## Abstract

Four-quadrature spectral interferometry using balanced coherent detection enables single-shot full-field characterization of complex-shaped optical waveforms with near quantum-limited sensitivity. A 90° optical hybrid places a waveform in four-quadrature-phases with a stronger, well-characterized reference pulse. Measurement of the four spectra with an integrating two-dimensional detector array followed by balanced detection uniquely determines the signal field. Balanced coherent detection provides common-mode noise rejection and coherent gain enabling near quantum-limited sensitivity. The single-shot waveform was temporally gated from a 100 ps periodic train of arbitrary optical waveforms generated using line-by-line pulse shaping on a 10 GHz optical frequency comb. The measurements show arbitrary waveforms with 200-ps record lengths, 500 GHz optical bandwidths and only 1200 detected photons.

©2009 Optical Society of America

## 1. Introduction

The capability to measure and generate ultrahigh-bandwidth, infinitely-long waveforms of any arbitrary shape in amplitude and phase will fundamentally impact science and technology. The generation of arbitrary optical waveforms scaling up to terahertz bandwidths [1–3], and equivalent measurement capability allows capture and diagnosis of complex Tb/s arbitrarily shaped optical waveforms. This is important in many fields of research, including optical communication [1–5], arbitrary microwave signal generation [6,7], atomic physics, and chemistry [8–10]. Furthermore, we desire such measurements to handle a single-shot (i.e., not based on an accumulated average of repeated traces), infinite record-length (capable of measuring truly arbitrary waveforms in a continuous stream), be quantum-limited (sensitive down to the photon-counting regime), and be phase-sensitive (measure both in-phase and quadrature components of the field).

Figure 1 shows a symmetrical and analogous description of the generation and measurement of arbitrary waveforms using ultra-stable GHz-spaced optical frequency combs (OFC). Starting with an OFC [11–13], a spectral demultiplexer (e.g., bulk-optic pulse shaper, arrayed-waveguide grating, etc.) isolates each comb line to a separate spatial location. Four-quadrature modulation [either amplitude and phase modulation, or direct in-phase (*I*) and quadrature (*Q*) modulation] is applied separately to each OFC comb line at the OFC repetition rate. The spectrally broadened comb lines are combined using a multiplexer with spectrally overlapping adjacent pass bands, producing the arbitrary optical waveform. The spectrum of the resulting waveform is continuous with the phase and amplitude fully specified across the entire bandwidth. This inherent parallelism allows for waveforms of ‘infinite bandwidth’. If the modulation is continuously applied, the waveform’s temporal duration is also ‘infinite’ (infinite only in theory). Static optical arbitrary waveform generation (OAWG), which typically uses DC amplitude and phase modulation, will produce waveforms at the OFC’s repetition rate with the OFC’s bandwidth; this is often referred to as OAWG via Fourier synthesis or line-by-line pulse shaping. The earliest implementations used integrated devices [14–16], then bulk optics [17–19] and more recently with high fidelity [20]. These current static OAWG implementations produce repetitive waveforms which through time-multiplexing [21] has, to a limited extent, increased the record length of the waveforms.

Optical arbitrary waveform measurement (OAWM), measures waveforms analogously to waveform generation using OAWG. Following Fig. 1, different portions of the arbitrary signal waveform are spectrally sliced by a demultiplexer with overlapping adjacent passbands. The spectral slices are then measured using four-quadrature homodyne and balanced coherent detection in the time-domain [22] against a reference comb line of identical center frequency. The reference comb line is provided by an OFC and spectral demultiplexer. With knowledge of the reference OFC phase and amplitude (obtainable by various multi-shot characterization techniques) the arbitrary waveform is precisely reconstructed without ambiguity. Both frequency domain and time-domain recording of the balanced detector outputs are possible. The time-domain records of the balanced detectors’ outputs allow calculation of infinite-length time records, while the parallel nature of the temporal records of each spectral region depicted in Fig. 1 enable scalability to infinite-bandwidth.

Balanced coherent detection provides a quantum-limited, phase-sensitive measurement over a large-dynamic range. Single-shot frequency-resolved optical gating (FROG) [23–25], heterodyne temporal imaging [26–28], spectral-interferometry for direct electric field reconstruction (SPIDER) [29–31], and spectral interferometery (SI) [32–38] cannot obtain the same total range of performance.

