Abstract

With the ever-increasing need for bandwidth in data centers and 5G mobile communications, technologies for rapid characterization of wide-band devices are in high demand. We report an instrument for extremely fast characterization of the electronic and optoelectronic devices with 27 ns frequency-response acquisition time at the effective sampling rate of 2.5 Tera-sample/s and an ultra-low effective timing jitter of 5.4 fs. This instrument features automated digital signal processing algorithms including time-series segmentation and frame alignment, impulse localization and Tikhonov regularized deconvolution for single-shot impulse and frequency response measurements. The system is based on the photonic time-stretch and features phase diversity to eliminate frequency fading and extend the bandwidth of the instrument.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

The global data traffic has been growing exponentially due to the popularity of bandwidth-hungry applications among mobile phone users, increased penetration of broadband technologies, and the adoption of cloud computing platforms [1]. The upcoming roll-out of fifth generation (5G) mobile communications and the future developments for the sixth generation (6G) will use millimeter wave (mmWave) bands to offer wide spectrum and multi-Gigabit-per-second data rates [2]. The optical fiber communication back-haul network is also impacted by the needs for high bandwidth and has met the demands by increasing the data rates. To accelerate the development of these high data rate technologies, we need fast methods to characterize the frequency response of high-speed devices at the production-level as well as for post-production debugging and maintenance. Long test times increase test costs at the production line [3]. With increasing frequencies, conventional test equipment cost, size, and power consumption must scale up proportionately. Real-time digitizers capable of recording mmWave signals are very expensive and consume high power. Low power and cost-effective instrumentation systems will be needed to meet the demand for production level testing of mmWave and optoelectronic devices.

Vector network analyzers (VNA) are used to measure frequency response of high bandwidth radio frequency (RF) devices during production-level testing and also for post-production debugging and maintenance. Conventional VNAs utilize frequency sweeping employing a harmonic mixer to perform frequency domain measurements of electronic components [4,5]. An optical vector analyzer employing single-sideband modulation for measuring the spectral responses of optical devices employing a frequency-swept RF source has also been reported [6]. The swept-frequency techniques avoid the need for real-time digitizers but result in longer measurement times which would increase test throughput and cost.

Time-domain impulse response measurement followed by digital fast Fourier transformation (FFT) is a powerful method to obtain the magnitude and phase frequency response of devices. But since the response of high-frequency devices is beyond the bandwidth of most real-time digitizers, an auto-correlator is typically needed to capture and digitize the output response. This results in a slower measurement technique and requires complex hardware [7]. The test times for high-speed devices can be significantly reduced if we can directly measure the fast response in real-time.

Time domain network analysis (TDNA) has been used to measure frequency-domain network parameters [8]. TDNA technique typically applies the test stimulus (a voltage step signal) to the device under test (DUT) periodically for many cycles and uses an equivalent-time digitizer (or digital sampling oscilloscope) to sample the response every cycle. The sampling rate of the digitizer is very low (typically 100 kHz to 10 MHz) and the signal waveform is reconstructed digitally provided the signal analyzed is periodic or clock-synchronous. Equivalent-time sampling oscilloscopes need several cycles of sampled data to be able to digitally reconstruct the original signal with high fidelity followed by digital signal processing (DSP) to analyze the signal to measure the network parameters of the DUT. Timing jitter and time base distortion significantly affects the performance of TDNA techniques and needs to be mitigated [8–10]. In addition, TDNA techniques are also slower due to the lower sampling rates and suffer in terms of accuracy to capture dynamics that occur at the nanosecond level.

Photonic time-stretch [11–13] was originally developed to overcome the speed and resolution limitations of high-speed analog-to-digital converters (ADC) [14–16]. Recently, photonic time-stretch has been used to perform fast single-shot characterization of photo-diodes and electro-optic modulators (EOM) [17–19]. The so-called SiNA (Single-shot Network Analyzer) enables extremely fast one-shot measurement of the complex (amplitude and phase) response of wide-band electronic devices. However, this instrumentation system required manual intervention for various calibration tasks. Additionally, the frequency response of the instrument was not calibrated, i.e. the response of the DUT contained the impulse response of the instrument. Manual intervention reduces the measurement throughput and the lack of impulse response calibration limits the accuracy especially when the bandwidth of the DUT is close to the bandwidths of the components used in the instrumentation system. Manual processing tasks including segmentation and localization of individual impulse responses from the measured continuous time series which limited the size of the data set that could be collected. Furthermore, the bandwidth of the time-stretch processor was limited by the dispersion penalty to 28 GHz. However, it is known that dispersion penalty is not a fundamental limit and can be overcome by methods such as phase diversity [20], single sideband modulation [21,22], and coherent detection [23] followed by digital backpropagation [24].

Photonic time-stretch has enabled the development of various high-throughput, real-time instruments for science, medicine, and engineering applications. Time-stretch technology has been employed for the discovery of “rare events” such as Optical Rogue Waves [25] and soliton explosions [26], the birth of laser mode locking events [27] and internal dynamics of soliton molecules [28]. Time-stretch was used to directly observe the relativistic electron bunch microstructures with sub-picosecond resolution in a storage ring accelerator [29–31]. It has enabled the record throughput of instruments such as serial time-encoded amplified microscopy (STEAM) [32–34] and high-speed quantitative phase imaging for label-free detection of cancer cells in blood with a sensitivity of one cell in a million [35,36]. A time-stretch accelerated processor (TiSAP) was used to perform real-time, in-service signal integrity analysis of 10 Gigabit/s streaming video packets for the first time on a commercial optical networking platform [37] and for ultra-wideband single-shot instantaneous frequency measurements [38]. Various research groups across the world have adopted time-stretch as a technique for the characterization of ultrafast phenomena and for increasing the resolution limits of high-speed ADCs [39–49].

In this paper, we propose a new single-shot Tera-sample-per-second device analyzer to overcome the limitations of existing techniques by using (i) a regularized deconvolution algorithm for on-line calibration, (ii) an automated time series segmentation and frame alignment to allow capture of large set of impulse responses, (iii) optical phase diversity for eliminating the frequency fading phenomenon known as the dispersion penalty in time-stretch systems. We have outlined means to ensure the instrument has a linear response. We also demonstrate, for the first time, extremely fast characterization of electronic devices using the time-stretch device analyzer.

2. Implementation

2.1 Tera-sample-per-second device analyzer

The Tera-sample-per-second device analyzer is a photonic time-stretch data acquisition system in which the impulse response of the DUT and the time aperture of the time-stretch system are synchronized because they are both created using the same laser pulse. The stimulus signal generation in time-stretch device analyzer is performed using a pulsed supercontinuum source. The pulsed supercontinuum source is a mode-locked laser centered at 1560 nm producing a short ~500 fs optical pulse at a repetition rate of 37 MHz. The optical pulse from the laser is power divided (90/10) and applied to a high-speed photodiode, PD1 (30 GHz), through a variable delay to generate an electrical impulse signal as shown in Fig. 1. This high bandwidth electrical impulse signal is used as the stimulus to test the wide-band electrical DUT. The impulse response of the electrical DUT is captured by the time-stretch data acquisition system.

 figure: Fig. 1

Fig. 1 Time-stretch device analyzer for single-shot measurement of the complex response of an electronic device under test (DUT). Phase diversity has been implemented using a dual-drive EOM with different chirps in outputs to produce 90° phase shifts between them, which enables overcoming the limitation in RF bandwidth caused by dispersion penalty. Bias controller enables the EOM operates in the linear region. D1, D2: dispersive elements; PD1, PD2, PD3, PD4: photo-detectors; PC: polarization controller; C1, C2: optical circulators; DSP: digital signal processing.

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The time-stretch data acquisition system modulates the impulse response of the DUT onto a pre-chirped optical pulse from the pulsed continuum laser. A single-arm, dual drive, and dual output high-speed intensity EOM with 4 V half-wave voltage, Vpi (measured at low frequency) is used to modulate the pre-chirped laser pulse with the impulse response of the DUT applied to its RF input port. These EOMs can operate in excess of 100 GHz [16,50]. To avoid nonlinear distortion in order to obtain an accurate measurement of the DUT impulse response, it is important to maintain the bias of the modulator at the quadrature (Vπ/2) voltage using a feedback bias controller. This technique utilizes the fact that at the quadrature bias, the modulator does not produce any second-order intermodulation distortion. A low frequency (out of the signal band) pilot tone is added to the signal and the bias is adjusted such the second harmonic of the pilot tone is minimized. Third order distortions are minimized by avoiding the modulator to be overdriven.

The mode-locked laser pulse is pre-chirped via a dispersive element such as a dispersion compensating fiber (DCF), D1 (−20 ps/nm). The pre-chirped pulse defines the time aperture of the time-stretch acquisition system. A bias controller employing a monitor photo-diode, PD2, is used to operate the EOM in the linear region. The two outputs of the modulator consist of the impulse response of the electrical DUT modulated on the pre-chirped laser pulse and its 90° out of phase signal respectively. These two outputs from the modulator with 90° out of phase are utilized to implement phase diversity to overcome the bandwidth limitation due to the dispersion penalty [20]. Both the modulator outputs undergo time-stretching by the same amount by a bidirectional transmission through the DCF, D2 (−984 ps/nm), enabled by optical circulators, C1 and C2. The two time-stretched signals from both ends of D2 are converted into electrical signals by photo-diodes, PD3 and PD4. The electrical signals are digitized using a two-channel digitizer to perform the measurements of the frequency response of the DUT. The repetition rate of the laser is 37 MHz and therefore, the impulse response measurement of the DUT is performed with a single-shot acquisition time of 27 ns. The backend digitizer is a real-time oscilloscope with a sampling rate of 50 Giga-sample/s, so the effective sampling rate of our time-stretch digitizer is 2.5 Tera-sample/s [15].

We note that the photonic time-stretch technique is an analog optical link with a broadband pulsed laser as the optical carrier and highly dispersive fiber. It is different from a time lens which is the temporal equivalent of a spatial lens and requires the input signal to be dispersed. The lack of low loss and wideband electrical dispersion technology prohibits the use of the time lens for slowing down electronic signals [12].

2.2 Eliminating frequency fading using phase diversity time-stretch

In a time-stretch system, when double-sideband modulation is used to modulate RF signals onto the optical pre-chirped carrier, a frequency-fading phenomenon due to dispersion occurs [12]. Dispersion introduces different phase shifts to the two sidebands and results in constructive and destructive interference beating of the sidebands with the carrier at the photo-detector. The destructive interference creates nulls in the RF transfer function of the time-stretch system as shown in Fig. 2, which is a simulation of the effect without any attenuation or detector bandwidth limits. The overall effect known as dispersion penalty is to limit the 3-dB RF bandwidth unless advanced time-stretch techniques such as phase diversity (discussed below), single sideband or backpropagation are used.

 figure: Fig. 2

Fig. 2 A simulation of RF fading due to dispersion penalty in a time-stretched data acquisition system. This RF fading would limit the 3-dB analog bandwidth of the device analyzer. The RF fading due to dispersion penalty can be overcome by using phase diversity employing an EOM that produces two outputs that have complementary fading characteristics and combining them [20]. This mitigates the effect of the dispersion penalty and extends the bandwidth of the time-stretch device analyzer.

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The intensity Mach-Zehnder modulator (MZM) with different chirps in outputs can be used to mitigate dispersion penalty with phase diversity. It can be implemented either using a single-arm, dual drive, and dual output (see Fig. 1, output1 and output2) MZM or a dual drive, single output MZM biased at two different quadrature points. Both these types of modulators can generate 90° phase shift in the two outputs and thereby have complementary fading characteristics. Figure 2 shows the simulation of such a time-stretch optical link with two complementary RF transfer functions for the output ports. The transfer functions of the two output ports H1(ω) and H2(ω) in such an MZM employing phase diversity can be written as [20]:

H1(ω)=cos(ω2β2L2Mπ4)
H2(ω)=cos(ω2β2L2M+π4)
where ω is the angular frequency, β2 is the dispersion parameter of the dispersive fiber, L is the fiber length and M is the stretch factor. The complementary fading characteristics of the two output ports can be combined to overcome the frequency fading effects of the time-stretch device analyzer. The maximum ratio combining (MRC) algorithm by weighted summation is used to combine the signals from the modulator ports. The weights for the summation depend on the signal-to-noise ratio (SNR) for that port. The combined output to recover the original RF signal in a phase diversity time-stretch system is given by [12]:

Y(ω)=H1(ω)Y1(ω)+H2(ω)Y2(ω)|H1(ω)|2+|H2(ω)|2

As seen from the simulation in Fig. 2, after applying the MRC algorithm to the two outputs of a single-arm, dual drive, and dual output MZM, the effect of the dispersion penalty is mitigated. This method extends the bandwidth of the time-stretch device analyzer beyond the dispersion penalty limits.

We note that digital equalization may be used to remove the dispersion penalty for device bandwidths below the first null in the RF transfer function. This can be implemented by a division operation in the frequency domain using DSP without adding hardware complexity. But this method is limited in bandwidth and does not utilize the true power of time-stretch. This method also requires avoiding the nulls in the frequency domain of the transfer function where signal power is too low, and the mathematical division operation is not defined due to division by zero. The null frequencies in the transfer function can be tuned by changing the stretch-factor or the input time aperture by adjusting the dispersion values, D1 and D2. The stretch-factor sets the trade-off between the input aperture size and the bandwidth. The phase diversity technique does not suffer from these limitations.

3. Automated DSP algorithms

The automated DSP for calibration is designed to (a) segment the time sequences into individual frames and align the frames in time, (b) identify the impulse and the proper frame length from the time-domain data sequence, (c) apply asymmetrical windowing and zero-padding (to improve the performance and resolution in frequency domain), and (d) perform Tikhonov regularization in the frequency domain to remove the instrument impulse response from DUT-instrument convoluted impulse response. The algorithm steps for automated calibration are listed below:

Perform the measurements

  • 1) Measure pulse envelope without any modulation, referred to as “Envelope.”
  • 2) Measure the instrument without DUT, this is the “Instrument” response.
  • 3) Measure the instrument with DUT, this is the total impulse of DUT and instrument, referred to as “DUT-and-Instrument.”

Prepare time domain signals

  • 4) Perform time series segmentation and frame alignment.
  • 5) Apply moving average filter to reduce noise by lowering the sampling rate.
  • 6) Regularize all the measured responses.
  • 7) Find the start and end sample points of the measured responses.
    • a) Find the maximum value in the responses within the selected frame, to locate the peak position.
    • b) Set the total number of sample points (20ns divided by the sample rate). Allocate them 1/2 to the left, and 1/2 to the right of the peak.
  • 8) Recover instrument responses (with and without DUT) by dividing them by the Envelope.
  • 9) Remove the artifacts in the recovered response, caused by the division operation at the tails, by locating the artifacts and truncating them.

FFT and deconvolution

  • 10) Apply asymmetric Hamming window to both recovered responses.
  • 11) Perform zero padding on both recovered responses after windowing.
  • 12) Perform FFT on both recovered responses to obtain the frequency response.
  • 13) Apply Tikhonov regularized deconvolution to calibrate the DUT-and-Instrument response for the Instrument response.
  • 14) Apply IFFT to obtain the calibrated time-domain impulse response of the DUT.

