1. Introduction

An important goal in quantum information science and technology is complete control of photonic states. Beyond the polarization and transverse spatial degrees of freedom, the time-frequency degree of freedom is largely an untapped quantum resource [1–5]. Orthogonal temporal modes (TMs) are defined by the complex longitudinal wave-packet shape functions of pulsed modes of light [6]. They form a discrete set of field-orthogonal functions representing a continuous-variable degree of freedom that is in principle infinite dimensional, and create a promising resource for quantum information science [7]. To exploit fully their use in a quantum network requires the ability to unitarily demultiplex arbitrary TM components from a light beam with near-unity efficiency and mode discrimination (i.e., no crosstalk). A device capable of such operations is known as a quantum pulse gate (QPG) [8, 9]. The ideal QPG must satisfy two conditions: (a) It must fully separate the desired TM component from the others without loss of photons, and (b) it must avoid contamination from orthogonal TM components in the “wrong” TM channels. When both of these are met, the QPG is said to have unit selectivity [10].

There exists a fundamental limit to selectivity of QPGs based on traveling-wave interactions in media with simple dispersion profiles, which enforces a trade-off between the two aforementioned conditions [10–13]. The best performing QPG proposed to date is based on temporal-mode interferometry (TMI) that performs pulsed, cascaded frequency conversion with multiple passes through standard dispersive nonlinear optical media (three-wave mixing in crystals or four-wave mixing in fiber) [14, 15]. The technique is a close relative of Ramsey interference of photons in a frequency-converting interferometer [15–17]. In TMI, TMs from all participating carrier-frequency bands coherently reinteract at every stage, resulting in a selectivity enhancement that overcomes the single-stage maximum [18]. The gain in selectivity is very significant for even two-stage schemes [19], and is predicted to improve asymptotically with the number of stages [15]. While TMI has been shown to operate as predicted, it presents practical difficulties due to coupling losses and engineering/manufacturing constraints for integration.

A QPG that can multiplex and demultiplex field-orthogonal optical pulses is closely related to devices studied for coherent optical code division multiple access (OCDMA) employing second-harmonic generation of phase-structured ultrafast pulses [20, 21]. Like the other implementations of coherent demultiplexers mentioned above, the TM selectivity of this scheme is limited by traveling-wave phenomena during nonlinear frequency conversion [21]. In a different research arena, optical-cavity-enhanced atomic-ensemble or solid-state quantum memories are known theoretically to have TM-selective qualities for coherent optical storage [22–24]. Finally, great technical advances have been made in design and fabrication of nonlinear-optical micro-ring resonators [25], and these have been employed for frequency conversion between telecom and visible bands via sum-frequency generation [26–29].

By combining insights from all the diverse areas discussed above, we have arrived at a means of using an optical micro-cavity with a large difference in finesse for two frequency bands participating in nonlinear frequency conversion by sum-frequency generation (SFG), to create a high-selective QPG. It functions by instigating multiple passes through a cavity-based nonlinear optical medium. The scheme mimics the TM-selective behavior predicted for cavity-based atomic quantum memories [24]. This all-optical “dichroic-finesse cavity” scheme provides a simple, realistic way to create a near-ideal add/drop (multiplexer/demultiplexer) device in a low-loss integrated-optics platform for use in quantum optical networks. By passing the control and signal pulses through the same frequency-converting medium many times, the device effectively performs TMI with a near-infinite number of stages, which explains its high TM selectivity. The proposed system operates without the need for atomic vapors or doped crystals. The weak signal pulse may be in a single-photon state (or any low-number Fock state), or in any other quantum state, such as squeezed vacuum. It can be temporally reshaped during the read-in and read-out process. The proposed scheme, while challenging to construct, relies only on already proven technology.

2. Design principles

Figure 1 shows a schematic diagram of the proposed micro-ring cavity system. S(t) and C(t)are the resonant cavity mode amplitudes of the two frequency bands that will interact via the optical nonlinearity of the medium, which uniformly fills the cavity. The intracavity control field Ω(t) can be that of a single strong, coherent laser pulse if the process is utilizing three-wave mixing, or a combination (product) of two pulse amplitudes if four-wave mixing is used. For definiteness, we will consider sum-frequency generation by three-wave mixing, but all results apply to four-wave mixing as well. For sum-frequency generation the frequencies are related by ω_s + ω_Ω = ω_c. For convenience we refer to ω_s as “red” and ω_c as “blue.”