Here, we demonstrate a *single*-*shot* implementation of OAWM (SS-OAWM) by performing four-quadrature spectral interferometry with balanced detection (FQSI) to measure a 500-GHz bandwidth, 200 ps arbitrary optical signal waveform *S _{g}* against a well characterized reference pulse

*R*. The arbitrary waveform is prepared using static OAWG on a 10 GHz OFC followed by high-extinction temporal gating to select a 200 ps measurement window. A bulk-optic spectrometer and a two-dimensional, integrating electron-bombarded charge-coupled device (EB-CCD) camera record the four spectra required to perform FQSI. The combination of the fast time gate and slow detector mimics the measurement of a single 200-ps window of an infinite-length OAWM measurement. Since the time gates and camera operate at 20 Hz (limited by camera’s analog-to-digital conversion rate), the intensity and phase of the signal waveform are updated in near real-time.

_{g}For the SS-OAWM measurements, the time-gated reference *R _{g}* contained either 350,000 photons (40 fJ) or 4,000,000 photons (500 fJ) (measured at the camera) depending on the measurement. The time-gated signal

*S*contained between 1,200 photons (150 aJ) and 80,000 photons (10 fJ). For comparison, some of the most sensitive FROG apparatus [39] are able to measure repetitive pulse trains with 124 aJ/pulse, but the measurement requires billions of pulses to retrieve the signal. Ultimately, comparisons to other techniques are not as meaningful as demonstrating quantum-limited measurement sensitivity.

_{g}## 2. Implementation

Figure 2 shows a diagram of the SS-OAWM experimental arrangement consisting of six major elements; a 10 GHz OFC source, a waveform shaper, high-extinction time gates, a 90° optical hybrid, a four-channel bulk-optic spectrometer, and an EB-CCD camera to record the spectra. These elements, and how FQSI retrieves the signal waveform, are described in the following sections. An example single-shot measurement of a shaped waveform with an energy of only 5 fJ is shown in Fig. 2(k).

#### 2.1. Optical frequency comb source

The OFC is generated via strong electro-optic amplitude and phase modulation of a tunable single-frequency laser (1551 nm) by a dual-electrode Mach-Zehnder modulator (T.DEH1.5-40-ADC, Sumitomo-Cement Co.) and two phase modulators (PM-DV4-40-PFA-PFA-LV, EOSPACE) [12, 20, 40]. In this configuration, over 50 uniform amplitude (±2 dB) lines with 10 GHz spacing are generated. The characteristic quadratic spectral phase of the OFC output is removed by propagation through 700 m of single mode fiber. Optical filtering with a 3-nm full-width at half maximum (FWHM) bandpass filter helps to further flatten the comb and provides a nearly transform-limited 2-ps pulse repeating every 100 ps. Figure 2(i) shows the OFC output used as both the reference waveform R for single-shot OAWM, and as the input to the static OAWG waveform shaper to generate a repetitive signal waveform S for later measurement. To increase the available reference pulse energy, a pre-gate (i.e., pulse picker) is used to select every eighth pulse of *R*(*t*) before it is amplified by an erbium-doped fiber amplifier (EDFA) with an output power of +15 dBm.

#### 2.2. Waveform shaper

The waveform shaper [Fig. 2(d)] is a 96 channel, 10-GHz silica arrayed-waveguide grating (AWG) with 64 of the channels looped back to the inputs [41]. Each wavelength channel has thermo-optic amplitude and phase modulators (<10 ms response) to adjust the intensity and phase of the input frequency comb on a line-by-line basis. The channel insertion loss is 13–15 dB while the extinction ratio of the amplitude modulators is typically 20–30 dB, and the phase modulators can provide over 2*π* rad of peak phase modulation at a heater power of 730 mW. Figure 2(j) shows a SHG-FROG measurement of a shaped waveform illustrating how time domain waveforms are crafted by setting the amplitude and phase of each spectral line. More complex waveforms, which span the full 100 ps OFC period, are created by large line-to-line spectral phase and amplitude variations.

