1. Introduction

Coherent communication systems based on the use of phase and amplitude controlled signals are being introduced due to their increased tolerance to noise and as a means of increasing the information spectral density [1]. This in turn means that phase controlled sources and phase detection methods are of great importance for both digital and analog systems. Integrated circuits with optical phase-locked-loops are under investigation [2]. Optical injection locking [3,4] has been shown to be very effective as a means to enhance the performance of semiconductor lasers through increasing the resonance frequency [5], reducing the linewidth [6] and the transmission chirp [7]. Recently, injection locking was used to realise a phase preserving limiter for 10 Gb/s phase encoded signals [8]. Theoretical and experimental studies have revealed the complex nature of these systems outside of the stable locking regime which can include multi-stability, pulsations, oscillations and chaos [9].

To date a complete analysis of the phase over the entire injection ratio range has not been performed as it is usually the frequency detuning limits which are considered due to their experimental accessibility. This paper theoretically investigates the variation in the phase difference between the injected and injection electric fields during stable locking (SL) using parameters appropriate to long wavelength Vertical Cavity Surface Emitting Lasers [10]. Based on the numerical stability of fixed point solutions to the rate equations we find that the range of stable phases is not fixed but varies non-linearly with increasing injection strength. The limits of these stable phases can be separated into three distinct regions with the transitions between these regions related to certain bifurcation points. The change in detuning of fixed phase values with injection ratio is investigated, in particular a locked phase value of zero. Some lines of fixed phase values are seen to gain stability after becoming tangent to the negative frequency locking boundary. At any injection ratio the detuning with zero phase will feature the maximum variation in photon and carrier number from their free running values and is easily located experimentally as the minimum of the voltage variation with detuning of an injected laser operated with a constant current. It is shown how to determine the coupling rate between the electric fields using this knowledge. Furthermore, the dynamic response at this detuning is described for different injection ratios. The modulation response parameters are calculated as a function of detuning and injection strength showing reduced frequency response induced by differences between the injected frequency and the modified cavity resonance.

2. Injection locked rate equations

The photon number, phase and carrier number of an injection locked system have been widely investigated [4, 11–13]. (The photon number can be measured through the optical power, the phase by interferometric means and the junction voltage is related to the logarithm of the carrier number and is readily measurable with detuning). The rate equations which describe an injection locked laser are [4]:

Numerical integration of the rate equations gives the variation of S, ϕ and N with detuning for a given injection ratio and the calculated S and N values as functions of detuning are presented using grey dots for R = −27 dB in Fig. 1. For detunings less than ∼ −5 GHz and greater than 0 GHz, these data points are scattered and disjoint as the laser is not in stable locking state. Between −5 GHz and 0 GHz they form a continuous curve. The continuous and smooth region is associated with the laser being in a stable locking configuration where the values of S and N remain constant in time. During stable locking the carrier number is generally lower than when free running while the photon number conversely increases as the laser acts as a regenerative amplifier of the input light [14]. In the unstable regions the laser oscillates at one or more frequencies which are non-linearly dependent on the detuning. The time series is calculated only for a finite period of time (200 ns) with only the final value shown accounting for the non-continuous nature of the data points at neighboring frequency detunings in these regions. While these regions are often discussed from a dynamical perspective the effect on S and N is rarely reported.

 

Fig. 1 (a) Photon, S, and (b) carrier number, N, as a function of frequency detuning from the free running cavity resonance for an injection ratio of −27 dB after a numerical integration. The end points of the selected time series signals at each detuning are indicated by grey dots while the maximum, minimum and average of those time series are shown using blue, green and black solid lines respectively. All possible fixed point values of S and N as calculated from the stationary analysis are traced out by a dashed line with stable points marked with red dots. S_fr and N_fr are the free running values of S and N respectively.

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The maximum, minimum and average values of the oscillations in S and N in the unstable regions measured after the timescale for relaxation oscillations are also plotted in Fig. 1. These bounds converge smoothly at the positive edge of the stable locking range as the amplitude of the oscillations shrink. The average carrier number is reduced below the free running value but, we note, this is not sufficient to indicate a stable locked state. It is the average of the carrier number curve which corresponds to the measured DC voltage across the device [10]. The negative edge of the SL region is abrupt due to a different mechanism governing the oscillations at this boundary, see Section 3. Distinct peaks and troughs near −6.5 GHz indicate that the laser enters and exits several areas of different higher order dynamics within a small detuning span.