 

Fig. 1 Schematic of dichroic-finesse cavity filled with second-order nonlinear optical material. The signal field is S, the pump (control) field is Ω, the cavity-trapped frequency-converted field is C. The converted field C(t) is not shown exiting the cavity, as this occurs on much longer time scales.

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The theoretical model presented below predicts that for a given temporal shape of the control field Ω(t) inside the cavity, only a single temporal mode of the incoming (red) signal field, called the target mode, will be frequency up-converted, creating narrowband (blue) light that is trapped in the cavity in the C(t) mode. All TMs temporally orthogonal to the target mode will be transmitted into the S_out(t) beam at the original (red) carrier frequency. The trapped blue light can subsequently be left to leak slowly from the cavity at later times, or it can be rapidly read out (ejected) from the cavity by applying a subsequent control pulse, which converts it back to red.

The temporal widths of the control field and the signal input S_in(t) must be much longer than the cavity round-trip time, but much shorter than the cavity lifetime for the C(t) mode. This leads to negligible leakage of the converted C(t) amplitude from the cavity during the process. The C(t) mode then becomes analogous to a coherent spin wave (for example) in a cold atomic ensemble; thus the system may be viewed as a quantum memory. The cavity-coupling coefficients are assumed to have values that put S(t) in the “bad-cavity” limit. This finesse differential across the frequency bands is the key feature necessary to break the interaction symmetry between S(t) and C(t), and yield efficient TM multiplexing, and leads to the name dichroic-finesse cavity.

The solutions to the coupled-mode equations of motion can be expressed as linear integral scattering relations between input and output temporal modes using Green’s functions, which are functions of both an input-mode time argument and an output-mode time argument. For the process to be mode discriminatory, the Green functions should be separable in their time arguments, which is impossible for time-stationary processes whose Green functions depend only on time differences. A key requirement for achieving Green-function separability in previous (cavityless) approaches has been a large difference in the group velocities between the various frequency bands [10, 12, 14, 15]. This is required because orthogonal TMs can share very similar (even identical) temporal features in local time slices. For the QPG to perform different transformations on these two TMs, the full global temporal-mode structure needs to be surveyed by the device, as the effect (depletion/enhancement/phase-shift) on any given time slice should depend on features in all other time slices. Differing group velocities cause pulsed-mode field amplitudes from different time slices to convect through each other, providing an effective means of carrying local mode information across different time slices. TMI operates by causing convecting pulses to overlap in spacetime over multiple stages, with the interaction being semi-perturbative during each stage. This avoids coherent-propagation ringing effects [30] induced by cascaded second-order nonlinearity [10, 20, 21], and ensures Green function separability even at high conversion efficiencies [18].

In contrast, our proposed scheme ensures inter-pulse convection by confining the TMs of one of the bands in physical space as the other TMs pass through it. This design uses a large difference in finesse across the frequency bands, and works for controls and signals with arbitrary relative group velocities, which is another advantage over (cavityless) traveling-wave QPG implementations.

3. Theory

We analyze the case of a nonlinear waveguide forming a resonant ring cavity, assuming frequencies that are phase matched for the control field Ω(t), signal S_in(t), and converted C(t). The key to is to have the cavity input-output coupling be frequency dependent, while still requiring that both signal and converted modes have high finesse. With a very long, smooth input signal pulse, this allows the bad-cavity limit (only) for the S-field, meaning it tends to leak from the cavity relatively rapidly. Concurrently, we assume the cavity has much higher finesse for the C-field than for the S field. Then, we can frequency convert the short, “red” input pulse S_in(t) into a long-lived, resonant mode C(t) at the “blue” frequency ω_c trapped within the cavity before it slowly (exponentially) leaks out into the travelling-wave guided mode.

Within the adopted parameter ranges, the system is well described using the standard input-output theory of Collett and Gardiner [31]. The approximations leading to this formalism require very weak coupling of the cavity modes to external freely propagating modes, and spectral widths of all signals significantly narrower than the free-spectral range of the cavity [32].