#### 2.3. High-extinction time gating

The EB-CCD camera’s minimum exposure time is 20 µs, therefore the camera integrates 2×10^{6} pulses from the 10 GHz OFC if external time gating is not used. To create a single-shot measurement, a single pulse from *R*(*t*) and a single measurement window from *S*(*t*) are temporally gated to create *R _{g}*(

*t*) and

*S*(

_{g}*t*). During a 20 µs period, the ratio of the energy of all the pulses in the camera gating window to the energy of one pulse is 53 dB. Each time gate is composed of two cascaded 10 GHz, > 55 dB-extinction ratio, Mach-Zehnder modulators (AX-6K5-10-PFA-PFAP-R4, EOSPACE) for a total extinction ratio of >110 dB for a continuous-wave (CW) signal and >90 dB over the entire 500 GHz OFC bandwidth providing a ~40 dB buffer to ensure only a single event is measured. The DATA and $\overline{\mathrm{DATA}}$ outputs from a pulse-pattern generator (<30 ps rise/fall time) provide the RF signals driving the modulators. The extinction ratio was verified by turning off the RF-drive signal to the modulators and measuring no detectable power on the EB-CCD camera with -100 dBm incident average optical power.

#### 2.4. 90° optical hybrid

The 90° optical hybrid (MINT 2×8, Kylia) [Fig. 2(f)] places *R _{g}*(

*t*) both in-phase and in quadrature-phase with

*S*(

_{g}*t*). It produces balanced output pairs for the in-phase component (consisting of

*I*

_{+}and

*I*

_{-}) and quadrature-phase component (consisting of

*Q*

_{+}and

*Q*

_{-}) so that

The optical hybrid maintains phase quadrature within ±1° for the optical bandwidths measured and thus does not contribute significant systematic errors.

#### 2.5. High-resolution four-channel spectrometer and EB-CCD camera

The spectrometer [Fig. 2(g)] consists of four fiber collimators stacked vertically followed by a cylindrical-optics telescope to horizontally expand the beams so that they fill the grating. After diffracting off of the grating, a cylindrical lens focuses the spectral components onto the EB-CCD camera (MOSIR950, Intevac). Figure 2(h) shows a measurement of |*R _{g}*(

*ω*)|

^{2}with gate duration of 1 ns indicating a spectrometer resolution sufficient to clearly resolve 10-GHz spectral lines (3.9 GHz FWHM measured spectral resolution). Figure 2(g) shows a measurement of |

*R*(

_{g}*ω*)|

^{2}with a 40 ps gate duration. The lack of optical modes in the spectrum indicates that only a single pulse is incident on the camera.

The spectra are measured by the EB-CCD camera at an update rate of 20 Hz for quasi-real-time display of the measured waveform’s spectral intensity and phase. The EB-CCD camera’s high sensitivity is achieved through multiplicative gain. An infrared-sensitive photocathode generates electrons which are accelerated into a silicon CCD array by a large electrical potential. This accelerating potential can be switched with a rise and fall time of <10 µs, providing an optical shutter with near infinite contrast and ≥20 µs duration. The photocathode quantum efficiency is 30% and the multiplicative gain is 110, for an overall gain of 30 at 1550 nm. The gain is linear to within 1% over the measurement range and any spatial nonuniformity in the camera’s gain is compensated during data post-processing. The count produced by a single photon is larger than the read noise and dark noise (at 20 µs exposure) of the camera, leaving the optical signal as the dominant noise source.

#### 2.6. Four-quadrature spectral interferometry with balanced detection for single-shot OAWM

FQSI measures the spectral amplitude and phase of an arbitrary signal relative to a well characterized reference where the signal and reference are required to spectrally overlap. FQSI is analogous to four-quadrature time-domain balanced detection [22], and it is a more comprehensive form of dual-quadrature spectral interferometry (DQSI) [34,37,38]. However, unlike DQSI, FQSI includes balanced detection to remove common intensity terms. Eliminating the common intensity terms increases the upper end of the dynamic range by removing a mathematical ambiguity from the retrieval which occurs when the signal is comparable to, or stronger than, the reference. The balanced detection improves the lower end of the dynamic range by removing common noise terms, thereby increasing the measurement sensitivity, and in the strong reference limit, it provides quantum-limited sensitivity. Furthermore, the calculations to retrieve the full signal field are simplified with the use of FQSI when compared to DQSI.

