Abstract

A coupled resonator optical waveguide (CROW) bottle is a bottle-shaped nonuniform distribution of resonator and coupling parameters. This Letter solves the inverse problem for a CROW bottle, i.e., develops a simple analytical method that determines a CROW with the required group delay and dispersion characteristics. In particular, the parameters of CROWs exhibiting the group delay with zero dispersion (constant group delay) and constant dispersion (linear group delay) are found.

© 2014 Optical Society of America

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References

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  1. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, Opt. Lett. 24, 711 (1999).
    [CrossRef]
  2. F. Morichetti, C. Ferrari, A. Canciamilla, and A. Melloni, Laser Photon. Rev. 6, 74 (2012).
    [CrossRef]
  3. M. Notomi, Rep. Prog. Phys. 73, 096501 (2010).
    [CrossRef]
  4. M. Sumetsky and B. Eggleton, Opt. Express 11, 381 (2003).
    [CrossRef]
  5. A. M. Prabhu and V. Van, Opt. Commun. 281, 2760 (2008).
    [CrossRef]
  6. H. C. Liu and A. Yariv, Opt. Lett. 37, 1964 (2012).
    [CrossRef]
  7. M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, Opt. Express 18, 26505 (2010).
    [CrossRef]
  8. M. Sumetsky and Y. Dulashko, Opt. Express 20, 27896 (2012).
    [CrossRef]
  9. M. Sumetsky, Phys. Rev. Lett. 111, 163901 (2013).
    [CrossRef]
  10. M. Sumetsky, “Dispersionless low-loss miniature slow light delay lines based on optical fibers,” Optical Fiber Communication Conference (in press).
  11. This assumption corresponds to the minimal ripple group delay behavior, which is required in most applications.
  12. P. A. Braun, Theor. Math. Phys. 41, 1060 (1979).
    [CrossRef]
  13. P. A. Braun, Rev. Mod. Phys. 65, 115 (1993).
    [CrossRef]
  14. L. D. Landau and E. M. Lifshitz, Mechanics, 3rd ed. (Butterworth-Heinemann, 1976).
  15. R. Gorenflo and S. Vessella, Abel Integral Equations, Analysis and Applications, Lecture Notes in Mathematics (Springer, 1991), Vol. 1461.
  16. M. Sumetsky, “A Kac CROW delay line,” arXiv:1305.7359v2 (2013).
  17. S. Ramachandran, Fiber Based Dispersion Compensation (Springer, 2007).

2013 (1)

M. Sumetsky, Phys. Rev. Lett. 111, 163901 (2013).
[CrossRef]

2012 (3)

2010 (2)

2008 (1)

A. M. Prabhu and V. Van, Opt. Commun. 281, 2760 (2008).
[CrossRef]

2003 (1)

1999 (1)

1993 (1)

P. A. Braun, Rev. Mod. Phys. 65, 115 (1993).
[CrossRef]

1979 (1)

P. A. Braun, Theor. Math. Phys. 41, 1060 (1979).
[CrossRef]

Assefa, S.

Braun, P. A.

P. A. Braun, Rev. Mod. Phys. 65, 115 (1993).
[CrossRef]

P. A. Braun, Theor. Math. Phys. 41, 1060 (1979).
[CrossRef]

Canciamilla, A.

F. Morichetti, C. Ferrari, A. Canciamilla, and A. Melloni, Laser Photon. Rev. 6, 74 (2012).
[CrossRef]

Cooper, M. L.

Dulashko, Y.

Eggleton, B.

Ferrari, C.

F. Morichetti, C. Ferrari, A. Canciamilla, and A. Melloni, Laser Photon. Rev. 6, 74 (2012).
[CrossRef]

Gorenflo, R.

R. Gorenflo and S. Vessella, Abel Integral Equations, Analysis and Applications, Lecture Notes in Mathematics (Springer, 1991), Vol. 1461.

Green, W. M. J.

Gupta, G.

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Mechanics, 3rd ed. (Butterworth-Heinemann, 1976).

Lee, R. K.

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Mechanics, 3rd ed. (Butterworth-Heinemann, 1976).

Liu, H. C.

Melloni, A.

F. Morichetti, C. Ferrari, A. Canciamilla, and A. Melloni, Laser Photon. Rev. 6, 74 (2012).
[CrossRef]

Mookherjea, S.

Morichetti, F.

F. Morichetti, C. Ferrari, A. Canciamilla, and A. Melloni, Laser Photon. Rev. 6, 74 (2012).
[CrossRef]

Notomi, M.

M. Notomi, Rep. Prog. Phys. 73, 096501 (2010).
[CrossRef]

Prabhu, A. M.

A. M. Prabhu and V. Van, Opt. Commun. 281, 2760 (2008).
[CrossRef]

Ramachandran, S.

S. Ramachandran, Fiber Based Dispersion Compensation (Springer, 2007).

Scherer, A.

Schneider, M. A.

Sumetsky, M.

M. Sumetsky, Phys. Rev. Lett. 111, 163901 (2013).
[CrossRef]

M. Sumetsky and Y. Dulashko, Opt. Express 20, 27896 (2012).
[CrossRef]

M. Sumetsky and B. Eggleton, Opt. Express 11, 381 (2003).
[CrossRef]

M. Sumetsky, “Dispersionless low-loss miniature slow light delay lines based on optical fibers,” Optical Fiber Communication Conference (in press).

M. Sumetsky, “A Kac CROW delay line,” arXiv:1305.7359v2 (2013).

Van, V.

