Miriam Greis
email: miriam.greis@codecentric.de

Hyunyoung Kim
email: hyunyoung.kim@imag.fr

Andreas Korge
email: andi-korge@web.de

Albrecht Schmidt
email: albrecht.schmidt@um.ifi.lmu.de

Céline Coutrix
email: celine.coutrix@imag.fr

SplitSlider: A Tangible Interface to Input Uncertainty

Keywords: Tangible User Interface, Uncertainty, Input Modality, Dial, Slider. 1

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Inputting data into a system while being uncertain about it is a common task. For instance, in a usage survey at the train station, we might get asked about how often we take the train. If we do not take the train regularly, it is difficult to answer a precise value between like "daily" and "never". Our answer might rather be "between every 6 months and weekly, and most often every 2 months". Other examples range from satisfaction surveys in airports to availability inquiry or medical questionnaires, where we might hesitate between 54kg and 56kg to input our weight. Currently, respondents are forced to answer a single, precise value even when they are uncertain. However, allowing respondents to express their uncertainty about their answer enables the people who requested the questionnaire to have precise and reliable data. For example, intraparticipant uncertainty can increase transparency and reliability of the data [START_REF] Morss | Communicating Uncertainty in Weather Forecasts: A Survey of the U[END_REF][START_REF] Roulston | A laboratory study of the benefits of including uncertainty information in weather forecasts[END_REF], produce relevant results [START_REF] Boukhelifa | Uncertainty Visualization: Why Might It Fail?[END_REF][START_REF] Greis | Input Controls for Entering Uncertain Data: Probability Distribution Sliders[END_REF] and help to make better decisions [START_REF] Joslyn | Uncertainty forecasts improve weather-related decisions and attenuate the effects of forecast error[END_REF][START_REF] Roulston | A laboratory study of the benefits of including uncertainty information in weather forecasts[END_REF], e.g., when designing a fidelity program for a company. Therefore, input mechanisms should allow expressing uncertainty together with the most probable input value.

However, there is a tradeoff between the usage simplicity and the ability to express the uncertainty. Interactions with questionnaires are recommended to be easy to use and discoverable while providing sufficient information [START_REF] Fink | How to conduct surveys: A step-by-step guide[END_REF]. Respondents often stay novices when responding to a survey, as they rarely answer to the exact same questionnaires twice. Hence survey companies either choose to offer a simple interface to encourage as many respondents as possible, or a more complex interface able to gather richer data from fewer participants but with the help of surveyors. Corresponding examples can be found in public transportation surveys, where clients are offered either happy/angry faces buttons in airports, or a surveyor approaching respondents with tablets and multiple questions. Allowing respondents to input uncertainty can increase data reliability, but it can also increase the survey's interface complexity and cost.

This motivates us to balance the simplicity and the ability to express the uncertainty. To achieve this goal, we explore Tangible User Interfaces (TUIs) as a way to offer users with a flexible compromise between simplicity and the ability to express the uncertainty. TUIs are a good candidate to offer a flexible compromise between simplicity and expressiveness, as they were found more discoverable than GUIs [START_REF] Horn | Comparing the use of tangible and graphical programming languages for informal science education[END_REF][START_REF] Lim | The anatomy of prototypes: Prototypes as filters, prototypes as manifestations of design ideas[END_REF] and better foster the simultaneous adjustments of parameters than GUIs, even more than multitouch GUIs [START_REF] Ma | Using a Tangible Versus a Multi-touch Graphical User Interface to Support Data Exploration at a Museum Exhibit[END_REF]. TUIs are already used to answer questionnaires, e.g., VoxBox [START_REF] Golsteijn | VoxBox: A Tangible Machine that Gathers Opinions from the Public at Events[END_REF] and happy/angry faces tangible buttons in satisfaction surveys in airports [START_REF]Happy or Not[END_REF]. However, these TUIs do not yet allow users to express their uncertainty about their answer.

To balance the simplicity and the ability to express the uncertainty, we explore the design of physical dials and sliders that can capture users' value and uncertainty on one device. We first use Morphees+ [START_REF] Kim | Morphees+: Studying Everyday Reconfigurable Objects for the Design and Taxonomy of Reconfigurable UIs[END_REF] features to design five controls based on standard tangible dials and sliders and that can input both value and uncertainty. We then present low-fidelity prototypes and conduct a focus group study to find design requirements for uncertain, tangible input. Following these findings, we present the most promising design: the SplitSlider. The SplitSlider's thumb supports entering one value (1-thumb slider) and can be split to additionally enter a probability distribution (2-and 3-thumb slider). We implemented and evaluated a functional tangible prototype. The results of our study show that the use of SplitSlider as 1-thumb slider is discoverable. The use of SplitSlider as 3-thumb slider is not as discoverable as the 1-thumb mode, but after its explanation, it was found easy to use to express the uncertainty. SplitSlider allows users to choose on the fly between its discoverable use as a standard 1-thumb slider or the ability to express their uncertainty as a 3-thumb slider.

Related Work

In the following, we present related work addressing the communication of uncertainty, graphical interfaces supporting users to input uncertainty and tangible user interfaces (TUIs) that could support uncertain input.

Uncertainty and Its Communication

Uncertainty, as described by Pang et al. [23], includes statistical variations or spread, errors and differences, minimum and maximum range values, noise, or missing data.

Studies show that decision-making is better supported by capturing, modeling, and visualizing uncertain data [START_REF] Boukhelifa | Uncertainty Visualization: Why Might It Fail?[END_REF][START_REF] Joslyn | Uncertainty forecasts improve weather-related decisions and attenuate the effects of forecast error[END_REF][START_REF] Morss | Communicating Uncertainty in Weather Forecasts: A Survey of the U[END_REF][START_REF] Roulston | A laboratory study of the benefits of including uncertainty information in weather forecasts[END_REF]. For instance, communicating uncertainty can increase transparency and reliability of weather forecasts [START_REF] Morss | Communicating Uncertainty in Weather Forecasts: A Survey of the U[END_REF][START_REF] Roulston | A laboratory study of the benefits of including uncertainty information in weather forecasts[END_REF]. Pang et al. [23] also name three steps of data processing in which uncertainty can be introduced: acquisition, transformation and visualization. In data acquisition, uncertainty is inevitable due to inexact measurements. During transformation, the original data can be altered by a human or an algorithm. Lastly, visualization may introduce uncertainty as it does not usually use the same media as the original data. We focus on the transformation step, where humans alter the data when answering questionnaires. We especially tackle the input of the possible spread of an input value.

Users answer questionnaires in many different ways. Among the most common methods, we find pen and paper, e.g., in trains, and computer systems. The interfaces to computer systems include TUIs, e.g., in airports with happy/angry buttons to give feedback about a service, or GUIs, e.g., to fill forms on the Web. Previous work proposed GUIs for users to express their uncertainty [START_REF] Greis | Input Controls for Entering Uncertain Data: Probability Distribution Sliders[END_REF]. Among others, they compared 1-thumb, 2-thumb and 3-thumb sliders to input uncertain data:

• The 1-thumb slider allows users to input a probability distribution by moving the single thumb: users move the peak of the distribution, while the standard deviation, skew and kurtosis are fixed. • The 2-thumb slider allows users to input a probability distribution by independently moving two thumbs: users move the minimum and maximum values, while the peak of the distribution stays in the middle of these two values. • The 3-thumb slider allows users to input a probability distribution by independently moving three thumbs: users move the minimum, the maximum and the peak values of the distribution.

Our study focuses on gradual transition between certain and uncertain TUI inputs in questionnaire settings. We consider the discoverability of interactions and the compromise between ease of use and expressiveness. The previous study did not evaluate the discoverability of each slider, as each slider was first explained to participants. The study rather focused on ease of use. Even though the graphical 3-thumb slider was experimentally found the best compromise between easiness and the ability to express uncertainty [START_REF] Greis | Input Controls for Entering Uncertain Data: Probability Distribution Sliders[END_REF], it offers a fixed compromise between easiness and the ability to express uncertainty. Moving the three thumbs in sequence was found cumbersome by the participants, compared to the 1-thumb slider. In addition, the 1-thumb slider was better suited for users with little knowledge in statistics, while the 3-thumb slider was better suited for users with more expertise in statistics. We aim at supporting all levels of expertise in statistics.