The four-channel, high-resolution spectrometer [Fig. 2(h)] measures the spectral intensity of the four optical hybrid outputs, *I*
_{+}(*ω*), *I*
_{-}(*ω*), *Q*
_{+}(*ω*), and *Q*-(*ω*), described by Eq. (1). From these four measurements, and an independent measurement of *R _{g}*(ω), the in-phase and quadrature-phase components of the signal spectrum,

*S*(

_{g}*ω*), are determined through elemental vector subtraction (balanced detection) and division by the known reference spectrum so that

where *I*
_{+}(*ω*) and *I*
_{-}(*ω*) are given by

and *Q*
_{±}(*ω*) have the same form as *I*
_{±}(*ω*) with the last term replaced by ±2Im{*S _{g}*(

*ω*)

*R**

_{g}(

*ω*)}. Note that no Fourier transforms are required to obtain the spectral description however, as usual, the time domain representation may be obtained from a Fourier transform.

In the experiment, the reference magnitude |*R _{g}*(

*ω*)| is measured directly and the reference phase ∠

*R*(

_{g}*ω*) is estimated by an SHG-FROG measurement of

*R*(

*ω*) because it is far too weak to be directly measured (-100 dBm average power). The FROG error,

*G*was less than 0.002 (128×128 array) for these reference waveform measurements, indicating successful retrieval. The estimation of the phase is assumed accurate because

*R*(

*ω*) is well contained within the reference time-gate,

*G*(

_{r}*t*), minimizing any distortion that may occur at the gate edges.

In practice, the optical hybrid does not have uniform transmission for *R* or *S* to each of the four outputs. The unequal transmission reduces the rejection of the common intensity terms of Eq. (3) (i.e., |*S _{g}*(

*ω*)|

^{2}, |

*R*(

_{g}*ω*)|

^{2}) during balanced detection. Therefore, to improve the rejection, additional signal processing (calibration) is performed. First, the transmission characteristics of the optical hybrid are measured as a function of frequency to yield eight amplitude transmission coefficients;

*r*(

_{i}*ω*) for

*R*and

*s*(

_{i}*ω*) for

*S*, where

*i*=1 to 4 corresponds to the hybrid outputs. Second, to improve the power balance of

*R*(

_{g}*ω*), the signal is blocked and the four spectral intensities are measured. If we assume that

*R*(

_{g}*ω*) is stable and constant, then these spectra are equivalent to the first term in Eq. (3) (i.e., |

*R*(

_{g}*ω*)|

^{2}) and may be directly subtracted from the four measured spectra

*I*

_{±}(

*ω*) and

*Q*

_{±}(

*ω*) to produce new terms

*I*′

_{±}(

*ω*) and

*Q*′

_{±}(

*ω*). Third, to power balance

*S*(

_{g}*ω*), the four measured spectra are divided by the

*s*(

_{i}*ω*) coefficients. Then, the full measurement with calibrations is given by

where *k _{I}* and

*k*compensate for the unequal gain of

_{Q}*I*and

*Q*components and are given by

#### 2.7. Presentation of single-shot OAWM data

Coherent detection is sensitive to both the linear spectral phase and the constant phase between *S*(*ω*) and *R*(*ω*), and environmental variations cause the absolute phase difference to drift from between measurements. Therefore, for all measurements, the phase is referenced to zero at the maximum of the temporal intensity. Statistics of 200 measurements show the peak-to-peak variation of the amplitude and phase, or *I*(*ω*) and *Q*(*ω*), as a lightly shaded region. Dark shaded regions show ±*σ* where *σ* is the standard deviation around the measured mean.

## 3. Single-shot OAWM results and performance

The fidelity and performance of single-shot OAWM were established through five demonstrations; (1) A comparison of single-shot OAWM to an independent multi-shot (repetitive) SHG-FROG measurement of a complex waveform, (2) reproducible measurements of a waveform as it is scanned through the entire time-gated measurement window, (3) extraction of the optical gating function *G*(*t*), (4) single-shot OAWM amplitude and phase measurements as feedback to the OAWG waveform shaper to generate complex waveforms, and (5) exploration of the sensitivity and dynamic range of the measurement.

