Mathieu Gonod
email: mgonod@cgfl.fr

Miguel Angel Suarez

Carlos Chacon Avila

Vage Karakhanyan

Clément Eustache

Julien Crouzilles

Samir Laskri

Jean-François Vinchant

Léone Aubignac

Thierry Grosjean
email: thierry.grosjean@univ-fcomte.fr

Characterization of a miniaturized scintillator detector for time-resolved treatment monitoring in HDR-brachytherapy

Introduction

High dose-rate brachytherapy (HDR-BT) [START_REF] Crook | Hdr prostate brachytherapy[END_REF][START_REF] Viswanathan | American brachytherapy society consensus guidelines for locally advanced carcinoma of the cervix. part ii: high-dose-rate brachytherapy[END_REF] delivers high dose of radiation in few fractions to a tumor while sparing the surrounding organs at risk. The prescribed dose is delivered from a stepping radioactive source of high activity which sequentially treats the target volume, thereby involving steep dose gradients in space and time. As a consequence, HDR-BT imposes stringent operational conditions to achieve a treatment delivery that reliably follows the Treatment Planning System (TPS) calculation (Damore et al 2020[START_REF] Kim | Dosimetric impact of prostate volume change between ct-based hdr brachytherapy fractions[END_REF][START_REF] Simnor | Justification for inter-fraction correction of catheter movement in fractionated high dose-rate brachytherapy treatment of prostate cancer[END_REF][START_REF] Viswanathan | American brachytherapy society consensus guidelines for locally advanced carcinoma of the cervix. part ii: high-dose-rate brachytherapy[END_REF][START_REF] Mayadev | A failure modes and effects analysis study for gynecologic high-dose-rate brachytherapy[END_REF][START_REF] Kertzscher | In vivo dosimetry: trends and prospects for brachytherapy[END_REF]. Any deviations of the dwell time or position of the source regarding the planned values can lead to serious errors in the dose distribution [START_REF] Kim | Dosimetric impact of prostate volume change between ct-based hdr brachytherapy fractions[END_REF][START_REF] Pantelis | Evaluation of a tg-43 compliant analytical dosimetry model in clinical 192ir hdr brachytherapy treatment planning and assessment of the significance of source position and catheter reconstruction uncertainties[END_REF][START_REF] Tanderup | Consequences of random and systematic reconstruction uncertainties in 3d image based brachytherapy in cervical cancer[END_REF], Buus et al 2018). Afterloader malfunctions, erroneous source activity, as well as movements of organs or applicators, can also be responsible for wrong dose administration.

To guarantee that a dose distribution is delivered accordingly to expectations, time-resolved in vivo dosimetry (IVD) has been envisioned for time-resolved treatment verification and error detection [START_REF] Fonseca | In vivo dosimetry in brachytherapy: requirements and future directions for research, development, and clinical practice[END_REF], Verhaegen et al 2020). Among time-resolved IVD strategies [START_REF] Fonseca | In vivo dosimetry in brachytherapy: requirements and future directions for research, development, and clinical practice[END_REF], scintillating material coupled to an optical fiber has demonstrated performances in the time-resolved monitoring of the dose rate [START_REF] Lambert | A plastic scintillation dosimeter for high dose rate brachytherapy[END_REF] In the steep spatial and temporal dose gradients of a stepping HDR-BT source, time-resolved IVD faces the challenge of combining high spatial resolution and high readout rate. High spatial resolution limits volume averaging effect near the radioactive source. As an example, the dose rate varies by 18 % across a 1-mm broad detector positioned at 1 cm from the center of an Ir-192 source. This dose rate variation reduces to about 4 % across a 0.24-mm broad detector. At the same time, a readout rate of at least 10 Hz is necessary to identify errors in a treatment with a dwell time as short as 0.1 s. Shrinking the detection volume while increasing readout speed is a challenge as the signal-to-noise ratio (SNR) of the detection process is strongly degraded, leading to higher uncertainties in the treatment monitoring.

The replacement of plastic scintillators by inorganic scintillators (of higher luminescence yield) at the tip of a 0.5-1mm outer diameter plastic fiber allowed for dramatic SNR enhancement in IVD for BT [START_REF] Wang | Implementation of gan based real-time source position monitoring in hdr brachytherapy[END_REF][START_REF] Guiral | Design and testing of a phantom and instrumented gynecological applicator based on gan dosimeter for use in high dose rate brachytherapy quality assurance[END_REF][START_REF] Andersen | Characterization of a fiber-coupled luminescence dosimetry system for online in vivo dose verification during brachytherapy[END_REF][START_REF] Johansen | Time-resolved in vivo dosimetry for source tracking in brachytherapy[END_REF][START_REF] Belley | Real-time dose-rate monitoring with gynecologic brachytherapy: Results of an initial clinical trial[END_REF][START_REF] Johansen | Dwell time verification in brachytherapy based on time resolved in vivo dosimetry[END_REF][START_REF] Kertzscher | Inorganic scintillation detectors for 192ir brachytherapy[END_REF], Jorgensen et al 2021a). As a result, time-resolved IVD has been recently reported with a detection volume as small as 0.08 mm 3 (0.4 mm large and 0.5 mm long) and at a 20 Hz readout rate(Jorgensen et al 2021a). Despite their energy dependence which requires corrections in a direct dose rate measurement (Jorgensen et al 2021a), such detectors can lead to accurate estimation of the instant dose via source tracking methods [START_REF] Guiral | Design and testing of a phantom and instrumented gynecological applicator based on gan dosimeter for use in high dose rate brachytherapy quality assurance[END_REF].

As an alternative approach, the concept of a miniaturized scintillator detector (MSD) has been recently introduced from the harnessing of inorganic scintillators to a narrow 125-µm diameter silica fiber with a microfabricated photonic antenna [START_REF] Suarez | Miniaturized fiber dosimeter of medical ionizing radiations on a narrow optical fiber[END_REF]. A MSD with a detection volume as small as 0.016 mm 3 (0.25-mm diameter and 0.48-mm long ellipsoid) has been demonstrated in external beam radiotherapy [START_REF] Gonod | Miniaturized scintillator dosimeter for small field radiation therapy[END_REF]. The MSD has been shown to overcome the stem effect, and despite its water nonequivalence (both scintillator and fiber), it minimizes electron fluence perturbation due to its extreme compactness.

In this paper, we demonstrate the potential of the MSD concept for source tracking and dose rate monitoring in HDR-BT. Our detector consists of a 0.017 mm 3 detection cell (0.28 mm diameter and 0.43 mm long ellipsoid) attached to the end of a 110-µm diameter silica optical fiber connected to a standard sCMOS camera. After specification of the MSD in terms of SNR, linearity, repeatability and temperature dependence, the detector is calibrated for source tracking. Calibration process is decorrelated from the potential source positioning errors of the afterloader by displacing the detector, rather than the source, with a high precision 2D translation stage. Finally, the MSD is tested in clinical HDR-BT conditions. Dwell position, dwell time and dose rate verification are performed in this study.

Material and methods

Fiber probe

The MSD shown in Fig. 1(a) consists of a 0.017 mm 3 scintillating cell (0.28-mm diameter and 0.43 mm long ovoid) attached to the end of a 110-µm outer diameter (100-µm core diameter) and 2-meter long biocompatible fiber from SEDI-ATI. Prior to scintillator-tofiber attachment, the fiber tip is specifically tapered to produce a "leaky-wave" photonic antenna [START_REF] Kraus | Antennas and wave propagation[END_REF][START_REF] Suarez | Miniaturized fiber dosimeter of medical ionizing radiations on a narrow optical fiber[END_REF] aimed at enhancing the optical coupling efficiency between the scintillating material and the fiber. Terbium-doped gadolinium oxysulfide (Gd 2 O 2 S:Tb) is chosen as the scintillator for our MSD. This material is known to emit visible light upon exposure with good efficiency, stability and linearity at room temperature [START_REF] Qin | Embedded structure fiber-optic radiation dosimeter for radiotherapy applications[END_REF][START_REF] Hu | Characterization of fiber radiation dosimeters with different embedded scintillator materials for radiotherapy applications[END_REF][START_REF] Alharbi | Dosimetric characterization of an inorganic optical fiber sensor for external beam radiation therapy[END_REF]. One other advantage of Gd 2 O 2 S:Tb is the very low temperature dependence of its luminescence intensity (O'Reilly et al 2020). Gd 2 O 2 S:Tb belongs to the family of inorganic (water nonequivalent) scintillators, whose superior X-ray to light conversion efficiency is in favor of a higher accuracy in IVD, at the expense of an energy dependence of the detector that need to be corrected [START_REF] Wang | Implementation of gan based real-time source position monitoring in hdr brachytherapy[END_REF][START_REF] Guiral | Design and testing of a phantom and instrumented gynecological applicator based on gan dosimeter for use in high dose rate brachytherapy quality assurance[END_REF][START_REF] Andersen | Characterization of a fiber-coupled luminescence dosimetry system for online in vivo dose verification during brachytherapy[END_REF][START_REF] Johansen | Time-resolved in vivo dosimetry for source tracking in brachytherapy[END_REF][START_REF] Belley | Real-time dose-rate monitoring with gynecologic brachytherapy: Results of an initial clinical trial[END_REF][START_REF] Johansen | Dwell time verification in brachytherapy based on time resolved in vivo dosimetry[END_REF][START_REF] Kertzscher | Inorganic scintillation detectors for 192ir brachytherapy[END_REF], Jorgensen et al 2021a). The optical fiber is surrounded by an opaque 0.9 mm black hytrel cladding to be protected from background light of the treatment room. Both the scintillating cell and the last 1.5 cm of the fiber extend beyond the extremity of the opaque cladding to be directly in contact to the phantom.