The weak quantum signal fields within the cavity are represented by annihilation operators S(t) for the “red” input field and C(t) for the “blue” converted field, and satisfy commutators [C(t), C^†(t)] = 1, [S(t), S^†(t)] = 1. The input fields immediately outside of the coupling mirror are S_in(t), C_in(t), which satisfy $[A_j(t),A_k^{†}(t^{\prime })]={\delta }_{jk}\delta (t-t^{\prime })$, where A ∈{S, C} and j, k ∈ {in, out}, with S_out(t), C_out(t) being the outgoing fields.

We take Ω(t) to be the intracavity control field in an auxiliary mode, which is trivially related to the incident field Ω_in(t) depending on any arbitrary finesse condition (for example, in the “bad-cavity” limit, it is simply proportional Ω_in(t)). We absorb the square-root of the control field energy into a nonlinear interaction parameter α such that Ω(t) is square-normalized to one. We assume both signal fields are exactly resonant with their cavity modes and there is no phase mismatch for the SFG process. Then the equations of motion within the cavity are [27, 33]:

In the following, we assume there is no external input to the C mode, so C_in(t) is omitted.

Equations 2 and 3 are linear in field operators (although nonlinear with respect to the control field, here an undepleted coherent state). Therefore they can describe the Heisenberg-picture operator dynamics of any quantum state of light. In the case that only a single signal photon is present throughout the system, the variables can be interpreted as Schrodinger-picture state amplitudes [34, 35].

The first crucial assumption is that the cavity out-coupling rate γ_s for the input channel is large compared to the rate at which all the fields vary -set by ${\tilde{\gamma }}_s$, ${\tilde{\gamma }}_c$ and α, so we can apply the “bad-cavity” approximation to S(t). By setting ∂_t S(t)→ 0, we get

The second crucial assumption is that the cavity has very high finesse (is a very “good” cavity) for the C-band $({\overline{\gamma }}_c\approx 0)$, and the entire process takes place well before any amplitude from C(t) has leaked out. For times shorter than 1/γ_c, we can drop the ${\tilde{\gamma }}_{c}C(t)$ term. Then the solution to Eq. 5 is

4. Demonstration of high selectivity

This dichroic-finesse cavity scheme is not only highly TM discriminatory, but is also highly efficient, under the assumption that there are negligible internal dissipative losses, that is ${\tilde{\gamma }}_s={\gamma }_s$. In this case, the efficiency is unity if the output field S_out (t) is zero. In the case that the input is given by Eq. 7, the total unconverted signal energy (photon number) behaves as $W_{\mathrm{o}ut}={\displaystyle {\int }_{-\infty }^{\infty }dt|S_{\mathrm{o}ut}(t){|}^2\to \exp (-2f_s)}$, the trend toward zero being achieved with increasing f_s. This prediction is valid only up to a certain value of control field strength, beyond which the system is driven out of the bad-cavity regime and the conversion efficiency degrades, as we discuss below.

To verify the scheme operates as a high-selectivity quantum pulse gate, we solve the more accurate Eqs. 1 and 2 numerically. Unless stated otherwise, the control field is taken to be Gaussian, Ω(t) = (2/π)^1/4 exp[−(t /(10\ τ_cav)-3²], with τ_cav being the cavity round-trip time. Time units are relative to the duration of this dimensionless control pulse. The first goal is to show that the near-optimal input pulse, designed according to Eq. 6, leads to efficient transfer of incoming energy into the frequency-converted cavity mode. (We continue to assume negligible internal cavity loss.)

Figure 2 shows simulation results in the dichroic-finesse cavity scenario, for the case γ_s = 10.1 and γ_c = 0.010, or dimensionless cavity lifetimes 1/γ_s = 0.09, 1/γ_c = 99.99. The value of nonlinear coupling, dependent on control pulse energy, is optimized to be α = 5.5. As expected, larger values begin to drive the system out of the bad-cavity regime and worsen the conversion efficiency (not shown). The integrated signal input energy equals 1.

 

Fig. 2 Numerical simulations of amplitude versus time for (a, b) Gaussian signal input and (c, d) the optimal input temporal mode. The input signal S_inand the control pulse Ω are multiplied by 0.8 for convenient plotting. Parameters for both cases: α = 5.5, γ_s = 10.1, γ_c = 0.01. τ_cav is the cavity round-trip time.