To support all levels of expertise in statistics, we explore deformable TUIs that allow both certain and uncertain inputs. Instead of using the one-, two-, three-thumbed sliders, we suggest different deformation of both dials and sliders to learn the advantages and disadvantages of the interactions with them.

Communicating Input Uncertainty with TUIs

Even though TUIs have not explicitly addressed the problem of inputting uncertain data, prior work could be used for this purpose. Coutrix et al. suggest a resizable tangible slider [START_REF] Coutrix | Shape-Change for Zoomable TUIs: Opportunities and Limits of a Resizable Slider[END_REF] to compromise between travel time of the thumb and input accuracy. The design could be also used to input uncertain continuous data. By interpreting the size of the slider as the certainty (smaller size means less precision and thus more uncertainty), this slider would be able to represent a Gaussian distribution (uncertainty ≈ standard deviation; value ≈ mean).

The Inflatable mouse [START_REF] Kim | Inflatable Mouse: Volume-adjustable Mouse with Air-pressure-sensitive Input and Haptic Feedback[END_REF] could be also used for uncertain input. The mouse has an elastic chamber, and users can squeeze it to zoom a map in/out or to change their scroll settings. For uncertain input, users could squeeze the mouse to express their certainty. The device can be used when users are inputting 2D data (e.g., X-Y coordinate values) with 1D uncertainty (same standard deviation in both dimensions).

In our work, we choose to systematically explore the design space of uncertain input TUIs from scratch, rather than starting from the previous work. By doing so, we consider more interactions than extending [START_REF] Coutrix | Shape-Change for Zoomable TUIs: Opportunities and Limits of a Resizable Slider[END_REF] and squeezing [START_REF] Kim | Inflatable Mouse: Volume-adjustable Mouse with Air-pressure-sensitive Input and Haptic Feedback[END_REF] only.

Design Exploration for Uncertain Input TUIs

To design deformable tangible input controls that allow the quantitative input of uncertainty, we generated variations of conventional continuous interfaces: dials and sliders. We keep their rotational or linear control to input value and added other modalities to input uncertainty. As an idea generation tool, we used the Morphees+ taxonomy [START_REF] Kim | Morphees+: Studying Everyday Reconfigurable Objects for the Design and Taxonomy of Reconfigurable UIs[END_REF]. The taxonomy describes all possible system-actuated shape-changes and manual deformations in a systematic way. The generated designs can express uncertainty in two ways: 1) all the dials and Expandable Slider (Fig. 1 and Fig. 2-left) can input uncertainty as an amount centered around the value (i.e., 0-100%), and 2) Split Slider (Fig. 2-right) can input a range of estimated values, independently of the central value. When the device was to input an amount of uncertainty, we mapped the larger surface or open space with more uncertainty and less pressure with more uncertainty for design coherency.

Dial-Based Designs

Dials are common TUIs for continuous variables. They can have bounded and unbounded input ranges depending on parameters and user needs. In this paper, we choose to use unbounded ranges to emphasis their difference with sliders, which have bounded input ranges. The usual interaction of turning dials changes the best estimate value (≈ mean of a desired input). The extended interactions inspired by Morphees+ change uncertainties (≈ standard deviation of a desired input, see Fig. 1).

Expandable Dial (Fig. 1-left) was driven by Morphees+' Area feature, which describes changes in area size on a surface. The dial's diameter can be increased or decreased, resulting in the area change at the top of the dial. The larger diameter the dial has, the more uncertainty the users are inputting. When the users are 100% sure (no uncertainty), they can decrease the diameter to the minimum. Pressure Dial (Fig. 1-right) can be pressed downwards in addition to the rotation, similarly to Button+ [START_REF] Suh | Button+: Supporting User and Context Aware Interaction through Shape-Changing Interfaces[END_REF]. It was driven by the Strength feature of Morphees+, i.e. the force needed to move a control point to another position. The stronger the dial is pressed, the more certain the input is.

Slider-Based Designs

Sliders are widely used in both graphical and tangible user interfaces for ranged variables. The standard interaction of sliding the (central) thumb changes the best estimate value (≈ mean of a desired input). The extended interactions inspired by Morphees+ change uncertainty (≈ standard deviation of a desired input or range of a desired input).

Expandable Slider (Fig. 2-left) was inspired by Morphees+'s Area feature as was the Expandable Dial. It has one thumb that can be expanded along the slider axis to communicate uncertainty. The center of the thumb represents the best estimate value, and the size of the thumb represents the amount of uncertainty.

Split Slider (Fig. 2-right) was driven by Morphees+'s Modularity, which describes an object's ability to be split into multiple pieces. It has a thumb that can be split into two or three thumbs. In the one-thumb mode, it functions as a standard slider and can input a single, precise value. When the thumb is split into two, the two thumbs input a range of uncertainty, and the center of the range is the input value. When the thumb is split into three, the middle thumb inputs a value while the outer thumbs enter the range of uncertainty.

Study Evaluating Our Designs

In this section, we aim at learning design criteria for uncertain input TUIs. We first implement low-fidelity prototypes of the explored design. We then conduct a focus group study where the participants evaluate the low-fidelity prototypes.

Low-Fidelity Prototype Implementation

We implemented low-fidelity prototypes of all the design explorations, except for the pressure dial, where we used a consumer product [24] (Fig. 3d). We had six prototypes as we had two different prototypes for the Expandable Dial, one that could be stretched (Fig. 3a) and one that could be squeezed (Fig. 3b). Here we briefly describe how they are implemented, and how they work.

Expandable Dial: Stretching Design. (Fig. 3a) We use a Hoberman mechanism [START_REF] Associates | [END_REF] to create an expandable circular shape. The center of the mechanism is connected to a rotational axis for value input. At the external corners, there are concave disks where users put their fingers on. Users can slightly press them and stretch the fingers inwards or outwards for inputting uncertainty.

Expandable Dial: Squeezing Design. (Fig. 3b) We use an aerospace design [START_REF] Guest | Inextensional Wrapping of Flat Membranes[END_REF] to fold a paper sheet into a condensed shape. The tension of the paper keeps the dial expanded, and users squeeze it to change the diameter.

Pinch Dial. (Fig. 3c) We create six equilateral prisms from a laser cutter and connect them into a hexagon, roughly resembling the round shape of a dial. They are held together with adhesive tape on the outer faces. We place a bent piece of plastic sheet between two prisms to act as a spring to open the dial. Users can close the dial by pinching the device.

Pressure Dial. (Fig. 3d) We used a PowerMate Bluetooth [24] for the design. It has a spring inside to push the upper rotational part back up when pressed. The device has only two states of heightpressed and not-pressed.

Expandable Slider. (Fig. 3e) We place a bent piece of plastic sheet between two thumbs of a laser-cut slider. The plastic sheet forced the thumbs to separate. Users can squeeze the thumbs for more certain input.

Split Slider. (Fig. 3f) There are three thumbs on a laser-cut slider. The thumbs have grooves on them, and users can place a U-shaped plastic piece to combine two thumbs together. Hence, the number of thumbs can be changed between one (certain input), two (range of uncertainty), and three (e.g., range of uncertainty and median).

Focus Group Study Design

To evaluate our design explorations, we conducted two focus group interviews. For this, we provided the low-fidelity prototypes for the explored designs.

Participants. We recruited twelve participants (10 male, 2 female) between 20 and 34 years old (M = 24.92, SD = 3.65) via personal invitations in order to have varying backgrounds such as social sciences and natural sciences. Their degrees ranged from A-level to M.Sc.

Task and Procedure. We had two participant groups. Each focus group interview lasted about 30 minutes. We took pictures and recorded the audio during the interviews. The participants were asked to sign the consent form and provide their demographic information. The task aimed to evaluate the low-fidelity prototypes when inputting value and uncertainty. The participants were grouped in three pairs, and we introduced our prototypes to the participants. Each pair was assigned a scenario to work for, e.g., inputting an uncertain time. Then each pair was randomly given one of the low-fidelity prototypes, so that three prototypes were given in total. They were asked to write down advantages, disadvantages, improvements and suitability of their respective prototypes for their scenario. Each pair then presented their findings to the others. We then repeated this step with the remaining three prototypes.