3.1 Time series segmentation and frame alignment

The single-shot time-stretch device analyzer has an extremely fast frame rate of 37 MHz. Being able to capture many instances of the impulse response can be utilized to collect statistical fluctuations of the DUT. To do so, one needs to automatically segment individual frames from the longtime series captured by the digitizer in the presence of drift in the laser repetition rate. This process of segmenting the longtime series into individual synchronized frames and aligning the frames into a 2D matrix is shown in Fig. 3.

 figure: Fig. 3

Fig. 3 Raw time series data collected by the time-stretch data acquisition system. Time-stretch can be used to perform ultra-fast single-shot measurements of frequency response from a single pulse as well as collect statistical fluctuations of the DUT from the raw time series data of multiple pulses. For studying statistical fluctuations of the DUT, (a) the frames from time series data are collected and (b) undergo automated jitter removal, segmentation and alignment.

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The raw digitized time series data first undergoes down-sampling by applying a moving average filter to reduce the noise and then split (segment) into individual frames. The approximate frame size is only used as the initial condition and is based on the known repetition rate of the pulsed continuum laser. The starting sample point in the first frame is identified using the minimum sample point, and the endpoint in the last frame is found using the repetition rate. An extremely accurate repetition rate is estimated by taking the signal sequence with start and end points as time-domain data, performing a fast Fourier transform on it and applying digital band-pass filtering to identify the frequency with the maximum spectral power (the fundamental frequency of the time series). The frame size is then corrected for the drift in the laser repetition rate by matching the first and last frames.

The automated algorithm can perform segmentation of the measured real-time data and can make full use of a massive amount of raw data captured by our instrument. The algorithm is adopted for the present application from our earlier work on time-stretch imaging for cancer cell classification [36]. It identifies the accurate start and end points of a single frame, in spite of the drift in the laser repetition rate. This is performed by an optimization algorithm. Using the correct frame size, the longtime series captured by digitizer is accurately split into segmented frames.

3.2 Envelope correction and localization of the impulse response

In the single shot network analyzer [17,18], both the instrument impulse response and the DUT impulse response are modulated onto the pre-chirped laser envelope. In order to eliminate the effect of the laser pulse envelope on instrument and device impulse responses, each frame is divided by the laser envelope. The latter is obtained by performing a measurement without any impulse input to the EOM. The instrument response is measured by bypassing the DUT (see Fig. 1). In other words, the output of the PD1 is directly inputted to the RF port of the EOM.

The location of the instrument and DUT impulse responses are identified by using a prominent associated feature. This feature is the main peak and subpeaks associated with the impulse, the location of which is identified by a search algorithm. An asymmetric Hamming window centered on the maximum point is then applied to the identified data in order to reduce frequency leakage of FFT. Zero-padding is also implemented to increase FFT frequency resolution.

3.3 Tikhonov regularization for removing the instrument response

The time-stretch device analyzer shown in Fig. 1 can be treated as a linear time-invariant system if all the components used are operated in the linear region. Non-linearities in the instrumentation system are contributed by the EOM bias offsets, dispersion induced non-linearity, photo-detector and electronic device non-linearities, and optical non-linearities. We operate the EOM in the linear region using a bias controller, the photo-diode in the unsaturated region, and the optical power is relatively low to reduce nonlinear effects in the dispersive elements.

The time-domain response measured by the time-stretch device analyzer shown in Fig. 1 is the convolution of the impulse response of DUT with the impulse response of the instrument (the output in the absence of DUT). The time-domain response can therefore be expressed as:

y(t)=h(t)x(t)
where h(t) is the impulse response of the DUT, x(t) is the response of the instrument, which includes the impulse response of the photo-diode and the EOM, and y(t) is the instrument output due to input h(t).

According to Eq. (4), the impulse response of the DUT can be measured by performing a deconvolution operation in time-domain, which is far simpler to implement by a division operation in the frequency domain. Improved measurement accuracy can be achieved by removing the impact of the instrument response. The calibrated DUT frequency response for continuous-time signals is then given by,

H(s)=Y(s)X(s)

In practice, however, the spectral division rarely works. It is very common to have denominators of zero for some frequencies which cause H(s) to be undefined. At other frequencies, H(s) will be non-zero but it may be extremely small and hence dominated by noise. The division operation by a small noisy function will lead to noise amplification. Tikhonov Regularization, also known as Ridge Regression in statistics, is a powerful method for solving such inverse problems. For discrete-time signals, the Tikhonov regularized frequency response of DUT is given by [51]:

H(z)=X(z)Y(z)(X(z)X(z)+λ)Δt
where the Tikhonov regularization parameter λ is a small positive real number, and ∆t is the sampling period. The denominator in Eq. (6) will be non-zero, and by properly selecting the parameter, λ [52], the frequency response is estimated with reduced noise and the time-domain response can be recovered. In the previously reported single-shot network analyzer [17,18], the measured impulse response of the DUT also included the photo-diode and EOM response which needs to be eliminated for accurate measurement. In this paper, we solve this important problem using the regularized deconvolution and the improved system to measure frequency responses of electronic devices.

Regarding the choice of the regularization parameter, the Tikhonov regularization minimizes the following objective function [51]:

minXHY22+λH22
This objective function in Eq. (7) is a weighted sum of a term that measures how well the model H fits the data Y and a term that measures the energy of the model H. Tikhonov regularization is effectively picking the smallest energy signal that fits the data reasonably well, with the relative balance of these two factors controlled by the regularization parameter λ. In our DSP, we found that when λ is 10 times larger than the min(X), the DUT time-domain response can be recovered smoothly and the result is not sensitive to λ value. There are many methods for picking the regularization parameter λ [52].

4. Results

4.1 Phase diversity

A 40 Gb/s, dual drive MZM (Fujitsu FTM7937EZ) is used for achieving phase-diversity generating both phase and amplitude modulation. When D1 is −20 ps/nm, and D2 is −984 ps/nm for testing the device under test, the first null in the RF frequency response of the time-stretch optical link occurs at ~56 GHz, which exceed the EOM bandwidth (40 GHz). To generate plots showing the effect of dispersion penalty and phase diversity, D1 was changed to −120 ps/nm. The input frequency of the signal input into the RF port biased was swept and the measurements were performed for two different quadrature points (Quad + and Quad-) to emulate phase diversity. The MZM output ports have a 90° optical phase shift resulting in complementary RF transfer functions in the frequency domain.

The experimental results of using phase diversity to compensate for the RF power fading caused by dispersion penalty in the time-stretch system is shown in Fig. 4. On applying the MRC algorithm described in Eq. (3), the effect of the frequency nulls due to dispersion penalty are removed. The roll-off in the measured transfer function at higher frequencies is a spurious effect and is mostly due to attenuation in the signal generator output power and the electro-optic modulator as explained below.

 figure: Fig. 4

Fig. 4 The measured transfer function of the phase diversity time-stretch system. The plot shows the electrical RF power. The outputs exhibit complementary fading characteristics. The frequency fading is removed after applying the maximum ratio combining (MRC) algorithm in Eq. (3). The remaining 15 dB of roll-off is not a property of the time-stretch systems. It is due to the roll-off in the electro-optic modulator as well as the RF generator. The half wave voltage of the EO modulator increases with frequency resulting in a measured reduction in the optical signal of 5 dB (10 dB electrical RF power). In addition, the RF signal generator power has 5 dB of roll-off beyond 25 GHz (measured) resulting in the observed 15 dB roll-off. This roll-off is device-dependent and is not an intrinsic property of the time-stretch device analyzer.

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We note that Fig. 4 shows the electrical RF power. The overall ~15 dB of roll-off in the electrical power (7.5 dB optical) is not a property of the time-stretch systems. It is due to the roll-off in the electrooptic modulator as well as the RF generator. The half wave voltage, Vpi, of the electro-optic modulator increases with frequency. In the device used in our experiments, this leads to a reduction in the optical signal of 5 dB (10 dB electrical RF power). In addition, the RF signal generator has a 5 dB of roll-off in RF power beyond 25 GHz. The roll-offs in the electrooptic response and the RF generator combine to results in the observed 15 dB roll-off.

4.2 Electronic amplifier measurements

The time-stretch device analyzer instrument impulse response is shown in Fig. 5(a). This instrument response has to be deconvolved from the measured impulse response of the DUT. Two wide-band electronic amplifiers were tested using the Tera-sample/s time-stretch device analyzer at an extremely fast single-shot acquisition time of 27 ns using the proposed automated DSP for calibration. The devices tested were an electronic low noise amplifier (Picosecond Pulse Labs’ 5840A) with 8.5 GHz bandwidth and wide-band microwave power amplifier with a wide analog bandwidth of 20 GHz (Analog Devices’ HMC870LC5). The peak drive voltages from PD1 and DUT were ~180 mV and ~800 mV respectively, and harmonics terms were ignored due to a small-signal modulation. The estimated ratio of peak drive voltage of the output of PD1 to Vpi of the modulator (modulation depth) was 0.045.

 figure: Fig. 5

Fig. 5 The measured instrument (a) impulse response and (b) frequency response. The measured instrument response is a convolution of the impulse response of the 30 GHz photodiode (PD1) with the time-stretch system.

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The time-domain impulse responses of the wide-band amplifiers shown in Fig. 6(b) and 6(d) are obtained from the inverse FFT of their measured complex frequency responses. From Fig. 6(a) and 6(c), the frequency responses of the two wide-band electronic amplifiers tested using the time-stretch device analyzer and the proposed automated DSP algorithm are in good agreement with the specifications provided by the respective manufacturers [53,54].

 figure: Fig. 6

Fig. 6 Single-shot frequency response measurements of high bandwidth electrical devices at an effective sampling rate of 2.5 Tera-sample/s with a low effective timing jitter of 5.4 fs enabled by the time-stretch device analyzer. The (a) frequency response and (b) time-domain impulse response of a low noise amplifier. The (c) frequency response and (d) time-domain impulse response of a wide-band power amplifier. The acquisition time for the frequency response measurements at single-shot is 27 ns.

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5. Discussion

The time-stretch device analyzer has an input capture aperture of 360 ps (length of the chirped pulse before the EOM) which sets a low-frequency cutoff at ~3 GHz which is dependent on the dispersion of the D1 and the repetition rate of the continuum laser. This lowest operating frequency of the instrument can be tuned according to the application by proper choice of laser bandwidth and dispersive elements D1 and D2. Because high-frequency devices have a flat frequency response at low frequencies, the data can be extrapolated below the cutoff.

5.1 Timing jitter

Inter-pulse optical jitter in the pulsed continuum laser and electronic jitter in the back-end ADC sampling-clock aperture are the sources of jitter in time-stretch systems and can impact the single-shot device analyzer. Time-stretch reduces the effective signal frequency by the stretch-factor M and thereby significantly reduces the effect of digitizer clock jitter. Because clock jitter is the main limitation in high-speed ADCs [55,56], time-stretch enhances the digitizer’s performance as it relates to the sampling clock jitter. The total effective jitter for a time-stretch system is given by [17,18]:

τj,eff=τj,laser2+(τj,clockM)2
where τj,laser is the laser inter-pulse jitter and τj,clock is the electronic clock jitter of the digitizer used. When measurements are performed using data from multiple laser pulses, the instrument jitter would be dominated by the continuum laser jitter but reduced by the stretch factor [12].

The time-stretch device analyzer operated in single-shot measurement mode uses the data from only one laser pulse. The inter-pulse laser jitter is therefore absent in single-shot operation. Therefore, the effective timing jitter for single-shot operation is given by [17,18]:

τj,singleshot=τj,clockM

The sampling-clock aperture jitter for the digitizer used is 270 fs (rms) and the stretch-factor used was 50. Therefore, the effective timing jitter for single-shot measurements is ~5.4 fs (rms), which is extremely low compared to state-of-the-art electronic digitizers [56]. The Tera-sample/s time-stretch device analyzer enables extremely low jitter single-shot time-domain measurements compared to conventional methods, where the jitter added by the instrument must be very accurately estimated and corrected.

The minimum effective timing jitter that can be achieved will be limited by the maximum stretch factor which is limited by the loss of second dispersive element, D2, and the reduction of optical power because the energy is spread over longer time. The attenuation in dispersive fiber can be eliminated using distributed Raman amplification inside the dispersive element to boost the signal-to-noise ratio of the time-stretch system [16]. Indeed, stretch factors as high as 250 with total dispersion of approximately 10ns/nm have been demonstrated with this technique [16]. The ultimate value will be limited by the accumulation of noise during amplification.

5.2 Envelope correction

The pulse envelope correction method used here does not take into account the pulse-to-pulse fluctuations on the continuum laser pulse. A differential detection method using balanced photo-detectors and a transimpedance differential amplifier can be employed to solve this problem. This method will in turn also significantly increase the dynamic range as well as improve SNR of the impulse response measurements by the instrument.

5.3 Bandwidth

The effect of RF power fading by dispersion penalty is eliminated in the time-stretch device analyzer by employing optical phase diversity, which has been proved to be an effective method for dispersion penalty calibration. The practical bandwidth for the presented experimental setup of the instrument is limited by the EOM, which has a nominal value of 40 GHz. However, these modulators can be operated at over 100 GHz due to their gradual roll-off [16]. Moreover, free-space electro-optic sampling using an EO crystal can be used to extend the analog bandwidth of the proposed instrument to several THz [29–31,57].

5.4 Accuracy

The frequency resolution is limited by the FFT window size which is improved by zero padding. Accuracy is the achievable effective number of bits (ENOB) which is the dynamic range in log scale. The dynamic range has two contributions: (a) the optical time-stretch frontend, and (b) the electronic digitizer backend. The dynamic range of the photonic frontend is limited by the third order distortion due to the nonlinear transfer function of the MZ modulator and optical nonlinearities, if any [12]. The results in [12] showed that the distortion-limited dynamic range of the optical frontend to about 60 dB (around 9 bits). In practice however, the overall dynamic range is usually limited by the backend digitizer. This is usually about 6 effective number of bits (~38 dB) for high speed digital oscilloscope such as the 50 Giga-sample/s unit used in our experiments. This modest dynamic range is the price paid for single-shot performance.

5.5 Applications

The Tera-sample/s time-stretch device analyzer demonstrated can meet the fast test and measurement requirements for the design and development of high-bandwidth electrical, optoelectronic, and electro-optic devices. The presented instrument has the potential to enable faster testing of mm-wave devices for 5G and next-generation wireless, free-space, and fiber communication technology. With the automated DSP calibration technology presented here, this technology performs high throughput testing as required in production environments. Time-stretch device analyzer can be used for material characterization applications such as measuring the complex permittivity which has paved the way for new non-invasive bio-medical and industrial applications [58,59].

6. Conclusion

We demonstrated an ultra-fast single-shot measurement of the frequency responses of high-bandwidth electronic devices using a time-stretch device analyzer with an automated DSP algorithm. The proposed instrumentation system employed phase diversity with time-stretch data acquisition to eliminate the effect of dispersion penalty and extend the bandwidth of the instrument. An automated segmentation algorithm followed by averaging was applied to improve the dynamic range of the instrument and the results were consistent with the 3-dB bandwidth of devices from the datasheet. Compared to the traditional frequency response measurement techniques for wide-band RF devices, our proposed single-shot device analyzer has an extremely fast frequency-response acquisition time of 27 ns at an effective sampling rate of 2.5 Tera-sample/s. The proposed Tera-sample/s device analyzer features an ultra-low effective timing jitter of 5.4 fs. These results demonstrate that the proposed time-stretch system powered by automated DSP is a powerful instrument to test the impulse and frequency response of ultra-wide bandwidth electronic devices at many orders faster speeds than the conventional network analyzers. The time-stretch device analyzer presented in this work can be used for single-shot frequency response and shot-to-shot fluctuation measurements of high bandwidth optoelectronic and electro-optic devices.