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Figures 2(a) and 2(b) show the results for the case that the input (“red”) signal shape S_in,Gaussian is identical in shape to the control pulse Ω, not the optimal case. The “red” cavity mode S reaches a maximum of about 0.2 before rapidly decaying. The converted “blue” cavity mode amplitude, plotted as −iC, reaches a value 0.8 before it begins a slow exponential decay into the output channel C_out. The “red” output channel S_out shows significant leakage and thus poor storage efficiency. The unconverted signal energy W_out equals 0.36 in this case with a non-optimized input pulse shape.

Figures 2(c) and 2(d) show the case that the input signal shape is given by Eq. 6, which is predicted to be near optimal. In this case the unconverted signal energy W_out equals 0.016, meaning less than 2% of the incoming pulse is not initially trapped in the frequency-converted cavity mode. Correspondingly, the trapped “blue” mode amplitude reaches a value near 1.0 before it begins a slow exponential decay. This means that a properly designed input pulse can achieve high storage efficiency, analogously to results found in atomic-based quantum memories [24].

Any TM orthogonal to the near-optimal mode given, by Eq. 6, is predicted to pass through the cavity system and not frequency convert (or only weakly convert). Figure 3 shows numerical solutions of Eqs. 1 and 2 for two such modes, and, indeed the conversion efficiency is very small for each. Orthogonal modes, denoted mode 1, mode 2, and so on, are constructed numerically using a Gram-Schmidt procedure starting from the optimal mode used earlier in Fig. 2.

 

Fig. 3 Numerical simulations of amplitude versus time for two temporal modes that are orthogonal to the optimum TM used in Fig. 2. Both remain nearly completely unconverted. Same parameters and plotting as in Fig. 2. τ_cav is the cavity round-trip time.

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Figure 3(a) shows as the dashed curve the “red” input mode 1, which resembles a Hermite-Gaussian-1 function. The converted “blue” cavity mode amplitude −iC reaches a value −0.7 then rapidly returns to a small value around −0.1 before beginning a slow exponential decay into the output channel C_out. The unconverted “red” output channel S_out in Fig. 3(b) shows large leakage. The unconverted signal energy W_out equals 0.98 in this case, that is it remains nearly completely unconverted. Figure 3(c) shows as the dashed curve the “red” input mode 2, which resembles a Hermite-Gaussian-2 function. The converted “blue” cavity mode amplitude −iC oscillates and rapidly returns to a near-zero value, and the “red” output channel S_out in Fig. 3(d) shows large leakage. The unconverted signal energy W_out equals 0.99 in this case, again consistent with expectations.

The simulations support our proposal for an all-optical cavity-based demultiplexer: we have shown that given a Gaussian control pulse, there is one optimal signal TM that can be frequency converted efficiently and stored for a time in the cavity, while any mode that is temporally orthogonal to the optimal mode is not frequency converted and passes through the system. The contrast between energy conversion and nonconversion is about 50:1 for the parameters and pulses sed, far better than any single-stage QPG based on traveling-wave SFG. Marginal increases of efficiency and discrimination could be achieved by numerically finding the “truly optimum” input shape determined by Eqs. 1 and 2, but this would only serve to strengthen the present case that a dichroic-finesse cavity can act as a near-ideal QPG.

From a different perspective, if we choose any particular targeted “red” signal input TM that we wish to convert to “blue” and store in the dichroic-finesse cavity, we can design the control field that optimizes its conversion and trapping, while not converting any orthogonal signal TM. The condition that ensures near-100% conversion of the “red” input pulse is S_out(t) = 0, which from Eq. 3 implies $S_{\text{in}}(t)=\sqrt{2{\gamma }_s}S(t)$. Then using the bad-cavity approximation for S, as given by Eq. 4, leads straightforwardly to ∂_t C(t) = K(t)C(t) and $S_{\mathrm{i}n}(t)={\mu }^{*}\sqrt{2K(t)}C(t)$, where $K(t)=f_s|\Omega (t){|}^2$ and µ = i exp{i arg[Ω(t)]}. From these, one can derive the design equation for K(t):

An equation of this form also appears in the context of optical storage in cavity-enhanced atomic quantum memories, where it is called the “impedance matching condition.” [22]. The resulting solution for the control field for optimal storage is (See the Appendix.)

