1704 IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 15, NO. 12, DECEMBER 2003 Raman Amplifier Model in Single-Mode Optical Fiber Idan Mandelbaum and Maxim Bolshtyansky Abstract—Equations to describe the process of Raman amplification in single-mode optical fiber and Stokes and anti-Stokes spontaneous emission generation with conservation of photon number are derived from first principles. The numerical simulation of Stokes and anti-Stokes spontaneous emission is in good agreement with experimental results. The importance of the sometimes-omitted anti-Stokes spontaneous emission terms in wavelength-division-multiplexed situations for optical signal-tonoise ratio is demonstrated by the simulation of light propagation through transmission fiber with and without Raman amplification. Index Terms—Optical fiber amplifiers, optical fiber communication, Raman scattering. figuration and counterconfiguration, which includes effects due to group velocity, Rayleigh backscattering fiber loss, as well as Stokes and anti-Stokes spontaneous emission. The equation conserves photon number in all modes and, thus, corrects the problem in  and , that gives negative power of ASEs in our simulations. In addition, the model is compared to experimental results to verify the validity of this approach. A derivation of the model is given in Section II; the experimental results are given in Section III. Finally, we conclude in Section IV. I. INTRODUCTION II. MODELING ISTRIBUTED Raman amplification can be found in both long-haul and ultralong-haul optical networks and is becoming an enabling technology for ultrawide-band communication, with over 100-nm flat bandwidth already reported . In addition, it can be used for increasing the bandwidth of Erbium-doped fiber amplifiers in hybrid systems . Raman amplifiers are being studied for use in metro applications as a flexible and simple way to supply gain anywhere in the communication spectrum. Thus, the accurate modeling of Raman amplification becomes extremely important for modern communication system design. The problem of Raman amplifier modeling has been discussed in scientific literature for quite some time. Stolen et al.  describe modeling of Stokes amplified spontaneous emission (ASE) and multiple Stokes shifts in single-mode fiber. Kidorf et al.  give detailed equations for a Raman amplifier model applied to single-mode optical fiber, in which an attempt has been made to conserve photon number. Achtenhagen et al.  verify the gain predicted by this model and add Rayleigh backscattering and wavelength scaling approximation to account for effective area. Perlin et al.  add anti-Stokes spontaneous emissions. Berntson et al.  give a similar model to , but gives correct the equation for spontaneous absorption of signal photons into the fundamental mode. This letter, to the best of our knowledge and for the first time, explicitly gives the Raman propagation equation in both cocon- Consider the Raman interaction between two arbitrary bands and with single-mode polarwith infinitesimal bandwidth ized light in both bands. The longer wavelength band will be referred to as the signal and the shorter wavelength band as the pump. Two processes describe the Raman interaction between the two wavelengths . The first process is responsible for Raman gain and Stokes spontaneous emission generation as well as pump depletion and is referred to here as the Raman emission process. The second process is responsible for antiStokes spontaneous emission generation and is referred to here as Raman absorption. Raman emission is the process whereby a pump photon generates a signal photon and a phonon at a frequency that is the difference between the pump and signal frequencies. This process has a rate of , where and are the photon densities at the pump and signal , respectively, is the frequencies within the bandwidth number of phonons at the frequency that is the difference between the pump and signal frequencies, is the annihilation is the creation operator with subscripts deoperator, and is a polarization as scribing pump, signal, and phonons. well as frequency-dependent emission probability coefficient, that will be related later to the Raman gain coefficient. Raman absorption is the process where a signal photon and a phonon combine to produce a photon at the pump frequency, with a rate of , where is some frequency and polarization-dependent absorption probability coefficient that will be related later to the Raman gain