Abstract

We experimentally investigate the initial chirp dependence of slow and fast light effects in a semiconductor optical amplifier followed by an optical filter. It is shown that the enhancement of the phase shift due to optical filtering strongly depends on the chirp of the input optical signal. We demonstrate ~120° phase delay as well as ~170° phase advance at a microwave frequency of 19 GHz for different optimum values of the input chirp. The experimental results are shown to be in good agreement with numerical results based on a four-wave mixing model. Finally, a simple physical explanation based on an analytical perturbative approach is presented.

© 2009 Optical Society of America

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References

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  1. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature 397, 594-598 (1999).
    [CrossRef]
  2. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, "Superluminal and slow light propagation in a room-temperature solid," Science 301, 200-202 (2003).
    [CrossRef] [PubMed]
  3. P. C. Ku, F. Sedgwick, C. J. Chang-Hasnain, P. Palinginis, T. Li, H. -L. Wang, S. -W. Chang and S. -L. Chuang, "Slow light in semiconductor quantum wells," Opt. Lett. 29, 2291-2293 (2004).
    [CrossRef] [PubMed]
  4. Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005).
    [CrossRef] [PubMed]
  5. N. V. Jespersen, and P. R. Herczfeld, "Optical techniques for reconfiguring microwave phased arrays," IEEE Trans. Antennas Propag. 38, 1054-1058 (1990).
    [CrossRef]
  6. J. Capmany, B. Ortega, D. Pastor, and S. Sales, "Discrete-time optical processing of microwave signals," J. Lightwave Technol. 25, 702-723 (2005).
    [CrossRef]
  7. J. Mørk, R. Kjær, M. van der Poel, and K. Yvind, "Slow light in a semiconductor waveguide at gigahertz frequencies," Opt. Express 13, 8136-8145 (2005).
    [CrossRef] [PubMed]
  8. H. Su, and S. L. Chuang, "Room-temperature slow light with semiconductor quantum-dot devices," Opt. Lett. 31, 271-273 (2006).
    [CrossRef] [PubMed]
  9. C. J. Chang-Hasnain, and S. L. Chuang, "Slow and fast light in semiconductor quantum-well and quantum-dot devices," J. Lightwave Technol. 24, 4642-4654 (2006).
    [CrossRef]
  10. P. K. Kondratko and S. L. Chuang, "Slow-to-fast light using absorption to gain switching in quantum-well semiconductor optical amplifier," Opt. Express 15, 9963-9969 (2007).
    [CrossRef] [PubMed]
  11. F. Öhman, K. Yvind, and J. Mørk, "Voltage-controlled slow light in an integrated semiconductor structure with net gain," Opt. Express 14, 9955-9962 (2006).
    [CrossRef] [PubMed]
  12. F. G. Sedgwick, B. Pesala, A. V. Uskov, and C. J. Chang-Hasnain, "Chirp-enhanced fast light in semiconductor optical amplifiers," Opt. Express 15, 17631-17638 (2007).
    [CrossRef] [PubMed]
  13. J. Mørk, F. Öhman, M. van der Poel, Y. Chen, P. Lunnemann, and K. Yvind, "Slow and fast light: controlling the speed of light using semiconductor waveguides," Laser Photonics Rev. 2, (2008) (to be published).
  14. F. Öhman, K. Yvind, and J. Mørk, "Slow light in a semiconductor waveguide for true-time delay applications in microwave photonics," IEEE Photon Technol. Lett. 19,1145-1147 (2007).
    [CrossRef]
  15. W. Xue, Y. Chen, F. Öhman, S. Sales, and J. Mørk, "Enhancing light slow-down in semiconductor optical amplifiers by optical filtering," Opt. Lett. 33, 1084-1086 (2008).
    [CrossRef] [PubMed]
  16. H. Su, P. K. Kondratko, and S. L. Chuang, "Variable optical delay using population oscillation and four-wave-mixing in semiconductor optical amplifiers," Opt. Express 14, 4800-4807 (2006).
    [CrossRef] [PubMed]
  17. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, "Observation of ultraslow light propagation in a ruby crystal at room temperature," Phys. Rev. Lett.  90, 113903-1-4 (2003).
    [CrossRef]
  18. F. Koyama and K. Iga, "Frequency chirping in external modulators," J. Lightwave Technol. 6, 87-93 (1988).
    [CrossRef]
  19. G. H. Smith, D. Novak, and Z. Ahmed, "Overcoming chromatic-dispersion effects in fiber-wireless systems incorporating external modulators," IEEE Trans. Microwave Theory Tech. 45, 1410-1415 (1997).
    [CrossRef]
  20. A. V. Uskov, and C. J. Chang-Hasnain, "Slow and superluminal light in semiconductor optical amplifiers," Electron. Lett. 41, 55-56 (2005).
    [CrossRef]
  21. W. Xue, S. Sales, J. Mørk, and J. Capmany, "Widely tunable microwave photonic notch filter based on slow and fast light effects," IEEE Photon Technol. Lett., DOI: 10.1109/LPT.2008.2009468 (to be published).

