Processed far field amplitude and phase images for typical flat apodized (a) χ = 1, and (b) χ = 2 devices. The amplitude image is annotated with critical points identified by the metrics: 1) D86 beam perimeter, and 2) measurement circle through max intensity. Algorithm identified nulls, null centroid, intensity centroid, and intensity max point are also marked. Phase image is corrected for spherical and linear tilt phase components. The circular ripples in the phase image are interferogram “print-through” artifacts from the phase recovery processing. They typically appear in areas of low signal-to-noise ratio.
4. Results
Of the 48 FGC devices fabricated, 40 devices (83%) were functional. No significant difference in yield was observed for any of the variants.
Prior to imaging, the coupling match was optimized at the input coupler and the total power in the vortex beam from each output FGC device was measured. Given the overall power transfer measurement uncertainty we were able to achieve (±0.5 dB), there was no significant difference in radiated power between any of the four types of FGC devices. Moreover, there was no significant difference in radiated power between the FGC devices and conventional Bragg devices fabricated on the same chip with similar apodization and driven the same way. Total loss through back-to-back couplers was about 10 dB, implying a single coupler loss of 5 dB, or about 32% efficiency.
Observed values for several performance metrics are shown in Table 2. These metrics include amplitude ripple and phase error, which are computed about the measurement circle, as well as the full-angle D86 divergence angle, and the angular null split radius.
Table 2. Mean measured vortex fidelity metrics of implemented FGC devices. The number following the symbol ± is the sample standard deviation of the measurements.
Amplitude ripple and phase error specify the quality of the vortex beam. Amplitude ripple is computed as the standard deviation of the amplitude (σA) over a finely sampled set of points about the measurement circle, normalized by the root-mean-square averaged amplitude over that same circle. Phase error σP is computed as the standard deviation of the difference between the measured phase and ideal vortex phase at points on the measurement circle.
The so called D86 divergence is a common beam profiling metric practical to measure in non-Gaussian beams such as the vortex beams emitted by the FGC. It is defined here as the full conical angle containing (e2 − 1)/e2 ≈ 86.5% of the beam power. In the case of a Gaussian beam, the D86 divergence is identical to the usual full angle divergence defined by the beam “edge” where the electric field amplitude is decreased to 1/e relative to its value on the beam axis. Non-Gaussian beams diverge with a beam diameter a fixed ratio [30] larger than their embedded Gaussian component. Thus, measuring divergence of non-Gaussian beams remains useful for characterizing their shape through space.
Null split radius is an important measurement for OV beams with |χ| > 1. It can be defined generally as the angular radius of a circle centered at the centroid of the set of phase singularities, that has least mean squared distance to all those points. In the χ = 2 case, it equals simply half the angular distance between the two singularities.
The theoretical σA for the apodization functions is 0.26 for exponential and 0.18 for flat. σP is theoretically zero. The best phase and amplitude ripple measured was from the χ = 2 flat apodized devices. These had σA = 0.15 ≈ 1.3 dB and σP = 9°. Typical data is shown in Fig. 6.
Fig. 6 Measured far-field amplitude (left) and phase (right) plotted versus measurement circle angle for a typical flat apodized χ = 2 device.
In all χ = 2 devices a single, second-order vortex null did not appear; in its place, two, closely spaced first order vorticies were observed, as seen in Fig. 5(b). This so-called “null splitting” is inevitable [31], as it can be caused by even the slightest coherent background intensity contaminating the vortex light. Null split and null shift can be viewed as specific manifestations of charge spectral crosstalk to the χ = 0 charge. In Table 2 the measured radius for the null split circle (in this case, half the distance between the two nulls) is tabulated for the χ = 2 devices. For χ = 1 devices, coherent background results in a shift of the location of the single null, but this shift is difficult to measure for lack of a good reference point.
4.1. Charge spectrum
Charge spectrum, aχ is defined [32]
where u(ρ, φ) is the measured complex amplitude and phase distribution of the beam at cylindrical radius ρ and angle φ. Charge spectrum at each χ represents an overlap projection onto ideal spiral harmonics, exp (iχφ), that have normalized magnitude. As such, it serves as an estimate of achievable multiplexing performance. In this work we computed charge spectrum using only the complex data about the measurement circle. As such, the spectrum represents the mode overlap between the measured field and a χ-order “perfect” vortex [33] of that radius.The charge spectrum, aχ, at χ equal to the device design charge represents the mode match loss between the measured device and a perfect vortex at the measurement circle. Table 3 gives the mean measured values of the spectra at these points. The theoretical mode mismatch inherent in the apodization is 0.31 dB for exponential and 0.14 dB for flat. For aχ at χ not equal to the device design charge, the spectrum value represents crosstalk leakage to those other charges. Figure 7 plots the overall charge spectra at near-in charges calculated from phase and amplitude measurements of the four types of implemented FGC devices. Outside the near-in range shown, the spectral sidelobes continue to fall off monotonically.
