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# Transmissive silicon photonic dichroic filters with spectrally selective waveguides

## Abstract

Many optical systems require broadband filters with sharp roll-offs for efficiently splitting or combining light across wide spectra. While free space dichroic filters can provide broadband selectivity, on-chip integration of these high-performance filters is crucial for the scalability of photonic applications in multi-octave interferometry, spectroscopy, and wideband wavelength-division multiplexing. Here we present the theory, design, and experimental characterization of integrated, transmissive, 1 × 2 port dichroic filters using spectrally selective waveguides. Mode evolution through adiabatic transitions in the demonstrated filters allows for single cutoff and flat-top responses with low insertion losses and octave-wide simulated bandwidths. Filters with cutoffs around 1550 and 2100 nm are fabricated on a silicon-on-insulator platform with standard complementary metal-oxide-semiconductor processes. A filter roll-off of 2.82 dB nm−1 is achieved while maintaining ultra-broadband operation. This new class of nanophotonic dichroic filters can lead to new paradigms in on-chip communications, sensing, imaging, optical synthesis, and display applications.

## Introduction

Integrated optical filters are one of the most important and widely used building blocks in photonic systems. Many applications in optics utilize interferometric on-chip filters enabled by the silicon photonics technology including microring resonators1,2,3,4, Bragg or arrayed waveguide gratings (AWGs)5,6,7,8, contra-directional couplers9,10, and photonic crystal filters11,12,13 for in-band filtering. On the other hand, with recent advances in integrated supercontinuum14,15,16,17,18 and second/high-harmonic generation19,20,21,22,23, on-chip handling of ultra-broadband optical signals is becoming increasingly important. Being able to arbitrarily and effectively filter and combine hundreds-of-nanometers-wide signals is crucial for on-chip integration of key functionalities including f − 2f24,25,26 and f − 3f17 interferometry, spectroscopy27,28, and wideband wavelength-division multiplexing29,30. Yet, limited optical bandwidths of the aforementioned interferometric filters typically preclude their use in applications that require octave-wide bandwidths.

For general spectral combining or separation in ultra-broadband photonics, free-space dichroic filters are more suitable as they achieve bandwidths as wide as 1000 nm31. Interference within alternating layers of optical coatings with high index contrast allows dichroic filters to provide highly reflective, wideband, and flat-top pass-bands in short- or long-pass configurations. However, free-space dichroic filters are limited in scalability and are not suitable for optical systems requiring tens or hundreds of optical filters at precise wavelengths. Although integrated filters are better suited for such scalable and precise optical filtering, achieving similar flat-top responses in ring resonators32 or Mach–Zehnder interferometers33 results in higher insertion losses. Despite several examples of on-chip filters with stop-bands up to hundreds of nanometers34,35, an efficient spectral filtering device with transmissive, flat-top, and ultra-wideband short- and long-pass outputs is yet to be demonstrated.

Recently, various asymmetric couplers have been used for on-chip polarization and mode-division handling. Coupling between fundamental/higher-order modes36,37, transverse-electric (TE)/transverse-magnetic (TM) modes38,39,40, or combinations of higher-order and cross-polarization modes41,42 have been demonstrated. Waveguide asymmetry has also been achieved with the use of sub-wavelength gratings, where the refractive index is periodically modulated in the direction of propagation43,44 to create fiber-to-chip couplers45, waveguide crossings46, and polarization splitters47. Despite their typical use for polarization or mode-division purposes, similar asymmetric couplers can also be used in wideband spectral filtering by taking advantage of the wavelength dependence of evanescent field overlap48,49. In this coupling scheme, while mode-evolution of the propagating field allows for broadband operation, the coupling coefficient κ is primarily influenced by waveguide dispersion and therefore has a weak wavelength dependence. As a result, in contrast to the sharp spectral transitions in interferometric devices like ring resonators, AWGs, or Fabry–Perot cavities, these adiabatic spectral filters exhibit slow pass-band-to-stop-band transitions (estimated ~0.07 dB nm−1 in ref. 48 and ~0.04 dB nm−1 in ref. 49). Similar asymmetric Y-junctions have also been demonstrated in the past for wavelength separation, with the use of different waveguiding materials50,51,52 or sub-wavelength gratings53,54. Although most of these asymmetric coupler-based filters can separate wavelengths sufficiently far apart, they have not been able to simultaneously demonstrate an ultra-wideband response and a fast spectral roll-off. Therefore, an ideal integrated filter that combines the low-loss, broadband operation of adiabatic structures with the sharp spectral characteristics of interferometric devices is highly desirable.

In this article, we address the aforementioned challenges in optical filters by presenting the theory, design approach, and experimental results for broadband, low-loss, integrated, 1 × 2 port, transmissive dichroic filters that simultaneously achieve single-cutoff operation, octave-wide optical bandwidths, and sharp filter roll-offs. First, we introduce the concept of spectral selection in asymmetric waveguides by investigating possible cross-sections for spatial separation of guided modes. Then we model the transmission behavior in these waveguides using a coupled mode description. We show that a fast filter roll-off is expected when the individual propagation constants of two adjacent waveguides are matched at a single wavelength and are highly mismatched otherwise. This allows the cutoff wavelength to remain a separate design parameter independent of the propagation length along the coupler. We then design the transitions to the desired cross-sections, fabricate the devices in a standard complementary metal-oxide-semiconductor (CMOS) foundry, and experimentally verify the short- and long-pass filter characteristics. Demonstrated devices achieve low-loss operation and the sharpest filter roll-offs reported to date with octave-wide simulated bandwidth. Our results suggest that single-cutoff broadband filters created with the methods described here can be adapted in many integrated photonic platforms to create highly customizable and efficient optical devices such as band- or all-pass filters as well as ultra-broadband multiplexers and couplers.

## Results

### Coupled-mode description of spectral selectivity

In order to describe spectral selectivity in a given waveguide cross-section, we first analyze the wavelength dependence of field distribution in a system of two coupled waveguides: Waveguide A (WGA) and Waveguide B (WGB). Under the weak coupling assumption55, the guided modes of the combined structure (supermodes) can be approximated by linear combinations of the individual waveguide modes (ψA and ψB for WGA and WGB, respectively). For waves carrying power in the same direction, the propagation of these supermodes can be described by a set of coupled differential equations56, which can be rewritten as

$$\left[ {\begin{array}{*{20}{c}} {\bar \beta + \delta } & { - \kappa ^ \ast } \\ { - \kappa } & {\bar \beta - \delta } \end{array}} \right]\psi _ \pm = \beta _ \pm \psi _ \pm$$
(1)

where κ is the coupling coefficient between the two waveguides, βA and βB are the individual propagation constants for ψA and ψB respectively, $$\bar \beta = \left( {\beta _{\mathrm{A}} + \beta _{\mathrm{B}}} \right)/2$$, and $${\delta=(\beta_{\mathrm{A}}-\beta_{\mathrm{B}})/2}$$. The fundamental supermode, also known as the quasi-even mode, is described by the normalized eigenvector corresponding to the larger eigenvalue β+ and is expressed as

$$\psi _ + = \frac{1}{{\sqrt 2 }}\left[ {\begin{array}{*{20}{c}} { - \sqrt {1 + \delta /S} } \\ {\sqrt {1 - \delta /S} } \end{array}} \right]$$
(2)

where S2 = δ2 + |κ|57. Once WGA and WGB are sufficiently separated, they can be treated as two outputs of an optical filter with their respective spectral transmissions. For the quasi-even mode input described above, the transmission for output port A, corresponding to WGA, is then the fraction of the optical power carried by ψ+ that remains in ψA given by

$$\left| {T_{\mathrm{A}}} \right| = \left| {\frac{{\psi _{\mathrm{A}} \cdot \psi _ + }}{{\psi _ + \cdot \psi _ + }}} \right|^2 = \frac{1}{2}\left( {1 + \frac{\gamma }{{\sqrt {1 + \gamma ^2} }}} \right) = \frac{1}{2}\left( {1 + {\mathrm{sin}}\left( {\mathrm{arctan}\,\gamma } \right)} \right)$$
(3)

where we defined the dimensionless quantity γ = δ/|κ|.

