2012 Vol. 37, No. 7 / July 2020 / Journal of the Optical Society of America B Research Article

Sensitivity comparison of free-space andwaveguide Raman for bulk sensingJérôme Michon,* Derek Kita, AND Juejun HuDepartment ofMaterials Science and Engineering,Massachusetts Institute of Technology, Cambridge,Massachusetts 02139, USA*Corresponding author: jmichon@mit.edu

Received 20 April 2020; revised 15 May 2020; accepted 18 May 2020; posted 18 May 2020 (Doc. ID 394973); published 15 June 2020

The sensitivity advantage of waveguide-enhanced Raman spectroscopy (WERS) over free-space Raman, measuredby the signal-to-noise ratio, is well established for thin molecular layer sensing, which traditionally relies on con-focal Raman setups. However, for bulk liquid or gas samples, WERS must be benchmarked against nonconfocalRaman configurations. We use ray tracing to calculate the power collection efficiency of several model free-spacesystems, such as microscopes and probes, encompassing both single-objective and dual-lens systems. It is shownthat considering only the focal volume of the source beam or the confocal volume of the microscope significantlyunderestimates the collected power from free-space Raman systems. We show that waveguide-based systems canstill outperform high signal collection free-space systems in terms of both the signal collection efficiency andsignal-to-noise ratio. ©2020Optical Society of America

https://doi.org/10.1364/JOSAB.394973

1. INTRODUCTION

Waveguide-based chemical sensors have shown sensitivityadvantages over their free-space counterparts thanks to anability to define a large interaction volume that scales with thewaveguide length and to concentrate the electric field in thevicinity of high-index contrast waveguides [1–9]. Waveguidesensors also benefit from the advantages inherent in integrateddevices such as miniaturization, robustness, and manufac-turing scalability. Waveguide-enhanced Raman spectroscopy(WERS) has attracted significant interest as a method leveragingthese benefits for Raman spectroscopy. Several demonstrationshave explored ways to improve performance through materialselection and waveguide design [10–16]. Signal-to-noise ratio(SNR) considerations showed that WERS yields a sensitivityimprovement of several orders of magnitude compared to con-focal Raman systems, particularly in the case of thin molecularlayer sensing [17–19].

However, many applications of Raman spectroscopy, suchas in-line measurement of chemical processes [20–25] or rawmaterials analysis [26–29], involve measuring bulk liquid orgas samples. In these cases, signal collection is not restrictedto the focal or confocal volume. Therefore, using the confocalvolume to estimate the signal collection efficiency of a Ramansystem considerably underestimates the collected power anddoes not allow for adequate benchmarking of WERS sen-sors [30]. While the collection volume has been extensivelystudied in confocal systems (e.g., for the purpose of precisemapping [31,32] or depth profiling [33–35] when varyingparameters such as the pinhole shape and size), such signal

collection from outside the confocal volume has not beenproperly assessed to quantify the sensitivity of different Ramantechniques.

In this work, we compare the bulk sensing performance ofWERS sensors and generic free-space Raman microscope andprobe systems by rigorously computing the power collectionefficiency of free-space setups. Our ray tracing approach prop-erly accounts for the contribution to the collected power fromany point within the bulk sensing region, not just from thefocal volume of the source beam. We evaluate various modelfree-space systems and find that increased confocality (i.e., col-lection mostly from the focal volume), comes at the expense ofmaximized collected power. We show that WERS sensors candisplay better SNR, and thus sensitivity, than free-space sensors,despite the background due to the waveguide core.

In Section 2, we first present the computational approachused to calculate the collection efficiency of free-space systems.We apply this method to exemplary model free-space setupsinvolving either one or two lenses in Section 3, and comparethe collected power and confocality of these setups. We alsodraw a comparison to another commonly adopted Ramantechnique, surface-enhanced Raman spectroscopy (SERS). Wethen use these results to compare the SNR of state-of-the-art,waveguide-based, and free-space systems in Section 4.

