Timing jitter from the optical spectrum in semiconductor passively mode locked lasers

Ricardo Rosales; Kamel Merghem; Anthony Martinez; Francois Lelarge; Alain Accard; Abderrahim Ramdane

doi:10.1364/OE.20.009151

1. Introduction

Semiconductor passively mode locked lasers (MLLs) are capable of emitting stable optical pulse trains in the absence of an external reference clock signal. Their compact size, ease of fabrication and low power consumption, make them interesting for a variety of applications including high bit rate optical time division multiplexing, clock recovery and millimeter wave generation. For all these applications low timing jitter is necessary in order to fulfill bit error rate and phase noise requirements. Understanding and quantifying this parameter is therefore of primary importance. Characterization of timing jitter is commonly performed by means of an electrical spectrum analyzer (ESA) after photodetection of the laser pulse train as described in [1]. This method is however restricted to relatively low repetition frequencies due to the limited bandwidth of currently available photodiodes. Besides, the model in [1] relies on the assumption of stationary timing fluctuations which is not the case in passively MLLs [2], where the timing jitter exhibits a diffusion-like behavior characterized by a diffusion constant. Timing jitter can also be measured by the cross correlation technique presented in [3], however the high power levels required by the second harmonic generation nonlinear crystal limit its applicability.

It is known that timing jitter is closely related to the phase noise of the optical modes under different types of ML regimes [4–11]. In [4], an expression relating the optical mode linewidths to that of the photocurrent spectrum was in particular obtained by using soliton perturbation theory, which is justified when the steady state pulse is closed to a soliton. This relation was later shown in [5] to be in agreement with experimental investigations on a two-section semiconductor passively MLL emitting at 1.3 μm, suggesting the possibility of timing jitter extraction from measurements of the laser modal linewidths, circumventing the limitations of the characterization methods previously described. A theoretical asymptotic analysis in [6] reveals scaling laws for the optical linewidth as a function of mode number, depending on specific laser dynamics, also demonstrating the timing jitter to optical linewidth relation. In this paper, a simple general formalism is proposed yielding an explicit expression relating the timing jitter diffusion constant to the modal linewidths of semiconductor lasers exhibiting the passively ML regime. We present for the first time experimental results validating these relations in single section quantum dash (QDash) MLLs, in which the ML phenomenon is attributed to enhanced nonlinear effects in these structures [12]. As no cavity losses are introduced by a saturable absorber section and thanks to the high modal gain available in QDash based active layers [13], the cavity length in these devices can be made short enough to achieve high repetition rates with reported values of up to 340 GHz [14]. This calls for a means to estimate the timing jitter which does not require pulse train direct photodetection. This is the case of the method here presented which readily becomes an attractive alternative.

2. Theory

Considering only the effects of phase fluctuations induced by quantum noise, the complex electric field in a semiconductor passively MLL can be written as:

E (t) = [\sum_{n = 1}^{N} E_{n} e^{j [ω_{n} t + θ_{n} (t) + ϕ_{n}]}] + c . c

where

E_{n}

,

ω_{n}

,

θ_{n} (t)

and

ϕ_{n}

are respectively the real amplitude, angular frequency, phase noise and static phase, of each of the

N

longitudinal modes. Amplitude noise has been neglected since phase noise dominates close to

ω_{n}

[8]. Under ML regime:

\frac{d}{d t} [(2 ω_{n + 1} - ω_{n} - ω_{n + 2}) t + 2 θ_{n + 1} (t) - θ_{n} (t) - θ_{n + 2} (t) + 2 ϕ_{n + 1} - ϕ_{n} - ϕ_{n + 2}] = 0

for

n = 1

to

N - 2

and

N \geq 3

, which results in all the modes being phase correlated and equally separated, i.e.:

θ_{n + 1} (t) - θ_{n} (t) = Δ θ (t)

ω_{n + 1} - ω_{n} = ω_{r}

with

ω_{r}

the angular repetition frequency such that the pulse train period

T = 2 π / ω_{r}

. Equation (3) can also be written as:

θ_{n} (t) = θ_{c} (t) + (n - n_{c}) Δ θ (t)

for

n = 1

to

N

, and with

n_{c}

a given mode number with corresponding phase noise

θ_{c} (t)

. The complex electric field then becomes:

