Real-time complex temporal response measurements of ultrahigh-speed optical modulators

Yongwoo Park; Tae-Jung Ann; José Azaña

doi:10.1364/OE.17.001734

1. Introduction

Direct monitoring of the temporal phase and amplitude response of optical modulators (i.e. electro-optic modulators, electro-absorption modulators, all-optical nonlinear switches etc.) to external drives is an essential procedure for diagnosing and optimizing the signal transmission performance in high-speed optical communication systems. Several techniques have been recently investigated for this purpose [1–3]. Although an indirect characterization of the temporal modulator under test already provides critical parameters, such as the voltage-induced phase shift, and the chirp parameter, for full optimization of the signal modulation process [4,5], a direct real-time characterization technique displaying the modulator amplitude and phase temporal response to a given excitation is of essential importance, e.g. for immediate adjustment of the modulated signal conditions. Here, by ‘real-time’ we refer to the frame rate at which the measured profiles are displayed, which should be faster than the video rate frequency (> 15 frames per second). Another potentially interesting feature of a time-domain signal characterization technique is that of “single-shot” capability, enabling the measurement of non-repetitive events.

Direct detection techniques using interferometric methods have been demonstrated [2]. Specifically, these techniques have enabled the characterization of modulation bandwidths up to 10-MHz. A technique for direct characterization of GHz-bandwidth modulators has been also reported [3]. Recently, we have demonstrated a simpler characterization technique using a stretched pulse interferometer [1], which allows direct detection of the complex modulator response. The proposed approach is based on the use of a temporally stretched input optical pulse followed by conventional interferometric detection of the modulated signal either in the time or in the frequency domain. We anticipated that the time-domain characterization method could have the potential to provide real-time and single-shot diagnosis capabilities. The setup demonstrated in [1] allowed us to characterize optical modulators (not in real-time) with a maximum modulation bandwidth of 2.5-GHz, limited by the photodetector bandwidth (time-domain method) or by the resolution of the spectral measurement (frequency-domain method). Considering the time/spectral resolutions of conventionally available photodetectors/spectrometers, it would be very challenging to increase the measurable modulation bandwidth beyond the value reported in Ref. [1] as well as to achieve real-time diagnosis of GHz-bandwidth modulators with single-shot capability using this previous basic setup.

In this paper, we report a significant improvement in our direct modulator characterization method [1], particularly in terms of measurable modulation bandwidth and acquisition speed. The bandwidth limitation of our original setup has been overcome by use of a ‘common-path temporal image magnification process’, which only requires the insertion of a second highly-dispersive element in the measurement setup combined with the pre-chirping of a probe pulse by the first low-dispersive element. This temporal magnification process drastically relaxes the required speed of photodetection and electronic processing, thus enabling real-time measurement with single-shot capability of ultrahigh-speed modulation events using a commercially available photodetector and oscilloscope. This phenomenon can be interpreted as a time-domain equivalent of the well-known ‘lensless imaging’ process in spatial optics [6]. A similar temporal image magnification technique has been reported for high speed analog-to-digital conversion in microwave photonics [7]. This method is in turn related with the ‘temporal microscope’ concept based on a so-called ‘time-lens’ [8]. It is important to note that our proposed method allows undistorted time stretching and subsequent recovering of the complex information of the modulation, whilst these previously reported techniques [7, 8] enabled only that of the signal amplitude.

We provide here a detailed theoretical analysis on the processes leading to temporal imaging (magnification) of the complex modulation information, which enables us to derive the key design specifications, particularly in terms of the required dispersion coefficients and input pulse bandwidth, to optimize the measurement system for a particular modulation signal specification. It is emphasized that the detailed analysis provided here reveals that the two used dispersions and the input pulse bandwidth (especially the dispersion of the first stretcher) need to be carefully selected, according to the targeted modulation bandwidth, to ensure that the complex modulation information can be properly ‘magnified’ and subsequently reconstructed using spectral interferometry. As a proof-of-concept experiment, complex temporal reconstruction of an impulse modulation is demonstrated on an EO modulator with a bandwidth higher than 40GHz at 30 frames per second.

