This paper presents a detailed analysis of the performance of a recirculating delayed self-heterodyne (R-DSH) method for high-resolution laser lineshape measurement. For increasing the delay time the R-DSH method utilizes circulation of light in a heterodyne ring interferometer (HRI) containing a frequency shifter, delay fiber, and fiber amplifier. It is shown both theoretically and experimentally that unwanted higher order frequency-shifted components induce distortion in the beat signal spectra, which significantly limits the maximum number of circulations. An effective technique is proposed and demonstrated for reducing the distortion by using optical filtering at the HRI output. Furthermore, a practical limit on the number of circulations is investigated by comparing the shape of observed beat signal spectra with theory. It is shown that the maximum delay is limited to about 180 km even with the use of the optical filtering technique.
© 2012 OSA
The spectral purity of laser sources is one of the key factors that influence the performance of digital coherent transmission systems. Even with the use of digital carrier phase estimation, higher order modulation formats impose stringent requirements on the laser linewidth. For example, square 16- and 64- quadrature amplitude modulation formats require the linewidths below 120 and 1.2 kHz, respectively, at a 40-Gb/s data rate . Recently, narrow-linewidth external cavity laser diodes (ECLDs) [2,3] and fiber lasers [4,5] have been developed and are expected as promising light sources not only for the above application but also for sensing and spectroscopy. A thorough characterization of the spectral purity is an indispensable task in order to meet the requirement from the above applications.
The delayed self-heterodyne (DSH) method, developed by Okoshi et al. in 1980 , has been a standard technique for measuring not only laser lineshape, but also Allan deviation  and frequency modulation (FM) noise power spectral density (PSD) [8,9]. For linewidth measurement the DSH method requires an optical fiber with a delay time much longer than the laser’s coherence time. To relax the requirement on the length of a delay fiber, I proposed the recirculating DSH (R-DSH) method , in which a heterodyne ring interferometer (HRI) containing a delay fiber and frequency shifter was employed instead of a heterodyne Mach-Zehnder interferometer. This configuration allows for an increase in the delay time by making use of multiple transmissions in the HRI. In order to compensate for the propagation and coupling loss in the HRI, Dawson et al.  employed an erbium-doped fiber amplifier (EDFA) and observed the beat signal generated after 30 times circulation. Although a resolution below 1 kHz was reported for linewidth measurements , the accuracy of the measured spectral lineshape has not been reported so far.
This paper reports a theoretical and experimental investigation on the performance of the R-DSH method for laser lineshape measurements. First, I point out the influence of unwanted higher order frequency-shifted components that are simultaneously detected with a photodetector placed at the HRI output. In Sections 2, theoretical beat signal spectra are calculated taking account of the contribution from the higher order frequency-shifted components. Section 3 describes an experiment for lineshape measurement of an external cavity laser diode (ECLD). From the comparison of theoretical and experimental results, it is shown that the higher order frequency-shifted components induce distortion in the beat signal spectrum. In Section 4, a new and effective technique is proposed and demonstrated for reducing the distortion of beat signal spectra by the use of optical bandpass filter (OBPF) that rejects unwanted frequency-shifted components. Next, a detailed analysis is presented on the maximum delay achievable with the R-DSH method, which is described in Section 5. The beat signals obtained with the R-DSH method are influenced by various factors such as amplified spontaneous emission (ASE) from the EDFA, delay time fluctuation due to mechanical and thermal disturbances to the fiber, noise of the photodetector and amplifier, etc., leading to a decrease in measurement accuracy. To address this issue, beat signal spectra measured for various delays are compared with the theoretical spectra that are calculated from the FM noise PSD . From this comparison, a practical limit in the maximum delay is estimated. Finally, a brief summary is given in Section 6.
