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Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 6 — Mar. 20, 2006
  • pp: 2398–2403
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Integral equation approach for the analysis of high-power semiconductor optical amplifiers

Young Jin Jung, Pilhan Kim, Jaehyoung Park, and Namkyoo Park  »View Author Affiliations


Optics Express, Vol. 14, Issue 6, pp. 2398-2403 (2006)
http://dx.doi.org/10.1364/OE.14.002398


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Abstract

For the first time, an integral equation approach for the numerical assessment of Semiconductor Optical Amplifiers (SOAs) is proposed. Performance comparisons between the suggested formulation and the traditional transfer matrix method are carried out in terms of the computation costs in solving the multi-wave mixing process in bidirectional, high-power SOAs. Computation efficiency improvement by more than an order of magnitude was observed with the proposed formulation, achieving better accuracy at equivalent spatial resolution.

© 2006 Optical Society of America

1. Introduction

Steady-state, frequency-domain analysis is one of the most powerful approaches in the precise assessment of nonlinear signal interactions (such as four-wave mixing) in Semiconductor Optical Amplifiers (SOAs). To find the solution of the given problem, traditional approaches rely on the coupled differential equations for the SOAs [1

1. C. Y. J Chu and H. Ghafouri-Shiraz, “Analysis of gain and saturation characteristics of a semiconductor laser optical amplifier using transfer matrices,” J. Lightwave Technol. 12, 1378–1386 (1994). [CrossRef]

, 2

2. S. L. Zhang and J. J. O’Reilly, “Modelling of four-wave-mixing wavelength conversion in a semiconductor laser amplifier,” Physical Modelling of Semiconductor Devices, IEE Colloquium on, 5/1-5/6 (1995).

, 3

3. H. Lee, H. Yoon, Y. Kim, and J. Jeong, “Theoretical study of frequency chirping and extinction ratio of wavelength-converted optical signals by XGM and XPM using SOA’s,” IEEE J. Quantum Electron. 35, 1213–1219 (1999). [CrossRef]

]. In the real implementation, the intensity / phase values of the waves are sequentially calculated along the propagation axis, with the corresponding carrier density distributions at each gain segments of the SOA. When a large number of signal waves or gain segments is necessary for the increased spectral resolution / convergence of the signal, much higher degree of calculation efforts are often required, occasionally with repetitive iteration procedures (e.g., for bi-directional SOAs [4

4. M. J. Connelly, “Wideband semiconductor optical amplifier steady-state numerical model,” IEEE J. Quantum Electron. 37, 439–447 (2001). [CrossRef]

]).

Based on coupled differential equations, other types of optical amplifiers (e.g., fiber Raman amplifier, FRA) [5

5. B. Min, W. J. Lee, and N. Park, “Efficient formulation of Raman amplifier propagation equations with average power analysis,” IEEE Photonics Technol. Lett. 12, 1486–1488 (2000). [CrossRef]

] traditionally have also experienced same fundamental difficulties in their numerical assessment - slow convergence and limited accuracy. Recently, a meaningful advance has been made for the assessment of FRA solution, drastically reducing the computation cost. Based on a novel integral/matrix formulation for the fiber Raman-amplifier problem, more than two orders of reduction in the computation cost was demonstrated; also enabling the successful application to the inverse problem of FRA gain design [6

6. N. Park, P. Kim, J. Park, and L. K. Choi, “Integral form expansion of fiber Raman amplifier problem,” Opt. Fiber Technol., 11, 111–130 (2005). [CrossRef]

, 7

7. J. Park, P. Kim, J. Park, H. Lee, and N. Park, “Closed integral form expansion of Raman equation for efficient gain optimization process,” IEEE Photonics Technol. Lett. 16, 1649–1651 (2004). [CrossRef]

]. In this approach, solutions were sought along the iteration axis - starting with the analytically determined, seed wave values (defined over all the amplifier segments), and then iteratively multiplying two-dimensional transfer matrices (segment and wave, elements determined from lower-order iteration results).

In this paper, we apply the integral equation formalism to the multi-wave, bi-directional high-power SOA problem including nonlinear interactions, and also address the key differences between SOA and FRA in the equations and solution searching process. Results show greater than 30~500 times improvement in computational efficiency, when compared to the traditional differential equation-based approaches for the SOA solution.