E (t) = [\sum_{n = 1}^{N} E_{n} e^{j [ω_{n} t + θ_{c} (t) + (n - n_{c}) Δ θ (t) + ϕ_{n}]}] + c . c

and the laser intensity:

\begin{matrix} I (t) \propto {| E (t) |}^{2} \\ = \sum_{m = 1}^{N - 1} \sum_{l = 1}^{N - m} 2 E_{m + l} E_{l} \cos [m ω_{r} t + m Δ θ (t) + (ϕ_{m + l} - ϕ_{l})] + \sum_{m = 1}^{N} E_{m}^{2} \\ = \sum_{m = 1}^{N - 1} \sum_{l = 1}^{N - m} 2 E_{m + l} E_{l} \cos [m ω_{r} (t + Δ t_{r} (t)) + (ϕ_{m + l} - ϕ_{l})] + \sum_{m = 1}^{N} E_{m}^{2} \end{matrix}

with

Δ t_{r} (t) = Δ θ (t) / ω_{r}

being the timing fluctuations in the pulse position, commonly referred to as timing jitter. It has been experimentally shown in two section quantum well [3] and single section QDash [15] passively MLLs, through cross correlation measurements, that the variance of

Δ t_{r} (t)

grows linearly with time which implies that timing jitter undergoes a random walk. By the central limit theorem,

Δ t_{r} (t)

is normally distributed at large

t

, therefore it can be described by a Gaussian random walk with zero mean and variance

〈 {| Δ t_{r} (t) |}^{2} 〉 = D t

, with

D

a diffusion coefficient. Substituting

Δ θ (t) = ω_{r} Δ t_{r} (t)

in Eq. (6) yields:

E (t) = [\sum_{n = 1}^{N} E_{n} e^{j [ω_{n} t + θ_{c} (t) + ω_{r} (n - n_{c}) Δ t_{r} (t) + ϕ_{n}]}] + c . c

The phase fluctuations in each mode, and therefore $θ_{c} (t)$ , are affected by amplified spontaneous emission noise and hence undergo a random walk process [11], which again for large $t$ will be normally distributed. From Eq. (8), the optical spectrum ${| E (ω) |}^{2}$ can be calculated by taking the electric field autocorrelation function $R (τ)$ and Fourier transforming the result:

\begin{array}{l} {| E (ω) |}^{2} = \int_{- \infty}^{\infty} R (τ) e^{- j ω τ} d τ = \int_{- \infty}^{\infty} \sum_{n = 1}^{N} E_{n}^{2} e^{j ω_{n} τ} e^{- \frac{1}{2} [〈 {| θ_{c} (τ) |}^{2} 〉 + ω_{r}^{2} {(n - n_{c})}^{2} D + 2 ω_{r} (n - n_{c}) 〈 θ_{c} (τ) Δ t_{r} (τ) 〉]} e^{- j ω τ} d τ \\ \propto {\sum_{n = 1}^{N} {{[\frac{1}{2} Δ ω_{θ_{c}} + \frac{1}{2} ω_{r}^{2} {(n - n_{c})}^{2} D + ω_{r} (n - n_{c}) γ_{θ_{c}, Δ t_{r}}]}^{2} + {(ω - ω_{n})}^{2}}}^{- 1} \end{array}

with

γ_{θ_{c}, Δ t_{r}}

the slope at large

t

of the correlation term

〈 θ_{c} (τ) Δ t_{r} (τ) 〉

. Consequently, the optical spectrum consists of

N

Lorentzian lines centered at

ω_{n}

with full width at half maximum (FWHM) linewidths:

Δ ω_{n} = Δ ω_{m i n} + ω_{r}^{2} {(n - n_{m i n})}^{2} D

with

n_{m i n} = n_{c} - γ_{θ_{c}, Δ t_{r}} / ω_{r} D

, and

Δ ω_{m i n} = Δ ω_{θ_{c}} - γ_{_{θ_{c}, Δ t_{r}}}^{2} / D

. Equation (10) shows that the diffusion coefficient

D

affects the linewidth of the modes; hence an estimation of timing jitter is readily available from the optical spectrum by measuring the mode linewidths around

n_{m i n}

. It can be noted that correlation of

θ_{c} (t)

and

Δ t_{r} (t)

affects the first and zero order terms in Eq. (10), resulting in a frequency shift of the mode with minimum linewidth relative to the case in which both random processes are statistically independent (

γ_{θ_{c}, Δ t_{r}} = 0

), in agreement with [9]. As the value of

D

is unambiguously determined from the second order term in Eq. (10), it can be extracted without requiring any knowledge of the correlation between

θ_{c} (t)

and

Δ t_{r} (t)

. It should be noted, however, that these random processes are more likely to be uncorrelated if only quantum noise is considered as the temporal position of a pulse should not affect its optical phase changes [8].