Acknowledgments

The work was conducted at the Photonics Laboratory in the University of California, Los Angeles (UCLA). The authors would like to thank Prof. Brian Borchers at New Mexico Institute of Mining and Technology, New Mexico, USA, for the advice on Tikhonov regularization.

Disclosures

BJ, CKL, and AMM: (P).

BJ: Roguescope (I).

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References

  • View by:

  1. Cisco Visual Networking Index, “The Zettabyte Era: Trends and Analysis,” https://www.cisco.com/c/en/us/solutions/collateral/service-provider/visual-networking-index-vni/vni-hyperconnectivity-wp.html .
  2. Y. Xing and T. S. Rappaport, “Propagation measurement system and approach at 140 GHz-moving to 6G and above 100 GHz,” in 2018 IEEE Global Communications Conference (GLOBECOM), (IEEE, 2018), pp. 1–6.
    [Crossref]
  3. M. Agrawal and K. Chakrabarty, “Test-cost modeling and optimal test-flow selection of 3-D-stacked ICs,” IEEE Trans. Computer-Aided Design Integr. Circuits Syst. 34(9), 1523–1536 (2015).
  4. F. Ziadé, A. Poletaeff, and D. Allal, “Primary standard for S-parameter measurements at intermediate frequencies (IFs),” IEEE Trans. Instrum. Meas. 62(3), 659–666 (2013).
    [Crossref]
  5. V. Teppati and A. Ferrero, “A Comparison of Uncertainty Evaluation Methods for On-Wafer S-Parameter Measurements,” IEEE Trans. Instrum. Meas. 63(4), 935–942 (2014).
    [Crossref]
  6. S. Pan and M. Xue, “Ultrahigh-resolution optical vector analysis based on optical single-sideband modulation,” J. Lightwave Technol. 35(4), 836–845 (2017).
    [Crossref]
  7. D. Derickson, Fiber Optic Test and Measurement (Prentice Hall Press, 1997).
  8. P. Ferrari and G. Angenieux, “Calibration of a time-domain network analyzer: A new approach,” IEEE Trans. Instrum. Meas. 49(1), 178–187 (2000).
    [Crossref]
  9. J. Verspecht and K. Rush, “Individual characterization of broadband sampling Oscilloscopes with a nose-to-nose calibration procedure,” IEEE Trans. Instrum. Meas. 43(2), 347–354 (1994).
    [Crossref]
  10. J. A. Jargon, P. D. Hale, and C. M. Wang, “Correcting sampling oscilloscope timebase errors with a passively mode-locked laser phase locked to a microwave oscillator,” IEEE Trans. Instrum. Meas. 59(4), 916–922 (2010).
    [Crossref]
  11. A. S. Bhushan, F. Coppinger, and B. Jalali, “Time-stretched analogue-to-digital conversion,” Electron. Lett. 34(9), 839–841 (1998).
    [Crossref]
  12. A. M. Fard, S. Gupta, and B. Jalali, “Photonic time-stretch digitizer and its extension to real-time spectroscopy and imaging,” Laser Photonics Rev. 7(2), 207–263 (2013).
    [Crossref]
  13. A. Mahjoubfar, D. V. Churkin, S. Barland, N. Broderick, S. K. Turitsyn, and B. Jalali, “Time stretch and its applications,” Nat. Photonics 11(6), 341–351 (2017).
    [Crossref]
  14. A. S. Bhushan, P. V. Kelkar, B. Jalali, O. Boyraz, and M. Islam, “130-GSa/s photonic analog-to-digital converter with time stretch preprocessor,” IEEE Photonics Technol. Lett. 14(5), 684–686 (2002).
    [Crossref]
  15. Y. Han and B. Jalali, “Photonic time-stretched analog-to-digital converter: fundamental concepts and practical considerations,” J. Lightwave Technol. 21(12), 3085–3103 (2003).
    [Crossref]
  16. J. Chou, O. Boyraz, D. Solli, and B. Jalali, “Femtosecond real-time single-shot digitizer,” Appl. Phys. Lett. 91(16), 161105 (2007).
    [Crossref]
  17. C. K. Lonappan, A. M. Madni, and B. Jalali, “Single-shot network analyzer for extremely fast measurements,” Appl. Opt. 55(30), 8406–8412 (2016).
    [Crossref] [PubMed]
  18. C. K. Lonappan, “Time-Stretch Accelerated Instrumentation for High-Speed Signal Analysis and Ultra-fast Device Characterization,” PhD diss., UCLA (2018).
  19. Z. Bai, C. K. Lonappan, A. M. Madni, and B. Jalali, “Time-stretch Network Analyzer for Single-shot Characterization of Electronic Devices,” in CLEO: Applications and Technology, (Optical Society of America, 2019), pp. AF3K–6.
  20. Y. Han, O. Boyraz, and B. Jalali, “Ultrawide-band photonic time-stretch A/D converter Employing phase diversity,” IEEE T. Microw. Theory 53(4), 1404–1408 (2005).
    [Crossref]
  21. Y. Han and B. Jalali, “Time-bandwidth product of the photonic time-stretched analog-to-digital converter,” IEEE Trans. Microw. Theory Tech. 51(7), 1886–1892 (2003).
    [Crossref]
  22. A. M. Fard, B. Buckley, S. Zlatanovic, C. S. Bres, S. Radic, and B. Jalali, “All-optical time-stretch digitizer,” Appl. Phys. Lett. 101(5), 051113 (2012).
    [Crossref]
  23. B. W. Buckley, A. M. Madni, and B. Jalali, “Coherent time-stretch transformation for real-time capture of wideband signals,” Opt. Express 21(18), 21618–21627 (2013).
    [Crossref] [PubMed]
  24. J. Stigwall and S. Galt, “Signal reconstruction by phase retrieval and optical backpropagation in phase-diverse photonic time-stretch systems,” J. Lightwave Technol. 25(10), 3017–3027 (2007).
    [Crossref]
  25. D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450(7172), 1054–1057 (2007).
    [Crossref] [PubMed]
  26. A. F. J. Runge, N. G. R. Broderick, and M. Erkintalo, “Observation of soliton explosions in a passively mode-locked fiber laser,” Optica 2(1), 36–39 (2015).
    [Crossref]
  27. G. Herink, B. Jalali, C. Ropers, and D. Solli, “Resolving the build-up of femtosecond mode-locking with single-shot spectroscopy at 90 MHz frame rate,” Nat. Photonics 10(5), 321–326 (2016).
    [Crossref]
  28. G. Herink, F. Kurtz, B. Jalali, D. R. Solli, and C. Ropers, “Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules,” Science 356(6333), 50–54 (2017).
    [Crossref] [PubMed]
  29. E. Roussel, C. Evain, M. Le Parquier, C. Szwaj, S. Bielawski, L. Manceron, J.-B. Brubach, M.-A. Tordeux, J.-P. Ricaud, L. Cassinari, M. Labat, M.-E. Couprie, and P. Roy, “Observing microscopic structures of a relativistic object using a time-stretch strategy,” Sci. Rep. 5(1), 10330 (2015).
    [Crossref] [PubMed]
  30. C. Szwaj, C. Evain, M. Le Parquier, P. Roy, L. Manceron, J.-B. Brubach, M.-A. Tordeux, and S. Bielawski, “High sensitivity photonic time-stretch electro-optic sampling of terahertz pulses,” Rev. Sci. Instrum. 87(10), 103111 (2016).
    [Crossref] [PubMed]
  31. C. Evain, E. Roussel, M. Le Parquier, C. Szwaj, M.-A. Tordeux, J.-B. Brubach, L. Manceron, P. Roy, and S. Bielawski, “Direct observation of spatiotemporal dynamics of short electron bunches in storage rings,” Phys. Rev. Lett. 118(5), 054801 (2017).
    [Crossref] [PubMed]
  32. K. Goda, K. K. Tsia, and B. Jalali, “Serial time-encoded amplified imaging for real-time observation of fast dynamic phenomena,” Nature 458(7242), 1145–1149 (2009).
    [Crossref] [PubMed]
  33. K. Goda, A. Ayazi, D. R. Gossett, J. Sadasivam, C. K. Lonappan, E. Sollier, A. M. Fard, S. C. Hur, J. Adam, C. Murray, C. Wang, N. Brackbill, D. Di Carlo, and B. Jalali, “High-throughput single-microparticle imaging flow analyzer,” Proc. Natl. Acad. Sci. U.S.A. 109(29), 11630–11635 (2012).
    [Crossref] [PubMed]
  34. A. Ayazi, K. Goda, J. Sadasivam, C. K. Lonappan, D. R. Gossett, E. Sollier, A. Fard, S. C. Hur, S. H. Kim, J. Adam, C. Murray, C. Wang, N. Brackbill, D. Di Carlo, and B. Jalali, “Real-time image processor for detection of rare cells and particles in flow at 37 MHz line scans per second,” in Imaging, Manipulation, and Analysis of Biomolecules, Cells, and Tissues XI, vol. 8587 (International Society for Optics and Photonics, 2013), p. 858713.
  35. A. Mahjoubfar, C. Chen, K. R. Niazi, S. Rabizadeh, and B. Jalali, “Label-free high-throughput cell screening in flow,” Biomed. Opt. Express 4(9), 1618–1625 (2013).
    [Crossref] [PubMed]
  36. C. L. Chen, A. Mahjoubfar, L.-C. Tai, I. K. Blaby, A. Huang, K. R. Niazi, and B. Jalali, “Deep Learning in Label-free Cell Classification,” Sci. Rep. 6(1), 21471 (2016).
    [Crossref] [PubMed]
  37. C. K. Lonappan, B. W. Buckley, D. Lam, A. M. Madni, B. Jalali, and J. Adam, “Time-stretch accelerated processor for real-time, in-service, signal analysis,” in IEEE Global Conference on Signal and Information Processing (GlobalSIP), (IEEE, 2014), pp. 707–711.
    [Crossref]
  38. D. Lam, B. W. Buckley, C. K. Lonappan, A. M. Madni, and B. Jalali, “Ultra-wideband instantaneous frequency estimation,” IEEE Instrum. Meas. Mag. 18(2), 26–30 (2015).
    [Crossref]
  39. B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias, and J. M. Dudley, “Real-time full bandwidth measurement of spectral noise in supercontinuum generation,” Sci. Rep. 2(1), 882 (2012).
    [Crossref] [PubMed]
  40. T. Konishi, K. Tanimura, K. Asano, Y. Oshita, and Y. Ichioka, “All-optical analog-to-digital converter by use of self-frequency shifting in fiber and a pulse-shaping technique,” J. Opt. Soc. Am. B 19(11), 2817–2823 (2002).
    [Crossref]
  41. T. Makino, H. Furukawa, M. H. Asghari, P. Trinh, B. Jalali, X. Wang, T. Kobayashi, W. S. Man, K. S. Tsang, and N. Wada, “High resolution single-shot time stretch spectroscopy with wavelength demultiplexer at one billion frames per second,” in CLEO: Science and Innovations, (Optical Society of America, 2017), pp. STu4I–6.
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  58. E. Piuzzi, C. Merla, G. Cannazza, A. Zambotti, F. Apollonio, A. Cataldo, P. D’Atanasio, E. De Benedetto, and M. Liberti, “A comparative analysis between customized and commercial systems for complex permittivity measurements on liquid samples at microwave frequencies,” IEEE Trans. Instrum. Meas. 62(5), 1034–1046 (2013).
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2018 (4)

2017 (4)

S. Pan and M. Xue, “Ultrahigh-resolution optical vector analysis based on optical single-sideband modulation,” J. Lightwave Technol. 35(4), 836–845 (2017).
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A. Mahjoubfar, D. V. Churkin, S. Barland, N. Broderick, S. K. Turitsyn, and B. Jalali, “Time stretch and its applications,” Nat. Photonics 11(6), 341–351 (2017).
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G. Herink, F. Kurtz, B. Jalali, D. R. Solli, and C. Ropers, “Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules,” Science 356(6333), 50–54 (2017).
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C. Evain, E. Roussel, M. Le Parquier, C. Szwaj, M.-A. Tordeux, J.-B. Brubach, L. Manceron, P. Roy, and S. Bielawski, “Direct observation of spatiotemporal dynamics of short electron bunches in storage rings,” Phys. Rev. Lett. 118(5), 054801 (2017).
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2016 (7)

G. Herink, B. Jalali, C. Ropers, and D. Solli, “Resolving the build-up of femtosecond mode-locking with single-shot spectroscopy at 90 MHz frame rate,” Nat. Photonics 10(5), 321–326 (2016).
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C. Szwaj, C. Evain, M. Le Parquier, P. Roy, L. Manceron, J.-B. Brubach, M.-A. Tordeux, and S. Bielawski, “High sensitivity photonic time-stretch electro-optic sampling of terahertz pulses,” Rev. Sci. Instrum. 87(10), 103111 (2016).
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C. L. Chen, A. Mahjoubfar, L.-C. Tai, I. K. Blaby, A. Huang, K. R. Niazi, and B. Jalali, “Deep Learning in Label-free Cell Classification,” Sci. Rep. 6(1), 21471 (2016).
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M. Lei, W. Zou, X. Li, and J. Chen, “Ultrafast FBG interrogator based on time-stretch method,” IEEE Photonics Technol. Lett. 28(7), 778–781 (2016).
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W. Zou, H. Zhang, X. Long, S. Zhang, Y. Cui, and J. Chen, “All-optical central-frequency-programmable and bandwidth-tailorable radar,” Sci. Rep. 6(1), 19786 (2016).
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C. K. Lonappan, A. M. Madni, and B. Jalali, “Single-shot network analyzer for extremely fast measurements,” Appl. Opt. 55(30), 8406–8412 (2016).
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F. Saltarelli, V. Kumar, D. Viola, F. Crisafi, F. Preda, G. Cerullo, and D. Polli, “Broadband stimulated Raman scattering spectroscopy by a photonic time stretcher,” Opt. Express 24(19), 21264–21275 (2016).
[Crossref] [PubMed]

2015 (5)

F. Xing, H. Chen, S. Xie, and J. Yao, “Ultrafast three-dimensional surface imaging based on short-time Fourier transform,” IEEE Photonics Technol. Lett. 27(21), 2264–2267 (2015).
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M. Agrawal and K. Chakrabarty, “Test-cost modeling and optimal test-flow selection of 3-D-stacked ICs,” IEEE Trans. Computer-Aided Design Integr. Circuits Syst. 34(9), 1523–1536 (2015).