By setting the phase to discrete values between −π and π Eqs. (4) and (6) can be solved simultaneously generating a set of all the possible (S, ϕ, N) steady state solutions, referred to as fixed points, which can be either stable or unstable. Most of the fixed points calculated here are unstable and so are not observed experimentally. Their values are overlaid on the numerical integration results in Fig. 1 using dashed lines tracing out an elliptical region. Those that are calculated to be stable in Section 3 below are shown with red dots. A stable fixed point means that if the system is at that fixed point and experiences a small perturbation such as a momentary increase in N due to current driver noise it will return to that fixed point. Experimentally this is observed as stable locking. In contrast, once the system is perturbed from an unstable fixed point it will not return but will instead switch to another fixed point or oscillation. For example, a detuning of −2.5 GHz cuts the ellipse in Fig. 1(b) at two points with one stable (red dot) and one unstable (dash). This means that at this detuning the system will tend to the stable fixed point at N ≃ 4.45 × 10⁶. At higher values of α there are regions of multistability where two stable attractors can coexist which are outside the scope of this work. Extensive discussions on fixed points, stability and bifurcations can be found in [9].

3. Stability analysis

The stability of the fixed points of a dynamical system can be determined by a method which involves linearization of the rate equations [13]. Applying this method to all the possible steady state solutions determines limits as to which phases are actually stable. These phase limits can be broken into three separate regions depending on the injection ratio namely S1, S2 and S3, Fig. 2(a). Under weak injection (S1) the phase limits are those expected from the steady state solution of (2) namely ϕ_lower = −π/2 − arctan(α) and ϕ_upper = π/2 − arctan(α). These limits can be found by considering the range of the principal values of the arcsine function which is −π/2 ≤ϕ ≤ π/2. Under medium strength injection (S2) the limits vary non-linearly with injection ratio and no simple analytical expressions for them exist but they depend strongly on α. In the strong injection regime (S3) the limits reach ϕ_lower/upper = ∓π/2. These phase boundaries in the limit of strong injection were previously pointed out in [15]. At most injection ratios the stable locking region is bound by a single (ϕ_lower, ϕ_upper) pair marked by red triangles and black circles, respectively, in Fig. 2. However, injection ratios between −52 dB and −50 dB shows two disjoint areas of stable phases, Fig. 2(b). At low and high injection ratios (S1, S3) there is an absolute difference of π radians between the locking limits while in S2 the difference varies with injection ratio reaching a minimum for these parameters of approx. 5π/8 − arctan(α) at R=−36 dB.

 

Fig. 2 (a) The range of stable phases as determined from a stability analysis. Under weak injection the stable phase is limited by ±π/2 − arctan(α) whereas at high injection ratios it is limited by ±π/2. Two separate regions of stable phase occur near −51 dB. The minimum phase span is approx. 5π/8 − arctan(α) and occurs near −36 dB. (b) A zoom of the region near −51 dB which has two separate areas of stable phase.

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Given steady state values for S, ϕ and N the corresponding Δω can be calculated for that fixed point using Eq. (5) and a map of the stable locking region in (R, Δω) space can be generated. The stable injection limits are indicated using black circles in Fig. 3 and are shown across two graphs to highlight features at different frequency detuning scales. For the α value used here, fixed points lose stability through two different mechanisms. Firstly a sinusoidal oscillation, called a limit cycle, is created at a stable fixed point as it changes to an unstable one. This process is referred to as a Hopf bifurcation (HB). The number of frequencies in the limit cycle and their frequencies can change via torus and period doubling bifurcations (not shown) [9]. The second mechanism through which the system loses stability is a Saddle-Node bifurcation (SN). In this process a stable fixed point collides with an unstable fixed point so that both are annihilated. In the system considered in this work the injected laser follows a stable limit cycle when fixed point stability is lost at a SN bifurcation, for example in the region including the point (−50, −2) in Fig. 3(a). The HB and SN lines are represented by blue and red lines respectively, and were generated using the bifurcation software AUTO [16] and match the limits generated by this stability analysis. The shape of the positive detuning SN and HB curves near −50 dB reflects the splitting of the range of stable phases as shown in Fig. 2(b). A large portion of the detunings within the stable locking region, in particular in S1 and S2, correspond to multiple fixed points. However only one of those fixed points results in stable locking, see Fig. 1. The frequency detuning as a function of injection ratio for selected values of ϕ are plotted in Fig. 3. At very low injection ratios, Fig. 3(a), ϕ = ±π/2 − arctan(α) form the locking boundaries (SN) while at high injection ratios, Fig. 3(b), ϕ = ±π/2 are the locking boundaries (HB) as identified in Fig. 2(b). At intermediate injection ratios, the positive boundary is the HB curve and the negative boundary is the SN curve.

