Photon number resolution without optical mode multiplication

2.1 Salisbury screen design for a single-layer SNSPD

A typical bare meander nanowire used for fabricating SNSPDs absorbs light weakly and, thus, is unsuitable for efficient light detection. Even placing such an SNSPD in the Salisbury screen geometry, where a reflector – metallic mirror of distributed Bragg reflector (DBR) is placed beneath the SNSPD with a λ/4-spacer in between, Figure 2A, does not allow to reach the total light absorption. We demonstrate this in Figure 2, where we calculate the optical response of such a structure assuming a meander nanowire made of niobium–titanium nitride (NbTiN) film with a standard filling factor f = 0.5 and incident light polarization parallel to the meander (grating) slits. In this case, the meander can be treated within the effective medium approximation as a uniform film with effective permittivity ε eff = ε film f + ε slit 1 − f , where ɛ_film and ɛ_slit represent the permittivity of the superconductor nanowire film (NbTiN) and that of the nanowire surrounding region (dielectric matrix), respectively [52, 53]. In Figure 2B, we calculate how the intensity reflection R and absorption A = 1 − R coefficients (transmission is negligible) of the Salisbury screen structure at telecom wavelength of λ = 1550 nm change during deposition of the nanowire layer (solid line) on top of the spacer with the refractive index n_sp = 1.5. Reflection is suppressed and absorption reaches unity for the nanowire thickness of 15 nm. This is the point of maximum absorption of the SNSPD layer. Further increase of the SNSPD layer thickness induces a decrease in absorption. The circular diagram in Figure 2C provides a useful representation of this behavior. Here, the complex amplitude reflection r coefficient of the structure is plotted for continuously increasing thickness of the SNSPD layer (representative thicknesses are indicated by symbols as in Figure 2B). The thickness of the nanowire affects mostly the magnitude of r, which reduces to nearly zero for 15 nm thick nanowire layer. The reflection and absorption spectra of the Salisbury screen structure with the 15 nm thick nanowire are shown in Figure 2D, where the optical constants for the reflector and spacer are assumed wavelength-independent while the dispersion of NbTiN film is accounted for. The complex relative permittivity of NbTiN film was taken from Ref. [54] where it was derived for a 5 nm thick film, the typical thickness of optimized SNSPDs (SEM images in Figure 1). The characteristic oscillations seen in the optical response are related to the spacer thickness optimized for absorption at 1550 nm.

Figure 2:

Salisbury screen SNSPD design. (A) The nanowire detecting layer (Det) is separated from the reflector (Refl) by a λ/4-spacer, thus placing the nanowire at the anti-node of the standing wave. In the calculations, the mirror is treated as a semi-infinite layer with a reflectivity of 99.9%. (B) The optical response of the structure-reflection R and absorption A, is calculated while sweeping the thickness D of the nanowire. (C) The corresponding amplitude reflection r coefficient is shown in the circular diagram, where the radial (angular) coordinate defines the magnitude (phase) of r. In (B) and (C), empty circles and squares mark the beginning (D = 0 nm) and end (D = 50 nm) of the deposition, respectively, while diamonds mark the optimal thickness. (D) The reflection R and absorption A spectra are calculated for an optimal SNSPD thickness of D_opt = 15 nm. The nanowires are assumed to be carved out of a NbTiN film with permittivity ε film = 4.21 + i 3.87 2 at 1550 nm.

Although full absorption could be achieved with NbTiN thickness of 15 nm, practically, the superconducting films need to be thinner than 10 nm, as the absorbed photon cannot break superconductivity in thicker layers, and the internal efficiency drops. As a result, the Salisbury screen is replaced by a resonant cavity in detectors with high system efficiency at the expense of sandwiching SNSPD between complex multilayered structures or extreme phase sensitivity [55, 56]. Alternatively, nanophotonic structures such as dielectric gratings and photonic crystals [57, 58] could be used to achieve perfect absorption in thin SNSPDs. So far, these structures have been largely unexplored in superconducting detector implementation.

2.2 Total absorption by partially absorbing nanowire layers

The optical response of a bare nanowire meander is shown in Figure 3. Assuming single side (traveling wave) illumination, as in Figure 2, we calculate the intensity transmission T_tr (red line), reflection R_tr (blue line), and absorption A_tr = 1 − T_tr − R_tr (black line) coefficients of the nanowire layer as a function of its thickness, Figure 3B. All calculations are done using the transfer matrix method (see Methods below) within the effective index approximation (see Supplementary Materials for comparison with finite element method simulations). The optical response shown in Figure 3B reveals that the maximum absorption is just 50%, which is a fundamental limit of traveling wave absorption of any subwavelength absorptive layer [59].