2. COMPUTATIONAL APPROACH

The generic setup under consideration to evaluate the powercollection efficiency of free-space Raman consists of a source

0740-3224/20/072012-09 Journal © 2020Optical Society of America

Research Article Vol. 37, No. 7 / July 2020 / Journal of the Optical Society of America B 2013

Fig. 1. Schematic of a model free-space Raman setup for cal-culation of the collection efficiency. Here a two-lens, forward- orbackward-collection (diascopic or episcopic) system is illustrated,but our calculation applies to arbitrary excitation beam profiles andcollection optics configurations.

beam, a sensing region where the source interacts with theanalyte(s), collection optics to gather the emitted Raman light,and a spectrometer to analyze the light, as represented in Fig. 1.Taking the paraxial approximation, the input beam, comingstraight from the laser or shaped by input optics [36,37], canbe modeled by a Gaussian beam with power P0, beam waistw0 at beam center z= z0 (here z denotes the coordinate alongthe propagation direction), and wavelength λ. The Rayleighrange of the beam is then zR = πw

20/λ, its numerical aperture

(NA) is NAbeam = λ/πw0, and, assuming that the attenuationin the sensing region is negligible, its intensity at any point ofcoordinates (x , y , z) is given by

I (x , y , z)=I0

1+ ζ 2exp

[−

2r 2

w20(1+ ζ 2)

], (1)

with I0 = 2P0/πw20 , r =

√x 2 + y 2 the distance to the beam

axis, and ζ = (z− z0)/zR . The collection optics is made byone or several lenses that are each characterized by local length,numerical aperture, and diameter. Finally, the spectrometer isdescribed by the geometry, the étendue (area and NA), and theposition za of its input aperture, an entrance slit or equivalentlya fiber that will then deliver light to the spectrometer. We notethat we are not making an explicit distinction between confocaland nonconfocal systems, since the confocal pinhole can betreated in effect as the input aperture to the spectrometer.

The Raman power emitted by the analyte into a solid angle�at any point of the sensing region is [38]

d P = ρd V · σ I (x , y , z) ·�, (2)

with σ the differential Raman cross-section of the analyte(sometimes noted as dσ/d�= σtot/4π , withσtot being the totalRaman cross-section) and ρ the analyte concentration. Thispower is collected only if the emitted light eventually reachesthe spectrometer aperture and falls within the acceptance angleof the spectrometer, which translates to a position-dependentsolid angle �coll(x , y , z) within which the scattered Ramanlight can be collected. Apparently, the collection solid angle�coll(x , y , z) scales with the étendue of the spectrometer. Thetotal collected Raman power is computed by integrating Eq. (2)over the volume of the sensing region Vs ,

P =∫

Vs

ρσ I (x , y , z)�coll(x , y , z)dV , (3)

where I (x , y , z), given by Eq. (1), is a property of the inputoptics. The collection optics, on the other hand, is described by�coll(x , y , z).

We calculate �coll(x , y , z) using a ray-tracing approach.This approach holds as long as ray optics can be used to describethe system’s behavior. Since we only consider emitted pho-tons in the far field, this is, in general, a valid assumption [39].Diffraction would, however, become significant for very smallapertures or pinholes. For each infinitesimal scattering volumed V = dr · r dϑ · dz centered around the point (x , y , z), andeach direction of azimuthal and longitudinal angles (θ, φ), weconsider the ray emitted from that point along that directionand use the conjugation relations of the lenses of the collectionoptics to calculate the position of the intersection of the ray withthe spectrometer aperture plane. A number of conditions arethen checked to determine whether the ray is actually collected:(1) It must reach and pass through all lenses in the imagingsystem, (2) It must reach the aperture plane at a position withinthe aperture, and (3) It must reach the aperture with an anglesmaller than the relevant numerical aperture (NA). This analysisyields a binary function2(x , y , z, θ, φ):

2(x , y , z, θ, φ)={

1 if the ray is collected,0 otherwise.

(4)

The collection solid angle is then obtained with

�coll(x , y , z)=∫2(x , y , z, θ, φ) sin(θ)dθdφ. (5)

To properly compare different systems, we define the effectiveinteraction length leff ≡ P/ρσ as an analyte-agnostic metricthat depends only on the characteristics of the optical systemand can be readily calculated from Eq. (3). Although the con-cept of effective height of the scattering volume (also calledintegrated collection efficiency) has been defined in previousworks and its value measured experimentally for a few differ-ent systems [40–42], it has not yet been used, to the best ofour knowledge, to systematically compare the performance offree-space systems.

