Abstract

We propose and demonstrate wavelength conversion of spectrum-sliced broadband amplified spontaneous emission light sources based on hybrid four-wave mixing (HFWM) in highly nonlinear, dispersion-shifted fibers (HNL-DSFs). The theory of HFWM between coherent pumps and incoherent signal is analyzed. The degenerate HFWM is demonstrated experimentally in a 1-km-long HNL-DSF, where the coherent pump light is provided by a tunable cw laser source and the incoherent signal light is spectrum-sliced from a broadband amplified spontaneous emission light source. A conversion efficiency of about -20.4 dB and a bandwidth of about 38 nm are measured. The experimental result agrees well with the theoretical analysis.

©2006 Optical Society of America

1. Introduction

Spectrum splicing in which narrow bands are filtered from broadband sources is an efficient technique to obtain multiple channel carriers in optical wavelength division multiplexed (WDM) communication systems [1–5]. It avoids the need for laser sources with well-defined wavelengths. The performance of spectrum-sliced systems has been severely limited in the past by the low available power of broadband sources until rare earth-doped fibers [6], erbium-doped fibers [7], or semiconductor optical amplifiers [8] provided the potential to realize high power, broadband amplified spontaneous emission (ASE) light sources.

Wavelength conversion is an essential operation in WDM networks. In the past decade, wave-mixing effects-based wavelength conversions have been regarded as the promising methods because of their high speed, high capacity, strict transparency, and low noise. They can be roughly classified according to the nonlinear media: (a) in quasi-phase-matched crystals, wavelength conversion has been demonstrated by difference-frequency generation (DFG) [9], cascaded second-harmonic generation and DFG [10], and cascaded sum-frequency generation and DFG [11]; (b) in semiconductor optical amplifiers, wavelength conversion has been performed through degenerate or nondegenerate four-wave mixing (FWM) [12]; (c) in highly nonlinear fibers, many techniques based on FWM have been proposed, such as broadband wavelength conversion [13, 14], polarization-independent wavelength conversion [14], tunable channel-selective wavelength conversion [15], and wavelength exchange between two signals [16]. However, all of these investigations focused on the coherent light source, especially the monochromatic laser source serving as a single channel signal. The wavelength conversion of spectrum-sliced incoherent light sources is also very urgent to WDM networks but it is scarcely investigated by far.

In this paper, we demonstrate the wavelength conversion of 2-nm-wide incoherent ASE spectrum that is sliced from a broadband light source through hybrid FWM (HFWM) in a 1-km-long highly nonlinear, dispersion-shifted fiber (HNL-DSF). The principle of HFWM between coherent and incoherent lights is analyzed and the performance of the wavelength conversion configuration is evaluated in terms of conversion efficiency and bandwidth.

2. Theoretical analysis

In wavelength conversion of incoherent signals, the pumps are preferred to be coherent lights in order to obtain an efficient converted signal. The fields of the pumps are expressed as

Epizt=Epiexp(αz2)exp(j2πfpitjβpiz),(i=1,2)

where Epi is the input pump field, fpi is the pump frequency, βpi is the pump propagation constant, and α is the fiber loss coefficient.

The incoherent spectrum-sliced ASE light that is used as the signal light in HFWM interaction is composed of numerous independent frequency components. The incoherent signal is described as [17]

Eszt=Es0fszexp(j2πfstjβsz)dfs

and

Es0fsz=Es0(fs)exp(αz2)

where fs is the signal frequency, βs is the signal propagation constant, E s0(fs,z) is the slowly varying amplitude of one frequency component in the incoherent signal and E s0(fs) is its input value at the beginning of the fiber.

Here we call the FWM interaction between coherent and incoherent lights as HFWM. According to the inhomogeneous wave equation, the frequency component of the converted signal E F0(fF, z) that is excited by the signal frequency component E s0(fs, z) follows [18]

dEF0fFzdz=αEF0fFz2+j(4π2nλF)(Dχ)Ep1Ep2Es0*(fs)exp(3αz2+jΔβz)

Solving Eq. (4), the field of the converted frequency component at the end of the fiber is obtained as

EF0fFL=j(4π22λF)(Dχ)eαL2Ep1Ep2E*s0(fs){[e(jΔβα)L1](jΔβα)}

Expressing the interaction lights by their powers, the power of the converted frequency component is written as

PF0fFL=(1024π6n4λF2c2)(Dχ)2(Leff2Aeff2)eαLPp1Pp2Ps0(fs)η(Δβ)

where L is the fiber length, Leff is the effective length, Aeff is the effective mode area of the fiber, and χ is the third-order nonlinear coefficient. In Eq. (6), the degenerate factor D = 1, 3, or 6 corresponding to three, two or none of the frequencies f p1,f p2, and fs are the same. The HFWM efficiency dependent on the phase mismatch ηβ) has been given in Ref [19]:

η(Δβ)={α2[α2+(Δβ)2]}[1+4eαLsin2(ΔβL2)(1eαL)2]

where Δβ is the phase-matched term. It can be obtained by expressing the propagation constants with their Taylor expansions at the zero-dispersion frequency f 0. Neglecting the fifth-order and higher order series, the phase-matched term is expressed as

Δβ=βp1+βp2βsβF
=πλ43c2dDcdλ[(fp1f0)3+(fp2f0)3(fsf0)3(fFf0)3]
[πλ52c3dDcdλ+πλ62c3d2Dcdλ2][(fp1f0)4+(fp2f0)4(fsf0)4(fFf0)4]

Now we will analyze the effective nonlinear coefficient in HFWM. Since the incoherent frequency components are independent, it is reasonable to suppose that the polarization states of incoherent lights are towards all the directions with a uniform distribution in a narrow frequency domain. In nonbirefringent fibers, the nonlinear coefficient only depends on the angles between the polarization states of the signal and the pumps [18]. Supposing that the two pumps have the same polarization state, the effective nonlinear polarization is

PNLeff=(Dχxxxx)Ep1Ep2Es0(12π)02π(4cos2θ+53)

where θ is the angle between the polarization components and the pumps. Contrasting Eq. (9) with the conventional nonlinear polarization, the effective nonlinear coefficient is written as

χeff=χxxxx(12π)02π(4cos2θ+53)=0.71χxxxx

Using Eq. (10), the power of the converted signal is obtained by performing integration on each frequency component shown in Eq. (6):

PFfFL=(1024π6n4c2)(Dχeff)2(Leff2Aeff2)eαLPp1Pp21λF2Ps0(fs)η(Δβ)dfs

In wavelength conversion configurations, conversion efficiency is an important parameter to evaluate the conversion properties. It is often defined as the ratio of the converted signal power with respect to the input signal power. In the HFWM wavelength conversion we consider,it is described as

η=10log[PFfFLPsfs0]

where the signal power is written as Ps(fs, 0) = ∫ P s0(fs)d fs.