The value of $D$ also determines the spectrum of the laser intensity ${| I (ω) |}^{2}$ , or more commonly called the RF (Radio Frequency) spectrum, which can be calculated straightforwardly from Eq. (7) in the same way as the optical one. Skipping the details, it is found to consist of a sum of $N - m$ Lorentzian lines centered at $m ω_{r}$ for $m = 1$ to $N - 1$ . At any given value of $m$ , each line will have identical FWHM linewidths given by:

Δ ω_{R F_{m}} = m^{2} ω_{r}^{2} D = m^{2} Δ ω_{R F_{1}}

with

Δ ω_{R F_{1}} = ω_{r}^{2} D

. Each line corresponds to the beating between a pair of modes separated by

m

modes in the optical spectrum. We notice that the diffusion coefficient

D

could be directly derived from

Δ ω_{R F_{1}}

, this is however not possible in practice at relatively high repetition frequencies due to the limited bandwidth of the photodiode. A relation between both the mode linewidths and those of their beatings can be established by combining (10) and (11) to yield:

Δ ω_{n} = Δ ω_{m i n} + \frac{Δ ω_{R F_{m}}}{m^{2}} {(n - n_{m i n})}^{2}

or equivalently:

Δ ω_{n} = Δ ω_{m i n} + Δ ω_{R F_{1}} {(n - n_{m i n})}^{2}

which was also obtained and experimentally observed in [5] for a two section passively MLLs emitting at 1.3 μm.

In order to be able to quantify the timing jitter from measurements of the mode linewidths, various basic aspects are first reviewed, some of which are also addressed in e.g [16,17]. Timing jitter is quantified as the standard deviation $σ$ of $Δ t_{r} (t)$ , hence, in a passively MLL, it will be given by:

σ = \sqrt{〈 {| Δ t_{r} (t) |}^{2} 〉} = \sqrt{D t}

Despite the lack of convergence as $t \to \infty$ , the timing jitter from pulse $i$ at time $t_{i}$ to pulse $i + N$ at time $t_{i} + N T_{r}$ can be defined. This yields:

σ (N) = \sqrt{D N T_{r}}

For $N = 1$ this gives the pulse to pulse timing jitter $σ_{p p}$ that is sometimes used in practice. Timing jitter can also be quantified in the frequency domain as:

σ = \sqrt{〈 {| Δ t_{r} (t) |}^{2} 〉} = \sqrt{\int_{- \infty}^{+ \infty} S_{Δ t_{r}} (ω) \frac{d ω}{2 π}} = \sqrt{\int_{- \infty}^{+ \infty} \frac{D}{ω^{2}} \frac{d ω}{2 π}}

Clearly, the PSD of $Δ t_{r} (t)$ does not converge as $ω \to 0$ , which is expected from the lack of convergence in the time domain as $t \to \infty$ and which is a consequence of the absence of a reference clock signal. The timing jitter can however be quantified between two given frequencies $ω_{1}$ , $ω_{2} > 0$ to yield the pulse to clock timing jitter, that is, the jitter relative to a perfect RF oscillator, also called r.m.s. integrated timing jitter:

σ_{p c} = \sqrt{\int_{ω_{1}}^{ω_{2}} \frac{D}{ω^{2}} \frac{d ω}{2 π}} = \sqrt{\int_{ω_{1}}^{ω_{2}} \frac{Δ ω_{R F_{1}}}{ω_{r}^{2} ω^{2}} \frac{d ω}{2 π}}

The above result is related to the commonly used formula for estimating the timing jitter from the single side band phase noise of the laser intensity as measured by an ESA, or L(f). In fact, if $Δ ω_{R F_{1}} / 2 < < ω < < ω_{r}$ , it can be approximated by [18]:

L (ω) \approx \frac{Δ ω_{R F_{1}}}{2 ω^{2}}

and by inserting this result into Eq. (17), we obtain the well known equation:

σ_{p c} = \frac{T}{2 π} \sqrt{\int_{ω_{1}}^{ω_{2}} 2 L (ω) \frac{d ω}{2 π}} = \frac{T}{2 π} \sqrt{\int_{f_{1}}^{f_{2}} 2 L (f) d f}

which demonstrates that the value of timing jitter commonly measured in practice from L(f) can be estimated from the MLL optical spectrum.