 

Fig. 3 (a) Frequency detuning of fixed values of ϕ with injection ratio for selected phases. The limits of the stable locking area (black dots) are calculated through the stability analysis. Hopf (HB) and Saddle-Node (SN) lines are shown in blue and red respectively. (a) At low injection ratios the SL region is limited by ±π/2 − arctan(α). (b) Detuning under stronger injection. The phase condition at the boundaries changes with injection until eventually the SL region is bounded by the ±π/2 line. Other phase curves stop varying in the detuning ordinate.The points marked with red crosses are cusp bifurcation points (labeled C) and codimension-two Fold-Hopf, also known as Zero-Hopf, bifurcation points (labeled FH) [9].

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These results show that the phase limits set by Eq. (5) in isolation are only valid at very low injection ratio (S1). There, the stable locking region is symmetrical with respect to detuning but the carrier and photon number are not so. For example, the asymmetry in N is shown in Fig. 4(a) where N is plotted against detuning for several injection ratios near the S1/S2 border at approx −52 dB. For ratios in S1, N > N_fr at detunings near the positive locking boundary. In S2 the locking range loses any symmetric properties it had as it splits into two and the separated positive region shrinks and disappears.

 

Fig. 4 (a) Variation in the carrier number with detuning at low injection ratios. The positive, Δω₊, and negative, Δω₋, locking widths as well as detuning at which the carrier number equals the free running carrier number, Δω₀, are highlighted for an injection ratio of −58 dB and α = 2. (b) Testing the correctness of $\left|\mathrm{\Delta }{\omega }_{-}/\mathrm{\Delta }{\omega }_0\right|=\sqrt{1+{\alpha }^2}$ for α = 0 to 3.5. This asymmetry is valid only for injection ratios in the S1 region for each α, as shown in Fig. 2(a) for α = 2.

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Experimental determination of the linewidth enhancement factor using optical injection methods has been reported on multiple occasions. Given the information presented in this work concerning the phase boundaries it is clear that the widely used expression to measure α in S2 [17, 18] via the asymmetry in the locking boundaries based on an expression derived using the fixed phase limits of S1 is invalid. Others more appropriately use power [19] and voltage [20, 21] variations in the S1 region. In general agreement with those works the expression required for this measurement is explicitly derived and discussed. Furthermore it is shown how the coupling rate, k, can be estimated in the same measurement. Estimates for k between 36 GHz and 1500 GHz have been used with some justifications [4, 22]. An interesting discussion on the effect of laser design on the coupling rate is given in [23]. The coupling rate is an important experimental parameter needed to relate observed dynamics to theoretical maps where the injection strength is usually non-dimensionalised [9] and has been used as a free parameter to fit experimental data to theory [24]. The value of k = 400 GHz used here was derived on a similar basis [10]. Discussions on the upper limits of the validity of Eqs. (4) to (6) in terms of injection strength usually focus on the injection ratio but must necessarily also include k which is often overlooked. A low coupling rate with strong injection can have a similar net effect to a high coupling rate with medium injection as it is always the product $k\sqrt{S_{\mathit{inj}}}$ which appears in the Eqs. (4) to (6). For instance these equations were used to successfully model cavity responses at an experimental injection ratio of +8 dB [25] which is close to the maximum value of +12 dB considered in this work.