Results

Design Requirements of Uncertain Input TUIs. Using thematic analysis [START_REF] Braun | Using thematic analysis in psychology[END_REF], we identified preliminary requirements for designing tangible devices for uncertain input. The requirements confirm previous work for rotational and linear input [START_REF] Kim | KnobSlider: Design of a Shape-Changing UI for Parameter Control[END_REF]. It shows that the general purposes of the input interfaces are similar. However, introducing uncertain input to these devices revealed a new requirement that were not considered for dials and sliders and that we present here. There are in total seven requirements, which we grouped in four categories.

Fast manipulation.

R1 Simultaneous input. In general, the participants liked to input both value and uncertainty simultaneously as this allowed fast manipulation. For instance, they could change the diameter of both Expandable Dial prototypes while rotating them. However, they complained that the Pinch Dial was difficult to pinch when the opening is no longer between the thumbs and index fingers.

Precise manipulation. R2 Small intervals. Dials were preferred for small intervals in value inputs, because they could have small control display gain. On the other hand, the input ranges of the sliders were limited by the slider sizes, hence a larger control display gain.

R3 No interference between inputs. Participants preferred not to have interference between the value and uncertainty inputs, i.e., sometimes they wanted to change either value or uncertainty and did not want to change the other by accident. E.g., when participants changed uncertainty (stretching or squeezing) with Expandable Dial, it could rotate the device slightly. When users grasped Pinch Dial, it applied some pressure on it, and it could cause unwanted change on uncertainty. R4 Easy finalization. The participants needed to finalize the inputs easily. With all designs, users needed another input interface (e.g., button) to finalize their inputs after changing value or uncertainty. It would require users either: to use another hand to press the button while the users are manipulating the devices, or to take off their hands from the devices and then press the button. We considered the second interaction was easier, as users do not have to keep the tension on the device (e.g., squeezing or pressing the devices) while pressing the button. It allows users to take off their hands from the devices and rest between manipulating the devices. For instance, with the Stretching Design of the Expandable Dial or Split Slider, the participants were able to take off their hands from it and then finalize their inputs.

Observable uncertainty for speed and precision. R5 Visual feedback. The participants preferred explicit visual information on the amount of uncertainty. For instance, the participants liked that Split Slider's thumb intervals supported fast and precise interaction through clear visual feedback on the uncertainty. R6 Force feedback. Participants also liked that Pinch Dial and Expandable Slider supported fast and eyes-free interaction through force feedback on uncertainty. The force feedback provided additional information on the visual feedback. The participants also mentioned that using more pressure for more certainty feels intuitive.

Supporting statistical knowledge.

R7 Supporting both experts and novices. The participants liked that the Split Slider supported both novice and expert users through the input mode change between 1 and three thumbs. Novice users with limited knowledge of statistics could use the one-or two-thumb modes. Expert users could use the three-thumb mode, allowing input of probability distributions such as normal distribution.

Evaluation of the Designs. We evaluated the 5 designs based on the design requirements and user comments from the focus group study (see Table 1). The fulfillment of the requirements was sometimes dependent on implementations. For instance, the spring of the Pressure Dial was strong, and the participants complained that it is hard to keep the pressure while rotating it. In this case, we clarify that the advantages and disadvantages are related to the implementation methods and discuss other implementation methods that could fulfill the requirements and used for future implementations. 3b) were both through squeezing, it was difficult for the participants to squeeze the Pinch Dial when the open space was not between the thumb of the index finger. With the Pressure Dial, it was not easy to adjust the level of the pressure on it while rotating. It could be improved by replacing the spring to a mechanism that allows a wider range of pressure such as memory foam. The Split Slider did not allow simultaneous input of value and uncertainty, because it had separate thumbs for them. R2 Small intervals. The dial-based designs had boundless input. It allows inputting small intervals and thus users can enter precise values. The slider-based designs had bounded ranges and resulted inevitably in bigger intervals than the dial-based designs. This can be mitigated with a longer input axis for the slider. R3 No interference between inputs. Only the Split Slider fulfilled this requirement, because it had separated thumbs. Both prototypes of the Expandable Dials could cause unwanted change in the value or uncertainty, when the users were changing uncertainty (stretching or squeezing), or value (rotating). It was the same with the Pinch Dial (squeezing and rotating), Pressure Dial (pressing and rotating) and Expandable Slider (squeezing and sliding). R4 Easy finalization. The Expandable Dial -Stretching Design (Fig. 3a) could keep both the angle and the diameter even when the users took off their hands. The Split Slider thumbs stayed still when the users were not manipulating them. They allowed the users to rest their hands while finalizing the inputs (e.g., pressing another button for finalization). On the other hand, the prototypes with spring mechanismsthe Squeezing Design of the Expandable Dial, Pinch Dial, Pressure Dial, and Expandable Slider changed the amount of uncertainty to an initial amount when the users were not manipulating them. Hence the users needed to keep the tension on the device to preserve the wanted amount of uncertainty and would need another hand to press a button for finalization.

R5 Visual feedback. The slider-based designs could clearly show the relative amounts of uncertainty. The users could see the ranges of the possible input on the axis of the sliders. The users could see the uncertainty between the thumbs of the Expandable Slider or between the outer thumbs of the Split Slider. The users could see the amount of uncertainty relatively to the ranges. The boundless dials did not provide such clear visual feedback. For instance, it was not clear for the users whether Pinch Dial was 50% or 60% open. R6 Force feedback. This requirement was mutually exclusive with the R4, easy finalization of inputs. The spring mechanisms of the Pinch Dial, Pressure Dial and Expandable Slider could give force feedback but also caused unwanted movement when finalizing input. R7 Supporting both experts and novices. The Split Slider was the only design that had two input modes for uncertainty: ranged input with 2-thumb mode and probabilistic distribution input with 3-thumb mode.

The evaluation shows that the Split Slider performs the best as an uncertain input TUI. It allowed no interference between the value and uncertainty input, and the finalization of the input was easy. It provided a clear visual feedback on the amount of uncertainty.

The feature that the slider could have between one and three thumb(s) supported users with both limited and good knowledge of statistics. We decide to build a high-fidelity prototype of the Split Slider to evaluate its ability to capture uncertain input.

SplitSlider: A Tangible Slider for Uncertain Input

We implement SplitSlider, a tangible interface that allows both certain and uncertain input (Fig. 4, Fig. 5). We use three off-the-shelf sliders as sensors (Bourns PSM 100mm). They are placed next to each other in a box, with dimensions W168×L68×D52.5 mm. This gives an illusion that the device is a single slider with three thumbs. We use 3D-printed thumbs to cover the width of all the three sliders. We insert small magnets (Ø5 mm, height 2 mm, strength 520g) on both sides of the thumbs to give haptic feedback on the (un)combination of the thumbs, and to hold the thumbs together. The outer thumbs are slightly higher than the middle thumb. When the three thumbs are combined, they create a concave shape that most of slider thumbs have. Each thumb's long edges are cut in order to let users easily put fingers between combined thumbs and split them. Each thumb can travel around 88 mm. We use an Arduino UNO to connect the prototype to a computer.

Evaluation of the SplitSlider

The SplitSlider aims at balancing simplicity and ability to express the uncertainty. In order to measure how discoverable it is, we first asked participants to use it without being provided any explanation. Then, in a second phase, after having explained SplitSlider to participants, we measured its perceived ease of use and its ability to express the participants' uncertainty. Note that we decided to focus on the 1-thumb and 3-thumb modes only in this study to keep the study simple.

Study Design

We asked participants to use the working SplitSlider to answer public transportation survey questions (see Table 2). The questions were to have continuous and quantitative answers, which our interface accommodates.

