Measurement of the Henry factor over large optical bandwidth is carried out in a single step without any filtering, using a technique based on the sinusoidal phase modulation method. This fast technique was successfully applied to a directly modulated Fabry Perot laser to obtain simultaneously the linewidth enhancement factor (LEF) of 14 longitudinal modes. It is also well suited for electro-absorption modulators (EAM) for which the α-factor is determined over 15 nm optical bandwidth. A very good agreement is found with the well established fiber transfer function method.
© 2011 OSA
The phase-amplitude coupling factor or Henry factor  is a fundamental parameter that determines many important characteristics of an optical transmitter for fiber optics communication. The existence of the so called Henry factor, or linewidth enhancement factor (LEF), induces a linewidth broadening of longitudinal modes of the transmitter. When the optical intensity of the emitter is modulated, this results in a frequency modulation (so called ‘chirp’) that interacts with the fiber chromatic dispersion and sets an upper limit to the product squared bit rate times propagation length. This parameter also determines the resistance to optical feedback of lasers . Different methods have been devised to measure the α-factor and their use depends on the type of transmitter, consisting either of a directly modulated laser or an external modulator. The Hakki-Paoli method can only be applied to Fabry-Perot (FP) lasers and it gives the LEF only below threshold. Recently a method based on the linewidth power ratio was proposed to determine the LEF of a FP laser above threshold , however it does not give the LEF versus injection current and cannot be applied to external modulators. Another technique relies on optical injection locking but does not provide a direct measurement of the LEF [4,5]. Similarly to the linewidth power ratio method, the optical injection locking technique is only applicable to laser but not to external optical modulators. The RF current modulation method enables to determine the LEF but it applies to only one optical carrier at a time . The well established fiber transfer function (FTF) method [7,8] exploits the interaction between the chirp and the fiber chromatic dispersion but it requires to filter out each individual longitudinal mode in the case of multimode Fabry-Perot lasers. It is therefore highly desirable to have a relevant, fast and direct method to simultaneously measure the LEF of multiple longitudinal modes of an optical transmitter.
In this paper, we present a novel technique based on the sinusoidal optical phase modulation method  to extract directly, simultaneously and together with a high sensitivity the LEF of each longitudinal mode of FP lasers. A simple expression is derived that yields both the value and the sign of the LEF. Moreover, the extraction is fast because it is a non-iterative method. This technique was successfully applied to a directly modulated FP laser to measure simultaneously the LEF of 14 longitudinal modes covering the FP optical spectrum. We also applied our method to an electro-absorption modulator (EAM) to determine the α-factor over 15 nm optical bandwidth and a very good agreement is found with the FTF method. As shown later, we stress here that this technique is adapted for even larger optical bandwidth.
2. Principle and experimental set-up
In small signal modulation regime, the complex amplitude of the spectral components at the output of the device under test (laser or external modulator), by keeping only the first order components for a modulation index, can be written :10]:
For a decade or so, several methods have been proposed to determine the complete temporal response of short pulses (i.e. both the amplitude and the phase) used in optical telecommunications [9, 11–18]. These results are obtained in the temporal domain by fast Fourier transform of the complex spectrum measured with one of these techniques [9, 11–18]. Among these methods, we have chosen the one that relies on the sinusoidal modulation of the phase of the signal under test  because it offers several advantages: easy implementation, high sensitivity, use of an electrical signal equivalent to that of the modulation of the device under test (DUT), and direct algorithm (i.e. non-iterative method) for obtaining the complex spectrum of the transmitter.
Figure 1 represents a schematics of the experimental set-up of this technique. The DUT is modulated by a sine wave supplied by a RF generator at a modulation frequency . An RF variable attenuator allows the control of the modulation index so as to keep it within the small signal modulation regime. The optical signal to analyze is sent to a fiber coupled LiNbO3 phase modulator that is also modulated at by the same oscillator. The inherent large optical bandwidth of LiNbO3 phase modulator is well suited to measure the Henry factor of devices exhibiting smaller optical bandwidth, e.g. electroabsorption modulator. An optical delay line is placed between the DUT and the phase modulator in order to control the delay between the optical and electrical signals at the input of the phase modulator. Instead of making use of a delay line in the optical domain, a RF phase shifter could be used on one of the electrical paths as in reference . The signal at the output of the modulator is sent to an optical spectrum analyzer (OSA) whose resolution and dynamic range are suitable to resolve the optical carrier and the modulation sidebands. As indicated in , the method relies on the measurement of 4 optical spectra corresponding to 4 different delays equally spaced by .