Optical readout

The optical signal from the MSD is recorded with a standard sCMOS camera (Zyla model from Andor Technology) whose sensitivity is maximum near the emission wavelength of the scintillators. The camera is placed in the control room, it is connected to the detector via a ten-meter fiber patch cable (with FC/PC connectors). To image the fiber output, the camera is equipped with a 35 mm objective (Fujinon HF35SA) and operates with an integration time of 0.08 s. The camera is driven under Labview in a Windows environment. Each image of the fiber output is automatically integrated over a specific region of interest, leading to an analog 12.5 Hz electric signal directly proportional to the MSD optical signal. Prior to photodetection, the Cerenkov signal present in the fiber upon exposure (the stem effect) is removed with a narrow chromatic filter (544/24 nm band pass filter from Semrock) positioned in front of the camera [START_REF] Gonod | Miniaturized scintillator dosimeter for small field radiation therapy[END_REF]. We verified that the detected stem effect is negligible in the presence of the chromatic filter, by use of a scintillator-free tapered fiber moving close to the source.

Brachytherapy system

HDR irradiation is realized with a MicroSelectron afterloader equipped with a 9.1 Ci Ir-192 HDR source. Reference dose rates were calculated using the AAPM-TG43 protocol [START_REF] Perez-Calatayud | Dose calculation for photon-emitting brachytherapy sources with average energy higher than 50 kev: report of the aapm and estro[END_REF]. These reference values are those applied to the TPS.

Phantom

Irradiations are conducted in a 40×30×30 cm 3 water tank. Two opposite polymetyl methacrylate (PMMA) walls of the tank are perforated to allow the source catheter to cross the tank horizontally (Fig. 1(b)). Tight seals fill the opening between catheter and tank walls to avoid water leakage. To be maintained straight in water, the source catheter is fixed in a V-groove milled in a piece of solid water immersed in the water bath. The water tank ensures a 10 cm scatter material all around the source. No temperature control of the water bath is performed during experiments. Water temperature ranges from 17 • to 19 • during acquisitions. To realize detector specification and calibration, the end of the fiber probe is mounted onto a solid-water/PMMA holder connected to a high precision 2D translation stage distributed by Thorlabs (Fig. 1(c)).

In our experiments, the axes of the source catheter and fiber probe are parallel (Fig. 1(d)). The coordinate frame of the set-up is arbitrarily chosen to be linked to the detector. The origin of the frame coincides with the point-like scintillation cell of the MSD. The z-coordinate of the source is controlled either by the afterloader or one axis of the 2D translation stage used for the probe positioning. The x-coordinate is solely controlled by the second orthogonal axis of the 2D motorized stage.

Detector specification and calibration

Detector specification and calibration is realized by moving the detector around the radioactive source, which remains static within its catheter. The SNR, dose linearity and repeatability of the detector are assessed with the probe positioned 10 mm away from the source along the (0x)-axis. The linearity of the MSD is tested as a function of the absorbed dose. The repeatability of the MSD is analyzed over ten irradiations of 30 seconds. Between two exposures, the sources is moved back into the afterloader and repositioned near the detector.

To evaluate the energy dependence of the MSD, the probe is scanned over the horizontal (xz)-plane containing the source, with the source maintained at a fixed position within the catheter. During raster scan, an image is accumulated by steps of 2 mm and 1 mm along the (0x) and (0z), respectively, with an integration time at each probe position of 0.16 s. Note that, by virtue of cylindrical symmetry of the source radiation, the dose distribution in the orthogonal (xz) and (yz) planes are expected to be identical. The acquisition procedure starts by positioning the detector at 10 mm from the source. The detector is then moved away from this reference position by 37 mm along (0x) and 32.8 mm along (0z) with the motorized stage. Finally, a raster scan is launched with the probe progressively approaching the source. The resulting image spreads over 41×66 mm 2 , ranging from 6 mm to 47 mm along (0x) and -32.8 mm to 32.8 mm along (0z). The detected optical signal is converted into a dose rate using the conversion rule D exp = aS + b, where a and b are constants [START_REF] Andersen | Characterization of a fiber-coupled luminescence dosimetry system for online in vivo dose verification during brachytherapy[END_REF]. The conversion parameters a and b are chosen to minimize the difference between the measured and planned dose rates across the overall acquisition area. The planned dose rate is calculated for all source-probe distances by use of the AAPM TG-43 dose parameters (Perez et al 2012).

The detector is calibrated along 5 lines parallel to the (0z)-axis and spaced by 6, 10, 20, 30 and 40 mm from the central axis of the source catheter. To this end, the probe sequentially scans over these lines by 1-mm steps and 125 to 375 images are integrated per probe position. The scan ranges from -55 mm to +5 mm along (0z). An interpolation is then applied to the experimental profiles to achieve a 0.1-mm sampling rate.

2.6. Dwell position, dwell time and dose rate verification 2.6.1. Measurements The MSD is tested using a 22-needle treatment protocol of prostate cancer. In our study, the 22 dose deliveries are realized with the source stepping within the same catheter (Fig. 1(b)). Each treatment sequence relies on a 12-to-16 dwell position protocol based on 2.5 mm steps and dwell times ranging from 0.1 to 11 seconds. The overall treatment is repeated at 5 different spacings x between the source and the probe: 6, 10, 20, 30 and 40 mm (Fig. 1(d)). Therefore, for each source-probe spacing x, the source travels 22 times within the same catheter shown in Fig. 1(b). Each of the 22 source displacements within the catheter follows a specific dwell position sequence. Information on the phantom and experimental set-up is detailed in section 2.4.

Dwell position monitoring

At each time of the treatment monitoring, the position z of the source along the catheter (z < 0) is determined from the readout signal and the source activity using the calibration table of the detector. The sourceprobe spacing along (0x) being precisely known (with an accuracy of 6.5 µm (Thorlabs Webpage)), the source-probe distance r is deduced from x and |z| (Fig. 1(d)). The displacement time of the source between two successive dwell positions being of the order of a few tens of milliseconds [START_REF] Fonseca | Hdr 192ir source speed measurements using a high speed video camera[END_REF], the resulting rise and fall times in the staircase readout signal are described by one or two acquisition points. To ensure that these transient phases are not included in the dwell position verification, the first and last acquisition points of each dwell position are omitted.

During treatment delivery, the MSD delivers a temporal signal S(t). At each instant t = kτ , where k ∈ N and τ is the acquisition time of the camera, the instant source position is deduced from the readout signal S as follows.

First, function f k is defined as:

f k (z) = C(z) -S k , (1)
where C(z) and S k are the calibration curve and readout signal at the k th time step, respectively. z is the space coordinate along the source catheter, sampled at 0.1 mm. The stepping source being located only at negative z-coordinates z < 0 (the origin of the z-axis is bound to the point-like scintillating cell), function f k shows a single minimum across the entire range of source positions. The instant source position Z k then corresponds to the z-coordinate of the minimum of f k . The source-probe spacing x being precisely known, we can deduce the instant source-probe distance r from Z k using the formula (cf. Fig. 1(d)):

r = x. 2 + |Z k | 2 , (2)
In section 3, dwell position verification will be reported in terms of r and x, rather than x and Z k ). Coordinate Z k can however be deduced from r and x for each source position using Eq. 2.

2.6.3. Dwell time verification HDR-BT monitoring leads to staircase time traces whose edges delimit two successive dwell positions of the source. Being equal to the delay between two consecutive edges, dwell times can be estimated from edge identification in the detection signal. Convolution techniques [START_REF] Canny | A computational approach to edge detection[END_REF][START_REF] Guiral | Design and testing of a phantom and instrumented gynecological applicator based on gan dosimeter for use in high dose rate brachytherapy quality assurance[END_REF] or signal derivative algorithms [START_REF] Johansen | Dwell time verification in brachytherapy based on time resolved in vivo dosimetry[END_REF] have shown good performances in dwell time verification. Here, we develop an alternative approach which minimizes noise while avoiding time averaging process which can decrease accuracy in dwell time determination. The temporal edges of the staircase signal S are identified using function F defined as:

F i = [S i+1 -S i ] SD N i √ S i , (3)
where S i is the i th point of the time trace (i is the ratio between the measurement time and the acquisition rate) and S i+1 -S i is the difference between two consecutive points. SD N i is a standard deviation calculated over N consecutive points of the time trace, centered with respect to the i th point. In this study, N is equal to 5. √ S i quantifies the fluctuation of the detected signal due to the camera noise. The shot noise being the main limiting factor of the camera sensitivity (verified in Fig. 3 (d)), these fluctuations can be described by a Poisson statistics where the variance of the detected signal matches the signal intensity. We also have :

SD N i = 1/N i+(N -1)/2 k=i-(N -1)/2 (S k -S) 2 , (4)
where

S = 1 N i+(N -1)/2 k=i-(N -1)/2 S k , (5)

Dose rate retrieval

The dose rate (D meas ) in an individual dwell position is obtained from the measured dwell position and source activity by applying the AAPM TG-43 dose parameters (Perez et al 2012). The relative offset to the planned dose rate ∆D/D = D meas /D T P S -1 is then calculated. D T P S is the planned dose rate for each dwell position of the TPS.

Uncertainty budget

Following Ref. (ISO,1995), uncertainties are categorized in two types, namely, Type A when data is treated via any valid statistical method, and Type B which involves scientific judgment based on available information. An uncertainty u of Type B is evaluated using the expression : u = (v max -v min )/ (12), where v max and v min are the maximum and minimum values, respectively. The combination u tot of a series of uncertainties (u i , i ∈ N ) reads u tot = i u 2 i .

Results

Detector specification and calibration

The SNR, dose linearity and repeatability of the MSD are analyzed in Table 1. The SNR varies from 205 to 52 at source-probe spacings x of 6 mm and 40 mm, respectively, with a source activity of 9.1 Ci. The linearity coefficient of the MSD exceeds 0.999 for all values of x.