As an independent variable, we had the two phases of the study: the phase before and the phase after the explanation of the prototype, to see if the use of the slider is discoverable and participants can use the uncertain input function without explanation. At the end of each phase, the participants filled in a feedback questionnaire, including a "Usability Metric for User Experience" (UMUX) questionnaire [START_REF] Finstad | The Usability Metric for User Experience[END_REF]. We used the same Likert scale for all these feedback questions, ranging from 1 (Strongly Disagree) to 7 (Strongly Agree). The participants then answered the Berlin Numeracy Test [START_REF] Cokely | Measuring risk literacy: The Berlin numeracy test[END_REF]. We did it at the end of the experiment not to bias the participants.

Participants

For the study, we recruited 12 participants (7 males, 5 females). Their ages ranged from 16 to 59 years (M = 35.25, SD = 16.09). None of them had seen or heard of the prototype prior to their participation. The participation was voluntary, and no compensation was given.

Apparatus

We implemented the software using C++ and openFrameworks. It showed a user interface that displays the questions for the user study and gives feedback of the device interpretation in a form of a gradient visualization on a distant screen (Fig. 6). The slider was taped on the table to prevent it from moving. We placed a keyboard on the left side to use its spacebar to finalize the input on the prototype. A 1440×900px (~287×180mm) laptop was positioned approximately 50 cm behind the slider, displaying the questions of the transportation survey and a graphical representation of the participant's answer below the questions. The slider on the screen had 1000px width (~199 mm), and 10mm movement of a SplitSlider thumb moved a cursor on the screen around 114px (~23mm). One question was displayed at once. The feedback questionnaire was prepared on a separate laptop. The participants swapped between a block of public transportation survey and a block of feedback questionnaire. As advised in previous work [START_REF] Matejka | The Effect of Visual Appearance on the Performance of Continuous Sliders and Visual Analogue Scales[END_REF], landmarks were not present on the slider's scale.

Tasks and Procedure

The participants signed a consent form and completed a demographic questionnaire. They were given a brief overview of the study procedure. The study was divided into two phases, and the participants answered six questions of the transportation survey per phase. The questions' order was pseudo-randomized. The study took around 30 minutes per participant.

• Phase 1: Participants used the prototype to answer the first six transportation survey questions. No explanation of the prototype was provided but they were asked to explore it. The three thumbs were combined at the beginning of the study. Afterwards, they filled in the feedback questionnaire. • Prototype explanation: The instructor explained the operating mode of the prototype to the participants. This included deterministic input with the one-slider mode, the meaning of each individual thumb when split, and an example of how these options could be used. • Phase 2: The participants answered the last six transportation survey questions with the prototype. Following the task, the participants filled in the identical feedback questionnaire that they already completed after Phase 1.

Results

The SplitSlider as a 3-thumb slider is not discoverable without explanation to most people. Only one third of the participants were able to discover the splitting function and used it to input uncertainty in Phase 1 (before explanation) (4/12, 33.3%, Fig. 7a). Among the other eight participants, only one managed to split the thumb into three at the last question of Phase 1, but he tried them for ~8 seconds and then put them back together to answer the question. In Phase 2 (after explanation), all participants used three thumbs to answer questions. Although the questions were calling for similar amount of uncertainty in Phase 1 and in Phase 2, the participants expressed different amount of uncertainty in both phases. Fig. 7b shows how much variance (i.e., uncertainty) was expressed in the two phases in a range of [0, 1000].

In Phase 1 the participants used the 3-thumb mode less than in Phase 2. Fig. 7c shows how often the different thumb modes were used in the two phases. A chi-square test of independence showed that participants used the 3-thumb mode significantly more often after the explanation (p < 0.001), showing that the prototype is not self-explanatory.

The SplitSlider is easy to use, both in 1-thumb and 3-thumb modes. The median UMUX score was 87.5 (Q0=33.3, Q1=75, Q3=92.7, Q4=95.83, mean=82.29) in the first phase, with a minor increase to 89.58 in the second phase (Q0=29.17, Q1=83.33, Q3=96.88, Q4=100, mean=85.75, see Fig. 8). Both scores are interpreted as excellent [START_REF] Bangor | Determining What Individual SUS Scores Mean: Adding an Adjective Rating Scale[END_REF], and there was no significant difference between the scores (p=0.366). This shows that the SplitSlider was found easy to use, whether expressing uncertainty or not.

The major criticisms in the qualitative feedback were the lack of smoothness of the thumb movement and the too large minimum interval between the thumbs. Removing the magnets could solve the problem.

Discussion and Future Work

The impact of the low fidelity of the prototypes on the results of a qualitative study should be further investigated. The aim of the first study was to find the design requirements and evaluate the low-fidelity prototypes, and we decided not to make obvious improvements, such as making the pressure of the Pressure Dial gradual to make the pressing interaction easy. Although the participants did not make any comments about the gradual pressing and the use of low-fidelity prototypes essential in HCI to evaluate the designs [START_REF] Dow | Parallel prototyping leads to better design results, more divergence, and increased self-efficacy[END_REF][START_REF] Lim | The anatomy of prototypes: Prototypes as filters, prototypes as manifestations of design ideas[END_REF], it would be interesting to explore if different prototypes would have resulted different design requirements.

The suggested design requirements are preliminary and should be evaluated. For example, R4 Easy finalization and R6 Force feedback were mutually exclusive, and we do not know which one is more important to novice or expert users. A future evaluation of the requirements can give priority to one over the other requirements.

The UMUX scores of the SplitSlider allow indirect comparison between the SplitSlider and its graphical counterpart by Greis et al. [START_REF] Greis | Input Controls for Entering Uncertain Data: Probability Distribution Sliders[END_REF]. The UMUX questionnaire has strong correlation with SUS scores (higher than 0.8 [START_REF] Finstad | The Usability Metric for User Experience[END_REF]), and the SUS was used in [START_REF] Greis | Input Controls for Entering Uncertain Data: Probability Distribution Sliders[END_REF]. The mean UMUX score of the SplitSlider was 85.75 in phase 2, and the mean SUS score of the graphical 3-thumb slider was 72.5. It shows the possibility that the SplitSlider may offer better usability than the graphical slider. A future study should investigate this comparison and explore different strengths of the two interfaces, such as discoverability, ease of use, accuracy and precision of user inputs.

The results of this work can be improved, refined and used. Our results could be improved by conducting further studies. First, the visualization of the user study was based on [START_REF] Matejka | The Effect of Visual Appearance on the Performance of Continuous Sliders and Visual Analogue Scales[END_REF]. We chose not to mention the meaning of the three thumbs, in order to let users freely interpret them, and include users with limited statistical knowledge. We could have added minimum value, most probable value and maximum value to balance between confusion and simplicity.

Our results can be refined in three directions. First, future work can check if participants' answers to the transportation survey were accurate by interviewing them before or after the study. Second, the SplitSlider can be compared to future alternative techniques to express uncertainty. Third, future work can explore how to improve its discoverability. Further design cues such as feedforward and affordances should be explored to better invite the user to explore the possibilities of the SplitSlider.

Our results could be used as an inspiration for future design of TUIs supporting the expression of uncertainty together with the input value. Sliders and dials have each their own benefits and drawbacks and are both widely used [START_REF] Joslyn | Uncertainty forecasts improve weather-related decisions and attenuate the effects of forecast error[END_REF]. Future work should further explore how to introduce the ability to express uncertainty in other common widgets such as buttons or dials. Our dials designed were not preferred in our focus group study. Further work can explore other deformation features of Morphees+ and also consider using uncertain input dials in different tasks, such as tasks that require more accuracy in value input (primary input through precise rotation [START_REF] Kim | KnobSlider: Design of a Shape-Changing UI for Parameter Control[END_REF]) and less accuracy in uncertainty input (secondary input).

Conclusion

In this paper, we aimed at balancing the simplicity and the ability to express the uncertainty. We explored the design of physical dials and sliders that can capture users' value and uncertainty on one device. We first used Morphees+ [START_REF] Kim | Morphees+: Studying Everyday Reconfigurable Objects for the Design and Taxonomy of Reconfigurable UIs[END_REF] features to design 5 controls that can input both value and uncertainty, which are based on standard tangible dials and sliders. We then presented low-fidelity prototypes and conducted a focus group study to find design requirements for uncertain, tangible input. Following these findings, we presented the most promising design: the SplitSlider. The SplitSlider's thumb supports entering one value (1-thumb slider) and can be split to additionally enter a probability distribution (2-and 3-thumb slider). We implemented and evaluated a functional tangible prototype. The results of our study show that the use of the SplitSlider as 1-thumb slider is discoverable. The use of the SplitSlider as 3thumb slider is not discoverable, but after explanation, it was found easy to use to express the uncertainty. The SplitSlider allows users to choose on the fly between its use as a standard 1-thumb slider or the ability to express their uncertainty as a 3-thumb slider. We envision that the SplitSlider to be the next standard TUI for uncertain input.

