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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 8 — Apr. 17, 2006
  • pp: 3427–3432
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A novel algorithm for a multi-cavity Raman fiber laser

Junhe Zhou, Jianping Chen, Xinwan Li, Guiling Wu, Wenning Jiang, Changhai Shi, and Yiping Wang  »View Author Affiliations


Optics Express, Vol. 14, Issue 8, pp. 3427-3432 (2006)
http://dx.doi.org/10.1364/OE.14.003427


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Abstract

In this paper a Raman Fiber Lasers (RFLs) with several embedded cavities are studied. A novel algorithm is proposed to solve the coupled equations describing the optical power evolution in a RFL. By using some invariant constants as the boundary condition at the output end, the problem of solving ordinary differential equations (ODEs) with guessing boundary value is translated into a two-boundary-condition ODE problem. The algorithm is based on Newton-Raphson method and proved rather fast and stable. Quantitative analysis is performed based on the algorithm.

© 2006 Optical Society of America

1. Introduction

2. Numerical model

The power evolution of the pump and Stokes waves in the optical fiber can be characterized by the following coupled equations [11

11. G. P. Agrawal, Nonlinear Fiber Optics, (Academic Press, New York,2001).

]:

±Pi±z=j=1,jing(νj,νi)Pi±(Pj++Pj)αiPi±
(1)

with the boundary condition of:

P1+(0)=Pp
P1(L)=RL1P1+(L)
Pi+(0)=R0iPi(0)Pi(L)=RLiPi+(L)(i=2n)

where n is the number of waves including the pump wave and the Stokes wave, Pi and Pj are the powers of the ith and jth wave propagating along the fiber, +/- stands for the wave forward/backward propagation, g(νi ,νj ) is the Raman gain coefficient between frequency νi and frequency νj :

g(νi,νj)={gr(νiνj)2Aeff(νi>νj)(νjνi)gr(νjνi)2Aeff(νi<νj)
(2)

where A eff is the effective area of the fiber, factor 2 stands for the consideration of the polarization effect, gr(νi - νj ) is the Raman gain spectrum measure at the frequency of νi which has a peak at the frequency of 14T Hz, R is the reflectivity of corresponding mirrors, subscript 0 and L stand for the input and output ends.

We did not include the amplified spontaneous emission (ASE) and the Raleigh backward scattering in Eq. (1). They may slightly affect the output power near the threshold; however, their effect is negligible when the input pump power is far beyond the threshold power.

From Eq. (1), we may easily obtain the relation of Pi+ Pi = Ci , where Ci is a constant along the z axis. By dealing with the boundary condition and resorting to Pi+ Pi = Ci , one can obtain

(P1(L))2RL1P1(0)=Pp
Pi+(L)Pi+(0)=1R0iRLii=2n
(3)

Since Pi+(0) and Pi(0) are correlated, there are only n unknown inputs,Pi+(0)(i = 2⋯n) and Pi(0). We treat(P1(L))2RL1P1(0)and Pi+(L)Pi+(0)(i=2n) as outputs. Now the problem is to find suitable input values that lead to corresponding outputs satisfying Eq. (3). To clarify the fact, we rewrite the mathematical relation mentioned above into the following equations:

output(L)=F(P(0))
(4)

where:

output(L)=((P1(L))2RL1P1(0)P2+(L)P2+(0)Pn+(L)Pn+(0))
P(0)=(Pi(0)P2+(0)Pn+(0))

the target output is:

tragetoutput(L)=(Pp1R02RL21R0nRLn)
(5)

The New-Raphson method can be used to obtain the solution that gives the target output with assistance of the Jacobi matrix. The Jacobi matrix J is defined as J(i,j)=outputiPj.

According to its definition, it can be obtained by the following means:

First, the output of the steady state should be stored. Then one should increase the ith element of the input Pi (0) by a small disturbance ΔPi (0) and obtain the output. By subtracting the state output from the output and dividing the result by ΔPi (0), the ith column of the matrix is obtained.

The procedure of Newton-Raphson method is as follows:

  1. Initial guessing values are given at the input.
  2. The coupled equation is integrated by Runge-Kouta method, and the error between the output and the target output Δoutput(L) is obtained.
  3. The Jacobi matrix is calculated and the input is updated by adding ΔP(0) = J-1Δoutput(L)
  4. Iteration stops if the error is below a small threshold, else, go to procedure A.