2008 (2)

J. Mørk, F. Öhman, M. van der Poel, Y. Chen, P. Lunnemann, and K. Yvind, "Slow and fast light: controlling the speed of light using semiconductor waveguides," Laser Photonics Rev. 2, (2008) (to be published).

W. Xue, Y. Chen, F. Öhman, S. Sales, and J. Mørk, "Enhancing light slow-down in semiconductor optical amplifiers by optical filtering," Opt. Lett. 33, 1084-1086 (2008).
[CrossRef] [PubMed]

2007 (3)

2006 (4)

2005 (4)

A. V. Uskov, and C. J. Chang-Hasnain, "Slow and superluminal light in semiconductor optical amplifiers," Electron. Lett. 41, 55-56 (2005).
[CrossRef]

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

J. Capmany, B. Ortega, D. Pastor, and S. Sales, "Discrete-time optical processing of microwave signals," J. Lightwave Technol. 25, 702-723 (2005).
[CrossRef]

J. Mørk, R. Kjær, M. van der Poel, and K. Yvind, "Slow light in a semiconductor waveguide at gigahertz frequencies," Opt. Express 13, 8136-8145 (2005).
[CrossRef] [PubMed]

2004 (1)

2003 (1)

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, "Superluminal and slow light propagation in a room-temperature solid," Science 301, 200-202 (2003).
[CrossRef] [PubMed]

1999 (1)

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature 397, 594-598 (1999).
[CrossRef]

1997 (1)

G. H. Smith, D. Novak, and Z. Ahmed, "Overcoming chromatic-dispersion effects in fiber-wireless systems incorporating external modulators," IEEE Trans. Microwave Theory Tech. 45, 1410-1415 (1997).
[CrossRef]

1990 (1)

N. V. Jespersen, and P. R. Herczfeld, "Optical techniques for reconfiguring microwave phased arrays," IEEE Trans. Antennas Propag. 38, 1054-1058 (1990).
[CrossRef]

1988 (1)

F. Koyama and K. Iga, "Frequency chirping in external modulators," J. Lightwave Technol. 6, 87-93 (1988).
[CrossRef]

Ahmed, Z.

G. H. Smith, D. Novak, and Z. Ahmed, "Overcoming chromatic-dispersion effects in fiber-wireless systems incorporating external modulators," IEEE Trans. Microwave Theory Tech. 45, 1410-1415 (1997).
[CrossRef]

Behroozi, C. H.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature 397, 594-598 (1999).
[CrossRef]

Bigelow, M. S.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, "Superluminal and slow light propagation in a room-temperature solid," Science 301, 200-202 (2003).
[CrossRef] [PubMed]

Boyd, R. W.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, "Superluminal and slow light propagation in a room-temperature solid," Science 301, 200-202 (2003).
[CrossRef] [PubMed]

Capmany, J.

J. Capmany, B. Ortega, D. Pastor, and S. Sales, "Discrete-time optical processing of microwave signals," J. Lightwave Technol. 25, 702-723 (2005).
[CrossRef]

W. Xue, S. Sales, J. Mørk, and J. Capmany, "Widely tunable microwave photonic notch filter based on slow and fast light effects," IEEE Photon Technol. Lett., DOI: 10.1109/LPT.2008.2009468 (to be published).

Chang, S. -W.

Chang-Hasnain, C. J.

Chen, Y.

W. Xue, Y. Chen, F. Öhman, S. Sales, and J. Mørk, "Enhancing light slow-down in semiconductor optical amplifiers by optical filtering," Opt. Lett. 33, 1084-1086 (2008).
[CrossRef] [PubMed]

J. Mørk, F. Öhman, M. van der Poel, Y. Chen, P. Lunnemann, and K. Yvind, "Slow and fast light: controlling the speed of light using semiconductor waveguides," Laser Photonics Rev. 2, (2008) (to be published).

