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Abstract
We investigate the generation and control of the reverse saturable absorption (RSA) and optical limiting (OL) at microwave (mw) range in high-Q single-crystal diamond mechanical resonator (DMR) embedded with many nitrogen-vacancy (NV) centers. The strain-induced acoustic modes enable mechanical manipulation of NV centers. On the basis of strain-coupling mechanism, it is shown that the saturable absorption (SA) switches to the RSA by applying the acoustic field, leading to induce the OL in the diamond through the cross-Kerr effect. We demonstrate that the OL characteristics such as, threshold, efficiency, and dynamic range can be controlled by changing either the intensity or frequency of the acoustic field. Moreover, we show that this optical limiter can amplify noiselessly the low intensity of the mw field input to the sensors and also attenuate any gain-induced noise and increase in the intensity of the mw field if it exceeds the intensity threshold. In addition, it is shown that by increasing either the number of NV centers or length of the diamond, the optical limiter can be more efficient. The physical mechanism of the OL establishment is explained using the analytical expressions, which are in good agreement with the numerical results. Our proposed acoustic-induced optical limiter can be a scheme for protecting different optical and electronic devices in mw range, remote sensing, navigation, communications, microwave heating and thermo/laser therapy.
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1. Introduction
Microwave photonics is broadly characterized as the investigation of high-speed photonic devices designed to operate within the microwave or millimeter wave frequencies which find application in both microwave and photonic systems [1]. This interdisciplinary field has played a significant role in scientific developments and new technology in the last few decades [2] which covers generation, process and control of significant phenomena such as broadband wireless communications, navigation, radar, sensing, imaging and instrumentation. In recent decades, control of mw and optical phenomena through laser-induced coherence has achieved great attention in atomic physics and quantum optics. The coherent control of optical properties such as intensity of the applied fields has led to phenomena such as all-optical switching [3,4] and optical limiting (OL) [5,6]. Microwave and optical instruments such as human eye, signal sensors and cameras are highly sensitive to intense illumination. Generally, a photonic sensor, often referred to as an optical sensor, is a device that utilizes the principles of photonics to detect and measure various physical properties or phenomena. These sensors are designed to convert electromagnetic signals into electrical signals, enabling the measurement of parameters such as intensity, wavelength, phase or polarization. For example, fiber Bragg grating, as a type of optical sensors, functions as a wavelength-selective filter and are sensitive to changes in temperature, strain, or other environmental conditions [7,8]. Applications of optical sensors based on intensity measurement span a wide range of fields, including industrial automation, environmental monitoring, biomedical sensing and telecommunications. The sensitivity of these sensors to changes in light intensity makes them valuable for tasks such as proximity sensing, object detection, and real-time monitoring of dynamic conditions. Nowadays, the utilization of high-power microwave signals constitutes a pivotal strategy in technology, where certain passive or active controlling sensors may be susceptible to damage or saturation from the high-power microwave signals. Consequently, safeguarding the microwave sensors within seekers, altimeters, and radars against high-intensity signals necessitates consideration of OL devices. Optical limiter is a nonlinear phenomenon consisting of attenuation and control of light intensity of any wavelengths. An optical limiter protects sensitive optical components by attenuating intense laser radiation above a specific optical threshold, while transmitting lower intensities of light. Optical limiters cover a wide range of organic and inorganic nonlinear materials [9–12]. These optical limiters, which are in the category of inherent limiters, have some limitations and cannot be used for all optical devices with different sensitivities. For this purpose, the design of optical limiters with the ability to control their optical characteristics has always been of interest. Recently, atomic optical limiters have been presented in which the optical threshold, dynamic range and the efficiency can be controlled coherently [13]. Most recently, Mahmoudi et al. have introduced an atomic based optical limiter whose optical properties can be controlled by a mw field [14]. In the previous works, atomic systems derived from a gas ($^{87}$Rb) were considered to form atomic optical limiters. It has been demonstrated that the optical phenomena in gas systems may be influenced by some factors such as Doppler broadening affected by atoms velocity.