#### 3.1. Single-shot OAWM comparison

Figure 3 shows statistics of 200 single-shot OAWM measurements of the OFC passing through the waveform shaper without any modulation applied compared to a multi-shot SHG-FROG measurement averaged for over 100 s (10^{11} shots). The SHG-FROG measurement is of the 100 ps repetitive waveform, S(t), therefore its spectrum has discrete lines spaced at 10 GHz. The single-shot OAWM is of the gated waveform *S _{g}*(

*t*), therefore it should have a continuous spectrum, yet it shows the interference of multiple waveforms within the 200 ps measurement window. The single-shot OAWM measurement is displayed as a spectral density of photons/10 GHz to relate the intensity to the number of photons collected per OFC spectral line. The statistics shown are the minimum to maximum (peak-to-peak) deviation and ±σ of the 200 shots. Figure 3(a) shows the directly measured

*I*(

*ω*) and

*Q*(

*ω*) components of

*S*(

_{g}*t*) which compare well to those obtained from the SHG-FROG measurement of

*S*(

*t*). Fig. 3(b) shows the intensity and phase representation for a better understanding of the waveform’s spectral shape. The random phase jumps occurring every 10 GHz are from uncompensated path lengths in the waveform shaper channels. Figure 3(c) shows a time-domain comparison of the SHG-FROG measurement of S(t) to a gated single-shot OAWM measurement of

*S*(

_{g}*t*). The time-domain apodization results from the combined effect of the optical time gate,

*G*(

*t*), and an indirect gate,

*G*(

_{s}*t*) imposed by the finite spectrometer resolution. Nevertheless, both measurements replicate all sub-pulses and the complex phase structure across the entire measurement window. Figure 3(d) presents an additional comparison between SS-OAWM, SHG-FROG and OSA measurements of a transform-limited waveform with Gaussian spectral shape. The excellent match between SHG-FROG and single-shot OAWM is indicative of its accuracy. The small deviations shown in Fig. 3(d) probably arise since slightly different signals are being compared, the ungated signal

*S*(

*t*) to the gated signal

*S*(

_{g}*t*).

#### 3.2. Scanned time-gate measurement

Point (2) was demonstrated by measuring the same waveform accurately over the entire measurement window (200 ps). The effective measurement window, *G _{m}*(

*t*), is comprised of the optical gate,

*G*(

*t*), and the temporal apodization imposed by the spectrometer resolution. Figure 4(a,b) shows the temporal intensity and spectral intensity as an arbitrary waveform is manually delayed with respect to the the time gate,

*G*(

*t*), using a fiber-coupled adjustable optical delay line. As the optical delay is increased, pulse ‘A’ enters the measurement window,

*G*(

_{m}*t*), at

*t*=-100 ps, and pulse ‘B’ exits the window at

*t*=100 ps. At each delay, all waveform features are accurately reproduced in the single-shot OAWM measurement. The structure between the OFC lines is vital and informative to obtain accurate measurement across the entire 200 ps window. Nulls between adjacent OFC lines indicate periodicity. At delay ‘D1’ when pulse ‘A’ and ‘B’ are at

*t*=±50 ps within the measurement window, there are nulls across most of the spectrum indicating 10 GHz periodicity. At this same delay, the features at

*t*=0 ps appear only once corresponding to the areas of the spectrum without nulls (

*f*=-90 GHz and

*f*=-15 GHz). Likewise, at delay ‘D2’ when pulse ‘A’ is centered in the window at

*t*=0 ps, the location of the spectral nulls are reversed. The reproducibility of the waveform during the sweep indicates the measurement technique is stable and accurate over the entire measurement window.

As mentioned previously, two gating functions affect the single-shot OAWM measurement. First, the direct gate, *G*(*t*), from the high-extinction ratio modulators and second, the finite resolution of the spectrometer imposes an artificial temporal gate, *G _{s}*(

*t*).

*G*(

_{s}*t*) is determined by inverse Fourier transforming the spectrometer’s resolution line shape (i.e., convert the optical spatial mode associated with a single-frequency source into a spectral shape). The total effective measurement time-gate,

*G*(

_{m}*t*), is equal to the product of

*G*(

_{s}*t*) and

*G*(

*t*). Figure 4(c) shows

*G*(

_{m}*t*) measured by picking the maximum intensity value occurring at each time from Fig. 4(a).

*G*(

_{m}*t*) is centered around

*t*=0 and both positive and negative times contain unique waveform information.

