## Self-validating technique for the measurement of the linewidth enhancement factor in semiconductor lasers |

Optics Express, Vol. 20, Issue 5, pp. 4979-4987 (2012)

http://dx.doi.org/10.1364/OE.20.004979

Acrobat PDF (840 KB)

### Abstract

A new method for measuring the linewidth enhancement factor (α-parameter) of semiconductor lasers is proposed and discussed. The method itself provides an estimation of the measurement error, thus self-validating the entire procedure. The α-parameter is obtained from the temporal profile and the instantaneous frequency (chirp) of the pulses generated by gain switching. The time resolved chirp is measured with a polarization based optical differentiator. The accuracy of the obtained values of the α-parameter is estimated from the comparison between the directly measured pulse spectrum and the spectrum reconstructed from the chirp and the temporal profile of the pulse. The method is applied to a VCSEL and to a DFB laser emitting around 1550 nm at different temperatures, obtaining a measurement error lower than ± 8%.

© 2012 OSA

## 1. Introduction

1. C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. **18**(2), 259–264 (1982). [CrossRef]

1. C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. **18**(2), 259–264 (1982). [CrossRef]

*α = (4π/λ)dn/dN·(dG/dN)*, where

^{−1}*λ*is the emission wavelength and

*dn/dN*and

*dG/dN*are the derivatives of the refractive index,

*n*, and of the gain,

*G*, with respect to the carrier density,

*N*, respectively.

2. I. D. Henning and J. V. Collins, “Measurements of the semiconductor laser linewidth broadening factor,” Electron. Lett. **19**(22), 927–929 (1983). [CrossRef]

3. Z. Toffano, A. Destrez, C. Birocheau, and L. Hassine, “New linewidth enhancement determination method in semiconductor lasers based on spectrum analysis above and below threshold,” Electron. Lett. **28**(1), 9–11 (1992). [CrossRef]

4. C. Harder, K. Vahala, and A. Yariv, “Measurement of the linewidth enhancement factor alpha of semiconductor lasers,” Appl. Phys. Lett. **42**(4), 328–330 (1983). [CrossRef]

5. F. Devaux, Y. Sorel, and J. K. Kerdiles, “Simple measurement of fiber dispersion and of chirp parameter of intensity modulated light emitter,” J. Lightwave Technol. **11**(12), 1937–1940 (1993). [CrossRef]

6. M. Osinski, D. F. G. Gallagher, and I. H. White, “Measurement of linewidth broadening factor in gain-switched InGaAsP injection lasers by CHP method,” Electron. Lett. **21**(21), 981–982 (1985). [CrossRef]

8. P. Lazaridis, G. Debarge, and P. Gallion, “Time-bandwidth product of chirped sech(2) pulses: application to phase-amplitude-coupling factor measurement,” Opt. Lett. **20**(10), 1160–1162 (1995). [CrossRef] [PubMed]

9. R. Hui, A. Mecozzi, A. D'Ottavi, and P. Spano, “Novel measurement technique of alpha factor in DFB semiconductor lasers by injection locking,” Electron. Lett. **26**(14), 997–998 (1990). [CrossRef]

10. Y. Yu, G. Giuliani, and S. Donati, “Measurement of the linewidth enhancement factor of semiconductor lasers based on the optical feedback self-mixing effect,” IEEE Photon. Technol. Lett. **16**(4), 990–992 (2004). [CrossRef]

11. M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor lasers - an overview,” IEEE J. Quantum Electron. **23**(1), 9–29 (1987). [CrossRef]

13. T. Fordell and A. M. Lindberg, “Noise correlation, regenerative amplification, and the linewidth enhancement factor of a Vertical-Cavity Surface-Emitting Laser,” IEEE Photon. Technol. Lett. **20**(9), 667–669 (2008). [CrossRef]

12. T. Fordell and A. M. Lindberg, “Experiments on the linewidth-enhancement factor of a Vertical-Cavity Surface-Emitting Laser,” IEEE J. Quantum Electron. **43**(1), 6–15 (2007). [CrossRef]