Figure 4(a) shows that in the S1 region the locking range is symmetric and that the carrier number can experience both a positive and negative deviation from N_fr. Inspection of Eq. (6) shows that N will be equal to N_fr only for cos(ϕ) = 0. Following from Fig. 2(a) the only stable phase inside the locking range which satisfies this is ϕ = −π/2. At this phase Eq. (5) reduces to

The detuning with injection ratio for phase values between −π/2 − arctan(α) and π/2 are plotted in Fig. 5. The plotted difference in phase is non-linear with the smallest step sizes near ±π/2. The stability of the phase at each injection ratio is not indicated but it varies with injection strength as expected from Fig. 2. For injection ratios between −30 dB and −5 dB, the phase condition on the negative detuning SN line (negative locking boundary) adjusts to allow the smooth graduation in phase to π/2 radians as depicted in the lower inset to Fig. 5. The evolution of a particular phase (ϕ = 1.178 rads) is highlighted to show how, at low R, it represents an unstable phase before gaining stability as it becomes tangent to the SN curve and remains stable at higher injection ratios. Note a similar change happens on the positive detuning SN curve which has a phase of −π/2 − arctan(α) over most of its length but decreases toward −π near the cusp marked C in Fig. 3(b). The positive and negative detuning SN lines are joined together by a curve which crosses the stable locking region whose phase condition is not discussed here.

 

Fig. 5 (a) Frequency detuning with injection ratio for fixed ϕ between −π/2 − arctan(α) and π/2 where the phases difference between the chosen ϕ is non-linear. The black dots show the stable locking boundaries. The cross section at +8 dB (upper inset) shows that, under strong injection, the majority of the π phase change happens near the ϕ = 0 point near −25 GHz. The lower inset shows the phase condition on the Saddle Node line smoothly changing from π/2 − arctan(α) to π/2 as phase lines of increasing value become tangent to it. The trajectory of one curve is highlighted by dashing.

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Under strong injection the majority of the π phase change across the locking region occurs within a narrow detuning range offset to negative detuning as shown in the upper inset of Fig. 5 for an injection ration of +8 dB. The detuning at which ϕ = 0 is always stable and converges to value of ∼ −25 GHz for the parameters used here (thick black line). Setting ϕ = 0 in Eq. (5) gives an expression for the detuning in terms of the steady state photon number:

Equation (6) shows that at all injection ratios, the zero phase point corresponds to the maximum deviation in the photon and carrier numbers from their free running values. The reduced N during stable locking (Fig. 1(b)) becomes the threshold carrier number and the addition of the injection term in the photon number rate equation can be considered as a phase dependent reduction in the cavity losses, γ_p, it being a positive carrier independent term which is proportional to a photon number ( $\sqrt{S_{\mathit{inj}}S}$). The phase and carrier number dependence on detuning for several injection ratios are shown in Fig. 6 with the zero phase point marked on each curve. Under weak injection, the zero phase point is at the negative detuning boundary but tends to a more central position in the stable locking region as the injection strength increases. Thus, at any injection ratio, an absolute reference phase (ϕ = 0) can be found using just a DC voltage measurement as a function of detuning. Applying the same S ≃ S_fr approximation in S1 the coupling rate can be estimated using the detuning of the minimum of the carrier variation:

 

Fig. 6 Plots of (a) phase and (b) carrier number with detuning for multiple injection ratios. The points at which ϕ = 0 are marked with black dots in both plots and always correspond to the minimum of the carrier number.

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4. Small signal analysis at ϕ = 0

Under unstable optical injection light will be emitted at both the injected frequency and the resonance frequency of the optical cavity. Once locked, emission will be at the injected frequency but the resonance of the cavity still remains. Generally during stable locking, there will be a reduction in the carrier number in the active region, increasing the refractive index. The optical path length in turn increases leading to a red shift in the cavity resonance frequency. The extent of the red shift can be quantified as $\mathrm{\Delta }{\omega }_{\mathit{cav}}=\frac{1}{2}\alpha G_0(N-N_{th})$ where N_th is the threshold carrier number. This effect on the phase of the laser is included in the rate equations by the first term on the right hand side of Eq. (2).