Among the various nonlinear mechanisms [15–17], reverse saturable absorption (RSA) has been the main nonlinear phenomenon responsible for the OL [18]. The absorption cross section of the excited state in RSA materials is larger than the ground state, leading the excited state to constantly depopulated. In materials exhibiting RSA [19], the absorption increases by increasing the intensity of the input light, leading to attenuate the transmission of the output light. In saturable absorption (SA) materials, the absorption decreases with the increase of the intensity of the input light. SA materials have applications in producing photonic diodes [20] and mode locking [21,22]. Despite intrinsic RSA materials, coherently inducing RSA in atomic systems have recently provided promising method for coherent control of optical phenomena such as OL [23].
On the other hand, nitrogen vacancy (NV) center consists a substitutional nitrogen atom and adjacent vacancy in diamond. NV centers in diamond represent a promising platform in solid state physics for investigating spin-based quantum systems interacting with their environment [24] and for studying nonlinear optical phenomena [25–27]. A long electronic spin coherence time in NV centers [28,29] makes them an interesting candidate for quantum information processing [30–32] and precision metrology. The spin states of the ground state triplet of the NV centers generate quantum structures that realize spin-phonon interaction along with mw fields [33,34]. A phonon produced in the lattice by a motion of the diamond mechanical resonator (DMR) can be absorbed by the NV center, resulting in coupling the NV center with the phonon. However, the NV center and acoustic field coupling can be realized through two mechanisms: magnetic field induced coupling [35,36] and the strain induced coupling [37–41]. The noteworthy point is that the coherent coupling of NV centers with acoustic field is done by mechanically driving electric dipole forbidden transition in degenerate levels of the spin triplet ground states of the NV centers [42]. Using this fact, Mahmoudi et al. showed that the birefringence can be induced in NV centers by strain-mediated acoustic wave in a DMR, and the rotation of a mw field polarization can be controlled as well [43]. Various studies have addressed the optical properties of NV centers in DMR while interacting with strain-generated acoustic wave and have shown that NV centers can be a sensitive host for the investigation and mechanical control of optical phenomena [44–46]. Recently, an integrated acousto-optic platform on thin film lithium niobate has been introduced in which acoustic waves in the microwave domain converts to optical light by a generalized acousto-optic interaction [47]. In addition, the acoustic vibration has been used to generate the microwave field via low-frequency Raman scattering [48]. Thus, NV centers in diamond as a solid state system can provide more stable hosts to generate controllable optical limiters than atomic systems. Moreover, the mechanical control of the intensity of the acoustic field by the voltage of a piezoelectric can be much simpler than the coherent control of the optical characteristics of an optical limiter.
In this work, we investigate the generation and control of the RSA and OL for mw range through the interaction of strain-induced acoustic field with spin triplet ground states of the NV centers embedded in a DMR. It is shown that the SA switches to the RSA by acoustic field originated from the strain in diamond, leading to induce the OL in NV centers through the cross-Kerr effect. We demonstrate that the optical threshold, efficiency and the dynamic region of the presented optical limiter at mw range can be controlled by the intensity and frequency of the acoustic wave. Unlike our previous atomic optical limiters, the current optical limiter can compensate for the low intensity of the mw field entering the sensors by noiselessly amplifying the mw field input to the optical limiter. Moreover, we show that any noise and possible increase in the input mw field intensity due to the gain is attenuated by the acoustic-induced limiter if it exceeds the intensity threshold. It is displayed that by increasing the number of NV centers in diamond, the optical limiter becomes more efficient. An analytical expression is presented to describe the physics of induced OL behavior in NV centers, which is in good agreement with the numerical results. Finally, a theoretical Z-scan experiment is employed to confirm the obtained OL. The acoustic induced optical limiter can be a rather interesting scheme for protecting the optical devices in mw range, remote sensing, navigation, communications , mw heating and thermo/laser therapy.