To investigate the energy dependence of the MSD, a 2D field map is plotted near the radioactive source (Fig. 2(a)). This image reveals the steep dose gradient of the source in 2D. From a basic signal-to-dose conversion, we find that our measurement deviates from the expected dose rate by 5.2 ± 4.7 (1SD) % across the scan area, with minimum and maximum discrepancies of 0.006% and 24.8 %, respectively (Fig. 2(b)). Therefore, a direct measurement of the dose rate with our MSD requires space-dependent correction factors.

Dwell position verification

Our study combines 1384 dwell positions with source-probe distances r (cf. discrepancy in source-probe distance is of 0.023±0.077 mm (1SD). For each value of r T P S , we also report the SD of r exp (Fig. 3(d)), i.e., the SD of the distribution of instant positions measured at a rate of 0.08s for each planned distance r T P S . We superimpose to these experimental points the theoretical SD curve obtained by converting the predicted shot noise of the camera into distance uncertainty via the calibration curves of Fig. 2(c). The shot noise is statistically described by a variance matching the amplitude of the readout signal (Poisson statistics).

Dwell time verification

The dwell time verification is performed at source-probe spacings x of 6, 10 and 20 mm, leading to an analysis of 966 dwell positions. The dwell time distribution is reported in Fig. 4(a). The fraction of the dwell times smaller than 1 second, between 1 and 5 seconds, and larger than 5 seconds is of 36%, 59% and 5%, respectively. The identification rate of the dwell positions by the MSD is reported in Table 2. 94% of the dwell positions are identified by the detector. Beyond 0.5 seconds, all dwells are successfully monitored. At the limit of 0.1 second, none are identified. The identification rate increases by a factor about 5 (from 17% to 86 %) when the dwell time increases from 0.2 to 0.3 s. Dwell times of 0.4 and 0.5 s are identified with a rate of 91% and 95%, respectively. In Fig. 4(c), we plot the difference ∆T between the measured and planned dwell times (T exp and T T P S , respectively). ∆T takes a discrete form imposed by the 0.08second integration time of the camera. The fourteen values of ∆T that exceeds 0.24 seconds (see the gray regions of Fig. 4(c)) are due to time lags in the data acquisition system, because the camera and data storage are both driven with the same computer (under Windows environment). Lags are evidenced in the time traces by a narrow time window where the noise vanishes and the signal follows a systematic artefactual temporal pattern. When the resulting artefactual points in the figure are removed, the average value of ∆T is statistically calculated to be 0.005 ± 0.060 s (1SD). We see from the inset of Fig. 4(c) that 95 % of the identified dwell times are within one integration time of the camera. Fig. 4(d) reports the SD of T exp versus the planned values T T P S . We calculate the SD of the measured dwell times at each source-probe spacing, and also the average SD for all three spacings. This allows to estimate the uncertainty of the dwell time measurement even if a single dwell position is observed per value of spacing x. The values of the SD that exceed 0.1 s are due to the above-mentioned time lags of the data acquisition system. When these artifacts are filtered out, the SD is found to be of 0.026 ± 0.024 s (1SD).

Dose rate retrieval

The experimentally determined dose rate per dwell position is reported in Fig. 5. The deviation to the planned dose rates is smaller than 3% across the entire range of sourceprobe distances. The average error is calculated to be of 0.14 ± 0.7 % (1SD). The larger offsets are observed at the shorter source-probe distances where the dose gradient is maximum. The uncertainty of the dose rate measurement is shown in Fig. 5(b) to be approximately comprised between 1% and 3%.

Uncertainty budget

Table 3 provides an uncertainty budget for our study. The positional uncertainty arising from the characterization of the repeatability is defined from Fig. 2(d). The variation of optical signal due to temperature fluctuations of the crystal (in the range of ± 1 • ) is defined from (O'Reilly et al 2020). This temperature-induced signal uncertainty is converted into positional uncertainty using the calibration plots of Fig. 2. Owing to our spectral filtering process, the stem effect is shown to be negligible. With our motorized stage, positional accuracy along (0x) is better than ±0.01 mm and it has been fixed to ±0.1 mm along (0z) from our probe-to-source alignment process along the catheter (using this stage). Since the initial positioning of the probe is realized with a ruler, the positioning accuracy of the detector along (0x) by the operator is set to ±0.5 mm (i.e., half a graduation of the ruler). This contribution only applies to the determination of the dose rate uncertainty.

Discussion

Detector specification and calibration

At a source-probe spacing of 20 mm, our MSD shows a SNR of 89, which is 3.2 times smaller that the SNR reported with the fiber detector of Jorgensen et al (Jorgensen et al 2021a) operating at 0.05 s acquisition rate. Assuming optical photometers whose SNR is mainly limited by the shot noise, an operation at 0.08 s of the two detectors would involve a SNR decrease of 4.4 for the MSD. As noted in the introduction, shrinking the detector increases the spatial resolution at the expense of a decrease of the SNR.

Our MSD shows a scintillation volume and a cross-section of its fiber core that are reduced by factors of 4.7 and 25 regarding Jorgensen's detector (Jorgensen et al 2021a), respectively. In that context, a decrease of the SNR by only 4.4 validates the concept of a nano-optical antenna as scintillator-to-fiber optical interface [START_REF] Suarez | Miniaturized fiber dosimeter of medical ionizing radiations on a narrow optical fiber[END_REF].

To be fully reliable, such a comparison should involve identical photometers for the two probes.

Note that the advantage of miniaturizing a fiber dosimeter is threefold. First, in the steep spatial dose gradients of a HDR-BT source, a smaller sensitive volume reduces volume averaging effect near the source. As an example, the dose rate varies by 7.6 % across a 0.4-mm broad detector (cf. Jorgensen et al 2021a) positioned at 1 cm from the center of an Ir-192 source. This dose rate variation reduces to about 4 % across our 0.24 mm broad detection cell. Second, when using inorganic scintillators as a probe, a smaller sensitive volume reduces the electron fluence perturbation, which minimizes the perturbation of the therapeutic process. Finally, shrinking the overall detector (scintillation + fiber) opens the prospect of a new multiprobe architecture in an individual catheter or needle. As an alternative to Therriault's approach (Therriault et al 2013), multipoint detectors would be obtained by cascading along (0z) the various probes engineered at the end of a fiber bundle. These multiprobe detectors would be completely free from cross-talk between the detection channels. With the detector approach proposed by Jorgensen et al. (Jorgensen et al 2021a), such multiprobe architectures would be too large to be inserted in a BT catheter or a BT needle.

The linearity of the MSD is excellent and agrees well with preceding studies of Gd

Dwell position verification

The average discrepancy to the planned source-probe distances is of 0.023±0.077 mmAs a comparison, Guiral et al. [START_REF] Guiral | Design and testing of a phantom and instrumented gynecological applicator based on gan dosimeter for use in high dose rate brachytherapy quality assurance[END_REF] reported a value of 0.11±0.7 mm over 900 dwell positions in a water phantom. In their detection system, four inorganic scintillator detectors based on 0.36 mm 3 scintillation cell were connected to four photomultiplier tubes operating in photon counting regime at a rate of 0.1 second. Despite a 21fold smaller scintillation volume and the use of a standard camera of lower detection efficiency, the measurement accuracy is significantly better with the MSD.

The standard deviation of our dwell position measurements remains below 0.8 mm over the full range of source-probe distances of this study (Fig. 3(d)). Johansen et al. [START_REF] Johansen | Time-resolved in vivo dosimetry for source tracking in brachytherapy[END_REF] reported a standard deviation in between 2 mm and 3 mm over 3239 dwell positions distributed in a similar range of source-probe distances. The detection system involved a fiber-coupled Al 2 O 3 :C crystal and a photomultiplier tube of larger sensitivity than our standard camera. In the case of Johansen et al. [START_REF] Johansen | Time-resolved in vivo dosimetry for source tracking in brachytherapy[END_REF], measurements were realized in vivo during prostate HDR-BT, which may tend to diminish positioning accuracy regarding our highly controlled measurements conducted in a water phantom. Moreover, their analysis focused on dwell times longer than 0.7 s. The ability of their system to monitor source position over short dwell times could not be assessed.

Figure 3 (d) shows that the sensitivity of our camera is the main limiting factor of the dwell position verification. The accuracy of the dwell position monitoring could be increased by use of a photomultiplier tube or an avalanche photodiode, which provide higher SNR capability than a sCMOS camera. However, these highly sensitive photometers are more expensive, can involve kV supply voltage and their detection performances can be seriously degraded after misuses, which are likely with end-users that are non-specialists in optics. By improving in-fiber coupling of the scintillation with a concept of nano-optical antenna [START_REF] Suarez | Miniaturized fiber dosimeter of medical ionizing radiations on a narrow optical fiber[END_REF], we show that state-of-theart dwell position verification in HDR-BT is achievable with a standard photometer that is more compatible with clinical protocols.