To illustrate and test this design prediction, consider as a target input signal any of the orthogonal Hermite Gaussians, $H{G}_n(t)=H_n(t)e^{-t^2/2}/\sqrt{2^n{\pi }^{1/2}n!}$. We numerically solve Eqs. 1 and 2 using Eq. 9 as the control field, and plot the results in Fig. 4, where Figs. 4(a) and 4(b) show the results for S_in(t) = HG₀(t) and HG₁(t), respectively. The value of the control strength parameter α is again optimized to the value 5.5. The shapes of the control fields before the signal begins turning on are arbitrary, and set by the value 10⁻⁷ of the parameter q. The unconverted signal energy in case (a) is 0.004, and in case (b) is 0.015, showing excellent conversion and trapping of the targeted input TMs.

 

Fig. 4 Illustrating the effectiveness of the control field Ω_opt (t) to efficiently convert and store the targeted “red” input mode S_in(t). (a) S_in,1(t) = HG₀(t), (b) S_in,2(t) = HG₁(t). In both cases, using the designed control field drives the converted cavity mode amplitude −iC to near its maximum possible value of 1.0. In both cases: α = 5.5, γ_s = 10.1, γ_c = 0.01, q = 10⁻⁷. τ_cav is the cavity round-trip time.

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The ability to selectively convert amplitudes of specific TM components into relatively long-lived cavity modes enables us to treat the nonlinear cavity as a short-term memory or buffer register. This can enable coherent interactions between TM components of pulses arriving at different times, as well as temporal reshaping of ω_s-band temporal modes.

As shown in Fig. 5(a), a “write” control pulse (black curve) converts the “red” Gaussian input signal (dashed red curve) into a stored “blue” cavity mode (blue curve). Subsequently, a modified “read” control pulse (green curve) depletes the stored “blue” cavity mode and regenerates the signal field, now with a modified temporal shape. In Fig. 5(b), the corresponding output fields are shown; the weak “blue” leakage (dashed blue curve) is suppressed, and the regenerated “red” output signal pulse (dashed red curve) has a shape similar to that of a “read” control pulse.

 

Fig. 5 (a) “Write” and “read” control fields being applied to the same cavity with a relative time delay. The input mode S_in(t) gets fully captured into a high-Q (by 3 orders of magnitude) cavity mode C(t). (b) The read-out control pulse for these parameters recovers 93% of the amplitude into the S_out(t) mode, whose TM shape can be controlled by the shape of the read-out control field. τ_cav is the cavity round-trip time.

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5. Discussion

If we choose the unit time scale in the simulations as 100 ps, the duration of the control pulse in Fig. 2 is 166 ps. Assuming a round-trip time of 10 ps and group velocity one-half the vacuum speed of light, gives a cavity round-trip length 150 µm. The cavity leakage parameters (γ_s = 10.1, γ_c = 0.01) correspond to rates 1.01 × 10¹¹s⁻¹and 1.0 × 10⁸s⁻¹ (and cavity-field lifetimes 10 ps and 10 ns) respectively. For carrier wavelengths 1550 nm and 775 nm for S and C modes, respectively, the “dichroic” cavity quality factors needed are Q_s = ω_s/2γ_s = 6020 and Q_c = ω_c /2γ_c = 1.2 × 10⁷. Finally, the internal dissipative losses need to be much smaller than γ_s, γ_c. These values, along with the needed control power, are within range of achievable values for whispering-gallery resonators [36] and possibly planar-waveguide micro-rings coupled evanescently to an external waveguide [25].

The dimensionality of the TM subspace [7] that such a system can effectively operate on is entirely determined by the Q-factors at ω_s and ω_c. For analytical convenience, one can decompose said subspace in a Hermite-Gaussian (HG) pulsed-mode basis. Slowly varying TMs are largely composed of the lower-order HG components with smaller bandwidths, whereas TMs with more complex temporally-varying structures possess higher-order HG contributions. The slowest allowable S_in(t) pulse is limited by how large Q_c is, as the conversion needs to occur before C(t) escapes the confines of the cavity. The largest bandwidth for S_in(t) is capped by Q_s, as the S-field needs to obey the “bad-cavity” approximation (bandwidth of S_in(t) needs to be smaller than the cavity linewidth at ω_s). The TM subspace dimensionality is therefore monotonically linked to the ratio Q_c /Q_s. Decreasing Q_s, however, would require more pump power to achieve the desired f_s value for efficient conversion.