2. Theoretical framework
The considered system shown in Fig. 1(a) is a single-crystal DMR that is embedded with many NV centers. The NV centers in the diamond are highly sensitive to the deformation of the lattice and can sense the lattice vibrations. Motion of the DMR caused by a piezoelectric changes the local strain at the position of an NV center. After producing a single phonon in the lattice, the NV center ensembles can perfectly absorb and emit it. Figure 1(b) shows the quantum system of the three-level closed-loop system of the ground state of one NV center. The ground state of the NV center has a spin-triplet structure. The spin state $|^{3}A,m_s=0\rangle$ (labeled by $|1\rangle$) experiences a zero-field splitting by $D_0 /2\pi$=2.87 GHz from the degenerate spin states $|^{3}A,m_s=\pm 1\rangle$ (labeled by $|\pm 1\rangle$) due to spin-spin interaction. A mw probe field $\vec {E}=\hat {x} E_{m} exp[-i(\omega _{m}t-k_{m}z)]+c.c$ with the Rabi frequency $\Omega _{m}=(\vec {\mu }_{31}.\hat {\epsilon }_{m})E_{m}/\hbar$ couples the state $|1, m_{s}=0\rangle$ to $|3, m_{s}= +1\rangle$. A coupling mw field with the Rabi frequency $\Omega _{c}=(\vec {\mu }_{21}.\hat {\epsilon }_{c})E_{c}/\hbar$ is applied to excite the transition $|1, m_{s}= 0\rangle \leftrightarrow |2, m_{s}= -1\rangle$. Finally, a strain-induced acoustic field drives the transition $|2, m_{s}= -1\rangle \leftrightarrow |3, m_{s}= +1\rangle$ with the Rabi frequency $\Omega _{s}=(\vec {\mu }_{32}.\hat {\epsilon }_{s})E_{s}/\hbar$. Note that the transition between $|2\rangle$ and $|3\rangle$ is dipole-forbidden ($\Delta _{m_s}$=2) and cannot be driven by a coherent field. The key is that a strain field can coherently couple this transition. It should be noted that since the strain field changes the electron density of the diamond, the acoustic field is modeled by an effective electric field. The intensity of the strain-induced electric field can be easily controlled by the voltage of a piezoelectric. Xu has shown and simulated the distribution of the electric field created in the silicon LED as a result of changing the density of electrons and their energy by applying voltage [49]. The interaction Hamiltonian of the applied fields and NV centers in the diamond considering the dipole and rotating wave approximation [50] is given by
Fig. 1. (a) Schematic of the DMR with many NV centers, (b) Energy levels of spin triplet ground states of a NV center, where a probe (coupling) mw field with Rabi frequency $\Omega _{p}(\Omega _{m})$ excites the transition $|1, m_{s}=0\rangle \leftrightarrow |3(2),m_{s}= +1(-1)\rangle$. In addition, a strain-induced acoustic field with Rabi frequency $\Omega _{s}$ drives the transition $|2, m_{s}=-1 \rangle \leftrightarrow |3, m_{s}=+1\rangle$. $D_0 /2\pi$=2.87 GHz is zero-field splitting.
3. Results and discussion
Now, we are going to investigate the absorption evolution of the mw probe field while passing through the three-level closed-loop system of NV center’s ground state embedded in the DMR by numerically solving the Eq. (2). Throughout the results, the system parameters are scaled by the effective ground state decoherence rate $\Gamma =2\pi \times 1.2$ MHz [52]. Figure 2(a) shows the absorption of the mw probe field as a function of the intensity of the probe field passing through the NV centers for different values of the strain-induced acoustic field. The used parameters are I $_{c}$= 8.15 GW/cm$^2$, $\gamma _{31}=\gamma _{32}=\gamma _{3d}=\gamma _{21}=\Gamma$, $\Gamma _{2d}=0.01\Gamma$, $\Delta _{m}=\Delta _{c}=\Delta _{s}=0$ and $\alpha l=50$. It is seen that in the absence of the acoustic field, the absorption of the probe field decreases with increase of the input intensity. Figure 2(a) displays that the SA governs the system when there is no vibration in the lattice. As the acoustic field is switched on, it is shown that the RSA is dominant in NV centers, and absorption takes an increase trend with the increase of the mw probe field intensity. Moreover, it is shown that gain happens in NV centers for some values of the mw and acoustic fields. The gain that the light experiences can be harmful because it increases the intensity of the light. It may also add noise to the light, which can disrupt the information contained in the light. It is demonstrated that the absorption behavior of the mw probe field in NV centers depends on the lattice vibration, and the acoustic field can switch SA to the RSA in NV centers ensemble in the DMR.
Fig. 2. Absorption evolution (a) and transmission (b) of the mw probe field versus its input intensity for different values of the acoustic field intensity. The used parameters are I $_{c}$= 8.15 GW/cm$^2$, $\gamma _{31}=\gamma _{32}=\gamma _{3d}=\gamma _{21}=\Gamma$, $\Gamma _{2d}=0.01\Gamma$, $\Delta _{m}=\Delta _{c}=\Delta _{s}=0$ and $\alpha l=50$.