Dwell time verification

With a 100% identification rate of the dwell times longer than 0.5 s, the proposed detection system shows promising perspectives for dwell time verification. The performances detailed in Table I are demonstrated over a 40-mm broad and 40-mm long cylindrical measurement volume around the probe axis. Such a volume encompasses the full range of source-probe distances considered in vaginal HDR-BT (typically, x <20 mm and z <40mm) [START_REF] Belley | Real-time dose-rate monitoring with gynecologic brachytherapy: Results of an initial clinical trial[END_REF]. If a larger measurement range is needed, e.g. for covering the maximum treatment volume in prostate HDR-BT (a 60 mm cubic volume), a photomultiplier tube or an avalanche photodiode, rather than a standard camera, could be used to enhance the SNR of the detection system. We however prefer to avoid this detection strategy which is less suitable for a clinical use. An alternative approach would consist of increasing the scintillation volume while keeping a standard camera for the luminescence detection. In that case, problem of volume averaging as well as a perturbation of the therapeutic process may arise. This solution would thus need further investigations to be validated. Finally, multiple miniaturized fiber probes could be inserted in the treatment region. By using four MSD in four dedicated catheters spaced 40 mm apart, one could ensure the performances shown in Table 2 across a 80×80 mm 2 area in the (x,y)-plane perpendicular to the catheter axes. Guiral et al [START_REF] Guiral | Design and testing of a phantom and instrumented gynecological applicator based on gan dosimeter for use in high dose rate brachytherapy quality assurance[END_REF] showed the feasability of a four-probe system and reported the ability with this architecture to extend the source tracking to x and y coordinates. The use of a multiprobe configuration in an individual catheter, as first suggested by Therriault et al. (Therriault et al 2013), may also extend the monitoring capability along the (0z)-axis (i.e., the catheter axis). In our case however, the multipoint detector would be obtained by cascading along (0z) the different probes engineered at the end of a fiber bundle. As an example, a fiber bundle of 7 fibers being only 0.375 mm in diameter with our approach, the resulting multiprobe would be narrow enough to be inserted in a catheter.

Dwell time verification is also a matter of edge detection in a staircase signal. Based on a signal averaging, the algorithm developed by Johansen et al. [START_REF] Johansen | Dwell time verification in brachytherapy based on time resolved in vivo dosimetry[END_REF] allows for efficient noise filtering but it lessens the slope of the edge between two successive dwell positions, thereby reducing the identification probability of the shorter dwell times. Their resulting identification rate, which was of 89 % for dwell times longer than 1 s, dropped down to 25 % for shorter dwell times. We find that 100 % and 83% of the dwell times longer and shorter than 1 s are identified. This could be partly explained by our algorithm which does not affect the edge slope (Fig. 4(b)), at the expense of a lower noise filtering potentially leading to edge-detection limitations for the larger source-probe distances. Edge-detection algorithms based on artificial intelligence could extend the dwell time monitoring capability further away from the probe while keeping time edges unaffected in the data post-treatment.

To overcome lag problems, the camera and data recording should be independently driven by two different systems, such as a micro-controller or a FPGA for the camera and a computer for the data recording. When the artefactual points are filtered out, the SD of the distribution of measured dwell times reduces to 0.06 s, which matches the SD obtained by Johansen et al. [START_REF] Johansen | Dwell time verification in brachytherapy based on time resolved in vivo dosimetry[END_REF]. Guiral et al. obtained standard deviations of 0.09 s and 0.08 s with four-detector architectures embedded in a plastic phantom and in an applicator, respectively [START_REF] Guiral | Design and testing of a phantom and instrumented gynecological applicator based on gan dosimeter for use in high dose rate brachytherapy quality assurance[END_REF]. With their imaging panel, Fonseca et al. reported a maximum absolute deviation of 0.2 s in a water phantom [START_REF] Fonseca | Online pretreatment verification of high-dose rate brachytherapy using an imaging panel[END_REF]. Note that Guiral et al. and Fonseca et al. limited their study to dwell times of the order of 5 s and 10 s, respectively. The performances of their systems to monitor sub-1s dwell times could not be assessed. Johansen et al. performed in vivo monitoring of dwell times shorter and longer than 1s in patient treatments. Reaching similar accuracy with a MSD coupled to a standard camera validates our miniaturization approach. Our MSD indeed shows a 32-fold smaller scintillation cell coupled to a fiber whose waveguiding area is 25 times tighter. Moreover, the interdwell spacing is of 3 mm for Johansen et al. and only 2.5 mm in our study, leading to noticeably smaller edges in our case given the steep dose gradients in play.

The good repeatability of our measurements is confirmed in Fig. 4(d) as the SD for all dwell time measurements remain below 0.1 s. The values larger than 0.1 s are due to the time lags of the detection system. As another indicator of the temporal accuracy of our MSD, 95 % of the identified dwell times are within one integration time of the camera (0.08s). Johansen et al. [START_REF] Johansen | Dwell time verification in brachytherapy based on time resolved in vivo dosimetry[END_REF] reported that 97% of the measured dwell times of patient treatments (prostate cancer) were within one sampling point of 0.15 s. In the monitoring of prostate treatment with a flat panel detector, Smith et al. obtained 29 % (78 %,respectively) of the dwell times matching the planned values within one (two, respectively) integration time of their system (0.067 s; 0.13 s, respectively) [START_REF] Smith | An integrated system for clinical treatment verification of hdr prostate brachytherapy combining source tracking with pretreatment imaging[END_REF].

Dose rate retrieval

Relying on water nonequivalent scintillators, our MSD requires a space-dependent correction factor to accurately determine the dose rate (see Fig. 2(b)). The dose rate determination from dwell position (using AAPM TG-43 dose parameters) is more accurate by one order of magnitude than a direct dose rate measurement (Fig. 5(a)). This indirect approach can be seen to bring the required correction factors. The increase of dose rate offset (up to 3%) at the smaller source-probe distances is due to the steeper spatial gradients of the dose rate close to the source, leading to higher uncertainties in the dose rate determination process. This error level is however 8 times below the discrepancy observed at the same distance range from a direct measurement (Fig. 2(b)). The SD of the distribution of experimental points being smaller than 3% over the full range of source-probe distances (Fig. 5(b)), the time-resolved dosimetry performances of the MSD are fully compatible with IVD for HDR-BT.

Clinical prospects of the MSD

Our MSD is compatible with clinical applications as it is made of biocompatible materials and it is compact enough to be easily inserted in a catheter, a BT applicator or a BT needle. As a preliminary step, we successfully positioned a MSD at the center of a 1-mm diameter sealed encapsulation tube of biocompatible PEEK material.

Note that the use of a motorized stage to position the probe ensures an accurate assessment of the intrinsic performances of the probe by minimizing manipulation errors. For future clinical use in HDR BT, we will follow the fiber-based treatment verification approaches already proposed in gynecologic BT [START_REF] Belley | Real-time dose-rate monitoring with gynecologic brachytherapy: Results of an initial clinical trial[END_REF] and prostate BT [START_REF] Johansen | Time-resolved in vivo dosimetry for source tracking in brachytherapy[END_REF]. In gynecologic BT, the source and the probe can be inserted in two different catheters which belong to the same applicator. In that case, the source-probe spacing x can be precisely known, as in the present study. In prostate BT, the problem is more complex as the source and probe are inserted in different independent needles that are manually implanted in the patient. In that case, the x and z coordinates (see Fig. 1(d)) can become noticeably coupled since the needle implant are rarely perfectly parallel, due to operational errors. In that case, the detection strategy suggested by Johansen et al [START_REF] Johansen | Time-resolved in vivo dosimetry for source tracking in brachytherapy[END_REF] is particularly interesting as it enables to find from IVD both x and z coordinates of the source relatively to the probe. 3D positioning strategies by triangulation in a multiprobe architecture may also be an alternative solution.

Conclusion

In this phantom-based study, we have demonstrated an IVD system for HDR-BT based on a miniaturized scintillator dosimeter (with high-Z Gd 2 O 2 S:Tb scintillators) coupled to a standard sCMOS camera. Given its high level of miniaturization and high detection efficiency, such a detector combines high spatial resolution, high detection speed, and low perturbation of the therapeutic process even with inorganic water nonequivalent materials. The use of a standard camera, instead of a photomultiplier tube or an avalanche photodiode of higher sensitivity, is motivated by its better scalability for clinical use. The lower SNR of these photometers is compensated by the concept of a MSD relying on an interface in between the scintillators and the fiber aimed at improving in-fiber collection efficiency of the X-ray-induced luminescence. Our MSD shows excellent linearity and repeatability and prior studies have reported a negligible temperature dependence of Gd 2 O 2 S:Tb in the context of a medical application (O'Rreilly et al 2020). The average offset to the planned dwell positions is of 0.023 ± 0.077 mm over 1384 dwell positions distributed within the source-probe distance range of 6-54 mm. The standard deviation of the measured source-probe distances varies from 0.04 mm at 6 mm to 0.8 mm at 54 mm. 94 % of the 966 dwell positions at or below the source-probe spacing x = 20 mm are successfully identified with a 100% detection rate for dwell times exceeding 0.5 s. 95 % of the identified dwell times are measured with an accuracy of one acquisition time of the camera (0.08 s). The average offset to the planned dwell times is of 0.005 +/-0.060 s, over values ranging from 0.2 s to 11 s (dwell times of 0.1 s have not been identified with our 0.08 s detection rate). The standard deviation of the measurements for each planned dwell time does not exceed 0.1 s. The dose rate retrieval at the probe position from the dwell position monitoring (using the AAPM TG-43 formalism) leads to an average relative error between experimental and planned values of 0.14 ± 0.7 %, with a standard deviation ranging from 1% to 3% from 6 to 54 mm source-probe distances. Our detector of unmatched compactness is thus totally adequate for IVD in HDR-BT. Future effort will be focused on multiprobe architectures to broaden the monitoring volume and extend source tracking to the three space dimensions [START_REF] Guiral | Design and testing of a phantom and instrumented gynecological applicator based on gan dosimeter for use in high dose rate brachytherapy quality assurance[END_REF]. The use of a camera, instead of a single pixel photodiode or a photomultiplier tube, is well adapted to the simultaneous parallel detection required in a multiprobe architecture. Time-resolved treatment monitoring in a patient is also a future goal to assess performances of our detector in IVD.

  al 2007, Therriault et al 2011, Therriault et al 2013, Linares et al 2019, Andersen et al 2009, Belley et al 2018, Johansen et al 2019, Kertzscher et al 2011, Jorgensen et al 2021a) as well as dwell time and position verification of the stepping source (i.e., source tracking) (Wang et al 2014, Guiral et al 2016, Johansen et al 2018, Linares et al 2020, Johansen et al 2019, Jorgensen et al 2021b), which represent clinically relevant information (Fonseca et al 2020).