13. T. Fordell and A. M. Lindberg, “Noise correlation, regenerative amplification, and the linewidth enhancement factor of a Vertical-Cavity Surface-Emitting Laser,” IEEE Photon. Technol. Lett. **20**(9), 667–669 (2008). [CrossRef]

14. K. Y. Lau, “Gain switching of semiconductor injection lasers,” Appl. Phys. Lett. **52**(4), 257–259 (1988). [CrossRef]

15. F. Li, Y. Park, and J. Azaña, “Complete temporal pulse characterization based on phase reconstruction using optical ultrafast differentiation (PROUD),” Opt. Lett. **32**(22), 3364–3366 (2007). [CrossRef] [PubMed]

16. A. Consoli, J. M. Tijero, and I. Esquivias, “Time resolved chirp measurements of gain switched semiconductor laser using a polarization based optical differentiator,” Opt. Express **19**(11), 10805–10812 (2011). [CrossRef] [PubMed]

## 2. Method description

*ν(t)*, (i.e. the chirp), can be expressed as a function of the output intensity,

*P(t)*by [17

17. R. Tucker, “High-speed modulation of semiconductor lasers,” J. Lightwave Technol. **3**(6), 1180–1192 (1985). [CrossRef]

*transient*chirp term, as it is dominant during the rising and falling edge of the optical pulse. The second term, proportional to

*P(t)*, is the

*adiabatic*chirp term and it takes into account the dependence of the laser frequency on the output power, due to the effect of the non linear gain compression on the threshold carrier density. Thus, if

*P(t)*and

*ν(t)*are known, fitting to Eq. (1) allows the extraction of α and κ. Under gain-switching operation, only the first spike of the relaxation oscillations is excited and therefore the adiabatic term in Eq. (1) can be neglected, since the steady state of the output is not reached [17

17. R. Tucker, “High-speed modulation of semiconductor lasers,” J. Lightwave Technol. **3**(6), 1180–1192 (1985). [CrossRef]

*ν(t)*is proportional to

*(P(t))*and the

^{−1}dP(t)/dt*α*-parameter can be obtained from a simple linear fit.

*ν(t)*as a function of

*(P(t))*is clearly shown in Fig. 1 and the value of the α-parameter used in simulations is recovered from the slope of the curve after linear fitting.

^{−1}dP(t)/dt*P(t)*and

*ν(t)*for the comparison with the independently measured spectrum, by the following simple procedure. First, the pulse optical phase,

*φ(t)*, is obtained by numerical integration of the measured instantaneous frequency

*ν(t)*, as:

*φ(t)*=

*∫2πν(t)dt + φ*, where

_{0}*φ*is a non relevant initial phase. The signal complex field

_{0}*E(t)*is then easily calculated as

*j*denotes the imaginary unit. Finally, the pulse spectral intensity is reconstructed as the squared magnitude of the Fourier transform of the signal complex field.

_{S}as:where

*I*and

_{m}(f)*I*are the measured and reconstructed pulse intensity spectra, respectively, and the integrals extent over the entire frequency range. As it will be described in detail in section 4, the experimental value of ε

_{r}(f)_{S}is a clear indication of the quality of the TRC measurements and therefore of the accuracy of the extracted α-parameter.

## 3. Experimental results

_{BIAS}and a Pulse Pattern Generator (Anritsu MP1808) providing a current pulse train. The lasers were temperature controlled with a Thermo Electric Cooler and the measurements were performed at different temperatures between 20° C and 40° C. The driving conditions were chosen in order to excite only the first spike of the relaxation oscillations. I

_{BIAS}was set close to the threshold current I

_{TH}(I

_{BIAS}= 1.1 I

_{TH}) and the electrical excitation pulse widths were 250 ps and 125 ps for the VCSEL and the DFB laser, respectively. More details on the experimental set-up and on the pulse properties of a similar GS VCSEL can be found in [19

19. A. Consoli, I. Esquivias, F. J. L. Hernandez, J. Mulet, and S. Balle, “Characterization of gain-switched pulses from 1.55-µm VCSEL,” IEEE Photon. Technol. Lett. **22**(11), 772–774 (2010). [CrossRef]