In future work we are thus interested in its ability to accurately input uncertainty and compare it with traditional TUIs and other designs such as dial-based ones in different context.

Pinch Dial (Fig. 1 -

 1 middle) includes an open space between the center and one side of the circumference of the dial. The open space can be closed or opened, making the dial looks as a full circle or a sector from the top of the dial. This is inspired by the Closure feature in Morphees+. A larger open space allows to input more uncertainty.

Fig. 1 .

 1 Fig. 1. The dial-based interfaces, enabling both value and uncertainty input. The usual rotation interactions are to input value and the extended inputs (grey arrows) are to input uncertainty. Expandable Dial: users increase the diameter to express more uncertainty. Pinch Dial: users increase the open spacebetween the center and one side of the circumferencefor more uncertainty. Pressure Dial: users enter their uncertainty by adjusting the pressure at the top of the dial (less pressure = more uncertainty).

Fig. 2 .

 2 Fig. 2. The slider-based interfaces, enabling both value and uncertainty input. The usual interaction of sliding thumbs is to input value, and the additional interactions (grey arrows) are to input uncertainty. Expandable Slider: users increase the width of slider thumb to increase uncertainty. Split Slider: users split the thumbs into two or three, to input the range of uncertainty.

Fig. 3 .

 3 Fig. 3. Low-fidelity prototypes of the explored uncertain input dials/sliders designs. A) Expandable Dial -Stretching Design, B) Expandable Dial -Squeezing Design, C) Pinch Dial, D) Pressure Dial, E) Expandable Slider, F) Split Slider.

Fig. 4 .

 4 Fig. 4. The schematic of the SplitSlider device with dimensions.

Fig. 5 .

 5 Fig. 5. Working prototype of SplitSlider. Left: One-thumb mode to input a single value. Right: Separated thumbs for communicate both a value and a range of uncertainty.

Fig. 6 .

 6 Fig. 6. (a) Experimental setup used by the participants. (b) Close-up of the distant visual feedback.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Thumb usage of the SplitSlider, comparing Phase 1: before explanation and Phase 2: after explanation of the three thumbs. (a) Number of participants that answered the questionnaires with three thumbs in each phase. (b) Amount of variance the participants used to answer the questionnaire.

Table 1 .

 1 Evaluation of the designs based on the design requirements.

		Expandable	Pinch Dial	Pressure Dial	Expandable	Split Slider
		Dial			Slider
	R1. Simultaneous input	✓			✓
	R2. Small intervals	✓	✓	✓
	R3. No interference between inputs					✓
	R4. Easy finalization	✓				✓
	R5. Visual feedback				✓	✓
	R6. Force feedback		✓	✓	✓
	R7. Supporting both experts and novices					✓

R1 Simultaneous input. The Expandable Dial and Expandable Slider fulfilled the requirement. The participants could change the diameter of the Expandable Dial and thumb size of the Expandable Slider while changing the value. Although the interactions of changing uncertainty with the Pinch Dial and the Expandable Dial -Squeezing Design (Fig.

Table 2. Questions used to evaluate the SplitSlider, with their respective min and max values.

Question

Min Max


Abdulhakim Ahmed

Cécile Doubre

Sylvie Leroy

Derek Keir

Carolina Pagli

James Hammond

Crustal Structure across the Western Afar margin from the Uplifted Plateau to the Rift Axis using Receiver Functions

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


V Vakarin
email: vevakarin@gmail.com

W N Ye

J Ramirez

Q Liu

J Frigerio

A Ballabio

G Isella

L Vivien

C Alonso-Ramos

P Cheben

J M Ramírez

D Marris-Morini

Ultra-wideband Ge-rich silicon germanium mid-infrared polarization rotator with mode hybridization flattening

Ultra-wideband Ge-rich silicon germanium mid-infrared polarization rotator with mode hybridization flattening