Figure 1 .

 1 Figure 1. (a) Magnified optical image of the MSD. (b) Photograph of the source catheter in the water tank. The catheter is embedded in a piece of solid water to be maintained straight. (c) Photograph of experimental setup with the detector attached to a holder made with solid water. The holder is movable thanks to a 2D motorized translation stage. (d) Schematic of the irradiation configuration in the water tank, top view.

  Fig. 1(d)). The source and the detector are aligned along the catheter axis (z=0 mm). SNR: Signal-to-Noise Ratio, DTR: Deviation to perfect repeatability (SD), LC: Linearity Coefficient regarding the absorbed dose.

Figure 2 .

 2 Figure2. (a) 2D field map acquired with the MSD near the radioactive source. We plot the intensity of the detected optical signal accumulated by our probe during raster scan. (b) Mismatch between the measured and expected dose rates (as a percentage of the expected dose rate). (c) Normalized field profiles plotted at source-probe spacings of 6, 10, 20, 30 and 40 mm. These profiles serve as calibration curves of the MSD for source tracking.

Figure 2 Figure 3 .

 23 Figure2(c) shows five radiation profiles along (0z) acquired with the MSD positioned at x equal to 6, 10, 20, 30 and 40 mm. These plots are used as calibration curves of the MSD for source tracking as they provide the signal-to-position conversion required for time-resolved dwell position verification. To ensure a low fluctuation level in these reference plots, the integration time for each probe position of the scan (spaced 1 mm apart from each other) varies from 10 seconds at x=6 mm to 30 seconds at x=40 mm.

  Fig.1(d)) spanning from 6 to 55 mm. The distribution of dwell positions over the full range of rvalues is represented in Fig.3(a).Fig. 3(b) shows one of the 22 dwell-position sequences of the treatment at a source-probe spacing x = 6 mm. We report the experimentally determined source-probe distance r in time, together with the planned values from the TPS.Fig.3(c) represents the mismatch to the planned source-probe distances. Each point corresponds to the difference between the planned distance r T P S and the average value rexp obtained from measurements over one or more dwell positions. The average

Figure 4 .

 4 Figure 4. (a) Distribution of the dwell times defined by the TPS: number of dwell positions per one-second time intervals. (b) Temporal-edge detection function F for the treatment shown in Fig 3(b). (c) Offset ∆T between the experimentally determined and expected dwell times (T exp and T T P S , respectively) versus the expected dwell time T T P S . The different shapes and colors of the data points are used to identify the source-probe spacing x, see inset of (d). The fourteen erroneous points located in the gray regions of the figure are due to lags in the data acquisition setup. The statistics of the white central region is calculated after removing these erroneous points. Inset: histogram of ∆T for all 966 dwell positions. (d) SD of T exp as a function of T T P S .

Figure 5 .

 5 Figure 5. (a) and (b) Average and SD of the relative deviation (in %) between the experimentally determined and planned dose rates ( Dexp and D T P S , respectively), versus r T P S . Dexp is the average value of the measured dose rate over one or more dwell positions. The offset is reported for the five source-probe spacings x (see legend of (b)).

  2 O 2 S:Tb-based fiber dosimeters used for monitoring external beam radiotherapy (Qinet al 2016, Alharbiet al 2018, Huet al 2018, Gonodet al 2021). Note from O'Reilly et al (O'Reillyet al 2020) that under fixed irradiation conditions, Gd 2 O 2 S:Tb detectors do not display any significant intensity variations at temperatures ranging from 15 to 40 • C. The Gd 2 O 2 S:Tb fiber detectors are thus promising candidate for in vivo dosimetry, showing no significant signal variations from room to body temperature.

Table 1 .

 1 Specification of the MSD at five different values x of the spacing between the source and the probe (cf.

		z = 0 mm
	x (mm) SNR DTR (%)	LC
	6	205	0.49	>0.999
	10	147	0.68	>0.999
	20	89	1.12	>0.999
	30	66	1.52	>0.999
	40	52	1.92	>0.999

Table 2 .

 2 Identification rate of the dwell positions with the MSD, as a function of the planned dwell time. These results are obtained for source-probe spacings x of 6, 10 and 20 mm.

	Dwell time (s) Number of dwells Identified dwells Identification rate (%)
	0.1	24	0	0
	0.2	30	5	17
	0.3	36	31	86
	0.4	33	30	91
	0.5	39	37	95
	>0.5	804	804	100
	All	966	907	94

Table 3 .

 3 Uncertainty budget for the dwell time and position measurements, and the retrieved dose rate.

			x=10 mm, z=-2 mm	x=20 mm, z=-2 mm
			Dwell	Dose rate	Dwell	Dose rate	Dwell time
			position	uncertainty	position	uncertainty uncertainty
			uncertainty	(%)	uncertainty	(%)	(s)
			(mm)		(mm)
	Detector	Repeatability	0.045	0.197	0.15	0.152	0.058
			(Type A)		(Type A)		(±0.1)
		Crystal temp.
		(±1 • C)
		(0z)	0.058	0.228	0.289	0.296	-
			(±0.1)		(±0.5)		-
		(0x)	0.012	0.259	0.023	0.0264	-
			(±0.02)		(±0.04)		-
		Stem effect	0	0	0	0	-
		Integration	-	-	-	-	0.023
		time					(0-0.08)
		Total	0.074	0.397	0.326	0.324	0.062
	Motorized	Position/(0x)	0.006	0.129	0.006	0.066	-
	stage		(±0.01)		(±0.01)		-
		Position/(0z)	0.058	0.228	0.058	0.059	-
			(±0.1)		(±0.1)		-
		Total	0.058	0.265	0.058	0.089	-
	Human	Position/(0x)	-	6.455	-	3.302	-
		(± 0.5 mm)
	Afterloader	Source	0.173	1.102	0.173	0.170	0.058
		position/(0z)	(±0.3)		(±0.3)		(±0.1)
	Overall		0.197	6.566	0.374	3.332	0.085

Acknowledgments

The authors are indebted to Karine Charriere, Lionel Pazart and Thomas Lihoreau for helpful discussions. This work is supported by the French Agency of Research (contract ANR-18-CE42-0016), the SAYENS Agency, the EIPHI Graduate School (contract ANR-17-EURE-0002) and the Region "Bourgogne Franche-Comte". This study is also partly supported by the French RENATECH network and its FEMTO-ST technological facility.


Mengjia Wang

Zhijin Huang

Roland Salut

Miguel Angel Suarez

Huihui Lu

Nicolas Martin

Thierry Grosjean
email: thierry.grosjean@univ-fcomte.fr

Converting between circularly polarized waves and longitudinal fields with an individual plasmonic nanohelix

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

I. INTRODUCTION

Controlling the polarization vector of light provides a key approach to manipulating light-matter interaction on the nanoscale. In turn, tailoring the local optical response of subwavelength structures brings new prospects to light polarization control. Recent advances in plasmonics and metamaterials led to numerous novel polarization systems combining the merits of high compactness, broadband operation and new functionalities [1]. So far, polarization control mainly refers to the transverse vector components of a propagating light wave. At small scale however, the 3D nature of light cannot be avoided and a longitudinal optical component arises and can even become dominant for strongly confined fields [2]. Here, we demonstrate the polarization properties of a single plasmonic helix positioned in a nonradiative plasmonic focus showing dominant longitudinal electric field component (perpendicular to the focusing plane, parallel to the helix axis). So far, helical nanoantennas were excited with transversely polarized propagating waves. Here, we study the polarization properties of a single plasmonic helix positioned in a nonradiative plasmonic focus showing dominant longitudinal electric field component (perpendicular to the focusing plane, parallel to the helix axis) [4].

II. RESULTS

We positioned a helical nanoantenna at the center of a circular slit which perforates a thin gold film on a glass substrate (Fig. 1(a)). The nanoantenna consists of a left-handed four-turn gold-coated carbon helix (core-shell structure [3,4]) lying on a 100-nm cylindrical pedestal. Two helix configurations, with a varying or constant pitch, are fabricated (see Figs. 1(b) and (c), respectively). The circular slit is 16-µm in diameter and 200-nm wide.

To demonstrate the generation of circular polarization from a longitudinal electric light field, we project a focused radially polarized beam onto the back of our sample. We then image the overall structure in the far-field with a microscope objective coupled to an infrared camera. The polarization state of the emerging light is analyzed with a rotating quarter-wave plate followed by a linear polarizer (i.e., a circular analyzer). Without circular analyzer, the image shows an individual spot at the center ETOPIM 12, Besanc ¸on, France, July 4 th -July 8 th 2022 where I RCP and I LCP are the intensities of the right and left-handed circularly polarized components of the outcoming light, respectively. For the helix of a constant pitch, the DOCP remains larger than 0.99 within a wavelength range of 80 nm. By breaking the helix periodicity, i.e., by using the varying-pitch helix, we extend the spectral invariance of the nanoantenna to a broader wavelength range exceeding the 180-nm bandwidth of our laser. We estimate that the efficiency with which the constant and varyingpitch helices convert incoming surface plasmons into freely propagating waves reaches 26% and 24%, respectively.

We also demonstrate the reciprocal configuration of incoming circularly polarized waves converted back into longitudinal fields [4].

III. CONCLUSION

We demonstrate an individual helical nanoantenna to convert between the circular polarization of a freely propagating wave and longitudinal electric fields in the near-field, thus providing a new degree of freedom in light polarization control. Being non-resonant, the helical nanoantenna operates over a broad spectral range.