3. Simulation results and discussion

We have calculated a Raman laser with three embedded cavities. The gain media is a piece of dispersion shifted fiber (DSF) with the length of 500m. The structure is illustrated in Fig. 1. We choose the pump wavelength at 1318.8 nm, so that the high power Yag laser can provide enough pump power at this wavelength [12

12. Yoko Inoue and Shuichi Fujikawa, “Diode-pumped Nd:YAG laser producing 122W CW power at 1.319μm,” IEEE J.Quantum Electron. 36, 751–756 (2000). [CrossRef]

]. The three cavities can be formed by six fiber Brag gratings (FBGs) and the pump power is reflected back by another FBG at 1318.8nm to have it fully utilized.

Fig. 1. The structure for RFL with several embedded cavities.

The pump power and the lasing power evolution along the fiber are illustrated in Fig. 2. From the figure we can see that the boundary condition of Eq. (1) is satisfied automatically. The error is less then 1e-4. The calculation of the solution takes less than one second using a conventional personal computer, i.e., Intel Pentium 4, 2.0 GHz. The powers of the Stokes waves at 1405nm and 1485nm maintain almost constant during the propagation. This is caused by two reasons. Firstly, the net gains of the forward and backward propagated waves at the two wavelengths are close to 0 dB, hence, the powers at the input end and the output end are almost equal. Secondly, the fiber length is rather short so that the pump power is not fully depleted.

Fig. 2. Pump power and lasing power evolution along the fiber.

Figure 3 shows the relationship between the input pump and the output lasing power. It can be seen that there is almost linear relationship between the input pump power and the output lasing power. The threshold pump power is about 0.6W. It is worth noting that the algorithm proposed above is only suitable for calculation when the pump power is beyond the threshold power, i.e., when there exists lasing power. Else, the calculated power will be negative. One may simply adjust the lasing power to zero when the negative power value occurs.

Fig. 3. The output lasing power at 1580nm versus input pump power at 1318.8nm.

In Fig. 4, the reflective coefficient at the lasing wavelength at the output end varies and so does the output lasing power. The pump power is fixed to be 1W. It can be seen that the laser reaches the maximum output when the reflectivity is around 75%.

Fig. 4. The output lasing power versus reflectivity at 1580nm.

4. Conclusion

We theoretically investigated the Raman fiber laser with several embedded cavities. Using some simple relations, we obtain some constants at the output end and therefore turn the ODEs with unknown values at two boundaries into ODEs with definite boundary conditions and no additional unknown constants. Based on the Newton-Raphson method, the coupled equations are solved. Our results show that the algorithm is stable and efficient.

Acknowledgments

This work is partially supported by NSFC (ID: 60377013, 90204006, 60507013), Ministry of Education, China (ID:20030248035) and STCSM(ID: 036105009)

References and links

1.

S. Namiki and Y. Emori “Ultrabroad-band Raman amplifiers pumped and gain-equalized by wavelength-division-multiplexed high-power laser diodes,” IEEE J. Sel. Top. Quantum Electron. 7, 3–16 (2001). [CrossRef]

2.

J. AuYeung and A. Yariv, “Theory of CW Raman oscillation in optical fibers,” J. Opt. Soc. Amer. 69, 803–807)(1979). [CrossRef]

3.

P. N. Kean, B. D. Sinclair, K. Smith, W. Sibbett, C. J. Rowe, and D. C. J. Reid, “Experimental evaluation of a fibre Raman oscillator having fibre grating reflectors,” J. Mod. Opt. 35, 397–406 (1988). [CrossRef]

4.

M. Rini, I. Christiani, and V. Degiorgio, “Numerical modeling and optimization of cascaded Raman fiber lasers,” IEEE J. Quantum Electron. 36, 1117–1122 (2000). [CrossRef]

5.

N. Kurukitkoson, H. Sugahara, S. K. Tusitsyn, O. N. Egorova, A. S. Kurkov, V. M. Paramonov, and E. M. Dianov, “Optimization of two stage Raman converter based on phosphosilicate core fiber: Modeling and experiment,” Electron. Lett. 37, 1281–1283 (2001). [CrossRef]

6.

Michael Krause and Hagen Renner, “Theory and design of double-cavity Raman Fiber Lasers,” IEEE J. Lightwave Technol. 23, 2474–2483 (2005). [CrossRef]

7.

Bumki Min, Won Jae Lee, and Namkyoo Park. “Efficient formulation of Raman amplifier propagation equations with average power analysis,” IEEE Photon. Technol. Lett. 12, 392–394 (2002).