2. Influence of higher-order frequency-shifted components - Theory
Figure 1(a) shows a schematic of the laser lineshape measurement apparatus based on the R-DSH method. The output light from a laser under test is input to an HRI, which consists of a directional coupler, frequency shifter, delay fiber, EDFA, and OBPF for rejecting the ASE. A part of the laser light reflected from the directional coupler is directly coupled to a photodiode (PD) and is used as a local oscillator (LO) for heterodyne detection. In each round-trip in the HRI, the laser light is given a frequency shift fs and a delay τd and is amplified to compensate for the propagation, coupling, and insertion losses. The EDFA gain is adjusted to prevent the self-oscillation of the HRI. Therefore, the laser light after n-times (n is an integer) circulation in the HRI has the relative frequency shift and delay of nfs and nτd, respectively, with respect to the input light. At the PD output multiple heterodyne beat signals with the frequency of nfs are generated simultaneously, which can be separately detected and analyzed with an RF spectrum analyzer. As compared with the conventional DSH method, n-times enhancement in the delay time is realized by detecting the beat signal at nfs.
Although the beat signal at frequency nfs is generated mainly from the LO light and the frequency-shifted component after n-times circulation, there are additional contributions from the pairs of the frequency-shifted components with the frequency difference of nfs. In the following, a theoretical analysis is presented for evaluating the distortion of the beat signal spectra arising from these unwanted frequency-shifted components.
The electric field of the laser output light is expressed by with constant amplitude A, center frequency ν0, and phase fluctuation ϕ(t). The laser FM noise ν(t) can be related to ϕ(t) by . The FM noise PSD Sν(f) is given by the Fourier transform of the autocorrelation function (ACF) Rν(τ) of ν(t) and is expressed as
The electric field of HRI output can be expressed asFig. 1(b), the heterodyne beat signal at frequency nfs is generated not only from the mixing of the LO and frequency-shifted components at ν0 and ν0 + nfs, respectively, but also from the mixing of the higher-order frequency-shifted components such as those with frequencies of ν0 + fs and ν0 + (n + 1) fs.
First, I derive the expression for the beat signal spectrum in the absence of the contributions from the higher-order frequency-shifted components, which corresponds to the conventional DSH method. The beat signal IDSH(t) at the PD output is given byEq. (3) expressed asEq. (4) is given byEq. (1) [8,9,12].Equations (6) and (7) state that beat signal spectra SDSH(f) can be derived from FM noise PSD Sν(f), which will be used to calculate the theoretical spectra in the next sections.
Next, I derive the beat signal spectrum SR-DSH(f) observed with the R-DSH method taking account of the contribution from the higher-order frequency-shifted components, and relate it to the SDSH(f) observed with the DSH method. For simplicity, I consider only the lowest order contribution resulting from the mixing of the frequency-shifted components with frequencies ν0 + fs and ν0 + (n + 1) fs. The phasor of the beat signal is given byEq. (4) and the second term represents the mixing between the frequency-shifted components. The ACF RR-DSH(τ) of Eq. (8) can be expressed in terms of the ACF RDSH(τ) of Eq. (6) byEq. (9) represent the contributions from the higher-order frequency-shifted components. The beat signal spectrum SR-DSH(f) corresponding to Eq. (9) is given byEq. (10)), which will be described in Section 4.
3. Influence of higher-order frequency-shifted components - Experiment
To verify the distortion of the beat signal spectra an experiment was carried out using the setup schematically shown in Fig. 1(a). The light source under test was an ECLD module (Redfern Integrated Optics, RIO0175) is used with an output power and center wavelength of 15 mW and 1552.52 nm, respectively. The module consists of gain chip and planar lightwave circuit including a waveguide with a Bragg grating [2,3]. The FM noise PSD of the laser is consistent with the oscillator noise model consisting of white and flicker noise, which is expressed as9]. Equation (11) is used for calculating the beat signal spectra using Eqs. (6) and (7).
A high-resolution laser linewidth analyzer (Inter Energy, ITS-7401) is used for lineshape measurement, which comprised of an HRI and a PIN PD with an integrated RF amplifier. The HRI consists of 50/50 directional coupler, 100-MHz acousto-optic frequency shifter, 20-km optical fiber, EDFA, and 0.5-nm bandpass filter. In the absence of the EDFA, the single-pass transmission loss of the HRI including that of the directional coupler is 13.6 dB. DSH beat signals detected with the PD are analyzed by using a vector signal analyzer (VSA: Agilent, 89640A), which consists of frequency down-converter, analog-to-digital converter, and digital signal processor.