Above analysis shows the principle of the HFWM interaction between a coherent pump and an incoherent signal. As an ideally special example, assuming the spectrum slice of the ASE light source is infinitely narrow and only one frequency component is left, the HFWM will be very similar to the coherent FWM.

3. Experiment

Figure 1 shows the experimental setup to realize the wavelength conversion of the spectrum-sliced ASE broadband light source. A degenerate HFWM is performed in this experiment. A tunable cw laser (Agilent 8164A) is used as the pump, which is amplified by an EDFA. The signal is provided by a broadband ASE light source (Opticwave BLS-C ASE Light Source, its 3-dB spectrum width is about 40 nm) and its spectrum is sliced using a tunable band-pass filter (Santec OTF-40M, Tunable region: 1545–1565 nm) whose bandwidth is about 2 nm. The polarization state of the pump is adjusted using a polarization controller. The pump and the spectrum-sliced signal are coupled into a 1-km-long HNL-DSF through a 90:10 coupler. The HFWM is generated in the HNL-DSF and the converted signal is output from the end of the fiber.

 figure: Fig. 1.

Fig. 1. Experimental setup to demonstrate wavelength conversion of spectrum-sliced ASE broadband light source. EDFA: erbium-doped fiber amplifier, PC: polarization controller, BPF: band-pass filter, HNL-DSF: highly nonlinear dispersion-shifted fiber, OSA: optical spectrum analyzer.

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4. Results and discussion

The observed wavelength conversion spectrum of spectrum-sliced ASE broadband source is shown in Fig. 2. The pump wavelength is tuned to 1543 nm that is near the zero-dispersion wavelength (ZDW) of the HNL-DSF. The pump power after the coupler is measured to be about 12.7 dBm. The spectrum of the broadband ASE light source is sliced about 2 nm as the signal, and the central wavelength is set as 1548 nm. The peak power of the incoherent signal is measured to be around -15.9 dBm. As shown in Fig. 2, the converted signal is centered at about 1538 nm and has a spectrum width of about 2 nm. After eliminating the based power of the EDFA, the peak power of the converted signal is measured to be about -36.3 dBm. The conversion efficiency obtained here is about -20.4 dB. It seems a little low to practical applications due to the limitation of the low pump power used in this experiment. Fortunately, the efficiency has great potential to be improved by increasing the pump power since the efficiency is proportional to the square of the pump power according to Eqs. (11) and (12).

 figure: Fig. 2.

Fig. 2. Observed wavelength conversion spectrum of spectrum-sliced ASE broadband source. The pump light is set at 1543 nm, the signal light is centered at 1548 nm with a spectral width of about 2 nm, and the converted signal is generated around 1538 nm.

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In order to study the polarization property of this wavelength conversion configuration, the polarization controller is adjusted to change the polarization state of the pump with the signal unchanged. Since the frequency components of the incoherent light are independent, numerous polarization states are included in a very narrow spectrum region within the optical spectrum analyzer resolution. As a result, the observed spectrum shape of the converted signal almost keeps constant. The conversion efficiency can be obtained by detecting the peak power instead of the total converted power for convenience if the incoherent signal spectrum is narrow. When the polarization controller is tuned freely, the converted peak power fluctuates less than 0.5 dB, equivalent to the level of that caused by pump instability. This wavelength conversion shows polarization-independent performance.

Figure 3 shows the normalized conversion efficiencies of theoretically calculation and experimentally measurement. In our calculation, the parameters of the HNL-DSF are as follows: the loss coefficient is 0.77 dB/km, the ZDW of the HNL-DSF is 1542.5 nm, the third-order dispersion is 0.019 ps/km/nm2, and the forth-order dispersion is -9×10-5 ps/km/nm3. The measurement is performed by fixing the pump light at 1543 nm and adjusting the central wavelength of the filter to slice different spectra from the broadband ASE light source. Limited by the wavelength region of the filter, the measured efficiency is given only between 1545 nm and 1565 nm. The principle of the measured efficiency agrees well with the theoretical prediction. In Fig. 3, the 3-dB conversion bandwidth is read out to be about 38 nm, from 1524 nm to 1562 nm.

The conversion bandwidth deeply depends on the wavelength difference between the pump and the ZDW of the HNL-DSF. Figure 4 shows the measured and calculated bandwidths as functions of the pump wavelength. The measured bandwidth is very consistent to the calculated one when the pump is far away from the ZDW. The bandwidth is broadened when the pump is adjusted near the ZDW, but the difference between the theoretical and experiential results is enhanced. The reason lies in that the ZDW typically varies along the fiber [20] and a little local dispersion will be remained in a narrow wavelength region near the ZDW. Unfortunately, the bandwidth is greatly sensitive to the remained local dispersion. Since the conversion bandwidth is predicted with an ideal ZDW assumption in the simulation, the measured conversion bandwidth degenerates in comparison to the theoretical calculation in the 1-nm-wide region around the ZDW.

 figure: Fig. 3.

Fig. 3. Normalized conversion efficiency versus the signal wavelength. The opened circles are the measured values and the solid line is the theoretically result.