We verified the validity of our approach and of its imple-mentation by simulating the experimental setup from [42],which uses an input beam withw0 = 80 µm at λ= 514.5 nm,a Canon 50 mm f /1.2 objective, and a 250× 2000 µm2 slit.Our calculations yield an effective length∼10% larger than thereported measured value, which may be explained by reducedcollection due to aberrations and the additional optics used forbeam routing. We also checked that, under conditions wherethe throughput of the system is limited by the collection aper-ture’s size or NA, the computed collected power scales with theétendue of the spectrometer.

3. RESULTS FROM MODEL SYSTEMS

We use this approach to evaluate the power collection efficiencyof free-space Raman systems, measured by their effective inter-action length leff.

The collection étendue (given by E = π S ·NA2, with S thecollection area) influences several key characteristics of a system,including its collection efficiency and resolution. In free-space

2014 Vol. 37, No. 7 / July 2020 / Journal of the Optical Society of America B Research Article

Fig. 2. (a) Effective interaction length of a single-lens system as a function of beam center position (distance from the objective, on one side) andaperture position (distance from the objective, on the other side), for an objective with fo = 2 mm and NA= 0.9 in air, and an input beam withw0 =

2 µm at λ= 785 nm. The black line corresponds to the position of the beam center image according to the thin lens formula. (b) Collection solidangle over the entire sensing region for the maximum of (a): z0 =−4 mm and za = 4 mm. The dashed white line shows the image of the aperture.(c) Effective interaction length of a single-lens system as a function of objective focal length and input beam waist. The beam center and aperture posi-tion were set at their optimal positions of z0 =−2 fo and za = 2 fo , and the objective diameter at 40 mm. The yellow cross indicates the parametersused for (a), (b), and the rest of the computations.

in particular, the limiting component (i.e., with the smallestétendue) is the spectrometer. As our metric of interest is thecollection efficiency, which grows with the system’s étendue, wefix the aperture characteristics to study the influence of otherparameters without a loss of generality. A representative valueis obtained by looking at typical collection fibers and slits usedin Raman spectroscopy. Raman probes rely on collection fiberswith diameter and NA ranging from 50 µm and 0.2 to 200 µmand 0.5 [43,44]. The étendue indicates that such fibers areequivalent to slits with an area and NA from 10× 1000 µm2

and 0.1 to 25× 2000 µm2 and 0.4. To get an upper bound onthe collection efficiency, we assume in all of the following thelargest of these values: a large circular aperture with diameter200 µm and NA= 0.5 (E ≈ 25,000 µm2

· sr). We note that,while even larger slits are possible, this represents a generousassumption as the trade-off between throughput and resolu-tion for grating spectrometers often requires the use of smallerapertures. This constraint will be discussed in more detailin Section 4.

A. Bulk Free-Space Raman

We first consider the simplest case of a single objective usedto couple Raman scattering to the spectrometer (Fig. 1 with-out the tube lens). Such setups may be useful in applicationswhere simplicity and compactness are paramount, such as onunmanned aircraft vehicles (UAVs) [45,46]. In Fig. 2(a), we plotthe value of leff while varying the position of the aperture andthe position of the beam center along the optical axis (z axis).We find that, for a given beam center position, the collectedpower is maximized when the aperture is located at the imageposition of the beam center [black line on Fig. 2(a)], such thatthe region with the maximum collection solid angle, shown onFig. 2(b), overlaps with the region of maximum Raman emissionintensity, near the beam center. Furthermore, the highest valueis obtained for a beam center position of z0 =−2 fo (with z= 0set at the position of the objective), corresponding to an aperturelocated at za = 2 fo . For macroscopically uniform samples,bulk sensing demands objectives with a longer focal length fo