Fig. 1 :

 1 Fig. 1: (a) SEM image of a fabricated (b) and (c) SEM images of two plasmonic helices constant pitch length, respectively. (d) DOCP spectra of the outcoming optical waves obtained with nanoantennas homogeneous and varying pitches. (e) and (f) Optical images obtained at λ= 1.56 µm with the varying-pitch helical nanoantenna after inserting (e) a left-handed and (f) a right-handed circular analyzer in front of the camera. Image size: 25 × 25 µm 2 .


Yannick Augenstein
email: yannick.augenstein@kit.edu

Matthieu Roussey

Thierry Grosjean

Emiliano Descrovi

Carsten Rockstuhl

Inverse design of cavities for Bloch Surface Waves interfaced to integrated waveguides

Keywords: effective index, surface waves, inverse design, Purcell effect

The design of functional elements for Bloch Surface Waves (BSW) is challenging because of the relatively low index contrast offered by the respective platforms. Here, we design a supporting photonic nanostructure that extracts as much light as possible from a quantum emitter into a waveguide in an integrated BSW architecture. The inverse problem is solved using topology optimization. Emphasis is put on discussing the algorithm's emerging strategies for the design to enhance the Purcell factor, the coupling efficiency, or both for different index contrasts. Fully three-dimensional simulations of an explicit device show the benefits of our devices and pave the way for integrating such unconventional photonic elements into future fully-integrated BSW devices.

Introduction

Bloch Surface Waves (BSW) are evanescent eigensolutions to Maxwell's equations, propagating parallel to the interface between an isotropic medium and a truncated one-dimensional photonic crystal (1DPC) [START_REF] Meade | Electromagnetic bloch waves at the surface of a photonic crystal[END_REF][START_REF] Yeh | Electromagnetic propagation in periodic stratified media. i. general theory[END_REF]. Interestingly, they exhibit the highest amplitude close to the 1DPC surface and they can be either TE or TM polarized [START_REF] Chen | Strong polarization transformation of bloch surface waves[END_REF]. The one-dimensional photonic crystal is typically implemented as a layer stack of dielectric materials with an alternating refractive index. In particular, BSWs are operated in the frequency domain of the band gap of the one-dimensional photonic crystal. This guarantees an exponentially decaying field in the homogeneous medium (e.g. air, water, etc.) and an oscillating amplitude with an exponentially decaying envelope in the layered medium, resulting in long propagation lengths. Although the underlying physics is rather different, BSWs bear many similarities to other guided surface modes, such as propagating surface plasmon polaritons [START_REF] Zayats | Nano-optics of surface plasmon polaritons[END_REF][START_REF] Barnes | Surface plasmon-polariton length scales: a route to subwavelength optics[END_REF], in that they are localized to the interface. The field localization is less pronounced since the effective index of the guided modes will not exceed those of the indices of the involved materials and tends to be relatively small in practical situations. However, BSWs offer in exchange the fantastic opportunity to reach propagation lengths in the order of many hundreds of micrometers or even millimeters, while still having the field largely extending in the outer medium [START_REF] Wang | Diffraction-free bloch surface waves[END_REF][START_REF] Vosoughi Lahijani | Centimeter-scale propagation of optical surface waves at visible wavelengths[END_REF]. This is possible because dissipation losses in the all-dielectric platform do not limit their propagation lengths. Instead, the propagation length is frequently only limited by the finite number of layers used to build the photonic crystal [START_REF] Dubey | Experimental investigation of the propagation properties of bloch surface waves on dielectric multilayer platform[END_REF]. In fact, a limited number of layers causes some leakage radiation due to evanescent tunneling toward the substrate beneath the multilayer, but it can be considered as an engineering problem to suppress it.

The ability to guide light over macroscopic distances in an integrated architecture enables a plethora of applications that serve societal needs [START_REF] Sinibaldi | Direct comparison of the performance of bloch surface wave and surface plasmon polariton sensors[END_REF][START_REF] Yu | Manipulating bloch surface waves in 2d: a platform concept-based flat lens[END_REF]. Examples of functional devices explored in the past range from on-chip information processing components [START_REF] Doskolovich | Spatial differentiation of bloch surface wave beams using an on-chip phase-shifted bragg grating[END_REF] to sensing platforms [START_REF] Rivolo | Real time secondary antibody detection by means of silicon-based multilayers sustaining bloch surface waves[END_REF][START_REF] Kuai | Real-time measurement of the hygroscopic growth dynamics of single aerosol nanoparticles with bloch surface wave microscopy[END_REF][START_REF] Wang | Directional imbalance of bloch surface waves for ultrasensitive displacement metrology[END_REF]. Such sensors can even be deposited on the end-facets of fibers to be extremely flexible [START_REF] Scaravilli | Grating-coupling-based excitation of bloch surface waves for lab-on-fiber optrodes[END_REF][START_REF] Scaravilli | Excitation of bloch surface waves on an optical fiber tip[END_REF]. A crucial component to exploit BSWs in integrated circuits is their confinement into a second dimension, i.e., the one tangential to the substrate interface and normal to the propagation direction [START_REF] Wang | Bloch surface waves confined in one dimension with a single polymeric nanofibre[END_REF]. This is usually achieved by patterning the terminating layer of the one-dimensional photonic crystal, frequently called the functional layer, such that for the operational frequency, the BSWs experience slightly different effective indices in the structured and unstructured functional layer, respectively [START_REF] Yu | Investigation of ultra-thin waveguide arrays on a bloch surface wave platform[END_REF]. This permits to achieve waveguides and resonators with lateral confinement [START_REF] Perani | Bloch-surface-wave photonic crystal nanobeam cavity[END_REF][START_REF] Perani | Long-range bloch surface waves in photonic crystal ridges[END_REF], but also more complicated functional devices that can be realized upon exploitation of the generally small index contrast [START_REF] Dubey | Ultra-thin bloch-surface-wave-based reflector at telecommunication wavelength[END_REF].

Nevertheless, the tiny index difference, which is frequently only in the order of ∆n ≈ 0.1, constitutes a prime design challenge [START_REF] Kim | Subwavelength focusing of bloch surface waves[END_REF][START_REF] Kim | Multiple self-healing bloch surface wave beams generated by a two-dimensional fraxicon[END_REF]. It implies that the direct application of conventional optical components, e.g., even just a lens, is not straightforward, as the design of these components usually hinges on a larger index contrast. Consequently, and entirely in the spirit of the current special issue to which this article aims to contribute, the design of functional components for an integrated BSW architecture is a wonderful playground to explore the application of novel computational approaches for inverse design [START_REF] Molesky | Inverse design in nanophotonics[END_REF]. Only with such strategies, it is possible to identify feasible designs that serve a well-defined purpose while fully respecting the constraints imposed by the limited index contrast. While a large variety of different methodologies exist to solve the inverse problem [START_REF] Piggott | Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer[END_REF][START_REF] Jin | Inverse design of compact multimode cavity couplers[END_REF][START_REF] Schneider | Benchmarking five global optimization approaches for nano-optical shape optimization and parameter reconstruction[END_REF][START_REF] Baxter | Plasmonic colours predicted by deep learning[END_REF][START_REF] Bayati | Inverse designed metalenses with extended depth of focus[END_REF][START_REF] Wang | Inverse design of digital nanophotonic devices using the adjoint method[END_REF][START_REF] Minkov | Inverse design of photonic crystals through automatic differentiation[END_REF][START_REF] Jiang | Simulator-based training of generative neural networks for the inverse design of metasurfaces[END_REF][START_REF] Augenstein | Inverse design of nanophotonic devices with structural integrity[END_REF][START_REF] Barry | Evolutionary algorithms converge towards evolved biological photonic structures[END_REF][START_REF] Lin | End-to-end nanophotonic inverse design for imaging and polarimetry[END_REF][START_REF] Zhou | Inverse design of metasurfaces based on coupled-mode theory and adjoint optimization[END_REF][START_REF] Deng | Neural-adjoint method for the inverse design of all-dielectric metasurfaces[END_REF][START_REF] Bennet | Analysis and fabrication of antireflective coating for photovoltaics based on a photoniccrystal concept and generated by evolutionary optimization[END_REF][START_REF] Getman | Broadband vectorial ultrathin optics with experimental efficiency up to 99% in the visible region via universal approximators[END_REF][START_REF] Bayati | Inverse designed extended depth of focus meta-optics for broadband imaging in the visible[END_REF][START_REF] Hassan | Topology optimization of dispersive plasmonic nanostructures in the time-domain[END_REF][START_REF] Wang | Advancing statistical learning and artificial intelligence in nanophotonics inverse design[END_REF], we exploit topology optimization here.

We apply topology optimization to contribute to an essential step for the further development of a BSW-based integrated photonic architecture. Our exploration concerns the extraction of light emitted from an embedded source into a BSW-integrated circuit. The circuit consists of, for simplicity, the form of a waveguide. The embedded source, considered as an electric dipolar emitter, is assumed to be externally pumped and shall release as much energy as possible into the waveguide. The structure at stake in our consideration is a finite domain surrounding the emitter that can be structured and shall direct as much light as possible into the waveguide. Two aspects must be kept in mind that deny simple solutions and prompt the application of advanced tools to solve the inverse design problem. First, the coupling efficiency needs to be maximized. The coupling efficiency is the fraction of the emitted light that finds its way into the waveguide. Second, the supporting structure can contribute to a Purcell enhancement, i.e. the dipole-waveguide coupler must be resonant. The Purcell enhancement measures the increased local density of states to extract the radiation from the emitters. It is defined as the power extracted from the source in the presence of the supporting photonic structure normalized to the power extracted from the source in free space (or some other referential structure). The higher the Purcell enhancement [START_REF] Novotny | Principles of nano-optics[END_REF] and the quicker the light is extracted, the faster the emitter can be excited again to release a further photon. The scientific challenge that we wish to explore concerns the discussion of how a suitable supporting photonic structure should look to transfer as much light from a quantum emitter into a waveguide through coupling to the supported BSWs assuming a dipole excitation in saturation regime. Emphasis is put on studying that effect depending on the index contrast the BSW platform offers to shed light on the interplay between the two effects that dominate the response. Topology optimization is used for the inverse design of the supporting photonic structure.