Generation of the RSA makes the NV centers ready to show the OL behavior. In Fig. 2(b), the transmission of the mw probe field is plotted versus the intensity of the input mw field for different values of the acoustic field intensity. Figure 2(b) displays that when the acoustic field is switched off, the transmission increases with the increase of the input field intensity. It is obvious that the increase in the intensity may affect and damage the mw and optical instruments. In the presence of the acoustic field, the intensity of the transmission of the mw probe field attenuates in the RSA region. In addition, it is shown that the OL threshold can be controlled by changing the intensity of the acoustic field. An investigation on Fig. 2(b) shows that the ability of the acoustic-induced optical limiter grows for the large values of the acoustic field, leading the optical limiter to be more efficient. Moreover, the initial transmittance of the optical limiter decreases by increasing the intensity of the acoustic field. It should be noted that low initial transmittance can be important for delicate optical devices. It is also shown that whenever an increase happens in the transmission of the mw probe field due to the gain, the optical limiter allows the low intensity of the mw field to be amplified and attenuates the intensity and noise of the transmission if it exceeds the optical threshold Fig. 2(b) demonstrates that the strain-induced acoustic field has a major role in inducing OL and controlling the characteristics of the optical limiter for various optical elements in mw range with different sensitivities.
In Fig. 3, density plot of the absorption of the mw probe field is plotted as a function of the acoustic field and the intensity of the mw probe field. The dependance of the absorption evolution of the mw probe field on the acoustic field is clearly seen in Fig. 3. The increasing or decreasing trend of absorption behavior can determine the SA or RSA in this figure. In Fig. 3, two areas including negative absorption (area 1) and positive absorption (area 2) can be seen, and these areas are separated by a black line. This line that shows zero absorption is called the electromagnetically induced transparency (EIT) line. Area 1 stands for gain area, and the output mw field may be amplified and also accompanied by noise. It is clear that this area can be an unsuitable (suitable) area when the intensity of the mw field entering the sensors is dangerous (low). However, the area 2 is the region of positive absorption. Considering the increasing absorption trend in area 2, one can expect OL in this area.
Fig. 3. Density plot of the absorption of the mw probe field as a function of the acoustic field intensity and the intensity of the mw probe field. The other parameters are those taken in Fig. 2.
The transmission slope of the probe field can provide useful information about the performance of the optical limiter. It is clear that the transmission slope is positive when the SA is dominant in the system where the transmission is increasing. In contrast, the slope becomes negative for the OL case when the transmission decreases. In Fig. 4, a density plot of transmission slope is displayed as a function of the acoustic field and intensity of the input mw probe field. Figure 4 works like a map so that one can see the areas of probe and acoustic field intensities where OL occurs. Also, the efficiency of the optical limiter can be extracted for different intensities of acoustic field. In Fig. 4, the major role of the strain-induced acoustic field is clearly seen in the OL behavior of the NV centers in diamond. Figure 4 is divided into three parts, which are separated by black lines that indicate zero slope. In areas 1 and 3, the transmission slope is positive. These areas are not suitable for optical limiting since SA dominates in these areas, and the intensity of the output field increases with the increase of the intensity of the input field. In area 2, the transmission slope is negative. In this area, the transmission of the mw probe field takes a downward trend, and NV centers in diamond show OL behavior. The difference in the amount of the transmission slope shows that the efficiency and threshold of the optical limiter can be controlled by the acoustic field. Moreover, area 1 of Fig. 4 corresponds to the area from Fig. 3 where the mw probe field experiences gain. However, as can be seen, area 1 of Fig. 4 is smaller than area 1 of Fig. 3. The area enclosed between these two areas has a negative transmission slope, which indicates that only the gain occurring in this enclosed area can be controlled or attenuated by the presented optical limiter.
Fig. 4. Density plot of the transmission slope of the probe field as a function of the acoustic field intensity and intensity of the input mw probe field. The used parameters are those taken in Fig. 2.