than in imaging and mapping, as an increased depth of field isdesirable to maximize the amount of light collected. However,for a given objective size, a longer focal length results in a smallerNA, which limits the collection. For a fixed 40 mm diameterlens, Fig. 2(c) shows that this trade-off between the depth offield and NA leads to a roughly constant collection efficiencywithin a certain range of focal length. The efficiency becomeslimited by the finite aperture of the lens for fo > 20 mm.Overall, we find a maximum effective interaction length of0.67 mm. We again note that 40 mm represents a generousassumption for the lens diameter, as industrial applications suchas in-line monitoring often require the use of smaller lenses[20–23]. In practice, the choice of the beam waist (or spot size)of the input beam obeys trade-offs and ultimately dependson the specific application. For example, confocal Raman orhigh-resolution imaging require high-divergence beams witha waist down to a few hundred nanometers, while wide-waist,low-divergence beams are needed for better spatial averaging orto reduce photodamage [47]. For the collected power however,Fig. 2(c) shows that varying the beam waist in the 1-50 µmrange has little (<10%) influence on a system’s maximumpower collection efficiency. We thus assume an input beamwith w0 = 2 µm at λ= 785 nm in the following. We notethat all the results presented here were calculated with lensessurrounded by air on both sides, but the observed trends remaintrue for any other configuration (e.g., an objective immersedin water on the sensing side and air on the spectrometer side).Corresponding lengths in other configurations can be calculatedusing the interfaces’ refractive power [48]. For the objectiveconsidered in Fig. 2(a), the values fo = 2 mm, NA= 0.9 inthe air–lens–air configuration become fo ,water = 3.97 mm,fo ,air = 2.98 mm, NAwater = 1.2 in the water–lens–airconfiguration.

The majority of Raman setups (e.g., microscopes andprobes), however, comprise at least two lenses: A tube lens isadded after the objective to focus the collected light onto thespectrometer aperture (as shown in Fig. 1). The tube lens istypically chosen to have a large focal length ft and the aper-ture is located at its focal point. The overall magnification

Research Article Vol. 37, No. 7 / July 2020 / Journal of the Optical Society of America B 2015

Fig. 3. (a) Effective interaction length leff and ratio P /Pfocal (withPfocal =

∫|z−z0|≤zR

Pslice(z)dz the power collected from the focalvolume) as a function of the tube lens focal length ft . (b) Power col-lected per slice Pslice/ρσ as a function of slice position (distance fromthe objective), for various ft . The beam center position was set atz0 =− fo and the aperture at za = ft .

is then given by ft/ fo . Here we study the influence of thetube lens on the setup’s performance and set the objective’sspecifications at fo = 2 mm and NA= 0.9, as in Fig. 2. Thetube length (distance between the objective and the tube lens)typically ranges from 160 to 200 mm [49] and has a mini-mal influence on the system performance. For this study, itwas set to 200 mm. Figure 3(a) shows the influence of thefocal length of the tube lens. We find that the effective lengthdecreases with the increasing focal length of the tube lens.This effect is rationalized by considering the power collectedfrom each slice along the propagation of direction, definedas Pslice(z)=

∫ρσ I (x , y , z)�coll(x , y , z)dxdy , such that

P =∫

Pslice(z)dz. We see in Fig. 3(b) that increasing the focallength of the tube lens reduces the amount of light collectedfrom outside the focal volume [red curve converging toward 1on Fig. 3(a)]. It is thus apparent that the addition of a long-focallength tube lens naturally increases the confocality of the setup,even without a pinhole. However, since this increased spatialselectivity comes with a decreased total signal collection, thisconfirms that confocal free-space systems do not provide thebest bulk sensing performance.

B. Surface-Enhanced Raman

The effective interaction length is also a primordial conceptin surface-enhanced Raman spectroscopy (SERS), where itunderlies the determination of the enhancement factor (EF)[41]. The majority of SERS measurements with substrate ensurethat only a single layer (or a few layers) of target molecules areprobed, either by functionalizing the SERS surface or ensuringthat there is only monolayer adsorption on the SERS surfacebefore measurement, thereby automatically restricting theinteraction volume to this monolayer [50–52]. However, thisis not the case when probing bulk, nondried liquids, in whichthe target molecules are not restricted to the area in close contactwith the SERS substrate [53–55]. This raises the question ofthe relative contribution of the SERS-enhanced signal and thebulk signal. Similar to the standard Raman case, our approachallows for precise, nonarbitrary calculation of the effectiveinteraction length. The setup under consideration for the SERScalculation is presented in Fig. 4(a). An enhancement layer isintroduced, corresponding to the thin region on the top of theSERS substrate where the Raman signal is enhanced. This layeris characterized by its thickness tSERS (typically a few nanome-ters, due to the r−10 distance dependence [50]) and the value ofthe enhancement factor ESERS, assumed for simplicity to be uni-form through the layer and absent elsewhere. The sensing regionis restricted to the space between the enhancement layer and theoptics. This amounts to including an extra factor E (x , y , z) inEq. (3), such that