Chuang, S. L.

Chuang, S. -L.

Dutton, Z.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature 397, 594-598 (1999).
[CrossRef]

Hamann, H. F.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

Harris, S. E.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature 397, 594-598 (1999).
[CrossRef]

Hau, L. V.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature 397, 594-598 (1999).
[CrossRef]

Herczfeld, P. R.

N. V. Jespersen, and P. R. Herczfeld, "Optical techniques for reconfiguring microwave phased arrays," IEEE Trans. Antennas Propag. 38, 1054-1058 (1990).
[CrossRef]

Iga, K.

F. Koyama and K. Iga, "Frequency chirping in external modulators," J. Lightwave Technol. 6, 87-93 (1988).
[CrossRef]

Jespersen, N. V.

N. V. Jespersen, and P. R. Herczfeld, "Optical techniques for reconfiguring microwave phased arrays," IEEE Trans. Antennas Propag. 38, 1054-1058 (1990).
[CrossRef]

Kjær, R.

Kondratko, P. K.

Koyama, F.

F. Koyama and K. Iga, "Frequency chirping in external modulators," J. Lightwave Technol. 6, 87-93 (1988).
[CrossRef]

Ku, P. C.

Lepeshkin, N. N.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, "Superluminal and slow light propagation in a room-temperature solid," Science 301, 200-202 (2003).
[CrossRef] [PubMed]

Li, T.

Lunnemann, P.

J. Mørk, F. Öhman, M. van der Poel, Y. Chen, P. Lunnemann, and K. Yvind, "Slow and fast light: controlling the speed of light using semiconductor waveguides," Laser Photonics Rev. 2, (2008) (to be published).

McNab, S. J.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

Mørk, J.

J. Mørk, F. Öhman, M. van der Poel, Y. Chen, P. Lunnemann, and K. Yvind, "Slow and fast light: controlling the speed of light using semiconductor waveguides," Laser Photonics Rev. 2, (2008) (to be published).

W. Xue, Y. Chen, F. Öhman, S. Sales, and J. Mørk, "Enhancing light slow-down in semiconductor optical amplifiers by optical filtering," Opt. Lett. 33, 1084-1086 (2008).
[CrossRef] [PubMed]

F. Öhman, K. Yvind, and J. Mørk, "Slow light in a semiconductor waveguide for true-time delay applications in microwave photonics," IEEE Photon Technol. Lett. 19,1145-1147 (2007).
[CrossRef]

F. Öhman, K. Yvind, and J. Mørk, "Voltage-controlled slow light in an integrated semiconductor structure with net gain," Opt. Express 14, 9955-9962 (2006).
[CrossRef] [PubMed]

J. Mørk, R. Kjær, M. van der Poel, and K. Yvind, "Slow light in a semiconductor waveguide at gigahertz frequencies," Opt. Express 13, 8136-8145 (2005).
[CrossRef] [PubMed]

W. Xue, S. Sales, J. Mørk, and J. Capmany, "Widely tunable microwave photonic notch filter based on slow and fast light effects," IEEE Photon Technol. Lett., DOI: 10.1109/LPT.2008.2009468 (to be published).

Novak, D.

G. H. Smith, D. Novak, and Z. Ahmed, "Overcoming chromatic-dispersion effects in fiber-wireless systems incorporating external modulators," IEEE Trans. Microwave Theory Tech. 45, 1410-1415 (1997).
[CrossRef]

O’Boyle, M.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

Öhman, F.

J. Mørk, F. Öhman, M. van der Poel, Y. Chen, P. Lunnemann, and K. Yvind, "Slow and fast light: controlling the speed of light using semiconductor waveguides," Laser Photonics Rev. 2, (2008) (to be published).

W. Xue, Y. Chen, F. Öhman, S. Sales, and J. Mørk, "Enhancing light slow-down in semiconductor optical amplifiers by optical filtering," Opt. Lett. 33, 1084-1086 (2008).
[CrossRef] [PubMed]

F. Öhman, K. Yvind, and J. Mørk, "Slow light in a semiconductor waveguide for true-time delay applications in microwave photonics," IEEE Photon Technol. Lett. 19,1145-1147 (2007).
[CrossRef]

F. Öhman, K. Yvind, and J. Mørk, "Voltage-controlled slow light in an integrated semiconductor structure with net gain," Opt. Express 14, 9955-9962 (2006).
[CrossRef] [PubMed]

Ortega, B.

