## Abstract

Complex envelope measurement using coherent linear optical sampling with mode-locked sources is investigated. It is shown that reliable measurement of the phase requires that one of the optical modes of the mode-locked laser be locked to the optical carrier of the data signal to be measured. Carrier-envelope offset (CEO) is found to have negligible effect on the measurement. Measurement errors of the intensity profile and phase depend on the pulsewidth and chirp of the sampling pulses as well as the detuning between the carrier frequencies of the data signal and the center frequency of sampling source.

© 2004 Optical Society of America

## 1. Introduction

Techniques for the measurement of high-speed temporal waveforms have advanced in response to demand in optical communication systems where per-channel data rate has approached 160 Gb/s and
beyond. Traditional electronic measurement techniques using sample-and-hold circuits have given way to nonlinear optical sampling techniques [1]–[5]. In nonlinear optical sampling higher temporal resolution is achieved because a short optical pulse provides the gating function.
Eye diagram measurement up to 500 Gb/s has been demonstrated [6]. This approach suffers from poor sensitivity because of the inherent low efficiency
of nonlinear optical mixing. Recently, a new approach to optical waveform measurement based on coherent homodyne linear optical sampling has been demonstrated. Since ultra-short optical
pulses are used in coherent linear optical sampling, it also provides high temporal resolution. Eye diagram measurement up to 640 Gb/s has been demonstrated [7]. Because coherent homodyne sampling is a linear process, measurement sensitivity (3·10^{3}mW^{2}) three orders of magnitude better than nonlinear
optical techniques (10^{6}mW^{2}) is possible [7]. Furthermore, coherent linear optical sampling offers the capability of measuring
both the intensity and phase of optical signals, providing a timely diagnostic tool for investigating advanced modulation formats such as differential (quadrature) phase shift keying
(DPSK/DQPSK) [8,9] before and after amplified fiber transmission where transmission impairments such as
amplified spontaneous emission (ASE) can accumulated. In [10], the constellation diagram of DPSK signals at 10 and 40 Gb/s was measured
experimentally. A deterministic rotation of the constellation diagram was observed and attributed to carrier-envelope offset (CEO) of the mode-locked laser. In this paper, we analyze the
coherent linear sampling process and point out the actual mechanism for the rotation of the constellation diagram. In addition, we establish the requirements for the sampling pulses to
eliminate this rotation and to achieve desired measurement accuracy for the phase and envelope profile.

## 2. Analysis of linear optical sampling

The schematic diagram of linear optical sampling is shown in Fig. 1(a). The phase and intensity of the optical data signal,
*E*_{d} (*t*), are measured by observing its interference with two orthogonal quadratures of the sampling pulse,
*E*_{s} (*t*). In the upper branch, the balanced detectors measure the interference of the data signal and sampling pulse
Re[*E*_{s} (*t*)**E*_{d} (*t*)]. The sampling pulse is delayed in phase by
*π*/2 before interfering with the data signal in the lower branch. The balanced detectors in the lower branch therefore measure the interference signal
Re[{*E*_{s} (*t*)*e*
^{jπ/2}}**E*_{d} (*t*)]=Im[*E*_{s}
(*t*)**E*_{d} (*t*)]. Therefore, the quadrature interference signals in the two branches together determine the
complex quantity *E*_{s} (*t*)**E*_{d} (*t*) completely (except a
proportionality constant). High temporal resolution can be achieved even with low speed detectors by using a short sampling pulse, for example, from a mode-locked laser, which functions
essentially as a gated local oscillator of coherent detection [7,10]. The interference between the data signal
and sampling pulse occurs only during this gated duration.

The electric field of the mode-locked sampling pulse, *E*_{s} (*t*), can be expressed as

where *ω*_{s,m} is the angular frequency of the mode, *T*_{s} is the period of the sampling pulse,
*a*_{m} and *ϕ*_{m} are the amplitude and phase of the *m*_{th}
mode. The parameter 0<*α*<2*π/T*_{s} is a frequency offset that describes the CEO [11].
It is straightforward to verify that a nonzero offset leads to pulse-to-pulse relative shifts between the pulse envelope and the optical carrier. The sampling pulse train can also be
decomposed into a slowly varying envelope around an optical carrier of center frequency *ω*_{sc} =2*πm*
_{0}/*T*_{s} as

As expected, the slowly-varying envelope (the summation in the brackets) is periodic with period *T*_{s} . The above expression can be rewritten by
introducing a complex envelope function *ε*_{s} (*t*) for a single pulse as

For simplicity, let’s assume that data pulse train is also periodic, for the time being, with a period *T*_{d} , given by

where *ε*_{d} (*t*) is the slowly-varying complex envelope profile for a single pulse,
*ω*_{d} is the carrier frequency, *ϕ*_{do} is a constant initial phase, and
*T*_{d} is the period of the data pulse.