*G*(

_{m}*t*) does not have a broad flat region and its rise and fall times are relatively long, therefore we choose to define the usable temporal width as the 1% (-20 dB) points (±100 ps). Figure 3(c) justifies this definition since the phase is correctly measured by single-shot OAWM at ±100 ps even though the signal is significantly attenuated (thereby decreasing the signal-to-noise ratio). Although not used for the data presented here, it is straightforward to calibrate out the effect

*G*(

_{m}*t*) has on the signal.

To illustrate point (3), the optical gating applied to the signal, *G*(*t*), is extracted from *G _{m}*(

*t*) using a measurement of the spectral resolution limited apodization,

*G*(

_{s}*t*). Figure 4(c) shows a prediction of

*G*(

_{s}*t*) determined by measuring the spectrometer intensity response to a CW laser and then performing a Fourier transform of the spectral intensity. Since

*G*(

_{m}*t*)=

*G*(

*t*)

*G*(

_{s}*t*), and both

*G*(

_{m}*t*) and

*G*(

_{s}*t*) are known,

*G*(

*t*) can be extracted. Figure 4(d) compares the extracted

*G*(

*t*) to an independent measurement of the

*G*(

*t*) performed by measuring a gated CW signal on a fast sampling oscilloscope. The accurate and full extraction of

*G*(

*t*) from the measurement, is direct evidence that all waveform energy falling within

*G*(

*t*) is completely characterized.

#### 3.3. Demonstration of waveform shaping and high-end dynamic range

To demonstrate (4), static OAWG generated arbitrary waveforms are shaped using single-measurement feedback from single-shot OAWM to the WS. The amplitude and phase values of each spectral line of the OFC are estimated by picking the amplitude and phase in the spectrum where there would be a discrete spectral line. Figure 5(a,b) shows Waveform 1; a shaped waveform which is constructed from two pulses with opposite linear and third order spectral phase. The required spectrum to create this waveform has rapid spectral intensity variations and spectral phase variations. This waveform has zero intensity at ±50 ps indicating it does not span the entire OAWG shaping window (100 ps).

As discussed earlier, balanced detection removes the common intensity terms and eliminates an ambiguity from the signal retrieval that occurs when the signal is stronger than, or comparable to, the reference. This increases the upper end dynamic range by allowing signals of any strength with respect to the reference. Figure 5(c,d) shows Waveform 2; a telecommunications on-off-keyed waveform at 180 Gb/s to demonstrate measurement of a waveform that is quasi-continuous over the entire measurement window, and to demonstrate point (5); the extended dynamic range obtained using balanced detection. The spectrum required to create this waveform has a strong peak (20,000 photons/10 GHz) which is stronger than *R*(*ω*) (9,000 photons/10 GHz) at *f*=0 GHz and weaker than *R*(*ω*) across the rest of the bandwidth (1,000 photons/10 GHz). Across the entire dynamic range (>20 dB) the phases of weak spectral components and strong spectral components are retrieved. The time domain waveform clearly follows the target bit-sequence and spans the entire temporal window.

#### 3.4. Demonstration of sensitivity and low-end dynamic range

The single-shot sensitivity was investigated by repeatedly measuring a shaped waveform, first with high energy (~80,000 photons, 10 fJ) and then with low energy (~1200 photons, 150 aJ). The statistics of the measurements were then compared to simulations of a shot-noise limited version of the waveform and similar statistics indicate near quantum-limited performance. We chose a complex waveform (i.e., time-bandwidth product of 100=500 GHz×200 ps) with Gaussian apodization and cubic spectral phase to demonstrate the technique’s versatility. Figure 6 shows the statistics (i.e., ±σ width) of the intensity and phase fluctuations of 200 single-shot OAWM measurements (dark shading) of a waveform with 10 fJ [Fig. 6(a,b)] and 150 aJ [Fig. 6(c,d)] compared to simulated statistics (light shading) for quantum-limited fluctuations of waveforms with the same energy. To calculate the simulated waveform’s statistics, quantum noise (shot noise) is added to each field-quadrature in each spectral bin (1 GHz wide). Following [42], each mode of the electromagnetic field at a specific frequency (spectral line for *S* or spectral bin for Sg) has a quantum noise variance of ¼ per field quadrature when expressed as a photon number. At 10 fJ the single-shot OAWM measurement shows shot-to-shot variations dominated by either excess noise on *S _{g}* or possibly systematic errors. The time-domain intensity fluctuations are 4% compared to the quantum-limited fluctuations of 1%. At 150 aJ, the waveform fluctuations from quantum noise, unbalanced |