J. Capmany, B. Ortega, D. Pastor, and S. Sales, "Discrete-time optical processing of microwave signals," J. Lightwave Technol. 25, 702-723 (2005).
[CrossRef]

Palinginis, P.

Pastor, D.

J. Capmany, B. Ortega, D. Pastor, and S. Sales, "Discrete-time optical processing of microwave signals," J. Lightwave Technol. 25, 702-723 (2005).
[CrossRef]

Pesala, B.

Sales, S.

W. Xue, Y. Chen, F. Öhman, S. Sales, and J. Mørk, "Enhancing light slow-down in semiconductor optical amplifiers by optical filtering," Opt. Lett. 33, 1084-1086 (2008).
[CrossRef] [PubMed]

J. Capmany, B. Ortega, D. Pastor, and S. Sales, "Discrete-time optical processing of microwave signals," J. Lightwave Technol. 25, 702-723 (2005).
[CrossRef]

W. Xue, S. Sales, J. Mørk, and J. Capmany, "Widely tunable microwave photonic notch filter based on slow and fast light effects," IEEE Photon Technol. Lett., DOI: 10.1109/LPT.2008.2009468 (to be published).

Sedgwick, F.

Sedgwick, F. G.

Smith, G. H.

G. H. Smith, D. Novak, and Z. Ahmed, "Overcoming chromatic-dispersion effects in fiber-wireless systems incorporating external modulators," IEEE Trans. Microwave Theory Tech. 45, 1410-1415 (1997).
[CrossRef]

Su, H.

Uskov, A. V.

F. G. Sedgwick, B. Pesala, A. V. Uskov, and C. J. Chang-Hasnain, "Chirp-enhanced fast light in semiconductor optical amplifiers," Opt. Express 15, 17631-17638 (2007).
[CrossRef] [PubMed]

A. V. Uskov, and C. J. Chang-Hasnain, "Slow and superluminal light in semiconductor optical amplifiers," Electron. Lett. 41, 55-56 (2005).
[CrossRef]

van der Poel, M.

J. Mørk, F. Öhman, M. van der Poel, Y. Chen, P. Lunnemann, and K. Yvind, "Slow and fast light: controlling the speed of light using semiconductor waveguides," Laser Photonics Rev. 2, (2008) (to be published).

J. Mørk, R. Kjær, M. van der Poel, and K. Yvind, "Slow light in a semiconductor waveguide at gigahertz frequencies," Opt. Express 13, 8136-8145 (2005).
[CrossRef] [PubMed]

Vlasov, Y. A.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

Wang, H. -L.

Xue, W.

W. Xue, Y. Chen, F. Öhman, S. Sales, and J. Mørk, "Enhancing light slow-down in semiconductor optical amplifiers by optical filtering," Opt. Lett. 33, 1084-1086 (2008).
[CrossRef] [PubMed]

W. Xue, S. Sales, J. Mørk, and J. Capmany, "Widely tunable microwave photonic notch filter based on slow and fast light effects," IEEE Photon Technol. Lett., DOI: 10.1109/LPT.2008.2009468 (to be published).

Yvind, K.

J. Mørk, F. Öhman, M. van der Poel, Y. Chen, P. Lunnemann, and K. Yvind, "Slow and fast light: controlling the speed of light using semiconductor waveguides," Laser Photonics Rev. 2, (2008) (to be published).

F. Öhman, K. Yvind, and J. Mørk, "Slow light in a semiconductor waveguide for true-time delay applications in microwave photonics," IEEE Photon Technol. Lett. 19,1145-1147 (2007).
[CrossRef]

F. Öhman, K. Yvind, and J. Mørk, "Voltage-controlled slow light in an integrated semiconductor structure with net gain," Opt. Express 14, 9955-9962 (2006).
[CrossRef] [PubMed]

J. Mørk, R. Kjær, M. van der Poel, and K. Yvind, "Slow light in a semiconductor waveguide at gigahertz frequencies," Opt. Express 13, 8136-8145 (2005).
[CrossRef] [PubMed]

Electron. Lett. (1)

A. V. Uskov, and C. J. Chang-Hasnain, "Slow and superluminal light in semiconductor optical amplifiers," Electron. Lett. 41, 55-56 (2005).
[CrossRef]

IEEE Photon Technol. Lett. (2)

W. Xue, S. Sales, J. Mørk, and J. Capmany, "Widely tunable microwave photonic notch filter based on slow and fast light effects," IEEE Photon Technol. Lett., DOI: 10.1109/LPT.2008.2009468 (to be published).