When one is interested in measuring the constellation diagram of the data signals, the period of the sampling pulse is chosen to be exactly equal to the proper integer multiple of the
period of the data signal. On the other hand, when one is interested in measuring the complex (intensity and phase) envelope of the data, it is convenient to make the period of the
sampling pulses intentionally deviate slightly from an integer multiple of the period of the data signal so that successive sampling pulses sample the data pulses at different locations as
shown in Fig. 1(b). When we assume that the peaks of the data pulse and the sampling pulse are aligned at *t*=0 in Fig. 1(b), the off-set, *t*_{o} (*k*), for the *k*_{th}
sampling pulse is given by *t*_{o} (*k*)=[(*kT*_{s}
+0.5*T*_{d} )mod*T*_{d} ] - 0.5*T*_{d} where mod is the
modulus function.

With the above notations, the measured complex envelope by the *k*_{th} sampling pulse can be expressed as

$$={\int}_{\left(k-0.5\right){T}_{s}}^{\left(k+0.5\right){T}_{s}}{\epsilon}_{s}{\left(t-k{T}_{s}\right)}^{*}{\epsilon}_{d}\left(t-k{T}_{s}+{t}_{0}\left(k\right)\right){e}^{j\left(\left({\omega}_{d}-{\omega}_{\mathit{sc}}\right)t+{\varphi}_{\mathit{do}}\right)}\mathit{dt}$$

where the integration is due to the slow detector. Now let’s assume that the carrier frequency of the data signal falls within the spectrum of mode-locked laser at an arbitrary position
near one of the modes as shown on the Fig. 1(c) so that *ω*_{d}
=*ω*_{s,n} +ΔΩ, where ΔΩ is the carrier frequency offset. Then Eq. (5) becomes

$$={e}^{j{\varphi}_{\mathit{do}}}{\int}_{-0.5{T}_{s}}^{0.5{T}_{s}}{\epsilon}_{s}{\left(t\prime \right)}^{*}{\epsilon}_{d}\left(t\prime +{t}_{o}\left(k\right)\right){e}^{j\left[\frac{2\pi \left(n-{m}_{0}\right)}{{T}_{s}}t\prime \right]}{e}^{j\Delta \Omega \left(t\prime +k{T}_{s}\right)}\mathit{dt}\prime $$

$$={e}^{j{\varphi}_{\mathit{do}}}{e}^{j\Delta \Omega k{T}_{s}}{\int}_{-0.5{T}_{s}}^{0.5{T}_{s}}{\epsilon}_{s}{(t\prime )}^{*}{\epsilon}_{d}\left(t\prime +{t}_{o}\left(k\right)\right){e}^{j\Delta \omega t\prime}\mathit{dt}\prime $$

where we have used the fact that $\mathrm{exp}\left\{j\frac{2\pi \left(n-{m}_{0}\right)}{{T}_{s}}k{T}_{s}\right\}=1$ and Δ*ω*=*ω*_{d} -*ω*_{sc} is the carrier frequency detuning between the
carrier frequencies. Let’s assume that the sampling pulsewidth (Δ*t*_{s} ,FWHM) is much smaller than the pulsewidth
(Δ*t*_{d} ) of the data signal so that the envelope and phase of the data pulse are almost constant during sampling. Then Eq. (6) becomes

$$={\epsilon}_{d}\left({t}_{o}\left(k\right)\right){e}^{j{\varphi}_{\mathit{do}}}{\tilde{\epsilon}}_{s}{\left(\Delta \omega \right)}^{*}{e}^{j\Delta \Omega k{T}_{s}}$$

where *ε*~_{s} (*ω*) is the Fourier transformation of the slowly varying envelope *ε*_{s} (*t*). Equation (7) is the main results of the analysis of the linear optical sampling process. There are several important conclusions that can be derived from Eq. (7).