2. Operation principle

A schematic of the operation principle is shown in Fig. 1. Temporal stretching of an ultra-short pulse (e.g. induced by linear dispersion in a linearly chirped fiber Bragg grating, LCFG [9]) enables us to map the ultra-fast complex temporal modulation along the stretched pulse duration in both the time domain and the spectral domain [1], taking advantage of the time-frequency mapping (TFM) induced by the pulse stretching and modulation processes [10]. In particular, the modulated temporal waveform, ĉ(t_R), can be written as ĉ(t_R) = â_c(t_R) · ŝ(t_R), where ŝ(t_R) is the complex modulation to be tested and â_c(t_R) is the temporally stretched pulse waveform (notice that in this notation, the used functions are the temporal complex envelopes of the corresponding optical waveforms; t_R refers to the reference time axis in which the average group delay of the considered system is obviated). The stretched pulse â_c(t_R) is assumed to be obtained by first-order linear dispersion of an input transform-limited ultrashort optical pulse â_i(t_R) through a first-order dispersive medium (LCFG), characterized by the following dispersion coefficient: $\ddot{Φ}$ ₁ = [-∂²Φ(ω_opt)/ω_opt]_{ω_opt=ω₀} , where ω_opt is the optical angular frequency variable and ω ₀ is the carrier optical frequency (ω= ω_opt -ω ₀ is the base-band frequency variable). This dispersion process can be mathematically described in the time domain using the temporal impulse response of the stretcher, ĥ₁(t_R) ∝ exp[jt ² _R/(2 $\ddot{Φ}$ ₁)] [9], in particular:

\begin{matrix} {\hat{a}}_{c} (t_{R}) = {\hat{h}}_{1} (t_{R}) * {\hat{a}}_{i} (t_{R}) \\ \propto {\hat{h}}_{1} (t_{R}) \cdot \int {\hat{a}}_{i} (τ) \exp [j τ^{2} / (2 {\ddot{Φ}}_{1})] \exp [j t_{R} τ / {\ddot{Φ}}_{1}] dτ \\ = {\hat{h}}_{1} (t_{R}) \cdot {\hat{a}}_{s} (t_{R}) \end{matrix}

where the asterisk ‘*’ denotes convolution and â_s(t_R) represents the integral function. This first temporal stretching mechanism is intended to map the temporal complex modulation information ŝ(t_R) into the frequency domain, i.e. for TFM. It has been previously shown [10] that in order to achieve this TFM process, the dispersion introduced by the stretcher needs to be sufficiently small so that to satisfy the following condition [10]:

Fig. 1. Concept diagram of the time-to-frequency conversion process and the temporal image magnification used for the proposed modulator direct time-response measurement in real time.

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∣ {\ddot{Φ}}_{1} ∣ ≪ \frac{8 π}{Δ ω_{s}^{2}}

where Δω_s is approximately given by the modulation bandwidth (i.e. bandwidth of ŝ(t_R).

The condition in Eq. (2) indicates that the square root of the first dispersion needs to be much smaller than the fastest temporal feature of the modulation. Under condition (2), the field spectrum at the modulator output, Ĉ(ω), is proportional to the temporal modulation signal [10]:

\hat{C} (ω) \propto \exp (j \frac{{\ddot{Φ}}_{1} ω^{2}}{2}) \cdot {[{\hat{a}}_{s} (t') \cdot \hat{s} (t')]}_{t' = {\ddot{Φ}}_{1} ω}

Eq.(2) provides an upper limit for the applicable dispersion for accurate TFM. The input dispersion also needs to satisfy a second condition to ensure that the time width of the temporally stretched pulse â_c(t_R) is longer than the total duration Δt_s of the complex modulation signal ŝ(t_R) to be measured. This is necessary to ensure that the full modulation waveform is copied on the optical pulse envelope. This translates into the following additional condition for the input dispersion:

∣ {\ddot{Φ}}_{1} ∣ (\frac{2 π}{Δ t}) > Δ t_{s}

where Δt is an estimate of the full time-width of the ultrashort input pulse â_i(t_R) before stretching by the first dispersive element (in Eq. (4) it is assumed that the input optical pulse is nearly transform limited). Notice that in most practical cases Δt_s > Δt, and as a result, Eq. (4) necessarily implies that the first dispersion ∣ $\ddot{Φ}$ ₁ ∣ is sufficiently higher than Δt ² so that the input pulse 9], i.e. â_s(t_R) ≈ Â_i(ω′ = t_R/ $\ddot{Φ}$ ₁)-Hence, the input dispersion needs to be selected according to the conditions in Eq. (2) (upper limit) and Eq. (4) (lower limit), i.e.