15. F. Li, Y. Park, and J. Azaña, “Complete temporal pulse characterization based on phase reconstruction using optical ultrafast differentiation (PROUD),” Opt. Lett. **32**(22), 3364–3366 (2007). [CrossRef] [PubMed]

*P(t)*, and the differentiated,

*Q(t)*, pulse intensity temporal profiles by applying the following expression:where

*A*is the differentiator slope in the frequency domain at the input signal frequency and

*Δf*is the frequency difference between the differentiator resonance frequency and the input signal frequency (see [15

15. F. Li, Y. Park, and J. Azaña, “Complete temporal pulse characterization based on phase reconstruction using optical ultrafast differentiation (PROUD),” Opt. Lett. **32**(22), 3364–3366 (2007). [CrossRef] [PubMed]

16. A. Consoli, J. M. Tijero, and I. Esquivias, “Time resolved chirp measurements of gain switched semiconductor laser using a polarization based optical differentiator,” Opt. Express **19**(11), 10805–10812 (2011). [CrossRef] [PubMed]

*P(t)*and

*Q(t)*were measured with a fast photodiode and an oscilloscope (20 GHz bandwidth). The signal is amplified with an Erbium Doped Fiber Amplifier (EDFA) before entering the interferometer. The spectra of the original and differentiated pulses are measured with an Optical Spectrum Analyzer (OSA), Ando AQ-6315A, and the ratio between the two spectra is used to determine the differentiator slope

*A*.

*P(t)*and the chirp

*ν(t)*as obtained with the VCSEL at 25 °C. The expected negative frequency variation during the pulse, very similar to the theoretical results in Fig. 1, is clearly observed. The predicted linear dependence of the chirp

*ν(t)*versus

*(P(t))*is also apparent in this figure and confirms the assumption that the adiabatic chirp term of Eq. (1) can be neglected. The linear fitting yields a value of 5.7 for the α-parameter in this case.

^{−1}dP(t)/dt_{S}in this case is 5.3%.

_{S}as it is discussed in section 4.

11. M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor lasers - an overview,” IEEE J. Quantum Electron. **23**(1), 9–29 (1987). [CrossRef]

_{BIAS}was varied accordingly (I

_{BIAS}= 1.1·I

_{TH}), while the electrical pulse amplitude, duration and duty cycle were not changed. Ten measurements were performed at each temperature for each device. Figure 4 shows the average values of the α-parameter obtained at each temperature for the DFB and for the VCSEL. The obtained values range between 3.5 and 3.7, and between 5.7 and 6, for the DFB and the VCSEL, respectively. Therefore, within the experimental error the values of the α-parameter are constant as expected. The reproducibility of the results was good, as the standard deviation of the measurements was better than 4% in all the cases. The error bars shown in Fig. 4 were obtained from the spectral error ε

_{S}of each measurement, as explained in section 4, combined with the standard deviation of the ten measurements performed at each temperature.

## 4. Error estimation

11. M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor lasers - an overview,” IEEE J. Quantum Electron. **23**(1), 9–29 (1987). [CrossRef]

13. T. Fordell and A. M. Lindberg, “Noise correlation, regenerative amplification, and the linewidth enhancement factor of a Vertical-Cavity Surface-Emitting Laser,” IEEE Photon. Technol. Lett. **20**(9), 667–669 (2008). [CrossRef]

12. T. Fordell and A. M. Lindberg, “Experiments on the linewidth-enhancement factor of a Vertical-Cavity Surface-Emitting Laser,” IEEE J. Quantum Electron. **43**(1), 6–15 (2007). [CrossRef]

**23**(1), 9–29 (1987). [CrossRef]

_{S}, and to relate ε

_{S}with the α-parameter error by means of numerical simulations. We identify three main error sources in our implementation: (i) the jitter and noise in the temporal profiles of the original and differentiated pulses, (ii) the uncertainty in the determination of the differentiator slope

*A*and (iii) the non idealities of the optical differentiator. For the pulse durations considered in the experiments, the finite bandwidth of the photodiode and oscilloscope (20 GHz) does not affect the error calculation, as confirmed by simulations.