F. Öhman, K. Yvind, and J. Mørk, "Slow light in a semiconductor waveguide for true-time delay applications in microwave photonics," IEEE Photon Technol. Lett. 19,1145-1147 (2007).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

N. V. Jespersen, and P. R. Herczfeld, "Optical techniques for reconfiguring microwave phased arrays," IEEE Trans. Antennas Propag. 38, 1054-1058 (1990).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

G. H. Smith, D. Novak, and Z. Ahmed, "Overcoming chromatic-dispersion effects in fiber-wireless systems incorporating external modulators," IEEE Trans. Microwave Theory Tech. 45, 1410-1415 (1997).
[CrossRef]

J. Lightwave Technol. (3)

J. Capmany, B. Ortega, D. Pastor, and S. Sales, "Discrete-time optical processing of microwave signals," J. Lightwave Technol. 25, 702-723 (2005).
[CrossRef]

F. Koyama and K. Iga, "Frequency chirping in external modulators," J. Lightwave Technol. 6, 87-93 (1988).
[CrossRef]

C. J. Chang-Hasnain, and S. L. Chuang, "Slow and fast light in semiconductor quantum-well and quantum-dot devices," J. Lightwave Technol. 24, 4642-4654 (2006).
[CrossRef]

Laser Photonics Rev. (1)

J. Mørk, F. Öhman, M. van der Poel, Y. Chen, P. Lunnemann, and K. Yvind, "Slow and fast light: controlling the speed of light using semiconductor waveguides," Laser Photonics Rev. 2, (2008) (to be published).

Nature (2)

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature 397, 594-598 (1999).
[CrossRef]

Opt. Express (5)

Opt. Lett. (3)

Science (1)

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, "Superluminal and slow light propagation in a room-temperature solid," Science 301, 200-202 (2003).
[CrossRef] [PubMed]

Other (1)

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, "Observation of ultraslow light propagation in a ruby crystal at room temperature," Phys. Rev. Lett.  90, 113903-1-4 (2003).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

Calculated RF phase shifts of P (black), P +1 (blue) and P -1 (red) induced by changing the normalized input optical power S from 0.01 to 1.6 as a function of the chirp parameter Δθ indicating the initial phase difference.

Fig. 2.
Fig. 2.

(a) Experimental set-up. (b) The green line shows the measured transmission curve of the DA-MZM. The blue lines show the corresponding initial optical phase difference Δθ, calculated using Eq. (6), as a function of the bias voltage.

Fig. 3.
Fig. 3.

Measured (markers) and simulated (solid lines) RF phase shifts of the detected microwave modulation as a function of the input optical power for different values of the initial chirp, realized by changing the bias voltages of the DA-MZM. In all cases the red-shifted sideband is blocked before detection. (a) 21°≤Δθ≤67° and (b) 112°≤Δθ≤158°. The reference phase is chosen at the input optical power of -10dBm for each of the bias voltages.

Fig. 4.
Fig. 4.

Measured RF power versus input optical power for (a) 21°≤Δθ≤67° and (b) 112°≤Δθ≤158°. (c) and (d) show the corresponding simulated results for the same values of the chirp parameter. In all cases the red-shifted sideband is blocked before detection.

Fig. 5.
Fig. 5.

Measured (markers) and simulated (solid lines) phase shifts of the detected microwave modulation as a function of input optical power for different bias voltages of the DA-MZM. (a) 21°≤Δθ≤67° and (b) 124°≤Δθ≤158° are for the case of blocking the blue-shifted sideband and (c) is without optical filtering.

Fig. 6.
Fig. 6.

(a). Representation of the complex power P +1 (δL) in the complex plane for different values of the initial phase difference Δθ. The thick black arrows indicate the reference points corresponding to minimum input optical power. Upon increasing the power the arrows rotate towards the dotted blue arrows, which correspond to the maximum input optical power. (b). The normalized phase shift δ φ+1(δL) as a function of Δθ (the length of the SOA is 10 μm).