The natural resonance of a free running laser is due to the coupling of the carriers and the photons as the system settles to its steady state. The frequency of these relaxation oscillations can be calculated to be $f_{r,fr}=\raisebox{1ex}{$\omega $}\!\left/ \!\raisebox{-1ex}{$2\pi $}\right.=\sqrt{G_0{\gamma }_p{S}_{fr}/4{\pi }^2}\simeq 2.9$ GHz for the current parameters. The overall relaxation rate is influenced by the introduction of a second resonance at a frequency related to the difference between the frequencies of the shifted cavity resonance and the injected light, |δf_res| [12]. Due to the non-linear change in N across the stable locking region, this difference frequency also changes non-linearly. This model for the resonance frequency under injection has been expanded upon by taking carrier pulsations [26] and non-linear gain into account [27].The non-linear gain term influences the resonance frequency through the amplification of the relaxation oscillation sidebands around the shifted cavity resonance. In particular the sideband at lower frequencies relative to the emission peak is seen to experience a detuning and injection strength dependent preferential gain which influences the overall resonance response. For the purposes of this discussion which focuses on the qualitative changes around the zero phase point these non-linear terms are not considered. A graphical method to understand the behaviour of the cavity resonance was recently proposed in [25].

Using the method outlined in [28] the resonance frequency, f_r, of the system under a small signal perturbation to the bias current can be calculated at each steady state point. At frequency detunings very close to the negative locking boundary, these perturbations can cause the laser to pulsate rather than oscillate. The variation of f_r as a function of detuning is plotted for injection ratios of −35 dB, −24 dB and −13 dB in Fig. 7. The minimum of f_r occurs near the zero phase point and decreases in magnitude with increasing injection strength. The highest values for f_r are calculated at the positive detuning boundary where very low damping rates (not shown) are also found. Injection at these detunings lead to isolated, high amplitude peaks in the frequency response. However, both resonance frequency and damping rate need to be relatively high in order to achieve a wide modulation bandwidth [26].

 

Fig. 7 The resonance frequency, f_r, of the optically injected laser calculated using a numerical analysis of the system and the frequency difference between the cavity resonance and injected light, |δf_res|, at injection ratios of −35 dB, −24 dB and −13 dB. The estimate of f_r from |δf_res| becomes more accurate with increasing injection strength as the effect of the optical injection becomes stronger than the intrinsic carrier/photon resonance. The value of the free running resonance frequency f_r,fr is indicated.

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The frequency difference, |δf_res|, between the optical injection and cavity resonance is calculated using the sin term in Eq. (2) and plotted as a function of detuning using dashed lines in Fig. 7. At each injection ratio this difference is zero at the zero phase point as the cavity resonance has red-shifted to exactly match the frequency of the injected light. However it is clear that |δf_res| only matches the actual resonance frequency f_r under stronger injection conditions. The value of f_r at the zero phase point with increasing injection strength is shown in Fig. 8(a). Under weak injection the resonance frequency of the laser at this detuning is due to the intrinsic photon/carrier interaction. At an injection ratio (≃ −35 dB) the resonance frequency begins to decrease as the injection effect becomes dominant and is zero by −16 dB. Unexpectedly, at a small span of higher injection ratios (2 – 5 dB) f_r becomes non-zero again. A more detailed picture is found by examining the behaviour of all the roots of the determinant of the Jacobian matrix of the system after the small-signal perturbation. These usually have the form of two complex roots, 0.5γ ± if_r, where γ is the damping rate and one real root, f_p, known as the real pole. Once f_r becomes zero the damping of the oscillations can no longer exist so the three roots will be referred to as the triple real poles.

 

Fig. 8 (a) The resonance frequency at the zero phase point as a function of injection ratio. At ratios < −35 dB the system retains a resonance at f_r,fr. As injection strength increases that resonance drops in frequency to 0 GHz except for a region around 3 dB where it returns approximately to f_r. (b) The various modulation parameters at the zero phase point as a function of injection ratio. Where the resonance frequency has gone to 0 GHz the damping rate no longer exists and instead splits into extra two real poles. These branch apart and then recombine at 2 dB where the resonance becomes non-zero again. At higher ratios one of the extra real poles gets very large while the other converges to the frequency of the conventional real pole.