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 figure: Fig. 4.

Fig. 4. Conversion bandwidth versus the pump wavelength. The closed squares are the measured results and the solid line is the theoretical calculation.

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5. Conclusion

Wavelength conversion of the spectrum-sliced ASE broadband light source based on HFWM in HNL-DSFs is proposed and demonstrated. The theoretical model of this configuration is analyzed and the conversion performance is measured experimentally. The experimental result agrees well with the theoretical analysis. In this experiment, the conversion efficiency is measured to be about -20.4 dB and the conversion bandwidth is about 38 nm. It provides a useful method for wavelength conversion operations of incoherent light sources.

Acknowledgments

This work is supported by the Postdoctoral Science Foundation of China (2005037007), the National Natural Science Foundation of China (60478003), and the Specialized Research Fund for the Doctoral Program of Higher Education (20040003064).

References and links

1. J. S. Lee, Y. C. Chung, and D. J. DiGiovanni, “Spectrum-sliced fiber amplifier light source for multichannel WDM applications,” IEEE Photonics Technol. Lett. 5, 1458–1461 (1993). [CrossRef]  

2. P. D. D. Kilkelly, P. J. Chidgey, and G. Hill, “Experimental demonstration of a three channel WDM system over 110 km using superluminescent diodes,” Electron. Lett. 26, 3671–1673 (1990).

3. D. K. Jung, S. K. Shin, C.-H. Lee, and Y. C. Chung, “Wavelength-division-multiplexed passive optical network based on spectrum-slicing techniques,” IEEE Photonics Technol. Lett. 10, 1334–1336 (1998). [CrossRef]  

4. J. H. Han, S. J. Kim, and J. S. Lee, “Transmission of 4×2.5-Gb/s spectrum-sliced incoherent light channels over 240 km of dispersion-shifted fiber with 200-GHz channel spacing,” IEEE Photonics Technol. Lett. 11, 901–903 (1999). [CrossRef]  

5. K. Akimoto, J. Kani, M. Teshima, and K. Iwatsuki, “Super-dense WDM transmission of spectrum-sliced incoherent light for wide-area access network,” J. Lightwave Technol. 21, 2715–2722 (2003). [CrossRef]  

6. R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Efficient superfluorescent light sources with broad bandwidth,” IEEE J. Sel. Top. Quantum Electron. 3, 1097–1099 (1997). [CrossRef]  

7. N. S. Kwong, “High-power, broad-hand 1550 nm light source by a tandem combination of a superluminescent diode and an Er-doped fiber ampifier,” IEEE Photonics Technol. Lett. 4, 996–999 (1992). [CrossRef]  

8. D. D. Sampson and W. T. Holloway, “l00 mW spectrally-uniform broadband ASE source for spectrum-sliced WDM systems,” Electron. Lett. 30, 1611–1612 (1994). [CrossRef]  

9. M. H. Chou, J. Hauden, M. A. Arbore, and M. M. Fejer, “1.5-mm-band wavelength conversion based on difference-frequency generation in LiNbO3 waveguides with integrated coupling structures,” Opt. Lett. 23, 1004–1006 (1998). [CrossRef]  

10. S. Gao, C. Yang, and G. Jin, “Flat broadband wavelength conversion based on sinusoidally chirped optical superlattices in lithium niobate,” IEEE Photonics Technol. Lett. 16, 557–559 (2004). [CrossRef]  

11. C. Q. Xu and B. Chen, “Cascaded wavelength conversions based on sum-frequency generation and difference-frequency generation,” Opt. Lett. 29, 292–294 (2004). [CrossRef]   [PubMed]  

12. D. F. Geraghty, R. B. Lee, M. Verdiell, M. Ziari, A. Mathur, and K. J. Vahala, “Wavelength conversion for WDM communication systems using four-wave mixing in semiconductor optical amplifiers,” IEEE J. Sel. Top. Quantum Electron. 3, 1146–1155 (1997). [CrossRef]  

13. O. Aso, S. Arai, T. Yagi, M. Tadakuma, Y. Suzuki, and S. Namiki, “Efficient FWM based broadband wavelength conversion using a short high-nonlinearity fiber,” IEICE Trans. Electron. 6, 816–823 (2000).

14. T. Tanemura and K. Kikuchi, “Polarization-independent broad-band wavelength conversion using two-pump fiber optical parametric amplification without idler spectral broadening,” IEEE Photonics Technol. Lett. 15, 1573–1575 (2003). [CrossRef]  

15. K. Inoue, “Tunable and selective wavelength conversion using fiber four-wave mixing with two pump lights,” IEEE Photonics Technol. Lett. 6, 1451–1453 (1994). [CrossRef]  

16. K. Uesaka, K. K. Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments,” IEEE J. Sel. Top. Quantum Electron. 8, 560–568 (2002). [CrossRef]  

17. Y. S. Jang and Y. C. Chung, “Four-wave mixing of incoherent light in a dispersion-shifted fiber using a spectrum-sliced fiber amplifier light source,” IEEE Photonics Technol. Lett. 10, 218–220 (1998). [CrossRef]  

18. K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “cw three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978). [CrossRef]  

19. K. Inoue, “Four-wave mixing in an optical fiber in the zero-dispersion wavelength region,” J. Lightwave Technol. 10, 1553–1561 (1992). [CrossRef]  

20. M. Eiselt, R. M. Jopson, and R. H. Stolen, “Nondestructive position-resolved measurement of the zero-dispersion wavelength in an optical fiber,” J. Lightwave Technol. 15, 135–143 (1997). [CrossRef]  

References

  • View by:

  1. J. S. Lee, Y. C. Chung, and D. J. DiGiovanni, “Spectrum-sliced fiber amplifier light source for multichannel WDM applications,” IEEE Photonics Technol. Lett. 5, 1458–1461 (1993).
    [Crossref]
  2. P. D. D. Kilkelly, P. J. Chidgey, and G. Hill, “Experimental demonstration of a three channel WDM system over 110 km using superluminescent diodes,” Electron. Lett. 26, 3671–1673 (1990).
  3. D. K. Jung, S. K. Shin, C.-H. Lee, and Y. C. Chung, “Wavelength-division-multiplexed passive optical network based on spectrum-slicing techniques,” IEEE Photonics Technol. Lett. 10, 1334–1336 (1998).
    [Crossref]
  4. J. H. Han, S. J. Kim, and J. S. Lee, “Transmission of 4×2.5-Gb/s spectrum-sliced incoherent light channels over 240 km of dispersion-shifted fiber with 200-GHz channel spacing,” IEEE Photonics Technol. Lett. 11, 901–903 (1999).
    [Crossref]
  5. K. Akimoto, J. Kani, M. Teshima, and K. Iwatsuki, “Super-dense WDM transmission of spectrum-sliced incoherent light for wide-area access network,” J. Lightwave Technol. 21, 2715–2722 (2003).
    [Crossref]
  6. R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Efficient superfluorescent light sources with broad bandwidth,” IEEE J. Sel. Top. Quantum Electron. 3, 1097–1099 (1997).
    [Crossref]
  7. N. S. Kwong, “High-power, broad-hand 1550 nm light source by a tandem combination of a superluminescent diode and an Er-doped fiber ampifier,” IEEE Photonics Technol. Lett. 4, 996–999 (1992).
    [Crossref]
  8. D. D. Sampson and W. T. Holloway, “l00 mW spectrally-uniform broadband ASE source for spectrum-sliced WDM systems,” Electron. Lett. 30, 1611–1612 (1994).
    [Crossref]
  9. M. H. Chou, J. Hauden, M. A. Arbore, and M. M. Fejer, “1.5-mm-band wavelength conversion based on difference-frequency generation in LiNbO3 waveguides with integrated coupling structures,” Opt. Lett. 23, 1004–1006 (1998).
    [Crossref]
  10. S. Gao, C. Yang, and G. Jin, “Flat broadband wavelength conversion based on sinusoidally chirped optical superlattices in lithium niobate,” IEEE Photonics Technol. Lett. 16, 557–559 (2004).
    [Crossref]
  11. C. Q. Xu and B. Chen, “Cascaded wavelength conversions based on sum-frequency generation and difference-frequency generation,” Opt. Lett. 29, 292–294 (2004).
    [Crossref] [PubMed]
  12. D. F. Geraghty, R. B. Lee, M. Verdiell, M. Ziari, A. Mathur, and K. J. Vahala, “Wavelength conversion for WDM communication systems using four-wave mixing in semiconductor optical amplifiers,” IEEE J. Sel. Top. Quantum Electron. 3, 1146–1155 (1997).
    [Crossref]
  13. O. Aso, S. Arai, T. Yagi, M. Tadakuma, Y. Suzuki, and S. Namiki, “Efficient FWM based broadband wavelength conversion using a short high-nonlinearity fiber,” IEICE Trans. Electron. 6, 816–823 (2000).
  14. T. Tanemura and K. Kikuchi, “Polarization-independent broad-band wavelength conversion using two-pump fiber optical parametric amplification without idler spectral broadening,” IEEE Photonics Technol. Lett. 15, 1573–1575 (2003).
    [Crossref]
  15. K. Inoue, “Tunable and selective wavelength conversion using fiber four-wave mixing with two pump lights,” IEEE Photonics Technol. Lett. 6, 1451–1453 (1994).
    [Crossref]
  16. K. Uesaka, K. K. Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments,” IEEE J. Sel. Top. Quantum Electron. 8, 560–568 (2002).
    [Crossref]
  17. Y. S. Jang and Y. C. Chung, “Four-wave mixing of incoherent light in a dispersion-shifted fiber using a spectrum-sliced fiber amplifier light source,” IEEE Photonics Technol. Lett. 10, 218–220 (1998).
    [Crossref]
  18. K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “cw three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
    [Crossref]
  19. K. Inoue, “Four-wave mixing in an optical fiber in the zero-dispersion wavelength region,” J. Lightwave Technol. 10, 1553–1561 (1992).
    [Crossref]
  20. M. Eiselt, R. M. Jopson, and R. H. Stolen, “Nondestructive position-resolved measurement of the zero-dispersion wavelength in an optical fiber,” J. Lightwave Technol. 15, 135–143 (1997).
    [Crossref]

2004 (2)

S. Gao, C. Yang, and G. Jin, “Flat broadband wavelength conversion based on sinusoidally chirped optical superlattices in lithium niobate,” IEEE Photonics Technol. Lett. 16, 557–559 (2004).
[Crossref]

C. Q. Xu and B. Chen, “Cascaded wavelength conversions based on sum-frequency generation and difference-frequency generation,” Opt. Lett. 29, 292–294 (2004).
[Crossref] [PubMed]

2003 (2)

T. Tanemura and K. Kikuchi, “Polarization-independent broad-band wavelength conversion using two-pump fiber optical parametric amplification without idler spectral broadening,” IEEE Photonics Technol. Lett. 15, 1573–1575 (2003).
[Crossref]

K. Akimoto, J. Kani, M. Teshima, and K. Iwatsuki, “Super-dense WDM transmission of spectrum-sliced incoherent light for wide-area access network,” J. Lightwave Technol. 21, 2715–2722 (2003).
[Crossref]

2002 (1)

K. Uesaka, K. K. Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments,” IEEE J. Sel. Top. Quantum Electron. 8, 560–568 (2002).
[Crossref]

2000 (1)

O. Aso, S. Arai, T. Yagi, M. Tadakuma, Y. Suzuki, and S. Namiki, “Efficient FWM based broadband wavelength conversion using a short high-nonlinearity fiber,” IEICE Trans. Electron. 6, 816–823 (2000).