D. Lam, B. W. Buckley, C. K. Lonappan, A. M. Madni, and B. Jalali, “Ultra-wideband instantaneous frequency estimation,” IEEE Instrum. Meas. Mag. 18(2), 26–30 (2015).
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E. Roussel, C. Evain, M. Le Parquier, C. Szwaj, S. Bielawski, L. Manceron, J.-B. Brubach, M.-A. Tordeux, J.-P. Ricaud, L. Cassinari, M. Labat, M.-E. Couprie, and P. Roy, “Observing microscopic structures of a relativistic object using a time-stretch strategy,” Sci. Rep. 5(1), 10330 (2015).
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A. F. J. Runge, N. G. R. Broderick, and M. Erkintalo, “Observation of soliton explosions in a passively mode-locked fiber laser,” Optica 2(1), 36–39 (2015).
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2014 (2)

V. Teppati and A. Ferrero, “A Comparison of Uncertainty Evaluation Methods for On-Wafer S-Parameter Measurements,” IEEE Trans. Instrum. Meas. 63(4), 935–942 (2014).
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X. Zeng, A. Fhager, Z. He, M. Persson, P. Linner, and H. Zirath, “Development of a time domain microwave system for medical diagnostics,” IEEE Trans. Instrum. Meas. 63(12), 2931–2939 (2014).
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2013 (5)

E. Piuzzi, C. Merla, G. Cannazza, A. Zambotti, F. Apollonio, A. Cataldo, P. D’Atanasio, E. De Benedetto, and M. Liberti, “A comparative analysis between customized and commercial systems for complex permittivity measurements on liquid samples at microwave frequencies,” IEEE Trans. Instrum. Meas. 62(5), 1034–1046 (2013).
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A. M. Fard, S. Gupta, and B. Jalali, “Photonic time-stretch digitizer and its extension to real-time spectroscopy and imaging,” Laser Photonics Rev. 7(2), 207–263 (2013).
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F. Ziadé, A. Poletaeff, and D. Allal, “Primary standard for S-parameter measurements at intermediate frequencies (IFs),” IEEE Trans. Instrum. Meas. 62(3), 659–666 (2013).
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B. W. Buckley, A. M. Madni, and B. Jalali, “Coherent time-stretch transformation for real-time capture of wideband signals,” Opt. Express 21(18), 21618–21627 (2013).
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A. Mahjoubfar, C. Chen, K. R. Niazi, S. Rabizadeh, and B. Jalali, “Label-free high-throughput cell screening in flow,” Biomed. Opt. Express 4(9), 1618–1625 (2013).
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2012 (3)

K. Goda, A. Ayazi, D. R. Gossett, J. Sadasivam, C. K. Lonappan, E. Sollier, A. M. Fard, S. C. Hur, J. Adam, C. Murray, C. Wang, N. Brackbill, D. Di Carlo, and B. Jalali, “High-throughput single-microparticle imaging flow analyzer,” Proc. Natl. Acad. Sci. U.S.A. 109(29), 11630–11635 (2012).
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B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias, and J. M. Dudley, “Real-time full bandwidth measurement of spectral noise in supercontinuum generation,” Sci. Rep. 2(1), 882 (2012).
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A. M. Fard, B. Buckley, S. Zlatanovic, C. S. Bres, S. Radic, and B. Jalali, “All-optical time-stretch digitizer,” Appl. Phys. Lett. 101(5), 051113 (2012).
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2010 (1)

J. A. Jargon, P. D. Hale, and C. M. Wang, “Correcting sampling oscilloscope timebase errors with a passively mode-locked laser phase locked to a microwave oscillator,” IEEE Trans. Instrum. Meas. 59(4), 916–922 (2010).
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2009 (1)

K. Goda, K. K. Tsia, and B. Jalali, “Serial time-encoded amplified imaging for real-time observation of fast dynamic phenomena,” Nature 458(7242), 1145–1149 (2009).
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2007 (3)

J. Stigwall and S. Galt, “Signal reconstruction by phase retrieval and optical backpropagation in phase-diverse photonic time-stretch systems,” J. Lightwave Technol. 25(10), 3017–3027 (2007).
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D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450(7172), 1054–1057 (2007).
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J. Chou, O. Boyraz, D. Solli, and B. Jalali, “Femtosecond real-time single-shot digitizer,” Appl. Phys. Lett. 91(16), 161105 (2007).
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2005 (1)

Y. Han, O. Boyraz, and B. Jalali, “Ultrawide-band photonic time-stretch A/D converter Employing phase diversity,” IEEE T. Microw. Theory 53(4), 1404–1408 (2005).
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2003 (2)

Y. Han and B. Jalali, “Time-bandwidth product of the photonic time-stretched analog-to-digital converter,” IEEE Trans. Microw. Theory Tech. 51(7), 1886–1892 (2003).
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Y. Han and B. Jalali, “Photonic time-stretched analog-to-digital converter: fundamental concepts and practical considerations,” J. Lightwave Technol. 21(12), 3085–3103 (2003).
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2002 (2)

A. S. Bhushan, P. V. Kelkar, B. Jalali, O. Boyraz, and M. Islam, “130-GSa/s photonic analog-to-digital converter with time stretch preprocessor,” IEEE Photonics Technol. Lett. 14(5), 684–686 (2002).
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T. Konishi, K. Tanimura, K. Asano, Y. Oshita, and Y. Ichioka, “All-optical analog-to-digital converter by use of self-frequency shifting in fiber and a pulse-shaping technique,” J. Opt. Soc. Am. B 19(11), 2817–2823 (2002).
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2000 (1)

P. Ferrari and G. Angenieux, “Calibration of a time-domain network analyzer: A new approach,” IEEE Trans. Instrum. Meas. 49(1), 178–187 (2000).
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1999 (1)

R. H. Walden, “Analog-to-digital converter survey and analysis,” IEEE J. Sel. Areas Comm. 17(4), 539–550 (1999).
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1998 (1)

A. S. Bhushan, F. Coppinger, and B. Jalali, “Time-stretched analogue-to-digital conversion,” Electron. Lett. 34(9), 839–841 (1998).
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1994 (1)

J. Verspecht and K. Rush, “Individual characterization of broadband sampling Oscilloscopes with a nose-to-nose calibration procedure,” IEEE Trans. Instrum. Meas. 43(2), 347–354 (1994).
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Adam, J.

K. Goda, A. Ayazi, D. R. Gossett, J. Sadasivam, C. K. Lonappan, E. Sollier, A. M. Fard, S. C. Hur, J. Adam, C. Murray, C. Wang, N. Brackbill, D. Di Carlo, and B. Jalali, “High-throughput single-microparticle imaging flow analyzer,” Proc. Natl. Acad. Sci. U.S.A. 109(29), 11630–11635 (2012).
[Crossref] [PubMed]

C. K. Lonappan, B. W. Buckley, D. Lam, A. M. Madni, B. Jalali, and J. Adam, “Time-stretch accelerated processor for real-time, in-service, signal analysis,” in IEEE Global Conference on Signal and Information Processing (GlobalSIP), (IEEE, 2014), pp. 707–711.
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Agrawal, M.

M. Agrawal and K. Chakrabarty, “Test-cost modeling and optimal test-flow selection of 3-D-stacked ICs,” IEEE Trans. Computer-Aided Design Integr. Circuits Syst. 34(9), 1523–1536 (2015).

Allal, D.

F. Ziadé, A. Poletaeff, and D. Allal, “Primary standard for S-parameter measurements at intermediate frequencies (IFs),” IEEE Trans. Instrum. Meas. 62(3), 659–666 (2013).
[Crossref]

Angenieux, G.

P. Ferrari and G. Angenieux, “Calibration of a time-domain network analyzer: A new approach,” IEEE Trans. Instrum. Meas. 49(1), 178–187 (2000).
[Crossref]

Apollonio, F.

E. Piuzzi, C. Merla, G. Cannazza, A. Zambotti, F. Apollonio, A. Cataldo, P. D’Atanasio, E. De Benedetto, and M. Liberti, “A comparative analysis between customized and commercial systems for complex permittivity measurements on liquid samples at microwave frequencies,” IEEE Trans. Instrum. Meas. 62(5), 1034–1046 (2013).
[Crossref]

Asano, K.

Ayazi, A.

K. Goda, A. Ayazi, D. R. Gossett, J. Sadasivam, C. K. Lonappan, E. Sollier, A. M. Fard, S. C. Hur, J. Adam, C. Murray, C. Wang, N. Brackbill, D. Di Carlo, and B. Jalali, “High-throughput single-microparticle imaging flow analyzer,” Proc. Natl. Acad. Sci. U.S.A. 109(29), 11630–11635 (2012).
[Crossref] [PubMed]

Barklund, A.

J. Mallari, C. Wei, D. Jin, G. Yu, A. Barklund, E. Miller, P. O’Mathuna, R. Dinu, A. Motafakker-Fard, and B. Jalali, “100Gbps EO polymer modulator product and its characterization using a real-time digitizer,” in Optical Fiber Communication Conference, (Optical Society of America, 2010), p. OThU2.
[Crossref]

Barland, S.

A. Mahjoubfar, D. V. Churkin, S. Barland, N. Broderick, S. K. Turitsyn, and B. Jalali, “Time stretch and its applications,” Nat. Photonics 11(6), 341–351 (2017).
[Crossref]

Bhushan, A. S.

A. S. Bhushan, P. V. Kelkar, B. Jalali, O. Boyraz, and M. Islam, “130-GSa/s photonic analog-to-digital converter with time stretch preprocessor,” IEEE Photonics Technol. Lett. 14(5), 684–686 (2002).
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A. S. Bhushan, F. Coppinger, and B. Jalali, “Time-stretched analogue-to-digital conversion,” Electron. Lett. 34(9), 839–841 (1998).
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Bielawski, S.

C. Evain, E. Roussel, M. Le Parquier, C. Szwaj, M.-A. Tordeux, J.-B. Brubach, L. Manceron, P. Roy, and S. Bielawski, “Direct observation of spatiotemporal dynamics of short electron bunches in storage rings,” Phys. Rev. Lett. 118(5), 054801 (2017).
[Crossref] [PubMed]

C. Szwaj, C. Evain, M. Le Parquier, P. Roy, L. Manceron, J.-B. Brubach, M.-A. Tordeux, and S. Bielawski, “High sensitivity photonic time-stretch electro-optic sampling of terahertz pulses,” Rev. Sci. Instrum. 87(10), 103111 (2016).
[Crossref] [PubMed]

E. Roussel, C. Evain, M. Le Parquier, C. Szwaj, S. Bielawski, L. Manceron, J.-B. Brubach, M.-A. Tordeux, J.-P. Ricaud, L. Cassinari, M. Labat, M.-E. Couprie, and P. Roy, “Observing microscopic structures of a relativistic object using a time-stretch strategy,” Sci. Rep. 5(1), 10330 (2015).
[Crossref] [PubMed]

Blaby, I. K.

C. L. Chen, A. Mahjoubfar, L.-C. Tai, I. K. Blaby, A. Huang, K. R. Niazi, and B. Jalali, “Deep Learning in Label-free Cell Classification,” Sci. Rep. 6(1), 21471 (2016).
[Crossref] [PubMed]

Boyraz, O.

J. Chou, O. Boyraz, D. Solli, and B. Jalali, “Femtosecond real-time single-shot digitizer,” Appl. Phys. Lett. 91(16), 161105 (2007).
[Crossref]

Y. Han, O. Boyraz, and B. Jalali, “Ultrawide-band photonic time-stretch A/D converter Employing phase diversity,” IEEE T. Microw. Theory 53(4), 1404–1408 (2005).
[Crossref]

A. S. Bhushan, P. V. Kelkar, B. Jalali, O. Boyraz, and M. Islam, “130-GSa/s photonic analog-to-digital converter with time stretch preprocessor,” IEEE Photonics Technol. Lett. 14(5), 684–686 (2002).
[Crossref]

Brackbill, N.

K. Goda, A. Ayazi, D. R. Gossett, J. Sadasivam, C. K. Lonappan, E. Sollier, A. M. Fard, S. C. Hur, J. Adam, C. Murray, C. Wang, N. Brackbill, D. Di Carlo, and B. Jalali, “High-throughput single-microparticle imaging flow analyzer,” Proc. Natl. Acad. Sci. U.S.A. 109(29), 11630–11635 (2012).
[Crossref] [PubMed]

Bres, C. S.

A. M. Fard, B. Buckley, S. Zlatanovic, C. S. Bres, S. Radic, and B. Jalali, “All-optical time-stretch digitizer,” Appl. Phys. Lett. 101(5), 051113 (2012).
[Crossref]

Broderick, N.

A. Mahjoubfar, D. V. Churkin, S. Barland, N. Broderick, S. K. Turitsyn, and B. Jalali, “Time stretch and its applications,” Nat. Photonics 11(6), 341–351 (2017).
[Crossref]

Broderick, N. G. R.

Brubach, J.-B.

C. Evain, E. Roussel, M. Le Parquier, C. Szwaj, M.-A. Tordeux, J.-B. Brubach, L. Manceron, P. Roy, and S. Bielawski, “Direct observation of spatiotemporal dynamics of short electron bunches in storage rings,” Phys. Rev. Lett. 118(5), 054801 (2017).
[Crossref] [PubMed]

C. Szwaj, C. Evain, M. Le Parquier, P. Roy, L. Manceron, J.-B. Brubach, M.-A. Tordeux, and S. Bielawski, “High sensitivity photonic time-stretch electro-optic sampling of terahertz pulses,” Rev. Sci. Instrum. 87(10), 103111 (2016).
[Crossref] [PubMed]

E. Roussel, C. Evain, M. Le Parquier, C. Szwaj, S. Bielawski, L. Manceron, J.-B. Brubach, M.-A. Tordeux, J.-P. Ricaud, L. Cassinari, M. Labat, M.-E. Couprie, and P. Roy, “Observing microscopic structures of a relativistic object using a time-stretch strategy,” Sci. Rep. 5(1), 10330 (2015).
[Crossref] [PubMed]

Buckley, B.

A. M. Fard, B. Buckley, S. Zlatanovic, C. S. Bres, S. Radic, and B. Jalali, “All-optical time-stretch digitizer,” Appl. Phys. Lett. 101(5), 051113 (2012).
[Crossref]

Buckley, B. W.

D. Lam, B. W. Buckley, C. K. Lonappan, A. M. Madni, and B. Jalali, “Ultra-wideband instantaneous frequency estimation,” IEEE Instrum. Meas. Mag. 18(2), 26–30 (2015).
[Crossref]

B. W. Buckley, A. M. Madni, and B. Jalali, “Coherent time-stretch transformation for real-time capture of wideband signals,” Opt. Express 21(18), 21618–21627 (2013).
[Crossref] [PubMed]

C. K. Lonappan, B. W. Buckley, D. Lam, A. M. Madni, B. Jalali, and J. Adam, “Time-stretch accelerated processor for real-time, in-service, signal analysis,” in IEEE Global Conference on Signal and Information Processing (GlobalSIP), (IEEE, 2014), pp. 707–711.
[Crossref]

Cannazza, G.

E. Piuzzi, C. Merla, G. Cannazza, A. Zambotti, F. Apollonio, A. Cataldo, P. D’Atanasio, E. De Benedetto, and M. Liberti, “A comparative analysis between customized and commercial systems for complex permittivity measurements on liquid samples at microwave frequencies,” IEEE Trans. Instrum. Meas. 62(5), 1034–1046 (2013).
[Crossref]

Cassinari, L.