*P(t)*in Eq. (1). In consequence, around 400 temporal traces have been acquired and averaged at each measurement, in order to minimize the contribution of stochastic fluctuations at the turn-on of the GS laser and of the detector noise.

*A*and

*Δf*are experimentally obtained from the original and differentiated pulse intensity spectra at each measurement. Small fluctuations of the laser spectrum or of the differentiator transfer function introduce errors in the estimation of

*A*and

*Δf*. On the other hand, the value of

*Δf*does not actually affect the α-parameter measurement, as

*Δf*contributes with a constant shift to the chirp

*ν(t)*and it does not modifies the slope of the plot of

*ν(t)*versus

*(P(t))*Thus, the uncertainty of the differentiator slope

^{−1}dP(t)/dt.*A*is considered as the main source of error in our experiments, and we consider that the fluctuations in the differentiator response give rise to the standard deviation of 4% in a set of measurements in nominally identical conditions previously commented.

20. F. Li, Y. Park, and J. Azaña, “Linear characterization of optical pulses with durations ranging from the picosecond to the nanosecond regime using ultrafast photonic differentiation,” J. Lightwave Technol. **27**(21), 4623–4633 (2009). [CrossRef]

*ν(t)*and consequently in ε

_{S}and in the α-parameter. The importance of this error is discussed further in the text.

_{S}with the α-parameter error the entire experimental technique was simulated. The initial optical pulse was obtained from the standard rate equations considering estimated laser parameters which provided pulses similar in shape and duration to those experimentally measured. The simulated pulses were filtered by the calculated frequency response of the birefringent interferometer, and then the chirp and the α-parameter were calculated using Eqs. (1) and (3). The spectral error was calculated from the comparison between the initial pulse spectrum and the pulse spectrum reconstructed from chirp, taking into account the spectral response of the OSA filter. The error in the α-parameter ε

_{α}in percent units is defined as:where α

_{RE}is the value of the α-parameter introduced in the rate equations of the laser and α

_{rec}is the recovered value after simulation of the entire measurement set-up.

*A*in the simulation results is illustrated in Fig. 5 , where ε

_{α}and ε

_{S}are shown as a function of

*A/A*, where

_{NOM}*A*is the value used in Eq. (3) to recover the chirp of the simulated pulse and

*A*is the nominal value extracted from the linearization of the interferometer transfer function. The spectral error ε

_{NOM}_{S}is shown for different values of the OSA filter bandwidth, while ε

_{α}is independent of the OSA bandwidth, as the α-parameter is obtained from the temporal intensity and chirp profiles. The minimum errors, obtained as expected for

*A/A*= 1, are greater than zero due to the non ideality of the implemented optical differentiator. For our particular polarization interferometer and pulse characteristics the minimum error in the α-parameter is around 1%.

_{NOM}_{α}increases almost linearly by increasing or decreasing the value of

*A*, as it can be easily understood from Eqs. (3) and (1). In our experimental conditions, the first term below the square root in the right hand side of exp. (3) is much greater than the second term, and therefore an increase of

*A*correspond to a decrease of the extracted chirp. Equation (1) shows that a decrease of the chirp corresponds to a decrease of the calculated (or measured) α-parameter. Similarly, a decrease of

*A*results in a value of the α-parameter higher than the original one.

_{S}becomes smaller, because the original and reconstructed spectra are increasingly smoothened. As it is shown in Fig. 5, the largest values of ε

_{S}are obtained in the case of a filter bandwidth of 0.1 pm, corresponding to the case in which the OSA resolution does not affect the calculation of ε

_{S}. For a large bandwidth of 1 nm, ε

_{S}is very small (~0%) in the entire range of

*A*, due to the high smoothing of the spectral features, resulting in an unacceptable error underestimation.

_{S}is slightly larger or equal to ε

_{α}in the considered range of

*A/A*, resulting in an upper bound to the errors obtained in the α-parameter measurements.

