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Phase ripple correction: theory and application Josh A. Conway,* George A. Sefler, Jason T. Chou, and George C. Valley Electronics and Photonics Laboratory, The Aerospace Corporation, 355 South Douglas, El Segundo, California 90245, USA *Corresponding author: [email protected] Received January 28, 2008; revised March 27, 2008; accepted April 3, 2008; posted April 9, 2008 (Doc. ID 92226); published May 13, 2008 Spectral phase ripple associated with novel dispersive devices can distort broadband optical signals. We present a digital postprocessing algorithm to correct for this distortion by exploiting the static deterministic nature of the ripple. This algorithm is demonstrated with empirical data for several systems employing chirped fiber Bragg gratings (CFBGs). We employ this technique in a photonic time-stretch system incorporating CFBGs, improving the signal fidelity by 9 dB. Simulations and experiments show that this algorithm, which can be reduced to a simple interpolation and matrix multiplication, also mitigates additive noise. We see that the act of distortion correction yields signal fidelity superior to that of an ideal dispersive element. © 2008 Optical Society of America OCIS codes: 060.5625, 060.3735.

The combination of chromatic dispersion and short broadband pulses enables applications ranging from sensing and metrology [1–4] to optical processing [5–8]. These typically require dispersions of hundreds to thousands of picoseconds of group delay per nanometer of optical bandwidth. While such a large dispersion can be achieved through conventional SMF-28 or a dispersion compensating fiber (DCF), these require long fibers and hence, large loss and nonlinear effects. Recently many novel dispersive devices have been investigated with large dispersion and shorter lengths [9,10]. Most significantly, chirped fiber Bragg gratings (CFBGs) have been developed with large dispersion and custom spectral profiles. Their length, typically ⬃4 orders of magnitude shorter than the equivalent SMF-28, dramatically reduces loss and nonlinearities. In the CFBG the delay of each spectral component is controlled by varying the period of the Bragg grating along the length of the fiber. Design and manufacturing imperfections create errors in this delay (the origin of phase ripple) that can cause severe amplitude distortion in the time-domain signal [11]. Herein we present a postprocessing algorithm that corrects this signal distortion using only the measured time-domain intensity and knowledge of the static system parameters. This allows us to demonstrate the use of a CFBG in a time-domain spectroscopy application and in a photonic time-stretch system. Finally, through simulation and experiment, we show the astonishing result that the use of the CFBG and postprocessing yields signal integrity superior to that of a dispersive element with ideal phase response. In this Letter, we define phase ripple as the residual phase distortion in the spectral domain caused by a dispersive device after linear, quadratic, and third-order dispersion are removed. Phase ripple is typically a small perturbation several orders of magnitude smaller than the overall phase imparted by the device but can yield a major effect. When considered in the time domain, it is clear why this is the case [12]. Simple Fourier analysis shows that the impulse response of sinusoidal phase ripple will create time-delayed copies of the original signal whose mag0146-9592/08/101108-3/$15.00

nitude is independent of the unperturbed impulse response. Arbitrary ripple can then be constructed from these various sinusoidal components. The effect is that of a low-Q Fabry–Perot cavity, mixing components of the time-domain signal. Upon square-law detection, this interference creates sharp artifacts, greatly distorting the overall signal in analogy to speckle in the spatial domain. Our experiments demonstrate the severity of phase ripple on the time-domain signal. As illustrated in Fig. 1(b), a 1.5 ps broadband 共1545– 1570 nm兲 pulse from a fiber laser (Precision Photonics) propagates into a 2000 ps/ nm CFBG (Proximion). This CFBG is approximately 10 m long with over a 35 nm optical bandwidth. The length necessitates stitching into the grating writing process, which contributes to phase ripple. The total phase of the CFBG is plotted in Fig. 2(b). Subtracting the cubic and lower-order terms yields the phase ripple shown in Fig. 2(b). To prevent adjacent stretched pulses from overlapping in time, the CFBG is followed by a 3 nm bandpass filter centered at 1550 nm (Bookham). The optical signal is detected by a high-speed photodetector (Discovery, 12 GHz analog bandwidth) and digitized by an 8 bit real-time oscilloscope (Tektronix 7204, 16 GHz analog bandwidth). The measured signal, plotted in Fig. 1(a), is averaged over 64 traces to mitigate the effects of noise. This signal is strongly structured with frequencies up to the detection limit.

Fig. 1. (Color online) (a) Plot of the measured time-domain optical intensity data. Also shown are simulation results with and without phase ripple and the measured data after ripple correction. (b) Experimental setup. © 2008 Optical Society of America

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Fig. 2. (Color online) (a) Measured loss versus wavelength for the CFBG. (b) Vendor supplied CFBG total phase and phase ripple. Note the scale difference of 3 orders of magnitude.