The process described herein, if lossless, is entirely unitary. This renders the results valid for quantum states of light, and enables novel exploitation of time-frequency entangled resources. On the other hand, the unconverted S_out (t) mode shapes are distorted versions of the corresponding S_in(t), thus limiting certain simple implementations of the scheme for applications that require shape preservation of unconverted modes. The mapping, however, unitarily preserves mutual orthonormality, enabling compensation through appropriate modifications to subsequent control-field pulses.

High-Q microcavities, be they rings, toroids, or whispering gallery resonators, are a pressing topic for both linear and nonlinear integrated nanophotonics [25, 27, 37–39]. The largest Q-factor achieved for silicon resonators is of order 10⁸ [39], whereas cavities made of nonlinear materials can reach 10⁷at certain wavelengths [25, 27]. The dichroic-finesse scheme requires a difference in Q-factor for the signal and “storage” bands by at least 3 orders of magnitude. Evanescently coupled microresonators are naturally dichroic in their finesse due to differential evanescent mode overlaps at widely separated frequencies. This dichroism can be further enhanced through interferometric/phase-matching means, or utilization of multiple coupling channels [28] (Fig. 6(a)), or accessing orthogonal polarization modes via appropriate phase matching of the nonlinear coupling [26, 36, 37] (where one polarization couples to the external waveguide better than the other).

 

Fig. 6 (a) Schematic for optical microresonator coupled to different guided fields. (b) Two microresonators mutually optically coupled through a third racetrack waveguide. The control fields can be shaped to adiabatically transfer amplitude. (c) Microresonators in a bus network topology. Different resonators can store different components of a qudit, either encoded in time bins or an overlapping temporal-mode space. Individually addressing each microresonator enables custom connectivity graphs, functioning as a quantum RAM (random-access memory) buffer, or indeed, simulating a scalable quantum internet.

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In contrast to the intuitive wave-packet shaping approach to coupling physically separated quantum nodes through flying optical qubits, a slowly varying set of control fields can be applied to a pair of nodes to adiabatically transfer quantum information amongst them [40–42], akin to STIRAP in stimulated Raman scattering [43]. Transfer of quantum information between cavities connected by photonic waveguides via application of write/read control pulses would provide for a platform for simulating such systems (Fig. 6(b)). The ability to address nodes of one’s choosing for coupling through a preset network of waveguides is an unresolved problem, with unique challenges (Fig. 6(c)). However, such a platform can provide utility to the quantum information processing community.

6. Conclusions

In summary, the newly proposed scheme offers several functionalities crucial for quantum information science. It can multiplex and demultiplex orthogonal temporal modes of light with high TM discrimination and efficiency. It is reconfigurable in real time to target any chosen TM by altering the shape of the control field that drives the sum-frequency generation. The efficiency of TM demultiplexing is tunable in real time by altering the intensity of the control field, giving the possibility to create and measure single-photon states that are superpositions of two time-bin states. It can be used as a short-time all-optical memory, the storage time being limited by cavity Q and losses. And it can be used to reshape optical pulses via the read-in, read-out process.

The vast majority of time-stationary optical processes satisfy the Fourier constraint, Δω_P Δt_P ≈ 2π, where Δω_P and Δt_P are the bandwidth and processing (interaction, or read-out) time of the process. (Rare exceptions may occur in systems lacking Lorentz (time-reversal) reciprocity [44].) Our system, being time-nonstationary, has the useful property that both the bandwidth and the storage time, while not being bound by Δω_PΔt_P ≈ 2π, are tunable in real time. The read-in bandwidth Δω_P for the input channel is set by the shape of the control field Ω(t), while the read-out time Δt_P in the output channel is set either by the natural decay time of the narrow C mode resonance (which is much narrower than that of the input S mode resonance) or by the duration of the outgoing red signal pulse in cases where a read control pulse is employed.