Instead of using the general formalism developed in  we modified it to the specific case of our signals given by (1). The optical signal at the output of the component can be written as:
In the spectral domain, at the optical frequency , by assuming a low phase modulation amplitude (typically few tenth of radians) and keeping the only three first terms of the Fourier series development of (i.e. the DC, and terms), the electric field has the following expression:
The spectral intensity measured by the OSA will be:
A similar calculation done for shows that:
Then, we define two quantities:
Finally, in our case, according to (1), as , from (3), we can deduce:
Hence, from the simple measurement of the 4 spectral intensities corresponding to the 4 different optical spectra related to 4 different optical delays and using Eqs. (9) and (11), we can directly and simultaneously extract the Henry factor of all longitudinal modes.
3. Results for Fabry Perot lasers
Figure 2 shows the results of the measurements of as a function of different longitudinal modes of the device under test for two different bias currents. The active region of this laser consists of 9 layers of InAs QDashes embedded in 40 nm thick InGaAsP (Q1.17) barriers . The modulation frequency is 12 GHz and the applied RF power yields a modulation index of about 2.5%. The inset shows the emission spectrum in a linear scale.
All the measurements of αH in Fig. 2 are obtained from only 4 spectra, by applying simultaneously formulas (9) and (11) to each of the 12 and 14 longitudinal modes at 30 mA and 40 mA respectively. An example of measurements of these 4 spectra around one of the longitudinal mode is depicted in Fig. 3 , where the vertical scale is expanded so as to highlight the amplitude variations of the 2 sidebands as a function of the optical delay. The LEF amounts to ~3.3 at 1543 nm and increases with wavelength up to ~3.7 at 1553 nm for 40 mA, which is attributed to the variation of the differential gain with wavelength as already observed in bulk and QW based lasers . These values are consistent with a previous work where the LEF was measured to be ~4 in a similar laser structure based on InAs/InP QDashes .
Compared to methods based on RF current modulation  or the fiber transfer function (FTF) , our proposed technique is fast because it does not require filtering out each individual mode. Besides, for lower power modes, the FTF technique requires to perform high number of averages  which implies time consuming. Moreover, our method allows to characterize the Henry factor of FP lasers against current, unlike the linewidth power ratio technique which yields an average value over the current .
4. Results for electroabsorption modulators
Our method is also suitable to determine the chirp of external modulators. This is especially important if one aims at implementing an integrated laser with an electro-absorption modulator (EAM) that enables transmission beyond the chromatic dispersion limit for example. Usually the chirping behaviour of the EAM is predicted by measuring the absorption spectrum versus bias values and using Kramers-Krönig integrals: the best operating point is deduced from the Henry factor versus wavelength detunings from the excitonic peak . The sinusoidal phase modulation method is very attractive because it allows a simple and direct measurement of the chirp of an EAM versus wavelength.
Therefore, we now applied our method to an optical transmitter that consists of an electroabsorption modulator with an external laser. We generally make use of a tunable laser diode at the input of the EAM in order to make this type of measurements. Here, we prefer to take advantage of the wide optical spectrum of the previously tested FP laser (~15 nm) so as to use it as a simple continuous source. The objective is to exploit all the different longitudinal modes independently in order to get a measurement of as a function of the wavelength by making only 4 spectral measurements. The EAM we test here consists of a AlGaInAs/GaInAs structure with 10 QW .
Figure 4 depicts the measurement of αH for two bias voltages (−1.0 and −1.6 V). The RF signal is at 10 GHz and its incident power on the DUT equals – 15 dBm only, which illustrates the sensitivity of the method. For - 1.6V, we notice that becomes negative for wavelength smaller than 1562 nm.
We finally applied our method to determine the of the transmitter as a function of voltage for a given wavelength and compare the results with the FTF method (Fig. 5 .). A very good agreement between these two curves over a large span of [ + 8 to −10] permits to validate this new technique.
For all results presented in this paper, the incident RF power on the phase modulator is + 4 dBm which leads to a value of 0.30 rad. For RF powers leading to rad, we observed differences that are no more negligible (more than 5% deviation between the FTF method and our method). This is attributed to the approximation by the only three first terms of the Fourier series development of done in part 2 which is no more valid under higher amplitude phase modulation.
This is further illustrated in Fig. 6 which shows the accuracy of the method for amplitude phase modulation ψ as low as 0.06 rad at a fixed bias voltage. Variation of the measured Henry factor for ψ in the range of 0.06-0.4 is less than ± 2%. A typical value of 0.30 rad for is consequently a good trade-off between accuracy and sensitivity for this method.
In conclusion, we present a novel method based on sinusoidal phase modulation to determine the Henry factor of FP lasers without extracting each individual mode using a band pass filter. We also applied our method to determine the chirp of an EAM versus the wavelength over more than 15 nm optical bandwidth. This method yields very good agreement with the well-established fiber transfer function technique. It is furthermore capable of analyzing very wide optical bandwidth devices as it is only limited by the phase modulator optical response (typically few tens of nm). This technique will be of great interest for e.g. the characterizations of WDM systems such as optical frequency combs where each channel is independently modulated.
The authors would like to thank C. Kazmierski and F. Lelarge for providing the EAM and QDash laser respectively.
References and links
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