_{NOM}## 5. Conclusions

## Acknowledgments

## References and links

1. | C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. |

2. | I. D. Henning and J. V. Collins, “Measurements of the semiconductor laser linewidth broadening factor,” Electron. Lett. |

3. | Z. Toffano, A. Destrez, C. Birocheau, and L. Hassine, “New linewidth enhancement determination method in semiconductor lasers based on spectrum analysis above and below threshold,” Electron. Lett. |

4. | C. Harder, K. Vahala, and A. Yariv, “Measurement of the linewidth enhancement factor alpha of semiconductor lasers,” Appl. Phys. Lett. |

5. | F. Devaux, Y. Sorel, and J. K. Kerdiles, “Simple measurement of fiber dispersion and of chirp parameter of intensity modulated light emitter,” J. Lightwave Technol. |

6. | M. Osinski, D. F. G. Gallagher, and I. H. White, “Measurement of linewidth broadening factor in gain-switched InGaAsP injection lasers by CHP method,” Electron. Lett. |

7. | J. Jeong and Y. K. Park, “Accurate determination of transient chirp parameter in high speed digital lightwave transmitters,” Electron. Lett. |

8. | P. Lazaridis, G. Debarge, and P. Gallion, “Time-bandwidth product of chirped sech(2) pulses: application to phase-amplitude-coupling factor measurement,” Opt. Lett. |

9. | R. Hui, A. Mecozzi, A. D'Ottavi, and P. Spano, “Novel measurement technique of alpha factor in DFB semiconductor lasers by injection locking,” Electron. Lett. |

10. | Y. Yu, G. Giuliani, and S. Donati, “Measurement of the linewidth enhancement factor of semiconductor lasers based on the optical feedback self-mixing effect,” IEEE Photon. Technol. Lett. |

11. | M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor lasers - an overview,” IEEE J. Quantum Electron. |

12. | T. Fordell and A. M. Lindberg, “Experiments on the linewidth-enhancement factor of a Vertical-Cavity Surface-Emitting Laser,” IEEE J. Quantum Electron. |

13. | T. Fordell and A. M. Lindberg, “Noise correlation, regenerative amplification, and the linewidth enhancement factor of a Vertical-Cavity Surface-Emitting Laser,” IEEE Photon. Technol. Lett. |

14. | K. Y. Lau, “Gain switching of semiconductor injection lasers,” Appl. Phys. Lett. |

15. | F. Li, Y. Park, and J. Azaña, “Complete temporal pulse characterization based on phase reconstruction using optical ultrafast differentiation (PROUD),” Opt. Lett. |

16. | A. Consoli, J. M. Tijero, and I. Esquivias, “Time resolved chirp measurements of gain switched semiconductor laser using a polarization based optical differentiator,” Opt. Express |

17. | R. Tucker, “High-speed modulation of semiconductor lasers,” J. Lightwave Technol. |

18. | L. A. Coldren and S. W. Corzine, |

19. | A. Consoli, I. Esquivias, F. J. L. Hernandez, J. Mulet, and S. Balle, “Characterization of gain-switched pulses from 1.55-µm VCSEL,” IEEE Photon. Technol. Lett. |

20. | F. Li, Y. Park, and J. Azaña, “Linear characterization of optical pulses with durations ranging from the picosecond to the nanosecond regime using ultrafast photonic differentiation,” J. Lightwave Technol. |

**OCIS Codes**

(060.4510) Fiber optics and optical communications : Optical communications

(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology

(120.5050) Instrumentation, measurement, and metrology : Phase measurement

(140.5960) Lasers and laser optics : Semiconductor lasers

(250.7260) Optoelectronics : Vertical cavity surface emitting lasers

(140.3538) Lasers and laser optics : Lasers, pulsed

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: January 11, 2012

Revised Manuscript: February 2, 2012

Manuscript Accepted: February 7, 2012

Published: February 13, 2012

**Citation**

Antonio Consoli, Borja Bonilla, Jose Manuel G. Tijero, and Ignacio Esquivias, "Self-validating technique for the measurement of the linewidth enhancement factor in semiconductor lasers," Opt. Express **20**, 4979-4987 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-5-4979