Equation (3) describes the distribution of power between any two coupled waveguides in general. Here the waveguide geometry can be engineered for the desired δ and κ or for a target spectral transmission. For spectral filtering and combining, we can exploit the wavelength dependence of γ. For instance, at the wavelength where γ = 0, power is evenly distributed between WGA and WGB, marking the 3 dB cutoff where WGA and WGB are phase matched. In general, the sign of γ is determined by the magnitudes of βA and βB. When βA > βB, δ > 0, γ > 0, and |TA| > 1/2. As the difference βA − βB grows, δ becomes increasingly positive, |TA| approaches 1, and majority of the optical power in ψ+ is confined within WGA. As soon as γ drops below 0, the optical power in ψ+ is primarily confined in WGB, as ψ+’s spatial profile rapidly moves from WGA to WGB. As a result, two coupled waveguides act as a spectral filter with a sharp transition and yet an extremely wide operation bandwidth.

Consequently, the key to achieving a dichroic filter response is to use two waveguides that are only phase matched at the filter cutoff (λC). The mismatch of propagation constants outside of λC is accomplished by waveguides with different group indices and is essential for a good extinction ratio between the two output ports. Two examples of these spectrally selective waveguide cross-sections are given in Fig. 1. Both examples consist of two waveguides whose guided modes are coupled via their evanescent fields. In Fig. 1b, WGB possesses spectral characteristics equivalent to a waveguide with a smaller refractive index core, resulting in a smaller group index. This idea of mimicking a continuous waveguide using segments smaller than the wavelength is similar to a sub-wavelength grating. The difference here is that instead of using the grating duty cycle to create dissimilar waveguides47,54, we independently engineer the effective and group indices with greater flexibility by controlling the widths and separations of individual segments. Unlike many asymmetric couplers36,37,38,39,40,41,42, the phase matching condition here is satisfied between the fundamental TE modes of the two waveguides, instead of between TE/TM or fundamental/higher-order modes. As such, for the proposed designs, the group index difference is instrumental for realizing phase mismatch at wavelengths outside of the cutoff and therefore is key for a spectrally selective cross-section.

Major electric field components of TE quasi-even supermodes in Fig. 1c, d indicate the evolution of the mode profile between the two waveguides, as γ changes from positive to negative (see Methods). Since Eq. (3) describes power distribution in an arbitrary coupled waveguide geometry, any two waveguides satisfying δ = 0 at one wavelength can be used to create a similar spectrally selective waveguide cross-section. The design in Fig. 1b is more advantageous for silicon-on-insulator (SOI) platforms due to its single-material and single-step lithography process. For the rest of the analysis, we therefore focus only on this specific cross-section. In Fig. 1e, we plot the effective indices of the individual fundamental TE modes of WGA and WGB for this geometry. With a wider WGA, due to the larger propagation constant, the filter cutoff at which βA = βB shifts to longer wavelengths. The supermode effective indices are plotted in Fig. 1f, showing the evolution of the quasi-even supermode ψ+ from ψA to ψB.

In principle, one can design WGB with any number of core segments and with any widths and gaps: The ideal solution is a flat or bulk-like effective index profile that can maximize |δ| when λ ≠ λC. One option is to use many core segments to mimic a bulk-like material, as the combination of closely spaced, sub-wavelength waveguides acts a single waveguide with a lower effective index. In reality, design choices are generally limited by the minimum widths and gaps that can be reliably fabricated in CMOS processes. Here we used a three-segment design to achieve a large βA − βB difference while ensuring reliable fabrication of individual segments.

### Power roll-off around the cutoff wavelength

The evolution of power in the quasi-even mode from WGA to WGB depends on γ’s rate of change with respect to wavelength. As γ = δ/|κ|, γ’s wavelength dependence is determined by the change in the effective indices and the coupling coefficient as a function of wavelength. In Fig. 2a, we plot δ(λ) and |κ(λ)| for the waveguide cross-section in Fig. 1b, with a separation of g = 750 nm between WGA and WGB. δ(λ) is highly dependent on wavelength and, by definition, undergoes a sign change around λC. On the other hand, the coupling coefficient |κ(λ)| is only a slowly varying function of λ, as it mainly depends on the overlap of the evanescent fields of ψA and ψB. The weak wavelength dependence of |κ(λ)| shown here explains the slow filter roll-offs for devices relying purely on waveguide dispersion. Since |κ(λ)| is monotonically increasing and only slowly varying, γ(λ) and δ(λ) exhibit similar spectral dependences. Therefore, the sign change in γ(λ) results in a maximum-to-minimum transition in |TA| where γ(λ = λC) = 0 marks the 3 dB cutoff.

In Fig. 2b, we plot the expected transmission response |TA| for the cross-section in Fig. 1b. The analytical result plotted using the coupled mode solution in Eq. (3) shows excellent agreement with the simulated result obtained from ratios of Poynting vector fluxes. Using this transmission response, the expected extinction ratio between the two output ports is calculated as a function of |γ| and shown in Fig. 2c. As predicted, the extinction ratio grows rapidly with the initial increase of |γ|, as the optical mode shifts from one waveguide to the other. With subsequent increase of |γ|, the extinction ratio (in dB) only grows logarithmically, since most of the optical power is already contained within one waveguide.

The spectral change in |TA| towards 1 or 0 on either side of the cutoff depends on how fast the magnitude of γ can grow with respect to wavelength. As |κ(λ)| is only slowly varying, γ(λ)’s magnitude is mainly influenced by δ(λ). Therefore, in order to approach 1 or 0 transmission on the left or right of the cutoff, one must design WGA and WGB to achieve maximum effective index difference when λ ≠ λC. Since the propagation constants βA and βB must match at λC, the effective index difference for λ ≠ λC is only possible with waveguides with different group indices. With a larger group index difference, although a faster transition from 1 to 0 is expected, phase matching at the desired λC becomes increasingly challenging. One can also increase the magnitude of γ by separating WGA and WGB with a larger gap, effectively reducing κ for all wavelengths. Evolving the quasi-even mode to a final cross-section with a large WGA − WGB separation without losing power to other modes requires longer transitions. As a result, the separation of WGA and WGB presents a design trade-off between device length and performance.