E. Roussel, C. Evain, M. Le Parquier, C. Szwaj, S. Bielawski, L. Manceron, J.-B. Brubach, M.-A. Tordeux, J.-P. Ricaud, L. Cassinari, M. Labat, M.-E. Couprie, and P. Roy, “Observing microscopic structures of a relativistic object using a time-stretch strategy,” Sci. Rep. 5(1), 10330 (2015).
[Crossref] [PubMed]

Cataldo, A.

E. Piuzzi, C. Merla, G. Cannazza, A. Zambotti, F. Apollonio, A. Cataldo, P. D’Atanasio, E. De Benedetto, and M. Liberti, “A comparative analysis between customized and commercial systems for complex permittivity measurements on liquid samples at microwave frequencies,” IEEE Trans. Instrum. Meas. 62(5), 1034–1046 (2013).
[Crossref]

Cerullo, G.

Chakrabarty, K.

M. Agrawal and K. Chakrabarty, “Test-cost modeling and optimal test-flow selection of 3-D-stacked ICs,” IEEE Trans. Computer-Aided Design Integr. Circuits Syst. 34(9), 1523–1536 (2015).

Chen, C.

Chen, C. L.

C. L. Chen, A. Mahjoubfar, L.-C. Tai, I. K. Blaby, A. Huang, K. R. Niazi, and B. Jalali, “Deep Learning in Label-free Cell Classification,” Sci. Rep. 6(1), 21471 (2016).
[Crossref] [PubMed]

Chen, H.

F. Xing, H. Chen, S. Xie, and J. Yao, “Ultrafast three-dimensional surface imaging based on short-time Fourier transform,” IEEE Photonics Technol. Lett. 27(21), 2264–2267 (2015).
[Crossref]

Chen, H. J.

Chen, J.

W. Zou, H. Zhang, X. Long, S. Zhang, Y. Cui, and J. Chen, “All-optical central-frequency-programmable and bandwidth-tailorable radar,” Sci. Rep. 6(1), 19786 (2016).
[Crossref] [PubMed]

M. Lei, W. Zou, X. Li, and J. Chen, “Ultrafast FBG interrogator based on time-stretch method,” IEEE Photonics Technol. Lett. 28(7), 778–781 (2016).
[Crossref]

Chen, L.

Chou, J.

J. Chou, O. Boyraz, D. Solli, and B. Jalali, “Femtosecond real-time single-shot digitizer,” Appl. Phys. Lett. 91(16), 161105 (2007).
[Crossref]

Churkin, D. V.

A. Mahjoubfar, D. V. Churkin, S. Barland, N. Broderick, S. K. Turitsyn, and B. Jalali, “Time stretch and its applications,” Nat. Photonics 11(6), 341–351 (2017).
[Crossref]

Coppinger, F.

A. S. Bhushan, F. Coppinger, and B. Jalali, “Time-stretched analogue-to-digital conversion,” Electron. Lett. 34(9), 839–841 (1998).
[Crossref]

Couprie, M.-E.

E. Roussel, C. Evain, M. Le Parquier, C. Szwaj, S. Bielawski, L. Manceron, J.-B. Brubach, M.-A. Tordeux, J.-P. Ricaud, L. Cassinari, M. Labat, M.-E. Couprie, and P. Roy, “Observing microscopic structures of a relativistic object using a time-stretch strategy,” Sci. Rep. 5(1), 10330 (2015).
[Crossref] [PubMed]

Crisafi, F.

Cui, Y.

W. Zou, H. Zhang, X. Long, S. Zhang, Y. Cui, and J. Chen, “All-optical central-frequency-programmable and bandwidth-tailorable radar,” Sci. Rep. 6(1), 19786 (2016).
[Crossref] [PubMed]

D’Atanasio, P.

E. Piuzzi, C. Merla, G. Cannazza, A. Zambotti, F. Apollonio, A. Cataldo, P. D’Atanasio, E. De Benedetto, and M. Liberti, “A comparative analysis between customized and commercial systems for complex permittivity measurements on liquid samples at microwave frequencies,” IEEE Trans. Instrum. Meas. 62(5), 1034–1046 (2013).
[Crossref]

De Benedetto, E.

E. Piuzzi, C. Merla, G. Cannazza, A. Zambotti, F. Apollonio, A. Cataldo, P. D’Atanasio, E. De Benedetto, and M. Liberti, “A comparative analysis between customized and commercial systems for complex permittivity measurements on liquid samples at microwave frequencies,” IEEE Trans. Instrum. Meas. 62(5), 1034–1046 (2013).
[Crossref]

Di Carlo, D.

K. Goda, A. Ayazi, D. R. Gossett, J. Sadasivam, C. K. Lonappan, E. Sollier, A. M. Fard, S. C. Hur, J. Adam, C. Murray, C. Wang, N. Brackbill, D. Di Carlo, and B. Jalali, “High-throughput single-microparticle imaging flow analyzer,” Proc. Natl. Acad. Sci. U.S.A. 109(29), 11630–11635 (2012).
[Crossref] [PubMed]

Dias, F.

B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias, and J. M. Dudley, “Real-time full bandwidth measurement of spectral noise in supercontinuum generation,” Sci. Rep. 2(1), 882 (2012).
[Crossref] [PubMed]

Dinu, R.

J. Mallari, C. Wei, D. Jin, G. Yu, A. Barklund, E. Miller, P. O’Mathuna, R. Dinu, A. Motafakker-Fard, and B. Jalali, “100Gbps EO polymer modulator product and its characterization using a real-time digitizer,” in Optical Fiber Communication Conference, (Optical Society of America, 2010), p. OThU2.
[Crossref]

Dong, X.

Dudley, J. M.

B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias, and J. M. Dudley, “Real-time full bandwidth measurement of spectral noise in supercontinuum generation,” Sci. Rep. 2(1), 882 (2012).
[Crossref] [PubMed]

Erkintalo, M.

Evain, C.

C. Evain, E. Roussel, M. Le Parquier, C. Szwaj, M.-A. Tordeux, J.-B. Brubach, L. Manceron, P. Roy, and S. Bielawski, “Direct observation of spatiotemporal dynamics of short electron bunches in storage rings,” Phys. Rev. Lett. 118(5), 054801 (2017).
[Crossref] [PubMed]

C. Szwaj, C. Evain, M. Le Parquier, P. Roy, L. Manceron, J.-B. Brubach, M.-A. Tordeux, and S. Bielawski, “High sensitivity photonic time-stretch electro-optic sampling of terahertz pulses,” Rev. Sci. Instrum. 87(10), 103111 (2016).
[Crossref] [PubMed]

E. Roussel, C. Evain, M. Le Parquier, C. Szwaj, S. Bielawski, L. Manceron, J.-B. Brubach, M.-A. Tordeux, J.-P. Ricaud, L. Cassinari, M. Labat, M.-E. Couprie, and P. Roy, “Observing microscopic structures of a relativistic object using a time-stretch strategy,” Sci. Rep. 5(1), 10330 (2015).
[Crossref] [PubMed]

Fard, A. M.

A. M. Fard, S. Gupta, and B. Jalali, “Photonic time-stretch digitizer and its extension to real-time spectroscopy and imaging,” Laser Photonics Rev. 7(2), 207–263 (2013).
[Crossref]

A. M. Fard, B. Buckley, S. Zlatanovic, C. S. Bres, S. Radic, and B. Jalali, “All-optical time-stretch digitizer,” Appl. Phys. Lett. 101(5), 051113 (2012).
[Crossref]

K. Goda, A. Ayazi, D. R. Gossett, J. Sadasivam, C. K. Lonappan, E. Sollier, A. M. Fard, S. C. Hur, J. Adam, C. Murray, C. Wang, N. Brackbill, D. Di Carlo, and B. Jalali, “High-throughput single-microparticle imaging flow analyzer,” Proc. Natl. Acad. Sci. U.S.A. 109(29), 11630–11635 (2012).
[Crossref] [PubMed]

Ferrari, P.

P. Ferrari and G. Angenieux, “Calibration of a time-domain network analyzer: A new approach,” IEEE Trans. Instrum. Meas. 49(1), 178–187 (2000).
[Crossref]

Ferrero, A.

V. Teppati and A. Ferrero, “A Comparison of Uncertainty Evaluation Methods for On-Wafer S-Parameter Measurements,” IEEE Trans. Instrum. Meas. 63(4), 935–942 (2014).
[Crossref]

Fhager, A.

X. Zeng, A. Fhager, Z. He, M. Persson, P. Linner, and H. Zirath, “Development of a time domain microwave system for medical diagnostics,” IEEE Trans. Instrum. Meas. 63(12), 2931–2939 (2014).
[Crossref]

Galt, S.

Genty, G.

B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias, and J. M. Dudley, “Real-time full bandwidth measurement of spectral noise in supercontinuum generation,” Sci. Rep. 2(1), 882 (2012).
[Crossref] [PubMed]

Goda, K.

K. Goda, A. Ayazi, D. R. Gossett, J. Sadasivam, C. K. Lonappan, E. Sollier, A. M. Fard, S. C. Hur, J. Adam, C. Murray, C. Wang, N. Brackbill, D. Di Carlo, and B. Jalali, “High-throughput single-microparticle imaging flow analyzer,” Proc. Natl. Acad. Sci. U.S.A. 109(29), 11630–11635 (2012).
[Crossref] [PubMed]

K. Goda, K. K. Tsia, and B. Jalali, “Serial time-encoded amplified imaging for real-time observation of fast dynamic phenomena,” Nature 458(7242), 1145–1149 (2009).
[Crossref] [PubMed]

Godin, T.

P. H. Hanzard, T. Godin, S. Idlahcen, C. Rozé, and A. Hideur, “Real-time tracking of single shockwaves via amplified time-stretch imaging,” Appl. Phys. Lett. 112(16), 161106 (2018).
[Crossref]

Gossett, D. R.

K. Goda, A. Ayazi, D. R. Gossett, J. Sadasivam, C. K. Lonappan, E. Sollier, A. M. Fard, S. C. Hur, J. Adam, C. Murray, C. Wang, N. Brackbill, D. Di Carlo, and B. Jalali, “High-throughput single-microparticle imaging flow analyzer,” Proc. Natl. Acad. Sci. U.S.A. 109(29), 11630–11635 (2012).
[Crossref] [PubMed]

Gupta, S.

A. M. Fard, S. Gupta, and B. Jalali, “Photonic time-stretch digitizer and its extension to real-time spectroscopy and imaging,” Laser Photonics Rev. 7(2), 207–263 (2013).
[Crossref]

Hale, P. D.

J. A. Jargon, P. D. Hale, and C. M. Wang, “Correcting sampling oscilloscope timebase errors with a passively mode-locked laser phase locked to a microwave oscillator,” IEEE Trans. Instrum. Meas. 59(4), 916–922 (2010).
[Crossref]

Han, Y.

Y. Han, O. Boyraz, and B. Jalali, “Ultrawide-band photonic time-stretch A/D converter Employing phase diversity,” IEEE T. Microw. Theory 53(4), 1404–1408 (2005).
[Crossref]

Y. Han and B. Jalali, “Time-bandwidth product of the photonic time-stretched analog-to-digital converter,” IEEE Trans. Microw. Theory Tech. 51(7), 1886–1892 (2003).
[Crossref]

Y. Han and B. Jalali, “Photonic time-stretched analog-to-digital converter: fundamental concepts and practical considerations,” J. Lightwave Technol. 21(12), 3085–3103 (2003).
[Crossref]

Hanzard, P. H.

P. H. Hanzard, T. Godin, S. Idlahcen, C. Rozé, and A. Hideur, “Real-time tracking of single shockwaves via amplified time-stretch imaging,” Appl. Phys. Lett. 112(16), 161106 (2018).
[Crossref]

He, J. B.

He, Z.

X. Zeng, A. Fhager, Z. He, M. Persson, P. Linner, and H. Zirath, “Development of a time domain microwave system for medical diagnostics,” IEEE Trans. Instrum. Meas. 63(12), 2931–2939 (2014).
[Crossref]

Herink, G.

G. Herink, F. Kurtz, B. Jalali, D. R. Solli, and C. Ropers, “Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules,” Science 356(6333), 50–54 (2017).
[Crossref] [PubMed]

G. Herink, B. Jalali, C. Ropers, and D. Solli, “Resolving the build-up of femtosecond mode-locking with single-shot spectroscopy at 90 MHz frame rate,” Nat. Photonics 10(5), 321–326 (2016).
[Crossref]

Hideur, A.

P. H. Hanzard, T. Godin, S. Idlahcen, C. Rozé, and A. Hideur, “Real-time tracking of single shockwaves via amplified time-stretch imaging,” Appl. Phys. Lett. 112(16), 161106 (2018).
[Crossref]

Hu, S.

Huang, A.

C. L. Chen, A. Mahjoubfar, L.-C. Tai, I. K. Blaby, A. Huang, K. R. Niazi, and B. Jalali, “Deep Learning in Label-free Cell Classification,” Sci. Rep. 6(1), 21471 (2016).
[Crossref] [PubMed]

Hur, S. C.

K. Goda, A. Ayazi, D. R. Gossett, J. Sadasivam, C. K. Lonappan, E. Sollier, A. M. Fard, S. C. Hur, J. Adam, C. Murray, C. Wang, N. Brackbill, D. Di Carlo, and B. Jalali, “High-throughput single-microparticle imaging flow analyzer,” Proc. Natl. Acad. Sci. U.S.A. 109(29), 11630–11635 (2012).
[Crossref] [PubMed]

Ichioka, Y.

Idlahcen, S.

P. H. Hanzard, T. Godin, S. Idlahcen, C. Rozé, and A. Hideur, “Real-time tracking of single shockwaves via amplified time-stretch imaging,” Appl. Phys. Lett. 112(16), 161106 (2018).
[Crossref]

Islam, M.

A. S. Bhushan, P. V. Kelkar, B. Jalali, O. Boyraz, and M. Islam, “130-GSa/s photonic analog-to-digital converter with time stretch preprocessor,” IEEE Photonics Technol. Lett. 14(5), 684–686 (2002).
[Crossref]

Jalali, B.