*R*(

_{g}*ω*)|

^{2}noise, and shot-to-shot variations of

*R*are dominant. The unbalanced reference noise spans the entire spectrum and is indicated by

_{g}**B**in Fig. 6(d). Its effect on the temporal waveform is a narrow pulse occurring at 0 ps indicated by

**A**in Fig. 6(c). However, the close comparison between measurement and the quantum-limited simulation indicate near quantum-limited performance. A future improvement to the measurement system will increase the rejection of the common mode noise on |

*R*(

_{g}*ω*)|

^{2}, and possibly add a fifth channel to the spectrometer to measure the fluctuations of

*R*. These two improvements may provide true quantum-limited performance.

_{g}## 4. Extending single-shot to infinite-duration waveforms

To this point, the single-shot OAWM experiments involved an integrating detector array (EB-CCD) and the static amplitude and phase modulation OAWG settings on each line of the OFC. Under this setting, OAWM showed the capability to achieve single-shot measurement over a 200 ps time window. However, dynamic quadrature modulations on the OFC can create time-varying arbitrary waveforms of infinite record-length and equivalently, temporally resolved, four-quadrature coherent detection of bandlimited contiguous spectral regions on OAWM can extend the measurement to infinite record-length. Thus for OAWM, the integrating detector array can be replaced with a fast detector array and the time gates removed. The detector’s bandwidth must then be equal to, or greater than, half of the optical bandwidth of a channel in the spectral filter (spectrometer).

Until recently, static OAWG was thought to be extendable to infinite record-length OAWG by building the infinite length waveform in the time-domain by arranging waveforms adjacent in time with near static modulations for one OFC period and then switching to a new modulation for the next waveform period [43]. Instead, the approach presented here Fourier transforms the extended-duration waveform to obtain the complex spectrum and then stacks bandlimited spectral slices adjacent in frequency together to build the waveform in the frequency domain. This approach naturally favors the ability of optical devices for their wavelength selectivity and time-domain electro-optic modulation for its unlimited duration.

Oppositely, stacking waveforms adjacent in time as in the time-domain approach does not favor either since the required near-instantaneous changes in electrical modulation at each waveform change causes near-infinite width modulation on each comb line. Then to maintain the waveform’s fidelity, the spectral multiplexer must combine the modulated comb lines without filtering (e.g., *N*×1 star coupler), which leads to very large losses when dealing with many comb lines.

To achieve high-fidelity infinite duration OAWG and OAWM, the design of the multiplexers and demultiplexers shown in Fig. 1 are extremely important. The multiplexer and demultiplexer which operate on modulated comb lines must be designed to have some overlap of the adjacent channels so that minimal information is lost in the spectral space between channels. To the contrary, the demultiplexers operating on the unmodulated OFC must strongly isolate a single comb line to a single waveguide so that modulation is not applied to other comb lines and coherent detection is performed between a spectral region and only the single comb line of interest. Such multiplexers can be built using arrayed-waveguide gratings with amplitude and phase shifters on each arm [44].

## 5. Conclusion

The demonstrations shown above establish that single-shot OAWM using FQSI and balanced detection is able to accurately and reproducibly characterize the full field of arbitrary optical waveforms. Measurements of complex single waveforms with only 1200 photons and a time-bandwidth product of 100 (200 ps×500 GHz) established the technique’s generality and that the measured waveform complexity is limited only by the spectrometer’s optical bandwidth and resolution. The data show that the measurement sensitivity is nearly quantum limited, thus providing a unique opportunity to view an optical waveform’s statistics. Single-shot OAWM provided complete optical waveform information for precise field control in OAWG.

## Acknowledgments

The authors would like to thank Jay Lowell, Erich Ippen, Bill Jacobs, and Enrique Parra for their constant encouragements and enlightening discussions. This work was supported in part by DARPA and SPAWAR under OAWG contract HR0011-05-C-0155.

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