Once the appropriate waveguide geometry is determined for the desired spectral response, efficient transitions from a single-core rectangular waveguide to the determined cross-section must be designed. For low insertion-loss and a wide bandwidth, a single-mode TE input at any given wavelength must stay in the quasi-even mode throughout these transitions. According to coupled local-mode theory, power lost to unwanted modes can be minimized by decreasing the rate of change of the dielectric constant along the propagation length58, or equivalently, by using longer transitions59. To this end, we designed the slowly varying waveguide transitions illustrated in Fig. 3a for evolution of the quasi-even mode through the device. Transitions for conversion of a single waveguide input to a three-segment waveguide, development of a new adjacent waveguide segment (WGA), slow separation of the newly developed segment, and conversion of the three-segment structure back into a single rectangular waveguide have been implemented in sections numbered through , respectively. In all transitions, we used a 100 nm minimum waveguide width and spacing, as dictated by the fabrication design rules. For two sets of filters with cutoff wavelengths in the C-band and around 2100 nm, we simulated the transmission of the quasi-even mode through each section using the eigenmode expansion (EME) method (see Methods), as plotted in Fig. 3b–d. We chose L1 = L4 = 200 μm, L2 = 260 μm, and L3 = 900 μm in the C-band filters to minimize insertion loss while avoiding excessively long transitions to reduce losses due to propagation. Here, at the end of section , WGA and WGB have been separated to a final gap of g = 2 μm, in order to minimize any further coupling between the two waveguides and allow convenient connections to downstream devices. For the filters with cutoff wavelengths around 2100 nm, using the determined waveguide widths and gaps, appropriate coupling was obtained with lengths L1 = L4 = 200 μm, L2 = 350 μm, and L3 = 1100 μm through similar EME analyses.

In the proposed devices, performance metrics and the lengths of transitions and heavily depend on the specific choices of the waveguide widths and gaps. While it may be possible to use a wider gap gB for ease of fabrication, satisfying the phase matching condition at the same wavelength then requires either wider individual segments for WGB or a narrower WGA. Wider segments in WGB more strongly confine the propagating mode and reduce the coupling coefficient. As a result, an exponentially longer section would be required to separate the propagating mode for λ < λC, significantly increasing the device footprint. This may be mitigated by optimized adiabatic transitions with nonlinear tapers. On the other hand, compensating for larger gB with a smaller wA reduces the aspect ratio of WGA. This may be tolerable for filters with wider WGA designed for use at longer wavelengths. However, for the C-band filter with wA = 318 nm, reducing wA further could potentially increase coupling to TM modes in the standard 220 nm SOI platform and result in a higher insertion loss at the short-wavelength output port.

Designed filters were fabricated in a standard CMOS fabrication facility (see Methods). A scanning electron micrograph of a fabricated device is shown in Fig. 3e where WGA and the multi-segment WGB are clearly resolved. For these fabricated filters with the dimensions above, we demonstrate the operation bandwidth in Fig. 4. Eigenmode expansion simulations at a wide range of wavelengths once again confirm that the input stays in the fundamental TE mode as it propagates through the designed couplers and is routed to the corresponding output port. Figure 4a, b show that the extinction ratio at short wavelengths is limited by L2, as majority of this short wavelength input couples to WGA in section . This is in agreement with the inverse wavelength dependence of κ, confirming the need for a significantly larger L2 for the same $${\int}_{L_1}^{L_1 + L_2} \kappa \,\mathrm{d}z$$ at shorter wavelengths. On the other hand, the extinction ratio at much longer wavelengths is limited by the final lateral separation of the two output waveguides. Figure 4g–i show the guided TE mode progressively widening with wavelength and increasing the amount of power coupled to WGA. With the above choices of lengths, widths, and gaps, the simulation results demonstrate the predicted octave-spanning operation bandwidth of the proposed filter from 1300 nm to over 2800 nm.

### Characterization and analysis of filter response

To test the fabricated filters, we used single-mode fibers, polarization controllers, and inverted facet tapers for on- and off-chip coupling of tunable, continuous-wave laser sources (see Methods). Figure 5a, b show the measured and simulated spectral responses of the filters. The measured spectrum at the short- and long-pass output ports confirm the predicted single cutoff response. The mismatch between the measurement and the design targets can be explained by the thickness and width variations due to fabrication. Since waveguide geometry influences propagation constants of the guided modes, minor differences between simulated and measured cutoff wavelengths are expected as in all integrated photonic filters. Operation of the C-band filter in Fig. 5a was captured by imaging the scattered light at wavelengths below and above the cutoff through a microscope objective. Overlaid images in Fig. 5c show the separation of the two inputs as they are routed to the short- and long-pass output ports.

The insertion loss through the filters in the pass-bands were measured to be approximately 1 dB for the short-pass ports and <2 dB for the long-pass ports. Higher loss in the long-pass output can be attributed to the increased interaction of the propagating mode with the sidewalls, due to the design of WGB. This sidewall interaction together with the coupling coefficient increase at longer wavelengths may also contribute to the slightly reduced extinction ratio in the long-pass output, due to scattering and coupling to the adjacent WGA. Extinction ratios of approximately 10 dB are measured for the C-band filter in Fig. 5a. For the 2100 nm filter in Fig. 5b, the measured extinction ratios were over 15 and 17 dB for the short- and long-pass ports, respectively. Power roll-off for the filters were calculated to be as high as 2.82 and 0.64 dB nm−1 by differentiating the transmission responses in Fig. 5a, b, respectively. Owing to the slowly varying transitions of waveguide cross-sections, the propagating mode in the dichroic filters demonstrated here does not rely on evanescent coupling to adjacent waveguides. Our measured roll-offs are therefore 10–70 times sharper than integrated filters relying on evanescent couplers48,49.

We can shift the cutoff wavelength by modifying the waveguide widths, which in turn changes the propagation constants in WGA and WGB. As δ(λ) is modified, the cutoff wavelength at which γ = 0 is also shifted. This cutoff dependence on waveguide widths can be utilized to design and fabricate filters with different cutoff wavelengths. For instance, to shift the response to longer wavelengths, one can either increase wA, or increase gB, resulting in a higher βA or a lower βB. Under both conditions, the wavelength at which ψA and ψB are phase matched shifts toward longer wavelengths. This spectral dependence on waveguide geometry is useful as it allows for adjustment of the cutoff without having to modify the adiabatic structures or the lengths L1 − L4.

To demonstrate the shift of the cutoff wavelength, a set of three filters were designed to have cutoffs around 1550 nm. In addition to nominal choice of wA = 318 nm, filters with wA = 312 nm and wA = 324 nm were also fabricated. For this set of filters, we kept the previously chosen dimensions of WGB constant at wB = 250 nm and gB = 100 nm. The expected cutoff shift is verified by measurements shown in Fig. 5d. Similarly, for longer wavelength applications, another set of three filters were designed in the 2100 nm spectral range, with wA = 486 nm, wA = 494 nm, and wA = 502 nm. For this set of filters with longer cutoff wavelengths, we maintained the dimensions of wB = 350 nm and gB = 100 nm for WGB and plotted the measured transmission data in Fig. 5e. For both cases, the EME simulations predicted similar extinction and roll-offs as before, at different cutoff wavelengths. It is important to note that, with these measurements, changes only in the width of WGA are considered, while WGB’s width remains the same. For a correlated increase or decrease in the widths of WGA and WGB, βA and βB would both change in the same direction. The resulting cutoff wavelength shift would then be much smaller and can be estimated from the derivatives of propagation constants with respect to widths and wavelength.