A. Mahjoubfar, D. V. Churkin, S. Barland, N. Broderick, S. K. Turitsyn, and B. Jalali, “Time stretch and its applications,” Nat. Photonics 11(6), 341–351 (2017).
[Crossref]

G. Herink, F. Kurtz, B. Jalali, D. R. Solli, and C. Ropers, “Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules,” Science 356(6333), 50–54 (2017).
[Crossref] [PubMed]

C. L. Chen, A. Mahjoubfar, L.-C. Tai, I. K. Blaby, A. Huang, K. R. Niazi, and B. Jalali, “Deep Learning in Label-free Cell Classification,” Sci. Rep. 6(1), 21471 (2016).
[Crossref] [PubMed]

G. Herink, B. Jalali, C. Ropers, and D. Solli, “Resolving the build-up of femtosecond mode-locking with single-shot spectroscopy at 90 MHz frame rate,” Nat. Photonics 10(5), 321–326 (2016).
[Crossref]

C. K. Lonappan, A. M. Madni, and B. Jalali, “Single-shot network analyzer for extremely fast measurements,” Appl. Opt. 55(30), 8406–8412 (2016).
[Crossref] [PubMed]

D. Lam, B. W. Buckley, C. K. Lonappan, A. M. Madni, and B. Jalali, “Ultra-wideband instantaneous frequency estimation,” IEEE Instrum. Meas. Mag. 18(2), 26–30 (2015).
[Crossref]

A. Mahjoubfar, C. Chen, K. R. Niazi, S. Rabizadeh, and B. Jalali, “Label-free high-throughput cell screening in flow,” Biomed. Opt. Express 4(9), 1618–1625 (2013).
[Crossref] [PubMed]

B. W. Buckley, A. M. Madni, and B. Jalali, “Coherent time-stretch transformation for real-time capture of wideband signals,” Opt. Express 21(18), 21618–21627 (2013).
[Crossref] [PubMed]

A. M. Fard, S. Gupta, and B. Jalali, “Photonic time-stretch digitizer and its extension to real-time spectroscopy and imaging,” Laser Photonics Rev. 7(2), 207–263 (2013).
[Crossref]

A. M. Fard, B. Buckley, S. Zlatanovic, C. S. Bres, S. Radic, and B. Jalali, “All-optical time-stretch digitizer,” Appl. Phys. Lett. 101(5), 051113 (2012).
[Crossref]

K. Goda, A. Ayazi, D. R. Gossett, J. Sadasivam, C. K. Lonappan, E. Sollier, A. M. Fard, S. C. Hur, J. Adam, C. Murray, C. Wang, N. Brackbill, D. Di Carlo, and B. Jalali, “High-throughput single-microparticle imaging flow analyzer,” Proc. Natl. Acad. Sci. U.S.A. 109(29), 11630–11635 (2012).
[Crossref] [PubMed]

K. Goda, K. K. Tsia, and B. Jalali, “Serial time-encoded amplified imaging for real-time observation of fast dynamic phenomena,” Nature 458(7242), 1145–1149 (2009).
[Crossref] [PubMed]

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450(7172), 1054–1057 (2007).
[Crossref] [PubMed]

J. Chou, O. Boyraz, D. Solli, and B. Jalali, “Femtosecond real-time single-shot digitizer,” Appl. Phys. Lett. 91(16), 161105 (2007).
[Crossref]

Y. Han, O. Boyraz, and B. Jalali, “Ultrawide-band photonic time-stretch A/D converter Employing phase diversity,” IEEE T. Microw. Theory 53(4), 1404–1408 (2005).
[Crossref]

Y. Han and B. Jalali, “Time-bandwidth product of the photonic time-stretched analog-to-digital converter,” IEEE Trans. Microw. Theory Tech. 51(7), 1886–1892 (2003).
[Crossref]

Y. Han and B. Jalali, “Photonic time-stretched analog-to-digital converter: fundamental concepts and practical considerations,” J. Lightwave Technol. 21(12), 3085–3103 (2003).
[Crossref]

A. S. Bhushan, P. V. Kelkar, B. Jalali, O. Boyraz, and M. Islam, “130-GSa/s photonic analog-to-digital converter with time stretch preprocessor,” IEEE Photonics Technol. Lett. 14(5), 684–686 (2002).
[Crossref]

A. S. Bhushan, F. Coppinger, and B. Jalali, “Time-stretched analogue-to-digital conversion,” Electron. Lett. 34(9), 839–841 (1998).
[Crossref]

C. K. Lonappan, B. W. Buckley, D. Lam, A. M. Madni, B. Jalali, and J. Adam, “Time-stretch accelerated processor for real-time, in-service, signal analysis,” in IEEE Global Conference on Signal and Information Processing (GlobalSIP), (IEEE, 2014), pp. 707–711.
[Crossref]

J. Mallari, C. Wei, D. Jin, G. Yu, A. Barklund, E. Miller, P. O’Mathuna, R. Dinu, A. Motafakker-Fard, and B. Jalali, “100Gbps EO polymer modulator product and its characterization using a real-time digitizer,” in Optical Fiber Communication Conference, (Optical Society of America, 2010), p. OThU2.
[Crossref]

Jargon, J. A.

J. A. Jargon, P. D. Hale, and C. M. Wang, “Correcting sampling oscilloscope timebase errors with a passively mode-locked laser phase locked to a microwave oscillator,” IEEE Trans. Instrum. Meas. 59(4), 916–922 (2010).
[Crossref]

Jin, D.

J. Mallari, C. Wei, D. Jin, G. Yu, A. Barklund, E. Miller, P. O’Mathuna, R. Dinu, A. Motafakker-Fard, and B. Jalali, “100Gbps EO polymer modulator product and its characterization using a real-time digitizer,” in Optical Fiber Communication Conference, (Optical Society of America, 2010), p. OThU2.
[Crossref]

Kang, J.

Kelkar, P. V.

A. S. Bhushan, P. V. Kelkar, B. Jalali, O. Boyraz, and M. Islam, “130-GSa/s photonic analog-to-digital converter with time stretch preprocessor,” IEEE Photonics Technol. Lett. 14(5), 684–686 (2002).
[Crossref]

Konishi, T.

Koonath, P.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450(7172), 1054–1057 (2007).
[Crossref] [PubMed]

Kudlinski, A.

B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias, and J. M. Dudley, “Real-time full bandwidth measurement of spectral noise in supercontinuum generation,” Sci. Rep. 2(1), 882 (2012).
[Crossref] [PubMed]

Kumar, V.

Kurtz, F.

G. Herink, F. Kurtz, B. Jalali, D. R. Solli, and C. Ropers, “Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules,” Science 356(6333), 50–54 (2017).
[Crossref] [PubMed]

Labat, M.

E. Roussel, C. Evain, M. Le Parquier, C. Szwaj, S. Bielawski, L. Manceron, J.-B. Brubach, M.-A. Tordeux, J.-P. Ricaud, L. Cassinari, M. Labat, M.-E. Couprie, and P. Roy, “Observing microscopic structures of a relativistic object using a time-stretch strategy,” Sci. Rep. 5(1), 10330 (2015).
[Crossref] [PubMed]

Lacourt, P. A.

B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias, and J. M. Dudley, “Real-time full bandwidth measurement of spectral noise in supercontinuum generation,” Sci. Rep. 2(1), 882 (2012).
[Crossref] [PubMed]

Lam, D.

D. Lam, B. W. Buckley, C. K. Lonappan, A. M. Madni, and B. Jalali, “Ultra-wideband instantaneous frequency estimation,” IEEE Instrum. Meas. Mag. 18(2), 26–30 (2015).
[Crossref]

C. K. Lonappan, B. W. Buckley, D. Lam, A. M. Madni, B. Jalali, and J. Adam, “Time-stretch accelerated processor for real-time, in-service, signal analysis,” in IEEE Global Conference on Signal and Information Processing (GlobalSIP), (IEEE, 2014), pp. 707–711.
[Crossref]

Larger, L.

B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias, and J. M. Dudley, “Real-time full bandwidth measurement of spectral noise in supercontinuum generation,” Sci. Rep. 2(1), 882 (2012).
[Crossref] [PubMed]

Le Parquier, M.

C. Evain, E. Roussel, M. Le Parquier, C. Szwaj, M.-A. Tordeux, J.-B. Brubach, L. Manceron, P. Roy, and S. Bielawski, “Direct observation of spatiotemporal dynamics of short electron bunches in storage rings,” Phys. Rev. Lett. 118(5), 054801 (2017).
[Crossref] [PubMed]

C. Szwaj, C. Evain, M. Le Parquier, P. Roy, L. Manceron, J.-B. Brubach, M.-A. Tordeux, and S. Bielawski, “High sensitivity photonic time-stretch electro-optic sampling of terahertz pulses,” Rev. Sci. Instrum. 87(10), 103111 (2016).
[Crossref] [PubMed]

E. Roussel, C. Evain, M. Le Parquier, C. Szwaj, S. Bielawski, L. Manceron, J.-B. Brubach, M.-A. Tordeux, J.-P. Ricaud, L. Cassinari, M. Labat, M.-E. Couprie, and P. Roy, “Observing microscopic structures of a relativistic object using a time-stretch strategy,” Sci. Rep. 5(1), 10330 (2015).
[Crossref] [PubMed]

Lei, M.

M. Lei, W. Zou, X. Li, and J. Chen, “Ultrafast FBG interrogator based on time-stretch method,” IEEE Photonics Technol. Lett. 28(7), 778–781 (2016).
[Crossref]

Lei, Z.

Li, M.

Li, W.

Li, X.

M. Lei, W. Zou, X. Li, and J. Chen, “Ultrafast FBG interrogator based on time-stretch method,” IEEE Photonics Technol. Lett. 28(7), 778–781 (2016).
[Crossref]

Liberti, M.

E. Piuzzi, C. Merla, G. Cannazza, A. Zambotti, F. Apollonio, A. Cataldo, P. D’Atanasio, E. De Benedetto, and M. Liberti, “A comparative analysis between customized and commercial systems for complex permittivity measurements on liquid samples at microwave frequencies,” IEEE Trans. Instrum. Meas. 62(5), 1034–1046 (2013).
[Crossref]

Lin, Z.

Linner, P.

X. Zeng, A. Fhager, Z. He, M. Persson, P. Linner, and H. Zirath, “Development of a time domain microwave system for medical diagnostics,” IEEE Trans. Instrum. Meas. 63(12), 2931–2939 (2014).
[Crossref]

Liu, M.

Lonappan, C. K.

C. K. Lonappan, A. M. Madni, and B. Jalali, “Single-shot network analyzer for extremely fast measurements,” Appl. Opt. 55(30), 8406–8412 (2016).
[Crossref] [PubMed]

D. Lam, B. W. Buckley, C. K. Lonappan, A. M. Madni, and B. Jalali, “Ultra-wideband instantaneous frequency estimation,” IEEE Instrum. Meas. Mag. 18(2), 26–30 (2015).
[Crossref]

K. Goda, A. Ayazi, D. R. Gossett, J. Sadasivam, C. K. Lonappan, E. Sollier, A. M. Fard, S. C. Hur, J. Adam, C. Murray, C. Wang, N. Brackbill, D. Di Carlo, and B. Jalali, “High-throughput single-microparticle imaging flow analyzer,” Proc. Natl. Acad. Sci. U.S.A. 109(29), 11630–11635 (2012).
[Crossref] [PubMed]

C. K. Lonappan, B. W. Buckley, D. Lam, A. M. Madni, B. Jalali, and J. Adam, “Time-stretch accelerated processor for real-time, in-service, signal analysis,” in IEEE Global Conference on Signal and Information Processing (GlobalSIP), (IEEE, 2014), pp. 707–711.
[Crossref]

Long, X.

W. Zou, H. Zhang, X. Long, S. Zhang, Y. Cui, and J. Chen, “All-optical central-frequency-programmable and bandwidth-tailorable radar,” Sci. Rep. 6(1), 19786 (2016).
[Crossref] [PubMed]

Luo, A. P.

Luo, Z. C.

Madni, A. M.

C. K. Lonappan, A. M. Madni, and B. Jalali, “Single-shot network analyzer for extremely fast measurements,” Appl. Opt. 55(30), 8406–8412 (2016).
[Crossref] [PubMed]

D. Lam, B. W. Buckley, C. K. Lonappan, A. M. Madni, and B. Jalali, “Ultra-wideband instantaneous frequency estimation,” IEEE Instrum. Meas. Mag. 18(2), 26–30 (2015).
[Crossref]

B. W. Buckley, A. M. Madni, and B. Jalali, “Coherent time-stretch transformation for real-time capture of wideband signals,” Opt. Express 21(18), 21618–21627 (2013).
[Crossref] [PubMed]

C. K. Lonappan, B. W. Buckley, D. Lam, A. M. Madni, B. Jalali, and J. Adam, “Time-stretch accelerated processor for real-time, in-service, signal analysis,” in IEEE Global Conference on Signal and Information Processing (GlobalSIP), (IEEE, 2014), pp. 707–711.
[Crossref]

Mahjoubfar, A.

A. Mahjoubfar, D. V. Churkin, S. Barland, N. Broderick, S. K. Turitsyn, and B. Jalali, “Time stretch and its applications,” Nat. Photonics 11(6), 341–351 (2017).
[Crossref]

C. L. Chen, A. Mahjoubfar, L.-C. Tai, I. K. Blaby, A. Huang, K. R. Niazi, and B. Jalali, “Deep Learning in Label-free Cell Classification,” Sci. Rep. 6(1), 21471 (2016).
[Crossref] [PubMed]

A. Mahjoubfar, C. Chen, K. R. Niazi, S. Rabizadeh, and B. Jalali, “Label-free high-throughput cell screening in flow,” Biomed. Opt. Express 4(9), 1618–1625 (2013).
[Crossref] [PubMed]

Mallari, J.

J. Mallari, C. Wei, D. Jin, G. Yu, A. Barklund, E. Miller, P. O’Mathuna, R. Dinu, A. Motafakker-Fard, and B. Jalali, “100Gbps EO polymer modulator product and its characterization using a real-time digitizer,” in Optical Fiber Communication Conference, (Optical Society of America, 2010), p. OThU2.
[Crossref]

Manceron, L.

C. Evain, E. Roussel, M. Le Parquier, C. Szwaj, M.-A. Tordeux, J.-B. Brubach, L. Manceron, P. Roy, and S. Bielawski, “Direct observation of spatiotemporal dynamics of short electron bunches in storage rings,” Phys. Rev. Lett. 118(5), 054801 (2017).
[Crossref] [PubMed]

C. Szwaj, C. Evain, M. Le Parquier, P. Roy, L. Manceron, J.-B. Brubach, M.-A. Tordeux, and S. Bielawski, “High sensitivity photonic time-stretch electro-optic sampling of terahertz pulses,” Rev. Sci. Instrum. 87(10), 103111 (2016).
[Crossref] [PubMed]

E. Roussel, C. Evain, M. Le Parquier, C. Szwaj, S. Bielawski, L. Manceron, J.-B. Brubach, M.-A. Tordeux, J.-P. Ricaud, L. Cassinari, M. Labat, M.-E. Couprie, and P. Roy, “Observing microscopic structures of a relativistic object using a time-stretch strategy,” Sci. Rep. 5(1), 10330 (2015).
[Crossref] [PubMed]

Merla, C.

E. Piuzzi, C. Merla, G. Cannazza, A. Zambotti, F. Apollonio, A. Cataldo, P. D’Atanasio, E. De Benedetto, and M. Liberti, “A comparative analysis between customized and commercial systems for complex permittivity measurements on liquid samples at microwave frequencies,” IEEE Trans. Instrum. Meas. 62(5), 1034–1046 (2013).
[Crossref]

Merolla, J. M.

B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias, and J. M. Dudley, “Real-time full bandwidth measurement of spectral noise in supercontinuum generation,” Sci. Rep. 2(1), 882 (2012).
[Crossref] [PubMed]

Miller, E.

J. Mallari, C. Wei, D. Jin, G. Yu, A. Barklund, E. Miller, P. O’Mathuna, R. Dinu, A. Motafakker-Fard, and B. Jalali, “100Gbps EO polymer modulator product and its characterization using a real-time digitizer,” in Optical Fiber Communication Conference, (Optical Society of America, 2010), p. OThU2.
[Crossref]

Motafakker-Fard, A.