Figure 2(a) shows the beat signal spectra obtained with the 20-km delay corresponding to n = 1 in Eq. (10), where curves with red and blue lines correspond to the theoretical (Eq. (12) shown below) and experimental results, respectively. The total delay of the HRI is adjusted so that the product fsτd is nearly equal to a positive integer. In this case Eq. (10) reduces toEquation (12) indicates that the magnitude of spectrum distortion depends on the ratio E2/E0. The curves A - E in Fig. 2(a) represent the results for different values of E2/E0, which is adjusted by changing the gain of the EDFA. The values E0 and E2 are estimated by observing the HRI output with a scanning Fabry-Perot interferometer. It can be seen that the undulation of the beat signal spectra increases with increasing the EDFA gain and that the experiment results agree well with the theoretical calculation. From these results it can be concluded that there is a restriction in the EDFA gain in order to avoid spectrum distortion, which limits the maximum number of circulation in the lineshape measurement using the R-DSH method, which will be discussed in Section 5.
4. Reduction of spectrum distortion by optical filtering
In this section I propose and demonstrate an effective technique for reducing the spectrum distortion discussed in the previous section. The basic idea is to remove unwanted frequency-shifted components by the use of an optical filter. Figure 3 shows the schematics of two feasible approaches. As shown in Fig. 3(a) a Fabry-Perot interferometer with the free spectral range of nfs can be used for selectively transmitting the lights with the frequencies of ν0 and ν0 + nfs while rejecting other frequency-shifted components. For stable operation the resonant frequency of the interferometer should be locked to that of the incident light. This scheme is not suitable when the number n of circulations is small, because higher order frequency-shifted components at ν0 + knfs (k = 2, 3, …) are transmitted at the same time. Moreover, this configuration is not flexible due to the difficulty in tuning the free spectral range over wide frequency range.
Figure 3(b) shows the schematic of the alternative realization, in which an OBPF placed at the HRI output is used to selectively transmit a specific frequency-shifted component. The laser output is divided into two and the first part is input to the OBPF. The second parts is mixed with OBPF output producing a beat signal at the specific frequency. This configuration overcomes the difficulties associated with the scheme in Fig. 3(a). For rejecting the unwanted frequency-shifted component the OBPF bandwidth should be much smaller than the frequency shift.
The experiment is carried out using the configuration shown in Fig. 3(b). As the OBPF a fiber Fabry-Perot interferometer (FFPI: Micron Optics, FFP-FI) is used with the transmission bandwidth of 8.53 MHz. The free spectral range of the FFPI is 5.04 GHz, which is wider than the bandwidth of the HRI output light. To avoid the drift of the resonant frequency the FFPI is locked to that of the transmitted light using a tunable filter controller (Micron Optics, FFP-C).
Figure 2(b) sows the heterodyne beat signal spectra obtained with the configuration shown in Fig. 3(b), where curves with red and blue lines correspond to the theoretical (Eq. (7)) and experimental results, respectively. The operating conditions of the HRI are the same as those of Fig. 2(a). It can be seen that the measured shapes of the beat signal spectra agree well with the theoretical results and are almost unchanged against the variation of the EDFA gain, which demonstrates the effectiveness of the proposed technique.
5. Estimation of the maximum delay
Although the R-DSH method with optical filtering effectively suppresses spectrum distortion caused by the higher-order frequency-shifted components, the circulating lights in the HRI suffer from attenuation and are influenced by amplified spontaneous emission from the EDFA and path length fluctuation due to thermal and mechanical disturbances. To clarify the influence of these factors, lineshape measurements are carried out as a function of the delay and the results are compared with the theory calculated using Eqs. (6), (7), and (11). Since the beat signal spectra obtained with the DSH method can be precisely predicted from the FM noise power spectral density [8,9,12], the accuracy of lineshape measurements can be verified by the comparison.