The detuning of the strain-induced acoustic field can be another coherent parameter to control the features of the optical limiter. In Fig. 5, the transmission evolution of the output mw probe field versus the intensity of the input mw probe field is plotted for different values of the acoustic field detuning. The used parameters are I $_{s}$= 0.58 GW/cm$^2$, I $_{c}$= 8.15 GW/cm$^2$ and $\Delta _{c}=0$. The multi-photon resonance condition is also satisfied as $\Delta _{m}=\Delta _{s}+\Delta _{c}$. In general, the detuning of a coherent field can be altered by changing the frequency of that field. Figure 5 illustrates that the efficiency and OL threshold of the optical limiter are controllable by changing the detuning of the acoustic field. In addition, it is displayed that if the intensity of the mw probe field exceeds in a harmful way, the transmission of the probe field can be attenuated to any desired value by changing the detuning of the acoustic field. Furthermore, the transmission may occasionally exceed its normalized limit due to the of gain ocured in the NV centers. Generally, the presence of gain in the absorption spectrum can be accompanied by advantages and disadvantages in the light emitted from the system. Gain can add some noise to the output light, which disrupts the information contained in the light. In addition, the gain induced in the system can also increase the intensity by amplifying the light. We know that the optical limiters are featured to have high transmittance for low intensity of input light and low transmittance for intense power. Figure 5 shows that the strain-induced limiter can amplify the mw field. Since the generated gain occurs in a small intensity range, the transmission of the low input intensity enhances. Thus, when the intensity of the mw field entering the system is low, the presented limiter can intensify the mw field by noiseless amplification through the gain [53]. However, increasing the light intensity beyond the permissible threshold can also damage the optical instruments. It is shown in Fig. 5 that if the amplified output light is dangerous, it is quickly reduced.
Fig. 5. Transmission of the mw probe field versus its input intensity for different values of the detuning of the acoustic field. The used parameters are I $_{s}$= 0.58 GW/cm$^2$, I $_{c}$= 8.15 GW/cm$^2$, $\Delta _{c}=0$ and $\Delta _{m}$ follows the multi-photon resonance condition as $\Delta _{m}=\Delta _{s}+\Delta _{c}$ .
Now, we are going to present an analytical expression of the imaginary part of the transition coherence $\rho _{31}$. An analytical expression delineates the physics of the phenomenon and the role of the different parameters in generating the OL in the system. Under steady state and when the applied fields are in resonance, the analytical expression obtained from Eq. (2) is given by
Equation (2) shows that the cross-Kerr nonlinearity by the acoustic and coupling fields is the dominant effect in establishing the OL in NV centers in diamond. It is seen that the acoustic field acts on the system through the five-photon transition term $\Omega _{m}\Omega _{c}^{2}\Omega _{s}^{2}$. This term is done through the transitions $|1\rangle \xrightarrow {\Omega _{m}}|3\rangle \xrightarrow {\Omega _{s}}|2\rangle \xrightarrow {\Omega ^{*}_{c}}|1\rangle \xrightarrow {\Omega _{c}}|2\rangle \xrightarrow {\Omega ^{*}_{s}}|3\rangle$. It is noteworthy to explain that the Eq. (2) is in good agreement with our numerical results.
The density of NV centers inside the DMR can also be considered as another scenario to control the optical limiter. The greater the number of NV centers, the more effective the strain-induced acoustic field can interact with the NV centers. Figure 6 shows the transmission of the probe field versus input intensity for different values of the resonant absorption $\alpha l$. Note that $\alpha l$ is a function of the NV centers density as well as the length of the DMR in which the NV centers are embedded. It is displayed in Fig. 6 that the efficiency and optical threshold of the acoustic field-induced optical limiter can be also controlled by changing the numbers of the NV centers inside the diamond.
Fig. 6. Transmission of the mw probe field versus its input intensity for different values of the resonant absorption $\alpha l$.