\frac{Δ t_{s} \cdot Δ t}{2 π} < ∣ {\ddot{Φ}}_{1} ∣ ≪ \frac{8 π}{Δ ω_{s}^{2}}

Under these conditions, the field spectrum at the modulator output will be proportional to the complex temporal modulation signal under test with an envelope defined by the input pulse spectrum [see Eq. (3)], i.e. Ĉ(ω) ∝ Â_i(ω) · ŝ(t′ = $\ddot{Φ}$ ₁ · ω).

To illustrate the derived design equations, in particular Eq. (5), simulations have been performed for three different conditions using the discrete Fourier transformation: ∣ $\ddot{Φ}$ ₁∣ ≤ Δt_s-Δt/(2π) [Fig. 2(a)], Δt_s-Δt/2π < ∣ $\ddot{Φ}$ ₁∣ ≪ 8π/Δω ² _s [Fig. 2(b)], and ∣ $\ddot{Φ}$ ₁∣ ≤ 8π/Δω ² _s [Fig. 2(c)]. In these simulations, the stretched optical pulse was defined as a Gaussian spectrum of the form Â_c(ω) ∝ exp(j $\ddot{Φ}$ ₁ ω ²/Δ ²)exp(j $\ddot{Φ}$ ₁ ω ²/2) , where Δω = 1/Δt. The modulation waveform intensity was also a Gaussian function: ŝ(t) ∝ exp(-t ²/Δt ² _s) (The square of the waveform is shown with dashed, red lines in Fig. 2). We assumed that the input pulse width and the modulation time width were fixed to Δt =250 fs and Δt_s = 5 ps, respectively. Figure 2 (solid, blue curves) shows the normalized energy pulse spectrum after temporal modulation, ∣Ĉ(ω)/Â_i(ω) ∣² for three different dispersion values: 0.2-ps² [Fig. 2(a)], 2.8-ps² [Fig. 2(b)], and 12.75-ps² [Fig. 2(c)], respectively. For the representation of the spectral profile along the time axis, the time-frequency scaling defined above (t_R = $\ddot{Φ}$ ₁ · ω) has been used. Considering that the factors in Eq. (5) Δt_s-Δt/2π and 8π/Δω ² _s are 0.2-ps² and 16-ps², respectively, the second simulated case would be likely an optimized dispersion for the considered parameters. This is clearly confirmed by the simulation results shown in Fig. 2. A truncation of the TFM waveform can be observed in Fig. 2(a) (blue line). As anticipated above, this distortion is caused by the fact that the duration of the dispersed optical pulse is shorter than the modulation full time width, i.e. the used dispersion is too low to satisfy the condition in Eq. (4). As a result, only a temporal portion of the input modulation waveform is mapped into the output spectrum.

Fig. 2. Modulation waveform in intensity (dashed red lines) and the imaged waveform via the TFM process for an insufficient amount of dispersion (a), an optimum dispersion value (b), and an excessively high dispersion (c)

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For completeness, we estimated the optimum dispersion range as a function of the modulation time width; for this purpose, the accuracy of the TFM process was calculated using the following cross-correlation (CC) coefficient: [11]

C = \frac{\int_{- \infty}^{+ \infty} ∣ \hat{S} (t) ∣ \cdot {∣ \hat{C} (t) / \hat{A} (t) ∣}^{2} dt}{{[\int_{- \infty}^{+ \infty} {∣ \hat{S} (t) ∣}^{2} dt \cdot \int_{- \infty}^{+ \infty} {∣ \hat{C} (t) / \hat{A} (t) ∣}^{4} dt]}^{1 / 2}}