Figure 3:

A single-layer NbTiN SNSPD absorber. (A) Free-space nanowire absorber with variable thickness D. (B) Optical response is evaluated under traveling (intensity transmission T_tr, reflection R_tr, and absorption A_tr coefficients) and standing (intensity absorption A_coh coefficient) wave illumination as a function of the nanowire thickness D. (C) The corresponding amplitude transmission t_tr and reflection r_tr coefficients are shown in the circular diagram, where the radial (angular) coordinate defines the magnitude (phase) of t_tr and r_tr. In (B) and (C), empty circles and squares mark the beginning (D = 0 nm) and end (D = 100 nm) of the deposition, respectively, while diamonds mark the optimal thickness. (D) The transmission T_tr, reflection R_tr, and absorption A_tr and A_coh spectra are calculated for an optimal thickness of D_opt = 30 nm. Similar calculations for NbN and MoSi SNSPDs are provided in Supplementary Materials.

On the other hand, this fundamental limit can be overcome when the partially absorbing film is placed in a standing wave appearing as a result of interference of two counter-propagating coherent beams, as in Figure 1A. When two beams are in phase, the anti-node (maximum electric field) of the standing wave coincides with the nanowire, resulting in enhanced absorption (coherent absorption regime) [37, 42, 48]. The coherent absorption coefficient, A_coh, of the nanowire for this regime is shown by the green line in Figure 3B. Compared to traveling wave absorption, absorption of the standing wave doubles at the optimal thickness, D_opt = 30 nm, to reach 100%. The nature of this enhancement is clearly seen in the circular diagram in Figure 3C. Here, the complex amplitude transmission t_tr and reflection r_tr coefficients are plotted for continuously increasing film thickness. Amplitude transmission and reflection coefficients are of equal amplitudes but opposite phases at the optimal film thickness (marked by diamonds): t_tr ≈ 0.5 and r_tr ≈ −0.5. The amplitudes of outgoing waves on both sides of the SNSPD are given by t + r and thus cancel, resulting in perfect light absorption. Remarkably, such a mechanism is effective in an extremely broad range of wavelengths, resulting in coherent absorption greater than 93% throughout the entire spectral range and greater than 99% between 600 and 2200 nm, as shown for the film of optimal thickness in Figure 3D. This is in stark contrast with the narrowband operation of the Salisbury screen due to the presence of a spacer. Conversely, when the input waves are out of phase, the amplitudes of the outgoing waves, given by ±(t − r), equal the amplitudes of the input waves and absorption is completely suppressed (coherent transmission regime) [37, 48]. In this case, the nanowire absorber is positioned at the node of the standing wave, where the electric field vanishes.

The coherent absorption regime is equivalent to the Salisbury screen geometry. In the latter case, a standing wave results from interference between the incident and reflected on the reflector waves. Due to the λ/4-spacer and π-phase shift on the reflector, the nanowire is always placed at the anti-node. Thus, while the Salisbury screen and counter-propagating geometries provide the same maximum light absorption, the counter-propagating geometry provides the additional flexibility to control the absorption level between zero and unity by changing the mutual phase of two interfering beams. This may be of interest for feed-forward protocols or self-configuring optical networks [60, 61]. At the same time, the interferometric configuration of light detection (inset in Figure 1A) relies on a well-defined mutual phase of interfering beams and, thus, is subject to the thermal and mechanical noise of the interferometer. Stabilization of the interferometer phase [47] will be required in this case. Conversely, the efficiency of light absorption in the Salisbury screen geometry is insensitive to phase fluctuations. In all cases, the light source noise and the quantum state of light do not affect the efficiency of light absorption (see Note B in Supplementary Materials).