A comparison with simulation illustrates that the sharp structure is due to phase ripple. To provide input to the simulation, the optical power spectral density (PSD), plotted in the inset to Fig. 1(a), is acquired at the system output using an optical spectrum analyzer. In a spectral measurement, the PSD is unaffected by phase ripple. The smooth PSD suggests that the sharp structure is due to the spectral phase. Through digital processing, this spectrum is mapped to a slowly varying envelope to represent the frequency-domain electric field. Note that this output envelope, stripped of phase information, is used as the input to the simulation. This field is then multiplied by the manufacturer-supplied CFBG transfer function as determined by an optical vector analyzer (Luna Technologies) with spectral resolution of approximately 3 pm. The phase response of the optical filter is measured using an optical vector analyzer, and its transfer function is multiplied by the field. The signal is then inverse Fourier transformed to the time domain and multiplied by its complex conjugate. As a control, the identical simulation is performed with an ideal quadratic phase response in place of the grating transfer function. Both results are plotted in Fig. 1(a). The agreement between measurement and simulation demonstrates that the large modulation depth and high-frequency distortions are due to phase ripple. Although the peak-topeak ripple is significantly smaller than the total phase delivered by the grating, this phase ripple is more than a small perturbation. The severity of the problem arises from the mixing of time-domain components of the field, each with their own base-band phase. The square-law detection then amplifies the time-delayed pieces of the field through the mixing process akin to heterodyne detection. Before the photodiode, however, this is a linear system. If the complex field were known, the removal of phase ripple would be trivial. The key to our technique is that the complex field is known from the system characteristics that create the distortion and the measured optical intensity. The impulse response of the CFBG is static, and the input pulse is an excellent approximation of an optical impulse. Thus the time-domain phase can be found directly from simulation. The simulated phase is combined with the square root of the measured intensity to obtain the complex field. The intensity is linearly interpolated to provide for much finer sampling to generate the 3 nm bandwidth spectral support. The field is then

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Fourier transformed, divided by the transfer function of the phase ripple, and inverse transformed back to the time domain. Multiplication of this signal by its complex conjugate gives the final corrected signal. This technique is applied to the measured data of Fig. 1(a) to obtain the corrected signal plotted as the top curve. The critical alignment of intensity and phase was facilitated by the sharp features provided by the ripple itself. The corrected signal shows no sharp features, and the noise is on the order of the oscilloscope’s least-significant bit. Because this process yields the complex field, we may calculate the optical spectrum strictly from time-domain intensity measurements. We note that the simulated phase is a well-behaved parameter that has minimal dependence on the spectrum used as the simulation input. The utility of this technique extends far beyond the spectroscopy application described above. It can be applied to systems with arbitrary dispersion and to highly modulated signals. To demonstrate this potential, we apply this algorithm to a photonic timestretch system [5] employing a CFBG as its second dispersive element. Photonic time-stretch technology relies on a wavelength-to-time mapping that allows for temporal stretching of an rf signal. The system, depicted in Fig. 3(b), begins with a short pulse identical to that discussed above. The first dispersive element consists of two DCF segments with a total dispersion of 1000 ps/ nm (Avanex) that map the various wavelengths within the pulse to time. The rf signal, a 3 GHz sine wave, is applied to the optical carrier by a zero-chirp Mach–Zehnder modulator (EOSpace) with a modulation depth of 0.28. By using the complimentary modulator outputs, the pulse envelope and relative intensity noise can be removed through postprocessing. This is achieved by delaying one output by half of the laser pulse period and recombining the two arms with a directional coupler. This step loses more than 50% of the optical power and is necessary, because only a single CFBG was available. Finally, the modulated signal traverses the CFBG and is detected and digitized with the same components described above. This combination of dispersive elements yields a stretch factor of 3, and the 1 ns rf period is evident from the real-time data plotted in Fig. 3(a).

Fig. 3. (Color online) (a) Representative single-shot experimental data from a single modulated optical pulse. The pulses have been temporally aligned in postprocessing. Also shown is the recovered signal after ripple correction and filtering, with the double-sided arrow indicating the region of interest. (b) Schematic of the experimental setup.