Figures 4(a) shows the beat signal spectra obtained without the use of optical filtering. In each curve, red and blue lines correspond to the theoretical and experimental results, respectively. The EDFA gain is adjusted to relatively small values (< 7 dB) to prevent spectrum distortion discussed in section 3, and the influence of amplitude spontaneous emission is negligible. Curves A, B, C, D, and E in Fig. 4(a) represent the beat signal spectra obtained with 2, 3, 4, 5, and 6 circulations corresponding to 40-, 60-, 80-, 100-, and 120-km delay, respectively. Although the shapes of measured spectra agree well with the theory below 80-km delay, the discrepancy between the theory and experiment increases with further increasing the delay at the pedestal parts of the spectra. The SNRs of the beat signals are 70.7, 54.8, 41.9, 30.5, and 22.2 dB for curves A, B, C, D, and E, respectively, with the noise floor contributed by the that of the PD and RF amplifier, which is measured in the absence of optical input to the PD. Therefore, the maximum delay of the conventional R-DSH method is limited by the SNR of the beat signal as imposed by the EDFA gain for avoiding spectrum distortion.
Figures 4(b) shows the beat signal spectra obtained with the use of optical filtering. Curves A, B, C, D, and E are obtained with 40-, 100-, 160-, 220-, and 280-km delay corresponding to 2, 5, 8, 11, and 14 circulations, respectively. It can be seen that the shapes of measured spectra agree well with the theory up to 220-km delay, which demonstrates the effectiveness of optical filtering. However, there is a large discrepancy between the theory and experiments for 280-km delay. It can be seen that the measured results exhibits overall spectral broadening in contrast to the curve E of Fig. 4(a), which exhibits increase at the pedestal parts of the spectrum. The SNRs of the beat signals are 82.9, 84.2, 55.0, 55.1, and 56.0 dB for curves A, B, C, D, and E, respectively. Therefore, the spectral shape is not influenced by the SNR of the beat signal. Since the shape of the beat signal spectra for 280-km delay is almost unchanged against the variation of the EDFA gain, the observed spectral broadening is not associated with the fiber nonlinearity, such as nonlinear phase noise induced by amplified spontaneous emission , which was shown to induce an increase at the pedestal part of the field spectra . As described below, the discrepancy is mainly caused by the mechanical and thermal fluctuations of the delay fiber. By comparing Figs. 4(a) and 4(b), it can be seen that the optical filtering offers longer delay as compared with the conventional R-DSH method, which is effective for improving the resolution of lineshape measurements.
For DSH-based lineshape measurements the delay time of the interferometer should be larger that the coherence time of the laser under test. Therefore, a larger number of circulations will be required to apply the R-DSH method to lasers with a higher spectral purity. Figure 5 shows the 3-, 10-, and 20-dB linewidth of the beat signal spectra as function of the delay. Solid lines represent the theoretical results that are evaluated from the beat signal spectra calculated using Eqs. (6), (7), and (11). Owing to the existence of flicker noise component in the FM noise PSD, the linewidth observed with the DSH method gradually increase with increasing the delay time . Blue circles in Fig. 5 correspond to the experimental results obtained with the conventional R-DSH method. Due to the limited EDFA gain for preventing spectrum distortion, the maximum number of circulation is limited to 7 (140-km delay). Although the 3-dB linewidth agree well with the theory up to 7 circulations, 10- and 20-dB linewidth exhibits large deviations, which is caused by a reduced SNR of the beat signals. By the use of optical filtering at the HRI output, the maximum number of circulations can be substantially increased as indicated by the red circles in Fig. 5. It can be seen that the measured 3-, 10-, and 20-dB linewidth agrees well with the theory up to 9 circulations (180-km delay). However, the discrepancy between the theory and experiments gradually increases with increasing the delay above 200 km, which can be mainly attributed to mechanical and thermal fluctuations of the delay fiber inducing phase variation of the beat signal at low frequencies. From the results of Fig. 5, it can be concluded that the delay of the R-DSH method is limited below 180 km even with the use of optical filtering.
In summary a detailed analysis was presented on the performance of the recirculating delayed self-heterodyne (R-DSH) method for high-resolution laser lineshape measurement. It was shown that the unwanted higher order frequency-shifted components induce distortion in the beat signal spectra, which significantly limits the maximum number of circulations. An effective technique was demonstrated for eliminating the distortion by using optical filtering. It was shown that the maximum delay is limited to about 180 km even with the use of optical filtering technique.
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