J. Mallari, C. Wei, D. Jin, G. Yu, A. Barklund, E. Miller, P. O’Mathuna, R. Dinu, A. Motafakker-Fard, and B. Jalali, “100Gbps EO polymer modulator product and its characterization using a real-time digitizer,” in Optical Fiber Communication Conference, (Optical Society of America, 2010), p. OThU2.
[Crossref]

Murray, C.

K. Goda, A. Ayazi, D. R. Gossett, J. Sadasivam, C. K. Lonappan, E. Sollier, A. M. Fard, S. C. Hur, J. Adam, C. Murray, C. Wang, N. Brackbill, D. Di Carlo, and B. Jalali, “High-throughput single-microparticle imaging flow analyzer,” Proc. Natl. Acad. Sci. U.S.A. 109(29), 11630–11635 (2012).
[Crossref] [PubMed]

Mussot, A.

B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias, and J. M. Dudley, “Real-time full bandwidth measurement of spectral noise in supercontinuum generation,” Sci. Rep. 2(1), 882 (2012).
[Crossref] [PubMed]

Niazi, K. R.

C. L. Chen, A. Mahjoubfar, L.-C. Tai, I. K. Blaby, A. Huang, K. R. Niazi, and B. Jalali, “Deep Learning in Label-free Cell Classification,” Sci. Rep. 6(1), 21471 (2016).
[Crossref] [PubMed]

A. Mahjoubfar, C. Chen, K. R. Niazi, S. Rabizadeh, and B. Jalali, “Label-free high-throughput cell screening in flow,” Biomed. Opt. Express 4(9), 1618–1625 (2013).
[Crossref] [PubMed]

O’Mathuna, P.

J. Mallari, C. Wei, D. Jin, G. Yu, A. Barklund, E. Miller, P. O’Mathuna, R. Dinu, A. Motafakker-Fard, and B. Jalali, “100Gbps EO polymer modulator product and its characterization using a real-time digitizer,” in Optical Fiber Communication Conference, (Optical Society of America, 2010), p. OThU2.
[Crossref]

Oshita, Y.

Pan, S.

Persson, M.

X. Zeng, A. Fhager, Z. He, M. Persson, P. Linner, and H. Zirath, “Development of a time domain microwave system for medical diagnostics,” IEEE Trans. Instrum. Meas. 63(12), 2931–2939 (2014).
[Crossref]

Piuzzi, E.

E. Piuzzi, C. Merla, G. Cannazza, A. Zambotti, F. Apollonio, A. Cataldo, P. D’Atanasio, E. De Benedetto, and M. Liberti, “A comparative analysis between customized and commercial systems for complex permittivity measurements on liquid samples at microwave frequencies,” IEEE Trans. Instrum. Meas. 62(5), 1034–1046 (2013).
[Crossref]

Poletaeff, A.

F. Ziadé, A. Poletaeff, and D. Allal, “Primary standard for S-parameter measurements at intermediate frequencies (IFs),” IEEE Trans. Instrum. Meas. 62(3), 659–666 (2013).
[Crossref]

Polli, D.

Preda, F.

Rabizadeh, S.

Radic, S.

A. M. Fard, B. Buckley, S. Zlatanovic, C. S. Bres, S. Radic, and B. Jalali, “All-optical time-stretch digitizer,” Appl. Phys. Lett. 101(5), 051113 (2012).
[Crossref]

Rappaport, T. S.

Y. Xing and T. S. Rappaport, “Propagation measurement system and approach at 140 GHz-moving to 6G and above 100 GHz,” in 2018 IEEE Global Communications Conference (GLOBECOM), (IEEE, 2018), pp. 1–6.
[Crossref]

Ricaud, J.-P.

E. Roussel, C. Evain, M. Le Parquier, C. Szwaj, S. Bielawski, L. Manceron, J.-B. Brubach, M.-A. Tordeux, J.-P. Ricaud, L. Cassinari, M. Labat, M.-E. Couprie, and P. Roy, “Observing microscopic structures of a relativistic object using a time-stretch strategy,” Sci. Rep. 5(1), 10330 (2015).
[Crossref] [PubMed]

Ropers, C.

G. Herink, F. Kurtz, B. Jalali, D. R. Solli, and C. Ropers, “Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules,” Science 356(6333), 50–54 (2017).
[Crossref] [PubMed]

G. Herink, B. Jalali, C. Ropers, and D. Solli, “Resolving the build-up of femtosecond mode-locking with single-shot spectroscopy at 90 MHz frame rate,” Nat. Photonics 10(5), 321–326 (2016).
[Crossref]

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450(7172), 1054–1057 (2007).
[Crossref] [PubMed]

Roussel, E.

C. Evain, E. Roussel, M. Le Parquier, C. Szwaj, M.-A. Tordeux, J.-B. Brubach, L. Manceron, P. Roy, and S. Bielawski, “Direct observation of spatiotemporal dynamics of short electron bunches in storage rings,” Phys. Rev. Lett. 118(5), 054801 (2017).
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E. Roussel, C. Evain, M. Le Parquier, C. Szwaj, S. Bielawski, L. Manceron, J.-B. Brubach, M.-A. Tordeux, J.-P. Ricaud, L. Cassinari, M. Labat, M.-E. Couprie, and P. Roy, “Observing microscopic structures of a relativistic object using a time-stretch strategy,” Sci. Rep. 5(1), 10330 (2015).
[Crossref] [PubMed]

Roy, P.

C. Evain, E. Roussel, M. Le Parquier, C. Szwaj, M.-A. Tordeux, J.-B. Brubach, L. Manceron, P. Roy, and S. Bielawski, “Direct observation of spatiotemporal dynamics of short electron bunches in storage rings,” Phys. Rev. Lett. 118(5), 054801 (2017).
[Crossref] [PubMed]

C. Szwaj, C. Evain, M. Le Parquier, P. Roy, L. Manceron, J.-B. Brubach, M.-A. Tordeux, and S. Bielawski, “High sensitivity photonic time-stretch electro-optic sampling of terahertz pulses,” Rev. Sci. Instrum. 87(10), 103111 (2016).
[Crossref] [PubMed]

E. Roussel, C. Evain, M. Le Parquier, C. Szwaj, S. Bielawski, L. Manceron, J.-B. Brubach, M.-A. Tordeux, J.-P. Ricaud, L. Cassinari, M. Labat, M.-E. Couprie, and P. Roy, “Observing microscopic structures of a relativistic object using a time-stretch strategy,” Sci. Rep. 5(1), 10330 (2015).
[Crossref] [PubMed]

Rozé, C.

P. H. Hanzard, T. Godin, S. Idlahcen, C. Rozé, and A. Hideur, “Real-time tracking of single shockwaves via amplified time-stretch imaging,” Appl. Phys. Lett. 112(16), 161106 (2018).
[Crossref]

Runge, A. F. J.

Rush, K.

J. Verspecht and K. Rush, “Individual characterization of broadband sampling Oscilloscopes with a nose-to-nose calibration procedure,” IEEE Trans. Instrum. Meas. 43(2), 347–354 (1994).
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Sadasivam, J.

K. Goda, A. Ayazi, D. R. Gossett, J. Sadasivam, C. K. Lonappan, E. Sollier, A. M. Fard, S. C. Hur, J. Adam, C. Murray, C. Wang, N. Brackbill, D. Di Carlo, and B. Jalali, “High-throughput single-microparticle imaging flow analyzer,” Proc. Natl. Acad. Sci. U.S.A. 109(29), 11630–11635 (2012).
[Crossref] [PubMed]

Saltarelli, F.

Solli, D.

G. Herink, B. Jalali, C. Ropers, and D. Solli, “Resolving the build-up of femtosecond mode-locking with single-shot spectroscopy at 90 MHz frame rate,” Nat. Photonics 10(5), 321–326 (2016).
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J. Chou, O. Boyraz, D. Solli, and B. Jalali, “Femtosecond real-time single-shot digitizer,” Appl. Phys. Lett. 91(16), 161105 (2007).
[Crossref]

Solli, D. R.

G. Herink, F. Kurtz, B. Jalali, D. R. Solli, and C. Ropers, “Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules,” Science 356(6333), 50–54 (2017).
[Crossref] [PubMed]

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450(7172), 1054–1057 (2007).
[Crossref] [PubMed]

Sollier, E.

K. Goda, A. Ayazi, D. R. Gossett, J. Sadasivam, C. K. Lonappan, E. Sollier, A. M. Fard, S. C. Hur, J. Adam, C. Murray, C. Wang, N. Brackbill, D. Di Carlo, and B. Jalali, “High-throughput single-microparticle imaging flow analyzer,” Proc. Natl. Acad. Sci. U.S.A. 109(29), 11630–11635 (2012).
[Crossref] [PubMed]

Stefani, A.

B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias, and J. M. Dudley, “Real-time full bandwidth measurement of spectral noise in supercontinuum generation,” Sci. Rep. 2(1), 882 (2012).
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Stigwall, J.

Sun, S.

Sylvestre, T.

B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias, and J. M. Dudley, “Real-time full bandwidth measurement of spectral noise in supercontinuum generation,” Sci. Rep. 2(1), 882 (2012).
[Crossref] [PubMed]

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C. Evain, E. Roussel, M. Le Parquier, C. Szwaj, M.-A. Tordeux, J.-B. Brubach, L. Manceron, P. Roy, and S. Bielawski, “Direct observation of spatiotemporal dynamics of short electron bunches in storage rings,” Phys. Rev. Lett. 118(5), 054801 (2017).
[Crossref] [PubMed]

C. Szwaj, C. Evain, M. Le Parquier, P. Roy, L. Manceron, J.-B. Brubach, M.-A. Tordeux, and S. Bielawski, “High sensitivity photonic time-stretch electro-optic sampling of terahertz pulses,” Rev. Sci. Instrum. 87(10), 103111 (2016).
[Crossref] [PubMed]

E. Roussel, C. Evain, M. Le Parquier, C. Szwaj, S. Bielawski, L. Manceron, J.-B. Brubach, M.-A. Tordeux, J.-P. Ricaud, L. Cassinari, M. Labat, M.-E. Couprie, and P. Roy, “Observing microscopic structures of a relativistic object using a time-stretch strategy,” Sci. Rep. 5(1), 10330 (2015).
[Crossref] [PubMed]

Tai, L.-C.

C. L. Chen, A. Mahjoubfar, L.-C. Tai, I. K. Blaby, A. Huang, K. R. Niazi, and B. Jalali, “Deep Learning in Label-free Cell Classification,” Sci. Rep. 6(1), 21471 (2016).
[Crossref] [PubMed]

Tanimura, K.

Teppati, V.

V. Teppati and A. Ferrero, “A Comparison of Uncertainty Evaluation Methods for On-Wafer S-Parameter Measurements,” IEEE Trans. Instrum. Meas. 63(4), 935–942 (2014).
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C. Evain, E. Roussel, M. Le Parquier, C. Szwaj, M.-A. Tordeux, J.-B. Brubach, L. Manceron, P. Roy, and S. Bielawski, “Direct observation of spatiotemporal dynamics of short electron bunches in storage rings,” Phys. Rev. Lett. 118(5), 054801 (2017).
[Crossref] [PubMed]

C. Szwaj, C. Evain, M. Le Parquier, P. Roy, L. Manceron, J.-B. Brubach, M.-A. Tordeux, and S. Bielawski, “High sensitivity photonic time-stretch electro-optic sampling of terahertz pulses,” Rev. Sci. Instrum. 87(10), 103111 (2016).
[Crossref] [PubMed]

E. Roussel, C. Evain, M. Le Parquier, C. Szwaj, S. Bielawski, L. Manceron, J.-B. Brubach, M.-A. Tordeux, J.-P. Ricaud, L. Cassinari, M. Labat, M.-E. Couprie, and P. Roy, “Observing microscopic structures of a relativistic object using a time-stretch strategy,” Sci. Rep. 5(1), 10330 (2015).
[Crossref] [PubMed]

Tsia, K. K.

K. Goda, K. K. Tsia, and B. Jalali, “Serial time-encoded amplified imaging for real-time observation of fast dynamic phenomena,” Nature 458(7242), 1145–1149 (2009).
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Turitsyn, S. K.

A. Mahjoubfar, D. V. Churkin, S. Barland, N. Broderick, S. K. Turitsyn, and B. Jalali, “Time stretch and its applications,” Nat. Photonics 11(6), 341–351 (2017).
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Verspecht, J.

J. Verspecht and K. Rush, “Individual characterization of broadband sampling Oscilloscopes with a nose-to-nose calibration procedure,” IEEE Trans. Instrum. Meas. 43(2), 347–354 (1994).
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R. H. Walden, “Analog-to-digital converter survey and analysis,” IEEE J. Sel. Areas Comm. 17(4), 539–550 (1999).
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K. Goda, A. Ayazi, D. R. Gossett, J. Sadasivam, C. K. Lonappan, E. Sollier, A. M. Fard, S. C. Hur, J. Adam, C. Murray, C. Wang, N. Brackbill, D. Di Carlo, and B. Jalali, “High-throughput single-microparticle imaging flow analyzer,” Proc. Natl. Acad. Sci. U.S.A. 109(29), 11630–11635 (2012).
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Wang, C. M.

J. A. Jargon, P. D. Hale, and C. M. Wang, “Correcting sampling oscilloscope timebase errors with a passively mode-locked laser phase locked to a microwave oscillator,” IEEE Trans. Instrum. Meas. 59(4), 916–922 (2010).
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Wei, C.

J. Mallari, C. Wei, D. Jin, G. Yu, A. Barklund, E. Miller, P. O’Mathuna, R. Dinu, A. Motafakker-Fard, and B. Jalali, “100Gbps EO polymer modulator product and its characterization using a real-time digitizer,” in Optical Fiber Communication Conference, (Optical Society of America, 2010), p. OThU2.
[Crossref]

Wetzel, B.

B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias, and J. M. Dudley, “Real-time full bandwidth measurement of spectral noise in supercontinuum generation,” Sci. Rep. 2(1), 882 (2012).
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Wong, K. K. Y.

Xie, S.

F. Xing, H. Chen, S. Xie, and J. Yao, “Ultrafast three-dimensional surface imaging based on short-time Fourier transform,” IEEE Photonics Technol. Lett. 27(21), 2264–2267 (2015).
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Xing, F.

F. Xing, H. Chen, S. Xie, and J. Yao, “Ultrafast three-dimensional surface imaging based on short-time Fourier transform,” IEEE Photonics Technol. Lett. 27(21), 2264–2267 (2015).
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Xing, Y.

Y. Xing and T. S. Rappaport, “Propagation measurement system and approach at 140 GHz-moving to 6G and above 100 GHz,” in 2018 IEEE Global Communications Conference (GLOBECOM), (IEEE, 2018), pp. 1–6.
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Xu, W. C.

Xue, M.

Yao, J.

H. J. Chen, M. Liu, J. Yao, S. Hu, J. B. He, A. P. Luo, W. C. Xu, and Z. C. Luo, “Buildup dynamics of dissipative soliton in an ultrafast fiber laser with net-normal dispersion,” Opt. Express 26(3), 2972–2982 (2018).
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F. Xing, H. Chen, S. Xie, and J. Yao, “Ultrafast three-dimensional surface imaging based on short-time Fourier transform,” IEEE Photonics Technol. Lett. 27(21), 2264–2267 (2015).
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Yu, G.