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Values of f_r, 0.5γ, f_p and the triple real poles at the zero phase point are plotted as a function of injection ratio in Fig. 8(b). It is 0.5γ which is plotted here as that is what appears in the root of the determinant. Once the resonance frequency goes to zero near −16 dB two new real poles branch away from the damping curve before coming back together as the resonance frequency becomes non-zero again. In this regime the single real pole increases in value until it begins to saturate at approx. 15 GHz. Above 6 dB the damping rate breaks into two again with one branch increasing to large values while the other converges to the value of the single real pole.

The link between the 0.5γ and f_p curves around the zero phase point is best illustrated by plotting all of the small signal parameters versus frequency detuning at selected injected ratios in Fig. 9. At −13 dB, Fig. 9(a), the triple real poles smoothly join the four disconnected ends of the 0.5γ and f_p curves while at −5 dB, Fig. 9(b), the two curves are isolated from each other with two of the real poles branching from and returning to the 0.5γ curve. The transition between the two configurations is smooth. Figure 9(c) shows the situation at 3 dB where the non-zero resonance frequency and the other curves remain continuous across the zero phase point.

 

Fig. 9 The modulation response parameters of the system as a function of detuning for injection ratios of (a) −13 dB, (b) −5 dB and (c) 3 dB emphasizing different configurations of the reals poles at ϕ = 0. (a) The triple real poles smoothly join the damping and single real pole curves together. (b) The curves no longer join and the two of the triple real poles now branch away from the damping curve while the third one continues as the single real pole as at detunings away from the zero phase point. (c) The resonance frequency is nonzero so the damping and real pole curves are continuous and single-valued.

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The shape of the frequency response is highly dependent on all of the modulation parameters at a given detuning. For a wide modulation bandwidth, a flat frequency response curve with a high resonance frequency is desirable. In general it is difficult to achieve both of these as a flat response curve needs a high γ. The damping rate is maximum around the zero phase point where the resonance frequency is obviously lowest while it is minimized at the positive locking boundary where the resonance frequency can grow to very large values. This leads to frequency responses which are either flat but without a high resonance frequency or responses which quickly roll off in amplitude to below the −3 dB level but then feature an intense but solitary resonance peak for detunings near the positive locking boundary. It is also important to consider the effect of the real pole which will cause a drop in the frequency response near its characteristic frequency. A real pole frequency near 0 GHz will reduce the DC value of the response which is used to determine the −3 dB point while a high real pole frequency could reduce the maximum achievable bandwidth. A median value is desirable where the resonance and the real pole are suitably offset so that the attenuation is compensated for by the resonance and the response remains flat. Such an optimization by varying the injection ratio and frequency detuning has been discussed [29] where the dip due to the real pole is minimized to −3 dB of the DC response in order to maximize the modulation bandwidth. No large resonance peak is recorded at this configuration as the damping rate must be large enough that the peak due to the resonance is sufficiently wide and intense to compensate for the real pole attenuation.

5. Conclusion

The locked phase of an optically injection locked semiconductor laser was analysed in-depth to reveal three qualitatively different behaviours as a function of injection ratio. In each region different phase limits apply at the stable locking boundaries with analytical expressions available for weak and strong injection. Under medium strength injection the phase limits vary in a non-linear way and no closed forms for the limits were found. The frequency detuning with injection ratio for fixed values of the phase also vary non-linearly but a quasi-static distribution is found under strong injection. Frequency detunings at which the phase is zero are of particular interest and correspond to the maximum change in photon and carrier number. Simultaneous determination of the coupling rate and the linewidth enhancement factor from a single measurement of the voltage as a function of detuning under weak injection was discussed. Regularly considered a fitting parameter, the importance of the coupling rate can be often overlooked but it is necessary for accurate comparison of theoretical and experimental studies. This method suggested here allows a good estimate for k and facilitates better fitting of other experimental parameters which are harder to measure. The response to a small signal bias modulation at the zero phase detunings is different to the rest of the locking range and includes interconnected damping and real pole curves and both zero and non-zero resonance frequencies. The relationship between these response parameters at the zero phase point has also been presented and discussed.