8.

Xueming Liu, H. Y zhang, and Y. L Guo. “A novel method for Raman amplifer propagation equations,” IEEE Photon. Technol. Lett. 15, 392–394 (2003). [CrossRef]

9.

Xueming Liu and B. Lee, “A fast and stable method for Raman amplifier propagation equations,” Opt. Express. 11, 2163–2176 (2003). [CrossRef] [PubMed]

10.

F. Leplingard, C. Martinelli, S. Borne, L. Lorcy, D. Bayart, F. Castella, P. Chartier, and E. Faou, “Modeling of multiwavelength Raman fiber lasers using a new and fast algorithm,” IEEE Photon. Technol. Lett. 16, 2601–2603 (2004). [CrossRef]

11.

G. P. Agrawal, Nonlinear Fiber Optics, (Academic Press, New York,2001).

12.

Yoko Inoue and Shuichi Fujikawa, “Diode-pumped Nd:YAG laser producing 122W CW power at 1.319μm,” IEEE J.Quantum Electron. 36, 751–756 (2000). [CrossRef]

OCIS Codes
(140.0140) Lasers and laser optics : Lasers and laser optics
(190.5650) Nonlinear optics : Raman effect

ToC Category:
Nonlinear Optics

History
Original Manuscript: January 3, 2006
Revised Manuscript: April 7, 2006
Manuscript Accepted: April 8, 2006
Published: April 17, 2006

Citation
Junhe Zhou, Jianping Chen, Xinwan Li, Guiling Wu, Wenning Jiang, Changhai Shi, and Yiping Wang, "A novel algorithm for a multi-cavity Raman fiber laser," Opt. Express 14, 3427-3432 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-8-3427


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References

  1. S. Namiki and Y. Emori "Ultrabroad-band Raman amplifiers pumped and gain-equalized by wavelength-division-multiplexed high-power laser diodes," IEEE J. Sel. Top. Quantum Electron. 7, 3-16 (2001). [CrossRef]
  2. J. AuYeung and A. Yariv, "Theory of CW Raman oscillation in optical fibers," J. Opt. Soc. Am. 69, 803-807 (1979). [CrossRef]
  3. P. N. Kean, B. D. Sinclair, K. Smith, W. Sibbett, C. J. Rowe, and D. C. J. Reid, "Experimental evaluation of a fibre Raman oscillator having fibre grating reflectors," J. Mod. Opt. 35, 397-406 (1988). [CrossRef]
  4. M. Rini, I. Christiani, and V. Degiorgio, "Numerical modeling and optimization of cascaded Raman fiber lasers," IEEE J. Quantum Electron. 36, 1117-1122 (2000). [CrossRef]
  5. N. Kurukitkoson, H. Sugahara, S. K. Tusitsyn, O. N. Egorova, A. S. Kurkov, V. M. Paramonov, and E. M. Dianov, "Optimization of two stage Raman converter based on phosphosilicate core fiber: Modeling and experiment," Electron. Lett. 37, 1281-1283 (2001). [CrossRef]
  6. M. Krause and H. Renner, "Theory and design of double-cavity Raman Fiber Lasers," J. Lightwave Technol. 23, 2474-2483 (2005). [CrossRef]
  7. B. Min, W. J. Lee, and N. Park. "Efficient formulation of Raman amplifier propagation equations with average power analysis," IEEE Photon. Technol. Lett. 12, 392-394 (2002).
  8. X. Liu, H. Y Zhang, and Y. L Guo. "A novel method for Raman amplifer propagation equations," IEEE Photon. Technol. Lett. 15, 392-394 (2003). [CrossRef]
  9. X. Liu and B. Lee, "A fast and stable method for Raman amplifier propagation equations," Opt. Express. 11, 2163-2176 (2003). [CrossRef] [PubMed]
  10. F. Leplingard, C. Martinelli, S. Borne, L. Lorcy, D. Bayart, F. Castella, P. Chartier, and E. Faou, "Modeling of multiwavelength Raman fiber lasers using a new and fast algorithm," IEEE Photon. Technol. Lett. 16, 2601-2603 (2004). [CrossRef]
  11. G. P. Agrawal, Nonlinear Fiber Optics, (Academic Press, New York, 2001).
  12. Y. Inoue and S. Fujikawa, "Diode-pumped Nd:YAG laser producing 122W CW power at 1.319μm," IEEE J.Quantum Electron. 36, 751-756 (2000). [CrossRef]

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