Introduction

The mid-infrared (mid-IR) spectral region comprising the wavelength range from 2 to 20 µm is of significant interest for many applications, including medical diagnosis [START_REF] Seddon | A prospective for new mid-infrared medical endoscopy using chalcogenide glasses[END_REF], biosensing [START_REF] Yu | Mid-IR biosensor: detection and fingerprinting of pathogens on gold island functionalized chalcogenide films[END_REF], astronomy [START_REF] Labadie | Mid-infrared guided optics: a perspective for astronomical instruments[END_REF], security [START_REF] Han | On-chip midinfrared gas detection using chalcogenide,glass waveguide[END_REF], etc. This is particularly because unique molecular and biological absorption features can be found in this spectral range. Commercially available mid-IR spectroscopic systems are bulky and expensive and integrated solutions are being investigated to develop miniaturized, low cost, mass produced and broadband systems. In this context, numerous platforms have been investigated: III-V materials [START_REF] Gilles | Mid-infrared wavelength multiplexer in InGaAs/InP waveguides using a Rowland circle grating[END_REF][START_REF] Roux | Low-loss orientationpatterned GaSb waveguides for mid-infrared parametric conversion[END_REF], chalcogenide glasses [START_REF] Lin | Demonstration of high-Q mid-infrared chalcogenide glass-on-silicon resonators[END_REF][START_REF] Gutierrez-Arroyo | Optical characterization at 7.7 µm of an integrated platform based on chalcogenide waveguides for sensing applications in the mid-infrared[END_REF] and a family of Si compatible platforms such as silicon on insulator (SOI) [START_REF] Mashanovich | Low loss silicon waveguides for the mid-infrared[END_REF], suspended Si [START_REF] Penadés | Suspended silicon waveguides for long-wave infrared wavelengths[END_REF], Si on sapphire [START_REF] Baehr-Jones | Silicon-on-sapphire integrated waveguides for the mid-infrared[END_REF][START_REF] Zou | Mid-infrared holey and slotted photonic crystal waveguides in silicon-on-saphire for chmical warfare simulant detection[END_REF], Germanium (Ge) on Si [START_REF] Nedeljkovic | Germanium-on-silicon waveguides operating at midinfrared wavelengths up to 8.5 μm[END_REF][START_REF] Malik | Germanium-onsilicon mid-infrared arrayed waveguide grating multiplexers[END_REF][START_REF] Malik | Germanium-on-Silicon Planar Concave Grating Wavelength (de)multiplexers in the mid-Infrared[END_REF][START_REF] Malik | Ge-on-Si and Ge-on-SOI thermo-optic phase shifters for the mid-infrared[END_REF], Silicon Germanium (SiGe) [START_REF] Brun | Low loss SiGe graded index waveguides for mid-IR applications[END_REF][START_REF] Sinobad | Mid-infrared octave spanning supercontinuum generation to 8.5 μm in silicon-germanium waveguides[END_REF] and Ge-rich SiGe on graded buffer [START_REF] Ramírez | Graded SiGe waveguides with broadband low-loss propagation in the mid infrared[END_REF][START_REF] Ramírez | Ge-rich graded-index Si 1x Ge x waveguides with broadband tight mode confinement and flat anomalous dispersion for nonlinear mid-infrared photonics[END_REF][START_REF] Vakarin | Ultra-wideband Ge-rich silicon germanium integrated Mach-Zehnder interferometer for midinfrared spectroscopy[END_REF][START_REF] Vakarin | Wideband Ge-Rich SiGe Polarization-Insensitive Waveguides for Mid-Infrared Free-Space Communications[END_REF][START_REF] Liu | Mid-infrared sensing between 5.2 and 6.6 µm wavelengths using Ge-rich SiGe waveguides[END_REF][START_REF] Liu | Integrated broadband dual-polarization Ge-rich SiGe mid-infrared Fourier-transform spectrometer[END_REF][START_REF] Serna | Nonlinear Properties of Ge-rich Si 1-x Ge x Materials with Different Ge Concentrations[END_REF][START_REF] Marris-Morini | Germanium-based integrated photonics from near-to mid-infrared applications[END_REF]. A plethora of photonic components has been implemented on these platforms: waveguides, cavities [START_REF] Lin | Demonstration of high-Q mid-infrared chalcogenide glass-on-silicon resonators[END_REF], Mach-Zehnder interferometers and spectrometers [START_REF] Vakarin | Ultra-wideband Ge-rich silicon germanium integrated Mach-Zehnder interferometer for midinfrared spectroscopy[END_REF][START_REF] Liu | Integrated broadband dual-polarization Ge-rich SiGe mid-infrared Fourier-transform spectrometer[END_REF], (de)multiplexers [START_REF] Malik | Germanium-onsilicon mid-infrared arrayed waveguide grating multiplexers[END_REF], reconfigurable circuits [START_REF] Malik | Germanium-on-Silicon Planar Concave Grating Wavelength (de)multiplexers in the mid-Infrared[END_REF] and modulators [START_REF] Cao | High-speed silicon modulators for the 2 μm wavelength band[END_REF], to name a few. Moreover, the implementation of light sources has been studied and supercontinuum generation has successfully been implemented on different platforms [START_REF] Kou | Mid-IR broadband supercontinuum generation from a suspended silicon waveguide[END_REF][START_REF] Yang | Mid-IR supercontinuum generated in low-dispersion Ge-on-Si waveguides pumped by sub-ps pulses[END_REF][START_REF] Sinobad | Mid-infrared octave spanning supercontinuum generation to 8.5 μm in silicon-germanium waveguides[END_REF]. Among Si compatible materials, SiGe and Ge-rich SiGe alloys are of particular interest because of their large transparency window in mid-IR [START_REF] Ramírez | Graded SiGe waveguides with broadband low-loss propagation in the mid infrared[END_REF][START_REF] Vakarin | Ultra-wideband Ge-rich silicon germanium integrated Mach-Zehnder interferometer for midinfrared spectroscopy[END_REF][START_REF] Liu | Integrated broadband dual-polarization Ge-rich SiGe mid-infrared Fourier-transform spectrometer[END_REF] and strong nonlinear optical effect [START_REF] Serna | Nonlinear Properties of Ge-rich Si 1-x Ge x Materials with Different Ge Concentrations[END_REF]. In this work, we exploit Ge-rich SiGe alloys to build an ultra-wideband polarization rotator. While various photonic building blocks have been developed for the mid-IR, comparatively less attention has been paid to the on-chip polarization control [START_REF] Chiang | Broadband mid-infrared polarization rotator based on optically addressable LCs[END_REF][START_REF] Guo | Mid-Infrared Silicon-on-Sapphire Polarization Rotator[END_REF][START_REF] He | Mid-infrared polarisation rotator based on an asymmetric Ge-strip-on-si waveguide[END_REF]. For example, polarization management is important for free space communications where the polarization of the incoming signal is not known [START_REF] Vakarin | Wideband Ge-Rich SiGe Polarization-Insensitive Waveguides for Mid-Infrared Free-Space Communications[END_REF]. Polarization rotators based on different approaches are widely used in classical telecommunication bands [START_REF] Xu | Ultra-broadband silicon polarization splitter-rotator based on the multi-mode waveguide[END_REF][START_REF] Chen | Highly efficient silicon optical polarization rotators based on mode order conversions[END_REF][START_REF] Fan | A compact, broadband slot waveguide polarization rotator[END_REF][START_REF] Majumder | Ultra-compact polarization rotation in integrated silicon photonics using digital metamaterials[END_REF]. The polarization rotator is also an important component for polarization agnostic photonic components such as arrayed waveguide gratings [START_REF] Takahashi | Polarization-insensitive arrayed-waveguide grating wavelength multiplexer on silicon[END_REF] for on-chip signal processing. Furthermore, using both orthogonal polarizations on-chip appears advantageous as photonic components can be designed with ultimate performance depending on the polarization of light, while nonlinear processes such as harmonic generation can take advantage of polarization diversity for mode phase matching. In this paper, first we introduce a new approach and corresponding design rules to obtain a wideband polarization rotator. The performance of the device is first estimated with an analytical approach based on modal analysis and then optimized with 3D eigen mode expansion (EME) propagation simulations performed with a commercial solver [START_REF] Solutions | Lumerical Solutions Inc[END_REF]. Finally, a tapered geometry is introduced, yielding an ultra-wideband operation and a large fabrication tolerance of ± 100 nm.

Broadband rotator waveguide

The mid-IR polarization rotator is designed for the Ge-rich SiGe platform shown in Fig. 1(a), with a top 2-µm-thick Si 0.2 Ge 0.8 layer and an 11-µm-thick graded index transition to the Si substrate, with Ge concentration decreasing from 0.79 to 0. Fabrication-wise the graded index can be implemented by means of epitaxial growth where the Ge content is tuned in the buffer all along the growth direction [START_REF] Ramírez | Graded SiGe waveguides with broadband low-loss propagation in the mid infrared[END_REF][START_REF] Vakarin | Ultra-wideband Ge-rich silicon germanium integrated Mach-Zehnder interferometer for midinfrared spectroscopy[END_REF][START_REF] Vakarin | Wideband Ge-Rich SiGe Polarization-Insensitive Waveguides for Mid-Infrared Free-Space Communications[END_REF][START_REF] Liu | Integrated broadband dual-polarization Ge-rich SiGe mid-infrared Fourier-transform spectrometer[END_REF][START_REF] Liu | On-chip Bragg grating waveguides and Fabry-Perot resonators for long-wave infrared operation up to 8.4 µm[END_REF][START_REF] Ramírez | Broadband integrated racetrack ring resonators for long-wave infrared photonics[END_REF]. Several photonic building blocks and devices have been demonstrated in this platform, including wideband waveguides [START_REF] Ramírez | Graded SiGe waveguides with broadband low-loss propagation in the mid infrared[END_REF], ultrawideband Mach-Zehnder interferometers [START_REF] Vakarin | Ultra-wideband Ge-rich silicon germanium integrated Mach-Zehnder interferometer for midinfrared spectroscopy[END_REF][START_REF] Vakarin | Wideband Ge-Rich SiGe Polarization-Insensitive Waveguides for Mid-Infrared Free-Space Communications[END_REF] and Fourier-transform waveguide spectrometers [START_REF] Liu | Integrated broadband dual-polarization Ge-rich SiGe mid-infrared Fourier-transform spectrometer[END_REF], Fabry-Perot [START_REF] Liu | On-chip Bragg grating waveguides and Fabry-Perot resonators for long-wave infrared operation up to 8.4 µm[END_REF] and racetrack resonators [START_REF] Ramírez | Broadband integrated racetrack ring resonators for long-wave infrared photonics[END_REF].