The standing wave picture also suggests a simple way to achieve total light absorption from nanowire layers of arbitrarily small thickness. Let us assume the nanowire layer of optimal thickness D_opt = 30 nm is “cut” into multiple sublayers. The sublayers positioned in the same anti-node of the standing wave would interact equally with the electric field (unlike the case of a thick absorber stretching across different regions of the standing wave). Equivalently, each sublayer may be positioned at different anti-nodes of the standing wave while experiencing the same interaction with the electric field. Thus, the optical response of this distributed structure would be identical to that of the original single-layer nanowire absorber. For example, let us consider a distributed structure consisting of three sublayers (three SNSPDs), Figure 4A, of thickness D_opt/3 = 10 nm each. For practical implementation, let us also assume that the SNSPDs are separated by dielectric spacers of thickness d_sp = λ/(2n_sp) rather than free space. Such spacers are known in thin film optics as “absentee” layers, as they do not change the reflection of the structure beneath [62]. Figure 4B shows how the optical response of the structure changes as the three nanowire sublayers (Det₁, Det_2,, Det₃ – solid lines) and the two dielectric spacers (Sp₁, Sp₂– dashed lines) are deposited consecutively. The completed structure is characterized by the same optical response as the original single-layer SNSPD of optimal 30 nm thickness (cf. Figure 3B).

Figure 4:

Three-layer distributed detector. (A) Distributed detector consists of three nanowire layers – Det₁, Det_2, and Det₃ and two λ/2-spacers Sp₁ and Sp₂. The intensity (B) and amplitude (C) coefficients are evaluated continuously as layers are deposited, starting with Det₁. In (B) and (C), solid (dashed) lines correspond to the deposition of the nanowire layers (spacers). Empty circles and squares mark the beginning and end of the deposition. (D) The spectrum of the entire structure.

The contribution of each layer of the distributed absorber can be understood from the circular diagram in Figure 4C. While the nanowire sublayers change the magnitudes of t_tr and r_tr, each spacer induces a π-phase shift for transmission and a 2π-phase shift for reflection, without changing the transmission and reflection magnitudes. Thus, the response of the entire multilayer structure is t_tr ≈ 0.5 and r_tr ≈ −0.5, as required to achieve coherent perfect absorption. Unlike coherent perfect absorption in the single nanowire layer with optimal thickness, however, the optical response of the multilayer structure strongly depends on wavelength, Figure 4D. Such dependence becomes more pronounced as the number of layers increases, an issue that in thin film optics is commonly addressed by introducing achromatizing layers [62].

Overall, distributed absorbers provide an alternative architecture to achieve total absorption in SNSPDs without the need for complex DBR or optical cavity structures that are fairly intolerant to the thickness of their components [63]. Furthermore, when implemented with independent on–off or PNR detectors, distributed coherent perfect absorption allows photon number resolution methods without spatial or temporal multiplexing.

2.3 Absorption uniformity across constituent detectors

Let us consider an array of equally spaced nanowire sublayers placed at the anti-nodes of a standing wave, Figure 1A. In the regime of coherent perfect absorption, this arrangement allows complete absorption of multiple photons distributed across the SNSPD array, which may be employed for photon number resolution.

For accurate photon number resolution, the number of constituent detectors in the array should be large enough to reduce the probability of absorbing more than one photon in the same constituent detector. For instance, to resolve a two-photon state with a probability of 90%, one would need at least 10 detectors (see discussion below). Thus, one should maximize the number of sublayers into which the optimal absorber thickness, D_opt, that guarantees coherent perfect absorption is split. Practically, this can be achieved by reducing the thickness of each nanowire sublayer (although, practically, it is challenging to fabricate high quality superconducting films thinner than 5 nm), or by decreasing the filling factor of the nanowire meanders in each sublayer absorber. In Figure 5A, the optimal thickness is plotted as a function of the filling factor f (solid blue line). Empty circles mark D_opt = dN_Det, where d = 5 nm is the thickness of each nanowire sublayer and N_Det = 5, 10, 15 is the number of layers (detectors) into which D_opt can be “cut”. The horizontal coordinates of these points (dashed vertical lines) define the filling factor of the nanowire meanders. On the same graph, we plot the absorption of a single bare nanowire sublayer (5 nm thick) under traveling wave illumination as a function of the filling factor (dashed black line). For instance, a distributed detector can be built out of five, ten or fifteen nanowire sublayers with a thickness of 5 nm each and a filling factor of 0.61, 0.30, and 0.20, respectively. A single bare nanowire with these parameters would absorb just 28%, 17%, and 12% of light.

Figure 5:

Performance of a multilayer distributed detector. (A, left vertical axis): Optimal thickness (solid purple line) of a single-layer SNSPD as a function of the filling factor. (A, right vertical axis): absorption (traveling wave) of a 5 nm thick nanowire as a function of the filling factor (dashed black line). Empty circles mark parameters for fabricating five-, ten-, and fifteen-layer distributed detectors. (B)–(D) For these three cases, circular diagrams and absorption per layer (under a standing wave illumination) are shown.

