Abstract

We treat the nonlinear phase shift response in the weak perturbation limit as a linear digital filter that can be synthesized into the values of its poles and zeros and mapped onto an optical architecture. This procedure results in a significant enhancement in the nonlinear sensitivity with a response that is robust to frequency changes within the filter passband. A precompensation technique can be used to reduce distortions under strongly driven nonlinear operation to achieve a larger phase shift. We also show that nonlinear sensitivity improves with increasing filter group delay and can be increased within constant linear bandwidth by use of higher-order filters.

© 2003 Optical Society of America

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References

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  1. J. E. Heebner and R. W. Boyd, Opt. Lett. 24, 847 (1999).
    [CrossRef]
  2. P. P. Absil, J. V. Hryniewicz, B. E. Little, P. S. Cho, R. A. Wilson, L. G. Joneckis, and P.-T. Ho, Opt. Lett. 25, 554 (2000).
    [CrossRef]
  3. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, Nature 397, 594 (1999).
    [CrossRef]
  4. M. D. Lukin and A. Imamoglu, in Quantum Electronics and Laser Science (QELS), Vol. 40 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), p. 137.
  5. S. Mookherjea and A. Yariv, Phys. Rev. E 65, 026607 (2002).
    [CrossRef]
  6. S. Blair, Opt. Lett. 27, 613 (2002).
    [CrossRef]
  7. Y. Chen and S. Blair, “Nonlinear phase shift of cascaded microring resonators,” J. Opt. Soc. Am. B (to be published).
  8. C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach (Wiley, New York, 1999).
    [CrossRef]
  9. K. Jinguji, J. Lightwave Technol. 14, 1882 (1996).
    [CrossRef]
  10. C. K. Madsen and J. H. Zhao, Appl. Opt. 36, 642 (1997).
    [CrossRef] [PubMed]

2002 (2)

S. Mookherjea and A. Yariv, Phys. Rev. E 65, 026607 (2002).
[CrossRef]

S. Blair, Opt. Lett. 27, 613 (2002).
[CrossRef]

2000 (1)

1999 (2)

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, Nature 397, 594 (1999).
[CrossRef]

J. E. Heebner and R. W. Boyd, Opt. Lett. 24, 847 (1999).
[CrossRef]

1997 (1)

1996 (1)

K. Jinguji, J. Lightwave Technol. 14, 1882 (1996).
[CrossRef]

Absil, P. P.

Behroozi, C. H.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, Nature 397, 594 (1999).
[CrossRef]

Blair, S.

S. Blair, Opt. Lett. 27, 613 (2002).
[CrossRef]

Y. Chen and S. Blair, “Nonlinear phase shift of cascaded microring resonators,” J. Opt. Soc. Am. B (to be published).

Boyd, R. W.

Chen, Y.

Y. Chen and S. Blair, “Nonlinear phase shift of cascaded microring resonators,” J. Opt. Soc. Am. B (to be published).

Cho, P. S.

Dutton, Z.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, Nature 397, 594 (1999).
[CrossRef]

Harris, S. E.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, Nature 397, 594 (1999).
[CrossRef]

Hau, L. V.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, Nature 397, 594 (1999).
[CrossRef]

Heebner, J. E.

Ho, P.-T.

Hryniewicz, J. V.

Imamoglu, A.

M. D. Lukin and A. Imamoglu, in Quantum Electronics and Laser Science (QELS), Vol. 40 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), p. 137.

Jinguji, K.

K. Jinguji, J. Lightwave Technol. 14, 1882 (1996).
[CrossRef]

Joneckis, L. G.

Little, B. E.

Lukin, M. D.

M. D. Lukin and A. Imamoglu, in Quantum Electronics and Laser Science (QELS), Vol. 40 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), p. 137.

Madsen, C. K.

C. K. Madsen and J. H. Zhao, Appl. Opt. 36, 642 (1997).
[CrossRef] [PubMed]

C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach (Wiley, New York, 1999).
[CrossRef]

Mookherjea, S.

S. Mookherjea and A. Yariv, Phys. Rev. E 65, 026607 (2002).
[CrossRef]

Wilson, R. A.

Yariv, A.

S. Mookherjea and A. Yariv, Phys. Rev. E 65, 026607 (2002).
[CrossRef]

Zhao, J. H.

C. K. Madsen and J. H. Zhao, Appl. Opt. 36, 642 (1997).
[CrossRef] [PubMed]

C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach (Wiley, New York, 1999).
[CrossRef]

Appl. Opt. (1)

J. Lightwave Technol. (1)

K. Jinguji, J. Lightwave Technol. 14, 1882 (1996).
[CrossRef]

J. Opt. Soc. Am. B (1)

Y. Chen and S. Blair, “Nonlinear phase shift of cascaded microring resonators,” J. Opt. Soc. Am. B (to be published).

Nature (1)

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, Nature 397, 594 (1999).
[CrossRef]

Opt. Lett. (3)

OSA Trends in Optics and Photonics Series (1)

M. D. Lukin and A. Imamoglu, in Quantum Electronics and Laser Science (QELS), Vol. 40 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), p. 137.

Phys. Rev. E (1)

S. Mookherjea and A. Yariv, Phys. Rev. E 65, 026607 (2002).
[CrossRef]

Other (1)

C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach (Wiley, New York, 1999).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Magnitude and phase response of a digital bandpass filter that produces an ideal nonlinear phase shift.

Fig. 2
Fig. 2

Intensity transmission (top), phase (middle), and group-delay (bottom) response for an optical ring lattice implementation of an ARMA digital filter with four poles and four zeros. The in-band phase change is approximately 4π, with a bandwidth of δν=100 GHz. Top left-hand inset, one stage of an ARMA ring lattice filter. The other two insets show details of the ripples in the peaks. The path-length difference (ϕt) and the coupling ratio (given by a transmission coefficient tt=1-κt) between the two arms determine the location of the zero, while the circumference (C=50 µm+ϕr/kfn) of the ring and coupling coefficient (tr=1-κr) determine the location of the pole. The parameters for our mapping are stage 0, κt=0.1471, ϕt=κr=C0=0; stage 1, κt=0.5018, ϕt=0.9510π, κr=0.0633, C1=50.004 µm; stage 2, κt=0.4982, ϕt=-0.4753π, κr=0.0633, C2=49.996 µm; stage 3, κt=0.1873, ϕt=0.4060π, κr=0.0845, C3=50.001 µm; and stage 4, κt=0.9398, ϕt=0.6824π, κr=0.0845, C4=49.999 µm.

Fig. 3
Fig. 3

Nonlinear phase shift response of four-stage ARMA filter (top) of Fig. 2. The response is calculated at frequencies of νm (λm=500 nm, solid curve), νm-δν/4 (dashed curve), and νm+δν/4 (dotted curve). The nonlinear response is enhanced 17 times over the bulk material of equal group delay (thick solid curve).

Fig. 4
Fig. 4

Nonlinear phase shift response of the four-stage design, parameterized by the degree of precompensation. The precompensation coefficients β are labeled in the figure.

Fig. 5
Fig. 5

Scaling of nonlinear response with group delay. The graph plots n2Iπ/4 and n2Iπ versus group delay for a constant passband width of 500 GHz, where group delay is increased by increasing the number of stages. Designs with 2, 4, 6, and 8 stages are shown. The best-fit scalings to these curves are 1/kgd (Iπ/4) and 1/kgd1.3 (Iπ).

Equations (2)

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n2eff=1LeffkfΔΦIin1LeffkfπIπ,
δCi=-βkgd,i2i=1Nkgd,i2,

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