1999 (1)

J. H. Han, S. J. Kim, and J. S. Lee, “Transmission of 4×2.5-Gb/s spectrum-sliced incoherent light channels over 240 km of dispersion-shifted fiber with 200-GHz channel spacing,” IEEE Photonics Technol. Lett. 11, 901–903 (1999).
[Crossref]

1998 (3)

D. K. Jung, S. K. Shin, C.-H. Lee, and Y. C. Chung, “Wavelength-division-multiplexed passive optical network based on spectrum-slicing techniques,” IEEE Photonics Technol. Lett. 10, 1334–1336 (1998).
[Crossref]

M. H. Chou, J. Hauden, M. A. Arbore, and M. M. Fejer, “1.5-mm-band wavelength conversion based on difference-frequency generation in LiNbO3 waveguides with integrated coupling structures,” Opt. Lett. 23, 1004–1006 (1998).
[Crossref]

Y. S. Jang and Y. C. Chung, “Four-wave mixing of incoherent light in a dispersion-shifted fiber using a spectrum-sliced fiber amplifier light source,” IEEE Photonics Technol. Lett. 10, 218–220 (1998).
[Crossref]

1997 (3)

D. F. Geraghty, R. B. Lee, M. Verdiell, M. Ziari, A. Mathur, and K. J. Vahala, “Wavelength conversion for WDM communication systems using four-wave mixing in semiconductor optical amplifiers,” IEEE J. Sel. Top. Quantum Electron. 3, 1146–1155 (1997).
[Crossref]

R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Efficient superfluorescent light sources with broad bandwidth,” IEEE J. Sel. Top. Quantum Electron. 3, 1097–1099 (1997).
[Crossref]

M. Eiselt, R. M. Jopson, and R. H. Stolen, “Nondestructive position-resolved measurement of the zero-dispersion wavelength in an optical fiber,” J. Lightwave Technol. 15, 135–143 (1997).
[Crossref]

1994 (2)

D. D. Sampson and W. T. Holloway, “l00 mW spectrally-uniform broadband ASE source for spectrum-sliced WDM systems,” Electron. Lett. 30, 1611–1612 (1994).
[Crossref]

K. Inoue, “Tunable and selective wavelength conversion using fiber four-wave mixing with two pump lights,” IEEE Photonics Technol. Lett. 6, 1451–1453 (1994).
[Crossref]

1993 (1)

J. S. Lee, Y. C. Chung, and D. J. DiGiovanni, “Spectrum-sliced fiber amplifier light source for multichannel WDM applications,” IEEE Photonics Technol. Lett. 5, 1458–1461 (1993).
[Crossref]

1992 (2)

N. S. Kwong, “High-power, broad-hand 1550 nm light source by a tandem combination of a superluminescent diode and an Er-doped fiber ampifier,” IEEE Photonics Technol. Lett. 4, 996–999 (1992).
[Crossref]

K. Inoue, “Four-wave mixing in an optical fiber in the zero-dispersion wavelength region,” J. Lightwave Technol. 10, 1553–1561 (1992).
[Crossref]

1990 (1)

P. D. D. Kilkelly, P. J. Chidgey, and G. Hill, “Experimental demonstration of a three channel WDM system over 110 km using superluminescent diodes,” Electron. Lett. 26, 3671–1673 (1990).

1978 (1)

K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “cw three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
[Crossref]

Akimoto, K.

Arai, S.

O. Aso, S. Arai, T. Yagi, M. Tadakuma, Y. Suzuki, and S. Namiki, “Efficient FWM based broadband wavelength conversion using a short high-nonlinearity fiber,” IEICE Trans. Electron. 6, 816–823 (2000).

Arbore, M. A.

Aso, O.

O. Aso, S. Arai, T. Yagi, M. Tadakuma, Y. Suzuki, and S. Namiki, “Efficient FWM based broadband wavelength conversion using a short high-nonlinearity fiber,” IEICE Trans. Electron. 6, 816–823 (2000).

Chen, B.

Chidgey, P. J.

P. D. D. Kilkelly, P. J. Chidgey, and G. Hill, “Experimental demonstration of a three channel WDM system over 110 km using superluminescent diodes,” Electron. Lett. 26, 3671–1673 (1990).

Chou, M. H.

Chung, Y. C.

D. K. Jung, S. K. Shin, C.-H. Lee, and Y. C. Chung, “Wavelength-division-multiplexed passive optical network based on spectrum-slicing techniques,” IEEE Photonics Technol. Lett. 10, 1334–1336 (1998).
[Crossref]

Y. S. Jang and Y. C. Chung, “Four-wave mixing of incoherent light in a dispersion-shifted fiber using a spectrum-sliced fiber amplifier light source,” IEEE Photonics Technol. Lett. 10, 218–220 (1998).
[Crossref]

J. S. Lee, Y. C. Chung, and D. J. DiGiovanni, “Spectrum-sliced fiber amplifier light source for multichannel WDM applications,” IEEE Photonics Technol. Lett. 5, 1458–1461 (1993).
[Crossref]

DiGiovanni, D. J.

J. S. Lee, Y. C. Chung, and D. J. DiGiovanni, “Spectrum-sliced fiber amplifier light source for multichannel WDM applications,” IEEE Photonics Technol. Lett. 5, 1458–1461 (1993).
[Crossref]

Eiselt, M.

M. Eiselt, R. M. Jopson, and R. H. Stolen, “Nondestructive position-resolved measurement of the zero-dispersion wavelength in an optical fiber,” J. Lightwave Technol. 15, 135–143 (1997).
[Crossref]

Fejer, M. M.

Gao, S.

S. Gao, C. Yang, and G. Jin, “Flat broadband wavelength conversion based on sinusoidally chirped optical superlattices in lithium niobate,” IEEE Photonics Technol. Lett. 16, 557–559 (2004).
[Crossref]

Geraghty, D. F.

D. F. Geraghty, R. B. Lee, M. Verdiell, M. Ziari, A. Mathur, and K. J. Vahala, “Wavelength conversion for WDM communication systems using four-wave mixing in semiconductor optical amplifiers,” IEEE J. Sel. Top. Quantum Electron. 3, 1146–1155 (1997).
[Crossref]

Han, J. H.