E (x , y , z)={

ESERS for 0≤ z≤ tSERS,

1 otherwise.(6)

Fig. 4. (a) Schematic of the simulated SERS setup. (b) P /PSERSlayer

as a function of the SERS EF and the SERS layer thickness.

2016 Vol. 37, No. 7 / July 2020 / Journal of the Optical Society of America B Research Article

On Fig. 4(b), we show the ratio of the total collected powerto the power collected only from the enhancement layer,PSERS layer =

∫ tSERSz=0

∫x ,y ρσ E (x , y , z)I (x , y , z)�coll(x , y , z)

dxdy dz, for different layer thicknesses and enhancement fac-tors. The results indicate a strong dependence of the collectedpower on the parameters of the enhancement layer. At a low EFand/or thin enhancement layer, the total power is up to 2,000×greater than the power from the enhancement layer, indicatingthat the signal generated by the bulk of the analyte sample (asopposed to the enhancement layer only) dominates the SERSsignal. In these cases, SERS may, therefore, not offer as largean advantage as suggested by the enhancement factor. Beyondrestating the need for careful characterization of the scatteringvolume, our approach thus outlines the specific conditionswhere single-layer (or few layer) coverage is required to trulybenefit from the SERS enhancement.

C. Waveguide Raman

The derivations to calculate the power collection efficiency ofwaveguide-based Raman systems have been described previ-ously [15,16]. The maximum power collected after the sensingregion depends on the collection scheme: forward or backwardcollection. It is a function of the waveguide mode profile, propa-gation loss α, and length of the sensing waveguide lwg. In thebackscattered collection configuration, the Raman power isgiven by

Pwg output,x = Pwg input ×1

2ρxσxηx

1− e−2αlwg

2α, (7)

where ρ is the molecular density, σ is the differential Ramancross-section, ηx = λ

2n2g

∫∫x |E|

4/(∫∫

c s ε|E|2)2

, Pwg input is thesource power at the start of the sensing region, and the factor 1/2accounts for the fact that only half of the Raman scattered light iscoupled into the backward-propagating mode. The subscript xdenotes integration over either the top cladding with the analyte(a) or the waveguide core (c). Both η and α can be computedfrom mode profile simulations [56], allowing for straightfor-ward calculation of the collected power in waveguide-basedRaman systems. For the fundamental TM mode of a 200 nmthick, 450 nm wide silicon nitride (SiN) strip waveguide, we getηa = 0.23 sr and ηc = 0.41 sr. Losses down to<0.1 dB · cm−1

have been reported by multiple foundries [57–59].

4. SENSITIVITY COMPARISON

The effective interaction length results we obtained from thediscussions above allow us to compare the relative performanceof WERS and free-space Raman. The figure of merit for Ramanspectroscopy is the SNR, which ultimately defines the limit ofdetection (LoD). Neglecting long-term drifts, which can beremoved by calibration, and the dark noise from the detectorwhose value (∼fW) is small compared to the powers consideredhere (∼pW), the sensitivity of Raman spectroscopy is limited byshot noise from the Raman background. The SNR for WERSsensors with high on-chip source rejection, such that the fibercontribution is negligible, is given by [16]

SNRWERS =

√γWERS Psource

1− e−2αlwg

4α

×ρaσaηa√∑

(ρiσi) ηa + ρcσcηc

, (8)

where γWERS are the total losses through the WERS system,Psource is the source power, and the subscript i denotes all speciesin the sensing region (not only the analyte of interest). We omit-ted the integration time and the responsivity of the detectorsince it is reasonable to assume that they are the same for thedifferent configurations under consideration here. Free-spaceRaman is free of any background from the waveguide core andfibers,

SNR f−s =PRaman, analyte of interest√

PRaman, all analytes

=√γ f−s Psourceleff

ρaσa√∑(ρiσi )

, (9)

where γ f−s is the total loss through the system, and leff is theeffective interaction length calculated in Section 3.