For both sets of filters, simulated and measured cutoff wavelengths are listed in Table 1 and show good agreement. The reason behind the slight mismatch is sensitivity of effective indices and the resulting phase matching point to fabrication variations, similar to many other silicon photonic devices. This is an inherent result of large derivatives of the propagation constants with respect to waveguide dimensions and is a common occurrence in high-index contrast platforms such as Si/SiO2 or SiN/SiO2. Here the difference between the simulated and measured cutoff wavelengths is about 5 nm on average and can be attributed to the photolithography-induced waveguide geometry changes. Many other devices that rely on phase matching also exhibit similar discrepancies as previously reported for Bragg gratings5,6, AWGs7,8, and contra-directional couplers9. For all these devices, precise spectral alignment is typically achieved with thermal tuning, as detailed in the following section. It may also be possible to accurately achieve the desired cutoff by designing WGA and WGB with group indices that are only slightly different, for instance, by using sub-wavelength gratings. The designed cross-section would then have a slowly changing |γ| as a function of wavelength and therefore a cutoff that does not depend on waveguide dimensions as much as before. The resulting trade-off between the filter roll-off and cutoff accuracy would need to be evaluated for the needs of the specific application.

### Thermal tunability of filter cutoff

The spectrally selective waveguide geometry allows for the cutoff wavelength to be thermally tuned, making use of the thermo-optic effect and thermal expansion in Si and SiO2. Here tunability is enabled by the imbalance in the thermal dependences of the effective indices of individual waveguide modes ψA and ψB. According to eigenmode solutions at different temperatures, the temperature dependence of βA is approximately twice as large as that of βB. This difference in thermal shifts arises from the respective geometries of the two waveguides: WGA is a strip waveguide confining the majority of the field within its core. In contrast, WGB is a slot waveguide60 where a significant amount of field remains in the SiO2 gaps between the waveguide cores. Together with SiO2’s much lower thermo-optic and thermal expansion coefficients than those of Si, this waveguide geometry explains the smaller thermal dependence of βB.

In Fig. 6a, we analyze the cutoff wavelength dependence on temperature by plotting δ as a function of temperature and wavelength. For these simulations, we consider the thermo-optic and thermal-expansion effects for both waveguides (see Methods). The expected thermal dependence is calculated from the change of cutoff wavelength with respect to temperature and is equal to 101.5 pm K−1 or −12.8 GHz K−1. This is experimentally confirmed in Fig. 6b where we plot the normalized filter transmission at temperatures from 12.5 to 55.0 °C and the 3 dB cutoff at each temperature. From these cutoff wavelengths, the thermal shift was measured to be −14.4 GHz K−1. In the fabricated structures, the thermo-optic and the thermal expansion coefficients of the SiO2 cladding are dependent on the specific deposition parameters used. Together with the fabrication-induced waveguide geometry changes, any deviation in the thermal properties from the literature values used here61,62 may explain the mismatch between the simulated and measured thermal tuning efficiencies. As temperature is recorded at the substrate, thermal impedance of the buried oxide separating the waveguide and the substrate may also play a role in the difference.

The actual range of thermal tunability highly depends on the specific thermo-electric cooler (TEC) setup including the sizes of the TEC and the substrate, as well as the quality of the TEC–heatsink thermal contact. The measured tuning efficiency of −14.4 GHz K−1, however, confirms that demonstrated dichroic filters can be spectrally aligned to their design targets like many other systems with ring resonators63,64,65 or Bragg gratings66,67 with similar tuning efficiencies. The energy required for thermal tuning may be reduced with the use of microheaters directly integrated on top of sections and , using a process with metal heater layers. These integrated heaters would enable a larger ΔT and resulting Δλ and also allow for multiple filters to be independently tuned.

## Discussion

The approach presented here provides a clear methodology to design highly selective, broadband, single-cutoff, transmissive, and low-loss optical filters based on the novel spectrally selective waveguide designs. These waveguides uniquely combine the broadband response of adiabatic transitions with the sharp spectral responses typically observed in interferometric filters. Since the transmission response in Eq. (3) depends on the propagation constants in the two waveguides, it is possible to design and integrate similar filters at visible or mid-infrared wavelengths using other core materials, such as Si3N4, Al2O3, or TiO2. Unlike devices that rely on propagation length to transfer power to an adjacent waveguide, the mode-evolution-based design here allows the cutoff wavelength to be determined independent of coupler length.

In theory, the proposed spectrally selective waveguide cross-section can provide much larger extinction ratios than measured with the fabricated devices, as predicted in Fig. 2c. From a design perspective, the measured extinction is determined by how effectively the input is evolved to the fundamental TE mode at the output. This explains the improved extinction ratio for the 2100 nm filters, due to their longer adiabatic transitions made necessary by their wider waveguides. The extinction ratio can be improved further by replacing the linearly tapered adiabatic sections and with transitions optimized for a constant adiabaticity measure $$\left\langle {\psi _{\mathrm{A}}\left| {\frac{\partial }{{\partial p}}} \right|\psi _{\mathrm{B}}} \right\rangle /\delta$$ for a given parameter p such as the waveguide widths or gaps57,68. Specifically, section can be optimized for the width of WGA, which would include a nonlinear increase as it tapers up to its final width of wA. Similarly, the lateral separation g between WGA and WGB in section can also be optimized, which would result in a faster increasing g as coupling reduces toward the end of the transition. These modified adiabatic couplers can greatly enhance coupling efficiency and extinction ratio while minimizing device length and suppressing fringing in the stop-bands. More generally, improvements may also be possible when using a cross-section similar to the one shown in Fig. 1a or by using lower index-contrast waveguides, due to reduced interaction of the guided mode with the sidewalls. Waveguide roughness may also contribute to the lower extinction ratio of the short-wavelength filter. Consequently, sidewall passivation, shorter transitions enabled by optimized adiabatic couplers described above, or the use of other waveguiding materials can also enhance the extinction ratio. However, material choices may be limited since phase matching at the cutoff wavelength poses strict restrictions on the material indices.

The mismatch between the measured cutoff wavelength and the design target is primarily due to fabrication-induced width and height changes that alter propagation constants, similar to other devices such as Bragg reflectors and ring resonators. More precise cutoff shifts may be possible with the use of wider waveguides at the expense of introducing higher-order modes, as the change in effective index gradually decreases with increasing width. Nevertheless, the expected and measured cutoff wavelength for varying wA agree well. Additionally, thermal tuning of the cutoff wavelength is also demonstrated, as this is commonly required in reconfigurable optical systems. An accurately controlled wavelength shift is significant when designing band-pass filters that can be achieved by cascaded combinations of tunable short- and long-pass filters described here. Devices demonstrated in this article can be used in many optical systems including on-chip f − 2f and f − 3f interferometers, ultra-broadband multiplexing and demultiplexing systems in communications, efficient pump couplers to laser cavities, optical signal processing applications where arrays of arbitrarily wide band-pass filters are needed, and spectroscopy measurements that require spectral separation of excitation and output signals closely spaced in wavelength. Optical environmental sensors such as those used in index or concentration measurements or distance and velocity sensing like light imaging, detection, and ranging can also benefit from the filters described here, as they transduce wavelength selectivity to other measured quantities.

## Methods

### Finite difference eigenmode simulations

Waveguide modes and effective indices were simulated using the Lumerical’s MODE software with a maximum discretization of 10 nm. Finer mesh regions with 5 nm maximum discretization were used for the waveguides themselves. The coupling coefficient was calculated from the eigenvalues in Eq. (1) using 4|κ|2 = (β+ − β)2 − 4δ2 and the propagation constants obtained from the eigenmode simulations. Transmission efficiencies plotted in Fig. 3b–d were extracted from the power transferred between the quasi-even input and quasi-even output modes as simulated by the EME method in MODE. In the direction of propagation, a total of 30 modes were solved every 5 μm in sections and and every 15 μm in region .