Fig. 7 Mean charge spectrum calculated from phase and amplitude measurements of the four types of implemented FGC devices. The dark line is the mean measured value for all devices of that type. The shading denotes the sample standard deviation of the measurements.
Table 3. Mean measured mode mismatch of implemented FGC devices at their design charges. The number following the symbol ± is the sample standard deviation of the measurements.
Worst case adjacent charge rejection (ACR) for non-apodized FGC devices is very low. In contrast, the flat apodization achieves better than 20 dB ACR. Theoretically, because of their symmetric amplitude distribution, flat devices should have infinite ACR. The asymmetric exponential apodization has a theoretical ACR of 17.3 dB.
Also of interest is opposite charge (+χ versus −χ) rejection (OCR). Most measured devices have reasonable rejection of the opposite charge, with the the flat, χ = +1 device being the exception because of an anomalously high sidelobe near χ = −1. Theoretical OCR for χ = 1 devices is 18 dB for exponential and flat devices. Theoretical OCR for χ = 2 is over 35 dB for exponential and flat. Measured OCR and ACR results are summarized in Table 4.
Table 4. Mean adjacent charge rejection (ACR) and opposite charge rejection (OCR).
4.2. Discussion
The estimated 32% (5 dB) basic FGC radiation efficiency is typical [2] for simple SOI grating couplers with no efficiency optimizations. There are many loss mechanisms that affect grating couplers, conventional or FGC. Perhaps half of the overall 5 dB loss is attributable to the lost substrate wave. Substrate power could be recovered with blazing or a substrate mirror. Another good fraction of the loss is attributable to the apodization design efficiency, η, given in Table 1, which ranges from 0.7 dB to 1.7 dB.
More significant with respect to the FGC is the mode mismatch loss. Mode loss adds on top of radiation loss, and is related specifically to the vortex mode. Table 3 shows the non-apodized (i.e. exponential) devices have almost 1 dB more mode mismatch as compared to the flat devices. Mode mismatch stems from amplitude and phase ripple. As we see from Table 2, the exponential devices have higher amplitude and phase ripple. As expected, the flat devices have significantly smaller amplitude ripple than the exponential devices. Phase linearity for flat devices is excellent, suggesting that most of their residual mode mismatch is caused by the amplitude ripple exclusively.
The charge spectra show clearly that exponential devices have higher multiplexing crosstalk levels. This is not surprising. Without apodization, most of the radiation scatters somewhat off axis from early grooves, causing significant asymmetry in intensity toward the transverse (x) direction. Flat apodization is significantly better, producing a symmetric and more-constant amplitude illumination.
There is considerable evidence that device fabrication variation is the main source of measurement variance. The mean ratio between maximum and minimum eigenvalue reported by the IIC algorithm is 2 × 104, suggesting that measured amplitude and phase variances are not caused by noise or other data disturbances. Re-measurement of devices resulted in repeatability an order of magnitude below the inter-device σP and σA, An exception to this is data from non-apodized (exponential) devices in regions that fall below the noise floor.
The measured, full-angle divergence, Θ, is consistent with a 12×12 um device size. We can estimate the aperture diameter D at the device using the elementary formula D = 4bλ /(πΘ), where b is a vortex beam waist expansion factor (b = 1.324 for a Laguerre-Gaussian (LG) mode of χ = 1 [30]). For example, assuming the vortex is approximately LG, the Θ = 18.6° full angle divergence measured for the flat χ = 1 device implies an aperture of approximately 8 um. The 1/e width of the TE mode in 12 um waveguide is 7 um.