where we recall that ŝ(t) ∝ exp(-t ²/Δt ² _s) is the modulation temporal waveform intensity (which is assumed to be Gaussian), and [∣Ĉ(ω)/Â_i(ω)∣²]_{ω=t/ $\ddot{Φ}$ ₁} is the normalized pulse energy spectrum at the modulator’s output after the proper TFM scaling. The values calculated from Eq. (6) quantify the normalized level of similarity between the modulation intensity waveform and the TFM spectral densities, where the ideal coincidence is given by 1. For example, the CC value for the results presented in Fig. 2(b) was ~ 0.998 whereas the CC value for the results in Fig. 2(c) was ~ 0.967. Figure 3 shows an estimate of the optimum dispersion range for the first temporal stretcher (φ₁) that guaranties a sufficiently high accuracy in the TFM process; in particular, the dispersion upper and lower boundaries that ensure a CC higher than 0.998 has been numerically traced as a function of the modulation time width (FWHM), assuming a fixed input pulse spectral bandwidth of 3 nm. Here, the input pulse and the modulation waveform were identical to those used in the numerical examples shown in Fig. 2. Random noises with amplitudes lower than 10% of the maximum spectral density have been also included in the TFM process by adding random fluctuations along the input pulse spectrum. The simulation result shown in Fig. 3 clearly reveals that the optimum dispersion range becomes more limited as faster modulation time features need to be resolved. In particular, for the example shown here, the optimum dispersion range becomes very tight for resolving time features in the picosecond regime; the dispersion conditions can be relaxed by use of an input optical pulse with a broader spectral bandwidth (essentially, this would reduce the value of the lower dispersion limit curve).

Following with our analysis of the measurement system in Fig. 1, the modulated waveform ĉ is made to interfere with the reference waveform with a proper relative time delay ζ; the output waveform can be approximated by the following expression (using a similar derivation to that leading to Eq. (9) in Ref. [9]):

\hat{i} (t_{R}) \infty {\hat{h}}_{1} (t_{R}) \cdot {{\hat{A}}_{i} (ω') \cdot \hat{s} (t_{R}) + {\hat{A}}_{i} (ω' - Ω) \cdot \exp (- j 2 ω' ζ)}

where we recall that ω′ = t_R/ $\ddot{Φ}$ ₁, and Ω. = 2ζ/ $\ddot{Φ}$ ₁ is the frequency shearing induced by the time delay ζ. In Eq. (7), is(t_R) ≡ î(t_R)/ĥ₁(t_R) is the waveform envelope at the output of the interferometer except for the quadratic phase factor ĥ₁(t_R). Obviously, the amplitude of this interference signal (interference amplitude pattern) is ∣î(t_R)∣ ∝ ∣ŝ(t_R)∣. The fastest temporal feature of the resulting interference amplitude pattern is essentially fixed by the temporal modulation signal to be measured ŝ(t_R) (in particular, for a proper phase reconstruction, the effective bandwidth of the interference pattern should be larger than that of the modulation signal). If the modulation bandwidth is too high, it will not be possible to resolve the temporal features of the resulting interference signal using conventional high-speed photodetection. Given that the interference amplitude pattern is also mapped in the frequency domain [see Eq. (3)], spectral-domain detection could be used as a potential alternative [1]. A main drawback of this latter approach is that in order to satisfy the frequency resolution specifications of the used spectral detection instrument, the required input dispersion may be too low (the higher the modulation bandwidth, the lower the required dispersion amount, see Eq. (2)), which would effectively limit the temporal extension of the measurable modulation waveform. This problem would be even more significant if a fast spectrometer is used (e.g. for real-time acquisition of the spectral interference pattern) as this equipment typically offers a much lower spectral resolution than other available spectral detection instruments (e.g. OSA).

Fig. 3. Dispersion upper/lower boundaries of the first temporal stretcher ensuring a TFM accuracy higher than 99.8% as a function of the modulation time width (FWHM), assuming a fixed input optical spectrum of 3 nm

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In this present work, the above mentioned limitations have been overcome by use of a second pulse stretching element (highly-dispersive LCFG) after the modulation process, leading to an effective undistorted time-domain magnification of the temporal interference pattern. This idea allows us to capture the interference pattern using time-resolved detection equipment with an operation bandwidth significantly lower than that of the original interference pattern (as determined by the bandwidth of the complex modulation waveform under test). The output interference waveform î_{out put} (t_R) after the second linear dispersive medium -characterized by a temporal impulse response ĥ₂(t_R) ∝ exp[jt ² _R/(2 $\ddot{Φ}$ ²)] -can be expressed as follows [12]:

{\hat{i}}_{out put} (t_{R}) \propto \hat{i} (t_{R}) * {\hat{h}}_{2} (t_{R}) \approx {\hat{h}}_{2} (t_{R}) \cdot \hat{I} (ω ″ \equiv t_{R} / {\ddot{Φ}}_{2})

where Î(ω″) ≡ ℱ{î_s · ĥ₁(t_R) · ĥ₂(t_R)} and ℱ is the Fourier transform operator. The output interference amplitude pattern is thus given by ∣î_{out put}(t_R)∣ ∝ ∣Î(ω″)∣, where:

\hat{I} (ω ") \infty {\hat{I}}_{s} (ω ") * {\hat{H}}_{12} (ω ")

where Ĥ₁₂(ω′) = ℱ {ĥ₁(t_R)·ĥ₂(t_R)} = exp[-j ω″²/(4α)], and α = 1/(2 $\ddot{Φ}$ ₁) + 1/(2 $\hat{Φ}$ ₂). Thus, from Eq. (9), it can be concluded that applying the second dispersion after the modulation simply induces an undistorted temporal stretching of the original interference amplitude pattern ∣î_s(t_R)∣ (so-called temporal image magnification process, TIM) by a time-domain magnification (stretching) factor

M = ({\ddot{Φ}}_{1} + {\ddot{Φ}}_{2}) / {\ddot{Φ}}_{1}

An important consideration in deriving Eq. (9) is that the following approximation has been used: (Δω ²/16) ≪ π where Δω is the full bandwidth of the original interference pattern is(t_R); under this approximation, the quadratic phase factor in the integral function in Eq. (9) can be neglected. Considering that Δω is of the order of the modulation bandwidth Acos, the reader can easily prove that under the conditions stated above (in particular, when the condition in Eq. (2) for the first dispersion is satisfied), this approximation is always valid, regardless of the value of the second dispersion, as long as the two dispersions used in the system have the same sign.

In summary, the relative dispersion amount between the second and the first stretching media defines the temporal image magnification (stretching) factor (see Eq. (10)), which effectively reduces the frequency bandwidth of the complex temporal features of the modulation signal (features to be resolved) without any loss of information. This is the mechanism used to capture the high-speed interference pattern by time-resolved detection equipment with a detection bandwidth significantly lower than that of the original temporal interferogram. A high magnification factor can be achieved using a small amount of dispersion in the first stretcher and a large amount of dispersion in the second one. In fact, a relatively small dispersion in the first stretching element is required to achieve the desired TFM of the high-speed modulation temporal features (as expressed by Eq. (3)). It is important to emphasize again that the maximum detectable frequency of the modulation signal is fixed by the dispersion of the first dispersion element (as determined by Eq. (2) to ensure an accurate TFM process). The second stretching element does not improve the temporal resolution of the TFM process but does lower the frequency of the interference pattern to be measured, thus enabling its detection by a time-resolved instrument with a relatively lower bandwidth.

To reconstruct the amplitude and phase profiles from the acquired interference waveform one should first keep in mind that the interference intensity that is measured at the photodetector output is actually proportional to the square of the waveform defined by Eq. (9). Following this squaring process, a similar interference equation to that defined by Eq. (1) in Ref. [2] is obtained. The well-known Fourier-Transform-Spectral-Inteferometry (FTSI) algorithm [13] can then be used to extract the amplitude and phase profiles from the measured interferogram. As detailed elsewhere [9], the dispersion introduced by both the first and the second linear temporal stretchers, including their respective phase noise and errors caused by the nonlinear dispersion slopes, can be precisely calibrated using the Hilbert transformation compensation method, which essentially requires a single pre-calibration measurement with the modulator turned off.