The article is structured into three sections. In Section 2, we discuss the setting of our inverse design problem and detail the used computational techniques. In Section 3, we present the results and discussion. We systematically study the details of the optimized devices initially in a 2D setting but verify the optimizations in full 3D simulations. The discussion orients on the question of how the details of the optimized structure vary by changing the possible index contrast. In Section 4, we conclude on our results.

Optimization setup

We aim to optimize functional photonic devices that can extract light emitted by a dipole-like source that couples into a surface mode sustained by a dielectric multi-layer stack into a waveguide (see Fig. 1) such that it can be used for, e.g., on-chip light sources. However, the system under consideration is relatively large, rendering an optimization using full-wave 3D simulations impractical. Instead, we transform the problem into a 2D one by using an effective index method, which has been applied successfully in the past to the simulation and design of functional devices for manipulating BSWs [START_REF] Baghbadorani | Guided modes in photonic crystal slabs supporting bloch surface waves[END_REF][START_REF] Augenstein | Inverse photonic design of functional elements that focus bloch surface waves[END_REF][START_REF] Stella | Enhanced Directional Light Emission Assisted by Resonant Bloch Surface Waves in Circular Cavities[END_REF] due to the low refractive index contrast of such systems.

In the proposed setup, the dipole source is oriented along the x direction and emits light with a Gaussian spectrum centered at a wavelength of λ 0 = 570 nm placed at the center of the spacer region (black circle). We define a circular design region as shown in Fig. 1 with a radius of 5.7 µm and define our objective function F as

F = |α 0 | 2 , ( 1
)
where α 0 is the mode coefficient of the fundamental forward-propagating TM-polarized mode (in-plane electric field) of the waveguide, normalized such that |α 0 | 2 corresponds to the power carried in that mode. We use a standard filter-and-project parametrization [START_REF] Wang | On projection methods, convergence and robust formulations in topology optimization[END_REF][START_REF] Zhou | Minimum length scale in topology optimization by geometric constraints[END_REF][START_REF] Lazarov | Length scale and manufacturability in density-based topology optimization[END_REF] and use the method of moving asymptotes (MMA) [START_REF] Svanberg | A Class of Globally Convergent Optimization Methods Based on Conservative Convex Separable Approximations[END_REF] as implemented in nlopt [51] (version 2.7.1) for constrained, nonlinear topology optimization. The designs are discretized on a 2D grid with a resolution of 100 px µm . To parametrize the designs, the dimensionless design variables ρ ∈ [0, 1] are first low-pass filtered using a Gaussian kernel to impose a minimum feature size of 50 nm. Afterward, a soft-thresholding function is applied to promote binary solutions in the optimization:

ρ = tanh( β /2) + tanh(β (ρ -0.5)) 2 tanh( β /2) . ( 2
)
The parameter β dictates the strength of the binarization and is increased in multiple steps during the optimization process to yield final, binary designs.

To prevent the optimization from reverting to less binary designs, we constrain the gray indicator

c g ( ρ) = 1 n n i=1 4ρ i (1 -ρi ) (3)
suggested in [START_REF] Sigmund | Morphology-based black and white filters for topology optimization[END_REF] to be smaller than a threshold γ, which is determined from the design directly after increasing β. After projection, a circular mask is applied to the design variables such that the device is constrained to a circular geometry. The device geometries are then linearly interpolated to their respective refractive indices, and the simulations are performed using Meep [START_REF] Oskooi | Meep: A flexible free-software package for electromagnetic simulations by the FDTD method[END_REF]. The sensitivities of the objective function w.r.t. the parametrization are obtained using Meep's adjoint module [START_REF] Hammond | High-performance hybrid time/frequency-domain topology optimization for large-scale photonics inverse design[END_REF], and the sensitivities of the parametrization w.r.t. the optimization variables ρ are obtained using autograd [START_REF]HIPS/autograd[END_REF].

We then formulate the optimization problem as:

max ρ |α 0 | 2 (4a) s.t. α 0 = S Ẽ * (r) × H0 (r) + Ẽ0 (r) × H * (r) • n dA (4b) c g ( ρ) -γ ≤ 0 (4c) 0 ≤ ρ ≤ 1 , (4d)
where Ẽ, H in Eq. 4b are the Fourier-transformed fields at the target wavelength of 570 nm at the mode monitor (c.f. Fig. 1) and Ẽ0 , H0 are the fields corresponding to the fundamental mode in the waveguide (obtained using the eigenmode solver MPB [START_REF] Johnson | Block-iterative frequency-domain methods for maxwell's equations in a planewave basis[END_REF]).

The effective index of the surface mode without the functional layer is assumed to be n eff,0 = 1.019, and we perform the optimization for different effective index contrasts ∆n eff ranging from 0.01 to 0.13 in increments of 0.01. For the device with the highest index contrast ∆n eff = 0.13, the surface mode with the functional layer has an effective index of n eff,1 = 1.149, which corresponds to the material platform presented in [START_REF] Stella | Enhanced Directional Light Emission Assisted by Resonant Bloch Surface Waves in Circular Cavities[END_REF].

The 2D simulations in the optimization are performed using MPI-parallel Meep (version 1.23.0) on a desktop computer with a 10-core CPU (Intel Core i9-10900 @ 2.80 GHz) at a resolution of 50 px µm .

Results and discussion

Optimization and evaluation in 2D

By running multiple optimizations for different effective refractive index contrasts in the range of ∆n eff = 0.01 to 0.13, we can numerically study x (µm) -5 0 5

x (µm) -5 0 5

x (µm) the implications on achievable device performance resulting from different conceivable surface-wave sustaining material platforms.

The results of the optimization for four selected devices and the corresponding electric field intensities (log scale) are shown in Fig. 2.

For all designs, we observe that the optimization forms a kind of distributed Bragg reflector (DBR), most notably in the upper half-space of the design. Such a solution to the optimization problem is reasonably intuitive, as the absolute power coupled into the waveguide, regardless of any additional coupling elements, is directly proportional to the power emitted by the source. It is thus a natural and expected solution that the optimization would converge towards designs that, at least in part, form an electromagnetic cavity. The coupling of the power emitted by the source to the waveguide is, of course, not only governed by the total power emitted but also by the efficiency with which the device can guide the field towards the waveguide. This efficiency, in turn, is limited by the refractive index contrast available for steering the electromagnetic fields. As the refractive index contrast becomes larger, the device can more efficiently make use of finer spatial variations in the patterned layer to guide the light, an observation that is evident when examining the different devices in Fig. 2.

The device for ∆n = 0.01 features a mostly adiabatic transition from the DBR in the upper half-space of the design to a tapered waveguide-like element. That is a clear indication that light emitted in the upper half-space should be back-reflected, but otherwise it is primarily of importance to increase the coupling efficiency towards the waveguide by steering the light into the right direction. A possible enhancement of the emission due to some cavity effect is not really visible. But that is also not surprising considering the very low index contrast that leads to cavities with small quality factors.

The device for ∆n = 0.13 exhibits a more intricate and non-intuitive design. The central part of its design consists of multiple sub-wavelength elements, and the waveguide-like structure in the lower half-space has made way for a more intricate design. In the essence the upper part continues to leave the impression as acting like a DBR but the lower part, i.e. the part between the source the waveguide, clearly balances our requirements now in terms of a good coupling efficiency and a high enhancement of the extracted power.

To quantify the performance of the optimized devices, we investigate the Purcell enhancement of the source in the spacer region and the coupling efficiency into the waveguide. We obtain spectra in the range (570 ± 30) nm for all optimized devices in Fig. 3 and we show the obtained values at the target wavelength of 570 nm separately in Fig. 4 is calculated as the ratio of power flow from the source in the device ("cavity") and a homogeneous medium (n eff,1 ). The power flow is obtained by placing a box of flux monitors with an edge length of 0.5 µm around the source and accumulating the appropriate Fourier-transformed fields. Similarly, the coupling efficiency is obtained as the ratio of the power contained in the fundamental mode of the waveguide with

P mode = |α 0 | 2 (5)
and the power emitted by the source. The optimized devices clearly show a Purcell enhancement at the target wavelength, with the ∆n = 0.01 device having a Purcell factor of approximately 2, increasing exponentially with increasing ∆n (c.f. Fig. 4) up to the ∆n = 0.13 device with a Purcell enhancement of around 70. We also see that the designs with higher index contrast exhibit better coupling efficiency, with the ∆n = 0.13 design achieving the highest coupling efficiency at 54 %. Notably, the higher index contrast designs exhibit much more significant wavelength specificity than the lower index designs, which show a more broadband response. This reinforces the previous consideration that due to the higher index contrast, the optimization can more specifically tailor the "guiding performance" of the device to the target wavelength, which at very low index contrasts yields only diminishing returns. Further, we observe that the coupling efficiency increases substantially between ∆n = 0.01 and ∆n = 0.04 and then slowly saturates at higher contrasts at values around 0.5. This is likely owed to the fact that, while there is a significant gap in performance between the devices optimized for extremely low index contrasts (∆n ≈ 0.01) and those at moderate contrasts (∆n ≈ 0.1), the index contrast of all optimized devices is still relatively low. Thus, a significant portion of the radiation emitted by the dipole is lost in the upper half-space of the design. These insights mirror the discussion that was phenomenologically made when looking into the different designs obtained for a different index contrast.