First, the CEO does not play any role in the measurement of the complex envelope of the data signal. This is because linear sampling is proportional to the interference between the optical
carriers, integrated over the sampling pulse. This interference signal depends on the optical carrier frequency detuning only and is independent of CEO. This is clear as the parameter
*α* is absent from Eq. (7). Second, when there is a carrier frequency offset, there will be a deterministic phase error for
the measured signal. Consider the special case when the data pulses are uniform and the sampling pulse period is an integer multiple of the period of the data pulses. When the carrier
frequency offset is zero, ΔΩ=0, the measurement will yield an identical results for every sampling pulse as it should be as shown in Fig. 2(a).
However, when ΔΩ≠0, the measurement will yield a different result for each sampling pulse which is rotated by phase angle of ΔΩ*T*_{S} between
successive sampling pulses, as shown in Fig. 2(b). It should be pointed out that the magnitude of the data signal will not be affected by this
misalignment. The constellation diagram can be obtained without this deterministic rotation of phase by locking one of the modes of mode-locked laser to the optical carrier of the data
signal. The periodic interference pattern between a mode-locked laser and a cw laser has been observed when one of the modes of the mode-locked laser is locked to the cw laser [12]. This rotation of the measured phase can potentially be corrected using a proper software correction scheme [10], assuming that the frequencies of the data signal and the sampling source stay fixed during measurement. Third, the measured intensity (∝|*χ*
(*k*)|^{2}) will be proportional to the spectral power density of the sampling pulse at the carrier frequency of the data signal as shown in Eq. (7). A quantitative example is given in Section 3 for the case of Gaussian pulses. This dependence of the measured intensity on the carrier
frequency of the data signal should be taken into consideration when comparing measured results of different wavelength-division multiplexed (WDM) channels using the same sampling
source.

## 3. Effects of chirp and finite pulsewidth of the sampling source

In this section, we describe the effects of the finite pulsewidth and linear chirp of the sampling pulses on the measurement of the complex envelope by retaining perturbation terms up to
(Δ*t*_{s} /Δ*t*_{d} )^{2}, which were ignored in Eq.
(7). As an example, we consider the case of Gaussian pulses for both sampling pulses and data pulses. The slowly varying envelope of the Gaussian sampling pulse is

where *C* is the chirp parameter of the sampling pulse and the energy of a pulse is |*ε*
_{s0}|^{2}. The data pulses are also Gaussian but with no chirp. Let’s assume that one of the modes of the mode-locked laser is locked to the optical carrier
of the data signal so that ΔΩ=0 in Eq. (6).

It is straightforward to show that if the sampling pulsewidth is much smaller compared to the pulsewidth of the data signal, the measured pulsewidth is

where Δ*ω*_{s} is the spectral bandwidth (FWHM) of the sampling pulse. Eq. (9b) shows that the
broadening of measured pulsewidth, Δ*t*_{d} ′-Δ*t*_{d} , is inversely proportional to the square of the
spectral bandwidth of sampling pulse and the pulsewidth of the data signal. For a fixed spectral bandwidth, chirp of the sampling source doesn’t affect the measurement result. On the other
hand, Eq. (9a) indicates that, for a fixed sampling pulsewidth, as chirp of the sampling pulse increases (bandwidth increases), the error in
the measurement decreases. This is because the interference signals at the leading edge and trailing edge of the sampling pulse oscillate in time due to chirp. When integrated with a slow
detector, the oscillating interference signals average to zero effectively narrowing the sampling pulsewidth.

As mentioned in the previous section, the intensity (∝|*χ* (*k*)|^{2}) of the complex envelope is proportional to the spectral power density of the
sampling pulse at the carrier frequency of the data signal. For the Gaussian pulses considered here, the measured intensity is proportional to

when a carrier frequency of the data signal is detuned from the center frequency of the Gaussian sampling pulse. A carrier frequency detuning of about
0.2Δ*ω*_{s} leads to a 10% reduction in measured intensity.

Assuming that the sampling pulsewidth is much smaller than that of the data signal, the measured phase error Φ(*t*_{o} (*k*)) can be
calculated from the phase of *χ* (*k*) in Eq. (6).

$$=\frac{32{\left(\mathrm{ln}\phantom{\rule{.2em}{0ex}}2\right)}^{3}\phantom{\rule{.2em}{0ex}}C{{t}_{o}}^{2}\u2044{{t}_{d}}^{2}-16{\left(\mathrm{ln}\phantom{\rule{.2em}{0ex}}2\right)}^{2}\Delta \omega \phantom{\rule{.2em}{0ex}}{t}_{o}}{{\Delta {\omega}_{s}}^{2}\Delta {{t}_{d}}^{2}}$$

The first term describes the quadratic phase measurement error due to chirp and the second term describes the linear phase measurement error due to carrier frequency detuning.