J. H. Han, S. J. Kim, and J. S. Lee, “Transmission of 4×2.5-Gb/s spectrum-sliced incoherent light channels over 240 km of dispersion-shifted fiber with 200-GHz channel spacing,” IEEE Photonics Technol. Lett. 11, 901–903 (1999).
[Crossref]

Hanna, D. C.

R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Efficient superfluorescent light sources with broad bandwidth,” IEEE J. Sel. Top. Quantum Electron. 3, 1097–1099 (1997).
[Crossref]

Hauden, J.

Hill, G.

P. D. D. Kilkelly, P. J. Chidgey, and G. Hill, “Experimental demonstration of a three channel WDM system over 110 km using superluminescent diodes,” Electron. Lett. 26, 3671–1673 (1990).

Hill, K. O.

K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “cw three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
[Crossref]

Holloway, W. T.

D. D. Sampson and W. T. Holloway, “l00 mW spectrally-uniform broadband ASE source for spectrum-sliced WDM systems,” Electron. Lett. 30, 1611–1612 (1994).
[Crossref]

Inoue, K.

K. Inoue, “Tunable and selective wavelength conversion using fiber four-wave mixing with two pump lights,” IEEE Photonics Technol. Lett. 6, 1451–1453 (1994).
[Crossref]

K. Inoue, “Four-wave mixing in an optical fiber in the zero-dispersion wavelength region,” J. Lightwave Technol. 10, 1553–1561 (1992).
[Crossref]

Iwatsuki, K.

Jang, Y. S.

Y. S. Jang and Y. C. Chung, “Four-wave mixing of incoherent light in a dispersion-shifted fiber using a spectrum-sliced fiber amplifier light source,” IEEE Photonics Technol. Lett. 10, 218–220 (1998).
[Crossref]

Jin, G.

S. Gao, C. Yang, and G. Jin, “Flat broadband wavelength conversion based on sinusoidally chirped optical superlattices in lithium niobate,” IEEE Photonics Technol. Lett. 16, 557–559 (2004).
[Crossref]

Johnson, D. C.

K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “cw three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
[Crossref]

Jopson, R. M.

M. Eiselt, R. M. Jopson, and R. H. Stolen, “Nondestructive position-resolved measurement of the zero-dispersion wavelength in an optical fiber,” J. Lightwave Technol. 15, 135–143 (1997).
[Crossref]

Jung, D. K.

D. K. Jung, S. K. Shin, C.-H. Lee, and Y. C. Chung, “Wavelength-division-multiplexed passive optical network based on spectrum-slicing techniques,” IEEE Photonics Technol. Lett. 10, 1334–1336 (1998).
[Crossref]

Kani, J.

Kawasaki, B. S.

K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “cw three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
[Crossref]

Kazovsky, L. G.

K. Uesaka, K. K. Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments,” IEEE J. Sel. Top. Quantum Electron. 8, 560–568 (2002).
[Crossref]

Kikuchi, K.

T. Tanemura and K. Kikuchi, “Polarization-independent broad-band wavelength conversion using two-pump fiber optical parametric amplification without idler spectral broadening,” IEEE Photonics Technol. Lett. 15, 1573–1575 (2003).
[Crossref]

Kilkelly, P. D. D.

P. D. D. Kilkelly, P. J. Chidgey, and G. Hill, “Experimental demonstration of a three channel WDM system over 110 km using superluminescent diodes,” Electron. Lett. 26, 3671–1673 (1990).

Kim, S. J.

J. H. Han, S. J. Kim, and J. S. Lee, “Transmission of 4×2.5-Gb/s spectrum-sliced incoherent light channels over 240 km of dispersion-shifted fiber with 200-GHz channel spacing,” IEEE Photonics Technol. Lett. 11, 901–903 (1999).
[Crossref]

Kwong, N. S.

N. S. Kwong, “High-power, broad-hand 1550 nm light source by a tandem combination of a superluminescent diode and an Er-doped fiber ampifier,” IEEE Photonics Technol. Lett. 4, 996–999 (1992).
[Crossref]

Lee, C.-H.

D. K. Jung, S. K. Shin, C.-H. Lee, and Y. C. Chung, “Wavelength-division-multiplexed passive optical network based on spectrum-slicing techniques,” IEEE Photonics Technol. Lett. 10, 1334–1336 (1998).
[Crossref]

Lee, J. S.

J. H. Han, S. J. Kim, and J. S. Lee, “Transmission of 4×2.5-Gb/s spectrum-sliced incoherent light channels over 240 km of dispersion-shifted fiber with 200-GHz channel spacing,” IEEE Photonics Technol. Lett. 11, 901–903 (1999).
[Crossref]

J. S. Lee, Y. C. Chung, and D. J. DiGiovanni, “Spectrum-sliced fiber amplifier light source for multichannel WDM applications,” IEEE Photonics Technol. Lett. 5, 1458–1461 (1993).
[Crossref]

Lee, R. B.

D. F. Geraghty, R. B. Lee, M. Verdiell, M. Ziari, A. Mathur, and K. J. Vahala, “Wavelength conversion for WDM communication systems using four-wave mixing in semiconductor optical amplifiers,” IEEE J. Sel. Top. Quantum Electron. 3, 1146–1155 (1997).
[Crossref]

MacDonald, R. I.

K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “cw three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
[Crossref]

Marhic, M. E.

K. Uesaka, K. K. Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments,” IEEE J. Sel. Top. Quantum Electron. 8, 560–568 (2002).
[Crossref]

Mathur, A.

D. F. Geraghty, R. B. Lee, M. Verdiell, M. Ziari, A. Mathur, and K. J. Vahala, “Wavelength conversion for WDM communication systems using four-wave mixing in semiconductor optical amplifiers,” IEEE J. Sel. Top. Quantum Electron. 3, 1146–1155 (1997).
[Crossref]

Namiki, S.