Here, Z-scan technique is introduced to investigate the feature of the OL behavior. The Z-scan technique devised by Sheik-Bahae et al. provides a reliable and straightforward method for measuring the nonlinear absorption coefficient as well as the OL behavior of a large variety of materials [54,55]. In this method, a Gaussian laser beam is propagated as the input probe field. The transmittance of the sample through the aperture is recorded as a function of the position, Z, of the nonlinear sample with respect to the focal point (z=0). This focal point is created by the convergence of a Gaussian light by a lens in open aperture Z-scan configuration. The intensity of the Gaussian laser probe field with the spot size of the beam $w_{0}$= 1.5 cm can be given by
where $I_{0}$ and $w(z)= w_{0}\left [1+\left (z / z_{0}\right )^{2}\right ]^{1 / 2}$ are the initial intensity of the incident probe field and the beam radius, respectively. Moreover, $z_{0}=\pi w_{0}^{2} / \lambda$ specifies the rate of the diffraction length for the Gaussian probe beam. Typically, the increase or decrease in light intensity is easily realized with moving the sample around the focal point. By bringing the sample closer (farther) to the focal point, the increase (decrease) in light intensity can be slowly imposed on the sample. Thus, recording a peak at the focal point in the output light intensity profile transmitted from the sample means that we are in the SA region. On the contrary, observation a dip at the focal point stands for establishing the OL in the sample.Figure 7 displays the intensity profile of the output mw probe field measured by the theoretical Z-scan experiment to investigate the obtained results of Fig. 2. The wavelength of the mw Gaussian probe field is 10.4 cm corresponding to the transition $|1\rangle \leftrightarrow |3\rangle$. In Fig. 7(a), the Z-scan measurement is presented for different values of acoustic field intensity in the case that the input mw probe field intensity is I$_0$ = 2.3 GW/cm$^2$. Figure 7(a) shows that when the strain-induced acoustic field is switched off, the Z-scan take a peak form. This peak indicates that the SA is the dominant phenomenon in the system when there is no lattice vibration. By applying the acoustic field, it can be seen that the Z-scan measurement turns into dip. Moreover, it is shown that the dips become deeper by increasing the intensity of the acoustic field, representing that the optical limiter is more efficient with increase the acoustic field. In Fig. 7(b), we present the Z-scan measurement for different values of the input mw probe field for the case that I $_s$ = 0.58 GW/cm$^2$. The intensities of the focal point are selected from Fig. 2 in such a way that OL and SA have occurred in the system. It is shown in Fig. 7(b) that for intensities I$_0$ = 1 GW/cm$^2$, 2 GW/cm$^2$ and I$_0$ = 4 GW/cm$^2$ where OL has happened in Fig. 2, the intensity profile of Z-scan takes a dip form at the focal point. For intensities I$_0$ = 10 GW/cm$^2$, 13 GW/cm$^2$ and I$_0$ = 15 GW/cm$^2$, a peak is seen at the focal point of the intensity profile, indicating the dominance of the SA in the NV centers in diamond. The measurements of the theoretical Z-scan experiment is in good agreement with our numerical results, and the performance of the acoustic field-induced optical limiter in the range of microwave fields is confirmed.
Fig. 7. Z-scan experiment for the obtained results of OL in Fig. 2, where (a) Z-scan measurement for different values of the acoustic field for a mw probe field intensity I$_{0}$ = 2.3 GW/cm$^2$ and (b) for different values of the input mw probe field intensity for an acoustic field intensity I $_{s}$ = 0.58 GW/cm$^2$ using input Gaussian probe field wavelength 10.4 cm.
4. Conclusion
In summary, we investigated the coherent generation and control of the RSA and OL via strain-induced acoustic field for mw field range in a DMR embedded with many NV centers. It was shown that in the absence of the acoustic field, the SA is governed in the NV centers. When acoustic field is applied to the diamond, the RSA is induced in the system through the cross-Kerr effect, and NV centers show OL behavior. We demonstrated that the optical threshold, dynamic range and efficiency of the optical limiter can be controlled by intensity and frequency of the acoustic field. It was illustrated that the acoustic-induced optical limiter can amplify noiselessly the low intensity of the mw field entering the sensors through the gain. In addition, we showed that the current optical limiter can attenuate any gain-induced noise and increase in the intensity of the microwave field if it exceeds the optical threshold. Moreover, it was shown that by increasing the number of NV centers, the optical limiter can be more efficient. The physics of generating and controlling the OL behavior of NV centers in mechanical resonator was presented by an analytical expression which was in good agreement with the numerical results. Finally, the theoretical Z-scan experiment was employed to confirm the obtained results. The proposed acoustic-induced optical limiter can be a useful tool for protecting different optical and electronic devices in mw range, remote sensing, navigation, communications, mw heating and thermo/laser therapy.
Disclosures
The authors declare no competing financial and non-financial interests.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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