Comparison between 2D and 3D

So far, we have examined the performance of the optimized devices in 2D, i.e., in the setting of the effective index approximation of the surface mode that they were optimized for. However, it is crucial to test whether this approximation holds for the optimized devices in the 3D setting that was envisioned. To do this, we adopt the dielectric multi-layer stack from [START_REF] Stella | Enhanced Directional Light Emission Assisted by Resonant Bloch Surface Waves in Circular Cavities[END_REF]. The material platform consists of a sequence of Silica and Tantalia slabs deposited on a glass substrate (n glass = 1.5), with a final layer of PMMA into which the device geometry is patterned. The layer stack consists of ten alternating layers of Ta 2 O 5 (n Ta 2 O 5 = 2.08, d Ta 2 O 5 = 95 nm) and SiO 2 (n SiO 2 = 1.46, d SiO 2 = 137 nm), an additional layer of Ta 2 O 5 , and a layer of SiO 2 with a thickness of 127 nm. The final PMMA layer has a refractive index of n PMMA = 1.48 with a thickness of 75 nm. All materials are assumed to be free of dispersion across the wavelength range in question ((570 ± 30) nm). The effective index of the surface mode at 570 nm supported by this system without the PMMA layer is n eff,0 = 1.019 and n eff,1 = 1.149 with the PMMA layer. This corresponds to the effective indices for which the ∆n = 0.13 device was designed. We extrude the optimized 2D design 75 nm in the z-direction to use it as the functional PMMA layer. The x-polarized dipole source is embedded at the center of the spacer region inside of the PMMA layer 35 nm beneath the surface. The 3D simulation is then run using FDTD (Meep) at a resolution of 80 µm -1 parallelized over 76 CPU cores (2x Intel Xeon Platinum 8368 @ 2.4 GHz), taking 16.5 hours. Analogous to the 2D simulations previously, the power flow from the dipole is monitored in all spatial directions using flux monitor planes with an edge length of 0.5 µm for both the homogeneous bulk PMMA as well as the patterned medium and both the Purcell enhancement as well as the coupling efficiency are calculated as before.

The comparison between the full-wave 3D simulation and that of the corresponding 2D effective index simulation is shown in Fig. 5. While there are apparent differences between both approaches, we see a marked increase in the Purcell enhancement and the coupling efficiency at the target wavelength between the 2D effective index simulation and a 3D full-wave simulation. The device was optimized for an effective index contrast of ∆n = 0.13, which corresponds to the material platform presented in [START_REF] Stella | Enhanced Directional Light Emission Assisted by Resonant Bloch Surface Waves in Circular Cavities[END_REF].

of the optimization in the 3D results. While the Purcell factor in 2D was calculated to be 70, the Purcell enhancement in 3D is reduced to a factor of 23. This is not surprising, as the effective index simulations entirely neglect out-of-plane scattering losses. A large part of the radiation emitted by the source is emitted out-of-plane, which is neither captured by the patterned PMMA layer in 3D nor accounted for in the 2D simulations. However, the analytically constructed DBR structure from [START_REF] Stella | Enhanced Directional Light Emission Assisted by Resonant Bloch Surface Waves in Circular Cavities[END_REF] achieves a Purcell factor of 32. As our optimized device represents a compromise between the Purcell enhancement and the coupling efficiency, it is expected that such a device will exhibit a lower enhancement since energy is continually extracted from the system. This indicates that while there is a mismatch in terms of absolute numbers between the 2D and 3D simulations, the devices optimized using the effective index method achieve reasonably good performance in 3D. A similar argument can be made for the observed coupling efficiency, which is 0.54 in 2D and 0.28 in 3D, in that a significant portion of the light emitted from the dipole is lost to out-of-plane scattering, i.e not coupled to BSW, which in turn limits the amount of power that can be coupled into the BSW waveguide. However, since the structure is only patterned in the PMMA layer, the degrees of freedom even in the 3D setup are effectively two-dimensional as there is no patterning in the z-direction. As such, the additional losses incurred in the 3D case can be assumed to be of similar magnitude between different devices when switching between the 2D effective index setup and the full 3D geometry, indicating that the optimized structures are still close to optimal even in the 3D case.

There is also the aspect of practicality -a single 3D simulation at sufficiently high resolution takes around 16 hours to complete on a highperformance computer. To calculate the sensitivities via the adjoint method, an additional simulation is needed at every iteration, increasing the time for a single gradient-based update step to around 32 hours if done with a 3D full-wave simulation. A typical optimization of the presented devices needs a few hundred such iterations, rendering a full optimization entirely infeasible. On the other hand, the effective index simulations take roughly 30 seconds per simulation on a mid-range desktop computer while yielding designs that still exhibit good performance. A possible compromise could be the optimization of devices using the effective index method and refining the converged solution for a few steps using a full-wave simulation. However, even just ten of such refinement steps would take on the order of weeks in the demonstrated setup and would be beyond the scope of this work.

There exists also the possibility of tackling the stated design problem with different approaches, as there has been much exciting work on the inverse design of photonic devices, both in 2D and 3D, in recent years. However, we note that the computational cost in our optimization stems almost exclusively from the cost of performing the simulations. While many alternative optimization schemes exist, none of them (to our knowledge) remove the cost of numerical simulations.

Gradient-free methods such as swarm or evolutionary optimization rely on extensive sampling of the solution space, which means doing many numerical simulations [START_REF] Baxter | Plasmonic colours predicted by deep learning[END_REF][START_REF] Jin | Complex inverse design of meta-optics by segmented hierarchical evolutionary algorithm[END_REF][START_REF] Barry | Evolutionary algorithms converge towards evolved biological photonic structures[END_REF][START_REF] Bennet | Analysis and fabrication of antireflective coating for photovoltaics based on a photoniccrystal concept and generated by evolutionary optimization[END_REF]. These methods can perform very well when gradients are not unavailable (e.g., for discrete optimization problems), but generally need to sample the objective function more often than gradientbased methods simply due to having less information about the objective function.

Data-driven approaches, such as machine learning, tend to "front-load" the computational cost of performing the simulations into the generation of a dataset with which, e.g., a fast surrogate solver is trained [START_REF] Makarenko | Robust and scalable flat-optics on flexible substrates via evolutionary neural networks[END_REF][START_REF] Lim | Maxwellnet: Physics-driven deep neural network training based on maxwell's equations[END_REF][START_REF] Chen | High speed simulation and freeform optimization of nanophotonic devices with physics-augmented deep learning[END_REF]. This can be useful when one needs to perform similar tasks many times, or when the dataset can be repurposed for a variety of tasks, but the trade-off is that the number of samples required for generating a sufficiently large dataset usually exceeds the number of simulations one would need to perform to solve only one, or a few, of these tasks.

Thus, we argue that all such methods are fundamentally bound by the speed with at which one can perform the relevant simulations. In our work, we propose using effective index simulations in 2D instead of full-wave 3D simulations as a means to speed up the optimization process by orders of magnitude, at the cost of sacrificing some accuracy in the simulation results.

Conclusions

We have demonstrated the use of topology optimization for the inverse design of devices that efficiently extract light from a dipole source into a waveguide in a low index contrast setting, as is typical for dielectric systems with surface modes. In our approach, we approximate the 3D system with an effective index method and numerically compare the performance of one of the optimized devices with its expected performance in an established experimental platform, showing a significant increase in both the Purcell enhancement and the coupling efficiency in the 3D case. The proposed devices open the door to the integration of spontaneous light emission with integrated photonic circuits and present a new way of manipulating the propagation of BSWs.

Figure 1 :

 1 Figure1: Full sketch of a prototypical material platform that supports BSWs (left). A functional layer is patterned on top of a dielectric multi-layer structure sitting on a substrate in this conceptual setting. The optimization setup for the 2D effective index simulation (right) consists of a circular design region with a radius of 5.7 µm and a central spacer region with a radius of 0.4 µm and fixed material distribution. The x-polarized dipole source is embedded in the center of the spacer region. A waveguide with a width of 0.3 µm extends out from the lower edge of the design region. The optimization aims to maximize the power coupled from the dipole into the fundamental TM-polarized mode of the connected waveguide.

2 Figure 2 :

 22 Figure 2: Optimized device designs (top row) and corresponding electric field intensities (bottom row) in log scale at the target wavelength of 570 nm. Shown here are four selected devices designed for different effective index contrasts (columns). Note that a short length of waveguide is included for clarity but it is not part of the design region.

Figure 3 :Figure 4 :

 34 Figure 3: Spectra of Purcell enhancement (left) and coupling efficiencies (right) for all designs as obtained from the 2D effective index setup in which the devices were optimized.

Figure 5 :

 5 Figure 5: Comparison of the Purcell enhancement (left) and the coupling efficiency (right)between the 2D effective index simulation and a 3D full-wave simulation. The device was optimized for an effective index contrast of ∆n = 0.13, which corresponds to the material platform presented in[START_REF] Stella | Enhanced Directional Light Emission Assisted by Resonant Bloch Surface Waves in Circular Cavities[END_REF].

  . The Purcell enhancement

		10 2						0.5				∆n 0.13
												0.12
	Purcell enhancement	10 0 10 1					Coupling efficiency	0.2 0.3 0.4				0.11 0.1 0.09 0.08 0.07 0.06 0.05 0.04
												0.03
								0.1				0.02 0.01
		10 -1	540	555	570 λ (nm)	585	600	540	555	570 λ (nm)	585	600

Acknowledgments Y.A. and C.R. acknowledge support from the German Research Foundation within the Excellence Cluster 3D Matter Made to Order (EXC 2082/1 under project number 390761711) and by the Carl Zeiss Foundation. M.R. acknowledges the Academy of Finland Flagship PREIN, decision 320166. Part of this work was performed on the HoreKa supercomputer funded by the Ministry of Science, Research and the Arts Baden-Württemberg and the Federal Ministry of Education and Research.