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### References

- C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron.18(2), 259–264 (1982). [CrossRef]
- I. D. Henning and J. V. Collins, “Measurements of the semiconductor laser linewidth broadening factor,” Electron. Lett.19(22), 927–929 (1983). [CrossRef]
- Z. Toffano, A. Destrez, C. Birocheau, and L. Hassine, “New linewidth enhancement determination method in semiconductor lasers based on spectrum analysis above and below threshold,” Electron. Lett.28(1), 9–11 (1992). [CrossRef]
- C. Harder, K. Vahala, and A. Yariv, “Measurement of the linewidth enhancement factor alpha of semiconductor lasers,” Appl. Phys. Lett.42(4), 328–330 (1983). [CrossRef]
- F. Devaux, Y. Sorel, and J. K. Kerdiles, “Simple measurement of fiber dispersion and of chirp parameter of intensity modulated light emitter,” J. Lightwave Technol.11(12), 1937–1940 (1993). [CrossRef]
- M. Osinski, D. F. G. Gallagher, and I. H. White, “Measurement of linewidth broadening factor in gain-switched InGaAsP injection lasers by CHP method,” Electron. Lett.21(21), 981–982 (1985). [CrossRef]
- J. Jeong and Y. K. Park, “Accurate determination of transient chirp parameter in high speed digital lightwave transmitters,” Electron. Lett.33(7), 605–606 (1997). [CrossRef]
- P. Lazaridis, G. Debarge, and P. Gallion, “Time-bandwidth product of chirped sech(2) pulses: application to phase-amplitude-coupling factor measurement,” Opt. Lett.20(10), 1160–1162 (1995). [CrossRef] [PubMed]
- R. Hui, A. Mecozzi, A. D'Ottavi, and P. Spano, “Novel measurement technique of alpha factor in DFB semiconductor lasers by injection locking,” Electron. Lett.26(14), 997–998 (1990). [CrossRef]
- Y. Yu, G. Giuliani, and S. Donati, “Measurement of the linewidth enhancement factor of semiconductor lasers based on the optical feedback self-mixing effect,” IEEE Photon. Technol. Lett.16(4), 990–992 (2004). [CrossRef]
- M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor lasers - an overview,” IEEE J. Quantum Electron.23(1), 9–29 (1987). [CrossRef]
- T. Fordell and A. M. Lindberg, “Experiments on the linewidth-enhancement factor of a Vertical-Cavity Surface-Emitting Laser,” IEEE J. Quantum Electron.43(1), 6–15 (2007). [CrossRef]
- T. Fordell and A. M. Lindberg, “Noise correlation, regenerative amplification, and the linewidth enhancement factor of a Vertical-Cavity Surface-Emitting Laser,” IEEE Photon. Technol. Lett.20(9), 667–669 (2008). [CrossRef]
- K. Y. Lau, “Gain switching of semiconductor injection lasers,” Appl. Phys. Lett.52(4), 257–259 (1988). [CrossRef]
- F. Li, Y. Park, and J. Azaña, “Complete temporal pulse characterization based on phase reconstruction using optical ultrafast differentiation (PROUD),” Opt. Lett.32(22), 3364–3366 (2007). [CrossRef] [PubMed]
- A. Consoli, J. M. Tijero, and I. Esquivias, “Time resolved chirp measurements of gain switched semiconductor laser using a polarization based optical differentiator,” Opt. Express19(11), 10805–10812 (2011). [CrossRef] [PubMed]
- R. Tucker, “High-speed modulation of semiconductor lasers,” J. Lightwave Technol.3(6), 1180–1192 (1985). [CrossRef]
- L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (John Wiley & Sons, Inc., 1995).
- A. Consoli, I. Esquivias, F. J. L. Hernandez, J. Mulet, and S. Balle, “Characterization of gain-switched pulses from 1.55-µm VCSEL,” IEEE Photon. Technol. Lett.22(11), 772–774 (2010). [CrossRef]
- F. Li, Y. Park, and J. Azaña, “Linear characterization of optical pulses with durations ranging from the picosecond to the nanosecond regime using ultrafast photonic differentiation,” J. Lightwave Technol.27(21), 4623–4633 (2009). [CrossRef]

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