### Device fabrication

All filters were fabricated using 300 mm SOI substrates with a 225 nm-thick Si layer and a 2 μm-thick buried oxide in a CMOS foundry with 65 nm technology node. SOI was patterned using 193 nm immersion photolithography and reactive ion etching. The Si layer was fully etched to form the strip waveguides. This was followed by an oxidization step to passivate the sidewalls, which reduced the Si thickness to 220 nm. A 4 μm-thick top cladding of SiO2 was chemical vapor deposited. Dicing trenches were etched on the die borders, and the wafer was diced to individual dies.

### Optical characterization setup

Filters were characterized using three continuous-wave laser sources that are tunable from 1484–1625, 1860–2010, and 2040–2400 nm. Single-mode fibers SMF-28 or SM-2000 were used with a fiber polarization controller to couple TE light on- and off-chip through inverted facet tapers.

### Prediction and measurement of thermal shift

For temperature dependence, eigenmodes in WGA and WGB were solved for by modifying the refractive indices and dimensions of the core and cladding materials using the following thermo-optic and thermal expansion coefficients: dnSi/dT = 1.86 × 10−4 K−1 69, dnSiO2/dT = 0.95 × 10−5 K−1 61, αSi = 2.61 × 10−6 K−1 70, and αSiO2 ≈ 5 × 10−7 K−1 62. In addition to the 5 nm meshes, edges of the waveguides and the gaps in between were spatially discretized with 2 nm resolution to accurately capture the small changes in the refractive index as a function of temperature. The simulation was centered at the free-running operation temperature of T0 = 22.6 °C as measured by the TEC with ~1 mW optical input power. Eigenmode simulations were performed with wavelength and temperature steps of Δλ = 0.5 nm and ΔT = 1 K, respectively. For measurement, the temperature was controlled by using a closed-loop feedback-driven TEC placed in contact with the chip substrate.

### Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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## Acknowledgements

This work was supported under the DARPA DODOS project, contract number HR0011-15-C-0056. The authors thank Dr. Robert Lutwak and Professor Franz X. Kärtner for helpful discussions and Gary Riggott and Di Zhu for their assistance with scanning electron microscopy. N.L. acknowledges a fellowship from Agency of Science, Technology and Research (A*STAR), Singapore.

## Author information

Authors

### Contributions

E.S.M. conceived the idea of spectrally selective waveguides; performed the theoretical analysis; designed, simulated, and characterized the devices; and wrote the manuscript. N.L. took the SEM images; and revised the manuscript review response. M.R. took the infrared images. C.V.P. coordinated the mask layout. C.V.P., A.R., and N.S. helped organize the manuscript. D.V., E.P.I., L.A.K., and M.R.W. supervised the project. All authors edited and revised the manuscript.

### Corresponding author

Correspondence to Emir Salih Magden.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Magden, E.S., Li, N., Raval, M. et al. Transmissive silicon photonic dichroic filters with spectrally selective waveguides. Nat Commun 9, 3009 (2018). https://doi.org/10.1038/s41467-018-05287-1

• Accepted:

• Published:

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text:
# 4.4 Analysis of Skiplists

In this section, we analyze the expected height, size, and length of the search path in a skiplist. This section requires a background in basic probability. Several proofs are based on the following basic observation about coin tosses.

Lemma 4..2   Let be the number of times a fair coin is tossed up to and including the first time the coin comes up heads. Then .

Proof. Suppose we stop tossing the coin the first time it comes up heads. Define the indicator variable

Note that if and only if the first coin tosses are tails, so . Observe that , the total number of coin tosses, can be written as . Therefore,

The next two lemmata tell us that skiplists have linear size:

Lemma 4..3   The expected number of nodes in a skiplist containing elements, not including occurrences of the sentinel, is .

Proof. The probability that any particular element, , is included in list is , so the expected number of nodes in is .4.2 Therefore, the total expected number of nodes in all lists is

Lemma 4..4   The expected height of a skiplist containing elements is at most .

Proof. For each , define the indicator random variable

The height, , of the skiplist is then given by

Note that is never more than the length, , of , so

Therefore, we have

Lemma 4..5   The expected number of nodes in a skiplist containing elements, including all occurrences of the sentinel, is .

Proof. By Lemma 4.3, the expected number of nodes, not including the sentinel, is . The number of occurrences of the sentinel is equal to the height, , of the skiplist so, by Lemma 4.4 the expected number of occurrences of the sentinel is at most .

Lemma 4..6   The expected length of a search path in a skiplist is at most .

Proof. The easiest way to see this is to consider the reverse search path for a node, . This path starts at the predecessor of in . At any point in time, if the path can go up a level, then it does. If it cannot go up a level then it goes left. Thinking about this for a few moments will convince us that the reverse search path for is identical to the search path for , except that it is reversed.

The number of nodes that the reverse search path visits at a particular level, , is related to the following experiment: Toss a coin. If the coin comes up as heads, then move up and stop. Otherwise, move left and repeat the experiment. The number of coin tosses before the heads represents the number of steps to the left that a reverse search path takes at a particular level.4.3 Lemma 4.2 tells us that the expected number of coin tosses before the first heads is 1.

Let denote the number of steps the forward search path takes at level that go to the right. We have just argued that . Furthermore, , since we can't take more steps in than the length of , so

We can now finish as in the proof of Lemma 4.4. Let be the length of the search path for some node, , in a skiplist, and let be the height of the skiplist. Then

The following theorem summarizes the results in this section:

Theorem 4..3   A skiplist containing elements has expected size and the expected length of the search path for any particular element is at most .

#### Footnotes

....4.2
See Section 1.3.4 to see how this is derived using indicator variables and linearity of expectation.
... level.4.3
Note that this might overcount the number of steps to the left, since the experiment should end either at the first heads or when the search path reaches the sentinel, whichever comes first. This is not a problem since the lemma is only stating an upper bound.
opendatastructures.org

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text:
# Teachers

(Redirected from Teaching)

Teachers are players who can help others learn commands and advance skills beyond the limits of guild teachers. In the case of non-primaries this peak is reached pretty fast. When skills become guild-maxed you have to rely on self-teaching (which costs a lot of XP) or on player teachers (which costs more than guild advancing, but less than self-teaching). The higher the other player's teaching bonus is, the less XP you have to spend.

## Learning skills

### From players

The teacher's bonus (ETB - Effective teaching bonus) has to be higher than your bonus for you to learn from them at all. Their effective teaching bonus is the average of their bonus in the skill and their bonus in the relevant teaching skill (up to twice the bonus in the taught skill). For example, if a teacher has a bonus of 200 in people.teaching.magic and 400 in magic.spells.offensive (ma.sp.of), their ETB in ma.sp.of will be (200 + 400) / 2 = 600 / 2 = 300. As long as the student has less than this bonus in ma.sp.of, they will be able to learn that skill from the teacher.

Teaching cost is based on your level, your bonus (on unmodified stats), and the teacher’s ETB.

A teacher can find their teaching bonus in a skill with:

teach bonus <skill>

To learn a skill from a player, the syntax is:

teach me <skill|n/max levels of skill> from <living>

If they have autoteach on, this will make them automatically offer to teach you the skill. This will show you how much xp it will cost, or will tell you that you don't have enough experience. (Note that the xp isn't actually spent yet.) Follow it up with:

learn <skill> from <living>

Specifying the skill without specifying the number of levels will have them offer to teach you one level. The "teach me max <skill> from <living>" version will have them offer to teach you as many levels as you can learn with the amount of xp you have.