O. Aso, S. Arai, T. Yagi, M. Tadakuma, Y. Suzuki, and S. Namiki, “Efficient FWM based broadband wavelength conversion using a short high-nonlinearity fiber,” IEICE Trans. Electron. 6, 816–823 (2000).

Nilsson, J.

R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Efficient superfluorescent light sources with broad bandwidth,” IEEE J. Sel. Top. Quantum Electron. 3, 1097–1099 (1997).
[Crossref]

Paschotta, R.

R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Efficient superfluorescent light sources with broad bandwidth,” IEEE J. Sel. Top. Quantum Electron. 3, 1097–1099 (1997).
[Crossref]

Sampson, D. D.

D. D. Sampson and W. T. Holloway, “l00 mW spectrally-uniform broadband ASE source for spectrum-sliced WDM systems,” Electron. Lett. 30, 1611–1612 (1994).
[Crossref]

Shin, S. K.

D. K. Jung, S. K. Shin, C.-H. Lee, and Y. C. Chung, “Wavelength-division-multiplexed passive optical network based on spectrum-slicing techniques,” IEEE Photonics Technol. Lett. 10, 1334–1336 (1998).
[Crossref]

Stolen, R. H.

M. Eiselt, R. M. Jopson, and R. H. Stolen, “Nondestructive position-resolved measurement of the zero-dispersion wavelength in an optical fiber,” J. Lightwave Technol. 15, 135–143 (1997).
[Crossref]

Suzuki, Y.

O. Aso, S. Arai, T. Yagi, M. Tadakuma, Y. Suzuki, and S. Namiki, “Efficient FWM based broadband wavelength conversion using a short high-nonlinearity fiber,” IEICE Trans. Electron. 6, 816–823 (2000).

Tadakuma, M.

O. Aso, S. Arai, T. Yagi, M. Tadakuma, Y. Suzuki, and S. Namiki, “Efficient FWM based broadband wavelength conversion using a short high-nonlinearity fiber,” IEICE Trans. Electron. 6, 816–823 (2000).

Tanemura, T.

T. Tanemura and K. Kikuchi, “Polarization-independent broad-band wavelength conversion using two-pump fiber optical parametric amplification without idler spectral broadening,” IEEE Photonics Technol. Lett. 15, 1573–1575 (2003).
[Crossref]

Teshima, M.

Tropper, A. C.

R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Efficient superfluorescent light sources with broad bandwidth,” IEEE J. Sel. Top. Quantum Electron. 3, 1097–1099 (1997).
[Crossref]

Uesaka, K.

K. Uesaka, K. K. Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments,” IEEE J. Sel. Top. Quantum Electron. 8, 560–568 (2002).
[Crossref]

Vahala, K. J.

D. F. Geraghty, R. B. Lee, M. Verdiell, M. Ziari, A. Mathur, and K. J. Vahala, “Wavelength conversion for WDM communication systems using four-wave mixing in semiconductor optical amplifiers,” IEEE J. Sel. Top. Quantum Electron. 3, 1146–1155 (1997).
[Crossref]

Verdiell, M.

D. F. Geraghty, R. B. Lee, M. Verdiell, M. Ziari, A. Mathur, and K. J. Vahala, “Wavelength conversion for WDM communication systems using four-wave mixing in semiconductor optical amplifiers,” IEEE J. Sel. Top. Quantum Electron. 3, 1146–1155 (1997).
[Crossref]

Wong, K. K. Y.

K. Uesaka, K. K. Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments,” IEEE J. Sel. Top. Quantum Electron. 8, 560–568 (2002).
[Crossref]

Xu, C. Q.

Yagi, T.

O. Aso, S. Arai, T. Yagi, M. Tadakuma, Y. Suzuki, and S. Namiki, “Efficient FWM based broadband wavelength conversion using a short high-nonlinearity fiber,” IEICE Trans. Electron. 6, 816–823 (2000).

Yang, C.

S. Gao, C. Yang, and G. Jin, “Flat broadband wavelength conversion based on sinusoidally chirped optical superlattices in lithium niobate,” IEEE Photonics Technol. Lett. 16, 557–559 (2004).
[Crossref]

Ziari, M.

D. F. Geraghty, R. B. Lee, M. Verdiell, M. Ziari, A. Mathur, and K. J. Vahala, “Wavelength conversion for WDM communication systems using four-wave mixing in semiconductor optical amplifiers,” IEEE J. Sel. Top. Quantum Electron. 3, 1146–1155 (1997).
[Crossref]

Electron. Lett. (2)

P. D. D. Kilkelly, P. J. Chidgey, and G. Hill, “Experimental demonstration of a three channel WDM system over 110 km using superluminescent diodes,” Electron. Lett. 26, 3671–1673 (1990).

D. D. Sampson and W. T. Holloway, “l00 mW spectrally-uniform broadband ASE source for spectrum-sliced WDM systems,” Electron. Lett. 30, 1611–1612 (1994).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (3)

R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Efficient superfluorescent light sources with broad bandwidth,” IEEE J. Sel. Top. Quantum Electron. 3, 1097–1099 (1997).
[Crossref]

D. F. Geraghty, R. B. Lee, M. Verdiell, M. Ziari, A. Mathur, and K. J. Vahala, “Wavelength conversion for WDM communication systems using four-wave mixing in semiconductor optical amplifiers,” IEEE J. Sel. Top. Quantum Electron. 3, 1146–1155 (1997).
[Crossref]

K. Uesaka, K. K. Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments,” IEEE J. Sel. Top. Quantum Electron. 8, 560–568 (2002).
[Crossref]

IEEE Photonics Technol. Lett. (8)

Y. S. Jang and Y. C. Chung, “Four-wave mixing of incoherent light in a dispersion-shifted fiber using a spectrum-sliced fiber amplifier light source,” IEEE Photonics Technol. Lett. 10, 218–220 (1998).
[Crossref]

S. Gao, C. Yang, and G. Jin, “Flat broadband wavelength conversion based on sinusoidally chirped optical superlattices in lithium niobate,” IEEE Photonics Technol. Lett. 16, 557–559 (2004).
[Crossref]