coefficient. The rate equation for the pump and signal photons can be written as D Manuscript received May 30, 2003; revised July 18, 2003. I. Mandelbaum was with the JDS Uniphase Corporation, Trenton, NJ 08638 USA. He is now with the Department of Electrical Engineering, Columbia University, New York, NY 10027 USA (e-mail: [email protected]). M. Bolshtyansky is with the JDS Uniphase Corporation, Trenton, NJ 08638 USA. Digital Object Identifier 10.1109/LPT.2003.819760 1041-1135/03$17.00 © 2003 IEEE (1) MANDELBAUM AND BOLSHTYANSKY: RAMAN AMPLIFIER MODEL IN SINGLE-MODE OPTICAL FIBER Using the values for and as propagation equation , this equation can be written (2) where is the group velocity of pump or signal, denoted by the subscript. Converting (2) from photon number to power density equations for the depolarized pump and signal 1705 As presented in (3) and (4), these terms represent only interaction between the two wavelength bands under consideration, thus, these equations cannot be used directly and should be integrated over all other wavelengths. The third term in (3) and (4) is responsible for the addition of spontaneously emitted photons at the signal and pump wavelength into the fundamental mode. Equations (3) and (4) are extended for the multiwavelength case and combined to form one general equation, in which multiple channels with optical bandwidths are included in the codirection and counterdirection. Here, the channels should cover all the bandwidth under investigation, which includes the bands where only ASE is generated and no signals are present (3) (4) where is Planck’s constant, is the Boltzmann constant, is the temperature, and are the pump and signal frequencies, are the Raman gain and the fiber respectively, and and effective areas, respectively, which are defined in . In the notation of this letter, is always taken to be the highest of the two frequencies. The summation over in (3) and (4) represents the spontaneous emission into all modes of the fiber, including radiation modes or a continuous spectrum that is not supported by the fiber, for which case the summation becomes an integration. Each mode is considered to have two polarization states and denotes optical power within bandwidth in both states, with unit of watts per hertz in SI. In the derivation of (3) and (4), it is assumed that 1) the phonons obey the Bose–Einstein distribution and are local, because at optical frequencies they cannot efficiently propagate through glass; 2) spontaneous events occur in two polarization states, thus additional factors of two for the . The second and third term in (3) and (4); and 3) , the Raman last assumption is justified because if gain becomes temperature-dependent. In , it is shown that the Raman gain is independent of temperature between 77 K and 300 K, which justifies the assumption. Some of the terms in (3) and (4) are the result of photon number conservation. These are the second terms in (3) and (4). The contribution due to conservation of photon number can be seen as a temperature and wavelength-dependent loss, but it is not power-dependent. This contribution is the loss present in the fiber due to the conversion of signal and pump photons into spontaneously emitted Raman frequency shifted photons. (5) where the subscript represents the th wavelength, the superscripts of and represent the codirection and counterdirection, respectively, and and are the wavelength-dependent loss and Rayleigh backscattering coefficients, respectively . In (5), includes the loss due to Raman emissions into all modes, as is usually measured, which means the corresponding terms in (4) are included in . If needed, the sums over the frequencies can be converted into integrals for a continuous spectrum. III. RESULTS In order to check the validity of (5), the fiber loss and Raman gain, as well as the Rayleigh backscattering coefficient are measured for 13-km Lucent TrueWave RS transmission fiber at room temperature. These parameters are directly used in the simulation, and have not been adjusted to best fit experimental data. The fiber is then pumped with 13 mW at 1560 nm. The comparison of the simulated and experimental results is shown in Fig. 1. Extremely good agreement between experiment and simulation can be seen. The model, which does not account for anti-Stokes spontaneous emission, would have no spontaneous emission for wavelength shorter than 1560 nm and, thereby, not agree with the experimental results. The values of the group velocity used were obtained from group index and dispersion data supplied with the fiber. 