The number of sub-layers and their individual absorption are tightly interlinked with the absorption uniformity achievable throughout the detectors. Since the optical thickness of each constituting absorber is finite, absorption is not uniformly distributed. Non-uniform absorption would then result in some detectors having higher multi-photon absorption probability, which effectively reduces the photon number resolution of the array.

To quantify the uniformity of light absorption, we introduce the absorption non-uniformity parameter Δ as the standard deviation of the absorption per layer from uniform absorption:

where A_coh and A_coh,i are the total coherent absorption of the distributed detector and coherent absorption of the ith layer, respectively, and N_Det is the number of the detecting layers in the structure. By dividing Δ by Δ max = A coh N Det N Det − 1 which holds when all the absorption happens in one constituent detector (maximum absorption non-uniformity), one can define the normalized non-uniformity parameter:

The parameter Δ_norm varies between zero for uniform absorption, A_coh,i = A_coh/N_Det, and unity for maximally non-uniform absorption.

Figure 5B–D shows the circular diagrams and absorption per layer (under a standing wave illumination) of representative distributed detector structures with the number of sublayers N_Det = 5, 10 and 15. The circular diagram for the five-layer detector, Figure 5B (left), shows that the entire structure fulfills the required optical response of t_tr ≈ 0.5 and r_tr ≈ −0.5 with conventional 5 nm thick nanowire detectors with 0.61 filling factor. The absorption uniformity of such structure is shown in Figure 5B (right). As the total thickness of five sublayers (25 nm) is well within the subwavelength approximation, the absorption is uniform, with negligible non-uniformity parameter Δ_norm = 0.0008. Increasing the number of sublayers to ten requires a reduction of the filling factor of each nanowire absorber to 0.30 (Figure 5A) to achieve the required optical response, Figure 5C (left). Absorption remains rather uniform throughout the ten sublayers, with Δ_norm = 0.0022, Figure 5C (right). Note that, for any distributed detector with an even number of layers, transmission and reflection coefficients are in phase, t_tr ≈ −0.5 and r_tr ≈ −0.5. Such a detector absorbs out-of-phase counter-propagating beams or λ/4-shifted standing wave [48, 49, 51]. For a fifteen-layer distributed detector composed of nanowire sublayers with a filling factor of 0.20 (Figure 5A), deviation from the subwavelength approximation is still tolerable. While the phase difference between t_tr and r_tr slightly decreases, Figure 5D (left), the total coherent absorption reaches 98% with uniform absorption per layer, Δ_norm = 0.0040, Figure 5D (right).

To increase the number of layers even further, an additional optimization would be required where the more general condition t_tr ≈ ±r_tr < 0.5 can be reached with detectors of larger total thickness. The latter condition still guarantees perfect absorption but not perfect transmission. At the same time, a distributed detector can be used as a unit cell for spatial or temporal multiplexing, increasing the total number of detectors by order of magnitude. In the former case, a 3D detector can be assembled with multiplication in all three spatial coordinates.

In addition to PNR capability, a multilayered distributed structure can substantially improve the temporal performance of single-photon detection in two ways. First, constituent nanowire sublayers have a lower filling factor and, thus, shorter nanowires. This, in turn, decreases the kinetic inductance of the nanowire, shortening the detectors’ recovery time (recovery time is inversely proportional to the kinetic inductance). Second, multiple sublayers prevent two consecutive photons from being absorbed within the same sublayer allowing higher photon flux to be registered. Also, such detectors can be used to measure light correlation functions: for instance, the g1 function can be measured in the interferometric configuration, as in the inset in Figure 1A with an additional delay line in one arm of the interferometer, or the g2 function can be measured by recording coincidence counts between different detecting layers as the function of electronic delay.

2.4 Photon-number resolution with a distributed detector

In this section, we first reproduce the framework for analyzing PNR detection in conventional (incoherent) schemes, like spatial and temporal multiplexing, following Refs. [64, 65]. Then, we apply this procedure to coherent detection with a distributed detector and analyze in detail the performance of the ten-layer distributed SNSPD introduced in the previous section.