Passive integrated polarization rotators typically rely on one of the three approaches: mode evolution [START_REF] Watts | Integrated mode-evolution-based polarization rotators[END_REF], cross polarization coupling based on a directional coupler [START_REF] Fei | Ultracompact polarization splitter-rotator based on an asymmetric directional coupler[END_REF], and rotator waveguide [START_REF] Alonso-Ramos | Polarization rotator for InP rib waveguide[END_REF]. Due to the ultralow birefringence of our Ge-rich SiGe waveguides [START_REF] Vakarin | Wideband Ge-Rich SiGe Polarization-Insensitive Waveguides for Mid-Infrared Free-Space Communications[END_REF], the first two approaches would be difficult to implement, and we use the rotator waveguide strategy instead. Our polarization rotator waveguide is based on a two-step geometry (see Fig. 1(a)), yielding hybrid modes with tilted optical axis while obviating the need for slanted etching or multi-trench geometries. We denote W and H the width and depth of the first etch step and K and J the width and depth of the second etch step in the rotator waveguide. The rotator waveguide is characterized by the optical axis tilt, φ, and the half-beat length, L π :

2 2 | | arc tan( ), | | Ez d Ey d φ Ω Ω Ω = Ω   (1) 1 2 , 2 | | eff eff L n n π λ = - (2)
where E z and E y are vertical and horizontal electric field components, Ω is the modal area, and n eff1 and n eff2 are the effective indices of the first two rotator waveguide modes. From these two parameters, the polarization conversion efficiency is calculated:

2 2 sin (2 )sin ( ), 2 R L PCE L π π φ = ( 3
)
where L R is the length of the rotator waveguide section. From Eq. (3) it follows that perfect rotation, i.e. PCE = 1, is achieved when φ = 45° and the rotator length equals the half-beat length L R = L π . The length of the rotator is fixed after fabrication. Hence, the bandwidth of the device is primarily determined by the wavelength dependence of the optical axis tilt and the half-beat length. Here we propose a novel design method that harnesses adiabatic index variation in the graded Ge-rich SiGe platform to minimize the wavelength dependence of the rotator waveguide, yielding an ultra-wideband operation. Here we propose a new design method to maximize the bandwidth of the rotator. The method comprises the following three main steps: (i) the optical axis tilt, φ, is calculated for a given wavelength as a function of the rotator waveguide geometry by selecting different combinations of W and H while scanning the parameters K and J, (ii) for each pair W-H an hybridization curve is defined following the K-J points yielding φ = 45° (see Fig. 1(b)) and fitted with a generalized hyperbola equation, and finally (iii) the wavelength dependence of φ and L π is calculated for the points near the vertex of each hyperbola, identifying the point corresponding to the broadest wavelength response as our optimum design.

If the proposed L-shaped waveguide had the same bottom and top cladding, then the hybridization curves would follow a linear trend, corresponding to the points where the geometry is perfectly symmetric along a 45-degree-tilted axis. However, the geometry considered in our work has a highly inhomogeneous cladding, with a top air cladding (n air = 1) and a bottom SiGe cladding (n SiGe >3). Then achieving 45-degree-titled hybrid modes requires balancing waveguide asymmetry and mode confinement distribution. This results in hybridization curves that do not follow the ideal linear trend. By calculating the influence of device geometric parameters on φ and L π we have found that the hybridization K-J curves for φ = 45° follow a generalized hyperbola equation:

1 , ( 1)

d g K e mJ = - + (4)
where d, e, g and m are the fitting coefficients. We also found that for each hyperbola the bandwidth is maximized for points near the vertex. Exploiting this behavior, it is possible to substantially reduce the number of geometries to be analyzed to find the optimal design. Specifically, geometrical parameters of the rotator waveguide are only sparsely scanned at a fixed wavelength to find the proper fitting coefficients for the hyperbola, while the full wavelength scan are only performed for points near the vertex. To further illustrate this method, in Fig. 1(b) we plot the hybridization curves for four different combinations of W and H. In these examples named A, B, C and D the width W varies while H is fixed to 4 µm to assure the compatibility with other Ge-rich SiGe photonic building blocks [START_REF] Ramírez | Graded SiGe waveguides with broadband low-loss propagation in the mid infrared[END_REF][START_REF] Vakarin | Ultra-wideband Ge-rich silicon germanium integrated Mach-Zehnder interferometer for midinfrared spectroscopy[END_REF][START_REF] Vakarin | Wideband Ge-Rich SiGe Polarization-Insensitive Waveguides for Mid-Infrared Free-Space Communications[END_REF][START_REF] Liu | Integrated broadband dual-polarization Ge-rich SiGe mid-infrared Fourier-transform spectrometer[END_REF][START_REF] Liu | On-chip Bragg grating waveguides and Fabry-Perot resonators for long-wave infrared operation up to 8.4 µm[END_REF][START_REF] Ramírez | Broadband integrated racetrack ring resonators for long-wave infrared photonics[END_REF]. It can be observed that the hybridization curves are well-fitted by the hyperbolas. In Figs. 1(c) and 1(d) we compare the wavelength dependence of φ and L π for different geometries, within the hybridization curves. First, if we compare the three geometries within hyperbola C (c 1 , c 2 , c 3 ) we can see that the point c 2 , near the vertex, yields the largest bandwidth. On the other hand, it is apparent that the point c 2 provides a substantially wider response than a 2 (near the vertex in hyperbola A). The extremes of each hyperbola correspond to highly asymmetric waveguide configurations with large etching depth, J, and small width, K. The vertex of each hyperbola corresponds to the region where the modes are hybridized with the most symmetric geometry. This symmetry optimization is directly related to the bandwidth increase. In ideal rotator waveguides with geometric symmetry along the 45 degree axis, perfect hybridization is ensured by geometrical symmetry, providing a broad wavelength operation. In our case, with highly asymmetric claddings, a specific mode confinement distribution is required to achieve perfect hybridization. This confinement distribution varies with the wavelength. Then, generally speaking, the bandwidth of the hybridization is proportional to the level of geometrical symmetry and inversely proportional to the variation of the mode confinement with the wavelength. That is the reason why the bandwidth is maximized for the vertex of a given hyperbola and why different hyperbolas lead to substantially different bandwidth behavior. Hence, by comparing geometries near the vertex of each hyperbola, a solution converging to optimal design can be found. Following the proposed methodology and considering W<6 µm and H = 4 µm for good fundamental mode confinement while avoiding highly multimode behavior and compatibility with other photonic building blocks on Ge-rich SiGe platform, the optimal geometry was found for W = 5 µm, H = 4 µm, J = 1.73 µm and K = 3.09 µm (point c 2 in Fig. 1(b)). To support this statement and illustrate the systematic bandwidth optimization at the vertex we calculate the PCE for all points along hyperbolas A, B, C and D and we plot for these hyperbolas the bandwidth for which the PCE>90% (Fig. 2(a)) and PCE spectral evolution for the vertices a 2 , b 2 , c 2 and d 2 (Fig. 2(b)). As shown on Fig. 2(a) the bandwidth is systematically maximized at each vertex of hyperbolas A, B, C and D. Moreover, the best results are given by the vertex c 2 . This better spectral behavior can be clearly seen by comparing the spectral evolution of all the vertices (Fig. 2(b)): the bandwidth increases from the vertex a 2 up to the c 2 (optimal design that we selected) and drops after with the vertex d 2 . As shown in Fig. 2(b), the optimized design provides a calculated PCE>90% for a bandwidth higher than 3.8 µm, i.e. more than 18-fold improvement compared with state-of-the-art broadband near infrared polarization rotators [START_REF] Xiong | Fabrication tolerant and broadband polarization splitter and rotator based on a taper-etched directional coupler[END_REF]. In summary, the remarkably large bandwidth of our polarization rotation is the result of: i) optimization of waveguide symmetry by operating in the vertex of the hybridization hyperbola and ii) minimization of the confinement variation with wavelength by harnessing the adiabatic index variation in the graded SiGe geometry. From calculation point of view, the number of tested hyperbolas depends on: i) fixed dimension range in W and H, H being usually fixed by compatibility with other photonic components, (ii) the sensitivity of the waveguide to dimensional variations which is linked to the chosen photonic platform. It is important to point out that the proposed design strategy is a substantial improvement compared to the classic design approach where the device performances have to be scanned for each combination of geometrical parameters.