The SNR ratio between waveguide and free-space Raman isthen

SNRWERS

SNR f−s=

√γWERS

γ f−s

1

leff

1− e−2αlwg

4α

×ρaσaηa√∑

(ρiσi ) ηa + ρcσcηc

√∑(ρiσi )

ρaσa. (10)

We plot this ratio on Fig. 5 for a model system, iso-propyl alcohol (IPA) in water as a function of the sensingwaveguide length and the weight percentage of IPA(wt%=mIPA/(mIPA +mwater)× 100), for the 819 cm−1

and 2882 cm−1 Raman peaks of IPA and a SiN WERS platform.The free-space effective interaction length was set at 0.67 mm,the maximum found on Fig. 2(c) with a E ≈ 25,000 µm2

· sraperture.

The Raman cross-sections of IPA are σIPA,819 = 7.92×10−31 cm2/sr and σIPA,2882 = 2.96× 10−30 cm2/sr [60].From [61], we get ρSiNσSiN,819 = 1.2× 10−9(sr · cm)−1, andwe infer ρSiNσSiN,2882 = 2× 10−11 (sr · cm)−1 from the spec-trum provided on Fig. 2(a) of [61]. We estimate the Ramancross-section of water at 819 cm−1 and 2882 cm−1 by compar-ing the full Raman spectrum of water [62,63] to the measuredcross-section of the 3400 cm−1 band, σwater,3400 = 1.21×10−30 cm2/sr [64]; therefore, σwater,819/σwater,3400 ≈ 0.07,yielding σwater,819 = σwater,2882 = 8.47× 10−32 cm2/sr andρpure waterσwater = 2.8× 10−9 (sr · cm)−1. These cross-sectionsare calculated for a source wavelength of 785 nm, assuming 1/λ4

scaling of the cross-section.The other parameters used for Fig. 5 are λ= 785 nm,

Psource = 500 mW, α = 0.1 dB/cm, ηa = 0.23 sr, ηc = 0.41 sr,ρpure IPA= 7.87×1021cm−3, and ρpure water = 33.36×1021cm−3.The WERS losses include fiber-chip coupling (∼1 dB [65,66]),propagation loss, and reflections at components, for an esti-mated total of γWERS = 5 dB. The free-space losses due to fiber

Research Article Vol. 37, No. 7 / July 2020 / Journal of the Optical Society of America B 2017

Fig. 5. Ratio of WERS and free-space SNR, given by Eq. (10),for IPA in water at Raman shift of (a) 819 cm−1 and (b) 2882 cm−1.The isoline for ratio= 1 is highlighted in black. (c) Influence of thewaveguide core material on the SNR of WERS, for IPA in water at819 cm−1.

coupling and misalignment are estimated at γ f−s = 3 dB. Themaximum waveguide length considered is 1/α ≈ 40 cm.

We find that WERS offers a higher SNR than free-spaceRaman for sufficiently long waveguides. Longer waveguidesimprove the SNR of WERS as they yield a higher collectedRaman power, according to Eq. (7). The relative performanceof WERS also improves at higher analyte concentrations whenthe analyte signal overcomes the background signal from thewaveguide core, which is slightly higher than the backgroundfrom water due to the larger Raman cross-section of SiN. Theinfluence of the background signal is highlighted by the com-parison of Figs. 5(a) and 5(b), as the SiN Raman cross-sectionis much lower at 2882 cm−1 than at 819 cm−1. Selecting awavenumber region with a reduced signal from the waveguidecore decreases the waveguide length necessary to reach a givenSNR level by as much as an order of magnitude. The maximumimprovement through waveguide core material engineeringis quantified on Fig. 5(c), showing an 8× achievable enhance-ment. The improvement plateaus for waveguide core materialswith adequately small Raman cross-sections as the backgroundsignal would become negligible compared to the signal fromthe analyte. While this analysis suggests that the cross-section ofSiN at 2882 cm−1 would be sufficiently low, characteristic peaks

of analytes may be located in any wavenumber region and thusan improved SNR across the entire spectrum of Raman shifts isdesirable. As SiN and other amorphous materials are limited bytheir inherently broadband Raman activity with a cross-sectiononly decreasing at high Raman shifts [67], a solution for genericWERS detection of ultra-low concentrations and/or weaklyRaman active analytes could reside in crystalline materials suchas silicon carbide (SiC) or sapphire (Al2O3). These materialspossess a very low Raman background over a wide wavenumberrange, with only localized Raman peaks that can be readilyisolated [68–71].