3. Experiments

3.1. Proof-of-concept experiments

As proof-of-concept experiments, we tested a Lithium Niobate EO intensity modulator (Z-cut, alpha parameter ~ 0.6) with EO bandwidth ≥ 35GHz (E-O space Inc.). A schematic of the used experimental setup is shown in Fig. 4. An ultrashort pulse (FWHM optical bandwidth ~ 2.7 nm, pulse time-width < 1 ps), generated from a wavelength tunable mode-locked fiber laser (Pritel Inc.) operating at 16.7MHz repetition rate and at an optical wavelength of ~1544 nm, was temporally stretched by linear reflection in a low dispersion LCFG with a bandwidth of 10-nm centered at 1545nm. The dispersion coefficient of this first LCFG was 170 ps/nm ( $\ddot{Φ}$ ₁=215 ps²) corresponding to the dispersion of 10-km single-mode fiber. The stretched pulse with ≈431-ps FWHM time duration was amplified and split by a 70/30 fiber coupler. The optical modulator under test was located in one of the coupler’s arm. The other coupler’s arm was used as the reference. The pulse average power in the reference arm was 33.3 μW. The pulse average power at the output of the modulator was varied depending on the applied bias voltage (< 100 μW). After combining the two signals through a 50/50 fiber coupler, the generated interferograms were temporally stretched with the second LCFG (2000 ps/nm, 42-nm bandwidth, Proximion Inc.) for common-path temporal image magnification by a factor of M = 12.7 (=2170/170). This interference was finally acquired in the spectral domain using an OSA (resolution = 10pm) and in the time domain using a sampling oscilloscope with an optical sampler of ~20 GHz 3dB-bandwidth. The reconstructed intensity ∣ŝ∣² and phase ϕ(t) profiles of the modulation signal from the spectral and the time domain methods are shown in Fig. 5 and Fig. 6, respectively. An input electric pulse with 28 ps at FWHM was used as the input excitation in the EO modulator; this pulse, shown in the inset of Fig. 5, was generated by simply converting the ultrashort optical pulse from the fiber laser into an electric signal using a high-speed photodiode. It is important to note that the applied electric pulse width was not sufficiently short to measure the impulse response (IR) of the EO modulator since according to the EO modulator bandwidth (> 40-GHz), the modulation pulse width for an IR measurement should be shorter than ~ 10 ps. Thus the temporal complex response evaluated in the experiment reported here would be actually the convolution of the applied electric pulse and the modulator’s IR rather than the IR itself. The peak instantaneous transmission by the impulse modulation was measured using CW light from a laser diode and was determined to be 6% of the maximum transmission of the E-O modulator at a certain bias voltage. Considering that in our experiments Δt · Δt_s/2π ≈ 0.71 ps² and 8π/Δω ² _s ≈ 500 ps², the dispersion of the first stretcher (215-ps²) was in the optimum range as discussed in Section 2 [Eq. (5)].

Fig. 4. Practical experimental setup for the direct real-time E-O impulse response measurement.

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There was a fairly good agreement between the results obtained from time-domain and spectral-domain measurements. However, more detailed features could be observed from the time domain method (Fig. 6) compared to the spectral domain approach (Fig. 5), which clearly evidenced the effectiveness of the proposed LCFG-induced temporal image magnification procedure for increasing the measurable modulation bandwidth. The ‘smoothing’ effect in the spectrum approach may be attributed to insufficient spectral resolution of the used spectrometer (FWHM linewidth > 10 pm). Note that complex time responses with features as fast as ≈ 35 ps (FWHM) were successfully characterized. The measurement accuracy is also shown in terms of the standard deviation for five consecutive reconstructions of the amplitude and the phase profiles; see error bars in Fig. 6(a). Considering that the peak transmission is only 6% of the maximum, this accuracy evidences the high sensitivity of the interferometric phase measurement system. Finally the instantaneous frequency chirp (Fig. 6(b)), (1/2π)∂ϕ/∂t, was numerically derived from the reconstructed phase profile.

Fig. 5. Spectral-domain detection: temporal intensity and phase responses of the intensity modulator to the 28ps-FWHM electric pulse (inset)

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Fig. 6. Time-domain detection: (a) temporal intensity and phase responses for the same modulation and (b) corresponding frequency chirp.

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3.2. Real-time reconstruction of the modulation response (amplitude, phase, and instantaneous frequency response)

Amplitude and phase reconstruction of the modulation response was demonstrated at video rates, i.e. at 30 frames-per-second. The realization of this real-time operation strongly depends on the performance of a high-speed digitizer. Here, we used a commercially available 10-bit digitizer (Agilent Inc., DC252) having a sampling rate of 8 Giga samples per second and a 3-dB bandwidth of 3 GHz. Note that a similar measurement at video rates may be practically implemented using a high-speed real-time oscilloscope which can provide a higher sampling rate than that of the digitizer. To be able to resolve the fastest temporal features of about 28-ps of the electric impulse (corresponding bandwidth > 35GHz) using the available 3GHz digitizer, the temporal image magnification should be at least higher than 12. For this reason, we increased the temporal magnification factor in our system up to 21 by replacing the first linear stretcher with a 100 ps/nm LCFG (Teraxion inc., TF-CUS-1552.5-9nm) operating in the 1545nm ~ 1559nm wavelength range. The center wavelength of the laser pulse was adjusted accordingly. This magnification system allows stretching the time scale of the waveform to be measured (interference pattern), thus effectively increasing the achieved sampling frequency, which is actually given by the nominal sampling rate of the digitizer multiplied by the magnification factor M. As discussed in detail above, the maximum detectable modulation bandwidth does not depend solely on the system magnification factor. The measurable bandwidth should be determined counting on both the magnification and the first stretcher dispersion limiting the time resolution (modulation bandwidth) of the TFM process. The applied dispersion was designed again to be within the optimal dispersion range, as given by Eq. (5) above.