Design of complete polarization rotator

We now consider the full polarization rotator schematically shown in Fig. 3(a), comprising input and output waveguides, tapers and rotator waveguide section. The input and output waveguides and the rotator waveguide section have a width of W = 5 µm and etch depth H = 4 µm. The tapers, of length Lt, are designed to minimize insertion loss due to mode mismatch between the input/output waveguide and the rotation waveguide. The rotation section has a length L R and the cross-section optimized in the design procedure outlined in the previous section. We studied the influence of the taper length on the rotator performance by means of EME propagation simulations. We calculate extinction ratio (ER) and insertion loss (IL) as following:

where T TE-TM denotes the calculated transmittance for transverse electric (TE) to transverse magnetic (TM) polarization and T TE-TE the transmittance for TE to TE polarization, from the input port to the output port. In Figs. 3(b) and 3(c) we show the ER and IL calculated as a function of the taper length, Lt, for a wavelength of 7.5 µm. Note that propagation through the taper will induce a phase shift that needs to be compensated in the rotator section. Hence, for each value of the taper length, the length of the rotation region, L R , was adjusted to maximize the ER. As can be seen, increasing the taper length reduces the insertion losses. On the other hand, long tapers result in an uneven excitation of the hybrid modes in the rotator waveguide that degrades the attainable ER. Therefore, we chose Lt = 100 µm and L R = 1660 µm as a tradeoff to ensure ER above 25 dB and IL below 0.1 dB. We have calculated the wavelength dependence of the ER and IL for the proposed rotator, with Lt = 100 µm and L R = 1660 µm. We considered a propagation loss of 2-3 dB/cm, determined in a previous experimental study [START_REF] Ramírez | Graded SiGe waveguides with broadband low-loss propagation in the mid infrared[END_REF], including its spectral dependence. As shown in Figs. 4(a) and 4(b) the polarization rotator yields a peak ER of 30 dB with IL below 0.5 dB. The rotator shows a remarkable broadband operation with an extinction ratio of 15 dB and 10 dB for bandwidths of 2 µm and 3.8 µm, respectively while the insertion loss is lower than 1 dB in the wavelength range from 5 µm to 9µm. This is a substantial improvement compared to previously reported polarization rotators in the near-infrared [START_REF] Watts | Integrated mode-evolution-based polarization rotators[END_REF][START_REF] Alonso-Ramos | Polarization rotator for InP rib waveguide[END_REF][START_REF] Xiong | Fabrication tolerant and broadband polarization splitter and rotator based on a taper-etched directional coupler[END_REF].

For comparison, in Fig. 4, we also show the ER for the rotator waveguide geometry a 2 (see Fig. 1(b)), with Lt = 30 µm and L R = 2000 µm. This geometry leads to a maximum ER of 42 dB which is higher than for the flattened hybridization geometry. However, this geometry yields a substantially narrower wavelength range, with 15 dB and 10 dB ER bandwidth of only 240 nm and 310 nm, respectively.

Optimizing fabrication tolerances and bandwidth by tapered design

An important parameter to be considered for a polarization rotator is its tolerance to fabrication errors. Considering all possible waveguide dimensions variations, we found that the critical parameter affecting the ER is the etch depth J. The rotator designed in the previous section requires etch depth fabrication accuracy better than ± 50 nm for ER>10 dB over a bandwidth of 3.8 µm. While tolerances of ± 50 nm are well within the capabilities of standard fabrication processes for silicon wire waveguides, it is difficult to achieve this accuracy for Ge-rich Si-Ge platform with the etch depth of a few micrometers. To overcome this limitation, we propose to implement a rotator waveguide section with a tapered width K N (see Fig. 5(a)). Tapered rotators are known to improve fabrication tolerances and bandwidth because the width variation in the tapered rotation region partially compensates the deviations from the geometry with the tilt angle of φ = 45° [START_REF] Xiong | Fabrication tolerant and broadband polarization splitter and rotator based on a taper-etched directional coupler[END_REF][START_REF] Ding | Fabrication tolerant polarization splitter and rotator based on a tapered directional coupler[END_REF]. Here we start with the structure designed in the previous sections and implement tapering by applying an offset -f in the beginning of the rotation region and + f at the end of the rotation region K N . The taper length Lt is maintained 100 µm. The EME propagation simulations have been performed for f = 100 nm, 200 nm, 300 nm and 400 nm, for wavelengths between 4 µm and 9 µm. For each offset value, the rotator length, L R , is optimized to maximize the ER. The value of f = 200 nm has been chosen as a good compromise between ER and tolerances to fabrication errors. The wavelength dependence of the optimized design (f = 200 nm, L R = 1695 µm) is shown in Fig. 5(b). This design yields a maximum ER of 27 dB at the peak wavelength, an ER higher than 15 dB in a bandwidth of 1.8 µm (between 6.5 and 8.3 µm wavelength) and an ER higher than 10 dB for the wavelength range 4.3 -8.65 µm. Interestingly, this configuration can support fabrication errors up to ± 100 nm on all dimensions of the rotation region while ensuring an operational bandwidth of at least 600 nm (for ER>15 dB), (ii) an ER>10 dB for the wavelength range 5 -8 µm. This is shown in Fig. 5(b), specifically the ER as a function of the wavelength for etch depth errors of ± 100 nm (from the nominal value J N = 1.73 µm). It can also be seen that the impact of under-etching affects more the 15 dB bandwidth (i.e. the spectral range with ER>15 dB) than over-etching. This is a two-fold tolerance improvement compared to the non-tapered design. If we compare the bandwidth of the tapered rotator design to the original, we gain 550 nm on 10 dB bandwidth which is already bigger than 3.5 µm and we lose 200 nm on 15 dB bandwidth. Consequently, the tapered design offers two advantages: simultaneous improvement of bandwidth and of fabrication tolerance.

Conclusion

In summary, we designed an ultra-wideband polarization rotator based on Ge-rich SiGe waveguide platform in the mid-IR wavelength range with an unprecedented bandwidth. An extinction ratio of at least 15 dB is obtained over a wavelength range of 2 µm, or 10 dB over 3.8 µm wavelength range. This was achieved by hybridization flattening, making optical axis tilt wavelength independent over a broad spectral range by judicious optimization of geometrical parameters. The geometry was optimized by modal analysis and EME simulations were performed to evaluate the device performances. The polarization rotator tolerance to fabrication errors has been studied and a tapered Ge-rich SiGe rotator with fabrication tolerances up to ± 100 nm was designed. The tapered rotator preserves the ultrawideband behavior with losses below 1 dB in the full operational range with an ER exceeding 15 dB (10 dB) within 1.8 µm (3 µm) bandwidth. This study paves the way for implementing on-chip polarization management in the mid infrared spectral region over a broadband wavelength range.

Fig. 1 .

 1 Fig. 1. (a) Ge-rich SiGe waveguide cross section used for polarization rotation, K and J are the width and the height, respectively, of the etched trench. The mode profile corresponds to K = 3.09, J = 1.73 µm and W = 5, H = 4 µm. (b) Mode hybridization curves: K as a function of J for φ = 45°. The points a 2 , b 2 , c 2 , d 2 represent the vertices of the hyperbolas A, B, C and D respectively. The extreme geometries on the hyperbola C are represented by c 1 and c 3 points. (c) Tilt angle φ evolution as a function of the wavelength: for configuration a 2 (red curve), b 2 (blue curve), c 2 (green curve), d 2 (black curve) and for geometries c 1 and c 3 (green dashed and green square, respectively). (d) Lπ as a function of the wavelength for a 2 (red curve), b 2 (blue curve), c 2 (green curve), d 2 (black curve) and for geometries c 1 and c 3 (green dashed and green square, respectively).

Fig. 2 .

 2 Fig. 2. (a) Bandwidth evolution for hyperbolas A, B, C and D: red, blue, green and black curves respectively. (b) PCE as a function of wavelength for the vertices a 2 , b 2 , c 2 and d 2 : red, blue, green and black curves respectively. The flattened configuration corresponds to the vertex c 2 .

Fig. 3 .

 3 Fig. 3. (a) Schematic view of the polarization rotator. (b) Extinction ratio as a function of the taper length calculated for 7.5 µm wavelength. (c) Insertion loss as a function of the taper length calculated for 7.5 µm wavelength.

Fig. 4 .

 4 Fig. 4. (a) Polarization rotator extinction ratio as a function of the wavelength. Broadband design (geometry c 2 , green curve). Red curve shows the ER of a rotator with no hybridization flattening (geometry a 2 ). Cyan dashed and solid lines indicate ER = 15 dB and ER = 10 dB, respectively. (b) Insertion loss in the rotator as a function of the wavelength for the optimized geometry c 2 .

Fig. 5 .

 5 Fig. 5. (a) Schematics of the tapered polarization rotator. K N = 3.09µm. (b) Extinction ratio as a function of the wavelength for nominal tapered rotator with f = 200 nm (green curve) and the effect of over-etching (orange curve) and under-etching (blue curve).

Funding

European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (639107-INsPIRE).