A. M. Prabhu and V. Van, Opt. Commun. 281, 2760 (2008).
[CrossRef]

Vessella, S.

R. Gorenflo and S. Vessella, Abel Integral Equations, Analysis and Applications, Lecture Notes in Mathematics (Springer, 1991), Vol. 1461.

Vlasov, Y. A.

Xia, F.

Xu, Y.

Yariv, A.

Laser Photon. Rev. (1)

F. Morichetti, C. Ferrari, A. Canciamilla, and A. Melloni, Laser Photon. Rev. 6, 74 (2012).
[CrossRef]

Opt. Commun. (1)

A. M. Prabhu and V. Van, Opt. Commun. 281, 2760 (2008).
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

Phys. Rev. Lett. (1)

M. Sumetsky, Phys. Rev. Lett. 111, 163901 (2013).
[CrossRef]

Rep. Prog. Phys. (1)

M. Notomi, Rep. Prog. Phys. 73, 096501 (2010).
[CrossRef]

Rev. Mod. Phys. (1)

P. A. Braun, Rev. Mod. Phys. 65, 115 (1993).
[CrossRef]

Theor. Math. Phys. (1)

P. A. Braun, Theor. Math. Phys. 41, 1060 (1979).
[CrossRef]

Other (6)

L. D. Landau and E. M. Lifshitz, Mechanics, 3rd ed. (Butterworth-Heinemann, 1976).

R. Gorenflo and S. Vessella, Abel Integral Equations, Analysis and Applications, Lecture Notes in Mathematics (Springer, 1991), Vol. 1461.

M. Sumetsky, “A Kac CROW delay line,” arXiv:1305.7359v2 (2013).

S. Ramachandran, Fiber Based Dispersion Compensation (Springer, 2007).

M. Sumetsky, “Dispersionless low-loss miniature slow light delay lines based on optical fibers,” Optical Fiber Communication Conference (in press).

This assumption corresponds to the minimal ripple group delay behavior, which is required in most applications.

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Figures (4)

Fig. 1.
Fig. 1.

(a) Reflecting CROW. (b) Illustration of variation of the CROW parameters, fast in the darker region and slow in the lighter region.

Fig. 2.
Fig. 2.

(a) Illustration of a bottle CROW and relation between coupling parameters δn, eigenfrequency shifts Δn, and effective potentials Un+ and Un. (b) A symmetric bottle CROW with δn determined from Eq. (10) and Δn=0. (c) A bottle CROW with Δn=2δn determined from Eq. (14) [see also Fig. 4(b)]. The classically allowed region is darkened.

Fig. 3.
Fig. 3.

(a), (d) Group delay of the Kac CROW determined by Eq. (10) and uniform CROW, respectively. (b), (e) Transmission power of a Kac CROW determined by Eq. (10) and uniform CROW, respectively. Red bold curve at the bottom of (b) and (e) is the power spectrum of the input Gaussian pulse. Black bold curves at the top of spectra are the result of averaging over the 0.03ΔB interval. (c), (f) Propagation of the Gaussian pulse through the Kac CROW determined by Eq. (10) and uniform CROW, respectively. Dashed curve, input signal; solid curve, output signal. Inset: comparison of the normalized input and output signals.

Fig. 4.
Fig. 4.

(a) Plot of function n(δ)/N defined by Eq. (14) and (b) Plot of the inverse function δ(n)/δ1. (c) Group delay and (d) transmission power of a CROW bottle determined by Eq. (14). Red bold curve at the bottom of (d) is the power spectrum of the input Gaussian pulse. Black bold curves at the top of spectra in (c) and (d) are the result of averaging over the 0.03ΔB interval. (e) Propagation of the dispersed Gaussian pulse through this device. Dashed red curve, undispersed Gaussian pulse; solid red curve, input dispersed signal; solid blue curve, output dispersion-compensated signal. Inset: comparison of the normalized undispersed Gaussian pulse (dashed curve) and output signal (solid curve).

Equations (15)

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(Δ1Δνi2(Γ+γ))C1+δ1C2=1,δn1Cn1+(ΔnΔνi2γ)Cn+δnCn+1=0,δN1CN1+(ΔNΔνi2γ)CN=0,n=2,3,N1,
T(Δν)=1iΓC1,τ(Δν)=12πIm(dln(T(Δν))dΔν).
Cn(Δν)=A[(ΔνΔ(n))24δ2(n)]1/4cos(φn(Δν)),φn(Δν)=nNturnarccos(Δν+i2γΔ(n)2δ(n))dn+ϑ,
Un±=U±(n)=Δ(n)±2δ(n),
ΔB=U+(1)U(n)=4δ1.
T(Δν)=a(Δν)cos(φ2(Δν))+b(Δν)sin(φ2(Δν))a*(Δν)cos(φ2(Δν))+b(Δν)sin(φ2(Δν)),a(Δν)=12(Δ1Δν+iΓ),b(Δν)=12[4δ12(Δ1Δν)2]1/2.
Δν0=Δ1,Γ=2δ1.
τ(Δν)=1π0Nturndn(ΔνΔ(n))24δ2(n).
n(δ)=24δ24δ12τ(ν)νdνν24δ2,
δn=δ12n216τ02.
Δn=2δn.
n(δ)=44δ4δ1τ(ν)ν1/2dνν4δ,
τ(Δν)=τ0(12ΔνΔB).
n(δ)=Nf(x),x=δ/δ1,f(x)=1.746·[(x23x28)ln(1+1x11x)+(123x4)1x].
Δν0=Δ1=Δ2,Γ=23/2δ1,δ2=21/2δ1.

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