According to the postulate of quantum mechanics, the measurement procedure is described by a set of measurement operators and a corresponding transformation of the quantum state after the measurement. When the latter is of no interest, as in the case of light detection where photons are absorbed, the POVM (“positive operator-valued measure”) formalism is widely used [66]. For a single on-off detector, like SNSPD, the POVM set (set of operators describing the measurement process) consists of two operators [64]

which are associated with the corresponding outcomes of the measurement: zero and one count. Here 0 is the vacuum state and n ̂ is number operator for the input light. The term : e − n ̂ : originates from the photocounting formula (true photon statistics), and the symbol “:…:” stands for the normal ordering of operators. The POVM set is complete, π ̂ 0 + π ̂ 1 = 1 ̂ , which is required for probabilities of getting zero, p ( 0 ) = π ̂ 0 , and one, p ( 1 ) = π ̂ 1 , counts to sum up to unity. Here, … denotes averaging over the quantum state of the input light. For a realistic detector with the efficiency η (ignoring dark counts), these probabilities are modified as

For spatial multiplexing, where N such on-off detectors monitor the input light, the probability of getting k counts is [64]

where uniform illumination is assumed. The same formula holds for the detection of N pulses in temporal multiplexing. In the Fock basis, Eq. (2) can be written (for η = 1) as [64]

By averaging over the input state of light, expressed in a Fock basis as well, this formula allows to find the photocounting statistics. Alternatively, the detector can be described by a set of conditional probabilities P(k|m) of getting k counts given m input photons. Again, by using the Fock state representation, the photocounting statistics can be derived for any state from these probabilities. The expression for P(k|m) for a set of on–off detectors was derived in Ref. [65] as:

where the effect of non-ideal efficiency of constituent detectors is accounted for. Equation (4) is equivalent to (2) and (3) if the number state m is used for quantum averaging in (2) and (3).

The above formalisms are valid for the distributed detector as well, with one exception. As discussed earlier, the coherent detection scheme guarantees perfect and uniform absorption of the incoming light, irrespective of the absorption of constituent detectors. Therefore, while in Eqs. (2)–(4) the constituent detectors are described by (1) with (absorption) efficiency η < 1 (or even ≪1), for the entire distributed detector η = 1 (assuming unity coupling and intrinsic efficiencies, as discussed above). In Figure 6, we compare the performance of coherent and incoherent detection schemes at detecting two- and three-photon number states. The solid black line shows the probability of getting the correct measurement – two, Figure 6A, and three, Figure 6B, counts for the distributed detector. This result also holds for spatial and temporal multiplexing with 100% efficient detectors. The vertical gray dashed line corresponds to the detector composed of ten constituent detectors. The probability of getting the correct measurement is 90% and 74% for two and three-photon detection, respectively. The probability drops to 81% and 64% (69% and 54%) for the incoherent detection with the efficiency of each constituent detector of 95% (dashed blue line) and 90% (dotted red line), accordingly. As the efficiency of each constituent detector in the ten-layer distributed detector is just 17%, we also show the performance of the incoherent scheme composed of such constituent detectors (dash-dotted green line). The probability of correct detection, in this case, is just about 2.6% for a two-photon and 0.4% for a three-photon detection. For all described detection scenarios, we plot counts distribution on the right-hand side of Figure 6A and B. The difference between the upper left (gray) and bottom right (green) distributions, corresponding to detection with the same constituent detectors but arranged according to coherent and incoherent detection schemes, clearly demonstrates the advantage of coherent detection over conventional multiplexing schemes when the efficiency of individual detectors is not unitary.

Figure 6:

Photon number resolution with coherent and incoherent detection. (A, left): The probability of getting two clicks assuming detection of a two-photon number state as a function of the number of constituent detectors. Efficiency of each constituent detector is 100% (solid black line), 95% (dashed blue line), 90% (dotted red line), and 17% (dash-dotted green line). (A, right): The distribution of detection clicks for these four scenarios. (B) Same but for detection of a three-photon number state. For conventional incoherent schemes: the probability of correct photon number measurement drops exponentially with the efficiency of constituent detectors. For a coherent scheme with a distributed detector: efficient photon number resolution can be implemented with inefficient constituent detectors.

Another test for the efficiency of photon number resolution is the observation of non-classicality in photon number distributions for quantum states of light, such as the squeezed vacuum state. The true distribution is,

where ξ is the squeezing parameter and n is the number of photons. Accounting for the detectors’ response (4), we evaluate count distribution, Figure 7, for the four scenarios discussed above assuming ξ = 1. The non-classicality of the state is apparent for the ten-layer distributed detector, Figure 7A, especially for low-count events: probabilities of one (1.9%) and three (3.8%) counts are noticeably lower than probabilities of two (17.4%) and four (5.6%) counts which are close to the true distribution. For the incoherent detection schemes with 95% and 90% efficient constituent detectors, Figure 7B and C, the distribution blurs, hiding the nonclassical features of the squeezed state’s photocounting statistics. Finally, for the incoherent detection with 17% efficient detectors, the true distribution completely vanishes, and approaches the distribution of the Glauber coherent (classical) state.