Fig. 7.
Fig. 7.

Calculated phase shifts of P +1 induced by increasing S from 0.01 to 1.6 as a function of the SOA length for (a) -10º<Δθ<90º and (b) 90º<Δθ<170º. (The increment between curves is 10º)

Equations (21)

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

{ E 0 z = γ 0 E 0 E 1 z = γ 0 E 1 + ξ 1 [ E 0 2 E 1 + E 0 2 E + 1 * ] E + 1 z = γ 0 E + 1 + ξ + 1 [ E 0 2 E + 1 + E 0 2 E 1 * ]
γ 0 = 1 2 [ g sat ( 1 ) α ] , g sat = Γ g 0 1 + S , S = E 0 2 P sat , and
ξ ± 1 = 1 2 g sat P sat 1 + S ± α Ω τ s ( 1 + S ) ± i Ω τ s ( 1 + S ) 2 + ( Ω τ s ) 2
P D = P 0 + ( P + 1 + P 1 ) e i Ω t + c . c .
P 0 E 0 2 , P + 1 E 0 * E + 1 , P 1 E 1 * E 0 , and P + 1 + P 1 P
{ d P 0 dz = ( γ 0 + γ 0 * ) P 0 d P dz = ( γ 0 + γ 0 * ) P + ( ξ + 1 + ξ 1 * ) P 0 P d P + 1 dz = ( γ 0 + γ 0 * ) P + 1 + ξ + 1 P 0 P d P 1 dz = ( γ 0 + γ 0 * ) P 1 + ξ 1 * P 0 P
{ E 0 ( 0 ) = E 0 ( 0 ) exp ( i θ 0 ) E + 1 ( 0 ) = E 1 ( 0 ) = E + 1 ( 0 ) exp ( i θ 1 )
{ P ( 0 ) = 2 E 0 ( 0 ) E + 1 ( 0 ) cos ( Δ θ ) P + 1 ( 0 ) = E 0 ( 0 ) E + 1 ( 0 ) [ cos ( Δ θ ) + i sin ( Δ θ ) ] P 1 ( 0 ) = E 0 ( 0 ) E + 1 ( 0 ) [ cos ( Δ θ ) i sin ( Δ θ ) ]
{ E 0 ( 0 ) = A 2 [ 1 + exp ( i V DC V π π ) J 0 ( V V π π ) ] E 0 ( 0 ) exp ( i θ 0 ) E + 1 ( 0 ) = i A 2 exp ( i V DC V π π ) J + 1 ( V V π π ) E + 1 ( 0 ) exp ( i θ 1 ) E 1 ( 0 ) = i A 2 exp ( i V DC V π π ) J 1 ( V V π π ) E + 1 ( 0 ) E + 1 ( 0 ) exp ( i θ 1 )
P ( δL ) = P ( 0 ) + Δ P
= 2 E 0 E 1 cos ( Δ θ ) [ 1 + ( g sat a + γ 1 ) δL + i β 1 δL ]
P + 1 ( δL ) = P + 1 ( 0 ) + Δ P + 1
= E 0 E 1 cos Δ θ { [ 1 + ( g sat a + γ 1 + α β 1 ) δL ] + i [ tan Δ θ + ( β 1 α γ 1 + ( g sat a ) tan Δ θ ) δL ] }
P 1 ( δL ) = P 1 ( 0 ) + Δ P 1
= E 0 E 1 cos ( Δ θ ) { [ 1 + ( g sat a + γ 1 + α β 1 ) δL ] + i ( tan ( Δ θ ) + [ β 1 + α γ 1 ( g sat a ) tan ( Δ θ ) ] δL ) }
γ 1 = g sat ( 1 + S ) S ( 1 + S ) 2 + ( Ω τ s ) 2
β 1 = g sat Ω τ s S ( 1 + S ) 2 + ( Ω τ s ) 2
δ φ + 1 ( δ L ) = arg { P + 1 ( δL ) } S arg { P + 1 ( δL ) } S ref
arctan { cos 2 ( Δ θ ) [ β 1 α γ 1 ( γ 1 + α β 1 ) tan ( Δθ ) ] δL }
cos ( Δ θ ) = 0 Δ θ = ± 90 °
or tan ( Δ θ ) = β 1 α γ 1 γ 1 + α β 1

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