3. Device

The devices used for the experiments were single section QDash based MLLs. The active region in these devices consists of nine layers of InAs QDashes separated by InGaAsP barriers. From this structure, buried ridge stripe waveguide lasers were processed with a ridge width of 1.5µm. The as-cleaved lasers have a total length of ~330, 1000 and 3800 μm yielding repetition frequencies of ~130, 40, and 11.2 GHz respectively. Threshold currents ranged from 10 mA to 70 mA with corresponding slope efficiencies varying from 0.1 W/A to 0.08 W/A per facet as shown in Fig. 1 . Modal gain and internal losses were determined at 50 cm-1 and 18 cm-1 respectively. A more detailed description on these devices can be found in [12,13].

Fig. 1 Light-current characteristics of the three QDash lasers with cavity lengths (a) 330 μm (b) 1000 μm and (c) 3800 μm.

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4. Experimental results

Figure 2(a) shows the optical spectrum of the 1000 μm long laser when biased at 150 mA, from which a total of $N = 30$ modes have been numbered. In order to verify the mode phase correlation and hence ML operation, the RF spectrum of all $N - 1 = 29$ mode beatings between consecutive pairs of modes ( $m = 1$ ) were measured. Figure 2(b) is a plot of the superposition of all these beatings. All beating spectra are very well fitted by lorentzian functions. The average beating linewidth is $Δ ω_{R F_{1}} / 2 π = 89 kHz$ with a standard deviation of 5.3 kHz, while the average center frequency is $ω_{r} / 2 π = 39.6 GHz$ with a standard deviation of 18.2 kHz. These small deviations demonstrate the mode phase correlation between all $N$ modes and thus validate Eq. (3) and Eq. (4); consequently the laser is under ML regime. The lorentzian shape of all the lines in the RF spectrum for $m = 1$ also confirms that $Δ t_{r} (t)$ is in fact a Gaussian random walk process.

Fig. 2 (a) Optical spectrum and (b) RF spectrum of the 29 consecutive mode beatings (m = 1) of the 1000 μm long laser under ML operation when biased at 150 mA.

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The linewidth of each of the $N$ modes of the optical spectrum in Fig. 2(a) was then measured by the self-heterodyne technique [19]. Each longitudinal mode was individually filtered and split into two paths. The mode frequency in one path is offset using an acousto-optic modulator. On the second path, a fiber is inserted to introduce a delay exceeding the coherence time of the mode light so that the combining beams interfere as if they were independent. All the resulting self-mode beating spectra fitted very well lorentzian functions, as can be seen in Fig. 3(a) for several modes, confirming the lorentzian shape of the modes as expected. In Fig. 3(b) a plot of the measured linewidths as a function of mode number is shown. The solid line is a parabolic fit $Δ ω_{n} / 2 π = 6.5 + 0.092 {(n - 12)}^{2} MHz$ , from which $Δ ω_{R F_{1}} / 2 π = 92 kHz$ is extracted with a standard error of ~3 kHz, which very much agrees with the measured linewidth of each beating in Fig. 2(b), verifying Eq. (13). We can also see from the fit that $n_{m i n} = 12$ which is the number of the less noisy mode with linewidth $Δ ω_{θ_{c}} / 2 π = 6.5 MHz$ , and this is what the linewidth of all the modes should be in the absence of timing jitter. The diffusion coefficient is also extracted from the parabolic fit to give $D = 0.0092 fs$ . By using Eq. (15) and Eq. (17), the pulse to pulse and the pulse to clock timing jitter from, e.g.16 MHz to 320 MHz [20] were then estimated at $σ_{p p} = 15.16 fs$ and $σ_{p c} = 166 fs$ respectively.

Fig. 3 (a) Self-mode beating spectra with lorentzian fits for modes 11, 16, 21 and 26. (b) Measured mode linewidths as a function of mode number and parabolic fit for the 1000 μm long laser under ML operation when biased at 150 mA.

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In order to verify Eq. (11), the linewidths of the RF spectrum for $m \geq 1$ have been measured from the laser with cavity length 3800 μm, yielding a repetition frequency $ω_{r} / 2 π = 11.2 GHz$ , when biased at 200 mA. Figure 4(a) shows the measured RF spectra for $m = 1$ to 4, superimposed with perfectly lorentzian fits. The inset is a plot of the full span RF spectrum from 10 to 50 GHz. Figure 4(b) shows the expected quadratic dependence of $Δ ω_{R F_{m}}$ on $m$ , as evidenced by the parabolic fit $Δ ω_{R F_{m}} / 2 π = 48.5 m^{2} kHz$ . From this we determine $D = 0.062 fs$ , $σ_{p p} = 74.12 fs$ and from 100 kHz to 4 MHz $σ_{p c} = 5.61 ps$ . To verify the agreement of the previous result with Eq. (19), the L(f) was measured by an ESA and plotted in Fig. (5) . By integrating the measured curve from e.g. 100 kHz to 4MHz, a value of $σ_{p c} = 5.63 ps$ was obtained, in agreement with the previously calculated value. The L(f) as measured by an ESA will then yield a correct value of $σ_{p c}$ as long as it decays at −20 dB/dec (see Eq. (18)).