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The stretched rf signal is recovered by taking the difference of the two modulator outputs divided by their sum, which removes the pulse envelope. This signal was then low-pass filtered at 3.3 GHz (equivalent to 10 GHz in the unstretched regime). Modulator wavelength dependencies were corrected, and finally a sine-fit test over six rf periods was used to quantify the signal integrity through the signal to noise and distortion ratio (SNDR). 1 ␮s of real-time data was recorded (34 laser pulses), yielding a mean SNDR of 26.7 dB. To test the algorithm, we then used the same data and processing with ripple compensation added to the first step. In this case, we first multiply the square root of the interpolated measured intensity by the complex phase factor from simulation. We then Fourier transform the signal and multiply it by the inverse transfer function of the CFBG, thereby backpropagating the signal to the modulator. The pulse envelope is removed and processed as above, moving the low-pass filter point to 10 GHz. After the addition of ripple compensation a mean SNDR of 36.0 dB was obtained, a 9.3 dB improvement over the uncorrected signal. Although the modulation is “chirp-free,” propagation of the modulated signal through a dispersive medium creates phase modulation. This perturbation on the total phase was corrected by first solving for the rf signal. The system was then simulated to find the output phase with this modulator input. With this improved phase, the rf signal was again recovered. This generated a mean SNDR of 36.4 dB, a total improvement of 9.7 dB over the uncorrected signal. The issue of temporal alignment provides a strong constraint on ripple correction. In the data processing above, many sharp ripple artifacts were used to align the signal to within 10 ps of the simulated phase. This timing was further verified by a drop in SNDR of approximately 0.05 dB with a 10 ps timing shift, half the sampling period of our oscilloscope. A 100 ps shift induces a drop of 1.75 dB. As a point of reference, the identical experiment is conducted with 2000 ps/ nm DCF replacing the CFBG. The fiber amplifier gain is increased by 1.3 dB to overcome DCF losses. Again 1 ␮s of data is collected and analyzed, yielding a mean SNDR ratio of 31.7 dB. Astoundingly, this device, which has no phase ripple, achieves a SNDR that is 4.7 dB worse than that of the corrected CFBG signal. These results are verified through extensive simulation. In this case, the systems both with and without phase ripple are simulated, and zero-mean white Gaussian noise is added to the output intensity. We equate the DCF case with the ideal quadratic phase response. The noise amplitude was fixed such that the mean SNDR matches that of the experiment using the DCF as the second stage. This noise amplitude is the same for both the DCF and the CFBG cases. Averaging over 200 laser pulses, the simulation gave a mean SNDR of 31.8 dB for the perfect dispersive element, 25.4 dB for the CFBG with uncorrected ripple, and 34.3 dB for the CFBG with ripple

correction. The simulations provide further insight into the origin of the corrected CFBG’s large gain in signal integrity. The act of backpropagating the signal to the modulator is seen to shape the additive noise out of band. This algorithm can be applied identically to the DCF case, although temporal alignment is complicated by the lack of alignment marks. To stretch rf signals beyond the analog limit of the detector, the system of Fig. 3(b) is modified by removing the second DCF component. This reduced the dispersion of the first stage to 330 ps/ nm, increasing the stretch factor to 7.1. The frequency of the rf tone is increased to 20 GHz. 1 ␮s stretches of real-time data were recorded for both the case of the CFBG and the case of the DCF as the second dispersive stage. Processing techniques identical to the previous experiment are used, and a sine-fit test is taken over six rf periods. Without ripple correction, the CFBG case yields a 23.3 dB mean SNDR, and the correction algorithm increases this to 34.3 dB. For the DCF case, the fiber amplifier gain was increased by 2.4 dB and yielded a 20.6 dB mean SNDR. Once again, the CFBG system with ripple correction shows superior signal fidelity to the other cases. We demonstrate, through simulation and experiment, a digital postprocessing routine that removes the effects of phase ripple. While standard digital image reconstruction protocols lead to noise amplification [13], our technique corrects ripple while mitigating noise. Thus this algorithm promotes CFBGs from the dominant source of signal degradation to the preferred dispersive device. This work was supported by DARPA under SSC San Diego grant N66001-07-1-2007. References 1. D. R. Solli, J. Chou, and B. Jalali, Nat. Photonics 2, 48 (2007). 2. R. E. Saperstein, N. Alic, S. Zamek, K. Ikeda, B. Slutsky, and Y. Fainman, Opt. Express 15, 15464 (2007). 3. Y. Park, T.-J. Ahn, J.-C. Kieffer, and J. Azaña, J. Phys. Earth 15, 4598 (2007). 4. J. Chou, Y. Han, and B. Jalali, IEEE Photon. Technol. Lett. 16, 1140 (2004). 5. Y. Han and B. Jalali, J. Lightwave Technol. 21, 3085 (2003). 6. B. Bortnik, I. Y. Poberezhskiy, J. Chou, B. Jalali, and H. R. Fetterman, J. Lightwave Technol. 24, 2752 (2006). 7. R. E. Saperstein, N. Alic´, D. Panasenko, R. Rokitski, and Y. Fainman, J. Opt. Soc. Am. B 22, 2427 (2005). 8. J. Stigwall and S. Galt, J. Lightwave Technol. 25, 3017 (2007). 9. V. S. Ilchenko, M. Mohageg, A. A. Savchenkov, A. B. Matsko, and L. Maleki, Opt. Express 15, 5866 (2007). 10. R. Kashyap, Fiber Bragg Gratings (Academic, 1999). 11. C. Scheerer, C. Glingener, G. Fischer, M. Bohn, and W. Rosenkranz, in International Conference on Transparent Optical Networks (IEEE, 1999), p. 336. 12. M. Sumetsky, B. Eggleton, and C. M. de Sterke, Opt. Express 10, 332 (2002). 13. R. C. Puetter, T. R. Gosnell, and A. Yahil, Annu. Rev. Astron. Astrophys. 43, 139 (2005).