1706 Fig. 1. Simulation versus measurement for ASE generated by 1560-nm signal at 300 K. IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 15, NO. 12, DECEMBER 2003 Fig. 3. Comparison of OSNR between models with anti-Stokes and without anti-Stokes spontaneous emissions in a system with 51-channel C -band in 100-km SMF-28 Fiber at 300 K. 1529.55 nm. It is interesting to note that, according to our equations, when the temperature is 0 K, there is no anti-Stokes spontaneous emissions generated, and therefore, the model with and without anti-Stokes give the same results. IV. CONCLUSION Fig. 2. Comparison of OSNR between models with anti-Stokes and without anti-Stokes spontaneous emissions in a system with two pumps at 1430 and 1454 copropagating with 51 channels. C -band in 100-km SMF-28 Fiber at 300 K. The inset shows the ON–OFF gain of the system. We have presented a model for Raman amplifiers, which includes Raman emission and absorption processes, effects due to group velocity, and that preserves photon number. In addition, fiber loss and Rayleigh backscattering have been added and the model has been extended to support multiple pump and signal wavelengths at both copropagating and counterpropagating directions. We also experimentally demonstrated good agreement of the model and the experiment in describing spontaneous emissions at anti-Stokes frequency shift. The importance of the correct accounting of the anti-Stokes ASE for OSNR calculation is also demonstrated. REFERENCES In order to demonstrate the importance of anti-Stokes spontaneous emission in system applications, the optical signal-to-noise ratio (OSNR) is calculated in pumped and un-pumped configuration. Using (5), and turning ON and OFF the anti-Stokes spontaneous emission term, OSNR for a 100-km span of Corning SMF28 with 51 channels in the -band (1529.55–1569.59 nm) is compared in pumped and un-pumped configurations. The total input signal power is 20 dBm with flat channel distribution. Input OSNR of 100 dB is used, with a detector bandwidth of 100 GHz. For the pumped case, a pair of copropagating pumps at 1430 and 1454 nm with powers 300 and 262.5 mW, respectively, is used in order to achieve a reasonably flat spectrum at the output. The ON–OFF gain, for this configuration, is shown in the inset of Fig. 2. Comparing the simulated results for the two different OSNRs at room temperature, which correspond to the different models in the pumped configuration, is shown in Fig. 2. The figure shows 2.17-dB discrepancy in the OSNR for the lowest signal wavelength used, 1529.55 nm. The difference is much more significant for the case where there is no pump and only the signals are present, as shown in Fig. 3. In that case, a 20-dB difference between the two noise figures can be observed at  H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijaona, “Pump interactions in a 100-nm bandwidth Raman amplifier,” IEEE Photon. Technol. Lett., vol. 11, pp. 530–532, May 1999.  M. Bolshtyansky, J. DeMarco, and P. Wysocki, “Flat, adjustable hybrid optical amplifier for 1610 nm–1640 nm band,” in Tech. Dig. OFC 2002, Anaheim, CA, Paper ThJ5.  R. H. Stolen, C. Lee, and R. K. Jain, “Development of the stimulated Raman spectrum in single-mode silica fibers,” J. Opt. Soc. Amer. B, vol. 1, pp. 652–657, 1984.  M. Achtenhagen, T. G. Chang, B. Nyman, and A. Hardy, “Analysis of a multiple-pump Raman amplifier,” Appl. Phys. Lett., vol. 78, pp. 1322–1324, 2001.  V. E. Perlin and H. G. Winful, “On distributed Raman amplification for ultrabroad-band long-haul WDM systems,” J. Lightwave Technol., vol. 20, pp. 409–416, Mar. 2002.  A. Berntson, S. Popov, E. Vanin, G. Jacobsen, and J. Karlsson, “Polarization dependence and gain tilt of Raman amplifiers for WDM systems,” in Proc. OFC 2001, Baltimore, MD, Paper MI2-1.  A. Yariv, Quantum Electronics, 3rd ed. New York: Wiley, 1989.  I. Flammer, C. Martinelli, and P. Guenot, “Raman gain prediction in germano-silicate single mode fibers,” in Proc. OAA 2002, Vancouver, Canada, Paper OMC2.  S. A. E. Lewis, S. V. Chernikov, and J. R. Taylor, “Temperature-dependent gain and noise in fiber Raman amplifiers,” Opt. Lett., vol. 24, pp. 1823–1825, 1999.  P. B. Hansen, L. Eskildsen, A. J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiGiovanni, “Rayleigh scattering limitations in distributed Raman pre-amplifiers,” IEEE Photon. Technol. Lett., vol. 10, pp. 159–161, Jan. 1998.