2. Formulation

We start from the familiar coupled wave equation for SOA [8

8. M. A. Summerfield and R. S. Tucker, “Frequency domain model of multiwave mixing in bulk semiconductor optical amplifiers,” IEEE J. Sel. Top. Quantum Electron. 5, 839–850 (1999). [CrossRef]

],

daldz=12gl(N)[(1)alm=cpp,shb,ch(Δωij(1jβm)εm1+jΔωijτmai*ajak)]γscal2
(1)

Fig. 1. Schematic diagrams comparing the multi-wave SOA analysis method based on (a) coupled differential equation, and (b) suggested integral equation method

The integral equation formulation is now obtained by integrating Eq. (1) over the propagation axis, z ; and is given by

al(z)=al(0)+0z12gl(N)[(1)alm(Δωij(1jβm)εm1+jΔωijτmai*ajak)γscal2dz]
(2)

To solve the above equation, we assume an adiabatic process and utilize the iteration procedure. After n iterations, Eq. (2) becomes

aln(z)=al(n1)(0)+0z12gl(N(n1))(1)al(n1)m(Δωij(1jβm)εm1+jΔωijτmai(n1)*aj(n1)ak(n1))γscal(n1)2dz
(3)

Further, dividing the SOA length with x units of elemental segments (of length Δz, Fig. 1), we now successfully convert Eq. (3) into a matrix form (for y number signals for each direction),

An=F(n1)T
(4)

where n is the iteration number, and A, F, and T are defined as,

An=AnAn,Fn=FnFn,T=TT
An=a1n(0)a1n(xΔz)ayn(0)ayn(xΔz),An=a1n(0)a1n(xΔz)ayn(0)ayn(xΔz)
(5)
Fn=a1n(0)f(0)f(xΔz)a1n(0)f(0)f(xΔz),Fn=f(0)f((x1)·Δz)a1n(xΔz)f(0)f((x1)·Δz)ayn(xΔz)
f(z)=(12gl(N(z))[(1)al(z)m(Δωij(1jβm)εm1+jΔωijτmai*(z)aj(z)ak(z))]γscal(z)2)×Δz
f(z)=(12gl(N(z))[(1)al(z)m(Δωij(1jβm)εm1+jΔωijτmai*(z)aj(z)ak(z))]γscal(z)2)×Δz
(6)
T=1111101212121201211100121100012110000012,T=11111121212120111120111201120001200000
(7)

Based on above construction, the numerical solution of the SOA problem can be obtained from the following process [also refer Fig. 1(b)].

Step 0. Initialize A 0 for all segments, assuming transparent SOA.

Step I. Calculate F 0 using A 0

Step II. Obtain A 1=F 0 T

Step III. Repeat step I & II for higher numbers of n, with Eq. (6) and A n=F (n-1) T

The practical implementation of the above process to the SOAs is different from the integral equation for the FRAs, and here we note - 1: the convolution process in Eq. (3) can be treated as a vector product in the Fourier domain (with an additional inverse Fourier transform step) for faster calculation. 2: the carrier density N (absent for FRAs) needs to be calculated for each segment, and 3: for sufficiently small signal, variations in the signal phase have to be ignored (when amplitude of the wave is too small, the phase becomes too sensitive to the amplitude change caused by interactions with other waves, and leads to oscillation).

3. Results

Fig. 2. Comparison of the result with previous publication (Ref. [8])
Fig. 3. L-I curves (forward output power) obtained with differential / integral equation method

Still, for SOAs operating at higher driving currents and larger input signal power, different response curves were observed for those two methods, when tested with the same number (10) of SOA segments (Fig. 3 : forward/backward input power = 10 / 8 dBm at 1549.2/1550.8 nm. 0.04 nm resolution. SOA input/output coupling loss = 4 dB). Signal power/carrier density distributions in SOA at 500 mA of driving current is also shown in Fig. 4. With the finite difference method, more than 2,000 SOA segments were necessary to reach the final solution. However, with the proposed integral equation method, only 10 SOA segments were necessary to obtain the final, converged solution with much less computation time.

Fig. 4. field distribution (left) and carrier density (right) in the SOA (Dark square + dotted line: finite difference method with 10 segments, Hollow circle + dotted line: proposed integral equation method with 10 segments, Solid line: finite difference method, 2000 SOA segments)
Fig. 5. Forward / backward signal power distributions in the SOA cavity Upper graphs Forward (left) / Backward (right) waves solved with differential equation Lower graphs : Forward (left) / Backward (right) waves solved with integral equation

Fig. 6. Convergence error (forward output power) and required computation time plotted as a function of number of SOA segments (driving current = 500mA. Dash: finite difference method [4], Dash dot: Runge-Kutta, Solid line: proposed integral equation method)

4. Conclusion

For the first time, we demonstrated that it is possible to apply the integral equation method to the analysis of multi-wave, high power, bi-directional SOAs. Solutions were sought along the iteration axis from the analytically-determined seed wave and subsequential multiplication of transfer matrices. Comparisons are carried out in terms of computation expenses between the suggested formulation and traditional differential-equation approach. Results show greater than 30~500 times improvement in the computation efficiency for the proposed method, when compared to the traditional/improved Runge-Kutta coupled differential equation methods. The application of the proposed formulation can be found in the faster and precise characterization of SOA response curves, for example, calculating the L-I curve of high-power SOAs, which involve wide range (numbers) of operating condition/driving currents.