N. S. Kwong, “High-power, broad-hand 1550 nm light source by a tandem combination of a superluminescent diode and an Er-doped fiber ampifier,” IEEE Photonics Technol. Lett. 4, 996–999 (1992).
[Crossref]

D. K. Jung, S. K. Shin, C.-H. Lee, and Y. C. Chung, “Wavelength-division-multiplexed passive optical network based on spectrum-slicing techniques,” IEEE Photonics Technol. Lett. 10, 1334–1336 (1998).
[Crossref]

J. H. Han, S. J. Kim, and J. S. Lee, “Transmission of 4×2.5-Gb/s spectrum-sliced incoherent light channels over 240 km of dispersion-shifted fiber with 200-GHz channel spacing,” IEEE Photonics Technol. Lett. 11, 901–903 (1999).
[Crossref]

T. Tanemura and K. Kikuchi, “Polarization-independent broad-band wavelength conversion using two-pump fiber optical parametric amplification without idler spectral broadening,” IEEE Photonics Technol. Lett. 15, 1573–1575 (2003).
[Crossref]

K. Inoue, “Tunable and selective wavelength conversion using fiber four-wave mixing with two pump lights,” IEEE Photonics Technol. Lett. 6, 1451–1453 (1994).
[Crossref]

J. S. Lee, Y. C. Chung, and D. J. DiGiovanni, “Spectrum-sliced fiber amplifier light source for multichannel WDM applications,” IEEE Photonics Technol. Lett. 5, 1458–1461 (1993).
[Crossref]

IEICE Trans. Electron. (1)

O. Aso, S. Arai, T. Yagi, M. Tadakuma, Y. Suzuki, and S. Namiki, “Efficient FWM based broadband wavelength conversion using a short high-nonlinearity fiber,” IEICE Trans. Electron. 6, 816–823 (2000).

J. Appl. Phys. (1)

K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “cw three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
[Crossref]

J. Lightwave Technol. (3)

K. Inoue, “Four-wave mixing in an optical fiber in the zero-dispersion wavelength region,” J. Lightwave Technol. 10, 1553–1561 (1992).
[Crossref]

M. Eiselt, R. M. Jopson, and R. H. Stolen, “Nondestructive position-resolved measurement of the zero-dispersion wavelength in an optical fiber,” J. Lightwave Technol. 15, 135–143 (1997).
[Crossref]

K. Akimoto, J. Kani, M. Teshima, and K. Iwatsuki, “Super-dense WDM transmission of spectrum-sliced incoherent light for wide-area access network,” J. Lightwave Technol. 21, 2715–2722 (2003).
[Crossref]

Opt. Lett. (2)

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Figures (4)

Fig. 1.
Fig. 1. Experimental setup to demonstrate wavelength conversion of spectrum-sliced ASE broadband light source. EDFA: erbium-doped fiber amplifier, PC: polarization controller, BPF: band-pass filter, HNL-DSF: highly nonlinear dispersion-shifted fiber, OSA: optical spectrum analyzer.
Fig. 2.
Fig. 2. Observed wavelength conversion spectrum of spectrum-sliced ASE broadband source. The pump light is set at 1543 nm, the signal light is centered at 1548 nm with a spectral width of about 2 nm, and the converted signal is generated around 1538 nm.
Fig. 3.
Fig. 3. Normalized conversion efficiency versus the signal wavelength. The opened circles are the measured values and the solid line is the theoretically result.
Fig. 4.
Fig. 4. Conversion bandwidth versus the pump wavelength. The closed squares are the measured results and the solid line is the theoretical calculation.

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

E pi z t = E pi exp ( αz 2 ) exp ( j 2 π f pi t j β pi z ) , ( i = 1,2 )
E s z t = E s 0 f s z exp ( j 2 π f s t j β s z ) d f s
E s 0 f s z = E s 0 ( f s ) exp ( αz 2 )
d E F 0 f F z dz = α E F 0 f F z 2 + j ( 4 π 2 n λ F ) ( D χ ) E p 1 E p 2 E s 0 * ( f s ) exp ( 3 αz 2 + j Δ βz )
E F 0 f F L = j ( 4 π 2 2 λ F ) ( D χ ) e αL 2 E p 1 E p 2 E * s 0 ( f s ) { [ e ( j Δ β α ) L 1 ] ( j Δ β α ) }
P F 0 f F L = ( 1024 π 6 n 4 λ F 2 c 2 ) ( D χ ) 2 ( L eff 2 A eff 2 ) e αL P p 1 P p 2 P s 0 ( f s ) η ( Δ β )
η ( Δ β ) = { α 2 [ α 2 + ( Δ β ) 2 ] } [ 1 + 4 e αL sin 2 ( Δ βL 2 ) ( 1 e αL ) 2 ]
Δ β = β p 1 + β p 2 β s β F
= π λ 4 3 c 2 d D c d λ [ ( f p 1 f 0 ) 3 + ( f p 2 f 0 ) 3 ( f s f 0 ) 3 ( f F f 0 ) 3 ]
[ π λ 5 2 c 3 d D c d λ + π λ 6 2 c 3 d 2 D c d λ 2 ] [ ( f p 1 f 0 ) 4 + ( f p 2 f 0 ) 4 ( f s f 0 ) 4 ( f F f 0 ) 4 ]
P NLeff = ( D χ xxxx ) E p 1 E p 2 E s 0 ( 1 2 π ) 0 2 π ( 4 cos 2 θ + 5 3 )
χ eff = χ xxxx ( 1 2 π ) 0 2 π ( 4 cos 2 θ + 5 3 ) = 0.71 χ xxxx
P F f F L = ( 1024 π 6 n 4 c 2 ) ( D χ eff ) 2 ( L eff 2 A eff 2 ) e αL P p 1 P p 2 1 λ F 2 P s 0 ( f s ) η ( Δ β ) d f s
η = 10 log [ P F f F L P s f s 0 ]

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