Fig. 4 (a) RF spectrum for m = 1 to 4 and corresponding lorentzian fits. Inset: full span RF spectrum. (b) Measured RF linewidths as a function of m with parabolic fit for the laser with cavity length 3800 μm when biased at 200 mA.

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Fig. 5 Measured L(f) for the laser with cavity length 3800 μm when biased at 200 mA and −20 dB/dec fit.

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The experimental results in the previous section have demonstrated the validity of a method for determining the timing jitter of passively MLLs from measurements of the individual mode linewidths. We now apply this method to estimate the timing jitter and the RF linewidth of a 130 GHz single section QDash laser. ML operation in this device is evidenced by the intensity autocorrelation function of the laser as it exhibits a pulsating behavior when biased at 300 mA, with a period corresponding to the inverse of the repetition frequency and pulse widths of ~850 fs having high extinction ratios, as shown in Fig. 6(a) . The corresponding optical spectrum is shown in Fig. 6(b) from which 13 modes have been numbered and filtered for measuring their corresponding linewidths. Figure 6(c) shows the mode linewidth evolution with mode number and the expected parabolic behavior. The parabolic fit $Δ ω_{n} / 2 π = 8.53 + 0.093 {(n - 7)}^{2} MHz$ gives $D = 8.76 \times 10^{- 4} fs$ , $σ_{p p} = 2.6 fs$ and from 16 MHz to 320 MHz $σ_{p c} = 51.3 fs$ .

Fig. 6 (a) Intensity autocorrelation, (b) optical spectrum, and (c) measured mode linewidths for the laser with cavity length 330 μm when biased at 300 mA.

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In order to evaluate the limits in repetition rate for the applicability of the proposed method, estimations of the RF linewidth as a function of measured number of modes have been performed, from which corresponding relative errors have been calculated. Figure 7 illustrates this for the laser with repetition rate of 40 GHz, with errors being determined relative to the RF linewidth value directly measured from the photocurrent spectrum. This figure suggests that it is not required to perform the measurements of the entire number of available modes in order to obtain similar relative error values, which can be kept smaller than ~10% by measuring at least six of the available modes. This gives an idea on the maximum repetition frequency that the method could be applied to. In this example, six available modes is equivalent to performing the measurements every five modes, corresponding to a separation in frequency of ~200 GHz, for which an estimation in RF linewidth with a relative error of about 11% was obtained. Hence, application of the method might be possible at repetition frequencies up to 200 GHz yielding similar relative errors provided six modes are available for linewidth measurements. Timing jitter at even higher repetition rates could however be estimated if more modes are available. A lower limit in repetition frequency for the applicability of the method is also imposed since precise linewidth measurements on multiple lines of the optical spectrum becomes more difficult as the lines become closer to each other. If the self-heterodyne technique is used, this limit will then be determined by the minimum bandwidth of commercially available optical filters of ~0.1 nm in order to individually separate the optical modes, setting a repetition frequency limit down to ~10 GHz. Other linewidth measurement techniques, such as the heterodyne detection with narrow linewidth reference lasers, may however be implemented, allowing a lower limit in repetition frequency depending on the specific instrumentation used.

Fig. 7 Relative error as a function of number of measured modes for the 1000 μm long laser.

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5. Summary

An analysis of the optical and intensity spectra of semiconductor lasers under passive mode locking was presented. This has allowed simple relations to be established between the mode linewidths, RF linewidths, L(f) and timing jitter. The use of these relations becomes particularly interesting at high repetion frequencies where direct measurement of the RF linewidths, L(f) and timing jitter is limited by the photodiode and the electronics instrumentation bandwidth. The analytical results have been experimentally validated for single section QDash passively MLLs at repetition frequencies of up to 130 GHz. As the presented theory is based on a phenomenological approach, the results can be applied regardless the MLL configuration.

Acknowledgments

This work was supported in part by the French National Research Agency through the project TELDOT.

References and links

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Timing jitter from the optical spectrum in semiconductor passively mode locked lasers

Abstract

1. Introduction

2. Theory

3. Device

4. Experimental results

5. Summary

Acknowledgments

References and links

Cited By

Figures (7)

Equations (19)

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