Numerical simulation was performed to confirm the analytic results presented above. The pulsewidth of the data signal was set to be 6.25 ps. The data pulse doesn’t have any chirp and the phase was constant. The peaks of the envelope were normalized to unit value to compare the envelope profiles. The chirp of the sampling pulse and optical carrier frequency detuning were varied while the energy of the sampling pulse was fixed in the simulation. Two spectral bandwidths at 3.769 nm [Fig. 3(a)] and 1.767 nm [Fig. 3(b)] were used. For Fig. 3(a), the sampling pulsewidth was 0.9375 ps with no chirp, which is much smaller than the data pulsewidth and thus the analytical results of the previous section apply. The results of numerical simulation agree well with analytical results. For example, pulsewidth broadening was 69.9 fs and 70.3 fs, respectively, using numerical simulation and Eq. (9). The phase variations within the pulsewidth of the data signal were 0.011 radian with chirp only, 0.037 radian with detuning only, and 0.038 radian with both chirp and detuning.

For Fig. 3(b), the sampling pulsewidth was 2 ps when there was no chirp and 3.6 ps when *C*=-1.5. The normalized carrier frequency
detuning, Δ*ω*/Δ*ω*_{s} , was set to either zero or 0.09. Even though the sampling pulsewidth was large compared with that of the
data pulse, the qualitative features predicted analytically remain valid. For the parameters chosen, the measured data pulse was broadened by only 5% even though the ratio of sampling
pulsewidth to data pulsewidth is rather large. A temporal shift of the intensity profile was observed when there were both carrier frequency detuning and chirp in the sampling source. The
phase variations within the pulsewidth of the data signal were 0.048 radian with chirp only, 0.074 radian with detuning only, and 0.095 radian with both chirp and detuning. Comparing the
two chirp-only cases, it is verified that the phase error was reduced with increased sampling pulse bandwidth. In practice, linear chirp in the source can easily be compensated. The
example here indicates that high-order chirp, which cannot be easily compensated, will result in phase measurement error. Simulation results for other pulse envelope profiles (such as
raised cosine) agree qualitatively with the case for Gaussian pulses.

## 4. Discussions and conclusions

In conclusion, we have analyzed the coherent linear optical sampling process. It is found that the measured intensity is proportional to the spectral power density of the sampling pulse at
the carrier frequency of the data signal. Carrier frequency offset of the sampling source doesn’t affect intensity measurement. Carrier frequency detuning results in a decrease in the
measured intensity. Furthermore, in combination with chirp, carrier frequency detuning will lead to a temporal shift of the intensity profile. Reliable phase measurement requires that one
of the modes of the sampling source be locked to the optical carrier of the data signal. Carrier frequency offset leads to a deterministic rotation of the phase angle between successive
sampling periods for the constellation diagram measurement. Carrier frequency detuning (linear chirp) results in a linear (quadratic) phase error in the measurement of the complex envelope
for Gaussian sampling pulses. Therefore it is very desirable to lock one of the modes of the mode-locked source to the carrier frequency of the data to remove carrier frequency offset.
This has the added benefit of reducing the phase measurement error caused by random phase noise of the sampling source as briefly discussed below. Linear chirp should be compensated and
high-order chirp should be minimized in the sampling source to reduce the phase measurement error. The sampling pulse width should be sufficiently small
(Δ*t*_{s} /Δ*t*_{d} <0.2) to ensure negligible broadening and temporal shift of the measured
intensity profile.

In this paper we only investigated deterministic effects of the sampling source on linear optical sampling, which lead to measurement errors. The noise performance of the sampling source,
on the other hand, will determine the intensity and phase measurement sensitivity. A complete analysis is beyond the scope of this paper. Here we will describe qualitatively the effect of
noises in the sampling source. First, the intensity measurement sensitivity is determined by fluctuations of the energy of the sampling pulse. From the Eq. (7), the measured intensity,∝|*χ* (*k*)|^{2}, is proportional to envelope |*ε*_{d}
(*t*_{o} (*k*))|^{2} of the data pulse and also to the spectral power density |*ε̃*_{s} (Δ*ω*)|^{2}, which is proportional to the peak power of the sampling pulse. So the peak power fluctuations of the sampling pulses will directly affect the
measurement of the intensity. However, the peak power of the sampling pulse can be monitored by a detector and this error may be reduced theoretically. Timing jitter of the sampling pulses
will be added to the jitter of the data signal in a statistically independent manner. Second, the phase measurement sensitivity is determined by the statistical characteristics of the
phase noise of the sampling source and measurement duration. For the special case where each mode of the mode-locked laser has the same phase noise (linewidth) because of coherence, the
sampling pulse can be expressed as