References and links

1.

C. Y. J Chu and H. Ghafouri-Shiraz, “Analysis of gain and saturation characteristics of a semiconductor laser optical amplifier using transfer matrices,” J. Lightwave Technol. 12, 1378–1386 (1994). [CrossRef]

2.

S. L. Zhang and J. J. O’Reilly, “Modelling of four-wave-mixing wavelength conversion in a semiconductor laser amplifier,” Physical Modelling of Semiconductor Devices, IEE Colloquium on, 5/1-5/6 (1995).

3.

H. Lee, H. Yoon, Y. Kim, and J. Jeong, “Theoretical study of frequency chirping and extinction ratio of wavelength-converted optical signals by XGM and XPM using SOA’s,” IEEE J. Quantum Electron. 35, 1213–1219 (1999). [CrossRef]

4.

M. J. Connelly, “Wideband semiconductor optical amplifier steady-state numerical model,” IEEE J. Quantum Electron. 37, 439–447 (2001). [CrossRef]

5.

B. Min, W. J. Lee, and N. Park, “Efficient formulation of Raman amplifier propagation equations with average power analysis,” IEEE Photonics Technol. Lett. 12, 1486–1488 (2000). [CrossRef]

6.

N. Park, P. Kim, J. Park, and L. K. Choi, “Integral form expansion of fiber Raman amplifier problem,” Opt. Fiber Technol., 11, 111–130 (2005). [CrossRef]

7.

J. Park, P. Kim, J. Park, H. Lee, and N. Park, “Closed integral form expansion of Raman equation for efficient gain optimization process,” IEEE Photonics Technol. Lett. 16, 1649–1651 (2004). [CrossRef]

8.

M. A. Summerfield and R. S. Tucker, “Frequency domain model of multiwave mixing in bulk semiconductor optical amplifiers,” IEEE J. Sel. Top. Quantum Electron. 5, 839–850 (1999). [CrossRef]

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(060.4510) Fiber optics and optical communications : Optical communications
(140.4480) Lasers and laser optics : Optical amplifiers
(250.5980) Optoelectronics : Semiconductor optical amplifiers

ToC Category:
Optoelectronics

History
Original Manuscript: January 11, 2006
Revised Manuscript: March 3, 2006
Manuscript Accepted: March 7, 2006
Published: March 20, 2006

Citation
Young Jin Jung, Pilhan Kim, Jaehyoung Park, and Namkyoo Park, "Integral equation approach for the analysis of high-power semiconductor optical amplifiers," Opt. Express 14, 2398-2403 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-6-2398


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References

  1. C. Y. J Chu, H. Ghafouri-Shiraz, "Analysis of gain and saturation characteristics of a semiconductor laser optical amplifier using transfer matrices," J. Lightwave Technol. 12, 1378-1386 (1994). [CrossRef]
  2. S. L. Zhang and J. J. O’Reilly, "Modelling of four-wave-mixing wavelength conversion in a semiconductor laser amplifier," Physical Modelling of Semiconductor Devices, IEE Colloquium on, 5/1-5/6 (1995).
  3. H. Lee, H. Yoon, Y. Kim, J. Jeong, "Theoretical study of frequency chirping and extinction ratio of wavelength-converted optical signals by XGM and XPM using SOA’s," IEEE J. Quantum Electron. 35, 1213-1219 (1999). [CrossRef]
  4. M. J. Connelly, "Wideband semiconductor optical amplifier steady-state numerical model," IEEE J. Quantum Electron. 37, 439-447 (2001). [CrossRef]
  5. B. Min, W. J. Lee, and N. Park, "Efficient formulation of Raman amplifier propagation equations with average power analysis," IEEE Photonics Technol. Lett. 12, 1486-1488 (2000). [CrossRef]
  6. N. Park, P. Kim, J. Park, L. K. Choi, "Integral form expansion of fiber Raman amplifier problem," Opt. Fiber Technol.,  11, 111-130 (2005). [CrossRef]
  7. J. Park, P. Kim, J. Park, H. Lee, and N. Park, "Closed integral form expansion of Raman equation for efficient gain optimization process," IEEE Photonics Technol. Lett. 16, 1649-1651 (2004). [CrossRef]
  8. M. A. Summerfield, and R. S. Tucker, "Frequency domain model of multiwave mixing in bulk semiconductor optical amplifiers," IEEE J. Sel. Top. Quantum Electron. 5, 839-850 (1999). [CrossRef]

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