J. Mallari, C. Wei, D. Jin, G. Yu, A. Barklund, E. Miller, P. O’Mathuna, R. Dinu, A. Motafakker-Fard, and B. Jalali, “100Gbps EO polymer modulator product and its characterization using a real-time digitizer,” in Optical Fiber Communication Conference, (Optical Society of America, 2010), p. OThU2.
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Zambotti, A.

E. Piuzzi, C. Merla, G. Cannazza, A. Zambotti, F. Apollonio, A. Cataldo, P. D’Atanasio, E. De Benedetto, and M. Liberti, “A comparative analysis between customized and commercial systems for complex permittivity measurements on liquid samples at microwave frequencies,” IEEE Trans. Instrum. Meas. 62(5), 1034–1046 (2013).
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Zeng, X.

X. Zeng, A. Fhager, Z. He, M. Persson, P. Linner, and H. Zirath, “Development of a time domain microwave system for medical diagnostics,” IEEE Trans. Instrum. Meas. 63(12), 2931–2939 (2014).
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Zhang, C.

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W. Zou, H. Zhang, X. Long, S. Zhang, Y. Cui, and J. Chen, “All-optical central-frequency-programmable and bandwidth-tailorable radar,” Sci. Rep. 6(1), 19786 (2016).
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W. Zou, H. Zhang, X. Long, S. Zhang, Y. Cui, and J. Chen, “All-optical central-frequency-programmable and bandwidth-tailorable radar,” Sci. Rep. 6(1), 19786 (2016).
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F. Ziadé, A. Poletaeff, and D. Allal, “Primary standard for S-parameter measurements at intermediate frequencies (IFs),” IEEE Trans. Instrum. Meas. 62(3), 659–666 (2013).
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Zirath, H.

X. Zeng, A. Fhager, Z. He, M. Persson, P. Linner, and H. Zirath, “Development of a time domain microwave system for medical diagnostics,” IEEE Trans. Instrum. Meas. 63(12), 2931–2939 (2014).
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A. M. Fard, B. Buckley, S. Zlatanovic, C. S. Bres, S. Radic, and B. Jalali, “All-optical time-stretch digitizer,” Appl. Phys. Lett. 101(5), 051113 (2012).
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M. Lei, W. Zou, X. Li, and J. Chen, “Ultrafast FBG interrogator based on time-stretch method,” IEEE Photonics Technol. Lett. 28(7), 778–781 (2016).
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W. Zou, H. Zhang, X. Long, S. Zhang, Y. Cui, and J. Chen, “All-optical central-frequency-programmable and bandwidth-tailorable radar,” Sci. Rep. 6(1), 19786 (2016).
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Appl. Opt. (1)

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J. Chou, O. Boyraz, D. Solli, and B. Jalali, “Femtosecond real-time single-shot digitizer,” Appl. Phys. Lett. 91(16), 161105 (2007).
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D. Lam, B. W. Buckley, C. K. Lonappan, A. M. Madni, and B. Jalali, “Ultra-wideband instantaneous frequency estimation,” IEEE Instrum. Meas. Mag. 18(2), 26–30 (2015).
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IEEE J. Sel. Areas Comm. (1)

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M. Lei, W. Zou, X. Li, and J. Chen, “Ultrafast FBG interrogator based on time-stretch method,” IEEE Photonics Technol. Lett. 28(7), 778–781 (2016).
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F. Xing, H. Chen, S. Xie, and J. Yao, “Ultrafast three-dimensional surface imaging based on short-time Fourier transform,” IEEE Photonics Technol. Lett. 27(21), 2264–2267 (2015).
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A. S. Bhushan, P. V. Kelkar, B. Jalali, O. Boyraz, and M. Islam, “130-GSa/s photonic analog-to-digital converter with time stretch preprocessor,” IEEE Photonics Technol. Lett. 14(5), 684–686 (2002).
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IEEE T. Microw. Theory (1)

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IEEE Trans. Computer-Aided Design Integr. Circuits Syst. (1)

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IEEE Trans. Instrum. Meas. (7)

F. Ziadé, A. Poletaeff, and D. Allal, “Primary standard for S-parameter measurements at intermediate frequencies (IFs),” IEEE Trans. Instrum. Meas. 62(3), 659–666 (2013).
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V. Teppati and A. Ferrero, “A Comparison of Uncertainty Evaluation Methods for On-Wafer S-Parameter Measurements,” IEEE Trans. Instrum. Meas. 63(4), 935–942 (2014).
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E. Piuzzi, C. Merla, G. Cannazza, A. Zambotti, F. Apollonio, A. Cataldo, P. D’Atanasio, E. De Benedetto, and M. Liberti, “A comparative analysis between customized and commercial systems for complex permittivity measurements on liquid samples at microwave frequencies,” IEEE Trans. Instrum. Meas. 62(5), 1034–1046 (2013).
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X. Zeng, A. Fhager, Z. He, M. Persson, P. Linner, and H. Zirath, “Development of a time domain microwave system for medical diagnostics,” IEEE Trans. Instrum. Meas. 63(12), 2931–2939 (2014).
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A. Mahjoubfar, D. V. Churkin, S. Barland, N. Broderick, S. K. Turitsyn, and B. Jalali, “Time stretch and its applications,” Nat. Photonics 11(6), 341–351 (2017).
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G. Herink, B. Jalali, C. Ropers, and D. Solli, “Resolving the build-up of femtosecond mode-locking with single-shot spectroscopy at 90 MHz frame rate,” Nat. Photonics 10(5), 321–326 (2016).
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D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450(7172), 1054–1057 (2007).
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Opt. Express (4)

Opt. Lett. (1)

Optica (1)

Phys. Rev. Lett. (1)

C. Evain, E. Roussel, M. Le Parquier, C. Szwaj, M.-A. Tordeux, J.-B. Brubach, L. Manceron, P. Roy, and S. Bielawski, “Direct observation of spatiotemporal dynamics of short electron bunches in storage rings,” Phys. Rev. Lett. 118(5), 054801 (2017).
[Crossref] [PubMed]

Proc. Natl. Acad. Sci. U.S.A. (1)

K. Goda, A. Ayazi, D. R. Gossett, J. Sadasivam, C. K. Lonappan, E. Sollier, A. M. Fard, S. C. Hur, J. Adam, C. Murray, C. Wang, N. Brackbill, D. Di Carlo, and B. Jalali, “High-throughput single-microparticle imaging flow analyzer,” Proc. Natl. Acad. Sci. U.S.A. 109(29), 11630–11635 (2012).
[Crossref] [PubMed]

Rev. Sci. Instrum. (1)

C. Szwaj, C. Evain, M. Le Parquier, P. Roy, L. Manceron, J.-B. Brubach, M.-A. Tordeux, and S. Bielawski, “High sensitivity photonic time-stretch electro-optic sampling of terahertz pulses,” Rev. Sci. Instrum. 87(10), 103111 (2016).
[Crossref] [PubMed]

Sci. Rep. (4)

B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias, and J. M. Dudley, “Real-time full bandwidth measurement of spectral noise in supercontinuum generation,” Sci. Rep. 2(1), 882 (2012).
[Crossref] [PubMed]

C. L. Chen, A. Mahjoubfar, L.-C. Tai, I. K. Blaby, A. Huang, K. R. Niazi, and B. Jalali, “Deep Learning in Label-free Cell Classification,” Sci. Rep. 6(1), 21471 (2016).
[Crossref] [PubMed]

E. Roussel, C. Evain, M. Le Parquier, C. Szwaj, S. Bielawski, L. Manceron, J.-B. Brubach, M.-A. Tordeux, J.-P. Ricaud, L. Cassinari, M. Labat, M.-E. Couprie, and P. Roy, “Observing microscopic structures of a relativistic object using a time-stretch strategy,” Sci. Rep. 5(1), 10330 (2015).
[Crossref] [PubMed]

W. Zou, H. Zhang, X. Long, S. Zhang, Y. Cui, and J. Chen, “All-optical central-frequency-programmable and bandwidth-tailorable radar,” Sci. Rep. 6(1), 19786 (2016).
[Crossref] [PubMed]

Science (1)

G. Herink, F. Kurtz, B. Jalali, D. R. Solli, and C. Ropers, “Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules,” Science 356(6333), 50–54 (2017).
[Crossref] [PubMed]

Other (15)

C. K. Lonappan, B. W. Buckley, D. Lam, A. M. Madni, B. Jalali, and J. Adam, “Time-stretch accelerated processor for real-time, in-service, signal analysis,” in IEEE Global Conference on Signal and Information Processing (GlobalSIP), (IEEE, 2014), pp. 707–711.
[Crossref]

A. Ayazi, K. Goda, J. Sadasivam, C. K. Lonappan, D. R. Gossett, E. Sollier, A. Fard, S. C. Hur, S. H. Kim, J. Adam, C. Murray, C. Wang, N. Brackbill, D. Di Carlo, and B. Jalali, “Real-time image processor for detection of rare cells and particles in flow at 37 MHz line scans per second,” in Imaging, Manipulation, and Analysis of Biomolecules, Cells, and Tissues XI, vol. 8587 (International Society for Optics and Photonics, 2013), p. 858713.

C. K. Lonappan, “Time-Stretch Accelerated Instrumentation for High-Speed Signal Analysis and Ultra-fast Device Characterization,” PhD diss., UCLA (2018).

Z. Bai, C. K. Lonappan, A. M. Madni, and B. Jalali, “Time-stretch Network Analyzer for Single-shot Characterization of Electronic Devices,” in CLEO: Applications and Technology, (Optical Society of America, 2019), pp. AF3K–6.

D. Derickson, Fiber Optic Test and Measurement (Prentice Hall Press, 1997).

Cisco Visual Networking Index, “The Zettabyte Era: Trends and Analysis,” https://www.cisco.com/c/en/us/solutions/collateral/service-provider/visual-networking-index-vni/vni-hyperconnectivity-wp.html .

Y. Xing and T. S. Rappaport, “Propagation measurement system and approach at 140 GHz-moving to 6G and above 100 GHz,” in 2018 IEEE Global Communications Conference (GLOBECOM), (IEEE, 2018), pp. 1–6.
[Crossref]

T. Makino, H. Furukawa, M. H. Asghari, P. Trinh, B. Jalali, X. Wang, T. Kobayashi, W. S. Man, K. S. Tsang, and N. Wada, “High resolution single-shot time stretch spectroscopy with wavelength demultiplexer at one billion frames per second,” in CLEO: Science and Innovations, (Optical Society of America, 2017), pp. STu4I–6.

B. Murmann, “ADC Performance Survey 1997-2019,” http://www.stanford.edu/~murmann/adcsurvey.html .

T. Jiang, C. K. Lonappan, C. Evain, C. Szwaj, E. Roussel, M. Parquier, S. Bielawski, and B. Jalali, “Time Stretch Single-shot Terahertz Analyzer,” B. Am. Phys. Soc. (2019).

J. Mallari, C. Wei, D. Jin, G. Yu, A. Barklund, E. Miller, P. O’Mathuna, R. Dinu, A. Motafakker-Fard, and B. Jalali, “100Gbps EO polymer modulator product and its characterization using a real-time digitizer,” in Optical Fiber Communication Conference, (Optical Society of America, 2010), p. OThU2.
[Crossref]

C. Richard, Aster, Brian Borchers, and Clifford H. Thurber. Parameter estimation and inverse problems. Elsevier, 2018.

P. C. Hansen, “The L-curve and its use in the numerical treatment of inverse problems,” in Computational Inverse Problems in Electrocardiology, P. D. Johnston, ed. (WIT Press, 2001), pp. 119–142.

Picosecond Pulse Labs, Ultra-Broadband Amplifiers 5840A Datasheet, https://www.oequest.com/getDatasheet/id/2116-2116.pdf .

Analog Devices, MZ Optical Modulator Driver, DC-20 GHz HMC870LC5 Datasheet, https://www.analog.com/media/en/technical-documentation/data-sheets/hmc870.pdf .

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Figures (6)

Fig. 1
Fig. 1 Time-stretch device analyzer for single-shot measurement of the complex response of an electronic device under test (DUT). Phase diversity has been implemented using a dual-drive EOM with different chirps in outputs to produce 90° phase shifts between them, which enables overcoming the limitation in RF bandwidth caused by dispersion penalty. Bias controller enables the EOM operates in the linear region. D1, D2: dispersive elements; PD1, PD2, PD3, PD4: photo-detectors; PC: polarization controller; C1, C2: optical circulators; DSP: digital signal processing.
Fig. 2
Fig. 2 A simulation of RF fading due to dispersion penalty in a time-stretched data acquisition system. This RF fading would limit the 3-dB analog bandwidth of the device analyzer. The RF fading due to dispersion penalty can be overcome by using phase diversity employing an EOM that produces two outputs that have complementary fading characteristics and combining them [20]. This mitigates the effect of the dispersion penalty and extends the bandwidth of the time-stretch device analyzer.
Fig. 3
Fig. 3 Raw time series data collected by the time-stretch data acquisition system. Time-stretch can be used to perform ultra-fast single-shot measurements of frequency response from a single pulse as well as collect statistical fluctuations of the DUT from the raw time series data of multiple pulses. For studying statistical fluctuations of the DUT, (a) the frames from time series data are collected and (b) undergo automated jitter removal, segmentation and alignment.
Fig. 4
Fig. 4 The measured transfer function of the phase diversity time-stretch system. The plot shows the electrical RF power. The outputs exhibit complementary fading characteristics. The frequency fading is removed after applying the maximum ratio combining (MRC) algorithm in Eq. (3). The remaining 15 dB of roll-off is not a property of the time-stretch systems. It is due to the roll-off in the electro-optic modulator as well as the RF generator. The half wave voltage of the EO modulator increases with frequency resulting in a measured reduction in the optical signal of 5 dB (10 dB electrical RF power). In addition, the RF signal generator power has 5 dB of roll-off beyond 25 GHz (measured) resulting in the observed 15 dB roll-off. This roll-off is device-dependent and is not an intrinsic property of the time-stretch device analyzer.
Fig. 5
Fig. 5 The measured instrument (a) impulse response and (b) frequency response. The measured instrument response is a convolution of the impulse response of the 30 GHz photodiode (PD1) with the time-stretch system.
Fig. 6
Fig. 6 Single-shot frequency response measurements of high bandwidth electrical devices at an effective sampling rate of 2.5 Tera-sample/s with a low effective timing jitter of 5.4 fs enabled by the time-stretch device analyzer. The (a) frequency response and (b) time-domain impulse response of a low noise amplifier. The (c) frequency response and (d) time-domain impulse response of a wide-band power amplifier. The acquisition time for the frequency response measurements at single-shot is 27 ns.

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

H 1 ( ω )=cos( ω 2 β 2 L 2M π 4 )
H 2 ( ω )=cos( ω 2 β 2 L 2M + π 4 )
Y( ω )= H 1 ( ω ) Y 1 ( ω )+ H 2 ( ω ) Y 2 ( ω ) | H 1 ( ω ) | 2 + | H 2 ( ω ) | 2
y( t )=h( t )x( t )
H( s )= Y( s ) X( s )
H( z )= X ( z ) Y( z ) ( X ( z ) X( z )+λ )Δt
min XHY 2 2 +λ H 2 2
τ j,eff = τ j,laser 2 + ( τ j,clock M ) 2
τ j,singleshot = τ j,clock M

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