You can also "teach me <skill|n/max levels of skill> from all", which will cause everyone in the room with autoteach on to offer to teach you that skill.

### From yourself

To self-teach, the syntax is:

teach <skill|n/max levels of skill> to me

This will show you how much xp it will cost, or will tell you that you don't have enough experience. (Note that the xp isn't actually spent yet.) Follow it up with:

learn <skill> from <living>

You can't self-teach commands.

## Learning commands

### From players

To learn a command from a player, the syntax is:

teach me <command> from <living>

If they have autoteach on, this will make them automatically offer to teach you the command. Follow it up with:

learn <command> from <living>

You can also "teach me <command> from all", which will cause everyone in the room with autoteach on to offer to teach you that command.

It doesn't cost xp to learn a command.

### From NPCs

To query a guild teacher about new commands say:

say teach me please

At least some teachers do not respond to some languages. If the teacher doesn't respond check that you're asking in the language of the area where the teacher is located.

Guild teachers typically only teach commands to members of their guild.

## Causes for failure

• Insufficient skills: you can't learn a skill from someone whose effective teaching bonus isn't higher from yours. You also can't learn a command from them unless you have the prerequisite skills and they have high enough skills to teach you.
• Busyness: Someone can't teach you if they're busy. If you get a message that they're too busy, wait until they finish whatever commands they're doing.
• Idleness: Someone can't teach you if they've been idle for five minutes or more.
• Moving rooms: Teaching will be interrupted if you're no longer in the same room with them (in this case the xp is not spent).
• No autoteach: If someone doesn't have autoteach on, then in order to learn from them, they have to actively offer to teach you the skill or command.

## Finding a teacher

Many teachers put their skills on player-run sites. Some also put skill bonuses in their finger information.

If you want to learn from a particular person who isn't online, a good plan is to add them as a friend (with "friend add <friend>") and then turn your friend inform on ("inform friend"). This will cause you to get a message when they log on or off.


text:
What I Did at the (Maths) Office

Andrew Stacey

2021-05-04

1 Introduction

Back when I was an academic Mathematician, a common question at a social gathering (after the usual opening gambit of "I was never very good at Maths at school.") was "What do you actually do all day? Add up big numbers?"1. I'd invariably answer with something about trying to invent new Mathematics, but then the conversation would get bogged down by a question about where the ideas come from. I found that hard to answer. I might say that I was working on jubjub birds because I'd come across it when studying bandersnatches, which I'd been led to from my study of slithy toves. And so on. By the time I got to anything resembling the "real world", my interlocutor had inevitably moved on.

1If the person was trying to show off, they'd say "multiply".

But there is one result of mine that I know the story of extremely well and it is short enough to tell. It recently resurfaced on the internet, and in a bit of chatter about it I considered explaining the story behind it. The result itself may not be an extremely exciting one, but the purpose of this is not to sell you the result but to show how it arose naturally and developed as I also developed as a Mathematician.

2 The Result

Here's an attempt at explaining the result. Start with Pascal's triangle, only left justify it.

 $\begin{array}{c}1\\ 1& 1\\ 1& 2& 1\\ 1& 3& 3& 1\\ 1& 4& 6& 4& 1\\ ⋮\end{array}$

Now pick a square of numbers which starts somewhere on the left hand side. Empty slots are considered to be $0$. For example,

 $\begin{array}{cccc}1& 2& 1& 0\\ 1& 3& 3& 1\\ 1& 4& 6& 4\\ 1& 5& 10& 10\end{array}$

Here's the result: this matrix has determinant $1$.

3 The What has What?

If that seemed fine up until the last sentence, this section is for you. If you're fine with matrices and determinants, feel free to skip to the next section (though you might find it useful to know how I'm going to talk about them so skimming through this section might be of some value).

Definition

A matrix is a rectangular grid of numbers.

We put brackets around the grid to indicate that we consider it as a single object. Examples are:

 $\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right],\phantom{\rule{thickmathspace}{0ex}}$

If you read a book with matrices in it, you will probably be told that a matrix is something more impressive2. It isn't. This is it. However, a matrix can represent many different things and that's the reason why they are such an important part of Mathematics. Many problems in disparate areas of Mathematics can be phrased in terms of matrices and then solved using a set of standard techniques.

2Especially if that book is the novelisation of The Matrix.

As far as we're concerned, a rectangular grid of numbers is as deep as we need to go.

The determinant is something a little more complicated, but we don't need a deep understanding of it. It only makes sense for so-called square matrices, which are matrices with the same number of rows as columns.

Rough Definition

The determinant of a square matrix is a number that summarises that matrix. It is calculated from all of the entries of the matrix by a series of multiplications, additions, and subtractions.

Thus the result says that if you take a square from the left-hand side of the left-justified Pascal's triangle and "summarise" it, you always get the number $1$.

4 Why?

The story starts when I was teaching an undergraduate topic called Linear Algebra. This is the topic that studies matrices and builds the standard techniques used to study them. As part of teaching the course, I had to set problems for the students to practise the techniques. This involved devising matrices with certain properties. I could build these matrices from certain ingredients, one of which was a square matrix. If that square matrix had determinant $1$ then the steps involved in solving the problem would be numerically simpler (in that there would be no fractions) and so would be better suited for early practice of the methods. So I needed a ready source of square matrices with determinant $1$ and digging them out of Pascal's triangle was quite an easy method of obtaining them3.

3For completeness, I should say that there are better ways than using Pascal's triangle.

One important aspect to note here is that I didn't need a proof of the result at this time. Whenever I used this result in building a problem, I would take a matrix out of the side of Pascal's triangle and then verify it had determinant $1$ by putting into a computer algebra system.

So I was in the position of knowing something to be true for practical purposes, but not needing to know it to be true for all purposes.

I expect that I figured out a simple proof for small matrices ($2×2$ and $3×3$), but the fact that I don't have a firm memory of it shows that it wasn't very important to me at the time.

The story now skips forward a couple of years (and several thousand miles) to a time when I was sitting round a lunch table with a group of Mathematicians. One of these happened to remark that in their latest work they needed a particular result about determinants of matrices drawn from Pascal's triangle. My memory is a little hazy, but I think they said that they didn't have a proof but had verified it in many cases. The result was not quite the same as mine: instead of taking a contiguous block, their method needed a matrix taken from every other line. Also, the resulting determinant was a power of $2$. Nevertheless, it was sufficiently similar that it brought to mind my old result and I wondered if they were two cases of a more general result.

What is important to note here is that now someone else is interested in the same, or nearly the same, result. So it becomes not a neat trick for making problems but something that someone else would like to know about. This then gives me an incentive to figure out what's really going on, and to ask whether the result that I knew worked for practical purposes was actually always true.

At the same time, in my research then I was learning about something called numerical polynomials. This had absolutely no connection with the ideas that had led to my discovery but had come from my work in Algebraic Topology. However, it turned out to be a key ingredient in the proof of the result. As it's a key ingredient, I'll explain it in the next section.