A dynamic movie displaying the direct complex impulse response measurement (in terms of amplitude, phase, and instantaneous frequency) of the (Z-cut) modulator excited by the electric impulse is shown in Fig. 7(Media 1) when a continuous increase of bias voltage was applied to the modulator. A similar phase variation to that shown in Fig. 5 and 6 can be observed in the video when the corresponding bias voltage is reached. As expected, discrete π-phase jumps can be also observed for bias levels at which the half-wave voltage of the tested intensity modulator is reached, in good agreement with our previous results shown in Ref. [1] for a lower bandwidth intensity EO modulator. It is worth noting that instantaneous frequency measurements have been reported already using the frequency discrimination technique [14–16]. Real-time acquisition has been demonstrated using a fast router [15]. However, notice that the sensitivity of the frequency discrimination strongly depends on the filter bandwidth and the slope so that a sensitive acquisition is only possible over a narrow frequency range. Another indirect real-time measurement technique for characterizing the complex information of temporal modulation has been also demonstrated based on the use of linear spectrograms [17]. It is worth noting that in general, any arbitrary optical pulse characterization technique that can provide the phase reconstruction of the light modulated from a CW light source could also be used for monitoring the complex modulation information [3, 17]. However, we emphasize again that the technique presented in this paper provides the following key advantages: (i) real-time measurement with significantly faster acquisition rates than in any previously proposed method (the repetition rate of the reconstruction process was close to 1000-frames/s even though the effective display rate for presentation was only 30 frames/s), (ii) more detailed and accurate characterization of the instantaneous phase, including discrete phase shift features, and (iii) the capability of providing single-shot measurements of non-repetitive events.

4. Conclusions

A direct real-time measurement method for accurately reconstructing the amplitude and phase time response of high-speed optical modulators with single-shot capability has been proposed and experimentally demonstrated. This technique is based on pulse interferometry combined with time-frequency mapping processes using fiber linear dispersion. Significant improvement of the measurable time features (10-times faster than in our previous report [1]) has been achieved by employing a ‘common-path temporal image magnification’ mechanism based on

Fig. 7. A dynamic movie displaying the impulse response of the E-O modulator excited by a 28-ps electric pulse in terms of amplitude (top graph), phase (middle graph), and instantaneous frequency (bottom graph). The horizontal axis is a triggered time in ns. (Media 1)

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The concept presented here could be applied for real-time, single-shot characterization of a variety of optical modulation processes, including EO modulators, electro-absorption modulators and nonlinear optical switches. Moreover, this same principle could be used for other interesting applications, including ultrashort optical pulse characterization (Indeed, the results presented here are on the characterization of a complex temporal optical signal obtained by modulation of CW light.) and diagnosis of a variety of nonlinear mechanisms with complex temporal responses, such as Four-Wave-Mixing (FWM), Cross-Phase-Modulation etc.

Acknowledgments

The authors would like to thank Dr. Julien Magné and Prof. Sophie LaRochelle, Département de Génie Electrique et de Génie Informatique, Université Laval, Québec City for the fabrication of the used low-dispersion LCFG. This work was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada through an Strategic Project Grant and by the Fonds Québécois de la Recherche sur la Nature et des Technologies (FQRNT).

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Real-time complex temporal response measurements of ultrahigh-speed optical modulators

Abstract

1. Introduction

2. Operation principle

3. Experiments

3.1. Proof-of-concept experiments

3.2. Real-time reconstruction of the modulation response (amplitude, phase, and instantaneous frequency response)

4. Conclusions

Acknowledgments

References and links

Supplementary Material (1)

Cited By

Figures (7)

Equations (13)

Optics Express