Figure 7:

Measurement of photon statistics of the vacuum squeezed state (with the squeezing parameter ξ = 1). Bar charts show the distribution of counts for incoherent detection with ten constituent detectors with the efficiency of 100% (A), 95% (B), 90% (C), and 17% (D). Dashed bars represent the true distribution. For incoherent detection with 17% efficient constituent detectors (D), the true distribution (dashed columns) is completely wiped in the measured distribution (green). For coherent detection with the same constituent detectors, the gray distribution in (A), close to the true distribution, holds.

2.5 Robustness against fabrication imperfections

Importantly, a multilayered distributed detector is robust against fabrication imperfections. Let us illustrate in detail the case of a five-layer detector. We ran a statistical (Monte-Carlo) simulation for the ensemble of one thousand distributed detectors (samples), allowing for the thickness of each layer – nanowire and spacer, to randomly fluctuate within a ±5% interval around the nominal values. As shown in Figure 8, in spite of the realistic variation in layers thickness, the amplitude transmission and reflection coefficients are still grouped around the optimal values of equal magnitudes and opposite phases, Figure 8A. As a result, almost three-quarters of the samples still show complete absorption (within 99%–100%), and all of them absorb more than 95% of light, Figure 8B. The absorption non-uniformity parameter Δ_norm varies insignificantly among the samples but is less than 0.03 in most of the cases, Figure 8C, revealing an anti-correlation between the absorption of the different nanowire sublayers: the decrease in absorption of one (or few) of the sublayers is compensated by the increase in absorption of other sublayers. Absorption of each sublayer falls in a narrow distribution around the nominal value, as shown in the inset of Figure 8C.

Figure 8:

Statistical simulation of fabrication imperfection for the five-layer distributed detector. An ensemble of one thousand samples is generated where, for each sample, the thickness of nanowire sublayers and spacers is chosen randomly with fluctuation ±5% around the nominal values. (A) Optical response under traveling wave illumination within the ensemble. Due to the asymmetry of the structure, the reflection coefficient is slightly different for illumination from the left, r_tr (blue dots), and right, r tr ′ (black dots), sides. (B) Fluctuation of coherent absorption in the ensemble. The inset shows the coherent absorption of each sample. (C) Fluctuation of the absorption non-uniformity in the ensemble. The inset shows the fluctuation of the absorption per each nanowire sublayer in the ensemble.

2.6 Generalized requirements for distributed absorbers

In previous sections, we built distributed detectors by starting with a single-layer absorber of the optimal thickness taking into account superconducting material and nanowire design parameters. This single-layer absorber was “cut” into multiple sublayers placed at different anti-nodes of the standing wave. Under these assumptions, the standing wave picture guarantees that the distributed detector retains the optical response of the original layer with optimal thickness, as well as uniformity of absorption. In general, for a distributed detector with M-sublayers of any nature operating in the counter-propagating geometry (as in Figure 1A), the amplitude transmission (t), reflection (r), and intensity absorption (A) coefficients of each sublayer should satisfy the following conditions (Supplementary Materials):

Here, the sublayers are assumed to be of a subwavelength thickness and spaced by λ/2-spacers. The optical response (5) can be achieved by proper design or adjusted empirically. This result is a generalization of what was presented above for multilayered SNSPD detectors. For instance, according to (5), each constituent detector of a five-layer distributed detector should absorb 28% of light under traveling wave illumination. This is precisely the value we estimated for the five-layer SNSPD detector (Figure 5A).

Similarly, the constituent sublayers of a K-layer distributed detector operating in the Salisbury screen geometry (as in Figure 1B) should possess the following amplitude transmission (t′), reflection (r′), and intensity absorption (A′) coefficients (Supplementary Materials):

Comparison of (5) and (6) reveals that, for distributed detectors with the same number of sublayers, K = M, sublayers should be less absorptive in the Salisbury screen geometry than in the counter-propagating configuration. Equivalently, given a certain optical response of the constituent detectors, half as many detectors are required in the Salisbury screen geometry, K = M/2.