5 Newton's Polynomials

There is a formula for the terms in Pascal's triangle. The $k$th term in the $n$th row is given by:

$\left(\begin{array}{c}n\\ k\end{array}\right)=\frac{n!}{k!\left(n-k\right)!}=\frac{n\left(n-1\right)\left(n-2\right)\dots \left(n-k+1\right)}{k\left(k-1\right)\left(k-2\right)\dots 1}$
 1

(Note that we're zero-based here.) This is also the number of ways of choosing $k$ objects from a set of $n$.

We can think of this as a function by varying the inputs. It's common to fix $n$ and allow $k$ to vary, which gives us the entries across a row. But for our purposes, we will fix $k$ and allow $n$ to vary which will give us the entries down a column. We can actually make sense of this for non-integers by using the extreme right-hand side of Equation (1), so we define:

 ${b}_{k}\left(x\right):-\frac{x\left(x-1\right)\left(x-2\right)\dots \left(x-k+1\right)}{k\left(k-1\right)\left(k-2\right)\dots 1}$

When $n\ge k$ is a positive integer, we have that ${b}_{k}\left(n\right)=\left(\begin{array}{c}n\\ k\end{array}\right)$. The first few are:

 $\begin{array}{rl}{b}_{0}\left(x\right)& =1\\ {b}_{1}\left(x\right)& =x\\ {b}_{2}\left(x\right)& =\frac{1}{2}x\left(x-1\right)=\frac{1}{2}{x}^{2}-\frac{1}{2}x\\ {b}_{3}\left(x\right)& =\frac{1}{6}x\left(x-1\right)\left(x-2\right)\end{array}$

These are known as Newton's polynomials4.

4Newton used them in his generalised binomial theorem.

These functions have some very nice properties. In this, and the following, we'll restrict $n$ to integers (positive and negative) and $k$ to positive integers (including $0$) while $x$ can be any real number.

1. For any $n$, ${b}_{k}\left(n\right)$ is an integer.

2. ${b}_{k}\left(x\right)$ is a polynomial of degree $k$.

3. For $k>0$ and $0\le n, ${b}_{k}\left(n\right)=0$.

4. For any $k$, ${b}_{k}\left(k\right)=1$.

5. For any $k$, the values of ${b}_{k}\left(n\right)$ for $n$ positive is the $k$th column of the left-justified Pascal's triangle.

That last property suggests that we could consider extending Pascal's triangle upwards using the ${b}_{k}\left(x\right)$ at negative integers, and indeed this makes sense to do but we shan't pursue it here.

These properties make the family $\left\{{b}_{k}\left(x\right)\right\}$ a very special set of functions. Although they typically have fractions for coefficients, they "happen" to map integers to integers. Such polynomials are called numerical polynomials. The other properties mean that any polynomial with that property can be written as a sum of these functions, where each appears an integral number of times.

In technical terms, the $\left\{{b}_{k}\left(x\right)\right\}$ form a basis for the $ℤ$–module $Int\left(ℤ\right):-\left\{p\in ℚ\left[x\right]:p\left(ℤ\right)\subseteq ℤ\right\}$.

Regardless of whether you read or skipped that last sentence, there is a crucial point: if $p\left(x\right)$ is a polynomial of degree $k$ with rational coefficients which happens to take integers to integers then there are integers ${a}_{0},\dots ,{a}_{k}$ such that:

 $p\left(x\right)={a}_{0}{b}_{0}\left(x\right)+{a}_{1}{b}_{1}\left(x\right)+\dots +{a}_{k}{b}_{k}\left(x\right)$

An important point to notice here is that we don't use any ${b}_{j}\left(x\right)$ of degree higher than our original polynomial.

The reason I was learning about these polynomials is that they play a very important role in Algebraic Topology. They describe, in a natural way, a particular object that I was studying at the time. Although there were lots of connections to binomials, and so to Pascal's triangle, the context meant that I rarely thought of ${b}_{k}\left(x\right)$ as a function of numbers – I was putting in much more complicated objects in place of $x$. So although I was studying the pieces I needed to prove my result, the connection wasn't obvious until I was prompted to look at it more closely by the conversation with the other Mathematicians.

6 Assembling the Proof

I'm not going to give the full proof here, but I will sketch it out for those reading this who might be interested. I've tried to keep it elementary, but experience with determinants and matrix multiplication will be helpful in following it.

When we take a square matrix from Pascal's triangle, we are choosing particular rows and then taking the first few elements from each one. Since we're taking the first elements, each row is given by taking the polynomials ${b}_{0}\left(x\right)$ to ${b}_{k}\left(x\right)$ and evaluating them on the row number. So if our rows are ${r}_{0}$ to ${r}_{k}$, the matrix is:

 $\left[\begin{array}{cccc}{b}_{0}\left({r}_{0}\right)& {b}_{1}\left({r}_{0}\right)& \dots & {b}_{k}\left({r}_{0}\right)\\ {b}_{0}\left({r}_{1}\right)& {b}_{1}\left({r}_{1}\right)& \dots & {b}_{k}\left({r}_{1}\right)\\ ⋮& ⋮& & ⋮\\ {b}_{0}\left({r}_{k}\right)& {b}_{1}\left({r}_{k}\right)& \dots & {b}_{k}\left({r}_{k}\right)\end{array}\right]$

(Because we're zero-based, this matrix has $k+1$ rows and columns.)

In the cases we're examining, the ${r}_{i}$ have a very specific pattern. They are of the form ${r}_{i}=p+qi$ where $p$ and $q$ are integers. So our matrix is:

 $\left[\begin{array}{cccc}{b}_{0}\left(p\right)& {b}_{1}\left(p\right)& \dots & {b}_{k}\left(p\right)\\ {b}_{0}\left(p+q\right)& {b}_{1}\left(p+q\right)& \dots & {b}_{k}\left(p+q\right)\\ ⋮& ⋮& & ⋮\\ {b}_{0}\left(p+qk\right)& {b}_{1}\left(p+qk\right)& \dots & {b}_{k}\left(p+qk\right)\end{array}\right]$

(In actual fact, we can deal with more variation than this, but this will be enough to get the idea across.)

Here's the trick. Rather than thinking of that as a matrix of numbers, we look at it as a matrix of functions. To do this, we just need to add in a judicious choice of variable in each cell:

 $\left[\begin{array}{cccc}{b}_{0}\left(p+q{x}_{0}\right)& {b}_{1}\left(p+q{x}_{0}\right)& \dots & {b}_{k}\left(p+q{x}_{0}\right)\\ {b}_{0}\left(p+q{x}_{1}\right)& {b}_{1}\left(p+q{x}_{1}\right)& \dots & {b}_{k}\left(p+q{x}_{1}\right)\\ ⋮& ⋮& & ⋮\\ {b}_{0}\left(p+q{x}_{k}\right)& {b}_{1}\left(p+q{x}_{k}\right)& \dots & {b}_{k}\left(p+q{x}_{k}\right)\end{array}\right]$

We get our original matrix back by setting ${x}_{i}=i$.

Now for each $j$, ${b}_{j}\left(p+qx\right)$ is still a polynomial of degree $j$ which takes integers to integers. Therefore it can be written as a sum of ${b}_{0}\left(x\right)$ up to ${b}_{j}\left(x\right)$. Let us write:

${b}_{j}\left(p+qx\right)={a}_{0j}{b}_{0}\left(x\right)+{a}_{1j}{b}_{1}\left(x\right)+\dots {a}_{jj}{b}_{j}\left(x\right)$
 2

and note that this holds for any $x$.