It is also of interest to consider the case of systems withconstrained étendue such that only one mode is supported,because it allows for a minimum spectrometer volume at agiven resolution. This condition corresponds to diffraction-limited systems with E = λ2 [72]. For integrated platforms,it is naturally satisfied by single-mode waveguides, and theresults used earlier are thus unchanged. In free-space, the spec-trometer’s aperture must be reduced to meet this condition.We use the dimensions of a single-mode fiber for the 780–970 nm range, a 4.4 µm diameter, and NA= 0.13 (yieldingE = 0.807 µm2

· sr≈ 900 nm2· sr), to calculate the effi-

ciency for the optimal system found on Fig. 2(c), and findleff = 0.0032 mm. Using the parameters of Fig. 5(b), it leadsto a SNR ratio between waveguide and free-space Raman of upto 64.15. As expected, reducing the aperture’s étendue leads toa degraded collection efficiency for free-space systems, and acomparatively better performance by waveguide-based systems.

In practice, however, free-space systems are not limited bythe volume of the spectrometer, but by the trade-off betweenthroughput and resolution. The aperture is thus chosen to beas large as allowed by the desired resolution, which may rangefrom <5 cm−1 for high-resolution Raman to >20 cm−1 forlower end Raman. Since the exact resolution of a spectrom-eter depends on a variety of parameters, we use commerciallyavailable systems to estimate the typical resolution of a gratingspectrometer that uses the slit specifications used in our cal-culations (E ≈ 25,000 µm2

· sr). For such slits, the AvantesAvaSpec-HERO [73], B&W Tek Glacier X [74], and OceanInsight QEPro Raman [75] have resolutions ranging from 0.6to 2.2 nm, which around 785 nm corresponds to 9− 43 cm−1.This confirms that we assumed a slit larger than those realis-tically used in high-performance Raman systems, and thusobtained an upper bound to these systems’ collection efficiency.We note that the throughput-resolution trade-off is relaxed forinterference-based and Hadamard spectrometers [76–78].

5. CONCLUSION

Our study demonstrates the advantage of waveguide-basedRaman over free-space Raman for bulk chemical sensing,thanks to the superior power collection efficiency of long wave-guides that outweighs the background signal increase due tothe waveguide core material. This background Raman signal,intrinsically present in amorphous materials, strongly influencesthe performance of WERS. Crystalline materials with verylow Raman cross-sections may thus represent a path towardhigh-sensitivity, compact WERS sensors. The dependence of

2018 Vol. 37, No. 7 / July 2020 / Journal of the Optical Society of America B Research Article

the free-space collection efficiency on the size of the spectrom-eter’s aperture also highlights the benefits of waveguides as acollection platform, allowing for minimum spectrometer size,or equivalently maximum resolution.

Furthermore, our calculations of the free-space collectionefficiency confirm the intuitive trade-off between localizedcollection (confocality) and overall power collection efficiency.They also highlight the need for precise definition and evalu-ation of the volumes under consideration in free-space systems,including SERS. It notably cannot be assumed a priori that thecollected signal from bulk samples originates only from the focalvolume of the input beam or the confocal volume of the collec-tion optics, which would result in a severe underestimation ofthe collected power.

Our ray-tracing model could be improved in several waysby, for example, implementing the Fresnel equations instead ofassuming thin lenses or taking aberrations into account. Othertypes of lenses such as ball lenses also could be included in thecomparison, although we expect their collection efficiency to belower due to aberrations. This model is nonetheless sufficient tocompare free-space setups and benchmark them against WERSsensors.

Funding. Deshpande Center for Technological Innovation,Massachusetts Institute of Technology.

Acknowledgment. The authors thank Tian Gu forinsightful discussions.

Disclosures. The authors declare no conflicts of interest.

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