where *ϕ*_{N} (*t*) is a random process. An additional phase *ϕ*_{N}
(*kT*_{s} ) will be added to the measured phase of the data signal. The sensitivity of phase measurement will be given by
<${\varphi}_{N}^{2}$>,
which is related to the linewidth of modes as <${{\varphi}_{N}}^{2}$>=Δ*ν* (1/*f*
_{2}-1/*f*
_{1})/*π* where Δ*ν* is the linewidth of each optical mode of the sampling source and *f*
_{1} (*f*
_{2}) is the upper (lower) frequency limit of the measurement [13]. The upper limit is determined by the sampling pulsewidth. The lower limit
is the inverse of the observation period.

## References and links

**1. **M. H. Chou, I. Brener, G. Lenz, R. Scotti, E. E. Chaban, J. Shmulovich, D. Philen, S. Kosinski, K.
R. Parameswaran, and M. M. Fejer, “Efficient wide-band and tunable
midspan spectral inverter using cascaded nonlinearities in LiNbO_{3} waveguides,” IEEE Photon. Technol. Lett. **12**, 82–84 (2000). [CrossRef]

**2. **S. Diez, R. Ludwig, C. Schmidt, U. Feiste, and H.
G. Weber, “160-Gb/s optical sampling by gain-transparent four-wave mixing in a semiconductor optical amplifier,”
IEEE Photon. Technol. Lett. **11**, 1402–1404 (1999). [CrossRef]

**3. **S. Kawanishi, T. Yamamoto, M. Nakawawa, and M. M. Fejer, “High sensitivity waveform measurement with optical sampling
using quasi-phasematched mixing in LiNbO_{3} waveguide,” Electron. Lett. **37**, 842–844 (2001). [CrossRef]

**4. **J. Li, J. Hansryd, P. O. Hedekvist, P. A. Andrekson, and S.
N. Knudsen, “300 Gb/s eye-diagram measurement by optical sampling using fiber-based parametric amplification,”
IEEE Photon. Technol. Lett. **13**, 987–989 (2001). [CrossRef]

**5. **S. Nogiwa, H. Ohta, Y. Kawaguchi, and Y. Endo, “Improvement of sensitivity in optical sampling
system,” Electron. Lett. **35**, 917–918 (1999). [CrossRef]

**6. **J. Li, M. Westlund, H Sunnerud, B. Olsson, M. Karsson, and P. A. Adnrekson, “0.5-Tb/s eye-diagram measurement by optical sampling using
XPM-induced wavelength shifting in highly nonlinear fiber,” IEEE. Photon. Technol. Lett. **16**, 566–568 (2004). [CrossRef]

**7. **C. Dorrer, C. R. Doerr, I. Kang, R. Ryfand, and P. J. Winzer,
“High-sensitivity high-resolution sampling using linear optics and waveguide optical hybrid,” OFC 2004, PDP18.

**8. **A. H. Gnauck, S. Chandrasekhar, J. Leuthold, and L. Stulz, “Demonstration of 42.7-Gb/s DPSK receiver with 45 photons/bit
sensitivity,” IEEE Photon. Technol. Lett. **15**, 99–101 (2003). [CrossRef]

**9. **R. A. Griffin, R. L. Johnstone, R. G. Walker, J. Hall, S. D. Wadsworth, K. Berry, A. C. Carter, and M. J. Wale, “10 Gb/s optical differential quadrature phase shift key (DQPSK)
transmission using GaAs/AlGaAs integration,” OFC 2002, FD6-1.

**10. **C. Dorrer, J. Leuthold, and C. R. Doerr,
“Direct measurement of constellation diagrams of optical sources,” OFC 2004, PDP33.

**11. **S. T. Cundiff, “Phase stabilization
of ultrashort optical pulses,” J. Phys. D: Appl. Phys. **35**. R43–R59 (2002). [CrossRef]

**12. **A. Takada and W. Imajuku, “Linewidth narrowing and optical phase control of mode-locked semiconductor ring
laser employing optical injection locking,” IEEE Photon. Technol. Lett. **9**, 1328–1330 (1997). [CrossRef]

**13. **R. F. Kalman, J. C. Fan, and L. G. Kazovsky,
“Dynamic range of coherent analog fiber-optic links,” J. Lightwave. Technol. **12**, 1263–1277 (1994). [CrossRef]