There is a way of putting these numbers into matrices which results in the expression:

 $\left[\begin{array}{cccc}{b}_{0}\left(p+q{x}_{0}\right)& {b}_{1}\left(p+q{x}_{0}\right)& \dots & {b}_{k}\left(p+q{x}_{0}\right)\\ {b}_{0}\left(p+q{x}_{1}\right)& {b}_{1}\left(p+q{x}_{1}\right)& \dots & {b}_{k}\left(p+q{x}_{1}\right)\\ ⋮& ⋮& & ⋮\\ {b}_{0}\left(p+q{x}_{k}\right)& {b}_{1}\left(p+q{x}_{k}\right)& \dots & {b}_{k}\left(p+q{x}_{k}\right)\end{array}\right]=\left[\begin{array}{cccc}{b}_{0}\left({x}_{0}\right)& {b}_{1}\left({x}_{0}\right)& \dots & {b}_{k}\left({x}_{0}\right)\\ {b}_{0}\left({x}_{1}\right)& {b}_{1}\left({x}_{1}\right)& \dots & {b}_{k}\left({x}_{1}\right)\\ ⋮& ⋮& & ⋮\\ {b}_{0}\left({x}_{k}\right)& {b}_{1}\left({x}_{k}\right)& \dots & {b}_{k}\left({x}_{k}\right)\end{array}\right]\left[\begin{array}{cccc}{a}_{00}& {a}_{01}& \dots & {a}_{k0}\\ 0& {a}_{11}& \dots & {a}_{k1}\\ ⋮& ⋮& & ⋮\\ 0& 0& \dots & {a}_{kk}\end{array}\right]$

(If you know about matrix multiplication you can verify that this is correct.)

Two crucial observations are now needed. The first is to determine the numbers ${a}_{jj}$. The simplest way to do this is to compare the coefficients of the highest power of $x$. On the right-hand side of Equation 2 only the term ${a}_{jj}{b}_{j}\left(x\right)$ contributes because all of the other terms have degree strictly less than $j$. On the left-hand side, the coefficient turns out to be ${q}^{j}$ times the coefficient of ${x}^{j}$ in ${b}_{j}\left(x\right)$. Hence ${a}_{jj}={q}^{j}$.

The second observation is that when setting ${x}_{i}=i$, the terms ${b}_{j}\left({x}_{i}\right)$ all vanish for $j>i$ and are equal to $1$ for $j=i$. We therefore have:

 $\left[\begin{array}{cccc}{b}_{0}\left(p\right)& {b}_{1}\left(p\right)& \dots & {b}_{k}\left(p\right)\\ {b}_{0}\left(p+q\right)& {b}_{1}\left(p+q\right)& \dots & {b}_{k}\left(p+q\right)\\ ⋮& ⋮& & ⋮\\ {b}_{0}\left(p+qk\right)& {b}_{1}\left(p+qk\right)& \dots & {b}_{k}\left(p+qk\right)\end{array}\right]=\left[\begin{array}{cccc}1& 0& \dots & 0\\ {b}_{0}\left({x}_{1}\right)& 1& \dots & 0\\ ⋮& ⋮& & ⋮\\ {b}_{0}\left({x}_{k}\right)& {b}_{1}\left({x}_{k}\right)& \dots & 1\end{array}\right]\left[\begin{array}{cccc}{q}^{0}& {a}_{01}& \dots & {a}_{k0}\\ 0& {q}^{1}& \dots & {a}_{k1}\\ ⋮& ⋮& & ⋮\\ 0& 0& \dots & {q}^{k}\end{array}\right]$

The last piece of the jigsaw relies on a couple of nice properties about determinants. I said earlier that a determinant was like a summary of a matrix. It can be quite complicated to calculate, but there are certain properties that help. In particular, when we have an equation as above then we can work out the determinant of the left-hand side by working out the determinants of the two on the right-hand side and multiplying the resultant numbers. The other useful property is that when we have a lot of zeros, as in both of the matrices on the right-hand side, the determinant is just the product of the terms on the diagonal from top left to bottom right.

So our determinant calculates as:

 ${1}^{k+1}\cdot {q}^{0}\cdot {q}^{1}\cdot {q}^{2}\cdots {q}^{k}={q}^{\frac{1}{2}k\left(k+1\right)}$

The full theorem is even more general than this, and can be found in the original paper.

7 Conclusion

Alongside explaining the idea of the actual result, what I wanted to show in this article was that theorems almost never come out of "thin air".

In this case, the result started out as something that I needed for a practical reason. Even though it was as part of my job, it was the teaching aspect rather than the pure research so I wasn't bothered about proving it I just needed something that worked.

Once someone else got interested, it is as if it switched from the teaching aspect to the research aspect. I still needed the right tools to attack the problem, though, and so it was partly the happy coincidence of that someone else getting interested at the same time as me learning of the tools I needed to prove it.

So theorems don't come out of "thin air", they arise far more naturally. Sometimes, as in this case, out of spotting a pattern and wanting to understand whether it is a genuine pattern or not.

Lastly, although the result itself is not particularly ground-breaking it was a very useful proof to work through because it taught me a lot about Newton's polynomials that I might not have learnt so easily in my research context. By taking them out of the rather complicated environment of my research, I was able to focus on learning about them.

8 Epilogue

Although I wrote this up, I never actually got round to publishing it – I was more concerned with getting my "serious research" published and the effort required to figure out a suitable journal for this, or even finding out if it was original, seemed too much. So it lingered on my website and CV, until I left academia and my academic website disappeared. Many years later, Alexander Bogomolny of Cut the Knot tweeted a result that looked like a special case of what I'd proven so I mentioned it to him. Someone dug up the paper from the wayback machine and linked it on that site. At that point, I thought about writing up the story of that paper as it felt like one where I could remember all the salient details. It's taken me a little while to get it up on this website, though!


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# Quasi-isometry classes of elementary amenable groups

Is there any elementary argument showing that there exist uncountably many distinct quasi-isometry classes of elementary amenable groups? How about solvable groups?

For amenable groups it follows from the result of Grigorchuk (proved in the 80's) stating that there are uncountably many groups of intermediate growth with pairwise incomparable growth functions.

-

@Denis: to any f.g. group $G$, you can associate the set $N(G)$ of ultrafilters $\omega$ such that $Cone(G,\omega,(1/n))$ is a real tree. So $N(G)$ is a QI-invariant of $G$. I think the idea is to show that elementary amenable groups achieve continuum many sets $N(G)$. It's not obvious (and I haven't checked) but I guess you can do this. – YCor Dec 25 '11 at 22:33


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1. Jan 15, 2006

### m00c0w

I'm trying to calculate the radius of the circle $$x^2 - 2ax + y^2 - 2ay = 0$$
I completed the square and got to $$(x-a)^2 + (y-a)^2 = 2{a}^2$$
That would mean the radius is $$\sqrt{2{a}^2}$$ right? Which would be $$\sqrt{2}\sqrt{a^2} \Rightarrow a\sqrt{2}$$... but my textbook quotes the answer as $$\sqrt{2a}$$ Did I do something wrong or is my textbook incorrect?

2. Jan 15, 2006

### arildno

Are you sure you are reading your textbook? It might be that the answer is $\sqrt{2}a[/tex] but a little blurred so it is not clear where the square root sign ends- it looks like [itex]\sqrt{2a}$.
On the otherhand that is exactly why most texts would use the clearer $a\sqrt{2}$ as you do.