As already noted, null split and null shift can be viewed as specific manifestations of charge spectral crosstalk to the χ = 0 charge. It’s arguable that uncorrected null-shift in the exponential apodized devices, and the consequent misalignment of the measurement circle, contributes to the higher than expected σA and σP observed in these devices. Assuming a strong test and weak reference beam have identical waist parameters, a χ-order null will split into a series of unit charge vorticies spaced uniformly on a ring of radius r0 according to
where w is the waist radius of the beams [34] and Ib/Iχ is the background contamination ratio. Using Eq. (5) along with the measured null-split data yields estimates of the coherent background intensity that, in rough order of magnitude, correspond to the charge spectral crosstalk measured at χ = 0.A basic fact about gratings is that the diffraction angle varies with wavelength. Consequently, the primary effect expected from varying wavelength through the FGC is a change in radiated beam angle, equivalent to a linear phase tilt across the mode field, degrading mode overlap and thereby reducing coupling efficiency. Elementary analysis of this effect for the 12 um square FGC grating predicts about 40 nm of bandwidth will produce ±π/2 phase error at the vortex maximal ring. Indeed, conventional grating couplers on SOI designed for 1550 nm wavelength light achieve about a 40 nm 1 dB bandwidth to a single-mode fiber mounted at a fixed angle [2]. Moving beyond first order analysis, complete, detailed characterization of fiber coupling response versus wavelength requires consideration of both the grating coupler intrinsic properties and the fiber parameters, as well as possible design approaches for dispersion compensation [35,36].
In the work presented herein, all measurements were made using an available unmodulated fixed-tuned laser with a wavelength of 1549.12 nm. Studying the effect of varying wavelength on FGC mode mismatch using enhanced instrumentation is an important area for future empirical investigation.
5. Conclusion
We have designed, simulated, fabricated and tested forked grating coupler (FGC) devices. The FGC is a simple, small, low profile interface between optical vorticies and photonic ICs. Optical testing of fabricated devices has demonstrated the basic viability of the FGC, along with the viability of focused gratings to reduce the feed length, and apodization by means of grating groove width modulation to improve multiplexing crosstalk performance. Flat apodized devices achieve better than 20 dB multiplexing crosstalk isolation in many configurations. With opposite-charge multiplexing (analogous to bipolar signaling or BPSK), near 25 dB crosstalk isolation is achieved for most devices. The best case was 28.9 dB of opposite charge rejection for the flat apodized χ = 2 devices.
Although useful performance was obtained from these prototypes, further refined designs can have even better performance. Many of the design enhancements that have been developed for traditional fiber grating couplers can be applied to the FGC. These enhancements include: substrate mirrors, groove blazing, sub-wavelength features, single-etch designs, normal-incidence designs, and squint-angle optimization.
Based on the relatively tight sample standard deviations observed for the measured charge spectra on many devices, and the very repeatable amplitude and phase images, it seems likely that device fabrication tolerance is not a significant cause of imperfection in the OAM beams. Simulation results confirm this, as similar vortex imperfections were observed both in measured and simulated results. Rather, the WKB approximation made in the design procedure, specifically as incorporated in Eq. (1) and Eq. (3), is a strong candidate for the cause of OAM fidelity aberration. This approximation assumes the propagation parameters βeff and αeff vary slowly with respect to the wavelength. The FGC grating is a small (≈ 7λ) yet significantly varying structure; such an approximation is unlikely to be accurate. Attempts to further refine apodization with more complex functions will further degrade the WKB approximation.
A possible way forward for better vortex fidelity is to use an iterative refinement algorithm with electromagnetic simulation that seeks to optimize desirable performance metrics. Designers of traditional grating couplers have used this approach with good results [1]. Such an algorithm could be used to minimize critical crosstalk terms, or to improve efficiency, or both.
Apodization functions with tapered tails (e.g. Hamming and Gaussian) promise better performance than flat apodization. Unfortunately, the constraint imposed by Eq. (3) makes these functions more difficult to realize with a short focus taper. It also should be considered that tapered tails will require narrow grating grooves that may violate e-beam feature size rules. A sub-wavelength duty-cycle scheme may be required to work around this constraint.
Specific to the FGC, many new areas can be explored, such as bandwidth improvement, support for other scalar and vector polarizations, and a demonstration of FGC coupling direct to/from high-index-ring vortex fiber. Perhaps the most important future direction is to develop multi-mode FGCs to drive multi-mode multiplexed “bus” waveguide.
Funding
Corning Corporation; National Science Foundation (NSF) Industrial Innovation and Partnerships (IIP) (1068028) for the Industry/University Cooperative Research Center for Metamaterials.
Acknowledgments
Lori A. Lepak, Ph.D. and Luke K. Rumbaugh, Ph.D. provided valuable consultation and assistance.
References and links
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